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Unified Constitutive Laws of Plastic Deformation

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Unified Constitutive Laws of Plastic Deformation Edited by A. S. Krausz and K. Krausz Department of Mechanical Engineering University of Ottawa Ontario, Canada

Academic Press San Diego

New York

Boston

London

Sydney

T o k y o Toronto

This book is printed on acid-free paper. Q Copyright 9 1996 by ACADEMIC PRESS, INC. All Rights Reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. A c a d e m i c P r e s s , Inc. A Division of Harcourt Brace & Company 525 B Street, Suite 1900, San Diego, California 92101-4495

United Kingdom Editionpublished by Academic Press Limited 24-28 Oval Road, London NW1 7DX Library of Congress Cataloging-in-Publication Data Unified constitutive laws of plastic deformation / edited by A.S. Krausz, K. Krausz. p. cm. Includes index. ISBN 0-12-425970-7 (alk. paper) 1. Deformations (Mechanic)--Mathematical models. 2. Plasticity-Mathematical models. 3. Dislocations in crystals--Mathematical models. I. Krausz, K. TA417.6.U57 1996 620.1' 123--dc20 96-2097 CIP PRINTED IN THE UNITED STATES OF AMERICA 96 97 98 99 00 01 MM 9 8 7 6 5

4

3

2

1

Contents Contributors Preface xi

~ l

ix

Unified Cyclic ViscoplasticConstitutive Equations: Development, Capabilities,and Thermodynamic Framework J. L. Chaboche I. II. III. IV. V.

List of Symbols 1 Introduction 2 A Cyclic Viscoplastic Constitutive Law Capabilities of the Constitutive Model Thermoviscoplasticity 33 Conclusion 61 References 63

4 20

Dislocation-Density-RelatedConstitutive Modeling Yuri Estrin I. II. III. IV.

~ 1

Introduction 69 One-Internal-Variable Model Two-Internal-Variable Model Conclusion 103 References 104

72 91

ConstitutiveLaws for High-TemperatureCreep and Creep Fracture R. W. Evans and B. Wilshire

Contents I. Introduction 108 II. Traditional Approaches to Creep and Creep Fracture 109 III. The 0 Projection Concept 117 IV. Analysis of Tensile Creep Data 123 V. Creep under Multiaxial Stress States 132 VI. Creep under Nonsteady Loading Conditions VII. Conclusions 150 References 151

~ ] 1

143

Improvements in the MATMOD Equations for Modeling Solute Effects and Yield-Surface Distortion Gregory A. Henshall, Donald E. Helling, and Alan K. Miller I. Introduction 153 II. Modeling Yield-Surface Distortions 160 III. Simulating Solute Effects through Short Range Back Stresses 189 IV. Using the Models 214 V. Summary 221 References 224

The Constitutive Law of Deformation Kinetics A. S. Krausz and K. Krausz I. II. III. IV.

Introduction 229 The Kinetics Equation 234 The State Equations 247 Measurement and Analysis of the Characteristic Microstructural Quantities 256 V. Comments and Summary 270 References 277

A Small-Strain Viscoplasticity Theory Based on Overstress Erhard Krempl I. Introduction 282 II. Viscoplasticity Theory Based on Overstress

282

Contents III. Discussion References

294 316

Anisotropic and InhomogeneousPlastic Deformation of PolycrystallineSolids J. Ning and E. C. Aifantis I. Introduction 319 II. Constitutive Relations for a Single Crystallite 321 III. Texture Effects and the Orientation Distribution Function 322 IV. Texture Tensor and Average Procedures 324 V. Texture Effect on the Plastic Flow and Yield 327 VI. Inhomogeneous Plastic Deformation 332 References 339

~ ~ l

Modeling the Role of Dislocation Substructure during Class M and Exponential Creep S. V. Raj, I. S. Iskovitz, and A. D. Freed List of Symbols 344 I. Introduction 347 II. Class M and Exponential Creep in SinglePhase Materials 355 III. Substructure Formation in NaC1 Single Crystals in the Class M and Exponential Creep Regimes 371 IV. Microstructural Stability 403 V. Nix-Gibeling One-Dimensional Two-Phase Creep Model 411 VI. Development of a Multiphase Three-Dimensional Creep Model 419 VII. Summary 428 Appendix 428 References 430

~ ~ l

Comments and Summary K. Krausz and A. S. Krausz Index

451

~

VII

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Contributors Numbers in parentheses indicate the pages on which the authors' contributions begin.

E. C. Aifantis (319), Center for Mechanics of Materials and Instabilities, Michigan Technological University, Houghton, Michigan 49931 and Aristotle University of Thessaloniki, Thessaloniki 54006, Greece J. L. Chaboche (1), MECAMAT, ONERA, 92320 Chatillon, France Yuri Estrin (69), Department of Mechanical and Materials Engineering, The University of Western Australia, Nedlands, Western Australia 6907, Australia R. W. Evans (107), Interdisciplinary Research Centre in Materials for High Performance Applications, Department of Materials Engineering, University of Wales, Swansea SA2 8PP, United Kingdom A. D. Freed (343), Lewis Research Center, National Aeronautics and Space Administration, Cleveland, Ohio 44135 Donald E. Helling (153), Hughes Aircraft, E1 Segundo, California 90245 Gregory A. Henshall (153), Lawrence Livermore National Laboratory, University of California, Livermore, California 94551 !. S. Iskovitz (343), Ohio Aerospace Institute, Cleveland, Ohio 44135 A. S. Krausz (229, 443), University of Ottawa, Ottawa, Ontario, Canada K1N 6N5 K. Krausz (229, 443), University of Ottawa, Ottawa, Ontario, Canada K1N 6N5 Erhard Krempl (281), Mechanics of Materials Laboratory, Rensselaer Polytechnic Institute, Troy, New York 12180 Alan K. Miller (153), Lockheed-Martin Missles and Space, Palo Alto, California 94304 J. Ning (319), Center for Mechanics of Materials and Instabilities, Michigan Technological University, Houghton, Michigan 49931 S. V. Raj (343), Lewis Research Center, National Aeronautics and Space Administration, Cleveland, Ohio 44135 B. Wilshire (107), Interdisciplinary Research Centre in Materials for High Performance Applications, Department of Materials Engineering, University of Wales, Swansea SA2 8PP, United Kingdom

ix

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Preface The constitutive law of plastic deformation expresses the effects of material behavior and properties for stress analysis in the design of manufacturing technology and product service behavior, for materials testing, and for the maintenance of structural and machine components. The book represents the state of the art, but the editors do not rule out other concepts of constitutive laws. There are many different facets of the same problem and as many answers; the right one is the one that gives the most practical solution, the one that best serves the specific problem. The selection of the best solution can be ensured with a complex procedure that involves analysis of material cost, performance characteristics other than plastic deformation, marketing concerns, financial decisions, etc. No single book can give a full presentation of all of these issues or guidance that addresses all of these concerns. In this volume, we focus on the technical aspects of the constitutive laws of plastic deformation. During fabrication the major manufacturing processes subject the workpiece to plastic deformation. Examples of these processes are forging, coining, extrusion, metal cutting, bending, and deep drawing. During service many structural and machine components are subjected to plastic deformation: pressure tubes and turbine blades creep; the service lifetime of springs is affected by stress relaxation, which in turn is controlled by plastic deformation, and thus fatigue is controlled by it; and crack growth is associated with plastic deformation. In addition, many other service conditions require an understanding of plastic deformation (Figure A). Efficient maintenance and materials testing depend on information derived from the constitutive laws. All of these activities are carried out with the assistance of computers and depend ultimately on the understanding and ingenuity of the design and operating engineer. The end result of these activities is to achieve cost efficiency while ensuring a marketable, competitive product. Within the bounds of this book the authors present their understanding of the constitutive laws and the application of these laws to this purpose. It is clear from these chapters that further work must be done; plastic deformation is a very complicated process.

xi

xii

Preface

The manufacturing process and product performance diagrams give a condensed schematic of the design aspects. The dark boxes indicate aspects that are served by the constitutive law of plastic deformation.

Constitutive laws serve to enhance our understanding of the mechanisms that control plastic deformation, as well as the need to represent behaviors and processes for the development of improved material characteristics--to tailor them for better performance. Clearly, there are a variety of causes to serve, and a variety of constitutive laws are needed. These laws do not contradict each other when they are developed within the principles of the other engineering sciences: these laws must be economical for the purpose that they serve. For instance, it would be wrong to base the design of bridges on the effects of atomic interaction energies and the applied forces acting on these atoms however true it may be that these control plastic deformation. This approach would be inappropriate, extremely uneconomical---extremely wrong. On the other hand, consideration of the effects of the microstructure obviously requires representation of the microscopic and submicroscopic conditions of the structure and the processes that occur at these levels. These concepts are very much embedded in science and engineering. It is well known that in the design of structures and machine elements, linear elasticity is usually considered, but in the design for fluctuating loads, where energy absorption is critical, the nonlinear hysteresis effect must be considered. Nature is one, but it has many facets to be examined and the one chosen must give the optimum condition for the specific purpose. It is in this context that we present the contributions to this volume.

Preface

xiii

The editors express their thanks to D. Grayson, J. Bunce, and D. Ungar of Academic Press for their kind and competent assistance in the preparation of this book. It has been a pleasure working with them. Much appreciation is due to the authors of the chapters for their contributions--their collaboration made our editing job easy. K. Krausz A. S. Krausz

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1 Unified Cyclic Viscoplastic Constitutive Equations: Development, Capabilities, and Thermodynamic Framework J. L. Chaboche

ONERA 92320 Chatillon, France

LIST OF SYMBOLS General notation X scalar functions, parameters, or variables X second- or fourth-rank tensors X' deviator of the second-rank tensor X scalar product of vectors or contracted tensor product tensorial product contracted twice |174174 tensorial products IIxlIM generalized second invariant o f X : IIXIIM = (X : M : 1 ) 1/2 ci time derivative of a O/Ox partial derivative d/dx total derivative 6 Kroenecker delta I fourth-rank unity tensor fourth-rank deviatoric operator Idev Specific variables or functions u a D

"back strain" or state variable associated with back stress aging state variable isotropic "drag" state variable drag stress

Unified Constitutive Laws of Plastic Deformation Copyright @ 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

2

J. U Chaboche

e s

A n v f2, ~p ~r p q r R S or, or' T VT u Wp Ws X, X' Y Z

strain tensor elastic strain tensor plastic strain or viscoplastic strain tensor thermodynamic state potential dissipation potential (rates) dual dissipation potential (forces) dissipation yield function hardening modulus second invariant overstress and back stress plastic multiplier elastic stiffness tensor direction of plastic flow direction of the back-stress rate viscoplastic potential static recovery potential accumulated plastic strain heat flux vector isotropic "yield" state variable yield stress increase entropy stress tensor, stress deviator temperature temperature gradient internal energy plastic work stored energy back stress tensor and its deviator yield stress thermodynamic force associated with aging variable a

~11

INTRODUCTION

s

q~ 4~* f h J, JT

The constitutive equations c o n s i d e r e d here, m a i n l y those devoted to metallic materials, are essentially d e v e l o p e d with the objective of the inelastic analysis of structural c o m p o n e n t s . Initially, they are based on the concepts of c o n t i n u u m m e c h a n i c s , w h e r e a particular representative v o l u m e e l e m e n t of material can be c o n s i d e r e d as submitted to a m a c r o s c o p i c a l l y u n i f o r m stress, neglecting the microstress/microstrain i n h o m o g e n e i t i e s at the m i c r o s c a l e (but not their effects). The physical facts, the precise role of the dislocations, their arrangements, and their evolution are c o n s i d e r e d m o r e in detail in several other chapters of this book. Here, we concentrate on a m a c r o s c o p i c description of the various processes, m a k i n g reference to the microstructural events as often as possible (at least qualitatively). The application d o m a i n s are limited to the quasistatic d e f o r m a t i o n of metallic materials (strain rate b e t w e e n 10 - l ~ and 10-1), especially u n d e r cyclic loading

Chapter 1 UnifiedCyclic Viscoplastic ConstitutiveEquations 3 conditions. The constitutive equations are written in their small strain form. Also, high-temperature conditions will be considered, as well as loading under varying temperatures. By "unified viscoplastic constitutive equations," we mean the nonseparation of the plastic (rate-independent) and creep (rate-dependent) parts of the inelastic strain. Moreover, the considered viscoplastic equations are based on a general framework consistent both with classical plasticity (elastic domain, yield surface, loading/unloading condition) and with thermoviscoplasticity without an elastic domain. Then rate-independent conditions will be obtained consistently as a limit case of the general viscoplastic scheme. The theoretical development of viscoplasticity has its origin in the works of Bingham and Green (1919), Hohenemser and von Prager (1932), Oldroyd (1947), Malvern (1951), Odqvist (1953), Stowell (1957), and Prager (1961), whose models do not contain evolving internal stage variables. The field started to gain momentum in the mid-1960s when internal state variable models began to appear in the theories of Perzyna (1964) and Armstrong and Frederick (1966). With the increased availability of the computer, rapid advances were made in the 1970s through the modeling efforts of Bodner and Partom (1975), Hart (1976), Kocks (1976), Miller (1976), Ponter and Leckie (1976), Chaboche (1977), Krieg et al. (1978), and Robinson (1978). Further refinements were introduced thoughout the 1980s, for example, by Walker (1981), Bruhns (1982), Lowe and Miller (1984), Krempl et al. (1986), Robinson (1983), Nouailhas (1987), and Henshall and Miller (1990). Comparative reviews of constitutive theories in cyclic plasticity or viscoplasticity have been given in the recent years by Chan et al. (1984), Miller (1987), Chaboche (1989a), Ohno (1990), and McDowell (1992). Several of these theories are presented in the present book. Thermodynamic treatments for viscoplasticity have been developed by Rice (1971), Geary and Onat (1974), Valanis (1980), Cristescu and Suliciu (1982), Lema~tre and Chaboche (1985) and Malmberg (1990a). Although this listing is by no means complete, it does provide the reader with a representative bibliography of the work done in the field of viscoplasticity for initially isotropic metallic materials. This chapter is divided into three main parts. We present first a general form of the unified viscoplastic constitutive equations in Section II.A. In that case, the material is considered as initially anisotropic. Section II.B restricts the equations to the initially isotropic material, while the two next sections introduce the limiting cases of rate-independent theory and of a creep theory. The determination procedure is briefly indicated in Section II.E, taking advantage of some closed-form solutions for the rate-independent case. In Section II.F, we discuss the relations between current constitutive theory and models based on multisurface approaches. In the second part, Section III, the capabilities of the viscoplastic constitutive equations and their main developments are illustrated on the basis of two particular t

4

J.L. Chaboche

polycrystalline materials. Various complicated processes can be modeled, including Bauschinger effects, creep, relaxation, strain rate effects, monotonic hardening, cyclic hardening or softening, static recovery effects at high temperature, creepplasticity interaction, and ratcheting effects. In the last part, Section IV, we discuss the various techniques by which the plastic and viscoplastic constitutive equations can be introduced into a thermodynamic theory with internal variables (Lema~tre and Chaboche, 1985). Then, the consequences of the thermodynamic treatment are examined, especially in terms of stored and heat-dissipated energies during (visco)plastic flow (Section IV.C). Some comparisons are made with published experiments. In the last Section (IV.D), we discuss the application of constitutive models under varying temperature conditions, based on the thermodynamic theory and also the modeling of metallurgical effects, such as aging, that are induced by temperature changes.

A CYCLIC VISCOPLASTICCONSTITUTIVE LAW This section is devoted to the presentation and development of a set of constitutive equations based on the combination of kinematic hardening and isotropic hardening with both "yield" and "drag" effects. The unified viscoplastic framework is chosen, but the limiting cases of rate-independent plasticity and stationary creep are also discussed. The equations are first written in their general anisotropic form (initial and fixed material symmetries), then particularized for the initially isotropic material, in the frequently used form presented previously by Chaboche (1977), Chaboche and Rousselier (1983), and Chaboche and Nouailhas (1989b). In these constitutive equations, small displacements and rotations are considered. In a Cartesian reference configuration, the strain e is taken to be composed of elastic ee (reversible~including thermal strain) and inelastic or plastic ep(irreversible) parts such that E "-- E e -'[- •p

(2.1)

and there is no inelastic strain in the stress-free virgin state.

A. The General Framework 1. The Viscoplastie Potential and the Hardening Variables The constitutive laws are developed in the framework of unified viscoplasticity, considering only one inelastic strain. We assume the existence of a viscoplastic potential in the stress space. Its position, shape, and size depend on the various hardening variables. We limit ourselves to the case where the potential is a given

Chapter I

Unified Cyclic Viscoplastic Constitutive Equations

5

function of the viscous stress (or overstress)

The shape of the equipotentials is given by the choice of the "distance" J in the stress space that will be discussed below. In Eq. (2.2), the variables X, Y, and D are the "internal stresses" or hardening variables (in the stress space). The theory uses a combination of kinematic hardening, represented by X, the backstress tensor, and isotropic hardening, described by the evolution of the yield stress Y and the drag stress D. The use of a yield stress introduces and elastic domain, corresponding to stress states where f -- J ( o - - X ) - Y < 0

(2.3)

In that case, given by the MacCauley brackets (in (2.2)), the viscous stress a~ is taken as zero. The elastic domain can be reduced to a point by choosing Y -- 0. In the old version of our model (Chaboche, 1977), isotropic hardening was present in the yield stress only, with a constant drag stress. On the other hand, many models use an evolving drag stress with Y -- 0 or a combination, such as Y -- D, in the viscoplastic theory of Perzyna (1964). In Section II.B.2 we will also use such a combination, with only one independent variable R, assuming that Y -- k + R and D -- K + mR, k and K being the initial values of Y and D, respectively. In the general case of an initially anisotropic material (single crystal, metal matrix composite, laminated steel), we formulate the distance in the stress space by introducing a fourth rank tensor M, as follows" J ( o " -- X ) - - [(o" -- X ) " M " (o" - X ) ] 1/2

(2.4)

The viscoplastic potential, i.e., the function G, can be particularized in various forms, as discussed previously by Chan et al. (1984) and Chaboche (1989a). The viscoplastic strain rate is given by the normality assumption:

kp =

00"

00"

-O

J (o" - X )

For convenience, we define the modulus of the plastic strain rate through the following norm:

--][~p[]- [ ~ p "

M-1

" ~p] 1/2

(2.6)

Then p is called the accumulated plastic strain, and we easily check that it obeys the evolution equation (2.7)

6

J.L. Chaboche

We also denote as n the direction of the plastic strain rate: kp = fin

2. The Evolution Equations for Hardening Variables (Isothermal Case) The rate equations for hardening variables obey a generic format that incorporates an hardening term, a dynamic recovery term, and a static recovery term (in the isothermal case). If we denote the genetic variable as x, we write JC -- Hh~:h- H d ~ d - Hs

(2.8)

where Hh, Ha, and Hs are hardening or recovery functions, and ~h and ~d are linear combinations of the plastic strain rate. The back stress X is decomposed into independent variables Xi, each of which obeys the same rule. As shown in previous studies (Moosbrugger and McDowell, 1989; Watanabe and Atluri, 1986), two or three of such variables are sufficient to describe, very correctly, the real materials. The whole set of equations is given as follows: (2.9)

X-- EXi i X i ---

Ni

" F.p - - ~ i ( X i ,

-- hyp-

b = hap-

R)Qi

" X i P -- Si (Xi,

R)Qi

" Xi

(2.10)

ry(R)Rp-

Sy(R)R

(2.11)

rd(D)Dp-

Sd(D)D

(2.12)

The yield stress (variable R) and the drag stress D are considered here as independent, as in Freed et al. (1991). In the above rate equations, the hardening term is proportional to the plastic strain rate or to its modulus/). The dynamic recovery term is also proportional to ,b and is either a linear or nonlinear function of the variable itself. The static recovery term is a nonlinear (and temperaturedependent!) function of the variable. The influence of the isotropic hardening on the back-stress rate equation can also be taken into account. The initial anisotropy is still in effect, with the fourth-order tensors Ni and Qi playing a role in the hardening and recovery terms of the back-stress rate equation. Moreover, in order to improve ratcheting modeling, we introduce the notion of a threshold in the dynamic recovery term for the back stresses (see Chaboche, 1991 or Chaboche et al., 1991). The function q~i is particularized into r

r R) = ~ ( J T ( X i ) JT(Xi)

-- X t i ) m

(2.13)

where the MacCauley bracket ( ) is zero when JT(Xi) ~ X l i . For the definition of the distance, we may also use an anisotropy effect with the fourth-order tensor T

Chapter 1 UnifiedCyclic Viscoplastic Constitutive Equations

7

(equal to or different from M): JT (Xi) = (Xi : T : Xi) 1/2

(2.14)

3. Remarks 9 For kinematic hardening, the first presentation of the dynamic recovery term was done by Armstrong and Frederick (1966). Such a term is used in many cyclic constitutive models. 9 The material-dependent functions in Eqs. (2.2), (2.10), (2.11), and (2.12) also depend on temperature. They will be defined in the applications to isotropic materials (Section II.B.2). 9 The evolution equations for hardening variables are given by Eqs. (2.10)(2.12) in the isothermal case. As discussed in Section IV.B.2, additional terms proportional to the temperature rate must be incorporated for anisothermal situations. 9 The fourth-order tensors M, Ni, Qi, and T are considered as constants for a given material, describing its initial anisotropy and obeying its symmetries. In some theories, not considered here, they can play the role of internal variables (like M in the theory by Zaverl and Lee, 1978). 9 In the "radial return" model proposed by Burlet and Cailletaud (1987), the fourth-order tensor Qi also depends upon the direction of the plastic strain rate n, with

Qi

=

/']i I

+ (1

-/]i)n

| n

(2.15)

where Oi i s a material-dependent scaling parameter. For/]i ~--- l, we recover the classical dynamic recovery term (whose direction is given by Xi). For/']i - - 0, we have a purely radial return, collinear with kp = ~bn (and proportional to Xi : n). 9 An application of the preceding model has been done by Nouailhas (1990a) for single crystals used in turbine blades of modern aeroengines. In that case, due to the cubic symmetries of the microstructure, each of the tensors M, Ni, Qi presents only 2 degrees of freedom (two independent coefficients). In these applications, neglecting the isotropic hardening (R = 0), the model was able to describe well the various monotonic and cyclic responses under tension-compression and tensiontorsion, for different specimen orientations like (001), (011), (111). A similar model has also been recently applied for metal-matrix composites (EI Mayas, 1994).

B. Application to Initially isotropic Materials ]. Restriction to Isotropy In the isotropic material, the fourth-order tensors that appear in Eqs. (2.4) and (2.10) must degenerate into identity tensors, constructed from the second-rank

I]

J.L. Chaboche

identity tensor 1. We assume the following choices: M:~31-

~1(1 | 1) -- ~Idev3

2

Ni -- s C i I

(2.16) T = M

Qi = y i l

(2.17)

where I is the fourth-rank symmetric identity tensor, expressed from the Kronecker delta symbol: (2.18)

Ii j kl --~ 1 (~ik~j l -Jr- ~il~jk)

The components of the fourth-rank tensor 1 | 1 (| denotes the tensorial product) are Uijk.l = 6ijS~l. From these choices the distance in the stress space can now be expressed as y(,~

-

x)

[~3 (O.t

=

(2.19)

__ X t) . (o.t __ S t ) ] 1/2

where or' and X' are the deviators of stress and back-stress tensors, respectively. We directly deduce the constant volume for plastic strain from the normality rule

(2.5):

2 J(o" - X)

~P --" 00"

~

n

(2.20)

Moreover, we define M -1 in (2.6) by M -1 -- 2Idev, with M " M -l -- Idev, where Idev is the fourth-rank deviatoric identity tensor such that Idev : er = a ' . With this choice, the norm (2.6) of the strain rate is written as usual: 2

) ~/2

Let us note the direction of the plastic strain rate as 3

o'I-X

f

n = 2 J(o" - X)

(2.22)

2 with kp - - / ) n and ~n 9n -- 1. The back-stress rate equation reduces to

Xi :

2

-~Ci~p - y i q b ( R ) ( J ( X i )

Xi

- X l i ) m J ( X i ) p - }"iSi(Xi' R ) X i

(2.23)

and Xi is identical to its deviator X~, provided Tr(Xi) = 0 for some initial conditions. 2. Particular Choice of Material Functions For the viscoplastic potential, we usually assume a power function. In fact, for application to a large domain in strain rate, it is often necessary to use more

Chapter 1

Unified Cyclic Viscoplastic Constitutive Equations

9

complicated functions. Several possibilities were compared by Chan et al. (1984) and by Chaboche (1989a). Here we limit ourselves to a sum oftwo power functions:

n + 1

-D

+ n2 -+- 1

(2.24)

The modulus for the plastic strain rate is then expressed as the sum of two terms: -~

+

~

(2.25)

The parameter k* serves to fix the units for the strain rate. It will depend on temperature, as well as exponents n and n2. The parameter ~ serves to scale the asymptotic effect for high strain rates, given by the second term with an exponent n2 significantly larger than n (in the applications we can chose n2 as 3n). The function q~(R) in Eq. (2.22), which introduces a coupling between kinematic hardening and isotropic hardening, was introduced first by Marquis (1979). Its form is taken as the one deduced from the endochronic theory (Valanis, 1980; Watanabe and Atluri, 1986; Chaboche, 1989a): ~b(R) =

1

1 + R//~

The other functions are chosen as power functions, so that the rate equations for the hardening variables are now (Nouailhas, 1989) 2 Xi Xi -- - ~ C i ~ p - ~'i(~(g)(J(Xi) - Xli) j ( X i ) ~3 (2.26)

- Ys; [ J (Xi) ]mi- 1Xi

k -- b ( Q - R ) p - y r ] R - Qr[mrsign(R- Qr) D - b'(Q'- D)p- diD-

Q'rlm'rsign(D- Q'r)

(2.27) (2.28)

A further particularization is obtained when only one isotropic variable R is selected, the drag stress being considered as depending explicitly on R by Y -- k + R

D -- K + coR

(2.29)

This particular case corresponds to b' -- b, Yr' - - Yr(-Dl-mr, Q' - coQ + K, and Q'r - - c o Q r -k- K. The viscoplastic constitutive equations are completely defined by Eqs. (2.9), (2.20), and (2.24)-(2.28), and the definition of the viscous stress is o'v ' - J (o" - - X ) - R - k. Let us note the following stress decomposition when

10

J.L. Chaboche

o

0 strain rate

/

Ep

~

y/ l /

P

I

Ov=O-g

r

(a) Stress decomposition in uniaxial tension--compression; (b) schematics of the limiting case of rate-independent plasticity.

inverting the viscoplastic equation

o.=ZXi-.]-Ie--l-k--t-(g-ql-coe)at-l(.-.~)ln i

(2.30)

where n is the direction of plastic strain rate. In the particular case of a simple power viscosity function (second term in (2.24) canceled), the function G '-1 in Eq. (2.30) is expressed as (j~/k*) 1/'. Figure 1 illustrates the stress decomposition (2.30) for uniaxial tension-compression. The various material parameters of this constitutive model are n, K, k for the viscosity function; n2, ~ for the limiting viscosity function, Ci, Yi, Xti for the kinematic (strain) hardening; mi, Ysi, for the kinematic (time) recovery; b, Q for the isotropic (strain) hardening; mr, Yr, Qr for the isotropic (time) recovery; and co for the drag effect. Some of these parameters must be temperature-dependent, especially the ones related to the viscosity and static recovery effects, but this aspect will be discussed in Section IV.D. 3. Additional Effects

As shown in Section III.C below, isotropic hardening serves to describe the cyclic hardening or cyclic softening processes that are observed on many polycrystalline

Chapter 1

Unified Cyclic Viscoplastic Constitutive Equations

11

materials (before stabilization). Neglecting the static recovery effects, the yield stress Y will increase progressively from k to k + Q (Q > 0 for cyclic hardening) during the successive cycles, as a function of the accumulated plastic strain. Subsequently, the drag stress can also be changed. After stabilization (to R = Q), additional cyclic hardening (or softening) is not possible. However, several experiments, especially on stainless steels, show that the asymptotic stress value after cyclic hardening depends on the prior history. Moreover, the amount of cyclic hardening is clearly dependent on the applied plastic strain range (see Chaboche et al., 1979, for example). The kinematic or isotropic variables cannot describe plastic strain memorization: Kinematic hardening is evanescent in nature, and isotropic hardening saturates toward a unique value. The introduction of new internal variables that memorize the prior maximum plastic strain range was first proposed by Chaboche et al. (1979). The concept uses a "memory" surface in the plastic strain space, which Ohno (1982) called the "cyclic nonhardening range". F - - ~2 J ( / c p - ~') - / 9

< 0

(2.31)

where ~2 J gives the distance between tensors gp and ~" in the strain space. Plastic flow inside the domain does not change the "memory state," that is, ~" and p. Change of the nonhardening domain is allowed only if F = 0 and if n* : •p > 0, where n* denotes the unit normal to the surface. The chosen evolution rule for p and ~" is defined by /5 = r / H ( F ) ( n : n*)/) -- x/~/2(1 - ~ ) H ( F ) ( n 9n*)n*~b

(2.32) (2.33)

The coefficient r/was introduced by Ohno (1982) in order to induce a progressive memory. The particular value r/-- 1/2 was used for the first developments under reversed cyclic conditions (Chaboche et al., 1979). For a constant plastic strain range (under tension-compression), Eqs. (2.32) and (2.33) lead to the following stabilized response: p = Aep/2

~" = epmoy "- (8pmax q- 8pmin)/2.

(2.34)

For ~ = 1/2, memorization is instantaneous and stabilization occurs after one cycle. A progressive memory is given by ~ < 1/2 ( r / = 0.1, for example). The dependency between cyclic plastic flow and the plastic strain range is introduced by considering an asymptotic isotropic hardening Q in Eq. (2.27) depending on p. For instance, Q = QM + (Q0 - QM)e -2up

(2.35)

12

J.L. Chaboche

This introduces a dependency between the size of the elastic domain R and the memory parameter p. We have now three additional coefficients: r/, #, and QM. Another commonly observed effect, especially for austenitic strainless steels, is the additional hardening under nonproportional multiaxial loadings. Out-of-phase loading conditions reveal much larger resistance to plastic flow than do in-phase conditions (Lamba and Sibebottom, 1978; McDowell, 1983; Nouailhas et al., 1983; Cailletaud et al., 1984; Krempl and Lu, 1984; Ohashi et al., 1985). These effects are increased both by the multiaxiality factor and the phase difference, but they play their role slowly and are partially memorized (some part is evanescent). Different approaches have been proposed to describe such effects (McDowell, 1983; Nouailhas et al., 1985; Benallal and Ben Cheikh 1987; Benallal and Marquis, 1987; Bodner, 1987; Krempl and Yao, 1987; Tanaka et al., 1987). They are generally based on some in-phase/out-of-phase loading indices. A simple and powerful approach was proposed by Benallal and Marquis (1987), who used the direction cosines of the back stress and back-stress rate. The effect interacts with the flow rule by increasing the saturation limit of isotropic hardening in a similar way to the method of strain-range memorization. The model is not detailed here, but is completely consistent with the present constitutive law (it does no modify the normal response under proportional loads).

C. The Rate-IndependentLimitingCase One interesting property of the considered viscoplastic unified constitutive law is that it degenerates into a rate-independent plasticity theory when the viscous stress ov becomes negligible. Practically, we can use two procedures to obtain (numerically) such a limiting behavior (static recovery is obviously neglected here): (1) Decrease the drag stress D to zero (K = o9 --+ 0), which eliminates the rate dependency in Eq. (2.30). This procedure, with a constant exponent n, has been followed by Benallal and Ben Cheick (1987) for the Inconel 718 superalloy. (2) Increase exponent n (toward infinity) and decrease the drag stress in order to approach a low (but nonzero) viscous stress that varies with the strain rate, but just slightly (not varying if n = cx~). Because the first procedure leads to difficult numerical problems when implemented in finite-element codes for structural application purposes (see Golinval, 1989), we prefer the second one, where an approximately constant viscous stress still exists. This procedure is illustrated in Fig. 1b for a power dependency such as

/,_/,,

()n ~ ~-2

(2.36)

Chapter 1

Unified Cyclic Viscoplastic Constitutive Equations

13

When n is increasing indefinitely, the viscous stress tends toward the constant a*, giving rise to a rate-independent plasticity condition lim p -- ~ 0 ,-+oc [ undetermined

if f - - J ( a - X ) - Y < a * if f - - J ( a - X ) - Y -- a*

In classical rate-independent plasticity, we use the notion of yield surface, which is now identical to the boundary of the elastic domain f--J(cr-X)-R-k*_0

(2.37)

where k* -- k + a* is modified in order to incorporate the part due to the viscous stress. In that case, the normality rule (2.5) or (2.20) is replaced by

Of _ ~n

(2.38)

kp - - J~00"

where J~ is the unknown plastic multiplier. Its determination follows from the consistency condition f -- f -- 0 when plastic flow occurs continuously, leading to ~'--hlH(f)(Oo'0f'&)

(2.39)

where h is the tangent plastic modulus, expressed from the rate equations of the state variables (neglecting the static recovery). Equation (2.39) contains the Heaviside function in order to impose f -- 0 for plastic flow and the MacCauley bracket to indicate that plastic flow ceases immediately (J~ -- 0) if unloading takes place (6" directed inside the elastic domain). Let us note that the above equations of rate-independent plasticity are valid for cases where the plastic tangent modulus is positive. Softening materials need slightly different procedures.

D. The LimitingCreep Law In some theories (see, for instance, Freed and Walker, 1993), the rate equations for hardening variables are chosen in order to obtain degenerated viscoplastic constitutive equations for steady-state creep that are identical to the classical power secondary creep law (Odqvist's creep law). This is not the method chosen here, but we can demonstrate that our constitutive equations degenerate approximately to Odqvist's law, at least for low creep stresses 9 We assume a given constant stress state (uniaxial or multiaxial) producing creep 9 We assume the existence of a steady state (stationnary creep), with a constant plastic strain rate (for any components). We first demonstrate that at steady state each back-stress Xis is necessarily collinear to the applied stress: Xis -- ~icr

(2.40)

14

J . L . Chaboche

The result is obtained by observing the back-stress rate equation (2.26) at steady state, w i t h ~i~i --- 0 ( w e have posed 4' -- 1 for convenience):

0-

Ci J(tr - X)

-

Yi

< lilx ] si ,xi,,milxi 1

J(Xi)

At steady state J (tr - X) and J ( X i ) are constants, as well as ~b. We can rearrange the equation above as

Ci o" - ~ Xj Bi

J (tr - X)

= Xi

(2.42)

in which we have posed

Bi

- - Yi

1

XI i >

J (Xi)

~b + Ysi[J (Xi)]mi-

1

The direct summation leads to the proportion X = ~ Xi = ~tr. Reapplying (2.42), we immediately obtain the result (2.40). Now we assume the simple power function for the viscoplastic strain rate. At steady state the equations to be checked are

[Ci - yi(~iJ(o')

--

b(Q -~rJ(o'))Ps

Xli)]Ps -" Ysi(~iJ(tr)) m~ -- yr(~rJ(O')

-- Q r ) mr

(2.43)

[ g s = ( ( 1 - ~ - ' ~ J ~ J -n~ r ) JD( ~ in which we have posed R = ~rJ (o') at stationary state. Let us note that ~i and ~r can depend on the modulus of the applied stress. The above system can be solved numerically for ~:i and ~r for any applied stress, which leads to the steady-state solution. In order to obtain the Odqvist law for steady-state creep analytically, we need the following additional assumptions: 9 Assume for simplicity that the drag stress is constant D -- K. 9 Neglect the initial value of the yield stress (k = 0), which is often the case at high temperature. 9 Consider the same value for exponents in the viscosity and static recovery functions (n = m i = m r ) . This is also a common choice, for example, in the theories by Miller (1976) or by Freed and Verrilli (1988) and Freed and Walker (1993). 9 Assume the static recovery of the isotropic hardening as complete asymptotically (Qr = 0).

Chapter I

Unified Cyclic Viscoplastic Constitutive Equations

15

Under these particularizations, the above system of equations reduces to

Ps __(]--~j--~r'~ (J(cr))" K

;

n --

Ysi~in

Ci

Yr (~r) n

b(Q-~rJ(o'))

Xli )

-- Yi <~i J ( o ' ) --

(2.44) These equalities can be checked with constant values for ~i and ~r only if the stress state is sufficiently small, allowing the steady-state values for Xi and R to be small compared to their asymptotic high rate values Ci/yi 4- Xli and Q. If we assume ~iJ(cr) << Xli + Ci/yi and ~rJ(o') << Q, we obtain the following equalities:

PS (J(cr)),

(1--~~j--~rln__ -

Ysi~.n

__

-

Ci . 1

?Jr

-

-'~?

-

(2.45)

We can easily solve these equations to find (neglecting threshold Xli):

1 ( C i ~ 1/n ~i -- Ks

Ys--i/

~r--1

(~)

1In

~

(2.46)

where we have posed Ks -- K + }-]~j(Cj/?,sj) 1/" + (bQ/yr) 1/n. We obtain ~bs (J(o')/Ks) n from (2.43), which corresponds exactly to Odqvist's law. Then we have effectively proved that our viscoplastic constitutive model degenerates explicitly to the classical power creep law at steady state, at least for low stresses.

E. Determination Procedure of the Material Parameters The determination procedure of the viscoplastic constitutive law takes advantage of two specificities: 9 The rate-independent version can be determined first with a convenient selection of the experimental data. The viscous and static recovery effects are determined in a second step. 9 The response of the hardening models may be obtained analytically for uniaxial monotonic or cyclic conditions (in the rate-independent case). 1. Closed-Form Solutions

For uniaxial conditions (or proportional multiaxial loadings) and neglecting the static recovery effects, the rate equations (2.26) and (2.27) can be written as Xl/)

Xi -- Ci~p- Vie(R) 1 - ]Xi] -- b(Q - R)]gp]

Xil~p]

(2.47) (2.48)

16

J.L. Chaboche

When we neglect threshold in the dynamic recovery term ( X l i - - 0 ) and consider ~p(R) = cp as a constant, we can easily integrate the equations from any initial condition ep0, X0i, R0 (Lema~tre and Chaboche, 1985). Ci

X i -- 13~

-~-

(

Ci) X0i- v~ exp(--vq~Fi(e p - -

R = Q + (R0 - Q)exp(-blep - ep0l)

6pO))

(2.49) (2.50)

where v = sign (kp) gives the sense of plastic flow. The solution for R can also be taken as a function of the accumulated plastic strain (provided R ----0 for p = 0): R -- Q(1 - e x p ( - b p ) )

(2.51)

The equation for the back stress can be applied half-cycle by half cycle, with v = 1 or - 1 . Provided R is slowly varying at each cycle, we can accept the approximation ~p = const within each half-cycle and update its value at the end. Where the threshold X l i is not negligible, the closed-form solution can also be obtained by separating several domains (not given here). 2. Determination Procedure

Obviously, the way by which we will determine the constants of a material model depends very significantly on the available experimental data. In many cases we need a preliminary selection of the significant data for each part of the constitutive model, considering separately the various effects that can be observed (depending on the strain rate, temperature, time, etc.): 9 the kinematic hardening response, present especially for stabilized conditions (at a given strain rate); 9 the isotropic hardening that occurs between the initial (monotonic) and the stabilized conditions; 9 the viscosity effects evoked by the effect of strain rate on the stress response or by the beginning of creep or relaxation curves; and 9 the static recovery effects that play a significant role only for sufficient durations (and at sufficiently high temperature). The determination method can use the trial-and-error technique or the automatic mean-square technique. However, the best method is, in fact, a mixture of the two, such as that used in the program AGICE (Automatic Graphic Identification of Constitutive Equations), which was developed at ONERA (Chaboche et al., 1991). This program helps in the identification process by combining the possibilities of automatic error-minimization methods with the interactive-graphic control of the results. The user guides the minimization procedure, checks the validity of the results by comparison with the available data, and fills in the missing data with additional facts about the general properties of the materials.

Chapter 1 Unified Cyclic Viscoplastic Constitutive Equations

17

In practice, the identification strategy for a complete set of constitutive equations can be performed as follows: (1) Begin with stabilized data without hold time (cyclic curve and stabilized loops). With these data, identify the kinematic hardening parameters within the rate-independent model. (2) Possibly, introduce "isotropic hardening memory effects," but only if this appears necessary according to the computation-test comparisons. (3) Identify the viscosity from stabilized data, with varying strain rates or from the beginning of creep or relaxation tests. (4) If the data with hold time are available only for monotonic loading (relaxation or creep under tension), it is necessary first to identify the difference between monotonic state and cyclic state from the isotropic hardening parameters (with or without memory effect). (5) Possibly, if justified by the temperature level and provided data with long hold times are available, identify the parameters of the static recovery models. Note: At any time, it is possible to go back to the identification of parts already

processed, either by leaving certain parameters free (these parameters can then be modified by the minimization process) or by releasing them manually from time to time.

F. Relations to Multisurface Approaches As has been pointed out on several occasions, the developed viscoplastic constitutive equations have some similarities with many other models, either in terms of the linear/nonlinear kinematic hardening for the hardening/dynamic recovery/static recovery format, or in terms of the viscous stress or overstress or for the choices of the viscosity functions. More specific analogies with models developed by Ohno and co-workers (Ohno, 1982; Ohno and Kachi, 1986) were also mentioned earlier in this paper with respect to the strain-range memory. Also, the threshold in the dynamic recovery term has some similarities with the model proposed by Ohno and Wang (1992), as pointed out by Chaboche (1994). In the current section, focusing on the rate-independent limit case, we want to recall the model as a two-surface plasticity model and discuss the specific advantages of the two ways of introducing the kinematic hardening. Let us consider first the simple case where just two back stresses are present, one of them obeying the linear rule. We also neglect the threshold and assume the isotropic case of Section II.B: X -~- X l + X2 2 u X1 -- ~ C l k p },'lXl~b

X2-

2 C2ep

(2.52) (2.53) (2.54)

18

J.L. Chaboche

The two rules can be combined as (assuming X2(0) -- ep(0) -- 0)" ]~I[2 -- g2 C2~p

~I[--~2 (C1 + C2)~p - }/1 (X - X2)/3

(2.55)

If we do consider the possible evolutions of X around X2, which supposed to be fixed for the moment, we can easily demonstrate the existence of the following bound: fx = J (X - X2) -

C1 -~-C2

)'1

< 0

(2.56)

Moreover, using the Schwarz inequality and relation (2.56), we can obtain the notion of a bounding limit surface for the stress centered on X2" j~ -- J ( t r

-

X2)

-

(kl +

R) < 0

kt - - k +

C1 -+-C2

yl

(2.57)

As demonstrated for the first time by Marquis (1979) this non-linear kinematic hardening rule (NLK) is a two-surface plasticity theory, obeying Mroz's translation rule. In order to demonstrate this property, let us define as tr~ the image point on the bounding surface, corresponding to the outward normal collinear with the direction n of plastic flow. We can write 3

t r ' - X'

3

tr't - X'2

(2.58)

~p = / ~ n = ~/~ k +~---R-- -- 2/~ k~ + R Expressing X' from the last equality and replacing (2.55) (recall X = X'), we obtain X - Yl (Or'l -- O") ~b,

kp

and X in the rate equation (2.59)

which corresponds exactly to the translation rule proposed initially by Mroz (1967). The difference is that Mroz was using a family of surfaces with constant hardening modulus (a different treatment for the consistency condition). Here, we have only two surfaces: the yield surface given by (2.37) and the bounding surface (2.57) which has its own linear translation rule. Such a two-surface plasticity model has been used several times (see, for example, Krieg, 1975; Dafalias and Popov, 1975; Ohno, 1982; McDowell, 1985; Moosbrugger and McDowell, 1990). In the case of more than two back stresses, the generalization of the above demonstration leads to the geometric construction of "nested surfaces," as proposed by Wang and Ohno (1991), the (simultaneous) movement of each one being obtained by a similar "image point" concept. We do not express this decomposition in detail: it just gives a geometric illustration of the nonlinear kinematic rule defined in Section II.B.2. Let us discuss now the additional degrees of freedom that are allowed if the constitutive model is defined in terms of the two surface presentation. The translation

Chapter 1 UnifiedCyclic Viscoplastic Constitutive Equations

19

rule of the center of the yield surface is defined by the position and size of the bounding surface and by the image point orl on the bounding surface. The two additional possibilities (compared to the N L K rule) are as follows: 9 The considered direction n*, which defines the image point, can be different from that of plastic flow n. Tseng and Lee (1983) proposed the deviatoric stress rate; other combinations were discussed by McDowell (1987), M o o s b r u g g e r and McDowell (1990), or by Voyiadjis and Kattan (1990). 9 The distance between the stress state and the image point, defined as a -J ( o r l - - o r ) , may be used explicitly as in several models, and (2.59) above can the replaced by X - g 2 K(a)/,,/b where u -

(2.60)

3 (or~ - o r ' ) / ~ is the direction given by the translation rule (2 u 9u -- 1).

The increase in degrees of freedom is evident if we note that (2.53) corresponds to the particular case K ( a ) - y 1 6 . Another interesting point concerns the generalization of the two-surface approach to the anisotropic description introduced in Section II.A. If we define the yield and the bounding surfaces by f - [(or - X) 9M " (or

-

X)]

1/2

-

k < 0

fl -- [(orl - Xt) " M/ 9(orl - X1)] 1/2 - kl < 0

(2.61) (2.62)

where X and X / a r e the centers of the two surfaces, and k and k/are their sizes, we have, with Mroz's translation rule for the back stress X and the linear rule for Xz, X - Y(orl - or)/)

(2.63)

Xl - N I - b p

(2.64)

where ort is the image point, with the same outward normal such that n --

Of 0or

=

Ofz 0ort

=

M " (or - X) k

=

Mz " (ort - Xl) kl

(2.65)

Replacing orz and or in (2.63) as taken from (2.65), we obtain X -- y ' [ ( k l M / 1 -- k M -1) " n -- (X -- X,)]/3 This rule is identical to the one of Section II.A for two back stresses, the second one being linear: X1 -- N " l~p - y X l p

(2.66)

X1 -- N l ' k p

(2.67)

with N - g(kzMi -1 - k M - 1 ) . We immediately notice that the dynamic recovery

20

J.L. Chaboche

term is of an isotropic form (no material directionality, Q1 -- I), which is less general than the rules (2.10) proposed in Section II.A. Another development of multisurface models can be done by introducing discrete memory surfaces that store particular events in the cyclic loadings and allow a better description of closure effects for small subcycles. Such approaches can be found in Mroz (1981), Chaboche (1989b,c), and Trampczynski and Mroz (1992). Finally, we can conclude this section by discussing of the respective advantages of the two approaches: 9 The two-surface theory has more flexibility for the definition of the translation direction, with various possibilities for the "image point" and has more degrees of freedom for the nonlinear evolution of the tangent hardening modulus K (3) as a function of the distance to the image point. Also special memorization effects can be introduced. 9 The NLK rule with a differential equation is more general for the use of initially anisotropic material (fixed anisotropy) including both directionalities of the hardening and dynamic recovery terms, and can be integrated analytically for proportional loading conditions (see Section II.D), provided the tangent modulus is linear in 3.

CAPABILITIESOF THE CONSTITUTIVE MODEL The constitutive equations developed in Section II have been applied systematically on many mono- and polycrystalline materials. Their possibilities can be illustrated by selecting materials and applied loading conditions in an increasing order of complexity. In the next subsections, we first apply simplified versions of the model in order to describe the simple stabilized cyclic behavior. Then we show the influence of isotropic hardening and strain-range memorization in the case of stainless steels. At high temperature their rate-dependent behavior is described by combinations of viscosity effects and static recovery effects. More complicated situations are also modeled with special cyclic loadings where creep and plasticity interact. Finally, the difficult problem of ratcheting is addressed and the capabilities of the model are illustrated in one case.

A. Normal Viscosityand KinematicHardening In order to illustrate the modeling in a simple case, we select the high-temperature cyclic behavior of cast IN 100 superalloy, which is used for turbine blades. In that case, we have the following simplifications: 9 Cyclic softening (very rapid and limited on this alloy) is neglected so that the material behavior is cyclically stable.

Chapter 1 UnifiedCyclic Viscoplastic Constitutive Equations 21

Stabilized cycles under stress control or elongation control for superalloy IN 100, at 800 and 900~ (O 9 VV) Test results; ( ~ ) model.

9 Only one back stress is used, with a nonlinear kinematic hardening that saturates for low strains (1%). 9 Static recovery is neglected (the creep durations considered are sufficiently short). 9 The viscosity function is the simple power function. With such a version, the material is described at a given temperature by only five coefficients: n, K, k, C1, yl. Figure 2 shows four typical creep (short hold

22

J.L. Chaboche

Cyclic curves at various frequencies and hold times for superalloy IN 100 at 1000~ (A 9Q O [--1)Test results; ( , -) model. time to 165 s under constant stress, both in tension and compression) and cyclic relaxation (330 s hold time under constant tensile strain curves). We note the clear kinematic character of the response, with a large Bauschinger effect (no elastic limit under compression after tension and vice versa). Not shown in the figure, is the fact that primary creep is regenerated at each new stress hold, both in tension and in compression. The "cyclic curves" of Fig. 3 demonstrate the very large influence of viscosity, as evidenced by the differences observed and modeled for various frequencies and various hold times (under strain or stress control).

B. Cyclic Hardening Cyclic hardening or cyclic softening correspond to slow changes during plastic flow, inducing, respectively, a progressive increase or decrease of the elastic limit as the plastic strain accumulates. Due to the rapid evanescence of kinematic hardening (which changes rapidly at each new plastic loading), the only way to describe cyclic hardening is to use the isotropic hardening variable R (assumed to be the only independent variable). The chosen evolution equation is (2.27), neglecting here the static recovery (Yr = 0). Thanks to the linear recall term, it can be easily integrated as a function of the accumulated plastic strain, giving Eq. (2.51). Depending on the sign of the asymptotic value Q, we shall describe cyclic hardening (Q > 0) or cyclic softening (Q < 0). The coefficient b is related to the rapidity of saturation of the process.

Chapter 1

Unified Cyclic Viscoplastic Constitutive Equations

23

Cyclic hardening for 316 stainless steel at room temperature (normalized scale). From Goodall et al. (1980).

The effect of the evolution of R is evident in the constitutive response in three places: 9 As 9 As function 9 As

a change in the yield limit Y = k + R. an evolution of the tangent modulus in the kinematic hardening, by the r given by (2.25). a change of the drag stress D -- K + mR.

These three choices will respond slightly differently. In particular, the third one will introduce a rate dependency of the stress range increase due to cyclic hardening. Assuming the first choice, we can simulate the evolution of the maximum stress for a tension-compression test under strain control, as shown in Fig. 4, taken from Goodall et al. (1980) for 316 L stainless steel at room temperature. In that case we have O-M -- CrMo

R

CrMs ~ O-Mo

Q

= 1 - exp(-bp)

(3.1)

In fact, as discussed in Section II.B.3, the stainless steels show strain-range memory effects. The corresponding model, with proper memory variables, has been introduced. Figure 5 shows the capability of this model of the isotropic hardening (in addition to three back stresses) to describe the cyclic stress-strain responses for 316 stainless steel at room temperature. The test is made under increasing strain ranges. Each time, we observe an additional cyclic hardening even if it was saturated at the previous step. It demonstrates also the memorization effect when the strain ranges are decreased. Other modeling capabilities have been pointed out previously (Chaboche et al., 1979).

24

J.L. Chaboche

l

Step • cyclictest on 316•L stainlesssteel at•room temperature,showingthe strainrange memorization effect.

C. LimitingCase of High Strain Rates In some materials, especially in stainless steels in the intermediate-temperature range, plasticity and creep effects are easier to treat separately by partitioning the inelastic strain into instantaneous plastic and creep components. This separation into two independent processes is well illustrated by the large inelastic strains during the transient rapid loading, followed by very low creep strains for several thousand hours under constant stress (Chaboche and Rousselier, 1983). However, the various coupling effects between creep and plasticity are also evident (Goodall et al., 1980; Walker, 1981; Ohashi et al., 1985). Two methods can be used: (1) Separate the plastic and creep strains but combine the corresponding hardening equations (Kawai and Ohashi, 1987; Contesti and Cailletaud, 1989). This permits freedom for the modeling, but leads to increasing computational difficulties. (2) Use the unified approach which incorporates, with a different viscoplastic potential, a smooth transition between the time-dependent regime (viscoplastic) and the quasi-time-independent one.

Chapter 1

Unified Cyclic Viscoplastic Constitutive Equations

Test r oCreep (SP sheet)L - Relaxation

25

(~o=240 E~o=3.9% (]~o= 170 (~o=150 Epo=0.246% (~o= 83

Limitation of the viscous stress for high strain rates. Example of 316 L stainless steel at 600~ with two asymptotic relations.

In the unified viscoplastic constitutive law, we follow the second approach, introducing a limitation for viscosity effects that plays a role under high strain rates (Nouailhas, 1987). In fact, many viscoplastic models incorporate similar limitations by exponential or hyperbolic functions. These modifications have been discussed and compared systematically by Chaboche (1989a). Here we limit the viscous effects by the second power function in (2.24). Figure 6 shows the case of 316 stainless steel (17-12 SPH) at 600~ It demonstrates three regimes: rapid-loading (Ep > 10 - 6 s - a ) ; intermediate, with a normal viscosity exponent (n = 24); and low rates, where static recovery takes place. Another limiting function, with an exponential factor, is also plotted on the figure.

D. High TemperatureViscosityand Static Recovery At high temperature, all the thermally activated processes are effective. The viscosity plays a significant role as well as all the thermal (or static) recovery effects. In this section, we demonstrate the capability of the unified constitutive model to describe both the monotonic and cyclic conditions, the tensile and creep loadings, even for long-term creep. The cyclic relaxation applications are taken from Nouailhas (1989). This is done by using two back stresses and the functions of the model, corresponding to the following facts:

26

J.L. Chaboche

Simulation of creep tests on 316 L stainless steel (17-12 SPH) at 600~ 9 the "limited viscosity," which allows us to describe the fast transition between creep and plasticity and which compensates the quasi-strain-rate insensitivity at high strain rates (Nouailhas, 1987); 9 the plastic strain memory observed in cyclic loadings (Chaboche et al., 1979); 9 the time recovery effects, which play an important role at this temperature as well as under monotonic loadings and under cyclic loading conditions with hold times (Nouailhas, 1989; Delobelle, 1988), and 9 the coupling for isotropic hardening with the kinematic hardening and the drag stress. Figure 7 gives the predictions of a set of creep tests. In this diagram, the acceleration of the creep strains for long times shows clearly the presence of time recovery effects. Another illustration is given in Fig. 8, which represents the prediction of a creep test at various levels. A good agreement is observed between data and calculations for the creep strains as well as for the partial recovery at zero stress. Figure 9a shows the stress response after cyclic saturation for straincontrolled tests (~" = 4 x 10 -3 s -1) with tensile hold times. The decrease of the maximum stresses (O'max) when the hold time increases corresponds to the recovery of the isotropic hardening. The recovery of the kinematic variables, together with the viscous part of the model, allows a good prediction of the relaxed stresses (O'mi n represents the stress at the end of the hold time in the stabilized cycle). An

Chapter 1

Unified Cyclic Viscoplastic Constitutive Equations

27

Prediction of the model for a varying-levels creep test on type 316 S. S at 600~ total strain versus time; (b) partial recovery at o- -- 0.

(a)

illustration of the shapes of the hysteresis loops from the first cycle to the steady state is given in Fig. 9b.

E. Viscous Ratcheting The ratcheting effect, which is observed under unsymmetric tension-compression or tension-torsion loadings, constitutes a difficulty for many models. But in fact the analysis of experimental results suggests two types of ratcheting according to the loading conditions. In the case of tension-compression loadings, where a reverse plastic flow occurs at each transient, the ratcheting effect in the model is mainly governed by the kinematic hardening variables. This point has been studied in detail by Chaboche and Nouailhas (1989a) and is considered in Section III.G. The second kind of loading, which also leads to the appearance of a progressive deformation, is the case of zero-to-tension loadings (no compression). In that situation, the unloading remains approximately elastic and the observed accumulated strain seems to be much more a creep effect than a cyclic plasticity effect. For that reason, we called this effect viscous "ratcheting." It is interesting to note that a similar distinction can be made concerning the mean stress relaxation observed under strain-controlled tests (with a nonzero mean strain) (Nouailhas, 1990a/b). The following calculations show the response of the viscoplastic model for viscous ratcheting. The test results, from Ruggles and Krempl (1987), have been

28

J.L. Chaboche

Cyclic relaxation on 316 L stainless steel (17-12 SPH) at 600~ time; (b) complete simulation of three conditions.

(a) Influence of hold

obtained on type 304 stainless steel at room temperature. Figure 10a shows the imposed loading: after a monotonic tension up to e -- 1% (point A), the specimen is cycled between O'A and zero with a period of 2 x 21 s per cycle. After 1000 cycles, a creep loading of 700 s is conducted at O'A. Then the specimen is unloaded and reloaded at strain-controlled points E, F, G, H. A last sequence under stresscontrolled conditions is then performed (points H, I) at low stress rate. The predictions were performed with a very simple version of the viscoplastic model, since only two kinematic variable were introduced and no isotropic, hardening has been considered. Figure 10b shows the response of the model corresponding to the test plotted in Fig. 10a. A good agreement is observed for the ratcheting deformation as well as for the subsequent creep and tension loadings. The influence of the stress rate is also well predicted as shown in Fig. 10c, which

i | [ t l l l : ( J | l ] l Viscous ratcheting on 304 stainless steel at room temperature: (a) Special ratcheting test, from Ruggles and Krempl (1987); (b) prediction of the test; (c) influence of stress rate on the ratcheting under repeated conditions.

30

J.L. Chaboche

plots the measured and calculated ratcheting strains for the same loading pattern with different periods. In these tests, it has been found that the ratio between the accumulated deformation and the subsequent creep strain increases when the period is increased. It suggests that the repeated loading conditions are near-monotonic creep conditions. This is why the nonlinear kinematic rule, which overpredicts the ratcheting strains at low mean stresses, gives quite good predictions for these repeated loading situations.

F. Creep-Plasticity Interaction This last application has been done in the context of a European benchmark on the constitutive equations (White, 1987). One of the exercises was the prediction of the test shown in Fig. 11 a. The specimen is loaded under strain controls (k -4 • 10 -4 S - 1 ) up to (9"1 --- 170 MPa (point A). It is then held at constant stress O"1 until a strain of 0.4% is added in creep (point B is recorded). The specimen is then reversed under strain control (k = 4 x 1 0 - 4 S-1) over a strain range of 0.6% (point C), recorded, and immediately reloaded at the same strain rate until the stress equals o'~ (point D). The same loading is then repeated between points B, C, and D. The material is a type 316 S. S. (UKAEA cast 83) and the temperature of the test was 625~ In these loading conditions, the most important effect to predict is the creepplasticity interaction that occurs at each cycle. For this application, only the initial yield stress (coefficient k) has been fitted on this test. Predicted stress-strain curves for cycles 1 to 39 are given in Fig. 11 b. Calculated strain at cr - 170 MPa is e A = 1.026% (then e B = 1.426% and ec = .826%) against e ' g - - 1.04% in the experiment. The isotropic hardening observed at each cycle by the increasing stress range is clearly predicted, but slightly underestimated. Creep curves for cycles l, 2, and 6 are plotted in Fig. 12a. Cycle 1 is in good agreement with the test. In cycles 2 and 6, creep strains are lower than in the test. The creep strain reaches the maximum value at the third cycle, then it slowly decreases as shown in Fig. 12b. A similar curve is obtained for the evolution of the creep duration (Fig. 12b). The complete description of this test shows a very good adequacy of the model for such a complex loading.

G. Ratcheting Effects Several studies have shown that the very limited ratchet strains under quasireversed cyclic tension-compression stresses (low mean stresses) are overpredicted by the classical nonlinear kinematic hardening models (see Chaboche and Nouailhas, (1989a), for instance). An adequate solution to this problem was to introduce the concept of a threshold in the dynamic recovery term (Eq. (2.26)). Comparison of Figs. 13 and 14

Chapter 1

Unified Cyclic Viscoplastic Constitutive Equations

31

Plasticity-creep interaction on 316 stainless steel (UKAEA cast 83) at 625~ Stress-strain curves (data from White, 1987); (b) model simulation.

(a)

~|(itij:tai|li

32

J.L. Chaboche

Creep results in plasticity-creep interaction for 316 stainless steel at 625~ (a) Creep curves; (b) cyclic creep strain and cyclic creep duration vs. number of cycles.

shows the capability of a model with such a threshold to capture both the low tangent modulus for large strains (Fig. 13) and the limited amount of ratcheting (Fig. 14). The simulations were performed (Chaboche, 1991) within the rateindependent limiting case of our viscoplastic model, including three back stresses, one with a threshold. The response to normal (reversed) cyclic loads was not significantly modified by the threshold. The application to another material, the Inconel

Chapter I

Unified Cyclic Viscoplastic Constitutive Equations

33

cr(MPa) Calc. 600

n

O

§

+

0

Tests 4.,,

450

300

.r

150

0

I

I

I

i

0.04

0.08

0.12

0.16

s

0.20

Description with the complete model of the monotonic tensile curve in the range 0-20% (true stress-true strain). Type 316 stainless steel, room temperature.

718 superalloy (used in turbine disks), was also satisfactory (Chaboche et al., 1991). Later, Ohno and Wang (1993) proposed a modified model, introducing the concept of "a critical state of (dynamic) recovery." W h e n identified on the same data base, the two modifications act quite similarly, both under uniaxial or multiaxial loadings (Chaboche, 1994). The current tendency, in order to improve the modeling capability, is to use a larger number of back stresses (say, 4 to 8). As shown, for instance, by Chaboche et al. (1991), the kinematic model can be constructed and determined in such a way that the number of material-dependent coefficients is not increased (it is even decreased compared to 3 back stresses).

TH ERMOVISCOPLASTICITY The elasto viscoplastic constitutive equations must be used under varying temperature conditions. Most of the structural applications have to be performed for

34

J.L. Chaboche

0.02

L ~p

0.016

0.012

Test

0.008

0.004 I--

0

t-'-- . . . . i

~

-"' I

n

I 1200

C o m p l e t e Model

,

I

,

N cycles I , 2400

I_

i:I[(RIJ,'IqBIR Description with the complete model of the ratcheting test. Stress range 400 MPa;

mean stresses of 20, 40, 60, and 80 MPa (load control). Experimentaldata on 316 stainless steel at room temperature from Goodman (1983).

complex thermomechanical loadings. In the field of metal manufacturing or metal forming, we must also take into account the thermomechanical coupling effects. In this section, we present first the general concepts of a classical thermodynamic framework and discuss the consequences in terms of constitutive models. We examine the various ways by which these models can satisfy the thermodynamic requirements, underscoring the possible modeling of stored and heat-dissipated energies. At the end, we discuss the use of constitutive models under varying temperature conditions and the problems associated with temperature history effects.

A. Thermodynamics with Internal Variables In general, the constitutive equations are based on one of the two following thermodynamic concepts:

Chapter 1

Unified Cyclic Viscoplastic Constitutive Equations

35

9 The present state of the material depends upon the present values and the past history of the observable variables only (total strain, temperature history), leading to hereditary theories. 9 The present state of the material depends upon the present values of both observable variables and a set of internal state variables. We follow the second kind of approach, introducing the state law to govern the reversible processes and the notion of a dissipative potential for irreversible processes. All the theory is developed in terms of "local" state equations, the state of a material point being assumed independent of the state at neighboring points. 1. The State L a w

The present state of the material is characterized by the observable variables, thermoelastic strain Ee, temperature T, thermal gradient V T, and by the internal state variables denoted ak. The observable variable is the total strain; but for nonviscous elastic strain, provided the use of the strain partition (2.1), which is valid for small strains, it is sufficient to consider the thermoelastic strain as the independent observable variable. We choose the Helmoltz free energy as the thermodynamic potential: -- u -

(4.1)

ST

where u is the internal energy, S is the entropy, and ~p is assumed to depend on all the independent state variables, (4.2)

1// - - lp(E:e, ak, T, V T)

Taking into account the conservation of energy, the second principle of thermodynamics reduces to the Clausius-Duhem inequality: rr:k-

~-

1 5'T - - - q . VT > 0 T

(4.3)

Therefore, by (4.2) and (2.1), Eq. (4.3) becomes

o"

01// 0Ee +o''kp

" ~e--

SJf -

T-

-~ -

Of/.

-~a ak

"

OVZ

1 -- - ~ q - V T > 0

Ot

(4.4)

If we assume the evolution equations for the plastic strain with the internal variables in the form ~p - - ~ p ( g e , aj, T , VT)

dk -- ~lk (Ee, a j,

T, VT)

(4.5)

36

J.L. Chaboche

where dk is zero for any instantaneous reversible change, we can follow the classical arguments of Coleman and Gurtin (1967). Inequality (4.4) must be valid for arbitrary reversible changes in the observable variables ee, T, OV T/Ot, so that the three first terms must vanish independently: (a) o" --

Oee

(b) S =

(c)

OT

OVT

= 0

(4.6)

The free energy does not depend on the thermal gradient. Equation (4.6a) is the expression for the thermoelastic law, which reduces to Hooke's law for isothermal conditions. By analogy, we express Ak --

(4.7)

Oak

as the thermodynamic forces, or affinities, conjugate to the thermodynamic displacements ak. The remaining terms in (4.4) correspond to the dissipation inequality 1

0"" ~.p- Akdk -- ~ q - V T

>_ 0

(4.8)

Classically, this inequality is separated into the intrinsic and thermal dissipation inequalities (Truesdell and Noll, 1965): (a) tr

9 ,l~p

-

Akdk >_0

1

(b) - : q . I

V T >_ 0

(4.9)

Usually, the evolution of heat flux is represented by Fourier's equation q = -k.

VT,

(4.10)

where the symmetric tensor for thermal conductivity k must be positive definite in order to satisfy the thermal dissipation inequality. 2. The Dissipative Potential: Standard Generalized Materials

In standard associated plasticity, Hill's postulate of maximum dissipation, written in the stress space, leads to the quasi-convexity of the yield surface and defines the direction of the plastic strain rate as the outer normal to the yield surface at the present stress state. The same applies for viscoplasticity, leading to the normality to the equipotential surface. Considering thermoviscoplasticity, it is possible to generalizes such a postulate in order to check the second principle automatically. Materials (or constitutive theories) obeying such a principle are called "standard generalized materials" (Halphen and Nguyen, 1975). In the dissipation inequality (4.8), we observe

Chapter 1 UnifiedCyclic Viscoplastic Constitutive Equations

37

the duality between rates of dissipative variables ( k p , - - a k , - - q / T ) and the corresponding thermodynamic forces (cr, A~, VT). Following Germain (1973), and Germain e t al. ( 1 9 8 3 ) , we assume the existence of a potential of dissipations in the space of rate of dissipative variables q~ -- q~ (~p, dk)

(4.11)

Here, the space does not incorporate the heat flux, which was already taken into account, leading to Fourier's equation (4.10). q~ is supposed to be positive and convex in its variables, and is assumed to contain the origin--that is, (p (0, 0) = 0. The thermodynamic forces are obtained as o- =

A~ =

Okp

(4.12)

O(--dk)

Applying to (p, the Legendre-Fenchel transform, we obtain the complementary dissipative potential (p* in the generalized space of thermodynamic forces (o', Ak) ~b* = min [o" : ~p kp, --ak

A~d~

-

(p(kp, -d~)] = ~p*(o', A~)

(4.13)

Equations (4.12) are then replaced by -kp -- 00"

ak =

(4.14)

OAk

Such a generalized normality, in the space (o', A~) confers interesting properties on the constitutive equations. In particular, the second principle is automatically verified due to the positiveness and convexity of ~b*. Provided 4~*(0, 0) = 0, the intrinsic dissipation (4.9a) can be written as cr " k p -

A~d~

-- o'"

0---~ + A ~ - ~

>_

(4.15)

>_ 0

3. R e m a r k s

9 In fact, the convexity of tp* is not necessary. A sufficient condition is the convexity of the domains tp _< tp*(o', A~) _< tp*, for every positive value ~b*. 9 The state law has been written with the Helmholtz free energy. Exactly the same results would be obtained by using other thermodynamic potentials, such as the complementary Gibbs free energy (obtained from ~ through the LegendreFenchel transformation): ~ * ( ~ , a~, T, VT) = inf[~(ee, a~, T, VT) Ee

-

cr : e e ] = u -

ST

-

cr : e e

(4.16) Also, the consequences of the thermodynamic framework as discussed below would remain valid.

38

J.L. Chaboche

4. Consequences of the General Treatment a. From the State Law In recent years the evolution equations for the internal variables of viscoplastic models have been extended to include the rates of the observables variables (strain or stress and temperature) in order to improve predictions (for example, see Krempl et al., 1986; Freed and Walker, 1990; Ramaswamy et al., 1990). This requires appropriate considerations when deriving constitutive restrictions from the Clausius-Duken inequality, an aspect studied by Lubliner (1973) for evolution equations that contain the rates linearly. Let us assume that the evolution equations for the internal variable contain linear terms in the rates of the observable variables be and ~r: d k =

ak(Ee, aj, T) +

I-Ik(ee, aj, T)

9 b e .qt_

Mk(t~e, aj, T)7"

Here we have omitted the thermal gradient for the sake of simplicity. reconsider the Coleman arguments, we now find the state law in the form (a) or =

O'e OOoa -t-

=---Hk

OOoa

(b) S -

7---M~,

(4.17) If we (4.18)

These relationship, which were first derived by Lubliner (1973), characterize the reversible portion of a process. The first term in the fight-hand side of these equations is the classical thermoelastic contribution to stress and entropy, respectively. The second term, which is nonclassical, exists because of reversible phenomena present in the evolution of internal state through the 7" and b e contributions, respectively, that are introduced in Eq. (4.17). The state equations (4.18) are clearly no acceptablemat least not the first onemif we want to consider the thermoelastic response of a Hookean material as characterized by one of the following equations: e e - - S : or + o ~ 0 or - - A

" (ee

-

c~0)

(4.19)

where 0 = T - To, To the initial temperature, c~ the thermal expansion tensor, and S and A the compliance and stiffness fourth-rank tensors, respectively. Clearly, the second term in Eq. (4.18a) introduces an undesirable coupling with the irreversible changes associated with ak. We could illustrate this inconvenience in the simple uniaxial isothermal case, considering only one variable and Hk as the identity tensor (this is the case, for example, in the viscoplasticity theory based on overstress, proposed by Krempl et al., 1986). The variable A = 0 ~ / 0 a represents the equilibrium or back stress and Eq. (4.18a) reduces to cr = E g e -+- A. Figure 15 shows the corresponding nonconventional definition for the elastic strain. Several tentatives were done, through variable changes or other considerations, to eliminate the unconvenient restrictions from Eqs. (4.18). In each case

Chapter 1 UnifiedCyclic Viscoplastic Constitutive Equations 39

I I I /

!

E!e

|

The nonconventional definition of elastic strain corresponding to the introduction of a stress rate term in the evolutionary equations for the internal variables. From Chaboche, J. Appl. Mech. 40 (1993), published by ASME. (see Freed et al., 1991), the proposed models that include the elastic strain rate (or the stress rate) in the evolution equation for the internal variables either do not respect the thermodynamic restriction as deduced from the C o l e m a n Gurtin (1967) treatment, or can be identically reduced (by a variable change) to a form without such a rate term (identical for both the mechanical and dissipative responses). The same conclusions were issued from the very complete study made by Malmberg (1990 a-b). This is why we exclude both the temperature rate and the elastic strain rate (or stress rate) from the rate equations for independent internal variables (see Chaboche, 1993). Let us note that the rate of the dependent variables, the thermodynamic forces, can have such rate terms, provided the coupling terms are introduced in the thermodynamic potential. Such an example will be given in Section IV.D.2 for the temperature rate. b. F r o m the Dissipative Potential In the framework of classical viscoplasticity, the notion of standard generalized materials implies some interesting consequences about the nature of internal state variables. Let us assume the simplest behavior with a power viscoplastic potential and no recovery at all (dynamic and static) for hardening variables. Let us also assume that the dissipative potential

40

J.L.Chaboche

has exactly the same form:

~)*--- ~ O ( J ( ~ n+l

n+l __ D

O(o'v) n+l

n't-1

D-

(4.20)

where the back stress X, the yield stress increase R, and the drag stress D are the thermodynamic forces associated with some internal variables denoted respectively as c~, r, and 3. The generalized normality indicates that k p = 0q~* 3 ~ ~,7 = 2 / ' g(o- - x ) &=

a4,* 0X = k p

; =

(~v) n with/, = ~4,*

k=

n cr~fi n+lD

OR = ]~

~-

(4.21) (4.22) (4.23)

Therefore, from Eqs. (4.22), the internal variable associated with the kinematic hardening is the plastic strain, and the one corresponding to isotropic hardening described by the evolution of the yield stress is the accumulated plastic strain. For the isotropic hardening through the drag stress, assuming the free energy to be quadratic in 3, from (4.23) we obtain 32 proportional to the accumulated viscoplastic work f cr~]~ dt. Let us remark that several theories that do not consider the yield stress are based on the plastic work for describing isotropic hardening (for example, the constitutive equations developed by Bodner (1987)).

B. Introduction of the Constitutive Equations in the Thermodynamic Framework Several methods have been used to write the viscoplastic constitutive equations in the general thermodynamic framework. To some extent, they can be considered equivalent, but it is interesting to note some conceptual differences concerning the use of the notion of dissipative potential. The considered constitutive equations are of the form indicated in Section II.A. For convenience, we treat the case with only one back stress, but all the developments apply immediately for several back stresses. 1. No Use of a Dissipative Potential

This was the way chosen by Chaboche (1978) and Freed et al. (1991) in order to introduce the non-linear kinematic hardening rule in the thermodynamic frame work. It is presented here for the more general anisotropic case. The state variables are ee, o~, r, and 3. The viscoplastic potential is assumed to be of the form given by Eqs. (2.2) and (2.4). We note that the viscous stress is f = o'~ = J ( c r - X ) - R - k ,

Chapter 1

Unified Cyclic Viscoplastic Constitutive Equations

41

and the normality rule gives: 0f2 kp =

pn

0tY =

(4.24)

with n -- M " (or - X)/J(o" - X) and p -- G ' ( f / D ) . The only thermodynamic requirement to be checked is the second principle; that is, from the results of Section IV.A.1 and Eq. (4.%): 4> -- cr 9kp - X 9 & - Ri" - D 6 > O. In order to meet this inequality and to recover the evolution equations (2.10)-(2.12) for X, R, and D (under isothermal conditions), we assume the following rate equations for the hardening variables: & - - kp -- r

i" = p -

(4.25)

9C~p -- r s 9o~

B(r)rp-

Bs(r)r

(4.26)

- cBs

(4.27)

-- c p - c B ' ( 6 ) S p

The role of the small parameter c will be explained below. In each equation, we recognize easily (1) the linear hardening term, (2) the dynamic recovery term (proportional to the state variable and to the modulus of the plastic strain rate), and (3) the static recovery term (proportional to the state variable), r and r s can depend explicitly on R and c~ (or X). Here, choice of the thermodynamic potential that governs the relations between thermodynamic forces (X, R, D) and state variables (c~, r, 3) will be the decoupled quadratic form (other options for the isotropic hardening will be discussed in Section IV.C.2)" 1

1r - - ~ E e "

1

1

1

A ' E e -~- ~c~ 9N 9o~ + -2hyr 2 + x - - h d ~2 z z 2c

(4.28)

From the classical state law, we obtain cr = A ' E e

X-

N'c~

(4.29)

1 R -- hyr

(4.30)

D = -hd6 c

Incorporating (4.29) and (4.30) into Eqs. (4.25)-(4.27), we express the rate equations for the thermodynamic forces as N ' Y ' " N -1 " X p -

-- N'kp-

k -- h y p -

B

L) -- h a p - c B '

R p-

-~a

N" 1-'s" N -1 " X

Bs

R

D p - cBs

(co) ~

(4.31) (4.32)

D

(4.33)

42

J.L. Chaboche

The complete identification with the constitutive equations (2.10)-(2.12) of Section II.A.2 follows immediately with 1-' = ~b(X, R)N -1 "Q 9N B

and

1-'s -- S(X, R)N -1 "Q" N

-- ry(R)

(co) ~d

Bs

-- Sy(R)

(co) ls o,

-- -rd(D)c

B~ ~

(4.34)

-- c

Now, in order to check the second principle, we decompose the dissipation

(I) = (I) 1 + (I)2 + (I)3 ~ 0, using the rate equations (4.25)-(4.27) and the above identification (4.34):

(I)1 -- ( o r - X ) ' ~ p ,t,2 =

[ 4,<x,~)x

R ] ~ - cD]9

" N -~ 9Q

ndD2] i' x + r y ( R ) - :nyR2 - - + crd(D~-7--

~3 -- S(X, R)X" N -1 9Q" X + S y ( R ) 7

R2

ny

(4.35)

D2 -F c S d ( O )

h--7

Assuming X 9N-1 . Q . X as a definite positive quadratic form, and the functions ry, rd, Sy, Sd as positive functions, we immediately check (I) 2 >__ 0 and ~3 _> 0. The first part can be expressed after easy manipulations: ~1 - [ J ( o r - X) - R - k l p + (k - c D ) p

(4.36)

The first term is always positive or/5 -- 0. The second term is always positive if k - cD > 0. This is the only condition to be checked for the drag stress evolution. It is the role of coefficient c to be sufficiently small to meet the requirement for the maximum value of D (given by Dmaxrd(Omax) -- hd). It follows that the use of a drag stress and the thermodynamic requirement needs to introduce also a (constant) yield stress k > 0, the value of which can be as small as we want. 2. Use of a Dissipative Potential Different from the Yield Surface This is a method useful for the rate-independent plasticity case (Lemai'tre and Chaboche, 1985). We assume also that static recovery effects are neglected. As the thermodynamic requirements are easy to meet for the parts of the constitutive model concerning the isotropic hardening, yield and drag (see Section IV.C.2), we limit the discussion to the non-linear-kinematic hardening (the case of the linear hardening, which is no problem at all, was already treated in Section IV.A.4). The yield surface is given by the boundary f = 0 of the elastic domain expressed by (2.3). However, in the generalized space of thermodynamic forces (or, X), and

Chapter I

Unified Cyclic Viscoplastic Constitutive Equations

43

R being neglected here, the indicatrice function of the convex dissipative potential is assumed to be of the form F - - f + g x - r -1 x - -

J ( t r - X ) - k + g X ' l - ' '1X

(4.37)

The generalized normality rule is written as OF --

kp -- J~ 00"

OF & -- - 2 - -

(4.38)

0X

Here, the fourth-order tensor r must depend on X, if we want to recover the constitutive equation (2.10). From the normality rule we have (let us note that -- ~b with definition (2.6)) 1 0F & -- ep - r ' X ~ b - ~ X " 0X "X~b

(4.39)

We continue to assume a quadratic form for the thermodynamic potential (4.28), so that X relates to c~ linearly" X-

N " c~

(4.40)

Combining (4.39) and (4.40), we find the following rate equation: 1 OF . X ) r q- ~ 3--~

]K-h'kp-h"

9X/)

(4.41)

as compared with Eq. (2.10) in the rate-independent case: X-

(4.42)

N" ep - q ~ ( X ) Q ' X p

The identification of the two dynamic recovery terms between (4.41) and (4.42) is not straightforward in general, so we limit ourselves to two particular cases" 9 If r is a constant tensor, and q5(X) - 1, the identification easily gives r = N -1 9 Q. We have only to assume that the fourth-order tensors N, Q, and r correspond to positive definite quadratic forms. 9 If r is dependent on X, we assume r to be collinear with the fourth-order identity tensor and Q to be equal to N: r-

r(J(X))I

The equality of the two dynamic recovery terms can be solved if we know the Xl function q5(X). For example, using q5(X) -- (1 - J-N5 )' we easily check that

J (X) + 2

J (X)

44

J.L. Chaboche

In the two cases provided, we have used the generalized normality rule, so the second principle is automatically verified. This can be checked with (I) - - O" " ~ p - - X "

& =

(o" -

X)

9 ~ p -~- ( I ) ( X ) X

"Q" X,b

9N -1

= [J(tr - X) - k]p + [k + ~p(X)X 9N -1 9Q " Xlp > 0

(4.43)

Let us note the main consequences of the above choices, which were made to meet the thermodynamic requirements: 9 In terms of the flow equation, the above theory can be called "standard and associated plasticity" because we meet the normality to the elastic domain in the stress space. 9 In terms of the generalized normality, the above theory can be called "generalized standard but nonassociative in the generalized space" because the generalized normality applies to a surface that is not identical with the elastic limit (it is only identical in the subspace of stesses). 3. U s i n g the State V a r i a b l e s as P a r a m e t e r s in the P o t e n t i a l

In the case of viscoplasticity theory, the method proposed by Chaboche (1983) and by Germain et al. (1983) was "to add" and "to subtract" terms in the potential. The dissipative potential is chosen directly as the sum of the viscoplastic potential (assume here a power form for simplicity) and of a static recovery potential: q~* -- s =

n+l

+ f2r(JT(X))

(4.44)

" Q ' X - ~ c ~1' Q ' N ' c ~

(4.45)

D-

c r ~ = J ( o ' - X ) - k + g X ' N -1

1

JT(X) -- (X " N-1

.

Q. x)

1/2

Here, the internal state variable ct plays the role of a parameter (and we have assumed that R -- 0). The generalized normality rule stands: -

kp - -

3~p* 30"

=p

3X

M " (tr - X) J (or - X)

= ~p -- N -1 " Q

=pn 9X/~

with , b -

< >n (7 v

-~-

1 3Qr N -1 "Q 9x JT 3JT

JT(X)

(4.46) (4.47)

Always with the same form of the thermodynamic potential, the rate equation for X becomes X = N'ep-

Q'X~b-

1 3ff2r JT 8JT Q - X

(4.48)

which is easily identified with Eq. (2.10) of Section II.A (when ~p = 1) with the proper choice of function f2r.

Chapter 1

Unified Cyclic Viscoplastic Constitutive Equations

45

Let us remark that, provided X = N : c~, we have always cr~ = J ( o " - - X ) - k. The viscous stress follows directly from the expression of the elastic domain. It can be said that the above method corresponds to a generalized standard and associative rule (even in the generalized space). However, the procedure of treatment of cry, adding and subtracting the "same term," is quite artificial. 4. The Complete Standard Generalized Normality

This corresponds to a modified model used recently by Ladeveze (1992). In this case, we do accept modification of the expression for the elastic domain, but apply the generalized normality rule in its standard and associated form. The consequence is that the mechanical response is not identical to the classical model of Section II.A. Some reidentification of the constants of the model is also necessary. This modification can be presented either in the rate-independent plasticity or the viscoplasticity case. We follow the first situation. The elastic domain, here identical to the indicatrice function of the dissipative potential, is expressed as F--f=J(o--X)-k+~X.N-

1

1

"Q-X_<0

(4.49)

j) -- ~.

(4.50)

and the generalized normality rule is written as kp --

. M " (o- - X) ~ Of __ ~. OX J (er - X)

&_~,Of

0X

- - ~p -- s - 1

" Q " Xp

(4.51)

which leads immediately to the searched hardening rule (2.10), provided we still have X = N : c~. The second principle is also immediately checked. The main difference is that the size of the elastic domain now varies as a function of X: k * = k - ~ X ' 1N -

1

"Q'X

(4.52)

Clearly, the initial value k must be high enough in order that k* > 0 even when X is saturated at its maximum value (in modulus). This model then differs significantly from the previous ones in terms of the mechanical response. 5. The Use of a "Multicriterion" Formalism

This is the last way we want to mention [it was followed by Benallal (1989)]. It corresponds to a generalization to viscoplasticity of the method used in Section IV.B.2. Also, the possibility for static recovery is considered. Now the viscoplastic potential is expressed like the function F in Section IV.B.2; and, independently, we define a static recovery potential (we also consider here the case where function 4~ is a constant): f = J (o- - X) - k

(4.53)

46

J.L. Chaboche ~'~p --

f -+- ~1 JT2 (X)

~"2r --- ~"2r(JT(X))

(4.54)

with JT(X) = (X : N -1 : Q : X)1/2. The normality rule is expressed as

~P _ j~ 0~"2p00.-~- 0~"2r0----~

(4.55)

CI~ -- --~ 09----~p

(4.56)

O~"2r

OX

OX

Obviously, Of2r/OCr = O, so there is no recovery term for the plastic strain rate. Moreover, the plastic multiplier J~ is free. In the viscoplasticity theory, we can express it as any function of f~ D, for example, for the power law

/ I) n

~. -- \

(4.57)

In the rate-independent plasticity case, ~ would be obtained, as usual, from the consistency condition f = f = 0. Such a method is called a "multicriterion" formalism because it is exactly this form of expression that is used in plasticity with multiple criteria (here the multiplier for the static recovery term is taken as 1). Finally, we find the searched evolution equation for X, after taking into account the relation X = N : c~ in (4.56):

& -- ~p -- N -1 " Q " X,b X-- N'kp-

Q'X,b-

1

0~rN-1 . Q . x

JT(X) OJT S(X)Q 9X

(4.58) (4.59)

Obviously, the second principle is still verified. The present way of introducing the constitutive equations in the thermodynamic framework is the most elegant. In fact, it corresponds to the use of a Lagrange multiplier technique used, for instance, by Hansen and Schreyer (1992) or by Cescotto et al. (1995). The results are general and match the other methods in both the mechanical response and the dissipative aspects (which we will discuss in the Section IV.C). This method will probably be the one we shall use in future developments of the theory.

C. Stored and Heat-Dissipated Energies Once introduced into the thermodynamic framework, the viscoplastic constitutive equations allow us to predict the energy dissipated as heat and the energy stored in the material during the viscoplastic flow. Their sum is obviously the plastic power or the external energy supplied by the material: Wp -- ~0 t o" " ep d r

(4.60)

Chapter 1 UnifiedCyclic Viscoplastic ConstitutiveEquations 47 Obviously, the energy dissipated as heat is given by integration of the dissipation equations such as (4.9a) or (4.35). In a recent article (Chaboche, 1993), the stored energies associated with kinematic hardening and isotropic hardening were determined on the basis of the thermodynamic framework. Only the main results are reproduced below. Let us begin by reviewing the classically observed experimental facts.

1. Stored Energy as Measured Experimentally Since the work of Taylor and Quinney (1934) and of many others, it has been well known that only a small part of the plastic work done is stored in the material. From the very complete review by Bever et al. (1973) we may extract the general trends as follows: 9 The stored energy Ws is a very small fraction of the plastic work Wp at large strains (5-100%), especially in pure metals. Values between 1 and 15% are commonly reported. 9 In fact, the ratio of stored energy over the plastic work is greatly dependent on the strain level. It is often observed (still for significant strains) that the ratio Ws/Wp decreases with the plastic strain. 9 The level of stored energy is larger for alloys and complex materials that have defects and complex dislocation arrangements. This is especially true for polycrystalline metals used in industrial applications [see the recent study by Chrysochoos (1985, 1987) on three engineering materials]. In such cases, the ratio Ws/Wp may vary between 50 and 20% for moderate strains. 9 In some cases, nonmonotonic evolution of the ration F : W~/Wp has been reported, with an increasing evolution for small strain (lower than 0.5-2%), as shown in Fig. 18 for 316 stainless steel (Chrysochoos, 1987; Beghi et al., 1986). 9 Very few experiments have been made under cyclic strains. The study made by Halford (1966) on OFHC copper shows that at each reversal a part of the stored energy is released, the effect of which is clearly related to the kinematic nature of a part of the hardening.

