Challenges In Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 2: Proceedings of the 2013 Annual Conference on Experimental

Bonnie Antoun · H. Jerry Qi · Richard Hall · G.P. Tandon Hongbing Lu · Charles Lu · Jevan Furmanski Alireza Amirkhizi Editors

Challenges In Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 2Proceedings of the 2013 Annual Conference on Experimental and Applied Mechanics

Conference Proceedings of the Society for Experimental Mechanics Series

Conference Proceedings of the Society for Experimental Mechanics Series

Series EditorTom ProulxSociety for Experimental Mechanics, Inc.,Bethel, CT, USA

For further volumes:

http://www.springer.com/series/8922

http://www.springer.com/series/8922

Bonnie Antoun • H. Jerry Qi • Richard Hall • G.P. TandonHongbing Lu • Charles Lu • Jevan Furmanski • Alireza Amirkhizi

Editors

Challenges In Mechanicsof Time-Dependent Materialsand Processes in Conventionaland Multifunctional Materials, Volume 2

Proceedings of the 2013 Annual Conference on Experimentaland Applied Mechanics

EditorsBonnie AntounSandia National LaboratoriesLivermore, CAUSA

H. Jerry QiUniversity of ColoradoBoulder, COUSA

Richard HallAir Force Research LaboratoryWright-Patterson AFB, OHUSA

G.P. TandonUniversity of Dayton Research InstituteDayton, OHUSA

Hongbing LuUniversity of Texas-DallasDallas, TXUSA

Charles LuUniversity of KentuckyPaducah, KYUSA

Jevan FurmanskiLos Alamos National LaboratoryLos Alamos, NMUSA

Alireza AmirkhiziUniversity California San DiegoLa Jolla, CAUSA

ISSN 2191-5644 ISSN 2191-5652 (electronic)ISBN 978-3-319-00851-6 ISBN 978-3-319-00852-3 (eBook)DOI 10.1007/978-3-319-00852-3Springer Cham Heidelberg New York Dordrecht London

Library of Congress Control Number: 2013945393

# The Society for Experimental Mechanics, Inc. 2014This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights oftranslation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformation storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed.Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose ofbeing entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permittedonly under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained fromSpringer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under therespective Copyright Law.The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specificstatement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor thepublisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, withrespect to the material contained herein.

Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com)

www.springer.com

Preface

Challenges in Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials,Volume 2: Proceedings of the 2013 Annual Conference on Experimental and Applied Mechanics represents one of eight

volumes of technical papers presented at the SEM 2013 Annual Conference & Exposition on Experimental and Applied

Mechanics organized by the Society for Experimental Mechanics and held in Lombard, IL, June 3–5, 2013. The complete

Proceedings also includes volumes on: Dynamic Behavior of Materials; Advancement of Optical Methods in ExperimentalMechanics; Mechanics of Biological Systems and Materials; MEMS and Nanotechnology; Experimental Mechanics ofComposite, Hybrid, and Multifunctional Materials; Fracture and Fatigue; Residual Stress, Thermomechanics & InfraredImaging, Hybrid Techniques and Inverse Problems.

Each collection presents early findings from experimental and computational investigations on an important area within

Experimental Mechanics, the Mechanics of Time-Dependent Materials and Processes being one of these areas.

This track was organized to address time (or rate)-dependent constitutive and fracture/failure behavior of a broad range of

materials systems, including prominent research in both experimental and applied mechanics. Papers concentrating on both

modeling and experimental aspects of time-dependent materials are included.

The track organizers thank the presenters, authors, and session chairs for their participation in and contribution to this

track. The support and assistance from the SEM staff is also greatly appreciated.

Livermore, CA, USA Bonnie Antoun

Boulder, CO, USA H. Jerry Qi

Wright-Patterson AFB, OH, USA Richard Hall

Dayton, OH, USA G.P. Tandon

Dallas, TX, USA Hongbing Lu

Paducah, KY, USA Charles Lu

Los Alamos, NM, USA Jevan Furmanski

La Jolla, CA, USA Alireza Amirkhizi

v

Contents

1 Micromechanics of the Deformation and Failure Kinetics of Semicrystalline Polymers . . . . . . . . . . . . . . . . . . . . . . . 1J.A.W. van Dommelen, A. Sedighiamiri, and L.E. Govaert

2 Stress-Relaxation Behavior of Poly(Methyl Methacrylate) (PMMA)

Across the Glass Transition Temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Danielle Mathiesen, Dana Vogtmann, and Rebecca Dupaix

3 The Effect of Stoichiometric Ratio on Viscoelastic Properties of Polyurea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Zhanzhan Jia, Alireza V. Amirkhizi, Kristin Holzworth, and Sia Nemat-Nasser

4 Dynamic Properties for Viscoelastic Materials Over Wide Range of Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21T. Tamaogi and Y. Sogabe

5 Spatio-Temporal Principal Component Analysis of Full-Field Deformation Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Srinivas N. Grama and Sankara J. Subramanian

6 Master Creep Compliance Curve for Random Viscoelastic Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Jutima Simsiriwong, Rani W. Sullivan, and Harry H. Hilton

7 Processability and Mechanical Properties of Polyoxymethylene

in Powder Injection Molding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49J. Gonzalez-Gutierrez, P. Oblak, B.S. von Bernstorff, and I. Emri

8 Constitutive Response of Electronics Materials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Ryan D. Lowe, Jacob C. Dodson, Jason R. Foley, Christopher S. Mougeotte,David W. Geissler, and Jennifer A. Cordes

9 Analytical and Experimental Protocols for Unified Characterizations in Real Time

Space for Isotropic Linear Viscoelastic Moduli from 1–D Tensile Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Michael Michaeli, Abraham Shtark, Hagay Grosbein, Eli Altus, and Harry H. Hilton

10 High Temperature Multiaxial Creep-Fatigue and Creep-Ratcheting Behavior of Alloy 617 . . . . . . . . . . . . . . . . . 83Shahriar Quayyum, Mainak Sengupta, Gloria Choi, Clifford J. Lissenden, and Tasnim Hassan

11 Metastable Austenitic Steels and Strain Rate History Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99Matti Isakov, Kauko Ostman, and Veli-Tapani Kuokkala

12 Measurement Uncertainty Evaluation for High Speed Tensile Properties

of Auto-body Steel Sheets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109M.K. Choi, S. Jeong, H. Huh, C.G. Kim, and K.S. Chae

13 Effect of Water Absorption on Time-Temperature Dependent Strength of CFRP . . . . . . . . . . . . . . . . . . . . . . . . . . . 121Masayuki Nakada, Shuhei Hara, and Yasushi Miyano

14 Stress and Pressure Dependent Thermo-Oxidation Response

of Poly (Bis)Maleimide Resins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129Nan An, G.P. Tandon, R. Hall, and K. Pochiraju

15 Comparison of Sea Water Exposure Environments on the Properties

of Carbon Fiber Vinylester Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139Chad S. Korach, Arash Afshar, Heng-Tseng Liao, and Fu-pen Chiang

vii

16 Low-Density, Polyurea-Based Composites: Dynamic Mechanical Properties and Pressure Effect. . . . . . . . . . 145Wiroj Nantasetphong, Alireza V. Amirkhizi, Zhanzhan Jia, and Sia Nemat-Nasser

17 Haynes 230 High Temperature Thermo-Mechanical Fatigue Constitutive Model Development . . . . . . . . . . . . 151Raasheduddin Ahmed, M. Menon, and Tasnim Hassan

18 Temperature and Strain Rate Effects on the Mechanical Behavior of Ferritic Stainless Steels . . . . . . . . . . . . . 161Kauko Ostman, Matti Isakov, Tuomo Nyyss€onen, and Veli-Tapani Kuokkala

19 Modeling and Simulation in Validation Assessment of Failure Predictions

for High Temperature Pressurized Pipes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167J. Franklin Dempsey, Vicente J. Romero, and Bonnie R. Antoun

20 Unified Constitutive Modeling of Haynes 230 for Isothermal Creep-Fatigue Responses . . . . . . . . . . . . . . . . . . . . . 175Paul Ryan Barrett, Mamballykalathil Menon, and Tasnim Hassan

viii Contents

Chapter 1

Micromechanics of the Deformation and Failure Kinetics

of Semicrystalline Polymers

J.A.W. van Dommelen, A. Sedighiamiri, and L.E. Govaert

Abstract An elasto-viscoplastic two-phase composite inclusion-basedmodel for themechanical performance of semicrystalline

materials has previously been developed. This research focuses on adding quantitative abilities to the model, in

particular for the stress-dependence of the rate of plastic deformation, referred to as the yield kinetics. A key issue

in achieving that goal is the description of the rate-dependence of slip along crystallographic planes. The model is used

to predict time-to-failure for a range of static loads and temperatures. Application to oriented materials shows a distinct

influence of individual slip systems.

Keywords Micromechanical modelling • Polyethylene • Semicrystalline polymers • Structure–property relation • Yield

kinetics

1.1 Introduction

Both short and long-term failure of polymers are known to originate from usually rapid development of local irreversible

(plastic) strain, manifesting itself in crazing and/or necking that ultimately results in loss of the structural integrity of the

product. The mode of failure can be either brittle, characterized by fragmentation of the product, or ductile, involving the

development of large localized plastic deformation zones accompanied by (more stable) tearing phenomena. The mode of

failure and the time-scales on which they occur are strongly influenced by the molecular weight distribution of the polymer,

the macromolecular orientation and the thermal history, i.e. factors that are directly connected to processing conditions. The

latter is particularly true for semicrystalline polymers in which structural features, such as the degree of crystallinity, crystal

type, size and orientation, that strongly influence their mechanical properties, may vary drastically depending on subtle

details of the manner in which the polymer is shaped into the final product. In particular, shear flow significantly accelerates

crystallization kinetics by increasing the amount of nuclei and generates an anisotropic morphology by inducing orientation.

The mechanical behaviour of semicrystalline polymeric materials, consisting of both amorphous and crystalline domains,

depends strongly on the underlying microstructure (e.g. [1, 2]). Their elastic and viscoplastic behaviour depend on many

factors such as the percentage crystallinity, the initial crystallographic and morphological texture and the mechanical

properties of the individual phases. The ability to predict the mechanical properties of polymer products is uniquely linked to

the capability to understand and predict the elasto-viscoplastic behaviour resulting from the underlying microstructure.

Semicrystalline materials with oriented microstructures will behave anisotropically, which can play a crucial role in the

performance and failure of polymer products.

Several experimental and modelling studies (e.g. [3–9]) have been dedicated to characterization and understanding of the

viscoplastic behaviour and the evolution of texture of semicrystalline polymers. A previously developed

micromechanically-based model for the constitutive behaviour of semicrystalline polymeric material [10] accounts for

both crystallographic and morphological texture, the latter corresponding to the orientation distribution of the lamellar

interface normals. The basic element in this model was a layered two-phase composite inclusion, comprising both a

crystalline and an amorphous domain as developed by Lee et al. [6] for rigid viscoplastic semicrystalline materials.

J.A.W. van Dommelen (*) • A. Sedighiamiri • L.E. Govaert

Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands

e-mail: [email protected]

B. Antoun et al. (eds.), Challenges In Mechanics of Time-Dependent Materials and Processes in Conventionaland Multifunctional Materials, Volume 2, Conference Proceedings of the Society for Experimental Mechanics Series,

DOI 10.1007/978-3-319-00852-3_1, # The Society for Experimental Mechanics, Inc. 2014

1

mailto:[email protected]

A three-level modelling approach was used to study intraspherulitic deformation and stresses for semicrystalline

polyethylene [11] and to predict the response of tensile specimens obtained at different angles with respect to the extrusion

direction of the material [12] in a qualitative sense.

The current research focuses on adding quantitative abilities to the micromechanical model, in particular for the

stress-dependence of the rate of plastic deformation, referred to as the yield kinetics. A key issue in achieving that goal is

the description of the rate-dependence of slip along crystallographic planes. The slip kinetics have been re-evaluated and

characterized using a hybrid numerical/experimental procedure, based on the results for isotropic HDPE, loaded at various

strain rates and temperatures. Because of the isotropy of the material, additional assumptions for the properties of these slip

systems are required. Finally, the isotropically characterized model is applied to oriented polyethylene to investigate the

potential of oriented system for characterizing the full slip kinetics of a semicrystalline polymer.

1.2 Model Description

The constitutive behaviour of semicrystalline material is modelled by an aggregate of two-phase composite inclusions, see

Fig. 1.1. This composite inclusion model, which is discussed in detail in [10], is concisely summarized in this section. Each

inclusion consists of a crystalline and an amorphous phase. A microstructural elasto-viscoplastic constitutive model is

defined for both the crystalline and the amorphous phase.

The crystalline domain consists of regularly ordered molecular chains. The response of these domains is modelled as

anisotropic elastic in combination with plastic deformation governed by crystallographic slip on a limited number of slip

planes [2–13], which are shown in Fig. 1.1 and for which a rate-dependent crystal plasticity model is used. In the model, the

plastic deformation rate is given by the summed contribution of all physically distinct slip systems:

Lp ¼XNα¼1

_γαðταÞPα (1.1)

where Pα ¼~s α~n α is the Schmid tensor of the αth slip system, Lp is the plastic velocity gradient tensor, and where the

constitutive behaviour of the slip systems is defined by the relation between the resolved shear stress τα and the resolved

shear rate _γα, which is referred to as the slip kinetics.

The amorphous phase of semicrystalline polymeric material consists of an assembly of disordered macromolecules,

which are morphologically constrained by the neighbouring crystalline lamellae. The elastic deformation of the amorphous

domains is modelled by a generalized neo-Hookean relationship. Furthermore, a viscoplastic relation based on an associated

flow rule is used, in combination with an eight-chain network model to account for orientation-induced strain hardening

[14–16]. In particular, the viscoplastic behaviour of the amorphous phase is characterized by a relation between the effective

shear rate and the effective shear stress _γaðτaÞ, referred to as the yield kinetics of the amorphous phase.

The mechanical behaviour at the mesoscopic level is modelled by an aggregate of layered two-phase composite inclusions

as was proposed by Lee et al. [6, 7] for rigid/viscoplastic material behaviour. Each separate composite inclusion consists of a

crystalline lamella which is mechanically coupled to its corresponding amorphous layer. The stress and deformation fields

Fig. 1.1 (a) Aggregate of two-phase composite inclusions and (b) slip systems of the crystalline phase of polyethylene

2 J.A.W. van Dommelen et al.

within each phase are assumed to be piecewise homogeneous, however, they differ between the two coupled phases.

The inclusion-averaged deformation gradient and the inclusion-averaged Cauchy stress are defined as the volume-weighted

average of the respective phases. To relate the volume-averaged mechanical behaviour of each composite inclusion to the

imposed boundary conditions for an aggregate of inclusions, a hybrid local–global interaction law is used [10].

1.3 Yield Stress and Time-to-Failure for Isotropic Material

In order to predict both the short- and long-term failure of polymers, quantitative predictions of the yield kinetics of these

materials are required. The present work is directed towards the prediction of the yield and post-yield behaviour in

semicrystalline polymers at different strain rates. A critical factor is the stress-dependence of the rate of plastic deformation,

the slip kinetics, which is the mechanism underlying time-dependent, macroscopic failure. The kinetics of macroscopic

plastic flow strongly depend on the slip kinetics of the individual crystallographic slip systems. Therefore, an accurate

quantitative prediction requires a proper description of the rate-dependence of slip along crystallographic planes. As a first

step in achieving this goal, an Eyring flow rule is used for each slip system [17, 18], see Fig. 1.2.

An activation energy is included in the slip kinetics in order to predict the temperature dependence of the kinetics of yield.

In order to predict the response in both tension and compression, a non-Schmid effect (i.e. a dependence on the normal stress

σαn acting on the slip system) is included in the slip kinetics [18], which for a single process is given by:

_γα ¼ _γα0 exp �ΔUα

RT

� �sinh

τα

τc0

� �exp

μασαnτc0

� �: (1.2)

The yield kinetics of the amorphous phase is described with a similar relation, where instead of the non-Schmid effect, a

pressure dependence is introduced.

The re-evaluation of the slip kinetics is performed using a combined numerical/experimental approach taking into

account uniaxial compression and tension data of isotropic HDPE, for different strain rates and temperatures, see Fig. 1.3.

The slip kinetics used to obtain these predictions were given in Fig. 1.2 (in absence of a normal stress on each slip system).

In Fig. 1.4, experimentally obtained data for the tensile yield kinetics and time-to-failure under creep conditions are

shown for polyethylene, indicating the presence of a second processes, in addition to the α-relaxation mechanism, at higher

temperatures. The kinetics of each slip system and the kinetics of the amorphous phase used in the model (as given in

Fig. 1.2) account for both processes.

10−5 10−4 10−3 10−2 10−10

5

10

15

20

shear rate [s−1]

reso

lved

she

ar s

tres

s [M

Pa]

{110}<110>, {110}[001]

(010)[001], (010)[100]

(100)[010]

amorphous

(100)[001]

25�C

80�C

Fig. 1.2 Slip kinetics at

different temperatures

1 Micromechanics of the Deformation and Failure Kinetics of Semicrystalline Polymers 3

Also shown in Fig. 1.4 is a prediction of the temperature dependence of the macroscopic yield kinetics and time-to-failure

with the micromechanical model with refined kinetics of crystallographic slip and deformation of the amorphous phase.

Both the yield kinetics and time-to-failure of isotropically oriented material are described well by the micromechanical

model for the range of temperatures and strain rates or applied loads, respectively.

1.4 Oriented Material

The mechanical response of extruded and drawn semicrystalline materials, in which a stacked lamellar morphology is

commonly observed, depends on the direction of loading with respect to the direction of flow. Plastic deformation and failure

are, therefore, both anisotropic. The predictive ability of the micromechanical model, including the characterization of the

kinetics of crystallographic slip and amorphous yield based on isotropic material, is next evaluated for oriented high-density

polyethylene. The initial morphology of the material is generated based on pole figures from wide-angle X-ray diffraction

experiments, which show a strong alignment of molecular chains with the drawing direction for specimens produced with a

large draw ratio (λ ¼ 6), see Fig. 1.5.

Uniaxial loading of an aggregate of 500 composite inclusions with the orientation distribution shown in Fig. 1.5 and with

slip kinetics as characterized for oriented material reveals slip activity on particularly the chain slip systems when the

loading direction is aligned with the original drawing direction of the material [19]. In contrast, loading perpendicular to the

0 0.2 0.4 0.6 0.80

5

10

15

20

25

engineering strain [−]

engi

neer

ing

stre

ss [M

Pa]

10−3s−1 , 25�C

10−2s−1 , 80�C10−1s−1 , 80�C10−2s−1 , 60�C10−2s−1 , 25�C

10−3s−1 , 25�C10−1s−1 , 25�C

10−4s−1 , 25�C10−3s−1 , 50�C10−4s−1 , 50�C10−4s−1 , 80�C10−5s−1 , 80�C

0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

a b

True strain [−]

Tru

e st

ress

[MP

a]

Fig. 1.3 (a) Tensile and (b) compressive response of isotropic HDPE. Markers indicate experimental results and lines are predicted by the

micromechanical model [18]

10−5 10−4 10−3 10−2 10−10

5

10

15

20

25

30a b

25oC50oC65oC80oC

strain rate [s−1]

yiel

d st

ress

[MP

a]

101 102 103 104 105 1060

5

10

15

20

25

30

time−to−failure [s]

appl

ied

stre

ss [M

Pa]

25�C50�C65�C80�C

Fig. 1.4 Temperature dependence of (a) yield kinetics and (b) time-to-failure of HDPE in tension. Markers indicate experimental results and lines

are predicted by the micromechanical model [18]


original drawing direction leads to slip activity at macroscopic yield on transverse slip systems. Consequently, modifying the

slip kinetics such that the chain slip systems have a larger shear yield stress for a given shear rate, affects the macroscopic

response of the material when loaded in the original drawing direction, see Fig. 1.6a, which effectively leads to an enlarged

macroscopic anisotropy. Analogously, increasing the transverse slip kinetic while keeping the chain slip systems unchanged

Fig. 1.5 Equal area projection pole figures of the principal crystallographic and lamellar orientation distributions for HDPE with a draw ratio of 6.

The draw direction is vertical (MD)

Fig. 1.6 The effect of modified slip kinetics on the yield stress for different loading angles [19]. Dashed grey lines show the model prediction

corresponding to the modified kinetics and solid black lines give the prediction corresponding to the original kinetics (a) Modified chain slip

kinetics, (b) modified transverse slip kinetics


leads to an increased macroscopic yield stress when loaded in the transverse direction relative to the original loading

direction, effectively decreasing the anisotropy of the material. This shows the potential of oriented systems for unambigu-

ously determining the yield kinetics of individual slip systems. In doing so, however, also the presence of a potentially

oriented amorphous phase should be dealt with. For more information, see [19].

1.5 Conclusions

The current research focuses on adding quantitative abilities to a micromechanical constitutive model for semicrystalline

polymers, in particular for the stress-dependence of the rate of plastic deformation, referred to as the yield kinetics. A key

issue in achieving that goal is the description of the rate-dependence of slip along crystallographic planes. The slip kinetics

have been re-evaluated and characterized using a hybrid numerical/experimental procedure, based on the results of uniaxial

compression and tension of isotropic HDPE, at various strain rates. The temperature dependence of the kinetics of yielding

and time-to-failure are described well by the model.

The next step in this research on micromechanics of semicrystalline polymers is to validate the qualitative predictive

capabilities for materials with oriented microstructures, which will behave anisotropically. This step might require a re-

evaluation of the difference in kinetics between different slip systems, which cannot be distinguished based on isotropic

microstructures. The potential of anisotropic systems for characterization of individual slip systems has been demonstrated.

This step will be crucial for the use of such structure–property relationships for predicting performance and failure of

polymer products. Furthermore, the model currently does not yet include the pronounced dependence on lamellar thickness

that is experimentally observed and that may be included through the kinetics of crystallographic slip [20].

References

1. Lin L, Argon AS (1994) Structure and plastic deformation of polyethylene. J Mater Sci 29:294–323

2. G’Sell C, Dahoun A (1994) Evolution of microstructure in semi-crystalline polymers under large plastic deformation. Mater Sci Eng A

175:183–199

3. Parks DM, Ahzi S (1990) Polycrystalline plastic deformation and texture evolution for crystals lacking five independent slip systems. J Mech

Phys Solids 38:701–724

4. Dahoun A, Canova R, Molinari A, Philippe MJ, G’Sell C (1991) The modelling of large strain textures and stress–strain relations of

polyethylene. Textures Microstruct 14–18:347–354

5. Bartczak Z, Cohen RE, Argon AS (1992) Evolution of the crystalline texture of high density polyethylene during uniaxial compression.

Macromolecules 25:4692–4704

6. Lee BJ, Parks DM, Ahzi S (1993) Micromechanical modeling of large plastic deformation and texture evolution in semicrystalline polymers. J

Mech Phys Solids 41:1651–1687

7. Lee BJ, Argon AS, Parks DM, Ahzi S, Bartczak Z (1993) Simulation of large strain plastic deformation and texture evolution in high density

polyethylene. Polymer 34:3555–3575

8. Nikolov S, Raabe D (2006) Yielding of polyethylene through propagation of chain twist defects: temperature, stem length and strain-rate

dependence. Polymer 47:1696–1703

9. Gueguen O, Ahzi S, Makradi A, Belouettar S (2010) A new three-phase model to estimate the effective elastic properties of semi-crystalline

polymers: application to PET. Mech Mater 42:1–10

10. Van Dommelen JAW, Parks DM, Boyce MC, Brekelmans WAM, Baaijens FPT (2003) Micromechanical modeling of the elasto-viscoplastic

behavior of semi-crystalline polymers. J Mech Phys Solids 51:519–541

11. Van Dommelen JAW, Parks DM, Boyce MC, Brekelmans WAM, Baaijens FPT (2003) Micromechanical modeling of intraspherulitic

deformation of semicrystalline polymers. Polymer 44:6089–6101

12. Van Dommelen JAW, Schrauwen BAG, Van Breemen LCA, Govaert LE (2004) Micromechanical modeling of the tensile behavior of oriented

polyethylene. J Polym Sci, Part B: Polym Phys 42:2983–2994

13. Argon AS (1997) Morphological mechanisms and kinetics of large-strain plastic deformation and evolution of texture in semi-crystalline

polymers. J Comput-Aided Mater Des 4:75–98

14. Arruda EM, Boyce MC (1993) A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials. J Mech Phys

Solids 41:389–412

15. Boyce MC, Montagut EL, Argon AS (1992) The effects of thermomechanical coupling on the cold drawing process of glassy polymers. Polym

Eng Sci 32:1073–1085

16. Boyce MC, Parks DM, Argon AS (1988) Large inelastic deformation of glassy polymers. part I: rate dependent constitutive model. Mech

Mater 7:15–33

17. Sedighiamiri A, Govaert LE, Van Dommelen JAW (2011) Micromechanical modeling of the deformation kinetics of semicrystalline

polymers. J Polym Sci, Part B: Polym Phys 49:1297–1310


18. Sedighiamiri A, Govaert LE, Kanters MJW, Van Dommelen JAW (2012) Micromechanics of semicrystalline polymers: yield kinetics and

long-term failure. J Polym Sci, Part B: Polym Phys 50:1664–1679

19. Sedighiamiri A, Govaert LE, Senden DJA, Van Dommelen JAW A micromechanical study on the deformation kinetics of oriented

semicrystalline polymers, in preparation

20. Argon AS, Galeski A, Kazmierczak T (2005) Rate mechanisms of plasticity in semicrystalline polyethylene. Polymer 46:11798–11805


Chapter 2

Stress-Relaxation Behavior of Poly(Methyl Methacrylate) (PMMA)

Across the Glass Transition Temperature

Danielle Mathiesen, Dana Vogtmann, and Rebecca Dupaix

Abstract Characterizing Poly(methyl methacrylate) (PMMA) across its glass transition temperature is essential for

modeling warm deformation processes such as hot embossing. Its mechanical properties vary significantly across the

glass transition as well as with strain rate. Several previous models have attempted to capture this behavior utilizing uniaxial

compression experimental data with limited success. In this work, compression experiments including stress relaxation at

large strains are conducted to aid researchers in developing better models. Multiple temperatures, final strains, and strain

rates are examined to characterize the material across values found in typical hot-embossing processes. It was found that the

amount of stress relaxed is highly dependent on the temperature and strain at which it is held. With this data, a model can be

developed that will accurately capture stress relaxation with the final goal of being able to simulate hot embossing processes.

Keywords PMMA • Glass transition • Stress relaxation

2.1 Introduction

Hot embossing or nanoimprint lithography is a process that is used to impose micro- and nano- scale surface features on a

polymer. Applications of hot embossing include the molding of microchannels or optical arrays [1]. The polymer is heated

past its glass transition temperature, Tg, and a finite deformation is applied to the stamp. For a period of time, this position is

held at the original elevated temperature, allowing the polymer to flow and fill in the stamp. Next, it is cooled and unmolded

simultaneously as shown in Fig. 2.1. The inherent sensitivity of the polymer’s mechanical behavior near the glass transition

combined with the many process variables involved make predicting the process outcome challenging. Polymers are known

to be highly sensitive to both temperature change and strain rate near the glass transition. In addition, the final strain and hold

time during the process significantly affect the outcome [2]. As a result, it is difficult and expensive to develop an optimized

process through experiments alone and a predictive material model has the potential to greatly improve the process.

Hot embossing is already embraced as a low-cost environmentally friendly fabrication technique and the ability to optimize

the process will only increase its attractiveness.

At the conditions at which hot embossing is performed, polymers are highly sensitive to strain rate and variation of

temperature [2]. One polymer commonly used in hot embossing is Poly(methyl methacrylate) (PMMA). PMMA is an

amorphous thermoplastic with a glass transition temperature of approximately 105–110 �C that makes it ideal for the process

of hot embossing [3]. While previous experiments have shown that its capability in hot embossing [1, 3], optimization of the

fabrication technique is challenging to develop a large-scale production of these devices. Several models have been

developed to try and capture the behavior of PMMA around glass transition temperature [4–8]. The majority of the models

developed have been based on uniaxial compression experiments [4–7]. While the models are largely able to capture the

behavior in uniaxial compression, they are still unable to predict the correct amount of spring-back present in hot embossing.

It is believed that they fail to capture the behavior because none adequately capture stress relaxation of the polymer.

Stress relaxation is present in the process of hot embossing during the hold period after the deformation is applied. To

accurately design a model, experimental stress relaxation data of PMMA in compression at small and large strains is needed.

D. Mathiesen • D. Vogtmann • R. Dupaix (*)

Scott Laboratory, The Ohio State University, 201 West 19th Ave, 43210 Columbus, OH, USA




9


The majority of stress relaxation data available on PMMA is based on the works of McLoughlin and Tobolsky that found a

master relaxation curve for PMMA at small tensile strains [9]. Others have also performed stress relaxation experiments on

PMMA at small tensile strains [8], multiaxial compression [10], and torsion [11] but little to no data is found at large

compressive strains during relaxation. It is important to study the relaxation behavior of PMMA at large compressive strains

because in hot embossing, large local compressive strains will occur that need to be accurately captured during a finite

element simulation. The purpose of these experiments is to provide necessary data to quantify the amount of stress relaxation

present in PMMA at temperatures, strain rates, and final strains found in hot embossing to better predict the spring-back.

2.2 Experimental

PMMA cylinders were cut from commercial sheet stock supplied by Plaskolite, Inc. to an initial height and diameter of 8.8

and 10 mm respectively. An Instron 5869 screw driven materials testing system was used in conjunction with an Instron

3119–409 environmental chamber to heat the samples to the specified temperature. An Instron 5800 controller running

Instron Bluehill software controlled the load frame. Displacement of the upper compression plate was controlled and the

force recorded with a 50 kN load cell. Using the displacement and force data, true stress and true strain were calculated,

using the initial dimensions of the sample and the assumption that volume remained constant. Samples were tested using a

ramp-hold loading history, where the ramp was a constant true strain rate followed by a hold at a specific final strain. Two

loading strain rates (�1.0/min and �3.0/min), three final true strains (�0.5, �1.0, �1.5) and five temperatures (95–135 �C)composed the testing matrix. All samples were placed in a dessicant chamber at least 24 h prior to the test to control the

amount of moisture present. Teflon sheets were placed between the compression plates and sample to reduce friction.

WD-40 was applied between the compression plates and Teflon film to provide additional lubrication. Each sample was

placed in the pre-heated environmental chamber for 30 min prior to testing to ensure the entire sample was at the testing

temperature. To ensure repeatability, each test was run twice.

2.3 Results

At temperatures less than Tg PMMA behaves as a viscoelastic solid. There is an initial region of elastic behavior followed by

a small period of strain softening as evident in Fig. 2.2. The strain softening is attributed to aging of the polymer, which

decreases the free volume of the polymer and thereby causes an elevated yield stress at small strains. If the polymer were

heated past Tg and quenched the free volume would increase and the strain softening would no longer be present [12]. As the

temperature is increased the strain softening effects diminish and are no longer present at temperatures greater than Tg as

shown in Fig. 2.3. Another temperature dependent region is the elastic portion at small strains. When the temperature is

greater than transition, the elastic region essentially disappears and the polymer behaves more fluid-like. After the initial

yield, the material begins to exhibit strain hardening attributed primarily to molecular orientation. At higher temperatures,

the amount of strain hardening decreases as shown in Fig. 2.4.

During the hold period, stress relaxation occurs and is highly dependent on temperature and the strain at which it is held.

At temperatures less than Tg there is a large initial drop in the stress. After this initial drop, the polymer continues to relax,

although the rate of relaxation remains at a lower, more constant rate. At temperatures near the glass transition, there is still a

small region of an initial drop once the hold period begins, however it is less severe than at lower temperatures. Similar to the

Fig. 2.1 (a) The polymer and stamp are heated to a temperature above glass transition (b) The stamp is lowered and pressure is applied to the

polymer forcing the polymer to flow and fill in the voids (c) The polymer and stamp are cooled briefly while still in contact then separated with

cooling continuing

10 D. Mathiesen et al.

low temperatures, it reaches a smaller, more constant relaxation rate after this initial drop. Much less of an initial drop is

present in the stress at the beginning of the hold period at temperatures greater than transition. For a given held strain and

temperature, strain rate does not affect the final relaxation value. As shown in Figs. 2.5, 2.6 and 2.7 the higher strain rate

values cause a larger initial drop in the stress than a lower strain rate at the same held strain. However, the steady stress it

relaxes to is approximately the same as its lower strain rate counterpart.

To quantify the amount of stress-relaxation, the percent relaxed is calculated to give an idea of relative relaxation

amounts. The percent relaxed is calculated for each temperature and held strain by subtracting the steady relaxation stress

from the maximum stress achieved before the hold period and dividing by the maximum stress. At temperatures less than

0 50 100 150 200 2500

10

20

30

40

50

Time (sec)

Str

ess

(MP

a)

ε=−1.5, dε/dt=−1.0

ε=−1.0, dε/dt=−1.0

ε=−0.5, dε/dt=−1.0

Fig. 2.2 Stress versus time at 95 �C for samples loaded at a rate of �1.0/min and held for 180 s. Each sample was held at a different final strain:

�0.5, �1.0, and �1.5

0 50 100 150 200 2500

0.5

1

1.5

2

Time (sec)

Str

ess

(MP

a)

ε=−1.5, dε/dt=−1.0

ε=−1.0, dε/dt=−1.0

ε=−0.5, dε/dt=−1.0

Fig. 2.3 Stress versus time at 135 �C for samples loaded at a rate of �1.0/min and held for 180 s. Each sample was held at a different final strain:

�0.5, �1.0, and �1.5

2 Stress-Relaxation Behavior of Poly(Methyl Methacrylate) (PMMA) Across the Glass. . . 11

glass transition, holding at a small strain will cause more stress-relaxation than if it were held at a larger strain. This is true

only for temperatures less than transition as evident in Fig. 2.8. At temperatures around glass transition, 105–110 �C, thetrend begins to decrease and the percent relaxed at different final strains becomes about the same. At temperatures greater

than glass transition, 125 �C and 135 �C, smaller hold strains cause less stress relaxation than at large strains, opposite of

what was found at low temperatures.

The relaxation effects can be explained by looking at how polymer molecules move depending on temperature and strain.

At temperatures less than Tg the polymer molecules do not flow readily and take larger amounts of time to rearrange

0 50 100 150 200 2500

10

20

30

40

50

Time (sec)

Str

ess

(MP

a)

95 �C

135 �C

125 �C

110 �C

105 �C

Fig. 2.4 Stress versus time for samples loaded at a rate of �1.0/min and held at a strain of �1.5. Each sample was at a different temperature:

95 �C, 105 �C, 110 �C, 125 �C, or 135 �C

0 50 100 150 200 2500

0.5

1

1.5

2

2.5

Time (sec)

Str

ess

(MP

a)

ε=−1.5, dε/dt=−1.0

ε=−1.0, dε/dt=−1.0

ε=−0.5, dε/dt=−1.0

ε=−1.0, dε/dt=−3.0

Fig. 2.5 Stress versus time at 125 �C for samples loaded at a rate of either �1.0/min or �3.0/min and held for 180 s. Each sample was held a

different final strain: �0.5, �1.0, or �1.5


themselves to reduce the stress. At small strains, the polymer molecules have not begun to align, like they do at large strains.

This lack of alignment allows them to rearrange to obtain a low stress state during the hold period. The highly oriented

molecules at large strains do not rearrange easily, which causes them to reach a higher steady stress state. At temperatures

greater than Tg the molecules flow readily and are immediately able to arrange themselves into as low of a stress state as

possible, even during loading. Therefore, at low strains they have already arranged themselves with the lowest stress-state so

there is little stress to relax. While at large strains, more imperfections in the arrangement have accumulated that can be

relaxed during the hold period. This effect also explains the large amount of relaxation that occurs with high strain rates.

0 50 100 150 200 2500

5

10

15

20

25

Time (sec)

Str

ess

(MP

a)

ε=−1.5, dε/dt=−1.0

ε=−1.0, dε/dt=−1.0

ε=−0.5, dε/dt=−1.0

ε=−1.0, dε/dt=−3.0



0 50 100 150 200 2500

2

4

6

8

10

12

14

Time (sec)

Str

ess

(MP

a)

ε=−1.5, dε/dt=−1.0

ε=−0.5, dε/dt=−1.0

ε=−1.0, dε/dt=−3.0

ε=−1.0, dε/dt=−1.0




When looking at Figs. 2.5, 2.6 and 2.7, when a faster strain rate is used, a larger amount of stress is generated prior to the hold

period compared to its low strain rate counterpart. This is because the polymer is not allowed the time necessary for it to

arrange into its lowest stress state. Once it is allowed to relax, it approaches the same steady relaxation stress as that of the

low strain rate sample.

2.4 Conclusions

With this new data, a new constitutive model can be developed to capture the behavior present. This model will need to

capture the stress relaxation behavior at multiple strains, strain rates, and temperatures in order for it to successfully predict

hot embossing processes. Additional data examining how PMMA behaves during cooling and actual hot embossing

processes need to be obtained so the material model can be validated for use in a finite element program. However, with

this new data, a material model can be developed that will predict stress relaxation behavior of PMMA.

References

1. Narasimhan J, Ian P (2004) Polymer embossing tools for rapid prototyping of plastic microfluidic devices. J Micromech Microeng 14:96–103

2. Dupaix RB, Cash W (2009) Finite element modeling of polymer hot embossing using a glass-rubber finite strain constitutive model. Polym

Eng Sci 49(1):531–543

3. Lu C, Cheng MM-C, Benatar A (2007) Embossing of high-aspect-ratio-microstructures using sacrificial templates and fast surface heating.

Polym Eng Sci 47:830–840

4. Anand L, Ames NM (2006) On modeling the micro-indentation response of an amorphous polymer. Int J Plast 22:1123–1170

5. Palm G, Dupaix RB, Castro J (2006) Large strain mechanical behavior of poly(methyl methacrylate) (PMMA) Near the glass transition

temperature. J Eng Mater Technol 128:559–563

6. Ghatak A, Dupaix RB (2010) Material characterization and continuum modelling of poly(methyl methacrylate) (PMMA) above the glass

transition. Int J Struct Chang Solids Mech Appl 2(1):53–63

7. Singh K (2011) Material characterization, constitutive modeling and finite element simulation of polymethly methacrylate (PMMA) for

applications in hot embossing. Dissertation, Ohio State University

90 100 110 120 130 14045

50

55

60

65

70

75

80

85

Temperature �C

Per

cent

Rel

axed

ε=1.5

ε=1.0

ε=0.5

Fig. 2.8 Percent of stress relaxed versus temperature for samples loaded at a rate of �1.0/min and held at �1.5, �1.0 or �0.5 final strain. The

percent relaxed is the difference between the maximum stress and the steady relaxation stress, divided by the maximum stress for a given

temperature and held strain


8. Pfister LA, Stachurski ZH (2002) Micromechanics of stress relaxation in amorphous glassy PMMA part II: application of the RT model.

Polymer 7:419–7427

9. McLoughlin JR, Tobolsky AV (1952) The viscoelastic behavior of polymethyl methacrylate. J Colloid Sci 7(6):555–568

10. Qvale D, Ravi-Chandar K (2004) Viscoelasatic characterization of polymers under multiaxial compression. Mech Time-Depend Mater

8:193–214

11. Takahshi M, Shen MC, Taylor RB, Tobolsky AV (1964) Master curves for some amorphous polymers. J Appl Polym Sci 8:1549–1561

12. Hasan OA, Boyce MC, Li XS, Berko S (1993) An investigation of the yield and postyield behavior and corresponding structure of poly(methyl

methacrylate). J Polym Sci 31:185–197


Chapter 3

The Effect of Stoichiometric Ratio on Viscoelastic

Properties of Polyurea

Zhanzhan Jia, Alireza V. Amirkhizi, Kristin Holzworth, and Sia Nemat-Nasser

Abstract Polyurea is a commonly utilized elastomer due to its excellent thermo-mechanical properties. In this study, the

polyurea is synthesized using Versalink P-1000 (Air Products) and Isonate 143 L (Dow Chemicals). The diisocynate blocks

generally assemble into hard domains embedded in the soft matrix, creating a lightly cross-linked heterogenous nano-

structure. We seek to evaluate the effect of the stoichiometric ratio of the two components on the viscoelastic properties of

the resultant polyurea. By altering the ratio, polyurea samples with different stoichiometric variations are made. In order to

approximate the mechanical properties of polyurea for a wide frequency range, master curves of storage and loss moduli are

developed. This is achieved by time-temperature superposition of the dynamic mechanical analysis (DMA) data, which is

conducted at low frequencies and at temperatures as low as the glass transition. Furthermore, in order to access the effect

of the stoichiometric ratio on the relaxation mechanisms in the polyurea copolymer system, continuous relaxation spectra

of all the stoichiometric variations are calculated and compared.

Keywords Polyurea • Stoichiometric variation • Master curves • Time-temperature superposition • Relaxation spectrum

3.1 Introduction

Polyurea is a commonly utilized elastomer derived from the reaction of a diisocyanates component and a diamine

component. The polyurea system used in this study is synthesized using the Versalin P-1000[1], which is a polytetramethy-

leneoxide-di-p-aminobenzoate and the Isonate 143 L[2], which is a polycarbodiimide-modified diphenylmethane

diisocyanate. This type of polyurea is a lightly cross-linked segmented copolymer, with the cross-links formed by the biuret

structure; it has hard domains embedded in the soft domain, which forms the heterogeneous nano-structure [3]. The hard

domains with high glass transition temperature (Tg) are mainly composed of the diisocyanate blocks; and the soft domain

with low Tg is composed of the flexible chains of the diamine component [4]. The nano-structure of polyurea is modified by

adjusting the stoichiometric ratio of the diisocyanate component and the diamine component. The commonly used

stoichiometric ratio for synthesizing this polyurea is 1.05. The 5 % extra isocyanate ensures the polymerization is complete

and the resultant polyurea is lightly cross-linked. In this study, seven stoichiometric ratios, as shown in Table 3.1, are studied

for this polyurea system. Dynamic mechanical properties of polyurea for all the stoichiometric ratios are characterized by

using the dynamic mechanical analysis (DMA). In order to approximate the mechanical properties in a wide frequency

range, master curves are developed using the DMA data. Furthermore, Continuous relaxation spectra are approximated to

study the effect of the stoichiometric ratio on the relaxation mechanisms and the molecular phenomena that underlie them.

Z. Jia (*) • A.V. Amirkhizi • K. Holzworth • S. Nemat-Nasser

Department of Mechanical and Aerospace Engineering, Center of Excellence for Advanced Materials, University of California,

9500 Gilman Drive, La Jolla, 92093-0416 San Diego, CA, USA




17


3.2 Sample Fabrication

Polyurea samples are fabricated by mixing the two components Versalink P-1000 and Isonate 143 L together in the

predetermined stoichiometric ratios. Versalink P-1000 and Isonate 143 L are yellow liquids at room temperature. The two

components are degassed separately under high vacuum (1 Torr) for 1 h before being mixed together. A magnetic stirrer was

used through the degassing process. The degassed components are thoroughly mixed together under vacuum by the magnetic

stirrer for 5 min before being transferred into the Teflon molds using syringes. Seven types of polyurea samples with

different stoichiometric ratios are fabricated and the stoichiometric ratios of the isocyanate and amino components are listed

in Table 3.1. The samples are cured for 2 weeks in an environmental chamber in which the relative humidity is controlled at

10 %. The DMA samples are cut from the cast polyurea with the nominal dimensions of 3 � 10 � 20 mm.

3.3 Dynamic Mechanical Analysis

Dynamic mechanical analysis is conducted on the TA Instruments Dynamic Mechanical Analyzer 2,980; the data is

collected on the corresponding software for the TA instrument data acquisition. The samples for DMA test are 3 mm in

thickness, 10 mm in width and 20 mm in length. The accurate dimensions of the samples are measured and used as inputs for

the software to calculate the modulus. In the DMA test, the sample is cantilevered at both ends with a free length of 17.5 mm

in between, as shown in Fig. 3.1. One cantilevered end is fixed in all degrees of freedom and the other moves vertically with

its displacement following a sine wave form. The amplitude of the sine waves is 15 μm, and the frequencies are 1, 2, 5, 10,

20 Hz. The temperature range of the test is from �80 �C to 50 �C, with 3 �C increment for each step. The thermal soaking

time is 3 min before the mover starts the frequency sweep. The storage and loss moduli are measured at each temperature

point and for five different frequencies. The relation between the complex, storage and the loss moduli is E� ¼ E0 þ iE

0 0,

where E� is the complex Young’s modulus, and E� ¼ σ�=E�.

3.4 Master Curves

In order to approximate the polyurea mechanical properties in a wide frequency range, time-temperature superposition

(TTS) is applied using the DMA data. For many viscoelastic materials, an increase in temperature is nearly equivalent to an

increase in time or a decrease in frequency in its effect on a modulus or compliance [5]. In this study, the reference

temperature is set at 0 �C, and the DMA data collected at various temperatures are shifted in both the modulus and

the frequency. The selection of the data for TTS is kept above the glass transition temperature, which is around �60 �C,since below Tg, the TTS is different from that above Tg[6]. The storage and loss moduli data are first rescaled by Tref/T. Then

the horizontal shift factor is measured for both storage and loss moduli for each temperature aiming to construct the

smoothest curve by using the data tested at different temperatures; the average shift factor from the storage and loss moduli

are used to develop the mater curve. The shift factor represents the distance the experimental moduli needed to be shifted in

Table 3.1 Stoichiometric ratios

of the diisocyanate and the

diamine components

Stoichiometric ratio

0.90

0.95

1.00

1.05

1.10

1.15

1.20

18 Z. Jia et al.

the frequency axis so that they are equivalent to the modulus at the reference temperature. And the shift factor is a function of

temperature; in the current studied temperature range, the curve of the shift factor can be well fitted by the William-Landel-

Ferry (WLF) equation [5], Eq. 3.1, where C1 and C2 are two fitting parameters and the temperatures are absolute.

log að Þ ¼ �C1ðT � Tref ÞC2 þ T � Tref

(3.1)

3.5 Continuous Relaxation Spectra

In theory, the relaxation spectra can be used to calculate the relaxation/compliance moduli as [5]

E0 ¼ Ee þ

ð1

�1Φω2τ2=ð1þ ω2τ2Þdlnτ; (3.2)

E00 ¼

ð1

�1Φωτ=ð1þ ω2τ2Þdlnτ; (3.3)

where Φ is the relaxation spectrum, τ is the relaxation time, ω is the circular frequency and Ee is the equilibrium storage

modulus; the constitutive relation of linear viscoelasticity can be further derived for various types of deformations [5]. Using

the relaxation spectra approximated from experimental data, such a procedure is only a rough estimation of material

properties, since in practice the inverse calculation from the experimental storage and loss moduli to relaxation spectrum is

not as convenient. But the relaxation spectrum is still valuable for the approximation of the material properties and to study

the different aspects of the viscoelastic behavior and the molecular phenomena that underlie them. Using the method by

Williams and Ferry [7, 8], relaxation spectra of all stoichiometric ratios are calculated and compared.

Fig. 3.1 DMA test setup

3 The Effect of Stoichiometric Ratio on Viscoelastic Properties of Polyurea 19

3.6 Results and Discussion

3.6.1 Dynamic Mechanical Analysis

In order to compare the storage and loss moduli of polyurea with different stoichiometric ratios, the storage and loss moduli

of each index are normalized by the corresponding moduli of the index 1.05. The moralized DMA result shows that as the

stoichiometric index increases from this value, the storage and loss moduli increase drastically. However, as the index is

reduced below 1.05, the storage and loss moduli decrease moderately. The differences are more pronounced at high

temperatures. At low temperatures around Tg, the various ratios do not show much difference; when the temperature

increases, the difference of moduli increases until it reaches the maximum value at round�25 �C for the storage moduli and

at around 10 �C for the loss moduli; as the temperature continues increasing, the difference of moduli slightly decreases.

Detailed data is presented elsewhere [9].

3.6.2 Master Curves

The resultant master curves developed from TTS of the DMA data covers a frequency range from 10�2 to 1010 Hz. When the

frequency is lower than 107 Hz, higher stoichiometric index yields both higher storage and loss moduli. When the frequency

is higher than 107 Hz, the storage moduli tend to approach a saturating value, and the loss moduli decrease as the

stoichiometric index decreases.

3.6.3 Relaxation Spectra

The relaxation spectra which cover more than ten decades of relaxation time are calculated for all the stoichiometric ratios.

In the log-log scale plot, the relaxation spectra, each with a different slope, cross at ω ¼ 10�7 Hz, and the slopes flatten as the

stoichiometric ratio increases. This change shows the contribution from the relaxation mechanism in the polyurea spreads

out to a broader frequency range when the content of hard domains increases as well as when the density of the cross-links

increases.

Acknowledgement This work has been supported by the Office of Naval Research (ONR) grant N00014-09-1-1126 to the University of

California, San Diego.

References

1. Air Product Chemicals, Inc (2003) Polyurethance specialty products, Air Products and Chemicals, Allentown

2. The Dow Chemical Company (2001) Isonate 143L, Modified MDI, Dow Chemical, Midland

3. May-Hernandez L, Hernandez-Sanchez F, Gomez-Ribelles J, Sabater-i Serra R (2011) Segmented poly (urethane-urea) elastomers based on

polycaprolactone: structure and properties. J Appl Polym Sci 119:2093–2104

4. Fragiadakis D, Gamache R, Bogoslovov RB, Roland CM (2010) Segmental dynamics of polyurea: effect of stoichiometry. Polymer

51(1):178–184

5. Ferry JD (1980) Viscoelastic properties of polymers. Wiley, New York

6. Knauss WG, Zhu W (2002) Nonlinearly viscoelastic behavior of polycarbonate. I. Response under pure shear. Mech Time-Depend Mater

6:231–269

7. Ferry JD, Williams ML (1952) Second approximation methods for determining the relaxation time spectrum of a viscoelastic material. Relax

Spectr Viscoelastic Mater 347–353

8. Williams ML, Ferry JD (1952) Second approximation calculations of mechanical and electrical relaxation and retardation distributions. J Polym

Sci XI(2):169–175

9. Holzworth K, Jia Z, Amirkhizi AV, Qiao J, Nemat-Nasser S Effect of isocyanate content on thermal and mechanical properties of polyurea.

Polym Under Rev

20 Z. Jia et al.

Chapter 4

Dynamic Properties for Viscoelastic Materials Over

Wide Range of Frequency

T. Tamaogi and Y. Sogabe

Abstract The purpose of this study is to examine the dynamic properties for viscoelastic materials over the wide range of

frequency by measuring the change of waveform propagating in the bar. The viscoelastic properties in the frequency up to

around 15 kHz can be evaluated by the impact test that the striker bar collides with the specimen bar mechanically. The

properties of the high frequency area are different from those of the low frequency area. However, the impact tests using

the striker bar cannot determine the dynamic characteristics in the high frequency range. The propagation tests using the

wave packet generated by the ultrasonic transducers were performed. The attenuation and dispersion properties were

examined by using the ultrasonic transducers having several characteristic frequencies within 25–200 kHz. It was found

that the dynamic properties in the low frequency range could be identified as a three-element model based on the elementary

theory. On the other hand, a five-element model based on the three-dimensional theory had to be applied in the high

frequency range.

Keywords Dynamic properties • Propagation • Viscoelastic • Ultrasonic • Transducer

4.1 Introduction

Polymer materials are widely used in various fields because of their impact resistance or the vibration control. The

deformation of the materials remarkably depends on the time or strain rate. It is, therefore, important to understand

the dynamic properties of viscoelastic materials. The viscoelastic theory is generally applied to their impact behavior.

The characteristics of viscoelastic medium are controlled by the different mechanical properties according to the various

frequencies. It was shown that the dynamic properties for polymethyl methacrylate (PMMA), which were generally used in

longitudinal impact tests for viscoelastic materials, could be approximated as a three-element model by the longitudinal

impact test [1–3]. Because the geometrical dispersion due to three-dimensional deformation will be caused by the high

frequency component involved in a wave, the three-dimensional theory should be employed to process the experimental data

of wave propagation [4, 5].

In this work, PMMA bars are tested as a typical specimen. The propagation tests using the wave packets generated by the

ultrasonic transducers having several characteristic frequencies (ultrasonic propagation tests) as well as the longitudinal

impact tests were carried out. The attenuation and dispersion properties for PMMA in the low frequency area were examined

by the longitudinal impact tests, while in the high frequency area were done by the ultrasonic propagation tests.

T. Tamaogi (*)

Depertment of Mechanical Engineering, Niihama National College of Technology, 7-1 Yakumo-cho,

792-8580 Niihama, Ehime, Japan


Y. Sogabe

Depertment of Mechanical Engineering, Ehime University, 3 Bunkyo-cho, 790-8577 Matsuyama, Ehime, Japan



21


4.2 Atteunation and Dispersion for Viscoelastic Bar

4.2.1 Elementary Theory

In case of a thin and uniform viscoelastic bar, let �εð x; ωÞ be the Fourier transform of a strain-time relation εð x; tÞ.When the material is linear viscoelastic, the following equation can be obtained [2]:

�εð x; ωÞ ¼ �εð0; ωÞ � expf�ðαþ ikÞgx; (4.1)

where x, ω and i are the coordinate along the rod axis, angular frequency and imaginary unit, respectively.

The attenuation coefficient α and wave number k are the functions of ω, and are related to the complex compliance as

α2ðωÞ ¼ 1

2ρω2 �J�1ðωÞ þ J�ðωÞj j� �

; (4.2)

k2ðωÞ ¼ 1

2ρω2 J�1ðωÞ þ J�ðωÞj j� �

; (4.3)

where J�ðωÞj j ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiJ�1

2ðωÞ þ J�22ðωÞ

q, ρ is the material density. The complex compliance J�ðωÞ, which represents one of the

viscoelastic properties of the material, is defined as

J�ðωÞ ¼ J1�ðωÞ � i J2

�ðωÞ: (4.4)

The phase velocity CðωÞ, which corresponds to dispersive properties, is given by

CðωÞ ¼ ω

kðωÞ : (4.5)

4.2.2 Pochhammer-Chree Theory

Consider a stress wave propagating in an infinite cylindrical elastic bar. The equation of motion is written in the following

vector form:

ρ@2u

@t2¼ ðλþ 2μÞgradΔ� 2μrot Ω; (4.6)

where u denotes the displacement vector, λ and μ are the Lame coefficients, Δ ¼ divu, 2Ω ¼ rotu, ν is Poisson’s ratio.

