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Modeling and Simulation for Material Selection and Mechanical Design

edited by George E. Totten

Lin Xie

Kiyoshi Funatani

G.E. Totten & Associates, LLC Seattle, Washington, i7.S.A

Solidworks Corporation Concord, Massachusetts, U.S.A

IMST Institute Nagoya, Japan

M A R C E L

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ENGINEERING A Series of Textbooks and Reference Books

Founding Editor

L. L. Faulkner Columbus Division, Battelle Memorial Institute

and Department of Mechanical Engineering The Ohio State University

Columbus, Ohio

1. Spring Designer's Handbook, Harold Carlson 2. Computer-Aided Graphics and Design, Daniel L. Ryan 3. Lubrication Fundamentals, J. George Wills 4. Solar Engineering for Domestic Buildings, William .A. Himmelman 5. Applied Engineering Mechanics: Statics and Dynamics, G. Boothroyd and

C. Poli 6. Centrifugal Pump Clinic, lgor J. Karassik 7. Computer-Aided Kinetics for Machine Design, Daniel L. Ryan 8. Plastics Products Design Handbook, Patf A: Materials and Components; Patf

6 : Processes and Design for Processes, edited by Edward Miller 9. Turbomachinery: Basic Theory and Applications, Earl Logan, Jr.

10. Vibrations of Shells and Plates, Werner Soedel 1 I. Flat and Corrugated Diaphragm Design Handbook, Mario Di Giovanni 12. Practical Stress Analysis in Engineering Design, Alexander Blake 13. An lntroduction to the Design and Behavior of Bolted Joints, John H.

Bickford 14. Optimal Engineering Design: Principles and Applications, James N. Siddall 15. Spring Manufacturing Handbook, Harold Carlson 16. Industrial Noise Control: Fundamentals and Applications, edited by Lewis H.

Bell 17. Gears and Their Vibration: A Basic Approach to Understanding Gear Noise,

J. Derek Smith 18. Chains for Power Transmission and Material Handling: Design and Appli-

cations Handbook, American Chain Association 19. Corrosion and Corrosion Protection Handbook, edited by Philip A.

Schweitzer 20. Gear Drive Systems: Design and Application, Peter Lynwander 2 1 . Controlling In-Plant Airborne Contaminants: Systems Design and Cal-

culations, John D. Constance 22. CAD/CAM Systems Planning and Implementation, Charles S. Knox 23. Probabilistic Engineering Design: Principles and Applications, James N.

Siddall 24. Traction Drives: Selection and Application, Frederick W. Heilich 111 and

Eugene E. Shube 25. Finite Element Methods: An Introduction, Ronald L. Huston and Chris E.

Passerello

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

26. , Brayton Lincoln, and

27. Lubrication in Practice: Second Edition, edited by W. S. Robertson 28. Principles of Automated Drafting, Daniel L. Ryan 29. Practical Seal Design, edited by Leonard J. Martini 30. Engineering Documentation for CAD/CA M Applications, Charles S. Knox 31 . Design Dimensioning with Computer Graphics Applications, Jerome C.

Lange 32. Mechanism Analysis: Simplified Graphical and Analytical Techniques, Lyndon

0. Barton 33. CAD/CAM Systems: Justification, Implementation, Productivity Measurement,

Edward J. Preston, George W. Crawford, and Mark E. Coticchia 34. Steam Plant Calculations Manual, V. Ganapathy 35. Design Assurance for Engineers and Managers, John A. Burgess 36. Heat Transfer Fluids and Systems for Process and Energy Applications,

Jasbir Singh 37. Potential Flows: Computer Graphic Solutions, Robert H. Kirchhoff 38. Computer-Aided Graphics and Design: Second Edition, Daniel L. Ryan 39. Electronically Controlled Proportional Valves: Selection and Application,

Michael J. Tonyan, edited by Tobi Goldoftas 40. Pressure Gauge Handbook, AMETEK, U.S. Gauge Division, edited by Philip

W. Harland 41. Fabric Filtration for Combustion Sources: Fundamentals and Basic Tech-

nology, R. P. Donovan 42. Design of Mechanical Joints, Alexander Blake 43. CAD/CAM Dictionary, Edward J. Preston, George W. Crawford, and Mark E.

Coticchia 44. Machinery Adhesives for Locking, Retaining, and Sealing, Girard S. Haviland 45. Couplings and Joints: Design, Selection, and Application, Jon R. Mancuso 46. Shaft Alignment Handbook, John Piotrowski 47. BASIC Programs for Steam Plant Engineers: Boilers, Combustion, Fluid

Flow, and Heat Transfer, V. Ganapathy 48. Solving Mechanical Design Problems with Computer Graphics, Jerome C.

Lange 49. Plastics Gearing: Selection and Application, Clifford E. Adams 50. Clutches and Brakes: Design and Selection, William C. Orthwein 51. Transducers in Mechanical and Electronic Design, Harry L. Trietley 52. Metallurgical Applications of Shock- Wave and High-Strain-Rate Phenom-

ena, edited by Lawrence E. Murr, Karl P. Staudhammer, and Marc A. Meyers

53. Magnesium Products Design, Robert S. Busk 54. How to Integrate CAD/CAM Systems: Management and Technology, William

D. Engelke 55. Cam Design and Manufacture: Second Edition; with cam design software

for the IBM PC and compatibles, disk included, Preben W. Jensen 56. Solid-state AC Motor Controls: Selection and Application, Sylvester Campbell 57. Fundamentals of Robotics, David D. Ardayfio 58. Belt Selection and Application for Engineers, edited by Wallace D. Erickson 59. Developing Three-Dimensional CAD Software with the ISM PC, C. Stan Wei 60. Organizing Data for ClM Applications, Charles S. Knox, with contributions

by Thomas C. Boos, Ross S. Culverhouse, and Paul F. Muchnicki


61. Computer-Aided Simulation in Railway Dynamics, by Rao V. Dukkipati and

62. fiber-Reinforced Composites: Materials, Manufacturing, and Design, P. K. Mallick

63. Photoelectric Sensors and Controls: Selection and Application, Scott M. Juds

64. finite Element Analysis with Personal Computers, Edward R. Champion, Jr., and J. Michael Ensminger

65. Ultrasonics: Fundamentals, Technology, Applications: Second Edition, Revised and Expanded, Dale Ensminger

66. Applied finite Element Modeling: Practical Problem Solving for Engineers, Jeffrey M. Steele

67. Measurement and Instrumentation in Engineering: Principles and Basic Laboratory Experiments, Francis S. Tse and Ivan E. Morse

68. Centrifugal Pump Clinic: Second Edition, Revised and Expanded, lgor J. Karassik

69. Practical Stress Analysis in Engineering Design: Second Edition, Revised and Expanded, Alexander Blake

70. An Introduction to the Design and Behavior of Bolted Joints: Second Edition, Revised and Expanded, John H. Bickford

71. High Vacuum Technology: A Practical Guide, Marsbed H. Hablanian 72. Pressure Sensors: Selection and Application, Duane Tandeske 73. Zinc Handbook: Properties, Processing, and Use in Design, Frank Porter 74. Thermal fatigue of Metals, Andrzej Weronski and Tadeusz Hejwowski 75. Classical and Modern Mechanisms for Engineers and Inventors, Preben W.

Jensen 76. Handbook of Electronic Package Design, edited by Michael Pecht 77. Shock- Wave and High-Strain-Rate Phenomena in Materials, edited by Marc

A. Meyers, Lawrence E. Murr, and Karl P. Staudhammer 78. Industrial Refrigeration: Principles, Design and Applications, P. C. Koelet 79. Applied Combustion, Eugene L. Keating 80. Engine Oils and Automotive Lubrication, edited by Wilfried J. Bartz 8 1 . Mechanism Analysis: Simplified and Graphical Techniques, Second Edition,

Revised and Expanded, Lyndon 0. Barton 82. fundamental Fluid Mechanics for the Practicing Engineer, James W.

Murdock 83. Fiber-Reinforced Composites: Materials, Manufacturing, and Design, Second

Edition, Revised and Expanded, P. K. Mallick 84. Numerical Methods for Engineering Applications, Edward R. Champion, Jr. 85. Turbomachinery: Basic Theory and Applications, Second Edition, Revised

and Expanded, Earl Logan, Jr. 86. Vibrations of Shells and Plates: Second Edition, Revised and Expanded,

Werner Soedel 87. Steam Plant Calculations Manual: Second Edition, Revised and Expanded,

V. Ganapathy 88. Industrial Noise Control: Fundamentals and Applications, Second Edition,

Revised and Expanded, Lewis H. Bell and Douglas H. Bell 89. finite Elements: Their Design and Performance, Richard H. MacNeal 90. Mechanical Properties of Polymers and Composites: Second Edition, Re-

vised and Expanded, Lawrence E. Nielsen and Robert F. Landel 91. Mechanical Wear Prediction and Prevention, Raymond G. Bayer


92. Mechanical Power Transmission Components, edited by David W. South and Jon R. Mancuso

94. Engineering Documentation Control Practices and Procedures, Ray E. Monahan

95. Refractory Linings Thermomechanical Design and Applications, Charles A. Sc hac h t

96. Geometric Dimensioning and Tolerancing: Applications and Techniques for Use in Design, Manufacturing, and Inspection, James D. Meadows

97. An lntroduction to the Design and Behavior of Bolted Joints: Third Edition, Revised and Expanded, John H. Bickford

98. Shaft Alignment Handbook: Second Edition, Revised and Expanded, John Piotrowski

99. Computer-Aided Design of Polymer-Matrix Composite Structures, edited by Suong Van Hoa

100. Friction Science and Technology, Peter J. Blau 1 0 1 . lntroduction to Plastics and Composites: Mechanical Properties and Engi-

neering Applications, Edward Miller 102. Practical Fracture Mechanics in Design, Alexander Blake 103. Pump Characteristics and Applications, Michael W. Volk 104. Optical Principles and Technology for Engineers, James E. Stewart 105. Optimizing the Shape of Mechanical Elements and Structures, A. A. Seireg

and Jorge Rodriguez 106. Kinematics and Dynamics of Machinery, Vladimir Stejskal and Michael

ValaSek 107. Shaft Seals for Dynamic Applications, Les Horve 108. Reliability-Based Mechanical Design, edited by Thomas A. Cruse 109. Mechanical Fastening, Joining, and Assembly, James A. Speck 1 10. Turbomachinery Fluid Dynamics and Heat Transfer, edited by Chunill Hah 1 1 1. High-Vacuum Technology: A Practical Guide, Second Edition, Revised and

Expanded, Marsbed H. Hablanian 1 12. Geometric Dimensioning and Tolerancing: Workbook and Answerbook,

James D. Meadows 1 13. Handbook of Materials Selection for Engineering Applications, edited by G.

T. Murray 114. Handbook of Thermoplastic Piping System Design, Thomas Sixsmith and

Reinhard Hanselka 1 15. Practical Guide to Finite Elements: A Solid Mechanics Approach, Steven M.

Lepi 1 16. Applied Computational Fluid Dynamics, edited by Vijay K. Garg 117. Fluid Sealing Technology, Heinz K. Muller and Bernard S. Nau 1 18. Friction and Lubrication in Mechanical Design, A. A. Seireg 119. lnfluence Functions and Matrices, Yuri A. Melnikov 120. Mechanical Analysis of Electronic Packaging Systems, Stephen A.

McKeown 1 2 1 . Couplings and Joints: Design, Selection, and Application, Second Edition,

Revised and Expanded, Jon R. Mancuso 122. Thermodynamics: Processes and Applications, Earl Logan, Jr. 123. Gear Noise and Vibration, J. Derek Smith 124. Practical Fluid Mechanics for Engineering Applications, John J. Bloomer 125. Handbook of Hydraulic Fluid Technology, edited by George E. Totten 126. Heat Exchanger Design Handbook, T. Kuppan


127. for Product Sound Quality, Richard H. Lyon in Franklin E. Fisher and Joy R.

Fisher 129. Nickel Alloys, edited by Ulrich Heubner 1 30. Rotating Machinery Vibration: Problem Analysis and Troubleshooting,

Maurice L. Adams, Jr. 131. Formulas for Dynamic Analysis, Ronald L. Huston and C. Q. Liu 132. Handbook of Machinery Dynamics, Lynn L. Faulkner and Earl Logan, Jr. 133. Rapid Prototyping Technology: Selection and Application, Kenneth G.

Cooper 134. Reciprocating Machinery Dynamics: Design and Analysis, Abdulla S.

Rangwala 1 35. Maintenance Excellence: Optimizing Equipment Life-Cycle Decisions, edi-

ted by John D. Campbell and Andrew K. S. Jardine 136. Practical Guide to Industrial Boiler Systems, Ralph L. Vandagriff 137. Lubrication Fundamentals: Second Edition, Revised and Expanded, D. M.

Pirro and A. A. Wessol 138. Mechanical Life Cycle Handbook: Good Environmental Design and Manu-

facturing, edited by Mahendra S. Hundal 139. Micromachining of Engineering Materials, edited by Joseph McGeoug h 140. Control Strategies for Dynamic Systems: Design and Implementation, John

H. Lumkes, Jr. 141. Practical Guide to Pressure Vessel Manufacturing, Sunil Pullarcot 142. Nondestructive Evaluation: Theory, Techniques, and Applications, edited by

Peter J. Shull 1 43. Diesel Engine Engineering: Thermodynamics, Dynamics, Design, and

Control, Andrei Makartchouk 144. Handbook of Machine Tool Analysis, loan D. Marinescu, Constantin Ispas,

and Dan Boboc 145. Implementing Concurrent Engineering in Small Companies, Susan Carlson

Skalak 146. Practical Guide to the Packaging of Electronics: Thermal and Mechanical

Design and Analysis, Ali Jamnia 147. Bearing Design in Machinery: Engineering Tribology and Lubrication,

Avraham Harnoy 148. Mechanical Reliability Improvement: Probability and Statistics for Experi-

mental Testing, R. E. Little 149. Industrial Boilers and Heat Recovery Steam Generators: Design, Ap-

plications, and Calculations, V. Ganapathy 150. The CAD Guidebook: A Basic Manual for Understanding and Improving

Computer-Aided Design, Stephen J. Schoonmaker 151. Industrial Noise Control and Acoustics, Randall F. Barron 1 52. Mechanical Properties of Engineered Materials, Wole Soboyejo 153. Reliability Verification, Testing, and Analysis in Engineering Design, Gary S.

Wasserman 154. Fundamental Mechanics of Fluids: Third Edition, I. G. Currie 155. Intermediate Heat Transfer, Kau-Fui Vincent Wong 156. HVAC Water Chillers and Cooling Towers: Fundamentals, Application, and

Operation, Herbert W. Stanford Ill 157. Gear Noise and Vibration: Second Edition, Revised and Expanded, J.

Derek Smith


158. Handbook of Turbomachinery: Second Edition, Revised and Expanded, Earl Logan, Jr., and Ramendra Roy

1 59. Piping and Pipeline Engineering: Design, Construction, Maintenance, lnteg- rity, and Repair, George A. Antaki

160. Turbomachinery: Design and Theory, Rama S. R. Gorla and Aijaz Ahmed Khan

161. Target Costing: Market-Driven Product Design, M. Bradford Clifton, Henry M. B. Bird, Robert E. Albano, and Wesley P. Townsend

162. Fluidized Bed Combustion, Simeon N. Oka 1 63. Theory of Dimensioning: An lntroduction to Parameterizing Geometric

Models, Vijay Srinivasan 164. Handbook of Mechanical Alloy Design, George E. Totten, Lin Xie, and

Kiyoshi Funatani 165. Structural Analysis of Polymeric Composite Materials, Mark E. Tuttle 166. Modeling and Simulation for Material Selection and Mechanical Design,

George E. Totten, Lin Xie, and Kiyoshi Funatani

Additional Volumes in Preparation

Handbook of Pneumatic Conveying Engineering, David Mills, Mark G. Jones, and Vijay K. Agarwal

Mechanical Wear Fundamentals and Testing: Second Edition, Revised and Expanded, Raymond G. Bayer

Engineering Design for Wear: Second Edition, Revised and Expanded, Raymond G. Bayer

Clutches and Brakes: Design and Selection, Second Edition, William C. Orthwein

Progressing Cavity Pumps, Downhole Pumps, and Mudmotors, Lev Neli k

Mechanical Engineering Sofmare

Spring Design with an IBM PC, Al Dietrich

Mechanical Design Failure Analysis: With Failure Analysis System Software for the IBM PC, David G. Ullman


In Memoriam

During the preparation of this book, one of our most valued authors andmentors passed away on April 29, 2003. Professor George C. Weatherly(1941–2003) graduated from Cambridge University in 1966. He began hiscareer as a research scientist in the Department of Metallurgy at Harwell.In 1968 he moved to Canada where he worked for the University ofToronto for 22 years as a professor in the Department of Metallurgy andMaterial Science. In 1990 he became a professor of Materials Science andEngineering at McMaster University. He was Director of BrockhouseInstitute for Material Research from 1996–2001 and a Chair of theDepartment of Materials Science and Engineering. Dr. Weatherly haspublished over 200 papers in different areas of Materials Science. He wasFellow for the Canadian Institute for Mining and Metallurgy and Fellowof ASM International. George was a devoted scientist in the field of electronmicroscopy and an educator with a distinguished career at McMasterUniversity and the University of Toronto. He will be cherished by hisfriends, colleagues, and students for the richness of his life, his quiet humor,his humanity and care for others, and above all for his unfailing honesty.His contributions were many and are written clearly in the lives of thosewith whom he taught and worked. This book is dedicated to his memory.


Preface

In every industry survey, development and use modeling, and simulationtechnology are cited among the top five critical needs for manufacturingindustries to remain viable and competitive in the future. This is particularlytrue for materials and component design. To address this need, variousresearch programs are currently underway in government, academic, andindustry laboratories around the world. This book addresses a number ofselected, important areas of computer model development.

Effective material and component design procedures are vitally impor-tant with increasing pressures to improve quality at lower production costsfor all traditional industrial markets. Advanced design procedures typicallyinvolve computer modeling and simulation if the necessary algorithms aresufficiently advanced or by using advanced empirical procedures. The objec-tive is to be able to make design decisions based on numerical simulations asan alternative to time-consuming and expensive laboratory or productionexperimental process development. In fact, advanced engineering processesare becoming increasingly dependent on advanced computer modeling anddesign procedures.

This book addresses various aspects of the utilization of modeling andsimulation technology. Some of the topics discussed include hot-rolling ofsteel, quenching and tempering during heat treatment, modeling of residualstresses and distortion during forging, casting, heat treatment, mechanicalproperty prediction, modeling of tribological processes as it relates to thedesign of surface engineered materials, and fastener design. These chapterssummarize and demonstrate key numerical relationships used in computer


model development and their application at various stages in the materialproduction process.

In particular, the material covered in this text includes:

� Modeling and simulation of microstructural evolution andmechanical properties of steels during the hot-rolling process,calculation of metallurgical phenomena occurring in steel duringhot-rolling, and prediction of mechanical properties from micro-structure.

� Heat treatment processes such as quenching and tempering is anactive area for process model development. Models used tosimulate the kinetics of multicomponent grain boundary seg-regations that occur in quenched and tempered engineeringsteels are discussed. These models permit the evaluation of theeffect of alloying elements and various tempering parameters onhydrogen embrittlement, stress-corrosion cracking, and otherphenomenon.

� Of all the various problems associated with component design andproduction, none are more important that residual stress anddistortion. Chapter 3 discusses the metallo-thermo-mechanicaltheory, numerical modeling and simulation technology, couplingof temperature, inelastic behavior and phase transformation andsolidification involved with elastic-plastic, viscoplastic and creepdeformation as they relate to quenching, forging, and castingprocesses.

� Modeling and simulation of mechanical properties, in particular,material behavior during plastic deformation, low-cycle fatigue,creep, and impact strength. This discussion includes the impor-tance of the determination and implementation of adequate mate-rial data, consideration of inelastic material behavior, and theformulation of physically founded material models.

� Chapter 5 discusses the role played by physico-chemical interac-tions in modifying and controlling friction and wear of criticallyloaded tribo-couple surfaces during high-speed cutting operations.

� A comprehensive overview of one of the most important processesin manufacturing is presented in Chapter 6. Threaded fastenerselection and design is addressed with many equations and figuresincluded to aid in the design process.

Chapters 1 through 4 describe advanced computer modeling andsimulation processes to predict microstructures, material process behavior,and mechanical properties. Chapters 5 and 6 describe more empiricalprocess design procedures for tribological and fastener design.


This book will be an invaluable resource for the designer, mechanicaland materials engineer, and metallurgist. Thorough overviews of thesetechnologies seldom encountered in other handbooks for materials designare provided. The book is an excellent textbook for advanced undergraduateor graduate engineering courses on the role of modeling and simulation inmaterials and component design.

We are indebted to the vital assistance of various international experts.Special thanks to our spouses for their infinite patience with the varioustime-consuming tasks involved in putting this text together. We extendspecial thanks to the staff at Marcel Dekker, Inc. including RichardJohnson, Rita Lazazzaro, and Russell Dekker for their invaluableassistance. Without their assistance, this text would not have been possible.

George E. TottenLin Xie

Kiyoshi Funatani


Contents

PrefaceContributors

1 A Mathematical Model for Predicting MicrostructuralEvolution and Mechanical Properties of Hot-Rolled SteelsMasayoshi Suehiro

2 Design Simulation of Kinetics of MulticomponentGrain Boundary Segregations in the Engineering SteelsUnder Quenching and TemperingAnatoli Kovalev and Dmitry L. Wainstein

3 Designing for Control of Residual Stressand DistortionDong-Ying Ju

4 Modeling and Simulation of Mechanical BehaviorEssam El-Magd

5 Tribology and the Design of Surface-EngineeredMaterials for Cutting Tool ApplicationsGerman Fox-Rabinovich, George C. Weatherly,and Anatoli Kovalev


6 Designing Fastening SystemsChristoph Friedrich


Contributors

Essam El-Magd, Dr.-Ing.habil. Aachen University, Aachen, Germany

German Fox-Rabinovich, Ph.D., D.Sc. McMaster University, Hamilton,Ontario, Canada

Christoph Friedrich, Dr.-Ing. RIBE Verbindungstechnik GmbH, Schwa-bach, Germany

Dong-Ying Ju, Ph.D. Saitama Institute of Technology, Okabe, Saitama,Japan

Anatoli Kovalev, D.Sc. Physical Metallurgy Institute, Moscow, Russia

Masayoshi Suehiro, Dr.Eng. Nippon Steel Corporation, Futtsu-City,Chiba, Japan

Dmitry L. Wainstein, D.Sc. Physical Metallurgy Institute, Moscow, Russia

George C. Weatherly, Ph.D.y McMaster University, Hamilton, Ontario,Canada

yDeceased


1A Mathematical Model forPredicting MicrostructuralEvolution and MechanicalProperties of Hot-Rolled Steels

Masayoshi SuehiroNippon Steel Corporation, Futtsu-City, Chiba, Japan

I. INTRODUCTION

A model for calculating the mechanical properties of hot-rolled steelsheets from their processing condition makes it possible not only todesign chemical compositions and processing conditions of steels throughoff-line simulation but also to guarantee the mechanical properties ofhot-rolled steels through on-line simulation. From this point of view,some attempts have been made to develop a mathematical model forcalculating the evolution of austenitic microstructure of steels duringhot-rolling process and their transformations during cooling subsequentto hot-rolling [1–3]. The mathematical models basically consist of fourmodels for calculating metallurgical phenomena occurring in hot-stripmill and a model for predicting mechanical properties from the micro-structure of steel calculated by the metallurgical models. In thischapter, the basic idea and several applications of the mathematicalmodel will be presented.


II. THE OVERALL MODEL

Since mechanical properties of hot-rolled steels are determined by theirmicrostructure, a model for calculating the mechanical properties ofhot-rolled steels is composed of two kinds of models: one for calculatingmicrostructure of steels from their processing conditions, and the otherfor calculating their mechanical properties from their microstructure. Thereare several kinds of hot-rolled steel products: sheet and coil, plate, beam,wire, rod, bar, etc. Although the processing conditions are dependent uponeach process, each product is produced through the processes such as heat-ing, hot-working, and cooling.

Figure 1 shows the schematic illustration of a hot-strip mill. Hot-rolledsteel sheets are produced through slab reheating, rough hot-rolling, finishhot-rolling, cooling, and coiling. Table 1 shows the typical thickness andtemperature changes in this process and the metallurgical phenomena occur-ring through this process. In the slab-reheating process, transformationfrom ferrite and pearlite to austenite and grain growth take place. The

Figure 1 Schematic illustration of a hot-strip mill.

Table 1 The Changes in Thickness and Temperature of Steelsand Metallurgical Phenomena in a Hot-Strip Mill

ProcessThickness(mm)

Temperature(8C) Metallurgical phenomena

Slabreheating

250 1200 Transformation, grain growth,dissolution, and precipitation of

precipitatesRoughrolling

!40 1200–1000 Recovery, recrystallization,grain growth, precipitation

Finishrolling

!3 1000–850 Recovery, recrystallization,grain growth, precipitation

Cooling 3 — Transformation, precipitation

Coiling 3 600–700 Precipitation


recovery and recrystallization, and grain growth of austenitic microstructureoccur during and after rough and finish hot-rolling and the transformationfrom austenite to ferrite, pearlite, bainite, and martensite takes place duringcooling and coiling. In the case where steels include alloying elements thatform carbides or nitrides, precipitation of such carbides and nitrides takesplace and affects recovery, recrystallization, and grain growth in each pro-cess. Accordingly, in order to calculate the microstructural evolution ofhot-rolled steels, the model used to calculate recovery, recrystallization,grain growth during and after hot deformation, transformation kinetics dur-ing cooling and precipitation kinetics in each process is shown in Fig. 2.

III. BASIC KINETIC EQUATION

In recrystallization and transformation, a new phase forms and grows.These new phases continue to grow until they meet each other and stopgrowing. This situation is called hard impingement and can be expressedby using the Avrami type equation (4a,4b,4c)

X ¼ 1� expð�ktnÞ ð1Þ

or the Johnson–Mehl equation (5). In these equations, the concept ofextended volume fraction is adopted. By using this concept, the hard impin-gement can be taken into consideration indirectly. The extended volume

Figure 2 The structure of the model for calculating microstructural evolution and

mechanical properties of hot-rolled steels.


fraction is the sum of the volume fraction of all new phases without directconsideration of the hard impingement between new particles and is relatedto the actual volume fraction by

X ¼ 1� expð�XeÞ ð2Þ

where X is the actual volume fraction and Xe is the extended volumefraction.

The general form of the equation was developed by Cahn [6]. A briefexplanation is presented here. The nucleation sites of new phases would begrain boundaries, grain edges, and=or grain corners. In the case of grainboundary nucleation, the volume fraction of a new phase after some timecan be expressed as follows. Cahn considered the situation illustrated inFig. 3 and calculated the volume of the semicircle.

In his calculation, firstly, the area at the distance of y from the nuclea-tion site B is calculated. The summation of this area for all nuclei gives thetotal extended area. From this value, the actual area can be calculated. Theextended volume can be obtained by integrating the area for all distances.Finally, the actual volume fraction can be derived.

Figure 3 Schematic illustration of the situation of new phase at time t whichnucleates at time t at grain boundary B.


The area of the section at a plane A for a semicircle nucleated at aplane B is considered. The radius r at time t can be expressed as

r ¼ ½G2ðt� tÞ2 � y2�1=2 for y < Gðt� tÞr ¼ 0 for y � Gðt� tÞ ð3Þ

where G is the growth rate of new phase and t is the time when the new phasenucleates at plane B. In this calculation, the growth rate is assumed to be con-stant. From this radius, the extended area fraction dYe for the new phasesnucleated at time between t and t þ dt can be obtained as

dYe ¼ pIs dt½G2ðt� tÞ2 � y2� for y < Gðt� tÞdYe ¼ 0 for y > Gðt� tÞ ð4Þ

where Is is the nucleation rate at unit area. By integrating for the time t from0 to t, the extended area fraction at the plane A at time t can be obtained as

Ye ¼Z t

0

dYe ¼ pIs

Zt�y=G0

½G2ðt� tÞ2 � y2� dt ð5Þ

By exchanging y=Gt for x, this equation leads to

Ye ¼ pIsG2t31� x3

3� x2ð1� xÞ

� �for x < 1

Ye ¼ 0 for x > 1 ð6ÞThe actual area fraction of new phases at plane A, Y can be calculated usingYe from

Y ¼ 1� expð�YeÞ ð7ÞThe integration of Y for y from 0 to infinity gives the volume of new phasesnucleated at unit area of plane B,V0, as

V0 ¼ 2

Z10

Y dy ¼ 2Gt

Z10

1� exp �pIsG2t31� x3

3� x2ð1� xÞ

� �� dx

ð8ÞMultiplying V0 by the area of nucleation site, the extended volume fractionis obtained as

Xe ¼ SV0 ¼ b�1=3s fsðasÞ ð9Þ


where

as ¼ ðIsG2Þ1=3t; bs ¼ Is8S3G

¼ Ns

8S4G

fsðasÞ ¼ as

Z10

1� exp �pa3s1� x3

3� x2ð1� xÞ

� �� dx ð10Þ

and Ns the nucleation rate for unit volume. The actual volume fraction X isexpressed as

X ¼ 1� expð�b�1=3s fsðasÞÞ ð11ÞFrom this equation, two extreme cases can be considered. One is the casewhere as is very small and the other is extremely large. For these two cases,the equation becomes

X ¼ 1� expð�p=3NsG3t4Þ as51 ð12Þ

X ¼ 1� ð�2SGtÞ as41 ð13ÞEquation (12) is the same as the one obtained for the case of random nuclea-tion sites by Johnson–Mehl. This equation implies that the increase in thevolume of new phases is caused by nucleation and growth. On the otherhand, Eq. (13) does not include nucleation rate and it implies that thenucleation sites are covered by new phases and the increase in the volumeis dependent only on the growth of new phases. This situation is referredto as site saturation [6].

Cahn did this type of formulation for the cases of grain edge and graincorner nucleations. Table 2 shows all the extreme cases. For all cases, theincrease of the volume of new phases for the case of small as conforms tothe case of nucleation and growth and site saturation for the case of largeas The value of as increases when the nucleation rate is small when comparedto the growth rate. The early stage of reaction corresponds to small as and

Table 2 The Kinetic Equations Depending on the Modes and theNucleation Sites of Reaction in Accordance with Cahn’s Treatment

Nucleation site Nucleation and growth Site saturation

Grain boundary X ¼ 1� expð�p=3 _NNG3t4Þ X ¼ 1� expð�2SGtÞGrain edge X ¼ 1� expð�pLG2t2ÞGrain corner X ¼ 1� expð�ð4p=3ÞCG3t3Þ


the latter stage corresponds to large as. From Table 2, we can recognizethat the exponent of time depends on the mode of reaction and the typeof nucleation site for the case of site saturation. A comparison of thisinformation with the experimental results gives useful information on themode of reaction and the nucleation site. The equations in Table 2 canbe used for calculating actual reactions such as transformation andrecrystallization by introducing fitting parameters obtained fromexperiments [7].

IV. UTILIZATION OF THERMODYNAMICS FOR THECALCULATION OF TRANSFORMATION ANDPRECIPITATION KINETICS

As transformation and precipitation kinetics are closely related to phaseequilibrium, thermodynamics can be utilized for their calculation. In thissection, the method for utilizing thermodynamics for the calculation willbe explained.

For the consideration of kinetics, the Gibbs free-energy–compositiondiagram is much more useful and should be the basis. Figure 4 shows theGibbs free-energy–composition diagram for austenite and ferrite in steels.Chemical composition at the phase interface between ferrite and austeniteis obtained from the common tangent for free-energy curves of ferrite andaustenite. The common tangent can be calculated under the condition thatchemical potentials of all chemical elements in ferrite are equal to those inaustenite. This condition is expressed as

mai ¼ mgi ð14Þwhere m is the chemical potential, the suffix i represents all elements in thesystem and a and g indicate ferrite and austenite, respectively. In Fig. 4,the driving force for transformation from austenite to ferrite, DGm, is indi-cated as well. It can be calculated by

DGm ¼X

xai mgi � mai� � ð15Þ

where x is the fractions of elements. These values are necessary for the cal-culation of moving rate of the interface during transformation and precipi-tation. The Zener–Hillert equation [8,9], which represents the growth rate offerrite into austenite, is expressed as

G ¼ 1

2rDCga � Cg

Cg � Cað16Þ


where D is the diffusion coefficient of C in austenite, r the tip-radius ofgrowing phase, Cga, Ca, and Cg is the carbon content in austenite, ferriteat a=g interface and in ferrite apart from interface, respectively. The carboncontent at the interface can be calculated from the common tangent betweentwo phases as shown in Fig. 4. There is the other type of expression ofmoving rate of interface which is expressed as

v ¼ M

VmDGm ð17Þ

where M is the mobility of interface, and Vm is the molar volume. The driv-ing force in this equation can be calculated for multicomponent system byEq. (15). This calculation makes it possible to consider the effect of alloyingelements other than the pinning effect and the solute-drag effect. Details ofthe thermodynamic calculation have been published [10–12]. Recently, somecommercial software for the thermodynamic calculation have been used forthis type of calculation [13].

Figure 4 Gibbs free energy vs. chemical composition diagram.


V. BASIC MODELS

A. The Concept of the Model

As mentioned above, the overall model for predicting mechanical propertiesof hot-rolled steels consists of several basic models: the initial state modelfor austenite grain size before hot-rolling, the hot-deformation model foraustenitic microstructural evolution during and after hot-rolling, the trans-formation model for transformation during cooling subsequent to hot-roll-ing, and the relation between mechanical properties and microstructure ofsteels. In the case where steels include alloying elements which form precipi-tates, the model for precipitation is necessary. Precipitates affect all themodels mentioned here. In this section, these basic models will be explained[14,15].

B. Initial State Model

In this model, austenite grain sizes after slab reheating, namely before hotdeformation, are calculated from the slab-reheating condition. In steels con-sisting of ferrite and pearlite at room temperature, austenite is formedbetween pearlite and ferrite and it grows into ferrite according to decompo-sition of pearlite. After all the microstructures become austenite, the graingrowth of austenite takes place. We should formulate these metallurgicalphenomena to predict austenite grain size after slab reheating. In hot-stripmill, however, the effect of initial austenite grain size on the final austenitegrain size after multi-pass hot deformation is small. This can be due to thehigh total reduction in thickness by several hot-rolling steps in which therecrystallization and grain growth are repeated and the size of austenitegrain becomes fine. This means that the high accuracy is not required forthe prediction of the initial austenite grain size in a hot-strip mill. From thispoint of view, the next equation (14) can be applied

dg ¼ exp 1:61 ln KþffiffiffiffiffiffiffiffiffiffiffiffiffiffiK2 þ 1

p �þ 5

n oK ¼ ðT� 1413Þ=100 ð18Þ

where dg is the austenite grain size after reheating of slab and T is the tem-perature in K.

On the other hand, the initial austenite grain size affects the finalaustenite grain size in the case of plate rolling because the total thicknessreduction is relatively small compared to hot-strip rolling. In this case, thehigh accuracy of the prediction may be required and the model that isapplicable for this case has been reported [16]. Three steps are consideredin this model: (1) the growth of austenite between cementite and ferriteaccording to the dissolution of cementite, (2) the growth of austenite into


ferrite at aþ g two-phase region, and (3) the growth of austenite in the gsingle-phase region. The pinning effect by fine precipitates on grain growthand that of Ostwald ripening of precipitates on the grain growth of auste-nite are taken into consideration. This model is briefly explained in the fol-lowing paragraphs.

The growth of austenite due to the dissolution of cementites can beexpressed as

dðdgÞdt¼ Dg

c

dg

Cyg � Cga

Cga � Cað19Þ

where t is the time, Dgc the diffusion constant of C in austenite, and Cg, Cga

are the C content in austenite at g=y phase interface and g=a phase interface,respectively. In the a þ g two-phase region, the austenite grain size dependson the volume fraction of austenite, Xg, which changes according to tem-perature. This situation is expressed as

dg ¼ 3Xg

4pn0

� �1=3

ð20Þ

where n0 is the number of austenite grains at a unit volume when cementitesare dissolved. Grain growth occurs in the austenite single-phase region. Forgrain growth, it is necessary to consider three cases; without precipitates,with precipitates, and with precipitates growing due to the Ostwald ripening.There are equations which are formulated to theoretically correspond tothese three cases. They are summarized by Nishizawa [17]. The equationfor the normal grain growth is expressed as

d2g � d2g0 ¼ k2t ð21Þwhere k2 is the factor related to the diffusion coefficient inside the interface,the interfacial energy, and the mobility of the interface. With the pinningeffect by precipitates, the growth rate becomes

dðdgÞdt¼M

2sVR� DGpin

� �; DGpin ¼ 3sVf

2rð22Þ

where f is the volume fraction of precipitates and r is the average size ofprecipitates. When precipitates grow according to the Ostwald ripening,the average size of precipitates used in the Eq. (22) is obtained from

r3 � r30 ¼ k3t ð23Þwhere k3 is the factor related to temperature, interfacial energy and the dif-fusion coefficient of an alloying element controlling the Ostwald ripening of


precipitates. By this calculation method, it is possible to predict the growthof austenite grain during heating when precipitates such as AlN, NbC, TiC,and TiN exist in austenite [16].

C. Hot-Deformation Model

The hot-deformation model is required to predict the austenitic microstruc-ture before transformation through recovery, recrystallization, and graingrowth in austenitic phase region during and after multi-pass hot deforma-tion. Sellars and Whiteman [18,19] made the first attempt on this issue andthen several researchers [20–27] developed models to calculate recovery,recrystallization, and grain growth. These models are basically similar toeach other. In some models, dynamic recovery and dynamic recrystallizationare taken into consideration. The dynamic recovery and recrystallization arelikely to occur when the reduction is high for single-pass rolling or strain isaccumulated due to multi-pass rolling. They should be taken into considera-tion in finishing rolling stands of a hot-strip mill because, the inter-pass timemight be less than 1 sec and the accumulation of strain might take place.Here, the hot-deformation model will be explained based on the modeldeveloped by Senuma et al. [20].

In this model, dynamic recovery and recrystallization, static recoveryand recrystallization, and grain growth after recrystallization are calculatedas shown in Fig. 5. The critical strain, ec, at which dynamic recrystallizationoccurs is generally dependent upon strain rate, temperature, and the size ofaustenite grains. The effect of strain rate on ec is remarkable at low strainrate region [28]. One of the controversial issues had been whether thedynamic recrystallization took place or not when the strain rate is high suchas that in a hot-strip mill. Senuma et al. [20] showed that it takes place andthe effect of strain rate on ec is small at a high strain rate.

The fraction dynamically recrystallized, Xdyn, and can be expressedbased on the Avrami type equation as

Xdyn ¼ 1� exp �0:693 e� ece0:5

� �2 !

ð24Þ

where e0.5 is the strain at which the fraction dynamically recrystallizedreaches 50%. On the other hand, the fraction statically recrystallized canbe expressed as

Xdyn ¼ 1� exp �0:693 t� t0t0:5

� �2 !

ð25Þ


where t0.5 is the time when the fraction statically recrystallized reaches 50%and t0 is the starting time of static recrystallization.

The growth of grains recrystallized dynamically after hot deformation ismuch faster than normal grain growth in which grains grow according tosquare of time. This rapid growth was treated with different equations[19–24]. The reason why this rapid growth takes place might be caused bythe increase of the driving force for grain growth due to high dislocation den-sity [20], the change in the grain boundary mobility [24] or the annihilation ofthe small size grains at the initial stage [19].

In the case of multi-pass deformation, the strain might not be reducedcompletely at the following deformation due to the insufficient time intervaland the effect of accumulated strain on the recovery and recrystallizationshould be taken into consideration. This effect is remarkable for a hot-stripmill because of the short inter-pass time and for steels containing alloyingelements which retard the recovery and recrystallization. This effect canbe formulated by using the change in the residual strain [22,25] or the dislo-cation density [20,21,24]. In the modeling process, the accumulated strain is

Figure 5 Schematic illustration of microstructural change due to hot deformation.


calculated from the average dislocation density which is obtained by calcu-lating the changes in the dislocation density in the region dynamically recov-ered, rn, and in the region recrystallized dynamically, rs, according to timeindependently.

This method makes it possible to calculate the changes in grain sizeand dislocation density. Table 3 shows the summary of equations used inthe model developed by Senuma et al. The numbers of phenomena inTable 3 correspond to those in Fig. 5. Figure 6 shows an example of calcu-lation of the changes in grain size and dislocation density [14]. Figure 7shows the calculation result of the effect of the initial austenite grain sizeon the final microstructure in the finishing stands of a hot-strip mill, whichshows that the initial austenite grain size does not affect very much the finalgrain size. This model can be applied to the prediction of the resistance tohot deformation as well and it can contribute to the improvement of theaccuracy in thickness. In this method, the average values concerning thegrain size and the accumulated dislocation density are used taking the frac-tion recrystallized into consideration. This averaging can be applied to thehot-strip mill because the total thickness reduction is large enough to recrys-tallize their microstructure. In the case of plate rolling, the use of the averagevalues is unsuitable because the reduction at each pass is small and the totalthickness reduction is not enough to recrystallize the microstructure ofsteels. The model applicable to this case has been developed by dividingthe microstructure into several groups [26].

This type of modeling was carried out for Nb-bearing steels [19,21,25],Ti- and V-bearing steels [21], Ti- and Nb-bearing steels [22], Ti-, Nb-, and V-bearing steels [27] as well as C–Mn steels. In these steels, the recovery andrecrystallization are retarded by alloying elements. This retardation mightbe caused by the pinning effect due to fine precipitates or by the solute-drageffect. This effect can be considered by modifying the values of fitting para-meters from experimental data.

D. Transformation Model

1. Basic Idea of the Modeling

In the cooling process subsequent to hot-rolling, steels transform fromaustenite phase to ferrite, pearlite, bainite, and=or martensite phases. Trans-formation model predicts the microstructural change during cooling and thefinal microstructure of steels after cooling. The modeling of transformationkinetics can be performed by obtaining the parameters k and n in Avramiequation [29–31], formulating new equations corresponding to transforma-tion kinetics obtained experimentally [32], and adopting the nucleation andgrowth theory [33–36].


Table 3 Equations Used for Calculating Microstructural Change During Hot Deformation

Phenomena Calculation model

1. Critical strain fordynamic recrystallization

ec ¼ 4:76 � 10�4 expð8000=TÞ ðaÞ

Grain size of dynamically

recrystallized grain

ddyn ¼ 22600½_ee expðQ=RTÞ��0:27 ¼ Z�0:27;Q ¼ 63800 cal=mol ðbÞ

Fraction dynamically recrystallized Xdyn ¼ 1� exp½�0:693ððe� ecÞ=e0:5Þ2� ðcÞe0:5 ¼ 1:144 � 10�3d 0:28

0 _ee0:05 expð6420=TÞ ðdÞDislocation density in dynamicallyrecrystallized grain

rso ¼ 87300½_ee expðQ=RTÞ�0:248 ¼ 87300Z0:248 ðeÞrs ¼ rsoexp½�90 expð�8000=TÞt0:7� ðfÞ

2. Dislocation density re ¼ ðc=bÞð1� e�beÞ þ r0e�be ðgÞ

3. Grain growth of dynamicallyrecrystallized grain

dy ¼ ddyn þ ðdpd � ddynÞy ðhÞdpd ¼ 5380 expð�6840=TÞ ðiÞy ¼ 1� exp½�295_ee0:1 expð�8000=TÞt� ðjÞ

4. Grain size of statically recrystallized grain dst ¼ 5=ðSveÞ0:6 ðkÞSv ¼ ð24=pd0Þð0:491ee þ 0:155e�e þ 0:1433e�3eÞ ðlÞ

Fraction statically recrystallized Xst ¼ 1� exp½�0:693ððt� t0Þ=t0:5Þ2� ðmÞt0:5 ¼ 0:286 � 10�7Sv�0:5 _ee�0:2e�2 expð30000=TÞ ðnÞ

5. Change in dislocation density dueto recovery

rr ¼ re exp½�90 expð�8000=TÞt0:7� ðoÞ

6. Grain growth d 2 ¼ d 2st ¼ 1:44 � 1012 expð�Q=RTÞt ðpÞ


When using the Avrami equation, the fitting parameters, k and n, canbe obtained from the Avrami plot based on the isothermal transformationkinetics as shown in Fig. 8. The effect of chemical composition of steelsand the austenitic grain size before transformation on transformationkinetics can be taken into consideration by obtaining the dependence onthe values of k and n from experiments. The first attempt of this type ofmodeling was carried out by Kirkaldy [29]. In his study, the predictionof mechanical properties was also tried. In order to increase the generalityof the transformation model, it should be necessary to take the nucleation

Figure 6 Changes in grain size and dislocation density during hot rolling of sixpasses.


and growth theory into consideration. Utilizing the Johnson–Mehl typeequation [5] or Cahn’s equation [6]. Cahn’s equation can be recognized asbeing the most general one because the approximation of the equationsleads to the Avrami and Johnson–Mehl type equations. The modeling basedon the nucleation and growth theory [33] will be explained in the followingparagraphs.

Assuming that the nucleation site is the surface of grain boundariesand the rates of nucleation and growth are independent of time, the trans-formation rate can be expressed for two cases [6]. One is the case where boththe nucleation and the growth of new phase occur and the other is the casewhere only the growth of new phase occurs after nucleation sites are coveredby new phase. The first case is described by

X ¼ 1� exp �p=3ISG3t4� � ð26Þ

Figure 7 Effect of initial grain size on the change in grain size during hot-rolling ina hot-strip mill.


and the second case is described by

X ¼ 1� �2SGtð Þ ð27Þ

Transformation rates can be obtained by differentiating the equations as

dX

dt¼ 4

p3

�1=4ISð Þ1=4G3=4 ln

1

1� X

� �3=4

1� Xð Þ ð28Þ

dX

dt¼ 2SG 1� Xð Þ ð29Þ

For obtaining Eq. (28), the term of time is replaced by the fractiontransformed on the assumption of the additivity of transformation withregard to the changing temperature. Equation (29) essentially holds theadditivity of transformation. By using this type of equations, it is possibleto obtain fitting parameters from continuous cooling transformation

Figure 8 Avrami’s plot. T1, T2, and T3 show different temperatures.


kinetics. It should be noticed that it is difficult to obtain accurate TTT dia-grams of low-carbon steels which are the primary products in a hot-stripmill. This is due to their very rapid transformation kinetics and it causes dif-ficulty in determining the fitting parameters from TTT data. From this pointof view, this type of equation is very useful. In a case where it is possible toobtain the accurate TTT data, we can use the equations in Table 2 for deter-mining the parameters.

Transformations from austenite to ferrite, pearlite, and bainite can becalculated by using these equations. The start of each transformation isassumed as follows. The start of ferrite transformation is when the tempera-ture of steel drops to Ae3. Carbon content in austenite increases during fer-rite transformation and pearlite transformation starts when the carboncontent in austenite achieves the amount shown on the Acm line. Bainitetransformation starts when the temperature of steel drops to Bs temperature.Ae3 and Acm can be calculated from thermodynamics. Although Bs can becalculated on the same assumptions relating to T0 temperature [37], theequation obtained from experimental data is used in this model becausethe calculated Bs temperature still does not fit with experimental data.

2. Ferrite Transformation

The nucleation site of transformation from austenite to ferrite is mainlythe surface of austenite grains and its nucleation would be completedat the beginning of the transformation. Accordingly, the transformationfrom austenite to ferrite in the early stage is calculated by Eq. (28) andthat in the latter stage is calculated by Eq. (29) in this model. The changein transformation kinetics, i.e., from the nucleation and growth to the sitesaturation, is assumed when the transformation rates calculated by bothequations coincide with each other. Although the dissipation of incubationtime for nucleation is generally used for the condition of the start of phasetransformation, it is quite unclear theoretically. In this model, g=(a þ g)temperature, A3, is used for the starting condition of the calculation ofphase transformation. There are two methods for calculating g=(a þ g)temperature; one is the condition called para-equilibrium [38] and theother is ortho-equilibrium. In the ortho-equilibrium, all elements arepartitioned between ferrite and austenite; on the other hand, in the para-equilibrium condition, only carbon is partitioned. The idea of thepara-equilibrium comes from that the diffusion of carbon which occupiesinterstitial sites in steel which is much more rapid than other substitutionalalloying elements such as Mn, Si, and so on. There is an idea of NP–LE(No Partition–Local Equilibrium) [39], where interstitial atoms are parti-tioned between ferrite and austenite, the substitutional elements are locally


in the state of equilibrium between ferrite and austenite and the partitionbetween ferrite and austenite does not occur after transformation. Theortho-equilibrium should be used when the transformation occurs at arelatively high temperature and the cooling rate is very slow. The para-equilibrium and the LE–NP should be used when the transformationoccurs at a relatively low temperature as in rapid cooling. For the phasetransformation in the continuous hot-rolling mill where the cooling rateis rapid, the para-equilibrium or the NP–LE should be used. The A3 tem-perature calculated based on the para-equilibrium and will be used in thismodel. For this calculation, C, Si, and Mn concentrations which are com-monly included in steels are taken into consideration.

According to the classical nucleation theory, the nucleation rate, I, canbe described by equation [40]

I ¼ Nb�Z exp �DG�

kT

� �ð30Þ

where N is the number of nucleation sites, b� is the rate of solute atomsarriving at the surface of new phase, Z (the Zeldovich factor) characterizesthe annihilation of nucleation, T the temperature, k is the Boltzmann con-stant and DG� is the driving force for forming the nucleus with a critical size.Z and b� can be described by the following equations, respectively

Z ¼ aT�1=2 ð31Þ

b� ¼ bD ð32Þwhere D is the diffusion coefficient of solute atoms, a the variable related tointerfacial energy and b is the variable related to the nearest atomic distance.DG� is described by the equation

DG� ¼ cs3

DG2V

ð33Þ

where s is the interfacial energy, DGV is the free-energy difference betweenferrite and austenite and c is the configurational coefficient. DGV is calcu-lated by the thermodynamic parameters. Although there are many reportsconcerning interfacial energy, its value is still unclear. The values of Zand b� are also unclear. In this model, we introduced two parameters forthe nucleation rate as

I ¼ k1T�1=2D exp � k2

RTDG2V

� �ð34Þ


and they are evaluated from experimental data to fit the calculation result tothat obtained by experiment. As mentioned above, DGV can be calculatedwith thermodynamic parameters from chemical compositions of steels. Inthe equation, R is the gas constant and D is the diffusion coefficient of Cin austenite. The diffusion coefficient of C can be calculated by the equationreported by Kaufman et. al. [41] which is expressed as

D cm2=sec� � ¼ 0:5 exp �30Cg

� �exp �QD

RT

� �ð35Þ

QDðcal=molÞ ¼ 38300� 1:9� 105C2g þ 5:5� 10 ð36Þ

where Cg is the carbon content (mole fraction) in austenite.For growth rate, the Zener–Hillert equation [8,9], Eq. (16) is used. This

equation is formulated for the lengthening growth of needle-shaped ferritebased on the idea that the growth of ferrite is controlled by the diffusionof carbon in austenite. It has been reported that this equation well describesthe growth of ferrite and bainite phases into austenite [41]. The tip radius ofgrowing ferrite affects the carbon content in austenite and ferrite at phaseinterface and the carbon content is calculated by the method reported byKaufman et al. [41] in this model. The carbon content in austenite increasesduring the transformation to ferrite due to the small solubility of carbon inferrite and the increase in carbon content in austenite affects the growth rateof ferrite. The carbon content in austenite, Cg, can be calculated by theequation

Cg ¼ C0 � XFCa

1� XFð37Þ

where C0 is the initial carbon content, XF is the fraction transformed to fer-rite, and Ca is the carbon content in ferrite. By putting this value for Cg intoEq. (16), the change of the growth rate of ferrite by the progress of ferritetransformation can be considered. Parabolic growth equation, G ¼ at�1=2,can be used instead for Eq. (16). This relation between growth rate and timecan be obtained from experiment and the theory for isothermal transforma-tion kinetics. This is due to the change in carbon content ahead of interfaceinto austenite during transformation. This situation is considered by usingEqs. (16) and (37) [33].

Growth equations explained above can consider only the partition ofcarbon between ferrite and austenite during transformation. The partitionof substitutional elements like Mn and Si cannot be considered. Recently,Enomoto and Atkinson [42–44] and Agren [45] have analyzed the partitionof substitutional elements and its effect on transformation in detail. The cal-culation method by Agren [46] is based on the local equilibrium theory and


has shown that chemical composition in ferrite and austenite changes from‘‘para’’ to ‘‘ortho’’ during transformation.

3. Pearlite Transformation

In the case of low-carbon steel, transformation to pearlite occurs subsequentto that of ferrite and can be assumed to start when the carbon content inaustenite calculated by Eq. (37) reaches the extrapolated Acm line, as shownin Fig. 9. This starting condition of pearlite transformation was assumed tocorrespond to the site saturation case because the determination of kineticsby experiments for low-carbon steel is almost impossible. The equation for-mulated by Hillert [47] is used for the growth rate which is expressed as

GP ¼ kPSl

D Cga � Cgb� � ð38Þ

where Sl is the lamella spacing, Cgb the carbon content in austenite at theinterface between cementite and austenite, and kP is the constant. Thelamella spacing has a linear relation with the inverse of the undercoolingtemperature below Ae1, DT, and Eq. (38) can be expressed as

GP ¼ kPDTD Cga � Cgb� � ð39Þ

Figure 9 Starting condition of each transformation in the equilibrium diagram.


The effect of a chemical composition on the growth rate of pearlite isconsidered by using Cga, Cgb and DT calculated by the thermodynamic para-meters. As the transformation from austenite to pearlite is a eutectoid reac-tion, the change of the chemical composition in untransformed austeniteduring the progress of transformation does not occur. Equation (38) showsthat pearlite transformation is controlled by the volume diffusion of carbon.On the other hand, it has been recently reported that the mechanism of pear-lite transformation is somewhere between the volume diffusion of carbon inaustenite and the interfacial diffusion of substitutional elements [48].

4. Bainite Transformation

Transformation to bainite is assumed to start when steels are cooled to thebainite-start temperature, Bs. In this report, Bs formulated from the observa-tion of microstructures of steels (0.05–0.15 mass% C–0.5–1.5 mass%Mn–0–1.0 mass% Si steels) which are transformed isothermally is used,which is expressed as

Bs ¼ 717:5 � 425½mass%C� � 42:5½mass%Mn�ð�CÞ ð40ÞBainite transformation in low-carbon steels occurs subsequent to ferrite orpearlite transformation. No flexion point between ferrite and bainite trans-formations is observed in the transformation curve. This result indicatesthat bainite transformation subsequent to ferrite transformation can be trea-ted by the site saturation. The nucleation site in this case would be the inter-face between austenite and ferrite. The progress of bainite is calculated byEq. (29), and the transformation rate is calculated by the Zener–Hillertequation (Eq. (16)).

5. Summary of Equations and Parameters Used for the

Transformation Model

Table 4 shows the summary of equations and fitting parameters. Althoughthe fitting parameters were obtained from the experimental data of one steel,this set of equations can simulate transformations of steels with differentchemical compositions due to the application of thermodynamics to the cal-culation of the growth rate and the free-energy difference between ferriteand austenite.

Figures 10 and 11 show the calculation results of the effects of chemi-cal composition and austenite grain sizes on transformation kinetics [33].The calculation of the fraction of each phase after transformation can bedetermined. By introducing the thermodynamic parameter of other alloyingelements, the applicability would be easily extended [49]. Recently, the cal-culation of transformation for a 10 element system was reported [50]. This


Table 4 Equations and Parameters Used for the Transformation Model

TransformationBasic equation oftransformation rate

Factor corresponding tonucleation rateand growth rate Coefficient

Ferrite Nucleationand Growthdx

dt¼ 4:046 kt6=d

4g IG

3 �1=4

ln1

1� x

� �3=4

ð1� xÞ

I ¼ T�1=2D exp � k3RT DG2

v

� �

G� ¼ 1

2rDCga � Cg

Cg � Ca

k1 ¼ 17; 476

k2 ¼ 8:933� 10�12 exp21100

T

� �k3 ¼ ðcal3=mol3Þ ¼ 1:305� 107

Pearlite Site saturation

dx

dt¼ k2

6

dgGð1� xÞ

G ¼ DTDðCga � CgbÞ K2 ¼ 6:72� 106

Bainite G� ¼ 1

2rDCga � Cg

Cg � Cak2 ¼ 6:816� 10�4 exp

3431:5

T

� �

Note: dg: austenite grain size, D: diffusion coefficient of carbon in austenite, Cg: carbon content in austenite, Ca: carbon content in ferrite, Cga:

carbon content in austenite at g=a boundary, Cgb: carbon content in austenite at g/cerm boundary, DT: undercooling below Ae1, G�: Zener–Hillert

equation (the value was calculated with the method by Kaufman et al.), and r: radius of curvature of advancing phase.


extension of the model is possible only when alloying elements affect trans-formation kinetics through the change in phase stability. From the view-point of the effect of the chemical composition on transformationkinetics, other effects such as the solute drag, the pinning etc. should betaken into consideration. Kinsman and Aaronson [51] reported that trans-formation in C–Mo steels is retarded due to the solute drag-like effect. Thereis a report that Nb retards transformation due to solute-drag effect [52]. Thesolute-drag effect in Fe–C–X systems was investigated as well [53–56]. Thesolute-drag effect and others are not considered in the model. Instead ofthe application of these theories, the fitting parameters for the rates ofnucleation and growth are introduced [57]. The introduction of these the-ories into the mathematical models is the remaining problem.

6. The Calculation of Ferrite Grain Sizes After Transformation

The prediction of ferrite grain size is necessary for the calculation ofmechanical properties. The calculation can be carried out using the austenitegrain size before transformation and cooling rate [22]. This type of formulaeis useful for the calculation because the prediction can be carried out

Figure 10 The effect of chemical compositions on transformation kinetics.


without knowing the accurate transformation kinetics. It, however, could notbe applied for the case where cooling rate is not constant. The next equationcan be applied for the case where the cooling rate changes during cooling.

On the assumption that the shape of ferrite grains is spherical, theaverage ferrite grain size after cooling, da, has a relation with a fractiontransformed to ferrite and the number of ferrite grains and the relation isexpressed as

XF ¼ N4p3

da2

� �3

ð41Þ

where N is the number of ferrite grains in the unit volume and XF is the vol-ume fraction of ferrite. The equation can be transformed into

da ¼ 6XF

pN

� �1=3

ð42Þ

Since XF can be obtained from the transformation model explainedabove, the calculation of ferrite grain size can be carried out if we know

Figure 11 The effect of austenite grain size on transformation kinetics of 0.15%C–0.5%Si–1%Mn steels.


the number of ferrite grains. The formulation of the equation for calculatingthe number of grains is based on the idea that the number of ferrite grains isdetermined in the early stage of ferrite transformation.

Figure 12 shows the relationship between the number of ferrite grainsand the temperature at 5% transformation, T0.05, which was calculated bythe above-mentioned transformation model. This experiment was carriedout for (0.1–0.15) mass% C–0.5 mass% Si–(0.5–1.5) mass % Mn steels.The temperature T0.05 is used as a representative temperature indicatingthe early stage of ferrite transformation. This figure shows that the numberof ferrite grains is dependent upon the temperature at the early stage of fer-rite transformation and the austenite grain size before transformation.Based on this result, the number of ferrite grains N(mm�3) was formulated as

N ¼ 3:47� 10�11d�1:75g exp21430

T0:05

� �ð43Þ

where dg is the austenite grain size before transformation. Using Eqs. (42)and (43), we can obtain the equation expressing the ferrite grain size da as

da ¼ 5:51� 1010d1:75g exp � 21430

T0:05

� �XF

� �1=3ð44Þ

Figure 12 The relationship between the number of ferrite grains and thetemperature, T0.05.


where da and dg are in units of mm. Figure 13 compares the ferrite grain sizescalculated versus those observed. A good agreement between those calcu-lated and observed is found in this figure.

Umemoto et al. [58] have derived theoretical equations for ferrite grainsize in an isothermally transformed steel as

da ¼ 0:564ðI=GÞ�1=6d2=3g ð45Þfor the austenite grain edge nucleation of ferrite, and as

da ¼ 0:695ðI=GÞ�2=9d1=3g ð46Þfor the grain surface nucleation. In the present study, ferrite grain size isexpressed by Eq. (45), in which the relationship between da and dg is

da ¼ kd1:75=3g ð47Þ

Figure 13 Comparison between calculated and observed ferrite grain sizes.


where k is the coefficient depending on the transformation temperature andferrite fraction. This result indicates that the nucleation of ferrite transforma-tion in low-carbon steels takes place mainly at the surface of austenite grains.

7. The Effect of Residual Strain on Transformation Kinetics

The austenitic microstructure with which we start the calculation using thetransformation model is a fully recrystallized austenite. In the hot-rollingmill, the austenite before transformation could contain many imperfec-tions such as dislocations, deformation bands, and so on, and they mayaffect the nucleation rate and the growth rate. This effect would causedeterioration in predicting the accuracy of the microstructure after cool-ing. At this time, it is difficult to account for the effect of imperfectionson transformation perfectly on theory. In this section, we thus will con-sider it using the dislocation density calculated by the hot-deformationmodel will be considered [20].

Figure 14 Comparison between calculated and observed ferrite grain sizes withand without consideration of the effect of the dislocation density.


Figure 14 shows the difference between the calculated and theobserved ferrite grain sizes as a function of residual dislocation densities.The dislocation density was estimated from the hot-deformation model[20]. In this figure, open circles show the case when the above-mentionedmodel is used, i.e., the effect of the residual strain is not considered, whilesolid circles show the case when the effect of residual strain is consideredby the method which will be explained below. In the case where the effectof residual strain is not considered, the difference between the ferrite grainsizes measured and those calculated is small for low dislocation densityand large for high dislocation density. Using the observed ferrite grain sizes,T0.05 and Eq. (44), we can estimate the austenite grain size upon which thecorrect ferrite grain size for a whole range of dislocation density can begiven, and this austenite grain size can be called the effective austenite grainsize. Figure 15 shows the difference between the austenite grain size, dg, cal-culated from the hot-deformation model, and the effective austenite grain

Figure 15 Effect of the dislocation density on the effective austenite grain size.


size estimated as a function of the dislocation density. From this figure, thefollowing relationship can be obtained:

d effg ¼ dg=ð1þ 10�11r1:154Þ ð48Þ

where r is the dislocation density before transformation in cm�2. The ferritegrain sizes calculated from the transformation model using dg

eff instead of dgagree well with those observed as shown in Fig. 14. Figure 16 compares theferrite fractions calculated with those observed when either dg or dg

eff isused. The use of dg

eff provides better agreement than the use of dg. Theseresults prove that the effect of stored strain on transformation kineticscan be predicted quantitatively.

Figure 16 Comparison between the ferrite fractions calculated and observed whiletaking=not taking into account the effect of dislocation density.


Umemoto et al. [59] studied the effects of the residual strain on the rateof growth and nucleation separately in pearlite transformation and showedthat only the change in nucleation rate is caused by the stored strain. Inthe transformation model explained above, the effect of austenite grain sizeon transformation kinetics is considered by using dg. The change in the areaof nucleation sites and the change in the rate of nucleation and growth wouldbe taken into consideration by using Eq. (48), although it is not clear which isthe dominant factor. For quantitative evaluation of each of these effects,conducting an experiment similar to the work done by Umemoto et al. isnecessary, although it is difficult to conduct such an experiment with regardto low carbon steels because of their rapid transformation.

E. Precipitation Model

Precipitation of carbides and=or nitrides in austenite phase could affectrecovery, recrystallization, and grain growth after hot deformation, andtransformation during cooling. Precipitation in ferrite phase could affectmechanical properties. Accordingly, it is necessary to take precipitation intoconsideration in the model.

It is not necessary to consider the hard impingement for the model-ing of precipitation. For this reason, nucleation and growth can be calcu-lated independently and the amount, number, and size distribution ofprecipitates can be directly calculated. Although formulations based onAvrami’s equation have been reported [60–62], modeling based on thenucleation and growth theory has been attempted from the above pointof view [63–67]. Okamoto and Suehiro [67] reported a model which canbe applied from the beginning to the end (the Ostwald ripening) of preci-pitation. The feature of this model is in the calculation method of growthrate of precipitates. This calculation method will be explained briefly asfollows.

The velocity of the interface between matrix and precipitates can beexpressed from the flux balance of each chemical element as

v ¼ JNb

0CNb �b CNb¼ JC

0CC �b CC¼ JN

0CN �b CNð49Þ

where Jj is the flux of each element, 0Cj andbCj the content of element j in

precipitates and matrix at the interface between matrix and precipitates,respectively. For the calculation of the content of element j in matrix atthe interface, the local equilibrium condition is normally applied. In thismodel, the content of element j is calculated considering the radius of


precipitate using the next equation. This relationship is called theGibbs–Thompson effect

mNbCNj þ 2sVNbCN=R ¼ mMj ð50Þ

where the second term of the left side of this equation is the increase in theGibbs free energy, Esurf, due to the interfacial energy, s. The Ostwald ripen-ing of precipitates appearing at the latter stage of precipitation in which thesmall size of precipitates would dissolve and the large size of precipitateswould grow can be calculated by considering this term. Figure 17 showsthe isothermal section of equilibrium diagram of three element system.From this figure, the effect of the radius of precipitates can be understood.Figure 18 compares the calculation result of the average diameter and themole fraction of precipitates with the experiments. Figure 19 shows the cal-culation result of the average diameter and the number of precipitates whichshows that the average diameter increases with a half power of time at thebeginning of precipitation (region II) and with one-third power of time atthe latter stage (region IV). Region IV would correspond to the Ostwaldripening.

F. Relationship Between Strength andMicrostructure of Steel

The mechanical properties to be predicted are dependent upon the type ofproducts. Sheets and coils require YS (yield strength), TS (tensile strength),and El (elongation). Plates require toughness other than YS, TS and El.Wire and rods require TS and the reduction of section area. The predictionof these mechanical properties was carried out by formulating regressionequations for strengths with respect to chemical compositions and grainsizes of ferrite [68]. Other formulations for strength were based on thevolume fractions of ferrite and pearlite phases, ferrite grain sizes, andlamella spacing of pearlite for steels consisting of ferrite and pearlite inphases [69]. For toughness, some regression formulae based on the ferritegrain size and=or chemical compositions were reported [70–72]. Variousreports on this type of formulations are available [32–74].

Irvine and Pickering [73] showed that the tensile strength of ferrite-pearlite steel or bainite steel was determined from the transformation tem-perature of steels [73]. In their experiment, only the content of alloyingelements was a variable of the transformation temperature, but the resultindicated that the change of transformation temperature due to processingvariables such as cooling rate had a similar influence on the strength of steels


Figure 17 Isothermal section of equilibrium diagram of Fe–Nb–C system; (a) thebeginning and (b) the latter stage of precipitation.


consisting of ferrite, pearlite, and=or bainite. A recent study confirmed thisresult with regard to the accelerated cooled steels of a substantially ferritictransformation structure [75].

Therefore, the determination of a more general relationship betweenstrength and transformation temperature applicable to the individualmicroconstituent in a mixed microstructure is required. Since the strength

Figure 18 Comparison of the calculated results with the observed ones. Steel A:0.006%C–0.14%Nb–0.0022%N, steel B: 0.018%C–0.052%Nb–0.0041%N.


of steel is generally proportional to its hardness, by assuming the law ofmixture for hardness, the tensile strength, TS, can be expressed as

TS ¼ a½XFðHF þ bd�1=2a Þ þ XPHP þ XBHB� ð51Þ

where X is the fraction of each transformed phase (F: ferrite, P: pearlite, andB: bainite), H the hardness of each microconstituent, da the ferrite grain sizein mm, and a and b are the constants. If the relationship between strengthand transformation temperature for each constituent of steel is established,the tensile strength can be calculated from Eq. (51).

Figure 20 shows the relation between the hardness of each microcon-stituent and its average transformation temperature, TM, calculated from

TM ¼Z

T dX=

ZdX ð52Þ

Figure 19 Change in the size and the number of precipitates calculated by the

model.


where the transformation temperature, T, is obtained for an infinitely smalltransformation product, dX, from the transformation model. A linear rela-tionship is found between the hardness and average transformation tem-perature for both ferrite and bainite. Such a relationship is not found forpearlite, although it is presumed. This is probably because of the narrowtemperature range of transformation in the steels used. Silica has a strongeffect on solid-solution hardening while C and Mn have a very small effect.Further, the hardness does not depend on the cooling rate. From theseresults, the hardness of each microconstituent is expressed as

HF ¼ 361� 0:357TF þ 50½mass%Si�HP ¼ 175

ð53Þ

Figure 20 The relationship between the measured microhardness of each micro-

constituent and its calculated mean transformation temperature.


and

HB ¼ 508� 0:588TB þ 50½mass%Si� ð54Þwhere H is the hardness and T is the average transformation temperature(8C).

Other Methods of attempting to predict YS, TS, n-value, etc. havebeen reported. Tomota et al. [76] attempted the prediction of YS, TS, andn-value, etc. based on the prediction of flow stress curves. Shikanai et al.[77] and Iung et al. [78] utilized the analysis by the finite element methodin order to consider the effect of morphology of microstructure.

For steels containing chemical elements forming precipitates such asNb, Ti, V, etc., the precipitation hardening should be taken into considera-tion. It would be possible by using the precipitation model. It might bepossible to predict the precipitation hardening from the alloying elementsin solid solution at austenite region before cooling [79,80].

VI. PREDICTION OF STRENGTH OF HOT-ROLLEDSTEEL SHEETS

Using the model mentioned above, we can predict the strength of hot-rolledsteel sheets from its composition and processing conditions such as hot-roll-ing condition, cooling condition, and so on. Figure 21 shows the flow chartof the calculation.

Figure 22 compares the calculated and observed transformed fractionof each phase in 0.2 mass% C-0.2 mass% Si-0.5 mass% Mn steels hot rolledin a two-stand laboratory mill from 40 to 2.4mm by six passes after beingsoaked at 11008C for 30min. The microstructure is ferrite–pearlite in thesample (a) cooled at around 108C=sec and ferrite–bainite in (b) cooled ataround 608C=sec. The values calculated by the present model are in goodagreement with those measured.

Tensile strengths were calculated using Eqs. (51), (53), and (54) with theconstant a of 3.04 and the constant b of 2.55. An agreement between the cal-culated and the observed tensile strengths is good for various steels (C: 0.1–0.2 mass%, Si: 0.006-0.5 mass%, Mn: 0.5-1.5 mass%) as shown in Fig. 23.

The present integrated model has been applied to the prediction of themicrostructures and strengths of steel hot rolled in a production mill. 0.15mass% C-0.1 mass% Si-0.6 mass% Mn steel was hot rolled. In the hot-roll-ing, the finish rolling temperature and the coiling temperature varied length-wise as shown in Fig. 24. Figure 25 shows the calculated and observed ferritegrain size and ferrite fraction of the steel sheet. Figure 26 compares thestrengths calculated with those measured. These figures show that the values


calculated by this model have a good agreement with those measured. Theseresults indicate that this type of simulation could be a very efficient tool fordesigning chemical compositions and processing conditions in order toobtain required mechanical properties.

Many empirical equations have been developed to predict the strengthof hot rolled steel products. These equations combine the parameters of che-mical compositions such as C, Mn, and Si, fraction of each microconstituentand cooling rate. In the present model, Eqs. (51), (53) and (54) for calculationof the strength of hot-rolled steel products have no explicit parametric termsof C, Mn and cooling rate. Therefore, it seems as if the content of C and Mnand the cooling rate do not influence the strength of these products. As men-tioned above, the microstructure and the hardness of each microconstituentpredicted based on the hot-deformation and transformation models arestrongly affected by the C- and Mn-content and the cooling condition.The results shown in Fig. 20 indicate that the solid-solution hardening byC and Mn reported in the literature includes the variation of themicrostructure with the change in the transformation temperature. Since

Figure 21 Flowchart of the calculation of microstructural change and mechanical

properties of hot-rolled steels.


the cooling rate does not appear explicitly, the present model is suitablefor the prediction of strength of steels processed by thermomechanicaltreatment.

VII. APPLICATION OF THE MODEL TO THE PREDICTIONOF TEMPERATURE OF HIGH CARBON STEELSDURING COOLING AFTER HOT DEFORMATION

A. Modification of the Model to the Applicationto High Carbon Steels

Steels containing more than 0.3mass%C carbon (high -carbon steels) show aremarkable evolution of latent heat of transformation during cooling. This

Figure 22 Comparison of the calculated microstructure with that observed forsteels cooled at about (a) 108C=sec and (b) 608C=sec.


evolution makes on-line control of temperature difficult and thus affectsmechanical properties of final products. Thus, accurate calculations of tem-perature of steel during cooling prior to production in a mill are desirable toenable the control and the investigation of suitable processing conditions.In order to calculate the temperature accurately, the prediction of transfor-mation is particularly important.

Figure 23 Comparison between the calculated and observed tensile strengths of

various steels.


Figure 24 Finish rolling and coiling temperatures for experiments in production mill.

Figure 25 Variation of ferrite fraction and ferrite grain size from top to tail of the coilof 0.16mass%C–0.015 mass%Si–0.73mass%Mn steel hot rolled in a production mill.


As mentioned above, a mathematical model for predicting transforma-tion of low-carbon steels during cooling while taking ferrite transformationinto consideration has been developed because the principal transformationproduct in low-carbon steels is ferrite. In high-carbon steels, however, themain transformation product is pearlite, so that the development of a modelfor calculating pearlite transformation accurately is necessary. In this sec-tion, the model for high-carbon steels and its application [81] will beexplained.

B. Transformation Model

1. Start of Pearlite Transformation

The calculation procedure for ferrite and bainite transformations is thesame as that explained in Sec. 5. The treatment for pearlite transformationis modified.

The calculation of ferrite transformation which starts when the tem-perature drops to the equilibrium temperature Ae3 is calculated using ther-modynamic parameters. Transformation from austenite to ferrite iscontrolled by the volume diffusion of carbon into austenite so that carbonin austenite increases with the progress of ferrite transformation. This pointis explained in the Sec. 5.

Figure 26 Variation of tensile strength from top to tail of the coil of 0.16mass%C–0.015mass%Si–0.73mass%Mn steel hot rolled in a production mill.


Pearlite transformation is generally assumed to start when the carboncontent in austenite meets the extrapolated Acm line in the phase equilibriumdiagram. The observed ferrite fraction of carbon steels transformed isother-mally from austenite, however, indicates that the pearlite transformationstarts earlier than the above expectation at the temperature range below960K shown in Fig. 27. This deviation might be due to the distribution ofcarbon between ferrite and austenite at the g=a phase boundary becausethe carbon content in austenite at=near the phase boundary can exceed thevalue on the extrapolated Acm line below Ae1 temperature. In this model, thestart of pearlite transformation is dealt with based on this result.

2. Kinetics of Pearlite Transformation

For low-carbon steels, there has been no experimental data which clarifiesthe mode of pearlite transformation kinetics. In 0.5mass%C steels, however,the experimental results of pearlite transformation kinetics show that pear-lite transformation conforms to nucleation and growth case [81]. Accord-ingly, Eq. (28) is used and the fitting parameter [81] was obtained fromthe experimental results of 0.5mass%C steels as shown in Table 5.

Figure 27 Deviation of maximum ferrite fraction measured in isothermaltransformation experiments from the value calculated thermodynamically.