Remark: The stored energy at the initial state is not necessarily zero. In the present paper Ws and Wp denote the changes of stored energy and plastic work since the initial state.

2. Modeling Stored Energy The thermodynamic framework of Section IV.A can be used to demonstrate the following results (neglecting the static recovery effects and the drag stress evolution), which are illustrated schematically in Figs. 16 and 17: 9 The work done by the initial yield k and the work done by the viscous part a, of the stress are entirely dissipated as heat.

48

J.L. Chaboche

~ l l g [ l l ' , t J l g Heat-dissipated and stored energies for (a) linear kinematic hardening, (b) nonlinear kinematic hardening, and (c) isotropic hardening. From Chaboche, J. Appl. Mech. 40 (1993), published by ASME.

9 The linear kinematic hardening leads to a continuously increasing ratio W s / W p , which is not in accordance with experimental results. 9 The nonlinear kinematic hardening gives very low values for the ratio Ws/Wp. Moreover, this ratio first increases, then decreases as a function of the plastic strain (Fig. 17). 9 The same is true for the isotropic hardening simulated by Eqs. (4.28) and (4.26), when the free energy is considered quadratic in the variable r. However, using a different form of the free energy, with a term ~p = Q r + ( Q / b ) ( e -br - 1) instead of lpp - ~l hyr2, and considering B ( r ) -- Bs(r) --" 0

Chapter 1

Unified Cyclic Viscoplastic Constitutive Equations

49

) Linear l l l [ l l l l : l l l V l l Simulation of (a) the stored energy and (b) the ratio F = W s / W p . ( . . . . kinematic hardening; ( ....... ) isotropic hardening; and ( - - ) nonlinear kinematic hardening.

in Eq. (4.26), we have a different figure, with the work done by the stress R entirely stored in the material (Fig. 16c). In that case, the ratio Ws/Wp is increasing from Ro/(k + Q) if the initial value of R is given by R0 (Fig. 17).

3. Comparisons with Experiments We have selected experimental results showing both the stress-strain curve and the ratio Ws/Wp. There are engineering polycrystalline alloys, the steel XC 38, the austenitic stainless steel 316L, and the aluminum alloy 2024-T4 (Chrysochoos, 1985, 1987). All materials are tested at room temperature under uniaxial tension. A recent study (Chaboche, 1993) has shown the possibility of describing very correctly both the stress-strain behavior and the stored energy by superposing two nonlinear kinematic hardening models and two isotropic hardening variables (one of which is saturated). Figure 18 illustrates the comparison for the three materials. The evolution of the ratio F = Ws/Wp is well represented, both qualitatively and quantitatively. Other results and good comparisons have been obtained for copper single crystals and for the energies stored and released during cyclic loadings. All these results are encouraging. They confirm that:

50

J.L. Chaboche

i:l[~lll;,l:lllSN

Modeling of (a) tensile curves and (b) ratio F -- Ws/Wp of the stored energy to the plastic energy for three materials: XC 38 steel, 2024 aluminum alloy, 316 stainless steel. (. . . . . . o) Experiments from Chrysochoos (1987); ( ~ ) model. From Chaboche, J. Appl. Mech. 40 (1993), published by ASME.

Chapter 1 UnifiedCyclic Viscoplastic Constitutive Equations 51 9 The nonlinear kinematic hardening model is the only one able to give sufficiently low values of stored energy and decreasing evolutions of the ratio Ws/Wp as a function of plastic strain that are observed experimentally under monotonic tension. 9 The proposed thermodynamic framework is useful both for justifying the choices made for mechanical constitutive models and to quantify the energies that are dissipated as heat or stored in the material. This last aspect is very important for applications of constitutive models to the coupled situations of thermoviscoplasticity.

D. ConstitutiveEquationsunder VaryingTemperature The constitutive equations were presented in Sections II and III for isothermal conditions. When temperature is varying, monotonically or cyclically, several questions must be considered: 9 In what way are the "material constants" or functions dependent upon temperature? (Associated with this problem are the temperature rate terms in the evolution equations for the hardening variables.) 9 In some situations temperature history effects have been observed. How is it possible to model such effects? 9 Aging effects are also observed phenomena. How is it possible to introduce in a thermodynamically consistent way such "time-hardening" effects? All these aspects are briefly presented and discussed in the following subsections.

1. Influence of Temperature in the Constitutive Equations Until now, the constitutive equations were considered under constant temperature. Their determination can be done at several temperature levels by isothermal tests. In principle, the material parameters that have been introduced in previous sections in order to model viscoplasticity, hardening, and recovery can depend upon temperature. Two techniques can be used: (1) One in which each coefficient is varying as an unprescribed function of temperature (spline functions, or parabolic functions, etc.), and (2) one in which some of the temperature-dependent effects are prescribed through given functions of temperature that are physically related. The second method is especially useful for the viscosity and static recovery effects. As in several unified constitutive models (Miller, 1976; Walker, 1981; Freed and Walker, 1993), we can use a Zener-Hollomon (1944) functional dependence, where the magnitude of the plastic strain rate, ,b, is decomposed into the product of two functions

])(o', X, R, D, T) = O(T)Z(o', X, R, D)

(4.61)

52

J.L. Chaboche

Stationary creep behavior: (a) Aluminum data; (b) copper data. Reprinted from International Journal of Plasticity, Volume 9, A. Freed and K. P. Walker, Viscoplasticity with creep and plasticity bounds, Pages 213-242, Copyright (1993), with kind permission from Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK.

where 0 > 0 is like a thermal diffusivity and Z > 0 is referred to as the Zener parameter. This thermal diffusivity is often represented as an Arrhenius function of temperature, which is valid over a significant but specified temperature range (Miller, 1987; Freed and Walker, 1993). The Zener parameter is therefore a temperature-normalized function for the magnitude of plastic strain rate. In the present context, the Zener parameter can be identified with the viscous function G'(crv/D) of Eq. (2.7); but, when dealing with the power function, for example, we immediately have difficulties due to the great dependency of the apparent power exponent as a function of temperature. The way by which we can overcome this problem, followed by Freed and Walker (1993), for example, is to define the Zener-Hollomon function over a very large number of decades (say 24 decades in strain rate, for example, associated with 3 or 4 decades in stress), so that the temperature change is used to shift the representative point from a region with a large apparent exponent (in general for low temperature) to a region with a low exponent (for higher temperature). Figure 19 shows an example for aluminum taken from Freed and Walker (1993). The temperature dependency in the viscoplastic strain rate equation (2.7) is then introduced by

[cr~/D1

p = O(T)Z ~r*(T)

(4.62)

Chapter 1 UnifiedCyclic Viscoplastic ConstitutiveEquations 53 The choice of functions cr*(T) of temperature is used to shift the "local stressstrain rate" diagram along the general Zener-Hollomon curve. The procedure is also helpful for prescribing the influence of temperature on the static recovery terms. As discussed in Section II.D and by Freed and Walker (1993), some implicit relations do exist between the steady-state creep and the static recovery. However, the change of the apparent power exponent as a function of temperature as given by the Zener-Hollomon function is not necessarily well correlated with its change as a function of the strain rate which is needed to model the limiting strain rate effects (see Section III.C). Further studies of this problem are necessary.

2. Temperature Rate in the Hardening Rules Let us consider now the behavior of the models under varying temperature. In the present thermodynamic framework, the independent state variables are considered to be the "strain-like" variables appearing in the thermodynamic potential. They are independent of temperature in the following sense: If we consider a rapid temperature evolution that does not produce viscoplastic strains and where thermal recovery has insufficient time to take place, the internal state variables (c~i, r, 5) do not change. However, the (dependent) associated thermodynamic forces (Xi, R, D) can change instantaneously. In fact, the constitutive equations under varying temperature are easily deduced from the thermodynamic potential by the derivative of Eq. (4.7):

021P

dj nt-

A~: -- Oa~:Oa---~

"F

(4.63)

Oa~O T

This introduces an additional temperature term in the rate equation for back stresses. For example, in the case of the free energy (4.28), if we consider that the fourth-order tensor Ni and the hardening factors hy and hd depend upon temperature, the hardening rate equations become Xi -- (Xi)t=0 + ONi aT .N~I . Xi ~ k -- (R);r=0 + b -- ( D ) t = o -~

1

Ohy

hy(T) o r 1

Rir

Ohd

--

hd(T) OT

(4.64)

DT

The additional temperature rate term in the equation for the back stresses is specially necessary as discussed below. It was considered in our first publication on constitutive models (Chaboche, 1977). It is also present in Walker's model (1981) and, implicitly, in several other ones [as in Miller's model (Schmidt and Miller, 1981) where the back stress is implicitly proportional to Young' s modulus].

54

J.L. Chaboche

The presence of such temperature rate terms has led to some controversial discussions (see Hartmann, 1990; Ohno et al., 1989). It is in fact well accepted, as shown in recent reviews by Ohno (1990) and McDowell (1992). However, we want to emphasize the need for such terms, especially for the back stresses, even if the thermodynamic aspects are not considered. 9 On the physical level the true state is defined by the dislocation arrangement, and the plastic strain incompatibilities (from grain to grain) are all quantities directly associated with the plastic strain. For the same dislocation state, if we rapidly change the temperature, we do change Young's modulus, which immediately changes the stress fields associated to the various strain incompatibilities. 9 From the phenomenological point of view, we know that a great dependency of the 0.2% proof stress as a function of temperature is possible so that, for example, the proof stress at low temperature T1 is higher than the rupture stress at high temperature T2. If we imagine now a 0.2% monotonic tensile plastic strain at T -- T1, then a rapid unloading and a rapid temperature change to T = T2 (without any new plastic flow) upon reloading, it is impossible to accept that the plastic flow will not begin before 0-0.2(Tl). Clearly, the correct behavior will be to begin plastic flow for a stress around 0-0.2(T2). 9 The last argument is given by a simple schematic counterexample, where we assume only a linear kinematic hardening in the limit case of rate-independent plasticity under uniaxial tension-compression. We consider a cyclic strain control (mechanical strain), between e g and --eg. The strain increase is applied at the constant temperature T2. Then strain is kept constant and the temperature changed to T2 > Tz (which produces plastic flow because k2 < k 1). The strain is decreased to --eg at temperature T2, then maintained during the temperature change. The responses corresponding to the two hypotheses are easily obtained and shown in Figs. 20b and 20c. In the first case, X = C(T)ep is the state variable, while in the second one the state variable is c~ -- ep and X changes instantaneously during the temperature change. Clearly, the first assumption leads to the ridiculous result involving a "stress ratcheting" as illustrated in Fig. 20b by 60- = 0-5 0-1 : 0"9 - - 0"5 . . . . . although the first cycle is correctly closed with the second assumption. Figure 21 shows a more realistic comparison where two NLK models are superposed to the linear one. The coefficients at T1 and T2 are indicated on the figure. Here also we observe the nonadmissible result given by the first hypothesis. Similar discussion and counterexamples have also been given by Wang and Ohno (1991).

3. Temperature History Effects Such effects have to be defined if the prior temperature history changes the values of the independent internal state variables. They are of two different natures:

Chapter 1

Unified Cyclic Viscoplastic Constitutive Equations

55

- - l l [ t l l J l | |~[I.~ A schematic example in the rate-independent case and for a linear kinematic hardening:

(a) Mechanical strain and temperature controls; (b) the cyclic response when considering the back stress as the state variable; (c) the cyclic response when the state variable is oe (with the temperature rate term in the back stress rate equation). From Chaboche, J. Appl. Mech. 40 (1993), published by ASME.

The same simulations as in Fig. 20 for the superposition of two NLK variables to the linear kinematic hardening: (a) Without temperature rate; (b) with temperature rate. From Chaboche, J. Appl. Mech. 40 (1993), published by ASME.

56

J.L. Chaboche

9 If coefficients in the dynamic recovery terms (Yi for kinematic hardening, b for isotropic hardening) are dependent on temperature, we may observe temperature history effects in the sense that the values of the internal variables ( a and r) are different for two identical plastic strains with two different temperatures, provided that r i - - i~p - - Y i ~ i

i" - - p - b r p

P

(4.65) (4.66)

This kind of history effect has been discussed by Ohno and Wang (1992) and by McDowell (1992). However, this history effect is rapidly evanescent for the kinematic hardening and does not modify the asymptotic conditions for isotropic hardening. Ohno e t al. ( 1 9 9 0 ) have used such a dependency, together with the "memory of maximum strain range," to describe some temperature history effects in stainless steels. 9 If temperature history effects are significant and not rapidly evanescent, they may be described by modifying the temperature rate terms in equations like (4.64), which is not consistent with the thermodynamic framework. Another way is to introduce such history effects with specific internal variables, whose purpose is to describe the corresponding changes in the microstructure. This way was used, for instance, by Cailletaud (1979) in the context of y' dissolution and (re)precipitation in turbine blade superalloys. He used two internal state variables, related respectively to the volume fraction of precipitates (a temperature-related effect, with some delay) and the mean radius of precipitates (related to the present strength of the material). Another way, appropriate for aging effects, is discussed in the next subsection.

4. Modeling of Aging Effects By "aging effects" we generally mean some overhardening produced either by time (static aging) or by the strain rate (dynamic aging). The effects of the second kind have often been introduced in the constitutive models through a change of the drag stress as a function of the strain rate (they are related to the slowing down of mobile dislocations by the drag effect of atoms in solution). One of the most popular models of this kind was proposed by Miller, introducing an additional state variable (called Fsol) that adds to the normal drag stress, Fsol being related to the magnitude of the present plastic strain rate. Obviously, provided we have explicit one-to-one relations, such an effect can be considered as directly contained in the equation that defines the viscoplastic strain rate. Effects of the first kind, "time aging," correspond to a progressive increase of the hardness of the material, after a prior aging period at room or intermediate temperature. In terms of modeling, the difficulty here is associated with

Chapter 1 UnifiedCyclic Viscoplastic ConstitutiveEquations 57 the thermodynamic framework and the requirement to have an associated positive dissipation. In the normal treatment (Section IV.B) we have shown several times that the heat dissipation associated with static recovery is, by nature, positive (corresponding to a negative rate of the state variable). Obviously, when we try to model a "timehardening" effect, with a positive rate of the same kind of term, we immediately encounter a negative heat dissipation, which is thermodynamically unacceptable. One solution, that respects the thermodynamic framework, is to introduce specific additional variables, as in the work by Cailletaud (1979). We present briefly a model recently proposed for the aging of aluminum alloys (El Mayas, 1994; Chaboche et al., 1995). As in the model by Marquis (1989), we introduce an additional state variable for aging, a, that plays a role in the size of the elastic domain. Two specificities can be adduced: (1) The first key of the model is a temperature dependency of the elastic limit that incorporates a state variable a, with 0 < a < 1: k ( T , a) = k * ( T ) + C ( T ) a

(4.67)

For a = 1, we have the completely aged state, a = 0 corresponds to the completely annealed state, with the minimum possible values for the elastic limit (independently of the possible isotropic hardening). More specifically, in order to reduce the number of constants, we may assume the following dependency: k* (T) -- k ~ + ~k~ - koc C ( T ) Co

(4.68)

where k~ is the lowest value of k, k* is the value at high temperature, and k~ and Co are the values of functions k * ( T ) and C ( T ) at room temperature (see Fig. 22a). Obviously the function C is chosen as decreasing with temperature. (2) The second key is an evolution equation for the aging variable: h = m ( T ) [ a o c ( T ) - a]

(4.69)

where am (T) gives the asymptotic value of a with typically two regimes: a ~ = 1 at low temperature, so that we necessarily describe aging effect (a >_ 0), and k is increasing progressively. Conversely, at high temperature a ~ = 0, and we describe annealing (a _< 0). Figure 22b indicates schematically these two regions and the corresponding evolutions of a when starting from a completely aged material at room temperature: a stays equal to 1 during the rapid temperature increase; then a decreases at high temperature (provided we have a sufficient time). At this regime, k is may be not greatly affected, because C is sufficiently small (see Fig. 22a). After returning

58

J.L. Chaboche

a k* (T)= k~ k~ ko- k~

~

-9

.

.......

k(T)+

k o- k o koo k o- 1%o

ko-k~

a~--. ~ x~ \ C(T)= k ~.:c:...

o-

k~ [k(T)- k~ ]

ko

koc

................. ! .... "... ..... 1

~

b

T

T1

To

f

a

~=1 i aging effe

~ annealing

1 1

a~-O

To

Tc

T1

f

~ | [ l l l l l l l ~ l l (a) Definition of the elastic limit from the temperature and aging variable; (b) the two limits and the evolution of the aging variable.

to room temperature, the value of a has been decreased, so that the aging effect (a _> 0) takes place. As mentioned earlier, the difficulty concerning the thermodynamic acceptability appears during the aging period, with a > 0. The way proposed by E1 Mayas consists in using the free energy without coupling terms between r and a: 7t l~e %- ~/'p + 1//"a (a) with the elastic and plastic parts defined as usual. The aging part is quadratic in the difference a ~ - a: ~ a ( a ) -- -1~ L ( a ~ ( T )

--

a) 2

(4.70)

Chapter 1 Unified Cyclic Viscoplastic Constitutive Equations

59

so that the associated force is (using the same notation as Lemaitre and Marquis, 1992)"

ar Z --

= L(a

- a~c(T))

(4.71)

Oa

The aging effect is not introduced by Z in the expression for the elastic domain but by the state variable a itself, serving as a parameter (we do not write back stresses as it is not concerned with the present discussion): f

-

J (or) -

R - k -

C (T)a

(4.72)

the dissipation potential is assumed in the form -Jr- ~1 m(T) Z2

~b* -- ~ p ( f )

(4.73)

where ['2p is chosen as in any viscoplasticity theory. The generalized normality assumption gives

'~P =

Off)* a ~~p Of Of 00" -- Of 00" = j) O---~ Off/)*

i~-

O ~'2p -

h--

3~*

(4.74)

- p

OR

Of _--

m(T)

OZ

z

L

Combining (4.74c) and (4.71), we easily obtain the searched evolution equation (4.69) for a. Moreover, we write the dissipation as r

--

R?- Zh

O" " k p -

m(T) = J(o')p

-

Rp

+

Z 2

L

= [J(o')

-

R - k -

-+-Lm(T)[aec(T)

C(T)a]p

(4.75) + [k + C ( T ) a ] p

-a] 2

Each of the three terms is positive (p = 0 if f < 0). Let us note that, contrary to the model proposed by Marquis (1989), the constant L can be chosen freely in order to describe the heat dissipation associated with the aging effect. This simple model has been applied by E1 Mayas (1994) to a metal matrix composite made of 2024 aluminum alloy and 16% SiC (short fibers). By temperature incursions of short durations at 100, 150, 200, 250, and 300~ the evolution of the elastic domain (half its size k) has been measured, showing the effect of annealing at high temperature, which induces a much lower yield limit at lower temperature. (Fig. 23 illustrates this effect for various conditions, taken from Chaboche e t al.

60

J.L. Chaboche

Elastic limit evolution before and after temperature incursions, 2024 A1/SiC MMC.

un[am.np, m

Room-temperature evolution of the elastic limit after various temperature histories, 2024 A1/SiC MMC.

Chapter 1 UnifiedCyclic Viscoplastic Constitutive Equations

61

(1995). Points correspond to experimental measures, and lines to the predictions by the model. Moreover, as shown by Fig. 24, aging at room temperature after high-temperature incursions induces a progressive increase of the yield limit, with a tendency to recover (at least partially) the initial value. These two figures illustrate the capability of the model to describe aging effects and their correlation with prior high-temperature incursions.

CONCLUSION In this chapter, we have considered many aspects related to the thermoelastic viscoplastic constitutive equations. The discussion is mainly based on the macroscopic constitutive models that we have experienced for a long time at ONERA, for applications in several domains, in the framework of life prediction methods for components working at high temperature, such as gas turbine components (aeroengines) and nuclear plants. Although the general content of the chapter is supported by previously published models and experimental/theoretical comparisons, some parts have been specially developed. These are outlined in the brief summary below. The chapter was divided into three parts:

(1) The constitutive model, which was discussed first, is mainly concerned with cyclic loading conditions (room and high temperature) under small strains. In this part, we considered constant temperature conditions. Among the several aspects considered, we mention more specifically the following: a. The general anisotropic case (Section II.A). This has not been published elsewhere in this form, although some concepts were already used for constitutive models of turbine blade single crystals (Nouailhas, 1990b). Its particularization to the initially isotropic material corresponds to our classical model. b. The limiting case of rate-independent plasticity (Section II.B). This is also classical (Chaboche, 1989a), but some effort was made here to extract solutions for the limiting case of stationary creep (Section II.C). Contrary to what is done in some recent models (Freed and Walker, 1993), we did not impose the form of the creep law in the constitutive equations, but we demonstrate that they degenerate to the classical power creep law at steady state, at least for low stresses. c. The determination procedure. This discussion, which takes advantage of the rate-independent limiting case and of the closed-form solutions for uniaxial lodings, was already published (Chaboche, 1989a). However, we discussed here the respective advantages/disadvantages of the twosurface models compared to the present structure of equations.

62

J.L. Chaboche

(2) The capabilities of the constitutive models were examined in Section III, on the basis of several examples on several materials (IN 100 turbine blade superalloy, INCO 718 disk alloy, and 316 and 304 stainless steels for nuclear components). The following phenomenological facts were considered successively: the normal viscosity coupled with kinematic hardening, cyclic hardening or softening, domain of the high strain rates, modeling of static recovery effects, viscous ratcheting and cyclic ratcheting, and creep-plasticity interactions. All these results were published previously. (3) The thermodynamics with internal variables was used in order to introduce some thermodynamic constraints on the form of the constitutive equations and, more significantly, to justify and check the modeling capabilities under varying temperature conditions. The following aspects have been explored. a. The general framework was recalled, based on the introduction of two potentials (Section IV.A). Among several classical developments, we insisted once again on the nonadmissibility of stress rate terms in the evolution equations for kinematic hardening (points recently discussed by Freed et al., 1991; Chaboche, 1993; Malmberg, 1993). b. In Section IV.B, we summarized several methods used in the past in order to incorporate the mechanical constitutive models into the thermodynamic framework. Most of them are equivalent in terms of energies dissipated. The differences are mainly conceptual and related to the use of a generalized normality assumption. c. Stored and heat dissipated energies were evaluated in Section IV.C, based on studies done recently (Chaboche, 1993). It is shown that the constitutive models and the way they are incorporated into the thermodynamic framework are able to predict correctly the stored and heat dissipated energies on several materials. This was a global check of the modeling capabilities, important for applications (high rates) where the complete thermomechanical coupling has to be taken into account. d. The use of constitutive equations under varying temperatures was then discussed, addressing both the temperature dependency of the viscosity function and the need for a temperature rate term (with no degree of freedom) in the rate equations for the back stresses (also discussed recently by Wang and Ohno, 1991; McDowell, 1992; Chaboche, 1993). The problem of aging modeling and of its thermodynamic admissibility was also addressed. Few attempts have been made in the past to model such aging effects with thermodynamic considerations. Here, we propose a new way of modeling, based on a generalization of the model introduced by Marquis (1989). Several points have not been addressed in the present chapter. For example, multiaxial loading effects (nonproportional overhardening or multiaxial ratcheting), distortions of the yield surface (corner effects), and models with discrete memory

Chapter 1 Unified Cyclic Viscoplastic Constitutive Equations

63

surfaces. They are still subjects for very interesting and important researches. Other aspects that remain to be developed are the relations with microstructural facts (the physics side) or the models based on micromechanical approaches, where the state variables are written (and their evolution laws) at some microstructural level (grains, subgrains, slip systems, precipates, etc.). This is clearly a challenge for future researches.

REFERENCES Armstrong, R J., and Frederick, C. O. (1966). "A Mathematical Representation of the Multiaxial Bauschinger Effect," CEGB Rep. RD/B/N731. Central Electricity Generating Board., Berkeley, UK. Beghi, M. G., Bottani, C. E., and Caglioti, G. (1986). Irreversible thermodynamics of metals under stress. Res. Mech. 19, 365-379. Benallal, A. (1989). Thermoviscoplasticit6 et endommagement des structures. These Doctorat d'Etat, Universit6 Paris VI. Benallal, A., and Ben Cheikh, A. (1987). Constitutive equations for anisothermal elasto-viscoplasticity. In "Constitutive Laws for Engineering Materials: Theory and Applications" (C. S. Desai, E. Krempl, E D. Kiousis, and T. Kundu, eds.). Elsevier, New York. Benallal, A., and Marquis, D. (1987). Constitutive equations for non-proportional cyclic elastoviscoplasticity. J. Eng. Mater. Technol. 109, 326-336. Bever, M. B., Holt, D., and Titchener, A.L. (1973). The stored energy of cold work. In "Progress in Metal Science" (B. Chalmers et al., ed.), vol. 17. Pergamon Press, Oxford. Bingham, E. C., and Green, H. (1919). Paint, a plastic material not a viscous liquid; the measurement of its mobility and yield value. Proc. Am. Soc. Test. Mater. 19, 640-664. Bodner, S. R. (1987). Review of a unified elastic-viscoplastic theory. In "Unified Constitutive Equations for Creep and Plasticity" (A. K. Miller, ed.), pp. 273-301. Elsevier, London. Bodner, S. R., and Partom, Y. (1975). Constitutive equations for elastic viscoplastic strain-hardening materials. J. Appl. Mech. 42, 385-389. Bruhns, O. T. (1982). New constitutive equations to describe infinitesimal elasto-plastic deformation. Am. Soc. Mech. Eng., Pressure Vessels Piping Div. [Publ.] PVP-PB 82-PVP-71, p. 1. Burlet, H., and Cailletaud, G. (1987). Modeling of cyclic plasticity in finite element codes. In "Constitutive Laws for Engineering Materials: Theory and Applications" (C. S. Desai, E. Krempl, E D. Kiousis, and T. Kundu, eds.), pp. 1157-1164. Elsevier, New York. Cailletaud, G. (1979). Mod61isation m6canique d'instabilit6 microstructurale en viscoplasticit6 cyclique ~ tempdrature variable. These Docteur Ingdnieur, Universit6 Paris VI (N. T. ONERA No. 1979-6). Cailletaud, G., Kaczmarek, H., and Policella, H. (1984). Some elements on multiaxial behavior of 316L stainless steel at room temperature. Mech. Mater. 3, 333-347. Chaboche, J. L. (1977). Viscoplastic constitutive equations for the description of cyclic and anisotropic behavior of metals. Bull. Acad. Pol. Sci. Sdr. Sci. Tech. 25, 33. Chaboche, J. L. (1978). Description thermodynamique et phdnomdnologique de la viscoplasticit6 cyclique avec endommagement. These d'Etat, Universit6 Paris VI. Chaboche, J. L. (1983). On the constitutive equations of materials under monotonic or cyclic loadings. Rech. Adrosp. 1983-5, 31-43. Chaboche, J. L. (1989a). Constitutive equations for cyclic plasticity and cyclic viscoplasticity. Int. J. Plast. 5, 247-302.

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Chaboche, J. L. (1989b). A new constitutive framework to describe limited ratcheting effects. Plasticity 89, Tsu, Japan. Khan and Tokuda eds. Chaboche, J. L. (1989c). Un nouveau sch6ma d'6crouissage cin6matique ~ surfaces m6moires discr~tes. Rech. A~rosp. 4, 49-69 (French and English eds.). Chaboche, J. L. (1991). On some modifications of kinematic hardening to improve the description of ratcheting effects. Int. J. Plast. 7, 661-678. Chaboche, J. L., Nouailhas, D., Savalle, S. (1991). AGICE: Logiciel pour l'identification interactive graphique des lois de comportement. Rech. A~rosp. 3, 59-76 (French and English eds.). Chaboche, J. L. (1993). Cyclic viscoplastic constitutive equations. Parts I and II. J. Appl. Mech. 60, 813-828. Chaboche, J. L. (1994). Modeling of ratcheting: Evaluation of various approaches. Eur. J. Mech., 13(4), 501-518. Chaboche, J. L., and Nouailhas, D. (1989a). Constitutive modeling of ratcheting effects. J. Eng. Mater. Technol. 111,384-392, 409-416. Chaboche, J. L., and Nouailhas, D. (1989b). A unified constitutive model for cyclic viscoplasticity and its application to various stainless steels. J. Eng. Mater. Technol. 111,424--430. Chaboche, J. L., and Rousselier, G. (1983). On the plastic and viscoplastic constitutive equations. J. Press. Vessel Techol. 105, 153-164. Chaboche, J. L., Dang Van, K., and Cordier, G. (1979). "Modelization of the Strain Memory Effect on the Cyclic Hardening of 316 Stainless Steel," SMIRT-5, Div. L. Berlin. Chaboche, J. L., Nouailhas, D., Pacou, D., and Paulmier, P. (1991). Modeling of the cyclic response and ratcheting effects on Incone1718 alloy. Eur. J. Mech., A/Solids 10(1), 101-121. Chaboche, J. L., E1 Mayas, N., and Paulmier, P. (1995). Mod61isation thermodynamique des ph6nom~nes de viscoplasticit6, restauration et vieillissement, C. R. Acad. Sci. Paris, t. 320, Ser. 2, 9-16. Chan, D. S. Lindholm, U. S., Bodner, S. R., and Walker, K. P. (1984). A survey of unified constitutive theories. 2d Symp. NASA-LEWIS Non-Linear Constitutive Relat. High Temp. Appl., Cleveland, OH, NASA Conf. Publ. 2369, 1985. Chrysochoos, A. (1985). Bilan 6nerg6tique en 61astoplasticit6, grandes deformations. J. M~c. Th~or. Appl. 4, No. 5. Chrysochoos, A. (1987). Dissipation et blocage d'6nergie lors d'un 6crouissage en traction simple. Thbse de Doctorat d'Etat, Universit6 de Montpellier. Coleman, B. D., and Gurtin, M. E. (1967). Thermodynamics with internal state variables. J. Chem. Phys. 47, 597-613. Contesti, E., and Cailletaud, G. (1989). Description of creep plasticity interaction with non unified constitutive equations, application to an austenitic stainless steel. Nucl. Eng. Des. 116, 265-280. Cristescu, N., and Suliciu, I. (1982). Viscoplasticity. Martinus Nijhoff, The Hague, The Netherlands. Dafalias, Y. E, and Popov, E. P. (1975). A model of nonlinearly hardening materials for complex loading. Acta Mech. 21, 173-192. Delobelle, P. (1988). Sur les lois de comportement viscoplastique h variables internes. Rev. Phys. Appl. 23, 1-61. E1 Mayas, N. (1994). Mod61isation microscopique et macroscopique du comportement d' un composite matrice mEtallique. Th6se de Doctorat, Universit6 Paris VI. Freed, A. D., and Verrilli, M. J. (1988). A viscoplastic theory applied to Copper. In "The Inelastic Behavior of Solids: Models and Utilization" (G. Cailletaud et al., eds.), Proc. MECAMAT, pp. 27-39, Besanqon, France. Freed, A. D., and Walker, K. P. (1990). Model development in viscoplastic ratcheting. NASA Tech. Memo. NASA TM-102509. Freed, A. D., and Walker, K. P. (1993). Viscoplasticity with creep and plasticity bounds. Int. J. Plast. 9, 213-242.

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Unified Cyclic Viscoplastic Constitutive Equations

65

Freed, A. D., Chaboche, J. L., and Walker, K. P. (1991). A viscoplastic theory with thermodynamic considerations. Acta Mech. 90, 155-174. Geary, J. A., and Onat, E. T. (1974). Representation of non-linear hereditary mechanical behavior. Oak Ridge Natl. Lab. [Rep.] ORNL-TM (U.S.) ORNL-TM-4525. Germain, E (1973). "Cours de m6canique des milieux continus," Tome I. Masson, Paris. Germain, E, Nguyen, Q. S., and Suquet, E (1983). Continuum thermodynamics. J. Appl. Mech. 50, 1010-1020. Golinval, J. C. (1989). Calcul par 616ments finis de structures 61asto-viscoplastiques soumises h des chargements cycliques ~ haute temp6rature. Th6se de Doctorat, Universit6 de Li6ge. Goodall, I. W., Hale, R., and Walters, D. J. (1980). "On Constitutive Relations and Failure Criteria of an Austenitic Steel under Cyclic Loading at Elevated Temperature," IUTAM Symp. "Creep in Structures," Springer-Verlag, Leicester. Goodman, A. M. (1983). Development of constitutive equations for computer analysis of stainless steel components, 4th. Int. Seminar on Inelastic Analysis and Life Prediction in High Temperature Environment, Chicago. Halford, G. R. (1966). Stored energy of cold work changes induced by cyclic deformation. Ph.D. Thesis, University of Illinois, Urbana. Halpen, B., and Nguyen, Q. S. (1975). Sur les mat6riaux standards g6n6ralis6s. J. M~c. 14(1), 39-63. Hansen, N. R., and Schreyer, H. L. (1992). Thermodynamically consistent theories for elastoplasticity coupled with damage. In "Damage Mechanics and Localization," MD-Vol. 34/AMD-Vol. 142. ASME, New York. Hart, E. W. (1976). Constitutive relations for the nonelastic deformation of metals. J. Eng. Mater. Technol. 98, 193-202. Hartmann, G. (1990). Comparison of the uniaxial behavior of the inelastic constitutive models of Miller and Walker by numerical experiments. Int. J. Plast. 6, 189-206. Henshall, G. A., and Miller, A. K. (1990). Simplifications and improvements in unified constitutive equations for creep and plasticity. Part 1. Equations development. Acta Metall. Mater. 38, 2101-2115. Hohenemser, K., and von Prager, W. (1932). Uber die Ansfitze der Mechanik isotroper Kontinua. Z. Angew. Math. Mech. 12, 216-226. Kawai, R. D., and Ohashi, Y. (1987). Coupled effect between creep and plasticity of type 316 stainless steel at elevated temperature. In "Constitutive Laws for Engineering Materials: Theory and Applications" (C. S. Desai, E. Krempl, E D. Kiousis, and T. Kundu, eds.). Elsevier, New York. pp. 967-974. Kocks, U. F. (1976). Laws for work-hardening and low-temperature creep. J. Eng. Mater. Technol. 98, 76-85. Krempl, E., and Lu, H. (1984). The hardening and rate-dependent behavior of fully annealed AISI Type 304 stainless steel under biaxial in-phase and out-of-phase strain cycling at room temperature. J. Eng. Mater. Technol. 106, 376-382. Krempl, E., and Yao, D. (1987). The viscoplasticity theory based on overstress applied to ratcheting and cyclic hardening. In "Low-Cycle Fatigue and Elastoplastic Behavior of Materials" (K. T. Rie, ed.), pp. 35-48. Elsevier, London. Krempl, E., McMahon, J. J., and Yao, D. (1986). Viscoplasticity based on overstress with a differential growth law for the equilibrium stress. Mech. Mater. 5, 35-48. Krieg, R. D. (1975). A practical two surface plasticity theory. J. Appl. Mech. 42, 641-646. Krieg, R. D., Swearengen, J. CI, and Rohde, R. W. (1978). A physically-based internal variable model for rate-dependent plasticity. In "Inelastic Behavior of Pressure Vessel and Piping Components" (T. Y. Chang and E. Krempl, eds.), PVP-PB-028, pp. 15-28. ASME, New York. Ladeveze, P. (1992) "New advances in the large time increment method. In "New Advances in Com-

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putational Structural Mechanics" (E Ladeveze and O. C. Zienkiewicz, eds.), pp. 3-22. Elsevier, Amsterdam. Lamba, H. S., and Siddebottom, O. M. (1978). Cyclic plasticity for non-proportional paths. J. Eng. Mater. Technol. 100, 96-111. Lema~tre, J., and Chaboche, J. L. (1985). "M6canique des mat6riaux solides." Dunod, Bordas, Paris; English edition, Cambridge Univ. Press, Cambridge, UK. Lema~tre, J., and Marquis, D. (1992). Modeling complex behavior of metals by the state-kinetic coupling theory. J. Eng. Mater. Technol. 114, 250-254. Lowe, T. C., and Miller, A. K. (1984). Improved constitutive equations for modeling strain softening. Parts I and II. J. Eng. Mater. Technol. 106(4), 337-348. Lubliner, J. (1973). On the structure of the rate equations of materials with internal variables. Acta Mech. 17, 109-119. Malmberg, T. (1990a). "Thermodynamics of a Visco-Plastic Model with Internal Variables," KfK Rep. No. 4572. Kernforschungszentrum, Karlsruhe. Malmberg, T. (1990b). Thermodynamics of some viscoplastic material models with internal variables. Proc. Int. Conf. Mech., Phys. Struct. Mater., Thessaloniki. Malmberg, T. (1993). "Thermodynamic Consistency of Viscoplastic Material Models Involving External Variable Rates in the Evolution Equations for the Internal Variables," Rep. No. KfK 5193. Kernforschungszentrum, Karlsruhe, Karlsruhe, Germany. Malvern, L. E. ( 1951). The propagation of longitudinal waves of plastic deformation in a bar of material exhibiting a strain-rate effect. J. Appl. Mech. 18, 203-208. Marquis, D. (1979). Etude th6orique et v6rification exp6rimentale d'un mod61e de plasticit6 cyclique. Th~se, Universit6 Paris VI. Marquis, D. (1989). Ph6nom6nologie et thermodynamique: Couplage entre thermo61asticit6, plasticit6, vieillissement et endommagement. Th~se d' Etat, Universit6 Paris VI. McDowell, D. L. (1983). On the path dependence of transient hardening and softening to stable states under complex biaxial cyclic loading. In "Constitutive Laws for Engineering Materials: Theory and Application" (C. S. Desai and R. H. Gallagher, eds.), pp. 125-132. Elsevier, New York. McDowell, D. L. (1985). A two surface model for transient nonproportional cyclic plasticity. J. Appl. Mech. 52, 298-308. McDowell, D. L. (1987). An evolution of recent developments in hardening and flow rules for rateindependent, nonproportional cyclic plasticity. J. Appl. Mech. 54, 323-334. McDowell, D. L. (1992). A nonlinear kinematic hardening theory for cyclic thermoplasticity and thermoviscoplasticity. Int. J. Plast. 8, 695-728. Miller, A. (1976). An inelastic constitutive model for monotonie cyclic and creep deformation. J. Eng. Mater Technol. 98(2), 97-105, 106-113. Miller, A. K., ed. (1987). "Unified Constitutive Equations for Plastic Deformation and Creep of Engineering Alloys." Elsevier Applied Science, New York. Moosbrugger, J. C., and McDowell, D. L. (1989). On a class of kinematic hardening rules for nonproportional cyclic plasticity. J. Eng. Mater. Technol. 111, 87-98. Moosbrugger, J. C., and McDowell, D. L. (1990). A rate-dependent bounding surface model with a generalized image point for nonproportional cyclic plasticity. J. Mech. Phys. Solids 38, 627. Mroz, Z. (1967). On the description of the work-hardening. J. Mech. Phys. Solids 15, 163. Mroz, Z. (1981 ). On generalized kinematic hardening rule with memory of maximal prestress. J. Mech. Appl. 5, 241-260. Nouailhas, D. (1987). A viscoplastic modeling applied to stainless steel behavior. In "Constitutive Laws for Engineering Materials: Theory and Applications" (C. S. Desai, E. Krempl, P. D. Kiousis, and T. Kundu, eds.), Elsevier, New York. Nouailhas, D. (1989). Unified modeling of cyclic viscoplasticity: Application to austenitic stainless steels. Int. J. Plast. 5, 501-520. J

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Unified Cyclic Viscoplastic Constitutive Equations

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Nouailhas, D. (1990a). Un modble de viscoplasticit6 cyclique pour mat6riaux anisotropes h sym6trie cubique. C. R. Acad. Sci., t. 310, Ser. 2, 887-890. Nouailhas, D. (1990b). Prediction of ratcheting and creep effects at room temperature on 304 stainless steel. Res. Mech. 30, 223-242. Nouailhas, D., Policella, H., and Kaczmarek, H. (1983). On the description or cyclic hardening under complex loading histories. In "Constitutive Laws for Engineering Materials. Theory and Applications" (C. S. Desai and R. H. Gallagher, eds.). Elsevier, New York. Nouailhas, D., Chaboche, J. L., Savalle, S., and Cailletaud, G. (1985). On the constitutive equations for cyclic plasticity under nonproportional loading. Int. J. Plast. 1, 317-330. Odqvist, E K. G. (1953). Influence of primary creep on stresses in structural parts. Trans. R. Inst. Technol. Stockholm 66, 1-17. Ohashi, Y., Tanaka, E., and Ooka, M. (1985). Plastic deformation of behavior of type 316 stainless steel subjected to out-of-phase strain cycles. J. Eng. Mater. Technol. 107, 286-292. Ohno, N., (1982). A constitutive model of cyclic plasticity with a nonhardening strain region. J. Appl. Mech. 49, 721-727. Ohno, N. (1990). Recent topics in constitutive modeling for cyclic plasticity and viscoplasticity. Appl. Mech. Rev. 43 (11), 283-295. Ohno, N., and Kachi, Y. (1986). A constitutive model of cyclic plasticity for nonlinear hardening materials, J. Appl. Mech. 53, 395-403. Ohno, N., and Wang, J. D., (1992). Transformation of a nonlinear kinematic hardening rule to a multisurface form under isothermal and nonisothermal conditions. Int. J. Plast. 7, 879-891. Ohno, N., and Wang, J. D. (1993). Kinematic hardening rules with critical state of dynamic recovery. Parts I and II. Int. J. Plast. 9, 375-403. Ohno, N., Takahashi, Y., and Kuwabara, K., (1989). Constitutive modeling of anisothermal cyclic plasticity of 304 stainless steel. J. Eng. Mater. Technol. 111, 106. Ohno, N., Kawabata, M., and Naganuma, J. (1990). Aging effects on monotonic, stress-paused, and alternating creep of type 304 stainless steel. Int. J. Plast. 6, 315-327. Oldroyd, J. G. (1947). A rational formulation of the equations of plastic flow for a Bingham Solid. Proc. Cambridge Phil. Soc. 43, 100-105. Perzyna, E (1964). On the constitutive equations for work-hardening and rate sensitive plastic materials. Bull. Acad. Pol. Sci., S~r. Sci. Technol. 12(4), 199-206 Ponter, A. R. S., and Leckie, E A. (1976). Constitutive relations for the time-dependent deformation of metals, J. Eng. Mater. Technol. 98, 47-51. Prager, W. (1961). Linearization of visco-plasticity. Oesterr. Ing.-Arch. 15, 152-157. Ramaswamy, V. G., Stouffer, D. C., and Laflen, J. H. (1990). A unified constitutive model for the inelastic response of Ren6 80 at temperatures between 538 C and 982 C. J. Eng. Mater. Technol. 112, 280-286. Rice, J. R. (1971). Inelastic constitutive relations for solids: An internal-variable theory and its application to metal plasticity. J. Mech. Phys. Solids 19, 433--455. Robinson, D. N. (1978). A unified creep-plasticity model for structural metals at high temperature. Oak Ridge Natl. Lab. [Rep.] ORNL-TM (U.S.) ORNL-TM-5969. Robinson, D. N. (1983). Constitutive relationships for anisotropic high-temperature alloys. NASA Tech. Memo. NASA TM-83437. Ruggles, M., and Krempl, E. (1987). "Rate Dependence of Ratcheting of AISI Type 304 Stainless Steel at Room Temperature," Rep. MML 87-41. Rensselaer Polytechnic Inst., Troy, NY. Schmidt, C. G., and Miller, A. K. (1981). A unified phenomenological model for non-elastic deformation of type 316 stainless steel. Parts I and II. Res. Mech. 3, 109-129, 175-193. Stowell, E. Z. (1957). A phenomenological relation between stress, strain rate, and temperature for metals at elevated temperatures. Natl. Advis. Comm. Aeronaut., Tech. Notes NACA TN-4000. Tanaka, E., Murakami, S., and Ooka, M. (1987). Effects of strain path shapes on nonproportional cyclic plasticity. J. Mech. Phys. Solids 33, 559-575.

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D islocation-Density-Related Constitutive Modeling Yuri Estrin

Department of Mechanical and Materials Engineering

The University of Western Australia Nedlands WA 6907, Australia

~11

INTRODUCTION

Over a decade or so, the author has been involved with the development of unified constitutive models in which the dislocation density plays the role of an internal variable representing the microstructural state of a material. This approach to modeling the viscoplastic behavior of crystalline materials, which goes back to Kocks and Mecking (Kocks, 1976; Mecking and Kocks, 1981), has proved able to provide an excellent description of the mechanical response of metallic materials to unidirectional loading and, beyond that, to ensure a very good predictive capability of the constitutive equations. Recent developments have turned the model into a versatile tool for describing and predicting the mechanical behavior of a broad range of metals and alloys. While the skeleton structure of the prototype model (Kocks, 1976; Mecking and Kocks, 1981) was retained, new "modules" accounting for various metallurgical features or different deformation conditions have been included. Alongside these developments, new techniques for estimation of the model parameters have evolved. In its present state, dislocationdensity-related constitutive modelingmwhile continuing to benefit from the simplicity of microstructure-related model architecture and a relatively small number of adjustable parameters--is considered mature enough to be broadly used by engineers and designers in finite element codes including viscoplasticity. The aim of this overview is to present the current state of this type of constitutive modeling, Unified Constitutive Laws of Plastic Deformation Copyright (~) 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

69

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with all its variants and ramifications, and to provide examples of its experimental validation. A general frame for the model is given as follows. The components of the tensor of total strain rate Eij are given by the sum of the elastic and plastic components, kej and k/~, respectively" (1)

Eij -- Eij "+" ~p"

The elastic part obeys Hooke's law and reads 9e = Cijkl~Ykl, Eij

Cijkl = 1 +E

U ~(~ikC~jl

!)

I+V

~ij~kl),

(2)

where oij is the stress tensor and a dot denotes differentiation with respect to time t; E, v, and ~ij are Young's modulus, Poisson's ratio, and Kronecker's delta,

respectively. The plastic part of the strain rate tensor is expressed in the form of the L6vy-von Mises equation 3 kp EP ~- -2 O" Sij

(3)

Here sij is the deviatoric stress; the quantities E P ~ V/~EP~"p

(4)

CY -- W/3 -~ Sij Sij

(5)

and

denote the equivalent von Mises plastic strain rate and stress, respectively. This is a fairly general formulation (cf., e.g., Bodner, 1987; Estrin and Mecking, 1985). The use of the von Mises equivalent quantities implies plastic isotropy of the material. The specifics of the model enter through a particular form of the equation relating the equivalent plastic strain rate and the equivalent stress; this form is referred to as the kinetic equation (Kocks, 1976, 1987; Mecking and Kocks, 1981; Estrin and Mecking, 1984). The problem of constitutive modeling is thus reduced to operating with scalar, rather than tensorial quantities. In choosing a suitable form of the kinetic equation, one is then guided by the experimental data (or their model description) obtained in uniaxial tests. A suitable mathematical form for the kinetic equation suggested by Kocks (1976) is a power-law: EP -- Eo (O'l~ ) m.

(6)

Chapter 2 Dislocation-Density-RelatedConstitutive Modeling

71

Here 6 is an internal variable representing the state of the material; the quantities ko and m are material parameters. Equation (6) is just a convenient representation of the Arrhenius equation for thermally activated plastic flow by dislocation glide. Accordingly, the temperature dependence of the plastic strain rate is contained in the stress exponent m, which can be shown (Kocks et al., 1975) to equal Vcr/kB T, where V is the activation volume for the underlying thermally activated dislocation mechanism, kB is the Boltzmann constant, and T is the absolute temperature. In a first approach, the factor ko, which is proportional to the density of mobile dislocations, is considered constant (Kocks, 1976, 1987; Mecking and Kocks, 1981; Estrin and Mecking, 1984). This assumption will be relaxed in a modified version of the model. Another variant of the kinetic equation in which the Arrhenius form is preserved (Kocks et al., 1975), k p - - koo

exp - kB T

6-

'

was used by Follansbee and Kocks (1988). Here AGo is the value of the Gibbs free energy of activation at zero stress and the exponents p and q are fit parameters providing the k p vs. cr curve the required shape; as in the previous variant, the pre-exponential factor koo can be considered constant. An obvious deficiency of this formula is that in the limit of a --+ 0 it yields a finite plastic strain rate. In what follows we shall be using the kinetic equation in the form of Eq. (6). It should be noted that the equations thus formulated imply that a definite nonzero plastic strain rate corresponds to any value of stress, however small it may be. That is to say, we are dealing here with a model with no yielding and loading/unloading condition. A number of constitutive models (Kocks, 1976; Mecking and Kocks, 1981; Estrin and Mecking, 1984; Bodner, 1987; Follansbee and Kocks, 1988; Miller, 1987) share this property. The kinetic equation refers to a fixed microstructure, i.e., to a constant value of the internal variable 6. However, as the microstructure varies in the process of plastic deformation, a separate equation is needed to describe the evolution of 6. It can be written in a general form (Kocks, 1976) d6 = f ( 6 ; kp, T). deP

(8)

This equation suggests that the rate of change of the internal variable is determined by its current value, i.e., that no memory- or path-dependent effects are included. A provision for temperature and rate sensitivity was made. Once the concrete form of the function f is specified, the constitutive formulation is complete. A salient feature of the model just outlined is a strict separation of the dislocationglide-controlled kinetics of plastic flow for a certain fixed structural state from

72

YuriEstrin

the kinetics of structure evolution. This implies that in the overall plastic strain rate the contribution of the dislocation processes responsible for structure evolution (e.g., recovery processes) can be neglected as compared to that of dislocation glide. A suitable correction in the form of an additive term in Eq. (6) can be made once the recovery mechanism is specified. In what follows, this correction will be disregarded for the sake of simplicity. The above description implies that the quantity 8 is the sole internal state variable to represent the microstructural state of a material (Kocks, 1976, 1987; Estrin and Mecking, 1984). However, it can generally be expected that several internal structure variables, characterized by different rates of relaxation toward their steady-state values, are needed to describe the mechanical response of a material. One can consider the steady-state value of a particular internal variable as a dynamic one in that it is governed by the current values of slower internal variables still evolving toward their steady state (Estrin and Mecking, 1984). As deformation proceeds, more and more internal variables will reach steady state so that eventually, at large strains, the slowest-evolving among the internal variables will have "enslaved" the rest of them. In the above formulation, the total dislocation density was considered to be that slowest internal variable. The examples of modeling plastic deformation given below demonstrate that for continuous monotonic straining, a one-internal-variable model is sufficient. More sophisticated models with two and more internal variables (Estrin, 1991 a; Estrin et al., 1995) can be invoked if rapid changes in deformation conditions or cyclic loading are to be accounted for. In the following, we shall discuss different variants of the model, adding new modules if needed for particular applications, but keeping its unified general frame.

ONE-INTERNAL-VARIABLEMODEL A. General Framework The Kocks-Mecking model (Kocks, 1976; Mecking and Kocks, 1981) can be considered as a "protomodel" that has pointed the direction for later developments to be discussed in this chapter. This model, whose elements can be found in more recent theories, is based on the kinetic equation in the form of Eq. (6), where the internal variable 8 is related to the total dislocation density p" 6 = MotGb~.

(9)

Here G is the shear modulus, b is the magnitude of the dislocation Burgers vector, M is the average Taylor factor, and c~ is a numerical constant. The model implies that the strength of the material is determined by dislocation-dislocation interactions. All other sources of resistance to dislocation glide have been disregarded so far. Arguments justifying this choice of the internal variable are given by Kocks

Chapter 2

Dislocation-Density-Related Constitutive Modeling

73

(1976, 1987). Various models of dislocation glide resistance in an ensemble of dislocations of average density p lead to Eq. (9); and it is, in fact, obvious from dimensional considerations that the glide resistance must be given by G b / A , where A is the characteristic obstacle spacing in the glide plane. This quantity is proportional to the average dislocation spacing which scales with the inverse of the square root of the dislocation densitymthe only quantity with the dimension of length in the system under consideration. In a plastically deformed material, where dislocations are organized in a cell wall or subgrain boundary structure, all the relevant lengths, such as the dislocation spacing and hence the size of dislocation sources in the boundaries, will still scale with 1/~/-fi, and so will A (Kuhlmann-Wilsdorf, 1985). Equation (9) can thus be assumed to have general validity (Kocks et al., 1975). The Taylor factor M -- cr/r - - y P / g P , relating the shear stress r and the shear strain yP to the axial stress ~r and plastic strain e p, is used to account for polycrystallinity of the material (Kocks, 1976). The dislocation density will evolve in the process of plastic deformation to the backdrop of M varying as a result of texture evolution. Models of texture development can be used to account for this variation of M. However, in the current article the variation of M, which is slow compared to that of the dislocation density, will not be considered. The evolution equation for p can be derived following Kocks (1976). Two concurrent effects determine the variation of p: storage and recovery. The storage term can be written by expressing a shear strain increment, dsP/M, in terms of the dislocation density increment dp associated with immobilization of mobile dislocations at impenetrable obstacles after they have traveled a distance L, the mean free path. M d e p : b L dp

(10)

dp/ds p = M/bL.

(11)

or

To include dynamic recovery, one can consider the annihilation of stored dislocations--a process that involves their leaving the glide planes on which they are stored at impenetrable obstacles. This is furnished by cross-slip of screw dislocations or climb of edge dislocations. The first process is prevalent at low temperatures and the second at high temperatures, the boundary between the two temperature ranges lying somewhere between one-half and two-thirds of the melting temperature. Dislocation annihilation reactions are certainly of binary type, so the dynamic recovery rate (in terms of time derivative) is quadratic in dislocation density, while in terms of a derivative with respect to strain it is linear in p. Inclusion of a dynamic recovery term turns Eq. (11) into dp/ds p = M

(1

-~

--

k2D

)

,

(12)

74

YuriEstrin

where the recovery coefficient ke - k20

(k~_~o)--1/n

(13)

is dependent on strain rate and temperature. Here, k20 is a constant, while the temperature dependence is contained either in n (low-temperature range, in which o (high-temperature case n is inversely proportional to T, while e9o* is constant) or in e"* range, in which case kS is given by an Arrhenius equation with the activation energy for dislocation climb equal to that for self-diffusion, while n is a constant, typically about 3-5) (Kocks, 1976, 1987; Estrin and Mecking, 1984; Klepaczko and Chiem, 1986). It should be noted that in the low-temperature range the parameter n is related to the stacking fault e n e r g y ~ a quantity strongly influencing the cross-slip probability. A systematic investigation of this relation for face-centered cubic (fcc) metals has been carried out by Mecking et al. (1986). The magnitude of n in the low-temperature range is fairly large (n >> 1), but typically n is smaller than m, the exponent in the kinetic equation (Kocks, 1976; Estrin and Mecking, 1984; Klepaczko and Chiem, 1986). The recovery processes included in the above equations are of dynamic origin, i.e., they take place only during the deformation. Static recovery of the BaileyOrowan type, in which a decrement of the dislocation density is proportional to the corresponding time (rather than strain) increment, can also be included, e.g., in the form dp/de p-

M

-~ -kzp

- k---p"

A reasonable phenomenological ansatz for the static recovery coefficient r would be, for instance, r--r0

exp

~

sinh

kBT

'

implying that static recovery is driven by the stress determined by the square root of the current dislocation density. The parameters U0, fl, and r0 can be considered constant. While the static recovery term has been included here for generality, it will not be retained in the further discussion, as for most of the applications considered below, the results are consistent with the assumption that dynamic recovery is prevalent.

B. Coarse-Grained Single-Phase Material The form of the evolution equation for the dislocation density provides a possibility of incorporating in the model metallurgical characteristics and microstructural

Chapter 2

Dislocation-Density-Related Constitutive Modeling

75

features of a material. We begin with an idealized "structureless" matrix material that is coarse-grained and single-phase. The only kind of impenetrable obstacles to moving dislocations will be those related to the dislocation structure itself. As mentioned above, regardless of how the dislocations are arranged--be they completely random or organized in a cell or subgrain boundary structure--the mean free path L will be proportional to 1/~/-~. It is assumed, of course, that L is much smaller than the grain size of the material. Equation (12) then assumes the form dp/d~ p = M(kl~/7-

(16)

kzp)

used by Kocks (1976). Here kl is a constant and k2 is given by Eq. (13). The constitutive model comprising Eqs. (6), (9), (13), and (16) is so simple that it can be integrated analytically, at least for the case of uniaxial deformation with constant plastic strain rate (Kocks, 1976; Mecking, and Kocks, 1981) and for constant stress creep (Estrin and Mecking, 1984). For constant strain rate, the Palm-Voce equation (Palm, 1948; Voce, 1948), found empirically to describe the strain hardening behavior for a range of metallic materials, is recovered: o- - os = exp

.

O'i -- O'S

(17)

8tr

Here the subscript i denotes some initial point on the stress-strain curve, as is a saturation stress given by

O"S molGb( kl ) (8.~o) l/m(8.~--~o ) 1In --

- -

(~p) l/m(~p) l/n

= ~ * -

k20

--

--

8o

80

(18)

and 8tr is a characteristic transient strain, a parameter indicating the rate at which the stress a approaches its saturation value as. The transient strain is given by the expression

8tr

=

~ ;

(~)II

t~II = - M 2 o e G b k l 2

~

ko

.

(19)

The quantity (~II denotes the stage II hardening rate. Indeed, differentiation of the stress given by Eq. (17) with respect to plastic strain yields for the strain hardening coefficient | = (O~/OeP)+p (~) - - (~)II

1 - -o's

.

(20)

It is recognized that | is the limit value of the strain hardening rate for ~ --+ 0. For fcc materials, it can be estimated to be of the order of ~ - ~ of the shear modulus G, provided the material is not textured and M equals about 3 (Kocks, 1976; Estrin and Mecking, 1988). Typically, m >> 1, so a weak strain rate and

76

Yuri Estrin

O~ 0ii

r

0

Os

0

The | vs. or diagram following from the Kocks-Mecking model (schematic).

temperature dependence contained in the factor (k p/ko) 1/m may be disregarded. In the | diagram, the range of stage II hardening degenerates to a point (Fig. 1), while stage III is represented by a linear decrease of | with stress due to dynamic recovery. The saturation stress as bears a dependence on recovery characteristics. If, for simplicity, one does not distinguish between the scaling quantities ko and ko, which is particularly legitimate in the low-temperature range where both 1/n and 1/m are small compared to unity, Eq. (18) can be simplified to k p ) l/p as -- or* k--o

(21)

where the quantity or* has been defined in Eq. (18) and (22)

1/p = 1/n 4- 1/m.

For the case of creep at constant stress or, the set of constitutive equations (6), (9), (13), and (16) yields, upon integration (Estrin and Mecking, 1984, 1988), (kP)I/P - (kP) ~) ~ [)/kP' x 1/p __ /kp, 1/p -- e x p

( -ep - ep) Etr-C~Ee~

,

(23)

where the subscript i refers to some initial point on the creep curve. The steadystate strain rate k~ is given by k p -- ko ~

(24)

Chapter 2

Dislocation-Density-Related

Constitutive

Modeling

77

while the quantity creep m ~tr --

//

m + n

"

O"

|

(25)

describes the rate at which this steady state creep rate is approached. The case of creep at a constant load can be solved similarly (Estrin and Mecking, 1984). An important conclusion that can be drawn from the solutions for creep is that all but one of the model parameters needed to describe the creep behavior in this model can be easily assessed from constant strain rate tests. This one parameter is the exponent m in the kinetic equation (6). It is associated with the isostructural strain rate sensitivity of the flow stress and must be found from a test in which the microstructure does not change, e.g., from a strain rate jump test (Mecking and Kocks, 1981). Once exponent m is known, one can determine other model parameters entering Eqs. (23)-(25) for creep from the strain hardening behavior. This is most conveniently done by extrapolating the | diagram to zero stress--thus obtaining | to zero strain hardening rate--thus obtaining the saturation stress crsmand, using its strain rate dependence, the parameters n, or*, and 8o. Of course, one should be aware of the fact that a constant cross-head velocity or even a constant applied strain rate test is different from a constant plastic strain rate test, for which the constitutive equations were solved; but these differences can be taken into account in a parameter evaluation procedure. In the constitutive formulation that goes back to Kocks (1976) and Mecking and Kocks (1981), the mean free path of gliding dislocations was assumed to be solely determined by localized obstacles associated with dislocations. An extension of this model (Estrin and Mecking, 1984) considers other obstacles to dislocation glide, such as grain boundaries, particles of a second phase, cementite lamellae in a pearlitic structure, etc. Below, we consider modifications to the Kocks-Mecking model for several metallurgically relevant situations.

C. Grain Size Effects 1. High Density of Geometric Obstacles In the limit case where the density of geometric obstacles is much larger than that of the obstacles caused by other dislocations in the population, the mean free path L can be identified with the spacing d between these geometric obstacles. First consider a stabilized obstacle structure, such that d does not change in the course of deformation. Assume further that the recovery coefficient k2 considered in the previous section is not affected by the obstacles. If a direct effect of the geometric obstacles on the flow stress can be disregarded, their only influence will be through their effect on the rate of dislocation density evolution, i.e., through the strain hardening rate. The kinetic equation, then, is still given by Eq. (6), whereas

78

Yuri Estrin

the evolution equation for p is now written as (26)

d p / d e p = M ( k - k2p)

where k = (bd) -1 = const. A particular case to which this model will be applied is that of grain boundary hardening. One should realize, however, that the limit case in which the grain size is the smallest characteristic length in the structure applies for submicron grain sizes only, warranting that d < 10/~/-~. (The expression on the fight-hand side gives a typical cell or subgrain size.) With the modified constitutive model now given by Eqs. (6), (9), (13), and (26), simple integration yields

o"2

(o.~nod)~ = e x p

mo--~e^tr

for the case of constant plastic strain rate and

(kP)2/P--(ksm~ 's') (-~ [)/k~P"2/p fkmod\Z/p = exp --

eP--eP

)

(28)

/ creep\mod ~6tr

)

for the case of creep at constant stress. The saturation stress cr~ ~ is now given by o'~n~ = M o t G b

(k)|/2(k~)

1/pmOd

~2o

=--

(

(O'*)m~

~P k~

)

1/pm~

(29)

and the steady-state creep rate by ---(~p)mod = ~o

(30)

where 1/pm~ = 1 / m + 1/(2n).

(31)

The characteristic transient strains for the two modes of deformation are mod _ tr

etr/2

(32)

and / creep,rood ~etr )

respectively.

p (kM40t2b2]_ 1 ~cr ,

= --, m

,

(33)

Chapter 2 Dislocation-Density-Related Constitutive Modeling

79

At small plastic strain increments 6 p -- 8 p, Eq. (27) can be simplified to

b)

O'-- MotGbkl/2(~.~-~o)l/m (S p __ Ep ) 1/2 __ Mc~G -~

1/2

(~p __ 8p ) 1/2.

(34)

Here it has been assumed that O'i << O'~nod. It is interesting to note that (i) the modified model yields parabolic hardening, O" c~ v/eP - e p, and (ii) a grain size dependence of the Hall-Petch-type, O" (x 1/v/d, follows. It should be stressed that this Hall-Petch-type relation stems from the effect of the grain size on the strain hardening rate, i.e., on the derivative of stress, rather than on the stress itself.

2. Superposition of Dislocation-Related and Geometric Obstacles The above considerations apply if geometric obstacles outnumber the dislocationstructure-related ones. In a more general case, when a superposition of the immobilizing effects of both obstacle types is considered, the inverse obstacle spacing 1/L in Eq. (12) can be expressed as a linear combination of the inverse spacings of the two types of obstacles taken separately. In other words, it is assumed that the obstacle densities are additive, implying that the obstacles of different types are equivalent in their dislocation immobilization effect. The resulting evolution equation for p is a hybrid of Eqs. (16) and (26) (Estrin and Mecking, 1984): (35)

dp/d~ p = M (k nt- kl x/~ - k2p).

The "hybrid" model, which comprises Eqs. (6), (9), (13), and (35), yields upon integration ff

_

hybrid O's

O'i -- O'S

[ [ ff + ( 1 O'i

. hybrid "] (l-a:) -

_

K)o" s

K)O'S

- exp

hybrid Str

for the case of deformation with constant plastic strain rate and

ybndl, (

(kp) a/p _ [(kp) ] -- exp (~p) l / p _ [(~p)hybrid]l/p

)

~8tr/creep hybrid)

(36)

(37)

for the case of creep deformation at constant stress. Here the parameter K = 211 + (1 + 4kk2/k 2) 1/2]-1

(38)

80

YuriEstrin

has been introduced. Other quantities entering Eqs. (37) and (38) are

ohybrid S hybrid

err

- - O-S/K,

(39)

= etr/(2 --/r

(40)

1+--

,

(41)

~)-1 . o-

(42)

O"

and

( creep,~hybrid etr

)

creep(

= etr

1 + --

All characteristics of the hybrid model are expressed in terms of the corresponding quantifies of the Kocks-Mecking model, the relations involving the parameters x or

E = MotGbk/kl,

(43)

both being dependent on the constant k. A simple check shows that the equations of the hybrid model recover those of the Kocks-Mecking model (k ~ 0) and of the modified model (kl --~ 0). Knowing the mechanical response of the prototype coarse-grained material, one can predict the behavior of a grain-refined one. This becomes clear if one considers the stress dependence of the strain hardening coefficient | in a particular type of diagram, plotting the product tOo- against o-. It can easily be shown (Estrin and Mecking, 1988) that the following equation holds:

(tOo- )hybrid __ (tOo-)matrix .+. M3(otGb)Zk/2

(44)

where the first term on the fight-hand side, corresponding to the prototype coarsegrained (matrix) material (d ~ oo), is represented by an inverted parabola intersecting the abscissa at cr -- 0 and o- -- o-s. The second term leads to a parallel shift of the parabola along the ordinate axis by an amount inversely proportional to d (Fig. 2). While the constant k can be found from the magnitude of the shift, the slope of the parabola at o- = 0 yields the stage II hardening rate tOn, while the strain rate dependence of o-s can be used to determine the exponent p. An illustration of the grain-size dependence of strain hardening, as reflected in too- vs. o- diagrams, is given by the data of Narutani and Takamura (1991); (cf. Fig. 3). An interesting corollary of the model is a change of creep behavior with grain refining. First, with decreasing d and the concomitant increase in E, the transient strain reflecting the extent of primary creep decreases; cf. Eqs. (42) and (43). Second, the steady-state creep rate, Eq. (41), decreases as well. It should be noted that a deviation from a straight line in a 1og(kP) hybrid vs. log o- plot in the range of small stresses is predicted. The doubling of slope from p in the high-stress range (o-/E >> 1) to 2p in the low-stress range (o-/E << 1) predicted by the model may have the appearance of a threshold behavior.

Chapter 2 Dislocation-Density-RelatedConstitutive Modeling

81

01!

I1 0

/

<

"~

"--.

0/2

The |

\ o

o h:, a,/tc

o

diagram in the hybrid model including the effect of geometric obstacles

(schematic). The hybrid model thus presented can also be used to describe plastic deformation of other systems where the dislocation mean free path is constrained by microstructural elements acting as impenetrable obstacles to gliding dislocations. Examples are the deformation of the ferritic phase in a pearlite structure and of the metal phase in a metal-matrix composite with nonshearable reinforcement

10

X 104 Ni 295K 032/.zm 091pm

,,,,

0

1

2

3

4

O" (MPa) The grain size effect on strain hardening as seen in the | crystalline Ni (Narutani and Takamura, 1991).

.,

!

5

X 102 diagram for poly-

82

Yuri Estrin

particles. The equations discussed in this section can be adopted for such systems by identifying d with the average spacing between cementite lamellae and between reinforcement particles, respectively.

D. Particle Effects Precipitates or second-phase particles commonly act as geometric barriers to dislocation glide. The above formalism can be used, in a somewhat modified form, to account for particle effects on the deformation behavior. Deformation of materials containing shearable and nonshearable particles will be treated separately. 1. Nonshearable Particles

We consider a case of a material that contains a dispersion of nonshearable secondphase particles, such as noncoherent precipitates, oxide or carbide dispersoids, and the like. Assuming that the particle size and distribution remain unchanged during the deformation process, we can formulate a constitutive model that is still based on the kinetic equation (6). The mechanical threshold 8 will, however, be taken in the form 6 -~- O'disl +

MxGb/D

(45)

where O'disl =

MotGb~

(46)

and X is a constant of the order of unity. The last term represents the stress required for a dislocation to circumvent, by the Orowan mechanism, particles separated by a distance D in the glide plane. In addition to the direct effect that particles have on the flow stress through this term, they also enhance dislocation storage and thus increase the strain hardening rate. This second effect can be described using Eq. (35). However, as the recovery rate may also be affected by dislocation-particle interactions, we introduce a modifying factor f = f(cr, T). The nature of the interaction will not be discussed here, but it can generally be said that the interaction will reduce the recovery rate, e.g., by inhibiting detachment of dislocations from particles (ROssler and Arzt, 1990). The evolution equation for the dislocation density thus reads

dp/de p -- M(kD + kl~v/--fi- f k z p )

(47)

where kD = (bD) -1. It is seen that the model used in the previous section to describe grain size effects has undergone only minimal change. In view of the importance of particles in the enhancement of creep resistance, we look at the result of integration of the present constitutive equations for the case of creep at constant stress. The resulting strain dependence of the creep rate is still

Chapter 2 Dislocation-Density-Related Constitutive Modeling

83

{~P~hybrid described by Eq. (37), but the steady-state creep rate, ~ sJ , and the transient strain, (3creep~hybrid ','-'tr J , now assume the following form:

(~p)hybrid _ k p f p

N

1+

)-P

(48)

-MxGb/D and creep, hybrid gtr ) ---

?I(o-MxGb/D)( m + n

|

1 +

~ ~ - MxGb/D

)-1

.

(49)

Here, N = M ~ G b k D / k l . It should be noted that whereas the quasi-threshold behavior of the steady-state creep rate mentioned in connection with the grain size effect in the foregoing section is fairly modest, it can be very p r o n o u n c e d in the case under consideration. The main effect stems from a rapid drop of the steady-state creep rate when the stress approaches the level of M X G b / D , which is further accentuated by the stress-dependent factor fP whose role increases with decreasing stress (Estrin, 1991 b). In the above considerations, the particle space D was considered constant. The set of constitutive equations, Eqs. (6) and (45)-(47), can also be solved if the evolution of the quantity ko = ( b D ) -1, e.g., due to in situ precipitation or Ostwald ripening, is known. In this case, however, D has to be considered as an additional internal variable. Figure 4 shows an example of the application of this model

Creep data for Alloy 800 H deformed at T = 1173 K and cr = 25 MPa with and without aging pretreatment stabilizing the particle structure. The respective curves calculated without using any fit parameters are shown as dashed and solid lines (after Schwarze, 1991).

84

Yuri Estrin

to describing transient creep in Alloy 800 H, in which precipitation and subsequent Ostwald ripening of carbide particles occurred during creep deformation (Schwarze, 1991).

2. Shearable Particles

Modeling of deformation of a material containing shearable particles is also fairly straightforward (Y. Estrin and Y. Br6chet, unpublished). Keeping the kinetic equation in the form of Eq. (6), we now express the mechanical threshold stress as

t7 -~- ~disl + t~part,

(50)

where the dislocation-related part, ~disl, is given by Eq. (46) and the particle-related part, 8part, is taken in the form initia,

O'part -- Opart exp

(51)

"

e

Equation (51) is an a n s a t z that represents a decline in the glide resistance to dislocation motion with increasing shear. The characteristic strain for this mechanism of loss of glide resistance is denoted by ~. This quantity is determined by the local shear strain needed to completely shear a particle; i.e., it takes into account, in some phenomenological way, the degree of strain localization associated with particle shearing. Hence, to include particle effects, two new model parameters, ~pinitial art and g, have been introduced. The constitutive model is now specified by Eqs. (6), (46), and (50), together with an evolution equation for the dislocation density. As shearable particles have no effect on the dislocation mean free path, we choose the Kocks-Mecking evolution equation, Eq. (16), as a most representative one. At constant plastic strain rate, we obtain for the strain hardening coefficient

t~=OII

1-~

O'S

+

~tr

1 ) " initial ( ~ P ) e Opart ~o

1/m

_

exp

.

(52)

It is obvious from this expression that the Palm-Voce strain hardening behavior represented in the Kocks-Mecking model by Eq. (20) is modified by the exponentially decaying term in Eq. (52) and is recovered asymptotically. Depending on which of the two characteristic strains, ~tr = O'S//l~)II and g, is larger, the contribution of this particle-related term is positive or negative. Thus, for ~ < ~tr, shearable particles contribute a negative term to the strain hardening coefficient. The two parameters associated with the particle term can thus be found from measured deviations from the Palm-Voce behavior of the particle-free matrix material.

Chapter 2 Dislocation-Density-RelatedConstitutiveModeling 85 Integration of Eq. (52) for k p a--as--(ai

=

const is straightforward:

-- O's) exp ( - 8 p 8p)Etr -

_initial [ (__EP--sP) _exp(_~ -~-Opart exp e

8P-- EP) ]

(53)

Here

_initial _initial O'i -- Opart -~-Odisl '

_initial ~_initial( EP") 1/m Opart -- Opart 8o//

o.~nitial ~_initial(sP) 1/m art -- O-part ~--o

(54)

are the initial values of stress, the particle-related contribution to stress, and the dislocation-density-related contribution to stress at 8 p -- 8p, respectively. In the case of ~ < 8tr, the particle effect dies off rapidly, and the approach of saturation is described by the Palm-Voce-type equation,

O" -- O"S -- '~O'initial disl -- ~rs) exp /n Ep -- Ep ) 8tr

(55)

In the opposite case of 6tr < 8, the particle effect persists, and the saturation stream of the particle-free matrix, crs, is approached from above:

_initial exp O" -- O"S -- O-part

(~ 8p-Sp ) .

(56)

In the case of creep at constant stress, a look at the variation of the derivative

dlnSPdsP

---m{~II ) -[ 1-[

(~II +U

(

)

(sP)-I/P( o--8or ~-~

kP -1/p

] (--initial)

_1

Opart

exp

(-- EP -- EP) } ~

(57)

shows that the rate of approach of steady state is modified as compared to the Kocks-Mecking model, though the level of the steady-state creep rate itself is still given by Eq. (24). For sufficiently small ~, when ~-1 is dominant in the second term in the curly brackets, the effect of particle shearing will be a decrease of (negative) slope of the log kp vs. 8 p curve in the range of primary creep, for strains of the order of ~ itself.

86

YuriEstrin

Equations (52) and (57) provide a means of predicting strain hardening and creep behavior of a material containing shearable particles of a second phase and allow us to separate the effects of particles from those of the particle-free matrix.

E. Solute Effects Like second-phase particles, solutes can affect the flow stress, both directly and indirectly, through the strain hardening coefficient. It is therefore quite understandable that the dependence of the flow stress on the solute concentration usually increases as straining progresses, the saturation (or steady-state) stress being affected by the solutes most strongly. In this section we shall look into possible solute effects on hardening. The cases of stationary and mobile solutes will be treated separately. The treatment is not meant to be exhaustive. Rather, the two variants of the model considered below are to be viewed as an illustration of the principal possibility of incorporating solute effects in the general constitutive framework.

1. Stationary Solutes Proceeding again from the kinetic equation, Eq. (6), the mechanical threshold stress, 8, can be expressed as --- O'disl + ~sol

(5 8)

introducing the solute contribution, 6sol, to the mechanical threshold stress. Here we assume a square-root dependence of this term on the solute concentration, or we may assume a more general power-law dependence, O'sol --"

~'cr,

(59)

where ~. is a constant reflecting the magnitude of the dislocation-solutte interaction. In addition to this direct effect on the stress via the mechanical threshold stress, solutes exert an effect on the recovery rate, thereby influencing the rate of strain hardening. The recovery rate is affected because solutes have an effect on the stacking fault energy. The underlying mechanism is as follows. A decrease of the stacking fault energy with solute concentration facilitates dissociation of dislocations into partials and inhibits cross-slip. Quantitatively, this can be accounted for by expressing the exponent n in the recovery coefficient k2 [cf. Eq. (13)] as n-

B/T,

(60)

assuming the case of low-temperature deformation--in the sense defined in Section II.A. As discussed by Kocks (1976), the dependence of the quantity B on the stacking fault energy (per unit area), F, can be represented as B =

Gb3/kB al + a2I'/Gb '

(61)

Chapter 2 Dislocation-Density-Related Constitutive Modeling where a l and a 2 are numerical parameters. linearize F = F (c) with respect to c,

87

For dilute solid solutions, one can

F = F0 - g c ,

(62)

where both the stacking fault energy 1-'0 and the coefficient g are concentrationindependent. Combining Eqs. (60)-(62), one obtains nmatrix

t/matrix

B = ~ , 1 -

n =

(63)

r/c

1 -

r/c'

where Bmatrix _._

Gb 3/ kB

nmatrix =_

al q- a2I-'o/Gb '

Gb 3/ kB T

(64)

al q- a 2 I - ' o / G b '

and r/---

a2g/Gb

al + ae F 0 / G b

.

(65)

In the absence of an alloying effect on the stacking fault energy (g = 0), B given by Eq. (63) is coincident with B matrix. A convenient feature of this formulation is that the equations of the other variants of the m o d e l P i . e . , those for a coarse-grained, single-phase ("matrix") material as well as for fine-grained or particle-reinforced material--can be retained, except that the term 6sol -- Xc r [cf. Eq. (59)] must be added to the respective expression for the mechanical threshold stress and n must be taken in the form of the second of Eqs. (63). Consider, as a simplest example, the case when the solid solution effect is the only one to modify the behavior of the matrix. Then, the closed-form constitutive model is given by Eqs. (6), (13), (16), (58), (59), and (63)-(65). The strain hardening coefficient at constant plastic strain rate, | = (O0"/OF_,P)kp, assumes the form |

-- |

1

rrs(c)

+ |

(66)

crs(c)

Here O'so 1 ~

(67)

O-so 1

and

( EPI (1-rlc)/nmatrix+l/m as(c)

= a*

--

ko

--

O-S(kp

_ rlr matrix (68)

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Yuri Estrin

Upon integration, one obtains a Palm-Voce-like equation, o" -- o'S ( c ) = o'sol +

initia, -- O'S ,c, lexp(--

[o'ciisl

O'S (C) / {~)II

(69)

describing the evolution of the dislocation-related stress o ' d i s l = O" - - o ' s o l from its initial value, O-initial disl - - MotGb v/pinitial (kP/ko) 1/m , to a steady-state value given by Eq. (68). A twofold effect of solutes becomes evident from this expression. First, there is just an additive contribution, o'sol, to the stress which would have been given by o'disl in the solute-free matrix. Second, the saturation stress approached asymptotically--and with it the stress level at all strainsmis increased, this increase being more pronounced for larger solute concentrations. Of particular interest is the behavior of the saturation stress. It has been noticed repeatedly (cf. Mecking et al., 1988) that, for a given temperature, the plastic strain rate dependence of the steady-state stress in constant strain rate testing and the stress dependence of the steady-state creep rate fall on a common curve. This "uniqueness" of the steady-state behavior also follows from the constitutive models under discussion in this chapter. The stress exponent obtained from the slope of such a compound curve plotted in double logarithmic scale is used as an important parameter to characterize the material. (One application is in creep life assessment using the Monkman-Grant approach.) As seen from Eqs. (21) and (24), for the prototype Palm-Voce material describable by the Kocks-Mecking equation, this steady-state power exponent is equal to p = mnmatrix/(m + n matrix) and can be approximated by n matrix in a typical case of m >> n matrix. According to the above prediction, solute effects should change this slope to n = nmatrix/(1 - r/c) nmatrix(1 + r/c). Thus, starting from a straight line representing the steady-state matrix behavior in the log ~P vs. log o" diagram, one can obtain the corresponding diagram for a dilute alloy in two steps: (i) first, the diagram must be tilted to increase its slope by a factor of (1 + r/c); (ii) second, it has to be moved to the right by an amount of log(o" + o ' s o l ) - - log o" ~" o'sol/o'. (The solute effect has been assumed to be small). The smaller the stress, the larger is the amount of displacement involved. The resulting departure from the straight line that is produced in step (i) when step (ii) is performed may lead to a curve indicating an apparent threshold-like behavior.

2. Mobile Solutes (Dynamic Strain Aging) When solute atoms are sufficiently mobile to catch up with moving dislocations, a particular kind of dynamic interaction takes place. If mobile dislocations are being intermittently arrested on localized obstacles (which we shall assume to be associated with forest dislocations and to be surmountable with the aid of thermal activation), diffusion of solute atoms toward dislocations during these temporary

Chapter 2

Dislocation-Density-Related Constitutive Modeling

89

arrests will lead to additional pinning. As the average waiting time at a localized obstacle, tw, is inversely proportional to the plastic strain rate, (70)

t w -- ~ / E P ,

the additional pinning effect is a decreasing function of the plastic strain rate. The contribution of the solute effect to the mechanical threshold, Eq. (58), thus exhibits a negative rate sensitivity. Of course, the expression for this contribution is no longer given by Eq. (59), but rather reads (Estrin and Kubin, 1989; McCormick and Estrin, 1991) as follows:

exp[

}

In Eqs. (70) and (71), f2 is an elementary strain increment produced when all mobile dislocations have overcome a localized obstacle and moved by a distance equal to the mean free path (proportional to 1/V/-fi); Zr is a characteristic relaxation time associated with solute migration toward dislocations, and c and Cm are the nominal solute concentration and the maximum attainable concentration on a dislocation, respectively. The constant fl represents the magnitude of the solute pinning effect. It should be noted that f2 is proportional to the density Pm of m o b i l e dislocations, but as no evolution of Pm is included in the one-internal-variable formulation, the variation of f2 will be related solely to that of 1/~/-fi. We thus write f2 = co/~fi,

(72)

where co is proportional to the mobile dislocation density. The form of Eq. (71) is such that in the limit of small waiting (i.e., aging) times the Cottrell-Bilby 2/3 power law is recovered, while in the limit of large waiting/aging times the solute concentration at arrested dislocations reaches a saturation value (cf., e.g., McCormick and Estrin, 1991). The evolution equation for p is based on Eq. (16), while the effect of solutes on the cross-slip probability is included by introducing in the recovery term a factor f s o , = expQ

V~176 kBT

'

(73)

where the "activation volume" V is assumed to be constant and Crsol is given by Eq. (67) with 6sol defined by Eq. (71). That is to say, Eq. (16) is replaced with d p / d e p = kl V/-fi -- f s o l k z p .

(74)

It is recognized that although the choice of the form of the factor fso~ was made heuristically, it captures the effect of solutes on the recovery rate. In the limit

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YuriEstrin

of high strain rates, O'sol vanishes and fsol tends to unity, while in the opposite limit case of low strain rates, when O'solreaches its saturation value and cross-slip involves release of dislocations from their saturated solute atmospheres, fsol drops down to Sat = exp(-

oJ

~Vcm) kBT

(75)

"

The constitutive model for the case under consideration is given by Eqs. (6), (58), (67), and (71)-(74). Omitting the details of this dynamic strain aging (DSA) variant of the model we consider here only the concentration dependence of the steady-state stress, which reads

k~)

1-exp[-(~m)

( f2

(76)

where Crsolis given by Eqs. (67) and (71). It is seen that in the high strain rate limit the steady-state stress coincides with that for a solute-free material, i.e., with crs of the Kocks-Mecking model, while in the low strain rate limit it is equal to

o"D SA = o'S e x p ( fl VB Tc m

--b fl Cm ( ~.~_~) l / m

(77)

F. Influence of the Peierls Stress So far, no effects of the crystal lattice resistance (Peierls stress) have been considered. This implicit assumption of negligibility of the Peierls stress may be justified for fcc metals where the Peierls relief is low, but it has to be dropped if one of the variants of the model considered above is to be applied to a material with a high Peierls relief. For such a material, e.g., a bcc metal or a semiconductor, the effect of the Peierls relief can be taken into account by including the Peierls stress as an additive term in the mechanical threshold stress. Typically, this term plays a substantial role below approximately 0. 2Tm, where Tm is the melting temperature (Rauch, 1994). A further effect of the lattice resistance consists in the inhibiting action of the Peierls relief on the cross-slip probability. A modifying factor accounting for this effect should be included in the parameter k2 representing the recovery rate in the evolution equation for the dislocation density. The exact form of this modification will not be considered here, however.

Chapter 2 Dislocation-Density-RelatedConstitutiveModeling

91

TWO-INTERNAL-VARIABLEMODEL A. Outline of the Model As mentioned in Section I, a model employing just one internal variable is not sufficient for deformation histories involving rapid changes of the deformation path. A distinction between dislocation types whose densities evolve with different rates offers a means of devising a more flexible model for the purpose. Instead of using a single internal variable related to the total dislocation density p, we now consider two: the mobile (Pm) and the relatively immobile, or forest, density (pf). It can generally be expected that the evolution rate of Prn is faster (in terms of strain) than that of pf. The mechanical threshold stress in Eq. (6) is now given by the quantity Mot G b ~ and the factor eo is no longer considered constant, but rather is represented (Estrin, 1991a) as eo = (Pro (( = const.). The evolution equations for the two dislocation densities proposed by Estrin and Kubin (1986) read d p m / d e p = M ( - k - kl ~ / ~ - k3Pm -+- k4pf/pm)

(78)

and d p f / d e p = M ( k -+- k l ~ / ~ -

k2pf -Jr-k3Pm).

(79)

The origin of each term in these equations has been discussed in the cited paper. It is seen that the negative terms in Eq. (78), which represent the loss of the mobile density due to various dislocation reactions, reappear as positive terms in Eq. (79). Equation (79) is akin to Eq. (35), an additional term, k3Pm, corresponding to the immobilization of mobile dislocations due to binary reactions between them. The new parameter, k3, represents the efficiency of this type of reaction. Another new model parameter, k4, appears in the term corresponding to generation of mobile dislocations by the forest sources. In the following, both these parameters will be considered constant. It is interesting that although it is significantly more complex than the oneinternal-variable model, this extended version can still be handled analytically, provided the characteristic strain for the mobile density evolution can be considered small relative to that for forest density evolution. Using this "hierarchy" of the relaxation strains, the evolution equations can be integrated, as was done by Estrin and Kubin (1986). However, since the model is meant to be used for numerical calculations, e.g., with finite element codes, this analytical form is not of practical value and will not be considered here. With the new two-internal-variable formulation, the constitutive model assumes the following closed form (Estrin, 1991a; Braasch and Estrin, 1993). The kinetic

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YuriEstrin

equation reads ~p = ~

O" (~OO) mx y _ m / 2

'

(80)

where nondimensional internal variables, X =

Pm

Pf

and

Pmo

Y = ~,

Pfo

(81)

have been introduced. The quantities Pmo and Pfo denote the respective initial dislocation densities. The parameters = ~'Pmo

(82)

and ~o = M c ~ C b 4 ~ o

(83)

have been introduced. The evolution equations for X and Y follow from Eqs. (78) and (79): d X / d e p = q ( - C - C1 ~ d Y / d e p = C + C1 ~ / Y -

C3X + C 4 Y / X ) ,

C2 Y + C3X.

(84)

(85)

The parameters in these equations are related to the coefficients in Eqs. (78) and (79) in an obvious way: C = Mk/pfo,

C1 = M k l / ~ f f ~ o ,

C2 = M k z = Mk2o C3 -- M k 3 / q ,

k--~

'

C4 = M k 4 / P m o .

