FRACTURE MECHANICS OF METALS, COMPOSITES,

WELDS, AND BOLTED JOINTS Application of LEFM, EPFM, And

FMDM Theory

by

Bahram Farahmand, Ph.D. Boeing Technical Fellow

tt KLUWER ACADEMIC PUBLISHERS

Boston / Dordrecht / London

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Library of Congress Cataloging-in-Publication Data

Farahmand, Bahram. Fracture mechanics of metals, composites, welds, and bolted joints: application of LEFM, EPFM, and FMDM theory / by Bahram Farahmand.

p. cm. Includes bibliographical references and index. ISBN 0-7923-7239-5 (alk. paper) 1. Fracture mechanics. 2. Metals--Fracture. 3. Welded joints--Cracking. 4. Composite materials--Fracture. 5. Bolted joints. I. Farahmand, Bahram. II. Title.

TA409.F35 2000 620'.1126--dc21

00-048696

Copyright © 2001 by Kluwer Academic Publishers

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, mechanical, photo- copying, recording, or otherwise, without the prior written permission of the publisher, Kluwer Academic Publishers, 101 Philip Drive, Assinippi Park, Norwell, Massachusetts 02061.

Printed on acid-free paper.

Printed in the United States of America.

This book is lovingly dedicated to my beautiful wife, Vida, for her valuable advice throughout my long involvement with the book, to my mother, Gohartaj, who inspired me for higher education, and to my two beautiful children, Houman and Roxana, for their patience and understanding.

Preface

In the preliminary stage of designing new structural hardware to perform a given mission in a fluctuating load environment, there are several factors that the designer should consider. Trade studies for different design configurations should be performed and, based on strength and weight considerations, among others, an optimum configuration selected. The selected design must withstand the environment in question without failure. Therefore, a comprehensive structural analysis that consists of static, dynamic, fatigue, and fracture is necessary to ensure the integrity of the structure. Engineers must also consider the feasibility of fabricating the structural hardware in the material selection process. During the past few decades, fracture mechanics has become a necessary discipline for the solution of many structural problems in which the survivability of structure containing pre-existing flaws is of great interest. These problems include structural failures resulting from cracks that are inherent in the material, or defects that are introduced in the part due to improper handling or rough machining, that must be assessed through fracture mechanics concepts.

The importance of fatigue and fracture in nuclear, pressure vessel, aircraft, and aerospace structural hardware cannot be over- emphasized whenever safety is of utmost concern. This book is written for the designer and strength analyst, as well as for the material and process engineer, who is concerned with the integrity of the structural hardware under load-varying environments in which fatigue and fracture must be given special attention. The book is a result of years of both academic and industrial experiences that the author has accumulated during his work with nuclear, aircraft, and aerospace structures. However, the material contained in this book is sufficient to be applied to other industries, where fracture and fatigue are equally important. Moreover, the scope and contents of the book are adequate for use as a textbook for both graduate and undergraduate level courses in the mechanical, material, and aerospace engineering departments with emphasis given to the application of theory rather than the detail mathematical derivation of fracture parameters. Each chapter has several example problems that have been hand-picked from industrial experiences which the authors

have accumulated throughout the years in the field of fracture mechanics.

This book addresses the traditional fatigue approach to life evaluation of structural parts where it is assumed that the structure is initially free from cracks and, after N number of load cycles, the crack will initiate in some highly localized stressed areas. In contrast to the traditional fatigue approach, Linear Elastic Fracture Mechanics (LEFM) assumes the existence of a crack in the structural part in the most unfavorable location perpendicular to the applied load. Chapter 1 covers both traditional fatigue (stress to life, S-N, and strain to life, ~- N) and an overview of the field of fracture mechanics, which includes, the Griffith energy balance, the LEFM concept, Elastic-Plastic Fracture Mechanics (EPFM), Fracture Mechanics of Ductile Metals (FMDM), and the failure prevention concept. The content of Chapter 1 is informative enough for the reader to become knowledgeable with the development of fracture mechanics and its application to structural parts. In Chapters 2 and 3, the application of fracture mechanics in determining the life of a structure is fully discussed through the use of the stress intensity factor parameter, K. The critical value of K is called fracture toughness and is discussed in Chapter 2. The development of the fatigue crack growth curve (da/dn versus AK) is presented in Chapter 3.

In manufacturing space or aircraft structures, it is common practice for pieces of structure that are mated together in a manner strong enough to withstand the load environment while allowing the transfer of load from one segment of the structure to another. Chapter 7 fully discusses the stress concentration sites in a bolted joint that are the prime location for fatigue failure, where cracks can initiate from the threaded region or the periphery of the bolted joint. Welding is another commonly used technique join structural parts in space, aircraft and nuclear structure. A good quality weld can yield almost the same fatigue properties as the parent material. On the other hand, a poorly welded joint with an unacceptable amount of porosity, shrinkage, cavities, or incomplete fusion can be the source of crack initiation and premature failure of the structure. Chapter 6 discusses the Variable Polarity Plasma Arc (VPPA) and a new state-of-the-art technique called Friction Stir Welding (FSW) that are classified as fusion and non-fusion welding techniques, respectively.

In using the LEFM approach to evaluate the life of a part, crack tip yielding must be small and localized and no net section yielding is allowed in the part. Two fracture mechanics approaches are discussed in this book for analysis of tough metals where fracture behavior often extends beyond the elastic dominant regime. The first is called the EPFM theory and uses the J-integral concept first proposed by Rice

xvi

as a path independent integral based on the deformation theory of plasticity (Chapter 4). The second approach is called the FMDM theory. The crack tip plastic deformation defined by the FMDM theory is composed of two distinct regions: 1) the local strainability at the crack tip (the region of highly plastic deformation) and 2) the uniform strainability near the crack tip. The energy absorption rate for these two regions was calculated (see chapter 6) and used to extend the Griffith theory of fracture that originally was developed for brittle materials. In contrast to LEFM, the FMDM theory was shown to accurately correlate with test data for commonly used structural metals over a wide range of crack sizes at stresses above, as well as below the yield stress. The FMDM computer program is capable of generating the variation of fracture toughness as a function of the material thickness for ductile metals and requires only the stress-strain curve as an input.

In structural applications, the use of composites is sometimes advantageous over metallic material because of their light weight and higher stiffness. Damage tolerance and durability of composite material is not yet fully understood. Fracture initiation in composites is associated with defects such as voids, machining irregularities, stress concentrating, damage from impacts with tools or other objects resulting in discrete source damage, delamination, and non-uniform material properties stemming, for example, from improper heat treatment. After a crack initiates, it can grow and progressively lower the residual strength of a structure to the point where it can no longer support design loads, making global failure imminent. Chapter 8 discusses various modes of failure in composite materials and emphasis is given to the GENOA-PFA computer code that enables the engineer to analyze durability and damage tolerance in 2D and 3D woven braided stitched composite materials and structures.

The contents of this book represent a complete overview of the field of fatigue and fracture mechanics, a field that is continuously being advanced by many investigators. This book is divided into 8 chapters:

• Chapter 1. Overview of Fracture Mechanics and Failure Prevention

• Chapter 2. Linear Elastic Fracture Mechanics (LEFM) and Applications

• Chapter 3. Fatigue Crack Growth and Applications

• Chapter 4. Elastic-Plastic Fracture Mechanics (EPFM) and Applications

xvii

• Chapter 5. Fracture Mechanics of Ductile Metals (FMDM) Theory

• Chapter 6. Welded Joints and Applications

• Chapter 7. Bolted Joints and Applications

• Chapter 8. Durability and Damage Tolerance of Composites

Fracture properties for conducting fatigue crack growth and structural life analysis are included in Appendix A, which was extracted from the NASA/FLAGRO material library.

The author wishes to express his appreciation to Mr. David OIIodort (The Boeing Co.) for his editorial assistance with the entire manuscript, Dr. V. L. Chen (The Boeing Co.) for his comments to Chapter 8, Dr. Ares Rosakis (from California Institute of Technology) for his valuable comments to the EPFM concepts, and Mr. Bruce Young (McDermott Technology) for his comments to Chapter 4. He would also like to thank Mr. Doug Waldron (The Boeing Co.) for contributing a portion of Chapter 6, and Dr. Frank Abdi (Alpha STAR Corporation), Dr. Levon Minnetyan (Clarkson University), and Dr. Chris Chamis (NASA/Glenn Research Center) for their contributions to Chapter 8. Finally, the support of his family, especially his loving and devoted wife, children, and dear mother, is gratefully acknowledged. Their sacrifices made it possible to complete this book.

xviii

CONTENTS

CHAPTER 1 OVERVIEW OF FRACTURE MECHANICS AND FAILURE PREVENTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 High Cycle Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Low Cycle Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3 Stress and Strain at Notch (Neuber Relationship) . . . . . . . . . . . . . 19

1.4 Linear Elastic Fracture Mechanics (LEFM) and

Applicat ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.4.1 Application of LEFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.5 Elastic-Plastic Fracture Mechanics (EPFM) . . . . . . . . . . . . . . . . . . . . . 31

1.5.1 Path Independent J-integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

1.5.2 Crack Opening Displacement (COD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1.6 Failure Prevention and Fracture Control Plan . . . . . . . . . . . . . . . . . . . 35

1.6.1 Material Selection, Testing, and Manufacturing . . . . . . . . . . . . . . . . . 40

1.6.2 Non Destructive Inspection (NDI) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

1.6.2.1 Liquid Penetrant Inspection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

1.6.2.2 Magnetic Particle Inspection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

1.6.2.3 Eddy Current Inspection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

1.6.2.4 Ultrasonic Inspection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

1.6.2.5 Radiographic Inspection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

CHAPTER 2 LINEAR ELASTIC FRACTURE MECHANICS (LEFM) A N D APPLICATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.0 Introduction to Elastic Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.1

2.2

2.2.1

2.2.2

2.2.3

2.2.3.1

2.2.4

2.3

2.4

2.5

2.6

2.6.1

2.6.2

2.6.3

2.6.4

2 . 7 ¸

2.7.1

2.7.2

2.8

2.8.1

2.8.2

2.8.3

2.8.4

2.9

2.9.1

2.9.2

Griffith Theory of Elastic Fracture ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

The Stress Intensity Factor Approach, K ......................... 56

General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Crack Tip Modes of Deformation ................................... 56

Derivation of Mode I Stress Intensity Factor ..................... 58

Stress Intensity Factor For Combined Loading ................. 63

Critical Stress Intensity Factor ....................................... 65

Fracture Toughness .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Material Anisotropy and its Effect on Fracture Toughness...69

Factors Affecting Fracture Toughness ............................ 71

Residual Strength Capability of a Cracked Structure ......... 74

Residual Strength Diagram for Material with Abrupt Failure.76

The Apparent Fracture Toughness ................................. 78

Development of the Resistance Curve (R-Curve) & K R ..... 79

Residual Strength Diagram for Structure with Built-Up

Feature .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

Plasticity at the Crack Tip within Small Scale Yielding ........ 87

Plastic Zone Shape Based on the Von Mises Yield

Criterion .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

Plastic Zone Shape Based on Tresca Yield Criterion ......... 90

Surface or Part Through Cracks ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Stress Intensity Factor Solution for a Part Through Crack...92

Longitudinal Surface Crack in a Pressurized Pipe ............. 95

Part Through Fracture Toughness, Kie .......................... 96

The Leak-Before-Burst (LBB) Concept .. . . . . . . . . . . . . . . . . . . . . . . . . . 99

A Brief Description of ASTM Fracture Toughness Testing.102

Plane Strain Fracture Toughness (KIc) Test ................... 103

Standard Kic Test and Specimen Preparation ................ 103

viii

2.9.3

2.9.4

2.9.5

Plane Stress Fracture Toughness (Kc) Test ................... 107

M(T) Specimen for Testing K c .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

Grip Fixture Apparatus, Buckling Restraint, and Fatigue

Cracking .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

CHAPTER 3 FATIGUE CRACK GROWTH AND -APPLICATIONS ................................................................ 118

3.1

3.1.1

3.2

3.2.1

3.3

3.3.1

3.3.2

3.3.3

3.4

3.4.1

3.4.2

3.5

3.6

3.6.1

3.6.2

3.6.3

Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Stress Intensity Factor Range and Crack Growth Rate ..... 121

Crack Growth Rate Empirical Descriptions ..................... 122

Brief Review of Fatigue Crack Growth Testing ................ 127

Stress Ratio and Crack Closure Effect .......................... 132

Elber Crack Closure Phenomenon ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Threshold Stress Intensity Factor Range, z~Kth .............. 137

Newman Crack Closure Approach ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

Variable Amplitude Stress and the Retardation

Phenomenon .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Wheeler Retardation Model .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...... 156

Willenborg Retardation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

Cycle by Cycle Fatigue Crack Growth Analysis ............... 165

Environmental Assisted Corrosion Cracking ................... 167

Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

Threshold stress intensity factor ( K~c and KEAc) .............. 170

ASTM Procedures for Obtaining K~c or K~c . . . . . . . . . . . . . . . . 173

References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

ix

4.0

4.1

4.2

4.2.1

CHAPTER 4 ELASTIC-PLASTIC FRACTURE MECHANICS (EPFM) AND APPLICATIONS ............................................ 180

Overview ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

Introduction ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

Introduction to Griffith Energy Balance Approach ............ 183

The Relationship Between Energy Release Rate, G, and

Complience .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

4.3 The Path Independent J- Integral and its Application ........ 189

4.3.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

4.3.2 Derivation of Path Independent J- Integral ..................... 191

4.4 Comments Concerning the Path Independent J-Integral

Concept .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

4.5 J-Controlled Concept and Stable Crack Growth .............. 204

4.6 Experimental Evaluation of J-Integral and J~c Testing ....... 209

4.6.1 Multispecimen Laboratory Evaluation of the J-Integral

(Energy Rate Interpretation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

4.6.2 Single Specimen Laboratory Evaluation of the J-Integral...215

4.6.3 Advanced Single Specimen Technique Using the C(T)

Specimen ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

4.7 Determination of J~ Value Based on a Singie Specimen

Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

4.7.1 Validity Check for Fracture Toughness from the J-R

Curve .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

CHAPTER 5 THE FRACTURE MECHANICS OF DUCTILE METALS THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

5.0 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

X

5.1 The Extended Griffith Theory ...................................... 237

5.2 Fracture Mechanics Of Ductile Metals (FMDM) ............... 240

5.3 Determination of g, = ~)U~)c Term ................................. 241

5.4 Determination of the g2= ~)Uu/~)c Term ..... . . . . . . . . . . . . . . . . . . . . . . 244

5.4.1 Octahedral Shear Stress Theory (Plane Stress

Conditions) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

5.5 Octahedral Shear Stress Theory (Plane Strain

Conditions) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

5.6 Applied Stress, G, and Half Crack Length, c, Relationship.255

5.6.1 Determination of W and h Terms Separately ................ 255 U U

5.6.2 Applied Stress and Crack Length Relationship ............... 256

5.7 Mixed Mode Fracture and Thickness Parameters ............ 257

5.8 The Stress-Strain Curve ............................... : ............ 259

5.9 Verification of FMDM Results with the Experimental Data.259

5.10 Fracture Toughness Computation by the FMDM Theory...262

5.10.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

5.10.2 Fracture Toughness Evaluation for 2219-T87 Aluminum

Alloy .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

5.10.3 Fracture Toughness Evaluation for 7075-T73 Aluminum

Alloy .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

C H A P T E R 6 W E L D E D JOINTS AND A P P L I C A T I O N S ......... 274

6.0

6.1

6.2

6.2.1

Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

Welding of Aluminum Alloys ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

Variable Polarity Plasma Arc (VPPA) ..... . . . . . . . . . . . . . . . . . . . . . . . 277

Static and Fracture properties of VPPA weld .................. 281

xi

6.3

6.3.1

6.3.2

6.3.2.1

6.4

Friction Stir Welding (FSW) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

Static and Fracture Properties of FSW .. . . . . . . . . . . . . . . . . . . . . . . . . 290

Application of FSW to Space Structures . . . . . . . . . . . . . . . . . . . . . . . . 292

Metallurigical Examination of Fracture Surfaces .... . . . . . . . . . . 298

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

CHAPTER 7 BOLTED JOINTS AND APPLICATIONS ......... 304

7.1

7.2

7.3

7.3.1

7.4

7.5

7.6

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

Bolted Joint Subjected to Cyclic Loading . . . . . . . . . . . . . . . . . . . . . . . 305

Bolt Preload . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

Bolt Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310

Fatigue Crack Growth Analysis of Pads in a Bolted Joint...317

Riveted Joints ........ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328

Material Anisotropy and its Application in Bolt Analysis ..... 330

. . . . . . . . . . . . . . . . . . . . . . . . . . . 332

CHAPTER 8 DURABILITY AND DAMAGE TOLERANCE OF COMPOSITES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

8.1

8.2

8.2.1

8.3

8.3.1

8.3.2

8.3.3

8.3.3.1

Overview of Composite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

Overview of Textiles Composi tes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

Categorizat ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

Progressive Fracture Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342

Character izat ion of Composi te Degradat ion . . . . . . . . . . . . . . . . . . . 343

Composi te Simulation Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343

Progressive Fracture Analysis (PFA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

Computat ional Simulation Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

xii

8.3.3.2

8.3.3.3

8.3~3.4

8.3.3.4.1

8.3.3.4.2

8.3.3.4.3

8.3.3.4.4

8.3.3.4.5

8.3.3.4.6

8.3.3.5

8.3.3.5.1

8.3.3.5.2

8.3.3.5.3

8.3.3.6

8.3.3.7

8.3.4

8.3.5

8.3.6

8.3.6.1

8.4

8.4.1

8.4.2

8.4.3

8.4.4

8.4.5

Damage Tracking Process ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351

Failure Evaluation Approach ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

Damage Evolution Metrics .......................................... 354

Total Damage Energy Release Rate (TDERR) ............... 355

Damage Energy Release Rate (DERR) ........................ 357

Strain Energy Damage Rate (SEDR) ............................ 358

Equivalent far field stress (~e ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358

The length of crack opening (a) ................................... 359

Equivalent fracture toughness from DERR or SEDR ........ 359

Evaluation of Elastic Constants ................................... 359

Stitched Simulation Capability .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360

Woven Patterns ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

Fiber Arrangement .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

Finite Element Analysis in PFA ................................... 363

Simulation of Damage Progression .. . . . . . . . . . . . . . . . . . . . . . . . . . . . 364

Methodology of Mesh Refinement in Progressive Failure

Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

Simulation of Reshaping Braided Fiber Preforms to Assist

Manufacturing .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

Probabilistic Failure Analysis ............................ .......... 367

Probabilistic Evaluation of Composite Damage

Propagation ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

Composite Structural Analysis and Input and Output ...... 369

Composite Analysis under Static Loading .................... 370

Composite Analysis under Low-Cycle Fatigue Loading... 371

Composite Analysis under High-Cycle Fatigue Loading.. 372

Random Power Spectral Density Fatigue Loading ......... 375

Composite Analysis under Impact Loading .................. 376

xiii

8.4.6 Composi te Analys is under Creep Loading . . . . . . . . . . . . . . . . . . . 377

8.5 Conc lus ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379

APPENDIX A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384

INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402

xiv

Chapter 1

OVERVIEW OF FRACTURE MECHANICS AND FAILURE PREVENTION

1.0 Introduction

Between 1930 and 1950, a series of failures of several large structures, including pressure vessels, storage tanks, ships, gas pipe lines, bridges, dams and many welded parts alarmed government regulators search for more effective ways to prevent structural failures [1,2]. Most of the observed failures occurred under operating cyclic stress well below the yield value of the material, in a catastrophic manner, with high velocities and little or no plastic deformation. In- depth scientific investigation into the nature of these failures indicated that poor structural design practices (the presence of stress concentrations), insufficient material fracture toughness, residual stresses, lack of inspection, unaccounted variation in load spectrum and presence of corrosive environment, can each contribute to an accelerated crack growth that may result in catastrophic failure and possible loss of life. Structural failure prevention and potential savings can be obtained by focusing attention on a few major areas which have material and structural dependency. Tighter control over material properties (such as static strength and fracture toughness) throughout the manufacturing and assembly phases of the hardware, is a major factor, which contributes to prevention of structural failures. That is, materials may degrade their design properties due to improper handling, rough machining, localized permanent deformation during the manufacturing and assembly processes, prior to their service usage. In addition, more economical and effective methods of estimating the structural life, judicious use of Non-Destructive Inspection (NDI), using the latest NDI techniques, better understanding of the fatigue and corrosive environment, and implementation of a thorough quality control plan can all reduce the costs of fracture related accidents. A brief discussion on the failure prevention concept one that can be implemented by applying a sound fracture control plan is presented in section 1.6.

In designing components of commercial aircraft or space vehicles to deliver the maximum structural performance, and to reduce the possibility of failure, the induced stresses in the components due to the applied loads must fall below the material design allowables. However, the presence of unavoidable localized stress concentration sites can produce localized plastic deformation that are suitable locations for crack initiation and eventually cause instability of the part. In both aerospace and aircraft structures, it is common practice to join structural parts by fasteners (bolted and riveted joints), and in some cases through the welding process. Preventive measures must be taken to avoid fatigue failures where cracks can initiate from the weld due to incomplete fusion, shrinkage, cavities, porosity or from the heat affected zone (HAZ) adjacent to the weld which has reduced properties as compared with the parent material, or from the fastener joint with inherent stress concentration sites. Figure 1 illustrates how a crack could initiate from two possible critical areas, such as a bolted

I " ~ ~ , , , B o l t e d Joints & Welded Parts

/ Shank to Head Area

I Plate I,,..~lt ~ Heat Affected Zone I VVe,0od <H Z> I

i "eQi~.~J,Lv ~P'atel I i ~ ~ B I I

I '"°°°"°°'°n Weld Nugget -~

ln a pressurized container crack Crack can initiate and propagate will intiate and propagate in the In the threaded or shank to head

Z or from the weld due to prosity area

a hole in a bolted " X I / - / joint and propagate ~ / - a bolted joint

Figure 1: Crack initiation from a bolted joint or a welded part.

joint with a stress concentration site or from a weld in a heat affected zone where material properties have degraded considerably as compared with the parent material. Figure 2 shows a two pass plasma arc butt welded joint for 2219-T87 aluminum alloy that has passed the

2

HAZ Region (Failure in most cases Weld nugget initiate from the HAZ region)

Figure 2: HAZ and weld regions of a VPPA joint.

NDI inspection requirements. Static and fracture test results for 2219- T87 arc welded joint (figure 2), based on several standard specimens, indicated that design allowables of the welded region are inferior to the base metal [3]. All the regions of the weld are shown in the figure.

The presence of residual stresses in the welded part due to the contraction of the weld metal during cooling, together with high applied stresses (or fluctuating stresses), and poor fracture toughness, can shorten the life of the part and results in premature failure of the structure. A good quality welded joint, requires a qualified welder and t!ght control over deficiencies such as incomplete fusion, shrinkage, and porosity, that can be the source of crack formation. Chapter 6 discusses the welding concept, and it's application in space structural joints. Emphasis will be given to the pressurized structures, which have been categorized by many agencies as fracture critical components. Specifically, attention will be given to two types of modern welding process: Variable Polarity Plasma Arc (VPPA) and a new state-of-the-art computer-controlled welding method called Friction Stir Welding (FSW). The FSW process does not require localized melting of the two mating parts. In brief, a circular pin stirs through the two mating plates. Pin rotational and linear speeds are kept under tight control, while the temperature of the localized region of the joined parts are kept below the material melting point. This is in

3

contrast with the traditional welding process, in which material does undergo localized melting. The joined parts can be welded either next to each other as butt joint or in lap joint position. Figure 3 shows a

HAZ (a)

(b)

Figure 3: a) Cross section and b) Center of a friction stir weld.

butt welded joint, that was processed by the FSW technique for 2014- T6 aluminum alloy (cross section and center of the weld). The FSW technique was first established at The Welding Institute (TWI) at Cambridge, England and later the process was matured and implemented by Boeing aerospace for joining space structural hardware such as pressurized tanks.

There are two sources of stress concentration in a fastener joint that must be given special consideration (see figure 1). Presence of stress concentrations in a threaded region (and in some cases shank to bolt head area), which can be minimize by undergoing the rolling process. Rolling process induces compressive residual stresses in the threaded areas with increased root radius [4,5]. Fastener hole is another area of great concern when designing a bolted joint. Cracks

4

can initiate at the stress concentration sites, next to the periphery of a hole in a pad, and will grow in a stable manner when subjected to a cyclic load environment. These cracks will continue to grow until they reach their critical length (figure 4). Crack initiation from the periphery of a hole in a pad can be delayed through a mandrelizing or cold expansion process [6] which induces compressive residual stresses around the hole, see figure 4. In brief, the cold expansion process expands the edges of the hole plastically by using a mandrel and a split sleeve device. When the mandrel is drawn through a sleeve and the hole (the mandrel rod is slightly larger than the hole at one end), the surrounding metal is yielded beyond its elastic limit. After plastically deforming the region around the periphery of the hole, the surrounding elastic material tries to maintain its original position and therefore exerts compressive stresses on the expanded region, which effectively extends fatigue life (figure 4).

Cold working of a ~ I ~ • ~ ~ hole to improve the ; I ~, ~ 1 II I f='Que 'ife °' the '°int / _ ~ L.~ ~l I ~ _ _ _ . . ~

I Crack initiated from a l ~ ' bolted hole

~ " Improvement in fatigue life due to Manderelizing Load process

St~ss With

f ~ x

~ ~ " Without Mandrelizing Region I manderelizing (compressive field) [

Cycles v

Figure 4: Manderelizing process induces compressive residual stresses around the hole.

Chapter 7 provides a comprehensive review of bolted and riveted joint integrity. Included are some example problems in which linear elastic fracture mechanics is applied to life evaluation of threaded bolts and

5

pads. The NASAJFLAGRO computer code used for fatigue crack growth analysis is used to determine the number of cycles to failure when a jointed structure is subjected to a cyclic load environment.

1.1 High Cycle Fatigue

Currently there are two known methods that are practiced in the aircraft and space industries for assessing the life of a structural component exposed to load varying environments. The classical or traditional approach to fatigue crack growth uses either the stress to life approach (S-N diagram) when the number of cycles to failure is high (high cycle fatigue) or strain to life (~-N diagram) which applies to low cycle fatigue for determining the total life of a structural part. A more recent state-of-the-art approach to structural life evaluation is Linear Elastic Fracture Mechanics (LEFM), although its applicability is limited to small scale yielding (the LEFM and it's application is fully discussed in Chapters 2 & 3). Also, see section 1.4 for a brief introduction to this topic. Using the classical fatigue approach to estimate the life of a structural part, it is assumed that the structure is originally free from flaws. After N number of load cycle, microscopic cracks from a suitable location, will initiate along the slip surfaces (stage I) [7]. The initiated crack will grow in a stable manner through the slip planes of a few neighboring grains until it becomes perpendicular to the applied tensile load (Stage II). After some additional number of cycles, the growing crack will reach it's critical crack length and cause unstable crack propagation and eventually failure of the part, (figure 5). Under low applied load where the bulk of the structure is elastic, and the plastic zone size is small and localized, the total number of cycles to failure (from initiation to the end of stage II) can be evaluated for the material by conducting a series of constant amplitude tests in the laboratory in accordance with ASTM E- 466 Standards. It requires about 12 to 15 standard specimens to establish a meaningful S-N curve. For each stress level the minimum of three tests are needed and the average value is used as a point on the curve. The first three specimens are used where the applied cyclic stress is equal to about 70% of the static tensile strength of the material (this value must be below the elastic limit of the material). For this region of the S-N curve, the number of cycles to failure is expected to be about 10 ~ to 104 cycles. The last three specimens are used for stress level equal to about 25% of the static tensile strength of the

6

material. This value is close to the endurance limit of the material and the number of cycles to failure could be about 108 to 107 cycles. The

s.,e i'-

site for initiation of a crack

Crack eminated from the periphery of a hole where stress concentration exist.

The initiated crack will extend through the slip planes of different grains within the material and eventually becomes perpendicular to the applied load

Load I

Figure 5: Crack initiation (stage I) and stable propagation (stage II).

remainder of the test specimens are utilized for other regions of the S- N curve where stress levels are between these two limiting values. The results of these tests are recorded and plotted in the form of an S-N curve (applied stress, S, versus number of cycles to failure, N) called Wohler's diagram (the German railway engineer who devised and conducted the first fatigue failure investigations) [8]. Figure 6 shows a typical S-N diagram for 2219-T851 aluminum alloy that was generated through the ASTM standards. The S-N diagram is used in evaluating the number of cycles to failure of machinery equipment, ships, aircraft, space structures and bridges that are subjected to fluctuating loads. The S-N diagram is an acceptable approach to the total life assessment of a structural part provided that the service conditions of the part under study are parallel to the test conditions conducted in the laboratory. This is known as the similitude law. That is, for the same material the life of a structural part is the same as the life of a test specimen if both have undergone the same loading environment (figure 7). In figure 7, the number of cycles to failure for the area of interest can be determined by using an S-N curve that has been

7

generated through laboratory tests on standard specimens for the material under consideration.

iii ......... iii ........

"t---i!

r lCt l(~ LIFE, CTCU~

Figure 6: The S-N diagram for 2219-T851 aluminum alloy [10]

Load

TA Aircraft Load Environment sthP:/ioma;n:,

~-- Cycle r v v v v v v

= , . . . Area of interest for hfe assessment

Labrotary tests on standard used to simulate

environment for life assessment of the aircraft part

' v ' \ " - r

Figure 7: The life of a structural part is the same as the life of a test specimen if both have undergone the same loading environment

There are two methods for plotting S-N curves. The S-N diagram is plotted as either the actual stress, S, versus the logarithmic scale of cycles, N, (semi-logarithmic) or both S and N are plotted in the form of

8

a log-log plot of S versus N (logarithmic plot). The semi-logarithmic method is the more widely used diagram in engineering applications and most engineering handbooks have the semi-logarithmic plots of S versus N, (see fatigue diagram shown in figure 6 for the 2219-T851 aluminum alloy plate). For some materials, such as Ferrous and Titanium alloys, the logarithmic plot of the S-N curve is approximated by a straight line with additional straight horizontal lines seen in the region above 108 cycles that accounts for the endurance limit, which below this value failure will not occur and therefore the material possess infinite life (as illustrated in figure 8). In contrast with Ferrous and Titanium alloys, aluminum alloys do not exhibit a well defined endurance limit. In figure 6, the endurance limit for 2219-T851 is approximated by a stress level associated with 106 to107 number of cycles to failure.

LOG S ~ S R= Smin/Smax

/ L/~ A A /'~SmaXTim e

I V V V ! s''o

~ Region of infinite life

X " ~ X X ~K"

Endurance Limit Region x

stress magnitude below this value will not cause structural failure (note that the endurance limit is a function of stress ratio, R)

LOGN

Figure 8: S-N diagram and endurance limit (region of infinite life)

When structural parts are subjected to several load varying environments with different stress amplitudes, the total number of cycles to failure can be evaluated by the linear damage theory proposed by Palmgren and Miner [9]. This damage theory concept, states that the total accumulated damages (the total amount of fatigue crack growth) caused by all the load cycles of different stress magnitude, should be considered in the life assessment of a structural part. The fatigue damage contribution of each individual load

spectrum at a given stress level is proportional to the number of cycles applied at that stress level, N i, divided by the total number of cycles

required to fail the part at the same stress level (Nil). It is obvious that

each ratio can be equal to unity if cycles at the same stress level would continue until failure occurs (i.e. Ni l= Nfl). The Miner rule describing the total fatigue failure, in terms of partial cycle ratios, can be written:

N / Nfl + Ni2 / Nf2 + Nu3 / Nf~ + . . . . . =1 1

Equation 1 is a useful tool to determine the life of a given structure subjected to several cyclic load cases of different stress magnitude. Even though the method is simple and versatile, the oversimplified assumption that the damage summation described by equation 1 is linear and that no account is made of the sequence in which the crack tip is experiencing cyclic stresses, may yield an unconservative result. For example, the Miner's total life concept should yield shorter life if the crack tip experiences high stress cycles during its early stage of life followed by the smaller amplitude cycles rather than the other way around. The smaller stress cycles following the larger amplitude cycles, are more effective in damaging the structure (damage becomes more sensitive to load cycles as the crack advances and grows in size). It is possible that using the high amplitude cycles first and low cycles next, may result in a Palmgren-Miner sum less than 1, while using the low amplitude cycles first and high amplitude stress next may result in a damage summation greater than 1. Moreover, load interactions between high and low amplitude load cycles causing a retardation effect (i.e. a delay in damage growth) is not considered in Miner's rule.

Example 1 A fracture critical component of a reusable space structure is made of 2014-T6 rolled bar aluminum alloy, and is subjected to high cycle fatigue of variable load magnitude (see figure 9 for the 2014-T6 S-N diagram provided from reference [10]). Load environments for one service life of the structure consist of 1) all load activities prior to flight, 2) flight loads (it consists of take off, landing and abort landing), 3) loads due to on orbit activities, and 4) on orbit thermal fluctuation (see table 1). Use Miner's rule to determine if the structure can survive four service lives. Note that in table 1, all stress magnitudes are given in

10

terms of % of limit stress. The number of times they occur throughout the service life of the part, is also shown in table 1.

Table 1 Load Spectrum for the Reusable Space Structure

Steps % o f l imit # o f M a x load Min load load t imes (ksi) (ksi)

Pro-F l igh t 1 100 1000 35 21 2 80 2800 28 16.8 3 60 4000 21 12.6 4 40 5000 14 8.4

Flight 1 100 100 40 -40 2 90 500 36 -36 3 80 1000 32 -32 4 70 10000 28 -28 5 60 100000" 24 -24 6 50 800000 20 -20

On-Orbit 1 100 800 25 0 2 90 2000 23 0

O n - O r b i t T e m p e r a t u m 1 100 400000 10 -10 2 50 800000 5 -5

m Io . . . .

so . . . .

i I 0 , ..

]O . . . .

30 . . . .

mlO,

' r i ;~ n[.l~, = 1 4 o l i Kt ' - I .O J I " ' ; NOTE S T R E S S E S A N [ B A S I [ O ! ; ~ ; i S m t M M I ' , O I !1

..... =,, N(+ = = + ~ " z li ° :~',~ I .................... • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . + ........... ~.,....M.: . : - . +.! i" , ,-=. qe +-~-

• ~ : i : i • 0.=¢l i !

• \ , ~ : + 4 - . , w ii~l ~, ,: .......... -3~" ..,.. ................ i : ..i..+ ~.~.4 ......... :...r ..~ -:.~.+,.~: ...................... ; ~.~ =

• ~ ........ ~ ....... ,......-~N ........... ~.,~~I.i:t!ti ........... i!+:;+;I .o . ..~., ",~-, .,~..-"~= : ~.:.k~,-osol. .!~ ~ : ; I

", , , . ~ . - .~...J,-~ : ~ i # ' / . . . . . "'~. . . . . . . :~ii":.~# i Z i ~ i i i i ' -: ; . ] ! i i : i

........... • "-',~ ........ - ........ .,......'~. *.'*'.+~....~'"/-'"'.'.""~~,,".'.t! ........... u"t.'"'.~t~tl

i +i = o=~ : i : ' . : ~ i i , ! i ! i : . ] : : i ............................. ) . . . . . ~ . . . ; _ ~ ~ ~ . . + . ~ . . . . . . . . . . . . . . . . -....,.+~,I

:* ~ i i i I ~ ~ 2 ' i i ~ , i i i i i I ..................... : - . . . - : - . : - - ~ . . , . . . ~ d - I

i ' i ' f i • : : i~: ! i i ! +i t l • ' " ~+ i+J

.......... +....-+..--i.-.+-.-:-+ ............... :....;.--+--++ .; ...................... +-.-+--+4.+.+ ....... . - ' . . . - - . . . i - - . . + - . . + . i ~ . '

+ii i i!il + i . . . . . . io' . . . . . . . i o ' . . . . m ' i i f . . . . . . . i'~

r m ' l ; U C L I V C , CYCLES

F i g u r e 9 : T h e S-N d iag ram fo r 2014-T6 a l u m i n u m a l loy [10]

11

Solution An empirical equation has been established for 2014-T6 aluminum alloy (see MIL-HDBK-5 [10]) that can relate the number of cycles to

failure, Nf, to the peak applied stress, Smax. In the high-cycle fatigue regime, the number cycles to failure, Nf, for stress concentration factor

kt = 1, in terms of equivalent stress, Seq, is given in reference [10] by:

Log Nf = 21.49 -9.44 log (Seq) 2

where Seq = Smax (l-R) °87, and the stress ratio, R, is the ratio of

Table 2 N Smax Smin R Seq Iog(Seq) Iog(Nf) Nf NilNf 1000 35 21 0.6 18.943 1.27745 9.43089 2.7E+09 3,7E-07 2800 28 16,8 0.6 15.1544 1.18054 10.3457 2.2E+10 1.3E-07 4000 21 12.6 0.6 11.3658 1.0556 11.5251 3,4E+11 1.2E-08 5000 14 8.4 0.6 7.57719 0,87951 13.1874 1.5E+13 3.2E-10

100 40 -40 -1 63.6429 1,80375 4.4626 29013.4 0.00345 500 36 -36 -1 57.2786 1.75799 4.89455 78442.2 0.00637

1000 32 -32 -1 50.9143 1.70684 5.37743 238468 0.00419 10000 28 -28 -1 44.55 1.64885 5.92487 841150 0.01189

100000 24 -24 -1 38.1858 1.5819 6.55685 3604553 0.02774 800000 20 -20 -1 31.8215 1.50272 7.30432 2E+07 0.0397

800 25 0 0 25 1.39794 8.29345 2E+08 4.1E-06 2000 23 0 0 23 1.36173 8,63529 4,3E+08 4.6E-06

400000 10 -10 -1 15.9107 1.20169 10.146 1.4E+10 2.9E-05 800000 5 -5 -1 7.95536 0.90066 12.9878 9.7E+12 8.2E-08

S U M = 0.093

Smin/Smax. Using equation 2, the total number of cycles to failure, Nf, for all load events described in table 1 can be determined. The fatigue damage ratio N~/Nf, from all load cases can be calculated and they will be incorporated into equation 1 to assess structural failure. In table 2, quantities, Seq , N~, and partial cycle ratios (NJNf) are recorded for each step. The sum of all partial cycle ratios, SUM (NJNf) = 0.093, is reported in table 2. For four service lives, the sum of all partial cycle ratios is 4x0.093 = 0.372 <1. Therefore the space structural component can survive the high cycle fatigue environment.

t.2 Low Cycle Fatigue

When plastic deformation in structural parts is not localized and the magnitude of the fluctuating stress is no longer in the elastic range of the material, significant plastic straining occurs throughout the body,

12

especially in the highly localized areas at stress concentration sites, and the number of cycles to failure is expected to be relatively short. This failure mechanism is referred to as low cycle fatigue. The steam turbin, gas turbin, jet engines, and nuclear parts are constantly exposed to low cycle fatigue as well as high temperature environments and their structural integrity in terms of number of cycles to failure must be analyzed. Under low cycle fatigue and high temperature environment, the total damage result must consider the creep effect that results when materials are subjected to low frequency load at elevated temperature (low cycle fatigue under low frequency environment). The combined creep and fatigue damage effects on structural components can be written by simply adding the fractions of each damage separately [11]:

T_, Fatigue Damage Contribution + E Creep Damage Contribution =1 la

Low cycle fatigue failure (without creep effect), also referred to as the strain-controlled or strain-life (~,N) approach, can not be characterized by an S-N curve. Depending on material strength and ductility, the number of cycles to failure for low cycle fatigue life is usually between 100 to below 10000 cycles and for high cycle fatigue the number is above 10000. The number of cycles to failure (or number of cycles to initiation of a measurable crack) in the highly plastic region near the stress concentration location, can be estimated by strain-life prediction model via Neuber relationship [12], and cyclic stress-strain curve conducted under strain-controlled condition. The development of material cyclic stress-strain curve by connecting the tips of stable hysteresis loops of different magnitude can be obtained through testing in accordance with the ASTM procedures [13,14]. Each hysteresis loop represents one complete load-unload cycle of a constant strain amplitude with stress ratio R=-I. Figure 10 shows steps used for development of a complete cyclic stress-strain curve from three or more stable hysteresis loop of different strain magnitudes. Figure 11 shows a cyclic stress-strain curve that has been developed through laboratory for 2014-T6 aluminum alloy which steps and procedures were illustrated in figure 10. Material's response to cyclic loading varies depending on the nature of heat treatment and its initial mechanical condition. In general annealed materials will harden when subjected to strain-controlled cyclic conditions. The hardening effect can be observed in hesteresis loop as an increase in the stress range,

13

Figure 10: steps used for development of a cyclic stress-strain curve

8 0

7 0 -

6 0 -

5 0 "

4 0 -

3 0 -

2 0 -

10 -

0 .0

0 .0

f C Y C L I C S T R E S S - S T R A I N C U R V E F O R 2 0 1 4 - T 6 A L U M I N U M ( R E P R O D U C E D F R O M D O U G L A S A I R C R A F T C O M P A N Y )

i i i ~ i I i i i

• 01 .02 .03 .04 .05 .06 .07 .08 .09 0.1

STRAIN, INJIN,

Figure t1: Cyclic stress-strain curve for 2014-T6 aluminum

14

until the constant amplitude range is reached (stability condition), see figure 10. On the other hand, a material with prior cold working will have a tendency to become soft and in some material the hardening or softening effects are not observed.

The cyclic stress-strain curve of figure 11 can be represented mathematically in terms of its elastic and plastic components by:

As Act +(Act ) l /n' cr ~,) l /n ' - o r ~ =--+( 3

2 2E 2K' E

where cr and s are the stress and strain amplitude, K' is the cyclic strength coefficient, and n' is the cyclic strain hardening exponent. Equation 3 is used when it is desired to obtain the localized stress or strain field in the region of plastic deformation via the Neuber equation when subjected to monotonic or cyclic load environments (section 1.3). When structural parts are subjected to a given strain amplitude range, As, cycling under the strain control condition, it is possible that the number of cycles to failure, Nf, can be estimated for the part in

consideration. Combining the elastic and plastic components of the total strain-life amplitude (described by the Basqin and Coffin-Manson relationship), the total strain-life curve can be expressed as [15]:

= ~' f /E (2Nf) b + s'f (2Nf) c 4

where ~'f is the fatigue strength coefficient. Its value is equal to the

monotonic true fracture stress, ~f, corresponding to 2Nf =1. The value

of the true fracture strength, crf, when N=I is larger than the

engineering ultimate strength of the material, cr UIt, and can be

determined through laboratory testing. The use of 2Nf reversals to

failure as shown in equation 4, is common practice throughout the literature. The failure process as a result of a single load application to failure is considered as a half cycle, N. Thus, based on this notation, a complete cycle or full reversal can be designated as 2N.

The elastic portion of equation 4, where ~e =cr'f/E(2Nf) b, was

derived by Basquin [16], who in 1910 first proposed the straight line relation between the true cyclic stress amplitude and the number of

15

cycles to failure. The quantity b is called the fatigue strength exponent (Basquin's exponent) and, for most materials, varies between -0.05 and -0.12, with typical values between -0.085 to -0.1 [17,18]. The plastic strain amplitude components of equation 4, %= ~'f (2Nf) c , is

also related to the number of half-cycle reversals to failure, 2Nf, and is

called the Coffin and Manson law [19]. The quantity ~'f is called the

fatigue ductility coefficient and its value is equal to the true monotonic fracture strain, % associated with 2Nf =1. The fatigue ductility

coefficient, in terms of reduction in area, RA, can be expressed as:

~;'f = In [ I / ( I+RA)] 5

where RA is the ratio of the change in the cross section, z~dk, to the original area, Ao. The exponent c in equation 4 varies between -0.5 to - 0.8, with a typical value of -0.6 applied in the analysis [17,18]. In equation 4, the constants c~'f, b , ~'f, and c are all considered to be material properties. Figure 12 illustrates a typical elastic and plastic strain-life diagram described by equation 4 in terms of it's elastic and

I Low Cycle Fatigue I1~

i,,%% J/---- I11 VII L~] High Cycle C ' ~ • I I ~ - - ~ i I I ~ I I Fatigue

o AE ] , o

,,/r,i '~ ~ " ~ ~ g i I"~---~t I J I '1

~. ~ \ ~ =~ = III I / I

[.__t~ 2NT Reversal to Failure (Log Scale)

Figure 12: a typical elastic and plastic strain-life diagram ( Equation 4)

16

plastic components. For most metals, the strain-life curve can be developed by testing at different strain amplitudes between + 2.0 and +0.2 percent [20]. Usually a few specimens (between 9 and 13) are required to obtain the low-cycle fatigue strain-life curve of a metal. Figure 13 shows the total strain-life curve developed for 2014-T6 aluminum alloy by combining both the elastic and plastic components of equation 4. Note that in figure 13, the failure is defined as the number of cycles for a 0.01 inch flaw to initiate.

Strain - Life Diagram (R=0.1) For 2014-T6

1.00E+00 ',~;~;t

1.00E-03 ~_~:. ~ . ~ 00

1.00E+ 1.00E+ 1.00E+ 1.00E+ 1.00E+ 1.00E+ 1.00E+ 1.00E+ 1.00E+ 00 01 02 03 04 05 06 07 08

Cycle to 0.01 inch Flaw to Initiation

Figure 13: The total strain-life curve developed for 2014-T6 aluminum

The intercept of the two straight lines at 2Nf=l are ~'f and (~'f/E for the plastic and elastic components of equation 4, shown in figure 12. The slopes of the two lines, b and c, are shown separately in the same figure. From this figure, it can be seen that the two curves, corresponding to the elastic and plastic components of equation 4, are intersecting at a point called the "transition point," 2NT. For a number of cycles less than the transition point, the plastic component dominates; this represents the low-cycle regime (short lives). For a number of cycles longer than the transition point, the elastic component dominates the plastic one; this represents the high-cycle regime (long lives). At the intercept of the two lines, the plastic and elastic strain amplitude is the same, that is:

2N T = [(~'f E )/(~' f)]-l/(c-b) 6

17

Example 2 A part is made of 2014-T6 aluminum alloy with the following fatigue properties:

Modulus of elasticity: Tensile Strength: Fatigue Ductility Coefficient: Fatigue Strength Exponent: Fatigue Ductility Exponent:

10E+7 psi 55,000 psi 0.40 -0.10 -0.65

It is recommended by the customer to surface treat the part for better fatigue properties. Determine whether this treatment is necessary for a load environment where the part is cycling under fully reversed constant strain amplitude of 0.008 in./in.

Solution We would like to know if the applied fully-reversed constant strain amplitude of 0.008 is associated with low or high-cycle fatigue. It is important to realize that for high cycle fatigue most of the structural life is used for crack initiation (stage I). If the proposed surface treatment is helpful to prevent crack initiation, then the number of cycles to crack initiation will increase. However, for low-cycle fatigue, the number of cycles to crack initiation is short or even non-existent. Therefore, the surface treatment is not necessary for the latter case. By employing equation 6, the total number of cycles at the transition point can be computed:

2N T = [(~'f E ) / ~ ' f)]-l/(c-b)

[(107 x0.4 •55000 ]-1/(-0.65+0.1)

2N T = 2358 reversals

The total strain at the transition (where 2NT = 2358 reversals) can be computed by using elastic or plastic components of equation 4, since, at the transition point, the two expressions are equal:

~p =~'f (2Nf) c

18

0.4 (2358) -0.65 = 0.00256

The total strain at the transition point is smaller than the fully reversed applied constant strain amplitude of 0.008, thus it falls under the low- cycle fatigue classification. For this reason, the surface treatment is not recommended. It should be noted that by increasing the strength of materials through cold working or heat treatment may improve fatigue properties in the high-cycle fatigue regime, but make them worse in the low-cycle environment.

1.3 Stress and Strain at Notch (Neuber Relationship)

In evaluating the fatigue life of a part that contains stress concentration (as shown in figure 14), the actual stress and strain at the discontinuity must be computed. Finite element methods (FEM) can enable the analyst to transfer the applied load throughout the structure and distribute it in the form of stress across the body, including localized regions where stresses are considerably higher,

KaK s = K2t

where Kt is the theoretical stress

concentration factor when no Region of localized I plasticiticity is present. plastic deformation I

" ~ - - ~ Stress (, o' ,s ] , ~ concentration ~ ~ , . . site where K a = a/S and K~ = El(S/E)

Far field I Note that, strain, ~, and stress, a, S [ ] .~ - stress where t at the stress concentration site

K t =1 I are related via the cyclic stress- I strain curve by

) I s = a/E + (a/K) l /n"

Figure 14: Region of stress concentration and localized plastic deformation at a notch.

19

(such as sharp corners, holes, or the roots of a notch). These localized regions are the source of crack initiation that can lead into final failure of structural parts. NASTRAN and ABAQUS are two well-established codes, used extensively throughout the aerospace industries for linear and non-linear finite element analysis of structural parts with localized discontinuity. Another technique that can be applied to evaluate the stress concentration in a highly stressed region is an experimental approach called photoelasticity. In the elastic range, when the nominal stress, S, and local stresses, (~, are below the yield, the theoretical stress concentration factor (K t ) is:

K t = (~/S 7

In the presence of localized yielding near the notch, the stress concentration can no longer be defined by equation 7. Both the stress and strain concentrations in that locality must be addressed. Around 1960 significant progress has been made in understanding fatigue behavior at critical locations (notch tip) through consideration of local stress-strain response [21,22,23 ]. The strain and stress concentration factors for the case of localized yielding (when the far field stress is in the elastic range) can be formulated as:

Ka = a/S and K~ = E~/S 8

Based on Nueber's Rule [11], the product of the two quantities K(~ and Kc during the plastic deformation is related to the theoretical stress concentration (in the elastic range), K t , by:

Ko Kc = (Kt)2 9

Rearranging equations 8 and 9, it follows that:

(~s = (K t S) 2 / E 10

For a given nominal stress, S, the localized stress, (~, and strain, s, at the notch can be calculated, provided that the stress-strain equation is available for solving the two unknowns quantities (localized stress and

20

strain). When the applied load is monotonic, the stress-strain relationship can be expressed by:

= (~/E + (o/K) TM 11

See equation 3 for a cyclic stress-strain curve where:

~: = o"/E + (~/K') vn' 3

Note that, in the case of cyclic loading, the cyclic stress-strain curve defined by equation 3 can be replaced by hysteresis loops or by multiplying the cyclic stress-strain curve by 2. Equation 10 in terms of stress and strain ranges:

A(~ As = (K t AS) 2 / E 12

Example 3 Determine the stress and strain at the stress concentration point for the applied nominal stress of 50 ksi, shown in figure 15a. The values of n and K describing the monotonic stress-strain curve for SAE 4340 steel (annealed condition) are 0.1 and 110.0 ksi, respectively. The value of E for SAE 4340 steel is 3x104 ksi.

~ ~ I ~P

Kt=3

~P (a)

I I P(t) _ 10kips __ _ I

-10kips

(b)

Figure 15: Stress concentration site for Example 3.

21

Solution Using equations 10 and 11, computed. From equation 10:

the two unknowns, (~ and ~, can be

c s = (3x50)2/3x104 And from equation 11:

c = (~/3xl 04 + ((~/110) 1/°'1

Solving for (~ and ~ :

(~ = 70 ksi and ~ = 0.011

Employing equation 8, the stress and strain concentration can be obtained:

K~, = 70/50 K~,=1.4

K, = Es/S = 0.011(50/3x104) 1

K, = 6.43 where K~, K, = (Kt) 2= 1.4x6.43=9.0

Example 4 For the same problem, determine the number of cycles to failure, where the cyclic load, P(t), is shown in figure 15b. The following properties related to the material are given as:

Modulus of elasticity: 3E+4 ksi Tensile Strength: 180 ksi Fatigue Ductility Coefficient: 0.626 Fatigue Strength Exponent: -0.080 Fatigue Ductility Exponent: -0.73 n z 0.16 K t 238 ksi

Cross sectional area 0.75 in 2

Solution Using equations 3 computed as follows:

and 12, the two unknowns (~ and ~, can be

22

s = ~ /E + (~/K')l/n'and

AS can be computed by:

A~ As = (K t AS) 2 / E

AS = P(t)max/A = 20kips/0.75 in 2

AS = 26.66ksi

The local stress and strain associated with applied AS = 26.66 ksi can be found by solving the two equations 3 and 12:

A~ As = (K t AS) 2 / E

A~ As = (3x 26.66) 2 /3x104 A~ As = 0.213

Applying the trial and error approach to obtain A~ and As:

As = 90/3x104 + (90/238) ~/°'16 As = 0.0023 in./in.

AG = 90.0 ksi

The number of cycles to failure can be obtained by using equation 4:

s = ~ ' f /E (2Nf) b +s'f (2Nf) c

At the transition point the number of cycles (reversals to failure) is:

2N T = [(s'f E )/~,f)]- l /(c-b)

2N T = [(3x104 x 0.625)/180/] -1/(-0"73+008)

2N T =1222 cycles

The strain associated with 2N T = 1222 cycles is:

s =180/3xl 04 (1222)-008+0.625 (1222) -0.73

s = 0.007 in./in.

23

This value is larger than z~E = 0.0023 and therefore, falls into the high- cycle fatigue regime:

= (~'f/E (2Nf) b

0.0023 =180/3x104 (2Nf) -0.08

2Nf =10000 cycles

1.4 Linear Elastic Fracture Mechanics (LEFM) and Applications

Materials used in manufacturing structural components generally contain microscopic cracks. Such cracks have initiated in the structure in many ways prior to its service usage. For example, they may be introduced as small surface cracks during machining and manufacturing of parts, they may grow from defects in the parent metal (such as inclusion), from incomplete welds, or they may nucleate and grow in structure under fatigue loading. Their presence in brittle materials are unsafe when subjected to cyclic loading, but they can be also dangerous in some ductile metals when subjected to a high rate of loading or low temperature environment. Griffith assumed the existence of these small invisible cracks in his theory of brittle fracture in solids. It is the role of fracture mechanics to determine when these inherent flaws become critical, that is, when they will reach a size at which they will grow catastrophically at an operational stress below the material yield allowable.

Fracture mechanics theory began from the work of A. A. Griffith at the Royal Aircraft Establishment (RAE) in the United Kingdom. His theory as described in "The Phenomenon of Rupture and Flow In Solids" [24], is applicable to high strength elastic material which under static normal rate of loading and at room temperature, undergo little or no plastic deformation before fracture in a tensile test. The Griffith theory assumes the material contains crack-like defects and that work must be performed on the material to supply the energy needed to propagate the crack by creating two new crack surfaces. The Griffith equation describing the fracture stress, (~F, of a sharp elliptical crack of length 2c in an elastic material, was formulated with the mathematical help of Professor C. E. Inglis, 1913 [25] as:

24

=/2ET 13 (~ F V =c

where E is the material modulus of elasticity and T is the surface energy defined as the work done in creating two new crack surfaces by the breaking of atomic bonds. However, this theory is not adequate for determining the fracture strength of ductile material, because tough metals undergo considerable plastic deformations in the region at the crack tip prior to final fracture. Later, Irwin [26] and Orowan [27] pointed out that for ductile material to fracture, the stored strain energy is consumed for both the formation of two new cracked surfaces, plus the work done in plastic deformation around the crack tip. Irwin realized that for ductile materials, the energy required to separate crack surfaces (the breaking of atomic bonds) is negligible compared to the work done in plastic deformation at the crack tip prior to separation. Generally, for tough metals, the energy required for plastic deformation at the crack tip is significantly larger than the surface energy [27]. This is the reason why metallic materials have much greater resistance to fracture than brittle material. The fracture stress, aF, as a function of crack length for ductile metals, when the crack tip undergoes an appreciable amount of plastic deformation, can be written by modifying the Griffith theory [27] described by equation 13 as:

2E (T+y p) (~F = 14

n O

where yp is the plastic work consumed at the crack tip region and is the result of available energy used up in the process of plastic deformation. Chapter 5 discusses a new approach to the fracture mechanics of ductile metals, the FMDM Theory that can be used to evaluate the energy dissipation rate for plastic deformation taking place around the crack tip. In this approach, the energy dissipated by plastically straining material at the crack tip is of two kinds, a highly strained region at the crack tip and a uniformly straining one near the crack tip. These two energy rate terms are fully derived in Chapter 5.

The main development and application of fracture mechanics to structural parts took place in the USA at the Naval Research Laboratory (NRL) in 1950 under Professor George R. Irwin [28].

25

Fracture mechanics became an engineering discipline following the development of Linear Elastic Fracture Mechanics (LEFM). The characterization of crack tip stresses when applying the LEFM concept requires the use of a stress intensity factor parameter, K, which can address important cracking problems in many industries where the safety of people and structure is of utmost concern. Recently, it has become a requirement by space agencies that all primary structural parts of fracture critical hardware be evaluated for their structural integrity during their service life by applying linear elastic fracture mechanics concepts. The crack tip parameter K is a useful tool to calculate the number of cycles consumed during stable crack growth up to failure for different crack geometry and loading conditions when the'crack tip plasticity is confined to a small region at the crack tip.

LEFM is an extension of the Griffith Theory, modified to account to a limited extent for the presence of the plastic zone at the crack tip. To be able to correlate the LEFM results with test data, the energy consumed at the crack tip for the plastic deformatior~ must be considered. Irwin and Dugdale [29,30] estimated the size of the plastic zone formed at the crack tip independently by assuming that local crack tip stresses are equal to the yield stress of the material. In Dugdale's model, the shape of the plastic zone formed at the crack tip was considered to be in the form of a strip in front of the physical crack tip, whereas Irwin assumed that the plastic zone ahead of the crack tip is a circle (see section 1.5). In both models, the effective crack length (the original crack length plus the plastic zone size), rather than the physical crack length, was used to calculate the stress intensity factor. In applying the LEFM to structural parts, engineers must recognize that the plastic deformation in the material must be limited to a small region at the crack tip when evaluating residual strength capability or stable crack growth under cyclic loading. Moreover, the applied stress in the part must be below the material yield value and no net section yielding is allowed in the material. Chapter 2 provides an in-depth description of the stress intensity factor, K, and its application to structural parts under different crack geometries and loading conditions.

The critical value of stress intensity factor, K, is associated with the fracture toughness of the material. Fracture toughness can be defined as the material's ability to resist unstable cracking in a neutral environment. The critical stress intensity factor can be obtained through laboratory testing under the condition of plane strain, K~c, plane

26

stress, Kc, and for surface crack fracture toughness, K[e, under slow rate of loading and normal temperature. Ductile materials have the ability to absorb energy and deform plastically prior to final rupture, hence possessing high fracture toughness. Brittle material on the other hand fails at low stresses in the presence of small cracks. Their failure occur in a sudden manner with high velocities and over great distances, hence possessing low fracture toughness tending to have cleavage fracture. Certain materials may drop their fracture toughness value when exposed to low temperature, corrosive environments, or when subjected to a high rate of loading. For these reasons, special attention must be given to the material selection when designing fracture critical hardware. For example, a given alloy which possess sufficient ductility and high fracture toughness at room temperature, such as ferritic stainless steel, can have considerable reduction in its fracture toughness value when exposed to cold temperature. The NASA/FLAGRO material library [31] contains a list of some ferritic steels having appealing fracture toughness at room temperature and a considerable drop in toughness at cryogenic temperature.

1.4.1 Application of LEFM

To be able to assess any cracking problem, such as residual strength capability evaluation or the number of cycles to failure of structural parts in a load varying environment, the material fracture toughness value (critical value of K) must be available to the analyst. This is analogous to the static strength analysis requirement that the calculated stress, (~, induced in the structural part by the external load must always be compared with design allowables. Both the tensile yield, (~Yield' and the ultimate, O'ULT, strength of the material are needed for checking the calculated stress against these values. That is, the fracture analysis would be incomplete without having material fracture allowables provided from a reliable source. Values for material fracture toughness can be obtained through ASTM Standard testing procedures (ASTM E-399 & E-541) for the plane strain and plane stress conditions respectively. The residual strength capability diagram can then be constructed through a relationship that describes the fracture stress, (~, as a function of crack length, a, through critical stress intensity factor (fracture toughness, Kc):

27

K c

a - [3/- ~ 15

where [3 is the correction to the crack geometry.

The load-carrying capability or residual strength of a structure is significantly affected by the presence of cracks and it is substantially lower than the strength of the undamaged structure. In aircraft and aerospace industries, residual strength diagram can be constructed and utilized for the purpose of planning the inspection interval, prior to the total loss of the structural part during its service life. For a single load path design, where pre-existing crack grow in a stable manner, the residual strength capability of the structure or the ability of the structure to tolerate damage can be determined with ease. For fail- safe and multiple load path structures or structures with a crack arrest design (such as skin stiffened structure), the construction of a residual strength diagram requires calculation of a failure stress at which partial failure (one load path fails) occurs and to determine whether the redistributed load can be tolerated by the remaining structure. Section 1.6 under failure prevention discusses the single and multiple load path design and provides an example problem.

One of the most important applications of LEFM is in its ability to estimate the total number of cycles to failure that a part can withstand and relate it to the actual number of cycles that the structure will be exposed to. This failure prediction approach requires a good understanding of 1) all loading events that a component will experience and the number of times that each event will occur (the fatigue spectrum), 2) an equation that can relate the fatigue crack growth rate, da/dN, to the crack tip stress intensity, AK, 3) the material fracture toughness, and 4) some estimate of the initial flaw size. An empirical relationship between da/dN and AK can be obtained as the result of laboratory test data and can then be utilized to solve crack growth problems where the structural part has undergone the same loading conditions. Having the above information available to the analyst, the remaining life cycles (number of cycles for a crack to grow from its initial length, a, to its final length, a0 can be calculated.

28

The first empirical relationship between the crack growth rate (da/dN) and the stress intensity factor range (AK) was developed by Paris et.al. [32]. This relationship was expressed as:

da/dn = c (z~K) r" 16

where c and m were constants that were evaluated using test data. The Paris equation adequately described only the middle portion of the crack growth rate diagram and needed further modification to account for other factors. This relationship was later extended to account for other regions of the curve, to include parameters such as stress ratio, threshold and critical stress intensity factors, and the retardation effect, etc. Probably one of the most useful crack growth rate equations was developed for the NASA/FLAGRO computer program [31], a current state-of-the-art computer code, that is being used in the aerospace and aircraft industries for safe-life analysis of space structural parts. The constant amplitude crack growth rate relationship (da/dN, AK) for all regions of the crack growth rate curve (including the threshold stress intensity factor, z~Kth, and the critical regions, Kc) are represented in the NASA/FLAGRO computer code. Figure 16 shows a da/dn versus AK crack growth rate diagram for Custom 455 steel alloy which was conducted in accordance with the ASTM-E647 procedures [31]. In figure 16, three regions of the curve, the threshold, Paris (middle region), and accelerated region (upper portion of the curve) are clearly indicated. The Newman crack closure phenomenon approach [33] is implemented in the computer code to account for different stress ratios, R, under constant amplitude loading. Chapter 3 provides a detailed description of the crack growth rate concept, including a discussion on the NASAJFLAGRO computer code.

In constant amplitude loading conditions, where there is no interaction among cycles (baseline fatigue crack growth condition when dR/dt=0, and R=~ rn~n/~r,,x), the crack growth rate analysis is relatively simple to perform. However, the actual cyclic loading that most structures experience during their service life is by no means of constant amplitude and the crack growth rate delay or acceleration due to load interactions between low-high or high-low cycles must be taken into consideration. For example, when aircraft wings are subjected to gust and maneuver loads, the tensile overload or peak load forms a tensile plastic zone at the crack tip region larger than the subsequent

29

10" 2 ,

10"

10".

I~hlID R-reline ~ Retf [] n'CBO11ABO1A1 O.111 0.612 113 o GSSO2OABOIA1 0.100 0,TSO 22 A GSBO21AB01A1 0,100 0.750 C 21 v QSBDg2AB01A1 0.100 0.750 C 21

Alloy:. CUSTOM ~ STEEL C.,~dlOn: H10QO Environment: 70F Spt(:~m: C(T) O~lntltlon: T-L Frequent : 20.00

10 "1

10-2

u

10" Z

"o

10 4 -

10"7.

lO'*.

,10-~

10-4 ~P

¢lmm Pammll l~ S,m/% = 0.S

11~1,00O

ItOti l tu 0.75 • 104 C • 3E~ n 2.44 p 0 . ~ q O.g5

Yield • 1 ~ KIQ • 100

/~ • 1 , 1 0 4

% o.s AJ( o = 4

R~ 0.7

1 10 100

[ k s i ° S Q R T ( i n ) ]

Figure 16: da/dn versus z~K crack growth rate diagram for Custom 455 steel alloy [31]

constant amplitude cycles (from high gust load to low load). Upon load release, the overload induced plastic zone causes a crack tip compressive stress and delay in the crack growth rate, da/dN. Use of the constant amplitude crack growth rate equations to express variable tensile amplitude loading will give conservative results when the number of cycles to failure of the structural part is of interest. It should be realized a tensile overload followed by a compressive overload, where R =-1, may annihilate each others effect when dealing with crack growth rate delay or acceleration (figure 17). In chapter 3 two well known models for crack tip retardation due to tensile overload, called the Wheeler and Willenborg models, based on crack tip plastic deformation, with example problems are presented.

30

High Load (Tensile Overload) Load

Low Load A ~ L o w Load / Low Load

A A A A / \ A A A A / VA A A A Fvvvv v v v v v \ / v v v \ Constant Constant Constant Amplitude Load

High Load (Compressive Overloac

Cycle

Amplitude Load Amplitude Load

Figure 17: Low-high-low tensile and compressive overload

1.5 Elastic-Plastic Fracture Mechanics (EPFM)

Elastic-plastic fracture mechanics and its application to low and intermediate strength materials increased when it became clear that the use of linear elastic fracture mechanics (LEFM) is too conservative and not adequate to handle many structural problems in design that show large scale yielding at the crack tip region. The LEFM is based on small scale yielding, where crack tip plasticity is small and localized compared with crack length and other dimensions. This restriction severely limits the application of LEFM to alloys, which exhibit an appreciable amount of plasticity at the crack tip region. The stress intensity factor parameter, K, in most instances, can be used to assess crack tip behavior for low strength materials that manifest small scale yielding under low load environment and plane strain conditions. However, under plane stress condition, where excessive plastic deformation and net section yielding may take place, crack tip parameters other than K must be introduced to extend fracture mechanics to large scale yielding. Two concepts are currently employed when dealing with non-linear fracture mechanics. The path independent J-integral, was first introduced by Rice [34] as a parameter to describe the crack tip field, and Crack Opening Displacement (COD), was first presented by Wells [35], to relate the amount of opening of the two faces of a crack to the strain at the crack tip. Both methods can be utilized to extend fracture mechanics into the low strength tough materials regime.

31

1.5.1 Path Independent J-integral

In Chapter 4, it was shown that the calculated value of J using the deformation theory of plasticity (plastic deformation is not reversible upon unloading) is independent of any arbitrary path that is taken to envelope the crack tip. The calculated value of J was shown to be identical for two extreme contours, one taken close to the crack tip and the other at the plate perimeter, where cracked plate was loaded by external forces applied to the boundary. Note that, for contours very close to the crack tip full path independence may not be observed [36]. In figure 18a, the path independent J-integral for any contour, r, can be expressed as:

J'r [Wdy - T ° (au/0x) ds] 17

where W is the strain energy density and for the plane stress condition it is given by:

W = J'r [Gxd~x + "~xydyxy + Gyd~y] 18

T is the traction vector defined according to the outward normal along

JIc

Fitted line Blunted line

determining the JIc value

v

Aa (b)

Figure 18: the path independent J-integral for any contour, F.

the contour, F, u is the displacement vector, and ds is an element of arc length along F. For large scale yielding, where the material exhibits plastic deformation, the quantity W, expressed by equation 18,

32

can be evaluated by having a relationship established between stresses and strains. The strain energy density has a unique value when the cracked plate is subjected to monotonic load and if unloading is not permitted. Note that during any stable crack advancement, the newly formed crack edges are fully unloaded from the previous stresses and the crack tip is experiencing some strain hardening that can alter the results of the analysis. For this reason, the critical value

of J, designated by Jlc, is limited to the onset of crack growth when the crack tip experiences blunting. An in-depth discussion of multi-

specimen and single specimen techniques for obtaining the JIc value is

presented in chapter 4. Figure 18b illustrates the results of a JIc test by generating a J-R curve (analogous to the R-curve) by applying the ASTM procedures to a single specimen technique. The critical value of

J, (JIc), was obtained at the intersection of the blunted line and the fitted curve to the test data. The acceptability of the J value, when small amount of crack growth take place, under certain conditions called "J-controlled" is also discussed in chapter 4.

1.5.2 Crack Opening Displacement (COD)

For a sharp crack in an elastic material, the displacement of the crack edges due to the applied load will cause fast fracture when stresses at the apex of the crack reach a critical value. Within the small scale yielding, the presence of a localized plastic zone at the crack tip allows the displacement of the two crack faces, where the amount of crack tip displacement, 5, can be evaluated using Dugdale crack tip yielding, independent of stress intensity factor, K, (derived by Irwin) [29,30]. Wells [35] in 1961 was the first to establish a relationship between the critical value of crack tip displacement, 5c, and material fracture toughness, K~c. He observed that the amount of the crack tip opening, as a consequence of localized yielding can be related to the material fracture toughness and this concept can be used to further describe large scale yielding.

The amount of plastic deformation at the apex of a crack, due to an applied tensile load was formulated by Dugdale by assuming the extension of original crack length , 2a, to 2a e = 2(a+rp) through the formed plastic zone, where rp is the length of plastic zone per crack tip

33

(figure 19). The new elastic crack, 2(a+rp), is experiencing not only an applied stress of(> but is also subjected to a constant compressive

G

Crack tip opening displacement

~ ~ 2 a ~

2a e

T 33 33333- O"

Figure 19: Crack tip opening displacement and Dugdale's model

stress equal to the material yield stress, Gy. Dugdale's model [30] expressing the plastic zone size, 2rp, in terms of crack length, applied stress, and material yield value, ~ y, was written as

rp = ~a[(c/(~ y)2]/8 19a

which can be rewritten in terms of stress intensity factor, K, as

rp = =[(K/(~ y)2]/8 19b

and crack opening displacement, 5, was derived by Burdekin and Stone [37] using the plastic zone size expression derived by Dugdale:

(~ = ~(~2a/E(~ y 20

34

A description of COD method used to assess crack tip environment for large scale yielding is discussed in chapter 4.

1.6 Failure Prevention and Fracture Control Plan

Maintaining a trouble-free damage tolerant structure during it's operational life requires a continuing multi-disciplined quality control process, which must begin in the preliminary design phase and extend through manufacturing and into the operational planning of the structure. The purpose of implementing a Fracture Control Plan (FCP), is to control and prevent damage that could cause catastrophic failure of the structure due to pre-existing cracks. The design philosophy, material selection, analysis approach, testing, quality control, inspection and manufacturing are elements of the fracture control program that can contribute to failure prevention and assure a trouble- free structure. Figure 20 illustrates the main elements of this process [38].

Figure 20: Main elements of the fracture control program

To provide adequate damage tolerance and safety, the structure must be designed to withstand the environmental load throughout it's service life, even when the structure has pre-existing flaws or when part of the structure has already failed. To have sufficient safety and to ensure that unstable crack growth is not reached, two types of damage tolerant design concepts are available: single load path (slow

35

stable crack growth) and fail-safe design (also called multiple load path design). Figures 21 and 22 simply illustrate the slow crack growth and

n0,aoo, J , ,

Figure 21: A single load path structure (a slow crack growth concept)

Figure 22: A multiple load path structure (a fail safe design concept)

fail- safe design concepts, respectively. The slow crack growth design concept stated simply that a single load path structure must be designed such that any pre-existing cracks will grow in a stable manner during the part service life, and thus the growing crack will not

36

achieve a critical dimension that could cause unstable propagation. In this design philosophy, the damage tolerance is ensured by maintaining a slow stable crack growth rate during the life of the structure. Moreover, the residual strength capability of the structural part must always be kept above the applied stress. The chandelier structure shown in figure 21 is a simple illustration of a single load path concept. The failure of the rod (the main load carrying element) can damage the chandelier and probably result in loss of life. The structural integrity analysis must determine that the rod has sufficient margin of safety based on static analysis, and moreover, it has adequate life during its usage.

In the fail-safe design philosophy, the structure is designed such that partial failure of a structural component due to crack propagation is localized and safely contained or arrested. The damage tolerance in the fail-safe design category must be assured by allowing the partial structural failure, while the remaining undamaged structural components maintain their integrity and are able to operate safely under the remaining distributed load. In some cases it must be shown that all released fragments due to partial failure are contained and will not pose a catastrophic hazard to the surrounding. A fail-safe structure has a multiple load path feature in its design. Figure 22 illustrates a fail-safe designed structure where the same chandelier (see figure 21) is now attached to 4 rods. The failure of one rod produces load redistribution to other rods. By analysis, it must be shown that the remaining redistributed load will not cause damage to the chandelier.

An example of a fail-safe designed structure with crack arrest feature, that is common to all aircraft structural parts, is the skin- stiffened design configuration illustrated in figure 23. The longitudinal stiffeners (which act as a crack arrestor) are attached to the skin by a series of fasteners. Let a transverse center crack initiate and grow in the skin as shown in figure 23. The residual Strength capability of skin plate without having stiffeners (viewed as a single load path structure) when subjected to tension is plotted in figure 24. As the

37

~'~ ~ ~ L-Shaped stiffeneres =~ I ii~iiii~ (eiiiE d:~i faTihl !~SsS ii ~

Figure 23: The skin-stiffened design configuration (crack arrest concept)

stable crack increases in length due to the applied load, the residual strength capability (or fracture stress) of unstiffened plate reduces as illustrated in figure 24. For small crack length, the corresponding fracture stress is close to the ultimate stress of the material. When stable crack grows in length while it is exposed to fluctuating load

.=

.=

U.

Residual Strength C u r v e °c

A single load path structure. A crack of 2 : <.-~, ~ length 2c will go catastrophic when subjected to fracture

tress, ~c

I C Half Crack Length, c

Figure 24: The residual strength capability diagram (unstiffened plate)

38

environment, the strength capability of cracked structure continuously drops.

To establish the residual strength capability diagram of a cracked plate with a fail safe design feature (such as a skin stiffener panel shown in figure 25), more complicated analysis is needed [39]. The advancement of a transverse crack reduces the fracture stress of the stiffened panel. While crack length is still small compared with the

D

(/) o

.=

LI.

Residual Strength Curve

A multiple load path structure. A crack of length 2c will build up its residual strength at the

t o f stiIIners.

Location of stiffener I

oc _'>_ Unstiffened Plate ~ I ~ r ' - . Failure

Half Crack Length, c

%

Figure 25: The residual strength capability diagram of a cracked plate with stiffeners

distance between the stiffeners, the residual strength diagram follows the same path as if the structure was without the stiffeners. Larger displacement in the skin when loaded in tension is expected as the crack increases in length. Stiffeners must follow this larger displacement. As the crack becomes closer to the stiffeners, the stresses in the skin drops (lower stress intensity factor) at the expense of higher stresses developed by the stiffeners. The stiffeners become more effective in reducing the crack tip stress intensity factor when their distance is closer and vice versa.

In summary, the residual strength capability of a fail-safe structure must be evaluated to ensure the structural integrity of the part. In other words, failure can be prevented if the redistributed load falls below the residual strength capability of the structural part. To assess the residual strength of a fail-safe structure, 1) the stress level at which

39

partial damage occurs must be known, capability of the remaining structure redistributed load must be evaluated.

and 2) the load-carrying that must withstand the

1.6.1 Material Selection, Testing, and Manufacturing

Material selection is a critical element of the fracture control process in the early stage of design. Trade off studies are conducted between materials and the use of comparative property data is necessary in the selection process. Ultimate strength, yield strength, and other important static properties must be available from a reliable source (such as Military Handbook) for comparison purposes. Plane strain and plane stress fracture toughness (including the fracture toughness for surface cracks) and stress corrosion property data must be considered in the material selection process for the expected environment. The crack growth rate as a function of stress intensity factor (da/dN, AK data) is also required. If the environment is corrosive, the crack growth rate data (da/dt versus AK) should be available to the analyst. Mechanical and fracture properties with respect to the anisotropic nature of certain materials must also be considered. A comprehensive list of materials currently recommended by NASA for space hardware application (which may also be utilized by other industries) together with their static and fracture properties is found available in Appendix A. In this appendix, the ultimate and yield strength as well as plane strain, K~c, and plane stress, Kc, part through fracture toughness, K~e, and crack growth properties for the listed material are available.

Testing methods for each critical part and assembly (an assembly is a piece of hardware which is made of several parts) are also developed and incorporated as part of the fracture control process.

Manufacturing processes must be selected for critical parts such that they do not reduce the damage tolerance level required by the design. Gouges and surface scratches due to improper handling and machining must be avoided throughout the manufacturing process since their presence if undetected can reduce the life of the part. Tight control over processes and selection of proper non-destructive inspection procedures to maintain process quality are prime considerations of a fracture control plan for preventing structural

40

failure. An overview of different non-destructive inspection techniques is presented in section 1.6.2

1.6.2 Non Destructive Inspection (NDI)

Non-Destructive Inspection (NDI) can be defined as the use of non-intrusive methods to determine the integrity of a material or structure. Many non-intrusive methods have been developed to evaluate materials for property determination, to verify quality of workmanship, and to screen a component for the existence of flaws. A flaw, in this sense, can be considered as any nonconformity that exceeds a n established size criteria. Improper NDI inspection methods for flaw detection, or lack of interval inspections on fracture critical hardware can lead to premature failure of structure and loss of life.

Flaw detection is by far the most important aspect of NDI in regards to safe-life assessment of fracture critical parts. Fracture mechanics analysis assumes the existence of a maximum flaw size in the part that grows in a stable manner during its service life. NDI provides the assurance that a flaw larger than the identified maximum size does not exist in the part. From a safety perspective, the initial assumed crack lengths provided by the NDI methods are larger than any pre-existing flaw that could be present in the structure after inspection. However, the degree of conservatism as a result of the larger initial crack size assumption (used to evaluate the life) must be realistic enough not to impact the structural weight or cause unnecessary rejection of parts.