C. Method for Calculating Temperature of SteelsDuring Cooling

The transformation model developed was coupled with the model for calcu-lating the temperature of steels during cooling by a two-dimensional finiteelement method in order to take into consideration the evolution of latentheat [15]. The heat conduction equation used here is as follows:

mCaP

@T

@t¼ l

@2T

@x2þ l

@2T

@y2þQ ð55Þ

where m is the density of steel, l the heat conductivity, and CPa is the specific

heat of ferrite on the assumption of nonexistence of magnetic transforma-tion as shown in Fig. 28. Q is the rate of latent heat evolution accompanyingtransformation and can be formulated by Eq. (56), in which the latent heatis divided into that of lattice transformation, ql, and that of magnetic trans-formation, qm:

Q ¼ m ql@X

@tþ @

@tðqmXÞ

� �ð56Þ

where X is the fraction transformed and calculated by the above-mentionedtransformation model. The value of ql used is 16.7 J=g and qm is expressed as

qm ¼Z910T

CP � CaP

� �dT ð57Þ

where CP is the specific heat of steel.

Table 5 Equations and Parameters for Pearlite Transformation

of High-Carbon Steels

Basic equation oftransformation rate

Factor corresponding tonucleation rate and growth rates Coefficient

Nucleationand growth

I ¼ T�1=2D exp � k3

RTDG2V

� �k1 ¼ 2.01� 1013

k3¼2.27� 109 (J3=mol3)

S¼6=dgdX

dt¼ 4

p3

�1=4ðk1ISÞ1=4

G3=4 ln1

1� X

� �3=4

ð1� XÞ

GP ¼ DTD Cga � Cgb� �


D. Calculation Results

1. Accuracy of the Model for Predicting Temperature of Steels

During Cooling

Figure 29 shows examples of simulation in which the temperature and theprogress of transformation of 0.5mass%C steels on the run-out table of ahot-strip mill were calculated simultaneously for three different cooling con-ditions. In this calculation, the initial state and the hot-deformation modelswere also used and the average austenite grain size before transformationwas calculated as being about 15 mm. A good agreement between calculatedand measured temperatures was obtained. This implies that transformationkinetics on the run-out table are accurately predicted by the model.

2. Improvement of Productivity

To improve the productivity in a hot-strip mill, it is necessary to increase thetraveling speed of the hot-strip and intensify the cooling rate. Figure 30(a)shows the calculated results of temperature and transformation behaviorof 0.5mass%C steel of 2mm thickness cooled at a heat transfer coefficient,

Figure 28 Specific heat and latent heat of magnetic transformation of steel.


a, of 1670 kJ=m2hK for water cooling. In this calculation, the finish rollingtemperature is 1123K, the coiling temperature is 873K and transformationis completed before coiling. The traveling time through the run-out table canbe shortened from 11.5 to about 5 sec by increasing the heat transfer coeffi-cient up to 5020 kJ=m2hK, as shown in Fig. 30(b). The figure, however,shows the undesirable situation where the temperature of steel drops toabout 800K which is below the bainite-start temperature (about 823K for0.5mass%C steel) and bainite which deteriorates the quality of steel mightappear. This situation can be avoided by changing the cooling condition.Figure 30(c) shows the suitable cooling condition in which the water coolingis stopped just before the transformation start and restarted at about 20%transformation.

Figure 29 Simulation of temperature and progress of transformation of 0.5mass%C steel under different cooling conditions on run-out table of a hot-strip mill.


The results show that the calculation by the model can be used fordetermining a suitable cooling pattern on the run-out table for attaininghigh productivity without conducting an experiment in a production mill.

3. Effect of the Finish Rolling Temperature on

Mechanical Properties

In a hot-strip mill, the finish rolling temperature varies widthwise. This varia-tion affects the transformation behavior during cooling due to the changesin austenite grain size and dislocation density in austenite, and it alsochanges the temperature range of water cooling on the run-out table as well.Figure 31 shows the calculated cooling curves of 0.5mass%C steel coil of4mm thickness and 1m width at two different positions; the center alongthe width and the position of 12.5mm apart from the edge of strip. The tem-perature difference between these two positions is about 408C. By changingthe water-cooling condition, the temperature difference can be reduced asshown in Fig. 32. This change contributes to the reduction of the fluctuationof mechanical properties as shown in Fig. 33, in which the index of vertical

Figure 30 Temperature change of 0.5mass%C steel on the run-out table underthree different cooling conditions.


Figure 31 Temperature changes of steel at two different positions in the directionof width on the run-out table.

Figure 32 Temperature change of steel at two different positions in the direction ofwidth on the run-out table. Cooling condition was modified from that in Fig. 31.


axis, (Ae1�TP)1=2 where TP is the mean transformation temperature of pear-

lite, representing hardness of pearlite because the pearlite hardness dependson the lamella spacing of pearlite which depends upon the under-coolingfrom Ae1 temperature. Conditions A and B in Fig. 33 correspond to thosein Figs. 31 and 32.

This result indicates that the fluctuations of properties due to the var-iation of temperature along the width can be compensated by controlling thewater-cooling intensity.

4. Effect of Fluctuations of Coiling Temperature on

Mechanical Properties

Coiling temperature is fluctuated widthwise by the fluctuations of watercooling intensity even though the finish rolling temperature is constant,and this coiling temperature fluctuation affects the mechanical propertiesdue to the change in the transformation temperature. Figure 34 shows anexample of the calculated results for 0.5mass%C steel sheets with thickness

Figure 33 Changes in the index of hardness, (Ae1�TP)1=2, in which TP is the mean

pearlite transformation temperature, along the width under two different conditions.


of 4 and 6mm under the condition that the finish rolling temperature is1122K and the coiling temperature fluctuates between 843K and 933K. Thisresult indicates that fluctuations of the index of hardness, D(Ae1�TP)

1=2,depend on those of coiling temperature, DCT, and their relationship is influ-enced by the thickness of steel sheets. The fluctuations of transformationtemperature are mostly dependent on the cooling rate just before and=orat the beginning of transformation. Since the water-cooling intensity forthe steel of 4mm thickness is the same as that for steel of 6mm thick in thiscalculation, the water-cooling time for steel of 6mm thickness is longer thanthat for 4mm steel because of the mass effect. Hence, the change in coolingrate, which causes a certain value of DCT, becomes smaller as steel stripthickens. This is the reason why the thickness of steel sheets causes fluctua-tions in hardness.

Figure 35 shows the effect of traveling speed on the fluctuations oftransformation temperature. The calculation shown was carried out forthree different traveling speeds (300, 400, and 500mpm) with the finishrolling temperature of 1123K and the coiling temperature of between

Figure 34 Effect of thickness of steel on the relationship between the fluctuations

of coiling temperature, DCT, and those of the index of pearlite hardness,(Ae1�TP)

1=2.


853K and 913K. The fluctuations of the index of hardness do not vary inthe order of traveling speed; the fluctuations for 400mpm are the largestamong three traveling speeds. This variation is also related to the coolingrate before and during transformation. The faster the traveling speed, thelonger the water-cooling time, in the case where the intensity for watercooling is constant regardless of the traveling speed. This is due to thechange in the water-cooling temperature range depending on the finish roll-ing temperature, the coiling temperature, the traveling speed, and the thick-ness. In this calculation, the water-cooling time for 300mpm is too short toaffect the transformation behavior greatly. This is the reason why thefluctuations in transformation temperature for 400mpm are greatest inFig. 35.

These results depend on the chemical composition of steel and thecapacity of the hot-strip mill, such as the water-cooling intensity and thelength of the run-out table. Accordingly, the pre-calculation by this type

Figure 35 Effect of traveling speed of steel through the run-out table on therelationship between the fluctuations of coiling temperature, DCT, and those of the

index of pearlite hardness, (Ae1�TP)1=2.


of mathematical model is very useful for designing cooling facilities anddetermining cooling conditions for obtaining a more uniform qualityproduct.

VIII. CONCLUSION

In this chapter, mathematical models for predicting microstructural evolu-tion during hot deformation and subsequent cooling, and mechanical prop-erties from the resultant microstructure of steels were explained. Thesemodels include some empirical parameters although they are based ontheory. This is because mechanisms of some phenomena are still unclear;for instance, solute-drag effect on recrystallization and transformation.The model calculating mechanical properties from microstructure is muchmore phenomenological. To extend the applicability, the empirical para-meters should be replaced by those obtained from theories.

Although the models include some empirical parameters, they are veryuseful for investigating production conditions such as chemical composi-tions, processing conditions and so on. The accuracy of predicted mechan-ical properties is satisfactory. It is noted that these models should be widelyused for off-line simulation of designing steel compositions and processingcondition and on-line simulation for guaranteeing mechanical propertiesof steels.

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2Design Simulation of Kineticsof Multicomponent GrainBoundary Segregations in theEngineering Steels UnderQuenching and Tempering

Anatoli Kovalev and Dmitry L. WainsteinPhysical Metallurgy Institute, Moscow, Russia

I. INTRODUCTION

The basic factors controlling grain boundary segregations (GBS) in engi-neering steels are discussed. In contrast to single-phase alloys, in engineer-ing steels, the multicomponent segregation is developed simultaneouslywith undercooled austenite transformations and martensite decomposition.Based on these reasons, the influence of steel phase composition andkinetics on concurrent segregations is discussed. It is established that grainboundary enrichment by harmful impurities (S and P) is possible after car-bon and nitrogen segregation dissolution. Two models of GBS aredescribed. The dynamic model of segregation during quenching is basedon the solution of independent diffusion and adsorption–desorption equa-tions for various impurities in steel. The model of multicomponent segrega-tion under tempering considers the influence of alloying and temperingparameters on concentration and thermodynamic activity of carbon inthe a-solid solution.


II. GRAIN BOUNDARY SEGREGATION AND PROPERTIESOF ENGINEERING STEELS

Chemical composition and structure of the grain boundary influences var-ious properties of engineering steels. The following are some of these prop-erties: inclination to temper and heat brittleness, resistance to hydrogenembrittlement, corrosion, delayed fracture, and creep. The intergranularfracture is the main reason for decrease of many steel exploitation proper-ties. Application of modern physical experimental or calculation methodshas successfully helped in solving the old metallurgical problem of intergra-nular fracture.

The affinity of various kinds of intergranular brittleness is associatedwith two main unfavorable factors that decrease intergranular bonds. Theimpurity segregation to grain boundaries (GBS) and localization of internalmicrostresses are necessary and sufficient conditions that could initiateembrittlement [1].

Despite the common features, certain kinds of steel brittleness are dis-tinguishable from each other and are stipulated by complex interaction ofthese factors. The concentration of internal stresses on grain boundariescould be an effect of martensite transformation, hydrogen accumulation,or carbide precipitation; and grain boundary segregations could appear dur-ing the equilibrium or non-equilibrium processes of element redistribution insteel.

The concentration of microstresses on grain boundaries cause theinitiation of cracking and acts as the primary reason for brittleness. Theenrichment of grain boundaries by harmful impurities is a major and com-mon condition for development of various intercrystalline brittleness phe-nomena and it specifies crack propagation entirely along grain boundariesat low stresses.

The concept of intercrystalline internal adsorption [2] that was con-firmed by theoretical [3], and experimental work [4], the thermodynamic ana-lysis of chemical element interaction during equilibrium grain boundarysegregation [5], and investigations of quenched and tempered steel [6,7]. Thismade it possible to interpret the tempering embrittlement phenomenon.

Elemental impurities enrich grain boundaries in thin layers up toseveral atoms and change the type and value of interatomic bonds thatlead to intercrystalline fracture. Embrittlement power is commonlyattributed to the elements of the 3rd to 5th periods of groups IV toVI in the periodic system [1]. Sulfur, phosphorus, arsenic, selenium, tell-urium, antimony, bismuth, and oxygen are the most harmful impuritiesthat segregate in grain boundaries. The concentration in grain bound-aries could reach several atomic percentages exceeding the volume one


by several hundreds. Intercrystalline brittleness, as caused by GBS, dueto harmful impurities is observed, as a rule, in BCC metals andalloys. The austenite alloys are significantly more resistant to this kindof fracture.

The motonic increase of plasticity that is expected after martensitedecomposition due to tempering of engineering steels is disturbed by twoanomalies resulting in a relative decrease of impact strength. These anoma-lies are accompanied by intergranular fracture. Steels may be susceptible toembrittlement when they are heated for prolonged period in the temperaturerange 350–5508C, or when slowly cooled through it. Depending on the heat-treatment cycle, the phenomenon is called either temper embrittlement (3508embrittlement) or reversible temper embrittlement (5508 embrittlement).Common indications of embrittlement are a loss of toughness, segregationof harmful impurities to grain boundaries and the fracture path usuallyalong prior austenite grain boundaries, and the impact transition tempera-ture (FATT—the fracture appearance transition temperature) which is dis-placed towards higher values.

The irreversible temper embrittlement of low-alloyed steels (<5% ofalloying elements) is developed during tempering of quenched steel in thetemperature range 300–4008C.

The reversible temper embrittlement of medium-alloyed steels is devel-oped during tempering in the temperature range 500–6008C. This kind ofbrittleness is observed after tempering and slow cooling of annealed, nor-malized, or quenched steel.

It is now established that GBS of small impurities play a decisive rolein development of these phenomena.

A. Brittle Fracture of Steel After Quenching andMedium Tempering

The quenched medium-alloyed engineering steel is subjected to tempering inthe temperature range 250–4008C to achieve high strength and plasticity.But after tempering at 300–4008C, one can see an abnormal drop in impactstrength. At higher tempering temperatures, the impact strength increasesagain (see Fig. 1) [8].

The intergranular fracture of steel tempered at 300–4008C is due to theaction of two unfavorable factors: enrichment of grain boundaries by P dur-ing austenitization and formation of lamellar Fe3C particles along the pri-mary austenite grains and martensite packs (Fig. 2a, b). The P segregationsdecrease the cohesion within boundaries significantly, and carbides blockthe dislocation movement. This is the reason for peak stresses under plastic


Figure 1 Impact strength at room temperature of samples with V-shape cut forseveral melts of industrial steel 4340 after quenching (8508C, 1 hr) in oil and 1 hrtempering at various temperatures. (From Ref. 8.)

Figure 2 (a) Lamellar parts of cementite Fe3C on primary austenite grains and (b)

martensite packs boundaries. Steel 0.35C–1.5Mn–0.1P. Tempering at 3508C, 1 hr(TEM, replicas).


deformation and, consequently, formation of grain boundary cracks. Thebrittle crack develops, in this case, within boundaries of primary austenitegrains and martensite packs (Fig. 3a, b). The significant enrichment of grainboundaries by phosphorus is confirmed by Auger spectroscopy. The decisiverole of austenitization when compared with tempering in achieving equili-brium GBS is shown by low diffusion mobility of harmful impurities’ atoms(P, As, S, and Sb) for medium tempered steel. Calculations show that onlynitrogen has significant diffusion mobility in this temperature range which issufficient for diffusion at a distance of 10 mm for 1 hr at 3508C [1]. Thetemperature of steel for quenching is sufficiently high for intensive diffusionof P in austenite with the formation of equilibrium GBS [9], and quenchingfixes this enrichment. The level of P segregation depends on the austenitiza-tion temperature and increases when the temperature decreases below10508C. This is related to the decrease of P solubility in austenite. It is con-firmed by a significant decrease of the quenched steel delay fracture resis-tance with respect to temperature in the austenite region (Fig. 4).Phosphorous content in steel influences its embrittlement at 3508C temper-ing. Its concentration in GBS is several hundred times higher than in thevolume, and this harmful element segregates in austenite even at low con-centrations, about 0.01% mass, leading to a significant decrease in theimpact strength (Fig. 5).

Change of the relative energy of grain boundaries results in segrega-tion enrichment by impurities during austenitization. The interferometry isone such direct technique for grain boundary energy determination. For this

Figure 3 (a) Brittle fracture through primary austenite grains and (b) martensitepack boundaries in steel 0.35C–1.5Mn–0.1P. Tempering at 3508C, 1 hr (SEM).


purpose, the opening angle of GB slot (y) is determined on metallographicgrinds after vacuum etching at various temperatures. The grinds are auste-nitized in vacuum at high temperature, then subjected to interim cooling tothe desired temperature and then quenched at a high cooling rate. The rela-tive energy of GB is calculated by the equation

gbgs¼ 2 cos

y2

ð1Þ

where symbols b and s correspond to boundary and bulk, respectively.Figure 6 shows the change of relative surface energy with respect to the

interim cooling temperature during austenitization and P content in the0.35% C, 1.5% Mn. For the steel with low phosphorus content, the GBenergy regularly decreases slowly as the temperature increases within theaustenite region. Significant segregation enrichment of grain boundariesby phosphorus in the steel with 0.1% P does not show this dependenceand decreases the surface energy of GB with decreasing of the temperature.The decrease of P solubility in austenite with decreasing temperature assistsits adsorption on the GB and decreases significantly its surface tensionenergy. At the same time, one can observe another process: P enriches

Figure 4 Delayed fracture diagram of quenched steel 0.35C; 1.5Mn; 0.1P afteraustenitization at 1423K, 3 hr and 30min interim cooling to temperatures: 1, 1123K;

2, 1223K; 3, 1273K and cooling in water. (From Ref. 10.)


non-metallic inclusions. The dependence of P content on Mn sulfides in steel0.35% C, 1.5% Mn, 0.1% P from austenitization temperature is shown inFig. 7. When annealing temperature decreases, P redistributes between thebulk and grain boundaries enriching them and the non-metallic inclusions.

Steel grain size reduction can decrease significantly its tendency to tem-per embrittlement. Increase of the specific surface of the GB at dispersion ofthe steel structure decreases the concentration of the impurity in the grainboundary that leads to growth of steel brittle fracture resistance (Fig. 8) [3].Phosphorous segregations decrease the surface energy of intergranular cohe-sion. Using the approach proposed in Ref. [11], one can estimate the role ofphosphorus in change of surface energy of intergranular cohesion for devel-opment of this kind of embrittlement.

Figure 5 Temper embrittlement of steel Fe–0.3C–3.5N–1.7Cr: 1, high purity steel;2, 0.01 mass% P; 3, 0.03 mass% P; 4, 0.06 mass% P. (From Ref. 8.)


Figure 6 Change of relative surface energy of grain boundaries in dependence of

temperature of austenitization. Fe–0.35C–1.5Mn steel: 1, 0.03% P; 2, 0.1% P.

Figure 7 Phosphorus concentration in manganese sulfides in 0.35C–1.5Mn–0.1Psteel vs. austenitization temperature.


Influence of grain size on ductile–brittle transition temperature FATTis determined from the well known Petch–Hall equation

sf ¼ so þ Ky d�1=2 ð2Þ

where Ky ffi aðGgÞ1=2; a is a constant (1–3); g is the surface energy of inter-granular cohesion for generation of cracks on grain boundaries; G is theshear modulus.

It is described by the equation

dT50

d d�1=2ð Þ ¼ �1

bð3Þ

where b is determined as a tangent of inclination angle of straight lines inFig. 8. Taking into account that

Figure 8 FATT (T50) of steel vs. grain size d�1=2 and specific grain boundary surfaceP

S. Phosphorus content: 1, 0.03%; 2, 0.1%. (DT50¼T50(0.1% P)�T50(0.03% P)).


dT50

d d�1=2ð Þ ¼dT

dsfð�K0yÞ; ð4Þ

it is possible to use the change of the inclination angle of lines 1 and 2 inFig. 8 to determine the influence of P on GBS by the coefficient b andthe effective intergranular cohesion surface energy g that is proportionalto b2. The values of b for steel samples containing 0.03% and 0.1% P atthe temper embrittlement state are equal to �0.54 and 0.22 sec, respectively.Taking into account dependencies (2) and (3), one can conclude that effec-tive intergranular cohesion surface energy in steel with higher P concentra-tion decreases in g0:03=g0:1 ¼ b20:03=b

20=1 ¼ 6:04 times.

B. Ductile Intergranular Fracture of Overheated Steel

The ductile fracture of steel as well as brittle fracture could be characterizedby the lowest energy capacity. Such a fracture occurs when the inclusions arelocated along grain boundaries occupying a very large volume near theboundary. The intergranular microvoid fracture is observed in this casedue to overheating of the steel. The samples are exposed to high temperatureheating to dissolve inclusions. As a rule, the large and lamellar oxysulfidesthat did not embrittle steel are dissolved. After their dissolution, segregationof O, S, P, and precipitation, disperse particles during steel cooling occurs.In low-alloyed steels, precipitates could be sulfides of chromium and man-ganese (MnS, CrS), and aluminum nitride, AlN. These particles build adense network on grain boundaries. The fracture occurs at higher or roomtemperatures by intergranular microvoid coalescence at low stress intensity.The micrometer scale cavities nucleate on the intergranular fine dispersionof sulfide or nitride particles (Fig. 9). The segregation of harmful impuritiesis observed on grain boundaries in this case.

C. Reversible Temper Embrittlement

The reversible temper embrittlement (RTE) is observed in engineering steelalloyed by carbide-forming elements after quenching and high tempering(500–6008C). This phenomenon is developed in steels of industrial purity.It consists of a large decrease in the steel impact strength after slow cooling,but after rapid cooling at 6508C, the steel has a standard impact strength.The RTE phenomenon was identified for the first time in 1883 [12], whenblacksmiths observed that some steels had to be water quenched after tem-pering, to avoid embrittlement. This decrease of impact strength is notaccompanied by a change of physical or other mechanical properties of


steel. The duration of the exposure time within a definite range of temperingtemperatures (brittleness zone) plays a decisive role in the development ofRTE. The brittle fracture goes through the primary austenite grains(Fig. 10). The duration of the exposure time of the normalized or annealedsteel within the dangerous temperature range also leads to this kind ofembrittlement. Steels sensitive to RTE are subjected to rapid cooling from6508C during different heat treatments.

The GBS of phosphorus is the main reason for this kind of brittleness.The RTE of alloyed steels is mainly sensitive to two factors: the chemicalcomposition of the grain boundaries, and the mechanical and microstruc-tural parameters of the alloy. The direct correspondence of embrittlementkinetic features andGBSofPat steel temperinghasbeen established.Figure 11shows the iso-FATT curves for temper embrittlement of Ni–Cr steel [13].

Figure 9 Ductile fracture of steel through grain boundaries (SEM).


At a constant temperature of embrittlement tempering, the FATT increaseswith time. The embrittlement is reversible. It can be rejected by short heatingabove the nose of the C-curve in the ferrite range. Renewed aging and slowcooling of a de-embrittled steel in the critical temperature region gives re-embrittlement. These processes are accompanied by a redistribution ofimpurities on the grain boundaries.

III. FACTORS DETERMINING MULTICOMPONENTINTERFACE ADSORPTION IN ENGINEERING STEELS,AND THE METHODS OF ITS CALCULATION

The intercrystalline internal adsorption, or grain boundary segregationphenomenon, means the increased concentration of small impurities on

Figure 10 SEM image of intergranular fracture of Fe–0.35C–1.5Mn–0.1P steelafter quenching with temperature of 9508C, 1 hr and tempering at 5508C, 2 hr.


the grain boundary (compared to bulk) is caused by decreasing boundaryenergy. The energy of impurity segregation corresponds to energy gain inthe ‘‘bulk-boundary’’ system that accompanies the transition of one impur-ity atom from bulk to boundary. Thermodynamic description of this phe-nomenon is based on the Gibbs theory of the equilibrium segregation onfree surface. The decrease of redundant energy of GB is the thermodynamicstimulus to change its chemical composition compared to bulk. The impu-rities that decrease the energy of interfaces are surface-active. These ele-ments could form equilibrium segregations under favorable conditions.The impurities that increase surface tension escape from surface. Gibbs’adsorption isotherm for grain boundary segregation in a solid binary sys-tem, where the matrix obeys Raoult’s law and Henry’s law of diluted solu-tions, may be expressed

Gb ¼ Xc

kT

dgbdXc

ð5Þ

where Gb is the surplus concentration on the GB, mol=m2; Xc is bulk molepart of impurity; dgb=dXc is the tendency to adsorb.

Figure 11 Diagram of time–temperature FATT, 8C: (1) �60; (2) �55; (3) �50; (4)�45; (5) �40; (6) �35; (7) �30; (8) �25; (9) �20; (10) �15; (11) �10; (12) �5; (13) 0;(14) þ5; (15) þ10; (16) þ15; (17) þ20; (18) þ25; (19) þ30.


McLean [14] has developed a statistical model of intercrystalline inter-nal adsorption. The main points of this theory are the following:

1. A definite number of adsorption centers (one monolayer) arepresent on the GB. They have equal adsorption potential.

2. The impurity atoms are adsorbed independently on each of thecenters.

3. The adsorption decreases at temperature growth.

This model adequately describes the GBS process for two-componentsystem:

Xb

X0b � Xb

¼ Xc

1� Xcexp �DG

RT

� �ð6Þ

where Xb and Xc are the impurity concentrations on boundary and in bulk,respectively; X0

b is the ultimate equilibrium concentration of the impurityon boundary; DG is the segregation energy.

Hondros and Seah [15] have established the interrelation of GB enrich-ment and solubility limit X0

c

Xb

X0b � Xb

¼ Xc

X0c

exp�DGRT

� �ð7Þ

The enrichment factor, determined as the GB content=bulk concentrationratio, is of the order of magnitude 104 and 101 for impurity and alloying ele-ments, respectively (see Fig. 12).

A. Binding Energy of Impurities with Grain Boundaries

One can determine the adsorption energy as an alteration of the system freeenergy during transition of the dissolved atom from the grain bulk to boundary

E ¼ ðEb � E 0b Þ � ðEc � E 0

c Þ ð8Þwhere Eb and Ec are the free energy of the system with impurity atoms on thegrain boundary or in the bulk, respectively; Eb

0 and Ec0 are the free energies

of boundary and bulk in the pure solvent.The value of energy E is tied with the elastic (dimensional) and chemi-

cal interaction of impurity with boundary.The influence of dimensional discordance of atoms is accounted by the

Eshelby equation [16]

E ¼ 16

3pGr3ðs� 1Þ2 ð9Þ

where G is the shear modulus of the solvent; r is the solvent atom radius; s isthe ratio of impurity and solvent atoms radii.


The decrease of elastic distortion energy during the transition of theimpurity atom from ideal bulk lattice to distorted boundary lattice is thedriving force adsorption.

According to Ref. [17], one can describe the chemical part of adsorp-tion

Ech ¼ DZðe12 � e11Þ ð10Þ

where DZ is the difference of the atom coordination numbers in bulk andon boundary; eij is the energy of interaction between the nearest neighbors(1 for the solvent, 2 for impurity).

Figure 12 Calculated and experimental enrichment coefficients for surface

segregations. (From Ref. 15.)


The Fowler–Guggenheim theory [18] accounts for the interaction ofatoms adsorbed on GB:

Xb

X0b � Xb

¼ Xc

X0c

expE1 � ZoXb=X

0b

KT

� �ð11Þ

where Z is the coordination number for impurity atoms; o is the interactionenergy of nearest atoms.

The portion of segregation energy ZoXb=X0b corresponds to inter-

action of the same and different atoms on boundary, and depends on thedegree of neighbor centers filled. At o > 0, the atoms repulse mutually,and they attract at o < 0. The attraction increases the adsorption energysubstantially. Interaction of atoms on interface influences segregationenrichment and promotes formation of 2D phases with ordered atomicstructure.

B. Impurities’ Concurrence During Adsorption

None of the existing adsorption theories adequately describe the microme-chanism of impurities’ concurrence on the adsorption centers. This is relatedto the peculiarities of adsorption from gaseous phase to the free surface todescribe the grain boundary segregation mechanism [19–21]. According tothis point of view, all segregating elements (for example N, C, S, and P)occupy equal positions on GB, described by the ‘‘site competition’’ term.The peculiarity of GBS formation consists of diffusion of alloying elementsand impurities from bulk to interface. The migration mechanisms for substi-tial and interstitial impurities are different. The reason for this is that theadsorption centers on interface are different for these two kinds of impuri-ties. Therefore, the interstitial impurities (C, N) are located in interstices, butS, P, Sb, Bi, etc. occupy substitution position on the GB. Adsorption of anysurface-active impurity on GB decreases its free energy and lowers the ther-modynamic stimulus for adsorption of other impurity in a similar way. Thiscauses a site competition between atoms on the GB [22].

The concurrence of impurities A and B segregating at the same tem-perature and occupying the same positions in crystalline lattice (latticepoints or interstices) with different binding energies to GB was investigatedin Ref. [23].

The equilibrium concentration of competing impurities A and B couldbe calculated using equations:

XAb ¼

XAC expðEA

seg=kTÞ1þ XA

C expðEAseg=kTÞ þ XB

C expðEBseg=kTÞ

ð12Þ


XBb ¼

XBC expðEB

seg=kTÞ1þ XA

C expðEAseg=kTÞ þ XB

C expðEBseg=kTÞ

ð13Þ

where Eseg is the segregation energy; XbI is the GB concentration of element I

(atomic fraction). As one can see from these equations, at EAseg > EB

seg, theconcentration of element A decreases with increasing temperature.

In this case, the adsorption level of the element B reaches its maximumat critical temperature:

Tcr ¼EAseg

k lnðEAseg=ðEB

seg � EAsegÞXA

b Þð14Þ

The grain boundaries are enriched in this case by element B at lowtemperatures, and by element A at high temperatures.

C. Thermodynamic Calculations of theSegregation Energy

The segregation energy calculations are based on various models of solidsolution electronic structure or quasi-liquid model of grain boundary.

Thermodynamic properties of the solid solution determine firstly thesurface activity of small impurities in the formation of equilibrium GBS.The components of the alloy influence the energy of interaction with grainboundaries significantly. The parameters of such interaction are determinedby thermodynamic calculations, phase equilibrium diagram analysis, com-puter modeling of GBS. The heat of solution of different atoms in solidsolution is the thermodynamic measure of their interaction.

The model establishing interrelationship segregation energy and heatof solutions is proposed in Ref. [24]

Eseg ¼ FHsol � PðgA � gBÞV2=3A ð15Þ

where F and P are empirical coefficients; Hsol is the heat of solution of A inB [25–30]; gA, gB is the surface enthalpy of elements A and B [31,32]; VA isthe molar volume of A.

Using the liquid grain boundary model approximation, the segrega-tion energy of impurities (Eseg) could be determined from an analysis ofthe solidus and liquidus curves on phase equilibrium diagrams:

Eseg ¼ �KTLnðK0Þ ¼ �KTLnðCL � CSÞ ð16Þ


where K0 is the coefficient of equilibrium distribution of the elementbetween solid and liquid phases [33]; T is the melting temperature of thepure solvent; CL and CS are concentrations of impurity in the liquid andsolid solutions, respectively.

An experimental method for determination of binding energy ofimpurity atoms to grain boundary is used.

The analysis of large number of phase equilibrium diagrams has ledto the establishment of the basic property of two-component solid solu-tions consisting of periodic variation of the segregation formation energyof an element as a function of its location in the periodic table (atomicnumber). As seen in Fig. 13, the impurities could have positive or nega-tive surface activity, or be neutral. The elements So, Mo, Ni, and Co areneutral in two-component alloys with Fe. At the same time, it is wellknown that molybdenum is the surface-active element in steels and itreduces the tendency of steel to the reversible temper embrittlement. Thischange of surface activity is observed only in multicomponent alloys, andit is due to the mutual influence of elements on its thermodynamic activ-ity. The binding energy of an impurity to the GB depends significantly onboundary structure. The wide spectrum of Eseg exists for the given sub-stance analogous to the spectrum of the GB energy. This circumstanceexplains the wide dispersion of the segregation energy for various impu-rities that are listed in literature sources. Based on this reasoning, it isuseful, for segregation modeling, to apply the unified approach for deter-mination of the generalized characteristic of definite impurity segregationin definite solvent. The thermodynamic calculations of segregation energyare the most suitable way for its estimation. Auger electron spectroscopy(AES) for investigation of segregation kinetics on the free surface of poly-crystalline foils is a reliable experimental technique for the averaged Eseg

determination.The part of elastic and chemical interaction in GBS process could be

estimated experimentally based on concentration dependencies of segrega-tion energy. These dependencies were determined for alloys whose composi-tions are listed in Table 1.

The segregation energy was determined based on AES of equilibriumfree surface segregations of phosphorus. The polycrystalline foil sampleswere tempered at 823K for 4 hr in a work chamber of electron spectrometerESCALAB MK2 after quenching from austenitization temperature of1323K. The segregation energy of P was determined using Eq. (6) basedon surface and bulk impurity concentration. Figure 14 shows the depen-dence of the P segregation energy and its bulk content in alloy. For thediluted solid solutions, Eseg is independent of concentration or temperature.It is caused only by elastic distortions that are formed by impurity atoms in


Figure 13 Change of calculated Eseg of impurities in Fe-base alloys in accordance

to its number in periodic system. Calculations were based on Hsol in the followingpublications: (a) Refs. 25 and 26; (b) Refs. 27 and 28; (c) Ref. 33.


bulk and on the interface. As seen in Fig. 14, the elastic interaction energy ofthe P atoms with grain boundaries in iron is equal to 0.53 eV=at anddecreases significantly at molybdenum alloying to 0.24 eV=at in the alloyFe–3.1at.% Mo. Decrease of segregation energy of the impurity at its

Table 1 Composition (at.%) of the Fe–P and Fe–P–Mo Alloys

Chemical composition, at.%

C S P Mo

0.01 0.002 0.017 00.01 0.003 0.10 0

0.01 0.002 0.15 00.01 0.003 0.093 3.10.01 0.002 0.033 3.10.01 0.002 0.14 3.1

0.01 0.005 0.09 0.30.01 0.014 0.074 0.02

Figure 14 Change of Eseg of phosphorus with its volume concentration in Fe (1)

and Fe–3.1 at.% Mo–P alloys (2). Auger electron spectroscopy of free surfacesegregations at 823K.


volume concentration growth is caused by chemical pair interaction of theatoms in alloy.

Using the example of the Fe–P system, we could determine chemicalinteraction of elements by applying the approach proposed in Ref. [34].Analyzing the solidus and liquidus equilibrium (volume and GB) on theequilibrium phase diagram at three temperatures permits the constructionof a system of three equations that describe this equilibrium

kT qa � ln100� Xs

100� Xl

� �¼ X2

sW0 � X2

lW00 þ kqaTa ð17Þ

where k is the Boltzmann constant; Ta is the melting temperature ofFe; qa is melting entropy per atom divided by Boltzmann constant;W0 and W00 are the mixing energies in solid and liquid states; Xs andXl are the impurity concentration in solid and liquid phases at the tem-perature T.

Solving these equations for the phase diagram of Fe–P binary sys-tem [35], the sign and value of mixing energy in liquid phase equal0.425 eV=at were determined. The positive value (in accordance with phy-sical sense) means that binding force of P–P and Fe–Fe atoms is higherthan for Fe–P atoms:

W ¼ WFe�P � 1

2ðWFe�Fe þWP�PÞ ð18Þ

emphasizing the tendency for solid solution tendency for stratification orintercrystalline internal adsorption.

D. Effect of Solute Interaction in Multicomponent Systemon the Grain Boundary Segregation

Guttman has expanded the concept for synergistic co-segregation of alloy-ing elements and harmful impurities at the grain boundaries. His theory isvery important for analysis of steels and alloys that contain many impuri-ties and alloying elements. In accordance with the theory, the interactionbetween alloying elements and the impurity atoms could be estimated fromenthalpy of formation of the intermetallic compounds (NiSb, Mn2Sb, Cr3P,etc.). The alloying elements could influence on the solubility of impurities inthe solid solution. Only the dissolved fraction of the impurity takes part inthe segregation [36]. When preferential chemical interaction exists betweenM (metal) and I (impurity) atoms with respect to solvent, the energy of


segregation becomes functions of the intergranular concentrations of Iand M:

DGI ¼ DG0I þ

bbMI

CbYb

M �baMI

Ca XaM ð19Þ

DGM ¼ DG0M þ

bbMI

abYb

I �baMI

aaXa

I ð20Þ

where Cb and ab are the fractions of sites available in the interface for I andM atoms, respectively ðab þ Cb ¼ 1Þ; Yb is the partial coverage in the inter-face; Xa is the concentration in the solid solution a; bMI is the interactioncoefficient of M and I atoms in a-solid solution (a) or on the grain boundary(b). For a preferentially attractive M–I interaction, the bMIare positive andthe segregation of each element enhances that of the other. If the interactionis repulsive, the bMI are negative and the segregations of both elements willbe reduced. For a high attractive M–I interaction in the a-solid solution, theimpurity can be partially precipitated in the matrix into a carbide, or inter-metallic compound. The interface is then in equilibrium with an a-solutionwhere the amount of dissolved I, XI

a, may become considerably smaller thanits nominal content.

In the ternary solid solutions, the segregation of impurity (I) could belowered or neglected at several critical concentrations of the alloyingelement (M) whose value (CM

a) depends on surface activity of each compo-nent (ESeg

I,M) and interaction features of the dissolved atoms (bMI):

CMa ¼

EISeg

bMIðexpðEMSeg=RTÞ � 1Þ ð21Þ

The critical concentration of alloying element is accessible for segregation ofimpurity and alloying element EI;M

Seg > 0 and repulsion of different atomsbMI > 0; or without segregation of alloying element EM

Seg < 0 and withattraction of different atoms bMI < 0.

In this case, the dependence of ESegI,M on the dissolved element concentra-

tion is not taken into account. Indeed, for systems with limited solubility, thealteration of value and sign of segregation energy is possible at a definite con-tent of alloying element. The phase equilibrium diagram analysis allows thedetermination of mutual influence of components on their surface activity.

The equilibrium distribution of solute elements between solid andliquid phases in iron-base ternary system (distribution interaction coefficientK0) is known to be an important factor in relation to microsegregation dur-ing the solidification of steels. As it was shown above, these analogiesare useful for the prediction of GBS and for impurity segregation energy


determination in the given solvent. The K0 of some elements, especially inmulticomponent systems, is considered to be different from those in binarysystems because of the possible existence of solute interactions, but themechanisms are so complicated that detailed information has not yet beenobtained. Therefore, it would be very useful if the effect of an addition ofan alloying element on the distribution could be determined by the use ofa simple parameter.

Equilibrium distribution coefficient K0�1 of various elements in Fe–C

base ternary system is calculated from equilibrium distribution coefficient iniron-base binary systems [40–43]. In Fig. 15, the calculated results are com-pared with the measured values by various investigators. The changes of theK0�1 of P and S with various alloying elements are shown in Fig. 16(a, b) in

Fe–P and Fe–S base ternary system, respectively.These data could be applied for calculation of phosphorus segregation

energy change under the alloying element influence in Fe–Me–0.1at.% Palloys (Fig. 17) or for calculation of the segregation energy change ofalloying elements with concentration of carbon in Fe–0.1Me–C alloys(Fig. 18). For the growth of carbon volume content, the segregation energyof C and P decreases which means lowering of the segregation stimulus forthese elements.

Figure 15 Change of the equilibrium distribution coefficient of some elements withcarbon concentration in Fe–C-based ternary systems. (From Ref. 37.)


E. Kinetics of Segregation

The existing models of multicomponent adsorption do not analyze in detailthe kinetics of the process. But in reality, GBS forms only during a limitedtime of the heat treatment process. The difference of segregation level fromthe equilibrium one depends on temperature and time. At low temperaturesand limited time of heat treatment, segregation is controlled by diffusion. Asthe temperature increases, segregations with lower equilibrium concentra-

Figure 16 (a) Change of the equilibrium distribution coefficient of phosphoruswith the concentration of alloying elements; and (b) change of the equilibriumdistribution coefficient of sulfur with the concentration of alloying elements. Solid

line: a-phase. Chain line: g-phase. (From Ref. 38.)


tion are developed, but rich segregations dissolve. Distinguishing diffusionmobility and mutual influence of elements on their diffusion coefficientsdetermines much of their segregation ability. Amplification or suppressionof adsorption could be due to a kinetic factor. This peculiarity determinesthe fundamental factor of distinguishing adsorption from gas phase to freesurface when comparing it to intercrystalline internal adsorption: GBS iscontrolled by diffusion during heat treatment of steels and alloys.

Many GBS features in multicomponent systems cannot be predictedadequately using the equilibrium segregation thermodynamic accountingbasis. Particularly, the thermodynamic concept of the cooperative (synergis-

Figure 16 (Continued)


tic) adsorption of elements is disturbed when they do not segregate at thesame temperature. The concurrence of impurities at GBS could be tied notonly due to their attractive or repulsive interaction, but also with higher diffu-sion mobility of some impurities. In many cases, the determination inter-atomic interaction on grain boundary that is proposed in Guttmann’stheory has a significantly lower effect for segregation prediction than account-ing of mutual influence of elements on their thermodynamic activity in thegrain bulk.

Mutual influence of the alloy components on their surface activity iscaused by their interaction in solid solution in the bulk. The interactionon grain boundaries could be analyzed only for those elements thatsegregate in near temperature ranges. Many postulates of the thermo-dynamic theory of equilibrium grain boundary segregation could not beapplied simply for heat treatment of multicomponent alloys. This is espe-cially important for steels, which have complex phase transformations dur-ing treatment that accompany change of the solid solution composition.Auger electron spectroscopy permits the investigation of multicomponentadsorption kinetics. The composition of grain boundaries on the intercrys-talline fracture surface made under high vacuum is analyzed for this pur-pose. In these cases, the experimental modeling of GBS is widely used.The chemical composition of free surface of thin poly-crystalline foils thatare heated in situ is investigated using Auger electron spectrometers. TheP grain boundary adsorption isotherms for samples of three Fe–Cr–Mn

Figure 17 Change of Eseg of phosphorus with the concentration of alloying

elements in Fe–Me–0.1% P alloys.


steels after quenching from 1273K and tempering at 923K for 25min, 1 and2 hr with air cooling are presented in Fig. 19. Dissolution of Ti and V car-bonitrides after steel quenching promotes enrichment of the solid solutionby these elements. They have high values of Gibbs energy for phosphide for-mation, decrease the thermodynamic activity of phosphorus in solid solu-tion and reduce its GBS.

Most models of kinetics are classically analyzed in terms of the lawderived by McLean [14] for binary alloys

XbðtÞ � Xbð0ÞXb � Xbð0Þ ¼ 1� exp

4Dit

ðXb=Xai Þ2d2

" #erfc

2ffiffiffiffiffiffiffiDitpðXb=X

ai Þ

� �ð22Þ

Figure 18 Change of Eseg of alloying elements (Me) with the concentration ofcarbon in Fe–0.1% Me–C alloys.


where Xb(t) is the interfacial coverage of element, at time t; Xb(0) — is itsinitial value and Xb its equilibrium value as defined by Eq. (7); Xi

a — isits volume concentration; Di is the bulk diffusivity of i and d is the interfacethickness.

Assuming Xb=Xia¼ const, using Laplace transformation for (22), one

can obtain the approximate expression

XbðtÞ � Xbð0ÞXb � Xbð0Þ ¼

2Xai

Xbd

ffiffiffiffiffiffiffiffiffiffiFDtip

rð23Þ

where F¼ 4 for grain boundaries and F¼ 1 for free surface.The kinetics of segregation dissolution could be described by these

equations (22) and (23). But, in this case, the variables Xb(0) and Xb

exchange places. The influence of Mo, Cr, and Ni additions on kinetics ofP segregation has been studied in six Fe–Me–P alloys, whose base composi-tions are listed in Table 1. These materials were austenitized for 1 hr at1323K and quenched in water. The tempering of foils at 773K was carriedout in a work chamber of an electron spectrometer ESCALAB MK2 (VG).The kinetics of P segregation studied for Fe–Me–P alloys (Figs. 20–22) showthat equilibrium is reached within several hours. Based on the starting posi-tion of adsorption isotherms, the phosphorus diffusion coefficients in thesealloys were calculated using Eq. (22). The data are presented in Table 2.Molybdenum reduces significantly P surface activity and decelerates itsdiffusion. Nickel is not a surface-active element in carbonless alloys, Fe–P–Ni. It increases sharply P thermodynamic activity and equilibrium GBconcentration, and accelerates its diffusion. Chromium segregates poorly

Figure 19 Kinetics of P GBS in steel 0.3C–1.6Mn–0.8Cr–008P (1) with adds of

0.047Ti (2) or (0.07Ti and 0.026V) (3), quenched from 1273K and tempered at 923K.


in these alloys. It also, as Ni, increases diffusion mobility of P and its grainboundary adsorption.

The adsorption isotherms of various elements have non-monotonousshape in multicomponent alloys. The isodose thermokinetic diagramspresent the averaged information on segregation of all components. Such

Figure 20 Kinetics of P segregation on free surface in Fe–P–Mo alloys with

different relative concentration of Mo=P at 773K.

Figure 21 Kinetics of P and Cr segregation on free surface in Fe–0.04P–2.3 Cralloy at 773K.


diagrams for steel with 0.2–0.3% C alloyed by Cr, Mo, Ni, Mn, V, and Nbare presented in Figs. 23–27.

Chemical composition of steel is listed in Table 3. The T–t diagramsare obtained based on adsorption isotherms on free surface of foils that wereheat treated in Auger spectrometer ESCALAB MK2 at vacuum about10�10 Torr. The isodose curves characterizing time of definite segregationlevel access depending on temperature are shown in these diagrams. At ele-vation of an isothermal exposition temperature, the mobility of impuritiesincreases, and time for reaching of definite segregation level decreases.The lower branch of isodose curve means decrease of segregation formation

Figure 22 Kinetics of P segregation on free surface in Fe–P–Ni alloys withdifferent relative concentration of Ni=P at 773K.

Table 2 Composition (at.%) of the Fe–Me–P Alloys and Kinetics Characteristics

of P Free Surface Segregation, Deduced from the Segregation Kinetics

Chemical Composition, at.%Surface

activity Xb=Xia

Bulk diffusivity ofP DP � 10�18 (m2=sec)P Mo Ni Cr

0.07 0 0.9 0 285 5.340.03 0 3.1 0 2530 10480.04 0 0 2.3 1450 160

0.16 1.0 0 0 380 270.21 2.1 0 0 250 2.280.15 3.1 0 0 80 0.3


time at increasing temperature. With temperature increase, the solubility ofimpurity in solid solution increases, and its GB concentration reduces. It fol-lows that the probability to form the segregation with high impurity contentreduces, and time for such segregation increases extensively. The upperbranch of isodose curves corresponds to dissolution of rich segregationsand access to new equilibrium with lower impurity concentration. The

Figure 23 The isodose C-curves of multicomponent interface segregation in 0.3C–Cr–Mo steel (see Table 3). Auger electron spectroscopy of free surface segregations.

Figure 24 The isodose C-curves of multicomponent interface segregation in 0.2C–Cr–Mn–Ni–Si steel (see Table 3) under its tempering. Auger electron spectroscopy offree surface segregations.


adsorption patterns for engineering steels have common as well as indivi-dual features. As a rule, carbon segregates at temperatures lower than523K, nitrogen—in 523–623K range, phosphorus—in 523–823K range, sul-fur segregates at temperatures higher than 723K.

The substitual and interstitial element concurrence promotes blockingof adsorption centers by mobile impurities and impedes P segregation at

Figure 25 The isodose C-curves of multicomponent interface segregation in 0.3C–Cr–Mn–Nb steel (see Table 3) under its tempering. Auger electron spectroscopy of

free surface segregations.

Figure 26 The isodose C-curves of multicomponent interface segregation in 0.3C–Cr–Mn–V steel (see Table 3) under its tempering. Auger electron spectroscopy of freesurface segregations.


temperatures lower than 523–673K. The preferential enrichment of GB by Pand S becomes possible after dissolution of C and N segregations. The alloy-ing elements change significantly the segregation stability regions for var-ious elements. Fig. 28(a,b) shows the P adsorption isotherms in theinvestigated steels at 723K. Molybdenum sharply slows down the Psegregation formation. The differences in diffusion mobility of elementsand temperature intervals of segregation stability are the reasons for non-equilibrium enrichment of grain boundaries. The rich segregations areformed at the initial stage of isothermal exposition, and they are dissolvedafter longer exposition. Comparing the behavior of 0.3C–Cr–Mn–Nb (1)and 0.22C–Cr–Mn–Si–Ni (3) steels at 673K tempering, one can see thatsmall (lower than 20min) expositions 0.3C–Cr–Mn–Nb, and longer ones(about 1 hr 20min) are dangerous for 0.22C–Cr–Mn–Si–Ni steel. Analyzing

Figure 27 The isodose C-curves of multicomponent interface segregation in 0.3C–

Cr–Mn–Si–Ti steel (see Table 3) under its tempering. Auger electron spectroscopy offree surface segregations.

Table 3 Chemical Composition of Steels

No. Steel

Concentration of elements, wt.%

C Si Mn Cr V Al Ti Nb Ni Mo S P

1 0.3C–Cr–Mn–V 0.32 0.25 0.88 0.92 0.088 0.014 0.024 0 0 0 0.016 0.027

2 0.3C–Cr–Mn–Nb 0.29 0.33 1.04 1.07 0 0.007 0.036 0.025 0 0 0.014 0.027

3 0.3C–Cr–Mo 0.33 0.23 0.56 0.96 0.003 0.014 0.025 0 0 0.25 0.005 0.004

4 0.3C–Cr–Mn–Si–Ti 0.28 0.61 1.15 0.75 0 0.029 0.016 0 0.32 0 0.013 0.022

5 0.2C–Cr–Mn–Ni–Si 0.22 0.43 0.92 0.89 0 0.030 0.015 0 0.91 0 0.015 0.025


the thermokinetic diagrams for ternary Fe–Me–P alloys based on Eqs. (23)and (6), the mutual influence of elements on their binding energy to GB wasdetermined [36]

EPseg ¼ 20:6 þ 183CP

a � 4:8CAla � 7:2CMo

a � 3:4CNia � 7141CB

a

þ 4:9CCra � 444CS

a � 183EMoseg � 87EN

seg ð24ÞESseg ¼ 6:9 � 151CS

a � 1:5CAla þ 14:5CP

a � 39ESnseg ð25Þ

ENseg ¼ 16 � 2:6CAl

a þ 3CMoa þ 4:2CCr

a � 2625CTia þ 175EMo

seg ð26Þ

Figure 28 Influence of alloying on the kinetics isotherms of P free surfacesegregation at 723K. The following steels were investigated (see Table 3): 1, 3C–Cr–Mn–Nb; 2, 3C–Cr–Mn–Si–Ti; 3, 2C–Cr–Mn–Ni–Si; 4, 3C–Cr–Mo; 5, 3C–Cr–

Mn–V.


ECseg ¼ 7:9 � 1:4CAl

a þ 5CMoa þ 676CB

a þ 1:2CCra � 130EN

seg þ 116EPseg

ð27Þ

EMoseg ¼ �0:7 þ 32EN

seg � 28EPseg ð28Þ

ETiseg ¼ 17 þ 3CC

a � EPseg ð29Þ

EAlseg ¼ 1:4CAl

a ð30Þ

ESnseg ¼ 21; ENi

seg ¼ 14; EBseg ¼ 54; ECu

seg ¼ 20 kJ/mol

where EsegI is segregation energy of the I element, Ca

j is bulk concentration ofj impurity.

F. Stability of the Segregation

The equilibrium GBS dissolves as temperature increases. Analysis of thekinetic development of the equilibrium segregation level of P shown inFig. 29 gives the T–t plot of segregation directly. Obviously that segregationlevel close to the maximum exists only within a specific temperature range.This range is characterized by a maximum temperature stability Tmax, overwhich the intensive dissolution of the segregates is observed. This tempera-ture can be calculated by computer analysis of Eq. (7) at dCb

max=dT¼ 0.The temperature Tmax depends on Eseg and temperature dependencies ofsolubility limits, which can be determined from analysis of phase equili-brium diagrams [43].

Using these dependencies as a generalizing criterion, it is possible tosimplify the analysis of data on element segregation kinetics in iron alloys.The interrelationship of maximum temperature of stability (Tmax) of richequilibrium segregations and segregation energies of different elements ispresented in Fig. 30.

The common features of kinetics show the following groups:

1. enriching grain boundaries at low- and medium-temperingtemperatures—B, C, N, and Cu;

2. co-segregating with P at high tempering—P, Sn, Ti, and Mo;3. segregating at high temperatures—S and Al.

Phosphorus in Fe alloys has abnormally weak dependence of Tmax

on Eseg in reversible temper embrittlement temperature range. In other


words, this means that the temperature of P segregation stability in theRTE development interval weakly depends on segregation energy or alloycomposition. This circumstance is associated with the specific shape of thetemperature dependence of P solubility in Fe. The established regularityallows to explain the difficulties with rational alloying of engineering steelsfor RTE suppression.

G. Nature of Reversibility of Temper Embrittlement

The reversibility of temper embrittlement is usually associated with precipi-tation or dissolution of carbide phase at various modes of quenched steelheat treatment below Ac1 [44–46]. The complex character of multicompo-nent GB adsorption—namely interrelation of two opposite processes: con-currence between impurities, and their cooperative segregation—is nottaken into account using this approach.

Figure 29 The calculated segregation level of P as a function of temperature

according to Eq. (7). Tmax is the maximal temperature of stability of rich segregationlevel. (From Ref. 42.)


Figure 30 The interconnection of Tmax-segregation stability temperature andEseg-energy of impurities segregation in Fe-base alloys.


Figure 31 presents the thermokinetic diagram of element segregationin 0.35C–1.5Mn–0.1P–0.6Al steel. The chemical composition of free surfacesegregations was determined by AES for a set of isothermal conditions inthe spectrometer ESCALAB MK2 (VG). The temperature–time intervalof preferential segregation of chemical elements is the result of different dif-fusion mobility and binding energy of elements with GB. The temperatureinterval of P preferential segregation is caused by concurrence of this impur-ity with mobile interstitial elements C and N. This process determinestemperature and exposition necessary for RTE development. Direct investi-gation of grain boundary composition by AES confirms the conclusionabout the prevailing role of concurrent segregation in RTE. The composi-tion of several grain boundaries on brittle intercrystalline fracture of0.35C–Mn–Al steel after heat treatment: quenching from 1223K, temperingat 923K for 1 hr with rapid (a) and slow (b) cooling is presented in Fig. 32[47]. These data are in good correspondence with those in Fig. 31. Acceler-ated cooling of steel, does not provide enough time for the development ofsegregations with high P content, and GB are enriched by C. During slowcooling, phosphorus has enough time to enrich the grain boundaries. In thiscase, the carbon concentration on GB is sufficiently lower than at rapidcooling of steel. Carbon segregations are unstable at temperatures higherthan 500–673K, and they are dissolved. At slow cooling, P segregates tograin boundaries, decreasing the GB redundant energy. This circumstancelessens the thermodynamic stimulus for carbon segregation as the tempera-ture decreases. Carbon and phosphorus in steels are responsible for RTEdevelopment. They have high surface activity and diffusion mobility that

Figure 31 Thermo-kinetics diagrams of multicomponent segregation on free

surface in steel 0.35C–1.58Mn–0.1P–0.6Al.


predetermines their segregation on GB at heat treatment. Difference of dif-fusion mobility as well as difference of maximum temperature of segregationstability is the reason for preferential segregation of an impurity. This is thereason for the characteristic temperature region of RTE development.Reversibility (i.e. disappearance) of temper embrittlement is associatedwith full dissolution of rich GBS of phosphorus at high temperatures and

Figure 32 Chemical composition of GB in steel 0.35C–1.58Mn–0.1P–0.6Al (AES);(a) tempering at 923K, water cooling; (b) tempering at 923K, cooling with furnace.

(From Ref. 47.)


enrichment of GB by carbon at rapid cooling [48]. Undoubtedly, carbidetransformation, internal stresses, substructure transformations are veryimportant for RTE. One should take into account such circumstanceswhere kinetics of C and P segregation are dependent significantly on steelalloying.

IV. DYNAMIC SIMULATION OF GRAIN BOUNDARYSEGREGATION

A. Interface Adsorption During Tempering of Steel

1. Decomposition of Martensite

The common laws of multicomponent GBS and analysis of experimentaldiagrams on elements segregation kinetics in iron alloys are used to developthe computer models of these processes. The exact solution of McLean’sdiffusion Eq. (21) accounting for temperature dependant of diffusion andelement solubility is a complex problem. In low-alloyed steels, the concen-tration of surface-active impurities (S, P, and N) is rather small, and basedon this reason, it is possible to analyze the diffusion of each element sepa-rately. The model takes into account mutual influence of bulk and surfaceconcentration of elements with respect to segregation energies.

Carbon in solid solution has maximum influence on phosphorus GBSkinetics. Concentration of C in martensite changes significantly duringquenched steel tempering and mainly depends on alloying element content.Based on this reason, one should take into account the solid solution com-position altering segregation processes modeling during tempering.

Investigations of martensite tetragonality at alloyed steel tempering[6,7] are the basis for calculations of mutual influence of alloying elementson martensite decomposition kinetics and carbon content in solid solution.The carbon content change in solid solution during tempering of engineer-ing steels is well described by equation

DXCa

XCa ð0Þ

¼ 1� exp �KDot exp � Q

RT

� �n� �ð31Þ

where

DXCa ¼ XC

a ð0Þ � XCa ðtÞ ð32Þ

XaC(0) and Xa

C(t) are the carbon content in quenched steel and after a time t;Do is the carbon diffusion coefficient; Q is the activation energy associatedwith the interstitial diffusion of carbon atoms; K is the constant associated


with the nucleation; n is the constant independent of both temperature andXa

C(0); R is the gas constant and T is the temperature.Influence of C and alloying elements on parameters Q, K, and n in

Eq. (31) is determined for various steels. The activation energy Q in low-alloyed steel depends on the concentration of carbon and alloying elementsin solid solution:

Qðcal=molÞ ¼ 8571:5XCa þ AXMe

a þ 18; 000 ð33Þ

where XaC and Xa

Me are concentrations of C and alloying elements, mass%;A is a constant depending on alloying element. The values of coefficients inEq. (31) are presented in Tables 4 and 5. The diffusion activation energy of

Table 4 Coefficient A in Eq. (28) for Low-Alloying Engineering Steels

Coefficient A

Alloying element

Ni Si Mn Cr Mo

433.56 1,432.54 �726.35 �2,898.91 �971.51

Table 5 Influence of Carbon and Alloying Elements on Parameters Q, K,and n in Eq. (31)

Steel, wt.% Q, cal=mol LnK n

0.4C–0.24Ni 21,532 15.364 0.260.39C–3.0Ni 22,643 17.481 0.220.37C–5.6Ni 23,599 18,575 0.24

0.4C–0.32Mn 21,196 15.737 0.240.4C–1.32Mn 20,298 14.241 0.220.4C–2.43Mn 19,406 13.713 0.240.4C–0.2Cr 20,848 15.366 0.21

0.4C–2.1Cr 15,348 10.076 0.240.4C–3.6Cr 10,992 5.481 0.420.4C–6.7Cr 2,005 1.698 2.32

0.4C–0.37Si 21,929 16.351 0.190.38C–1.75Si 23,764 15.234 0.220.4C–2.75Si 25,368 10.050 0.15

0.4C 21,429 16.72 0.241.4C 30,000 40.881 0.071.2C–2.0Mo 26,343 29.768 0.08


carbon decreases on the growth of carbide-forming element (Mn, Cr, andMo) concentration. The contrary effect is observed for Ni and Si.Obviously, it is associated with the different influence of these elementson thermodynamic activity of carbon in ferrite. These dependencies arebasic for calculations of segregation kinetics of C since carbon is the elementthat influences on P segregation highly. The kinetics of carbon content insolid solution change during tempering of quenched steel 0.43C–2.43Mn(mass%) are shown in Fig. 33. These data are obtained by computer mod-eling using Eqs. (31–33) and those from Tables 4 and 5.

This model provides the possibility of calculating the influence ofalloying on cementite formation temperature interval, growth rate of itsparticles, and many other parameters of martensite decomposition at tem-pering [49].

Fig. 34 presents the calculation results of effective growth rate of Fe3Cnucleus at tempering of engineering alloyed steels. The calculations werecarried out using expression [49]:

VmaxR ¼ ð27D=256pÞN ð34Þ

where R is the cementite particle radius; N is the right part of Eq. (30). Man-ganese decreases martensite stability significantly promoting its decomposi-tion at low temperatures. Silicon, at a concentration greater than 1%,activates martensite decomposition at 700–800K and inhibits it at lower

Figure 33 Change of carbon concentration in solid solution with temperature and

time of tempering. Steel 0.43C–2.43Mn (mass%). Isodose curves for: 1, 1 at.% C; 2,0.5 at.% C; 3, 0.1 at.% C; 4, 0.05 at.% C; 5, 0.03 at.% C.


content. Chromium does not change the temperature of intensive cementitegrowth.

2. Calculation of Thermokinetic Diagrams of Impurities’

Segregation During Tempering of Steel

Modeling of multicomponent adsorption kinetics is carried out using asequence of computer calculations.

At the initial stage, thermodynamic characteristics of surface activityin Fe-base binary and ternary alloys are determined.

Analysis of phase equilibrium diagrams permits the determination ofthe impurity segregation energy Eseg

i and temperature dependence of ulti-mate solubility X8c. These two parameters are very important for determina-tion of equilibrium GB concentration of impurity Xi

b using Eq. (17).Examples of such calculations for binary and ternary alloys have been

presented. Mutual influence of alloy components on their surface activitycould be refined experimentally. The equations for binding impurity segre-gation energy with solid solution composition could be obtained by regres-sion analysis of multicomponent adsorption diagrams. These experimentsallow the determination of the effective diffusion coefficient of elements.The diagram of the equilibrium impurity concentration calculation on grainboundary in engineering steel is presented in Fig. 35.

Carbon concentration in martensite changes drastically during temper-ing, as it depends on chemical composition of steel, temperature, and duration

Figure 34 Change of effective growth rate of Fe3C nucleus with alloying of 0.4% C

steel: 1, unalloyed steel with 0.4C(mass%); 2, alloyed with 0.35Si; 3, alloyed with2.1Cr; 4, alloyed with 1.75Si; 5, alloyed with 2.43Mn.


of treatment. This factor influences on thermodynamic activity of all steelcomponents and on their energy of GB segregation.

The second important stage of GBS modeling includes calculationof C volume concentration in martensite Xa

C(T), depending on steelchemical composition Xa

i (0) and parameters of tempering. New segregationenergy values of each element at changing of treatment temperature ortime and new equilibrium GBS level have been calculated in this way (seeFig. 35).

The final stage of modeling includes a set of independent calculationsof various element diffusion to GB zone, and their desorption. The limitedcapacity of boundary and its effective width (about 0.5 nm) are shown. It isassumed that interstitial and substitial impurities occupy different positionson GB. Time t of reaching the definite concentration of impurity in segrega-tion Xb(t) at given temper temperature T is calculated by (22), and it is con-trolled by diffusion Di(T).

Adsorption in multicomponent system is accompanied by concur-rence: arrival of some surface-active impurity decreases GB energy and, inthis way, the thermodynamic stimulus for segregation of other impurities.Dissolution of segregations is observed at increasing temperature. Impuritydesorption to grain bulk is analogous to adsorption, however it is tied notwith concentration Xi(0) but with Xb(t), and it is also controlled by diffusionDi(t).

Figure 35 Calculation scheme of equilibrium impurity GBS. Xai (0) is the initial

concentration of ith element in the steel; XaC(T,t) is the running carbon concentration

in martensite during its tempering; Xbi (T) is the maximal equilibrium GBS of ith

element; Eiseg Fe–i is the segregation energy of ith element in two-component Fe–I

alloy; Eiseg Fe–i–j is the segregation energy of ith element in multicomponent alloy;

Di(T) is the diffusion coefficient of ith element in austenite, martensite, and ferrite.


The model is restricted to initially homogeneous bulk concentra-tions

Xibð0Þ ¼ Xi

a ð35Þ

The kinetics of segregation to surfaces or grain boundaries from thebulk are determined by volume diffusion of impurities with bulk concentra-tions Xi

a(t) which can be treated as a one-dimensional problem. Since bothbulk concentrations are very small, Arrhenius type diffusion coefficients:

Di ¼ Di0 exp � Qi

RT

� �ð36Þ

can be used which are independent of Xia(t). In the case of site competition,

the GB impurities concentration is

qi ¼ Xi

1�PJ Xjexp � Ei

KT

� �ð37Þ

The equations describing the time evolution of segregation for homo-geneous initial condition [60] are

XiðtÞ ¼ Xið0Þ þ 2X0

iffiffiffipp

d

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ t

0

Diðt0Þ dt0s

� 1ffiffiffipp

d

Z t

0

qiðt0ÞDiðt0ÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR t0t Diðt00Þ dt00

q ð38Þ

In the case of constant temperature (i.e. Di¼ const), Eq. (38) can besimplified:

XiðtÞ ¼ Xið0Þ þ 2ffiffiffiffiDpffiffiffipp

d½X0

i � qiðtÞ�ffiffitp ð39Þ

Diffusion coefficient for impurities in Fe and Fe-base alloys in ferriteinterval is present in Table 6.

The calculated diagrams of multicomponent adsorption in steels 0.3C–Cr–Mo, 0.3C–Cr–Mn–V, 0.3C–Cr–Mn–Si–Ti (see Table 3) are presented inFigs. 36–38. Comparing these diagrams with the experimental ones (Figs.24, 26, and 27), a good correlation of segregation kinetic featuresfor various elements is observed, that confirms the basic principles of theproposed model of GBS in steels. According to this model, the mainrole of carbide precipitation in GBS consists of changing solid solution


composition, as it is exactly this factor that controls mutual influence of ele-ments on their surface activity. Only such elements that segregate in neartemperature ranges mutually influence GBS. The computer calculations ofsegregation kinetic diagrams predict these effects with small changes of steelchemical composition. Figures 39 and 40 present the modeling data on influ-ence of sulfur content in 0.3C–Cr–Mn–Si–Ti steel on phosphorus segrega-tion kinetics. Sulfur and Phosphorus are strong surface-active elements,and they can compete at grain boundaries. Desulfurization of steel signifi-cantly slows down GBS of S. Indeed, P adsorption increases with a decreaseof S content. According to calculations (see Fig. 40), the time of 6% P GBSformation exceeds 4000 sec at a sulfur content more than 0.02 at.%. Thistime it is significantly longer than the usual duration of quenched steel

Table 6 Coefficients of Diffusion for Impurities in a-Fe and Steels

Solute System Temperature, K D, m2Sec�1 D0 (m2Sec�1)

Q,

kcal=mol Reference

C 0.3C–10Ni

Martensite

723–873 5.26 � 10�5 15.2 [50]

C a-Fe 623 5.3 � 10�14 [51]

P a-Fe 723 2.8 � 10�19 [52]

P a-Fe 748 7.7 � 10�19 [52]

P a-Fe 773 2.0 � 10�18 [52]

P a-Fe 798 4.8 � 10�18 [52]

P a-Fe 9.55 � 10�6 50.6 [53]

P Fe–2.1Mn 1.43 � 10�4 54.2 [53]

P Fe–Ni–P 0.51 � 10�4 55 [54]

P a-Fe 0.108 exp(�288=RT) [55]

P 0.1P–0.15Cr a 0.336 exp(�296=RT) [55]

P 0.1P–0.13Si a 18 exp(�329=RT) [55]

P 0.1P–0.17Mn a 0.235 exp(�292=RT) [55]

P 0.1P–0.14Mo a 3.23 10�6 exp(�434=RT) [55]

P 0.1P–0.14Ni a 43.9 exp(�336=RT) [55]

S Fe–3Si 973 1.7 � 10�2 61.2 [56]

Sn a-Fe 973–1303 5.4 55.5 [57]

Cr Fe–Cr 1048 2.33 � 10�4 57.1 [58]

Co Fe–6.8Co 903–1073 4.69 � 10�5 44.7 [59]

C Fe–0.79Si 803 3.6 � 10�12 [60]

C Fe–0.79Si 873 1.4 � 10�11 [60]

C Fe–0.79Si 973 1.9 � 10�10 [60]

C Fe–0.6Ni 803 4 � 10�4 [60]

C Fe–0.6Ni 873 16 � 10�4 [60]

C Fe–0.6Ni 973 6.9 � 10�4 [60]

C Fe–0.56Mo 873 7.2 � 10�4 [60]

C Fe–0.56Mo 923 21 � 10�4 [60]

C Fe–0.56Mo 973 55 � 10�4 [60]


tempering. Deeper cleaning of steel by S activates GB adsorption of P, dropsdown time of segregation formation, and increases the maximum tem-perature of segregation stability (see Figs. 39 and 40). Such calculationsare very useful for the design of optimal alloying and purification degree onharmful impurities, since they permit the determination of the influence ofalloying on ultimate concentration of harmful impurities.

Figure 36 The isodose C-curves of multicomponent interface segregation in 0.2C–Cr–Mn–Ni–Si steel (see Table 3) under its tempering. Computer simulation.

Figure 37 The isodose C-curves of multicomponent interface segregation in 0.3C–Cr–Mn–V steel (see Table 3) under its tempering. Computer simulation.


B. Interface Adsorption During Quenching ofEngineering Steels

Mathematical models of GBS [61] and phase transformations permit theanalysis of heat treatment with respect to the accompanying phenomenain a greater detail than that of a simple summary of the experimentalknowledge.

Figure 38 The isodose C-curves of multicomponent interface segregation in 0.3C–Cr–Mn–Si–Ti steel (see Table 3) under its tempering. Computer simulation.

Figure 39 Dependence of Tmax of P GBS as a function of sulfur containing in0.3C–Cr–Mn–Si–Ti steel under its tempering. Computer simulation.


The results of mathematical modeling provide backgrounds for rea-sonable planning of full-scale experiments when seeking for the opti-mum technological procedures and steel composition and they enable theextrapolation of the consequences of variations in the technological condi-tions even outside the boundary of the empirical experience we have available.

Interaction of GB segregation enrichment and phase transformationsduring heat treatment of steels in the austenitic region is hard to imagine.Nb and V carbonitride precipitation in microalloyed austenite, precipita-tion of free ferrite, change chemical composition of austenite, and influ-ence on GBS kinetics to a large extent. The experiments show that non-equilibrium grain boundary phenomena occur for a rather short time upto 100 sec. The minimum time of 5% volume fraction of Nb and V carbo-nitride precipitation is about 1000 sec [62,63]. Precipitation of free ferriteneeds from several seconds to several minutes depending on steel chemicalcomposition. Therefore, the non-equilibrium GBS in steels with a wideregion of undercooled austenite stability independently from phase trans-formations. This computer model has some limitations but redistributionof harmful impurities between grain bulk and boundaries permits the ana-lysis of steel quenching.

The modeling of non-equilibrium GB phenomena allows during inves-tigation of such short-time changes of chemical composition that could notbe measured experimentally and that has an extreme importance for modernheat-treatment processes with high heating and cooling velocities incontrolled media.

Figure 40 Dependence of time of 6 at.% GBS of phosphorus and sulfur as afunction of sulfur concentration in 0.3C–Cr–Mn–Si–Ti steel during its tempering at700K. Computer simulation.


1. Phase Transformations of Undercooling Austenite

At present, many computer models of evolution of structure and phase com-position of steels during quenching have been developed. Most of them arebased on physical models of phase transformations [64–66]. But physicalmodels cannot describe adequately all kinetic features of undercooled auste-nite transformations. The computer models based on regression analysis ofexperimental data can best predict steel phase composition changes duringsteel cooling.

It was introduced directly by Davenport and Bain [67] and the time–temperature-transformation (TTT) diagram was the predominant tool todescribe the isothermal decomposition kinetics of supercooled austenite.In most TTT diagrams, general S- or C-curves are used to represent thekinetics of a number of isothermal transformation products: ferrite, pearlite,upper bainite, lower bainite, and martensite. Conversely, many experimentalresults demonstrate that each type of transformation product has a separateC-curve.

To build a mathematical model, all TTT diagrams published in Refs.[68–71] were analyzed. The rationalization of the kinetics of isothermaldecomposition of austenite permitted the establishment of a metastableproduct (phase) diagram of a number of steels of different compositionswith 6% of total content of all alloying elements.

Figure 41 The presentation of C-curve on simulating TTT diagrams. (Scheme.)Parameters U and S correspond to Table 7.


Most isothermal transformations take place by nucleation at the aus-tenite grain boundaries, so the original austenite grain size will affect the iso-thermal decomposition kinetics of austenite.

From the total number of factors characterizing austenite matrix, thepresent day experimental knowledge allows only an approximate examina-tion of the statistically recrystallized proportion and estimation of the sizeof deformed austenite grains.

The grain growth kinetics satisfy the law [73]

dðtÞ ¼ d0 þ kt exp � Q

RT

� �ð40Þ

where d(t) is averaged grain size at moment t; d0 is initial grain size; Q is acti-vation energy; k is a constant.

The algorithm of calculating the size of austenite grains is described inRef. [73].

The procedure for calculation of the structural proportions of ani-sothermal decomposition of austenite at engineering steel cooling is givenin Tables 7 and 8 and shown in Fig. 41.

The cooling curve is approximated partially by a constant functionand at the individual time intervals Dt and the rate of decomposition is cal-culated as isothermal transformation corresponding to the mean tempera-ture of that interval. The required kinetic data are available from the TTTdiagrams [68–71] that can be digitized (see Table 8) by procedures shownin Fig. 41, using equation

S� S0

SN � S0¼ e1=2

U�U0

UN �U0

� �1=2exp � 1

2

U�U0

UN �U0

� �ð41Þ

where S¼ Int-time interval, s; U¼ 1000=(Tþ 273).Since it is necessary to distinguish between the parts of the C-curves

representing the formation of ferrite, pearlite, and bainite, only thosediagrams having readily distinguishable component curves were used inthe analysis.

The calculation method includes the effect of the size of austenitegrains on the kinetics of phase transformations. The main precondition isknowledge of this effect on the course of C-curves showing the start andend of transformations in the graph of isothermal decomposition of auste-nite for the relevant steel.


Table 7 The Algorithm of Calculation of the Structural Proportions

1. Temperature of start of transformations yAc3, yAc1, yBa, yMs2. t 0; y(0) y0; V1(0) 1; Vi(0) 0, i¼ 2, 3, 4, 5; i¼ 1-austenite,

2-ferrite, 3-pearlite, 4-bainite, 5-martensite2.1 Mean temperature at the interval of ht, tþDti

y¼ (y(t)þ y(tþDt))=2;if y <¼ yMs pass to 3;if y <¼min (yjs) then n j;

2.2 for i¼ 2, . . . , n carried out as follows:– calculation of the transformable proportion of austenite

Vmi(t) for i¼ 2:

Vm2(t)¼0; for y¼> yA3,Vm2ðtÞ ¼ V0

m2yA3�yyA3�yA1

; for yA1 < y < yA3;

Vm2ðtÞ ¼ V0m2; for y <¼ yA1;

V0m2 ¼ Sð%CÞ�Xð%CÞ

Sð%CÞ�Pð%CÞ ; for i > 2 and Vmi ¼ 1

– calculation of the start and end of transformation tsi, tfi and exponentki for yki¼ 6.127=ln (tsi=tfi); for ferrite k2¼ 1;

– the fictive volume fraction of the transformed proportionXi¼Vi(t)=[Vi(t)þVi(t) Vmi(t)];

– the fictive time of isothermal transformation required for reachingthe proportion Xi

t0i ¼ � tkisilnð1�XiÞbS

�1=ki;

– the fictive volume proportion of the structural component at timetþDt

Xiðti þ DtÞ ¼ 1 � exp �b2 tiþDttsi

�ki� �;

– the volume proportion of the structural component at time tþDtViðtþ DtÞ ¼ Xiðt0i þ DtÞ½ViðtÞ þ ViðtÞVmiðtÞ�;

2.3 the new value of residual content of austenite

V1ðtþ DtÞ ¼ 1�Pni¼2 Viðtþ DtÞ; if V1ðtþ DtÞ <¼ 0 the transfor-

mation is finished;2.4 t tþDt; pass over 2.1

3. The martensite transformation for yMf<¼ y<¼ yMs

V5(y)¼ (1�V2�V3�V4)[1�exp(�0.011(yMs�y))]4. The residual content of austenite at y<¼ yMf: Vi ¼ 1�P5

i¼2 Vi;I¼ 1, 2, 3, 4, 5—austenite, ferrite, pearlite, bainite, martensite; S(%C),P(%C), X(%C)�containing carbon in points S and P of Fe–C phase diagramand in the steel, consequently.