(86a) (86b) (86c)

The factor q - Pfo/Dmo,

(87)

which appears in Eq. (84), represents the ratio of initial values of the forest and the mobile dislocations. Note that the initial values of X and Y in the evolution equations are both equal to unity. If uniaxial deformation is considered, the form of Eqs. (81), (84), and (85) is to represent the tensile case. The set of Eqs. (80), (84), (85), together with Eqs. (1)-(5), define the twointernal-variable model in a general form. [Note the strain rate and temperature dependence of the coefficient C2; cf. Eq. (86b).] It is interesting that the oneinternal-variable formulation formally follows from this set of constitutive models if the parameters q and C3 are set equal to zero. The variants of the constitutive

Chapter 2 Dislocation-Density-RelatedConstitutive Modeling 93 equations discussed in the previous sections in the context of the one-internalvariable model and accounting for various microstructural effects can be translated into the two-internal-variable formulation by modifying the expression for the mechanical threshold stress, i.e., by replacing the factor y-m~2 in Eq. (80) with (y1/2 _+_6,/Cro)-m, where 6" is one of the possible contributions to the mechanical stress considered in connection with the one-internal-variable model. A further modification necessary if the respective effects are to be accounted for consists in the inclusion in the coefficient C2 [cf. Eq. (86)] of a factor (e.g., f or fsol) describing the associated decrease of the recovery rate. For lack of space, we do not go through the exercise of reformulating the two-internal-variable model for each of the individual cases considered for the one-internal-variable model. We note that the effect of grain boundaries or other geometric obstacles to dislocation motion (e.g., inclusions with no affinity for dislocations of the matrix), whose role is merely to limit the dislocation mean free path and thereby increase the dislocation storage rate without changing the recovery rate, is already accounted for by the term C in the evolution equations, Eqs. (84) and (85). The set of Eqs. (1)-(5), (80), (84), and (85), with the strain rate and temperature dependence of the coefficient C2 given by Eqs. (86b) and (13), comprise a unified elastic-viscoplastic constitutive model without a yield criterion that is sufficiently versatile for most applications involving monotonic loading. Recipes for inclusion of further micromechanical features have also been given. Before we turn to illustrating the descriptive and predictive capabilities of the model, we address the question of how the material parameters involved can be identified on the basis of test data.

B. Parameter Identification Technique (Evolution Strategy) Standard techniques of parameter evaluation for a model are based on minimization of an objective, or quality function, which is chosen a priori and which commonly makes use of least absolute values or least squares of the differences between the predicted and the measured data or of the maximum likelihood principle. However, when applied to viscoplastic models, most of the established algorithms of quality function minimization encounter serious difficulties as the topology of the quality functions is very complicated. Usually, a large number of local optima limit the applicability of the conventional gradient methods of optimization. An alternative approach that appears to be well suited for reliable determination of the global optimum in the parameter space is the so-called "evolution strategy" (Rechenberg, 1973; Schwefel, 1975). First attempts to use this approach in the context of viscoplastic modeling by Mtiller and Hartmann (1989) demonstrated its suitability for the purpose. In a recent work by Braasch and Estrin (1993) an evolution-strategy-based parameter evaluation technique for the two-internal-variable model described above

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YuriEstrin

was developed. By analogy with biological evolution, a set of the model parameters is looked upon as a set of"genes" in a chromosome of a living organism. According to a certain rule, a descendant belonging to the next generation inherits the set of his parents' genes, but with some "mutation": the value of each inherited parameter (gene) is multiplied by a random number chosen individually for each gene by using a random number generator. The particular variant of the evolution algorithm chosen is referred to as the (~., /z) strategy. A set of ~. predecessors is considered to collectively produce a set of # descendants (/z > ~.). The reproduction procedure is specified as follows. To create a descendant, two parents are chosen at random from among the group of )~ predecessors. Each gene of the descendant (i.e., each model parameter value) is taken to be inheritedwwith mutationmfrom one of these two parents taken at random. The standard deviation of the log-normal distribution function used in the random number generation for the mutation procedure is determined by the average of the corresponding standard deviations of both parents. It is either adopted without change, which is done in 50% of all cases, or after multiplication by 1.2 (in 25% of all cases) or division by 1.2 (in 25% of all cases). It should be noted that each gene is characterized by two numbers: a parameter value and the magnitude of the associated standard deviation. Both values evolve in the mutation process. Each new descendant created according to this procedure is checked with respect to its compliance with physical constraints on the admissible parameter values and is dropped if it does not comply. In this way,/x descendants are created in each generation of which )~ fittest ones are selected on the basis of the quality function to further participate in the reproduction in process. One of the greatest benefits of the procedure is that the system does not "get stuck" at one of the local minima, the entire space of physically admissible minima being searched for a global minimum. Coming closer to the optimum, the step size of the mutation procedure decreases until the optimum is found; in this final stage of the search it is beneficial to switch over to gradient method. Though there is no general rule regulating the choice of ~. and #, it turns out to be appropriate to take the number of predecessors to equal that of the parameters of the constitutive model and the number of descendants to be the square of that number. When running the evolution strategy on a computer with vector or parallel facility, one should use a value of # that is a multiple of the size of the vector register or of the number of parallel processors in order to take maximum advantage of the hardware. A very beneficial feature of our constitutive model is the clear physical and metallurgical significance of the model parameters. This makes it relatively easy to preselect the range of parameter values within which the parameter optimization procedure is carried out. Moreover, many of the parameters involved can be predetermined from a simple set of experiments. Thus, the isostructural stress exponent m can be found from the stress jump accompanying an "instantaneous" strain rate jump test. Furthermore, from a series of constant strain rate tests carried

Chapter 2 Dislocation-Density-RelatedConstitutiveModeling

95

out with different strain rates, one can identify the parameters C, C1, C20, and p. This is best done by using the | vs. cr representation of the data. First, such a representation permits a quick check of whether the two-internal-variable model is consistent with experimental data, i.e., whether the | vs. cr diagrams are given by inverted parabolas with a c o m m o n initial slope (~)II. Second, if this is the case, C1 can be found from this initial slope by using Eqs. (19) and (86a), C is obtained from the intercept with the ordinate axis by using Eq. (44) and (86a), and both the constant C20 and the exponent 1/p -- 1 / n + 1 / m are evaluated from the strain rate dependent intercepts with the abscissa by using Eqs. (18) and (86b). The last procedure requires only a parallel shift of the parabolas downward until they pass through the origin (to suppress the contribution of the term C) and the determination of the intercepts with the abscissa of these "reconstructed" matrix material diagrams. Although the simple relations to be used for these estimates of the parameters are valid only for the one-internal-variable version of the model, this procedure is still justified in the parameter preselection stage. Subsequent fine-tuning is then done using the evolution strategy algorithm described (Braasch and Estrin, 1993).

C. An Experimental Example: AI 1100 To illustrate both the capabilities of the two-internal-variable model and the efficiency of the above parameter evaluation technique, Braasch and Estrin (1993) considered experimental data of Brown et al. (1989) for the aluminum alloy A1 1100. In a first exercise, the parameters of the two-internal-variable model for this material were determined using four constant strain rate compression tests (Fig. 5) and three strain rate jump tests (Fig. 6) conducted at 623 K. Preselection of the parameter values was made using the parabolic fit described above. With the "preliminary" parameter values derived from this fit, the full analysis based on the evolution strategy was then made. The optimal set of parameters obtained in this way after less than 100 "generations" is as follows: C --0; C4 - -

C1 = 2.21 x 10;

1.20 x 102;

C20 =

m = 1.57 x 10; --4.01 x 104 s-l;

1.95 x 10; n = 1.22 x 10;

C3 = 3.20 x 10; q --7.17 x 10-1;

O'o = 3.33 x 10 MPa.

The deformation curves simulated with the two-internal-variable model using this set of parameter values are shown in Figs. 5 and 6, together with the experimental ones. The good agreement obtained is a demonstration both of the efficacy of the evolution strategy as applied to the model under consideration and of the good descriptive ability of the model itself. A more serious test of the value of a constitutive model is, of course, one in which test data that were not used in the parameter evaluation step are to be

96

YuriEstrin

Constant strain rate compression tests for AI 1100 (Brown et al., 1989) used for parameter evaluation. Calculated curves are shown as solid lines.

described without using any further fit parameter. For such a verification step, we considered data from a constant strain rate test and from a downward strain rate jump test of Brown et al. (1989). A comparison between the model predictions and the experimental curves (Fig. 7) shows that the predictive capability of the model is not worse than the descriptive one.

D. The Caseof Cyclic Deformation 1. Constitutive Equations As already mentioned, the two-internal-variable formulation is believed to be a good modeling tool for monotonic deformation. While the same constitutive frame can be used for the case of cyclic deformation conditions, it appears necessary to have a separate " m o d u l e " - - a variant of the model specially designed to describe the material behavior under cyclic loading conditions. The need for this is obvious if one takes a microstructure-based stand. Indeed, a particular channel-like dislocation structure (Fig. 8) that is formed under cyclic deformation (cf., e.g., Mughrabi, 1979) cannot be expected to produce the same macroscopic mechanical response as the cellular or subgrain structures characteristic of monotonic deformation. Still, in modeling the cyclic loading case, an attempt was made (Estrin et al., 1995) to keep the general constitutive frame as close to the above two-internalvariable model as possible. The microstructural input in the cyclic deformation

Chapter 2 Dislocation-Density-RelatedConstitutiveModeling 97

Strain rate jump tests on A1 1100 (Brown et al., 1989) used for parameter evaluation together with the data on Fig. 5. Calculated curves are shown as solid lines.

variant of the model is as follows. The dislocation structure is viewed as a system of parallel narrow walls with a high density of segmented edge dislocations (Fig. 8). These walls separate "channels"mregions of low dislocation density. Dislocation loops in the channels provide links between the adjoining walls--screw dislocation segments that are fairly mobile within the channels. A moving screw segment drags along the edge segments of the loop, thus changing the length (and hence the density) of the edges. When the edge segments are pressed against the walls by the applied stress, the dislocation density in the walls increases. As long as the edge segments are not woven into the dislocation structure of the walls, the edge density increment can be recovered by the motion of the screw segments in the opposite direction following a stress reversal. However, part of this density increment will be irreversibly trapped in the wall due to thermally activated jog formation on the edge segments. In the language of dislocation storage, these processes make it necessary to distinguish between truly immobilized and recoverable stored dislocation density. Another process that has to be taken into account in the cyclic deformation model is generation of mobile screw segments once the applied stress drops below the level of the internal stress acting in the walls. Immobilized edge segments will then bulge out between the pinning points, injecting mobile screw segments into the adjacent channels. The dislocation dynamic features discussed can be cast into the constitutive equations of the two-internal-variable model. The edge dislocation density immobilized in the walls can be identified with the internal variable Y. Equation

98

Yuri Estrin

A monotonic test and a downward strain rate jump test on AI 1100 (Brown et al., 1989) used for validating the model predictions.

(85) can be adopted as the evolution equation for Y. However, an additional internal variable, Pr, representing the density of dislocations trapped in the wall during deformation cycle, but partially recoverable upon stress reversal, has to be introduced. In nondimensional form, it is given by Z = Pr/Pri, where Pri is the initial value of Pr. The dislocation species thus introduced form a subset of Y, the total immobilized dislocation density. Taking into account that the storage rate of Z is inversely proportional to the average channel width which scales with 1/V t-Y, the evolution equation for Z can be written as (Estrin et al., 1995) Z

d Z __ C5 ~ / Y - C2 Z -- C6 ~--~. deP

(88)

Here, the term C2Z stands for the dynamic recovery rate in the walls, while the term C6Z/~ p represents the rate of loss of recoverable edge density due to thermally activated jog formation. This process is time-, rather than straindependent, which explains the appearance of kp in the denominator of this term. For a given temperature, the parameter C6 can be considered constant, its temperature dependence being given by an Arrhenius equation involving the activation energy for jog formation. An important feature of the cyclic deformation model is that the dislocation density Z is recovered upon each stress reversal. This is modeled by discrete updating (Estrin, 1991b; Estrin et al., 1995): Y:=Y-Z~ Z'--0

J

if 0 r - - 0 .

(89)

Chapter 2 Dislocation-Density-Related Constitutive Modeling 99

A dislocation wall-and-channel structure characteristic of cyclic deformation (after Mughrabi, 1979).

Modifications also involve the evolution equation for the mobile dislocation density in the channels. The above-mentioned injection of mobile dislocations into the channels that occurs when the stress drops below the internal stress level, Crov/-Y, leads to an additional term, transforming Eq. (84) into d X / d 6 p -- q (---C - C1 ~ / Y - C3X + C 4 Y / X + C~ Z ) ,

(90)

where

C-~ -

C7

if lal < c~0~/-Y

0

otherwise

(91)

Here, C7 can be considered constant. Finally, to account for the curvature of the mobile dislocation segments in the channels, a backstress O'back was introduced in the kinetic equation. Instead of

100

Yuri Estrin Total dislocation density

X+Y

Mobile dislocation density X

Immobile dislocation density Y

Recoverable

Non-recoverable

Z

Y-Z

Internal variables and their meaning in terms of dislocation densities (Estrin et al., 1995).

Eq. (80), we now have •p __ ~

(

)m

O" -- tTback

xy_m/2"

(92)

Cro An evolution of the backstress with the plastic strain was considered (Estrin et al., 1995) to obey the following equations: dO'back = C 8 ~ / ~ -

C90"back .

(93)

deP

Two new parameters, C8 and C9, have thus been introduced. The viscoplastic part of the constitutive model is now given by the set of Eqs. (85) and (88)-(93). This version of the model actually involves three internal variables an overview of which is given in Fig. 9. The three-internal-variable formulation can be reduced to the "canonical" two-internal-variable model if the specific cyclic deformation features are suppressed. Further details of the cyclic deformation module can be found in the cited paper by Estrin et al. (1995).

2. Experimental Validation of the Model In the same paper, the model was applied to describe cyclic deformation data for Alloy 800H (Estrin and Giese, 1993; Giese, 1994) and both monotonic and cyclic deformation data for Inconel 738 LC. The Inconel data were kindly provided by Juergen Olschewski of BAM, Berlin. In the case of Alloy 800H, seven cyclic tests conducted at 1123 K with different strain rates and plastic strain amplitudes were analyzed. Evaluation of the data showed that it was sufficient to apply the abridged variant of the model in which the internal variable X is not activated; i.e., q and C3 are set equal to zero. Furthermore, the parameter values C6 = 0 and ~ = 1

Chapter 2

Dislocation-Density-Related Constitutive Modeling

101

i l l [ l ~ l l l t I l O l Fit of the hysteresis stress vs. strain curve for A1 800 H tested at a nominal strain rate of 2 x 10 -3 s -1 and T = 1123 K with a plastic strain amplitude of 8 x 10 -3 (Estrin et al., 1995).

were used. The remaining nine parameters were allowed to vary in the evolution algorithm applied to the entire group of the tests considered jointly. The quality of the fit for an individual experiment of the group is shown in Fig. 10. The following values of the material parameters for Alloy 800H were found: m--5.97;

n = 13.25;

C20 = 523.5;

~ro = 325.3 MPa;

C8 = 7.39 x 104 MPa;

C = 0.0;

C9 = 1.14 x 103;

C1 -- 131.1; So = 1 s -1.

It should be noted that the value of C = 0 was not set deliberately, but rather came out of the parameter optimization procedure. This indicates that the presence of carbide particles does not affect the cyclic deformation behavior of the alloy. In the case of Inconel 738 LC, the data for five cyclic loading tests were "blended" together with the data for five monotonic and six strain rate j u m p tests in the parameter evaluation process. A cyclic experiment with phases of stress relaxation from the peak stress was spared to be used as a verification test in which no further parameter fit was admitted. All tests were done at T -- 1123 K. Figures 11 and 12 illustrate the fit of individual tests for the o p t i m u m parameter set obtained from all tests treated collectively. This parameter set was as follows: m = 15.73; C1 = 263.2;

n -- 17.95;

~ = 1 s-l;

C20 = 479.4;

C8 - - 8 . 0 x 104

MPa;

~o -- 852.6 MPa;

C5 -- 98.5; C9 =

C -- 356.7;

C6 = 6.35 x 10 -4 s-a;

3.30 x 102; ~o = 1 S- 1 .

102

Yuri Estrin

Comparison between experiment and model for strain rate jump tests on Incone1738 LC. The solid lines correspond to the full three-internal-variable model with provision for cyclic deformation; the dashed lines stem from a fit in which cyclic data were not included and the internal variable Z was not activated (Estrin et al., 1995).

i l [ t l l l l ~ ( l l l | l l Comparison between experiment and model for a cyclic test on Inconel 738 LC for the strain rate of 10 -3 s -1 and the plastic strain amplitude of 8 x 10 -3 (Estrin et al., 1995).

Chapter 2 Dislocation-Density-RelatedConstitutive Modeling 103

||[ti~lJ'|ai|| Verificationtest on Incone1738LC: Cyclic deformationwith phases of stress relaxation

from the peak stress (strain rate: 10 -3 s - l ; total strain amplitude: 6 • 10-3). The symbols represent the experimental data, while the solid lines correspond to the model prediction (Estrin et al., 1995).

Both the individual fits shown and, notably, the results of the verification test presented in Fig. 13 demonstrate that the model captures the deformation behavior of Incone1738 LC, allowing one to make reliable predictions of its behavior under more complex deformation conditions. More generally, the experimental examples of this section show that the extension of the model to describe cyclic deformation behavior was quite successful.

CONCLUSION In this chapter, we considered a dislocation-density-based constitutive description of plastic deformation of crystalline, primarily metallic materials which forms the basis of an isotropic elastic-viscoplastic constitutive model without a yield criterion or loading/unloading condition. Starting with the prototype Kocks-Mecking model which adequately describes the mechanical behavior of single-phase, coarsegrained materials under monotonic loading conditions, we followed the series of modifications required to permit inclusion of additional classes of materials or particular deformation conditions in the model. In the end, we have a versatile model that can account for a broad range of materials and effects. We presented a selection of experimental examples proving the good potential of the model, along

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YuriEstrin

with a newly developed technique for parameter evaluation based on the evolution strategy. The quintessential formulation of the model is given by its two-internal-variable form specified by Eqs. (1)-(5), (80), (84), and (85), while further ramifications are possible along the lines discussed when different varieties of the one-internalvariable formulation were considered. The set of Eqs. (1)-(5), (80), (84), and (85) is perhaps the best form of the model for which its implementation in finite element codes, such as, e.g., ABAQUS, could be recommended. It has been shown (Ronda, private communication; Braasch, private communication) to be well suited for that. For most applications involving monotonic loading conditions, it will be sufficient to further abridge this form of the model by setting the parameters q and C3 to zero. On the other hand, inclusion of particular features, such as particle or solute effects, is straightforward if the recipes given in this chapter are followed. Clear benefits of the model discussed include the simplicity of the constitutive equations, a relatively economical use of model parameters, and, notably, their direct relation to the principal dislocation processes determining the microstructural evolution of the material in the process of plastic flow. In summary, the model presented has matured to a degree that makes it a useful tool for structural analysis and life assessment of structural components as well as for simulation of metal forming operations, solidification, welding, and other processes. Further developments will include a broader implementation of the model in finite element codes.

ACKNOWLEDGMENTS The author is indebted to Heinz Mecking whose seminal ideas lent initial impetus to the workdescribed in this chapter. Useful discussions of various aspects of constitutive modeling with Fred Kocks, Ladislas Kubin, Yves Br6chet, Paul McCormick, and Harald Braasch are thankfully appreciated. JiJrgen Olschewski kindly provided experimental data on Inconel 738 LC. Financial support from the Australian Research Council is gratefully acknowledged.

REFERENCES Bodner, S. R. (1987). Reviewof a unified elastic-viscoplastictheory.In "Unified ConstitutiveEquations for Creep and Plasticity" (A. K. Miller, ed.), pp. 273-301. Elsevier, London. Braasch, H. (1995). Private communication. Braasch, H., and Estrin, .Y. (1993). Parameter identification for a two-internal-variable constitutive model using the evolution strategy. In "Material Parameter Estimation for Modern Constitutive Equations," MD-Vol. 43/AMD-Vol. 168, pp. 47-56. ASME, New York. Brown, S. B., Kim, K. H., and Anand, L. (1989). An internal variable constitutive model for hot working of metals. Intl. J. Plasticity 5, 95-130.

Chapter 2

Dislocation-Density-Related Constitutive Modeling

105

Estrin, Y. (1991a). A versatile unified constitutive model based on dislocation density evolution. In "Constitutive Modeling: Theory and Application," MD-Vol. 26/AMD-Vol. 121, pp. 65-75. ASME, New York. Estrin, Y. (1991b). Strain hardening and creep behavior of materials containing nonshearable secondphase particles. In "Microstructure and Mechanical Properties of Materials" (E. Tenckhoff and O. V6hringer, eds.), pp. 17-24. Deutsche Gesellschaft ffir Metallkunde, Germany. Estrin, Y., and BrEchet, Y., unpublished. Estrin, Y., and Giese, A. (1993). Steady state behavior of alloy 800H under cyclic deformation. Scr. Metall. Mater. 29, 1223-1228. Estrin, Y., and Kubin, L. E (1986). Local strain hardening and nonuniformity of plastic deformation. Acta Metall. 34, 2455-2466. Estrin, Y., and Kubin, L. E (1989). Collective dislocation behavior in dilute alloys and the Portevin-Le Ch~telier effect. J. Mech. Beh. Mater. 2, 255-274. Estrin, Y., and Mecking, H. (1984). A unified phenomenological description of work hardening and creep based on one-parameter models. Acta Metall. 32, 57-70. Estrin, Y., and Mecking, H. (1985). An extension of the Bodner-Partom model of plastic deformation. Int. J. Plast. 1, 73-85. Estrin, Y., and Mecking, H. (1988). Microstructural aspects of constitutive modeling of plastic deformation, In "Elastic-Plastic Failure Modeling of Structures with Applications" (D. Hui and T. J. Kozik, eds.), PVP-Vol. 141, pp. 181-191. ASME, New York. Estrin, Y., Braasch, H., and Br6chet, Y. (1996). A dislocation density based constitutive model for cyclic deformation. J. Eng. Technol. Mater., in press. Follansbee, E S., and Kocks, U. E (1988). A constitutive description of copper based on the use of the mechanical threshold stress as an internal state variable. Acta Metall. 36, 81-93. Giese, A. (1994). "Mechanisches Verhalten von ein- und vielkristallinem Aluminum und der Eisenbasislegierung INCOLOY 800H unter Wechselbelastung." Shaker-Verlag, Aachen, Germany. Klepaczko, J. R., and Chiem, C. Y. (1986). On rate sensitivity of FCC metals, instantaneous rate sensitivity and sensitivity of strain hardening. J. Mech. Phys. Solids 34, 29-54. Kocks, U. E (1976). Laws for work-hardening and low-temperature creep. J. Eng. Mater. Technol. 98, 76-85. Kocks, U. E (1987). Constitutive behavior based on crystal plasticity. In "Unified Constitutive Equations for Creep and Plasticity" (A. K. Miller, ed.), pp. 1-88. Elsevier, London. Kocks, U. F., Argon, A. S., and Ashby, M. F. (1975). "Thermodynamics and Kinetics of Slip," Prog. Mater. Sci., Vol. 19. Pergamon, Oxford. Kuhlmann-Wilsdorf, D. (1985). Theory of workhardening 1934-1984. Metall. Transa. A 16A, 19852091. McCormick, E G., and Estrin, Y. (1991). Constitutive modeling of dynamic strain ageing. In "Modeling the Deformation of Crystalline Solids" (T. C. Lowe et al., eds.), pp. 293-303. TMS, New York. Mecking, H., and Kocks, U. E (1981).Acta Metall. 29, 1865-1877. Mecking, H., Nicklas, B., Zarubova, N., and Kocks, U. E (1986). A "universal" temperature scale for plastic flow. Acta Metall. 34, 527-535. Mecking, H., Styczynski, A., and Estrin Y. (1988). Steady state and transient plastic flow in aluminum and aluminum alloy. In "Strength of Metals and Alloys" (E O. Kettunen, T. K. Lepist6, and M. E. Lehtonen, eds.), pp. 989-997. Pergamon, Oxford. Miller, A. K. (1987). The MATMOD equations. In "Unified Constitutive Equations for Creep and Plasticity" (A. K. Miller, ed.), pp. 139-219. Elsevier, London. Mughrabi, H. (1979). Microscopic mechanisms of metal fatigue. In "Strength of Metals and Alloys" (E Haasen, V. Gerold, and G. Kostorz, eds.), pp. 1615-1638. Pergamon, Oxford. Mfiller, D., and Hartmann, G. (1989). Identification of material parameters for inelastic constitutive models using principles of biological evolution. J. Eng. Mater. Technol. 11,299-305.

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Narutani, T., and Takamura, J. (1991). Grain-size strengthening in terms of dislocation density measured by resistivity. Acta Metall. Mater. 39, 2037-2049. Palm, J. H. (1948). Stress-strain relation for uniaxial loading. Appl. Sci. Res. Sect. A 1, 198-210. Rauch, E. F. (1994). The relation between forest dislocations and stress in bcc metals. Key Eng. Mat. 97-98, 371-376. Rechenberg, I. (1973). "EvolutionsstrategiemOptimierung technischer Systeme nach Prinzipien der biologischen Evolution." Friedrich Fromman Verlag, Stuttgart-Bad Cannstadt, Germany. Ronda, J. (1994). Private communication. R6ssler, J., and Arzt, E. (1990). A new model-based creep equation for dispersion strengthened materials. Acta Metall. Mater. 38, 671-686. Schwarze, E. (1991). Anwendung der Stoffgleichung 'Ein-Parameter-Modell' aut eine hochwarmteste Legierung. Ph.D. Thesis, RWTH Aachen, Germany. Schwefel, H.-P. (1975). Evolutionsstrategie und numerische Optimiesung. Ph.D. Thesis, Technische Universit~it Berlin, Germany. Voce, E. (1948). The relation between stress and strain for homogeneous deformation. J. Inst. Met. 74, 537-562.

3 Constitutive Laws for High-Temperature Creep and Creep Fracture R. W. Evans and B. WUshire

Interdisciplinary Research Centre in Materials for High-Performance Applications Department of Materials Engineering University of Wales Swansea SA2 8PE United Kingdom

The limitations of traditional mechanistic interpretations of power-law creep behavior and the deficiencies of conventional parametric procedures for long-term creep data estimation are discussed, demonstrating the advantages of a unified theoretical and practical approach to creep and creep fracture, termed the 0 Projection Concept. The 0 methodology introduces detailed materials constitutive laws relating stress~strain~time~temperature, which can be derived by micromodeling of the processes governing creep strain and creep damage accumulation at high temperatures. New data on the creep and creep fracture characteristics of a low-alloy steel under multiaxial stresses show that the 0 equations introduced to quantify tensile creep properties can be extended to predict and explain materials behavior under the complex nonsteady stress-temperature conditions experienced during high-temperature service. Moreover, the model-based 0 relationships can be presented in a computer-efficient form that Unified Constitutive Laws of Plastic Deformation Copyright (~) 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

107

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R.W.Evans and B. Wilshire

is ideally suited to modern engineering approaches to design and life extension of high-temperature components in electricity-generating stations and other large-scale plants.

INTRODUCTION Creep and creep fracture are often the life-limiting factors when components are required to operate for long periods under stress at elevated temperatures in aeroengines, electricity-generating stations, petrochemical plants, and other hightemperature service applications. In the case of components and structures subject to creep loading in power stations and large-scale plants, designs are normally based on the criteria that (a) the application of the load should not cause excessive short-term yielding, (b) deformation due to creep should not become excessive, and (c) creep rupture should not occur within the planned operational lifetime. The first of these criteria can be satisfied by ensuring that the component crosssections are large enough so that the service stresses do not exceed the yield stress (or proof stress) of the material at the operating temperatures. Yet, while data from standard tensile tests provide the information needed for design against rapid distortion or immediate failure, under long-term exposure conditions, creep failure can occur at stresses significantly lower than the yield stress at the service temperature. Long-term creep and stress-rupture tests must then be carried out to determine the stresses that the relevant materials can sustain at the creep temperatures without reaching some limiting creep strain or, more usually, without creep failure occurring within the planned design life. Although most high-temperature components and structures serve under complex nonsteady stress and temperature conditions, engineering designs are generally based on uniaxial creep data. Moreover, because modem electricity generation and petrochemical plants are usually designed for lives of 250,000 hours (over 30 years), equations relating stress/strain/time/temperature are required that allow the results obtained from uniaxial tests of relatively short duration to be extrapolated to provide accurate long-term design data. Many different relationships have been proposed for this purpose, but the equations developed can be employed with real confidence for extended extrapolation only if they are based on a sound fundamental understanding of the micromechanisms governing creep strain accumulation and eventual fracture of complex commercial creep-resistant alloys. For these reasons, the limitations of traditional theoretical and practical approaches to creep and creep fracture are reviewed in relation to the advantages offered by an alternative methodology, termed the 0 Projection Concept, which introduces unified constitutive laws quantifying the precise shape of individual creep

Chapter 3 ConstitutiveLaws for High-Temperature Creep and Creep Fracture 109 strain/time curves and the detailed dependence of the creep curve shape on stress and temperature (1-3). Using new as well as previously published data sets for 0.5Cr0.5Mo0.25V ferritic steel, a material widely selected for the steam pipework and other high-temperature components of power and petrochemical plants, the model-based 0 relationships are shown to allow (i) interpolation and extended extrapolation of short-term creep and fracture properties for accurate and cost-effective prediction of long-term design data, (ii) quantification of the behavior patterns displayed under the multiaxial stress states encountered during service, and (iii) analysis of material performance under nonsteady stress and temperature conditions, allowing reliable estimation of the remaining creep life of serviceexposed components for safe life extension of operational high-temperature plants.

TRADITIONAL APPROACHESTO CREEPAND CREEPFRACTURE While creep can occur at all temperatures above absolute zero, significant creep strains are usually recorded only when diffusion can take place at temperatures above about 0.4 Tm, where Tm is the absolute melting point. Under high-temperature creep conditions, most materials display normal creep curves, as illustrated in Fig. 1 for 0.5Cr0.5Mo0.25V ferritic steel in the normalized and tempered condition. After the virtually instantaneous strain observed on loading at the creep temperature (eo), the time-dependent or creep strain (e) accumulates with time (t), so that the total strain (ec) at any instant is ec : eo -+- e

(1)

The eo value is a function of stress (cr) and temperature (T), ~o = fl (or, T)

(2)

whereas e varies not only with stress and temperature but also with time, as e = f2(cr, T, t)

(3)

No general agreement has been reached on the form of the equation that should be adopted to describe the accumulation of creep strain with time, even when normal creep curves are recorded at high temperatures (Fig. 1). Consequently, the creep curve shape is commonly discussed in terms of the changes in creep rate, (6 = de/dt) throughout a test. After the initial extension (eo), the creep rate usually decays gradually during the primary stage until a minimum or secondary rate is attained (km), after which the creep rate accelerates during the tertiary stage

110

R. W. Evans and B. Wilshire

1.0

r t~

o

N ~

0.5

E t_

0 Z

0.0 ! 0.0

t

J

0.5

1.0

Normalized time Constant-stress creep curves for 0.5Cr0.5Mo0.25V ferritic steel, plotted as normalized strain (e/ef) against normalized time ( t / t f ) , where 6f and tf are the strain and time to failure. The tests were carried out at stresses of (a) 310, (b) 235, and (c) 175 MPa at 838 K. The points are experimental creep strain/time readings, whereas the solid lines were constructed using Eq. 27. For clarity, only about one in ten of the creep strain/time readings taken during the tests are shown for each curve.

which terminates in fracture. Thus, Eq. 3 is presented in differential form as k = f3(t7, T, t)

(4)

k -- f4 (a, T, e)

(5)

or

The secondary creep rate is then assumed to be the parameter of greatest significance so that, since km does not vary appreciably with time or strain during the secondary period, Eqs. 4 and 5 reduce to

km

=

fs(a, T)

(6)

This relationship can be further simplified by assuming that the variables are separate and independent, allowing Eq. 6 to be rewritten as

ks = u(a)v(T)

(7)

Chapter 3

Constitutive Laws for High-Temperature Creep and Creep Fracture

111

In this way, the high-temperature creep behavior of a material can be characterized by evaluating the function u(cr) from measurements of the variations of km with stress at constant temperature, while v(T) can be determined from the dependence of ~m on temperature at a fixed stress. When this approach is taken, the temperature dependence of ~m is often represented by an Arrhenius expression of the form

~m O(v(T) cx e x p ( - Q c / R T )

(8)

where Qc is the activation energy for creep and R is the universal gas constant, (8.31 J mo1-1 K-l), so Qc has units ofjoules per mole. Several different expressions could be used for u(~r) but, most frequently, the stress dependence of Em is described using Norton's law (4), as ~m (X U (O') (X O"n

(9)

where n is a constant. Combining Eqs. 7, 8, and 9 then leads to a power-law relationship of the form Em

--"

Acrn e x p ( - Q c / R T )

(lO)

where A is a constant.

A. Power-LawCreep of Pure Metals Power-law relationships are widely used to describe the creep properties of metallic and ceramic materials at high temperatures. However, the constants in power-law equations are found to vary in different stress/temperature regions; i.e., A, n, and Qc in Eq. 10 are themselves functions of stress and temperature. For instance, with pure metals, the stress exponent is generally considered to increase with increasing stress, from n ~ 1 at low stresses to n -~ 4 or 5 at high stresses, with n then increasing continuously as the stress is increased further into the so-called "power-law breakdown" regime. Furthermore, under conditions such that n ~ 1 or n -~ 4 or 5, the measured Qc value depends on the test temperature selected. Above about 0.7Tm, Qc is frequently reported to be close to the activation energy for lattice self-diffusion (QL), whereas Qc values of approximately 0.5QL are recorded at temperatures around 0.4 to 0.7 Tm. The problem of variable "constants" is usually overcome by assuming that different creep mechanisms, each associated with different values of n and Qc, control the creep behavior displayed in different stress/temperature regimes. When n -~ 4 or 5, creep is known to occur by diffusion-controlled generation and movement of dislocations, with Qc ~ QL when lattice diffusion is easy at high temperatures (T -~ 0.7Tm and above), decreasing to Qc -~ 0.5QL -~ Qcore when diffusion takes place preferentially along dislocation cores as lattice diffusion becomes progressively more difficult as the temperature decreases toward about 0.4Tm (where Qcore

112

R.W. Evans and B. Wilshire

is the activation energy for diffusion along dislocation cores, sometimes called pipe diffusion). The transition from n -~ 4 or 5 at high stresses to n -~ 1 at low stresses is generally interpreted in terms of a change in the dominant creep mechanism from dislocation processes to diffusional creep processes which do not involve dislocation movement. Diffusional creep requires the flow of vacancies (and counterflow of atoms) from grain boundaries under tension to those under compression. In the diffusional creep regime when n -~ 1, the stress-directed vacancy diffusion may occur through the lattice so that Qc -~ QL at temperatures approaching 0.7 Tm and above (5, 6), whereas Qc becomes equal to the activation energy for diffusion along grain boundaries (Qgb -~ 0.5 QL) as lattice diffusion becomes difficult as the creep temperature is decreased toward 0.4Tm (5). While dislocation creep is often considered to be grain-size-insensitive, diffusional creep theories (5-7) predict that the creep rate should increase strongly with decreasing grain diameter (d). For this reason, Eq. 10 is often modified to feature the grain-size dependence of the minimum or secondary creep rate, as ~m = A t c r n ( l / d ) m

exp(-ac/RT)

(11)

where A' (:/: A) is a constant and m is the grain-size exponent. In relation to Eq. 11, when dislocation processes determine the creep properties of pure metals at high stresses, n -------4 or 5 and m -------0, with Qc ~" QL at high temperatures and Qc "" a c o r e at temperatures ranging from around 0.4 to 0.7Tm. When diffusional creep processes are thought to be dominant at low stresses, n = 1, m = 2, and Qc = QL when lattice diffusion predominates at high temperatures (a process known as Nabarro-Herring creep), whereas n = 1, m - 3, and Qc = Qgb when stress-directed vacancy transfer occurs along grain boundaries at lower temperatures (a process referred to as Coble creep). Various dislocation and diffusional creep processes can therefore be defined, each characterized by a different set of values for n, m, and Qc, suggesting that measured n, m, and Qc values can be compared with theoretical predictions in order to identify the mechanism controlling the behavior patterns displayed under different creep exposure conditions. This view has been popularized through the construction of deformation mechanism maps, which purport to show the creep process that is dominant within different stress/temperature regimes (8).

B. Power-Law Creep of Particle-Hardened Alloys Pure metals and even single-phase alloys rarely prove adequate for practical applications where resistance to creep at elevated temperatures is a requirement for satisfactory service performance. Consequently, many commercial creep-resistant alloys have been developed, with composition tailored and processing operations controlled to introduce a dispersion of fine, relatively stable precipitates and/or insoluble particles.

Chapter 3

Constitutive Laws for High-Temperature Creep and Creep Fracture

11 3

I

,--

1 0 -4

--

10-5

_

10-6

-

n-

10-7 -

L. 0

E 10-8 E

.m

t'-

N

1 0 .9

_

n-1

10-1o _

10-11 _ 20

I

I

I

I

I

30

50

100

200

300

Stress (MPa)

The log cr/log Sm curve for 0.5Cr0.5Mo0.25V ferritic steel at 838 K. The solid line was predicted using the 0 relationships. Also included are the measured minimum creep rate values recorded over the stress range covered in order to determine the 0 values (A), together with long-term data obtained independently for the same batch of steel (O, II).

W h e n the h i g h - t e m p e r a t u r e creep properties of particle-hardened alloys are described using power-law relationships, the o b s e r v e d b e h a v i o r patterns differ significantly from those n o r m a l l y recorded for pure metals and m o s t single-phase alloys. As illustrated in Fig. 2 for 0 . 5 C r 0 . 5 M o 0 . 2 5 V ferritic steel, the results can be taken to show a transition from n -~ 1 to n ~ 4 as the stress is increased but, at even h i g h e r stress levels, an e x t e n d e d region can be defined with n = 12. As with pure metals, a change from n -~ 1 to n -~ 4 can be interpreted on the a s s u m p t i o n that the d o m i n a n t creep m e c h a n i s m c h a n g e s from diffusional creep processes at low stresses to dislocation m e c h a n i s m s as the stress is increased, with microstructural studies showing that creep occurs by diffusion-controlled generation and m o v e m e n t of dislocations in the ferritic alloy matrix in both the n = 4 and n = 12 regimes (9). For particle-hardened alloys, the c h a n g e f r o m n -~ 4 to n ~ 10 or m o r e has b e e n reported to occur at a stress close to the rapid yield stress of the alloy at the creep t e m p e r a t u r e (10). This observation can be explained on the basis that, on e n c o u n t e r i n g particles, dislocations m o v i n g in the alloy matrix m u s t s u r m o u n t

114

R.W. Evans and B. Wilshire

the obstacles by climb and cross-slip in the n ~ 4 regime, but can cut through or bow between the particles when n > 4 at stresses exceeding the yield stress of the material. Yet, while creep of 0.5Cr0.5Mo0.25V steel occurs by dislocation movement in the ferrite matrix in both the n ~ 4 and n ---- 12 regimes, the increase in n value is accompanied by a corresponding increase in Qc such that, when n -~ 12, the measured Qc value of ~ 600 kJ mo1-1 is far higher than the activation energy reported for diffusion in the ferrite matrix ( ~ 240 kJ mo1-1). These observations are fully consistent with the fact that, at stress levels giving easily measurable creep rates at high temperatures, n values up to 50 or more and Qc values many times larger than the activation energies for matrix diffusion have been widely reported for many dispersion-strengthened alloys. An early attempt to rationalize the anomalously large n and Qc values recorded for particle-hardened alloys (11) suggested that creep occurs not under the full applied stress (or) but under a reduced stress (or - cro), such that ~m - -

A*(cr -

tTo) p e x p ( - Q c / R T

)

(12)

with A* :fi A ~ A' and p ~ 4, with Qc representing the temperature dependence o f ~m at constant (~r - t r o ) rather than at constant cr as in the determination of Qc (Eqs. 10 and 11). With the cro approach, n ~ p and Qc ~ Qc when cro ~ 0 or when cro cx or, whereas n > p and Qc > QS when tro is large and decreases with decreasing temperature (11). This concept has been modified and extended considerably over recent decades, with tro now almost universally referred to as a "threshold stress." Indeed, several recent reviews have summarized the many recent experimental and theoretical studies directed to threshold stress ideas; but, despite their increasing complexity, no dislocation mechanisms based on threshold stress concepts have yet been evolved to satisfactorily explain the anomalous power-law creep behavior of particle-strengthened alloys (12, 13). Moreover, there is no physical justification for using relationships such as Eqs. 10, 11, and 12 for extrapolation of creep data into stress-temperature ranges that have not been investigated experimentally. With n decreasing with decreasing applied stress (Fig. 2), linear extrapolation of conventional log or/log ~m plots constructed using results obtained in tests of short duration seriously overestimates long-term performance, so alternative methodologies are required for accurate prediction of long-term creep data.

C. Parametric Proceduresfor Creep Data Extrapolation Under high-temperature creep conditions, as the creep strain accumulates gradually with time, grain boundary cavities and microcracks usually nucleate and develop to cause eventual failure. Many theoretical studies have been directed toward identification of the processes governing intergranular creep fracture; but, as with the micromechanisms evolved to explain creep strain accumulation, the

Chapter 3

Constitutive Laws for High-Temperature Creep and Creep Fracture

115

theories developed have yet to prove capable of quantitative prediction of long-term behavior. At high temperatures, the creep rupture life (tf) often varies inversely with the secondary creep rate (km), a phenomenon referred to as the Monkman-Grant relationship (14); thus, (13)

tf -- M/~m

where M can be constant over broad ranges of stress and temperature for a given alloy. Combining Eqs. 10 and 13 then gives Sm

--

(14)

M / t f = Act" e x p ( - Q c / R T )

so that, as with the secondary creep rate, power-law relationships can be used to describe the stress and temperature dependence of the creep life. However, with the secondary creep rate and the creep-rupture life linked through Eq. 13, the log ~r/log tf plots must curve because the log or/log Sm plots curve (Fig. 2), with the result that linear extrapolation of data derived from short-term tests overestimates not only long-term creep performance but also long-term stress-rupture properties (Fig. 3). An accurate knowledge of the stresses that engineering materials can sustain without creep fracture occurring within the planned design life is essential for design of high-temperature components and structures. So, with conventional

2"6

!

I

I

I

I

I

I

2.5 2.z~

~ 2.3

v~ 2"1 or 19 1.8 I-7

m

1.0

I

1.5

I

2.0

I

I

I

2.5 3.0 3.5 Logl0(Rupture Time, Hrs)

I

t~.O

/+5

The log or/log tf curves for 0.5Cr0.5Mo0.25V ferritic steel at 823, 848, and 873 K. The solid lines were predicted using the 0 relationships, whereas the error bars represent the scatter for various stress/temperature conditions in multilaboratory long-term stress-rupture data.

116

R.W. Evans and B. Wilshire 500 " 723 K

I

I

I

I

I

I

9 773 K o 823 K 98~8K 300 -- x 873 K

Z

200

0

t,,t..i t--

150

100

70

15

16

I

1

I

I

i

17

18

19

20

21

22

~=tog t - 17.3001T

Stress-rupture data for 0.5Cr0.5Mo0.25V ferritic steel showing that the results obtained at temperatures from 723 to 873 K can be superimposed onto a single line using the parametric relationship proposed by Orr et al. (15).

power-law relationships incapable of allowing accurate data extrapolation (Figs. 2 and 3), numerous attempts have been made to develop "parametric procedures," i.e., methods of plotting stress-rupture data using a "correlation parameter" that permits results obtained over a range of temperatures to be superimposed onto a single master curve. The use of these parametric procedures can be illustrated by considering the time-temperature parameter, P (or), introduced by Orr et al. (15), namely, P(cr) -- log tf

-

-

P~ T

(15)

where P is an arbitrary constant chosen so as to fit the data empirically. Figure 4 shows the superimposition of stress-rupture values obtained at temperatures from 723 to 873 K for 0.5Cr0.5Mo0.25V steel using Eq. 15. The marked curvature of this plot is hardly surprising, because Eq. 15 can be derived by rearranging Eq. 14 as tf

.

exp(-Qc/RT)

-- f*(cr)

where f * (or) is a stress function. Taking logs then leads to the expression P(cr) = l o g t f - ( Q c / R ) / T

(16)

which has the same form as Eq. 15. The plot in Fig. 4 therefore curves in the same way as standard log or/log tf based on power-law descriptions of creep-rupture properties (Fig. 3). Furthermore, the problem is not solved by using one of the

Chapter3 ConstitutiveLaws for High-TemperatureCreep and Creep Fracture 117 many other correlation parameters introduced with the aim of extrapolating stressrupture data (16, 17). The unknown curvatures of parametric plots, such as Fig. 4, make extrapolation of creep rupture lives extremely unreliable. Indeed, the uncertainties associated with conventional methodologies restrict extrapolation to three times the longest test figures available; i.e., only when tests lasting up to 30,000 hours have been completed can 100,000-hour estimates be considered reasonable. Consequently, expensive and protracted test programs must be undertaken to obtain reliable longterm design data.

THE 0 PROJECTIONCONCEPT Long-term design data for engineering steels have been supplied mainly by laboratories equipped with many constant-load stress-rupture machines. National and international standards have been laid down to cover the test procedures that should be adopted; but, even assuming that laboratories operate to these standards, the high degree of scatter in the long-term stress-rupture properties generated through multilaboratory test programs (Fig. 4) does not promote confidence in conventional data acquisition and analysis procedures. When standard stress-rupture tests are performed, each test quantifies only the rupture life (tf) and the strain to failure or creep ductility (ef), specifying the point of failure at a fixed stress and temperature. Yet, even when creep tests are carried out, only a few additional parameters are commonly monitored, such as the minimum or secondary creep rate in academic studies or the time to certain preselected strain levels for engineering purposes. Hence, these traditional procedures disregard the primary and tertiary stages, so that a major proportion of the information available from a creep curve is totally ignored. By not quantifying the rate of creep strain accumulation under different test conditions, so that detailed stress/strain/time/temperature relationships can be defined, full advantage cannot be taken of the powerful finite-element codes now being developed for high-temperature design. These arguments suggest that active consideration should be given to the adoption of alternative testing methodologies and data processing methods, with one option involving the analysis of the full creep strain/time curves obtained using high-precision constant-stress creep equipment (1). Although the techniques for achieving constant-stress conditions were established over half a century ago, few such machines are currently in use. Admittedly, with materials displaying very low creep ductilities, virtually identical creep curves will be recorded, irrespective of whether constant-stress or constant-load procedures are employed. In contrast, with materials of even modest ductility (5-10%), the continuous increase in stress that occurs as the specimen cross-section decreases with increasing creep strain

118

R.W. Evans and B. Wilshire ]A t

/

l

'-

A

,/ /

/

BI

/

B

Time -----,Schematic diagram illustrating differences in creep strain/time behavior found using

constant-stress (A and B) and constant-load test procedures (A' and B'). Differences between the constant-stress and constant-load curves are greater with the primary-dominated curves often displayed at high stresses (A and At) than with the tertiary-dominated curves generally exhibited at low stresses (B and B').

during a constant-load test can cause considerable distortions in true creep curve shape. In general, the primary stage becomes less dominant and the tertiary stage more pronounced with decreasing stress at a fixed creep temperature (Fig. 1). Under these circumstances, constant-load test methods cause serious curve shape distortions when significant primary strains are displayed at high stresses, with lower deviations from the tertiary-dominated curves typically observed at low stresses (Fig. 5). In aiming to use high-stress data to predict low-stress behavior patterns, projecting creep properties derived from short-term constant-load tests can therefore introduce significant errors, a problem that can be avoided by using constant-stress procedures. Provided only that high-precision creep curves are determined over sensible stress/temperature ranges, it should be possible to analyze constant-stress data in terms of constitutive equations that quantify the stress/strain/time/temperature behavior in a manner permitting reliable estimation of long-term creep and creep fracture properties. These ideas formed the basis for the initial studies that led to the development of the 0 Projection Concept ( 1 ) - - a n approach defining materials constitutive relationships that can be derived by modeling the processes governing creep strain and creep damage accumulation at high temperatures (2, 3).

A. Modeling of Primary Creep Behavior When creep occurs by diffusion-controlled generation and movement of dislocations, the multiplication of dislocations early in the creep life usually causes

Chapter 3 ConstitutiveLaws for High-TemperatureCreep and Creep Fracture 119 the material to strain-harden. Simultaneously, recovery processes such as climb and cross-slip allow the dislocations to rearrange into low-energy configurations. As the creep strain accumulates during the primary stage, the dislocation density increases and the dislocation arrangements become progressively less uniform. Dislocations attempting to move across the crystal therefore encounter barriers to continued movement, the strength of the barriers being dependent on the local dislocation density. This situation can be modeled (2, 3) on the basis that a dislocation moving under the action of a local stress, or, approaches a barrier of strength, 6. When a < 6, thermal fluctuations are needed to help the stress to continue movement. Let the magnitude of the thermal fluctuation needed to overcome the barrier be q*. The rate at which thermal fluctuations of this magnitude occur is then given by an exponential expression of the form e x p ( - q * / k T ) . Clearly, when a > 6, q* will be zero because the local stress is sufficiently high to allow the barrier to be overcome without the aid of thermal fluctuations. At the other extreme, when the local stress is zero (a -- 0), q* must be equal to the total thermal energy required to surmount the barrier (say, q* = A F ) . On this basis, for the general situation when 0 < a < 6, the thermal energy needed will be

q*--[AF(6-a)/6] The local creep rate will therefore depend on the frequency with which thermal fluctuations of this magnitude occur, so that de = B e x

1-

(17)

with the factor B containing terms that describe the general geometry of the local region, including the local dislocation density. The strengths of the barriers opposing dislocation movement are then altered progressively by work-hardening and recovery, as

d6

de =h--r dt dt

(18)

where h ( = d a / d e ) is the rate of work-hardening and r ( = d a / d t ) is the rate of recovery. Early in the creep process, 6 will not be very different from a , so that Eq. 18 becomes (6

-

a)

=

he

-

rt

(19)

Similarly, A F can be written as

A F -- 6 ba

(20)

where b is the Burgers vector and a is the activation area, i.e., the area of the slip plane swept out by the dislocation in overcoming the obstacle. Substitution of

120

R.W. Evans and B. Wilshire

Eqs. 19 and 20 into Eq. 17 gives

de = B exp dt

_ba -~

(he - rt)

]

(21)

Since (or - 8) is small, the exponential term can be approximated by the first two terms of a Taylor series, leading to the differential equation

de Bbah [ rba ] dt + kT . e = B l + - - ~ - . t

(22)

For the initial conditions e = 0 at t = 0, this differential equation has the solution, 8, ---

rba)}r k T I B -+- r ( 1 - exp ( -k-----~" t Bbah -b

+ ~t

(23)

In general form, Eq. 23 can be written as

e

"-- O1 (1

- e -~

(24)

+ kst

with the parameters describing the primary creep curve, 01 and 02, given by

O1 = B bah B +

and

rba

0 2 = kT

and with ks (= r~ h) representing the steady-state creep rate attained eventually when the rate of work-hardening is balanced by the rate of recovery. Differentiation of Eq. 24 then shows that the primary creep rate, kp (= k - ks) decreases as a linear function of the primary creep strain ep ( = e - kst), as ~p = 0102 e-O2t

-- 02ep

(25)

indicating that primary creep obeys first-order reaction-rate kinetics, with the rate constant given by the parameter 02. The meanings of the terms, 01 and 02, in relation to the primary creep curve should then be evident from Fig. 6a.

B. Modeling of Tertiary Creep Behavior Equation 24 describes a creep curve characterized by a creep rate that decays gradually until, at long times when the total primary strain approaches 01, a steady-state creep rate is attained (ks). The form of Eq. 24 is fully consistent with the constant-stress curve shapes displayed when, in the absence of any phenomenon that could cause an acceleration in creep rate, continuously decaying creep curves are recorded for materials such as single crystals and polycrystalline superpurity aluminum (2). However, under high-temperature creep conditions,

Chapter 3

Constitutive Laws for High-Temperature Creep and Creep Fracture

primary stage

1

tertiary stage

Cp: e 1( 1 - e-e2t)

ct : e3 (eeJ-1)

I

cJ u~ c~ r u

e2~deter m ines f the curvature

of primary creep

time

121

ethe 1 quantifies tatar primary strain / I

#-

--fracture

e~determines / II the curvature I of tertiary creep J e3 scales I ~_ ~ the tertiary I j.%," creep sfratn

=

time

!

The 0 Projection Concept envisages normal creep curves in terms of a decaying primary component (ep) and an accelerating tertiary component (et). The parameters 01 and 03 scale the primary and tertiary curves with respect to strain, whereas the terms 02 and 04 are rate parameters that quantify the curvatures of the primary and tertiary stages, respectively.

most polycrystalline metals and alloys exhibit normal creep curves, as illustrated for 0.5Cr0.5Mo0.25V steel in Fig. 1. When normal curve shapes are recorded, the acceleration in creep rate during the tertiary stage is usually associated with phenomena such as intergranular damage accumulation. Grain boundary cavities and microcracks generally form early in the creep life, the numbers and sizes of the cavities and cracks increasing with increasing creep strain. The onset of the tertiary stage does not then coincide with damage nucleation, but rather with the development of intergranular damage to an extent sufficient to cause an acceleration in creep rate. To provide a quantitative description of tertiary behavior, it is necessary to define the ways in which tertiary processes such as intergranular damage accumulation modify the creep curve shape predicted by Eq. 24. This equation indicates that, after a relatively rapid decay in creep rate during the early stage of a creep test, the creep rate decreases more slowly with increasing creep strain. Even when intergranular cavities and cracks nucleate early in the creep life, the initial damage levels will not be sufficient to have any significant effect on the initial rapid decay in creep rate. However, as the numbers and sizes of the cavities and cracks increase with increasing strain, the resulting increase in creep rate will eventually offset the gradual decay in creep rate toward ks. These events can be modeled in the manner suggested by Kachanov (18). This approach introduces a damage parameter, co, which is initially zero for undamaged material. The value of co then increases as the damage levels increase. Instead of approaching ks, the creep rate will therefore accelerate (Fig. 7a), with the tertiary creep rate (kt --- det/dt) given by d~t

dt

= ks (1 + co)

(26)

with the rate of damage accumulation & (= dco/dt)- C(det/dt) where C is a

122

R.W. Evans and B. Wilshire

a

E

Frocture j

!

as ~,--, 0,

i

s

f

time ~

time

In the absence of any processes capable of causing a tertiary acceleration, the creep rate will decay continuously, eventually reaching a steady-state value (ks) at long times when the primary creep strain approaches 01. However, the decaying primary rate can be offset by the tertiary acceleration caused by phenomena such as damage accumulation, giving a minimum creep rate (km) which may be significantly greater than the steady-state value (ks). (a) A pronounced tertiary stage is then found for creep-ductile materials which can sustain high damage levels before failure occurs, whereas (b) little or no tertiary stage is observed with creep-brittle materials.

constant and et (-- e - - E p ) is the tertiary creep strain. Integration replaces the "steady-state" term in Eq. 26 to give e = O1 (1 - e -02t) + 03 (e 04t

--

1)

(27)

where 03 (= 1/ C) scales the tertiary strain and 04 (= C~s) quantifies the curvature of the tertiary stage (Fig. 6b). Differentiating Eq. 27 then shows that the tertiary creep rate, kt, increases linearly with the tertiary creep strain, et, as Et "~- (93(94 eOat = 04(03 "t- Et)

(28)

demonstrating that the tertiary stage also obeys first-order reaction-rate kinetics, with the rate constant given by the parameter 04. Although Eq. 27 was derived on the basis that the tertiary stage is usually caused by damage accumulation, relationships of this form will therefore be obtained for any tertiary process obeying first-order kinetics (Eq. 28), with the rate parameter being process-specific.

C. Tensile Creep Failure While Eq. 24 describes the constant-stress creep curves for materials like superpurity aluminum when the creep rate decays gradually to a steady-state value (es), Eq. 27 must be used to represent normal curves showing clearly defined primary and tertiary stages. When normal curves are displayed, a minimum rate occurs when the decaying primary rate is offset by the acceleration caused by tertiary processes, such that the minimum rate (~m) may be greater than or equal to es, as illustrated in Fig. 7a. Yet, while Eq. 27 has been shown to accurately describe the

Chapter 3

Constitutive Laws for High-Temperature Creep and Creep Fracture

123

normal creep strain/time curves recorded for a wide range of materials (19-21), to provide a complete description of materials behavior at high temperatures, it is necessary to define some suitable failure criterion. During high-temperature creep exposure, most materials fail in an intergranular manner when the level of creep damage reaches some critical value. This critical damage level depends on the material and on the stress/temperature conditions imposed. However, under standard tensile-creep conditions, the damage levels generally increase with increasing tertiary strain (Eq. 26). Under these circumstances, the stress-rupture properties may be determined by defining the limiting creep strain or creep ductility; i.e., since the detailed shape of the creep strain/time curve is known, the time to fracture (tf) can be estimated as the time taken for the accumulated creep strain to reach the total creep ductility (el). This failure criterion is particularly convenient since it allows precise characterization of creep-ductile and creep-brittle behavior. Creep-ductile materials show pronounced tertiary periods (Fig. 7a), even though the creep-damage levels are increasing with increasing tertiary strain. In contrast, creep-brittle materials are not creep-damage-tolerant, so developing microcracks propagate rapidly to cause low-ductility failure, terminating the creep curve with little or no tertiary stage (Fig. 7b). To describe the truncated curve shapes exhibited by creep-brittle materials (when 8f/Ep ~ 1), Eq. 27 must be modified by expanding the tertiary exponential term by means of its Taylor series, as e = 01 (1 - e -~

+ 03(04t + (04t)2/2 . . . . . )

(29)

Since 04t << 1, higher-order terms can be ignored, giving 8 = 01 (1 - e -~

+ 0304t

(30)

Equation 30 then describes the truncated creep strain/time curves found for creep-brittle materials, while Eq. 27 quantifies the normal curve shape observed for creep-ductile materials (Fig. 7).

ANALYSISOF TENSILECREEPDATA Because of its widespread selection for the high-temperature pipework and other components in electricity plants and petrochemical installations, multilaboratory test programs have been carried out to provide long-term stress-rupture data for 0.5Cr0.5Mo0.25V steel in the normalized and tempered condition (22). However, the resulting international data sets are characterized by wide scatter bands, typically -t-20% in stress, which is equivalent to about an order of magnitude in creep life (Fig. 4). Since designs are based on the premise that failure must not occur within the planned operational life, the minimum property values are used, so designs are conservative.

124

R.W.Evans and B. Wilshire

Designers then encounter further problems caused by the fact that thermal and mechanical loading conditions vary throughout most components and the prevailing conditions usually change with time. Ideally, then, elastic-plasticviscoplastic finite-element procedures should be used to perform the necessary design calculations, providing a check on the relatively crude results achieved with such conventional design codes, as ASME B31.1. Unfortunately, utilization of advanced finite-element methods is limited by the lack of creep data for most commercial steels, plus the fact that the scatter in properties such as the time to 1% creep strain is even greater than for multilaboratory stress-rupture values. Indeed, the problems imposed by the lack of reliable long-term creep strain and creep life data for low-alloy structural steels prompted the initial studies that led to the development of the 0 Projection Concept (1).

A. Analysisof Creep Curve Shape For 0.5Cr0.5Mo0.25V ferritic steel in the normalized and tempered condition, constant-stress creep tests were carried out at temperatures from 808 to 868 K at stresses selected to give maximum creep lives of only about 1000 hours (2). Under all test conditions, normal creep curves were recorded, displaying clearly defined primary and tertiary components (Fig. 1). During each test, well over 100 creep strain/time readings were taken, allowing the best values of the four 0 parameters in Eq. 27 to be determined for each curve using a nonlinear least-squares curvefitting routine described elsewhere (2). As evident from Fig. 1, Eq. 27 accurately describes the shape of individual curves, with the results of the entire test program showing that each 0 parameter varied systematically with stress and temperature. At each creep temperature, a linear relationship was found between stress and In Oi (where i = 1, 2, 3, 4). For the strain-like parameters, 01 and 03 (Fig. 6), the linear stress/In 01 and stress/In 03 plots observed at different temperatures could be superimposed onto a single line by normalizing the creep stress by the rapid yield stress at the appropriate creep temperature (Cry); i.e., the parameters 01 and 03 are characterized by a temperature dependence comparable with that of the yield strength of the material (Fig. 8a). Moreover, for the rate parameters, 02 and 04, (Fig. 6), the stress/In 02 and stress/In 04 plots at each temperature could be superimposed by incorporating an Arrhenius term with a suitable activation energy (Fig. 8b). Thus, the variation of each 0 parameter with stress and temperature can be written as 01 = G1 exp H1 (a/Cry) 02 =

G2 e x p - [ ( Q 2 -

Hzcr)IRT]

03 -- G3 exp H3(cr/cry) 04 = G4 e x p - [ ( Q 4 -

H40)/RT]

Chapter 3

a

Constitutive Laws for High-Temperature Creep and Creep Fracture

-2

1

I

I

5

1

-3

m

9

-4

9Jo

9 9 ~

-5

9

..mO

~ 1 9

j~//

-6

c" m

125

1

9

0

-7

o/

-8-

-

-1

--

-2

-

-3

--

-4

[] -9

-

-

-10

-

-11

-

A / []

-12 0.6

O0 iD 9

A

808K 838K 868K

I

I

I

I

0.8

1.0

1.2

1.4

-5 1.6

Stress/Yield Stress

b

30

I

I

j

I

80

- 7O 25

-60

,,.-.,,

}-

nu

[]

./P

o 00 o

04 -

20

i--rr

50

+

o

P~

r 0 ")

X

r" n

15

-

40

-

30

-

20

x

_.=

-

z~

10 150

c~

I 200

I 250

oo

808K

iD

838K

AA

868K

I

I

3OO

350

Stress (MPa)

The stress and temperature dependences of (a) the strain-like parameters 01 and 03 and (b) the rate constants 02 and 04 for 0.5Cr0.5Mo0.25V ferritic steel at 808-868 K.

126

R.W.Evans and B. Wilshire

where Gi and Hi (with i = 1, 2, 3, 4) are constants for the material, while Q2 and Q4 are the activation energies associated with the rate parameters 02 and 04, respectively. For 0.5Cr0.5Mo0.25V ferritic steel, Q2 = 224 kJ mo1-1. This value is close to the activation energy for lattice self-diffusion in ferrite, consistent with TEM observations showing that creep occurs by diffusion-controlled generation and movement of dislocations in the ferrite matrix over the entire stress/temperature range investigated. A situation such that Q2 = Q4 is then found for materials such as polycrystalline copper, tested under conditions resulting in the tertiary stage being governed solely by the formation, growth, and linkup of grain boundary cavities (20). Since cavity development is strain-controlled, similar activation energies would be expected for the primary and tertiary rate constants. In contrast, with 0.5Cr0.5Mo0.25V ferritic steel, many processes influence the tertiary acceleration in creep rate. Certainly, intergranular damage accumulates during the tertiary stage, leading to intergranular failure. However, considerable changes in carbide size, type, and distribution also occur, particularly in low-stress tests (Fig. 9). Indeed, at low stresses, denuded zones form and grain boundary migration can take place, with migration leaving behind developing cavities (Fig. 9d). The tertiary curve shape is also affected by neck formation, especially when high ductilities are recorded in tests of short duration. All these processes affect the tertiary acceleration, but their relative importance depends on the stress and temperature conditions imposed. Since each process will be characterized by a different activation energy, the overall activation energy (Q4) must change with changing test conditions, in line with the results presented in Fig. 8b. As confirmed subsequently for a wide range of materials displaying normal creep curves (19-21), the results for 0.5Cr0.5Mo0.25V ferritic steel demonstrate that Eq. 27 provides an accurate description of the shape of individual creep curves. Moreover, Eq. 31 defines the variations of the 0 parameters with stress and temperature, so quantifying the variations in creep curve shape with changing test conditions. Thus, the 0 methodology introduces equations that precisely define stress/strain/time/temperature behavior in a manner fully consistent with the micromechanisms governing primary and tertiary creep characteristics.

B. Prediction of Power-Law Behavior Having established that the 0 relationships provide a model-based description of creep properties, the next logical step is to show that the 0 methodology offers a convincing alternative to traditional steady-state approaches to creep. This can be achieved by proving that the 0 equations can predict and explain all features of power-law behavior patterns (Eqs. 10 and 11).

Chapter 3

Constitutive Laws for High-Temperature Creep and Creep Fracture

127

Transmission electron micrographs showing (a) interphase VC carbides and (b) VC precipitates associated with dislocations in 0.5Cr0.5Mo0.25V ferritic steel in the normalized and tempered condition; in contrast, (c) H-type carbides form and (d) grain boundary migration occurs, leaving cavities at the original boundary locations after service exposure for 64,000 hours at around 838 K in electricity-generating plant.

W i t h Eq. 27, the creep rate r e a c h e s a m i n i m u m v a l u e after a t i m e (t') g i v e n by 1 01022 t' -- ~ In ~ 02 + 04 03 02

(32)

so that the minimum creep rate is defined as ~:m -- 0102 exp(--Ozt') + 0304 exp(O4t')

(33)

12B

R.W. Evans and B. Wilshire

From Fig. 8, the values of the four 0 parameters can be determined for a fixed stress and temperature, so that/~m can be calculated using Eqs. 32 and 33. However, the linearity of the plots in Fig. 8 suggests that the 0 data should allow not only interpolation but also reasonable extrapolation of creep properties. For this reason, in the initial study that led to the development of the 0 Projection Concept (1), 0 data derived from tests of relatively short duration were used to predict the log or/log ~m behavior over a wide stress range at 838 K (the normal operating temperature of UK power plants). In Fig. 2, the predicted log or/log ~m curve is compared with measured values determined from the short-term creep curves used to derive the 0 data, together with ~m values measured in long-term tests carried out independently for the same batch of 0.5Cr0.5Mo0.25V ferritic steel (23). Clearly, the predicted curve is in excellent agreement with the experimentally determined values (Fig. 2), showing that the 0 relationships quantify the long-term effects of progressive changes in microstructure even for steels known to exhibit marked changes in carbide dispersion during prolonged creep exposure (Fig. 9). When traditional power-law approaches are adopted, the smooth log or/log ~m curves are fitted by a series of straight lines with gradients preselected to match the n values expected for different creep mechanisms. Thus, in Fig. 2, straight-line segments can be chosen to show n -------12, n ~ 4, and n ---- 1 in progressively lower stress regimes. Moreover, using the 0 data to calculate Qc in Eq. 10 over this stress range, Qc ~ 500 kJ mo1-1 when n -~ 12, decreasing to Qc = 150 kJ mo1-1 when n ---- 1 at low stresses. Hence, in addition to allowing accurate prediction of measured long-term data (Fig. 2), the 0 methodology predicts the exact decrease in n and Qc observed with decreasing stress, even though the 0 data were derived by analysis only of short-term curves recorded at stresses such that n ~- 12. If different creep mechanisms became dominant in different stress/temperature regimes, as is commonly assumed to explain variations in n and Qc values, it would be impossible to extrapolate data recorded over a limited range of test conditions to predict the exact behavior patterns displayed in an entirely different stress/temperature regime. With 0.5Cr0.5Mo0.25V ferritic steel, over the range of test conditions covered in Fig. 8, creep occurs by diffusion-controlled generation and movement of dislocations in the ferrite matrix. Yet, extrapolation of the high-stress 0 data in Fig. 8 predicts the observed decay in n value toward n ~ 1 (Fig. 2), when creep is traditionally assumed to take place by diffusional creep processes. Although diffusional creep theories are particularly elegant (5-8), the experimental evidence supporting diffusional creep concepts is not convincing (24, 25), suggesting that essentially the same dislocation creep processes are dominant at all stress levels (3). This view is supported by TEM studies carried out for 0.5Cr0.5Mo0.25V steel after long-term creep exposure (9), which show that, while the dislocation density decreases progressively with decreasing applied stress, dislocation configurations similar to those found at high stresses are also observed in the so-called n -------1 regime at low stresses (Fig. 10). No valid objection to the

Chapter 3 ConstitutiveLaws for High-TemperatureCreep and Creep Fracture 129

~|[~ll~Illlol~ Transmissionelectron micrographs showing the dislocation arrangements developed (a) within the grains and (b) in grain boundary regions (dark field) for 0.5Cr0.5Mo0.25V ferritic steels after 50,000 hours exposure at approximately 40 MPa at 838 K.

0 methodology can therefore be raised on the assumption that the curvatures of log a~ log ~m plots are attributable to changes in creep mechanism. On this basis, measurements of parameters such as n and Qc in power-law relationships do not offer a meaningful procedure for identifying creep mechanisms or for constructing deformation mechanism maps. Instead, the changes in n and Qc with changing stress and temperature are simply consequences of the complex dependence of the minimum creep rate on creep curve shape and hence on the variations in curve shape with changing test conditions. These changes in curve shape can now be described quantitatively using the 0 relationships (Eqs. 27 and 31) which, in turn, can be discussed logically in terms of the processes governing primary and tertiary creep.

C. Prediction of Long-Term Design Data Although the predicted log a~ log ~m curve in Fig. 2 was derived by analysis of constant-stress creep curves with a maximum duration of little more than 1000 hours, the 0 relationships predict the measured creep rate values for conditions giving creep lives in excess of 100,000 hours. This compares with conventional parametric procedures which allow reasonable data estimates to be made only for test conditions giving creep lives up to three times the longest reliable results available. The 0 Projection Concept therefore offers a means of reducing dramatically the scale and costs of the long-term multilaboratory test programs currently needed to obtain engineering design data for high-temperature plant. Moreover, since the

1 30

R . W . E v a n s and B. Wilshire

0 relationships describe the full creep curve shapes, it is possible to predict various long-term creep strain or creep rate parameters, so that modern finite-element codes can be used for design calculations. However, the forms of the relationships in Eq. 31 are not convenient for interpolation and extrapolation of creep properties. For this reason, an alternative empirical method for analyzing 0 data has been evolved, which is efficient when programmed into modern computer-based design procedures (2). For 0.5Cr0.5Mo0.25V ferritic steel, as for many other materials (19-21), a linear relationship exists between stress and In Oi at each creep temperature (Fig. 8). Furthermore, since only narrow ranges of temperature are usually of practical significance (so that 1 / T is almost linear with T), the 0 data at any stress can be adequately described by linear plots of In Oi against temperature. Each 0 parameter can therefore be described by a general empirical function of stress and temperature, as In Oi = ai + bier + ci T + diet T

(34)

where ai, bi, ci, and di are constants for a material (with i = 1, 2, 3, 4). These coefficients can be evaluated by multilinear least-squares regression analysis of the stress/In Oi plots at various temperatures, with the values derived in this way for 0.5Cr0.5Mo0.25V steel listed in Table 1. Once the 16 coefficients associated with Eq. 34 are determined for a material (Table 1), the values of the four 0 parameters can be derived easily, so that any creep strain or creep rate parameter can be computed conveniently for any stress and temperature. For instance, minimum creep rate values can be computed using Eqs. 32 and 33. Similarly, parameters needed for practical purposes may also be determined rapidly; e.g., the time to reach some specified creep strain (ex) is obtained by solving (numerically) the equation 01(1

-

e -02t) --[- 0 3 ( e 04t -

1) - ex = 0

(35)

for t = (tx). Having computed this time, the creep rate at a strain of et is derived simply by substituting the value for tx into Eq. 33. Then, by placing ex = ~f, where

ABLE 1 Uniaxial Creep Constants ai ln01 ln02

-0.6934

bi x 102

0 . 5 4 3 7 • 101

ci

0.8221 x 10 -1 -0.1389

• 10 ~

ln03

-0.2096

x 102

-0.5958

x 10 -1

ln04

-0.2593

x 102

-0.1984

• 10 ~

8f

-0.5212

x 10 ~

-0.3637

x 10 -1

di

0 . 6 9 6 8 • 10 -1 -0.2675

- 0 . 6 7 5 8 x 10 - 4

• 10 -1

0.2056 x 10 -3

0 . 1 2 6 5 x 10 -1

0.1111 x 10 -3

0 . 6 3 2 9 x 10 - 2 -0.2954

x 10 - 2

0.2751 • 10 -3 - 0 . 4 8 8 9 x 10 - 4

Chapter 3 ConstitutiveLaws for High-TemperatureCreep and Creep Fracture 131 ef is the creep ductility so that tx = tf, Eq. 35 also allows rapid calculation of rupture lives. This method of computing stress-rupture data is particularly convenient since, for 0.5Cr0.5Mo0.25V ferritic steel, the creep ductility can also be described reasonably by a general linear function of stress and temperature (2), as

In E f

--

a + ba + c T + da T

(36)

with the coefficients a, b, c and d also listed in Table 1. Using this procedure for calculation of rupture lives, the predicted log a~ log tf curves in Fig. 3 are clearly well within the scatter bands characterizing the multidisciplinary stress-rupture data for this steel. However, because of the wide scatter bands, this result could not be defined as conclusive. So, in collaboration with a consortium of UK power industries, an extended study has been made to compare the creep life predictions made using the 0 methodology with measured data obtained independently for the same batches of various power-plant steels. The initial results of this extensive validation exercise have been produced for a 1CrMoV rotor steel, again showing an excellent correlation between the 0 predictions and the independently measured long-term stress-rupture data (21). The accuracy of the predictions made for various structural steels indicates that the 0 methodology permits accurate extrapolation over time scales considerably greater than the maximum permissible with traditional parametric methods. Yet, care should be exercised in using the 0 equations for extrapolation outside the temperature ranges covered to derive the experimental 0 data. For instance, with 0.5Cr0.5Mo0.25V ferritic steel, the carbide type and distribution differ markedly above and below about 900 K, so data obtained over the present temperature range of 808 to 868 K should not be employed to predict behavior patterns at substantially higher temperatures. Similarly, the gradual decrease in creep ductility with decreasing stress, as represented by Eq. 36, need not continue to very low stresses. Indeed, for several steels, there is evidence to suggest that the creep ductility decreases to a minimum and then increases again in tests of very long duration. This may not pose particular difficulties in the calculation of creep life because, with the primary stage becoming less pronounced and the tertiary stage more dominant with decreasing stress (Fig. 1), so that creep strain accumulates rapidly only late in tertiary stage in tests of long duration, variations in creep ductility should result only in relatively modest changes in the calculated values of the rupture life. As with any other forecasting technique, the accuracy of the 0 predictions must inevitably improve as the extent of the extrapolation decreases. In seeking to predict long-term creep properties, projections should therefore be based on comprehensive high-precision data sets determined over wide ranges of stress at several different temperatures. For this reason, no claim is made that the 0 methodology obviates the need for long-term test measurements, but merely that this predictive technique can radically reduce the scale and costs of the multilaboratory test

132

R.W.Evans and B. Wilshire

programs currently needed to obtain long-term design data. Even so, used sensibly, once the short-term 0 properties of a material are determined accurately, any creep strain, creep rate, or creep time parameter can be calculated for different stresses over the temperature ranges investigated, providing a highly computer-efficient materials engineering input to modern computer-based design codes.

CREEPUNDER MULTIAXIALSTRESSSTATES Many industrial components are required to operate under conditions of stress that are not simple uniaxial. These multiaxial states can be complex and can change with time due both to changes in external loading and to internal stress relaxation. In these circumstances, the designer needs to know the creep response of the material to the general stress tensor (although simplifying assumptions are often made in order to reduce the problem to one of approximately uniaxial stress). For ostensibly steady-state conditions, it will be necessary to understand the relationship between rupture life and the strain tensor. For more general design, numerical methods will be required, and these will need relationships similar to Eqs. 3 and 4 with the uniaxial quantities replaced by the relevant tensors. The important early work of Johnson and Henderson (26) suggested that as far as rupture life was concerned, metals and alloys could be divided into two classes, depending on their response to certain functions of the stress components. Thus, materials like aluminum had rupture lives that were dependent on the value of the effective stress, t~, whereas copper showed a relationship between life and the maximum principal stress, trl. Further work (27) has indicated a more complex dependence and other functions of the stress might be involved. In general, the rupture life, tf, is tf "-" (OlO" 1 -~- flJl + Y J2) -x

(37)

where c~, fl, y, and Z are experimental constants, Jl is the first stress invariant (proportional to the hydrostatic stress), and J2 is the second stress invariant (proportional to the von Mises effective stress). Thus for aluminum ct and fl are small whereas y is small for copper alloys. The experimental determination of the constants requires considerable experimentation and, even when they are available, there are serious problems with interpolation and extrapolation. These difficulties are of the same kind as those discussed in Section II.C except that the methodologies required are even less well known. For those situations where it is necessary to calculate the whole of the strain rate tensor, it is usual to proceed through a potential theory (28-31) to establish the dependence on stress state. Using some simplifying assumptions, including isotropy and the negligible effect of the third stress invariant, relationships like ! Eij - - ~,O'ij

(38)

Chapter 3

ConstitutiveLaws for High-Temperature Creep and Creep Fracture

133

can be established. Here kij and cr/'j are the components of the strain rate and deviatoric stress and )~ is a nonconstant proportionality factor. )~ is usually taken to be 3{/26 where ~ and ~ are the von Mises effective strain rate and effective stress, respectively. Considerable attention has then been devoted to the exact relationship between general stress and strain rate and some detailed theories involving internal stress parameters have been derived. The considerable simplification that the 0 projection methods have given to the understanding of uniaxial creep behavior suggest that it might be worthwhile exploring the possibility of extending the idea to multiaxial creep. It will be necessary to define (in 0 projection terms) methods for (a) the evaluation of the strain rate tensor for values of the stress tensor and for an arbitrary material history and strain state, (b) the evaluation of a suitable damage measure, and (c) the definition of a rupture criterion.

A. TestingMethods It is very difficult to devise a high-temperature testing technique that allows all three principal stresses to be set and controlled independently. A three-dimensional cruciform specimen has been proposed (32); however, it is very difficult and expensive to make, and it is not certain that the region of the specimen that experiences a uniform stress field is very large. Triaxial stress states have usually been obtained by testing suitably notched cylindrical bars in tension. A variety of notch designs have been used (33) but measurements on the creeping specimens are practically confined to rupture time. A major difficulty in using the test for creep strain determination is that the stress field at the notch is nonuniform and changes during creep because of relaxation effects. While numerical methods can be used to estimate the initial and steady-state stress fields, strain measurements are very difficult. In addition, if the material exhibits much creep damage (or extensive tertiary creep) steady-state stresses might not be attained at all. Most of the multiaxial creep work has been carried out under biaxial conditions. This inevitably precludes the use of genuinely triaxial states, but it is much easier to construct test specimens and to obtain reasonably uniform states of stress. In addition, meaningful strain measurements are possible, as well as estimates of rupture lives. Two-dimensional cruciform specimens have been used both for investigation of the tensile-tensile (34) and the tensile-compressive (35) regions of principal stress space. The major problem arises from the difficulty of designing specimens that have a sufficient volume of material with a uniform stress field, since it is difficult to prevent the development of stress concentrations at specimen corners during creep. The successful designs that have been produced are very expensive to machine. The most frequent biaxial testing method uses hollow tubes whose wall thickness is small compared to their diameter. A few investigations have used tensile

134

R.W. Evans and B. Wilshire

~ | [ I L I l l t m l _~ Loading system for a tubular tension-torsion test piece.

forces along the tube length together with internal pressure (36), but the combined tension (compression)-torsion test has proved most useful. The general loading system for such a test is shown in Fig. 11. The wall thickness of the tube is such that the stress system can be considered to be uniform and a suitable choice of gauge length allows both tensile and compressive axial loads to be applied. The stress system in the tube for given applied loads can be readily calculated. For a tube with internal and external radii rl and r2 with an initial gauge length l0 and with an axial load L and torque 1-', the various stresses with respect to the axes shown in Fig. 11 are as follows: 0"22 --"

7r(r 2 - r 2)

3F ~21 = eq2 = 2rr'r~{,~ -r~'i')

(39)

(40)

Chapter 3

Constitutive Laws for High-Temperature Creep and Creep Fracture

135

Mean stress: 0"m ~-

O'22

(41)

3

von Mises stress: c7 -- V/0"22 + 30"21

(42)

Principal stress:

Pl- 5-+

-T (43)

2 P3 -- 0 Since the principal stress and the von Mises effective stress are given by different relationships, it is possible to maintain one of these quantities constant and vary the other. Only stress states in the tension-compression quadrant of stress space can be investigated, but the stresses can range from pure tension through pure shear to pure compression. Since experimental elongation and twist can be measured, it is possible to determine various functions of the strain state. In particular the von Mises effective strain is given by g-

V//e22

4 -~- 3s

2

B. Experimental Program for 0.5CrO.5MoO.25V Steel Biaxial creep tests have been carried out on tubular specimens having internal and external diameters of 5 and 6 mm, respectively, and a gauge length of 10 mm. Tests were carried out under load control using a servo-hydraulic machine. Independent tensile and torque loadings were applied and the loading adjusted with tensile elongation in order to maintain true-constant stress states during testing. Specimen temperatures were controlled to better than 4-1 K and measurements of extension (or compression) and twist were recorded as a function of time using a suitably constructed extensometer. All tests were conducted at 838 K. The loading conditions used for testing were such as to maintain the 6 constant over all conditions. Thus, the tensile and shear stresses, 0"22 and 0"12, were adjusted so that the maximum principal stress pl changed at constant 6. Thus, testing conditions with respect to 0"22 and 0"12 were on the curve defined by the ellipse 6 2 -- 0"22 -t- 3o'?2

The detailed testing conditions are shown in Table 2.

1 36

R.W. Evans and B. Wilshire ABLE

Experimental Test Conditions for # = 250 MPa o'22

(712

o`m

o'22

! (712

Pl

Test

(MPa)

(MPa)

(MPa)

(MPa)

(MPa)

(MPa)

1 2 3 4 5 6

250 200 125 0 - 125 -200

0 86.4 125 144.3 125 86.4

83.3 66.7 41.7 0 -41.7 -66.7

166.7 133.3 83.3 0 -83.3 - 133.3

0 86.4 125 144.3 125 86.4

250 232.1 198.6 144.3 77.1 31.8

T h e effective strain (g)/time c r e e p c u r v e s o b t a i n e d f r o m the testing are s h o w n in Fig. 12. All creep c u r v e s s h o w e d a p r i m a r y ( d e c r e a s i n g creep rate) portion, but only those t e s t e d u n d e r c o n d i t i o n s o f positive h y d r o s t a t i c m e a n stress (tests 1, 2, 3, and 4) s h o w e d tertiary creep. Tests 5 and 6 s h o w e d a c o n t i n u o u s l y d e c r e a s i n g creep rate and w e r e t e r m i n a t e d b e f o r e fracture. In g e n e r a l , the tertiary c r e e p rates at a given

Effective strain/time curves for 0.5Cr0.5Mo0.25V steel tested at 838 K, effective stress of 250 MPa, and various maximum principal stresses from 32 to 250 MPa.

Chapter 3

Constitutive Laws for High-Temperature Creep and Creep Fracture

137

m|(llglltllllltll Ratio of the observed shear strain to tensile strain for various ratios of the shear stress to tensile stress.

time decreased with maximum principal stress and (where appropriate) rupture times increased sharply with decreasing principal stress for the same effective stress. For each of the curves shown in Fig. 12, the shear strain el2 is plotted against tensile strain e22 at a given time in Fig. 13. The different curves here represent different stress ratios o12/o22. It is clear that, regardless of whether the specimens were in primary or tertiary creep, the ratio of strains remained constant all the way to failure. This suggests that Eq. 38 is followed closely for this material and can confidently be used as a basis for constitutive equation construction.

C. 0 Analysis The shape of the creep curves in Fig. 12 suggests that they may be suitable for the determination of 0 parameters using standard techniques. Figure 14 shows the degree of fit observed when Eq. 27 is fitted to the creep curve obtained at a maximum principal stress of 232 MPa. In view of the extremely good fit, it is reasonable to carry out the analysis on all the multiaxial creep curves. Creep curves

138

R.W. Evans and B. Wilshire

||[glll't|lm

Experimentaland fitted creep curves for # = 250 MPa and Pl = 200 MPa.

with negative mean stresses do not show a tertiary (accelerating) component and model Eq. 27 must be replaced by Eq. 30. Thus, for the creep curves conducted at maximum principal stresses of 77 and 32 MPa, only estimates of the product 0304 are available, rather than the independent quantities. Figure 15 shows the variation of primary 0 constants (0102) with maximum principal stress for all tests conducted at an effective stress of 250 MPa. The amount of primary strain (as gauged by the value of 01) is about 1% in all cases; and, beating in mind the degree of experimental uncertainty associated with these small strains, it is very probable that the primary parameters are not a function of principal stress. However, uniaxial testing (where ~ = Pl) shows a dependence on stress, so it is clear that only the von Mises stress is important in governing primary behavior. The marked change in tertiary behavior with principal stress shown in Fig. 12 is reflected in the variation of the 0 parameters response for tertiary curve of shape (0304). For those tests where independent measures of 03 and 04 are possible (i.e., when mean stresses are not less than zero), In 03 decreases in a linear manner with increasing Pl whereas In 04 increases with Pl (Fig. 16). For all the tests shown in Fig. 12, it is possible to evaluate the product of 03 and 04 and this is also plotted

Chapter 3

Constitutive Laws for High-Temperature Creep and Creep Fracture

139

~ l l [ e L I l ' t J l , ' l Variation of 01 and 02 with maximum principal stress for ~ = 250 MPa.

in Fig. 16 as a function of Pl. Within the limits of the accuracy of the estimation procedures, the product 0304 is independent of the principal stress, even when the creep curve shape changes from a tertiary-dominated to a nontertiary form. These 0 relationships can be made more quantitative by a weighted least-squares analysis of the data available at the single temperature (838 K) available. The conditions are divided into two regimes, depending on the value of the mean stress. a) For O"m < 0, -

01 (1 - e -~

= O] (1 - e -~

+ 0304t

(44)

+ 0st

(k - 1, 2, 5)

In 0h - c~ + 1~6 b) Foram > 0 g - - O1 (1 - e -~

+ 03 (e ~

In Ok - - Otk + i l k # + Y k P l

-

1)

(k -- 1, 2, 3, 4)

The values of the various constants in Eqs. 44 and 45 are given in Table 3.

(45)

1 40

R . W . Evans and B. Wilshire

Variation of the tertiary parameters with maximum principal stress for # -- 250 MPa.