There are numerous NDI methods utilized for flaw detection in structural components [40]. Although many specialized methods are developed for specific materials and configuration, most techniques are variations on a few general methods which use visual enhancement of defects or measure some form of energy transmission through materials and its interaction with defects. The most prevalent of the NDI techniques commonly used in the detection of flaws in aerospace and aircraft components are: liquid penetrant, magnetic particle, eddy current, ultrasonic, and radiography. The purpose of this section is to briefly describe the different NDI methods, to explain in simple language the physical phenomenon of non-destructive inspection, it's application, and to discuss the variables affecting them.

41

1.6.2.1 Liquid Penetrant Inspection

Liquid penetrant inspection is the most widely applied NDI method for the detection of all types of surface flaws, porosity, shrinkage area, and similar types of surface discontinuities in both metals and nonmetallic materials [41,42]. Essentially an enhanced visual evaluation, penetrant inspection relies on capillary action to penetrate

A Surface Crack

Liquid penetrant applied to the surface of cracked specimen. Liquid will penetrate into the cracked surface by capillary action.

Thickne~ /

After cleaning the surface and developer applied to the cracked surface, reversed capillary action takes place by drawing the penetrant from the discontinuity.

Figure 26: Penetrant inspection relies on capillary action

into discontinuities open to the surface (figure 26). Excess dye is removed and a white powder developer is applied to the surface to act as a blotter to the trapped dye, thus enhancing flaw detection sensitivity. Figure 27 illustrates the steps used in conducting penetrant inspection on a part surface. Fluorescent dyes are most commonly used in penetrant fluids, providing high sensitivity for critical part inspections. Penetrant inspection is fast, portable, and easy to interpret, but does require that the flaw be open to the surface for detection. For metal components subjected to mechanical operations that may mask existing flaws, such as grinding or machining, etching may be required to remove the smeared metal prior to inspection.

42

1.6.2.2 Magnetic Particle Inspection

Magnetic particle inspection is another NDI method, that uses a fluorescent media to reveal the presence of a magnetic field and

Penetrant Inspection steps for detecting surface flaws

Penetrant Cleaner Apply the Clean the

Applycleanerthe ~ penetran_.l~ ~ surface & t _ . J (capillary dry (do not remove ~ , ~ \ ~ action) penetrant from opening)

/ I "\ \ \ ' \ 1 , , ' \ ~\ R

Developer - . . . . Clean the developer A Apply o,e .,---,- I , ,~1 (reverse ~ fluorescent \~/ surface to capillan/ -'--%, ~ light " - ~ remove action~ ~ ~ / / ~ developer ,/~\\\

• J ~%\ • "4~ / / I \ \ \

Figure 27: Penetrant steps for detecting surface flaws

therefore discontinuities. This NDI approach is only applicable to ferromagnetic materials, such as plane carbon or alloy steels and ferritic stainless steels. A magnetic field is generated on the part's surface by flowing electric current through the part or adjacent to it. Figure 28 shows the longitudinal and circumferential magnetic fields induced in a cylindrical part. Surface and near surface discontinuities cause variations in the magnetic field (provided that the discontinuity to be at right angles to the magnetic field) as illustrated in figure 28. The fluorescent magnetic oxide particles are then flowed over the part's surface and are attracted to the magnetic field variations, where they can be detected under ultraviolet light [43,44]. If a discontinuity is situated a short distance below the surface of the magnetized part, it will still cause the field to stray out. The magnetic particle inspection method is quick and simple in principle, however, several different magnetic field orientations and magnetization strengths may be required to thoroughly inspect the suspected flaw orientations. Demagnetization of the part due to the residual or remnant field is also required after the inspection is completed. Demagnetization is

43

required when subsequent machining or welding of the part must be performed. Magnetized chips from specimen may stick to the cutting tool and cause damage to the part. Some limitations regarding the magnetic inspection method are: 1) success of this method is directly

Applied ~ Circular Magnetic Field Current r---~__~

Applied Circular Cylinder Current

~ L o n g i t u d i n a l Magnetic Field |

Circular Magnet,." Circular Cylinder

Ci,c,,,fero°,,, Long,.~.a, ~ !1 ~I-,J~" I Crack Magnet¢ Field " ~ , ~ ,~ J

Longitudu'~al Crack r - -

Figure 28: Surface and near surface discontinuities cause variations in the magnetic field

related to the skill of the inspector, 2) the procedure is only applicable to ferromegnetic material, 3) the depth of inspection is limited to surface and subsurface defects only as opposed to internal flaws.

1.6.2.3 Eddy Current Inspection

Another non-destructive technique used to detect surface and near surface flaws is eddy current inspection. Eddy current is an electromagnetic technique which utilizes small diameter coils to induce an electric current in conductive materials. The eddy current induced in the material, in turn, generate its own magnetic field [45]. The magnitude, time lag, phase angle and flow pattern of the eddy current within the test material is detected as a change in the electrical characteristic of the inducing coil. The presence of defects cause significant disruptions in the eddy current flow, and results in impedance and voltage variations. A simple presentation of eddy current and the presence of defects that result in a reduction of current

44

flow (causing a change in the impedance of the test circuit) is shown in figure 29. Calibration is performed on reference defects of known size.

Test Coil Magnetic Field AC Current "~ lt~ ?(-~\ ~_~

L i I , • Indicator I I I

Direction of coil's ~-~J~ current field ~ ~ Direction of Eddy

~ current field

>q \.I Z Material with ~ I

discontinuity Distortion of Eddy Material without currents by material discontinuity discontinuity

Figure 29: The presence of defects cause significant disruptions in the eddy current flow

Eddy current inspection methods have been used extensively in the aircraft and aerospace industry for in-service inspections of components subjected to a fatigue environment. The high sensitivity of eddy current methods for crack detection provides a notable advantage over other NDI methods for this application. Conventional eddy current equipment designed for crack detection utilize a CRT screen which displays the phase and amplitude of the received signal on an impedance plane.

1.6.2.4 Ultrasonic Inspection

Ultrasonic evaluation methods are most prominently used to detect subsurface flaws in components. This capability permits ultrasonic evaluation of material in its raw stock form, thereby screening material containing rejectable flaws before a large investment in material processing is made. Ultrasonic evaluations require generating high frequency (1 MHz - 50 MHz) acoustic waves with a vibrating crystal transducer and then introducing them into a component [46,47]. A liquid or gel couplant is used to transmit the ultrasonic waves between the transducer and the material. As the ultrasonic wave encounters an

45

interface, such as at the couplant/test part surface, part of the wave is reflected back to the transducer while part is transmitted into the part (figure 30). The amplitude of the signal reflected from and transmitted by the liquid/solid interface is dependent upon the ratio of the acoustic impedance of the part and the coupling media. This reflectance and

Test Specimen

~ - Transducer

~-- Ultrsonic Wave I

B Defect

, , J 1 2 3

Front Surface

Back Surface

Defect

j/ ~ 1o 1 2

Back Surface

\

Oscilloscope Screen

Figure 30: Ultrasonic inspection prominently used to detect subsurface flaws in components

transmittal of the ultrasonic wave occurs at every subsequent interface (e.g. defects as well as the front and back surface of the test part). The amplitude and position in time of the reflected or transmitted signal is monitored producing a marked signal change when encountering material flaws. The signal can be represented as an analog trace on an oscilloscope screen as shown in figure l b where the reflected signal from front and back surfaces as well as the discontinuity are illustrated (pulse reflected method). The defect position on the oscilloscope screen must be adjusted with respect to the front and back surface pulse. In figure 1 the front surface pulse on the grid is marked "0" and back surface marked "3". The defect is situated close to the grid marked "2".

1.6.2.5 Radiographic Inspection

Radiography, or x-ray imaging, is another NDI method employed in the detection of subsurface flaws. This method relies on the differential absorption of x-ray photons as they propagate through a material. Some of the beam energy is removed when entering material which will be converted into heat (absorption) and the remain

46

is removed by scattering. In conventional film radiography, x-rays are generated via electrons striking a tungsten target and exit the x-ray tube in a conical beam [48]. The photons which penetrate the material are imaged on the underlying photographic film. Areas of greater thickness or density will absorb more of the penetrating photons. If the difference in density between a flaw and the surrounding material is significant this difference will appear on the x-ray film as a difference in film density. X-ray systems are usually sensitive to changes that result in an apparent change of at least 1 to 2% of the material thickness or density. This limits the flaw sensitivity to defects oriented parallel to the x-ray beam such as cracks, voids, and inclusions. X-ray inspections are of particular value to evaluations of welds and castings where defect type and orientation is favorable.

Table 4 shows the crack size assumptions used by NASA programs for several NDI methods currently applied in aircraft and aerospace structure. These flaw sizes are associated with a surface crack geometry only and are defined as the maximum flaw size that could escape reliable detection. Their values are used to evaluate the structural life of fracture critical parts when subjected to load varying environment [45].

The above clearly demonstrates the interconnection between design, testing, manufacturing, and NDI inspection, and their respective roles in maintaining the desired damage tolerance structure and reducing the incidence of fracture related failures and loss. The structural integrity of a hardware that contains many parts (such as an aircraft, a space structure or a pressure vessel) can be maintained by ensuring the implementation of an effective fracture control plan. For example, the implementation of a sound fracture control plan for a space structure is required by the regulating agency (NASA). Implementation of the plan and it's related activities is monitored by representatives of the procuring agency after its approval.

References

1. Eo R. Parker, "Brittle Behavior of Engineering Structures," John Wiley & Sons, 1957

2. R. P. Reed, J. H. Smith, B. W. Christ, "The Economic Effect of Fracture in the United States," SP 647-1, NBS, March 1983

47

3. B. Farahmand, "Static and Fracture Mechanics Properties of 2219- T6, VPPA Weld," Boeing, Huntington Beach, California, 1999.

4. T. R. Higgins, "Bolted Joints Found Better Under Fatigue, "Engineering News-Record, Vol. 147, pp. 35-36, Aug. 2, 1951.

5. R. A. Walker and G. Meyer, "Design Recommendations for Minimizing Fatigue in Bolts," Machine Design, Vol. 38, No 21, 15 Sep. 1966, pp. 182-186

6. Fatigue Technology Inc. (FTI), Extending the Fatigue Life of Metal Structures Material Testing, Seattle Washington USA.

7. S. S. Manson, "Metal Fatigue Damage- Mechanism, Detection, Avoidance, and Repair," STP 495, 1971, pp. 5-57.

8, "Wohler's Experiments on the Strength of Metals," Engineering, August 23, 1967, p. 160.

9. M. A. Miner, "Cumulative Damage in Fatigue," Journal of Applied Mechanics, Trans. ASTM, Vol. 12, September 1945, pp. A-159-164.

10. MiI-Handbook 5, Department of Defence, Washington, D.C.

11. H. O. Fuchs and R.I Stephens, "Metals Fatigue in Engineering," Wiley interscience Publication, 1980, Ch. 11.

12. H. Neuber, "Theory of Stress Concentration for Shear-Strained Prismatical Bodies with Arbitrary Non-linear Stress-Strain Law, Trans., ASME, Appl. Mech., December 1961, pp 544.

13. R. C. Juvinall, "Supplement to Engineering Considerations of Stress, Stain, and Strength," McGraw-Hill, 1967.

14. Morrow, "Cyclic Plastic Strain Energy and Fatigue of Metals," Internal Friction, Damping, and Cyclic Plasticity, ASTM STP 378, 1965, p. 45.

48

15. R. W. Smith, M. H. Hirschberg, and S. S. Manson, "Fatigue Behavior of Materials Under Strain Cycling in Low and Intermidiate Life Range," NASA, TN D-1574, April 1963.

16. O. H. Basquin, "The Exponential Law of Endurance Tests," Proc. ASTM. Vol. 10, Part II, 1910, p. 625.

17. J. F. Tavernelli and L. F. Coffin, Jr., "Experimental Support for Generalized Equation Predicting Low Cycle Fatigue," Trans. ASME, J. Basic Eng., Vol. 84, No. 4, Dec. 1962, p. 533.

18. S. S. Manson, discussion of reference 23, Trans. ASME J. Basic Eng., Vol. 84, No. 4, Dec. 1962, p. 537.

19. T. Endo and JoDean Morrow, "Cyclic Stress-Strain and Fatigue Behavior of Representative Aircraft, "Journal of Material, Vol. 4, No. 1, March, 1969, pp. 159-175.

20. J. A. Graham, Ed., Fatigue Design Handbook, SAE, 1968.

21. R. E. Peterson, Materials Research and Standards, MTRSA, Vol. 3, No. 2, Feb. 1963

22. Morrow, JoDean and Johnson, T. A., Material Reasearch and Standards, MTRSA, Vol. 5, No. 1, Jan. 1965

23. J. H. Crews, Jr., and H. F. Hardrath, Experimental Mechanics, EXMCA, Vol. 6, No. 6, 1966, p. 313

24. A. A. Griffith, "The Phenomena of Rupture and Flow in Solids," Philos. Trans., R. Soco Lond., Ser. A., Vol. 221, 1920.

25. C. E. Inglis, "Structure in a Plate due to Presence of Cracks and Sharp Corners," Trans. Inst. Naval Architects London, Vol. 60, p. 219, March, 1913.

26. G. R. Irwin, "Fracture Dynamics, " Fracture of Metals, ASM, 1948, pp. 147-166

27. E. Orowan, "Fracture and Strength of Solids," Rep. Prog. Physics, Vol. 12, 1949, pp. 185-232

49

28. G. R. Irwin, "Analysis of Stress and Strains Near the End of a Crack Traversing a Plate, " Trans. ASME, J. Appl. Mech. Vol. 24, 1957, p. 361.

29. G. R. Irwin, Fracture, Handbuch der Physik, VI, pp. 551-590, Springer-Verlag, Heidelberg, 1958.

30. D. S. Dugdale, "Yielding of Steel Sheets Containing Slits,"J. Mech. Phys. Solids, 8, 1960, pp. 100-108

31. Fatigue crack Growth Computer Programing, "NASA/FLAGRO, Version 3.0,"JSC-22267.

32. P. C. Paris, M. P. Gomez and W. E. Anderson, A Rational Analytic Theory of Fatigue, Trend Engng Univ. Wash., 13 (1), 1961, pp. 9-14.

33.. J. C. Newman Jr., "A Crack Opening Stress Equation for Fatigue Crack Growth," International Journal of Fracture, Vol. 24, No. 3, March 1984, pp. R131-R135.

34. J. R. Rice, "A Path Independent Integral and Approximate Analysis of Strain Concentration by Nothes and Crack," Jouranal of Applied MechanicsVol. 35, 1968, pp. 379-386.

35. A. A. Wells, "Application of Fracture Mechanics at and Beyond General Yielding, British Welding Research Ass. Rep. March 13, 1963

36. J. W. Hutchinson, "Singular Behavior at the End of Tensile Crack in a Hardening Material," Journal of Mechanics and Physics of Solids," Vol. 16, No. 1, 1968, pp. 13-31

37. F.M. Burdekin, and D.E.W Stone, "The Crack Opening Displacement to Fracture Mechanics in Yielding Materials," J. Starin Analysis, 1 (1966)pp. 145-153.

38. B. Farahmand, "Fatigue and Fracture Mechanics of High Risk Parts," Chapman & Hall, Ch. 5.

39. B. Farahmand, "Fracture Mechanics Manual," Boeing Company, R-35-SSC, 1990, pp. Ch. 2.

50

40. W. E. Schall, "Non-Destructive Testing," The Machinery Publishing Co., 1968.

41. Classroom Training Handbook, "Non-Destructive Testing, Liquid Penetrant," General Dynamics, Convair Division.

42. McMaster, R.C., Ed., Liquid Penetrant Tests, Nondestructive Testing Handbook, 2nd Edition, Vol. 2, American Society for Nondestructive Testing, 1982.

43. Classroom Training Handbook, "Non-Destructive Testing, Magnetic Particle," General Dynamics, Convair Division.

44. Schmidt, J.T., Skeie, K., Technical Eds., Maclntire, P., Ed., Magnetic Particle Testing, Nondestructive Testing Handbook, 2nd Edition, Vol. 6, American Society for Nondestructive Testing, 1989.

45. Classroom Training Handbook, "Non-Destructive Testing, Eddy Current," General Dynamics, Convair Division.

46. Ro W. Smilie, "Nondestructive Testing, Ultrasonic," NASA/General Dynamics, PH Division, Inc.

47. Birks, A.S., Green, R.E., Jr., Technical Eds., Maclntire, P., Ed., Ultrasonic Testing, Nondestructive Testing Handbook, 2nd Edition, Vol. 7, American Society for Nondestructive Testing, 1991.

48. Bryant, L.E., Technical Ed., Maclntire, P., Ed., Radiography and Radiation Testing, Nondestructive Testing Handbook, 2nd Edition, Vol. 3, American Society for Nondestructive Testing, 1985.

51

Chapter 2

LINEAR ELASTIC FRACTURE MECHANICS (LEFM) AND APPLICATIONS

2.0 Introduction to Elastic Fracture

In brittle fracture it is assumed that failure in an elastic material takes place when the available elastic energy is adequate to overcome the energy necessary to propagate a crack and to create new crack surfaces. At the instant of instability, the stresses at the apex of an elastic crack must have sufficient magnitude, capable of driving the crack to failure. In other words, one must assume the presence of either one single crack of sufficient length, or a group of smaller cracks in the material, that will eventually join together to form one large crack, capable of creating brittle fracture. The fundamental mechanism by which pre-existing flaws are formed, which ultimately grow and become critical upon application of increasing monotonic load, is not well defined. An understanding of elastic fracture behavior in material may be reached through a microscopic approach to fracture mechanics [1], where the formation of microcracks within grains of the material is assumed upon application of tensile load. The formation of microcracks within the grain, having length much smaller than the grain diameter, is a result of microscopic stress risers, which are created due to the presence of pile up dislocations forming cavities. These cavities are suitable locations for crack initiation [2]. When applied load increases in magnitude, microcracks continue to grow within the grain until they are halted at the grain boundary. Due to the presence of high crack tip stresses, the crack will easily overcome this barrier and penetrate into the neighboring grain. This process will continue until critical Griffith crack length is reached. It is also probable that the mechanism of crack formation would take place simultaneously within other grains in the material. Eventually, smaller cracks with correct orientation with respect to the applied load will merge together to form a larger crack, that could have potential for brittle failure. The growth of each crack through grains within the material consumes energy from the available energy and creates new crack surfaces. Griffith established a relationship between the fracture stress and critical crack length through a balance between the surface energy of the spreading crack and the release of elastic energy supplied by the applied stress field. Section 2.1 briefly discusses the Griffith theory of elastic fracture in brittle material. It's understanding is essential to linear elastic fracture mechanics concept, that allows for a limited amount of plasticity at the crack tip (small scale yielding). Modification to the Griffith theory for ductile metals is presented in chapter 5. Also in chapter 5, the energy dissipated at the crack tip for

large plastic deformation (in addition to the energy consumed in the development of new crack surfaces) is derived.

2.1 Griffith Theory of Elastic Fracture

Griffith's theory, outlined in the "Phenomena of Rupture and flow in Solids" [3] was derived from a consideration of balance between the release of the strain energy of the applied stress and the surface energy of the propagating crack. Brittle crack propagation under the fixed grip condition begins because the decrease in stored elastic energy of the system becomes available and is greater than the energy necessary to create two new crack surfaces. For brittle materials that do not exhibit plastic deformation at the crack tip, the critical condition for instability can be formulated as:

~'C [Ue -US ]=0 1

where U e is the elastic energy of the system per unit thickness available to create the new crack surfaces resulting from a sharp elliptical crack of length 2c in a plate of infinite size under a tensile stress, a, normal to the plane of the crack. The term U s is the elastic surface energy and is defined as the energy consumed per unit thickness in breaking atomic bonds to elongate the crack at the tip, and to create the new crack surfaces. In establishing a relationship between the applied stress (fracture stress, a) and critical half crack length, c, Griffith assumed the existence of an elliptical hole in the material with major sem-iaxis, c, with its orientation perpendicular to the direction of applied stress, a, see figure 1. The maximum stress generated at the apex of the major axis of an elliptical hole by the applied forces or by the elastic energy of the system, has been formulated by Inglis [4] as:

am~ = a (1+ 2c/b) 2a

where b is the minor sem-iaxis of the ellipse. The radius of curvature, p, at the apex of the ellipse is b2/a. For a small value of notch raduis, p, the maximum stress at the tip of a sharp ellipse of length 2c is given by:

a=,, = 2a ~/(c/p ) 2b

The amount of elastic displacement at the mouth of the crack, 2v, can be evaluated to formulate the elastic energy available by the applied remote stress, a. This can be accomplished by calculating the work done in elastically deforming the two crack surfaces. The total value of

53

2v = 4 o / E [(1 -v2) (c 2 - x2) 1/2

Figure 1: Griffith elliptical hole with major semi-axis perpendicular to the direction of applied stress.

displacement, 2v, at the mouth of the crack, in the direction of applied stress, (~, can be used to calculate the work done on the body. The total elastic displacement of the upper and lower crack faces, 2v, is given for the plane strain condition as [5]:

!

2v = 4°(1-v2)(c 2 - x 2 ) 2 3 E

where E and v are modulus of elasticity and Poisson's ratio, respectively. The elastic strain energy per unit thickness, U e, is defined in terms of displacement, 2v, as:

Ue = 2 7 ]-.-(2v)G dx O:Z

3a

After simplifying the integral of equation 3a by replacing 2v with equation 3, the quantity U e in its final form is given by:

xo2c 2 U e - - -

E 3b

54

and the elastic surface energy, U s, per unit thickness is given by [6] :

Us = 4cT 4a

where T is the surface tension of the material, that is, the work done in breaking the atomic bonds. For brittle material, the surface energy, T, can be written as [7]:

T = 13 Ec{o/20 4b

where C~o is the atomic spacing in Angstroms and 13 is 0.394 x 10 -8 inch/Angstrom. For an elliptical center crack of length 2c in an infinite plate loaded by a uniaxial far field stress, (~, the Griffith energy balance approach to elastic crack instability can be expressed as (applying equations 3b and 4a into equation 1)

rc~ 2c

E =2T 5

The expression on the left of equation 5 is the elastic energy release rate per crack tip (or the crack extension force, G [8]) and the expression on the right side is the energy absorption rate, per crack tip, for creation of new crack surfaces. The critical value of energy release rate, Gc, is a constant for truly brittle materials because T is a constant.

The Griffith energy balance approach described by equation 5 simply states that, when an ideally sharp crack of length 2c in an infinite plate is subjected to applied stress, (~, instability occurs by formation of two new crack surfaces. Equation 5 is applicable to the analysis of structural components where the condition of elastic crack instability as the result of monotonic tensile load is necessary. However, this approach is not applicable to other important loading cases, such as stable crack growth that occurs when a crack is subjected to cyclic loading. A fracture mechanics parameter that can characterize the crack tip behavior under cyclic loading environment, is available for fracture mechanics application and is called the stress intensity factor (designated by K). The stress intensity parameter is a useful tool that can address the cracking problems that are of interest in aircraft, pressure vessels, ships, and space vehicle structures, such as stable crack growth under fluctuating load, as well as the condition of instability when small scale yielding prevails. The two fracture parameters, elastic energy released rate, G, and the stress intensity factor, K, are linked together in the energy balance equation by Irwin [9] as:

55

~ 2c (c V/-~-c) 2 G - - - - - K2/E 5a

E E

The stress intensity factor approach for characterization of the crack tip environment, using linear elastic fracture mechanics, will be discussed throughout this chapter. The fatigue crack growth concept and structural life assessment using the stress intensity factor range, AK, will be presented in chapter 3

2.2. The Stress Intensity Factor Approach, K

2.2.1 General

As was mentioned in section 2.1, the Griffith energy balance approach to brittle crack problems, expressed as the energy release rate (designated by the letter G) leads to the same results as the stress intensity factor (see equation 5a). The stress intensity factor, K, characterizes the crack tip stress field under small scale yielding and it's applications to fatigue crack growth and life prediction problems make it a very important parameter in the field of linear elastic fracture mechanics. For this reason, it's derivation will be briefly discussed in section 2.2. The critical value of the stress intensity factor, called the "fracture toughness", is discussed in section 2.3. In section 2.6, methods of constructing the residual strength diagram for a single load path and built-up structures (such as skin stiffeners structures) is discussed. The resistance curve approach, known as the R-curve when plasticity is confined to a small region at the crack tip is also introduced.

The failure of structural metals when subjected to a fluctuating load environment, is in many instances, a result of their surfaces exposed to scratches during the machining process, from corrosion pitting, or through surface damage from improper handling, all of which could be a source of crack initiation. The behavior of these surface cracks and their growth can be assessed using the stress intensity factor parameter. The derivation of crack tip stress intensity factor for surface cracks is presented in section 2.8. The critical value of stress intensity factor for surface cracks is also presented in this section. Finally, a brief discussion of the plane strain, Klc, and plane stress, K c, fracture toughness testing is included in section 2.9.

2.2.2 Crack Tip Modes of Deformation

The stress fields and displacement modes at the crack tip can be classified into three types (see figure 2 for crack surface displacement in the x, y, and z directions):

56

Y Case (a) , ~

V X Z Case (c)

Opening Mode Sliding Mode Tearing Mode

Figure 2: Illustration of the three modes of crack surface displacement

I) The opening mode (Mode I) is characterized by displacement of the two crack surfaces moving directly apart from each other. In this case, the applied load and displacement, v, are perpendicular to the crack surfaces. The stress intensity factor corresponding to this mode is denoted by K I, see figure 2, case (a).

II) The shear or sliding mode (Mode II) occurs when the two crack surfaces are displaced by sliding over each other. The direction of the applied load and displacement, u, are parallel to the crack surfaces, see figure 2, case (b).

III) The tearing mode (Mode III) occurs when the crack surfaces slide over each other in a direction parallel to the leading edge of the crack, see figure 2, case (c).

Note that the symbols I, II, and III are Roman numerals which refer to the modes of fracture. The opening mode is shown by Mode I. The shear and tearing modes are represented by II and III, respectively.

The tensile opening mode, or Mode I, type of failure represents the most frequent type of separation that engineers design against and must be prevented. While failure of structural parts by Mode II or Mode III, where fracture is induced by shear stresses, is also possible, these types of fracture seldom occur during the service life of the part. In addition, most of the available test data generated for the critical value of the stress intensity factor, K, (fracture toughness) has been established for Mode I type failure, see appendix A. Therefore, throughout this book, attention will be given to the opening mode, Mode I, and its application to fatigue crack growth when evaluating the service life of a structural part.

57

2.2.3 Derivation of Mode I Stress Intensity Factor

Linear Elastic Fracture Mechanics (LEFM) is based on the application of classical linear elasticity to cracked bodies. The elasticity assumptions made in deriving the stress intensity factor are that displacements are small and the linearity between stresses and strains must be maintained. The assumption that the material is both homogeneous and isotropic is also included in the derivation of the stress intensity factor, K. The use of the crack tip stress intensity factor, K, for assessing behavior of small cracks is not recommended. For more information pertaining to anomalous behavior of a small crack, where crack dimensions are on the order of the material microstructure, the reader may refer to chapter 3.

The following steps are used in deriving the Mode I stress intensity factor. No attempt is made to include all the derivations. The reader may refer to References [9] and [10] for a more in-depth treatment of this topic if he so chooses.

1) The equilibrium equations of stresses 2) Hooke's law, and the strain-displacement relationship 3) The compatibility equations of strains 4) The Airy's and Westergaard stress functions 5) The use of boundary conditions for a sharp elliptical crack 6) The stress intensity factor equation for Mode ], and combined

loading.

In solving any two-dimensional stress problem, the equilibrium equations that describe the relationship between forces in the x and y directions can be written as [11,12]:

a(~ x & xy - - + = 0

ax

0~y &xy - - + = 0

6

From the above two equilibrium equations, there are three unknowns ((~x, %, ~xy) and two equations.

The stresses at each point in the body are related to the strains within the elastic range by Hooke's law as:

58

E~ x = o x -Very

E~y = Oy - w x

Ey xy = T

2 ( I + v ) x:v

7

where E and v are the modulus of elasticity and the Poisson's ratio, respectively.

The strain-displacement relationship can be written as:

0U

I ~ X = - - 0x

0v Y 0y

au av -- + Y xy ay ax

8

where u and v are displacement functions in the x and y directions respectively. From equation 8 the strain-compatibility equation is:

021xy 02 a.J 02 Ov 9a

In terms of strains, equation 9a becomes:

02y xy 02 02 - - - (~x) + 0-~x (Cy) Ox0y a2y

9b

From Equations 6, 7, and 9b, there are six equations and six unknowns, so that together with the boundary conditions, the state of stress at each point in a body subjected to the plane stress condition can be solved. Consider the Airy stress function (1)(x,Y) such that:

a 2 $(x, y) a 2 $(x, y) 02 $(x, y) Oy- ~ Oy ~ ~ =

02x 02x xy 0xay 10

59

Using equation 10 to replace ax, a., and -c_, it is easy to demonstrate that the equilibrium condition des'cribed ~y equation 6 is satisfied. Writing equation 7 in terms of the Airy stress function and substituting the results into the compatibility equation described by equation 9b, we obtain:

v 4 (~) = v 2 [v 2 (~)] 11

where the quantities V 4 (~) and V 2 (~) are called the biharmonic and harmonic functions, respectively. Any function that satisfies equation 11 is called an Airy stress function. Selecting an appropriate stress function, Z(z), that can satisfy all boundary conditions for a center crack panel in an infinite plate subjected to biaxial stress state (see figure 3), enables one to evaluate the crack tip stresses, a x, ay, and

a (=)

/ °x,xv and ~v are crack tip Y ~_ ~ ( _._~' ~ - - / stresses " ~r ~ -~ "~x ~

~ / Crack tip field under Mode I subjected ~ 1 ~ to biaxial far field tensile stress

Figure 3: Illustration of a biaxially loaded plate containing central crack

"~xy" Writing the Airy stress function in terms of its real and imaginary components of the second and first order integrals of the Westergaard stress function, Z(z) [10]:

=Re Z(z) + ylmZ(z) 12

m

where Z(z) = dZ(z)/dzand Z(z)=dZ(z)/dz. Applying equation 12 into equation 10 to obtain stresses (ax, %, and "Cxy), and by using the Westergaard stress function that can satisfy the boundary conditions of figure 3, the crack tip stresses can easily be derived. The function that fulfills these boundary conditions is [13, 14]:

60

oz 13 Z(z)- ~z2 _c 2

where z = re i° . After algebraic simplification, the crack tip stresses in their final form can be written as:

O'X= - - c o s O / 2 1 1 - s i n e / 2 s i n 3 0 / 2 ] 14a

a Y = 2x~n r - - c o s O / 2 [ 1 + s i n e / 2 s i n 3 0 / 2 ] 14b

XxY = s i n O / 2 c o s e / 2 c o s 3 0 / 2 14c

Note that, for the plane stress and the plane strain conditions, the quantity a z = 0 and a z = v (a x + ay) respectively.

The derivation of equation 14 is based on the assumption that the highly stressed region at the crack tip is localized and the quantity r is limited to a small zone at the crack tip where the term r/a=0. From equation 14, it can be seen that there is a stress singularity at r=0 and it implies that, as the distance r from the crack tip decreases, the magnitude of the stress increases and approaches infinity. We will show in section 2.7 that most aerospace materials will deform plastically and yield at the crack tip before the stress becomes large.

The crack tip stresses described by equation 14 can be written in terms of their radial and angular positions, (1H2rcr)qJ(0), and the quantity al~lrcc that defines the magnitude of the crack tip stresses in terms of applied stress, a, and half crack length, c. The second quantity, alq=c, is called the mode ] (or opening mode) stress intensity factor and is designated by the symbol K~ having units of (ksi'qin.). Thus, the stress intensity factor, K,, can relate the crack tip stresses (ax, ay, and ~xy) to parameters that are measurable quantities (a and ~/=c). The stress intensity factor should not be confused with the stress or strain concentration factor (K t or K~) or any other similar symbols.

The influence of external variables, such as the magnitude and method of loading, and the crack geometry, on the crack tip stresses

61

can also be described by the stress intensity factor. The general form of the stress intensity factor, including the effect of the crack geometry and loading condition, can be written as:

K,=i31 132 cr ~j-~ 15

where 131 and 132 are corrections to the loading and crack geometry, respectively.

The critical value of Kl=(~=~/=c is called the fracture toughness (Kc) and is a measure of the material's resistance to unstable cracking. Irwin [15] failure criteria simply states that if the level of the crack tip stress intensity factor exceeds a critical value, Kcr, unstable cracking will occur. This is analogous to the case of a static stress at a point that exceed the criteria for yield or ultimate strength. The critical stress intensity factor and the material's resistance to fracture (fracture toughness) are discussed in sections 2.2.4 and 2.3, respectively. A brief description of the ASTM procedures for the determination of plane strain and plane stress fracture toughness is presented in section 2.9.

The stresses defined by equation 14 were derived for the case of an infinite plate containing a sharp center through crack that is subjected to a remotely applied biaxial load. For the case of a uniaxial tensile load the quantity C~y(OO) must be subtracted from the crack tip stresses. However, because the crack tip stresses are much higher than the applied stresses, the effect is negligible. The remote stress (~x(OO) in the x direction is parallel to the crack surfaces and has little or no influence on Mode ! crack opening or the stability.

When the applied load is perpendicular to the crack surfaces, the variation of crack tip stresses (local stresses, (~x and ~y ) along the x axis, where 0 =0, is illustrated in figure 4. The reader should note that there is a local stress, (~x, even though the applied load is in the y direction. The induced stress, ~x, is as the result of straining in the x direction due to Poisson's effect. In addition, there is always a transverse stress (through the thickness), ~z, whose value is a maximum at the center of the crack tip. The magnitude of ~z drops at the two edges and becomes zero at +t/2, as shown in figure 4. For the plane stress condition, the quantity (~z is practically zero and for the plane strain and mixed mode the quantity (~z = V(Gx +~y).

The stress intensity factor solutions for several crack geometry cases that are important for use in aircraft and space vehicle applications, subjected to combined applied load are documented in reference [16,17].

62

u~

Ul

~3

T/i I° _ .egionofcrack,ip

-t/2 U2 v , ¢ ~ I~ localized stresses,

I 1 . ~ 1 °x and GY ~ . Crack tip stress, c~/ I I - J (plane stress

--~""-- -==---- - Far field stress Crack tip stress, o x

II,, Distance From Crack T ip

Figure 4: Stress variation in the region of crack tip

2.2.3.1 Stress Intensity Factor For Combined Loading

When a cracked plate is subjected to more than one type of induced mode I load (combined loading), the resultant stress intensity factor is additive. For example, for the case of a center through crack in an infinite plate that is subjected to an axial tensile load, P, and bending moment, M, the total or combined stress intensity factor characterizing the crack tip stresses can be written as:

K Total = KAxial + KBending

In general, a combined or superposition approach to the stress intensity factor of a crack plate subjected to different loading conditions, each of which produce mode I fracture, can be expressed as [17]:

K = (o o ~o + ~, J3, + o 2 1~ 2 + °3 1~3) 1 6

where % ~1, o2, o3 are the applied stresses due to tension, bending through the thickness, bending through the width, and pin bearing, respectively. Figure 5 case (a) illustrates different kinds of loading conditions which can induce a mode ! crack tip stress intensity factor. In equation 16, the quantities 130, 131, 132, and ~3 are the corrections to the crack geometries and loading conditions. One example of combined loading is a bolted joint containing plates that experience most of the load types described by equation 16 (local plate bending due to bolt tension, bearing stress due to shear, and tension due to far field stress), see figure 5 case (b).

63

Pin bearing

Case (a)

Bending Through the thickness Pure Tension Bending through

the width

Case (b) ( 1 ) ~ ,

Crack emanate from a hole

Combined Loading

-- Pure Tension (1) -- Bending Through

the thickness (2) -- Bending through

the width (3) -- Pin bearing (4)

Note : Load case number (3) does not apply to this crack geometry

Figure 5: Illustration of crack plates subjected to different loading conditions

The principle of superposition is helpful, not only to evaluate the stress intensity factor solution of a cracked plate of known geometry subjected to different loading conditions (as long as the crack geometry remains the same for all loading cases), but also to obtain the stress intensity factor solution for crack cases where the solution does not exist or is difficult to obtain [18]. The reader should pay attention to the fact that when dealing with combined loading it is understood that each individual load contributes to the opening mode I of fracture only. The effective stress intensity factor is the superposition of all the loads as described by equation 16. The K solution for the case of a circular shaft subjected to combined axial and torsional loading can no longer be described by the superposition criteria (shown by equation 16) because torsional applied load does not produce mode I fracture for the same crack geometry. The important point to remember is that at any point in an elastic body, no matter how complicated the geometry of the part or applied load is, the state of stress at a point in a body can be given by the principal stresses.

64

2.2.4 Critical Stress Intensity Factor

The critical value of the stress intensity factor is an important parameter in the field of fracture mechanics when dealing with structural failure resulting from unstable cracking. It is analogous to other critical values in structural analysis, such as tensile yield or ultimate strength of the material. The critical value of the stress intensity factor at which unstable crack propagation occurs is called the fracture toughness. In general, the critical value of K is thickness dependent and its laboratory value decreases as the material thickness increases. For a thick section of a given material in which the plastic deformation at the crack tip is constrained and negligible (plane strain condition), the critical stress intensity factor for mode ! at instability is designated by Klc. The failure criteria for unstable cracking when the plane strain condition prevails can be written:

K > Klc 17

Equation 17 simply states that abrupt failure occurs when the applied crack tip stress intensity factor, K, reaches or exceeds the material's fracture toughness, Klc. The plane strain fracture toughness, Kic, is dependent on the type of material, loading rate and temperature however, it is independent of crack length and part thickness. It is important to note that in performing a fracture mechanics analysis for the plane strain condition, both K l and Kic are needed and must be available to the analyst. This is analogous to the strength analysis requirement that the calculated limit stress (also called the working stress or service stress), (~, must always be compared with both the yield, (]'Yield, and the ultimate, CUlt, strength of the material. That is, the analysis would be incomplete without both material allowables.

As the material's thickness decreases, the constraint to plastic flow decreases, and the state of plane stress is reached. The fracture toughness associated with the minimum thickness is called the plane stress fracture toughness and is designated by K c. Fracture toughness for thicknesses smaller than the plane strain condition is thickness and crack length dependent. The failure criterion described by Equation 17 is valid when the plastic deformation at the crack tip is assumed localized and small (called small scale yielding), since the assumptions stated in the formulation of the stress intensity factor are based on linear elastic analysis, see section 2.2.2. A brief presentation of the ASTM testing method to obtain the plane strain and plane stress fracture toughness values of isotropic materials is made in section 2.9. The reader may refer to references [18,19] for a detailed description of ASTM testing practice.

65

Fracture toughness of low strength tough metals where plasticity at the crack tip is no longer localized can be obtained through the J- integral concept via a multispecimen or single specimen technique, described in chapter 4. Fracture Mechanics of Ductile Metals (FMDM) can evaluate material fracture toughness for both small and large scale crack tip yielding. The FMDM theory is presented in chapter 5 and is a valuable tool for obtaining both Kic and K c fracture toughness values without going through complicated and costly ASTM test procedures.

Once fracture toughness values are available, it is easy to determine the critical flaw size for a given stress level where structural instability takes place. When the load environment is fluctuating in nature, the fracture toughness data (critical value of stress intensity factor) is essential for a reliable crack growth analysis and should include both plane strain and plane stress as well as part through fracture toughness. These critical values are used in fatigue crack growth analysis to lay down the upper bound of stress intensity factor used in the fatigue crack growth curve. In section 2.3, the concept of fracture toughness is discussed and later, in section 2.8, surface cracks and their critical values (part through fracture toughness) are briefly reviewed. Extensive fracture toughness data for many aerospace alloys is available in Appendix A for use in fatigue crack growth analysis of structural components.

2.3 Fracture Toughness

Fracture toughness is defined as the resistance of the material to unstable crack growth in a non-corrosive environment. This parameter characterizes the intensity of the stress field at the locality of the crack tip when unstable cracking takes place. The plane strain fracture toughness, K[c, is believed to be crack length and thickness independent; however, it can vary as a function of temperature and strain rate. The letter I in the subscript is used throughout the literature to indicate the instability as the result of mode I crack propagation with cleavage type fracture (flat fracture surfaces) without shear lip.

For typical metals the plane strain fracture toughness, K[c, is associated with thick sections. Under the crack tip plane strain condition, the state of stress at the crack tip approaches a triaxial tensile stress. Under this circumstance, the crack tip yielding is constrained. Experimental data have shown that for homogeneous material under plane strain condition, where stress in the thickness direction is no longer small and negligible, the yielding occurs at a much higher stress level compared with uniaxial tension. The plane strain condition prevents the crack tip plastic deformation. In certain circumstances when principal stresses become almost equal to each

66

other in magnitude, the material will stay elastic with little plastic deformation up to failure. However, at the two extreme free edges, where there is no contribution from the stress in the thickness direction, the state of stress is biaxial and fracture occurs by a shear mechanism at a 45 degree plane with respect to the flat surfaces. At the center of a thick plate near the crack tip, where the plane strain condition exists, the principal stresses can be almost equal in magnitude. That is, the maximum shear stress, "~max, (difference between the principal stresses) is negligible, and thus does not allow plastic deformation to occur by the slip mechanism at the crack tip [20,21]. This phenomenon is expected to occur at the center of the material thickness and is shown in figure 6. The constraint to plastic deformation reduces for elements which are away from the center, and at the two free surfaces (where (~z =0) slanted failure occurs, see edge elements shown in figure 6. The material behavior in the locality of the crack tip through the part thickness (at the middle and the two

Crack Tip Behav io r

Center Element

Center elements are prevented from plastic deformation. Side elements are not constraint through the thickness and will deform plastically.

(plane strain)

Side Elements r ~ rk (plane stress) / Contraction

y .

Figure 6: Illustration of the state of stress at the crack tip for plane stress and strain conditions

ends) can be analogous to several tensile bar specimens that are situated next to each other. Those tensile bars at the two edges have the flexibility to deform and exhibit reduction of area (contraction) due to Poisson's effect. In the plane stress condition, where (~z=0 and ~z~0, the material behavior due to Poisson's effect, is illustrated by figure 6 whereby contraction in the z direction and consequently the formation of plastic zone are permissible. In the plane strain condition,

67

where ~z=O and az~O, the bars in the middle of the specimen are prevented from contraction, which results in developing stress in the z direction and therefore blocking the formation of plasticity.

Figure 7 shows the variation in the amount of flat fracture (failure by cleavage) as a function of thickness. For plane strain failure (the region of Kic ), the portion of the flat surfaces is much larger than the slanted sections. On the other hand, for thin sections where the state of stress at the crack tip is biaxial, the constraint to plastic deformation lessens and the failure is associated with plane stress. The amount of flat fracture under plane stress conditions reduces considerably. For

K c

c;

u) u)

J¢ ol o

I - K I c -

...I

¢J ¢11 t .

U . .=,.,

+ P l a n e stress thickness

(tO

~ Slanted fracture

t 3 > t 2 > t l ~ -" -- - tl I ~ / 1 / I

I t Flat fracture I ~1 / (cleavage)

3

Thickness, t - ~ % Flat Fracture ---~ Plane strain thickness

(tlc)

Figure 7: Variation of K c and amount of flat fracture versus thickness

other thicknesses, in which plane strain and plane stress are combined, the state of stress is termed mixed mode. Figure 7 also shows the variation in fracture toughness for three regions (plane strain, mixed mode, and plane stress) as a function of the material's thickness, with the amount of flat and slanted surfaces corresponding to each region. The asymptotic portion of the fracture toughness curve is associated with plane strain fracture toughness and is thickness independent (figure 7). For thicknesses smaller than the plane strain value (t < tic ), mixed mode fracture toughness is obtained and the maximum fracture toughness value, K c, corresponds to minimum thickness, t c, on the curve.

68

2.4 Material Anisotropy and its Effect on Fracture Toughness

In most materials, the fracture toughness varies with crack orientation and loading direction. This is due to the anisotropic nature of the material that evolved during its manufacturing process, cold rolling or heat treatment, see figure 8. The ASTM E-616 coding system for manufactured material having rectangular cross-section is such that the longitudinal direction which grains are elongated (due to the working process) is designated by L as shown in figure 8. The other two directions are the long transverse, T, and short transverse, S. In fracture mechanics, the ASTM coding system contains two letters when dealing with material fracture toughness and fatigue crack growth properties. The first letter denotes the loading direction and the second letter represents the direction of expected crack propagation (see figure 9). The same system is applicable to bar and hollow cylinders. The ASTM standard nomenclature relative to directions of

I Original width

Working direction, L

I A

Final width

F-

k Elongated grains after rolling

L = Longidutional Direction T = Long Transverse Direction S = Short Transvers(~ Direction

Figure 8: The anisotropic nature of the material that evolved during its manufacturing process

Mechanical working (elongated grain direction) for rectangular cross sections is:

L = Direction of maximum deformation (maximum grain elongation). This is the longitudinal direction of the rolling, extrusion, or forging process (figure 9).

T = Direction of minimum deformation. This is the direction of long transverse (figure 9).

69

S = Direction perpendicular to the plane of L and T. This is the direction of short transverse (through the thickness), figure 9.

S

Load in the T direction Lom ~ Load in the L direction T - S ~ Crack in the S direction

Crack in the T direction

T-L - ~ Load in the T direction S-L - - ~ Load in the S direction Crack in the L direction Crack in the L direction

Figure 9: ASTM crack plane orientation code designation for rectangular and cylindrical cross sections

For fracture properties determination using the ASTM testing procedures, the crack orientation code L-T indicates that the loading direction is in the longitudinal direction, L, and the direction of propagation is in the direction of long transverse, T. It is worthy to mention that the designated orientation code for non-cracked parts contains only one letter. For example, the letter L designates material static properties (not fracture properties) in the direction of maximum grain elongation.

For cylindrical sections where the direction of maximum deformation is parallel to the direction of principal deformation (for example: drawn bar, extrusions or forged parts with circular cross section), a similar system of nomenclature for the three directions is:

L= Direction of maximum deformation (longitudinal direction)

R = Radial direction

C = Circumferential or tangential direction

70

Figure 9 schematically represents the loading and crack propagation directions for cylindrical sections. For example, the two letter code L- R indicates that the loading is in the longitudinal direction (L) and the expected crack propagation is in the radial direction (R). It is important for the analyst to specify the crack and loading directions corresponding to a given fracture toughness for the part under consideration.

2.5 Factors Affecting Fracture Toughness

The variation of fracture toughness with respect to material thickness was discussed in the previous section and is depicted in figure 7. From this figure, it can be seen that several specimens with various thicknesses (including plane strain fracture toughness, Klc ) are needed to generate such a curve. A minimum of three fracture toughness tests per given thickness is required to establish a complete curve that can fully describe the variation of K c versus thickness, t. When data points are not available for analysis purposes, the following relationship may give somewhat reasonable results, as long as the plane strain fracture toughness, Kic, is available for the material under study. Irwin's equation describes plane stress fracture toughness, K c, written in terms of K]c, plate thickness, t, and yield strength of the material, (~Yie~d, as [22]:

].4 Kic Kc = Kic [1 +__( )2] 18 t a

Yield

Another useful empirical equation describing K c in terms of Klc is available in NASA/FLAGRO and is shown by equation 49 of section 2.9.

Other factors affecting the fracture toughness values and fatigue crack growth curve are mechanical working, loading rate, temperature, temperature rate and yield strength, see figure 10. From figure 10a, it can be seen that, for aluminum alloys the response of fracture toughness to temperature change is increasing as temperature decreases, whereas an opposite trend was observed to be true for most ferrous alloys possessing ferritic or martensitic microstructure [23,24]. Experimental test data obtained in the laboratory indicates that most aluminum alloys possess a higher fracture toughness value at the liquid nitrogen temperature (-320 o F) than at room temperature, see Appendix A [25]. This positive trend in fracture toughness value is desirable in aerospace, aircraft and pressure vessel structures, when a proof test at both room and liquid nitrogen are required. A higher fracture toughness at liquid nitrogen indicates that if a proof test at room temperature is successfully completed, there is no need to

71

T ~ (a) L

Temperature ~ (b) ~/emperature

"/A,u inu Ferrous Alloy

v

Loading Rate Loading Rate T (c)T Kc Ferrous Alloy Kc Aluminum Alloy

Tensile Yield Tensile Yield

Figure 10: Illustration of fracture toughness with temperature, loading rate and tensile yield value.

conduct an additional proof test at liquid nitrogen. Study on this topic for aluminum alloy (2014-T6), Maraging steel, Inconel 718, and Titanium alloys revealed that the stress to failure resistance under high cycle fatigue environment increases at the cryogenic temperature as compared to room temperature. At low cycle fatigue, however, the number of cycles to failure increases as temperature increases [26].

In the case of ferritic and martensitic steels, where the material possesses sufficient ductility and good fracture toughness at room temperature, cleavage or brittle fracture can occur at a service temperature below their transition temperature. This change in material properties is known as ductile-brittle transition (see figure 11) and was the cause of many cleavage failures occurring in ships, pressure vessels, bridges and tanks [27,28]. Apparently at lower temperature, the amount of slip representing ductile fracture behavior reduces. As the temperature approaches absolute zero, a slip mechanism tends to disappear totally. Therefore, the material becomes brittle and the fracture is of cleavage type.

The fracture toughness for aluminum alloys seems not to vary with loading rate, whereas ferrous alloys (such as ferritic and martensitic

72

steels) are shown to be sensitive to this parameter, as indicated in figure 10b. Under high rate of loading, the slip mechanism for ductile

Scale

Charpy Strike Specimen ~1'

Brittle

/ I I Transition ~hl temperature j ~.~eratu r b~e ~

Temperature

Strike mass

Figure 11: Illustration of Charpy impact energy versus temperature.

fracture is retarded and the material behaves in a more brittle manner. When combining the effects of temperature and rapid rate of loading for ferritic steel, the Charpy impact test (under different temperature environments) can provide information on the energy absorption for fracturing a notched specimen [29]. Figure 11 shows the variation of Charpy impact absorbed energy (representing the material's notch ductility) versus the temperature for most ferritic steels. Originally, the Charpy Impact Test was developed to relate the amount of energy absorbed by a material when loaded dynamically in the presence of a notch at room temperature. Later it was realized that the absorbed energy is a function of temperature for several ferritic steels. If several notched bar impact test specimens (shown in figure 11) made of ferritic low carbon steel are impacted by the pendulum of a Charpy Impact Test machine, the energy absorption value recorded on the machine shows a decrease in value when the temperature falls below room temperature, see figure 11. This can be an indication that the material has gone brittle (plane strain mode of failure). The low energy absorption value recorded by the Charpy Impact Test machine, when compared with room temperature, indicates brittle behavior of the material due to a low temperature environment. As a final remark, for the Charpy V-notched test to be meaningful, other parameters, such as thickness, the rate of loading, and specimen geometry, must be kept constant. For more information related to this topic, the reader may refer to ASTM E-23 "Standard Methods for Notched Bar Impact Testing of Metallic Materials,".

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In general, materials with Body Center Cubic (BCC) crystal structure (ferritic steel) show a reduction in the fracture toughness with an increase in rate of loading. Material with Face Center Cubic (FCC) crystal structures are not sensitive to the rate of loading. In addition, FCC materials do not exhibit brittle behavior when exposed to cold temperature. However, the fracture properties of ferritic and martensitic steels (which have BCC crystal structure), will degrade as the ambient temperature drops below room temperature. A comprehensive review related to the effect of temperature, loading rate, and plate thickness on fracture toughness is available in reference [29] by Rolf and Barsom.

The yield strength varies due to the heat treatment given to the material and this variation is illustrated in figure 10c. When an alloy is subjected to a different level of heat treatment, its fracture properties increase as the tensile yield value decreases. In selecting a high strength material with little ductility (increase in yield value after heat treatment) in order to reduce the size of the structural part and save weight, the engineer must be aware that the fracture toughness value has been reduced considerably. That is, as the ability of the material to absorb energy and deform plastically decreases, the size of a flaw that could initiate instability becomes very small.

Additional parameters that may effect the fracture toughness value and can be important when dealing with the fracture analysis of a structural part include:

• The coarse grain size may result in lowering the fracture toughness value.

Embrittlment (i.e. segregation of Phosphorous, P, Nitrogen, N, or possibly Sulfur, S, to the grain boundary causing an intergranular mode of fracture) due to microstructure or environmental contamination can result in lowering the fracture toughness value.

• Work hardening lowers the fracture toughness value by elevating the material tensile yield value and reducing the ductility.

2.6 Residual Strength Capability of a Cracked Structure

At the atomic level, the fracture phenomenon occurs when the bonds between atoms break. For materials with perfect crystalline structure, called "whiskers", the measured fracture strength is much higher than the value obtained in the laboratory by testing a typical standard tensile specimen. Typical aluminum alloys that are used commercially to produce aircraft or space vehicle parts fail at applied stress around 100 times or more lower than their theoretical strength values (this value can be approximated by El10, where E is the

74

modulus of elasticity of the material [30,31]). The differences in strength magnitude between the actual and theoretical values has been recognized by scientists for a long time, who believed that for a given alloy, the actual tensile strength obtained from a laboratory test should have a lower value than the theoretical value estimated by El10 derived based on the force necessary to break atomic bonds. This discrepancy was the result of imperfection and defects in the material, such as missing atoms, dislocations, grain boundaries, and cavities. Upon conducting tensile tests with glass fibers of fine diameter in which the probability of defects per volume is low, Griffith [33] showed a tensile strength value of 5x105 psi (= El10). He postulated that this result could be much lower if the diameter of the test specimen was significantly larger, due to the presence of more defects. Indeed, Weibull, in 1939, was the first to apply statistical methods to brittle material to explain the greater probability of fracture as a result of finding more cracks in a larger specimen than in a smaller one, leading to the possibility of the existence of a size effect on the fracture stress.

From a macroscopic point of view, the presence of a crack introduced in a structure due to manufacturing, machining, or improper handling, will significantly lower the strength of the structure as compared with the uncracked condition. The amount of strength that is left in a structure after crack initiation, that must withstand the service load throughout it's remaining design life, is called the residual strength. For uncracked structures, the load carrying capacity or residual strength capability is simply the ultimate strength of the material. When the applied load in an uncracked structure exceeds the ultimate strength of the material, failure will start to occur.

The existence of a crack in a structure will result in lowering the residual strength of the structure below the ultimate of the material. That is, when the load on the structure exceeds a certain value called the critical stress, o.cr, unstable cracking will occur. For brittle materials or material with low fracture toughness, unstable cleavage cracking is associated with fast fracture, causing complete fracture of the structural part. Total failure will occur in an explosive manner by releasing the elastic energy that had been stored by the external forces. Figure 12a illustrates the abrupt failure (brittle failure or fast fracture) of three specimens with crack lengths, a 1 >a 2 >a 3, in which the failure stress for each individual crack length is o.1 <o.2 <o'3, respectively. Figure 12b also illustrates the abrupt failure points for the same crack size and crack geometry but corresponding to a different material possessing a higher fracture toughness value, (Klc)2 > (K[c) 1. For comparison, the fracture points associated with figure 12b are also plotted in the same figure. It can be seen that the critical stress values (residual strength values) for the same crack lengths are

75

higher for the second material (o2> o1), since it has a higher fracture toughness value.

== .=

Fracture Behavior of Britt le Matedal

Crack Initiation & Propagation

0" 13 i ~ Fast Fracture

12 Fast Fracture

2ttal [ 1 _ Fast Fracture 0" 1 ~.-

0"3> 0"2 > a 1 a a 2 1

Crack Length, a (a)

al> a2> a 3 I, Initiation

I ~ Mat. 1

1 1 2 T ~ Fast Fracture

~. I - L ~ i l i l ~ F a s t F r a c t u r e

a31a lal I . _ v

Crack Length, a (b)

Figure 12: Illustration of abrupt failure for three crack sizes

For material with adequate fracture toughness however, the existing crack may exhibit plastic deformation at its tip, and grow in a stable manner until it becomes unstable. Figure 13 illustrates the variation of applied stress as a function of crack length, a+Aa, for three specimens with different initial crack lengths, a 1 >a 2 >a 2 within small scale yielding. The onset of stable crack growth is shown by the letter 1, and as the applied load increases, stable crack growth takes place at the crack tip. Upon further increasing load, critical crack length is reached and instability of material is expected, as shown by the letter F in figure 13. From figures 12 and 13, it can be concluded that the load carrying capacity of a cracked structure is a function of the crack size, a, material fracture toughness, Kic or K c, and also the crack correction factor, 13.

2.6.1 Residual Strength Diagram for Material with Abrupt Failure

The residual strength capability diagram for material with abrupt failure can be plotted by simply employing the equation of the stress intensity factor, K, to relate the critical applied stress, o.cr, to the critical crack length, acr, for a given crack geometry and by replacing the stress intensity factor at its critical value, Kcr, with the material's plane strain fracture toughness, Klc, obtained through testing. That is:

K ) Kcr = Kic, where K=I3 o. (=a) 1/2 19a

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Figure 14 shows the plot of the residual strength diagram from equation 19a, in terms of the critical stress, ~cr, versus the critical crack length, acr, for a center through crack structure where the correction factor 13=1. It is obvious from figure 14 that, as the crack length increases, the load carrying capacity of the cracked structure is reduced, and, if it falls below the maximum design stress level, failure can be expected.

== ==

<C

Fracture Behavior of Ductile Material

F 3 "- bility t~ v Insta

[3 F 2

I [ / , ~ ~ Instability 12 F 1 Instability

I, Initiation F, Propagation

a 3 a 2

Crack Length, a V

Figure 13: Illustration of tearing failure for three crack sizes

The same diagram can also be obtained experimentally if the applied stresses from the test data shown in figure 12 are plotted as a function of crack length, see figure 14. Note that for thick sections or brittle material, the onset of crack growth (as indicated by the letter I in figure 12) means fast fracture and failure of the structural part.

m IJ.

• Experimental Data ~--Crack Initiation & Propagation _ 13

I ]2 K[c = 13 ~ (=a) 1/2

--I - 11

I I I a31a21 al I ,,.-

v

Critical Crack Length

Figure 14: Residual strength diagram based on initial crack length

77

The residual strength diagram for the plane stress and mixed mode conditions can not be described in the same way as was described for the case of a brittle material with abrupt fracture behavior. For ductile materials, the stable crack extension first occurs at some stress level below the critical stress (figure 13). The residual strength diagram (within small scale yielding) for material with stable crack growth prior to final failure can be constructed either by apparent fracture toughness or the R-Curve approach [18,19,33,34,35] that will be introduced in section 2.6.2 and 2.6.3, respectively. The apparent fracture toughness approach will give conservative results when evaluating the residual strength of the material. The apparent residual strength curve will fall between two distinct curves, shown in figure 15. The data points with the letter ] represent the onset of stable crack extension and the data points with the letter F describe the final failure.

L Residual Strength Capability Curve

~ F3 • Test data

13 F ] 2 ~ 1 F 1

l ~ - - Final Failure

11 ~ - - ~ _ _ _ Onset of Stable Crack Growth

v

Critical Crack Length

Figure 15: Residual strength diagram for ductile metals

2.6.2 The Apparent Fracture Toughness

One of the methods of constructing the residual strength diagram for a ductile material subjected to low load environment, where crack tip plasticity is localized and plane stress condition exists, is based on the apparent fracture toughness approach. In this approach, the apparent fracture toughness, KAp P, can be calculated by using the initial crack length (represented by the letter D and final critical stress (as shown in figure 13 by the letter F). The apparent residual strength diagram falls between the two curves (defining the onset of stable crack extension and final crack length) represented by the dotted line shown in figure 16. The apparent residual strength diagram shown in figure 16 gives the lower bound estimate of fracture toughness for materials with tearing failure behavior since the failure criteria, K > Kcr

78

= KAp P, is based on the original crack length rather than the final crack length. If Kcr is evaluated based on the final crack length, then the R-Curve approach described in the next section must be employed.

IJ.

Residual Strength Capability Curve

F3 KFina I = 13 (~ (~acr)1/2

I ~ F2 KAp p = 13 a (~aOnset) 1/2

[ 3 \ , "~. • Test data

" ~ . . ~ ' ~ ' " ' ~ - K App ] 1 K Onset

v

Critical Crack Length

Figure 16: The residual strength diagram for KONSET, KApp, and Kcr

2.6.3 Development of the Resistance Curve (R-Curve) & K R

Another available method for obtaining fracture toughness and constructing the residual strength diagram when using a tough material with localized crack tip plasticity is the crack growth Resistance Curve (R-Curve) approach [19]. This method can be applied to a material that exhibits plastic deformation within small scale yielding at the crack tip. This type of behavior occurs in the plane stress condition or to ductile metals, where the crack extension is slow and stable prior to final failure. The failure criterion described by the apparent fracture toughness value was considered as a single parameter failure criterion where K _> Kcr = KAp P. The crack growth resistance approach for tearing type fracture is based on two fracture parameter criteria, which simply state that fracture will occur when the applied stress intensity factor, K, becomes equal to or greater than the material's fracture resistance, K R. Furthermore, fracture will also occur when the rate of change of applied K with respect to crack length becomes equal to or greater than the rate of change of K R with respect to crack length, that is:

aK ~K R K_>K R, and _ > - - 19b ~a ~a

79

In other words, the two failure criteria say that, at failure, when abrupt fracture occurs (K _> KR), the energy available to extend the crack becomes equal to or greater than the material resistance to crack

~K aK R >-- ). growth ( c~a aa

To obtain the plane stress fracture toughness, K c, for a material with tearing fracture behavior, a resistance curve, known as the R- Curve, must be constructed. Consider the variation of the stress intensity factor (up to the point of failure) with respect to total crack lengths for a given material's thickness, as plotted in figure 17a. In this figure, the calculated stress intensity factors for each original crack length, a 1 >a 2 >a 3 , correspond to the onset of stable crack growth, where KONSE T = 13 GONSET (~ ao )1/2 is shown as a dotted line.

A1• (a) T (b) '~ / F F2 F1 c 3 ~" F2 F1 ~ ~----~"~P'~---~- ~ e ~ I F3( - '=1 ; ', , . - - - oo,e,

I j - I i / I I I ; I ~ - - - - - 41~

Crack Length, a Crack Extension, Aa

Figure 17: Variation of stress intensity factor versus a) crack length and b) crack extension, Aa

Note that a o is the original crack length and is equal to the crack length at the onset of stable crack extension, as shown in figure 17a by the letter I. In addition, the variation of the calculated stress intensity factor corresponding to final failure (where the crack becomes unstable) as a function of the amount of stable crack extension, Aa, is plotted in figure 17b. Figure 17b presents the crack growth resistance curve or R-Curve which umbrella all crack growth resistance behavior that has been constructed for different original crack lengths (figure 17a). It can be concluded that the R-curve developed in figure 17 is independent of the initial crack length, but is dependent on the amount of crack extension, Aa.

80

In developing the R-curve for a given thickness, the K R value is evaluated by using the measured effective crack length and the critical load obtained through testing. The equation describing K R is [19]:

K R = (P/Wt) (=aeff)l/2 x f (a/W) 20

where P is the applied load corresponding to the fracture at instability, W is the width of the specimen, f (a/W) is the correction to the width, and aeff is the effective crack length. The effective crack length is the total crack length and is expressed as:

ae f f = a 0 + z~a + rp 21

where rp is the correction for the plastic zone (the estimation of the size and the shape of the plastic zone at the crack tip by using different yield criteria, (section 2.7). Please note that the R-Curve is supposed to be independent of the original crack length. However, when it is developed for a given crack length and thickness based on testing, it can be matched with the applied stress intensity factor curve to estimate the fracture toughness, K C, and the load necessary to cause unstable crack propagation (figure 18). The tangent point between the developed R-Curve and the applied stress intensity factor at a = a c, where K= K c, determines the fracture toughness, as shown in figure 18. At the tangency point, shown in figure 18, the two failure criteria described by equation 19b are met.

"6 #.

K (Applied) Curve

K = urve

Crack Length, a

Figure 18: Illustration of the R-curve where K R and K are tangent at the instability point

81

In general, the construction of a residual strength diagram involves the following steps:

Step 1) The relationship between the crack length, applied stress, and the stress intensity factor for the crack geometry under consideration (equation 19) must be known or developed.

Step 2) The appropriate fracture toughness values must be available for the material under consideration. Apply the failure criteria described in equation 19 by equating the critical stress intensity factor with the fracture toughness value ( K = Kcr > Kic or Kc)

Step 3) Construct the residual strength diagram by plotting the variation of the fracture stress, ~c, versus the critical crack size, acr, for the crack geometry under consideration.

2.6.4 Residual Strength Diagram for Structures with Built-Up Feature

The residual strength diagram for a simple unstiffened structure was discussed in the previous section. It was pointed out that for a single load path cracked panel (unstiffened structure) the residual strength capability diagram can be established by one failure criterion, either by, using the apparent stress intensity factor, where (~c = Kapparent/13~/~a o o r it can be expressed by (~c = Kc/13~/=ac (see figure 16). However, when the cracked panel has a crack arrest feature and is considered as a built-up structure, the failure criterion for establishing the residual strength diagram involves the analysis of each part that carries load and supports the structure (such as skin and stiffeners). A built-up structure normally require more than one failure criterion to assess the failure. Structural parameters that must be considered for skin-stiffened structure consist of the type of construction, panel geometry (stiffener spacing and orientation) and details of construction (stiffener geometry and attachment details). The residual strength analysis must take all of these parameters into consideration. A detailed analytical approach for assessing the residual strength diagram of a complex structure is available in references [36,37,38] with emphasis given to aircraft parts. A brief description of center crack panel behavior situated between two stringers (riveted to the panel) is discussed below.

Figure 19a shows the residual strength diagram for an unstiffened panel, where a crack in the plane stress condition will grow and become unstable. The onset of stable crack extension, as well as the region associated with unstable crack growth, are shown in figure 19a. When a cracked panel is stiffened with stringers (as shown in figure 19b) the crack tip stress intensity factor is reduced as the crack tip approaches the stringer. This is true because stringers have the ability

82

e-

4.A ¢/)

"0

(a)

tttttt 2a

Unstiffened Structure

,o,ta.,,., ! I sta"'o

a I'ni"='oo sta2e Crack L e n g t h , a

t l ' ttttttt

Stiffened Structure

Unstable Crack Growth J Stringer Center Line

Stable crack growth I

Crack Length, a

Figure 19: Residual strength diagrams for an unstiffened and stiffened panel

to take load from the skin, where K = I~/=a, and the quantity 13 is reduced (p<l) to reflect this behavior. Further crack growth takes place at higher stresses in the panel (as shown in figure 19b) until instability is reached.

The stringer failure can also take place when its stress reaches the material ultimate stress, %,. If (~ represents the induced stress in the panel, the quantity (~(~ will be the stringer stress (where ~ >1 and is a measure of the load transferred to the stringer), and failure will occur when c~(~ = %,. The variation of panel and stringer fracture stresses (residual strength diagram) as a function of crack length is shown in figure 20. In the case when crack length is small (2a << w, where w is stringer spacing), the crack tip stresses are not influenced by the stringers. Under these circumstances, both the crack initiation and instability curve follow the curves shown in figure 19a. As the crack length increases and approaches the stiffener in an unstable manner, the stresses in the stringers becomes large (all the load is carried by stiffeners), which cause failure of the stiffeners without stopping the unstable crack growth (see point A of figure 20). When the original crack length at point B is larger than point A, as shown in figure 20, stable crack growth starts at point I, and unstable crack growth (point F) can take place in the panel. Unstable crack growth will be arrested by the stiffener at point D. Further increase in load is required to bring both the stringer and the plate to failure.

83

tt

l Stringer center

I ~ , ~ Stringer Failure

Plate / | ' ~ (instab~ty) ~ ,.=

~ "~ Unstable Crack

Both stringer and plate fail

A I B I C I 2a<< w 2a 1 w

Crack Length, a

Figure 20: Residual strength diagram for a stiffened panel

For large crack length (almost equal to w), the presence of the stiffener is effective in reducing the crack tip stress intensity factor and to prevent crack instability in the system (see point C of figure 20). With increasing load, stable crack growth occurs until at point E (the same failure stress level as for a crack length associated with point B) fracture of the panel and the stiffener will occur.

Example 1 Establish the residual strength capability of an eccentric through crack in a finite plate as shown in figure 21. The crack geometry and loading conditions are also shown in the figure. Assume the flaw size obtained by two different inspection methods are: through cracks of length 1) 2c=1.5 in. and 2) 2c=2.4 in.

Solution From figure 21a, it is obvious that the stress intensity factor for the tip- A is more critical than tip-B. This is due to the width correction factor, 131, for side A versus 132for side B, where WI< W2 and therefore, 13~> 132. The failure criterion based on fracture toughness, described in section

84

2.6 by equation 19a, can be applied here to find the critical crack length for an applied stress of 20 ksi.

=.~---------J~ A, a - "u ~=j t = 0.3 in. W= 15in. W 1 = 2 in.

e =5.5 in.

Material ~ t Properties o'Yiel d = 70 ksi cUi t = 70 ksi Klc = 38 ksi (in.)1/2

c = 20 ksi

(a)

t

a = 20 ksi

(b)

Figure 21: A through crack in a finite plate for Example 1

Case 1 (2c=1.5 inches): The critical flaw size based on tip-A can be obtained through equation 19a as:

K= ~ 13w "V~c, where 13w = (sec'Kc/VV) 112

The width correction factor for this type of crack geometry is a function of distance, W1, and the amount of eccentricity, e, as shown in figure 21a. A numerical solution to the width correction factor for the above crack geometry is available in reference [16]. Another approach, more conservative but simpler to apply, was suggested by Kaplan and Reiman [39] where the total width for the crack tip-A is taken as twice the distance from the crack center to the edge of the plate, 2W1. In this case, the width for tip A is W=4 in. The width correction factor based on this assumption can be obtained as:

= (sec=a/W) 1/2= 1.09

K, = 20 (1.09) (3.14x0.75)1/2

K[ = 33.454 ksi (in.)1/2

The calculated stress intensity factor based on the initial crack length reported by inspection (2c=1.5 in.) is smaller than the critical stress intensity factor Kc = 38 ksi (in.) 1/2. The calculated critical crack length based on Kc = 38 ksi (in.) 1/2, (~ =20 ksi and W=4 in. is:

85

38 ksi (in.)1/2= 20 ksi (sec=a/W) 1~2 (3.14xc) 1/2 Solving for c:

c = 1.149 in. or 2c = 2.299 in. > 1.5 in. (by inspection)

The calculated critical crack length (2c = 2.299 in.) is larger than the crack length found by the inspection (2c = 1.5 in.). Therefore, the cracked plate will survive the load environment.

Case 2 (2c= 2.4 inches):

As shown by the analysis performed for the previous case, the calculated critical crack length 2c = 2.299 in. is smaller than the pre- existing crack reported by the second inspection. -Based on this assumption the crack at tip-A is critical and will propagate toward the edge of the plate. The new crack geometry is now a single edge crack with length c= 2.0 + 2.4/2=3.2 in., as shown in figure 21b. The stress intensity factor and width correction factor for an edge crack in a finite plate is available [16] and is given as

K, = o 13o ~/~c where13o = Y [0.752 + 2.02W + 0.37(1-sin13) 3] and Y = sec 13 [(tan 13)/13)] 1/2, and 13 = =c/2W

Calculating the quantities Y, 13 and the width correction 13o."

13= (3.14 x 3.2)•(2 x 15)=0.334 Y = sec (0.334) [(tan 0.334)/0.334] 1~2

13o = 1.081 [0.752 + 2.02(3.2/15) + 0.37 (1-sin0:334) 3= 1.398 K~= 20 x 1.398 x (3.14 x 3.2) 1/2= 88.63 > 38 ksi (in.) 1/2

From the above analysis, it is clear that the calculated stress intensity factor for the new crack geometry (a single edge crack) is much higher than the fracture toughness of the material. Therefore, as soon as tip-A becomes unstable and is arrested by the left free edge, the new formed crack geometry (a single edge crack) becomes unstable also.

In general, when analysis indicates that the residual strength capability of a given structural part is not adequate, it is recommended that either 1) the inspection method be revised to obtain a smaller initial crack length or 2) the magnitude of the applied stress, (~ be reduced at the expense of increasing the part thickness. However, by doing that the fracture toughness value for the new thickness is now reduced and the material will tend to approach the plane strain condition.

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2.7 Plasticity at the Crack Tip within Small Scale Yielding

Linear elastic fracture mechanics is based upon the assumption that the size of the plastic zone formed at the crack tip is negligible as compared to the crack length and plate thickness. That is, the crack tip plastic deformation is confined to a small region around the crack tip and the bulk of the structure is elastic. For metals which generally go through extensive plastic deformation at the crack tip prior to failure, the use of linear elastic fracture mechanics yields conservative results when solving a given crack problem. This is true because the applied load does work on the cracked body which is stored in the form of strain energy. For brittle materials, all of the available energy will be consumed in creating two new crack surfaces. In ductile material, a large portion of the available energy will be consumed in plastically deforming the material at the crack tip (in metallic materials the energy required for plastic deformation is approximately 103 times larger than the surface energy [40]). For this reason, tough metals possess much higher fracture toughness than brittle material.

One problem which arises in applying linear elastic fracture mechanics is that the calculated crack tip, stress approaches the very high value predicted by the quantity 1H2=r (shown by equation 14) whenever the term r ~ 0. In real situations, there will be a finite plastic zone (rp) ahead of a loaded crack where the material will yield prior to final failure. To evaluate the size and shape of the plastic zone, the crack tip elastic stresses must be available for mode ! loading (see equation 14). These stresses are responsible for bringing the material at the crack tip to yielding upon application of load. In addition, a failure criterion based on material yielding, such as the Von Mises yield criterion, is needed to evaluate the plastic shape and size at the crack tip within small scale yielding.

2.7.1 Plastic Zone Shape Based on the Von Mises Yield Criterion

Irwin was the first to address the crack tip plastic size and shape within small scale yielding. His simplified model [41] states that plasticity at the crack tip takes place when the mode ! tensile stress in that locality reaches the material yield value. In addition, Irwin assumed that the shape of the plastic zone is a circle, and thus the dependency of the plastic zone with respect to the angle, e, was not accounted for. The radius of the plastic zone, rp, when the crack tip tensile stress becomes equal to the material yield value, ayield, (for the case of 8 =0) for the plane stress condition is:

rp = (K!/ayield ) 2/2= 22

87

A more appropriate yield criterion must be employed to account for the size and shape of the plastic zone for all values of 0. The most commonly accepted yield criterion is based on the Von-Mises criterion (proposed in 1913 [42]), which simply states that, for yielding to occur, the maximum value of the distortion energy per unit volume in that material must reach the distortion energy per unit volume needed to yield the material in a tensile test specimen of the same material. In terms of principal stresses, the Von-Mises criterion can be written as:

2 (0"1- o'2) 2 + (o.2- o'3) 2 + (o.1- o'3) 2 = 2 0. e 23

The equivalent stress, 0.e, shown in equation 23, is calculated from triaxial stress state, where o.1, 0.2, and o.3 are principal stresses at a given point in the body. Based on Von-Mises criterion, yielding occurs when the quantity 0"e exceeds the monotonic yield value of the material. Note that the principal stresses, o.1, 0"2, and 0"3 are related to the crack tip stresses o.x, 0"y, and 0"z by the following relationships:

C x - ° Y °' x - ° ' Y /2 o ] = + [ ( _ _ . ) 2 +,[. 2xy ]] 24 2 2

~ x - + a Y ~ x-O'Y /2 a 2 - .+[( .) 2 +~ 2xy ]] 2 2

25

where % = 0 for the plane stress and o.3 = u (o.1 + 0.2) for the plane strain conditions and ~ is the Poisson's ratio. Substituting for the quantities o., o. and ~ from equation 14 of section 2.2.3, the

. . x y xy pnnclpal stresses, 0.1 and o.2' in terms of stress intensity factor becomes:

a I = (KJ~/2=r) cos el2 [1+ sin 6/2] 26

0" 2 = (KJ'~/2=r) cos el2 [1- sin 0/2] 27

Inserting the principal stresses from equations 26 and 27 into the Von Mises yield criterion (shown by equation 23 where o.e = o.Yield), an expression for the plastic zone radius, rp, as a function of 0 can be obtained. For the case of the plane stress condition, where o.3 = 0:

rp(e) = (KJo.Yield) 2 [1+ COS 0 + (3/2)sin 2 0]/4= 28

88

and for plane strain where (~3 = L) (c 1 + (~2):

rp(e) = (K~/(~Yield)2 [(1-2v)2 (1+ COS e) + (3/2)sin 2 e] /4= 29 ~

The plastic zone size, rp, for the case of e = 0 can be obtained based on the Von Mises yield criterion via equations 28 and 29 for the plane stress and plane strain conditions respectively. The plot of the non- dimensional quantity rp(e)/rp(0) versus the angle, e, is shown in figure 22a. Note that the plastic zone size, rp, for e = 0 is equal to the value of rp that was obtained from equation 22 derived by Irwin.

Shape of the Plastic Based on the Von Mises

~0 = 90

Plane Strain 1 . 2 ~ . . , ~

(a)

1,0

0 =0

Plane Stress Mode

Edge Surface Midsection

B/2

(b)

Plane strain condition at the midsection

Plane stress condition at the two edge surfaces

Figure 22: The variation of the plastic zone size and its shape through the thickness based on the Von Mises yield criterion

The non-dimensional quantity rp(e)/rp(0) for the plane stress condition can be written as:

rp(e)/rp(0) = 1/2 + (3/4) sin 2 e + (1/2) cose 30a

and for the plane strain :

89

rp(0)/rp(0) = (1-2v)2 ( l+cose)/2 + (3/4) sin2e 30b

As discussed in section 2.3, the fracture toughness is related to the amount of crack tip plastic deformation and varies with the material thickness. For a structural part with a given thickness, the shape and size of the plastic zone vary throughout the section. The variation of the plastic zone size and its shape through the thickness based on the Von Mises yield criterion is illustrated in figure 22b. At the free edges, where o.3 = 0, the plastic zone strongly resembles the plane stress case (see figure 22b). In the interior or the midsection region, the plastic zone shape and size corresponds to the plane strain condition.