Assuming axial symmetry, and applying the Fourier transformation and the correspondence principle [6] to Eq. 4.6, the

following equations for a viscoelastic medium on the cylindrical coordinate plane are deduced:

�ρω2Ur ¼ ðλ� þ 2μ�Þ @D@r

� 2iξμ�W

�ρω2Uz ¼ ðλ� þ 2μ�Þð�iξÞD� 2μ�@W

@rþW

r

� �9>>=>>;; (4.7)

where λ� and μ� are the complex Lame coefficients, the displacement �urðr; z;ωÞ ¼ Urðr;ωÞ � expð�iξzÞ and

�uzðr; z;ωÞ ¼ Uzðr;ωÞ � expð�iξzÞ, the volumetric strain �Δðr; z;ωÞ ¼ Dðr;ωÞ � expð�iξzÞ, the rotation vector�Ωθðr; z;ωÞ ¼ Wðr;ωÞ � expð�iξzÞ, ξðωÞ ¼ kðωÞ � iαðωÞ respectively. Solving the Eq. 4.7 in D and W, the Bessel’s

differential equations of order zero and one are obtained. The solutions can be expressed as follows:

22 T. Tamaogi and Y. Sogabe

Dðr;ωÞ ¼ A0J0ðprÞWðr;ωÞ ¼ A1J1ðqrÞ

); (4.8)

where J0, J1 are Bessel functions of order zero and one, A0 and A1 are the arbitrary functions of ω,p2 ¼ ρω2=ðλ� þ 2μ�Þ � ξ2, q2 ¼ ρω2=μ� � ξ2, respectively.

Considering stress free boundary conditions at the external surface of the bar, the following frequency equation results:

ðq2 � ξ2Þ2J0ðpaÞJ1ðqaÞ � 2p

aðq2 þ ξ2ÞJ1ðpaÞJ1ðqaÞ þ 4ξ2pqJ0ðqaÞJ1ðpaÞ ¼ 0; (4.9)

where a is radius of the bar. Solving Eq. 4.9 for ξ numerically, the attenuation coefficient αðωÞ, the wave number kðωÞ andthe phase velocity CðωÞ are given by

αðωÞ ¼ Re½�iξ� (4.10)

kðωÞ ¼ Im½�iξ� (4.11)

CðωÞ ¼ ω

kðωÞ ¼ ωIm½�iξ� (4.12)

4.3 Wave Propagation Tests

4.3.1 Experimental Method and Results of Longitudinal Impact Tests

A schematic diagram of a longitudinal impact test is given in Fig. 4.1. Polymethyl methacrylate (PMMA) is used for the

specimen. The length and diameter of the PMMA bar are 2000 and 15 mm, respectively. The striker bar is also made of

PMMA, and has a length and diameter of 20 and 15 mm. Four strain gages are situated at positions separated by equal

intervals (200 mm) at a distance from the impact face. The striker bar is launched by the air compressor, and impacts the

front end of the PMMA bar. Figure 4.2a shows the measured strain pulses. It is seen that the attenuation and dispersion

generate as the strain pulses propagate. The frequency spectrums and phase spectrums of each strain pulse are given in

Fig. 4.2b, c. The values of frequency spectrums of all waves coincide with each other when the frequency is 0 kHz. It can be

said that the waves propagate as the area keeps constant. It is found that the values of phase spectrums become negative, and

decrease in linear relation.

4.3.2 Experimental Method and Results of Propagation Tests UsingWave Packets Generated by Ultrasonic Transducer

Figure 4.3 indicates a schematic diagram of an ultrasonic propagation. The PMMA bar is also used for the specimen, and is

2000 and 15 mm in length and diameter. A transducer is attached to the one side of the specimen.

Fig. 4.1 Schematic diagram of longitudinal impact test

4 Dynamic Properties for Viscoelastic Materials Over Wide Range of Frequency 23

Four semiconductor strain gages are situated at positions separated by equal intervals (100 mm) at a distance from the

transducer’s side. The transducer is vibrated at the natural frequency by giving the voltage amplified with an AC amplifier.

The properties of the transducer used on experiments are enumerated in Tables 4.1a, b. The measured strain waves using the

ultrasonic transducer type ⑤ (50 kHz) is shown in Fig. 4.4a as a typical example. It is found that the attenuation and

dispersion generate as the waves propagate like the strain pulses measured on longitudinal impact test. Besides, the

frequency and phase spectrums of each wave are represented in Fig. 4.4b, c. The frequency spectrums have a lot of

frequency elements of in the frequency around 50 kHz. The values of 43–54 kHz, which are 70 % of the maximum values of

the frequency spectrums, are used for evaluation of the attenuation and dispersion properties.

4.4 Evaluation of Attenuation Coefficient and Phase Velocity

The strain wave propagating in a cylindrical viscoelastic bar based on the elementary theory is expressed by Eq. 4.1. On the

other hand, the strain wave propagating in the bar based on the Pochhammer-Chree theory is obtained by differentiating

�uzðr; z;ωÞ with respect to z.

εz r; z;ωð Þ ¼ @uz@z

¼ �iξUz r;ωð Þ exp �iξzð Þ ¼ �ðαþ ikÞUz r;ωð Þ exp � αþ ikð Þf gz: (4.13)

0 200 400 600 8000

500

1000

Time µs

Stra

in µ

m/m

0 5 10 150

2

4

6

8

Frequency kHz

Am

plitud

e se

c

Frequency Spectrum 1Frequency Spectrum 2Frequency Spectrum 3Frequency Spectrum 4

× 10-8

0 5 10 15-30

-20

-10

0

Phase angle 1Phase angle 2Phase angle 3Phase angle 4

Frequency kHz

Pha

se a

ngle

rad

iane1(t) e2(t) e3(t) e4(t)

a b c

Fig. 4.2 Experimental results on longitudinal impact test (a) Measured strain pulses (b) Frequency spectrums (c) Phase spectrums

AC amplifier

Functiongenerator

Specimen

Bridge box

Digital oscilloscope

1600

100 100 100 100

f 15

Transducer

Fig. 4.3 Schematic diagram of propagation test using wave packets generated by ultrasonic transducer


On the surface of the bar, r ¼ a:

εz ¼ ε0 ωð Þ exp � αþ ikð Þf gz; (4.14)

where ε0 ωð Þ ¼ �ðαþ ikÞUz a;ωð Þ. Therefore, the strain wave in a cylindrical viscoelastic bar is calculated by the same

equation. Using the least square method, αðωÞ and kðωÞ can be determined from the experimental data.

αðωÞ ¼P

zmP

log j�εmj � 4P

zm log j�εmj4P

z2m � Pzmð Þ2

kðωÞ ¼P

zmP

θm � 4P

zmθm

4P

z2m � Pzmð Þ2

9>>>>=>>>>;; (4.15)

where m indicates gage position number (1 ~ 4), j�εmj is absolute value of �εm, θm is phase angle argð�εmÞ, respectively. Then,the phase velocity can be given by Eqs. 4.5 or 4.12. Based on the Eqs. 4.15 and 4.5, the attenuation coefficient αðωÞ andphase velocity CðωÞ are obtained by connecting the experimental values of the longitudinal impact tests (up to 15 kHz) to

those of the ultrasonic propagation tests (from 25 to 200 kHz) as shown in Fig. 4.5a, b. The plots in the figure show average

experimental values, and the vertical bars indicate the standard deviation. The solid and dotted line are the predicted values

Table 4.1 Properties of ultrasonic transducers

(a) Ultrasonic transducers for process machinery (NGK SPARK PLUG CO., LTD.)

Type Frequency (kHz) Diameter of radial plane (mm) Length (mm) Capacitance

1 DA2228 28.17 20 92.4 1,250

2 DA2240 38.94 20 64.4 1,090

3 DA21560A 60.04 15 40.4 680

4 DA2275A 74.79 20 30.4 1,740

(b) Piezoelectric ceramics transducers (FUJI CERAMICS CORPORATION)

Type Frequency (kHz) Element diameter (mm) Length (mm) Capacitance (pF)

5 0.05Z 15D 50.00 15 26.20 136

6 0.075Z 15D 74.60 15 16.40 220

7 0.1Z15D 99.20 15 10.50 339

8 0.13Z10D 131.40 10 8.40 192

9 0.15Z20D 148.20 20 8.40 788

10 0.2Z15D 199.30 15 6.20 564

0 200 400 600 800Time μs

Stra

in μ

m/m

x=0.0m

x=0.1m

x=0.2m

x=0.3m

100μm/m

20 30 40 50 60 70 80

-150

-100

-50

0

Phase angle 1Phase angle 2Phase angle 3Phase angle 4

Frequency kHz

Pha

se a

ngle

rad

ian

20 30 40 50 60 70 800

1

2

3spectrum 1spectrum 2spectrum 3spectrum 4

Frequency kHz

Am

plitud

e se

c

× 10-3a b c

Fig. 4.4 Experimental results on propagation test using wave packets generated by ultrasonic transducer (a) Measured strain waves (b) Frequency

spectrums (c) Phase spectrums


calculated by using the viscoelastic models shown in Fig. 4.6a, b. The experimental dependence of the attenuation

coefficient and phase velocity in the frequency up to around 15 kHz is identified as the three-element model based on

the elementary theory, while the dependence from 25 to 200 kHz is approximated as the five-element model using the

Pochhammer-Chree theory. The viscoelastic values for the three-element model E1, E2 and η2 are 5.62GPa, 58.5GPa, and3.02 MPa·s. The viscoelastic values for the five-element model E1, E2, η2, E3 and η3 are 5.89GPa, 58.4GPa, 2.80 MPa·s,

122GPa and 0.39 MPa·s, respectively. It is found that the experimental and model’s predicted values up to 15 kHz

are almost identical. The evaluation of the attenuation and dispersion properties over 15 kHz is not enough even if the

five-element model is used. In contrast, their properties can be evaluated with high accuracy based on the Pochhammer-

Chree theory.

4.5 Conclusions

The conclusions obtained from the present study are summarized as follows:

• The attenuation and dispersion properties for viscoelastic material over the wide range of frequency were examined by

two kinds of propagation tests.

100 200

5

10

15

0Frequency kHz

Elementary TheoryPochhammer-Chree Theory

100 200

1000

2000

3000

4000

0

Frequency kHz

C m

/s

Elementary TheoryPochhammer-Chree Theory

impact testimpact test

a m

-1a b

Fig. 4.5 Experimental and analytical values obtained by connecting experimental values of longitudinal impact tests to those of propagation tests

using wave packets generated by ultrasonic transducers (a) Attenuation coefficient (b) Phase velocity

E2

E1

η 2η

E3 E2

E1

3η

2

a b

Fig. 4.6 Viscoelastic models for determining mechanical properties of PMMA material (a) Three-element model (b) Five-element model


• It was clarified that the viscoelastic properties in the frequency up to around 15 kHz could be evaluated by the

longitudinal impact test based on the elementary theory.

• It was confirmed that the viscoelastic properties in the low frequency area could be identified as a three-element model, in

contrast, a five-element model based on the three-dimensional theory had to be applied in the high frequency area.

References

1. Sackman JL, Kaya I (1968) On the determination of very early-time viscoelastic properties. J Mech Phys Solids 16(2):121–132

2. Sogabe Y, Tsuzuki M (1986) Identifidation of the dynamic properties of linear viscoelastic materials by the wave propagation testing. Bull

JSME 29(254):2410–2417

3. Lundberg B, Blanc RH (1988) Determination of mechanical material properties from the two-point response of an impacted linearly viscoelastic

rod specimen. J Sound Vib 126(1):97–108

4. Zhao H, Gary G (1995) A three dimensional analytical solution of the longitudinal wave propagation in an infinite linear viscoelastic cylindrical

bar. Application to experimental techniques. J Mech Phys Solids 43(8):1335–1348

5. Bacon C (1999) Separation of waves propagating in an elastic or viscoelastic hopkinson pressure bar with three-dimensional effects. Int J Impact

Eng 22(1):55–69

6. Flugge W (1975) Viscoelasticity, Springer-Verlag, p 159


Chapter 5

Spatio-Temporal Principal Component Analysis

of Full-Field Deformation Data

Srinivas N. Grama and Sankara J. Subramanian

Abstract Full-field displacements are the output of several non-contact experimental mechanics techniques such as the

Grid method or Digital Image Correlation (DIC). Although it appears that an enormous amount of data is available from such

measurements, such data are often highly redundant. In the past, orthogonal shape descriptors such as Zernike moments,

Fourier-Zernike moments (Patki and Patterson, Exp Mech 1:1–13, 2011) and Tchebicheff moments (Sebastian et al., Appl

Mech Mater 70:63–68, 2011) have been proposed to reduce dimensionality. We recently proposed the use of Principal

Components Analysis (PCA) to reduce the dimensionality of full-field displacement data, identify primary spatial variations

and compute strains without any a priori assumptions on the form the shape descriptors. In this work, we extend this idea to

time-dependent problems and investigate spatio-temporal PCA to identify evolution of the primary displacement patterns

with time in a deforming solid. The proposed method is applied to synthetic data obtained from a finite-element analysis of a

thin visco-plastic solder specimen subjected to cyclic shear loading. Progressive damage is introduced into the specimen

through the reduction of element stiffness at a specific location after pre-determined number of cycles. Displacement fields

collected at periodic intervals are analysed using spatio-temporal PCA and the possibility of inferring local damage from the

time-evolution of the eigenfunctions and their singular values is demonstrated.

Keywords Principal component analysis • Eigenfunctions • Rate-dependent materials • Damagedetection • Spatio-temporal

analysis

5.1 Introduction

Solder alloys typically find applications in electronics packaging as low cost electrical connections and mechanical support

between different components in the assembly. As the solder is in contact with materials of different coefficients of thermal

expansion and elastic moduli (typically silicon on one side and printed circuit board on the other), it experiences

predominantly shear stress due to thermal cycling [1]. Solders have low melting points in the range of 180 ∘ to 250∘ C

and therefore even at room temperature, visco-plastic deformation under such thermo-mechanical loads is of great concern.

Solder is softer compared to the material it joins and thus failure is most likely to occur in the solder joint due to thermo-

mechanical fatigue.

Fatigue behaviour of time-dependent materials like solders are often described using phenomenological models such as

Basquin’s model and Coffin-Mason model by curve fitting the experimental data. A definition for fatigue failure is however

required to obtain the parameters. However, there is no consensus in literature on this and various researchers have come up

with customized ways for defining fatigue life [2, 3]. For example, [3] defined fatigue life as the number of cycles

corresponding to 25 % of the maximum load. Continuum damage models have also been used to describe the evolution

of field variables and predict the fatigue behaviour of solders [4, 5]. Full-field experimental techniques such as Digital Image

Correlation enable continuous monitoring of local deformation fields, and the possibility of identifying fatigue damage

initiation and propagation by detailed analysis of these spatio-temporal deformation data.

S.N. Grama (*) • S.J. Subramanian

Department of Engineering Design, Indian Institute of Technology Madras, Chennai, India

e-mail: [email protected]; [email protected]



29



Since full-field deformations are typically available at tens of thousands of points in the field of interest, the resulting data

are of very high dimensionality. Therefore, if the data over several cycles are to be analyzed, it is critical to work with

parsimonious representations of such data. Shape descriptors such as Fourier-Zernike moments [6] and Tchebicheff

moments [7] have been proposed to achieve this objective. Recently, Principal Component Analysis (PCA) has been used

[8] for dimensionality reduction as well as to identify the dominant variations of displacements and minimise noise. In that

work it is also shown that strains may be obtained in a straightforward manner through the differentiation of the eigenvectors

identified using PCA. In the present work, we extend this idea to time-dependent problems in order to describe creep

deformation as well as progressive damage.

PCA is an orthogonal projection technique wherein the data in naive basis is projected onto its orthogonal subspace such

that the variance is minimised. Usually, the dimension of the principal subspace is lower than that of original data. PCA, also

known as Proper Orthogonal Decomposition, has been used previously in modal analysis to detect the location and severity

of the damage [9]. Galvanetto et al. [9] used the difference in principal orthogonal modes to detect the induced damage in a

finite element framework. Lanata et al. [10] used Proper Orthogonal Decomposition of nodal displacement evolution to

detect the insurgence of damage and determine its location and intensity for enabling structural health monitoring of bridges.

In the current work, a finite-element framework is used to simulate low cycle fatigue behaviour of solders along with the

introduction of progressive damage in the form of element stiffness reduction at a known location after pre-determined

interval of time. The severity of damage is increased linearly with time and the possibility of describing the relaxation

behaviour along with the detection of damage is investigated through the PCA of full-field displacements. As is customary in

detecting the change in deformation behaviour, PCA of the difference between the nodal displacements of consecutive

cycles at applied peak displacements (displacement controlled test) is performed to identify the change in the relaxation

behaviour and initiation of damage. In order to fix the time instant at which change in relaxation behaviour and/or occurrence

of damage, the shape of the dominant principal variations of nodal displacements are carefully examined. Damage or change

in relaxation behaviour is temporally localized when change in the evolution of prominent variations of differential

displacements are observed. Angle between the difference in eigenvectors of nodal displacements at peak loads is used as

a metric for this purpose. After change in relaxation behaviour or damage is detected, its spatial localization is achieved by

examining the prominent variations obtained from spatial PCA of the actual displacement fields at that instant. The details of

the finite element analysis and temporal and spatial localization of damage or change in relaxation behaviour are detailed in

the following sections.

5.2 Finite Element Simulation

5.2.1 Material Model

In this work we follow several other researchers [11, 12] in using the Anand model [13] to describe the visco-plastic

deformation of solder alloys. The main features of Anand model are as follows:

• No explicit yield condition

• Rate-independent plasticity and creep are unified in the same set of flow and evolution equations

• A single scalar internal variable is used to represent the isotropic resistance to plastic flow offered by the internal micro

structure of the material.

The flow equation has the following form

_Ep ¼ A exp � Q

Rθ

� �sinh ξ

~σ

s

� �1=m" #

(5.1)

where A is a pre-exponential factor, Q is activation energy, R is gas constant, θ is temperature in ∘K, m is strain rate

sensitivity of stress and ξ is stress multiplier

The evolution of the scalar internal variable, s is given by

_s ¼ h0

��1� s

s�

��a

sgn 1� s

s��

_Ep (5.2)

30 S.N. Grama and S.J. Subramanian

where h0 is a measure of strain hardening rate, a is strain rate sensitivity of hardening, s∗ represents a saturation value of

s associated with a set of given temperature and strain rate and is given by the equation

s� ¼ s_Ep

Aexp

Q

Rθ

� �� n(5.3)

where n is the strain rate sensitivity of deformation resistance and hats is a material parameter. Table 5.1 shows the material

parameters used in the present study wherein a constant temperature of θ ¼ 318∘ K is maintained throughout the analysis.

Although the initial value of the deformation resistance is dependent on the prior mechanical work done on the specimen and

also on temperature, for our analysis it is chosen as 7. 1986 MPa at θ ¼ 318∘ K [14].

5.2.2 Model Geometry and Boundary Conditions

A finite element model is built to simulate displacement-controlled simple shear under plane stress conditions (Fig. 5.1)

using quadratic elements in the commercial FEA package ABAQUSTM. Figure 5.1 shows the dimensions of the model and

shear loading is simulated by keeping face AD fixed and applying displacement of � 0. 2 mm along face BC.

Damage is progressively introduced into the model over region X (see Fig. 5.1) from the 10th cycle till the 20th cycle

through a linear reduction of element stiffness from 48 GPa to 2 GPa (Fig. 5.2). After performing a mesh convergence

study, a mesh model with 12,800 plane stress elements is chosen for analysis. The variation of load with number of

cycles (Fig. 5.2) shows a gradual relaxation, although the shape of the load displacement curve appears to change after the

introduction of damage.

Table 5.1 Material parameters

of Anand modelMaterial parameter A (1/s) Q/R (K) s (MPa) h0 (MPa) ξ m n a

Value 177,016 10,278.43 52.4 117,888.256 7 0.207 0.0177 1.6

Fig. 5.1 FE Model with

displacement control along

face BC to simulate shear

deformation for 20 cycles.

Damage is numerically

added into the FE model

in region X

5 Spatio-Temporal Principal Component Analysis of Full-Field Deformation Data 31

5.3 PCA of Full-Field Displacement Data

5.3.1 Temporal Localization of Change in Deformation Fields

The nodal displacement fields obtained at any time instant are arranged in 161 �81 matrices and PCA is performed through

Singular Value Decomposition (SVD) technique [15], following which we write

Lm�mSm�nRTn�n ¼ Um�n (5.4)

where L is a matrix of left eigenvectors, S is a matrix of singular values, R is a matrix of right eigenvectors andU is 161�81

matrix of X � displacement components. It is often seen that only a few, say p, singular values are dominant and therefore

the displacement matrix can be reconstructed using only these p left and right eigenvectors corresponding to just these

dominant singular values. Left and right eigenvectors have a clear physical significance – left eigenvectors correspond to the

prominent variations of the displacement field along Y � direction and right eigenvectors correspond to the prominent

variations along X � direction.

The main objective of this work is to investigate if PCA of full-field displacement data of a rate-dependent material may

be used to detect damage or changes in the relaxation behaviour. Since these events may be reflected in very subtle

displacement changes, at small changes, instead of working with the full displacement fields, we work with differential

displacement fields Δ Ui obtained by subtracting the displacement field Ui at peak displacement at the ith cycle from the

corresponding matrix Ui � 1 at the (i � 1)th cycle. The SVD of this differential displacement field is written as

ΔLm�mΔSm�nΔRTn�n ¼ ΔUi

m�n (5.5)

where ΔUi ¼ Ui � Ui�1 and Ui, Ui � 1 are the matrices of displacement field in X � direction at peak applied displacement

during the ith and (i � 1)th cycles respectively. As the primary displacement variable here is V i.e. displacement in

Y � direction, logs of singular values (LSV) plots and the evolution of change in eigenvector plots of Δ V fields are

examined first. Figure 5.3 shows a slight difference in the LSV plots ofΔV2,ΔV11 andΔV20, which is due to the relaxation

behaviour of solders and also due to the presence of damage. To understand in detail these differences, we examine

the evolution of the first ten left and right eigenvectors. It is worth mentioning that one of the central tasks in PCA is

in choosing the number of dominant singular values so as to enable accurate reconstruction, generate parsimonious data

Fig. 5.2 The evolution of load with time (top) indicates a gradual relaxation with increasing number of cycles. The bottom plot shows the

magnitude of change in the Young’s modulus in region X with time, which is a numerical manifestation of damage


and reduce noise. A detailed discussion on the selection of dominant singular values can be found in the book by Jolliffe

[16]. In the current work, the focus is less on reconstruction and more on examining the evolution of eigenvectors so as to

describe the time-dependent deformation behaviour of solders along with the detection of damage.

The evolution of 1st right eigenvector of ΔV displacement field with number of cycles is shown in Fig. 5.4. The first

dominant right eigenvector of ΔV field does not detect the change observed in the relaxation behaviour at ΔV6 (see Fig. 5.2,

where change in the relaxation rate is observed) nor does it show any effects of damage until ΔV19. This is physically

reasonable on the grounds that the first right eigenvector represents the most dominant variation along the Y � direction, and

the perturbation in displacement due to visco-plastic relaxation or damage must be suitably large before its effect can be seen

in this most dominant variation. However a significant change is found in the second cycle as the deformation behaviour

pattern of ΔV2 and ΔV3 are different – i.e. first peak displacement obtained after a quarter cycle, i.e. monotonic deformation

Fig. 5.3 LSV spectrum forΔV fields at ΔV2, ΔV11 and ΔV20 show some change in its singular values due to the relaxation behaviour observed in

solder and local induced damage behaviour

Fig. 5.4 The evolution of first right eigenvector of ΔV field indicates the change in the shape of eigenvector at ΔV2 and also the effect of damage

is clearly observed in the form of change in slope in the damaged region X at ΔV20


whereas the second peak displacement is obtained after a cyclic deformation. As observed in Fig. 5.2, hardening is

manifested in the form of change in eigenvector shape between 1st and 2nd cycles and relaxation behaviour is observed

from 2nd cycle onwards while the relaxation rate increases from 6th cycle. It is to be noted that the initial value of

deformation resistance, s is 7. 1986 MPa and it increases gradually for the first quarter cycle whereas the change in the

deformation resistance is not significant after first quarter cycle through the remaining 19 cycles. Also, at the last cycle, a

change in slope of the eigenvector is observed at the location of added damage as the severity of damage is high. Even

though the relaxation rate changes from 6th cycle onwards, its effect is not observed in the most dominant eigenvector

variation but is observed in a few less dominant eigenvectors.

Figure 5.5 shows the evolution of 7th right eigenvector ofΔV displacement field. It is clear that the eigenvector atΔV11 is

different from its previous cycles, which coincides well with the numerical addition of damage at the same time instant.

Close observation of ΔV20 eigenvector reveals that the change in slope of the eigenvector is found at the region of added

damage i.e from 3. 125 to 3. 5 mm. From the visual examination of the eigenvector plots, five groups can be identified. First

group contains the eigenvector at ΔV2, which indicates the change in the deformation behaviour from 1st cycle to 2nd cycle

wherein hardening is observed. Second group contains eigenvectors from ΔV3 till ΔV10 wherein visco-plastic deformation

dominates. Third group contains eigenvector at ΔV11 wherein damage in the form of stiffness reduction was added into the

numerical model. Fourth group contains eigenvectors from ΔV12 to ΔV18 wherein deformation behaviour includes the

relaxation phenomenon as well as local distortions due to the presence of damage. Fifth group contains ΔV19 and ΔV20

wherein the severity of damage becomes very high and is reflected in the eigenvector shape at the location of damage in the

form of change in slope.

Although we have discussed just the 1st and 7th dominant right eigenvectors of the differential displacement fields, ΔV,the evolution of all dominant eigenvectors needs to be closely examined to identify the presence of any change in the local

deformation behaviour. In order to do this in a systematic way, we define the angle θji as the angle between the jth

eigenvector of ΔVi and ΔVi � 1. This angle is conveniently used as a metric to quantify the change in the jth eigenvector.

If there is no significant change, the angle is close to zero. The evolution of θji for the first ten (j¼1:10) dominant

eigenvectors of ΔV displacement field are shown in Fig. 5.6. As a general trend it can be observed that the angle is initially

high and decreases suddenly, which is due to the fact that the ΔV2 and ΔV3 eigenvectors of ΔV displacement field are very

different due to the initial hardening and subsequent relaxation behaviour. Further, the angle increases at the θj6 when

( j ¼ 5, 6, 8, 9), which physically indicates the change in relaxation behaviour thus agreeing to the load history curve in

Fig. 5.2. Also, angle shows a peak at θj11 and θj

12 at (j ¼ 6, 7) eigenvectors which indicates the change in deformation

behaviour due to the introduction of progressive damage from the end of 10th peak applied displacement. Another general

trend is that all eigenvector angles start to increase in last few cycles which indicates the fact that sufficient damage has

occurred to cause changes in the deformation. The evolution of first ten left eigenvector angles of ΔV shown in Fig. 5.7,

is similar to that observed for right eigenvectors in Fig. 5.6.

Fig. 5.5 The evolution of 7th dominant right eigenvector of ΔV field indicates the change in the shape of eigenvector at ΔV2, ΔV11 and also the

effect of damage is clearly observed in the form of slope change in the damaged region at ΔV20


Corresponding plots of the eigenvector angles for first ten right and left eigenvectors of ΔU field are shown in Figs. 5.8

and 5.9 respectively. Peaks are observed at 6th and 13th cycles within the first five dominant left and right eigenvectors. The

peak found at 6th cycle can be attributed to the change in the relaxation behaviour observed at this cycle. From Figs. 5.6–5.9,

temporal localization of change in deformation behaviour can be made at 2nd, 6th, 11th, 13th cycle and towards the end of

the deformation. Thus, it appears that deformation events can be temporally localized by looking at the principal components

of the deformation fields. Next, we turn our attention to the task of spatial localization of these events.

Fig. 5.6 Evolution of θji (i¼1:20; j¼1:10) for right eigenvectors of ΔV field. Peaks can be observed at (j ¼ 3, 6, 11, 12) and in the final few

cycles of deformation

Fig. 5.7 Evolution of θji (i¼1:20; j¼1:10) for left eigenvectors ofΔV field. Peaks can be observed at ( j ¼ 3, 6, 11, 12) and in the final few cycles

of deformation, similar to that observed in Fig. 5.6


5.3.2 Spatial Localization of Deformation Events

After the time instant where a change in deformation is detected, the next step would be to locate the region of change in

deformation behaviour, if any. To this end, the singular values, left eigenvectors and right eigenvectors of the displacement

fields at 2nd, 6th, 11th or 13th and 20th cycles were calculated as per Eq. 5.4. The eigenvector plots at the above mentioned

cycles were visually examined for any abrupt change in slope. Figures 5.10 and 5.11 shows the right eigenvectors of V

displacement field and left eigenvectors of U displacement field at 2nd, 6th, 11th and 20th cycles respectively. In the

eigenvector plot of 20th cycle, an abrupt change in slope is observed in 5th, 7th and 9th eigenvectors from 3 to 3. 5 mm in

X direction, which is the region where damage was added in X direction. However, an abrupt change in slope is not observed

Fig. 5.8 Evolution of θji (i¼1:20; j¼1:10) for right eigenvectors ofΔU field. Peaks can be observed at ( j ¼ 3, 6, 13) and in the final few cycles of

deformation

Fig. 5.9 Evolution of θji (i¼1:20; j¼1:10) for left eigenvectors of ΔU field


at 6th and 11th cycle indicating the fact that the change in the kinematic deformation behaviour at these cycles is due to a

global phenomenon and not a local one, thus indicating the promise of the present technique in differentiating between

progressive global relaxation and sudden local damage. A abrupt change in slope is observed in 20th cycle from about 1. 9 to

3 mm in Y � direction in lower dominant eigenvectors (Fig. 5.11), which coincides with the region X where damage was

numerically added. As in Fig. 5.10, the eigenvectors at 6th and 13th cycles do not show any abrupt change in slope thus

indicating no damage in a gross sense. A similar trend is observed in the left eigenvectors ofU field and right eigenvectors of

V field and is not shown here for the sake of keeping the discussion brief.

5.4 Discussion

From the numerical simulation of low cycle fatigue and by performing PCA of full-field displacement data, it is seen that the

local deformation behaviour is captured in the eigenvectors obtained from the PCA of the differential displacement fields.

It is evident from Figs. 5.6 and 5.7 that the most dominant right and left eigenvector picks up damage only towards the end

Fig. 5.10 Right 1st, 3rd, 5th, 7th and 9th eigenvectors of V displacement field at 2nd, 6th, 11th and 20th cycles. Twentieth cycle eigenvectors

show an abrupt change in slope in the region of added damage as demarcated by the vertical lines

Fig. 5.11 Left 1st, 3rd, 5th, 7th and 9th eigenvectors of U displacement field at 2nd, 6th, 13th and 20th cycles. Twentieth cycle eigenvectors show

an abrupt change in slope in the region of added damage as demarcated by the vertical lines


of the deformation. However as the dominance of the eigenvector decreases, damage is picked up earlier since any small

deviation in the local deformation behaviour is likely to be reflected in less dominant eigenvectors. Although only a few

eigenvectors have been plotted, a similar trend is observed in lower dominant eigenvectors as well. It is worth noting that no

additional noise has been added to the current synthetic data; however, real experimental data are noisy, and care has to be

exercised in comparing less dominant eigenvectors since they are likely to be more severely affected by noise. This issue

will be investigated in detail in a future study.

Temporal localization of change in deformation is found at 11th cycle from the eigenvector angle plots of ΔV field but is

made at 13th cycle in angle plots of ΔU field. To understand this difference, another finite element analysis was performed

without the addition of damage. PCA of the differential displacement fields were performed and angle plots were used to

check if any change in angle is observed at 11th cycle and 13th cycle. For the sake of conciseness, only the evolution of angle

in right eigenvectors in ΔV field is shown in Fig. 5.12. From the comparison of Figs. 5.12 and 5.6, it is noted that the peaks

at 3rd, 6th, 11th, 12th cycles remain unchanged while the increase in angle towards the end of deformation is not observed in

Fig. 5.12 as no damage is added in this case. This confirms that the increase in angle towards the end of deformation is due to

the accumulation of damage whereas the peaks observed at 2nd, 6th, 11th, 13th cycles are due to the change in relaxation

behaviour.

It is to be highlighted that the visual examination of full-field displacement plots at every cycle does not clearly show the

location of damage even though stress and strain variation is found at the damaged region. This is due to the fact that the

changes due to damage and relaxation are small fractions of the overall deformation, and when the total deformation fields

are considered, these small changes are drowned out. However, these fine variations are revealed as abrupt changes in slopes

in the eigenfunctions of the differential displacement fields, thus enabling the detection of local damage.

5.5 Concluding Remarks

Spatio-temporal PCA of full-field deformation data has been demonstrated on synthetic data generated using finite-element

analysis and it is shown that this technique may be used in the following way:

1. Right eigenvectors of displacement field indicate the spatial variation of displacement in X direction, while left

eigenvectors indicate the spatial variation of displacement in X direction.

2. The temporal localization of change in deformation behaviour can be done by investigating the time evolution of the

eigenvector angles of the differential displacement fields.

Fig. 5.12 Evolution of θji (i¼1:20; j¼1:10) for right eigenvectors of Δ V field without the addition of damage. Peaks can be observed at 3rd, 6th,

11th, 12th cycles due to change in relaxation behaviour


3. Spatial localization of change in deformation behaviour can be performed by closely examining the shape of the

eigenvector plots of the displacement fields for any abrupt change in slope. For the data analysed, the change in slope

occurs consistently at the same location for lower dominant eigenvectors.

References

1. Rao RT (2001) Fundamentals of microsystems packaging. McGraw-Hill Professional, New York

2. Kariya Y, Otsuka M (1998) Mechanical fatigue characteristics of Sn-3.5Ag-X (X ¼ Bi, Cu, Zn and In) solder alloys. J Electron Mater 27

(11):1229–1235

3. Kanchanomai C, Miyashita Y, Mutoh Y (2002) Low cycle fatigue behavior and mechanisms of a eutectic Sn-Pb solder 63Sn/37Pb. Int

J Fatigue 24(6):671–683

4. Chen X, Chen G (2006) Constitutive and damage model for 63Sn37Pb solder under uniaxial and torsional cyclic loading. Int J Solids Struct

43(11–12):3596–3612

5. Aluru K, Wen FL, Shen YL (2011) Direct simulation of fatigue failure in solder joints during cyclic shear. Mater Des 32(4):1940–1947

6. Patki AS, Patterson EA (2011) Decomposing strain maps using Fourier-Zernike shape descriptors. Exp Mech 1:1–13

7. Sebastian C, Patterson E, Ostberg D (2011) Comparison of numerical and experimental strain measurements of a composite panel using image

decomposition. Appl Mech Mater 70:63–68

8. Grama SN, Subramanian SJ (manuscript under review) Computation of full-field strains using principal component analysis

9. Galvanetto U, Violaris G (2007) Numerical investigation of a new damage detection method based on proper orthogonal decomposition.

Mech Syst Signal Process 21(3):1346–1361

10. Lanata F, Del Grosso A (2006) Damage detection and localization for continuous static monitoring of structures using a proper orthogonal

decomposition of signals. Smart Mater Struct 15(6):1811–1829

11. Adams PJ (1984) Thermal fatigue of solder joints in micro-electronic devices. Master’s thesis, Massachusetts Institute of Technology

12. Wang GZ, Cheng ZN, Becker K, Wilde J (2001) Applying Anand model to represent the viscoplastic deformation behavior of solder alloys.

Trans ASME J Electron Packag 123(3):247–253

13. Anand L (1985) Constitutive equations for hot-working of metals. Int J Plast 1(3):213–231

14. Chen X, Chen G, Sakane M (2005) Prediction of stress-strain relationship with an improved anand constitutive model for lead-free solder

Sn-3.5Ag. IEEE Trans Compon Packag Technol 28(1):111–116

15. Golub GH, van Van Loan CF (1996) Matrix computations. The Johns Hopkins University Press, Baltimore

16. Jolliffe IT (2002) Principal component analysis, 2nd edn. Springer, New York


Chapter 6

Master Creep Compliance Curve for Random

Viscoelastic Material Properties

Jutima Simsiriwong, Rani W. Sullivan, and Harry H. Hilton

Abstract The objective of this study is to apply the time-temperature superposition principle (TTSP) to the viscoelastic

material functions that exhibit a large degree of variability to predict the long-term behavior of a vinyl ester polymer

(Derakane 441–400). Short-term tensile creep experiments were conducted at three temperatures below the glass transition

temperature. Strain measurements in the longitudinal and transverse directions were measured simultaneously using the

digital image correlation technique. The creep compliance functions were characterized using the generalized viscoelastic

constitutive equation with a Prony series representation. The Weibull probability density functions (PDFs) of the creep

compliance functions were obtained for each test configuration and found to be time and temperature dependent. Creep

compliance curves at constant probabilities were obtained and used to develop the master curves for a reference temperature

of 24 �C using the TTSP.

Keywords Creep compliance • Prony series • Time-temperature superposition principle (TTSP) • Vinyl ester polymer •

Weibull probability distribution

6.1 Introduction

Polymer-matrix composites (PMCs) have been increasingly adopted for primary and secondary load carrying members due

to their many advantageous properties (lightweight, high strength, high fatigue resistance, etc.). In some of these

applications, the deformations or loads applied to PMC structures are maintained relatively constant throughout their

service life. To accurately analyze or predict the long-term structural integrity of PMC structures, it is necessary to correctly

characterize their time and temperature dependent response, such as creep or stress relaxation, which are mainly governed

by the viscoelastic nature of the polymer matrix [1]. However, the inherent nature of these materials combined with an

inability to manufacture PMCs to high mechanical property specifications results in the usage of materials whose material

functions show very large degrees of scatter. In contrast to metals, for which the elastic properties are ensured to

within � 5 %, viscoelastic material properties, especially in high polymers and composite materials, generally exhibit a

large degree of variability. Such scatter can be on the order of 50–100 % and is normally attributed to insufficient quality

control in the manufacturing process [1]. The stochastic nature of viscoelastic properties, namely compliances or moduli,

has been illustrated extensively in the literature [2–6]; however, the variability in the viscoelastic behavior of polymers is

largely ignored and deterministic approaches are typically used in obtaining material functions. These deterministically

J. Simsiriwong (*) • R.W. Sullivan

Department of Aerospace Engineering, Mississippi State University, 39762 Mississippi, MS, USA


H.H. Hilton

Department of Aerospace Engineering, College of Engineering and Private Sector Program Division,

National Center for Supercomputing Applications, University of Illinois at Urbana-Champaign,

Urbana, IL 61801-2935, USA




41



obtained properties are subsequently used in the design and analysis of PMC structures. It has been shown that ignoring

statistical analysis in formulating viscoelastic material functions leads to false conclusions, thereby jeopardizing safety and

economy [6].

6.2 Experimental

6.2.1 Materials

Since the variability of the time-dependent properties of PMCs is mainly governed by the viscoelastic responses of the

polymer matrix [8], this study focuses on obtaining the statistical distributions of the creep compliance functions of a

neat polymer resin. A thermoset vinyl ester (VE) resin (Derakane 441–400, Ashland Co.) was selected for this study. Vinyl

ester polymers are commonly used as an alternative to epoxy or unsaturated polyester resins because of their low cost,

corrosion resistance, and high mechanical strength [9, 10].

6.2.2 Quasi-static Tensile Creep Tests

Tensile creep tests were performed on the VE polymer specimens according to the test matrix in Table 6.1. The stress level

of 60 % of the ultimate tensile stress, σu, was obtained by performing quasi-static tensile tests to determine the ultimate

strength of the polymer at 24 �C, 40 �C, and 60 �C. All tests were performed below the VE polymer’s glass transition

temperature Tg ¼ 135 �C, which was determined from dynamic mechanical analysis [11].

The specimens were subjected to the constant stress level of 60 % σu for 2 h. A total of ten creep tests at each temperature

(24 �C, 40 �C, and 60 �C) were performed on an INSTRON model 5869 compression/tension electromechanical testing

system with a 50-kN load cell. The higher temperature tests were performed inside an environmental chamber with an

optical quality-viewing window that was mounted to the electromechanical test system. Heating parameters were monitored

using a programmable controller with an accuracy of � 0.5 %.

The digital image correlation technique was used to obtain the strain measurements, through the LaVisonStrainMaster®

system, in the longitudinal and transverse directions simultaneously. The selected camera system is a high resolution 14-bit

charged-coupled device camera that has up to 16 million pixel spatial resolution and a recording capacity of 29 frames per

second. The images of the specimen were taken over the complete duration of the creep tests at a sampling rate of 24 Hz.

Figure 6.1 shows the variability in the experimental data of the VE specimens tested at 60 �C and at 60 % σu.The measured data was subsequently used to determine the creep compliance for each test using the generalized viscoelastic

constitutive equation with a Prony series representation.

6.3 Analytical

6.3.1 Creep Compliances Using a Prony Series Representation

In this study, the exponential Prony series was used to describe the viscoelastic response and material functions. The

Generalized Kelvin model (GKM), comprising of a finite number (1 � n � Npr) of Kelvin elements in series, was chosen as

the viscoelastic mechanical model for creep. The GKM creep compliance functions C11iiðtÞ can be expressed as [12]

Table 6.1 Test matrix for tensile

creep tests of vinyl ester polymer

(Derakane 441–400)Temperature

Stress level

60 % s (MPa)

24 �C 45

40 �C 42.1

60 �C 35.4

42 J. Simsiriwong et al.

C11iiðtÞ ¼ C011ii þ

XNprn¼1

Cn11ii 1� e�

tτn

� �(6.1)

where C011ii denotes the instantaneous creep compliance, Npr is the total number of elements, and Cn

11ii and τn are the Pronyseries coefficients and retardation times, respectively. The underscored indices indicate no summation with the indices

i ¼ 1, 2 representing the Cartesian longitudinal, and transverse directions, respectively. Using a Prony series representation,

the constitutive relation of an isothermal linearly viscoelastic material in the direction of loading x1 can be expressed as [13]

εiiðtÞ ¼ C011iiσ11ðtÞ þ

XNpr

n¼1

Cn11ii

ðt

0

e�t�t0τn@σ11ðt0Þ

@t0dt0

24

35 (6.2)

where εii are the experimental strains and σ11 are the experimental tensile stresses. In Eq. 6.2 the first term describes the

instantaneous elastic response and the second term represents the secondary viscoelastic creep. Additionally, for a complete

viscoelastic response, the starting transients were included in the loading phase and the constant loading was applied for the

steady-state phase by describing the applied tensile stress as

σ11ðtÞ ¼fσðtÞ 0 � t � t1 ðloading phaseÞσ0Hðt� t1Þ t � t1 ðsteady-state phaseÞ

((6.3)

where fσ(t) is the loading function to be selected, t1 is the time when the ramp loading is completed, σ0 is the applied constantstress, and H(t) is the Heaviside unit step function. The loading function fσ(t) was selected to satisfy the physical

interpretation of the actual loading phase, i.e., zero slope dfσdt ¼ 0 at t ¼ 0 and at t ¼ t1 as

fσðtÞ ¼XQq¼0

ϕqtq 0 � t � t1 (6.4)

In Eq. 6.4, ϕq are the constants obtained for times at q ¼ 0, 1, .., Q. Using the complete loading history (Eq. 6.3) and

prescribing the retardation times from the initial time t0 and the final time t1, Eq. 6.2 yields a system of linear equations that

are solved for the Prony series coefficients Cnii11 using the least squares optimization scheme [13]. Once the Prony

Fig. 6.1 Creep strain in (a) longitudinal and (b) transverse directions at 60 �C and 60 % σu

6 Master Creep Compliance Curve for Random Viscoelastic Material Properties 43

coefficients were determined, the analytical creep strain was calculated and compared to the experimental strain data.

A complete description of the determination of the creep compliance functions can be found in Ref. [14].

6.3.2 Statistical Analysis of Creep Compliances

The Weibull distribution was selected to develop the statistical distribution functions using the creep compliance values of

all ten tests at each time, as shown in Fig. 6.2a. The probability density function (PDF), f(x|w), of the Weibull distribution

that identifies the probability of the observed data vector x is given by

f ðxjγ; βÞ ¼ γ

β

x

β

� �γ�1

exp � x

β

� �γ� �0 � x � 1 (6.5)

where γ and β are the shape and scale parameters, respectively. The Weibull parameters of the creep compliances were

estimated using the maximum likelihood method at each time (0 � t � 7,000 s). The PDFs of the creep compliance

functions at the selected times of t ¼ 25 s, 500 s, 3,000 s, and 7,000 s are shown in Fig. 6.2b. As seen, at a constant

temperature and stress level, the probability distribution of the creep compliance functions is highly time-dependent and

shifts to the right for longer times. To further observe the time-dependent behavior of the PDFs, the Weibull parameters were

plotted as functions of time (0 � t � 7,000 s), as shown in Fig. 6.3a, b, and c for 24 �C, 40 �C and 60 �C, respectively. At alltemperatures, both Weibull parameters (γ and β) change significantly for t < 1,000 s, which is a result of the initial large

values of the creep compliances from the starting transients. For longer times (t > 1,000 s), the scale parameter β shows a

much greater time dependency than the shape parameter γ, which is reflected in the gradual increase of the creep compliance

values for all temperatures.

To investigate the effect of temperature on the probability distributions of the creep compliances, the PDF and the

corresponding cumulative density function (CDF) at all times were obtained. As an example, at t ¼ 3,000 s, the Weibull

PDFs and CDFs of the creep compliances for all temperatures are shown in Fig. 6.4a, b, respectively. These figures show that

as the temperature increases, the creep compliances C1111 also increases, as expected.

Fig. 6.2 (a) Creep compliance functions C1111 at 60�C and 60 % σu and (b) the corresponding Weibull PDFs at selected times


Fig. 6.3 Time dependence of the Weibull distribution parameters, γ and β, of the creep compliance functions C1111 at 60 % σu, (a) 24 �C,(b) 40 �C, and (c) 60 �C

Fig. 6.4 Weibull (a) PDF and (b) CDF of the creep compliance functions C1111 at 60 % σu at t ¼ 3,000 s


6.3.3 Creep Compliances Master Curves

The time-temperature-superposition principle (TTSP) was used to predict the long-term creep behavior of the VE polymer

from the short-term creep responses at all elevated temperatures. According to the TTSP, for linear viscoelastic materials,

the long-term creep responses at a reference temperature Tr can be obtained from the creep responses obtained from tests at

increased temperatures T by a change in time scale. This time-temperature equivalence is given by [15, 16]

Cðt; TÞ ¼ btCt

at; Tr

� �(6.6)

where at and bt are the horizontal and vertical temperature shift factors, respectively. From this empirical superposition

approach, the creep responses obtained from tests conducted at a specified stress level and at several temperatures can be

shifted horizontally and/or vertically to a reference temperature to generate a master curve.

To include the statistics of the measured data in the viscoelastic material functions, short-term creep compliance curves at

constant probabilities were obtained for all temperatures. For demonstration, the time-dependent creep compliance for the

probabilities of 0.5 and 0.75 are shown in Fig. 6.5a, b, respectively. Selecting 24 �C as the reference temperature, the creep

compliance curves for 40 �C and 60 �C were shifted horizontally using the reduced time factors at and vertically with shift

factors bt to obtain the master creep curves shown in Fig. 6.5c. The master curves extend the creep compliance function from

the actual test time of 7,000 s for each temperature to 1011 s for the reference temperature of 24 �C.

Fig. 6.5 Short-term creep compliance curves obtained at the selected probabilities of (a) P ¼ 0.5 and (b) P ¼ 0.75, and (c) the corresponding

master curves for a reference temperature of 24 �C


6.4 Conclusions

In this study, the variability in the creep response of a neat VE resin (Derakane 441–400) was included by formulating the

statistical distributions of the creep compliance functions using a 2-parameter Weibull distribution. The longitudinal and

transverse creep strains of the neat VE polymer were obtained experimentally from short-term creep tests at three

temperatures at 60 % σu using the digital image correlation technique. The measured data was subsequently used to

determine the creep compliances for each test configuration using the generalized viscoelastic constitutive equation with

the GKM Prony series representation. The statistical analyses were performed on the creep compliance functions and the

Weibull CDFs were obtained at each temperature. Creep compliance curves at constant probabilities were obtained from the

CDFs and used to develop the master curves for a reference temperature of 24 �C using TTSP with both horizontal and

vertical factors. The resulting creep compliance master curves include the statistical distribution of the experimental

viscoelastic strain response.

References

1. Beldica CE, Hilton HH (1999) Analytical and computational simulations of experimental determinations of deterministic and random linear

viscoelastic constitutive relations. In: Twelfth international conference on composite materials, Paris

2. Gnip IY, Vaitkus S, Kersulis V, Vejelis S (2011) Analytical description of the creep of expanded polystyrene (EPS) under long-term

compressive loading. Polym Test 30(5):493–500

3. Gnip IY, Vaitkus S, Kersulis V, Vejelis S (2010) Experiments for the long-term prediction of creep strain of expanded polystyrene under

compressive stress. Polym Test 29(6):693–700

4. Gnip IY, Vaitkus S, Kersulis V, Vejelis S (2008) Long-term prediction of compressive creep development in expanded polystyrene. Polym

Test 27(3):378–391

5. Barbero EJ, Julius MJ (2004) Time-temperature-age viscoelastic behavior of commercial polymer blends and felt filled polymers. Mech Adv

Mater Struct 11(3):287–300

6. Schwarzl FR, Zahradnik F (1980) The time temperature position of the glass-rubber transition of amorphous polymers and the free volume.

Rheol Acta 19(2):137–152

7. Hilton HH, Hsu J, Kirby JS (1991) Linear viscoelastic analysis with random material properties. Probab Eng Mech 6(2):57–69

8. Sullivan JL (1990) Creep and physical aging of composites. Compos Sci Technol 39(3):207–232

9. Liao K, Altkorn R, Milkovich S, Fildes J, Gomez J, Schultheisz C, Hunston D, Brinson L (1997) Long-term durability of glass-fiber reinforced

composites in infrastructure applications. J Adv Mater 28(3):54–63

10. Harper CA (2002) Handbook of plastics, elastomers, and composites. McGraw-Hill Professional, New York

11. Nouranian S (2011) Vapor-grown carbon nanofiber/vinyl ester nanocomposites: designed experimental study of mechanical properties and

molecular dynamics simulations, Department of Chemical Engineering. Vol. Ph.D. Dissertation, Mississippi State University, Mississippi

State, MS

12. Christensen RM (1982) Theory of viscoelasticity: an introduction. Dover, NY

13. Michaeli M, Shtark A, Grosbein H, Steevens AJ, Hilton HH (2011) Analytical, experimental and computational viscoelastic material

characterizations absent Poisson’s ratios. 52nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference.

Denver

14. Simsiriwong J, Sullivan RW, Hilton HH, Drake D (2012) Statistical analysis of viscoelastic creep compliance of vinyl ester resin. 53th AIAA/

ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, American Institute of Aeronautics and Astronautics Inc.,

Waikiki, 23–26 Apr 2012

15. Ferry J (1980) Viscoelasticity properties of polymers. Wiley, New York

16. Shaw MT, MacKnight WJ (2005) Introduction to polymer viscoelasticity. Wiley-Interscience, Hoboken


Chapter 7

Processability and Mechanical Properties of Polyoxymethylene

in Powder Injection Molding

J. Gonzalez-Gutierrez, P. Oblak, B.S. von Bernstorff, and I. Emri

Abstract Polyoxymethylene (POM) is considered a high performance engineering polymer with many applications

primarily in the automotive industry. Currently, POM has also found uses in powder injection molding (PIM) technology,

where it acts as a carrier medium for metal or ceramic powders during the injection molding process, it is later removed and a

metallic or ceramic piece is obtained after sintering. The main advantage of using POM in PIM technology is the faster

debinding process compare to polyolefin-based feedstock, since POM sublimates into its monomer directly when exposed to

an acid vapor. During the process of PIM, the binder has two contradictory requirements: viscosity should be as low as

possible when in the molten state, but mechanical properties in the solid state, like toughness, should be as high as possible.

One way to lower the viscosity is to use POM with lower molecular weights. In this work it has been observed that the

viscosity follows a power law function as with other linear polymers, while the fracture toughness follows an exponential

function of the average molecular weight. Therefore, a molecular weight can be chosen in a way that a compromise between

low enough viscosity and sufficient fracture toughness can be reached.

Keywords Molecular weight • Polyoxymethylene • Powder injection molding • Toughness • Viscosity

7.1 Introduction

Polyoxymethylene (POM) is a high–molecular weight engineering polymer of formaldehyde with hydroxyl ends stabilized

by esterification or etherification, sometimes also referred as polyacetal or less commonly as aldehyde resins [1]. POM is

distinguished from other engineer polymers in its crystallinity level that can be between 60 % and 90 % [2, 3]; such

high crystallinity induces very good mechanical properties such as high modulus, stiffness, fatigue, creep resistance and

hardness [3, 4]. These properties allow the use of POM as a structural material in many different applications [5]. The use of

POM is growing steadily in the automotive and electronics industry and it is supposed to replace metals in pressure loaded

parts, such as window regulator or gear wheels and pinions [6].

Nowadays, POM has also found uses in powder injection molding (PIM) and micro powder injection molding (μPIM),

which are versatile mass production methods for small complex shaped components of metal or ceramic [7–10]. In PIM,

POM acts as carrier medium for metal or ceramic powders during the injection molding process and it is later removed to

obtain a metallic or ceramic piece after sintering. The main advantage of POM in powder injection molding comes from the

instability of acetal linkages which results in rapid hydrolysis. Since the main chain of POM is composed of –CH2-O- bonds, the

methyl-oxygen bonds are easy to break under heat and oxygen [11], the process gets accelerated in the presence of acid vapors,

J. Gonzalez-Gutierrez (*) • P. Oblak

Center for Experimental Mechanics, Faculty of Mechanical Engineering, University of Ljubljana, Pot za Brdom 104, Ljubljana, Slovenia


B.S. von Bernstorff

BASF Aktiengesellschaft, Ludwigshafen, Germany

I. Emri

Center for Experimental Mechanics, Faculty of Mechanical Engineering, University of Ljubljana, Pot za Brdom 104, Ljubljana, Slovenia

Institute for Sustainable Innovative Technologies, Pot za Brdom 104, Ljubljana, Slovenia



49


this breakage results in a continuous depolymerization reaction yielding formaldehyde. The released formaldehyde and

formic acid formed through oxidation of formaldehyde can accelerate the depolymerization reaction; this process is usually

called the zipper mechanism [12]. This unzipping process represents a major advantage during the debinding process and it

is generally called catalytic debinding [13]. Catalytic debinding occurs at a significant higher rate than other dedinding

techniques, such as solvent and thermal, and greatly speeds up the PIM process [14]. Additionally and as previously

mentioned, POM in the solid state has good mechanical properties and thus makes for easier handling of molded parts, which

with other binders can be fragile or easily deformable.