It is now possible to develop a multiaxial stress constitutive relationship. Sections III.A and III.B have shown that the 0 creep strain equations represent a theoretical model that is dependent on creep strain to govern general deformation and damage accumulation to control tertiary creep rates. It is the basis of a constitutive relationship with two "internal variables." These are effective strain, g, and damage parameter, co, and it is the changes in these that allow creep processes to ABLE;

Multiaxial Creep Constants O'm

0

Otk

flk

~"k

<0 <0 <0

lnO1 In02 In03

- 0 . 1 1 0 1 • 102 - 0 . 1 6 9 8 • 102 - 0 . 3 0 9 8 • 102

0.2577 x 10 - ! 0.3339 x 10 -1 0.6566 • 10 -1

0 0 0

>0 >0 >0 >0

ln0j In02 ln03 ln04

-0.1101 -0.1698 -0.1063 -0.2063

0.2577 0.3339 0.5519 0.5970

• • • x

102 102 102 102

• • x •

10 - l 10 - l 10 -1 10 -2

0 0 - 0 . 2 1 6 6 x 10 -1 0.2615 • 10 -1

Chapter 3

Constitutive Laws for High-Temperature Creep and Creep Fracture

141

be followed. The model can thus be summarized as a "strain-hardening-damageinfluenced" procedure. It is now useful to identify those terms that can be experimentally determined. In the absence of damage, the creep rate equation becomes dg dt

--

0102

exp(-02t) + k0

(46)

where k0 is the natural steady-state creep rate governed by work-hardening and recovery (see Eq. 23). It is now possible to establish the value of k0 since it corresponds to the tertiary 0 equation term at zero damage (i.e., at zero times). Thus e0 = 0304

(47)

The strain-hardening rule thus specifies the current effective creep rate at an effective stress 6 and an effective strain g as dg

0304

(48)

01 (1 - exp(-0zt)) + 0304t = g

(49)

--

~

dt

"-- 0 1 0 2 0 " ( - - 0 2

t*)

-+-

where t* is the solution to the nonlinear equation

In this equation, the 0 values must be calculated at a maximum principal stress equal to the effective stress, since 01 and 02 show no principal stress dependence and at zero damage 03 and 04 behave similarly. Whereas Eq. 48 is useful at zero damage, it is inapplicable to those cases in which damage has occurred. Examination of Eq. 26 indicates that the damage growth parameter, C, is related to 03 by C =

1

03

(50)

In terms of the damage parameters, the overall damage for constant C (and hence 03) is co -- exp(C~0t - 1) -- exp(04t - 1)

(51)

and in differential form do) dt

= C~oexp(Ckot) -- 04 exp(04t) -- 04(1 + co)

(52)

and the overall creep rate equation is derived from Eq. 48 by the modification --

dg dt

--- 0102 e x p ( - - 0 z t * ) -k- 0304(09 + 1)

(53)

In the application of Eqs. 52 and 53, 03 and 04 must be calculated taking into account the current values of the maximum principal stress (the product 0304 will be the same for a given 6 regardless of the value of Pl). The creep strain constitutive

142

R.W. Evans and B. Wilshire

Variation of damage parameter at rupture with normalized maximum principal stress.

equation can now be built in differential form from Eqs. 48, 52, and 53. Once the effective stress and effective strain rate are known, the overall strain rate tensor, Eij c a n be derived from Eq. 38. While these equations allow the definition of creep rates and damage rates, they say nothing about failure criteria. The value of w is zero at zero damage and fracture occurs when w reaches some value. The failure value (.Of can be calculated from Eq. 51 for specimens which have failed and for which 04 and tf are available. Calculated values of (.Of are plotted against the maximum principal stress in Fig. 17. The maximum principal stress has been normalized for different testing temperatures by dividing by the appropriate Young's modulus. It is clear that rupture occurs when (.Of

> 2.92

•

10 - 9

()_3 Pl

(54)

and this factor can be included in the analysis to determine the point of creep collapse.

Chapter 3 ConstitutiveLaws for High-TemperatureCreep and Creep Fracture 143

CREEPUNDER NONSTEADY LOADING CONDITIONS The 0 Projection Concept provides model-based constitutive laws that accurately quantify creep and creep fracture properties, even under multiaxial stresses. However, in seeking to provide a comprehensive description of materials behavior at high temperatures, it is essential to demonstrate that the 0 relationships also introduce a theoretically valid basis for design and assessment of components that must normally operate for long periods under nonsteady stress and temperature loading conditions during service. In the case of electricity generation plants and other large-scale installations, high-temperature components are usually designed not on the concept of an ultimate life but on the criterion that failure must not occur within some stipulated "design life" under the anticipated operating conditions. Current design codes then provide guidance as to the procedures to be adopted in order to achieve the necessary minimum safe operating life, but designers encounter problems in relation to the available materials data. In particular, because of the high degree of scatter in traditional multilaboratory test results (Fig. 4), the minimum property values of the scatter band must generally be used for design calculations since failure must not occur within the planned life of plant. Yet clearly, most of the material supplied for construction of large-scale plant will be characterized by property values better than the lower limits of the scatter bands. As a result of assuming minimum property values, together with the fact that a safety factor of 1.3 to 1.6 in stress is commonly assumed at the design stage, large-scale plant tends to be "over-designed." Indeed, the view that the design process is unnecessarily conservative is supported by operating experience showing that service lives considerably greater than design expectations are often achieved in practice. Electricity-generating stations and other large-scale plants represent massive capital investments. Continued operation beyond the original design life, commonly taken as the endpoint of capital depreciation, then offers significant economic benefit. Many plants are now approaching the end of their planned life spans, so worldwide attention has been focused on refurbishment and life extension of existing installations. Unfortunately, failure of high-temperature components in power stations and petrochemical plants can lead to serious financial and safety problems. Hence, continued plant operation can be allowed only with the reasonable assurance that catastrophic failure will not occur. Plant life extension should therefore involve "fitness-for-purpose" evaluations which demonstrate that components have an adequate margin against failure during any planned period of future operation. Since the risk of failure increases with increasing plant age, an integrated plant condition monitoring plan is required to prevent escalating maintenance costs from offsetting the benefits of plant life extension.

144

R.W. Evans and B. Wilshire

A. Traditional Approaches to Plant Life Extension Components of operational large-scale plant vary in size, age, and type, so no single assessment procedure can be considered as providing an adequate assurance of continued safe operation. Even so, a general approach to safe life extension can be based on identification of the damage mechanism or mechanisms likely to cause component failure, together with component-specific information on the current damage state, a knowledge of the rules governing the rate of damage accumulation, and a valid definition of a suitable failure criterion that specifies the life-limiting damage level. In line with this general concept, for components operating in the creep regime, a three-stage life extension procedure is often adopted (37). Stage I involves a reassessment of the original design methods, using any new long-term materials data that may have become available since the plant was commissioned. This reassessment is based essentially on the use of the Robinson Life Fraction Rule (38), as t/= 1 i

(55)

tf

ti is the time at one temperature for the applied stress and tf is the lower limit of the scatter band of the stress rupture data at that stress and temperature for the relevant material. From a knowledge of the original design calculations, together with any plant records of operating temperatures and so on, it may be possible to estimate the approximate fraction of the component creep life that has already been exhausted. However, plant operating records are usually incomplete and certainly not comprehensive. Furthermore, this design reappraisal does not compensate for the original design conservation or for the assumption of minimum property values from the scatter band of the stress-rupture data. Consequently, this Stage I assessment is normally used to identify components "at risk," so that more detailed evaluations can be made, especially for critical components where failure has major safety or financial implications. Stage II assessment then focus on in situ component inspection. Many different inspection procedures can be selected, including checks on material composition, surface hardness testing, surface metallography, and replication to quantify the development of surface cracking. Yet, Stage II inspection methods provide only a qualitative indication of the state of a component, with the added concern that in situ surface examinations may supply information that is not representative of the bulk properties of the material. Stage I and II appraisals therefore serve predominantly to identify components having a damage state that justifies progression to Stage III assessment. Stage III procedures require sampling of components for laboratory microstructural studies and, particularly, for post-service-exposure stress-rupture testing. where

Chapter 3 ConstitutiveLaws for High-Temperature Creep and Creep Fracture 145 accelerated postexposure test data (1)

extrapolation to service temperature, T s

E %

Ts "

remanent life

-

_./.

Io9 tf With traditional post-exposure stress-rupture procedures for remanent life assessment, accelerated tests are usually carried out at the service stress but at higher temperatures. Extrapolation of the accelerated test data to the service temperature (T) then gives an estimate of the remaining life.

While the microstructural examination supplies additional qualitative information on component condition, the post-exposure stress-rupture tests are undertaken in an attempt to supply quantitative estimates of remanent life. Essentially, test specimens are machined from samples taken from the at-risk components. These specimens are generally loaded at the estimated service stress, but the stress-rupture tests are carried out over a range of temperatures above the normal plant operating temperature in order to reduce the maximum test duration to 1000 hours or so. As illustrated in Fig. 18, extrapolation of the short-term temperature-accelerated data to the service temperature then indicates the remaining life of the component. As with standard stress-rupture testing, there is generally scatter in the post-exposure test results, so that extrapolation is often not accurate. More importantly, the temperatures at which the accelerated tests are performed are necessarily well above the actual plant operating conditions, making the unwarranted assumption that the high testing temperatures d o not seriously modify the microstructures typically developed in steels during long-term plant exposure. With 0.5Cr0.5Mo0.25V ferritic steel, the carbide type and distribution above about 900 K are significantly different from those present at typical service temperatures of 820 to 840 K, so post-exposure tests undertaken at temperatures up to 1000 K are difficult to justify. To overcome this problem, the 0 methodology has been extended to introduce a new approach to remanent life estimation based on stress/temperature acceleration (39), with a firm theoretical foundation for the use of post-exposure creep testing provided by analyses of the behavior patterns displayed when stress-change experiments are performed during creep.

146

R.W. Evans and B. Wilshire

-3-0 t

i

i

I

I

I

-3.5 .-.,,

~u . I -~0

-

eY

z -~s rvi---t/1

"~ -s 0 C~ ...J

-5 S -6-0 0

I

I

I

I

I

-05

-10

-15

"20

"25

.30

STRAIN

i | [ l ~ e l , , t l l i l l Creep rate/creep strain curves for superpurity aluminum at stresses of (a) 4.031 and (b) 4.741 MPa at 573 K, also showing the transition periods observed when the applied stress was changed suddenly between these stress values.

B. StressChange Experimentsduring Creep Figure 19 shows the creep curves recorded for superpurity aluminum at 573 K, plotted to emphasize the variations in creep rate with increasing creep strain at stresses of 4.031 and 4.741 MPa. With superpurity aluminum, grain boundary cavities do not develop during creep so that, under constant-stress test conditions chosen to avoid grain growth and recrystallization, the creep rate decays gradually toward a steady-state value at high uniform strain levels (Eq. 24). Further tests were then carried out in which the applied stress was periodically increased or decreased between the stress limits of 4.031 and 4.741 MPa (2). The results presented in Fig. 19 are fully consistent with the dislocation creep concepts used to derive Eq. 24 (Section Ill.A). At any instant during creep, there are many dislocations moving in the crystal lattice, but the magnitude of the local internal stress opposing continued forward motion differs for each dislocation, depending on its position with respect to the other dislocations present. The behavior observed following a stress reduction during creep can then be explained through the classical work of Nix and co-workers (40-42). Immediately following a stress reduction from cr to CrR,the subsequent creep strain/time behavior depends on the magnitude of erR and cri, where cri represents the average internal stress. For stress reductions such that crR> cri, the effective stress governing the movement of most dislocations remains positive (CrR--cri), so the new creep rate immediately after the instantaneous specimen spring-back is low but still positive. With large stress

Chapter 3 ConstitutiveLaws for High-TemperatureCreep and Creep Fracture 147 reductions, the local internal stress generally exceeds the new reduced applied stress, forcing the majority of dislocations to reverse their direction of movement, so that a negative creep rate is observed immediately after the stress reduction. For some intermediate stress reduction, an overall zero creep rate is recorded after the specimen spring-back, when the numbers of dislocations that continue to move forward more slowly match the numbers forced to reverse their direction. When the stress is reduced from o- and the test continued at a new lower value (say, with O-R greater than o'i), the low creep rate immediately after the stress reduction gradually accelerates to the correct value for the lower stress because recovery processes progressively reduce the dislocation density and the average internal stress to the levels expected for O-R. Thus, the creep rate changes such that, eventually, the creep rate/creep strain curve following the stress reduction coincides with the curve obtained in an uninterrupted test conducted entirely at the lower stress level. Conversely, when the stress is increased during a creep test, the creep rate immediately after the stress increase is very high, but decreases as the dislocation substructure changes toward the configuration typical of the higher stress. Thus again, after a transition period, the creep curve recorded after a stress increase follows the curve expected for the higher stress (Fig. 19). As shown in Fig. 19 for superpurity aluminum, the transition period following a stress change can extend over a considerable time period; i.e., the time needed for the dislocation substructure to change to the configuration anticipated at new stress condition can represent a significant fraction of the total creep life, illustrating the stability of dislocation substructures developed during creep of pure metals. In contrast, with particle-hardened alloys, this transition period is relatively small. Provided only that the frequency of the stress changes is not so rapid that the time at any stress is within this small transition time, when the creep behavior becomes cycle-dependent, to a good first approximation, the transition period can be ignored with particle-hardened alloys. Under these circumstances, the 0 methodology provides a straightforward method of analyzing creep behavior under nonsteady loading conditions. If the stress-temperature conditions are changed from o-1/'1 to o-2/'2 (with o1 < o-2 and/'1 < T2), the 0 values for these conditions can be computed from Eq. 34 (provided only that the coefficients in Eq. 34 have been evaluated, as for 0.5Cr0.5Mo0.25V steel in Table 1). Once the 0 values are known, full creep curves can be constructed using Eq. 27. It is then necessary to define procedures for transferring from one curve to the other, allowing the creep rate after the stresstemperature change to be calculated. The situation is illustrated in Fig. 20, when two schematic curves are shown corresponding to o-1/'1 and o-2T2. It is possible to proceed from the first curve to the second along many paths, the extreme cases being referred to as time-hardening (A) and strain-hardening (B). With steels, studies of the effects of changes in stress and temperature have shown that the strain-hardening path is very nearly correct, so a simple constitutive relationship

148

R . W . Evans and B. Wilshire

z rY I---t,/) t.I..I ILl

rr, t..J

/ TIME

Schematic diagram showing a change in test conditions from trl Tl to tr2 T2.

can be constructed on this basis. Thus, if a strain el has been attained at O"1 T1, then the corresponding 0 values are 0i (with i = 1, 2, 3, 4) such that In Oi = h (crl 9Tl ) SO

kl = 0102 exp(-02tl) + 0304 exp(0atl)

(56)

where t~ is the root of O1 (1 - e -Ozt ) -k- 03 ( e 04t -

1) - e 1 = 0

When the imposed conditions are changed to 02 T2, the new creep rate k2 will be given by e2 = 0~0~ exp(-0~t2) + 0~0~ exp(0~t2)

(57)

where lnO[ = h(cr2T2) and t2 is the root of 01 (1 -- e -02t) q- O~ (e Oat --

1) -

el

= 0

In this way, the 0 Projection Concept provides the full materials constitutive relationship required for use with modem computer-based stress analysis methods, allowing design calculations to be performed for components experiencing complex, nonuniform, nonsteady stress and temperature loading conditions during service.

Chapter 3 Constitutive Laws for High-Temperature Creep and Creep Fracture

149

C. The 0 Approach to Remanent Life Assessment The 0 m e t h o d o l o g y offers advantages not only for design calculations, but also for life extension of high-temperature plant. Thus, for Stage I design r e a s s e s s m e n t exercises, once the 0 coefficients have been evaluated for the relevant steels (as in Table 1 for 0 . 5 C r 0 . 5 M o 0 . 2 5 V ferritic steel), m o d e r n finite-element methods can be e m p l o y e d to check the validity of the original design code calculations, which should result in better identification of at-risk components. Moreover, the 0 relationships introduce a new procedure for r e m a n e n t life estimation; but, in this case, post-exposure creep tests rather than post-exposure stress-rupture tests must be performed. The 0 approach to r e m a n e n t life assessment is illustrated in Fig. 21. In relation to this schematic diagram, assume that the stress-temperature conditions experienced by a c o m p o n e n t in service can be defined as Crl T1. Samples of material taken from the service-exposed c o m p o n e n t are then tested under conditions that accelerate failure, say cr2 T2 (with ~rl < cr2 and T1 < T2). Using the 0 relationships, full creep curves can be constructed for the service conditions (Crl Ta) and for post-exposure test conditions (cr2T2). The post-exposure creep test is then equivalent to a creep test in which the conditions are changed from (9"1 T 1 t o 0"2 T2, so the situation can be described using Eqs. 56 and 57. The creep strain/time data recorded in the postexposure test (shown as the curve AB in Fig. 21) corresponds to the final portion of the full creep curve expected for n o n - s e r v i c e - e x p o s e d material at cr2 T2 (shown

/

Creep Strain

B[~T2

"Sh/ /

/

E:j /

/

i

11

Life

Time

~ l [ l l l l : t l l l l l Schematic representation of the creep curve predicted for the stress/temperature conditions relevant to service (o17"1), together with a curve expected for the accelerated test conditions (o.2T2). The accelerated conditions are also used in a test designed to obtain a creep curve for a sample exposed for some extended time under service conditions. For the accelerated conditions (o.2T2), the curve for non-service-exposed material (OB) and for the service-exposed sample (AB) can be matched to determine the point A on the full curve, OB. This defines the creep strain e* accumulated during service, so that the remanent life at O"1 T1 can be derived.

150

R.W. Evans and B. Wilshire

as the curve OB in Fig. 21). In this way, the post-exposure test identifies the creep strain (e* at position A) accumulated during service, allowing the remanent life to be calculated as the time needed for the creep strain to increase from e* to ef during creep at or1T1. Remanent life estimates obtained using the 0 analysis are based on strainhardening rules (Eq. 56 and 57), whereas conventional stress-rupture procedures assume that the less realistic time-hardening rules apply (Eq. 55). For this reason, a study was made to compare the remaining life values determined using the 0 methodology with the results obtained with traditional methods (39). This comparison was straightforward since the information required for both methods could be 15roduced from the same set of tests; i.e., a full post-exposure creep test provides not only a creep strain/time record, but also the strain and time to failure as in a standard stress-rupture test. This comparison established that, for 0.5Cr0.5Mo0.25V steel, the two methods yielded similar average results, but the scatter bands of the estimates from the 0 analysis were far lower. Admittedly, adoption of the 0 approach requires high-precision constant-stress creep equipment rather than less expensive stress-rupture machines. However, the creep tests provide far more data, so fewer tests need be conducted to obtain good remanent life estimates. Moreover, the 0 methodology offers two distinct advantages. (a) In contrast to traditional post-exposure stress-rupture testing in which only the temperature is increased to accelerate failure, with the 0 approach, both the stress and temperature can be raised to cause failure in times less than about 1000 hours. With materials such as 0.5Cr0.5Mo0.25V steel, the temperatures selected for post-exposure testing need not then exceed 900 K, so that the carbide types and dispersions developed during prolonged exposure at service temperatures of around 820 to 840 K are not modified significantly during post-exposure testing. (b) Once the 0 analysis is performed for samples taken from operational components, the method allows stress/temperature contours to be computed to define the conditions that must be maintained in order to ensure a specified future period of continued safe plant operation. In this way, the 0 methodology has been used successfully to allow life extension even of components approaching the end of their useful life, allowing ordering and replacement of components to be scheduled safely and economically.

CONCLUSIONS (i) The 0 Projection Concept provides a theoretically sound description of creep behavior, based on micromodeling of the dislocation processes governing primary creep and the damage processes that can cause the creep rate to accelerate during the tertiary stage. The 0 relationships accurately describe the shape of

Chapter 3 ConstitutiveLaws for High-TemperatureCreep and Creep Fracture 1 51 individual creep curves and the variations in creep curve shape with changing stress/temperature conditions. Thus, full materials constitutive laws are provided to quantify stress/strain/time/temperature behavior, allowing interpolation and reasonable extrapolation of short-term constant-stress creep and creep fracture properties for accurate and cost-effective prediction of long-term engineering design data. (ii) Using the 0 relationships, all features of conventional power-law behavior patterns can be predicted, without the assumption that different creep mechanisms become dominant in different stress-temperature regimes. Although the "secondary" creep rate may be the simplest of creep measurements, standard power-law representation of the stress and temperature dependences of the secondary creep rate do not appear to offer a meaningful procedure for identifying the dominant creep process or for the construction of deformation mechanism maps. (iii) The 0 relationships introduced to define uniaxial tensile creep properties can be extended to analyze and interpret the creep and creep fracture data recorded under complex nonsteady stress and temperature conditions. For 0.5Cr0.5Mo0.25V steel, the creep deformation characteristics observed under multiaxial stresses can be precisely quantified in terms of the relationship between effective stress and effective strain rate, but a complete description of tertiary creep and fracture behavior requires recognition of the dominant role played by the maximum principal stress in determining creep damage accumulation. Even so, the 0 relationships can be presented in a computer-efficient form that is ideally suited to use with modem finite-element design procedures. (iv) Adoption of the 0 methodology offers major advantages for safe and costeffective life extension of operational high-temperature plant which is approaching the end of its original design life. By providing full materials constitutive relationships, modern computer-based design methods can be used to check the validity of the original design-code calculations in order to identify critical components at-risk of failure. Moreover, using high-precision post-exposure creep tests rather than traditional post-exposure stress-rupture testing, the 0 approach allows accurate determination of remanent creep life, so that component replacement can be scheduled economically.

REFERENCES 1. Evans, R. W., Parker, J. D., and Wilshire, B. (1982). In "Recent Advances in Creep and Fracture of EngineeringMaterials and Structures" (B. Wilshire and D. R. J. Owen, eds.), p. 135. Pineridge Press, Swansea, UK. 2. Evans,R. W., and Wilshire, B. (1985). "Creep of Metals and Alloys." Institute of Metals, London. 3. Evans,R. W., and Wilshire, B. (1993). "Introduction to Creep." Institute of Materials, London. 4. Norton, E H. (1929). "The Creep of Steel at High Temperature." McGraw-Hill, New York. 5. Coble, R. L. (1963). J. Appl. Phys. 34, 1679. 6. Nabarro,E R. N. (1948). "Reporton Conferenceon Strengthof Solids." Physical Society,London.

152 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

22. 23.

24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42.

R.W. Evans and B. Wilshire Herring, C. (1950). J. Appl. Phys. 21,437. Frost, H. J., and Ashby, M. E (1982). "Deformation Mechanism Maps." Pergamon, London. Williams, K. R., and Wilshire, B. (1981). Mater. Sci. Eng. 47, 151. Threadgill, P. L., and Wilshire, B. (1974). Met. Sci. 8, 117. Parker, J. D., and Wilshire, B. (1975). Met. Sci. 9, 248. Artz, E. (1991). Res. Mech. 31, 399. Reppich, B. (1993). "Materials Science and Technology" (R. W. Cahn, P. Haasen, and E. J. Kramer, eds.), Vol. 6, p. 311. VCH, Weinheim, Germany. Monkman, E C., and Grant, N. J. (1956). Proc. Am. Soc. Test. Mater. 56, 593. Orr, R. L., Sherby, O. D., and Dorn, J. E. (1954). Trans. Am. Soc. Met. 46, 113. Larson, E R., and Miller, J. (1952). Trans. ASME 74, 765. Manson, S. S., and Haferd, A. V. (1953). Natl. Advis. Comm. Aeronaut., Tech. Notes NACA TN 2890. Kachanov, L. M. (1958). IZv Akad. Nauk SSSR 8, 26. Brown, S. G. R., Evans, R. W., and Wilshire, B. (1986). Mater. Sci. Eng. 84, 147. Brown, S. G. R., Evans, R. W., and Wilshire, B. (1987). Mater. Sci. Technol. 21,239. Evans, R. W., Willis, M. R., Wilshire, B., Holdsworth, S., Senior, B., Fleming, A., Spindler, M., and Williams, J. A. (1993). In "Creep and Fracture of Engineering Materials and Structures" (B. Wilshire and R. W. Evans, eds.), p. 633. Institute of Materials, London. Johnson, R. E, May, M. J., Trueman, R. J., and Mickleraith, J. (1967). In "High Temperature Properties of Steels" p. 265. Iron and Steel Institute, London. Brown, R. J., Cane, B. J., Parker, J. D., and Waiters, D. J. (1981). In "Creep and Fracture of Engineering Materials and Structures" (B. Wilshire and D. R. J. Owen, eds.), p. 645. Pineridge Press, S wansea, UK. Wilshire, B. (1990). In "Creep and Fracture of Engineering Materials and Structures" (B. Wilshire and R. W. Evans, eds.), p. 1. Institute of Metals, London. Wolfenstine, J., Ruano, O. A., Wadsworth, J., and Sherby, O. D. (1993). Scr. Metall. Johnson, A. E., and Henderson, J. (1962). "Complex Stress Creep, Relaxation and Fracture of Metallic Alloys." H. M. Stationery Office, London. Hayhurst, D. R., Morrison, D. R., and Leckie, E. A. (1975). J. App. Mech. 97, 613. Marin, J. (1937). J. App. Mech. 4(2), 55. Soderberg, C. R. (1936). Trans. ASME 58, 733. Ziegler, H. (1967). In "Irreversible Aspects of Continuum Mechanics" (B. Broberg, J. Hult, and E Niordson, eds.), Almqvist & Wiksell, Stockholm. Rice, J. R. (1970). J. App. Mech. 37, 728. Hayhurst, D. R., and Felce, I. O. (1986). "Techniques for Multiaxial Creep Testing" (D. J. Gooch and I. M. How, eds.), p. 241. Elsevier, London. Loveday, M. S., (1986). In "Techniques for Multiaxial Creep Testing" (D. J. Gooch and I. M. How, eds.), p. 177. Elsevier, London. Hayhurst, D. R. (1972). J. Mech. Phys. Solids 20, 381. Morrison, C. J. (1986). In "Techniques for Multiaxial Creep Testing" (D. J. Gooch and I. M. How, eds.), p. 111. Elsevier, London. Lonsdale, D., and Flewitt, P. E. J. (1901). Proc. R. Soc. London, Ser. A 373, 491. Townsend, R. D. (1987). "Refurbishment and Life Extension of Steam Plant," p. 223. Inst. Mech. Eng., London. Robinson, E. L. (1952). J. Appl. Mech. 74, 777. Wilshire, B., and Evans, R. W. (1989). In "Life Assessment and Life Extension of Power Plant Components" (T. V. Narayanam, et al, eds.), Vol. 171, p. 217. ASME, New York. Ahlquist, C. N., and Nix, W. D. (1969). Scr. Metall. 3, 679. Solomon, A. A., and Nix, W. D. (1970).Acta Metall. 18, 863. Gibling, J. C., and Nix, W. D. (1977). Mater. Sci. Eng. 11,453.

Improvements in the MATMOD Equations for Modeling Solute Effects and Yield-Surface Distortion Gregory A. Henshall

Donald E. Helling

Lawrence Livermore National Laboratory

E1 Segundo, California

University of California

Hughes Aircraft

Livermore, California

Alan K. Miller

Lockheed Palo Alto Research Laboratory Palo Alto, California

INTRODUCTION Constitutive equations, together with the equations of equilibrium and compatibility, are used to predict the deformation response of solids subjected to external or internal loading. When these equations are integrated into advanced computer programs, such as finite-element codes, predictions of deformation can be made for a wide variety of structural and manufacturing purposes. Accurate predictions of deformation may be required as an end goal, for example, in manufacturing processes such as forging, or as an intermediate step in performing life predictions, such as thermomechanical fatigue of gas-turbine or power-plant components. In many cases, sufficiently accurate predictions of nonelastic deformation can be made by using simple empirical or purely scientific laws describing deformation over a limited regime and/or for simple materials. However, for other high-technology systems limited by deformation behavior, these simple models are inadequate. In general, these cases involve complex engineering materials subjected to complex loading and thermal histories. Such histories may include Unified Constitutive Laws of Plastic Deformation Copyright (~ 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

153

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G.A. Henshall, D. E. Helling, and A. K. Miller

large variations in stress, temperature, strain rate, and loading direction. In these cases, the deformation behavior may cut across the usual classifications, such as "creep" and "plasticity." For these problems, more complex unified constitutive equations are required, particularly if a single set of equations is to be applicable to a wide range of problems. To be successful, these equations must simulate a wide range of deformation behavior, and they must have a physical or microstructural basis if interactions among the various phenomena are to be accurately modeled and if one is to have confidence in extrapolating beyond the regimes for which experimental data are available. These requirements have led to the development of physical-phenomenological unified constitutive equations, as first pioneered by Hart (1970).

A. The MATMOD Family of Unified Constitutive Equations 1. The MATMOD Philosophy of Constitutive Modeling The MATMOD (MATerials MODel) family of unified constitutive equations was developed with the goal of predicting nonelastic deformation in engineering materials subjected to complex loadings, such as those present in leading-edge technological systems, e.g., gas-turbine engines, nuclear power plants, and modem steel mills. To achieve this goal, the MATMOD family was designed to simulate a broad range of phenomena, including: low-temperature plasticity, high temperature creep, cyclic deformation, strain softening, solute strengthening, and multiaxial deformation. Table 1 lists the phenomena treated by the two versions of MATMOD described in detail in this chapter. (Other phenomena, such as dynamic recrystallization, are treated in special versions of MATMOD). Unfortunately, this breadth reduces the accuracy with which any single phenomenon can be simulated since our knowledge is incomplete concerning many details of the mechanisms of deformation over this wide range of behavior. In addition, some classes of deformation are not included. These include nonelastic deformation processes not driven by dislocation motion, such as grain-boundary sliding and most diffusional creep mechanisms. The development of constitutive models coveting such a broad range of behavior was accomplished using the "physical-phenomenological" approach, in which "internal" state variables represent, in an approximate but physically meaningful manner, the controlling internal physical processes. The overall structure of each model and the functional interrelationships among its variables follow from an understanding of these physical causes and effects. Adoption of the "unified" approach, in which one set of equations treats all of the above phenomena, follows logically from the fact that they are all controlled by the same set of physical mechanisms. To provide the required accuracy, the specific equations in the models were derived by fitting the quantitative behaviors observed across entire classes of materials.

Chapter 4 Improvements in the MATMOD Equations

155

ABLE I

Phenomena Simulated by the MATMOD-4V-DISTORTION and MATMOD-BSSOL Constitutive Equations 1. General plasticity, including (a) essentially elastic behavior followed by gradual yielding (b) strain-rate sensitivity, including negative values (c) temperature sensitivity, including peaks and plateaus for alloys (d) interactions between solute strengthening and strain hardening 2. General creep, including (a) power-law and power-law breakdown steady-state creep (b) Class I behavior with a constant stress exponent approximately equal to 3 (c) change in stress exponent from 3 to 5 corresponding to the change from Class I to Class II behavior (d) dependence of the steady-state creep rate on solute concentration (e) the stress dependence of the primary creep strain 3. Multiaxial deformation, including (a) yield-surface expansion (b) yield-surface translation (c) yield-surface distortion (d) anisotropy

4. Cyclic stress-strain behavior, including (a) Bauschinger effect (b) cyclic hardening and softening (c) shakedown to a saturated condition of constant stress or strain amplitude (d) hysteresis loop asymmetry 5. Recovery, including (a) static recovery at high temperatures (b) dynamic recovery 6. Strain softening, including (a) unidirectional strain softening (b) directional strain softening (c) cyclic strain softening 7. Complex histories, including (a) stress changes (b) strain-rate changes (c) temperature changes (d) creep-plasticity interaction (e) yield stress plateaus in cold-worked materials 8. Precipitation strengthening 9. Interactions of all the above

A detailed description of the M A T M O D p h i l o s o p h y of p h y s i c a l - p h e n o m e n o l o g ical m o d e l i n g was given by M i l l e r (1987), so only a brief s u m m a r y is p r e s e n t e d here. As first s u g g e s t e d by Hart (1970) and now g e n e r a l l y agreed, the p r o p e r w a y to treat the history d e p e n d e n c e of nonelastic d e f o r m a t i o n is t h r o u g h a unified set of equations containing a single nonelastic strain variable, e. A "kinetic" equation is written to describe the d e p e n d e n c e of its time rate of change, k, on stress, temperature, and a small n u m b e r of internal, or "structure," variables that r e p r e s e n t the controlling microstructure. T h e s e internal variables evolve during d e f o r m a t i o n and are g o v e r n e d by auxiliary structure evolution equations w h i c h calculate their rates of change. The f r a m e w o r k of the m o d e l is therefore = f ( o , T, X)

(1)

Jf -- g(k, T, X)

(2)

w h e r e a is the stress, T is the absolute temperature, and X denotes the structure

156

G.A. Henshall, D. E. Helling, and A. K. Miller ABLE

Description of the MATMOD-BSSOL Structure Variables* Variable

Definition

RA

Short-rangeback stresses

RB

Long-range back stresses Isotropic strain hardening due to homogeneously distributed obstacles Isotropic strain hardening due to heterogeneously distributed obstacles Interactive solute strengthening

Fz Fsol

Microstructural significance Bowing of individual dislocations, small dislocation pile-ups Bowing of subgrain walls, large pile-ups Forest dislocations, small dislocation tangles, sessile dislocations Subgrains, cells, large dislocation networks, persistent slip bands Mobile solutes that interact with the current dislocation substructure

*Reprinted from G. A. Henshall and A. K. Miller, "Simplifications and improvements in unified constitutive equations of creep and plasticity. I. Equation development," Acta Metall. Mater. 38, 2101-2115 (1990), with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington OX5 1GB, UK.

variables. The choice of internal structure variables is guided by the underlying mechanisms of nonelastic deformation. In the M A T M O D approach, each structure variable represents one category of strengthening mechanism. For example, some mechanisms strengthen the material equally in all directions, notably hardening due to overall forest dislocation density and subgrains. Other mechanisms, such as dislocation pile-ups or bowed subgrain walls, produce internal stresses that strengthen or weaken the material directionally. The above mechanisms all develop as the material is strained while other mechanisms, such as grain boundary and solute strengthening, are "intrinsic" to the undeformed material. Thus, at least three types of structure variables are needed to represent these three different categories of strengthening. Fortunately, the number of strengthening mechanism categories is far fewer than the number of phenomena to be treated, so tractable models can be derived. An example of the correspondence between categories of strengthening mechanisms and structure variables is given for the M A T M O D - B S S O L equations (described in Section Ill) in Table 2. In addition to the choice of variables, deformation physics governs the general forms of Eqs. (1) and (2), for example, the first-derivative relationships. These equations are designed so that the behavior of the structure variables imitates what we believe to be the behavior of the actual physical variables under the same conditions. Once the mechanistically inspired forms of the equations have been determined, actual algebraic expressions (applicable to the entire class of materials) are then developed by fitting a variety of mechanical test data. It is emphasized that these phenomenological equations are derived so that the behavior of both the externally measured and the internal variables duplicates the observed response of

Chapter 4 Improvementsin the MATMODEquations 157 real materials. This gives the model a degree of physical meaning and reliability that is not present in simple empirical equations. In summary, the data fitting provides the means for obtaining accurate simulations for a variety of simple histories, while the use of deformation physics provides the means for obtaining reasonable predictions for complex loading histories. Of course, the resulting equations must be simple enough so that they are computationally efficient and, more importantly, so that the phenomenological constants contained within them can be evaluated from mechanical test data for materials of interest.

2. History of the MATMOD Family of Constitutive Equations As shown in the genealogy of Fig. 1, multiple versions of MATMOD have been developed over more than a decade. The original version of MATMOD developed by Miller (1975) contained one structure yariable for each of the three classes of strengthening mechanism. Faef, denoting:the "friction stress due to deformation," was the isotropic hardening variable, which evolved as nonelastic deformation built up during the deformation history. The variable representing directional hardening was termed R, denoting the "rest stress," and also evolved in a historydependent manner. Finally, the intrinsic hardening variable, Fsol, was included largely to account for the temperature and strain-rate dependent effects of solutes on the strengthening of alloys. Fsol depended only on the current value of these variables, and was not history-dependent. Unlike many unified models, temperature was included explicitly in the model through the Arrhenius-like variable, | This approach has the practical benefit that the material constants in the model are temperature-independent. In addition, | has physical significance. Both considerations are advantageous when simulating variable-temperature loading histories. Following Sherby and Miller (1979), the temperature-dependent | variable is defined so that the activation energy for deformation is constant at high temperatures and decreases linearly starting at a transition temperature, Tt, down to zero at 0 K: exp

-

exp

-~-

In ;

+ 1

9

T < Tt T>Tt

where k is Boltzmann's constant and Q* is the activation energy for creep at high temperature, usually equal to that for lattice self-diffusion. The next version of MATMOD, developed by Schmidt and Miller (1981), aimed at improving the simulation of solute strengthening, particularly in alloys for which interactions between (chiefly interstitial) solutes and isotropic strain hardening cause increases in the strain-hardening rate and flow stress at interme-

158

G.A. Henshall, D. E. Helling, and A. K. Miller

/'

ORIGINAL MATMOD EQUATIONS (A. K. Miller, 1975): 9F def (isotropic hardening) 9R (directional hardening) 9Fso I (solute drag - 1978)

SOLUTE STRENGTHENING IMPROVEMENTS

TRANSIENT HARDENING/SOFTENING IMPROVEMENTS

(316 SS; C. G. Schmidt, 1979): 9FsoI ~ Fsol,1 (non-interactive solute drag) ' ~ Fsol,2 (solute drag interactive with isotropic hardening)

("MATMOD-4V", pure metals; T. C. Lowe, 1983): 9R ~

R A (short range back stresses) R B (long range back stresses) 9Fdef --.~.Fdef, p (isotropic hardening due to forest dislocations) Fdef,~. (isotropic hardening due to subgrains/cells)

SIMPLIRCATIONS - BACK STRESS DUE TO SOLUTES

('MATMOD-BSSOL'; G. A. Henshall, 1987):

9IqA = f(solute content); solute clusters as pinning points against which dislocations can rapidly build up back stresses 9R B, Fdef,p, Fdef,~.- retained 9Fso I - more minor role than previously 9Utilizes "NONSS" (T. G. Tanaka, 1983) for numerical integration

RECRYSTALLIZATION ("MATMOD-ReX"; R. O. Adebanjo, 1987) ee

Fde 9 f :=> rate of recrystallization => Fdef

YIELD SURFACE DISTORTION ("MATMOD-4V-DISTORTION"; D. E. Helling, 1986) 9Mij (anisotropy coefficients) = f(R A, R B)

9Fdef :=> recrystallized grain size

The MATMOD constitutive equationsma genealogy.

diate temperatures. As shown in Fig. 1, this led to the creation of two historyindependent solute-strengthening variables in the model: Fsol,1, which was similar to the original Fsol, and Fsol,2, which interacted with Fde f to represent the synergistic interaction between moving solute atoms and strain hardening. Further developments by Lowe and Miller (1984a, b) improved the way in which the model simulates the effects of heterogeneities in the dislocation substructure.

Chapter 4 Improvementsin the MATMODEquations 159 (Note that only the effects of these heterogeneities are simulated; as for all versions of MATMOD, the equations represent the behavior of a single material point which is homogeneous on a coarser spatial scale). These efforts led to further refinement in the variables representing the classes of history-dependent strengthening within a new version of the equations, MATMOD-4V (denoting four strengthening variables). Fdef was split into two variables: Fdef,p and Fdef,~ (Fig. 1). The former represents isotropic hardening due to forest dislocations and other homogeneously distributed substructural features that evolve quickly with strain. The latter represents isotropic hardening due to large-scale, slowly evolving heterogeneous substructures, such as subgrain boundaries. Similarly, R was split into the variables RA and RB. RA represents directional strengthening due to rapidly evolving, small-scale substructures, such as dislocation pile-ups. RB represents directional strengthening due to slowly evolving, large-scale substructures, such as bowing subgrain boundaries. The equations embody the expected close relationships between RA and Fdef,p and between RB and Fdef,Z. One of the chief benefits of this new modeling approach was the ability to accurately simulate many strain-softening phenomena. Starting with the model of Schmidt and Miller (1981), dynamic recrystallization phenomena were added in the model of Adebanjo and Miller (1989). Here, the isotropic strain-hardening variable, Fdef, was used to control dynamic recrystallization events. The second time derivative of Fdef was used to determine the rate of recrystallization, while the value of Fdef itself was used to determine the recrystallized grain size. This specialized version of MATMOD is beyond the scope of this review.

B. Goals The goal of this chapter is to describe developments in the MATMOD family of unified constitutive equations since its last review by Miller (1987). In particular, we describe in detail two MATMOD versions which have the common theme of using two directional, or "back stress," variables. This formulation is shown to provide a flexible, coherent means of modeling both yield surface distortions and solute effects. In Section II, the use of two back stress variables is combined with an anisotropic yield function to model multiaxial deformation and yield surface distortions. Accurate simulations of these phenomena were not possible using MATMOD-4V, which was developed from experiments in which the direction of straining was always reversed along the same axis, e.g., tension-compression loading. Verification of the improved model is demonstrated through simulations and independent predictions of yield locus distortions under proportional and complex nonproportional loadings for 1100 A1, 70:30 brass, and 2024-T7 A1. Having demonstrated (from multiaxial experiments) the existence of separate short-range and long-range back stresses, we utilize the short-range back stresses to treat solute effects in Section III. This approach, together with other improvements, simplifies

160

G.A. Henshall, D. E. Helling, and A. K. Miller

the equations and reduces the number of material constants without significantly compromising their physical basis or capabilities. This version of the equations is verified through simulations of peaks and plateaus in the flow stress vs. temperature curves, steady-state creep behavior, and independent predictions for a variety of nonelastic deformation behavior for pure A1 and dilute binary A1-Mg alloys. Finally, we give brief descriptions of the techniques employed to numerically integrate the equations and the methods used to determine the material constants. Values of the material constants are tabulated for a variety of metals and alloys.

MODELING YIELD-SURFACE DISTORTIONS A. Introduction to Yield-Surface Distortions Yield surfaces describe the multiaxial flow behavior of a material. The traditional yield surface describes the boundary in stress space between elastic and nonelastic behavior. A variety of yield theories have been developed over the years to describe the shape of the yield surface for different materials and the evolution of these surfaces. The multiaxial yield behavior of annealed untextured polycrystalline ductile metals is usually well described by the von Mises yield criteria. The von Mises yield surface is plotted as a circle centered at the origin within coordinates of Crxx vs. ~/3r, or as an ellipse centered at the origin with coordinates of Crxx vs. O'yy (Fig. 2). The normal to the yield surface indicates the direction of the strain rate vector at that stress state (Fig. 3). The yield surface is experimentally measured using a consistent yield definition and a locus of points is then plotted in stress space. Common yield definitions

~xx

~xx

(a) von Mises yield criterion in normalized normal/shear stress coordinates. (b) von Mises yield criterion in normal/normalstress coordinates.

Chapter 4 Improvementsin the MATMODEquations 161

Yield Locus

~

The strain increment vector is normal to the yield locus.

include plastic strain offsets from the elastic line, the proportional limit (the first observed deviation from elasticity), or an extrapolation between the elastic and nonelastic portions of the flow curve. Nonelastic deformation of a ductile metal tends to produce a combination of expansion of the yield surface (classical isotropic hardening) and translation of the yield surface (classical kinematic hardening). Yield surfaces of very small strain offsets exhibit distortions from the elliptical yield surfaces predicted by the von Mises criteria. These distortions take the form of a sharpening of the yield surface in the direction of the previous direction of straining and a flattening of the yield surface on the opposite side. These distortions are more apparent when plotted in axx-x/-3r space. The accuracy of the hardening law depends not only on the material and its history, but also on the definition of yield. More sensitive yield definitions tend to measure less relative expansion than "coarser" yield definitions. For this reason, it can be useful to consider the evolution of nonelastic flow, or a family of yield surfaces corresponding to different strain offset definitions, rather than a single yield definition and yield surface. As the yield definition becomes more indicative of gross flow in the material, the yield surface tends to be better approximated by a von Mises surface again, with distortions due to the crystallographic texture of the deformed material (Stout et al., 1985). Yield-surface distortions have a technological relevance during metal-forming operations, for example. The most extreme yield-surface distortion could include a comer on the surface, which, if present, would mean that the direction of the strain

162

G.A. Henshall, D. E. Helling, and A. K. Miller

rate would not be uniquely defined at that point. If the strain rate direction is not known or consistent during a complex metal-forming operation, the final shape of the formed piece is not predictable. In addition, during complex nonproportional deformation, the accuracy of the von Mises approximation to the shape of the yield surface becomes less satisfactory in the small strain-offset regime. Yield-surface distortions have also proved useful in understanding the nature of deformation, offering insight into how these processes can be modeled. Since the distortions have directionality, it is logical to explore in what way they are related to the MATMOD variables that produce directional hardening, RA and RB.

B. Overview of Modeling Yield-Surface Distortions with MATMOD-4V-DISTORTION The original motivation for extending MATMOD to the multiaxial case was to increase its generality. The extrapolation of the uniaxial MATMOD model was initially performed by Lowe (1983) using the approach suggested by Rice (1970). In extending the uniaxial equations to six-dimensional stress/strain space, strain rate, stress, and the back stresses become six-dimensional vectors (Ei, O'i, R g i , and RBi, where i = 1-6) while Fdef, p, Fdef,;~, and Fsol remain scalars. A yield function similar to that proposed by Baltov and Sawczuk (1954) has been used. This function

3~2 = M i j ( c r i / E - R i ) ( c r j / E - Rj)

(4)

includes an anisotropy tensor, Mij, that has the ability to change the shape of the yield surface. This tensor is similar to that proposed by Hill (1948) to account for an initially anisotropic material. The strain rate is derived from a Prandtl-Reuss flow rule (Mendelson, 1968) which then implies von Mises-like behavior. The calculated nonelastic strain rate is proportional to the normal of the yield function (~i o~ igf2/i)cri), so that normality of the strain rate to the yield surface is automatic. In MATMOD, translation of the yield surface in stress space is modeled by the back stress variables while expansion of the yield surface is modeled by the evolution of the Fde f and Fsol variables. Lowe's improvements to MATMOD included splitting the back stress into short- and long-range components (Lowe, 1983; Lowe and Miller, 1984a, b). An important distinction between the short- and long-range back stresses is that the short-range back stresses change in direction and magnitude more easily (with less strain) than long-range back stresses. This suggests that yield surface distortions may also display long- and short-range effects that could be associated with the two back stress components. The MATMOD-4V equations are summarized in Table 3. The yield surface reflects directionality in two ways: the yield surface is translated and it is distorted. The center of the yield surface, or the location where k = 0, is equal to the sum of the two back stresses, while the size of the yield

0

9

0

i

I

II

.~

~

II

I

II

II

II

~

II

~

II

~

II

I I

i

7

.~

|

li

,

"~.,

.

~

I

'

~

II

~ ,

9- ~

I

,

..~

"U~

.~ ~ . ,

I

,

I'~

"~

,

II

I

r

I

7

i

!

II

,

|

164

G.A. Henshall, D. E. Helling, and A. K. Miller

surface is related to the sum of the Fdef and Fsol variables. The fact that the behaviors of the several MATMOD structure variables manifest themselves in several different ways permits identification of the structure variable behaviors through multiaxial test data.

1. Review of Key Experimental Findings An experimental study was initiated to investigate the evolution of yield-surface distortion to establish how these distortions could be modeled in a physically meaningful way and to provide further evidence of the existence of two separate back stress components (Helling et al., 1986; Helling, 1986). Critical tests were performed to (1) determine if yield-surface distortion can be related to back stresses and (2) verify the existence of separate long- and short-range back stresses. A feature of back stresses is that they evolve as a function of the strain path (and not the stress path or stress state, for example). Critical experiments were conducted in which the directions of the stress path, stress state, and strain path were different in order to distinguish which one controlled yield-surface distortion. Yield-surface measurements were made on 1100-O aluminum, 70:30 brass (single phase), and an overaged 2024-T7 aluminum that contained nonshearable precipitates. Testing was performed in coordinates of trxx-~/3r to facilitate observation of yield locus distortion. Tests were performed on thin-walled tubes to eliminate the effects of a stress/strain gradient across the specimen. The yield loci were determined under stress control. The selected prestress path was applied to the specimen, and the stress state was then dropped to the center of the yield surface for its measurement without completely unloading the specimen. Although the center of the yield surface is not known a priori, an estimate of the center is initially made based on experience. By continuously monitoring the strain during the test, it can be verified whether the center of the yield surface has been found. The center of the yield surface is the region in which behavior is completely elastic so that even creep strains are not produced. The yield surface is then measured by probing out in stress space until the 5 x 10 - 6 m/m von Mises yield definition is reached and again returning to the center of the yield surface. An advantage of the small yield definition is that the individual measurements (yield-surface probes) have a negligible effect on the yield surface so that one sample can be used for an entire yield locus measurement, thereby eliminating sample-to-sample variations. The evolution of yield-surface translation, expansion, and distortion is shown for the three materials in Figs. 4-6. The behaviors for all three materials are similar, but the yield surface of each material exhibits different proportions of translation, expansion, and distortion. The effects of yield definition are shown in Fig. 7. The small strain offset definition locus shows greater distortion and greater translation but less expansion than the back-extrapolated locus. The first nonproportional prestress/prestrain comparison involved measuring the yield loci produced by (a) tension followed by torsion, (b) torsion followed

Chapter 4 Improvementsin the MATMOD Equations 165

V-32" (MPa) i

y/V-3=

120r ~-~A3

' -2o

'

A1'

20

'

80'

INITIALYIELDLOCUS

O- (MPa)

Measured yield loci of 1100-O aluminum after shear prestress. (A1) No prestress; (A2) 4c3r = 58 MPa, y/~/3 = 0.012; (A3) ~/3r = 107 MPa, g/~/3 = 0.29. Reprinted from Helling et al. (1986, Fig. 2, p. 314), with permission from ASME.

by tension, and (c) proportional tension/torsion on three different samples. The final prestress state was the same for all three paths. The resulting yield surfaces for the 1100-O aluminum are shown in Fig. 8. All three surfaces exhibit similar overall behavior: they appear flattened and translated in the direction opposite to the final prestress state (and expanded). Although it is clear that the distortion is not related to the stress path (locus A4 is not flattened on the left, for example), the observed distortions are all also opposite to the final straining direction. For convenience, the stress and strain paths are inset in the figure. Additional tests were then conducted to separate the direction of the final stress state from the final straining direction. Figure 9 compares the yield surfaces measured after different prestress paths. Yield locus A7 was measured after (a) applying torsion to an effective stress of 60 MPa (-- ~/3r), (b) reducing this torsional stress to 25 MPa [which corresponds to the center of the yield surface after prestress (a) and which is shown dashed], and then (c) applying tension without further reduction of the torsional stress. This final stress path generated only axial strains. For this yield locus, then, the final stress state is not collinear to the final straining direction. This locus appears to be flattened on two sides, the left and the bottom. For comparison, locus A4 was measured (a) after applying torsion to an

166

G.A. Henshall, D. E. Helling, and A. K. Miller V-3-t" (MPa)

320

B4 i y/V-3=0.32

280,

200

B3

160

B2

-120

I

-40

.L

40

~'/V-3= O. 14

)"/V-3 = 0.024

120 1

O" (MPa)

INITIAL YIELD LOCUS

Measured yield loci of 70" 30 brass after shear prestress. (B 1) No prestress; (B2) q/3r = 167 MPa, y/~/3 = 0.024; (B3) ~/3r = 252 MPa, y/x/~ = 0.14; (B4) v/3r = 378 MPa, g/~/~ = 0.32. Reprinted from Helling et al. (1986, Fig. 3, p. 314), with permission from ASME. effective stress of 60 MPa, but then (b) applying tension without any reduction in the torsional stress. This locus is not as strongly flattened on the left side (opposite the tensile direction) because prestress leg (b) generated a significant amount of shear strain together with tensile strain. This is unlike prestress path (c) on sample A4, which generated tensile strains exclusively, producing the flattening of the locus on the left. This result suggests that the distortions of the loci are related to the strain path and, furthermore, that there may be more than one direction associated with the distortion. Figures 10 and 11 show similar results for the 7 0 : 3 0 brass and the 2024-T7 aluminum, which even more strongly suggest this association. A final set of tests was conducted to determine if it is possible for a yield surface to be distorted but not translated. The idea was to apply a torsional prestress and then reverse the torsion. By reversing the torsion, the short-range back stress would realign quickly into the new straining direction; but the long-range back stress, which would realign more slowly, would remain in the direction of the first prestress. If the two back stresses were equal in magnitude but oriented opposite

Chapter 4 Improvementsin the MATMOD Equations 167

!

v/-3 "L"(MPa) Jr

!

y~=o.o8 ,.'~f-'-."i OA4 , ( .--4---_.'~ OA3 o

INITIAL YIELD LOCUS

I -200

I

-',oo! ~

-',oo ! ~

! 1

o~ i

I

200

O-(MPa)

OA1

-200 Measured yield loci of 2024-T7 after shear prestress. (OA1) No prestress; (OA2) ~/-3r = 260 MPa, 7/~/3 -- 0.015; (OA3) ~v/'3r -- 310 MPa, V/~/3 = 0.039; (OA4) V/-3r -- 310 MPa, y/~/3 = 0.08. Reprinted from Helling et al. (1986, Fig. 4, p. 315), with permission from ASME.

to each other, the yield loci would be centered at the origin. If yield-surface distortions were associated with the back stresses, the yield surface would then be distorted. This would confirm that no single back stress was equal to zero. Figures 12-14 show the results of this test: the yield loci are, in fact, flattened on the top and bottom but are not translated from the origin. From these experiments we conclude that (1) yield-surface distortion is related to the strain path history, (2) it is possible to have a yield surface that is distorted in more than one direction, and (3) it is possible to have a yield surface that is distorted but not translated.

2. K e y M e c h a n i s t i c F e a t u r e s of Y i e l d - L o c u s D i s t o r t i o n s The experimental results described in the previous section suggest that it is reasonable to associate yield-surface distortion with both back stress components, RA and RB. First, the distortions are related to the strain path history, like the two back stresses. Second, distortions can occur simultaneously on more than one side of

168

G.A. Henshall, D. E. Helling, and A. K. Miller 4/3T (MPa) ~20 +

TORSION PRELOAD

BACK EXTRAPOLATED

YIELD POINTS -7

IOO

.4

9

|

29

.4

--

r

0

t 5 =156 OFFSET

\ /

YIELD LOCUS

-Izo VON

MISES

t-ao

-4o

I -40

40

ao

i 120 CT (MPo)

YIELD FUNCTION

" 120

Small and large strain-offset yield loci of 1100-O aluminum after a shear strain of 0.5. Reprinted from Int. J. Plasticity, Vol. l, M. G. Stout, E L. Martin, D. E. Helling, and G. R. Canova, "Multiaxial Yield Behavior of 1100 Aluminum Following Various Magnitudes of Prestrain,', Pages 163-174, Copyright (1985), with kind permission from Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington OX5 1GB, UK.

the yield surface. This suggests that the distortions are related to both RA and RB (not just one of them). This also implies that the distortions change direction at different rates, like the two back stresses. Finally, distortion is lost with coarser yield definitions when the yield-surface probe imparts enough strain (due to the coarse yield definition) to realign the back stresses. Although common to the three materials studied, the relative amount of yieldsurface expansion, translation, and distortion is material-dependent. In terms of MATMOD variables, this would be due to the relative amount of isotropic hardening modeled by Fde f and the kinematic hardening modeled by the back stresses. 3. Use of the Anisotropy Tensor to Model Yield-Surface Distortions Hill (1948) proposed modeling anisotropic initial yield surfaces by introducing constants (F-N) to modify terms in the following yield function: 2f(o'i) = F(o'] -- 0"2) 2 --[- G(cr3 -- o"1) 2 @ H(O'l -- 0"2)2 + 2L(cr4) 2 -I--2M(o'5) 2 Jr- 2N(o'6) 2 -- 1

(5)

Chapter 4 v/-3 "t" ( M P a )

Improvements in the MATMOD Equations

1 69

-'-

80-A4

-

1,6,#' ~

~,MPg)o_

[]

~'~ o.o~tp~ 'MPa)I\\-- \ , Iox\ o.o,3 I ~"-~ ~ , N -,o V/3 "r"(MPa),

•

,o

//'T l J_

60'

v'3 T (MPa) I 60 r t

03,~2~(MPa)I T 9~ . I o.o2~

i

~/~~ I 612~(MP') 0.0091-~,,

]

o.o,2 0.03

A comparison of the measured yield loci of 1100-O aluminum after three different stress paths to the same point in stress space: c~ = 62 MPa, v/-3r = 60 MPa. (A4) ~r followed by T; (A5) r followed by ~; (A6) proportional loading involving a ~r/v/3T ratio of 1. Reprinted from Helling et al. (1986, Fig. 7, p. 316), with permission from ASME.

Shih (1977) used a similar approach (first proposed by Baltov and Sawczuk, 1965) and used a variable anisotropy tensor, Mij, which was a function of prestrain. The multiaxial MATMOD equations also include this tensor, which allows initial anisotropy to be predicted. When the initial yield surface is isotropic, the tensor takes on isotropic von Mises values where M]I = M 2 2 = M33 = 1 and M44 = M55 = M66 = 3 and the other components are calculated assuming symmetry and invoking incompressibility.

C. Development of the Anisotropy Tensor in the MATMOD Equations 1. Modification of the Yield Function with the Anisotropy Tensor, M~/ To demonstrate how the anisotropy tensor components can produce anisotropic yield surfaces in the MATMOD equations, consider the case in which only simple shear (in the 6 direction) is applied. The strain rate in the shear direction, the only nonzero component, is calculated as follows (for R i = 0): ~:6 - -

X/-~66B|

sinh

v 'v16__._.~6~,()__6/r'__~)

(6)

170

G.A. Henshall, D. E. Helling, and A. K. Miller

X/V-3 o.o3~

v~'r

o.oi I

60 o" (MPa)

I

0.013

C

E]

A4 I

/

~

A2 i

t

2,

/~

I

-40

(7" ( M P a )

-20\ A1

~/~ r (MPa) A7 -40-

L e~ o . o o g r ~1 0.004

A comparison of measured yield loci of 1100-O aluminum after different stress/strain paths. (A4) r followed by or" (A7) control of stress to produce e followed by y. Reprinted from Helling et al. (1986, Fig. 8, p. 316), with permission from ASME.

This expression shows that the strain rate in the 6 direction increases with M66 , meaning that the material is softened in that direction when M66 increases. Therefore, yield will occur at a smaller value of 0.6 when M66 is increased or the yield surface is n a r r o w e r in that direction. Since M66 affects the strain rate only in the 6 direction, the yield surface b e c o m e s anisotropic if the other t e n s o r c o m p o n e n t s are not c h a n g e d . N o w if M66 is anisotropic (due to a back stress in the 6 direction), all other R i - - 0 and all stresses other than 0.1 and 0 6 are zero, then the effective strain rate m a g n i t u d e can be e x p r e s s e d as follows l: = f(0./E)

-

R) = [Mll(0.1/E)

2 + M66(0.6/E-

R6)2] 1/2

(7)

The effective strain rate in Eq. (7) is not influenced by an increased (anisotropic) 1The bar over (cr/E - R) indicates that this is an effective stress, calculated using the von Mises criteria.

Chapter

4

Improvements in the MATMOD Equations

1 71

v/-3 "r ( M P a )

170~---~

v/-37"(MPa) I 200

--l--

/

/

'{

~80 - 4 0

+

..;,

/1

-4o+

~ o-

I 0.069~f' o.o17r j

.

.

.

.

.

tMl'a)

.

,

40

"...~80 I . . / -~--

/

B5

l

~/v/3 / 170

8~ \

120

160

70~

~,/,r

170

200 O'(MPa)

<7"MP " I

a/

0.031p ! i E 0.O05

A comparison of the measured yield loci of 70" 30 brass after different stress/strain paths. (B5) r followed by a; (B6) control of stress to produce >, followed by e. Reprinted from Helling e t al. (1986, Fig. 9, p. 317), with permission from ASME.

M66 value when (cr6/E - R6) : 0 regardless of cq/E. This means that a constant mij t e n s o r can increase the strain rate on one side of the yield surface (and thereby produce a flattened side) without affecting the flow behavior in other directions. This behavior is illustrated in Fig. 15.

2. Calculation of M#

Given the experimental results described in Section II.A.I, we need to model the following principal features with the mij tensor: (1) Anisotropy is associated with both back stresses [Mij = f ( R A i , RBi)]; (2) anisotropy increases with increasing back stress (there is no anisotropy for an initial, annealed material when the back stresses are also zero); and (3) the anisotropy can occur in any direction in stress space. A more complete description of anisotropy could also include the anisotropy tensor due to a crystallographic texture whose evolution could also be modeled.

172

G.A. Henshall, D. E. Helling, and A. K. Miller

V/31," (MPa)

2601 = v-3 rOMP,)

~ 1

3oo-

I

I

-200

I

I

!

/

r/~ / ~so I

0.044 ~/r

o.o ,r, 0.01~]

-

I -i100 %\ % "~J O0

-- 0" (MPa)

~I~ '

_

~,/3r(MPa) 260 ~-

,?---OA6

t

II

/

/

/

/

/

I

I

200

:

300

O'(MPa)

-200 A comparison of the measured yield loci of 2024-T7 after different stress/strain paths. (OA5) r followed by or. (OA6) control of the stress to produce y followed by e. Reprinted from Helling et al. (1986, Fig. 10, p. 317), with permission from ASME.

The current model only treats the evolution of anisotropy of small strain-offset yield-surface distortion associated with dislocation anisotropy, and a constant crystallographic texture anisotropy. M A and M B are then defined as the anisotropy tensors resulting from short-range back stresses and long-range back stresses, respectively. Since the approach to modeling the small strain-offset yield-surface distortion is the same for modeling the effects of RA and RB, the approach will be outlined only for determining M A. Since the initial yield surface for an annealed material is undistorted (when the back stresses are also zero), it is assumed that the magnitude of the distortion is related to the magnitude of the effective back stress. The effective back stress is calculated directly from the von Mises relation and is referred to as R. The

Chapter 4

Improvements in the MATMOD Equations

173

v/3 T (MPa) 60 f

/

/

/

\

40

/

\

o'(MPa)

\\A2

-40~

\

l

I

20

\

-40

v/3 r (MPa)

i'

,; /20

' \4o

0.01~

E

O"(MPa)

-20

-40

l[l~llltIIPJ A comparison of experimental yield loci of 1100-O aluminum after a torsional prestress (A2) with that resulting from a slight reversed torsional prestress (A8). Reprinted from Helling et al. (1986, Fig. 11, p. 317), with permission from ASME.

effective anisotropy, termed M, which controls the magnitude of the distortion, is then related to the effective back stress. A given value of M should result in the same shaped yield surface, regardless of its orientation in stress space. Based on experimental data (Helling et al., 1986; Helling, 1986), the following function has been selected to relate the effective anisotropy to the effective back stress" ~.1- 1 + (KR) v

(8)

This function allows M to equal 1 when k equals zero and allows M to increase with R. For the three materials that have been studied, 0.25 _< V _< 1. Incompressibility and symmetry reduce the number of independent components in Mij from 36 tO 15. Rather than solving for 15 equations with 15 unknowns, we take a more intuitive approach to solve for the components of Mij. It is noted that for the special case of pure shear (in the 6 direction, for example), the only nonzero back stress is R 6 and the only anisotropic Mij component is M66. This

174

G.A. Henshall, D. E. Helling, and A. K. Miller

v/-3 "t"( M P a ) 160-

/ I I

/ ~

/

//

r

/

~ B2

\

-~ /

//

~'~. --

\

40"--

' { 18o ' ,2o/ ,' o '

\

x

. . . . . .

.... ""

'

i

X

170

~

-70p--

\

~

o'(MPa)

0-02'I-Y/v~ ~ 0.025 \ I-

i

8 o,)

' 40

~.v/3 "r (MPa)

\

,

120

'

r

~

-80

~/

/ / B1

-120-

A comparison of experimental yield loci of 70"30 brass after a torsional prestress (B2) with that resulting from a slightly reversed torsional prestress (B7). Reprinted from Helling et al. (1986, Fig. 12, p. 318), with permission from ASME.

direction makes it a convenient reference state for which it is easy to determine the size and shape of any yield surface since the back stress and anisotropy can be directly related to their effective values by the following relationships: R~ = R/V/3 M~6 = 31171

(9) (10)

where the superscript asterisk (*) indicates that these are values in this reference direction. All other Mi~)components remain at their isotropic values. The distortion of a yield surface in the 6 direction is completely described by M~6. A three-step approach is then taken to calculate Mij for the more general case. First, calculate M~6; this defines the correct size and shape of the yield surface in the 6 direction, which is used as a reference. Second, calculate M~j, the anisotropy

Chapter 4 4-3

Improvementsin the MATMOD Equations 175

T"( M P a ) 300 -

OA2

If-

7/v"-3 = 0 . 0 1 5

/

/

|

t

//

v/~r(MPa)

[]

200-

\

-

/ f100-"

I

\

E] O A 7

". 1oo:

~

-200

/

o'(MPa)

\

/\

I

\

\

~x

# -200

i -50 ~-

260

0.023 0.020

~/v~ t --

i

/

I /

I

200

300

O'(MPa)

,~

-

Bi[~IU~tiEm A comparison of experimental yield loci of 2024-T7 after a torsional prestress (OA2)

with that resulting from a slightly reversed torsional prestress (OA7). Reprinted from Helling et al. (1986, Fig. 13, p. 318), with permission from ASME.

tensor in principal coordinates, from M~6. Third, calculate Mij by transforming M[j into the material coordinates. Figure 16 shows an example of the first rotation in the 7r-plane, normalized by E, the elastic modulus, as is used in MATMOD. If tensile deformation produces a back stress of Ri, the effective back stress is equal to R, R~ can be calculated from Eq. (9), and M~6 can be calculated from Eq. (10). This reference yield surface is also shown in the figure. It has the correct size and shape, but is oriented along the a~ / E axis. The next step is to calculate M[j. The two yield surfaces shown in Fig. 16 have the same size and shape. By definition, equivalent points on these yield surfaces produce the same magnitude of strain rate from the M A T M O D equations. Equivalent points on the yield surfaces are points that, in the yr-plane, are the same distance from the back stress and have the same angle from the back stress. An

176

G. A. Henshall, D. E. Helling, and A. K. Miller

iso

>M~

13'1

l|[ll|l'tIl~

Effect of an anisotropic M66 component on a yield locus.

example is shown in Fig. 17. Points A and B are equidistant from R~ and Ri, respectively, and both are at an angle Y2 from the back stress. By defining Y2 as the angle between the back stress and the 2 axis, the point A can be expressed in terms of the principal coordinates as A ( t r 6 / E - g6)* = A ( t r 2 / E - g2) cos Y2

(11)

A ( t r a / E - R3)* = A ( c r 2 / E - R 2 ) s i n Y2

(12)

and

The effective stain rates that are calculated at these points should also be equal, so

(13)

(eriE- R)* = ( c r / E - R) which can be expanded to ~[M}j(o-i/g

I Ri),(o.j/g

_ g j ) * ] -- w / [ M ~ j ( o i / g

- gi)(oj/g-

ej)]

(14) Inserting the nonzero components [Eqs. (11)-(12)] into this expression gives [M~*6(A(O'E/E - R2)) 2 c0s2 Y2 -'[- M3*3(A(o'2/E -- R2)) 2 s in2 )'2 -1- M~*6(A(o'2/E -- R2)) sin y 2 ( A ( c r 2 / E -- [ M ~ 2 ( A ( o ' 2 / E -

-- R2)) cos y2] 1/2

R2))2] 1/2

(15)

Since Mj'3 and M~6 are isotropic, their values are 1 and 0, respectively. Therefore, M22 = M6"6 c0s2 )/2 -[- s in2 Y2

(16)

Chapter 4

Improvements in the MATMOD Equations

177

(~ HI

\

E

% %

%

% %

/

/

i

/

/

/

/

// Reference Yiel

(3"6 -g-

_

, E

// "Real" Yield Surface/ /

-g-\

/

\

\ % \

m|[ILlltlllPJm Projection of the back stress produced by tensile deformation ( R i ) into the 7r-plane along with its associated reference back stress (R~). l~III

E

A ((y2/E - R2) sin ~2

G6

(~II

E ~|[glll'tmt,J frame.

E

Rotation of a unit vector in the direction of at1 [ A ( a 2 / E -- R2)] into the reference

178

G.A. Henshall, D. E. Helling, and A. K. Miller

The other diagonal components in M[j can be solved with similar relationships, while the off-diagonal components are calculated from incompressibility. In general,

m[i = mg 6 c0s2 where

Yi is the angle between

~i + sin 2

(where i = 1, 2, 3)

Yi

(17)

the back stress and principal stress axis i, and

t 1 t t M12 = - ~ ( M l l + m~2-

t M33 )

(18)

, __ - ~ l ( M ~ 2 4- M33 , - Mll) , M~3

(19)

M22)

(20)

M13' - - - - g l ( M ' l l + M ; 3 -

The anisotropy tensor in the principal coordinates of the rr-plane, transformed into M~j in the material coordinates by

M[j, can be (21)

Mijkl = aimajnakoalpMmnop where and

Mijkt and Mm,,op are

tensor representations of

Mij

and

M~j, respectively,

aqr is the direction cosine between the q and r axes.

Although Mij does not distinguish between the front and the back sides of the yield surface, experimental data show that one side is typically more distorted than the other. The front of the yield surface may be distorted more or less than the back of the yield surface by using different constants in Eq. (8) for each side. The front of the yield surface is defined as the side in the direction of Ri. Since the strain rate is normal to the yield surface, the front and the back sides of the yield surface are easily distinguished by taking the dot product of the back stress and the nonelastic strain rate. The stress will be on the front of the yield surface if (RAi)(Ei)

> 0

(22)

< 0

(23)

or on the back of the yield surface if

(RAi)(Ei)

The final calculations for Mij a r e quite simple [Eqs. (8), (17)-(23)] and provide the desired modeling capabilities. The Mij t e n s o r s are functions only of the back stresses and the current stress state and do not depend on the direction of the current strain rate. Normality of the strain rate to the yield surface is maintained, as well as convexity of the yield surface. To combine the three anisotropy tensors, M A, M~, and M T (the anisotropy due to RA, RB, and a crystallographic texture, respectively), it is necessary to add their deviations from isotropy. This is expressed as M/TOTAL - -

M A + M/~ + M T -- 2M/i~~

where Mji~~ is the isotropic von Mises tensor (Helling and Miller, 1987).

(24)

Chapter 4

Improvements in the MATMOD Equations

179

M A - I + ( K , R A ) v'

MA,ii -- sin2 Y~a + MA COS2 Y

(for i = 1 to 3)

M A,O~ -- a ~ a jn a ko a ipM A,mnop iso

M o - MA, 0 + MB,0 + Mr,0 - 2 M 0 l|[~llS|ml

Calculation of

mij in the MATMOD-4V-DISTORTION model.

Figure 18 summarizes the complete equations used to calculate the anisotropy tensors from the back stresses and are substituted for the anisotropy tensor in Table 3 to complete the MATMOD-4V-DISTORTION model.

D. Predictive Capabilities of MATMOD-4V-DISTORTION 1. Simulations of Yield Loci during P r o p o r t i o n a l L o a d i n g

Yield-surface simulations are conducted to simulate the same sequence by which yield loci are experimentally measured. Experimentally, the sequence is (1) to prestress the material along the selected prestress path, (2) to partially unload the material to the center of the yield locus (the elastic region where the strain rate is zero), and then (3) to probe out in different directions in stress space, using a small strain-offset yield criterion. Yield locus simulations are performed with MATMOD in the same manner. The prestress is simulated (during which time the internal state variables are changing to reflect the deformation history) and the stress drop is simulated. Since the strain rate is zero at the center of the yield surface, the only changes in the MATMOD internal state variables are those caused by static recovery, which occur relatively slowly. In the yield-surface experiments of Helling et al. (1986), the stress was held at the center of the yield surface for approximately 30 minutes during the installation of strain gages before yield-surface-probe measurements began. The simulations also included this 30-minute hold period to allow this small amount of static recovery to occur. The values of all MATMOD variables at the end of this hold period are stored, and these variable values are then used at the start of each yield-surface probe. This makes the predicted yield locus independent of the probe sequence. Just as in experiments, the yield-surface probesmalbeit smallmcan affect the material (and the MATMOD variables). The yield-surface probes are performed along a prescribed angle in stress space from the center of the yield surface until the desired strain offset is reached. The entire yield surface is constructed by probing out in different directions in stress space.

180

G.A. Henshall, D. E. Helling, and A. K. Miller

The basic ability of MATMOD-4V-DISTORTION to predict the key features of distorted yield loci after proportional prestrains is shown in Fig. 19, which compares the predicted yield loci (of 7 0 : 3 0 brass) after a tensile prestrain to the experimental data of Shiratori et al. (1979) on 6 0 : 4 0 brass. The predicted yield loci have the same features as the experimental loci. The small strain-offset yield loci are translated and flattened on the side opposite to the prestrain. These characteristics are lost for larger strain-offset yield loci, which tend toward more isotropic behavior. The materials being compared are not identical, so the yield loci are not quantitatively identical. The effects of increasing prestrain are demonstrated in Fig. 20, which compares the MATMOD-4V-DISTORTION simulations of small strain-offset (5 x 10 -6 m/m) yield loci to experimental data on 1100-O aluminum after a shear prestrain. Features similar to those observed experimentally are modeled. The yield loci are expanded after the prestrain, they are translated in the direction of the prestrain, and they are flattened on the side opposite to the prestrain. Figure 21 shows the behavior of the MATMOD-4V variables during the shear prestrain and compares them to interpreted experimental data, which are shown as dashed lines (as discussed in Section IV). The short-range back stress, RA, increases rapidly and saturates after about 3% effective strain. The long-range back stress, RB, increases more slowly, and is still increasing after 25 % strain. The isotropic hardening variables, Fdef, p and Fdef,~ are summed and plotted together as Fde f in Fig. 21 b. The increase of these variables with increasing prestrain causes the predicted yield loci to expand. It is worth noting that the behavior of recent versions of MATMOD involving the buildup of two different back stresses over two different strain magnitude regimes (Fig. 21 a), combined with isotropic strain hardening, is similar to that of other recent models (Korhonen et al., 1987), based on Hart's original formulation (Hart, 1970). In these other models, initial back stresses build up by dislocation pile-ups against "weak barriers" and only "anelastic" strain occurs. With further increases in stress, leakage through the weak barriers occurs and dislocations pile up against strong barriers. Finally, leakage through the strong barriers causes "macroplastic" strain. Reversals in the loading direction affect the behavior in the Hart-based model in a manner similar to that in MATMOD; algebraic superposition of applied stresses and the two back stresses controls the direction of the dislocation motion/nonelastic strain rate. The buildup of the back stresses causes the yield surface to translate (with a yield surface centered at the vector sum of the two back stresses) and to distort the yield surface, primarily on the side opposite to the prestrain (or opposite the direction of the back stresses). Figure 22 shows the behavior of the MATMOD variables in the 30-minute static recovery period during which the stress is held in the center of the yield surface. Of

Chapter 4

Improvements in the MATMOD Equations

1 81

200 "

-

70:30 Brass Offsets noted (10 ~)

-

~

I00

t, x

-100

-200 -200 "

i

i

I

i

I

.

|

m

.

.

-100

9

.

|

0 o

.

t

|

100

200

(MPa)

1.5

'

'

''

I

'

'

'

'

I

'

60:40 Brass Offsets as noted 1.0

0.5

0.0I

2% - 1.0

...... -0.5

0.0

0.02% 0.5

1.0 Ca*)

~/O* ~a(llll~tmi,,ll Comparison of the predictions of MATMOD-4V-DISTORTION (with 70:30 brass material constants) against the data of Shiratori et al. (1979) (on 60 : 40 brass) for yield loci of various strain offsets, after a proportional prestrain.

1112

G . A . Henshall, D. E. Helling, and A. K. Miller

150f,

i

; ; ; i ; ' ; ' MATMOD-4V-DISTORTION

i

I i l

i

i

I I

l 100-0

I00

y / ~ = .25

50 = .017

-5oI~ -100

, , , ! . -50

.

l

l

9 l

9 !

0

,

9

9 l

50

100

(MPa)

,/'3 T (MPa) 120 ~/v/'~ = 0 . 2 8 9 A3

y/V3 "0.012

-80

-40

A2 ~[..,,/A 1 /

40

80

o" (MPa)

"1 INITIAL YIELD LOCUS 140 t

Comparison of MATMOD-4V-DISTORTION simulations of 5 x 10 -6 yield loci of 1100-O aluminum after proportional prestrain, against experimental data.

Chapter 4

4 I I I. . I . . . . . . . . . . '

i

. . . .

]

Improvements in the MATMOD Equations

. . . .

' ~ . . . . .

I

. . . . .

I

. . . .

. . . .

i

s

RB6

3

183

MATMOD-4V-DISTORTION

0

-

f

2

f

I

RA6

~'/DATA

[

MATMOD-4V-DISTORTION f

DATA

J 0.00

0.05

0.10

0.15

0.20

0.25

8

b n

-

ON

.-~

5

i

0t._. 0.0

0.1

0.2

0.3

E ~ | [ ~ 0 1 1 1 ~ l n Comparisons of MATMOD-4V-DISTORTION model predictions against experimental data: (a) For the evolution of back stresses in 1100-O aluminum; (b) for the evolution of Fdef in 1100-O aluminum.

184

G.A. Henshall, D. E. Helling, and A. K. Miller

RB6

Hold period "~

Unloading to center of yield surface

2

RA6 1

Fdef, p 6 oO i

o

Faa,

4 2

0

500

1000

1500

2000

Time (see) l | [ l l l | [ | l q U l Evolution of MATMOD-4V-DISTORTION variables for 30% prestrain, unloading to the center of the yield surface, and a 30-minute hold period to allow static recovery. 1100-Oaluminum.

note is the fact that RA recovers about 30%, while the other variables are unaffected. The recovery of RA produces yield loci that do not enclose the prestress point (shown as open squares in Fig. 20), which is also observed experimentally. 2. S i m u l a t i o n s of Yield Loci d u r i n g N o n p r o p o r t i o n a l L o a d i n g

Figure 23 illustrates the capability of MATMOD-4V-DISTORTION to simulate yield-surface distortion on different sides of the yield surface. The prestress path involves a prestress in shear, decreasing the shear stress into the center of the yield surface and then, while holding this shear stress constant, applying a tensile stress. This final stress leg produces primarily tensile strains. Enough tensile strain is generated to realign RA into the tensile direction, but not so much as to alter the orientation of RB which is still aligned in the shear direction. The two back-stress vectors are also shown in Fig. 23a. It can be seen that the center of the predicted

Chapter 4

Improvements in the MATMOD Equations

185

yield surface is at the vector sum of the back stresses and the yield surface is distorted opposite to each of the back stresses (on the left and the bottom). A simulation was also conducted involving a reversal of prestress direction (Fig. 24). In this simulation the two back-stress vectors are in opposite directions but are equal in magnitude. Since the vector sum of the back stresses is zero, the center of the predicted yield surface is at the stress origin. The predicted yield surface is heavily distorted on both the top and bottom; i.e., there are separate distortions due to RA and RB. MATMOD-4V-DISTORTION has displayed similar predictive capabilities for 7 0 : 3 0 brass and 2024-T7 aluminum (Helling, 1986; Helling and Miller, 1988). These other two materials experience less isotropic-type hardening and more of the back-stress-controlled kinematic type. The general features are the same as demonstrated here for 1100-O aluminum. 3. Effect of Yield-Surface Distortion on Previous M A T M O D Capabilities

The anisotropic mij t e n s o r has very little impact on other MATMOD capabilities, as desired. The front of the yield surfaces is usually isotropic (or nearly so), and so all capabilities to simulate proportional loading are unaffected. The behavior of MATMOD under reversed loading is primarily affected in the small-strain regime, meaning that the small-strain Bauschinger effect will be enhanced for a given value of the back stresses. If these effects are critical, the MATMOD constants may need to be refined to more accurately simulate cyclic effects. The general capability of the model to simulate these effects is unchanged by Mij. At larger strains, the mij tensor realigns in the straining direction as RA and RB realign, and the behavior reverts to that of the original MATMOD-4V equations. The uniaxial steady-state strain rate is affected very little. The steady-state strain rate can be written as B|

sinh{ / ss

~ d - ~ f . ; ; Fdef,Z)

Lowe (1983) has shown that the steady-state MATMOD state variables can be written as

where C = sinh -l(Ikssl/B|

RA, ss - - C / A 2

(26)

RB,ss -- C/A3

(27)

(Fdef, p,ss) p/2(p-1) = C / A 4

(28)

(Fdef,~.,ss) p/2(p-1) = C / A 5

(29)

1/" 9 Substituting Eqs. (26)-(29) into Eq. (25)

1 86

G . A . Henshall, D. E. Helling, and A. K. Miller

a 1100-0 50

r"

......

." ,,

L~

I

"'.

.,

~

9

.

:~

.''

."

25-~.

(.Pa)

o

.." ~

25t-

I

-"

MATMOD4V-DISTORTION Simulatim

-75

.. ! .... -5o ....

..I .... ! .... I,, 60 o (MPa)

~

-

-50!.

o

I .... 0 ....

G

.02 .0!

! ........ -~ ....

r/4~

~j ....

! .... 5o

o

. . . . . . . . . "':! .... !.... !

o

.01

.02 6

~5

(MPa).

V~TI (MPa~. 60tt1 !I A2~ -40

/,

S

40

%

J

20 -20 AI

20

1 40

-'-"f (7" (MPa) ;0 r

A7 -40

(MPa)

r/v~L e~

o.oo9FTI o.oo4

Comparison of MATMOD-4V-DISTORTION predictions for a 90 ~ corner in the nonelastic prestrain path. 1100-O aluminum.

Chapter 4

Improvements in the MATMOD Equations

187

75.,,,,i,,.,i,,,, ''''i''''i'''C-" MATMOD-4V-DISTORTION 50-"

1100-O

9

.'"

,

.

9

9

25

L

o

.""

~

"':

-

.017 .015

0

t~ .25 m

-50

--

-75"__,, -75

::!.,

-so

..

!,,

,,.

-~

c~

0

,,

..

!,

25

,,

,!

50

...,

75

(MPa)

v~ T (MPa) 60

///

f

o'(MPa)

40 -.

\

I I v/~r (MPa)

"'"

)/~

2o

~..._ _~._~:

-,o (--;o

'y/x/i,

0.0181 0.015 /

i 2 0 " \40 O"(MPa) ~.~IA1 J

~AB -40

t

~ | ( q , lJltlRll Comparison of MATMOD-4V-DISTORTION predictions for a reversed prestress path in 1100-O aluminum.

188

G.A. Henshall, D. E. Helling, and A. K. Miller

produces ~"-M(crss/E - C/A23)

C1/p

(30)

~/(C/A4) v + (C/As) v

where A23 -- (A: + A3)/AzA3 and v = 2 ( p Rearranging and replacing C, we get

1)/p.

9

es__~s_ B sinh |

(31)

where !

1

-A- =

A2 + A3

A2A3

1

+ ~

/

1

1

V (An) v -~- ~(As) v

(32)

m

These expressions show that M does slightly affect steady-state behavior. This influence is a function of M, Ai, and p. For the three materials studied to date, the maximum effect on the stress exponent n in Eq. (25) was only 2.3% over 15 orders of magnitude of kss/| The model also breaks down if M is decreased too much (as if to simulate hardening in a particular direction). Although the yield-surface distortion modeled here always caused M to increase, other types of yield-surface-distortion phenomena might be modeled with a decreased M (such as a crystallographic texture). As M decreases toward zero, 1/A in Eq. (32) approaches infinity. This would cause ess to go to zero regardless of the applied ass.

E. Summaryof MATMOD-4V-DISTORTION By allowing the anisotropy t e n s o r (Mij) to be a function of both the short- and long-range back-stress state variables, a wide variety of multiaxial phenomena can be simulated while retaining the model's ability to treat other complex behaviors. The model can realistically simulate small and large strain-offset yield loci which exhibit differing amounts of translation, expansion, and distortion. The distortions have been found to be dependent on both short- and long-range back stresses, thereby allowing the prediction of yield surfaces that are distorted on two different sides and yield surfaces that are distorted but not translated from the stress origin. In terms of the development of the MATMOD equations, the distortions provide direct evidence of the existence of two separate back-stress components which evolve at different rates. The measurement of small strain-offset yield surfaces allows the evolution of these two back stresses to be resolved and their relative contributions measured as discussed in Section IV.

Chapter 4 Improvementsin the MATMODEquations 189

SIMULATING SOLUTE EFFECTSTHROUGH SHORT-RANGE BACKSTRESSES As discussed in Section I.B, to reduce the complexity of the model and the number of material constants, the method of modeling solute effects was reexamined. Simply including the Fsol,1 and Fsol,2 variables (Fig. 1) of Schmidt and Miller (1981) within MATMOD-4V would have created a complex model containing 44 material constants, making their evaluation problematic. The results of experiments by Henshall and Miller (1989), which are summarized in the following section, were used to develop a simple and physically plausible treatment of solute strengthening. In particular, the short-range back stress (demonstrated from multiaxial experiments discussed in Section II) was used to treat plateaus in the curve of yield strength vs. temperature. A link was established between yieldstrength plateaus and "Class I" steady-state creep, which is characterized by a steady-state stress exponent of approximately 3 (Sherby and Burke, 1967). In many solute-strengthened systems, such as the A1-Mg system shown in Fig. 25, both phenomena evolve as the solute concentration increases, suggesting that they have a common physical origin.

A. Overview of Approach to Modeling Solute Effectswith MATMOD-BSSOL Table 4 and Fig. 26 give an overview of the manner in which the new model, MATMOD-BSSOL (Back Stresses from SOLutes), uses the effect of solutes on short-range back stresses to predict behavior in both the plasticity and creep regimes in a physically plausible manner (Henshall and Miller, 1990a). 1. Low-Temperature Plasticity In earlier versions of MATMOD, simulation of yield-strength plateaus was achieved using Fsol,1, which was based on the concept of thermally activated solute drag first proposed by Cottrell and Jaswon (1949). However, more recent analysis and experiments using A1-Mg alloys led Henshall and Miller (1989) to conclude that yield-strength plateaus are largely caused by the rapid evolution of short-range back stresses, not by thermally activated solute drag. Figure 26 summarizes the new modeling concept. Nonelastic deformation begins at low stress levels, probably corresponding to the microstrain-resolution proportional limit. As this early nonelastic strain (e N in Fig. 26) develops, it builds up short-range back stress, RA, very rapidly (Fig. 26, upper left), causing a rapid increase in the flow stress. Similar to the model of Korhonen et al. (1987) based on Hart's original formulation, this corresponds to the anelastic deformation regime in which the slope of the stress-

190

G.A. Henshall, D. E. Helling, and A. K. Miller

a

20

i

~, = 1.28 x 10 "4 s "1 s = 0.02%

15

o X

A I - 5.8 at. % Mg 9A I - 3 . 1 a t . % M g o Al-l.lat.%Mg A Pure aluminum []

a = 8

a aa@

10

B []

0

o

O0

0

~

0

0

&|

&

&.

&~

0 o

O

o

D @

0

&&&&&l ,

0

o

[]

: O o O O

200

400

0

600

800

Temperature (K) 10

-1

10 -2 A

~J

10-3

9A I - 0.52 at.% Mg

9AI

1.09at.%

Mg

9AI

3.25at.%

Mg

=t,.d

IX C (g !,._

ct} Q

10"

4

lO -5

(U '~

10 -6

'10 m Q OrJ

10

-7

lO -8

Pure

///15 .

.

.

600 K .

.

.

|

i

i

l

l

10

l

l

l

i

100

Steady State Stress, (;ss (MPa) /l[llll'|lil The evolution of yield-strength plateaus and Class I steady-state creep in AI-Mg alloys as the Mg concentration increases. (a) The modulus-compensated flow stress at a nonelastic strain of 0.02% as a function of temperature and Mg concentration. Data of Henshall and Miller (1990a). (b) The steady-state creep rate as a function of the applied stress and Mg concentration. Data of Oikawa et al. (1984). Reprinted from Acta Metall. Mater., Volume 38, G. A. Henshall and A. K. Miller, "Simplifications and improvements in unified constitutive equations of creep and plasticity. I. Equation development," Pages 2101-2115, Copyright (1990), with kind permission from Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington OX5 1GB, UK.