2.7.2 Plastic Zone Shape Based on Tresca Yield Criterion

The Tresca yield criterion [43] is based on the maximum shear stress theory, which simply states that a given structural component is safe when the maximum value of shear stress, ~max, in that component is smaller than the critical value. The critical shear stress value corresponds to the value of the shearing stress in a tensile test specimen of the same material as the specimen starts to yield. In

> o. > o.3' and uniaxial yield terms of principal stresses, when o.1 2 stress, o.Yield, the Tresca yield criteria can be written as:

"~max = I o.1 - o.31/2 = o.Yield/2 31

Forthe plane stress condition, where o.3 = 0, equation 31 in terms of the crack tip stress intensity factor can be written (see equations 24- 26) as:

o.1 = (K~/~/2=r) cos el2 [1 + sin el2] = o.Yield 26

Solving for the plane stress plastic zone size, r = rp:

rp(e) = (KJo.yield)2 COS (e/2)[l+sin (e/2)] 2 / 2= 32

and for the plane strain condition, where o.33 = t~ (o" 1 + o'2), the size of the plastic zone in terms of the crack tip stress intensity factor can be expressed as:

rp(e) = (KJo.yield)2 COS 2 (6/2)[(1-2v) - sin (~/2)] 2 / 2= 33

The shape of the plastic zone based on the Tresca yield criterion (as described by the above two equations), is different from Von Mises and is plotted in the figure 23.

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2.8 Surface or Part Through Cracks

In most structure, pre-existing cracks are found in the form of surface cracks (also called part through cracks) that initiate at surface discontinuities or emanate from a hole in the form of corner cracks,

Plane Stress

0 : 90

e = 0

Plane Strain

Figure 23: The shape of the plastic zone (Tresca yield criterion)

see figure 24. Surface scratches are introduced into the part as the result of surface machining, grinding, forming, or may be due to improper handling during manufacturing and assembling the hardware. These surface cracks may become through cracks during

2c (a)

i I , - (b)

Figure 24: Illustration of (a) a surface crack and (b) a corner crack from a hole

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the service life of the structural part before reaching their critical size. In other cases, embedded cracks found in welded parts will grow gradually to the surface and become surface cracks, which grow further in depth and length directions to become a through crack.

Surface cracks will grow in both length and depth directions. Therefore, in analyzing these crack geometries by linear elastic fracture mechanics, it is important to have an expression for the mode ! crack tip stress intensity factor, K~. When the load environment is fluctuating, the expression pertaining to surface crack stress intensity factor, K~, must be provided to the analyst if fatigue crack growth analysis for both the depth and length directions are needed. In section 2.8.1, the stress intensity factor equation for a surface crack in an infinite plate is formulated. The solution corresponding to one of the most commonly used surface crack geometries is discussed in sections 2.8.2 (a longitudinal surface crack in a pressurized pipe). The stress intensity factor solutions to part through fracture toughness for both the depth and length directions are covered in section 2.8.3. Finally, the concept of Leak-Before-Burst (LBB), which is a requirement for non-hazardous gas/liquid pressurized hardware, is introduced in section 2.8.4, together with example problems.

2.8.1 Stress Intensity Factor Solution for a Part Through Crack

The stress intensity factor, KI, for a surface crack (sometimes referred to as a thumbnail crack) in a plate subjected to uniform tensile load (as plotted in figure 24) can be formulated by using the stress intensity factor equation corresponding to an embedded elliptical crack in an infinite body subjected to uniform load, see figure 25 for the geometry and loading [44].

At any point, m, along the boundary of the elliptical crack, the mode I stress intensity factor can be written as:

K l = ((~V=a){[(a/c)2 cos2e + sin2e]114}/~) 34

where e is the angle that defines any point around the perimeter of the elliptical crack (figure 25). The quantity • is a complete elliptical integral of the second kind and is given by:

~12

(t) = .t" [1- (1-(a/c) 2 sin e] 1~2 dO 35 0

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An empirical expression that can describe the quantity (# of equation 35 for different crack depth to crack length aspect ratios, a/c, is given by [45]:

Y

Figure 25: Embedded elliptical crack geometry under mode I loading

(#2 = 1.0 + 1.464 (a/c) 1"6s 36a

(#2 = 1.0 + 1.464 (c/a) 1'65 36b

for a/c_<1 and a/c>1, respectively.

Using the K] solution for an embedded crack described by equation 34, the stress intensity factor equation at any point, m, around the periphery of a surface flaw in an infinite plate subjected to tensile load can be written as [44]:

K I = (1. l(~/rca){[(a/c) 2 cos2e + sin2e]l~4}/(# 37

The maximum and minimum stress intensity factors around the crack front are associated with the angles e = 90 ° and 0 = 0 o , respectively.

When the part through crack is a circular surface crack, as shown in figure 26, where the aspect ratio a/c=1, the value of the stress intensity factor around the crack front is a constant. From equation 37, the stress intensity factor corresponding to a circular crack (a/c=1) can be written as:

K I = (1.1 ~q=a)/(# 38

where the quantity (#=2.464 (see equation 36a for a/c=1). Accordingly, for a shallow crack, where a/c = 0.2 and e = 90 °, as shown in figure

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26, the stress intensity factor can be represented by equation 38, in which (t) takes the value of 1.102.

Circular Surface Crack

tI

~, a/c = 0.2

Figure 26: Illustration of circular and shallow cracks

The two previously mentioned aspect ratios of a/c=1 (a circular crack) and a/c=0.2 (a shallow crack) as shown in figure 26 are the two limiting aspect ratio cases that are widely used in fracture mechanics analysis. For example, NASA requires all fracture critical flight hardware (such as thin wall pressure vessel structure) to be examined for the safe-life requirement using these two limiting aspect ratios to ensure that the structural part can survive the expected load environment.

The correction factor, p, employed for the equation of stress intensity factor to account for the plate width, back surface correction and loading conditions of a part through surface crack (see equation 38), is provided by references [46,47] and can be written as:

K I = 13(a/c, a/t, c/w, O)((~/=a)/(t ) 39

where the correction factor 13 is:

13(a/c, a/t, c/w, 9) = [ M 1 + M 2 (a/t) 2 + M 3 (a/t) 4 ]*g* f(I) *fw 40

The back surface and width corrections fw = [sec ~ c/w (a/t) 1~2] lJ2. Other quantities in equation 40 for the case of a/c < 1, are given as:

94

M 1 = 1.13 - 0.09 (a/c)

M 2 = -0.54 + 0.89•(0.2 +a/c)

M 3 = 0.5 - [1/(0.65 + a/c)] + 14 (1 - a/c) 24

g = 1 + [ 0.1 + 0.35 (a/t) 2 ] (1 - sin 0) 2

f~ = [ (a/c) 2 cos 2 0 + sin 2 8] TM

and for the case of a/c >1, the correction factor parameters are:

MI= ~/c/a [(1 + 0.04 (c/a)]

M2= 0.2 (c/a) 4

M 3 = -0.11 (c/a) 4

g = 1 + [0.1 + 0.35(a/t) 2 (c/a)] (1 -sin e) 2

f~ = [(c/a) 2 sin 2 0 + cos 2 0] TM

It should be noted that equation 40 is valid within the limits of:

0_< a/c_<2, c/w<1/4,0<_0_<~ a/t<1.25(a/c+0.6) for 0<a/c<_0.2

a/t for 0.2_<a/c<oo

The stress intensity factor solutions for other surface crack geometries are also available in references [48]. In section 2.8.2, only the case of a longitudinal surface crack in a pressurized vessel, which can be applied for life evaluation of pressurized hardware will be discussed. Another important and commonly used part through crack geometry case is a corner crack emanating from the edge of a hole. This case is used typically in the analysis of fastened joints, and must be given special attention.

2.8.2: Longitudinal Surface Crack in a Pressurized Pipe

The stress intensity factor for a surface crack oriented in the longitudinal direction in a pressurized pipe has the form of equation 39. The U.S. Air Force and NASA require all pressurized containers to be considered as fracture critical or high risk parts. They must demonstrate an analytical life of not less than four times the service

95

life of the structure. A pressure vessel is defined as a container that stores fluid having a stored energy of at least 14240 ft-lbs or contains gas with pressure above 14.7 psi. It is therefore necessary to develop the stress intensity factor solution for a pressurized pipe or vessel with a longitudinal surface crack, with hoop induced stress acting perpendicular to the crack orientation causing mode ! failure. The correction factor, 13, for the stress intensity factor of a part through thickness crack in a pressurized pipe (shown in figure 27) with diameter D is [47]:

13(a/c, a/t, {))= 0.97 * [ Ml+M2(a/t)2+M3(a/t) 4 ]* g * f(I) * fc* fi 41

where the quantities in the equation 41 are:

MI= 1.13 - 0.09 (a/c) M 2 = -0.54 + 0.89•(0.2 +a/c)

M 3 = 0.5 - [1/(0.65 + a/c)] + 14 (1 - a/c) 24

g = 1 + [ 0 . 1 + 0 . 3 5 ( a / t ) 2 ] ( 1 - s i n e ) 2 f(1) = [ (a/c)2 c°s2 e + sin 2 e] TM

The stress intensity factors for the depth and length directions are associated with the angles 0 = 90 ° and C) = 0 o, respectively. The correction quantities fc = [(1 + k2)/(1 - k 2) + 1-0.5 ( a l t ) l / 2 ] [ t l ( D I 2 - t)] where k=l-2t/D and the value of fi = 1 for internal crack and 1.1 for external crack.

a = p ( D - 2t)12t

p = Internal

'} 2c

P

Figure 27: Part through longitudinal crack in a pressurized pipe with diameter D

2.8.3 Part Through Fracture Toughness, Kie

The critical value of the stress intensity factor for a part through crack is called the part through fracture toughness and is designated by the symbol K[e. Just like the plane strain and plane stress fracture

96

toughness for through cracks, the part through fracture toughness is also an important parameter to have when the number of cycles to failure associated with a given load environment is needed (see chapter 3 for an in-depth discussion of the fatigue crack growth mechanism).

The part through fracture toughness can be obtained through testing, see ASTM-E740 (lately it has been revised by the ASTM E08 Committee). The ASTM surface crack tensile test specimen for determining the part through fracture toughness value, Kie, is shown in figure 28 (SC(T) specimen). A dog bone shape specimen crack possessing crack depth, a, and crack length, 2c, is subjected to axial load as shown in the figure 28. Specimen with surface cracked subjected to bending can also be used for obtaining the Kie value and

r i 1 ~ t

LIi <2c

Figure 28: Illustration of ASTM part through crack SC(T) test specimen

it is called surface cracked bend specimen, SC(B). In both cases (SC(T) and SC(B)), specimens must have a uniform rectangular cross section in the test sections. Specimens with circular cross section can be used having diameter, D, (2.0 times larger than the crack length, 2c) with circumferentially oriented crack. The crack dimensions (a and 2c) of surface crack must resemble the actual surface crack geometry used in service and to be loaded in the same manner. Otherwise, the results of Kie test may not be conservative when it is applied for analysis. It should be noted that the results of the surface crack tests depends on the aspect ratio, a/c, crack depth to specimen thickness,

97

a/t, loading conditions, and amount of crack growth that occur during test.

The critical values of load at fracture, as well as the final crack length and crack depth after fracture are measured. If the stress intensity factor is determined based on final load and initial crack length it must be reported. Using equations 39 and 40, the critical value of stress intensity factor at fracture where K=KIe can be calculated. For SC(B) specimen the stress intensity factor equation must be available (see reference [48]). In both cases of tension and bending tests, the calculated stress intensity factor must is limited to cases where a<c and a<0.8t (specimen thickness, t). For the specimen loaded in tension, the maximum K value occurs at maximum crack depth, a. For specimen loaded in bending, the maximum K value occurs at the intersection of the crack tip, c, with the free surface. A typical test specimen configuration for aluminum alloys with 0.5 inch thick plate can be designed as follow:

Use 2c=1.5, a=0.2, W= 6.0, L=12, t=0.5, and total length of 18.0 inches.

In the case where the Kie value is not available for the fracture analysis, it can be approximated in terms of plane strain fracture toughness, Kic, for through cracks by the following equation [25].

Kie = Klc (1 + C k Kic/CyJeld ) 42

where C k is an empirical constant, that for most aerospace material, is equal to 1.0 (In.) -1/2. Equation 42 gives reasonably good results for a variety of isotropic materials under tensile loading. The value of the part through fracture toughness, Kie, for material with high Kic/Cyield ratio will be limited to 1.4Kic [25]. High value of Kie/Kic is probably due to the remaining ligament (t-a) being small to cause back surface plastic deformation [49,50]. For SE(B) specimen under bending load the ratio Kie/Kic ranges from 1.0 to 3.0. As a final note, for the surface crack test to be valid it is necessary that the crack depth and the remaining ligament satisfy the LEFM requirement. For tensile specimen the remaining ligament (t-a)>(K/co) 2 and applied stress c < Co (where co is the effective yield stress).

As was mentioned previously, there are two stress intensity factors that are associated with a part through crack, corresponding to the angles e = 90 ° and e = 0 °. The critical values of K I for the depth

direction, where e = 90 ° (also called a-tip), is given by equation 42 and

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is designated by Kie. The.critical value of K! for the length direction,

where e = 0 ° (c-tip), is approximated by 1.1Kie.

One of the most important applications of Kie is in pressurized tanks used in aerospace and nuclear structures where the requirement of Leak-Before-Burst (LBB) must be met. In the case of a pressurized tank that contains a hazardous fluid or gas, LBB is not desirable and the tank must instead demonstrate adequate safe-life during its usage. Section 2.8.4 briefly discusses the LBB concept.

2.8.4 The Leak-Before-Burst (LBB) Concept

In most materials, it is desirable to have crack stability even though the part through crack grows and becomes a through crack [51,52]. For this to happen, the following conditions at the onset of part through crack to through crack transition should be valid:

(KI)Part through crack = (KI)Through crack 43

Moreover:

and (KI)Part through crack < KIe

(KI)Through crack < Kc or Kic

44a

44b

Under these conditions, the part through crack (now it is a through crack) will grow until the stress intensity factor, K], becomes equal to or greater than K c or Kic. This condition is called leak-before-burst (LBB).

The leak before burst criteria described by equations 43 and 44 can be applied to a pressurized tank. The assumed surface crack (based on the NDE inspection technique, such as penetrant inspection) will grow due to fatigue and sustain load through the thickness. From a safety point of view, for a hazardous fluid or gas, neither leak-before- burst nor the condition of instability, where K] (at a-tip) > Kie are desirable. For a non-hazardous fluid or gas, however, leak before burst is required, as illustrated in figure 29.

If the criteria for leak-before-burst described by equation 43 and 44 do not hold and a leak-before-burst design is desired, the following modifications might be considered:

1) Thicken the structure to reduce the design stress

99

2) Change the material

Examples of fatigue crack growth problems related to pressurized vessels are demonstrated in chapter 3.

LBB for

v KI < Kie (K] )Surface Crack < (Kle)Through Crack

;

v

W Stability: K! < Kle Instability: (K[) > K c, Kic

Figure 29: A surface crack before and after transition to a through crack

Example 2 A pressurized cylindrical tank is subjected to 200 psi pressure, as shown in figure 30. As part of the safety requirement, the tank must undergo a proof test of 1.5 x operating pressure (300 psi). The flaw size obtained by performing standard penetrant inspection prior to the proof test indicated that the maximum pre-existing surface flaw that can escape the inspection is a circular crack that has 2c=0.15 in. with a/c=1 The pressurized tank is made of 2219-T87 aluminum alloy with Kic =30 ksi (in.) 1/2. Determine: 1) the tank will be leak-before-burst and 2) whether or not the through crack will be stable (assume Kle= 1.1 Kic ).

Solution The stress intensity factor solution to the crack geometry shown in figure 30 is given by equation 41. The correction factor to the stress intensity factor of the above crack geometry is:

13(a/c, a/t,0)= 0.97 * [ Ml+M2(a/t)2+M3(a/t) 4 ]* g * f~ * fc* fi

where a/t and the quantities M 1, M 2, M 3 for the case of a/c=1 can be calculated as follows:

MI= 1.13 - 0.09 (a/c) = 1.04

100

M 2 = -0.54 + 0.89/(0.2 +a/c) = 0.201

M 3 = 0.5 - [1/(0.65 + a/c)] + 14 (1 - a/c) 24 = -0.1

g = 1 + [ 0.1 + 0.35 (a/t) 2 ] (1 - s i n 0 )2 = 1

f(l) = [ (a/c)2 c°s2 e + sin 2 0] TM =1 The stress intensity factor for the depth direction is associated with an angle of 0 = 90 °.

Penetrant 2c = 0.15 With a/c

D = 25.0 in. p = 100 psi

H o o p Stress

= p (D - 2t)/2t

p = Internal

t = 0.10 in.

K Ic = 30 ks i (in. 1/2)

K c = 30 ks i ( in, 112)

Figure 30: A pressurized tank with a surface flaw (Example 2)

The quantit ies fc = [(1 + k2)/(1 - k 2) + 1-0.5 (a/t) l /2][t/(D/2 - t)] = 1.015 where k= l -2 t /D= 0.992. The value of fi = 1 for an internal crack and 1.1 for an external crack. Substituting these values into the stress intensity correction factor equation:

13(a/c, a/t, 0)= 0.97 * [1.04 + 0.201(0.75) 2 -0.1(0.75) 4] * 1 -1 ,1 .015 * 1.1=1.21

Moreover, the induced hoop stress due to 200 psi proof pressure is:

(~ = p(D-2t)/2t = 37200 psi = 37.2 ksi

K! = 13(a/c, a/t, c/w, e)((~/=a)/(t)

=1.21x37.2 x (2.14 x .075) 1/2 / 1.569= 13.92 ksi (in.) 1/2

where (z) 2 = 2.464

101

Note that the calculated stress intensity factor is much smaller than the part through crack fracture toughness value, Kie = 1.1Kic =33 ksi (in.) 1/2. Therefore, no catastrophic failure is expected, that is:

Kie = 33 ksi (in.) 1/2 > 13.92 ksi (in.) 1/2

Now let us assume the existing crack grows and becomes a through crack having length 2c=2t=0.2 in. (the part through crack is now leaking). To check for LBB, it is necessary to calculate the stress intensity factor for a through crack of length 2c=0.2 inch. From equation 44, the condition for stability can be written as:

(KI)Through crack < Kc

The equation for the stress intensity factor for a through crack in a cylinder [25] is:

K = (Col3o)~/~c

[30 = (1 + 0.52 ~. + 1.29x 2 _ 0.074x 3)1/2

where X = c/(Rt) 1/2

;L = 0.1/(12.5 x 0.1) 1/2 = 0.09

13o= (1 + 0.52 x 0.09 + 1.29 x 0 . 0 9 2 - 0.074 x 0 . 0 9 3 ) 1/2 = 1.04

K = (%13o)q=c = 37.2 x 1.04 x (2.14 x 0.1) 1/2 = 21.68 ksi (in.) 1/2

Because the calculated value of the stress intensity factor is smaller than the fracture toughness of the material, the tank would be leak before burst:

(KI)Through crack = 21.68 ksi (in.) 1/2 < Kc= 50 ksi (in.) 1/2

indicating that the through crack will be stable.

2.9 A Brief Description of ASTM Fracture Toughness Testing

The present method of determining the residual strength capability of a given structural part by applying the LEFM concepts requires one to obtain the critical value of the stress intensity factor (called the fracture toughness). Several laboratory tests performed in accordance with ASTM practice can generate the fracture toughness data. Sections 2.3 through 2.6 provide an in-depth discussion on plane strain and plane stress fracture toughness and their importance in generating the residual strength capability curve for single and multiple

102

load path structures. Plane stress fracture toughness is a function of thickness and crack length, it is therefore necessary to conduct many laboratory tests with various thicknesses and crack length when assessing the fracture behavior of structural materials. These tests require specimen preparation, surface finish and data collection (such as load-displacement data, stable crack growth and load measurements).

Standard test methods for plane strain fracture toughness of metallic materials, Kic, are briefly discussed in section 2.9.1 and the determination of plane stress and mixed mode fracture toughness, K c, by the Resistance Curve [19] is presented in section 2.9.2.

2.9.1 Plane Strain Fracture Toughness (Kic) Test

As was mentioned in section 2.3, the plane strain fracture toughness, Kic, is independent of thickness and crack length. To obtain a valid KIc value, the state of stress near the crack front must approach the triaxial tensile plane strain and plastic deformation will be halted. Moreover, the presence of any plasticity zone formed at the crack tip must be small compared to other specimen dimensions (such as crack size and plate thickness). The relationships between the crack tip plastic zone size, ry, and specimen size requirement to ensure elastic plane strain condition throughout the body are [18]:

rp = (KIc / O'Yield)2/6~ 45

a >_ 2.5 (Kic / (~Yield) 2 , B >_ 2.5 (KIc / ayi~d) 2 , W >_ 5.0 (KIc / (3'Yield) 2 46

where a, B, and W are the crack length, specimen thickness and width, respectively. The following calculation shows that the specimen thickness should be approximately 47 times the radius of the plane- strain plastic zone, rp, in order to meet the plane strain test specimen requirements. Dividing the crack length, a, or specimen thickness, B, from equation 45 by equation 46:

Specimen Thickness/Plastic-zone size = (a or B)/rp = 2.5(6=) = 47

The plastic zone shape and size within small scale yielding were derived in section 2.7 and the quantity rp based on the Von-Mises yield criteria was discussed also.

2.9.2 Standard Kic Test and Specimen Preparation

Two common types of standard specimen are available for Kic testing, namely the slow-bend test specimen and the Compact-

103

Tension Specimen (CTS or simply C(T)). The C(T) specimen is more commonly used for Kic testing than the slow-bend test specimen, see figure 31. Other specimen configurations are also available for plane strain fracture toughness testing and the reader may refer to the ASTM-E399 Standards. The corresponding dimensions for the C(T) specimen are shown in figure 31.

L

A 0 5 W

B = W/2 ± 0.01W

D < W/10

G = 0 .275W ±0.005W

HI2 = 0 .6W + 0 .005W L = 1.25 W ±0.01W

2R = 0 .25W ±0.005W

W = W + 0 . 0 0 5 W

Figure 31: Plane strain fracture toughness C(T) specimen dimensions

Prior to Kic testing, specimen dimensional measurements must be taken such that the thickness, B, is measured to 0.1%, and the crack length measurement after fracture is measured to the nearest 0.5%. The final crack length measurement after completion of the test should be the average of three measurements made at three positions along the crack front.

The crack is introduced in the specimen by a starter notch which extends by fatigue cracking. The purpose of fatigue cracking is to simulate a natural crack that can provide a satisfactory plane strain fracture toughness test result. The stress ratio associated with cyclically loading the notch is 0.1>R>-1 and the Kmax (for fatigue cracking)<60% Kic. It is common to fatigue-crack the C(T) test specimen by the amount 0.05W, where W is the width of the specimen.

Note that before the plane strain fracture toughness specimen can be machined, some estimate of the sizing of the C(T) test specimen

104

must be known. Table 1 provides the ASTM recommended crack length or thickness for the suggested ratio of yield strength to modulus of elasticity• For example, 2219-T851 aluminum alloys with yield strength of 53 ksi give a a,,,/E ratio of 0.0052. The recommended crack

• . r . . .

length or test specimen thickness, B, Is approximately 2.0 in. (table 1).

TABLE 1

Recommended Crack (;ys IE Length or Thickness

0.0050-0.0057 3.00 0.0057-0.0062 2.50 0.0062-0.0065 2.00 0.0065-0.0068 1.75 0.0068-0.0071 1.50 0.0071-0.0075 1.25 0.0075-0.0080 1.00 0.0080-0.0085 0.75 0.0085-0.0100 0.50 0.0100 & Greater 0.25

To establish that a valid Klc has been obtained, it is first necessary to calculate the conditional stress intensity factor, KQ, which can be determined through the construction of test records consisting of an autographic plot of the applied load, P, versus displacement, •, (P-A curve)• The conditional stress intensity factor, KQ, is given [18,53] as:

KQ = PQ [29.6 (a/W) 1/2 - 185.5 (a/W) 3~2 + 655.7 (a/W)s/2 - 1017(a/W) 7/2 + 638.9(a/W) 9'2]/BW1/2 47

where the applied load, PQ is the point on the P-A curve shown in figure 32. The point PQ is obtained by drawing a secant line with slope

+ I O r i g i n a l S l o p e / ,

. -L / / "qk 5 % O f f s e t

/ / / / PQ= Ps

f Displacement, ,&

L._ . .J

Figure 32: Load versus displacement for obtaining the apparent KQ

105

5% from the initial slope tangent to the linear part of the record. The obtained value of PQ on the P-A curve is compared with the Pmax value situated on the curve (see figure 33 for different types of load- displacement records) to calculate the apparent plane strain fracture toughness, KQ, in order to verify the conditions established by equations 45 and 46. In figure 33, the value of P5 is considered to be equal to PQ if there is no other load in the P-A record that precedes it and is higher than P5 (see case 1 of figure 34). Otherwise, the PQ load

• (1 ) , (2 ) . (3 )

~ P5=PQ "~'1'' ' Pmax ~ PQ Pmax T PQ-~ P5

/ / . ' / / / /; / , /"

Displacement, A ~ Displacement, A ~ Displacem t, A

Figure 33: Load versus displacement and the interpretation of KQ

is the higher value on the load displacement record as shown in cases 2 and 3 of figure 33. If the ratio of Pmax/PQ (the load Pmax is a load point situated above the P5 value) is less than 1.10, then the calculated KQ value from equation 47 should be used in equation 45 to compute the plastic zone size, ry. This value should be compared with specimen thickness, B, and crack length, a. If both B and a are larger than 47ry , then KQ = Kic. In addition to the above mentioned conditions for obtaining a val id-Kic value, the following two requirements are also important to remember when conducting plane strain fracture toughness testing:

It is necessary to perform a minimum of three tests for each material heat treatment

The loading rate should be between 30 ksi (in./min.) 1/2 to 140 ksi (in./min.) 1/2

The plane strain fracture toughness, KIc, value for several selected aerospace alloys are listed in Appendix A. The Kic values for

aluminum alloys may range as high as 46 and as low as 16 ksi (in) 1/2

106

for 7075-T63 and 2020-T651 aluminums, respectively. Plane strain fracture toughness as high as 200 ksi (in) 1/2 can be found in the NASA/FLAGRO material library among the stainless steel alloys (see Appendix A). In general, a high Kic is associated with ferrous alloys that have undergone appropriate heat treatment which can introduce sufficient ductility and high resistance to fracture in the material.

A Kic plane strain test specimen thickness as large as 12 in. or more is reported in the literature for alloys that possess high ductility and good resistance to fracture [54]. The specimen preparation for determining the plane strain fracture toughness value, when the thickness requirement is as high as 12 in. is costly and not feasible to implement. On the other hand, the FMDM theory can be used to evaluate the fracture toughness analy.tically, and thereby eliminate the difficulties associated with specimen preparation and time-consuming laboratory testing (see chapter 5).

Another important point that must be emphasized in conjunction with the plane strain fracture toughness value is its variation with respect to the grain orientation that evolves during the manufacturing process. The plane strain fracture toughness values for a material that has undergone a forging process are usually expected to be smaller in the T-L direction than in the L-T. For example, the KIc values for 7050-T7452 aluminum

alloy in the forged condition are 31 and 21 ksi (in) 1/2 in the L-T and T-L directions, respectively. As another example, the Kic values for 7050-

T76511 in the extruded condition are 30 and 24 ksi (in) 1/2 in the L-T and T-L directions, respectively (see Appendix A). From these examples, the analyst must ensure that the proper Kic value has been used in assessing the service life of the structural part. Failure to recognize the difference in material properties with respect to their orientation (and they may differ considerably) can lead to erroneous analysis results.

2.9.3 Plane Stress Fracture Toughness (Kc) Test

In contrast to plane strain fracture toughness, Kic, the critical stress intensity factor for plane stress and mixed mode conditions, K c, are thickness and crack length dependent. The variation of material fracture toughness, K c, versus specimen thickness was discussed in section 2.3, and is shown in figure 34 for 2219-T87 aluminum alloy !53]. From figure 34, it can be seen that the fracture toughness value increases as the thickness decreases. The increase in fracture toughness value is strongly dependent on the state of stress and the extent of plasticity formation in that region. For a thick section, the extent of the plastic zone through the thickness is at a minimum, and as the thickness decreases, the constraint to plastic flow decreases, untill the state of plane stress is reached. The fracture toughness

107

I 0 0 ,'

! t=

so ~ i o - - , - o . 2 , - o ~ . / : • ~ 8 0 . 4 5 m0.68 J : ~ - - 8,.1.00 e• 1.20

re... • • •~ : • - - - 8..1..50 m 1.60

J," A ' " - :

. ~ . %.. ,

: 4o Iml * ' • aO

2 0 K I ~ Nk'VVP, m = V * , - -

• Ke - Z 7 Ytdd - $8

- 0.54 ~Ak - 1

8 0 0 . 3 ! .0

M a t e r i a l T h i c k n e s s , t ( i n c h )

Figure 34: Variation of Kc as a function of material thickness for 2219- T87 aluminum alloy

associated with minimum thickness is called the plane stress fracture toughness, K c. The maximum K c value for 2219-T87 aluminum alloy is 70 ksi (in.) 1/2 and it is associated with a minimum thickness t =0.15 in. and half crack length a=1.5 to 1.8 in. (as shown in figure 34). The smallest K c value (with t=0.15 in.) in this figure is 44 ksi (in.) 1/2 and it applies to a half crack length of 0.24 to 0.38 in. (the crack tip must stay within small scale yielding to have a valid test).

To develop a curve similar to the one shown in figure 34 for another alloy or the same alloy with a different heat treatment, it is necessary to conduct many tests with cracked specimens of various thicknesses. Each specimen will provide a distinct R-Curve, different from the others and represents a material resistance to fracture for the thickness under consideration. The R-Curve represents the material's resistance to fracture during incremental stable slow crack growth, Aa, under monotonic increasing load (load control). As indicated in section 2.4.3, the R-Curve is constructed by plotting the material resistance to fracture, K c = K R, for different crack lengths, see figure 35. Because the R-Curve is proven to be independent of original crack length, ao, it can be plotted as a single curve (K R versus Aa) for a given thickness using a standard ASTM cracked specimen as recommended by ASTM E-561 (Standard Practice for R- Curve Determination).

108

O

Kfor o =oc ,. KR

Crack Length, a

Figure 35: Construction of R-curve and determination of Kc where the applied K and KR are tangent to each other

In the R-Curve concept, unstable crack extension occurs when the applied K becomes equal to or exceeds the crack growth resistance, K R, of the material. At the instability point, the crack growth resistance, K R, is equal to K c as shown in figure 35. That is, the plane stress fracture toughness, K c, is determined from the tangency between the R-Curve and the applied K curve (K versus physical or effective crack length, ao+Aa) of the specimen. The specimen configuration and dimensions specified by the ASTM testing procedure for conducting a valid R-Curve test are briefly discussed in section 2.8.2.1.

The variation of fracture toughness, K c, with material thickness for many aerospace alloys can be found in the NASA/FLAGRO material library. This variation is presented in the form of an empirical equation described in terms of plane strain fracture toughness, Kic, [55] as:

Kc= KIc [1 +Bk e-(Ak*t/B)2 ] 48

where B is the thickness that meets the plane strain condition and t is the thickness associated with the part that could be in the plane stress or mixed mode conditions with fracture toughness, K c, to be determined. The constants Ak and Bk are the curve fit parameters and are given in Appendix A for several aerospace alloys. Note that the curve fit relation described by Equation 48 is applicable to minimum crack length, as was shown in figure 34 for the case of 2219- T87 aluminum alloy with half crack length a=0.24 to 0.38 in.. The

109

plane stress fracture toughness values associated with minimum crack length will yield a minimum fracture toughness value that is conservative to use when evaluating the service life of a fracture critical component. Also note that large cracks in wide plate M(T) specimen when conducting fracture toughness tests (see section 2.9.4) yield higher fracture toughness values when compared with C(T) specimen.

2.9.4: M(T) Specimen for Testing K c

For determination of material resistance to fracture using the R- Curve approach, three types of specimen are recommended by the ASTM committee. The middle center cracked wide tension panel M(T) specimen, the compact specimen C(T), or the crack-line-wedge- loaded specimen C(W) are used to conduct the plane-stress fracture toughness test. Only the center cracked wide panel is discussed in this section, see figure 37. The reader may refer to E561 for the recommended specimens that can also deliver applied stress intensity factor, K, to the material. Section 2.9.5 briefly describes 1) the apparatus (grips and fixture) employed for uniformly transferring the applied load into the M(T) standard specimen, 2) the method of preventing buckling of the M(T) specimen, 3) specimen fatigue cracking and 4) load and specimen dimensional measurements.

As discussed in the previous section for the plane-strain fracture toughness case (per the ASTM E-399 test method), the planar dimensions of the specimens are sized to ensure that elastic conditions are met at all values of load in the net section of the specimen. In other words, the width, W, and crack size, a, must be chosen so that the remaining ligament is below net section yielding at fracture. The ASTM guide for designing an M(T) panel is such that a specimen with W= 27 ry width and crack length 2a=W/3 is expected to fail at a net section stress equal to the yield strength of the material. The selected width dimension must be larger than 27 ry in order to avoid net section yielding. Figure 36 is a list of minimum recommended M(T) specimen dimensions for an assumed Kmax/ayield ratio, where Kma x is the maximum K level obtained in the test and ayield is the 2% offset yield strength of the material. Note that for the plane stress condition, the value of ry is given in terms of Kma x and ayield by equation 22 as:

rp = (K I/(]yield) 2/2~ 49

For an M(T) specimen, the Kma x can be calculated (ASTM-E561) by:

110

Kma x = (P/WB) [x aeff sec (=aeff/W)]l/2 50

The grip fixture apparatus must distribute the applied load to the M(T) test specimen uniformly in order to obtain accurate readings for constructing a valid R-Curve. To ensure that uniform tensile stress is transferred from the pin grip fixture (when a single pin grip is used) into the specimen, the length between the two single pins on each end

2 Holes (W/3 Dia,) 2a n () 2a

IJ ~.-I d" ~..Ij ,~_ j -~_ __t. Iw/2 I 1.5w 1.5w I w/2 _=_1 I I I III IIi I

Figure 36: Standard M(T) specimen for obtaining Kc value

2.9.5: Grip Fixture Apparatus, Buckling Restraint, and Fatigue Cracking

should be 3W (as shown in figure 37). For example, for a test specimen with width W=3.0 in. and crack length 2a=1.0 in., the length

I ~ , ' ~ Crack Length, 0 2a ~-- G r i p ~ W

I t ~ 3W

®® ~® ® ® 2a ® ®

~® W>12 Grip/~" ®® ® inch ( ~

v 1 ,SW Figure 37: The pin grip fixture apparatus for W=3 and W=12 in. M(T)

specimen

111

requirement between the application of loads at the two ends is 9.0 in.. In the case when the specimen width, W, is wider than 12 in., the ASTM Standard requires use of multiple pin grips to uniformly distribute the load, with a length requirement of only 1.5W, as illustrated in figure 37. For example, for W=20 in., the length requirement is 1.5W=30 in.

When the test specimen is thin, buckling can develop during loading that may affect the accuracy of the test data. To prevent the test specimen from buckling, rigid face plates are required to be attached to both sides of the specimen. In the case of the M(T) specimen, the rigid plates should be affixed to the test specimen as illustrated in figure 38. The buckling restraint plates must be lubricated to allow for the lateral motion of the test plate and moreover, it must be attached to both sides of the test specimen in such a way that the crack measurement data readings are accessible and not blocked.

In order to simulate a natural crack in the M(T) specimen, the starting notch is first introduced in the specimen either by saw cutting or Electrical Discharge Machining (EDM). The starting notch length plus fatigue precrack should be between 30 to 40% of the specimen width, W, and situated in the center of the test specimen. The extended length associated with the fatigue precrack must not be less than 0.05 in.. Fatigue cracking may be eliminated if the cutting saw thickness can simulate the sharpness of a fatigue starter crack. The ASTM procedures for obtaining a valid fracture toughness value clearly indicate that extensive preparation both before and after testing are necessary.

M(T) Specimen I

Buckling restraints Plate

2a J !

~) Buckling restraints (~) ® Plate ~)

Figure 38: Buckling restraint plates affixed to the cracked plate

112

References

1. S. S. Manson, "Metal Fatigue Damage, Mechanism, Detection, Avoidance, and Repair,''ASTM, STP 495, 1971, pp. 61-115

2. B. L. Averback, D. L. Felbeck, G. T. Hahn, D. A. Thomas, "Fracture," Proceeding of an International Conference on the Atomic Mechanism of Fracture, April 12-16, 1959, pp. 1-160.

3. A. A. Griffith, "The Phenomena of Rupture and Flow in Solids," Philos. Trans., R. Soc. Lond., Ser. A., Vol. 221, 1920. 4. Inglis

5. H. Liebowitz, "Fracture, An Advance Treatise," Volume II, Acadamic Press, 1968, Ch. 1

6. H. L. Ewalds and R. J. H Wanhill, "Fracture Mechanics," Edward Arnold, 1986, Ch.2

7. A. S. Tetelman, and A. J. McEvily, JR., "Fracture of Structural Materials", John Wiley& Sons Inc., 1967, pp 39-48.

8. G. R. Irwin, "Analysis of Stresses and Strains Near the End of a Crack Traversing a Plate," Trans. ASME, Jo Appl. Mech.Vol. 24, 1957, p. 361.

9. C. P. Paris and G. C. Sih, "Stress Analysis of Cracks," in "Fracture Toughness Testing and Its Applications," ASTM STP No. 381, ASTM, Philodelphia, 1965.

10. H. M. Westergaard, "Bearing Pressures and Cracks," Transactions, ASME, Journal of Applied Mechanics, 1939.

11. S. P. Timoshenko, and J. N. Goodier, Theory of Elasticity, 3rd Edition, McGraw-Hill (1970).

12. N. I. Muskhelishvili, "Some Basic Problems of the Mathematical Theory of Elasticity," (1933), English Translation, Noordhoff (1953).

13. G. c. Sih, " On the Westergaard Method of Crack Analysis," International Journal of Fracture Mechanics, Vol. 2, 1966, pp. 628-631

14. J. Eftis, and H. Liebowitz, "On the Modified Westergaard Equations for Certain Plane Crack Problems," International Journal of Fracture Mechanics, Vol. 8, 1972, pp. 383-392

15. G. R. Irwin, Fracture Handbuch der Physik, Springer-Verlag, Heidelberg, VI, 1958,

113

16. H. Tada, P. C. Paris and G. R. Irwin, ed. Stress Analysis of Cracks Handbook, Del Research Corporation, Hellertown, Pa., 1973

17. G. C. Sih, Handbook of Stress Intensity Factors for Researchers and Engineers, Institute of Fracture and Solid Mechanics, 31, Series E. No 2, June 1964.

18. Annual Book of ASTM Standards, "Standard Test Method for Plane Strain Fracture Toughness of Metallic Materials," Vol. 03.01, 1999, pp. 413-443

19. Annual Book of ASTM Standards, "Standard Practice for R-Curve Determination," Vol. 03.01, 1999, pp. 494-506

20. A. S. Tetelman, and A. J. McEvily, JR., "Fracture of Structural Materials", John Wiley& Sons Inc., 1967, pp 132-139.

21. J. M. Craft, A. M. Sullivan, R. W. Boyle, "Effects of Dimensions on Fast Fracture Instability of Notched Sheet," Crack Propagation Symposium, Cranfield 1961, Paper 1.

22. M. F. Kanninen, and A. T. Hopper, "Advance Fracture Mechanics," Oxford Engineering cience Series, 1985, pp. 150-200

23. J. M. Barsom, "The Development of AASHTO Fracture Toughness Requirement for Bridge Steel," Engineering Fracture Mechanics, Vol. 7, No. 3, September 1975.

24. 19. J. M. Barsom, "The Development of AASHTO Fracture Toughness Requirement for Bridge Steel," American Iron and Steel Institute, Washington, D. C., February 1975.

25. Fatigue Crack Growth Computer Program "NASA/FLAGRO", Developed by R. G. Forman, V. Shivakumar, J . C. Newman. JSC- 22267A, January 1992.

26. H. O. Fuchs and R. I. Stephens, "Metal Fatigue in Engineering," John Wiley and Sons, 1980, pp. 217-255.

27. The Making, Shaping and Treating of Steel, Edited by H. E. McGannon, Ninth Edition, United State Steel, Pittsburgh, Pennsylvania, December, 1970.

114

28. American Association of State Highway and Transportation Officials (AASHTO) Material Toughness Requirement, Association General Offices, Washington, D. DC, 1972.

29. S. T. Rolfe and J. M. Barsom, " Fracture and Fatigue Control in Structures, Applications of Fracture Mechanics," Prentice-Hall, Inc., Englewood Cliffs, New Jersy.

30. A. S. Tetelman, and A. J. McEvily, JR., "Fracture of Structural Materials", John Wiley& Sons Inc., 1967, pp 3-82.

31. B. L. Averback, D. L. Felbeck, G. T. Hahn, D. A. Thomas, "Fracture," Proceeding of an International Conference on the Atomic Mechanism of Fracture, April 12-16, 1959, pp. 20-43.

32. R. A. Flinn, P. K. Trojan, "Engineering Materials and Their Applications", Third Edition, Houghton Mifflin Company, 1986, PP. 321-322.

33. W. S. Margolis and F. C. Nordquist, "Plane Stress Fracture Toughness of Aluminum Alloy 7475-1/2in. Plate, Temperes- T7651 and T7351 and of Aluminum Alloy 2024-1/8 in. Sheet - T81 and T62 Temper," general Dynamics, Forth Worth Div., F-16 Air Combat Fighter Technical Report TIS GA2300, CDRL A031, USAF Contract F33657-75-C-0310.

34. J. E. Srawley and W. F. Brown, "Fracture Toughness Testing Method," ASTM STP 381, (1965), PP. 133-195.

35. J. M. Krafft, A. M. Sullivan and R. W. Boyle, "Effect of Dimensions on Fast Fracture Instability of Notched Sheets," Cranfield Crack Propagation Symposium, Vol. 1, (1961), pp. 8-28.

36. H. Vlieger, "Residual Strength of Cracked Stiffened Panel," Engineering Fracture Mechanics, Vol. 5, 1973, 447-478.

37. T. Swift, and D. Y. Wang, "Damage Tolerant Design Analysis Method and Test Verification of Fuselage Structure, Air force Conference on Fatigue and Frcature," AFFDL-TR-70-144, 1970, PP. 653-683.

38. T. Swift, "Development of Fail-Safe Design Features of DC-10, ASTM STP 486, 1971, PP. 653-683.

39. M. P. Kaplan, and J. A. Reiman, "Use of Fracture Mechanics in Estimating Structural Life and Inspection Intervals," Journal of Aircraft, Vol. 13, No. 2, Feb. 1976, pp. 99-102.

115

40. E. Orowan, "Fracture and Strength of Solids," Rep. Prog. Physics, Vol. 12, (1949), pp. 185-232.

41. F. A. McClintock and G. R. Irwin, "Plasticity Aspects of Fracture Mechanics," ASTM STP 381, (1965) pp. 84-112.

42. W. Johnson, and P. B. Mellor, (1962), Plasticity for Mechanical Engineers, Van Nostrand, New York.

43. G. S. Spencer (1968), An Introduction to Plasticity, Chapman and Hall, London.

44. G. R. Irwin, "Crack Extension Force for a Part Through Crackin a Plate," Journal of Applied Mechanics, December 1962, pp. 651-654.

45. J. C. Newman, Jr. "Fracture Analysis of Surface-and Through Crack Sheets and Plates," Engineering Fracture Mechanics, Vol. 5 No. 3, Sept. 1973, pp. 667-685.

46. R. Mo Engle, Jr., "Aspect Ratio Variability in Part-Through Crack Life Prediction, " ASTM STP 687, Am. Society for Testing and Materials, 1979, pp. 74-88.

47. J. C. Newman, Jr. and I. S. Raju, "Stress Intensity Factor Equations for Cracks in Three Dimensional Finite Bodies," NASA TM 83200, NASA Langley Research Center, Aug. 1981.

48. J. C. Newman and I. S. Raju, "Analysis of Surface Cracks in Finite Plates Under Tension or Bending," NASA TP-1578, 1979.

49. w. g. Reuter, J. C. Newman, B. D. Macdonald, and S. R. Powell, "Fracture Criterion for Surface Cracks in a Brittle Material," Fracture Mechanics, ASTM STP 1207, American Society for Testing Material, 1994, pp. 614-635.

50. W. G. Reuter, N. C. Elfer, D. A. Hull, J. C. Newman, D. Munz, and T. L. Panontin, "Fracture Toughness Results and Preliminary Analysis for International Cooperative test Program on Specimen Containing Surface Cracks," Fatigue and Fracture Mechanics: ASTM STP 1321, Vol. 28, 1997.

51. ASTM Committee, "The Slow Growth and Rapid Propagation of Cracks," Material Res. and Standards, 1 (1961) pp. 389-394.

52, G. R. Irwin, "Fracture of Pressure Vessels," Materials for Missiles and Spacecraft, pp. 204-229, McGraw-Hill (1963).

116

53. J. E. Srawley, "Wide Range Stress Intensity Factor Expressions for ASTM E 399 Standard Fracture Toughness Specimens," International Journal of Fracture Mechanics, Vol 12, June 1976 p. 475.

54. B. Farahmand, "Fatigue and Fracture Mechanics of High Risk Parts,''Chapman and Hall, 1997, Appendix A

55. B. Farahmand, "Fatigue and Fracture Mechanics of High Risk Parts,''Chapman and Hall, 1997, 316-317

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Chapter 3

FATIGUE CRACK GROWTH AND APPLICATIONS

3.1 Introduction

The traditional approach for estimating the total life of a structural part, when it is subjected to a load varying environment, is to use the S-N diagram (alternating stress, S, versus the total number of cycles to failure, N). The S-N diagram for high cycle fatigue is constructed using the ASTM E-466 Standard. Several laboratory tests are conducted on highly polished standard specimens (free from surface defects) subjected to low amplitude fluctuating load, which closely simulate the load environment under consideration [1]. With the S-N approach, it is assumed that the structure is initially free from defect and the total life of the structure is defined as the sum of the load cycles associated with crack initiation and stable crack growth. The crack initiation (stage I) is always associated with localized regions with high stress concentration, where material is plastically deformed. For example, a crack can initiate in the vicinity of intermetalic particles, either by cracking of the particle itself or cracking at the interface of the particle- matrix zone [2]. This type of initiation mechanism is not the only one to occur in material. Surface deformation in the form of persistent slip bands (psbs) within some of the grains of metals can form which in cross section consists of extrusions and intrusions that are suitable localized regions for crack initiation [3,4]. The number of cycles required to generate a small crack along the active sli.p plane is dependent on the load amplitude, material grain size, and temperature. Under low load amplitude, the extent of stage I is large and most of the structural life is consumed for crack initiation. On the other hand, when load amplitude is raised to the low cycle fatigue regime, where the bulk of the structure is plastic, the number of cycles used for stage I is small and the entire structural life is expended for stable crack growth (stage II). In applying linear elastic fracture mechanics to determine the number of cycles to failure of a structural component subjected to a low load cyclic environment, it is assumed that the material contains pre-existing defects. Therefore, the number of cycles to crack initiation is disregarded and only those cycles associated with stable crack growth are considered in the life evaluation of the component.

The application of linear elastic fracture mechanics in estimating the life of structural hardware assumes that microscopic cracks have

already initiated in the material. In reality, this is often the case, because the pre-existing crack may have its origin in many ways. For example, cracks may be introduced during the manufacturing and assembling of structural parts, they may grow from defects in the parent metal, from incomplete welds, or from shrink cracks or other imperfections in weldments, surface pits or scratches during handling. The initial crack length associated with these cracks can be estimated conservatively based on a reliable Non-Destructive Inspection (NDI) method (see section 1.6 of chapter1). That is, the largest crack that could escape detection by non-destructive inspection can be assumed to exist in the material and it can be used as the initial crack size for life evaluation of the structural component. However, if the structural part contain crack like defects of considerable magnitude that can readily be detected by conventional NDI methods, the structural service life must be evaluated based on the detectable flaw size. In some cases, the part can be totally rejected if the detectable flaw size is large enough that it does not meet the NDI requirements.

The material as received from the vendor may contain defects of a small size, such as porosities (fine holes and pores in a metal, most commonly in welds and casting), inclusions (such as oxides, sulfides and silicates) or microcracks that can eventually lead to fracture. These inherent flaws are considerably smaller than the NDI capability to detect them and will not grow appreciably in service. Figure 1 illustrates the crack like defects distribution in the structural hardware throughout its service life. It can be seen that the initial flaw size associated with defects in material prior to fabrication stage (inherent

J

J

J

Maximum flaw size prior to fabrication, ama x

Critical crack

acr NDI

~ Defect size Capability ..~ (for different

NDI f ~. techniques)

Defect in material Defects introduced Defects introduced prior to fabrication dudng fabrication during service

Figure 1: Illustrating defects distribution in the structural hardware throughout its service life

119

flaws) are much smaller than NDI defects capability detection. From a safety viewpoint, the use of a longer initial crack length (obtained through NDI methods) is conservative, however, from a practical consideration, the longer crack size assumption should be realistic enough not to impact the weight or cause rejection of the part. It should be also noted that the initial crack size to be used to evaluate the crack growth behavior must not be so small that it would violate the concepts of fracture mechanics. That is, the concept of isotropic continuum must be obeyed, such that the flaw size can not be smaller than the grain size.

For aircraft and space vehicles, it is required to assume that cracks pre-exist in all primary structures which are classified as high risk or fracture critical parts. These cracks are not allowed to grow to a critical size at a specified load during their usage period. Therefore, it is necessary to predict the rate of growth associated with the assumed flaw (provided by the NDI technique), and to compute the number of cycles to failure so that it may be compared with the number of cycles used during their service life.

Different crack growth empirical equations capable of calculating the number of cycles to failure are discussed in section 3.2. Section 3.2 includes the crack growth equation employed in the NASNFLAGRO (NASGRO) computer code [5]. This program was initially developed for fracture analysis of space hardware and recently has been extended to crack growth analysis of aircraft structural problems. The NASA/FLAGRO computer program is currently the standard computer code for the NASA, the European Space Agency, the U.S. Air Force, and many aerospace companies. A brief review of the ASTM fatigue crack growth rate testing is also included in section 3.2. Later in section 3.3, the Elber and Newman crack closure concepts describing the effect of stress ratio on crack growth behavior are presented and are followed by a few example problems. Variable amplitude loading and the retardation concept (including example problems) are discussed in section 3.4.

Under sustained load (when applied load is static) the presence of a corrosive environment can be detrimental to high strength alloys with pre-existing cracks. Fracture instability can occur when the exposed time under aggressive environment is sufficient to grow the crack to its critical dimensions. The effect is even more detrimental to cracked structure when the combination of both a corrosive environment and repeated stress are acting simultaneously. Section 3.5 addresses environmental assisted cracking under plane strain and plane stress conditions when cracked structure is subjected to static load. A brief overview of ASTM determination of threshold value of environmental assisted cracking K,Esc and KEsc for plane strain and stress conditions, respectively are also presented in this section.

120

3.1.1 Stress Intensity Factor Range and Crack Growth Rate

It is the goal of fracture mechanics to estimate the total number of cycles that are required for the largest pre-existing crack of length, a i, to reach its final length, af, during the structural service life or period between inspections. The crack tip stress intensity factor was extensively discussed in chapter 2 and it is an extremely useful parameter to address crack growth behavior as long as the bulk of the material is elastic and plastic deformation is limited to a small region at the crack tip. The stress intensity factor range associated with maximum and minimum stresses of each cycle (Kma x -Kmi n) is used to describe the amount of crack advancement, Aa = af - a i, during AN number of cycles. In using fracture mechanics to describe fatigue crack growth, the minimum value of the stress intensity factor, Kmi n, in a cycle is taken to be zero when R < 0. The rate of crack growth, Aa/AN, in terms of the crack tip stress intensity factor range, AK, can be written as:

Aa - - = f(AK) 1 AN

Equation 1 simply states that the material's rate of crack growth, Aa/AN, is a function of the stress intensity factor range, AK. The function f (AK) can be obtained as the result of laboratory test data and can then be utilized to solve crack growth problems in which the structural part has undergone the same loading conditions (R=~min/~max). It should be emphasized that when the crack tip stress states at a given time are the same for two separate crack lengths and loading conditions, AK, the crack growth rate, Aa/AN, would be the same for the two crack cases. For example, consider the case of a wide center crack plate with initial crack length of 1.5 inch Subjected to a remote constant fluctuating load (R=0) of 15 ksi. The rate of crack growth for this crack geometry will be the same as if the crack length is 0.5 inch and subjected to a remote loading of 26 ksi (R=0):

AK = Kma x -Kmi n (note that ~min=0 and therefore, Kmi n =0)

AK = Kma x = [3 ~max (=a) 1/2

for ~ma x =15 ksi, Kma x = 15 x ( 14 x 1.5) 1/2 =32.6 ksi (inch) 1/2

for ama x =26 ksi, Kma x = 26 x ( 14 x 0.5) 1/2 =32.6 ksi (inch) 1/2

121

In the next section, several empirical relationships will be developed that can define the crack growth behavior shown by equation 1.

3.2 Crack Growth Rate Empirical Descriptions

All of the equations that presently describe the function I(z~K) are based on the trends developed by experimental data. In general, a centered crack specimen (also called a Middle Tension crack specimen, M(T), see figure 2) with original crack length, a o, is subjected to constant cyclic loading of a given stress ratio, R. Other crack geometries, such as a Compact Tension specimen, C(T), are used to generate crack growth data as shown in figure 2. The incremental crack length growth, (Aa), is periodically measured and recorded together with the number of cycles, (AN). From this information, the variation of crack advancement versus associated number of cycles is plotted (as depicted in figure 2) which represents the experimental fatigue crack growth test data. Figure 3 shows the variation of crack growth, ao+ Aa, versus the number of cycles for

M(T) Specimen C(T) Specimen

' m n

i ""

. l

/ Slope, da/dN, at a point (ai,Ni)

Number of Cycles, N

~ t ) A Omax A A/_ Cyc,e,

I V V V V -

• Experimental data recorded by measuring crack growth after N number of cycles

Figure 2: A schematic representation of fatigue crack growth data and slope at a given point

2014-T6 welded aluminum alloy tested in accordance with ASTM E- 647 practice [6] (the crack tip was situated in the heat affected zone). The slope of the curve (da/dN = Aa/AN) at a point shown in figure 2 is computed for any crack length, a= ao +Aa, along the curve and is called the fatigue crack growth rate or crack extension per cycles of loading. Currently there are two methods recommended by the ASTM for determination of the slope, Aa/AN. In the secant method of determining the slope, two data points are needed, whereas in the incremental polynomial method a minimum of five data points are

122

2 . 0 ( i

2.6~

i m m m B I

..,1 1 ,0t

. 40 0 500,000

c~cz~ (N)

i I I

J I

f

1,000~000

Figure 3: Variation of crack advancement versus number of cycles for 2014-'!"6 welded aluminum alloy (courtesy of Westmorland Company)

required to compute the slope at a point, da/dN [7], see figure 4. The computed slope and the corresponding stress intensity factor range at a point are to be used in generating the da/dN, AK curve. In the secant method of determining slope at the mid-point of two data

Secant Method (slope evaluated by connecting two adjacent data points by a straight line)

Incremental Polynomial Method (slope evaluated by polynomial fit to five, six or seven successive data points)

a i +z~a

ai +1 aave

a i

AK= 13(Omax" °rain )(~aave) 112 aj + z~a •

:la/dN=(a i+1 "ai)/(N i+t "Ni) •

aave = ( a i + a i + 1)/2 • •

Ni Ni +1

Number of Cycles

Figure 4: Determination of slope at a point by secant and incremental polynomial methods

points, one data point has to be sacrificed. That is, from the two data points, a~ and a=+l, as shown in figure 4, only one point is generated to

123

describe the da/dN, AK curve ((a~ +a~+1)/2). In the incremental polynomial method, a smooth curve (a parabola) is fit through five, seven, or nine data points for the determination of the slope, da/dN. With this method, more than two data points are lost when generating the da/dN, AK curve. Figure 5 shows typical crack growth rate data and the curve fit for 2024-T861 aluminum alloy (Plate & Sheet, T-L, Room Temperature) that was generated under stress ratio R=0.1 (taken from the NASNFLAGRO material library [5]).

10 -3

10 -4

U 10-6

C

Z

M

10 -6

10 -7

10 -8

R=O.1 I¢:b ~

Alloy: 2024 AL I c ~ Condition: T861 ~a Environment: LA ¢:~a Specimen: C(T) Orientation: T-L Frequency: 0.10

. . ~ Curve Parameters ,~m Smax/Oo = 0.3

. . ~ a = 1.5 K c = 37.751

~1~ m at thk=0.02 ~ - C = 3.09E-8

,~ n = 2.895 ~ 'a p = 0.5

q = I I - : ,2 / ~c = 19

A K = 1

B K = 1 / o~o = 2.2

Rcl = 0.7

I NK [ksi*SQRT (in.)] I z I I I I I I I I I I I I I I I I , , , , I

10 100

Figure 5: A typical crack growth data and the curve fit for 2024-T861 [7]

Numerous fatigue crack growth rate empirical and analytical relationships (da/dN versus AK) have been developed and are available in [8]. The earliest relationship describing the da/dN behavior was formulated by Paris at Lehigh University in 1960 [9] and is expressed as:

124

da/dN = c (AK) m 2

where c and m are constants that can be determined from the test data. The constants c and m describing the Paris relation for 2024- T861 are calculated as 4.8E-8 and 3.4, respectively. For most metals, the Paris constants c and m fall between c=10 -8 to 10-6 and m =3 to 5, respectively. Fatigue crack growth data presented in figure 5 were utilized to compute some of the constants used in the crack growth rate curve fit relation described by equation 4 for 2024-T861 aluminum alloy.

The Paris' relation described by equation 2 is applicable only to the middle region of the crack growth curve, where the variation of log (da/dN) with respect to log (AK) is linear, as indicated in region II of figure 6. In general, there are three regions associated with the crack growth curve. In region I, the crack growth rate (da/dN) is small (little amount of crack advancement, Aa, associated with a large number of cycles) and the corresponding stress intensity range, AK, approaches a minimum value called the threshold stress intensity factor, AKth, below which the crack does not grow. The value of AKth is not associated with da/dN=0; rather, it is associated with a cut-off growth rate of 4x10 -10 m/cycle assigned by ASTM- E647. In region III, the crack growth is rapid and accelerates until the crack tip stress intensity factor reaches its critical value. The critical value of stress intensity factor, K c, is shown in figure 6 as an asymptotic line to the crack

I I [gion If!

I/y %.;,, "J 85-90Y, I Region II I I

I / Log ( AK i)

Figure 6: Three regions of the fatigue crack growth curve

125

growth curve. The value of K c obtained through cyclic loading (the critical value of AK obtained from the da/dN curve) may be smaller than the monotonic loading case. It should be remembered that an invalid fracture toughness value (as a result of improper fracture toughness testing) incorporated into the crack growth equations can result in significant error when determining the life of the part in consideration.

The number of cycles that a cracked structural part spends in each region of the da/dN, AK curve is different. Under low load cyclic environment, more than 85% of the life of the structure is usually spent in region I, where the existing crack is short and the crack growth rate is small. Figure 6 illustrates the percent life of a typical aerospace alloy in regions I, II, and II1.

Utilizing the Paris crack growth relation described by equation 2 beyond its limit can result in life estimation error. Note that the Paris equation does not apply to regions I or III, where crack growth is slow or rapid. In addition, the effect of the stress ratio, R, on fatigue crack growth was not considered in equation 2.

Forman [10] formulated an empirical equation describing the crack growth behavior in regions II and III, including the effect of R:

da c(AK) n

dN (1-R)K c -(AK)

where c and n can be obtained through experimental data and K c is the fracture toughness of the material and is thickness dependent (see section 213 of chapter 2). Equation 3 was later modified by Forman- Newman-de Koning (FNK) to account for all the regions of the crack growth curve (including the threshold region), plus the stress ratio and crack closure effects [11,12]:

da c(1- f)nAK n (1- AKth )P

AK

dN ( l-R) n(! AK )q (1 - R)K c

4

where c, n, p, and q are empirically derived constants, R is the stress ratio, and AK and AKth are the stress intensity factor range and threshold stress intensity factor range, respectively. Note that in equation 4 the quantity AK/(1-R)=Kmax (maximum stress intensity factor in a cycle). The parameter f is called the crack opening function and it will incorporate the effect of plasticity induced closure behavior

126

on crack growth rate under constant amplitude loading. The Forman- Newman-de Koning (FNK) crack growth rate relation described by equation 4 is widely used in the aerospace industry for life estimation of fracture critical parts. A detailed discussion related to constant amplitude crack closure phenomenon and crack opening function, f, will be given in section 3.3.2

3.2.1 Brief Review of Fatigue Crack Growth Testing

The establishment of a complete da/dN versus AK curve (as shown in figure 5 for 2024 aluminum alloy) when evaluating the number of cycles to failure of machinery parts requires conducting steady-state fatigue crack growth rate testing in accordance with ASTM E-647, "Standard Test Method for Measurement of Fatigue Growth Rates". Two types of specimen configurations, Compact Tension, C(T), and Middle-Tension, M(T), are recommended for use in generating the da/dN versus AK plot, see figure 7. This test method involves cycling notched M(T) or C(T) specimens which have been precracked, while the incremental crack growth length, Aa, is periodically measured and recorded, together with the number of cycles, AN. Other specimen configurations may be used, provided

oia=o2sw _ _ ~ i [__ ~ - -# -

Y - r -1= I - o -~=_ J I I

-~ an---~ I I

z o 1.25w O.OlW "~1 W/20< B <W/4

(Recommended Thickness)

>I< I" w/2 r l - 1.5w 1.5w ~1" w/2 r[

Figure 7: Two types of specimen configurations, Compact Tension, C(T), and Middle-Tension, M(T)

127

that stress intensity factor solutions for the geometries are available.

In precracking the test specimen to simulate an ideal crack, the final Kmax value associated with precracking should not exceed the initial Kmax value which the test specimen will experience upon testing. Otherwise, the crack tip will be subjected to the retardation effect at the onset of testing. The retardation effect will slow the rate of crack growth due to overload and is discussed in section 3.4. This effect is particularly important when the threshold stress intensity factor range, AKth, is recorded. The amount of precracking should not be less than 0.05 of the width of the test specimen, W. Electrical Discharge Machining (EDM), milling, broaching or saw-cutting techniques can be used for notch preparation prior to precracking.

The incremental crack growth associated with N number of cycles is measured either visually or with some other optical technique, and the growth rate, Aa/AN, as a function of the stress intensity factor range, AK=Kmax-Kmin, is calculated. The fatigue crack length reading technique, irrespective of the method of measurement, must be capable of measuring as good as 0.002W. To be capable of reading crack extension visually to 0.002W, a low power traveling microscope with 50x magnification capability is recommended. It would be helpful, although not necessarily required, to apply reference marks at equal intervals on the test specimen prior to the start of testing to eliminate the potential for errors while reading the data.

Crack extension, da, results in a decrease in both load and plate stiffness. The inverse of stiffness is called compliance which for a cracked plate varies as a function of crack length. In figure 8 the plate compliance, C, is the inverse of the slope of applied load, p, versus displacement, v. The compliance method determines an analytical relationship between the inverse of the load-displacement slope, v/p, that has been normalized to elastic modulus, E, and specimen thickness, t, (Evt/p) and the normalized quantity of crack length over the specimen width, a/w, and is one non-visual technique that can be used to monitor the crack advancement as it is subjected to cyclic loading. In the compliance method [13], a clip gage is located at the mouth of the specimen and the displacement of the gage, together with the corresponding load, are measured, see figures 8 and 9. Depending on the location of data measurements on the specimen mouth, a theoretical relationship between the two dimensionless quantities Evt/p and a/w can be developed. For example, when the clip gage is situated at the edge of a C(T) specimen (as shown in figure 9 at locations A and B), the polynomial equation describing a/w as a function of Evt/p can be written [14] as:

a/w=1.001-4.6695(x)+18.4 (x)2-236.82(x) 3+1214.9(x)4-214 6(x) s 5

128

Where x=[(Evt/p) 112 + 1]-1

When using the compliance method, it is good practice to also check the crack measurements with an optical method at several intervals during the test. Multiple reading methods are advisable because it is possible that the clip gage could become loose at the mouth of the specimen due to specimen and machine vibration, which can introduce errors while data is collected.

Loading Machine ~ Specimen ~ice

oi.,,. eo nt Rea( ~ I

&, Figure 8: The compliance method of measuring load-displacement

•I ~~._~ain Gages

II Tensi°n

I] Cra~k I 'ipGage l ~ ~-

B A

Tensk

Figure 9: Clip gage set up for measuring load-displacement

129

The crack length readings recorded in figure 3 of section 3.2 (2014-T6 weld) must be arranged in such a way that the da/dN data can be evenly distributed with respect to stress intensity factor range, AK. In region I, where the incremental crack growth, Aa, is small, the associated number of cycles, AN, is large. The opposite is true in the region III, where the crack growth accelerates and the elapsed cycles, AN, are small. For the case of the C(T) specimen type, the even distribution of da/dN with respect to AK can be accomplished by measuring the incremental crack growth length, Aa, to be at most equal to or smaller than 0.04W value when 0.25<a/w_<04. When 0.40<a/w<0.6, the Aa measurement should decrease to 0.02W. In the region III with 0.60<a/w, the crack length measurement should be more frequent (Aa<0.01W). In the case of the M(T) test specimen, with Aa_<0.03W for 2a/W<0.6, and in the case of 2a/W>0.6, the reading should be Aa<_0.02W. Generally, the measured value of Aa= 0.01 in. is an acceptable value for both specimen types mentioned above. As an example, for a compact tension specimen having W=2.0 in. and total crack length a=0.45 in., the incremental crack growth length measurement, Aa, visually should be equal to or less than 0.08 in. where a/W=0.225. As the crack length increases and becomes equal to a=0.85 in. (where a/W=0.425), the crack length measurement interval will be every 0.04 in.

The results of crack growth tests are mostly thickness independent; however, materials may show thickness dependency in region III of the da/dN versus AK curve when crack growth accelerates and Kmax approaches the material fracture toughness. In selecting specimen size, adequate thickness is needed to avoid buckling. The presence of buckling can introduce error in the test data measurements. For the C(T) specimen type, the recommended thickness, B, should be larger than W/20 and should not exceed W/4, as shown in figure 7.

If an M(T) specimen is used to generate fatigue crack growth data (this type of specimen is used when data for the case of R<0 is of interest), the recommended thickness is W/8. For a thin M(T) specimen, lateral deflection may occur due to improper loading. Strain gages should be mounted on the specimen for detection of any bending-induced strain. Bending strain as high as 5% of normal strain may be acceptable in the specimen (ASTM E-647).

In both types of test specimen, the ratio of original crack length, a, to specimen width, W, must be such that net section yielding does not occur at all values of loading (LEFM limitation). That is, the specimen must be predominantly elastic, except in the localized region at the crack tip where small scale yielding criteria holds. To avoid net section yielding, an empirical relationship based on test results relating the specimen dimensions (width and crack length) to the material

130

yielding, Cyie~d, and the calculated maximum stress intensity factor, Kmax, is established [6]. For the C(T) specimen, net section yielding, can be avoided when (W-a)> 4/=(K,..x/(~y~e~d) 2 and for the M(T) specimen (W-2a)_> 1.25 (PmJt(~y~.~d), where Pr.ax is the maximum load in the cycle and t is the specimen thickness. The maximum stress intensity factor, Kr, ax, can be calculated for the C(T) specimen type by the following relationship [15]:

. Pmax (2 + a / w). Kma x = [ . . . . . j 13

t,l~w (] - ,, / w )

where 13=[0.886 + 4.64(a/VV) - 13.32(a/W) 2 +14.72(a/W)3 _ 5.6(a/W)'] and the stress intensity factor range,.AK is given by:

AK = [(Pmax - Pmin )(2 + a/w).] I[3 t ~ w ( l - a l w ) 312

6a

Accordingly for the M(T) specimen [16]:

AK = Pmax -Pmin ~a ]/2 [(~a)sec - - ] 6b tw w

The measured crack length to be incorporated into equations 6a and 6b must be an average of the measured value from both sides of the specimen to ensure load symmetry and that the material in consideration is isotropic.

In using the M(T) specimen type, the crack measurement must be the average value of two crack tips. Including back surfaces, a total of 4 measurements are required (for the C(T) specimen, only two meas(Jrements are needed). If crack growth directions at the crack tip are not perpendicular to the applied load to within + 20 °, the test must be discontinued and the data obtained is invalid.

Residual stresses can have significant influence in fatigue crack growth data, specifically in the region where the AKth value is of interest. The residual stress is added to the applied stress, which can either lower or raise the calculated value of the crack tip stress intensity factor. For example, when the magnitude of the compressive residual stress is above or equal to the applied stress, the crack growth rate data will be close to the threshold value since the crack tip stress intensity is not effective in causing any crack growth (Aa=0). This phenomenon can occur when the test specimen is machined from the weld region where post stress relief is impossible. Other parameters influencing the crack growth rate are:l) temperature, 2)

131

severity of corrosive environment, 3) influence of specimen thickness, 4) fatigue crack size (it is known that short cracks exhibit greater growth rate than long fatigue cracks, see section 3.3 for more discussion related to this topic).

In summary, fatigue crack growth testing is performed for a given stress ratio, R, to provide the analyst with the following information:

• Fatigue crack growth data can be used to determine the number of cycles to failure for an initial crack length in a given material subjected to a given cyclic load environment.

• For two or more materials under the same cyclic loading conditions (R), it can help to establish the selection of the material that provides the longest life. When the da/dN curve of a given material is shifted to left, shorter life is expected for the environment under study.

• The plot of fracture stress versus the number of cycles to failure can be used to establish the inspection interval, an important tool for quality assurance purposes.

• The effect of heat treatment, fabrication (material anisotropic) and environment on fatigue crack growth can be determined by conducting laboratory crack growth tests to generate data for different grain orientations.

3.3 Stress Ratio and Crack Closure Effect

Data from experimental testing conducted in the laboratory to generate the crack growth curve under constant amplitude loading clearly indicate that the stress ratio, R, and crack closure behavior have a significant effect on crack growth rate [5]. Figures 10 and 11 show the influence of the stress ratio on the crack growth curve for 6061-T6 and 7075-T651 aluminum alloys, respectively. From figures 10 and 11, it can be deduced that, for a given crack growth rate, da/dN, the stress intensity range, AK, is generally higher as R becomes smaller and approaches a negative value. In addition to the influence of the stress ratio on fatigue crack growth rate, the crack closure behavior has significant influence on the behavior of the growing fatigue crack and the rate at which it is advancing. The stress ratio and closure correction to the crack growth rate equation (da/dN versus z~K ) is usually done in one of two ways: 1) By replacing the stress intensity factor range, hK, with its corrected value (called effective stress intensity range, hKeff), and then the amount of crack growth is calculated simply by the modified Paris equation or 2) Using the apparent stress intensity factor range, z~K, but incorporating other effects separately, as shown by equation 4, where the stress ratio and

132

1o~.

lo-*.

Oaln IO JMABIOAm0tn I MSAB10A801R IdSABI 0AIJ01B~t

Pl.,Vlluo TNll 0.I00 0+~60 O.SO0 0.3S0 0.730 0 2 5 0

C~ddkm: r.w~mme*m: Speamm: OJk~iboa:

e M ! N,. 111 LA

UNK 0.00

R = 0 . 7 5

I td

U

R = 0 . 5

R = 0 . 1 , 1 0 "t

, 1 0 "a

~.~10 4.

! lo'-

10"7.

lo'.

,10-,1

8 o / % • 0.5

• 41M~1

d m k . O.S C - 1 4 ' 104

a • U p = OJ q m U

Yield • 41

m O.?'S " 10'4

AK [ksJ'SORT(In)] R d . 0.7

1 10 100

Figure 10: Influence of stress ratio, R, for 6061-T6 aluminum alloy [5]

closure effect is incorporated by the quantities (l-R)" and (l-f) n, respectively.

In this section, the Elber [17,18,19] and Newman [12] approaches to the crack closure phenomenon will be discussed. Discussion of crack closure by other mechanisms, such as oxide, roughness, or corrosion is not included in this book.

3.3.1 Elber Crack Closure Phenomenon

Wolf Elber [17,19] explained the effect of constant amplitude loading of various positive stress ratios, R, on crack growth rate behavior by using the concept of plastic yielding at the crack tip. Prior to his observation on crack closure behavior, it was believed by the experts in the field [16] that the two crack surfaces contact each other

133

I0'

10"1

10"

10"2

lo"

10 4 T

~o4 g

Z

10,4

lo 4

10 4

i04

1 10 100

Figure 11: Influence of stress ratio, R, for 7075-T65 aluminum alloy [S]

and close the crack at zero or under compressive stresses. This assumption is valid for an ideal crack where plastic deformation is limited to a localized region at the crack tip (an ideal crack was defined as a saw-cut crack of zero width with no plasticity in the wake of crack). Elber argued that, upon application of constant amplitude cyclic loading during the increasing tensile portion of the load, a plastic zone is formed. Fatigue crack growth tests on aluminum alloys generated in the laboratory by EIber have indicated that for constant amplitude loading, the formation of plasticity occurring at the crack tip remains behind as the crack grows in a stable manner. As illustrated in figure 12, the tensile load and crack tip plastic deformation will create compressive stresses which, when the specimen is unloaded, are distributed at the locality of the crack tip, together with the plastically deformed material left behind, cause them to contact prior to complete unloading. The compressive residual stresses are a result of permanent deformation that stretches the material, and upon load removal it has to fit the elastic material surrounding it. The misfit causes the elastic material to induce compressive residual stress in the plastic region local to the crack tip.

134

In figure 12, when the crack length was at location (1), it was surrounded by a plastic zone larger than any other previous position, while its corresponding crack length is smaller than location 2. At

Kmax

Kcl Krnin

K ASeff = Smax- scl •Keff =Kma x- Kcl

/ ! 1 I Stain Scl Smax

Unloading Stage of a Cycle

Plastic Zone Plastically deformed (Compressive material left behind Stresses)

The presence of plastic zone causes the crack to close at a higher load Scl>Smi n upon unloading

Stress

SmaxlA A A / \ sSc,17 ' - V - V-V- .

Cycle

Figure 12: Crack closure based on plastic zone and effective stress

location (2) after N number of cycles, a new plastic zone is formed that now is larger than location (1) because of its length and higher value associated with crack tip stress intensity factor. The crack tip plastic zone at all stages of advancing fatigue crack forms an envelope that contains all the previous plastic zone sizes and its presence is the cause of crack closure behavior. Figure 13 illustrates the crack tip plastic zone and the envelope of all plastic zones for an advancing crack subjected to cyclic load compared with an ideal crack with zero width.

An Ideal Crack (Saw-Cut Crack With Zero Width)

I Plastic Zone

I A fatigue crack with plastic zone enveloped at the two crack surfaces in the wake of advancing crack

Figure 13: Plastic deformation formed at crack surfaces in the wake of an advancing crack (compared with an ideal crack)

135

The plastic deformation helps to partially close the crack surfaces such that the crack will close at a stress level higher than the Smi n. This effect results in having a smaller stress intensity factor range value and, consequently, a smaller crack growth rate, as illustrated in figure12. The effective stress intensity factor range, in terms of the effective stress range (AS)eft, is written as [15]:

AK= 13 As (=a) 1/2

AKeff= 13 (AS)ef t (~a) 1/2 6

where (As )e f f = smax - scl and As = smax - smin. The quantity scl is the stress level at which the closure occurs (Smi n < Scl) and Kcl is the corresponding stress intensity factor (Kmi n < Kcl). In addition to the quantity Scl (the stress level at which the two crack surfaces are in contact upon unloading), equal attention must be given to another parameter, called the crack opening stress, Sop, (the stress at which the two crack surfaces are not in contact with each other upon reloading). Hence, the fatigue crack will advance only during the opening portion of the cycle. In order to be able to address the closure behavior and its influence on crack growth rate further theoretical and experimental work is needed to evaluate the quantities Sop, Scl and, moreover, the difference in load magnitude between them. In this book, it is assumed that the two quantities Scl and Sop are the same (in general, for a given material, the closure stress, Scl, has a different value than Sop [20]).

The effective stress range ratio, U, that corrects the apparent stress intensity factor range to account for closure behavior, can be written [17] as:

U= AKef f / AK =(AS)eff/(As)=(Sma x - Scl)/(Sma x - Smi n) 7

Elber's crack growth rate equation, in terms of the apparent stress intensity range, AK, and quantity U, can be expressed in terms of the Paris law as:

da/dN=C(UAK) n 8

Based on constant amplitude testing of 2024-T3 aluminum alloy sheet, a linear equation for U in terms of stress ratio, R, was established [17] that can be written:

U = 0.5 + 0.4 R -0.1<R<0.7 9a

136

This relationship was later modified by Schijve [21] as follows:

U= 0.55 + 0.35 R + 0.1R 2 9b

The quantity U described by equations 9a and 9b provides the correction for the closure effect (described in terms of stress ratio, R) on the crack growth relationship. From equations 8 and 9 it can be seen that the two quantities, effective and apparent stress intensity factor range, become equal when the stress ratio R increases. That is, the crack closure effect becomes less effective when the R value increases and approaches unity.

3.3.2 Threshold Stress Intensity Factor Range, AKth

In the limit when Kma x reaches the Kcl value, the effective stress intensity factor AKeff =0 and, therefore, no crack growth is expected. This does not necessarily imply that the stress intensity factor range, AK, is zero; it simply states that there is a minimum value of AK, called the threshold stress intensity factor range (AKth), in which the growing crack ceases to advance. As was mentioned previously (in section 3.2), zero growth is approximated by a cut-off growth rate of 4x10 -1° m/cycle assigned by ASTM-E647. The AK level at which the growth is zero (or when da/dN.~4xl0-1°m/cycle) is a function of the stress ratio.

The threshold stress intensity range, AKth, in terms of stress ratio, R, and the threshold stress intensity factor at R=0, AK0, in a simplified form was empirically expressed as [22]:

(AKth)R = (1-CoR) d AKo 10

Co and d are the fitting constants. By allowing the constant Co=1 the Klesnil and Lukas [23] equation is obtainable:

(AKth)R = (l-R) d AKo 11

When the fitting constant d=l, the Barsom [24] relationship is obtained. In cases where the two fitting constants are not available, the value of Co=d=1 will give conservative results for the threshold stress intensity factor, AKth for R _>0. In general, the fatigue threshold value, AKth, decreases with increasing stress ratio, R and becomes a constant at R < 0.