The feedstock material used in PIM has two main contradictory requirements; first, the feedstock should have low

viscosity at the molding temperatures (190–210 �C), and second, it should have good mechanical (e.g. high impact

toughness) properties in the solid state (>160 �C) before debinding. Currently available POM-based feedstock materials

fulfill the second requirement very well; however, the first condition, which is related to processability, is partially not meet

since neat POM has much higher viscosity than other binders based on polyolefins [15]. It has been suggested that the binder

should have a viscosity lower than 10 Pa s at a shear rate of 100 s�1 [16]; however, currently available POM-based binder has

a viscosity around 200 Pa s at the specified shear rate. One way to lower the viscosity of polymers is to lower their molecular

weight [17–19], thus in an effort to decrease the viscosity of binders used in PIM, POM materials with distinct molecular

weights have been synthesized and their viscosity and toughness have been investigated.

It is well known that the viscosity of polymeric systems is greatly influenced by their molecular weight [17], and it is also

known that the toughness of polymers is also dependent, among other things, on the molecular weight [18]. It has been

observed in a variety of polymers that both viscosity and toughness increase with molecular weight [19–26], but they do not

increase in a similar manner. Therefore, the goal of this paper is to determine the maximum molecular weight of POM that

will provide adequate viscosity (<10 Pa s) without compromising its toughness. The viscosity which is directly linked to the

processability of the feedstock in the injection molding machine, while the toughness is linked to the mechanical strength of

the molded parts that can influence the way these parts are handled before sintering.

7.2 Materials and Methods

7.2.1 Materials

For this investigation 9 POM copolymers with different average molecular weights were synthesized by BASF

(Ludwigshafen, Germany). The nomenclature and average molecular weight of all the POM materials used in this study

is shown in Table 7.1. Molecular weights were measured by the supplier using gel permeation chromatography (GPC).

7.2.2 Viscosity Measurements

Viscosity measurements in oscillatory mode were performed in a Haake MARS-II rotational rheometer (Thermo Scientific,

Germany). Frequency sweep tests were performed at 190 �C, using a truncated cone-plate geometry (diameter ¼ 20 mm,

gap ¼ 0.054 mm). Frequency was increased from 0.01 (0.0628) to 100 Hz (628.32 rad/s) in 25 increments equally spaced in

the logarithmic scale. All measurements were performed applying a shear stress of 100 Pa, previously determined to be

within the linear viscoelastic region of all materials. All viscosity measurements were performed six times per material.

Table 7.1 Average molecular

weight of POM copolymersCopolymer ID Average molecular weight, Mw, [g/mol]

MW0 10,240

MW1 24,410

MW2 36,340

MW3 60,500

MW4 81,100

MW5 92,360

MW6 109,000

MW7 129,300

MW8 204,400

50 J. Gonzalez-Gutierrez et al.

In this study, viscosity results are presented as the magnitude of the complex vicosity (|η*|), which is related to the constant

rotational viscosity (η) through the Cox-Merz rule [27], which has been previously determined to apply to POM.

7.2.3 Impact Toughness Measurements

Charpy tests were performed at room temperature in order to measure the impact toughness of all POM copolymers selected

for this investigation. Non-notched cylindrical specimens were prepared via twin screw extrusion in a PolyLab Haake OS

(Thermo Scientific, Germany). A glass tube (external diameter ¼ 9 mm, internal diameter ¼ 6 mm and length ¼ 200 mm)

was placed at the end of the extrusion die and filled with the extrudate up to a minimum length of 80 mm. The melt

temperature at the die was measured to be 190 �C. After extrusion, extrudates were left to cool to room temperature inside

the glass tube for at least 4 h before performing the impact tests. All measurements were repeated six times.


7.3.1 Viscosity

The magnitude of the complex viscosity as a function of angular frequency for all the different POM materials is shown in

Fig. 7.1. It is clear that asMw increases so does the viscosity, also it can be seen, that almost all of the materials investigated

display Newtonian behavior in the frequency range investigated. Only the material with the highest molecular weight

(MW8) shows a clear deviation from Newtonian to shear thinning behavior starting at approximately 6 rad/s. This is not

unexpected, since as the molecular weight increases, it is expected that the level of entanglement increases and the amount of

free volume decreases, which reduce the chain mobility and as a consequence increases the viscosity [21]. However, as the

frequency of excitation or shear rate increases these entanglements break and the viscosity starts to drop, i.e. shear thinning.

Polymers with higher Mw have a higher number of entanglements and therefore can be more susceptible to shear leading to

an onset of shear thinning at lower frequencies, as observed in Fig. 7.1.

0.01

0.1

1

10

100

1000

10000

0.01 0.1 1 10 100 1000 10000

Mag

nitu

de o

f the

Com

plex

Vis

cosi

ty [P

a s]

Angular Frequency [rad/s]

T = 190 �C

MW0 MW1MW2 MW3MW4 MW5MW6 MW7MW8

τ = 100 Pa

Fig. 7.1 Viscosity dependence on angular frequency and molecular weight for polyoxymethylene at temperature T of 190 �C with an applied

stress τ of 100 Pa

7 Processability and Mechanical Properties of Polyoxymethylene in Powder Injection Molding 51

Polyoxymethylene can be classified as a linear entangled polymer and it is well known that for this type of polymers the

shear Newtonian viscosity, η0 and the average molecular weight, Mw are related by a power law function of the form

proposed by Fox and Flory [19]:

η0 ¼ KMaw (7.1)

where the K parameter quantifies the temperature and pressure dependence of the Newtonian viscosity of molten polymers,

and a is related to the level of entanglement of the polymers. Figure 7.2 shows that the above equation applies also for POM.

The value of a has been reported for several polymers to be between 3.3 and 3.7 whenMw > Mc and a � 1 whenMw < Mc,

whereMc is a critical average molecular weight [20–22]. BelowMc the flow units are single macromolecules while aboveMc

the flow units are chain segments since the macromolecules are entangled [20]. As can be seen in Fig. 7.2, all the POM

materials investigated appear to be above the critical molecular weight, since the value of a is approximately 3.7; this was

expected since it has been estimated in the literature that the molecular weight for entanglement Me of POM is 3,100 g/mol

[28] and it is generally believed that Mc is between two and three times larger than Me [20–22]. In this particular study the

lowest molecular weight available is around 10,000 g/mol, which is more than three times the estimated molecular weight

for entanglement, Me.

With respect to the viscosity required for PIM (< 10 Pa s), it appears that one could select a POM material with an

average molecular below or equal to 36,340 g/mol, i.e. MW0, MW1 and MW2. However, the decision cannot be taken

without considering the solid mechanical properties of the polymer, in particular the impact toughness of the material, since

it is desirable that the molded part exhibits good toughness in order to be easily handled after injection molding without

fracturing.

7.3.2 Impact Toughness

It is known that the impact toughness of polymeric materials is highly dependent on the molecular weight. When the

molecular weight of polymers is increased, the mechanical response goes from brittle to ductile [29], i.e. the toughness

increases with molecular weight [23, 26]. For semi-crystalline polymers, this increase has been attributed to an increase in

density of inter-lamellar tie chains and chain entanglements, which give higher craze fibril strength and, hence, a higher

energy for fracture initiation is required [23]. Figure 7.3 shows that for POM, a similar behavior has been observed, as the

molecular weight increases the impact toughness increases: in the range between 10,240 and 36,340 g/mol the increase

is very small and it appears that a plateau is present between 24,410 and 36,340 g/mol; as the Mw increases beyond

y = 6E-17x3.7076

R2 = 0.9964

0.01

0.1

1

10

100

1000

10000

1000 10000 100000 1000000

New

toni

an V

isco

sity

[Pa

s]

Average Molecular Weight [g/mol]

Power law fit

T = 190 �Cτ =100 Pa

Fig. 7.2 Power law dependence of Newtonian shear viscosity and average molecular weight of POM at temperature T of 190 �C with an applied

stress τ of 100 Pa


36,340 g/mol the increases in toughness is very pronounced; and finally at molecular weights larger than 129,300 g/mol the

increases in toughness levels off. Similar behavior has been reported in other polymers with respect to their mechanical

strength [30]. It has also been reported that as the molecular weight increases beyond a very large molecular weight a

decrease in the fracture toughness can be observed as in the case of ultrahigh molecular weight polyethylene, thus toughness

is a non-monotonical function of molecular weight with a maximum [31]. In this particular case, the maximum was not

reached in the range of molecular weights investigated.

In order to select the POMmaterial to be used as part of the PIM binder, it is important to take into account the viscosity of

the material as well as its toughness. The viscosity should be as low as possible to allow easy molding, while the toughness

should be as high as possible to prevent damage to the molded part before sintering. As it can be seen in Fig. 7.4, the viscosity

increases much more rapidly than the toughness; viscosity increases approximately 6 orders of magnitude, while at the same

time the toughness increases only 3 orders of magnitude. Figure 7.4 also shows that the dependence of toughness (absorbed

energy by Charpy) with viscosity follows a similar shape as its dependence with average molecular weight (Fig. 7.3),

showing a plateau at the viscosity values between 1 and 3 Pa s, which correspond to an average molecular weight between

24,410 and 36,340 g/mol (MW1 and MW2); therefore by looking at these results it can be suggested that POMMW1 should

be used as the main component for the binder since it has three times lower viscosity than POMMW2, but the same level of

toughness. It is important to mention that the POM currently used as binder for PIM feedstock has a similar molecular weight

to MW5, thus if we select MW1 as the new binder we can expect a decrease in viscosity of almost 200 times, while a

decrease in toughness of approximately 10 times, which can be considered a significant improvement.

7.4 Conclusions

POM used as a binder for powder injection molding (PIM) has the major advantages that it can undergo catalytic debinding

which is much faster than other debinding processes and that the molded part has good mechanical strength (i.e. high impact

toughness). However, currently used catalytic binder has high viscosity that can bring difficulties to the injection molding

process. In this investigation the viscosity and toughness of different POM copolymers has been studied.

0.01

0.1

1

10

1000 10000 100000 1000000

Abs

orbe

d E

nerg

y by

Cha

rpy

[J]

Average Molecular Weight [g/mol]

T = RoomNon - notched specimenCylindrical specimen ≈ 6 mm

Fig. 7.3 Toughness dependence with average molecular weight for POM copolymers at room temperature on non-notched cylindrical specimens

with a diameter of 6 mm


It has been observed that both properties increase as the average molecular weight Mw increases. However, the viscosity

increases much more rapidly than impact toughness. Viscosity increases with molecular weight as a power law function,

with an exponent a ~ 3.7, as it has been reported for other polymers [20–22]. Therefore, for an increase in Mw of

approximately 20 times there is a viscosity increase of almost 50,000 times. The impact toughness, measured by Charpy

tests, increases approximately 150 times as theMw increased from 10,240 to 204,400 g/mol. The increase in toughness does

not follow a simple relationship with molecular weight and it appears that there is a plateau at small molecular weights.

With the information here gathered, it possible to suggest that a POM copolymer with an average molecular weight of

around 24,000 g/mol could be used as the main component of a binder used in PIM. As compared to the currently available

binder, using POM with the suggestedMw can lead to a decrease in viscosity of 200 times, while reducing toughness only by

10 times; this can be considered a significant improvement on the performance of POM-based binders for PIM.

References

1. Zhao R (2005) Melt blowing polyoxymethylene copolymer. Int Nonwovens J 14(2):20–24

2. Edidin AA, Kurtz SM (2000) Influence of mechanical behavior on the wear of 4 clinically relevant polymeric biomaterials in a hip simulator.

J Arthroplasty 15(3):321–331

3. Jauffres D, Lame O, Virgier G, Dore F, Chervin C (2007) Mechanical and physical characterization of polyoxymethylene processed by high-

velocity compaction. J Appl Polym Sci 106:488–497

4. Al Jebawi K, Sixou B, Seguela R, Vigier G (2007) Hot compaction of polyoxymethylene. II. Structural characterization. J Appl Polym Sci

106:757–764

5. Dziadur W (2001) The effect of some elastomers on the structure and mechanical properties of polyoxymethylene. Mater Charact 46:131–135

6. Luftl S, Archodoulaki VM, Glantschnig M, Seidler S (2007) Influence of coloration on initial material properties and on thermooxidative

ageing of a polyoxymethylene copolymer. J Mater Sci 42:1351–1359

7. Attia UM, Alcock JR (2012) Fabrication of hollow, 3D, micro-scale metallic structures by micro-powder injection moulding. J Mater Process

Technol 212:2148–2153

8. Schneider J, Iwanek H, Zum Gahr KH (2005) Wear behaviour of mould inserts used in micro powder injection moulding of ceramics and

metals. Wear 259:1290–1298

9. Krug S, Evans JRG, ter Maat JRR (2002) Differential sintering in ceramic injection moulding: particle orientation effects. J Eur Ceram Soc

22(2):173–181

10. Krug S, Evans JRG, ter Maat JRR (2001) Effect of polymer crystallinity on morphology in ceramic injection molding. J Am Ceram Soc

84(12):2750–2766

0.01

0.1

1

10

0.01 0.1 1 10 100 1000 10000

Abs

orbe

d E

nerg

y by

Cha

rpy

at R

oom

Tem

p.&

6m

m D

iam

eter

Sam

ples

[J]

Newtonian Viscosity at 190 �C [Pa s]

Fig. 7.4 Impact toughness measured by Charpy tests as a function shear Newtonian viscosity for POM materials


11. Zhao X, Ye L, Hu Y (2008) Synthesis of melamine-formaldehyde polycondensates as the thermal stabilizer of polyoxymethylene through

ultrasonic irradiation. Polym Adv Technol 19:399–408

12. Pielichowska K (2012) The influence of molecular weight on the properties of polyacetal/hydroxyapatite nanocomposites. Part 1.

Microstructural analysis and phase transition studies. J Polym Res 19:9788–9798

13. Krug S, Evans JRG, ter Maat JRR (2001) Transient effects during catalytic binder removal in ceramic injection moulding. J Eur Ceram Soc

21:2275–2283

14. Stringari GB, Zupancic B, Kubyshkina G, von Bernstorff B, Emri I (2011) Time-dependent properties of bimodal POM – Application in

powder injection molding. Powder Technol 208(3):590–595

15. Gonzalez-Gutierrez J, Stringari GB, Zupancic B, Kubyshkina G, von Bernstorff B, Emri I (2012) . Time-dependent properties of multimodal

polyoxymethylene based binder for powder injection molding. J Solid Mech Mater Eng 6(6):419–430

16. Mutsuddy BC, Ford RG (1995) Ceramic injection molding. Chapman & Hall, London, pp 1–26

17. Nichetti D, Manas-Zloczower I (1998) Viscosity model for polydisperse polymer melts. J Rheol 42(4):951–969

18. Kanai H, Sullivan V, Auerbach A (1994) Impact modification of engineering thermoplastics. J Appl Polym Sci 53:527–541

19. Fox TG, Flory PJ (1951) Further studies on the melt viscosity of polyisobutylene. J Phys Colloid Chem 55:221–228

20. Colby RH, Fetters LJ, Graessley WW (1987) Melt viscosity-molecular weight relationship for linear polymers. Macromolecules

20:2226–2237

21. Grosvenor MP, Staniforth JN (1996) The effect of molecular weight on the rheological and tensile properties of poly(ε-caprolactone). Int JPharm 135:103–109

22. Ajroldi G, Marchionni G, Pezzin G (1999) The viscosity-molecular weight relationships for diolic perfluropolyethers. Polymer 40:4163–4164

23. Andreassen E, Nord-Varhaug K, Hinrischen EL, Persson AM (2007) Impact fracture toughness of polyethylene materials for injection

moulding. Extended abstracts for PPS07EA, Gothenburg

24. Cazenave J, Seguela R, Sixou B, Germain Y (2006) Short-term mechanical structural approaches for the evaluation of polyethylene stress

crack resistance. Polymer 47:3904–3914

25. Cho K, Lee D, Park CE, Huh W (1996) Effect of molecular weight between crosslinks on fracture behavior of diallylterephthalate resins.

Polymer 37(5):813–817

26. Garlotta D (2001) A literature review of poly(lactic acid). J Polym Env 9(2):63–84

27. Cox WP, Merz EH (1958) Correlation of dynamic and steady flow viscosities. J Polym Sci 28:619–622

28. Plummer CJG, Cudre-Mauroux N, Kausch HH (1994) Deformation and entanglement in semicrystalline polymers. Polym Eng Sci

34(4):318–329

29. Han Y, Gang XZ, Zuo XX (2010) The influence of molecular weight on properties of melt-processable copolyimides derived from

thioetherdiphthalic anhydride isomers. J Mater Sci 45:1921–1929

30. Balta Calleja FJ, Flores A, Michler GH (2004) Microindentation studies at the near surface of glassy polymers: influence of molecular weight.

J Appl Polym Sci 93:1951–1956

31. Nakayama K, Furumiya A, Okamoto T, Yagi K, Kaito A, Choe CR, Wu L, Zhang G, Xiu L, Liu D, Masuda T, Nakajima A (1991) Structure

and mechanical properties of ultra-high molecular weight polyethylene deformed near melting temperature. Pure Appl Chem 63(12):

1793–1804


Chapter 8

Constitutive Response of Electronics Materials

Ryan D. Lowe, Jacob C. Dodson, Jason R. Foley, Christopher S. Mougeotte,

David W. Geissler, and Jennifer A. Cordes

Abstract Electronics in mission- or safety-critical systems are expected to survive a wide range of harsh environments

including thermal cycling, thermal ageing, vibration, shock, and combinations of the aforementioned stresses. The materials

used in these electronic systems are diverse and frequently change as the electronics industry rapidly innovates. These

materials are dual use, fulfilling both electrical and mechanical functions. Of particular interest are electronic materials

classes such as polymers (e.g., encapsulants/potting and packaging), composites (e.g., hard potting and printed circuit

boards), and interconnect materials (e.g., solder). Thus, predicting the operational response of electronics systems in harsh

environments requires understanding of the materials constitutive response to the environmental characteristics for all the

relevant materials. The paper estimates the rate-, temperature-, and pressure-dependent constitutive response of representa-

tive electronic materials. Experimental response of circuit boards, potting materials, and solder interconnects are measured

in low and intermediate strain rate dynamic tests. Traditional mechanical sensors (e.g. strain gages and accelerometers) are

complemented by non-contact techniques (e.g., laser velocimetery, high speed digital image correlation) to obtain high

fidelity experimental data on material response. Estimates of the corresponding constitutive parameters are calculated,

and observed features of the dynamic response are discussed.

Keywords Composites • Constitutive model • Dynamic testing • Electronic materials • Strain rate-dependent materials

Nomenclature

ρ Density

σ Stress

ε; _ε Strain strain rate

γ; _γ Shear strain shear strain rate

τ; _τ Shear stress shear stress rate

g Acceleration (due to gravity)

kB Boltzmann’s constant

Cp Heat capacity

E Elastic modulus

P Pressure

T Temperature

R.D. Lowe (*) • J.C. Dodson • J.R. Foley

U.S. Air Force Research Laboratory AFRL/RWMF, 306 W. Eglin Blvd., Bldg. 432, Eglin AFB, FL 32542-5430, USA


C.S. Mougeotte • D.W. Geissler • J.A. Cordes

U.S. Army Armament Research, Development, and Engineering Command



57


8.1 Introduction

In the worst case scenario, dropping a cell phone from ear level can result in a local acceleration of over 1,500 g on an

internal circuit board. The consequence of failure in consumer electronics is low, since most systems are expected to be

replaced after a couple years of use (or after an unintentional drop event). On the other hand, electronics in mission- or

safety-critical applications are expected to reliably function in a wide range of harsh environments over a lifespan measured

in decades. Safety-critical applications include aviation (flight computers and engine controllers), energy (nuclear power),

and medical (life support equipment); mission-critical applications include electronics for military and space flight. While

each of these applications has a unique combination of operational environments, they are generally much harsher in one or

more specific environments (such as thermal cycling and aging, vibration, and shock [1]) relative to commercial

applications.

The commercial electronics industry dominates the global market [2], so electronics for these applications are almost

entirely fabricated from commercial off-the-shelf (COTS) components due to the often prohibitive cost of developing

application-specific integrated circuits (ASICs). Electronics system designers instead rely on design and testing to improve

the robustness of electronics in these mission- and/or safety-critical applications with harsh environments. As an example,

the Joint Electron Devices Engineering Council (JEDEC) has developed board-level test standards for evaluating

components under unintentional shock loads experienced in commercial applications. One such standard for a drop shock

uses a 1,500 g half-sine pulse with 0.5 ms duration [3, 4] on standardized board configurations. Some recent studies that have

considered various aspects of drop shock testing are summarized below in Table 8.1. Several different aspects of the board-

level electronics environment and the associated tools have been investigated. A few trends are readily apparent in previous

work. FEA is a nearly ubiquitous tool used to model the dynamic response of electronics and infer damage mechanisms.

Secondly, dynamic testing and characterization experiments, such as modal analysis, are increasingly common for providing

experimental validation of these simulations. Finally, the need for material properties to support both of these approaches

has led to a focus on this area, even for the more benign requirements of commercial applications. However, there is little

evidence that existing models have been validated as predictive tools for the survivability of electronics in extreme

environments.

8.2 Materials and Constitutive Models

In order to evaluate and model the electronics performance in the extreme environments of these safety- and/or mission-

critical applications, the components and materials must be carefully evaluated for their multifunctional performance since

they simultaneously fulfill both electrical and mechanical functions. This is a significant challenge in complex electronics

systems since each discrete electronic component or assembly has a different function and correspondingly unique materials,

physics of operation, and failure modes [38]. Electronic materials (and application) can be categorized into different classes

of materials such as polymers (encapsulants/potting, packaging), ceramics (packaging), semiconductors (die), composites

(encapsulants/potting, printed circuit boards), and metallic interconnects (traces, solder). In order to predict the in situ

response of electronics systems in harsh operational environments, an understanding of these diverse materials’ response to a

wide range of environmental characteristics is required. This is typically accomplished by developing a mechanical

constitutive model to predict the material’s deformation as a function of pressure, stress, loading rate, and temperature.

Examples and commonly-used models are briefly discussed for these material classes below.

8.2.1 Solders

Solder interconnects provide the simultaneous mechanical and electrical connection between micro-scale integrated circuits

and the macro-scale printed circuit board. Solder interconnects have long been known to be a leading reliability concern for

circuit boards [39–42]. A crack through any of the tens of thousands of solder interconnects on a typical printed circuit board

(e.g., a computer motherboard) can render the entire board inoperable. The recent transition to lead-free electronics was

driven by the Reduction of Hazardous Substances (RoHS) declaration of the European Union in 2003 [43]. This has created

a large number of new solder compositions, the RoHS compliant Pb-free solder family, in addition to the traditional tin-lead

58 R.D. Lowe et al.

Table

8.1

Recentexam

plesofdropshock

testsandrelatedstudiesoncircuitboards

Lead

author

Year

Material

properties

Actual

versus

simulated

mass

Displacementor

strain

time

History

Qualitative

modal

(e.g.,shapes)

Quantitative

modal

(e.g.freqs.)

Qualitative

failure

location

FEA

error

(Experim

ent

versusFEA)

FEA

meshsize

validation

FEA

efficiency

Correlation

withlife

data

(Weibull)

References

Mishiro

2002

X[5]

Tee

2004

XX

XX

[6]

Tan

2005

XX

X[7]

Luan

2006

X[8]

Lall

2006

XX

X[8]

Wang

2006

XX

XX

[10]

Chong

2006

X[11]

Wong

2006

XX

[12]

Yeh

2006

X[13]

Lall

2007

XX

[14]

Jenq

2007

XX

X[15]

Syed

2007

[16]

Qu

2007

X[17]

Wong

2008

X[18,19]

Xu

2008

X[20]

Zhang

2008

XX

X[21]

Lee

2008

XX

XX

[22]

Long

2008

X[23]

Chou

2008

XX

[24]

Lall

2009

XX

XX

XX

[25,26]

Liu

2009

XX

X[27]

Yu

2009

X[28]

Wong

2009

XX

X[29]

Yu

2010

XX

XX

[30]

Amy

2010

XX

XX

X[31,32]

Nguyen

2011

XX

[33]

LeCoq

2011

XX

[34]

An

2011

XX

[35,36]

Anuar

2012

XX

[37]

8 Constitutive Response of Electronics Materials 59

(SnPb) solder family. These are shown below in Table 8.2 with red (SnPb) and green (Pb-free) shading, respectively.

Alloy names, composition, and constitutive models used for a selection of solders are also given.

Solders have been characterized over a wide range of conditions, including temperature and strain rates, from quasistatic

[44] to 3,000 s�1 [48]. Several constitutive models, as noted in Table 8.2, have been applied to capture the rate- and

temperature-dependent properties of solder. Among these, the Johnson-Cook and Anand models are the most common.

8.2.1.1 Johnson-Cook Model

The Johnson-Cook (J-C) model for the rate- and temperature-dependent response of metals [49] has been implemented for

solders [23]. The J-C model for flow stress in a material is given by

σe ¼ Aþ B εpe� �n� �

1þ C ln _ε�½ � 1� T�m½ �; (8.1)

where σe is the von Mises flow stress, εpe is the equivalent plastic strain, _ε� is the equivalent plastic strain (normalized to a

reference strain rate, i.e., _ε� ¼ _εpe= _ε0, where _ε0 is a reference strain rate, typically 1 s�1), T� is a the homologous temperature

(normalized to a reference temperature T0, i.e., T� ¼ T � T0ð Þ Tm � T0ð Þ= where Tm is the melting temperature), and A, B, C,

n, and m, are constants as described in Table 8.3.

The Johnson-Cook model is readily implemented in many analytic codes due to its simple form. The properties of solder

can be estimated from quasistatic or dynamic compression or tension experiments. Figure 8.1 shows the J-C model

prediction for the stress–strain relationship for three solders (Sn37Pb63, Sn96.5Ag3.5, and SAC305) using constants from

Qin et al. (see reference [44]) at a strain rate of _ε ¼ 0.001 s�1. While all three of the solder types have similar yield stress,

the post-yield strength varies significantly.

Table 8.2 Composition, nomenclature, and constitutive models used for solders

Alloy

Composition (%) Constitutive Model(s)[Reference(s)]Sn Pb Ag Cu Bi

SnPb 63 37 Johnson-Cook [44]

Sn62Pb36Ag2 62 36 2

Sn60Pb40 60 40

92.5Pb5Sn2.5Ag 92.5 5 2.5

SAC105 98.5 1 0.5

SAC205 97.5 2 0.5

SAC305 96.5 3 0.5Johnson-Cook [44]

Anand, Ramberg-Osgood [45, 46]

SAC405 95.5 4 0.5

SAC0307X 98.9 0.3 0.7 0.1

Sn -3.5Ag 96.5 3.5Johnson -Cook [ 44 ]

Anand [ 47 ]

Sn2.5Ag0.5Cu 97 2.5 0.5 Johnson -Cook [ 23 ]

Table 8.3 Parameters in the

Johnson-Cook constitutive modelSymbol Parameter

σe Effective flow stress

εpe Equivalent plastic strain

_ε� Effective plastic strain rate (normalized)

n Strain hardening exponent

m Temperature softening exponent

A Yield stress

B Strain hardening coefficient

C Strain rate coefficient

T� Homologous temperature (normalized)

60 R.D. Lowe et al.

The rate dependence of the Johnson-Cook model is illustrated in Fig. 8.2 below for Sn37Pb63 solder. The yield stress

increases with strain rate, which is modeled [44] using Eq. 8.1, i.e.,

σy ¼ A 1þ C ln _ε�ð Þ; (8.2)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

20

40

60

80

100

120

Solder stress-strain curves, J-C model

Strain, e [ ]

Str

ess,

σ(e

) [M

Pa]

Sn37Pb63Sn96.5Ag3.5SAC305

Fig. 8.1 Stress–strain plot for three common solder alloys: Sn37Pb63, Sn96.5Ag3.5, and SAC305. All three curves are at quasistatic strain rates

( _ε � 10�3 s�1)

Fig. 8.2 Stress–strain-strain rate surface predicted using the Johnson-Cook constitutive model and constants from [44]


where A and C re the same constants as Eq. 8.1. The significant increase in yield strength as a function of strain rate is evident

in Fig. 8.2; however, the underlying assumption is of constant elastic modulus.

8.2.1.2 Anand Model

The Anand model was developed to model the rate-dependent deformation of metals [50] and has since been used to

characterize the viscoplastic response of solder materials [47]. It is commonly used to model the rate-dependent response of

solders, especially at creep-like strain rates (see Table 8.2). The Anand model uses a stress equation and an internal variable,

s, that is proportional to the material’s resistance to plastic flow. The resulting stress equation [45] is

σ ¼ σ _εp; εp; T� � ¼ c s; (8.3)

where _εp and εp is the plastic strain rate and plastic strain, respectively, T is the temperature; c is a coefficient given by

c _εp; T� � ¼ 1

ξsinh�1 Z _εp; T

� �A

� �m( ): (8.4)

where Z is the Zener-Holloman parameter [51],

Z _εp; T� � ¼ _εpe

GkBT (8.5)

It can be shown that the evolution equation can be expressed as,

s ¼ s _εp; εp� � ¼ s� � s� � soð Þ 1�að Þ þ aþ 1ð Þ hoð Þ s�ð Þ�af gεp

h i 11�a

(8.6)

where h0 is the hardening constant, a controls the rate dependency, and s� is the reference (saturation) value of the

deformation resistance [45]. The s� parameter is defined as,

s� ¼ sZ _εp; T� �A

� �n; (8.7)

where s the deformation resistance coefficient (a material parameter). The parameters in the Anand model are summarized in

Table 8.4.


Anand constitutive modelSymbol Parameter

A Pre-exponential factor

ξ Normalization constant

G Activation energy

m Strain rate sensitivity

h0 Rate hardening coefficient

a Rate hardening power

s0 Initial value of s

s� Reference (saturation) deformation resistance

s Deformation resistance coefficient

n Rate deformation power

kB Boltzman’s constant (universal gas constant)

62 R.D. Lowe et al.

8.2.2 Polymers

Polymers are another common material in electronics, particularly in packaging and structural assemblies. Examples include

plastic packaged components, structural reinforcing materials (encapsulants/potting and underfills), and polymer composites

(printed circuit boards). The particular polymers used are as diverse as the applications [52], and several constitutive models

have been proposed to analytically describe the highly rate- and temperature-dependent response of polymers. Two such

models commonly used for electronics polymers, the Zerilli-Armstrong and Mulliken-Boyce models, are now briefly

discussed.

8.2.2.1 Zerilli-Armstrong Model

The Zerilli-Armstrong model [53] is a constitutive model for estimating stress via dislocation mechanics. It was originally

proposed to model the high-rate deformation of metals, but has since been used to predict the dynamic deformation of

polymers [54–57]. The Zerilli-Armstrong model captures temperature-, pressure -, and rate-dependent deformation due to

thermally activated displacement of material “flow units” under applied shear stress. The viscoelastic component of the

model is represented by a Maxwell-Weichert linear model in series with a nonlinear thermally-activated dashpot for the

viscoplastic component [57]. This is shown in Fig. 8.3.

The total stress is found from the sum of the deviatoric stresses in each network element; the stress evolution equation in

the kth element is given by

_σ0ðkÞij

2Gkþ σ 0

ijðkÞηk

¼ _ε0ij � _ε0ðpÞij ; k ¼ 1; 2; . . . ;N; (8.8)

where _ε0ij is the total deformation rate, _ε0ðpÞij is the viscoplastic rate, and Gk and ηk are the shear modulus and viscosity for the

kth component. The relaxation time ðτkÞ is a function of pressure and temperature and its evolution equation is

τk ¼ τ0;keHkkBT; (8.9)

where Hk ¼ Hk0 þ ApkP is the activation energy and Apk is the pressure coefficient of the kth element [57]. Equation 8.9 can

be written in a compact 1-D form i ¼ j ¼ 1ð Þ as

_σðkÞ11 þ σðkÞ11

τk¼ 3Gk

Kðε; TÞ þ 13Gk;0

_ε11 � _εðkÞ11

� (8.10)

where K is a temperature- and strain-dependent bulk modulus. The parameters in the Zerilli-Armstrong model are

summarized in Table 8.5.

2G1 2G2 2Gn

η1 η2 ηn

32

′ijσ

sij(1)′

sp (e (p) , e (p) )

sij(2)′ sij

(n)′sij

′

eij(p)

e (p)

Fig. 8.3 Maxwell-Weichert

element schematic of the

Zerilli-Armstrong constitutive

model for polymers


8.2.2.2 Mulliken-Boyce Model

The Mulliken-Boyce (M-B) model is a two phase (α and β) viscoelastic-viscoplastic model. It incorporates both the polymer

network stress (B) and the two phases of chain stress (A) due to the polymerization; this is shown schematically using

Maxwell-Weichert elements in Fig. 8.4. The M-B model captures the rate-and temperature-dependent behavior of polymers,

particularly the rate-dependent yield and post-yield material strength, with high accuracy [58]. This is important in the high

rate, high amplitude stress events encountered in the extreme environments (as discussed in previous sections) since these

conditions can lead to yield in structures, particularly polymers. Predicting the subsequent structural response due to

potentially large deformations is important for predicting electronic component or assembly survivability.

The total stress in the material is given by

σtotal ¼ σA;α þ σA;β þ σB; (8.11)

and the strain is

εt ¼ εe þ εp ¼ εα ¼ εβ: (8.12)

Using a uniaxial approximation to simplify the various contributions to the stresses (see [58] for a full treatment), a

nonlinear system of equations can be developed:

_yðx; tÞ ¼

_ε_σtα_σtβ_sα_sβ_γpα_γpβ_θ

266666666664

377777777775¼

_εEα _ε� _εpα

� �Eβ _ε� _εpβ

� hα 1� sα

sss;α

� _γpα;0 exp � ΔGα

kBT1� τα

sαþααP

� h ihβ 1� sβ

sss;β

� _γpβ;0 exp � ΔGβ

kBT1� τβ

sβþαβP

� h i2 _γpα;0exp � ΔGα

kBT

� sinh ταΔGα

sαþααPð ÞkBT�

2 _γpβ;0exp � ΔGβ

kBT

� sinh

τβΔGβ

sβþαβPð ÞkBT �

1ρCp

_ταγpα þ τα _γpα� �þ _τβγ

pβ þ τβ _γ

pβ

� h i

26666666666666666664

37777777777777777775

: (8.13)

This system of equations is readily solved using ordinary differential solvers. The constitutive parameters, listed in

Table 8.6, are then fit to experimental data at various strain rates and temperatures.

Fig. 8.4 Maxwell-Weichert

element schematic of the

Mulliken-Boyce constitutive

model for polymers


Zerilli-Armstrong constitutive

model

Symbol Parameter

Ap Pressure coefficient

G Shear modulus

H Activation energy

K Bulk modulus

τ Relaxation time

τ0;k Relaxation time (reference temperature)

η Element viscosity

64 R.D. Lowe et al.

Output from the M-B model is shown in Fig. 8.5 for an Epon 826/DEA epoxy system at varying strain rates _εð Þ from 10�3

to 1.4 � 104 s�1 using parameters identified in previous efforts [58]. The significant increase in the yield strength as well as

the post-yield softening and hardening behaviors are captured accurately.

8.3 Experimentation

While a constitutive model describes the physical response of a material to applied stresses, the parameters of the model

must be estimated from experimental data. There are two general approaches to characterizing materials in this way. The

first (and more common) approach is to perform a series of dynamic characterization experiments under controlled stress,

0 0.1 0.2 0.3 0.4 0.5 0.60

50

100

150

200

250

Strain [ ]

Str

ess

[MP

a]

Stress-Strain at Multiple Strain Rates for Epon 826/DEA Epoxy

IncreasingStrainRate

dε/dt ~ 10-3 s-1

dε/dt ~ 1 s-1

dε/dt = 1400 s-1

dε/dt = 3900 s-1

dε/dt = 1.4 x 104 s-1

Fig. 8.5 Stress–strain predictions from the Mulliken-Boyce model overlaid on experimental data (from [58]) for Epon 826 DEA at strain rates

from 10�3 to 14,000 s�1


Mulliken-Boyce constitutive

model

Symbol Parameter

κα _ε; θð Þ Bulk modulus (α phase)

κβ _ε; θð Þ Bulk modulus

μα _ε; θð Þ Shear modulus

μβ _ε; θð Þ Shear modulus

_γpα;0 _ε; θð Þ Pre-exponential factor for shear strain rate

_γpβ;0 _ε; θð Þ Pre-exponential factor for shear strain rate

ΔGα Phase activation energy

ΔGβ Phase activation energy

αα Pressure coefficient

αβ Pressure coefficient

hα Softening slope

sss;α Steady state preferred athermal shear stress

sss;β Steady state preferred athermal shear stress

sss=s0 Steady state preferred athermal shear stress ratio

CR Rubbery modulusffiffiffiffiN

pLimiting chain extensibility


strain, temperature, and strain rate conditions. The principle weakness of this approach is that typically one of the conditions

is allowed to vary while the others are held constant; Fig. 8.5 is an example of this kind of characterization over several

decades of strain rate. While this does not necessarily reproduce the operational conditions for the electronics, it provides an

extensive data set from which to estimate the material constants.

A second approach is to implement a fully featured model of a desired system in its operational configuration (or a related

surrogate) and estimate the properties by varying model parameters in the material submodel(s). Many commercial finite

element codes, such as Abaqus [59], include built-in optimization modules to systematically estimate material parameters.

The principle shortfall is that the specific material models that have been implemented and are available in a code library are

limited, and developing such models requires significant time and expertise. If it is possible for a system to be described by

analytic solutions, however, the relative accuracy of the analytic assumptions becomes the most important consideration.

Three experiments, representing both approaches, are discussed in this section. The first is a cantilevered electronics

board that is subjected to a displacement initial condition and transient response. This test represents an analytic case

(cantilevered/clamped-free beam) and is nicknamed the “diving board”. The second experiment involves dynamic uniaxial

loading of a simply supported round electronics board; the geometry and response of the system has led this particular

experiment to be nicknamed the “trampoline”. The third experiment is dynamic mechanical analysis (DMA) performed on

electronics materials.

8.3.1 “Diving Board” Transient Response Test

The “diving board” experiment was conceptualized as a simple test to provide quick insight into the dynamic response of

electronics components. The design was driven by the need for an analytic case (e.g., a cantilevered beam) that could be

easily modeled using finite element analysis software and experimentally implemented to validate these models. A clamped-

free beam was chosen since beam deflection solutions are readily available of varying complexity from many sources

(see, for example, references [60, 61]). The geometry is shown in Fig. 8.6 below.

The diving board test specimen is a glass fiber-reinforced epoxy composite G10 board (the composite for FR4 board)

[62]. An analytic solution exists for the static deflection with a known point load (P) applied to the beam: the analytic beam

deflection is

δmax ¼ � PL3

3E I0(8.14)

where δ is the net deflection, L is the unsupported beam length, E is the elastic modulus, and I is the moment of inertia. Using

isotropic values from a material datasheet (ρ ¼ 0.15 g/cm3, E ¼ 90 GPa, and ν ¼ 0.13) and a defined beam geometry

(width w ¼ 32.5 mm, thickness d ¼ 3.2 mm, and length L ¼ 144 mm), an applied load of 2.5 N results in 1.2 mm of

deflection.

The dynamic response of a simple cantilevered beam is also readily obtained. The pre-loaded board is abruptly released

(by severing a load-carrying filament) and allowed to freely vibrate. Since the initial displacement is instantaneously

released, the appropriate solution has no point load. The modal response of an isotropic clamped-free beam can be found in

many references (e.g., [63]), and the modal frequencies can be found from

fn ¼ An

2π

ffiffiffiffiffiffiEI

μL4

r; (8.15)

Clamped Boundary Condition

Applied Initial Displacement

Fig. 8.6 The “diving board”

test is a simple analytic case

(in this case a cantilevered

beam made of printed circuit

card material) that is readily

reproduced experimentally

66 R.D. Lowe et al.

where μ is the mass per unit length of the beam and An is a modal constant. The first five mode constants and the resulting

natural frequencies are listed in Table 8.7.

While useful for order-of-magnitude calculations, the results in Table 8.7 are useful for order-of-magnitude assessments

of the bending modes of an isotropic beam. However, G10 is a laminated composite material and its properties are expected

to be orthotropic. The orthotropic material constants for G10 are obtained from [64] and are listed in Table 8.8. Using these

orthotropic material properties, the mode shapes and frequencies are calculated using Abaqus and shown in Fig. 8.7.

Dynamic results from both experiments and simulations are shown in Fig. 8.8 for three different initial displacements:

1.2 (blue), 2.4 (red and green), and 3.6 mm (black). The frequency and simulations are accurate within 3 % for the 1st mode:

the first experimental mode is measured to be 68.6 Hz whereas the computational first mode frequency is 70.7 Hz.

The structural damping rate was also determined using the log-decrement method. The experimental damping ratio is

estimated to be ζ ¼ 0.008; this value is applied to the simulation which dramatically improves the fidelity of the response at

later times, i.e., when having accurate damping properties are critical to accurate predictions.

8.3.2 “Trampoline” Impact Response Test

The second test method, the so-called “trampoline” dynamic test, is a reverse Hopkinson bar experiment [65]. The

trampoline uses an electronics housing fixture that is compatible with other shock tests. A striker impacts a long incident

bar which transmits a dynamic compressive stress wave into the fixture supporting a circuit board assembly. This is shown

schematically in Fig. 8.10.

Standoffs attach the circuit board to a cast aluminum fixture, providing a direct load path into the test articles, which are

41 mm diameter circular printed circuit boards. Photographs of the experimental setup are shown in Fig. 8.11. The incident

bar has been instrumented with semiconductor strain gages and calibrated using a dispersion-correction technique [66]. An

example of the propagating stress waves along with annotated features and the estimated time-of-arrival at the fixture of the

transient stress waves are shown in Fig. 8.12. Utilizing the strain history in the incident bar provides a more accurate estimate

of the force-time history applied to the test fixture. A reference accelerometer (shown in Fig. 8.11b) is used to verify the local

acceleration due to the applied force.

Table 8.8 Orthotropic

material constants for G10FR-4 (isotropic plane xy) (Auersperg et al. 1997)

Temperatures [�C]

Properties �40 30 95 125 150 270

Ex (MPa) 24,252 22,400 20,680 19,300 17,920 16,000

Ey (MPa) 24,252 22,400 20,680 19,300 17,920 16,000

Ez (MPa) 2,031 1,600 1,200 1,000 600 450

vxy 0.02 0.02 0.02 0.02 0.02 0.02

vyz 0.1425 0.1425 0.1425 0.1425 0.1425 0.1425

vxz 0.1425 0.1425 0.1425 0.1425 0.1425 0.1425

αx (ppm/C) 16 16 16 16 16 16

αy (ppm/C) 16 16 16 16 16 16

αz (ppm/C) 65 65 65 65 65 65

Gxy (MPa) 662 630 600 500 450 441

Gxz (MPa) 210 199 189 167 142 139

Gyz (MPa) 210 199 189 167 142 139

Table 8.7 Analytic bending

modes of a cantilevered

(clamped-free) beam

Mode number n Modal constant An Natural Frequency fn

1 3.52 89.9 Hz

2 22.0 562 Hz

3 61.7 1.58 kHz

4 121.0 3.09 kHz

5 200.0 5.11 kHz


An OFV-332 Polytec laser vibrometer head [67] is used with the OFV-3020 high speed (20 m/s) controller/demodulator

to provide a non-contact measurement of the surface velocity of the board. Figure 8.13a shows the surface velocity time

history and equivalent acceleration of the center of the printed circuit board. The instantaneous acceleration

(unfiltered, shown in Fig. 8.13b) exceeds 30,000 g’s, indicating a far more severe local acceleration than anticipated. This

has further motivated the need to characterize electronic materials at high rates of loading since a rate-dependent increase in

stiffness (as described by a J-C or M-B model) is a possible contributing factor.

Vel

ocity

[in/s

]

Vel

ocity

[in/

s]

40

a b

20

0

0 0.005 0.015 0.000 0.005 0.010 0.0150.01

-20

-40

40.

20.

0.

-20.

-40.

Time [s]

Test 1 Data

v3_test1v3_test4v3_test7v3_test11

Test 4 DataTest 7 DataTest 11 Data

Time [s]

Fig. 8.8 Side-by-side comparison of velocity time histories from the diving board (a) experiment and (b) simulation using orthotropic material

properties. Results from three different initial displacements are shown: 1.2 (blue), 2.4 (red and green), and 3.6 mm (black)

Fig. 8.7 The first nine mode shapes of the “diving board” calculated using orthotropic material properties

68 R.D. Lowe et al.

Fig. 8.11 Photographs showing (a) initial experiment apparatus and a close-up (b) of the lower fixture with a mounted test board and reference

accelerometer

Fig. 8.10 Schematic of the trampoline experiment with electronics fixture and the circuit board under test

Fig. 8.9 Estimate of the damping using the log-decrement method from experimental and simulation results using orthotropic material properties


8.3.3 Dynamic Mechanical Analysis (DMA)

To complement the two dynamic tests, the material response of a common epoxy is evaluated using dynamic mechanical

analysis (DMA). Epon 828/DEA was chosen for study due its similarity to a previously studied material, Epon 826/DEA

[58]. The samples were cast and cured into samples that measured 60 mm long, 12.5 mm wide, and 3.2 mm thick. These

samples were tested in a dual cantilever configuration in a TA Instruments Q800 [68, 69] at frequencies of 1, 10, and 100 Hz

and a temperature range of�100 �C to 190 �C. The displacement was held constant at 15 μm for this analysis. Typical DMA

data for the Epon 828/DEA is shown in Fig. 8.14 at the measured frequencies of 1, 10, and 100 Hz. The frequency is

converted to strain rate ( _ε) using the equivalence relationship

Fig. 8.12 First wave analysis of strain gage observations. Solid lines are longitudinal strain, and dashed lines are bending strain

Fig. 8.13 (a) Printed circuit board surface velocity of five unique tests and (b) equivalent instantaneous acceleration (in g’s) of the center of theprinted circuit board in the trampoline experiment for a single test

70 R.D. Lowe et al.

_ε ¼ 4ωd0lg; (8.16)

where ω is the angular frequency (in rad/s), d0 is the amplitude of the displacement, and lg is the specimen gage length. The

glass transition in the range of 355–365 K is evident in the loss modulus. It is also interesting to note the presence of multiple

phases in the loss modulus [70]. This is expected to contribute heavily to the high damping (~0.01) observed in circuit board

materials and is currently being investigated.

8.4 Future Work

Simple elastic models – even with orthotropic material properties – are insufficient for harsh environments where high rate,

high amplitude stress loads are expected. We will continue to expand these results using the discussed techniques at higher

strain rates and over varying temperatures while implementing and estimating parameters in the constitutive models

reviewed in this work (and others as appropriate). Future efforts will focus on supplementing application-relevant test

methods with traditional characterization to improve model validity while implementing the probabilistic framework

introduced by Foley et al. [70].

8.5 Summary

Electronics for mission- and safety-critical environments must function in a wide range of environments. However,

the diverse materials encountered in electronics coupled with a rapidly changing marketplace creates great uncertainty in

predicting the dynamic response of materials to thermal and mechanical stresses. While several studies have examined

various aspects of the problem, a large number of electronics materials remain relatively uncharacterized in the harshest

environments. New experiments, including the diving board and trampoline experiments, were created to provide dynamic

200 250 300 350 400 4500

500

1000

1500

2000

2500

Temperature [K]

Mod

ulus

[M

Pa]

DMA data for Epon 828/DEA

f = 1 Hz

f = 10 Hz

f = 100 Hz

Increasing Frequency

Loss Modulus (x10)

Storage Modulus

Fig. 8.14 DMA data for Epon 828/DEA epoxy


mechanical loads on electronic assemblies representative of harsh operational environments. The trampoline test, in

particular, was able to achieve local accelerations above 30,000 g’s. The participating materials are also being examined

using traditional characterization experiments; initial DMA results for Epon 828/DEA epoxy are presented. Future work will

focus on supplementing application-relevant test methods with traditional characterization to improve model validity.

Acknowledgements The authors would like to thank the Air Force Office of Scientific Research and the Department of Defense for supporting

this research. Opinions, interpretations, conclusions and recommendations are those of the authors and are not necessarily endorsed by the United

States Air Force.

References

1. (2008) Test method standard for environmental engineering considerations and laboratory tests. MIL-STD-810G, Department of Defense

2. Custer W (2011) Business outlook for the global electronics industry. IPC Outlook (downloaded from www.ipc.org)

3. JEDEC (2003) JEDEC solid state technology association JESD22-B111. Board level drop test method of components for handheld electronic

products

4. JEDEC (2006) JEDEC solid state technology association JESD22-B103B. Vibration, Variable Frequency

5. Mishiro K, Ishikawa S, Abe M, Kumai T, Higashiguchi Y, Tsubone K-i (2002) Effect of the drop impact on BGA/CSP package reliability.

Microelectron Reliab 42(1):77–82

6. Jing-En L, Tong Yan T (2004) Analytical and numerical analysis of impact pulse parameters on consistency of drop impact results.

In: Proceedings electronics packaging technology conference. EPTC 2004. Proceedings of 6th, pp 664–670

7. Tan VBC, Tong MX, Kian Meng L, Chwee Teck L (2005) Finite element modeling of electronic packages subjected to drop impact. IEEE

Trans Compon Packag Technol 28(3):555–560

8. Jing-En L, Tong Yan T, Pek E, Chwee Teck L, Zhaowei Z, Jiang Z (2006) Advanced numerical and experimental techniques for analysis of

dynamic responses and solder joint reliability during drop impact. IEEE Trans Compon Packag Technol 29(3):449–456

9. Lall P, Panchagade DR, YueLi L, Johnson RW, Suhling JC (2006) Models for reliability prediction of fine-pitch BGAs and CSPs in shock and

drop-impact. IEEE Trans Compon Packag Technol 29(3):464–474

10. Wang Y, Low KH, Pang HLJ, Hoon KH, Che FX, Yong YS (2006) Modeling and simulation for a drop-impact analysis of multi-layered

printed circuit boards. Microelectron Reliab 46(2–4):558–573

11. Chong DYR, Che FX, Pang JHL, Ng K, Tan JYN, Low PTH (2006) Drop impact reliability testing for lead-free and lead-based soldered IC

packages. Microelectron Reliab 46(7):1160–1171

12. Wong EH, Seah SKW, van Driel WD, Caers JFJM, Owens N, Lai YS (2009) Advances in the drop-impact reliability of solder joints for mobile

applications. Microelectron Reliab 49(2):139–149

13. Yeh C-L, Lai Y-S, Kao C-L (2006) Evaluation of board-level reliability of electronic packages under consecutive drops. Microelectron Reliab

46(7):1172–1182

14. Lall P, Gupte S, Choudhary P, Suhling J (2007) Solder joint reliability in electronics under shock and vibration using explicit finite-element

submodeling. IEEE Trans Electron Packag Manuf 30(1):74–83

15. Jenq ST, Sheu HS, Yeh C-L, Lai Y-S, Wu J-D (2007) High-G drop impact response and failure analysis of a chip packaged printed circuit

board. Int J Impact Eng 34(10):1655–1667

16. Syed A, Seung Mo K, Wei L, Jin Young K, Eun Sook S, Jae Hyeon S (2007) A Methodology for drop performance modeling and application

for design optimization of chip-scale packages. IEEE Trans Electron Packag Manuf 30(1):42–48

17. Qu X, Chen Z, Qi B, Lee T, Wang J (2007) Board level drop test and simulation of leaded and lead-free BGA-PCB assembly. Microelectron

Reliab 47(12):2197–2204

18. Wong EH, Seah SKW, Shim VPW (2008) A review of board level solder joints for mobile applications. Microelectron Reliab 48(11–12):

1747–1758

19. Wong EH, Selvanayagam CS, Seah SKW, Driel WD, Caers JFJM, Zhao XJ, Owens N, Tan LC, Frear DR, Leoni M, Lai YS, Yeh CL (2008)

Stress–strain characteristics of tin-based solder alloys for drop-impact modeling. J Electron Mater 37(6):829–836

20. Xu ZJ, Yu TX (2008) Board level dynamic response and solder ball joint reliability analysis under drop impact test with various impact

orientations. In: Proceeding electronics packaging technology conference, 2008. EPTC 2008. 10th, pp 1080–1085

21. Zhang B, Ding H, Sheng X (2008) Modal analysis of board-level electronic package. Microelectron Eng 85(3):610–620

22. Lee Y-C, Wang B-T, Lai Y-S, Yeh C-L, Chen R-S (2008) Finite element model verification for packaged printed circuit board by experimental

modal analysis. Microelectron Reliab 48(11–12):1837–1846

23. Long W, Xingming F, Jianwei Z, Qian W, Jaisung L (2008) Dynamic properties testing of solders and modeling of electronic packages

subjected to drop impact. In: Proceeding electronic packaging technology & high density packaging, 2008. ICEPT-HDP 2008. International

Conference on, pp 1–7

24. Chou C-Y, Hung T-Y, Yang S-Y, Yew M-C, Yang W-K, Chiang K-N (2008) Solder joint and trace line failure simulation and experimental

validation of fan-out type wafer level packaging subjected to drop impact. Microelectron Reliab 48(8–9):1149–1154

25. Lall P, Lowe R, Suhling J, Goebel K (2009) Leading-indicators based on impedance spectroscopy for prognostication of electronics under

shock and vibration loads. In: Proceedings 2009 ASME interPack conference, IPACK2009, American Society of Mechanical Engineers, 19–23

Jul 2009, pp 903–914

72 R.D. Lowe et al.

http://www.ipc.org/

26. Lall P, Lowe R, Suhling J, Goebel K (2009) Prognostication based on resistance-spectroscopy for high reliability electronics under

shock-impact. In: Proceeding 2009 ASME international mechanical engineering congress and exposition, IMECE2009, American Society

of Mechanical Engineers, 13–19 Nov 2009, pp 383–394

27. Weiping L, Ning-Cheng L, Porras A, Min D, Gallagher A, Huang A, Chen S, Lee JCB (2009) Shock resistant and thermally reliable low Ag

SAC solders doped with Mn Or Ce. In: Proceedings electronics packaging technology conference, 2009. EPTC ’09. 11th, pp 49–63

28. Yu AM, Jun-Ki K, Jong-Hyun L, Mok-Soon K (2009) Tensile properties and drop/shock reliability of Sn-Ag-Cu-In based solder alloys.

In: Proceeding electronics packaging technology conference, 2009. EPTC ’09. 11th, pp 679–682

29. Wong EH, Mai Y-W (2009) The damped dynamics of printed circuit board and analysis of distorted and deformed half-sine excitation.


30. Yu AM, Kim J-K, Lee J-H, Kim M-S (2010) Pd-doped Sn–Ag–Cu–In solder material for high drop/shock reliability. Mater Res Bull 45(3):

359–361

31. Amy RA, Aglietti GS, Richardson G (2010) Board-level vibration failure criteria for printed circuit assemblies: an experimental approach.

IEEE Trans Electron Packag Manuf 33(4):303–311

32. Amy RA, Aglietti GS, Richardson G (2010) Accuracy of simplified printed circuit board finite element models. Microelectron Reliab 50(1):

86–97

33. Nguyen TT, Park S (2011) Characterization of elasto-plastic behavior of actual SAC solder joints for drop test modeling. Microelectron Reliab

51(8):1385–1392

34. Le Coq C, Tougui A, Stempin M-P, Barreau L (2011) Optimization for simulation of WL-CSP subjected to drop-test with plasticity behavior.