Chapter 4

Improvements in the MATMOD Equations

191

Summary of the Way in Which RA Controls Yield Strength and Creep Behavior in the MATMODBSSOL Equations* Internal physical process Rapid buildup of RA Onset of mechanical recovery of RA Flow stress dominated by RA at a level set by mechanical recovery Flow stress dominated by Fp because mechanical recovery keeps RA low Higher RA levels required to sustain mechanical recovery as dislocation substructure (Fp) develops Thermal recovery of RA Thermal recovery of RA by dislocation climb Thermal recovery of RA by viscous glide Drag of mobile solutes by dislocations

In "low-temperature plasticity" behavior "Anelastic" portion of cr-e curve Macroscopic yielding

In "high-temperature creep" behavior "Loading" strain Power-law breakdown

Yield strength plateau

Yield strength > plateau at low temperature

Creep rate in power-law breakdown regime

Strain hardening beyond yielding

Primary creep

Yield strength decreases at high temperature

Power-law creep regime 5-Power regime 3-Power regime

Yield strength peaks

Anomalous slopes of creep rate vs. stress curves

*Reprinted from G. A. Henshall and A. K. Miller, "Simplifications and improvements in unified constitutive equations of creep and plasticity. I. Equation development," Acta Metall. Mater. 38, 2101-2115 (1990), with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington OX5 1GB, UK.

strain curve is close, but not quite equal, to the true elastic modulus. When the back stresses become large enough, substantial mechanical recovery is triggered, greatly decreasing their net rate of increase with continued straining and corresponding to the macroscopic yield strength. Because the recovery is mechanical, it occurs at similar back-stress levels over a broad range in temperature (Fig. 26, upper right), corresponding to the yield-strength plateau (Fig. 26, lower left). The mechanical recovery rate is affected by the presence of immobile solutes, so that the plateau stress level rises with increasing solute concentration. If the material continues to strain and a dislocation substructure develops, the level of back stress at which the anelastic strain-hardening rate is in equilibrium with the mechanical recovery rate increases; this corresponds to macroscopic strain hardening (Fig. 26, upper left).

192

G.A. Henshall, D. E. Helling, and A. K. Miller

,..~.~

Mechanical recovery ~ l\ of RA ..........

RA RA

Alloy Pure Metal

/

.,.,,

Viscoussolute drag-controlled

Macroscopic .......I",~ yield s t r e n g t h ~ '~ Rapid workhardening of RA ~

Thermal recovery of RA Climb-controlled

~

~N

TI~N

+

~',,.,,,,,,,............: Alloy

Fp f9' l ( " ~ - )

TI~ N

Yield Strength \

\

/ Alloy

Pure Metal

%.,.... . . . . . . . . . . . . . . . . . .

N~

Creep Rate ~N

Log ( ~

/

J

Alloy / PureM e t a /

)

%.

",,

l Mechanical ,/Recovery

/

/ 9 Recovery: ,...-~3" Vir:C;US~~

/,':;/15C,mb-contro,~ ~..':

-T

N

ThermalRecovery:

Log ( Oss/E )

m|[ILIIltIP~nl Schematic illustration of how short-range back stresses, RA, control yield strength and creep behavior in the MATMOD-BSSOL equations. Fp. f - I (k/| represents the contribution of isotropic strain hardening to the flow stress. Reprinted from Acta Metall. Mater., Volume 38, G. A. Henshall and A. K. Miller, "Simplifications and improvements in unified constitutive equations of creep and plasticity. I. Equation development," Pages 2101-2115, Copyright (1990), with kind permission from Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington OX5 1GB, UK.

Chapter4 Improvementsin the MATMODEquations 193 Yielding, yield-strength plateaus, and solute strengthening in MATMODBSSOL are therefore controlled through the short-range back-stress variable, RA. The complicated Fsol,1 variable and its 11 material constants are not needed in MATMOD-BSSOL, affording a considerable simplification.

2. High-Temperature Creep Solutes are mobile at high temperatures and cannot pin dislocations when large stresses are applied. Therefore, solute mobility causes a transition from mechanically activated to thermally activated behavior, marked by an end to the yieldstrength plateau (Kocks, 1985; Heritier et al., 1974; Luton and Jonas, 1972). In the context of MATMOD-BSSOL, this transition to thermally activated flow means that high temperatures and solute mobility trigger thermal recovery of RA (Fig. 26, upper right). Unfortunately, theoretical and experimental information is not directly available regarding the thermal recovery of back stresses at temperatures above those associated with the yield-strength plateau. Therefore, the physical basis for the development of the thermal recovery term in the/~A equation comes from another source: steady-state creep of Class I solid solution alloys. Phenomenologically, the connection between recovery of back stresses and Class I steady-state creep is suggested by the fact that both plateaus in the stress-temperature curve and Class I steady-state behavior evolve as the solute concentration increases, as shown in Fig. 25. A simple explanation that unifies the low- and high-temperature behavior is that the large short-range back stresses, which cause flow stress plateaus at low temperature, are also responsible for the presence of Class I steady-state creep behavior (with a stress exponent close to 3) as they thermally recover at high temperature (Fig. 26, lower right). At even higher temperatures, the viscous drag forces dominating the 3-power creep regime are lost. Therefore, thermal recovery of short-range back stresses in this regime must be controlled by some other mechanism, which is probably related to the recovery of the dislocation substructure through dislocation climb. These additional climb-controlled recovery mechanisms lead to a different power-law dependence (close to 5) of the creep rate. In the description just preceding, thermal recovery of short-range back stresses has been associated mainly with power-law creep behavior. However, in the MATMOD-BSSOL approach, the same processes are also responsible for the drop-off in the yield strength at elevated temperature (Fig. 26, lower left). Thermal recovery prevents short-range back stresses from building up to high levels and thus reduces the macroscopic yield strength. Furthermore, in much the same unifying spirit, the mechanical recovery processes, which in the preceding have been associated mainly with low-temperature yield-strength behavior, are also responsible for power-law-breakdown creep

194

G.A. Henshall, D. E. Helling, and A. K. Miller

behavior. At high temperature-compensated creep rates, back stresses build up to the point where substantial mechanical recovery is activated (Fig. 26, lower right). The consequent reduction in back stresses (relative to those that would exist if only thermal recovery occurred) causes a corresponding reduction in applied stress, which is manifested as power-law breakdown.

3. Dynamic Strain Aging The increase in solute mobility that marks the end of the yield-strength plateau can also cause peaks in the flow stress vs. temperature curve and serrated yielding (dynamic strain aging), as the now-mobile solutes exert substantial drag forces on moving dislocations. All of these phenomena are modeled in MATMOD-BSSOL by using an interactive solute-strengthening variable (similar to Fsol,2 in previous models), which is simply denoted Fsol. Following Schmidt and Miller (1981), the interaction between mobile solutes and strain hardening is represented by multiplying Fsol with the isotropic strain-hardening variable. The treatment of low-temperature (immobile) solute strengthening via short-range back stresses and substructure evolution allows the equation governing the strain rate and temperature dependence of the solute-drag variable, Fsol, to be very simple, thereby greatly reducing the number of material constants.

B. Development of the Specific MATMOD-BSSOL Equations (in One Dimension) Development of MATMOD-BSSOL emphasizes breadth and simplicity. The equations employ the concepts developed for MATMOD-4V and earlier versions but, as described by Henshall and Miller (1990a), are not specifically based on earlier versions (except for Fsol). Phenomena specifically used to design the equations include peaks and plateaus in the stress-temperature curve, power-law (PL) and power-law breakdown (PLB) steady-state creep behavior, cyclic saturation at low temperatures, and directional strain softening. All of these phenomena have been represented in previous versions of MATMOD, but never in a single model. For simplicity, the equations are developed in their one-dimensional form, although the calculations presented in Section III.C were performed using a three-dimensional extension of the equations similar to that described in Section II (Henshall, 1987).

1. The Kinetic Equation The kinetic, or strain-rate, equation describes the nonelastic strain rate for a given (i.e., constant) stress, temperature, and microstructure, Eq. (1). When MATMOD was first developed, little was known about constant structure deformation, so steady-state creep phenomenology was used to derive the hyperbolic sine stress

Chapter 4 Improvementsin the MATMODEquations 195 dependence in the kinetic equation (Miller, 1976, 1987). This led directly to the sinh and sinh -1 terms in the recovery expressions of the structure evolution equations (Lowe and Miller, 1984a). Largely motivated by the desire to simplify these recovery expressions, the stress dependence in the kinetic equation was reexamined. Several experimental (Gibeling and Nix, 1982) and theoretical (Nix et al., 1985; Poirier, 1976; Barrett and Nix, 1965) investigations of constant structure deformation suggest an exponential dependence of strain rate on stress. Therefore, the stress dependence in the MATMOD-BSSOL kinetic equation was chosen to be similar to that derived by Nix et al. (1985):

~ot[exp(a/E)-

1]d

(33)

where E is Young's modulus and d is a constant. The strain rate depends exponentially on the stress, except at low stresses for which the " - 1 " becomes significant and forces the strain rate to equal zero when the stress equals zero. Raising the quantity [exp(a/E) - 1] to the d power provides flexibility in simulating a variety of constant structure data. The exponential form of Eq. (33) leads to recovery expressions containing logarithmic and exponential terms, instead of the sinh and sinh -1 terms present in previous versions of MATMOD. The temperature and microstructure dependencies of nonelastic deformation are included in the MATMOD-BSSOL kinetic equation following the methods used successfully in previous versions of MATMOD (Miller, 1987). Specifically, the effects of temperature are represented through the temperature-dependent elastic modulus and through the Arrhenius-like | parameter, Eq. (3). The dislocation substructure is represented in MATMOD-BSSOL using the four history-dependent structure variables present in MATMOD-4V (Section II) with the variables Fp and Fz in MATMOD-BSSOL corresponding to the variables Fdef,p and Fdef,Z in MATMOD-4V. In addition to these history-dependent variables, a single interactive solute-strengthening variable, Fsol, represents the presence of mobile solutes, as discussed in the previous section. A summary of the five variables used in MATMOD-BSSOL to represent the deformation substructure is given in Table 2. The complete one-dimensional MATMOD-BSSOL kinetic equation is then

-- B|

{

exp

1 -1 }d sgn[a/E-~3(RA+RB) ] [a/E _3(RA+RB), ~ v/Fz + Fp(1 + Fsol)

(34)

where B is a constant. Following Miller (1976), the signum function (sgn) and absolute values are required for cyclic simulations; they essentially create a onedimensional version of the full multiaxial Prandtl-Reuss type equation. For consistency in the material constants used for the one-dimensional and three-dimensional models, the absolute value of the back stresses in the one-dimensional version is multiplied by the factor 3/2, as discussed by Henshall (1987).

196

G.A. Henshall, D. E. Helling, and A. K. Miller

2. The RA Equation Development of the/~a equation is based on the assumption that short-range back stresses control athermal yielding and Class I steady-state creep in alloys. The/~a equation follows the same general format as that used successfully in MATMOD4V: (35)

R A = h -- rd -- rt

where h is the hardening rate, rd is the dynamic or strain-activated recovery rate, and rt is the thermally activated recovery rate. The expressions for h and rd are derived from an analysis of yield-strength plateaus, and rt is derived from an analysis of steady-state creep, particularly in Class I solid solution alloys. a. The Hardening and Dynamic Recovery Terms As described by Henshall and Miller (1990a), the hardening and dynamic recovery terms were derived using the concepts illustrated in Fig. 26 and Table 4. In the absence of recovery the hardening rate is assumed to be proportional to the strain rate. Following the concepts of athermal yielding proposed by Kocks (1985), Hirth and Lothe (1968), and Luton and Jonas (1972), the hardening term represents the evolution of shortrange back stresses as dislocations bow out between pinning points. The dynamic recovery term was designed in conjunction with the hardening term to produce athermal yielding at temperatures within the plateau regime. Physically, this term represents the mechanically activated breakaway of dislocation segments from their bowed configuration. In addition, short-range back stresses evolve because dislocation-dislocation interactions produce small pile-ups, dislocation tangles, and other features that contribute to the short-range back stress. These are the substructures that accounted for the evolution of RA in MATMOD-4V (Lowe and Miller, 1984a). RA should therefore increase (Fig. 26) as Fp evolves, since Fp represents these microstructural features in the model. For monotonic deformation, then,

1 J~A --- n l ~ -

[AI(RA)m']

. --~~

rt

(36)

where H1, A1, and m l are material constants. For large m l, dynamic recovery occurs very suddenly as RA increases, and RA saturates quickly with increasing strain. The rapid saturation of RA, and hence or, simulates dislocation breakaway and athermal macroscopic yielding (Fig. 26). Furthermore, by making A1 a function of solute concentration, the observed dependence of the yield strength on solute content can easily be included in the model (Fig. 26). The form of the interaction between/~a and Fp in Eq. (36) is simpler than that in MATMOD-4V, but this expression represents the physical link between the evolution of short-range back stresses and the dislocation substructure: as Fp increases, the dynamic recovery

Chapter 4 Improvementsin the MATMOD Equations 197 term in Eq. (36) decreases, and RA continues to evolve. The quantity (1/Fp) is assumed to enter linearly into Eq. (36) only because it significantly simplifies the effort in determining the material constants and it has proven to be sufficiently accurate (Henshall, 1987). Consideration of cyclic deformation requires that one more term be included in the dynamic recovery expression for RA. This addition concerns the representation of back stress and strain rate "colinearity." Implementation of this concept follows that described by Lowe and Miller (1986), resulting in

R A - H i e - a l [(31RAI)m~] 9RAIe____~]_ ?'t F~

(37)

The absolute value of RA is used in the first term of the dynamic recovery expression because RA can be negative during cyclic deformation; the quantity (RA) m~ is undefined if RA is negative and m l is not an integer. The term that is linear in RA causes mechanical recovery always to tend toward a state of RA = ' 0 . For reasons given earlier, the factor of 3/2 is present only in the one-dimensional form of the equation. b. The Thermal Recovery Term As discussed earlier and in more detail by Henshall and Miller (1990a), derivation of the thermal recovery term in the RA equation is based on the assumption that power-law steady-state creep in alloys, including the transition between Class I and Class II behavior at low stress, is controlled by thermal recovery of short-range back stresses. The expression derived from this assumption is also flexible enough to model the behavior of pure metals and alloys that do not exhibit a change in the steady-state stress exponent. The behavior of RA,ss and Crss/E (the subscript ss denotes the steady-state value) must be similar to that shown schematically in Fig. 27. Figure 27a illustrates the behavior of pure metals and Class II alloys, and Fig. 27b illustrates the behavior of Class I alloys, such as A1-Mg. Note that RA, ss is a large fraction of Osg/E in the power-law (PL) regime for the Class I alloy. A simple equation was phenomenologically derived that describes the temperature and strain-rate dependence of RA,ss for either type of behavior: 9

m 2

ess -- B~tD1RqlA,ss [1 - e x p ( - D 2 R A , ss)]

(38)

where D1, D2, ql, and m2 are constants. For simplicity, the temperature dependence in this equation is given by the | parameter defined in Eq. (3) (Henshall, 1987). In the Class I regime Crss/Eand Ra,ss are large, so that the exponential term is approximately zero, and k ~" R A, ql ss

(39)

If ql equals 3, the macroscopically observed stress exponent will equal 3, as desired (Fig. 27). In the Class II regime Oss/E and RA,ss are small, so that m2

m2

exp(-D2RA, ss) ~ 1 - D2RA, ss

(40)

198

G.A. Henshall, D. E. Helling, and A. K. Miller

a

%

aA, s~

--

.__ E

,o0 ( Bo, ss) 5

Iog(RA, ss) or Iog(-~---)

%

RA,ss"-'~ PLB ~

.,/'~-'-- PLB

,o0 Bo,

E | [ I ~ I I ' | I , N I Schematic illustration of the dependence of the steady-state flow stress and the steady-state value of RA, RA.ss, on the temperature-compensated steady-state strain rate: (a) Pure metals or Class II alloys; (b) Class I alloys, such as AI-Mg. Reprinted from Acta Metall. Mater., Volume 38, G. A. Henshall and A. K. Miller, "Simplifications and improvements in unified constitutive equations of creep and plasticity. I. Equation development," Pages 2101-2115, Copyright (1990), with kind permission from Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington OX5 1GB, UK.

Chapter 4

Improvements in the MATMOD Equations

199

and m2 /?(qlWm2) "~ D1R A,ss q~ [1 -- 1 + D2RA,ss ] ~ D1D21,A,ss

(41)

If m2 equals 2, the macroscopically observed stress exponent will equal 5, as desired (Fig. 27). The thermal recovery term in the RA equation is derived by combining Eq. (38) with the/~A hardening term under steady-state conditions at high temperature 9 The dynamic recovery term is neglected in this derivation because thermally activated processes dominate the recovery of short-range back stresses in the PL creep regime (Fig. 26). The/~A equation under these conditions is then /~A, ss -m- Hlkss - rt(RA,ss, T) = 0

(42)

Substituting Eq. (38) into Eq. (42) yields rt (RA,ss, T ) -

H1B|

- - e x p [ - D 2 p(q~+m2~''A,ss ]}

(43)

Following the methods used to derive previous versions of MATMOD (Miller, 1976), this equation can be generalized to include transient deformation because steady state is just one specific case of the more general behavior, and the equation presumably contains all of the relevant variables. Thus the subscript ss in Eq. (43) can be dropped. By substituting ]RAI for RA, again using the signum function and the 3/2 factor, Eq. (43) can be modified so that cyclic histories can also be simulated. The complete one-dimensional expression for thermal recovery of RA is given in Eq. (44):

rt -- H 1 B ~ ' D I [(3[RAI) ql] 3 x {1 - exp [--D2(TIRAI)m2]}

9sgn(RA)

(44)

3. The RB, P~, and Fp Equations Derivation of the remaining structure evolution equations involved only modeling concepts present in previous versions of MATMOD. The guiding principle in the development of the remaining structure evolution equations was to maintain the previous modeling capabilities while simplifying the equations and minimizing the number of material constants 9 To maintain capabilities, the physical bases of the equations were the same as those of earlier versions, particularly MATMOD-4V. To produce the simplifications needed, however, some of the assumptions used in deriving the MATMOD-4V equations were relaxed. For example, the assumption that all of the structure evolution equations must have separate thermal- and strainactivated recovery terms was dropped. Instead, a single recovery term containing both thermal and strain activation was used for the Fz and Fp equations. The kB equation contains only a single dynamic recovery term; thermal recovery of RB is controlled through its close coupling with Fz. As in previous versions

200

G.A. Henshall, D. E. Helling, and A. K. Miller

of MATMOD, steady-state creep behavior, cyclic saturation, and strain softening played a large role in the derivation of the remaining structure evolution equations. Detailed derivations of the J~B, Px, and Pp equations are given by Henshall and Miller (1990a). The resulting expressions are given in Fig. 28, which gives the complete MATMOD-BSSOL equations in one-dimensional form. Note that the sinh-! terms present in previous versions of MATMOD have been replaced by logarithmic terms since the kinetic equation has an exponential form in MATMODBSSOL, Eq. (34), instead of the earlier sinh form.

4. The Fsol Equation Fso 1 represents the increase in strength due to the interaction of mobile solutes with mobile dislocations. This interaction does not depend on the prior deformation history, but only upon the relative velocities of the dislocations and solutes (Schmidt and Miller, 1981; Henshall and Miller, 1989; Miller and Sherby, 1978). Thus, the Fsol equation is not an evolution equation. Rather, it gives the value of Fso~ as a function of the current strain rate and temperature. The Fsol equation in MATMOD-BSSOL has the same form as that originally proposed by Miller and Sherby (1978):

Fsol -- fsol,max exp [ - ( l ~

- l~

(45)

where Z is defined as Z --

I~1 O'sol

(46)

and Fsol,max,/~, and Zmax are material constants. Equation (45) is simply the equation for a log-normal distribution, and has the shape shown previously by Miller and Sherby (1978). Following the work of Schmidt and Miller (1981), | represents the temperature dependence of the interactive solute drag process. | is given by l~'soI - - e x p ( -Qs~ kT )

(47)

where Qso~ is the activation energy for the process controlling interactive solute drag. In previous versions of MATMOD, the equation for the interactive solutestrengthening variable, denoted Fsol,2 by Schmidt and Miller (1981), contained three separate terms and 12 material constants. This complexity was necessary largely to represent solute strengthening at low temperatures. In MATMODBSSOL, low-temperature solute strengthening is modeled through the back-stress terms, allowing the Fsol equation to have the simple form given in Eq. (45)-(47)

e=BO'

[io/E- ~-(RA + RB)I]_

{exp

!., IF~ +l~,p(l+Fsol).A

~. : . , ~- ~,

1}

3 (R A + RB)] sgn[o/E - ~.

[ql~l) ml]. "~'~__/' Fp

- HIBO' D1 [(3]RAI)q'] { 1 - exp[-D 2 (23-.[RA[)m2 ] } - s g n (RA)

Fp H'2 I~I- H'2 BO' {exp [A 2F (mll+l)

(I' I ~/d.

F

-

H4 C4 C5 Be' { exp [ A 4 ~

Fso1 - FsoLmLx exp

o,-{

-

o~(-~)

'

13

( l n { ( ' ~I~! ) 1/d+ 1 } ) ]

' --]

;

- 1 }n

Zu ~ e'so I

,~~,

"i'|[llll'lW,[|ll The MATMOD-BSSOLconstitutiveequations in one dimension. ReprintedfromActa Volume 38, G. A. Henshall and A. K. Miller, "Simplifications and improvements in unified constitutive equations of creep and plasticity. I. Equation development," Pages 2101-2115, Copyright (1990), with kind permission from Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington OX5 1GB, UK.

Metall. Mater.,

202

G.A.Henshall, D. E. Helling, and A. K. Miller

with just 4 material constants. In addition, Fsol plays a much more minor role than in previous versions of MATMOD, and may be omitted from the model for convenience in the numerical integration (Tanaka and Miller, 1988) without producing serious errors in the simulation of the temperature dependence of the flow stress. This is an important practical improvement.

5. Summary of the MATMOD-BSSOL Equations Derivation The emphasis in deriving the MATMOD-BSSOL equations has been to develop the simplest possible mathematical expressions that would be capable of simulating a wide range of deformation behavior in a physically meaningful way. Simplification in modeling solute strengthening through short-range back stresses represents the single most important improvement present in MATMOD-BSSOL (Table 4 and Fig. 26). This physically meaningful methodology gives MATMOD-BSSOL, fundamentally a rate-dependent model, the capability of simulating "athermal" or apparently rate-independent behavior, i.e., yield strength plateaus. It also allows for great simplification in the equation relating the interactive solute-strengthening variable, Fsol, to the current strain rate and temperature. Other simplifications not related to solute effects include the use of an exponential form of the kinetic equation and simplifications in the structure evolution equations by relaxing the assumption that all of the structure evolution equations must contain separate thermal- and strain-activated recovery terms. MATMOD-BSSOL is the first version of MATMOD that is capable of modeling both complex strain-softening phenomena and solute strengthening. The complete one-dimensional MATMOD-BSSOL equations are summarized in Fig. 28. These equations require that 26 material constants be determined for an alloy. This is a reduction of approximately 40% from what would have resulted if Fsol,1 and Fsol,2 had simply been added to MATMOD-4V to form a model with capabilities similar to those of MATMOD-BSSOL. Since the difficulty in determining the material constants is a nonlinear function of their number, this reduction in the number of constants is quite significant. It is also important to stress that these constants are valid for all temperatures and strain rates for which the model is valid. No variation of the material constants with temperature or other modifications for special conditions are needed.

C. Simulationsand Predictions The purpose of this section is to demonstrate the capabilities of MATMOD-BSSOL in simulating and predicting a variety of mechanical test data. All of the calculations presented here were made using material constants for alloys in the A1-Mg system. This system was chosen because of the availability of data and because solute effects are pronounced in these materials, thus providing the opportunity

Chapter 4 Improvementsin the MATMODEquations 203

to test the new methods used to represent these phenomena. The methods for evaluating these constants and their values for the A1-Mg alloys are briefly described in Section IV. The three-dimensional extension of the equations given in Fig. 28 was used to perform the calculations (Henshall, 1987). Section IV also briefly describes the numerical integration of these equations using the NONSS formalism developed by Tanaka and Miller (1988) and Miller and Tanaka (1988). The calculations presented here have been divided into two major categories: simulations and independent predictions. The former refers to cases in which the data being simulated have been used to determine some of the material constants. These results demonstrate the accuracy with which MATMOD-BSSOL simulates the phenomena that were specifically used to derive the equations, and focuses on the solute-related phenomena of flow stress plateaus and Class I steady-state creep. Independent predictions are calculations in which the experimental data have not been used to evaluate any of the material constants. These results demonstrate the extent to which the physics of nonelastic deformation has been properly represented in the equations. The predictive capabilities, therefore, are used to gauge the success of MATMOD-BSSOL in modeling complex deformation histories. 1. M A T M O D - B S S O L Simulations

a. Flow Stress Plateaus Figure 29 compares the MATMOD-BSSOL simulations of the stress-temperature curve (flow stress at constant nonelastic strain vs. temperature) at several nonelastic strains with the data of Henshall and Miller (1989) for A1-5.8 at% Mg. MATMOD-BSSOL accurately simulates the complex stress-temperature curves over the entire range of temperature and strain. The ability of MATMOD-BSSOL to simulate the stress-temperature behavior of pure aluminum for nonelastic strains ranging from 0.02% to 20% with a similar degree of accuracy has been demonstrated by Henshall (1987). The accuracy of the simulations in Fig. 29 is a result of modeling solute strengthening in a physically meaningful manner. Beginning with the 0.02% nonelastic strain curve, the plateau in the simulation is produced by the very rapid evolution and mechanically activated recovery of the short-range back stress variable, RA, as shown in Fig. 30. (RB is negligible for strains this small.) The rapid evolution of RA, which leads to the large values shown in Fig. 30 at 0.02% nonelastic strain, represents the pinning and athermal breakaway of dislocations from groups of immobile solute atoms, as discussed earlier. This plateau is accurately simulated by MATMOD-BSSOL because RA evolves rapidly relative to the isotropic hardening variables. In addition to a plateau, the data in Fig. 29 for 0.02% strain show a small peak in the curve near 400 K. This peak is caused by the interaction between mobile solutes and moving dislocations, and is accurately simulated through the Fsol variable, which represents this interaction. At temperatures above 450 K the stress decreases rapidly with increasing temperature. This behavior is accurately

204

G.A. Henshall, D. E. Helling, and A. K. Miller 60

,

'

'

I

. . . .

A I - 5 . 8 Z Mq T

0 x

LU

I

. . . .

e

I

. . . .

= 1.28 X 10 .4 $-1 MATMOD-BSSOL

40

J

J

DATA: o 207. C N x 107. s 02Z s O .027. s N

30 2O

O

O

O

D

10

0

O

200

400 TEMPERATURE (K)

600

800

The modulus-compensated flow stress as a function of temperature for A1-5.8 at% Mg deformed over a wide range of strains at an engineering strain rate of 1.28 x 10 - 4 s - 1 The MATMOD-BSSOL simulations of the data of Henshall and Miller (1989) are given by the solid lines. Reprinted from Acta Metall. Mater., Volume38, G. A. Henshall and A. K. Miller, "Simplificationsand improvements in unified constitutive equations of creep and plasticity. II. Behavior and capabilities of the model," Pages 2117-2128, Copyright (1990), with kind permission from ElsevierScience Ltd, The Boulevard, Langford Lane, Kidlington OX5 1GB, UK. simulated by the model because, as shown in Fig. 30, RA thermally recovers in this regime. The data at strains larger than 0.02% are also accurately simulated by MATMODBSSOL. The low-temperature athermal behavior of the 5.8% Mg alloy at intermediate strains is produced in MATMOD-BSSOL by the large values of RA and RB. Comparing Fig. 30 with the simulations in Fig. 29 shows that the total back stress is a large fraction of or/E for each strain and is relatively temperature-independent at low temperatures. The interactive solute drag peaks that are present in the stress-temperature curve at intermediate and large strains are accurately simulated in MATMOD-BSSOL through Fsol and its interaction with Fp in the kinetic equation, Eq. (34). The data show that the temperature at which the solute drag peak occurs, Tmax, decreases as the strain increases. The MATMOD-BSSOL simulations also exhibit this behavior because the back stresses begin to recover at lower temperatures with increasing strain (Fig. 30). Thus, at low temperatures and low strains the solute drag peak is superposed upon the athermal plateau, but at higher temperatures and higher strains the increase in stress due to interactive solute drag is partly canceled by thermal recovery of the back stresses. Thus, MATMOD-BSSOL predicts (not just simulates) the decrease in Tmax that occurs

Chapter 4 30

(D

. . . .

. . . .

I

. . . .

I

A I-5,8% M g

MATMOD-BSSOL

-

-

() =

BEHAVIOR:

-

1.28

X 1 0 "4 s "j

o-

20--

o

~ ~

]l~-

205

. . . .

.

1~1~1

I

Improvements in the M A T M O D Equations

207. E~N

~--

10zcN

O -

0.02~

--

~N

.

_

~

"~

fy

I0--

o

.

-

r

. . . . . . . . . .

I

0

.

.

.

.

.

.

.

.

.

I

.

I

200

400

600

I

800

TEMPERATURE (K]

~ll[l~|Jl|l{lll The total back stress, RA + RB, corresponding to the simulations shown in Fig. 29.

Reprinted fromActa Metall. Mater., Volume38, G. A. Henshall and A. K. Miller, "Simplifications and improvements in unified constitutive equations of creep and plasticity. II. Behavior and capabilities of the model," Pages 2117-2128, Copyright (1990), with kind permission from Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington OX5 1GB, UK.

as the strain increases. This capability, which earlier versions of MATMOD do not have, is evidence that the physics of solute strengthening has been properly represented in MATMOD-BSSOL. The simulations in Fig. 29 confirm the feasibility of simulating plateaus in the stress-temperature curve using the back-stress variables, and of simulating peaks in the stress-temperature curve at all strains with a single interactive solutestrengthening variable. The generality of this approach is supported by MATMODBSSOL simulations of the stress-temperature curves of mild steel over a wide range of strains (Adebanjo, 1987). The accuracy of these simulations is better than was possible using a previous version of MATMOD (Tanaka, 1983). b. S t e a d y - S t a t e C r e e p Figure 31 presents the MATMOD-BSSOL simulation of the steady-state creep data for pure aluminum, where | is used to compensate the strain rate for temperature, Eq. (3). The simulations are reasonably consistent with the data over 21 orders of magnitude in the temperature-compensated strain rate. The accuracy of the simulation, which is similar to that of MATMOD-4V (Lowe, 1983), demonstrates that the exponential form of the kinetic equation (and the resulting Fz and Fp equations) provides the capability to represent steady-state behavior over wide ranges of strain rate and temperature.

206

G.A. Henshall, D. E. Helling, and A. K. Miller 1030

' '" -

.... I

I

' ''"'"1

I

AI

Pure

I

8

"

mO

"

1024 -

-

0

" =

o Data

i018

m

& Grant

[ 5 7 3 K)

9 Data

or GxbeJlnq

[ 6 7 3 K)

=

r

oF S e r v l

(366 - 866 K] ~ Data oF Ferrlera & Stanq

o Data

or Luthy

et

al.

( l g 4 - 5 3 3 K)

-

MATMOD-BSSOL

j

1012

~oe " 10-5

I

~,,,~'~ I

I III

I

I

I

I

I

10-4

I

III

I

I

I

I

10-3

~sslE

Steady-statecreep data and the corresponding MATMOD-BSSOL simulationfor pure aluminum. The data are those of Luthy et aL (1980), Ferriera and Stang (1979), Servi and Grant (1951), and Gibeling (1979). Reprinted from Acta Metall. Mater., Volume 38, G. A. Henshall and A. K. Miller, "Simplificationsand improvementsin unifiedconstitutiveequationsof creep and plasticity. II. Behavior and capabilities of the model," Pages 2117-2128, Copyright(1990), with kind permission from Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington OX5 1GB, UK. /l[glll'tllllm

The M A T M O D - B S S O L simulation of the steady-state data for the A1-3% Mg alloy is presented in Fig. 32a. The data are accurately simulated, including the transition between Class I (slope of about 3) and Class II (slope of about 5) behavior. Corresponding to the simulation shown in Fig. 32a, Fig. 32b presents the behavior of the components of the stress provided by each of the MATMODBSSOL structure variables. ISO-p and ISO-)~ refer to the components of the flow stress due to the isotropic strain-hardening variables Fp and Fz, and are defined as follows:

e )l/d crIS~ criSO x/_ffx,ln[( k ~

ff~-7)

lid

+l

1

+ 1]

(48) (49)

Chapter 4

1011

:=--:

I010

r 09

-(O

207

........ ,/,,,

1012

CD

Improvements in the MATMOD Equations

F

/o

DATA:

o Horlta & L~nqdon~o n Olkc~uael: a l . S ~

I09 r

o

10 8 10 7 10 6

105

.....

I

. . . . . . . .

10 .4

I

,

,

,

i0"3

~ss/E

b

lO14

9

' .......

I

........

MATMOD-BSSOL

I

........

I

........

I

........

I--

BEHAVIOR

1012

-

lib

+ RA, ss

x RB, ss

0 m 1010 co

-CO

-

o

ISO-o

~

ISO-~

AI - 3 MQ

"

10 8

I0 6

T~ , ,,,,,,,I

10-7

, ,,,,,,,I

......

I

10-8 1 0 - 5 1 0 - 4 10-3 COMPONENTS OF THE STRESS

/ | [ I I I R t I t P l l (a) Steady-state creep data and the corresponding MATMOD-BSSOL simulation for A1-3% Mg. The data are those of Horita and Langdon (1985) and Oikawa et al. (1984). (b) The components of the flow stress for the simulations shown in (a). Reprinted from Acta Metall. Mater., Volume 38, G. A. Henshall and A. K. Miller, "Simplifications and improvements in unified constitutive equations of creep and plasticity. II. Behavior and capabilities of the model," Pages 2117-2128, Copyright (1990), with kind permission from Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington OX5 1GB, UK.

208

G.A. Henshall, D. E. Helling, and A. K. Miller 1015 DATA:

1012

o Pure AI AI-IZ Mq 9AI-3Z Mq o AI-5Z Mq MATMOD-BSSOL

(D 4r

"(,D

10g

1015

10.5

~If/15 10.4

10

Crss/E / l [ i l [ l l ' , | l t l l Fig. 33. Summaryof the steady-statecreep data and simulations for pure AI and AI-Mg alloys. The data are from the sources cited in Figs. 31 and 32, and from Oliver (1981) and Mills (1985). Reprinted from Acta Metall. Mater., Volume38, G. A. Henshall and A. K. Miller, "Simplifications and improvements in unified constitutive equations of creep and plasticity. II. Behavior and capabilities of the model," Pages 2117-2128, Copyright (1990), with kind permission from Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington OX5 1GB, UK.

Figure 32b shows that RA,ss dominates the flow stress throughout the Class I regime and the lower portion of the PLB regime. As discussed earlier, these large values of RA simulate the domination of solute drag in the Class I regime, which is represented by its effects on the short-range back stress. The use of RA to simulate Class I steady-state creep, instead of the Fsol terms used in earlier versions of MATMOD, allows M A T M O D - B S S O L to simulate a stress exponent that remains at a constant value of about 3 throughout the Class I regime. The transition from Class I to Class II creep is also simulated through the behavior of RA, as discussed previously. Although not clearly apparent in Fig. 32b, the slope of the RA, ss curve changes from 3 to 5 at kss/(9' approximately equal to 106. The capability of the M A T M O D - B S S O L equations to simulate the steady-state behavior of pure metals and Class I alloys is summarized in Fig. 33. The effects of solute concentration on the steady-state creep curve are accurately simulated over the entire range of temperature-compensated strain rate. In particular, as the solute concentration increases, M A T M O D - B S S O L correctly simulates the increase in

Chapter 4 Improvementsin the MATMODEquations 209 the stress- or strain-rate range over which Class I creep occurs. The steady-state simulations for the alloys are largely controlled by thermal recovery of RA, which dominates the flow stress in the Class I regime. One advantage of simulating Class I creep through RA, instead of through Fsol, is that the stress exponent in the Class I regime is then independent of solute concentration. As shown by Miller and Sherby (1978), using Fsol to produce Class I creep causes the stress exponent in the Class I regime to decrease as the solute content increases.

2. Independent Predictions a. Simple Loading Histories Since primary creep data were not used to evaluate any of the material constants, calculations of the primary creep curve are independent predictions of the model. Predictions of the amount of primary creep strain as a function of stress are shown to compare favorably with data for pure aluminum in Fig. 34a. Figure 34b presents a similar comparison for A1-5% Mg. MATMOD-BSSOL correctly predicts that the amount of primary creep is much smaller for the alloy than for pure aluminum. This behavior is predicted because the MATMOD-BSSOL structure variables and material constants governing the amount of primary creep are the same as those governing strain hardening during constant strain rate tests, since the underlying deformation mechanisms are the same. Thus, fitting constant strain rate tensile data at high temperature provides the quantitative information necessary to predict transient behavior during constant stress creep. One feature of the primary creep (and high-temperature tensile) data for A1-Mg alloys that is not predicted by MATMOD-BSSOL, however, is the inverted nature of the transient (Ahlquist and Nix, 1971). Inverted transient behavior, which is due to mobile dislocation density effects, was not included in MATMOD-BSSOL because the improvements in accuracy were judged to be insufficient to justify the added complexity of the equations (Henshall, 1987). As described by Henshall and Miller (1990b), the MATMOD-B SSOL equations also predict the temperature and strain regime over which negative strain-rate sensitivity occurs in A1-Mg alloys during monotonic deformation. The predicted range of temperature was shown to be consistent with that for which serrated yielding occurs. b. Complex Loading Histories The ultimate goal in developing an advanced constitutive model such as MATMOD-BSSOL is to be able to predict material response to complex loading histories. Demonstrations of this capability in MATMOD-B SSOL have been given by Henshall (1987) and Henshall and Miller (1990b). However, since these calculations do not focus on the new modeling concepts presented in this chapter, only a limited number of them are presented here to illustrate the extent to which the capabilities of the previous models are retained. A sudden change in the applied stress during creep is one example of complex loading. Figure 35a presents the MATMOD-BSSOL predictions of the stress-

21 0

G.A. Henshall, D. E. Helling, and A. K. Miller

a

0.3

Z

ST RESS (ksi)

~

'''I

....

I ....

Pure

I ....

I ....

I ....

AI 0

I--

(/) Ix. LU LU n,, 0 >.

o

0.2

0.1

-o / oQ/

n,' n

,,vn.n

0

",

0.05

o DATA OF AHLOU]ST & NIX o DATA OF GIBELING MATMOD-BSSOL PREDICTION

I .... 40 STRESS

, , , ~ .... 20

9 AI-

, 5%

9

,

9

I

O0 [MPa) ,

Mg

I

....

80

9

I

....

,

100

9

,

o

Data

of Mills

[3

MATMOD-BSSOL

9

(1985)

0.04 C 1=

,m

Prediction

Ik=

c/}

Q. @ @

0.03

L.

0 i_

r

0.02

E

Ik,,

n_

0.01

0.00

9

0

I

10

,

I

,

20

I

,

30

Stress

'

40

9

'

50

9

60

(MPa)

(a) The MATMOD-BSSOL prediction of the amount of primary creep strain for pure A1 as a function of the applied stress is compared with the data of Ahlquist and Nix (1971), Gibeling (1979), and Sherby et al. (1957). (b) Comparison of the primary creep strain data of Mills (1985) for A1-5% Mg with the MATMOD-BSSOL predictions for two different applied stresses. Reprinted from Acta Metall. Mater., Volume 38, G. A. Henshall and A. K. Miller, "Simplifications and improvements in unified constitutive equations of creep and plasticity. II. Behavior and capabilities of the model," Pages 2117-2128, Copyright (1990), with kind permission from Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington OX5 1GB, UK.

Chapter 4

Improvements in the MATMOD Equations 211

change data of Sherby et al. (1956). The steady-state creep rates computed by the model at large strains are not independent predictions because steady-state data were used to determine some of the material constants. The predicted strain rates during all of the transients, not just those following the stress change, are independent of the data. The transient prior to the stress change (cr = 27.6 MPa) and that for which the stress was not changed (or -- 13.8 MPa) are both accurately predicted. The predicted transient following the stress change has the proper curvature, but is somewhat too rapid. Considering the breadth of phenomena covered by MATMOD-BSSOL, however, this prediction is reasonably accurate. The reason that MATMOD-BSSOL predicts such a rapid transient is that RA, which dominates the stress, evolves very quickly following the stress change, as shown in Fig. 35b. The evolution of Fp following the stress change is also quite rapid and contributes to the error in the prediction. Qualitatively similar predictions are produced for stress increases or changes in the applied strain rate (Henshall, 1987). In all cases the model overpredicts the speed of the transients following a change in the boundary conditions, but the results are generally reasonable. It has also been shown that MATMOD-BSSOL retains the capabilities of MATMOD-4V to predict strain-enhanced recovery and strain softening of pure aluminum due to sudden decreases in the applied strain rate at high temperatures (Henshall, 1987). One important feature of MATMOD-BSSOL, and of previous versions of MATMOD, is that the low-temperature "plastic" strain rate is not artificially separated from the high-temperature "creep" strain rate. This unified approach enables the model to predict the interactions between creep and plasticity that occur when a material is deformed over a wide range of temperature. The ability of MATMODBSSOL to predict creep-plasticity interactions is demonstrated in Fig. 36. The open circles and the lowest line correspond to deformation of annealed pure aluminum at 298 K and the constant strain rate shown in the figure. The other symbols and the upper two lines correspond to constant strain rate deformation at 298 K following a 35% prestrain under constant stress creep conditions at 478 K. MATMOD-BSSOL properly predicts that the prior creep deformation increases the flow stress without significantly changing the strain-hardening rate. The reason that MATMOD-BSSOL is capable of predicting creep-plasticity interaction is that the same state variables are involved in both creep and plastic deformation. The high-temperature prestrain causes the structure variables to reach values that are larger than those for an annealed material. The increased values of the structure variables then cause the low-temperature flow stress to be larger than that for the annealed material.

D. Summaryof MATMOD-BSSOL MATMOD-BSSOL combines most of the capabilities of previous versions of MATMOD in one comparatively simple set of equations. It is the first version of MATMOD capable of modeling both solute strengthening and complex strain-

212

G . A . Henshall, D. E. Helling, and A. K. Miller

a

10o

. . . .

I

. . . .

I

. . . .

I

. . . .

I

. . . .

o, o Data oF Sherby, Trozera, and Dorn

w

-

w I-. 10-3 ,r

Z ,< I--

~o 1 0 - 6

MATMOD-BSSOL PREDICTIONS

27.6~ [4 ksi| 0 0 " 2Oo.

~

o= o ~ # I~ } 13.8 MP,

~ o

12 ksi)

I.-

o

._1 Z

o Z

P u r e

....

b

1.4 q 0

1.2

x

0.8

9"

~:<

.

.

.

I .... I .... I .... 0.1 0.2 0.3 NONELASTIC STRAIN

.

I

o

e==l

x

m

e-,i

.

I

.

0.175 0.150 0.125 0.100 0.075 0.050

i

2.0 1.5

~a.

1.0 0.5

--

.

.

.

.

.

I

.

.

.

.

I

0.6

x

_6" 0 -.9 x

.

0.5

Pure A I 478 K

2.5 o

.

I .... 0.4

1.0

0.4

q

AI

478 K

I0 "g

-

.

.

.

.

I

.

.

.

.

.

I

,

.

.

I

9

.

.

I

2.5 2.0 1.5 =

0

.

.

0.1

.

.

I

0.2 0.3 NONELASTIC STRAIN

.

.

I

0.4

i -

Chapter 4 Improvements in the MATMOD Equations 80

....

Pure

O0

2g8

K

~: =

2.8

I ....

I ....

i .... m

x 10 "3

--

0

8

o

e

40

rY

I---

c~

/ ~ _ 0

20

_ r

0

0

0

CREEP HISTORIES ( 4 7 8 K] o 4 ksi

TO e:

0

,,,,I

4

#

90 , 3 5

92 ksi TO 8 o 0 . 3 5

o ANNEALED -HATMOD-BSSOL

o

10

$'~ 0

00

Ix. :E (D oo LtJ

AI

2"13

.... I,,,,I .... 0.02 0.04 0.06 TOTAL STRAIN

2

o

0.08

i|[llll~t|EJ MATMOD-BSSOL predictions of the effects of prior creep straining at 478 K on the stress-strain curve for pure aluminum at 298 K. Data of Sherby and Dorn (1954). Reprinted from Acta Metall. Mater., Volume 38, G. A. Henshall and A. K. Miller, "Simplifications and improvements in unified constitutive equations of creep and plasticity. II. Behavior and capabilities of the model," Pages 2117-2128, Copyright (1990), with kind permission from Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington OX5 1GB, UK.

softening behavior, and t h e r e f o r e exhibits the b r e a d t h r e q u i r e d for s i m u l a t i n g c o m plex d e f o r m a t i o n histories. C o n s i d e r i n g this breadth, the a c c u r a c y o f the m o d e l is g e n e r a l l y good. S o l u t e - s t r e n g t h e n i n g p h e n o m e n a , p a r t i c u l a r l y p l a t e a u s and p e a k s in the curve o f stress vs. t e m p e r a t u r e and Class I s t e a d y - s t a t e c r e e p behavior, are a c c u r a t e l y s i m u l a t e d by M A T M O D - B S S O L . T h e s e capabilities s t e m l a r g e l y f r o m treating the effects o f solute c o n c e n t r a t i o n on the e v o l u t i o n o f the s h o r t - r a n g e b a c k stress. I n d e p e n d e n t p r e d i c t i o n s using the m o d e l h a v e also b e e n s h o w n to be c o n s i s t e n t with a w i d e variety o f m e c h a n i c a l test data. Specifically, the capability

(a) The effect of a decrease in the applied stress on the nonelastic strain rate of pure aluminum deformed at 478 K. The circles and squares represent the data of Sherby, et al. (1956), and the lines represent the MATMOD-BSSOL predictions. The circles and the dashed line represent the behavior at a constant stress of 13.8 MPa (2 ksi). The squares and the solid line represent the behavior for a stress decrease from 27.6 MPa (4 ksi) to 13.8 MPa (2 ksi). (b) The behavior of the MATMODBSSOL structure variables during the stress decrease prediction shown in (a). Reprinted from Acta Metall. Mater., Volume 38, G. A. Henshall and A. K. Miller, "Simplifications and improvements in unified constitutive equations of creep and plasticity. II. Behavior and capabilities of the model," Pages 2117-2128, Copyright (1990), with kind permission from Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington OX5 1GB, UK.

214

G.A. Henshall, D. E. Helling, and A. K. Miller

of MATMOD-BSSOL to model transient creep, creep-plasticity interaction, and changes in applied stress have been demonstrated. The ability of MATMODBSSOL to properly model these (and other) aspects of nonelastic deformation stems from the consistency between the internal behavior of the model and the physical processes believed to control nonelastic flow in metals. As with any complex model, however, there are some areas for improvement. For example, the transients predicted by MATMOD-BSSOL immediately following a change in boundary conditions are invariably too rapid. These errors are caused by the rapid evolution and domination of the short-range back-stress variable, RA. The stress relaxation predictions shown by Henshall and Miller (1990b) demonstrate another area in which improvements to the model are warranted.

USING THE MODELS A. Numerical Integration Numerical integration of the equations described in this chapter is required to generate simulations of material response to a given loading history. It is important to consider the techniques of numerical integration for reasons of computational economy and compatibility with structural mechanics codes in which the constitutive equations may be employed. The MATMOD-4V-DISTORTION equations are numerically integrated using the Gear Method for stiff differential equations. Such methods are necessary because of the inherent mathematical stiffness of any unified model, including MATMOD. The stiffness of these equations arises from the coupling of nonelastic and elastic strains to calculate the total strain, and the fact that k is a strong function of tr. Analogous problems arise in integrating the structure evolution equations, adding to the difficulty of the integration. To attack these problems and to provide a better interface with existing finite-element structural mechanics codes, the NONSS (NONlinear System Solver) method was developed (Tanaka, 1983; Tanaka and Miller, 1988; Miller and Tanaka, 1988) and employed for integration of the three-dimensional MATMOD-BSSOL equations (Henshall, 1987). Details of the methods used to integrate the MATMOD-4V-DISTORTION and MATMOD-BSSOL equations are provided elsewhere (Helling, 1986; Henshall, 1987; Miller, 1987; Tanaka, 1983; Tanaka and Miller, 1988). The Fortran programs embodying these numerical integration schemes are available for both models through any of the authors. Recently, the NONSS method was incorporated into a finite-element code by Chellapandi and Alwar (1996). They used the 23parameter Chaboche viscoplastic constitutive model and compared the computational efficiency of NONSS against their standard Self-Adaptive Forward Euler (SAFE) method. For their three most complicated cases including cyclic loading and behavior around notches, the ratios of computational time by SAFE to that

Chapter 4 Improvements in the MATMOD Equations 215 using NONSS were 8.47, 9.7, and 7.33, respectively. This indicates that considerable time can be saved in finite-element analyses involving advanced unified constitutive equations by employing appropriate numerical methods.

B. Evaluation of Material Constants 1. M A T M O D - 4 V - D I S T O R T I O N

Multiaxial yield locus measurements provide a means of directly measuring Fdef, p -~Fdef,)~ , which is indicated by the radius of the yield surface, and RA and RB, which are indicated by the location of the center of the yield surface. The relative contributions of Fdef, p and Fdef,3. c a n n o t be distinguished by yield surfaces, however. The evolution of Fdef, p + Fdef,z can be related to the change in radius of the measured yield loci during accumulated strain. The relationship between Fdef, p + Fdef,~. and the yield surface radius is easily determined by running several simulations to determine the relationship between the initial yield stress and Fdef,p -[- Fdef,~. The two back-stress components can be distinguished by analyzing nonproportional prestress yield loci data to determine the direction and magnitude of the back stresses when they are not collinear. The center of the yield locus is always at the vector sum of the back stresses. For example, Fig. 37 shows the yield surface measured after a 1.2% effective prestrain in the shear direction. An undistorted circular yield locus that fits best to the data is shown as dashed. The center of this circle is equal to the elastic modulus times the vector sum of the two back stresses, as shown in the figure. Figure 38 shows the measured yield locus translation in 1100-O aluminum as a function of prestrain. The relative contributions of RA and RB can then be determined by analyzing the yield loci after nonproportional prestresses when the two back stresses are not collinear. Figure 39 shows one example. The directions of the back stresses are indicated by the flattened sides of the locus. RA is associated with the flattening caused by the final prestrain increment while RB, which changes more slowly, is associated with the larger first prestrain increment. This indicates how much of the yield surface translation caused by the first prestrain increment is associated with RB, since it is assumed that RB was unaffected by the second prestrain increment. The separation of the two back stresses is then completed and is shown in Fig. 40. The additional assumptions made in generating this plot are that (1) RA will saturate before RB and (2) the slope of the RB vs. prestrain plot is constant beyond the 0-1% prestrain range over which data were obtained. The first assumption is simply a characteristic associated with RA; it changes more rapidly than RB and therefore saturates sooner. The second assumption is not critical, particularly at small strains, but is reasonable in light of the limited available data (Helling, 1986).

21 6

G . A . Henshall, D. E. Helling, and A. K. Miller

,/~ x (MPa)

'~ 40.

(RA + RB)'E

,-_q__20

- 40

-2

~ -.

~l_

~ -

"120

40

60

o (MPa)

Standard locus

-.40

~il[llll'tllt~

8o[-

Measurement of RA + RB after 1.2% prestrain in 1100-0 aluminum.

i

i

;

"

i

'

"

"

"

i

1100-0

60'--

401"-

§

_~-

~

i

-

t

!

J

/'-ir 20

0

0.0

/

0.1

0.2

0.3

Evolution of the sum of the back stresses in 1100-O aluminum as a function of prestrain, after proportional loading, as measured by a 5 x 10 -6 strain-offset yield definition.

Chapter 4

Improvements in the MATMOD Equations

217

~- x (MPa) 60

/ / ,, t I .40

/ I I

//

/

I (RB)'E

' :20;,,,;

(RA + RB)'E

(RAI'E

~

\.-,20-1Standard locus

l

1i

20 ~.

|

-40 1

~. ~ . . . . .

I!

I

!

Ix.

' 4'0 ~

O

,,o

a

J ..-

4"3"~

iL''l~ 0,009~1~

_

0,004 n|[llllilit~ll

Separation of back stresses in 1100-0 aluminum after a nonproportional prestrain.

The relationship between M and yield locus distortion is shown in Fig. 41. The degree of flattening of the yield locus is solely dependent upon M and is independent of its size or center. Therefore, plots such as Fig. 41 can be used to estimate the M needed to produce the amount of distortion found experimentally. On experimental yield loci, it is easiest to measure the ratio of the distance from the center of the yield surface to the radius of the best-fit undistorted circular yield locus. A plot of the behavior of M is shown in Fig. 42, which is then used to determine ~//for any measured yield locus. Plots of 2174vs. k are used to define the relationship between these two variables. A power law, Eq. (6), has been found to be suitable for the three metals studied. M A and MB are determined separately, as are the distortions on the front (usually a small distortion) and back of the yield locus. The remaining constants were evaluated following the procedures used for MATMOD-4V, as discussed by Lowe (1983) and Helling (1986). Table 5 summarizes the MATMOD-4V-DISTORTION constants that have been obtained for 1100-O aluminum, 70"30 brass, and 2024-T7 aluminum.

21 8

G.A. Henshall, D. E. Helling, and A. K. Miller

8o t

i

i i

i

L

6oit~

'

'

'

~

I

'

'

'

,

,1

RB RA

0

I

R A + RB

40

20 ~ 0 t:l 4.

'

4-

13

0.0

0.1

0.2

g

0.3

E

Separation of RA and RB in 1100-0 aluminum, determined after static recovery.

2. M A T M O D - B S S O L The material constants for the three-dimensional equations used in NONSS are the same as those for the one-dimensional equations given in Fig. 28 (Henshall, 1987). A key point is that none of the material constants is temperature or strain rate-dependent; for each material there exists a single set of constants. Thus, although the number of constants appears to be higher in MATMOD-BSSOL than in other advanced constitutive equations, the total number of constants is often fewer than for other models when several temperatures are of interest. The values of these constants for specific materials were determined largely by trial-and-error fits to a variety of mechanical test data. Although linear regression or other techniques would have been preferable, the complex interdependencies between the constants due to the structure variable interactions precluded the use of such techniques. However, subsequent to determination of the MATMOD-BSSOL constants in the original work, noteworthy progress has been made on more rigorous and computer-automated procedures for determining the material constants in unified constitutive equations (Bertram et al., 1993). Such procedures were applied to an early version of the MATMOD equations by Senseny et al. (1993). They used the B io Medical Data Processing statistical software package both to integrate the stiff differential equations for the histories seen in the experimental tests and to determine the set of material parameters that gives a minimum error (weighted sum of squared residuals criterion) for the simulations vs. the test data.

Chapter 4

2

,=

1

~

'

9

9

9

i

9

9

Improvements in the MATMOD Equations

i

9

I

9

!

t

J

!

9

9

9

--

0

.2i

,

, , ,

-2

I .... -1

....

0

I,,, 1

,I

2

o (MPa) |a(IIIIR||III

Simulation of the effect of M66 upon a constant strain-offset yield locus.

1.o

rM rlso

o.s

0.6

0.4

0.2

0.0

0.0

0.5

1.0

1.5

log (M) ~a[~wl~tmp~m Change in distortion ratio,

rM/rlSO as a function of M.

2.0

21 9

220

G . A . Henshall, D. E. Helling, and A. K. Miller

The MATMOD-4V-DISTORTION Material Constants Constant

1100-O Aluminum

70"30 Brass

2024-T7 Aluminum

Q* (cal/mol) Tt(K) B n p

35,500 461 7 • 10 l~ 5 2

40,000 630 1.5 • 108 4.5 1.5

35,500 461 7 • 101~ 5 2

A2 A3 An A5

14,500 7,000 6 • 107 1 • 107

6,000 450 1 • 101~ 1 • 101~

3,600 1,700 8.4 • 107 1 • 101~

H2 H3 H4 /-/5

.5 2.75 • 10 -3 7 • 10 -6 1 • 10 -6

20 6 • 10 -3 3.5 • 10 -7 3 • 10 -9

.2 5 • 10-2 6 • 10-7 1 • 10- l ~

C2 C3 C4 C5

.03 .002 2 • 10 -8 2 x 10 -8

2 x 10 -4 .002 2 • 10 -8 2 • 10 -8

.3 .002 2 • 10-8 2 • 10-8

P2 P3 P4 P5

.25 .1 .1 .5

.25 .1 .1 .5

.25 .1 .1 .5

K1 K2 K3 K4

0 1.9 • 105 48 4.8 • 105

1,770 5,000 - 133 5,230

0 15,400 0 19,800

Wl V2 V3 V4

.5 .5 .25 .25

1 1 1 .9

1) V V V

F~ef,p Fffef,~.

1 x 10 - l l 1 • 10 -11

4 • 10 -8 1 • 10 -15

1.17 • 10 -7 1 • 10-15

Fsol, 1 Fsol,2

1 • 10 -8 0

0 0

0 2.7

They showed a sixfold improvement

in a c c u r a c y u s i n g t h i s p r o c e d u r e

compared

to t h e p r i o r t e c h n i q u e . T h e d a t a r e q u i r e d to d e t e r m i n e t h e M A T M O D - B S S O L

constants for a partic-

u l a r m a t e r i a l a r e g i v e n in T a b l e 6. T h i s t a b l e s h o w s that, e x c e p t f o r the c o n s t a n t s t r u c t u r e t e s t s , all t h e e x p e r i m e n t s

n e e d e d to d e t e r m i n e

material constants are standard experiments

the MATMOD-BSSOL

and are easily performed.

Table 7

Chapter 4 Improvements in the MATMOD Equations

221

ABLE t~

Summary of the Equipment and Experiments Needed to Determine the MATMOD-BSSOL Material Constants*

Apparatus Creep machine

Tensile machine

Reverse torsion machine

Test type and approximate number required

Measure

Constants

Stress change at various temperatures (5-10) Temperature change (8-15) Constant-stress creep (10-15)

ecs as a function of Ores(pure solvent metal or alloy) kss (pure solvent metal) kss as a function of Crss (alloy)

B, d

Constant engineering strain-rate test (8-15)

e as a function of ~r at various temperatures (alloy)

A 1, A3, C4, H1, /-/2, /-/3, H4, m l, G2, fl Fsol,max, Zmax

Constant engineering strain-rate test (5-15)

e as a function of ~r at various temperatures and strain r a t e s could also use diffusion data

Qsol

Cyclic test at a constant engineering strain rate (2)

Cyclic hysteresis loop (alloy)

A3, C4, G2, /43, H4, q3

Q*, Tt A4, C2, C5, D1, 02, m2, n, ql

Notation: Subscript cs denotes constant structure; subscript ss denotes steady state. *Reprinted from G. A. Henshall and A. K. Miller, "Simplifications and improvements in unified constitutive equations of creep and plasticity. II. Behavior and capabilities of the model," Acta Metall. Mater. 38, 2117-2128 (1990), with kind permission from Elsevier Science Ltd.,The Boulevard, Langford Lane, Kidlington OX5 1GB, UK.

gives the values of the material c o n s t a n t s u s e d to p e r f o r m the cal cul at i ons for pure a l u m i n u m and three A 1 - M g binary alloys. T h e const ant s for pure a l u m i n u m and the 5.8% M g alloy w e r e o b t a i n e d f r o m best fits to the data. For the 1.15% and 3.1% M g alloys, m a n y of the m a t e r i a l constants w e r e e s t i m a t e d by interpolation from the values for pure a l u m i n u m and the 5.8% M g alloy (Henshall, 1987). This p r o c e d u r e significantly r e d u c e d the effort r e q u i r e d to evaluate the mat eri al constants. T h e M A T M O D - B S S O L material constants h a v e also b e e n d e t e r m i n e d for mild steel and two H S L A steels by A d e b a n j o (1987), and the const ant s for a single-crystalline p r e c i p i t a t i o n - s t r e n g t h e n e d n i c k e l - b a s e s u p e r a l l o y ( C M S X - 3 ) have b e e n d e t e r m i n e d by M i l l e r et al. (1988).

SUMMARY T h e M A T M O D (MATerials M O D e l ) family of unified constitutive e q u a t i o n s was d e v e l o p e d with the goal of p r e d i c t i n g n o n e l a s t i c d e f o r m a t i o n in e n g i n e e r i n g

222

G.A. Henshall, D. E. Helling, and A. K. Miller A,BLE

The MATMOD-BSSOL Material Constants for Pure Aluminum and AI-Mg Alloys* Constant a

Pure A1

A1-1.15% Mg

A1-3.1% Mg b

A1-5.8% Mg 6

B d Tt Q* E0 E1 E2

1.0 • 104 2.0 461 K 148 kJ/mol 1.16 x 104 -4.392 - 1 . 5 5 • 10 -3

1.0 x 104 2.0 461 K 148 kJ/mol 1.16 • 104 -4.392 -1.55 • 10 -3

1.0 • 104 2.0 461 K 148 kJ/mol 1.16 x 104 -4.392 - 1 . 5 5 • 10 -3

1.0 • 104 2.0 461 K 148 kJ/mol 1.16 • 104 -4.392 -1.55 • 10 -3

H1 A1 ml H2

5.0 5.00 • 1024 8.0 3.33 • 10 -4

5.0 3.08 • 1021 8.0 2.00 x 10 -5

5.0 5.52 • 1018 8.0 2.00 x 10 -5

5.0 5.00 • 1016 8.0 2.00 • 10-5

H3 A3 H4 C4

1.20 1.50 4.00 1.50

7.00 5.00 9.56 1.50

8.70 5.00 1.97 1.50

1.42 5.00 3.33 1.50

n D1 D2 ql m2 C2 A4 C5

5.0 6.00 • 1022 5.00 x 103 3.1 1.9 1.25 • 104 5.50 • 103 7.53 x 107

5.0 4.06 • 1015 2.00 x 108 3.1 1.9 6.00 x 103 1.22 • 104 7.53 • 107

5.0 1.26 • 1015 2.00 x 108 3.1 1.9 3.30 x 103 1.35 • 104 7.53 • 107

5.0 8.00 x 1014 2.00 x 108 3.1 1.9 2.00 x 103 1.80 • 104 7.53 x 107

G2 q3 Fsol,max Zmax

3.00 • 10 -5 0.5 m --m

2.00 x 10 -4 0.5 1.33 9.83 • 101~ 25.0 126 kJ/mol

2.40 • 10-4 0.5 3.75 9.83 • 10 l~ 25.0 126 kJ/mol

2.87 • 10 -4 0.5 7.00 9.83 • 101~ 25.0 126 kJ/mol

Qsol a

x • x •

10 -3 10 -8 10 -4 10 - 6

• • • x

10 -3 10 -9 10 -4 10 - 6

• • • x

10-3 10-9 10-3 10- 6

x • • •

10-2 10-9 10-3 10- 6

The temperature dependence of Young's modulus is given by E ( T ) = Eo + EI T + E2T 2

where T is in kelvins and E has units of megapascals. bConstants affecting cyclic deformation are approximate due to lack of data. *Reprinted from G. A. Henshall and A. K. Miller, "Simplifications and improvements in unified constitutive equations of creep and plasticity. II. Behavior and capabilities of the model," Acta Metall. Mater. 38, 2117-2128 (1990), with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington OX5 1GB, UK. m a t e r i a l s s u b j e c t e d to c o m p l e x l o a d i n g s .

To a c h i e v e this g o a l , t h e M A T M O D

f a m i l y w a s d e s i g n e d to s i m u l a t e a b r o a d r a n g e o f p h e n o m e n a ,

including low-

t e m p e r a t u r e " p l a s t i c i t y , " h i g h - t e m p e r a t u r e " c r e e p , " c y c l i c d e f o r m a t i o n , strain softening, solute strengthening, and multiaxial deformation. using the " p h y s i c a l - p h e n o m e n o l o g i c a l "

This was accomplished

a p p r o a c h , in w h i c h " i n t e r n a l " v a r i a b l e s

Chapter 4 Improvementsin the MATMODEquations 223 represent, in an approximate but physically meaningful manner, the controlling deformation processes. The overall structure of each model and the functional interrelationships among its variables follow from an understanding of these physical causes and effects. Adoption of the "unified" approach, in which one set of equations treats all of the above phenomena, follows logically from the fact that they are all controlled by the same set of physical mechanisms. To provide the required accuracy, the specific equations in the models were derived by fitting the quantitative behaviors observed across entire classes of materials. The focus of effort on a variety of deformation phenomena has led to several versions of MATMOD, as summarized in the genealogy given in Fig. 1. This chapter has focused on two later versions of the equations that address improvements beyond the earlier models. Specifically, additional effort was undertaken in simulating multiaxial deformation, particularly yield-surface distortions, and for combining the capability to model both solute effects and complex strain-softening behaviors within a single model. This latter model was kept as simple as possible so that the difficult job of determining the material-dependent constants for specific alloys would not become impossible. The key modeling concept that links these two new versions is the use of two back-stress variables, which interconnect a wide range of deformation behavior. Section II has described how the multiaxial MATMOD-4V equations have been modified to enable the prediction of distortions in the yield surface that are typically observed at small strain offsets following prestraining. These data indicate the presence of both long-range and short-range back stresses in several alloys. Distortion of the yield surface has been modeled using a Hill-type anisotropy tensor, M, to modify the yield function in the model. Allowing the coefficients (mij) in the Hill approach to be a function of both the short- and long-range back stresses (the kinematic hardening variables in the model) results in an evolution of anisotropy during deformation. This improved model, MATMOD-4V-DISTORTION, retains the capability to predict expansion and translation of the yield surface, as well as a wide variety of mechanical behaviors simulated by MATMOD-4V. Verification of the model was achieved through simulations and independent predictions of yield locus distortions under proportional and complex nonproportional loadings for 1100 A1, 70:30 brass, and 2024-T7 A1. Having demonstrated (from multiaxial experiments) the existence of shortrange and long-range back stresses, the short-range back stresses were then utilized to treat solute effects in a simple manner in the MATMOD-BSSOL equations, described in Section III. This approach is consistent with the experimentally observed behavior as well as most accepted theories and physical models of solute strengthening. In addition, it provides a physically plausible link between the lowand high-temperature behavior of solution-strengthened alloys. At the same time, strain softening and most of the previous MATMOD capabilities were retained. Other improvements, not related to solute effects, simplify the equations without significantly compromising their physical basis or capabilities. MATMOD-

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BSSOL has been verified through simulations of peaks and plateaus in the flow stress vs. temperature curves, steady-state creep behavior, and through independent predictions of a variety of nonelastic deformation behavior for pure A1 and dilute binary A1-Mg alloys. Finally, the use of these models has been discussed in terms of numerical integration and the evaluation of the material constants. Numerical integration of MATMOD-4V-DISTORTION is achieved through a Fortran program employing the Gear Method for integrating systems of stiff differential equations. MATMODBSSOL is integrated using the NONSS program, which was designed to interface efficiently with finite-element solid mechanics codes. Evaluation of the material constants for each model involves trial-and-error procedures through which model simulations are compared with a variety of mechanical test data. The values of these temperature-independent constants have been provided for several materials.

ACKNOWLEDGMENTS Development of the equations described in this chapter was supported at Stanford University through the U.S. Department of Energy, Office of Basic Energy Sciences, under Grant # DE-FG03-84ER45119. Discussions during that period with the authors' colleagues, including Terry Lowe, Charles Schmidt, Oleg Sherby, Michael Stout, and Toshimitsu Tanaka, were very helpful.

REFERENCES Adebanjo, R. O. (1987). Modeling the effects of recrystallization and non-proportional multidirectional strain paths on the flow behavior of steel. Ph.D. Dissertation, Stanford University, Stanford, CA. Adebanjo, R. O. and Miller, A. K. (1989). Modeling the effects of recrystallization on the flow behavior during hot deformation by modifying an existing constitutive model. Part I: Conceptual development of the MATMOD-ReX equations. Part II: Predictive and fitting capabilities. Mater. Sci. Eng. A l l 9 , 87-101. Ahlquist, C. N., and Nix, W. D. (1971). The measurement of internal stresses during creep of A1 and AI-Mg alloys. Acta MetalL 19, 373-385. Baltov, A., and Sawczuk, A. (1954). A rule of anisotropic hardening. Ukr. Mat. Zh. 17, 55-65. Baltov, A., and Sawczuk, A. (1965). A rule of anisotropic hardening. Acta Mech. 1, 81-92. Barret, C. R., and Nix, W. D. (1965). A model for steady state creep based on the motion of jogged screw dislocations. Acta Metall. 13, 1247-1258. Bertram, L. A., Brown, S. B., and Freed, A. D., eds. (1993). "Material Parameter Estimation for Modem Constitutive Equations," MD-Vol. 43/AMD-Vol. 168. ASME, New York. Chellapandi, P., and Alwar, R. S. (1996). Development of Non-Iterative and Self-Correcting Solution (NONSS) Method for viscoplastic analysis with Chaboche model. Int. J. Numer. Methods in Eng., in press. Cottrell, A. H., and Jaswon, M. A. (1949). Distribution of solute atoms round a slow dislocation. Proc. R. Soc. London, Ser. A 199 104-114.

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225

Ferriera, I., and Stang, R. G. (1979). The effect of stress reductions on the creep behavior and subgrain size in aluminum deformed at 573 K. Mater. Sci. Eng. 38, 169-174. Gibeling, J. C. (1979). The use of stress change experiments to study the mechanisms of elevated temperature deformation. Ph.D. Dissertation, Stanford University, Stanford, CA. Gibeling, J. C., and Nix, W. D. (1982). Anomalous and constant substructure creep transients in pure aluminum. In "Strength of Metals and Alloys" (R. C. Gifkins, ed.), pp. 613-618. Pergamon, Oxford. Hart, E. W. (1970). A phenomenological theory for plastic deformation of polycrystalline metals. Acta Metall. 18, 599-610. Helling, D. E. (1986). The incorporation of yield surface distortion into a unified constitutive model. Ph.D. Dissertation, Stanford University, Stanford, CA. Helling, D. E., and Miller, A. K. (1987). The incorporation of yield surface distortion into a unified constitutive model. Part 1: Equation development. Acta Mech. 69, 9-23. Helling, D. E., and Miller, A. K. (1988). The incorporation of yield surface distortion into a unified constitutive model. Part 2: Predictive capabilities. Acta Mech. 72, 39-53. Helling, D. E., Miller, A. K., and Stout, M. G. (1986). An experimental investigation of the yield loci of 1100-O aluminum, 70 : 30 brass, and an overaged 2024 aluminum alloy after various prestrains. J. Eng. Mater. Technol. 108, 313-320. Henshall, G. A. (1987). Solute enhanced back stresses and their role in a simplified phenomenological constitutive model for the nonelastic deformation of metals and alloys. Ph.D. Dissertation, Stanford University, Stanford, CA. Henshall, G. A., and Miller, A. K. (1989). The influence of solutes on flow stress plateaus, with emphasis on back stresses and the development of unified constitutive equations. Acta Metall. 37, 2693-2704. Henshall, G. A., and Miller, A. K. (1990a). Simplifications and improvements in unified constitutive equations for creep and plasticity. I. Equations development. Acta Metall. Mater. 38, 2101-2115. Henshall, G. A., and Miller, A. K. (1990b). Simplifications and improvements in unified constitutive equations for creep and plasticity. II. Behavior and capabilities of the model. Acta Metall. Mater. 38, 2117-2128. Heritier, B., Luton, M. J., and Jonas, J. J. (1974). The transition between thermally activated and athermal flow in polycrystalline alpha-zirconium. Met. Sci. 8, 41-48. Hill, R. (1948). A theory of the yielding and plastic flow of anisotropic metals. Proc. R. Soc. London, Set: A 193, 281-297. Hirth, J. P., and Lothe, J. (1968). "Theory of Dislocations." McGraw-Hill, New York. Horita, Z., and Langdon, T. G. (1985). High temperature creep of A1-Mg alloys. In "Strength of Metals and Alloys" (H. J. McQueen et al., eds.), Vol. I, pp. 767-772. Pergamon, Oxford. Kocks, U. F. (1985). Kinetics of solution hardening. Metall. Trans., A IliA, 2109-2129. Korhonen, M. A., Hannula, S., and Li. C. (1987). State variable theories based on Hart's formulation. In "Unified Constitutive Equations for Creep and Plasticity" (A. K. Miller, ed.), pp. 89-137. Elsevier, London. Lowe, T. C. (1983). New concepts in modeling strain softening. Ph.D. Dissertation, Stanford University, Stanford, CA. Lowe, T. C., and Miller, A. K. (1984a). Improved constitutive equations for modeling strain softening. Part I: Conceptual development. J. Eng. Mater. Technol. 106, 337-342. Lowe, T. C., and Miller, A. K. (1984b). Improved constitutive equations for modeling strain softening. Part II: Predictions for aluminum. J. Eng. Mater. Technol. 106, 343-348. Lowe, T. C., and Miller, A. K. (1986). Modeling internal stresses in the non-elastic deformation of metals. J. Eng. Mater. Technol. 108, 365-373. Luthy, H., Miller, A. K., and Sherby, O. D. (1980). The stress and temperature dependence of steady-state flow at intermediate temperatures for pure polycrystalline aluminum. Acta Metall. 28, 169-178.

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Luton, M. J., and Jonas, J. J. (1972). Solute strengthening at high temperatures in zirconium-tin alloys. Can. Metall. Q. 11, 79-90. Mendelson, A. (1968). "Plasticity: Theory and Application." Macmillan, New York. Miller, A. K. (1975). A unified phenomenological model for the monotonic, cyclic, and creep deformation of strongly work-hardening materials. Ph.D. Dissertation, Stanford University, Stanford, CA. Miller, A. K. (1976). An inelastic constitutive model for monotonic, cyclic, and creep deformation: Part I. Equations development and analytical procedures. J. Eng. Mater. Technol. 96H, 97-105. Miller, A. K. (1987). The MATMOD equations. In. "Unified Constitutive Equations for Creep and Plasticity" (A. K. Miller, ed.), pp. 139-219. Elsevier, London. Miller, A. K., and Sherby, O. D. (1978). A simplified phenomenological model for non-elastic deformation: Predictions of pure aluminum behavior and incorporation of solute effects. Acta MetaIL 26, 289-304. Miller, A. K., and Tanaka, T. G. (1988). NONSS: A new method for integrating unified constitutive equations under complex histories. J. Eng. Mater. Technol. 110, 205-211. Miller, A. K., Tanaka, T. G., and Adebanjo, R. O. (1988). "Application of the MATMOD-BSSOL Constitutive Equations to the Single Crystal Nickel-Base Superalloy, CMSX-3," Garrett Turbine Engine Company, Pheonix, AZ. Mills, M. J. (1985). A new theoretical interpretation of the high temperature deformation of solid solution alloys based on the steady state and transient creep properties of AI-5.5 at % Mg. Ph.D. Dissertation, Stanford University, Stanford, CA. Nix, W. D., Gibeling, J. C., and Hughes, D. A. (1985). Time-dependent deformation of metals. Metall. Trans., A 16A, 2215-2226. Oikawa, H., Honda, K., and Ito, S. (1984). Experimental study on the stress range of Class I behavior in the creep of AI-Mg alloys. Mater. Sci. Eng. 64, 237-245. Oliver, W. C. (1981). Strengthening phases and deformation mechanisms in dispersion strengthened solid solutions and pure metals. Ph.D. Dissertation, Stanford University, Stanford, CA. Poirier, J. P. (1976). On the symmetrical role of cross-slip of screw dislocations and climb of edge dislocations as recovery processes controlling high temperature creep. Rev. Phys. Appl. 11, 731-738. Rice, J. R. (1970). On the structure of stress-strain relations for time-dependent plastic deformation in metals. J. Appl. Mech. 37, 728-737. Schmidt, C. G. (1979). A unified phenomenological model for solute strengthening, deformation strengthening, and their interactions in Type 316 stainless steel. Ph.D. Dissertation, Stanford University, Stanford, CA. Schmidt, C. G., and Miller, A. K. (1981). A unified phenomenological model for non elastic deformation of Type 316 stainless steel. Part I: Development of the model and calculation of the material constants. Res. Mech. 3, 109-129. Senseny, P. E. Brodsky, N. S., and DeVries, K. L. (1993). Parameter evaluation for a unified constitutive model. J. Eng. Mater. Technol. 115, 157-162. Servi, I. S., and Grant, N. J. (1951). Creep and stress rupture of aluminum as a function of purity. J. Me. 141,909-916. Sherby, O. D., and Burke, P. M. (1967). Mechanical behavior of crystalline solids at elevated temperature. Prog. Mater. Sci. 13, 325-388. Sherby, O. D., and Dorn, J. E. (1954). An analysis of the phenomenon of high temperature creep. Proc. Soc. Exp. Stress Anal, 12, 139-154. Sherby, O. D., and Miller, A. K. (1979). Combining phenomenology and physics in describing the high temperature mechanical behavior of crystalline solids. J. Eng. Mater. Technol. 101, 387-395. Sherby, O. D., Trozera, T. A., and Dorn, J. E (1956). Effects of creep stress history at high temperatures on the creep of aluminum alloys. Proc. Am. Soc. Test. Mater. 56, 789-804.

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Sherby, O. D., Lytton, J. L., and Dorn, J. E (1957). Activation energies for creep of high-purity aluminum. Acta Metall. 5, 219-227. Shih, C. E (1977). "Plasticity Theories and Structural Hardening of Anisotropic Metals-Zircaloy," EPRI NP-500. Electric Power Research Institute, Palo, Alto, CA. Shiratori, E., Ikegami, K., and Yoshida, E (1979). Analysis of stress-strain relations by use of an anisotropic hardening plastic potential. J. Mech. Phys. Solids. 27, 213-229. Stout, M. G., Martin, E L., Helling, D. E., and Canova, G. R. (1985). Multiaxial yield behavior of 1100 aluminum following various magnitudes of prestrain. Int. J. Plast. 1, 163-174. Tanaka, T. G. (1983). A unified numerical method for integrating stiff time-dependent constitutive equations for elastic-viscoplastic deformation of metals and alloys. Ph.D. Dissertation, Stanford University, Stanford, CA. Tanaka, T. G., and Miller, A. K. (1988). Development of a method for integrating time-dependent constitutive equations with large, small, or negative strain rate sensitivity. Int. J. Numer. Methods Eng. 26, 2457-2485.

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The Constitutive Law of Deformation Kinetics A. S. Krausz and K. Krausz Faculty of Engineering

University of Ottawa Ontario, Canada

~11

INTRODUCTION

The optimum design of manufacturing processes and of service performance requires economical methods for the stress analysis of plastic deformation. Figure 1 illustrates the conceptual framework of the procedure. First, the overall engineering problem is outlined: What type of manufacturing processes should be considered; is the plastic deformation that accompanies crack growth of concern in the service of a machine element; is creep to be analyzed; etc. (AT&T, 1993). Once the overall task is established, the appropriate constitutive law has to be developed, expressed in mathematical terms, and stated in engineering quantities. With these, well-defined engineering measurements can be carried out and the constitutive law can be introduced in the finite-element or finite-difference stress analysis programs. Stress analysis requires the establishment of the stress and strain rate (or strain) fields that result from the imposed forces and shape changes in a specific material. In three dimensions, there are six stress and six strain rate components determined from five rigorously defined conditions (Metals Handbook, 1985, 1988; Fox, 1985; Boyle and Spence, 1951; Timoshenko and Goodier, 1950; Nadai, 1950; Jager, 1956; Sub and Turner, 1975). These form a system of differential equations which, together with the forces and displacements applied at the boundary, describe completely the deformation: (a) the equilibrium of stresses, a dynamic condition specified by the first law of Newtonian mechanics; Unified Constitutive Laws of Plastic Deformation Copyright (~) 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

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The role of the constitutive law in the development of the design procedure.

(b) the strain rates, a geometric condition specified by the velocity components; (c) the compatibility conditions, which state that the continuity of the solid is preserved; (d) the stress-strain rate condition, which relates strain rates and stresses with the Levy-von Mises condition; (e) the constitutive law.

A. The Constitutive Law = Kinetics Equation + State Equations The conditions listed in (a)-(d) are rigorous dynamic or geometric statements. It follows that the constitutive law should also be stated rigorously: deformation kinetics theory, based on statistical mechanics, expresses rigorously and fully two types of equations:

Chapter 5 The ConstitutiveLaw of Deformation Kinetics 231 1. The kinetics equation, which defines a. what characteristic material quantities need to be considered for the specific application, b. what role and effect these quantities have in the process, c. the functional form of the stress dependence of the work, d. the effect of temperature; 2. The state equations, which define the behavior and properties of the material quantitatively. These characteristics are much too complex, so only partial physical rigor can be expectedmas is the case in all engineering sciences. The state and the changes of the microstructural characteristics are described by phenomenological and empirical relations. Often, these are determined by testsmsimilarly to the practice of the theory of elasticity. The basic form of the constitutive law is expressed for materials a. b. c. d. e.

that are isotropic, that deform by shear only, in which hydrostatic pressure has no effect, that have the same behavior in compression and in tension, and in which the principal stresses and strains are collinear.

Alternative expressions were also developed for special cases: For example, for many polymers subjected to short-term deformation, or for rocks deforming over geological time, the effect of hydrostatic pressure must be considered; and for single-crystal turbine blades, the anisotropic behavior must be expressed. For these examples and for other forms the references may be consulted (McClintock and Argon, 1966; Miller, 1987; Kocks, 1987; Bodner, 1987; Krieg et al., 1987; Frost and Ashby, 1982; Krausz and Eyring, 1975). These constitutive laws are evolutionary: They express the evolution of the microstructure during plastic deformation. Efficient application demands that these constitutive laws should consider that plastic deformation is always rateand temperature-dependent--that they are unified constitutive laws. The unified evolutionary constitutive law consists of the following coupled equations. The kinetics equation defines the dependence of deformation rate on the 9 mechanical load, 9 environmental effects including the temperature, 9 microstructural state, 9 their time variation; and it is described with the relation k = f i ( ~ , e, T, Si, d / d t , Pi);

(la)

the state equations define the state of the microstructure, as it affects the kinetics equation, with the

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properties: microstructural characteristics that remain unchanged during plastic deformation, and state variables: microstructural characteristics that change with plastic deformation.

Each state variable is described with the relation

dSi dt

= fs(0., e, T, S, P, d / d t )

(lb)

That the stress 0", strain e, the temperature T, and the material state variable S may change with time is represented in Eqs. (la) and (lb) by the symbol d / d t , expressing the change of the microstructure with strain and at high temperature with time directlymthe evolutionary character of the constitutive law. Si is the state variable of the i th kind, representing the state of the microstructure, that is, the dislocation and vacancy densities and distribution, grain size and shape, etc., described by a set of functions that designate each of the state variables necessary for the adequate description of the problem at hand. Pi represents material properties of the ith kind, that is, the characteristic material quantities that do not change during deformation--chemical composition, lattice constant, Young's modulus, etc. In Eqs. (1) 0" is the equivalent stress 0. - - { ~1[(0.1 __ 0.2)2 "k- (0"2 -- 0"3) 2 + (0"3 -- O"1)21) 1/2

(2)

and k is the equivalent strain rate - - { ~2[ ( ~ 1 -- ~2) 2 + (~2 -- ~3) 2 q- (~3 -- ~ 1 ) 2 ] } 1/2,

(3)

where a l , 0"2, o'3 are the principal normal stresses, and k l, k2, k3 are the principal normal strain rates. As an example for the "visualization" of the concept of the constitutive law, the simple empirical kinetics relation expressed for tensile tests is often mentioned: -- (1 / C)0. n. For many materials, the power law and the logarithmic law may be used as empirical relations (Ashby and Frost, 1975; Bagley et al., 1995; Braasch et al., 1995; Evans and Wilshire, 1985; Freed and Walker, 1995; Frost and Ashby, 1982; Jager, 1956; Krausz and Eyring, 1975; Nabarro, 1967; Nadai, 1950; Suh and Turner, 1975; Garofalo, 1965; Metals Handbook, 1985, 1988; Krausz and Krausz, 1985a; Puchi, 1995; Sharpe, 1995; Yang and Mohamed, 1995). These models may have practical benefit because they are simple to use, but they also have disadvantages and may even hold dangers because their validity is severely limited. Semi-empirical and phenomenological constitutive laws were also developed

Chapter 5 The ConstitutiveLaw of Deformation Kinetics 233 (Miller, 1987; Kocks, 1987; Bodner, 1987; Krieg et al., 1987; Frost and Ashby, 1982; Krausz and Eyring, 1975) and employed with great success as discussed in the other chapters of this book.

The constitutive laws of plastic deformation describe the behavior of materials for the design of manufacturing technology and service performance. Thus the constitutive law has to: 9 be expressed with physically based and rigorously derived kinetics law; 9 be expressed with engineering quantities that are appropriate for the standard methods of stress analysis; 9 lead to the description of complex geometric and time-dependent loading conditions; 9 define the effect of temperature; 9 represent the material characteristics and their evolution during plastic deformation; 9 express the characteristic material quantities in terms of temperature and microstructural ranges that were not measured; 9 assure the most economical design, testing, manufacturing, service, and maintenance conditions; i.e., it has to facilitate the achievement of the optimum cost-benefit ratio. The physically based kinetics law combined with the evolutionary expression of the state variables satisfies these requirements. The core of the deformation kinetics constitutive law is the basic physical principle from which the theoretically rigorous description of the deformation kinetics expression--appropriate for the engineering problem--is derived. The deformation kinetics theory defines the characteristic material quantities, the appropriate material microstructural relations and their evolution, and the measurement of these material characteristics and their evaluation. The procedure results in the constitutive law, as shown in Fig. 2. Deformation kinetics is a coherent system of the constitutive law of plastic deformation. It is based on the kinetics equation that is derived from the fundamental principles of statistical mechanics (statistical thermodynamics). The conclusions defined in this system are rigorously valid. It has the same validity and role as the other branches of engineering science with which it is intimately related: solid mechanics, dynamics, thermodynamics, and diffusion. The essence of engineering sciences, physical theories, is that within their limits of validity the conclusions derived are rigorously established: Valid predictions and extrapolations can be made outside the observed and measured ranges with full confidence. This is what distinguishes exact sciences from empiricism. The physical principles of the kinetics equation are reviewed first, after which the state equations are discussed in the context of deformation kinetics theory. Then the analytical method that defines the constitutive law is presented, followed by a basic testing method that illustrates the application of the theory.

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The development of the constitutive law consists of the combination of the physically based deformation kinetics plus the description of the state and evolution of the characteristic microstructural quantities.

THE KINETICSEQUATION Because all plastic deformation processes are temperature- and time-dependent, their behavior is described by the kinetics equation of the constitutive law (Krausz and Eyring, 1971, 1975; A.S. Krausz and Krausz, 1983, 1987, 1993; K. Krausz and Krausz, 1988). The equation is derived from the axioms of statistical mechanics: The kinetics equation is the combination of the elementary rate constants in exactly defined kinetics systems.