(AKth)R = (l-R) AKo (for Co=d=1) 12

Equation 10 can be modified to account for the dependency of threshold stress intensity factor on crack size [25]. Frost [26,27], and

137

Usami [28,29] studied the effect of crack length on the threshold stress intensity factor and concluded that AKth decreases with decreasing crack length. Cracks in material can be classified as being long, short, or small in size. Long or large cracks are usually considered as being through the thickness cracks in a thick specimen. The crack growth behavior of large cracks is well established in terms of the closure phenomenon through the effective stress intensity factor, AKeff (see equation 8). In the threshold regime (region I), however, where crack growth is slow, the test measurements for long cracks can be sensitive to several parameters, such as overloading (that introduces the retardation effect), and inconsistency in data recording can occur.

Short cracks can be defined as being physically short in dimensions, such as. surface cracks with short crack length about 0.02 inch [26] in length. Cracks with short dimensions are less effected by closure phenomenon and will grow faster than corresponding long cracks subjected to the same stress intensity factor, AK. In general, crack closure effects increase with crack length. In addition, short cracks show a lower threshold value when compared with long cracks as shown in figure 14. The transition from short to long crack behavior has been reported by various sources [28,29,30] to be between 0.5 and 2 mm.

..I

Short Crack ) Behavior /

IT Lt / .~,,, Behav ,o ' Long C~ck

| AKth I AKth v

Log (AK)

Figure 14: Illustration of da/dn versus •K for short and long crack

Small cracks are defined as having dimensions as small as their material characteristic microstructure dimensions, such as the grain size. Their growth is controlled by grain boundaries and can not be described by the linear elastic fracture mechanics approach. There are several classes of small cracks which Ritchie and Lankford defined in their work "Small Crack Problems" [30]. They used mechanically, microstructurally, and physically small as useful qualifiers to classify

138

small fatigue cracks. For example, a crack is mechanically small when the crack length, a, is smaller than the plastic zone size, rp (a<rp). When the crack length is smaller than the grain size, dQ, it is called microstructurally small (a<d,). Other classes that fatigue cracks can be defined include physically small, when a<l mm and chemically small when a<10 mm.

Tanaka, Nakai and Yamashita [25] established a theoretical model to describe the small crack length effect on the threshold stress intensity factor, AKth, based on the crack-tip slip band blocked by the grain boundary concept (called the BSB model). The BSB model is based on the assumption that a small crack ceases to advance (the threshold condition) when the crack tip slip band has been arrested by the grain boundary and prevented .to grow into the next grain. The effect of crack length and grain size on the threshold stress intensity factor, AKth, was studied both theoretically and experimentally and a general relationship between AKth, crack length, a, and the intrinsic crack length, a 0 was established as:

AKth = [a/(a + a0)]1/2 x (AKth)R=0 13a

Equation 13a was later modified to account for the stress ratio, R [5]:

AKth = [ (4/=) tan -1 (l-R)] [a/(a + a0)] 1/2 x (AKth)R=0 13b

where (AKth)R=0 is the threshold stress intensity factor for R=0. The intrinsic crack length, a 0, defines the boundary between small and long cracks and represents the minimum initial crack size for conducting a meaningful life evaluation using fracture mechanics analysis. Based on the theoretical results and experimental data, an estimated value of 0.004 inch is assigned to a variety of steels and aluminums [31]. As was mentioned in 3.1, theminimum initial crack size used in the damage tolerance crack growth analysis must be larger than the material's grain size in order not to violate the isotropic continuum concept. The threshold stress intensity factor range incorporated in equation 4 for crack growth rate is a modification of equation 13b [32] and is approximated by the following empirical equation:

AKth=[a/(a + a0)]1/2 x (AKth)R=0 / [(1- t)/(1- A0)(1-R)] (l÷ct") 13c

where f is the Newman closure function, A 0 is a constant defined by equation 17 (section 3.3.2), and Ct, is an empirical constant. The intrinsic crack size, a 0, in equation 13c has a value of 0.0015 in. which

139

provides a good fit to the data points when threshold stress intensity factor as a function crack depth is plotted [33].

The minimum accepted crack size that can be used for conducting a meaningful crack growth analysis using LEFM can be estimated by employing the Kitagawa and Takahashi Diagram [34]. Kitagawa argued that, in many structural parts, the crack growth process is by the initiation and growth of cracks as small as 0.5 to 1.0 mm in length, below which their growth behavior deviates from the LEFM regime. Therefore, it is necessary to develop a method to observe the initiation and growth of such small cracks that may behave differently from the experimentally measured crack growth data associated with standard cracks. Kitagawa defined the threshold stress range, z~Sth, for a fatigue crack to be associated with a crack growth rate less than 2x109 mm/cycle (=8x10 11. in./'cycle). To studythe effect of crack length on the threshold stress intensity factor range, AKth, the experimental variation of z~Sth versus full crack length, 2a, was plotted for crack length smaller and !arger than 0.5 mm. The measured experimental data showed that for crack length larger than 0.5 mm, good correlation with the LEFM results was obtained (a straight line relationship with slope -1/2, see figure15). This observation (constant slope of -1/2) indicates that the threshold stress intensity factor, AKth, is independent of crack length and is a constant for long cracks where 2a>0.5 mmo For crack length smaller than 0.5 mm, a deviation from the straight line was observed, indicating that the LEFM concept may not be applicable to define the fatigue crack growth threshold behavior.

o = ...I

Endurence Limit for Uncracked Smooth Specimen (R = 0)

_ Non-Damaged Reg'on

~ - - ~ - - AS=AKTh / 13(~a)l/2

i c:c, i S=Lo AKTh - ,/2Log=a

i o a* a0 Log(a)

Figure 15: Kitagawa diagram for estimating the ao

In the Kitagawa Diagram, the boundary between small and large cracks was estimated to be at the onset of deviation from straight line behavior. The experimental data on HT-80 steel, shown in reference [30], indicates that the continuation of the trend deviation from the

140

straight line becomes gradually asymptotic to the fatigue limit of an uncracked smooth specimen, as shown in figure 15. Cracks smaller than a* are non-damaging cracks when subjected to an applied cyclic load equal to the endurance limit. Note that the Kitagawa Diagram is dependent on the stress ratio, R, and will shift to a lower stress level as R increases (also note that the endurance limit for most materials decreases with increasing R value, as discussed in chapter 1)

A minimum initial flaw size employed in the aerospace industry for evaluating the life of fracture critical components has been developed based on the capability of crack detection using standard NDI methods. The reader should realize that the crack size assumption by standard NDI methods is based on the concept of a maximum initial flaw size that may escape detection. If a smaller flaw size is required by the analyst, a special level NDI could be performed on the part. These values are much higher than the limiting value of a0= 0.5 mm (0.019 in.) observed in the Kitagawa Diagram.

Recently, the NASA has accepted the option of replacing the conventional fatigue approach (S-N curve) in estimating the life expectancy of non-fracture critical or low risk parts with a fracture mechanics approach that uses a part through crack of depth a=0.005 in. and length 2c=0.01 inch. This value is much smaller than the initial flaw size provided by standard NDI methods when performing safe-life analysis for primary space structural components. This method is preferred over the conventional fatigue approach (see chapter 1) because of its feasibility and a ready availability of material data as shown in appendix A. The flaw size assumption provided by the NASA (0.005 in.) is slightly larger than the material grain size and falls in the boundary between small and large cracks, and the concept of isotropic continum is obeyed.

3.3.3 Newman Crack Closure Approach

The crack closure behavior and its effect on fatigue crack growth rate under constant amplitude loading was discussed by Newman in his Doctoral thesis entitled "Finite Element Analysis of Fatigue Crack Propagation Including the Effect of Crack Closure" [20]. He later developed an analytical model to describe the crack closure phenomenon for both plane stress and plane strain conditions (the thickness influence on the crack closure behavior). The closure model was based on a strip yield model (Dogdale model) that was modified by Newman to incorporate the envelope of all the plastic zones on the wake of an advancing crack. In both plane stress and plane strain conditions, the closure model considers that the material is plastically deformed along the crack surfaces as the crack grows. Newman argued that fatigue cracks remain close during part of the cycle when unloading occurs. When the two crack surfaces are in contact with each other during the final part of the unloading portion of the cycle,

141

the material may undergo compressive yielding. Upon reversing the cycle as the specimen is reloaded, the crack produces tensile residual stresses at the crack tip which cause the crack to open (zero contact between the two crack surfaces) lower than or equal to the closure stress occurring in the previous portion of the cycle, Scl. The stress at which no surface contact is present upon reloading the specimen is called the crack opening stress, Sop. An equation was established by Newman that could define the crack opening stress, Sop, as a function of stress ratio, R, maximum stress amplitude, Sma x, and specimen thickness. The specimen thickness effects in closure behavior were defined in terms of a two dimensional closure model [31,35] by applying a constraint factor, o~, on yielding. The quantity a is used to describe material yielding in two and three dimensions through the flow stress, (~o (tensile yielding when ~o is positive and compressive yielding when (~o is negative). The flow stress, (~o, is taken as the average value of the tensile yield and the ultimate of the material. Crack tip yielding occurs depending on the biaxial or triaxial state of stress on that region. Under plane strain conditions, yielding occurs at a stress level much higher than the uniaxial yield stress obtained through laboratory testing. The effective yield stress value, a(~ o, could be as high as three times the uniaxial yield stress when a three- dimensional state of stress (plane strain) prevails. Therefore, the material will yield when the stress at the crack tip has reached the quantity aa o. For plane stress and plane strain conditions, the constraint factor, (~, takes the values of ~=1 and 3, respectively. High a values (2.5 or higher) have been assigned to materials with low K~J(~Yiel d ratio (low fracture toughness value such as high strength steels), whereas high K[J(~Yiel d ratios have 1.5 <a _< 2.0.

The closure effect on the crack growth rate was described by correlating the effective stress intensity range, AKeff, to the apparent stress intensity range, AK, through the crack opening stress, Sop, and stress ratio, R, as:

AKeff= U AK =[(1- Sop/Sma x )/(1-R)]AK 14

Equation 14 simply states that, given a crack growth rate, (da/dN) 1, obtained from test data under constant amplitude loading and a stress ratio, R 1 (for simplicity assume R 1=0.2) with the corresponding stress intensity factor range AK; for any other stress ratio, R 2, where (da/dN)l=(da/dN)2, the shift in the stress intensity factor range (AKeff) can be evaluated through the parameter U (for crack closure adjustment) described by equation 14, see figure 16. Experimental

142

- - I rUar~k lilto sW~trhe I

j T '~-' / I 1

- I - - dald.NN _ -

i i , - / / , / : i / / ( 7 ! '

l"Ifl: I I T / , , , -I''E-' CAK)2--~I I I

10 (AK) 1

Log (AK)

Experimental data

~--- (AK) 2

100

Figure 16: Shift in the AK can be evaluated through U (equation 14)

data shows that for most aluminum alloys, the stress ratio contribution that can shift the crack growth rate curves is negligible for a stress ratio, R, greater than 0.7 [36], see figure 17. This figure shows the experimental data obtained for A536 cast Iron with different stress ratios, R= 0.0, 0.1, 0.3, 0.5, and 0.7 that were curve fit by employing equation 4 which includes the Newman crack closure effects [5]. Equation 4 can be used to describe the crack growth rate with different stress ratio, R, where the effect of stress ratio can be expressed through the crack closure parameter, f, described by equations 22 and 23 and it provides excellent curve fit results.

For a center crack specimen under tension loading, the normalized crack opening stress, Sop/Smax, describing AKef f (that was expressed by equation 14 in terms of stress ratio, R, and constraint factor, ~) , can be written as [12]:

Sop/Sma x = [ Ao+ A 1 R+ A2R2+ A3R3 ] for R >0 15

and

Sop/Sma x = [ Ao+ A1R ] for -1< R<0 16

143

10"2, o m l o ~ 11Irk RIV

OD A I ~ A 1 0.0W ~ AIACS~4IM,B1 0.100 0,,100 241

A AIACSOAS01C, 0.100 0.~0 241 v AIAC~AB011~ 0.S00 0.~0 241 '0, A SAI~0AKIIA2 O.S~O 0,.,,10

10 4 ° • AIA(2~OABO1L1 O,llO0 0.300 241 ! • A ,Al~10Aa0184 0.700 0.~0 24t

ASTM ~ GR0 80-U.04 C.AST IRON : All CAST

Enve~.~n~ LA

• I ~ c m m.~

10"4. F ~ 2S.~

10 "1

10"2

,10",1

1 0 4 .

u

Z

104 . "o

10"7.

10 ~.

10"4

o Q

lo*,

uS

10 "s

c w v , ~ S.,ml % - 0.1

I I I N . 0.3 C 7E-I0 ,10 -4 n 2.1 : p O.S q 0. i

Yield Sl

• % 0.Ts 10.7 ~, oh , x , a

R I 0.1

10 ~J( [ I m i ' S Q R T ( i n ) ] 100

Figure 17: The stress ratio contribution that can shift the crack growth rate curves is negligible for a stress ratio, R, greater than 0.7

The coefficients Ao, A1, A2, and A 3 for Sop > Smi n are given by:

Ao= (0.825 - 0.34c~ + 0.05c~ 2 ) [ cos (~Sma x/2~o)] 1/°~ 17

A I = (0.415 - 0.071(~)X(Sma x /a o) A2= 1- A 0 - A 1 -A 3 A3= 2A0+ A 1-1

18 19 20

The flow stress, c o, is taken to be the average of the uniaxial yield stress and uniaxial ultimate tensile strength of the material:

(~o=(~Yield + oUlt)/2 21

144

The contribution of the crack closure effect on the FNK empirical relation described by equation 4 is described by the crack opening function, f. For plasticity induced crack closure, the parameter, f, can be written as [12]:.

f= Kop/Kma x = Max (R, Ao+ A1R+ A2R2+ A3R3 ) for R _>0 and,

22

f= Kop/Kma x =( Ao+ A 1R ) for -2 < R <0 23

where Kop is the crack opening stress intensity factor below which the crack is closed. For a stress ratio of R=0, the crack closure parameter, f, will have the maximum value given by equation 22.

Newman and Raju [35] studied the crack closure effect for surface and corner cracks under constant amplitude loading and variable stressratio, R. They have shown that better crack growth results at the c-tip can be obtained by multiplying AK by a crack closure factor (13R). The value of 13 R for the stress ratio R>0 is given by:

I~ = 0.9 + 0.2R 2 - 0.1R 4 24

and for R<0 the factor 13R = 0.9.

The FNK empirical equation describing the crack growth rate (see equation 4) is used in the NASA/FLAGRO computer code that was developed to provide an automated procedure for analysis of fracture critical parts of NASA space flight hardware and launch support facilities. In this computer program, the effect of the stress ratio is incorporated through the quantities (l-R)" and the crack closure function f by assuming a constant value of Sma x/Co =0.3. This value was selected because it is close to the average vaJue obtained through fatigue crack growth tests using various specimen types. For most aluminum alloys (2000 through 6000 series), c{ is chosen to have a value of 1.5 and for 7000 series (~=1.9. Having the two quantities Smax/(~o and c~ available, the crack closure effect for any other stress ratio, R, can be evaluated by utilizing equations 22 and 23. Table 1 presents a listing of all the constants that must be provided as input to equation 4 for 7075-T651 aluminum. Information contained in Table 1 was extracted from the NASA/FLAGRO material library. The quantities Ak and Bk are the fit parameters that define the plane stress fracture toughness, K,, in terms of the plane strain fracture toughness.

TABLE 1

°'Yield ~Ul KIc Kle AK BK C n p q DK 0 Rcl a SR 76, 85, 28, 38, 1.0, 1.0, 0.2E-7, 2.885, 0.5, 1.0, 2.0, 0.7, 1.9, 0.3

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The net section yielding check is confirmed by NASNFLAGRO when the net section stress exceeds the yield strength of the material. The NASA/FLAGRO computer code contains the net section yielding equations for each of the through crack cases subjected to different loading conditions. As soon as net section yielding occur, a warning is issued, however, fatigue crack growth analysis continues. It should be realized that the result of analysis when net section yielding occurs may be non-conservative, due to a reduction in material fracture toughness in the deformed zone ahead of the advancing crack. For the case of part through cracks, a net section yielding check for the through the thickness direction is performed when the sum of the crack length, a, and the crack tip plastic zone, p, becomes equal to or exceeds the part thickness (a+p > t). The plastic zone size calculation was discussed in chapter 2 and was given by equation 22. Before and after the net section yielding check, the calculated value of Kr, ax is always compared to Kla and when the total crack length exceeds the thickness (a+p > t) the quantity K,,,x (at the c-tip) is compared with the Kc value. In the length direction (c-tip), a net section yielding check is performed based on the net section stress if it exceeds the material yield value. Moreover, when a+p < t, the Kr,=( (at c-tip) is compared with 1.1 K~e and for a+p _> t, it is compared with the Kc value.

The NASA/FLAGRO (version 3.0) computer code has the following capabilities:

1)

2)

Safe-Life Analysis of structural components subjected to a constant amplitude cyclic load.

Critical Crack Size that will cause the structure to become unstable.

3)

4)

Stress Intensity Factor Solutions for new crack geometries and loading conditions.

Retardation effect due to load interaction described by the Generalized and Modified Generalized Willenborg models, Walker-Chang-Willenborg, Strip Yield Constant Closure model.

5) Improved threshold equation which describes Kth dependency on crack length.

NASA/FLAGRO has a rich material library containing fracture property data for many aerospace alloys subjected to varying heat treatment conditions. Appendix A provides fracture properties of most aerospace alloys obtained from the NASA/FLAGRO material library (see table 1 for the definition of the symbols). In addition, the program

146

has enhanced stress intensity solutions that cover various crack geometries under different loading conditions. For a more detailed description of the NASA/FLAGRO computer code, the reader may refer to reference [5]. Improvements to this code are in progress and as the technology advances, the program's capabilities will be enhanced accordingly.

Example 1 A pressurized cylindrical tank is made of Ti-4AI-4V alloy. Its fracture properties (based on the NASA/FLAGRO material library) are shown in figure 18. The tank is pressurized and de-pressurized 8 times per year during its service life. For operating pressure, Pop=600 psi; 1) perform a cycle by cycle analysis using equation 4 and compare the results of the first three cycles with the NASA/FLAGRO compute r code output (or any other available code), 2) Determine the final crack length at the end of the year. The original crack length based on penetrant inspection is given as a circular surface crack with ao=0.075 and 2c=0.15 inch. The purpose of part 1) of this example problem is to allow the reader to learn the steps necessary for computing the crack growth values based on the NASA/FLAGRO approach and accumulation of growth corresponding to each cycle.

D = 40 in. a = 0.075 in. c = 0.075 in. Pop = 600 psi

o = pr/t

P R = 0

v

Cycle

Material Properties From NASA/FLAGRO (Ti-6AL-.4V)

~Y, cUlt, KIc, Kle, Ak, Bk, C, n, p, q, DKo, Rcl, c~, SR

140, 150, 55, 45, 0.5, 1.0, 0.2E-8, 3.0, 0.25, 0.75,3.5, 0.5, 2.5,0.3

Figure 18: Material and crack geometry for the pressurized cylinder in Example 1

Solution Using equation 4, the amount of crack growth associated with each cycle can be computed. The stress intensity range, ,~K, for each cycle is calculated by employing equation 41 of chapter 2 for a longitudinal surface crack in a pressurized pipe:

13(a/c, a/t, 0 )= 0.97 * [ Ml+M2(a/t)2+M3(a/t) 4 ]* g * f~ *fc* fi * fx

where:

147

MI= 1.13 - 0.09 (a/c)

M 2 = -0.54 + 0.89•(0.2 +a/c)

M 3 = 0.5 - [1/(0.65 + a/c)] + 14 (1 - a/c) 24

g = 1 + [0.1 + 0.35 (a/t) 2 ] (1 - sin 0)2

f~ = [ (a/c) 2 cos 2 0 + sin 2 0] 1/4

fx = [1+1.464(a/c) -1.65]-112

The quantity fc = [(1 + k2)/(1 - k 2) + 1-0.5 ( a l t ) l l 2 ] [ t l ( D I 2 - t)] where k=l-2t/D and the value of fi = 1 for an internal crack. Note that the

quantity fx=(¢ )-1/2, as discussed in chapter 2.

The calculated values of M 1, M 2, M 3, fx, and fc for a/t=0.75 and a/c=1 using equation 41 are:

M1=1.04, M2=0.2016, M3=-0.1066 , fx = 0.6370 and fc=1.0053

The correction factor is 13=0.6956, where O=n/2. The stress intensity factor range, AK, for the case of R=0 (see figure 18) can be obtained as:

z~K=Kmax =13 Cmax(na) 1/2

=0.6956x(600x20/0.1)(nx0.075)112

=40.509 ksi (in) 1/2

Note that the applied stress for the first cycle (AN=l) is equal to the hoop stress, 120 ksi. When AN=I and R=0, equation 4 can be written as:

c(1-f)n AKn (l - AKth )P Aa- AK

hK (I - )q

KIe

148

The crack opening function, f, can be calculated from equation 22 by replacing the quantity Sma x /~ o by 0.3 (f =0.2745). Moreover, the threshold stress intensity range AKth = (AKth)R=0 and is designated by

AKo =3.5 ksi (in) 1/2 in the NASA/FLAGRO material library.

Simplifying the crack growth equation by supplying the appropriate values of the constants f, AKo, AKth, C, n, p, and q (see the material properties shown in figure 18), the amount of growth for the first cycle is given by Aa=0.00013 inch. The original crack depth is advanced by the amount of 0.00013 inch and the new crack length at the a-tip is now acting as the original crack depth for the next cycle (ao=0.075 + 0.00013=0.07513 inch).

The amount of growth in the length direction (c-tip) can be calculated in the same way as for the a-tip. Note that the equation of the stress intensity factor for the c-tip was formulated in chapter 2 (described by equation 41) by simply replacing the angle 0=90 ° with 0 °. The correction factor, 13, is slightly higher for 0=0 ° due to the dependent quantities g(0) and f¢ on the angle 0 (see equation 41). The calculated values of g and f¢ are 1.2968 and 1, respectively. The stress intensity range for the c-tip can be calculated as:

AK=Kma x =13 ~max(uc)l/2

0.9x120x(ux0.075)1/2

52.54 ksi (in) 1/2

Crack growth for the c-tip can be obtained from equation 4 in the same manner as was applied for the a-tip direction. However, the fracture toughness value for the c-tip, K[e, is now-replaced by 1.1K~e [5]. The closure effect for the c-tip was described by equation 24 and for the case of R=0 it takes the value of 13R=0.9:

c(1- f)n (0.9AK)n ( ! - AKth )P 0.9AK Ac=

0.9AK (1 )q

l.lKie

The calculated amount of growth at the c-tip is given by Ac=0.00025 inch. The original crack length at the c-tip is advanced by the amount of 0.00025 inch and the new crack length at the c-tip is now acting as the original crack length for the next cycle (Co=0.07525 inch).

149

The amount of crack growth in both the a and c-tip directions can be calculated for the next cycle in the same manner; however, the aspect ratio a/c for the new analysis must include the actual value, rather than the original value of a/c=1. For example, for the second cycle of pressurization and depressurization, the new aspect ratio a/c=0.07513/0.07525=0.9985 must be employed. The following shows the calculated values of the quantities that are used to evaluate the crack growth up to three pressure cycles for both the a and c-tips. In addition, the values of the amount of crack growth for each cycle are calculated by the NASA/FLAGRO computer code and presented for comparison.

Results of the hand analysis for Example 2

a/c MI M2 M3 g(90) 1 1.04014 0.2026 -0.10661 1

0.9985 1.03998 0.2015 -0.10596 1

0.99699 1.03998 0.2015 -0.10596 1

a/c M1 M2 M3 g(O) 1 1.04 0.20167 -0.1063 1.29688

0.9985 -2.52183 0.20211

0.99699 1.04006 0.20211

f~ 1

1

1

f, 1

-0.1063 1.29758 0.99925 0.63659 1.00537

-0.1063 1.2983 1.o0o14 0.63612 1.00536

fx fc ~ AK ~a 0.63706 1,00537 0.69563 40.5095 0.00013

0,63659 1:00537 0.69563 40.5458 0.00014

0.63612 1.00536 0,69552 40.5759 0.00014

fx fc ~ ~K ~c 0.63706 1.00537 0.90223 52.5406 0.00025

0,90182 52.6035 0.00025

0,90104 52.5584 0.00025

Final Crack Sizes by Hand Analysis

Cycle 1 Final Crack Sizes: a = 0.07513 , c = 0.07525, a/c = 0.9985

Cycle 2 Final Crack Sizes: a = 0.07527 , c = 0.07555, a/c = 0.9962

Cycle 3 Final Crack Sizes: a = 0.07541 , c = 0.07575, a/c = 0.9955

Results of NASAIFLAGRO (Cycle By Cycle)

Cycle 1 Final Crack Sizes: a = 0.0750690 , c = 0.0751050, a/c = 0.9995

Cycle 2 Final Crack Sizes: a = 0.0750833 , c= 0.0751263, a/c= 0.9994

Cycle 3 Final Crack Sizes: a = 0.0750876 , c = 0.0751476, a/c = 0.9993

In using NASA/FLAGRO with three cycles at once rather than cycle by cycle, the results are:

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Final Crack Sizes: a = 0.0750389 , c = 0.0756066, a/c = 0.99712

In the second part of this example problem, the final crack length at the end of the year (after 8 cycles) is:

FINAL RESULTS Critical Crack Size has NOT been reached. At Cycle No. 8.00 Crack Sizes: a = 0.760554E-01 , c= 0.766525E-01, a/c= 0.992210

Note: The crack growth analysis conducted by the NASA/FLAGRO computer code is not based on a cycle by cycle approach; rather, it is based on incremental growth, Aa=(l/n)ao. The advantages associated with this method are cost effectiveness, and speed. The analysis is performed by assuming the amount of incremental growth always to be 1/200 of the original crack size. The number of cycles associated with ao/200 is calculated by using equation 4. Therefore, the average amount of growth per cycle is:

(z~a)l cycle = (ao/200)/N

Discrepancies between the hand analysis and NASA/FLAGRO are therefore expected for the first few cycles until the calculated Aa value becomes equal to ao/200.

Example 2 Using data from the previous example problem 1) Determine the number of cycles needed to have leak before burst, 2) Find the stress value (magnitude of the stress due to pressure) for which leak before burst is not achievable (the leak before burst criteria was discussed in chapter 2).

Solution It would be difficult, if not impossible, to obtain the number of cycles associated with leak-before-burst by employing a hand analysis as demonstrated in the previous example. Using the NASA/FLAGRO computer code, however, the results can easily be provided to the analyst. The following are the results of the NASA/FLAGRO analysis for part 1 of this problem.

RESULTS Transition to a through crack occur at Cycle No. 55.82 where the crack length at the a-tip: a = 0.9382E-01 (t - 0.10 in.) and crack size at c-tip: c = 0.09042, a/c = 0.9322

For stress level associated with 132 ksi (internal pressure of 660 psi) leak before burst does not take place as indicated in the final results obtained through NASA/FLAGRO output (K>K~e).

151

FINAL RESULTS: Unstable crack growth, max stress intensity exceeds critical value: K max = 60.56, Kcr = 60.50 at Cycle No. 24.77. Crack Sizes: a = 0.818863E-01, c= 0.868883E-01, a/c= 0.942432

Example 3 An embedded crack was detected by X-ray inspection in the weld region of a pressurized liquid oxygen tank to be used for a space vehicle, as shown in figure 19. The detected crack was shown to be very close to the surface of the weld and conservatively considered as a surface crack having leng.th 2c=0.32 in. The crack is oriented in the circumferential direction ~n the shell region of the tank with cApplied=PR/2t (where P is the maximum operating pressure, R is the tank radius, and t is the thickness associated with the weld region in which the crack was detected). The da/dN-AK data curve for the base material is shown in figure 19. Use the NASA/FLAGRO computer code (or any other available fatigue crack growth computer code) to determine the number of pressurization cycles that the tank can

1E-2 70

35 " l l E

da/dN - zM< curve for the weld ~/ I j I - - "-~ I I

._ . 50% Reduction ~ . ~ / / ' / ,

' L . , . .,J

/ I I :

' g J I I

lo I 70 16o

Log ( t ~ )

1E-7

Figure 19: Crack growth curve for the welded region described in example 4

withstand before failure. Assume the weld properties are 50% lower than the parent material (stress ratio, R=0, P=35 psi, ((~Yield)parent = 55 ksi, ((~UIt)parent =60 ksi, tank radius, Radius= 72 in., and t=0.1 in.). The plane strain fracture toughness, K~c, for the weld is equal to 18

152

ksi~/in. The crack geometry was assumed to be a circular surface crack with total crack length 2c=0.14 and crack depth a=0.07 inch.

Solution To obtain the total number of pressurization cycles to failure in the weld region of the liquid oxygen tank, a 50% reduction from base metal properties must be considered (as is shown in figure 19). Furthermore, region I i I of the c u r v e is approximated by using the asymptotic line associated with 35 ksi~/in (this is equal to 50% of the parent material fracture toughness.

To use the NASA/FLAGRO computer code for determination of the number of cycles to failure, the fatigue growth rate constants used to establish the da/dN, AK curve must be computed from the data points shown in figure 19.

Material Properties

UTS YS Kle KIc Ak Bk Thk Kc 3 0 27.5 25.2 18.0 1.0 0.0 0.10 35

where K~e = 1.4K~c (see chapter 2). From da/dN, AK test data with 50% reduction in the properties:

da/dn Delta K

0.8000E-02 35.00 0.1000E-02 27.50 0.4000E-03 25.00 0.2000E-03 18.00 0.5500E-04 9.000

0.2200E-04 5.000

FINAL RESULTS: Transition to 1-d solution (TCO7) at cycle No. 504 of load step No. 1 of Block No. 1. Final crack size: c=0.0928 in. a/c=0.952.

3.4 Variable Amplitude Stress and the Retardation Phenomenon

Fatigue crack growth rate test data needed for life analysis (da/dN versus AK) are produced in the laboratory under the condition of constant amplitude loading. All of the fracture properties that are compiled in the NASA/FLAGRO material library [5] (shown in Appendix A) are generated under a constant amplitude loading condition. However, in real situations, most structural parts are

153

subjected to variable amplitude loading throughout their service life. Crack growth test data have shown that, under variable amplitude loading, there is delay (retardation) or acceleration in the amount of crack growth due to high or low loads. Therefore, the effect of high and low loads on the amount of crack growth must be addressed. For example, the loads that the wing of a transport aircraft encounter during its service usage are complex and the effect of stress interaction due to variable amplitude loading on crack growth rate is an important problem in aircraft design.

Experimental data have indicated that a high tensile load, followed by a constant amplitude load, will reduce or retard the rate of growth, as illustrated in figure 20. This phenomenon is called retardation [37]. Retardation or delay in the rate of crack growth results from the plastic deformation that occurs at the crack tip. The tensile overload produces large tensile plastic deformation (called the overload affected

Constant Amplitude o Load Retardation Due to High Load " " ' L / ' 4-

. ~ - - // ~"",./~llt/~ k High Ioading?v~le~~V /

¢J ao

Number of Cycles, N Figure 20: Decrease in rate of crack growth due to tensile overload

zone) and, upon removal of this load, the material in the vicinity is elastically unloaded. The plastic zone surrounding the crack tip however, experiences compressive stresses. Crack growth by subsequent smaller cycles have a retarded rate across the affected zone. The higher the magnitude of the tensile overload, the larger the retardation effect when a subsequent low cycle amplitude is applied.

Even though the plastic deformation under constant amplitude loading has a smaller dimension, it does not have any retardation effect at the crack tip. Constant amplitude plastic deformation contributes to the closure phenomenon described in 3.3, since crack surfaces close at a non-zero load level (Smin<Scl).

154

In general, structural parts undergo random loading during their service life. The load environment contains not only the tensile overloading, as shown in figure 20, but also other types of overloading, such as compressive and tensile-compressive overload (see figure 21), that effect the rate of crack growth. Negative or compressive overloads may have the opposite effect and tend to accelerate the crack growth rate [38,39]. It is worthy to mention that for pre-cracking the brittle material, when it is difficult to precrack under the normal tensile mode ! loading procedure, a high compressive load concept can be used to initiate the crack or to extend the pre-existing crack in the material. When a tensile overload is followed by a compressive overload, the effect of the retardation on the rate of growth is reduced or may be totally eliminated.

To adjust the crack growth rate due to tensile overload, the empirical crack growth equations, described in section 2, must be corrected. Various models are available that describe the retardation phenomenon. Two well known mathematical models, called the

Load

£ Random Loading

A AAA vv V

Cycle

Cycle

Load Load

IAAAA AA/ I VV v, Cycle i

Tensile Over Load Load Compressive Under Load

Cycle

Tensile & Compressive Overload

Figure 21: Illustration of the types of loading that occur in service life (tensile and compressive overloads)

Wheeler and Willenborg models, based on yield zone [40,41] are presently available and will be discussed in sections 3.4.1 and 3.4.2, respectively.

155

3.4.1 W h e e l e r Retardat ion Model

In this model, the crack growth equation is corrected to account for the retardation effect. The effect of the tensile overload on the rate of growth is through the crack growth reduction factor parameter, Cp [40]. The retarded crack growth rate, (da/dN)r, in terms of the apparent crack growth rate, (da/dN), can be written as:

(da/dN)r=Cp(da/dN)=C p f (AK) 25

The retardation factor, Cp, is related to the size of the plastic zones created by the tensile overload (the overload-affected zone, (rp)oL), the crack size at the onset of over load, a OL, and the plastic zone size for the current or subsequent cycles, (rp) n, as illustrated in figure 22:

Co:/aoL (rp)n ]m

+ ( rp)oL - an 26

Overload Affected Plastic Zone Size

Regular Cycles Plastic Zone Size

I ' (r pn) ° I I< ~-'

(r p)OL

Overload

A AA/ vv vv

t Regular Cycles

Figure 22: Illustration of plastic zones and the retardation effect as the result of overload

m is an empirical constant that can be tailored from variable amplitude test data to allow for reasonably accurate life predictions. Test data have indicated that the constant m depends on the material, crack size and the level of the applied overload. The power m was evaluated by Wheeler f o r D6AC steel and Ti-6AI-4V alloys to be 1.43 and 3.4 respectively. The quantity (an-aoL) is the amount of growth associated with n cycles consumed in the overload affected zone. Note that within the retardation zone the quantity aoL+(rp)oL>an+(rp)n . The plastic zone

156

sizes, (rp)oL and (rp) n, shown in figure 22, can be written in terms of the stress intensity factor and the tensile yield of the material by:

(rp)oL =(1/2~)[(K)oL/(~Yield] 2 27a

(rp)n=(l/2=)[(K)n/(~Yield] 2 27b

From equation 26 (see also figure 22), it can be seen that the Wheeler retardation model described by the parameter Cp is effective as long as the plastic zone size associated with the nth cycle, (rp) n, is within the larger zone (rp)oL- That is, when the current crack size reaches the end of the yield zone, an=(rp)oL, and the crack growth reduction factor parameter Cp=l.

3.4.2 Willenborg Retardation Model

The Willenborg model considers the effect of retardation on the rate of crack growth through the extent of the plastic zone formed by the overload [41]. The correction to the crack growth is handled through the use of an effective stress intensity factor that can be expressed as:

(Kn)ef f = K n- Kred 28

where the reduction stress intensity factor, Kre d, (also called the residual stress intensity factor, due to the compressive residual stress state caused by differences in load levels) as the result of overload is given as [41]:

Kred = (KoL)max [1-(an- aOL)/rpoL] 1/2 - Kmax,n 29

in which (KoL)max is the maximum stress intensity factor due to the overload and Kmax, n is the maximum stress intensity factor for the subsequent smaller cycle (see figure 22). When the current crack size has reached the boundary of the overload yield zone (a n- aoL=rpoL), the retardation effect becomes zero, Kre d =0. Any overload greater than the previous overload creates a new retardation effect that is independent from the preceding one.

Equation 28 in terms of maximum (Kmax)ef f and minimum (Kmin)eff effective stress intensity factors can be written as:

(Kmax,n)ef f = 13(Cmax, n- (~red)(=an) 1/2 30

157

(Kmin,n)ef f = 13((~min, n- ared)(=an) 1/2 31

and the effective stress intensity range is:

(AKn)ef f = (Kmax,n)ef f - (Kmin,n)ef f = AK n

32

From equations 30, 31 and 32, one can conclude that the effective stress intensity factor range, (AKn)eff, for a complete cycle is equal to the apparent stress intensity factor, AK n. That is, in the Willenborg retardation model, both quantities, the effective stress range (A(~)e ff and the apparent stress range, A(~ are the same:

(A~)eff =((:~max,n)eff - ((~min,n)eff = (~max,n-(~min,n 33

where the effective stress ratio, Ref f :

Reff = (Kmin)eff/(Kmax)ef f 34

Because the effective and apparent stress intensity range are equal in the Willenborg retardation model, the retardation effect is affected by the effective stress ratio described by equation 34. By calculating (AK)eff or AK and Reff, the amount of growth, da/dN, in the retardation zone can be computed by any crack growth equation in which the stress ratio, R, appears. For example, consider the Forman crack growth relation described by equation 4:

c (AKe.)n da / dN = 35

(1-Reff)Kc - AKeff

The Willenborg retardation effect can be handled cycle by cycle through equation 35. This equation is effective as long as Reff is positive. One limitation of the Willenborg retardation model (described by equation 29) is that, for the case of (KoL)max =2 x Kmax, n, at the application of the overload where a i =aOL, complete crack arrest is obtained and crack growth ceases completely. That is:

when an=aoL ~ Kre d = (KoL)max - Kmax, n =Kmax, n ... (Kn)eff = K n- Kre d =0

To account for an overload greater than twice the previous load, the Willenborg model was revised by Gallagher and Hughes [42] and is

158

known as the "Generalized Willenborg Model". Gallagher and Hughes introduced a parameter ~) into equation 29, such that:

Kred =(I) { (KoL)max [1-(an - aOL)/rpoL] 1/2 - Kmax,n} 36

where:

K 1- max, th

K i)= max,n

(SOL)so -1 37

The quantity Kmax,th is the threshold stress intensity factor associated with zero crack growth and (SoL)SO is the shutoff overload ratio that can cause complete retardation when the crack ceases to grow (da/dN=0):

(SoL)so=Overload/Underload=(KoL)max/Kmax,n 38

Example 4 An aircraft component is made of 7075-T651, L-T, 75F aluminum alloy (constant amplitude crack growth data as shown in figure 23a [5]) and is subjected to a variable load environment. The fluctuating load is repeated every three cycles and has maximum peak stresses of 18,14, and 10 ksi with R=0, as illustrated in figure 23b. Visual inspection shows that the maximum detected flaw size is a center crack with length 2a=1.0 inch. The thickness and width of the part are 0.25 and 10.0 inch, respectively. Use the Forman equation to calculate the amount of crack growth for the following cases:

1) No retardation effect, 2) Apply the retardation effect using the Wheeler model (assume m=1.4), 3) Use the Willenborg model, and Generalized Willenborg models. Conduct the above mentioned analysis for the first 6 cycles (Cycles associated with 18, 14, 10, and 18,14, 10 ksi). The tensile yield of the material is given as 65 ksi [5].

Solution The equation for the stress intensity factor of a through center crack was formulated in chapter 2:

Kmax = I~(~a) 1/2

For the first cycle, with (~=18 ksi:

159

Z

tll n, J:

o L9 ¢J

(.1

Crack Growth Data (7075-T651 Aluminum Alloy)

10 -1 Stress ~1

F V v v v ~ "/ I

I J 10 -2 Time Compact Tension Specimen

10 -3

10-4

10 "5 / (~ ~ r fl rt

I (Through Crack)

10 -6 I I I I I I I

10 20 30 40 50 60 70 Kma x , ksi (inch) l/Z

Figure 23a: Crack growth curve rate for example 5

Stress

Time Variable Amplitude Loading Repeated Every 18, 14, 10 ksi

Figure 23b: The fluctuating load environment for Example 5

Kmax=18x(~xO.5) 1/2 =22.55 ksi (in.) 1/2 (13=1)

160

The Forman crack growth equation for the case of R=0 can be written:

da c(Kmax )n

dn K c -Kma x

The constants c and n can be obtained by selecting two points in the linear region of the crack growth data (figure 23), as demonstrated in the example problem of 2.

c=1.13E-8, and n= 4.0

The Forman equation, in terms of constants c and n, becomes:

da 1.13E - 8(Kma x )4.0 q

dn K c -Kma x

1) No Retardation Effect:

The amount of growth for the first cycle (o=18 ksi) with ao=0.5 (growth at one tip) is:

(Aa) l=da=l. 13E-8 (22.55)4.0/(55-22.55)=0.0000900 inch

For the second cycle:

Kmax=14x(~x0.50009) 1/2 =17.54 ksi (in.) 1/2 for o=14 ksi

(Aa)2=da=l. 13E-8 (17.54)4.0/(55-17.54)=0.0000285 inch

For the third cycle:

Kmax=10x(~x0.500118) 1/2 =12.53 ksi (in.) 1/2 for o=10 ksi

(Aa)3=da=l. 13E-8 (12.53)4-0/(55-12.53)=0.0000065 inch

For the fourth cycle:

Kmax=18x(=x0.5001245) 1/2 =22.5567 ksi (in.) 1/2 for o=18 ksi

(Aa)4=da=1.13E-8 (22.5567)4.0/(55-22.5567)=0.0000902 inch

For the fifth cycle:

161

Kmax=14x(=x0.500215) 1/2 =17.5472 ksi (in.) 1/2 for a=14 ksi

(Aa)5=da=l. 13E-8 (17.5472)4-0/(55-17.5472)=0.0000286 inch

For the sixth cycle:

Kmax =10x(=x0.500244) 1/2 = 12.533 ksi (in.)1/2 for a= 10 ksi

(z~a)6=da= 1.13E-8 (12.533)4.0/(55-12.533)=0.00000656 inch

No Retardation Effect Original Crack Length (in.) Final Crack Length (in.)

0.5000000 0.5000901 0.5000901 0.5001180 0.5001180 0.5001245 0.5001245 0.5002150 0.5002150 0.5002440 0.5002440 0.5002506

2) Wheeler Retardation Model:

To apply the Wheeler model, the quantities associated with the retardation factor, Cp, should be calculated. In this example problem, the first cycle with 18 ksi is considered as the overload and will retard the growth of the subsequent cycles:

co:[ rpn ]m

aOk +(rp)oL - a n

where (rp)oL =(I/2=)[(K)oL/aYield] 2 and rpn=(l/2=)[(Kmax)n/ayield] 2

(rp)oL =(112=)(22.55/65)2=0.0191 inch, and

rpn=(112=)[17.5416512=0.01159 inch

The amount of growth associated with the first cycle is the same as the previous case (no interaction effect) where Cp=l:

Kmax=18x(=x0.5) 1/2 =22.55 ksi (in.) 1/2 for a=18 ksi

162

(Aa)l =da=l. 13E-8 (22.55)4.0•(55-22.55)=0.0000900 inch

To obtain (Aa)2 and (Aa)3, the quantity Cp for each case must be calculated. The values of aOL and a 1 are 0.5 and 0.50009 inch, respectively. The value of Cp for (Aa)2 is:

Cp=[0.01159/(0.5 +0.0191 - 0.50009)] 1.4 =0.5

(Aa)2 =Cp f (AK)=0.5x1.13E-8 (17.54)4.0/(55-17.54)=0.00001425 inch

and for the case of (Aa)3:

Kmax=l 0x(=x0. 5001043) 1/2 =12.5312 ksi (in.) 1/2

(rp)oL =(1/2=)(22.55/65)2=0.0191 inch, and

rpn=(l/2=)[12.5316512=0.005917 inch

Cp=[0.005917/(0.5 +0.0191 - 0.5001043)] 1.4 =0.195

(Aa)3=0.195x 1.13E-8 (12.5312) 4. 0/(55-12.5312) =0.000001268 inch.

and the amount of growth associated with the fourth cycle (no retardation) is:

(Aa)4=da=l. 13E-8 (22.5567)4.0/(55-22.5567)=0.0000902 inch

where Kmax=l 8x(=x0.50010557) 1/2 =22.5563 ksi (in.) 1/2

Accordingly, the values of (Aa)5 and (Aa)6 can be calculated as shown in the previous cases for (Aa)2 and (Aa) 3 by considering the effect of retardation as a result of 18 ksi overload:

(rp)oL =(1/2=)(22.5563/65)2=0.01917, and

rpn=(l/2n)[17.54516512=0.01160 inch

where Kmax=14x(=x0.50019577) 1/2 =17.545 ksi (in.) 1/2

Cp=[0.01160/(0.5 +0.01917 - 0.50019577)] 1.4 =0.611

163

(Aa)5=0.61 l x l . 13E-8 (17.545)4-0/(55-17.54)=0.00001746

and in the same way, the quantity (Aa)6 can be calculated:

Kmax=10x(=x0.500213) 1/2 =12.5326 ksi (in.) 1/2

(rp)oL =(1/2=)(22.5563/65) 2 =0.01917 inch, and rpn =

(1/2~)[12.5326/65] 2

=0.0059197 inch

Cp=[0.005919/(0.50010557 +0.01917 - 0.500213)] 1.4 =0.1945

(Aa)6--0.1945x 1.13E-8 (12.5326)4.0/(55-12.5326)=0.000001276 inch.

Wheeler Retardation Effect

Original Crack Length (in.) Final Crack Length (in.)

0.50000000 0.50009010 0.50009010 0.50010430 0.50010430 0.50010557 0.50010557 0.50019577 0.50019577 0.50021300 0.50021300 0.50021430

3) Willenborg Retardation Model:

The Willenborg correction to the crack growth is handled through the use of an effective stress intensity factor and the effective stress ratio. From equation 28:

(Kmax)eff = Kma x- Kre d , (Kmin)ef f =Kmi n- Kre d (where Kmin=0)

and Kred = (KoL)max [1-(ai - aOL)/rpoL] 1/2 - Kmax,i

From the above equations, it can be deduced that:

(Kmax)eff - (Kmin)eff = Kmax

which states that for the case of R=0, both the Willenborg and the "Generalized" Willenborg model can not be effective to correct the retardation effect. For any other stress ratio larger than zero, the

164

Willenborg retardation models can be used to address the retardation effect. Therefore, case 3 is identical to case 1 of this example problem. The following summary table summarizes the calculated crack growth for the first 6 cycles associated with 18, 14, 10, and 18,14, 10 ksi).

Summary Table For Example 5

No Retardation Effect Wheeler Retardation Effect

Original Crack Final Crack Original Crack Final Crack Length (in.) Length (in.) Length (in.) Length (in.)

0.5000000 0.5000901 0.50000000 0.50009010 015000901 0.5001180 0.50009010 0.50010430 0.5001180 0.5001245 0.50010430 0.50010557 0.5001245 0.5002150 0.50010557 0.50019577 0.5002150 0.5002440 0.50019577 0.50021300 0.5002440 0.5002506 0.50021300 0.50021430

3.5 Cycle by Cycle Fatigue Crack Growth Analysis

To prevent catastrophic failure due to the presence of undetected flaws that would result in the loss of life and the structure, it is important for the engineer to design components that provide structural integrity, even in the presence of undetected flaws. Therefore, it is essential to know whether the pre-existing flaw will grow to a critical length during its service life.

Due to the complexities in the crack growth analysis that emerge as a result of a large number of cycles (containing constant and variable amplitude load) together with complicated load and crack geometries, automated flaw growth programs are usually employed. Ideally, a flaw growth computer program should have libraries for 1) standard NDE flaw sizes, 2) stress intensity factor solutions for different crack geometries and loading conditions, 3) empirical fatigue crack growth equations, and 4) material properties. In addition, it should be efficient and easy to run ("user friendly").

Most crack growth computer programs that are tailored for the aircraft industry have the capability to perform variable amplitude fatigue life evaluation by employing one or both of the retardation models discussed in section 3.4.1t should be noted that, based on NASA requirements stated in NHB 8071-1, "Fracture Control Requirements for Payloads Using the Space Shuttle", the retardation effect is not allowed to be considered in evaluating the life of a fracture critical or high risk components that will be used as a payload in the

165

Space Shuttle. The cycle by cycle crack growth procedure generally consists of the following steps:

Step I .A standard NDE initial crack size, a 1, is assumed. As an example, for standard penetrant inspection, the initial crack size is a part through crack with depth a=0.075 and 2c= 0.15 inch.

Step 2. The stress intensity range for the first cycle can be determined, AK 1= 13Aa l(xa) 1/2, where Ac 1 is the stress range in the first cycle and 13 is the appropriate correction factor for crack geometry and loading conditions. Note that the crack growth analysis performed for a part through crack should consider the growth for both the depth (a-tip) and the length direction (c-tip).

Step 3. Determine the amount of growth for the first cycle, Aa 1, by using the appropriate crack growth equation that considers the effect of the stress ratio, R (for example, the Forman relationship described by equation 4) or directly from the da/dN versus AK curve. The amount of growth for the first cycle, Aa I , is:

Aal= (da/dN)l x 1

The new crack length will be a2= al+ Aa 1. If the crack geometry in consideration is a part through crack, it is preferable to have the da/dN versus AK test data for the crack geometry under study.

Step 4. The size of the plastic zone, rp, in the first cycle is calculated by equation 33 described in chapter 2. If the first cycle is considered as the overload cycle then the extent of the plastic zone is:

Z l= ((rp)OL)l + a 1 where al= aOL

where for the plane stress condition:

((rp)OL)l =l/2x(Kmaxl/(~Yield) 2

and for the plane strain condition:

((rp)oL)l=l/6x(Kmaxl/~Yield) 2

Step 5.The new crack length is now a 2 and the stress intensity range

for the second cycle AK2= 13Aa2(xa)l/2. If the new cycle has a load magnitude larger than the previous cycle, the new overload plastic zone must be calculated. On the other hand, if the subsequent cycle

166

has a smaller load magnitude, then the effect of retardation has to be considered. The extent of the yield zone for the new cycle is:

Z= a2+ (rp) 2

Step 6. If the extent of the plastic zone Z<Z 1, a correction for the retardation effect according to the Willenborg (equations 32 and 34) or Wheeler model (calculate the quantity C D from equation 26) is needed. For the Willenborg model the new crack length is:

a3= a2+ Aa 2 where Aa2= (da/dN) 2 x 1

The quantity (da/dN)2 is obtained by using the AKeff and Reff in the Forman crack growth equation. When using the Wheeler model, the stress intensity factor, (AK)2, is calculated and the retardation factor, Cp, is incorporated to calculate the new crack length:

a3= a2+ Aa 2 + Cp(da/dN)2 x 1

Step 7.If the extent of the plastic zone Z>Z 1 (no retardation effect), determine the quantity Aa 2 =(da/dN)2 x 1 and the new crack size, a 3, as:

a3= a2+ (da/dN)2 x 1

For the subsequent cycles, steps 6 and 7 can be repeated in such a way that the nth cycle with new crack length a n replaces a 2 and the subsequent cycle an+ 1 replaces a.

3.6 Environmental Assisted Corrosion Cracking

3.6.1 Introduction

Many industrial alloys fracture when subjected to sustained load while exposed to corrosive environment. The cracking mechanism under corrosive environment and static loading is referred to as stress- corrosion cracking (a special case of environmental assisted cracking). The mechanism for corrosive attack and the corrosion process has been attributed to the chemical reactions that take place at the crack tip by diffusion of active elements (such as Hydrogen) into the highly stressed region ahead of the crack [43]. This will result the embrittlement of a metal or alloy, where failure can occur by intergranular fracture. The embrittlement can be reflected in material as a reduction of ductility [44].

167

The corrosion mechanism is a result of establishing galvanic cells and the accompanying electrical currents, where the presence of two dissimilar electrodes (anode and cathode) are essential to establish potential difference. The galvanic reaction takes place when material differences in composition or differences in stress magnitude (for example the highly localized crack tip stresses) in the metal exist Lack of understanding the corrosive environment and its effect on the material can cause premature failure of structural components. A survey in this area indicated that the cost of corrosion induced damage is around ten billion dollars a year in the USA alone [45], thus engineers must understand the mechanism of corrosion and try to minimize structural damage from environmental attack.

Several alloys such as aluminum, copper, magnesium, and titanium are susceptible to stress corrosion cracking, therefore, it must be of significant concern since it has occurred in the critical components of aircraft and space structures. The embrittlement as a result of stress corrosion is worst when materials of higher strength (for example, AI-Li or 18 Ni maraging steels) are used in the future for weight saving [46] purposes.

Among most common types of corrosive environments that engineering materials are exposed to are high humidity air, water, and salt water. Air with high humidity content increases the electrical conductivity of the metal surface area and allows flow of larger electrical currents (helps to accelerate corrosion reaction). Another type of corrosive environment is a propellant stored in pressurized tanks used in launching space vehicles, where the liquid propellant can interact with the material under stress if time of exposure is sufficient to initiate the corrosion. When structural components are subjected to high cycle fatigue (uncracked specimen), any of the above environments can reduce the number of cycles to failure, as demonstrated in figure 24. However, under the low cycle fatigue condition, not much difference in the number of cycles to failure is expected because of insufficient time forcorrosion to be effective.

Corrosion can also affect some material by introducing localized pitting (loss of material) on their surfaces, which can act as the source of stress concentration and crack initiation. Under high cycle fatigue crack initiation cycles (stage I) are absent due to pit formation, and the remaining structural life (number of cycles to failure) is entirely associated with stable crack growth (stage II). A pit usually begins with the surface of discontinuity, such as an inclusion or from grinding marks where an oxygen concentration cell develops between the discontinuity (anodic area) and the surrounding material (cathodic area) [47]. In other cases, accumulation of dirt restricts the access of oxygen and that establishes an anode (with respect to clean surfaces that become cathodes) resulting in localized pitting (figure 25).

168

The environmental assisted corrosion cracking process has been noted to be a function of exposed time and it is highly dependent on the severity of the environment, the material quality, and the applied stress (or stress-intensity factor) level. When cracked structural parts are subjected to fluctuating load with stress ratio, R>0, the crack growth rate increases at the presence of a corrosive environment. In a non-corrosive environment, the crack growth rate is almost frequency independent, whereas recorded test data, under cyclic loading subjected to a corrosive environment, showed dependency to loading frequency [48].

Dry Air t,~ Hig 1 I

Water ;alt V

Number of Cycles (LOG N)

High Humadity

Salt Water

Figure 24: The effect of corrosive environment on the S-N diagram

(R .=

Clean Surface (Cathode) Dirt or surface contamination j

~ f ~ w w ~. J Anode ~ Anode I I Pits • I

Pits formation on the

Een w,. P,s Pits act as crack initiation site No Pits and will reduce structural life

Number of Cycles (LOG N)

Figure 25: An oxygen concentration cell develops between the discontinuity (anodic area) and the surrounding material (cathodic area)

169

3.6.2 Threshold stress intensity factor ( Kz~c and KE,c)

For cracked structures subjected to static stress and exposed to corrosive environments, the crack tip behavior can be addressed through the stress intensity factor parameter. The incremental crack growth and the stress-corrosion cracking rate, da/dt, have been noted to be controlled by the stress intensity factor, K. Experimental observation indicated that similar specimens with the same initial crack size but loaded to different stress levels (different initial K values) show different failure time while in corrosive environment. In figure 26, three

a1<02 <03 Agressive Environment

(¢rl, t 1) (G2, t2 ) ((~3, t3)

Figure 26: C(T) specimens subjected to different sustained stress levels

identical compact tension specimens are exposed to an aggressive environment while subjected to different sustained stress levels where GI<G2 < G3- It is obvious that the time to failure (time required for crack to become unstable) is different for each specimen. When a cracked specimen is initially loaded to the material fracture toughness level (plane strain, K~o, or plane stress, Ko, conditions), it is expected to fail in a short time. As the level of stress intensity factor drops, the rate of crack advancement with respect to the amount of time (da/dt) exposed to the corrosive environment reduces. The static stress level below which crack growth is not expected to occur for a long period of time is called the threshold stress intensity factor and is denoted by K~EA c or KE ̂c for environmental assisted cracking under plane strain and plane stress conditions (figure 27), respectively. Please note that when the applied load is kept constant during the stress corrosion cracking process (sustained load), the stress-intensity factor will gradually increase, due to the increasing crack length (at the threshold level however, no crack growth is expected to occur). As a result, the crack- growth rate (with respect to time), da/dt, in terms of stress intensity factor can be written as:

da/dt = f(K) 39

170

G Air Environment

m. Corrosive Environment

IEAC

o" Air Environment

~ Corrosive Environment

KEAC

v v

Crack Length Crack Length

Figure 27: The static stress level below which crack growth is not expected to occur (K!~ c and K~c ) for environmental assisted cracking

where f(K) can be determined through several laboratory tests using cracked specimens subjected to the environment under study. Note that equation 39 expresses the crack growth rate with respect to time and is analogous to da/dN described by equation 1. When the crack has grown to a size that K becomes equal to K~c or Kc, the specimen fails. In typical tests, specimens may be loaded to various initial K's such as K~, K2, and K3. The time to failure is recorded, giving rise to the typical data point (tl,K1). Crack growth rate information, da/dt, can then be obtained and a plot of da/dt versus K can be established. During the test, K will increase (as a result of crack extension in a corrosive media) from its initial value to its critical value (K~c or K~), whereupon final failure occurs. The times t2, and t3, represent the time to failure for higher K's, such as K2 and K3. This is shown schematically in Figure 28. The calculated stress intensity factor in all cases must be based on the linear elastic fracture mechanics (small scale crack tip yielding criteria shall be maintained).

The stress-corrosion threshold and the rate of growth under sustained load depend on the material and the environmental conditions. Data on K and da/dt can be found in the Damage Tolerant Design (Data) Handbook [49] for selected aerospace materials. Typical examples of K and da/dt data (plotted in log-log format) is shown in Figure 29 for 2024-T351 aluminum alloy. Specimens with various thickness (1.0, 1.2, and 2.0 inches) were tested in wet environment and the variation of da/dt versus kr~,~ were plotted.

As illustrated in Figure 28, a component with a given initial crack length, 2a, fails at a stress g ven by c = Kc/[13~/=af], when the fina crack length, 2af, has been reached due to a corrosive environment while specimen was subjected to sustained stress (~. If the level of some stress is such that crack growth does not take place after some specified time, then the quantity K = 13(~/~a~ can be referred to by K~EAC or KEAC for the plane strain and plane stress conditions, respectively.

171

Thus, prevention of stress corrosion cracking is the preferred design policy over controlling it, as is done for fatigue cracking. That means

KIc

K3

Kz KI

KIEAC

t3 tz I

0 KtEAC

No Failure t3 t 2 t ] Time to Failure

Stress Corrosion Cracking

Figure 28: A typical example of K]~c and time to failure variation

® ,cP

i::: I0 ' *

I

I0-1 - - =

VO'* I

K rosa ( M P A V ~ ) 4 tO 40 100

" - I ' I ' l , i q I * I ' I ' 1 ' 1 - WORM: 1 . I'TI.I

I = t . ~ l " E 0 1 c a

10

• --=

I , I , t . h I , I , h h ' ~ • 10

I

;I

K ~ ( M P A V ' ~ ) 4 tO ~ v ~

CORM: 1 . 2 " ' r H " ~

4 i t , ~00 I • 10 40 ~(

© ~ , 0 , ,o, ~o~. =. =-T. "1

: -f,0' 1o_. -I,,,

" -." t,," ,o" ~ !

" • ~ i 1o 4

-~ Io 4 104 I ,

to"* I 0 ~l l , I , l , h | * 1 , 1 , 1 ,

1 4 10 K m l ( l a l l ~ t m

I .

.-_:-

Figure 29: Typical examples of K and daldt data for for 2024-T351 aluminum alloy

172

that stress-corrosion critical components must be designed to operate at a stress level lower than (~ = [K[EA c or KEAo]/(~a,) "2. However, if stress corrosion can occur, it must be accounted for in life estimation analyses by using equation 39. If a structural part is subjected to cyclic loading with mean stress larger than zero (R>0), as well as corrosive environments, fatigue crack growth rate must be a combined effect. That is the stress corrosion cracking rate, da/dt, should be added on the fatigue crack growth rate, da/dn [49]:

da/dn + da/dt = f(AK) + f(K) 40

In service, stress-corrosion cracks have been found to be predominantly a result of residual stresses and/or a sustained load in the heat affected zone of welded hardware. For thick structural components with stress that can be developed in the thickness direction (S), stress corrosion cracking favors the short transverse direction (S-L) usually not the primary load path direction. In many materials, the (T-L) and (L-T) directions are not very susceptible to stress corrosion. Thus, the corrosion problem is generally limited to plane strain conditions with poor properties in the S direction.

3.6.3 ASTM Procedures for Obtaining KIE,c or KE, c

The laboratory evaluation of the environmental assisted cracking threshold stress intensity factor (K~A c or KEAc) for metallic material used in manufacturing aerospace vehicle, is based on the ASTM E 1681 Standard [50] and it is briefly discussed in this section. Prefatigued test specimens are subjected to corrosive environment and sustained load while crack advancement is measured periodically. A displacement gage to be located at the mouth of the crack can be helpful to detect crack growth during the testing. In this test method, either a bend bar or compact tension specimen with sufficient thickness that meets the ASTM requirements is used to determine the threshold quantities K~E, c or KE, c for plane strain or plane stress conditions, respectively. The quantity K~Es c is defined as the highest value of stress intensity factor at which crack growth is not observed for a given material and a given corrosive media, provided that the specimen thickness is adequate to meet the thickness requirement of the ASTM E-399. The laboratory value of KXEAc obtained in an aggressive environment must always be smaller than the value of Kic. It should be noted that the test specimen must be subjected to a fixed load and that the presence of cyclic loading will alter the results of the test significantly [50]. Another variable that can influence results of measured quantities K~EA c or KEA c is the specimen size [51,52]. It is found that smaller specimen yields a lower KEA c value.

173

Figure 30 shows a typical fixture containing a bend bar specimen and is subjected to a horizontal moment arm as a result of a constant dead weight load. The fixture in figure 30 must have the capability of handling multiple test specimens simultaneously in order to save time if repeated tests are required to check the validity of the data. For multispecimen environmental assisted corrosion tests in one single fixture, the fixture must be rigid enough to ensure that loads from other specimens do not alter the test result. Moreover, the applied load to each specimen must be within an accuracy of 1%. The ASTM E 1681 bend bar specimen configuration attached to the fixture of figure 30 is shown in figure 31a with planner dimensions that are given in terms of the quantity W. Furthermore, W for a bend bar specimen in terms of material thickness must fall between B<W<2B. A compact tension specimen C(T) for determining the quantities K,EAC or K~c is also shown in figure 31b.

Test Specimen In Corrosive ~ EnvironmeNt

/ \

\ /

Dead Weight i

I I i L=8W

Moment Arm

I Figure 30:A typical fixture containing a bend bar specimen and is

subjected to a horizontal moment arm for corrosion testing

The mode I stress intensity factor equation, KI, describing the crack tip phenomenon in corrosive media for the bend bar specimen is given as follows [50]:

K, = [M/(BVV3'2)] f(ao/W)] 41

where M is the moment due to the applied load on the crack plane and ao is the original crack length. The correction factor f(ao/W) for the bend bar specimen in terms of 1,= ao/W can be expressed as:

f(ao/W) =6 1,1/2{1.9878-1.3253 1, + (1-1,) 7 [-3.8308+10.1081(1, - 17.94151,2+16.82821,3-6.2241 1,4]}/(1-1,3/2) 42

174

and for the compact tension specimen:

K, = [P/(BWI'2)] f(ao/VV)] 43

where P is the applied load and the correction factor f(ao/VV) for a C(T) specimen is given by:

f(ao/W) = (2+~,)[0.886+4.64~( -13.3272 +14.7273 5.6~,4]/(1-~, ;/2) 44

I~ 4W ~1~

(a)

4W ...I v

m

1.2W

0,25W ~ " ~ . ~ .

(b) - ~ ' -I

w j 1.25W

B

Compact tension specimen C(T)

Figure 31: a) Standard beam specimen, b) Compact tension specimen

where ~,= aoNV. When the stress intensity factor for any of the above two crack geometries were calculated (see equations 41 or 43), corresponding to the threshold value, the following validity checks are required:

and (W-ao), B, ao > 2.5(K,EAC)/~Yie,d

(W-ao) >(4/~)(KEAc)/ayie,d

45a

45b

References

1. Annual Book of ASTM Standards, "Standard Practice for Conducting Force Controlled Constant Amplitute Axial Fatigue Tests Of Metallic Materials," Vol. 03.01, 1999, pp. 471-475

175

2. B. L. Avenback, D. K. Felbeck, G.T. Hahn, and D.A. Thomas, "Fracture," The MIT Press, 1959, pp. 1-66

3. S. S. Manson, "Metal Fatigue Damage- Mechanism, Detection, Avoidance, and Repair," STP 495, 1971, pp. 5-57.

4. H. L. Ewalds and R. J. H Wanhill, "Fracture,"Edward Arnold, 1986, Ch.12

5. Fatigue Crack Growth Computer Program "NASA/FLAGRO", Developed by R. G. Forman, V. Shivakumar, J . C. Newman. JSC- 22267A, January 1998.

6. Annual Book of ASTM Standards, "Standard Practice for Measurement of Fatigue Crack Growth Rates," Vol. 03.01, 1999, pp. 562-598

7. W. G. Clark Jr. and S. J. Hudak Jr., "Variability in Fatigue Crack Growth Rate Testing," Jornal of Testing and Evaluation, Vol. 3, No. 6, 1975, pp. 454-476.

8. D. W. Hoeppner and W. E. Krupp, "Prediction of Component Life by Application of Fatigue Crack Growth Knowledge," Lockheed California Company, Burbank, California, pp 8-9.

9. P. C. Paris, M. Po Gomez and W. E. Anderson, A Rational Analytic Theory of Fatigue, Trend Engng Univ. Wash., 13 (1), 1961, pp. 9-14.

10. R. G. Forman, V. E. Kearney, and R. M. Engle, Numerical Analysis of Crack Propagation in Cyclic Load Structures, J. bas. Engng. Trans. ASME, Ser D, 89, 459 (1967).

11. R. G. Forman and Mettu, S. R., "Behavior of Surface and Corner Cracks Subjected to Bending and Tensile Loads in Ti-6AI-4V Alloy," Fracture Mechanics: Twenty Second Symposium, Vol. 1, ASTM STP 1131, H. A. Earnest, a. Saxena, and D. L. McDowell, eds., American Society for Testing and Materials, Philadelphia, 1992, pp. 519-546.

12. J. C. Newman Jr., "A Crack Opening Stress Equation for Fatigue Crack Growth," International Journal of Fracture, Vol. 24, No. 3, March 1984, pp. R131-R135.

13. J. K. Donald, and D. W. Schmidth "Computer- Controlled Stress Intensity Gradient Techniques for High Rate Fatigue Crack Growth Testing, " Journal of Testing and Evaluation, Vol. 8, No. 1, Jan. 1980, pp. 19-24.

176

14. a. Saxena, and S. J. Hudak Jr., "Review and Extension of Complience Information for Common Crack Growth Specimens," International Journal of Fracture, Vol. 14, No. 5, October 1978.

15. J. C. Newman Jr. , "Stress Analysis of Compact Specimen Including the Effects of Pin Loading," Fracture Analysis (8th Conference), ASTM STP- 560, ASTM, 1974, PP. 105-121.

16. C. E. Fedderson, "Wide Range Stress Intensity Factor Expression for ASTM Method E 399 Standard Fracture Toughness Specimens," International Journal of Fracture, Vol. 12, June 1976, pp. 475-476.

17. W. Elber, "Fatigue Crack Prpagation," Ph.D. thesis, University of New South Wales, Australia, 1968.

18. W. Elber, "Fatigue crack Closure under Cyclic Tension," Engineering Fracture Mechanics, Vol. II, No. 1, Pergamon Press, July 1970.

19. W. Elber, "The Significance of Fatigue Crack Closure," Damage Tolerance in Aircraft Structure, ASTM STP 486, 1971, P. 230.

20.. J. C. Newman Jr, "Finite Element Analysis of Fatigue Crack Propagation Including the Effect of Crack Closure," UMI Dissertation Service, 1974.

21. J. Schijve, "The Stress Ratio Effect on Fatigue Crack Growth in 2024-T3 Alclad and the Relation to Crack Closure," Delft University of Technology Department of Aerospace Engineeing, Memorandum M- 336, August 1979.

22. J. Backlund, A. F. Blom, and C. J. Beevers, Fatigue Threshold, Fundementals and Engineering Applications, The Proceedings of an International Conference held in Stockholm, June 1-3, 1981.

23. M. Klesnil, and P. Lukus, "Effect of Stress Cycle Asymetry on Fatigue Crack Growth," Material Science and Engineering, Vol. 9, 1972, pp. 231-240.

24. J. M. Basom, "Fatigue Behavior of Pressure Vessels Steel, WRC Bulletin 194, Welding Research Council, New York, May 1964.

25. K. Tanaka~ Y. Nakai, M. Yamashita, "Fatigue Growth Threshold of Small Cracks," Internationa Journal of Fracture, Vol. 17, No. 5, October 1981, pp. 519-53

26. N. E. Frost, Proceedings of Institution of Mechanical Engineers, 173 (1959) pp. 811-835

177

27. N. E. Frost, Journal of Mechanical Engineering Science, 5 (1963) pp. 15-22

28. H. Ohuchida, S. Usami, and A. Nishioka, Transactions of Japan Society of Mechanical Engineers, 41 (1975) pp. 703-712

29. S. Usami, and S. Shida, Fatigue of Engineering Materials and Structures, 1 (1979), pp. 471-482.

30. Small Fatigue Cracks, Edited by R. O. Ritchie and J. Lankford "Proceeding of the Second Engineering Foundation International Conference/Workshop, Santa Barbara, California, January 5-10, 1986". A Publication of the Metallurgical Society, Inc.

31. Short fatigue Crack, Editors, K. J. Miller and E.R. de los Rios, ESIS, Publication 1

32. The Behavior of Short Fatigue Cracks, Editors, , K. J. Miller and E.R. de los Rios, EGF, Publication 1 (Colection of Papers and References in Crack Initiation).

33. T. C. Landley, "Near Threshold Fatigue Crack Growth: Experimental Methods, Mechanism, and Applications," Subcritical Crack Growth Due to Fatigue, Stress Corrosion, and Creep, L. H. Larsson, ed., Elsevier Applied Science Publications, New York, PP. 167-213, 1985.

34. H. Kitagawa, and S. Takahashi, "Applicability of Fracture Mechanics to Very Small Cracks or the Cracks in the Early Stage," Proc. Second International Conference on Mechanical Behavior of Materials, Boston, MA, 1976, pp. 627-631.

35. J. C. Newman Jr., and I. S. Raju, "Prediction of Fatigue Crack Growth Patterns and Lives in Three-Dimensional Cracked Bodies," Presented at the Sixth International Conference on Fracture, New Dehli, India, December 1984.

36. R. A. Schmidt, P. C. Paris, "Threshold for Fatigue Crack Growth Propagation and Effects of Load Ratio and Frequency," Progress in Flaw Growth and Fracture Toughness Testing, ASTM STP-536, American Society for Testing and Materials, Philadelphia, pp. 79-94.

37 J. Schijve and D. Broek, "Crack Propagation Based on a Gust Spectrum With Variable Amplitude Loading," Aircraft Engineering, 34, (1962) pp. 314-316.

38. J. Schijve, "The Accumulation of Fatigue Damage in Aircraft Materials and Structures," AGARDograph No. 157, (1972).

178

39. J. Schijve, "Cumulative Damage Problems in Aircraft Structures and Materials," The Aeronautical Journal, 74, (1970), pp. 517-532.

40. O. E. Wheeler, "Spectrum Loading and Crack Growth," J. Basic Engineering, 94D, (1972), pp. 181.

41. J. D. Willenborg, R. M. Eagle, and H. A. Wood, "A Crack Growth Retardation Model Using an Effective Stress Concept," AFFDL-TM-71- 1 FBR, (1971).

42. J. P. Gallager, and T. F. Hughes, "Influence of the Yield Strength on Overload Affected Fatigue-Crack-Growth Behavior of 4340 Steel," AFFDL-TR-74-27, (1974).

43. E. G. Coleman, D. Weinstein and W. Rostoker, Acat Met., Vol. 9, 1961, pp. 491.

44 A. J. Forty, and P. Humble, Phil. Mag., Vol. 8, 1963, pp. 247.

45. G. H. Cartledge, "Studies in Corrosion," Scientific American, Vol. 195, 1956, pp. 35.

46. A. S. Tetelman, and A. J. McEvily, JR., "Fracture of Structural Materials", John Wiley& Sons Inc., 1967, pp 424-428.

47. L. H. Van Vlack, "Element of Material Science," Addison-Wesley Publishing Company, 1964, pp. 334-371.

48. C. Q. Bowles, "The Role of nvironment, Frequency, and Wave Shape During Fatigue Crack Growth in Aluminum Alloys, "Delft University of Technology, Department of Aerospace Engineering, Report LR-270, May 1978.

49. Damage Tolerant Design Handbook, Metals and Ceramics Information Center, Battelle, Columbus, Ohio, 1975.

50. Annual Book of ASTM Standards, "Standard Test Method for Determining a Threshold stress Intensity Factor for Environment- Assisted Cracking of Metallic Materials Under Constant Load," Vol. 03.01, 1999, pp. 943-952.

51. R. A. Oriani, and P. H. Josephic, "Equilibrium Aspects of Hydrogen Induced Cracking of Steel,"Acta Metallurgica, Vol. 22, 1974, pp. 1065- 1074.

52. Characterization of Environmentally Assistec Cracking for Design ; State of the Art, NMAP-386, National Materials Advisory Board, National Academy Press, Washington DC, 1982.

179

Chapter 4

ELASTIC-PLASTIC FRACTURE (EPFM) AND APPLICATIONS

MECHANICS

4.0 Overview

As discussed in Chapters 2 and 3, the application of the LEFM approach to structural life assessment is limited to a low load environment where the bulk of the structure is elastic and crack tip plastic deformation is highly localized. In many structural parts that are made of low-strength, tough material, however, an appreciable amount of crack tip plastic deformation and stable crack growth (also called stable tearing or simply tearing) can occur prior to instability. Application of the LEFM theory, using the stress intensity factor, K, is not adequate to characterize the crack tip field behavior in the presence of large yielding and extensive stable crack growth. Fracture mechanics concepts other than the LEFM approach are therefore necessary to address structural integrity analysis of components that are ductile. Two fracture mechanics approaches are discussed in this book for the analysis of tough metals used in building aerospace, aircraft, and nuclear structures, where fracture behavior often extends beyond the elastic dominant regime. The first approach is called the Elastic-Plastic Fracture Mechanics (EPFM) theory and it uses the J- Integral concept first proposed by Rice (1968) as a path independent integral for characterizing crack tip stresses and strains [1]. The second approach is called the Fracture Mechanics of Ductile Metals (FMDM) theory (see chapter 5). The FMDM theory uses a modified Griffith energy balance concept that was initiated by Bockrath, and Glassco was later modified by Farahmand [2]. It utilizes the energy density (defined as the total area under the uniaxial stress-strain curve) for the metal under study to calculate the energy release rate at the crack tip and near the crack tip up to instability, as shown in Figure la. In contrast, the J-Integral uses the area under the load versus load line displacement (work done on a cracked specimen) to calculate the material fracture toughness, J~o, see Figure lb. Detailed descriptions of these two approaches to fracture mechanics analysis of low strength, tough metals are contained in chapters 4 and 5, respectively.

I v

Strain

i J'Integra r I I

I , Load Line Displacement

Figure 1: Illustrating a) the FMDM concept and b) the J-integral

4.1 Introduction

All metals and alloys that are used in designing aerospace and aircraft structural parts manifest some plastic deformation in the region of the crack tip when they are subjected to external applied load. In thick sections when the plane strain condition prevails and plastic deformation through the material thickness is constrained, the LEFM concept is proven to be a useful tool when predicting the life of the structural component. In the space and aircraft industries, structural weight is an important issue and must be given special attention when designing structural parts. Seldom it is seen that components used in designing aircraft or space structure are made of very thick material. In the plane stress and mixed mode conditions, however, the formation of plastic deformation at the crack tip is no longer constrained, considerable plastic deformation and stable crack growth at the crack tip prior to final failure can occur. The amount of plastic deformation at the crack tip is directly related to the material fracture toughness (the material's ability to resist cracking) and for a given material varies as a function of part thickness. Figure 2 shows the variation of material fracture toughness, Kc, for 2219-T87 aluminum alloy as a function of thickness for several crack lengths when plasticity at the crack tip is small and localized [3]. In figure 2, a valid plane strain fracture toughness value, K~c, was obtained under the strict dimensional requirements set forth by the ASTM E-399 Standards. For material thickness smaller than the required plane strain condition, some stable crack growth can occur under crack tip plastic deformation. A resistance curve approach (R-curve) based on the stress intensity factor, K, can be used to evaluate material fracture

181

70

60

I I~ 30

20

o 2219 - T87 Aluminum NIoyf L.T r 75 F M(T) Specimen Data [] Q • o

• g (a Is 112 crack length)

~ Curve f~ for a = 0.45 to 0.66 inch = o a = 0.24 to 0.38 inch

• a = 0.45 to 0.68 inch • a = 1.00 to 1.20 inch ~ 50 to 1 80 inch

KIc . . . . . . . . . .

'-' Thickness, t (Inch)

0.0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1.0 1.125

Figure 2: The variation of material fracture toughness, Kc, for 2219-T87 aluminum alloy as a function of thickness [3]

toughness, Ks, under slow stable crack growth where small scale yielding is dominant.

For tough material, the spread of crack tip plastic deformation is no longer localized and can cause yielding in the remaining ligament of the structural part. Application of the LEFM concept for fracture assessment of low strength tough material parts is inaccurate, whenever the crack tip field shows non-linear elastic behavior. Therefore, the path independent J-Integral as part of the EPFM solution must be implemented for the non-linear fracture mechanics analysis. The concept of the path independent J-Integral is based on an energy balance approach and the deformation theory of plasticity. For material which obey Hooke's law whose material crack tip behavior is linear elastic, the crack propagation mechanism is predominantly cleavage, the calculated J value is identical to the Griffith elastic energy release rate (crack extension force, G) [1].

Figure 3 illustrates the limitations and applicabilities of LEFM and EPFM in assessing crack tip behavior in structures. Note that when material is fully plastic or net section yielding exists in the cracked structure (or the crack tip plasticity is large), the LEFM is not applicable for fracture analysis (figure 3).