35. An T, Qin F (2011) Cracking of the intermetallic compound layer in solder joints under drop impact loading. J Electron Packag 133(3):

031004–031004

36. An T, Qin F, Li J (2011) Mechanical behavior of solder joints under dynamic four-point impact bending. Microelectron Reliab 51(5):

1011–1019

37. Anuar MA, Isa AAM, Ummi ZAR (2012) Modal characteristics study of CEM-1 single-layer printed circuit board using experimental modal

analysis. Procedia Eng 41:1360–1366

38. Amy RA, Aglietti GS, Richardson G (2009) Reliability analysis of electronic equipment subjected to shock and vibration – a review. Shock

Vib 16(1):45–59

39. Norris KC, Landzberg AH (1969) Reliability of controlled collapse interconnections. IBM J Res Dev 13(3):266–271

40. Goldmann LS (1969) Geometric optimization of controlled collapse interconnections. IBM J Res Dev 13(3):251–265

41. Dally JW, Lall P, Suhling JC (2008) Mechanical design of electronic systems. College House Enterprises, Knoxville

42. Lall P, Pecht M, Hakim E (1997) Estimating the influence of temperature on microelectronic reliability. CRC Press, Boca Raton

43. (2003) Restriction of the use of certain hazardous substances in electrical and electronic equipment directive (2002/95/EC). European Union

44. Qin F, An T, Chen N (2010) Strain rate effects and rate-dependent constitutive models of lead-based and lead-free solders. J Appl Mech

77:011008

45. Motalab M, Cai Z, Suhling JC, Lall P (2012) Determination of Anand constants for SAC solders using stress–strain or creep data.

In: Proceeding thermal and thermomechanical phenomena in electronic systems (ITherm), 2012. 13th IEEE Intersociety Conference on,

pp 910–922

46. Lall P, Shantaram S, Suhling J, Locker D (2012) Mechanical deformation behavior of SAC305 at high strain rates. In: Proceeding thermal and

thermomechanical phenomena in electronic systems (ITherm), 2012. 13th IEEE Intersociety Conference on, pp 1037–1051

47. Wang GZ, Cheng ZN, Becker K, Wilde J (2001) Applying Anand model to represent the viscoplastic deformation behavior of solder alloys.

J Electron Packag 123(3):247–253

48. Siviour CR, Walley SM, ProudWG, Field JE (2005) Mechanical properties of SnPb and lead-free solders at high rates of strain. J Phys D: Appl

Phys 38(22):4131

49. Johnson GR, Cook WH A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures.

In: Proceeding of the seventh international symposium on Ballistics, pp 541–547

50. Anand L (1982) Constitutive equations for the rate-dependent deformation of metals at elevated temperatures. J Eng Mater Technol 104(1):

12–17

51. Zener C, Hollomon JH (1944) Effect of strain rate upon plastic flow of steel. J Appl Phys 15(1):22–32

52. Frear DR (1999) Materials issues in area-array microelectronic packaging. J Mater 51(3):22–27

53. Zerilli FJ, Armstrong RW (1987) Dislocation-mechanics-based constitutive relations for material dynamics calculations. J Appl Phys 61(5):

1816–1825

54. Zerilli FJ, Armstrong RW (1999) Application of eyring’s thermal activation theory to constitutive equations for polymers. In: Furnish MD,

Chhabildas LC, Hixson RS (eds) Proceedings shock compression of condensed matter – 1999, American Institute of Physics, pp 531–534

55. Zerilli FJ, Armstrong RW (2000) Thermal activation based constitutive equations for polymers. Journal de Physique IV France 10(9):3–8

56. Zerilli FJ, Armstrong RW (2001) Thermal activation constitutive model for polymers applied to polytetrafluoroethylene. In: Furnish MD,

Thadhani NN, Horie Y (eds) Proceeding shock compression of condensed matter-2001, American Institute of Physics, pp 657–660

57. Zerilli FJ, Armstrong RW (2007) A constitutive equation for the dynamic deformation behavior of polymers. J Mater Sci 42:4562–4574

58. Jordan J, Foley J, Siviour C (2008) Mechanical properties of Epon 826/DEA epoxy. Mech Time-Depend Mater 12(3):249–272

59. (2009) Abaqus/CAE User’s Manual (version 6.7), DSS Simulia

60. Avalone EA, Baumeister T, Sadegh AM (2007) Marks’ standard handbook for mechanical engineers. McGraw-Hill, New York

61. Timoshenko S, Woinowsky-Krieger S (1959) Theory of plates and shells. McGraw-Hill, New York

62. IPC (1997) IPC-4101: specification for base materials for rigid and multilayer printed boards. IPC (Institute for Interconnecting and Packaging

Electronic Circuits), Northbrook

63. Harris CM (1996) Shock and vibration handbook. McGraw-Hill, New York


64. Auersperg J, Schubert A, Vogel D, Michel B, Reichl H Fracture and damage evaluation in chip scale packages and flip chip assemblies by FEA

and Microdac. In: Proceedings symposium application fracture mechanics electronic packaging, ASME, pp 133–139

65. Bell W, Davie N (1987) Reverse Hopkinson bar testing. Sandia National Labs, Albuquerque

66. Dodson JC, Lowe RD, Foley JR (2013) Dispersive wave propagation analysis and correction in Hopkinson bar experiments. In preparation for

the Journal of Shock and Vibration

67. (2010) OFV-552 Laser Vibrometer, Polytec

68. (2010) DMA Q800 Specifications, TA Instruments

69. (2011) TA Instruments Dynamic Mechanical Analyzer (TA284), TA Instruments

70. Foley JR, Jordan JL, Siviour CR (2012) Probabilistic estimation of the constitutive parameters of polymers. EPJ Web Conf 26:04041

74 R.D. Lowe et al.

Chapter 9

Analytical and Experimental Protocols for Unified Characterizations

in Real Time Space for Isotropic Linear Viscoelastic Moduli

from 1–D Tensile Experiments

Michael Michaeli, Abraham Shtark, Hagay Grosbein, Eli Altus, and Harry H. Hilton

Abstract It is shown that for linear isotropic elastic and viscoelastic materials a single type of 1–D set of tension

experiments with optical measurements supplies sufficient stress and strain data to completely characterize all moduli

(including Young’s, shear and bulk ones) and all compliances. This is accomplished directly in real time space without the

use of integral transforms and/or Poisson’s ratios and includes the complete history of loading and of displacements

including their build ups. Additionally, several approaches to the determination of instantaneous moduli from 1–D quasi-

static and dynamic experimental data are presented and evaluated.

Keywords Computational, experimental and analytical material characterizations • Instantaneous(elastic) moduli/compliances

• Loading history • Prony series • Starting transients

9.1 Introduction

The analytical developments and descriptions of experimental techniques are described in [1–6], while the starting transient

loading analysis may be found in the last reference. The least squares numerical protocols necessary to extract relaxation

moduli and creep compliance Prony series coefficients and relaxation times from the experimental data are presented in [3].

This paper concentrates on the direct determination in the real time space of shear and bulk relaxation moduli as well as all

other isotropic moduli and compliances from 1–D experimental tensile data. Additionally, analyses are offered to test

material linearity,

M. Michaeli

Lecturer in Mathematics, Department of Mathematics R&D – Structures Analysis Team, Bar-Ilan University,

Ramat Gan 52900, Israel

R&D – Structures Analysis Team, IMI, 1044Ramat Hasharon 47100, Israel


A. Shtark • H. Grosbein

R&D – Structures Analysis Team, IMI, 1044Ramat Hasharon 47100, Israel


E. Altus

Mechanical Engineering Department, Technion, Israel Institute of Technology, Haifa 32000, Israel


H.H. Hilton (*)

Professor Emeritus of Aerospace Engineering and Senior Academic Lead for Computational Structural/Solid Mechanics,

College of Engineering and Private Sector Program Division National Center for Supercomputing Applications,

University of Illinois Urbana-Champaign, 104 S. Wright Street, Urbana, IL 61801-2935, USA




75






Electrical resistance strain gages, which have seen active and valuable service for three quarters of a century are primarily

intended for testing in conjunction with metals and other very high modulus materials. These wire resistant gages have metal

matching moduli that create a difference of three to six orders of magnitude when bonded to high polymers. Consequently,

instead of measuring true strains in polymer specimen, strain gages provide substantial local reinforcements and induce

otherwise absent similar local strains [1–6].

These shortcomings can be overcome by using optical strain measurement techniques instead of strain gages as has been

done in the present paper and in [1–6], which provide experimental data for the illustrative examples. Two distinct high

speed camera systems, 24 FPS and 5,000 FPS,1 have been employed. The later one is particularly useful in capturing data

during the initial loading phases, which form integral parts of all the included analyses.

Similar experimental and analytical techniques have been utilized in where the experimental data has been additionally

statistically characterized to produce probability distribution functions and probabilities of occurrence for viscoelastic

material properties. The probabilistic approach leads to more realistic material property characterizations and, consequently,

to more reliable stress analyses of real materials which notoriously posses widely scattered statical responses. However,

the development of a sufficiently large statistical data base requires an extensive multiplicity of duplicate experiments on

non-repeatable viscoelastic specimen whose magnitude has not yet to be determined.

9.2 Analytical Issues

9.2.1 General Considerations

Consider an isotropic isothermal linear viscoelastic medium with a Cartesian coordinate system x ¼ xi ¼ { x1, x2, x3} and

where x1 is the loading tensile direction. The conventional Einstein tensor notation applies throughout the paper.

The experimental measurements of time t, stresses σ11(t) and two mutually perpendicular strains E11(t) and E22(t) obtainedfrom 1–D tension experiments will be interpreted to determine

• Instantaneous moduli E0, G0, K0 etc.

• All relaxation moduli and all creep compliances based on the complete sets of experimental data including the responses

during the unsteady loading phases and evaluations of the loading contributions

• An approximate material property linearity validation excluding and including the loading phases.

9.2.2 Instantaneous Moduli

The determination of the instantaneous or elastic moduli E0, Ei j k l0, G0 and K0 and/or their compliance counterparts from

quasi-static experimental data remains extremely problematic. This is due to the fact that the exact determination of these

moduli is fraught with pervasive difficulties arising from the inability to achieve trustworthy experimental measurements in

the neighborhood of t ¼ 0 from quasi-static experiments.

The following protocol suggest themselves for the determination of instantaneous moduli:

(a) “Convergence” of Prony series

Description: Perform least square (LSQ) with increasing number of terms in the compliance or moduli Prony series until

desired “convergence” of errors is reached.

Pros: Relatively simple repetitive calculations.

Cons: In the absence of uniqueness and existence theorems, there is no assurance that the process will converge to

the proper initial (instantaneous) values since there is no data measurable in the immediate t ¼ 0 +

neighborhood.

1 FPS ¼ frames per second, not feet per second which are denoted by fps.

76 M. Michaeli et al.

(b) Asymptotic expansions

Description: The functions describing relaxation moduli are analytically extended to t ! 0 and then fitted by LSQ

techniques to the 1–D quasi-static tension experimental data.

Pros: A straight forward computational effort.

Cons: Same as (a) above.

(c) Time derivative limits

Description: An application of L’Hopital’s rule resulting in

limt!0

Eðx; tÞ ¼ E0ðxÞ ¼ limt!0

σ11ðx; tÞE11ðx; tÞ ¼ lim

t!0

dσ11ðx; tÞdE11ðx; tÞ ¼

limt!0

@σ11ðx;tÞ@t

limt!0

@E11ðx;tÞ@t

(9.1)

Pros: In principle Eq. (9.1) should yield accurate results, but see Cons for negative aspects.

Cons: It is an established fact that slight statistical variations in function values results in huge errors in their

derivative. Previous experiments with a 24 FPS [7] provided insufficiently finely spaced data for accurate

evaluations of the derivatives in (9.1). The acquisition of a new 5,000 FPS camera [8] covers such difficulties

by sampling considerably more data points during significantly smaller time intervals.

(d) Dynamic wave experiments

Description: An additional experiment measuring the known elastic wave front velocity resulting from the impact of one

end of a uniform homogenous isotropic viscoelastic bar. The instantaneous modulus can be calculated from

measuring velocity and density values and using the relation

v ¼ffiffiffiffiffiffiffiffiffiffiE0=ρ

q(9.2)

Pros: An extremely accurate protocol for determining E0.

Cons: A relatively difficult and demanding experiment. The velocity is large and the bar needs to be short enough to

prevent cantilever beam deflections under its own weight. However, short bars will also initiate wave

reflections from the far end that will contaminate measurements of the wave after-flow.

9.2.3 Viscoelastic Poisson’s Ratios

While Poisson’s ratios (PR) [9, 9] have proven to be a powerful and most useful elastic material characterization parameter,

it has been demonstrated [11–17] that their viscoelastic six distinct counterparts are time, stress and stress history dependent

[4, 14–17]. Figure 9.1 displays the disparity of various PR time histories based on divers loadings and on their corresponding

measured experimental strains.

The popular Class III PR [15] defined in terms of Fourier transforms (FT) as [15]

νIIIij ðωÞ ¼ �

EjjðωÞEiiðωÞ

¼ �

Ð1�1

exp �{ω tð Þ Ðt�1

Cjjklðt� t0Þ @σklðt0Þ@t0 dt0 dt

Ð1�1

exp �{ω tð Þ Ðt�1

Ciimnðt� t0Þ @σmnðt0Þ@t0 dt0 dtwith i 6¼ j (9.3)

is time, stress and stress history dependent but additionally the FT constitutive relations based on its presence, have forms

that make them extremely computational intensive in real time space [17]. This means that their FT inverses produce double

convolution integrals

9 Analytical and Experimental Protocols for Unified Characterizations in Real Time Space. . . 77

E11ðtÞ ¼ðt

�1Cðt� t0Þ @σ11ðt

0Þ@t0

dt0 �ðt

�1ν12ðt� t0Þ

ðt0

�1Cðt� sÞ @σ22ðsÞ

@sds dt0

�ðt

�1ν13ðt� t0Þ

ðt0

�1Cðt� sÞ @σ33ðsÞ

@sds dt0

(9.4)

This PR is not even applicable in the special 1–D case when σ22(t) ¼ σ33(t) ¼ 0 since then it is still process dependent and

non-exportable to other types loadings and their time histories. Nor can this 1–D PR be generalized to multidirectional

loadings.

9.2.4 Direct Determination of Shear and Bulk Relaxation Modulifrom 1–DTension Experiments

The linear isotropic isothermal viscoelastic constitutive relations may be written in deviator and dilation forms as [18–24]

σijðx; tÞ � δij σðx; tÞ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}¼ stress deviator¼Sijðx;tÞ

¼ 2

ðt

�1Gðx; t� t0Þ @

@t0Eijðx; t0Þ � δij Eðx; t0Þ� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}¼ strain deviator¼Dijðx;t0Þ

dt0 (9.5)

σðx; tÞ ¼ σiiðx; tÞ3

¼ðt

�1Kðt� t0Þ @Eðx; t

0Þ@t0

dt0 and Eðx; tÞ ¼ Eiiðx; tÞ3

(9.6)

where δi j is the Kroenecker delta and with the understanding that all state variables are at rest for � 1 � t � 0 � .

A specific example of 1–D tensile tests of isotropic materials, can be conducted under the following conditions (see

Fig. 9.2).

σ11 ¼ σ11ðtÞ; σ22 ¼ σ33 ¼ 0 0 � t � t1 (9.7)

0.4

0.5

0.6

0.7

0.8

0.9

1

0.45

0.46

0.47

0.48

0.49

0.5

0 0.1 1 10 100 1000 10000

0.05 mm / sec

0.5 mm / sec

1 kgf / cm2 in 10 sec

1.85 kgf / cm2 in 2 sec

PO

ISS

ON

S'S

RA

TIO

FR

OM

RE

LA

XA

TIO

N D

AT

A

PO

ISS

ON

'S R

AT

IO F

RO

M C

RE

EP

DA

TA

LOG (TIME)

RELAXATION INITIAL RATE

CREEP CONSTANT STRESS

Fig. 9.1 PRs based on creep

and relaxation experiments [4]


@E11ðx; tÞ@t

¼

0 �1 � t � 0

f1EðtÞ 0 � t � t1

_Ec11 t � t1

8>>>>>><>>>>>>:

(9.8)

@E22ðx; tÞ@t

¼ @E33ðx; tÞ@t

¼

0 �1 � t � 0

f2EðtÞ 0 � t � t1

_Ec22 t � t1

8>>>>>><>>>>>>:

(9.9)

fiEð0Þ ¼ @fiEð0Þ@t

¼ @fiEðt1Þ@t

¼ 0; fiEðt1Þ ¼ _Ecii (9.10)

with@E11ðtÞ@t � 0, the applied tensile loading condition in this instance. Other types of 1–D loading, such as constant stress,

stress rate, stress, strain, creep, relaxation, etc., are equally applicable and can be used with this protocol subject to proper

interpretation (see Fig. 9.2).

Substituting (9.7), (9.8) into (9.5) and (9.6) yields

σ11ðx; tÞ|fflfflfflffl{zfflfflfflffl}¼ 3 S11ðx;tÞ = 2

¼ 2

ðt

�1Gðt� t0Þ @

@t0E11ðx; t0Þ � E22ðx; t0Þ½ �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

¼ 3D11ðx;t0Þ = 2

dt0 (9.11)

σ11ðx; tÞ|fflfflfflffl{zfflfflfflffl}¼ 3 σðx;tÞ

¼ðt

�1Kðt� t0Þ @

@t0E11ðx; t0Þ þ 2 E22ðx; t0Þ½ �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

¼ 3 Eðx;t0Þ

dt0 (9.12)

0

0.2

0.4

0.6

0.8

1

1.2

0 0.1 0.1 0.2 0.2

LOADING PHASE

STEADY-STATE PHASE

HEAVISIDE FUNCTION

ST

AT

E V

AR

IAB

LE

(S

TR

AIN

, S

TR

AIN

RA

TE

,S

TR

ES

S,

ST

RE

SS

RA

TE

)

TIME (s)

t1

Fig. 9.2 Typical start up

patterns at 0 � t � t1 forstate variables, Eqs. (9.8)

and (9.9)


Proper caution must be exercised when using (9.11) as it applies only to stress Sij and strain Dij deviators and meanstresses σ and strains ε in toto. For instance, adding the first and second of (9.11) yields the identity 0 ¼ 0 and gives no

information about E(t), the relaxation modulus equivalent to the elastic Young’s modulus. The proper relation in terms of

Fourier transforms of Eqs. (9.14) and (9.15) is

EðωÞ ¼ 3GðωÞ1þ GðωÞ=KðωÞ

or EðtÞ ¼ð1

�1

3GðωÞ1þ GðωÞ=KðωÞ

exp {ω tð Þ dω (9.13)

If the 1–D constitutive relations are desired, then σ(t) must be eliminated in (9.5) through substitution from (9.6). This

then results in

σ11ðx; tÞ ¼ðt

�1

4Gðt� t0Þ þ Kðt� t0Þ3|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

¼ E1111ðt�t0Þ6¼ Eðt�t0Þ6¼ E1122ðt�t0Þ

@E11ðx; t0Þ@t0

dt0 þðt

�1

�4Gðt� t0Þ þ 2Kðt� t0Þ3|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

¼ E1122ðt�t0Þ

@E22ðx; t0Þ@t0

dt0 (9.14)

0 ¼ðt

�1

�2Gðt� t0Þ þ Kðt� t0Þ3|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

¼ E2211ðt�t0Þ¼ E1122ðt�t0Þ

@E11ðx; t0Þ@t0

dt0 þðt

�1

2Gðt� t0Þ þ 2Kðt� t0Þ3|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

¼ E1111ðt�t0Þþ E1122ðt�t0Þ

@E22ðx; t0Þ@t0

dt0 (9.15)

which are the correct 1–D isotropic constitutive relations for ε22 ¼ ε33 and directly define the moduli Ei j k l(t) in real time

t space. The relaxation moduli can be represented by Prony series [25]

GðtÞ ¼ G1 þXNG

n¼1

Gn exp � t

τGn

� �with G0 ¼ G1 þ

XNG

n¼1

Gn (9.16)

with similar Prony series for the other moduli and compliances. The parameters NG and τnG are not necessarily equal to their

bulk modulus counterparts and each as separate entities.

Thus, the relaxation moduli G(t) and K(t) can be determined directly and independently from Eqs. (9.11) and (9.12) and

from the experimental data.

Space limitations do not permit the display of results, which will be included in a full length paper.

Acknowledgements Support from IMI at Ramat Hasharon, Israel; Technion, Israel Institute of Technology (IIT) at Haifa; and from the Private

Sector Program Division of the National Center for Supercomputing Applications (NCSA) at the University of Illinois at Urbana-Champaign

(UIUC) is gratefully acknowledged.

References

1. Shtark A, Grosbein H, Sameach G, Hilton HH (2007) An alternate protocol for determining viscoelastic material properties based on tensile

tests without use of Poisson ratios. In: Proceedings of the 2007 international mechanical engineering congress and exposition, Seattle. ASME

Paper IMECE2007-41068

2. Shtark A, Grosbein H, Hilton HH (2009) Analytical determination without use of Poisson ratios of temperature dependent viscoelastic material

properties based on uniaxial tensile experiments. In: Proceedings of the 2009 international mechanical engineering congress and exposition,

Lake Buena Vista. ASME Paper IMECE2009-10332

3. Michaeli M, Shtark A, Grosbein H, Steevens AJ, Hilton HH (2011) Analytical, experimental and computational viscoelastic material

characterizations absent Poisson’s ratios. In: Proceedings of the 52nd AIAA/ASME/ASCE/AHS/ASC structures, Structural Dynamics and

Materials (SDM) conference. AIAA Paper 2011-1809

4. Shtark A, Grosbein H, Sameach G, Hilton HH (2012) An alternate protocol for determining viscoelastic material properties based on tensile

tests without use of Poisson ratios. ASME J Appl Mech (accepted for publication). JAM08-1361

5. Michaeli M, Shtark A, Grosbein H, Hilton HH (2012) Characterization of isotropic viscoelastic moduli and compliances from 1–D tension

experiments. In: Proceedings of the 53rd AIAA/ASME/ASCE/AHS/ASC structures, Structural Dynamics and Materials (SDM) conference,

Anaheim. AIAA Paper ID 1212795


6. Michaeli M, Shtark A, Grosbein H, Altus E, Hilton HH (2013) A unified real time approach to characterizations of isotropic linear viscoelastic

media from 1–D experiments without use of Poisson’s ratios. In: Proceedings of the 54th AIAA/ASME/ASCE/AHS/ASC structures, Structural

Dynamics and Materials (SDM) conference, Boston. AIAA Paper ID 1512571

7. Anonymous (2011) www.gom.com/metrology-systems/digital-image-correlation.html

8. Anonymous (2012) www.visionresearch.com

9. Simeon-Denis P (1811) Traite de mechanique. Courcier, Paris

10. Simeon-Denis P (1829) Memoire sur l’equilibre et le mouvement des corps elastiques. Memoires de l’Academie Royal des Sciences de

l’Institut de France 8:357–570, 623–627

11. Lakes RS (1992) The time-dependent Poisson’s ratio of viscoelastic materials can increase or decrease. Cell Compos 11:466–469

12. Tschoegl NW, Knauss WG, Emri I (2002) Poisson’s ratio in linear viscoelasticity – a critical review. Mech Time-Depend Mater 6:3–51

13. Lakes RS, Wineman A (2006) On Poisson’s ratio in linearly viscoelastic solids. J Elast 85:45–63

14. Hilton HH, Sung Yi (1998) The significance of anisotropic viscoelastic Poisson ratio stress and time dependencies. Int J Solids Struct

35:3081–3095

15. Hilton HH (2001) Implications and constraints of time independent Poisson ratios in linear isotropic and anisotropic viscoelasticity. J Elast

63:221–251

16. Hilton HH (2009) The elusive and fickle viscoelastic Poisson’s ratio and its relation to the elastic–viscoelastic correspondence principle.

J Mech Mater Struct 4:1341–1364

17. Hilton HH (2011) Clarifications of certain ambiguities and failings of Poisson’s ratios in linear viscoelasticity. J Elast 104:303–318

18. Alfrey T Jr (1948) Mechanical behavior of high polymers. Interscience Publishers, Inc., New York

19. Brinson HF, Brinson LC (2008) Polymer engineering science and viscoelasticity: an introduction. Springer, New York

20. Christensen RM (1982) Theory of viscoelasticity – an introduction, 2nd edn. Academic Press, New York

21. Hilton HH (1964) An introduction to viscoelastic analysis. In: Baer E (ed) Engineering design for plastics. Reinhold Publishing Corp.,

New York, pp 199–276

22. Lakes RS (2009) Viscoelastic materials. Cambridge University Press, New York

23. Wineman AS, Rajakopal KR (2000) Mechanical response of polymers – an introduction. Cambridge, New York

24. Zener C (1948) Elasticity and anelasticity of metals. University of Chicago Press, Chicago

25. Prony Gaspard CFMR Baron de (1795) Essai experimental et analytique. Journal de l’Ecole Polytechnique de Paris 1:24–76

26. Anonymous (2010) www.instron.us/wa/home/default_en.aspx


http://www.gom.com/metrology-systems/digital-image-correlation.html

http://www.visionresearch.com

http://www.instron.us/wa/home/default_en.aspx

Chapter 10

High Temperature Multiaxial Creep-Fatigue

and Creep-Ratcheting Behavior of Alloy 617

Shahriar Quayyum, Mainak Sengupta, Gloria Choi, Clifford J. Lissenden, and Tasnim Hassan

Abstract Nickel based Alloy 617 is one of the leading candidate materials for intermediate heat exchanger (IHX) of

the next generation nuclear plant (NGNP). The IHX is anticipated to operate at temperatures between 800 �C and 950 �C,which is in the creep regime. In addition, system start-ups and shut-downs will induce low cycle fatigue (LCF) damages in

the IHX components. Hence, designing IHX using Alloy 617 for NGNP construction will require a detailed understanding of

the creep-fatigue and ratcheting responses. In this study, a broad set of multiaxial creep-fatigue and ratcheting experiments

are performed and the results are critically evaluated. Experiments are conducted by prescribing multiaxial loading histories

in axial and shear, stress and strain space at 850 �C and 950 �C with different strain rates and strain amplitudes.

Experimental results revealed that the axial strain ratcheting and cyclic hardening/softening responses of Alloy 617 vary

significantly with temperature levels, strain rates and strain amplitudes indicating the dependence of creep-fatigue and

ratcheting responses on these parameters. A unified constitutive model (UCM) based on the Chaboche framework is

developed and validated against the multiaxial experimental responses. UCM simulated responses are compared against

the experimental responses for determining the current state of material modeling and if modeling improvement are needed

for IHX design applications.

Keywords High temperature fatigue • Ratcheting-fatigue • Creep-fatigue • Mutliaxial ratcheting • Unified constitutive

modeling

Nomenclature

εx Axial strain

εc Strain amplitude

σx Axial stress

σxm Mean axial stress

σxc Stress amplitude

γxy Shear strain

τxy Shear stress

τc Shear stress amplitude

τm Mean shear stress

N Number of loading cycles

t Time

T Temperature

Nf Number of loading cycles to failure

S. Quayyum (*) • T. Hassan

Department of Civil, Construction and Environmental Engineering, North Carolina State University, Raleigh, NC, USA


M. Sengupta • G. Choi • C.J. Lissenden

Department of Engineering Science and Mechanics, The Pennsylvania State University, University Park, PA, USA



83


Ci Kinematic hardening parameter

Dγi Kinematic hardening evolution rate parameter

E Young’s modulus

p Cumulative inelastic strain

aγi Kinematic hardening evolution parameter

r Kinematic hardening static recovery parameter

s Deviatoric stress tensor

tr Trace

kγi Nonproportionality scaling parameter

εe Elastic strain tensor

ν Poisson’s ratio

α Back stress tensor

DR Rate constant

R Drag resistance

RAS Saturated value of the drag resistance

A Nonproportionaility parameter

R1 Maximum nonproportional yield surface evolution

R0 Maximum proportional yield surface evolution

kR Nonproportionality scaling parameter

γi Kinematic hardening parameter

γiAS Kinematic hardening evolution parameter

γi1 Maximum value of γi from nonproportional loading

γi0 Maximum value of γi from proportional loading

I Indentity tensor

J() Second invariant

K Rate dependent parameter

a Deviator of back stress

ai Deviator of back stress components

ε Strain tensor

bi Kinematic hardening static recovery parameter

bγi Kinematic hardening evolution parameter

cγi Kinematic hardening evolution parameter

σ Stress tensor

σ0 Yield stress

(•) Differential with respect to time

10.1 Introduction

High temperature gas cooled reactor (HTGR) design is dictated by the US Department of Energy (DOE) for the next

generation nuclear plants (NGNPs). This design will have outlet gas temperature in the range of 800–950 �C. However,the American Society of Mechanical Engineers (ASME) design code (Section III, subsection NH) doesn’t include design

provisions for any materials at this temperature range. Hence, one of the primary objectives of the NGNP initiative is to

conduct very high temperature testing of the IHX candidate materials in order to develop technology and tools for selection

of materials for high temperature nuclear application, design code development, licensing and construction activities. Even

though the HTGR design is yet to be finalized, various candidate alloys are under consideration. The primary candidate

materials for the intermediate heat exchanger (IHX) of the NGNP are the Alloys 617 and 230. Because of the available

material database, experience base, and available product forms, Alloy 617 is the top choice as the IHX material. This study

is developing multiaxial creep-fatigue and ratcheting data for Alloy 617 and an experimentally validated UCM to be able to

design IHX components using the ASME NH design by analysis methodology. Due to start up and shut-down, the IHX will

be subjected to thermo-mechanical low-cycle fatigue loading, with long hold periods at peak temperature within

850–950 �C, and pressure up to 8 MPa [1]. Meeting such a strenuous demand makes the design of the IHX among the

most challenging tasks of NGNP design development. In fact, the Independent Technical Review Group (ITRG) identified

IHX as a high risk NGNP component [2]. Toward overcoming this challenge, NGNP IHX Materials R&D Plan [3]

84 S. Quayyum et al.

recommended a long list of uniaxial experiments on Alloy 617 for developing design curves and a UCM. The state of stress

at critical locations of IHX and other HTGR components can be multiaxial, hence, it is imperative that multiaxial

experimental responses of Alloy 617 are developed and UCM is validated against these responses. The study reported

herein undertakes a broad set of multiaxial creep-fatigue and ratcheting experiments on Alloy 617 at very high temperatures,

thereby mimic the conditions of IHX critical locations [4]. Experiments are conducted prescribing different multiaxial

loading histories at 850 �C and 950 �C in the axial and shear, stress and strain space with different strain rates and strain

amplitudes. These multiaxial experimental responses of Alloy 617 are evaluated to understand the influence of multiaxial

loading, interactions between creep-fatigue-ratcheting failure responses, and effects of loading rates and strain amplitudes.

Based on these experimental responses a UCM is developed and validated. Validation of UCM against multiaxial creep-

fatigue-ratcheting responses ensures the robustness of the model.

10.2 Experimental Procedure

Alloy 617 specimens used in testing were machined in the form of thin-walled tubular specimens. The specimens were

machined from a 38 mm thick annealed plate with the axis of the specimen aligned in the rolling direction. The chemical

composition of the alloy is provided in Table 10.1. The gage section has an external diameter of 21 mm and an internal

diameter of 18 mm and has a total length of 305 mm (see Fig. 10.1 for more details).

The end sections are 30 mm in diameter so that they can be gripped in the hydraulic collet grips without crushing the tube.

Multiaxial experiments are conducted using a universal axial-torsion hydraulic testing system with�245 kN axial force and

�2,830 N-m torque capacity. The specimen was held in water cooled hydraulic collet grips in the vertical position. The axial

and shear strains were measured using high temperature biaxial extensometer. A 7.5 KW radio-frequency induction heater

was used to achieve the test temperatures of 850 �C and 950 �C. The induction heater sends out a high-frequency current tothree coils that were mounted on an adjustable-positioning-mechanism rig. These coils induced eddy currents on the

specimen surface and resistance in the specimen created joule heating. Thermocouples were spot welded to the specimen

to measure and control the test temperature. The temperature gradient was kept within 1 % of the desired temperature as

described in ASTM 606. Figure 10.2 shows the experimental setup with specimen gripped and extensometer mounted.

10.3 Test Type and Loading Histories

To investigate the multiaxial creep-fatigue and fatigue-ratcheting interaction of Alloy 617, hybrid strain controlled and

stress controlled loading histories were prescribed on the specimens. Twenty two experiments were performed for

investigating the effect of temperature, strain rate, strain amplitude and loading history on the multiaxial behavior of

Alloy 617. These experiments were conducted using two different strain rates (0.04 %/s and 0.1 %/s) and multiple strain

amplitudes (0.2 %, 0.3 % and 0.4 %) at 850 �C and 950 �C. Here strain amplitude refers to the equivalent strain amplitude.

To study the effect of loading history, three different types of loading histories MR1, MR2 and MOP were considered as

shown in Fig. 10.3.

The MR1 test involved applying a steady axial stress and symmetric shear strain cycles (Fig. 10.3a). The cyclic shear

strain could create cyclic hardening or softening under a small degree of loading non-proportionality. Due to inelastic

Table 10.1 Chemical composition of Alloy 617 in wt. % [5]

Ni Cr Co Mo C Fe Al Ti Si Mn Cu

Balalnce 21.9 11.4 9.3 0.08 1.7 1.0 0.3 0.1 0.1 0.04

Fig. 10.1 Schematic of the specimen with dimensions in mm

10 High Temperature Multiaxial Creep-Fatigue and Creep-Ratcheting Behavior of Alloy 617 85

interaction between the steady axial stress and shear strain, ratcheting of axial strain occurs. The MR2 loading history

involved applying a cyclic axial stress with a mean stress, dwell periods at the positive and negative peaks, and symmetric

shear strain cycles along a bow-tie path (Fig. 10.3b). The bow-tie loading path mimics stress history of piping under cyclic

bending [6]. The MR2 loading path also results in axial strain ratcheting. Since the axial stress is cyclic, fatigue damage is

more detrimental under MR2 than MR1. The cyclic shear strain could result in cyclic hardening or softening under

intermediate degree of loading non-proportionality which in turn influences axial strain ratcheting rate [7]. The MOP

loading path in Fig. 10.3c involves 100 axial strain cycles (Path I) followed by 100, 90� out-of-phase cycles (path II)

followed by 100 more axial strain cycles (path I). These tests demonstrate cyclic hardening-softening behavior under highest

degree of loading non-proportionality [8, 9]. Data from these tests would be needed for determination of the unified

constitutive model multiaxial or non-proportional parameters. Table 10.2 shows the test matrix of the multiaxial experiments

performed on Alloy 617 using these three loading paths.

Fig. 10.2 Experimental setup

Fig. 10.3 (a) MR1, (b) MR2 and (c) MOP loading histories prescribed in the multiaxial experiments on Alloy 617 specimens

Table 10.2 Test matrix of multiaxial experiments

Load path No of specimens Temp. (�C) Strain rate (%/s) Strain amp. (%)

MR1 9 850, 950 0.04, 0.1 0.2, 0.3,0.4

MR2 9 850, 950 0.04, 0.1 0.18, 0.4

MOP 4 850, 950 0.04, 0.1 0.2


10.4 Multiaxial Experiment Results

The MR1 and MR2 experiments were conducted by prescribing different temperatures, strain amplitudes and strain rates

under varying degree of loading nonproportionality to investigate the ratcheting-creep-fatigue life, ratcheting strain rates and

limits, mean-stress effect, cyclic hardening-softening and failure life of Alloy 617. The creep strain, axial strain ratcheting

and cyclic hardening-softening behavior of the material would be of particular interest. MOP experiments are primarily

performed to study the cyclic hardening-softening of Alloy 617 under highest degree of nonproportionality and to determine

the UCM parameters. Figure 10.4 shows typical responses of Alloy 617 under MR1 load path. In this figure, cyclic softening

in the shear stress–strain hysteretic response (Fig. 10.4a) and axial strain ratcheting (Fig. 10.4b) are observed. It is noted here

that in mutliaxial experiments failure is defined by peak stress drop below 80 % of the maximum stress. Carroll et al. [10]

used similar definition of fatigue failure in uniaxial experiments. The effects of various loading parameters on the multiaxial

loading responses of Alloy 617 are discussed below.

10.4.1 Effect of Temperature

To maintain its economic advantage over early generation reactor systems, the VHTR of NGNP may use helium at

temperatures higher than 900 �C and pressures up to 8 MPa for a design life of 60 years [10, 12]. Conceptual design

requires an outlet temperature of greater than 850 �C to efficiently generate hydrogen, with a maximum expected

temperature of 950 �C [5, 10, 13]. Hence, the multiaxial experiments were conducted at 850 �C and 950 �C to characterize

the material behavior in this temperature range which is in the creep regime for Alloy 617. The material showed ominously

different behavior at these two temperatures. Figure 10.5 shows equivalent stress amplitudes and axial strain ratcheting from

MR1 and MR2 loading experiments plotted against the number of loading cycles for different strain rates and strain

amplitudes.

It was observed that irrespective of strain rate, strain amplitude and loading path, at 850 �C, the material showed cyclic

hardening for the initial few cycles followed by cyclic softening, whereas at 950 �C, the material showed rapid initial

softening followed by gradual softening (Fig. 10.5a). Similar response was observed by Chen et al. [14], Rao et al. [15, 16]

and Burke and Beck [17] from uniaxial fatigue experiments on Alloy 617. It is interesting to note in Fig. 10.5a that the

equivalent stress amplitude did not stabilize to a steady state value, instead keep decreasing with increasing cycles. It is also

Fig. 10.4 Response of Alloy 617 from MR1 load path experiment: (a) equivalent cyclic stress–strain hysteresis response, (b) axial strain

ratcheting response


observed in Fig. 10.5a that the equivalent stress amplitude as well as the fatigue life of Alloy 617 is lower at 950 �Ccompared to 850 �C regardless of the strain rate and strain amplitude. The fatigue life reduces by a factor of more than two

when the temperature is increased from 850 �C to 950 �C. This reduction in fatigue life might be influenced by the increased

axial strain ratcheting rate with increase in temperature as shown in Fig. 10.5b. There is a sharp increase in the axial strain

ratcheting rate as the experiment temperature changes from 850 �C to 950 �C. The effect of temperature on the viscoplastic

material behavior can be observed from the elastic and plastic strain amplitude of the equivalent stress–strain hysteresis

loops in Fig. 10.6. The first cycle equivalent hysteresis loops fromMR2 experiments are plotted in Fig. 10.6. It is evident that

Fig. 10.5 (a) Equivalent stress amplitude and (b) axial strain accumulation plotted against the number of loading cycles for MR1 and MR2

experiments at 0.1 %/s strain rate, 0.2 % strain amplitude

Fig. 10.6 Equivalent shear stress–strain hysteresis loops from the first loading cycle at different temperatures and different strain rates for MR2

experiments with 0.4 % strain amplitude


at 950 �C, the hysteresis loop width is larger than that at 850 �C which indicates larger plastic strain amplitude at 950 �C.This in turn increases the axial strain ratcheting at higher temperature. Also note in Fig. 10.6 that the hysteresis loops at

850 �C from two strain rates are coinciding, indicating no rate effect, whereas for the 950 �C loops significant rate effect is

observed.

10.4.2 Effect of Loading History

Figure 10.5 illustrated the effect of loading history on the response of Alloy 617. The fatigue life was always lower for MR2

tests compared to MR1 tests. Under MR2 tests, Alloy 617 encountered axial stress fluctuation in presence of a nonzero axial

mean stress which induces higher rate of axial strain ratcheting than under MR1 tests. This signifies the effect of loading

history on the multiaxial ratcheting response of Alloy 617. It is also noted that the degree of non-proportionality in the MR2

test is higher than in the MR1 test, however its direct influence is yet to be clear. Also, understanding the influence of the

mean axial stress in the MR1 and MR2 tests on axial strain creep and thus on the axial strain ratcheting needs further study.

More analysis of the responses is underway to understand the effect of load history on fatigue life of Alloy 617. The MOP

loading path incorporated the highest degree of non-proportionality through the 100 cycles of 90� out-of-phase axial and

shear strain cycles (circular load path, Path II). Axial stress amplitudes from the MOP load history tests at 850 �C are plotted

in Fig. 10.7, where a marked discontinuous increase in the axial stress amplitude is observed immediately after the axial

strain cycle (path I) is changed to 90 � out-of-phase cycle (path II). This cross effect can be accounted for by a large

resistance brought about by a stable dislocation structure formed in the preceding proportional cycles to the dislocation

movement in the subsequent cycles in another direction [18]. Conversely, when path II cycle is changed to axial strain cycle,

abrupt change from highest nonproportional load path to low nonproportional path (because of small residual shear stress),

abrupt softening of Alloy 617 is observed at both the 850 C and 950 �C.

10.4.3 Strain Rate Sensitivity

The multiaxial experiments MR1, MR2 and MOP were conducted at two different loading rates (0.04 %/s and 0.1 %/s) to

investigate the effect of strain rate on the creep-fatigue-ratcheting response of Alloy 617. The equivalent shear stress

amplitudes and axial strain ratcheting from MR2 tests are plotted as a function of the number of cycles in Fig. 10.8a, b

respectively, to demonstrate the effects of strain rate on Alloy 617 at 850 C and 950 �C. The equivalent shear stress

Fig. 10.7 Axial stress

amplitude plotted against

number of loading cycles for

MOP experiments for 0.2 %

strain amplitude at 850 �C


amplitude is smaller at 0.04 %/s strain rate compared to that at 0.1 %/s strain rate regardless of temperature, strain

amplitude and loading path. At 850 �C, strain rate effect on the stress amplitude of Alloy 617 is small (almost rate-

independent) for all three loading histories. However, at both 850 �C and 950 �C slower loading rate leads to lower fatigue

life of the material for both MR1 and MR2 experiments. From the axial strain ratcheting plots of Fig. 10.8b, it is observed

that the axial strain increased linearly with the number of loading cycles. Slower loading (0.04 %/s) rate showed higher

axial strain ratcheting rate compared to faster loading rate at 950 �C. On the other hand, at 850 �C, the axial strain

accumulation rate seemed to be insensitive of the loading rate. The loading rate effect in the MOP tests is shown in

Fig. 10.7. In the MOP tests, effects of loading rate is only observed at 950 �C (not shown), where at 850 �C (Fig. 10.7)

Alloy 617 seems to be rate independent. The above results show that the slower loading rate at higher temperature is more

detrimental to Alloy 617. Rao et al. [15, 16] observed similar type of strain rate dependent behavior of Alloy 617 under

uniaxial fatigue loading. They explained that the low fatigue life at smaller strain rate was mainly attributed to the

continuous increase in inelastic strain with cycle. The effect of loading rate is also evident from the equivalent shear

stress–strain hysteresis loops plotted in Fig. 10.6, where it can be observed that at 950 �C, the plastic strain amplitude was

increased for slower loading rate compared to faster loading rate, whereas, at 850 �C the loop shape was insensitive to the

loading rate.

10.4.4 Strain Amplitude Dependence

Three strain amplitudes (0.2 %, 0.3 %, 0.4 %) were considered for the multiaxial experiments to determine the effect of

strain amplitude on the creep-fatigue-ratcheting response of Alloy 617. Figure 10.9a, b, c show the effect of strain amplitude

on the equivalent stress amplitude, axial strain ratcheting and hysteresis loops in the MR1 experiments respectively.

As expected, the equivalent stress amplitude and axial strain ratcheting of Alloy 617 are influenced by the strain amplitude

of the loading paths. With an increase in the strain amplitude, the creep-fatigue life reduced and the axial strain ratcheting

rate increased regardless of loading rate, temperature and loading history. An increase in strain amplitude signifies an

increase in plastic strain amplitude as shown in Fig. 10.9c, and this in turn increases the accumulation of axial strain

ratcheting rate (Fig. 10.9b). Consequently, the fatigue life decreases. It is noted that the effect of higher strain amplitude was

more detrimental at higher temperature, whereas the effect of higher strain amplitude was similar at both slower and faster

loading rates.

Fig. 10.8 (a) Equivalent stress amplitude and (b) axial strain accumulation plotted against number of loading cycles for different strain rates at

850 �C and 950 �C for MR2 experiments


10.5 Constitutive Model

One of the primary objectives of the multiaxial experiments on Alloy 617 is to develop and validate a unified constitutive

model against the experimental responses. The modeling framework of Chaboche [19] was chosen for the development of

the unified constitutive model. This model was modified and various new modeling features were incorporated to improve

the simulation of the experimental responses. The modified Chaboche model assumes the classical plasticity approach, i.e.

decomposition of strain (ε) into elastic (εe) and inelastic (εin) parts:

ε ¼ εe þ εin (10.1)

The elastic part of the strain component obeys Hook’s law as:

εe ¼ 1þ ν

Eσ� ν

Etrσð ÞI (10.2)

Fig. 10.9 (a) Equivalent stress amplitude and (b) axial strain accumulation plotted against number of loading cycles, and (c) equivalent

shear stress–strain hysteresis loops for the first loading cycle for different strain amplitudes at different temperatures for MR1 experiments at

0.04 %/s strain rate


where E and ν indicate Young’s modulus and Poisson’s ratio, respectively, σ and I are the stress and identity tensors,

respectively, and tr is the trace. To model the rate-dependent behavior at high temperature, the viscoplastic flow rule was

adopted:

_εin ¼ 3

2_p

s� a

J σ� αð Þ (10.3)

where (·) denotes the differentiation with respect to time, s and a are the deviators of the stress and back stress respectively.

Norton’s equation is used to express _p as in Eq. 10.4 and J(σ-α) is expressed following von-Mises as in Eq. 10.5. R(p) isthe isotropic hardening parameter, σ0 is the initial yield stress, and K and n are rate-dependent parameters.

_p ¼ J σ� αð Þ � RðpÞ � σoK

� �n

(10.4)

J σ� αð Þ ¼ 3

2s� að Þ : s� að Þ

� �12

(10.5)

Chaboche [19] proposed to use four back stress terms in the nonlinear kinematic hardening rule as in Eq. 10.6. The

kinematic hardening rule has dynamic recovery, static recovery and temperature rate terms as shown in Eq. 10.7.

a ¼X4i¼1

ai (10.6)

_ai ¼ 2

3Ci _εin � γiai _p� biJ aið Þr�1

ai þ 1

Ci

@Ci

@T_Tai (10.7)

In order to include the strain-range dependence into modeling, a strain memory surface of Chaboche et al. [20] was

considered in the UCM. The cyclic hardening/softening is modeled through the simultaneous evolution of both the isotropic

hardening parameter R and the kinematic hardening parameter γi. The evolution of R is obtained using Eqs. 10.8, 10.9, 10.10

and 10.11, where q is the plastic strain surface size, RAS(q) is the saturated value of the drag resistance Rwhich evolves based

on the rate constant DR. The maximum yield surface evolution R1 is obtained from a 90� out-of-phase strain-controlled

experiment, the maximum yield surface evolution R0 is obtained from a proportional strain-controlled experiment.

_R ¼ DR RASðqÞ � R� �

_p (10.8)

RASðqÞ ¼ A R1ðqÞ � R0ðqÞ� þ R0ðqÞ (10.9)

R0ðqÞ ¼ a1R 1� e�b1R q�c1Rð Þ �

(10.10)

R1ðqÞ ¼ kR R0ðqÞ� (10.11)

The evolution equations for γi are given in Eqs. 10.12, 10.13, 10.14 and 10.15, where γi1 and γi

0 are the maximum values

of γi for 90� out-of-phase and axial strain-controlled responses, respectively for the current plastic strain surface size. The

parameter γ4 is kept constant (not a function of q) because this parameter influences only the ratcheting rate and does not

influence the hysteresis loop shape.

_γi ¼ Dγi γASi ðqÞ � γi

� _p; for i ¼ 1; 2 and 3 (10.12)

γASi ðqÞ ¼ A γ1i ðqÞ � γ0i ðqÞ� þ γ0i ðqÞ (10.13)

γ0i ðqÞ ¼ aγi þ bγie�cγiq; for i ¼ 1; 2 and 3 (10.14)


γ1i ðqÞ ¼ kγi γ0i ðqÞ

� ; for i ¼ 1; 2 and 3 (10.15)

In order to improve the UCM simulation of various nonproportional cyclic and ratcheting responses, this study will

incorporate the nonproportional parameters of Tanaka [21] through a fourth order tensor C and an associated nonpropor-

tionality parameter A which are expressed as,

dC ¼ cc n� n� C �

_p (10.16)

A ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffitr CTC �

� nCTCn

tr CTC �

vuuut (10.17)

where cc is a rate parameter. The influence of the degree of loading nonproportionality will be included in the UCM through

Eqs. 10.9,10.10, 10.11, 10.12 and 10.13 using the parameter A. In Eq. 10.17, A ¼ 0 represents the proportional loading,

and the maximum value of A ¼ 1/√2 represents the highest degree of nonproportionality for the 90� out-of-phase loading.For intermediate degrees of nonproportionality, A varies between 0 and 1/√2.

10.6 Parameter Determination

The simulation of the experimental responses using the modified UCM needs determination of a number of material

parameters including rate-independent kinematic and isotropic hardening parameters, rate parameters, static recovery

parameters, strain range dependence parameters and nonproportionality parameters. The parameter determination scheme

is under development for implementation of the modified Chaboche model. The rate-independent kinematic hardening

parameters have been determined by using a set of strain-controlled, uniaxial experimental responses at 850 �C and 950 �C[22]. The rate parameters have been determined by using uniaxial experiments at different loading rates. Isothermal

experiments with strain hold times (relaxation) were used to determine the static recovery parameters of kinematic hardening.

10.7 Simulation

In this ongoing study, the determined parameters are only sufficient to simulate the experimental responses of MR1 tests.

MR1 tests had small degree of nonproportionality in comparison with MR2 and MOP tests. Hence, simulations using the

modified UCM without the nonproportional parameter should yield results which should be in close agreement with the

experimental responses from the MR1 tests. MR2 and MOP tests had intermediate and highest degree of nonproportionality

respectively, so simulations without the nonproportional parameter would have been erroneous. The comparison of

experimental responses with the corresponding simulations at 850 �C and 950 �C are shown in Figs. 10.10 and 10.11,

respectively. It can be observed that the simulated equivalent shear stress–strain hysteresis loop shape agreed very well with

the experimental responses at both temperatures. At 850 �C, the material showed initial hardening for few cycles followed by

softening, which was not observed in simulated responses, where the material reached a stable state after few initial cycles of

hardening. To incorporate this type of mixed hardening-softening behavior of the material at 850 �C, additional featuresneed to be included in the constitutive model. However, with the current state of the modified UCM, the simulated hysteresis

loop shapes resembled the experimental loop shapes very well at 850 �C up to about 1,000 cycle (Fig. 10.10b, c). At 950 �C,the material showed cyclic softening from the very beginning. Since the UCM parameters are not determined to simulate

cyclic softening, this feature of the response cannot be simulated. However, the simulated hysteresis loop shapes resembled

the experimental loop shapes very well as can be seen in Fig. 10.11a, b. The determination of the evolution of the kinematic

hardening parameters, the strain range dependent parameters and the nonproportionality parameters are underway, and once

these parameters are included in the simulation, the quality of simulation will be improved. Moreover, with the current state

of the UCM and parameter set, the axial strain ratcheting cannot be simulated well. The simulation over predicted the axial

strain ratcheting rate and hence is not shown here. To improve the ratcheting simulation, the multiaxial ratcheting parameter

proposed by Bari and Hassan [23] need to be included in the UCM.


Fig. 10.10 Comparison of

experimental and simulation

responses for MR1

experiment at 850 �C with

0.1 %/s strain rate and 0.2 %

strain amplitude. Equivalent

shear stress–strain hysteresis

loop in the (a) 1st loading

cycle, and (b) 1000th loading

cycle. (c) Equivalent shear

stress amplitude and mean

as a function of the number

of loading cycles


Fig. 10.11 Comparison of the

experimental and simulation

responses for MR1

experiment at 950 �C with

0.04 %/s strain rate and 0.4 %

strain amplitude. Equivalent

shear stress–strain hysteresis

loop in the (a) 1st loading

cycle, and (b) 25th loading

cycle. (c) Equivalent stress

amplitude and mean as

a function of the number

of loading cycles


10.8 Conclusion

A set of multiaxial experiments are conducted to investigate the creep-fatigue-ratcheting responses of Alloy 617 which will

aid in developing a unified constitutive model (UCM). Simulations of the experimental responses were performed using the

UCM developed. The results of the experiments and simulations led to the following conclusions:

• Multiaxial fatigue and ratcheting responses of Alloy 617 are significantly influenced by the temperature, strain rate, strain

amplitude and loading history.

• At 850 �C, the material showed cyclic hardening for the initial few cycles followed by cyclic softening, whereas at

950 �C, the material showed rapid initial softening followed by gradual softening. At 950 �C and slower loading rate

(0.04 %/s), there was a significant reduction in the fatigue life of Alloy 617. With an increase in the strain amplitude, the

decrease in fatigue life became more substantial. The axial strain ratcheting rate was high at higher temperature (950 �C)and slower loading rate (0.04 %/s).

• The loading history has strong influence on the fatigue and ratcheting responses of Alloy 617. The fatigue life of material

from MR2 tests was lower compared to MR1 tests. Since the axial stress was cycled in the MR2 tests the fatigue damage

in these tests was more significant compared to the MR1 tests. Moreover, the influence of loading nonproportionality was

evident from the cross hardening in the MOP tests.

• The simulated stress–strain hysteresis loop shape using the modified UCM shows good agreement with the experimental

responses in the initial few cycles.

• In the current state of the UCM, the stress amplitude saturates to a steady state value. However, experimental observations

revealed continued softening without saturation. Additional features need to be included in the UCM to capture this

behavior. Moreover, multiaxial ratcheting and nonproportional parameters need to be incorporated to improve the quality

of the cyclic and ratcheting response simulations of the MR1, MR2 and MOP experiments.

Acknowledgement The research is being performed using funding received from the DOE Office of Nuclear Energy’s Nuclear Energy

University Program.

References

1. Natesan K, Moisseytsev A, Majumdar S (2009) Preliminary issues associated with the next generation nuclear plant intermediate heat

exchanger design. J Nucl Mater 392(2):307–315

2. Independent Technology Review Group – INEEL/EXT-04-01816 (2004) Design features and technology uncertainties for the next generation

nuclear plant

3. INL-PLN-2804 (2008) Next generation nuclear plant intermediate heat exchanger materials research and development plan

4. Project Proposal-NGNP grant-09-288 Creep-fatigue and creep-ratcheting failures of Alloy 617: experiments and unified constitutive modeling

towards addressing the ASME code issues

5. Carroll L, Madland R, Wright R (2011) Creep-fatigue of high temperature materials for VHTR: effect of cyclic loading and environment,

Paper 11284. In: Proceedings of ICAPP 2011, Nice

6. Corona E, Hassan T, Kyriakides S (1996) On the performance of kinematic hardening rules in predicting a class of biaxial ratcheting histories.