Chapter 5 The Constitutive Law of Deformation Kinetics 235 This equation represents the effect of the stress, the temperature, and the characteristic material quantities on the strain rate explicitly and rigorously: it is expressed, as it has to be, in practical engineering design and test terms. Temperature- and time-dependent behavior is the consequence of thermal activation (Faucher and Krausz, 1980, 1981; A. S. Krausz and Krausz, 1983, 1984, 1985a, 1986, 1987, 1992, 1993; K. Krausz and Krausz, 1988a, b; Wu and Krausz, 1994; Krausz and Eyring, 1971; Ashby and Frost, 1975; Krausz, 1969, 1970, 1973; Hanley et al., 1974a, b; Garofalo, 1965; Evans and Wilshire, 1985; Folansbee et al., 1985; Hirth and Lothe, 1968; Zurek and Folansbee, 1995). All thermally activated processes are expressed by the elementary rate constants combined in kinetics relations and derived from statistical mechanics. Based on this theory, deformation kinetics describes the strain rate with the kinetics equation strain

rate

--

f ( k ) = f(stress, temperature, microstructure, their time variation)

(4)

The mechanisms of plastic deformation always consist of stepwise movements. The rate of the steps is defined by the elementary rate constant k derived rigorously from statistical mechanics principles. It is expressed as (Krausz and Eyring, 1975) 1 k -- v exp [ - AGJ;(W) k~

(5)

where v is the frequency factor, which is weakly temperature-dependent; for most engineering purposes, it can be considered as a constant quantity v ~ 6 x 1012 s-1 (Krausz and Eyring, 1975). k -------8.33 x 10 .5 J K -1 is the Boltzmann constant, a universal quantity. The contribution of the thermal energy content of the solid to the deformation process is A G* (W), a rigorously defined function of the work W, which, in turn, contributes to the deformation process by the applied stress field, and T is the absolute temperature. Hence [ AG*-W(cr) 1 k -- v exp kT "

(6)

In Eq. (6), A G* is the activation energy, the energy of the atomic bonds that have to be broken in the process of dislocation movement or vacancy migration; ~ is the deviatoric stress acting on the dislocation, or on the vacancy. When the mechanism consists of different types of elementary steps, their rate constants can be combined in either the parallel or consecutive systems, or, in the general case, the parallel combination of consecutive systems. Equations (5), (6), and the kinetics combinations were derived from the first principles of statistical mechanics (Krausz and Eyring, 1975; Hill, 1960; Krausz and Krausz, 1985b). The testing procedures and their evaluation for the determination of material behavior can be developed with standard engineering and research practices (Hanley et al.,

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1974a, b, 1977; Fox, 1985; Krausz and Krausz, 1985a). The establishment of the kinetics of the constitutive law is therefore a practical method. The behavior in plastic deformation is strongly dependent on the structure of the material. The characteristic material quantities and their role in plastic deformation are defined in the kinetics equation; and their value is determined from empirical, semi-empirical, mechanistic, or phenomenological equations requiring testing. During deformation, the microstructure changes and this change must be expressed in the constitutive law. Some of the changes can be described in terms that are derived from first principles, while others have to be expressed in phenomenological or empirical forms. Materials development, product design, manufacturing design, testing--they all have to employ the specific model that facilitates the overall economy of processing. That requires mathematical simplicity that is verified with physical arguments; deformation kinetics satisfies this requirement. In plastic deformation the concentration and distribution of microstructural defects change. Direct dislocation density measurements show that the density is often a linear function of the stress and strain. The effect of microstructural changes on the characteristic quantities of the material A G ~t, W (or), and the kinetics combinations can be measured with the methods of deformation kinetics (Krausz and Eyring, 1971, 1975; A. S. Krausz and Krausz, 1983, 1987, 1993; K. Krausz and Krausz, 1988a, b; Hanley et al., 1974a, b, 1977). To understand plastic deformation, it is necessary to consider it in three interrelated and overlapping regions: the atomic, the microscopic, and the macroscopic scales. To describe the process and the behavior well, the theory has to be developed from basic physical principles and expressed rigorously in terms of exactly defined quantities. The model that represents the theory for a specific research use or industrial application should be practical, that is, economical, for the purpose of its application. In the following section we discuss the physical basis of the elementary rate constants and the kinetics combination.

A. The Physical Processesat the Atomic Level" The Elementary Rate Constant When force is applied to a solid, the interatomic distances change: The solid deforms. Figure 3 shows the interatomic force between two atoms. When the applied force is increased, the interatomic distance also increases. After the maximum is reached, the atom pair separates even under decreasing force (Fig. 3a). Figure 3b is the integrated curve of Fig. 3a. It shows the relation between the interatomic distance and the energy that holds two atoms, any two atoms, together (Broek, 1978; Tetelman, 1967; Dieter, 1983; Glasstone, 1941; Lawn, 1973, 1975;

Chapter 5

The Constitutive Law of Deformation Kinetics

237

The interaction between two atoms: (a) The interatomic force as the function of the atomic distance; (b) the interatomic energy as the function of the atomic distance. The arrows indicate the distance between two atoms as they separate. This type of relation is valid in any material; only the exact quantitative relationship varies.

Thomson, 1980; Kelly, 1966). As the energy increases, the interatomic distance also increases. When a large enough separation is reached, the energy changes very little on any further increase of the distance. The energy input needed to break the bond, to separate the atoms completely, is the bond energy, A G*; this is the energy that has to be supplied to move a defect (vacancy, dislocation) by one or more atomic distances. In plastic deformation processes, the atoms at the defect zone move apart from their equilibrium locations until the atomic bonds between neighbors are broken and new bonds are established between new neighbors: This is the essence of all plastic deformation processes. When, on the removal of the force, the atoms return to their initial distances, the deformation is elastic. It is advantageous from a fundamental point of view to consider this in terms of energy: To produce elastic deformation, energy has to be supplied to the body in the form of work; on the release of the elastic deformation, the same amount of mechanical energy is recovered.

238

A.S. Krausz and K. Krausz

When the force is large enough to cause plastic deformation, the atomic configuration changes and previous neighbors are replaced by new ones. The configuration change remains after the force is removed: A permanent change of the atomic configuration has occurred which, on the macroscopic level, is visible as the change of the shape of the body. The essence of plastic deformation is that atomic bonds are broken and new bonds are established among different atoms. The difference between plastic and elastic deformation is significant: In the latter, only atomic distances change, but not neighbors. To bring about plastic deformation, enough energy has to be supplied to break the atomic bonds among neighbors. On the removal of the force, some of the invested energy remains in the body as heat and microstructural change, including surface energy and stored elastic energy. In the following, two well-known typical atomic bond-breaking and-establishing mechanisms will be considered: diffusion and dislocation movement (Figs. 4a and 4b). The change in distance between two atoms is simultaneous with the change in the interatomic energy. During elastic deformation, there is only a small change in the distance; on the release of the force, the atoms return to their equilibrium separation. In plastic deformation, the atoms separate until the bond breaks; this requires an amount of energy equivalent to the bond energy. This energy has to be supplied. If enough mechanical energy is supplied, the bond breaks and plastic deformation is instantaneous (to be precise, deformation proceeds at a rate that approaches asymptotically to the velocity of sound). When the applied force is less than what is needed for bond breaking, the thermal energy of the solid (the thermal vibration of the atoms) can make up the energy deficiency. The amplitude of the atomic vibrations is random, and it is related to the temperature: The higher the temperature, the larger the amplitude because the average thermal energy is greater. Because the vibrational amplitude fluctuates, the bond-breaking step has to wait until a large enough amplitude occurs. Consequently, plastic deformation is temperature-dependent; and because the waiting time depends on the occurrence of a large enough amplitude, it is also time- and rate-dependent. This is the physical cause of rate and temperature dependence. The amplitude fluctuation is a probabilistic event and is defined by statistical thermodynamics. The average frequency of the random occurrence of a large enough amplitude is expressed by the elementary rate constant k for all thermally activated processes as k = v exp -

AG~t (W) kT

(5)

where AG$(W) is the apparent activation energy, the energy supplied by the heat content of the solid, which is a rigorously defined function of the work W. The effect of these two sources of energy is the breaking of the bond. As shown before, when the apparent activation energy is expressed explicitly as the function of the

Chapter 5

The Constitutive Law of Deformation Kinetics

239

(a) The vacancy diffusion mechanism of plastic deformation. The empty circles indicate some of the atoms that move by breaking bonds (figure on the left) and established new bonds (figure on the right). The empty locations with broken bonds are the vacant lattice sites. (b) The dislocation mechanism of plastic deformation. When the dislocation moves, the bond between atoms b and e is broken and bond is established between atoms a and b. This representation is for edge dislocation, but essentially the same bond change is involved in the movement of screw dislocations.

w o r k , t h e r a t e c o n s t a n t is

k -- v exp -

AG~-W(a)] kT

"

(6)

I n E q . (6), A G * is t h e t r u e a c t i v a t i o n e n e r g y , t h e e n e r g y o f t h e a t o m i c b o n d s t h a t

240

A.S. Krausz and K. Krausz

The energy-reaction path relation when no mechanical energy is acting.

have to be broken in the process of dislocation movement or vacancy migration; cr is the deviatoric stress acting on the dislocation, or the vacancy. Note that AG~(W) is the Gibbs free energy; consequently, Eqs. (5) and (6) are rigorously valid physical expressions. During the elementary step of plastic deformation, the atoms are rearranged in the whole solid body; all atoms movemtheir relative distances change. The energy content of the solid depends on the configuration of all of the n atoms of which the body is composed. The distance-energy relation is represented by an (n + 1)-dimensional surface that is fully described by statistical mechanics (Krausz and Eyring, 1975). The expression, and of course the representation, is rather involved. For most purposes, it is the general practice to represent the energy-atomic distance change in the two-dimensional energy-activation distance coordinate system. Figure 5 shows its characteristic form. The activation distance represents the extent of the rearrangement of the atoms; the energy axis measures the thermal energy needed to break the bond. When the energy input into the atomic configuration change reaches the activated state, indicated with the symbol ~, the rearrangement process continues with a decrease of energy, as the figure illustrates graphically. Energy is then released back into the body in the form of heat, which causes the warming of the solid during plastic deformation. The consequences of warming are often negligible for most metals and ceramics, but may be quite significant in materials that are temperature- and hence rate-sensitive, as most polymers are. Figure 5 also demonstrates that not all the input energy is released at the end of the elementary step; some part remains in the solid in the form of structural change. This is always the case; all permanent change of atomic configuration is a change of microstructuremand plastic deformation is always a permanent configuration change.

Chapter 5

The Constitutive Law of Deformation Kinetics

241

At the atomic level, processes always proceed in the reverse direction as well (Krausz and Eyring, 1975; Krausz and Krausz, 1983; Hill, 1960; Hirth and Lothe, 1968; Frost and Ashby 1982; Mataya and Sackschewsky, 1994); that is, the elementary step of plastic deformation goes not only in the direction driven by the stress, but in the reverse direction as well, against the drive of the stress. The macroscopically observed result is still a deformation in the direction of the stress because the reverse activation rate is smaller. The rate theory describes the rate of the forward activation with the elementary rate constant as

kf -

v

exp -

= v exp -

AG$(W)f]

[~;

J

AG[- W ( o - ) f kT

(7)

(7a)

and the rate of reverse activation as kr -

v

exp -

-- v exp -

AG*(W)r] kT AGr* + W(cr)] kT '

(8) (Sa)

The operation of the forward-reverse pair is well known. An example is the diffusion process, where it is the very essence of the model for which the mass transport equation is expressed (Manning, 1968; Shewman, 1989; Jost, 1960; Jordan, 1980; Borg and Dienes, 1988; Crank, 1970; Mrowec, 1980). Note that the two-directional progress of dislocation movement is also well known (Ashby and Frost, 1975; Faucher and Krausz, 1980; Frost and Ashby, 1982; Hanley etal., 1974b; Hill, 1960; Hirth and Lothe, 1968; Krausz and Eyring, 1975; Nabarro, 1967). Inspection of Eqs. (7a) and (8a) show that kf > kr. Because the resultant rate is the algebraic sum of the two, the macroscopically observed rate is kf - kr. The change in the height of the energy barrier, due to the work, is shown in Fig. 6. Each activation event contributes to the strain rate the amount 9 a b k for a single dislocation segment, where oe is a geometric factor and b is the Burgers vector; 9 a o b k for vacancy migration mechanism, where or0 is the interatomic distance.

From a dislocation or vacancy density of p, the effect of the activation is p times that much. The strain rate that results from p dislocation segments or vacancies at each energy barrier is p a b k = p e ~ or p(~obk = p e ~ for the corresponding

242 A.S. Krausz and K. Krausz

energy-reaction path •forward and reverseTactivation h relation when thee mechanical energy W is acting. The energiesare indicated. mechanism: OE Ot

-- pfe~ -- pfgOv

-- pfe ~ v

pre~

exp[exp[-

AG~t (W)f] -kT

J

-

-

AG[ -kTW(~

preOV

exp[-AG~t(W)r lkT

_ preO v exp

AGr~ +kzW(O')r ].

In the high-stress and low-temperature region pfe~ >> pre~

(9)

and Eq. (9) re-

Chapter 5 The Constitutive Law of Deformation Kinetics 243 duces to exp[ - A G]$k(W)f T

Oe~Pfe~176

W(cr)f1/

AG{-

= pfeOvexp -

kT

"

(10)

This is the condition for many of the plastic deformation processes that occur in the temperature region below 1/4-1/3 of the homologous temperature. In the hightemperature region where recovery occurs, the energy barrier is (approximately) symmetric and the strain rate is

Ot -- ps~ exp = fie~

kT:

- ps~ exp -

k-T

AG; ) [ W(o-) sinh kT kT

exp(-

(11)

At small stresses Eq. (11) becomes

Ot

- ~ kT

= pe~

(12)

kT

For many mechanisms the work is a linear function of the stress

W(~) = V~; with this, Eq. (9) is

Ot

= pfe~ exp -

- prS~ exp -

kT

,

(9a)

Eq. (10) is

Ot

=

Pfe~ exp --

kT

'

(lOa)

and Eqs. (11) and (12) become

OeOt= fie~

exp(-

AG* kT ) s i n k ( V ~

(lla)

and

Ot = pe~

AG~ exp - k T ) (

~

(12a)

244

A.S. Krausz and K. Krausz

B. The Kinetics Combination of Rate Constants Many mechanisms are more complex than those consisting of a single energy barrier: To be precise, all mechanisms are the complex combinations of barriers (Krausz and Krausz, 1984; 1985b; Krausz and Eyring, 1975). It is a principle of physics that, when describing models, the simplest conditions and mathematical expressions have to be considered. In the case of plastic deformation, this principle leads to single-barrier representation that is sometimes adequate; however, many mechanisms require a combination of two or more barriers. For the analysis and description of multibarrier processes, the theory and methods of deformation kinetics have to be applied. An example is the Peierls-Nabarro model, which has to be expressed by a consecutive combination composed of the double-kink nucleation and spreading barriers. The variety of the energy-barrier combinations is as numerous as the variety of mechanisms. This may be perceived as leading to a bewildering array of combinations--and it is indeed so. However, a system can be recognized that renders the kinetics combinations manageable, even straightforward. All plastic deformation processes (with the exception of but a few seldom- encountered special cases) can be described by a system of kinetics combinations in which the energy barriers are combined in one of the following (Krausz and Eyring, 1975; Hanley et al., 1974a; Krausz and Krausz, 1985a): parallel sequences, consecutive sequences, or consecutive sequences combined in parallel branches. It is noted that the third combination corresponds to the most general description, so that the purely parallel and consecutive system types can be considered as the special cases. The consecutive and parallel systems of energy barriers and their combinations are shown in Fig. (7). Because the mechanisms in parallel barrier systems are independent of each other, the kinetics equation (which is the description of parallel mechanisms in terms of energy barriers) is simply the sum of the rate that each barrier contributes. Accordingly, i ---n

Oe 0---t- = ~"~(Pfe0kf- preOkr)i"

(13)

i=1

An example for parallel mechanisms is dislocation glide proceeding independently of vacancy migration in the intermediate-temperature range, in the region between low and high temperatures. In consecutive systems, each barrier may be overcome only if it is preceded by activation over the previously surmounted barrier, that is, the mechanisms are dependent on each other. The mathematical description is somewhat more involved than that for the parallel system. The kinetics equation of a two-barrier system with breaking activations only is the simplest form of the constitutive law

Chapter 5

The Constitutive Law of Deformation Kinetics

245

n:2

n=3

b

n=l

n=2

(a) The parallel system of three energy barriers; (b) the consecutive system of three energy barriers; (c) three-barrier consecutive sequences combined in two parallel branches.

of consecutive, dependent systems. The total time t necessary for the crossing of the two-barrier system (Krausz and Eyring, 1975; Krausz and Krausz, 1985b) is t - - tfl + tf2

(14)

where tfl is the average waiting time in front of the first barrier, i.e., the average time necessary for an activation to occur; and tf2 is the average waiting time at the second activation type in the forward direction. The rate constant is the reciprocal of the waiting time ki -- 1 / t i ; hence the frequency of activations over the twobarrier system is 1

. t

1

. . tn + tf2

1

. . 1/kn + 1/k~2

kflkf2

kfl + kf2

.

(15)

246

A.S. Krausz and K. Krausz

The strain rate is then Oe

pfl e~176

Ot = 1/pfle~

kf2

(15a)

+ 1/pf2e~

Frequently, the mechanism is such that plastic deformation occurs only after the crossing of both barriers, when Eq. (15a) reduces to Oe Ot

= pe ~

kflkf2

.

(15b)

1/kfl + 1/kf2

For the derivation of n-barrier consecutive systems, the references are offered (Krausz and Eyring, 1975; Krausz and Krausz, 1985b). It is very seldom, however, that more than two barriers are involved in the consecutive system. An example for this is the already mentioned Peierls-Nabarro mechanism when the double-kink nucleation is followed by a single-kink spreading event; when the effect of the subsequent kink spreading steps are not negligible, then, instead of the two-term barrier expression, a multibarrier system has to be considered. The derivation and the resultant constitutive law are given in the references. These are essentially similar to Eq. (14) but with a multiterm form. The most general expression of the kinetics equation is the parallel combination of consecutive energy barriers. Because each consecutive branch is independent of the others, the kinetics equation is the sum of the individual branches, each of which is expressed by the relation represented by Eq. (14) for two consecutive barriers. Accordingly,

0

0E 0t

i=1

0

)

Pfl E01# f 2 E O k f l kf2 - Prl erl Pr2erEkrl kr2 0 o " Pfle01kfl + Pf2e~ + Prlerlkrl + Pr2erEkr2 i

Extension to more than two barriers in sequence follows from this description and for systems in n parallel branches are indicated in the equation. Again, as an example, the medium-temperature range is mentioned, where both dislocation movement and vacancy migration are effectively contributing to plastic deformation. The kinetics equation describes the behavior and the properties of the material as the relation among stress, strain, temperature, microstructure, and their time variation. Radiation and chemical environmental effects may also be significant, particularly for polymeric materials. Because chemical reactions are typical thermally activated processes, their inclusion in the deformation kinetics description of plastic deformation follows immediately from the theory. In fact, the deformation kinetics method is an integral part of chemical kinetics. Examples are hydrogen embrittlement, hydride formation, and corrosion. A similarly direct introduction of radiation effects can also be implemented.

Chapter 5

The Constitutive Law of Deformation Kinetics

247

THE STATEEQUATIONS The essence of plastic deformation is, as stated earlier, that the atomic configuration changes permanently. This means that the microstructure changes. In all plastic deformation processes, the structure changes as the deformation progresses. There is a continuous change in the quantities that represent the state of the rnicrostructure: 9 true activation energy A G*; 9 activation volume V; 9 effective stress a; 9 concentration of the flow units, dislocation or vacancy density p; 9 contribution of each activation to the strain e~ 9 geometric factor c~; 9 kinetics. The consideration of thermally activated structural changes fits easily and naturally into the constitutive law of deformation kinetics. This is a significant aspect of the deformation kinetics theory because it provides a common system for the representation of the development of both deformation and structure: It facilitates the economy of testing, analysis, and the use of the constitutive laws for stress analysis. Nonthermally activated structural changes have to be included in the constitutive law according to the models that express these changes. For these, empirical, semi-empirical, phenomenological, or mechanistic relations have to be used when the theoretically rigorous models are lacking. Consequently, it is typical that the state of the microstructure is established by testing. Not because the basic physical principles are not knownmthey are well established, but the rigorous description of the state and the development of the microstructure from statistical mechanics and quantum mechanics is hindered by the weaknesses of the mathematics. The true activation energy A G* changes because the broken complex of atomic bonds is rearranged at the activation sites as a deformation step occurs, that is, as the structure changes. The measured activation energy gives valuable information on the character and the extent of the development of the microstructure. A similar consideration applies to the activation volume V. For both, theoretical and empirical arguments were developed and introduced in the evolutionary constitutive laws (Krausz and Krausz, 1985b; Krausz and Eyring, 1975). The deformation kinetics theory statement that the rate of plastic deformation is controlled by the work W implies that it is exercised not by the applied stress O'applied, but by the effective stress a , defined as the resultant of the applied stress and the internal stress ainternal. In the simple one-dimensional form it is expressed as O" "-" O'applie d - - O'intema 1.

248

A . S . Krausz and K. Krausz

The vectorial equivalent follows directly. The structural change, the increase of defect concentration, distorts further the crystal lattice, consequently increasing the interatomic energy, that is, it increases the internal stress field. The effect is perceived as work-hardening. Because work-hardening decreases the strain rate, low-temperature creep and stress relaxation tests can provide valuable information on the internal stress and can be used for the quantitative definition of the state of the microstructure (Raj and Farmer, 1995). Other methods, such as incremental strain rate changes, are also used. Work-hardening is not thermally activated, and is therefore outside the scope of the deformation kinetics theory. Howevermand this is significant---deformation kinetics provides the rigorous, exact, method for the measurement of the process. As noted, theoretical and empirical descriptions other than thermal activation concepts were also developed. As a first approximation, the change in the effective stress is defined as O'internal =

f(s)

and the internal stress is defined with the work-hardening coefficient H as the function of the strain. At small strain it reduces to O'internal --" H E .

(16)

It was shown that the resultant rate constant is not of the simple form that would follow from relation (16), but the simplicity of the expression often makes its use preferable. The strain rate Oe

Ot

(AG~t-Vcr)

= pe ~ exp -

kT

is then expressed as

Ot = pe~ exp -

A G ~t _ V (O'applied kT

-- O'internal)

]

]

[ A G * - V(crapplied- He) = ,oe~ exp This relation is only a simplified form of an average effect; the physical processes and the distribution of the defect structure are complex 9 The resultant rate constant is not of the simple form that would follow from the O ' i n t e r n a 1 ~-- He relation 9 Nevertheless, this expression is often used for the benefits afforded by its simplicity. In the high-temperature region, time-dependent recovery processes are also affecting the internal stress by reducing the internal stress field. The change in the flow unit density p expressed in the pre-exponential factor is a complex of thermally activated and other processes. The effect may be negligible because p is in the pre-exponential factor; or it may be decisive for the validity of the model, as for instance in the description of the yield drop effect. Model

Chapter 5 The Constitutive Law of Deformation Kinetics 249 building is not a trivial exercise; there is no templatemnot even a blueprint-but guidance is provided with the theory within a well-defined system. Only a thorough understanding of the theory of the constitutive laws combined with microstructural consideration leads to appropriate descriptions. The definition of the flow unit density change depends on the character of the process. For example, direct and indirect dislocation density measurements indicate that p can be considered as the function of the stress or as the function of the strain. For the description of these functional relations, empirical equations are convenient, and theoretical models of greater complexity were also developed. The effect of structural changes on the geometric factor ot is usually negligible. Small variations of the microstructure do not change the kinetics of the process; large structural changes, however, drastically alter the mechanism and with it the kinetics. The progression of thermally activated structural changes is defined by the appropriate combination of elementary rate constants. In the rate constants, the activation energy and the other quantities have the values characteristic of the specific structural state and change. They are represented by using the same physical principles and methods used for all thermally activated processes--as described above for plastic deformation: The constitutive law of deformation kinetics defines fully the characteristic material quantities, their place in the constitutive law, and their effect on the deformation. This is an exact expression. The values of the state variables and the properties have to be represented with empirical or theoretical expressions. When satisfactory empirical or theoretical expressions are not available, it is indispensable to determine the microstructural quantities by tests. Because these are fully and rigorously described by deformation kinetics (that is, all the necessary microstructural quantities are included and their meaning is rigorously stated by physical quantities), their measurements are well defined. It is therefore possible and necessary to design the tests so that they conform to the material testing methods routinely practiced in research and applied engineering. This ensures the economy of testing and the ease of their interpretation; that is, it renders them economical. The typical testing methods used for the establishment of the microstructural quantities are stress relaxation, creep, constant strain rate, constant stress rate, and cyclic stress or strain.

A. The Operational Equation The constitutive law represents the interrelation of strain, stress, temperature, and the microstructure. If follows that, within the validity range of a specific constitutive law, any deformation regime imposed on these gives a valid description of the process. It is therefore fully possible to convert from one type of imposed,

250

A.S. Krausz and K. Krausz

external, constraint to any other and have a valid description. This condition is of critical importance; the conversion is done by computer processing and no additional testing has to be done. Exploration of the structural effects and changes can be also carded out using computer simulation of the deformation kinetics constitutive law. The resulting economy is the very essence of the physically based constitutive laws: This is the purpose of all mature branches of science and engineering. The coupled equations of the constitutive law of deformation kinetics represent the kinetics and the state of the structure. The combination of these with the external constraints is the operational equation. The operational equation defines the condition for material testing and for manufacturing process design, and it represents the service conditions of the product. The determination of the material characteristics has to be carried out with a rigorous method of testing and analysis. The great variety of materials and structural states results in extremely wide ranges of material characteristic values. This condition demands the theory and method that ensures the greatest economy in testing and analysis: Deformation kinetics provides this. In testing programs the information on the structural characteristic quantities are often obtained by the establishment of the strain rate versus stress and the strain rate versus temperature relations. Other information, such as data that can be obtained in the measurement of the effect of pressure, is also used sometimes. Usually in the high-stress and low-temperature region, processes are controlled only by the forward activation step over a single energy barrier. The kinetics is then quite simple. It is this state from which the full analysis of all mechanisms should begin. The forward rate constant can be determined from the log(de/dt) versus cr and the log(de/dt) versus 1/ T coordinate systems. The behavior in these coordinate systems is then represented by straight lines because [Eq. (10a)] log~ --log(pfe~

1 AG[ 1 Vf 2.3 kT t 2.3 k-T ere

(17)

is a linear function of cr and 1/T. The slope of the line in the first coordinate system defines the activation volume Vf, and the intercept on the ordinate is

l~176

1 AGf 2.3 kT"

To obtain the true activation energy A G~, the intercept quantities have to be separated. To do this, the log ~ versus 1/ T relation has to be determined. It follows from the constitutive law, Eq. (10a), that the log(de/dt) versus 1/T relation is

de = In pfe~ -

In d t

A G ~ - Vfcr 1 --" k

T'

(17a)

ChaRier 5 The Constitutive Law of Deformation Kinetics 251

(a) The schematic representation of the stress dependence of strain in plastic deformation. The activation volume Vf is defined by the slope; the intercept on the rate axis defines the relation log(pfe~ - (1/2.3)(AG[/kT). (b) The Arrhenius-type plot. It demonstrates the determination of the apparent activation energy AGf~(W) and the pre-exponential factor pfe~

thus the slope defines the apparent activation energy A G[ - Vfcr. From this the true activation energy A G[ is readily available by analyzing the rates at several stress levels (Krausz and Krausz, 1985a; Krausz and Eyring, 1975); the pre-exponential factor pfe~ follows from the intercept. Figures (8a) and (8b) illustrate the procedure for the analysis of the activation volume and energy. The relations (17) and (17a) are represented in the semilogarithmic coordinate system with a straight l i n e m a very significant practical advantage for the evaluation of the test results. The characteristic material quantities Vf, A G [ , pf80 associated with the forward activation are obtained with these two plots. The analysis is greatly sim-

252

A . S . Krausz and K. Krausz

plified by the fact that in the semilogarithmic plot of deformation kinetics, the test points are evaluated with the linear functions of the stress and the reciprocal temperature. Simple computer analysis, or even visual methods, can be used for the determination of the characteristic quantities. This is shown in Figs. 8a and 8b. The effect of the reverse activation on the strain rate can be determined by the same method. The relation In

0

p r ~ r kr =

0 In p r ~ r 1)

AGr~ kT

Vr kT cr

is of the same type as (17a). It is of practical significance that only two points have to be specified in the fracture kinetics analysis of the stress and the temperature dependence for each of two activation types. For a full analysis 9 only a total of six quantities have to be determined--more than this is redundant; 9 simple methods--least square or even visual observation--can be used for the analysis because straight line fits are needed; 9 alternatively, the six quantities can be determined with computer analysis without plotting by using, for instance, the steepest descent method. The practicality and efficiency of the analysis must be contrasted with the doublelogarithmic-type plot where a sigmoidal curve has to be analyzed, requiring the use of complex many-point analyses (Hanley et al., 1974b; Krausz, 1978; Cekirge et al., 1976; Krausz and Eyring, 1975; Krausz and Krausz, 1985b). The typical strain rate versus stress behavior is shown in Fig. 9 for the full range of strain rates and stress. It is clear that at zero effective stress intensity (and often even higher than that) the strain rate has to be zero; consequently, in the semi-logarithmic coordinate system the curve has to approach asymptotically to the rate axis. The high strain rate dynamic region is not thermally activated; special test methods are needed for its measurement and it is of interest only in "explosive" service conditions. It follows that (1) the high rate approaches asymptotically to the limit value that corresponds to a significantly large fraction of the velocity of sound in the specific material; and (2) the slow rate approaches asymptotically to the threshold because zero rate in the semilogarithmic coordinate system is at minus infinity. Both limit conditions are asymptotic under any deformation conditions: The strain rate versus driving stress, must therefore have the typical behavior, or shape, as shown in Fig. 9. This, together with the linear relation for each rate constant in the semilogarithmic plot, simplifies very well the analysis and allows far-reaching conclusions to be drawn for materials research and product design. Figure 9 illustrates that all mechanisms of plastic deformation have the same characteristic shape; this means that the analyses of the constitutive law have similar techniques. The typical engineering interest is fully in the thermally activated range shown

Chapter 5

The Constitutive Law of Deformation Kinetics

253

The schematic representation of the strain rate versus stress relation in the semilogarithmic coordinate system. It represents the typical behavior of the single-energy-barrier mechanism. In this plot the relation is not linear at low stresses, the consequence of the increasing role of the reverse activation term which reduces the strain rate.

with the heavy curve; the very rapid, dynamic deformation requires special testing methods and thus is outside the validity of the deformation kinetics theory. Note that the work range of the dynamic condition is rather narrow; for most practical conditions, the thermally activated process dominates the full plastic deformation range. Investigations of a wide variety of materials and external conditions demonstrated that most deformation mechanisms are controlled by the kinetics combination of elementary rate constants with linearly stress-dependent work W. Occasionally, however a single rate constant controls the deformation. In these cases, the stress-intensity dependence of the mechanical energy W (a) is not a linear function, but is of the functional form that produces the behavior represented in Fig. 9. That is, we must allow for a nonlinear relation. The analytical method of deformation kinetics theory clearly identifies this case and defines the relation (Krausz and Krausz, 1994). Thus all macroscopic engineering quantities are represented as (a, e, T), and the microstructural quantities (AG*, V, p, e ~ are readily measured and calculated from the semilogarithmic plots--one of the major advantages of this representation, as opposed to the double-logarithmic plot used in the empirical powerfunction relations (Garofalo, 1965; Frost and Ashby, 1982; Nabarro, 1967; Suh and Turner, 1975). As noted earlier, the deformation kinetics theory provides a systematic method for the complete analysis of the constitutive law. The first term that represents the forward activation has been determined in the first stage of the analysis. The second,

254

A.S. Krausz and K. Krausz

~ l [ ~ l l , ' t l l I ~ The second stage of the development for the determination of the rate constants for a single-barrier mechanism. (a) The analysis of the reverse rate constant: the line is kr ---- ,Ore0 v exp(-(b) The assembly of the forward and reverse rate constants.

AG~r+ Vra/kT).

r e v e r s e - a c t i v a t i o n t e r m can n o w be d e t e r m i n e d b y r e a r r a n g i n g the constitutive law

k = pfeOve x p

(

--

kT

-

preOve x p

--

kT

into

-k-q-pfe~

exp(-

+ Vra ) AG~-_k.TVfa ) -- pre.~ e x p ( - A GSrkT

(9a)

Chapter 5

The Constitutive Law of Deformation Kinetics

255

The Arrhenius plot for the determination of the reverse rate constant. The true activation energy can be determined from the slope AG[

(W)/kT.

and taking its logarithm In - k +

pfe~

exp -

AG[kT- Vfcr

= lnpre~

-

kT

"

(18)

The left-hand side was determined in the first stage of the analysis described before, and the right-hand side of Eq. (18) is of the same type as Eq. (17). The analysis of the reverse-activation term can follow the same pattern that was show for the forward activation. It is now plotted, of course, in the In [ - k +

pfe~

- AG~-kTVfo )]

versus

cr

ln[-k +

pfe~

- AG[-kT Vfcr)

versus

1 -T

and

coordinate system (Figs. 10 and 11, respectively). No new method is then involved in the analysis of the reverse-activation term. Once this analysis is carried out, the constitutive law for the single-barrier mechanism is fully established. For the determination of the activation energy, the temperature dependence of the strain rate is measured. The slope of the line in the Arrhenius plot (Fig. 11) is equal to the apparent activation energy. There are various methods to establish from this the true activation energy A G*. The essential aspect of these techniques is that the work W is known from the stress plot; and because AG*(W) = AG* - W, we have A G * = A G *(W) + W. The activation energy of the reverse rate constant

256

A.S. Krausz and K. Krausz

~'i[gl|iltNIlUll The Arrhenius plot of the behavior over the full range of the single-energy-barrier mechanism. is determined by the method applied for the reverse stress dependence (that is, rearrangement to the rate equation to separate the reverse rate). The Arrhenius plot for the full deformation range represents the reverse as well as the forward activation and is shown in Fig. 12. The analysis of multibarrier systems follows the same method. First, the forward rate constant kf has to be isolated by analyzing the linear sector of the log k versus a relation. Following this, the constitutive law has to be rearranged, as shown above, to isolate the next term that must be established. Then, one after the other rate constant has to be isolated in the same way, be it part of a parallel or a consecutive mechanism. Exactly the same step-by-step process defines all terms. The analytical method of deformation kinetics works for any thermally activated mechanism, providing the means for computer processing. It is to be understood that the exercise of judgment may make shortcuts possible.

THE MEASUREMENTAND ANALYSISOF THE CHARACTERISTICMICROSTRUCTURALQUANTITIES A variety of material-testing methods are used for research and development, and several are included in industrial standards. The test methods and the analyses of deformation kinetics conform to these standards and are demonstrated in this section. Deformation kinetics theory and practice employ two test types for the evaluation of the characteristic quantities that represent the microstructure: the measurement of (1) the stress dependence of the strain rate and (2) the temperature

Chapter 5

The Constitutive Law of Deformation Kinetics

257

dependence of the strain rate. Deformation kinetics fully defines what the char-

acteristic microstructural quantities are, how they have to be measured, and how evaluated. The tests are carried out in conformity to standard research and industrial practice--a considerable practical advantage for planning and operational activities. The evaluation is in the semilogarithmic coordinate system, as discussed above and already widely employed for the Arrhenius analysis. This is emphasized here because the semilogarithmic coordinate system promotes the perception of the behavior and the analytical evaluation process. The uniform use of this coordinate system in the practice of deformation kinetics simplifies the applications and renders it eminently suitable for the evaluation of the constitutive law. It is an economical testing method. Kinetics theory provides a full description of thermally activated, i.e., rate- and temperature-dependent, plastic deformation and provides a coherent system for the expression of the constitutive law. It describes rigorously the thermally activated microstructural changes as well. The result is an evolutionary constitutive law. Physically based descriptions of plastic deformation consist of two fundamental conditions: 9 the internal constraint relation: the expression of the properties and behavior

of the material; 9 the external constraint relation: the load-displacement-thermal test or service conditions to which the specimen and the component are subjected.

Together they constitute the operational equation that fully defines the behavior. This section demonstrates the application of the operational equation in stress relaxation. Other test and service conditions (creep, constant strain rate, constant stress rate, cyclic loading, thermal and mechanical loads in service) are similarly well defined and described with the constitutive law of deformation kinetics (Krausz and Eyring, 1975; Hanley et al., 1974a, b, 1977).

A. Examplefor The Operational Equation" StressRelaxation In several manufacturing processes and in some service conditions components are subjected to stress relaxation. Stress relaxation also offers considerable advantages for material testing. A full relaxation test covers a very wide range of strain rates with the smallest extent of plastic deformation and hence the least microstructural changes; indeed, below the annealing temperature range, structural changes in stress relaxation are often small enough to be negligible. Stress-relaxation testing requires little operator time, and data can be logged and evaluated readily with a computer. It offers an economical, practical method for the measurement of the microstructural quantities for the determination of the constitutive law (Krausz and Eyring, 1975; Fox, 1979, 1985; Kula and Weiss, 1982).

258

A.S. Krausz and K. Krausz

In stress relaxation, the external constraint is constant applied strain. In this imposed condition the stress decreases because the flow units (dislocations or vacancies) move under the effect of the applied stress. The resultant plastic strain relaxes the elastic strain by an equal amount, which in turn reduces the stress because a change in elastic strain is always accompanied by a corresponding stress change proportionally to the elastic modulus. From these considerations it follows that in stress relaxation (1) the external constraint is such that no overall deformation occurs; (2) the total strain rate kT is expressed as the sum of the elastic kE and the plastic k strain rates; (3) elastic and plastic strain changes are of the same magnitude, and therefore the elastic strain k E has to be included in the analysis of the tests. It follows from points 1 and 2 that •T __ kE_~_k = 0;

hence k E -- --E. The operational equation is

kT = ~E + k -- E dt + pe~ exp -

kT

= 0

(19)

where E is Young's modulus. There are two methods for the determination of the stress-relaxation behavior and for the analysis of the microstructural quantities: defining (a) the time dependence of the stress and (b) the stress dependence of the stress rate. Method (a) follows from Eq. (19) directly: *

[Aa

-

-

k=pe ~

kT

"

Then, because a = Ee E,

A G ~" -- V (EE E -- 4 ) ]

[

= pe ~ exp -

kT

{ = pe~

AG:~-V[E(eT-e)-a~]} -

kT

hence the solution of the differential equation is fo ~ exp( -~-e

de = pe~ exp -

9 '

Chapter 5 The Constitutive Law of Deformation Kinetics 259 and

kT VE

exp V---~-Ee

11 -e--

Finally, because e

=

-E E =

kT

Ao- -- - - -

v

G*+V~

exp( In

VE

KI

pe ~

-

t+l

kT

-Ao'/E,

it follows that

VE

( AGJ;+Vo'~] ] exp t + 1

In

pe~

.

(20)

kT

In relaxation the stress changes as the function of 9 time 9 characteristic material quantities p, e ~ AG*, V, o-g 9 temperature Figure 13a demonstrates the time dependence of stress relaxation at three temperature levels for the single-rate-constant mechanism represented by Eq. (20). The stress change in the o- versus the log of time coordinate system is shown in Fig. 13. Note that the early curving is the consequence of the condition

VE k--~pe~

( - AG~kT + V~)

t < 1,

and that over longer periods the stress changes as the linear function of the logarithm of time. In the time region where

kT pe~

-

kT

t >> 1,

Eq. (20) becomes a linear function of the logarithm of time. In this region

kT

Ao- -- - ~

v

In

VE pe~

( A G J ; + V ~ ) 1 kT

exp -

kT

v

In t

as can be seen in Fig. 13. Figure 14 demonstrates that using this linear region the material quantities can be evaluated easily by measuring the slope -kT/V, from which the activation volume can be readily determined, and the intercept at

kT ln[VE~ (_ ---V- ~1 pe~ exp

AG~+Vo-~)

kT

'

which gives information of the temperature dependence of the mechanism and the product pe ~ Moreover, it leads to the determination of the activation energy with an Arrhenius-type analysis.

260

A . S . Krausz and K. Krausz

i|[llll~'(|liR

The time dependence of stress change in stress relaxation, below recovery temperature, controlled by the single-rate-constant constitutive law. Computer-generated with Eq. (20). The characteristic material quantities are normalized for the clarity of presentation: V E = 1, p e ~ ---- 1015, AG ~t = 1, Vcrd = 0.1. The heavy curve is at T = 293 K, the light at T = 303 K, and the dashed at T = 283 K. (a) The symbols represent the test results obtained in lead (Trouton and Rankine, 1904). (b) The stress change in the stress-log time coordinate system.

Chapter 5

The Constitutive Law of Deformation Kinetics

261

The dependence of stress change on the logarithm of time in stress relaxation as the function of temperature. Computer-generated with Eq. (20). The characteristic material quantities are normalized for the clarity of presentation: ps~ = 1015, AG* = 1, VE = 15, Vcr/ = 0.15. The heavy curve is at T = 293 K, the light at T = 303 K, and the dotted at T = 283 K. The figure demonstrates that the behavior is linear in the time region where V E/kT ps~ e x p ( - A G* + V cr~/kT)t >> 1.

M e t h o d ( b ) follows from Eq. (19) in a simplified form when the change in the internal stress is negligible as

...

dt

-- -Eps~

-- -Eps~

-

exp

(

-

kT

AG* + VcrI kT

exp

k-T

cr

"

(21)

Experiments carried out on polycrystalline and single-crystal tough pitch copper were analyzed for the determination of the stress-relaxation behavior and the constitutive law over the low-temperature range. Figure 15 (Krausz and Eyring, 1975; Krausz and Craig, 1966; Krausz e t a l . , 1976) demonstrates that the behavior is linear. Equation (20) was derived with method (a); the alternative is obtained with method (b), which defines the rate of stress change as the function of the stress, rather than that of the time, as given by Eq. (21) above. Figure 16 shows the behavior. The logarithm of stress rate is plotted with a different set of material characteristics and temperatures than in the time-dependence plots to demonstrate further their effects. The linear relation obtained in the defor-

262

A . S . Krausz and K. Krausz

||[l[ll't|llll A typical Act versus log t relation measured in tough pitch copper (Krausz and Eyring, 1975; Krausz, 1966; Krausz, et al., 1976).

mation kinetics coordinate system facilitates the evaluation of the microstructural quantities. Over the temperature, stress, and time range where the constitutive law can be approximated with a single rate constant, Eqs. (20) and (21) can be used to evaluate the activation volume and the activation energy from either. The determination

~ | ( l ~ l l t l m l g The dependence of the rate of stress change on the stress in stress relaxation as the function of the temperature and activation volume. Computer-generated with Eq. (21). The characteristic material quantities are normalized for the clarity of presentation: Epe~ -- 1014, AG ~t = 1, V = 1, crd --- 0.1. The heavy curve is at T = 293 K, and the light is at T --- 303 K; the dotted line illustrates the effect of the activation volume and is plotted with V - 1.5.

Chapter 5

The Constitutive Law of Deformation Kinetics

263

The dependence of the stress-change rate on the reciprocal temperature, the Arrhenius-type plot, of stress relaxation for the evaluation of the activation energy. The curves show the behavior as the function of time, activation energy levels, and activation volume. Computer-generated with Eq. (20). The characteristic material quantities are normalized for the clarity of presentation: V = 1, Eps~ = 1, AG$ = 0.9, cr/ = 0.1; heavy curve is at t = 102 and the light at t = 104. The dashed curve shows the effect of the activation energy. All data as for the heavy curve but AG ~t = 1.2; note the difference in the slope of the straight part. The dotted curve is with the same data as the light curve but with an activation volume of V = 2.

of the activation e n e r g y and the p r o d u c t p s ~ is f r o m the A r r h e n i u s plot. As is k n o w n , the A r r h e n i u s - t y p e plot is used for the evaluation of the activation energy. Its application to the description of the stress-relaxation o p e r a t i o n a l e q u a t i o n is d e m o n s t r a t e d in Fig. 17. F i g u r e 18 shows a typical set of test results o b t a i n e d in the investigation on the effect of the initial stress or0 in z o n e - r e f i n e d iron ( W r a y and H o m e , 1966). O b s e r v e that, as is well k n o w n (Krausz et al., 1976; Krausz, 1976; F a u c h e r and Krausz, 1980), the g r e a t e r the initial stress, the s m a l l e r the activation v o l u m e . C o m p a r i s o n of the analytically d e t e r m i n e d b e h a v i o r s h o w n in Fig. 14 with the m e a s u r e d b e h a v i o r verifies the d e f o r m a t i o n kinetics description. R e c a l l that a constitutive law, while rigorous, can be valid only in a r a n g e of external constraints and m i c r o s t r u c t u r a l s t a t e s - - j u s t as in any other theory. F i g u r e s 13 and 14 d e m o n s t r a t e that as the stress r e d u c e s near to the initial effective stress a~ ff = or0 - a~, the o p e r a t i o n a l e q u a t i o n leads to the m a t h e m a t i c a l but i m p o s s i b l e result that the stress is negative. This is the c o n s e q u e n c e of the c o n d i t i o n that

264

A.S. Krausz and K. Krausz

l | [ l [ l l ' , | l l l l l A typical example of the time dependence of stress relaxation when activation occurs over a single energy barrier and in the forward direction only. Zone-refined iron tested at 77 K (Wray and Home, 1966). The test results plotted from A to G are for the initial stresses as shown.

the the the ~_

single-term constitutive law was considered. This relation is valid only in high-stress and homologous temperature region where pfe~ ~ prF,Okr . As stress decreases, the reverse term becomes comparable to the forward term Pf~f0k f - Pr~'rokr, and both have to be included in the constitutive law (Krausz and Eyring, 1975). In the intermediate stress and temperature region the constitutive law for the single-energy-barrier model is k = pfe~

p:~

with

pf6 0 e x p

--

k----~ r pre r exp

kT

"

The operational equation for stress relaxation is then k =kE--~-k =

1 d~ E dt

+ pfe~

pre~

- 0

(22)

Chapter 5

The Constitutive Law of Deformation Kinetics

265

||[il'lllJ'tlii~]l

High temperature stress relaxation behavior controlled by diffusion mechanism and a single symmetric energy barrier in the low-stress region. Computer-generated with Eq. (28). The microstructural quantities are normalized for the clarity of presentation: N1/2 = -0.0051, (N2 + NI/M) = 1.1, N2/M + C = 59, and scaled by 0.02 for unit stress change.

and do"

dt

(23)

= - - E (/gfE0kf -- prE0kr).

It is of mathematical advantage in some cases to consider that the two applied energy terms can be expressed as exp - - ~

--l+--~+~\kT

/

o

.

o

and as exp - ~

Vrcr )

--1-

1 (Vr~ Vrcr kT + 2 \ k T

2 +

~

,

o

The operational equation (23) can be integrated numerically. As discussed in Section I, computer-generated investigations are economical means for the exploration of the behavior of the material. Figures 19 and 20 present computer-generated explorations of stress relaxation in the high-temperature-lowstress region for deformation controlled by the constitutive law (28). The temperature dependence of stress relaxation is shown in Fig. 20. Again, as shown in Fig. 19, the stress reduces to zero very rapidly at high temperatures. The temperature dependence of Eq. (29) is represented in the Arrhenius-type plot, Fig. 20. The inset demonstrates the behavior in the full-stress range.

266

A.S. Krausz and K. Krausz

reciprocal temperature

~l[ll~ll,'tJ[lll The temperature dependence of high-temperature stress relaxation. The figure demonstrates the effect of temperature, activation volume, activation energy, and recovery rate. Computergenerated with Eq. (29). The microstructural quantities are normalized for clarity of presentation: for all curves, or0 = 1, E p e ~ = 10 i~ or0/ -- 0.1, and R = 0.001; for the heavy curve, V = 1.2, AG ~t = 2.5, and t = 1; for the light curve, the same as the heavy curve but V -- 2; for the dotted curve, the same as the heavy curve but t = 5; for the dashed curve, the same as the heavy curve but AG ~t -- 2.75. The inset demonstrates a wider temperature range. A typical s t r e s s - r e l a x a t i o n b e h a v i o r o b t a i n e d in lead at r o o m t e m p e r a t u r e is s h o w n in Fig. 21 ( K r a u s z and E y r i n g , 1975; W i l s o n and G a r o f a l o , 1966; W i l s o n and W i l s o n , 1966). C o m p a r e this figure with Fig. 19. T h e s y m b o l s in Fig. 21 stand for the m e a s u r e d values and the c u r v e w a s c a l c u l a t e d with Eq. (28). Figure 21 d e m o n s t r a t e s that the b e h a v i o r d e t e r m i n e d w i t h the o p e r a t i o n a l m e t h o d is r e p r e s e n t e d well b y the c o n s t i t u t i v e l a w o f d e f o r m a t i o n kinetics. In h i g h - t e m p e r a t u r e d e f o r m a t i o n it is u s u a l l y n e c e s s a r y to e x p r e s s the constitutive l a w w i t h the r e v e r s e as w e l l as the f o r w a r d t e r m k - - p f ~ ~ - p r e ro kr. T h e s i m p l e s t f o r m o f this (in k e e p i n g w i t h the diffusion m e c h a n i s m o f h i g h - t e m p e r a t u r e

Chapter 5

The Constitutive Law of Deformation Kinetics

267

~ l l [ g l l ~ t l l l l l Stress relaxation behavior of high-purity lead at 298 K. The symbols represent the measured behavior, and the curve was determined with computer analysis from Eq. (28). The characteristic material quantities are the same as in Fig. 19, but with activation energy AG* = 2.8 eV. (Wilson and Garofalo, 1966; Wilson and Wilson, 1966; Krausz and Eyring, 1975).

deformation) is the symmetric energy barrier in which (Krausz and Eyring, 1975; Freed, 1991; Frost and Ashby, 1982; Nabarro, 1967; Hirth and Lothe, 1968; Evans and Wilshire, 1985) AG[-

AG[-

AG*",

Pf Vf-

m

gr-

Pr

m

8 0 ~ 8r0

P;

g.

80 ;

(24)

The symmetric energy barrier results in the often-considered hyperbolic constitutive law k -- 2pE~ exp -

k-----T sinh

kT

(25)

as can be ascertained by the substitution of Eq. (24) into the deformation kinetics constitutive law of the forward-reverse mechanism. As described in this section, during stress relaxation very little plastic deformation occurs and work-hardening is negligible; but in the high-temperature region, recovery does occur and the microstructure changes as a consequence. The kinetics equation is (25). The deformation kinetics model describes the condition dE T / d t -- 0 and the relation between the elastic and plastic strain is kE = --6; hence the operational equation is _

kT _ kE + k -- -~ d-t + 2 p e ~ exp - kT

sinh

+ R,)

kT

-- 0, (26)

268

A.S. Krausz and K. Krausz

where o . e f t

_ _ O. _

O.I

_~_Rt and R is the recovery rate. From this

do" = Ek E = -Ek = -2Epe~ exp

sinh k-T kr Consider now the low-stress region of high-temperature stress relaxation where the operational equation (26) reduces to dt

"

AG ~t) V (a - a~ + Rt) kT kT

k = 2pe~ exp

(26a)

and

da _2EpeOvexp at =

(

AG ~ t ) V ( a - a I + R t )

~

k-T

9

(27)

This is a standard first-order linear differential equation

d----t+ 2Epe~ exp - k---T- k--T-a = 2Epe~ exp

kT

--~-+-~--t

and the solution is o.=

exp(-f0t M dt) [ fot(Nlt + N2)exp( foot M dt)dt + C]

I N1 (Mt =

-ff

-

1) +

Nit2 5-

+ Nzt +

N1

M2

N2)_M C 1 [ 1 - e x p ( - M t ) ] (28)

where

M = 2Epe~ e x p ( -

kT

k-T

and

N1 = 2Epe~ e x p ( -

AGr

and

N--2Ep~ The negative exponential dependence of the stress on time is typical (Krausz and Eyring, 1975; Wilshire, 1985; Evans and Wilshire, 1985; Findley et al., 1989). Equation (28) describes the behavior when the mechanism is controlled

Chapter 5

The Constitutive Law of Deformation Kinetics

269

by a symmetric energy barrier. It represents the temperature dependence as well as the effects of the state and the changes of the microstructure: It is a complete representation of the effects of material and external constraints for this mechanism. Because sinh[ V ( a - R4t )+= k r

] sinh(V~) cosh[ V(RtkT-- ~

]

sinh V (RtkT- aI) ]

+cosh ~ in the high-temperature and high-stress range where

sinh(~-~) ~cosh(~-~), the equation

da _ Ee o _ -Ek -- -2Epe~ exp dt

k-T

sinh

kT

becomes do-

dt -- -2Epe~ exp - k----~ sinh kT

x {cosh[V/Rtk

+ sinhE

~-- -2Epe~ exp(- kT

exp V (RtkT- aI) 1 "

sinh

Hence the solution of the differential equation is f ~ sinh(~-~)da --_2Epe0v exp(

AG*k_T ) f0texpI V (RtkT- a~) I dt

and the operational equation is a

V

tanh-1 ex In tanh

_2EpeOl exp R

\ 2kT J kT

exp

kT

- 1

(29)

270

A.S. Krausz and K. Krausz

Stress-relaxation behavior of high-purity lead at 298 K. The symbols represent the measured behavior (Wilson and Garofalo, 1966; Wilson and Wilson, 1966; Krausz and Eyring, 1975). These operational equations explain the principles and the methods for the derivation of the constitutive law and the operational equation for the various other forms of the rate-controlling mechanism (Krausz and Eyring, 1975; Frost and Ashby, 1982; Nabarro, 1967; Hill, 1960; Hirth and Lothe, 1968; Evans and Wilshire, 1985). The resultant expressions provide the means for materials development and for the representation of the constitutive laws in the high-temperature region for design purposes with economical computer processing. For mechanisms controlled by a single energy barrier, the constitutive law is -- pf~Okf -- Prerokr.

(30)

Most of the material properties and the effects of the extemal constraints are represented well by this model---or at least adequately. Equation (30) was used by Wilson and Garofalo (1966) and Wilson and Wilson (1966) in the analysis of the stress-relaxation results obtained for a wide range of metals and alloys. In these studies good agreement was found with the symmetric energy barrier model (Fig. 22) kf~ - k ~ -- k ~

and

pf = p r -

p,

where k ~ = v exp -

kr

"

COMMENTS AND SUMMARY The development of industrial activity starts with the determination of the task, that is, with the determination of what is the interest of the activity--the market condition. Once this decision is made, the product is specified, raising the question

Chapter 5 The Constitutive Law of Deformation Kinetics 271

/|[tll~ll~||~ll| The overall place of the constitutive law in the industrial activity.

of the corresponding product and manufacturing process design function of plastic deformation. The determination of the constitutive law of plastic deformation enters at this stage. Obviously, just as the industrial interest was scrutinized in terms of the economy of the whole activity, the design (and with it, the decisions associated with the development and use of the constitutive law) has to fit in with the overall goal. That is, it has to be economical (Fig. 23). Space limits the full presentation of these fundamental considerations to the technical aspects of the constitutive law, although some references are made to the use of it in the economical sense. In passing, it is noted that more extensive attention is given to the overall concept in a forthcoming book by us. The need for the understanding and control of plastic deformation extends over a wide range. The basic processes occur at the atomic and microscopic levels and

272

A.S. Krausz and K. Krausz

The ranges of plastic deformation (after Broek, 1978).

these must be considered in materials research and developments. At the macroscopic level the solid is considered as a continuum, and mechanistic and continuum mechanics model were developed; much of mechanical and civil engineering interest is concentrated on this region. These extend over a size range of ten orders of magnitude: the constitutive law has to cover this full range (Fig. 24). The interest in the temperature range of the constitutive law is from - 5 0 to 800~ and is expanding with the development of new demands. The commonly used industrial materials range from metals and alloys, through ceramics, glasses, and polymers, to their compositesmtens of thousands of them. The constitutive law has to encompass all, and provide an economical means for the expression of the relations among strain, stress, temperature, the characteristic material quantities, and their variations with deformation, temperature, and time. A tall order. Obviously, a physically based constitutive law is sought that 9 can be expressed in practical engineering terms, 9 is economical to test, 9 can use materials data banks, and 9 can be processed with computer-aided design. Deformation kinetics fully satisfies these demands for time- and temperaturedependent plastic deformation processes. In the low- and medium-strain rate range plastic deformation is controlled by thermally activated mechanisms. The behavior is represented in the logarithm of

Chapter 5

The Constitutive Law of Deformation Kinetics

273

~ a [ t l l l ~ t B ~ The stress dependence of the strain rate in the semilogarithmic representation. Thermally activated processes dominate below the dynamic strain rate range; viscous drag mechanism and relativistic effects become effective in the high deformation rate and stress ranges.

strain rate versus stress coordinate system (Fig. 25) It is recognized that in the very high strain rate range thermal activation has a lesser role and other, dynamic mechanisms are operating. In the limit, relativistic effects constrain the dislocation velocity below the speed of sound, which it approaches asymptotically. Considerations have shown that this effect becomes above 0.25-0.3% of the velocity of sound. Between the two is a range where dislocation movement is controlled by viscous drag. When the load is high enough to supply all the energy that is needed at the sites where bonds have to be broken, the dislocation segment or the vacancy will move by one atomic distance (or by a distance that is a small integer multiple of it) solely under the effect of the mechanical energy and at the maximum rate this is possible for any change in the solid--at the velocity of the sound. When, however, the mechanical energy is not enough to supply the full amount of the bond energy, the dislocation or vacancy can move only if another energy source is available. This source is the thermal energy of the solid. The energy balance can be written as the expression of the first law of thermodynamics, which requires that for a process to take place the condition energy needed -- energy supplied has to be satisfied. Thus in plastic deformation, energy needed to break bonds = mechanical energy + thermal energy.

274

A.S. Krausz and K. Krausz

When this equality is satisfied at the atoms that are involved in the movement of the dislocation or the vacancy, the bonds break and an elementary step of plastic deformation occurs. This is the fundamental mechanism of all plastic deformation processes. The constitutive law is an energy expression because the rate of the process is determined by the frequency at which thermal energy is supplied to the sites where bond breaking occurs. It follows than that 9 dislocations and vacancies move exclusively in the direction and at the rate determined by the flow of the energy. The direction and rate of the movement of the flow units are defined by the constitutive law: When expressed in three-dimensional form, it represents the directional rate of plastic deformation. Dislocation or vacancy velocity is a vectorial quantity; the energy field is a scalar quantity from which the directionality is determined by the direction in which the largest thermal energy peaks A G ~(W) arrive at the greatest rate and the direction of the arrangement of defect structure where activation occurs. This statistical condition sets the direction and magnitude of the dislocation or vacancy velocity vectors. The fact that the process is defined by a field equation is of considerable significance. In amorphous materials there are no dislocations, but in these also the exchange of bonds is the basic mechanism of plastic deformation. Because all materials have the same type of bond energy-interatomic distance relation (Fig. 3b) and all plastic deformation processes consist of just the change of atomic bonds, the development and expression (as well as the application) of the constitutive law are the same for all. Thus the concepts and conclusions expressed by the theory of deformation kinetics are fully valid for all types of materials, be they metals, polymers, ceramics, glasses, or their composites. The same goes for minerals, ice, salt, and so on. This is a very important conclusion, and the consequences of this recognition are far-reaching. Deformation kinetics theory provides the means by which all materials can be considered within a common framework: There is no need to develop new and different theories of the constitutive law for different materials or for different conditions. The means are given by nature; exploitation of it affords great advantages and simplification in the use of constitutive laws. Deformation kinetics theory considers that plastic deformation is the consequence of permanent changes in atomic configuration. As noted, it follows from this recognition that the process is controlled by an energy balance condition and is expressed by the first law of thermodynamics 9 energy needed for the rearrangement of the atoms energy supplied;

Chapter 5 The Constitutive Law of Deformation Kinetics 275 in plastic deformation this takes the specific expression 9 bond energy = thermal energy + mechanical energy AG J;-- A G $ ( W ) + W(cr). Often, the work is a linear function of the stress: W = Va. Because the thermal energy A G ~(W) is supplied at the rate determined by the heat content of the material, the deformation is time- and temperature-dependent. This is a characteristic feature of plastic deformation together with its dependence on the state of the microstructure. These fundamental concepts must be expressed by the constitutive law in terms that are design quantities: stress, strain, temperature, microstructure, and their time variation. The constitutive law is then expressed as = f (a, s, T, microstructure, d/dt) The theory leads to the conclusion that the constitutive law is a well defined function of the rate constant k. The elementary rate constant k -- v exp -

AG ~ - W(o-) ] kT

J

is the elementary building block of the constitutive relations. It follows from the fundamental principles of statistical mechanics, from which the theory of deformation kinetics is derived, that at the atomic level plastic deformation also takes place in the direction against the driving force, causing the reverse movement of the flow unit as well as movement in the forward direction, which proceeds in the sense of the driving force. Accordingly, the rate constant takes two complementary forms: 9 the rate constant of the forward activation

I

kT

kr -- v exp -

kT

kf -- v exp 9 the rate constant of the reverse activation

"

All constitutive laws are the appropriate kinetics combinations of these ~ = f (k). Deformation kinetics theory leads to the conclusion that there are only three types of kinetics combinations: (1) parallel, (2) consecutive, and (3) parallel combination of consecutive systems. This recognition is of significant benefit in the development and application of the constitutive laws. Practically all types of plastic deformation processes in any material are defined by these. The simplicity introduced by this recognition is essential in working with the difficult conditions that the establishment and application of the constitutive laws present.

276

A.S. Krausz and K. Krausz

During plastic deformation both the concentration and distribution of microstructural defects change. Direct measurements show that the dislocation density is often a linear function of the stress and strain. The effect of microstructural changes on the representative characteristic quantities of the material A G J;, W (a), V, p, e ~ and the kinetics can be measured with the methods of deformation kinetics. Phenomenological relations were also developed. There are two methods for the study of the mechanical behavior of materials:

(1) Analysis. In analytical investigations, the behavior is studied by measuring the relations among strain, stress, temperature, and their time variation. The tests are designed according to a selected theory of the constitutive law and their interpretation is carded out in the context of this theory. The results are expressed with this specific constitutive law. (2) Synthesis. In the synthesis method, the behavior of models that are subject to exactly defined external and internal constraints are investigated. The result is the operational equation. In this the imposed internal constraints, expressed with the state variables, and the external constraints, the test or manufacturing and service conditions, are combined in the constitutive law model that is selected for the exploration of its behavior. Because the theory is based on physical principles the conclusions derived for specific models are rigorously valid within the limits of the correlation between the imposed conditions and the model: 9 The predictions and extrapolations derived from the synthesis are fully valid. The theory on which the analytical and synthesis developments are based has to be derived from physical principles. Deformation kinetics satisfies this requirement and is the theory of the analysis of the constitutive law and the synthesis of operational equations. Any deviation in the observed behavior is due to the existence of conditions that were not made part of the model and were not considered in the imposed internal and external constraints. Its predictions, or extrapolations, are permanently valid. Deviations do not contradict the validity of the model. In fact, these are valuable indicators of the existence of additional or differing conditions and give important guidelines for the further extension of the model and the investigations. Because the deformation kinetics analysis and synthesis can be made with computer processing, it provides an economical alternative to testing. In closing, competitive materials development and the design of manufacturing processes and product requires physically based constitutive laws that are rigorously derived from basic physical principles and expressed fully in well-defined quantitative engineering terms that are economical to apply. Deformation kinetics is developed in this sense.

Chapter 5

The Constitutive Law of Deformation Kinetics

277

REFERENCES Ashby, M. E, and Frost, H. J. (1975). The kinetics of inelastic deformation above 0~ In "Constitutive Equations in Plasticity" (A. S. Argon, ed.), pp. 117-148. MIT Press, Cambridge, MA. AT&T. (1993). "Testing to Verify Design and Manufacturing Readiness." McGraw-Hill, New York. Bagley, R. L., Jones, D. I. G., and Freed, A. D. (1995). Renewal creep theory. Metall. Mater. Trans., A. 26A, 829-843. Bodner, S. R. (1987). Review of unified elastic-viscoplastic theory. In "Unified Constitutive Equations for Creep and Plasticity" (A. K. Miller, ed.), pp. 273-302. Elsevier, London. Borg, R. J., and Dienes, G. J. (1988). "An Introduction to Solid State Diffusion." Academic Press, San Diego, CA. Boyle, J. T., and Spence, J. (1951). "Stress Analysis for Creep." Butterworth, London. Braasch, H., Dudeck, H., and Ahrens, H. (1995). A new approach to improve materials models. Metall. Mater. Trans., A 117, 14-19. Broek, D. (1978). "Elementary Engineering Fracture Mechanics." Sijthoff & Noordhoff, The Hague, The Netherlands. Cekirge, H. M., Tyson, W. R., and Krausz, A. S. (1976). Static corrosion and static fatigue of glass. J. Am. Ceram. Soc. 58, 265. Crank, J. C. (1970). "The Mathematics of Diffusion." Oxford Univ. Press (Clarendon), London. Evans, R. W., and Wilshire, B. (1985). Creep of metals and alloys. The Institute of Metals, In "Creep Behavior of Crystalline Solids" (B. Wilshire and R. W. Evans, eds.). Pineridge Press, Swansea, UK. Faucher, B., and Krausz, A. S. (1980). The kinetics of plastic flow. Z. Naturforsch., A 35A, 1152-1161. Faucher, B., and Krausz, A. S. (1981). Development of time dependent constitutive equations. Proc. Can. Congr. Appl. Mech., 8th, pp. 249-250. Findley, W. N., Lai, J. S., and Onaran, K. (1989). "Creep and Relaxation of Nonlinear Viscoelastic Materials." Dover, New York. Folansbee, E S., Regazzoni, G., and Kocks, U. F. (1985). High strain rate measurements of dislocation mobility." In "The Mechanics of Dislocations" (E. C. Aifantis and J. E Hirth, eds.), pp. 237-246. American Society for Metals, Metals Park, OH. Fox, A., ed. (1979). "Stress Relaxation Testing." ASTM, New York. Fox, A. (1985). Stress relaxation tension testing. In "Metals Handbook," 9th ed., Vol. 8, pp. 323-328. American Society for Metals, Metals Park, OH. Freed, A. D., and Walker, K. E, eds. (1991). High Temperature Constitutive Modeling--Theory and Applications, Proc. 1991 ASME Winter Annual Meeting (ASME MD-Vol. 26, Amd-Vol 121). Freed, A. D. and Walker, K. E (1995). Voscoplastic model development with an eye toward characterization. Metall. Mater. Trans., A. 117, 8-13. Frost, H. J., and Ashby, M. E (1982). "Deformation-Mechanism Maps." Pergamon, Oxford. Garofalo, E (1965). "Fundamentals of Creep and Creep Rupture in Metals" Macmillan, New York. Glasstone, S., Laidler, K. J., and Eyring, H. (1941). "The Theory of Rate Processes," McGraw-Hill, New York. Hanley, T. O'D, Krausz, A. S., and Krishna, N. (1974a). Thermally activated deformation. II. Deformation of sintered Iron. J. Appl. Phys. 45, 2016. Hanley, T. O'D., Krausz, A. S., and Maheshwari, D. (1974b). Thermally activated deformation of pure iron. Mater. Sci. Eng. 16, 155. Hanley, T. O'D., Krausz, A. S., and Krishna, N. (1977). Mechanical properties of sintered iron, Int. J. Powder Metall. Powder Technol. 13, 215-219. Hill, T. (1960). "Statistical Mechanics." Addison-Wesley, Reading, MA.

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Hirth, J. P., and Lothe, J. (1968). "Theory of Dislocations." McGraw-Hill, New York. Jager, J. C. (1956). "Elasticity, Fracture and Flow." Methuen, London. Jordan, P. C. (1980). "Chemical Kinetics and Transport." Plenum, New York. Jost, W. (1960). "Diffusion in Solids, Liquids, Gases." Academic Press, New York. Kocks, U. E (1987). Constitutive behavior based on crystal plasticity. In "Unified Constitutive Equations for Creep and Plasticity" (A. K. Miller, ed.), pp. 1-88. Elsevier, London. Krausz, A. S. (1969). A rate theory of strain relaxation. Mater Sci. Eng. 4, 193. Krausz, A. S. (1970). A rate theory analysis of the temperature dependence of dislocation velocity. Mater Sci. Eng. 6, 260. Krausz, A. S. (1973). The forming of superplastic alloys. J. Met. 15, 24. Krausz, A. S. (1976). A deformation kinetics analysis of the stress sensitivity, Mater. Sci. Eng. 26, 65. Krausz, A. S. (1978). The deformation and fracture kinetics of stress corrosion cracking, Int. J. of Fracture 14, 5-15. Krausz, A. S. and Craig, G. B. (1966). Variable stress creep in copper. Acta Metall. 14, 1807. Krausz, A. S., and Eyring, H. (1971). Chemical kinetics of plastic deformation. J. Appl. Phys. 42, 2382. Krausz, A. S., and Eyring, H. (1975). "Deformation Kinetics." Wiley Interscience, New York. Krausz, A. S., and Krausz, K. (1983). A review of the rate dependent plastic deformation processes and the associated constitutive laws. In "Constitutive Laws for Engineering Materials: Theory and Applications" (C. S. Desai and R. H. Gallagher, eds.), pp. 82-89. Elserier, New York. Krausz, A. S., and Krausz, K. (1984). Time and temperature dependent plastic flow and fracture: physical theory and engineering applications. Proc. Southeast. Conf. Theor. Appl. Mech. 12th, pp. 169-179. Krausz, A. S., and Krausz, K. (1985a) "Fracture Kinetics of Crack Growth." Kluwer Academic Publishers, The Hague, The Netherlands. Krausz, A. S., and Krausz, K. (1985b). On the physical processes of plastic deformation and fracture of ceramic materials." Mater. Sci. Monogr. 38B, 1239-1245. Krausz, A. S., and Krausz, K. (1986). The energy expression of the microstructure in thermally activated processes. In "NATO Advanced Research Workshop on Patterns, Defects and Microstructures in Nonequilibrium Systems" (D. Walgraef, ed.), pp. 300--308. Krausz, A. S., and Krausz, K. (1987). The constitutive laws of deformation kinetics. In "Constitutive Laws of Engineering Materials: Theory and Applications" (C. S. Desai, E. Krempl, P. D. Kiousis, and T. Kundu, eds.), p. 777. Elsevier, New York. Krausz, A. S., and Krausz, K. (1992). Deformation kinetics of plastic deformation: Physically based constitutive laws. Proc. Sym. Plas. Flow Creep, ASME Appl. Mech. Summer Conf., 1992. Krausz, A. S., and Krausz, K. (1993). On the deformation kinetics constitutive law of plastic deformation: The rate equation. Proc. 33rd Isr. Annu. Conf. Aviat. Astronaut., pp. 387-402. Krausz, A. S., and Krausz, K. (1994). The deformation kinetics theory of stress dependence. Acta Metall. Mater. 42(3), 937-945. Krausz, A. S., Craig, G. B., and Faucher, B. (1976). Volume d' activation en fonction de la m6thode de measure. Scr. Met. 10, 393. Krausz, K., and Krausz, A. S. (1984). The kinetics of time dependent plastic flow and fracture. In "Ceramic Materials" (R. E. Tressler, and R. C. Bradt, eds.), pp. 547-554. Plenum, New York. Krausz, K., and Krausz, A. S. (1988a). The unity of plastic deformation and crack growth. Res. Mech. Spec. Issue 23, 99-112. Krausz, K., and Krausz, A. S. (1988b). 23rd Annu. Meet. Soc. Eng. Sci. Spec. Sess. Krieg, R. D., Swearengen, J. C., and Jones, W. B. (1987). A physically based internal variable model for rate dependent plasticity. In "Unified Constitutive Equations for Creep and Plasticity" (A. K. Miller, ed.), pp. 245-268. Elsevier, London. Kula, E., and Weiss, V. (1982). "Residual Stress and Stress Relaxation." Plenum, New York.

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Lawn, B. R., and Wilshaw, T. R. (1975). "Fracture of Brittle Materials," Cambridge University Press, Cambridge, UK. Manning, J. R. (1968). "Diffusion Kinetics for Atoms in Crystals." Van Nostrand, Princeton, Nd. Mataya, M. C. and Sackschewsky, V. E. (1994). Effect of internal heating during hot compression on the stress-strain behavior of alloy steel. Metall. Mater Trans., A. 25A, 2737-2752. McClintock, E A., and Argon, A. S. (1966). "Mechanical Behavior of Materials." Addison-Wesley, Reading, MA. Metals Handbook. (1985). "Mechanical Testing," 9th ed., Vol. 8. American Society for Metals, Metals Park, OH. Metals Handbook. (1988). "Forming and Forging," 9th ed., Vol. 14. American Society for Metals, Metals Park, OH. Miller, A. K. (1987). "The Matmod equation." In "Unified Constitutive Equations for Creep and Plasticity" (A. K. Miller, ed.), pp. 139-220. Elsevier, London. Mrowec, S. (1980). "Defects and Diffusion in Solids." Elsevier, London. Nabarro, F. R. N. (1967). "Theory of Crystal Dislocations." Oxford Univ. Press (Clarendon), London. Nadai, A. (1950). "Theory of Flow and Fracture of Solids." McGraw-Hill, New York. Puchi, E. S. (1995). Constitutive equations for commercial-purity aluminum deformed under hotworking conditions. Metall. Mater Trans., A. 117, 20-27. Raj, S. V., and Farmer, S. C. (1995). Characteristics of a new creep regime in polycrystalline NiA1. Metall. Mater Trans., A 26A, 343-3553. Sharpe, W. N., Jr. (1995). ASME 1993 Nadai Lecture--Elastoplastic stress and strain concentrations. Metall. Mater Trans., A 117, 1-7. Shewman, E G. (1989). "Diffusion in Solids." McGraw-Hill, New York. Suh, N. E, and Turner, A. E L. (1975). "Elements of the Mechanical Behavior of Solids." McGraw-Hill, New York. Tetelman, A. S., and McEvily, A. J. (1967). "Fracture of Structural Materials," Wiley, New York. Thomson, R. (1980). Physics of fracture. In "Atomistics of Fracture" (R. M. Latanision and J. R. Pickens, eds.), pp. 167-208, NATO Conf. Ser., Plenum Press, New York. Timoshenko, S., and Goodier, J. N. (1950). "Theory of Elasticity." McGraw-Hill, New York. Trouton, P., and Rankine, A. O. (1904). Philos. Mag. [6] 8, 538. Wilshire, B., and Evans, R. W., eds. (1985). "Creep Behavior of Crystalline Solids," Pineridge Press, Swansea, UK. Wilson, J. F., and Garofalo, F. (1966). Mater. Res. Stand. 6, 85. Wilson, J. F., and Wilson, N. K. (1966). Trans. Soc. Rheol. 10, 399. Wray, E J., and Horne, G. T. (1966). Philos. Mag. [8] 13, 899. Wu, X. J., and Krausz, A. S. (1994). Kinetics formulation for low temperature plasticity. J. Mater Eng. Performance. Yang, S. T. and Mohamed, E A. (1995). On the characteristics of the threshold stress for superplastic flow in Zn-22 Pct A1. Metall. Mater Trans. A 26A, 493-496. Zurek, A. K., and Folansbee, E S. (1995). A comparison of shear localization susceptibility in U-0.75 Wt Pct Ti and W-Ni-Fe during high strain rate deformation, Metall. Mater Trans., A 26A, 1483-1490.

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6 A Small-Strain Viscoplasticity Theory Based on Overstress Erhard Krempl

Mechanics of Materials Laboratory Rensselaer Polytechnic Institute

Troy, New York

The viscoplasticity theory based on overstress (VBO) for small strain, isotropy, and isochoric inelastic deformation is introduced. VBO is a unified theory without a yield surface representing a solid. The total strain rate is the sum of the elastic and inelastic strain rates. The inelastic strain rate is an increasing function of the overstress, the difference between the stress and the equilibrium stress, which is a state variable of the theory. The overstress is a measure of rate dependence. The purpose of the kinematic stress, a second state variable, is to model work hardening (softening) in monotonic loading. The isotropic stress or rateindependent stress, the third state variable, is constant for cyclic neutral behavior. For cyclic hardening or softening, a growth law is needed. It also changes when high homologous temperature behavior is modeled. Asymptotic solutions f o r constant strain rate exist and are useful in the identification of rate-dependent and rate-independent contributions to the stress. Creep and relaxation behavior are delineated, and it is shown that a nonzero stress can be sustained at rest. This basic theory is augmented to model high homologous temperature deformation behavior, variable temperature, and anisotropy. The relation of the present approach to others is Unified Constitutive Laws of Plastic Deformation Copyright (~) 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

281

282 Erhard Krempl discussed, including some criticisms using thermodynamics. It is demonstrated that the concept of an equilibrium stress can also be found in the theory of linear viscoelastic solids.

~11

INTRODUCTION

Within the last two decades, new material models for inelastic deformation behavior of metals and alloys have appeared in the literature (see Miller, 1987, for a summary). They are intended to replace conventional creep and plasticity formulations and to reproduce the deformation behavior in monotonic and cyclic loadings. They do not concentrate exclusively on one mode of deformation, e.g., creep. Inelastic deformation behavior is considered rate-dependent, a notion that is shared by materials scientists who view inelastic deformation as a rate process. These models do not contain separate repositories for plasticity and creep and are known as unified constitutive equations. Most of them do not employ a yield surface and do not need loading and unloading conditions. The total strain rate is the sum of elastic and inelastic strain rates. The rate form of Hooke's law is the model for the elastic strain rate, and the inelastic strain rate is a function of stress and one or more state variables. Frequently, a first-order nonlinear differential equation is the growth law for the state variables. Collectively, they represent the evolving microstructure and other deformation features. The author and his students have, over the years, developed the viscoplasticity theory based on overstress (VBO) which is one of the state variable theories. The results of servo-controlled tests and materials aspects led to the development of this theory. In turn, the theory suggested experiments needed to confirm or controvert the model.

VISCOPLASTICITYTHEORY BASEDON OVERSTRESS A. ConstitutiveEquations 1. F l o w L a w

We assume small strain, isotropy, and incompressible inelastic deformation. The stress and strain tensors are tr and e, respectively, and s and e are their respective deviators. The flow law for VBO is __ l~el _~_

ein _-- 1 + V S -~ 3 s - g E 2 Ek[1-']

(1)

where E and v are the elastic constants; g is the deviatoric equilibrium stress; k[1-'] is the decreasing, positive viscosity function; and F is the overstress invariant

Chapter 6 A Small-Strain Viscoplasticity Theory Based on Overstress 283 defined by I-'2 -- 3 tr((S -- g)(s -- g)) The viscosity function has the dimension of time and represents a variable relaxation time that depends on the overstress. It is very large for zero overstress and decreases nonlinearly with increasing overstress. The volumetric deformation is elastic:

1 -2v el,k = - - 6 ~ 1 , E

(2)

We see from Eq. (1) that the inelastic strain rate is zero when s = g = 0 or when s = g. The quantity g can therefore be identified as the stress sustained at rest. The growth law for this quantity specifies the current value of this, generally path-dependent, state variable. It is called the equilibrium stress (the term "back stress" is also in use).

2. Growth Law for the Equilibrium Stress The purpose is to model the initial quasi-elastic regions and the final work hardening or softening with a transition in between. The growth law of the equilibrium stress g is, in effect, an interpolation between initial quasi-elastic growth and final inelastic flow (see Lee, 1989). We have for the equilibrium stress deviator g

eel g - ~p[r] 1 + v

+

2 ein~ (g - f) - ~ -3 / b[r]

-: (3)

where

~ 2 _ ~2 3 tr(~in~in)

and

b[r]-

A l/r[1-']- Et

(4)

Equation (3) can be written in stress formulation by using the definitions of the elastic and inelastic strain rates: _

~p[r]

E

s +

s-g) k[F]

F (g-f) Ek[r] b[r]

(5)

This version of the growth law presumes that the slope of the stress-strain curve is ultimately Et on the basis of total strain. For a general final slope of the stressstrain curve as represented by an arbitrary growth law for the kinematic stress f, it is best to introduce the kinematic stress and its growth i'. That changes Eq. (5) to 1 _ gr[F] E

s-~

s--g k[1-']

r (g--f) k[F] a

+

1This change was suggested by M. R. Eggleston (see Majors, 1993).

1+

i' E

(6)

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Erhard Krempl

The scalar A with the dimension of stress is the rate-independent or isotropic stress. When A is constant, cyclic neutral behavior is modeled. If cyclic hardening or softening is to be modeled, A has to be considered a state variable and a growth law must be specified. It also changes for high homologous temperature deformation modeling. The normalized positive shape function ~p/E is decreasing and ~p/E < 1.2 It influences the transition between initial quasi-elastic behavior 3 and full inelastic flow.

3. Growth Law for the Kinematic Stress

The purpose of the kinematic stress is to model work hardening in monotonic loading. When f = 0 in Eq. (6), the equilibrium stress-strain diagram ultimately becomes horizontal. The purpose of the kinematic stress f is to render this slope different from zero. It can be positive (work hardening) or negative (work softening). It will be shown later that the growth laws for g and f presented here correspond to the multiple back-stress growth laws of the Chaboche theory (see Chaboche, 1993). Although complicated growth laws may be used, the simplest growth law for the kinematic stress _

2/~tein

=

Et s -- g E k[r]

(7)

is sufficient here. Here, /~t "-- Et/(1 - E t / E ) is the tangent modulus at the maximum inelastic strain of interest. If another ultimate slope is modeled by a new growth law for the kinematic stress, Eq. (6) must be used. It will be shown below that Eq. (7) sets the slope of the stress-strain diagram at large times when the asymptotic solution is reached. This corresponds to the attainment of steady inelastic flow in a real test. Equations (1)-(4) constitute the basic VBO theory which models nonlinear rate sensitivity, creep, and relaxation, stress-strain diagrams with an initial slope of E and a final slope of Et based on total strain. In completely reversed strain control, the hysteresis loop closes after the first cycle and cyclic neutral behavior is modeled. We prefer to work with Eqs. (1), (2), (6), and (7). It is possible to consider other versions of the growth laws for the equilibrium stress; these are discussed below.

2An initial value of 1 would reduce the system of equations to a linear hypoelastic system. Thus only linear elastic behavior would be reproduced. 3VBO cannot reproduce exactly linear elastic regions. However, the graphs generated by solving the set of nonlinear differential equations to VBO can show linearity with the "correct" elastic slopes.

Chapter 6

A Small-Strain Viscoplasticity Theory Based on Overstress

285

B. Modified Growth Laws for the Equilibrium Stress 1. Additional Constant Additional modeling capabilities are possible by writing Eq. (6) as _ O[P] E

s+

E-

k[F]

k[1-']

a

+

1

E

t"

(8)

Here we have introduced the constant c < E with the dimension of stress that provides additional modeling capabilities. When c = E, Eq. (6) is recovered (see Majors and Krempl, 1994). All the asymptotic properties to be discussed below are valid for Eqs. (6) and (8). m

2. Direction of D y n a m i c Recovery Term In Eqs. (6) and (8), the "dynamic recovery term" is in the direction of g - f. It can be written so that the overstress determines its direction. We have

tg -- ~p[1-'] s +

E

-E

k[I']

k[F]

A

-+-

1

E

t"

(9)

where | -- v/~tr(g - f)(g - f) is the invariant of the difference between the equilibrium and the kinematic stress. Equation (9) appears to have essentially the same properties as Eq. (6) and can be used as a growth law instead of Eq. (6). This has been done by Lee (1989) and Lee and Krempl (199 l b) in the formulation of an orthotropic theory of viscoplasticity. Now, the dynamic recovery term changes sign with the overstress. Previously, the sign change occurred with the difference between the equilibrium stress and the kinematic stress. We will illustrate later the significant effects of this modification.

C. Application to Modeling 1. General R e m a r k s Equations (1), (2), (6), and (7) or (9) 4 are the constitutive equations. For homogeneous motions, they cannot be solved unless boundary conditions are specified. The model predictions or the responses are obtained by specifying either stress or strain as a function of time. Then, the constitutive equations become a set of nonlinear, coupled, nonautonomous ordinary differential equations. Closed-form solutions, as a rule, are not possible and numerical integration is needed. There are two major loading cases: (1) load (stress) control, where the stress is prescribed 4The growth law for the kinematic stress can also be different.

286

ErhardKrempl

as a function of time (included is the creep test for which the stress rate is zero), and (2) strain (displacement) control, which includes relaxation (the strain rate is zero). In addition to boundary conditions, initial conditions for stress or strain and for the state variables are needed. The initial conditions for stress and strain are easily obtained from the test conditions. For an annealed material, it is assumed that the state variables equilibrium and kinematic stress are zero at first and then evolve with deformation. As loading continues, these state variables grow and their final values will become the initial values for the next history. In general, the modeling of the influence of prior deformation rests with the initial conditions. 2. A s y m p t o t i c S o l u t i o n s

The constitutive equations admit asymptotic solutions for constant strain rate and large (mathematically infinite) time. These solutions can be obtained by converting the differential equations into integral equations and formally proceeding to infinite time (see Cernocky and Krempl, 1979; Chow, 1993). Alternatively, methods of stability analysis can be employed (see Stit~ti and Krempl, 1989). Experience has shown that with usual material functions, these solutions are useful at finite times and hence at small strains. The asymptotic solutions have very important physical interpretations and are thought to correspond to fully established inelastic flow in a material. Inelastic flow is established once the stress is sufficiently beyond the elastic region and when the tangent modulus is much smaller than the elastic modulus. Materials scientists call this region "flow stress." Formally, the limit 5 is for the overstress {S -- g} -- ~ 2(Ee-

{g}(1 - v))k[{F}]

=

~ 2 E{ein}k[{I"}]

(10)

In the last step, we have made use of {s} = {l~}. Equation (10) can be converted to a relation between the invariants

{r} = {~}Ek[{r}]

(11)

A nonlinear relation between the overstress and the inelastic strain rate is depicted. The viscosity function k controls the rate dependence. If k is a constant, the overstress (s - g) is linearly dependent on the inelastic strain rate. This is not true for metals (nor, in fact, for most ductile materials), and consequently, k has to be a decreasing function since the overstress normally increases with strain rate. If VBO is to represent a nearly rate-independent material, then 1-' should vary little with inelastic strain rate and the viscosity function must be highly nonlinear. Nearly rate-independent behavior can be modeled by VBO with an appropriate 5{a} designates the asymptotic limit of a.

Chapter 6 A Small-StrainViscoplasticityTheory Based on Overstress 2117 viscosity function. Experiments show that the inelastic strain rate can be varied by several orders of magnitude without significantly affecting the flow stress. If, on the other hand, the influence of inelastic strain rate on the flow stress level is very strong, a different k function is required. The viscosity function k is the repository for modeling rate dependence. There is also an asymptotic solution for the difference g - f. We obtain from Eqs. (5), (6), or (8) {g - f} - A {s - g____~}=

{r}

2AEk[{V }] {ein} 3

{r}

(12)

This relation is exact for Eqs. (6) or (8). For Eq. (5), we have used Et/E << 1 or E,t/E ~< 1. To arrive at Eq. (12), we also assumed {s} -- {g} = if}. For the Lee version of the growth law for the equilibrium stress, Eq. (9), the asymptotic solution is the scalar relation | -- A, which can also be obtained from Eq. (12). Equation (12) shows that the asymptotic limit for the difference of the equilibrium stress and the kinematic stress is rate-independent and equal to the isotropic stress A. Although there is no asymptotic solution for either the stress or the equilibrium stress unless the ultimate slope of f is zero, the difference (s - g), the overstress, and (g - f), the difference between the equilibrium and the kinematic stress, admit asymptotic solutions that are both in the direction of the inelastic strain rate. The limit for the overstress is rate-dependent, whereas the magnitude of {g - f} is rate-independent and equal to the isotropic or rate-independent stress A. The identity s = ( s - g) + ( g - f) + f

(13)

which holds all the time, and the relation (valid only in the asymptotic state) {s}-

{I~}- {f}

(14)

permit the sketch shown in Fig. 1. We see that the "flow stress," the asymptotic stress, consists of the kinematic part, the rate-independent contribution A, and the viscous part represented by the overstress. Each contribution can be adjusted. The kinematic contribution is controlled by the evolution law for the kinematic stress, and its purpose is mainly to set the ultimate slope, which can be positive (work hardening), negative (work softening), or zero (inelastic flow at constant stress, perfectly plastic behavior). The isotropic, rate-independent contribution A is added to the work hardening. Finally the rate-dependent overstress (effective stress) is added. The overstress can be very small to represent nearly rate-independent behavior, or it can be large for strongly rate-dependent materials. These phenomenological identifications exhibit a rough correspondence to microstructural mechanisms. At low homologous temperature, the rate dependence might originate from dislocation bowing and viscous resistance to dislocation glide.