182

Fully plastic

LEFM Not Applicable

Net section Large crack yielding tip plasticity

J~

r o - 2a

EPFM Not Applicable Fully plastic

u

Q

3

Applicable Applicable Small crack Large crack Small crack tip plasticity tip plasticity tip plasticity

, , , , , ,

ro<<2a r o ~ 2a ro<<2a

Figure 3: Illustrates the limitations and applicabilities of LEFM and EPFM in assessing crack tip behavior

4.2 Introduction to Griffith Energy Balance Approach

The Griffith's energy balance principle describing fracture instability in brittle material by cleavage mechanism, is based on energy input rate and the rate at which material absorbs or dissipates energy around the apex of the elastic crack, see figure 4. Unstable crack propagation due to the incremental crack growth, Aa, results when the energy input rate, G, (crack extension force) exceeds the rate of energy absorbed by the material at the crack tip (crack resistance force, R). The energy rate equation describing crack instability (G>R) in terms of component of available and consumed energy for a cracked plate subjected to external forces having thickness, B, can be written [5] as:

dW/dA + dUe/dA _> dU S IdA + dUp/dA la

183

G = R (stable but ready to go) G > R (propagat ion) G < R (stable)

t " Rate of energy input to drive

a crack (G)

Rate of energy absorbed to

propagate a crack (R)

a l a

I

Figure 4: Energy balance principle describing fracture instability

where A=2Ba (A is the total crack area at both crack ends). In terms of energy rate per unit thickness:

dW/da + dUe/da >_ dU S/da + dUp/da lb

where W is the work done on the cracked body by the external forces and it is equal to the applied load x displacement, U e is the available

elastic energy contained in the plate, U S is the energy absorbed for

the formation of two new crack surfaces, and Up is the absorbed energy for plastically deforming material at the crack tip. When there is no work done by the external forces (the case of fixed grip condition where the plate is stressed to a given stress level, a, and fixed at its ends) the term dW/da=0. The reader should note that for the fixed grip condition the energy required to extend the crack must be delivered by the stored elastic energy, U e. In this case, the available elastic energy

stored in the cracked plate decreases as the crack extends (-dUe/da).

For truly brittle materials, when growth of the elastic crack is not halted by plastic deformation (none of the available energy is consumed in the plastic deformation at the crack tip), the energy term Up is set equal to zero. Under the fixed grip condition, the energy balance equation for the case of elastic material (no plasticity at the crack tip) can be simplified as:

dUe/da > dU S/da lc

The energy balance rate equation described by equation lc states that

184

the change in elastic energy, dUe, that resulted to extend a crack is

consumed in creating two new crack surfaces. The maximum stress at the apex of the major axis of a sharp elliptical crack of length 2a in an infinite plate of thickness B was formulated by Professor C. E. Inglis [4] (see chapter 2) and the elastic energy per unit thickness, U e,

for brittle material was calculated under two loading conditions: a) fixed grips and b) tension stress is maintained at the boundary. In the case of fixed grip

U e = -~G2a2/E 2

where E is the Young's modulus of elasticity. Furthermore, the energy consumed at the crack tip (per unit of thickness) to propagate the crack and to create two new crack surfaces, U S , is given by [5]:

U S = 4aT 3

where T is the surface tension of the material, that is, the work done in breaking the atomic bonds and forming two new cracked surfaces. Therefore, the Griffith's energy balance equation, described by equation lc, defining crack instability in brittle materials can be rewritten as:

d (U e - U S )Ida >_ 0 G = ~(~2a/E >_ 2T 4

for the plane strain condition, E must be replaced by E/(1-v 2) where v is Poisson's ratio.

4.2.1 The Relationship Between Energy Release Rate, G, and Complience

Figure 5 shows the variation of total energy of a center cracked plate (U = U n + Ue+ U S - W) as a function of crack length, 2a. The

term U n is the elastic energy of a loaded uncracked specimen. The

minus sign is used to indicate that the work done on the plate is by the external forces. For the fixed grip condition, where energy for crack extension must be supplied by released elastic energy, W=0, the

185

Stressed to a value and fixed at its end

a l

,=,, ,,=,

I.-

Total energy of a loaded cracked plate (dU/da >0)

Onset of instability point , where dU/da = 0

pA l ~

, ° , E = , o . n . r g , o, , ==cr I loaded uncracked ~. / Y ' ~ I _ .

2a I ~ l o ~ . - . . . . ~ / / ",--p-- z a + a a I '~; ""~ .......... --~ V c,

CRACK LENGTH, 2a v O FIXED DISPLACEMENT

Figure 5: The variation of total energy of a center cracked plate as a function of crack length, 2a

equilibrium condition at the onset of instability (point I of figure 5) can be obtained by letting dU/da=0:

dU/da = d (U n + Ue+ U S )Ida = 0 5

Note that the term dU n Ida in equation 5 is zero since the quantity U n

is independent of the crack length. Therefore, equation 5 will reduce to the energy balance rate relationship described by equation 4. At any point A on the curve associated with crack length 2a< 2ac, (prior to the instability, I) the available energy in the plate is shown by the area OAC. At point A of figure 5, the amount of elastic energy is not sufficient to drive the elastic crack and therefore, the cracked plate is considered to be in a stable position (see equation la where G<R). If the length of the loaded crack at position A was extended by the amount of da, the available energy will drop to the OBC value. Thus, the amount of energy released is equal to OAB = (OAC - OBC) represented by the shaded area shown in figure 5. It is obvious that when the cracked plate is stressed to a higher value (point I), the available energy is sufficient to create unstable crack propagation when the half crack length a = ac,.

The amount of energy released due to crack extension, da, (as shown by the shaded area OAB) results in a drop of load and a corresponding decrease in plate stiffness. The inverse of stiffness is called compliance, and for a cracked plate varies as a function of

186

crack length. In figure 5 the plate compliance, C, is the inverse of the slope of the applied load, P, versus the displacement, (5. A clip gage device is usually mounted at the mouth of the cracked plate and the variation of the applied load versus crack mouth displacement that is related to the plate displacement, is plotted. In the linear regime, the applied load and the corresponding displacement are linearly related through the compliance, C, as

P = 5 / C 6

For the fixed grip condition the drop in the elastic energy, needed to extend the crack in a plate of thickness B (shown by the shaded area OAB), was provided by the total available elastic energy in the plate where U e = P 5 /2 (the area under the triangle OAC) . In terms of plate compliance, C

Ue = (p2 x C)/2B 7

The rate of elastic energy released for crack extension, da, to occur in a plate under the fixed grip condition can be written as:

dU e Ida = (p2 x dC/da)/2B

Note that in equation 8, for the fixed grip condition, the quantity dP/da = 0. Under the fixed end loading condition, no work was performed by the external forces on the cracked plate and all of the energy necessary to extend the crack was delivered by the elastic stored energy. It was mentioned that the stiffness of a cracked plate of length 2a decreases (drop in the applied load from PA to PB) as the total crack length extends by the amount of da (see figure 6, case a). If the applied load remains a constant, PA = P,, resulting in a displacement of d5 = (52 - 5, (as shown in figure 6, case b), the infinitesimal crack extension, da, takes place by releasing energy equal to the area OBD - OAC. In the case of constant load condition (case b), the cracked plate displacement, dS, results in the increase of elastic energy when crack extension takes place (area OBD > area OAC). The work done by the application of constant load on the cracked body to extend the crack is equal to P (52 - 51) represented by the area ABCD, case b. The energy release rate, G, for a cracked plate of thickness B, subjected to constant loading condition can be written

187

~ P

Loaded to P value and fixed at its end

PA ~ PA

o, PBli tp S 8 .B o,

o 6 o FIXED DISPLACEMENT

Constant load condition

~_.~ 2

~, w>>2a

_ _ A _ B I - q l ~ l -

, / / I as i

c l . ID ~

al 62 ,v DISPLACEMENT

Case (a) Case (b)

where

Figure 6: Illustrating fixed and constant applied load versus displacement

G = [d(W-Ue)/da]/B 9

dW/da = [P d8 Ida +6 dP/da]/B 10

Using the compliance relationship described by equation 6, which varies as a function of crack length (6 = P x C(a))

dW/da = [p2 x (dC/da) + C x P x (dP/da)]/B

The second term of equation 9, the elastic energy dUe/da, can be written as:

dUe/da = [P x (dS/da)/2 +(3 x (dP/da)/2]/B 8a

From equation 6 where 5 = P x C

dUe/da=[P 2 x (dC/da)/2 +P x C x (dP/da)/2+C x P x (dPIda)12]lB 8b

lOa

release rate,

188

The energy release rate, G, for the constant load condition (equation 9) in terms of its components described by equations 10a and 8b can be simplified as

G =[p2 x (dC/da) + C x P x (dP/da) .p2 x (dC/da)/2 -C x P x (dP/da)]/B

G =[p2 x (dC/da)]/2B 9b

Comparing the energy released rate, G, derived for two loading conditions (see equations 8 and 9b for the fixed grip and constant load cases, respectively.) it can be concluded that the quantity G is the same for both cases and is independent of the loading rate dP/da [6,7]. In both cases the compliance of the cracked plate increases as the crack extension takes place. The only difference between the two loading cases is that in the fixed grip case the elastic energy necessary to extend the crack is depleted from the total available elastic energy, Ue, and thus the quantity dUe/da is negative (drop in

load from PA to PB). In the constant loading condition (case b) the elastic energy available to extend the crack by the amount of da, is provided by the external force that is represented by the area ABCD (external work, PAX AS), see figure 6. In this case, the total amount of elastic energy that is available to extend the crack is positive (area OBD- area OAC) and thus the quantity dUe/da is positive.

4.3 The Path Independent J- Integral and its Application

4.3.1 Introduction

The two dimensional path independent concept of the J-Integral that encloses the crack tip front from one surface to the other crack surfaces is well documented [1] however; a brief overview of its concept and mathematical derivation is presented herein. Comments concerning the path independent J-Integral and its limitations are discussed in section 4.4. In section 4.5 the J-Controlled concept and stable crack growth, initiated by Paris and Hutchinson are presented. Detailed discussion on several experimental techniques for obtaining the J value is presented in section 4.6. Also in section 4.6, the multispecimen and single specimen approach for determination of J and the critical value of J at the onset of stable crack growth, J~c, are presented. Finally, in section 4.7 the application of the critical value of

189

the J-Integral based on the onset of crack extension, J~c, through a J-R resistance curve is discussed. The development of its value based on the ASTM testing requirements and limitations associated with its application are discussed.

In section 4.2 it was pointed out that for material where the crack tip plastic size is small and negligible, as compared to other crack dimensions, the rate of energy consumed for crack extension, dUe/da, is equal to the Griffith elastic energy release rate, G. It can be shown that for linear elastic material the Griffith energy release rate, G, and the J value based on the elastic energy change during crack propagation (see equation 26d), are identical. In terms of the crack tip stress intensity factor, K, where the plasticity is confined to a small region at the crack tip, the J-integral, J, and Griffith crack extension force, G, described by equation 4 are linked together and can be written [1] as:

J = G = K2/E ' 4a

where E' is equal to E and El ( l -v ~) for the plane stress and plane strain conditions, respectively. For low strength tough material where most of the available energy is consumed in the creation of an appreciable amount of plastic deformation at the crack tip, the Griffith energy release rate, G, (described by equation 4) or the stress intensity factor, K, expressed in chapter 2, are not satisfactory parameters for assessing any non-linear behavior at the crack tip. The load displacement diagram is linear in the elastic range (figure 7a) and non-linear at the crack tip region due to large plastic deformation. Figure 7b shows the crack tip non-linear behavior due to excessive plasticity when load versus crack opening displacement is recorded. The crack tip opening (blunting) and stable crack growth behavior in ductile metals (called stable tearing) when applied load is monotonically increasing can be separated into four regimes as shown in figure 7c. First, the initially sharp crack will undergo crack tip elastic displacement, and upon increasing load, crack tip blunting with a small amount of crack extension due to regional plastic deformation is expected (as illustrated in figure 7c and d). As the applied load increases, further crack growth (stable tearing) and blunting will occur and when the crack reaches its critical size, unstable crack propagation under load control condition will take place [8]: When the

190

(b)

"J Linear -J / / Non-Linear "g elastic ~= | / elastic

behavior ~ F behavior

Displacement Displacement

Growth due t to blunting (d)

~'~ IC~ ~ Instability Sharp crack II '-' J (Ioac= control) - ~ (no blunting)

m m Blunting (extension due to blunting)

~'~ I/TV I Start of crack initiation i=,," ~ - - j¢_ Crack extension i,.-- ; j , . ~ and stable crack

Crack Extension ~ ~ growth

(e) C . r a ~ d ~ p l ~ ~

45 o

Figure 7: Load displacement diagrams for linear and non-linear case

crack is blunting, the crack tip opening, St, (separation between the two crack faces at the mouth of the crack) is a measure of the deformation around the crack tip. In figure 7e the quantity 5t is defined as the intercept of the two lines drawn at 45 degree angles from the crack tip to the two crack faces representing the stretched position of it's original position for a given applied load [9]. The amount of deformation around the crack tip in a ductile material is a function of the applied load, specimen geometry, crack length, and material properties. The Crack Tip Opening Displacement (CTOD), St, and its critical value when fracture takes place are discussed extensively in references [10,11,12].

4.3.2 Derivation of Path Independent J- Integral

In 1968 Rice [1] defined the path independent integral for any arbitrary closed contour T" (as shown in figure 8a), not encompassing any singularity, where the result of the integral was found to be equal to zero, that is:

191

~ n, outward unit normal vector

T, traction vector ~ ~ _ ~ u, displacement vector

(b) ~2

Figure 8: Illustrating the path independent J-integral for any arbitrary closed contour F

J'r [Wdy - T • (au/c~x) ds] = 0 11

The closed integral presented by equation 11 is called a conservation integral. If now the integral is taken around a path as shown in figure 8b, surrounding the crack tip (for example the closed path ABCDEFA), the result of evaluating the Rice integral has a unique value for all contour ['1, F2, F3,.., and it is called the J-Integral. The proof can start by considering a two dimensional linear or non-linear elastic body subjected to mechanical loads as shown in figure 8. Note that, in a non-linear elastic body both strain and stress will follow the original stress-strain curve upon load removal with no plastic deformation left behind. In equation 11, the term W is defined as the strain energy density and it can be described as the work done per unit volume on the elastic body by the external forces. The strain energy density, W(e), in terms of stresses, c~j, and strains, e~ i, (for the selected contour, F) can be written as:

W = lr dW(c) = J'r oil d e q 12

For a two dimensional body the infinitesimal strain energy density, dW(~), in terms of stress and strain components can be formulated as:

192

dW(s) = Cxx d s xx + 2Cxy d s xy + cr~ d s yy 13

Using Green's theorem and replacing the line integral along the closed contour E" with the double integral for the area, S, enclosed by the path r , of case (a) the first term of the integral in terms of stress and strain components becomes:

.l'rW(x,y ) dy = J'.[s Cxx #lax [(#Ux/#X)] dx dy + j'.i's Oxy #lax [(#Ux/#y)

+ (#u~#x.)] dx dy+ .i'J's c~w #l#x [(#u~#y)] dx dy 13a

The second term in equation 11 contains the traction vector, T, having dimension of stress acting on a segment within the contour of the perimeter, ds, with outward normal, n, and creating a displacement du. In terms of its components (Tx and Ty in the x and y directions) tile traction vector can be expressed as (see figure 8):

T = % i + T y j = ( O x x . n x + o ~ . n y ) i + ( o ~ , . n x + O n . n y ) j

where n is the unit vector normal to the surface (figure 8). x and y coordinates, n can be written as

n = nx i + ny j

The components of the displacement rate vector, equation 11 in the x and y directions are:

14

In terms of

15

#u/#x, shown in

#u/#x = (#Ux/#X) i + (#uy/#x)j 16

Now the second part of the integral of equation 11 in terms of its components can be simplified as:

.1" r T • (au/#x) ds =

17

J'r{ [((:rx~ • n, + cxy. ny )i + ((:ryx. nx + (:r~. ny )j ],[(#ux/#x)i + (#u~#x)j]} ds

After rearrangement of terms .i'r [(~xx. (#Ux/~) + Cxy. (#uy/0x)] nxds

+ J'r [(~xy • (#Ux/~) + cw. (#uy/~)] ny ds

193

Substituting nxds = dy and ny ds = -dx and replacing the line integral with the double integral results in:

.[J" {~xx. a/ax [(aUx/aX)] + ~xy. ca/ax [(a%/ax)]} dxdy - .lj r {~xy. a/o~Y [(aUx/O~y)] + of f . a/ay [(auy/ay)]} dxdy 17a

Rewriting Equation 11 in terms of its components described by equations 13a and 17a, the final result after simplification yields J'r [VVdy - T • (au/c~x) ds] = 0 - ) a conservation integral around ['. Now consider the closed,path ABCDEF surrounding the crack tip as shown in figure 8 case (b). It can be argued that the line integral for this path is also zero as long as it is a closed path as discussed for the case (a). The line independent integral for the closed path ABCDEF can be written as

J =J'rl +.1",-2 = 0 11b

where the path of the first integral is taken by the path ABC and the second integral is DEF. Noting that there is no contribution by the two segments of crack faces AF and DC (shown in figure 8) since the two terms (dy) and the traction vector (T) seen in equation 11 are zero. From equation 11b it can be seen that the path of the two integrals must be opposite to each other:

J = J r l = - J r 2 11c

or changing the direction of the path as shown in figure 8 case (c)

J =J rl =J r2 (the same direction as r l )= j " [Wdy - T-(au/ax) ds] 1 ld

From equations 11c and 11d it can be concluded that the crack tip field can be characterized by enclosed curves, ['1, r2, r3,.., all equal to a value designated by J. Note that, no singularities in the area between the two enclosed curves are allowed.

NOW consider two extreme paths in which one is shrunk very close to the notch tip where traction T = 0, and the other case is shown in figure 9 where the J contour is selected by its outer boundary remote from the crack tip (defined by the rectangular path ABCDEFGH). For the case of T = 0, the J value is written by J" Wdy, where J is an

194

average measure of strain on the notch tip. For the alternate path (figure 9), the outward unit normal, n, perpendicular to the path, ds, is shown for the four sides of the path. The numerical J integral evaluated by this rectangular path utilizes the elastic load and displacement acted on the plate boundary and remote from the crack tip. That is, the crack tip behavior and the energy release rate expressed in terms of the J-integral can be defined by a remote path where its J-integral can easily be evaluated numerically by not focusing attention directly to the crack tip region (for example, the plate boundary, as shown in figure 9). This is analogous in LEFM to a crack tip field that may be characterized in terms of the stress intensity

nx= 0

ny = 1 Rectangular contour

; t ; G ' • F

n x = -1

ny=o dS = -dy

n

H

A

n "~ -

ds

Crack tip /

1~ d s = d y = O , T = O

+S B

n x = 0

ny=-I ds = dx

• --~- n

E - ~ X D

n x = l

r~_-o ds = dy

Figure 9: The closed path ABCDEF surrounding the crack tip

parameter (K = 13o-~/=a) computed from the far field stress, c, the crack length, a, and crack geometry, 13. In general, the concept of a path independent J-integral is used to characterize the crack tip linear and non-linear elastic behavior, and gives one an option of selecting any contour that can be situated either far from the crack tip (where the material is elastic) or close to the crack tip where material is plastically deformed when subjected to monotonic loading conditions. As the J contour become closer to the crack tip, the analysis results are subjected to more error. Kanninen et al [13] determined the J values

195

for different sets of integration paths that range from one very near the crack tip to one fairly remote from crack tip (J2, J3, .. ) where J2 was evaluated near the crack tip while the higher numbers (J3 . . . . ) corresponded to more remote contours. It was shown that the value of J2 begins to drop and level off after some crack extension while others continue to increase. Figure 10 shows the variation of applied J versus crack extension Aa for the two cases of remote and inside the region dominated by the crack tip field [13]. The experimental test results show that the value of the J-Integral calculated for the contour remote from the crack tip is in close agreement with the J measured experimentally usin 9 the load-displacement method (see section 4.6).

~O O-"

t (a ' j JT"

J2 O Experiment

v

Crack Extension, ~a

Figure 10: The variation of applied J versus crack extension •a for the two cases of J= and Js

The computed numerical value of J for a given crack geometry and loading condition must always be compared with the critical value of J in order to assess the failure. This is analogous to the stress intensity factor, K, whose calculated value for a given crack geometry and applied load was compared to its critical value, called the material fracture toughness, Kc. The critical value of J at the onset of crack initiation is designated by the J~c and it represents material fracture toughness and it's value can be obtained through the ASTM procedures. Therefore, the failure criteria in terms of the J-Integral crack tip parameter can be written as:

J -> Jl¢ 18

196

The J-integral under the deformation theory can characterize the near crack tip stress, cjj, and strain, ~ j, fields. Under both small scale yielding (where J= K ~2/E') and in the fully plastic regime, the crack tip stress and strain fields described by Hutchinson [14,15], Rice and Rosengren (HRR) [16], in terms of the J-integral have the form of:

cij = ao [J/(o~Go~oI. r)] ~/~1+n~ fij (e) 19

s~j = (Z(~o [J/(c{~o col, r)] nJll*n) gu (O) 20

where (~o is the material flow stress or effective yield stress (average of the yield and ultimate of the material), Co=(~o/E, In is a numerical constant whose value depends on the material stress-strain behavior, n is the strain hardening exponent, and o~ is a constant. Note that, equations 19 and 20 are analogous to the LEFM, where the crack tip field was expressed in terms of the stress intensity factor, K~, and radial and angular positions, see chapter 2.

For n = 1, the material behavior is linear elastic. As n increases and approaches infinity, the material behavior approaches the perfectly plastic condition. The dimensionless quantities f~j(0) and g~j (e) provide the variation of the stress and strain with respect to angular position around the crack tip. Equations 19 and 20 are capable of describing the near crack tip stress and strain behavior (for a stationary crack) when the material at the crack tip experiences either large or small-scale plastic deformation. They are referred to as the Hutchinson, Rice and Rosengren (HRR) fields. J is the strength or amplitude of the crack tip singularity in the power law format. Hutchinson, Rice and Rosengren's work shows that a Singularity, l/r, does exist that is dependent upon the material strain hardening exponent, n. From equation 19, when 0=0 and n = 1 (the case of linear elastic with small scale yielding), the crack tip elastic stresses become:

~ij = (i) (J/r)1/2 21a

Rewritting J in terms of the stress intensity factor where J = G = K ~2/E' (equation 21 a):

o i j - 13K~ ( l / r ) ~/2 21b

197

For ideally plastic material where n approaches infinity, the strain field described by equation 20 exhibits l / r singularity. The HRR stress- strain field described by equations 19 and 20 shows stronger strain singularity and much weaker stress singularity in the presence of extensive plasticity (large n) when compared with the linear elastic case described by equation 21b. That is, for an ideally plastic material where n = oo:

a~j = Cro f~j (e)l ,=~ 19a

t; i j = ~O 'o [ J / ( ( X . ~ o % | n r)] g i j ( 6 ) I n =oo l / r singularity 20a

To evaluate the crack tip parameter J, the two terms shown in equation 11 must be available. That is, the stress-strain relationship describing material behavior when subjected to monotonic load is required to calculate the strain energy density term W presented in equation 14. The analysis must be based on adapting a deformation theory of plasticity. Assume the material stress-strain relation has the Ramberg-Osgood form where strain, E, in terms of stress, c, can be written as:

E; i f t; o "" E~ / (3" 0 "1- OL ((~ / (3"o) n 22a

where n is the strain hardening index, (z is chosen to fit the stress- strain data, and (~o is the effective yield stress and Eo=ao/E. Applicability of equation 22a is restricted to monotonically increasing load and unloading is excluded for the above relationship to be valid. A slight amount of crack growth will alter the value of n. This restriction can be overcome under condition which will be called J-controlled growth. Small stable crack growth and its acceptability when J is evaluated as a crack tip parameter is discussed in sections 4.4 and 4.5.

Example Evaluate the J-Integral for a rectangular path remote from the crack tip (see figure 9) where the two terms in the Integral must first be numerically computed. Note that for the remote path ABCDEFGH of figure 9, the J-Integral is calculated by utilizing the load and displacement acting on the boundary. In figure 9 the uniaxial tensile loading is acting on the cracked specimen. If the path independent

198

integral defined by equation 11 is taken as a rectangular contour, the two terms J'r Wdy and J'rT • (c~u/ax)ds must be calculated for each segment of the path as shown in figure 9. In the plane stress condition where c~33 =0, the J-integral for each path (AB, BC, CD, EF, FG and GH) can be rewritten as:

.i" r [Wdy- T • (au/ax) ds] =2 { J'AB [ W -(~XX (aUx/aX)-(~xy (auy/o~x)] dy+ J'BC [(~XY (au~/ax)+%y (auy/ax)] dx+.i" cD [W-(~xx (aUx/O~X) - c~y(auy/ax)] dy} 23

The factor 2 in front of the bracket is due to the loading symmetrey (uniaxial loading) and symmetry due to crack plate geometry which allow the calculation to be carried out on only half of the paths, ABCD. In order to perform the integration described by equation 23, it is necessary to have a relationship between stresses and strains along the path. For example, the work density term, W, along the path AB reduces to j'AB Cyy dc~ because all stress components except a n are zero along this path (also true for paths CF and GH in the plane stress condition). The quantity c~yy is given by E ~ when applied load is below the yield and material behavior along the boundary is elastic (E is the material Young's modulus). The resulting expression for W in terms of strain components for path AB can be written as:

W = j'A8 ayy dc~ = E cw 2/2 24

Note that the traction vector along contour segments AB, CF and GH is zero because they are on free surfaces.

Along segments BC and FG the work density, W, does not contribute to the J-Integral since the term dy is zero (both BC and FG are parallel to the x axis). However, the traction components of equation 23 along contour BC and FG, where ny =1 and ny = -1 respectively, are Txx = 0, and Tyy = cw. The x component of displacement, u, is negligible and therefore Ux = 0. As the result, the integrand for paths BC and FG have c~yy (auy/o~x) as a quantity. Equation 23 can be simplified as:

.1" r [Wdy - T • (c~u/c~x) ds] = 2 x { JAB [E s~ 2/2] dy + J" Bc [+ c~ (auy/c~x)] dx + .1" co [E ~ 2/2] dy} = 4 x { JAa [E s~ 2/2] dy + 2 x j" Bc [ ~ (auy/ax)] dx 23a

199

4.4 Comments Concerning the Path Independent J-Integral Concept

It was mentioned in section 4.3 that by applying the potential energy definition to the J-integral, the two quantities J and energy release rate, G are equal within the linear elastic range. For small scale yielding, where the plastic zone is small and confined to a small region at the crack tip, the J-Integral analysis and LEFM are in agreement with each other (J = G = K2/E'). The computational value of J described by equation 11d (applying the deformation theory of plasticity) defines the amplitude of crack tip stress-strain fields (see equations 19 & 20) and, as was the case for LEFM, a critical value that provides a failure criterion for material fracture toughness must be assigned to J. That is, structural failure can occur when:

J (Computed or applied) = J (Critical)

The critical value of J under considerable plastic deformation, when a small amount of cleavage stable crack extension takes place, is designated by J~c- The quantity JJc must be independent of specimen geometry (as it was in the case of K~c) where it's value can be obtained through the ASTM testing procedure. The ASTM E-1820 (Standard Test Method for measurement of fracture toughness [17]) identifies three fracture toughness properties for metals. They are designated by Jc, J,c, and J. and all vary with the amount of crack extension, to be examined upon test termination. In this book the bulk of discussion will be centered upon the J,c value because it's application in space structure, as a failure criterion, is most common. Presently, there are several techniques that can determine the amount of crack growth at the onset of stable crack extension, Aa. These techniques are the loading/unloading compliance method [18,19], the electrical potential drop method [20,21], the ultrasonic method [22,23], and the acoustic emission method [24]. Some of these methods are more suitable for multi-specimen techniques than single specimen and are discussed in reference [20-24]. The unloading compliance is probably the most commonly used technique in determining the physical crack size and the amount of crack extension, Aa. Later in section 4.6, it can be seen that in the case of a single specimen technique, for establishing a J-R curve and determining the J,c value, the amount of stable crack growth

200

is best determined by the compliance technique. J-R curve using a single specimen technique is discussed in sections 4.7.

The three fracture toughness parameters (J¢, J,c, and Ju) which vary with the amount of crack extension are defined by the ASTM E- 1820 as: J¢ - a value of J at fracture instability under the crack tip plane strain condition prior to onset of significant stable crack extension (see figure 1 la). This quantity may be thickness dependent. J~c - a value of J under the crack tip plane strain condition when a small amount of stable crack extension at initiation of ductile cracking takes place (see figure 1 lb). J. - a value of J at fracture instability after significant stable crack extension (see figure 11c). It may be size and test specimen dependent.

J J J (B)

Jlc

v

Crack Extension, Aa

Jc

i (A) I I

(c)

= L

v

Figure 11: The three fracture toughness parameters (Jc, J,c, and J.) which vary with the amount of crack extension

Applying the value of J,¢ as a failure criterion to a tough metal may be too conservative to qualify as the elastic-plastic fracture toughness parameter. The J,¢ value corresponds to the material failure at the onset of crack initiation, which is associated with small amount of crack extension (in the middle of the part thickness) under plane strain condition, can be obtained by the ASTM E 1820 Standards [17]. For example, in structures such as nuclear pressure vessels where the operating temperature is above the ductile brittle transition point (ductile behavior), stable crack growth for the structural parts beyond the initiation is allowed as long as the amount of crack extension can be predicted conservatively. A small amount of crack growth for many

201

tough materials (by a millimeter or two) can increase the material fracture toughness to several times the initiation fracture toughness, J,c [25,26]. That is, in tough material a substantial margin of safety can exist with respect to final failure where extensive stable crack growth after crack initiation under monotonically rising load up to instability takes place. Figure 12 illustrates a typical crack extension curve (J-R) in which the fracture toughness at initiation, J,c, and stable crack growth prior to final failure is shown. In this plot the quantity J~¢ can be obtained at the intersection of stable crack growth line (fitted to the test data) and the blunting line.

4

JIc

J

/

' /__

~ Blunting Line

h , = v

Stable Crack Growth, Aa

Figure 12: Illustrates a typical crack extension curve (J-R) where the fracture toughness at initiation is shown

From a fracture mechanics point of view, the use of J,c as a failure criterion to predict the residual strength capability is as restrictive as using the Kj¢ when stable crack growth is expected in the structural components, For this reason, the ASTM has established procedures to develop a crack growth resistance curve analogous to the R-curve, called the J-R curve, which is a plot of the J-Integral (JappUed) versus crack extension Aa. In applying the J integral to establish a valid J-R curve, certain conditions must be followed that are fully described in section 4.7 based on loading and unloading compliance method where a single compact tension specimen is used.

In the case of LEFM, stable crack extension, Aa, implies negligible unloading near the crack tip. Crack tip unloading due to crack tip extension for small scale yielding is acceptable because the plasticity

202

is small and confined to a very localized region at the crack tip (see figure 13a). However, for large scale crack tip plastic deformation, the crack tip unloading due to incremental stable crack extension is not permissible if the deformation theory of plasticity (see equations 19

Linear Elastic Elastic-Plastic Deformation Incremental

Theory Theory

~ n g l / ,Loading A Loadin//~/~U n G (3" Unloading ol ,f,

,~ E; ~- E; ng v o dinga ~ Crack tip plasticity

Plasticity due to unloading Deformed 1 ~ ~ Z~~ne

K field where ~ J - Integral field where Gij = [K I 2~tr)̂ 0.5] ~ij aij = ~o [J I QSo r In) ] 11(1+n) ~ij

(a) (b)

Figure 13: Crack tip unloading due to crack tip extension for small and large scale yielding

and 20) is to be used to describe crack tip behavior for a stationary crack. Unloading in the plastic region leads to permanent crack tip deformation, resulting in a multivalued strain energy density, W, which can alter the result of the J quantity. The error introduced in calculating the J-Integral after unloading is a result of the fact that the material properties near the crack tip prior to unloading were different than after reloading (different singularity). Not taking into consideration the differences between strain fields (the existance of some strain hardening due to unloading) when crack extension takes place will alter the J-Integral results. It is important for the reader to note the distinction between the deformation theory of plasticity and the incremental theory of deformation. With the deformation theory, the crack tip deformation for both cases of linear and non-linear body is reversible under monotonic loading and unloading conditions as shown in figure 13b. That is, the crack tip stress and strain leading to fracture is reversible upon unloading (the stress-strain curve for

203

unloading is the same as for loading). In contrast in the incremental theory of plasticity, crack extension will cause the crack tip elastic unloading and non-proportional plastic loading which alters the strain hardening exponent used to define the stress and the strain field at the crack tip (see equations 19 and 20). With this in mind the incremental theory of plasticity is applicable to a stationary crack and rules out materials which show crack growth prior to final fracture (both theories must give equivalent results if the loading is monotonic with no unloading throughout the body). For this reason, the failure criterion defined by the J-Integral describing the large scale crack tip yielding must be limited to the plane strain condition where the crack initiation at the onset of cleavage fracture is visible as a small fiat region ahead of the crack tip. This restriction is similar to the K~¢ failure criterion, where crack extension was limitted to 2% of the amount of crack growth (an amount equivalent to 5% offset from the linear elastic). Moreover, the cracked specimen in the K~¢ test must have sufficient thickness, B, to prevent crack tip plastic deformation, and to meet the ASTM E-399 [27] size requirement expressed by:

e > 2.5 (Klc/CYield)2 25

Application of the J-Integral for characterization of the crack tip field for the onset of crack initiation, and J-controlled stable crack growth conditions, are discussed in section 4.5. In section 4.6, different experimental techniques and procedures for evaluating the J- Integral from the load versus load point displacement record have been proposed. The fracture toughness concept based on the onset of crack extension, J~c, and the ASTM procedure employed to determine it's value fora given material are addressed in section 4.7. From a safety standpoint, the use of the J~c value is conservative and can be satisfied, however the results of analysis (flaw size prediction for a given stress level) may not be realistic. In section 4.5, it has been shown that if the HRR field based on the deformation theory of plasticity (described by equations 19 and 20) increases in size more rapidly than it advances due to stable crack growth, the crack tip behavior can be described by the quantity dJ/da (the slope of the J- Integral versus the crack extension, •a, as shown in figure 12). This is called the J-controlled crack growth regime and it is discussed in the next section.

204

4.5 J-Controlled Concept and Stable Crack Growth

The Rice J-Integral theory [1] describing crack tip plastic behavior is based on the deformation theory of plasticity (i.e., material stress- strain curve in the nonlinear region, well beyond the elastic range, can be represented by a power law format) and its application is restricted to analysis of stationary cracks subjected to monotonic load where no unloading is allowed. The application of the path independent J Integral for crack tip analysis allows the analyst to select any convenient path surrounding the crack tip for the evaluation of the J integral when numerical methods via equation 11d are employed. It was also mentionecl that the J-Integral can be viewed as a prameter characterizing the intensity of the plastic stress-strain field surrounding the crack tip. The HRR crack tip stress strain fields in terms of J quantity (characterizing the intensity of the field) can be written as [14,15,16]

~ij = Kn jl/(l+n) r-l/(1.n) fij (0) 19a E;ij = Kn jn/(l+n) r-hill*n) gij (e) 20a

where •, = (~o((~oI,)"v('*n) (see equations 19 and 20). From these two equations it can be argued that for a given material when two cracks of different length have the same field intensity, J, then the crack tip stress-strain fields surrounding these two cracks must be identical. This is called the law of similitude, which is similar to the case of linear elastic fracture mechanics, where two cracks of the same intensity, K, must have the same crack tip field. Furthermore, what happens at their crack tips, such as small crack extension, Aa, must be identical for both crack cases [28]. For example, consider the case of two initial crack lengths of 1.0 and 0.5 in. length (in a wide center crack plate) subjected to remote constant amplitude fluctuating stresses of &(~ = 15 ksi and 26 ksi, respectively (R = 0):

AK = Kma x -Kmi n (note that AK = 13A(~ma x (=a) 1/2)

for (~max =15 ksi, and Cmi n =0, AK =15 x (4.14 x1.5) 1~2=32.6 ksi (in.) ~/2

and for ~max = 26 ksi, and Cmin = 0, AK=26 x (4.14 x0.5) 112=32.6 1/2

ksi(in.)

205

under the limitation where crack extension, z~a, would not disrupt the HRR crack tip fields. That is, the HRR field must increase in size more rapidly than it advances due to stable crack extension (figure 14). Hutchinson and Paris [29] established the theoretical basis for the use of the J-Integral of the deformation theory of plasticity, in the large scale yielding range, where small amounts of stable crack growth can be tolerated and analyzed. The mechanism of stable crack growth involves some elastic unloading and distinctly nonproportional plastic deformation (near the crack tip) which is not allowed by the deformation theory of plasticity, figure 14. They [29] argued that the use of the J-Integral in assessing material fracture toughness when stable crack growth occurs is permissible only under the restricted circumstances which were called the J-Controlled growth conditions. Hutchinson argued that in order for the J-Controlled growth condition to exist, it is essential that proportional plastic deformation occur everywhere except in a small region at the crack tip, where nonproportional plastic loading and elastic unloading occur. By differentiating equation 20a, the increment of strain, d ~j, due to simultaneous increase in crack length, da, and J is:

d ~ij = Kn J n/(l+n) r-n/(l*n)dJ/J [(n/n+1) gij ( 0 ) ] + K n J n/(l+n) r-n/(l*n) (da/r) * [n/(n+l) cos (O) + sin (O) a /~ gij (O)] 20b

IIc

line

D

Aa

Non-proportional plastic loading

Proportional loading (controlled by J-field)

r" I \ /

"-~l I "=-- Elastic unloading

Region of elastic unloading and non-proportional loading are embeded in J-dominated field

Figure 14: The HRR field must increase in size more rapidly than it advances due to stable crack extension

206

The first term corresponds to the proportional loading increment (d J/J) where d ~j ~ ~j, and the second term is not (da/r). The J-Controlled condition can be established when the following condition holds:

dJ/J >> da/r 26

The form established by equation 26 indicates that the HRR field increases in size more rapidly than it advances due to stable crack extension, Aa, as controlled by the second term. From figure 14 the quantity dJ/da=J/D where D is assumed to be related to the amount of crack extension with J value corresponding to 2JIc (J advances rapidly with small amount of Aa) From figure 14 the following relationship can be established:

l ID = l/J * (dJ/da) 27

From equations 26 and 27 one can conclude that D << r. In figure 14 the quantity R represents the size of the HRR field measured from the crack tip to its boundary and r is any value within the field. Therefore:

D << r < R 28

which states that a small amount of crack extension, Aa = D, is acceptable within the J independence concept if it does not disrupt the HRR crack tip fields, where Aa << R. The path independent J-Integral will remain independent of the integration path as long as crack extension, Aa, for a stationary crack is small and satisfies equation 28. Shih et. al. [25] have found almost the same value of J over a wide range of paths in their finite element analysis (having only 5% variation with 5% increase in crack length) under a fully yielded and plane strain compact tension specimen condition. For a fully yielded specimen with uncracked ligament, b, the condition for J-controlled growth (described by equation 27) can be rewritten as follow:

biD = b/J * (dJ/da) = co for b>>D=Aa -) (o>>1 29

Values of e) for steel alloys are calculated by Paris to fall between 0.1 and 100. A value of e)= 40 was found by Shih [25] when using a

207

compact tension specimen made of A533-B steel, where b, J,c, and dJ/da (after initiation) were applied to equation 29.

Material resistance to crack growth for most ductile metals is expected to increase when the material undergoes some small amount of stable crack growth. When small stable crack extension, Aa, under monotonic load takes place, the material fracture toughness due to higher applied load and longer crack length (a + Aa) increases. Within the linear elastic fracture mechanics regime, an R-Curve approach was established (see the ASTM E-561) to evaluate material resistance to fractu[e where the tangency point between the applied K curve and the experimentally generated R-Curve was designated as the material fracture toughness (for brittle material abrupt failure without crack extension is expected). This is illustrated in figure 15 where material resistance to fracture, K., versus effective crack length is plotted for a given plate thickness. Note that a given R-curve is usually associated with a given thickness (thickness dependent) however, it is independent of the original crack length. The applied K curves are plots of computed K versus physical crack length for different stress values which are crack geometry dependent. With the same analogy, the J-R curve can be developed under the J-Controlled condition (JR versus Aa curve provided by experiment) and the tangency point between the applied J curves and the J-R curve will determine the material fracture toughness.

KR Brittle Failure

Abrupt Failure

a O

KF

K C

V

Ductile Failure (Small Scale Yielding)

~pplied K Curve ~ ' - I ~ ~ R e s ~ Curve

l l s Aa =.._

a o

Figure 15: Illustrating material resistance to fracture, KR, versus effective crack length (brittle and ductile failure)

208

4.6 Experimental Evaluation of J-Integral and J,c Testing

The path independent J-integral was used originally as an analytical tool for crack tip field determination, when treated by a deformation theory of plasticity, and is applicable to material which exhibits large scale yielding at the region of discontinuity. Later, the J- integral was proposed as a fracture parameter whose critical value was used for fracture analysis of structural parts. The use of J when applying the deformation theory of plasticity restrict its use to any crack extension under monotonic loading since any unloading violate the theory of plastic!ty. The laboratory evaluation of the J-Integral and it's critical value at the onset of cleavage fracture, J~c, was initiated first by Begley and Landes [31] and Landes and. Begley [32] in 1972 in their work entitled "The J Integral as a Fracture Criterion" and "The Effect of Specimen Geometry on Jfc", respectively. Their experimental evaluation of J and the J~c values utilized the potential energy rate [33] concept that uses several identical standard specimens called the multispecimen technique. With the multispecimen energy rate technique, the load versus load line displacement curves are generated for each specimen at a fixed displacement. The J-Integral calculation can be accomplished through the graphical assessment approach by using the measurement of the area under the load deflection curve for each specimen. Later, the costly and time consuming multispecimen technique of Landes and Begley was replaced by a single specimen method developed first by Rice [34]. The evaluation of J and J~c (as a fracture parameter) by a single specimen was established based on a bend specimen with a deep notched pre-fatigued crack where the area under the load versus load line displacement curve was measured. Andrews et al. [35] in 1976 developed a somewhat different approach to the single specimen technique for the evaluation of J~c. They used the elastic compliance method and an electronic signal amplification instrument which allowed the load versus load line displacement to be recorded to establish a J-R plot. The single specimen technique was subsequently advanced by Joyce and Gudas [36] (1979) by implementing a computer-enhanced interactive system capable of producing the J-R curve and consequently an evaluation of the J~c value. In this method the crack extensions were measured through the compliance technique by loading and unloading the cracked plate after crack extension, and was used to generate the J-R curve. Sections 4.6.1

209

and 4.6.2 will discuss the multispecimen and a bend bar single specimen techniques respectively. The advanced single specimen technique proposed by Joyce and Gudas is presented in section 4.6.3. The validity of evaluating the plane strain fracture toughness, K~¢, from the Jt¢ test at failure for fully plastic behavior, when the thickness requirement for a valid K~c can not be met, is fully discussed by Landes in reference [26].

4.6.1 Multispecimen Laboratory Evaluation of the J-Integral (Energy Rate Interpretation)

The application of the path independent J-Integral to characterization of large scale yielding at the crack tip under monotonic loading without focusing directly at the crack tip region was proposed by Rice [1]. It has been shown that the crack tip field parameter, J, can be calculated analytically by using a stress-strain analysis of a closed contour taken remotely from the crack tip, with accurate results obtained when numerical techniques such as FEM solution employing an incremental plasticity is applied [13]. Another alternative interpretation of the Rice J-Integral for both linear and non- linear elastic material is the potential energy rate differences between two identical, two dimensional cracked specimens subjected to the same system of loading, consisting of traction or displacement prescribed on the boundary. The two specimens are identical in composition, geometrical shape but have different crack length a and a+Aa, as shown in figure 16a [33]. For a two dimensional elastic body of area, S, having an edge cracked with original length, a, and a boundary path [', the potential energy, U, normalized per material thickness, B, is defined as:

U/B = JA W dx dy - J rs T u ds 26a

where [', is that portion of the path F with traction vector T specified on the boundary and u is the displacement vector (see figure 9). The quantity W is the strain energy density or simply the work done per unit volume on the body in loading to a given condition. When loading is by imposed displacement (applied load on the body is such that the displacement vector, u, is a constant on the boundary), the second term of equation 26a (J rs T u ds) drops out and the potential energy, U, becomes equal to the quantity JAW dx dy (related to the area under

210

the load deflection curve shown in figure 16). The increase in potential energy by AU for a crack to extend by the amount of Aa is

U (a+ Aa) = Uo +AU = j'A (W+ AW)dx dy - J'rs T (u + Au) ds 26b

(a) Case (b) ] Case

Constant Constant I~ b""l I--'~--I I Case i_~,. ~ ~_ ~Displacement I~ I I ~ I L°adL~Se F t FAa I I I (It I L~II

i P ~'PAT A B I

I / - /U" i~i I

.m / / d " ~ j ~ I _ olY ÷,a !c !D I

~ ~ I Displacement, ~ Displacement, 8 J

Figure 16: Illustrating load deflection diagrams

The J integral can be interpreted as the potential energy difference between the two above identical bodies (having different crack length a and a+Aa) per unit crack extension normalized to material thickness, B. That is:

J=- 1__[ U o (a)-U(a+Aa)]

B &a 26c

For linear elastic behavior the J-Integral is considered as the available energy for crack extension, and for plastic behavior can be viewed as an energy comparison of two similar cracked bodies with different crack length that are identically loaded. Note that the interpretation of the J-Integral as the change of the deformation energy during crack propagation, da, can be written as:

J = - (dU/da)/B 26d

211

Whereas equation 26c describes the J-Integral in terms of differences in potential energy of two identical cracked specimens with crack length a and a+Aa. Finite element analysis indicated that different value of J may be obtained by applying equation 26c than its value calculated through equation 26d [37].

The load versus load point displacement diagram shown in figure 16 case (a), is constructed for two identical cracked specimens having two different crack lengths a and a+Aa under a constant displacement, 8. The area under each curve is the work done in loading to a given displacement, 8, and the shaded area between the two curves is related to the J value through JBda (see equation 26c). The calculated value of J for the case of constant load is also illustrated in figure 16 case (b) for two identical specimens with different crack lengths a and a+Aa. The area between the two load versus load point displacement curves (the shaded area shown in figure 16b) is also related to the J and its magnitude is almost the same as case (a) in which a constant displacement loading condition is applied to the test specimen. The additional triangular area embedded in the shaded area shown in figure 16b is small and negligible and therefore the two cases of 16 a & b are equal.

The experimental interpretation of energy rate approach defined by equation 26d (J=-(dU/da)/B) for obtaining the quantity J requires several identical cracked specimens and for this reason the method was referred to as the multispecimen technique [31,32]. The purpose of this experimental approach is to evaluate the quantity J via the energy rate equation for a fixed displacement. Under this type of loading condition, the second term-of equation 26a drops out and the potential energy, U, becomes equal to the area under the load displacement curve. The steps required to obtain the two quantities J and it's critical value, J~c, experimentally, at the onset of growth, through the multispecimen technique when using equation 26d are as follows:

1) Prepare several identical pre-fatigued test specimens with range of crack lengths. In the original Landes and Begley multispecimen test, bend bars specimens of different pre-fatigue crack lengths (ranged from 4.5 to 9.8 mm.) made of Ni-Cr-Mo-V Steel, where used.

212

2) Record the load, P, versus load point displacement, 5, at fixed displacement values of 51, 52, and (53 for each cracked specimen as shown in figure 17. Measure the area, A, under the load versus load point displacement associated with each displacement. The areas under the curve (A,, A2, A3, A,) represent the energy per unit thickness (U/B) delivered to the specimen. Note that, only the area A, associated with displacement 81 of crack length a, is shown in figure 17.

3) Plot the potential energy per unit thickness U/B (represented by the measured area A described in step 2) versus crack length for several values of constant displacement (51, 52, and 53) as shown in figure 17.

4) Calculate the slopes of the curves generated in step 3 for each crack length, -l/B(dU/da), and create J versus displacement curves for different crack lengths. Note that the quantity J = - l/B(dU/da). The displacement at the onset of crack initiation where

J = -lIB [dU/da] J 5 =constant

] = . . . .

Applied Load

t- LU

_= o

~ , ~ 1 < 62 <83 ....

l aa I ]'~--f-. 81 = I , I

a l a 2 a3 a4 Crack Length, a

• Crack Initiation

a l< a 2 <a 3 ....

,,/~J~.J~_~_ ~ "~a 2 ~ --~"~aa3 4

.~.-F~ I I "~z- su=n Enemy I I (Energy under

I I I I I I I 61 r ~ s a

Dis olaeement, (5

== JIc

-4

,51 ~2 C~3 Displacement, 8

Figure 17: The steps required to obtain the two quantities J and it's critical value, J,c, experimentally

213

load becomes maximum and drop off thereafter corresponds to the critical value of J and designated by J~c(See figure 17). The drop off load and its association with the onset of crack extension is true for small test specimens and may not be applicable to larger specimen sizes.

The start of crack initiation in the multispecimen technique is usually evaluated through visual/optical observation by a heat tenting process where a small amount of crack growth beyond initial crack length associated with the pre-fatigue crack had occurred. An example of a complete multispecimen testing procedure for evaluation of J and Jwc is shown in figure 18. The testing was conducted for an intermediate strength rotor steel, Ni-Cr-Mo-V alloy that utilizes equation 26d and follows steps 2 through 4, corresponding to cases A through C, described above. The thickness and width of the bend bar specimen were 12.0 and 20.0 mm, respectively. In case D of figure 18, the critical value of displacement versus crack length is plotted that corresponds to the critical value of J at the onset of crack initiation. An

Specimens made o f N i -Cr -Mo-V Steel

z .04 ~- Case ( A ) / ? ~,

d / / a =6.12 mm

,01 F / / , : ~7.9 mm -- _

0 0,1 02 0.3 0.4 0.5 0.6 0.7 Displacement, mm

~'.h> a _>3.6, Step,

~1 ,20 f e Crack I n i~ t Jon /

~ ' 1 6 ~ C..~se{C, / / . = 8 . 1 ,

t / / 0

0 .2 .4 .6 .8 Displacement, rnm

24

2O

12

8

4

\ Step 3

0,5 --\GOu\ = 0.635 turn

~ Case (B)

=o%\\

I I I I 2 4 6 8

Crack Length, rnm

Case (D)

I I I T 4 6 8 10

Crack Length, mm

Figure 18: An example of a complete mulUspecimen testing [31]

214

average value of 0.187 M Jim 2 for crack lengths ranging from 4.6 to 7.1mm. was obtained as shown in figure 18.

There are several disadvantages to determining the J value through the multiuspecimen technique. The main one is the number of specimens that are needed to generate the J versus displacement curve (usually it takes from 5 to 10 specimens or more and this is wasteful of material). Secondly, the graphical steps that are required to obtain either J or J~, values could introduce errors when calculation is performed. Finally, the material variability as a result of having several specimens could scatter the final values of J or J~, considerably, in addition, because all of the load versus load line displacement diagrams were taken beyond the point of onset of crack initiation, the exact determination of the crack extension point is made unclear.

4.6.2 Single Specimen Laboratory Evaluation of the J-Integral

Landes and Begley were the first to propose the energy rate interpretation of the J-Integral for obtaining experimentally the material fracture toughness, using the multi-specimen technique. The multi- specimen approach, the most explicit and exact method of measuring J requires 5 or more specimens and this was too cumbersome and wasteful of material for evaluation of the J~, parameter (see section 4.6.1). An improved method of determining the J~, parameter was established based on the work of Rice at al [34] in which a bend bar specimen with a deep edge crack was used to determine the quantities J and J~, (see figure 19). An alternate and equivalent definition of J, that was the, basis for the Rice approach, in terms of both constant displacement and constant load, was formulated as:

J = .lo v (-~3p/c~a) dv I constant v or .io P (-~/aa) dP I constant P 27

For a deep crack in a plate with uncracked ligament, b, subjected to a bending moment M (as shown in figure 19), the J definition of equation 27 in terms of angular displacement, e, can be formulated as:

J = (2 J'o e('~) Mde(w,))/Bb 28

where E)(w,) is the contribution of the angle change to the total bending

215

Bend Bar Specimen

P/2~lr= GageD'Splacement [--~-.._j PI2/8t = [ao+Z + p (W'ao)FP (W'ao)

+z

"ao)

M = PS/4 \ / , S=4W y F ~ ' q / / Sf2

Figure 19: A bend bar specimen with a deep edge crack

due to the introduced crack (specimen with crack, wc) and depends on material properties. The total angle of bending due to the applied moment, M, in terms of its components can be written as:

9tot= = O(.c) + O(w¢) 29

where etot= is equal to the amount of rotation due to the presence of crack, O(.c), and additional rotation as if the crack did not exist in the specimen, e(nc). For a deep bend bar cracked specimen where the contribution of e(nc) is negligible, the quantity e(v~)>> O(nc) and therefore etot= = e(~). Note that the quantity under the integral described by equation 28 (.i0 e(w¢) Mde(w~)) is equivalent to the area under the M versus e(w¢) diagram and is simply the work done in loading the cracked specimen in which the energy contribution due to no crack, U(.c), is eliminated. In the case where the bend test specimen does not have a deep crack (a/w <0.6), the energy contribution from no crack presence, U(.~), to the total energy, Utot.,, can not be ignored and must be evaluated. The quantity U(n¢) can be obtained by calculating the area under the moment, M, versus O(n¢) of a bend bar specimen with no crack presence.

216

Since it is difficult to obtain the plot of M versus e(~) in a bend bar specimen, Rice argued that the quantity J'o e(wc) Mde(wc) from equation 28 can be replaced by jo ~(wc) PdA (w~)and equation 28 can be re formulated as:

J = (2 fo ~ (w~) PdA (wJBb) 28a

where A is the line load displacement and A (wJP is the measure of compliance with crack presence in a bend bar specimen. The load point displacement (P, A) is difficult to measure accurately in the three point loading case because of the necessity of correcting for nonessential displacement [38]. That is, the option of measuring load point displacement directly as was in the case of C(T) specimen, is not available for-three point bend specimen. An easier and alternative measure of displacement, A, in a bend bar specimen is to obtain the displacement at the mouth of the crack, V, as shown in figure 19. The two quantities A and V are related through a similar triangles concept (figure 19) by the following relationship (assuming small bending angles, e)

A= V*S/4[ao +z + p (W - ao)] 30

The above assumption is valid when plasticity is restricted to the ligament, b, (the whole ligament is above the material tensile yield) and the two halves of the bar remain straight. The specimen is rotating with respect to the center of rotation situated in the plastic region of ligament, b, at a distance p (W - ao) below the original crack tip, ao. The rotational factor, p, is defined as the ratio of the distance between the crack tip and the center of rotation to the net ligament, W-ao, figure 19. The quantity p = 0.4 for a bend bar specimen with a ~ / > 0 . 4 5 was proposed by Sumpter and Turner [39].

The approximate formulation for the quantity J given by Rice, in terms of load versus load point displacement curve can be written as:

J = 2A/Bb 28b

where area under the load versus displacement, A, is equal to the total potential energy, Utot,,, for a deep crack in a bend bar specimen (Utot,, = U,~). Applying equation 28b to a single bend bar specimen to obtain J, by calculating the area under the curve, definitely has an advantage

217

over the exact method (multi-specimen technique) described by the energy rate method. Accordingly, the quantity J~c at the onset of crack initiation can also be determined by using a single specimen technique provided that the crack extension measurement at the critical point to be established. The difficulty associated with the critical point measurement for J~c is similar to the plane strain fracture toughness test (ASTM E-399) that 2% crack extension was set on the load displacement diagram when evaluating the Kt~. For the J,, test, the measuring point at the moment of crack extension can be dependent on specimen size and on the capability of the instrument used to detect/measure the~, initiation point. The problem of measuring the initiation point for the J~¢ parameter can be resolved if test data from several identical specimens (between 4 and 6 specimens are required [26]) were presented in the form of a resistance curve with J versus crack extension, Aa, plotted (J-Rcurve) as illustrated in figure 20. Each specimen was loaded to a different amount of displacement and then load is removed. The crack extension was marked for each broken specimen. Currently, there are two methods that can be used to measure crack extension reading. For materials that become brittle at low temperature, such as ferritic steel, the marking for crack extension, ~a, can be accomplished by first heat tinting the cracked specimen. Heat tinting of ferritic steel (for example 4340 steel) for 10 to 15 minutes at around 6500 F will oxidize the extended cracked surfaces. Specimens can subsequently broken in a liquid nitrogen environment (-3200 F) in which exposed oxidized surfaces are associated with the amount of extension which can be cleady distinguished from failure surfaces that have cleavage behavior (figure 20). However, for materials which are not sensitive to a low temperatureenvironment, such as aluminum alloys, the amount of crack extension can be measured by post fatigue cracking the specimen under low load cyclic environment (figure 20). Specimens were then broken and the crack extension, Aa, recorded. The area under the load versus displacement curve, A, was measured and the quantity J for each specimen based on equation 28b was calculated. The J-R curve can now be established by plotting the value of J for each specimen versus the amount of crack extension, Aa, measured by the methods described above. The intersection of a straight line fitted through the data recorded for each specimen (J, Aa) with the blunting line will determine the Jz¢ value (see figure 20). The blunting line is associated with the stretched zone where a sharp crack

218

originally in the unloading condition will blunt upon the application of load. The amount of blunting increases as the applied load increases until the crack extension at the tip of the original crack occurs (see figure 21). It is exactly at this moment that the fracture toughness parameter, J,c, is best defined. Landes and Begley studied the fracture appearance of A216 steel specimen by applying the tinting technique [40]. Careful examination of the two broken surfaces revealed the existence of two distinct regions, stretch zone and then material separation associated with crack extension, figure 21. Fractographic

Identical specimens, (1) through (4), were loaded to ~ifferent displacement levels and area under the curve was measured

Blunting Line

n , r - ~ _ ~ , < ~ I / Linear Fit

Displacement Displacement Crack Extension

Fractured Surfaces Pre-Fatigued Fatigued Notch Pre-Fatigued Stable Final Notch Stable Final

Region Tearing Fracture Region Tea~.,,q_ Region Fracture

a o Aa a o Aa - , - =i I-- =, (Heat Tinting Approach) (Fatiged Cracked Approach)

Figure 20: Illustrating a resistance curve with J versus crack extension

examination of the broken surfaces shows that the blunted regions appeared oriented at an angle approximately 45 o with respect to pre- fatigue and material separation zones surfaces. The formula that relates the crack advancement in the blunted region (apparent crack extension as compared to actual material separation) to the J parameter is illustrated in figure 21 and for ductile metals is given as:

J = 2(~,~. (Aa) 31

219

where G.ow is the flow stress that is the average of the material yield and ultimate stress. The measured amount of crack advancement for ductile metals is taken as half of the crack tip blunting, I/2 ~, (figure 7).

4.6.3 Advanced Single Specimen Technique Using the C(T) Specimen

The C(T) specimen has the advantage over the bend bar specimen when evaluating the area under the curve associated with load displacement diagram. In the case of single edge bend specimens, the location of the clip gage is at the mouth of the crack, as illustrated in figure 19. For the C(T) specimen, the displacement

i

f ~t = Crack tip opening displacement Crack

~a= Crack extension due to blunting Extension Stretch F Susequent

k extension) ueenl

/ j - -~r~- '~ sharp crack (no ioad & no blunting)I+- ao---~l

Crack Extension, &a

Figure 21: Illustrating the crack extension due to blunting and the definition of fracture toughness

measurement is done by applying the clip gage at the mouth of the specimen along the line of the applied load (figures 22a&b) as specified by the ASTM specification. To have a valid J-R curve for determining the material fracture toughness (critical value of Jtc), by using the advanced C(T) specimen technique, all ASTM requirements, including the accuracy of the apparatus and test equipment applied to measure the test data, must be met [17]. A clip gage (figure 22b) is required to evaluate the J value from the area under the load versus load line displacement (displacement gage of figure 22b is attached to

220

the mouth of the crack on load line locations at points A and B). Data reading must be within the working range of the clip gage when using the elastic compliance method to. evaluate the crack advancement including the original and final physical crack length. The applied load must also be measured by a load transducer capable of recording the load continuously.

The specimen configuration for a standard C(T) specimen is shown in figure 23 and has planer dimensions given in terms of W (a distance measured from load line application to the specimen edge).

Clip Gage Mounted I at the Load Line I - ' ~

i o . L , .

Load Application

(b)

Clip Gage A~

Figure 22: Illustrating the clip gage at the mouth of the specimen along the line of the applied load

The quantity W for J determination is based on the specimen thickness, B, where 2< W/B<4.

In using the compliance method for crack length measurement and to insure a straight crack front, it is recommended that side grooves are added to both crack edges along the crack ligament surfaces (figure 23). Not using the side grooving on the specimen, the crack reading measurements can be underestimated by loading/unloading technique due to non-uniform crack extension when compared with an optical average crack reading. Three dimensional finite element analysis have shown that the presence of side grooving help to make

221

. . . .

. . . .

2<~W/B ~< 4

o S :ve°l wO.0.1.,w i:

Figure 23: Side grooves are added to both crack edges of a C(T) specimen along the crack ligament surfaces

the J Integral at the crack tip more uniform across the specimen thickness and eliminates the low J value near the two surfaces [41,42]. The amount of thickness reduction (B-BN) due to side grooving should not exceed 0.25B as recommended by the ASTM. The net specimen

thickness, BN, is the minimum thickness measured at the roots of the side grooves (the quantity BN=B if no side grooves are present in the specimen). Side grooving is recommended to be performed on the specimen after precracking and must be aligned with the notch root.

To provide a satisfactory fracture toughness value, fatigue cracking the C(T) specimen at the V-notch is required to produce a natural crack prior to testing. Most of the steps for single specimen preperation (such as precracking, specimen crack length marking for initial and final crack length reading, and specimen geometry) are the same as for the multispecimen technique discussed in section 4.6.1. The amount of fatigue crack at the tip of the sharp V-notch should not be less than 0.05 inch. The load for introducing a fatigue crack at the end of the notch can be either under load control or displacement control and at all times net section yielding must be avoided. The prefatigue load magnitude should not exceed:

Pf = 0.4B(W- ao)2~yiewJ(2W+ao) 32

222

where ao is the original crack length and %=eld is the material yield value.

The procedure for establishing a J-R curve is to load and unload the C(T) specimen within the elastic range to estimate the new crack length and incremental crack advancement. The applied load must stay below the final prefatigue load, Pf, expressed by equation 32. Usually it takes more than 25 unloading steps to establish a J-R curve. Figure 24a shows a load versus load line displacement curve where a few loading-unloading steps are shown for illustration purposes. An enlarged portion of figure 24a (the region designated by A) is shown in figure 24b, where each loading and unloading has a distinct area associated with it which can be used to evaluate the J value. Figure 25 is an actual load versus load line displacement curve for 2219-T87 welded aluminum with almost 29 load-unloading steps. The slope of unloading for all steps appears to be almost parallel to each other (figure 24). However, a slight variation in slope is expected if the compliance method is used to determine the amount of crack growth.

ng and Unloading

"~ (a)

/ c,,on IS shown in figure 24b.

Load-Line Displacement

Load-Line Displacement

Figure 24: Illustrates load versus load line displacement curve for a few loading-unloading steps

223

1200

IOSO

9OO

?SO

~00

4S0

3OO

IS0

- . 0 1 0 .000 .010 .030 .030 .040 .OSO .040 .070 .080 .090 .100

Coo ( l , ' , )

Figure 25: An actual load versus load line displacement curve for 2219- T87 welded aluminum with almost 29 load-unloading steps

That is, as the crack advances due to loading and unloading, the compliance of the cracked plate increases and the slope of load versus load line displacement decreases. The reader may refer to section 4.7 and the ASTM-E1820 for more discussion on load vs. load- line displacement data gathering for the advanced single specimen technique.

4.7 Determination of J~c Value Based on a Single Specimen Test

The J-R curve is a plot of J versus crack extension for metallic materials that can be established by a single specimen technique under the J controlled growth condition. To generate a valid J-R curve for determining a valid J~¢ value (through loading/unloading of a single C(T) specimen), first the elastic and plastic components of J at a point corresponding to a given displacement v~ and load Pi must be calculated [34,17]. Other specimen geometries, such as single edge bend specimens, SE(B), and disk shape compact specimens, DC(T), are available and can be utilized for establishing the J-R curve.

The elastic component of J (Jel) is related to the stress intensity factor, K, through equation 4a:

224

Jel = K2 (1-v2)/E 4a

where K at load P, is given by:

Ig.= [P/(BBNW) lr2] f(a-/W) 33

The geometrical correction factor, f(a/W), for the current crack length, a~, (note that the f(a/W) value changes and the most recent one is used at each instant) is given by the following equation:

f(a~/V) = [(2+aJW) (0.886 + 4.64 (a/W) - 13.32(a/VV) 2 + 14.72(a~/V) 3 - 5.6(a~"V)')]/(1-a./VV) 3~2 34

The plastic component of J is given as:

Jpl = 11 [(Area),l]/Bnbo 35

where (Area)p~ is the area under the load versus load-line displacement curve as shown in figure 26, bo is given by (W-ao) and T1=2+0.522 boNV. For the J-R test, both elastic and plastic components of J at each point (Je~(~) and Jpt(~)) from the load versus load line displacement

Total Load-Line Displacement, v

Figure 26: Illustrates the area under the load versus load-line displacement curve

must be calculated. The elastic component of J is given by equations 4a and 33. Its plastic component, Jp~(~), at a point corresponding to a, v~ and P~ (see figures 24 and 27) can be written as [34,17]:

225

Jp,(i) = [Jp,(i-,) + (Tl(i.~)/b(~l))(~ ~, pt/BN)] X [1 -- "Y(i-1) (~a~ b(i.1))] 36

where ~ , p~= (Area)p~(~) - (Area)p,(~.~) and Aa~ = a(~) - a(,.~). The quantities TI(F1 ) and y(F1) are equal to (2.0+0.522 b(i.1)NV) and (1.0+0.76 b(i.1)/W), respectively. The quantity AApt is shown in figure 27 and is the increment of plastic area under the load versus load line displacement record between lines of constant displacement at points i and i-1. The value of Jpt(~) represents the total crack growth plastic J and can be obtained by simply adding the calculated Jp=(i-1) to the increment of J

P(i-1) P(i)

i f Plastic Area

M_ Area = ~(i)+ P(i-1)) x (vio I (i)- Vpl (i-1))/2

Vpl(i-1) Vpl(i)

Plastic Load-Line Displacement, V p l

Figure 27: An increment of plastic area under the load versus load line displacement

growth due to increment of load increase (P(j)- P(~)), that is:

(Area)pl(i) = [(Area)p~(~l) + (P(i)+ P(~.I)) x (Vpl (j)- vpw (~1))/2] 37

where (P(i)+ P(~l))x(vpl (~)- vpl (~1))/2 represents the area under the curve shown in figure 24, Vp~(~)is the plastic part of the load-line displacement and is equal to v (~)-(P ~ CLL (~)). The load line compliance CLL (~)= (Av/AP) ~. Tables la and lb represents the steps necessary to develop a J-R curve through a single C(T) specimen technique for 2219-T87 friction stir welded aluminum. Only steps 21 through 29 are shown here. In table l a, all the necessary steps for determining final crack length after each unloading and corresponding incremental crack growth through the compliance method are shown. Note that each step represents one unloading as shown in figure 24. Table lb shows

226

TABLE la

C a l p l l l i - l . l l l U l - o l l

Zld.f.~ld. C~llak Cml)~L1) - 0.93151 6 ~ * *

Ooload 14ad ; it.turf, o f ¥ # Jltalr~ o f

mmbo= u .Lo ,d lag unloa~Lm8

(5ha) (Aa)

V ~ ( 1 1 5 n s v l ~ l l l l

I1' - U . I I l u ~

g l P I I J . ) Czac~ ~ Czlak Gzm~h ~ f a e

/Iot~r,.~Lo~ ( ~ (An)

21 I l i . I .6116 l . l l l ~ i I T . ] ~ l l . ] 0 | . • T l l | . | • i T

~ 615.3 .063• ~,~2C1~-06 l l * i l l l * 6 ~ O*lJ~O O*O4~l

6 0 3 . 4 . O ~ T ~ . ~ - I S 1 0 . 2 6 51 . R9 O • • 6 ~ 0 , 0 4 ? 2

~4 ISl,l.2 .OST| ~. 3131-06 5~ . 2 i 6 ] .39 0 . i j 1 5 0.05~2

~ ] 675. ] .OiO0 ~. )T l u . O j 14.SO 5 i * O• O.• l?6 6*6J04

1 • 8 ~ . 2 . O ~ i 1 . 6 1 1 ~ • 6 . 1 1 I T . T • l .OOl4 1 . 6 ~ l

17 8 4 6 . 5 . 6 1 4 5 I • 6618 -06 l i , O I l • . $1 1 , 0 0 4 9 O , 0 1 5 1

l l 131 • 4 .~6 • • 4 | • l ~ ~nt-@l 1 0 0 . i 6 ~ O I . I I l . • O I T l . l T O l

I I 116.6 . I l l I . i l l 08-Ol l 115.11 l O l . i l 1.11.11il i . 0 7 i l

1 1 l * 4 t ~ v e ) l n d ~ u l m X l l : l l * ( ( l i k ~ l l l l F - k J l l ' i l l i . l l 3 O t l l . l l l C k l l l l )

- i l l . 0 I I (Jto/l l) " l - I I . I5928 (J4o/l l "3~:tO • l o l l IUIo~l) " 4 - 9 • I531• G4o/I) " I l l

• lie ~e.~Lee~ ~ o a Data I ~ • leo a~ 4~$ueI--~: Data 81~

TABLE lb

J * C A L C U ~ T Z ( N I (ASTIr 1 1 8 2 0 - 9 6 ) Not SpleLmm8 11~Lalutle8 Cb| e .~JO JLIJ | | | ~ t L ~ TLel4 | ~ e STIIOO.O It~k

~ . 4 V I o t t o e l ~ I e ~ o f ~ J J • J J CtlIZLILo6II lumber u ~ e ~ l L q u~JLo~JLq I~.~J~Le PLS~r.Lc b i l e Dofhm~lLo8 H m J L f L s d DJLoqu~LLflLod

( ~ ) ( ~ J ) ( I ~-Lb81 ( ~ * - l b l l l q * l ~ . ) ( l ~ - [ b l / l q , r C . ) ( [ l o ~ l l s q * r ~ * ) ( ~ - ~ s / 8 q . r t . ) ( | I S e J ) / | l f ° T L o b

21 . M l 4 916.9 N . 2 1 365.0 124o4 ~ . 6 $18,6 ~ l L f L ~ Z2 . 0 5 ~ 913.3 3 0 . 0 1 3 H . 0 12.5.1 $13.9 562.1 ~ L / f l e d

23 .n~q7 ~ ] . & 31.82 t t 0 . 1 | ~ . 4 $ ~ . $ ~ 9 o 0 ~ f L e d

26 ° ~ 1 8 6J3 .2 33.&7 421.0 121.$ ~ . 5 $9&.3 QaJd.LILod .0600 079.3 35.16 6 ~ . 9 15*.5 575.4 6~n.2 ~ f L ~

* ~ U *A1.2 31.13 61t .9 1 ~ . $ ~ . 3 667.5 Qa81£fLod 2T . ~ 4 3 1141.3 3 8 ° M 455.8 |~q ,7 619.5 6T$.2 Qom]LLfLe4 28 ° ~ 03.5.6 40.38 513.3 I~q°5 638.8 6M°7 Q~GLfLod n .0616 U t . 6 4,1.33 536.6 I~q.6 ~ . I ~ .6 ~ f L ~

J P l u ( L ) - (2',0,522(b(L)J • (Are8 p t - , . ( l ) ) l l t ee~ • b (L ) ) • ( I - ( 1 . 0 . 7 • • b ( L ) ~ ) • (a(L)-8(L-l))/b(L)l J n u ( L ) - (E (L ) '2 * I[lopoLeNa'2)))TJbduJ.u8

~ l ~ l J • . P(L) * W(&(L)/U) / Sq~(lBrolo * b t * VldCk)

J Do|(L) - J II, u t L a ( L ) • J PlJmtLa(L)

J Nod(L) o J IU~J¢L=(L) 4. J r lsor~c:(L) * Det l~ J(L)

uhero . Dotts J(L) o Delr, a J ( L * I ) ,, |I~o(L)IBo(f,)I • [ a ( t ) - s ( L - | ) ] x J PI .~ tJ~(L)

( ~ / D J L a q u a l ) - (15 * J D e f ) / | l l , YLe14 n u t be • bo

227

the calculation of other parameters such as area under the load versus displacement, and elastic and plastic components of J, including the total J value (J.j+Jp~).

When a sudden increase in displacement and a decrease in load is recorded, it is recognized as a fatigue crack growth instability arrest and is related to the material characteristic that causes an increase in cracked plate compliance. This phenomenon is referred to as pop-in crack propagation and will be evaluated only when the load rises with increasing displacement after its occurrence. The acceptability of pop- in is shown in figure. 28 based on the ASTM procedure. A line with 5% reduced slope (CF)from line CB that is parallel to the initial slope line OA and passes through the start of the pop-!n point, is drawn. If the fracture instability point due to pop-in is within the region of BCEF, it is considered as insignificant pop-in, figure 28a. Otherwise, the pop-in, BG, is significant (figure 28b). The fracture toughness for the case(a) corresponds to the load Pu or Pc of onset designated by B.

i

O

LPuorP c F/ j . A A B F . . . . . . .

/ / , , ' "' : / / l"1 '" Insignificant / / / / I

Clip Gage Displacement Clip Gage Displacement

Y (0.95BD)~

Sig~iant

(b )

Figure 28: The acceptability of pop-in based on the ASTM procedure

4.7.1 Validity Check for Fracture Toughness from the J-R Curve

Validity of the Jtc test from the J-R curve depends on several requirements that if they are not met the test must be repeated after corrections had been made. Proper usage of load apparatus (fixture, clevises, load alignment), load transducer, clip gage, specimen preparation and their accuracy are requirements that must all met. Test data must be examined prior to converting them to J versus crack

228

extension data (J-R curve). When final crack length measurements through loading/unloading were completed (assuming fracture instability did not occur), the final crack length readings must be measured optically and compared with compliance data. After final unloading, the crack specimen will be heat tinted (or fatigue cycling) as discussed in section 4.6.2. For optical measurements the specimen is broken to expose the crack surfaces for crack lengths readings (figure 29). Nine equally spaced points centered about the center of specimen

Fractured Surfaces (Fatiged Cracked Approach)

Pre-Fatigued Stable Fatigued Final Notch Region "re " R e ' '~ ~ . . . ~ . . Fracture

I Aa = ' ~ ~ = Side Groove

j specimen

L . . . . . . . N

J ~

l L Final Fatigued Front Notch (f=~al crack length)

I ~, ao

Pre-Fatigued Front (original crack length)

Side Groove ---,,-

Center line

Side Groove -----

Figure 29: Illustrates the method of measuring crack lengths

are selected at two locations. The first set of crack readings (nine of them) is associated with the first precracking step for measuring the original crack length, and the second nine readings is for final crack length after some stable crack growth. Figure 29 shows the method of measuring the original and final crack lengths. The procedure for measuring crack length recommends to use the nine equally spaced marks along the front of the original fatigue crack as well as the final physical crack length (after stable tearing) as indicated in figure 29. These nine readings are in such a way that are symmetric with respect to the center line (four readings at each side of the center mark). First

229

The generated data for fracture toughness, from load versus load- line displacement, when converted to J values versus crack extension must follow three steps:

1) All the J-R data must be bounded at one side by the coordinate axes and from the other side by Jmax and Aamax, as shown in figure 30

280

240

200

m

x C

160

C - - 120

80

4 0

I J m a x

I----

the two extreme marks are averaged and the results will be averaged with the remaining seven readings. None of the nine physical crack length readings for both original and final crack lengths may differ by more than 5% from the average value. Moreover, none of the nine physical crack extension measurements, Aap, must be less than the 50% of average crack extension. Also, the crack extension estimated by loading/unloading compliance method at the last unloading shall be compared with the measured optical crack extension reading. The difference between them must not exceed 15% Aap for crack extensions less than 0.2ba.

Crack Extens ion, in

I I I I I I 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

Figure 30: J-R curve box bounded by coordinate axes and from the other side by Jmax and Aam=x

(it is called the J-R curve box). The maximum value of the J-integral capacity, Jmax, for a specimen is determined by the smaller of bc~o/20 or Bc~o/20 and the maximum crack extension capacity of the specimen as Aama~ = 0.25bo (C~o is the flow stress). The two quantities b and bo

230

are the remaining ligaments based on the distance from the physical crack length, ap, and original crack, ao, to the back edge of the specimen, respectively. The first two above conditions require that the specimen is large enough so that net section yielding prevented (b>20J/(~o) and the minimum thickness requirement (B>20J/(~o) for crack extension to occur under the plane strain condition, respectively. The second requirement (Aar, ax=0.25bo) restricts the amount of ductile crack extension allowed.

2) After the J-R box based on the above two regional restrictions is established, some J values may turn out to be unacceptable, depending on the amount of crack extension. These data may also need some adjustment to establish the best estimate of initial crack length whenever the predicted initial crack length (initial crack size obtained from compliance measurement, aoq) matches the measured average fatigue crack length within 0.01 W. It should be noted that a tentative J value, JQ, is dependent on the aoq value used to calculate Aa~ quantities. Evaluation of J~c is required to evaluate and qualify the JQ. The intersection of the power law fit and the 0.2 mm (0.008 in.) offset line (parallel to the construction line) defines the quantity JQ, as shown in figure 31a. Note that construction lines are used for data qualification and only those data points that fall between the construction line, J=2~o Aa, and a second line parallel to it with an offset value of 1.5 mm. (having slope m>2(~o) and capped by J,r,, = bo(~O/15 are considered. The intersection of 1.5 mm. exclusion line with the power law fit curve defines the Aa,m, line (figure 31). At least one data point must lie between the first e~:clusion line (with 0.15 mm. offset) and a second exclusion line with 0.5 mm. offset as shown in figure 31b. These offset lines (0.15 and 1.5 mm.) bound the region of acceptable J values. The 0.15 mm. offset line ensures that the crack extension is at least 0.15 mm. and the 1.5 mm. line ensures that Aa is less than 0.06b.