Table 8 The Constants Blk for Calculations of the YL Parameters of C-Curves

Transformation YL

k

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Blk Dlk

Const C Mn Si Ni Cr Mo GS Const C Mn Si Ni Cr Mo GS

Ferrite-start T8C TOFS 727 �13 9 �17 22 3.5 0 229 �30 66 �22 �70 5 0TNFS 572 �33 44 �9 25 50 �1.01 48 57 �25 �15 �15 �25 3.90

Time IntOFS 10.13 2.5 2.84 0.46 4.8 5.9 �0.095 �1.92 2.80 �3.55 1.50 �3.00 �3.50 0.0096IntNFS �1.04 0.6 0.19 0.07 4.8 5.9 �0.038 �2.58 0.20 �1.10 �0.04 �2.55 �3.50 0.021

Pearlite-start T8C TOPS 727 �13 9 �17 22 3.5 0 0 15 �5 �5 22 0TNPS 572 �33 44 �9 25 50 1.01 48 32 �20 �5 �27 �23 3.20

Time IntOPS 10.13 2.5 2.84 0.46 4.8 5.9 0.095 �1.92 10.12 �3.55 3.30 �3.00 4.20 0.096IntNPS �1.04 0.6 0.19 0.07 4.8 5.9 0.038 �2.58 1.31 �0.20 0.20 �2.55 2.50 0.041

Pearlite-finish T8C TOPF 727 �13 9 �17 22 3.5 0 0 0 15 �5 �5 22 0TNPF 577 2 35 �7 53 58 �0.52 6 4 5 �5 �36 25 1.31

Time IntOPF 10.55 6.86 3 1.81 4.8 12.3 0.095 �0.03 14.08 �0.76 1.99 �3.00 �4.35 0.0096IntNPF 0.15 2.0 0.19 0.07 2.65 10.6 �0.038 �0.03 1.31 �1.56 0.65 �0.98 �4.35 0.021

Bainite-start T8C TOBS 570 �12 16 �12 20 �22 0 107 �12 –5 –5 –5 –40 0TNBS 485 �12 2 �7 40 �32 0.50 107 �30 �35 �5 �36 �2 0.37

Time IntOBS 3.52 2.14 1.07 0.05 2.3 3.1 0.070 �5.44 �1.46 �1.64 0.40 �3.49 �3.00 0.0626IntNBS �0.96 1.30 0.30 0.02 2.2 2.2 0.024 �1.40 �1.29 �1.64 �0.20 �2.04 �6.60 0.0248

Bainite-finish T8C TOBF 570 �12 16 �7 22 �22 0 76 �12 �5 �7 �5 �51 0TNBF 488 �48 25 �7 47 �61 �0.52 76 �30 �28 �7 �40 96 0.37

Time IntOBF 6.90 5.23 2.8 0.7 6.2 9.5 0.070 �7.73 �0.75 �5.44 0.40 �0.90 �6.88 0.0813IntNBF �0.15 3.00 2.8 0.7 3 5.9 �0.024 1.98 �1.17 �5.84 0.02 �2.47 �4.00 0.0340

MS TMS 539 �423 �30.4 0 �17.7 �12.1 �7.5 0 0 0 0 0 0 0 0

Ck

C1 C2 C3 C4 C5 C6 C7 C8

1 %C %Mn %Si %Ni %Cr %Mo 2

YL ¼P8

k¼1 Blk þDlk þ 8ð0:8� C2Þ�Ck½ ; GS—grain size (ASTM).


The program involves the calculation of temperatures of transforma-tions of bainite and twinned, athermal and lamellar martensite [74]

BS ¼ 720� 585:63ðCÞ þ 126:6ðCÞ2 � 66:34ðNiÞ þ 6:06ðNiÞ2

� 31:66ðCrÞ þ 2:17ðCrÞ3 � 91:68ðMnÞ þ 7:82ðMnÞ2

� 42:37ðMoÞ þ 9:16ðCoÞ � 0:125ðCoÞ2 � 36:02ðCuÞ ð42Þ

MTMS ¼420�208:33ðCÞ�72:65ðNÞ�43:36ðNÞ2�16:08ðNiÞþ0:78ðNiÞ2�0:025ðNiÞ3�2:47ðCrÞ�33:428ðMnÞþ1:296ðMnÞ2

þ30:0ðMoÞþ12:86ðCoÞ�0:2665ðCoÞ2�7:18ðCuÞ ð43Þ

MLMS ¼ 540� 356:25ðCÞ � 260:64ðN� 24:65ðNiÞ þ 1:36ðNiÞ2

� 17:82ðCrÞ þ 1:42ðCrÞ2 � 47:59ðMnÞ þ 2:25ðMnÞ2

þ 17:5ðMoÞ þ 21:87ðCoÞ � 16:52ðCuÞ ð44ÞMA

S ¼ 820� 603:76ðCÞ þ 247:13ðCÞ2 � 55:72ðNiÞ þ 3:97ðNiÞ2

� 31:1ðCrÞ þ 2:348ðCrÞ2 � 66:24ðMnÞ � 24:29ðMoÞ� 0:196ðCoÞ þ 0:165ðCoÞ2 � 31:88ðCuÞ ð45Þ

The size of ferritic grain is expressed as follows [75]

da ¼ 11:7þ 0:14dg þ 37:7V�0:5C ð46Þ

where dg is the size of austenitic grain, (mm); VC is the cooling speed,(8Cmin�1).

The interlamellar distance of pearlite can be estimated as follows [75]:

S ¼Xi

18:0DVPðyiÞ=ð996� yiÞ" #

=VP ð47Þ

where DVP(yi) is the volume proportion of pearlite transformed at yi tem-perature.

The thickness of the ferritic and carbide lamellae of pearlite is approxi-mately lf¼ 0.885S; lc¼ 0.115S

The size of martensitic and bainitic particles is identical with theoriginal size of the austenite grains. The course of the anisothermaldecomposition of austenite in several steels has been calculated by the


just-described method and by applying the digitized TTT diagrams ofTable 7.

2. Determination of the Kinetics of Carbonitride

Precipitation in Austenite

Microalloying of steels with Ti, V, Nb, and Zr affects decomposition ofsupercooled austenite, its recrystallization and grain boundary segregationsof harmful impurities. These changes of material properties are associatedwith carbonitride precipitation and changes of austenite chemical composi-tion. Based on these reasons, the modeling of the kinetics of carbonitrideprecipitation is important.

The nucleation time t of carbonitrides per unit volume N at any tem-perature T, can be expressed as [76]

t ¼ C expQ

RT

� �exp

B

T3ðLnKSÞ2 !

t ¼ C expQ

RT

� �exp

B

T3ðLnKSÞ2 !

ð48Þ

where C¼ 6 � 1013 for homogeneous nucleation; activation energy of Nbdiffusion Q¼ 270 kJ=mol;

B ¼ 16pg3V2mN0=3R

3 ð49ÞVm¼ 1.28 � 10�5m3=mol; g¼ 0.5 Jm�2; N0 are numbers of nucleus byradius R per molar volume Vm; KS is supersaturation [77]

LgðKSÞ ¼ �A

Tþ B ð50Þ

where thermodynamics parameters A and B for various carbides andnitrides are calculated in Ref. [78] and presented in Table 9. Thus, the cal-culation of carbonitride nucleation time necessary to reproduce the C-curves

Table 9 Thermodynamics Parameters in Eq. 50 (From Ref. 77)

Chemical compound

Parameter AlN VC VN TiC TiN NbC NbN ZrC ZrN

A 7,130 9,500 7,985 8,872 15,573 7,714 10,440 8,464 13,968

B 1.463 6.72 3.09 4.04 3.82 3.27 3.87 4.96 3.08


corresponds to start or finish of this transformation in austenite under heattreatment of steel.

3. Calculation of Thermokinetic Diagrams of Impurity

Segregation During Quenching of Steel

Computation of grain boundary multicomponent adsorption kinetics couldbe simplified for steels with high undercooled austenite stability. The GBSdevelops in this case in austenite in short time and has no dependence onphase and structure transformations at steel quenching. Enrichment of grainboundaries by various impurities as well as their desorption is treated as aresult of multicomponent diffusion of impurities from near-boundaryvolume to the boundary. Impurity binding energy with GB includesmutual influence of elements in grain bulk and on the boundary in accor-dance with Guttmann’s theory [Eqs. (18) and (19)]. Auger electron spectro-scopy is the technique for experimental investigation of GBS kinetics. Theseexperiments are basic for analysis of correlation of impurity segregationenergy with the content of other elements in the bulk and on boundaries(see Section 2.5, Eqs. (23)–(29).

Adsorption and desorption of impurities on GB (qi) at steel quenchingis modeled well using the equation

qi ¼ qið0Þ þ 2q0iffiffiffiffiffiffipdp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ t

0

Diðt0Þ dt0�s

1ffiffiffiffiffiffipdp

Z t

0

Ciaðt0ÞDiðt0ÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR tt0 Diðt00Þ dt00

q dt0 ð51Þ

where d is the grain boundary thickness; Di(t0) is the diffusion coefficient of

impurity which depends on the temperature and phase composition (auste-nite, martensite, and ferrite); in the case of adsorption

Cia ¼

CiGB

1�Pj CjGB

expð�Gi=kTÞ ð52Þ

where CGBi is the element i concentration on grain boundary; Ca

i is the con-centration of ith element in the adjacent bulk layer; Gi is segregation energy.Desorption is determined by GB concentrations of impurities, and in thiscase, the parameter Ca

i in Eq. (51) is equivalent to GB concentration XbI in

Eqs. (12) and (13).The change of temperature at cooling or isothermal exposition is

described by equation

TðtÞ ¼ ½Tð0Þ � Tð/Þ� expð�rtÞ þ Tð/Þ ð53Þ


Table 10 presents cooling rates r for heat-treatment processes. The blockdiagram of multicomponent intercrystalline adsorption model is shown inFig. 42. Adsorption of P, C, and S is determined by parameters K1, K2,K3, and their desorption by parameters K2, K4, K6. The parameters Ci

are equivalent to GB concentration of element i. This model allows the com-putation of the condition when there is change of GB composition in steelsand alloys at preselected arbitrary mode of cooling including isothermalexposition.

Given below are the examples of investigation of phosphorus and sul-fur grain boundary adsorption in Cr–Ni–Mo steel (see Table 11).

The components of steel mutually influence their diffusion mobilityand GBS activation energy. Based on this reason, one should take intoaccount the stochastic fluctuations of diffusion flows of various impuritieson GBS kinetics. For this purpose, the random fluctuation of diffusioncoefficients up to 30% of its mean value was used in the model. Figures 43and 44 present the GBS kinetics calculation results at cooling of various pur-ity steels cooling that were carried out using the stochastic model. As onecan see, the self-regulation of adsorption is observed which is developingdespite significant short-time oscillations of impurity concentration on grainboundaries. The significant non-equilibrium enrichment of GB by impuri-ties is observed at initial stage of the heat treatment. This effect is deter-mined by cooling velocity as well as impurities content. Increasing coolingvelocity from 0.001 to 1000K s�1 decreases the non-equilibrium GBS of Pand S. Formation of non-equilibrium rich GBS of harmful impurities atsmall cooling times could be established only by using computer modelingmethods. The experimental verification of such phenomena needs specialtechniques which allow to open grain boundaries: hydrogenation ofquenched samples or delayed fracture tests. Since these techniques areconducted in air and could not be applied in the vacuum chamber ofelectron spectrometer; for most of engineering steels, the regularities ofnon-equilibrium GBS formation at quenching could only be estimated by acomputer experiment.

Table 10 Cooling Rates for Some Metallurgical Technologies

Name of the treatment Cooling rates r (K sec�1)

Quenching 100–10

Controlled cooling 10Air cooling of hot-rolled metal 10–0.1Cooling with furnace 0.01Controlled cooling of large-size forging 0.001


Table 11 Chemical Composition of Cr–Ni–Mo Steels

Smeltingnumber

Chemical composition, mass%Time of austenitestability at 600 hrC S P Ni Cr Mo

82 0.38 0.027 0.054 3.95 3.0 0.51 2.083 0.38 0.01 0.006 4.02 3.0 0.50 2.0

Figure 42 Calculation scheme of three-component GBS (phosphorus, sulfur, and

carbon).


Figure 43 The change of GBS during quenching of Cr–Ni–Mo steel containing

0.027S and 0.054P (mass%). Computer simulation of fast (a) and slow (b) cooling.


Figure 44 The change of GBS during quenching of Cr–Ni–Mo steel containing0.01S and 0.006P (mass%). Computer simulation of fast (a) and slow (b) cooling.


Figure 45 Chemical composition of GB in Cr–Ni–Mo steel containing 0.027S and

0.054P (mass%) after austenitization at 1373K (30min), interim cooling up to 873Kand quenching in water (a) and in furnace (b). Auger electron spectroscopy ofintergranular fracture.


Figure 46 Chemical composition of GB in Cr–Ni–Mo steel containing 0.01S and0.006P (mass%) after austenitization at 1373K (30min), interim cooling up to 873Kand quenching in water (a) and in furnace (b). Auger electron spectroscopy of

intergranular fracture.


The validation of calculation reliability was done for steel composition82 and 83 (see Table 10) by Auger spectroscopy. The samples after austeni-tization at 1373K (30min) were in the interim cooled to 873K with furthercooling in water or with furnace cooling. The undercooled austenite in thissteel has high stability and does not transform in ferrite region for 2 hr.After cooling the samples had martensite–baintite structure. To investigatethe chemical composition of grain boundaries by Auger spectroscopy, spe-cial samples were crushed in the electron spectrometer ESCALAB MK2 atvacuum at about 10�8 Pa at temperature 83K. The fields with intercrystal-line fracture type were investigated on the fracture surface. The variationof phosphorus and sulfur content in GBS in Cr–Ni–Mo steel of several meltsafter heat treatment is shown in Figs. 45 and 46. At accelerated cooling theGB are significantly enriched by carbon. The P concentration in GBincreases only at slow cooling of samples, and P segregation is strongly sup-pressed in pure steel. A good correspondence of calculated and experimentalresults is observed for all cases to be analyzed.

The results of numerical modeling give information about the equili-brium and non-equilibrium character of a GB adsorption processes, whichare frequently unavailable from experiments. Moreover, these simulationmethods explain the phenomenon of reverse temper embrittlement as theresult of non-equilibrium concurrent GBS of carbon and phosphorus. Theseresults explain many questions in the multicomponent GB adsorptionkinetics in engineering steels that were dynamically developed in the last10 years. Further investigations in this direction are required especiallyfor competitive internal adsorption in engineering steels treated by usingnewest schemes of heat treatment.

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3Designing for Control ofResidual Stress and Distortion

Dong-Ying JuSaitama Institute of Technology, Okabe, Saitama, Japan

I. INTRODUCTION

Residual stresses in materials are often produced from metallurgical pro-cesses, such as casting, forging, welding and quenching processes, and soon. Usually, the production of residual stresses during metallurgical processdepends on changes of thermal sources and volume due to microstructure.Generally, residual stresses of two types can be considered, i.e., macro-resi-dual stresses and micro-residual stresses [1,2]. The macro-type depends onthe plastic deformation of solid materials due to rapid non-uniform cooling.And, the strain and deformation due to phase transformation and change ofmicrostructure are the sources of the micro-type residual stresses. We alsoknow that the distortion due to thermal and elastic–plastic deformationand strain as well as change of phase transformation and texture in manu-factured materials are important no matter the type of residual stress. There-fore, one of the many important problems is how to control and utilizeresidual stresses and distortion due to variations of macro- and microstruc-ture in materials for increasing and ensuring strength and quality of pro-ducts after metallurgical process.

To improve the mechanical properties of materials, it is importantto know how to raise compressive residual stress and reduce tensionresidual stress in materials as it is known that the compressive residualstress can increase fatigue strength of materials. Therefore, mea-surement method and analysis of residual stresses are always developed


as a conspicuous technology. Many measurement methods of residualstress such as x-ray diffraction, neutron diffraction, rapid drillingmethod, and so on have been developed in industrial technology [3–8].However, each measurement technique always has some limitationsand problems when used for measuring, for example, the x-ray methodonly measures residual stresses on surface of materials; on the otherhand, in the method of neutron diffraction, a major problem is howto measure the standard lattice spacing d0 of strain-free materials, whichconsider distortion and strain due to microstructure or texture. In orderto address thermal–mechanical behavior in the entire material process, aso-called metallo-thermo-mechanical theory is proposed by Inoue et al.[9–18]. Based on this theory, computer simulation of metallurgical pro-cesses can be used as a new technology. This is a critical component of theworldwide effort to develop virtual metallurgical capabilities in order toacquire a better understanding of processing operations and optimizeprocesses with a view to improving quality and reducing productioncosts. By using this technique, residual stress and distortion in metallur-gical process also can be predicted; hence some useful theory and simu-lation methods can be proposed and introduced in the simulations ofvarious metallurgical processes [19–23]. From these research results, wecan obtain useful knowledge to seek more perfect designing formetallurgical process.

In this chapter, the metallo-thermo-mechanical theory, numericalmodeling and simulation technology considered with coupling of tempera-ture and phase transformation or solidification as well as inelastic beha-vior involved with elastic–plastic, viscoplastic and creep deformation willbe introduced. The theory and simulation technology also are used inmetallurgical process, such as heat treatment, continuous casting, and ther-mal spray coating as well as forging for the control of residual stress anddistortion in these processing. This chapter not only presents various simu-lation methods and results of the residual stress, distortion and thermal–mechanical behavior as well as microstructure according to the peculiarityof each metallurgical process, but also the thermo-mechanical modelingthat was verified by comparison with the experimental data, such as themeasured temperature, residual stresses and distortion in various metallur-gical processes.

In the final part of the chapter, some conclusions and remarkswhich are used in the designing of metallurgical process for controlof residual stress and distortion are presented. From these con-clusions, we summarize many, yet expected problems and subjects infuture metallurgical process and material industry for reference ofresearch.


II. METALLO-THERMO-MECHANICAL MODELING FORCONTROL OF RESIDUAL STRESS AND DISTORTION

There are many physical and chemical phenomena that occur in metallurgi-cal processes. These phenomena do not always act alone on materials; themetallic structures incorporating phase transformation, temperature andstress=strain are also strongly coupled in the process. The metallo-thermo-mechanics theory are considered with the coupling effect on temperature,stress=strain and phase transformation fields as schematically illustratedin Fig. 1. The coupling effects are indicated as follows:

1. Thermal Stress. The thermal expansion caused by such atemperature gradient is restricted by the shape of a solid body,thus generating thermal stress.

2. Heat Generation due to Deformation. When stress=strain thatappears in the case of large inelastic deformation is applied tomaterials, the energy is partially discharged as heat.

3. Temperature-dependent Phase Transformation. Temperature is themajor factor, which determines phase transformation start time.However, in the case of diffusion-type transformations of ferrite,pearlite, and bauxite, the temperature history also affects the phasetransformation.

4. Latent Heat due to Phase Transformation. Latent heat generated inthe course of phase transformation affects the temperature field.

Figure 1 Metallo-thermo-mechanical coupling during processes involving phasetransformation.


5. Stress (or Strain)-Induced Transformation. The phase transforma-tion behavior is also affected by stress=strain existing in the solid.For example, pearlite transformation time is shortened undertensile stress, and vice versa. Martensite transformation is gen-erated even though a material is processed at a temperature higherthan the martensite transformation temperature under the appliedstress or strain.

6. Transformation Stress and Transformation Plasticity. Volumedilatation in the work is caused by the phase transformations.When this volumetric dilatation is inhomogeneous depending onthe complicated shape of the body, stress and strain are induced, itis defined as transformation stress and strain. The level of suchinduced stress is comparable to the thermal stress. The effect oftransformation plasticity is sometimes important.

The details of introducing the governing equations in the frame-work of thermo-mechanical behavior for describing temperature andstress=strain fields incorporating metallic structures in the heat treatmentprocess are already reported elsewhere, and are applied in welding, cast-ing and so on [24–31]. Based on the theory, a simulation program called‘‘HEARTS’’ [32,33] for heat treatment process was developed to predictthe temperature field, phase transformation, residual stress, and distor-tion during heat treatment process. The fundamental equations whichcan be applied in the metallurgical process are summarized in thefollowing.

A. Mixture Rule

When a material point undergoing a metallurgical treatment process isassumed to be composed of multi-phase structure, an assumption is madethat a material parameter w is described by the mixture law [34]

w ¼XNI¼1

wIxI andPNI¼1

xI ¼ 1 ð1Þ

where xI denotes the volume fraction for the Ith phase.

B. Heat Conduction Equation

The local energy balance or the first law of thermodynamics is usually givenin terms of the internal energy e¼ gþTZ þ (sijeij)=r, as


r _ee ¼ sij _eeij � @hi@xi

ð2Þ

with stress power sij _eeij . Here, r and hi are the density and heat flux, respec-tively. Z is the entropy of the thermodynamic state. Introducing the expres-sion for specific heat c¼T(Z=T), Eq. (2). is reduced to a heat conductionequation

rc _TT� @

@xik@T

@xi

� �� sij _ee

pij þ

XrIlI _xxI ¼ 0 ð3Þ

where k and lI denote the coefficients of heat conduction and the latent heatproduced by the progressive Ith constituent.

The boundary conditions of heat transfer on the inner surface areassumed to be

�k @T

@xini ¼ hðTÞðT� TwÞ ð4Þ

where h(T) is a function dependent on temperature. Tw denotes the heattransfer coefficient and the temperature of coolant on heat transfer bound-ary with unit normal ni, respectively.

C. Diffusion Equation of Carbon Content

Carbon content during carburizing process is arrived at by the diffusionequation

_CC ¼ @

@xi�D @C

@xi

� �ð5Þ

where C is the content in the position xi-direction, D is the diffusion con-stant determined by the boundary condition being specified by the reactionacross the surface layer

D@C

@xini ¼ hcðC� CsÞ ð6Þ

where hc and Cs are the surface reaction rate coefficient and the knowncontent of the external environment, respectively.

D. Constitutive Equation

In order to simulate distortion due to variations of temperature, phasetransformation, and inelastic deformation in metallurgical processes, many


constitutive models which are capable of representing the relation of stressand strain including macro- and microstructures are proposed and estab-lished [35–52]. A few constitutive equations to be used in simulation ofmetallurgical processes are introduced in this chapter.

Total strain rate _eeij is assumed to be divided into elastic, plastic, ther-mal strain rates and those by structural dilatation due to phase transforma-tion and creep such that

_eeij ¼ _eeeij þ _eepij þ _eeTij þ _eemij þ _eecij þ _eetpij ð7Þ

Elastic and thermal strains are normally expressed as

eeij ¼1þ nE

sij � nEskkdij ð8Þ

and

eTij ¼ aðT� T0Þdij ð9Þ

with Young’s modulus E, Poisson’s ratio n and thermal expansion coeffi-cient a, respectively. T0 is the initial temperature of material.

Strain rates due to structural dilatation and transformation plasticitydepending on the Ith constituent are

emij ¼XNI¼1

bIxIdij ð10Þ

and

_eetpij ¼XNI¼1

BIhðxIÞ _xxIsij and hðxIÞ ¼ 2ð1� xIÞ ð11Þ

where bI stands for the dilatation due to structural change. BI denotes thematerial parameters depending on phase transformation.

The plastic strain rate is reduced to the form when employing tempera-ture-dependent material parameters

_eepij ¼ l@F

@sij¼ GG

@F

@skl_sskl þ @F

@T_TTþ

XNI¼1

@F

@xI_xxI

!@F

@sijð12Þ

with a temperature dependent yield function

F ¼ Fðsij; ep; k;T; xIÞ ð13Þ


with hardening parameter k, where

1

GG¼ � @F

@epmnþ @F

@ksmn

� �@F

@smnð14Þ

The creep strain rate is assumed to follow a simple Norton creep law as

_eecij ¼3

2A1=m

c �ssðn�mÞ=m�eecðm�1Þ=msij ð15Þ

Here, sij, s� and e� are deviatoric stress, equivalent stress, and equivalentstrain, respectively. Either isotropic or kinetic hardening type of yieldfunction F is available to be used in the section. Symbol Ac, n and m arematerial parameters based on Norton creep law. Either isotropic or kinetichardening type of yield function F is to be used in this section.

In casting and welding processes, we often should consider inelasticdeformation incorporating the solidifying behavior of materials. In thesolidifying process of materials, there has been substantial development ofthe so-called ‘‘unified’’ theories of viscoplasticity in which all aspects ofinelastic behavior are intended to be represented by the same variable.Consider three types of constitutive models which have some possibilityof describing the experimental behavior obtained above. Generally, the totalstrain rate _eeij is expressed as the sum of the elastic strain _eeeij and the visco-plastic strain rate _eevpij , i.e., _eeij ¼ _eeeij þ _eevpij , where the elastic strain rate is givenby Hook’s law. The models for inelastic model are Perzyna’s model [42,43]based on the excess stress theory, and superposition model [52] which areintroduced below. The excess stress theory assumes that the viscoplasticstrain is produced as

_eevpij ¼ L cðFÞh i @F@sij

ð16Þ

where L denotes a viscosity constant of the material. c(F) is a function ofthe static yield function ðsij ; evpij ; k;TÞ in stress space involving inelasticstrain evpij , the hardening parameter k and temperature T. As a proposalof the viscoplastic constitutive theory, Perzyna has proposed four kinds offunctions for c(F), among which a special case c(F)¼F is employed to givethe viscoplastic strain rate _eevpij ,

_eevpij ¼3l2

3

2sklskl

� �1=2

�sy* +n

sij

ð3J2Þ1=2ð17Þ

where J2 is the second invariant of deviatoric stress sij, and sy denotes theyield stress. In order to deal with a viscous fluid, we choose l¼ 1=3 m with


the viscosity m. When the value of parameter n tends to unity, eqn. (17)which is reduced to a uniaxial stress–strain relation under monotonic ten-sion is then expressed as

_eevp ¼ lð sj j � syÞ ssj j ð18Þfrom which the parameter l, or viscosity m is to be identified.

When the Norton’s equation (e_c¼As�n ) is adopted to describe creepbehavior of material associated eqn. (18), the model can deal with morecomplex inelastic behavior as a superposition model.

E. Kinetics of Phase Transformation inMetallurgical Processes

In the case of quenching, two kinds of phase transformation are anticipated:one is governed by the diffusionless or martensite mechanism. From a ther-modynamic consideration, the formula for this type of reaction from auste-nite is assumed to be governed by modified Magee’s [53] rule as

xM ¼ 1� exp½fðT� TMÞ � jðsijÞ� ð19Þwith

jðsijÞ ¼ A1sm þ A2J1=22 ð20Þ

where TM is the martensite-start temperature under vanishing stress. Theparameters A1 and A2 can be identified if we have the data of the martensitictransformation depending on the applied stress.

The other type of phase transformation is controlled by diffusionmechanism, and the volume fraction of developing phase such as pearlitemay be expressed by modifying the Johnson–Mehl [54] relation as

xp ¼ 1� expð�VeÞ ð21Þwhere Ve is defined by

Ve ¼Z t

0

�ffðT; sijÞðt� tÞ3dt ð22Þ

Here, we separate the function f�(T, sij) into two independent functionsof temperature and stress as

�ffðT; sijÞ ¼ f1ðTÞf2ðsijÞ ð23Þ

Since the time–temperature-transformation TTT diagram under theapplied mean stress sm in logarithmic scale deviates from the one without


stress which is represented by the function f(T), the kinetic equation ofdiffusion type is often applied to the variations of pearlite or ferrite structurein quenching processes. An identification of the function f(T) can be madepossible by the use of some experimental data of the structure change.

In casting processes, it is important to determine a moving boundarydue to solidification as two phase structure with solidus and fluidus [55,56].However, when we consider solidification of an alloy material, the mixturelaw also can be applied in the casting simulation.

To identify the volume fraction of the solid phase xs during solidifica-tion appearing in eqns. (1) and (3), a phase diagram for the alloying systemis employed, and the volume fraction of the solid phase xs is assumed to bedetermined by the Scheil equation [57,58]:

xs ¼ 1� f�1ð1�k0Þ and xl ¼ 1� xs ð24Þ

Here, f is a dimensional function that is dependent on temperature:

f ¼ ðT� TsÞ=ms

ðT� TsÞ=ms þ ðTl � TÞ=mlð25Þ

where ml and ms are the gradients of the liquidus and solidus temperature Tl

and Ts with respect to the alloying element on the phase diagram, respec-tively, and k0 is a distribution coefficient representing the segregation effect.When the segregation effect is assumed to be neglected as in the presentarticle, Eq. (24) can be reduced to the simple form

xs ¼ðTl � TÞ=ml

ðT� TsÞ=ms þ ðTl � TÞ=mlð26Þ

which is called the lever rule.

F. Algorithm of Finite Element Analysis

The formulated finite element equation system, considering the couplingbetween increment of nodal displacement fDug and temperature fTg as wellas volume fraction of structure xI, can be expressed as

½P�f _TTg þ ½H�fTg ¼ fQðxIÞ; sijÞg ð27Þand

½KðuiÞ�fDuig ¼ fDFðT; xIÞg ð28ÞHere, [P], [H] and [K] represent the matrices of heat capacity, heat

conduction and stiffness, respectively, and the vectors fQg and fDFg areheat flux, and increments of thermal load. These equations are strongly


nonlinear, which are derived by the use of the expression of stress incrementvector as

fdsg ¼ ½D� fdeg � 3d�eec

2�ssfsg �

XNI¼1f�aag dT�

XNI¼1f�bbg dxI

!

þ 1

S0

@�ss2

@TdTþ

XNI¼1

@�ss2

@xIdxI

!fsg ð29Þ

where e�c and fsg are equivalent creep strain and deviatoric stress vector.S0 and [D] denote a parameter depending on material hardening and anelastic–plastic matrix based on Mises’ type yield function. Here, the func-tions depending on temperature and phase transformation a� and b� can bewritten as

f�aag ¼ @

@Tð½De��1fsgÞ þ

XNI¼1

aIxIf1g þ@aI@T

xI dTf1g� �

ð30Þ

and

f�bbg ¼ @

@xIð½De��1fsgÞ þ bIf1g ð31Þ

aI and bI are the coefficients of thermal expansion and dilatation due to Ithphase transformation, respectively. [De] denotes the elastic matrix ofmaterials.

In order to treat the unsteady coupled nonlinear FEM equationsdependent on time, a time integration scheme ‘‘step-by-step time integra-tion’’ method and a modified ‘‘Newton–Raphson’’ method are introducedin numerical calculation, while an incremental method is used for deforma-tion and stress analysis. Because the heat transfer coefficient is dependent onthe variation of the temperature on the boundary of heat transfer, we alsouse the time step of non-uniform cooling to calculate temperature, phasestransformation and deformation fields [59,60].

In the simulation of continuous casting, the finite element methodbased on the Eulerian coordinate is applied to analyze the velocity, displa-cement rate vector fug, strain rate _eeij and stress rate _ssij. Displacement ui,strain eij, and stress sij can be evaluated by integrating the rate _eeij, and _ssijalong flow line of velocities which have been verified by Ju and Inoue [61].

A finite volume computational method [62] to simulate large deforma-tion in forging is adopted in this chapter. The advantages of the finite ele-ment and the finite volume approaches are combined: it employs a fixedfinite volume mesh for tracking material deformation and an automaticallyrefined facet surface (material surface) to accurately trace the free surface of


the deforming material. This is particularly suitable for large three-dimen-sional material deformation such as forging since remeshing techniquesare not required. By means of this finite volume method, an approach basedon the metallo-thermo-mechanics to simulate metallic structure, tempera-ture, and stress=strain in the forging process associated with strain-inducedphase transformation has been developed. The material is considered aselastic–plastic and takes into account the phase transformation effects onthe yield stress. The temperature increase due to plastic deformation, heatconduction in the workpiece and dies, heat transfer between workpiece=die and ambient and thermal stress has been analyzed simultaneously.Strain-induced phase transformation, latent heat, transformation stress,and strain are included. This approach has been implemented in the com-mercial computer program MSC.SuperForge [62].

III. DESIGNING OF HEAT TREATMENT PROCESS FORCONTROL OF RESIDUAL STRESS AND DISTORTION

The purpose of the quenching and carburizing quenching of steel parts is toget desired metallurgical structures, hardness and strength of steel partsincreased by the cooling rate and change of microstructure in quenching.So, we need to decide the compromise between maximizing hardness andminimizing distortion must often be made. Therefore, it is very importantand difficult to select the proper quenching conditions, i.e., quenchanttemperature, the method, and the cooling intensity of the quenchant media,etc. [63–65]. In order to obtain the optimum conditions, computer simula-tion is very useful for the determination of the quenching conditions,because the application of metallo-thermo-mechanical theory is capable ofdescribing the interaction among temperature field, stress=deformation fieldand microstructure changes in quenching so that the simulation technologyhas been developed. By using this method based on continuum thermody-namics and the finite element method, several numerical simulations werecarried out for the quenching process [66–71]. The simulation of quenchingprocess is so complicated that most of the current research focused on thevalidity of numerical modeling for designing steel parts. On the other hand,to obtain improved mechanical properties and fatigue strength of machinecomponents, such as gears, and shafts carburizing–quenching process isoften used for surface hardening in the industry. However, carburizing–quenching is usually accompanied with carbon concentration and leadsto distortion, residual stresses, and hardness of steel parts. The effectof carbon content is especially significant on phase transformationbehavior.


The first part of this section will present the thermal–mechanical beha-vior included with residual stress and distortion by using quenching condi-tions on cylinder and ring of carbon steel, which are simulated by an FEMprogram ‘‘HEARTS’’ [32]. Several techniques [63,71] were used for the esti-mation of the heat transfer coefficients during water and polymer quench-ing. The effect of the heat transfer coefficients on the quench distortionand the simulation result will be discussed. The simulation accuracy willbe verified by the comparison between the experimental data of the distor-tion and the simulated results.

The second part of the section will present the simulation results ofthe residual stress field and distortion of a steel gear in carburizing–quench-ing process. The computer simulation is based on the metallo-thermo-mechanical theory involving the diffusion of carbon and considering theeffects of coupled temperature, phase transformation, and stress=strainfields on carbon component. In carburizing–quenching process, the fieldsof metallic structures and stress=strain as well as temperature affect eachother [67]. A series of theoretical models, taking into consideration theeffect of carbon diffusion and distribution is introduced. The accuracyof simulation is also verified by comparison with the experimental data.The predicted results validated the improvement of the hardness andstrength of the gear component in carburizing–quenching process. Basedon the comparison with the deformation experimental work, the thermo-mechanical predictions are found to be in good agreement with the experi-mental results.

A. Residual and Distortion in CarburizingQuenching of Cylinder

1. Experimental Method of Carburizing Quenching

Cylinder specimens of 20mm diameter and 60mm length were used to verifythe accuracy of the simulation. The carbon content of carburizing environ-ment was 1.3% and the aim is to reach 0.9% carbon content on the surfaceof the cylinder. To detect the temperature changes during the process, fourmetallic sheathed thermocouples were attached to the specimen which areshown in Fig. 2.

The entire carburizing–quenching process is shown in Fig. 3. Thespecimen was heated and carburized in a furnace and then quenched inthe oil bath. In the cooling process, the temperatures were detected by thethermocouples attached to the specimen, and the signals were fed into a per-sonal computer after the A=D conversion. The changes of the thickness andthe diameter of the specimen were measured with a digital indicator and amicrometer caliper.


2. Models of Simulation

The simulation model in carburizing–quenching process is a steel cylinder of20mm diameter, 60mm length and 0.45% carbon. It is assumed that themodel is located in a uniform coolant. Then, the finite element modelbelongs to an axisymmetrical problem. As an initial step of the heat treat-ment process, calculations for the heating and carburizing process were usedto simulate thermal stress field, thermal distortion as well as carbon distri-bution in the model. The quenching process of the model was started fromthe initial temperature 8508C and the model was cooled to 308C with oil.The heat transfer coefficients h during quenching were calculated as afunction of temperature by the methods mentioned below and were usedfor the simulation as the surface boundary condition in Eq. (4).

3. Identification of Heat Transfer Coefficients

It is important to determine the heat transfer coefficients in the quenchingprocess of metal parts for numerical simulation. However, it is rather diffi-cult to evaluate heat transfer coefficients during quenching of actual steel

Figure 2 The specimen and sheathed thermocouples.

Figure 3 The carburizing–quenching process.


parts which depend on not only the quenchant but also the shape, size,surface condition, and thermal properties of parts, etc. It is, therefore, verydifficult to evaluate the heat transfer coefficients in quenching of steel parts.Some approximate methods are estimating the coefficients from the coolingcurve data of standard probes which are used for evaluation of the coolingpower of liquid quenchants.

We have already reported the availability of the lumped-heat-capacitymethod for the estimation of the heat transfer coefficient from the coolingcurve data of the JIS silver probe (pure silver solid cylinder of 10mm dia-meter by 30mm length, Japanese industrial standard K 2242 [72]) whichhas a high thermal conductivity. A computer program ‘‘LUMPPROB’’based on the lumped-heat-capacity method was developed [73]. On the otherhand, it was confirmed that the inverse method is more suitable for estima-ting the heat transfer coefficients from quenching data of the ISO probe(Inconel 600 alloy solid cylinder of 12.5mm diameter by 60mm length,International standard ISO 9950 [74]), because of its low thermal conductiv-ity. We developed a computer program ‘‘InvProbe-2D’’ [75], which usesboth a lumped-heat-capacity method and a two-dimensional inverse methodwith the least residual method. In this section, we used these programsfor the estimation of the heat transfer coefficients during quenching.Furthermore, more precise heat transfer coefficients were estimated by atrial-and-error method, in which the calculation of cooling curves and mod-ification of the surface boundary condition were repeated until the simu-lated cooling curves gave good agreement with the measured coolingcurves of the steel specimen.

4. Carbon Diffusion and Distribution

Fig. 4(a) shows the changes of carbon content with time in different posi-tions during the carburizing process. The carbon content in the surface ofthe steel cylinder increases from 0.45% to 0.9% in 250min. Fig. 4(b) and(c) compares the difference of carbon content of the steel cylinder beforeand after the carburizing process. The carbon content decreases beingreserved in heating furnace for 35min after carburization, which showsthe effect of diffusion on carbon content distribution.

5. Heat Transfer Coefficients and Cooling Curves

The heat transfer coefficients used for the simulation are shown in Fig. 5.The coefficients are estimated by using the inverse method program‘‘InvProbe-2D’’ and the cooling curve data of the ISO Inconel 600 alloyprobe. The cooling curves that were calculated with these heat transfer


coefficients are shown in Fig. 6. The cooling curve of the center point in thesteel specimen shows good agreement with the measured one. And thosenear the boundary have a little difference from the measured one due tothe influence of boundary of the steel cylinder specimen.

6. Prediction of Martensite Distribution and Equivalent Stress

Martensite distribution and equivalent stress with consideration of thetransformation plasticity in carburizing–quenching process are predictedand shown in Fig. 7. From the result of Fig. 7(a), we know that martensiteis mainly generated near the surface due to the increase of the carbon

Figure 4 The distribution of carbon after carburizing and heating.

Figure 5 The heat transfer coefficient.


content. On the other hand, depending on measured hardness as shown inFig. 8, distribution of martensite after carburized–quenching also is verified.

7. Distortion During Quenching

Figure 9 shows the distortion of the calculated and the measured diameter ofthe steel cylinder after carburizing–quenching. Except for the influence ofsurface boundary condition, the calculated distortion of the center partof the cylinder is in good agreement with the measured value as shown inFig. 9. However, because identification of the heat transfer coefficient on thecorner of the cylinder is difficult, prediction of the distortion on the cornerremains to be solved.

Figure 6 Calculated and measured cooling curves in different position.

Figure 7 Distribution of (a) martensite and (b) equivalent stress.


8. Effect of Transformation Plasticity

To check the effect of the transformation plasticity on the simulation result,simulation results which were calculated with or without the transformationplasticity are compared. In Figs. 10 and 11, it is shown that residual stressesconsidering the transformation plasticity are much less than without consid-ering transformation plasticity. These results show the importance ofconsidering transformation plasticity in the simulation of carburizing–quenching process.

Figure 8 Measured hardness in center and surface.

Figure 9 The calculated and experimental diameter expansion of the steel cylinder

after carburizing–quenching.


B. Residual Stress and Distortion inCarburizing–Quenching of Gear

Based on the series of governing equations above, a finite element programcalled ‘‘HEARTS’’ was developed to predict the temperature field, carbondiffusion, phase transformation and distortion during carburizing–quench-ing process.

The simulation model in carburizing–quenching process is a JIS-SCM420 steel gear with edge circle diameter of 36mm, teeth number 16and module 2mm as seen in Fig. 12. Figure 13 shows the variation ofTTT-curves of the material when carbon content is changed to 0.8% by using

Figure 10 The calculated and experimental residual stress on the surface.

Figure 11 Comparison of residual stresses with and without consideration oftransformation plasticity.


the carburizing–quenching process as seen in Fig. 14. The identified heattransfer coefficient is shown in Fig. 15. It is assumed that the model(Fig. 16) is located in a uniform heating and cooling process as in Fig. 14.Then, the finite element model belongs to an axisymmetrical problem. Asan initial step of the heat treatment process, calculations for the heatingand carburizing process were used to simulate thermal stress field, thermaldistortion as well as carbon distribution in the model. The quenching processof the model was started from the initial temperature 8508C and the model

Figure 12 Illustration of gear specimen.

Figure 13 Variation of TTT curves depending on carbon component.


was cooled to 308C with oil. The heat transfer coefficients h during thequenching were calculated as a function of temperature by the methods men-tioned below and used for simulation as the surface boundary condition.

1. Carbon Diffusion and Distribution

Fig. 17 shows the changes of carbon content in different positions withtime during the carburizing process. The carbon content in the surface ofthe steel gear increases from 0.45% to 0.9% in 250min. Fig. 18(a) and (b)

Figure 14 Process conditions of carburizing–quenching.

Figure 15 Heat transfer coefficient depending on temperature.


compares the difference of carbon content of the steel gear before and afterthe carburizing process. The carbon content decreases since it is present inheating furnace for 35min after carburization, which shows the effect of dif-fusion on the carbon content distribution.

2. Prediction of Martensite Distribution

Martensite distributions in carburizing–quenching process are pred-icted and are shown in Fig. 19. Fig. 19 shows that martensite is mainly

Figure 16 FEM model of gear.

Figure 17 Variation of carbon contents in different position.


formed near the surface due to an increase of the carboncontent.

3. Residual Stresses

In order to predict the residual stresses in the gear, the simulation resultswere calculated taking into account the strain due to phase transformation.From Fig. 20, we come to know that the residual stresses are greater at thecorner of the gear surface. These results show the importance of consideringthe transformation plasticity in the simulation of carburizing–quenchingprocess.

Figure 18 Distribution of carbon after carburizing and heating.

Figure 19 Simulation of martensite distribution.


4. Distortion after Quenching

Figure 21 shows the distortion of the calculated and the measured diameterof the steel gear after carburizing–quenching. And except for the influenceof surface boundary condition, the calculated distortions of the center partof the gear are in good agreement with the measured value as shown inFig. 22. However, because identification of the heat transfer coefficient onthe corner of the gear is difficult, prediction of the distortion on the cornerremains to be solved.

Figure 20 Equivalent stress.

Figure 21 Deformation of gear.


IV. DESIGNING OF CONTINUOUS CASTING FOR CONTROLOF RESIDUAL STRESS AND DISTORTION

In order to optimize material metallurgical processes, such as casting andwelding during solidification, it is important to determine the residual stressafter these processes for evaluating the mechanical properties and strengthof materials and to optimize the operative conditions in manufacturing[76–78]. However, the formation of residual stress not only depends on thethermo-mechanical behavior and processing effect due to the variation ofin micro-=macro-structures during the manufacturing of materials, but alsoon the interaction of heat conduction, change of phase transformation dueto solidification, and stress=strain. Especially, to describe variation of themechanical behavior from viscous fluid to solid and the mixture domainin the material due to solidification, a unified constitutive model incorporat-ing with solids and fluids of metal is proposed [64].

In order to solve the above-mentioned problems, this chapter presentssome developments of the thermo-mechanical theory and numerical analysismethod incorporating solidification of material to simulate the residual

Figure 22 Deformation on surface of gear teeth.


stress formation during casting. A unified inelastic constitutive relationshipcapable of describing both elastic–viscoplastic solids and viscous fluids toapply simulation of the casting process was proposed and verified by experi-mental and numerical results. On the other hand, a proposal based on thefinite element method to couple temperature, stress fields as well as defor-mation during solidification was presented. Depending on the simulationsof the continuous or semi-continuous casting, the mechanism of the residualstress formation during these casting processes can be represented. Thethermo-mechanical modeling was also verified by a comparison with theexperimental data, such as the measured residual stress and variation tem-perature in casting. Vertical semi-continuous direct chill casting process isone of most efficient methods to produce ingots of aluminum alloys andother metals. It is beneficial for optimizing the operating conditions to simu-late thermo-mechanical field in the solidifying ingot. So many reports havebeen published concerning such analyses of the temperature distributionincorporating solidification by finite element method, but a few papers treatthe induced stress=strain field. Simulations of thermal stress in continuouscasting slab were made by using elastic–plastic constitutive models [79,80],and viscoplastic stresses [81–84] were simulated based on the solidificationanalysis by Williams et al. [31]. However, in their studies, the influence ofcasting speed was neglected, so that the numerical simulation along withthe variation of casting conditions could not be realized. In order to solvethis problem, Ju and Inoue [62] proposed a numerical simulation methodby the Eulerian coordinate, and application to the continuous casting pro-cess of steel slab was performed.

A. Residual Stress Formation DuringSemi-continuous Casting

The aim of this section is to apply the coupled method of temperature andstresses incorporating solidification developed for semi-continuous directchill casting of aluminum alloys. When the bottom block plate is locatedat the upper position and the length of the growing ingot is small, the tem-perature, liquid–solid interface, and stresses in the ingot vary with time,both in the sense of space and of material. However, when the ingotbecomes long enough, the physical field in the upper part is regarded tobe time-independent or steady in the spatial coordinate fixed to the system.In the first part of this section, a steady heat conduction equation with heatgeneration due to solidification is formulated in a spatial coordinate systemwhen considering the material flow. A numerical calculation for thetemperature in the solidifying ingot as well as the simulation of the locationof liquid–solid interface is carried out by a finite element technique.


Most metallic materials at low temperature may be treated as anelastic–plastic solid. However, if they are heated beyond the melting point,the materials can be regarded as a viscous fluid, and they behave in a time-dependent inelastic manner at high temperature close to the melting tem-perature. Therefore, a unified constitutive model needs to be establishedto describe the elasto-plastic and viscoplastic behavior of the solidified partof the ingot as well as the viscous property of the liquid state. Taking intoaccount the effects of such phenomena, a modification of Perzyna’s consti-tutive model similar to the one in other sections is presented in the secondpart of this section, and some experimental results of the viscosity appearingin the model are presented for a Al–Zn type alloy. By using the model,elastic–viscoplastic stresses are calculated for the ingot to establish theresidual stress distribution, and are verified by the measured data from ahole-drilling strain-gauge technique.

Finally, results of a numerical simulation are presented on the influ-ence of operating conditions on temperature and stresses, such as ingot size,casting speed, and initial temperature, to provide fundamental data for opti-mizing the operating condition.

1. Finite Element Model and Casting Conditions

The theory and the procedure developed above are now applied to the simu-lation of the vertical semi-continuous direct chill casting process shownschematically in Fig. 23. The material treated is a Al–Zn type alloy with5.6% zinc and 2.5% magnesium. A quadrilateral finite element mesh pat-tern of 600 elements with 1941 nodes illustrated in Fig. 24 is employed forboth analyses of temperature and stress fields.

The boundary condition for heat conduction is assumed in such a waythat the temperature of the meniscus of molten metal is prescribed to be w0,and that heat is insulated along the central line and the bottom of ingot aswell as the surface contacted with the refractory. The cylinder facing themold is regarded as the boundary Sq on which heat flux is given. The otherpart of the surface Sh is given by a heat transfer boundary due to the coolingof water. Figure 25 depicts the measured heat flux q absorbed by the mold,and heat transfer coefficient h depending on flow rate of water TW is shownin Fig. 26.

Other data used for temperature calculation incorporated with solidi-fication are shown in Table 1. Characteristic results of calculated tempera-ture and residual stresses for an ingot of 1m in length with the diameter of240mm are compared with experimental data to verify the method. Simula-tions in other cases of different operating conditions such as casting velocity,size of the ingot, and cooling rate are also made.


2. Results of Simulation

Figure 27 indicates a bird’s eye view of the isothermal representation of cal-culated temperature distribution. The lines denoted by Tl and Ts in the fig-ures are the liquidus and solidus temperature, respectively. The data isreplotted to give the temperature change at the center and on the surface,while the circles represent the measured temperature by thermocouples asshown in Fig. 28. The fact that the calculated temperature on the surfacecoincides well with the measured data may indicate the validity of the simu-lation method.

Change of the volume fraction xs at four characteristic points alongthe distance from meniscus is represented in Fig. 29, which may give infor-mation on the progressing mode of solidification and thickness of solidifiedshell.

Figure 23 Schematic view of semi-continuous vertical direct chill casting.


Figure 24 Finite element mesh pattern for semi-continuous vertical direct chillcasting.


Simulated results of temperature and mode of solidification presentedin the above analysis of stresses was carried out on the entire area of theingot, including the molten metal, using the finite element method basedon the elastic–viscoplastic constitution model. The displacement mode isdepicted in Fig. 30. The contour of radial, tangential, and axial stress dis-tributions sr, sy, sz are shown in Fig. 31. The contours of stress distributionare represented in Fig. 32.

Examples of the calculated stress distribution by elastic–viscoplasticconstitutive model and the one by time-independent elastic–plastic modelare shown in Fig. 33. When compared with each other, the viscoplastic stressanalysis gives smaller results, at least on the surface, than elastic–plasticstresses. The stresses are found to be generated at the location where thesolidification starts (see Figs. 31 and 32), and the radial distributionbecomes steady owing to the flatter temperature distribution. In order toexamine this effect, the stress distributions at several locations are givenin Fig. 34, in which the distribution at the end of the ingot (Fig. 34d) canbe regarded as residual stresses. Circles in the figure indicate the experimen-tally measured residual stresses by a hole-drilling strain gauge method [7].The fact that the experimental data coincide well with the analytical resultssuggests the validity of the simulation procedure based on viscoplasticity.As is seen in Fig. 34 where the shear stresses are insignificant in the area

Figure 25 Variation of heat flux from top of the mold.


of molten state, the normal stresses sr, and sz (see Figs. 31 and 33) areregarded to be hydrostatic stresses, which means that the constitutiveequation employed here reveals to modify deformation of the liquid.

3. Simulations for Other Operating Conditions

This procedure may be applied to other cases of operating conditions.Hereafter, the focus is on the effect of temperature and stress distributionson the casting speed, size of the ingot and cooling condition. Figures 35and 36 represent examples of temperature profiles for different castingspeed and the radius of ingots with various cooling rate. The effect of dis-charge of cooling water on thickness of the mussy zone at the center issummarized in Fig. 37. The effects of casting speed and ingot radius onthe stress s are represented in Figs. 38 and 39, respectively, and the rela-tion between stresses and casting speed is summarized in Fig. 40. Thesimulated stresses varying with the discharge of cooling water are alsoplotted in Fig. 41. The calculated results shown above seem to simulatethe characteristic of temperature and stresses depending on the operating

Figure 26 Distribution of heat transfer coefficient depending on discharge of

cooling water.


conditions. If the data of such a simulation at different operating condi-tions are accumulated as shown in Figs. 35–39, the possibility of optimiz-ing the design of the system would be realized.

B. Residual Stress Formation During Strip ContinuousCasting by Twin-Roll Method

The continuous strip casting technology by twin-roll method is a promisingtechnology that not only saves energy, but also reduces production costs inthe manufacturing of material. However, there are many difficulties in

Table 1 Material Properties of Aluminum 7075

Physical item Value Unit Temperature range

Heat conductivity kl¼ 0.0425; cal=(mm 8C)ks¼ 0.0125þ 0.0583T

Density r¼ 2.8� 10�3 g=mm3

Specific heat cl¼ 0.015þ 0.00173T

cs¼ 0.083þ 0.00267T cal=(g 8C)Latent heat due ls¼ 93.16

to solidification cal=gCasting speed V¼ 80.0 mm=min

Liquidus

temperature

Tl¼ 638 8C

Solidus temperature Ts¼ 600 8CGradient of

liquidus line

ml¼ 3.69 8C=%

Gradient of solidus

line

ms¼ 9.09 8C=%

Young’s modulus El¼ 500.0

Es¼ 75000.0 – 76.3T MPa

Poisson’s ratio n¼ 0.33

Viscosity m¼ 3700.0 MPa s T� 3468C� 0.007178T2

21.1698T 10160.7

T 3468C

Initial yield stress sv0¼ 2.0 MPa T� 3468C�v0 150.0 – 0.428T T < 3468C

Hardening

coefficient

H0 ¼ 350.0 – 0.13333T

H0 ¼ 330.0–1.683(T�150)H0 ¼ 0.132

MPa T 1508C150 < Ta < 3468CT� 3468C

Thermal expansion

coefficient

al¼ 33� 10�6

as¼ 21.8 � 10�6

þ 0.2T � 10�7l=8C

Dilatation due to

solidification

b¼ 7.5%


Figure 27 View of the calculated temperature profile.

Figure 28 Temperature variation at the center and surface of the ingot.


controlling the quality of the strip because of the existence of the deforma-tion of the strip itself, due to thermal expansion or thermal stress. There aretwo key points: firstly, if the solidification is completed before the liquidreaches the minimum clearance point between the rolls, then the stripwill occur at a fixed gap. Hence, one of the key points is controlling of

Figure 29 Volume fraction of solid along the distance from meniscus.

Figure 30 View of deformation.


Figure 31 View of stress distributions.


solidification. Another key point is that the viscoplastic deformation incor-poratingmaterial flowmust be considered in this thermo-mechanical process.

1. Continuous Casting System by Twin-Roll Method

The twin-roll continuous casting system is schematically illustrated inFig. 42(a). In this process, molten metal is between the two rolls rotatingin opposite directions with same angular velocity. The level of the moltenmetal is always kept constant by overflowing the excess molten metalfrom the nozzle. As soon as the molten material is poured into the rolls,solidification takes place on the roll surface, which is cooled by circulatingwater inside the roll. Therefore, the problem then is to find this steadysolidification profile and the distribution of temperature and fluid velocitiesin both the liquid phase and the solid phase. On the other hand, due to thesymmetry to the central line, half of the model shown in Fig. 42(b) is treatedfor the analysis.

Figure 32 Iso-stress contours.


2. Analytical Models and Parameters

The procedure developed above is now applied to the simulation of thethin slab casting process under the same operating conditions. The resultsare summarized as follows. Figure 43 represents the finite element descri-tization of the whole region of the strip and roll. The surface of roll as wellas the contacted boundary with the roll and strip is assumed to belong to

Figure 33 Calculated stresses by (a) elastic–viscoplastic model and (b) elastic–plastic models.


heat transfer boundary and the surface of the strip to the heat radiationboundary.

In order to verify the numerical analysis method proposed inSection 2, continuous casting of SUS304 steel is taken into considerationin this section. In continuous casting process of SUS304, the thicknessof the slab is 1mm, and two casting speeds are used Vc¼ 400 and600mm=sec.

3. Calculated Results

Simulated results of steady temperature field both in the strip and roll isshown in Fig. 44(a) and (b) for the casting speeds of 400 and 600mm=sec.

Figure 34 Stress distribution in several sections of ingot.


sec. Temperature distribution along the central line and surface is plottedin Fig. 45. The fraction of solid phase in solidified region of the strip withliquidus and solidus temperatures Tl¼ 14608C and Ts¼ 13998C is depictedin Fig. 46. In the early stage of rapid cooling by the roll, the solidifiedshell is seen to grow gradually toward the central part. The change incasting speed is known to affect the distribution of temperature and shellthickness. As the casting speed becomes faster, the position of the liquidusline moves downstream. Figs. 45 and 46, shows that the temperaturedistribution on surface of slab at the roll outlet presents violent fluctuation

Figure 35 Isothermal lines by effect of casting speed.


due to latent heat generation by the solidification domain. When the castingspeed Vc is raised, the temperature fluctuation towards a small range ispresent on the surface of the slab.

The distribution of the horizontal stress sx from meniscus to the kis-sing point at the roll outlet is represented in Fig. 47. From these figures,the high-level stress field by rapid cooling is found to occur near the surfaceof the shell, and the equivalent stress distribution on the surface of the slabat roll outlet also presents violent fluctuation due to the temperaturefluctuation. The equivalent stress analysis results based on several constitu-tive equations are plotted in Fig. 48. In the section of slab at the roll outlet,calculated equivalent stress by three types of constitutive equations is

Figure 36 Isothermal lines by effect of ingot radius.


approximated to the same distribution. However, when the casting speed islower, the effect of creep strain is evident.

V. DESIGNING OF THERMAL SPRAY COATINGFOR CONTROL OF RESIDUAL STRESS

Plasma-spray coating is an important method by which a new functionallayer can be produced on material surface [85,86]. In the spray coatingprocess, solidified particles form thin lamellae whose microstructuredepends on the particle cooling rate during solidification. Indeed, lamina-tion of several kinds of materials deposited by laser thermal spray coatingleads to an increase of the strength of the structural components, such asplates, and cylinders, especially in the case when exposed to severe tempera-ture gradient. However, the residual stresses at an interface between multi-layer after coating play an important role that affects thermo-mechanicalproperties of the materials, since it is associated with the growing liquidboundary due to the supply of the molten material and also from themoving liquid–solid interface depending on the progress of phase trans-formation. It is also necessary to consider the effect of coupling between

Figure 37 Relation between thickness of mussy zone at the center depending ondischarge of cooling water.


temperature and stress=strain fields and the phase transformation, orsolidification in this case. Some open problems still remain in the develop-ment of a mathematical model capable of treating the growing boundaryof molten state and a moving interface between liquid and solid phases aswell as the evaluation of residual stresses.

There are three complicated aspects to be considered: The first one isthat the spray coating process is nonlinear problem with respect to time andlocation, and the second is that the solidifying process plays an importantrole in the simulation of temperature and stresses field. The third is howto deal with the difference in mechanical and physical properties of thematerial in each layer. To solve the first and second problems, a schemefor numerical analysis by the finite element method has already been pro-posed by the authors [86].

This chapter shows the simulation of temperature and stress=straininduced in the layers of the plate with progressive domains based on themetallo-thermo-mechanics, and a finite element scheme is proposed toevaluate the change of both fields. Heat input due to successively pouring

Figure 38 Stress distribution by effect of casting speed.


molten metal is introduced in the heat conduction equation coupled withmechanical work and latent heat generation by solidification. Simulta-neously, a method of stress analysis using an elastic–viscoplastic constitutiverelationship capable of describing the mechanical behavior of both solid andliquid is proposed.

Examples of the numerical calculation of temperature with solidifica-tion mode and residual stresses in the multi-layer plate in the course of thespray coating are presented, and the validity of the calculated results is dis-cussed in comparison with the x-ray experimental results.

A. Experimental Procedure of Spray Coating

Generally, from the viewpoint of continuum thermomechanics, the spraycoating process of material has many complicated phenomena which areaffected by the interaction between temperature, stresses, and solidificationincorporating rapid cooling, and the state in the solidifying material termed

Figure 39 Stress distribution depending on effect of ingot diameter and coolingwater rate.


as ‘‘mushy zone’’ which is a mixture of solid and liquid phases. Figure 49illustrates an example of spray coating system, which is developed to fabri-cate laminated plate. In most cases of the spray coating process, after thespray layer is solidified completely, the other kind of material is poured ontothe substrate so that the inner boundary grows and the interface of moltenstate moves toward spray layer direction.

In the spray coating process, solidifying material in the growingdomain undergoes a variation of mechanical quantities, such as mass,momentum and energy as well as the change of material properties due tophase transformation from liquid to solid state. The complicated interactionbetween temperature and inelastic deformation, in this case, is to be takeninto consideration.

B. Numerical Model and X-Ray ResidualStress Measurement

1. Modeling of Numerical Simulation

To verify the validity of the theory and the procedure stated above, simula-tion of the fields of temperature and solidification mode and the stress

Figure 40 Relation between stresses at the center of ingot and casting speed.


distribution are performed over the course of the spray coating process oflaminated plates with two layers. The, assumption is made that both endsof laminated plate are constrained during the spray coating, and that thelaminated plates are large enough, so that a growing model of two-dimen-sional finite element is proposed to interpret the experimental phenomenaof the spray coating process and to compute the interfacial thermo-mechan-ical behavior between the impinging particles and the surface of substratejust beneath them. A model of the spraying layer represented by a flat diskof 30mm diameter and 0.5mm thickness of initially uniform temperaturewhich is put into contact with the substrate elements is shown in Fig. 50.When the numerical analysis is started, the growth elements incorporatedwith the impinging particle from the spraying direction were put into thesubstrate. In this analysis, the growth model is represented by an axisymme-trical problem. Thermal flux entering and leaving each element as well as thelatent heat liberated within the elements themselves during the solidificationprocess is evaluated and the resulting element temperature is computed aftereach successive time increment.

To simulate the spray coating processes, it is assumed that the differ-ent materials are successively poured into the substrate, i.e., the different

Figure 41 Relation between stresses at the center of ingot and discharge of coolingwater.


material is supplied after the material in the outer layer is completely solidi-fied. On the outer surface along spraying direction, the heat radiationboundary conditions was set on the initial step of the coating, and heattransfer condition was set on the cylinder surface of the axisymmetricalmodel, respectively.

C. Properties and Coating Conditionof Specimen Materials

Laminated layers are deposited on a stainless steel (SUS304) substrate of5mm thickness. In this work, the layer thickness and layer materials pro-duced were: 0.50mm for stainless steel. The specified thickness was obtainedwhen spraying was performed 10 times. Table 2 shows the components ofthese wires which are generally used for carbon dioxide arc welding. Table3 shows the conditions of the spray coating process.

The thermo-physical properties of the wire materials used for tempera-ture and stress calculation incorporated with solidification are shown inTables 4 and 5, respectively [69]. It is assumed that the properties ofsubstrate are the same as those of the wire. The heat transfer coefficient

Figure 42 View of (a) twin-roll casting system and the (b) model for simulation.


for air on the surface of the model is chosen to be h¼ 2.78 � 10�3

(cal=(mm2 sec deg)). The thermal radiation coefficient G¼ 7.028 � 10�7

(cal=(mm secK)) is used for the model.Depending on the inelastic constitutive model of Section 2, the inelas-

tic strain rate can be given by a viscoplastic relationship. Here, the viscositywhich described the viscoplastic model of wire material (stainless steel) isshown in Fig. 50.

D. X-Ray Residual Stress Measurement

To measure the residual stresses in the spray coating process, CrKa charac-teristic x-rays were used. Diffraction planes and angles were (2 1 1) and2y0¼ 1568 for the stainless steel SUS304. Surface roughness of the coatedlayer was about 6.5 mm. These values might be too large for x-ray stress mea-surement to provide reliable results. However, the parallel beam methodcould give stress values with sufficient accuracy on such rough surface.Before x-ray stress measurement, electropolishing conducted to remove an

Figure 43 Mesh of finite element model.


oxide-film from the layers. Stresses were measured parallel and perpendicu-lar to the spray traveling direction. The x-ray measuring conditions areshown in Table 6.

The full width at the half-maximum method was used to determinepeak positions. We measured the residual stresses in a phase with 2 1 1 dif-fraction and g phase with 2 2 0 diffraction. The stresses were obtained by thesin2c method. The c-diffractometer method was used for the measurementof residual stresses.

E. Verification and Discussion of Simulation Results

An example was used for simulating thermo-mechanical behavior and resi-dual stress during the spray coating. Figure 51 shows the variation of tem-perature distribution on the central element of spraying surface and outsidesurface of the layer. From these results, we arrive at the temperature differ-ence between the central element and the outside surface of the sprayinglayer due to the solidifying process. The distribution of temperature onthe total domain dependent on time is shown in Figs. 52 and 53. The volumefraction of solid and the variation of the solidified thickness are depicted inFigs. 54 and 55, respectively. These figures, show that the temperature onthe central element tends to decrease slowly by the latent heat generation

Figure 44 Distribution of temperature in strip and roll.


due to solidification and also by the heat supply by the successively pouredmaterial followed by the rapid temperature decrease at the end of solidifica-tion. Difference in heat conductivity due to the temperature differencereveals the influence on the cooling rate and mode of solidification.

As for the results of stress analysis, residual stress is represented in thefollowing figures, in which increasing stress reduces fluctuation or jumpdepending on the solidification and growing domain. Distribution ofresidual stresses sr on the radial direction is shown by the lines in Fig. 56.

Figure 45 Temperature variations at center and surface of strip.

Figure 46 Distribution of solid fraction.


The data are compared with the experimental results represented by thesame condition of the process. Relatively reasonable agreement betweenboth values is seen even in the region with fluctuation of stresses on theinterface boundary between the spraying layer and the substrate. The distri-butions of residual stress sy and sz are shown in Figs. 57 and 58. Here, thejump behavior of stresses on the interface is presented by these simulatedresults. Thus, it is important to reveal the damage of the spraying layerbased on the theory and numerical method.

VI. DESIGNING OF FORGING PROCESS FORCONTROL OF INTERFACIAL STRESS

A. Basic Description

A typical industrial metal component may be manufactured by forgingand heat treatment. In the design of forging processes, information such

Figure 47 Distribution of stress sx.


Figure 48 Dependence of constitutive relationship on stress distribution.


as material flow in the dies, level of die fill, defects, strain, stress, tem-perature distribution in the work pieces and the dies, and forging forceis necessary. In the subsequent heat treatment operations, informationon combination of microstructure, residual stresses, and dimensionalaccuracy in the final product is also very important. Such informationmay be obtained by numerical simulation [87–90].

The currently available commercial (and also academic) codes of for-ging are usually based on the finite element method. For a forging processwhere the metal is plastically deformed under high pressure into machineparts with high-strength performance, the finite element method sometimesexhibits weaknesses which must be carefully monitored

� Finite element meshes usually become over-distorted; hence, auto-remeshing is then necessary to complete the simulation. But theauto-remeshing technology for three-dimensional problems is notso robust and is also very time consuming.

Figure 49 Schematic diagram of laser spraying.


� Even for two-dimensional elastic–plastic problems, the remeshingmay lead to an erroneous result [70]. Each remeshing step willinvolve a loss in volume, which is not acceptable for a considerableforging simulation.

On the other hand, the finite element method is suitable for stimulat-ing the heat treatment process after forging because it is principally a rela-tively small deformation process [88]. But when deformation-induced phasetransformation during the process of metal forming (large deformation likeforging) is taken into account, the finite element method will meet the sameproblems of remeshing and loss of volume.

In this section, a method to simulate forging and subsequent heattreatment processes is described. The advantages of the finite element andthe finite volume approach are combined: it employs a finite volume meshfor tracking material deformation and an automatically refined facet surface(material surface) to accurately trace the free surface of the deforming mate-rial. This is particularly suitable for large three-dimensional material defor-mation such as forging because remeshing techniques are not required.

Figure 50 Coefficient of viscosity based on viscoplastic model.

Table 2 Composition of the Substrate and Wire (wt%)

C Mn Si Cr Ni

Substrate 0.06 – – 18 0.4Wire 0.04 1.90 0.46 20.1 8


This finite volume method provides an approach based on the‘‘metallo-thermo-mechanics’’ to simulate metallic structure, temperatureand stress=strain coupled in the heat treatment process such as quenching.The material surface utilizes the advantages of the finite element methodto ensure a level of accuracy for the small deformation process in the frameof the finite volume method. This approach also provides latitude for imple-mentation of phase transformation analysis coupled with forging process inthe future.

B. Validation Example

In this section, we consider forging followed by subsequent quenching of anS45C carbon steel rod used in a ship engine. The outline of the basic numer-ical model is shown in Fig. 59.

In the first process, the rod is forged from a round billet of 524mmdiameter and 500mm length between two closed dies. A three-dimensional model is necessary for the simulation. In the second process,the rod goes through the water spray quenching. The problem is an axialsymmetric model.

(1) In the forging process, the initial temperature of the billet is12508C. Since the material is above the recrystallization temperature, theinfluence of strain upon flow stress is insignificant, while the influence of

Table 3 Conditions of Spray Coating

Laser output 3.0 (KW)Wire feed 5.1 (m=min)Inner gas pressure 5.0 (kg=cm2)Out gas pressure 5.0 (kg=cm2)

Spraying distance 100 (mm)Nozzle Double nozzleAssist gas Argon

Table 4 Thermal Properties of Wire

Heat conductivity (cal=(mm s deg) k¼ 1.83� 10�3þ1.02� 10�6TDensity (g=mm3) rs¼ 7.9� 10�6; rl¼ 7.4� 10�6

Specific heat (cal=(g deg)) c¼ 65.2þ 9.77� 10�6TLatent heat (cal=g) ls¼ 65.0Temperature of solidus and liquidus (deg) Ts¼ 1432; Tl¼ 1449


strain rate becomes increasingly important. The flow stress for hot forging isdefined as follows:

sy ¼ C_eeM ð32Þwhere C and M depend on equivalent strain and temperature listed in theliterature [90].

The heat transfer between workpiece and die plays an important rolein hot forging. Several methods are available to determine the contact heattransfer based on data reported in the literature. Here, 5000 W=(m2K) areused according to Ref. [88]. Other material properties of heat conduction inthe workpiece and dies and the heat transfer between the workpiece=die andambient are also taken from Ref. [88] as seen in Table 7.

The punch velocity is described in Fig. 60, shows the different stagesof forging, holding, and ejecting. Figure 61(a) and (b) shows a quartermodel of the rod in its initial position and at the end of the ejection stage.

Table 5 Mechanical Properties of Wire

Young’s modulus (MPa) Es¼ 2.5� 105�1.73� 102T; El¼ 2264Yield stress (MPa) sy0¼ 280.35–1.4Tþ 5.75� 10�3T2�1.1� 10�3T3

þ 9.8� 10�9T4� 4.14 � 10�12T5þ6.7� 10�16T6

Poisson’s ratio n¼ 0.35Thermal expansion (L=deg) as¼ 12.76� 10�8; al¼ 12.67� 10�12

Dilatation due to

solidification (%)

b¼ 0.25

Table 6 X-ray Measuring Conditions

Characteristic X-ray CrKa

Tube voltage 35 kVTube current 9mA

X-ray optics Parallel beamDivergence slit 0.58Receiving slit 0.58Sampling time Unifixed

Filter VDiffraction plane a Phase 211

� Phase 220

Irradiated area 2 � 8mm2


The simulated shape of the rod is in good agreement with the experiment.The burr can be clearly seen here and is also confirmed by experiment.

The residual stress and equivalent plastic strain obtained at the end ofthe ejecting stage is depicted in Fig. 62. The sample points are taken fromthe center to the surface in the middle of the rod along the punch movingdirection. The residual stress will be taken into account in the subsequentquenching process as the initial condition.

The temperature in the center of the rod increases during forgingbecause of heat generation due to mechanical work. But the temperature

Figure 51 Temperature variation on center and surface of layer.

Figure 52 Temperature distribution at t¼ 0.35 sec.


near the surface drops because of die chilling. In the ejection stage, the tem-perature near the surface may rise again slightly since the heat transfer fromthe workpiece to ambient is much lower than that of the die, so that the sur-face can be heated again at the center. Figure 63 represents the temperaturevariation with time at the center and the surface in the middle of the rodduring forging.

(2) In the quenching process, the temperature range is from 9008C toroom temperature (208C). The material at 9008C is in the austenite phase.

Figure 53 Temperature distribution at t¼ 0.75 sec.

Figure 54 Volume fraction of solid phase at t¼ 0.35 sec.


Fig. 64, shows that when the rod is continuously cooled down, the austenitecan transform into pearlite through diffusion transformation. If the coolingrate is high enough, austenite may transform into martensite when the tem-perature is decreased below the martensite transformation temperature. Theproperties of the material are highly dependent upon the content of eachphase and are found in Ref. [34].

To obtain a relatively effective cooling medium for water sprayquenching, 9100W=(m2K) are used as the heat transfer coefficient, and

Figure 55 Volume fraction of solid phase at t¼ 0.75 sec.

Figure 56 Distribution of residual stress sr.


208C is used as a constant ambient temperature. Figures 63 and 64 showthe temperature changes with respect to time taken at the center to thesurface in the middle of the rod during quenching and illustratesthe volume fraction of metallic structures formed. Figures 65 and 66

Figure 57 Distribution of residual stress sy.

Figure 58 Distribution of residual stress sz.


illustrate equivalent stress and axial residual stress distribution fromthe center to the surface in the middle of the rod after quenching,respectively.

In Fig. 66, two cases are shown for axial residual stress distribution:one case is quenching without taking into account the residual stressfrom the forging process; and the other is quenching, taking account of

Figure 59 Outline of basic numerical model.

Table 7 Material Properties of Heat Conduction

Heatconductivity

Specificheat

Heat transferwith ambient

Die 31W=mK 470 J=kgK 20W=m2KWorkpiece 35W=mK 800 J=kgK 52W=m2K

Figure 60 Punch velocity of forging, holding, and ejecting.


Figure 61 The rod in its (a) initial position and at the (b) end of ejecting stage.

Figure 62 Residual stress at the end of ejecting stage.

Figure 63 Cooling curve.


the residual stress from the forging process. Experimental results measuredby the Sacks method are also shown in the same figure. In both cases, theaxial residual stress is in tension near the center and in compression nearthe surface. Because of the effect of the residual stress from the forging pro-cess, less tension near the center and more compression near the surface wereobtained as seen in Fig. 66. The axial residual stress coming from the for-ging process does not make a big difference with the quenching calculation,

Figure 64 Volume fraction of metallic structures.

Figure 65 Effective stress after quenching.


because the axial residual stress coming from the forging process is quitesmall.

VII. CONCLUSION AND REMARKS

A general discussion on the framework of thermo-mechanical theory incor-porating phase transformation and solidification is described in this chapter,and a series of practical analytical schemes based on the finite elementmethod are introduced. These schemes are applied to simulate thermo-mechanical fields in quenching, casting, coating, and forging processes.

A coupling method to simulate phase transformation and solidifica-tion and temperature as well as stress distribution in these metallurgical pro-cesses is formulated, and the implementation by finite element calculation ispresented in this chapter as examples of the application of the theory andprocedure developed in the handbook. Elastic–plastic constitutive equationsand a modification of Perzyna’s viscoplastic constitutive relationship wereused, which reflects actual phase transformation and solidification duringthese metallurgical processes. The results of simulation are also verified bycomparison with experimental data, and application of the technique toother operating conditions to obtain the fundamental data of optimumdesign of the system.

From the discussion of theory, simulation method and results pre-sented in the chapter, the conclusions and remarks obtained are summarizedas follows:

Figure 66 Axial residual stress after quenching.


1. Comparing with calculated results and experimental data fortemperature, distortion and residual stress in these metallurgicalprocesses, the metallo-thermo-mechanical theory and simulationmethod proposed in Section II are verified.

2. Effects of cooling curves, distortion, and residual stresses on theoccurrence of the phase transformation in quenching are provedby simulations of quenching and carburizing-quenching processes.

3. It is important to identify the heat transfer coefficients ofquenchants with respect to the quenching process are obtainedfrom simulation results.

4. The unified inelastic constitutive equation may describe the stressand deformation in the whole region of the solidifying processincluding liquid and solid state.