288

Erhard Krempl

~

Schematic showing the stress contributions in the asymptotic state.

As the temperature increases, diffusion processes enhance time (rate) dependence. The rate-independent or isotropic stress can be a measure of dislocation density and of the obstacles to dislocation movement. The equilibrium stress, identified as the generally p a t h - d e p e n d e n t stress that can be sustained at rest, 6 is a measure of the defect structure and of the elasticity of the material. The equilibrium stress has an additional function of m o d e l i n g elastic regions; for this reason, we have included an elastic strain rate term or stress rate term in the growth laws for the equilibrium stress. A discussion of the overstress concept and its relation to other approaches is given by K r e m p l (1987, 1995b).

3. Asymptotic Properties in Stress Space The asymptotic solutions have been discussed previously in a one-dimensional setting. It is instructive to elucidate their properties in a three-dimensional context and in relation to classical theories and associated properties such as normality and other concepts. Equation (13) is valid all of the time and can be used to locate the stress s in deviatoric stress space as shown in Fig. 2.

6VBO is a model of a solid. It can support a nonzero stress at rest after deformation. Many of the unified models are fluids, so their rest stress is mathematically zero. However, in most practical cases, the approach to zero is so slow that zero stress is reached only at very long times, which are outside the time of practical interest.

Chapter 6

A Small-Strain Viscoplasticity Theory Based on Overstress

289

n

-

f

Composition of stress in deviatoric stress space.

In general, the stress and the overstress have different directions. The same is true for the overstress and the difference between the equilibrium stress and the kinematic stress as seen in Fig. 2. In the asymptotic state, both quantities have the same direction given by the asymptotic value of the inelastic strain rate; see Eqs. (10) and (12). They emanate from the center f of the sphere with radius I[s - fll = (F + A). It is then possible to draw spheres (circles) around the point f with the radii (F + A) and A as shown in Fig. 3. Then the inelastic strain rate is along the radius and is normal to the surface of the sphere centered at the point f. We see that normality holds in the asymptotic state. When a creep or relaxation test is started after the asymptotic solution is reached in loading, the inelastic strain rate maintains that direction for the duration of the

Alignment of stresses in deviatoric stress space when the asymptotic solution holds.

290

Erhard Krempl

creep or relaxation test. First, the direction is maintained at the start of the creep or relaxation test since the inelastic strain rate is continuous for a jump in strain (stress) rate. Furthermore, all of the terms in the growth laws for the equilibrium stress [Eqs. (15) or (18) below] and for the kinematic stress initially have the same direction, which is maintained throughout the entire creep or relaxation tests; see Fig. 3. In a creep test the stress is fixed and the equilibrium surface changes its radius, following the scheme shown in Fig. 3. On the other hand, for the relaxation test, the stress, the equilibrium stress, and the kinematic stress move. Either an increase or a decrease is possible. From Eq. (18) below, we infer that in most cases the stress and the kinematic stress change in the same direction, but with opposite sign. This discussion shows that the normality rule holds in the asymptotic state and in subsequent creep or relaxation tests. In other conditions, normality cannot be expected.

4. Modelingof Creep For creep, the stress is constant, say So. This condition is inserted into the constitutive equations to yield

3 s0-g 2 Ek[I"o]

Ig_ ~r[1-'0](s0-g E

F0 g-f)

k[F0]

k[F0]

A

( +

~[F0])t, 1

(15)

E

L't SO -- g

E k[Fo] where the invariant F0 has to be calculated using the constant stress So. For creep, there is no asymptotic solution for the overstress; but for {g} = {i'}, {g - f} can be obtained as [g - f} =

{so

-

gJ - {fJk[{r0J]

F0

A

(16)

and using Eq. (7) {g_f}=

So gA Fo

_

1-

~

s0g

Fo

A

(17)

The asymptotic values of { g - f} obtained in creep and in strain control are almost equal; cf. Eq. (12). This result is valid for the modeling of most ductile materials. 7 Although the evolution of the equilibrium stress is different in creep and 7Their tangent modulus in the inelastic range is almost always much smallerthan the elastic modulus.

Chapter 6 A Small-StrainViscoplasticityTheory Based on Overstress 291 in monotonic loading, the differences are small. As a consequence, it is possible to assume that the evolutions in monotonic loading and creep are approximately equal. Since the stress is constant in creep, creep behavior is controlled by the evolution of the equilibrium stress g; see Eq. (15). The influence of the slope of g on creep behavior is shown schematically in Figs. 4a-4c. Depending on the evolution of g as determined by f , primary, secondary, or tertiary creep can be modeled. Positive slopes of the stress-strain diagram produce only primary creep (Fig. 4a). For the model to yield secondary creep, the slope of f must become zero (see Fig. 4b). A negative slope of g and a stress level greater than g will model tertiary creep (see Fig. 4c). When the Lee version of the growth law for the equilibrium stress, Eq. (9), is used instead, the qualitative behavior of the model is not changed. The slope of the real stress-strain diagram in the inelastic region determines the tangent modulus to be used in the growth law for the kinematic stress. This same tangent modulus appears now in the creep equations. We deduce that there can be only primary creep if there is a positive tangent modulus in a tensile test. Also, the creep strain accumulation predicted by the model in regions with nearly elastic slopes is small, and only primary creep will be reproduced. This behavior is frequently called "cold creep." It is known that at elevated temperature real materials can exhibit a positive tangent modulus and secondary and tertiary creep at stress levels in the quasilinear region of the stress-strain diagram. This behavior cannot be modeled with the present low-temperature version of VBO, as demonstrated above. It will be shown later that a modification of the growth laws that accounts for the effects of diffusion (static recovery) prevalent at high homologous temperature will remedy this shortcoming.

5. Modeling of Relaxation For relaxation the strain rate is zero and eel : _ e i n ~__ S(1 + v) / E. The constitutive equations reduce to

3

s-g

. _ _ .

2 (1 + v)k[1-'] _ 7r[r] E

1

2v s - g

2(1 + v) k[1-']

t

g-f a k

+

1

[F]

t

(18)

E

Ets-g E k[1-'] The solution of Eqs. (18) yields the relaxation curve. The stress rate is zero when s - g -- 0 since the right-hand sides of all three equations of Eqs. (18) are

292

Erhard Krempl

Creep behavior: (a) Primary creep only; (b) primary and secondary creep; (c) primary, secondary, and tertiary creep.

Chapter 6 A Small-Strain ViscoplasticityTheory Based on Overstress 293 m

r I11 J4 4J

Strain

Motion of stresses during relaxation.

zero at this point. However, nothing can be said about the stress at which the relaxation terminates. Since the equilibrium stress rate and the kinematic stress rates have opposite signs, and since the relaxation rates of the stress and of the equilibrium stress normally have the same sign, the termination point of relaxation is between the values that g and f had reached at the beginning of the relaxation test. s Relaxation stops at a nonzero stress that depends on prior history. The motion of the stresses is indicated in Fig. 5. It should be noted that the evolution of the slope of the equilibrium stress-strain curve, in contrast to creep, does not affect the relaxation behavior since relaxation takes place at constant strain. There is also no qualitative difference between the uniaxial relaxation behavior predicted by the different growth laws for the equilibrium stress.

6. Aftereffect A special case of the creep or relaxation test is the one performed after prior loading at zero stress. When the stress is kept at zero, Eq. (15) applies. For positive equilibrium stress values, its rate and that of the kinematic stress are negative. The strain rate is also negative. Recovery is simulated. By an appropriate growth law for the equilibrium stress, the recovered strain can be made very small so that metallic material behavior is simulated. When the strain is kept constant at zero stress, Eq. (18) shows a positive stress rate for positive equilibrium stresses, and the theory predicts upward relaxation. 8This behavior is in costrast to the deformationtheory of VBO (see Cernocky and Krempl, 1980), the original Perzyna theory (Perzyna, 1963), and the viscoplasticity theory proposed by Korzen (1988) and Haupt and Lion (1995), where the equilibrium stress is stationary during relaxation.

294

Erhard Krempl

The upward relaxation is observed in corresponding tests on real materials when unloaded from tension; see Figures 14 and 15 of Kujawski et al. (1980). It appears that the model is capable of reproducing anelastic behavior without any additional assumptions.

DISCUSSION A. Relationto Other Models Although there are many subtle differences between the present overstress theory and the "classical overstress theory" of Perzyna (1963), the expressions for the inelastic stress rate are very close. The inelastic strain rate of Eq. (1) can be written as

ein _. _3 F[F] s ~ - g 2 toF

(19)

where to is a time constant and F[F] is an increasing dimensionless function. The relation to Eq. (1) is established by 1

Ek[F]

=

F[r]

(20)

t0F

Equation (19) is very close to the corresponding expression of the Perzyna theory. The difference is the absence of a yield surface and the direction of the inelastic strain rate. It is in the direction of the stress deviator for the Perzyna model. Also, by this manipulation, the "variable" relaxation time k[F] is replaced by the constant to and the dimensionless function F[I-']. A detailed discussion of various overstress approaches can be found in Krempl (1987). Equation (19) can also be used to demonstrate that VBO contains the widely used Norton creep law as a simplification. To this end, we require that the equilibrium stress be zero and that F[F] be a power law. The constitutive equation is then l+v

3

s

6 = ---E-s + ~ Fir]t0 r

(2~)

where F is to be calculated with zero equilibrium stress. Such a constitutive equation would always model relaxation to zero stress, secondary creep only, and, ultimately, horizontal stress-strain curves. The hysteresis loop, without a Bauschinger effect, would close after one cycle. Furthermore, no distinction between rate-dependent and rate-independent contributions to the stress could be made. Such a model would have limited applicability.

Chapter 6 A Small-StrainViscoplasticityTheory Based on Overstress 295 This choice also eliminates the possibility of modeling some unusual creep phenomena, as discussed by Krempl (1987, 1995b). 9 The theory is close to the overstress version of the Chaboche model (Chaboche, 1993). If the yield surface is reduced to zero size, then the flow laws of Chaboche and VBO are identical. Setting qJ[1-'] = B in Eq. (8) and omitting the stress rate term, we obtain

g-b

(2ein _ ~ g )

f

( B )

1-2

f

(22

where b = (B/E)c. In Chaboche's theory, multiple back stresses are used. Now we set g -- fg + and identify the new quantities as state variables of the Chaboche theory. We obtain from Eq. (22) ~--b(~e

in- ~A) (23)

_ f_ 2/~tein _ ~b(~_ f) 3

A

For constant b, the first of Eqs. (23) is the Frederick-Armstrong law. The second is the growth law for the second back stress f, which admits an asymptotic solution { ~ - i'} -- 0. The slope of the second back stress f is ultimately equal to that of the kinematic stress f; and in the asymptotic state, the difference between these two quantities is 2 J~tA { +in } b

(24)

Even with a constant shape function ~p, the present growth laws for the equilibrium and the kinematic stresses are equivalent to three back stresses in the Chaboche theory. The shape function provides additional modeling capabilities in VBO.

B. Properties of the Present Theory 1. General Properties The present overstress theory (VBO) has its origin in the investigation of the nonlinear viscoelastic solid (Cernocky and Krempl, 1979). In this paper, the asymptotic solutions were investigated, and it was concluded that an overstress dependence endowed the model with many desirable properties. At that time, 9Creep rate at the same stress level is found to be larger on loading than on unloading and creep rate need not increase with an increase in stress level (see Krempl, 1987, 1995b, and the papers quoted).

296

Erhard Krempl

the equilibrium stress was a unique function of strain. The model represented a deformation theory of viscoplasticity (see Cernocky and Krempl, 1980). To remedy this shortcoming, the equilibrium stress was made a state variable with an evolution equation (see Krempl et al., 1986). A three-dimensional version appeared in Yao and Krempl (1985). 1~ Nishiguchi et al. (1990) published a finitedeformation version. Majors and Krempl (1994) modified the growth law for the equilibrium stress to allow for an arbitrary ultimate slope of the kinematic stress. The present theory cannot reproduce exactly linear elastic regions since the inelastic strain rate, however small, is always present. That is also a property of the Bodner theory developed previously (Bodner and Partom, 1975). However, the graphs generated by plotting the results of the numerical integration show regions where the stress-strain curve is straight, with a slope corresponding to the relevant elastic modulus. Elastic regions can therefore be modeled with sufficient accuracy. Perfect linear elasticity is, in itself, an idealization. Even at very low stresses, vibrations die out in vacuum due to internal damping, i.e., nonelastic behavior. The absence of exactly elastic regions has the advantage that recovery of strain at zero stress, creep in the linear portion of the stress-strain diagram, and other unusual phenomena can be modeled without difficulty. Also, within VBO there are no loading and unloading conditions and no separate stress or strain space formulations. As explained previously, only the initial and boundary conditions are needed for finding the behavior predicted by the model. In accordance with other "unified" theories, no separate repositories for creep and plasticity are found in VBO. The distinction between plasticity and creep constitutive equations in mechanics is historical in origin. Plasticity and creep theory were developed separately. Technological needs required the combination of the two fields in the late 1950s. They were then implemented in stress analysis computer programs and used in the design of nuclear reactors and jet engines, to name just a few applications Materials science considers plasticity a rate-dependent process and does not use the concept of rate independence. Inelastic deformation is caused by dislocation movements and other changes in the defect structure, and these deformation mechanisms are the physical bases for plasticity and creep alike. On a phenomenological level, the equivalence of creep and "plastic" strains has been demonstrated by suitable experiments by Ikegami and Niitsu (1985).

1~ this paper, the growth law for the equilibrium stress was inconsistently formulated with regard to Poisson's ratio. This inconsistency did not affect the correlations and predictions. A consistent formulation is found in Nishiguchi et al. (1990).

Chapter 6

A Small-Strain Viscoplasticity Theory Based on Overstress

297

2. Termination of Relaxation

In the deformation theory of VBO (see Cernocky and Krempl, 1979, 1980), the equilibrium stress depends only on strain. As a consequence, the equilibrium stress did not change during relaxation and relaxation terminated at the equilibrium stress after infinite time. This property is also shared by the theory of Korzen (see Korzen, 1988; Haupt and Lion, 1995). For VBO this is not the case; the equilibrium stress changes during relaxation. There is some experimental evidence that suggests a change in the equilibrium stress during relaxation. Figure 7a of Majors and Krempl (1994), reproduced here as Fig. 6, shows that the stress at the end of relaxation tests of equal duration depends on the strain rate that existed prior to the start of the relaxation. A smaller stress is obtained at the end of the relaxation test started from the high-strain-rate stress-strain curve than from the relaxation test started from the low-strain-rate curve. In each of the tests the inelastic strain rate is approximately equal at the end of relaxation and implies a constant overstress; see Eq. (18). It follows that the equilibrium stress is different at the end of the two relaxation tests. This fact can, in principle, be modeled with VBO; see Figure 7b of Majors and Krempl (1994). Comparable experimental results are found in polymers (see Figures 4 and 5 of Bordonaro and Krempl, 1992). For room-temperature relaxation tests on alloys, the results are not as clear (see Figure 5 of Krempl, 1979; Figure 5 of Kujawski and Krempl, 1981). The results of Haupt and Lion (1995) are in agreement with a constant equilibrium stress. It appears that a nonunique equilibrium stress is suggested by most of the experiments.

Repeated relaxation of equal duration starting from stress-strain curves with different strain rates. Material: Modified 9Cr-IMo steel at 538~ Note the termination points.

298

Erhard Krempl

3. Influence of Equilibrium Stress Growth Laws In this section, the difference in the predictions of VBO for the differently formulated growth laws of Eq. (6), called the Yao version, and of Eq. (9), called the Lee version, will be illustrated using numerical experiments. For each type of loading, the numerical experiments are performed using Eq. (6) or Eq. (9). No other changes are introduced. Note that the difference in the two versions is only the direction of the dynamic recovery term in the growth law for the equilibrium stress. The material constants are taken from Yao and Krempl (1985) and pertain to the 6061 A1 alloy in T6 heat-treatment condition. Figure 7 shows the behavior predicted in repeated loading and unloading in strain control at a constant strain rate magnitude. Both versions predict the same stress-strain behavior. Figure 8 shows the same graph, but this time the variation of the state variables is also plotted. It is seen that the path of the equilibrium stress closely follows that of the stress. The kinematic stress is small but is responsible for modeling the small work-hardening slope. Krempl and Gleason (1996) demonstrate that no difference exists between the two versions in proportional cycling. Since the isotropic stress is constant, cyclic neutral behavior is modeled under completely reversed strain-controlled conditions. The situation changes when nonproportional loadings are considered. Figure 9 depicts the behavior in 90 ~ out-of-phase cycling. The two growth laws predict a 350 cr Yao I o Lee

300 250 200 150 100 50

0

0.000

0.005

I

0.010

I

0.015

E

0.020

I

0.025

0.030

0.035

Stress-strain behavior during repeated loading and unloading predicted by the Yao and Lee versions of VBO.

Chapter 6 A Small-Strain ViscoplasticityTheory Based on Overstress 299

Same as Figure 7 with the evolution of the state variables. considerably different behavior. Yao and Krempl (1985) show that the Yao version models real experiments very well. Thus for this A1 alloy, the predictions of the Lee version would not be acceptable. To analyze the differences, Eq. (6) and Eq. (9) are specialized for the axialtorsion loading to yield

1/s g~-2~+~

F [s12--g12 (l~r r 2+

gll = ~-Sll-+--7K

(l~r) 1-T

Et)g12-~- -

Sll-gll (~nt_(1 ~ ) Et) r

2

-E-

A

f12 lCr1 e

g11-f11~ r -

A

(25) (26)

for the Yao model. For the Lee version we have

g12

1/s

-- -~S12 -t-

.

,~11 = ~$11 +

S12 -- g12

(27)

k

Sll -- gll

k

Et

(28)

In the above, the argument of the shape function ~ and the viscosity function k has not been written for brevity. Figure 9 shows that the differences originate at a point during unloading where the respective overstress component appears to change sign and the respective stress rate component is negative. Both stress components are present at this point.

300

Erhard Krempl

The predictions of the Yao and Lee models in 90 ~ out-of-phase cyclic loading: (a) Axial response; (b) torsional response.

Therefore, the invariants are not zero. At this point, the second term on the fighthand sides of Eq. (27) or (28) is zero. This is not the case for Eq. (25) or (26), where the term proportional to the respective equilibrium stress minus the kinematic stress component contributes negatively and increases the magnitude of the respective equilibrium stress rate component. At a zero respective overstress component,

Chapter6 A Small-StrainViscoplasticityTheory Based on Overstress 301 a greater magnitude of the respective equilibrium stress rate exists for the Yao version than for the Lee version. This difference continues until the equilibrium stress minus the kinematic stress components change sign, which is close to zero stress; see Fig. 8 as a typical example of the evolution of the kinematic stress. It is seen that the direction of the dynamic recovery term can have a significant effect on the prediction of the model. Burlet and Cailletaud (1987) have a model similar to the Lee version. The influence of the direction of the dynamic recovery term was also investigated by Freed et al. (199 l a, b, c), and it is stated that the two versions predict "vastly different transient responses" (Freed et al., 199 lb, p. 166). Here we have shown that the steady-state response can also be very different. Other instances of different predictions are given by Krempl and Gleason (1996). The examples presented here point to the importance of considering not only uniaxial loadings but also the biaxial situation when a model is to represent material behavior in complex loadings. Although the Yao version was well suited for modeling the behavior of the A1 alloy, the situation may be completely different for another alloy. Only the comparison between prediction and experimental results can give confidence in the validity of a model under complex loadings.

C. Extensionsof the PresentTheory 1. Cyclic Hardening (Softening) The model introduced here represents cyclic neutral behavior since the isotropic (rate-independent) stress is constant. In a uniaxial simulation, the hysteresis loop closes after one cycle in completely reversed strain control. For the modeling of cyclic hardening or softening including extra hardening in out-of-phase loading, additions to the theory are needed. This can be done in the following way. The origin of the hardening or softening phenomenon has to be clarified first. In Fig. 1, three possible contributions to the stress can be identified: the kinematic stress, the rate-independent (isotropic) stress, and the rate-dependent overstress. In general, it is not known which of the three components changes with cyclic loading. Since the slopes of the monotonic and the cyclic stress-strain diagrams are not significantly different, the contributions must come mainly from the other two terms. As a first approximation, it is permissible to leave the tangent modulus /~t unaffected by cyclic loading. To find out which of the other two components is contributing, experimental data or microstructural reasoning is needed. Experiments on stainless steel at room temperature have shown that the hardening is predominantly rate-independent (see Krempl, 1987; Krempl and Lu, 1984; Haupt and Lion, 1995). Consequently, a growth law for the isotropic or rateindependent stress A was formulated to model hardening. The growth law distinguishes proportional loading and nonproportional loading, which causes the extra

302

ErhardKrempl

hardening. To model the extra hardening in nonproportional loading, a term is added to the growth law that vanishes for proportional loading. The latest versions of this very complicated modeling task that is not yet completely solved and that must also be accomplished by conventional theories have recently been published (see Krempl and Choi, 1992; Choi and Krempl, 1993). A review of the methods of modeling cyclic hardening was given by Ohno (1990). Some materials show cyclic hardening followed by softening. Apparently, this feature has not yet been modeled in a consistent way. It is entirely possible that the changes in the cyclic behavior are caused by a change in the rate-dependent behavior, by the rate-independent behavior, or both. It is not known a priori where the changes come from. Only suitable tests can distinguish between the mechanisms. Alternatively, microstructural reasoning may help to decide which component is changing. However, the writer has not seen such a reasoning. If only the rate-dependent properties are affected, a drag stress should be introduced. The schematic in Fig. 1 is useful to illustrate the properties of VBO and the experiments that are used to find the nature of hardening. To this end, we perform tests with the model and with the real alloy. Suppose that a virgin specimen is subjected to a tensile test with strain rate cycling between two values of the strain rate. Then a diagram similar to Fig. 1 results, where the flow stress alternates between two values. II The state variables cannot be seen in a real experiment. A second, identical specimen is now subjected to cyclic loading until saturation occurs. The specimen is unloaded to zero stress and strain. Then the strain rate cycling experiment follows. In a real test, we could observe any of the following: (1) A stress level change for the curve with the high strain rate and a change in the distance of the flow stresses of the two stress-strain curves obtained at different strain rates. (2) A stress level change for the curve with the high strain rate and no change in the distance of the flow stresses of the two stress-strain curves obtained at different rates. (3) No change of stress level of the curve obtained at the high strain rate but a change in the distance of the flow stresses. The first outcome indicates that rate-independent and rate-dependent mechanisms are responsible for the changes. The second is indicative of rate-independent changes, and the third suggests rate-dependent changes only. Figure 3 of Krempl (1987) depicts the case of hardening due to the rateindependent mechanisms or the isotropic stress A. Similar experiments are performed by Krempl and Lu (1984) and Haupt and Lion (1995). !1Figure 3 of Krempl (1987) shows such a test.

Chapter 6 A Small-StrainViscoplasticityTheoryBased on Overstress 303

2. High Homologous Temperature In the discussion of creep behavior, it was mentioned that the present theory permits only the modeling of "cold creep." It can represent the hardening due to inelastic deformation, but has no repository to account for the effects of diffusion that counteract these effects as temperature increases. Essential elements of high homologous temperature deformation behavior can include the following (see Majors and Krempl, 1994): (1) region (2) (3) (4)

Primary, secondary, and tertiary creep at stress levels in the quasi-elastic of the stress-strain diagram Cyclic softening Strain rate history effect Influence of rest periods at zero stress after inelastic deformation

Of these properties, the last one is frequently cited and taken to be important. The others are not as well known, but are crucial for modeling, especially item 1. To model the effect of diffusion, the Bailey-Orowan format is usually adopted. When this is done, Eq. (6) _ ap[F] ( s -~ s - g E k[1-']

F (g-f)) k[1-'] a

+ (1

~ [ F ] ) i ' - R[~,]g E

(29)

~/3gijgij

changes by the addition of the static recovery term R[~,] where ~, is the effective overstress. The function R is positive and increasing andV-reduces the equilibrium stress rate. Note that Eq. (29) is no longer homogeneous of degree 1 in the rates; therefore, time now has an influence on the evolution of g. The recovery term is usually made such that its influence is significant only at long test durations. It does not affect the short-time behavior. The addition of this term increases the modeling capability. It is now possible to model primary and secondary creep at stress levels in the quasi-elastic region of the stress-strain diagram. However, only cyclic neutral behavior can be reproduced. It is not possible to model cyclic softening using Eq. (29). Also, it cannot model the strain rate history effect, and there is no permanent effect of rest time on the subsequent stress-strain diagram (see Majors and Krempl, 1994, for details). To reproduce the other effects listed above, a softening of the isotropic stress A is needed. To this end, a recovery modified inelastic strain path length was introduced by Majors and Krempl (1994). It controls the decrease of the isotropic stress. It is shown that this model now incorporates all the effects listed above. Detailed modeling of uniaxial loading for a modified 9CrlMo steel at 538~ is presented by Majors and Krempl (1994). The same approach with a changed growth law for the isotropic stress was also used to model tensile and creep behavior of alloy 800H at homologous tempera=

304

ErhardKrempl

ture exceeding 0.7 (see Tachibana and Krempl, 1995). It is shown that primary, secondary, and tertiary short-term and long-term creep are modeled together with the stress-strain behavior at regular testing speed with one set of constants. An example is given in Fig. 10 for alloy 800H. Tachibana and Krempl (1995) also consider the transition from the solid to the fluid state as the homologous temperature approaches 1. Upon an increase of the temperature to the melting point, the constitutive equation has to change to that of a fluid. Although data are scarce, it is generally assumed that the molten metal can be represented by a Newtonian fluid. VBO permits this transition (see Tachibana and Krempl, 1995). Theories with a yield surface set the yield limit using data from the tensile test, say the proportional limit. Within this limit, elasticity is the deformation mode according to plasticity theory. However, when creep is considered, substantial inelastic deformation can take place within this limit. Therefore, a discrepancy arises. Plasticity demands elastic behavior, and creep test results indicate significant inelastic behavior at stress levels that are elastic according to plasticity. Theories without a yield surface do not have this conceptual difficulty.

3. Variable Temperature So far, all discussions have concentrated on constant temperature. The extension to variable temperature and isotropy is accomplished as follows. The fight-hand side of Eq. (2) is augmented by the thermal strain rate kkk = ~

E

~rkk + 3 o t T

(30)

where ~r is the temperature rate and c~ is the coefficient of thermal expansion. We allow for temperature-dependent material properties and postulate that the elastic deformation remains path-independent even when the material constants depend on temperature. This postulate is implemented in the first term on the fight-hand side of Eq. (30). In general, Poisson's ratio is not very sensitive to temperature; the elastic modulus normally is. Differentiation of the first term on the right-hand side of Eq. (30) gives rise to a term multiplied by the temperature rate. The deviatoric flow law, Eq. (1), is also rewritten as d (l+v) E s

i~-- dt

3 s-g -f 2 E k [ F ]

(31)

so that the elastic deformation is again path-independent. All material constants in the viscosity function can now also depend on temperature.

Chapter 6

A Small-Strain Viscoplasticity Theory Based on Overstress

5

305

0

4

0

A

o

E

~,

i

~,

L

1-'

o

4~

4

test data temp. str. (~ (MPa) [ ] 950 25 0 950 23 ,/k 950 20 simulation

95O i

0

500

1000

a

T i m e (hr) 150

test data [ ] 750 (~ 0 850 Z& 950

100

i

1500

20OO

simulation ~ s t r e s s . . . . . g

d

o

o

o

o

o

o

o

c

12. U) U~

. . . . . . . . .

IL_

Or)

50

/

I f 1 ~

0

13

i~.

/

//

,

~''-

/ / /// // _-///t "" "-"

,

,

I

0.2

,

,

,

I

0.4

,

,

,

I

,

,

0.6

,

I

0.8

,

,

,

1

Total Strain (%)

The modeling of (a) long-time creep behavior at 950~ and (b) short-time tensile tests at 750, 850, and 950~ at a strain rate of 5.0E - 5 (l/s) for alloy 800H.

306

Erhard Krempl

The only change to the growth law for the equilibrium stress is a change in the ~p[F] eel) + ~r ~ ( 1-i-~)e ap[1-'] el in elastic contribution. We replace the elastic term by (1-~ Eq. (3) and by ( - ~ s ) + ~r a--~(~--~)s in Eq. (6) to ensure path independence of the elastic deformation. The invariant F does not explicitly depend on temperature. The extra terms that are multiplied by the temperature rate have their origin in the assumption that elastic deformation is path-independent. These terms arise naturally in VBO. In other theories, they are introduced to improve the prediction in thermomechanical loading (see Lee and Krempl, 199 l a, and the discussion on thermodynamics later in this paper). We have assumed that all material constants can be temperature-dependent. This does not mean that all have to be dependent on temperature. Indeed, Tachibana and Krempl (1995) have shown that some material constants do not have to depend on temperature in the modeling of creep and tensile behavior of alloy 800H at very high homologous temperature. This method of introducing temperature dependence allows for flexibility because arbitrary functions of temperature can be implemented.

4. Anisotropy After several attempts, the anisotropic VBO was formulated for orthotropy by Lee and Krempl (1991b). It was extended to general anisotropy by Chow (1993). In this case, vector notation of the stress and strain tensors is appropriate and we write for the flow law --

d ~-~(C-1 o') + K -l (tr - g) + c~T

(32)

where C -1 is the positive definite elastic compliance matrix and K -~ - Rk-1Kd 1 is the inelastic compliance matrix, which can be written as the product of the diagonal compliance matrix K d 1 given as 1

gxx

1

Kyy 1

Kd 1 =

k:[1-']

Kzz

(33)

1

Kyz 1

Kzx

1

Kxy

where the Kij are stiffnesses and k[F] is the viscosity function with the dimension of time. The matrix R~-1 contains the inelastic Poisson ratios (for details, see Lee and Krempl, 1991b).

Chapter

6 A Small-StrainViscoplasticity Theory Based on Overstress 307

To have a formulation similar to the one used in the isotropic case, we follow T.-L. Sham (personal communication, 1990) and write R = K -1Kxxk[F]

(34)

so that the flow law now reads d

(o- - g)

k -- _ - = ( c - l o ") + + R ~ F dt

+ c~jr

(35)

where + = F/(Kxxk[F]) is the inelastic strain rate invariant and the overstress invariant F = ~ / ( R ( ~ - g))t(R(o" - g)). A superscript "t" denotes the transpose 9 The last term in Eqs. (32) and (35) is the thermal contribution, with T denoting the temperature rate and c~ the coefficient of thermal expansion vector. It is shown by Lee and Krempl (1991b) that the growth laws for the state variables written in stress formulation can be applied unchanged. In Eqs. (6) and (7), the invariant F defined above has to be used; and in Eq. (9), the invariant | has to be replaced by the anisotropic version given by | = x/(H(cr - g))t(H(o" - g)). The dimensionless matrix H has the appropriate symmetry of the problem and corresponds to that of R. The modeling of inelastic incompressibility rests with the matrix R. Special conditions apply for certain components (see Lee and Krempl, 1991b). Applications of the anisotropic VBO have been made to composites (see Krempl and Hong, 1989; Yeh and Krempl, 1993; Lee and Krempl, 1990).

D. ThermodynamicConsiderations VBO and almost all "unified models" were developed considering isothermal motions only. There is a strong experimental background that supports the development of the theory. With suitable assumptions and additions, the behavior under varying temperatures can also be modeled. This has been done for VBO (see above). Many investigators have adopted the method of Coleman and Gurtin (1967). After making an assumption about the independent variables in the free enthalpy or free energy function, the Coleman-Gurtin formalism is employed to show that the second law or the dissipation inequality is not violated. If a violation is found, the constitutive equations are deemed deficient. Another advantage of the method is to derive the constitutive equation from the thermodynamic formalism. Using this line of reasoning, Chaboche (1991 a, b, Sect. II.B) has stated that "In some theories, reversible changes produce instantaneous changes of the internal variables (in the viscoplasticity theory based on overstress by Yao and Krempl, 1985, for example)." The use of stress rate or elastic strain rate " . . . . is either unacceptable in order to meet the thermodynamic principles or reducible to the

308

ErhardKrempl

classical form . . . . . " Chaboche (1993, before Sect. III.A.2) has a similar statement: "In our theory we do not accept such instantaneous changes for the internal variables." Using a different approach, Malmberg (1993), p. 127) finds "... that the thermoleastic strain-stress relation (Hooke's law) and the evolution equation for the equilibrium stress are in conflict with each other because of thermodynamic reasons: An appropriate Gibbs function does not exist." Two authors find, for different reasons, that the elastic growth of the equilibrium stressmthe term multiplied by the elastic strain rate in Eq. (3) or by the stress rate in Eqs. (5), (6), or (8)mis considered to be responsible for the conflict with thermodynamics. a. Chaboche's Analysis. The elastic growth of the equilibrium stress was introduced by Krempl et al. (1986) to improve the modeling of the quasi-elastic regions; see their Figures 1 and 2, reproduced here as Figs. 1 la and 1 lb. If the stress rate term were to be omitted, then a less realistic modeling of the quasielastic regions would be accomplished, the modeling would be akin to Fig. 1 l a. None of the asymptotic properties are influenced by this term; see the sections on Asymptotic Solutions and Asymptotic Properties. If inelastic deformation is the main object of modeling, this elastic contribution to the growth law can be omitted. The stress rate term in the growth law for the equilibrium stress is specific to VBO. Since the state variables are normally considered repositories of the changing microstructure, they should grow only when the microstructure changes, according to conventional reasoning. This happens when inelastic deformation takes place. Therefore, other unified theories do not include the stress rate term. In VBO, the equilibrium stress has a dual function. It serves the traditional role of a "regular" state variable, but is also a repository for the modeling of quasielastic regions. It is for this reason that the stress rate term was introduced. It was mentioned above that this term has no influence on the asymptotic properties representative of fully established plastic flow. Chaboche ( 1991 a, b, 1993) shows that the inclusion of"instantaneous reversible changes of the internal variables," meaning the stress rate term in the growth law for the equilibrium stress, leads to a nonclassical form of Hooke's law; see his Figure 1. To perform the thermodynamic formalism, the dependence of the free energy or free enthalpy on independent variables must be assumed. But no guidance exists as to which variables should be chosen. Since the thermodynamic formalism is intended to yield the constitutive properties of the material under consideration, the independent variables should reflect these properties. One key element of viscoplasticity is the rate dependence of inelastic deformation. We note that in Eq. (3) of Chaboche (1991a, b) and in Eq. (11) of Chaboche (1993), the free energy depends on the elastic strain, on the state variables, on the temperature, and on the temperature gradient. We consider here only isothermal motions, and therefore the temperature dependence can be omitted. Since the evolution of the state variables is rate-independent in the Chaboche theory, his

Chapter 6

A Small-Strain Viscoplasticity Theory Based on Overstress

309

250 2001

~

ar

100

a

0.0

0.4 '

018

1'.2

STRAIN (%)

1.'6

2.0

250 200

(3"

13.. 150 ILl 100~ n" I--50

k U

0

0.0

0.4

0.8 1.2 STRAIN (%)

1.6

2.0

~ | [ ~ . e l ~ ( m i m Simulation showing the influence of the elastic term in the growth law for the equilibrium stress: (a) Without the elastic growth law term; (b) with the elastic growth law term. From Krempl et al. (1986).

choice of the independent variables in the free energy function does not include a repository for rate-dependent effects! It is therefore not possible to derive a rate-dependent theory using the usual thermodynamic formalism. We consider the VBO theory as described in the main text. Then the evolution of the kinematic stress and of the equilibrium stress are rate-independent 12 as are the state variables in the Chaboche theory. In this respect, both theories are equivalent.

12In the transient region, the evolution of the equilibrium stress can be rate-dependent. The asymptotic state is, however, rate-independent.

31 0

Erhard Krempl

The absence of a repository for rate dependence in the free energy function makes the criticism not applicable to VBO. Also, in Eq. (59) of Chaboche (1993), temperature rate terms are postulated that are not derived from thermodynamics and are contrary to the statement "In our theory we do not accept such instantaneous reversible changes for the internal variables"; see p. 815 of Chaboche (1993). Indeed it is stated by Freed et al. (1991 a, c, p. 534) that "From the perspective of material science [5], the physically correct, internal, state variable has an evolution equation with n o sij or T dependence." Temperature rate terms are used to model real material behavior as demonstrated by numerical experiments in Figure 6 of Chaboche (1993). 13 In VBO, the temperature rate term follows from the path independence of elastic behavior and is not introduced separately. In VBO and in the Chaboche theory, the temperature rate term is needed for realistic modeling under thermomechanical loading. b. Malmberg's Analysis. In his criticism of VBO, Malmberg (1993) starts from a different viewpoint. In the introduction, an analysis of general growth laws for the state variables is performed. For a very general class of growth laws given by Eq. (1.6), it is stated that "Starting off from the state 14 A1 = (Ckl,, T1, otpl)15 and following the same path of the external variables in the (Ckl, T1,)-space in the reverse direction, which is always possible, one reaches the original initial state A0. In this sense the path is reversible but path dependent" (Malmberg, 1993, p. 8). This property is apparently maintained when his analysis is applied to viscoplasticity. It is stated on page 80 of Malmberg (1993) that " . . . . all results of Sect. II.B.4 can be used if the following renaming is done . . . . . " Sect. II.B.4 (on p. 61) deals with the thermodynamic consistency of assumed strain-stress relations that are used later to show the inconsistency of the equilibrium stress growth law of VBO. It appears that the property of reversibility is still maintained in the analysis applied to VBO. Furthermore, it is postulated that the strain-stress relation is of the form Emn = ~mn[O'kl, T, Otp]

(36)

see Eq. (2.172) of Malmberg (1993). It is easy to show that VBO does not share the two properties. Since it models hysteresis, the solutions of VBO do not return to their original state upon reversal of the loading path; see Figure 12. 13In our model, the temperature rate term is introduced by the postulate of path independence of elastic deformation in both the flow law and in the growth law for the equilibrium stress (see Lee and Krempl, 1991a, b). In the Chaboche theory, the temperature rate term is a separate assumption. Figure 6 of Chaboche (1993) and Figure 9 of Lee and Krempl (1991 a) demonstrate the same effect via numerical experiments--the drifting of thermomechanical hysteresis loops in the absence of the temperature rate term. 14That had been reached by the path (Ckl, T). 15C is the right Cauchy-Green tensor, T is the absolute temperature, and C~p are the state variables.

Chapter 6

A Small-Strain Viscoplasticity Theory Based on Overstress

311

Strain

~|[l~ll~t:liiPJ Hysteresis the plasticity and viscoplasticity.

Suppose that the schematic stress-strain diagram of Fig. 12 is created in strain control at constant temperature. The strain reaches a maximum at point B. Subsequently, the strain path is reversed and instead of returning to A, the solution of V B O - - i n this case the stress--reaches point C. At that point, the strain is zero and has returned to the initial value, but the stress has not. The back stress or equilibrium stress shares this property of the stress. VBO does not represent reversible behavior as discussed by Malmberg. In this respect, VBO does not differ from other theories of viscoplasticity. For all of them, the back stress has the properties sketched in Fig. 12. Also, the isotropic variable of most other viscoplasticity theories is nondecreasing and is not reversible. Equation (2.172) of Malmberg (1993), or Eq. (36), does not describe the properties of VBO. For fixed temperature and for a fixed stress and rate-independent evolution of the state variables, as is the case for VBO, 16 only one strain can be obtained from Eq. (36). This is not true for VBO and viscoplasticity in general, where different strains may result for a given stress due to rate dependence; see Fig. 13. Also, the uniqueness of Eq. (36), and hence invertibility, is not guaranteed for non-work-hardening and work-softening materials. Invertibility is frequently invoked as a necessary condition in the analysis of Malmberg. The premises of Malmberg's criticism do not apply to VBO. 16See footnote 12.

312

Erhard Krempl

Fast Slow

Equilibrium

stress

Strain

il[llelSmllll Rate sensitivity in viscoplasticity.

E. The Overstress Concept in Viscoelasticity The flow law given in Eq. (1) shows that the second term, the inelastic strain rate, depends only on the overstress. The drag stress has not been introduced. The reasons have been given by Krempl (1987, 1995b) and are to be found in experiments described above. At low homologous temperature, we expect that a nonzero stress can be maintained after prior inelastic deformation for an indefinite period of time. This motivates the introduction of the equilibrium (back) stress, which is considered a measure of defect density and elasticity of the alloy. The argument presented by Krempl (1995b) shows that, in the absence of rate-dependent hardening or softening, the drag stress is redundant. Consequently, only the back stress was used in the formulation. It is generally not known that the overstress (effective stress) has been used not only in materials science, state variable theories, and viscoplasticity, but also in viscoelastic solids. The standard linear solid (sls) is the simplest linear viscoelastic solid that shows relaxation and creep. It is called a solid because it is capable of maintaining a nonzero stress for infinite time in relaxation motions, i.e., when the strain rate is zero. The mechanical model of the sis is depicted in Fig. 14. From the properties of the springs and the dashpot, the constitutive equation of the sis is usually written in terms of stress cr and strain e as E2

k +--er/

t7

o

E1

r/a

--+-

(37)

where E1 and E2 are the elastic moduli of spring 1 and spring 2, respectively. The quantity ~ is the viscosity coefficient of the dashpot with the dimension of

Chapter 6

A Small-Strain Viscoplasticity Theory Based on Overstress

'vvv Spring Spring 1

313

2

Dashpot

F-

~ll[ql'LIIll,lllln~n Standard linear solid.

stress time, a n d a - - E1/(E1 + E 2 ) is the modulus ratio. A superposed dot denotes time differentiation. Rearranging the terms in Eq. (37) results in the overstress format 6

g - -- + E1

o

-

aEze

_

--

a~7

ke1 + ki n

(38)

The term o- - a E z e is the overstress. When the numerator of the second term is set equal to zero, the equilibrium response, i.e., the response as the rates approach zero, is obtained. In this motion, the dashpot is not activated and the response is that of two springs in series. It is also seen that the inelastic strain rate is proportional to the overstress. Following the methods outlined above, it can be demonstrated that all stressstrain curves for finite strain rate become parallel to the equilibrium response when the asymptotic solution is reached. Then there is a linear relation between the strain rate and the overstress. As a consequence, a tenfold increase of the strain rate increases the overstress tenfold. The infinitely fast response is elastic, with the slope corresponding to the modulus of the spring in series. This behavior is sketched in Fig. 15. For creep, the stress rate is equal to zero and the creep response is obtained from Eq. (38) as k = or0 - a E 2 e at/

__ ki n

(39)

For relaxation, the strain rate equals zero, so that 6

E1

= --

o

-

a Ezeo at/

(40)

The subscript zero indicates that the quantity is kept constant. The creep strain rate and the stress rate in relaxation depend linearly on the overstress. The images of the creep and relaxation curves in the stress-strain plane are horizontal and

314

Erhard Krempl m W W

Fast

Fast strain rate

Slow strain rate

g

~ a E 2 Strain

I|[~|l~'l~ Schematicshowingthe responsesof standardlinearsolidin monotonicloading.

vertical lines, respectively. These motions stop when the equilibrium stress is reached. This will happen after infinite time. The sls has many important properties that are useful for viscoplasticity. However, the linear dependence of the overstress on strain rate is not realistic for metals and alloys, and therefore a nonlinearity was introduced in Eq. (1). Also, a sls can reproduce only primary creep, and therefore nonlinear evolution laws for the equilibrium stress are needed and are introduced in VBO. A systematic description of the "evolution" from the sls to VBO is given by Krempl (1995a). It is now possible to introduce an overstress-dependent viscosity factor and to make the evolution of the equilibrium response nonlinear and dependent on history. If this is done, the viscoplasticity theory based on overstress results. Details are discussed by Krempl (1995a).

E Outlook The overstress concept has many origins, and we have presented the main features of VBO that are exclusively based on the overstress concept. Although not shown here (but see Krempl, 1995b), many results of transient tests seem to be paradoxical. But when the equilibrium stress and a suitable growth law are postulated, the paradoxical results can be easily explained. This capability is used as one justification for using the overstress in VBO. Indeed, we have constructed tests to verify or controvert the existence of the equilibrium stress. No test result is known that would contradict the existence of an equilibrium stress.

Chapter 6 A Small-StrainViscoplasticityTheory Based on Overstress 315 The development of VBO started with nonlinear viscoelasticity, the equivalent of the deformation theory of plasticity for rate dependence. The overstress dependence was introduced on the basis of the asymptotic solutions (see Cernocky and Krempl, 1979). It was then taken over for VBO described here. In VBO, there are no stress or strain space formulations. For homogeneous motions, the boundary conditions need to be specified and the system of nonlinear, nonautonomous differential equations needs to be solved. The boundary conditions can be in terms of stress or in terms of strain. Also, no loading or unloading conditions are needed; the specification of the boundary conditions is sufficient. Arriving at a form similar to the original viscoplasticity theory by an unusual approach had added considerable flexibility, and a much wider variety of phenomena can now be modeled. Included are primary, secondary, and tertiary creep; relaxation; cyclic hardening (softening); high homologous temperature; and thermomechanical loadings. As long as we stay with isotropy, the "flow law" [Eq. (1)] need not be changed. Rate sensitivity is always present. The different phenomena can be accommodated by adding or deleting terms from the growth laws of the state variables. A hierarchical concept applies (see Navayogarajah et al. 1992). An isotropic finite deformation version of VBO has been proposed by Nishiguchi et al. (1990). VBO evolved over the years in a back-and-forth between experiment and theory. The experiment brought unexpected results, and then the theory had to be modified. In turn, the theory suggested certain properties that had to be checked experimentally. Although phenomenological, the present theory considers the underlying microstructural mechanisms. The influences of these mechanisms are "integrated out" in experiments by the specimen, and its response reflects the combined actions of all the active mechanisms. The theory is based on new experiments that showed unusual phenomena. 17 It deviates in some respects from generally accepted notions. VBO has provided acceptable correlations and predictions (see Yao and Krempl, 1985; Krempl and Yao, 1987; Krempl and Ruggles, 1990; Krempl and Choi, 1992; Choi and Krempl, 1993; Majors and Krempl, 1994; Tachibana and Krempl, 1995, for the isotropic case). Application of the anisotropic theory to single-crystal modeling was performed by Choi and Krempl (1989). A forward gradient method for the time integration of the basic VBO was proposed by Sham and Chow (1989) and Chow (1993).

17It started with the observationof rate dependence at low homologoustemperature. This was known as early as the turn of the century (see Ludwik, 1909), but was not the subject of active research of the Mechanics communityuntil recently.

316

Erhard Krempl

ACKNOWLEDGMENTS Most recently, this research was supported by the Department of Energy, Grant No. DE-FG0286ER13566. The paper was partially written while the author held a Senior Humboldt Fellowship in Germany at the Technische Hochschule Darmstadt with Professor E G. Kollmann and at the Universitiit Erlangen with Professor G. Kuhn.

REFERENCES Bodner, S. R., and Partom, Y. (1975). Constitutive for elasticviscoplastic strain-hardening materials. J. Appl. Mech. 42, 385-389. Bordonaro, C. M., and Krempl, E. (1992). The effect of strain rate on the deformation and relaxation behavior of 6/6 Nylon at room temperature. Polym. Eng. Sci. 32, 1066-1072. Burlet, H., and Cailletaud, G. (1987). Modeling of cyclic plasticity in finite element codes. In "Constitutive Laws of Engineering Materials: Theory and Applications" (C. S. Desai, E. Krempl, P. D. Kiousis, and T. Kundu, eds.), pp. 1157-1164. Elsevier, New York. Cernocky, E. P., and Krempl, E. (1979). A non-linear uniaxial integral constitutive equation incorporating rate effects, creep and relaxation. Int. J. Non-Linear Mech. 14, 183-203. Cernocky, E. P., ~nd Krempl, E. (1980). A theory of viscoplasticity based on infinitesimal total strain. Acta Mech. 36, 263-289. Chaboche, J.-L. (199 la). "Thermodynamically Based Viscoplastic Constitutive Equations. Theory Versus Experiment," Tire a Part No. 1991-213. ONERA, Chatillon, France. Chaboche, J.-L. (1991b). In "High Temperature Constitutive Modeling: Theory and Application" (A. D. Freed and K. P. Walker, eds.), MD-Vol. 26/AMD Vol. 121, pp. 207-226. ASME, New York. Chaboche, J.-L. (1993). Cyclic viscoplastic constitutive equations. Part I: Thermodynamically consistent formulations. J. Appl. Mech. 60, 813-821. Choi, S. H., and Krempl, E. (1989). Viscoplasticity theory based on overstress applied to the modeling of cubic single crystals. Eur. J. Mech. A/Solids 8, 219-233. Choi, S. H., and Krempl, E. (1993). Viscoplasticity theory based on overstress: The modeling of biaxial cyclic hardening using irreversible plastic strain. ASTM Spec. Tech. Publ. STP 1191, 259-272. Chow, H. W. (1993). A finite element method for an incremental viscoplasticity theory based on overstress. Ph.D. Thesis, Rensselaer Polytechnic Institute, Troy, NY. Coleman, B. D., and Gurtin, M. E. (1967). Thermodynamics with internal state variables. J. Chem. Phys. 47, 597-612. Freed, A. D., Chaboche, J.-L., and Walker, K. P. (1991 a). On the thermodynamics of stress rate in the evolution of backstress in viscoplasticity. In "Plasticity and its Current Applications" (J. P. Boehler and A. S. Khan, eds.), pp. 531-535. Also NASA TM 103794. Freed, A. D., Chaboche, J.-L., and Walker, K. P. (1991 c). A viscoplasticity theory with thermodynamic considerations. Acta Mech. 90, 155-174. Freed, A. D., Chaboche, J.-L., and Walker, K. P. (1991b). On the thermodynamics of stress rate in the evolution of backstress in viscoplasticity. NASA Tech. Memo. NASA TM 103794, p. 32. Haupt, P., and Lion, A. (1995). Experimental identification and mathematical modeling of viscoplastic material behavior. Continuum Mech. Thermodyn. 7, 73-96. Ikegami, K., and Niitsu, Y. (1985). Effects of creep prestrain on subsequent plastic deformation. Int. J. Plast. 1, 331-345. Korzen, M. (1988). Beschreibung des inelastischen Material-verhaltens in Rahmen der Kontinuumsmechanik: Vorschlag einer Materialgleichung vom viskoelastischen-plastischen Typ. Disser, tation Technische Hochschule Darmstadt, Darmstadt.

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317

Krempl, E. (1979). An experimental study of room-temperature rate sensitivity, creep and relaxation of Type 304 Stainless Steel. J. Mech. Phys. Solids 27, 363-375. Krempl, E. (1987). Models of viscoplasticity. Some comments on equilibrium (back) stress and drag stress. Acta Mech. 69, 25-42. Krempl, E. (1995a). From the standard linear solid to the viscoplasticity theory based on overstress. Proc. ICES 95, Int. Conf. Comput. Eng. Sci. Mauna Laui, Hawaii, (S. N. Atluri, G. Yagawa, and T. A. Cruse, eds.), pp. 1679-1684, Springer-Verlag, Berlin/New York. Krempl, E. (1995b). The overstress dependence of inelastic rate of deformation inferred from transient tests. Mater. Sci. Res. Int. 1, 3-10. Krempl, E., and Choi, S. H. (1992). Viscoplasticity theory based on overstress: The modeling of ratchetting and cyclic hardening of AISI Type 304 Stainless Steel. Nucl. Eng. Des. 133, 401-410. Krempl, E., and Gleason, J. M. (1996). "Isotropic Viscoplasticity Theory Based on Overstress (VBO). Numerical Experiments on the Influence of the Direction of the Dynamic Recovery Term in the Growth Law of the Equilibrium Stress," Int. J. Plast., to appear. Krempl, E., and Hong, B. Z. (1989). A simple laminate theory using the orthotropic viscoplasticity theory based on overstress. Part I: In-plane stress-strain relationships for metal matrix composites. Compos. Sci. Technol. 35, 53-74. Krempl, E., and Lu, H. (1984). The hardening and rate-dependent behavior of fully annealed Type 304 Stainless Steel under biaxial in-phase and out-of-phase strain cycling at room temperature. J. Eng. Mater. Technol. 106, 376-382. Krempl, E., and Ruggles, M. B. (1990). The interaction of cyclic hardening and ratchetting for AISI Type 304 Stainless Steel at room temperature. II. Modeling with the viscoplasticity theory based on overstress. J. Mech. Phys. Solids 38, 587-597. Krempl. E., and Yao, D. (1987). The viscoplasticity theory based on overstress applied to ratchetting and cyclic hardening. In "Low-Cycle Fatigue and Elasto-Plastic Behavior of Materials" (K. T. Rie, ed.), pp. 137-148. Elsevier, New York. Krempl, E., McMahon, J. J., and Yao, D. (1986). Viscoplasticity based on overstress with a differential growth law for the equilibrium stress. Mech. Mater. 5, 35-48. Kujawski, D., and Krempl, E. ( 1981). The rate (time)-dependent behavior of Ti-7 A1-2Cb- 1Ta Titanium Alloy at room temperature under quasi-static monotonic and cyclic loading. J. Appl. Mech. 48, 55-63. Kujawski, D., Kallianpur, V., and Krempl, E. (1980). An experimental study for uniaxial creep, cyclic creep and relaxation of AISI Type 304 Stainless Steel at room temperature. J. Mech. Phys. Solids 28, 129-148. Lee, K.-D. (1989). An orthotropic theory of viscoplasticity based on overstress and its application to laminated metal matrix composites. Ph.D. Thesis, Rensselaer Polytechnic Institute, Troy, NY. Lee, K.-D., and Krempl, E. (1990). Thermomechanical, time-dependent analysis of layered metal matrix composites. ASTM Spec. Tech. Publ. STP 1080, 40-55. Lee, K.-D., and Krempl, E. (1991 a). Uniaxial thermomechanical loading. Numerical experiments using the viscoplasticity theory based on overstress. Eur. J. Mech. A/Solids 10, 173-192. Lee, K.-D., and Krempl, E. (1991b). An orthotropic theory of viscoplasticity based on overstress for thermomechanical deformation. Int. J. Solids Struct. 27, 1445-1459. Ludwik, E (1909). "Elemente der Technologischen Mechanik." Berlin. Majors, E S. (1993). Modeling of rate-dependent deformation of metals and alloys using the viscoplasticity theory based on overstress: Behavior of modified 9Cr-lMo Steel at 538~ and deformation induced anisotropy at finite strains. Ph.D. Thesis, Rensselaer Polytechnic Institute, Troy, NY. Majors, E S., and Krempl, E. (1994). The rate-dependent behavior of modified 9Cr-lmo Steel at 538~ Modeling using the isotropic viscoplasticity theory based on overstress. Mater. Sci. Eng. A186, 23-24.

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Malmberg, T. (1993). Thermodynamic consistency of viscoplastic material models involving external variable rates in the evolution equations for the internal variables," Kernforschungszentrum Karlsruhe Report, KfK 5193, Karlsruhe, Germany, September. Miller, A. K., ed. (1987). "Unified Constitutive Equations for Plastic Deformation and Creep of Engineering Alloys." Elsevier, New York. Navayogarajah, N., Desai, C. S., and Kiousis, P. D. (1992). Hierarchical single surface model for static and cyclic behavior of interfaces. J. Eng. Mech. 118, 990-1011. Nishiguchi, I., Sham, T.-L., and Krempl, E. (1990). A finite deformation theory of viscoplasticity based on overstress. Part I. Constitutive equations. J. Appl. Mech. 112, 548-552. Ohno, N. (1990). Recent topics in constitutive modeling of cyclic plasticity and viscoplasticity. AppL Mech. Rev. 43, 283-295. Perzyna, P. (1963). The constitutive equation for rate-sensitive plastic materials. Q. Appl. Math. 20, 321-332. Sham, T.-L., and Chow, H. W. (1989). A finite element method for an incremental viscoplasticity theory based on overstress. Eur. J. Mech., A/Solids 8, 415-436. Stit~ii, M., and Krempl, E. (1989). A stability analysis of the uniaxial viscoplasticity theory based on overstress. Comput. Mech. 4, 401-408. Tachibana, Y., and Krempl, E. (1995). Modeling of high homologous temperature deformation behavior using the viscoplasticity theory based on overstress (VBO): Part I. Creep and tensile behavior. J. Eng. Mater. Technol. 117, 456--461. Yao, D., and Krempl, E. (1985). Viscoplasticity theory based on overstress: The prediction of monotonic and cyclic proportional and nonproportional loading paths of an aluminum alloy. Int. J. Plast. 1, 259-274. Yeh, N.-M., and Krempl, E. (1993). The influence of cool-down temperature histories on the residual stresses in fibrous metal matrix composites. J. Compos. Mater. 27, 973-995.

Anisotropic and Inhomogeneous Plastic Deformation of Polycrystailine Solids J. Ning*

E. C. Aifantis t

Center for Mechanics of Materials

Aristotle University of Thessaloniki,

and Instabilities

Thessaloniki 54006, Greece

Michigan Technological University Houghton, Michigan

INTRODUCTION Anisotropy and inhomogeneity are two inherent characteristics of the plastic deformation of polycrystalline solids. The anisotropy results mainly from crystallographic texture due to the rotation of grains into preferred orientations as well as from morphological texture due to the nonequiaxial shape, size, and arrangement of highly deformed grains. The inhomogeneity results mainly from inhomogeneous slip within the grains, as well as from constraints imposed for deformation compatibility at grain boundaries. For a comprehensive understanding of these characteristics, it is necessary to study systematically the material properties at three different length scales: (i) the properties of the slip system at the microscale; (ii) the properties of the single crystal at the crystallite scale; and (iii) the overall properties of the polycrystal at the macroscale.

*On leave from the Institute of Applied Mechanics, Southwestern Jiaotong University, People's Republic of China. t Also Center for Mechanics of Materials and Instabilities, Michigan Technological University, Houghton, Michigan, 49931. Unified Constitutive Laws of Plastic Deformation

Copyright (~) 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

319

320

J. Ning and E. C. Aifantis

Crystal plasticity theories (e.g., Taylor, 1938; Bishop and Hill, 1951; Hill, 1965; Kocks, 1970; Asaro, 1983) have been extensively used to describe crystallographic texture and corresponding polycrystal stress-strain curves. While these theories are in general capable of modeling the evolution and spatial distribution of textures, they may not lead to accurate predictions for the stress-strain curves, as they disregard the effect of microstructure at the crystallite scale. The relevant calculations are often complicated because of the large number of grains involved in the averaging procedures used to cast the microscopic flow equations into macroscopic stress-strain relations. To account for the effects of microstructure at the crystalline scale, a self-consistent approach has been developed (e.g., Kroner, 1958; Budiansky and Wu, 1962; Hill, 1965; Hutchinson, 1970; Berveiller and Zaoui, 1979). However, the results of this approach depend strongly on the particular assumptions for the internal stress; therefore, it does not provide a universal description for the plastic deformation of polycrystalline solids. In the search for a more effective technique, a compromise method has been developed in recent years by incorporating the crystal plasticity formalism into a continuum approach (e.g., van der Giessen and van Houtte, 1992; Rashid, 1992; Dafalias, 1993; Prantil et al., 1993). In this approach, an orientation distribution function (ODF) is introduced to model the orientation state of the grains and thus successfully describe the planar plastic flow and yield behavior of polycrystalline solids. However, its applicability is limited to two dimensions (2-D) and the relevant calculations are sensitive to a priori assumptions pertaining to the structure and configuration of the slip systems. Here, we propose an alternative method by incorporating the ODF formalism into the scale invariance approach (Aifantis, 1984, 1987) originally developed for establishing contact between micro- and macroplasticity equations. This theory is based on the configuration of single slip to define the proper kinematic and constitutive equation at the microscale, as well as on a maximization procedure and a scale invariance argument to recast the microscopic plastic relations for the single slip into macroscopic relations for a plastically deformed medium. Various models of phenomenological plasticity are thus obtained for the flow rule and the evolution equation for the back stress, with the extra dividend of providing a corresponding evolution equation for the plastic spin (e.g., Bammann and Aifantis, 1987; Zbib and Aifantis, 1988; Shi et al., 1990; Ning and Aifantis, 1994). However, these studies do not consider the effect of texture at the crystallite scalemthe task undertaken in the present paper. To interpret the inhomogeneity and patterning phenomena routinely observed in plastic deformation of polycrystalline solids, higher-order gradients in dislocation densities (microplasticity) and strain (macroplasticity) have been introduced into the standard equations of dislocation dynamics and plastic flow (Aifantis, 1984, 1987). These gradients provide an internal length scale, as well as a stabilizing mechanism for the goveming differential equations in the material softening regime, where deformation instabilities occur and the associated spatiotemporal

Chapter 7 Anisotropicand InhomogeneousPlastic Deformationof Polycrystalline Solids 321 deformation patterns take place. Among other things, shear band thicknesses and stress-strain curves in the softening regime can be obtained within this approach (e.g., Zbib and Aifantis, 1989; Muhlhaus and Aifantis, 1991; Vardoulakis and Aifantis, 1991; Oka et al., 1992). Other researchers have also resorted to higherorder strain gradient descriptions, such as Coleman and Hodgdon (1985) and more recently Fleck et al. (1994). Various aspects of the strain gradient approach will be discussed in this paper with emphasis on the physical interpretation of the gradient coefficients, as well as the comparison between the various strain gradient theories proposed. The purpose of this chapter is to provide a formalism for conveniently describing the anisotropy and inhomogeneity characteristics associated with the plastic deformation of polycrystalline solids. In Section II, the constitutive relations of a single crystal are developed based on the scale invariance approach at the micro and crystallite scales. In Section III, the orientation distribution function (ODF) and its general solution are introduced to describe the orientation state of the grains. In Section IV, an ODF-based texture tensor is introduced into the averaging procedure to include the effect of texture on the overall plastic properties of the polycrystal. This effect is then discussed in detail for different deformation modes in Section V. Finally, in Section VI, various aspects of the gradient approach are elaborated upon. In particular, estimates for the gradient coefficient are provided by employing the self-consistent approach. The capability of the gradient approach to address size effects is also illustrated. The material of this chapter dealing with the incorporation of texture effects into the scale invariance approach and the interpretation of the gradient coefficients, including size effects, has briefly been reviewed in two recent articles by Aifantis (1995a, b).

CONSTITUTIVE RELATIONSFOR A SINGLE CRYSTALLITE When a single crystallite undergoes plastic deformation, there is at least one slip system being activated. Following the scale invariance approach, one starts from the microscopic configuration of a single-slip system defined by two unit vectors (n, u), where n denotes the normal to the slip plane and u is in the slip direction. In this system, the following basic equations hold for the plastic strain rate D p and the dislocation or back (internal) stress T D (Aifantis, 1984, 1987): D p -- ~,PM, T D = tmM + tnN,

T D = S - T L,

(1) (2)

where S is the Cauchy stress and T D is the lattice or effective stress. The parameters tm and tn are functions of the plastic strain history, commonly expressed in terms of the equivalent plastic strain y p (97p = ~ / 2 D P 9D P ) . [For the case to be elaborated

322

J. Ning and E. C. Aifantis

upon here, we will assume that tm and tn are zero (S = T L) because only the crystallographic texture is being considered.] The orientation tensors M and N are defined by M = ~ I (n | r, + v | n),

N = n @ n.

(3)

The lattice of the crystallite rotates, with the rate of the slip system defined by the antisymmetric tensor w given by r

= W-

WP;

W p = '~P~"~,

(4)

where W is the spin of the continuum or vorticity and the orientation tensor f~ is given by 1 (Is | n - n |

v).

(5)

What is still undetermined is the counterpart of the flow rule (1) at the crystallite scale and the appropriate yield condition associated with it. This was accomplished by adopting a microscopic yield conditions of the form r = tr(SM) = K ( y P ) ,

(6)

which is directly motivated by Schmid's law for crystallographic slip. To identify the representative direction of the orientation M, a maximization procedure (Aifantis, 1984, 1987) is developed by maximizing the corresponding microscopic plastic work rate w p = tr(SD p)

(7)

subject to the constraints tr M = 0 and tr M e = 1/2 implied by the definition of M. Such a maximization procedure yields the plastic flow and yield condition for a representative single crystallite by the relations D p ~--

f'----~PS';

2,/7

~

-- K ( y P ) .

(8)

where the deviatoric Cauchy stress S' = S - l(tr S)I. If there is no texture (the orientation state of grains is isotropic), the above relations reduce to the conventional J2 flow model of isotropic hardening polycrystal plasticity.

TEXTURE EFFECTSAND THE ORIENTATION DISTRIBUTION FUNCTION In order to cast the above constitutive relations of the single crystallite into an overall stress-strain relation for the polycrystalline aggregate, the effect of grain orientation must be taken into account. The orientation of a crystallite may be specified by three Euler angles (Bunge, 1972), which can be described by a texture vector a with unit length in a Cartesian coordinate system. Denoting by cp the angle between a and the slip direction v, and by b the unit vector normal to a, the

Chapter 7

Anisotropic and Inhomogeneous Plastic Deformation of Polycrystalline Solids

323

orientation variables in Eqs. (3) and (5) read M - ~ l [ s i n 2r

@ b - a | a) + cos 2~0(b | a + a | b)]"

(9) (10)

a(a|174 Upon combination of these two equations, we have

(11)

f~ = sec 299[M(b | b) - (b | b)M]. Thus, the corresponding plastic spin WP of the crystallite reads W p = ~,(AD p -- DPA),

(12)

where )~ = sec 2~0 and A = a | a. For a polycrystalline aggregate, the orientation state of the grains can be described by an orientation distribution function (ODF) O(a, t) (Bunge, 1972; Clement, 1982). This function presents the probability that a grain is oriented along a at time t, and is the solution of a Fokker-Planck equation with a diffusion coefficient of zero (Altan et al., 1989) O--7-

(13)

§ -=--(bi~) -- O, oa/

where ai are the components of the unit vector a. Because all possible orientations constitute a unit sphere, the ODF 7t(a, t) has the following properties: ~p(0, 4~; t) -- 7t(zr - 4~, 0 + 7r; t);

/

7t(a, t) da

1,

(14)

where the integral is defined by

J' ( ) da - f2~ f~( JO

) sin ~ d ~ dO.

(15)

JO

It is noted that, for planar plastic flow, Eqs. (13)-(14)1 reduce to the widely used (e.g., Clement, 1982; van der Giessen and van Houtte, 1992; Rashid, 1992; Dafalias, 1993; Prantil et al., 1993) equations

0~r _-- ___0 ( 0 ~ ) , Ot

~ ( 0 , t) -- ~p(0 + re, t).

(16)

O0

If the orientation state of the grains is initially random, then the initial condition is specified from condition (14)2 as

1 (0, ~; 0) = 47r'

(17)

for the 3-D plastic flow case, and as

1 ~p(0, 0) = --, 7t"

for the 2-D planar plastic flow case.

(18)

324

J. Ning and E. C. Aifantis

According to these initial conditions, the solution of Eq. (13) is found to be of the form (Dinh and Armstrong, 1984) 1

~(a, t) = ~-~-[GTG 9A]

-3/2

(19)

for the 3-D plastic flow case, and 1

~(a, t) = --[GTG. A] Jr

-1

(20)

for the 2-D planar plastic flow case. The tensor G is defined by G = 0X/0x, where the vectors X and x denote the initial and current coordinates of a material point in the aggregate and the superscript T denotes the transpose of a tensor. For any planar steady flow, the velocity gradient tensor L reads L =

[k

k2

-k

kl]

'

(21)

and then it turns out from Eq. (20) that 1

~p(O,t) = --[1 - y sin20 + y2sin20] -1, Jr

(22)

for simple shear k = k2 = 0, k l - - y / t ) , while ~p(0, t) = --l [e-2Dt sin20 q_ e2Dtcos20] 1, 7/"

(23)

for rolling (k = - D , kl = k2 = 0). It is noted that Eqs. (22) and (23) can also be obtained from the single-slip model (van der Giessen and van Houtte, 1992) and the double-slip model (Rashid, 1992; Dafalias, 1993) when the two slip systems coincide with each other; however, this requires a much more complex derivation not based on the simple form of the general solution given by Eq. (19).

TEXTURETENSORAND AVERAGEPROCEDURES Although the ODF ~ (a, t) is able to describe the orientation state of the grains, it does not account for the material symmetry and anisotropic properties reflected in constitutive equations. This could be accomplished, in principle, through the introduction of the so-called orientation tensors

Bij = .~ aiaj l p ( a , t) da,

(24)

i

Cijkl -- ~ aiajakallP(a, t) da. J

(25)

Chapter 7 Anisotropicand InhomogeneousPlastic Deformationof Polycrystalline Solids 325 The second- and fourth-order tensors represent, respectively, the second and fourth moments of ODF ~p. They are invariant under orthogonal transformations and they can be used for the description of both 2-D and 3-D deformation. In deriving appropriate constitutive relations for a polycrystalline aggregate from the properties of a single crystallite, Taylor's assumption and a volume or orientation average procedure (see, e.g., Kroner, 1961" Asaro and Needleman, 1985; Rashid, 1992; Dafalias, 1993) are commonly used. This type of average does not consider the effect of morphological texture, especially the fact that the orientations of large grains have a more pronounced effect than those of small grains. In order to take this effect into account, a fourth-order tensor (texture tensor) with the symmetric properties (26)

Kijkl -- Kjikl -- Kijlk -- Kklij,

is introduced as a weighting function in the average expression for the plastic stretching. Because the tensor should be transversely isotropic on this texture vector a, it thus reads

Kijkl -- klaiajakal + k2(aiaj~kl + akal~ij) --t- k3(aiak6jl + aial6jk -]-ajakSil + ajal~ik) + k4Sij~kl -]--k5(~ikSjl +

~il~jk),

(27)

where ~ij is the Kronecker delta. The coefficients ki (i -- 1. . . . . 5) are material parameters, which may depend on the plastic strain history, temperature, and the grain size and shape to account for the effect of morphological texture. In the following analysis, they are assumed to be constant for simplicity. For an incompressible material, they are subjected to the constraints kl + 3k2 -+-4k3 -- 0,

(28)

3k2 + 3k4 + 2k5 -- 0,

(29)

kl -k- 2k2 q-4k3 = 0,

(30)

k2 nt- 2k4 -k- 2k5 = 0,

(31)

for the 3-D plastic flow, and

for 2-D plastic flow, in order to conform with the deviatoric property

Kijkk = 0

(32)

implied by the incompressibility condition. Thus, only three independent material parameters are involved. The overall plastic flow rule is then obtained by the "averaging" requirement g KS ' ~P(a, t) da. f KDPTt(a, t) da - / 2~/-] p

(33)

326

J. Ning and E. C. Aifantis

According to Taylor's assumption (D p = IS)P), the above procedure gives I7)P ---

PP

2~/_)_ (K)- 1S',

(34)

where the superscript - 1 denotes the inverse of a tensor. The overall Cauchy stress deviator S' and texture tensor (K) are defined as

S' - J KS'~p(a, t) da;

(35)

(K) = f K~p(a, t) da.

From Eqs. (24)-(27), the tensor (K) reads

(K)ijkl = klCijkl -+- k2(Bijr -'l- r Bkl) -~- k3(Bik~jl -'1- Bilr + Bjlr + ka3ij~kl + k5(~ik~jl + ~il~jk).

-'t- Bjkr (36)

It is noted that when kl -- k2 -- 0 and Ks -- 1/2, the above average procedure reduces to the one used in the previous works (e.g., Kroner, 1961; Asaro and Needleman, 1985; Rashid, 1992; Dafalias, 1993). In view of the definition for the effective strain and the microscopic yield condition (6) as well as the scalar invariance argument, the overall yield condition reads = {~l S'(K)-I.

(K)-l S'} 1/2 =

K(yP).

(37)

It is noted that Eq. (37) is similar to the previous phenomenological description for the yield behavior of metallic and granular material in using the concepts of the "fabric" and "transformed" or "modified" stress tensors (Oda et al., 1982; Satake, 1983; Mehrabadi and Nemat-Nasser, 1983; Boehler, 1987; Tobita, 1988; Karafillis and Boyce, 1993). Therefore, Eq. (37) may be regarded as a micromechanical justification of these phenomenological models. Because the plastic spin is essentially connected with the crystallographic texture, the overall plastic spin ~A/'Pis obtained by the following average procedure:

/

W p --

WP~(a,

t) d a ,

(38)

which, together with Eq. (12), gives W P = ) ~ [ n o p - I)PB].

(39)

Equation (39) indicates that the overall plastic spin results from the noncoaxiality between the second-order orientation tensor and the plastic strain rate. This expression for the overall plastic spin is also obtained by Prantil et al. (1993).

Chapter 7

Anisotropic and Inhomogeneous Plastic Deformation of Polycrystalline Solids

327

TEXTURE EFFECTON THE PLASTIC FLOW AND YIELD In this section, the present approach is used to describe the texture effect on the plastic flow and yield behavior of polycrystalline solids for 2-D and 3-D deformations. In the following analysis, the elastic strain is neglected due to the finite deformation involved (D ~, I)P, ~ ~ ))P).

A. SimpleShear The strain rate and the vorticity read

o,[0,1 -2- 1

w [011

0 '

2- - 1

0 '

(4o)

and the corresponding ODF is given by Eq. (22). The flow rule (34) gives 6;1 -- 2{klCl112 -+- (k2 -+- 2k3)B12},

(41)

O'22 -'- 2{klC2212 -+- (k2 -+- 2k3)B12},

(42)

612 -- 2{klC1212 + k3 + ks},

(43)

where o'ij -- Sij/K. Without loss of generality, the isotropic hardening parameter tc is assumed to be constant for simplicity. According to Eqs. (28)-(29), it is easy to prove that (44)

Cl112 -- B12 - C2212,

implying that -' 0-11

--

-' n 0-22 .

(45)

Equation (45) is a basic relation for planar simple shear deformation, such as that occurring in a thin-walled tube under fixed end torsion (e.g., Ning and Aifantis, 1994). In the following calculation, we assume that #33 -- 0; therefore, 0-11-' ---611 -- --0"22

--0"22.

The corresponding plastic flow behavior is shown in Fig. 1. The results point out that the evolution of texture in simple shear leads to a nonmonotonic shear stress-strain response and brings about the development of axial stresses. The yield surface is described by the following relation: (h 2 + h2)6"21 + 2 ( h l h 2 -+- h2h3)(~lla12 -+- (h 2 + h~)672 -- 4, where h i -- ( K ) 1 2 1 2 / A ; h3-

(K)llll/A;

h2---(K)lll2/A,

A-

(K)llll ( K ) 1 2 1 2 -

(K)~l12.

(46)

328

J. Ning and E. C. Aifantis 1.5

I

'

I

'

1

'

I

,

I

ff'12

1.o

NO 0.5 11

Z

o.o

ff'22 -0.5

,

0

I

,

l

I

,

2

3

I

4

Shear strain Plastic flow behavior in simple shear.

It is noted that Eq. (46) is similar to the following widely used yield condition (Hill, 1950) for transversely isotropic materials: [(5G + F) cos 2 2 ~ + 2M sin 2 2~]~21 + (5G + f - 2M) sin 4~611c312 + [(5G + F) sin 2 2 ~ + 2M cos 2 2~]~22 -- 1,

(47)

where the parameters G, F, and M are material constants, and 9 is the angle between the axis of transverse isotropy and the coordinate x l. However, Eq. (47) is difficult to use, especially when the orientation of the transverse axis is not known. Figure 2 illustrates how the yield surface rotates and distorts with the increase of shear strain. Corresponding to the perfectly random distribution of the grains, the initial yield surface is a circle. However, the yield surface will rotate and distort with the evolution of texture. In the calculation, the values of the material parameters were taken as kl = 0.8, k2 = 0.2, and k5 -- 0.7.

B. Rolling The strain rate and vorticity read

o i -o0 D01 '

W = 0,

(48)

where D is a positive constant and the corresponding ODF is described by Eq. (23). The flow rule (34) gives o'11 - - 2 k l C l 1 2 2 + k2 -

2k5,

(49)

~22 = - 2 k l C l 1 2 2

-k2

-+- 2k5,

(50)

612 = k1(C1222 -

Cl112) = 0.

(51)

Chapter

7

Anisotropic and Inhomogeneous Plastic Deformation of Polycrystalline Solids

329

Y i e l d s u r f a c e s in s i m p l e s h e a r at d i f f e r e n t s h e a r strains.

Figure 3 presents the stress-strain response of a polycrystalline solid with a perfectly random distribution of the grains. The values of the material parameters were taken as kl = 0.8, k2 = 0.2, and k5 = 0.8 in the calculation. It is clearly seen that the texture may also result in deformation hardening. It is also noted that the shear stress 812 = 0 during rolling. In the analysis of the double-slip model (Rashid, 1992; Dafalias, 1993), a nontrivial solution for the shear stress is o b t a i n e d m a fact rather unexpected for an initially isotropic material.

1.0

9

,

'

I

0.6 _ _ . . . . .

'

I

'

I

'

I

,

I

'

I

,

0"22

0.2 t~ t~

-0.2 m

0

Z

0-11 -0.6 -

-1.0

0.0

'

I

0.5

,

I

1.0

,

I 1.5

,

2.0

2.5

Tension strain (D't) Plastic flow b e h a v i o r o f the m a t e r i a l in rolling.

3.0

330

J. N i n g a n d E. C. A i f a n t i s

C. Three-Dimensional Plastic Flow 1. Uniaxial Tension

The strain rate, vorticity, and ODF are given by the following relations:

-~/2

1

6=

r

0

o 0

-~ 0

0] 0 -~/2

1 qg, g) = ~-~-[e~ sin 2 goCOS 2 0

W=O,

,

(52)

e -2~ sin 2 r sin 2 0 + e ~ COS 2 qg]-3/2.

-+-

(53)

The flow rule (34) gives 2 O'ij = "---'~{ (g)ijll vJ

-- 0.5(g)ij22

(54)

- 0.5(g)ij33},

which, together with Eqs. (27) and (33), implies -' - 0"22

--

2~

1

=

- - 2 o '-' 33 '

-' -- 0 O'ij

and

(i # j).

(55)

Figure 4 depicts the plastic flow behavior of the material under uniaxial tension where the values of the material parameters were taken as kl = 0.8, k2 = 0.2, and k5 - 0.8 in the calculation. The results indicate that the development of texture will first lead to softening, followed by hardening after the orientations of the majority of grains are aligned in the preferred direction.

2.0

I

'

I

"

1

"

1

1.5

0"22

0

1.0 c~

o

N .w-i

o

z

0.5

0.0

Orll

-0.5

-1.0

, 0

I

1

,

I

2

,

1

3

,

1" . . . . .

4

Tension strain P l a s t i c f l o w b e h a v i o r in u n i a x i a l t e n s i o n .

",-. . . . . . 5

Chapter 7 Anisotropic and Inhomogeneous Plastic Deformation of Polycrystalline Solids 2.

Tension-Torsion

Deformation

In this case, the strain rate and vorticity are given by I)-- 2

1~ 0

2~ 0

0

;

W--

~

[010]

-~

-1 0

0 0

0 0

.

331

(56)

The corresponding O D F is then found to be of the form (0, q), t) -- ~ 1 [ e ~ sin 2 q9 COS 2

-k-

4932

~--(e

0 nt-

2~7 eU 2 (e_~ _ eU2) sin2 99 sin 20

-~ -- e ~/2) nt- e

-2~)

-3/2 sin 2 q) sin 2 0 + e ~ COS2 qg] (57)

The flow rule (34) gives 2 ~ij -- ~--p{kl

(Cij22 --0.5Cijll -0.5Cij33)~ -Jr-klCijl2~ -Jl-k2(Bklbkl)(~ij

+ 2k3(BikDkj + DikBkj) + 2k5Dij}.

(58)

The yield surfaces at different axial strains for a fixed strain rate ratio r / - - p / ~ 2 are shown in Fig. 5. In the calculation, the values of the material parameters were taken as kl -- 0.8, k2 -- 0.2, and k5 -- 0.6. The results show clearly that the yield surface will distort and rotate with texture evolution.

Yield surfaces in tension-torsion deformation.

332

J. Ning and E. C. Aifantis

In conclusion, it has been demonstrated that the evolution of texture depends strongly on the deformation modes; and, in turn, the texture results into different plastic flow and yield behavior for the polycrystalline solid under different loading conditions.

INHOMOGENEOUS PLASTICDEFORMATION For finite deformations, the effect of material inhomogeneity becomes pronounced and has a dominant influence on the constitutive and instability behavior of polycrystalline solids. In order to account for this effect, a gradient theory of plasticity was developed by Aifantis (1984, 1987, 1992, 1994a, b) and co-workers (e.g., Zbib and Aifantis, 1989; Muhlhaus and Aifantis, 1991; Vardoulakis and Aifantis, 1991; Oka et al., 1992) by introducing a gradient-dependent expression for the flow stress. The simplest form reads (59)

r - - K ( y ) -- c V 2 ~ ,

where (r, y) are the equivalent stress and strain invariants, x ( y ) is the usual homogeneous part of the flow stress, and the gradient coefficient c measures the heterogeneous or nonlocal character of the hardening/softening mechanisms. Various generalizations of Eq. (59) are possible so as to include rate and temperature dependence for example, by allowing, the homogeneous part of the flow stress x to depend on ~, and 0; i.e., (60)

r = K ( y , ~', 0 ) -- c V 2 ~ .

Another generalization would be possible by allowing the inhomogeneous part of the flow stress to include strain gradient terms up to second degree as well; i.e., r ~- K ( y , ~', 0 ) -- c V 2 y

-- c V ) /

9V ~ .

(61)

In certain situations, one may also include a dependence o n V4y if the inclusion of second-order terms alone does not ensure the desired stability properties in the softening regime. In this connection, one could also consider including higherorder gradients of the equivalent strain rate ~,, as well. For a comprehensive understanding of the nature of the gradient coefficient c, certain microscopically based arguments have already been advanced for its theoretical estimation. These estimates depend on the scale of interest and the particular deformation mechanisms involved. For granular materials, a Taylor series expansion for the average strain has led (Muhlhaus and Aifantis, 1991; Vardoulakis and Aifantis, 1989, 1991) to the conclusion that the coefficient c scales with the grain size R via a relationship of the form c - - C R 2,

(62)

Chapter 7

Anisotropic and Inhomogeneous Plastic Deformation of Polycrystalline Solids

333

where the "scaling" coefficient C is proportional to the material plastic modulus and a geometric factor (1/8 for 2-D and 1/10 for 3-D). The Taylor series expansion argument for the average strain was recently utilized by Zbib (1994) within self-consistent framework for particulate composites to obtain a relationship like Eq. (62), but with C proportional to the matrix shear modulus. The self-consistent approach can also be used, in principle, to obtain relations of the form of Eq. (62) for a variety of continuously distributed microcracks or aligned fibers.

A. Self-ConsistentApproach and Its Relation to the Gradient Approach Let us assume that the local and overall constitutive equations are respectively given by the expressions Y = f(r);

~ = F(2),

(63)

where the relations between the local quantities (y, T) and the overall quantities (~7, 7)are (Kroner, 1958, 1961) F(~) -- V

f ( r ) dV;

r -- ~ + tint,

(64)

where V denotes the volume of the polycrystalline aggregate. The internal stress Tint is assumed to be of the form Tint ~--- - - 0 / # ( y -- ~ ) ,

(65)

where c~ is a material parameter and # is the shear modulus. For a polycrystalline aggregate, the internal stress actually arises from the deformation incompatibilities between grains. With a correlation length equal to the grain size d, Eq. (65) gives Tint ~'~ O//~d 2 V 2)/.

(66)

By using Eshelby's approach and approximating the grains by spheres, it is possible to show that a relationship of the form of Eq. (66) can also be derived via a more detailed argument based on the self-consistent approach. In fact, the self-consistent approach for simple shear gives the following result: r = f" - / 3 A y,

(67)

where Ay = y -- 97,/~ = or#(1 -- 2S1212), and $1212 denotes the appropriate component of the Eshelby tensor. Different values for c~corresponding to different assumptions for the internal stress are given in Table 1. If the internal stress Tintfollows an elastic-plastic interaction law, an incremental form of the self-consistent approach should be used. Then the parameter ot turns

334

J. Ning and E. C. Aifantis il~fiH., Different Values for c~ Model

Assumption

ot

Taylor Sachs Kroner-Budiansky-Wu

y = ? and "tint :)/: 0 "t -- ~ and "tint --- 0 "tint based on an elastic interaction between the matrix and inclusion

or 0

"tint based on elastic incompatibilities

Lin

1 2 (1 - 2S1212)

out to be (Hill, 1965; Hutchinson, 1970) a function of the elastic-plastic tangential modulus of the polycrystalline aggregate c~ =

h(7 - 5v') [ 6 / z ( 4 - 5v') + 15h(1 - v')](1 - 2 S 1 2 1 2 ) '

(68)

with

v

' =

vh + #(1 + v) 9 h + 2/z(1 + v ) '

d~? h = ~ d~p'

(69)

or a function of the secant modulus of the polycrystalline aggregate (Berveiller and Zaoui, 1979) 1 ot = 1 + 3 1 z / 2 H '

H -- 6e/g p.

(70)

By using the definition of average strain ~)P - - ~1-

J~v y P ( x -~- r) d V

(71)

and a Taylor series expansion y P ( x -~- r) -

1 yP (x) + V y p 9r + ~ V (2) ~P 9r | r + . . . ,

(72)

we have ~)P ~

y P -+- -R' ~2 V 2 y p "

(73)

It then turns out that the gradient coefficient c has the form g 2 C

(74)

Chapter 7

Anisotropic and Inhomogeneous Plastic Deformation of Polycrystalline Solids

335

ABLE

Values for the Gradient Coefficient c for 10 # m < R < 100/zm Material

/z (GPa)

v

Icl (N) (KBW)

Icl (N) (Lin)

Structural steel Copper and its alloys Aluminum and its alloys

70-80 40-60 26-30

0.33 0.33-0.35 0.31-0.34

0.373-37.26 0.213-32.31 0.137-16.06

0.7-80 0.4-60 0.26-30

where R denotes an internal length scale defining the radius of the spherical elementary volume of the polycrystalline aggregate. It should be of the same order of magnitude as the maximum size of the material inhomogeneity. The parameter /3 and h are given by the relations 7 - 5v fl = ot/z 15(1 - v)

d~ '

h --

d))P

.

(75)

It is noted that when h >>/3, Eq. (74) yields the same result as that given by Muhlhaus and Aifantis (1991); and when h << fl, Eq. (74) gives a result similar to that obtained by Zbib (1994).

B. Estimates for the Gradient Coefficient c In the case of h <
(76)

and recall Considere's criterion that the strain at the maximum load equals the strain hardening exponent g = n.