3) Data points must fall between the 0.5 and 1.5 mm. exclusion lines and the remaining data points can be anywhere inside the exclusion region as highlighted in figure 31b (regions of data qualification). The intercept of the power law curve with the 0.15 mm. and 1.5 mm. lines will indicate Aami n and Aalim, , respectively (figure 31). At least five valid data points must exist in the valid region (cross-hatched area), otherwise additional tests must be run.

231

In summary, the following requirements must be achieved as the validity checks when determining the J~c value from a J-R diagram:

1) None of the nine original physical crack measurements may differ by more than 5% of the average value.

2) None of the nine final physical crack measurements may differ by more than 5% of the average value.

80C

60C

"-' 40C

J(

20C

0.0

Construction

~ regress'~ line/ 11.5mm. / ~ . ' ~ - ; - - ' ~ ~"~ ~ / I J /A~ . . . . . / i E~c~us~ n jQ ~ . . ~ . ~ . . , ~ u : u ~ / 11.5mm. n

t l imit

0.0 I 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2 .5 /

Z~amin Crack Extension, mm. Crack Extension, mm.

Figure 31: Construction lines and definition of regions for data qualifications

3) None of the nine physical crack extension measurements must be smaller than 50% of Aap.

4) The initial crack size obtained from compliance measurement, aoq, may not differ from ao by more than 0.01W.

5) The estimated final crack extension must be within +15% of Aap for crack extensions.

6) The number of regression points for curve fit must be at least 8. 7) Use the method of least squares to determine a linear regression

line described by (InJ=lnCl+C21n(Aa/k)) where k=l.0mm. (0.0394 in.), and power coefficient C2<1.0.

8) The specimen thickness associated with side grooving, Bnet, should be greater than 25JQ/a0.

9) The initial ligament, bo, should be greater than 25JQ/ao (to meet size independent fracture toughness requirement).

10) The slope of power law regression line at AaQ<ao.

232

R e f e r e n c e s

.

.

.

.

.

.

.

.

.

J. R. Rice, "A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notch and Cracks," Journal of Applied Mechanics, June, 1968, pp. 379-386.

B. Farahmand, "Fatigue and Fracture Mechanics of High Risk Parts," Chapman & Hall, 1997, ch. 6

B. Farahmand, :"Fatigue and Fracture Mechanics of High Risk Parts," Chapman & Hall, 1997, pp. 169

C. E. Inglis, "Structure in a Plate due to Presence of Cracks and Sharp Corners," Trans. Inst. Naval Architects London, Vol. 60, p. 219, March, 1913.

A. S. Tetelman, and A. J. McEvily, JR., "Fracture of Structural Materials", John Wiley& Sons Inc., 1967, pp 39-48.

G. R. Irwin, Fracture, Handbuch der Physik, Vl, pp. 551-590, Springer-Verlag, Heidelberg, 1958.

J. L. Sanders, "On the Griffith-lrwin Fracture Theory," ASME, Trans., Vol. 27 E, 1961, pp. 352-353.

C. F. Shih, et al., "Methodology for Plastic Fracture," General Electric Corporate Research and Development Division Report to the EPRI on RP601-2, EPRI NP 1735, Schenectday, N. Y. March, 1981.

J. R. Rice and D. M. Tracey, "Computational Fracture Mechanics," Numerical and Computer Methods in Structural Mechanics (Ed. S. J. Feves et al.), Acadamic Press, N. Y., 1973, pp. 585-623.

10. J. N. Roberson and A. S. Tetelman, "The critical Crack Tip Opening Displacement and Microscopic Fracture Criteria for Metals, UCLA RE 7360, University of California, Los Angeles, Ca, 1973.

233

11. C. F. Shih, "Relationship Between the J-integral and the Crack Opening Displacement for Stationary and Extending Cracks," General Electric Co. TIS Report No. 79crd075, April 1979.

12. M. G. Dawes, "Elastic-Plastic Fracture Toughness Based on the COD and the J-Contour Integral Concepts," Elastic Plastic Fracture, ASTM STP 668, ASTM 1976, pp. 307-333.

13. M. F. Kanninen and et al, "Elastic-Plastic Fracture Mechanics for Two Dimensional Stable Crack Growth and Instability Problems," Elastic-Plastic Frcature, STP 668, ASTM, 1977, pp. 121-150.

14. J. W. Hutchinson, Journal of the Mechanics and Physices of Solids, VOL. 16, 1968, pp. 13-31.

15. J. W. Hutchinson, Journal of the Mechanics and Physices of Solids, VOL. 16, 1968, pp. 337-347.

16. J. R. Rice, and G. F Rosengren, Journal of Physices and Mechanics of Solids, Vol. 16, 1968, pp. 1-12.

17. Annual Book of ASTM Standards, "Standard Test Method for Measure of Fracture Toughness," Vol. 03.01, 1999, pp. 981-1013.

18. A. Saxena and S. J. Hudak, "Review and Extension of Complience Information for Common Crack Growth Specimen," International Journal of Fracture Mechanics" Vol. 14, October 1978, pp. 453- 468.

19. M. G. Vassilaros, J. A. Joyce, and J. P. Gudas in Fracture Mechanics: Twelfth Conference, ASTM STP 700, American Society for Testing Materials, 1980, pp. 251-270.

20. E. M. Hackett, M. T. Kirk, and R. A. Hayes, " An Evaluation of J-R Curve Testing of Nuclear Piping Material Using The Direct Potential Drop Technique," NURG/CR-4540, U.S. Nuclear Regulatory Commission, August 1986.

21. B. Marandent, G. Sanz. "Flaw Growth and Fracture," ASTM STP 631, American Society for Testing Materials, 1977, pp. 462-476.

234

22. S. J. Klima, D. M. Fisher and R. J. Buzzard, Journal of Testing and Evaluation, Vol. 4. No. 6, 1976, pp. 397-404.

23. J. H. Underwood, D. C. Winters, and D. P. Kendall, "The Section and Measurements of Cracks," The Welding Institute, Cambridge England, 1976, pp. 31-39

24. H. Takahashi, M. Khan, K. Shimomura, and M. Suzuki, in Proceeding, Fourth International Accoustic Emission Symposium, High Pressure Institute, Japan, Tokyo, 1978, Session 8, pp. 24-25.

25. C. F. Shih, H.G. deLorenzi, and W. R. Andrews, "Studies on Crack Initiation and Stable Crack Growth," Elastic Plastic Fracture, ASTM STP 668, pp. 65-70.

26. J. A. Begley, and J. D. Landes in Fracture Analysis, ASTM STP 560, American Society for esting Material, 1974, pp. 170-180

27. Annual Book of ASTM Standards, "Standard Test Method for Plane Strain Fracture Toughness of Metallic Materials," Vol. 03.01, 1999, pp. 413-443.

28. B. Farahmand, "Fatigue and Fracture Mechanics of High Risk Parts,''Chapman and Hall, 1997, pp. 179-176.

29. J. W. Hutchinson, and P.C. Paris, "Stability Analysis of J- Controlled Crack growth," Elastic Plastic Fracture, ASTM STP 668, pp. 37-64.

30. B. Farahmand, "Fracture Prperties of 1460 Russian Alloy," Boeing Technical Report, 1977, pp. 60-70.

31. J. A. Begley, and J. D. Landes, "The J Integral as a Fracture Criterion," Fracture Toughness, ASTM STP 415, 1971, pp. 1-23.

32. J. D. Landes and J. A. Begley,"The Effect of Specimen Geometry on J~c," Fracture Toughness, ASTM STP 415, 1971, pp. 24-39.

33. J. R. Rice, in Fracture, An Advance Treatise, Vol. II, Acadamic Press, pp. 191-308

235

34. J. R. Rice, P.C. Paris, and J. G. Merkle, "Some Further Results of J-Integral Analysis and Estimates, Progress in Flaw Growth and Fracture Toughness Testing," ASTM STP 536, PP. 231-245, 1973.

35. W. R. Andrews, G. A. Clark, P. C. Paris, and D. W. Schmidt, "Single Specimen Tests for J~c Determination," Mechanics of Crack Growth, ASTM STP 590, 1976, PP. 27-42.

36. J. A. Joyce, J. P. Gudas, "Computer Interactive J~c Testing of Navy Alloys," Elastic Plastic Fracture, ASTM STP 668, 1979, PP. 451- 468.

37. J. D. Landes, H. Walker, and G. A. Clarke, "Evaluation of Estimation Procedures Used in J-Integral Testing," Elastic-Plastic Fracture, ASTM STP 668, 1977, pp. 266-287.

38. A. A. Willoughby, and S. J. Garwood, "On the Unloading Complience Method of Driving Single Specimen R-Curve in Three Point Bending," Elstic-Plastic Fracture, Vol. 2, ASTM STP 803, 1983, pp. 373-397.

39. J. D. G. Sumpter, and C. E. Turner, "Method for Laboratory Determination of Jc," Cracks and Fracture, ASTM STP 601, 1976, pp. 3-18.

40. J. D. Landes and J. A. Begley, "The Results From J-Integral Studies: An Attempt to Establish a JIC Testing Procedure.

41. M. Nevalainen, and R. H. Dodds, Jr., "Numerical Investigation of 3D Constraint Effects on Brittle Fracture in SE(B) Bending and C(T) Specimens," UILU-ENG-95-2001, Department of Civil Engineering, University of Illinois, Urbana, IL.

42. C. e. Shih, H. G. DeLorenzi, and W. R. Andrew, "Studies on Crack Initiation and Stable Crack Growth," Elastic-Plastic Fracture, ASTM STP 668, 1979, pp. 65-120.

236

Chapter 5

THE FRACTURE MECHANICS OF DUCTILE METALS THEORY

5.0 Introduction

Materials selected by engineers to manufacture fracture critical hardware must have sufficient toughness and exhibit some amount of plastic deformation and stable crack growth at the crack tip region prior to their final failure. Dependability of materials employed in designing components of aircraft or aerospace structure is directly related to the material toughness and its ability to resist cracking during its service usage. The present approach to the life assessment of high strength or low fracture toughness materials is to use linear elastic fracture mechanics, which utilizes the stress intensity factor, K, as the crack tip parameter. In chapter 2, it was shown that the crack tip behavior under small scale yielding could adequately be described by the stress intensity factor, K, to characterize the crack tip stress and strain environment when subjected to external applied load. However, in the presence of large plastic deformation, the use of K to describe the crack tip behavior is too conservative. The amount of energy consumed at the crack tip for plastic deformation is largely due to material resistance in that region prior to final failure, which is not properly accounted for in the linear elastic fracture mechanics analysis. Irwin and Orowan [1,2] independently observed that for tough material, the amount of energy dissipated at the crack tip for plastic deformation, Up, is much larger than the energy consumed for the formation of two crack surfaces, Us. It should be noted that in the Griffith theory of brittle fracture, it was assumed that all of the available energy is depleted for the formation of crack surfaces. The general principle from which the Griffith Theory [3] was derived for elastic crack propagation is not limited to ideally brittle materials, such as glass. The principle applies as well when dissipative mechanisms, such as plastic deformation, are present. Irwin and Orowan showed that Griffith's energy balance principle can be extended to apply to ductile metals where the crack tip exhibits considerable plastic deformation.

5.1 The Extended Griffith Theory

The energy terms that contribute to the extended balance equation for crack extension can be written as:

Griffith energy

c~[UE -Us-Up]/Cqc=0 1

where UE is the available energy, Us is the surface energy of the crack per unit thickness, and Up is the energy consumed per unit thickness in plastically straining material at the crack tip region. For low strength/tough metals, where Up >> Us, the expression for surface energy, Us, can be omitted from equation 1. Therefore, crack instability or complete fracture will occur if

c~[UE-Up]/c~c _>0 2

In chapter 4, Elastic Plastic Fracture Mechanics (EPFM), it was shown that when small scale yielding condition prevails, the Griffith crack extension force, G, and the J integral are equal. The Griffith and Irwin relation for elastic crack propagation, and the J-integral at the onset of crack extension are related by:

Gc = ~a2clE = Jc = K2J E 3

The quantity to the left is the crack extension force at the instability, Gc, and the quantity to the right of Equation 3 is the critical value of the stress intensity factor, Kc, called fracture toughness, which for thick sections (the plane strain condition), is designated by K~c.

Within the linear elastic range, the path independent J-Integral is equivalent to both the crack extension force, G, and the stress intensity factor, K, via equation 3. As was discussed in chapter 4, the crack tip behavior under both small and large plastic deformation can be addressed through the path independent J-Integral. The plastic portion of the J integral (J.~), which can be related to the rate of energy consumed at the crack tip'for plastic deformation, can be evaluated by simply measuring the plastic area under the load versus load line displacement curve (Begely and Landes [4]), see figure 1. One of the

Clip . P

/ / Area under the _1 I / load-displacement I ~.j

V curve I

Load Line Displacement

• ameter

stress-strain ¢1 :turervSTtrain I I

I I I I 1 J

Strain, c

Figure 1: The area under the load versus load line displacement and a typical stress-strain curve

238

practical applications of the path independent J-Integral is its flexibility for allowing the analyst to select any path far from the crack tip (such as the plate boundary) to be substituted for a path near the crack tip region that would be otherwise difficult to analyze. However, the laboratory evaluation of its critical value through the multispecimen or single specimen techniques makes this approach unattractive. Yet another energy related approach, called the "Fracture Mechanics of Ductile Metals" (abbreviated by FMDM), accounts for the presence of large plastic deformation at the crack tip and is presented in detail in this chapter. The FMDM theory is based on the extended Griffith's energy balance concept and is capable of calculating the residual strength capability of structural components by using the area under the full uniaxial stress-strain curve for the ductile material under study (figure 1). Thus, one of the major differences between the EPFM and the FMDM approach to ductile fracture is the use of the area under the load versus displacement curve. In the former case, the area under the curve represents the variation of applied load versus load line displacement of a cracked specimen, while in the latter case, the area denotes the variation of load (normalized with respect to the original cross sectional area) versus elongation (normalized with respect to the gage length) for an uncracked specimen. Information obtained from a full stress-strain curve can be used to assess the crack tip environment under mode I.

In order to verify the results and accuracy of the FMDM theory, several comparisons were made with the experimental data provided from different reliable sources. The correlation between the FMDM theory and the experimental data with various alloys is presented in section 5.9. In addition, the fracture toughness values for several alloys were calculated by the FMDM approach and the results were compared with the ASTM fracture toughness testing (section 5.10).

To appreciate the FMDM theory, the reader may refer to chapters 2 and 4 pertaining to the complexities associated with obtaining material fracture toughness (K~c, J~c, and K~) based on the ASTM Standards and Procedures. It should be realized that for construction of a residual strength capability diagram for a given structure, using the linear elastic fracture mechanics approach, it is required to have the knowledge of the material fracture toughnes~ obtained from laboratory testing. Fracture toughness can link the cdtical crack length to the fracture stress. Whereas with the FMDM theory, the residual strength diagram can be obtained directly from an equation that describes the crack length as a function of applied stress for the crack geometry in consideration. Thus, there is no need to acquire material fracture toughness when this approach is employed. However, by having fracture stress and the corresponding Crack length, material fracture toughness can be obtained (see section 5.10). Therefore, expensive and time consuming procedures that are prerequisite to a

239

valid test, such as specimen preparation (machine starter slot, fatigue cracking the specimen prior to testing), the method of load application, load reading and crack measurement, etc., can be avoided.

5.2. Fracture Mechanics Of Ductile Metals (FMDM)

The FMDM theory assumes the fracture characteristics of a metal, local to the crack tip are directly related to its ability to deform. Crack tip straining from the FMDM position is of two kinds, local strainability at the crack tip, the region of highly plastic deformation and uniform strainability near the crack tip. Thus, fracture behavior can be characterized by two energy released terms representing the absorbed energy at and near the crack tip. Both terms can be shown to be determinable :from the full uniaxial stress-strain curve. The two deformation regions are illustrated in figure 2. The total energy per unit

Area associated with the start of necking up to fracture (highly strained region)

¢,~ P Uniform t,J~n- f / I straining urjR0rml A center crack

~trau~ha in a wide plate

, f

I v

STRAIN,

Region of u n i f o r m

-.__._~)~_._..__~a i ni ng ~ ..~ . f

strained L LEFM region Region

Figure 2: The crack tip plastic zone and different region of the stress- strain curve

thickness absorbed in plastic straining of the material around the crack tip, Up, can be written as:

Up = U F + U U 4

where UF and Uu are the energy absorbed per unit thickness in plastic straining of the material beyond the ultimate at the crack tip and below the ultimate stress near the crack tip, respectively. Equation 1, in terms of UF and Uu, described by equation 4, can be rewritten as:

240

~[UE- Us- UF - Uu]/~c = 0 5

where gl=auF and g2=aUu are the rates at which energy is absorbed ac ac

in plastic straining beyond the ultimate stress at the crack tip and below the ultimate stress near the crack tip, respectively.

To obtain the residual strength capability of a cracked plate, (the magnitude of an applied stress which corresponds to a given critical crack length), it is therefore necessary to determine the energy absorption rates for the two plastic regions formed around the crack tip (gl and g2). Sections 5.3 and 5.4 will discuss the theoretical approach and assumptions used in obtaining the auF and auu terms for the

ac ac plane stress condition. The plane strain and mixed mode conditions are discussed later in sections 5.5 and 5.7, respectively.

5.3. Determination of gl = aUFlac Term

To obtain the energy absorption rate for the highly strained region at the crack tip, gl=c~UF/ac, equation 5 in terms of its components can be rewritten as:

2 aU F + c~Uu =~ c = 2T+ 6

E ac ac

where T is the surface tension of the material, that is, the work done in breaking the atomic bonds and its value was given in chapter 2 [5] by:

aUs/aC = 2T = 13 E~ 7

where c~ is the atomic spacing in angstroms, E is the material modulus of elasticity, and 13 is the correction factor (3.94x10 -10 inch/angstrom). The quantity C~UF/C~C in equation 6, is set equal to WFhF, where WF is equal to the unrecoverable energy density (energy per unit volume) represented by the area under the plastic uniaxial engineering stress- strain curve from the stress at which necking begins to the stress at fracture for an uncracked tensile specimen (see figure 3 for different types of stress-strain curves). For a given alloy, the area under the stress-strain curve from the stress at which necking begins to the stress at fracture is a constant for a wide range of thicknesses, and therefore, the value of WF is a constant. It is assumed that essentially all of the energy represented by WF is absorbed at the crack tip in a single dominant coarse slip band. The quantity hF is the effective height of the highly strained portion of the small region at the crack tip

241

as a result of W F. Its minimum value is considered to be equal to the effective height of a coarse slip band oriented to make an angle of 45 °

W u = 0

and WF=0

I

STRAIN

t~

v

B I

/ I and I f I WF=0 I

! f

STRAIN

t/)

C

I WF=O STRAIN

I " u * " I " ~ I

r l I I I I t I~

STRAIN

F i g u r e 3: T y p i c a l s t r e s s - s t r a i n c u r v e s f o r d i f f e r e n t m a t e r i a l s

with the plane of the crack under monotonic load, see figure 4. The size of coarse slip bands at high stress and at room temperature is approximately 10 micrometers [6,7], and therefore:

hF = 0.000556 in. 8a

An empirical relationship based on test data [8,9] has been developed for tough metals with large necking strains (see, for example, case C of figure 3) that gives a higher hF value and better correlation with the test data than indicated by Equation 8a. It is assumed that the quantity hF is directly related to the energy per unit volume, WF, for the material in consideration, and was formulated as:

hF = ~'(E2/a%) WF 8b

where 7=(8/~)(:z.hmin, o" u is the material ultimate allowable stress, hm~,=0.000556 in. (see equation 8a) and (~ is material atomic spacing. The atomic spacing, (z, for Aluminum, Titanium alloys, and Iron and Nickel are 2.86 and 2.48 Angstrom, respectively. The quantity WF in equation 8b is the area under the stress strain curve from necking up to fracture (see figure 3) and is approximately equal to:

WF =~-UF s PN 9

242

where 5UF is the neck stress and its value is taken at the centroid of the piastic energy bounded on the top by the stress-strain curve from

Direction of

I

I ~ Monotonic ~., , , , . . , . , -~'~ I L o a d

Figure 4: The quantity hE is the effective height of the highly strained portion of the small region at the cra(~k tip as a result of WE

the beginning of necking to fracture and on the bottom by a straight line from the beginning of necking to fracture. For material with a negligible amount of necking the neck stress FUF = (cu, + OF)/2. The plastic straining, ePN' (equation 9) at the onset of necking is fully discussed in reference [9]. Equation 6 in terms of hF, FUF, and ePN can be expressed as:

~ o 2c aU U = 2 T + h F ~ - U F S P N + ~

E 8c 6a

In constructing the residual strength diagram for a given material, the fracture stress vades as a function of crack length. That is, the magnitude of applied stress increases as the critical crack length decreases. For fracture stress equal to the ultimate allowable stress of the material, Cru, the corresponding crack length is designated by c u, which can be viewed as a constant for a given material thickness. In this case, the contribution of the g2 = aUu/aC term becomes small and can be considered as negligible [7]. Equation 6a can be simplified as:

E 10 Cu = (2T + kh F ~-UF 8PN) 2

np.c U

243

where Cu is the material ultimate allowable stress and !~ and k are the correction for material thickness (see section 5.9). As was discussed in chapter 3, for structural life evaluation when the load environment is cyclic, the initial crack size assumption (the inherent flaw that pre- existed in the material prior to its usage) must be available in order to conduct a meaningful fracture mechanics analysis. Thus, the maximum flaw size associated with the residual strength equal to the material ultimate allowable stress can be obtained through equation 10 and the full stress-strain diagram for the material under consideration. The quantity c u is a function of material thickness. That is, the possibility of finding a larger flaw size to be present in a thicker material (having a larger volume) exists.

From equation 10, one can also notice that for material with no necking, where the quantity hF ~'uF cp, is small and negligible, the critical flaw size at ultimate is small. That is, brittle material with little or no plastic deformation at the crack tip, can tolerate much smaller initial flaw size at failure. Conversely, ductile metals will undergo an appreciable amount of straining and necking prior to their final failure, and thus can tolerate a much larger flaw size.

5.4 Determination of the g= = aUulac Term

The quantity aUu can be expressed as Wuh u, where W u is the ac

unrecoverable energy density represented by the area under the plastic uniaxial stress-strain curve from the elastic limit stress, ~L' to the ultimate allowable stress, c u, for an uncracked specimen. Equation 6a becomes:

2

E -~-~c _ 2T+hF 6UF 8 pN +Wuh u 6b

The effective height of the volume in which W u is absorbed is h u. The expression for aUu is derived in sections 5.4 and 5.5 independently

c~c from Wuh u terms for the two cases of plane stress and plane strain condition, respectively. Later in section 5.6, the two quantities W u and h u are derived separately. Steps taken to derive the energy rate term, g2, for the plane stress condition are as follows:

• Obtain the true crack tip stresses ((~Tf' CTe' and CT,e) in terms of applied stress and crack length in polar coordinate by using the linear elastic fracture mechanics. These stresses are responsible for material deformation at the crack tip region.

244

• Use the octahedral shear stress theory [5] to correlate the biaxial tension stress with the true uniaxial tension stress, aT.

• Apply the crack tip uniaxial tension strain, ST, in the energy absorption rate equation, to obtain the aUu term (or Wuhu).

ac

5.4.10ctahedral Shear Stress Theory (Plane Stress Conditions)

By applying the Linear Elastic Fracture Mechanics concept [10], the crack tip true stresses for plane stress conditions (aTz = O, XTez = 0 and tTzr = O) can be written in polar coordinates as:

aT, (r,e) = - ~ q-'p(e)

O'TO (r,e) = ~ - we(e ) 11

where: ~"/r(e) = (5cos0/2 +cos 30/2)/4

We (0) = (3cos 0/2 +cos 30/2)/4 12

W,o (0) = (sinO/2 +sin30/2)/4

In equation 11, it is assumed that the true stresses are approximately equal to the engineering stresses within the elastic range.

The equation for octahedral shear stress, tOOT, (also known as the strain energy of distortion) satisfactorily correlates states of biaxial tension stress (aTr and ate) with uniaxial tension stress, aT, and is used here to relate the stresses in the zone just ahead of the crack tip (mode I) to the uniaxial tension stress, aT [11]. The octahedral shear stress, TOCT, measures the intensity of stress that is responsible for bringing a solid substance into the plastic state. The Equation for octahedral shear, in terms of r and e, at the crack tip region is:

"COC T = 1/3 [(aT,- O'TO) 2"1" ( a T e - O'Tz) 2 + (~Tz " (~Tr) 2

+ 6 (XT~O 2 + "~T0Z 2 + 1;TZr2)] I/2 13

For p lane s t r e s s w h e r e arz = 0, XTOZ = 0 and XTZ, = 0 the quanti ty XOC T becomes:

] 1:OC T = ~ [((3"Tr- O'TO)2 + (]'T0 2 4" (3"Tr2+ 60"Tr02] I/2 14

245

- {~- , [1 - (aTdaT~) + (~To/~T,) 2 + 3 (~T~/~'r,) 2] ,~.~} 3

For simple uniaxial stress, ~T, where (~vo and (~v~ are zero:

15

1:OC T = o. T 3

16

Equating Toc T from equation 16 for uniaxial stress to "COC T (equation 15) for plane stress:

(3" T -~-. O'Tr [l - (O'TgO'Tr) -I- (O'T0 /O'T0 2 4. 3 (O'Tr~/G'Tr) 2] i/2 1 7

Equation 17, in terms of equation 11, can be simplified as:

O. T = O.Tr { l . [~jO(O)/I.IJr(0)] + [LIJo(O)/kiJr((~)] 2 4" 3 [~Jro(0)/l"I'/r(O)] 2} li2 18

and, (~t = Ctr {[~'/r2(O) "~Jr(0) LI/o(O) +LI/o2(0) +3 ~I'tro2(0)] '/2}/UX/r(0 ) 19

The quantity inside the bracket is a function of angle, e, and it can be set equal to:

• o~(0) = [~r2(0) -~P~(0) ~o(0) +%2(0) +3 LP,o2(0)] '/'- 20

Substituting equation 20 in equation 19:

o- T - O'Tr [~I~oc(0)/~IJr(0)] 21

The true crack tip uniaxial stress, aT, in terms of applied stress, cy, and crack length, c, by substituting for (~Tr from equation 11, can be written as:

Ug [ E_ ,1- %c(o)] (3" T =

V r 22

and in terms of strain, ~V, equation 22 becomes:

23

Equations 22 and 23 are derived from linear elastic fracture mechanics and define the variation of true stress and strain at the crack tip up to the elastic limit of the material, CTL. Within the limit of small scale yielding the quantity (~V can be obtained from equation 22

246

with reasonable accuracy. When an appreciable amount of straining occurs at the crack tip, the variation of uniform true straining, ~T, can also be described by equation 23, and it's variation with respect to crack tip distance (from rL to ru) is shown by Figure 5.

Region of uniform

----.._~L_..._~ ain i n g ~ - ~ -- Acenter crack ( ~ ~ . . ~

"2Y... I_ LEF. strained Region ST region

~ru ~T = {a (c/2r)0"51E} (Uniform straining)

~n.

A"~nter crack ru r (distance from crack tip)

Y r

rL

Figure 5: Uniform straining near the crack tip from limit up to the material ultimate

The quantity ~V, as described by equation 23, can be used to determine the rate at which plastic energy is absorbed in the larger zone associated with uniform straining. An element inside this zone that can contribute to energy absorption rate as the crack grows is shown in figure 6. The quantity g2 = 0Uu/0C can be written as:

OC O~ TP TP rdrde =2fo e,' I'~ua T - - r d r d e 24 g2 = aUulOc = f(A) (~ T OC OC

where ec is the angle at which the plastic energy absorption rate vanishes, i.e. the angle at which no contribution to plastic deformation is made at the crack tip. Replacing the total strain expressed via Equation 23 into Equation 24 (by assuming the total strain is approximately equal to the plastic strain, ~T ~ ~TP ):

~U U ~ T • " = 2 I~" j'~l. o T rdrde 25 Oc ,u Oc

247

When the magnitude of the applied stress is below the material yielding with no net section yielding, the limits of the integral described by equation 25 are correct. However, when the applied stress is in

/ • CrTr 0 aTr

/ c3r d c ~ ' ~ . a f l dc

o:o_ x

Figure 6: Illustration of an element, dA, chosen in the plastic zone for analysis

such a way that net section yielding occurs in the material, the limit of the integral in equation 25 can be rewritten as:

aU u O~T = 2 .l'00c I r

ac r u ~ T ac rdrd0 26

where r can be smaller than rL. Note that when the applied stress becomes larger than the elastic limit stress of the material, the plastic deformation contribution at the crack tip due to the energy absorption rate term associated with the uniform plastic region (shown in figure 5) is less and only a portion of it is included in Equation 26. Throughout this chapter the magnitude of the applied stress is in all cases below the elastic limit of the material. The reader should note that one of the limitations associated with the LEFM is that the formation of the crack tip plasticity must be limited to the small region at the crack tip and the bulk of the material must be elastic. Otherwise, the LEFM can not be utilized to assess the residual strength capability of a part when large scale yielding takes place. With the present approach, large scale yielding at the crack tip can be assessed by the two energy dissipation terms (gl and g2), determined by applying the extended Griffith approach.

248

From Equation 25, the quantity can be simplified as follows: ~ T 8c

c3s T o "t'oc (O ) ~ ' e oc (0) Or o -2,5E oc

+ + - -

Oc ac o 27

Under constant load conditions, the stress would be constant as the aa

crack begins to grow catastrophically. In this case, - - term would be 0c

zero and the last quantity in equation 27, &r e T would vanish. i

Oc a From simplification of equation 27:

as T 1 - S T [ - -

ac 2c

- - Woc (0) 1 8 r ~ a O

] 28 2r ~c U/oc(O) ac

ar ~0 From figure 6, the quantities - - and - -

0c 8c can be replaced by:

a r a0 sin0 - - = - cos0, and - - - ac ac r

Substituting equation 29 in equation 28:

29

& T ST

¢9c 2r

But, r/c <<1, therefore,

de T

ac

0 - - q 'oc ( 0 )

r - - [-- + cosO + 2 sinO

c ~ o c (0 )

S T - - [ cos0 + 2 sin0 2r

0 - - ~e oc (0)

] Woc (e )

30

31

249

Let:

f(O)= [cosO +2sin0 - - "r'oc ( 0 )

q"oc (O ) 32

Substituting equation 32 in equation 31"

~ T _ ST f(O) ac 2r

33

Substituting equation 33 in equation 25, the energy absorption rate near the crack tip can be written as:

aU u oc rL E--T-T f(o) rdrd0 =2 I" J a T 2r

a C 0 r u

0 c r L

J" f a T c T f(0) drd0 o r U

34

34a

For metals whose true-stress vs. true plastic-strain curve can be approximated by the Ramberg-Osgood equation (when fitted at a

TL

and aTU), the true stress in terms of true plastic strain can be written a s :

I t;

aT =OTU [ TP ]n

TPU

35

The quantity e is the true plastic strain at ultimate and n is the strain TPU

hardening coefficient (for perfectly plastic material, the value of n= oo) that can be described by:

In(TPU) TPL n = 36

I n ( T U ) G TL

250

where ~TPL is the true plastic strain at the elastic limit. In the region of appreciable plastic strain, where ~TP = ST, equation 35 can be rewritten a s :

]

~ T=(~ TU[ s T ]n TU

37

Substituting equation 37 in equation 34:

n+l c~Uu Oc rL ~: T n

' =OTU~TU .1" I [ ] ac 0 r u E TU

f (0) dr dO 38

From equation 23, the ratio of the quantity ST/CTU can be replaced by its equivalent:

TU 39

Substituting equation 39 in equation 38:

n+l ~U U % rL

=~ TU ~ TU J" J" [ r u ] 2n f(O) drd0 40 ~ C o r u C

Integrating:

~U u

ac

n-1 eJ" 2n-1 U r ~u - C T U sTu ---:r [ ( - - ) 2 n ] f (0) dO on ru

n-1 aU U n o ,. - - - - - o TU~TU [( rL ) 2n -1] J" 2r U f(0) dO 41

~c n -1 r u o

From test data for plane-stress and infinite width, an empirical relationship has been established that can describe the r u value for ductile metals [12]:

251

G 2ru =Q ( 1 ) [1-( T ) ]n+l Wo2c(0)

CT c TU

42

where Q= h(E;TF E:TL)/E.ru (the value of h=0.000556 in., see equation 8. For finite width plate, W, the correction factor for equation 42 is as follows:

Q 1 cr n+ l T )] ~2c (O) 2r u y~.,,, ( ) [1- ( ~T G TU

42a

where the width correction, Y, for a center cracked panel of length 2c is expressed as [13]:

1 Y = [ 1 ]

~ 2 c ( c o s - - - )

2 W

Substituting equation 42 in Equation 41 and noting that the ratio of

(rL/ru)1/2= ~rUFTL "

aU U n ° 'T n+ l e TFE TL I , - - - - - c r T U eTu [ 1 - ( ) ] h [ J

ac n-1 a TU E TU G T 43

n-1 0 c TU n qJ2.. (0) [( ) - l ] f ,,~ f(0) dO TL 0

By numerical evaluation, the thickness parameter, 13, for plane-stress is:

0¢.

13 : J" ~Jo2c(O) f(O) dO = 1.3 o

44

The thickness parameter term, 13, is used here to show the distinction

between the energy absorption rate, dULl, defined by equations 43 8c

and 50 (defined in section 5.5) for the two conditions of plane strain

252

and plane stress, respectively, (see equations 44 and 51a which describe the 13 values).

5.5. Octahedral Shear Stress Theory (Plane Strain Conditions)

Using the crack tip stresses described by Irwin for the plane strain condition ( see equation 11) where:

aTZ = (aTr +aT0), aT0Z = 0, and aT~Z =0

The octahedral shea r s t ress for the plane strain condition (see equation 14 for the plane s t ress case) becomes :

"~OCT = 1/3 [(aT,- ate)2 + (aTe " V a T r" MaTe) 2 + (aTz - aTr) 2

+ 6 "l;2Tr o + (aTr -- V a T r - V a T)2] 4 5

or :

XOC T = (~/2/3) [(1-v+v2)(a2rr + a2T0) -- (1 +2V--2V 2) aTr aTO+3X2TrO 45a

For plastic straining, v= 0.5, then:

Xoc T = ('V213)[(314)(a2Tr + a2ro) -- (3•2) aT0/aT, +3(XTJaT~) 211/2

or :

xOCT=(q2/3)aT~ [(3/4)(I+ a2TO/a2Tr ) -- (3/2) ate/aTr +3(XTJaTr) 2] lt2 45b

As before, equating equation 45b to the simple uniaxial stress, where aTO=0 and ) amr e =0:

or:

and

Let:

aT= aTr [(3/4)(1 + a2Te/a2Tr ) -- (3•2) ate/aTr "l'3('~T4aTr) 2] 112 46

(::~r-'aTr {(3/4)[1 +~'/2O(O)/~/2r(O)] -(3/2)[W0(0)/~,(0)]+3 ['~,e(O)/~,(O)] 2} '/~ 46a

(~t=[at,/q',(O)] {(3/4)[ q~2r(O)+qao(O)]-- 3/2[q%(O)~Vr(O)]+3[q',)(O]2} 'a 47

q~oc(0) = {(3/4)[ q~2,(O)+q'2o(0)]- 3/2[q~o(O)q'r(0)]+3[q~m(0] 2 } '/~ 48

253

For the plane strain condition, the energy absorption rate in the larger zone can be simplified the same way as was done for the plane stress condition, see equation 25. The quantity 2r u from test data for plane- strain and infinite width:

(IT 2r u =hQ( I ) [ ]_( )n+112 2(0 ) s T OTU x

49a

For a finite width plate, a correction factor, Y, for equation 49a is introduced (similar to the linear elastic fracture mechanics) for a center crack panel (see equation 42a):

°T )n+l 2r U = Q ( ~ i )[1-( ]2q j2(0) Y ~T °TU n

49b

The energy absorption rate in the larger zone, c3Uu/aC, in its final form for infinite width is given by:

aU u n

ac n+ l cr ~ TF c TL.], cr TUSTU [1_ ( T ) n + l ] h [

cr TU c TU ~ T

n-1 4Cl

TU n 2 "[c [( ) -I]---~-_ ~o2c(e)f(O) de 50 TL x 0

By numerical evaluation, the thickness parameter, 13, for plane-strain is condition (note that this value is different from the plane stress case discussed before):

2 ( o ) f ( o ) d o =0. 27 ~ = 2

g o

51a

Note that the value of 13 for a thickness, t, greater than the maximum thickness for plane stress (to) but less than the thickness for plane- strain (this condition is called mixed mode when the value of 13 falls between the plane stress and plane strain conditions) is empirically formulated by:

t n t - t n = [,.3 t ÷ 51b

254

5.6 Applied Stress, a, and Half Crack Length, c, Relationship

5.6.1 Determination of W u and h u Terms Separately

The quantity W u in equation 10a can be evaluated from the area

under the stress-strain curve taken from the stress equal to the limit stress to the ultimate of the material:

oTU

W u = J" aTd~Tp 52 aTL

From equation 37:

~TP = 8TPU[ cT ]n 53 aTU

Taking the derivative:

da T dSTp = n~TPU[ aT ]n-1

~TU aTU

Substituting equation 54 in equation 52:

~TU a T W U = I n~TPU(aTU)nd:~T

e~TL

Integrating:

54

55

n °T )n + lc~ laTu 56 WU = n + 1CTPU(aTU TUJ~.

WU = n (~, TL)n+ 1] n + 18TPU~TU[I - aTU

57

Replacing quantities shown in equations 57 by Wu into equation 50:

n-1 c~Uu n + 1WU h [.e TF 8 TL ],[(8 TU ) n 1113 c~c n - 1 ~ TU 8 T ~ TL

58

255

where 13 is given by equation 50. The quantity hu defining the height of the plastic deformation in the uniform strain region near the crack tip can be formulated as follows:

aU U ac - wu hu 59

From equation 59 and equation 58:

n-1 n + I c TF c TL ~ TU n h u - h [. ]*[( ) n - 1 c TU c T ~ TL

-I]13 60

5.6.2 Applied Stress and Crack Length Relationship

From equation 6, we can write:

no 2c =2T + WFh F + Wuh U

E

E c = 2 [2T +WFhF +Wuhu] 61

For a thin plate (plane stress conditions, 13 = 1.3) of infinite width:

WF =OTUF[~TPF -~TPU]=OUF~PN 62

Substituting equations 57, 60, and 62 into equation 61, the fracture stress, c, corresponding to a half crack length, c, in a wide plate of infinite width (where 2c >> W), under plane-stress condition can be written as:

C = ~ E

2 n o r8 TF t: TL {2T+8-UF~pNhF+ ~TU~TU[I_( T )n+ l ] h 1 L

n-1 cr TU E TU ~ T

n-1 c TU n

[( ) - l ] l~ } E TL

63

For mixed mode fracture where material thickness fall between the plane strain and plane stress conditions, equation 63 for wide plate becomes:

256

E C ~ - -

na 2p.

n ~ T n+ l f ETF 8 TL {2T+5UF~pNhFk + c TU ~ TU [1 - ( ) ] h , ]

n-1 ~ TU ~ TU ~ T

n-1 TU n [( ) -1 ]p}

¢ T L

and for a finite width condition:

64

C - E

=~ 2~y2 {2T + 5UFCpNhFk + - - a T ) n + l c TF ¢ "1" n cTUETU [ !_ ( ] h [ ~

n-1 y6 ~TU STUC"

n-1 TU n [( )

c T L

-1]~}

64a

(See section 5.7 for a detailed description of the thickness parameters 13, k and I~).

Equations 63 (plane stress) and 64 (mixed mode) simply state that, for a given crack length 2c, the corresponding fracture stress, a, can be easily computed, provided that the full stress-strain curve for the material under study is available. The plot of fracture stress versus half crack length, c, is called the residual strength capability diagram and is extremely important in the field of fracture mechanics. From the residual strength data obtained from equation 64, the material resistance to fracture for different thicknesses and crack lengths can be computed.

5.7 Mixed Mode Fracture and Thickness Parameters

Expressions for k, 13and p. for mixed mode fracture (shown in equation 64) are described as follows:

A shear lip is formed in plane-stress fracture, whereas plane-strain fracture is characterized by a flat fracture surface. It is assumed that the energies released and absorbed in plane-stress and plane-strain are in proportion to these thicknesses, as shown in figure 7.

Let to be the maximum thickness for plane-stress fracture. The thickness corrections are applied to two strained regions at the crack tip. The thickness parameters, k and 13, are used for the two highly

257

Shear lip Flat surface surface (plane strain (plane s t r e s s ) ~ ~

Figure 7: Shear lip and flat fracture surfaces

strained regions at the crack tip and near crack tip respectively (see equation 6). Moreover, the thickness correction factor for available energy is designated by ~t which can be express as follow:

~l~c 2c - 2T + WFhFk + E Wuhul3 65

where the two quantities k and ~t are defined as:

t - t 0 k = [--~- + 2(--~---)] 66

and:

2 t - t0 = [ + ( l - v 67

The minimum values of k and ~ are equal to unity when t=t o. This is associated with the maximum plane stress condition where complete shear lip prevails. Furthermore, the thickness parameter, 13, for the mixed mode condition in the uniform strained region near the crack tip, was described by equation 51b, and its maximum and minimum values are1.3 and 0.127 for the plane stress and plane strain conditions, respectively.

258

5.8. The Stress-Strain Curve

The short-time uniaxial full-range tension test is used to determine the mechanical properties of structural metals. Information obtained from this test can be used to calculate the fracture properties of structural metals. As was shown in Figure 3, a typical stress-strain curve for a ductile metal shows that the metal undergoes initial elastic straining, then uniform plastic straining and finally local plastic straining (necking). The stress-strain curve in Figure 3A is typical of brittle material, such as glass or high strength steels, where ~YJe~d = Cu,. These materials have low resistance to fracture when the two quantities WFhF and Wuhu are zero. The curve in figure 3B is representative of metals which do not "neck", such as cross-rolled beryllium. These alloys generally have lower resistance to fracture (WFhF =0) as compared to material that exhibits necking. The curve in Figure 3C is typical of many structural metals~ and the curve in Figure 3D, which shows very little uniform plastic straining, is typical of some precipitation hardened steels.

In a uniaxial tension test, strains are obtained from deflections measured by a gage that is usually longer than the necked portion of the test specimen. The typical gage length employed for most uniaxial tension tests has a 2 in. length. This average strain must be corrected to obtain the actual local strain in that part of the test specimen. The correction can be made by multiplying the local average strain by the length of the gage and dividing by the thickness, t s, of the tensile test specimen. The correction for the actual true "neck" strain and the procedure to obtain the true "neck" strain, epn, was discussed in [9],

Typical full-range uniaxial tensile stress-strain curves should be obtained from reputable and reliable sources. If such curves are not available, five or six full-range tensile stress-strain curves should be obtained through testing by applying the ASTM-E8 Standards and Procedures. Either flat or round test specimens are suitable. Typical stress-strain curves for many aerospace alloys can be found in MIL- HDBK-5, Metallic Materials and Elements for Aerospace Vehicle Structures.

5.9 Verification of FMDM Results with the Experimental Data

The following are the calculated residual strength capability data provided by the FMDM theory for several aerospace alloys compared with test data gathered from several reliable sources. Both finite and infinite cracked plate cases were used to check the analysis. In chapter 2, the ASTM standard procedures for determination of fracture toughness, including standard specimen preparation, surface finish, and pre-fatigue cracking the specimen, as well as data gathering, are briefly discussed. In section 5.10, the FMDM approach for determination of fracture toughness for 2219-T87 and 7075-T73

259

aluminum alloys are discussed and the calculated results are compared with the test data obtained through the ASTM procedure.

To employ the FMDM theory for constructing the residual strength diagram (fracture stress as a function of half critical crack length), it is necessary to have the full range stress-strain curve for the material under study. The full range stress-strain curve must be obtained from reliable sources. The results of the analysis conducted by the FMDM theory indicated exceptionally good agreement with experimental data. In figure 8, the stress-strain curve for 2219-T87 aluminum alloy (with plate thickness ts=0.125 in.) is plotted from MIL-HDBK-5. The pertinent information is extracted from this curve as input to the FMDM equation to obtain the variation of fracture stress as a function of half critical crack lengtl~, as shown in figure 9. The experimental data plotted in figure 9 was extracted from reference [14]. Excellent correlation between the test data and the FMDM results were observed. As another example, HP-9NI-4CO-0.2C Steel [15,16] was selected and information needed for generating a residual strength diagram was taken from the stress-strain curve shown in figure 10. Figure 11 is the constructed residual strength capability curve that describes the variation of the fracture stress as a function of half critical crack length. Good agreement between the test data and the FMDM result can be seen. Additional data that can validate the results of the FMDM analysis are available in reference [8].

1211 I',

ilo j/ /:: o I I

S t ra in ~ , I n / i n .

Figure 8: The stress-strain curve for 2219oT87 aluminum alloy

260

'200 F r a c t u r e S t r e s s , 2 2 1 9 - T 8 7 A l u m i n u m A l l o y

1 ~ U l t i m a t e S t r e s s

6 0

50 40

• Test Data 30

2 0

I n f i n i t e w i d t h

0.02 0.03 "0.04 0.1 0.2 0.3 0.4 1.0 2 3 4 5

C r i t i c a l Ha l f C rack Leng th , c, I nch

Figure 9: Residual strength diagram for 2219-T87 aluminum alloy

00"~T]'eS$ . . . .

8 0 [ - [ Forglng, R0om 0=0.5 inch, RAm 0.667

60

140

20

00

80

60

40

20

Stress-Sb'ain Curve, HP 9N1-4CO-0.20C Steel

Strain, s, Inlln . - ~

0.0 0.02 0.04 0.06 0.08 0.10 0.12 0.14

Figure 10: The stress-strain curve for HP-9NI-4CO-0.2C steel

261

30¢

200

100

*,00

200

Applied Stress,

Fracture Stress, H.P. 9NI-4C0-0.20C Steel

Ultimate Strength

Infinite Width

• Test Data

o.o2 0.03 o.o4 o.os 0.1 o.2 0.3 o.4 o.5 1.0 2.0

Initial Half Crack Length, c, Inch.

Figure 11: Residual strength diagram for HP-9NI-4CO-0.2C steel

5.10: Fracture Toughness Computation by the FMDM Theory

5.10.1 Introduction

As mentioned previously, plane stress fracture toughness testing is costly and time consuming because of it's dependency on the material thickness and the crack length. This dependency necessitates having standard specimens with different thicknesses that must be prepared based on the ASTM procedures. The specimen preparation for fracture toughness testing includes the machine starter slot, fatigue cracking the specimens prior to testing, method of applying 10ad to the specimen, load recording, and crack length measurements, all of which are elements that can contribute to a costly and time-consuming process. The following scenario, which can often happen in the aircraft and aerospace industries, clearly illustrates the need for knowing the variation in fracture toughness as a function of material thickness. For example, consider a situation in which the fracture mechanics analysis of a component of an aircraft structure shows that a part can not survive the load varying environment. Thus, there is a need to increase the part thickness, tl, in order to reduce the applied stress, (~. A new fracture mechanics analysis of the part with increased thickness (t2 > tl) requires a corrected fracture toughness value that corresponds to the new assigned thickness, t2. With the FMDM theory, the Kc value can easily be computed for the material, provided that the material

262

stress-strain curve from reliable sources, such as MIL-HDBK-5, is available to the analyst.

Equation 64 provides the variation in fracture stress, c, with respect to the crack length, c. Incorporating the fracture stress, a, and half critical crack length, c, into the crack tip stress intensity factor equation (for a center crack panel in an infinite plate or wide plate), the obtained K value will correspond to the fracture toughness. The stress intensity factor equation for a cracked plate having thickness t and width W (no net section is allowed) can be written as:

K c = c (~c sec (=c/W)) 1/2 where (~< (~Yi,J~ 68

The analyst must remember that the computed value of the fracture stress from the FMDM theory should not fall above the material yield value nor is net section yielding allowed when using equation 68 for the fracture toughness computation. That is:

((~c)net-section= (~c [W/(W-2c)] < (~Y~,~d 69

Sections 5.10.2 and 5.10.3 contain example problems pertaining to computation of the fracture toughness by the FMDM theory for 2219- T87 and 7075-'1"73 aluminum alloys which have stress-strain curves taken directly from MIL-HDBK-5.

5.10.2: Fracture Toughness Evaluation for 2219-T87 Aluminum Alloy

The stress-strain curve for 2219-T87 aluminum alloy (sheet and plate, longitudinal direction, room temperature) is shown in figure 8. The FMDM residual strength capability data generated by the FMDM computer code (see equation 64) is plotted in figures 12 through 15 for plate thicknesses of t= 0.1, 0.2, 0.4 and 0.8 inches, respectively. Using equation 68, the variation in calculated fracture toughness, Kc, with respect to material thickness (t=0.1, 0.2, 0.4, 0.8, 1.0, 1.4 and 2.0 inches) can be plotted (figure 16). The FMDM residual strength capability diagram for other thicknesses (t=l.0, 1.4, and 2.0 inches) are not shown here, however they are included in figure 16 for fracture toughness plot.

Because fracture toughness is a function of crack length, two arbitrary values of crack length were chosen. One is associated with a fracture stress cc =80% of ay,e~d, and the other one with half crack length c=1.5 inches. Note that, the total crack length, 2c=3.0 inches, was used in the analysis in order to correlate with the test data generated by reference [17] (see figure 17). Two other intermediate crack lengths were also used and the corresponding fracture toughnesses are shown in figure 16. The reader should remember that the larger the original crack

263

100

m 10

"g

<

~ ._ Inf inite

• W= 20

W= 3 W= 6 W= 12

2219-T87 (t'=- 0.1 Inch)

W = Width of The Plate

1 I I I I 0.01 0.1 1 10 100

Half Crack Length, inch

Figure 12: The residual s t rength capabi l i ty d iagram for 2219-T87 generated by the FMDM computer code ( th ickness, t=0.1 in.)

U) "o ._m D. Q. <

100

10

l n f i n i t e

0

2219-T87 (t= 0.2 Inch) W= 6 W= 12

W = Width of The Plate

1 I I I I 0.01 0.1 1 10 100

Half Crack Length, inch

Figure 13: The residual s t rength capabi l i ty d iagram for 2219-T87 generated by the FMDM computer code ( thickness, t=0.2 in.)

264

"3 =¢ W .= 4.1

1= Q

: = O. o. <

100

10

Infinite Plate

• ¢

.

W:- 6 W: 12 2219-T87 (t=- 0.4 inch)

W : Width of The Plate

1 I I I I 0.01 0.1 1 10 100

Ha l f C rack Length , inch

Figure 14: The residual strength capability diagram for 2219-T87 generated by the FMDM computer code (thickness, t=0.4 in.)

m

.=

<

100

10

lnflnite

W= 12 ~ 6

2219-T87 It= 0.8 inch)

W : 20

W : Width of The Plate

1 I I I I 0.01 0.1 1 10 100

Hal f Crack Length, inch

Figure 15: The res idual s t reng th capabi l i ty d iagram for 2219-T87 generated by the FMDM compu te r code ( th ickness, t=0.8 in.)

265

80" • 2 2 1 9 - T 8 7 A l u m i n u m A l l o y

70 • • • A

"~ 60 • • • • o • •

: ~ 5 0 • • • & •

4 0 • • o • • = • i 30 • x • =

2 o . • ---, M i n i m u m C r a c k Length O, • --~ Intermediate Crack Length

lo • 41, --, Max Crack Length (2c=3.0 inches) u. 0 I I

0.5 1

• • • • •

• • • • •

I I

1.5 2

Ma te r i a l T h i c k n e s s , t, (Inch)

Figure 16: Variation of material fracture toughness vs. thickness for 2219- T87 Aluminum alloy

70 - -

6 0 -

I ,o_

" i 40 --

3 0 -

2 0 -

0.0

[ ] [] 2219 - 1"87 Aluminum Alloy, L-T~ 75 F M(T) Specimen Data

• [ ]

• ~ (a Is 112 c r a c k l e n g t h )

• Curve fit for a = 0.45 to 0.68 inch • o a = 0.24 to 0.38 inch

• a = 0.45 to 0.68 inch • a = 1.00 to 1.20 inch

~ = [] a = 1.50 to 1.80 inch

K0c

T h i c k n e s s , t (inch)

I I I I I I I I

0.125 0.25 0.375 0.5 0.625 0.75 0.875 1.0 1.125

Figure 17: Variation of Kc as a function of thickness for 2219-T87 aluminum

length, the higher is the computed fracture toughness value, Kc. Therefore, the minimum fracture toughness is expected to be associated with a half critical crack length having fracture stress of ~c =80% of (~Y~eJd. A minimum fracture toughness value, corresponding to the minimum crack length, where Cc =0.8x~y~e~d, gives a conservative result when

266

conducting crack growth rate analysis of a structural component. In most real cases, the initial crack length assumption in the structure is small, so that the minimum fracture toughness is an acceptable value to use in the analysis. Also note that the fracture toughness values shown in Appendix A are the lower bound K~ values based on minimum initial crack size (the apparent fracture toughness). The fracture stress corresponding to the minimum crack size should not cause plastic deformation in the bulk of the test specimen.

Figure 17 shows the variation of the experimentally obtained fracture toughness, K~, vs. material thickness for 2219-T87 aluminum alloy taken from Reference [17]. Fracture toughness curves shown in figure 17 correspond to half crack lengths c=0.24 to 0.38, 0.45 to 0.68, 1.0 to 1.2 and 1.5 to 1.8 inches. Both figures (16 and 17) clearly illustrate that the fracture toughness value increases with decreasing thickness and, moreover, that Kc is a function of crack length, c. Once again, fracture toughness values calculated by the FMDM theory are in good agreement with the test data.

5.10.3: Fracture Toughness Evaluation for 7075-T73 Aluminum Alloy

The pertinent information extracted from the stress-strain curve, from computer output (figure 18) for 7075-T73 aluminum alloy (sheet and plate, longitudinal direction at room temperature) . These data were used to construct the FMDM residual strength capability curve. Using the FMDM Computer code the residual strength diagrams were plotted in figures 19 through 22 for different plate thicknesses t= 0.1, 0.2,0.4 and 0.8 inches, respectively. Equation 68 was utilized to calculate the material fracture toughness and to establish the variation of K c, vs. material thicknesses (t=0.1, 0.2, 0.4, 0.8, 1.0, 1.4 and 2.0 inches) for different crack lengths, 2c (figure 23). No attempt was made to include all the residual strength capability diagrams for the thicknesses that were used to generate figure 23.

A minimum fracture toughness value corresponding to the minimum crack length is assigned to a fracture stress equal to 80% of the material yield strength. Also, a half crack length c=1.5 in. was selected as the upper limit of the fracture toughness (two other intermediate crack lengths were also used and the corresponding fracture toughnesses are shown in figure 23). Please note that the upper limit of the fracture toughness was arbitrarily chosen to be associated with a half critical crack length, c=1.5 in. A higher fracture toughness value for this material can be obtain by using a larger crack length. The fracture toughness dependency of 7075-1"73 aluminum alloy on the crack length and the material thickness is also shown in figure 24, in which data points were

267

Aluminum Alloy, 7075-T73, Plate, L-T Dir., Room Temp. The following mechanical properties were obtained from fiat tensile test bars.

A~m. Spacing. g l t S~. YId. S~. Rup. S~. Neck S~.

2.860 83.000 73.000 75.500 80.900

Poisn. R. UIt Strn. Rup. Strn. Thk. TsL Sp. Gage L.

0.3300 0.0880 0.12 0.0625 2.000

Computed Basic Data

t o WF EPN STU STF 0.146 68.82 0.850 90.037 152.18

Atomic spacing is in Angstroms;

E

10300.00

I! 17.89

stresses, modulus and WF are in ksi; 0t, thicknesses, gage length, widths and crack sizes are in inches.

Neck Str. is the average stress from the beginning of necking to rupture of the tensile test specimen.

UIt. Strn. is the strain at the beginning of necking.

t o is the maximum thickness of a through cracked test specimen for plane- stress fracture.

WF is the density of plastic energy under the stress-strain curve from strair at the beginning of necking to strain at rupture.

EPN is the corrected neck uniaxial plastic tensile strain. (corrected from gage length).

STU is the uniaxial true ultimate tensile stress.

STF is the uniaxial true rupture tensile stress.

n is the exponent in the Ramberg-Osgood relation for uniaxial true plastic tensile strain.

Str. is the computed gross area fracture stress.

Figure 18: Portion of the FMDM computer output for 7075-T63

268

w" ¢) 2 u) "o =~ o. Q. <C

100

10

Thickness t=.l inch

7075-T63 Infinite

~ , ~ ~ ~ t w : 16 i." ~, e

1 I I I I 0.01 0.1 I 10 100

Half Crack Length, inch

Figure 19: The residual strength capability diagram for 7075-T63 generated by the FMDM computer code (thickness, t--0.1 in.)

100

<

Thickness t=.2 Inch

Inf inite Plate

I I I l

0.01 0.1 1 10 100

Half Crack Length, inch

Figure 20: The residual strength capability diagram for 7075-T63 generated by the FMDM computer code (thickness, t=0.2 in.)

269

J<

U) "O

Q . ,<

100

10

Thickness t=.4 inch

Infinite Plate

W = 16.0 In.

1 I I I I 0.01 0.1 1 10 100

Half Crack Length, inch

F igure 21: The residual s t rength capabi l i ty d iagram for 7075-T63 generated by the FMDM compute r code ( th ickness, t=0.4 in.)

,=¢

(n

v., U l "O ._o Q. <

100

10

Thickness t=-.8 inch

Infinite Plate

W = 16.0 In.

1 1 I I I 0.01 0.1 1 10 100

Half Crack Length, inch

Figure 22: The residual s t rength capabi l i ty d iagram for 7075-T63 generated by the FMDM compute r code ( th ickness, t=0.8 in.)

270

120"

100"

;80-

:60-

~40-

20-

0 0

• • •

• -~ Minimum Crack Length 0, • -~ Intermediate Crack Length • -, Max Crack Length (2c=3.0 inches)

I I I I I I 0.2 0.4 0.6 0.8 1 1.2

M a t e r i a l T h i c k n e s s , t ( I n c h e s )

Figure 23: Variation of material fracture toughness vs. th ickness for 7075- T63 A luminum alloy

120

tOlD-

BO-

40.

ZO-

O 0

| I I | I I I

O O

O

* : , ° ,** .

0 0 0

I ~..~ ,~. pNiC

• ~ Cstlf)

~a • ~ CdaSdls) 0141 • 4-~I ¢aln

Rke I . T 5 to h e I . = S I

KIo- 3 I . M Yll° 6~ , le I

I a'2 e:+ e'+ e:e

7075-1"73 Aluminum Alloy, L-T, 75F

i I I .; I l . e 1 . 2 1 . 4 I S I . B 2 . e

K~

Mnterlal Th loknNs , t ( inches)

Figure 24: Variation of Kc as a function of thickness for 7075-T63 aluminum

271

obtained through laboratory tests conducted in accordance with the ASTM Standards. Both figures (23 and 24) clearly illustrate that the fracture toughness is dependent on the material thickness (decreases as thickness increases) and the crack length, 2c, (increases as crack length increases).

References

1. G. R. Irwin, "Fracture Dynamics" Fracture of Metals, ASM, (1948), PP. 147-165.

2. E. Orowan, "Fra~:ture and Strength of Solids," Rep. Prog. Physics, Vol. 12, (1949), pp. 185-232

3. A. A. Griffith, "The Phenomena of Rupture and Flow in Solids," Philos. Trans., R. Soc. Lond., Ser. A., Vol. 221, 1920, p. 163.

4. J. A. Begley, and J. D. Landes, "The J Integral as a Fracture Criterion," Fracture Toughness, ASTM STP 415, 1971, pp. 1-23.

5. A. S. Tetelman, and A. J. McEvily, JR., "Fracture of Structural Materials", John Wiley& Sons Inc., 1967, pp 39-48.

6. A. H. Cottrell. Dislocations and Plastic Flow in Crystals. Oxford, at the Claredon Press, 1958, p. 3.

7. B. L. Avenback, D. K. Felbeck, G.T. Hahn, and D.A. Thomas, "Fracture," The MIT Press, 1959, pp. 1-66

8. G. E. Bockrath and J. B. Glassco, "A Theory of Ductile Fracture", McDonnell Douglas Astronautics Company, Report MDC G2895, August 1972, Revised April 1975.

9. B. Farahmand, "Fatigue and Fracture Mechanics of High Risk Parts,"Chapman and Hall, 1997, Appendix B

10. G. R. Irwin, "Analysis of Stresses and Strains Near the End of a Crack Traversing a Plate," Trans. ASME, J. Appl. Mech.Vol. 24, 1957, p. 361.

11. A. Nadai. Plasticity. McGraw-Hill Company, 1960, pp 103, 260.

12. J. B. Glassco, B-1 Fracture Analysis by Ductile Fracture Method, Rockwell International Report TFD-76-1368-L, 1977.

13. C. E. Feddersen, Discussion, ASTM STP 410, 1967, pp. 77-79

272

14. B. Farahmand, G.E. Bockrath, "A Theoretical Evaluating The Plane Strain Fracture Toughness Journal Of Fracture Mechanics, March 1996.

Approach For " Engineering

15. B. Farahmand, "A New Analytical Approach To obtain The Plane Stress Fracture Toughness Value By The FMDM Theory "The Fourth Pan American Congress of Applied Mechanics, Buenos Aires, Argantina, January 5-9, 1995.

16. B. Farahmand, "Fatigue and Fracture Mechanics of High Risk Parts,"Chapman and Hall, 1997, pp. 389-312

17. Fatigue Crack Growth Computer Program "NASNFLAGRO", Developed by R. G ~" Forman, V. Shivakumar, J . C. Newman. JSC- 22267A, January 1993.

273

By: B. Farahmand, Boeing Technical Fellow and D. Waldron, Boeing Technical Associated Fellow

Chapter 6

WELDED JOINTS AND APPLICATIONS

6.0 Introduction

Aluminum alloys have been the mainstay of aerospace hardware since the 1960's because of their high strength-to-weight ratio. A prime aerospace and aircraft industry goal has always been to provide the necessary structural strength at a minimum weight, thus the drive is to achieve the maximum strength-to-weight ratio. From well established and published static and fracture properties, one can pick an aluminum alloy that can provide highest mechanical allowables, favorable fracture toughness, and adequate fatigue crack growth resistance when subjected to load varying environment. In addition, the alloy must have the best performance when exposed to other environments, such as a high rate of loading, extreme operational temperatures, including cryogenic or elevated temperatures. Engineers must also consider the feasibility of fabricating the hardware in the material selection process.

In manufacturing space or aircraft structures, it is common practice that pieces of structure are mated together in some manner strong enough to withstand the load environment throughout their service usage. There are several types of joints, which allow the transfer of load from one segment of a structure to another. Bolted and riveted joints have already been presented in chapter 7 with several practical example problems related to life determination of jointed parts. It was pointed out in chapter 7 that a fastener joint has inherent stress concentration sites that are the prime locations for fatigue failure where cracks can initiate either from the threaded region or from the periphery of the bolted hole. Welding is another acceptable technique for joining structural parts, which if done properly under a tightly controlled environment, can yield almost the same fatigue properties as the parent material. On the other hand, a poorly welded joint with

an unacceptable amount of porosity, shrinkage, cavities, or incomplete fusion can be the source of crack initiation and failure of the part when subjected to a fluctuating load environment. Table 1 shows a list of several structural failures that took place in a welded joint as a result of poor quality welding with probable defects. There are total of 11 weld related failures that led to the destruction of expensive hardware and, in some cases to a loss of life [1].

TABLE 1

Three Vierendeel Truss Bridges, Albert Canal (in Hasseit, Herenthals-Oolen, and Kaulille), Belgium, 1938 through 1940. Duplessis Bridge, Three Rivers, Quebec, Canada, January 31, 1951. Spherical hydrogen storage tank, Schenectady, New York, February 1943. Spherical ammonia tank, Pennsylvaia, March 1943. Spherical pressure vessel, Morgantown, West Virginia, January 1944. Cylindrical gas pressure vessel, Cleveland, Ohio, October 1944. Five oil storage tanks in Russia, December 1947. Crude oil storage tank, Midwest area, United States, February 1947. Two oil storage tanks, Fawley, England, February and March 1952. Failures of three empty storage tanks in Europe during 1952. Penstank at the Anderson Ranch Dam, Boise, Idaho, January 1950.

6.1 Welding of Aluminum Alloys

Welding has been a mainstay within the aerospace industry; the more complex the structure, the more vital the welding process is to the overall structure. Since their implementation in the industry, methods of welding aluminum alloys have been under constant development in an effort to produce joints with high quality and attractive static and fatigue properties. Aluminum alloys, in specific cases, could be classified as one of the most demanding alloys for welding. These alloys are more susceptible to porosity than any other alloy used for structural applications. Porosity or pores that are the

275

source of crack initiation can occur by the absorption of evolved gases and chemical reactions. Tenacious oxide layers that form instantly on the surface of the aluminum alloy creates unique issues that greatly influence the quality of weldments. Many other unique issues associated with welding of aluminum alloys have created tremendous development activities throughout the aerospace industry. One must also understand that the heat produced from the welding process itself greatly influences the properties of the aluminum alloy.

In addition to classifying aluminum alloys into categories by major alloying elements, the varieties of alloys are also classified into two major classifications of, 1) Non-Heat-Treatable (NHT), and 2) Heat- Treatable (HT) [2]. To fully understand the behavior of these alloys during welding one should differentiate the effects of heat from normal plate processing. The only difference between them is the time at temperature is much shorter during welding than for normal plate processing.

In general, the NHT alloys will gain strength from a combination of cold working and the alloying elements, but when these alloys are re- exposed to elevated temperatures, they lose strength by recovery, recrystallization, and grain boundary melting. While normal processing will subject the material to heat treat cycles in terms of hours, metallurgical reactions will take place during the first seconds of a weld thermal cycle. Regardless of what temper the alloy begins with, there will always be a Heat-Affected Zone (HAZ) that is a direct result of the temperatures reached during welding. The HAZ is the base metal which is subjected to elevated temperature during welding to cause recrystallization and therefore has different mechanical properties than the base metal.

The HT alloys usually gain strength from subjecting alloying elements at specific thermal cycles. Unique tempers can be obtained by creative thermal cycles to produce desired strength. It is well understood that welding creates thermal cycles that influence base metal mechanical properties. In general, the higher the heat input, the greater the impact on mechanical properties. This holds true for either NHT or HT alloys. The welding processes are very important with respect to heat input.

276

For the purpose of this chapter, welding processes are classified into two major groups: 1) fusion weld and 2) non-fusion weld methods. While the fusion welding methods have long been the foundation for large pressurized structures, recent developments have resulted in new non-fusion weld methods being utilized. This chapter limits the static and fracture properties comparison of two major welding processes, one being the widely accepted fusion welding method of Variable Polarity Plasma Arc (VPPA) while the second is the recently developed non-fusion friction stir welding (FSW) process. Both methods are used for welding critical aerospace components and have experienced tremendous success, with the later enjoying the most positive impact on the welding industry. In section 1.2 the VPPA welding technique and its application to space structural hardware is discussed. The development of static and fracture properties of 2219- T87 VPPA weld, based on the ASTM Standard, is presented for room and cryogenic temperatures. Later in section 1.3, the FSW welding technique is introduced, together with a brief description of hydraulic pressurization to failure test of a subscale tank made of 2014-T6 aluminum alloy. The subscale tank was fabricated with the longitudinal FSW butt seam welds made down each side of the tank and a circumferential FSW lap seam joining the caps (2219-T87) is presented. Static and fracture properties of 2014-T6 FSW weld for this alloy are also discussed in section 1.3.

6.2 Variable Polarity Plasma Arc (VPPA)

The VPPA welding process has been used with considerable success on conventional aluminum alloys. The optimum process results have been achieved using a single pass keyhole technique. Additional fill passes, in general, tend to reduce joint strength, increase distortion, and increase the risk of introducing internal weld defects [3]. The VPPA weld process is a variation of the plasma arc welding (PAW) technique [4], in which the polarities during welding are independently controlled to a customized setting. The ability to control independently both the straight (DC-electrode negative), reverse (DC- electrode positive) frequency and current settings makes this process attractive for aluminum welding while maintaining the benefits of the plasma arc process.

Plasma arcs can be characterized as a high temperature, partially ionized column of gas that is produced by flowing an inert gas through

277

an electric field. This electric field is created from a non-consumable tungsten electrode (cathode) to an electrically conductive work piece (anode). A plasma column is found in both PAW and the gas tungsten arc welding (GTAW) process. The ionized gas, being a good conductor of electricity, is also affected by the magnetic fields created in the arc region. This column is formed and transferred to the work piece primarily by magnetic pumping of the plasma arc; however,

/ me~e - ~/Beca~le

' ~ Shield Gas P

Figure 1: GTAW and PAW standard torch configuration by forcing the plasma gas through a constricting orifice

PAW employs further columnization by forcing the plasma gas through a constricting orifice with prescribed openings (Figure 1).

In an unconstricted arc, as in the GTAW process, the flow of current interacts with the magnetic induction, creating an inward directional force per volume of arc material. As the arc expands away from the electrode, the arc column area increases and arc pressure decreases. By constricting the arc, as in the PAW process, the outward expansion is eliminated, creating a high-temperature region (10,000 to 20,000 K) with greater ionization and current densities (Figure 2). Treating the constricting orifice as a frictionless, constant- area duct and applying the conservation of momentum, a forceful jet of plasma is created. Plasma jet velocities have been calculated and cited by several investigators in the field [3,4] from 300 to 2000 m/sec for PAW compared to the 80 to 150 m/sec for the GTAW process.

278

This ability to create a high temperature, high-velocity plasma column enables the arc to penetrate completely through a molten weld

100 100 100

Figure 2: Profile of typical arcs with greater ionization and current densities

puddle. This condition is commonly referred to in the welding industry as a keyhole. It is a circular-shaped hole through the work piece that is created when the plasma column passes over the area and solidification occurs as the molten pool flows around the sides of the keyhole to the rear and joins together. Both laser and electron beam welding employ keyhole welding; however, the keyhole formed during a plasma weld differs from these processes. The keyhole produced from a laser beam or electron beam is maintained by the pressure of the vaporizing base material; whereas PAW maintains keyhole pressure from the plasma column force of the arc itself (figure 2). This results in a much wider keyhole and less penetration capability than the beam welding processes. Relative to arc welding processes, PAW offers a means of keyhole welding without the high costs associated with beam welding.

Constant development activities have sought to optimize and expand the use of VPPA since its introduction into production in 1979. Significant effort has been expended to achieve thicker welds, increase travel rates, improve mechanical as-welded properties, and

279

most recently the use of VPPA to join aluminum-lithium alloys [5]. In all these cases, a threshold has been established and properties well understood.

One of the major applications of VPPA welding technique is in space rocket pressurized hardware for manufacturing the liquid oxygen cylindrical shape tank, and two domes assembly as a fuel container, used during the launch environment (see figure 3). Briefly, with this approach 3 rectangular curved plates (2219-T87 aluminum alloy) are welded together by the VPPA method along their lengths by butt welding process. The two open ends of the cylindrical tank are closed by two spherical domes through the VPPA technique icircumferentiaily) where cylinder anddome (2219-T6 aluminum alloy)

Figure 3: VPPA welding technique is used in space rocket pressurized hardware

are lap welded together. After completion of VPPA welding, where cylindrical tank and spherical domes are joined together, all parts will be inspected radiographically (along the length of the weld including the circuferential weld, where cylinder and domes are mating

280

together). Finally as part of the requirements, the tank assembly will be proof tested (due to the presence of welds) to verify its structural integrity.

6.2.1 Static and Fracture properties of VPPA weld

Static properties of 2219-T87 VPPA weld material were established based on the ASTM-E08 standards at room and liquid Nitrogen (-320F) temperature environments [6]. The -320°F static tests were conducted solely to check for any material degradation in cryogenic environment as compared to room temperature. Test coupons were taken transverse to the weld (as shown in figure 4) and material yield, ultimate, modulus of elasticity and % elongation were recorded in all cases. Prior to static properties testing, the welded plate was radiographically standard inspected for the presence of any

Welded P~~td ~ Hoear~t tHffeCtAz)ed

Location of static failure Stress Concentration

I ""~"~ I HAZ

Base Metal Unshared Static Test Coupon

Figure 4: Test coupons were taken transverse to the weld for static properties

unacceptable defects. More than 50 specimens were pulled to failure at each temperature environment and, in all cases, the failure occurs at the boundary of the base metal and the HAZ region (figure 4). The observed failure location was expected in the boundary region

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mentioned above, due to the presence of stress concentration at the toe (its magnitude depends on the geometry of the weld), reduced properties in the HAZ, and smaller cross sectional area, where the crown was kept in the unshaved condition (see figure 4). Table 2 shows the A-Basis static properties of 2219-T87 VPPA weld for room and -320°F temperatures with zero mismatch. The mismatch is defined as the amount of thickness difference between the two mating joints as illustrated in figure 5 [7]. From table 2, it can be seen that the static properties of the welded joints are reduced considerably (in the weld regions) as compared to the base metal. The presence of mismatch will reduce material allowables considerably. The variation

Base Metal Base Metal

VPPA Weld

/_. A

/ _ I Thickness, tA A

Thickness variation tB-tA=At % Mismatch=(m/t)xl00 Where t is the thickness of thickest member

Figure 5: Illustrating the mismatch between two mating plates

of material yield and ultimate (FTU and FTY) as a function of the amount of mismatch is plotted in figures 6 and 7 for -320°F and room,

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45 40 35

~ 3o o 2 5

20 15 ,= -

10 5 0

Material Al lowables as a Function of Mismatch @ -320F Temperature

(2219-T87 VPPA Weld)

0 0.05 0.1 0.15

Mismatch, inch

Figure 6: Static properties can be reduced as a function of mismatch amount in the welded joint (-320°F)

Material Allowables as a Function of

Mismatch @ Room Temperature

(2219-T87 VPPAWeld)

b

¢ l

E

0 l 0m

0=

45 40 35 30 25 20 15 10 5 0

0 0.05 0.1 0.15

Mismatch, inch

Figure 7: Static properties can be reduced as a function of mismatch amount in the welded joint (room temperature)

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respectively. These figures clearly indicate that static properties can be reduced as a function of mismatch amount in the welded joint. The designers must establish the static allowables to be used in their analysis based on the weld allowables indicated in table 2, with an additional reduction in the allowables due to the presence of plate mismatch (figures 6 and 7).

Fracture toughness tests for 2219-T87 VPPA welded material were conducted [8] based on the ASTM-E 561 requirements. The R- curve technique (presented in chapter 2) was applied to Compact Tension C(T) specimens to generate the Kc value. As part of the requirement, all specimens were examined by radiographic inspection prior to fracture toughness testing to check for the presence of any

TABLE 2 Static Properties, 2219-T87 (A-Basis)

(Zero %Mismatch) VPPA Weld

Ultimate Yield (FTU) (F'P0

Room 39.9 ksi 21.1 ksi -3200F 41.5 ksi 25.4 ksi

Base Metal Room 63.0 ksi 50.0 ksi -3200F 74.3 ksi 58.0 ksi

porosity or defects. Two regions of the weld (weld center and the HAZ) were tested (see figure 8) at room and -320°F temperature environments. The -320°F fracture toughness test was conducted to check for any weld fracture property degradation in the cryogenic environment as compared to room temperature. It should be mentioned that the C(T) specimens were shaved in the weld region (the weld crown was removed and flushed) and the EDM notch was situated in all cases parallel to the weld direction (figure 8). To estimate the plane strain fracture toughness value without using the thickness that meets the ASTM E-399, additional tests based on the ASTM-1820 standards were conducted and the quantity Kj~c was calculated [9]. In this test method, a single specimen technique was used to generate the J~c data at room and -320°F temperature environments (see chapter 4 for an in-depth discussion related to this

284

C(T) SPECIMEN FOR ASTM E 561 TESTING

Cmck in the HAZ Region / - - A VPPA Weld Center

0.,

~ "~'~ D = 0.25W

I..= W "-1 p , v

I_ 1.25 w I" ~1

n

1.2W

Figure 8: C(T) specimens were shaved in the weld region prior to testing

topic). The results of fracture toughness data obtained at room and - 320°F temperatures are shown in table 3. In addition to fracture toughness testing, the variation of material hardness across the weld (2219-T87 parent material, weld, and the HAZ regions) were measured, by KNOOP microhardness number, and is plotted in figure 9. The results of hardness testing in the weld region clearly indicated

l m

t10

E

N N

P-

W ~ 7*

Q a

I

-70

n ° B 0 " 0 ~

B • i a m • • No

• ° m • ° i10 • n

° o o oo ° °

• o N • i

• . = • • • u

~" Center Line

-IO -SO -40 -30 .ZO .10 0 10 20 30 40 SO a n ?0

Figure 9: Variation of material KNOOP hardness across the weld

285

that KNOOP microhardness values in the center of the weld are reduced as compared with the HAZ region. Based on the observed hardness variations, one may conclude that the weld center fracture toughness value can be higher than the HAZ region (material fracture toughness is shown to be inversely related to hardness [10]).

In conducting structural life analysis, it is necessary to obtain the fatigue crack growth diagram (da/dn versus AK) for both room and - 320°F temperature environments (see chapter 3). In all cases, the weld and the HAZ regions were tested based on the ASTM E-647 Standards. Figure 10 shows crack growth fatigue curves for the two regions of the VPPA weld that were conducted at room temperature.

da/dN VS. DK FOR ROOM TEMPERATURE (R = 0.1)

1.00E-04

I •

' t :~.

: 7!: i 1.00E-05

• , ~,..~:, ~ ~,,~

1.00E-07 i i , ~!; :~: ! : i , , i:~:,

1.00E-06

DK, KSI ( IN. )*0.5

Figure 10: crack growth fatigue curves for the two regions of the VPPA weld (room temperature)

Conservatively, the minimum values of fracture toughness and da/dn versus AK data were recorded and documented for analysis purposes. It should be noted that the fatigue crack growth data for

286

2219-T87 weld at -320°F were always superior to room temperature for both regions of the weld.

TABLE 3

Fracture Toughness, Kc, 2219-T87

VPPA Weld Base Metal Weld HAZ

Room 40 36 5 6 -3200 47 42 7 0 Note: The above data is baed on C(-I-) ~ecimens

6.3 Friction Stir Welding (FSW)

6.3.1 Introduction

Friction Stir Welding (FSW) can achieve high quality welds in aluminum alloys that are of interest to the aerospace industry (e.g. alloy series 2000, 6000, 7000 and numerous aluminum-lithium alloys). The low distortion solid-phase welds exhibit metallurgical and mechanical properties, including fatigue, which are superior to conventional fusion welds achieved by arc processes. Weld property data showed a dramatic improvement over conventional arc processes and additional studies have concluded that the FSW process could significantly improve weld quality and lower production costs associated with welding [11,12,1 3].