Int J Plast 12:117–145

7. Hassan T, Taleb L, Krishna S (2008) Influence of non-proportional loading on ratcheting responses and simulations by two recent cyclic

plasticity models. Int J Plast 24:1863–1889

8. Lamba HS, Sidebottom OM (1978) Cyclic plasticity for nonproportional paths. Parts 1 and 2: comparison with predictions of three incremental

plasticity models. J Eng Mate Technol 100:96–111

9. Hassan T, Kyriakides S (1994) Ratcheting of cyclically hardening and softening materials: II. Multiaxial behavior. Int J Plast 10(2):185–212

10. Carroll L, Cabet C, Wright R (2010) The role of environment on high temperature creep-fatigue behavior of alloy 617, PVP2010-26126.

In: ASME 2010 pressure vessels and piping conference, ASME, Washington

11. Ren W, Swindeman R (2009) A review on current status of Alloys 617 and 230 for Gen IV nuclear reactor internals and heat exchangers.

J Press Vessel Technol 131:044002

12. Charit I, Murty KL (2010) Structural materials issues for the next generation fission reactors. J Mater 62(9):67–74

13. Wright JK, Carroll LJ, Cabet C, Lillo TM, Benz JK, Simpson JA, Lloyd WR, Chapman JA, Wright RN (2012) Characterization of elevated

temperature properties of heat exchanger and steam generator Alloys. Nucl Eng Des 251:252–260

14. Chen X, Sokolov MA, Sham S, Erdman DL III, Busby JT, Mo K, Stubbins JF (2013) Experimental and modeling results of creep-fatigue life of

inconel 617 and Haynes 230 at 850�C. J Nucl Mater 432:94–101

15. Rao KBS, Schiffers H, Schuster H, Nickel H (1988) Influence of time and temperature dependent processes on strain controlled lowcycle

fatigue behavior of Alloy 617. Metallurgical Trans A 19A:359–371

16. Rao KBS, Meurer HP, Schuster H (1988) Cree-fatigue interaction of inconel 617 at 950�C in simulated nuclear reactor helium.

Mater Sci Eng A 104:37–51


17. Burke MA, Beck CG (1984) The high temperature low cycle fatigue behavior of the Nickel Base Alloy IN-617. Metallurgical Trans A

15A:661–670

18. Tanaka E, Murakami S, Ooka M (1985) Effects of strain path shapes on non-proportional cyclic plasticity. J Mech Phys Solids 33(6):559–575

19. Chaboche JL (1989) Constitutive equations for cyclic plasticity and cyclic viscoplasticity. Int J Plast 5(3):247–302

20. Chaboche JL, Dang-Van K, Cordier G (1979) Modelization of the strain memory effect on the cyclic hardening of 316 stainless steel.

In: Proceedings of the fifth international conference on SMiRT, Div. L, Berlin

21. Tanaka E (1994) A nonproportionaility parameter and a cyclic viscoplastic constitutive model taking into account amplitude dependences and

memory effects of isotropic hardening. Eur J Mech A13:155–173

22. Pritchard PG, Carroll L, Hassan T (2013) Constitutive modeling of high temperature uniaxial creep-fatigue and creep-ratcheting responses of

Alloy 617, submitted for review at ASME 2013 pressure vessels and piping conference (Paper no. PVP2013-97251), ASME, Paris

23. Bari S, Hassan T (2002) An advancement in cyclic plasticity modeling for multiaxial ratcheting simulation. Int J Plast 18:873–894


Chapter 11

Metastable Austenitic Steels and Strain Rate History Dependence

Matti Isakov, Kauko Ostman, and Veli-Tapani Kuokkala

Abstract This paper addresses a previously relatively little discussed topic related to the plasticity of metastable austenitic

steels, namely the strain rate history dependence. In this concept, the mechanical response of a material is not necessarily

determined only by the current deformation conditions, such as temperature and strain rate, but also by the previous values of

these variables. From the microstructural point of view, strain rate history effects are a direct manifestation of the variations

in the microstructural evolution during plastic deformation. For metastable austenitic steels, which can undergo strain-

induced phase transformation from austenite to martensite, strain-rate history effects can be notably large. The purpose of

this paper is to present and discuss the experimental methods and test procedures the authors have found applicable for the

studies of the strain rate history dependence of a metastable austenitic stainless steel EN 1.4318. Special emphasis is put on

studying the strain rate history dependence at high strain rates, which is complicated by the dynamic nature of the tests and

the lack of closed loop control. The presentation is concluded with examples of test results that demonstrate the relevance of

the research topic.

Keywords Strain rate path • Tensile Hopkinson split bar • Recovery test • Transformation induced plasticity

11.1 Introduction

Metastable austenitic stainless steels show complex dependence on strain rate, temperature, and the history of deformation.

This can be related to the large microstructural changes taking place during plastic deformation. At suitable conditions, the

originally austenitic microstructure can transform partially or almost completely to α’-martensite resulting in a notable

increase in the strain hardening capability of the material [1–9]. Previous studies [1–9] have identified temperature as one of

the most important parameters in describing the stability of a certain steel composition. Studies [10–14] on the effects of

strain rate indicate that both the α’-martensite transformation rate and the strain hardening rate decrease with increasing

strain rate. The most common explanation seems to be that the transformation is suppressed by deformation induced heating

[10–14], which takes place already at relatively low strain rates due to the low thermal conductivity of these heavily alloyed

steels. However, the aforementioned studies are based on the comparison of data obtained from tests carried out at different

strain rates but keeping the strain rate constant in an individual test. As previously noted [15], these studies have one key

limitation, i.e., the strain rate sensitivity of the material is obtained under conditions of varying microstructure and

temperature. In each individual test the microstructure and material temperature evolve in a manner characteristic to the

imposed strain rate. Therefore these tests may not reveal all the strain rate dependent characteristics of metastable austenitic

steels. Concerning practical applications, this can be a major limitation. For example, many structural components undergo

cold forming during their manufacturing process. This cold forming subjects the material to plastic deformation at a certain

strain rate leading to microstructural changes, which then affect the subsequent behavior of the material during use and may

result in a notably different response than expected based on the tests carried out on the as-received (non-cold formed)

material.

M. Isakov (*) • K. Ostman • V.-T. Kuokkala

Department of Materials Science, Tampere University of Technology, P.O. Box 589, FI-33101 Tampere, Finland




99



The analysis presented in this paper is based on the classical division of the flow stress into two components, the athermal

component σA and thermal component σ* (Eq. 11.1):

σ ¼ σA strð Þ þ σ� str; T; _εp� �

(11.1)

In Eq. 11.1 the athermal component σA represents the strain rate and temperature independent part of the flow stress, i.e.,

the resistance of those glide obstacles, which the dislocations are able to overcome only under external stress. The thermal

component σ* represents those obstacles that can be overcome with the aid of thermal fluctuations, i.e., this component

represents the strain rate and temperature dependent part of the flow stress. Due to the statistical nature of thermal

fluctuations, σ* (the required external stress) increases with increasing strain rate and/or decreasing temperature. As denoted

in Eq. 11.1 by the str-symbol, both these components are dependent on the material microstructure.

A key point of the following analysis is that, as noted previously [16], the evolution of both components in Eq. 11.1 can be

temperature and strain rate dependent. In experimental measurements this is seen as the dependence of the strain hardening

rate on both temperature and strain rate (Eq. 11.2):

θ ¼ dσ

dεpðstr; T; _εpÞ ¼ dσa

dεpðstr; T; _εpÞ þ dσ�

dεpðstr; T; _εpÞ (11.2)

When the strain hardening rate is dependent on temperature and strain rate, the material behaves in a strain-rate history

dependent manner. In that case the value of the strain rate sensitivity, such as the commonly used semi-logarithmic strain

rate sensitivity parameter β (Eq. 11.3), depends on the manner it is evaluated (i.e., based on rapid strain rate changes versusbased on the comparison of constant strain rate tests carried out at different strain rates).

β ¼ ΔσΔlog10 _εp

(11.3)

The purpose of this paper is to study the existence of strain rate history effects in a metastable austenitic stainless steel EN

1.4318. Tests have been carried out at low strain rates in isothermal conditions and at high strain rates, where adiabatic

heating takes place. For the high rate tests techniques based on the Tensile Hopkinson Split Bar (THSB) method are used.

These techniques involve rapid upward jumps in strain rate with the use of a low rate prestraining device built into the THSB

setup, as well as a specimen recovery method in the THSB for downward jumps. The discussion ends with a consideration of

the structure of a numerical model capable of describing strain rate history effects in the studied material.

11.2 Methods

The test material used in this study is austenitic stainless steel EN 1.4318 (AISI 301LN) produced by Outokumpu Stainless

and supplied as a 2 mm thick sheet in the 2B-delivery condition (cold rolled, solution annealed, pickled and skin passed).

Table 11.1 presents the chemical composition of the test material provided by the steel producer as well as the grain size

determined using optical microscopy.

Figure 11.1 presents the specimen geometry used in this study. The geometry was originally chosen for THSB testing

based on the study of Curtze et al. [17]. The grip sections of the specimen varied according to the requirements of the test

setup, i.e., in conventional THSB testing short grip sections were used while the long grip sections with bolt holes were used

in the low strain rate, low-to-high strain rate jump, and in the recovery THSB tests. The specimens were prepared by cutting

with CO2 laser from the sheet so that the specimen loading axis was aligned parallel to the transverse direction of the rolled

sheet.

Low rate tensile testing at and below the strain rate of 100 s�1 was carried out using an Instron 8800 servohydraulic

materials testing machine. Load was measured using a 100 kN Instron load cell and specimen strain with a 6 mm gauge

length extensometer. The servohydraulic materials testing machine was used also in the low rate tests of specimens

Table 11.1 Composition in weight percent and grain size of the test material

C Si Mn Cr Ni Mo Cu N Fe Mean intercept length (μm) ASTM GS

0.023 0.48 1.19 17.4 6.5 0.1 0.22 0.138 bal. 14 9

100 M. Isakov et al.

prestrained at a high rate with the recovery THSB apparatus. High strain rate testing near the strain rate of 1,000 s�1 was

carried out using the THSB apparatus at the Department of Materials Science of Tampere University of Technology.

The constant strain rate test setup was essentially the same as described in references [18] and [19]. Figure 11.2 shows the

components and dimensions of the setup. The setup consists of an incident bar made of tempered steel and a transmitted bar

made of an aluminum alloy. The sheet specimen is glued with cyanoacrylate adhesive into slits machined to the ends of the

bars. Compressed air is used to propel a steel striker tube against the flange at the free end of the incident bar. Figure 11.2

depicts also the momentum trap bar arrangement used in the recovery tests, which are discussed below. The elastic waves in

the bars are measured with strain gauge pairs attached to the bars so that wave overlapping does not take place during the

test. The strain gauge signals are amplified with Kyowa CDV-700A signal conditioners and recorded with a Yokogawa

DL708E digital oscilloscope. Dispersion correction of the waves is carried out with a procedure described in [20]. Classical

HSB wave analysis is used to calculate the specimen stress-strain behavior based on the force acting on the transmitted bar

end (for stress) and the relative motion of the bar ends (for strain rate and strain).

In order to facilitate a large strain rate change directly from low strain rates to the high strain rate region, the THSB

apparatus was fitted with a capability of prestraining the specimen in the strain rate range of 10�4 . . . 10�3 s�1. The low

strain rate loading setup consists of an electric motor and a spindle attached to the incident bar as well as a rigid clamp at the

far end of the transmitted bar. The main design principle was to minimize the possible effects of the low rate loading

equipment on the generation and measurement of the stress waves during the subsequent high rate THSB test. The

transmitted bar clamp is attached to the free end of the transmitted bar so far from the strain gauges that the transmitted

wave measurement is not interfered by wave reflections from the clamp. As shown in Fig. 11.3, low rate loading is

transferred from the motor/spindle –combination to the incident bar with a special fixture. The fixture is attached to the

incident bar with a M8 bolt so that the bolt transfers the quasi-static tensile loading but, when the striker hits the flange, the

bolt can slide freely and no dynamic loading is transferred through the fixture. This way the subsequent THSB test proceeds

in a manner similar to conventional THSB testing. Furthermore, the loading fixture collapses in a controlled way at the end

of the test so that the incident bar is stopped by the shock absorber shown in Fig. 11.3.

A 6 mm gauge length extensometer is used to directly measure the strain in the specimen gauge section. The extensometer

is manually removed from the specimen a few seconds prior to the start of the dynamic loading. The amount of additional

strain in the specimen between the removal of the extensometer and the start of the dynamic loading is monitored by a linear

4

8

25 mm

R2

22

812

42

RD

TDND

Fig. 11.1 Specimen geometry used in the study. Sheet thickness is 2 mm

3000 mm

Transmitted barØ22 (AA 2007)

Momentum trap tube outer Ø30, inner Ø27(low-alloy steel)

Removable flange(aluminum)

800

800800

500

4000 mm

20

Momentum trap bars Ø22 (tempering steel)

Incident bar Ø22(tempering steel)

Striker tube outer inner Ø22 (tempering steel)

Ø32,

Ø32

Fig. 11.2 Tensile Hopkinson split bar apparatus used in the high rate testing. Setup is shown in the recovery test configuration. For loading until

fracture the momentum trap bars are removed and a 1,600 mm long striker is used

11 Metastable Austenitic Steels and Strain Rate History Dependence 101

variable differential transducer (LVDT) attached to the incident bar. Throughout the test, the load acting on the specimen is

measured using the strain gauges attached to the bars. During testing it was observed that the specimen attachment by gluing

did not provide necessary long-term strength during the quasi-static loading phase. The best combination of joint strength and

signal quality was found by using both the cyanoacrylate adhesive and M8 bolts through the holes shown in Fig. 11.1.

To minimize impedance differences, a steel bolt and an aluminum bolt were used in the incident and transmitted bars,

respectively. It should be noted, however, that even with this setup some signal quality and sensitivity was lost since the sharp

yield peak observed for prestrained and room temperature aged specimens could not be detected when the bolts were used.

As noted above, the high rate loading phase following the low rate prestraining proceeds similarly to conventional THSB

testing. However, as Fig. 11.4 shows, the preloading introduces some differences in the wave measurements. The incident

and transmitted waves travelling in the bars are superimposed on the strain created by the preloading. The reflected wave

travels in an unloaded bar, because as the striker hits the flange, it disconnects the low rate loading device from the incident

bar. This effectively creates an unloading wave following the incident wave. Therefore, the strain gauge measures directly

the amplitude of the reflected wave, but for incident and transmitted waves the strain corresponding to the preload needs to

be subtracted from the signal in order to calculate the bar end velocities. In contrast, when the forces acting on the bar ends

are calculated, the preload has to be taken into account.

Previous investigators have proposed two different methods to recover the specimen in a THSB test. The first method relies

on a rigid protective fixture around the specimen [21, 22], while the second method is based on the complete removal of the

residual waves from the bars [23–26]. Often the latter method is preferable, since it leaves the specimen free from any

surrounding objects so that for example high speed photography or specimen temperature control devices can be used

similarly to conventional THSB tests. Furthermore, the bars are free of any additional fixtures between the strain gauges and

the specimen, which might affect the wave propagation in the bars and thus complicate the analysis of the wave data. After the

first loading sequence in a THSB test, two residual waves exist in the bars: the reflected wave in the incident bar and

the transmitted wave in the transmitted bar. The transmitted wave changes its sign to compression when it reflects from the

Incident bar

Free movement

Free movement (slide bearing)

Shock absorber

Spindle M8 bolt

Fig. 11.3 Equipment for low rate prestraining prior to the high rate loading in the THSB setup

0 500 1000 1500 2000-1.5

-1

-0.5

0

0.5

1

1.5

Time ( s)0 500 1000 1500 2000

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Time ( s)

Bar

str

ain

x 10

-3

Bar

str

ain

x 10

-3

a b

Fig. 11.4 Example of the recorded waves in a THSB test involving low rate prestraining. It should be noted that the bars in the setup have

dissimilar material properties


free end of the transmitted bar. The amplitude of the wave is proportional to the specimen stress during the first loading, so its

ability to cause further plastic deformation is very limited. In principle the compressive wave could cause buckling of the

specimen, but given the small length to cross-sectional area ratio of a typical THSB specimen, this situation is easily avoided,

when at least a portion of the transmitted wave is extracted. In contrast, the residual wave in the incident bar can have a

magnitude comparable to the original loading wave, and therefore its extraction is critical for a successful recovery test.

Nemat-Nasser et al. [23, 24] described a method to extract the reflected wave from the incident bar. In this method

a momentum trap bar is placed in front of the incident bar so that a carefully set gap exists between them before the test.

The principle of this method is that the gap closes precisely at the same instant as the incident wave formation ends. When

the wave reflected from the specimen end of the incident bar reaches the flange-momentum trap bar interface, it is

transmitted to the momentum trap bar instead of being reflected back towards the specimen. The crucial part and the

main challenge of this method is the control of the gap width. If the gap is too narrow, the incident bar hits the trap bar

already during the striker impact, which results in a disturbance in the incident wave and a possible total detachment of the

trap bar from the incident bar. If the gap is too wide, it will not close completely until the reflected wave causes further bar

motion. This, however, leads to incomplete extraction of the reflected wave from the bar. Although in theory the exact width

of the gap can be calculated based on the striker speed, in practice experimental uncertainties in the gap width setting and the

striker speed control can easily lead to a series of trial-and-error experiments and very low number of successful tests. Van

Slycken [25] addressed this problem by building an electromagnet based actuator system, which was able to adjust the gap

width just a couple of milliseconds prior to the impact of the striker. However, even this method relied on the accuracy of the

determination of the striker speed.

In this study another solution to the above described problem was used. This method is illustrated in Fig. 11.5. The idea is

to completely remove the need for the gap by adding an additional momentum trap bar in contact with the first momentum

trap bar. As shown in Fig. 11.5, when the striker hits the flange, a compression wave is imparted to the first momentum trap

bar and a tensile wave in the incident bar. Assuming that the incident bar is sufficiently long, the sequence of events is the

following: the compression wave in the first trap bar is completely transferred to the second trap bar, where it reflects as a

wave of tension and causes the separation of the two traps from each other (steps 2 and 3 in Fig. 11.5). This, however, leaves

the first trap bar free from any wave motion and in contact with the incident bar. At this point the situation corresponds to the

method presented by previous investigators [23–26] with an ideal gap width. When the wave reflected from the specimen

end of the incident bar reaches the flange end of the incident bar, it is completely transferred to the momentum trap bar

leaving the incident bar free from any residual waves (steps 4–6 in Fig. 11.5). The use of two momentum trap bars provides a

failsafe method to extract the reflected wave, since it is insensitive to the striker speed. The main drawback of the method is

that, as discussed above, part of the wave created by the striker impact is transferred to the momentum trap bar, which

necessitates the use of higher striker speeds than usually. Furthermore, amplitude variations may appear in the incident

wave, if the contact between the momentum trap and the incident bar is not perfect in the beginning of the test.

Figure 11.2 illustrates the implementation of the above described method in the THSB apparatus. The diameter of the

momentum traps equals the diameter of the incident bar. The application of the basic wave theory shows that this is needed

for a successful transfer of the reflected wave to the momentum trap. However, because of the impedance disturbance caused

by the flange, there exists a finite ring-up time, during which some wave reflection takes place at the flange-incident bar

1.

T

2.

6.

5.

4.

3.

C

C

C

Fig. 11.5 Illustration of the

reflected wave trapping with

the use of two momentum

traps. Open arrows denoterigid body motion. Closedarrows denote the stresswaves, “C” for compressional

and “T” for tensional wave


boundary. This means that the early part of the reflected wave cannot be removed from the bar. However, this ring-up time is

in the order of microseconds due to the small length of the flange. A removable flange made of aluminum and a momentum

trap tube made of low alloy steel were added to the transmitted bar in order to trap also the transmitted wave. The dimensions

and the material of the momentum trap were selected to minimize the impedance mismatch between the trap and the

transmitted bar. Slide bearings made of Teflon keep the trap centered around the bar. All momentum traps in the setup are

brought into a controlled stop after the test by viscous shock absorbers.

Figure 11.6 shows an example of a THSB test with a successful trapping of the residual waves. As can be seen, apart from

the small peak at the beginning of the incident wave, the initial waveform corresponds to an ordinary THSB test. The

specimen was fixed to the bars using M8 bolts. This inevitably leads to an increase in signal noise compared to specimen

attachment by gluing, but enables the specimen to be removed after the test without any mechanical or thermal loading,

which is the main purpose of the recovery test. As can be seen in Fig. 11.6, the reflected wave is almost completely trapped

into the momentum trap bar. Small residual waves, which can be related to the imperfect contact between the bars exist, but

their amplitude is too low to cause any further plastic deformation in the specimen. As seen in Fig. 11.6, also in the

transmitted bar the residual waves are below the elastic (and buckling) limit of the specimen.


Figure 11.7 presents examples of stress versus strain and strain hardening rate versus strain curves obtained at different strainrates at the initial temperature of +24 �C. Tests were also repeated at the temperatures of�40 �C and +80 �C (the results are

presented in [27]). The behavior of the test material changes drastically over the studied relatively narrow temperature range.

At +80 �C the material shows nearly parabolic behavior with continuously decreasing strain hardening rate except at very

high strains. At lower temperatures a distinct “S”-shaped flow curve is observed. As seen in Fig. 11.7, first the strain

hardening rate decreases to a very low value, then rapidly increases to a very high value, and again decreases until necking of

the specimen starts. Similar behavior was observed at�40 �C, but the changes in the strain hardening rate were more distinct

and took place at lower plastic strains. As was noted in the Introduction, this kind of behavior has been related to the strain-

induced austenite to α’-martensite phase transformation [1–9].

0 1000 2000 3000 4000 5000

-0.8

-0.4

0

0.4

0.8

Time ( s)

Incident bar

0 1000 2000 3000 4000 5000

-0.1

0

0.1

0.2

Time ( s)

Transmitted bar

Bar

str

ain

x 10

-3B

ar s

trai

n x

10-3

Fig. 11.6 Example of the recorded waves in a recovery THSB test. It should be noted that the bars in the setup have dissimilar material properties


It was observed [27] that at +80 �C the shape of the stress versus strain curve is relatively independent of strain rate, whileat lower temperatures notable changes take place, i.e., the maximum of strain hardening rate decreases and its occurrence is

shifted to higher strains. A closer examination of Fig. 11.7 shows that at strains below 0.1 the difference in the strain

hardening rate between different strain rates is quite small and becomes evident only at higher strains. It should be noted that

in the THSB tests the specimen strain was calculated based on the bar end motion, which easily causes an overestimation of

the specimen strain. This overestimation depends on the amount of additional deformation occurring outside the gauge

section, which in turn depends on the amount of strain hardening occurring within the gauge section. However, the THSB

results follow the trend observed already at 100 s�1, i.e., increasing strain rate decreases the strain hardening capability of the

test material at low temperatures.

Similar effects of strain rate on the behavior of metastable austenitic stainless steels have been reported previously

[10–14]. There seems to exist a general agreement that the strain hardening capability of these materials is decreased due to

the deformation-induced heating, which suppresses the austenite to α’-martensite phase transformation. Similar conclusions

can be drawn based on the results of this study. Based on in-situ measurements and numerical simulations [27] it was

confirmed that notable heating occurs already in the low strain rate region below 100 s�1.

Figure 11.8 shows the effect of a strain rate jump on the stress versus strain and strain hardening rate versus strain curvesin a jump test from 2 · 10�4 s�1 to 103 s�1. As can be seen, the post-jump strain hardening rate seems to follow the strain

hardening rate observed in the constant strain rate test at the higher rate rather than be affected by the lower strain rate prior

to the jump. This is especially evident at higher plastic strains, where the strain hardening rate rapidly decreases when the

strain rate is suddenly increased. This kind of behavior was observed also in jump tests from 2·10�4 s�1 to 100 s�1 carried out

with the servohydraulic materials testing machine.

0 0.1 0.2 0.3 0.4 0.50

200

400

600

800

1000

1200

1400

1600

True plastic strain

Tru

e st

ress

(M

Pa)

+24 °C

0 0.1 0.2 0.3 0.4 0.50

1000

2000

3000

4000

5000

6000

True plastic strain 1

Tru

e st

rain

har

deni

ng r

ate

(MP

a)

+24 °C

2x10-4 s-1

100 s-1

103 s-1

2x10-4 s-1

100 s-1

103 s-1

a b

Fig. 11.7 Material behavior at different strain rates when the strain rate is held constant during the deformation: (a) stress versus strain (b) strainhardening rate versus strain

0 0.1

a b

0.2 0.3 0.4 0.50

200

400

600

800

1000

1200

1400

1600

True plastic strain

Tru

e st

ress

(M

Pa)

+24 °C

↑

↑

0 0.1 0.2 0.3 0.4 0.50

1000

2000

3000

4000

5000

6000

True plastic strain

Tru

e st

rain

har

deni

ng r

ate

(MP

a)

+24 °C

↑ ↑2x10-4 s-1

103 s-1

2x10-4 s-1

103 s-1

Fig. 11.8 Material behavior in tests where the strain rate is suddenly increased during the test: (a) stress versus strain (b) strain hardening rate

versus strain. Black arrows denote the points of strain rate increase from the lower to the higher


Figure 11.9 presents the behavior of the test material in a test where the material was subjected to a high rate loading

up to true plastic strain of 0.1 with the recovery THSB apparatus, unloaded, and subsequently deformed at a strain rate of

2·10�4 s�1 with the servohydraulic materials testing machine. Similar behavior as in Fig. 11.8 can be observed, i.e., the strain

hardening rate is more dependent on the instantaneous value of the strain rate rather than on its previous values. However,

at the initial stages of the low rate deformation (between 0.1 . . . 0.13 of plastic strain in Fig. 11.9) the strain hardening rate

remains at a low value until it increases to its highest value at around 0.2 of plastic strain. This maximum value is somewhat

higher than that observed in the constant strain rate test (red curve in Fig. 11.9). This implies that already below 0.1 of plastic

strain there are differences in the microstructural evolution between different strain rates. These differences are probably

related to the nucleation of the strain-induced α’-martensite and its effects on the strain hardening rate. Microstructural

studies to reveal these differences are in progress [28].

A general view in the literature [10–14] is that deformation-induced heating is the reason for the suppression of the

deformation-induced martensitic phase transformation and decrease in the strain hardening capability with increasing strain

rate. This conclusion has been justified by comparing the temperature increase at high strain rates with the temperature

sensitivity of the material behavior observed at low strain rates. As noted above, this view is supported also by the constant

strain rate tests of this study. However, macroscopic heating cannot explain the immediate reduction in the strain hardening

rate observed after a sudden strain rate increase (Fig. 11.8). Shortly after the strain rate jump the bulk material temperature

should still be close to room temperature and the phase transformation should therefore readily take place and maintain the

strain hardening rate at a high level. As the deformation proceeds at the high strain rate, one would then expect to see a

gradual decrease in the strain hardening rate due to gradual material temperature increase. It is therefore evident that some

other mechanism than macroscopic adiabatic heating is responsible for the decrease of the strain hardening rate.

In terms of phenomenological modeling of the results in Fig. 11.8, a rather simple model can be applied as a first

approximation. The simplicity is based on the observation that the strain hardening rate seems to depend more on the current

value of strain rate than on its history. The model presented here is based on the concept of athermal and thermal components

of flow stress discussed for example by Klepazcko and Chiem [16]. Taking Eq. 11.1 as the basis for this approach and using

the strain rate sensitivity parameter β to account for the changes in the thermal part of the flow stress (with respect to a certain

nonzero reference strain rate), the strain rate history can be accounted for by integrating the strain hardening rate with respect

to plastic strain:

σðεp; _εpÞ ¼ σ0 þðεp

0

@σ

@εpðεp; _εpÞ dεp þ βlog10

_εp_εref

(11.4)

In Eq. 11.4 σ0 corresponds to the initial yield strength at the reference strain rate ( _εref ). Depending on the deformation

conditions, parameter β can be a function of plastic strain. This simple model does not explicitly include temperature as a

variable, but some temperature-effects are implicitly included in the strain rate dependence of the strain hardening rate.

A fully non-isothermal model would inherently contain cross-terms between the strain rate and temperature sensitivities of

both the strain hardening rate and instantaneous strain rate sensitivity β. The form of Eq. 11.4 is suitable for modeling the

0 0.1 0.2 0.3 0.4 0.50

200

400

600

800

1000

1200

1400

1600a b

True plastic strain

Tru

e st

ress

(M

Pa)

+24 °C

↓

0 0.1 0.2 0.3 0.4 0.50

1000

2000

3000

4000

5000

6000

True plastic strain

Tru

e st

rain

har

deni

ng r

ate

(MP

a)

+24 °C

↓

2x10-4 s-1

103 s-1

2x10-4 s-1

103 s-1

Fig. 11.9 Material behavior in a test where the strain rate is suddenly lowered in the test (discontinuous loading): (a) stress versus strain (b) strainhardening rate versus strain. Black arrow denotes the point of strain rate decrease from the higher to the lower


results presented in Fig. 11.8, i.e., for upward strain rate jumps. This is a direct consequence of the observed strain rate

history independent behavior of the strain hardening rate at the high strain rate. For this reason the strain hardening, which in

this case is the only source of history dependence, can be modeled with the simple plastic strain based integral term

presented in Eq. 11.4. This approach, however, has obvious short-comings in terms of modeling the transient behavior

following the strain rate change in the downward jumps. In order to account for the transients, the strain hardening rate itself

should be modeled as history dependent.

11.4 Conclusions

Metastable austenitic stainless steel EN 1.4318 shows strain rate history dependence of flow stress when deformed near room

temperature. However, the strain hardening rate is less history dependent and mainly dependent on the current value of strain

rate. This feature can be utilized in the numerical modeling of these steels. The experimental results presented in this paper

show that macroscopic adiabatic heating cannot solely explain the reduction of the strain hardening rate at high strain rates.

Acknowledgements M.Sc. Turo Salomaa and M.Sc. Jari Kokkonen are gratefully acknowledged for their technical help. This study was

conducted with the support from the FIMECC Ltd. (Finnish Metals and Engineering Competence Cluster) Demanding Applications Program.

References

1. Talonen J (2007) Effect of strain-induced α’-martensite transformation on mechanical properties of metastable austenitic stainless steels.

Doctoral thesis, Helsinki University of Technology

2. Lecroisey F, Pineau A (1972) Martensitic transformations induced by plastic deformation in the Fe-Ni-Cr-C system. Metallurgical Trans

3:387–396

3. Martin S, Wolf S, Martin U, Kruger L, Jahn A (2009) Investigations on martensite formation in CrMnNi-TRIP steels. DOI: 10.1051/esomat/

200905022

4. Narutani T, Olson GB, Cohen M (1982) Constitutive flow relations for austenitic steels during strain-induced martensitic transformation.

Journal de Physique 43:C4-429–C4-434

5. Spencer K, Conlon KT, Brechet Y, Embury JD (2009) The strain induced martensite transformation in austenitic stainless steels Part 2 – Effect

of internal stresses on mechanical response. Mater Sci Technol 25:18–28

6. Byun TS, Hashimoto N, Farrel K (2004) Temperature dependence of strain hardening and plastic instability behaviors in austenitic stainless

steels. Acta Mater 52:3889–3899

7. Guntner CJ, Reed RP (1962) The effect of experimental variables including the martensitic transformation on the low-temperature mechanical

properties of austenitic stainless steels. Trans Am Soc Metals 55:399–419

8. Kruger L, Wolf S, Martin U, Martin S, Scheller PR, Jahn A, Weiss A (2010) The influence of martensitic transformation on mechanical

properties of cast high alloyed CrMnNi-steel under various strain rates and temperatures. J Phys: conference series 240, Article number 012098

9. Rosen A, Jago R, Kjer T (1972) Tensile properties of metastable stainless steels. J Mater Sci 7:870–876

10. Lichtenfeld JA, Mataya MC, Van Tyne CJ (2006) Effect of strain rate on stress-strain behavior of Alloy 309 and 304L austenitic stainless steel.

Metallurgical Mater Trans A 37A:147–161

11. Talonen J, Nenonen P, Pape G, H€anninen H (2005) Effect of strain rate on the strain-induced γ ! α´-martensite transformation and

mechanical properties of austenitic stainless steels. Metallurgical Mater Trans A 36A:421–432

12. Larour P, Verleysen P, Bleck W (2006) Influence of uniaxial, biaxial and plane strain pre-straining on the dynamic tensile properties of high

strength sheet steels. Journal de Physique IV 134:1085–1090

13. Talyan V, Wagoner RH, Lee JK (1998) Formability of stainless steel. Metallurgical Mater Trans A 29A:2161–2172

14. Andrade-Campos A, Teixeira-Dias F, Krupp U, Barlat F, Rauch EF, Gracio JJ (2010) Effect of strain rate, adiabatic heating and phase

transformation phenomena on the mechanical behaviour of stainless steel. Strain 46:283–297

15. Ghosh AK (2007) On the measurement of strain-rate sensitivity for deformation mechanism in conventional and ultra-fine grain alloys. Mater

Sci Eng A 463:36–40

16. Klepaczko JR, Chiem CY (1986) On rate sensitivity of f.c.c. metals, instantaneous rate sensitivity and rate sensitivity of strain hardening.

J Mech Phys Solids 34:29–54

17. Curtze S, Hokka M, Kuokkala V-T, Vuoristo T (2006) Experimental analysis of the influence of specimen geometry on the tensile Hopkinson

split bar test results of sheet steels. In: Proceedings of the MS&T 2006 conference, Cincinnati, 15–19 Oct 2006

18. Hokka M (2008) Effects of strain rate and temperature on the mechanical behavior of advanced high strength steels. Doctoral thesis, Tampere

University of Technology

19. Curtze S (2009) Characterization of the dynamic behavior and microstructure evolution of high strength sheet steels Doctoral thesis, Tampere

University of Technology

20. Vuoristo T (2004) Effects of strain rate on the deformation behavior of dual phase steels and particle reinforced polymer composites.

Doctoral thesis, Tampere University of Technology


21. Lezcano RG, Essa YE, Perez-Castellanos JL (2003) Numerical analysis of interruption process of dynamic tensile tests using a Hopkinson bar.

Journal de Physique IV 110:565–570

22. Essa YES, Lopez-Puente J, Perez-Castellanos JL (2007) Numerical simulation and experimental study of a mechanism for Hopkinson bar test

interruption. J Strain Anal Eng Design 42:163–172

23. Nemat-Nasser S, Isaacs JB, Starret JE (1991) Hopkinson techniques for dynamic recovery experiments. Proc R Soc A 435:371–391

24. Nemat-Nasser S (2000) Recovery Hopkinson bar techniques. In: ASM handbook, vol 8, mechanical testing and evaluation, 1st printing,

ASM International, Materials Park

25. Van Slycken J (2008) Advanced use of a split Hopkinson bar setup application to TRIP steels. Doctoral thesis, Ghent University

26. Huang W, Huang Z, Zhou X (2010) Loading and unloading split Hopkinson tension bar technique for studying dynamic microstructure

evolution of materials. Adv Mater Res 160–162:891–894

27. Isakov M (2012) Strain rate history effects in a metastable austenitic stainless steel. Doctoral thesis, Tampere University of Technology.

Available online at http://URN.fi/URN:ISBN:978-952-15-2919-1

28. Isakov M, Ostman K, Kuokkala V-T unpublished research


http://urn.fi/URN:ISBN:978-952-15-2919-1

Chapter 12

Measurement Uncertainty Evaluation for High Speed Tensile

Properties of Auto-body Steel Sheets

M.K. Choi, S. Jeong, H. Huh, C.G. Kim, and K.S. Chae

Abstract Evaluation of the crashworthiness is one of the important issues in the automotive industry. In order to evaluate

the crashworthiness of auto-body structure, finite element method has been conducted for crash analysis. Generally, strain

rates distribution of auto-body structure in the car crash ranges from 0.001/s to 500/s. Since material properties of steel

sheets depend on the strain rates, the dynamic behavior of sheet metals must be examined and applied to the finite element

model appropriately. This paper is concerned with the evaluation of measurement uncertainty of high speed tensile

properties of auto-body steel sheets. Obtaining procedure of the true stress�true strain data at intermediate strain rates is

properly designed for the experiment and data acquisition. The measurement uncertainty of the true stress is evaluated

considering sources of uncertainties of input quantities and their associated sensitivity coefficients. A combined standard

uncertainty is evaluated from not only the uncertainties of the input quantities but also influence factors of high speed tensile

tests. The results show that the measurement uncertainty evaluation procedure has been successfully applied to high speed

tensile properties.

Keywords Measurement uncertainty • High speed tensile properties • Intermediate strain rates • Auto-body steel sheets

• Uncertainty evaluation

12.1 Introduction

The auto-body design is usually performed to achieve lightweight design with enhanced crashworthiness by numerical

analysis. At car crash, the strain rates in an auto-body structure are distributed in a wide range such that the maximum strain

rate reaches to about 500/s while the minimum strain rate is near quasi-static [1–4]. Such variation of strain rates has a

significant effect on the material properties of auto-body steel sheets. Generally the true stress of a steel sheet increases as the

strain rate increases. The material properties of auto-body steel sheets with the variation of strain rates need to be measured

with an appropriate measurement procedure for accurate numerical analysis.

Many researchers have studied experimental methods to identify mechanical properties of materials at intermediate strain

rates. Recently, servo-hydraulic machines are employed in tensile tests at intermediate strain rates. Huh et al. developed a

servo-hydraulic machine for high speed tensile tests at strain rates ranging from 0.1/s to 200/s. Strain rate effects on the

M.K. Choi • H. Huh (*)

School of Mechanical, Aerospace and Systems Engineering, Korea Advanced Institute of Science and Technology,

Daeduk Science Town, 305-701 Daejeon, South Korea


S. Jeong

Hyundai Heavy Industries, 1,000, Bangeojinsunhwan-doro, 682-792 Dong-gu, Ulsan, South Korea

C.G. Kim • K.S. Chae

Korea Research Institute of Standards and Science, 267 Gajeong-ro, 305-430 Yuseong-gu, Daejeon, South Korea



109


tensile properties of auto-body steel sheets are investigated with the testing machine considering temperature variations

[5–7]. Durrenberger et al. conducted tensile tests of auto-body steel sheets at a wide range of strain rates in order to formulate

a visco-plastic constitutive model which describes strain rate dependency [8]. Beguelin et al. investigated strain rate

sensitivity to the yield stress and the drawn stress of polymeric composite materials at intermediate strain rate using

a servo-hydraulic testing machine [9].

High speed tensile properties mentioned above have to be acquired considering the measurement uncertainty as well as

the reliability and the traceability of the experiment. The standardized test and verification methods for high speed tensile

tests have not be established yet while those for quasi-static tensile tests have been established. Bahng et al. tried to establish

the traceability chain in the measurement of mechanical properties through international round-robin tests with development

of certified reference materials and uncertainty evaluation [10]. After the guide to the expression of uncertainty in

measurement (GUM) was published [11], some of researchers conducted estimation of uncertainties in tensile properties.

Lord and Morrell discussed some practical issues and estimated uncertainty sources associated with the tensile test that

needs to be considered to acquire reliable values for the Young’s modulus [12]. However, uncertainty sources in determining

the true stress with respect to the true strain have not been investigated at intermediate strain rates.

This paper deals with the measurement uncertainty evaluation for high speed tensile properties of auto-body steel sheets.

Tensile tests were conducted with a servo-hydraulic testing machine and a high speed camera at intermediate strain rates

ranging from 0.1/s to 100/s. The measurement procedure properly follows the ISO standard method [12, 14] which uses

a servo-hydraulic machine and proposes indirect displacement measurement methods such as a use of a laser extensometer

and digital image analysis with a high speed camera. The measurement uncertainty of the true stress is evaluated considering

sources of uncertainties of input quantities and their associated sensitivity coefficients. Fianally, a combined standard

uncertainty of the true stress data with respect to the true stain is evaluated from not only the uncertainties of the input

quantities but also influence factors of high speed tensile tests according to the law of uncertainty propagation.

12.2 Measurement Procedure of High Speed Tensile Properties

A servo-hydraulic high speed material testing machine was used in high speed tensile tests, which is shown in Fig. 12.1.

The machine has a maximum stroke velocity of 7,800 mm/s, a maximum stroke displacement of 300 mm, and its maximum

measurable load is 30 kN.

Since the distribution of the strain and the strain rate is dependent on the specimen geometry, it is necessary to determine

the specimen dimensions for high speed tensile tests. Huh et al. proposed appropriate specimen dimensions of auto-body

steel sheets for high speed tensile tests. They investigated distributions of the strain and the strain rate in a specimen by finite

element analysis and proposed specimen dimensions which were confirmed by experiments [5]. The dimension and shape of

specimen is shown in Fig. 12.2.

The overall measurement procedure is summarized in Fig. 12.3 as a flow chart. The initial width and thickness of the

specimen are measured by digital vernier calipers. Tensile speed of lower grip of the testing machine is adjusted for

imposing the strain rate on the specimen accurately. After this adjustment, the specimen is installed on the upper jig in the

testing machine and then the tensile testing is operated. The load data is acquired by a load cell and DAQ board. The load

data is then smoothened with the FFT filter in order to eliminate oscillation signals from high speed tensile tests. The

oscillation is mostly induced by the load-ringing phenomenon. In order to reduce the oscillation signals due to the load-

ringing phenomenon, the load signals are transformed to frequency domain and higher frequency components are removed

by a FFT filter. The initial and deformed lengths of the specimen with respect to the time are acquired by digital image

analyses using capture images from the high speed camera (FASTCAM SA4, Photron). A length in the captured image is

defined as a distance between two designated points. Square grids are marked on the specimen surface with uniform spaces

of 1 mm and the designated points can be prescribed in the captured image. Change of the distance is measured by counting

the number of pixels between the two designated points in images. The actual length of the distance is calculated by

calibration of high speed camera images with microscope images which contains standard scale of 1 mm length. Lastly,

the load and displacement data are synchronized with time and the load–displacement data is converted to the true

stress–true strain data. The whole measurement procedure was repeated five times at the same condition to observe the

reproducibility of the test.

110 M.K. Choi et al.

12.3 Measurand, Input Quantities and Influence Factors

of High Speed Tensile Properties

In the uncertainty evaluation of the tensile properties, it is important to specify a measurand, input quantities and influence

factors of the test. A measurand is an object being measured. It is determined by measurement results of input quantities and

affected by influence factors of the test. Input quantities and influence factors can be sources of uncertainties and they have to

be accounted in the tensile test with a systematically established evaluation procedure. The measurand in this measurement

is the true stress with respect to the true strain at a prescribed strain rate. The true stress is defined by dividing the applied

force by the current cross-sectional area as shown in Eq. 12.1 where σt denotes the true stress, F and Ad stand for the applied

load to the specimen and the cross-sectional area at deformed state respectively. The true stress is determined by

measurement results of the input quantities: the load; the initial thickness; the initial width; and the initial and deformed

length in the gauge section. The current cross-sectional area is related to the initial dimensions as shown in Eq. 12.3 by the

conservation of mass as shown in Eq. 12.2. In the case of a thin specimen the true stress can be defined as the measurand with

five measurable input quantities taking account of the thickness and width to the cross-sectional area at a deformed region.

σt ¼ F

Ad(12.1)

Fig. 12.2 Dimension

of the tensile specimen

Fig. 12.1 High speed

material testing machine

(HSMTM)

12 Measurement Uncertainty Evaluation for High Speed Tensile Properties of Auto-body Steel Sheets 111

ρ0A0l0 ¼ ρdAdld; ρ0 ¼ ρd ¼ constant (12.2)

σt ¼ F

Ad¼ F

A0

A0

Ad¼ F

A0

ldl0¼ F

A0

1þ ld � l0l0

� �¼ F

t0w0

1þ ld � l0l0

� �¼ f ðF; t0;w0; l0; ldÞ (12.3)

l0 and ld are the initial and deformed length between two designated points in the gauge section. t0 and w0 are the initial

thickness and width of the gauge section in a specimen. These are input quantities to be used to determine the measurand.

In addition to input quantities, the true stress is also affected by influence factors such as the FFT filter smoothing process,

the strain rate change during the test, and a deviation of results in repeated tensile tests. The load data can be distorted when

oscillation signals from high speed tensile test are reduced by the FFT filter smoothing process. As tensile properties of

auto-body steel sheets are generally depend on the strain rates, the strain rate change during the test is also a influence factor

of the test. Lastly, there’s a lack of uniformity of the tested specimen, we should consider a deviation of results in repeated

tensile tests as a influence factor of the test.

12.4 Analytic Model for the Measurement Uncertainty of the High Speed Tensile Properties

When a measurand of Y is defined as a function of a set of input quantities of Xi, the combined uncertainty of the measurand

is calculated by combining all of the uncertainty components according to the input quantities as explained below:

Y ¼ f ðX1;X2; � � � ;XnÞ (12.4)

um2 ¼

Xni¼1

@f

@Xi

� �2

ui2 ¼

Xni¼1

ci2ui

2 (12.5)

Fig. 12.3 Measurement procedure to obtain high speed tensile properties


where um is the combined standard uncertainty of the measurand, and ui is the measurement uncertainty of Xi and

corresponding sensitivity coefficient ci is derived by taking the partial derivative of f. The combined standard uncertainty

is defined as the square root of summation of squared measurement uncertainty components associated with the input

estimates. The true stress is determined with five input quantities as mentioned in Eq. 12.3 and the combined standard

uncertainty is calculated with the measurement uncertainties of the input quantities and their sensitivity coefficients.

The sensitivity coefficients are defined by taking partial derivative of the true stress by corresponding input quantities.

The combined standard uncertainty of the measurand is calculated as shown in Eq. 12.6.

um ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXni¼1

ci2ui2

s

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi@σt@F

� �2

u12 þ @σt@t0

� �2

u22 þ @σt@w0

� �2

u32 þ @σt@l0

� �2

u42 þ @σt@ld

� �2

u52

s(12.6)

In evaluation of the measurement uncertainty of the true stress data, uncertainty components are added considering the

FFT smoothing process, the strain rate change during the test and the deviation of results in repeated tensile tests as well as

the uncertainties of input quantities. Therefore, the combined standard uncertainty of the true stress data is calculated as

shown in Eq. 12.7

uc ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffium2 þ uf 2 þ u _ε2 þ ur2

q(12.7)

where uf denotes the uncertainty from the FFT smoothing process, u _ε stands for the uncertainty by the strain rate change

during the test and ur stands for the experimental standard deviation in the repeated measurements. Uncertainty components

are summarized in Table 12.1 according to their uncertainty sources associated with the measurement procedure for the true

stress.

12.5 Measurement Uncertainty Evaluation for the Applied Load

To evaluate the measurement uncertainty for the applied load, a calibration error of load cell and noise in the signal

amplification have to be considered as well as the limitation of resolution in data acquisition system. Uncertainty evaluation

in the load measuring system is conducted by carrying out a standard calibration test. The standard calibration test is carried

out with a deadweight force standard machine which was calibrated by Korea research institute of standards and science

(KRISS) and certified by intercomparisons with the national metrology institute of Japan (NMIJ) and the physikalisch-

technische bundesanstalt (PTB, Germany). The result evaluated from the standard test represents overall uncertainty of the

load measuring system which includes a load cell, an amplifier and a DAQ board.

Results from the standard calibration test show that the maximum value of the expanded uncertainty was estimated as

0.620 % when the applied load is 5 kN. Since the maximum load is usually observed in the in the high speed tensile tests for

Table 12.1 Uncertainty sources in measurement of the high speed tensile properties

Measurement

Symbol of input

quantities

Uncertainty

components Source of uncertainty

Measurement of load F u1 Uncertainty of load measuring system

Measurement of thickness

and width

t0 u2 Determination of dimensions using a vernier calipers

w0 u3 Deviation in repeated measurement

Reliability of vernier calipers

Measurement of length l0 u4 Determination of length by digital image correlation

ld u5 Limit of resolution in digital images

Uncertainties from the calibration procedure of high-speed

camera images

Signal processing – uf Signal distortion by the FFT filtering

Strain rate change – u _ε Strain rate change during the test

Repeated experiments – ur Deviation in repeated measurements


auto-body steel sheets ranging from 1 to 10 kN, it is rational evaluation to adopt 0.620 % as the representative expanded

uncertainty value. The absolute value of the measurement uncertainty is calculated by multiplying the relative uncertainty by

the applied load as shown in Eq. 12.8 where Ul and k denote the expanded uncertainty and the coverage factor of the load

measuring system respectively. u1 denotes the measurement uncertainty of the load data and c1 is the associated sensitivity

coefficient. The sensitivity coefficient is calculated by taking partial derivative of the true stress with respect to the load as

shown in Eq. 12.9.

u1 ¼ F� Ul

k(12.8)

c1 ¼ @σt@F

¼ 1

t0w0

1þ ld � l0l0

� �(12.9)

12.6 Measurement Uncertainty Evaluation for Initial Dimensions

The measurement uncertainties of the initial thickness and width are evaluated considering the deviations in the repeated

measurements as well as the limitation of accuracy of a measuring device. The thickness and the width are measured by

digital vernier calipers. The specification of the vernier calipers indicates that the maximum error is 0.02 mm. The maximum

error implies a range which is defined by the upper and lower bounds of the indicated value. The indicated value can be

located with the equal probability in the range and the measurement uncertainty is then calculated using the rectangular

probability distribution. The value of the maximum error is assumed as a half-width of the probability distribution and the

measurement uncertainty of the vernier calipers can be calculated as shown in Eq. 12.10.

uv ¼ 0:02ffiffiffi3

p ¼ 0:0115 ½mm� (12.10)

The thickness and the width of a specimen were determined as the mean value of the repeated measurements. The

measurements are repeated nine times at three different locations along the longitudinal direction of the specimen.

The measurement uncertainties are evaluated as shown in Eqs. 12.11 and 12.12 where u2 and u3 stand for the standard

uncertainties of the initial thickness and the initial width respectively. The associated sensitivity coefficients are calculated

by taking derivatives of the true stress with respect to the initial thickness and the initial width as shown in Eqs. 12.13

and 12.14

u2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�u22 þ uv2

p¼ 0:0135 ½mm� where �u2 ¼ s�t0 ¼

st0ffiffiffin

p ¼ 1ffiffiffin

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

n� 1

Xni¼1

t0ji � �t0� �2s

(12.11)

u3 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�u32 þ uv2

p¼ 0:0133 ½mm� where �u3 ¼ s�w0

¼ sw0ffiffiffin

p ¼ 1ffiffiffin

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

n� 1

Xni¼1

w0ji � �w0

� �2s(12.12)

c2 ¼ @σt@t0

¼ � F

t0ð Þ2 � w0

� ldl0

(12.13)

c3 ¼ @σt@w0

¼ � F

t0 � w0ð Þ2 �ldl0

(12.14)

where s�t0 and s�w0denote the experimental standard deviation of the mean of the initial thickness and the initial width

respectively.


12.7 Measurement Uncertainty Evaluation for the Initial and Deformed Length

Deformation of a specimen was captured by digital images using the high speed camera during the tensile test. The initial

and deformed length of specimen were calculated by counting the number of pixels between two designated points in the

captured images using digital image analysis. The number of pixels has to be converted in the real scale which has a physical

dimension. The actual length between the two designated points is then estimated by multiplying the number of pixels to the

actual dimension of a pixel in the captured image. The actual length of l is defined with four input quantities as shown in

Eq. 12.15

l ¼ P1 rclmP4

� �where rc ¼ P3

P2

(12.15)

where P1 is the number of pixels between the designated points. The actual dimension of a pixel is calibrated by scaling the

high speed camera image using a reference length of a stage micrometer with microscope observations. P2 denotes the size of

a grid in pixels in a high speed camera image and P3 denotes that of the same grid in a microscope image. rc is defined

by dividing P3 by P2 resulting that rc implies the scale ratio between a high speed camera image and a microscope image.

lm stands for the actual length of the stage micrometer. P4 is the number of pixels corresponding to a length of the stage

micrometer in a microscope image.

usc stands for the measurement uncertainty of a length and is estimated by combining uncertainties of associated input

quantities as shown in Eq. 12.16. Uncertainty components are calculated with separate observations and associated

sensitivity coefficients are defined by taking derivatives of the length with respect to P1, rc, P4 and lm as shown in

Eq. 12.17. Uncertainty components are tabulated for determination of the initial and deformed length and associated

input quantities as shown in Table 12.2.

usc ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

@l

@P1

� �2

us12 þ @l

@rc

� �2

us22 þ @l

@P4

� �2

us32 þ @l

@lm

� �2

us42

s(12.16)

cs1 ¼ @l

@P1

¼ rclmP4

; cs2 ¼ @l

@rc¼ P1

lmP4

; cs2 ¼ @l

@rc¼ P1

lmP4

; cs3 ¼ @l

@P4

¼ �P1 rclm

P42

� �; cs4 ¼ @l

@lm¼ P1 rc

1

P4

� �(12.17)

The selected pixel only defines a range where the point locates since a pixel has a finite size due to the limitation of

resolution in the digital images. The limitation of resolution has to be accounted as one of the uncertainty components

considering the size of a pixel. If the size of a pixel is a, the half-width of the gradation is determined by a/2 and the

measurement uncertainty of defining a location in pixels is calculated as shown in Eq. 12.18 using the rectangular probability

distribution.

up ¼ a

2� 1ffiffiffi

3p ¼ affiffiffiffiffi

12p (12.18)

Table 12.2 Uncertainty components of the length measured by image analysis

Measurement

Symbol of input

quantities

Measured

value Source of uncertainty

Measurement

uncertainty

Number of pixels between designated points in the

high-speed camera image

P1 [pixel] 474 Limit of resolution in digital images us1 ¼ 0.408

Scale ratio between high-speed camera image and

microscope image

rc½pixel=pixel� 11.76 Limit of resolution of high-speed camera

image and microscope image

us2 ¼ 0.144

Observed difference in repeated estimation for

18 grids

Length of a reference scale in the microscope

image

P1 [pixel] 482.9 Limit of resolution in microscope image us3 ¼ 0.410

Deviation of repeated measurement

Actual length of a reference scale [1 mm] lm [mm] 1.0 Uncertainty of a reference scale us4 ¼ 0.003


In order to determine P1, two different points need to be designated in a captured image. us1 denotes the measurement

uncertainty of P1 and is evaluated by counting up twice as shown in Eq. 12.19.

us1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiup2 þ up2

q¼ 1ffiffiffi

6p ¼ 0:408 ½pixel� (12.19)

Uncertainties in determination of P2 and P3 are accounted in the same way of P1 considering the width of a pixel.