5. In the simulation of continuous casting, the development ofstresses from solidifying domain is presented. On the other hand,the effects of distortion on solidification also shown to be animportant factor.

6. In the simulation of coating process, the jump behavior of stresseson the interface between substrate and spraying layer is shown.Thus, it is important to reveal the damage of the spraying layer basedon the metallo-thermo-mechanical theory and numerical method.

7. The advantage of finite volume technique over the finite elementmethod in the simulation of forging was shown. From this point ofview, the finite volume method is expected to be a powerful tool inthe simulation of metal forming processes.

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4Modeling and Simulationof Mechanical Behavior

Essam El-MagdAachen University, Aachen, Germany

With the rapid increase of the capacity and speed of computers, workstations, and even personal computers, numerical methods can now beapplied to solve easily many complex engineering problems, for example,in the fields of metal forming, strength of materials, and reliability studiesof parts, components or systems. Some of the conventional methods ofstress analysis, such as photo-elasticity, have nearly disappeared. Also, theinterest in analytical methods like the elementary theory of plasticity andthe slip line theory is decreasing to some extent with the increasing accuracyof the numerical methods.

At first, great effort had to be done in developing the numericalmethods themselves to insure stability, convergence, and accuracy of thecomputation process. In the mean time, many powerful codes that carryover the major part of this responsibility are commercially available.Especially for engineers, the main field of activities changed in anotherdirection, namely to the development of adequate constitutive equationsthat describe well the behavior of the material in the macroscopic, micro-scopic, and even in the atomistic scale. From the practical point of view, itis not urgently required to increase numerical accuracy by some 0.1%,while the material data for plastic deformation processes deviate by morethan 2% from reality or when the data for creep or fatigue life scatter by afactor between 1

2 and 2.There are still several important improvements to be done in the

numerical codes, for example, in order to achieve an accurate considerationof large deformations. However, the determination and implementation ofadequate material data seem to be one of the most important tasks. Due


to lack of material data, the computations were carried out in the past mostlyassuming an elastic material behavior using the common values of the mod-ulus of elasticity and the Poisson ratio, even if it was well known that thematerial behavior is inelastic under the conditions considered. Complex pro-cedures were developed in order to estimate the inelastic behavior using theelastic computational results. This is changing monotonically towards theconsideration of the inelastic material behavior with current codes.

The formulation of the material law is decisive for the experimentaleffort and costs needed to determine its parameters. Empirical relationsmay be helpful when only few variables of the process are consideredand when these variables vary within limited ranges. Otherwise, physicallyfounded material models may be more suitable, as they a priori definethe tendency of the relations to be determined. However, the experimen-tal determination of each parameter of these models according to itsexact physical meaning may require such a great effort that this proce-dure remains restricted to academic research activities. For the practicalapplication, a compromise is gaining increasing interest, according towhich the functions are taken from physical and microstructure-mechan-ical models, but their parameters are determined by curve fitting of theexperimental data.

In the following, some examples are represented for modeling andsimulation of the material behavior during plastic deformation, low cyclefatigue, creep, and impact strength.

I. PLASTIC BEHAVIOR

Flow curves represent the relationship between the true stress and thetrue stain during plastic deformation at constant strain rate and temper-ature. They are usually determined in tensile, compression, or torsion tests.

When a cylindrical rod of an initial length L0 and initial cross-sec-tional area A0 is loaded by a force F, its dimensions change to L and A.If the force is further increased by an increment dF, the length increasesby dL and the area decreases by dA. The corresponding increments ofthe engineering stress and strain are defined by dS¼ dF=A0 and de ¼dL=L0, while the increments of the true stress and strain increments aredefined by ds ¼ dF=A and de ¼ dL=L. At an arbitrary time point duringthe test, the stresses and strains are given by

s ¼ F=A0; e ¼ ðL� L0Þ=L0

s ¼ F=A; e ¼ lnðL=L0Þð1Þ


If the deformation is uniformly distributed along the bar andvolume constancy can be assumed, the true and the conventional stressesand strains are related by s ¼ Sð1þ eÞ and e ¼ lnð1þ eÞ. In addition tothe physical relevance, the use of true stresses and strains allows for: (a) asimple addition of the strains of different deformation steps e ¼P ei, (b)a simple formulation of the plastic volume constancy by exx þ eyyþezz ¼ 0, and (c) equal absolute values in tension and compression if speci-men length is increased or decreased by the same factor. Under serviceconditions, engineering materials are usually subjected to a multiaxialstress state. On the other hand, material data are determined in labora-tory tests under almost uniaxial loading. For comparison, an equivalentuniaxial stress is to be define for the multiaxial case. In the case of iso-tropic incompressible materials, the equivalent stress is given according tovon Mises by

seq ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðsxx � syyÞ2 þ ðsyy � szzÞ2 þ ðszz � sxxÞ2

2þ 3 t2xy þ t2yz þ t2zx �s

ð2Þ

No plastic deformation takes place, as long as this equivalent stress islower than the flow stress sY of the material. When the loads are soincreased that the equivalent stress reaches the flow stress, the materialstarts to yield. For the strain state, which is represented by the plasticstrain tensor epij , an equivalent strain increment is defined by

depeq¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

3ðdepxxÞ2þðdepyyÞ2þðdepzzÞ2h i

þ1

3ðdgpxyÞ2þðdgpyzÞ2þðdgpzxÞ2h ir

ð3Þwhere gij ¼ 2eij with i 6¼ j.

The plastic strains epij , which are the irreversible response of the mate-rial to the applied stresses, depend not only on the current values of thestresses, but also on the total loading history. For isotropic incompressiblematerials, the plastic strain increments are proportional to the deviatoricstresses, i.e., the normal stresses reduced by their mean value ðsxx þ syyþszzÞ=3 and the shear stresses unchanged, or in short form Sij ¼ sij�ð1=3Þdijskk. If the material follows the Mises yield criterion, the plastic strainincrements are given by

depij ¼3

2

depeqseq

Sij ð4Þ


A. Flow Curves

In order to characterize the strain hardening behavior of metallic materialsduring plastic deformation, one has to determine experimentally the relationsY ¼ sYðeeq; _eeeq;TÞ that defines the dependence of the flow stress sY on theplastic parts of the equivalent strain eeq, the equivalent strain rate _eeeq and onthe temperature T. Flow curves are defined as the relation sY ¼ sYðeeqÞdetermined for _eeeq ¼ const: at a constant temperature. They are often deter-mined in compression test, taking into consideration the influence of fric-tion. They are also to be determined in tension test up to the ultimateforce assuming uniform deformation.

1. Empirical Relations

The flow curves are almost described by power laws. The oldest of theserelations, introduced in 1909 by Ludwik [1], is given by

sY ¼ K0 þ Ken ð5Þ

This relation allows a good description of the flow curves of materialshaving a finite elastic limit. For a plastic strain ðe ¼ 0Þ, the flow stress equalsK0. It leads, however, to an infinite value for the slope of the curve @sY=@eat the yield point. A simplified form of this equation

sY ¼ Ken ð6Þwas suggested by Hollomon [2]. Because of its simplicity, it is till now themost common relation applied for the description of the flow curve. How-ever, no yield point is considered by this relation as sY ¼ 0 for e ¼ 0. Espe-cially for materials with a high yield point or materials previously deformed,the flow stress cannot be described well by this relation in the region of smallstrains. A more adequate description is achieved by the Swift relation [3]

sY ¼ KðBþ eÞn ð7Þ

For e ¼ 0, a yield point is considered with a value of sY ¼ KBn. Analternative description

sY ¼ aþ b½1� expð�ceÞ� ð8Þwas introduced byVoce [4] and is well applicable for the range of small strains.

Figure 1 shows the optimum fit achieved by the four equations (5–8)for the flow curves of an austenitic steel at different temperatures in therange of relatively small strain up to 0.2. The figure shows that the Swiftrelation and the Voce-relation describe well the flow curves in the relative


small strain range considered. Also, a good description is achieved by theLudwik relation when the parameter K0 is considered as an arbitrary con-stant and is chosen to be much smaller than the actual yield point.

2. Microstructure Mechanical Relations

In one-parametrical models, the flow stress depends only on the total dislo-cation density r which is considered as the single internal variable of thematerial according to

s ¼ aGbffiffiffirp ð9Þ

where G is the shear modulus, and b is the Burger vector. If the rates of crea-tion and annihilation of dislocation are known, an evolution equation canbe determined for the flow stress.

Mecking and Kocks [5] introduced the following evolution equationfor the total dislocation density:

drde¼ k1

ffiffiffirp � k2r ð10Þ

and determined the slope of the flow curve by

dsde¼ k2

2

k1k2aGb

� s� �

ð11Þ

Figure 1 Description of the flow curves of the austenitic steel X6CrNi18-11 at_ee ¼ 8� 10�4 sec�1 by empirical relations.


If the parameters k1 and k2 are considered to be constants, the flowcurves follow by:

s ¼ s0 þ ðs1 � s0Þ 1� exp �e=e�ð Þ½ � ð12Þ

This equation is identical with the empirical Voce relation. In therange of relatively small strains, it fits the experimental data very well. How-ever, it fails to describe the flow curves in the range of high strains becausethe experimental results for the flow stress do not asymptotically approach adefinite value [6].

The following modification can be suggested, to yield an evolu-tion equation that describes well the strain hardening in the range of highstrains. The parameter k1 ¼ 1=ðl ffiffiffi

rp Þ, where l is the dislocation free path.

This parameter can be considered as a function of strain and may beexpressed as k1 ¼ kð1þ ceÞ. The evolution equation of the flow stressbecomes

dsde¼ k

2aGbþ kc

2aGbe� K2

2s

� �ð13Þ

Figure 2 Flow curve of the austenitic steel X8CrNiMoNb16-16, described by Eqs. (13)and (14).


The solution of this differential equation is

s ¼ C1 þ C2eþ C3 1� expð�C4eÞ½ � ð14Þwhere C1 is the yield stress, C2 ¼ kc=ðk2aGbÞ, C3 ¼ kð1þ 2c=k2Þ=ðk2aGbÞ;and C4 ¼ k2=2. This equation is identical with the empirical relation intro-duced in Ref. [7]. It is found to give the optimum fit for the experimentalresults of several materials (Fig. 2). However the determination of its para-meter needs some more effort. It should be mentioned that also the empiri-cal Swift relation given by Eq. (5) fits well the experimental data in thisstrain range.

B. Influence of Strain Rate and Temperature

Figure 3a shows an example for the influence of increasing temperature onthe flow stress for given values of strain and strain rate [8]. Considering theslope ds=dT , three different temperature ranges can be defined: (A) range oflow temperatures, between absolute zero and about 0.2 of the absolute melt-ing point, where the influence of the temperature on the flow stress is great.The material behavior is governed by thermally activated glide, (B) range ofintermediate temperatures between 0.2 and 0.5 of the absolute melting tem-perature. Only a slight influence of strain rate and temperature on the flowstress is usually observed in this range, and (C) range of temperatures higherthan 0.5Tm in which the flow stress depends highly on the temperaturesbecause of the dominance of diffusion-controlled deformation processes.

The influence of the strain rate variation [9] is represented in Fig. 3b.Three different strain rate ranges can also be recognized according to the

Figure 3 (a) Temperature influence on the yield stress of NiCr22Co12Mo9 at_ee ¼ 3� 10�4 sec�1 [8]. (b) Influence of stain rate on shear yield stress of mild steel [9].


variation of @s=@ ln _ee: (I) range of low strain rates with only a slight influ-ence of the strain rate due to athermal glide processes, (II) range of inter-mediate and high strain rates with relatively high strain rate sensitivitydue to thermal activated glide mechanisms, and (III) range of very highstrain rates where internal damping processes dominate and a very highstrain rate sensitivity is observed. The boundary between the ranges (I)and (II) depends on the temperature. Overviews concerning the mechanicalbehavior under high strain rates are represented, e.g. in Refs. [10,11].

To estimate the mechanical behavior over wide ranges of strain rateand temperature, constitutive equations must be established taking the timedependent material behavior into consideration. A visco-plastic behavior isoften assumed by using, for example, the Perzyna equation [13]

_eeij ¼_SSij

2mþ 1� 2v

2E_sskkdij þ 2ghFðFÞi @f

@sijð15Þ

where m is the shear modulus, f is square root of the second invariant of thestress deviator Sij and F ¼ (f=k) �1 is the relative difference between f andthe shear flow stress k ¼ sF=

ffiffiffi3p

. The function FðFÞ is often estimated usingsimple rheological models assuming FðFÞ ¼ F and leading to linear relationof the type s ¼ sFðeÞ þ Z_ee which is acceptable for metals only at strain rates>103 sec�1.

1. Empirical Relations

Different empirical relations could be implemented in Eq. (15). WithFðFÞ ¼ expðF=aÞ � 1 or FðFÞ ¼ F 1=m, the corresponding relations betweenstress and stress rate in the uniaxial case are identical with the empirical rela-tions introduced 1909 by Ludwik [14]

s ¼ sFðe;TÞ 1þ a lnð1þ _ee=aÞ½ � ð16Þ

s ¼ sFðe;TÞ 1þ ð_ee=a�Þm�½ ð17Þ

The influence of temperature on the flow stress is also described by dif-ferent relations of the type s ¼ sðe; _eeÞf ðT=TmÞ where Tm is the absolutemelting point of the material, such as

s ¼ s0ðe; _eeÞ exp �bT=Tm½ � ð18Þor according to Ref. [15]

s ¼ s0ðe; _eeÞ 1� ðT=TmÞv½ � ð19Þ


On applying such empirical relations, the flow stress is usuallyrepresented by s ¼ ftðeÞf2ð_eeÞf3ðTÞ as a product of three separate func-tions of strain, strain rate and temperature. This is a rough approxima-tion especially in the case of moderate strain rates of _ee < 103 sec�1.However, the basic problem is that nearly all the parameters of theseempirical equations can only be regarded as constants only within rela-tively small ranges of e, _ee, and T. The determination of the functionalbehavior of the parameters requires a great number of experiments.Therefore, constitutive equations based on structure-mechanical modelsare gaining increasing interest as they can improve the description ofthe mechanical behavior in wider ranges of strain rates and temperatureand may, if carefully used, allow for the extrapolation of the determinedrelations.

2. Structure-Mechanical Models

The macroscopic plastic strain rate of a metal that results from the accumu-lation of sub-microscopic slip events caused by the dislocation motion isgiven by

_ee ¼ brmv=MT ð20Þ

In this equation, the Burger vector b and the Taylor factor MT areconstants for a given material whereas the mobile dislocation density rmis mainly a function of strain. The relation between the dislocation velocityv and the stress was experimentally determined for several materials [16]. Itcan be represented in the range of low stresses by a power lawv ¼ v0ðs=s0ÞN. At very high stresses, the dislocation velocity approaches

asymptotically the shear wave velocity cT and s ¼ avn=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� ðv=cTÞ2

q.

3. Athermal Deformation Processes

In the range of intermediate temperatures and low strain rates (combinedranges B and I), and at relatively low temperatures, i.e., less than 0.3 ofthe absolute melting point Tm, the influence of strain rate and temperaturedepends on the _ee-range of the deformation process.

Below a specific value of the strain rate, that depends on tempera-ture, only a slight influence of strain rate and temperature on the flowstress is observed. In this region I, athermal deformation processes aredominant, in which the dislocation motion is influenced by internal longrange stress fields induced by such barriers as grain boundaries, precipi-tations, and second phases. The flow stress varies with temperature in the


same way as the modulus of elasticity. The influence of strain rate can bedescribed by

s ¼ CEðTÞEðT0Þ

_eem ð21Þ

where E is the modulus of elasticity and m is of the order of magnitudeof 0.01.

4. Thermally Activated Deformation

In the ranges of low temperatures (A) and intermediate to high strain rates(II), the dislocation motion is increasingly influenced by the short rangestress fields induced by barriers like forest dislocations and solute atomgroups in fcc-materials or by the periodic lattice potential (Peierls-stress)in bcc materials. If the applied stress is high enough, these barriers canimmediately be overcome. At lower stresses, a waiting time Dtw is requireduntil the thermal fluctuations can help to overcome the barrier. A part of thedislocation line becomes free to run, in the average, a distance s� until itreaches the next barrier within an additional time interval Dtm. The meandislocation velocity is given by v ¼ s�=ðDtw þ DtmÞ.

The waiting time Dtw equals the reciprocal value of the frequencyn of the overcoming attempts. If the strain rate is lower than ca. 103

sec�1, it can be assumed that Dtw 4Dtm. The relation between strain rateand stress is then given by _ee ¼ _ee0ðeÞ exp �DG=kT½ � where _ee0 ¼brmn0s

�=MT. The activated free enthalpy DG depends on the differences� ¼ s� sa between the applied stress and the athermal stress accordingto kT lnð_ee0=_eeÞ ¼ DG ¼ DG0 �

RV �ds� where V� ¼ bl�s�=MT is the

reduced activation volume.For given stress and strain, the value of T lnð_ee�=_eeÞ is constant for all

temperatures and also for all strain rate values between _ee0 exp½�DG0=ðkTÞ� and _ee0. This means that the increase of stress at constant strain withdecreasing temperature or with increasing strain rates is the same, as longas the values of DG ¼ kT lnð_ee�=_eeÞ are equal in both cases.

Depending upon the formulation of the function V�ðs�Þ; differentrelations for _ee ¼ _ee0ðsÞ were proposed in Refs. [17–21]. The mostcommon are the relation introduced by Vohringer [19,20] and by Kockset al. [21]

_ee ¼ _ee0ðeÞ exp �DG0

kT1� s� sa

s0 � sa

� �p� �q� �ð22Þ


and that by Zerilli and Armstrong [22,23]

s� sa ¼ DG0

V�0exp � b0 þ

k

DG0ln

_ee0_ee

� �� T

� �ð23Þ

5. Transition to Linear Viscous Behavior

At strain rates higher than some 103 sec�1, the stress is high enough to theextent that Dtwvanishes. Only the motion time Dtm is to be considered. Thedislocation run with high velocity throughout the lattice and dampingeffects dominate. The dislocation velocity v ¼ s�=Dtm can then be given byv ¼ bðt� thÞ=B according to Ref. [24]. The flow stress follows the relation

s ¼ shðe;TÞ þ Z_ee ð24Þwith Z ¼MTB=ðb2NmÞ. This relation is validated experimentally in Ref. [9]as well as by Sakino and Shiori [25], as shown in Fig. 4a. A continuous tran-sition takes place, when the strain rate is increased from the thermal activa-tion range (II) to the damping range (III). This can be described in twodifferent ways: regarding the dislocation velocity to be equal tov ¼ s�=ðDtw þ DtmÞ, the strain rate can be represented by

_ee ¼ _ee0 expDG0

kT1� s� sa

s0 � sa

� �p� �q� �þ xs� sh

� ��1ð25Þ

where x is a function of strain. Alternatively, the continuous transition canbe described by an additive approximation. The stress is regarded to be thesum of the athermal, the thermal activated and the drag stress components.According to this approximation, s � sa þ sth þ Z_ee where sth is the thermalactivated component of stress determined from Eq. (22) or (23).

6. Diffusion-Controlled Deformation

In the range of high temperatures (C), the deformation is governed by strainhardening and diffusion-controlled recovery processes

_ee ¼ _ee0sG

�nexp � Q1

RT

� �ð26Þ

At very high temperatures and low stresses

_eed;e ¼ 14sOkT

1

d2Dv 1þ pdDB

dDv

� �ð27Þ


Figure 4 Dependence of flow stress on the strain rate. (a) In the range of dampingcontrolled deformation, described by Eq. (24) [25]. (b) In the transition range

between thermally activated and damping controlled deformation ranges, describedby Eq. (25) [26].


C. Material Laws for Wide Ranges of Temperaturesand Strain Rates

Material laws that describe the flow behavior over very wide ranges of tem-peratures and strain rates are needed for the simulation of several deforma-tion processes, such as high-speed metal cutting. In this case, differentphysical mechanisms have to be coupled by a transition function. Fig. 5shows the dependence on the stress with the strain rate at different tempera-tures for a constant strain. Three main mechanisms can be distinguished: (a)diffusion-controlled creep processes with _eecr / sNðTÞ in the region (1) of lowstrain rates and high temperatures, (b) dislocation glide plasticity withs / _eemðTÞpl in the region (2) of intermediate temperatures and strain rates,and (c) viscous damping mechanism with s ¼ sG þ Zð_ee� _eeGÞ in the regionof very high strain rates _ee > 1000 sec�1 in the region (3).

1. Visco-plastic Material Law

For a continuous description over the different ranges, the strain rates haveto be combined [27] to obtain

_ee ¼ 1�Mð Þ _eekr þ _eepl� �þM_eedamping ð28Þ

with the transition function M ¼ 1� exp½�ð_ee=_eeGÞm�. The complete strainrate range can be described by

_ee ¼ 1�Mð Þ ss0 T; eð Þ� �N Tð Þ

þ ssH T; eð Þ� �1=m Tð Þ !

_ee�

þMs� sG T; eð Þ

Zþ _eeG T; eð Þ

� �ð29Þ

with _ee� ¼ 1 sec�1. The parameters and functions s0(T, e), sH(T, e), m(T),N(T), and Z have to be determined by curve fitting in the individual regions(1)–(3), whereas the parameters sG and _eeG are determined requiring that thederivative @s=@ _ee follows a continuous function in the transition region:sG ¼ sHð_eeG=_ee�Þm and _eeG ¼ ðmsH=ZÞ1=ð1�mÞ. The values of the parameterused are given in Ref. [28].

An exception of the rule of the reduction of flow stress with increasingtemperature is the influence of dynamic strain hardening observed in ferriticsteel at temperatures between 2008C and 4008C, where the flow stressincreases towards a local maximum. It is caused by the interaction betweenmoving dislocations and diffusing interstitial atoms. The additional stresscan be described by Ds ¼ að_eeÞ exp½�fðT � bð_eeÞÞ=cð_eeÞg2�, With this addi-tional term, the dependence of flow stress of steel Ck45 (AISI 1045) on tem-perature and strain rate is determined [28] and represented in Fig. 6.


Figure 5 True stress vs. true strain rate at different temperatures [27]. Markers:experimental results, curves: calculations according to Eq. (29).


Figure 6 Influence of temperature and strain rate on the flow stress of unalloyedsteel with 0.45% C. (From Ref. 27.)


2. Adiabatic Softening

Flow curves determined in the range of high strain rates are almost adiabatic,since the deformation time is too short to allow heat transfer. The majorpart of the deformation energy is transformed to heat while the rest isconsumed by the material to cover the increase to internal energy dueto dislocation multiplication and metallurgical changes. On strain increaseby de, the temperature increases according to

dT ¼ k0:9

rcs de ð30aÞ

where the factor 0.9 is the fraction of the deformation work transformed toheat, s is the current value of the flow stress which is already influenced bythe previous temperature rise and k is the fraction of energy remaining in thedeformation zone. At low strain rate, there is enough time for heat transfersout of the deformation zone and the temperature increase is negligible. Inthis case, k ¼ 0. On the other hand, the deformation process is almost adia-batic at high strain rate and k ¼ 1. A continuous transition from the isother-mal deformation under quasi-static loading to the adiabatic behavior underdynamic loading can be achieved considering k as a function of strain rate inthe form.

Figure 7 Quasi-static and adiabatic flow curves of unalloyed fine grained steel.


kð_eeÞ ¼ 1

3þ 4

3parctan

_ee_eead� 1

� �ð30bÞ

The transition strain rate _eead depends on the thermal properties of thematerial. If the temperature of the surroundings is the room temperature, _eeadis around 10þ1 sec�1.

As the flow stress usually decreases with increasing temperature, theflow curve shows a maximum (Fig. 7). A thermally induced mechanicalinstability can take place leading to a concentration of deformation, alocalization of heat and even to the formation of shear bands.

An overview of different criteria for the thermally induced mechanicalinstability is presented in Ref. [29]. The adiabatic flow curve can be deter-mined numerically for an arbitrary function sðe; _ee;TÞ for the shear stresswhich has been determined in isothermal deformation tests. In order toobtain a closed-form analytical solution demonstrating the adiabatic flowbehavior, the simple stress–temperature relation s ¼ sisoðe; _eeÞCðDTÞ canbe used [30,31]. In this case, the change of temperature can simply be deter-mined by separation of variables and integration. For example,

s ¼ sisoðe; _eeÞ 1� mT� T0

Tm

� �; s ¼ siso exp � 0:9km

rcTm

Zsiso de

� �ð31Þ

s¼ sisoðe; _eeÞ exp �bT�T0

Tm

� �; s¼ siso 1þ 0:9kb

rcTm

Zsiso de

� ��1ð32Þ

Tm is the absolute melting point of the material, �rr and�cc are the mean valuesof density and specific heat in the temperature range considered. Aroundroom temperature, the product rc lies between 2 and 4MPa=K for mostof the materials. For a rough approximation, it can be assumed that(rcTm=0.9) � 3Tm in MPa using Tm in K.

Many experimental investigations e.g. Ref. [32] were carried out inorder to determine the temperature dependence of the flow stress. Up to ahomologous temperature of 0.6, the stress–temperature relation can bedescribed better by Eq. (35) than by Eq. (34), showing values of b between1 and 4. Therefore, only Eq. (35) will be considered in the following discus-sion. If the isothermal stress can be simply described by

siso � KenFð_eeÞ ð33Þ

the flow stress, determined in an adiabatic test with constant _ee, is thengiven by


sad ¼ KenFð_eeÞ 1þ kað1þ nÞTm

Ke1þnFð_eeÞ� ��1

ð34Þ

where a ¼ 0:9b=ð�rr�ccÞ. The parameter a can be considered as approximatelyconstant represented by its mean value over the deformation process whichis of the order of magnitude of 1K=MPa. The flow curve shows a maximumsmax at the critical strain ec, where

ec ¼ nð1þ nÞTm

k aKFð_eeÞ� �1=ð1þnÞ

; smax ¼ KFð_eeÞ1þ n

nð1þ nÞTm

k aKFð_eeÞ� �n=ð1þnÞ

ð35Þ

and the parameters K and a can be estimated by

K ¼ 1þ n

Fð_eeÞsmax

enc; k a ¼ nTm

smaxecð36Þ

the remaining unknown parameter n can be determined by fitting the curvethe adiabatic flow curve [12].

Similar to the process of neck formation in a tensile specimen, the exis-tence of a stress maximum leads to mechanical instability. Especially afterreaching the stress maximum, a great part of the specimen is unloaded elas-tically causing further deformation localization. In dynamic torsion tests,the deformation localization leads to a heat concentration and hence ahigher local temperature rise and a high shear strain concentration. Coffeyand Armstrong [33] introduced a global temperature localization factorwhich is the ratio of the plastic zone volume to the total specimen volume.The influence of inhomogeneity on the strain distribution has been demon-strated by using a simple model [34] which represents the torsion specimenby two slices, a reference one and another slice with slight deviations instrength or dimensions. Furthermore, the deformation localization couldbe traced during the torsion test by observing the deformation of grid lineson the specimen surface by means of high-speed photography [35,36].

The influence of adiabatic softening can be illustrated in the case ofcompression test at high strain rates.

Due to friction between the cylindrical specimen and the loadingtools, a compression specimen becomes a barrel form during the test. Inan etched cross-section of a quasi-statically tested specimen, two conicalzones of restricted deformations can often be recognized after quasi-staticupsetting. The deformed geometry is symmetrical about the midplane(Fig. 8). An FE-simulation is carried out for a compression test with _ee¼0.001 sec�1 considering stain hardening according to Eq. (8) and frictionat the upper and lower surfaces by a coefficient m ¼ 0.1. The computa-tional results indicate that the maximum values of equivalent stress as well


as that of the equivalent strain lie in the center of the cylindrical compres-sion specimen (Fig. 9).

In the case of dynamic compression with a strain rate of _ee ¼5000 sec�1, only one compression cone exists (Fig. 10). This is due to theinfluence of the mass inertia forces, which cannot be neglected at such highstrain rates. The loads proceed in the form of mechanical waves propagatingthrough the material. The stress at the upper surface which is impacted by ahammer is much lower than the stress at lower part of the specimen due towave superposition after reflection from the lower surface which contactsthe fixed tool. In the FE simulation, the mass inertia forces have to be con-sidered. The deformation process is adiabatic. The local temperatureincrease is computed according to Eqs. (30a) and (30b) and its influenceon the local flow stress is considered according to Eq. (32). The computationverifies the non-symmetry of the stress and strain distributions (Fig. 11). Inthe regions of high strain values, the critical strain is exceeded and the stressdecreases with increasing strain. Therefore, low stress values are determinedin these regions.

3. Extrapolation to Very High Strain Rates

A large deformation concentration is expected in piercing and cutting pro-cess. The shear process can be dealt with as localized plastic deformationprocess. In the deformation zone, very high strains and high strain ratesarise. The experimental tests are carried out using the so-called hat speci-mens, which is loaded by a compression force at its the upper surface

Figure 8 Etched longitudinal section of a cylindrical compression specimen of steel9SMnPb36 loaded quasi-statically (_ee ¼ 0:001 sec�1).


Figure 9 Quasi-static compression test on cylindrical specimens with _ee ¼0:001 sec�1. (a) Etched section of a SMnPb-steel. (b) Distribution of Mises’ stress.(c) Distribution of the equivalent plastic.


(Fig. 12a). The combination of experiment and finite-element simulationallows examining the possibility of extrapolation of materials laws to therange of very high strains and strain rates [28]. In addition, valuable infor-mation can be obtained for the optimization of the width B in shear

Figure 10 Etched longitudinal section of a cylindrical compression specimen ofArmco iron loaded dynamically (_ee ¼ 5000 sec�1).

Figure 11 Distribution of Mises’ stress and equivalent plastic strain in a

compression specimen after dynamic test with _ee ¼ 5000 sec�1, friction coefficientm ¼ 0.1.


processes of different materials. The specimens are tested dynamically usinga split Hopkinson bar arrangement with a mean test velocity of 34m=sec.The force as well as the displacements are recorded as functions of time.

The process is simulated using a commercial explicit code taking massinertia forces into consideration. Coulomb friction is considered with a coef-ficient of m ¼ 0.1. As loading conditions, the experimentally determined

Figure 12 Hat-shaped specimen [37] and network of the shear zone [28].


Figure 13 Distribution of the equivalent stress and the equivalent stress in theshear zone of a hat-shaped specimen [28]. (a) Equivalent strain. (b) Equivalent stress

in MPa according to von Mises.


displacement–time function of the upper surface is applied to upper thenodes. In order to reduce the total number of elements, the lower die is idea-lized using the so-called infinite elements.

The material law is determined in compression tests at differenttemperatures with strain rates up to 7500 sec�1. As discussed above, onecan assume a linear viscous behavior according to s ¼ shðeÞ þ Z_ee, when_ee > 2000 sec�1 and the damping mechanism dominates. The simulationshould examine the accuracy of the reproduction of the force–displacementcurves determined experimentally for this geometry.

The distributions of the von Mises equivalent stress and equivalentstrain, represented in Fig. 13, show a great non-uniformity. High strain con-centrations exist at the two diagonally opposite corners of the deformationzone. In these regions, the strain rate is so high that the influence of adiabaticsoftening is more than compensated and high stress values are determinedthere.

Examples for the force–displacement curves determined experimen-tally and computed by the FEM are shown in Fig. 14 for different valuesof the shear zone width B. The deviation of the computed curves from theexperimental ones is relatively small. Therefore, it can be assumed thatthe material law determined in the range of high strain rates _ee > 2000 sec�1

Figure 14 Force–displacement curves of the hat specimen. Markers: experimentalresults, curves: FE simulation. (From Ref. 28.)


can be extrapolated to much higher strain rates assuming the dominance ofthe viscous damping mechanism according to Eq. (24). This result is consis-tent with the experimental results of Sakino and Shiori (Fig. 4a).

Such material laws allow the simulation of different metal forming aswell as metal cutting processes. They can be validated by high-speed metalcutting tests [38]. Structural damage during high rate tensile deformationcan be accounted for by introducing a damage function [39].

II. CYCLIC DEFORMATION BEHAVIOR

A. Phenomenological Approach

If a specimen is extended with a constant strain rate _ee0, the stress increasesfirst according to Hooke’s law of elasticity till the elastic limit is reached.Then, a plastic deformation begins accompanied with a non-linear harden-ing. After reaching an arbitrary total strain e1tot ¼ e1el þ e1pl, the strain ratechanges to (�_ee0 ). At first, the material is unloaded elastically and is thencompressed until a total strain of (�e1tot ), as represented in Fig. 15.

It can be clearly observed that the plastic compression begins at anelastic limit R

=e , whose absolute value much smaller than the initial value

Figure 15 Stress–strain diagram of an experiment with a change of the loadingdirection.


Re. In general, it can be stated that a previous tensile deformation reducesthe compression elastic limit. Also, a compression deformation reducesthe subsequent elastic limit under tension. This phenomenon,known asBauschinger effect, is characteristic for the behavior of the material undercyclic loading in the low cycle fatigue range. With further cyclic loading,the stress range increases usually due to strain hardening (Fig. 16). If thematerial is highly pre-deformed or hardened, a cyclic softening takes placeand the stress range decreases with increasing number of cycles.

The rate of change of the stress range Ds decreases with the number ofcycles and approaches a stationary value and the hysteresis loop remainsunchanged.

The strain hardening phenomena under cyclic loading can be classifiedin two terms:

(a) Isotropic deformation resistance sF, that includes the yield pointas well as the isotropic change of the flow stress. It increases (ordecreases) monotonically with the number of cycles, dependingupon the specific plastic deformation work

Rs de, or on the

accumulated plastic strainP

dej j. Its variation with strain can beconsidered as a result of the increase of the density of immobile

Figure 16 Stress–strain diagram of an austenitic steel under cyclic loading with aconstant range of the total strain of De ¼ 0:0066 at 6508C.


dislocation with an additional influence of the changing micro-scopic residual stress state.

(b) Kinematic hardening or internal back stress si that depends onthe direction of the deformation and the loading history andaccounts for the Bauschinger effect. It may result from thereversible interactions of mobile dislocations with obstacles, suchas in the cases of pile ups or bowing between particles.

Figure 17 shows the influence of these stress parts in the biaxial caseon the form of the Mises-ellipse. The isotropic hardening leads to an equalincrease of the ellipse in all directions, while the kinematic hardening shiftsthe ellipse in the loading direction. Usually, both of these hardening typesare to be expected during plastic deformation.

During uniaxial cyclic deformation, the influence of the strain rate canbe considered in two ways:

s� sið Þ2¼ s2F Fð_eeplÞ ð37aÞ

s� sið Þ2 ¼ sF þCð_eeplÞ� 2 ð37bÞ

If the direction of the strain rate, and hence its sign, is suddenly chan-ged, the sign of the isotropic material resistance sF changes at once. In con-trast, the value of internal back stress si changes gradually with increasingdeformation approaching asymptotically a stationary value sis with thesame sign as the strain rate.

In the first cycle, the stress equals sF0 þ si0, at the beginning of the plas-tic deformation where si0 is approximately equal to 0 for annealed materials.With increasing strain, both of sF and si increase approaching the stationary

Figure 17 Mises’ ellipse after: (a) isotropic hardening, (b) kinematic hardening,

and (c) mixed mode hardening.


values sF1 and sis. On reaching the maximum strain of Detot=2, the maxi-mum stress is given by smax1 ¼ sF1 þ si1. If the loading direction is changedfrom tension to compression, the isotropic material resistance changes from(þsF1) to (�sF1) at once. The material is first unloaded and the stress dropsby the amount of sF1. With further reduction of length, plastic compressionstart when the stress is reduced by 2sF1. During this short time, the internalback stress si remains unchanged at the value si1. After the beginning of plas-tic compression, it starts to decrease gradually approaching a new stationaryvalue ð�sisÞ that corresponds to the new strain rate of (�_ee ).

Each time when the strain rate changes from (þ_ee) to (�_ee) in an arbi-trary cycle, the stress drops during the elastic deformation by DsF ¼ 2sFand then gradually by Dsi during the plastic deformation of the half cycle.On the next reverse (�_ee) to (þ_ee), a stress decreases first by DsF and then gra-dually by Dsi. The stress ranges is Ds ¼ DsF þ Dsi. The maximum and theminimum stress smax ¼ sF þ Dsi=2 and smin ¼ �sF � Dsi=2.

The range DsF is defined as the difference between the maximum stressand the elastic limit in the subsequent compression phase. Especially in cyc-lic deformation, it is rather difficult to exactly determine the elastic limit,i.e., the transition point between the elastic and plastic deformation rangesbecause this transition is almost gradual. However, this point can be easilyestimated if the hysteresis loop (Fig. 18a) is differentiated and ds=detot isrepresented as a function of s (Fig. 18b). Starting at minimum stress (point1), the slope of the curve remains approximately constant and equals themodulus of elasticity till point 2 is reached. Then the slope decreases to a

Figure 18 (a) Hysteresis loop, and (b) the derivative with respect to the total strainas a function of the stress.


relatively low value at point 3 of the maximum stress. The stress–straincurve rotates to point 4 of maximum strain. The slope decreases to �1,changes to þ1 and decreases again to the value of the modulus of elasticity.This value should remain unchanged till point 5, where plastic compressionbegins and the slope decreases again till reaching point 6. Between point 6(minimum stress) and point 1 (minimum strain), the value of slope changesto þ1 then to þ1 and decreases to the value of the modus of elasticity. Therelation between ds=detot and s can be linearized in the plastic rangesbetween the points 2 and 3 as well as between points 5 and 6. The intersec-tion of these linear relations with the elastic relation ds=detot ¼ E defines thevalue of range DsF of the isotropic material resistance. The range of theinternal back stress is the defined by Dsi ¼ Dsþ DsF:

Repeating the procedure represented in Fig. 18 for the different cycles,one obtains Fig. 19a. For each half cycle, the value of DsF and the linearrelation between ds=de and s can be determined.

The isotropic component sF as well varies monotonically and continu-ously with increasing number of cycles. However, it can be assumed that itsvalue is constant within arbitrary half cycles and changes only at the begin-ning of the next one, if the total number of cycles is great enough. In this case

dside¼ ds

deð38Þ

within each half cycle. The linear relation

dsidetot

¼ E 1� si � si0sis � si0

� �ð39Þ

can be written for the internal back stress as well. The stationary value sisvaries with the number of cycles.

The isotropic material resistance sF as well as the stationary value sisof the internal back stress si are represented in Fig. 19b as functions of theaccumulated strain.

These relations may be described by

sF ¼ sF0 þ sF1 � sF0ð Þ 1� exp �CFeaccð Þ½ � ð40Þsis ¼ sis0 þ sis1 � sis0ð Þ 1� exp �Cieaccð Þ½ � ð41Þ

yielding the evolution equations

dsFdeacc

¼ CF sFs � sFð Þ ð42Þ

dsisdeacc

¼ Ci sis1 � sisð Þ ð43Þ


Within each half cycle, si approaches the stationary value sis as shownin Fig. 20a. The transition range from the elastic to the plastic deformationranges is fairly good described by Eq. (39). On the other hand, relativelylarge deviations are observed at low values of dsi=detot, which correspondsto high stress and plastic strain values. A more accurate description is

Figure 19 (a) Description of the derivative ds=detot by a linear function of stress inthe transition range between the ranges of elastic and plastic deformation. (b) sF andsis as function of the accumulated plastic strain.


achieved by considering two internal variables for the internal back stressinstead of only one as considered here [40].

With depl ¼ detot � dsi=E, the derivative of the internal back stresswith respect to the plastic strain can be obtained from Eq. (39) as

dsidepl¼ E

sis � si0si � si0

� 1

� �ð44Þ

A comparison of this relation with the experimental results is repre-sented in Fig. 20b.

B. Constitutive Equation of Cyclic Behavior

In contrast to these experimental facts, serious simplifications are usuallymade to reduce the number of the material parameters involved incomputation. The hyperbolic function of Eq. (8) is simply linearizedyielding

dsidepl¼ C� gsi ð45Þ

Figure 20 Relation between stress derivative and internal back stress si. (a)Derivative with respect to the total stress. (b) Derivative with respect to plastic strain.


where C and g are material constants. The stationary value sis is consideredconstant assuming that the material follows the Masing-rule and Eq. (5) isreduced to

sis ¼ C=g ð46Þ

Lemaitre and Chaboche [41,42] introduced a non-linear isotropic=kinematic hardening model, which provides predictions that are near tothe experimental evidence. This model is applicable for isotropic incom-pressible materials. The yield surface is defined by the function

F ¼ fðsij � XijÞ � s0 ¼ 0 ð47Þ

where s0 is the yield stress that is equivalent to the isotropic material resis-tance sF and Xij is the tensor of the internal back stress denoted sis in theuniaxial case. The function f ðsij � XijÞ equals the equivalent Mises stresswhen the back stress X is taken into consideration:

f ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3

2ðs0ij � X0ijÞðs0ij � X0ijÞ

rð48Þ

where s0ij is the deviatoric stress tensor and is the X 0ij deviatoric part of theback stress tensor. The associated plastic flow is given by

_eeplij ¼@F

@sij�_ee_eepl ¼ 3

2

ðs0ij � X0ijÞf

�_ee_eepl ð49Þ

where _eepl represents the rate of plastic flow and _�ee�eepl is the equivalent plasticstrain rate

�_ee_eepl ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi2

3_eeplij _ee

plij

rð50Þ

The size of the elastic range, s0, is a function of the equivalent plasticstrain �eepl and the temperature. For a constant temperature, it is written simi-lar to Eq. (40) as

s0 ¼ s 0 þQ1 1� expð�b�eeplÞ� �� ð51Þ

where s 0j is the yield surface size at zero plastic strain, and Q1 and b areadditional material parameters that must be determined from cyclicexperiment.


The evolution of the kinematic component of the model, when tem-perature and field variable are neglected, is defined as

_XXij ¼ 2

3C_eeplij � Xij

�_ee_eepl ¼ Cs0ij � X0ij

s0� gXij

� ��_ee_eepl ð52Þ

where C and g are material parameters.

C. Application to Life Assessment

The assessment of the fatigue life under cyclic elasto-plastic deformationrequires an accurate determination of the strain ranges of the individualloading cycles in the region of maximum local deformation. For this reason,FE simulation is often needed especially when the analytical solutions arenot available or when they include unacceptable simplifications.

For example, the fatigue life of notched machine parts is often predictedusing approximation formulas [43–45] that have been driven using the Neu-ber rule [46]. However, the accuracy of these methods remains lower than thatof the inelastic FE analysis, when adequate materials lows are implemented.

Figure 21a illustrates the distribution of the axial stress sxx in anotched 3-point bending specimen. The mesh is built of three-dimensionalcontinuum solid elements. Around the notch, a refined mesh is chosen sothat the critical zone at notch root would cover several elements. The mate-rial considered is the AlZnMgCu alloy AA7075. The material parametersdetermined in uniaxial cyclic experiments are: sj0¼ 310MPa, Q ¼ 75MPa,b ¼ 36.6, C ¼ 14,844MPa and g ¼ 86.3. As a loading condition, a line loadwith a total compressive force F is applied at the midlength of the upper sur-face. The force follows a sinusoidal time function with Fmin=Fmax ¼ 0:1. Thespecimen is supported at two parallel lines on the lower surface with a spanwidth of 80mm. The computational results of the local strain components atnotch root are represented in Fig. 21b as functions of time.

For an arbitrary loading cycle, the time functions eijðtÞ are used todetermine an equivalent strain range D�ee for the cycle according to differentapproaches [47–52].

With an additional damage accumulation rule, a representative peri-odically repeated strain range can be computed which should lead to thesame fatigue life.

Experimentally, fatigue life is determined as the number of cycles, atwhich a technically detectable crack is initiated. Often, a direct currentpotential drop system [53,54] is used to determine the potential drop acrossthe notch as time function. This is then converted to a relation betweencrack length and number of cycles to initiate cracks.


Figure 21 (a) Distribution of the bending stress sxx in a 100� 20� 10mm3

specimen with a notch radius or 6mm. (b) FE computation results for the variationof local strain components at notch root.


The equivalent strains, determined according to Ref. [48] for differentnotch radii vs. the number cycles to fracture are plotted in Fig. 22.

The experimental results obtained for alternating tension–compressionloading on smooth bars are also represented in Fig. 22 and are described bythe usual function Detot ¼ Depl þ Deel with Depl according to the Manson–Coffin equation and Deel following a similar one:

Detot ¼ aN�bi þ cN�di ð53ÞThe results of the notched bending specimens, obtained by the combi-

nation of FE analysis, strength hypothesis, and experimental life determina-tion, lie in a scatter band around the uniaxial data.

D. LCF of Metal Matrix Composite Materials

The fatigue life of metal matrix fiber composites is found to be stronglyreduced in the range of low cycle fatigue due to the formation of kink bands,at which fatigue cracks initiate.

A cross-section of one of these materials is represented in Fig. 23a.This composite consists of a pure copper matrix and continuous fibers of

Figure 22 Representative strain range as function of the number of cycle to failure.

(From Ref. 54.)



the austenitic steel X5CrNi18-12. It is used mainly in the electrical industryas a contact material. The mean fiber diameter dm ¼ 200 mm and the volumefraction equals 40%.

Under alternating elasto-plastic strains, the slim fibers buckle plasti-cally within the softer matrix during the compression phase of the loadcycle and then expand to the straight form in the tension phase. How-ever, if the material is unloaded and the elastic part of the deformationdiminishes, the fibers do not remain completely straight. This promoteslateral deflection in the next compression cycle and the fiber bucklingincreases from one cycle to the other. The fiber takes an S-shape inthe first few cycles. With increasing number of cycles, bending localiza-

Figure 23 Continued

Figure 23 Copper reinforced by austenitic steel fibers. (a) Etched cross-section. (b)Initiation of fatigue cracks. (c) Crack growth.

"


tion takes place at two points of each fiber, and a kink band is observed,which is inclined to the load direction [59–59]. Within the kink band, thematrix suffers high cyclic shear deformation, which may cause crackinitiation. If the fibers are brittle, fiber fracture occurs at the kink bandboundaries.

Figure 23b shows the initiation fatigue cracks in a specimen subjectedto alternating low cycle fatigue loading. The specimens were tested straincontrolled with an alternating total strain value by a strain rate_ee ¼ 0:0017 sec�1 [60].

A two-dimensional idealization is chosen in order to get a clear ideaabout the deformation process even when the results are only of a qualita-tive character. The material is supposed to consist of plain layers of cupperand austenitic steel. To simulate buckling, an imperfection must beintroduced to allow for mechanical instability. This is done by bringing inan inclination with a small angle b (for example 28) which may representa deviation between the load and the fiber directions resulting from anon-accuracy of the specimen geometry or a non-alignment of the testingmachine axis.

The computed distribution of the equivalent stress is shown in Fig. 24afor a small elastic tensile strain of 0.001. The volume fraction of the hardermaterial component equals 40%. Due to the difference in the modulus ofelasticity of the material components, higher stresses arise in the elementsof the stiffer materials and they appear brighter in the plot. The gradientof the stress in the lateral (horizontal) direction is related to bending causedby stretching of the inclined network. Fig. 24b shows the stress distributionafter reaching the maximum strain of 0.024 in the first cycle. The displace-ments are exaggerated in the plot. Both the material components areplastically deformed. The harder material appears in the mean brighter thanthe softer one.

Fourteen cycles later, the mesh and the stress distribution look com-pletely different, as shown in Fig. 24c and d for the time instants of reachingthe maximum compressive strain of the 14th cycle and the maximum tensilestrain of the next cycle. The originally softer materials show higher isotropichardening due to the greater amount of accumulated strain. An inclinedkink band can be easily recognized with high stresses in the matrix.

The successive fiber buckling with increasing number of cycles can bemore obviously observed under pulsating compressive stresses. A character-istic phenomenon is the initiation of an inclined shear band accompanied bya reduction in the specimen deformation resistance, deformation localiza-tion in the matrix, and hence a reduction in the fatigue life.

Figure 25 shows a comparison between the 2D-FEM results for thedeformed mesh with the fiber configuration in longitudinal section of


Figure 24 The von Mises equivalent stress distribution in the deformed mesh. Thedarker the element, the lower the stress. Displacements are exaggerated. (a) Under

partial elastic loading. (b) At the maximum elasto-plastic tensile strain of the firstcycle. (c) At maximum compression strain of the 14th cycle. (d) At the maximumtensile strain of the 14th cycle.


specimens that have been subjected to the same number of cycles in pul-sating compression LCF-tests. In both the experiment and simulation, thespecimens are subjected to a compressive stress pulsating periodicallybetween �450MPa and –45MPa. The influence of friction at the topand the bottom of the specimen is not taken into consideration in thesimulation. The fibers are continuous with a volume fraction of 20%.The fibers do not appear as continuous dark lines in the optical photo-graphs because they do not necessarily lie completely in the plane of theprepared section.

The similarity of both configurations allows a better understanding ofthe failure mechanism as far as the history of deflections, local deforma-tions, residual stresses, multiaxiality, and similar values can be followedup through out the test simulation.

Figure 25 FEM-plots and etched longitudinal sections of cylindrical specimenssubjected to pulsating compressive loads up to: (a) 135 cycles, (b) 157 cycles, and (c)

284 cycles.


III. CREEP BEHAVIOR

In contrast to plasticity, a long-time high-temperature creep exposure causesa continuous change in the constitution of the materials. Beside hardeningby the increasing dislocation density, several microstructural events takeplace such as initiation of subgrains, precipitation, ripening, and coagula-tion of particles, oxidation, high-temperature corrosion or even phase trans-formation. The creep strain is accompanied by a slowly increasing damageprocess that covers a great fraction of the creep life.

Constitutive equations based on a combination of overstress concept[62,62] and threshold stress concept [5,63] allow an adequate descriptionof the materials behavior if successive damage is taken into consideration.The current value of the strain rate depends on the current values effectivestress, which is the difference between the applied stress s and the internalback stress si, the particle deformation resistance sp, the material creepresistance sF and the degree of damage D. For long-time creep under lowstresses, the creep rate can be represented by

_ee ¼ Cs� si

sF 1�Dð Þ� �n

sgnðs� siÞ ð54Þ

where in case of high-temperature creep

C ¼ _ee0 exp � Q

RT

� �ð55Þ

This relation is applicable for true stress and true strain rate. If thecreep tests are carried out with a constant force, the applied engineeringstress value is to be multiplied by the factor ð1þ eÞ in order to accountfor the increase of true stress due to reduction of area. The engineeringstrain rate is to be divided by the same factor.

In many cases of the modeling of high-temperature material behavior,it is highly recommended to use the same set of equation to describe time-dependent creep and the time-independent plastic behavior. As discussedabove, the low cycle fatigue can be described by

s� sisF 1�Dð Þ ¼ 1 ð56Þ

With decreasing strain rate, a transition takes place towards a time-dependent creep behavior described by Eq. (54), which can be rewritten inthe form.


s� sisF 1�Dð Þ ¼

_eeC

� ��1=nð57Þ

A continuous transition function may be given by

s� sisF 1�Dð Þ ¼

_ee_ee0

� ��m�=nþ1

!�1=m�ð58Þ

as represented in Fig. 26.In contrast to low cycle fatigue behavior, with its relative short life, the

parameter sF is considered in the case of long-time creep exposure as a con-stant reference stress and can be set equal to 1MPa. In this case, only theoverstress concept is considered.

The kinematic hardening may include several components [64]:sd þ sS þ sp. The first term sd accounts for the variation of the dislocationdensity. An additional material resistance sS that accounts for subgrain for-mation can be taken into consideration, considering the material as a com-posite consisting of hard subgrain boundaries and soft subgrain interior [65].The particle stress sp accounts for the interaction between mobile disloca-

Figure 26 Transition between the ranges of time-dependent (creep) and time-independent (plasticity) ranges. (From Ref. 40.)


tions and precipitates. It depends on the mean distance between the particleswhich may change in the course of creep exposure (Figs. 27).

For engineering construction and life assessment, it is required toreduce the number of parameters to the amount that is essential for thedescription of the mechanical behavior and that can be determined with atolerable experimental effort. Therefore, most engineering materials maybe described well using si ¼ sd þ sS with a unique function of strain. Theparticle is then treated separately and eq. (54) is rewritten as

_ee ¼ Cs� si � spsF 1�Dð Þ� �n

sgnðs� siÞ ð59Þ

A. Damage Function

Different formulations can be applied for the damage function D. Kachanovand Rabotnov [67,68] introduced the relation

dD

dt¼ a

ð1�DÞp ; D ¼ 1� 1� t

tf

� �1=ðrþ1Þð60aÞ

where tf is the fracture time (Fig. 28). Other applicable functions are

dD

dt¼ bDm; D ¼ ðt=tfÞM ð60bÞ

dD

dt¼ cþ gD; D ¼ expðgtÞ � expðgt0Þ

expðgtfÞ � expðgt0Þ ð60cÞ

As the damage increases very rapidly with time in the late tertiarystage, a more accurate description can be achieved by formulating damageas a function of strain instead of the time function used above. In this case, amodified Kachanov and Rabotnov relation:

D ¼ 1� ½1� ðe=efÞ�1=ðmþ1Þ ð61aÞor further function such as

D ¼ ðe�efÞ1�ð1�nÞ ð61bÞ

D ¼ ½expðbe=efÞ � 1��½expðbÞ � 1� ð61cÞcan be applied.


Figure 27 Main factors affecting internal back stress. (a) Dislocations. (b)Subgrain boundary. (c) Particles. (From Ref. 66.)


B. Internal Back Stress

If the applied stress is reduced from s0 to sR at a time point t0, the straindecreases directly by ðs0 � sRÞ=E to the strain value e0, and starts to increaseor to decrease depending on the value of the reduced stress sR. After a tran-sition time, the steady state is reached again and the strain increases withnormal strain rate expected for the stress sR. The relation between the sub-sequent creep strain e� e0 and the time t� t0 after the partial unloading isshown in Fig. 29. There is a certain value of the reduced stress sR at whichthe creep rate is reduced to 0 immediately after load reduction. This value ofsR is equal to the current value of the internal back stress si according toEq. (54). For materials with negligible particle resistance, it is equal to theinternal back stress.

If the particle stress sP and the additional substructure resistance sScan be neglected, the internal back stress si is only influenced by the evolu-tion of the dislocation density and can be set equal to sd. In this case, theevolution of the internal back stress in the course of creep exposure underconstant or varying stresses and temperatures can be formulated [70]. Evenif the assumptions mentioned do not exactly hold, such a formulation canhelp in determining the creep behavior as a phenomenological description.

Figure 27 Continued


Figure 28 Creep damage. (a) Crack and voids initiated in the tertiary range. (b) Description of damage as a function of creep

strain.


Similar to Eq. (9), the internal stress component related to dislocationdensity is defined as sd ¼ aGb

ffiffiffirp

. With the evolution Eq. (10) of the dislo-cation density [5], the variation of the internal back stress is described by therelation

dside¼ C1

e1sis � sið Þ ð62Þ

which is validated experimentally, e.g. in Ref. [69]. In this equation,sis is thequasi-stationary value of the internal back stress and e1 is the correspondingcreep strain. Under constant stress and temperature, the internal back stressincreases in the primary stage according to

sisis¼ 1� exp �C1

ee1

� �ð63Þ

with e1 as the strain at the end of the primary creep stage. This relationshipis shown in Fig. 30 for different materials stresses and temperatures.

The quasi-stationary value sis depends on the applied stress (Fig. 29).With increasing creep stress, sis increases approaching a saturation valuesiss. The experimental data for the secondary creep rate are usually welldescribed by the Norton–Bailey relation _ees ¼ AsN [71]. On the other hand,_ees should follow the relation _ees ¼ C s� sisð Þn. It can be shown that

Figure 29 Creep strain after partial unloading as a function of time. (From

Ref. 69.)


s=siss < N=ðN� nÞ; sississ¼ s

siss1� n

N

N� n

N

ssiss

� �N=n�1" #ð64aÞ

s=siss � N=ðN� nÞ; sississ¼ 1 ð64bÞ

as shown in Fig. 31.In these equations, N is the usual Norton–Bailey stress exponent

N ¼ @ ln _ee=@ ln s. The parameter n depends on temperature. It can beset equal to 4 in the temperature range of the practical service conditionsof high-temperature engineering materials. The saturation value sissdepends on the temperature and can be described by the exponentialfunction:

sissðTÞ ¼ k0 expðb=TÞ ð65Þwhere k0 and b are material constants. As a rough approximation, sisscan be considered as proportional to the creep strength Rm=1000 for afracture time of 1000 hr and set equal to about 1:4Rm=1000.

Figure 30 Development of the internal back stress in the primary creep stage.(From Ref. 70.)


After a sudden increase of the applied stress, the internal back stressstarts to increase gradually approaching the quasi-stationary value (Fig 32a).If the applied stress is then reduced to the original value, a gradual reductionof the internal back stress towards the corresponding quasi-stationary valueis observed (Fig. 32b).

As the strain rate depends on the difference between the applied loadand the internal back stress, a load enlargement leads to a very high strainrate that reduces gradually to normal value. In the same way, a load dropcauses a severe strain rate reduction, even to negative strain rate values whens declined to a value lower than si.

Figure 33a shows the creep rate curve of the austenitic 18=11 Cr–Ni-steel under cyclic creep loading. The stress is changed periodically between150 and 125MPa. The period is equal to 96 hr. The influence of fatigue canbe neglected and the material behavior can be described as a pure cycliccreep.

Under cyclic stress, the influence of the creep strain transients is foundto reduce the creep life. Compared with life values calculated by the lineardamage accumulation rule

Figure 31 Relation between the quasi-stationary internal back stress and theapplied stress.


Figure 32 Variation of the internal back stress after a sudden change of the applied

stress. (a) Load increase. (b) Load reduction. (From Ref. 70.)

Figure 33 Description of strain transients under pulsating stress by the overstress

model. (From Ref. 70.)


X t

tf¼ L ð66Þ

the value of L decreases to about 0.6 in case of cyclic stress at constant tem-perature and about 0.8 in the case of pulsating temperature under constantstress [72].

C. Influence of Particles

1. Behavior of Dispersion-Strengthened Materials

Dispersion-strengthened materials are usually produced by powder metal-lurgical techniques, especially mechanical alloying. They include very fineoxide or carbide particles embedded within the grains. The particles obstaclethe dislocation motion and increase the resistance to deformation. Thestrength depends mainly on the size and the volume fraction of the disper-soids as well as on the consolidation process and the matrix material [73,74].Contrary to precipitation hardening, the dispersoids are thermally stableand do not ripen or coagulate during long-time high-temperature exposure[75,76]. Therefore, such materials are predestined for applications underhigh-temperature creep conditions. Their behavior is studied under tensileand compressive loads, e.g. in Refs. [77–79]. Fig. 34a shows the creep ratecurves of the Aluminum alloy AlSi20 which was produced from its powderwithout any additions by cold pressing and hot extrusion. This materialincludes an oxygen content of less than 0.5mass%. The correspondingcurves in Fig. 34b are determined for a dispersion-strengthened versionAlSi20C1O2 produced by mechanical alloying. Carbon powder is addedto the matrix powder and the mixture undergoes intensive milling beforecold pressing and hot extrusion. The material includes a volume fractionof 4% of Al2O3 and 4% of Al4C3 as dispersoids with a mean particle sizeof 150 nm.

The creep rate can be described by _ee ¼ f ðs� spÞ where sp is the addi-tional resistance to deformation caused by the particles (Fig. 35a).

The following simple relation can be used for the estimation of theminimum creep rate

_ees ¼ Cs� sp

E

h iNexpð�Q=RTÞ ð67Þ

where E is the modulus of elasticity at the creep temperature, N is the Nor-ton–Bailey stress exponent of the matrix material, Q is the activation energyfor self-diffusion of the base element of the matrix. Fig. 35b shows that thecreep strength is increased by a constant value which depends only on thevolume fraction of the particles and their morphology.


Figure 34 Creep rate curves of Aluminum Alloy AlSi20 [78]: (a) without, and(b) with dispersoids.


Figure 35 Influence of dispersoids on creep stress [78], for arbitrary values of:

(a) minimum creep rate, and (b) creep life.


The particle resistance may be estimated by sp � sO where sO is theOrowan stress given by

sO ¼ 0:84MG

4pð1� nÞ2b

L� dln

L� d

2b

� �ð68Þ

In this equation, M is the Tailor factor, G is the shear modulusdepending on temperature, L is the mean distance between particles, andb is the Burger vector.

A precise description of the experimental data in the range of lowcreep stresses and high fracture times allows a model introduced by Reppichet al. [80]. According to this model, the additional particle resistance spdepends not only on the particle morphology but also on the applied stress.In the range of high creep stresses, sp approaches an upper limit s�p, whichcan be set equal to the Orowan stress sO. With decreasing creep stress, thedislocation can overcome the particle resistance by partial climb, and theparticle resistance is assumed to be directly proportional to the appliedstress. Fig. 36 shows the relation between the relative particle resistancesp=s�p as a function of the normalized creep stress s=s�p. A continuous

Figure 36 Increase of additional particle resistance with increasing creep stress.(From Ref. 79.)


transition from the range of low stresses to the range of higher once may bedescribed by a function of the type

sps�p¼ 1� exp � s

s�p

" #m !" #1=mð69Þ

The upper limit of the particle resistance can be set equal to the Oro-wan stress: s�p � sO.

2. Precipitation Hardening

The high-temperature creep behavior of precipitation hardenable industrialalloys is influenced by the kinetics of the precipitation and the ripening pro-cesses. Creep specimens are found to exhibit a longer creep life after solutiontreatment compared to those ones additionally aged before testing [81]. Thisphenomenon is attributed to the precipitation of fine particles during theearly stages of creep [82], which strengthen the material and reduce the creeprate (Fig. 37). With increasing the creep time, the particle coarsening leadsto an increase of the interparticle spacing and to an acceleration of creepstrain rate.

Figure 37 Precipitates and dislocations in Alloy 800HT. (From Ref. 66.)


The increase of the size of an existing precipitate can be consideredaccording to the Oswald-ripening mechanism. However, the precipitationprocess under creep loads is rather complex and is expected to be dependentnot only on the temperature but also on the change in the degree of super-saturation of the matrix as well as on the defect structure and dislocationdensity. Therefore, the rate of precipitation is assumed to depend on creepconditions.

The particles strengthen the material by exerting an internal, or thresh-old, stress sp on the moving dislocations. According to the theory of Brownand Ham, reported by Martin [83], it is assumed that the high-temperatureyield stress of particle-hardened material is controlled by local climb of dis-locations over the particles

sp ¼ S

Lð70Þ

where S is a material constant and L is the planar interparticle spacing.From geometrical considerations, it can be shown that the planar interpar-ticle spacing L varies with Vp and d according to the relationshipL ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

p=ð6VpÞp

d. The particle stress can be rewritten as

sp ¼ffiffiffi6

p

rS

ffiffiffiffiffiffiVp

pd

ð71Þ

An analytical model [84] that considers the influence of a continuousprecipitation process accompanied by particle coarsening on the creep beha-vior of metallic materials will be discussed briefly in the following lines. Thevolume of particles per unit volume V p that precipitate out of the supersa-turated matrix can be written as

Vp ¼ Npp6d3 ð72Þ

where Np is the number of nuclei or the number of carbide particles per unitvolume and d is the average particle diameter. The number of growing nucleiper unit volume, Np, that form during precipitation out of a supersaturatedsolid solution is known to depend on the degree of supersaturation, the tem-perature, and the defect structure, especially the dislocation density. At agiven temperature, the value of Np can be written as

Np ¼ Np0 þ fðtÞgðtÞ ð73Þwhere Np0 is the initial number of nuclei and can be neglected in case ofsolution annealed materials, f(t) is a function of the temperature and thedefect structure, g(t) is a function of the remaining supersaturation, which


decreases during the precipitation process. During the primary creepstage, the dislocation density depends on the creep strain and f(t) canbe written as

fðtÞ ¼ f½a1f1ðTÞ þ a2ðsn2e�Q2=RTÞM2 �tgM ð74Þwhere a1 , a2, n2, M2, and M are constants, s, t, and T are stress, time,and temperature in Kelvin, respectively, Q2 is the creep activation energyand is close to the activation energy of self-diffusion. The first part of f(t)including f1(T) represents the nucleation under static conditions when nostress is acting on the material while the second part represents the effectof the applied creep stress.

As precipitation progresses, the supersaturation of the matrixdecreases. The function g(t) can be represented as a hyperbolic functionof time

gðtÞ ¼ 1þ ðBtÞn½ ��m ð75Þwhere n and m are parameters that control the rate of precipitation, B is afunction of stress and temperature. The increase in particle diameter d withtime due to Oswald ripening can be represented by the function [85]

d ¼ dm0 þ Ct� 1=m� ðCtÞ1=m ð76Þ

where d0 is the original particle diameter and may be neglected, m is a con-stant whose value depends on the particle growth mechanisms and C is afunction of temperature given by C ¼ C0 expð�Q1=RTÞ, where Q1 is theactivation energy of the coarsening process, e.g. for carbon in iron. Basedon the experimental results, the first part of the function f(t) in Eq. (74) isexpected to be small compared with the second part so that it may beneglected for creep loading. With this approximation, the volume fractionVp can be written as

Vp ¼ ðp=6Þ½a2ðsn2e�Q2=RTÞM2 �MC3=m tMþ3=m

1þ ðBtÞn½ � m ð77Þ

With increasing time, Vp approaches asymptotically a final value Vp1

that depends only on the initial supersaturation, but not on a function oftime or stress, M þ ð3=mÞ must be equal to mn and the volume fractionVp can be written as

Vp ¼ Vp1ðBtÞn

1þ ðBtÞn� �m

ð78Þ


As M must be positive, mn > 3=m. Taking m ¼ 3, mn > 1. Based onthe above assumptions, the parameter B can be expressed as a function ofstress and temperature as follows:

B ¼ B0sna exp � Qa

RT

� �ð79Þ

where Qa ¼ ð3Q1=mþM2MQ2Þ=ðmnÞ is an apparent activation energy with-out a specific physical interpretation and na ¼ n2M2M=ðmnÞ. SubstitutingEqs. (76) and [78] in eq. (71), sp can be written as

sp ¼ S

C1=m

ffiffiffiffiffiffiffiffiffiffiffi6

pVp1

rBtð Þn

1þ Btð Þn� �m=2

t�1=m ð80Þ

The condition for maximum strengthening due to precipitation,_ee ¼ _eemin, is achieved when sp reaches a maximum value,spmax, after timet�, thus

t� ¼ nmm� 2

2

� �1=n1

Bð81Þ

Figure 38 Time to minimum creep rate of Alloy 800HT for different creep stressesand temperatures. (From Ref. 84.)


Figure 39 (a) Experimental data (markers) and (b) model results (curves) for thecreep rate of Alloy 800HT as a function of time or creep strain. (From Ref. 66.)


Figure 40 Creep rate curves of AA2024 in two heat treatment conditions. (a) Highly coarsened precipitates. (b) Solutionannealed. (From Ref. 86.)


spmax ¼ S

C1=m

ffiffiffiffiffiffiffiffiffiffiffi6

pVp1

r1þ 2

mmn � 2

� ��m=2

t��1=m ð82Þ

Considering the stress dependence of the parameter B, the followingrelationships are determined:

t� ¼ A0s�naeQa=RT ð83Þ

This relation is found to fit well experimental results for precipitationhardening austenitic steel (Fig. 38) and Aluminum AA2024.

The variation of sp with creep time is given by the relationship

spspmax

¼nmm2

ðnmm=2Þ � 1þ ðt=t�Þ�n� �m=2

t

t� ��1=m

ð84Þ

Figure 39 shows a comparison between experimental results andmodel results for Alloy 800HT. Similar results, obtained for the AluminumAlloy AA2024, are presented in Fig. 40. In order to determine the matrixbehavior, the material is over-aged for a long span of time until the precipi-tates coagulate. The particle diameter and the distance between them are soincreased that the particle stress can be neglected. This matrix behavior isdescribed by Eq. (54). For the solution annealed condition, the influenceof the particle stress is considered using Eqs. (59) and [84].

For practical applications, some of the parameters can be set equal tocertain values. Assuming that ripening takes place on the basis of volumediffusion, the parameter m is to be set equal to 3. Furthermore, the variationof the volume fraction of the precipitates with time, according to Eq. (78),can be well described with m ¼ 2 as far as the parameter n can be determinedby best fit of the experimental results (Fig. 41). With these values, the par-ticle stress can be given by

spspmax

¼ 3n3n � 1þ ðt=t�Þ�n

t

t� ��1=3

ð85Þ

D. Simulation of Creep Microcrack Growth

Under high-temperature creep exposure, failure takes place due to the initia-tion and growth of inter-crystalline cavities, in the form of voids or inter-crystalline microcracks. In the case of high temperatures, low stresses andsmall grain size of the material, the inter-crystalline void initiation, growthand coalescence is the dominant damage mechanism. Several well-founded


models were introduced to estimate the rate of void growth. At lower stres-ses, the voids grow by diffusion of atoms out of the void surface into thegrain boundary. At higher stresses, the void growth takes place by a creepdeformation of the surrounding material [87–94].

On the other hand, higher creep stresses can be allowed by loweringtemperature or by increasing grain size. In this case, wedge type inter-crys-talline microcracks are observed. A successive damage is caused by thegrowth of the existing cracks as well as by the continuous initiation ofnew ones. Most of these microcracks are extended over only few grainboundary facets (Fig. 42).

In contrast to the long technical cracks, whose growth rate can be cal-culated by the methods of the non-linear fracture mechanics, there are only afew investigations about the initiation and growth of wedge type inter-crys-talline creep microcracks. The micrograph of Fig. 42 shows a representativeexample for the form of the wedge type inter-crystalline creep microcracks.Few typical shapes and a preferred orientation can be identified. The micro-cracks start usually at grain triple points and extend along such grain bound-aries, that are nearly perpendicular to the direction of the applied stress. Allthe different crack shapes can be explained by the fact that the initiation andgrowth of the cracks result from the grain boundary sliding.

Figure 41 Variation of the particle stress with time according to Eq. 85.


Figure 42 (a) Microcracks in a longitudinal section of a creep specimen of steelX6CrNi18-11 after creep fracture (s ¼80MPa, T ¼ 7008C). (b) The role of grainboundary sliding in crack initiation.


Figure 42 Continued


The estimation of the rate of growth of such microcracks can beachieved mainly by two methods: (a) the determination of the statistical dis-tribution of the microcrack length and its variation with the creep time, and(b) finite-element simulation of the creep behavior regarding the material asa composite consisting of grains separated by thin grain boundary layer withdifferent properties.

1. Statistical Model

The fundamental concept of the statistical model is that a small fraction ofshort cracks and a high fraction of long cracks are expected when thegrowth rate is high, and vice versa [95–98]. In order to achieve reliableresults using statistics, a great number of cracks have to be classified. Overa period of several years, about 50,000 cracks were classified in the steel X6CrNi18-11 and more than 60,000 cracks in the steel X8CrNiMoNb16-16 fordifferent temperatures and stresses.

Based on the results of metallographic investigations, the followingassumptions are introduced: (a) A crack grows quickly along the grainboundary from one triple point to the next, where it rests for a longer timebefore it grows again to the next triple point, (b) The crack length is alwaysan integral multiple n of grain boundary facets and (c) every crack isinitiated in the length class n ¼ 1 and grows step by step to next higherlength classes.

Let Z be the total number of cracks per unit area and Yn the number ofcracks having a length n. In a time unit, Vn,n þ 1 cracks grow out of thelength class n into the next higher length class (n þ 1). In the same time,Vn � 1,n cracks grow from the lower length class (n � 1) into the consideredclass n. The mean rate of growth is given by dn=dt ¼ Vn;nþ1=Yn and the rateof change of dYn=dt ¼ Vn;nþ1 � Vn�1;n. As all cracks initiate with the lengthn ¼ 1, the rate V0;1 represents the rate of crack initiation and must be equalto rate dZ=dt of the increase of the total crack number. Therefore, followingrelation can be deduced:

Vn;nþ1 ¼ dZ

dt�Xni¼1

dYi

dtð86Þ

Denoting the fraction of cracks with the length of n by Xn, so thatdYn ¼ XndZ þ ZdXn, the mean rate of growth ðdn=dtÞn of the cracks ofthe length class n can be written as

dn

dt

� �n

¼ 1�Pni¼1 Xi

Xn

� �1

Z

dZ

dt

� �� 1

Xn

Xni¼1

dXi

dt

!¼ FðnÞGðtÞ �Hðn; tÞ

ð87Þ


The first term denoted F(n) is determined by the statistical distribution ofthe microcrack length. This distribution can be described by

Xn ¼ ð1� qÞqn�1 ð88Þas represented in Fig. 43 for two different austenitic steels. Deviations aremainly observed in the range of long microcracks and low population. Thesedeviations can be avoided by adding a second term including a Leibnitzseries, but it will be neglected here.

The function FðnÞ is then given by

F ¼ q

1� qð89Þ

The parameter q depends on stress, temperature, and the constitutionof the material, but not on crack length. Therefore, approximately no influ-ence of the crack length on the growth rate arises from this term.

The function G, which is the relative rate of crack initiation, dependson the material and the creep conditions. In order to determine this func-

Figure 43 Statistical distribution of the inter-crystalline creep microcracks.


tion, several creep tests are to be carried out until different stages of damagein tertiary creep stage are reached. Using the results of metallographic inves-tigations and digital image analyzing systems, the function Z(t) is found tobe described adequately by the Kachanov–Rabotnov-relation given ineq. (60a), as well as by the empirical relation.

ðZ=ZfÞ ¼ exp½�gðtf � tÞ=tf� ð90Þwhere the index f indicates the value at fracture (Fig. 44). According to thisrelation

G ¼ g

tfð91Þ

with g depending onmaterial constitution, stress, and temperature. The quan-tity G remains constant during the creep test, as long as Eq. (90) isvalid. The functionH is determined by the rate of change of the statistical dis-tribution (Fig. 45) and can be written as HðtÞ ¼ �n _qq=ð1� qÞ, where q(t) isdescribedbyapower lawaccording toq ¼ qf ðt=tf Þm.Hence,Hcanbewrittenas

H ¼ nmt

q

1þ qð92Þ

Figure 44 Number of cracks Z per unit area, related to its value at fracture as afunction of the life fraction.


The rate of growth reads

dn

dt¼ g

tf

qfðt=tfÞm1� qfðt=tfÞm 1þ m

gt=tfn

� �ð93Þ

Just before fracture (t � tf ), the creep crack growth is

dn

dt¼ g

tf

qf1� qf

1þ mgn

� �ð94Þ

The model indicates that the ratio between the growth rates of long and shortcracks is smaller than that expected by applying the fracture mechanicsconcept.

2. FEM Simulation

The grain boundary can be considered as a thin amorphous layer surround-ing the grains. Its thickness equals few atomic distances. At low tempera-tures, the grain boundary is harder than the crystalline grains. Withincreasing temperature, the grain boundary becomes more softer than the

Figure 45 Decrease of the fraction of short microcracks (n ¼ 1) with increasingcreep exposure time.


Figure 46 Idealization of the grain=grain boundary combination.

Figure 47 (a) Regular and (b) randomly modified idealization.


grains. At higher temperatures, the grain boundary behaves as a viscouslayer with much higher strain rate sensitivity than the grains. In the FEManalysis, two different material elements are used for the idealization ofgrains and grain boundaries with different material parameters (98), asshown in Fig. 46.

The creep behavior of the grain interior and the grain boundary layersis described by the Norton–Bailey creep law

_ee ¼ Cðs=s�ÞN ð95Þ

with s� just equal to the stress unit. The parameters C and N are set approxi-mately equal to the values determined for the entire in the secondary creepstage, neglecting the influence of the grain boundaries in this stage due totheir small volume fraction.

The grain boundary zone can be considered as a linear viscous New-ton solid. Its stress exponent is set equal to unity as first approximation. Asuitable thickness and the parameter C of the grain boundary layers aredetermined iteratively. Their values are varied till the fracture time com-puted for different creep stresses coincides with the experimentallydetermined values.