FSW is an innovative, yet simplistic, aluminum welding process that was originally invented and patented by The Welding Institute (TWI) at Cambridge, England in 1991 and even within its infancy, the process is advancing rapidly. The ease with which it produces high quality welds at a low cost makes it attractive to the aerospace industry. Aluminum materials are joined by plunging a non- consumable rotating pin into the adjoining plates and subsequent heating caused by the rotating pin tool elevates the temperature of the local weld region high enough to plasticize the aluminum material. Through mechanical forces, the heated material is extruded from the front of the pin to the back as the pin traverses the length of the joint (see figure 11). The combination of the frictional heat and mechanical

287

SI

Sufficient downward force to maintain registered contact

Leading edge of the rotating tool

T the rotating tool Retreating sloe of weld

Figure 11: The heated material is extruded from the front of the pin to the back as the pin traverses the length of the joint

working produces a solid-phase joint. Because no melting takes place, the weld is left in a fine grain wrought structure and other problems associated with liquid to solid transformation (porosity, solidification cracking, residual stresses) are all minimized, if not eliminated. Benefits are also realized due to the low heat input and no-melting feature of the process. Increased mechanical properties, less critical pre-weld surface tool preparation and cleaning, weld joint fit-up is similar to that of arc welding and in some cases less critical, process variables are also minimal compared to that of arc welding.

While these benefits are indeed attractive, the process could not be accepted into critical pressurized structures with load bearing weld joints without operational testing of aerospace hardware. However, one must have some basic understanding of the process as it relates to metallurgical issues. While the benefits of the low heat input from the FSW process are indeed a dramatic improvement over other arc welding processes it still produces heat sufficient enough to cause considerable weld and heat affected zone softening. In section 1.3.3, a brief description of a subscale tank welded by the FSW technique that has gone under hydraulic pressure testing is discussed.

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The FSW process relies on two basic modes of heat generation during welding operations. The first mode is that of the friction at the interface of the tool shoulder surface and base material surface. The second mode is from the plastic deformation of the weld region caused by the interaction of the pin within the base material. The process of plunging the shoulder and pin into the base material produces high strain rates and elevated temperatures. This cycle of high strain at high temperature applied during the welding operation alters the metallurgical and mechanical properties of the material immediately adjacent to the joint line and within the weld nugget itself. A moving column of hot material is "stirred" around the pin and consumes the weld joint interface, along with disrupting and dispersing any aluminum surface oxides. The weld consolidates and cools as the tool travels along the joint line but the resultant heat generates a heat affected zone (HAZ). Within this HAZ near the weld nugget, a thermo- mechanically affected zone (TMAZ) is also produced, as shown in figure 12. The FSW process generates a weld nugget that is broader

TMAZ HAZ ~ I¢ WeldNugget ~J , / /

I T h i c k n e s s

\ / Figure 12. Micrograph of a FSW in 2014-1"6 aluminum and schematic

illustration of weld zone microstructure.

at the shoulder interface region and a smaller well-defined weld nugget region in the center of the material. The weld nugget profile is very much dependent upon the shape and profile of the pin that

289

penetrates into the material being welded. All regions are considered part of the weld microstructure (figure 12).

Currently, the FSW technique is a preferred method by Boeing Space Center over the VPPA welding process for manufacturing the pressurized tanks used for space vehicles. Longitudinal friction stir butt welding along the length of the tank can be performed with much faster speed and better weld quality, free from defects, when inspected ultrasonically. The circumferential FSW is not yet mature, but small scale laboratory welding shows good quality weld with attractive properties.

6.3.2 Static and Fracture Properties of FSW

2014-T6 FSW static and fracture properties were established for room and cryogenic temperatures based on the ASTM Standards. It should be noted that these properties vary as a function of several variables, such as pin geometry and plate thickness. Test coupons for static and fracture properties were taken transverse to the test specimen, as shown in figure 13. For room and -320°F temperature

Base Metal i ~ Base Metal

Failure Location ~ , . ~ ,

FSW Region ~ ~ Static Specimen

c ~ Specimen

.,q--Q

Crack is situated

~""~ in the weld

l ~ l Crack is f ~ situated

Specim d

(a)

(b)

(c)

Figure 13: Test coupons for static and fracture properties

290

environments, more than 50 specimen were used in each case. All specimen failed in the boundary of the base metal and the HAZ regions (as shown in figure 13a). Material properties based on static tests indicated that the FSW yield and ultimate stresses (FTY, FTU) at -320°F were superior to room temperature by 40 and 20 percent, respectively. Furthermore, the FSW static properties at both temperature environments were above the VPPA values and lower than the base metal (see table 2).

Fracture toughness and fatigue crack growth data for 2014-T6 FSW were generated based on the ASTM requirements. Both cases of room and -320°F temperatures were examined for the two regions of the weld. C(T) specimens (with 0.25 inch thickness)were used to establish the resistance curve (R-curve) in accordance with the ASTM- E561 requirements. Based on three C(T) specimens, an average of 65 ksi~/in, was calculated at room temperature. Figures 13b&c show the C(T) specimens where crack situated in the weld and HAZ regions, respectively. In addition to C(T) specimen, a wide panel M(T) specimen (see figure 14) was prepared [14] to conduct the fracture toughness test at room temperature and an EDM notch was situated in the FSW nugget region. An R-curve was established based on the

Total length of specimen

I, 1276mm "(

- . . . . . . . . .

[L 10 _ 1, 1150mm

1750mm

Figure 14: A wide panel M(T) specimen was prepared to conduct the fracture toughness test [14]

291

ASTM-E561 requirements and a fracture toughness value of Kc=97 ksi ~/in. was reported. The M(T) specimen planar dimensions, including the original crack length, are shown in figure 14. It should be mentioned that the final crack length at the onset of instability was almost 13.5 inches (original crack length a0=10.8 in.), yet net section yielding was not observed.

The fracture toughness value obtained through the M(T) specimen [14] gave a considerably higher fracture toughness value when compared with the C(T) specimen, as was expected [15]. Moreover, the FSW fracture toughness value reported by using the M(T) specimen (provided by [!4]) was also higher than the 20!4-T6 base metal that was extracted from the NASA/FLAGRO material library. However, the fracture toughness value recorded for the FSW by the C(T) specimen (based on the average result of three specimens) was lower than the base metal fracture toughness value provided by the NASA/FLAGRO. It should be mentioned that in most cases the reported fracture toughness data provided by NASNFLAGRO material library is based on the apparent value and moreover, the M(T) cracked plate dimensions used in generating fracture toughness data by NASA/FLAGRO are considerably smaller than the cracked specimen shown in figure 14.

6.3.3 Application of FSW to Space Structures

The following study is an example to characterize the process as it relates to alloys and hardware used in aerospace. The tank design is shown in Figure 15. The aim of the design was (a) to ensure hydraulic -pressure testing Of the longitudinal FSW butt seam welds made down each side of the tank cylinder body and (b), to avoid the need for a special collapsible frame to support the end caps during operation to FSW lap seam join the caps to the cylinder barrel [16].

The design of the endcaps was chosen to ensure that critical hydraulic pressure occurs in the thinner cylinder material where the butt seam welds are evaluated. The wide edge of each end cap enabled two parallel FSW lap seam joints equally spaced on 50mm centers. The 50mm rim thickness more than adequately overcomes the necessary welding tool reaction force without distorting the cylinder shape and provides a stable platform for attaching the tank rotation mechanism required for the end cap welding operation.

292

F ~ (2")

rat

I I ~ (74")

lJ - - J

I All dimensions in mm (inches)

Figure 15: Pressurized tank design with 2014-1"6 cylinder and 2219- T87 endcaps (FSW)

Aluminum alloy 2014-T6, 6mm thick was used for the tank cylinder and alloy 2219-T87 for the end caps. Custom built and or modified equipment was used to withstand the process variables of FSW. The welding head movement can be adjusted to apply selected vertical downward forces to overcome FSW tool vertical reaction forces.

A special table was designed and built to support two curved panels that were required to be butt welded together to form the cylinder. To rotate the tank end caps and cylinder assembly together beneath the FSW tool to make the connecting lap seam welds, a rotation tool was designed and built. The tank was rotated underneath a stationary FSW head assembly during welding.

The longitudinal weld joints were made with a square butt type joint and circumferential weld joints were made using a lap type joint. In both cases, a smaller tack weld made from FSW was used to assist in holding the plates together during main welding operations. Both weld configurations are shown in Figures 16 and 17.

Separate FSW pin tool designs were utilized for each joint configuration (e.g. butt welds, lap welds and the tack welds). The tool configuration used for friction stir welding will vary from user to user, but a general description of the design obstacles associated with each joint configuration is presented that may be useful to the reader.

To eliminate the risk of surface mismatch and parting of the faying surfaces, due to thermal buckling, individually loaded finger clamps

293

Base metal

Figure 16: Schematic of typical butt FSWjoint (full penetration, 6mm. thick 2014-T6 aluminum alloy)

Base rr letal

Figure 17: Schematic of typical lap FSW joint (6mm. thick 2014-'1"6 to 6mm. thick 2219-T87 aluminum alloy

running the length of the weld seam would be the ideal solution. Since this option was outside the scope of this study, the decision was taken to partial penetration tack weld the seams by FSW. Tack welds were made to a depth of 4mm and a nominal width less than the main FSW weld to assure full consumption of the tack weld region.

Each panel was roll formed prior to welding and two panels were held together with steel bands to form a cylinder while tack welds were made. Following tack welding, the bands were removed and plugs of 2014-T6 were inserted into the exit hole left at each tack weld when retracting the pin tool. Each longitudinal weld was made consuming the tack welds and plug material and subject to radiographic and dye penetrant examination.

294

Following the longitudinal welding, the cylinder was trimmed at each end for proper fitup of the end caps. Each cylinder end bore was measured and its end cap diameter machine turned to achieve a tight interference fit. To establish and assure good contact between the lapping faces of the tank cylinder skin and the end cap throughout the entire and continuous lap welding operation, a series of equally spaced FSW lap seam tack welds were made around the circumferential seam.

The completed tank assemblies were prepared for hydrostatic cyclic and burst testing. The end caps were made with threaded connections for hydraulic pressure feed and air bleed points. The tank was then instrumented with fifty foil resistance strain gauges installed on the internal and external surface at locations adjacent to the longitudinal seams and end closure welds as detailed in Figures 18 and 19. All gauges were of the two element biaxial type of 120-ohm resistance, and were matched to the thermal expansion coefficient of

Figure 18: Strain gages locations along the vertical and circumferential FSW weld

the aluminum alloy to minimize apparent thermal strains. Gauges adjacent to the weld were of 2 or 3mm-gauge length and remote locations away from the welds used 6mm gauge lengths and wired to remote instrumentation.

295

Internal and external gauges were positioned back-to-back. Internal strain gauges were also coated with Bostik sealant for water

Figure 19: Strain gage pattern for the FSW weld in the middle of cylinder

resistance. Wires for the internal gauges passed through nozzles set in one end of the tank.

The data logging system comprised a conventional bridge completion unit and a National Instruments multichannel logging unit interfaced to a PC computer. A pressure transducer mounted in the pressure feed line adjacent to the vessel was also connected to the logging system. The unit was programmed to provide real-time displays of pressure and strain readings.

Hydraulic pressurization was provided by an air-hydro pump unit capable of a maximum supply rate of 5 liters per minute. During the test, the tank was supported on wooden cradles with longitudinal welds at the 3 o'clock and 9 o'clock positions. The tests were conducted at ambient temperature.

The principal objective was to establish the performance of longitudinal weld seams. It was decided to minimize stresses at the lap welds between the cylinder and end caps by installing a

296

circumferential steel band around the tank at each end just inboard on the lap welds. These bands had the effect of providing local support to the shell, thus minimizing the bending stresses imposed on the lap welds.

After installation in the test bay, the tanks were filled with water and air was removed through the bleed point. A nominal pressure of 50psi was applied and the system checked for leaks. The pressure was then returned to zero and the instrumentation balanced to zero on all channels. This procedure ignores the small pressure head between the top of the tank and pressure transducer level.

Pressurization in steps of 50 psi was applied to 300psi and, thereafter, in 25 psi steps to 350 psi. Pressure and strain values were recorded continuously at a rate of five samples per second throughout. After a short dwell at 350psi, pressure was returned to zero. Inspection of the tank and test bay confirmed that no leaks had occurred.

Strains at all gauges showed some initial non linearity, due to local bending of the shell as it conformed to a true circular cross section. The majority of gauge locations exhibited linear elastic behavior with a return to zero strain on unloading. However, the softer zones or heat affected zones (HAZ) on each side of the longitudinal seams were strained plastically in the hoop direction, resulting in permanent strain offsets on unloading.

Strains measured adjacent to the end cap welds were assessed at the end of the first pressure cycle. Behavior in this area is a compound effect of local bending of the shell adjacent to the inboard lap weld, and, because of the evidence of permanent offsets in some channels, it was decided to tighten the support straps for the final pressurization to burst.

Forty pressure cycles to a maximum of 315psi were then conducted. All strain channels were logged as previously. The first cycle was applied with incremental pressure steps and a rapid return to zero pressure. The remaining 39 cycles were applied continuously with a minimal pressure of about 30psi. Despite the extent of plasticity shown on the first pressurization, no significant additional plastic

297

deformation occurred under cyclic conditions. The tank and test bay were periodically inspected for leaks; none were found.

The support bands were tightened at a nominal internal pressure of 50psi. The pressure was increased steadily to 300psi, and thereafter at a more gradual rate until failure, which occurred at 588psi. The tank separated along one of the longitudinal seam welds.

Figure 20 shows the tank after removal from the test bay. Failure was by complete separation along one longitudinal seam weld, the other weld remaining intact. The failure had clearly progressed along the length of the longitudinal seam and deviated into the first lap weld at each end.

Figure 20: Tank failure alongthe FSW region following burst pressure (fracture of longitudinal weld)

6.3.3.1 Metallurigical Examination of Fracture Surfaces

The fracture surfaces were examined in detail using a binocular microscope. The failure was ductile along the full length of the seam, as evidenced by the shear lip at the inner surface, and by the extent of necking of the section. There was also a shear lip-like feature at the weld root, extending approximately lmm from the inner surface, but it did not appear to be associated with significant plasticity. Based on the fracture appearance and the positions of maximum bulging and failure opening, the most likely sites of primary fracture origin were established. At a number of points on the fractures, especially

298

adjacent to the surface of the weld, the fracture characteristics tended to reflect the material flow lines created by the FSW process. Apart from these, there were no undesirable fracture features of special interest.

Sections taken transverse to the failed seam at the likely failure origin were prepared using standard metallographic techniques. Figure 21 shows a light macrograph after etching in Keller's agent, of the fractured seam clearly illustrating the shear lip at the weld cap and the angled feature at the root, which had been observed on the surface, with failure predominantly on the radial plane at mid section.

Sections of fracture surfaces taken were cleaned ultrasonically in acetone and examined in a scanning electronic microscope. Figure 22 shows that failure was by the ductile mechanism of microvoid coalescence: on the angled fracture at the root and on the radial fracture plane at mid-section, respectively. The size of the microvoids was much finer on the angled feature at the root than on the other areas of the fracture surface.

Figure 21: Sections taken transverse to the fracture using standard metallographic techniques

6.4 Summary

This characterization study into the manufacture of representative aerospace hardware using the FSW process clearly identified areas of manufacturing and behavior of the process itself under simulated load

299

environments. Clearly the tank assembly completed with FSW longitudinal and circumferential welds performed satisfactorily without leakage under both static and cyclic pressurization. The tank assembly subject to final burst pressure also clearly showed favorable characteristics of the FSW process.

Assuming a nominal hoop stress of Pr/t, where P is the internal pressure, r is the internal radius, and t is the shell wall thickness, and taking the results of transverse weld tensile tests for FSW in this alloy, the failure pressure was anticipated to be between about 420psi for general yielding and 660psi based on UTS. The difference between the estimated and actual failure pressure can be attributed to bending in the vicinity of the weld joint. Of interest is that the HAZ zone close to the parent material on either side of the nugget did not fail despite

300

the evidence of plastic strains in the region, the failure occurring near the HAZ edge close to the weld nugget.

At low pressures, when the strains were fully elastic, there was evidence of elastic stress concentration at the weld. This effect is likely to be due to the shape configuration, i.e. the narrow flat zone at the weld, and the fact that the weld thickness is very slightly less than that of the parent material. Also compounding the bending issue was the local flat region due to the pre-weld rolling operation on either side of the weld joint. In order to examine the elastic stress distribution adjacent to the seam welds, strains recorded prior to the onset of detection of significant yielding on any gauge were converted to stress using the standard relationship for a biaxial strain field as illustrated in Equation 1.

E (E; Hoop +1.)• Axial) f f Hoop = l - ~------T

E Hoop) (~A*:ial = l-/-~O 2 ([~ Axial+I.)E:

where c and ~ are stress and strain components respectively; E is the elastic modulus (70kN/mm 2 );

is Poisson' s ratio (0.33 assumed). (I)

Inspection of the results illustrated the local elevation of hoop stresses at the weld resulting from the bending effects referred to earlier. It is also clear that a small hoop bending component was present even at a distance of 100mm away from the weld centerline. Taking an average nominal hoop stress of 74N/mm 2 , which compares well with the nominal value based on the tank dimensions and Pr/t, which is estimated to be 77N/mm 2.

Examination of the axial stress results indicated a longitudinal bending component, which increases to a maximum at the weld. This suggests a slight inward bowing along the length of the seam weld in the as-welded condition, the bending stresses resulting from the elimination of this bowing under internal pressure. During the first pressurization cycle of both tanks, hoop strain illustrated extensive yielding of the HAZ near the weld edge. The behavior of these zones

301

under subsequent cyclic pressurization indicates that shakedown occurred on the first cycle, after which no further significant deformation was observed.

R e f e r e n c e s

1. E. R. Parker, "Brittle Behavior of Engineering Structures," John Wiley & Sons, 1957.

. A. C. Nunes, and et al., "Variable Polarity Plasma Arc Welding on Space Shuttle External Tanks," Welding Journal 63, Vol. 9, 1984, pp. 27-35.

. A. C. Nunes, and et al., "The Variable Polarity Plasma Arc Welding Process: Its Application to the Space Shuttle External Tank-First Interim Report," NASA Technical Memorandum, TM-82532, Marshal Space Flight Center.

4. J.C. Metcalfe, and M. B. C. Quigley, "Heat Transfer in Plasma Arc Welding," Welding Journal 54, Vol. 3, 1975, pp. 99-103.

. Space Shuttle External Tank Program, Founded under US government contracts, NAS-800016 & NAS-36200, 1988 and 1999, respectively.

. B. Farahmand, Static Properties of 2219-T87 VPPA Weld for Delta III Rocket," Technical Memorandum, Boeing Space Cenetr, Huntington Beach, Ca, 1988.

. L. Johnson, P. L. Hinkelday, "Investigation into Effect of Peaking and Mismatch on 2219 aluminum VPPA Weld Strength," NASA EST Report 826-2312, 1990, pp. 1-88.

. B. Farahmand, Plane Stress Fracture Toughness Tests of 2219- T87 VPPA Weld for Delta III Rocket," Technical Memorandum, Boeing Space Center, Huntington Beach, Ca, 1988.

. B. Farahmand, J-R Fracture Toughness Tests of 2219-T87 VPPA Weld for Delta III Rocket," Technical Memorandum, Boeing Space Center, Huntington Beach, Ca, 1998.

302

10. S. T. Rolfe and J. M. Barsom, " Fracture and Fatigue Control in Structures, Applications of Fracture Mechanics," Prentice-Hall, Inc., Englewood Cliffs, New Jersy.

11.C.J. Dawes and W. M. Thomas, "Friction Stir Process for Aluminum Alloys," Welding Journal 1996, Vol. 3, pp. 41-45.

12. B. K. Christner, and G. D. Sylva, "Friction Srir Weld Development for Aerospace Applications, International Conference on Advances in Welding Technology, Clumbus, Ohio, 1996, pp. 311-320.

13. M. W. Mhoney, "Friction Stir Welding-Unparalleled Potential for Welding Aluminum Aerospace Structures," Welding and Joining, Jan. 1997.

14. TWI Report for the Boeing Company, "R-Curve Fracture Toughness of Friction Stir Welds In Delta II Developed Tank, 621897," January 1999.

15. B. Farahmand, Plane Stress Fracture Toughness Tests of 2014-T6 FSW Weld for Delta II Rocket," Technical Memorandum, Boeing Space Center, Huntington Beach, Ca, 1999.

16. D. J. Waldron, R. W. Roberts, C. D. Dawes, P.J. Tubby, "Friction Stir Welding-A Revolutionary New Joining Method," SAE, 1997.

303

Chapter 7

BOLTED JOINTS AND APPLICATIONS

7.1 Introduction

There are several types of joints that are used to connect structural parts. The most (~ommon kinds are threaded fasteners (bolts and screws) and riveted joints. These joints are primarily used to provide continuity of structure and transfer of internal load from one member to another. Welded joints and their applications in space structures, specifically pressurized tanks, were discussed in chapter 6. A fastener joint can be viewed as a source of stress concentration, so that preventative measures must be taken to minimize structural failure from cracks that can initiate from highly stressed regions in the joint. The following are descriptions of a few documented fatigue-related failures in which defects initiated in riveted and bolted joints were the source of stress concentration [1]:

Water Standpipe, Gravesend, Long Island, New York, October 7, 1886 Gasholder, Brooklyn, New York, December 23, 1898 Water Standpipe, Sanford, Maine, November 17, 1904 Molasses Tank, Boston, Massachusetts, January 15, 1919. The tank contained 2,300,000 gallons of molasses that caused 12 deaths and 40 injuries. Crude Oil Storage Tank, Ponca City, Oklahoma, December 1921. Eight Riveted Crude Oil Tanks, South and Midwest United States, 1930-1940 Oil Storage Tank, Midwest United States, December 14, 1943

In this chapter, bolted joints will be discussed and the emphasis will be given to the integrity of bolts and pads (the two mating plates being fastened together) when they are subjected to a fluctuating load environment. Riveted joints are briefly presented in section 7.5 and several categories of failure which may occur in this type of joint are discussed.

Many structures, such as buildings, bridges, space vehicle, nuclear hardware and airplanes, rely heavily upon bolted joints for their structural integrities. The factors that are important to consider in designing and analyzing a sound joint are the bolt material, pad material, bolt pattern and the degree of preloading (or prestressing) the bolt in tension. The effect of bolt preloading on bolted joint fatigue life will be addressed in detail with typical example problems applicable to fracture critical hardware. Experimental data based on a series of high cycle fatigue tests (S-N diagram) indicated that the number of cycles to failure of a preloaded bolt in a joint improves considerably as the magnitude of preloading increases. In addition to the high cycle fatigue approach (S-N), attention will be given to the application of linear elastic fracture mechanics to the fatigue lifeevaluation of structures with bolted joints subjected to a cyclic load environment.

7.2 Bolted Joint Subjected to Cyclic Loading

Fatigue failure of bolts in a bolted joint that are exposed to a fluctuating load environment usually occurs at the threaded locations where the bolt and nut are engaged [2,3,4]. However, bolt failures can also take place in the thread-to-shank run out region and, in some cases (less probable), shank to bolt head area where a stress concentration due to the geometrical change can exist, as shown in figure 1. Experimental data on threaded bolts subjected to rotating- bending tests [3] suggests that about 80 to 85 percent of threaded fasteners fail in the threaded region at the root of the first thread where the nut and bolt are engaged. A larger root radius reduces the stress concentration at the root and increases the fatigue life of the bolt [5,6], A larger root radius allows smoother flow of stress in that region, thus reducing the degree of stress concentration, Kt. Figure 2 shows the variation of theoretical stress concentration factor, Kt, with respect to thread root radius [7]. A significant drop in stress magnitude can be obtained when a generous thread root radius is applied to the threads. Test data indicated that as little as 0.002 inches in root diameter increase the fatigue life cycle by an order of magnitude [8]. Other experimental studies have shown that thread processing (by rolling the threads) improves the fatigue life, and is the preferred method over machining the roots [9,10], figure 3. Rolled threads will introduce compressive residual stresses at the stress concentration sites and also increase the root radius at thread roots that help to extend fatigue

305

d a

~ u

\ First thread engagment (Almost 85% of failure)

Bolt to head junction Thread to shank r u n out

Figure 1: PrObable locations where a threaded bolt fails

Root Radius (in.)

0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008

0.009 0.010

Threads with fiat roots (high stress concentration)

3.9 3.4 Threads with large radius 3.1 (low stress concentration) 2.8 2.7 2.5 2.3 2.2

Figure 2: The variation of theoretical stress concentration factor, Kt, with respect to thread root radius

life o f bolts in a joint. For this reason, the rolling process is exceptionally beneficial in reducing stress concentration sites and preventing crack initiation in that locality. Threaded bolts used in fastening fracture critical hardware are required to undergo a rolling process in order to avoid fatigue failure at the roots during their service life. In addition to the rolling process for reducing the stress concentration in the threaded region, it is also good practice to distribute the load among those few threads which are engaged with the nut. This is helpful for avoiding the possibility of crack initiation in the first engaged thread, and therefore tends to prolong the fatigue life of the bolted joint. The designer should be aware that the first few bolt

306

Threads with sharp V roots (high stress concentration)

Threads with large radius (low stress concentration)

Threads with flat roots (reduced stress concentration)

Threads subjected to rollklg process

(Compressive residual stress)

Figure 3: Rolled process greatly improves fatigue life

threads that are engaged with the nut carry the major part of the load. Figure 4 shows the variation of stress concentration factor in the threaded region where the nut and bolt are engaged [11]. Photoelastic studies of Reference 11 clearly indicate that when the nut and bolt are in uniform contact, they will shift almost all of the load to the lower threads. From figure 4 it can be concluded that bolt failure always takes place at the first bolt thread (the region of contact with the nut), where stress concentration is maximum. This condition can be improved by increasing the height of the nut, hence better distribution of load. A larger the nut height will spread the load over more engaged bolt threads, thus increasing the fatigue life. Figure 5 shows the results of some fatigue tests where the increase in number of cycles to failure as a function of nut height for NAS 1069 nuts of various strength levels (200 to 140 ksi) is plotted [8].

7.3 Bolt Preload

A tensile preloaded bolt is also considered to be desirable in preventing fatigue failure and increasing the life of a bolt in a joint. A preloaded bolted joint reduces the amplitude of the fluctuating load at the expense of increasing the mean stress when it is exposed to cyclic loading, as illustrated in figure 6 [12]. Preloading a bolt results in reducing the continued action of unloading and reloading that could lead to fatigue failure in the threaded area where stresses are

307

Conventional nut with maximum stress concentration factor K t = 3.85

_ _ _ _ r ° Z ° f 2 u 2 _

"1= 0--

0.0 1.0 2.0 3.0 4.0

Figure 4: The variation of stress concentration factor in the threaded region where the nut and bolt are engaged

considerably higher than the average stress in the unthreaded region. The preload process can be implemented in a bolted joint by an applied torque, usually performed by torquing the nut upon the bolt to clamp the nut and bolt head contact faces. This process will tighten the joint and prevent any gapping between the two mating plates. A loose bolted joint without sufficient preload will have a much shorter life due to a larger load amplitude in a load varying environment.

The presence of preload increases the bearing pressure between the two mating surfaces, which results in increasing the frictional resistance between the pads and, moreover prevents any side movement along the joint (figure 6). Figure 7 shows several S-N high cycle fatigue curves for different amount of applied preload. Better fatigue life (number of cycles to failure) is expected for a given stress level as the percent preload increased. For the preload to be effective, the placement of a preload indicator washer (at the side where the torque will be applied) and the presence of sufficient lubricant (to minimize the friction between the nut and the bolt) are extremely important in providing a sound bolted joint.

308

120

"~ 105

0 8 90 i o

~ 7s o

~ 60 . ~

I,L

E 45 -1

c ° ~

• 30 "o

0.0

Strength Level (1000 psi)

,40/ /

/

/

/ ~ " " 185

200 ~ / . . . v

3116 3/8 112 Nut Height (in.)

Figure 5: The increase in number of cycles to failure as a function of nut height for NAS 1069 nuts

SI

Load

~ / ~ ~ ~ Cycles

Tension Tension • A

.=ar

Sh,

Tens ion Tension

Figure 6: The presence of preload increases the bearing pressure between the two mating surfaces

309

200

180

z 160

W n, 1 4 0 <

LU 120 r~ <

100 "I'-

Z 80

~ 6o

~ 4o

~ 2O

, ~ - . . . ; " - , , . m l l u u m : ; - - _ " = m n n m l

li -L ;i unnllllni I l l l l . q l i ~ . M I I I I l l l l I INNIl i !mL~III~Nh;.!!Ul nmnllNInl~mm~.lllNN~! mmNIINNIImI,.~Klli~IINBN Ilmmlllnlll l: lmk~llNi:~nil mmll l lNI l~L~m~nl l l~ iN INNINNNIImMmmLHNINN~II I n l l l l l l m N i ~ l l i g l n l . n l lunn lmin~ inmnl iz~n l l l N i i i l l n l i ~ n n l n ~ M i i l l i l l l n n i i l l i ! l l l l l l l l l l l l l l l l l l l l ~ l l l l l l l l l m i l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l I l U l I l l l l I I I I I I I I l l U l I I I I I l I I I l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l

l l l U l l | n i i n l l l

! l l l l l l m l l l l l l l I I I I l i i ~ m l i i l i l ] i I I I I I I l i i l l l l l l l I I I I I I l i i l l l l l l l l I l i i ; ' . l~_l l l~gE[i l I I I I I I I m m l l l l l l l l I I I I I I I B I I I I I I I I I I I I I I m m l l l l l l l I I I I I I I B I I I I I I I

l i l l l l l B U i i i i i ; : ~ l l l l l m m l l l l l l l l I I I I Im i~_~¢~ i . ' : ~ I I I I I B I B I I I I I I I l i l l l z ~ m a l l l l ; i l I l l l l l l l l l l l l l l I l l U l I I I I I l I I I I l l l l l l l l l l l l l l I l l U l I I I I l I I I I

~0%

BO%

50%

30%

10%

1,000 10,000 100,000 1,000,000 t0,000,000

CYCLES TO FAILURE

Figure 7: Several S-N high cycle fatigue curves for different amount of applied preload

As was mentioned previously, a high percentage of failures in a bolted joint occur in the threaded region where bolt and nut are engaged. However, a second source of failure is in the shank at the thread run out region and, again is directly related to a high stress concentration at this location. Thread rolling will eliminate fatigue failure in the shank- to-thread regions that are not engaged with the nut (see figure 1).

7.3.1 Bolt Analysis

Consider a bolt in a bolted joint with maximum fluctuating stress designated by (Pt)App,ed and a stress ratio R=0. The calculated life of this bolt is higher in the presence of preload. Higgins [12], in his experimental work with high strength steel bolts, concluded that a preload could increase the fatigue life considerably compared with similar joints that are not preloaded when subjected to high cycle fatigue (figure 7). The following bolt analysis introduces some of the factors that are involved in assessing the fatigue life of a bolt. Emphasis is given to the application of linear elastic fracture mechanics to crack growth analysis of a bolted joint when a crack can initiate in the most critical regions, where stresses are maximum under

310

load varying environment. In aircraft and aerospace structural pans, when critical components (in the primary load path) are fastened together, it is customary to inspect the parts prior to their service usage for the presence of any manufacturing defects which would cause rejection of the part. Yet an assumption of a pre-existing crack has to be made using an initial flaw size based on the NDI inspection method and a fatigue crack growth analysis of the joint is conducted to determine the number of cycles to failure.

When a bolt is subjected to an alternating tensile applied load, (Pt)applied, and a preload, Ppreload, (see figure 6), the total bolt load, Pb, can be obtainedthrough the following relationship [13]:

where: Pb=Ppreload+[K(Pt)applied] x Factor of Safety

Ppreload =T/Dp.

1

T is the mean wrenching applied torque, and D is the nominal bolt diameter. The quantity p. is the friction or torque coefficient at the nut- to-bolt assembly, and it's value is dependent on the degree of lubricantion at the nut-to-bolt assembly. An average value of ~ = 0.2 is commonly used, based on a coefficient of friction of 0.15. The joint stiffness factor, K, shown in equation 1 (K reduces the amplitude of cyclic loading) is given by the following equation:

K=C/[L(1 +AmEm/AbEb)]

where: L= t l + t2 + tshim, C=(tl + t2 )/2 + tshim

and Am and Ab are the compressed joint area and the bolt shank area, respectively. Eb is the modulus of elasticity in tension of the bolt and Em is the modulus of elasticity in compression of the mating material. The equations describing Am, Ab, and Em are:

Am=~(B2-D2hole) where B=Dhead+L (tan30)/2 4

and Dhead is approximately 1.6 Dbolt. Furthermore:

311

and: Ab = =/4 (Dbolt) 2

Em=(tl + t2 + tshim)/[tl/El+t2/E2+tshim/Eshim]

Let us for simplicity assume that the bolt shown in figure 6 is subjected to a fluctuating stress, (Pt)App,ed, equal to 30 percent of the material ultimate allowable with a stress ratio of R=0 and a preload stress of 50 percent of the material ultimate allowable. Other pertinent information related to the bolt, plate geometry and material properties needed to calculate the joint stiffness factor, K, are: Dbolt = 0.25 inch, tshim = 0, t l = t2 = 0.319 inch, E1 = E2 = 10.8E+6 psi, Eb=29.1E+6 psi,

CULT=1.1E+5 psi, and Dhead = 1.6Dhole, with factor of safety =1.7.

L = t l +t2 +t shim

C = t1/2 + t2/2 + t shim tstack = L B = Dhead + 0.5 tstack (tan 30) Am = p/4 (B2- Dhole2) Ab = p/4 (Dbolt) 2 K = C/[L(1 + AmEm/AbEb)]

0.638 in

0.319 in

0.638 in 0.542 in 0.217 in2 0.049 in2 0.189<1

Note that the stiffness factor, K, will reduce the amplitude of cyclic loading by 0.189.

Equation 1 for maximum alternating bolt stress becomes:

O" b =0.50"UL T +0 .189x l .4X0.30"UL T la

In terms of ultimate stress, (3"ULT:

O b "-0.50UL T +0.0793 O'UL T lb

The second quantity in equationlb is fluctuating between zero and 0.0793 aUL T (R=0). Figure 8 illustrates the cyclic loading that the bolt will experience when it is subjected to mean stress (~M =59.34 ksi > preload stress of 55 ksi. It can be concluded that, in the presence of preload, the mean stress acting on the bolt increases, but the amplitude of the fluctuating stress, Oa=7.34 ksi, becomes smaller than

312

the applied stress prior to preload (0.15 O'UL T =17.5 ksi), which results in increasing the fatigue lifetime of the bolt.

r (~UL

I-- I b m (~a = 4.34 ksi Jb-- (~max = 63.69 ksi I

aUL ~ ~ . 7 / _ _ ~__.~ M =59.34ksi ~-- 55 ksi

2 l , , , , . . . , '~' PreloadLevel (~max = 33 ksi

~ " - - \ GM = 16.5 ksi i

• Time

Figure 8: Illustrating the effect of preload on a bolt subjected to fluctuating load

Example 1 illustrates the effect of preload on service life of a bolt. Fatigue crack growth analysis of pads in a joint is discussed in section 7.4. Examples 2 and 3 will show the analysis approach to evaluate the integrity of pads in a bolted joint by assuming the pre-existence of a corner crack emanating from the bolt holes. The final results, obtained by using the NASA/FLAGRO computer code, are included.

Example 1 A fracture critical bolt (shown in figure 9) is subjected to a fluctuating load of (Pt)applied=50001b with stress ratio R=0. The bolt is made of A286 high strength steel with fracture properties shown in the table 1 (see chapter 3 and Appendix A for the definition of these parameters). The threads are rolled to minimize the stress concentration. A standard level Eddy Current inspection is performed on the bolt. The assumed initial flaw size in the shank area is a part through crack having half crack length c=0.075 inch and crack depth a=0.064 inch, see figure 9. Evaluate the life of the bolt for an applied preload of 0.5 CUL T, where the joint stiffness factor, K=0.25. Compare the results of the analysis with the no preload case. The factor-of-safety for the load environment is 1.7.

Note: The rolling process will introduce compressive residual stresses at the threaded roots. For this reason, the crack size assumption and fracture analysis in this example problem is performed in the shank

313

't'_ '1' Do=0.44 in

Di 2r i

44444

S¢08 Model

SC07 Model

,

:i® Figure 9: A surface crack shown on the shank area of a bolt (Example

area where the probability of failure is assumed to be higher (see the additional note provided at the end of this example problem).

TABLE 1 Material Properties for A286, 200 ksi, (Forged Rod, L-R)

6Yiel d cui Kic Kle A K B K C n p q DK 0 Rcl (~ SR 190 200 100 140 1.0 0.5 0.3E-8 2.1 0.25 .25 3.5 0.2 3.0 0.3

Solution From equation 1, the bolt load, in terms of the preload and the applied fluctuating load, can be written as:

Pb=Ppreload+[K(Pt)applied]X1.4

where K(Pt)applied=12501b and cpreload=0.5auLm . The load variations that the bolt is subjected to for the two above cases (with and without preload) are shown in figure 10. The crack geometry used for crack growth analysis is a thumbnail crack as shown in figure 9 (designated by SC07 in the NASA/FLAGRO computer code versions 2 and 3) [14]. The crack depth length, a, in terms of crack length 2c, was expressed in terms of the diameter D=2r=0.44 inch [14] as:

a = D[tan(2c/D) + 1 - (tan2(2c/D) + 1 ) 1/2]/2

314

m GUL

(200ksi)

0.750"UL (150ksi)

0.5CrUL (100ksi)

0.25O'uL (50ksi)

GM = 105.76 ksi

R=0.67

Preload=lO0 ksi)

GM = 23.04 ksi

R=O.O

(No Preload Case) Cycle

Figure 10: The cyclic load environment with and without preload

A fracture mechanics analysis for both the preloaded (K=0.25) and non-preloaded (where K=I) cases was conducted and the final results from the NASA/FLAGRO output are shown here. The results clearly indicate that a longer life is expected in the presence of a preload.

FATIGUE CRACK GROWTH ANALYSIS

PROBLEM TITLE Bolt Analysis Without Preload

GEOMETRY MODEL: SC07-Part-circular Surface crack on cylinder circular plane Cylinder Diameter, D = 0.4400

FLAW SIZE: a (init.) = 0.6461E-01 c (init.) = 0.75E-01

MATERIAL A286 (200ksi Bolt material)

Forg. rod, L-R Material Properties

315

UTS YS Kie Kic A286 200.0 190.0 140.0 100.0

Material Crack Growth Eqn Constants C n p q DKo Rcl Alpha Smax/SIGo 0.3D-08 2.1 0.25 0.25 3.50 0.20 3.00 0.30

See chapter 3 and Appendix A for the definition of material constants.

FINAL RESULTS Net-section stress exceeds the Flow stress. (Flow stress = average of yield and ultimate) At Cycle No, 91537.03 Crack Size a = 0.203986 Corresponding semi-crack length, c = 0.750000E-01

The final results (Net-section stress exceeds the Flow stress) represent the final failure where the material ability to handle the load is reduced and total separation is imminent (failure of the bolt in two parts). When the total length of the advancing crack is in such a way that the applied load can cause net section (W-a) yielding, linear elastic fracture mechanics can not be used further to describe the crack tip stress field and the corresponding crack growth rate analysis.

Based on the above results, the number of cycles to failure for the bolt when it is not preloaded is shown to be 91537. However, the life of the bolt is expected to increase by applying Cpreload = 0.5 CUL T to the bolt. This is true because upon application of preload, some of the applied load is absorbed by the jointed plates in compression, and only a portion of it will be affected by the bolt, K(Pt)applied, where K=0.25. Also note that the magnitude of stress in the bolt as a result of preload should not be high enough to cause net section yielding. The following is the final result of the analysis conducted using NASA/FLAGRO where the magnitude of the applied fluctuating stress is reduced by the joint stiffness factor K=0.25 (Cmax=l 11.5, and amin=100 ksi).

BOLT ANALYSIS (WITH PRELOAD)

FINAL RESULTS: Net-section stress exceeds the Flow stress. (Flow stress = average of yield and ultimate)

316

at Cycle No. 832977.38: Crack Size a = 0.132176, crack length, c = 0.750000E-01.

The fracture analysis of the bolt with preload described in this example problem clearly indicates that the number of cycles to failure has increased by 9 times as compared to the previous analysis where preload was not incorporated.

Final Note: If a thorough inspection is possible to ensure that the process of rolling the threads is properly implemented and effective, then the area of concern would be in the shank and bolt head-to-shank region. However," because 100% inspection at the roots is not possible, it is recommended that bolt fracture analysis be performed in the threaded region only. Note that in analyzing the threaded region (using an SC08 crack model in the NASA/FLAGRO computer code version 2.0 or any other available code, see figure 9), the far field stress will be based on the minor diameter which will give a higher preload stress as compared with the shank region.

7.4 Fatigue Crack Growth Analysis of Pads in a Bolted Joint

The failure of one bolt in a bolted joint that contains more than one bolt will not necessarily cause the total destruction of the joint. If by analysis it can be shown that the remaining bolts can withstand the redistributed load during their service life, the joint has a fail-safe design feature. Even though the bolts in a given joint might be classified as having a fail-safe design and the failure might be prevented by redistribution of load to the remaining bolts, fatigue cracking may initiate at other locations in the joint and cause a detrimental effect to the structural parts fastened together. For example, a crack may initiate from a hole where two plates are bolted together by a four bolted joint pattern, as illustrated in figure 11. The initiated crack from one of the holes in the joint may grow to a critical size due to the fluctuating load environment, and cause the complete separation of the two structural parts.

When a bolted joint like the one shown in figure 11 is subjected to a fluctuating load environment, the tension and shear forces applied to the bolts will induce localized bending and bearing stresses on the plate. The calculated induced stresses are useful information to determine whether the joint can maintain its integrity during its usage

317

Structural parts are bolted together

by a four bolt pattern O ~ I . 0

Crack can initiate

~ f r o m a h o l e

Figure 11: A four bolted joint where the two plates are fastened together

period. Figure 11 illustrates the possibility of a crack emanating from a hole in a joint as a result of localized cyclic bearing and bending stresses. The following analysis briefly describes the methodology used in evaluating the structural integrity of a bolted joint when it is subjected to a resultant force and moment (Fx, Fy, Fz, Mx, My, Mz), applied at the centroid of the bolt pattern shown in figure 12. Using the bolt load analysis approach described by Bruhn in reference [16], the maximum resultant bolt tension load, Pt, and shear, V, can be calculated (figure 12). The bearing stress that is effective for growing the crack is given by:

(~br=V/Dt 5

where D and t are the diameter of the hole and plate thickness, respectively. Moreover, the bending stress induced in the pad (mating plates) due to the axial load on the bolt, Pt, also contributes in growing the crack from a hole and can be calculated as a bath tub channel fitting (case a), angle fitting (case b), or plate bending approach (case c), depending on the geometry of the joint, [17] (see figure 13).

For the simple case of plate bending (case c), the induced plate bending stress, Cbendin Q, to be employed in the NASA/FLAGRO computer code, can be estimated for the region designated as ABCD by the following relationship (see figure 14):

318

J

Bolted Joint A A /~ A IVVVVl

Crack can initiate from a hole

Time

F x, Fy, F z M x, My, M z

Figure 12: The axial and shear induced loads at a bolt

O'bending=6M/bt 2 6

The following example shows the analysis approach for determining the number of cycles to failure of a crack that has already initiated from a hole in a joint (it was detected by the NDI inspection) that contains a four bolt pattern.

(1) The Bending Plate (Flange) (2) Plane of The Web

(a) (3) Integerated Stiffnere Plate Bath Tub (Channel) Fitting (4) Integerated Stiffnere Plate

' P t Pt - - ' ~ P t ~

~ ~, 0 t o -'~k. L--~ o,,

Figure 13: The three types of fitting design used in designing a joint

319

Plate Bending Pt ~ ~

Dhead ~ - - - - - - ~ - - ~ - - - - ~ " ~ J ~ ~ O -~ kX"

Where b=2d + D head

Applicable to the Region of ABCD

Figur e 14: Evaluating bending stress in a plate subjected to tensile bolt load

Example 2 A bolted joint with a four bolt pattern (situated in the flange of an I- beam), shown in figure 15, is subjected to forces and moments at its centroid location. The bolt is made of A286 high strength steel (figure 16) with fracture properties for bolts and plates shown in tables 1 and table 2, respectively. The analysis indicated that the maximum fluctuating bolt tension load Pt-25001b with resultant shear load V=2000 Ib (stress ratio, R=0). To improve fatigue life and avoid gapping, the bolts are required to be preloaded as high as 0.35 GUIt. The load spectrum for one service life is shown in table 3.

1) Determine if the bolts can survive 4 service lives. Consider the existence of a part through crack in the shank location with surface crack depth and length of a=0.04 and 2c=0.1 inch, respectively, and 2) assume a corner crack of length c=0.05 and depth a=0.05 inch is emanating from the hole in the joint (see figure 16). Determine if the joint can survive a minimum of four service lives. The far field tension stress O'mension=10 ksi with stress ratio of R= -1 (see figure 15). The joint stacking thickness and other required information are: t l = t2 =

TABLE 2 Material Properties for 2219-T851 L-T

aYiel d cUi Kic Kle A K B K C n p q DK 0 Rcl a SR 53 65 33 46 1.0 0.5 0.12E-73.2 0.5 1.0 2.5 0.7 1.5 0.3

320

0.4 in., tshim = 0, Dbolt = 0.5 in., Dhead = 0.8 in. and E1=E2=10.8E+6 psi, Eb=29.1E+6 psi, (~ULT =1.99E+5 psi, d=2D.

TABLE 3 Load Spectrum For Example 2

Steps Cycles Cyclic Stress (%Limit) Minimum Maximum

1 100 0.0 100 2 300 0.0 90 3 400 0.0 80 4 700 0.0 70 5 1000 0.0 60 6 2000 0.0 50 7 280O 0.0 40 8 5500 0.0 30 9 9000 0.0 20 10 15019 0.0 10 11 28853 0o0 7

, ~ ~ ~ ~ ! ~ S t r e s s j ~ i ~ o

,och \ Bolted JointWith ~ O ~ I [ ~-~F~ " t Four Bolt Pattern I ~ . ~ / ~ d ~ t l P ~ J ~ a • _ ~ . ~4~=I " ' ' "[

Pad l(t ,~ i r ' ~ ( t 2) a

Figure 15: A bolted joint and crack geometry (Example 2)

321

m

Figure 16: A surface crack in a preloaded bolt (Example 2)

Solution The stiffness factor, K, for this joint can be calculated.by employing equation 3:

L = t l + t2 + tshim C = t1/2 + t2/2 + tshim tstack = L B = Dhead + 0.5 tstack (tan 30) Am = ~1/4 (B 2 - Dhole 2) Ab = =1/4 (Dbolt)2 K = C/[L(1 + AmEm/AbEb)]

0.800 in 0.400 in 0.800 in 1.014 in 0.611 in 2 0.196 in 2 0.232

The alternating tensile stress in the bolt can be calculated with the presence of induced preload stress of apreload = 0.3x200=60ksi. From equation 1, the total bolt stress can be written as:

abolt=apreload + Kaapplied

where the maximum and minimum alternating bolt stresses (based on Pt=25001b) are 62.96 and 60 ksi, respectively. The final result pertaining to fracture analysis of bolts subjected to the fluctuating load environment (described in the load spectrum of table 4) is shown for four service lives as:

FINAL RESULTS: All Stress Intensities are below the Fatigue Threshold. NO growth in Four Service Life

322

Crack Size a = 0.455928E-01 Corresponding semi crack length, c = 0.500000E-01

Steps

TABLE 4 Load Spectrum

( % Limit Stress) Cycles Min Max

1 100.00 60.00 62.96 2 300.00 60.00 62.66 3 400.00 60.00 62.36 4 700.00 60.00 62.07 5 1000.00 60.00 61.78 6 2000.00 60.00 61.48 7 2800.00 60.00 61.18 8 5500.00 60.00 60.88 9 9000.00 60.00 60.59 10 15019.00 60.00 60.29 11 28853.00 60.00 60.20

The final result indicates that all the calculated stress intensity factors are below the threshold stress intensity factor, Kth, (described in chapter 3) and thus no growth is expected when the bolt is subjected to the fluctuating load environment shown in table 4. The induced tensile stress in the bolt due to a 25001b varying load is not effective in driving the crack above the threshold value.

To examine the integrity of the pad in the joint, the bending stress, O'bend ing , and bearing stresses, O'br, should be calculated to check if the corner crack is stable during its four service lives. From equation 6:

(:]'bending = 6M/bt2 where M=Ptd = 500x2D = 2500 in-lb. From figure 14: b= (4D+Dhead) = 2.8 in.

(:3"bending -" 6x2500/(2.8x0.42) = 33.48 ksi From equation 5, the bearing stress, O'br:

O'br " V/Dt O'br =2000/(0.5x0.4)=10.0 ksi

FINAL RESULTS: Unstable crack growth, max stress intensity exceeds critical value:

K m a x = 57.44 K re f= 0.0000 K c r = 50.60"

323

At the End of Two Service life: Crack Sizes: a = 0.381844, c = 0.294645 , a/c= 1.29594

The above results indicate that unstable crack growth occurs when the pads in the bolted joint have been exposed to only two service lives (associated with the number of cycles used in the two load spectrum). Even though the bolts in this joint could survive four service lives when subjected to the fluctuating load environment, the pads will not survive a minimum of four service lives. The designer is advised to make one of the following changes in order to ensure a safe joint.

1) Reduce applied loads by redesigning the joint. 2) Use a special level NDE inspection to obtain a smaller flaw size. 3) Consider changing the pad material.

Example 3 A low risk structural component (the box shown in figure 17 is holding a fliud line) is attached to a primary piece of hardware (a high risk part) by a four bolt pattern (bolt diameter D=0.25 in. and plate thickness t=0.1 in.). The fluid line (not shown in the figure) and the box have a total mass of 5 Ibm and are subjected to 50 ft./sec 2 acceleration (in the x and y directions) in a cyclic load environment. Use NASA/FLAGRO or any other computer code to determine the integrity of the bolted interface during its service life (one service life is equal to 4 times the number of cycles shown in the load spectrum, table 5). Both the primary and secondary structures are made of 2219-T851 aluminum alloys with fracture properties as shown in table 2. Assume a corner crack of length 0.05 in. with a/c=1 pre-existed at the hole.

Steps

Load TABLE 5

Spectrum For Example 3 % Limit Load

Cycles Max% Min%

1 100.00 100.0 -100.0 2 500.00 90.00 -90.00 3 1000.00 80.00 -80,00 4 2000.00 70.00 -70,00 5 3000.00 60.00 -60.00 6 4000.00 50.00 -50.00 7 5000.00 40.00 -40.00 8 10000.00 30.00 -30.00 9 12000.00 20.00 -20.00 10 15000.00 10.00 -10.00

324

Figure 17: Loads and crack geometry described in Example 3

Solution The forces induced in the x and y directions due to 50 ft./sec 2 acceleration are Fy =Fx =5x50=250 lb. These forces carry moments Mx and My at the centroid of the four bolt pattern, as shown in figure 18. The reactions at the bolted joints due to Fx, Fy, Mx, and My can be calculated as follows:

The maximum tension load is taken by bolt number 1, where the reactions from Mx and My contributions are added together (91.66 + 55=147.66 Ib). The bending stress induced on the plate (t=0.10 in.) due to tension on the bolt can be determined by using an equation describing the bending stress for case (c) of figure 13, see also figure 14. From equation 6:

M=147.66x(1.5/2)=110 in.-Ib (figure 19) b=2x0.75+1.6x0.25=1.9 in., and t2=0.010 in. 2 cbending=6M/bt2=(6x110)/(1.9x0.010)=34736 psi

325

Y

R R R= F x/4=250/4=62.51b

My=F x xL

R= (F x x L)la

62.5x2.2/1.5=91.66

~ Y

R= Fy/4=250/4=62.51b

Mx= Fy x L

X

~ 2 3~" R

R= (Fy x L)/b

62.5x2.2/2.5=55 Ib

I~y

Figure 18: Bolt load due to forces and moments applied to the centroid of the bolt pattern

The bearing stress due to shear forces will also contribute to crack growth. The resultant of the shear forces for the bolt at location 4 is (see equation 5 and also figure 20):

R=V=(91.662 + 552)1/2=107 Ib abr=V/Dt =107/(0.25x0.1)=4276 psi

M=Pt xd

a=6M/bt 2 Where b=2d + D head

Figure 19: Bolt tension load and plate bending (Example 3)

326

i = "

Figure 20: Shear resultant and the induced bearing stress (Example 3)

FATIGUE CRACK GROWTH ANALYSIS

GEOMETRY MODEL: CC02-Corner crack from hole in plate (2D) Plate Thickness, t = 0.1000 Plate Width, W = 2.0000 Hole Diameter, D = 0.2500 Hole-Center-to-Edge Dist., B = 0.5000 Poisson sratio = 0.30

FLAW SIZE: a (init.) = 0.5000E-01 c (init.) = 0.5000E-01 a/c (init.) = 1.000

MATERIAL MATL 1:2219-T851 Pit & Sht; L-T; LA, DA Material Properties: UTS YS Kle Klc Ak Bk Thk Kc

65.0 53.0 47.0 33.0 1.00 1.00 0.100 65.7

Crack Growth Eqn Constants C n p q DKo Rcl Alpha

0.119E-7 3.156 0.50 1.00 3.00 0.70 1.50 Sma~SIGo

0.30

FINAL RESULTS:

327

Critical Crack Size has NOT been reached. at Cycle No. 15000.00 of Load Step No. 10 Step description: of Block No. 4 of Schedule No. 1 Crack Sizes: a =0.665341E-01, c =0.183923, a/c = 0.361750

The result of the analysis indicated that the part can survive the load environment with a final crack length a=0.0665 in. (initial crack length was 0.05 in.) and a/c=0.37.

7.5 Riveted Joints

Riveted joints are used extensively in aircraft and aerospace industries whenever two or more structural sheets are connected together to allow the transfer of load across a jointed interface. Figure 21 illustrates a portion of an aircraft wing whose skin is attached to stiffeners by multiple rivets. A corner crack (or an edge crack) is initiated at the stress concentration site around the periphery of the riveted hole that can propagate due to a load varying environment through the skin and stiffeners (figure 21). When considering the crack propagation and residual strength capability of a structure in the sheet material, the problem is the plane stress one. In chapter 2, the residual strength capability diagram for stiffened and unstiffened structures was

Crack initiated at the periphery of riveted stiffener~ r ~ l I , ~ " ~ hole & propagate through stiffener and skin

Skin ~ I I I I I

A rivet " - ' ~ , p / ) Crack growth

M J i

continued damage from stiffener (initiated

Tata, ,.i,ur. ifene, • consequently the whole structure

~ J

Figure 21: Illustrating a portion of an aircraft wing whose skin is attached to stiffeners by multiple rivets

328

discussed. Emphasis was given to the presence of a crack in the skin region that propagated toward the stiffener (an arrest condition). However, crack initiation can take place in many cases at the riveted joint where a stress concentration exists.

The strength of a riveted joint that is composed of many rivets depends on the number of rivets used in that joint. The strength of each joint with only one rivet depends on joined material, the fastener material, the degree of stress concentration at the rivet head to shank area, the edge distance, and the ratio of the shank diameter to the sheet thickness. Several types of failure are common in a riveted joint when structure is subjected to external applied load. Engineers should design a jointed structure based on the type of failure that might be encoUntered during its usage (see figure 22) which is directly related to the type of loading. A shear joint is used extensively in aerospace structure. Two types of failure are common in a shear joint; the shear of the rivet shank, and the bearing or compressive failure of the alloy in the area of rivet to sheet bearing. Shear failure occurs in the riveted shank region and is shown in figure 22a. Bearing failure is shown in figure 22b, and is a function of e/D (where e is the edge distance from the rivet center to the plate edge and D is the hole diameter). A short edge distance can cause bulging of the edge of the sheet due to the joint force acting toward the sheet edge that produces crushing or bearing failure. Allowables based on bearing values can be obtained when the edge distance is sufficient to prevent edge bulging (e/D>2). Formulas used for static failure conditions associated with applied load P (illustrated in figure 22) are as follows:

Shear Failure (case a) Bearing Failure (case b) Tension Failure (case c)

Ps = n(D2xult)/4 Pbr = DtO'br Pt = (b-D)tcult

Where O'br , "£Ult and ~u, are the bearing, shear and tensile ultimate stresses, respectively. Let us assume a riveted joint has b=1.5 in., D=0.25 in., t=0.15 in., and edge distance e=0.5 inch. Material allowables for both skin and rivet are, cu, = 60 ksi, Xu,t = 36 ksi, and O'br

=90 ksi. The lowest monotonic load that will cause failure can be determined as follows:

Pt = (b-D)t~ult = (1.5-0.25)x0.15x60= 11250 lb. Ps = n(D21;utt)/4 = 3.14x(0.25)2x36/4 = 1766 lb. Pbr = DtCbr = 0.25X0.15X90= 3375 lb.

329

SI

P

A Riveted Joint

P

Riveted Joint Failure J)

P

Bearing Failure (b)

Tension failure in sheet matal (C)

P w I P

-'-"[ e ]"-- Edge distance

Figure 22: Different types of rivet failure that might be encountered during part usage

The joint is expected to fail under an applied load of 1766 lb by shearing the rivet. Extensive discussion related to failure of riveted joints in aircraft structural parts when subjected to cyclic loading is available in references 17,18,19.

7.6 Material Anisotropy and its Application in Bolt Analysis

Almost all of the beams used in structural hardware have their orientations in such a way that the elongated grain direction (L- direction) is along the length of the beam. The long transverse, T, and short transverse, S, directions are associated with the web and the width of the flange, respectively (as illustrated in figure 23 for a beam that has been machined to have an I-shaped cross sectional area). Fracture toughness test data on most alloys has shown lower values in the T-L direction as compared to the L-T direction, as indicated in Appendix A for many materials shown in the NASA/FLAGRO material library [14] (see section 2.4 of chapter 2 for standard nomenclature

330

relative to directions of mechanical working (grain direction) for rectangular sections). For 2219-T851 aluminum alloy, the fracture toughness values for the L-T and T-L directions are almost equal;

Odginal Beam A bored joint with four bolts Flange (before machining) ~ ~ I t I

'-Y a/2

-., ,C<) " °

(Kc)T'L<(Kc)L-T L-T Web ~ "

Figure 23: The material anisotropy, crack orientation, and fracture toughness for an i beam

however, for other materials, such as 2124-T851 aluminum alloy, the difference is no longer small. The analyst must use the appropriate value of fracture toughness for the life evaluation of the part under study.

When a beam is mechanically bolted to another structure (bolted joints at the flange or web locations), it is important to select the correct material orientation when evaluating the usage life of the part under study. In most cases, when the beam is subjected to far-field induced bending and tension stresses (as shown in figure 23), the L-S and L-T crack orientations (for the flange and the web, respectively) are considered to be the appropriate orientation to use for fracture analysis of the joints, even though the (Kc)T-L and (Kc)S-L values are lower than (Kc)L-T and (Kc)L-S, respectively. For this reason, in the case of a flange, as well as the web, it is recommended that the analyst orient the assumed initial flaw (crack emanating from a hole) perpendicular to the far field stress when conducting safe-life analysis (figure 23). It should be noted that the S-L and T-L crack orientations are unaffected by the far field induced stresses. However, when dealing with localized induced stresses such as bearing stress at the

331

hole of a joint (no contribution due to far-field stress), the material orientation must be considered. When using L-T or L-S fracture toughness values, less conservative results (a larger number of cycles to failure) can be expected.

References

1. E. R. Parker, "Brittle Behavior of Engineering Structures," John Wiley & Sons, 1957.

2. S. M. Arnold, "Effect of Screw Threads on Fatigue," Mechanical Engineering, Vol. 65, pp.497-505, July, 1943.

3. A. Schwartz Jr., "New Thread Form Reduces Bolt Breakage," Steel, Vol. 127, pp. 86-94, September, 4, 1950.

4. A. F. C. Brown and V. M. Hickson, "A Photo Elastic Study of Stresses in a Screw Threads," Proc. IME, Vol. 18, 1952/1953, p. 605.

5. A. M. Smith, "Screw Threads, The Effect of Method of Manufacture on the Fatigue Strength, Iron Age, Vol. 146, pp. 23-28, Aug. 22, 1940.

6. F. B. Stulen, H. N. Cummings, W. C. Schulte, "Preventive Fatigue Failures," Part 1 Through part 3, Vol. 33, No. 9-11, 1961.

7. McDonnell Douglas Space Division, "Fasteners and Joints," Structural Analysis, 1970, pp. 13-8.

8. S. S. Manson, Metal Fatigue Damage, Mechanism, Detection, Avoidance, and Repair," ASTM, STP 495, 1971.

9. J. O. Almen, "On Strength of Highly Stressed, Dynamically Loaded Bolts and Studs," SAE Journal, Vol. 52 No. 4, pp. 151-158, April, 1944.

10. R. B. Heywood, "Designing Against Fatigue of Metals," Reinhold, New York, 1962.

11. V. M. Farres, "Design of Machine Elements," 4 ~ Ed., MacMillan Co. 1965.

332

12. T. R. Higgins, "Bolted Joints Found Better Under Fatigue," Engineering News-Record, Vol. 147, pp. 35-36, Aug. 2, 1951.

13. "Preloaded Bolts and Screws," Lockhead Aircraft Corporation, Report No. 2072, March 1, 1950, pp. 1-7.

14. Fatigue Crack Growth Computer Program "NASNFLAGRO Version 2.0", Developed by R. G. Forman, V. Shivakumar, J . C. Newman. JSC-22267A, January 1996.

15. E. F. Bruhn, "Analysis and Design of Flight Vehicle Structure," Jacobs Publishing Inc., pp. D1-D14.

16. "Tension Type Fittings," Lockhead Report No. 2072, Lockhead California Company, Nov. 1, 1968, pp. 1-12.

17. USAF Damage Tolerant Design Handbook for Aircraft Structures, AFWAL-TR-82-3073, May 1984.

18. D. Broek, "Elementary Engineering Fracture Mechanics," Kluwer Acadamic Publisher, 1986, Chapter 16.

19. B. Farahmand, "Fracture Mechanics Manual," R-35-SSC, Boeing Space Division, 1990, Chapter 2.

333

Frank Abdi, Alpha STAR Corporation and

Levon Minnetyan, Clarkson University

and Chris Chamis NASA~Glenn Research Center

CHAPTER 8

DURABILITY AND DAMAGE TOLERANCE OF COMPOSITES

8.1 Overview Of Composites In certain structural applications the use of fiber composites is

advantageous over metallic structures (lighter weight, and higher stiffness). Polymer matrix composite (PMC) materials with continuous high modulus graphite fibers are very effective under high-cycle fatigue loading due to the ability of fibers to transmit and disperse the high-frequency vibration loads. A well-configured PMC structure can achieve practically infinite fatigue life, carrying relatively high loads. PMC structures subjected to high cycle fatigue usually experience damage initiation by transverse tensile failures (perpendicular to applied loading direction) in the matrix. As a result, a large number of microscopic cracks appear in the matrix, parallel to the fibers. The structural response properties and resistance to damage propagation usually remain intact at the presence of a large number of microscopic transverse tensile cracks [1]. At the end of fatigue life, fiber fractures occur at stress concentration sites, such as interlaminar boundaries. The resulting local delaminations indicate onset of the damage propagation and failure stages. Composite failure is not predictable with a higher reliability compared to metallic structures due to the large number of material parameters and structural elements that contribute to the composite load redistribution and load carrying capability. The cumulative uncertainty from the large number of structural parameters

that are present in PMC materials is less affected by the uncertainty associated with any one material variable. As a result, PMC structures can be designed with a higher degree of reliability. Composites are also well suited for damage tolerant design by the proper selection of fiber reinforcement patterns at damage critical regions. Composite structures are better able to redistribute stresses and redirect load paths around both discrete and distributed damage zones. PMC materials are particularly effective for damage tolerance due to flexibility of the polymer matrix that allows for small local adjustments of fiber alignment to help reduce stress concentrations [2]. Fiber reinforcement configurations can be designed for the purpose of containing accidental damage. Hybrid composites that use both high-modulus and intermediate modulus fibers are effective in reducing stress concentrations as well as increasing damage energy absorption capability of PMC materials. High modulus fibers are typically made from layered anisotropic graphite, whereas intermediate modulus fibers are usually made from homogeneous materials such as fiberglass or Kevlar.

Polymer matrix composite (PMC) materials in the form of braided, woven or stitched laminates are increasingly being used in critical aircraft structural parts that must be damage tolerant for reasons of safety, reliability, maintainability and supportability. This means that PMC structures must be durable and reliable under service conditions that can expose aircraft structure to many types of damage mechanisms. Basic to the evaluation of durability and reliability is the analysis of fracture initiation and progression under static or fluctuating thermo-mechanical loading, and fast rate of loading in variable environments (e.g., humidity, engine exhaust, or sea air).

Fracture initiation is associated with defects such as voids, machining irregularities, stress concentrating design features, damage from impacts with tools or other objects resulting in discrete source damage (DSD), and non-uniform material properties stemming, for example, from improper heat treatment. After a crack initiates it can grow and progressively lower the residual strength of a structure to the point where it can no longer support design loads making global failure imminent. The processes of fracture initiation and subsequent progressive growth have large probabilistic elements stemming from the complexities introduced in PMC materials by the presence of

335

multiple components and their interactions. Add to this the multiplicity of design options arising from the availability of numerous choices of fibers, fiber coatings, fiber orientation patterns, fiber preform variations, matrix materials, and constituent material combinations and there results a large array of design parameters to be considered making numerical analysis difficult. As a consequence, costly and time-consuming experimental testing has been primarily relied upon to evaluate design iterations of PMC structures.

Engineering applications of composites require diverse material properties and fiber reinforcement configurations. Engineered composites are classified into categories based on (a) fiber/matrix constituent materials and (b) fiber reinforcement configurations. Fiber/matrix constituent based differences in composites generally address use temperature of the structure. At relatively low temperatures, ranging from subzero to 150°C, polymer matrix composite (PMC) materials are usually selected. Metal Matrix Composite (MMC) materials are indicated for higher temperature applications, ranging up to 1,000°C. Ceramic Matrix Composite (CMC) materials are considered for temperatures above 1000°C such as in engine combustion chambers. Fiber configuration based classifications distinguish between short fiber and continuous fiber composites. The continuous fiber composites are usually subdivided into straight fiber laminates that have only in-plane fiber orientations and braided or woven textile composites that have both in-plane and out-of-plane fiber orientations. Straight fiber laminates are used extensively in aircraft structures. However, the potential for delamination is a major problem because the interlaminar strength is matrix dominated. Textile composites that do not have distinct laminae are not susceptible to delamination, but their strength and stiffness are sacrificed due to fiber curvatures.

The automation process of composite manufacturing emphasizes quality and cost of the end product. The type of process (i.e. braiding textile, pultrusion) is selected to provide a cost effective means of manufacturing two and three dimensional fiber reinforcement configurations used in the design of PMC structural components.

The selection of the manufacturing process is based on the cost comparison of different processes, shown in Figure 1.

336

The bar-graph in Figure 1, that was originated by Krolewski and Gutkowski (1986), indicates that pultrusion is the lowest cost manufacturing process. Nevertheless, the shape of some parts, such as a wind turbine blade does not lend itself to pultrusion, and therefore, the process of choice for wind turbine blades is filament winding or braiding.

O o

MANUAL.AUTOMATED ROBOTIC TAPE ROBOTIC PULTRUDE FILAMENT CUTI'ING TRANSFER ' LAYUP LAYUP WINDqNG

Figure 1 Cost Comparison of seven different manufacturing processes

8.2 Overview of Textile Composites

Textile composites for engineering structures draw on many traditional textile forms and processes. Therefore the processes and architecture that can yield composites with the high performance required for aerospace structures will be summarized. These textiles are generally those that most effectively translate stiff, strong yarns into stiff, strong composites.

A textile composite has internal structure on several scales. At the molecular scale, both the polymer matrix and the fibers exhibit structural details that profoundly affect strength and stiffness. Matrix properties are determined-by polymer chain morphology and cross- linking, among other things. Carbon fibers, which are often the preferred choice in aerospace materials, owe their axial stiffness and strength to the arrangement of carbon atoms in oriented graphitic sheets. On a coarser scale, typically 1 mm., lots of ~ 103 -104 fibers are bundled into yarns or tows. Within the finished composite, each tow behaves as a highly anisotropic solid entity, with far greater stiffness and strength along its axis than in transverse directions. Because tows are rarely packed in straight, parallel arrays, stresses and strains often possess strong variations from tow to tow. Thus composite mechanical properties such as elasticity can only be considered approximately uniform on scales that are even larger still,

337

say ~10 mm or higher, where the effects of the heterogeneous structure at the tow level are averaged out. Finally, the textile forms part of an engineering structure, perhaps the stiffened skin of a wing or fuselage. Since the engineering structure itself usually has some dimensions as small as ~10 mm, the fabrication of the composite material and the fabrication of the engineered structure may no longer be considered distinct operations. To fabricate the textile composite is to fabricate the structure.

Figure 2 illustrates scales in one textile process. The shown part is an integrally formed skin/stiffener assembly. The first processing step is the formation of yarns from fibers. In the second step, the yarns are woven into plain weave cloth. The cloths are then laid up in the shape of the skin and stiffener and stitched together to create an integral preform. Finally, the composite part is consolidated by the infiltration of resin and curing in a mold.

oty I,.do~

- l i ! i Yaurn Textile (ThousMcls ot FI I~e) P m c , ~

Thk:knen ol Thickness ol on:k)r I0 pm o~4¢ 1 mm

Resin T m

Iqnld Mllchlno . ~ Operations

Molding

Thlcknmm ol omlwr I - 10 mm

WlleNng Tmei lk~ ~

Figure 2. Steps in the production of a textile composite structure

The fabrication method of Figure 2 also illustrates fairly high utilization of the axial stiffness and strength of the fibers. The fibers in the skin are arranged approximately in-plane, straight, and with reasonably high volume fraction. High in-plane composite stiffness and strength can therefore be expected. Certain other traditional textiles do not achieve this. For example, many knitted fabrics loop

338

yarns in highly curved paths, rather than aligning them; and, because of the openness of the fabric, can achieve only moderate fiber volume fractions. Composite stiffness and strength are consequently inadequate for airframes. For similar reasons, fiber mats and discontinuous reinforcements are usually unattractive to airframe designers. These materials will be excluded from further consideration in this section.

Figure 2 also exemplifies reinforcement that is heterogeneous on the scale of the structure. The length of the stitches varies with the thickness of the flange of the rib, and their spacing is not far below the thickness of either the rib itself or the flanges. Just as the material and the structure are fabricated simultaneously, so in this case they must be analyzed simultaneously. Dealing adequately with the fiber architecture in determining stress distributions requires analyzing the external geometry of the part itself.

8.2.1 Categorizations

Figure 3 introduces the most important groups of textile forms that are candidates for airframes. Many of these were investigated in NASA's ACT [3] Program. The left column categorizes textiles according to the machines and processes used in creating them. High- performance composite structures have been created using all the processes listed with conventional textile machinery having been modified in many cases to handle the high modulus fibers needed in airframes and to reduce costs through automation. In modeling their macroscopic properties, all 2D and many 3D textile composites can be considered to function as laminates, with relatively minor allowance for their textile nature, even though the routes to their manufacture are very different from conventional tape lay-up. Most textile composites for skin or sheet applications are designed for high in-plane stiffness and strength. This requires the majority of fibers lie in plane, relatively few can be dedicated to through-thickness reinforcement without unacceptable loss of in-plane properties. In the case of most sheet applications, damage tolerance and delamination resistance require that a modest volume fraction of through-thickness fibers be introduced to provide load bearing capacity along all three axes. However, the volume fraction of through-thickness fiber reinforcement

339

needed to suppress delamination increases with increasing curvature because the resultant through-thickness stretch tension increases.

The similarity of braided textile geometry to that of woven textiles allows some perception of braided fiber reinforced material performance to be obtained based on experimental characterizations of woven fiber reinforced PMC materials. Thus, 3D braided PMC composites would be expected to show similar results to those obtained on extensive testing of various angle and orthogonal interlock 3D woven composites. These tests [3] showed 3D woven composites to possess an extraordinary combination of strength, damage tolerance, and notch insensitivity in compression, tension, and in monotonic and cyclic loading. Also, the performance of 3D woven

~ I Ou~i-Laminar

W e a v e s ~ U n l w e a v e s

. . ~ Laye r - t o -Laye r --1 ~ .~ -Ang le In ter look ~ j QuaM-Larninar

" ~ T h r o u g h - T h l o k n a s s I in elasb'c regime 31) J only

' ~ ' ~ Or tho gons l I n t e r l o c k

Mul t i -Ax ia l Warp Kn i t ~ - B l a s 7

2D ~ l Ouasi-Laminar /

Bra ids "-1

3D ~ Car tes ian 4-Step Nonlarntnar SUtchlng 7 Quasi-laminar J

- - ~ Mul t i -S tep

N o n - W o v e n 3 Nonlaminar N-Dl re~Uonal

Figure 3 Categories of Textile Composites for Airframe Applications (Courtesy of Handbook of Analytical Methods of Textile Composite Version 1.0

January, 1996)

composites was found to far outstrip that of conventional 2D laminates or stitched laminates in many important regards. Thus, 3D braided fiber reinforced PMCs can be expected to exhibit similar, superior performance to 2D braided PMC materials. Further, tow size, strength

340

and distributions in space of geometrical flaws are found to be key design and manufacturing parameters for controlling performance characteristics of 3D woven PMCs, and can also be expected to be important parameters for 3D braided PMCs.

Classification into 2D and 3D composites depends on whether the fibers preformed alone (in the absence of matrix) can transport loads continuously in three or only two linearly independent directions. Thus a 2D composite has distinct layers, which may be separated without breaking fibers. Of course, yarns in textile composites that are defined to be 2D may still follow path segments with components in the through-thickness direction, as in a laminated plain weave; but surfaces may be defined through which no tows pass and which separate the composite into layers. This cannot be done for a 3D composite.

In modeling their macroscopic properties, all 2D and many 3D textile composites can be considered to function as laminates, with relatively minor allowance for their textile nature, even though the routes to their manufacture are very different from conventional tape lay-up. Most textile composites designed for skin or sheet applications fall into this category. When high in-plane stiffness and strength are demanded, the majority of fibers must lie in-plane; relatively few can be dedicated to through-thickness reinforcement without unacceptable loss of in-plane properties. And indeed for most sheet applications, damage tolerance and delamination resistance require modest volume fractions of the through-thickness fibers. Textile composites that behave in most ways like laminates are called "quasi-laminar."

In structures where substantial triaxial stresses exist, the optimal reinforcement will no longer be a laminate with moderate through- thickness reinforcement. Instead, fibers will be arranged with roughly equal load bearing capacity along all three axes of a Cartesian system. Such textile composites are called "nonlaminar".

Nonlaminar textiles are often manufactured to respond to complex part geometry and triaxial loads, for example the union of a skin and stiffening element of a short beam with approximately equiaxed cross- section. But even a curved plate designed as a laminate with through- thickness reinforcement must be considered nonlaminar if its

341

curvature is sufficiently high. As the curvature increases, greater through-thickness tension is generated by applied bending moments, and so much through-thickness reinforcement is needed to suppress delamination that the part loses its laminar character. Figure 3 shows how the main classes of textiles may be categorized as quasi laminar or nonlaminar. Occasionally, experience with damage modes obliges a 3D categorization of a textile as nonlaminar in describing its passage to ultimate failure, even though in the elastic regime it is clearly quasi- laminar.

Whether a textile is quasi-laminar or nonlaminar is a crucial question in choosing an appropriate approach to modeling its properties. Quasi-laminar composites can generally be modeled accurately by some modification of standard laminate theory. Nonlaminar textiles require a model that computes the distribution of stresses in all tows, which is a much more difficult problem.

8.3 Progressive Fracture Methodology

Extensive efforts to identify the various modes of damage in composite materials have been undertaken in recent years. The primary finding of most of these investigations was that macroscopic fracture was usually preceded by an accumulation of the different types of microscopic damage and occurred by the coalescence of this small-scale damage into macroscopic cracks. Additionally, it was generally found that analyses based on classical fracture mechanics did not adequately model the damage effects and did not provide a satisfactory degree of predictive capability [4,5].

Damage stability is influenced by both local factors, such as constituent material properties at the location of damage, and global factors, such as structural geometry and boundary conditions. The interaction of these factors, further complicated by the numerous possibilities of material combinations, composite geometry, fiber orientations, and loading conditions, renders the assessment of composite damage progression very complex. This complexity makes it difficult to identify and isolate all significant parameters affecting damage stability without a model based computer code capable of incorporating all factors pertinent to determining structural fracture

342

progression, fundamental to evaluating the durability and life of composite structures [6-14].

Predicting crack propagation in PMC structure is further complicated by the existence of a multiplicity of design options arising from the availability of numerous choices of fibers, fiber coatings, fiber orientation patterns (Figure 4), matrix materials, constituent material combinations, and hybridizations [15]. The resulting large array of design parameters, presents a logistical problem that complicates and prolongs design optimization and certification processes and adds significantly to the cost of composite parts.

8.3.1 Characterization of Composite Degradation

Damage progression in a fiber composite structure will usually be initiated by matrix cracking due to tensile stresses transverse to the fiber orientation. Braid separation or debonding will occur due to inter- braid and, intra-braid stretch tension, shear stresses, and excess relative rotation of contiguous braids. Damage growth may involve matrix crushing due to transverse compression or fiber compressive failures due to longitudinal compression of a braid. Fiber compressive failures may be assisted by the loss of matrix stiffness. Longitudinal compressive failure of a braid will usually propagate immediately into structural fracture. Fiber tensile failures may also occur but most composite structures are able to withstand some amount of fiber tensile failures prior to structural fracture [16].