Determination of rc is carried out for 18 grids in a specimen, which are located side-by-side along the longitudinal direction.

rc is estimated as the mean value of the measurement results for 18 grids and the measurement deviation is accounted into the

measurement uncertainty. The measurement uncertainty of rc is evaluated considering the uncertainties in determination

of P2 and P3 as well as the standard deviation as shown in Eq. 12.20. Sensitivity coefficients are calculated as shown

in Eq. 12.21

us2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficr12ur12 þ cr22ur22 þ �us22

pwhere �us2 ¼ src ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

n� 1

Xni¼1

rcji � �rc� �2s

(12.20)

cr1 ¼ @rc@P2

¼ � P3

P22; cr2 ¼ @rc

@P3

¼ 1

P2

(12.21)

where us2, ur1 and ur2 denote the measurement uncertainties of rc, P2 and P3 respectively.

P4 refers to the length of the stage micrometer which is utilized as a reference scale in a microscope image.

The measurement uncertainty of P4 which is denoted as us3 is estimated considering the limitation of resolution and the

standard deviation of repeated measurements as shown in Eq. 12.22. s �P4stands for the experimental standard deviation of the

mean of P4.

us3 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiup2 þ up2 þ �us32

qwhere �us3 ¼ s �P4

¼ sP4ffiffiffin

p ¼ 1ffiffiffin

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

n� 1

Xni¼1

P4ji � �P4

� �2s(12.22)

lm refers to the length of the stage micrometer and us4 denotes the measurement uncertainty of the length of the stage

micrometer. Us stands for the expanded uncertainty of the stage micrometer and is given according to its calibration test by

Korea research institute of standards and science(KRISS). Results of calibration test show that the measurement uncertainty

was estimated to be 0.003 mm.

The measurement uncertainty in the initial and deformed lengths between two designated points was calculated by

Eq. 12.17 combining all of the uncertainty components. Locations of the designated points are shown in Fig. 12.4. When the

initial length is 10.19 mm, the measurement uncertainty of measured initial length is estimated to be 0.142 mm resulting in

u4 ¼ 0.142 [mm]. The measurement uncertainty of a deformed length is 0.170 mmwhen the measured length was 11.78 mm

resulting in u5 ¼ 0.170 [mm]. The sensitivity coefficients of the initial length and the deformed length are calculated with

the following equations.

c4 ¼ @σt@l0

¼ F

t0 � w0

� ld

�l02

(12.23)

c5 ¼ @σt@ld

¼ F

t0 � w0

� 1l0

(12.24)

12.8 Measurement Uncertainty Evaluation for Influence Factors

The true stress is also affected by influence factors such as the FFT filter smoothing process, the strain rate change during the

test, and a deviation of results in repeated tensile tests. To evaluate the measurement uncertainties of the high speed tensile

properties, the measurement uncertainties by influence factors of the high speed tensile test should be considered.


Signal distortion is investigated in a FFT smoothing method by comparing smoothened results to the original data of a

reference load curve. Difference between the smoothened result and the original data is accounted as the standard

uncertainty in the signal processing procedure. The load data is prepared by adding the extracted oscillated signal to the

reference load data. The FFT smoothing is applied to the generated load data as shown in Fig. 12.5a. The difference between

the smoothing result and the reference load data is estimated as shown in Fig. 12.5b. The maximum difference is about

0.71 % and the maximum value is regarded as the standard uncertainty of the performed FFT smoothing process.

During the tensile test, the strain rate on the specimen gradually decreases as a parallel region of the specimen increases

by its deformation. Change of the strain rate on the specimen as shown in Eq. 12.25 was measured using high speed camera

images and the measurement uncertainty of the strain rate change can be obtained using Eq. 12.26.

Fig. 12.4 Locations of designated points in the specimen: (a) designated points at the initial state; (b) designated points at a deformed state

0 1 2 3 4 5

a b

0.0

0.5

1.0

1.5

2.0 Generated load dataSmoothing result

Lo

ad [

kN]

Time [msec]0.0 0.5 1.0 1.5 2.0

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

Dif

fere

nce

bet

wee

nre

fere

nce

an

d F

FT

res

ult

[%

]

Time [msec]

Max difference = 0.71%

Fig. 12.5 Result of an FFT smoothing process: (a) generated load data and the smoothing result; (b) difference between the reference load data

and the smoothing result


_εðtiÞ ¼ ΔlΔt

1

li(12.25)

usr ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficΔl2uΔl2 þ cΔt2uΔt2 þ cli

2uli2 þ udiff 2

qwhere cX ¼ @ _ε

@X(12.26)

uΔl refers to the measurement uncertainty of distance change at a certain time ti and it contains the measurement uncertainty

of deformed length, uΔt denotes to the measurement uncertainty of time duration and sources of uncertainty is the limitation

of resolution of frame speed. uli refers to the measurement uncertainty at certain distance and udiff denotes to difference

between target strain rate and measured strain rate. As the tensile properties of auto-body steel sheets are generally depends

on the strain rates, the true stress can be a function of the strain rates. Lim�Huh model, which is shown as Eq. 12.27,

indicates constitutive equation considering strain rate and the measurement uncertainty of the true stress by the strain rate

change can be obtained using this model. Equation 12.28 shows how the measurement uncertainty of the true stress by the

strain rate change is evaluated.

σðε; _εÞ ¼ σrðεÞ � 1þ qðεÞ � _εm1þ qðεÞ � _εrm where qðεÞ ¼ q1

ðεþ q2Þq3 (12.27)

u _ε ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi@σ

@ _ε

� �2

usr2

s¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiσrðεÞ � m � qðεÞ � _εm�1

1þ qðεÞ � _εrm� �2

usr2

s(12.28)

Tensile tests were repeated five times to determine a representative value of the true stress. The representative value is

determined as the mean of the repeated measurements. The standard uncertainty is then estimated as the experimental

standard deviation as shown in Eq. 12.29. In this repetition, the experimental standard deviation implies the reproducibility

of the measurement since different tensile specimens are used for the repeated tests.

ur ¼ sσt ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

n� 1

Xni¼1

ðσtji � �σtÞ2s

(12.29)

Up to this point, the sources of uncertainties are evaluated for the measurement of input quantities as well as influence

factors of the high speed tensile test. The combined standard uncertainty of the true stress data is computed including all of

the uncertainties as shown in Eq. 12.7.

For an auto-body steel sheet of DP590, the true stress data is measured at intermediate strain rates ranged from quasi-

static to 100/s and the associated measurement uncertainties are evaluated for the true stress data measured. Measured true

stress-true strain data of DP590 are shown in Fig. 12.6 at the strain rate of 1/s, 10/s and 100/s respectively. The measurement

uncertainties of the data are evaluated with respect to the true strain as shown in Fig. 12.7. The absolute value of the

measurement uncertainty is represented along with the relative value which is calculated by dividing the absolute value by

0.00 0.05 0.10 0.15 0.200

100

400

500

600

700

800

900 DP590 1.0t

Tru

e st

ress

[M

Pa]

True strain

100/sec10/sec1/sec

Fig. 12.6 Measured true

stress of DP590 (1.0 t) at

various strain rates


measured true stress. In the evaluation, the measurement uncertainty increases with respect to the true strain and the strain

rate since the measured load is increased by the strain hardening and strain rate hardening effects of the material.

12.9 Conclusion

This paper proposes a procedure to evaluate the measurement uncertainty for the true stress data obtained from the high

speed tensile test. The measurement procedure is presented to obtain the true stress data including a load measuring system

and a digital image correlation process to quantitatively measure the deformation of tensile specimens. The measurement

uncertainty is also evaluated for the signal processing, the strain rate change and a deviation of results in repeated tensile

tests. In order to estimate the combined standard uncertainty an analytic model is established according to the law of

uncertainty propagation. The true stress data of auto-body steel sheets of DP590 were measured as an example of the

proposed measurement procedure.

References

1. Yoon JH, Huh H, Kim SH, Kim HK, Park SH (2005) Comparative crashworthiness assessment of the ULSAB-AVC model with advance high

strength steel and conventional steel. proc IPC 13:724–747

2. Huh H, Lim JH, Song JH, Lee KS, Lee YW, Han SS (2003) Crashworthiness assessment of side impact of an auto-body with 60TRIP steel for

side members. Int J Automot Techn 4:149–156

0.00 0.05 0.10 0.15 0.200

5

10

15

20

a

c

bC

om

bin

ed s

tan

dar

d u

nce

rtai

nty

of

the

tru

e st

ress

[M

Pa]

True strain

Combined standard uncertainty, Uc [MPa]

0

2

4

6

8Relative standard uncertainty, Uc /σ [%]

Rel

ativ

e st

and

ard

un

cert

ain

tyo

f th

e tr

ue

stre

ss [

%]

max uc=13.07 [MPa]

max uc=14.10 [MPa]

max uc=13.56 [MPa]

max uc /σ=1.71 [%]

max uc /σ=1.74 [%]

max uc /σ=1.83 [%]

0.00 0.05 0.10 0.15 0.200

5

10

15

20

Co

mb

ined

sta

nd

ard

un

cert

ain

tyo

f th

e tr

ue

stre

ss [

MP

a]

True strain

0

2

4

6

8

Rel

ativ

e st

and

ard

un

cert

ain

tyo

f th

e tr

ue

stre

ss [

%]

0.00 0.05 0.10 0.15 0.200

5

10

15

20

Co

mb

ined

sta

nd

ard

un

cert

ain

tyo

f th

e tr

ue

stre

ss [

MP

a]

True strain

0

2

4

6

8

Rel

ativ

e st

and

ard

un

cert

ain

tyo

f th

e tr

ue

stre

ss [

%]


Relative standard uncertainty, Uc /σ [%]


Relative standard uncertainty, Uc /σ [%]

Fig. 12.7 Standard uncertainty of measured true stress of DP590 (1.0 t): (a) strain rate: 1/s; (b) strain rate: 10/s; (c) strain rate: 100/s


3. Yoshitake A, Sato K, Hosoya Y (1982) A study on improving crashworthiness of automotive parts by using high strength steel sheets.

SAE Technical Paper: 980382

4. Mahadevan K, Liang P, Fekete J (2000) Effect of strain rate in full vehicle frontal crash analysis. SAE Technical Paper: 2000-01-0625

5. Huh H, Kim SB, Song JH, Lim JH (2008) Dynamic tensile characteristics of TRIP-type and DP-type steel sheets for an auto-body. Int J Mech

Sci 50:918–931

6. Huh H, Lim JH, Park SH (2009) High speed tensile test of steel sheets for the stress–strain curve at the intermediate strain rate. Int J Automot

Techn 10:195–204

7. Huh H, Lee HJ, Song JH (2012) Dynamic hardening equation of the auto-body steel sheet with the variation of temperature. Int J Automot

Techn 13:43–60

8. Durrenberger L, Klepaczko JR, Rusinek A (2007) Constitutive modeling of metals based on the evolution of the strain-hardening rate. J Eng

Mater–T ASME 129:550–558

9. Beguelin P, Barbezat M, Kausch HH (1991) Mechanical characterization of polymers and composites with a servohydraulic high speed tensile

tester. J Phys III 1:1867–1880

10. Bahng GW, Kim JJ, Lee HM, Huh YH (2010) Establishment of traceability in the measurement of the mechanical properties of materials.

Metrologia 47:32–40

11. BIPM, IEC, IFCC, ISO, IUPAC, IUPAP, OIML (1995) Guide to the expression of uncertainty in measurement. International Organization for

Standardization, Geneva

12. Lord JD, Morrell RM (2010) Elastic modulus measurement–obtaining reliable data from the tensile test. Metrologia 47:32–40

13. ISO (2009) Metallic materials–Tensile testing: 1. Method of test at room temperature ISO 6892–1. International Organization for

Standardization, Geneva

14. ISO (2011) Metallic materials–Tensile testing method at high strain rates, part 2: Servo-hydraulic and other test systems ISO 26203–2.

International Organization for Standardization, Geneva


Chapter 13

Effect of Water Absorption on Time-Temperature

Dependent Strength of CFRP

Masayuki Nakada, Shuhei Hara, and Yasushi Miyano

Abstract A general and rigorous advanced accelerated testing methodology (ATM-2) for the long-term life prediction of

polymer composites exposed to an actual loading having general stress and temperature history has been proposed.

The tensile and compressive static strengths in the longitudinal and transverse directions of two kinds of unidirectional

CFRP under wet condition are evaluated using ATM-2. The applicability of ATM-2 can be confirmed for these static

strengths. The effect of water absorption on the time and temperature dependence of these static strengths can be

characterized by the viscoelastic behavior of matrix resin.

Keywords Carbon fiber reinforced plastics • Water absorption • Strength • Time-temperature dependence • Viscoelasticity

13.1 Introduction

Carbon fiber reinforced plastics (CFRP) are now being used for the primary structures of airplanes, ships and others, in

which the high reliability should be kept during the long-term operation. Therefore, it would be expected that the accelerated

testing methodology for the long-term life prediction of CFRP structures exposed under the actual environments of

temperature, water, and others must be established.

We have proposed a general and rigorous advanced accelerated testing methodology (ATM-2) which can be applied to

the life prediction of CFRP exposed to an actual load and environment history based on the three conditions. One of these

conditions is the fact that the time and temperature dependence on the strength of CFRP is controlled by the viscoelastic

compliance of matrix resin [1]. The formulations of creep compliance and time-temperature shift factors of matrix resin are

carried out based on the time-temperature superposition principle (TTSP). The formulations of long-term life of CFRP under

an actual loading are carried out based on the three conditions.

In this paper, the tensile and compressive static strengths in the longitudinal and transverse directions of two kinds of

unidirectional CFRP under wet condition are evaluated using ATM-2. The applicability of ATM-2 and the effect of water

absorption on time and temperature dependence of these static strengths are discussed.

13.2 Advanced Accelerated Testing Methodology

ATM-2 is established with following three conditions: (A) the failure probability is independent of time, temperature and

load history [2]; (B) the time and temperature dependence of strength of CFRP is controlled by the viscoelasticity of matrix

resin. Therefore, the TTSP for the viscoelasticity of matrix resin holds for the strength of CFRP; (C) the strength degradation

of CFRP holds the linear cumulative damage law as the cumulative damage under cyclic loading.

M. Nakada (*) • Y. Miyano

Materials System Research Laboratory, Kanazawa Institute of Technology, 3-1 Yatsukaho, 924-0838 Hakusan, Japan


S. Hara

Graduate School, Kanazawa Institute of Technology, 7-1 Ohgigaoka, 921-8501 Nonoichi, Japan



121


The master curve of static strength can be shown by the following equation based on ATM-2.

log σf t0; T0;Pfð Þ ¼ log σf0 t0

0; T0ð Þ þ 1

αlog � ln 1� Pfð Þ½ � � nr log

D� t0; T0ð ÞDc t00; T0ð Þ� �

(13.1)

The first term of right part shows the reference strength (scale parameter for the static strength) at reduced reference

time t0’ under the reference temperature T0. The second term shows the scatter of static strength as the function of failure

probability Pf based on condition (A). α is the shape parameter for the strength. The third term shows the variation by

the viscoelastic compliance of matrix resin which depend on temperature and load histories. nr is the material parameter. The

viscoelastic compliance D* in Eq. 13.1 can be shown by the following equation,

D� t0; T0ð Þ ¼ ε t0; T0ð Þσ t0; T0ð Þ ¼

Ð t00Dc t0 � τ0; T0ð Þ dσ τ0ð Þ

dτ0 dτ0

σ t0; T0ð Þ ; t0 ¼ðt0

dτ

aT0 T τð Þð Þ (13.2)

where Dc shows the creep compliance of matrix resin and σ(τ’) shows the stress history. t’ is the reduced time at T0, aToshows the time-temperature shift factor of matrix resin and T(τ) shows the temperature history. The viscoelastic compliance

D* under constant deformation rate loading (static loading) can be shown by

D� t0; T0ð Þ ¼ Dc t0 2= ; T0ð Þ (13.3)

13.3 Experimental Procedures

Two kinds of unidirectional CFRP laminates were employed in this study. One is the T300/EP which consists of carbon fiber

T300 and epoxy resin 2,500 (Toray). The laminates were cured by autoclave technique at 135 �C for 2 h and then post-cured

at 160 �C for 2 h. The aging treatment for post-cured specimen was conducted at 110 �C for 50 h. The Wet specimens by

soaking the aged specimen (Dry specimen) in hot water of 95 �C for 121 h for 1 mm thick specimen in longitudinal direction,

95 �C for 144 h for 2 mm thick specimen in longitudinal direction and 95 �C for 121 h for 2 mm thick specimen in transverse

direction were respectively prepared. The water content of all of wet specimen was 1.9 wt%. The other is the T700/VE which

consists of carbon fiber T700 unidirectional non-crimp fabric (Toray) and vinylester resin Neopol 8250 L (Japan U-PICA).

The laminates were molded by vacuum assisted resin transfer molding technique and then cured at room temperature for

24 h. The post-cure is conducted at 150 �C for 2 h. The Wet specimens by soaking the aged specimen (Dry specimen) in hot

water of 95 �C for 25 h for 1 mm thick specimen in longitudinal direction, 95 �C for 50 h for 2 mm thick specimens in

longitudinal and transverse directions were respectively prepared. The water content of wet specimen was 0.5 wt% for 2 mm

thick specimen, and 0.7 wt% for 1 mm thick specimen.

The dynamic viscoelastic tests for the transverse direction of unidirectional CFRP were carried out at various frequencies

and temperatures to construct the master curve of creep compliance for matrix resin. The static tests for typical four

directions of unidirectional CFRP were carried out at various temperatures to construct the master curves of static strength

for unidirectional CFRP. Longitudinal tension tests were carried out according with SACMA 4R-94. Longitudinal bending

tests under static and fatigue loadings were carried out according with ISO 14125 to get the longitudinal compressive static

strengths. Transverse bending tests were carried out according with ISO 14125 to get the transverse tensile static strengths.

Transverse compression tests under static and fatigue loadings were carried out according with SACMA 1R-94.


13.4.1 Viscoelastic Behaviour of Matrix Resin

The left side of Fig. 13.1 shows the loss tangent tan δ for the transverse direction of two kinds of unidirectional CFRP (Dry

specimen) versus time t, where t is the inverse of frequency. The right side shows the master curve of tan δ which is

constructedby shifting tan δ at various constant temperatures along the logarithmic scale of t until they overlapped each other,

122 M. Nakada et al.

for the reduced time t’ at the reference temperature T0 ¼ 25 �C. Since tan δ at various constant temperatures can be

superimposed so that a smooth curve is constructed, the TTSP is applicable for tan δ for the transverse direction of two

kinds of unidirectional CFRP. The master curve of tan δ for Wet specimens can be also constructed as shown in Fig. 13.1.

The TTSP is also applicable for tan δ under wet condition. The master curve of tan δ is shifted to the left side by water

absorption as shown in Fig. 13.1.

The left side of Fig. 13.2 shows the storage modulus E’ for the transverse direction of two kinds of unidirectional CFRP

(Dry specimen) versus time t. The right side shows the master curve of E’ which is constructed by shifting E’ at variousconstant temperatures along the logarithmic scale of t using the same shift amount for tan δ and logarithmic scale of E’ untilthey overlapped each other, for the reduced time t0 at the reference temperature T0 ¼ 25 �C. Since E’ at various constanttemperatures can be superimposed so that a smooth curve is constructed, the TTSP is applicable for E’ for the transverse

direction of two kinds of unidirectional CFRP. The master curve of E’ for Wet specimens can be also constructed as shown

in Fig. 13.2. The TTSP is also applicable for E’ under wet condition.The time-temperature shift factor aTo(T) which is the horizontal shift amount shown in the upper portion of Fig. 13.3 can

be formulated by the following equation,

log aT0ðTÞ ¼ΔH1

2:303G

1

T� 1

T0

� �H Tg � T� �þ ΔH1

2:303G

1

Tg� 1

T0

� �þ ΔH2

2:303G

1

T� 1

Tg

� �� 1� H Tg � T

� �� (13.4)

Fig. 13.1 Master curves of loss tangent for transverse direction of unidirectional CFRP (a) T300/EP (b) T700/VE

Fig. 13.2 Master curves of storage modulus for transverse direction of unidirectional CFRP (a) T300/EP (b) T700/VE

13 Effect of Water Absorption on Time-Temperature Dependent Strength of CFRP 123

where G is the gas constant, 8.314 � 10�3 [kJ/(K•mol)],ΔH1 andΔH2 are the activation energies below and above the glass

transition temperature Tg, respectively. H is the Heaviside step function.

The temperature shift factor bTo(T) which is the amount of vertical shift shown in the lower portion of Fig. 13.3 can be fit

with the following equation:

log bT0ðTÞ ¼X4i¼0

bi T � T0ð Þi" #

H Tg � T� �þ X4

i¼0

bi Tg � T0� �i þ log

TgT

" #1� H Tg � T

� �� (13.5)

where bi are the fitting parameters.

The creep compliance Dc of matrix resin was back-calculated from the storage modulus E’ for the transverse direction oftwo kinds of unidirectional CFRP using [3]

DcðtÞ � 1=EðtÞ; EðtÞ ffi E0ðωÞjω!2 πt= (13.6)

and approximate averaging method by Uemura [4].

The master curves of back-calculated Dc of two kinds of matrix resin are shown in Fig. 13.4. The master curve of Dc can

be formulated by the following equation,

logDc ¼ logDc;0ðt00; T0Þ þ logt0

t00

� �mg

þ t0

t0g

!mr" #

(13.7)

Fig. 13.3 Shift factors of storage modulus for transverse direction of unidirectional CFRP (a) T300/EP (b) T700/VE


where Dc,0 is the creep compliance at reduced reference time t’0 and reference temperature T0, and t’g is the glassy reduced

time on T0, and mg and mr are the gradients in glassy and rubbery regions of Dc master curve. Parameters obtained from the

formulations for aTo(T), bTo(T), and Dc are listed in Table 13.1.

13.4.2 Master Curves of Static Strengths for Unidirectional CFRP

Figures 13.5 and 13.6 show the master curves of static strengths for longitudinal tension X, longitudinal compression X’,transverse tension Y and transverse compression Y’ for Dry andWet specimens of two kinds of unidirectional CFRP obtained

from the strength data at various temperatures by using the time-temperature shift factors aTo shown in Fig. 13.3. The solid

and dotted curves in these figures show the fitting curves by Eq. 13.1 using the master curves of creep compliance of matrix

resin in Fig. 13.4. The parameters obtained by formulation are shown in Table 13.2.

From these figures, the static strengths of two kinds of unidirectional CFRP decrease with increasing time, temperature

and water absorption. The time, temperature and water absorption dependencies of static strength of unidirectional CFRP are

different with the loading direction. Figure 13.7 shows the relationship between the static strength of two kinds of

Fig. 13.4 Master curves of creep compliance for matrix resin calculated from the storage modulus for the transverse direction of unidirectional

CFRP (a) T300/EP (b) T700/VE

Table 13.1 Parameters for

master curve and shift factors of

creep compliance for matrix resin

T300/EP T700/VE

Dry Wet Dry Wet

T0 [�C] 25 25 25 25

Tg [�C] 110 65 110 (90)

Dc0 [1/GPa] 0.337 0.351 0.337 0.339

t’0 [min] 1 1 1 1

t’g [min] 1.54E06 2.34E03 1.36E06 (1.80E04)

mg 0.0101 0.0348 0.00893 0.0195

mr 0.405 0.466 0.373 (0.373)

ΔH1 [kJ/mol] 132 150 128 129

ΔH2 [kJ/mol] 517 547 576 –

b0 1.65E-02 0.150 3.24E-04 �8.77E-03

b1 �1.86E-03 �1.39E-02 �1.99E-04 1.45E-04

b2 6.64E-05 4.26E-04 8.64E-06 9.63E-06

b3 �8.29E-07 �4.71E-06 �1.75E-08 –

b4 3.81E-09 1.73E-08 �1.71E-10 –


Fig. 13.5 Master curves of tensile and compressive strengths in the longitudinal direction of unidirectional CFRP (a) T300/EP (b) T700/VE

Fig. 13.6 Master curves of tensile and compressive strengths in the transverse direction of unidirectional CFRP (a) T300/EP (b) T700/VE


unidirectional CFRP and the viscoelastic compliance of corresponding matrix resin. The slop of this relation corresponds to

the parameter nr in Table 13.2. The slop depends on the loading direction while that changes scarcely with water absorption.It is cleared from these facts that the time, temperature and water absorption dependencies of static strength of unidirectional

CFRP can be determined by the viscoelastic behavior of corresponding matrix resin.

13.5 Conclusion

A general and rigorous advanced accelerated testing methodology (ATM-2) for the long-term life prediction of polymer

composites exposed to an actual loading having general stress and temperature history has been proposed. The tensile and

compressive static strengths in the longitudinal and transverse directions of two kinds of unidirectional CFRP under wet

condition are evaluated using ATM-2. The applicability of ATM-2 can be confirmed for these static strengths. The time,

temperature and water absorption dependencies of static strength of unidirectional CFRP can be determined by the

viscoelastic behavior of matrix resin.

Table 13.2 Parameters for

master curve of static strength

of unidirectional CFRP

T300/EP T700/VE

Dry Wet Dry Wet

X σs0 [MPa] 1700 1675 2174 1881

nr 0.0762 0.0528 0.0633 0.122

α 14.7 20.7 22.9 22.7

X0 σs0 [MPa] 1446 1535 1416 1389

nr 0.316 0.356 0.738 1.01

α 10.0 7.18 6.92 21.1

Y σs0 [MPa] 121 90.6 47.9 34.4

nr 0.387 0.371 0.362 0.436

α 7.04 7.97 14.7 32.4

Y0 σs0 [MPa] 156 131 164 136

nr 0.0868 0.130 0.713 1.06

α 5.68 11.4 5.92 31.2

Fig. 13.7 Static strength of unidirectional CFRP versus viscoelastic compliance of matrix resin (a) T300/EP (b) T700/VE


Acknowledgements The authors thank the Office of Naval Research for supporting this work through an ONR award with Dr. Yapa Rajapakse

as the ONR Program Officer. Our award is numbered to N000140611139 and titled “Verification of Accelerated Testing Methodology for

Long-Term Durability of CFRP laminates for Marine Use”. The authors thank Professor Richard Christensen at Stanford University as the

consultant of this project and Toray Industries, Inc. as the supplier of CFRP laminates.

References

1. Miyano Y, Nakada M, Cai H (2008) Formulation of long-term creep and fatigue strengths of polymer composites based on accelerated testing

methodology. J Compos Mater 42:1897

2. Christensen RM, Miyano Y (2006) Stress Intensity Controlled Kinetic Crack Growth and Stress History Dependent Life Prediction with

Statistical Variability. Int J Fract 137:77

3. Christensen RM (1982) Theory of viscoelasticity, 2nd edn. Dover Publications, Inc, Mineola, p 142

4. Uemura M, Yamada N (1975) Elastic Constants of Carbon Fiber Reinforced Plastic Materials. J Soc Mater Sci 24(156), Japan


Chapter 14

Stress and Pressure Dependent Thermo-Oxidation Response

of Poly (Bis)Maleimide Resins

Nan An, G.P. Tandon, R. Hall, and K. Pochiraju

Abstract Thermo-oxidative degradation in high temperature resins can be accelerated by increased oxygen pressure and

the stress in the material. While the oxygen pressure increases the adsorbed oxygen concentration on the surface, stress

affects the diffusivity. We describe a comprehensive sorption, oxidation (diffusion-reaction) and stress evolution model for

polymers and composites and apply the model for oxidation growth prediction in (bis)maleimide resins. The model

framework uses a pressure-dependent boundary sorption model and stress-dependent diffusion model. Evolution of

thermo-oxidative degradation in a commercially available resin and composite system is experimentally characterized

and the results are used for identifying model parameters and model validation. The key contributions are the characteriza-

tion and validation of a single parameter for stress-diffusion coupling. The parameter value is identified at one stress-level

and the oxidation behavior at higher level is simulated. The results show that the effect of substrate stress and the oxygen

pressure can be effectively simulated using the developed framework.

Keywords Bismaleimide • Material Behavior Parameters • Oxidation • Pressure-Accelerated • Stress-Accelerated

14.1 Introduction

In order to observe the long-term behavior in a shorter time scale, the degradation behavior of these materials can

be accelerated in a controlled and coupled manner without introducing extra degradation mechanisms or ignoring

essential mechanisms [1]. Two most commonly utilized acceleration methods are elevated temperature [2, 3] and elevated

pressur [3]. Stress-accelerated aging [4] is seldom used. The elevated temperature can increase the diffusion and reaction

processes in the polymer matrix and accelerate the oxidative aging. However, higher temperature can affect the rate of

degradation by increasing the thermal stress in polymer composites caused by differences in the thermal expansion

coefficient of the constituents. Additionally, the elevated temperature may promote degradation processes that do not

occur at normal application temperatures. For instance, during the experimental study of PMR-15 resin specimens aged at

343ºC (compared to the near application temperature of 288ºC) in an inert argon environment, substantial weight

loss percentage was observed which is attributed to the non-oxidizing thermal aging [1]. Thus, there is likely a change in

the thermal aging mechanism of the specimens aged at the elevated temperature above the glass transition temperature

of the matrix resin. In addition, the anomalous degradation behavior and the nonlinearity effects of degradation rate will

make the lifetime prediction difficult [1, 2].

N. An (*) • K. Pochiraju

Department of Mechanical Engineering, Stevens Institute of Technology, Hoboken, NJ 07030, USA

G.P. Tandon

University of Dayton Research Institute, Dayton, OH 45469-0060, USA

AFRL/RXCC, Wright Patterson Air Force Base, Dayton, OH 45433, USA

R. Hall

AFRL/RXCC, Wright Patterson Air Force Base, Dayton, OH 45433, USA



129

The elevated pressure is another practical method for degradation acceleration of HTPMCs. It involves increased air

pressure, such as increased oxygen fraction or increased total pressure at relatively lower temperature. From Fig. 2.1, the

oxidation degradation can be observed to be much faster in the elevated pressure environment than that in the ambient

pressure environment for neat resins (PMR-15) [3, 5]. The similar trend was also observed for composite materials [2, 6]. It

should be noted that the results indicated the same thermo-oxidative degradation mechanisms for glass-reinforced epoxy

resin occurring in both air and oxygen [5]. Another experimental study upon the pressure-accelerated aging of PMR-15 resin

showed a nearly two-fold increase both in the rate of volume change and in the weight loss rate at a 0.414 MPa pressured air

than at ambient-pressured air [3]. However, mechanical testing reveals that those specimens aged in the pressurized air

environment hold much lower failure strain and larger strength reduction rate compared to those aged in normal ambient

pressure, as shown in Fig. 2.2. Thus, the elevated pressure is likely to induce significant increase in mechanical property

degradation despite its good capability in accelerated oxidative degradation.

The stress-accelerated aging method is relatively complicated compared to the other two methods, as crack/damage

growth is always associated with the oxidation process. In terms of the stress loading mechanisms, external applied stress

showed the same effect on oxidation rate as the residual stress [7]. A study on the long-term thermal aging of bismaleimide

neat resin showed that the addition of mechanical stress has an acceleration effect in oxidation growth [4]. It is also indicated

that only minor decrease in the tensile strength were observed in the stress-accelerated aging resins compared to that aged in

ambient air pressure. Another long-term oxidation study with a PMR-15 laminates showed greater weight loss in the cross-

ply laminates than that in the unidirectional laminates [8]. The greater weight loss is attributed to the residual stresses at both

the fiber-matrix micromechanical level and the ply level as the stress-induced bond rupture and the chain scission reactions

will accelerate the chemical reactions and enhance the diffusion. It can be concluded so far that compared to the other two

acceleration mechanisms, stress-accelerated aging mechanism is more favorable as it introduces neither additional anoma-

lous aging mechanism nor major change in mechanical strength degradation.

14.2 A Coupled Stress-Diffusion Model for Oxidation of Polymers

A chemo-mechanics based mechanism for modeling oxidation and damage is utilized to describe the diffusion/oxidation

where a three-layer thermo-oxidation model is used to describe the oxygen diffusion-reaction mechanisms for the constitu-

ent domain of the aged material as shown in Fig. 14.1. An oxidation state parameter ϕ is introduced for the material based on

which three states of the material can be identified as the oxidized polymer phase (typically near the surface, region I), the

active reaction zone (where a mix of oxidized and un-oxidized polymer exists, region II), and an un-oxidized polymer

(typically in the interior of the specimen, region III). ϕ is initially set to be ϕ ¼1 representing the un-oxidized state and

ϕ ¼ ϕox for completely oxidized state. The thermo-oxidation modeling for high temperature material system is summarized

in the following for the completeness of the model discussed in the present study.

We adopt the model from Aifantis [9] who derived the stress-diffusion coupling framework and applied to gas diffusion

in metals and extend it with the oxidation reaction rate term in the continuity equation. The model comprises of a linear

elastic solid subjected to a strain field (e) and the diffusing gas modeled as a perfectly elastic ideal fluid [9–11]. The elastic

behavior of the solid is assumed to be linear and that the stress (σ) and strain fields (e) satisfy equilibrium, stress-strain

Fig. 14.1 Three-zone model for thermo-oxidation propagation in neat resin specimen

130 N. An et al.

http://dx.doi.org/10.1007/978-3-319-00852-3_2

http://dx.doi.org/10.1007/978-3-319-00852-3_2

relations and the boundary conditions, independent of the diffusing gas concentration (C). Furthermore, we assume that the

constitutive relation for the polymer is oxidation state (ϕ) dependent and can be written as follows:

σ ¼ λðϕÞtrðeÞIþ 2μðϕÞe (14.1)

The oxidation dependence is incorporated with suitable homogenization of the two constants for the properties in the

active reaction zone as given in Eq. 14.2. The effect of the diffusing gas on the substrate polymer is assumed confined to the

oxide conversion only and that the concentration of the diffusing oxygen does not cause property changes.

λ ¼ λox ϕ ¼ ϕoxð Þ; μ ¼ μox ϕ ¼ ϕoxð Þ;λ ¼ λun ϕ ¼ 1ð Þ; μ ¼ μun ϕ ¼ 1ð Þ;

λ;1

μ

� �φ

¼ φ� φox

1� φox

λun;1

μun

� �þ 1� φ

1� φox

λox;1

μox

� �(14.2)

We extend the relation as derived and parameterized by Aifantis [9] to include consumption due to the oxidation

reactions. After several simplifications, the mass balance, diffusive force-flux relations, and momentum balance reduce to

the relationship shown in Eq. 14.3. The diffusion behavior is controlled by an oxidation state and temperature diffusivity

parameter, D(ϕ,T) and the coupling parameter N determines the influence of strain on diffusion.

@C x; tð Þ@t

¼ ½D ϕ; Tð Þ þ Ne�r2C x; tð Þ � Nre � rCðx; tÞ � R� C x; tð Þ;ϕð Þ (14.3)

Dð1; TÞ ¼ DunðTÞ 1�1ox

1�1ox

� �þ DoxðTÞ 1�1

1�1ox

� �(14.4)

where DunðTÞ ¼ Dun0 exp �Eun

a =RT� �

and DoxðTÞ ¼ Dox0 exp �Eox

a =RT� �

and Ea are the activation energy parameters [12].

The notations un and ox utilized in either superscripts or underscripts are for the unoxidized state ð1 ¼ 1Þ and unoxidized

state ð1 ¼ 1oxÞ, respectively. These notations will be used in the rest of the paper with model parameters whenever needed

for clarity. R� C;1; Tð Þ� represents the oxidation reaction rate as a function of oxygen concentration in the material and is

influenced by the temperature T, available oxygen concentration C, and oxidation state variable 1,

R� C;1; Tð Þ ¼ 1�1ox

1�1ox

R C; Tð Þ (14.5)

dϕ

dt/ R� C;ϕ; Tð Þ (14.6)

where ϕox is the oxidation state where the reaction terminates after complete conversion of the polymer [4]. The reaction

kinetics is modeled following Abdeljaoued [12] and Colin et al. [13]

R C; Tð Þ ¼ R0ðTÞ 2βC

1þ βC1� βC

2 1þ βCð Þ� �

(14.7)

where R0 is the maximum reaction rate when the reaction is not oxygen-starved and β is a non-dimensional parameter. The

saturation reaction rate is temperature dependent and an Arrhenius relationship is assumed for the temperature dependence,

R0ðTÞ ¼ R00 exp � Ra

RT

� �(14.8)

where R00 is the rate constant and Ra is the activation parameter.

14 Stress and Pressure Dependent Thermo-Oxidation Response of Poly (Bis)Maleimide Resins 131

14.2.1 Stress-Diffusion Coupling

A method for the characterization of coupling coefficient in Eq. 14.3 for a high temperature resin system is described in this

section. The hypothesis explored here is that there is a linear relationship between apparent diffusivity and applied stress

which was established for hydrogen and carbon diffusion in metals [9]. A linear relationship between the apparent diffusion,

D�, and stress σ was identified based on considerable diffusivity data for H and C diffusion in nickel and iron and at a variety

of temperatures (300 K~1,173 K) and loading conditions (uniaxial tension and hydrostatic pressure). We tested if this

hypothesis can hold for apparent diffusivity of oxygen in polymeric resins.

Due to the presence of the oxidation reaction, diffusivity measurement is challenging. Therefore, we show that the

oxidation layer size (S) correlates linearly with the diffusivity of the oxidized material and may be used as a substitute for

diffusivity in establishing the stress coupling. The oxidation layer size (S) depends upon the both the diffusivity and reaction

kinetics of oxidation rather than diffusivity alone. Figure 14.2 shows the dependence of oxidation layer thickness (S) on the

diffusivity (Dox) of the oxidized zone. The figure shows the oxidized layer size at 200 h with the diffusivity of oxidized

region varied parametrically in a simulation. The results indicate that as the diffusivity of the oxidized region increases, the

oxidized region grows with linear proportionality. Parametric analysis of oxidation growth on the reaction rate within a

relatively broad range (from 5.5 to 0.5 mol=m3� in) indicated that the oxidized layer growth has little dependence on the

reaction rate [1]. Since Eq. (14.3) defines an apparent stress-accelerated diffusivity and the oxidation layer size is linearly

proportional to the diffusivity as shown in Fig. 14.2, we can identify a single coupling coefficient as shown in Eq. (14.9).

S / Dox

D�ox ¼ Dox þ NoxE

S�

S� �D�

ox

Dox� 1 ¼ NoxE

Dox

S�

S� 1 ¼ k�σ (14.9)

For a one-dimensional state of stress, k� ¼ Nox= DoxEoxð Þ, with Eox being the modulus of the oxidized zone at the aging

temperature. Although the oxidation growth in neat resin is a complex function of proportionality constant α, diffusivities ofDox and Dun, and reaction rate R0ðTÞ, the approximation that the oxidation zone is only dependent on the diffusivity of the

oxidized zone may be reasonable for the following reasons. The proportionality constant α relates the molar reaction rates

with observed weight loss with respect to aging time. According to the experimental results by Ripberger et al. [3] the initial

weight loss rate is much more rapid than rates at longer aging times and approaches a constant rate for longer aging times.

Furthermore, we know that the oxidation growth is predominantly controlled by the diffusivity of the oxidized region and is

far less sensitive to the diffusivity of the unoxidized material. Direct measurement of diffusivity in polymers is complicated

as diffusion at low temperature is very low and the temperature acceleration leads to the onset of oxidation reaction.

Therefore, alternative method is needed to find effective diffusivity.

140

120

100

80

60

40

20

0

Oxi

dato

n th

ickn

ess(

S)

(mm

)

Diffusivity(D*ox) (mm2/min)

0 10.5 1.5 2 2.5

XDox

Fig. 14.2 Relationship

between diffusivity in

oxidized zone and oxidation

layer size

132 N. An et al.

14.3 Parameters Characterizing the Oxidation Behavior of Bismaleimide

14.3.1 Diffusion-Reaction Behavior

In the absence of stress coupling, the diffusion-reaction equation with oxidation state (ϕ) dependent diffusivity and reactionrate parameters is given in Eq. 14.10.

@C x; y; z; tð Þ@t

¼ D 1; Tð Þr2C x; y; z; tð Þ � R� C;1; Tð Þ (14.10)

where D is the diffusivity which depends on both temperature and oxidation state variable1. The parameters for the model

are therefore determined in two phases. First the diffusivity, reaction rates and the conversion behavior is determined and

then the stress-coupling parameter is deduced from experimental observations. The diffusivity and reaction rate constants for

unoxidized bismaleimide were obtained from literature [13] and the values for the oxidized matrix were inferred by

matching the experimentally observed oxidation growth profiles. Table 14.1 shows the dissolved oxygen concentration

(Cs), reaction rate constant (Ro), concentration dependence parameter (β in Eq. 14.7) and oxidation state (ϕox) where the

polymer is considered completely oxidized. Further details of model parameter determination were published in earlier

work [1]. The elastic properties were obtained from literature and datasheets for the material.

14.3.2 Pressure Acceleration

Changing the boundary concentration can simulate varying the effect of oxygen pressure. The oxidation growth in HTPMCs

under pressure can be simulated based on the existing model. Figure 14.3 shows results of the oxidation growth history under

two different pressure conditions, 413.7 kPa (60 psi) and 551.6 kPa (80 psi), respectively. The symbols represent the

experimental data while the solid curves represent the simulation results. The simulation agrees well with the experimental

measurement for both pressure conditions. It can be seen that oxidation growth increases with the pressure level.

The pressure acceleration is simulated by computing the dissolved oxygen concentration at the boundary using Henry’s

law. The saturation concentration for dissolved oxygen (Cs) for BMI exposed to air at atmospheric pressure is seen to be 7.3

mol/m3. We simulated the effect of increased pressure using the solubility of O2 in BMI and by adjusting the saturation

concentration for the two increased air pressures. The saturation concentrations used for the simulation are 4.1 and 5.4 times

the concentration in air at atmospheric pressure. The saturation concentration is seen to initially accelerate the oxidation

zone size but has little influence after long-term exposure. The long-term exposure is controlled by the diffusivity in the

oxidized zone, which is not affected by the increased pressure of exposed oxygen.

Table 14.1 Parameters used for modeling oxidation of BMI resin

Parameter Value Description

Cs (mol/m3) 7.3 Oxygen concentration

R0 (mol/m3-min) 5.76 Reaction rate constant

β m3=molð Þ 0.078947 Non-dimensional parameter

α m3=molð Þαmin ¼ 1e� 7;

αmax ¼ 1e� 5

Time-dependent proportionality parameter between

oxidation reaction rate

and oxidation state parameter

tswitch (h) 250 Time where physical aging influences conversion (αmin)

φox 0.2 Oxidation state variable for oxidized matrix

D0 (mm2/min) 1.62e–4

1.94e–4

Diffusivity of the unoxidized matrix (φ ¼ 1)

Diffusivity of the oxidized matrix (φ ¼ φox)

E (MPa) 4,600 Elastic modulus

v 0.3 Poisson’s ratio

k� 0.24 Coupling constant for stress assisted diffusivity


14.3.3 Stress Acceleration

Figure 14.4 compares the oxidation layer growth in BMI neat resins aged at 177 �C for 1,000 h (experimental data were

obtained from [14]) subjected to a uniform strain corresponding to a stress of 13.8 MPa tension to that without external

tension loading. Under both conditions, the oxidation layer growth can be fitted with a power law function where S*

represents the oxidation thickness of material under tension stress while S represents the oxidation thickness of material

under no external loading. The coupling coefficient can be derived from Eq. 14.9 and the oxidation layer sizes corresponding

Fig. 14.3 Oxidation thickness measured under two partial pressures

Fig. 14.4 Variation of oxidation thickness with stress

134 N. An et al.

to no stress and 2 Ksi load conditions. Note that we make the assumption that the entire specimen is under constant strain,

which must be examined in future work due to the presence of oxidation-induced strains.

The evolution of the coupling coefficient, k*, extrapolated to long-term aging (500 days) is shown in Fig. 14.5. We may

also infer that using the long-term asymptote for the value of k* provides an estimate for the coupling coefficient. We

estimate that the value of k* ¼ 0.24.

We further examined the dependence of the oxidation profile on value of parameter k* as shown in Fig. 14.6. The

oxidation profiles based on different values of k* were compared to the experimental data shown by the round symbols.

The triangle symbols represent the experimental value of oxidation growth without tensile loadings as a reference. The solid

curves in red, black and blue colors represent the oxidation growth when k* equals to 0.18, 0.21, and 0.24, respectively. It

can be seen when k* is in the range of 0.18–0.24, the simulated oxidation value is very close to the experimental data.

Especially when k* ¼ 0.24, the simulation result matches most with the experimental data. In comparison, as represented by

the dashed curve of k* ¼ 1.2, the simulated oxidation value is far beyond the reasonable range of oxidation growth.

Therefore, k* ¼ 0.24 is reasonable for oxidation prediction in BMI resins under tensile loading.

Fig. 14.5 Long-term

behavior of the coupling

coefficient as determined

from the extrapolation of

experimental data

Fig. 14.6 Sensitivity of

oxidation thickness prediction

to the value of k*. The tension

load applied corresponds to a

stress of 13.79 MPa (2 Ksi)


In order to investigate the ability to predict the oxidation layer growth at a different stress level (5 Ksi) with k* set at 0.24,and to test the validity of oxidation prediction for BMI resins, we compared our simulation results (solid curves) with

experimental data (symbols). The results are shown in Fig. 14.7. The simulation for σ ¼ 34.47 MPa slightly over predicts

the oxidation layer size. The discrepancy in the predictions can be from two assumptions. The strain in the oxidizing resin is

assumed to be equal to the applied strain. As oxidation induces shrinkage strain, the actual strain state is not only triaxial but

may be locally different from the applied strains. Further noting that the experimental oxidation measurements do not

account for erosion of surface oxide for measurements taken after hundreds of hours of aging, the correlation between

experiments and simulation values is seen to be reasonable. Further comparisons may be warranted in areas of high stress

such as regions around the crack tip.

14.4 Concluding Remarks

This paper presents a comprehensive model for the prediction of stress and pressure accelerated oxidation growth for

bismaleimide resins. The phenomenological coefficients required to model the behavior have been obtained from matching

the model response with experimental observations. The pressure acceleration is considered by changing the boundary

adsorption characteristics with Henry’s law as the basis. The simulations show that the pressure dependence on oxidation

growth can be predicted based on changes to the saturation molar concentration at the surface. The stress-acceleration is

treated with a strain-dependent diffusivity parameter. The diffusion reaction system is also modified to include a coupling

term between hydrostatic strain and concentration fields. The resulting system has been simplified to a single coupling

coefficient, which can be estimated from oxidation growth observations. Using stress-accelerated oxidation growth results at

one stress-level, the coupling coefficient is estimated for BMI. The oxidation growth is predicted at a second stress-level.

Given the behavior of the fiber reinforcements, the phenomenological coefficients determined for BMI resins may be used

for modeling the oxidation behavior at the composite lamina and laminate scales.

Fig. 14.7 Comparison of

simulations and experiments

for two load levels

136 N. An et al.

References

1. Schoeppner GA, Tandon GP, Pochiraju KV ((2007) Chapter 9: predicting thermo-oxidative degradation and performance of high temperature

polymer matrix composites. In: Kwon Young W, Allen David H, Talreja Ramesh R (eds) Multi-scale modeling and simulation of composite

materials and structures. Springer, New York, pp 373–379. ISBN 978-0-387-36318-9

2. Tsotsis TK, Lee SM (1998) Long-term thermo-oxidative aging in composite materials: failure mechanisms. Compos Sci Technol 58:355–368

3. Tandon GP, Ripberger ER, Schoeppner GA (2005) Accelerated aging of PMR-15 resin at elevated pressure and/or temperature.

In: Proceedings of the SAMPE 2005 symposium and exhibition, Seattle

4. Accelerated test methods for durability of composites, High Speed Research Materials Durability (Task 23), High Speed Research Program,

1998

5. Ciutacu S, Budrugeac P, Niculae I (1991) Accelerated thermal aging of glass-reinforced epoxy resin under oxygen pressure. Polym Degrad

Stab 31:365–372

6. Tsotsis TK, Keller S, Lee K, Bardis J, Bish J (2001) Aging of polymeric composite specimens for 5000 hours at elevated pressure and

temperature. Compos Sci Technol 61:75–86

7. Popov AA, Krysyuk BE, Zaikow GY (1980) Translation of vysokomol soyed. Polymer science USSR, 22

8. Bowles KJ, Jayne D, Leonhardt TA, Bors D (1993) Thermal stability relationships between PMR-15 resin and its composites. NASA Tech

Memo(106285)

9. Aifantis EC (1978) Diffusion of a gas in a linear elastic solid. Acta Mech 29:169–184

10. Aifantis EC (1982) On the theory of stress-assisted diffusion I. Acta Mech 45:273–296

11. Aifantis EC (1975) Thermomechanical modelling for gaseous diffusion in elastic stress fields. PhD thesis, University of Minnesota

12. Abdeljaoued K (1999) Thermal oxidation of PMR-15 polymer used as a matrix in composite materials reinforced with carbon fibers.

Ecole Nationale Superieure des Arts et Metiers, Paris

13. Colin X, Marais C, Verdu J (2001) Thermal oxidation kinetics for a poly(bismaleimide). J Appl Polym Sci 82:3418–3430

14. Tandon GP (2012) Chapter 9: characterization of thermo-oxidation in laminated and textile composites. In: Tandon GP, Schoeppner GA,

Pochiraju KV (eds) Long-term durability of polymeric matrix composites. Springer, New York, pp 345–382


Chapter 15

Comparison of Sea Water Exposure Environments

on the Properties of Carbon Fiber Vinylester Composites

Chad S. Korach, Arash Afshar, Heng-Tseng Liao, and Fu-pen Chiang

Abstract Composites used in infrastructure and structural applications can be exposed to environmental conditions

initiating degradation in the composite due to stress, UV radiation, moisture and chemical effects. Combined exposure of

UV radiation and sea water creates synergistic degradation, and is generated from cyclic exposure to the individual

conditions. Here, three separate exposure systems are used to age carbon fiber-reinforced vinylester composites: UV

radiation, salt spray, and humidity environmental chambers; full sample immersion in salt and sea water conditions; and

outdoor exposure in a tidal pond. Characterization of the time-dependent changes in the mechanical strength and modulus of

the coupons is performed for each environment and IR spectroscopy is used to assess chemical changes in the vinylester

matrix. Comparison between the conditions will be discussed in the context of long-term outdoor exposure with accelerated

laboratory conditions.

Keywords Carbon fiber • Composites • Environmental degradation • Sea water • Vinylester

15.1 Introduction

Carbon fiber-vinyl ester composites are extensively used in the marine industry and offshore structures. This is due to the

high strength-to-weight ratio and the corrosion resistance of these composites. Carbon fiber is a hydrophobic material and

aqueous environment doesn’t change its mechanical properties. Vinylester resin also has a high corrosion resistance

and ability to withstand water absorption. However, long term environmental exposure of polymeric composites is a great

concern. The effect of different environmental exposure on the polymeric composites has been investigated in recent years

[1–4]. Studies show that the degradation of mechanical properties of polymeric composite can be more severe when two

or more different exposure systems affect synergistically on the composite materials. Combined effects of ultraviolet

radiation and moisture at elevated temperatures have shown an increased degradation of mechanical properties for epoxy

resin composites [5]. However, different environmental exposures have not been addressed for vinylester resin composites.

The effects of UV radiation, salt spray, and humidity in weathering chambers, Indoor immersion of samples in salt and

sea water and outdoor exposure to ultraviolet, humidity and sea water on the carbon fiber-vinylester composites are

investigated here.

C.S. Korach (*) • A. Afshar • H.-T. Liao • F. Chiang

Department of Mechanical Engineering, Stony Brook University, Stony Brook, NY 11974, USA




139


15.2 Materials and Methods

15.2.1 Material

Carbon-fiber reinforced vinylester unidirectional composite laminates (Graphtek LLC) were used for all experiments and

conditions. Composite laminate sheets with nominal thickness of 1.4 mm were machined using a diamond wet saw into two

sizes: (1) 12.5 � 77 mm (width � length) for flexural testing with [0�] fiber direction, and (2) 25 � 152 mm with [0�] fiberdirection in the length for fracture testing. Two 5 mm notches were machined in specimen size (2) at the midpoint of the

length with a diamond saw to create double edge notch (DEN) fracture specimens.

15.2.2 Exposure Conditions

The first group of samples were exposed to 800 and 2,000 h of combined and individual accelerated aging before

characterization using two chambers: (i) Moisture and heat in a Tenney Benchmaster BTRS temperature and humidity

chamber, and (ii) UV Radiation/Condensation in a Q-Lab QUV/se accelerated weathering chamber (Fig. 15.1). UV

radiation simulates natural sunlight using fluorescent UV bulbs at a 340 nm wavelength. Intensity is monitored by real-

time UV irradiance sensors. Temperature is controlled using a blower. Condensation is provided by water evaporation

which condenses on the sample surfaces. One-half the samples are rotated between the two chambers every 24 h to create a

combined effect between controlled constant temperature and humidity and the cyclic UV radiation/Condensation

condition. The conditions in the chambers remained constant for the duration of the exposure. In the temperature and

humidity chamber, moisture was set at 85 % relative humidity (RH) and temperature at 35 C. In the QUV chamber, the UV

radiation and Condensation conditions cycled every 3 h. For the UV cycle, the UV irradiance was set at 0.6 W/m2 at 60 C,

and the Condensation cycle was set at a temperature of 50 C. The second group of samples was immersed in distilled water,

salt water, and sea water for 6 months and 1 year durations at room temperature in closed containers in a dark environment

(Fig. 15.1). The third group of samples was mounted on a frame with four different levels which was placed in a tidal pond

for 1 year (Fig. 15.1). Based on the position of the samples on the ladder frame, samples were exposed to varying amounts

of sea air (which contains sunlight and salt-spray) and sea water. The four levels of the frame are as follows: The upper

level (Row 1) samples always remained out of water and the samples were exposed to the sea air only for the entire

duration. The samples in the second level (Row 2) were exposed to 18 h of sea air and 6 h of sea water immersion for each

day; two cycles of 9 h sea air followed by 3 h sea water. The samples in the third level (Row 3) were exposed to 6 h of sea air

and 18 h of sea water immersion for each day; two cycles of 3 h sea air followed by 9 h sea water. The samples in the lower

level of the frame (Row 4) are immersed for the entire duration in sea water. Five specimens per condition were used in the

experiments.

Fig. 15.1 Environmental exposure performed in the laboratory with accelerated aging chambers (Left, QUV/se ultraviolet radiation and

condensation chamber); in room temperature water immersion (Center); and outdoors in a tidal pond (Right)

140 C.S. Korach et al.

15.2.3 Mechanical Testing

Three point bending tests were performed on the composite samples following the ASTM D790 [6] standard using a screw-

driven mechanical loading frame (TiraTest 26005) with a 0.5 kN load cell. The tests determined flexural strength and

modulus of the composites. Specimen sizes were 77 � 12.5 � 1.4 mm (L � W � H). Support geometry followed ASTM

D790, with the support span set for 60 mm resulting in a span/thickness ratio of ~ 43. A crosshead rate of 4.25 mm/min. was

used to give a strain rate of 0.01 mm/mm/min. Energy release rate was determined using the DEN specimens in a hydraulic

mechanical loading frame (Instron 8501). A gage length of ~ 102 mm was used with the edge notches in the center of the

gage. A crosshead rate of 2 mm/min. was used, and load at first failure was recorded. Tensile modulus of an un-notched

specimen was measured with the recorded load and incrementing with strain gages to compute the tangent modulus.