In order to avoid all grain boundaries having the same orientationfracture simultaneously, the size of each individual element in the networkis stochastically changed by adding random values to the grain nodecoordinates (Fig. 47). The network determined in this manner has to be con-sidered as a quarter of the idealized body and to be symmetrically mirrored,so that no additional anisotropy is induced. The whole network can also berotated by an angle between 0 and 608 to exclude preferred orientations forcrack initiation.

Two different crack initiation criteria are tried out: a strain criterionand a stress one. According to the strain criterion, a crack initiates as soonas the equivalent creep strain reaches a critical value. In this case, the grainboundary element is not totally eliminated but its thickness is reduced by afactor of 1=1000. Such a weakened element behaves during further deforma-tion like a crack. The second criterion which is based on the maximum prin-ciple stress or the maximum shear stress instead of the equivalent strain isfound to be non-applicable because the experimentally determined stress-lifefunction could not be achieved with this criterion.

With increasing extension of the whole mesh under constant loadforces, the crack opening criterion is fulfilled first at a single grain boundaryfacet. The next crack opens at a different grain far from the first crack, butat a place where the orientation of the grain boundary facet is favorable.After the initiation of several individual cracks having a length of one grain


boundary facet, the total creep extension of the mesh is high enough toinduce crack growth along the neighboring facets which are steeply inclinedto the load direction. In this way, cracks of length class n ¼ 2 initiate at dif-ferent locations. With further growth, the individual cracks start to coales-cence resulting in a great additional extension of the mesh. Fracture isconsidered to take place as soon as the total creep extension of the meshreaches a predefined value, and the computation is stopped.

Figure 48 presents the ratio of the number of cracks Z to that of cracksat fracture Zf as a function of relative strain e=ef determined by the finite-element simulation and by the creep experiment. The comparison showsthat most of the data from the finite-element simulation lie in the same scat-ter band as those of the experimental investigation.

Figure 49 shows that the fraction X1 of short cracks having a length ofone grain boundary facet slightly increases with increasing nominal stressesas determined in experiments and by the finite-element simulation.

With these results, the main reason for crack initiation and growthseems to be the relatively high local strains, and not the local stress, inthe neighborhood of the grain boundaries. Metallographic investigationconfirms the existence of such deformations in the neighborhood of the

Figure 48 Comparison between experimental data and computational results forthe increase of the number of cracks with increasing creep strain.


grain boundaries. The same procedure can be applied to estimate creepdamage and fracture time of the technical geometries. An example is repre-sented for the simulation of the creep behavior of a notched specimen [99].Figure 50 shows the deformed mesh as well as a metallographic section.

IV. BEHAVIOR UNDER IMPACT LOADING

During plastic deformation at high strain rates, such as in the case of forgingor automobile crash conditions, the mechanical behavior of metallic materi-als is influenced by mass inertia forces, an increased strain rate sensitivity,and the adiabatic character of the deformation process. An accuratesimulation of the material behavior needs an adequate consideration ofthe mechanical wave propagation, steeping and reflection phenomena[100]. Under impact loading, stress waves propagate through out the mate-rial and the strain distribution is time dependent and highly non-uniform.

Figure 51 shows a long rod with a free front surface at x ¼ 0. The farend of the rod is fixed. The cross-sectional area is constant over the wholelength and is denoted as A. If a force F is applied quasi-statically on the free

Figure 49 Influence of creep stress on the fraction of short microcracks at fracture.


front surface, one can assume that the stress F=A induced is uniformly dis-tributed over the whole rod. On the other hand, if the rod is impacted, e.g.by a hammer at the front surface, the mass inertia forces cannot beneglected. The rod front is pushed forward by a velocity v. An arbitrarycross-section at a distance x from the free end dose not start immediatelyto move with the same velocity, before all masses between the front surfaceand the cross-section considered have been accelerated to the velocity v. This

Figure 50 Creep microcrack initiation and growth in a notched specimen.


needs a certain time interval Dt. The longer the distance x, the longer thetime interval. This explains why displacements, strains, and stresses propa-gate throughout the material in the form of mechanical waves with the char-acteristic wave properties, such as reflection at surfaces.

At an arbitrary time point, the cross-section at the distance x is dis-placed by u. At a neighboring cross-section x þ dx, the displacement isuþ ð@u=@xÞdx. The strain in the material element dx is given by e ¼@u=@x. The forces acting on the element are �As and A½sþ ð@s=@xÞ dx�.The mass inertia force is rA dxð@2u=@x2Þ. Therefore,

Að@s=@xÞ dx ¼ rA dxð@2u=@x2Þ ð96ÞIn the case of elastic behavior,

@s@x¼ E

@s@x¼ E

@2u

@x2ð97Þ

and the following differential equation is obtained for the local displace-ment:

@2u

@t2¼ E

r@2u

@x2ð98Þ

Any function fðx� ctÞ or fðxþ ctÞ fulfills this condition, when

c ¼ffiffiffiffiE

r

sð99Þ

A certain value of the displacement u� ¼ fðx0 � ct0Þ that is observedat the distance x0 at the time point t0 arises at the distance x0 þ Dx after

Figure 51 Material element in an impacted bar.


the time interval Dt, yielding fðx� ctÞ ¼ f½xþ Dx� cðtþ DtÞ� and hence,Dx ¼ cDt. Therefore, c is the propagation velocity of the longitudinal wave.If the load is applied in the lateral direction or if the load is a torsionmoment, a transversal wave is induced that propagates with a velocity ofcT ¼

ffiffiffiffiffiffiffiffiffiG=r

p, where G is the shear modulus. When plastic deformation takes

place, the modulus of elasticity E and the shear modulus G are to bereplaced by the tangent modules

c ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi@s=@e

r

s; cT ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffi@t=@gr

sð100Þ

While these equations are essential for analytical modeling, they have not tobe necessarily externally considered in the numerical simulation when ade-quate computation codes are used. These codes must account for the massof the material, for example, by considering point masses lumped at thenodes of the finite elements. Beside the FEM, the finite difference methodand the method of characteristics are often applied.

A. Non-uniformity of Strain Distribution

If a tensile specimen is chosen too long or the impact energy input is rela-tively low, the local strain at the impacted specimen end is found experimen-tally to be much lower than that measured at the far end of the specimen.Such phenomena can be explained by an FE-simulation using a code fortransient dynamic problems. The loading time function and the idealizationof the impact tensile test arrangement are shown in Fig. 52. The material isconsidered as strain hardening and strain rate sensitive.

Immediately after loading the specimen, an elastic and a plastic wavepropagate along the axial direction of the specimen. The elastic wave ismuch faster than the plastic one. An elastic deformation propagates alongthe specimen to the far specimen head, where the elastic wave reflects. Itruns back towards the near specimen head, where it reflects again. This

Figure 52 Input load time function and idealization of impact tension test.


process is repeated many times during the propagation of the plastic wave,representing an elastic vibration superimposed plastic deformation process.

The plastic wave propagates first throughout the specimen and is thenreflected from the far specimen head. Due to superposition of the advancingand the reflected wave, high stresses and strains are induced at the far end ofthe specimen. If the impact energy is completely consumed by the plasticdeformation, a permanent non-uniform strain distribution remains in thespecimen (Fig. 53).

With increasing impact energy, the plastic wave can run several timesalong the specimen, reflecting at both ends, before the impact energy W iscompletely consumed by the plastic deformation of the material. In thiscase, the strain distribution is approximately uniform over the whole gaugelength (Fig. 54).

B. Fiber Composites Under Dynamic Compression

Under quasi-static compressive loading of a composite material, the slimfibers buckle within the softer matrix leading to a global plastic bendingof the work piece. In order to simulate this behavior, an imperfection is

Figure 53 Variation of the distribution of the plastic strain in a tensile specimen atdifferent time point after dynamic loading.


introduced in form of a small inclination of 28 of the specimen axis, whichshowed that the specimens must buckle in the form observed experimentally(Fig. 55).

Under dynamic loading, the specimens are found to get the usual bar-rel form of compression specimens. However, etched sections show that thefibers have undergone a buckling process. When the mass inertia forces aretaken into consideration by regarding the material mass to be lumped at thenodes, the material behavior can first be understood [101,102]. A specimenbuckling needs a lateral motion of the upper and lower contact surfaces inopposite directions. This motion is now obstacled by the inertia forces in theradial direction. The fibers buckle without causing a global bending of thespecimen (Fig. 56).

C. Dynamic Notch Sensitivity

With increasing deformation rate, the strain rate sensitivity increases. Thisacts stabilizing on a tensile deformation process. As soon as neck formationstarts, the local strain rate in the neck zone increases rapidly. The local flow

Figure 54 Computational and experimental results for the local strain distributionalong an impact–tension specimen of steel X6CrNi18-11, tested by different values

W of impact energy.


Figure 55 Fiber buckling under quasi-static loading of copper reinforced by 45%volume fraction of austenitic steel fibers with 0.2mm diameter. (From Ref. 101.)


stress increases as well, so that higher tensile forces are needed for the con-tinuation of extension. Other specimen regions undergo additional deforma-tion, so that the uniform elongation increases with increasing strain ratesensitivity and extension rate. On the other hand, the adiabatic character

Figure 56 Fiber buckling in a composite material under dynamic loading. (From

Ref. 101.)


of the deformation process reduces the flow stress and promotes instability.Mass inertia in the lateral direction arises in connection with radial accelera-tion due to the reduction of area. This causes the initiation of either lateraltensile or lateral compressive stresses depending on the time function of spe-cimen elongation.

In addition to these ductility considerations, an increased notch sensi-tivity is observed under dynamic loading. One of the reasons is that the localfracture strain decreases with increasing strain rate. This will be discussedlater on in this chapter. The other reason lies in the interaction between

Figure 57 Idealization of perforated plates.


Figure 58 Stress distribution around voids at different time points after impact loading: (a) t¼10ms, smax¼598MPa, (b) t¼18 ms,smax¼647MPa, and (c) t¼24ms, smax¼661MPa.


mechanical waves and notches. Figure 57 shows the idealized part of a per-forated plate used in a study of the wave notch interaction [102,103]. Theholes are chosen as circular or elliptical with different axes ratio and orien-tation. Also, the distance between the holes is variable.

The variation of the stress distribution with increasing time, numeri-cally computed, shows the propagation of the mechanical wave throughthe material (Fig. 58). Stress peaks are observed at the notch roots, beforethe maximum loading stress reaches this points. High stress values remainat the peaks, even when the maximum lading stress has passed through.Compared with the notch effect under quasi-static loading, the dynamicnotch effect is characterized by higher stress and strain concentrations,greater strain gradients, lower stress relief by neighboring voids and lowerinfluence of the orientation in the case of elliptical voids.

V. DUCTILE FRACTURE

The ductile fracture results usually from nucleation, growth, and coales-cence of microvoids, that initiate mostly around inclusions. In accordanceto its appearance of the fracture surface, ductile fracture can be classifiedinto two cases [105]. In the case of softer materials, void nucleation at inclu-sions followed by marked void growth with internal necking and shear frac-ture of the intervoid matrix. The fracture surface shows a structuredconfiguration of dimples often orientated perpendicular to the loading direc-tion (Fig. 59). In case of high strength materials, shear fracture takes placewithout distinctive void growth. The matrix fails due to instabilities likeshear bands forming between voids resulting in fracture with nearly nonecking, promoted by low strain hardening material, high stress multi-axiality, and regions of high porosity [106].

A. Failure Criteria

Beside macromechanical empirical failure criteria [107,108], several meso-scopic mechanical models are introduced. The failure criterion is definedby the local failure strain �eefðsm=�ssÞ as a function of the ratio of the localmean stress sm to the equivalent stress �ss.

For the nucleation of microvoids, different models have been deducedconsidering an energy criterion [109–111], critical stress [112–114], or criticalstrain [115–119].

Rice and Tracy [120] deduced a closed-form solution for the rate-of-change of the mean radius of a void, in an ideal plastic material, as afunction of the current value of the radius and of the ratio between the meanstress and the effective stress


Figure 59 SEM micrograph of fracture surface of highly over aged AluminumAA7075 after tensile impact loading at room temperature. (From Ref. 104.)


�ss ¼ const:;1

R

dR

de¼ 0:28 expð3sm=2�ssÞ ð101Þ

Hancock and Mackenzie [121] showed that the failure strain is assumed tobe inversely proportional to the relative cavity growth rate (d lnR=de). Thestrain at fracture can be deduced from the Rice and Tracy criterion and beexpressed as

ef ¼ en þ a exp �3sm=ð2�ssÞ½ � ð102Þ

where en is the effective strain before void nucleation. The Rice and Traceymodel has been used, e.g., in Ref. [122] and was verified by Thomason[123–125] in numerical simulations. Experimental results of Marini et al.[126] showed that the factor 0.28 of Eq. (101) should be replaced by highervalues according to the volume fraction of inclusions. In Ref. [121], the localplastic strain which leads to coalescence of cavities was found to be highlyinfluenced by the volume fraction of inclusions fN. Using special treatmentsfor ferritic steels, different residual sulfur-concentrations were realized byHolland et al. [127] which were found to affect the fracture strain(Fig. 60a). These results were described by the modified relation

ef ¼ en þ a exp �bsm=�ss½ � ð103Þ

where instead of the factor 3=2, a parameter b is introduced with values ran-ging between 5 and 23. The degree of purity had a drastic influence on en,which was affirmed by the investigation of further materials and treatments(Fig. 60b).

Based on the models of McClintock [128] and of Rice and Tracey,Gurson [112] deduced a yield function for materials with randomly distrib-uted voids of a volume fraction f. In this model, the flow rule according toMises is extended by two additional terms including the porosity f. In moredetailed investigations carried out by Tvergaard and Needleman [129–131],the Gurson model is modified yielding a plastic potential in the form

f ¼ 3

2s2YSijSij þ 2q1f

� coshq2skk2sY

� �� 1þ ðq1f �Þ2h i

¼ 0 ð104Þ

In this equation, Sij is the stress deviator given by Sij ¼ sij � dijskk=3 wheredij is the second order unit tensor. sY is the yield stress of the matrix and skkis the sum of the normal stress components. f � is a function of the volumefraction f of the voids according to

f fc; f � ¼ f ð105aÞ


Figure 60 Influence of sulfur content in steel on: (a) the local effective strain atfracture as a function of ratio of mean stress to flow stress, and (b) the fracture strainfor high multiaxiality as a function of the true strain in the neck zone of unnotched

specimens, (Z¼ reduction of area at fracture). (From Ref. 107.)


f > fc; f � ¼ fc þ 1

q1� fc

� �f� fcfF � fc

ð105bÞ

where fc is the volume fraction at the beginning of void coalescence and fF isthe volume fraction at fracture. The rate-of-change _ff of the void volumefraction, is considered as the sum of three different contributions: (1) thegrowth rate of existing voids, which is proportional to (1�f ) and to thelocal strain rate, (2) the nucleation rate of new voids depending on the effec-tive strain rate �_ee_ee in the matrix, and (3) the nucleation rate of new voids whichis proportional to the rate of change of the mean stress sm ¼ dijskk=3. Whenthe third contribution is neglected, the following relation is used for the evo-lution of fv:

_ff ¼ _ffgrowth þ _ffnucleation ¼ ð1� f Þdij _eeplij þ A�_ee_eepl ð106aÞA non-zero value of A is only used if �eepl exceeds its current maximum in thetime increment considered. In this case

A ¼ fNffiffiffiffiffiffi2pp

sNexp � 1

2

�eepl � eNsN

� �� ð106bÞ

where fN is volume fraction of particles thatmay nucleate voids, eN is themeanvalue strain for nucleation, and sN is the corresponding standard deviation.

B. Influence of Strain Hardening andStrain Rate Sensitivity

In an early study on the growth of cavities by plastic deformation of the sur-rounding material, McClintock [128] deduced a closed-form analytical solu-tion for the rate-of-growth of cylindrical cavities of elliptical cross-sectionwith the semi-axes a and b in a strain-hardening material which is

s ¼ Ce�n;1

R

dR

de¼

ffiffiffi3p

2ð1� nÞ sinhffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3ð1� nÞp2

sa þ sb�ss

" #ð107Þ

Where R¼(aþb)=2 is the mean cross-sectional radius and sa and sb are thenormal stresses in the direction of the ellipse axes.

Because of its simplicity, the Hancock–Mackenzie relation is alsoapplied to the range of high strain rates after introducing correction factorsconsidering the influences of strain rate. Carroll and Holt [132] introduced avisco-plastic modification of the Hancock–Mackenzie model. Johnson andCook [133] considered the strain rate sensitivity as well as the influence ofthe temperature on the local fracture strain

ef ¼ D1 þD2 expðD3sm=�ssÞ½ � 1þD4 lnð_ee=_ee0½ � 1þD5T=Tm½ � ð108Þ


with _ee0 ¼ 1 sec�1 and Tm the absolute melting point. As sm=�ss; _ee and Tchange during deformation, it is assumed that fracture takes place when adamage parameter D ¼ S½De=efðsm; _ee;TÞ� reaches the value of 1.

A failure criterion for void growth considering non-linear visco-plasticbehavior of a strain-hardening and rate-sensitive material can be obtainedusing an analytical solution [134]. The void growth is to be determined bymeans of flow stress described by �ss ¼ K�eenð�_ee_ee=_ee�Þm, with equivalent stress�ss, equivalent strain rate _�ee�ee, reference strain rate _ee� ¼ 1 sec�1, equivalent plas-tic strain �ee and the material constants K, n and m. A spherical void of radiusR is considered to exist at the center of a metallic sphere (Fig. 61). At theouter radius L of this hollow sphere, a radial stress component srL is actingwhich is set equal to the mean stress sm ¼ ðs1 þ s2 þ s3Þ=3, which leads to avisco-plastic deformation of the material and hence to an increase in voidvolume. For an arbitrary void radius r, the tangential strain rate is givenby _eet ¼ _rr=r. Under consideration of the plastic volume constancy,

Figure 61 Spherical void growing in a hollow sphere matrix.


�_ee_ee ¼ 2_rr=r. Regarding the continuity condition r2 _rr ¼ R2 _RR, the equivalentstrain rate can be rewritten as _�ee�ee ¼ ðR=rÞ3 _�ee�eeR ¼ ðL=rÞ3 _�ee�eeL, and the correspond-ing equivalent stress �ss ¼ �ssLð�eeR=�eeLÞnð_�ee�eeR=_�ee�eeLÞmðR=rÞ3ðmþnÞ follows from�ss ¼ �ssL for _�ee�ee ¼ _�ee�eeL. In order to determine the distribution of the radial stress,the condition of equilibrium @sr=@r ¼ �2ðsr � stÞ=r is taken into considera-tion. According to the von Mises yield criterion, st � sr ¼ �ss where st is thetangential and sr is the radial stress component. With the boundary condi-tion sr ¼ 0 for r ¼ R, a closed-form analytical solution is deduced for therate of radius increase reading

1

R

dR

d�eeL¼ 1

2

3ðmþ nÞ2ð1� fmþnÞ

ðsrÞL�ssL

� �1=ðmþnÞð109Þ

with f ¼ ðR=LÞ3, which is approximately equal to the volume fraction ofvoids. At the outer radius of the sphere (r ¼ L), the values of sr L, �ssL and_�ee�eeL can be regarded as equal to sm, �ss and _�ee�ee, which are determined for theconstruction element geometry considering the material as continuum. iff51, the failure criterion is given by

�ss ¼ K�eenð�_ee_ee=_ee�Þm; ef ¼ en þ a3ðmþ nÞ

2

sm�ss

� ��1=ðmþnÞð110Þ

In the cases of high temperatures or very high strain rate, this relation can beapplied using n¼0 and m ¼ 1 as a special case

ef � en _ee;Tð Þ þ asm=�ssð Þ ð111Þ

C. Growth of Microcracks

In order to increase the strength of engineering materials, several strength-ening mechanism are adopted. Beside precipitation hardening, the strengthof the matrix is increased by alloying elements. During plastic deformation,microcavities initiate in two different ways. (a) At low temperatures andhigh strain rates delamination takes place at the interface between matrixand particles leading to microcrack formation (Fig. 62a). (b) At higher tem-peratures or lower strain rates, particles fracture causing a microcavity thatelongates with further plastic deformation (Fig. 62b).

In order to consider damage by both cavitation mechanisms, a newmodel is introduced in Refs. [104,135]. In analogy to the Avrami theoryof the kinetics of phase change [136], the following assumptions are madefor the initiation and growth of microcavities. Precipitations and inhomo-geneities embedded in a matrix can be interpreted as active nuclei for void


Figure 62 SEM Micrograph of microcavitations in Aluminum AA7075 T7351 after impact tensile loading ð_ee � 3500 sec�1Þ. (a)Penny-shaped microcracks at room temperature perpendicular to loading direction. (b) Microvoid at 150 8C parallel to loadingdirection. (From Ref. 104.)


and crack initiation. The total number of particles representing possiblenuclei for damage decreases with increasing global strain due to cavitationinitiation at some of them. Around each cavitation, a region of reducedstresses and strains exists (hatched areas in Fig. 63) in which no furthercavitations can initiate. This region is spherical with radius r in case ofpenny-shaped cracks and ellipsoid in case of microvoids which can beapproximated by a cylinder with a constant radius a, which is equivalentto the mean particle diameter and a length of l.

It can be assumed that the number of new cracks initiated per unitstrain is proportional to the number of remaining particles lying outsidethe relieved regions. The size distribution of cracks in impacted speci-mens was determined by Curran et al. [137]. It was found that the lin-ear crack growth rate dr=de is not a function of the current value of theradius r but only proportional to the relative nucleation rate of newsmall cavitations. In the case of penny-shaped microcracks, the sphericalregion of relieved stresses and strain grows spherically with a constantradial rate dr=de. In the case of microvoids, the cavitation radiusremains constant, but its length changes with a constant rate (one-dimensional growth). The degree of damage is proportional to therelieved volume fraction, so that the fraction of damaged area DðeÞreads

DðeÞ ¼ C 1� exp � ee� �k� ��

ð112Þ

Figure 63 Particles (spots) acting as nuclei for cavity initiation. (a) Penny-shapedmicrocracks with spherical regions of reduced stresses (hatched areas) growingspherical. (b) Microvoids with cylindrical regions of reduced stresses growing onedimensional. (From Ref. 104.)


with the material constant C and e�, which is proportional to thatstrain, at which first damage occurs. The exponent k was found to beequal to 4 in the case of microcracks and 2 in the case of microvoids.As an application, this damage model was used to describe the flowcurves of Aluminum AA7075 [138] measured in impact tensile tests atroom temperature (microcracks) and 1508C (microvoids) (Fig. 64).

Figure 64 Experimental (marker) and computational (curves) results at differentmean strain rates de=dt for tensile specimens of Aluminum Alloy AA7075 T7351.

(From Ref. 138.)


D. Starting Point of Ductile Fracture

In order to determine the failure criterion, which is defined by the local fail-ure strain �eefðsm=�ssÞ as a function of the ratio between local mean stress smand local equivalent stress �ss, tensile tests on differently notched specimensmay be carried out. The time functions of specimen elongation measuredexperimentally can be applied as a boundary condition to FE computationsin order to determine the local values of stresses and strains along the nar-rowest cross-section, which is assumed to be critical for fracture initiation(Fig. 65).

As a result, the time-dependent distributions can be determined, as it isshown for two examples in Fig. 66 [135]. The analysis shows that in case ofunnotched or smoothly notched bars, both the maximum equivalent plasticstrain and degree of multiaxiality lie at the specimen axis (radius¼ 0),whereas in case of a sharply notched specimen, the maximum equivalentplastic strain is reached in the notch root, where the degree of multiaxialityshows a minimum. Therefore, it can be stated that, in case of unnotchedbars, the starting point of fracture lies at the specimen axis. On the other

Figure 65 FE Simulation of dynamic tensile test on a notched bar of AluminumAA7075 (explicit code). (From Ref. 139.)


hand, the starting point of fracture in case of notched specimens is found todepend on the notch radius. In bars with sharp notches, cracks are firstinitiated at the notch root. However, with increasing notch radius, thelocus of the crack initiation is shifted to the specimen axis. Therefore, nogeneral statement can be made for notched specimens and at first the start-ing point of fracture is considered to be unknown. Each point in the mini-mum cross-section is regarded to be a potential starting point for cracks.Therefore, the stresses and strains have to be checked along a radius inthe minimum cross-section.

Using the experimentally determined specimen extension at fracture astermination time point for the FE-simulation, the corresponding values ofthe local equivalent plastic strain �eeðrÞ and degree of multiaxialitysmðrÞ=�ssðrÞ are computed for the different notch geometries at the differentGauss-integration points along the radius r of the specimen in the narrowestcross-section D0, as it is shown in Fig. 67 for Aluminum AA7075 highly over

Figure 66 FE simulation results for the distribution of plastic strain andmultiaxiality in the notch cross-section of impact tension tests on specimens ofAluminum Alloy AA7075, highly over-aged. (From Ref. 104.)


aged [138]. The results are represented by a continuous curve for each geo-metry (symbols). Each point of a curve represents a location r along theradius in the smallest cross-section. Only one point of each curve fulfillsthe failure criterion for ductile fracture, so that the envelope of all curvesrepresents the failure criterion. For quasi-static loading (Fig. 67a), the envel-ope is described by the Hancock=Mackenzie relation and for dynamic load-ing (Fig. 67b) by Eq. (111).

The comparison between the failure criterions determined for quasi-static and dynamic loading is represented in Fig. 68b in addition to that

Figure 67 Local equivalent plastic strain at fracture as a function of the degree of

multiaxiality ðsm=svÞ along the specimen radius at the narrowest cross-section ofdifferently notched specimen of aluminum AA7075 highly over aged under (a) quasi-static, and (b) dynamic loading. (From Ref. 138.)


determined for the T7351 heat treatment condition (Fig. 68a). Under quasi-static loading, the local effective plastic strain for a given degree of multiaxi-ality is higher than in the case of dynamic loading.

E. Transition to Brittle Fracture

With strain rate and multiaxiality increasing, the local stress peaks becomeso high that they can reach the microscopic cleavage fracture strength s�f ofthe material. Brittle fracture is expected, when the local value of the maxi-mum principal stress s1 exceeds s�f over a characteristic distance xc whichdepends on the microstructure of the material [140]. The transient tempera-ture Tt from ductile to brittle fracture is shifted to higher values due to theincrease of the maximum normal stress and can reach the current local tem-perature during the deformation process causing transition to brittle frac-ture (Fig. 69).

The influence of the multiaxiality M ¼ sm=�ss on the maximum normalstress can be demonstrated by the simple case of proportional stresses withtwo equal principal stresses: sII ¼ sIII ¼ asI. With the mean stresssm ¼ ð1þ 2aÞsI=3 and the effective stress �ss ¼ ð1� aÞsI; the maximum nor-mal stress follows by eliminating a:

sII ¼ sIII ¼ asI; sI ¼ 2

3þ sm

�ss

�� ssðT;�ee;�_ee_eeÞ ð113Þ

Figure 68 Failure criteria for Aluminum Alloy AA7075 in the (a) T7351, and (b)highly over aged condition under quasi-static and dynamic loading (elongation rate

v ¼ 18m=sec) at room temperature. (From Ref. 138.)


Figure 69 The transition temperature shift due to an increase in multiaxialityM ¼ sm=�ss, prestrain e and rate of elongation.

Figure 70 Influence of deformation rate and strength on the transitiontemperature shift [141]. (a) J-integral-temperature curves for steel 15NiCuMoNb5.(b) Transition temperature shift as a function of yield strength. (c) Measured andcalculated values for the transition temperature as a function of the machine ram

velocity v.

"



If this is the case, the brittle fracture condition is simply assumed to bes�f � sI ¼ 0. The microscopic cleavage strength s�f can be considered as pro-portional to the modulus of elasticity EðTÞ. The transition temperature Tt

from brittle to ductile fracture can be determined by the intersection ofthe functions s�f ðTÞ and �ssðTÞ for given values of multiaxiality M, prestraine, and strain rate _ee. A variation of these parameters results in a shift of thetransition temperature which is determined by

DT ¼ ð@sI=@MÞDMþ ð@sI=@eÞDeþ ð@sI=@ _eeÞD_eeðds�f =dTÞ � ð@sI=@TÞð114Þ

where ds�f =dT � dE=dT . However, this equation seems to overestimate thetransition temperature shift. An alternative procedure for the determinationof the effect of the loading rate on the transition temperature, whichdescribes the experimental results more accurately, was introduced by Falkand Dahl [141]. This procedure needs the knowledge of a single value Tt1 forthe transition temperature at a known loading rate as well as the relationbetween flow stress, strain rate, and temperature determined, e.g. in tensiontests. According to their analysis, the transition temperature for anotherstrain rate is determined by the intersection of the function mðTÞ andTt1 þ DT=ð@m=@TÞT¼0 where m ¼ @ ln s=@ ln _ee is the strain rate sensitivity.According to this method, the transition temperature shift can be expressedby

DT ¼ @m

@T

� �T¼0� @m

@T

� �� 1 @m

@ ln _eeD ln _ee ð115Þ

Figure 70 shows experimental results determined and their description byEq. (115).

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5Tribology and the Design ofSurface-Engineered Materialsfor Cutting Tool Applications

German Fox-Rabinovich and George C. WeatherlyMcMaster University, Hamilton, Ontario, Canada

Anatoli KovalevPhysical Metallurgy Institute, Moscow, Russia

I. INTRODUCTION

The materials for cutting tools have traditionally been chosen for their excel-lent hardness and wear resistance under the extreme service conditions (highstresses and temperatures) associated with a high-speed machining opera-tion. The interaction between the tool and workpiece was once thought tolie wholly in the domain of the mechanical or physical response of the sys-tem to these conditions, and little attention was paid to the role of chemicalinteractions between the tool and the machined part. Recent research haschallenged this viewpoint, and it is now realized that extensive physical aswell as chemical interactions can occur, both between the tool and the work-piece, as well as with the surrounding atmosphere. These considerationshave become of even greater importance with the current drive to highermachining rates, often under conditions where a lubricant is not or cannotbe used, that marks modern manufacturing trends.

The theme of this chapter is the role played by physico-chemicalinteractions in modifying and controlling the friction and wear of thetribo-couple (i.e., the critically loaded surfaces of the cutting tool and theworkpiece) during high-speed cutting operations. The chapter is dividedinto three sections. In the first section, the characteristic features of


friction and the role of ‘‘self-organizing systems’’ in helping to control thewear processes are described. A ‘‘self-organizing system’’ is one thatresponds to the external mechanical, thermal, and chemical forces witha positive feedback loop that leads to an improvement in the wear char-acteristics of the couple. Two types of stable secondary structures formedat the surface of the tool have been identified in ‘‘self-organizing sys-tems’’. They are usually oxide films, either highly plastic or refractoryand less plastic that form under machining conditions by reaction ofthe tool material with oxygen.

The second section develops these ideas of self-organization for somecommon tool materials, and shows how they can be understood andexploited for alloys such as high-speed tool steels (HSS), cemented car-bides, and cermets. A deep level of understanding of the complex inter-actions that lead to the formation of stable secondary structures has comefrom the use of techniques such as Auger spectroscopy and electron energyloss spectroscopy, which have been extensively used to study the wear cra-ters formed during machining. These studies, when coupled to more con-ventional wear and friction experiments, clearly demonstrate the positiverole of secondary structures in reducing the wear rate in the initial (run-in) phase of wear. In addition, the formation of secondary structures isshown to prolong the steady-state wear regime, with positive benefits onthe overall life of the cutting tool.

In the third section, a number of recent trends to enhance the perfor-mance of cutting tools are discussed. These include the use of monolithic ormulti-layered coatings, substrate modification, surface-engineered tools,and multi-layered self-lubricating coatings. Throughout the discussion, therole of secondary structures is highlighted, and the concept of a ‘‘smart’’coating that can respond to the cutting environment (with a positive feed-back) is proposed. Finally we propose that any future development ofimproved cutting tools will depend on a better understanding of the natureof the secondary structures. Examples are given as to how these improve-ments might be exploited for high-speed machining operations.

II. TRIBOLOGICAL ASPECTS OF METAL CUTTING

A. Cutting Tool Wear Mechanisms

Metal cutting is associated with mechanical and thermal processes thatinvolve intensive plastic deformation of the workpiece ahead of the tooltip, and severe frictional conditions at the interfaces of the tool, chip, andthe workpiece. Most of the work of plastic deformation and friction is


converted into heat. In cutting, about 80% of this heat leaves with the chip,but the other 20% remains, increasing the temperature of the tool. The pres-sure at the nominal contact area during cutting is approximately103MN=m2, i.e., high pressure dominates, and extreme local temperatures(up to 13008C) can be encountered, especially during the high-speed cuttingof steels [1]. The surface of the tool continuously comes into contact withvirgin chip=workpiece material which has been unaffected by the environ-ment, e.g. by oxidation. The freshly produced chips may interact chemicallywith the tool material. The contact friction behavior at the tool–workpieceinterface has been shown to be adhesion related. The current understandingof this aspect of cutting is focused on the friction of a clean (in a physico-chemical sense) and oxidized surface of a tool and a workpiece through theiradhesive interaction [2]. At high cutting speeds (typical of modern opera-tions), the mechanical and physical conditions at the tool–metal interfaceare far removed from the ‘‘classical friction’’ situation.

These severe service conditions can lead to a very intensive, surfacedamaging wear of cutting tools. Several mechanisms of tool wear have beenidentified, namely adhesive wear, crater wear on the top rake face of thecutting tool due to chemical instability, including diffusion and dissolution,and abrasive wear. Some authors have also discussed electrochemical andde-lamination wear of HSS tools. Adhesive wear is caused by the formationof welded asperity junctions between the chip and the tool face. The sub-sequent fracture of the junctions by shear leads to microscopic fragmentsof the tool material being torn out and adhering to the chip or the work-piece. This kind of wear may occur at the flank face in low-speed cuttingwhen the contact temperatures are low [3]. Abrasive wear is caused by hardparticles of carbides or oxides in the work material or by highly strain-hardened fragments detached from the unstable built-up edge of the tool.

Wear due to chemical instability is very important in high-speedmetal cutting because of the high temperatures prevailing at the contactingsurfaces; this includes both diffusion wear and solution wear. Diffusionwear is characterized by material loss due to the diffusion of atoms ofthe tool material into the workpiece. Solution wear describes the wearmechanism that takes place when the wear rate is controlled by the disso-lution rate rather than by convective transport. An oxidation reaction withthe environment can produce a scaling of the cutting edges [1]. As shownin Fig. 1, for a given workpiece material, adhesive wear is found mainly atlow cutting temperatures, corresponding to low cutting speeds. Wear dueto chemical instability, including effects such as diffusion and oxidation,appears at high cutting speeds. Abrasive wear occurs under all cuttingconditions. This type of wear is important for low-speed machining of castiron and steels, but is less important for cutting conditions where wear due


to chemical instability prevails. However, this mechanism of wear is veryimportant for high-speed machining when oxidation of the workpiece sur-face leads to the formation of oxide particles which can be detached fromthe surface of the tool.

B. Tribological Compatibility and the Developmentof Materials for Surface-Engineered Cutting Tools

The increasing demands of modern manufacturing technologies for high-speed machining have spawned the development of new advanced materialsfor use as cutting tools, possessing a high level of productivity and wearresistance. The development of new cutting tool materials can be consideredto be a typical problem of engineering optimization. In this process, an inte-grated engineering and physical approach is taken to the problem of devel-oping novel wear resistant materials. The key concept, and one that formsthe basis for the arguments to be developed in this review, is focused onthe issue of tribological compatibility of two surfaces.

Tribological compatibility is related to the capacity of the two surfacesto adapt to each other during friction, providing wear stability withoutdamage to the two components for the longest (or given) period of time.

Figure 1 Schematic diagram of tool wear mechanisms appearing at differentcutting temperatures corresponding to the parameters of cutting. (From Ref. 15.)


In our case, the problem of compatibility is reduced to the development ofmaterials for the ‘‘cutting tool–workpiece tribopair’’. The goal is to achievestable tool service and a predictable rate of tool wear with a given set of cut-ting parameters. In this interpretation, compatibility implies an integratedoptimization, both from an engineering (minimal wear rate) and physical(self-organizing) point of view.

An understanding of compatibility follows from the concepts of whatwe might call ‘‘state-of-art’’ tribology. Modern tribology, an interdisciplin-ary science based on mechanics, physics, chemistry, materials science, metal-lurgy, etc., is a very complex subject. Models that generalize knowledgein this area of science and might be acceptable for engineering applicationsare critically needed. Since friction is a process of transformation and dissi-pation of mechanical energy into other kinds of energy, an energy-basedapproach is the most effective one from our point of view. In this reviewchapter, the feasibility of using an irreversible thermodynamic approachto the problem of the design of cutting tool materials is explored. The con-cepts of tribology can be used in the development of a simple algorithm forthe development of novel materials. The solution can be implemented usingthe following steps:

(1) an assessment of the friction surface, including the analysis of self-organizing phenomena at the surface of the tribopair

(2) the development of an optimal engineering solution (in our case—thedevelopment of an advanced material), ensuring the compatibility ofthe surfaces under severe friction, and

(3) testing of the system to verify its compatibility.

C. Self-Organizing of Tribosystems

Friction is a dissipative process, with part of the mechanical energy (thework done by the external loading system) being expended by the accu-mulation of energy into surfaces and the generation of heat. The energydissipation during friction leads to the accumulation of high plastic strainsin the surface layers and the formation of an ultrafine-grained orientedstructure [4]. This raises the free energy of the contact zone, by creatinga high density of structural imperfections in the surface layers (the activa-tion phenomenon). The activation process transforms the surface layersinto an unstable or metastable state. From the point of view of thermo-dynamics, a transition to an equilibrium state is natural. Therefore, theactivation process may be followed by passivation, i.e., a reduction in thefree energy of the material as a result of interaction with the environment


and the generation of protective secondary structures (formed by friction).A tribosystem can be considered to be an open thermodynamic systemthat exchanges energy, matter, and entropy with its environment. Forthese systems, the second law of thermodynamics is still operative, buta more complex and general behavior is found compared to the more clas-sical case discussed in standard textbooks. According to the principlesdeveloped by I. Pigogine, the second law does not eliminate the possibilityof highly organized dissipative structures being formed in an open tribo-system. In these systems, when the excursion from equilibrium exceedssome critical value (typical for cutting), the process of material orderingcan proceed by the spontaneous formation of a self-organizing dissipativestructure [5,6].

During friction and wear, structural adaptations of the materials ofthe tribo-couple evolve in response to the external conditions imposedby the cutting system, leading in many cases to drastic structural changesin the surface layers of the materials. These changes include many of thecharacteristic properties of the friction surfaces and the near-surfacelayers (e.g., geometrical parameters, microstructure, physico-chemical,and mechanical properties). The structural adaptation is completed inthe initial stage of life of the tribosystem, i.e. during the running-in stage.Although evolving in a step-by-step fashion and becoming increasinglycomplicated during the stage, the secondary structures eventually stabilizefor a given tribopair and conditions of friction. When the characteristicsof the surface layers become optimal, the running-in phase is completed,and the parameters of friction (i.e. the coefficient of friction and the wearrate) stabilize [7].

During the self-organizing period, of the process of screening takesplace [8]. The phenomenon of screening reflects the coordination of therates of destructive and recovery processes in the friction zone. This istypical of a pseudo-stationary state where the processes of activationand passivation of the surface layers are in a dynamic balance. Externalreactions usually result in the destruction of the screening phase, butthese reactions and the associated process of matter exchange with theenvironment may provide for its reactivation. With the correct correlationof these processes, the state of the system is stable, corresponding to aminimum in the rate of entropy production. Then, according to the prin-ciple of screening, any kind of interaction of the surfaces or destructionof the base metal should be eliminated. The contact area will be con-trolled by interactions of the secondary structures, and stable friction(wear) conditions prevail as long as the dissipative structures associatedwith friction are self-adjustable.


1. Secondary Structures

The self-organizing phenomenon (SO) is characterized by the formation ofthin (from several nm up to a micron thick) films of the secondary structures(SSs) at the friction surface. These are generated from the base material bystructural modification and=or by interaction with the environment. It hasbeen estimated that 90–98% of the work of friction can be accumulatedin the secondary structures, with no more than 2–10% in the primary struc-tures. Thus, secondary structures represent an energy sink for the preferen-tial dissipation of the work of friction [4,5,9,10].

The synergistic processes of adaptation to the extreme deformation,thermal and diffusive conditions associated with cutting can be concentratedin the thin layer of SSs. The self-organizing of the tribosystem is often accom-panied by a kinetic phase transition. In this case, all the interactions are loca-lized in a thin surface layer, the depth of which can be lowered by an order ofmagnitude than that typically associated with damage phenomena. The rateof diffusion and chemical reactions may also increase substantially, while thesurface layers may become ductile. In addition, the solubility of many ele-ments might be increased and non-stoichiometric compounds might form.

There are two kinds of secondary structures: superductile and super-strong [10]. Secondary structures of the first type (SS-I) are observedafter structural activation that is marked by an increase in the density ofatomic defects at the surface. The structures are supersaturated solid solu-tions formed by reaction with elements from the environment (most often,oxygen). The reaction between oxygen and the substrate during the self-organizing phase can be very different from the classical case encounteredin regular oxidation experiments. SS-I are similar to Beilby layers, havinga fragmented and textured structure, aligned in the shear direction, and theyare often free of dislocations [10]. In these secondary structures, the materialmay be superplastic (due to a very fine grained or amorphous-like structure)with an elongation up to 2000%. The amorphous-like structure of SS-I mayalso lead to a decrease in the heat conductivity of the surface, an importantconsideration in controlling friction of cutting tools [15]. Fragmentation ofSS-I is often detected under severe friction conditions. From an energeticpoint of view, SS fragmentation is another mechanism whereby the structureof the surface layers can be modified.

Secondary structures of the second type (SS-II or tribo-ceramics con-taining a higher percent of elements such as oxygen) are formed by ther-mally activated processes. The SS-II are usually non-stoichiometriccompounds and, as a rule, they contain a deficit of the reactant. However,under heavy loading conditions (in particular, during high-speed cutting),non-stoichiometric compounds with an excess of the reactant have been


observed [4]. Secondary structures of this type (SS-II) exhibit a very high hard-ness. It is thought that one of the benefits they bestow is to accommodatethe stress associated with cutting by elastic rather than plastic deformation.The adaptation of the tribosystem in this case relies upon the high hardnessof the thin surface film formed during cutting. On the other hand, destruc-tion of the surface of the tool might be linked to the poor fracture toughnessof certain SS-II structures.

2. Principles of Friction Control

For the purposes of this paper, the entire diversity of processes that takeplace during friction can be divided into two groups: quasi-equilibrium,steady-state processes (encountered during normal friction and wear) anda non-equilibrium, unsteady state, associated with surface damage processes(Fig. 2). Surface damage is usually observed in the initial (running-in) and

Figure 2 Diagram of wear and friction process: (I) region of non-steady process(running-in stage); (II) region of quasi-equilibrium (steady-state) process (normalwear stage); (III) surface damage region (catastrophic or avalanche-like) wear stage.(1) Regular wear curve; (2) widening of the region of stable (normal) wear; (3)

minimizing wear intensity of non-steady process (in the running-in stage); (4)combination of two main methods of wear control.


final (avalanche-like) stages of wear. During the period of service undernormal friction and wear conditions, no macroscopic damage of the contactsurfaces can be observed. Normal friction corresponds here to the comple-tion of the self-organizing processes discussed above and the transition to astable stage of wear.

Friction control in this context implies the existence of a stable tribo-system, which resists any instability leading to damage below the surfacelayers [4]. The transition from a thermodynamically non-equilibrium condi-tion to a more stable, quasi-equilibrium condition is connected to the accel-erated formation of a beneficial surface structure formed as a result ofself-organizing. From the point of view of self-organizing, both naturaland synthetic processes can be considered during friction. Therefore, it isnecessary to try and control (or modify) the synthetic processes to encouragethe evolution of those natural processes that lead to a minimum wear rate.The problem of compatibility includes developments that ensure the stabili-zation of the friction and wear parameters, in particular, by the selection ofappropriate materials.

3. The Features of the Self-Organizing Process During Cutting

To understand the features of the self-organizing phenomenon during cuttingone should understand the processes occurring at the contact surface of the‘‘cutting tool=workpiece’’ tribosystem over a range of cutting speeds [11,12]).Studies of cutting of a structural medium-carbon steel (�1040) have shownthat the tool life can vary dramatically (Fig. 3). At low speeds of cutting,up to 50m=min, lying in the domain of cutting with HSS tools, intensivebuild-up formation takes place. The machined material transfer at the toolsurface is frequently observed during the metalworking of commonstructural steels. At the same time, oxygen (from the air) penetrates into thecutting zone. Chip fragments containing Fe will react with oxygen and car-bon at the tool surface and form a boundary layer of both iron carbidesand oxides. This leads to a built-up edge. The formation of a built-up layercan be considered to be the result of self-organization of the ‘‘the tool-machined part’’ tribosystem at low cutting speeds. The built-up layer is a dis-sipative structure or composite ‘‘third body’’, which consists of heavilydeformed and refined machining material as well as oxides, nitrides and othercompounds generated during cutting. The built-up layer is similar in manyways to a composite material. The ‘‘ceramic-like’’ built-up layer offerssignificant protection to the tool surface. However, the stability of a built-uplayer as a dissipative structure is very low, especially when cemented carbidetools are used. For example, the adhesive interaction of tungsten carbidegrains with a machined part can lead to microcrack formation. These cracks


are generated at the interface of the phases due to the cyclical stress action atthe points of adhesion with the workpiece, and leads to separation of carbidegrains from the tool surface. The principal mechanism of wear (and hence,the dissipation of energy) of cemented carbide tool surface layers is the for-mation of microcracks and ‘‘drop-out’’ of the carbide grains. The failure ofthe built-up layer results in significant tool surface damage. The formation ofa built-up layer demonstrates that the self-organizing of a system may some-times lead to a minimization of the wear, but under other conditions, when adissipative structure is unstable it may also lead to increased wear (for cemen-ted carbide tools).

At cutting speeds higher than 50m=min (used without a coolant), theprocess of seizure intensifies. On the surface of the tool face close to the cut-ting edge, a zone of plastic contact with a high friction coefficient will beformed [3]. It results in the formation of a thin layer of heavily worked mate-rial at the tool=chip interface. The wear rate of the carbide tool is reducedwith an increase in the cutting speed up to 50–80m=min. At the optimumcutting speeds for carbide tools (50–80m=min), the contact processes atthe tool surface become nearly constant, the formation of the built-up layerdecreases, and a flow zone forms [3]. The formation of the flow zone with anincrease in the cutting speed is an outcome of self-organizing of thetribosystem, leading to a stabilization of friction. In this regime, the tungstencarbide grains undergo considerable fragmentation. This reduces the wearrate, due to a decrease in the volume of the spalled fragments.

With a further increase in cutting speeds (more than 100–150m=min),the wear rate again increases, due to an intensification of diffusion processes

Figure 3 Tungsten carbide tool life vs. cutting speed. Turning test data acquiredwith 1040 steel. Parameters of cutting: speed (m=min): 10–125; depth (mm): 1.0; feed

(mm=rev): 0.2.


and the separation of carbide grains from the tool. These high cutting speedslie in the domain of application of cermets and ceramic cutting tools.

One of the principal features associated with cutting is a rapid increasein the dislocation density near the surface, with the deformation beinglocalized in a thin layer. Under extreme cutting conditions, this layer canundergo dynamic recrystallization, leading to a very fine-grained structure.This structure appears to be very stable during wear, as it has the ability toboth effectively accumulate and dissipate the energy [11–14]. The increase indislocation density in this local volume is accompanied by activation of thesurface layers of the tool leading to further interactions with the environ-ment. This can result in the formation of passivated surface structures(i.e., solid solutions of oxygen in the metal or tribo-oxides), which mightcontrol any damaging adhesive interactions at the ‘‘cutting tool – work-piece’’ interface. Fragmentation of the carbides (or metal matrix) followedby subsequent formation of an energy-absorbing oxygen-containing surfacefilm (at optimal cutting speeds) can also be considered to be an example of a‘‘cutting tool=workpiece’’ self-organizing phenomenon.

We conclude that a number of complicated self-organizing phenomenaare encountered in a ‘‘cutting tool–workpiece’’ tribosystem. The self-organizing occurs at two different levels: (1) the macrolevel (the chip–work-piece interface), and (2) the microlevel (the workpiece–tool interface). At thefirst level, self-organization is associated with the formation of dissipativestructures such as a built-up or flow zone (under different cutting condi-tions). At the second level, the self-organization is exhibited in the formationof dissipative structures such as thin films of secondary structures that maycontrol the process of tool wear.

Two advantages can be realized if the correct tool–workpiece combi-nation can be found. Firstly, stable cutting can be achieved and the wearrate of the tool can be reduced. This leads to an improved workpiece quality[e.g., a better surface finish, improved dimensional accuracy [15]]. Secondly,the generation of specific stable secondary structures could significantlyreduce both friction and surface damage. This can also lead to an increasedmachining productivity and improvements in the cutting tool life. The pro-cesses at both levels are interdependent. The present study is focused on phe-nomena that occur at the workpiece=tool interface, because these processesare critical to the tool life and improved manufacturing productivity.

III. MAJOR CUTTING TOOL MATERIALS

It is possible to classify cutting tool materials according to their differentcharacteristics and domains of application.


A. Universal Cutting Tool Materials

The principal focus of this paper is on the so-called ‘‘universal’’ cutting toolmaterials and their modifications (HSS and HSS-based materials andcemented tungsten carbides). These materials are used (more or less success-fully) is a broad range of applications, not only for the machining of steels,but also for many other materials. A major trend in the metallurgical designof these cutting tool materials is the application of standard (or slightlymodified) universal tool materials using different methods of surfaceengineering.

There are several ways to improve the wear resistance of regular toolmaterials. The first one is by refining the structural components of tradi-tional tool materials using the methods of powder metallurgy (powderHSS and fine-grained cemented carbides). The application of powder metal-lurgy HSS tools results in less surface damage under conditions of adhesivewear [17]. This is most important for the relatively brittle high cobalt HSSand super-HSS tool materials. Other methods are also available, for exam-ple, the refining of carbide particle sizes (and martensite grains in the case ofhigh-speed steels) as a result of a surface laser treatment.

In comparison to tool steels, cemented carbides [18] are harder andmore wear resistant but they also exhibit a lower fracture resistance. Onthe other hand, they have lower thermal conductivities than HSS. In recentyears, WC–Co alloys with submicron carbide grain sizes have been devel-oped for applications requiring more edge strength and minimal surfacedamage. Typical applications include a wide variety of solid carbide dril-ling and milling tools. Mixed tungsten–titanium–tantalum carbides areused for steel machining to resist chemical (diffusion) wear. Tungsten car-bide diffuses rapidly into the chip surface (Table 1) but a solid solution of

Table 1 Dissolution Rate of Refractory Compounds vs. Temperature Relativeto TiC [18]

Dissolution rate at,8C

Material 100 500 1,100

WC 1.1 � 1010 5.4 � 104 3.2 � 102

TiC 1.0 1.0 1.0TaC 2.3 1.2 8.0 � 10�1

TiB2 9.9 � 101 8.5 2.8TiN 1.0 � 10�8 1.8 � 10�3 2.2 � 10�1

HfN 2.5 � 10�12 3.8 � 10�5 2.5 � 10�2

Al2O3 1.1 � 10�24 8.9 � 10�11 4.1 � 10�5


tungsten carbide and titanium carbide resists this type of chemical wear.Titanium carbide is more brittle and less abrasive resistant than tungstencarbide (Table 2). For this reason, tungsten carbide alloys have a betterwear resistance for machining of cast iron when abrasive wear is signifi-cant. The amount of titanium carbide added to the tungsten carbi-de=cobalt alloys is limited to about 30%. It is obvious that during thehigh-speed cutting of steels, a surface phase transformation, resulting inthe formation of oxygen-containing, stable secondary structures of theTi–O type (see Table 2), might occur. The mechanism for Ti–O formationis described in some detail below for tribological materials. There is adearth of information in the literature about the self-organization of thesealloys during cutting, but it is clear that the formation of protective stablesecondary structures can result in a significant tool life increase at elevatedcutting speeds.

B. Specific Cutting Tool Materials

Specific cutting tool materials (e.g., ceramics, diamonds, or cubic boronnitride) have a unique serviceability but a limited domain of application.

Table 2 Relative Resistance of Different Chemical Compounds Against

Abrasive Wear and to Dissolution in Iron at 7008C [16]

Material

Relative wearresistance againstabrasive wear Material

Relative wearresistance againstdissolution in iron

SiC 0.004 Al2O3 0.0000WC 0.008 TiO2 0.0000

Si3N4 0.030 TiO 0.0000Al2O3 0.075 HfN 0.0009HfN 0.28 TiN 0.018HfC 0.34 HfC 0.035

ZrC 0.79 TiCo0.75O0.25 0.32TiC 1.0 ZrC 0.36TaC 1.0 TiC 1.0

TiCo0.75O0.25 1.3 TaC 1.1HfB2 1.6 NbC 1.9NbC 2.2 TiB2 5.3

TiO2 2.2 Si3N4 250TiO 2.8 WC 5,200Mo2C 110 Mo2C 12,000

TiN 170 SiC 24,000


They are mainly used for the machining of non-ferrous alloys or hard-to-machine steels and alloys. Such materials as diamond and cubic boronnitride have a high wear resistance due to their ability to dissipate the energygenerated during friction into thin surface layers of atomic dimensions.

The most important cutting tool materials of this type are ceramics[19] such as alumina or silicon nitride (Si3N4). The main advantages thesematerials offer for high-speed machining are their excellent hot hardness,chemical inertness, oxidation resistance, and ability to play a role as a ther-mal barrier due to their extremely low thermal conductivity at high tempera-tures [4]. The main drawback to ceramics is their low fracture toughness andthe interactions found with ceramics such as SiAlON with certain grades ofsteel. Surface-engineered ceramics and functionally graded materials are themost advanced materials of this class for future application. The excellentstability of ceramics suggests the composition of the surface layer of a func-tionally graded tool material should be one that generates alumina at thesurface on interaction with the environment. This will give a critical changein the cutting conditions due to minimal interaction with the workpiece,increased service stability, and heat flow redistribution from the surface ofthe tool to the chip and the surrounding environment.

C. Tribological or Adaptive Cutting Tool Materials

The characteristic feature of these materials is the formation of stable sec-ondary protective structures during cutting. These structures lead to a con-centration of the interaction between the tool and the workpiece into a thinsurface layer that prevents further damage to the surface. Cermets are typi-cal representatives of this class of material.

Cermets usually contain titanium carbide or titanium carbo-nitrides(and recently more complex titanium–molybdenum–carbon–nitrogen andtitanium–tungsten–carbon–nitrogen compounds) as the hard refractoryphase, comprising approximately 30–85% by volume of the tool [15]. Themetallic binder phase can consist of a variety of elements such as nickel,cobalt, iron, chromium, molybdenum, and tungsten.

The crater and the flank wear resistance of titanium carbide and tita-nium carbo-nitride cermet tool materials are superior to those of conven-tional cemented carbide (WC) tools. Nitrogen additions to the hard phaselead to a higher wear resistance [15]. Titanium carbo-nitrides are the pri-mary materials used for cutting tool applications. Titanium nitride andcubic boron nitride are excellent cermets when they are combined with ahard binder metal. Cermets are more wear resistant and allow for highercutting speeds than tungsten carbides. The main drawback to traditionalcermets is a lack of toughness and thermal shock resistance, but additions


of molybdenum carbide and tantalum=niobium carbides have broadenedtheir application range [20]. Cermets possess many of the characteristicsof a tool material that is capable of filling the gap between conventionalcemented carbides and ceramics.

1. Frictional and Wear Behavior and Self-Organization

of Adaptive Cutting Tool Materials

As noted above, one of the characteristic features of cermets is associatedwith the formation of a thin layer of a protective secondary structure posses-sing some lubricity at the tool surface. The formation of a stable secondarystructure results in an excellent surface finish and close tolerance on longerproduction runs [15], due to self-organization on cutting [21]. These attrac-tive properties can be illustrated by comparing the shape of wear curves ofcemented carbide and cermet tools. Cermets have a better adaptability asshown by the lower wear value during the running-in phase and more stablecutting conditions at the normal wear stage, due to the formation of thesesecondary structures (Fig. 4) [15]. The stability of the secondary structuresis controlled by the thermodynamic stability of the compounds formed atthe tool–workpiece interface. The thermodynamic stability of the com-pounds can be assessed by the enthalpy of formation of various cutting toolmaterials [15].

Figure 4 Wear comparison between cemented carbide and cermet cutting toolswith 4135 alloy steel. (From Ref. 15.)


Two types of cermets are used for cutting tools:

– those with a ductile metal matrix where the refractory phasecontent is more than 50%. The typical application of thesematerials is as turning inserts for high-speed cutting;

– those with a hard steel matrix where the matrix content is less than30%. The typical application of these materials is as end mills formoderate cutting speeds.

One of the problems of traditional tool materials, especially at low andmoderate cutting speeds when adhesion wear predominates, is connectedwith the low dissipative properties of the materials. The wear resistance isstrongly correlated to the relaxation properties of the material, especiallyat the unstable stage of wear. If the tool material has a structure that isunable to effectively transform and dissipate the energy generated by fric-tion (a concern with the majority of traditional tool materials), damagingsurface relaxation processes (e.g., adhesion to the machined part or crackformation on the tool surface) dominate during cutting. Under these condi-tions, the thin surface films of SSs may become unstable and lose their abil-ity to protect the metal surface against excessive wear. The SS that form onthe metal substrate must possess favorable relaxation properties to avoidthis situation [22].

Powder HSS, modified by the addition of titanium compounds, canadapt to the conditions of cutting and have better relaxation properties(compared to tungsten carbides), giving tribological compatibility underlow-speed cutting conditions. The metallurgical design of these materialsis based on the application of the principles of screening by self-organiza-tion, as discussed earlier. The screening effect prevents the direct interactionof the cutting tool and workpiece with the consequent destruction of the toolmetal. The localization of the tool=workpiece interactions in the thin surfacelayers of the secondary structures prolongs the tool life.

Presently, tool materials based on HSS made by sintering and hotextrusion of powders also contain between 10% and 50% of high-meltingcompounds (e.g., Sandvik Coronites, deformed composite powder materi-als—DCPM [23–25]). A characteristic feature of these tool materials is thereaction of refractory compounds (carbides, carbonitrides or nitrides of tita-nium) during cutting, leading to the formation of oxygen-containing surfacelayers (secondary structures). The wear resistance of these materials hasbeen studied with regard to changes in the composition and structure atthe surface of the tool during operation. The materials studied includeM2 and T15 types of HSS as well as DCPM with an addition of 20%TiC. The wear resistance was assessed by the turning of a 1040 carbon steelusing a tool that had indexable tetragonal inserts (with side length 12mm).


The cutting parameters used in the tests were at a speed of 55–70m=min, adepth of cut of 0.5mm and a feed rate of 0.28mm=rev.

The frictional properties of the tribopair under analysis were deter-mined with the aid of an adhesiometer whose design is described elsewhere[20]. One rotary sample of the material to be studied was sandwichedbetween two polished samples made of a 1040 steel (of hardness HRC 30or HB 180, see Fig. 5). To simulate typical machining conditions, the surfaceof the samples was heated by means of an electrocontact method to tem-peratures ranging from 1508C to 5008C. A standard load of 2400N, wasapplied, leading to extensive plastic strain at the contact. The adhesion com-ponent of the friction coefficient responsible for wear at low and mediumcutting speeds (typical for HSS tools) [3] was used as a measure of the fric-tion (t). This parameter t was defined as the ratio of the resistance to shearof the adhesion bonds (formed between the sample made of the tool material

Figure 5 Schematic diagram of friction test apparatus. (1) Specimen made ofmachined 1040 steel; (2) specimen made of DCPM; (3) driving rope; (4) driving disk;

(5) electrical contact wires; (6) isolation system.


and the workpiece under test) to the short-time tensile yield strength of thesofter contact body at the test temperature. The value of t is simply a mea-sure of the resistance of the joint to shear. The friction condition at the sur-face of a cutting tool will be similar to that for which the value of t wasmeasured.

The results of the wear resistance tests are given in Fig. 6. As can beseen, the wear resistance of HSS tools is 2.0–3.5 times lower than that ofDCPM tools. This reduction was associated with a significantly lower fric-tion parameter of DCPM compared to HSS (Fig. 7), and a broadening ofthe range of normal friction (Fig. 6). Within the normal friction range,the rate of wear for DCPM is much lower than that for HSS (Fig. 6, curves1–3). While the hardness and heat resistance values of the HSS T15 andDCPM are similar, the wear resistance of the latter is significantly higher(Table 3). In our opinion the lower wear intensity of the DCPM-tool mate-rial is related to the presence of titanium carbides in the structure and theirsubsequent transformation to oxygen-rich compounds during cutting.

When studied by secondary ion mass spectroscopy (SIMS), the analy-sis of typical wear craters revealed the formation of oxygen-containingphases. The data in Fig. 8 demonstrate that the transformation of titaniumcarbide into an oxygen-containing phase starts in the initial stage of wear(during the running-in process, Fig. 8a). With further operation, there isincreased surface oxide formation at the bottom of the wear crater. Thisprocess is accompanied by stabilization of the wear processes (Fig. 6 and8b,c) and an expansion of the normal friction range. Evidently, this is deter-mined by the phenomenon of self-organization that is connected with theemergence of secondary structures (titanium–oxygen compounds), whichplay the role of stable solid lubricants [25].

Figure 6 The dependence of the flank wear value of cutting tools on the cutting

time: (1) HSS M2; (2) HSS T15; (3) DCPM. Turning test data acquired with 1040steel. Parameters of cutting: speed (m=min): 55; depth (mm): 0.5; feed (mm=rev):0.28.


Figure 7 Impact of the test temperature on the frictional and wear characteristicsas determined from wear contact tests for the DCPM with a 20% TiC addition.


The microstructure of an as-polished lap section made across the cut-ting tool face in the wear zone is shown in Fig. 9(a). The corresponding dis-tributions of Ti, O, and C along the direction I–I, as obtained by Auger

Table 3 Properties of HSS and HSS-Based DCPM materials [23]

Heat Treatment Physico-mechanical Properties

Material

Hardening

temperature

8C

Temperature of

tempering,

8C

Hardness after

heat treatment,

HRC

Bending

strength,

MPa

Impact

toughness,

kJ=m2

Thermal

stability,

8C

M2 1,220 Triple

treatment

at 560 8C

63–65 3,200 400 610

T15 1,240 Triple

treatment

at 560 8C

67–68 2,400 220 645

HSS-

based

DCPM

1,210 Triple

treatment

at 560 8C

69–70 2,000 80 655

Figure 8 Mass spectra of a wear crater in a cutting tool made of HSS-based

DCPM with a 20% TiC addition, determined as a function of the cutting time: (a)4min; (b) 20min; (c) 24min.


Figure 9 Microphotograph of tool friction surface with films of secondarystructures: (a) general view of the surface using secondary electrons; (b) distributionof oxygen close to the ‘‘built-up-crater’’ contact surface (SI, intensity of signals,

arbitrary units).


electron spectroscopy, are given in Fig. 9(b). In the left part of themicrograph (a), a build-up of 1040 C steel can be seen. The right part ofthe micrograph shows the distribution of dispersed hardening phases inthe HSS-based DCPM. Angular (dark) particles of titanium carbide (lessthan 8 mm in cross-section) as well as dispersed tungsten and molybdenumcarbides (less than 0.2–1.5 mm in diameter) are uniformly distributed inthe HSS matrix. In the surface layers of the tool material, we can observea zone of intense plastic deformation less than 5 mm in depth. There, dis-persed particles of a titanium-containing phase have been drawn out parallelto the wear surface, forming a discontinuous film. The titanium carbides inthe wear zone have been transformed into oxides (Fig. 8 and [23]). Titaniumoxides are known to be much more plastic than titanium carbides, account-ing for the plastic deformation of the particles in the surface layers of theHSS-DCPM on cutting.

These results are confirmed by Auger-spectroscopy. Fig. 9(b) repre-sents the distribution of the intensity of the characteristic Auger KLL linesfor O, C, and the LMM (418 eV) line of Ti along the I–I direction inFig. 9(a). The analysis volume includes the built-up layer (of 1040 steel),the built-up layer=wear crater boundary, and the DCPM volumes beneaththe wear crater. At the interface, the titanium compounds show an increasedconcentration of oxygen and a decreased carbon content. The observedchange in chemical composition is related to the instability of titanium car-bide. Due to the high cutting temperatures (in excess of 4508C) and pro-nounced affinity for oxygen, titanium adsorbs the latter from theenvironment and forms thin films of oxygen-containing compounds, inagreement with the SIMS data presented in Fig. 8. The total plastic defor-mation of these particles at the wear surface is greater than 600%. The crys-tal structure of these compounds is believed to differ from the titaniumoxides that would be obtained under equilibrium conditions (see below).

An understanding of the self-organizing phenomenon is criticallyimportant for the development of advanced tool materials. A major interestin these studies (from the point of view of materials science) is the nature ofthe secondary structures forming under severe cutting conditions. Accordingto the principles of current tribology, one of the main methods to controlfriction is the creation of stable secondary structures at the tool surface.The more stable are the secondary structures, the greater will be the tool life.The development of protective secondary structures can be manipulated byalloying or by surface treatment technologies. The type of secondary struc-tures formed during cutting depends strongly on the conditions of cuttingand the type of the material under analysis.

A detailed study of the physico-chemical parameters of the SSs formedduring cutting using a tool made of DCPM was done using AES, ELS, and


EELFS methods. To interpret the atomic structure of the tool wear surface,data obtained in this work by the EELFAS method were compared to TiCand TiO2 standards. Fig. 10 presents the Fourier transform of data obtainedon analyzing the extended electron energy loss fine structure (EXELFS) fortitanium carbide (TiC) with a cubic (B1) structure. The positions of the mainpeaks (Fig. 10a) are consistent with the interatomic distances for a (1 0 0)plane in the cubic lattice of titanium carbide (see Fig. 10b).

Figure 10 (a) Fourier transform of EELFS close to the line of back-scatteredelectrons for TiC specimen, Ep¼1500 eV; (b) cubic lattice of titanium carbide (¼Ti;�¼C).


Fig. 11(a) shows data for TiO2 with the rutile (C4) structure. We canidentify the type of bonds by using partial functions F(R) obtained from theanalysis of the fine structure of spectra close to the characteristic Auger linesof oxygen and titanium. By comparing these data with those given inFig. 9(a), we can see that TiO2 has a more complex crystalline structurethan TiC. This explains the greater number of F(R)-function peaks.The positions of the main peaks are again in good agreement with the

Figure 11 (a) Fourier transform of EELFS close to the line of back-scattered

electrons for TiO2 specimen with rutile structure, Ep¼1500 eV; (b) cubic lattice oftitanium oxide (¼Ti;�¼O).


interatomic distances for a (1 0 0) plane in the TiO2 lattice. The completeanalysis of all the peak positions by the Fourier transform method showsthat the interatomic distances O–O and Ti–O in the secondary structuresare different from those discussed in the literature (see Fig. 11b). Thismay be related to a deviation from stoichiometry, or the interatomic dis-tances, measured from an analyzed volume that is only several angstromsthick near the surface, are different from the equilibrium values.

The evolution of the atomic structure in the surface films on the wearcrater of the cutting tool is well illustrated by the data given in Fig. 12. Theoxygen-containing films in the wear crater are significantly enriched withtitanium and oxygen after only 5min of cutting (see Fig. 12a–d). As thistakes place, a periodicity in the arrangement of atoms of various types isobserved both in the nearest coordination sphere and at greater interatomicdistances, up to approximately 7 A (see Fig. 12b).

As noted above, the interatomic distances in these oxygen-containingfilms differ from those observed in equilibrium titanium oxides, includingrutile (compare with Fig. 11). The very thinnest films may be 2D (two-dimensional) phases whose atomic structure is close to the supersaturateda-solid solutions of oxygen in titanium. After 15min of cutting, the degreeof long-range order is reduced, while the intensities of peaks from higherorder coordination spheres are less pronounced (see Fig. 12c). After30min of cutting (Fig. 12d), the translational symmetry at large interatomicdistances disappears, and peaks at R> 4 A are lost.

The adaptability of the surface layer to external thermo-mechanicaleffects is the physical basis of such evolution. The surface is gradually con-verted to an amorphous state during the wear process (after cutting times ofabout 15min). When a steady-state condition is reached, i.e., after the devel-opment of the SS is completed, the surface generates amorphous-like filmshaving an effective protective function. The lattice instability of the solidsolution of oxygen in titanium finally leads to complete amorphization ofthe water surface. A similar effect was observed earlier from EELFS datafor TiN-coatings on worn cutting tools [26].

Typical EELS spectra of TiC, TiN, TiO2 obtained with a 30.0 eV pri-mary electron beam are shown in Fig. 13. The elastic peak has a 30meVFWHM (full width at half maximum). The high-resolution structures ofthe spectra are represented at 1000� magnification after normalizing.The experimental curves are approximated by Gaussian peaks in each spec-trum in the range of 1–9 eV energy loss.

In the series of titanium compounds TiC–TiN–TiO, the number of 4selectrons in the atomic sphere of the metal decreases from the carbide to theoxide. These electrons are transferred to the 2p orbital of the non-metalatom, and the band energy is lowered due to the increasing Ti–X attraction


Figure 12 Fourier transform of EELFS close to the line of back-scattered electrons for a cutting tool made of DCPM aftercutting times of: (a) initial stage; (b) 5min; (c) 15min; (d) 30min.


in this series. Accordingly, it is more likely that on being excited, electronswould pass from the 2p orbital to the 3d orbital. The peak observed at6.8 eV corresponds to the X 2p!Ti 3d transition (Fig. 13). The intensityof lines at about 6.8 eV is increased in the series TiC–TiN–TiO2. The Ti3d orbital is hybridized with X 2p orbitals, but the energy of this pd orbitalinteraction is shifted downward along the energy scale as one moves fromTiC to TiO, according to Eb(Ti–C)¼4.5 eV; Eb(Ti–N)¼4.8 eV, and Eb(Ti–O)¼6.89 eV [27].

The observed lineshift at 6.7 eV on the electron energy loss spectra fortitanium compounds is in qualitative agreement with these data (see Fig.13a–c). A partial distribution analysis of the valence electrons enables oneto better understand the particular features of chemical bonds in titaniumcompounds and on the wear surface of cutting tools. The lines at 1.6 eVin the electron energy loss spectra reflect intraband transition t1g ! t2g inthe p-band, while those at 3.1 eV correspond to transitions 2t2g ! 3eg inthe s band of Ti-atoms. The reduced intensity of lines at 3.1 eV in the seriesTiO2 ! TiN!TiC is due to the decreased contribution of the ds-electronsof Ti to covalent-ion bonds. This arises when the 4s-electrons of Ti aretransferred into the 2p orbital of a non-metal atom in a titanium compound:

Figure 13 Representative EELS spectra of energy loss region 0–10 eV below theelastic peak of (a) TiC, (b) TiN, (c) TiO2. The spectra were obtained using a 30.0 eV

primary electron beam.


this orbital is more completely filled in the case of oxygen than in eithernitrogen or carbon. For this reason, when the 3d Ti- and 2p X orbitalsare hybridized, the contribution of the ds-electrons of Ti is less pronouncedin the oxide and more expressed in the nitride and carbide. Consequently,the oxide has considerably less strength and hardness than the carbide ornitride [27].

The interaction of Ti–Ti atoms is realized at the expense of dp-elec-trons. The metallic nature of this compound is related to the high densityof dp-electrons. As seen in Fig. 13, the intensity of t1g ! t2g transitions inthe p-band is relatively insignificant in TiC, but it is much higher in TiO2.This implies that the density of conduction electrons is low in TiO2 but itis higher for TiC and TiN. This is consistent with the electrical conductivitydata of these compounds, which is extremely low for the dielectric TiO2, but16,400 (Ohmm)�1 for TiC and TiN, respectively [27].

The replacement of carbon with oxygen in titanium compounds wasshown to change their properties significantly. Thus, the oxidation of TiCat 823K for 30min influences the electronic structure of the material, theelectron spectrum acquiring some features specific to TiO2 (see Figs. 14a

Figure 14 Representative ELS spectra: (a) after oxidation of TiC by heating up to823K for 30min in air; (b) wear surface of DCPM cutting tool after 5min of

operation; (c) wear surface of DCPM cutting tool after 30min of operation. Thespectra were obtained using a 30.0 eV primary electron beam.


and 13c). After oxidation of TiC, the intensity of the lines at 6.7 and 1.6 eV issubstantially enhanced. On oxidation, the titanium carbide loses its metallicproperties and acquires those of a dielectric. In this case, we observe areduced concentration of conduction electrons and a localization of the elec-tron density both in the metal and non-metal atoms. This is shown by theincreased intensity of the peak at 1.6 eV corresponding to p-states in the3dd-band of titanium (see Fig. 13a).

It was noted earlier that an intense oxidation of TiC could be observedduring the operation of a DCPM tool [20,22]. In this case, the nature of thephase transformation differs significantly from that found on heating a TiCstandard up to 823K for 30min. Figure 14(b) and (c) presents electronenergy loss spectra from the wear crater after 5 and 30min of DCPM-tooloperation.As thewear time increases, thespectradisplayasomewhat increasedintensity of peaks at 6.8 and 3.1 eV. Peaks corresponding to plasmon losses(p1 and p2) appear, while the peak at 1.6 eV is significantly attenuated.

The thin SS films in the wear crater of the cutter are associated withthe formation of supersaturated solid solution of oxygen in titanium dueto the oxidation of titanium carbide. In this case, we observe an increasein the electron density in the 2p orbital of the non-metal (peak 6.8 eV) aswell as an enhanced filling of the ds-electron band of titanium atoms (peak3.1 eV). These effects are similar to those encountered in the model oxida-tion of TiC (Fig. 14). There are, however, substantial differences. As the cut-ting time increases, the effects brought about by the crystalline structureof phases become significantly weakened in the electron spectrum. Thesplitting of the 3d orbital into p and s-states degenerates, the intensity oft1g! t2g transitions is reduced as well as the density of p-electrons whichare related to the long-range Ti–Ti bonds in the lattice (along the diagonalsin the (1 0 0) planes). These distinctive features of the electron structure arerelated to amorphization and to the increasing role of short-range inter-atomic bonds. Of considerable interest is the appearance of plasmon losspeaks p1 and p2 in the spectra of Figs. 14(b,c) due to the growing concen-tration of conduction electrons. The delocalization of p-electrons close tothe titanium atoms enhances the metallic nature of bonds in the amorphousfilms developed on the friction surface. These specific traits of electronic andatomic structural change might help to explain the unique mechanical prop-erties in the secondary structures of the first type. The high wear resistanceand good frictional properties of DCPM tools are associated with complexstructural and phase transformations on the surface, among them TiC oxi-dation and the development of thin protective amorphous films. The SSs aresaturated or supersaturated (amorphous) solid solutions of oxygen intitanium, whose electron structure is characterized by a high density of con-duction electrons giving metallic characteristics.


These results show that the SSs formed during cutting not onlyincrease the DCPM-tool life but also change friction characteristics as well.The amorphous-like secondary structures of the first type behave like a solidlubricant with enhanced tribological properties [4].