Generally, composite structures are much more damage tolerant compared to homogeneous structures due to the inherent fail safe characteristics (layered material). A local material failure in a PMC component will not usually mean immediate structural failure as is expected in metals with low fracture toughens. It is, therefore, important to have the capability to quantify the level of structural safety after damage initiation and as damage growth takes place. The relationship between certain damage characteristics and remaining reliable life need to be established for in-service structural health monitoring of structural components. Thermal, hygral, and environmental effects on structural response need to be quantified in order to define a framework for the development of reliable nondestructive test methods.

343

8.3.2 Composite Simulation Software

Numerous FEA programs currently exist, (Table 1) and have been applied to evaluate the durability and damage tolerance (D&DT). These programs use discretization and numerical approximation to simulate structural and material response. However, very few of these programs take advantage of the many recently developed specialized techniques of modeling material behavior. Even fewer consider the complex highly coupled response to structural loads exhibited by anisotropic materials. In the case of 2D/3D composite (woven/laminated/stitched/braided), the magnitude of complexity is even greater due to vast differences in constituent material behaviors. Experience proved that conventional FEA does not permit sufficient flexibility and control over the computation to enable realistic simulations in many interesting 3D composite cases. They reported that it is difficult to treat stitched resin film infusion (S/RFI) simulation with the ABAQUS based code [17]. This is due to: 1) limited through thickness modeling capability, 2) inability to predict behavior of failed

B;axial High modulus Multiloyer Triax~ol woven .woven woven woven

NNNN Warp knit Warp kni t W|WK WIWK

Tubular Tul~lor f l a t braid Flat braid braid braid L I ward

L[ warp

F|ber mat $titchbonded Adhesive XYZ Laid-in LI warp, weft bonded syStem

Weft kni t Weft kni t Weft kn i t Weft kn i t L| we f t L| w a r p L I wef t ; c

MWK~/-4S MWK 01.-4~ MWK ~-45/90 PMWK 0 / ) : .A~lgO

Square Square Fashioned Fashioned braid braid braid braid

LI warp LI w a r p

NT~Oe4139

Figure 4. Woven, Knitted, Braided and Non-woven Fabric Structure.(Courtesy of Textile Structural Composite- T. Chou, and F. Ko)

344

stitched elements and the surrounding composites, 3) lack of complete control over small iterations which cause the stress concentration of discrete source damage (DSD) to become too large, 4) inability to predict very local failure caused by DSD. It has also been reported that the STAG computer program (see Table 1) has difficulty mapping the FEA stress results into the fracture mechanics simulation of FRAN3D [17]. DYNA3D [18] lacks capability to model layered composite behavior, and therefore an orthotropic material model or empirical equation of state is assumed. A number of attempts have been made to computationally analyze durability and damage tolerance (D&DT) in fiber reinforced composite materials and structures. The resulting computer codes, as with GENOA-PFA (Huang 1998) code, generally utilize simplifying assumptions, semi- empirical relations derived from closed form approximate solutions

Table 1 Available Finite Element Program Suited for Composite

Name DYNA/

Developer Lawrence Livermor e

Type of Analysis Non- linear

Progressive Yes/Meta Fracture I

Local/global No Analysis

Adaptive Mesh No Material No Degradation Local No Crippling/Micro Buckling layered No Analysis Stitched No Composite No Element

Buckling Yes Probabilistic No Verified No DSD D&DT Accuracy Fair

GENOA STAG- NASTRAN ABAQUS MARC FRANC

!Alpha NASA- MSC HKS MARC STAR- Lockheed

Linear non-linear Linear Linear Linear non-linear non-linear non-linear Y e s Yes Limited Limited Limited

Yes No No No No

Yes No No No No Yes No No No No

Yes No No No No

Anisotropic No No No No

Yes NO No No No Thickness No No No No Integration point Yes Yes Yes Yes Yes Yes No No No No Composite/ Metal Only Limited No No Metal High Fair Fair- Fair Fair

345

correlated with test data. These codes are typically fast, cost effective, and easy to use but lack the capabilities of GENOA-PFA software to: simulate: 1) dynamic loading responses,2) 3D fiber geometries, 3) woven/braided fiber configurations, 4) through the thickness fiber stitching, 5) degradation of fiber and matrix mechanical/physical properties, 6) damage tracking, 7) creep, and 8) fatigue. Also, these codes do not have GENOA-PFA's adaptive mesh capability for automatically refining an FEM mesh at stress concentration locations to assure simulation accuracy, and the capability to incrementally integrate FEM with macro and micro mechanical basics. The use of GENOA-PFA capabilities are presented in Table 2. A comparison of GENOA-PFA's simulation capabilities with other availabieD&DT simulation software codes is presented in Table 1. Table 3a presents the advantages and disadvantages of D&DT prediction methods.

Table 2. GENOA-PFA Vs. Other Durability and Damage Tolerance (D&DT) Solutions [19]

Integrated GENOA D&DT Solution Other D&DT Solutions Damage initiation/growth and residual strength predictions. Modeling of fiber orientation, matrix, and Stitching. Manufacturing defects: voids, fiber waviness, fiber misalignment and cure residual stress). Degradation of material properties (stiffness, and strength) under service condition. Detailed stress analysis, photo-elastic fringe pattern. Automated adaptive FEM meshing, and crack growth monitoring. Sensitivity analysis of design parameters to failure criteria. Static, Fatigue, impact and creep predictions.

Provide for design (preliminary, and detailed) parametric studies and preliminary structural analysis of resulting designs, detailed designs. Can be used for empirical prediction of damage growth and residual strength (semi-empirical). Accuracy guarantee is at best limited to range of test variables).

8.3.3 Progressive Fracture Analysis (PFA)

The modeling of PMC in PFA considers the physics and mechanics of PMC materials and structure by integration of a hierarchical multilevel macro-scale (lamina, laminate, structure) and micro scale (fiber, matrix, and interface) simulation analyses (Figure 5). The modeling involves: 1) ply layering methodology utilizing FEM models

346

with through-the-thickness representation, 2) simulation of effects of material defects and conditions (e.g., voids, fiber waviness, and residual stress) on global static, cyclic fatigue, and creep, 3) including material nonlinearities (by updating properties periodically) and geometrical nonlinearities (by Lagrangian updating), 4) simulating crack initiation, and growth to failure under static, low/high cyclic fatigue, random fatigue (power spectrum density), creep, and impact loads, 5) progressive fracture analysis to determine durability and damage tolerance, 6) identifying the percent contribution of various possible composite failure modes involved in critical damage events, and 7) determining sensitivities of failure modes to design parameters (e.g., fiber volume fraction, ply thickness, fiber orientation, and adhesive-bond thickness) [20]. The concept of the PFA is shown in Table 3b. PFA progressive failure analysis software is now ready for use to investigate the effects on structural responses to PMC material degradation from damage induced by static, cyclic (fatigue), creep,

Table 3a. Advantages and Disadvantages of D&DT Prediction Methods

Method Advantages Disadvantage Genoa Progressive Fracture - A New Approach

A) Reduce Experimental Testing B) Reduce Design Time C) Reduce Design Cost D) Computer Code Available E) Verified Accuracy F) Most Powerful Of Methods

Requires Significant Computer Resources

Simplified A) Rapid Analysis Accuracy Is Limited Equations [21] B) Promotes Design Optimization R-Curve, (See Well Established Method A) Little Predictive Capability Chapter 2) For Fracture Propagation In PMC

B) Requires Extensive Testing C) R-Curves Are Part Specific a) Linear Elastic

Fracture Mechanics (LEFM, See Chapter 2)

A) Accurate Prediction Of Tensile Strength If Matrix Cracking And Delamination Are Minimal B) Good For Long Damage Cuts

A) Computational Simulation Method B) Indicates Structural Resistance To Damage Propagation.

Damage Energy Release Rate (DERR)

b)

Not Good If Matrix Cracking And/Or Delamination Are Significant Needs Case Specific Fracture Toughness Parameters

Non-Linear Analytical Study Suggests A) Insufficient Experimental Response [ 2 2 ] Accurate Prediction Of Stiffened Verification

Shell Response To Damage B) Limited Effort In This Area

347

Table 3b. Concept and functionality of GENOA-PFA

Methodology Updated/Total Lagrangian Material Property Degradation At Fiber/Matrix

Functionality Geometrical Non-Linearity Material Non-Linearity

Adaptive Meshing Singularity Conditioning Mixed Iterative Fem Minimize Residual Error Fourteen Failure Mechanism Flexibility For Crack Growth (3d)

Percent Contribution Of Failure Modes To Fracture Identify Fracture For Each Mode Strain Energy Rate - Local And Global Damage And Fracture Monitoring

and impact loading in 2D/3D PMC structures subjected to hygrothermal environments. Its use will significantly facilitate targeting

I I , . , -I ~ tP3ER iNTERFACE ~ PmNbil~

ANALYSIS / , 4 ~ OOMA~

COMPOSITE CI~ISTITUENT

Figure 5. GENOA, a Software For Structural Analysis of Polymer Matrix Composites, Utilizes a Hierarchical Multi-Level Approach on Macro and Micro

Scales

348

design parameter changes that will be most effective in reducing the probability of a given failure mode occurring.

Development of 2D/3D PMC progressive failure analysis took into account requirements and verification criteria defined by key industry sources to help designers meet manufacturing and FAA requirements. Key PFA simulation applications are shown in Table 4. The developed code demonstrated achievement of the overall goal of providing a validated PFA tool that an analyst or designer can use to accurately and rapidly evaluate durability and damage tolerance by progressive failure analysis in design of PMC structure and thereby reduce design cost and time to market. Specific efforts have been directed to modify an ex is t ingPFA precursor code to: 1)Model a variety of 2D/3D woven/braided/stitched laminate fiber architectures; 2) Improve flexibility and portability by modularization and standardization; 3) Refine FEM meshing as necessary at any iteration; 4) Simulate progressive fracture under static, cyclic fatigue, and impact loading; 5) Simulate reshaping of braided fiber preforms to assist manufacturing; 6) Perform probabilistic failure analyses; 7) Generate equivalent woven/braided/stitched composite material properties (Figure 6); 8) Perform virtual testing; 9) Provide interface capability to commercially used software (i.e. NASTRAN, and PATRAN); 10) Improve graphics for visualization of simulated results; 11) Provide the capability to show results in animated graphics (movie) form; 12) Update User's and Theoretical manuals; 13) Allow Porting of PFA software to Unix (HP, SGI, IBM), and NT operating systems.

8.3.3.1 Computational Simulation Strategy

The GENOA-PFA code, used for computational simulation, is an integrated, open-ended (Figure 6), stand alone computer code utilizing 1) micro and macro composite mechanics analysis (Figure 7), 2) finite element method (FEM) analysis, and 3) damage evaluation methods. Calculated material stiffness values are input to the finite element analysis module that models composite materials with anisotropic brick and thin shell elements. As shown in Figure 8, GENOA-PFA utilizes an iterative analysis approach to simulate damage accumulation in a structure. The overall evaluation of composite structural durability is carried out in the damage-tracking module that incrementally evaluates composite material degradation in a structure

349

Table 4. PMC Progressive Failure Simulation Key Multi-Disciplinary Features Under Load Spectrum (Static, Low Cycle Fatigue, High Cycle Fatigue,

Random Fatigue (PSD) Creep, Impact) Conditions

Features Functionality Durability

And Damage

Tolerance-

Virtual Testing

Probabilistic Failure Analysis

Manufacturing of Preform

Composite Net Part Shape

Equivalent Laminate

Properties of 2D/3D

woven/braided/ Stitched

• Damage initiation and location (fiber, matrix ) within a lamina • Percent of contributing failure mechanisms • Failure location, and fracture path within lamina, and structure • Residual strength after damage • Prediction of life cycle • Prediction of S-N curves, da/dn, and fracture toughness • Stress intensity factor • Prediction of material property degradation cycle • Animated graphics of PFA process • Contour of global strain/stress at initiation, propagation, and failure • Plot of far-field applied load vs. Deflection (deflectometer) • Plot of applied load vs. Strain (strain gage) • Photo-elastic fringe simulation (isochromatic, and isoclinic) • Local and global energy release rates vs. Applied loads representing

acoustic emissions • Plot of crack length vs. Applied load to show the fracture toughness • Plots of stress vs. Strain at selected locations • Predictions of static failure from discrete source damage • Prediction fatigue damage initiation at multiple sites • Prediction of required tests based on sensitivity of failure criteria • Movie play of virtual testing process • Uncertainty evaluation of material strength to material parameters • Sensitivities of design requirements to design parameters. • Predicting degree to which design parameters contributed to failure • Cumulative distribution functions for failure strength evaluation • Probability of time to failure • Margin of safety predictions • Fiber orientation and volume fraction changes • Attainable best fit to a shape, • Minimize occurrences of failure (buckling, fiber wrinkling) • Multiple preforms of sizes interleaved with woven sheet strips • Transferring fiber orientation directly to design process software • Animated graphics of manufacturing process • Equivalent laminate moduli, moisture property, thermal property,

and heat conductivity • Degradation of material properties due environments (moisture,

thermal), or manufacturing (voids, defects, residual strains) • Plot of ply strength vs. Ply stress

subjected to a specified load spectrum. The composite damage- tracking module evaluates damage initiation/progression in a structure based on the FEM analysis results and failure criteria that guide the

350

synthesis of structural stress redistribution due to material degradation. The damage-tracking module also relies on a composite mechanics module (GEN-PMC) for material property data needed to characterize lamina and laminate micromechanical behavior.

The FEM module used in GENOA-PFA was originally derived from the MARC analysis FEM code developed over 20 years ago. This software development lineage results in GENOA-PFA being a nodal based finite element code. The nodal approach enables the analyst to use nodal interpolation for coordinates, displacements, strains, and stresses. The FEM module library contains 4-node plate and 8-node brick elements suitable for use in structural analyses. The 4-node plate element is most suited for extended analysis since the use of Reissner-Mindlin theory in its formulation allows accounting for transverse shear deformations in a structure.

PvM i casc dioo i .AESTRO [ R.P.PEE. , I Parallel Message Distribution Dynamics Load Static Load ITesting Facility Pass ng Table Ba ance Balance [ Usage

Parallel/Processor Modules

'User 1 t GENOA Executive

Translator

>

Post-Processor Module I Graphics

User nterface

, Data ' :Sor t ing

, J , Report , . . . . . . . . J

I ~ t ltDurability And Reliability! !MC I lAnalysis ~ t t ~

Polymer Metal Matrix Ceramic Matrix Progress ive Probabilistic Composite Composite Composite Failure Material Strength Analyzer Analyzer Analyzer Analyzer SimulatorAsc.o73A

Figure 6. Architecture of GENOA Durability &Reliability Software System

8.3.3.2 Damage Tracking Process

Damage tracking is carried out in the damage progression module that keeps track of composite degradation for the entire structure. The

351

. ~ . - iNTER PI.y

..... . . < _ ,

Icxl / , , ~ , J \ , 121

"~" ~. / r~,~ ,,?~' °.~ "k. RE.RESE,,.,,,, E INTRAPLy COMPOSITE STRESSES Io¢) pty STRESSES Io~) IIEGION

~ ~ = - ~ ~ = ~ ~' , ~ _ o ~ Q ] J~-- STRESSES~ IAI~ STRESSES/' (BI~ S11~ESSES {JBl~ m ~ q ~ f , - _ . PLY MICROCRACK

' ° " ' ° " ' ~ J , - " I ' HI' I:I-'.{:-_-T.'~.~"~'~R x PLY MICROSTRESSES REPRESENTATIVE INTRAPLY REGION

Figure 7: GENOA-PFA Calculation of Ply Microstresses Through Composite Stress Progressive Decomposition

1. Cc~ute me~anic~ mod~e - Calculates cx:z'npo~te ardstropic materlai proper~es at laminate and lar~na levels

2. Firit eetement analysis rnodule - Anisotr opic th~ and Ihin sl~l eSernents to mo:ld cornposit~

3. Damagetrad~ng module - Irii~a~m~proojession

4. (?~n~tatJons made by rnicromech anics equa~m in the cornpo~te mechanics rnodute utiSze congruent ~flne~ and strenglh paramele~ ¢btaned h'om a malenal data bank

(a)

_ ~ -

Figure 8. A Schematic Diagram of the Principal Elements of the GENOA Progressive Failure Analysis (GENOA-PFA) S/W Package

3 5 2

damage progression module relies on the composite mechanics module for micromechanic and macromechanic analysis and uses the FEM analysis module to obtain generalized stresses and displacements in a composite structure.

Figure 9 shows an example of GENOA-PFAs damage tracking sequence as the load on a structure is increased. A damage equilibrium state is defined to exist when an incremental load increase does not either initiate or exacerbate damage. As the load is increased a point is reached (Point 1 in Figure 9) where there is an assessment of initial composite material damage based on the 14 failure criteria. According to the operative failure criteria, material properties are then degraded for Use in FEM iterations to reevaluate the new damaged structure at the damage initiation load. The applied load at a given damage event is maintained and FEM iterations continued as damage accumulates (Points 2, 3, and 4 in Figure 9) until an equilibrium damage state is again reached or until global structural failure occurs (Point 5 in Figure 9).

GEN-PMC is called before and after each FEM analysis to update composite properties based on the fiber and matrix constituent characteristics and the state of damage in the composite lay-up. Through-the-thickness laminate properties computed by GEN-PMC consider the elastic moduli of membrane terms, bending terms, membrane-bending coupling terms, bending gradient terms, and shear terms. The finite element analysis module accepts the GEN-PMC generated properties for each node of a structural model under evaluation and then performs a structural analysis to determine the effect of a given load increment on the generalized nodal force resultants and deformations. The new values so determined are supplied back to the GEN-PMC module that evaluates the nature and amount of local damage, if any, in the plies of the composite laminate.

8.3.3.3 Failure Evaluation Approach

GENOA's approach to failure evaluation involves comparison of computed constituent properties with criteria of stress limits, distortion energies, degree of relative ply rotation, global scalar-damage, and global damage energy release rate (DERR), that is analogous to the Griffith energy release rate described in chapters 2, and 4. Of the 14 local failure criteria in Table 5 used by GEN-PMC to evaluate damage,

353

the first 12 are stress limits computed by the micromechanical equations in GEN-PMC based on a material's constituent stiffness and strength values [24]. In addition to the 12 failure criteria based on stress limits, interply delamination due to relative rotation of plies and a modified distortion energy (MDE) failure criterion that takes into account combined stresses are considered. If ply damage is predicted due to longitudinal tensile or compressive failure, ply stiffness is reduced to zero at the damaged node. On the other hand, if ply damage is predicted due to transverse tensile, compressive, or shear failures, only the matrix stiffness is degraded and the longitudinal tensile stiffness of fibers is retained. Progressive damage and fracture evaluations are carried out by imposing failure criteria locally within unit sub-volumes with reference to the local coordinates of the composite materials. At each individual load step, stresses in stitching and in-plane sub-volumes obtained through the composite microstress analysis are assessed according to distinct failure criteria (Table 5).

Failure Index Process/Description i. Ply failure indicates initial damage • Properties are degraded * Reconstitute a new FEM reanalysis is

conducted under the same load 2. Damage accumulation • More degradation, possible composite ply

failure 3. Damage stabilization • No additional damage, with structure in

equilibrium 4. Damage propagation 5. Analysis stops at nodal fracture, crack

opening, adaptive meshing reconstructs a newmesh, material properties re- calculated, and equilibrium achieved

Load iIy Load Increment

< Displacement

Figure 9. Damage Tracking Expressed in Terms of Load vs. Displacement

8.3.3.4 Damage Evolution Metrics

In addition to the 14 failure conditions, five functions are calculated as follows: 1) Percent Damage Volume versus Load - representing margin of safety (Figure 10(a)), 2) Damage Energy Release Rate (DERR) - representing acoustic emission, and inspection, and non destructive evaluation (Figure 10(b)), 3) Total Damage Energy Release Rate (TDERR) - representing global failure (Figure 10(c)), 4)

354

Strain Energy Damage Rate (SEDR), 5) Equivalent Fracture Toughness (Figure 10(d)).

8.3.3.4.1 Total Damage Energy Release Rate (TDERR)

The measure of global fracture toughness is defined in terms of the TDERR, a scalar damage variable that is equal to the amount of damage energy expended for the creation of unit damage volume in the composite structure. The magnitude of TDERR varies during progressive degradation of the composite structure under loading, reflecting the changes in the fracture resistance of the composite. The TDERR function is useful for assessing the overall degradation of a given structure under a prescribed loading condition, and the rate of its increase provides a measure of structural propensity for fracture.

Assessment of failure with TDERR is reflected by maximum and

4",. HH-.-t-[- " - = f - - - - ~ - r -÷z~-4- .~ ~- ~-~.-r , ! i ~ ! , ~ , . ~ . i " r I i - "~ ! i i ! i

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- T ~ , = , , . . . . . , . . . . . , . . . . . q

C Y C L E

a) Percent damage Volume Vs. Load Spectrum Spectrum

+cGI~ b ~ - l ? : ;

:~ t ~ i:. :t:~-! ~,4:~-:i: ::~.~:i:H:~: ~

} i i i i , I }'. i i ! i ' . l ' . ~ i , i ! } ! i ! ~

C~LE

c) TDERR Vs. Load Spectrum

- ' ; " ~ {i "I 1 - ! ~ . . . . . . . . i 'T"h- "i" i P ' T - f " t " -

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-, -- ............... :+:{:;:H'fi: -H-~-.;F::~::,

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d) Stress Intensity Factor Vs. Load Spectrum

Figure 10. Damage Evolution Metrics as derivatives of Energy Release Rate

355

minimum values. A local maxima (stick point) represents a build up of energy and initiation of damage in a composite structure. Typically, at the stage of damage initiation, there is a high rate of energy release that dissipates a significant portion of the strain energy stored in the composite structure. A local minimum (slip point) represents the end of fracture stability in a damaged composite under the applied loading (Figure 10 (c)). Computation of the TDERR during progressive fracture allows evaluation of the composite fracture toughness and the degree of imminent failure. The TDERR is computed as the ratio of damage energy to the corresponding damage volume that is generated:

TDERR= Total Damage Energy/Total Damage Volume

Table 5. Fourteen Damage Modes Considered In GENOA 125, 26]

M o d e o f Fai lure Descr ipt ion Longitudinal Tensile Fiber tensile strength and the fiber volume ratio. Longitudinal Compressive

Transverse Tensile

Transverse Compressive

Normal Tensile Normal Compressive

In Plane Shear (+)

In Plane Shear (-)

Transverse Normal Shear (+)

I. Rule of mixtures based on fiber compressive strength and fiber volume ratio

2. Fiber microbuckling based on matrix shear modulus and fiber volume ratio, and

3. Compressive shear failure or kink band formation that is mainly based on ply intralaminar shear strength and matrix tensile strength.

Matrix modulus, matrix tensile strength, and fiber volume ratio. Matrix compressive strength, matrix modulus, and fiber volume ratio. Plies are separating due to normal tension Due to very high surface pressure i.e. crushing of laminate Failure due to Positive in plane shear with reference to laminate coordinates Failure due to negative in plane shear with reference to laminate coordinates Shear Failure due shear stress acting on transverse cross section that is taken on a transverse cross section oriented in a normal direction of the ply

Transverse Normal Shear (-) Shear Failure due shear stress acting on transverse cross section that is taken on a negative transverse cross section oriented in a normal direction of the ply

Longitudinal Normal Shear (+) Shear Failure due shear stress acting on longitudinal cross section that is taken on a positive longitudinal cross section oriented in a normal direction of the ply

Longitudinal Normal Shear (-) Shear Failure due shear stress acting on longitudinal cross section that is taken on a negative longitudinal cross section oriented in a normal direction of the ply

Modified Distortion Energy Modified from Distortion Energy combined stress Criterion failure criteria used for isotropic materials Relative Rotation Criterion Considers failure if the adjacent plies rotate

excessively with respect to one another

356

Assuming locally linear elastic properties prior to damage, the damage energy can be computed as follows:

DamageEnergy=ZZ(0.SS2 / Ej)~ i j

Where Sj is the local composite strength associated with the damage mode, Ej is the elastic modulus corresponding to Sj, and V~ is the volume of damage. Typically, at the stage of damage initiation, there is a high rate of energy release that dissipates a significant portion of the strain energy stored in the composite structure. After the first burst of energy release, the TDERR usually drops, indicating a significant reduction in fracture stability of the damaged composite under the applied loading.

8.3.3.4.2 Damage Energy Release Rate (DERR)

The DERR (Figure 10(b)) is defined as the rate of work done by external forces during structural degradation with respect to the damage produced in the structure and is equal to the incremental amount of damage energy expended for the creation of unit damage volume in the composite structure. The DERR can be used to evaluate structural resistance to damage propagation at different stages of loading. The magnitude of DERR varies during progressive degradation of the composite structure under loading, reflecting the changes in the fracture resistance of the composite. The traverse of a local minimum value of the DERR during the progression of fracture typically precedes a very high rate of damage propagation and generally predicts the imminence of total failure. After an energy release the DERR usually drops, indicating a significant reduction in fracture stability of the damaged composite under the applied loading.

The damage energy release rate (DERR), is defined as the ratio of incremental damage energy to the corresponding incremental damage volume that is generated,

Damage Energy= ZY (0.5 * S2Fj / EFj)*VFi

"}- E2 (0.5 * S2Mj/EMj)* VMi 2

+ Zg (0.5 * S2U / Eu)*Vli

357

The above sums are over i (1,2,3) and j (1,2,3), respectively. Where S's, E's, and V's are defined as:

SFj is the fiber strength, j can be 11, 22, 33, 12, 23, 13. EFj is the fiber modulus, j can be 11, 22, 33, 12, 23, 13. VFi is SM~ is EM~ is VMi is S U is E~j is V~ is

the volume corresponding to the failed fiber. the fiber strength, j can be 11, 22, 33, 12, 23, 13. the fiber modulus, j can be 11, 22, 33, 12, 23, 13. the volume corresponding to the failed matrix. the fiber strength, j can be 11, 22, 33, 12, 23, 13. the fiber modulus, j can be 11, 22, 33, 12, 23, 13. the volume corresponding to the failed interface.

Therefore, damage energy release rate is defined as: DERR = Incremental Damage Energy / Incremental Damage Volume

8.3.3.4.3 Strain Energy Damage Rate (SEDR)

In formal fracture mechanics methods the change in the strain energy with damage is an important parameter. Computation of the damage energy expended, UD, is based on summation of the local strengths of the exhausted failure modes. The incremental strain energy can be computed by subtracting the expended incremental damage energy from the incremental work done by external forces, or dU=dW-dUD. The SEDR would be defined as dU/dvD.

8.3.3.4.4 Equivalent far field stress Ge

In linear elastic fracture mechanics (LEFM), the fracture toughness is related to the crack opening length, a, and the magnitude of far field stress through critical crack length, a, (see chapter 2) as:

K~, = f(a/w)c~ 4 ~ a. 3

The far field stress is the component of normal stress perpendicular to the crack and the quantity of f(a/w) is the correction factor. For homogeneous materials, crack extension is perpendicular to the direction of maximum normal stress (except when the direction of crack growth is modified by boundary conditions). However, for fiber composites, the direction of crack propagation depends on the composite lay-up as well as the direction of stress. Also the

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orientation of far field stress may be different than the orientation of stress at the fractured node. Under the circumstances, the average in-plane normal stress in the composite laminate may be an acceptable substitute for the equivalent far field stress % the quantity (~e is therefore computed as the average of ox and % in plane normal stresses of all nodes of the composite structure.

8.3.3.4.5 The length of crack opening (a)

The LEFM fracture toughness is related to the crack length, a, and the magnitude of far field stress, Ge, or Kc,=Go ~Jxa. Therefore, to directly evaluate an equivalent fracture toughness parameter K~, GENOA computes the crack extension length, a. When computing the fracture volume, coordinates of nodes adjacent to a fractured node are determined. The crack extension length, a, is computed as the sum of the distances from the fractured node to the adjacent nodes.

8.3.3.4.6 Equivalent fracture toughness from DERR or SEDR

This approach presents an alternative method to compute a measure of fracture toughness instead of K:r. From LEFM, the relation between the fracture toughness Kcr and the strain energy damage rate (SEDR) that may be taken equivalent to Ge, is Kcr2=Ge*E, where E is the elastic modulus. The equivalent SEDR computed by GENOA, Ge, may be assumed as G~ = DERR or Ge = SEDR. This allows evaluation of an equivalent fracture toughness parameter Ke as K~= ~/(DERR*E~) or K~=~/(SEDR*Ee), where E~ is an equivalent elastic modulus obtained by transforming the nodal composite modulus to the direction perpendicular to the crack path. However, since in GENOA cracks open simultaneously in two directions from the fractured node, therefore, E~ may be taken as Eaverag e for the fractured node.

8.3.3.5 Evaluation of Elastic Constants

Elastic constants are essential for stress analysis of composites. The stiffness averaging method that was developed by [27] is widely used to predict the deformation characteristics of a composite with three-dimensional reinforcement from the known mechanical properties of its components. However, utility of the stiffness averaging method is limited to the prediction of overall stiffness properties of a structure for elastic analysis. Stiffness averaging does

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not retain the information on the spatial configurations of fiber reinforcements. Therefore, the effects of damage and degradation cannot be directly taken into account for a specific ply and the results of structural analysis cannot be decomposed to the micromechanics level for detailed analysis of damage progression. In the present program the stiffness averaging method and classical laminate theory were combined to predict the properties for S/RFI composites, For these composite laminates, each ply consists of both in-plane fibers and through-the-thickness stitching or braid fibers. Accordingly, each ply is divided into two sub-volumes, one consisting of only in-plane fibers and the other consisting of only stitching fibers. The properties of each sub-volume in its local coordinate system are computed from the properties of fiber and matrix, based on composite micromechanics theory, and then transformed to structural directions. The ply properties are obtained by adding the contributions of each sub-volume using the following stiffness averaging equation:

1 N [Ec]= ~-./Z 1 l//[Ri]" [Ei][Ri] 4

where: Ec is the composite ply stiffness V~ is the calculated volume of the i-th subvolume N is the number of subvolumes E~ is the stiffness for the i-th subvolume R~ is the coordinate transformation matrix for the i-th subvolume

After computation of the ply properties from the properties of subvolumes, the properties of the entire stitched laminate are obtained by applying laminate theory as for unstitched composites.

8.3.3.5.1 Stitched Simulation Capabil ity

GENOA's verified algorithm for simulating stitched PMC materials was used in the composite mechanics module (3D PMC) by Alpha STAR to simulate S/RFI materials. The S/RFI composite is divided into a series of unit cells (Figure 11) with both the fiber and stitch segments idealized as linear in the unit cells. The modified PMC3 module computes S/RFI [28] stress limits by adding the oriented contribution of each stitch to each strength (longitudinal or transverse tension, compression or shear) component by tensor transformations in the absolute value.

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"1', Fiber yarn segment Z

I

\ \ \ \

f Figure 11. Idealized Fiber Stitch Segment in Unit Cell of 3D Braided

Composites [29]

8.3.3.5.2 Woven Patterns

Xu et al. (1994) [30] provided complete statements of the rather complex sequencing of through-the-thickness yarns. Figure 12 shows the three typical types of weave in woven composites, namely, (a) layer to layer, (b) through the thickness angle interlock, (c) orthogonal interlock weaves.

In woven composites [31], the stuffers and fillers alternate in layers through thickness. The stuffers and fillers form a coarse 0°/90 ° array shown in Figure 12 for most woven composites. Nevertheless, the method allows for arbitrary orientation of stuffers and fillers in the X-Y plane. The through the thickness reinforcement, or warp weavers, may be oriented in any direction with reference to the 3D composite coordinate axes. Stitched composites may be modeled by stitch fibers that are oriented perpendicular to the X-Y plane and parallel to the Z- axis of the composite.

8.3.3.5.3 Fiber Arrangement

Complete understanding of arrangement of fibers, including stuffer,

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Body w a r p w eave r

F i l l e r (Wef t )

, J Stuffer

W a r p w eave r

(b)

(c)

@ i t

t l l ! '

Figure 12. Schematics of (A) Layer-to-layer Angle Interlock, (B) Through-the- Thickness Angle Interlock, and (C) Orthogonal

Interlock Weaves.

filler, and warp weaver fibers, is essential to predict the engineering properties accurately. Woven composites are typically composed of ns layers of stuffers and ns+l layers of fillers through the thickness as shown in Figure 13.

In order to estimate the elastic properties of 3-D woven composites, an approach based on modified laminate theory and orientation averaging is used. First of all, composites are divided into plies so that every ply contains either stuffer or filler fibers as shown in Figure 14. Since the weaver fibers go through the entire thickness, each ply must contain some weaver fibers for woven composites.

z

f l . . . . . z ,

w

x

Figure 13. Representative Layer Sequence of Fillers and Stuffers Through the Thickness, With the Layer Thickness Tf, and Ts For the Case Shown, ps =2

Figure 14. Schematic of Ply Division in 3D Woven Composite

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For orientation averaging, a three-dimensional fiber arrangement can be considered as composed of two types of structures. The baseline structure formed by fibers in X-Y plane direction, such as stuffer fibers and filler fibers, and the interwoven or braided structure composed of warp weaver fibers that penetrate the thickness of composites. Therefore each ply contains two types of domains, the primary domains which consist of baseline fibers, and the weaver domains which consist of weaver fibers, the occupying volume fractions of these domains are denoted by fp and fw respectively.

Each domain is characterized by an orientation along which the fibers within it are presumed to lie. All of the fibers in the primary domains are assumed to be parallel:to X-Y plane and each ply can consist of the primary fibers in the same direction. Warp weaver fibers are assumed to be piecewise straight and their orientations are defined by the directional angles to the X,Y, and Z axes.

Let [£(~')] denote the stiffness matrix for domain o~. Based on orientation averaging method, the composite stiffness matrix [Ec] of 3-D woven composite can be predicted by the following equation:

where [E(")] denotes [E(~')] transformed from domain c~ material coordinate system into the composite coordinate system and f(, denotes volume fraction of domain o~.

8.3.3.6 Finite Element Analysis in PFA

After the micromechanics analysis module generates the elastic properties for a composite (Figure 7), the finite element analysis module is called to analyze the structural response. In general, the type of finite element model used depends on the complexity of the structure and the availability of computer resources. There are two possible choices for the analysis of composite structures. One is using anisotropic three-dimensional solid elements such as hexahedral or brick elements that accept the computed three dimensional composite properties directly. However, the modeling of a practical composite structure with three-dimensional brick elements

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is usually impractical because it requires huge computer resources. The second option is to use anisotropic shell elements that use the composite plate/shell element properties. The use of anisotropic plate or shell elements to represent through-the-thickness properties of the woven/braided/stitched composite is more efficient computationally than using three-dimensional elements. Therefore, implementation was focussed on the use of plate/shell elements for finite element modeling. The finite element module accepts the force-deformation relations computed by the composite mechanics module, and carries out a stress analysis to generate the generalized membrane, bending and out-of-plane shear stresses, namely Nx, Ny, Nxy, Mx, My, Mxy, Qxz, Qyz for each node. The generalized stresses are supplied back to the composite mechanics module for the computation of local ply/stitch stresses and failure analysis.

8.3.3.7 Simulation of Damage Progression

After each finite element stress analysis, failure criteria are used to evaluate possible failure within each sub-volume of each ply at each node of the composite structure. Once the damage modes at each node are determined, a damage index is created to record the damage information for each node. The damage index contains the node number, the ply number, and the list of damage criteria that have become activated. When a new failure occurs at a sub-volume, the damage index is updated accordingly. The properties of each node- domain are degraded according to the nodal damage index.

If there is no damage after a load increment, the structure is considered to be in equilibrium and an additional load increment is applied. If damage occurs or escalates, the composite properties affected by the damage are degraded, the computational model is reconstituted with an updated finite element mesh and material properties, and the structure is reanalyzed under the same load increment. After reanalyzing, if there is any additional damage, the properties are degraded further and the structure is reanalyzed. This cycle continues until no further damage occurs.

The damage progression module keeps a detailed account of composite degradation for the entire structure. It also acts as the master executive module that directs the composite mechanics

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module to perform micromechanics and macromechanics analysis and synthesis functions, and calls the finite element module with thick shell analysis capability to model woven/braided/stitched laminated composites for global structural response.

Realization of PFA required verification of the software system with experimental test results of fiber reinforced composite primary structures. This was recently accomplished by Alpha STAR with key industry partners in the major aerospace composite programs.

8.3.4 Methodology of Mesh Refinement in Progressive Failure Analysis

Mesh refinement for PFA software was conducted to improve simulation accuracy. In order to avoid unduly increasing FEM computation time, a simulation was begun with a computation time saving coarse FEM mesh that was subsequently refined in only those regions delineated by elements connected to damaged nodes. Thus, when nodes suffer damage according to the damage criteria, those elements connected to the damaged nodes are refined generating smaller elements to represent the damage regions. When all modes of composite resistance fail at a node, that node is deleted and new detached nodes are created at the same point. The number of new nodes created in place of deleted node is equal to the number of elements that connected to the deleted node. This approach allows tracking growth without loss of accuracy due to improper meshing.

Two approaches to mesh refinement were used in modification of PFA. In one approach each element connected to a damaged node is divided into five elements in place of the original element. In the second approach each element connected to a damaged node is divided into three elements in place of the original element. In either approach if any node connected to newly generated elements is damaged, those elements are further refined. This process is continued until final fracture occurs.

8.3.5 Simulation of Reshaping Braided Fiber Preforms to Assist Manufacturing

Simulation of novel, emerging manufacturing method of reshaping braided tubular preform, was developed as extension to PFA to provide a link between the computer generated knowledge base,

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analysis software and automated machine tools. The method is based on a braided preform of commingled thermoplastic yarns in the form of a tublar preform that is made to conform to a net shaped mandrel. This very promising, economical process of generating shaped fiber preforms for composites involves three steps: 1) braiding a simple, commingled fiber tubular preform, 2) reshaping the preform over a mandrel of a desired shape, and 3) processing (heating)the reshaped preform to flow the resin matrix material around the high strength fibers. For a rational design, it is necessary to predict 1) fiber orientations and volume fraction changes, 2) the necessary forces/strain tQ best fit onto a shaped mandrel, and 3) occurrences of failure (buckling, fiber wrinkling) in the preform after shaping. A durability and damage tolerance analysis was conducted to demonstrate the importance of the fiber angle change tracking [32]. Since the reshaping operation can significantly change fiber orientations, it becomes critical to evaluate the extent of this change, its effect on composite performance, and the extent to which the changes can be negated by adjusting the braid angles of the tubular textile preform. The software allows attainment of the best preform net fit to a desired shape without incurring buckling and fiber wrinkling/crimping. The method of approach is by: 1) iterative FEM analysis utilizing composite micromechanics; 2) prediction of the effects of changes of braid/weave angle on composite micromechanical properties; and 3) use of an iterative contact algorithm for analysis of preform conformance to a shaped tool.

The simulation of a simple automation process to get a bent mandrel starts with a strait mandrel is progressively bent using small enough steps to keep stability so that no user intervention is required (apart from fixing the step-sizes). Figure 15 shows an example of such a process for a bent cone. The boundary conditions between two steps are equivalent, whether normal loads or normal displacements are used. This process has been evaluated using experimental test results provided by GE The preform reshaping simulation tool was developed by Alpha STAR in support of the NASA funded AST GE project and verified against experimental GE results of reshaping preforms for conical bent and elliptical components.

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K

i

o o s o s =o ~s z o z6 s

a) Fitting on Straight Mandrel

~s

ao

z s

l o

o s i o i s ~ ~s =o a~ ~ 4 s

(b) Intermediate Steps

o s l o i s ~ ~s ~ a s 4o ¢s

(e) Final Shape

Figure 15. Automatic Fitting Of the Bent Cone Fitting Preform

The GENOA code has provided the needed capabilities for the fiber preform reshaping process for large, complex, PMC components. This effort has been verified to simulate reshaping of tubular fiber textile preforms (e.g, GE90 Turbine blade, bent cone, cylinder, bent cylinder) in a cooperative exploratory effort with General Electric Aircraft Engine. The Sandia National Laboratory 26 ft long and 4 ft wide Windmill best fit simulation has been verified against actual hardware developed by Goldsworthy Associates to be within four percent of the actual measured data by simulating use of multiple reshaped preforms of different sizes with commingled fibers.

8.3.6 Probabilistic Failure Analysis

Alpha STAR has developed GENPAM, a computer software for probabilistic composite structural analysis that integrates probabilistic methods, finite element methods, and composite mechanics. The software has been integrated as part of the GENOA suite of codes [33]. Available probabilistic methods include: 1) Monte E;ado Simulation (ME;S), 2) advanced reliability algorithms and 3)importance sampling methods. ME;S, traditionally used for reliability assessment, is deemed computationally too expensive for large structures or structures with complex behaviors. Extensive effort has been devoted to development of new, more computationally economic probabilistic algorithms for advanced reliability and importance sampling methods in the GENPAM program are a direct result of ten years of probabilistic structural analysis research funded by NASA.

The GENPAM code is constructed such that any real value in the input file of the progressive failure deterministic analysis can be selected as a random variable (Table 6). An interface module was

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developed that can interface with any deterministic code as long as the uncertainties are one of the real values shown on the original deterministic input file. Integration with many commercial or in-house computer codes becomes transparent. Thus, integration effort is minimized and simplified. Various responses can be selected to be analyzed probabilistically, CDF/PDF functions and sensitivities to design random variables. The types of responses that can be specified are: 1) Type 1: Displacement responses specified by setting node numbers and the degree-of-freedom direction numbers, and 2) Type 2: Material responses specified by setting node number, layer number, and the 20 material properties and their combinations. The 20 material properties/responses that can be extracted for each ply node are the following [34, 35]:

Table 6. Geometrical and Material Uncertainties Considered In GENOA

l. Longitudinal strain 11. Transverse strain 2. In plane shear strain 12. Longitudinal stress 3. Transverse stress 13. In plane shear stress 4. Longitudinal tensile strength 14. Longitudinal compressive strength 5. Transverse tensile strength 15. Transverse compressive strength 6. In plane shear strength 16. MDE failure criterion 7. Hoffman's failure criterion 17. Interply delamination failure criterion 8. Fiber crushing criterion (compressive 18. Delamination criterion (compressive

strength) strength) 9. Fiber microbuckling criterion 19. Longitudinal normal shear stress

(compressive strength) 10. Transverse normal shear stress 20. Transverse normal shear strength

8.3.6.1 Probabilistic Evaluation of Composite Damage Propagation

The objective is probabilistically assessing the effect of design variable uncertainties on structural response and residual strength after damage and fracture. The methods and corresponding computer codes are applicable to the uncertainty in the material properties, fabrication parameters, geometry, boundary conditions, and loads acting on composite structures. For a probabilistic evaluation of damage and fracture progression, an integrated probabilistic analysis code is used in conjunction with progressive damage simulation. The probabilistic analysis code considers the uncertainties in material properties as well as in the composite fabrication process and global structural parameters. The effects on the fracture of the structure of

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uncertainties in all the relevant design variables are quantified. The composite mechanics, finite element structural simulation, and Fast Probability Integrator (FPI) have been integrated into the probabilistic analysis code. FPI, contrary to the traditional Monte Carlo Simulation, makes it possible to achieve orders-of-magnitude computational efficiencies that make probabilistic analysis acceptable for practical applications. Therefore, a probabilistic composite assessment becomes feasible which cannot be done traditionally, especially for composite materials and structures that have a large number of uncertain variables. A probabilistic analysis cycle starts with defining uncertainties in material properties at the most fundamental composite scale, i.e., fiber/matrix constituents. The uncertainties are progressively propagated to those at higher composite scales subply, ply, laminate, and structural. The uncertainties in fabrication variables are carried through the same hierarchy. The damaged/ fractured structure and ranges of uncertainties in design variables, such as material behavior, structure geometry, supports, and loading are input to the probabilistic analysis module. Consequently, probability density functions (PDF) and cumulative distribution functions (CDF) can be obtained at the various composite scales for the structure response. Sensitivities of various design variables to structure response are also obtained. Input data for probabilistic analysis is generated from the degraded composite model available as progressive damage and fracture stages are monitored. Additionally, the design variables with probabilistically defined uncertainties and the response parameters that are to be analyzed probabilistically are user selectable.

8.4 Composite Structural Analysis and Input and Output

The fundamental categories of composite structural analysis available in computational simulation are: 1) static loading, 2) low- cycle fatigue loading, 3) high-cycle fatigue loading, 4) random power spectral density fatigue loading, 5) impact loading, 6) creep loading, and 7) probabilistic evaluation of composite damage propagation. Requirements of input data have both common and distinct features for each category of structural analysis. The common features of input data are associated with the definition of composite laminate structure, the finite element model, and the boundary conditions. The laminate structure is defined in the input data by entering the ply schedule, use and cure temperatures, moisture content, fiber angles, ply thickness, fiber and matrix constituent code words identifying databank

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properties, fiber volume ratio, void volume ratio, out-of-plane fiber tow ratios and material types, if any.

The finite element model is defined by entering nodal coordinates and laminate type identifier for each node, and the element connectivities. The local nodal coordinate systems are defined by the finite element code such that the local z axis is in the normal direction of the laminate. The local y axis is defined from the cross-product of the local z and global x axes. The local x axis is defined from the cross-product of the local y and local z axes. The boundary conditions are defined by entering the node number and the degree of freedom direction number corresponding to a displacement boundary restraint. Uniform displacement requirements are implemented by duplicate node constraints that combine the degrees of freedom of two or more nodes. The distinct input requirements and corresponding output results from each type of analysis are as follows:

8.4.1 Composite Analysis under Static Loading

The input data for static analysis must define the loading. Concentrated forces are defined at nodes and corresponding to the degree-of-freedom directions in which they act. Pressures may be applied on the upper and lower surfaces of Mindlin type thick shell finite elements. Applied pressures are defined as positive if they act on the upper surface of the laminate (upper surface of the laminate is that penetrated by the positive local z axis). Pressures applied on the lower surface are input as negative. Forces and pressures are gradually increased by the progressive fracture analysis code as local damage modes are evaluated and tracked. If the initially applied forces and pressures are greater than those corresponding to damage initiation, they are reduced to a lower level to enable gradual damage tracking. Computational simulation is carried out using an incremental loading iterative approach. The load incrementation rate is controlled by two user supplied parameters that specify the maximum number of nodes allowed to sustain damage and the maximum number of nodes allowed to experience through-the-thickness fracture during a single iteration.

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The output from static analysis includes laminate through-the- thickness stress resultants, laminate surface strains, ply level stresses, constituent level microstresses, and the progression of damage characterized by the damage index, as a function of loading, at each ply of each node. The output data can be processed by a graphical user interface (GUI). Stress and damage parameters can be plotted on the finite element model for quick and effective visualization of structural damage locations.

8.4.2 Composite Analysis under Low-Cycle Fatigue Loading

Low-cycle fatigue loading is defined as a cyclic loading with a maximum load that is below the static ultimate structural failure load. Cyclic loads are usually applied slowly such that inertial effects are not important. An example of low-cycle fatigue loading is represented by pressurization-depressurization cycles of an aircraft fuselage. Each cycle of a low-cycle fatigue loading is applied as a static loading. A cyclic degradation coefficient 13 is required as an input parameter to characterize the degradation of the analyzed composite parts. Computational simulation under low-cycle fatigue repeats the same load amplitude, but increases the number of load cycles until damage initiation, progression, and ultimate failure stages are simulated. Cyclic loading degradation is evaluated by the composite mechanics module. Cyclic loads are superimposed over static, thermal, and hygral loads. Static ply strengths in the longitudinal tensile/compressive, transverse tensile/compressive, and shear components are degraded due to changes in temperature, moisture, and also with the number of cycles, according to the cyclic degradation coefficient. The cyclic ply strength degradation factor is computed as F=1-13 log N. Cyclic load ratios are defined at each node as the ratio of actual local load magnitude to that which would cause a ply stress to reach its limit. For each ply of each node cyclic load ratios are computed for Nxx, Nyy, Nxy, Mx~, Myy, and Mxy loads with regard to the upper and lower limits of ~1, ~22, and a~2 stresses. For each stress limit, the static load ratio and the six cyclic load ratios are added to obtain the sum of load ratios. If the sum of load ratios is greater than 1.0 in absolute value, failure is predicted. The failure mode is determined according to the ply stress limit that is exceeded by the load ratio sum that is greater than one.

Cyclic load amplitudes are defined at nodes and corresponding to the degree-of-freedom directions in which they are acting. Nonzero

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average stresses are modeled by superimposing a static load over a cyclic load. Cyclic pressure loads may also be applied on the upper and lower surfaces of finite elements. Under cyclic fatigue simulation, force and pressure amplitudes are kept constant but the number of cycles is increased by the progressive fracture analysis code as local damage modes are evaluated and tracked. Computational simulation is carried out by incrementing the number of loading cycles. The output from cyclic analysis includes laminate through-the-thickness stress resultants, laminate surface strains, ply level stresses, constituent level microstresses, and the progression of damage characterized by the damage index, as a function of the number of cycles, at each ply of each node. The output data can be processed by a graphical user interface (GUI). Stress and damage parameters can be plotted on the finite element model for quick and effective visualization of structural damage locations.

8.4.3 Composite Analysis under High-Cycle Fatigue Loading

High-cycle fatigue loading is defined as a cyclic loading with a maximum load that is much below the static ultimate structural failure load. Cyclic loads are usually applied dynamically such that inertial effects are important. An example of high-cycle fatigue loading is the dynamic loading cycles applied on a composite airfoil due to base accelerations or harmonic pressures produced by an acoustic environment. A cyclic degradation coefficient 13 is required as an input parameter to characterize the degradation of the analyzed composite material. Computational simulation under high-cycle fatigue requires the input definition of harmonic loads. The computational simulation cycle begins with the definition of constituent properties from a materials databank. The composite mechanics module is called before and after each finite element time-history analysis. Prior to each finite element analysis, the composite mechanics module computes the composite properties from the fiber and matrix constituent characteristics and the composite layup. The finite element analysis module accepts the composite properties that are computed by the composite mechanics code at each node and performs a time-history dynamic structural analysis for a time increment. After a finite element analysis, the computed generalized nodal force and moment time histories are supplied to the composite analysis module that evaluates the nature and amount of local

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damage, if any, in the plies of the composite laminate. The evaluation of local damage due to cyclic loading is based on simplified mathematical models embedded in the composite mechanics module. The step of the simulation process is shown in Figure 16.

MICROMEC~CS: PLY PRiPER'r~.~

M AC ROM~CNANICS: I . A M ~ A T E

Figure 16. Low/high Cycle Fatigue Process Figure 17. Random Fatigue Process

The fundamental assumptions in the development of these models are the following: (1) Fatigue degrades all ply strengths at approximately the same rate. (2) All types of fatigue degrade laminate strength linearly on a semilog plot including: (a) mechanical (tension, compression, shear, and bending); (b) thermal (elevated to cryogenic temperature); hygral (moisture); and combinations (mechanical, thermal, hygral, and reverse-tension compression). (3) Laminated composites generally exhibit linear behavior to initial damage under uniaxial and combined loading. (4) All ply stresses (mechanical, thermal, and hygral) are predictable by using linear laminate theory.

Individual ply failure modes are assessed by using margins of safety computed by the composite mechanics module via superposition of the six cyclic load ratios. The cyclic loads that are considered are the Nxx, N~, Nxy, in-plane loads and Mxx, M~, and Mxy flexural moments. The lower and upper limits of the cyclic loads, the number of cycles, and the cyclic degradation coefficient 13 are supplied to the composite mechanics module at each node for the computation of a complete failure analysis based on the maximum stress criteria. The cyclic degradation coefficient 13 has been determined to be in the range of 0.01 to 0.02 for graphite/epoxy composites. Time-history dynamic analysis of composite structures subjected to cyclic excitation are conducted using the modal basis. Computed nodal stress resultant time-histories are used to assess the maximum and minimum values of the local load cycles and frequencies at each node. The composite mechanics module with cyclic load analysis capability

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evaluates the local composite response at each node subjected to fluctuating stress resultants. The number of cycles and the time required to induce local structural damage are evaluated at each node. After damage initiation, composite properties are reevaluated based on degraded ply properties and the overall structural response parameters are recomputed. Iterative application of this computational procedure results in the tracking of progressive damage in the composite structure subjected to cyclic load increments. Computational simulation cycles are continued until the composite structure fractures. The number of cycles for damage initiation and the number of cycles for structural fracture are identified in each simulation. After damage initiation, when the number of load cycles reaches a critical level, damage begins to propagate rapidly in the composite structure. After the critical damage propagation stage is reached, the composite structure experiences excessive damage or fracture that renders it unsafe for continued use.

The generalized stress-strain relationships are revised locally according to the composite damage evaluated after each finite element time-history analysis. The model is automatically updated with a new finite element mesh having reconstituted properties, and the structure is reanalyzed for further deformation and damage. If there is no damage after a cyclic load increment, the structure is considered to be in equilibrium and an additional number of cycles are applied leading to possible damage growth, accumulation, or propagation. Simulation under cyclic loading is continued until structural fracture.

In general, overall structural damage may include individual ply damage and also through-the-thickness fracture of the composite laminate. The computational simulation procedure uses an accuracy criterion based on the allowable maximum number of damaged and fractured nodes within a simulation cycle. If too many nodes are damaged or fractured in the simulation cycle, the number of cycles are reduced and analysis is restarted from the previous equilibrium state. Otherwise, if there is an acceptable amount of incremental damage, the number of cycles is kept constant but the constitutive properties are updated to account for the damage from the last simulation cycle. After a valid simulation cycle in which composite structural degradation is simulated with or without damage or fracture, the structure is

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reanalyzed for further damage and deformation. When there is no indication of further damage under a load, the structure is considered to have endured the previously applied cyclic loading. Subsequently, additional sets of cyclic loads are applied leading to possible damage growth, accumulation, or propagation. Analysis is stopped when global structural fracture is predicted.

The type of damage growth and the sequence of damage progression depend on the composite structure, loading, material properties, and hygrothermal conditions. A scalar damage variable, derived from the total volume of the structure affected by various damage mechanisms is also computed as an indicator of the level of overall damage induced by loading. This damage variable is useful for assessing the overall degradation of a given structure under a prescribed loading condition. The rate of overall damage growth during damage progression may be used to evaluate the propensity of structural fracture with increasing loading.

Under high-cycle fatigue dynamic force and pressure amplitudes are kept constant but the duration of loading is increased by the progressive fracture analysis code as local damage modes are evaluated and tracked. The output from cyclic analysis includes laminate through-the-thickness stress resultants, laminate surface strains, ply level stresses, constituent level stresses, and the progression of damage characterized by the damage index, at each ply of each node. The output data can be processed by a graphical user interface (GUI). Stress and damage parameters can be plotted on the finite element model for visualization of stress concentrations and structural damage locations.

8.4.4 Random Power Spectral Density Fatigue Loading

Fatigue under power spectral density (PSD) loading is defined as a random cyclic loading with varying frequency content. PSD fatigue loads are defined as forces, pressures, or accelerations. The input data file defines the load/acceleration levels corresponding to the considered frequency ranges. The loads are applied dynamically such that inertial effects are taken into account. Structural analysis is carried out in the frequency domain by the finite element code via superposition of the component harmonic analyses. Results are

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numerically integrated in the frequency domain. The finite element results consist of the mean-square stress resultants at nodal points. The PSD module determines the dominant frequencies in the mean- square response and interprets the results as equivalent nodal cyclic loads for durability analysis. An example of PSD random fatigue loading is the dynamic shaking of a composite airfoil due to base accelerations. Computational damage tracking under PSD fatigue loading follows closely the methods used under high-cycle fatigue evaluations. The step of the simulation process is shown in Figure 17.

8.4.5 Composite Analysis under Impact Loading

GENOA progressive failure analysis software is able to simulate composite structures under impact and compression after impact (CAI) loading. Simulation extends to the prediction/verification under the impact loading of impact load versus time, damage, and fracture pattern of a composite sandwich foam structure and prediction/verification of damage pattern, residual strength under CAI loading. An example is composite foam sandwich specimen under impact and CAl. A composite foam core sandwich panel (Skin: G30- 500/R3676, adhesive; FM-300, foam core: Rohacei 200WF), was simulated [36]. GENOA progressive failure analysis was used to predict the impact process, the impact damage (Figure 18), and subsequent damage propagation in the panel under pure compressive load. The impact test showed that the foam core in the middle of the panel was damaged over a 5x5-inch area by the impact. No impact damage occurred in the skin surface. Simulation results were compared with experimental impact and subsequent test in compression, The post impact evaluation showed the presence of minimal damage on the skin, and core. The post CAI evaluation showed crippling effect and the separation of skin from the core on the lower skin for both test and simulation.

Input data for simulation under impact loading requires the definition of the initial velocity and mass of the impacting object, finite element model of the impacting object, as well as that of the impacted composite structure. Analysis options include (1) dynamic analysis including inertial effects both in the impacted composite structure and the impacting object and (2) pseudo-static analysis where only the

376

Simulated Fracture Pattern (5 by 5)

Figure 18. Comparison of the Fracture Pattern between test and simulation

inertial forces of the impacting object are considered. Simulation is based on an energy balance approach where the reduction of kinetic energy of the impacting object is balanced by the work done by the impacting force. Impact energy is imparted to the composite structure as strain energy, kinetic energy, and damage energy. Results include local damage in the composite structure, stresses, displacements, and velocities with time from the initiation of impact.

8.4.6 Composite Analysis under Creep Loading Creep rupture effects are important for ceramic matrix composites

(CMC) at elevated temperatures under tensile stresses. Creep effects are evaluated by considering the effects of time exposure to stresses. Semi-empirical multi-factor interaction strength degradation with time model is used to quantify the creep rupture response of composite structures. A refined composite mechanics module based on fiber substructuring is used for the simulation of CMC structures. Figure 19 shows the unit cell and also it's slice by slice representation.. The CMC module is based on software developed to specifically simulate aspects unique to ceramic matrix composites. It incorporates a fiber substructuring technique that provides: 1) accurate micromechanical representation of interfacial conditions, both around the fiber and through-the-thickness properties, and 2) greater detail of stress distribution within a ply. This module generates the properties (A, B, D

377

Figure 19. Ceramic Matix Composite Unit Cell Representation

matrix) of ceramic matrix composites from properties of the constituents (fiber, matrix, and fiber-matrix interface). The generated structural properties are used in the FEM module. CMC module can also be used to determine constituent level stresses, such as fiber, matrix and interface stresses. The FEM module performs finite element analysis to determine the generalized stresses over the entire ceramic composite structure. These generalized stresses are referred back to the CMC module for iterative determinations of composite microstresses.

The PFA module keeps a detailed account of composite degradation over the entire structure and also acts as an executive control module to direct the CMC module to perform micromechanics, macromechanics, and structural property synthesis functions. The damage progression module also calls FEM module with anisotropic thick shell analysis capability to model laminated composites for global structural response. After each finite element analysis, the CMC module is called to compute constituent stresses. These stresses are compared to failure criteria for evaluation of possible fiber, matrix, and interface failure in each ply at each node of the ceramic composite structure. Once the state of damage at each node is assessed, a damage index is created to record the damage information at each damaged node. The damage index contains the node number, the ply number, and the list of damage criteria that have become activated. Fibers, matrices, and interfaces are each assigned their own damage index to record the corresponding damage information. When a new

378

failure occurs in a fiber, matrix, and interface, its damage index is updated. The properties of fibers, matrices and interfaces are degraded according to their damage index. For example, if a fiber is damaged due to longitudinal tensile failure criteria, its corresponding modulus (E,) is degraded to almost zero.

8.5 Conclusions From the general perspective of the integrated computational

evaluation method, the following conclusions are drawn: 1) The assessment and design of fiber composite structures may be improved by an evaluation of the composite damage initiation and damage propagation mechanisms. 2) In general, composite structures demand a detailed understanding of the stress states and material degradation characteristics for effective design. 3) Computational simulation, with the use of established composite mechanics and finite element methods, can be used to simulate the various stages of degradation in a composite structure, such as damage initiation, damage growth, damage accumulation, damage propagation, and structural fracture. 4) Computational simulation is based on constituent level material properties. However, even if the fiber and matrix properties are not explicitly defined, they can be determined from the given composite ply properties using a system identification procedure. 5) Significant design parameters such as composite constituent stiffness, strength, and effective ply configurations may be identified with the help of a composite mechanics module and computational simulation. 6) The progressive damage tracking procedure is flexible and applicable to all types of constituent materials, structural geometry, and loading. Hybrid composites and homogeneous materials, as well as binary composites can be simulated. 7) The present computational simulation procedure provides a new general methodology to investigate damage propagation and progressive fracture for composite structures.

References

l) Chamis, C.C. (1969), Minnetyan, L., "Progressive Fracture of Polymer Matrix Composite Structure a New Approach" 14th Annual Energy-Source Technology Conference and Exhibition, Houston Texas, January 1992.

379

2)

3)

4)

5)

6)

7)

8)

9)

lo)

Darwin Moon, Frank Abdi, Alpha Bill Davis,. Discrete Source Damage Tolerance evaluation of s/RFI stiffened panels, {Sampe 99}, Long Beach, Ca, May 1999

Cox, B. N, Flanegan, G., "Handbook of Analytical Methods for Textile Composites Binary Model Of Textile Composites," Rockwell Science Center, Version 1, January, 1996.

Damage-Tolerance and Fatigue Evaluation of Structure, Advisory Circular 25.571-1A, Federal Aviation Administration, Washington, DC, 1986.

Composite Aircraft Structure, Advisory Circular 20-107A, Federal Aviation Administration, Washington, DC, 1984.

Minnetyan, L., Murthy, P.L.N., and Chamis, C.C., "Composite Structure Global Fracture Toughness via Computational Simulation," Computers & Structures, Vol. 37, No. 2, pp.175-180, November 1990.

Irvine T. B. and Ginty C.A., "Progressive Fracture of Fiber Composites," Journal of Composite Materials, Vol. 20, March 1986, pp. 166-184.

Minnetyan L., Chamis C.C., and Murthy P.L.N., "Structural Behavior of Composites with Progressive Fracture," Journal of Reinforced Plastics and Composites, Vol. 11, No. 4, April 1992, pp. 413-442.

Minnetyan L., Murthy P.L.N., and Chamis C.C., "Progressive Fracture in Composites Subjected to Hygrothermal Environment," International Journal of Damage Mechanics, Vol. 1, No. 1, January 1992, pp. 60-79.

Minnetyan, L., Chamis, C. C., and Murthy, P. L. N., "Structural Durability of a Composite Pressure Vessel," Journal of Reinforced Plastics and Composites, Vol. 11, No. 11, November 1992, pp. 1251-1269.

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12)

~3)

14)

15)

16)

Minnetyan, L., and Chamis, C.C., "Pressure Vessel Fracture Simulation," Presented at the ASTM 25th National Symposium on Fracture Mechanics, Lehigh University, Bethlehem, Pennsylvania, June 28-July 1, 1993, Published in ASTM STP 1220, Fracture Mechanics: 25th Volume, August 1995, pp. 671-684.

Minnetyan, L., Rivers, J. M., Chamis, C. C., and Murthy, P. L. N., "Discontinuously Stiffened Composite Panel under Compressive Loading," Journal of Reinforced Plastics and Composites, Vol. 14, No. 1, January 1995, pp. 85-98.

Minnetyan, L., and Chamis, C. C. 1997. "Progressive fracture of composite cylindrical shells subjected to external pressure." ASTM J. Compos. Technol. and Res., 1 9(2), 65-71.

Chamis, C.C. (1969) "Failure Criteria for Filamentary Composites," Composite Materials Testing and Design: ASTM STP 460, American Society for Testing and Materials, Philadelphia, pp 336-351.

Chou, T, W, Ko, F. K, "Textile Structural Composites," Elsevier, 1989, pp. 16.

Minnetyan, L., Murthy, P.L.N. and Chamis, C. C. 1992. "Progressive Fracture in Composites Subjected to Hygrothermal Environment," International Journal of Damage Mechanics, Vol. 1, No. 1, pp69-70.

17) Cox, B. N, Carter, W. C, and Fleck, N. A., "A Binary Model Of Textile Composites Formulation," Acta Metall. Mater, Vol, 42, No. 10 pp. 3463-3479, 1994.

]8) Hallquist, J. O, Whirley, R. G., "DYNA3D User's Manual," November, 1982.

19) F. Abdi, R. Lorenz., Hadian, J., "Progressive Fracture of Braided Composite Turbomachinery Structures," NASA SBIR Phase I Report No. NAS3-27334, 1996.

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20) F. Abdi, Minnetyan, L., "Progressive Fracture of Braided Composite Turbomachinery Structures," NASA SBIR Phase II Report No. NAS3-97041, 1999.

21) Murthy, P. L. N., and Chamis, C. C. (1986). "Integrated composite analyzer (ICAN): Users and programmers manual." NASA Tech. Paper 2515, Nat. Aeronautics and Space Admin., Washington, D. C.

22) Starnes, J. H, and Britt, V. O, Rankin, C. C. "Nonlinear Response of Damaged Stiffened Shells subjected to Combined Internal Pressure and Mechanical Loads" AIAA 951462-CP.

23) Nakazawa, S. "The MHOST Finite Element Program, 3-D Inelastic Analysis Methods for Hot Section Components," Volume I - Theoretical Manual, NASA Contract Report CR-182205, August, 1991.

24) Murthy, P.L.N., Chamis, C.C. 1986. "Integrated Composite Analyzer (ICAN): Users and Programmer's Manual," NASA Technical Paper 2515, 1986.

25) B. N. Cox, "Failure Models For Textile Composites," NASA Contractor Report 4686, Contract No. NAS1-19243, August, 1995.

26) Chamis, C. C., Murthy, P. L. N., and Minnetyan, L. (1996). "Progressive fracture of polymer matrix composite structures." Theoretical and Applied Fracture Mech., 25(1), 1-15.

27) Kregers, A. F. and Melbardis Y. G. 1978. "Determination of the Deformability of Three-Dimensional Reinforced Composites by the Stiffness Averaging Method," Polymer Mechanics, No. 1 pp3- 8.

28) Abdi, F., Davis, B., and Kedward, K., "Analytical and Experimental Verification of DSD in S/RFI Commercial Aircraft" Contract No, NAS1-20546-6XY016021, June 7, 1999.

382

29) Huang, D. (1998). "GENOA Progressive Failure Analysis Program: Computational Simulation of Three-Dimensional Fiber Reinforced Composites, Volume I-Theoretical Manual" AlphaSTAR Corporation, Long Beach, California, September 1998.

30) Xu, J., Cox, B.N., McCIockton, M.A., and Carter, W.C. 1992, "A Binary Model of Textile Composites: II The Elastic Regime," Acta Metallurgica Materiatia, Vol.43, No. 9, pp3511-3524.

31) Huang, D. "Computational Simulation of Damage Propagation in Three-Dimensional Woven Composites, A Doctoral Thesis, Clarkson University 1997.

32) Abdi, F. Goldsworthy and Associates "Blade Manufacturing Process Technology Development of Preform Composite Net Shapes," December 1998.

33) Shah, A. R. Shiao, M. C. Nagpal, V. K., and Chamis. C. "Probabilistic Evaluation of Uncertainties and Risks in Aerospace Components. Chapter 10, AIAA 1992.

34) Liu, W. K. and Belytschko, T. "Stochastic Computational Mechanics for Aerospace Structures, Chapter 7, AIAA 1992.

35) F. Abdi, R. Lorenz., "Concurrent Probabilistic Simulation of High Temperature Composite Structural Response", NASA Lewis SBIR Phase II Final REPORT, 1995.

36) Huang, D.), Abdi, F., Khatiblou, M. " Progressive Failure Analysis (PFA) and Verification of Composite Test Panel Under Impact and Tension After Impact (TAI) Loading Using GENOA" Contract No. FW809FAH, December 1999.

383

APPENDIX A

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INDEX

A

Accumulative damage, 10 Abrupt failure, 76 Advanced single specimen, 220 Airy stress function, 59 Anisotropy

-beam analysis, 330 -material, 69

Anti buckling device, 111, 112 Applications of

-FSW welds, 280 -VPPA welds, 292-301

Apparent fracture toughness, 78 Area under load displacement, 217 ASTM-E-399, 27 ASTM E-466, 6 ASTM E-561,27 ASTM E-616, 69 ASTM E-647, 122 ASTM E-740, 97 ASTM E-1681, 174 ASTM E-1820, 200, 201,224 Available energy, 53, 184

B

Back surface correction, 95 Basquin exponent, 13-14 Beam analysis, 330 Bend bar J specimen, 216 Bolts joint analysis

-bearing stress, 318

-fatigue crack growth, 317 -pad analysis, 317-328 -pad bending, 319, 320 -preloaded bolt, 307

Brittle failure (cleavage), 24, 52, 53 Butt weld, 2

C

Categorization of composites, 339 Center crack, 110 Charpy test, 71,72 Circular surface crack, 94 Closure Phenomenon, 29

-Elber, 133 -Newman, 141 -Raju, 136

Coffin-Manson law, 15 Combined stress intensity, 63 Comments on J, 200 Comments on Miner's rule, 10 Compact tension specimen, 103 Compliance method, 128 Compliance of cracked specimen, 185, 187 Composites, 334 Composites analysis

-cyclic load, 371-375 -creep, 377 -static, 370 -impact, 376

Composite degradation, 343

Composite simulation software, 344, 349 Constant amplitude fatigue test, 127-130 Constant displacement, 186 Constant load, 187 Conventional fatigue

-High cycle fatigue, 6- 12

-low cycle fatigue, 12- 23 Corrosion testing, 173-175 Correction

-Back surface, 95 -Plastic zone, 25, 33, 87 -Thickness, 68, 71,257 -Width, 62, 81,225, 252

Crack arrest feature, 37 Crack closure effect, 132 Crack extension force, 55 Crack initiation concept

-PSB, 118

Crack growth analysis -bolts, 304-317 -pads, 317-328

Crack growth rate, 122 -ASTM testing, see

testing -cycle by cycle, 165 -equations, 124, 126 -NASA/FLAGRO, 126 -stress ratio effect, 132 -Threshold value, 125,

137 Crack in a pressurized pipe, 95 Crack opening displacement, 31 Crack opening function, 145 Crack opening stress intensity, 58-61

Crack tip -blunting, 190 -modes, 56 -plasticity, 88, 240 -straining, 240 -stresses, 61

Creep effect, 13, 377 Critical initial flaw size, 256, 257 Critical stress intensity factor, 65 C(T) specimen, 104 Curve

-cyclic stress-strain, 14 -stress-strain, 242, 259

Cycle by cycle fatigue crack growth, 165 Cyclic stress-strain curve, 14

D

Damage tolerance, 334, 346 Damage energy release rate, 357 Damage evaluation, 354 Damage tracking process, 351 Deformation

-plastic, 88, 240 -Modes, 57 -theory of plasticity, 203

Delamination, 334, 336, 339 Determination of

_o~UF energy term, ac

241

244

.o~U v energy term, o~c

-W F, 242 -W u, 255

403

Development of -R-curve, 79-80

Diagram -S-N, 8-9 -E-N, 16 -residual strength

capability, 28, 39, 74, 57 -Kitagawa, 140

Ductile -brittle transition, 72, 73 -fracture (see EPFM

and FMDM theory) Dugdale's model, 26, 33 Durability and damage tolerance, 334

E

Eddy current, 44, 45 Effective crack size, 81 Effect of crack size

-crack closure, 133 -threshold stress

intensity, 139 Effect of thickness

-fracture toughness, 68, 71,257 Effect of temperature

-fracture toughness, 71, 72 Effect of rate of loading

-fracture toughness, 73 Effect of yielding

-fracture toughness, 73 Effective crack length, 81 Effective stress intensity, 136, 142 Effective stress range, 136 Elastic constants, 359 Elastic energy release rate, 55, 185

Elastic fracture, 52 Elber crack closure, 133 Endurance limit, 9 Energy balance approach, 53, 183 Energy per unit volume, 180, 241 Energy rate interpretation of J, 210 Energy release rate, 183-189, 355 Environmental corrosion, 167 EPFM, 31,182 Equilibrium equation, 53, 58, 186 Examples

-Chapter (1), 10,18, 21, 22

-Chapter (2), 84, 100 -Chapter (3), 147, 151,

152, 159 -Chapter (4), 198 -Chapter (6), 292 -Chapter (7), 313, 320,

324 Experimental evaluation of J, 209 Extended Griffith theory, 237

F

Factors influencing -fracture toughness, 71 -threshold stress

intensity, 139 Fail safe design, 36 Failure criterion

-Octohedral shear stress, 245, 253

-Von-Mises, 88 Failure evaluations, 353

404

Failure prevention, 35 Fatigue crack growth, 118 Fatigue ductility coefficient, 16 Fatigue strength coefficient, 15 Fatigue

-high cycle, 6-12, 310, 372

-low cycle, 12-23, 371 FEM of PFA, 361 Fiber arrangements, 361 Fixed grip condition, 184, 186

Flaw size -elliptical, 92 -embedded, 93 -size inspection (see

NDI) FMDM theory, 25, 180, 237 Forman equation, 126 Fracture

-brittle, 24, 25 -stress, 24, 256

Fracture control plan, 35 Fracture toughness, 26, 66

-apparent, 78 -by FMDM, 282 -equivalent, 359 -plane strain, 103 -plane stress, 107 -surface crack, 97 -testing (see testing),

102-112 Friction stir welding, 3, 287-301

G

GENOA computer code, 351- 353, 356 Griffith theory, 24,53, 183 Grip fixture apparatus, 111

H

Heat affected zone, 2, 276, 289 High cycle fatigue, 6-12, 309, 310 HRR field, 197 Hysteresis loop, 13

Incremental theory, 203 Inspection (see NDI) Irwin plastic zone, 87 Irwin theory of fracture, 25 Initiation

-fatigue crack, 7, 118

J J-Integral, 31,180 J-Controlled, 205-207 JIc, 33, 202, 224 JIc testing, 209 J-R curve, 33, 202, 219

K

Kitagawa diagram, 140 Keyhole welding, 279

I.

Landes and Begely experiment, 209, 212-214 Leak-before-burst, 99 LEFM, 24, 58 Linear cumulative damage, 9 Load line displacement, 220 Loading/unloading steps, 223 Longitudinal surface crack, 95 Low cycle fatigue, 12-23

405

M

Magnetic particle inspection, 46-47 Mandrelizing, 4 Manson, 15 Manufacturing process, 40 Material selection, 40 (see also appendix A) Material anisotropy, 69 Mesh refinement, 365 Metallurgical examination, 298- 299 Miner's rule, 9 Mismatch in welds, 282 Mixed mode fracture, 68, 257 Modes of deformation, 57 M(T) specimen, 110 Multispecimen technique, 210 Multiple load path design, 36

N

NASA/FLAGRO computer code, 29, 120 Net section yielding, 146 Neuber's rule, 19 Newman crack closure, 141 Non destructive inspection (NDI), 41,119

-Eddy current, 44-45 -magnetic particle, 43-

44 -penetrant, 42-43 -radiographic, 46-47 -ultrasonic, 45,46

O

Octahedral shear stress, 245, 253

Opening mode, 57 Opening stress, 136 Overload, 30

P

Power

curve,

Paris Law, 29 Part through cracks

-fracture toughness, 96 -stress intensity factor,

92 Path independent integral, 189, 191-194 Penetrant inspection (see NDI) Persistent slip bands (PSB), 118 Pitting Phenomenon, 168 Plane strain fracture toughness, 66-68, 103 Plane stress fracture toughness, 66-68, 107 Plastic zone shape

-Irwin, 87 -Tresca, 90 -Von-Mises, 88 law representation -cyclic stress-strain

-stress-strain curve, Plate bending, 319-320 Polymer matrix composite, 335 Pop-in concept, 228 Potential energy differences, 211 Power spectrum density, 375 Pre-cracking, 104, 128, 222 Preloaded bolts, 307 Probabilistic failure, 367, 368 Properties of welded joints

-FSW, 290-292 -VPPA, 281-289

406

Progressive fracture, 342 Progressive fracture analysis, 346 Pultrusion, 337

Q

R

Radiographic inspection, 46-47 Ramberg-Osgood equation, 198, 250 Reshaping braided fiber, 365 Residual strength capability, 28,39, 74, 82, 257 Retardation effect, 30

-Wheeler model, 30, 153

-Willenborg model, 30 Reversal to failure, 15 R-curve, 79, 108 Rolled threads, 4, 305-307

S

Safe-life, Side grooving, 221-222, 229 Simulative law, 8 Single specimen technique, 215-220, 224 Slope method

-polynomial, 122-123 -secant, 123

Small cracks, 138 Specimen preparations, 103- 104 Stitched simulation, 360 Stage I and stage II crack growth, 7, 118 Stored elastic energy, 184 Strain-controlled, 15

Strain energy damage rate, 358 Strain energy density, 192

Strain-life prediction, 15 Stress and strain at notch, 19 Stress concentration factor, 19, 3O6 Stress 172 Stress intensity factor,

-derivation, 58-61 -equation, 61 -range, 121 -surface crack, 92

Stress function -Airy, see Airy

Stress ratio effect, 132 Stress-strain curve, 242,259 Structural composite analysis, 369 Superposition principle, 63 Surface cracks, 91

-stress intensity factor, 91

-longitudinal in a pipe, 95 -fracture toughness, 97

Surface energy, 25, 55

corrosion cracking, 167-

T

Tearing failure, 77, 180, 181 Testing

-crack growth test, 127 "J ~c test, 224-232 -K= c test, 103 -Kc test, 107 -KEA c test, 173-175 -K~EAc test, 173-175 -S-N test, 6

407

-specimen, 103, 104, 110 Textile composites, 337 Theoretical stress concentration factor, 19, 306 Thickness effect, 68, 71,257 Threaded bolt, 305-306 Threshold stress intensity factor, 137, 170 Total damage energy, 355, 356 Transition point, 17 Tresca yield criterion, 90 Triaxial stress state, 67 Two and 3D composites, 341

LI

Ultrasonic inspection, 45, 46

V

Validity check for JIc, 228-232 Variable amplitude stress, 153 Verification of FMDM result, 259 Von-Mises yield criterion, 88 Variable polarity plasma (VPPA) weld, 3, 277-287 VPPA weld properties

-FSW, 281-287 -VPPA, 283-287

W

Welded joints failure, Welded joints, 274-301 Welding of aluminum, 275 Weld mismatch, 282 Westergaard stress function, 60 Wheeler model, 156

Willenborg model, 157 W~hler's diagram, 7 Woven pattern, 361

X

X-Ray inspection, 46-47

Y

Yield criterion -Tresca, 90 -Von-Mises, 88

Yield strength effect, 74

Z

408