15.3.1 Mechanical Flexural Response

Samples (all [0�] fiber direction) exposed to different environmental exposures have been characterized by three point

bending to determine flexural modulus and residual strength (ASTM D790) [6]. All exposure conditions showed an

insignificant difference in the flexural modulus when compared to the unconditioned specimens, and within experimental

error (Figs. 15.2 and 15.4). Results showed the flexure strength decreases up to 18 % for both indoor immersion and outdoor

exposure (Figs. 15.3 and 15.5). There is minimal difference in residual flexural strength of samples immersed in distilled,

salt and sea water. The decrease in flexural strength is attributed to degradation of the fiber-matrix interface due to moisture

absorption causing swelling and loss of integrity, and perhaps more significantly, microcracking on the surface of the

vinylester matrix. The data from outdoor exposure shows higher degradation in flexural strength of samples immersed in sea

water for a longer period of time. No significant changes in flexural modulus and flexural strength have been observed

between 6 month and 1 year of indoor immersion or outdoor exposure. On the contrary, in the environmental chamber a

substantial decrease in flexural strength of samples with 2,000 h exposure time compared to 800 h exposure time has been

observed (Fig. 15.5). This result is believed to be due to the elevated temperatures in the laboratory chambers (50–60 C)

versus the immersion (21 C) and outdoor exposure environments. This result implies a rapid reduction in the flexural

0

20

40

60

80

100

120

Undegraded Distilled Salt water Sea water

Ben

ding

Mod

ulus

(G

Pa)

Indoor Immersion - 0 Degreea b

6 month 1 year

0

20

40

60

80

100

120

Undegraded Row 1 Row 2 Row 3 Row 4

Ben

ding

Mod

ulus

(G

Pa)

Outdoor Expsoure - 0 Degree

6 month 1 year

Fig. 15.2 Flexural modulus of CF/VE composites exposed to (a) indoor immersion, and (b) outdoor environmental exposure

15 Comparison of Sea Water Exposure Environments on the Properties of Carbon Fiber Vinylester Composites 141

strength of samples at the beginning of a long-term exposure to different environmental conditions compared to the rest of

the exposure time, which is dependent on the exposure temperature. Samples exposed to the tidal conditions showed a

decrease in flexural strength with more exposure to sea water, though the large data scatter makes drawing significant

conclusions on the effect of the amount of sea air versus sea water unsubstantiated.

0

20

40

60

80

100

120

Unconditioned Heat and Humidity UV-Heat andHumidity

Salt Spray UV-Salt Spray

Ben

din

g M

od

ulu

s (G

Pa)

800 hours 2000 hours

Fig. 15.4 Flexural modulus for 0� CF/VE composites exposed to environmental chamber conditions

0.0

0.5

1.0

1.5

2.0

2.5

Undegraded Distilled Salt water Sea water

Fle

xura

l Str

engt

h (G

Pa)

Indoor Immersion - 0 Degreea b

6 month 1 year

0.0

0.5

1.0

1.5

2.0

2.5

Undegraded Row 1 Row 2 Row 3 Row 4

Fle

xura

l Str

engt

h (G

Pa)

Outdoor Expsoure - 0 Degree

6 month 1 year

Fig. 15.3 Flexural strength for CF/VE composites exposed to (a) indoor immersion, and (b) outdoor environmental exposure


15.3.2 Fracture Strength

Energy release rate of longitudinal fracture of samples after 6-month outdoor exposure and indoor immersion were

characterized with the method of Nairn [7] that utilizes the shear-lag model. Samples were all 0� composites with double-

edge notches (DEN) machined across the fibers with a diamond saw. To compute the energy release rate, the tensile

modulus of an un-notched specimen was measured by instrumenting with strain gages then computing the tangent

modulus, which was found to be 152 GPa. All indoor immersion and outdoor exposure conditions showed a decrease

in the energy release rate compared with the unconditioned specimens (Fig. 15.6). For the indoor immersion conditions,

0

100

200

300

400

500

600

700

a b

UC FilteredWater

Sea Water Salt Water

Ene

rgy

Rel

ease

Rat

e, G

L (J

/m2 )

Ene

rgy

Rel

ease

Rat

e (J

/m2 )

0

100

200

300

400

500

600

700

UC Row 1 Row 2 Row 3 Row 4

Fig. 15.6 Energy release rate for longitudinal fracture of samples exposed to 6 months of (a) indoor immersion (b) outdoor tidal exposure

0

0.5

1

1.5

2

2.5

Unconditioned Heat and Humidity UV-Heat andHumidity

Salt Spray UV-Salt Spray

Fle

xura

l Str

eng

th (

GP

a)

800 hours 2000 hours

Fig. 15.5 Flexural strength for 0� CF/VE composites exposed to environmental chamber conditions

15 Comparison of Sea Water Exposure Environments on the Properties of Carbon Fiber Vinylester Composites 143

which compare the effects of three different water environments at room temperature, the salt water condition had the

lowest energy release rate. Though, when all three conditions are compared the differences are not statistically significant,

due to the large scatter in data. The full immersion in sea water showed a decrease in the energy release rate of ~ 65 %

(with a large data scatter) of the unconditioned samples, where the sea water immersion at room temperature had a

decrease of ~ 51 %. This difference may be attributed to the range of temperatures the outdoor samples were exposed to.

Results of the energy release rate for the outdoor exposures varied by vertical position in the sample frame. The two rows

which underwent combined sea air/sea water exposure (Rows 2 and 3) showed a fracture energy which was lowest for

Row 2 (~63 %) which was exposed to 75/25 (sea air/sea water). This may be explained by the synergistic degradation

occurring for the UV intensive sea air and sea water exposure, but when samples are primarily exposed to sea water (as in

Row 3), the synergistic degradation is less. Synergistic degradation has been found to cause significant surface damage to

the vinylester resin, which may cause weakening of the fiber-matrix interface at the notch and erosion of the matrix; these

damage mechanisms can cause high, local stress concentrations that decrease the energy release rate. It is interesting to

note that the sea air (Row 1) and Row 3 specimens had similar release rates. Row 4 (sea water only) had a lower release

rate, though large amounts of data scatter. As a comparison, the energy release rate for specimens exposed to the

environmental chamber conditions (not shown) for 2,000 h had the lowest value for the combined UV/Salt-Spray case

(73 % decrease in the unconditioned values) [8]. The discrepancy between the chamber conditions and the outdoor and

immersion conditions can be explained by the temperature difference, and indicates the importance temperature has on

accelerated aging of polymer composites.

15.4 Conclusions

The Carbon fiber-reinforced vinylester unidirectional composites were characterized by long-term outdoor exposure to sea

air and sea water; indoor immersion in distilled, salt and sea water; and exposure to UV radiation, salt spray, and humidity in

accelerated weathering chambers. The study shows that the effect of various type of exposure on the flexural modulus of [0�]fiber direction samples is minimal. However, the flexural strength has been diminished significantly. This may be because of

degradation of the fiber-matrix interface due to hydrolysis and plasticization during moisture absorption which deteriorates

the load transfer mechanism between matrix and fibers. Extensive microcracking on the surface of the composite samples

induced by UV is another factor in diminishing of flexural strength of samples exposed to long-term UV radiation. The

energy release rate was computed for 0� DEN samples in tension, where a decrease in energy release rate was found to occur

for all samples subjected to different environmental exposure in a tidal pond and full immersion at room temperature. The

largest decrease was found for the case of 25 % sea air/75 % sea water exposure. When comparing the accelerated

degradation from the environmental chambers to the indoor immersion and outdoor exposure, it becomes clear that the

elevated temperature in the chambers had a significant effect on the resulting mechanical properties, though trends followed

similar paths.

Acknowledgements The authors respectfully acknowledge the support from Drs. Yapa D.S. Rajapakse and Airan J. Perez from the Office of

Naval Research through grant #N000141110816.

References

1. Weitsman YJ (2006) Anomalous fluid sorption in polymeric composites and its relation to fluid-induced damage. Compos Part A 37:617

2. Chin JW, Aouadi K, Haight MR, Hughes WL, Nguyen T (2001) Effects of water, salt solution and simulated concrete pore solution on the

properties of composite matrix resins used in civil engineering applications. Polym Compos 22(2):282

3. Liau WB, Tseng FP (1998) The effect of long-term ultraviolet light irradiation on polymer matrix composites. Polym Compos 19(4):440

4. Grant TS, Bradley WL (1995) In-situ observations in SEM of degradation of graphite/epoxy composite materials due to seawater immersion.

J Compos Mater 29:852

5. Kumar BG, Singh RP, Nakamura T (2002) Degradation of carbon fiber-reinforced epoxy composites by ultraviolet radiation and condensation.

J Compos Mater 36:2713

6. ASTM D790 (2000) Standard test method for flexural properties of unreinforced and reinforced plastics and electrical insulating materials.

West Conshohocken, ASTM International, West Conshohocken, PA

7. Nairn JA (1988) Fracture mechanics of unidirectional composites using the shear-lag model model II: experiment. J Compos Mater 22:589

8. Korach CS, Chiang FP (2012) Characterization of carbon fiber-vinylester composites exposed to combined UV radiation and salt spray,

In: ECCM 15 - 15th European conference on composite materials,, Venice, 24–28 June 2012


Chapter 16

Low-Density, Polyurea-Based Composites: Dynamic Mechanical

Properties and Pressure Effect

Wiroj Nantasetphong, Alireza V. Amirkhizi, Zhanzhan Jia, and Sia Nemat-Nasser

Abstract In this study, we explore the fabrication, characterization and modeling of low-density polymeric composites to

understand their acoustic responses. Polyurea is chosen as the matrix of the composites due to its excellent properties and

advantages, i.e. blast mitigation, easy casting, corrosion protection, abrasion resistance, and various uses in current military

and civilian technology. Two low-mass-density filler materials of interest are phenolic and glass microballoons. They have

significant differences in their mechanical properties and chemical interactions with the matrix. Ultrasonic tests are

conducted on samples with different volume fractions of fillers and variable pressure. Computational models based on the

methods of dilute randomly distributed and periodically distributed inclusions are created to improve our understanding of

low-density polymer-based composites and serve as tools for estimating the dynamic mechanical properties of similar

composite material systems. The experimental and computational results are compared. The results are expected to facilitate

the design of new elastomeric composites with desirable densities and acoustic impedances. These new composites will be

useful in developing layered metamaterial structures. Furthermore, we seek to find out whether such inclusions may

substantially affect the time-dependent response of the composite by introducing new resonant modes.

Keywords Polyurea • Phenolic microballoon • Glass microballoon • Acoustic impedance • Dilute randomly distributed

inclusions • Periodically distributed inclusions

16.1 Introduction

Polymeric composites with hollow spherical inclusions are also known as syntactic foamed plastics. They are composed of a

polymer matrix, called the binder, and a distributed filler material of hollow spherical particles, called microspheres,

microcapsules or microballoons.

In the present work, polyurea is chosen as the matrix of the composites due to its excellent blast-mitigating capabilities.

Polyurea is a block copolymer formed from a chemical reaction of diisocyanates with polyamines [1]. The reaction is

generally very fast and insensitive to humidity and low temperatures [2]. This viscoelastic material is stable and very tough,

making it a popular material for coating applications. Structures selectively coated with polyurea can potentially absorb

more blast energy before failure, enhancing their dynamic performance [3–6]. To modify polyurea, we consider two filler

materials: glass and phenolic microballoons. The glass microballoons have high strength and good chemical and temperature

resistance [7, 8]. The phenolic microballoons have lower strength and environmental resistances, but also have a relatively

low density, making them an excellent choice for reducing weight. Additionally, the polymeric microspheres have more

flexibility than the glass microspheres. They deform more readily under pressure and still return to their initial geometry

after the pressure is released [7, 8]. Damage of microspheres under pressure could be related to the buckling of the spherical

shells [9, 10]. The tensile and compressive strength of syntactic foams with polymeric microspheres could be improved by

adding fibrous materials [11].

W. Nantasetphong (*) • A.V. Amirkhizi • Z. Jia • S. Nemat-Nasser

Department of Mechanical and Aerospace Engineering, Center of Excellence for Advanced Materials,

University of California, 9500 Gilman Drive, San Diego, La Jolla, CA 92093-0416, USA




145


Syntactic foams are used in a wide variety of applications. On boats, submarines and undersea structures, they can be used

to reduce the reflection of sonar waves. Due to their low mass density, they can be tailored to have acoustic impedance

comparable to water. When the impedance of an object is matched to that of the surrounding medium, incident acoustic

waves will completely transfer from the medium to the object.

In this work, two kinds of composites (polyurea with glass microballoons and polyurea with phenolic microballoons)

were studied with two objectives in mind: (1) to investigate the effect of inclusion content and hydrostatic pressure on

acoustic impedance, and (2) to develop models to estimate the effective impedance. The frequency range of study is in the

ultrasonic regime (0.5–1.5 MHz).

16.2 Experimental Details

16.2.1 Material Characteristics

Polyurea (PU) serves as the matrix component in the studied composite materials. It was synthesized from the reaction of

Isonate 143 L, a polycarbodiimide-modified diphenylmethane diisocyanate [12], and Versalink P-1000, an oligomeric

diamine [13], in a nearly stoichiometric ratio of 1.05:1. Ideally, the total number of isocyanate groups would equal the total

number of hydroxyl groups in order to achieve a complete chemical reaction. In practice, to ensure that the reaction was

completed and produced some cross-linking between the hard domains formed from semi-crystallization of the diamine

molecules, a slight excess of Isonate 143 L (+0.05) was used. The amount of excess Isonate 143 L was estimated through

weight measurements of the containers before and after processing. The density of polyurea is 1.1 g/cm3 [14].

Phenolic resin or phenol formaldehyde (PF) is synthetic polymer obtained by the reaction of a phenol or substituted

phenol with formaldehyde. It is very well known due to its excellent Fire Smoke Toxicity (FST) properties, retention of

properties after long-term exposure to high temperatures, and excellent electrical and chemical resistance [15]. Its bulk

density is 1.28 g/cm3 [16]. In this study, it is used in the shape of thin-shelled microballoons. Its apparent density is

0.227 g/cm3. This thermosetting plastic serves as a filler material in the fabrication process of polyurea-with-phenolic-

microballoons syntactic foam.

K1 glass microballoons (from 3 M) are soda-lime-borosilicate engineered hollow glass microspheres. They have a high

strength-to-weight ratio, low alkalinity and high water resistance. They are useful for increasing strength and stiffness, while

reducing weight. The bulk density of glass is around 2.23–2.53 g/cm3. The apparent density of microballoons is around

0.1–0.14 g/cm3 [16].

16.2.2 Preparation of Composites: Polyurea-with-Phenolic-Microballoonsand Polyurea-with-Glass-Microballoons

Due to the short gel time of polyurea at room temperature, pheonolic microballoons were first added to Versalink P-1000 in a

predetermined volume fraction. The mixture was mixed by hand to prevent the vacuum from pulling out the microballoons,

and then mixed for an hour using a magnetic stirrer under vacuum (1 Torr absolute pressure) until most of the trapped air

bubbles were gone. The second component, Isonate 143 L, was degassed and stirred for an hour separately. After the

degassing process, the Isonate 143 L was added into the mixture of Versalink P-1000 and phenolic microballoons, and all of

the components were thoroughly mixed together under vacuum for 5 min. The resultant mixture was then transferred into a

Teflon mold and allowed to cure at room temperature for 24 h in an environmental chamber at 10 % relative humidity. The

samples were then removed from the mold and allowed to cure unrestrained for 2 weeks in the chamber before testing. They

were unrestrained to prevent the formation of residual stresses. A similar procedure was also used for the fabrication of

polyurea-with-glass-microballoons compositesxx3014ty isensitye ach lower than its bulk densitycroballoonballoons. sfy

desire experimental condition.

146 W. Nantasetphong et al.

http://en.wikipedia.org/wiki/Phenol

http://en.wikipedia.org/wiki/Formaldehyde

16.2.3 Ultrasonic Measurement Under Quasi-Hydrostatic Pressure

Direct contact measurements were used to measure the speed of longitudinal waves in the composites. The ultrasonic

measurement apparatus consisted of a PC-based computer containing a Matec TB-1000 Toneburst Card, two Panametrics

videoscan longitudinal transducers (V103 Panametrics-NDT OLYMPUS), an attenuator box, and a digital oscilloscope

(Tektronix DPO 3014). As shown in Fig. 16.1, a toneburst signal of a specific frequency was sent from the card to the

attenuator box and the generating transducer. The received signal was sent directly to the oscilloscope where the amplitude

and travel time were measured. To find the longitudinal wave speed and attenuation, two tests were performed using two

different sample thicknesses. Longitudinal wave couplant was applied to all contact surfaces. For each test, the sample was

confined and compressed in a closed cell (Fig. 16.2), creating quasi-hydrostatic conditions due to the nearly incompressible

Channel1 2 3 4

Matec TB-1000 card withPC-based controls

Attenuator box

Transducers

Oscilloscope

Fig. 16.1 Ultrasonic measurement apparatus

Top Piston

Sample

Collar

Bottom Piston

Transducer

Transducer

Potentiometer

Assembry

Fig. 16.2 Components and assembly of the compression cell

16 Low-Density, Polyurea-Based Composites: Dynamic Mechanical Properties and Pressure Effect 147

nature of the polymeric composite. The cell was placed in an environmental chamber where the temperature was controlled

(Fig. 16.3.). Compression was generated by an Instron load frame (Model 1332) controlled by a MTS 407 controller

(Fig. 16.3). Sample displacement and force were recorded. Tests were conducted at 20 �C and pressures ranging from 1 to

10 MPa in 1 MPa increments. The speed of the longitudinal waves was determined by measuring the difference between the

times of travel through two samples of different thicknesses. The attenuation was measured from the transmitted wave

amplitudes of these two tests. The results were used to calculate storage and loss longitudinal moduli for each material type.

Composite density was calculated at each pressure. With a known density and longitudinal modulus, the acoustic impedance

of the composite was calculated.

16.3 Modeling Details

16.3.1 Composite with Dilute and Randomly Distributed Hollow Spherical Inclusions

The moduli of the matrix and particles are known. The effective moduli of the composite are calculated by considering the

change in strain energy in a loaded homogeneous body due to the insertion of inhomogeneities. With the appropriate choice

of admissible stress or strain fields, combined with the minimum complementary energy and minimum potential energy

theorems, the bounds for the moduli can be obtained [18, 19]. The structure of the model is shown in Fig. 16.4.

The predictions of this model are compared with the experimental data at low volume fraction of microballoons.

16.3.2 Composite with Periodically Distributed Spherical Inclusions

This model was first introduced by Nemat-Nasser et el. [20]. The unit cell of the model is shown in Fig. 16.5. To estimate the

overall moduli of the composite, the total elastic energy per unit cell, which depends on the properties of its constituents, is

equated with the corresponding elastic energy of a homogenized model with uniform properties throughout the entire

volume. For higher volume fractions of microballoons, such a periodic model considers the interaction of inclusions with

each other to a limited extent.

MTS Controller

TemperatureController

EnvironmentalChamber

Instron Load Cell

Computer for recodingdisplacement and force signals

Hydrauliccylinder

Fig. 16.3 Temperature and force-displacement control units


Acknowledgments This research has been conducted at the Center of Excellence for Advanced Materials (CEAM) at the University of

California, San Diego. This work was partially supported through the Office of Naval Research (ONR) grant N00014-09-1-1126 to University

of California, San Diego.

References

1. Roland CM, Casalini R (2007) Effect of hydrostatic pressure on the viscoelastic response of polyurea. Polymer 48:5747–5752

2. Broekaert M, Pille-Wolf W (2000) The influence of isomer composition and functionality on the final properties of aromatic polyurea spray

coatings. In: Proceedings of the Utech 2000 conference, Netherlands Congress Centre, The Hague

X

Y

Z

X

Y

Z

A unit cell

Fig. 16.5 A composite with periodically distributed spherical inclusions and its unit cell

X

Y

Z

Fig. 16.4 A composite with dilute and randomly distributed hollow spherical inclusions

16 Low-Density, Polyurea-Based Composites: Dynamic Mechanical Properties and Pressure Effect 149

3. Mock W, Balizer E (2005) Penetration protection of steel plates with polyurea layer. Presented at polyurea properties and enhancement of

structures under dynamic loads, Airlie

4. Amini MR, Isaacs JB, Nemat-Nasser S (2006) Effect of polyurea on the dynamic response of steel plates. In: Proceedings of the 2006 SEM

annual conference and exposition on experimental and applied mechanics, St Louis, 4–7 June 2006

5. Bahei-el-din YA, Dvorak GJ (2006) A blast-tolerant sandwich plate design with a polyurea interlayer. Int J Solids Struct 43(25–26):7644–7658

6. Tekalur SA, Shukla A, Shivakumar K (2008) Blast resistance of polyurea based layered composite materials. Compos Struct 84:271–281

7. Lee SM (1993) Handbook of composite reinforcements. Wiley, New York, pp 252–261

8. Shutov FA (1986) Syntactic polymer foams. Adv Polym Sci 73–74:63–123

9. Huston R, Joseph H (2008) Practical stress analysis in engineering design. CRC Press, Boca Raton, pp 31.1–31.8

10. Krenzke MA, Charles RM (1963) The elastic buckling strength of spherical glass shells, Report 1759, David Taylor Model Basin, Department

of the Navy

11. Huang YJ, Wang CH, Huang YL, Guo G, Nutt SR (2010) Enhancing specific strength and stiffness of phenolic microsphere syntactic foams

through carbon fiber reinforcement. Polym Compos 31(2):256–262

12. The Dow Chemical Company (2001) Isonate 143L; modified MDI. Dow Chemical, Midland

13. Air Products Chemicals, Inc. (2003) Polyurethane specialty products. Air Products and Chemicals, Allentown

14. Qiao J, Amirkhizi AV, Schaaf K, Nemat-Nasser S (2011) Dynamic mechanical analysis of fly ash filled polyurea elastomer. J Eng Mater

Technol 133(1):011016

15. Pilato L (2010) Phenolic resins: a century of progress. Springer, New York

16. Ashby MF (2009) Materials and the environment: eco-informed material choice. Elsevier, Oxford, pp 318–319

17. 3M™, Glass Bubbles K Series S Series

18. Hashin Z (1962) The elastic moduli of heterogeneous materials. J Appl Mech 29:143–150

19. Lee KJ, Westmann RA (1970) Elastic properties of hollow-sphere-reinforced composite. J Compos Mater 4:242–252

20. Nemat-Nasser S, Iwakuma T, Hejazi M (1982) On composites with periodic structure. Mech Mater 1:239–267


Chapter 17

Haynes 230 High Temperature Thermo-Mechanical Fatigue

Constitutive Model Development

Raasheduddin Ahmed, M. Menon, and Tasnim Hassan

Abstract Service temperatures of propulsion turbine engine combustor components can be as high as 1,800 �F. This inducesa thermo-mechanical fatigue (TMF) loading which, as a result of dwell periods and cyclic loadings, eventually leads to failure

of the components via creep-fatigue processes. A large set of isothermal and anisothermal experiments have been carried out

on Haynes 230, in an effort to understand its high temperature fatigue constitutive response. Isothermal experiments at

different loading strain rates show that the material can be considered to be rate-independent below and at 1,400 �F. However,isothermal strain hold experiments show stress relaxations below and at 1,400 �F. The out-of-phase strain-controlled TMF

experiments show a mean stress response. A Chaboche based viscoplastic constitutive model with various features is under

development with the final objective of predicting the strains in an actual combustor liner in service through finite element

simulation for fatigue lifing. Temperature rate terms have been found to improve hysteresis loop shape simulations and static

recovery terms are essential in modeling stress relaxation at temperatures where the behavior is overall rate-independent.

It is anticipated that the new modeling feature of mean stress evolution will model the experimentally observed

thermo-mechanical mean stress evolution.

Keywords Constitutive modeling • Thermo-mechanical fatigue • Stress relaxation • Haynes 230 • Mean stress

17.1 Introduction

Nickel-base superalloys have been favored in the high temperature service zones of gas turbine engines owing to their

excellent mechanical properties at elevated temperatures. The particular nickel-base superalloy of interest in the current

research is Haynes 230 which is used in airplane turbine engines. Turbine engine operation subjects combustor components

to thermo-mechanical cyclic loading with temperature fluctuating between room temperature to as high as 1,800 �F. Thefluctuation of temperature leads to the creation of “hot spots”, which are areas of considerably higher temperature than

surrounding areas. The geometry of the structure results in constraints which leads to compressive strains in the vicinity of

the hot spot as the hot spot tries to expand. Thus, in combustor liners out-of-phase (OP) thermo-mechanical fatigue (TMF) is

the phenomenon of primary interest. In OP TMF increase in temperature increases the magnitude of the compressive strain.

Airplane flight causes the turbine engine materials to be repeatedly subject to dwell periods at the compressive strain peak of

the thermo-mechanical fatigue loading. This leads to phenomenon such as creep-fatigue interaction which is a life-limiting

factor through processes of crack initiation, propagation and failure.

An accurate description of cyclic stress–strain responses during service is essential for the development of reliable life

prediction techniques for critical high temperature components in the aerospace, nuclear power, chemical and automobile

industries. The lifetime prediction of a component is usually performed by post-processing the stress and strain responses

from a finite element analysis [1]. Thus, reliable life prediction techniques of these components depend upon the accuracy of

R. Ahmed (*) • T. Hassan

Department of Civil, Construction, and Environmental Engineering, North Carolina State University, Raleigh, NC 27695-7908, USA


M. Menon

Honeywell Aerospace, Phoenix, AZ 85034, USA




151




constitutive models used for stress–strain computation under service loading. This calls for the development of an

appropriate robust constitutive model capable of describing the phenomenon which occurs due to the interaction of cyclic

plasticity and cyclic creep. The core of numerous studies found in literature has been models based on the Chaboche model

[2–4] framework as it enables to reliably describe a wide range of inelastic material behaviors such as cyclic hardening/

softening and stress relaxation for various steels and nickel-base alloys. Material parameters from the model are determined

from a carefully selected set of experiments representing a range of monotonic and cyclic, relaxation and creep tests.

The simulation of mean stresses with cycles under anisothermal conditions and its modeling is a challenging problem.

This evolutionary behavior of mean stress with cycles in nickel-base alloys was first reported by Yaguchi et al. [5, 6], who

observed the evolution in mean stress for isothermal experiments with hold times (dwell periods), and then with

anisothermal experiments but with no hold times. It has been shown that it is difficult to express the shift of the stress–strain

hysteresis loop under anisothermal conditions by conventional constitutive equations [8].

A wide set of experiments on Haynes 230 (HA230) coupons were carried out as a part of this study. Isothermal

experiments at different loading strain rates and with hold times, and thermo-mechanical experiments, both out-of-phase

and in-phase, will be discussed in this paper. These are followed by a description of the constitutive model and its features.

For details of parameter determination steps the readers are referred to Barrett et al. [9]. Finally, we present the simulations

of the thermo-mechanical fatigue tests.

17.2 Experimental Study

Strain-controlled experiments have been performed with strain ranges similar to that estimated in combustion liners during

service conditions. Temperature gradients that exist in service conditions of combustor liners cause non-uniform thermal

expansion (hot spots) which results in an essentially strain-controlled load cycle [10].

17.2.1 Experimental Procedure

The nickel-base polycrystalline superalloy Haynes 230 was received as bulk rods in solution annealed conditions.

The chemical composition of the material is summarized in Ahmed et al. [11]. The specimens were machined to a dog-

bone shape with a gage length of 0.63 in. and diameter of 0.25 in. at the gage location using a low-stress grinding technique.

A large number of experiments were conducted in an effort to characterize the material behavior of Haynes 230 under

fatigue loading. A total of 120 isothermal tests and 16 thermo-mechanical fatigue tests were carried out as part of the study.

These tests were uniaxial as combustor liners in service experience primarily uniaxial loading conditions. Isothermal low

cycle fatigue tests were performed using universal, servo-hydraulic fatigue testing machines. For testing at higher

temperatures heating was achieved through an induction heating system with thermocouples aligned for checking tempera-

ture uniformity. The low cycle fatigue tests were performed according to ASTM E606-04. Isothermal low cycle fatigue tests

with symmetric, axial strain controlled loading histories were performed at various strain rates and thenwith hold times (dwell

periods) at a constant strain rate. The loadingwaveformwas triangular for experiments without hold times, and trapezoidal for

experiments with hold times. Three loading strain rates of 0.2 cpm (cycles per minute), 2 and 20 cpm were prescribed in the

experiments at temperatures ranging from 75 �F to 1,800 �F. The imposed strain ranges varied from 0.3 % to 1.6 %.

The thermo-mechanical fatigue (TMF) tests were performed according to ASTM E2368-10. Cooling was achieved

through an airflow system to ensure the correct temperature gradient. Both out-of-phase and in-phase thermo-mechanical

fatigue tests were carried out. The thermal cycle had a minimum temperature of 600 �F and a maximum temperature between

1,400 �F and 1,800 �F. Hold times of 2 min. or 20 min. were imposed. The strain ranges varied from 0.25 % to 0.7 %.

17.2.2 Isothermal low Cycle Fatigue Strain Rate Effect

Responses from isothermal uniaxial cyclic strain-controlled low cycle fatigue experiments at three different loading strain

rates were used to investigate the effect of viscosity at seven different temperatures in the range 75–1,800 �F. A symmetric

strain-controlled loading history was prescribed. The experimental responses showed that for temperatures up to and

including 1,400 �F the material behavior was essentially rate-independent as the different loading strain rates do not have

152 R. Ahmed et al.

much of an effect on the stress amplitude responses (Fig. 17.1a). The material was found to behave rate-dependently at

temperatures greater than and including 1,600 �F. At 1,600 �F a saturation of stress amplitude responses for higher strain

rates was observed (Fig. 17.1b). The responses at 2 and 20 cpm were found be almost the same as the stress amplitude values

were very close to each other. At 1,800 �F the typical positive strain rate dependence observed for viscous materials was

obtained (Fig. 17.1c). Yaguchi et al. [6] also found a nickel-base alloy IN792 to be rate-independent at lower temperatures

(<1,292 �F) and rate-dependent at the higher temperatures (1,472 �F).

17.2.3 Isothermal Stress Relaxation

Isothermal experiments with strain holds at the maximum compressive peak were carried out to serve a twofold purpose.

First, it was to observe the effect of viscosity at the different temperatures. And second, it served as a lead-up to the more

complex thermo-mechanical experiments which followed where the temperature was also varied along with the strain

controlled path with hold times. Figure 17.2 shows the effect of temperature on the normalized relaxed stresses. The

normalized relaxed stresses have been calculated by normalizing the relaxed stress with the peak compressive stress of each

cycle. It is clear that with increase in temperature we see increase in the normalized stress. At the higher temperatures

(>1,600 �F) for which we know the material behaves in a viscous manner we see significant stress relaxation with hold

times. At the lower temperatures of 1,200 �F and 1,400 �F though we found that the material is mostly rate-independent at

the different loading strain rates we still see some amount of stress relaxation. This behavior has been seen by other

researchers as well [10]. The stress relaxation is greater at 1,400 �F than at 1,200 �F.

0

20

40

60

1 100 10000

Str

ess

(ksi

)

Cycle (log scale)

a b c

Cycle (log scale)

T = 1400�F T = 1600�F T = 1800�F

0.2 cpm

2 cpm

20 cpm

0

20

40

1 100 10000

Str

ess

(ksi

)

0.2 cpm2 cpm20 cpm

0

20

40

1 100 10000

Str

ess

(ksi

)

Cycle (log scale)

0.2 cpm

2 cpm

20 cpm

Fig. 17.1 Stress amplitude responses of Haynes 230 under different strain rates and temperatures: (a) 1,400 �F, (b) 1,600 �F, (c) 1,800 �F(1 ksi ¼ 6.895 MPa)

0

0.3

0.6

0.9

0 200 400 600

No

rmal

ized

rel

axed

str

ess

Cycle

1200F 1400F

1600F 1800F

tH = 120s

Fig. 17.2 Normalized stress relaxation at different temperatures for 120 s compressive strain hold

17 Haynes 230 High Temperature Thermo-Mechanical Fatigue Constitutive Model Development 153

17.2.4 Thermo-Mechanical Fatigue Test Hysteresis Loops and Mean Stress Evolution

The thermo-mechanical fatigue tests were designed to simulate combustor engine operating conditions, i.e. temperature and

strain excursions at critical locations in the components, as closely as possible. The prescribed thermo-mechanical loading

histories are shown in Fig. 17.3. Out-of-phase and in-phase experiments have been conducted to better understand the

material behavior and gradually enhance the robustness of the constitutive model.

Figure 17.4 shows the hysteresis loops at different cycles for typical out-of-phase thermo-mechanical fatigue experiments

at maximum cycling temperatures of 1,500 �F and 1,800 �F. The minimum temperature for all thermo-mechanical fatigue

experiments conducted is 600 �F. Hysteresis loops have been shown up to the half-life cycle of each experiment. Stress

relaxation occurs at the compressive hold which is also the maximum temperature hold. For both the experimental responses

the hysteresis loops shift in stress space in the tensile direction. The hysteresis loops in case of in-phase thermo-mechanical

fatigue experiments shift in stress space in the compressive direction. Thus for thermo-mechanical fatigue experiments the

mean stresses evolve in the direction opposite to that of the hold. This evolutionary behavior of mean stress with cycles was

reported by Yaguchi et al. [5, 6], who observed the evolution in mean stress for isothermal experiments with holds, and then

with anisothermal experiments but with no holds. In the present study the phenomenon has been observed for anisothermal

conditions with hold. Figure 17.5 shows evolution of mean stress with cycles for the out-of-phase thermo-mechanical fatigue

experiments shown in Fig. 17.4. As already discussed the mean stress evolves in the tensile direction for out-of-phase

thermo-mechanical fatigue experiments. It is interesting to observe that despite the difference in stress amplitudes due to

different maximum cycling temperatures the mean stress evolutions are remarkably similar. Similar trends were observed

for the other out-of-phase experiments as well as the in-phase thermo-mechanical fatigue experiments. Mean stress

evolutions were also seen in isothermal experiments with compressive holds. The mean stresses for all the isothermal

experiments were however much smaller in magnitude compared to the thermo-mechanical fatigue experiments.

a b

time

Tem

pera

ture

Tmax

Tmin

time

Str

ain

Strainrange

StrainrangeS

trai

n

timetime

Tem

pera

ture

Tmax

Tmin

Fig. 17.3 Thermo-mechanical fatigue loading histories prescribed in the experiment. (a) Out-of-phase TMF, (b) In-phase TMF

-50

-25

0

25

50

75

-0.6 -0.4 -0.2 0

Str

ess

(ksi

)

Strain (%)a

T: 600-1500�F, 120s hold T: 600-1800�F, 120s hold

1

2

50

500

-40

-20

0

20

40

-0.3 -0.2 -0.1 0

Str

ess

(ksi

)

Strain (%)b

1250500

Cycle number: Cycle number:

Fig. 17.4 Out-of-phase thermo-mechanical fatigue hysteresis loops for two tests: (a) 600–1,500 �F, (b) 600–1,800 �F (1 ksi ¼ 6.895 MPa)

154 R. Ahmed et al.

17.2.5 Thermomechanical Fatigue Plastic and Elastic Strain Shift

Thermo-mechanical fatigue experiments display an accumulation of inelastic strain with cycles. This type of behavior was

also reported by Zhang et al. [7] for a martensitic steel. Zhang et al. suggested that the plastic strain accumulates as a result of

the non-instantaneous plastic strain rate dropping to zero. Figure 17.6a shows the hysteresis loops for the first 20 cycles of a

thermo-mechanical fatigue test in a plot of stress against mechanical strain. The test is an out-of-phase thermo-mechanical

fatigue one with compressive holds. The control temperature is varied from 600 �F to 1,700 �F at a total mechanical strain

range of 0.3 %. The test is strain controlled and as a result there is no shifting of the hysteresis loops in total mechanical strain

space. However, when the same plots are made with respect to inelastic strain (Fig. 17.6b) and elastic strain (Fig. 17.6c) we

see hysteresis loops shifting in strain space. The inelastic strain accumulates in the compressive direction while the elastic

strain accumulates in the tensile direction for out-of-phase thermo-mechanical fatigue tests. This means that the total

mechanical strain is still maintained as the control mechanical strain and thus the strain decomposition is always satisfied.

The underlying cause for the plastic and elastic strain accumulation with cycles is the difference in material properties, in

particular, the elastic modulus, at different temperatures as shown in Fig. 17.6c.

0

20

40

60

0 1000 2000 3000

Str

ess

(ksi

)

Cycle

1500F 0.5 Stress Amplitude

1500F 0.5 Mean Stress

1800F 0.25 Stress Amplitude

1800F 0.25 Mean Stress

Fig. 17.5 Out-of-phase thermo-mechanical fatigue amplitude and mean stress responses with cycles for two maximum cycling temperatures

(1 ksi ¼ 6.895 MPa)

-40

-20

0

20

40

-0.33 -0.22 -0.11 0

Str

ess

(ksi

)

Strain (%)

Tmin: 600�F

Tmax: 1700�F

Δε: 0.3 %

2 min hold

a b c

-40

-20

0

20

40

-0.6 -0.4 -0.2 0

Str

ess

(ksi

)

Inelastic Strain (%)

-40

-20

0

20

40

60-0.15 0 0.15 0.3 0.45

Str

ess

(ksi

)

Elastic Strain (%)

Fig. 17.6 Evolution of hysteresis loops for the first 20 cycles of a thermo-mechanical fatigue test in (a) mechanical strain space, (b) inelastic strain

space and (c) elastic strain space (1 ksi ¼ 6.895 MPa)


17.3 Constitutive Model

17.3.1 Constitutive Equations

A constitutive model is under development to simulate the wide range of experimental phenomena observed. A Chaboche

type modified viscoplastic constitutive model has been adopted [3]. Various features included have been shown previously

successful in modeling many different phenomena observed in cyclic plasticity [12]. Decomposition of strain (ε) into elastic(εe) and inelastic (εin) part has been assumed Eq. 17.1. The elastic part obeys Hooke’s law as shown in Eq. 17.2. Here E and νindicate Young’s modulus and Poisson’s ratio, respectively, σ and I are the stress and identity tensors, respectively, and tr isthe trace.

ε ¼ εe þ εin (17.1)

εe ¼ 1þ ν

Eσ� ν

Etrσð ÞI (17.2)

A unified viscoplastic model has been chosen as it allows the modeling of rate-dependent behavior, an important feature

particularly at higher temperatures. The inelastic strain rate is expressed as,

_εin ¼ 3

2_p

s� a

J σ� αð Þ (17.3)

where (·) denotes the differentiation with respect to time, s and a are the deviators of the stress and back stress, respectively.

_p and J σ� αð Þ are expressed as shown in Eqs. 17.4 and 17.5 respectively. σo is the yield stress, and, K and n are rate-

dependent parameters.

_p ¼ J σ� αð Þ � σoK

� �n

(17.4)

J σ� αð Þ ¼ 3


� �12

(17.5)

A Chaboche nonlinear kinematic hardening rule with four superimposed back stresses is chosen Eq. 17.6. The kinematic

hardening rule describes the evolution of the back stress and has the features of dynamic recovery, static recovery and

temperature rate terms. The static recovery term is essential for the simulation of rate-dependent behavior such as stress

relaxation and creep [5]. Simulations of the stress relaxation behavior at half-life under dwell condition, is one of the most

important deformation behaviors in terms of creep-fatigue damage analysis of the actual components. Temperature rate

terms are required in the kinematic hardening rule for obtaining stable conditions [13].

a ¼X4i¼1

ai

_ai ¼ 2


ai þ 1

Ci

@Ci

@T_Tai (17.6)

17.3.2 Strain Range Dependence

The effect of strain range on the shape of the hysteresis loops has been shown in Barrett et al. [9]. The importance of

capturing the shape of the hysteresis loops as closely as possible has been shown [12] to have an impact in the overall

simulation quality. Strain range dependence is modeled by considering a strain memory surface which memorizes the prior

156 R. Ahmed et al.

largest plastic strain range [12–14]. The radius and center of the strain memory surface are q and Y respectively. The

memory surface equation is given by Eq. 17.7 and the evolution equations of q and Y are given by Eqs. 17.8 and 17.9

respectively. Material constant η can be determined from uniaxial response and are related to the stabilized plastic strain

amplitudes. H(g) is the Heaviside step function. The kinematic hardening dynamic recovery parameters γi of Eq. 17.6 are

varied with cycles and are functions of q. The evolutions of γi are according to Eqs. 17.11 and 17.12.

g ¼ 2

3εin � Y� �

: εin � Y� �� 1

2

� q ¼ 0 (17.7)

_q ¼ ηHðgÞ n : n�h i½ � _p (17.8)

_Y ¼ffiffiffiffiffiffiffiffi3 2=

p1� ηð ÞHðgÞ n : n�h in�½ � _p (17.9)

n� ¼ffiffiffi2

3

rεin � Y

q(17.10)


� �_p (17.11)

γASi ðqÞ ¼ aγi þ bγie�cγiq (17.12)

17.3.3 Mean Stress Evolution

It has been shown by Yaguchi et al. [5, 6] that the evolution of mean stresses is difficult to express using conventional

kinematic hardening rules. Equation 17.13 is the modified kinematic hardening rule of Eq. 17.6 through incorporating a

second order tensor Yb in the dynamic recovery term (the second term). This form of the dynamic recovery was first

introduced by Chaboche-Nouailhas [15] however its behavior and evolution is different in the Yaguchi et al. model. The

driving force of Yb Eq. 17.14 is assumed to be rate/time-dependent deformation as the dislocation networks generally form

under creep conditions.

_ai ¼ 2

3Ci _εin � γi ai � Ybð Þ _p� biJ aið Þr�1

ai þ 1

Ci

@Ci

@T_Tai (17.13)

_Yb ¼ �αb Ystai

J aið Þ þ Yb

J aið Þf gr (17.14)


The parameters for the model have been determined from a broad set of experimental responses. The steps in determining

the parameters using a genetic algorithm based method have been outlined in Barrett et al. [9]. Strain-controlled uniaxial

isothermal experiments without any strain hold times are used to determine the rate-independent kinematic hardening

parameters. Experiments at different loading rates as well as the isothermal creep data is used for rate-dependent parameter

determination. Isothermal experiments with strain hold times (relaxation) are used to determine the static recovery

parameters of kinematic hardening. Finally, from the thermo-mechanical responses the temperature dependence parameters

are finalized.


17.5 Simulations

17.5.1 Thermo-Mechanical Fatigue Simulations

The first set of simulations highlight the importance of the temperature rate terms in the kinematic hardening rule.

Figure 17.7 shows the simulation of the half-life hysteresis loop for a thermo-mechanical fatigue test with and without

temperature rate terms. The strain range for the out-of-phase thermo-mechanical fatigue test is 0.3 % and the temperature

cycle is from 600 �F to 1,700 �F. The model cannot simulate the mean stress evolution in the out-of-phase thermo-

mechanical fatigue tests as can be seen in Fig. 17.7. While the shape of the hysteresis loop is captured, the simulation

loops are not shifting with cycles in the tensile direction to match the mid-life hysteresis loop. This is because of the

deficiency in simulating the shift in hysteresis loop (mean stress evolution). In these simulations, strain range dependence

was considered with the full evolutionary behavior of the kinematic hardening dynamic recovery parameter γ according to

Eqs. 17.7, 17.8, 17.9, 17.10, 17.11, 17.12 and 17.13. Simulation improvements can be achieved by considering that the

evolution of the kinematic hardening parameter γ at each temperature is influenced by the initial state at the particular

temperature (γi,inst). Thus a weighted average of the evolved γi,evol and the instantaneous γi,inst is determined using,

γi ¼ f �γi;inst þ 1� fð Þ�γi;evol (17.15)

In this modified modeling, f is the weighted average fraction which is a material parameter. A low value of f ¼ 0.1 shows

an improvement in the simulation of the hysteresis loop in terms of the shape and size as shown in Fig. 17.8a. Once again the

simulation mean stress does not match with the experimental value. Figure 17.8b illustrates this aspect more clearly. The

simulation and experimental hysteresis loops have decent agreement in stress amplitude, but the experimental mean stress

shows a significant evolution which cannot be reproduced by the simulation. Thus, a proper simulation of the thermo-

mechanical fatigue responses requires a modeling feature capable of describing the observed mean stress evolution. It is

anticipated that the modification of the kinematic hardening rule as suggested by Yaguchi et al. [5, 6] Eqs. 17.13 and 17.14

may lead to the desired mean stress simulations. Figure 17.9 shows the simulation of the first 20 hysteresis loops for the same

out-of-phase thermo-mechanical fatigue experiment plotted as stress versus plastic strain. The hysteresis loops are seen to

shift in the compressive direction in plastic strain space. This agrees well with the experimental observation described in

Fig. 17.6b.

17.6 Conclusions

Experimental responses show that Haynes 230 behaves rate-independently at temperatures including and less than 1,400 �F.The material behavior is rate-dependent at 1,600 �F and higher. The isothermal stress relaxation experiments revealed that

despite being overall rate-independent at 1,200 and 1,400 �F the material shows stress relaxation at peak compressive strain

-40

-20

0

20

40

60-0.3 -0.2 -0.1 0

Str

ess

(ksi

)Strain (%)

T = 600-1700�FΔεx = 0.3%

Experiment

Simulation withouttemp. rate term

Simulation withtemp. rate term

Fig. 17.7 Simulation of

half-life hysteresis loop for

a thermo-mechanical fatigue

test (600–1,700 �F) with and

without temperature rate terms

(1 ksi ¼ 6.895 MPa)

158 R. Ahmed et al.

hold. This stress relaxation can be modeled using the static recovery term in the kinematic hardening rule. At temperatures

including and greater than 1,600 �F the stress relaxation was modeled using a combination of rate-dependence and static

recovery.

Reliable lifing techniques are dependent upon the correct description of thermo-mechanical stress–strain behavior.

Thermo-mechanical fatigue (TMF) experiments, in-phase or out-of-phase lead to mean stress evolution with cycles in a

direction opposite to the strain hold. Yaguchi et al. [5, 6] reported this evolution for isothermal experiments with hold times

and then for anisothermal experiments without hold times. To the authors knowledge this mean stress evolution has not been

previously reported for anisothermal conditions with hold times (Fig. 17.3). Conventional kinematic hardening rules with or

without the temperature rate term have been found to be incapable of describing the mean stress evolution in TMF

experiments. It is anticipated that the simulation of mean stress evolution in the TMF experiments of Haynes 230 can be

improved by adding the kinematic hardening modeling feature Eqs. 17.13 and 17.14 proposed by Yaguchi et al. [5, 6].

It was observed that hysteresis loops shift in inelastic and elastic strain with cycles as a result of the change in elastic

modulus with temperature. The plastic and elastic strain offset each other such that the total mechanical strain equals the

prescribed strain.

Thermo-mechanical fatigue simulations were challenging in obtaining the correct hysteresis loop shape as well as the

experimentally observed mean stress evolution. The hysteresis loop shape was simulated fairly well using a new modeling

concept of weighted average evolution for the backstress kinematic hardening parameter.

Acknowledgments The authors are grateful to Honeywell Aerospace for the financial support of the project. All experiments were conducted by

Element in Cincinnati, Ohio.

-40

-20

0

20

40

60-0.3 -0.2 -0.1 0

Str

ess

(ksi

)

Strain (%)ab

Δεx=0.3%,T = 600-1700�FtH = 120s

0

10

20

30

40

0 2000 4000

Str

ess

(ksi

)

Cycle

Experiment:Amplitude stressSimulation:Amplitude stressExperiment: MeanstressSimulation: MeanstressExperiment

(cycle 3047)

Simulation

Fig. 17.8 (a) Simulation of half-life OP TMF hysteresis loop using weighted average for kinematic hardening dynamic recovery parameter;

(b) Simulation of stress amplitudes and mean stresses (1 ksi ¼ 6.895 MPa)

-40

-20

0

20

40

-0.6 -0.4 -0.2 0

Str

ess

(ksi

)

Inelastic Strain (%)

Fig. 17.9 Simulation of the first 20 cycles of thermo-mechanical fatigue 600–1,700 �F experiment (1 ksi ¼ 6.895 MPa)


References

1. Szmytka F, Remy L, Maitournam H (2010) New flow rules in elasto-viscoplastic constitutive models for spheroidal graphite cast-iron. Int J

Plasticity 26(6):905–924

2. Chaboche JL (2008) A review of some plasticity and viscoplasticity constitutive theories. Int J Plasticity 24(10):1642–1693

3. Chaboche JL (1989) Constitutive equations for cyclic plasticity and cyclic viscoplasticity. Int J Plasticity 5(3):247–302

4. Chaboche JL (1986) Time-independent constitutive theories for cyclic plasticity. Int J Plasticity 2(2):149–188

5. Yaguchi M, Yamamoto M, Ogata T (2002) A viscoplastic constitutive model for nickel-base superalloy, part 1: kinematic hardening rule of

anisotropic dynamic recovery. Int J Plasticity 18(8):1083–1109

6. Yaguchi M, Yamamoto M, Ogata T (2002) A viscoplastic constitutive model for nickel-base superalloy, part 2: modeling under anisothermal

conditions. Int J Plasticity 18(8):1111–1131

7. Zhang Z, Delagnes D, Bernhart G (2002) Anisothermal cyclic plasticity modelling of martensitic steels. Int J Fatigue 24(6):635–648

8. Wang JD, Ohno N (1991) Two equivalent forms of nonlinear kinematic hardening: application to nonisothermal plasticity. Int J Plasticity 7

(7):637–650

9. Barrett PR, Menon M, Hassan T (2012) Constitutive modeling of Haynes 230 at 75-1800 �F. ASME pressure vessels & piping conference

(Paper no. PVP2012-78342), Toronto, 15–19 July 2012

10. Almroth P, Hasselqvist M, Simonsson K (2004) Viscoplastic-plastic modelling of IN792. Comput Mater Sci 29(4):437–445

11. Ahmed R, Menon M, Hassan T (2012) Constitutive model development for thermo-mechanical fatigue response simulation of Haynes 230.

ASME pressure vessels & piping conference (Paper no. PVP2012-78221), Toronto, 15–19 July 2012

12. Krishna S, Hassan T, Ben Naceur I (2009) Macro versus micro-scale constitutive models in simulating proportional and nonproportional cyclic

and ratcheting responses of stainless steel 304. Int J Plasticity 25(10):1910–1949

13. Chaboche JL (2008) A review of some plasticity and viscoplasticity constitutive theories. Int J Plasticity 24(10):1642–1693

14. Nouailhas D, Cailletaud G, Policella H (1985) On the description of cyclic hardening and initial cold working. Eng Fract Mech 21(4):887–895

15. Chaboche JL, Nouailhas D (1989) Constitutive modeling of ratcheting effects – part 2: possibilities of some additional kinematic rules. ASME

J Eng Mat Tech 111:409–416

160 R. Ahmed et al.

Chapter 18

Temperature and Strain Rate Effects on the Mechanical

Behavior of Ferritic Stainless Steels

Kauko Ostman, Matti Isakov, Tuomo Nyyss€onen, and Veli-Tapani Kuokkala

Abstract To gain knowledge about the applicability of ferritic stainless steels in exhaust pipes and other high temperature

applications, mechanical testing of EN 1.4509 (ASTM S43932) and EN 1.4521 (ASTM 444) was conducted at 600 �C and

800 �C. Tensile tests with short high temperature exposure were carried out to determine the material properties in the as-

received condition. To study the high temperature service performance and the effects of possible microstructural changes

during long-term high temperature exposure, tensile tests were performed for samples that had undergone a 120 h furnace

heat treatment at 600 �C. As an example of the effect of exposure time, serrated flow was observed in the tests for as-received

EN 1.4509, which indicates dynamic strain aging. The effect, however, disappeared after the 120 h heat treatment,

suggesting that notable microstructural changes take place at high temperatures. Also fatigue and high strain rate tensile

tests were conducted on the test materials to reveal the effects of high temperature exposure on their properties, microstruc-

ture and service performance.

Keywords Ferritic • High temperature • Precipitation • Steel • Strain rate

18.1 Introduction

Due to the fluctuating price of nickel, ferritic stainless steels are studied as substitutes for austenitic stainless steels.

Simultaneously there is a need for higher efficiency in combustion processes, which often leads to higher temperatures.

Thus, one important application area for ferritic stainless steels is the exhaust pipes of power plants and vehicles, where high

temperature performance of the materials is crucial.

Tensile properties of ferritic stainless steels EN 1.4509 and EN 1.4521 were characterized in this study. Tensile tests were

performed with different strain rates. In addition to room temperature tests, these experiments were also performed at low

(�50 �C) and high (600 �C) temperatures. These tests were made to gain information about the performance of as-received

materials as a function of temperature. To study the possible microstructural changes and service performance at high

temperatures, part of the experiments were performed on samples heat treated for 120 h at 600 �C in a laboratory furnace.

18.2 Materials

Two ferritic stainless steel grades, EN 1.4509 (ASTM S43932/441) and EN 1.4521 (ASTM 444), were studied in this

research. The materials were of normal production quality and received as cold rolled and heat treated 2 mm sheets.

Their surface was pickled and skin passed. The chemical compositions of the materials are presented in Table 18.1. The

main difference between the two materials is that EN 1.4509 does not contain molybdenum whereas EN 1.4521 contains

2.1 % of Mo.

K. Ostman (*) • M. Isakov • T. Nyyss€onen • V.-T. Kuokkala

Department of Materials Science, Tampere University of Technology, Korkeakoulunkatu 6, 33720 Tampere, Finland




161


18.3 Experiments

Tensile test specimens were prepared by laser cutting from 2 mm thick steel sheets. The sample geometry contained a 4 mm

wide and 8 mm long gage section and roundings with a 2 mm radius at the ends. A slightly modified sample geometry was

used in the fatigue tests, i.e., a 2 mm long gage section and 5 mm radius end roundings.

Low and medium strain rate tensile tests were performed with a conventional servo-hydraulic materials testing machine.

High strain rate tests were performed with a tensile Hopkinson split bar (THSB) system, where the impact was created with a

striker tube around the incident bar. The striker was propelled with pressurized air and impacted on a flange at the end of

incident bar, thus creating an elastic tensile wave which propagated into the sample. Similar samples were used with the

hydraulic machine and the THSB to make comparison of the results more feasible.

Fatigue tests were conducted with a servo-hydraulic materials testing machine under constant load amplitude control.

Symmetric sinusoidal tension-compression loading was used with zero mean load. The cycling frequency was 10 Hz.