Additional alloying of the DCPM might be beneficial. For example,the partial substitution of titanium carbide by aluminum oxide, which isstable under cutting leads to a decrease in the friction coefficient (Fig. 15)

Figure 15 Impact of the test temperature on the wear and friction characteristicsas determined from wear contact tests for the DCPM with 15% TiC þ5% Al2O3;20% TiCþ2% BN and 20% TiCN additions.


and in an increase in the wear resistance of the tool (Fig. 16). The decreaseof the friction coefficient when Al2O3 is added is important not only as itincreases the wear resistance but also because it lowers the cutting tempera-ture at the tool surface [28,29]. Alloying often cannot be implemented by thetraditional metallurgical methods since this may induce an undesirablechange in the properties of the cutting material. We took a differentapproach by making small additions of low-density compounds, whichare relatively unstable at the operational temperatures. This allowed us touse this compound in relatively small quantities (up to 2w%) with minimalpossible impact on the bulk properties. The solid lubricant (hexagonal BN)was chosen as the additional alloying compound [28]. The high probabilityof oxygen-containing secondary phases formed from BN during cutting wasalso taken into account. The possibility that TiC and BN might oxidize andgenerate thin surface oxide films for exploitation in cutting tools can beassessed by a thermodynamic approach [27].

Secondary ion mass spectroscopy investigations have shown that oncutting DCPM with a boron nitride addition, oxygen-containing com-pounds develop at the wear-crater surface, associated with a set of paralleldisassociation reactions of BC, BN, TiC leading to the formation of BO,TiO and TiB,N. Figure 17a–c presents spectra of the positive and negative

Figure 16 Wear curves of friction contact materials: (1) DCPM with 20% TiC;(2) DCPM with 20% TiC and 2% BN; (3) DCPM with 15% TiC and 5% Al2O3;(4) DCPM with 20% TiCN. Turning test data acquired with 1040 steel. Parameters

of cutting: speed (m=min): 90; depth (mm): 0.5; feed (mm=rev): 028.


ions obtained upon analyzing the chemical and phase composition of a BN-doped carbide steel, investigated at various depths beneath the crater sur-face. In the volume closest to the tool surface (0.15 mm), there is an increasein intensity of peaks O�, BOþ, TiOþ and a decrease in the intensity of peaksBN�, BCþ and TiC2

þ compared to data gathered at a greater depth(0.6 mm). Weak peaks corresponding to TiBN and TiBO also appear (seeFig. 17a–c). As a rule, the observed ions in the SIMS method cannot berelated directly to the compounds encountered in the analyzed regions,but they provide a useful ‘‘fingerprint’’ that allows one to identify thecompound.

Figure 18 presents the SIMS data from different depths beneath thewear crater surface for a specimen of the same material. Comparing thesedata with those shown in Fig. 17c, one can see that an appreciable weaken-ing of the intensity of the peak BCþ, the disappearance of peak TiC2

þ, aswell as pronounced enhancement of peaks BOþ, TiBNþ and TiBOþ, arecharacteristic of the formation of thin surface films. These are compoundsof titanium and boron with oxygen and nitrogen, formed as a result ofthe carbide and nitride reaction with the atmosphere.

The degree to which particular crystalline structures can develop sec-ondary structures depends on the composition of the tool material.A comparative analysis of the nearest atomic neighbors using EELFS

Figure 17 Mass spectra of secondary ions of BN-doped DCPM specimen after4min cutting determined at different depths beneath the surface of a wear crater:(a) 0.5 mm; (b) 0.15 mm; (c) at the surface.


spectroscopy was done to demonstrate this phenomenon for the phases thatform on the wear crater surfaces of M2 high-speed steel, DCPM and BN-doped DCPM tools. Figure 19a–c show Fourier transforms obtained fromdata collected from the surface of wear craters in HSS and DCPM alloyedwith either TiC or TiC with BN. The F(R) functions feature pronouncedpeaks in the range 1–2, 4–5 and 7–8 A for all cases. Using partial F(R) func-tions obtained by analysis of the fine structures close to the Auger lines of C,B, and Ti, it was possible to interpret the nearest-neighbor interatomicbonds. It was found that Fe–O bonds are typical of the HSS sample, whileB–O and Ti–B bonds are observed at the wear crater of the BN-dopedDCPM sample.

The results of these investigations have shown that the composition ofthe tool material not only determines the composition of the phases devel-oped at the friction surface in the cutting zone, but it also exerts an influenceon the perfection of the crystalline structure of the new phases. The thinnestfilms of iron oxide formed on the HSS-tool friction surface are crystalline,

Figure 18 Change in absolute and relative values of the peak intensity of thesecondary ion mass spectra for the BN-doped DCPM tool at different depthsbeneath the surface of the wear crater developed after 4min cutting.


as shown by the pronounced F(R) function maxima at R � 4.3 A and R �7.4 A (see Fig. 19a). These are secondary structures of the second type, i.e.,oxides whose composition is close to being stoichiometric, [4,10].

As seen in Fig. 19(b and c), tools made of a TiC-containing composi-tion of DCPM and of a [TiC, BN]-doped DCPM posses very different sec-ondary structures in the cutting zone, which are quite distinct from thesurface films formed at the HSS-tool surface (Fig. 19a). The surface ofthe DCPM tool reveals the development of secondary structures of the firsttype, i.e., supersaturated solid solutions having an amorphous-like structure[4]. This is shown by the attenuation of the peak intensity of the Fouriertransforms in the vicinity of the coordination spheres at R � 4–5 A andR � 7–8 A. BN-additions to DCPM enhance the amorphization effect foroxygen-containing phases. This is clearly shown by the attenuation of thepeak intensity in the vicinity of 4–5 A (see Fig. 19c in comparison withFig. 19b).

Thus, both mass spectroscopy and the EELFS data indicate that sec-ondary, oxygen-rich, amorphous structures develop on the surface of theBN-doped powder sample. Judicious alloying of a carbide steel throughBN-doping is beneficial for the development of complex compounds ofTiBxOy that appear on the tool surface along with simpler compounds of

Figure 19 Fourier transforms of extended fine structure within the range of 250 eVclose to the line of elastic scattering of electrons with primary energy E¼1500 eVfrom the wear crater surface in a cutting tool made of (a) M2 HSS; (b) DCPM;(c) BN-doped DCPM.


the TiO family. The amorphization of secondary structures probablydepends on the DCPM composition and on increases in the level of BN-alloying. One can see that the wear resistance of this material is increasedby 80% compared to the carbide steel having a base composition with20% TiC (Fig. 16). This suggests that alloying enhances the stability ofthe secondary structures developed during friction. This is promoted bythe presence of BO-type compounds that act as liquid lubricants [at elevatedcutting temperatures [28]] and promote the stability and the self-organiza-tion of the complex compounds. This is of paramount importance for toolwear resistance. The thickness of the stable secondary structures layer doesnot exceed 0.1–0.15 mm (Fig. 18).

Finally, it is possible that such alloying of DCPM provides both areduction in the friction coefficient and a broadening of the normal wearstage. In our opinion, the same goal can also be achieved in HSS-basedDCPM by the substitution of TiC with TiCN. Figures 15 and 16 demon-strate that this substitution is extremely effective, decreasing the frictioncoefficient to abnormally low values (in the range of 0.03–0.05) at a servicetemperature of 500–550 8C, and significantly increasing the tool life (Fig. 16).In this case, the self-organization mechanism differs somewhat from the pro-cess found for materials alloyed with BN.With TiCN the diffusional transferof nitrogen into the chip arises from dissociation of TiCN during cutting [26].The increased nitrogen concentration on the contact surface of the chip is adirect consequence of the mass transfer of nitrogen, formed by dissociationof nitrides and carbo-nitrides. Such mass transfer takes place under theextreme temperature and stress conditions encountered in the friction zone.

Therefore, the selection of titanium carbo-nitride for cermets and tita-nium nitride as the hard phase in Sandvik Coronites seems to be completelyreasonable from the standpoint of tribological compatibility. The behaviorof titanium nitride should be comparable to a carbo-nitride [24,30] and thishas been confirmed experimentally by the analysis of the self-organizingphenomenon of PVD TiN coatings (see below).

Another important factor is the thermal stability of DCPM com-pared to cemented carbides. The importance of the thermal stability of thistype of material is evident at elevated cutting speeds. One way to improvethis property is to increase the volume fracture of the hard phase in theDCPM; e.g., Sandvik Coronite has 50% of the hard phase (TiN) [30]and its wear resistance is higher than either HSS or cemented carbides(Fig. 20). So the benefits of an adaptive material are obvious from thestandpoint of both wear resistance and tribological compatibility. However,the application of these materials is currently limited, although the pro-blems encountered might be partially solved by improved surface-engineering techniques. For example, state-of-the-art coating technologies


together with the use of adaptive materials could be developed for func-tionally graded tool materials, i.e., materials whose structure and proper-ties might be tailored from the core to the surface. That might result ina significant increase in tool life.

D. Functional Graded Materials for Cutting Tools

There are two pressing problems that should be addressed in the develop-ment of materials for cutting tools. The first one is the choice of economic-ally alloyed tool materials, due to the severe shortage of core metals suchas tungsten. This problem becomes more and more pressing. The secondproblem is to find the optimum combination of different, and even contra-dictory, properties in the same cutting material. New innovative materialsare needed. The material of interest will have to combine such diverseproperties as high strength and toughness at elevated temperatures withexcellent wear resistance. A recent development is the production of afunctionally graded material, whose properties (wear resistance andstrength) vary from the core to the surface. This class of functionallygraded materials (FGM) combines a high wear resistance at the surface

Figure 20 Comparative tool life of the cutting tool materials. End mill test data.Machined material—1040 steel. Parameters of cutting: speed (m=min): 21; depth

(mm): 3.0; width (mm): 5; feed (mm=flute): 0.028; cutting with coolant.


with high strength and toughness of the core. The intermediate layer hasgraded properties that lie between those of the core and the surface.Promising developments in this field have been reported by the Laboratoryof Materials Processing and Powder Metallurgy of the Helsinki Universityof Technology. They developed a novel functionally graded material hav-ing a surface ceramic layer, a graded WC-cermet composition with highcrack resistance and a cemented carbide core with excellent toughness. Ahigh crack resistance parameter value (K1C¼25MPam1=2) at a hardnessof 1500HV (typical for tool steels with half the hardness) was found(http:==www.hut.fi=Units=LMP=).

There are two methods used for FGM processing. The first one,noted above, is a surface-engineering method. This method has uniquepossibilities and versatility. But other methods, principally those basedon powder metallurgy, are also widely used. Combinations of thesetwo methods have recently been put into practice. Thus, tools manufac-tured by ordinary sintering processes, having high-toughness cementedcarbide substrates with high wear resistant ceramic coatings and func-tionally graded interlayers show excellent wear resistance [31]. Function-ally graded materials can have superior wear resistance, resistance tofracture, and good thermal shock resistance in the comparison to con-ventional cermets, with a beneficial compressive residual stress distribu-tion [32]. Ceramics have also been recently developed with functionallygraded structures. In order to combine high hardness and high tough-ness, graded ceramics of Al2O3þTiC (surface)=Al2O3þTi (inner core)and sialon (inner core) have been successfully developed [33]. Function-ally graded ceramic tools can exhibit better cutting performance thanregular ceramic tools [34]. Functionally graded materials have also beensuccessfully used for milling applications [35]. Functionally graded pow-der materials are normally used for high-speed cutting, but they can alsobe successfully employed in the domain of HSS tools applicationparticularly with functionally graded cement carbide and hard PVDcoatings [36]

III. TRENDS IN THE DEVELOPMENT OFSURFACE-ENGINEERED TOOL MATERIALS

Surface engineering has recently become one of the most effective ways ofimproving the wear resistance of tool materials. The principal beneficialeffects associated with surface engineering for this application are shownin Table 4.


A. Monolithic and Multi-layered PVD TitaniumNitride Coatings

The most widely used approach for the surface engineering of tools is basedon titanium nitride coatings deposited by physical vapor deposition meth-ods (PVD). These coatings display a favorable combination of properties,such as good adhesion to the substrate, high hot hardness, excellent chemi-cal stability, and improved oxidation stability (up to 550–6008C), resultingin an increased resistance to solution wear. The ability to improve the con-tact conditions at the cutting edge (i.e., a reduction of the tool–chip inter-actions) has been reported, leading to lower friction and decreasedtemperatures at the surface of the tool [1]. However, monolithic TiN coat-ings have a critical weakness. Following the discussion in the previous sec-tions, it is clear that coatings whose properties can be tailored from thesubstrate to the top surface are required. Unfortunately it is almost impos-sible to combine such divergent properties as good adhesion to a HSS sub-strate coupled with minimal workpiece interactions in a monolithic TiNcoating. In addition, high hardness and the possibility of energy dissipationwithout coating failure are often mutually exclusive properties [37]. One ofthe solutions to this problem is to adopt a classical metallurgical designapproach. For example, multi-layered coatings or coatings with a metal-based sublayer could be applied [1]. However technical problems have beenencountered in the deposition of these coatings. An alternative approach is

Table 4 Improved Performance of Surface-Engineered Cutting Tools [1]

Favorable effect

Improved performance

of cutting tools Practical advantages

Reduced friction Lower heat generation,

Lower cutting forces

Increased productivity,

improved workpiecequality (better surfacefinish, improved dimen-

sional accuracy), in-creased cutting tool life

Reduced adhesion to the

workpiece surface

Less material transfer

from the tool surfaceImproved diffusion

barrier and chemical

stability properties

Reduced diffusion Increased productivity,increased cutting

tool lifeIncreased hardness Reduced cutting tool flank

wear and abrasive wearIncreased cutting tool life


to vary the parameters of deposition, using a regular PVD unit, to optimizethe coating properties and coating design.

A study designed to optimize the deposition parameters for TiN coat-ings [38] determined that the nitrogen pressure is the most critical processparameter responsible for changes in the coating structure and propertiesof the films. For the deposition conditions described in Ref. [38], an increasein the nitrogen pressure up to 0.4–0.6 Pa leads to stoichiometric TiN(53 at.% N2). Further increases in the nitrogen pressure lead to a decreasein the nitrogen concentration of the film (to 43 at.% N2, Fig. 21a). This isprobably caused by a decreased intensity of the plasma-chemical reactionas a result of a reduction in the flux and energy of the impinging ions.The phase composition changes from a-TiþTi2N at very low nitrogen pres-sures to TiN at higher nitrogen pressures (0.4–0.6 Pa).

The structural parameters of the film also depend on the nitrogen con-centration in the coating. The lattice parameter and subgrain size (or equiva-lently the dislocation density) are related to the nitrogen concentration inthe coating (Fig. 21b–d) and have maximum values at a composition corre-sponding to stoichiometric TiN. An increase in the nitrogen pressure leadsto a pronounced axial texture (Fig. 21e), with (1 1 1) planes in the TiN beingparallel to the substrate surface. The (1 1 1) axial texture increases to 95% asthe gas pressure is raised to 0.6 Pa and remains practically unchanged withsubsequent pressure increases. The residual compressive stress in the coatingshows similar trends (Fig. 21f), with the stress increasing from 200 to1300MPa as the gas pressure is raised to 0.6 Pa. The rate of increase inthe compressive stress slows with a further increase in the nitrogen pressureup to 2.6 Pa, reaching a maximum value of 1600MPa. It is important to rea-lize that the trends shown by these structural-dependent parameters dependprimarily on the deposition conditions. An increase in the nitrogen pressureover 1.3 Pa decreases the ion energy, and the effective temperature at theTiN crystallization front decreases. At a nitrogen pressure in excess of1.3 Pa, the coatings form under conditions similar to those encountered inthe balanced magnetron sputtering process (for the PVD method). Theseconditions can be indirectly characterized by the deposition rate, whichshould not exceed 5–6 mmhr�1. A decrease in the deposition rate enhancesboth the axial texture and the magnitude of the residual stresses. The mostprobable cause of the high compressive residual stress found in thin con-densed films deposited at nitrogen pressures greater than 1.3 Pa is a highdensity of point defects [39]. An increase in the nitrogen pressure alsodecreases the crystallite dimensions (Fig. 21f).

The microhardness of the coatings depends on their phase composi-tion. The maximum microhardness (H0.5¼45GPa) is achieved when thetwo-phase a-TiþTiN composition changes into the three-phase composition


a-Ti, Ti2N, and TiN. The Palmquist toughness of the coatings (Fig. 22a) is astructure-sensitive characteristic. The maximum toughness and plasticity(determined from the hysteresis in indentation testing) correspond to a sin-gle phase coating having a stoichiometric TiN composition. Any deviation

Figure 21 The dependence of the structural characteristics of TiN PVD coatingson the nitrogen pressure.


from stoichiometry causes a decrease in the Palmquist toughness, particu-larly when a second phase is formed in the coating (e.g., Ti2N). This resultseems unexpected at first, but we should remember that the coating shouldbe regarded as a quasi-brittle material, whose toughness is determined pri-marily by its crack propagation resistance. The optimum nitrogen con-centration corresponds to the largest lattice parameter [40], although the

Figure 22 The dependence of the properties of TiN PVD coatings on the nitrogen

pressure.


microhardness of the coating with this structure is not relatively high. Inaddition to the intrinsic mechanical properties of the coating, the level ofthe residual compressive stress is important for crack initiation and propa-gation. The value of the residual stress is about �800MPa in stoichiometricTiN coatings (Fig. 21f). The fracture resistance also appears to depend onthe columnar grain size, which again can be controlled if balanced deposi-tion conditions can be achieved. The adhesion of the coating to the substrateand the shear load resistance (i.e. the cohesion of the coating) both decreaseas the nitrogen pressure is increased. The nitrogen atoms (ions) in theplasma scatter the Ti ions, so that the net effect of Ti ion bombardmentof the coating is reduced as the nitrogen gas pressure rises. As noted earlier,both the axial texture and the residual stress gradient at the coating–substrate interface increase as the pressure rises, and consequently the adhe-sion of the coating falls (Fig. 22b). The coating wear resistance during drysliding friction under high loads (Fig. 22c) reaches its maximum value inthe three-phase field area a-TiþTiNþTi2N.

Cutters made of a high-speed steel with TiN coating are used underadhesive wear conditions [3]. The tool life of TiN coated parts then dependson the friction coefficient under cutting conditions that are close to gallingor seizure during wear testing (Fig. 22, e).

The overall conclusion of this study is that the optimum combinationof properties of the coating for adhesion wear is obtained at low depositionrates (for the PVD method) of 5 mmhr�1 and a stoichiometric compositionof TiN. This can be achieved by optimizing the deposition parameters. Inthis case, the hardness and toughness increase, while the shear resistancedecreases. A coating with the optimum structure will crack by shear failureat or near to the surface of the coating rather than forming deep cracks lead-ing to a catastrophic failure of the whole tool. However, at the same time,the shear stress resistance of the coating should be strong enough to allowfor easy flow of the chip (the value of cohesion should be about 0.2). Sincea monolithic TiN coating usually has a low adhesion to the substrate, adhe-sive sublayers are necessary to achieve high efficiency from this type ofcoating.

A number of principles guiding the selection of the processing of TiNmulti-layer coatings for adhesive wear conditions can be proposed from thisstudy. The coating should have at least three sublayers:

(1) An adhesion sublayer, deposited with substoichiometric nitrogen.These deposition conditions provide the maximum kinetic energy ofthe ions and a low nitrogen concentration in the layer (up to 35%). Atthe same time the (1 1 1) axial texture should not exceed 50%, while theresidual stress at the coating–substrate interface should be low (not


more than 200MPa). When this combination is achieved, the adhesionof the sublayer is high.

(2) A transition layer deposited with a gradual increase in the nitrogenpressure to provide:(a) development of an axial (1 1 1) texture (from 48% up to 100%)

from the substrate to the top layer;(b) a residual compressive stress increase from 200MPa in the

adhesion layer up to 1700MPa; and(c) a gradual transition from a three-phase structure a-TiþTiNþTi2N

to single phase TiN.(3) A working (contact) layer, deposited under high nitrogen pressures and

balanced conditions (the deposition rate for the CAPDP method is 5–6.5 mmhr�1). In addition the deposition conditions at this stage shouldbe chosen to yield stoichiometric TiN, having a nearly perfect axialtexture, a high residual compressive stress (more than 2000MPa), and afine columnar grain size containing minimal ‘‘droplet’’ phases.

A multi-layer coating is required to meet these diverse requirements.This coating offers many advantages (in comparison to monolithic coatings)in satisfying the broad range of mechanical properties needed in these appli-cations. It has high adhesion to the substrate but low adhesion to theworkpiece (i.e. a minimal friction coefficient), a high microhardness(H 0.5¼35GPa) and a high toughness (more than 50 Jm�2, see Table 5). Thisfavorable blend of structural and mechanical properties has many advan-tages for wear resistance during cutting operations (Table 5).

Table 5 Comparative Characteristics of TiN Coatings Deposited by the PVDMethod on the Coating Deposition Process [38]

Parameters

PVD method

Coating

design

Microhardness

(GPa)

Palmquisttoughness

(Nm�2)

Coefficientof adhesion

to the

substrate

Wear resistance

on cutting

Regular arc

deposition

Monolithic

coating

25.0 26.0 0.5 1.0

Multi-layercoating

30–35 50–60 0.8 1.5–2.0

Filtered arcdeposition

Multi-layercoating

35–37 150–200 0.8 2.0–2.5


The same multi-layer coating could be used for filtered PVD coat-ings with the additional advantages offered by this technology [41]. Thesesystems not only eliminate the ‘‘droplet’’ phase from the coating, butthey can also be used to control the deposition conditions so that anexcellent microstructure and properties are obtained in the film. An extre-mely fine-grained structure (the grain size is 10 nm in comparison to agrain size of several microns in regular coatings) can be achieved [42],with excellent mechanical properties. The nano-grained structure of thefilm is due to:

(1) the higher ionization rates achievable in filtered arc-evaporatedplasmas, which are thought to enhance the nucleation rate and depressthe growth rate of coarse crystals; and

(2) a lower overall deposition rate for the filtered arc PVD process thatresults in a drop in the temperature at the growth front of the film.

Although lower deposition rates are found with this process and canlead to a loss of productivity, these disadvantages can be offset by improve-ments in the quality of the film. The hardness and Palmquist toughness of aTiN coating deposited by this method can be increased up to 35–37GPa(instead of 25GPa) and 150–200 Jm�2 (instead of 26 Jm�2), respectively.The adhesion of this coating is also very high (kadh¼0.8, see Table 5). Inaddition, the wear resistance of filtered coatings is usually much better thanregular coatings (Table 5). The principles outlined above for a multi-layeredTiN coating can also be successfully applied to filtered coatings.

1. Frictional Wear Behavior and Self-Organization of

TiN PVD Coatings

Hard TiN coatings act as surface lubricants by inhibiting the adhesion of thetool surface to the workpiece [1]. The friction parameter of this coating isalso low at the operating temperature (Fig. 23). The wear behavior changeswhen TiN coatings are applied to cutting tools (Fig. 24a) [43]: the initial rateof wear (during the running-in stage) is significantly lower and the range ofnormal wear is expanded. As a result, both reduced friction and wear con-trol can be achieved (see Fig. 2). To explain the enhanced wear characteris-tics imparted by TiN PVD coatings, a study of the self-organizationbehavior of the tool was performed [26]. Protective secondary structures(SS-I) of the Ti–O type form at the surface of hard PVD TiN coatings dur-ing cutting. The transition from the running-in stage to the normal wearstage is marked by the development of a supersaturated solid solution ofoxygen in titanium (Fig. 25a–c). At the same time, as the friction drops,


Figure 23 The dependence of the frictional parameters of TiN PVD coatings ontemperature.


Figure 24 Tool face wear vs. time. Turning test data (a, cutting speed¼70m=min;b, cutting speed¼90m=min; machined material, 1040 steel; depth (mm), 1.0mm;feed, 0.28 mm=rev): (1) M2 HSS; (2) M2þ ion nitriding; (3) M2þPVD TiN

coatings; (4) M2þ ion nitridingþ PVD TiN coatings; (5) T15 HSSþ ion nitridingþPVD TiN coatings.


the wear rate of the tool is reduced and the process enters the steady-statestage. Titanium oxide has a high resistance to friction [44] andcutting (Table 2) and readily fulfills a protective role for the underlyingTiN coating.

The nature of the thin surface layer formed at the surface of the coat-ings was studied by using EELFS analysis. An analysis of the fine structureobtained from the surface of the wear crater at different stages of operationin coated HSS cutting tools was used to follow the changes in the structureof the surface layers (see Fig. 26a–c). When the cutting period was 30 sec, i.e.at the running-in stage, the surface features in the EELFS spectrum agreedwith crystalline titanium nitride. The characteristic signature of TiN is apeak at R2¼2.0 A as well as a peak at more remote interatomic distances(at about 4–5 A, see Fig. 26a). When the cutting lasts for 180 sec, titaniumoxide develops in the coating, the degree of remote order in the crystallattice is reduced and the coating structure appears to amorphize. Thisis shown by the appearance of a peak at R2¼2.20 A (RTiþRO¼1.45þ0.73¼2.18 A) and by the attenuation of peaks at more remoteinteratomic distances (see Fig. 26b). When catastrophic wear occurs (Fig.26c), at a cutting time of 2100 sec, the coating is destroyed while the steelsurface is exposed. This is shown by the change in the form of the Fourier

Figure 25 SIMS spectra of TiN and ‘‘quasi-oxide’’ Ti–O films as a function of the

service time of M2 HSS tool with coatings: (a) cutting time¼15 sec; (b) cutting time ¼90 sec; (c) cutting time ¼ 120 sec.


Figure 26 Fourier transform from EELFS analysis of a wear crater, TiCrN

coating on nitrided T15 steel: (a) cutting time ¼ 30 sec; (b) cutting time ¼ 180 sec;(c) cutting time ¼ 2100 sec.


transform. The first peak is now located at a distance R1¼1.7 A, while thesecond peak is at a distance R2¼2.65 A. These peaks correspond approxi-mately to the length of C–Fe and Fe–Fe bonds (RCþRFe¼0.51þ1.26¼1.77 A; RFeþRFe¼1.26þ1.26¼2.52 A). The spectrum shown in Fig. 26c istypical of the BCC-lattice of T15 high-speed steel.

B. Tribological Properties and the Metallurgical Design ofSurface-Engineered Tools

The study of the wear resistance of coated tools demonstrates that the pro-tective role of the coating is most efficient when the effects of the work ofcutting can be localized in the near-surface region of the coating [43]. Cur-rent coating technologies achieve this goal by modifying the energy distribu-tion (from the tool surface into the chip), and by promoting the self-organization of the tool. This is done in two ways: (1) by surface engineeredand self-lubricated coatings for low and moderate speed machining, and (2)by the use of hard or superhard coatings, that can act as thermal barriersand form very stable ‘‘tribo-ceramics’’ at the surface during high-speedcutting.

1. Surface-Engineered or Duplex Coatings

The principal application of these coatings [45] is for cutting at low speeds,when HSS and DCPM tools are used. It is desirable to deposit the hardcoating, not directly onto the steel substrate but rather onto an engineeredsublayer, so that a gradual change in properties at the coating–substrateinterface, i.e., a functionally graded material, is realized. This sublayer canbe obtained by different technologies, e.g. ion nitriding. Usually such coat-ings will then include both a nitrided sublayer and a hard PVD coating.

The nitrided sublayer has two roles. It prevents intensive plastic defor-mation of the substrate (HSS or DCPM) and cracking of the PVD coatingthat might be caused by deformation of the underlying substrate, while atthe same time, it provides an additional thermal barrier [43]. The advantagesof HSS cutting tools with surface-engineered coatings are shown schemati-cally in Fig. 27.

However the structure of the nitrided sublayer must be optimized induplex coatings to achieve the best tool life. The duration and temperatureof the process are the most important parameters in ion nitriding [38]. Theion current density should not be high, preferably about 3Am�2. Theexperimental data given below were obtained when the surface tempera-ture during nitriding was about 500–5308C. At this temperature, rapid


Figure 27 Schematic diagrams of composite cutting tools: (a) HSSþTiN PVDcoating; (b) HSSþ surface-engineered coatings.


nitrogen diffusion occurs. The dependence of the structure and propertiesof an M2 tool steel on the nitriding time is shown in Fig. 28. Ion bom-bardment leads to the formation of a defective structure in the surfacelayers, which enhances nitrogen diffusion. During the fist 10–20min ofnitriding, a saturated solid solution of N is formed. After 30min of nitrid-ing a supersaturated solid solution of N is obtained at the surface(Fig. 28a). The most pronounced changes in the lattice parameter and linebroadening of the (2 1 1) reflection occur after 0.5–2.0 hr of nitriding(Fig. 28b). A further increase in the nitriding time from 2 to 4 hr has littleeffect on either the lattice parameter or the line broadening. Nitrides areobserved after about 2–4 hr. (The formation of a nitride using X-ray

Figure 28 The time dependence of the structural characteristics and properties of

the ion nitrided sublayer of a surface-engineered coating: (1) M2 HSS; (2) D2 toolsteel; (3.1) nitrided layer of a cutting tool; (3.2) un-nitrided layer of the die steel.


diffraction can be detected when the concentration of the nitride isapproximately 5%.) The first nitride to be detected by x-ray diffractionin this study is the e-phase (W,Fe) 2–3N, while after 4 hr of nitriding, boththe e and g0 (W,Fe) 4N phases are detected. After 4 hr of nitriding, thenitrides can be clearly detected by optical metallography as a network ofthin, needle- or lath-shaped particles.

It is known that the presence of tungsten,molybdenumand chromium inthe solid solution of the steel can lead to the formation of a high density of finenitrides with a marked increase in the hardness. When the nitriding time isincreased to 2hr or more, mixed (Cr,W,Mo) nitrides will also nucletlate. Thesenitrides are very finely dispersed and hence are difficult to detect by X-ray dif-fraction, but they contribute significantly to the increased hardness (Fig. 28d).

The coefficient of plasticity of a nitrided M2 steel changes according tothe data shown in Fig. 28e. This coefficient (determined from an indentationtest) is highest (52%) when the hardness is low, and conversely decreases (to48%) when the hardness is high. (The Palmquist toughness for nitridedsteels cannot be used to give a meaningful measure of the fracture resistanceas the depth of the nitrided layer changes as nitriding proceeds.) The plasti-city of the nitrided layer is sensitive to the microstructure. When there are nonitrides in the layer, the plasticity coefficient is proportional to the nitrogensaturation. The N content in this zone can be characterized by the latticeparameter of the a-phase (Fig. 28a). As the nitrogen concentration (and lat-tice parameter) in the surface layer rises, there is a corresponding decrease inthe plasticity, and vice versa. A low plasticity is correlated with an increasedlattice deformation of the solid solution, associated with the dissolution ofN into the iron lattice, as shown by the line broadening of the (2 1 1) reflec-tion of the nitrided martensite (Fig. 28b). In addition, some influence on theplastic properties is exerted by residual stresses, which are formed in the sur-face layer during nitriding (Fig. 28c). The residual stresses are high when thenitrogen content in the nitrided layer increases and extensive precipitationoccurs on cooling. The volume of the surface layer increases on nitridingand as a result compressive residual stresses are formed. This effect is typicalfor M2 grade steels.

High compressive stresses in the nitrided layer of a M2 steel lead toincreased hardness and plasticity, and inhibit cutting edge-flaking duringthe tool life. It is important that the level and sign of stresses formed inthe nitrided layer are similar to those in the adhesion sublayer of multi-layer coatings. then, the stress gradient between the nitrided substrateand the coating is low and the adhesion is improved. The service propertiesof the nitrided layer also have a high structural sensitivity. The longest toollife of nitrided HSS steels is obtained with an a-solid solution structureand is at least double that of un-nitrided tools. The tool life increases with


the nitrogen content in the layer, which, as noted earlier, can be monitoredby the change in the lattice parameter of the nitrided martensite (Fig. 28f).After nitrides have precipitated, the tool life decreases as a result of flakingat the cutting edge, caused by a decrease in the plasticity of the surfacelayer. The formation of a residual compressive stress also plays some rolein flaking, as these stresses are highest with a N solid solution.

In addition to the structure of the surface-engineered coating, the nat-ure of the coating–substrate interface is also of great importance. The adhe-sion of the coating is one of the principal factors (together with the thermalstability) determining the tool life. The interface must be free from brittlecompounds (such as oxides, nitrides, etc.) formed in the hardening processor during interaction with the environment. Several studies suggest thatthe surface of the tool should be polished to remove surface nitrides formedafter the ion treatment [50]. Surface cleaning is also effective when ion etch-ing is used, but the etching must be performed very carefully. The cuttingedges of a sharp tool should not be rounded, the surface roughness shouldnot increase and the tool dimensions should be kept to a close tolerance. Allthis is the subject of technological optimization, but with care, excellentresults can be achieved [45].

a. Friction and Wear Behavior and the Features of Self-Organizing ofSurface-Engineered Coatings. A surface-engineered coating can act as a‘‘protective screen’’ at the surface of a cutting tool (Fig. 27). Duringsteady-state wear, a gradual, but controlled wear of the coating takes place.All these advantages became even more obvious when surface-engineeredcoatings are applied. Tests done at increased cutting speeds (90m=min)(forfor HSS tools) enhance all the thermal processes associated with cutting.Under these conditions, the heat-insulating effect of a hard TiN coating isdiminished, the protective function of the coating is reduced, plastic defor-mation of the steel substrate can occur, and the stability of cutting is disrup-ted. All these trends can be seen in the data presented in Figs. 24b and 29.Hardening an M2 steel by a surface-engineered coating can be employedto counteract these effects. The wear value is considerably lower and thezone of stable cutting process is significantly broader (Fig. 29, curve 4).The best results are achieved when a substrate material (T15 HSS) havinga high heat resistance is used. The dissipation of energy is channeled intoprocesses other than surface damage, i.e. compatibility of the tool andworkpiece is realized to a great degree. The coating plays the role of a pro-tective screen for the contact surfaces. It should be emphasized that thesuccessful fulfillment of this function, however, is possible only when theexternal thermo-mechanical effects are localized in the coating layer. Studiesof coatingwear have shown that the intensive self-organizingprocess observedduring cutting only occurs when a surface-engineered coating was used.


The practical results of duplex surface-engineered coatings, i.e. a func-tionally graded tool material, are quite impressive. This material combines ahigh surface wear resistance (hard coating) and high core toughness (HSS).The tool life is increased by a factor of five to ten times [46], while at thesame time, the metalworking productivity can be increased by a factor oftwo to four. The cutting speeds of high-speed steel tools with duplex coat-ings (when cutting ordinary construction grades of steel) can be as high as130–150mmin�1. These cutting speeds are found with carbide tools onlyunder certain limited cutting conditions.

2. Multi-layered, Self-Lubricating Coatings

For transient or surface damaging friction conditions (e.g., during therunning-in or avalanche-like stages of wear), the efficiency of hard

Figure 29 Tool flank wear vs. time. Turning test data (cutting speed ¼ 90m=min;machined material, 1040 steel; depth of cutting ¼ 0.5mm; feed rate ¼ 0.28mm=rev.):(3) M2þPVD TiN coatings; (4) M2þion nitridingþPVD TiN coatings; (57) T15HSSþ ion nitridingþPVD TiN coatings.


coatings becomes questionable due to their brittleness. During themachining of several types of alloys (e.g. stainless steels or nickel-basedalloys), unstable conditions can dominate and surface damaging mechan-isms become prevalent. In this case, the ability of a thin surface layer toprotect the surface, well as dissipate most of the energy generated duringcutting, thereby minimizing the cracking of the tool, becomes criticallyimportant. This is a practical application of the universal principle of dis-sipative heterogeneity [47].

For the most demanding cutting applications a third type of coating—the self-lubricated hard coating—has been developed. A typical example ofthis type of development is the multi-layer coating, TiAlN–MoS2, with twoenergy-dissipating mechanisms built into the microstructure [48]. The first isassociated with the formation of an oxygen-containing secondary structure(SS-I) that readily forms at the surface of the hard coating (TiAlN) andplays the role of a solid lubricant. The second is associated with the thinMoS2 lubricating layer. A second example of a similar technology is theuse of nano-composite nc-TiN–BN coatings [49]. These coatings give goodresults at moderate cutting speeds. Following the earlier discussion, it seemslikely that a high-alloyed Ti–B–O secondary structure of the first type (SS I,see above) and B2O3 both form. The boron oxide plays the role of a liquidlubricant at the temperatures of cutting [50].

The most important phase of the self-organizing process is associatedwith the running-in stage of wear. During this stage of self-organization, thewear process gradually stabilizes and finally transforms to a stable (or nor-mal) stage [7]. It is very important to prevent surface damage and promoteintensive self-organization at the surface during the running-in stage of wearusing the phenomenon of screening [4,7]. The less surface damage at thebeginning of the normal stage of wear, the longer will be the tool life(Fig. 2).

Hard coatings are brittle and susceptible to extensive surface damageduring this running-in stage. Frequently, much of the hard coating is alreadydestroyed at this phase, prior to the start of the stable (normal) stage ofwear, where the wear rate can be lowered by an order of magnitude dueto the self-organizing of the system (Fig. 2). The initial surface damage oftenleads to a dramatic decline in the wear resistance of the coating. For this rea-son, a top layer with high anti-frictional properties is a critical component,and can be used to protect the surface of the hard coating. This is one of themost important goals for wear resistant coatings, especially at low and mod-erate cutting speeds, and for handling hard-to-machine materials whereadhesive wear dominates. This can be achieved by applying self-lubricated,multi-layer coatings. These structures have many complex microstructuralfeatures that contribute to energy dissipation [e.g. the TiAlN–MoS2 (or


MoST) coatings [51,52], discussed earlier]. One of the most effective com-mercial coatings of this type is the multi-layered TiAlN=WC-C hard lubri-cant coating developed by Balzers [53]. The main advantage of thiscoating is a very low initial wear rate, during the running-in stage of wear(Fig. 2) that leads to a significant increase in the tool life (Fig. 30). Recently,several oxides such as WO3, V2O5, and TiO2 [54] were found to exhibit goodtribological properties at elevated temperatures. All these oxides containcrystallographic shear planes with low shear strengths at high temperature[44]. They are promising materials as solid lubricants for elevated tempera-ture applications, and can be deposited by PVD methods.

The service performance of multi-layered coatings with an anti-frictiontop layer is characterized by the wear curves shown in Fig. 31. The top (anti-frictional) layer leads to a decrease in flank wear as soon as the running-instage is completed, and the tool life is significantly increased (Figs. 2 and31). Unfortunately, not every mode of the running-in phase leads to theoptimum self-organization [4,47], because damaging modes are also possi-ble, especially during cutting. Thus, the goal of friction control is to preventserious surface damage at the running-in stage and transform the tribosys-tem from its initial state into a self-organizing mode. If this can be achieved,

Figure 30 Tool life of end mills with advanced coatings. Machined material, 1040

steel. Parameters of cutting: speed (m=min): 21; depth (mm): 3.0; width (mm): 5; feed(mm=flute): 0.028; cutting with coolant.


the effective volume of interaction between the tool and the workpiece candrop by orders of magnitude. For severe conditions of cutting, the effectivethickness of the interaction volume at the self-organizing stage is in therange of 0.1–1.0 mm [4,55,56]. The high anti-frictional nature of the surfacelayer is necessary to achieve these goals [57].

An alternative type of anti-frictional surface layer has been success-fully applied for hard coatings—the Z-DOL layer [58,59]. Z-DOL[a 0.5% solution of perfluorine polyester acid (Rf-CH2OH) in freon 113]

Figure 31 Tool flank wear vs. time. Turning test data (with and without coolant).Cutting speed ¼ 90m=min; machined material, steel 1040; depth of cutting¼ 0.5mm,

feed rate¼ 0.25mm=rev.: (1) M2þ ion nitridingþPVD TiN coatings (with coolant);(2) M2þ ion nitridingþPVD TiN coatings (without coolant); (3) M2þ ionnitridingþPVD TiN coatingsþZ-DOL anti-frictional layer (with coolant); (4)M2þ ion nitridingþPVD TiN coatingsþTiþN layer, modified by ion mixing

(without coolant); (5) M2þ ‘‘smart’’ coating with programmable change ofproperties, combining coatings 3 and 4 (with coolant).


can be deposited by dipping the part into a boiling solution. The physico-chemical properties of Z-DOL are shown in Table 6. The thin film consistsof a close-packed molecular mono-layer, that provides an even coating to arough tool surface. This coating has a high adsorption ability and due to itslow thickness it also has high adhesion to the substrate and penetrates intopores. The surface energy of oils contained in the typical coolant regularlyused for machining is higher than the surface energy of the Z-DOL film. Asa result of the molecular interaction of the oil and Z-DOL film, the latterfilm is not sheared from the surface of the cutting tool during the first stagesof cutting. The principal function of the top anti-frictional layer (Fig. 32) isto increase the adaptability of cutting tools with hard nitride coatings.

The two surfaces are separated by a layer of oil that prevents seizureand wear during the initial stages of the tool service. Studies of surface-engi-neered coatings (a PVD TiCrN hard coating and a top layer of Z-DOL)deposited on a HSS substrate in contact with a 1040 steel show that the fric-tion characteristics are improved at the service temperature (5008C, Fig. 23).The tool life data (Fig. 31a) reflect a very low pattern of surface damage atthe running-in stage of wear, leading to a marked improvement in the overalltool performance.

3. ‘‘Smart’’, Multi-layered Wear Resistant Coatings

Similar problems of friction control at service conditions leading to sur-face damage arise when the wear process changes from the normal to the‘‘avalanche-like’’ stage. As noted above, cutting tools made of HSSusually operate under conditions of adhesive wear, where seizure mightoccur, accompanied by a rapid increase in the wear intensity [56]. Prolon-gation of the normal friction and wear stage, however, is quite feasible,even if seizure is a problem. This can be achieved by applying an

Table 6 Physico-chemical Propeties of Z-DOL [58,59]

Property Value

Molecular mass 2,194

Average number of units in the molecule 12Molecular dimensions of SAM 5nmDensity 1,560 kg=m3

Thickness of epilamon layer 5–2,500 nm

Load-carrying capacity 3GPaMaximum service temperature 723K


additional sublayer to the multi-layered coating at the surface of the toolsubstrate. This layer should combine anti-frictional properties with anability to generate protective secondary structures at the coating/substrateinterface.

One way to create these layers is by ion modification (ion alloying) ofthe surface of the tool. ‘‘Triplex’’, multi-layered coatings have been studied[60]. The coating in this study was deposited using three separate units. Ahigh-speed M2 steel was first nitrided using the glow discharge method.This was followed by ion implantation, prior to the application of a hard(Ti, Cr) N coating deposited by the PVD method. The (Ti, Cr) N coatingwas chosen for its high wear resistance [61]. Before applying the PVD coat-ing, the samples were implemented at room temperature with 60 keV ions toa total flux of 4� 1017ions=cm2. Sixteen different ions were chosen forstudy. Prior to ion implantation, the surface of the samples was etchedby argon ions and surface contamination was controlled during implanta-tion by the use of a cold trap that maintained a background pressure ofabout 2� 10�6 Torr.

The sixteen elements selected for this work can be grouped as

Figure 32 Schematic diagram of multi-layered ‘‘smart’’ PVD coatings for cutting

tools with a programmable change of properties:(1) anti-frictional layer (Z-DOL);(2) hard TiN PVD coating; (3) additional sublayer formed by (TiþN) ion mixing;(4) nitrided sublayer; (5) HSS substrate.(I) Running-in stage of wear; (II) normal

wear (steady-state) stage; (III) catastrophic or avalanche-like stage of wear.


follows:

(1) elements forming stable protective surface films under frictionalconditions [4], e.g. O, N, and Cl;

(2) non-metals (e.g., B, C, Si) forming compounds with goodtribological properties when they interact with base materials andelements in the environment; and

(3) metals including:(a) low-melting point elements (in particular In, Mg, Sn, Ga) used

as lubricants or anti-friction materials;(b) metals with a hexagonal lattice and anti-frictional properties

[62,63](c) metals (Al, Cr) that form stable oxide films during cutting, with

good anti-frictional properties, and a low coefficient of thermalconductivity; and

(d) metals (Ag, Cu) known to have a low coefficient of friction,and low mutual solubility when in contact with steel, nickel andtitanium alloys (Fig. 33) [63].

In addition, the study was extended to study surfaces subjected totreatments with:

– four types of anti-friction alloys used to improve conditions ofsliding friction, viz. Zn þ Al(9%) þ Cu(2%), Cu þ Pb(12%) þSn(8%), Pb þ Sn (1%) þ Cu (3%) and Al þ Sn(20%) þ Cu(1%) þSi(0.5%) [28];

– Zr þ N, W þ C, W þ N, Ti þ N, Al þ O, to create layers with ahigh wear and oxidation resistance.

The wear of these coatings was studied while turning 1045 carbonsteels at a cutting speed of 70m=min, a cutting depth of 0.5mm and a feedrate of 0.28mm=rev. with and without a coolant. The flank wear of tetrago-nal, indexable HSS inserts with multi-layered coatings was studied; when theflank wear exceeds 0.3mm, the cutting tool loses its serviceability [3]. Theeffectiveness of ion modification was determined by comparing the cuttingtime to reach a specified depth of wear of tools with multi-layered coatings(i.e., those having both surface-engineered coatings and ion modification)with identical surface-engineered coatings prepared without the additionalstep of ion modification. Adhesion was determined using the scratchmethod. Friction coefficients were determined with the aid of a speciallydesigned adhesiometer shown in Fig. 5 .

The results of these tests, summarized in Table 7, demonstrate to alarge extent that the influence of the implanted elements on the tool life isdetermined by the cutting conditions. The operational temperature during


high-speed cutting is at least 6008C. If a coolant is used, the temperature issignificantly reduced (by not less than 1008) [63]. Thus, the effects of implan-tation vary, depending on whether the cutting is aided by coolants or not.Ion implantation significantly affects the tool life [64], but the change in toollife is caused by a complex combination of interacting factors. The factorsthat are important in this context are:

– the formation of liquid phases or low-melting point eutectics whichact as lubricants;

– the development of amorphous, oxygen-containing films with lowcoefficients of friction and thermal conductivity; and

Figure 33 The mutual solubility of metals based on binary phase diagram data[69]. The data are characterized in ranges running from IV—low mutual solubility toI—high mutual solubility.


Table 7 Tool Life of Cutters with a Modified Surface Layer

(Ion Implemnetation and Ion Mixing)

N of group

(subgroup) Material

Element

composition

Coefficient of

PVD-coating

adhesion to

modified

surface base

Durability coefficient

on cutting

Without

coolant

With

coolant

Surface modified by ion implantation

1 Elements with

high affinity

for oxidation

O 0.25 0.9 1.25

N 0.41 2.0 1.83

I 0.7–0.8 3.2 0.7

Cl 1.8

2 Non-metals B 0.6 1.2 0.65

C 0.6 1.7 0.83

Si 0.7 0.6

3 Metals

a Low-melting In 0.6 2.4 2.1

Mg 0.25 3.0 0.08

Sn 0.6 0.8 0.7

Ga 2.0

b With hexagonal

lattice

Co 0.5 1.8 0.13

c Forming stable

oxides

Al 0.4 0.15 1.3

Cr 0.6 0.2 1.2

d With low

coefficient

of friction

Cu 0.55 1.0 2.5

Ag 0.4 3.1 2.7

Surface modified by anti-friction materials

4 Zn–Al–Cu

9–1,5

GOST

21437–75

(Russia)

Zn þ Al (9%)

þ Cu (2%)

0.44 1.98 —

Bronze 8–12 Cu þ Pb (11%)

þ Sn (9%)

0.4 0.95 —

(Continued)


a reduction in the adhesion of the tool surface to the processedmaterial and, at the same time, an increased adhesion of the hardPVD-coating to the modified base material.

The data from Table 7 show that a class of anti-frictional alloys, widelyused to improve the conditions of sliding friction [28,62], can double the toollife. However, this method of increasing the tool life, i.e. one that primarilydepends on a reduction in the strength of the adhesion bonds between the tooland workpiece, is not the most efficient, as the adhesion of the coating to themodified surface was found to be rather low. This precludes their usage, asde-cohesion of a coating cannot be tolerated in practical applications.

Implanting elements such as indium, silver and nitrogen enhances thetool life by a factor of 2–3 (see Table 7) for a range of cutting conditions(with and without cooling). These results are consistent with the observationthat indium and silver show little interaction with iron, and find use as solid-state lubricants (Fig. 33). Nitrogen implantation probably leads to the for-mation of an amorphous film with improved tribological characteristics [65].Ion modification of the tool surface with the other elements studied led tounstable or negative effects, i.e. a reduction in tool life and=or poor adhe-sion between the hard coating and the substrate.

Table 7 (Continued)

N of group

(subgroup) Material

Element

composition

Coefficient of

PVD-coating

adhesion to

modified

surface base

Durability coefficient

on cutting

Without

coolant

With

coolant

Babbitt BK2

GOST

1320–74

(Russia)

Pb þ Sn (1.5%) 0.35 0.6 —

Al–Sn–Cu

AO20–1

GOST 14113–69

(Russia)

Al þ Sn (20%)

þ Cu (1%)

þ Si (0.5%)

0.3 0.4 —

Surface modified by ion mixing of wear resistant elements

5 AlþO 0.4 3.0 —

TiþN 0.6 4.0 2.5

ZrþN — 0.53 —

WþN 0.4 0.4 —

WþC 0.4 1.33 —


The most beneficial element in this study was indium. The life of thetool was found to be a maximum, with or without the use of a coolant(see Table 7). At the same time, the adhesion between the coating andindium-modified surface of the tool was sufficient to ensure a reliable toolperformance. Indium is a surface-active metal and usually displays a low tri-bological compatibility with traditionally machined alloys based on steel,nickel, and titanium [63]. Because of this, the wear peculiarities of In-con-taining coatings have been comprehensively investigated [64].

Scanning electron microscopy and x-ray microanalysis were used tostudy surface-engineered cutting tools, composed of an ion-doped HSS sur-face, nitrided by a glow discharge technique, with a hard PVD coating overthe In-modified layer (Fig. 34a). Figure 34 shows the microstructure of a 58angle lap specimen (including the surface-engineered coating), taken in theSEM with the back-scattered electron signal which is sensitive to the meanatomic number. Separate layers of the multi-layered coating (dark for TiNand gray for the In-rich sublayer) can be seen in the back-scattered electronimage. The thickness of this zone is about 6.0 mm, so that the true depth ofthe modified (gray) layer is about 0.3 mm. It is probably a Fe-layer contain-ing implanted Ar (as a result of etching by Arþ after nitriding) and In. Thepresence of W in the tool steel increases the intensity of the x-ray In Ka

radiation and the background emission. This matrix effect influences theapparent emission volume of In Ka radiation and degrades the accuracyof measurement of the In distribution. In addition, surface heating (up to

Figure 34 Microstructure of the multi-layered HSS-base (Ti, Cr)N coating with anIn-modified surface (ion implantation). 600� magnification. (a) Microstructure ofthe angle lap section of the multi-layered coating (SEM image); (b) distribution of

elements along the II direction (x-ray microanalysis).


5008C) during (Ti, Cr) N deposition will modify the as-implanted In profile,which is expected to be about 0.3 mm in depth [66].

Following the x-ray microanalysis, the intensity ratios of the character-istic lines Lb=La for an In standard (99.99% purity) and the nitrided speci-men were found to be 0.63 and 0.97, respectively. Changes in the intensityof the characteristic x-ray fluorescence are frequently observed when pureelements and their chemical compounds are compared [67]. In this study,clusters of In–N are thought to develop in the zone of In implantation. SIMSdata (Fig. 35) demonstrated that the ratio of the In concentration in a freestate or present as clusters was approximately 10:1.

To explain how the implanted indium influences the tool life, the fol-lowing factors were investigated:

(1) the dependence of the friction coefficient on temperature;(2) the distinctive features of indium oxidation in the wear zone (as

investigated by SIMS); and(3) the development of oxides on heating specimens with an In-

modified surface.

Figure 35 Secondary ion mass spectra from the wear zone of the cutting tool(cutting time is 30min).


The temperature dependence of the friction coefficient demonstratedthat In improves the frictional properties of HSS (Fig. 36), by acting as alubricant and reducing the shear strength (t) of the adhesion bonds devel-oped in the tribo-couples. This factor, however, is probably insufficient to

Figure 36 Impact of test temperature on the frictional properties of surface

modified HSS cutting tools.


account for the twofold increase in the tool life of cutters having an In-mod-ified surface. Mass-spectrometric analysis of the wear zone (Fig. 35) suggeststhat the role of In is more complicated. Apart from metallic indium, thewear zone reveals the presence of indium oxide, coming from both In andIn–N dissociation and reaction during the wear process.

X-ray photoelectron spectroscopy (XPS) was used to study thechanges in the shape of the In 3d5=2 lines in the electron spectra after oxi-dation. Fig. 37a–d presents the spectra obtained before and after heatingthe specimens to 823K, with exposure times of 0, 0.5, 15, and 20min,

Figure 37 Change in the shape of the In 3d5=2 line from the photoelectron

spectrum taken from the HSS surface after ion implantation and oxidation at 823Kfor: (a) 0min; (b) 5min; (c) 15min; and (d) 20min. Pressure of oxygen in thechamber ¼ 2.5� 10�6 Pa.


respectively. The position of the In 3d5=2 peak in the starting sample cor-responds to a binding energy of 444.8 eV. Deconvolution of the spectrafrom the oxidized sample gave an additional peak, initially located atabout 445.7 eV and, after a 25-min exposure, at 445.8 eV. These higherbinding energies correspond to the formation of the oxide, In2O3. Therelative intensity of this line compared to In 3d5=2 (the ratio of IIn2O3=IIn)was 23% in the initial state (Fig. 37a), increasing to 41% after 25min(Fig. 37d).

Figure 38 presents the change in the relative concentration of In2O3 onthe surface of HSS specimens during heating at 423K, 623K, and 823K fortimes up to 25min. At 823K, oxidation of the implanted indium risesquickly and saturates after about a 25min exposure. The edge of a cuttingtool runs at a temperature of about 773K (5008C) during normal operations.These conditions suffice for oxidation of a fraction of the implanted indium.However, not all the indium is oxidized, as a part remains dissolved in solidsolution in the iron matrix.

As the hard overlay (TiCr) N-coating is worn away, typically at thetransition from the normal to catastrophic wear stage [43], the In-modifiedlayer becomes exposed at the friction surface. This usually coincides with the

Figure 38 Change in the relative concentration of In2O3 on the surface of a HSSspecimen after In-implantation and oxidation at temperatures: (1) 423K, (2) 623K,and (3) 823K.


point where the protective PVD-coating detaches from the contact face ofthe tool. Under conditions of high load and high temperature, partial oxida-tion of In will probably start before the complete destruction of the PVD-coating. Since normal friction is characterized by minimal depth of damageof the contact surface [4], even a relatively thin modified layer can enhancethe tool life. Indium improves the frictional properties of the surface andreduces the sticking intensity over the friction surface. In addition, an oxy-gen-containing amorphous In–O film, formed by interaction with the envir-onment, is likely to enhance favorable friction conditions in the contact zoneof a cutting tool. Indium lies in the same group as Al in the periodic table,and probably forms oxygen-containing phases having a low coefficient ofthermal conductivity. These protect the tool surface, enhance the thermalconditions of cutting, and delay the onset of catastrophic wear. Thus, theinfluence of In is twofold: on the one hand it acts as a metal lubricant; onthe other, it forms protective oxygen-containing phases. Indium enhancesboth the self-organization of the system and extends the stage of normaland stable wear, in accord with the principal laws of friction control [4].

The data presented in Table 7 show that the highest wear resistanceafter the triple surface treatment is achieved when transition metals togetherwith nitrogen are used to modify the surface by ion mixing. Compoundssuch as TiN, ZrN, WN, WC, Al2O3 do not appear to form, the implantedions remaining in solid solution. The best wear resistance is shown by a1 mm thick layer modified with Ti and N (Fig. 39a and c). At the same time,implantation can lead to amorphization of the surface layer (see Fig. 40).The decrease of the peaks intensity at Fourier transform at remote inter-atomic distances shows that this modifies the wear mechanism due to adelay in surface crack propagation [68].

The diffusion of the implanted nitrogen into the chip and the reverseflux of oxygen into the tool surface lead to a partial replacement ofimplanted nitrogen by oxygen during cutting (Fig. 39c). The rapid forma-tion of a protective secondary structure takes place (Fig. 41), since the initialstructure of the surface after mixing is similar to the structure of the filmsformed at the friction surface as a result of the self-organizing process.Ion mixing can produce thin surface layers with a fine, so-called, nano-crystalline structure [68]. As noted above, the secondary structures have asimilar microstructure, an amorphous supersaturated solid solution of oxy-gen (coming from the environment) having been formed by reaction with themetal component of the tool material [4,55]. Ion mixing enhances this pro-cess, which naturally evolves in the tribosystem during the self-organizingstage and results in the formation of a stable secondary structure. In thefinal stage of wear, oxygen from the environment penetrates through thenumerous pores and cracks in the PVD coating to the surface of the


Figure 39 Distribution of chemical elements close to the ‘‘built-up=wear crater’’

interface: (a) Initial stage of the Ti þ N layer, as modified by ion mixing. (b) After acutting time of 120 sec. Surface of (Ti, Cr)N PVD coating. (c) After a cutting time of600 sec. Surface of the Ti þ N layer, modified by ion mixing.


modified layer. Since this layer contains a very high density of point defects[68,69], the reaction with oxygen is rapid (Fig. 39). As the PVD coatingwears, this oxygen-rich layer can act to screen the ion modified (Ti þN) sur-face and protect it against subsurface damage. Thus, when the hard coatingis completely worn away, protective secondary structures (SSs) have already

Figure 40 Fourier transform from EELFS analysis of the surface of the Ti þ N

layer, as modified by ion mixing.

Figure 41 SIMS spectra of the surface of the Ti þN layer, modified by ion mixing:(a) initial stage of the Ti þN layer, as modified by ion mixing; (b) after a cutting timeof 600 sec.


formed on the near-surface layers. These secondary structures (SSs) delaythe transformation to the avalanche-like stage of tool wear, and thetribosystem can again revert to a stable state (i.e. to a normal pattern ofwear). From our point of view, this is the most beneficial effect of the mod-ified Ti þ N layer on the wear behavior.

The behavior of multi-layer coatings illustrates an important principlethat can be used to design effective materials for cutting tools. The moreenergy dissipation channels that can be built into the microstructure forthe transitional (non-steady) stage of tool wear, the longer will be the toollife. These channels can operate simultaneously in the same stage of the wear(e.g. during the running-in stage), but also subsequently, using multi-layercoatings, when the wear process transforms from one stage to another. Aftercompletion of the first stage of wear and exhaustion of the correspondingchannel of energy dissipation in the top layers, the next layer of a multi-layered coating can be used to control the ‘‘avalanche-like’’ wear with alter-native channels of energy dissipation [37]. In this way, a multi-layered,‘‘smart’’ coating can be developed where each layer fulfills a given functionat a definite stage of wear (Fig. 32), leading to high serviceability over a widerange of operating conditions. This concept has been widely used for protec-tive coatings, corrosion control [70], and, as shown above, this concept canbe extended to wear resistant coatings. At the stable stage of wear, the coat-ing must have adequate strength and toughness, and a stable SS. In theunstable stage(s), the coating must have sufficient energy dissipation chan-nels to prevent surface damage. A multi-layer coating that includes an adap-tive top layer, composed either of oxides with favorable friction properties(such as WO3, V2O5) or a self-lubricated layer, a working layer of asuperhard or self-protecting coating, and an anti-frictional sublayer formthe basis for future ‘‘smart’’ coatings.

4. SuperHard (nano-composite=superlattice) andSelf-Protecting Coatings

An alternative way of improving the performance of cutting tools reliesupon the deposition of multi-component compounds. Recent improvementsin the lifetime of cutting tools have been achieved by the development oftitanium aluminum nitride (Ti,Al) N coatings (see Fig. 30). The results ofmilling tests with TiAlN coatings have demonstrated that the wear behavioris improved by lowering the running-in wear and increasing the duration ofthe period of normal wear. This can be achieved when all the interactionsbetween the tool and workpiece are localized in a thin surface layer, i.e. astriking demonstration of ‘‘tool–workpiece’’ compatibility. Films such asTiAlN with a Ti=Al ratio of 1.0 [71,72] display a unique combination of


properties, viz. a high hardness at elevated temperature together withthermal and chemical stability (i.e., stability to diffusion, dissolution intothe chip and oxidation stability). Considerably, more heat is dissipated viachip removal. An extremely important advantage of (Ti, Al) N coatings istheir oxidation stability up to 850–9258C [73], due to the formation of stableoxide films (a mixture of rutile and alumina) [74,75], see Table 8. Stabletribo-ceramic films (SS-II) can then be formed on the surface during cutting[76] and limit the diffusion of the coating material into the workpiece. Thesecompounds are probably a mixture of alumina and rutile, as found in theoxidation of titanium aluminides [77,78].

Another recent development is the application of compounds thatensure stable friction and wear under high-speed=high-stress cutting condi-tions. Two methods have been developed for this application. The first isbased on the use of advanced multi-component coatings, i.e. the so-called‘‘superhard’’ coatings, with a room temperature hardness in excess of40GPa, and excellent oxidation resistance up to 10008C. Under high-speedmachining conditions the surface of the tool can reach 10008C, so the coat-ing should be stable at this temperature [68]. The development of nano-com-posite coatings with very fine grains (about 10 nm or less) illustrates thepotential of this class of material. The mechanical behavior of nano-compo-site materials may be controlled by the response of the grain boundary,because the number of atoms in the grain can be comparable to that inthe boundary regions. Grain boundary sliding can replace dislocation climband glide as the dominant plastic deformation mechanism. These materialscan be prepared only by methods that simultaneously ensure a high rate ofnucleation and a low rate of growth. Magnetron sputtering and filtered arcdeposition can be used for the production of nano-crystalline films [42,68].

Table 8 Oxidation Stability of the Compound (PVD,

CVD coating) [74]

Coating

Loss of oxidation stability

(max working, T 8C)

TiC 400

Ti(C,N) 450TiN 550ZrN 500=600CrN 650CrC 700TiAlN(50:50) 850

TiC þ Al2O3 1,200


In both methods, a highlyionized plasma ensures rapid crystal nucleation onthe one hand and very fast cooling rates on the other. The kinetic energy ofthe bombarding ions is transferred into very small volumes, of atomicdimensions, and the cooling rate of the film is high [68]. These are highlynon-equilibrium processes.

The most familiar type of superhard coating is nc-MeN=a-nitride,where Me¼Ti, W, V, Zr or other transition metals and a-nitride is anamorphous Si3N4 [79]. The high hardness is associated with the forma-tion of isolated nanocrystals of the nitride phase dispersed in an amor-phous matrix, so that dislocation motion and grain boundary slidingare suppressed. The second type of superhard nano-composite coating,nc-MeN=metal [68,80], relies on a combination of soft and hard materialssuch as Cu, Ni, Y, Ag, and Co with TiN (or other nitrides) [81]. Super-hard nano-composite films have a high hot hardness (beneficial for flankwear), a high resistance against crack formation, and increased thermaland chemical stability (beneficial for crater wear resistance), as shownby their high oxidation stability up to 11508C [82]. As noted before,the unique properties of these coatings, in particular the nc-TiN=a-Si3N4 or TiAlSiN coatings [83], are associated with the small dimensionsof the nanocrystals (1–10 nm). Strong segregation effects can lead to athermodynamic stabilization of the grain boundaries, with a high energyof activation for grain coarsening [79]. Coatings such as nc-TiN=a-Si3N4;nc-TiAlSiN, and nc-TiN-ncBN [83–87] are promising materials for cuttingtool application. TiAlN coatings could also be superhard [68] and showexcellent wear resistance at high-speed cutting.

Superlattice or multi-layer coatings with a superlattice period rangingfrom 5 to 10 nm have also been developed for cutting tool applications. Thebi-layers in these superlattice structures can be metal layers, nitrides, car-bides, or oxides of different materials or combinations of these compoundssuch as TiAlN=NbN; TiAlN=CrN or TiAlN=VN [88–90]. The mechanism ofhardening in these coatings is associated with the restriction of dislocationmotion across an interface or within the layer itself, due to the suppressionof the normal dislocation source and multiplication effects encountered inbulk materials [88–90].

If we summarize all the known data of superhard coatings and ion-mixed structures and compare them to the properties of dissipative struc-tures, it is apparent that the self-organization in these systems at extremelynon-equilibrium conditions of the coating deposition process can be asso-ciated with the formation of a stable, nano-structured material [91,92].A novel material (in the form of a thin film) is created at the surfacewhose characteristics are very similar to the extreme properties of the dis-sipative, secondary structures associated with friction. The material is both


very hard and as chemically stable as SS-II, while at the same time, itsstructure is similar to the amorphous state of SS-I. Thus, it is possibleto create an ‘‘artificial’’ material, possessing unique, but previously unat-tainable properties, and a new level of tool performance under the extremeconditions of high-speed cutting. This is an exciting yet practical realiza-tion of the underlying principles of friction control.

Further improvements in the use of coatings for high-speed machiningwill depend on a better understanding of the stable tribo-ceramics that format the surface of superhard nano-composites. To achieve this research goal,the elements of the coating composition that have an ability to act synergis-tically must be investigated. All known commercial coatings (e.g. TiN,TiCrN, TiAlN) and even ‘‘state-of-the-art’’, superhard coatings probablygenerate only tribo-ceramics such as rutile or mixtures of rutile and aluminaor rutile and chromia that possess limited stability at high-speed cutting.The generation of a stable continuous film of alumina on the surface couldbe a goal for the development of new coatings for high-speed cutting appli-cations. It has been proven indirectly that a thin layer of alumina wasformed on the surface of PVD TiAlN coatings, and multi-layered coatingswith excellent properties have been reported [93].

Recently, some companies (e.g. Bulzers, CemeCon) have started todeposit an aluminum-rich layer (65–75 at.%) in the coating to ensure the for-mation of a protective alumina layer on cutting [94]. The formation of an alu-mina-like SS-II on the surface of the coating during cutting might enhancethe tool life. For this type of coating, other alloying components in the hardcoating might act synergistically to promote the formation of stable tribo-ceramics. A promising composition of this type is based on TiAlCrN coatings[95]. It is known that certain ternary TiAlCr alloys form a very stable aluminalayer during high-temperature oxidation, in contrast to TiAl alloys where theoxide that forms is non-protective [96]. Several authors have reported on thebeneficial properties of these coatings for high-temperature applications [97].TiAlCrYN and TiAlN=CrN superlattice coatings show excellent oxidationresistance [90,98,99] while the former demonstrated promising wear resis-tance at elevated temperature [100]. The addition of Y drastically reducesthe grain size [101] and leads to superhard coatings. In ternary TiAlCr alloys,chromium forces the other elements to act synergistically, and forms a protec-tive alumina film at the surface. The development of the next generation ofcoatings for high-speedmachining could combine the properties of superhardcoatings, as outlined above, with an ability to generate a protective aluminalayer during cutting (the principle of self-protection).

An alternative technology is the use of stable ceramic coatings (e.g.alumina or zirconia) [102–104]. These ceramics are the most stable and wearresistant materials for high-speed cutting applications. Unfortunately, these


ceramics are brittle, but when they are used as thin films in coatings, thisproblem can be mitigated.

In this chapter, we have described how current coating technologiesmight be exploited to develop and bring new tribological tool materials tothe marketplace, based on the concept of a functionally graded microstruc-ture. We believe that in the near future this combination of surface engineer-ing and the further optimization of tribological materials will increase thewear resistance of cutting tools and lead to an increased productivity underthe extreme conditions encountered in high-speed machining.

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6Designing Fastening Systems

Christoph FriedrichRIBE Verbindungstechnik GmbH, Schwabach, Germany

This chapter gives a compressed, but comprehensive and clearly structuredview of the design of threaded fastening systems as an example for fasteningof components in general. Formulae are given to transfer the aspects to ownproblems, but there is no space to derive them, so they are proposed andconcentrated in figures. The aim is to present an effective engineering toolfor optimized fastening of components, especially if made of materials thisbook deals with. To realize this, some basic concepts are discussed to aidthe reader.

This chapter will provide the designer and engineer with a newapproach to threaded fastening systems.

I. NATURE OF FASTENING SYSTEMS IN GENERAL

Nearly every component has to be fastened to another, no matter of whichmaterial it is made. Engineering of fastening systems plays a fundamentalrole for all engineered products all over the world. But more importantly,the optimization of components due to more power, more reliability, lowervolume and weight as well as lower cost, because this leads to new require-ments for the fastening system: it has to work with new materials (oftenwith low strength and significant thermal expansion behavior), highertemperatures, it must transmit all loadings within a small material volume,it has to work without failure for a long period and must guarantee agood appearance for the operating time. In many cases, the fastening systemhas to be extremely low priced, because there is almost no money left tofasten the ‘very carefully designed and expensive components’ in anadequate way. Besides this, the fastening system has to function well even


if misuse occurs, so that an accident by overloading-failure is avoidedwhenever possible.

To meet these different requirements, a lot of interdisciplinary physi-cal, chemical, and engineering knowledge is necessary as well as an effectivemethod to bring these disciplines together. Therefore, the aim of this chapteris to provide a guideline for optimized fastening systems, shown with boltedjoints as an example. But, the same aspects are also valid for other fasteningtechniques.

This chapter is written for the use in an engineer’s daily life. So, theprocess of designing and a comprehensive point of view are very important,to deal with some particular technical details, which are already investigatedvery extensively and documented.

Another point is that a fast design process has become more and moreimportant in the competitive industry—to realize this an early integration offastening engineering in the design process is necessary—this chapter showsthe reasons for these aspects.

Figure 1 presents a fastening system based on the view of general sys-tems theory, which was founded by the Society for General SystemsResearch in the year 1954 in London [6].

Each product is a component system, and if it has fastened compo-nents, at least two components are included (nos. 1 and 2 in Fig. 1). Eachcomponent is characterized by its geometry, its material, and its productionprocess, which leads to inhomogeneities in material properties or limited tol-erances as well as limited surface finish.

Of importance is that the whole product (component–system) has tobe optimized for maximum functionality due to the end-user. Also, the load-ing during the product life cycle—mechanical, thermal or reactive—isrelated to the entire product.

Any connection between two or more components is a fasteningsystem, which can be also characterized as a subsystem of the product.Figure 1 shows two very often used fastening techniques of engineering(bolted joint and welding). The basic difference between both the fasteningprocesses is that bolted joints need an additional fastening element (3),whereas welding needs working materials [e.g. filler rod or protective gas,(3)] and much more energy. Of course, a bolted joint can be disassembledand reassembled easily in contrast to weldings—this is oneimportant reason for the worldwide practice of using screws and bolts(and if we think about increasing recycling requirements, the importanceof fastening systems with the possibility for easy disassembly will grow inthe future).

In practice, often specific boundary conditions of the manufacturersare decisive, and based on this, the type of fastening system is selected. These


boundary conditions are named in Fig. 1, e.g. existing assembly equipmentor existing qualified and experienced personnel.

Finally, the fastening technique must be suitable for the product.Relevant aspects are summarized in Fig. 1. For example, compatibility ofthe fastening technique with the components (e.g. sufficient materialsstrength to transmit the preload force of a bolt or use of weldablematerials).

The following sections focus on threaded fastening systems as anexample for heavy-duty fastening of components with high relevance

Figure 1 Fastening system as a subsystem of product and typical characteristics.


and reliability. The experience with threaded fastening systems overmany decades and detailed calculation methods are useful base for everydesign engineer. For other fastening techniques like shaft keys, adhesivebonding, clamping devices, pins, or retaining rings, see Refs.[3,4,5,8,55,65,68].

II. CHARACTERISTICS OF THREADED FASTENINGS

For understanding the design process of a threaded fastening system, wehave to look closely at the different screw design sections A–D and compo-nent design sections E, F like that shown in Fig. 2. These are: engaged screwthread, free screw thread, screw shank and screw head resp. clamped partand nut thread component. If a screw (3) is tightened to a certain preloadFp, a closed flow of preload is generated. If the bolted joint is operating overits period of life cycle, a superposition of residual preload and additionaloperating load is acting.

This flow causes tensile stress in sections B and C. Compressivestress is generated in the clamped part (2 resp. F) and a combination of com-pressive stress as well as shear stress occurs in the nut thread component(1 resp. E) and in sections A and D.

This inhomogeneity of stresses is the reason why the behavior ofbolted joints is very complex and has to be designed carefully. The sche-matic diagram to the right of Fig. 2 shows another stress inhomogeneitycaused by different cross-sections of the screw shank and stress concentra-tion effects of the notched thread geometry. The result is that at theposition of the first bearing screw thread flank a significant stress concent-ration in the range of 5–8� due to the existing average stress level. This isvery important, especially for dynamic behavior of bolted joints. Since thisalso causes the beginning of local material plastification of the screw in thefirst bearing thread flank, even if the screw is tightened only to 1=5 to 1=8of its yield limit. Therefore, the material for screws has to be ductile (as alower limit from experience the fracture toughness of a screw has to be lar-ger than 5%).

Table 1 gives an overview over the six basic design criteria of athreaded fastening system, which have to be valid in any case. These criteriaare dependent on the entire fastening system, and not merely on the screwitself. For example, the stability of a preloaded bolted joint over the periodof use, depends on all stressed components within the flow of preload (no.1 in Table 1, comp. also Fig. 2). According to no. 2 in Table 1, thedynamic loading capacity of a bolted joint is especially influenced by the


local stress peaks and these local stress peaks can be determined by thedesign engineer. A screw shank is always under high level tensile stressso a fastening design must avoid screw material conditions which are sen-sible for stress corrosion cracking (no. 3 in Table 1, e.g. screws made ofhigh-strength aluminum alloy with wrong heat treatment without safe sur-face protection from electrolytes during operation). Self-loosening effectsare highly dependent on the vibrational (transversal) loading of the fasten-ing system and on the contact conditions between components and screw;so this is also a significant system behavior (no. 4, in Table 1). Numbers 5and 6 in Table 1 are self-explanatory.

A. Screw—Basic Details

Today, many details for design of screws are known. For standardizednomenclature of screw geometries, see ISO 1891 [26], for concepts of thread

Figure 2 Flow of preload, design sections, and stress concentrations of a boltedjoint.


system see Ref. [36], for general tolerances, see Ref. [35], for technicaldrawings see Refs. [38,49]. Figure 3 gives an overview of the basic detailsof an optimized screw which have been established for determining themechanical behavior.

Table 1 Basic Design Criteria for Threaded Fastening Systems

No. Criteria Remarks

1 Sufficient preload

for the period of use

Includes worst case of tightening,

worst case of friction in thread and head

support contact zones as well as worst

case of relaxation effects (see also Fig. 18).

2 Sufficient loading capacity

to resist all operating loads


Includes static and dynamic loading

capacity, includes screw thread, screw shank,

screw head; material properties must be stable

over the period of use (e.g. no hydrogen-

embrittlement), see also Fig. 50.

3 Sufficient corrosion resistance


Includes all kinds of corrosion, also stress

corrosion cracking under tensile stress; for

combination of different materials in a

bolted joint galvanic corrosion is very

important, see also Table 5.

4 No self loosening Includes both, extreme decrease of preload

and loosening of the screw; only a few

solutions give a safe prevention from

self-loosening, see also Figs. 41 and 76.

5 Easy assembly=disassembly

and high stability of assembly

process

Includes screw drive design (Fig. 35),

accessibility of fastening system, assembly

device, tightening method (Fig. 18); these

factors are responsible for a guaranteed high

preload and, therefore, are responsible for

the function of the fastening system; assembly

cost is, in any case, a very important cost

factor, see also Fig. 41.

6 Low cost target for the

fastening system

Includes not only low price for screw, but

also for total life cycle of product=fastening

system (e.g. also cost for storage and

logistics, assembly cost, failure cost, cost of

maintenance and repairing, quality cost; as

a rough approximation purchasing cost of

screw is under 20% of total cost for fastening

system), see also Ref. [65] and Fig. 56.


For a given thread size, thread type, and screw material, the shanktype, underhead fillet, and drive type have to be designed in such a mannerso that the ‘‘screw design principle’’ is achieved. This means that if loadingor overloading the screw, a quasistatic screw failure always has to takeplace in screw section B or C referred to Fig. 2. Fatigue failure has to belocated in section B or between A and B (first load bearing thread flankof screw).

The most used thread type is metric thread of the coarse series [definedin ISO 68 [40], ISO 724 [42], ISO 965 [47]]. This thread geometry has a flankangle of 608 and is defined by its diameter and its pitch—therefore, all ISOscrew threads use the designation: ‘‘diameter’’ � ‘‘pitch-value’’.

The selection of shank type influences the elastic resilience of the screwand therefore affects the entire fastening system. The higher the screw resi-lience without changing of the clamped part, the lower is the additionalstressing of the screw under external operating force. This advantage is uti-lized for a screw with wasted shank. The opposite is true for an increased

Figure 3 Basic details of screws established for high-duty bolted joints. (FromRef. 17.)


shank which is used to obtain an additional centering function of the screw.A full shank is typical for screws with cut thread whereas threaded shankand reduced shank are typical for screws with rolled thread. To design ascrew with high resilience, good yielding, and plastification behavior as wellas low cost, a threaded shank should always be the first choice. Wastedshank and increased shank are relatively expensive because normally a turn-ing operation is needed.

The screw surface is responsible for appearance as well as for both cor-rosive and frictional behavior—therefore, the screw surface is very impor-tant for the designed operational behavior.

Support diameter under screw head and thread engagement deter-mines the surface contact pressure at the two contact zones of a screw.The thread engagement also influences the thread stripping load—thisstripping load can destroy the nut thread or screw thread depending onthe material strengths and diameter tolerances. It always has to be higherthan the failure load of screw section B or C, so thread engagement hasto be designed with sufficient length.

The four most important geometries for high duty bolted joints aretaken: hexagon, bihexagon, triple square and hexalobular. The plain sup-port type is used most often, but the others, countersunk and ball section,provide a better self-loosening resistance and a centering function of thescrew.

Options like washers, thread ends, and cone points or adhesivesagainst self-loosening are not covered in Fig. 3 (but see Fig. 41). All thedescribed aspects are also valid, if a headless stud with nut is used insteadof a screw, then a second thread engagement between stud and nut has tobe considered (Fig. 46).

The basic screw details are treated more precisely in sections below. Inthis chapter, the reader would get information on general interactions ofscrew details and design criteria.

1. Standard Thread Geometry for Existing Nut Thread

When using a screw, the selection of thread type is a fundamental aspect forall mechanics of the bolted joint. Normally, a worldwide compatibility isimportant—this is the reason for the extensive use of metric thread system.Figure 4 proposes the metric thread system as an example of high loadingcapacity thread. The basic dimensions are characterized like those shownon left side of Fig. 4 by a 608-thread angle and a fundamental triangle withheight H (see also Ref. [42]). The external screw thread flank tips are cutwith a width of an eighth of the pitch P. The internal nut thread flank tipsare cut to a quarter of the pitch P. For the basic dimensions, the major


diameter (d, D), the pitch diameter (d2, D2) and the minor diameter (d1, D1)are the same for external and internal thread. The pitch diameter is definedfrom an imaginary cylinder whose external surface cuts a screw threadwhere the width of the ridge and the groove of the thread are equal [36].

To provide a solution that can be put into practice, the basic threadprofile has to be added by tolerances and radii at thread roots, which isshown in right side of Fig. 4. Then any diameter must be distinguished inminimum and maximum value and corresponding external=internaldiameters are different (external: small letters, internal: capital letters, e.g.D > d).

In ISO 965 [47], the detailed tolerance positions and tolerance fieldsare standardized. The right side of Fig. 4 also provides the tolerance posi-tions from ISO 965: e, f, g, h for the bolt thread and G, H for the internalnut thread, whereas the combination of g, H is most used. Combinationswith even more clearance are used for bolts with thick coatings, e.g. forenhanced corrosion protection. The tolerance field is between 4 and 7, nor-mally 6 is used—that means a combination of 6g for external screw threadand 6H for internal nut thread.

The notation of symbols in Fig. 4 is done strictly according to ISO 965.Table 2 provides numeric values for some selected thread sizes

M5–M30.In the left column, after each thread size, the pitch value is added. For

coarse series, the pitch value is written in parenthesis. From the table, it is

Figure 4 Metric thread system for fastening.


Table 2 Numeric Values of Metric Thread Diameters with Tolerance and Nominal Stress Area

No.

Nominal

Size

Screw thread (External)

Tolerance

(ISO 965)

dmin

(mm)

dmax

(mm)

d2min

(mm)

d2max

(mm)

d3min

(mm)

d3max

(mm)

As

(mm2)

1 M5 (� 0.80) 6g 4.826 4.976 4.361 4.456 3.869 3.995 14.2

2 M6 (� 1.00) 6g 5.794 5.974 5.212 5.324 4.596 4.747 20.1

3 M8 (� 1.25) 6g 7.760 7.972 7.042 7.160 6.272 6.438 36.6

4 M10 (� 1.50) 6g 9.732 9.968 8.862 8.994 7.938 8.128 58.0

5 M12 (� 1.75) 6g 11.701 11.966 10.679 10.829 9.602 9.819 84.3

6 M12� 1.50 6g 11.732 11.968 10.854 10.994 9.930 10.128 88.1

7 M12 (� 1.75) 6e 11.664 11.929 10.642 10.792 9.565 9.782 84.3

8 M12 (� 1.75) 4h 11.830 12.000 10.768 10.863 9.691 9.853 84.3

9 M16 (� 2.00) 6g 15.682 15.962 14.503 14.663 13.271 13.508 156.7

10 M30 (� 3.50) 6g 29.522 29.947 27.462 27.674 25.306 25.653 560.6

11 M30� 2.00 6g 29.682 29.962 28.493 28.663 27.261 27.508 580.4

Nut thread (internal)

Tolerance

(ISO 965)

Dmina

(mm)

Dmaxb

(mm)

D2min

(mm)

D2max

(mm)

D1min

(mm)

D1max

(mm)

12 M5 (� 0.80) 6H 5.000 5.1 4.480 4.605 4.134 4.334 —

13 M6 (� 1.00) 6H 6.000 5.12 5.350 5.500 4.917 5.153 —

14 M8 (� 1.25) 6H 8.000 8.15 7.188 7.348 6.647 6.912 —

15 M10 (� 1.50) 6H 10.000 10.2 9.026 9.206 8.376 8.676 —

16 M12 (� 1.75) 6H 12.000 12.2 10.863 11.063 10.106 10.441 —

17 M12� 1.50 6H 12.000 12.2 11.026 11.216 10.376 10.676 —

18 M12� 1.50 7H 12.000 12.25 11.026 11.262 10.376 10.751 —

19 M16 (� 2.00) 6H 16.000 16.4 14.701 14.931 13.835 14.210 —

20 M30 (� 3.50) 6H 30.000 30.7 27.727 28.007 26.211 26.771 —

21 M30� 2.00 6H 30.000 30.7 28.701 28.925 27.835 28.210 —

aalmost no root radius Rnmax.bvalues for guidance with root radius Rnmax=P=16.


obvious that the permissible difference between minimum and maximumdiameter using a tolerance grade of 6 is about 1–3% of the minimum value(relatively high deviations for small thread size). For a tolerance grade of 7,this deviation is about 2–6% (lines 17, 18 of Table 2); for a tolerance gradeof 4, this deviation is about 0.5–2% (lines 5, 8).

For M12 and M30 also, the influence of pitch value is considered (linenumbers (5, 6); (10, 11); (16, 17); (20, 21); coarse and fine series). For boltthread, it is typical that the diameters dmin and dmax do not vary much, butthe flank diameters d2min and d2max and especially the root diameters d3min

and d3max are different between coarse and fine thread series. This leads toa larger cross-section of a threaded screw shank with fine pitch than withcoarse pitch (compare also nominal stress areas As in right column of Table2). A fine pitch also leads to a small helix angle (lead angle or pitch angle),which produces a higher preload for given thread size and has a higherself-loosening resistance, which is more sensible for thread flank deforma-tions and defects as a result of transportation and handling (see also Fig. 55).

The influence of tolerance position can be estimated from lines (5, 7) ofTable 2 (comparison between 6g and 6e for same pitch value). Between thesetwo lines, the corresponding diameters are about 40 mm smaller for e-posi-tion than for g-position.

More information about thread dimensions can be found in Refs.[3,51,56,62] or other handbooks dealing with threaded fastenings.

Figure 5 compares ISO metric thread system with ANSI unified threadsystem. Both systems have the same basic thread profile. The effect on the leftside of Fig. 5 is due to coarse series and the right side deals with fine series.

The designation of metric thread system starts with ‘M’ and followsthe nominal screw diameter and the pitch (for standard coarse series thepitch is usually not written). The unified thread system begins with ‘UNC’resp. ‘UNF’, followed by the nominal screw diameter and the number ofthreads per inch. The two scales show the thread sizes that have similardimension.

If comparing the pitch values between metric and unified for corre-sponding diameters, metric system always has smaller values for same series.

2. Thread Rolling Screws for Medium-Strength Materials

Producing a bolted joint involves different steps like those shown in Fig. 6a.For this reason, the price per screw piece is less important, relative to thetotal cost of a threaded fastening system is done. From this point ofview, if a screw can produce its nut thread by itself during assembly, it isa significant advantage for the situation of total cost, because the nut threadcomponent can be produced with lower expense (Fig. 6, part b). For casting


components, often the manufacturing of a pilot hole can be integrated in thecasting process; therefore the drilling or stamping operation is avoided. As aconsequence, the production lines can work more quickly and with higherproductivity.

Another point, which will be more and more important in the future, isthe extensive use of cooling lubricant or cutting oil in production lines. Here,thread rolling screws also offer the possibility to eliminate such environmen-tally unfriendly fluids.

Thread rolling screws produce their nut thread by material rollingwithout chipping. In this case, the screw is both the thread rolling tool forthe nut thread and also the fastening element to generate the preload. Toobtain this in an effective way, the rolling screws at least have a formingpoint with lobularity and reduced diameter (Fig. 7). In most cases lobularityinvolves a screw geometry with three forming zones, therefore, the name‘trilobular screws’ (see cross-section in Fig. 7). The difference between cir-cumferential circle and minimum cross-section contour is about 4% of thescrew diameter. For more information about thread rolling screws, seeRefs. [57,63].

If the rolling screw is designed in the right way, a replacement by ascrew with standard thread is possible after disassembling the rolling screw.To obtain this, the thread diameters of the rolling screw must be oversized

Figure 5 Comparison of metric thread system with unified thread system.


compared with the thread standard to eliminate elastic resilience of the nutthread flanks.

The screw material must provide a high strength, so normally har-dened steel alloys may utilize an additional surface treatment (case harden-ing, induction hardening). The permissible strength of the nut threadmaterial is limited—as a rough approximation, to the strength of the nutthread material can reach half of the screw strength at forming point.Therefore, maximum nut thread material strength is about 600–700MPa(87–101 ksi). Of course, the nut thread material must provide sufficient

Figure 6 Comparison of procedures for producing a bolted joint using screws for(a) existing nut thread and (b) rolling screws.


ductility and formability. The flanks can be formed by the screw—here arough approximation is > 5% fracture toughness in tensile test, so manysteel alloys are covered. Aluminum alloys are recommended materials forthread rolling screws, whereas brittle materials such as gray cast iron ormaterials with hexagonal crystal structure lead to a low process stabilityduring assembly. Magnesium alloys (also hexagonal structure) thread roll-ing screws can be used, if a careful adaptation of all influences is considered(see also Fig. 64).

The pilot hole can be drilled, stamped, or cast. The pilot diameter is inthe range of the screw flank diameter d2. The tolerance of the pilot diameter(normally tolerance field and grade H11 is used) influences the forming tor-que necessary to bring the screw to head contact. This forming torque canbe decreased by a low friction film on the screw surface (app. half of theforming torque with suitable low friction film than without).

Table 3 illustrates the diameter of the pilot hole as well as a range offorming torque and tightening torque dependent on the screw strength forhigh-duty bolted joints with thread rolling screws. The tightening torquemust be distinguished by using the rolling screw in blind hole or bringingthe forming point out of thread engagement during tightening. This deter-mines the residual forming torque which cannot be utilized for preload gen-eration. In a blind hole, the tightening torque must be higher for producingthe same preload compared to a situation with through hole and outstand-ing forming point of the screw (see also Fig. 67).

The rolled nut thread has two significant technical characteristics. (1)There is almost no clearance between the screw thread and the nut thread,so the prevailing torque is rather high and the behavior against self loosen-ing is improved after retightening (clearance-like theoretical profile inthe left side of Fig. 4). (2) The nut thread material is strain hardened after

Figure 7 Basics of rolling screws—geometry and materials.


rolling of the thread flanks; so that the local strength of the nut thread mate-rial is increased which can enhance the loading capacity of the nut thread.

3. Thread Types for Low-strength Materials

Materials with low tensile strength with respect to a low yield point are notable to transmit high preload forces using high contact pressure. If compo-nents made of those materials with an external or internal thread, a large

Table 3 Selected Characteristic Values for Assembly of Thread Rolling Screws [63]

No.Threadsize

Pilotdiameterof hole(mm)

Formingtorque(Nm)

Propertyclass ofscrew

Tighteningtorque

blind hole(Nm)

Tighteningtorque

through hole(Nm)

1 M5 (� 0.80) 4.50–4.65 1.5–3.5 8.8 6.4 5.510.9 9.3 8.0

12.9 10.9 9.42 M6 (� 1.00) 5.45–5.60 2.5–6.0 8.8 11.6 9.9

10.9 16.3 14.0

12.9 18.9 12.93 M8 (� 1.25) 7.40–7.60 7–15 8.8 27.3 23.4

10.9 38.9 33.3

12.9 45.2 38.74 M10 (� 1.50) 9.30–9.50 15–30 8.8 53.6 45.9

10.9 78.8 67.5

12.9 91.4 78.35 M12 (� 1.75) 11.10–11.35 25–52 8.8 91.4 78.3

10.9 134.4 115.212.9 157.5 135.0

6 M12 � 1.50 11.20–11.40 27–55 8.8 96.6 82.810.9 141.8 121.512.9 165.9 142.2

7 M16 (� 2.00) 15.10–15.40 55–115 8.8 231 19810.9 336 28812.9 399 342

Remarks: Length of thread engagement must be sufficient to avoid stripping of nut thread

(app. 2–2.5� diameter of screw); forming torque depends on various parameters, such as pilot

diameter, surface conditions, lubrication; optimized assembly of rolling screws must be

determined with tests experimentally; for screws with hardened surface, the tightening torque of

core strength is relevant; thread rolling in thin sheet metals with extruded rim hole is possible,

if the pilot diameter is reduced by app. 4% of the value from this table.


flank engagement is required. This can be realized either by deep threadengagement and=or large radial overlapping.

Figure 8 illustrates the influence of thread angle a with respect to flankangle b. Part (a) demonstrates the situation for metric thread geometry witha¼ 608. An axial preload Fp leads to corresponding radial Force Fr, depen-dent on the flank angle b. For same axial preload Fp, the radial part Fr canbe reduced, either by asymmetric flank geometry (part b of Fig. 8) or smallthread angle (part c of Fig. 8).

A small thread angle leads to thin screw thread flanks. This is no pro-blem if the tensile strength of screw is much higher than that of the nutthread material.

In case of forming the nut thread during assembly for the same over-lapping distance x, a small thread angle (part c in Fig. 8) reaches only �60%of area A compared to (a) and (b) of Fig. 8. This area A determines thematerial volume of nut thread which has to be deformed during thread roll-ing for a radial flank engagement (distance x).

Usually these aspects are reasonable for thread flanks with smallthread angle for nut thread materials with low tensile strength, such asaluminum, magnesium, or even plastics as a rolling screw.

Figure 9 shows as an example a thread rolling screw for magnesiumcomponents with reduced thread angle, so that the deformed volume ofnut thread for a given flank engagement is minimized. Figure 10 shows an

Figure 8 Details of thread flank geometry—thread types suitable for low-strength

materials.


Figure 9 Special thread geometry for thread rolling in magnesium components, a5mm screw made of aluminum RIBE63, total thread engagement from screw pointto free thread 8mm.

Figure 10 Asymmetric thread profile.


example for an asymmetric thread profile, which leads to reduced radialforce Fr under preload.

Figure 11 shows another example for a thread geometry with lowthread angle. This screw can be used for fastening of components withthread rolling in both plastic materials and light metals. The forming torquefor screw of dimension M5 is � 4Nm in aluminum. For further perfor-mance of such thread geometry, see Fig. 75.

4. Elastic Screw Elongation Under Tensile Loading

Any mechanical loading of an elastic material leads to a correspondingdeformation. For a screw, which is a component with different cylindric sec-tions, the most important loading is the preload in axial direction. The screwelongation under tensile loading is defined by the term ‘axial elastic resili-ence of screw ds’. This means the ratio of length-changing Dl over theapplied axial force F.

‘

Figure 11 A screw with thread geometry for thread rolling in both plasticmaterials and light metals.


Figure 12 emphasizes the situation schematically. The upper screwis unloaded, so it has its original length and diameter values are d, d3, dsh,lsh, lft, te respectively. The fixed point of the mechanical model isdefined for the nut thread component lying on the screw axis. In con-trast to the upper screw, the lower screw is loaded by a symmetric axialforce F under head (F can be the sum of preload Fp and axial operatingload Fax). This leads to a screw-elongation Dl, which is contributedmainly by four parts of ds: the resilience of head dh, the resilience ofscrew shank dsh, the resilience of free thread dft and the resilience ofengaged thread det. The commonly used calculation formulae are alsoshown in Fig. 12 after Ref. [70]. It is clear that these formulae only givean approximation, especially for dh and det (not taking into considera-tion the height of head as well as of drive type or tolerances betweennut thread and screw thread or length of thread engagement or modulusof elasticity of nut thread component En).

Figure 12 Definition and determination of the axial elastic resilience of a screw.

(From Ref. 17.)


B. Nut Thread Component

The nut thread component can be a standardized nut or a mechanicalcomponent with nut thread. The main difference between these for the samematerial and same thread geometry is a variation of the stiffness of the nutthread component under mechanical loading as Fig. 13 shows.

The hexagon nut of part (a) in Fig. 13 shows shortening and wideningof its contour under an axial force F. Therefore, the height of a standard nutshould be 1� nominal diameter of screw. At this height, two chamfers areincluded (�0.8 � pitch P of thread). This height is necessary for overelastictightening without nut thread failure (if the nut thread material has samestrength as screw thread material, e.g. same property classes; see also Fig.18). The height of 1� d is also defined in modern standards for nut geome-try such as ISO 4161 [32]. For a combination of different materials betweenthe screw thread and nut thread, see Fig. 33. Screw threads manufactured inhigh series production can have incomplete thread flanks in a length rangeof (2–3) � pitch of thread. The screw should extend through the nut for thisdistance.

In contrast, part (b) of Fig. 13 contains a nut thread produced in abulk material of a component. Caused by the much higher stiffness underloading F this configuration shows almost no deformation. The relevantthread engagement te has to be measured excluding a chamfer and excludingany screw point and reduced additionally by 1 � P for incomplete threadflanks.

The local axial stress distribution on the nut thread component isdrawn schematically for both cases (a) and (b) in the middle of Fig. 13.

Figure 13 Stiffness of a nut thread component under mechanical loading. (a)Standard nut and (b) bulk material with a nut thread.


It is significant that a stress peak occurs in the first bearing nut thread flank(same as in first bearing screw thread flank). This means that the material ofnut thread should have a sufficient ductility to compensate for this stresspeak without cracking or failure. For brittle materials, this stress peak isthe critical aspect. In this case, large thread engagement should be used toreduce the stress peak or consider actions as shown in Fig. 41 for improvingthe fatigue loading capacity.

Often nut thread components are used, which do not provide sufficientthread engagement for screw failure in the case of overloading (see designprinciple, Fig. 2). In those situations, the (nut) thread is stripped and thedesign engineer has to maximize the number of full engaged thread flanksand the materials strength of nut thread component. Besides this, the tigh-tening specification has to be determined carefully, and the documentationfor production and field service must be performed accurately (see aspects ofquality management, Fig. 55).

For optimized fastening solutions, special types of nut thread compo-nents exist (Fig. 14). The use of a nut thread insert is suitable for high axialpreload which has to be transmitted to a soft nut thread material (e.g. pre-vention of relaxation). The reason for the improvement compared to usewithout the insert is that the contact zone of the soft material is increasedsignificantly by the outer diameter of the insert. But always remember that

Figure 14 Special types of nut thread components.


an insert is an additional fastening element which adds to the cost,logistics, and assembly. For more information regarding inserts, see Ref.[14].

Staking nut, shear nut, and blind rivet nut are designed for thin wallednut thread components which provide no material for engaged threadflanks. The staking nut [64] has the advantage of self-centering, the shearnut [58] provides no deformation=forming of nut thread material, andthe blind rivet nut [69] needs only one side access to the component (see alsoblind rivet, Fig. 70). An important field of application for special types ofnut thread components is nonweldable materials, where weld nuts cannotbe used (e.g. metallic foam structures or reinforced fiber material for light-weight design).

Another possibility for thin walled components is using a clip nut(most established for fastening of plastic components) or direct thread tap-ping into rim holes of metal sheets with a screw having a small diameter atthe forming point (pilot hole is only �85% of nominal screw diameter d;technique belongs to thread rolling screws). A clip nut has to be made ofhigh-strength material (e.g. spring steel) because there is only one flankengaged between screw thread and nut thread. For this reason, the screwmaterial requires high tensile strength to obtain high failure torques forthe threaded fastening system.

C. Clamped Part

A clamped part has to be provided with a through hole and on both sides of asupport area to transmit the preload by using clamping force between thescrewheadandnut thread component.The geometric requirements are simple.In ISO 273 [28], the size of clearance holes in clamped parts are standardized.

But the stress distribution in the clamped part under preload andunder loading during operation is very difficult and varies over a wide rangewhich is dependent on the geometry, material, and type of loading. There-fore, numeric calculations like FEM are important to analyze the local stres-sing of clamped parts. In the next step, the numeric results must betransferred to the analytical calculation of the particular threaded fasteningsystem.

The following linear model provides a fundamental approach to stres-sing and deforming of clamped parts (Fig. 15). If a clamped part has thinwalls like a tube with Dp < da, the whole cross-section of the clamped partis stressed homogeneously (sketches (a) and (b) of Fig. 15). Betweenunloaded and loaded situation, the clamped part shows a shortening Dl,which can be obtained from Dl¼ dpF. In this case, the axial elastic resilience


can be calculated from dp¼ lc=(Epp(Dp2� dh

2)=4). Sketch (c) of Fig. 15 givesan example for a component with more detailed geometry. The localstress in the clamped part under axial load F is dependent on the particularlocation within the clamped part. To obtain the same simple linear calcula-tion procedure as with (a) of Fig. 15, the real stress distribution is reduced toa virtual diameter Dsub with homogeneous stress distribution. Dsub repre-sents the same shortening as with inhomogeneous stress distribution andleads to a substituting area Asub for calculation of axial elastic resiliencedp¼ lc=(EpAsub).

An extreme situation is given for bulk materials (sketch (d) in Fig. 15,e.g. large component compared to the screw dimension). The resilience ofthe clamped part is small compared to (a) of Fig. 15 but substitutingdiameter Dsub is still limited.

The complete set of formulae is given in Fig. 15. Normally, the resili-ence of clamped part dp is much smaller than that of the of screw ds.

Figure 15 Axial elastic resilience dp of clamped part. (From Refs. 17, 70.)


Some general rules for designing surroundings of the screw forclamped parts are:

1. Use cost-optimized tolerances for through-hole diameter.2. Through hole must not have any burrs.3. Avoid geometrical mismatch between through hole and underhead

fillet of screw.4. Make the clamping length as long as possible (small load factor F).5. Avoid overstressing the clamped part by excessive contact pressure

compared to the yield point of clamped part Rp0.2p (see also Fig. 39).6. Do not use washers between screw head and clamped part, except

special requirements to avoid surface damaging of clamped part orreduction of contact pressure.

7. Use the right design of clamped part for accessibility of screw, nut,wrench or bit.

D. Screw Assembly

Effective screw tightening involves significant static loading of the entirebolted joint prior over the operating period. Therefore, the loading andstressing of the screw under tightening conditions has to be considered indetail.

An experienced and accepted approach is shown in the mathematicformula in Fig. 16. Eight steps summarize the calculation of screw tighten-ing with a torque load for a given stressing of the screw shank by the equiva-lent one-dimensional stress seq. In this case, the axial stress sax is equal toseq divided by ks (step 1). The equivalent stress should be taken from90% of yield point of screw material for torque-controlled tightening. Foryield-point controlled tightening, it is the yield point itself and for angularcontrolled tightening, it is approximately the mean value of yield pointand ultimate tensile strength of screw material.

The factor ks depends on the screw and thread geometry and on thefrictional coefficient mt of the thread d0 is the minimum of the diameter ofthe nominal stress area or the diameter of the screw body (important if usinga wasted shank). For a metric screw with highest stressing in the screwthread, ks can be simplified as shown in step 1 of Fig. 16.

From this axial stress sax and the relevant stress area A0, the axial pre-load Fp acting in the screw shank can be calculated (step 2). Then, step 3gives the corresponding thread torque Tt which is generated by the leadangle j of the thread profile and the thread friction coefficient mt.

The result of step 4 is the head frictional torque caused by frictioncoefficient mh. The reason for this head frictional torque is the sliding of


head during tightening under the preload Fp with an effective mean bearingdiameter Deb. Deb has to be calculated separately and is dependent on thehead type (see Fig. 38).

Step 5 finally results in the necessary assembly torque Ttot. This torquevalue has to be applied to the screw drive to obtain the given stress seqand this torque leads to the calculated preload Fp. After step 5 for the cal-culated geometry, material, and friction of the bolted joint, three maincharacteristics are determined: stressing seq, preload Fp, and assemblytorque Ttot.

As additional information, step 6 allows the calculation of the maxi-mum torsional stress tmax. For calculation of tmax, a polar section modulus

Figure 16 Loading and stressing of a screw under tightening conditions withtorquings (From Refs. 17, 18.)


Wp for full plastified cross-section is used (Wp¼ pdo3=12). Step 7 offers the

lead angle of the thread profile j. Step 8 formulates the theory of maximumdistortion energy for producing a material failure (this is also called ‘‘vMisestheory of failure’’). This is the background-formula for step 1 to combineequivalent stress and axial stress. General information about theories of fail-ure can be found in Ref. [3].

In general, the friction coefficient m is defined as the ratio of normalforce acting over produced tangential frictional force in a sliding motionof two bodies (Fig. 17). The frictional force is always directed against thedirection of motion. For a screw, the normal force is the preload Fp. Thetangential force can be formulated as mt�Fp in the thread contact zone andas mh�Fp in the head support area. These tangential forces cause frictionaltorques, because of the radii of thread and head contact zones due to screwaxis (diameters Deb resp. d2, see also Fig. 16). Therefore, the frictional

Figure 17 Definition of friction coefficients mh and mt.


coefficients define the part of preload, which acts tangentially in the contactareas of a screw.

Table 4 proposes classes of frictional coefficients valid for boltedjoints, based on the VDI 2230 guideline [70] and experience [17]. If no exactvalue is available, one can select a value from this table which is valid forlow surface roughness. But one must always remember that the friction co-efficient depends on complex influences like materials surfaces, lubricationincl. homogeneity, hardness ratio of the two surfaces in contact, local stresspeaks or stress distribution in contact zone, tolerances for contact geometryas well as tightening level and number of (re) tightenings. A selection tablecan only provide rough approximations. The supplier of screws can provideinformation related to friction behavior.

In practice, all parameters for calculations of Fig. 16 have deviations.Main influences are based on minimum and maximum strength of screwmaterial (e.g. heat treatment process) as well as minimum and maximumfriction coefficients (e.g. roughness and lubricant). Geometry is usually veryprecise, so tolerances from diameters are not significant for screw tightening.

This situation is shown schematically in Fig. 18 with two correspond-ing diagrams for highest material stressing in the screw shank. The uppercase A refers to conditions with minimum friction mmin (both, mtmin andmhmin) and maximum screw strength Rmsmax. On the abscissa axis, the

Table 4 Values for Guidance of frictional Coefficients mt and mh inClasses A–E [17,70]

mt, mh (—) Characteristics=Typical examples

A 0.04–0.10 Hard polished surfaces, thick lubrication with wax or grease,high pressure lubricants, anti-friction coatings, e.g. polishedmagnesium and screw with PTFE-low friction coating and

MoS2, no peak pressure at edges of support areaB 0.08–0.16 Commonly used conditions with defined friction by optimized

lubricants, such as oil, wax, grease for fasteners; suitable for

ferritic steel metallic blank, phosphate, zinc and microlayersurfaces as well as nonferrous metals with relevant lubricant

C 0.14–0.24 Usual conditions with only thin or inhomogeneous lubricant,

austenitic steel screws with suitable lubricant; zinc, zinc alloy,and nonelectrolytical applied surfaces without lubricant

D 0.20–0.35 Austenitic steel with oil, rough surfaces and Zn=Ni coatingwithout lubricant

E 0.30–0.45 Austenitic steel, aluminum, and nickel alloys blank withoutlubricant


rotation angle u is drawn; on the ordinate axis the preload Fp and the assem-bly torque Ttot are drawn. If the screw is tightened with Ttott, for case A acertain preload Fptmax is generated (due to mechanics of screw assembly inFig. 16). With the same torque Ttot1 and same screw in case B only a preloadof Fptmin is obtained. For this reason with a very precise torque a significantpreload deviation can occur. The ratio Fptmax=Fptmin is app. in the rangeof 2 for torque controlled tightening—currently overelastic tighteningmethods with lower preload deviation are established for high-duty boltedjoints.

One method is yield point controlled tightening. By measuring andevaluating the gradient of torque-increasing over the increasing ofrotation-angle, the screw will be tightened exactly to the beginning of

Figure 18 Basics of screw tightening for applying an assembly torque.


plastification of the screw shank (transition of the strong gradient of thetightening curve in Fig. 18 to the low gradient in the range of plastifiedscrew).

Another possibility to reach high tightening levels is using the angularcontrolled tightening method (also called ‘‘turn-of-the-nut-method’’): Afterapplying a snug-torque Ts an additional, fixed defined tightening angle Du isadded, so the screw is plastified to a certain grade in any case (comp. mark-ings in Fig. 18).

For yield point controlled tightening and angular controlled tighteningthe ratio of Fpymax=Fpymin resp. Fpanmax=Fpanmin is about 1.1–1.3. The devia-tion in practice is reduced drastically. For this reason, the greatest advan-tage of overelastic tightening methods is a significant increase of theminimum preload and a slight increase of the maximum preload. But onemust always note the resulting torque value can vary extremely for overelas-tic tightening methods, because torque is no controlled parameter.

Some hints for selection of parameters considering deviations in prac-tice are: for calculating the highest preload (related to the highest screwstressing) always take minimum friction coefficients and maximum screwstrength. This is relevant for maximum contact pressure under head). Ifthe lowest preload has to be determined, maximum friction coefficientsand lowest screw strength are relevant. To obtain maximum assemblytorque for overelastic tightening method, take maximum friction coefficientsand highest screw strength. This is relevant for maximum screw driveloading.

If new tightening devices have to be designed for a production linewith screw assembly, these devices should be able to apply a high torquevalue for angular controlled tightening. In practice, more than the doubletorque limit should be designed compared to torque controlledtightening.

E. Loading During Operation

1. Mechanical Loading

If a threaded fastening system is tightened, then screw, clamped part, andnut thread component are loaded mechanically by the flow of preload with-out external force (Fig. 19). The preload leads to head contact pressure pchbetween screw head support and clamped part surface as well as to threadcontact pressure pct at engaged thread flanks. Between clamped part andnut thread component, the component contact pressure pcc is generated(important for sealing). Following considerations due to force—elonga-tion-behavior which are based on Ref. [70], details are discussed in Refs.[7,67,72].


If a threaded fastening system is loaded by mechanical forces, thesecan act into the direction of screw axis (Fax) or transversal axis (Ft).Transversal forces produce shear loading of the bolted joint. If no transmis-sion of transversal forces by contour interaction is possible, they have to besmaller than mcc�Fp to avoid (micro)slipping between clamped part and nutthread component. Then, a transversal force does not load the screw.

The mechanics of screw loading illustrated in Fig. 20 suggests thesimplification of a threaded fastening system with a ‘‘spring-model’’.

Figure 19 Tightening and loading of threaded fastening system. (From Ref. 17.)

Figure 20 Spring model of a threaded fastening system. (From Ref. 17.)


This idealized model reduces all elastic contributions within the system torigid bodies and two springs with defined resilience: The screw shank ismodeled as one tensile spring with ds, the clamped part is represented bya compressive spring with elastic resilience dp.

Before tightening, all ‘‘springs’’ are unloaded (left side of Fig. 20).After tightening, usually the tensile spring of the screw is elongated muchmore than the compression spring of the clamped part (right of Fig. 20).If an external axial force Fax is induced within the clamping length lc, theinducing factor n determines which part of the clamped part is additionallyloaded (towards the screw head) and which part is unloaded by Fax (towardsthe nut thread component). These parts of additional loading and unloadingby an external axial force Fax influence the relevant elastic resiliences of dsand dp, if the fastening system is loaded. Therefore the resiliences varybetween tightening and operating, if n < 1.

Fig. 20 leads to the following force–elongation diagram shown inFig 21. The diagram shows on the x-axis the elongation of screw(left of ‘‘0’’) and clamped part with clamping length lc (right of ‘‘0’’). Onthe y-axis, the corresponding preload Fp in the screw shank is drawn. For

Figure 21 Force–elongation-characteristics of screw and clamped part.


the stable tightening level Fp0, a (positive) screw elongation of Fp0�ds and atclamped part an (negative) elongation of Fp0�dp is generated.

The representative curves of screw and clamped part are linear up tothe yield point of each material. Here, the stable tightening level Fp0 is com-pletely within the linear range. If screw or clamped part show plastification,each nonlinear behavior has to be considered for force–elongation diagram(degressive dashed lines in Fig. 21).

If a tensile external axial force Fax is applied to the fastening system,on the one hand, the screw is loaded additionally by nfFax and on the otherhand the clamped part is unloaded by (1� nf)Fax, because the two springsare a parallel arrangement. The consequence is that Fax reduces the residualclamping load and increases the tension in the screw shank, but always onlya part of Fax acts in any ‘‘spring’’.

The additional operating force of screw (nfFax) besides the load factorf is dependent on the inducing factor n. For this reason, Fig. 22 gives someexamples for the value of n, which are approximations. Some referencespropose a calculation of n [70], but an analytical solution is usually a lotof work, and a simple approximation often gives the same range in practice.

Figure 22 Examples for approximation of inducing factor n (From Ref. 70.)


Numeric calculations like FEM are very suitable to determinenf¼Faxscrew shank=Fax external directly for a given geometry by selecting thenodes of the screw shank cross-section for Faxscrew shank and all nodes, whichare loaded externally for Fax external. With the result of nf, the analyticalcalculation can be continued; therefore, FEM can be used to consider allinfluences from geometry and inhomogeneous stress distribution (e.g. forclamped part).

The determination of the inducing factor n is an example, to show thatvery detailed design modifications lead to significant changing in screwloading. In general, it is valid that a small inducing factor n decreases theadditional operating force of screw (interesting for increasing the fatigueloading capacity of the fastening system), and reduces also the residualclamping force under axial loading with an operating force (compare alsoFig. 21).

If no numeric calculation is done, the load factor f can be approxi-mated with the analytical model of Fig. 23, see also Ref. [70,72]. This loadfactor can be calculated from f¼ dp=(dsþ dp), if the axis of screw, clampedpart centerline and external axial force Fax is the same. If these axes havedifferent positions, additional bending of the screw and clamped partoccurs, so that the elastic resiliences and in consequence the load factor fare changed.

Figure 23 Linear model for determination of load factor F. (From Ref. 17.)


For the model shown in Fig. 23, the force Fax, the distances of axes sand a, the through-hole diameter dh as well as the elastic resiliences ds and dpfrom Figs. 12 and 15 and the substituted area Asub must be known. Fromthese, the substituted diameter Dsub can be calculated. This constantdiameter corresponds to Asub for the same resilience dp. The model is assum-ing a linear stress distribution s(x) within Dsub.

For the use of Fig. 23, it is necessary that the real stress distribution issimilar to the linear distribution in the model. The size of the clampedpart may not be much larger than Dsub, so the moment of inertia Ifull keepsvalid.

Then, the moment of inertia Ifull can be obtained and as a next step fcan be calculated. Ifull does include the cross-section area of the screw,because the screw gives also a bending resistance during loading with Fax.

After tightening, any threaded fastening system shows relaxationeffects. This short time relaxation often is called ‘seating’: it leads to a pre-load reduction as demonstrated in Fig. 24. Important influence for this isthe roughness and strain hardening of all surfaces in contact zones betweenscrew, clamped part(s) and nut thread component as well as the direction ofmechanical loading due to a normal vector on the contact area. Under con-tact pressure, the high surface spots are deformed axially which leads to aseating distance fz of the fastening system and in consequence to a reductionof preload down to a stable preload level Fp0.

Significant short time relaxation always occurs if the fastening systemis partially overloaded, such as when thread engagement is too small

Figure 24 Preload reduction by seating (short time relaxation).


(see Fig. 33) or if contact pressure under the head is too large (see Fig. 39),material mismatch (e.g. material strength of clamped part is too low) or geo-metric mismatch (e.g. nonperpendicular nut thread or screw head, oversizedunderhead fillets). The approximational equation for fz given in Fig. 24 canbe used if there is no partial overloading.

An eccentric loading of a threaded fastening system can lead to com-ponent separating. Figure 25 demonstrates this for an external force Fax act-ing with a distance a from the axis of symmetry 0–0 of clamped part. Theconfiguration of Fig. 25 is the same as in Fig. 23.

Figure 25 Mechanics of component separating as a result of eccentric loading byFax (From Ref. 17.)


There exists a point of tilting on one side of clamped part; on the oppo-site side, of the first component separating occurs. With the given valuesFp0, Fax, s, a, dh, Dp and f after calculating the area Ap of clamped part inthe contact zone between components and the moment of inertia Ip, the pre-load for first separating Fps can be estimated for a given axial force Fax. If thepreload Fp0 is larger than Fps, then component separation does not occur forloading with Fax.

On the other hand, if a stable preload after tightening Fp0 is given, Fax-

crit determines the beginning of component separating, if Fax > Faxcrit. Thisleads to two cases indicated in Fig. 25. Case 1 is determined by elastic screwloading regarding the force–elongation diagram of a threaded fastening sys-tem. The additional operating load of screw Fsa is equal to nFFax. Case 2refers to the situation of a beam lever system, built by Fp and Fax and thelength values a, s, Dp.

Component seperation must be avoided (case 2) because it leads toextensive additional loading of the screw Fsa and to early failure either bystatic overloading or by fatigue fracture. But in some cases, for optimizedcomponents with high resilience dp and with exactly defined tightening byloading, a partial component separation can be allowed without problems(e.g. bolted joints at lightweight piston rods). For more details regardingcomponent separation under eccentric mechanical loading, see Refs. [67,70].

Figure 26 Preload behavior for overelastic tightening.


Figure 26 explains the preload behavior for overelastic tightening ofscrew. The corresponding force–elongation diagram illustrates the screwplastification with a degressive curve for exceeded elastic limit under the ten-sile and torsional stressing during tightening.

The first preload level after tightening Fp1 is reduced to the stable pre-load Fp0 by the reason of seating effects. Besides this, a general aspect is thatafter tightening a screw the torsional stress is reduced significantly—to app.30–50% of the torsional stress under applied torque. This leads to anincreased elastic limit of screw and leading to a higher preload limit duringoperating compared to tightening. A screw, which was tightened overelastic,can be loaded by a large operating force Fax. In practice, there is almost nodifference between the tightening methods due to the loading capacity dur-ing operation (for dynamic loading, see also Fig. 52).

Up to now, no time dependence ofmechanical load is considered. Fig. 27displays the effects for an alternating axial force Fax. For positive axial forceFax (tensile loading), the preload in the screw shank will be increased andthe clamping forcewill be reduced, producing the same effect as for static load-ing. If a threaded fastening system is loaded axially, the preload in the screwshank is not the same as the clamping force between components.

For a negative axial force �Fax, just the opposite aspects are true: thepreload in the screw shank will be reduced and the clamping force between

Figure 27 Preload behavior for mechanical dynamic axial loading.


the components will be increased. In this case, by the negative axial loading�Fax, a plastification of the clamped part can be generated which does notoccur during tightening and leading to relaxation effects that are not accep-table during operation.

But overall, also for dynamic axial loading, the screw has to bear onlythe part (nfFax) due to the complete axial force Fax. For a well-designedthreaded fastening system, this part normally should be smaller than 10–20% of Fax.

2. Thermal Loading

Often, a threaded fastening system must be used at different temperatures,e.g. tightening at room temperature (t1) and operating at elevated tempera-ture (t2). If screw material and material of clamped part have different ther-mal properties like Young’s modulus (Es,Ep) or thermal expansioncoefficient (as, ap) or if the properties are temperature-dependent in the rangeof temperatures applied, the preload Fp varies, and this can be significant.

The design engineer must check if the thermal loading of the paricularthreaded fastening system does not lead to overloading by preload increas-ing or missing of clamping force by preload reduction.

Figure 28 shows a linear approximation of the temperature-dependentpreload change DFp. Again, the screw is tightened to its stable preload levelFp0 at temperature t1. The temperature change DT¼ t2 � t1 leads to thermalelongations at screw and clamped part Dl1, Dl2 and to changed elastic con-stants Es, Ep. Therefore, the force-elongation diagram is modified so that, a

Figure 28 Approximation of preload changing by thermal loading. (FromRef. 17.)


preload change DFp is generated. Influences from nut thread component areneglected because the main part of the preload is transmitted by the firstthread flanks, therefore only a very short expansion length is relevantcompared to the clamping length lc.

This preload change DFp can be positive or negative. It is positive, ifYoung’s moduli are constant and the clamped part has a larger thermalexpansion coefficient than the screw (typical for threaded fastening systemswith light metals and steel screws). For example, it is negative for titaniumscrews and steel components.

A positive preload change DFp can result in a screw failure (static frac-ture of screw by too large yielding=plastification). For example, in anextreme relaxation of preload by plastification of clamped part or screw,a negative DFp can result in a component separation and finally in a fatiguefailure of screw.

The preload change demonstrated in Fig. 28 is valid for the same tem-perature of screw and clamped part (steady state); during heating up ordown a peak difference in temperature can occur, which generates evenmore preload change. The equation indicates what can be done to minimizeDFp: reduce the thermal expansion mismatch (ap� as), reduce temperaturedifference DT, maximize for given clamping length lc both the resiliencesds and dp (e.g. by low Young’s moduli).

This means that in practice the positioning of screws away fromextreme hot or cold places using the same materials for screw and clampedpart (e.g. Al-screws for Al-components) and using long thin walled distancetubes (e.g. for pipe constructions).

Figure 29 proposes a fundamental example for thermal loading withnumeric values. A description of the situation is given with the sketch on theleftside of Fig. 29. A screw with nominal diameter d and support diameter dais tightened with a clamped part with through-hole diameter 1.1d, then heatedto a temperature difference between tightening and operating of DT. Thisgenerates a preload change DFp which results in an axial stress change Dspin the screw shank and also in a change of contact pressure under head Dpch.

The diagram contains values for ferritic steel screws and aluminumscrews combined with a clamped part made of aluminum or magnesium(Young’s moduli are set to constant for this calculation). The highest ther-mal stress increase takes place for steel screw with magnesium component. Ifapplying DT¼ 1008C, this combination has about 250MPa stress increasewhich means 170MPa contact pressure increase. If a standard ISO 4014screw is used only 65MPa contact pressure increase using a flange head withda¼ 2d will be obtained. The result from this thermal stress increase can bethe plastification of clamped part and leading to extensive relaxation; seealso examples in Fig. 66.


The reason that the contact pressure pch is relevant for the higheststressing of clamped part is that a flange head should always be used ifthe clamped part is sensible for thermal preload change (e.g. low creepingstrength at elevated temperatures like for magnesium alloys).

3. Reactive Loading

Another important group of loadings to threaded fastening systems is char-acterized with the term ‘‘reactive loading’’. This term consists of chemicalreaction (all kinds of corrosion) or aging (embrittlement of materials, e.g.by heat=radiation or radioactive effects with defect generation in grainstructure of materials). Reactive loading effects always are time dependent,so the design has to take into account the life time of the threaded fasteningsystem and the period of use.

The primary aspect is corrosion, because almost all technical materialshave a corrosive behavior which must be considered. Aging effects in manycases can be avoided by using suitable materials.

Types of corrosion can be divided into three main groups: chemicalcorrosion (area corrosion), galvanic corrosion (electrochemical corrosion),and selective corrosion (e.g. stress corrosion cracking). Details are charac-terized in Table 5. Other types of corrosion exist like fretting, crevice corro-sion, or microbiological corrosion [72], but in this chapter, only basicaspects and general hints are provided. For more details, see the literature,e.g. Ref. [1,3].

Figure 29 Linear approximation of thermal stress increase and thermal contactpressure increase. (From Ref. 16.)


When designing for corrosive behavior of different material surfaces,Table 6 with normal potential, measured against a standardH-electrode (flat electrode, 258C, 1M-solution of ions in the electrolyte) isused for theoretical estimation of suitable metal combinations. But galvaniccorrosion is determined by system behavior so that any table can only pro-vide a tendency not quantitative information.

Metals with low (negative) potential are called anodic (base metals,likely to corrode). The materials with high potential are called cathodic(noble metals, unlikely to corrode). The existing corrosion current in a gal-vanic cell is determined by the combination of the metals. For a minimumcorrosion activity, the design engineer should combine materials with lowdifference in electrochemical potential. One can conclude that the idealsituation would be a screw made of the same material as the clamped part.Besides the corrosive stability, this also has almost no thermal loading underchanging temperature (see Fig. 29).

Exceptions are passivated metals (indicated with �). They build a thinoxide layer on their surface which has a dense structure and, therefore,

Table 5 General Types of Corrosion

1. Chemical corrosion. Chemical reaction of the material surface with electrolyte;the metal dissolves in a corrosive liquid until either it is consumed or the liquid issaturated (in practice, the ‘‘liquid’’ also can be humid air atmosphere, possiblywith solvents of compounds, such as SO2 or salt at sea coasts).

2. Galvanic corrosion. Chemical reaction of two electrically coupled metals usingan electrolyte as transmitter for electrons (electrochemical cell). Then, thecorrosion rate of the less corrosion-resistant metal is increased significantly.

Therefore, this type of corrosion normally shows high corrosive speed, but thecorrosion-rate depends on many parameters, such as potential-difference,temperature, purity, grain structure, convection=diffusion, influence of corro-

sion-products, ratio of cathodic and anodic areas, geometry. In practicegalvanic corrosion is always a subject, if only one of two coupled metals isattacked and if the attack is reduced with increasing distance from the

borderline between the two materials.3. Selective Corrosion. Chemical reaction located within a part of a material. This

corrosion type is typical for alloys where different elements=phases withdifferent sensibility for corrosive media exist e.g. dezincification of brass. Stress

corrosion cracking is an intercrystalline reaction at grain boundaries, induced bythe existing mechanical loading of special material=electrolyte=environment-combinations. Examples for this are stainless steel and chloride-electrolyte

(seawater) or some high-strength aluminum alloys and electrolyte with salt-solvent. Another type of selective corrosion is the so-called ‘‘hydrogen-embrittlement’’ of high-strength steels (see also text).


insulates against further corrosion current. This thin oxide layer is stable(utilized for protective functions of Zn-coatings, Al-alloys, stainless steels).These metals can be either in ‘‘active’’ or in ‘‘passive’’ state. This is animportant mechanism that demonstrates why also some base materials withquite negative electrochemical potential do not corrode. This is because theyhave their own integrated ‘‘protective coating’’ by passivation.

In Refs. [3,50], a table of galvanic series for sea-water is given whichincludes not only pure metals, but also metal alloys (Table 7). This list refersto an environment other than the laboratory values from Table 6. Compar-ing both the tables, the main sequence is the same, but single combinationslead to a varying potential difference.

Steels with high tensile strength over approximately 1200MPa includethe danger of hydrogen-induced embrittlement (see also Refs. [50,71]). Thismeans a brittle behavior without deformation in the event of failure which isproduced by a typical intercrystalline fracture along the grain boundaries withloosened grains (Fig. 30). As a preventive action, a production process withminimizing exposure to hydrogen atoms (electroplating) and tempering forH-effusion is established. But hydrogen diffusion into steel also can take place

Table 6 Guidance Values for Electrochemical Potential

of Metals, Measured Against Standard H-electrode [1,3,72]

Metal Chemical Pure(Active, if Passivating) Potential (mV)

Li �3,050Mg �2,370Ala �1,660Tia �1,630Mn �1,630Zna �760Cra �740Fe �440Nia �240Sn �140Pb �130H 0Cua 340

Ag 800Pta 1,200Au 1,500

ametal is able to passivate.


during operation, e.g. H atoms from corrosion reactions. As a result, screwsmade of steel should be coated nonelectrolytically for class 12.9 or higher.

The data is shown in Fig. 31 suggests fundamental corrosion mech-anisms of threaded fastening systems. The characteristics are printedto each part in the figure itself. If the screw material is a base metal andthe component material is noble chemical, the screw material corrodes(e.g. steel screw in copper component). If the difference of electrochemicalpotential is opposite the component corrodes (e.g. steel screw in magnesiumcomponent). This is shown in part (a) of Fig. 31 (left and right). Any corro-sion product like oxide generates a limited appearance and can increase thespeed of corrosion. For a further state of corrosion, destroying the supportarea leads to extreme relaxation because the residual original materialcannot bear the initial preload from tightening. In general, the first stepof corrosion is relevant for appearance, the second step of corrosion isrelevant for preload function.

Part (b) of Fig. 31 demonstrates the same situation for a coatedcomponent and coated screw with internal drive. An internal screw drivecan collect electrolyte, and therefore, is set to a severe corrosive environ-ment. This is the reason why often screws with internal drive configura-

Table 7 Galvanic Series for Seawater from [3,50] in part,

Measured Against Saturated Calomel Reference Electrode (SCE)

Metal=AlloyRange of potential

(mV)

Titanium �40 to þ40Ni–Fe–Cr-alloys �30 to þ30Ni–Cu-alloys �150 to �30Silver �150 to �100Platinum þ180 to þ230Stainless steels, active �300 to �50Stainless steels, passive �550 to �350Copper �350 to �250Brass �400 to �270Cast iron �730 to �590Low-carbon steel �730 to �590Low-alloy steel �610 to �580Aluminum alloys �1000 to �750Zinc �1200 to �900Magnesium �1650 to �1580Bronze Cu–Sn �320 to �240Measured against SCE, flow of seawater 2.4–4.0msec; temperature 5–308C


Figure 30 SEM image of fracture from screw M12-12.9; loosened grains are typical for hydrogen embrittlement, magnification1000� .


tion begin to corrode within their screw drive. Coatings separate thefunction of the component into (1) surface protection, and (2) mechanicalperformance of the bulk material under the coating. But the most impor-tant point is that coatings always have defects and can be damaged—and

Figure 31 Fundamental aspects of corrosion for threaded fastening systems.


then the protection is partly reduced. For coated screws with high corro-sion resistance, a hexagon drive configuration should be avoided by thereason of the high bit contact pressure and possibly high edge-deforma-tion of screw (see also Fig. 36).

Coating systems make the chemical corrosion complex (four materi-als in Fig. 31b), which can react: two bulk materials, two coating materi-als). The noble material does not corrode (compare damaged componentcoating and the resulting local corrosion). Coating systems for screw pro-tection must provide a high quality adhesion, because they have to workunder extreme mechanical surface pressure (explanations of Fig. 31b).Besides electrolytical coatings, there are also very effective nonelectrolyti-cal coating-systems for enhanced corrosion protection of steel screwsknown. For established suppliers of nonelectrolytical coatings, see Refs.[54,12,13,53,54]); for standardization see Ref. [23].

Part (c) of Fig. 31 focuses on electrical insulation as amechanical way toprevent from corrosion. Remarks are given in the figure. Part (d) summarizesgeneral aspects for corrosion of threaded fastening systems in practice.

III. DESIGN STRATEGY FOR THREADED FASTENINGS

For realizing an optimized threaded fastening system, an effective develop-ment procedure is necessary. Figure 32 demonstrates this with a flowdiagram by distinguishing three columns: calculation=design, verification,and realization. The main topics of calculation=design are: tightening=operating (determination of loadings the bolted joint has to bear), screw,clamped part, nut thread component (specifications of all parts of the boltedjoint), and design analysis (engineering results based on theory and experi-ence). If the design analysis meets the requirements and is proposing a reli-able function of the bolted joint, the verification column is started.Prototypes are the very first practical realization of the theoretical design.With these parts, the laboratory tests and the field tests can be done, ifthe prototypes are representative for series production.

The realization column contains assembly process (parameters oftenare determined by assembly process capability as a result of laboratorytests), purchase, series production and field service. Today, basic aspectsof quality management are teamwork, documentation of results and his-tory, failure modes and effects analysis as well as feasibility reviews. Theseconcepts can be transferred to several topics of Fig. 32 (only drawn fordesign analysis and prototypes, because here they are necessary in any case,see also Ref. [19]).


A. Determination of Screw Material and Surface

In general, materials for screws must provide both high strength toobtain sufficient preload and compatibility to the environment of operat-ing. This paragraph provides a compressed overview over the mainaspects important for material selection of screws. Details of certainmaterials can be found in Refs. [1,3]. For the selection of suitable screwsurface, specialized coating companies can offer established and enhancedsolutions, e.g. Refs. [13,53,54].

As an introduction to screw material selection, Table 8 gives a 10-ques-tion list for checking the qualifications.

Figure 32 Development process of a threaded fastening system.


1. Operating Environment and Material-Related Standards

The operating environment determines the materials that are suitable.Table 9 gives fundamental selection criteria and a few examples for alloys(for established materials, see Tables. 10–12). Only when standard materialscannot be used should special solutions be considered. In this case, the sup-plier of fastening elements can give support, e.g. Refs. [2,62]. The Europeanstandard EN 10269 provides steels and nickel alloys for fasteners at elevatedor low temperatures with temperature-related properties [11]. As a roughestimation, the material strength at limiting temperature of the material isapproximately half of the strength at room temperature. The European des-ignation system for steels is defined in standard EN 10027. The Vickershardness test procedure is defined in standard ISO 6507 [39]. Electrolyticalsurface coatings for fastening elements are defined in ISO 4042 [31] (types ofcoatings, coating thickness, tolerances, hydrogen-embrittlement, designa-tions of coating systems), nonelectrolytical coatings for fastening elementsare defined in ISO 10683 [23]. Detection of hydrogen embrittlement is dealtin ISO 15330 [25]. Surface discontinuities are proposed and evaluated inISO 6157 [37].

Table 8 Check List for Screw Material Selection

No. Question for Theoretical Checking of the Selected Screw Material

1 Is the screw material suitable for sufficient preload (material strength high

enough)?2 Is the screw material suitable for required dynamic loading (notch-sensitivity,

material fatigue behavior)?3 Is the screw material suitable for operating temperature?4 Is the thermal expansion coefficient of screw material suitable for permitted

change of preload under temperature?5 Is the screw material resp. screw surface suitable for corrosion requirements

(climate, fluids=electrolytes, material contacts)?6 Is the screw material suitable for tightening (adhesion, friction in mechanical

contacts)?7 Is the screw material suitable for screw manufacturing (availability of

raw material, forming, cutting, heat treatment, large batch productionrequirements)?

8 Has the screw material good-natured behavior if overloading (ductility resp.significant plastification before fracture, no embrittlement)?

9 Has the screw material sufficient long-term properties under tensile stress(stable grain-structure, no creeping, no embrittlemement)?

10 Is the screw material economic?


Table 9 Fundamental Selection of Screw Material and Screw Surface

No.Characterization of

environmentExamples of experienced materialsfor high-strength screws and bolts

1 Dry and temperature< app. 3008C

Low alloyed- or carbon steel (C35, C45, 35B2, 20Mn5, 42CrMo4), painted or othercorrosion protection coating suitable forthis temperature range

2 Humid þ (salt) solutionþ temperature< 3008C

Low alloyed- or carbon steel (C35, C45,35B2, 20Mn5, 42CrMo4), enhanced cor-rosion protection coating suitable for

this temperature range3 High temperature up to

app. 6008CHigh alloy steel with Cr-, Ni- or Mo-

content (42CrMo4, 42CrMo5-6,

X5CrNi18-10, X22 Cr MoV12-1), ferri-tic or austenitic

4 High temperature over app.

6008CHeat-resistant steels with high Cr-content

and alloyed elements Ti, Nb (10CrNi-MoMnNbVB15-10-1, X6 NiCrTi-MoVB25-15 -2 ) ; N i -ba s e a l l oy s(NiCrTiAl20)

5 Low temperatures underapp. �508C

Austenitic steels with sufficient Cr- and Ni-content (X5CrNi18-10, X2CrNiMoN17-13-3)

6 Long time appearancerequirements

Screws made of stainless steel (X5CrNi18-10), chemical inert materials like tita-nium (TiAlV6-4), steel with multilayer

coatings7 Chemical reactive compo-

nents like magnesium,lithium as clamped part

or nut thread component

Screws made of component material, screwsmade of passivating=chemical inert me-tals, steel with multilayer coatings

8 Fastening of light metalcomponents, e.g. made

from aluminum or mag-nesium

Screws made of aluminum (AA 6013, AA6056, AA 7075; matching of corrosive

behavior and high thermal expansioncoefficient, [16])

9 Small volume and extreme

lightweight designs

Highest screw strength over 1400MPa up to

app. 2000MPa (use in aviation- andaerospace-industry, special productionrequirements necessary and high contact

pressures must be acceptable; 38NiCr-MoV7-3, X2NiCoMo18-8-5) or light-weight materials for screws (TiAlV6-4,tensile strength app. 1100MPa)


But one must always remember that any screw is only one part of afastening system (Fig. 1) and the behavior of the entire system is relevantfor function and reliability.

For the time-dependent behavior of a threaded fastening system withinthe operating period, the superposition of mechanical, thermal, and reactiveloading is important (for example strength of screw at high temperature).Besides corrosion for long operating periods, the material creep can becritical (permanent increase of screw elongation under preload, which leadsto drastically reduced clamping force in threaded fastening systems). Creepshould be considered if the operating temperature of a material is higherthan 1=3 of its melting temperature. Face-centered cubic crystal systemshave lower creeping resistance than other crystals. Creep can be redu-ced=eliminated by special creep-resistant alloys.

Second, for long operating periods, the case of misuse can=will beprobable, especially if the fastening system has to be disassembled or

Table 10 Important Properties of Screws Made of Carbon Steel or Alloy

Steel, Defined in ISO 898 [46], for Design Purpose and Details, Refer to Standard

Property

class

Minimum

tensile

strength

Rm (MPa)

Minimum Proof

resp. yield

stress

ReL, Rp0.2

(MPa)

Minimum

elongation

after

fracturea (%)

Minimum

vickers

hardness

HV 10

Maximum

vickers

hardness

HV 10

Example

for suitable

material

(not

defined

in ISO 898)

3.6 330 190 25 95 250 C35, 35B2

4.6 400 240 22 120 250 C45, 35B2

4.8 420 340 14 130 250 C45, 35B2

5.6 500 300 20 155 250 C45, 35B2

5.8 520 420 10 160 250 C45, 35B2

6.8 600 480 8 190 250 C45, 35B2

8.8b 800 640 12 250 320 35B2,

19MnB4

8.8c 830 660 12 255 335 35B2,

19MnB4

9.8 900 720 10 290 360 35B2,

19MnB4

10.9 1,040 940 9 320 380 37Cr4,

34CrMo4

12.9 1,220 1,100 8 385 435 37Cr4,

42CrMo4

aMeasured at unnotched, cylindric sample from screw with length of cylinder¼ 5�diameter of

cylinder.bd 16mm.cd < 16mm.


retightened by nonprofessionals (e.g. wheel bolts of cars, Fig. 73). Also,misuse has to be tested during verification of the design (Fig. 32).

2. Established Materials for Screws

If searching for established screw materials, three main groups can be found:low alloyed- or carbon-steels (mostly used, ISO 898 [46]), stainless steels(ISO 3506 [29]) and nonferrous metals for screws (ISO 8839 [44]). In ISO898 and ISO 3506, only grades for groups of materials are specified. Besidesthis, in ISO 7085 [41], mechanical properties of case hardened and heat trea-ted screws and in ISO 2702 [27] mechanical properties of heat treated tap-ping screws are defined.

The well-known property classes of screws (3.6, 4.6, 4.8, 5.6, 5.8, 6.8,8.8, 9.8, 10.9, 12.9) are defined in ISO 898 are only valid for screws made ofcarbon steel or alloy steel (definition of property classes: first number: mini-mum tensile strength Rmmin of material=100 in N=mm2; second number:10 � ratio of proof stress Rp0.2 over tensile strength Rmmin). ISO 898 doesnot apply to high temperatures above 3008C or low temperatures under�508C. Table 10 summarizes the important properties defined in ISO 898.

Another material group is also well established: screws made of stain-less steels. Related properties for fasteners are defined in ISO 3506 [29].

Table 11 Some Properties of Screws made of Stainless Steels, Defined in

ISO 3506 [29], for Design Purpose and Details, Refer to Standard

Steel

group

and

grade

Minimum

tensile

strength

Rm

(MPa)

Minimum

proof stress

Rp0.2

(MPa)

Minimum

elongation

after

fracturea

(%)

Minimum

vickers

hardness

HV 10

Maximum

vickers

hardness

HV 10

Example for

suitable Material

(not defined

in

ISO 3506)

A2-70 700 450 �24 — — X5CrNi-18-9,

X5CrNi1816

A2-80 800 600 �16 — — X5CrNi-18-9,

X5CrNi1816

A4-70 700 450 �24 — — X5CrNiMo17-12-2,

X2CrNiMo17-13-3

A4-80 800 600 �16 — — X5CrNiMo17-12-2,

X2CrNiMo17-13-3

C1-70 700 410 �8 220 330 X12Cr13

C1-110 1,100 820 �8 350 440 X12Cr13

C3-80 800 640 �8 240 340 X19CrNi16-2

F1-60 600 410 �8 180 285 X3Cr17, X6CrNb12

ain ISO 3506 originally measured at manufactured screw as elongation over total length in mm.


There are three groups of stainless steels: austenitic (A), martensitic (C), andferritic (F). Each group can have different steel grades, which are distin-guished by different digits following the characteristic letter. Then, a thirdvalue is added to indicate 1=10 of the tensile strength of the fastener.For example, after ISO 3506, the designation for an austenitic screw of steelgroup 2 with a tensile strength of 700MPa is A2-70.

Table 12 Nonferrous Metals for Screws from ISO 8839 [44], for

Design Purpose and Details refer to Standard

Symbol MaterialDiameterrange

Minimumtensilestrength

Rm

(MPa)

Minimumproofstress

Rp0.2

(MPa)

Minimumelongation

after

fracturea

(%)

CU1 CU-ETP orCU-FRHC(ISO 1337)

M39 240 160 14

CU2 CuZn37 (ISO 426) M6 440 340 11

M7–M39 370 250 19CU3 CuZnPb39-3

(ISO 426)M6 440 340 11

M7–M39 370 250 19CU4 CuSn6 (ISO 427) M12 470 340 22

M13–M39 400 200 33

CU5 CuNiSi1(ISO 1187)

M39 590 540 12

CU6 CuZnMnPb-40-1 M7–M39 440 180 18

CU7 CuAlNiFe10-5-4(ISO 428)

M13–M39 640 270 15

AL1 AlMg3 (ISO 209) <M10 270 230 3M11–M20 250 180 4

AL2 AlMg5 (ISO 209) <M14 310 205 6M15–M36 280 200 6

AL3 AlMgSiMn1

(ISO 209)

<M6 320 250 7

M7–M39 310 260 10AL4 AlCuMgSi4

(ISO 209)<M10 420 290 6M11–M39 380 260 10

AL5 AlZnMgCu0.5 <M39 460 380 7AL6 AlZnMgCu5.5

(ISO 209)<M39 510 440 7

atest specification according ISO 898-1.


Table 11 contains some properties of screws made of stainless steelsregarding ISO 3506. Austenitic steels cannot be hardened and are usuallynonmagnetic. Alloys of steel grade A2 are most frequently used (kitchenequipment, apparatus industry), but they are not stable in environmentswith chlorides (e.g. swimming pools or chemical devices). Alloys of gradeA4 are the so-called ‘‘acid proof steels’’ with molybdenum as alloy ele-ment to increase corrosion resistance, to a certain extent, also againstchloride ions (used for chemical industry, food industry, ship-buildingindustry).

Steels of martensitic grades C1 and C3 can have higher strength thanaustenitic steels and can have relatively higher proof stress Rp0.2, but theyhave a limited corrosion resistance, so they are widely used in machines withhigh loading and controlled environment, such as pumps and turbines. Fer-ritic steels of grade F have a permanent ferritic grain structure at room tem-perature, so they cannot be hardened, but they are magnetic. They are analternative for steels of grade A2.

For all situations, where ISO 898 and ISO 3506 cannot offer suitablematerials for screws or bolts, the materials of ISO 8839 should be checked.Table 12 proposes the nonferrous metals of this standard which are used forelectrical contacts (screws made of copper, brass), special corrosive condi-tions, lightweight design or constructional elements (screws made of alumi-num). AL5 and AL6 of Table 12 can be sensitive for stress corrosioncracking, depending on their grain structure. Currently, additional alumi-num alloys for screws are available, which provide high strength withoutstress corrosion cracking (e.g. alloys 6013 and 6056, in work standards oftencalled AL9, see also Refs. [15,16]).

B. Determination of Screw Thread Size

The screw thread size normally is the main parameter used to determine theinitial preload of a threaded fastening system. The other parameters are inmany cases preselected, such as screw material (determined by environ-ment), assembly method (determined by assembly line, field maintenanceor philosophy), and frictional situation (determined by surfaces in contact).

But the design engineer always has to distinguish both initial preload(generated during tightening, see also Fig. 18) and residual preload (stablepreload level during operating, see also Fig. 24). The initial preload canbe calculated in a detailed manner, the residual preload strongly dependson the material’s behavior and the local contact conditions. Therefore, thisvalue often is estimated from experience or if necessary measured (preloadmeasurement by ultrasonics or strain gauges).


Table 13 Estimated Preload Level for Different Metric Screw Types

Preload Level (kN), Metric Screw Thread

Tensile strength and yield strength ratio of screw

Threadsize

NominalAs (mm2)

Rm (MPa) 300 400 400 500 500 600 800 900 1,000 1,100 1,200 1,400kR (�) 0.6 0.6 0.8 0.6 0.8 0.8 0.8 0.8 0.9 0.9 0.9 0.9

M1 0.458 0.06 0.08 0.11 0.10 0.14 0.17 0.22 0.25 0.31 0.34 0.38 0.44M2 2.069 0.28 0.38 0.50 0.47 0.63 0.75 1.01 1.13 1.42 1.56 1.70 1.98M3 5.000 0.68 0.91 1.22 1.14 1.52 1.82 2.43 2.74 3.42 3.76 4.10 4.79M4 8.800 1.20 1.61 2.14 2.01 2.68 3.21 4.28 4.82 6.02 6.62 7.22 8.43

M5 14.20 1.94 2.59 3.45 3.24 4.32 5.18 6.91 7.77 9.7 10.68 11.7 13.6M6 20.10 2.75 3.67 4.89 4.58 6.11 7.33 9.78 11.00 13.7 15.12 16.5 19.2M8 36.60 5.01 6.68 8.90 8.34 11.1 13.4 17.8 20.0 25.0 27.5 30.0 35.0

M10 58.00 7.93 10.6 14.1 13.2 17.6 21.2 28.2 31.7 39.7 43.6 47.6 55.5M12 84.30 11.5 15.4 20.5 19.2 25.6 30.8 41.0 46.1 57.7 63.4 69.2 80.7M12 � 1.5 88.10 12.1 16.1 21.4 20.1 26.8 32.1 42.9 48.2 60.3 66.3 72.3 84.4

M14 115.4 15.8 21.0 28.1 26.3 35.1 42.1 56.1 63.1 78.9 86.8 94.7 110.5M14 � 1.5 124.6 17.0 22.7 30.3 28.4 37.9 45.5 60.6 68.2 85.2 93.7 102.3 119.3M16 156.7 21.4 28.6 38.1 35.7 47.6 57.2 76.2 85.7 107.2 117.9 128.6 150.1

M16 � 1.5 167.3 22.9 30.5 40.7 38.1 50.9 61.0 81.4 91.5 114.4 125.9 137.3 160.2M18 192.5 26.3 35.1 46.8 43.9 58.5 70.2 93.6 105.3 131.7 144.8 158.0 184.3M18 � 1.5 216.2 29.6 39.4 52.6 49.3 65.7 78.9 105.2 118.3 147.9 162.7 177.5 207.0M20 244.8 33.5 44.7 59.5 55.8 74.4 89.3 119.1 134.0 167.4 184.2 200.9 234.4

M22 303.4 41.5 55.3 73.8 69.2 92.2 110.7 147.6 166.0 207.5 228.3 249.0 290.5M24 352.5 48.2 64.3 85.7 80.4 107.2 128.6 171.5 192.9 241.1 265.2 289.3 337.6


M24 � 2 384.4 52.6 70.1 93.5 87.6 116.9 140.2 187.0 210.3 262.9 289.2 315.5 368.1M27 459.4 62.8 83.8 111.7 104.7 139.7 167.6 223.5 251.4 314.2 345.7 377.1 439.9

M30 560.6 76.7 102.3 136.3 127.8 170.4 204.5 272.7 306.8 383.5 421.8 460.1 536.8M36 (� 4) 816.7 111.7 149.0 198.6 186.2 248.3 297.9 397.2 446.9 558.6 614.5 670.3 782.1M36 � 3 864.9 118.3 157.8 210.3 197.2 262.9 315.5 420.7 473.3 591.6 650.8 709.9 828.2M36 � 2 914.5 125.1 166.8 222.4 208.5 278.0 333.6 444.8 500.4 625.5 688.1 750.6 875.7

M36 � 1.5 940.3 128.6 171.5 228.7 214.4 285.9 343.0 457.4 514.5 643.2 707.5 771.8 900.4M39 975.8 133.5 178.0 237.3 222.5 296.6 356.0 474.6 534.0 667.4 734.2 800.9 934.4M48 1,475 201.7 269.0 358.6 336.2 448.3 538.0 717.3 806.9 1,009 1,110 1,210 1,412

M56 2,032 278.0 370.6 494.2 463.3 617.7 741.2 988 1,112 1,390 1,529 1,668 1,946M64 2,678 366.4 488.5 651.4 610.7 814.2 977 1,303 1,466 1,832 2,015 2,198 2,565M80 4,490 614.3 819.1 1,092 1,024 1,365 1,638 2,184 2,457 3,071 3,379 3,686 4,300

M90 5,594 765.3 1,020 1,361 1,275 1,701 2,041 2,721 3,061 3,826 4,209 4,592 5,357M100 6,998 957 1,277 1,702 1,596 2,128 2,553 3,404 3,830 4,787 5,266 5,744 6,702

Boundary conditions: (1) Yield point controlled tightening; (2) Friction �tot¼ 0.16; (3) As is smallest area of cross-section; (4) proper screw section

design, so failure is located at threaded cross-section, (no thread stripping, no head stripping).

Notes (1) For torque controlled tightening in practice, the preload can be reduced (app. � 0.7); (2) for utilization of �eq¼ 90%; of Rp0.2, multiply

relevant preload by 0.9; (3) yield strength ratio kR¼Rp0.2=Rm; (4) for angular controlled tightening, multiply relevant preload by

[1þ 0.3(1� kR)=kR].


Table 14 Estimated Preload Level for Different Unified Screw Types

Preload level (kN), Unified Screw Thread

Threadsize

NominalAs (mm2)

Tensile strength and yield strength ratio of screw

Rm (MPa) 300 400 400 500 500 600 800 900 1,000 1,100 1,200 1,400kR (�) 0.6 0.6 0.8 0.6 0.8 0.8 0.8 0.8 0.9 0.9 0.9 0.9

1=4–20 20.5 2.80 3.74 4.99 4.67 6.23 7.48 10.0 11.22 14.02 15.42 16.8 19.65=16–18 33.8 4.62 6.17 8.22 7.71 10.3 12.3 16.4 18.5 23.1 25.4 27.7 32.43=8–16 50.0 6.84 9.12 12.2 11.4 15.2 18.2 24.3 27.4 34.2 37.6 41.0 47.9

7=16–14 68.6 9.38 12.5 16.7 15.6 20.9 25.0 33.4 37.5 46.9 51.6 56.3 65.71=2–13 91.5 12.5 16.7 22.3 20.9 27.8 33.4 44.5 50.1 62.6 68.8 75.1 87.69=16–12 117.0 16.0 21.3 28.5 26.7 35.6 42.7 56.9 64.0 80.0 88.0 96.0 112.0

5=8–11 146.0 20.0 26.6 35.5 33.3 44.4 53.3 71.0 79.9 99.9 109.9 119.8 139.83=4–10 215.0 29.4 39.2 52.3 49.0 65.4 78.4 104.6 117.6 147.1 161.8 176.5 205.97=8-9 298.0 40.8 54.4 72.5 67.9 90.6 108.7 144.9 163.1 203.8 224.2 244.6 285.41–8 391.0 53.5 71.3 95.1 89.1 118.9 142.6 190.2 214.0 267.4 294.2 320.9 374.4

1 1=4–7 625.2 85.5 114.0 152.0 142.5 190.0 228.1 304.1 342.1 427.6 470.4 513.1 598.71 1=2–6 906.4 124.0 165.3 220.4 206.7 275.6 330.7 440.9 496.0 620.0 682.0 744.0 868.01 3=4–5 1,226 167.7 223.6 298.1 279.5 372.6 447.2 596.2 670.8 838.4 922.3 1,006 1,174

2–4 1=2 1,613 220.6 294.2 392.3 367.7 490.3 588.4 784.5 882.6 1,103 1,214 1,324 1,5453–4 3,852 526.9 702.5 936.7 878.2 1,171 1,405 1,873 2,108 2,634 2,898 3,161 3,6884–4 7,148 978 1,304 1,738 1,630 2,173 2,608 3,477 3,912 4,889 5,378 5,867 6,845

5–4 11,484 1,571 2,095 2,793 2,618 3,491 4,189 5,586 6,284 7,855 8,640 9,426 10,9971=4–28 23.5 3.21 4.29 5.72 5.36 7.14 8.57 11.4 12.86 16.07 17.68 19.3 22.55=16–24 37.4 5.12 6.82 9.10 8.53 11.4 13.6 18.2 20.5 25.6 28.1 30.7 35.8


3=8–24 56.6 7.74 10.3 13.8 12.9 17.2 20.6 27.5 31.0 38.7 42.6 46.5 54.27=16–20 76.6 10.5 14.0 18.6 17.5 23.3 27.9 37.3 41.9 52.4 57.6 62.9 73.4

1=2–20 103.0 14.1 18.8 25.0 23.5 31.3 37.6 50.1 56.4 70.5 77.5 84.5 98.69=16–18 131.0 17.9 23.9 31.9 29.9 39.8 47.8 63.7 71.7 89.6 98.6 107.5 125.45=8–18 165.0 22.6 30.1 40.1 37.6 50.2 60.2 80.3 90.3 112.9 124.1 135.4 158.03=4–16 241.0 33.0 44.0 58.6 54.9 73.3 87.9 117.2 131.9 164.8 181.3 197.8 230.8

7=8-14 328.0 44.9 59.8 79.8 74.8 99.7 119.7 159.5 179.5 224.4 246.8 269.2 314.11–12 428.0 58.6 78.1 104.1 97.6 130.1 156.1 208.2 234.2 292.8 322.0 351.3 409.91 1=4-12 692.3 94.7 126.3 168.4 157.8 210.4 252.5 336.7 378.8 473.5 520.9 568.2 662.9

1 1=2–12 1,020 139.5 186.0 248.1 232.6 310.1 372.1 496.1 558.1 697.7 767.4 837.2 976.81 3=4–12 1,413 193.3 257.7 343.6 322.1 429.5 515.4 687.2 773.1 966.4 1,063 1,160 1,3532–8 1,787 244 326 435 407 543 652 869 978 1,222 1,345 1,467 1,711

3–8 4,200 575 766 1,021 958 1,277 1,532 2,043 2,298 2,873 3,160 3,447 4,0224–6 7,465 1,021 1,362 1,815 1,702 2,269 2,723 3,631 4,085 5,106 5,616 6,127 7,1485–6 11,871 1,624 2,165 2,887 2,707 3,609 4,331 5,774 6,496 8,120 8,932 9,744 11,368

Boundary conditions: (1) Yield point controlled tightening; (2) Friction �tot¼ 0.16; (3) Proper screw section design, so failure is located at threaded

cross-section (As is smallest area of cross-section; no thread stripping, no head stripping).