In the low temperature tests the samples were cooled with nitrogen gas that flowed through a heat exchanger immersed in

a liquid nitrogen bath. The gas flow rate was controlled with a PID controller to maintain the desired test temperature. The

sample was enclosed in a chamber whereto the nitrogen gas was directed. In high temperature tests the samples were heated

with magnetic induction by running alternating electric current through a coil surrounding the specimen.

Heat treatments of the samples were performed in a conventional laboratory furnace. The temperature of 600 �C was

selected because it is rather near to the high end of the usable temperature range of these materials. The heat treatment time

of 120 h was selected to be long enough so that most of the possible microstructural changes would have taken place and

saturated. Thus, the samples would represent the true service performance of the material at that temperature. After the heat

treatment the samples were allowed to cool down to room temperature with the furnace, which took several hours.

18.4 Results

Figure 18.1 presents the tensile test results for EN 1.4509 and EN 1.4521 sheets at the strain rate of 0.001/s. It is evident that

for both materials the strength decreases when temperature increases. At 600 �C the materials lose about 170 MPa of their

strength when compared to room temperature. At 800 �C the materials are too weak to most load carrying applications.

At �50 �C, room temperature and 600 �C EN 1.4521 has about 70–80 MPa higher strength than EN 1.4509.

Figure 18.2 presents the tensile test results at 600 �C for samples that have been heat treated for 120 h at 600 �C.Figure 18.2 also includes results for the as-received samples at the strain rate of 0.001/s at 600 �C. The heat treatment clearly

reduces the strength of both materials. All these tests were done so that there was first a 30 min holding time at 600 �C and

the sample loading was started immediately after that. It is evident that the 120 h furnace heat treatment causes

microstructural changes that do not yet take place during the 30 min hold period. The shape of the curves shows that the

heat treatment reduces the strain hardening capability of the test materials, i.e., the ultimate tensile strength is lower and it is

achieved at smaller strains.

Figure 18.3 shows a part of Fig. 18.2 enlarged. It can be seen that as-received EN 1.4509 deforms with serrated plastic

flow, but the heat treatment almost completely removes this behavior. When strain rate is increased to 0.1/s the plastic flow is

very smooth.

Figure 18.4 presents the tensile test results at room temperature for the as-received and 120 h at 600 �C heat treated

samples. The heat treatment notably increases room temperature strength of these materials. When these results are

compared with Fig. 18.2, it can be seen that the effect of heat treatment is completely opposite to the change of properties

at 600 �C.Figure 18.5 presents the W€ohler (S-N) curves of the fatigue behavior for the studied ferritic stainless steels in the as-

received and heat treated states. Results are similar as in the room temperature tensile tests (Fig. 18.4), i.e., the heat treatment

seems to increase the fatigue endurance.

Table 18.1 Chemical composition (mass %) of studied materials [1, 2]

EN ASTM Carbon Chromium Nitrogen Titanium Molybdenum Niobium Silicon Manganese Nickel

1.4509 S43932 0.02 18 0.02 0.11 0 0.39 0.5 0.5 0.2

1.4521 444 0.02 17.8 0.02 0.12 2.1 0.37 0.5 0.5 0.2

162 K. Ostman et al.

Fig. 18.2 Tensile test curves at 600 �C for the as-received samples and for the 120 h at 600 �C heat treated samples. (a) EN 1.4509 (b) EN 1.4521

Fig. 18.3 Enlarged presentation of the tensile test results at 600 �C. (a) EN 1.4509, where serrated flow is visible especially in the as-received

condition. (b) EN 1.4521

Fig. 18.1 Tensile test curves for the as-received samples at the strain rate of 0.001/s at �50 �C, room temperature, 600 �C and 800 �C. (a) EN1.4509 (b) EN 1.4521

18 Temperature and Strain Rate Effects on the Mechanical Behavior of Ferritic Stainless Steels 163

18.5 Discussion

Juuti et al. have conducted metallographic studies for EN 1.4509 [3] and EN 1.4521 [4] and found that titanium nitrides

(TiN), niobium carbides (NbC) and Laves phases are present in the ferritic microstructure of these steels. Laves phases are

secondary phases, which in EN 1.4509 are composed of iron, niobium, and silicon (FeNbSi) and in EN 1.4521 of iron,

molybdenum and silicon (FeMoSi). There is the difference in the Laves phase composition because EN 1.4521 contains

molybdenum and EN 1.4509 does not [3–5].

The calculated equilibrium volume fractions of the precipitates show that titanium nitrides and niobium carbides remain

stable in the temperature range 0–1,000 �C. Molybdenum (FeMoSi) and niobium (FeNbSi) containing Laves phases will

precipitate at temperatures below about 700 �C, although at low temperatures the reaction kinetics may make the reaction

extremely slow [3–5].

SEM micrographs showed that during heat treatments the size and number of TiN and NbC particles remained rather

stable even at 800 �C. Also the Laves phase was quite stable at 450 �C, but during long heat treatments at 650 �C its

size increased and also new particles nucleated. It was found that the nucleation occurred predominantly at grain boundaries.

The amount of Laves phase was higher after 120 h heat treatment at 650 �C than at 800 �C, which indicates that the Laves

phase is not stable at the higher temperatures [3, 4].

Fig. 18.4 Tensile test curves at strain rates ranging from 0.001/s to 1,000/s at room temperature for the as-received and 120 h at 600 �C heat

treated samples. (a) EN 1.4509 (b) EN 1.4521

Fig. 18.5 W€ohler curves compiled from the results of fatigue tests at 10 Hz at room temperature for the as-received and 120 h at 600 �C heat

treated samples. (a) EN 1.4509 (b) EN 1.4521


Precipitation in ferritic stainless steels may have a pronounced effect on the high temperature performance of the

material. For example in EN 1.4521, the formation of the Laves phase may remove molybdenum from the ferritic matrix

causing a decrease in corrosion resistance [4]. Changes in the size and distribution of the precipitates also affect the

mechanical properties. The as-received states of EN 1.4509 and EN 1.4521 contain notable amounts of precipitates [3, 4].

According to the results obtained in this study, they seem to have a precipitation hardening effect. That could explain the

changes caused by the heat treatment on the mechanical properties. At the same time, Cr, Nb and Ti have the ability to solid

solution harden the alloy [5]. Thus, while the changes in the size and amount of the Laves phase change the precipitation

hardening effect, also the depletion of free solid solutes from the ferritic matrix has an effect on the strength and strain

hardening ability of the alloy.

The mechanical tests made in this study show (Fig. 18.3) that the as-received EN 1.4509 has a high tendency to serrated,

unstable, plastic flow. This phenomenon, also known as the Portevin–Le Chatelier (PLC) effect, indicates that dynamic

strain aging takes place during the tensile test. This behavior can be explained by the dislocations that create local strain

variations in the microscopic scale. These strain gradients promote the diffusion of free solute atoms to the vicinity of

dislocations, which can be locked in place until an increase in the external load mobilizes them again. When the moving

dislocations encounter the next obstacle, such as a precipitate or other dislocations, they are momentarily halted and the

diffusion process repeats itself and the dislocations are again locked in place [6–8].

The experiments revealed that as-received EN 1.4509 shows strain aging behavior in tensile tests conducted at 600 �C at

the strain rate of 0.001/s. When the 120 h heat treatment at 600 �C was made before the tensile test, dynamic strain aging was

clearly reduced. Dynamic strain aging requires effective diffusion, which needs a high content of free solute atoms,

especially carbon [8]. These mechanical tests support the findings of metallographic studies [3–5] showing that part of

free solute atoms form precipitates and thus vanish from the matrix. Because carbon is an interstitial atom and thus one of the

most easily diffusible constituent, it is likely that nucleation and coarsening of NbC precipitates have an effect on the

dynamic strain aging. Also diffusing nitrogen can take part in the static [9] and dynamic strain aging [6] processes, and

therefore if free nitrogen is consumed by the TiN precipitates, it might also reduce the serrated flow. Tensile tests at 600 �C at

strain rates 0.001/s and 0.0003/s on the heat treated samples showed small amounts of serrated flow. The test at the strain rate

of 0.1/s did not reveal such a phenomenon indicating that the loading rate was too high for the solute atoms to diffuse and

keep up with the pace of dislocations.

18.6 Conclusions

The tensile properties of ferritic stainless steels EN 1.4509 and EN 1.4521 were characterized at different temperatures and

strain rates. The experiments were done both on as-received production quality samples as well as on samples that had been

heat treated for 120 h at 600 �C. It was found that the heat treatment has a pronounced effect on the strength and strain

hardening behavior of the test materials. The results support the metallographic studies and give new information about the

precipitation behavior of ferritic stainless steels. However, it still remains unclear why the heat treatment improves the room

temperature properties but impairs the high temperature properties. In the high temperature tests the dislocations have more

thermal energy to overcome the glide obstacles. It might be that the heat treatment increases the size and/or number of

thermal glide obstacles, thus increasing the room temperature strength. Simultaneously the size and/or number of athermal

glide obstacles decreases, which reduces the high temperature strength.

Acknowledgements This research work has been done within DEMAPP (Demanding Applications) research program which is part of FIMECC

Ltd. (Finnish Metals and Engineering Competence Cluster). Financial support has been obtained from Tekes (Finnish Funding Agency for

Technology and Innovation).

References

1. Outokumpu Stainless AB (2008) Avesta research centre: steel grades. Properties and Global Standards

2. Outokumpu Stainless OY (2010) Inspection certificates for EN1.4509 and EN1.4521 Unpublished

3. Juuti TJ, Karjalainen LP, Heikkinen E-P (2012) Precipitation of Si and its influence on mechanical properties of type 441 stainless sSteel. Adv

Mater Res 409:690–695

4. Juuti T, Karjalainen P, Rovatti L, Heikkinen EP, Pohjanne P (2011) Contribution of Mo and Si to laves-phase precipitation in type 444 steel and

its effect on steel properties. In: Proceedings of 7th European stainless steel conference, Como, 21–23 Sep 2011

18 Temperature and Strain Rate Effects on the Mechanical Behavior of Ferritic Stainless Steels 165

5. Sello MP, Stumpf WE (2011) Laves phase precipitation and its transformation kinetics in the ferritic stainless steel type AISI 441. Mater Sci

Eng A 528:1840–1847

6. Yilmaz A (2011) The Portevin–Le Chatelier effect, a review of experimental findings. Sci Technol Adv Mater 12:1–16

7. Bross S, H€ahner P, Steck EA (2003) Mesoscopic simulations of dislocation motion in dynamic strain ageing alloys. Comput Mater Sci 26:46–55

8. Choudhary BK (2013) Influence of strain rate and temperature on serrated flow in 9Cr–1Mo ferritic steel. Mater Sci Eng A 564:303–309

9. Barisic B, Pepelnjak T, Math MD (2008) Predicting of the Luders’ bands in the processing of TH material in computer environment by means of

stochastic modeling. J Mater Process Technol 203:154–165


Chapter 19

Modeling and Simulation in Validation Assessment of Failure

Predictions for High Temperature Pressurized Pipes

J. Franklin Dempsey, Vicente J. Romero, and Bonnie R. Antoun

Abstract A unique quasi-static temperature dependent low strain rate finite element constitutive failure model has been

developed at Sandia National Laboratories (Dempsey JF, Antoun B, Wellman G, Romero V, Scherzinger W (2010) Coupled

thermal pressurization failure simulations with validation experiments. Presentation at ASME 2010 international mechani-

cal engineering congress & exposition, Vancouver, 12–18 Nov 2010) and is being to be used to predict failure initiation of

pressurized components at high temperature. In order to assess the accuracy of this constitutive model, validation

experiments of a cylindrical stainless steel pipe, heated and pressurized to failure is performed. This “pipe bomb” is

instrumented with thermocouples and a pressure sensor whereby temperatures and pressure are recorded with time until

failure occurs. The pressure and thermocouple temperatures are then mapped to a finite element model of this pipe bomb.

Mesh refinement and temperature mapping impacts on failure pressure prediction in support of the model validation

assessment is discussed.

Keywords Thermal plasticity ductile failure validation

19.1 Experimental Validation Tests

An experimental test program [1] was established to validate a quasi-static thermal elastic–plastic ductile failure constitutive

model [2, 3]. To do this, a simple pipe geometry was selected with dimensions of 3 in. in diameter and 14 in. long. The mid-

section is machined down to a 0.02 in. wall thickness with thicker end sections. The pipe is made from 304 L stainless

extruded tube stock, annealed then machined. It is heated at the center and water-cooled on the ends. The pipe ends are held

fixed during the experiment.

The tests are designed to heat and pressurize the pipe until catastrophic failure occurs. Figure 19.1 shows a typical test

setup, heating and repetitive failure samples. Twenty strategically located intrinsic thermocouples are tack-welded to the

pipe to measure and characterize heating through analytical interpolation the temperature profile. The test and analysis suite

includes a combination of applied temperature and pressure ramps and holds to fail the pipe as shown in Fig. 19.1.

J.F. Dempsey (*) • V.J. Romero

Sandia National Laboratories, Albuquerque, NM 87185, USA


B.R. Antoun

Sandia National Laboratories, Livermore, CA 94551-0969, USA



167


19.2 Materials Characterization

In order to model this validation experiment the materials are characterized, first by machining out round tensile specimens

from the original extruded/annealed tube stock, then performing tensile load-deflection measurements through failure.

Knowing the initial neck area of the specimen, engineering stress–strain is computed and reported. From this, true stress vs.

log strain can be computed using an iterative curve fitting algorithm that captures the tensile necking response, shown in

Fig. 19.2. As the true stress and true strain are calculated, a tearing parameter is also calculated. At the point of failure, a

critical tearing parameter is computed. This process is repeated for elevated temperatures to define a thermal elastic plastic

response through failure. This set of temperature dependent tensile material data, along with temperature dependent young’s

moduli, poisons ratio and yield strength is embedded into a thermal elastic plastic constitutive material model of the pipe

bomb with failure being defined by the critical temperature dependent tearing parameter [2].

19.3 Finite Element Model

A thermal-mechanical finite element model of the pipe bomb is created to simulate the validation experiments of Fig. 19.1.

Figure 19.3 shows this model. The 304 L stainless tube contains an inner slug mass to fill the internal void and minimize the

potential energy of the gas volume at failure. An Inconel heating shroud heated by high temperature radiant lamps is used to

deliver uniform heating to the pipe bomb. The shroud is modeled and located beside the center region of the pipe as shown.

As the pipe bomb is heated and pressurized in the thin section and at the hot spot, the ductile material starts bulging toward

the heat source. As the material continues to heat, it softens and bulges further toward the heat source until failure initiation

occurs. Subsequent to the initiation of failure, a loud release of gas energy is heard and failure propagation happens (see

failed specimens I Fig. 19.1).

At the point of failure initiation, a physical quasi-static instability exists. The heated material softens and begins to

separate from the body of the pipe bomb. Because of this, the quasi-static finite element solver [3] becomes unstable, the

modeled stiffness matrix becomes ill conditioned and the solver cannot continue. At this point, the plastic strain has

Fig. 19.1 Pipe bomb test setup and conduct with failure modes shown. Combinations of pressure and temperature ramps/holds were used in a

variety of test sequences to produce the failure specimens

168 J.F. Dempsey et al.

Fig. 19.2 Tensile tests performed for material characterization

Fig. 19.3 Finite element model of the pipe bomb used for (1) two-way thermal-mechanical coupling and (2) temperatures mapped from

experiments for solid mechanics calculations

19 Modeling and Simulation in Validation Assessment of Failure Predictions for High Temperature Pressurized Pipes 169

increased exponentially, forcing the solver to take smaller and smaller time steps to maintain equilibrium. In this “pressure

loaded case” and at the time of failure, time steps can be on the order of nanoseconds, plastic strains in excess of 100 % with

critical tearing parameters reaching five or greater. The inability of the quasi-static solver to continue is judged to define

initiation of failure.

19.4 Mesh Sensitivity and Convergence

In order to validate this model to experimental data, uncertainties must be quantified. First, a mesh sensitivity study must be

performed to determine what element densities are sufficient to predict failure due to high temperature pressurization. Then

the error due to mapping discrete thermocouple temperatures to a full temperature field on the finite element mesh must be

assessed

An approach of mesh doubling starting with one element through the wall thickness was used to show that mesh

convergence on failure pressure was possible. The main difficulty here is that the models are large and they are statically

unstable at the time of failure.

Figure 19.4 shows the finite element model used to perform the mesh sensitivity study. As shown, a one quarter symmetry

pipe bomb finite element model is created. The center region uses uniform 1:1:1 aspect hex elements. It was meshed with

one, two, four and six elements through the thickness, with commensurate mesh densities in the other directions to preserve

nearly 1:1:1 aspect hex elements. Element counts started at 32,368 for the one through the thickness (1tt) model to 7.4

million elements at six through the thickness (6tt). At 1tt, a failure pressure of 1,069 psi was calculated. As the number of

elements through the thickness increased, the predicted failure pressure decreased. Unfortunately, at 6tt, the run was not

completed due to its size. Run times exceed 16 days using 400 processors on a highly parallel computer server [4].

Figure 19.5 shows the results of mesh convergence for 1tt, 2tt, 4tt and 6tt models. Failure pressure vs. number of elements

through the thickness is plotted. In the limit, a Richardson’s extrapolation predicts a failure pressure of 797 psi based on a 1.8

empirical order of convergence from the results of the 1tt, 2tt, 4tt meshes. An associated estimate for numerical solution

uncertainty is 21 psi or +�2.5 % of the failure pressure from the 4tt mesh.

Fig. 19.4 Mesh refinement variations used to perform solution convergence


In addition, a mesh refinement/element quality study was performed on the tensile test material characterization models.

Here, the sensitivity of element aspect ratios, element refinement and hour glassing controls were analyzed. The results of

the study determined these quantities have little influence on the true stress/true strain response but are important at the point

of tensile failure. Typically, the load controlled quasi-static pipe bomb calculations become unstable and never get to this

point, unlike a displacement controlled event.

19.5 Thermocouple Mapping Error

Two models were used to assess the error due to thermocouple mapping from experiment to the finite element model. A fully

coupled two-way thermo-mechanical model was used to simulate the experiment (Fig. 19.3). In this simulation, called a

nearby problem, shroud temperatures pipe pressures and end displacements were taken from the experiment. Emissivities

were estimated and thermal convection was not modeled. A coupled thermo-mechanical response was produced whereby

temperature outputs were recorded at the thermocouple locations to simulate an experiment. This temperature information

was then mapped, via quasi-Hermite bi-cubic interpolation [5, 6], to the pipe bomb model for a simulation comparison to

understand the mapping error.

Figure 19.6 shows the results of the temperature mapping error study. A comparison is shown from the exact temperature

field (coupled simulation) and the interpolated temperature field (mapped). The temperature plots are very similar around the

hot spot but not exactly the same. A temperature difference calculation was made to study the differences between exact and

interpolated simulations as shown. The nature of a cubic spline interpolation will produce an exact mapping of temperatures

at defined thermocouple extraction points but some interpolation error will exist between them. The art of this interpolation

scheme is to minimize the impact of failure pressure prediction with acceptable errors in temperature interpolation. Both

front (hot) and back (cold) views are depicted. This study showed that up to a 5.9 % (48 psi) error in predicted failure

pressure can be expected if not corrected for temperature interpolation error.

Fig. 19.5 Mesh dependence of calculated failure pressure results


19.6 Conclusions

Mesh refinement was found to be an important aspect in predicting pressurization breach of heated ductile pressurized pipes.

Without careful attention to element aspect ratios, a successful mesh convergence on failure pressure was improbable. Use

of at least four elements through the thickness in the heated failure region is recommended but this still produces a solution

uncertainty of 21 psi or + �2.5 % of the failure pressure when compared to the Richardson’s extrapolation. Addition

elements through the thickness will approach a better solution but element counts are very high and computational resources

become challenging.

Thermocouple mapping from experiment to model was done in order to minimize failure pressure prediction error and to

better assess the thermal elastic plastic constitutive model by excluding the requirement to estimate thermal convection and

emissivity physics. A fully coupled thermo-mechanical simulation is easily done but induces more error than mapping error

because the physics cannot be modeled accurately. The error involved in mapping experimentally measured temperatures

from 20 discrete thermocouple locations to the finite element mesh showed that up to a 5.9 % (48 psi) error in predicted

failure pressure can be expected if not corrected for temperature interpolation error.

Final validation of the thermo-mechanical constitutive model is still in progress. A complete uncertainty quantification

study will include material characterization variations, impact of minor wall thickness variations on failure pressure, further

mesh density sensitivities and validation simulations of experiments.

Acknowledgements Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly

owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract

DE-AC04-94AL85000.

Temp (K) Temp (K) Temp (K) Temp (K)

Temp (K)4225

7

-10

-27

-45

-62

-80

-97

1008 1008

829

650

470

291

991

817

642

468

293

829

650

470

291

991

817

642

468

293

Y

Z

X Y Y

Z Z

X X X

Z

Y

Fig. 19.6 Thermocouple mapping error quantification


References

1. Antoun B (2009) Sandia National Laboratories C6 L3 milestone report: material characterization and coupled thermal-mechanical experiments

for pressurized, high temperature systems, 11 Sept 2009

2. Wellman GW (2012) A simple approach to modeling ductile failure. Sandia National Laboratories report SAND2012-1343 printed

3. Adagio 4.24 User Guide, Sandia National Laboratories report SAND2011-1825, printed March 2011

4. Red Sky computing resource, Sandia scientific, engineering and high performance computing, ~505 Tflops peak, 2823 nodes, 22,584 cores,

2012

5. Romero V, Dempsey JF, Wellman G, Antoun B, Sherman MModel validation and UQ techniques applied to a stainless-steel constitutive model

tested on heated pipes pressurized to failure. Sandia National Laboratories report in preparation

6. Romero V, Dempsey JF, Wellman G, Antoun B (2012) A method for projecting uncertainty from sparse samples of discrete random functions─example of multiple stress–strain curves. In: 14th AIAA non-deterministic approaches conference, Honolulu, 23–26 Apr 2012


Chapter 20

Unified Constitutive Modeling of Haynes 230 for Isothermal

Creep-Fatigue Responses

Paul Ryan Barrett, Mamballykalathil Menon, and Tasnim Hassan

Abstract Lifing analysis and design of high temperature components, such as, turbine engines, needs accurate estimation of

stresses and strains at failure locations. The structural integrity under these high temperature environments must be

evaluated through finite element structural analysis. The structural analysis requires a robust constitutive model to predict

local stresses and strains. The robustness of a new constitutive model can be validated by predicting stress and strain

responses for a broad set of loading histories representative of local structural responses. The experimental database

encompasses low cycle creep-fatigue experiments for a nickel-base superalloy, Haynes 230, under symmetric, uniaxial

strain-controlled loading histories which include isothermal with and without hold times, with and without a mean strain, at

temperatures ranging from 75 �F to 1,800 �F. A unified viscoplastic model based on nonlinear kinematic hardening

(Chaboche type) with several added features, such as strain range dependence and static recovery will be critically evaluated

against the experimental responses. This study will especially evaluate various flow rules, like, Norton, sine hyperbolic, and

creep-plasticity interaction models on the viscoplastic simulation. Simulations from the modified model are compared to the

experimental responses to demonstrate the strengths and weaknesses.

Keywords Haynes 230 • High-temperature creep-fatigue • Viscoplasticity • Unified constitutive modeling • Flow rules

20.1 Introduction

High-temperature systems like gas turbine engines and nuclear power reactors are rich in material complexities because of

fatigue and creep interactions. As a result of start-up and shut-down cycles, the nature of the loading induces repeated

thermomechanical stresses that gradually degrade the materials. The high-temperature components may experience

temperatures up to 1,800 �F (982 �C), which inherently changes the material structure and behavior of the component

characterizing the complexity of material behavior. The material complexities in high temperature components are

manifested as time-dependent processes such as creep, oxidation, dynamic strain aging, creep-fatigue, thermomechanical

fatigue and cyclic creep or ratcheting. In order to substantially improve current design methodologies, it is essential to

understand these complex material phenomena, under broad loading conditions and high temperature environments. The

structural integrity under these harsh environments must be evaluated in FEA software so that the design ensures safety,

reliability, and performance. However, the structural design and analysis requires an adopted model to predict locally

stresses and strains. Therefore, the development of a unified, robust constitutive model that can reliably predict stresses and

strains under a broad set of loading histories is essential. An advanced constitutive model helps in improving accuracy and

fidelity in prediction of stress and strain redistribution in components under service.

In the present study, the component application of interest is a gas turbine engine, in which the combustor liners of this

engine are fabricated from sheets of Haynes 230. During turbine engine operation, thermomechanical cyclic loads induce an

abundance of complex viscoplastic responses at life limiting locations and initiate low-cycle fatigue cracks. Haynes 230 is a

P.R. Barrett (*) • T. Hassan

Department of Civil, Construction and Environmental Engineering, North Carolina State University, Raleigh 27695-7908, NC, USA


M. Menon

Honeywell Aerospace, 85034 Phoenix, AZ, USA



175


solid-solution-strengthened alloy, a Ni-Cr-W-Mo superalloy, which possesses excellent high temperature strength and

outstanding resistance to oxidation in these severe environments. Mechanical behavior of fatigue and creep-fatigue

interactions at elevated temperatures has been studied over the last two decades [1–11]. Superalloys, ferritic steels, and

stainless steels form a class called Austenitic Carbide Precipitating (ACP) alloys which are similar from a materials

modeling point of view [12]. This commonality amongst the alloys allows for a constitutive model development that can

characterize a broad range of alloys employed in various applications. Our model investigation requires an experimental

database for validation. The experimental characterization of isothermal, low cycle fatigue of Haynes 230 is presented over a

scope of test parameters: temperature, strain range, strain rate, and hold times helps reveal the macroscopic complexities

caused by subtle microstructural changes. Macroscopic constitutive models describing cyclic viscoplasticity must be able to

describe the hysteretic, irreversible nature of cyclic deformation in a consistent thermodynamic framework.

The development of these constitutive equations, in which the present state of the material depends only on the present

values of the observable macroscopic quantities like stress or strain and a set of internal-state variables, such as, kinematic

and isotropic variables must be consistent with the thermodynamics of irreversible processes [13]. Our constitutive model

development is applied in a hierarchical framework based on unified theories similar to the Chaboche model [14–20]. These

models employ a hardening superposition based approach where kinematic, isotropic, and thermal-recovery (time-recovery)

hardening are inherently connected. Using this hardening superposition the complexity of the model develops this hierar-

chical framework which allows for description of time-dependent processes such as creep, fatigue, and dynamic strain aging

related to high temperature exposure along with other multiaxial and nonproportional characterizations that develop.

Modern constitutive models are increasingly complex, thus, it is critical to adopt an automated parameter determination

that uses specific experimental responses. A brief summary of an automated parameter determination involving a hybridized

optimization process is also presented. Finally, the simulation capabilities of our model are evaluated.

20.2 Experimental Procedures

20.2.1 Test Material and Metallography

The Haynes 230 alloy was received as bar stock in solution annealed conditions. Microstructural grains are manifested as

non-straight boundaries whereas inside these boundaries annealing twins characterized by straight boundaries develop

internally. The Haynes 230 alloy also contains precipitates which manifest themselves as particles inside the matrix. The

sample was electrolytically etched in a solution containing hydrochloric acid (HCl) and hydrogen peroxide (H2O2) to reveal

the microstructural features. The precipitates were tungsten-rich primary carbides of the stoichiometric composition M6Ctype, where M denotes the metallic atom and C represents the carbon contribution. Grain size measurements were performed

on the heat-treated specimen in which microstructural images at 100� magnification were compared with the standard

ASTM plates, according to ASTM E112-10. The average grain size was found to be around 60 μm. The nominal chemical

composition of the alloy was presented in Barrett et al. [22].

20.2.2 Low Cycle Fatigue Testing

Low cycle fatigue tests were performed on universal, servo hydraulic testing machines. The frames were outfitted with

commercially available software used to control the test and collect data. For elevated temperature testing of either

isothermal or non-isothermal, heating is achieved through an induction heating system. In accordance with ASTM standards,

LCF testing followed ASTM E606-04. LCF tests with symmetric, axial strain-controlled loading histories were performed

isothermally with and without hold times, with or without a mean strain, as well as varying strain rates and strain ranges, at

temperatures ranging from 75 �F to 1,800 �F. Continuous strain-controlled axial cycling had a triangular waveform with

cyclic frequencies of 0.2, 2, and 20 cycles per minute (cpm) at imposed strain ranges varying from 0.30–1.60 %. In order to

explore the scope of test parameters relevant to engine operation, experiments were divided into groups to differentiate these

independent and dependent testing parameters. Isothermal LCF tests under strain control were conducted at a constant

frequency of 20 cpm, with a strain ratio (min/max) of �1.0, and temperatures ranging from 75 �F to 1,800 �F comprised

Group 1 (G1). It should be noted that some of these tests were conducted with sinusoidal control waveforms along with a

different cyclic frequency in order to control the strain better. These isothermal, continuous LCF tests of G1 allow for the

176 P.R. Barrett et al.

investigation of the effect of temperature and strain range on the hysteresis curve and Bauschinger effect, including the

cyclic stress–strain behavior as well as the isothermal LCF lifing (Fig. 20.1a). The next sequence of tests, similar to G1, was

conducted except with three rates at a cyclic frequency of 0.2, 2, and 20 cpm at ‘fixed’ strain ranges. These ‘fixed’ strain

ranges were established from plotting the measured plastic strain range versus the total strain range at half-life for the various

temperatures tested in G1, in which a baseline level of 0.20 % ‘fixed’ plastic strain range, Δεxp was chosen. These particular

tests comprise Group 2 (G2) experiments which were critical in understanding the effect of loading rate or strain rates,

determining the rate dependency of HA230 (Fig. 20.1a). In order to investigate the effect of hold time, a series of tests Group

3 (G3) were designed such that interrupted, isothermal LCF with hold times imposed at the peak compressive strain were

conducted at the ‘fixed’ strain ranges of G2 and a fixed loading rate (ramping time) of 20 cpm. The temperatures ranged from

1,200 �F to 1,800 �F. Hold periods, tH, of 60 and 120 s were imposed. Operating temperatures are in the range where creep

deformation occurs so that creep-fatigue interaction can be studied (Fig. 20.1b).

20.3 Experimental Results and Discussion

20.3.1 General Hysteresis Characteristics

Hysteresis curves at different temperatures reveal different stress evolutions (Fig. 20.2a, b). Moreover, the hardening/

softening evolution of the HA230 can be characterized by analyzing the stress amplitude responses in Fig. 20.3a, b. The

results indicate that cyclic deformation and the developed stresses are highly sensitive to temperature under low cycle

fatigue conditions. At room temperature, HA230 cyclically hardens initially followed by cyclic softening (Fig. 20.3a).

Whereas, at 400–1,200 �F, the material continues to cyclically harden without any sign of stabilization (Fig. 20.3b). For

1,200–1,400 �F, the rate of cyclic hardening gradually reduced, whereas for 1,600 �F and above either cyclic hardening or

cyclic softening is observed (not shown). The rapid hardening evolution found between 400 and –1,400 �F seems to be

related to the time-dependent effects of dynamic strain aging, which strengthens the strain hardening mechanisms [4–9]. The

cyclic stress–strain response of the material is governed by these three hardening regimes that are more pronounced with

higher strain range amplitudes. The above described cyclic evolutionary responses correlate to dislocation and particle

interaction during plastic deformation. Overall, Haynes 230 alloys exhibits an inherent strain range as well as temperature

dependence which will be of importance in modeling the hysteretic phenomena.

20.3.2 Effect of Strain Rate

The rate dependence of loading on the HA230 responses is characterized through Group 2 (G2) experiments. In general,

viscoplastic materials exhibit loading rate effects as well as creep and stress relaxation. Viscoplasticity relates the temporal

growth of permanent deformations. Rate effects can occur due to the time-dependent nature of the deformation. The stress

amplitudes, σxa, and mean stresses, σxm, against number of cycles from symmetric, strain-controlled experiments for loading

rates of 0.2, 2, and 20 cpm for temperatures 800 �F and 1,800 �F are depicted in Fig. 20.4a, b. At the lower temperatures,

75–400 �F (now shown), the stress amplitude responses are essentially superimposed indicating the effect of loading rate is

not present. However, in the temperature regime 800–1,400 �F (Fig. 20.4a), the stress amplitudes for lower loading rates (i.e.

lower strain rates) are greater in most cases. In essence, over this temperature domain, negative rate sensitivity of the stress

response is caused by the effect of dynamic strain aging [4–9]. The dynamic strain aging temperature regime for HA230 is

thus believed to be between 800–1,400 �F. Lastly, rate-dependent responses can be observed for 1,600–1,800 �F(Fig. 20.4b), where stress amplitude responses are higher for higher strain rates.

a b

t t

ex exεa

εa

Fig. 20.1 Isothermal LCF,

symmetric axial strain cycling

(a) without holds Groups 1–2

and (b) with strain holds

Group 3

20 Unified Constitutive Modeling of Haynes 230 for Isothermal Creep-Fatigue Responses 177

20.3.3 Role of Dynamic Strain Aging

Dynamic strain aging has been found to occur in the intermediate temperature domain for austenitic carbide precipitating

(ACP) alloys, which encompass Ni base superalloys, like Haynes 230, Co base superalloys, like Haynes 188, FeNi base

super stainless steels, like Haynes HR-120, and ferritic steels, like 9Cr-1Mo [4–9]. All of these alloys are austenitic,

delivered solution treated, that develops similar physical mechanisms and time dependent processes [12]. While most

researchers understand the creep, oxidation, and metallurgical instabilities that arise because of the time dependent nature

caused by elevated temperature, few have tried to understand the phenomenon of dynamic strain aging. Some of the

macroscopic manifestations of DSA include, strong cyclic hardening, negative strain rate dependency, and serrated yielding.

Macroscopic evidence of DSA, including the aforementioned phenomena has been depicted already in Figs. 20.2b, 20.3b

a b

-100

-50

0

50

100

-1 -0.5 0 0.5 1

(ksi)

HA230/G1T = 75�F

(20 cpm)

-100

-50

0

50

100

-0.75 -0.5 -0.25 0 0.25 0.5 0.75

Cycle

2

501000

10000

Dex = 1.4%

HA230/G2T = 800�F

(2 cpm)

Dex = 1.0%

sx(ksi)sx

ex (%) ex (%)

Fig. 20.2 Isothermal LCF strain-controlled, without holds, hysteresis responses from G1 (a) Δεx ¼ 1:4% T ¼ 75�F and G2 (b)

Δεx ¼ 1:0% T ¼ 800�F

a b

-20

20

60

100

1 100 10000 1000000

(ksi)

Log(N)

HA230/G1HA230/G2

T = 75∞FT = 800∞F

Δεx = 1.60 :: σxa

Δεx = 1.60 :: σxa

Δεx = 1.20 :: σxa

Δεx = 1.00 :: σxa

Δεx = 0.80 :: σxa

Δεx = 0.60 :: σxa

Δεx = 1.40 :: σxa

Δεx = 1.20 :: σxa

Δεx = 1.00 :: σxa

Δεx = 0.80 :: σxa

Δεx = 0.60 :: σxa

σxm

σxm

σxm

σxm

σxm

σxm

-20

20

60

100

1 100 10000 1000000

σxm

σxm

σxm

σxm

σxm

sx

Fig. 20.3 Haynes 230 strain range dependence upon cyclic stress evolution, with stress amplitude (σxa) and mean responses (σxm) (G1) withconstant isothermal temperature: (a) T ¼ 75�F and (b) T ¼ 800�F


and 20.4a. In these figures, the strong cyclic hardening over all strain ranges (Fig. 20.3b) as well as the negative strain rate

sensitivity in the cyclic stress evolution (Fig. 20.4a) supports the DSA claims above. The occurrence of DSA features

appears only within the temperature regime of 800–1,400 �F. Dynamic strain aging presents a serious modeling challenge.

20.3.4 Effect of Strain Holds

The introduction of a hold time in a strain controlled, low cycle fatigue test causes stress relaxation, what is referred in the

literature as creep-fatigue interaction [1–3]. The influence of the material’s viscosity in the hold time creep-fatigue test is

obtained through the plastic deformation induced during the hold. This influence is inherently linked to the time-recovery

effects corresponding to a slow restoration of the crystalline structure. The Haynes 230 material which experiences cyclic

hardening for temperatures below 1,800 �F also undergoes increased stress relaxation as the cycles evolve. Naturally, one

would assume that an increase in the relaxed stress would be directly related to the hold time of the creep-fatigue interaction;

however, it can be shown that the hardening/softening patterns of the material state influences the relaxed stresses during

cyclic deformation. In order to quantify the effect of the hold time for various temperatures tested, we have calculated a

normalized relaxation stress at each cycle by subtracting the compressive stress after the hold from the compressive stress

before the hold to obtain Δσr and have subsequently scaled this stress difference, by the compressive stress peak before the

hold. At the higher temperatures (>1,600 �F) for which we know the material behaves in a viscous manner we see significant

stress relaxation with hold times. At the lower temperatures of 1,200 �F and 1,400 �F though we found that the material is

mostly rate-independent at the different loading strain rates we still see some amount of stress relaxation. Another important

trend is that higher normalized stresses occur with increasing temperature (Fig. 20.5a). The examination of the hysteresis

responses for each temperature uncovers additional characteristics unique to the cyclic relaxation tests. Macroscopically

stress relaxation is manifested in the hysteresis loops as a differential stress drop at the imposition of the peak compressive

strain hold (Fig. 20.5b).

20.4 Unified Viscoplasticity

A unified viscoplastic constitutive model is under development in the study in order to account for the interactions between

creep and plasticity due to inherent viscous responses caused by elevated temperatures. A modified Chaboche based

viscoplastic constitutive model has been chosen with various features [14–20]. The scale of modeling is macroscopic.

At the macroscale, the homogenized continuum approach neglects local heterogeneities and characterizes the material

behavior through both observable and internal state variables. Therefore, a representative volume element of a material is

a b

-20

20

60

100

1 100 10000

(ksi)

N

HA230/G2T = 800∞F

HA230/G2T = 1800∞F

-10

10

30

50

1 10 100 1000 10000

(ksi)

N

0.2 cpm σxa

2 cpm σxa

20 cpm σxa

σxm

σxm

σxm

σxm

σxm

σxm

0.2 cpm σxa

2 cpm σxa

20 cpm σxa

sx sx

Fig. 20.4 Haynes 230 strain rate dependence with stress amplitude and mean responses for G2, isothermal temperatures: (a) T ¼ 800 �F and (b)

T ¼ 1,800 �F


subjected at a uniform macroscopic state. The present state of the material depends only on the present values of the

observable macroscopic quantities like stress or strain and a set of internal-state variables, such as, kinematic and isotropic

variables that are consistent with the thermodynamics of irreversible processes as shown in Fig. 20.6.

The basic assumption of small strains is applied with typical ingredients of any plasticity model of strain decomposition

Eq. 20.1, generalized Hooke’s law Eq. 20.2, and the normality rule for viscoplastic flow Eq. 20.3. Strain decomposition of

strain (ε) into elastic (εe) and inelastic (εin) part has been assumed,

ε ¼ εe þ εin (20.1)

The elastic part obeys Hooke’s law in 3D,

εe ¼ 1þ ν

Eσ� ν

Etrσð ÞI (20.2)

where E and ν indicate Young’s modulus and Poisson’s ratio, respectively, σ and I are the stress and identity tensors,

respectively, and tr is the trace. A unified viscoplastic model has been chosen as it allows the modeling of rate-dependent

behavior, an important feature particularly at higher temperatures. The inelastic strain rate is expressed as,

_εin ¼ 3

2_p

s� a

J σ� αð Þ (20.3)

where (·) denotes the differentiation with respect to time, s and a are the deviators of the stress and back stress, respectively.

J σ� αð Þ is expressed as shown in Eq. (20.4).

a b

0

0.3

0.6

0.9

0 200 400 600

No

rmal

ized

rel

axed

str

ess

N

1200F1400F1600F1800F

HA230/G3tH = 120s

HA230/G3T = 1400∞F

tH = 120s

-100

-50

0

50

100

-0.5 -0.25 0 0.25 0.5

(ksi)

ex (%)

Dex = 0.64%

Cycle

110100500

sx

Fig. 20.5 (a) Normalized stress relaxation of Haynes 230 for different temperatures at a constant hold time tH ¼ 120 s, and (b) Isothermal LCF

strain-controlled, with holds, hysteresis responses from G3

Internal Variablesaj

s,e,T

RVE

Fig. 20.6 Macroscopic constitutive modeling for unified viscoplasticity


J σ� αð Þ ¼ 3


� �12

(20.4)

The plastic strain rate norm determines the type of flow rule one adopts considering associative viscoplasticity for

normality of viscoplastic flow. Three plastic strain rate norms are presented [20] in the classical Norton’s rule for secondary

creep, Eq. 20.5, sine hyperbolic form of the Norton’s rule, Eq. 20.6, and the product logarithm form of the Norton’s rule,

Eq. 20.7. σo is the yield stress, and, K and n are rate-dependent parameters. Each flow rule differs in the algebraic form of the

viscoplastic strain rate norm as a function of the Norton’s power law rule.


� �n

(20.5)

_p ¼ A sinhJ σ� αð Þ � σo

K

� �n� �(20.6)


� �n

eαJ σ�αð Þ�σo

Kh in (20.7)

A hardening superposition accounting for kinematic, isotropic, and thermal-recovery hardening is adopted. The modified

Chaboche model is a superposition of the Armstrong-Frederick rule (1966) with added features [14–20]. The kinematic

hardening rule comprises strain hardening, dynamic recovery, and static recovery Eq. 20.8. Static recovery provides creep

and thermal recovery for low strain rates. Simulations of the stress relaxation under strain holds, is one of the most important

deformation behaviors in terms of creep-fatigue damage analysis of the actual components.

a ¼X4i¼1

ai

_ai ¼ 2


ai þ 1

Ci

@Ci

@T_Tai

i ¼ 1 to 4 (20.8)

The importance of capturing the shape of the hysteresis loops as closely as possible has been shown in [20, 20] to have an

impact in the overall simulation quality. Strain range dependence is modeled by considering a strain memory surface which

memorizes the prior largest plastic strain range. The radius and center of the strain memory surface are q and Y respectively.

The memory surface equation is given by Eq. 20.9 and the evolution equations of q and Y are given by Eq. 20.10 and 20.11

respectively. Material constant η can be determined from uniaxial response and are related to the stabilized plastic strain

amplitudes. H(g) is the Heaviside step function. The kinematic hardening dynamic recovery parameters γi of Eq. 20.8 are

varied with cycles and are functions of q. The evolutions of γi are according to Eqs. 20.13 and 20.14.

g ¼ 2

3εin � Y�

: εin � Y� � �1

2

� q ¼ 0 (20.9)

_q ¼ ηHðgÞ n : n�h i½ � _p (20.10)

_Y ¼ffiffiffiffiffiffiffiffi3 2=

p1� ηð ÞHðgÞ n : n�h in�½ � _p (20.11)

n� ¼ffiffiffi2

3

rεin � Y

q(20.12)


� _p (20.13)

γASi ðqÞ ¼ aγi þ bγie�cγiq (20.14)



The parameters for the model have been determined from a broad set of experimental responses. The steps in determining

the parameters using a hybridized genetic algorithm have been outlined in Barrett et al. [22]. Strain-controlled uniaxial

isothermal experiments without any strain hold times are used to determine the rate-independent kinematic hardening

parameters. Experiments at different loading rates as well as the isothermal creep data is used for rate-dependent parameter

determination. Isothermal experiments with strain hold times (relaxation) are used to determine the static recovery

parameters of kinematic hardening.

20.6 Simulations

20.6.1 Rate-Dependence

The plastic strain rate norm (Eqs. 20.5, 20.6 and 20.7) determines the type of flow rule one can adopt in simulating the

rate-dependent behavior of a material. The relation between this viscous stress and the plastic strain rate norm is usually

highly nonlinear. In our study, we explored three different flow rules. Each flow rule is intended to control the magnitude of

viscoplastic flow. Also, in all cases the viscoplastic deformation occurs when the von-Mises stress exceeds the yield stress

given by the viscous stress. The case of rate-independent plasticity can be deduced from the flow rule as a limiting case for

all three rules. Therefore, the transition between rate-independent plasticity and rate-dependent viscoplasticity can easily be

handled numerically. Both the sine hyperbolic flow and the product logarithm of the Norton’s rule have an additional

parameter that controls the saturation of stress amplitudes at high strain rates. At 1,800 �F positive rate dependent behavior is

exhibited with cyclic softening for Haynes 230 (Fig. 20.4b) and the simulations of the modified Chaboche model are

presented for the Norton’s flow rule only to show the strength of the adopted model. Figure 20.7 shows that through the rate-

dependent parameters we can simulate hysteresis loops from different strain rates at 1,800 �F for a half-life cycle. Similar

simulations are obtained for each temperature.

20.6.2 Strain Range Dependence

The importance of capturing the shape of the hysteresis loops as closely as possible has been shown to have an impact in the

overall simulation quality. Strain range dependence is modeled by considering a strain memory surface which memorizes

the prior largest plastic strain range. The kinematic hardening parameters γi are varied with cycles and are functions of the

size of the strain memory surface. This strain memory surface size stabilized to half the width of the stabilized hysteresis

loop. The modeling capability of strain range dependence allows one to specify, for a particular temperature, an evolution

equation for the Chaboche parameters as a function of the strain range which is physically linked to the hysteresis responses.

The modeling capability of the cyclic stress–strain behavior for stress amplitudes at different strain ranges and a specific

temperature are presented in Fig. 20.8a. The simulations for, T ¼ 800 �F, perform fairly well in capturing the overall

hardening behavior of the material. Hysteresis loops for the initial and stabilized cycles are shown in Fig. 20.8b. The strain

range dependence modeling through the Chaboche kinematic parameters are critical in fidelity of the hysteresis loops

whereby it enables one to robustly capture hysteresis loop shape and size. The hysteresis cycles shown in Fig. 20.8b reflect

the fidelity one can achieve when strain range dependence is properly modeled. For all temperatures in the range

75–1,800 �F, the simulations perform fairly well in capturing the hysteresis responses.

20.6.3 Creep-Fatigue Stress Relaxation for Strain Holds

The modeling of stress relaxation has been performed for isothermal cyclic strain controlled experiments with strain holds at

the peak compressive strain. The accurate modeling of stress relaxation is very important to reliably predict creep-fatigue

interaction. For temperatures at which the material behavior is overall rate-independent (� 1,400 �F) but still shows stressrelaxation for strain holds the stress relaxation has been modeled through the static recovery term in the Chaboche kinematic


a b

0

50

100

1 100 10000

(ksi)

N

HA230/G1T = 800∞F HA230/G1

T = 800∞F

Δεx = 1.60 :: σxa

Δεx = 1.20 :: σxa

Δεx = 0.80 :: σxa

1.60 :: Sim

1.20 :: Sim

0.80 :: Sim

-100

-50

0

50

100

-1 -0.5 0 0.5 1

(ksi)

ex (%)

1Sim2000Sim

Dex = 1.2%sx

sx

Fig. 20.8 (a) Haynes 230 simulation of stress amplitudes with cycles at 800 �F for different loading strain ranges and (b) simulation of hysteresis

loops at 800 �F for initial and half-life cycles

a b

c

-30

-15

0

15

30

-0.4 -0.2 0 0.2 0.4

(ksi) (ksi)

(ksi)

ex (%) ex (%)

ex (%)

1000

Sim

HA230/G2T = 1800∞FDex = 0.39%

0.2 cpm

HA230/G2T = 1800∞FDex = 0.39%

2 cpm

HA230/G2T = 1800∞FDex = 0.39%

20 cpm

-30

-15

0

15

30

-0.4 -0.2 0 0.2 0.4

1045

Sim

-30

-15

0

15

30

-0.4 -0.2 0 0.2 0.4

1000

Sim

sx sx

sx

Fig. 20.7 Simulation of hysteresis loops at 1,800 �F for different strain rates: (a) 0.2 cpm, (b) 2 cpm, (c) 20 cpm


hardening rule (Eq. 20.8). The simulations at 1,400 �F for two hold times of 60 and 120 s are shown in Fig. 20.9. The

corresponding hysteresis loop simulations for the same temperature are shown in Fig. 20.10. The simulations describe the

experimental responses well. For higher temperatures (� 1,600 �F) the simulation of stress relaxation used both the rate-

dependence of the flow rule (Eq. 20.5) and static recovery (Eq. 20.6). the simulation of the normalized relaxed stresses reveal

the quality of the overall simulation as the simulation of the peak compressive stresses are inherently present. The simulation

results for higher temperatures are not shown; however, the quality of simulations is comparable in terms of fidelity of the

model.

20.7 Conclusions

Mechanical testing allows for constitutive model development in the design of high temperature systems experiencing a

range of loading histories. The loading histories strive to replicate in-service conditions. A comprehensive experimental

database allows one to evaluate the predictive capabilities of any constitutive model. In the model it is important to capture

the various damage mechanisms and relevant material complexities associated with the time-dependent nature of the process

a b

-60

-30

0

30

60

-0.4 -0.2 0 0.2 0.4

(ksi)

ex (%) ex (%)

HA230/G3T = 1400∞FDex = 0.64%tH = 120s

HA230/G3T = 1400∞FDex = 0.64%tH = 120s

1

Sim-Ini-60

-30

0

30

60

-0.4 -0.2 0 0.2 0.4

600

Sim-Final

sx

Fig. 20.10 Simulation of hysteresis loops for low cycle fatigue with 120 s hold time at 1,400 �F: (a) 1st cycle (b) 600th cycle

-30

-20

-10

00 200 400 600

Dsr (k

si)

N

HA230/G3T=1400�F

Experiment – 60s hold

Simulation – 60s holdSimulation – 120s hold

Experiment – 120s hold

Fig. 20.9 Haynes 230 simulation of relaxed stresses at 1,400 �F for hold times of 60 and 120 s


including the coupling effects of creep, plasticity, and environmental degradation. The modified Chaboche model proposed

in the paper accounts for these microstructural characterizations in a phenomenological manner. The hardening superposi-

tion of kinematic, isotropic, and thermal-recovery (time-recovery) hardening enables one to accurately predict these material

changes. The modeling features of strain range dependence; temperature dependence as well as thermal recovery have been

modeled and validated against the Haynes 230 experimental database. The simulations perform fairly well for the

isothermal, strain-controlled experiments at all temperatures. The importance of strain range dependence has proven to be

irreplaceable in achieving a desired robustness in the model. However, the opportunities of incorporating other advanced

model features are available in order to enhance the current state of the art modeling. Some nuances related to dynamic strain

aging as well as post-peak softening are some of the isothermal challenges. An isothermal modeling for these ACP alloys at

high temperatures seems to be another area of research that needs to be further investigated in order to fully understand the

complexity of design of high temperature systems experiencing thermomechanical loaded components. Overall, the fidelity

of current simulations is promising in leading to the ultimate objective of life prediction of fatigue.

Acknowledgements The financial support of Honeywell Aerospace is gratefully acknowledged. All experiments were conducted by Element in

Cincinnati, Ohio.

References

1. Rao K, Meurer H, Schuster H (1988) Creep-fatigue interaction of inconel 617 at 950�C in simulated nuclear reactor helium. Mater Sci Eng A

104:37–51

2. Rodriguez P, Rao K (1993) Nucleation and growth of cracks and cavities under creep-fatigue interaction. Prog Mater Sci 37:403–480

3. Rodriguez P, Mannan SL (1995) High temperature low cycle fatigue. Sadhana 20(1):123–164

4. Bhanu Sankara Rao K, Schiffers H, Schuster H, Nickel H (1988) Influence of time and temperature dependent processes on strain controlled

low cycle fatigue behavior of alloy 617. Metallurgical Trans 19(A):359–371

5. Mannan SL (1993) Role of dynamic strain ageing in low cycle fatigue. Bull Mater Sci 16(6):561–582

6. Valsan M, Sastry DH, Rao K, Mannan SL (1994) Effect of strain rate on the high temperature low cycle fatigue properties of a nimonic PE-16

superalloy. Metallurgical Mater Trans A 25(A):159–171

7. Rao K, Castelli MG, Ellis JR (1995) On the low cycle fatigue deformation of Haynes 188 superalloy in the dynamic strain aging regime.

Scripta Metallurgica et Materialia 33(6):1005–1012

8. Rao K, Castelli MG, Allen GP, Ellis JR (1997) A critical assessment of the mechanistics aspects in Haynes 188 during low-cycle fatigue in the

range 25�C to 1000�C. Metallurgical Mater Trans A 28(A):347–361

9. Chen LJ, Liaw PK, Mcdaniels RL, Klarstrom DL (2003) The low-cycle fatigue and fatigue-crack-growth behavior of Haynes HR-120 alloy.

Metallurgical Mater Trans A 34(A):1451–1460

10. Lu YL, Chen LJ, Wang GY, Benson ML, Liaw PK, Thompson SA, Blust JW, Browning PF, Bhattacharya AK, Aurrecoechea JM, Klarstrom

DL (2005) Hold time effects on low cycle fatigue behavior of HAYNES 230® superalloy at high temperatures. Mater Sci Eng A 409

(1–2):282–291

11. Mannan SL, Valsan M (2006) High-temperature low cycle fatigue, creep–fatigue and thermomechanical fatigue of steels and their welds. Int J

Mech Sci 48(2):160–175

12. Hasselqvist M (1999) TMF crack initiation lifing of austenitic carbide precipitating alloys. ASME Turbo Expo 2004:1–9

13. Germain P (1973) Cours de mecanique des milieux continus, vol I. Masson, Paris

14. Frederick PJ, Armstrong C.O (1966) A mathematical representation of the multiaxial bauschinger effect, Report RD/B/N731, CEGB, Central

Electricity Generating Board, Berkeley

15. Chaboche JL (1986) Time independent constitutive theories for cyclic plasticity. Int J Plast 2:149–188

16. Chaboche JL (1989) Constitutive equations for cyclic plasticity and cyclic viscoplasticity. Int J Plast 5(3):247–302

17. Chaboche JL (1991) On some modifications of kinematic hardening to improve the description of ratcheting effects. Int J Plast 7(7):661–678

18. Bari S, Hassan T (2000) Anatomy of coupled constitutive models for ratcheting simulation. Int J Plast 16:381–409

19. Bari S, Hassan T (2002) An advancement in cyclic plasticity modeling for multiaxial ratcheting simulation. Int J Plast 18(7):873–894

20. Chaboche JL (2008) A review of some plasticity and viscoplasticity constitutive theories. Int J Plast 24(10):1642–1693

21. Krishna S, Hassan T, Ben Naceur I (2009) Macro versus micro-scale constitutive models in simulating proportional and nonproportional cyclic

and ratcheting responses of stainless steel 304. Int J Plast 25(10):1910–1949

22. Barrett PR, Menon M, Hassan T (2012) Isothermal fatigue responses and constitutive modeling of Haynes 230, 2012 ASME PVP, Toronto,

15–19 July 2012


Documents

Challenges In Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 2: Proceedings of the 2013 Annual Conference on Experimental