Notes (1) For torque controlled tightening in practice, the preload can be reduced (app.� 0.7); (2) for utilization of �eq¼ 90% of Rp0.2, multiply

relevant preload by 0.9; (3) yield strength ratio kR¼Rp0.2=Rm; (4) for angular controlled tightening, multiply relevant preload by

[1þ 0.3(1� kR)=kR].


The difference between initial and residual preload is caused by con-tact plastification (seating) or relaxation (material creeping, especially athigh temperatures).

1. Minimum Initial Preload

The minimum preload required is responsible for the selection of screw size.For a given screw strength and assembly method, the minimum level isgenerated for maximum friction coefficient. Therefore, in Table 13 formetric screw thread geometry, a friction coefficient mtot¼ 0.16 is assumed(relevant values of mtot see Table 4). The listed preload levels are reachedfor yield point controlled tightening. Using the legend, preloads for othertightening methods can be calculated. To achieve a preload level for a fric-tion coefficient mtot¼ 0.08, multiply relevant value of table by 1.15.

From the preload level of Table 13, with formulae of Fig. 16, the corre-sponding torque values can be obtained. But for torque controlled tighteningone must remember that the smallest torque corresponds to the smallest fric-tion coefficient and this has to be specified for assembly specification. If ascrew with high friction is tightened with the specified torque of low friction,the generated preload is reduced (see aspect 1 of legend from Table 13).

Table 13 refers to screws for existing nut thread. If thread rollingscrews (Fig. 6) are used, the preload level is reduced by some percentagebecause of the higher thread friction (app. �5%). For generating nearlythe same preload with thread rolling screws as with same screws for existingnut thread, the tightening torque has to be increased significantly (see alsoRef. [63] or Fig. 67).

Table 14 gives the same information as Table 13 for unified screwthread geometry. For designations of screw threads, see Fig. 5.

How does one find the required minimum initial preload Fp0?As a rule,the initial preload should be at least five times the maximum operating loadof the screw (nfFax) added by 10% for relaxation loss. This rule applies tostable threaded fastening systems without creeping effects. The initial pre-load must prevent any component from separating (guaranteeing sealingfunction, see also Fig. 49; avoiding of increasing load factor f, see alsoFig. 25), microsliding (fretting, self-loosening) or significant relaxation(continued preload loss with possible failure in consequence).

2. Boundary Conditions in Practice

For selection of screw size, handling during operation and field maintenanceis an important consideration. As an example, a screw of dimension M6 orhigher can normally be hand tightened by workers without danger of


overtightening. A screw up toM12 can be tightened=retightened with normalwrenches and moderate manpower. Screw dimensions between M6 and M12can be used by nearly every person without special qualification=trainingand=or special equipment.

The design engineer can always decide if a few large screws or moresmall screws are used to achieve the summarized preload. An increasednumber of small screws has the advantage of better stress homogeneity inthe components, better sealing of flanges, reduced local separating of com-ponents with low stiffness under operating load. But a multi-screw-fasten-ing-system needs a detailed calculation of the loading of each particularscrew and a defined tightening sequence during assembly.

Screws, which need exact preload, should be tightened by yield pointcontrol or angular control (see also Fig. 51).

Finally, the requirements from deproliferation have to be met. Thenumber of different parts which have to be purchased, stored, and managed,has to be minimized. This means, consolidating similar screws due to screwlength, screw diameter, screw head, screw material, and screw surface.

C. Determination of Screw Geometry

If the screw thread size is known, several additional geometry details have tobe determined. These are screw head, length of thread engagement, screwbody, thread length, and other design options. This chapter shows the fun-damental aspects for design decisions.

1. Thread Engagement

If the design principle from Fig. 2 is valid, the thread engagement requires aminimum value temin, and thread stripping of screw or nut cannot happen.

Figure 33 points out the result from calculations regarding the VDI2230 guideline [70] for metric thread series (thread standard, see Fig. 4).The diagram illustrates the relative minimum thread engagement temin=dover tensile strength of nut thread component Rmn for different propertyclasses of screw. Details are printed in the diagram. Generally speaking,the required length of thread engagement increases with increasing screwstrength Rms and decreasing nut strength Rmn. This diagram has two dimen-sions of interpretation: for thread engagements higher than the relevanttemin, no thread stripping will occur and in any case the screw shank will fail(direction of ordinate-axis). If the relevant point for temin on the selectedhyperbolic curve is located in the tangential section, the screw thread willstrip for engagements smaller than temin. If the relevant point for temin is


in the gradient section, for engagements lower than temin, the nut thread willstrip (direction of abscissa).

A special characteristic is important for low-strength nut thread com-ponents (e.g. made of magnesium): for example, a nut thread with a tensilestrength Rmn of only 250MPa requires theoretically as a minimum threadengagement of �2 � d (d: nominal diameter of screw thread) with a screwof class 8.8; whereas only �1 � d are necessary for a screw made also ofmoderate screw strength (Rms¼ 400MPa, which could be a screw made ofaluminum). As a rule, the screw strength Rms should not exceed 1.0� –2.5� the strength of nut thread component Rmn from the point of threadengagement.

The diagram in Fig. 33 does not consider incomplete thread flanks atthread end of screw and does not include any safety factors, so the values fortemin in practice should be multiplied with 1.3–1.5. For experimental verifi-cation of Fig. 33, see Fig. 58.

2. Screw Body and Thread Length

Figure 34 describes the fundamental possibilities to design a screw shank(a)–(e). Normally, a screw shank has two different cross-sections: the

Figure 33 Calculation of thread engagement vs. strength of nut thread material.(From Ref. 16.)


threaded cross-section with area As and the unthreaded cross-section witharea Ab resp. A2 (area of cross-section with flank diameter d2).

The length of a threaded shank with rolled thread flanks as shown in(a) is limited by the length of the rolling die for screw production. A fullshank (b) has a constant outer diameter in the range of the nominal screwdiameter d. Such a screw possesses good self-centering behavior throughholes. An exactly defined centering function can be realized with anincreased shank (c). A reduced shank (d) often is an optimum betweenscrew-weight, -cost, and -function, because the reduced shank has a dia-meter in the range of thread flank diameter d2, so the screw production linecan be made effective. A wasted shank (e) gives a high screw resilience withlow additional screw force under loading; here the body diameter dB shouldbe made as long as possible (a guiding diameter is necessary under head andthe transition between different diameters has to be designed with large radiifor avoiding of stress concentrations). As a guideline, a shank type (a) or (d)should be taken whenever possible.

The clamping length is the distance between head support and start ofthread engagement and the plastification length is the length of free shankunder preload with smallest cross-section As or Ab. Therefore, lc is the same

Figure 34 Clamping length lc and plastification length lp of threaded fasteningsystem from Ref. 18.)


for all screw types (a)–(e). In contrast, the plastification length varies fromlp¼ lc at type (a) to smaller values at types (b)–(e). By reason of the signifi-cant area difference between As and Ab resp. A2 at the same screw, only thesmallest cross-section will plastify under tensile load; this smallest cross-sec-tion comes to failure before the other cross-section gets plastified dependenton the materials ratio of Rp0.2s over Rms.

3. Screw Head

The screw head includes two important aspects: (a) type of screw drive, and(b) type of support area. The type of screw drive is responsible for capabilityof assembly process, the type of head support area is influencing thedesigned function of the fastening system.

Figure 35 presents the established and widely used types of screw drivegeometries. They are distinguished by external and internal types. Forexternal and internal geometry four designations are important: hexagon,bihexagon, triple square, and hexalobular. If considering internal configura-tions also, cross-recess drives (e.g. [21]) and slotted screw drives are ofinterest. These two geometries are dominant for small screws without high

Figure 35 Basics of screw drive selection.


preload because they cannot provide high torque values which can be trans-mitted reliably between bit and screw.

The most common screw drive globally is the hexagon geometry.This is important for components which have to work and must berepaired in areas without technical experience. This drive type is suitablefor high torque values if there is only a small clearance between bit=wrenchwrench and screw and if the drive has no damage. Using an open wrenchas a rough estimation, only half of the torque compared with a ring span-ner can be applied with reliability. The reason is that when using an openwrench, only two flanks are used for torque transmission. Since six driveflanks and a small contact angle between bit and screw for the line contact,the hexagon drive may lead to damaging the surface of the screw, espe-cially if the screw is coated for corrosion protection or if worn bits areused. In Fig. 35, these aspects lead to a sum of 9 assessment-points fromthe 20 possible.

A significant improvement of drive torque loading capacity and relia-bility is achieved with 12 flanks (bihexagon and triple-square drive geome-tries). A bihexagon drive geometry is created by two hexagon drives,which have the same center point and an angular misfit of 308. A triplesquare drive geometry is created by three square contours, which have thesame center point and an angular misfit of 308 each.

A hexalobular drive geometry [established by Textron-Camcar underthe designation TORX#] consists of one (small) convex and one (large) con-cave contour radius, which are alternately combined [22]. This leads tosmooth contact pressure between screw and bit as well as small-sized outerbit diameters for compact design structures. There is no significant differ-ence in using this design compared to bihexagon or triple square, except thatthe same maximum drive diameter, the hexalobular drive geometry has alower drive section modulus against torsional failure. Triple square orbihexagon drives should be used.

For the internal drive configurations, the same comments are valid.Compared to the external configurations with the same head diameter,the drive flanks are smaller and the internal configurations are stressed toa higher level for same torque transmission. The bit is much smaller whichis very positive for the accessibility. In most cases for internal bihexagon, tri-ple square or hexalobular drive, the bit determines the torque limit, not thedrive of the screw. Internal drive configurations usually have lower weightof the screw head than external drives, but internal drives can lead to headstripping under preload, if their bit-engagement is too deep. On the otherhand, a minimum bit engagement is necessary for reliable assembly process.These two influences determine the height of head for screws with internaldrive.


Slotted screws are only relevant for applications with low requirementsfor screw tightening. They have a cam-out-reaction under torque loadingand the blade of the screw driver can have a radial misalignment, whichleads to damage of screw, screw driver and possibly of component surface.Cross-recess drives are an obvious improvement over the slotted screws inlow torque applications like screws for fastening wooden constructions orplastic components. They provide a radial alignment between screw anddriving bit, but the negative cam-out-reaction is significant. The life timeof cross-recess bits is quite short.

Of course, there exists many other drive systems for special require-ments, such as Square drive, Multispline, Hexapol#, Triwings#, Clutch-

Figure 36 Contact conditions of high torque screw drives. (From Ref. 17.)


type#, Torx-Plus#, or Polydrive#, which often are trademarks of differentcompanies.

Figure 36 demonstrates the contact conditions of high torque screwdrives from Figure 35 in a more detailed manner. In any case, the tolerancesituation is important for the torque loading limit of the drive. The clearancein Fig. 36 is oversized in order to emphasize that all screw drives have singlecontact lines at each drive flank, if they are undeformed (only contact pointsin drawn cross-sections).

The applied torque Ttot leads at each drive flank to a circumferentialforce Fc, which can be divided into a normal part Fn (torque transmission)and a tangential part Ft (contact sliding and in consequence flank wear). Foran ideal drive geometry, this Fc can be calculated as shown in Fig. 36.

Between Fc and Ft, one can measure the contact angle E. This value is308 for hexagon and bihexagon drive, 458 for triple square (see also Fig. 63)and about 608 for hexalobular drive geometry. A small contact angle meanshigh contact sliding under torque loading. This is the reason for surfacedamaging of the screw area engaged to the bit as well as the reason for wearof the bit flanks.

Figure 36 confirms that a bihexagon drive has the same contact con-ditions as a hexagon geometry, but the increased number of engaged flankslowers the Fc at each single flank. The triple square and hexalobular drivesystems have an increased contact angle, so they should be taken as adesigned screw drive system today, if no advantages of other drive systemsare predominant.

If the clearance between bit and screw drive contour is too large, thebit life time decreases significantly and the danger of screw drive damagingoccurs.

Often for small-volume-designs, the space for screw head and theaccessibility for bit are limited. Figure 37 compares the space requirementsof three screw head designs with hexalobular drive type for same thread dia-meter d and same support diameter da. Part (a) refers to an external config-uration, which is characterized not only by high stiffness of the screw head,but also by large height requirements. The bit for driving the screw normallyhas a largest diameter up to 2.0d as the head support diameter da.

If using a standard design with internal configuration (b) the height ofscrew head is reduced to �80% of (a). Also, the size of the screw drive flanksis reduced to only � 60% of (a). This can cause problems if the screw hashigh material strength and if the screw is tightened to high preload levelbeyond the screw material yield limit. In this case, the cross-section of thedriving bit exceeds its fatigue limit, so that the life time of the bits isdecreased drastically. Another aspect of internal drive configuration is theratio of screw head height and length of bit engagement. This ratio has to


be large enough for a given head geometry, so that no head stripping occursunder preload.

An optimized design with low height of head, large screw drive flanksand deep bit engagement is proposed in part (c) of Fig. 37. Even if the height

Figure 38 Types of support area and calculation of effective bearing diameter.

(From Ref. 17.)

Figure 37 Comparison of screw head designs for same diameters d and da.


of head is only 0.7d, no head stripping occurs under preload due to the rea-son of the conical head-shank-transition. The large length of bit engagementguarantees a high assembly process capability. The large size of screw driveflanks leads to a long bit life time for any tightening method. The internalconfiguration offers an easier drive accessibility by a small bit diameter com-pared to the external configuration of (a).

Another important design aspect of screw head is the type of supportarea. Figure 38 displays three established types of support area betweenscrew head and clamped part. Each type has its own calculation for theeffective bearing diameter Deb [72]. This diameter Deb represents the virtualdiameter, where the circumferential force produced by the contact frictioncan be concentrated for calculation; it influences the head frictional torqueTh directly (see Fig. 16).

A plain support type is used as a standard; it is easy to manufacture andrequires no special geometrical matching of screw and clamped part. Largehead support diameters da are suitable for low surface contact pressure(see also Figure 39) and for covering large clearance holes. Countersunk-and ball-section-support types provide a centering function between screw

Figure 39 Required relative support diameter for given maximum contactpressure. (From Ref. 16.)


axis and position of clamped part. Therefore, the positioning tolerance ofsuch multi-screw-fastenings has to be precise (e.g. wheels of vehicles).

Countersunk- or ball-section-support types have to be tightened withdifferent torque values to obtain the same preload depending on the effectivesupport diameter Deb. Normally the assembly torque Ttot is increased by�15% compared to the plain support type and similar other boundary con-ditions. However, this approximation cannot replace a detailed calculation.By reason of the increased head frictional torque Th, countersunk and ball-section-support types provide an enhanced safety against self-loosening.

If using countersunk- and ball-section-support types, the tolerances ofcountersunk angle, the ball diameter of the screw, and the clamped partmust fit together. In any case, a full bearing area in the support contact isguaranteed [see also ISO 7721 [43]]. If using plain support type with clampedparts of high strength in the range of the screw strength or higher, thedetailed geometry of the screw support area should be designed in a slightlyconcave manner, so that the contact diameter is defined clearly.

If using thin sheet materials, significant angle tolerances between screwaxis and support area or rough surfaces. The effective bearing diameter Deb

in practice can differ from the theoretical calculations regarding Fig. 38.A measurement of Deb in experiment is recommended.

Important for design of screw geometry is the support diameter da.Figure 39 illustrates the dependence of required minimum head support dia-meter for a given permitted maximum surface pressure (plain support type).The three functional curves belong to different property classes of screws(tensile strength values 1200, 800, and 400MPa; for property classes, seealso Table 10). The lowest curve represents a screw made of low-strengthmaterial like aluminum. For example, a given maximum surface pressureof 100MPa in the contact zone between head and surface of clamped partmeans a very large relative head diameter of 3.2 � d if using a bolt with astrength of 1200MPa and a relative diameter of only 2 � d if using a boltof strength 400MPa (e.g. made from aluminum).

This diagram makes it clear that only screws with flange head and alow screw strength can reach the demand for low surface pressure withacceptable head diameter. This is required for materials of clamped partwith limited loadability and significant creep behavior, e.g. magnesium com-ponents at elevated temperatures.

The permitted contact pressure pchperm for the particular materialclamped should not be exceeded to avoid any excessive plastic deformationin the head contact area, even if the bolted joint is in operation (this wouldresult in a loss of preload). As a rough estimation, the permitted contactpressure pchperm should not exceed the minimum of (Rp0.2þRm)=2, eitherof the screw material or of the clamped part material. This can only be done


if no creep occurs. In that case, experiments must determine the contactpressure limit pchperm, which does not lead to significant preload loss. Oftenit is about half of Rp0.2.

If a high-strength screw with only low assembly torque to limit the lowpermitted head contact pressure is used, the danger of misassembly occurs(missing information for the right handling in field service, perhaps byunauthorized workers).

Figure 40 finally emphasizes the result of a torque-preload measure-ment over tightening angle done at RIBE laboratory with two slightly dif-ferent contact angles for head support (with plain, but concave bearinggeometry). Part (a) belongs to a support angle of only 18 between screw headsupport area and spot face of clamped part. Here, the maximum preloadreaches about 57 kN and the maximum tightening torque increases up to107Nm. Part (b) contains a screw with support angle of 38, which leadsto almost the same maximum preload of 55 kN, but to a very high maximumtightening torque of 168Nm (as a result of metallic adhesion between screwand clamped part caused by severe local stress peaks at support diameterregion).

This means that very small changes of the head support geometry orsurface can lead to significant changes in tightening behavior, especially ifoverelastic tightening methods are applied (see also Fig. 18). Besides this,Fig. 18 confirms that in spite of the large difference in friction for both situa-tions (a) and (b), the preload is almost the same—this result can be achievedonly with overelastic tightening methods.

Figure 40 Influence of head support details on tightening behavior.


D. Design Options

1. Established Main Types

Besides the rules of basic mechanics for threaded fastening systems, a largenumber of design options exist. The most important options are proposedbelow. The design engineer has to select the correct options important forhim. Of course, some options are suitable for more than one design targetand others are very specialized.

Figure 41 indicates design options for three optimization targets (a)–(c):improved assembly, improved fatigue limit, and avoiding of self-loosening.

For all optimization targets, the most important actions are listed.Figure 41 is self-explanatory, but some aspects are discussed in a moredetailed manner in the following lines. Thread ends for finding the bestnut thread are most important for short screws which have no significantself-alignment by the through hole of the clamped part (clamping length lcunder 1 � d, see also Ref. [21]). The automated handling of screws is mucheasier if the screw exhibits a center of gravity location with dominatingshank-weight so that the screws tend to fall ‘‘head-up’’. The mathematicalcondition given in Fig. 41 is a rough approximation for guidance. Forfurther details, see Ref. [21].

The fatigue limit of a bolted joint depends on all the parts of the fas-tening system. One action to improve the fatigue behavior is to increase thematerial limit of the screw (materials selection, etc. local residual stresses,etc. rpar; or to reduce the additional operating force of the screw duringoperating by increasing the elastic resilience of the screw. Examples aregiven in Fig. 41. Another aspect is the reduction of stress concentrationswhich appear most at first bearing thread flank (see also Fig. 2).

Self-loosening can happen under high preload if microsliding in thecontact zones of head support and thread contact appears (e.g. largedynamic transversal loading, see also Fig. 76). A head support with lockingteeth is a very effective action to prevent self-loosening without influencingthe preload level of the fastening system. Washers or similar additional ele-ments with ribs or teeth are usually of no help against self-loosening.

Thread flank clamping as an alternative is based on ‘zero-clearance’between nut thread and screw thread flanks, but a clamping torque reducesthe acting preload after tightening. Adhesives also eliminate flank clearancebetween nut thread and screw thread without a clamping torque (but with athread frictional coefficient mt of about 0.20). Adhesives have a limited oper-ating temperature of �2508C depending on the adhesive material.

Three other design targets can be: improving preload, reducing weight,and avoiding unauthorized disassembly. Established design options to meetthese requirements are demonstrated in Figure 42.


Figure 41 Design options for bolted joints due to assembly, fatigue, and self-loosening.


Figure 42 Design options for bolted joints due to preload, weight, and disassembly.


For improving the preload acting in the fastening system, the mainactions are increased screw diameter (1), increased screw strength (2) andselecting an optimized tightening method (3, see also Fig. 18). For compo-nents made of low-strength materials, the time dependency of preload isimportant (relaxation effects, see also Fig. 66). In this case often, it is moreuseful to reduce the preload retention during time of operation than increas-ing the initial assembly preload which is decreased extremely over time (4 inFig. 42).

For reducing weight in part (b) of Fig. 42, six actions are mentioned. Ofcourse, first the weight of the fastening element can be minimized, e.g., byusing an aluminum screw instead of steel screws. This is very positive espe-cially if light metal components with low-strength and high thermal expan-sion coefficient have to be fastened. A secondary weight saving effect isthat for low-strength nut thread components an aluminum screw offers areduced minimum thread engagement with chance for a small component size(see also Fig. 33 for screw with a strength of 400MPa and section 4 in Fig. 42).

For components with high strength and in consequence high loadbearing limit, the use of a high-strength screw is suitable. The size of thishigh-strength screw can be reduced compared to a screw made of materialof a lower property class (2). The same effect can be realized with a bettertightening level of the screw (3). Here also, the design limits of screw andcomponents must be sufficient so that, the tightening method is fundamentalfor a design analysis (3 in column (a) and Fig. 50). Additional actions forreducing weight can be minimizing screw head volume and, if necessary, ahollow screw shank (interesting for large screws, which are not stressed upto their loading limit).

The right column in Fig. 42 shows actions for avoiding unauthorizeddisassembly of a threaded fastening system. For using a special screw drive(2), the compatibility with maximum torque level during tightening and withavailable tightening tools in production and field service has to be checked.A shear-off-drive is designed, so that the drive is sheared-off if the ultimatetightening torque is generated. Then the screw can be disassembled by exten-sive mechanical work resulting in destroying the screw. Applications includelocking devices. An important aspect is the missing corrosion protection inthe broken shear plane of the screw.

Another way to avoid an unauthorized disassembly is using a combi-nation of thread rolling screw, adhesives, and large thread engagement. Thescrew can be tightened, but the screw drive is not suitable to transmit such ahigh torque which would be necessary for disassembly. This solution is usedfor components with safety relevance (which may not be opened, e.g., con-trol units for anti-locking-brake-systems).


From the point of screw mechanics, washers should be avoided becausethey always lead to more surface contacts in the fastening system with rough-ness and in consequence with possible preload retention (seating, relaxation).But washers are useful to prevent surface damage of component surface byrotating screw head during tightening (e.g., for fastening of painted compo-nents). Another reason for using washers can be providing a high-strengthcontact surface for the screw head, which transmits the preload to the low-strength component material (reduction of contact pressure for the compo-nent). then, the washer needs a thickness of about 20% of the nominal screwdiameter and a hardness, which is in the range of the screw or higher.

To avoid self-loosening of the screw, any washer design must bechecked very critically because only a few washer geometries can guaranteethis (see Fig. 41).

2. Special Elements for Threaded Fastenings

Two groups of fastening elements which have a strong growth for newdevelopments of components are visible in Figs. 43 and 44. Staking elementsare of great importance for automated generation of a screw thread or nutthread in thin sheet metal components without sufficient material for threadengagement. Figure 43 contains self-explanatory details for a staking boltwith additional characteristics for use. In contrast to welding bolts or nuts,

Figure 43 Principle of staking bolt; system RIMS. (From Ref. 64.)


staking elements produce no thermal loading of the component and, there-fore, can be used for nonweldable materials or components with surface fin-ishing. Besides this, they need no welding equipment, but can be integratedin stamping tools or deep-drawing tools of existing production lines, so theyare very economic high-duty fastening elements.

If the screw is combined with a sleeve, this preassembled product canbe inserted to the clamped part with interference fit tolerance, so the screw ispart of the component and cannot get lost. Significant advantages areachieved if used with components with integrated screws: no searching forscrews, no falling of screws, no separate fastening elements (purchase, sto-rage, logistics), better positioning of component (especially for overheadassembly), and no mixing up of different screws (e.g. screw lengths). AsFig. 44 shows, these elements can offer the additional functions of preloadtransmission by the sleeve itself (e.g., for low-strength components made ofplastic, see also Fig. 65) or acoustic insulation (reduction of structure-bornenoise in engines or machines, Fig. 45).

3. Characteristics of Studs

Often, instead of screws with head, as an alternative, headless studs withnuts are used. On the one hand, studs have their advantages in easy posi-tioning of clamped part by the stud itself (field service and repair) and insmall space requirements for inserting the nut instead of a long screw.

On the other hand, the screw has the advantage that only one fasteningelement for one fastening system is used, which means well-defined assembly

Figure 44 Principle of screw combined with sleeve, system Rifixx. (From Ref. 61.)


and a high total process capability, especially for high preload levels (e.g.overelastic tightening). Figure 46 summarizes the most important aspectscomparing studs and screws.

4. Characteristics of Washers

Both washers and flange heads can be designed with the same support dia-meter for the component (Fig. 47). The design engineer has to decide whichgeometry must be selected for the fastening solution. In Fig. 47, the mostimportant aspects are summarized. In practice, the main reasons for usingwashers are the prevention of damage of (painted) surfaces, the reductionof contact pressure at clamped parts with low material strength (e.g. plastics)or covering large through holes. For high-duty threaded fastening systems,always a washer head (flange head) of the screw should be considered by rea-son of the enhanced assembly process capability. Flange heads of screws canbe produced economically up to (2.5–3) � nominal screw diameter.

Figure 45 Example of Rfixx-plus-element. (From Ref. 61.)


In Fig. 47, the outer support diameter is drawn equally for washer andflange head. But for the assembly behavior only the sliding diameter da isrelevant. The maximum preload requires different tightening specificationsbetween washer and flange head. Washers often have chamfers. The

Figure 46 Comparison of screw and stud with nut.


maximum washer diameter has to be placed on the clamped part. But it ispossible that a washer rotates on the clamped part during tightening. Thisleads to changing assembly behavior of the threaded fastening system (seealso Fig. 48).

Figure 48 demonstrates the assembly behavior of a screw with captivewasher, tightened on a clamped part made of aluminum with low surfaceroughness of the machined spot face, where the washer with almost the samewasher diameter as the screw head diameter is supported. By reason of theseboundary conditions, during tightening, it is possible that for a low preload,

Figure 47 Comparison of screw with washer and screw with flange head.


the washer rotates on the component and for higher preloads the screwrotates on the washer. But the contact zones under screw head change(different roughness, different materials, and different lubrication), whichleads to a significant change of the head frictional torque (recorded in themeasuring diagram of Fig. 48 over preload in screw shank).

For automated assembly procedures, this can cause error signals of thetightening device, especially if the screw is tightened with yield control.

5. Characteristics of Sealings

If a gas- or liquid-tight threaded fastening system is required, three contactshave to be considered: (1) head contact sealing, (2) component contact seal-ing, (3) thread contact sealing (Fig. 49). Often, number (2) is most important.

For the design engineer, the task of selecting the right ratio betweenclamping length lc and screw distance x is an important one. This ratio x=lcshould be smaller than 10 in order to minimize critical zones for leaking.The mean nominal contact pressure for a gasket should be larger than2MPa. In order to obtain the same tightening behavior with and withoutsealing, a gasket should be as thin as possible (e.g., spring steel sheet withsome mm polymer-coating to fill the roughness of technical surfaces).

For further details, see Fig. 49. For sealing technology with liquid gas-kets and adhesives, see Ref. [52].

E. Loading vs. Loading Capacity—Design Analysis

1. General Procedure

A design analysis has to guarantee that no failure of the fastening systemcan happen. This is only possible by comparing maximum loading (of screw,

Figure 48 Assembly behavior of screw and washer with similar support diameter;measured data.


clamped part, and nut thread component) and loading capacity of thesecomponents in a bolted joint; and this has to be done for both tighteningand operating situations. Distinguishing tightening and operating is impor-tant for considering different stress states and different temperatures (t1, t2).

Figure 50 proposes a fundamental approach to this for a given data ofscrew, clamped part, and nut thread component. The design requirement is asufficient preload Fpnec (often this preload is based on technical experience).

Besides this, also time- and temperature-related limits are input datafor design analysis. These are, on the one hand, material strength values ofscrew (static Rmst1, Rmst2 and dynamic saspermt2), clamped part (Rmpt1,Rmpt2), and nut thread component (Rmnt1, Rmnt2), on the other, force and

Figure 49 Fundamental aspects of sealing with threaded fastening systems.


Figure 50 Principle of comprehensive design analysis for threaded fasteningsystem. (From Ref. 17.)


pressure limits, which are also dependent on the specific geometry (Ttotpermt1,pchpermt1, pchpermt2, Fthreadpermt1, Fthreadpermt2, Fheadpermt1, Fheadpermt2, andFtranspermt2). For example, the permitted contact pressure under head aftertightening at temperature t1 (pcht1) and during operating at temperature t2(pcht2) must not reach extensive plastification of the bearing surface, whichis characterized by pchpermt1 and pchpermt2.

Ttotpermt1 is the maximum torque during tightening which can be trans-mitted by the screw drive without problems. Fthreadpermt1 and Fthreadpermt2 arethe maximum preloads before thread stripping of the bolted joint occurs(stripping can take place at both, nut thread or screw thread dependingon the tolerances and material strengths, see also Fig. 33). Fheadpermt1 andFheadpermt2 are the maximum preloads before head stripping of the screwtakes place. During operation, acting transversal forces have to be lowerthan Ftranspermt2.

The first loading of the fastening system is done during tightening. So,the minimum and maximum assembly preload Fpamin and Fpamax have to becalculated (of course, dependent on the tightening method). If the assemblypreload is known, also the tightening torques Ttotamin and Ttotamax forassembly can be determined.

From these tightening preloads and tightening torques, the loadingswith bullets under (2) are applied during operation, e.g., axial static ordynamic force. This leads to the results of minimum and maximum operat-ing preload Fpomin and Fpomax. One important aspect, especially for tighten-ing methods with screw plastification, is the reduction of torsional stressafter taking away the tightening torque (about �10% to �30% of the high-est torsional stress under torquing conditions). This increases the axial load-ing limit of the screw, so also plastified screws can bear significantadditional loads in a threaded fastening system. Besides this, a chemical sta-bility is assumed in general for this design analysis.

The design criteria under (3) compare all relevant loading values withthe corresponding loading limits (forces, stresses, pressure, and torque).Only if all criteria are valid, the bolted joint is designed safety. But some-times, certain criteria are not important; so the number of relevant criteriacan vary (e.g., only low level tightening makes stripping, contact pressure,drive torque limit, and screw overloading uncritical, so most of the designcriteria can be neglected). The design criteria are distinguished for axial=transversal force and design section compatibility (see also Fig. 2). Theselection and assessment of the design criteria are an engineering task.

If the threaded fastening system only works at room temperature, notemperature influence has to be considered (t1 and t2 are missing). If the fas-tening system operates at various temperatures, the highest and lowest tem-perature have to be considered, so then t3 occurs.


2. Preload Deviation in Practice

For engineering design, it is important to consider that the initial preloadcan vary significantly for the same screw specification, depending on thetightening method, the deviation of friction coefficient in contact zonesand for overelastic tightening methods depending on the deviation of screwstrength.

Figure 51 gives an example for a threaded fastening system with screwM8-10.9. The diagram specifies the torque-preload-behavior for four casesA–D, calculated with formulae from Fig. 16. The cases A–D distinguish inde-pendent deviations of frictional coefficients mth and mh from 0.08 to 0.16 as

Figure 51 Torque-preload-behavior of screw M8-10.9.


well as deviation in screw strength from 1000 to 1200MPa (10.9). This resultsin four different linear functions between tightening torque and preload. Oneach curve, there are three markings (rhomb for an equivalent one-dimen-sional stress seq¼ 0.9Rp0.2 during tightening, triangle for correspondingseq¼Rp0.2 and quadrangle for seq¼Rm).

Now, if the screw is tightened with torque control in the range of 16–20Nm (see gray field in Fig. 51), the generated preload can vary from 8 to22 kN (8 kN for case D and minimum torque of 16Nm; 22 kN for case Aand maximum torque of 20Nm). Please note this is a ratio of maximumpreload over minimum preload of almost 3. In practice, this means thatfor this fastening system and this tightening specification, only 7 kN areguaranteed at minimum.

On the other hand, for case A, the yield point of the screw is achieved atabout 24Nm (position of triangle), so for this situation, the tightening torquecannot be increased significantly. As a result, in general, the disadvantage oftorque controlled tightening is that the tightening torque Ttot must be speci-fied for lowest possible torque value (case A in Fig. 51). For other combina-tions of deviations, this gives a poor preload Fp (e.g. case D in Fig. 51).

The difference of overelastic tightening compared to torque control isoutlined for yield point controlled tightening in Fig. 51. If yield control isused, for every screw the beginning of plastification is detected, so everyscrew, is tightened to its triangle marking. Then the preload is generatedin the range from 18 to 27 kN (ratio maximum preload over minimum pre-load is reduced significantly from 3 for torque control to �1.5!; arrows withdashed lines). In practice, this means a slightly increase of maximum preloadand an extensive increase of minimum preload (see also Fig. 18). The smallerdeviation of preload must lead to higher deviation in torque values (Ttot is inthe range of 24–43Nm in Fig. 51). One should never worry about changingtorque values if overelastic tightening is used; the preload is safe, if the screwstrength and the friction are as specified.

3. Dynamic Loading Capacity

The dynamic loading capacity of threaded fastening systems depends on alot of details regarding the entire joint-like value of external alternatingloading, stiffness and resilience, eccentricity, symmetry, component separat-ing, thread geometry, residual stresses, occurring stress peaks, manufactureof screw, and finally besides others also fatigue strength of screw material.Therefore, an exact determination is only possible by experiment resp.measurement of the original system in the particular application. Testingof the dynamic behavior of a designed structure covers the most oftenperformed tests.


Generally speaking, the maximum stress concentration factor of about8 (at the first bearing thread flank of screw, see Fig. 2) reduces the screwmaterial fatigue limit by a theoretical factor of 8 compared to resultsobtained with cylindrical samples without notch geometry effects (oftenlisted in engineer’s handbooks. For a first approximation of the screwfatigue limit without additional information, take the sample value of thecylindrical screw diameter and divide by 10. This often is necessary forscrews made of nonferrous metals).

If no data for fatigue-limit are available, as a rough approximation,the sample fatigue limit of steels is about half of the tensile strength andthe fatigue limit of aluminum is about one-third of the tensile strengthregarding the same sample for axial loading, see Ref. [48].

Reference [70] gives an empirical relationship between steel screw fati-gue limit and screw diameter as reported in Fig. 52. The diagram, on the onehand, distinguishes between thread rolling before and after heat treatment,and on the other, considers the preload dependence, correlated by the termFp=(Rp0.2�As)—a ratio up to 0.7 belongs to screw tightening without plasti-fication, a ratio of 0.8 belongs to yield point controlled tightening and aratio of 0.9 belongs to angular controlled tightening with significant plasti-fication of screw shank.

The diagram shows two general aspects: (1) overelastic tightenedscrews still have a significant fatigue limit, and (2) the fatigue limit of screws

Figure 52 Fatigue limit sasperm0 of steel screws depending on screw diameter d.(After Ref. 70.)


with rolled thread after heat treatment depends significantly on the preloadlevel (reason: strain hardening and residual stresses from thread rolling arenot compensated by a new grain structure from heat treatment, so nonlinearprofiles from loading stresses and residual stresses are superposed).

The test principle for determining axial fatigue load saspermt2 is definedin ISO 3800 [30] or more detailed in DIN 969 [10] for threaded fastening ele-ments. Normally, the screw shank is the location of fatigue failure, but theclamped part or nut thread component can also end in fatigue failure, e.g.,thin sheet metals as clamped part and a screw head with locking teeth. If thelocation of fatigue failure is at screw head fillet, significant bending of screwis probable (see also Fig. 54).

Minimizing fatigue problems can be realized by reducing the screwstressing (e.g., larger screw size, lower additional force for screw in fasteningsystem), proper screw section design (see Fig. 2, e.g., sufficient radius athead-to-shank-fillet, perpendicularity of screw axis and head support,smooth transition of each discontinuity at screw shank, such as differentdiameters of screw, running out of thread to unthreaded shank), no overlap-ping of stress concentrations (e.g., chamfer at clamped part under head or at

Figure 53 Typical cross-section of screw M18 � 1.5-12.9 failed in fatigue, result oftesting procedure according ISO 3800, stress amplitude sa¼ 80MPa, mean stress555MPa, symmetric axial force without bending moment.


first nut thread flank, avoiding corrosive pittings at thread flanks or athead-shank-transition). The most established actions to increase the fatiguelimit of the screw itself are discussed in Fig. 41.

Figure 53 contains a cross-section of a screw M18 � 1.5 which hasfailed in fatigue. A plane fracture zone can be seen at outer regions of thecross-section (area of crack initiation and crack propagation) and anunplane fracture area in the center of the cross-section (residual area ofrapid failure under preload).

Figure 54 in contrast to Fig. 53, gives an impression of a fatigue failurewith significant bending moment under external loading. This result wasobtained with a transversal vibrational test (see also Fig. 76). Now, the areaof crack initiation and crack propagation with ‘‘cycle lines’’ is clearly differ-ent from the residual fracture area. The size of this second part of cross-sec-tion gives information whether the acting preload at the event of failure washigh or not (fatigue failures often are induced by wrong tightening or loss ofpreload caused by relaxation or self-loosening).

F. Aspects of Quality Management

The overall objective for quality aspects of a threaded fastening system isto guarantee sufficient preload. This preload normally is not specifieddirectly. Due to this reason, a large number of details must fit together,which have to be realized by different responsibilities. Figure 55 demon-strates seven main groups of authorities which have to give their contri-bution to quality of the fastening system. The drawn boxes make clearthat every authority has its objective, its risk and takes actions due todifferent criteria. Besides this, it is important for clarifying failures thateach authority in most cases belongs to different business units or com-panies, so various communication interfaces exist, which have to workwithout deficit or error. So, from the point of organization, a ‘‘fasteningmanager’’ is recommended.

A few more quality aspects are:

� The calculation of threaded fastening systems in any case is anapproximation because numerous details have to be estimated likereal screw loading in the system, local fatigue strength of screw,material inhomogeneities, real external loading spectrum or others.By these uncertainties, the compatibility of a designed fasteningsystem with experience from former solutions is valuable, andcritical calculations have to be verified by experimental testing. Butproper designing due to guidelines of this chapter avoids a lot offailure situations and reduces testing expense.


� From the viewpoint of manufacturing, it is very important torealize that the (mass) production of screws is characterized by ahigh degree of automization and a high precisement due togeometry tolerances and material tolerances of the fasteningelement (e.g. see tolerances in Fig. 4).

� The lifetime of products tends to be higher; so the degree of long-term quality due to relevant fastening systems also has to beenhanced. The most important aspects are corrosion and relaxa-tion as well as fatigue strength. These properties cannot be testedduring component production.

� Optimized components with high material utilization have strongrequirements to their fastening systems. So, a componentoptimization always has to include the fastening systems as earlyas possible. In most cases, high quality and expensive componentscannot use low-quality and low-price fastening elements.

In practice, often problems during tightening resp. assembly exist. Thereasons may be:

1. wrong size (diameters, lengths, positions),

Figure 54 Fatigue failure of screw with bending moment under preload fromvibrational testing with transversal displacement� .


2. wrong alignment (axis of screw, clamped part through hole andnut thread),

3. wrong thread (screw thread or bolt thread tolerances, crossedthread flanks, deformed flanks, wrong nut thread depth),

4. local materials mismatch (chamfers, burrs, dirt in holes, chips),5. wrong=inprecise screw drive,6. wrong material strength (screw, clamped part, nut thread

component), and7. wrong lubrication (screw surface, cleanliness of components).

All assembly problems lead to reduced preload after tightening oroverstressing of screw. Therefore, it is important that these problems aredetected by the assembly process (e.g., tightening device with specific evalua-tion routines). Assembly problems can be the reason of failure of well-designed fastening systems during operation. Therefore, for analysis of

Figure 55 Authorities responsible for reliable function of a threaded fastening

system.


every screw failure during operation, the assembly process has to be ana-lyzed too.

Always, three general reasons for failures during operation have to beconsidered:

1. wrong initial preload (tightening process, poor design),2. wrong residual preload (relaxation by creeping of materials,

gaskets), and3. overloading mechanically, thermal or reactive (too high operating

load with plastification or sliding, too high temperature withcreeping or decreasing of strength, too strong environment withsignificant corrosion).

G. Cost Accounting of Fastening Systems

Always cost accounting of a fastening system has to be done due to life cycleof the component system (product) because only this life cycle cost can becompared to the customer value. Then, all boundary conditions of Figs. 1and 32 have to be included and evaluated monetarily—the fastening elementis only one contribution to this life cycle cost. Figure 56 proposes a funda-mental approach to total cost accounting, which takes into account the maintypes of cost related to a fastening system.

Options for cost optimizing are thread rolling (Fig. 7), standard mate-rials (Fig. 10), coarse tolerances for geometry (Table 2). But one mustalways remember that the guaranteeing of reliable function has to be ofhigher priority than the cost for a technical system. Otherwise, the producthas no customer value and therefore no market.

IV. EXAMPLES OF DESIGN

A. Fastening with Optimized Initial Preload

Traditionally, screws are tightened with torque control (see Fig. 19). Thetightening torque Ttot is specified for the conditions with lowest friction.This is shown on the left side of Fig. 57 (as a supplement to Fig. 51) for asteel screw 8.8, tensile strength of 800MPa. The first case A considers alow friction situation with coefficients of mt¼ mh¼ 0.08. This screw at20Nm tightening torque is stressed up to 0.9 � Rp0.2 (rhomb marking)and produces a preload of almost 20 kN. Because of high screw stressing,the torque specification cannot be increased over 20Nm if yielding orbreaking of screw has to be avoided in any case.

For the same screw in large series assembly lines, the frictional situa-tion can change to mt¼ mh¼ 0.16 (curve B). Then, for the same tightening


torque of 20Nm, a preload of roughly 9 kN is achieved because the highfrictional torque in screw head contact Th consumes the main part of tigh-tening torque. This influence of friction on the preload has to be consideredcarefully for designing bolted joints with torque controlled assemblymethod.

On the right side of Fig. 57, the corresponding diagram for anenhanced aluminum screw with tensile strength of 400MPa is drawn. It is

Figure 56 Comprehensive cost accounting of fastening system.


obvious that the lower screw strength leads to lower torque values on the x-axis if the screw is stressed up to Rp0.2 or Rm. But the resulting initial preloadof this diagram is in the range of 12–14 kN and, therefore, exceeds the mini-mum preload of the steel screw. The enhanced aluminum screw with lowtorque and low strength yields the same performance with 1=3 of weightand other significant advantages (see Refs. [15,16]) including low requiredthread engagement, stable corrosive behavior, excellent thermal fit to lightmetal components made of aluminum or magnesium.

This result is achieved with two additional actions:(1) The screw must provide an effective and reliable low friction film

which reduces the frictional coefficients to the range of 0.08–0.12. Such alow friction film requires greater performance than normal lubrication forestablished steel screws.

(2) The screw must be suitable for yield point- or angular controlledtightening. Then, the materials utilization is much better and the minimumpreload is increased significantly. To obtain this, the screw requires a

Figure 57 Tightening diagram and preload level for steel screw and aluminumscrew. (From Ref. 15.)


defined minimum plastification before fracture, therefore, the manufacturingprocess must be optimized. In principle, using yield point controlled tighten-ing, the screw cannot be overloaded. Using angular controlled tightening,the screw cannot be overloaded, if the snug torque is low enough (e.g.10Nmþ 908 in Fig. 57). Figure 26 explains why also a screw tightened withangular control can be loaded additionally.

As a side-effect, the diagrams in Fig. 57 make it clear that torque-con-trolled tightening (and a steel screw for aluminum components) is an ‘‘old-fashioned’’ and not very optimized solution from the viewpoint of engineer-ing threaded fastening systems. The diagrams confirm that a torque valuementions ‘‘nothing’’ about the preload acting in the joint.

B. Fastening with Small Thread Engagement

For every component design, the necessary minimum thread engagementneeds space and therefore generates component weight. A minimization ofthread engagement is required. Figure 33 contains the basic mechanicsand suggests the use of relatively high-strength nut thread material or theuse of relatively low-strength screw material.

Figure 58 Measured low minimum thread engagement for AluformTM screws M6

(From Ref. 15.)


Figure 58 gives a practical verification of the calculation from Fig. 33for a pull-out-test with RIBE-Aluform# screws [60] M6 (Rm > 400MPa)engaged to an aluminum nut thread plate (Rm¼ 300MPa) with certainlengths of thread engagement te. The bar diagram shows the maximumpull-out-forces Fzmax in the event of failure. The upper level of �9 kNbelongs to a tensile screw breaking in the screw shank. The lower levelbelongs to nut thread stripping.

The transition begins exactly at the point 0.7 � d which is also pre-dicted by Fig. 33 (do not forget to consider chamfer of 1 � P in Fig. 33).So, indeed Aluform# screws have a reliable behavior against stripping alsofor low-strength nut thread components and low thread engagements te,which would never be fulfilled with a steel screw 8.8 or higher.

C. Fastening of High-strength Components

High-strength components provide the chance for small threaded fasteningsystems because contact pressure at screw head as well as thread flanks canbe in the range between minimum of Rp0.2 and Rm of the materials in con-tact. Besides this, hard surfaces are almost unaffected by roughness chan-

Figure 59 Screw head design with small head diameter for tightening on hard

surface.


ging with adhesive or abrasive wear meachanisms, so that the frictionalsituation is constant over a wide range of preload.

Figure 59 demonstrates a small-diameter screw head design for tigh-tening on hardened component surface in the range of Rm¼ 1400MPa(for high-strength screw materials see Table 9). Such a screw of dimensionM11 � 1.5 and a screw tensile strength of 1150MPa produces an initial pre-load of about 60 kN. This leads to a mean contact pressure of 950MPa(compare also diagram in Fig. 39 and explanations related). In addition,for such design, the lubrication of the screw is of significant importance.These screws offer the possibility for small space flanges and in consequencefor a compressed design of component. In contrast to light metal com‘ponents, these high-strength materials possess significant mass density,but can be used for an extreme compact design.

The same aspects are valid for thread engagement but at leastte¼ 0.8 � d should be realized to avoid stripping of screw thread flanks(compare asymptotic behavior of Fig. 33 for high nut thread strengthRmnut). If fastening high-strength components, precise support geometriesand small contact roughness is required, then peak contact pressure isavoided, which can be the origin of crack propagation and fatigue failureof the component.

Threaded fastenings with components made of high-strength materialsprovide the possibility for meeting small space requirements (low threadengagement, small head diameter, small screwing boss diameter at clampedpart). If this is combined with overelastic tightening and reliable lubrication,then also lightweight fastening is possible.

D. Fastening of Components Made of Brittle Materials

‘‘Brittle’’ means that a material has a low ductility before fracture, whichleads to a sudden rupture without plastified deformation in the case of ten-sile testing or overloading a component. As a guide, the fracture toughnessfrom tensile test is for brittle materials smaller than 3–5%). For such acomponent (e.g., made of magnesium, titanium with hexagon crystal struc-ture or cast iron with high carbon content or ceramic materials), not onlyis the mean stressing important, but also all local stress peaks have to beminimized.

Therefore, thread engagement of a screw in brittle materials should beincreased by at least þ20% due to Fig. 33 because of the inhomogeneousstress distribution up to the event of fracture, which is not compensatedby local plastification of the nut thread flanks.

Brittle materials of low strength (cast iron, magnesium) tend to pro-duce increased adhesive-abrasive wear in the screw head contact zone


during tightening (Fig. 60). This results in increased roughness, particlesand undefined contact conditions, so the head frictional torque isincreased and—if the screw is tightened by torque control—the preloadis reduced significantly. To avoid this, the lubrication of screws for brittlematerials should be enhanced.

Use of thread rolling screws in brittle materials is critical. Figure 61presents an example from thread rolling with a high-performance threadrolling screw M8 (induction hardened forming point) in high-strength duc-tile gray iron GGG 50. The result is that particles of nut thread material areproduced in an unacceptable amount. They lead to poor nut thread qualityas well as screw thread damage and therefore to insufficient process capabil-ity for series production (see torquing diagram in Fig. 61 with temporarybreakdown of torque curve).

E. Fastening of Light Metal Components

Light metals, such as aluminum and magnesium, are characterized mechani-cally by low strength and high thermal expansion coefficient. Especially in

Figure 60 Adhesive–abrasive wear of head support area after angular controlledtightening of screw M14� 2-11.9, specification 150Nmþ 908þ 908þ 908, preloadapp. 90 kN, surface gray cast iron GG25.


the field of automotive and transportation as well as for design of hand-held-equipment, these materials are used more and more.

For the design of threaded fastening systems with light metal compo-nents, it is important that:

1. Realization of sufficient thread engagement for low-strength nutthread material (Fig. 33, for steel screw app. te¼ (2.5–3)d and foraluminum screws te¼ (1–1.5)d). This guarantees reliable tighteningand avoids component damage by wrong tightening=repairing.

2. Realization of sufficient head contact area for low contact pressureunder screw head (use of screws with flange head, at least da¼ 2d,Fig. 39). This helps to minimize creeping problems. An additionalaction is to design the screwing-boss-diameter in the range of (2–3)d.

3. Applications with aluminum components or aluminum screwsshould use enhanced lubrication because frictional coefficients arehigh for contacts of materials with cubic face-centered crystalstructure (e.g., aluminum, nickel, and austenitic steel). Foruncertain frictional situation, one must use yield point controlled

Figure 61 Thread rolling in high-strength cast iron as an example for critical

application.


tightening or angular controlled tightening (Fig. 26) and a screwwith threaded screw shank (Fig. 3).

4. If operating at elevated temperatures, a special adaptation ofthermal fit for minimization of thermal stress increase is necessary(often screws made of aluminum are an effective alternativecompared to steel screws, Fig. 29).

5. For aluminum components, thread rolling screws are widely used(Fig. 7, Fig. 62).

6. For corrosion stability, use enhanced corrosion protection for steelscrews or use aluminum screws (Fig. 63).

Figure 64 contains the fastening of a magnesium component withthread rolling screw made of aluminum in two columns: the left side refersto five repetitions of screwing with same screw into the same nut threadhole. The right side refers to the situation where the same screw is screwedinto a new pilot hole without nut thread for each repetition.

The images of the screwing bosses confirm a high quality of the pro-duced nut thread in magnesium for both columns of Fig. 64. The diagramis very detailed due to the formation of positive torque and prevailing torque(negative). All values are proposed for 1 to 5 screwing operation.

Figure 62 Trilobular stud M6 � 32 for thread rolling in aluminum component,dry lubrication, maximum thread forming torque 3Nm, tightening torque for stud

10þ 0.5Nm, thread engagement 11mm, casted pilot hole with diameter 5.4–5.6mm.


The result is that for the left column the torque values are reduced dueto repetitions. For the right column, the values are increased. But also forthe right column of Fig. 64, the forming torque does not exceed the halfof the tightening torque, which is acceptable.

If using magnesium components, the reduction of forming torque forretightening into the same screwing boss is higher compared to steel or alu-minum components.

Finally, Fig. 65 illustrates a fastening solution for very soft materialsas clamped part in a threaded fastening system with aluminum metal foam.The length of sleeve is undersized due to the length of clamped part, so thatthe level of compression=preload in the foam is defined. The wall thicknessof the sleeve has to be adapted to the total preload of the screw. This designhas no problem with contact pressure under the screw head or sliding of thescrew head during tightening because the sleeve has a flange support area.

Some impressive examples that the initial tightening preload is notequal to the residual preload after a long time of operation are shown inFig. 66 which result from several thermal exposure tests with differentbolted magnesium components and different screw dimensions. Magnesiumis very sensitive for creep at elevated temperatures. All details due to the testconditions are given in Fig. 66. During the tests, there was no additionalmechanical loading applied.

Figure 63 Appearance of aluminum component with blank aluminum screw aftersalt spray test: only the component (alloy Al–Mg–Cu) shows white corrosion

products, no galvanic corrosion between screw and component; for details of triplesquare drive see Fig. 36.


Figure 64 Fastening of magnesium component with thread rolling screw;

retightening behavior for two different situations (left and right).


Figure 65 Fastening of aluminum metal foam with screw–sleeve-combination, see

also Fig. 44.

Figure 66 Thermal decrease of preload by relaxation of threaded fastening systemswith magnesium components at elevated temperatures; measured data. (From Ref.16.)


Every case, from 1 to 7, shows the initial preload for steel screw (lightbar) and aluminum screw (dark bar) before and after thermal exposure.For case 1 (screw M10), this means 32 kN resp. 19 kN before thermalexposure and 5 kN resp. 9 kN after exposure. The other cases confirmsimilar behavior. The most extensive preload relaxation occurs for fasteningsystems with the widely usedmagnesium alloy AZ91 (high-strength alloy with9wt.% aluminum and 1wt.% zinc; relatively stable corrosive behavior; lowcreeping resistance at temperatures above 1208C). Cases 3 and 5 confirmwhere the steel screw leads to almost no preload after thermal exposure—here aluminum screws are the only solution for reliable fastening systems.

In general, for these situations, an aluminum screw always gives ahigher residual preload than a steel screw.

F. Fastening of Components with Thread Rolling Screws

What is the difference between thread rolling screws and screws for existingnut thread in practice? The fundamental principle of thread rolling screws is

Figure 67 Functional behavior of thread rolling screw compared to a screw forexisting nut thread used for the same application, see also Ref. 63.


emphasized in Fig. 7. Figure 67 demonstrates the behavior of tighteningtorque Ttot and preload Fp compared between metric screw for existingnut thread and thread rolling screw of same nominal diameter M8. Part(a) refers to a metric screw in machined nut thread—there is no forming tor-que for the first eight revolutions until the screw head is in contact with theclamped part surface. Then, the preload is generated for further screw turn-ing. To obtain a preload of 15 kN, a tightening torque of 28Nm is necessaryfor case (a). A similar situation states case (b), but there exists a forming tor-que during the first 12 revolutions of �10Nm. This forming torque isreduced to �3Nm, if the forming point of the screw is turned outside ofthe nut thread. Therefore, to obtain a preload of 15 kN, the tightening tor-que has to be increased to 31Nm. For case (c), the forming torque in thesituation of head contact is �8Nm, so the tightening torque for a preloadof 15 kN has to be increased to 35Nm.

As a result, thread rolling screws need a higher tightening torque, ifthey have to generate the same preload as a screw for existing nut thread(with related dimension). Therefore, the maximum preload in event offailure is not as high as for screws without thread rolling function, butthe level of failure torque is in the same range. Overall, the preload dif-ference is not very significant. Critical application of thread rollingscrews, see Fig. 61.

G. Fastening of Sheet-metal Components

More and more sheet metal designs are used for automated production ofcomponents with large lot sizes. These components should also be fastenedautomated. This can be done in two ways: (1) use of thread rolling screwsfor sheet metals, and (2) use of staking elements for generatinga high-duty thread at thin walled components (for principle seeFig. 43).

Thread rolling screws for sheet metals are based on the principleshown in Fig. 7 but in contrast to bulk materials sheets offer only a verylow thread engagement. Therefore, the diameter of pilot hole is as smallas possible, so the screw must provide a small thread tip diameter. Screwsas shown in Fig. 68 can be used with and without an extrusion aroundthe pilot hole (rim-hole, see also Ref. [63]). For an estimation of pilot holediameter, see Table 3.

Another tendency to fasten sheet-metal components is using modernriveting systems such as self-piercing rivets (Fig. 69) or high-strength blindrivets. Self-piercing rivets are the first choice if thermal joining technologieslike welding are not suitable or not possible and if liquid-tight fastenings with


Figure 68 Cross-section of thread rolling screws M6 for sheet metals.

Figure 69 Example of self-piercing rivet made of high-strength aluminum byRIBE; ideal solution for high reliability and easy recycling of fastened light metalcomponents. (From Ref. 16.)


low precisement in positioning of the fastening-element are required.Another advantage is the very fast and fully automized assembly process.

Figure 70 demonstrates the principle of blind rivets as a two-piece-fastening-element (rivet body and stem). The clamping force of the rivetis generated by deforming the rivet body under tensile load Fax of therivet stem, which leads finally to a breaking of the stem at the desiredbreaking area. Figure 70 displays an expanding of the rivet shank—another possibility is bulbing the rivet shank. Blind rivets are first choice,if only one-side-access to the clamped parts exists (chassis structures, e.g.in vehicle-, transportation-, and aerospace-industry). A blind rivetrequires a minimum of surface preparation before setting (low roughnessrequirements). The residual stem is important for high shear strength andhigh fatigue strength of the joint.

Figure 71 shows an example for a bulbing blind rivet (in contrast toblind rivets with expanding head) which is applied to aluminum sheet metals(sheet thicknesses 2 and 3mm). This kind of blind rivet produces a high levelclamping force.

In general, riveting systems always have material deformation eitherof clamped part and=or of rivet, so high-strength rivets have to be engi-neered for each particular application (size and number of rivets, geometrytolerances, and setting parameters). For more information regarding rivet-ing systems, see Ref. [59]. For standardization of blind rivets, see Ref. [24].

Figure 70 Simplified principle of blind rivets (break-stem-system).


H. Fastening of Components, Which NeedExtreme Reliability

A typical example for threaded fastening systems which must guaranteeextreme reliability, is wheel bolts for fastening of wheels to car suspensions.These bolts are exposed to extreme operating conditions, such as a largenumber of retightenings, poor exactness of tightening torque, severe saltenvironment, elevated and low temperatures.

One corrosion test procedure with strong corrosive loading is theRIBE-acid-salt-spray-test. Figure 72 shows the corrosion test result forwheel bolts with two surface coatings—the nonelectrolytical standard coat-ing (left) failed in this test already after 96 hr exposition with starting sub-strate corrosion, which normally occurs after 1000 hr. The right side ofFig. 72 demonstrates an enhanced corrosion protection by a nonelectroly-tical multilayer coating with silicate top layer. After the same test durationno substrate corrosion can be found. Coatings for enhanced corrosion pro-tection must provide a very dense structure and an additional electricalinsulation.

The ball-section-support geometry with relatively small ball sectiondiameter dbs results in a cone-clamping with significant head frictional tor-

Figure 71 Blind rivet application, RIBE Ribulb#.


que, which avoids any self-loosening (for calculation of ball-section-headsupport, see Fig. 38).

Figure 73 demonstrates the measured tightening behavior of a screwcorresponding to the right side of Fig. 72. The diagram contains the tighten-ing torque Ttot and preload Fp over the applied tightening angle n for a newbolt as fabricated. The bolt was tightened on steel surfaces.

Normally, the bolt is tightened with a specification of 150Nmþ 20Nm.From the diagram, we can see that with this specification the bolt is utilizedonly to �50% of the possible tightening torque. The other 50% are reservesfor wrong tightening or misuse as well as tightening on light metal surfaces(e.g. aluminum wheels).

The friction coefficients mtot are calculated for different tighteningangles—they show only a slightly change, but an increase of friction isobvious for large tightening angles—such a change of friction coefficientis typical for angular controlled tightening.

Wheel bolts only can be tightened by torque control in practicebecause of unprofessional workers and light metal wheels, which wouldbe damaged by the too-high preload of angular controlled tightening.

I. Fastening with Reduced Total Cost

As considered in Fig. 56, only the total cost is relevant for monetaryassessment of a fastening system. One typical example is given in Fig. 74

Figure 72 Corrosion of wheel bolts after severe testing with salt spraying andadditionally daily wetting with phosphoric acid (pH¼ 2, RIBE-test).


Figure 73 Tightening behavior of wheel bolt with multilayer coating for enhancedcorrosion protection with integrated lubrication, measured data.

Figure 74 Thread rolling screw for reducing total cost of fastening system.


which compares the main types of cost between metric screw for existingnut thread and thread rolling screw of the same nominal diameter M6.As a conclusion, thread rolling screws should always be used if the func-tional properties are sufficient (preload level, large thread engagement, nodisassembly=repair by unprofessionals, limited number of retightenings).

An example for significant deproliferation is shown in Fig. 75.The same thread rolling screw is used for (nut thread) componentsmade of three materials: aluminum, magnesium, and plastics. Thiscould be realized by a reduced thread angle a similar to Fig. 11, byvarying the pilot diameter in the nut thread component from 4.7mmdown to 4.0mm and by a special dry lubrication of the screw. If onlylow quality of the fastening system is required, one specified tighteningtorque can be used for the three cases (Ttot¼ 3.8þ 0.4Nm). Note thatthe thread engagement te is relatively small in this fastening system.

Other practical aspects of total cost saving for designing threadedfastening systems are given in Table 15. In no case, can all aspects berealized for the same application. Therefore, the design engineer hasthe responsibility to set priorities. More information due to cost-optimi-zation of fastening systems can be found in Ref. [66].

Figure 75 Thread rolling with same screw geometry in different materials.


Table 15 Aspects of Cost Saving for Design of Threaded Fastening Systems

No. Aspect Explanation=Remark

1 No oversizing of screw; use of

high screw strength, if a highcontact pressure under head ispossible

Gives smaller space requirements

and in consequence smaller com-ponents

2 Designing an easy and reliable

assembly process (preassembledscrews like in Fig. 44, threadrolling screws like in Fig. 74,

suitable screw drive like inFig. 36)

Assembly process is very impor-

tant for the total cost of afastening system, so the assem-bly process should analyzed pre-

cisely (e.g. time required forpressembly=inserting of screwsand washers, cost for additional

lubrication, lifetime of bits)3 Use of overelastic tightening

methods, whenever possibleThis is increasing the minimumpreload significantly (see also

Fig. 51)4 Use of optimized high quality

fastening elements with exactspecification and low deviations

in functional properties

The design has to consider theworst case of properties, so lowquality of screw leads to poor

performance of fastening sys-tem or oversized components(see influences in Fig. 40 or 51)

5 Use of coarse geometric tolerancesfor fastening system at rightplace (e.g. through-hole dia-

meter, length of screw, lengthof external thread, height ofscrew head, etc.)

Makes production of screw andcomponents only as precise asnecessary

6 Design of clamped part and nut

thread component as simple aspossible

For example, no separate machin-

ing of screw head support (spot-face of clamped part) for use ofscrew with locking teeth under

head; use of rolling screws forductile materials

7 Use of synergy effects by design of

standard classes of fasteningsystems

Deproliferation (minimization of

number of different fasteningelements)

8 Use of comprehensive design pro-

cess after Fig. 32 includingcalculations and guidelinesfrom this book chapter

Gives short development period

with reduced efforts for testing,modifications and repairing

(Continued)


J. Fastening Without Self-loosening

For many fastening systems, it is important that a self-loosening failure can-not happen. Figure 76 shows the characteristics of commonly used actionsagainst self-loosening of threaded fastening systems. The diagram contains

Table 15 Continued

No. Aspect Explanation=Remark

9 Intensive and permanent coopera-

tion between supplier and userof fastening elements and fas-tened components

Most functional effects of fasten-

ing systems are based on theinteraction between screw andcomponent, so all responsiblepeople must work together

from the first developmentstage on to open up the poten-tial for optimization—this as-

pect is very important for future10 Use of enhanced logistics to mini-

mize overhead cost for manage-

ment, delivery and transport offastening elements

For example, purchase of higherlot size reduces cost of transport

and gives reserves for suddenneeds, using electronic orderingsystems to minimize managing

cost

Figure 76 Self-loosening of threaded fastening systems.


results from vibration measurements with the well-known Junkers config-uration (see sketch in Fig. 76 or Refs. [7,72]). The Junkers test procedureis performed with a very extensive loading in order to compare different fas-tening solutions.

For this test, the screw is tightened up to a certain initial preload (here40 kN). Then, a predefined transversal displacement x is applied and thepreload-behavior over the number of cycles is recorded. A threaded fasten-ing system without actions against self-loosening fails after a short numberof cycles. In most cases, a system with serrated washers or cone washersfails in a short period of testing. Only three kinds of prevention are relevantfor high transversal loading: (1) washers with optimized locking teeth onboth sides, (2) using adhesives in thread contact zone, or (3) using a flangehead screw with optimized locking teeth under head. The use of a ball-section-head support can be an action against self-loosening (compareFig. 72).

The self-loosening-behavior is strongly dependent on the entire fasten-ing system, such as stiffness, tolerances, materials, tightening level, numberof contact zones, level of loading. For a detailed assessment, each systemmust be evaluated individually.

V. LIMITED WARRANTY

All data are given with best knowledge and references. For design purposealways refer to referenced standards. Because of various influences on thebehavior of a threaded fastening system, the author does not warranty thecorrectness of the results in using this chapter.

VI. APPENDIX A

A. Notation

Variable Unit Explanation

d mm Major diameter of external screw threadd1 mm Minor diameter of external screw threadd2 mm Pitch diameter of external screw thread

d3 mm Root diameter of external screw threadd0 mm Relevant diameter for calculation of screw assembly

(loaded diameter of screw with smallest cross-section;

db or diameter of nominal stress area)

Continued


Appendix A (Continued)


D mm Major diameter of internal nut thread

D1 mm Minor diameter of internal nut threadD2 mm Pitch diameter of internal nut threadDeb mm Effective mean diameter of bearing surface between screw

head and clamped partDsub mm Substituting diameter, which represents the resilience of

clamped part dp by a tube of constant diameter Dsub

Dp mm Diameter of clamped partP mm Pitch of thread (axial displacement of one flank for one

rotation of 3608 around the screw axis)

H mm Height of fundamental triangle of thread profileR mm Root radius of external thread profileRnmax mm Maximum root radius of internal thread profileda mm Maximum support diameter of screw head

Di mm Minimum support diameter of clearance hole, which is incontact with the screw head

dbs mm Ball section diameter for screw head with ball section

geometrydb mm Diameter of screw body (unthreaded shank)dsh mm Screw shank diameter unthreaded

dh mm Diameter of clearance hole (through hole)x mm Distance transversal to screw axisl mm Length in direction of screw axis

f mm Axial deformation of threaded fastening systemDl mm Change of screw length under tensile forcelsh mm Length of unthreaded screw shanklft mm Length of shank with free thread flanks under tensile

loadingte mm Length of thread engagementlc mm Clamping length

lp mm Plastification length (length of smallest cross-section withinclamping length of a threaded fastening system)

a mm Length transversal to screw axis, which describes the

distance between bending axis of clamped part and axisof external axial force Fax

s mm Length transversal to screw axis. which describes thedistance between bending axis of clamped part and axis

of through hole in clamped partsd2 mm Largest length of a screw drive in a plane transversal to

screw axis (e.g., width across corners for hexagon drive)

Continued




fz mm Seating distance, which leads to preload reduction caused

by roughness of technical surfaces in contactArea Values

A mm2 Area in general

A0 mm2 Circular area corresponding to d0As mm2 Nominal stress area for tensile loading of screw thread;

As¼ [0.5(d2þ d3)]2p=4 for metric thread system

Ah mm2 Contact area at head support between bearing surfacesAn mm2 Nominal circle area from thread sizeAsub mm2 Substituted area relevant for resilience of clamped part

Moment of Inertia and Polar Section ModulusIfull mm4 Moment of inertia for bending of clamped part and screw

shank together, used for calculating of load factor undereccentric loading

Ip mm4 Moment of inertia for bending (clamped part), used forcalculating of component separating under eccentricloading

Wp mm3 Polar section modulus. used for calculating of torsionalstress of screw thread from thread torque Tt;Wp¼ pd3=12

Angle Valuesa 8 Thread angleb 8 Flank angle (in most cases b¼ a=2)j 8 Lead angle (pitch angle)

8 Rotation angle resp. tightening angleg 8 Angle of cone section at countersunk head of screwe 8 Contact angle at screw drive flank

Values of Axial Resilienceds mm=N Linear elastic axial resilience of entire screwdh mm=N Linear elastic axial resilience of screw head

dsh mm=N Linear elastic axial resilience of screw shankdst mm=N Linear elastic axial resilience of free threaddet mm=N Linear elastic axial resilience of engaged thread

dp mm=N Linear elastic resilience of clamped partdst1 mm=N Elastic resilience of screw at temperature t1dst2 mm=N Elastic resilience of screw at temperature t2dpt1 mm=N Elastic resilience of clamped part at temperature t1dpt2 mm=N Elastic resilience of clamped part at temperature t2

Continued




Temperature-related Values

t1 8C Temperature of tighteningt2 8C Temperature of operatingDT 8C Change of temperature (between tightening and operating)

as K�1 Linear thermal expansion coefficient of screw materialap K�1 Linear thermal expansion coefficient of clamped part

material

Force ValuesF kN Force in generalFp kN Preload acting in screw shank

DFp kN Change of preload (e.g. by thermal expansion or relaxation)Fp0 kN Stable preload after tightening and sort time relaxationFps kN Separating preload for given eccentrical loading with axial

force Fax

Faxcrit kN Maximum axial force for given preload Fp0, beforecomponent separating occurs

Fsa kN Additional axial force of screw under external axial force

Fax

Ftangenial, Ft kN Tangential force related to the screw axisFaxial, Fax kN Axial force related to the screw axis

Fpnec kN Necessary preload for safe working of bolted jointFthreadpermt1 kN Permitted axial stripping force of nut thread at temperature

t1Fthreadpermt2 kN Permitted axial stripping force of nut thread at temperature

t2Fheadpermt1 kN Permitted axial stripping force of screw head at temperature

t1Fheadpermt2 kN Permitted axial stripping force of screw head at temperature

t2Ftranspermt2 kN Permitted transversal force at temperature t2Fpamin kN Minimum preload after tightening of bolted jointFpamax kN Maximum preload after tightening of bolted jointFpomin kN Minimum preload of bolted joint for operating

Fpomax kN Maximum preload of bolted joint for operatingFpanmin kN Minimum preload for angular controlled tighteningFpanmax kN Maximum preload for angular controlled tighteningFpymin kN Minimum preload for yield point controlled tightening

Fpymax kN Maximum preload for yield point controlled tighteningFptmin kN Minimum preload for torque controlled tighteningFptmax kN Maximum preload for torque controlled tightening

Continued




Fc kN Circumferential force at screw drive flank

Ft kN Tangential part of Fc, tangential forceFn kN Normal part of Fc

Torque Values

Ttot Nm Total torque (torque applied on screw drive for assembly)Ttott Nm Total torque for torque controlled tighteningTt Nm Thread torque during tightening

Th Nm Head frictional torque during tighteningTtotpermt1 Nm Maximum permitted total torque, which can be transmitted

by the screw drive without problems during tightening

Ts Nm Snug torque for angular controlled tighteningStress and Pressure Values

sax MPa Axial stress in screw shankseq MPa One-dimensional equivalent stress from maximum distor-

tion energy theory (vMises theory) for reducing com-bined stresses to one uniaxial stress value (e.g. reducingthe combination of tensile- and shear stress in the screw

shank to one value)sa MPa Axial stress amplitude in generalsast2 MPa Axial stress amplitude of screw at temperature t2 modulated

by dynamic operating force of bolted jointsaspermt2 MPa Permitted axial stress amplitude of screw at temperature t2

modulated by dynamic operating force of bolted joint

sasperm0 MPa Fatigue limit at room temperature (ISO 3800: sa)tmax MPa Maximum shear stress in screw shankpch MPa Pressure in contact area Ah

pchmaxt1 MPa Maximum pressure in contact area Ah at temperature t1pchmaxt2 MPa Maximum pressure in contact area Ah at temperature t2pct MPa Pressure in helical contact area of thread engagement

normal to loaded thread flanks

pcc MPa Component contact pressure between clamped part and nutthread component

Rmst1, Rmst2 MPa Tensile strength of screw at temperature t1 resp. t2Rmnt1, Rmnt2 MPa Tensile strength of nut thread component at temperature t1

resp. t2Rmpt1, Rmpt2 MPa Tensile strength of clamped part at temperature t1 resp. t2Rp0.2s MPa Proof stress with 0.2% plastic strain under conditions of

tensile testEs MPa Young’s modulus of screw material (modulus of elasticity)Ep MPa Young’s modulus of clamped part material (modulus of

elasticity)

Continued




Est1, Est2 MPa Young’s modulus of screw material at temperature t1 resp.

t2Ept1, Ept2 MPa Young’s modulus of clamped part material at temperature

t1 resp. t2Frictional Coefficients

mt — Thread frictional coefficientmh — Head frictional coefficient

mtot — Total frictional coefficientmcc — Frictional coefficient of component contact between

clamped part and nut thread component

Ratio Values and Numbers— Load factor; ratio between external axial force of bolted

joint and additional axial force in screw shankn — Inducing factor; defines, in which position the external axial

force is applied to the fastening system; n¼ 0–1nx — Number of transversal displacements for vibration testingks — Stress factor for relationship between sax and seq (based on

von Mises theory of failure)

AbbreviationsFMEA Failure modes and effects analysisSOP Start of production

FEM Finite element methodPTFE Poly tetra fluoro ethyleneMoS2 Molybdenum disulfide

Me Chemical symbol for ‘metal’SCE Standard calomel electrode (reference potential for mea-

surement of corrosion current)

x-axis Abscissay-axis Ordinate


B. Conversion of Units

ACKNOWLEDGEMENTS

The author thanks Mr. H. Meier and Mr. T. Riehl for performing and eval-uating a large number of investigations and testings, Mr. W. Thomala forvaluable discussions, Dr. G. E. Totten, Dr. K. Funatani, Dr. L. Xie, andMr. R. Johnson for engaged management of the editorial process, RIBEGmbH Schwabach for support with photographs and experimental resultsand finally my family for support with plenty of time during preparingthe manuscript.

Metric

unit

English

unit

To convert metricto english

multiply by

To convert englishto metric

multiply by

Length mm in 0.039370 25.400

ma ftb 3.2808 0.3048Area mm2 in2 0.001550 645.16

m2 ft2 10.764 0.092903

Volume mm3 in3 0.000061024 16387m3 ft3 35.315 0.028316

Mass g lb 0.0022046 453.59

Force Nc lbf 0.22481 4.44822Torque Nm lbf ft 0.7376 1.3558

Nm lbf in 8.8508 0.1130Pressure=stress MPa KSId 0.145037 6.89479

Nmm2 KSId 0.145037 6.89479Temperature 8C 8F 8F¼ (1.8 8C)þ 32 8C¼ (8F�32)=1.8a1m 1000mm.b1 ft 12 in.c1N 9.81�gf.d1 ksi 1000 lbfin2.

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