(77)

For the aluminum alloy All 100 the mechanical properties at yield and maximum load are listed in Table 3. The corresponding values of the gradient coefficient estimated from different self-consistent models are shown in Table 4. Table 4 shows that the plastic modulus h has little influence on the value of c as predicted by the K , B - W (Kroner, 1958, 1961; Budiansky and Wu, 1962) and Lin (1954) models. This is due to the fact that these models are based on the assumption that the internal stress from elastic interactions between grains; thus, plasticity effects are not accounted for properly, as the elastic shear modulus is much larger than h. In contrast, h has a strong influence on the value of c predicted

336

J. Ning and E. C. Aifantis ABLE ~!

Mechanical Properties for Aluminum Alloy A11100 g

n

K (MPa)

h (GPa)

H (GPa)

# (GPa)

0.002 0.242

0.242 0.242

146 146

5.1 0.035

21 0.428

26 26

by the B - Z (Berveiller and Zaoui, 1979) and H - H (Hill, 1967; Hutchinson, 1970) models. It is also noted that the value of c given by the H - H model at e -- 0.242 is almost equal to zero, as this model implies different signs of the gradient coefficient for the hardening and softening regimes. For a rigid-plastic polycrystalline solid, in particular, this model gives R2 c = -1.5h~. 10

(78)

It is noted that in the analysis of Sluys and de Borst (1994) of the dispersive properties of gradient-dependent and rate-dependent media, the values of the gradient coefficient were taken as c = 50,000 (N), for h = - 2 (GPa), and c = 100,000 (N) for h = - 4 (GPa) respectively. According to Eq. (78), these values of c correspond to very large internal characteristic lengths; i.e, R = 1.291 (cm) and R -- 1.826 (cm), respectively. The magnitude of the internal characteristic length R should be established either by experiments determining the maximum size of material inhomogeneities or by micromechanics analyses through characterizing the evolution of microstructure. For example, if we assume that all grains are spheres of the same volume and properties and neglect the effect of texture, we have

"YP--Ylj~vYPdV-"Nvgl~uf•

(79)

where N denotes the number of the grains and 13g denotes the volume of an individual grain. Equation (79) suggests that the internal characteristic length R may be taken as R = where d denotes the grain size. It should be pointed out that,

d/2,

ABLE

Values of the Gradient Coefficient Icl for 10/zm < R < 100/zm K-B-W 0.002 0.242

0.188-18.791 0.137-13.735

Lin 0.31-31.10 0.26-26.04

B-Z 0.08-8.00 0.03-3.08

H-H 0.072-7.235 0.0005-0.053

Chapter 7

Anisotropic and Inhomogeneous Plastic Deformation of Polycrystalline Solids

337

although the self-consistent approach cannot give an unambiguous and unique value for the gradient coefficient c, it provides a good estimate for its magnitude and its dependence on other physical parameters, which can, in effect, serve as a guide for determining the value of this coefficient by experiments. In concluding this section, we provide a brief discussion for the gradient coefficient based on dislocation arguments. The deforming material is, in general, regarded as composed of two phases: a soft phase and a hard phase. At the early stage of plastic deformation, dislocations are able to move easily through the soft region, where the obstacle spacing is large, and tend to collect in the hard region, where the dislocation density is higher (Kocks, 1975; Nix et al., 1980). As a consequence, the hard region becomes harder and the substructure becomes more heterogeneous. In the soft phase, the average stress ~'s available to drive dislocations is less than the applied stress ~'. It may be obtained by subtracting the long-range back stress 38 from the applied stress. ~

-

~ -

rB,

(80)

which corresponds to the local relation (Li, 1968) "tTs - - T + Tint.

(81)

In the hard phase, the average stress available to drive dislocations should be larger than the applied stress; thus, the appropriate local relation in the hard phase is of the form (82)

Th - - T -- Tint,

Upon combination of Eq. (81) and Eq. (82), the following relation can be assumed for the resultant stress of the polycrystal: r = 2 ( f s -- fh)Z'int = 2 -+-Ot/Z d 2 ( f s -

f h ) V 2 y p,

(83)

where fs and fh are the volume fractions of the soft and hard phases, respectively (fs + fn = 1). Equation (83) suggests that the value of the gradient coefficient is negative in the strain hardening regime and positive in the strain softening regime.

C. Size Effects in Plastic Deformation of Polycrystalline Solids Experiments have shown the existence of size effects in plastic deformation of polycrystalline solids. For example, it has recently been observed in torsion tests that specimens with smaller diameters exhibit stronger deformation behavior than specimens with larger diameters (Fleck et al., 1994). The reason behind this phenomenon is that the inhomogeneity of the deformation is no longer negligible when the internal characteristic length scale of microstructure becomes comparable to the geometric size of the specimen. It will be shown that the gradient-dependent

338

J. Ning and E. C. Aifantis

plasticity of Eq. (61) can effectively be used to describe the size effects in plastic deformation of polycrystalline solids. Recently, a theory of strain gradient plasticity proposed by Fleck, et al. (1994) and further elaborated upon by Fleck and Hutchinson (1993) was used to describe the aforementioned size effects. The theory is physically motivated by Ashbly's concept of "geometric necessary dislocations," whose density scales directly with the gradient of plastic strain. It is a nonlinear generalization of Cosserat couple stress theory; but, in contrast to other plasticity theories of this type (e.g., Muhlhaus and Vardoulakis, 1987), which present difficulties in the definition of yielding, it has an appealing and much less formal structure. This is due to the fact that Fleck et al. are essentially using the same mathematical procedure and philosophy in developing their strain gradient plasticity structure as that followed earlier in the original gradient plasticity theory based on Eq. (61). This, however, is established indirectly by using the formalism of plastic flow potentials and allowing an invariant of strain gradient to enter into their constitutive dependence. While this practice of utilizing potentials and variational principles is appropriate for problems involving no dissipation and states close to thermodynamic equilibrium (e.g., nonlinear elasticity or its proper extension and interpretation to deformation theory of plasticity), its validity may be questioned for a general flow theory of plasticity. Moreover, one still has to deal with the difficulty of asymmetric stressesma difficulty that does not seem to prohibitive in the authors' formulation. In fact, the authors do not address the question of asymmetric stress and its experimental validation, but report, instead, their experimental results obtained by twisting high-purity (99.99%) copper wires of varying diameter. They find a size effectmi.e., the torsional strength depends on the diameter of the wire for a fixed amount of surface strainmand this is attributed directly to the varying intensity of the linear strain gradient along the radius of the wires. They claim that their first strain gradient plasticity theory captures the observed size effect, but they comment that the original second-order strain gradient plasticity (e.g., Zbib and Aifantis, 1989; Muhlhaus and Aifantis, 1991) based on Eq. (61) does not do so. However, this comment is not correct, since the second-order gradient term does not vanish during their torsional experiments (as the authors seem to imply), where polar coordinates should be used to evaluate the Laplacian of strain entering into Eq. (61). For a cylindrical bar with radius a under pure torsion, the strain field reads ?, = rcp

(0 < r < a),

(84)

where q9 is the angle of twist per unit length of the bar. Substituting Eq. (84) into Eq. (61) and using the corresponding equilibrium equation, we obtain Q - 2rr

g0a3 Ys~ 3+n

c~

a - c

2

a

'

(85)

where Q is the applied torque and Ys is the surface shear strain of the bar. In the

Chapter 7

Anisotropic and Inhomogeneous Plastic Deformation of Polycrystalline Solids

339

35O 300 250

150 100 50 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Surface strain ~

Size effect described by gradient approach.

derivation of Eq. (85), we assumed that x ( y ) = x o y n with K0 = 117 (MPa) and n --- 0.2, as determined from the torsion data. For the two sets of experimental data obtained for wires of diameters a = 85 # m and a = 15 # m (Fleck et al., 1994), respectively, the theoretical predictions based on Eq. (89) are depicted in Fig. 6, by choosing ? -- 0.006 (N) and c -- - 0 . 0 1 (N). The self-consistent estimate from the K - B - W model predicts a value of the coefficient c in the range - 0 . 0 1 3 ~ - 0 . 5 0 5 N for grain size in the range 5 ~ 2 5 / z m .

ACKNOWLEDGMENTS The support of the Air Force Office of Scientific Research under grant Nos. AFOSR-91-0421 and AFOSR-95-1-0208 is gratefully acknowledged. Partial support from the National Science Foundation under grant No. NSF/MSS-9310476 and from the Commission of the European Communities under contract No. ERBCHB CT 92-0041 is also acknowledged.

REFERENCES Aifantis, E. C. (1984). On the microstructural origin of certain inelastic models. J. Eng. Mater. Technol. 106, 326-330. Aifantis, E. C. (1987). The physics of plastic deformation. Int. J. Plast. 3, 211-247. Aifantis, E. C. (1992). On the role of gradients in the localization of deformation and fracture. Int. J. Eng. Sci. 30, 1279-1299. Aifantis, E. C. (1994a). Gradient effect at macro, micro, and nano scales. J. Mech. Behav. Mater. 5, 335-353.

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J. Ning and E. C. Aifantis

Aifantis, E. C. (1994b). Spatio-temporal instabilities in deformation and fracture. In "Computational Material Modeling" (A. K. Noor and A. Needleman, eds.), AD-Vol. 42/PVP-Vol. 294, pp. 199-222, ASME, New York. Aifantis, E. C. (1995a). From micro- to macro-plasticity: The scale invariance approach. J. Eng. Mat. Tech. 117, 351-355. Aifantis, E. C. (1995b). Pattern formation in plasticity. Int. J. Eng. Sci. 33, 2161-2178. Atlan, M. C., Advani, S. G., Guceri, S. I., and Pipes, R. B. (1989). On the description of the orientation state for fiber suspensions in homogeneous flows. J. Rheol. 33, 1129-1155. Asaro, R. J. (1983). Micromechanics of crystals and polycrystals. Adv. Solid Mech. 23, 1-115. Asaro, R. J., and Needleman, A. (1985). Texture development and strain hardening in rate dependent polycrystals. Acta Metall. 33, 923-953. Bammann, D. J., and Aifantis, E. C. (1987). A model for finite deformation plasticity. Acta Mech. 69, 97-117. Berveiller, M., and Zaoui, A. (1979). An extension of the self-consistent scheme to plastically-flowing polycrystals. J. Mech. Phys. Solids 26, 325-344. Bishop, J. F. W., and Hill, R. (1951). A theory of the plastic distortion of a polycrystalline aggregate under combined stresses. Philos. Mag. [7] 42, 414-427. Boehler, J. P. (1987). "Applications of Tensor Functions in Solid Mechanics." Chapters 1-7. SpringerVerlag, Wien. Budiansky, B., and Wu, T. T. (1962). Theoretical predication of plastic strains of polycrystals. Proc. U.S. Nat. Congr. Appl. Mech., 4th, pp. 1175-1185. Bunge, H. J. (1972). Some applications of Taylor's theory of polycrystal plasticity. Krist. Tech. 5, 145-175. Clement, A. (1982). Prediction of deformation texture using a physical principle of conservation. Mater. Sci. Eng. 55, 203-310. Coleman, B. D., and Hodgdon, M. L. (1985). On shear bands in ductile materials. J. Ration. Mech. Anal. 90, 219-247. Dafalias, Y. F. (1993). Planar double-slip micromechanical model for polycrystal plasticity. J. Eng. Mech. 119, 1260-1284. Dinh, S. M., and Armstrong, R. C. (1984). A rheological equation of state for semiconcentrated fiber suspensions. J. Rheol. 28, 207-227. Fleck, N. A., and Hutchinson, J. W. (1993). A phenomenological theory for strain gradient effects in plasticity. J. Mech. Phys. Solids 41, 1825-1857. Fleck, N. A., Muller, G. M., Ashby, M. E, and Hutchinson, J. W. (1994). Strain gradient plasticity: Theory and experiment. Acta Metall. 42, 475-487. Hill, R. (1950). "The Mathematical Theory of Plasticity." Oxford University Press, Oxford. Hill, R. (1965). Continuum micro-mechanics of elastoplastic polycrystals. J. Mech. Phys. Solids 13, 89-101. Hill. R. (1967). The essential structure of constitutive laws for metal composites and polycrystals. J. Mech. Phys. Solids 15, 79-95. Hutchinson, J. W. (1970). Elastic-plastic behavior of polycrystalline metals and composites. Proc. R. Soc. London, Ser. A 319, 247-272. Karafillis, A. P., and Boyce, M. C. (1993). A general anisotropic yield criterion using bounds and a transformation weighting. J. Mech. Phys. Solids 41, 1859-1886. Kocks, U. F. (1970). The relation between polycrystal deformation and single-crystal deformation. Metall. Trans. 1, 1121-1143. Kocks, U. E (1975). Constitutive relations for slip. In "Constitutive Equations in Plasticity" (A. S. Argon, ed.), Chapter 3, pp. 81-115. MIT Press, Cambridge, MA. Kroner, E. (1958). Berechnung der elastischen Konstanten des Vielkristalls aus den Konstanten des Einkristalls. Z. Phys. 151,504-518.

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Anisotropic and Inhomogeneous Plastic Deformation of Polycrystalline Solids

341

Kroner, E. (1961). Zur plastischen verformung des vielkristalls. Acta Metall. 9, 155-161. Li, J. C. M. (1968). Kinetics and dynamics in dislocation plasticity. In "Dislocation Dynamics" (A. R. Rosenfeld et al., eds.), pp. 87-116. McGraw-Hill, New York. Lin T. H. (1954). A proposed theory of plasticity based on slips. Proc. U.S. Nat. Congr. Appl. Mech., 2nd, pp. 461-468. Mehrabadi, M. M., and Nemat-Nasser, S. (1983). Stress, dilatancy and fabric in granular materials. Mech. Mater. 2, 155-161. Muhlhaus, H. B., and Aifantis, E. C. (1991). A variational principle for gradient plasticity. Int. J. Solids Struct. 28, 845-857. Muhlhaus, H. B., and Vardoulakis, I. (1987). The thickness of shear bands in granular materials. Geotechnique 37, 271-283. Ning, J., and Aifantis, E. C. (1984). On anisotropic finite deformation plasticity. Parts I & II. Acta Mech. 106, 55-85. Nix, W. D., Gibeling, J. C., and Fuchs, K. E (1980). The role of long-range internal back stresses in creep of metals. ASTM Spec. Tech. Publ. STP 765, 301-321. Oda, M., Nemat-Nasser, S., and Mehrabadi, M. M. (1982). A statistical study of fabric in a random assembly of spherical granules. Int. J. Numer. Anal. Methods Geomech. 6, 77-94. Oka, E, Yashima, A., Adachi, T., and Aifantis, E. C. (1992). Instability of gradient dependent viscoplastic model for clay saturated with water and FEM analysis. Appl. Mech. Rev. 45, 103-111. Prantil, V. C., Jenkins, J. T., and Dawson, E R. (1993). An analysis of texture and plastic spin for planar polycrystals. J. Mech. Phys. Solids 41, 1357-1382. Rashid, M. M. (1992). Texture evolution and plastic response of two-dimensional polycrystals. J. Mech. Phys. Solids 40, 1009-1029. Satake, M. (1983). Fundamental quantities in the graph approach to granular materials. In "Mechanics of Granular Materials: New Models and Constitutive Relations" (J. T. Jenkins and M. Stake, eds.), pp. 9-19. Elsevier, Amsterdam. Shi. M. F., Gerdeen, J. C., and Aifantis, E. C. (1990). On finite deformation plasticity with directional softening. Part I. One-component model. Acta Mech. 83, 103-117. Sluys, L. J., and de Borst, R. (1994). Dispersive properties of gradient-dependent and rate-dependent media. Mech. Mater. 18, 131-149. Taylor, G. I. (1938). The mechanism of plastic deformation of crystals. Proc. R. Soc. London, Ser. A 145, 362--415. Tobita, T. (1988). Contact tensor in constitutive model for granular materials. In "Micromechanics of Granular Materials" (M. Satake and J. Jenkins, eds.), pp. 263-270, Elsevier, Amsterdam. van der Giessen, E., and van Houtte, P. (1992). A 2D analytical multiple slip model for continuum texture development and plastic spin. Mech. Mater 13, 93-115. Vardoulakis, I., and Aifantis, E. C. (1989). Gradient dependeny dilatancy and its implications in shear banding. Ing. Arch. 59, 197-208. Vardoulakis, I., and Aifantis E. C. (1991). A gradient flow theory of plasticity for granular materials. Acta Mech. 87, 197-217. Zbib, H. M. (1994). Strain gradients and size effects in nonhomogeneous plastic deformation. Scri. Metall. Mater. 30, 1223-1226. Zbib, H. M., and Aifantis, E. C. (1988). On the concept of relative and plastic spins and its implications to large deformation theories. I & II. Acta Mech. 75, 15-33, 35-56. Zbib, H. M., and Aifantis, E. C.(1989). A gradient-dependent flow theory of plasticity: Application to metal and soil instabilities. Appl. Mech. Rev. 42, 295-304.

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8 Modeling the Role of Dislocation Substructure during Class M and Exponential Creep S. V. Raj

I. S. Iskovitz

Materials Division

Ohio Aerospace Institute

NASA Lewis Research Center

NASA Lewis Research Center

Cleveland, Ohio

Cleveland, Ohio

A. D. Freed

Materials Division NASA Lewis Research Center Cleveland, Ohio

The different substructures that form in the power-law and exponential creep regimes for single-phase crystalline materials under various conditions of stress, temperature, and strain are reviewed. The microstructure is correlated both qualitatively and quantitatively with power-law and exponential creep as well as with steady-state and non-steady-state deformation behavior. These observations suggest that creep is influenced by a complex interaction between several elements of the microstructure, such as dislocations, cells, and subgrains. The stability of the creep substructure is examined in both of these creep regimes during stress and temperaturechange experiments. These observations are rationalized on the basis of a phenomenological model, where normal primary creep is interpreted as a series of constant-structure, exponential creep rate-stress relationships. The implications Unified Constitutive Laws"of Plastic Deformation

343

344

s. v. Raj, I. S. Iskovitz, and A. D. Freed

of this viewpoint for the magnitude of the stress exponent and steady-state behavior are discussed. A theory is developed to predict the macroscopic creep behavior of a singlephase material using quantitative microstructural data. In this technique the thermally activated deformation mechanisms proposed by dislocation physics are interlinked with a previously developed multiphase, three-dimensionaL dislocation substructure creep model. This procedure leads to several coupled differential equations interrelating macroscopic creep plasticity with microstructural evolution.

LIST OF SYMBOLS A AHT ALT A1 A2 A* a B

B2

Bij b

q Deft

Dijkl Dl DO D01 D0p d de des ds dsb

Dimensionless constant in the power-law equation (3) Dimensionless constant in the high-temperature climb equation (5) Dimensionless constant in the low-temperature climb equation (6) Constant in the exponential creep equation (4) Constant in Eq. (22b) Activation area for a thermally activated deformation process Lattice parameter Dimensionless constant in the exponential creep equation (4) Constant in Eq. (22b). Tensorial representation of the back stress Burgers vector Density of extended jogs Effective diffusion coefficient for lattice and pipe diffusion Elastic moduli tensor Lattice self-diffusion coefficient Frequency factor for diffusion Frequency factor for lattice diffusion Frequency factor for pipe diffusion Grain size Cell size Stress-dependent stacking fault width Subgrain size Diameter of new subgrains nucleated by localized migration of a preexisting subboundary (= fllFsb/r) Applied or global strain tensor Local deviatoric strain tensor in phase r

fij AF fcb fh fs G

Local deviatoric strain rate tensor in phase r Tensorial representation of the forward stress Helmholtz free energy associated with moving a dislocation past an obstacle in Eq. (11) Volume fraction of cell boundaries Volume fraction of hard regions Volume fraction of soft regions Shear modulus

Chapter 8 AGcs h Iklmn

K K1

k L Lh LN Ls L* M m

N

We n nt

Pj p Qc

Qcs Qg Q1 Q0 Q01 Qp QclQNa+ q R Si

:i

Sijkl

T rc rm To

t tc V* Vc

Xij x ol olc

Modeling the Role of Substructure during Class M and Exponential Creep

Activation free energy for cross-slip at zero stress in Eq. (10) Dislocation spacing in a cell or subboundary Identity tensor Constant relating the subgrain size and the applied stress in Eq. (16) Constant relating dislocation spacing in the cell or subgrain boundaries with the stress in the hard regions Boltzmann's constant (1.38 x 10 -23 J K -1) General representation of the cell (dc) or subgrain (ds) size Average dimensions of the hard regions Dislocation spacing in a network Average dimensions of soft regions Activated length for cross-slip Taylor factor Subgrain size exponent in Eq. (16) Number of cell or subboundaries Number of dislocations in a piled-up array in Eq. (8) Creep stress exponent Effective creep stress exponent jth material property Constant in Eq. (4) obtained from the stress dependence of the dislocation density. In general, p ~ 2 or 3 for most materials Activation energy for creep Activation energy for cross-slip mechanisms Activation energy for the obstacle-controlled glide process Activation energy for lattice-self-diffusion Maximum activation energy for cross-slip in Eq. (9b) Maximum activation energy for cross-slip in Eq. (10) Activation energy for dislocation core diffusion Activation energy for C1- diffusion in NaC1 Activation energy for Na + diffusion in NaC1 Stacking fault energy exponent Universal gas constant (8.314 J mo1-1 K -1) Microstructural parameters l, 2, 3 . . . . . i Deviatoric stress in phase ~b Deviatoric stress rate tensor in phase ~b Eshelby deviatoric stress tensor Absolute temperature Absolute transition temperature above which Qc "~ Q1 and below which Qc < Q1 Absolute melting temperature Reference temperature in Eq. (57) time Characteristic time for a pair of edge dislocations to climb a distance h / 2 before annihilation Activation volume for creep Climb velocity General representation of a local tensorial strain or stress field in phase 4~ Global tensorial strain or stress field Spatial coordinate Geometric constant in the Taylor equation (20) Coefficient of thermal expansion

345

346 Otkl or4~ Oll ol !

3 31 ~2

3' F Fsb

Y yh ys ~,ph

~,ps E E EHT ELT Ecl Eg Emn E0 K Ah

As u

VD P Pci Pcb ph Psi Psb Pr p~ o"

O'b O'e crf Crcb O'ci O'mn

S.V. Raj, I. S. Iskovitz, and A. D. Freed Coefficient of thermal expansion tensor Constant in the Taylor equation relating the strength of a phase 4~ with the dislocation density in the phase Geometric constant in Eq. (12) Constant in Eq. (10) dependent on the stacking fault energy Shape factor of a cell or subgrain (13 = 2(4 - 5v)/15(1 - v) for spheroids) Constant determining the size of newly formed subgrains nucleated by subboundary migration Constant in Eq. (38) equal to about 1000 Constant in Eq. (8) equal to about 1.0 and 2.0 for low and high stacking fault energy materials Stacking fault energy Subboundary surface energy Shear strain Shear strain in the hard regions Shear strain in the soft regions Shear strain rate Plastic shear strain rate in the hard region Plastic shear strain rate in the soft region Kronecker delta Normal strain Normal uniaxial creep or strain rate Normal creep rate for high-temperature climb Normal creep rate for low-temperature climb Dislocation-recovery--controlled creep rate in Eq. (24) Strain rate associated with thermally activated glide in Eq. (24) Creep or strain rate tensor Instantaneous creep strain on loading Bulk modulus Average distance traveled by a dislocation in the cell wall before being stopped Average distance traveled by a dislocation in the cell interior before being stopped Prandtl-Reuss rate parameter given by Eq. (53) Poisson's ratio Debye frequency Dislocation density Dislocation density within cells Dislocation density within cell boundaries Dislocation density in the hard regions Dislocation density within the subgrains Dislocation density in the subboundaries Total dislocation density (e.g., Pr = Psi -k- Psb) Dislocation density in phase ~b Normal uniaxial applied stress Back stress on a dislocation in the cell or subgrain interior or soft region General representation of the effective stress acting on a dislocation within in cell or subgrain Forward stress acting on a cell or subgrain boundary Effective stress acting on a cell boundary (= a + crf) Effective stress acting on a dislocation in the cell interior (= cr - ~rb) Applied stress tensor

Chapter 8 O'obs CrPLB r rb rf r h

f~ l-s x x0 f2

~11

Modeling the Role of Substructure during Class M and Exponential Creep

347

Obstacle strength Power-law breakdown stress Applied shear stress Shear back stress acting on the dislocations in the soft regions Forward shear stress acting on the hard regions Effective shear stress acting on the hard regions (= r + rf) General representation of the shear stress in the hard (q5 = h) or soft (~b = s) regions Shear strength of an obstacle in the hard or soft region Effective shear stress acting in the soft regions (= r - rb) General representation denoting hard (h) or soft (s) phase Lh/L s Initial value of X at the beginning of deformation Atomic volume

INTRODUCTION

A. Kineticand Microstructure Evolution Laws Creep is defined as the time-dependent deformation of a material at any absolute temperature, T, under the action of either a slowly time-variant or a timeinvariant montonic applied stress. Although a material can creep at any temperature above absolute zero, measurable time-dependent deformation occurs at elevated temperatures--typically above 0.3 Tin, where Tm is the absolute melting temperature of the material. Considerable data, accumulated over several decades, have enabled a detailed characterization of the creep process in terms of its stress, temperature, and microstructural dependence. As a result, several models have attempted to formulate constitutive laws to express the creep rate, k, in terms of cr or r, T, Pj, and Si, where cr and r are the applied normal and shear stress, respectively; Pj is t h e j t h material property (e.g., lattice parameter, a, atomic volume, S2, elastic moduli, Burgers vector, b, and diffusion frequency factor, Do); and Si is the i th microstructural parameter (e.g., grain size, d, dislocation density, p, cell size, dc, subgrain size, ds, and stacking fault energy, F). It should be noted that Si is also referred to as an internal state variable in the plasticity literature. A more general formulation can express the tensorial creep rate, kmn, in terms of a tensorial stress state, ~mn. The role of microstructure in influencing creep behavior has been studied extensively in a large number of materials, as evidenced by the numerous review articles (Schoeck, 1957; Sherby, 1962; McLean, 1966; Sherby and Burke, 1967; Bird et al., 1969, Mukherjee et al., 1969; Lagneborg, 1972; Weertman, 1975; Takeuchi and Argon, 1976; Blum, 1977, 1993; Nix and Ilschner, 1980; Myshlyaev, 1981; Bendersky et al., 1985; Nix and Gibeling, 1985; Orlovfi and 0adek, 1986; Caillard and Martin, 1987; Blum et al., 1991; Longquan and Northwood, 1993; Yoshinaga, 1993; Raj, 1994) and texts (Garofalo, 1965; Gittus,

348

S.V.Raj, I. S. Iskovitz, and A. D. Freed

1975; Poirier, 1985) that have followed the course of development of the subject over the last four decades. These advances suggest that deformation behavior can be represented by a generalized kinetic or rate equation (Frost and Ashby, 1982)

= f(cr, T, Si, Pj)

(1)

and a generalized microstructure evolution equation describing the rate of change of the microstructure, 57i, during deformation

Si -- g(cr, T, Sj, Pj)

(2)

It is important to note that the kinetic equation (1) describes deformation for a constant microstructure, whereas the microstructure evolution equation (2) represents the manner in which the microstructure changes during the course of deformation. Equations (1) and (2) are fairly general and they are applicable to both low- and high-temperature deformation under either cyclic or monotonic stresses. This generality argues for the development of a unified deformation model that describes flow behavior under different stresses states at any temperature for conditions where thermally activated dislocation processes are important. In reality, the problem is complex, in part because the present understanding of the dislocation mechanisms controlling deformation is incomplete in many instances and in part because the role of microstructure and its evolution on deformation processes is difficult to quantify. Thus, specific deformation characteristics and microstructure may vary from one material to another. Nevertheless, the last four decades of research have shown that there are many similarities in the deformation behavior and microstructural evolution in several materials, ranging from simple metals to the more complex geological rocks (Frost and Ashby, 1982). These similarities have permitted a general understanding of the deformation behavior of materials. The emphasis of the present chapter is on dislocation-controlled flow processes, except for Harper-Dorn creep (Harper and Dorn, 1957; Harper et al., 1958), occurring under uniaxial deformation. The mechanism for Harper-Dorn creep is poorly understood at present and this process is not considered. In contrast to classical approaches, which artificially treat low- and high-temperature deformation as unrelated flow behavior, the current treatment views deformation from a broader global perspective, as a material response to a set of external variables resulting from a corresponding evolution in the microstructure. Dislocations within a material move, rearrange, and annihilate themselves to form low-energy dislocation substructures (LEDS), thus achieving steady-state conditions in accordance with the laws of irreversible thermodynamics. In principle, this steady-state condition and the corresponding microstructure is achievable under all deformation conditions at all temperatures above 0 K if other factors, such as mechanical and microstructural instability, do not intervene. However, in practice the rate of recovery may be extremely slow so that steady-state deformation conditions and an equilibrium microstructure may not be observed in the time frame of a typical experiment.

Chapter 8

Modeling the Role of Substructure during Class M and Exponential Creep

349

B. Types of Flow Behavior Figure 1 shows schematics of the different types of tensile creep curves, plotted either as creep strain, e, against time, t, (A) or k vs. e (B), as is generally observed in creep tests. The corresponding tensile cr-e curves (C) obtained from constant strain rate tests are also shown, since the latter tests are often used for generating steady-state k-o- plots in studies relating to the high-temperature deformation of materials. Some special cases, such as serrated yielding and dynamic recrystallization, are not discussed. Although the constant strain rate and creep data often correspond closely at high temperatures, it is important to recognize that the transient dislocation substructures formed in these two tests are likely to be vastly

or = constant l

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Schematic plots of creep strain vs. time (A), creep rate vs. creep strain (B), and stress vs. strain (C) illustrating typical (a) normal, (b) sigmoidal, and (c) inverse primary creep and the corresponding stress-strain transients observed under constant stress and constant strain rate deformation conditions. Regions I, II, and III represent the primary, secondary, and tertiary regions.

350

S.V. Raj, I. S. Iskovitz, and A. D. Freed

different. This difference arises from the fact that the thermodynamic path by which the substructure evolves toward a steady-state configuration will generally be different in constant stress and constant strain rate tests. A typical creep curve consists of several stages (Fig. 1): an instantaneous strain on loading; a primary creep region where the creep rate changes with time (I); a region of linear, steady-state flow or a secondary creep regime in which is independent of strain and time (II); and a period of accelerating creep in the tertiary stage resulting from mechanical (i.e., external or internal necking) or microstructural (e.g., dynamic recrystallization and grain growth) instability (III). The duration of secondary creep will be short if tertiary creep occurs early. Tertiary creep is generally absent in compression creep if microstructural instability does not occur. However, the creep rate can continue to decrease with strain or time during constant load compression creep due to an increase in the cross-sectional area. Normal primary creep, typical of class M (metal) type (Yavari et al., 1981) or Class II (Sherby and Burke, 1967) behavior, is illustrated by Fig. 1a, where the creep rate decreases with increasing strain in the primary region as the material work hardens. In the absence of significant recovery, the material will continue to harden until the creep rate is immeasurably small. However, significant recovery occurs in most materials at intermediate and high temperatures, 1 and the material tends toward steady-state behavior where the work-hardening rate balances the recovery rate. Under these conditions, all microstructural parameters attain quasiequilibrium so that Si "~- 0. Sigmoidal (Fig. l b) and inverse (Fig. l c) primary creep are often observed in materials exhibiting class A (alloy) type (Yavari et al., 1981) or class I (Sherby and Burke, 1967) behavior, where a solute atmosphere around a dislocation tends to inhibit its motion (Weertman, 1957). In some instances, steady-state flow is attained almost immediately on loading and the creep curve is linear until tertiary creep intervenes. Inverse and sigmoidal primary creep occur when the initial mobile dislocation density is low, which results in a low initial creep rate. The creep rate increases with strain during inverse primary creep until a steady-state value is obtained (Fig. l c). Sigmoidal primary creep, which is intermediate between inverse and normal primary creep, involves an initial region of increasing creep rate to a maximum value, followed by a period of decreasing creep rate until steady-state creep is achieved (Fig. 1b). Most creep mechanisms fall into three broad categories: diffusion, power-law, and exponential creep. Two diffusion creep mechanisms involving Newtonian 1The terms 'intermediate' and 'high' as used in the creep literature strictly refer to the temperature ranges above 0.3 Tm for which Qc < Q~ and Qc ~ QI, respectively, where Qc and Q1 are the activation energies for creep and lattice self diffusion, respectively. For simplicity, the temperature ranges T ~ 0.3-0.7 Tm and T > 0.7 Tm are generally assumed to define the intermediate and high temperature creep ranges, respectively, since Qc ~ QI when T > 0.7 Tm for most materials.

Chapter 8 Modelingthe Role of Substructure during Class M and Exponential Creep 351 viscous flow with k proportional to cr (Nabarro, 1948" Herring, 1950; Coble, 1963) occur at low values of normalized stress, ~ / G , where G is the shear modulus and typically o-/G _< 5 x 10 -6. In this case, strain is produced by the diffusion of atoms from grain boundaries under compression to those under tension, with vacancies flowing in the opposite direction (R. Raj and Ashby, 1971). NabarroHerring creep (Nabarro, 1948; Herring, 1950) occurs when the diffusion path is through the lattice, in contrast to Coble creep (Coble, 1963), where the diffusion path is along grain boundaries. A third Newtonian creep mechanism, which was discovered by Harper and Dorn (1957; Harper et al., 1958), involves a poorly understood dislocation-controlled process. In contrast, power-law creep occurs at intermediate stresses, typically 5 x 10 -6 _< ~ / G < crpLB/G, where crPLB is the power-law breakdown (PLB) stress. In this case, the creep rate depends on the applied stress through a power-law relation with k proportional to ( ~ / G ) n, where n is the stress exponent, typically varying between 4 and 5 for class M behavior (Bird et al., 1969). Strain is produced by a sequential motion of dislocations through glide and climb, where the slowest mechanism is the rate-controlling step. Since the power-law relation is a generic equation that can often be used to describe almost any set of creep data plotted double-logarithmically, the term "power-law creep" is not very descriptive of the characteristics of the deformation mechanism. Instead, the terms "class M creep" and "class A creep" will be used in this paper to describe class M or class A material behavior, respectively. The class M creep relation breaks down at high stresses when ~ / G > ~PLB/G and the stress dependence of k is better described by an exponential relationship with k proportional to exp(B~/G), where B is a constant. Unlike the creep mechanisms dominant in the diffusion and class M creep regimes, where the diffusion of atomic and defect species is important, exponential creep mechanisms are generally nondiffusiona! in nature. Since the major objective of the present chapter is to model the role of dislocation substructure during class M and exponential creep, class A and Newtonian viscous creep mechanisms are not addressed in this review. Although class M creep has been studied extensively in several materials (Sherby and Burke, 1967; Bird et al., 1969; Mukherjee et al., 1969), considerably less attention has been paid to the mechanisms dominant in the exponential creep region and the microstructures that form under these conditions. Noting that many engineering applications require materials to operate at intermediate temperatures (Table 1) (Nix and Gibeling, 1985), where the diffusion rates are slower than at higher temperatures, the rate-controlling mechanisms may often be nondiffusional in nature and similar to those dominant in the exponential creep region (Raj and Langdon, 1989, 1991 a, b). Despite the potential importance of characterizing exponential creep behavior, this subject has been largely ignored, primarily because of the difficulty in modeling non-steady-state creep. Since creep data are seldom acquired over a very large range of stresses and temperatures in any one

352

S.V. Raj, I. S. Iskovitz, and A. D. Freed ABLE 1

Typical Operating Temperatures in Engineering Applications a

Typical materials

Typical temperatures (K)

T~ Tm

Cr-Mo-V steels

825-975

0.45-0.50

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650-750

0.35-0.40

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850-950

0.45-0.55

Application Rotors and piping for steam turbines Pressure vessels and piping in nuclear reactors Reactor skirts in nuclear reactors

aFrom Nix and Gibeling (1985).

investigation, detailed microstructural characterization of the transition from class M to exponential creep is virtually nonexistent for most materials. As a result, our understanding of the microstructural changes occurring during this transition is poor, so it is difficult to formulate realistic microstructure-based creep models. Nevertheless, in comparison to previous attempts (Weertman, 1955, 1957, 1975; McLean, 1966; Lagneborg, 1972; Takeuchi and Argon, 1976), a fundamental change occurred in the philosophy and approach adopted toward the development of microstructure-based creep models when Nix and Ilschner (1980) developed a two-phase microstructural composite creep model to predict deformation behavior in the class M and exponential creep regimes. It is interesting to note that Mughrabi (1980) and Mughrabi and Essmann (1980) also presented a similar approach in modeling low-temperature behavior and fatigue at the same conference. Although the Nix-Ilschner model (1980) is conceptually incorrect (Raj and Langdon, 1991 b), it provided the foundation for the development of current creep models. A significant step in this direction occurred with the publication of two important papers by Nix and Gibeling (1985) and Nix et al.) (1985), which attempted to model creep using a set of coupled kinetic and evolution equations describing low- and high-temperature deformation mechanisms. Other developments and refinements followed (Vogler and Blum, 1990; Hofmann and Blum, 1993; Zhu and B lum, 1993). Typically, these models assume that the microstructure consists of alternate regions of long subboundaries, or "hard" phases, and subgrain interiors, or "soft" phases, which must deform in such a manner that compatibility is maintained between the two phases. When considered with similar modeling efforts for low-temperature uniaxial (Mughrabi, 1983; Argon and Haasen, 1993) and cyclic (Mughrabi, 1981, 1987) deformation, it becomes apparent that, at least in principle, this approach to modeling low- and high-temperature deformation behavior can lead to the development of a unified model describing various deformation behavior of a material.

Chapter 8 Modelingthe Role of Substructure during Class M and Exponential Creep 353 Despite their potential advantages, the two-phase deformation models are still largely limited in scope for several reasons. First, they assume that subgrains form soon after loading; therefore, they do not take into account the large microstructural variations that actually occur during primary creep and in the exponential creep region. Second, they unrealistically assume that subgrains are one-dimensional in nature. Third, they assume that the volume fractions of the subboundaries and dislocation densities do not change with strain or time. Fourth, they assume the presence of one type of substructure, i.e., cells or subgrains. In reality, as discussed in Section III.A.2, both cells and subgrains can form during creep. In a recent development, Freed et al. (1992) attempted to address some of these issues. They extended the one-dimensional composite models to the more realistic three dimensions using an Eshelby analysis for isotropic spheroidal cells or subgrains and allowed the microstructural terms to vary with deformation. The approach was general enough to consider a variant microstructure of cells and subgrains coexisting together although, for simplicity, the analysis was presented for one type of substructure. It was demonstrated that the shape factor,/~, of the cell or subgrain introduces an additional second-order tensorial elastic strain term to describe the local strain compatibility between the soft and hard regions in the crystal. The equations reduce to the Voigt approximation used by Nix and others in the one-dimensional case when/~ - - 0 . In principle, this three-dimensional model, termed dislocation substructure viscoplasticity (DSV), should lead to the development of a unified deformation theory to describe plastic deformation for any stress state with additional refinements. In contrast to the microstructure-based creep theories, viscoplasticity models ignore the role of microstructure on creep behavior (Kocks, 1987). Instead, the material is treated as a "black box," with stress and temperature as the primary input variables and creep strain and creep rate as the main output variables. The internal state variables are introduced to account for transient effects, such as the Bauschinger effect and cyclic hardening under fatigue-type loading situations. This approach is more useful in predicting the material response in engineering design, whereas the microstructural theories permit sound material design and selection. However, both approaches to creep modeling are limited in that the viscoplastic models have to make arbitrary assumptions regarding Si while current microstructure-based creep theories cannot easily be scaled up to predict the macroscopic creep behavior of an engineering component. Therefore, the question arises: Can a model be developed by coupling the creep equations from viscoplasticity with those from dislocation physics? If so, it should be possible, at least in theory, to predict the macroscopic creep response of an engineering component or test specimen at one extreme and the evolving deformation microstructure at the other by using the principles of finite-element analysis (FEM), viscoplasticity, and dislocation physics. The fundamental objective of this chapter is to develop such a modeling technique. In its final form the

354

S.V. Raj, I. S. Iskovitz, and A. D. Freed

model should predict the time evolution of the shape of a deforming body and the corresponding internal stresses and dislocation microstructures for macroscopic stress states produced during bend, cyclic, multiaxial, and uniaxial tests. However, owing to the complexity of the problem and the amount of computational time required, the present goals are rather modest and tackle only uniaxial deformation. A focused review of uniaxial compression creep data and microstructural observations on NaC1 single crystals is presented here with the primary purpose of using the information in developing the creep model. No attempt is made to specifically review the creep properties of other materials. The use of NaC1 single-crystal data in this study was influenced by a number of factors critical to the present modeling effort. Since it is a single-phase material, which exhibits class M creep behavior, additional complications arising from solid-solution effects and second-phase particles are avoided. In addition, grain boundary effects are eliminated for single crystals. More importantly, a considerable amount of creep data, as well as qualitative and quantitative microstructural data, are available on this material, varying over a very large range of stresses and temperatures corresponding to class M and exponential creep (Ilschner and Reppich, 1963; Le Comte, 1965; Stokes, 1966; Blum and Ilschner, 1967; Heard, 1972; Poirier, 1972a, b, Eggeler and Blum, 1981; Carter and Hansen, 1983; Wawersik, 1984, 1985; Raj and Pharr, 1986, 1989, 1992; Nadgomyi and Strunk, 1987; Przystupa et al., 1987; Raj et al., 1989, 1991; Raj and Freed, 1992; S. V. Raj and G. M. Pharr, unpublished research, Rice University, Houston, TX, 1986). Figure 2 schematically illustrates the approach adopted in this work. The essential idea used in developing this mesoscopic creep model is to couple the microscopic kinetic and evolution equations for different thermally activated defor-

InDUt Stress, temperature, material constants ~

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i ~ I ll,I

10-2

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I

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1,1,11 l t 10-6 Dimensions (m)

I

I i l,lil, 10..7 110_9

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Schematic block diagram of the coupling between dislocation mechanics, dislocation substructure viscoplasticity, and finite-element analysis required to study the interaction between the macroscopic and microscopic stresses, strains, and strain rates.

Chapter 8 Modelingthe Role of Substructure during Class M and ExponentialCreep 355 marion mechanisms with the macroscopic stresses, strains, strain rates, and timedependent microstructural variables using the DSV model (Freed et al., 1992). A solution of the equations from the DSV model at different points on an engineering component or a test specimen using FEM analysis then gives the macroscopic response in terms of stresses, strains, and creep rates. This coupling between the macro and the micro states of the component allows a two-way flow of information, as shown in Fig. 2. In essence, this approach promises to be a powerful analytical tool for understanding the deformation behavior of solids under uniaxial, multiaxial, and cyclic loading conditions if the details of the dislocation mechanisms are well characterized for the dominant deformation processes. However, this paper provides only the theoretical basis for coupling the DSV model and dislocation mechanics. The dislocation model considered here is applicable only to the simple one-dimensional cell model postulated by Nix and Gibeling (1985) and Nix et al. (1985). A more detailed treatment of the dislocation processes governing cell and subgrain formation and their subsequent deformation is required before the DSV model can be applied to the general three-dimensional case. Moreover, in order to keep the problem tractable, the extension of the computation to study the deformation of an entire test specimen using numerical analysis and FEM is not considered here. For simplicity, grain boundary effects are neglected, so the present solutions are valid strictly for single crystals. The solutions are valid for polycrystalline materials only if the strain contributions from grain boundary effects, such as grain boundary sliding and grain boundary migration, are small. This chapter is divided into several sections. First, a general discussion of the current understanding of the class M and exponential creep mechanisms is presented for single-phase class M materials. Second, the characteristics of the dislocation substructures formed during class M and exponential creep are discussed. Third, the stabilities of these microstructures after stress and temperature changes are examined. Fourth, a modified version of the one-dimensional, twophase Nix-Gibeling creep model (1985) is presented. Fifth, the development of the DSV model is treated.

CLASSM AND EXPONENTIALCREEPIN SINGLE-PHASEMATERIALS The steady-state creep rate at intermediate values of normalized stress corresponding to the class M creep regime is usually given by (Bird et al., 1969) = a(DoGb/kT)(a/G)

n exp(-Qc/RT)

(3)

where b is the Burgers vector, k is Boltzmann's constant, R is the universal gas constant, and A is a dimensionless constant. Typically, the experimental values

356

S.V.Raj, I. S. Iskovitz, and A. D. Freed

of the stress exponents n are about 4 to 5 and Qc ~ Ql for a large number of materials exhibiting class M behavior (Sherby and Burke, 1967). A point that may be noted here is that the observation Qc ~ Ql is not accepted by all investigators. For example, Poirier (1978, 1979) has suggested that Qc > Q1 at very high temperatures and low stresses, and that Qc ~ Q1 is only an approximation of true deformation behavior in certain stress and temperature ranges. This viewpoint has been questioned by Sherby and Weertman (1979) and Nix and Ilschner (1980). At high values of normalized stresses, the power-law relation given by Eq. (3) breaks down when or/G > (rr/G)PLB, where (rr/G)PLB is the normalized powerlaw breakdown stress. The creep rate then exhibits an exponential dependence on normalized stress over a large range of stresses and temperatures as k -- A l ( c r / G ) p e x p ( B c r / G ) e x p ( - Q c / R T )

(4)

where A1 and p are constants, where p reflects the stress dependence of the dislocation density so that p ~ 2 (Bird et al., 1969; Takeuchi and Argon, 1976). It is generally accepted that class M creep is controlled by dislocation climb (Bird et al., 1969; Takeuchi and Argon, 1976; Nix and Illschner, 1980). As first pointed out by Weertman (1975), theories for dislocation climb creep always predict a stress exponent of n = 3 if no special assumptions are made, irrespective of the microstructural basis for the model. Therefore, it was hypothesized that there is a natural or universal creep law which predicts that n = 3 for class M creep. In reality, experimental observations consistently reveal that n > 3 for most materials undergoing class M creep (Bird et al., 1969). In contrast, the exponential creep law is even more poorly understood in terms of dislocation processes operating in this regime. Some of the proposed mechanisms are discussed below.

A. Assessmentof Dislocation Core Diffusion-Controlled Creep Mechanisms If the activation energy is plotted as a function of temperature, it is often seen that Qc ~ Q1, where Ql is the activation energy for lattice diffusion, for T > 0.7 Tm and Qc < Q1 for T < 0.7 Tm (Sherby and Burke, 1967; Ltithy et al., 1980). Interestingly, early research suggested that Qc ~ 0.5-0.6 QI for some materials at intermediate temperatures (Fig. 3) (Barrett and Sherby, 1964; Robinson and Sherby, 1969; Ltithy et al., 1980). Sherby et al. (Barrett and Sherby, 1964; Robinson and Sherby, 1969; Liithy et al., 1980) rationalized these observations on the basis that creep behavior at these temperatures is also controlled by dislocation climb, where vacancy diffusion now occurs along dislocation cores instead of through the lattice, so that Qc ~ Qp, where Qp is the activation energy for dislocation pipe or core diffusion. Thus, creep processes controlled by dislocation climb at intermediate and high temperature were termed low-temperature (LT) and high-temperature (HT) climb, respectively (Frost and Ashby, 1975, 1977, 1982). Since dislocation core

Chapter 8 Modelingthe Role of Substructure during Class M and Exponential Creep 357

Variation of the activation energy for creep, normalized by the activation energy for lattice self-diffusion, with the homologous temperature for different materials showing the probable dominance of low-temperature climb at low and intermediate temperatures. (Reprinted from Acta Metall., Volume 28, H. Ltithy, A. K. Miller, and O. D. Sherby, The stress and temperaturedependence of steady-state flow at intermediatetemperaturesfor pure polycrystalline aluminum, Pages 169-178, Copyright (1980), with kind permission from Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK.)

diffusion is dependent on the dislocation density (Hart, 1957; Shewmon, 1963) and p is proportional to a 2, the predicted stress exponent for low-temperature climb is (n 4- 2) (Robinson and Sherby, 1969; Sherby and Young, 1975). Thus, the creep rate equation for low-temperature climb can be specifically expressed as eLT = ALT(DopGb/kT)(a/G) n+2 e x p ( - Q p / R T )

(5)

and that for high-temperature climb as SHT = AHT(DolGb/kT)(a/G) n e x p ( - Q 1 / R T )

(6)

where D01 and D0p are the frequency factors for lattice and pipe diffusion, respectively, and AnT and ALT are dimensionless constants. Figure 4 schematically illustrates the regions of dominance of high-temperature climb, low-temperature climb, and exponential creep on a plot of normalized creep rate, kkT/DeffGb, against a / G , where Deff is an effective diffusion coefficient representing the combined effects due to lattice and core diffusion. The two sets of curves shown in Fig. 4 represent the creep behavior of high and low stacking fault energy materials. Sherby and Burke (SB) (1967) first suggested that the power law breaks down at almost a constant value of ~/Deff ~ 1013 m -2 (i.e., kkT/DeffGb ~ 10 -8) for many materials, and this criterion is indicated in Fig. 4. Although the low-temperature climb process predicts a stress exponent of (n + 2) and Qc ~ Qp, Spingarn et al. (1979) concluded from a reevaluation of

358

S.V. Raj, I. S. Iskovitz, and A. D. Freed

Schematic illustration of normalized creep rate vs. normalized stress for high and low stacking fault energy pure metals showing the regions of class M creep (high-temperature and low-temperature climb) and exponential creep. (Reprinted fromActa Metall., Volume 37, S. V. Raj and T. G. Langdon, Creep behavior of copper at intermediate temperatures. I. Mechanical characteristics, Pages 843-852, Copyright (1989), with kind permission from Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK).

the creep data for several materials that increases in the stress exponents did not correlate with Qc ~ Qp. Furthermore, the predicted creep rates based on theories for core diffusion were about 20-400 times slower than the experimental values for four metals: A1, Cu, Ni, and W. Similarly, Kassner and Oldani (1988) and Kassner (1989) concluded that the Qc vs. T plot for Ag does not show any plateau corresponding to Qc "~ Qp. More recently, a detailed study on the creep of Cu also failed to reveal any evidence of low-temperature climb-controlled creep in this material (Raj and Langdon, 1989, 1991a, b). In this case, the activation energy was observed to decrease linearly with increasing values of or/G, and it was demonstrated that this stress dependency of Qc could not be attributed to a transition from high- to low-temperature climb with increasing normalized stress. Furthermore, it was demonstrated through a compilation of Qc values published

Chapter 8 Modelingthe Role of Substructure during Class M and Exponential Creep 359 in the literature that the activation energy data for Cu do not exhibit the simple step-function transition shown in Fig. 3 (Raj and Langdon 1989). Similarly, a plot of Qc against T~ Tm for NaC1 polycrystals (Le Comte, 1965) and single crystals (S. V. Raj and G. M. Pharr, unpublished research, Rice University, Houston, TX, 1986) shows no clear evidence of a plateau at intermediate temperatures (Fig. 5). Instead, Qc increases monotonically with increasing homologous temperature. The horizontal broken lines show the values for Qcl- and QNa+, where Qcl- and QNa+ are the average activation energies for the lattice diffusion of C1- and Na + ions, respectively, in NaC1 determined by replotting diffusion data obtained from several sources (Harrison et al., 1958; Barr et al., 1960, 1965; Beniere et al., 1970, 1977; Downing and Friauf, 1970; Nelson and Friauf, 1970; Rothman et al., 1972; Mitchell and Lazarus, 1975; Ho and Pratt, 1983). Figure 5 also shows the predicted variations in the temperature dependence of Qc, assuming there is a transition from low- to high-temperature climb with increasing temperature for ~ / G = 10 -4 and 10 -3. The mean values of Qc are given by (Langdon and Mohamed, 1977) Qc -- ( QI~HT -~- Qp~LT) / (~HT -Jr ~LT)

(7)

The calculated values of Qc were obtained using values of Qp ~ 155 kJ mo1-1 (Ho and Pratt, 1983) and Q1 = Qcl- ~ 230 kJ mo1-1 (Raj and Pharr, 1989).

Temperature dependence of the experimental activation energies for creep of polycrystalline (Le Comte, 1965) and single crystalline (S. V. Raj and G. M. Pharr, unpublished research, Rice University, Houston, TX, 1986) sodium chloride showing an absence of a plateau at intermediate temperatures. The expected trend from low- to high-temperature climb is also shown. (Curves were calculated, based on the model of Ltithy et al., 1980.) The activation energies for the lattice self-diffusion of C1- and Na+, and that for pipe diffusion are represented by the horizontal broken lines.

360

s.v. Raj, I. S. Iskovitz, and A. D. Freed

Therefore, an increasing body of evidence suggests that creep involving dislocation core diffusion is not a valid concept for many materials. It can then be inferred from Fig. 4 that in the absence of low-temperature climb, there is a direct transition from class M to exponential creep. This implies that the mechanism(s) dominant in the exponential creep region is (are) identical to those controlling creep at intermediate temperatures, where the transition from class M to the exponential creep involves a changeover from a diffusional to a nondiffusional process(s). Therefore, the term "class M creep" as used in this chapter will specifically refer to high-temperature climb-dominated creep. Alternatively, the low values of Qc < QI observed at intermediate temperatures have been attributed to one or more nondiffusional processes (Dorn and Jaffe, 1961; Jaffe and Dorn, 1962; Poirier, 1976; Caillard and Martin, 1987; Raj and Langdon, 1991b). Most of these mechanisms predict a stress-dependent activation energy, which leads to exponential creep behavior similar to Eq. (4) or, if a power-law relation is used to analyze the data, to values of n >> 4. Among the several mechanisms that have been proposed (Caillard and Martin, 1987; Raj and Langdon, 1991b), those involving the cross-slip of screw dislocations (Friedel, 1959, 1964, 1977; Dorn and Jaffe, 1961; Jaffe and Dorn, 1962; Poirier, 1976; Caillard and Martin, 1987; Carrard and Martin, 1987, 1988) and obstacle-controlled glide (Nix and Ilschner, 1980; Raj and Langdon, 1991b) appear to be important at intermediate temperatures and high stresses.

B. NondiffusionalCreep Mechanisms Several nondiffusional creep mechanisms have been proposed, and these are reviewed elsewhere (Caillard and Martin, 1987; Raj and Langdon, 1991 b). However, only two of these--namely, cross-slip and obstacle-controlled glide mechanisms m appear to be relevant in most instances (Caillard and Martin, 1987; Nix and Illschner, 1980; Raj and Langdon, 1991 b). These are discussed below.

1. Cross-Slip Mechanisms The cross-slip of screw-oriented dislocations can lead to a decrease in the dislocation density through the annihilation of screw dislocations of opposite signs. Thus, it is possible for cross-slip to act as a recovery process in addition to its role as a multiplicative mechanism. The mechanism is complex, and no good creep model exists to describe the process. As a result, there is some controversy as to whether this process is dominant at intermediate and high temperatures (Poirier, 1976, 1978, 1979; Sherby and Weertman, 1979). This controversy hinges on a fundamental question: Do edge and screw components of a dislocation loop move independently or sequentially during deformation? The conventional viewpoint, first advocated by Weertman (1955), assumes that the edge and screw components

Chapter 8 Modelingthe Role of Substructure during Class M and Exponential Creep 361 move sequentially, with the rate of climb of edge dislocations being slower than the rate of cross-slip of screw dislocations. Therefore, the creep rate was controlled by dislocation climb in the Weertman model. The justification for this rationale lies in the fact that Qc ~ Q1 for experimental data reported for several materials in the class M creep regime (Sherby and Burke, 1967; Bird et al., 1969). In contrast, Poirier (1976, 1978) reanalyzed some of the published creep data for many metals and concluded that Qc ~ Q1 only in certain stress and temperature ranges and that Qc > Q1 at higher temperatures and lower stresses. He attributed these high values of Qc to creep controlled by cross-slip of screw dislocations, and proposed a creep model in which climb of the edge and cross-slip of screw components were assumed to occur as independent mechanisms (Poirier, 1976). The model predicts that Qc < Q1 at very high stresses and low temperatures. Theories involving cross-slip fall into two broad categories, depending on whether the partial dislocations constrict along a length equal to that of a FrankRead source (Schoeck and Seeger, 1955; Wolf, 1960; Ptischl and Schoeck, 1993) or along a length less than the Frank-Read source (Friedel, 1959, 1964, 1977; Escaig, 1968a, b; Bonneville and Escaig, 1979; Duesbery et al., 1992) prior to cross-slip onto the new slip plane. In each case, the unit dislocation dissociates into its partials in the cross-slip plane. The former process is energetically unfavorable by comparison with the latter, so it is generally not important. Two variations of the second process exist, both of which are based on the Friedel model (Friedel, 1959, 1964, 1977). Jaffe and Dorn (1962) were the first to propose that the values of Qc < Q1 observed at intermediate temperatures (Fig. 3) could be rationalized using the Friedel cross-slip model. They assumed that the partial dislocations constrict over a critical length of about 4dcs, where des is the stress-dependent stacking fault width. The activation energy for cross-slip, Qc~, for this mechanism is (Jaffe and Dorn, 1962) Qcs - (O.O14fl'Gb3/Np)(G/a)~176

~

(8)

where fi' is a constant which is approximately 1.0 and 2.0 for low and high stacking fault energy materials, respectively, and Np is the number of dislocations in a piledup array. Equation (8) predicts a nonlinear dependence of Qcs on stress. In the second version, Escaig and others (Escaig, 1968a, b; Poirier, 1976; Bonneville and Escaig, 1979) assumed that the partial dislocations cross-slip at preexisting constrictions. However, as the constricted dislocation cross-slips and dissociates spontaneously into its partials in the cross-slip plane, two constricted nodes are formed at the intersection of the primary and cross-slip planes. The movement of these nodes away from each other as the partials glide on the new slip plane provides the driving force for cross-slip. An analysis of this process results in a complex expression for Qcs which depends on the magnitudes of the local stresses acting on the partials. A simplified relationship for this process leads

362

S.V. Raj, I. S. Iskovitz, and A. D. Freed

to a linear dependence of Qcs on stress given by (Escaig, 1968a, b; Poirier, 1976) Qcs -- ( G Z b 4 / 1 8 7 5 F ) [ l n ( G b / 1 4 . 5 F ) ] ~

-

3bo'/2F)

(9a)

or

Qcs = Q0(1 - cr V*)

(9b)

where Q0 is the maximum activation energy for cross-slip given by Q0 - 5 x 10-4[(Gb 3(Gb/I")][ln(O.O7Gb/1-')] 0.5

(9c)

and the apparent activation volume, V*, is given by (9d)

V* = 3bQo/ZF

Note that V* -- A'b, where A* is the activation area for deformation. Application of Eqs. (8) and (9) to creep data obtained on polycrystalline Cu (Fig. 6) (Raj and Langdon, 1991 b) showed no good agreement between theory and experiment. For Cu, Qc decreases linearly with increasing values of or/G, and the

I

I

300

.

.

250 7 "5 200 E "3

"-6 150 O L O

I

.

Qe = 210 kJ mo1-1 m

-k

\.

'\

100

\. ~'--

0 50

.

Cu d = 250 ~m T = 673 K

Present investigation Jaffe and Dorn (1962) Escaig (1968b)

Qc"

Qcs Qcs ....

I

~ " - "

0

~

" "

~

I

5

13 =1 ~ ~

j 7 13' = 2

~'.-.11,~

9

I

I

10 15 ~/G (xl 04)

I

20

25

Comparison of the experimental activation energies for copper with those predicted by the cross-slip models. The values of/3' -- 1 and 2 are for low and high stacking fault energies, respectively. (Reprinted from Acta Metall., Volume 39, S. V. Raj and T. G. Langdon, Creep behavior of copper at intermediate temperatures. II. Surface microstructural observations, Pages 1817-1822, Copyright (1991), with kind permission from Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK.)

Chapter 8 Modelingthe Role of Substructure during Class M and Exponential Creep 363 nonlinear trend predicted by Eq. (8) is inconsistent with experimental observations (Fig. 6). Although Eq. (9) predicts this linear dependence of Qc, it underestimates the experimental values by more than a factor of 3 in the case of Cu (Fig. 6) (Raj and Langdon, 199 l b). In the case of NaC1, the Jaffe-Dorn cross-slip model given by Eq. (8) is reasonably close to the experimental data for Np ~ 5 and/~' = 1 (Fig. 7) (S. V. Raj and G. M. Pharr, unpublished research, Rice University, Houston, TX, 1986). In contrast, the Escaig model underestimates Qc by a factor of 4-20. Experimental observations of wavy slip morphology in A1 (Carrard and Martin, 1987, 1988), Cu (Raj and Langdon, 1991a) and NaC1 (Wawersik, 1984; Haasen et al. 1986; Skrotzki and Haasen, 1988) clearly suggest that cross-slip processes are important during creep. In the case of Cu, it was observed that wavy slip occurs at high stresses and high temperatures close to the transition region between class M and exponential creep (Raj and Langdon, 1991a). In this case, since wavy slip was observed only after a finite amount of deformation, it appears that a critical combination of stress, temperature, and strain conditions must be satisfied for these slip features to occur during creep. If wavy slip can be solely attributed to the cross-slip of screw dislocations, the existence of these criteria can be qualitatively understood if it is assumed that the local stress required to constrict the partial dislocations to form a unit dislocation attains a critical value only at high temperatures and high strains. This criterion is more likely to be attained in high stacking fault energy materials, such as A1, than in materials with lower values of F, such as Cu.

400 350

V A -.

I

300 O

250

I

NaCI

I

I

Single specimen tests Multiple specimen tests Jaffe and Dorn (1962) Escaig (1968b) Obstacle-controlled glide ~--DCI- = 230 kJ mo1-1 - -

E 200

"-6 150 O 100 50 0 0.0

2.0

4.0 6.0 cr/G (xl 04)

8.0

10.0

Comparison of the stress dependence of the experimentalactivation energies for creep of NaC1 single crystals with those predicted by the Escaig-Poirier cross-slip creep model (Escaig, 1968b; Poirier, 1976) and the obstacle-controlled creep model. (Reprinted with permission from Nix and Ilschner, 1980.)

364

S.V. Raj, I. S. Iskovitz, and A. D. Freed

In the absence of good theoretical models for cross-slip, empirical equations are often used to force a fit through the data, where it is necessary to make an a p r i o r i assumption that cross-slip is the dominant deformation mechanism in order to obtain the fitting constants. Nix and Gibeling (1985) and Nix et al. (1985) expressed the activation free energy for cross-slip, A Gcs, as exp(AGcs/kT)

= exp(-Qol/kT){exp(rbA*/kT)

- 1}

(10)

where a01 is the activation energy for cross-slip at zero stress. The e x p ( r b A * / k T ) term in Eq. (10) represents cross-slip events in the direction of the resolved shear stress while -1 represents the activation energy for dislocation motion in the opposite direction. Nix and Gibeling (1985) and Nix et al. (1985) assumed that A* = or' QOl/Gb, where c~' is a constant depending on the stacking fault energy of the material. Both Q01 and ~' are adjustable parameters that must be determined from experimental data assuming that cross-slip is the dominant mechanism. Analyzing the data in Fig. 7 for a cross-slip process and assuming typical values of G, Q01 ~ 185 kJ mo1-1 for Nacl if the probability ofbackjumps is assumed to be zero (i.e., the - 1 is deleted from Eq. (10)). Since A* ~ 300b 2 for cross-slip (Conrad, 1964; Evans and Rawlings, 1969; Bonneville and Escaig, 1979), the magnitude of c~' ~ 500-950 between 300 and 973 K. 2. Obstacle-Controlled Glide of Dislocations

Nix and Ilschner (1980) suggested that exponential creep is dominated by thermally activated, obstacle-limited dislocation glide (Kocks et al., 1975; Frost and Ashby, 1982), where a gliding dislocation is obstructed by immobile or "forest" dislocations in single-phase materials. Dislocation motion is limited by the cutting of these obstacles. The rate equation for this mechanism, assuming "rectangular" obstacles, is given by (Nix and Ilschner, 1980) = (4 x l O l 2 / M 3 ) ( ~ / G ) Z e x p [ - A F / k T { 1

- (~ - ~b)/~obs}]

(11)

where M is the Taylor factor, A F is the Helmholtz free energy required to move a dislocation past an obstacle without aid from an external stress, ~b is the back stress on the dislocation, and ~obs is the obstacle strength. The quantity ~ --~b represents an effective stress on the dislocation. The magnitude of AF, which depends on the type of obstacle, is typically 0 . 2 - 1 . O G b 3 for medium-strength obstacles such as forest dislocations (Forest and Ashby, 1982). Typically, values of A F = 0.4-0.6Gb 3 appear to describe the experimental data on Cu (Fig. 8) (Raj and Langdon, 199 l b) and NaC1 (Fig. 7) (S. V. Raj and G. M. Pharr, unpublished research, Rice University, Houston, TX, 1986) reasonably well. The magnitude of ~obs is given by (Taylor, 1934) O'obs =

(OtlM)Gb(p) ~

(12)

Chapter 8 Modelingthe Role of Substructure during Class M and Exponential Creep 365 I

Qg"

250 Y

E

200

I

I

Cu d = 250 ~m T = 673 K

Present investigation Obstacle-controlled glide Nix and Ilschner (1980)

ec"

300 m

I

Qe = 210 kJ mo1-1

_--x~.

~.O 150 ot.. o 100 O3 o 50

0

I

5

I

10

I

15

~r/G (xl 0 4)

I

20

25

Comparison of the experimental activation energies for copper with those predicted by the obstacle glide-controlledcreep model. (ReprintedfromActa Metall., Volume 39, S. V. Raj and T. G. Langdon, Creep behavior of copper at intermediatetemperature. III. A Comparison with theory, Pages 1823-1832, Copyright (1991), with kind permission from Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK.)

where Ot 1 is a geometric constant equal to about 0.5 when M -- 2 (Bird et al., 1969; Mecking and Kocks, 1981). Alternatively, Eq. (11) can be formulated in terms of V* with

V * - - AF/aob~

(13)

Qc - AF{ 1 - (or - Crb)/erobs}

(14)

Equation (11) gives

As shown in Fig. 7 (S. V. Raj and G. M. Pharr, unpublished research, Rice University, Houston, TX, 1986) and Fig. 8 (Raj and Langdon, 1991b), the magnitudes of Qc determined from Eq. (14) are in reasonable agreement with the experimental data for NaC1 and Cu, respectively. These predicted curves were calculated assuming Orb ~ 0.5or for Cu and NaC1. Figures 7 and 8 suggest a direct transition from Qc ~ Q1 to Qc < Q1 with increasing cr/G for both these materials, which is consistent with the observed transition from class M to exponential creep (Figs. 9 and 10).

366

S.V. Raj, I. S. Iskovitz, and A. D. Freed

10-4

I

"-

'

I

' I'1'

I

'

I

' I'll_

10-5

10-6

10-7 .o (.9 a

-

S h e r b y and BurkeO~]al~ /~ (1967) ~ (~//

It

_-_~z__ 973 K lO -8 - - - ~ - - - 623 K

,~

Cu

10-9

d = 250 l.tm

10-10

10-11

4.3 1.0

10-12 L 10-4

9 V [3 A 9 II 9

T (K) 623 673 723 773 823 873 923 973

10-3 ~r/G

10-2

Normalized creep rate vs. normalized stress for copper showing the transition from class M creep with n ~ 4.3 to exponential creep. (Reprinted from Acta Metall., Volume 39, S. V. Raj and T. G. Langdon, Creep behavior of copper at intermediate temperatures. III. A Comparison with theory, Pages 1823-1832, Copyright (1991), with kind permission from Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK.)

3. Effect of Stacking Fault Energy on Class M Creep The effect of stacking fault e n e r g y on the creep b e h a v i o r of metals and alloys has b e e n studied e x t e n s i v e l y u n d e r class M creep conditions (Barrett and Sherby, 1965; D a v i e s et al., 1965; S h a l a y e v et al., 1969; Singh D e o and Barrett, 1969;

Chapter 8

Modeling the Role of Substructure during Class M and Exponential Creep

367

1012

HaCI 108

10q .~9 100

(.D

.~

lo-Zl

10-8

(,001) E=0.2 T (K) (5 373

[] Z~ 0

V 0 o

'o X

I~

473 573 673 773

II .=I a= I0

873 973 t598t,lo23 1.0 5.0

10-12 10-1G 10-6

10-5

lO-q

O/G

10-3

10-2

10-1

~a[~llj~|m[i]i Normalized creep rate vs. normalized stress for NaC1 single crystals showing the transition from class M creep with n ~ 5.0 to exponential creep at (o-/G)pLB ~ 3 x 10 -4 (Raj et al., 1989; Raj and Pharr, 1989). (Reprinted from Raj et al., Mater. Sci. Eng., A l l 3 , 1989, 161-175.)

Vandervoort, 1969; Oikawa and Karashima, 1971; Johnson et al., 1972; Kozyrskiy et al., 1972a, b; Mohamed and Langdon, 1974; Okrainets and Pishchak, 1979; Yang et al., 1987). As a result, it is now well established that the creep rate decreases by a factor of approximately 6000 for about an order magnitude decrease in E (Barrett and Sherby, 1965). Stacking fault energy influences both dislocation climb (Weertman, 1965; Argon and Moffatt, 1981; Kong and Li, 1993) and crossslip (see Section II.B. 1) so that as 1-'/Gb decreases, the rate of climb and cross-slip decrease correspondingly. The exact manner by which E affects the parameters in the class M creep Eq. (3) is poorly understood at present. However, most theoretical treatments of the problem assume that the extended dislocations form a constriction prior to cross-slip or climb so that the effect of E enters the creep equation through its influence on the constriction energy. At present, both theory (Argon and Moffatt, 1981; Argon and Takeuchi, 1981; Kong and Li, 1993) and experiment (Barrett and Sherby, 1965; Mohamed and Langdon, 1974) suggest that the A parameter in Eq. (3) is a function of ( F / G b ) q, where q is the stacking fault energy exponent.

368

S.V.Raj, I. S. Iskovitz, and A. D. Freed

Barrett and Sherby (1965) first proposed that ~ cx 1-'3"5 in the class M creep regime based on their experimental observations on pure metals. In a later extension of these results to other metals and alloys, Mohamed and Langdon (1974) observed that q ~ 3. Several other values of q have also been reported in the literature with q varying between 1 and 4 (e.g., Oikawa and Karashima, 1971). In addition, alternative expressions to the power-law relation have been proposed. For example, Oikawa and Karashima (1971) observed ~ cx e x p ( F / G b ) for several copper alloys. However, the power-law relation with a value of q ~ 3 is more commonly accepted in the literature. 4. Power-Law Breakdown Criterion

Since both the climb and cross-slip velocities are dependent on F/Gb (Poirier, 1976; Argon and Moffatt, 1981; Kong and Li, 1993), it is expected that recovery processes, and hence the rate of evolution of the steady-state substructure, should decrease with decreasing values of F / G b . Thus, nondiffusional creep mechanisms are likely to dominate at lower values of normalized creep rate with decreasing values of F / G b . As shown in Fig. 11, this is indeed the case when experimental creep data for a number of face-centered cubic (fcc) metals with F/Gb varying between 2.3 x 10 -2 for A1 to 2 x 10 -3 for Ag are compared with each other on normalized plots (Raj, 1986; Raj and Langdon, 1991 b). These observations reveal that the normalized creep rates for A1 are higher than those for Ag by as much as three to four orders of magnitude (Fig. 11). An important outcome of these results is that the power law breaks down at a constant value of ( t T / G ) P L B '~ 5 • 10 -4 irrespective of the stacking fault energy. Similarly, (or/G)PLB ~'~ 3 • 10-4 for NaC1 single crystals (Fig. 10) (Raj and Pharr, 1989) and 5.5 x 10 - 4 for an A1-3%Mg alloy (Wang et al., 1993). In contrast, the normalized creep rate, (~kT/D1Gb)PLB, where Dl is the diffusion coefficient for lattice self-diffusion, at which the powerlaw relation deviates from n ~ 4.5 for the pure metals, decreases with decreasing F/Gb as (~kT/DIGb)PLB oc (F/Gb) 35 (Fig. 12) (Raj and Langdon, 1991b). This exponent is similar to the values q = 3.0-3.5 observed in the class M creep regime (Barrett and Sherby, 1965; Mohamed and Langdon, 1974). Three important points can be made regarding Figs. 11 and 12. First, the existence of an almost constant value of (or/G)PLB for these materials (Fig. 11) is significant since a number of substructural features depend on normalized stress (Bird et al., 1969; Takeuchi and Argon, 1976; Raj and Pharr, 1986a). Therefore, the results shown in Fig. 11 appear to imply that the power law breaks down, at least in part due to a change in one or more of these microstructural characteristics. Second, the decrease in (~kT/D1Gb)PLB with decreasing (F/Gb) (Fig. 12) can be attributed to the increasing dominance of nondiffusional creep mechanisms, such as obstacle-controlled glide, over high-temperature climb at lower values of normalized creep rates in the low stacking fault energy metals. This is to be

Chapter 8

Modeling the Role of Substructure during Class M and Exponential Creep

104

i ,l't'l'l

~ '~'J'PI

i ,I I1'1'! i

369

! ,1 '1,1, I i ,i 'l'i' AI

102

"--

POWER-

POWER,-

LAW CREEP

100

,=" 0 "x

"

10-2

~=~ 10-6

.a

"i

10- 4 __

I--

LAW BREAKDOWN

b ,

i Ag

Cu

_

w

m

AI--x \

10 -10 - -

///

10-12 __

"']

i

Ni

~-Cu !---'Au__ Ag

I1/

1,0

--

--

I

--

10-111 __

lO-16

I ,1,1~1,1 10-6

10-5

I n lvhlll

i ~111~1~1 ! =l=l,lJl

10-1t

10-3

10-2

I a i~l~lJ 10-1

0/13 Normalized creep rate vs. normalized stress for several fcc metals showing that the deviation from a class M creep relation with a stress exponent of about 4.5 first occurs at (cr/G)PLB ~ 5 X 10 -4 (Raj, 1986; Raj and Langdon, 1991b. (Reprinted from Scr. Metall., Volume 20, S. V. Raj, The effect of stacking fault energy on the creep power-law breakdown criterion in FCC metals, Pages 1333-1338, Copyright (1986), with kind permission from Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK.)

expected since it is well established that the extent of recovery decreases with decreasing stacking fault energy. The significant implication of this observation is that the transition temperature, Tc, above which Qc ~ Q1 in a plot of Qc against T~ Tm is expected to increase as the stacking fault energy decreases. Indeed, as shown in Table 2 (Raj and Langdon, 199 l b), this appears to be the case when the values of Tc are compared with 1-'/Gb for A1, Cu, and Sn; but it is cautioned that more experimental data are required to confirm this trend. Third, the Sherby-Burke criterion, which predicts that the power law breaks down at an almost constant

370

S.V. Raj, I. S. Iskovitz, and A. D. Freed

10-8

I

i

I llWll

10-9

I

I

~lil-

Ni 0

-

u

w

en

10-10

Cu

l

--I ,,t

0 Pb

A

l-JIr "W

1.0

10-11

10"12

Au

3.5

~

-AgC

10-13 ~ 10-3

/ I

(O/G)pLB = 5 X 10-q

I

I

~lllll

10-2

I

=

I Jill

10-1

(I/fib) Variation of the normalized creep rate at which the power-law breaks down for several fcc metals (i.e., at ((7/G)PLB ~ 5 • 10 -4) with the normalized stacking fault energy. (Reprinted from Acta Metall., Volume 39, S. V. Raj and T. G. Langdon, Creep behavior of copper at intermediate temperatures. III. A Comparison with theory, Pages 1823-1832, Copyright (1991), with kind permission from Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK.)

ABLE~ Effect of Stacking Fault Energy on the Transition Temperature for Several Materials a Material AI Cu Sn

I-" (mJ m -2)

F/Gb

Tc/Tm

200 55 3

23 7 0.6

0.55 >0.72 0.85

Note: Tc represents the transition temperature at which Qc "~ Ql in a plot of Qc against T. aFrom Raj and Langdon ( 1991 b).

Chapter 8 Modelingthe Role of Substructure during Class M and Exponential Creep 371 value of k/D1 ~ 1013 m -2 (or equivalently, ( k k T / D 1 G b ) p L B ~ 10 -8) (Sherby and Burke, 1967), appears to be valid only for high stacking fault energy materials (17' > 150 Jm -2) (Fig. 12) (Raj, 1986).

SUBSTRUCTURE FORMATION IN NaCI SINGLE CRYSTALS IN THE CLASSM AND EXPONENTIAL CREEP REGIMES Any attempt to develop realistic creep models must rest on a reasonable understanding of the processes leading to the formation and evolution of dislocation substructures during creep as well as the effect of stress, temperature, and strain on the microstructure. These modeling efforts require detailed qualitative and quantitative characterization of the microstructure to provide a truer picture of creep. In particular, a microstructural characterization of both the class M and the exponential creep regions is essential in understanding why the power-law relation breaks down. For example, it is not clear at present whether the transition to an exponential stress dependence involves a sudden breakdown in the subgrain microstructure (Pharr, 1981) or whether it is due to a more gradual transformation in the substructure from a subgrain morphology present at high temperatures and low stresses to a microstructure consisting of dislocation tangles and cells characteristic of low-temperature deformation (Michel et al., 1973; Challenger and Moteff, 1973; Kestenbach et al., 1976, 1978).

A. Effectof Normalized Stresson Creep Substructure 1. Distinction between Cells and Subgrains The terms "cells" (C) and "subgrains" (SG) have been traditionally used to describe the morphologies of certain low-energy dislocation substructures formed as a result of the clustering of a uniform distribution of dislocations. The distinction between cells and subgrains is generally made with respect to differences in the appearance of the cell walls and the subgrain boundaries (Thompson, 1977). Cells consist of broad, diffused boundaries containing dislocation tangles. In contrast, subboundaries (Sb) are narrow and well defined, where the boundaries have a larger misorientation than the cell walls. Figure 13a shows examples of these two substructural features in an etched NaC1 crystal after creep. The cell walls are resolvable into individual etch pits, unlike the subordinates. Each individual etch pit represents the point of termination of the dislocation line at the free surface. The cell boundary misorientation angle has been estimated to be about 0.1 ~ (Raj and Pharr, 1989). A close examination of Fig. 13 reveals that there are very few dislocations inside the cells. Two types of

372

S.V. Raj, I. S. Iskovitz, and A. D. Freed

/ l [ l ~ l l ~ | l l l l Creep substructure in a NaCI single crystal specimen deformed to (a) a true strain of

0.9 at 873 K under a normalized stress of 2.2 x 10 - 4 showing cells (C) and subboundaries (Sb); (b) a true strain of 0.37 at 973 K under a normalized stress of 7.5 x 10-5 showing cells (C), and primary (P) and secondary (S) subboundaries. (Reprinted from Raj and Pharr, Mater. Sci. Eng., A122, 1989, 233-242.)

subboundaries, primary (P) and secondary (S), have been identified (Fig. 13b) (Raj and Pharr, 1989). In comparison to the primary subboundaries, where the etch pits along the boundary cannot be distinguished individually at high magnification, the secondary subboundaries consist of individually distinct, but partially overlapping, etch pits. Thus, the primary subboundaries have a larger misorientation angle than the secondary subboundaries, which in turn are more misoriented than the cell boundaries. Dislocation substructures may not always conform to these ideal definitions, so a precise identification of the microstructure may not be possible in some

Chapter 8 Modelingthe Role of Substructure during Class M and Exponential Creep 373 instances. Additional difficulties may arise due to the limitations of the observation technique used for studying the microstructures. For example, a cell boundary can be misidentified as a subboundary in transmission electron microscopy (TEM) owing to the smaller field of view in this technique. For this reason, dislocation substructures are best studied using etch pit techniques (Takeuchi and Argon, 1976; Blum, 1993), although TEM has a decided advantage in the measurements of the boundary misorientation angles. As a result, there is some confusion in the use of these terms, especially in the earlier literature, where the terms "cells" and "subgrains" are used interchangeably to describe the same substructural feature (e.g., Feltham and Sinclair, 1962-1963).

2. Qualitative Descriptions of the Microstructures Formed in the Class M and Exponential Creep Regions The fact that (o'/G)PLB is similar for NaC1 single crystals (Fig. 10) (Raj and Pharr, 1989) and for several fcc metals (Fig. l l) (Raj, 1986; Raj and Langdon, 1991b) is interesting because many creep microstructural parameters, such as ds and p, are stress-dependent. This observation suggests that the transition from class M to exponential creep with increasing values of normalized stress, or/G, and decreasing homologous temperatures, T~ Tm, occurs when there is some critical transformation in the microstructures. Although a wide body of compiled information is now available on creep substructure (Sherby and Burke, 1967; Bird et al., 1969; Takeuchi and Argon, 1976; Nix and Ilschner, 1980), most of these observations were conducted in a limited range of stresses and temperatures primarily corresponding to the class M creep regime. As a result, these early studies do not generally provide much information regarding the exponential creep region, although they form the basis of our current understanding of class M creep. Limited microstructural studies have been conducted on AISI 316 stainless steels in the class M and exponential creep regions (Michel et al., 1973; Challenger and Moteff, 1973; Kestenbach et al., 1976, 1978). These observations suggest that the equiaxed subgrain microstructure, formed at low stresses and high temperatures corresponding to the class M creep region, is replaced by a uniform distribution of dislocations at high stresses and low temperatures well within the exponential creep regime (Fig. 14) (Kestenbach et al., 1976, 1978). This transformation from a subgrain microstructure occurs gradually, passing through several intermediate stages involving the formation of other substructural features, such as elongated subgrains and cells (Michel et al., 1973; Kestenbach et al., 1978). Similar observations were also reported for NaC1 single crystals deformed in the [001] direction under a constant compressive stress (Raj et al., 1989; Raj and Pharr, 1989). Large equiaxed primary subgrains were observed at 973 K and at a value of or/G -- 7.5 x 10 -5 (Fig. 13b) corresponding to the class M creep region in Fig. 10. Two other microstructural features are also visible within the primary

374

S.V.Raj, I. S. Iskovitz, and A. D. Freed 1150

I

V

9 Garofalo et al. (1963) U Barnby (1966) O@ Hopkin and -Taylor (1967) [] 9 Cuddy(1970) (3 9 Challenger and _ Moteff (1972) A A 9 Kestenbach et ai. (1976)

1100

1050

1000

|

p-

19

9

A

950 A

9

900

m

c3, 850

800

~ - - Su bg rai ns ~

@ Mixture ~

I

100

I

r

200

O/~

<> Cells

300

Variation in the creep substructure in 316 stainless steels at different stresses and temperatures. (Reprinted with permission from Kestenbach et al. (1976) and Metall. Trans.)

~|[lllJltl|lg

subgrains: secondary subboundaries, which divide the primary subgrains by forming random interconnecting networks, and equiaxed cells, which are present within the secondary subgrains. These microstructures were generally clean and well defined in comparison to those produced at lower temperatures and higher stresses. True steady-state creep behavior was observed under these conditions, so these microstructures represent steady-state creep substructures (Raj et al., 1989; Raj and Pharr, 1989). These secondary boundaries and cellular features are also evident in the published micrographs of creep-deformed LiF (Streb and Reppich, 1973) and NaC1 (Eggeler and Blum, 1981) single crystals. The primary and secondary boundaries also from at a higher value of o'/G = 10 -4, as observed in the sections transverse (Fig. 15a) and parallel (Figs. 15b and 15c) to the stress axis. The longitudinal section was cleaved from the crystal, and the cleavage facets are visible in Figs. 15b and 15c. Equiaxed primary subgrains, subdivided by secondary subboundaries and cells, are visible in Fig. 15a. Similarly, a single primary subboundary and several interconnecting secondary subboundaries can be seen in Fig. 15b. A high-magnification view of region B in Fig. 15b shows that the cell boundaries consist of a loose configuration of etch

9

0"Q e...

9

c~

0~

O

~.

~:l[q~m;tml,.91

Microstructure of a specimen deformed to a true strain of 0.35 at 973 K under a normalized stress of about 10 -4 showing sections (a) transverse; and (b) and (c) parallel to the stress axis" (c) is a high-magnification view of region B in (b). An increase in the cell width (e.g., at A) is evident in (a). (Reprinted with permission from Raj and Pharr, Mater. Sci. Eng., A122, 1989, 233-242.)

Chapter 8 Modelingthe Role of Substructure during Class M and Exponential Creep 377 pits, whereas those in the secondary subboundaries are present as distinguishable, but nonresolvable etch pits (Fig. 15c). The similar microstructures observed in the transverse (Fig. 15a) and longitudinal (Figs. 15b and 15c) sections clearly demonstrate that these substructural features are representative of the bulk and are not due to any polishing artifact, as suggested by Vogler et al. (1991). More importantly, cells have been reported to form in a large number of materials both in compression and tension creep, thereby suggesting that they are not specific to NaC1, testing conditions, or metallographic preparation techniques (Gupta and Strutt, 1967; Hasegawa et al., 1970, 1971; Feltham and Sinclair, 1972; Streb and Reppich, 1973; Goel et al., 1983; Soliman et al., 1983; Ginter et al., 1984; Lee and Nam, 1988). A comparison of Figs. 13b and 15a reveals a significant increase in the width of a few cell boundaries within some of the primary subgrains with an increase in or/G (e.g., A in Fig. 15a). Such widening of the cell boundaries becomes increasingly predominant with increasing normalized stresses. This increase in the width of the cell boundaries appears to be correlated with a corresponding decrease in the number of secondary subboundaries and a tendency toward non-steady-state creep behavior (Raj et al., 1989; Raj and Pharr, 1989). An example of this microstructural variation is shown in Fig. 16a for a specimen deformed at a value of c r / G = 2.2 x 10-4; here it is seen that, while the primary subgrains are still equiaxed, the cell boundaries within many of them are extremely coarse (e.g., A in Fig. 16a), in contrast to those shown in Fig. 13b. In this case, secondary subboundaries were rarely observed. However, narrow cell boundaries and secondary subboundaries are evident within some of the primary subgrains in other regions of the specimen (e.g., B in Fig. 16a). It is clear from a comparison of Figs. 13b and 16a that the steady-state microstructure involves a refinement of the cell boundaries and the formation of secondary subboundaries. It is important to note that these microstructures are formed under class M creep conditions when n ~ 5 (Fig. 10), although no real steady state was attained when or/G > 10 -4. These observations

~l[llll:tgl[tll Variation in the creep substructure in NaC1 single crystals deformed at different values of normalized stress. (a) or/G ~ 2.2 x 10-4: substructure in the class M creep region showing that the cell boundaries are wider in subgrain A than in subgrain B, although both primary subgrains are equiaxed. (b) cr/G ~ 3.3 x 10-4: elongated (A) and equiaxed (B) subgrains containing coarser and narrower cell boundaries, respectively, are visible close to (cr/G)PLB. (C) cr/G ~ 4.0 x 10-4: elongated (A) and equiaxed (B) primary subgrains showing different internal cellular and dislocation microstructures within them. The elongated subgrain boundaries are oriented along the (110) crystallographic direction, and these subgrains exhibit a "ladder-like" appearance. (d) ~r/G ~ 7.5 x 10-4: microstructure of the exponentialcreep region showing long subboundaries; light patches are visible in an otherwise uniform distribution of dislocations. (e) or/G ~ 1.4 x 10-3: uniform distribution of dislocations (Raj et al., 1989; Raj and Pharr, 1989). (Reprintedwith permissionfrom Raj, Whittenberger, and Pharr, Mater. Sci. Eng., All3, 1989, 161-175.)

378

S.V.Raj, I. S. Iskovitz,and A. D. Freed

contradict the commonly held opinion that subgrain formation is a sufficient condition for the occurrence of steady-state creep (Vogler et al., 1991" B lum, 1993). Similarly, Mo single crystals did not exhibit steady-state creep behavior even after 42% strain, despite the fact that a subgrain microstructure had developed during deformation (Clauer et al., 1970). The first evidence of a change in the appearance of the primary subgrains occurs at a value of or/G = 3.3 x 10 -4, which is close to the point of transition from class M to exponential creep (Fig. 10). In this case, some of the subgrains are more elongated (e.g., A in Fig. 16b) than others (e.g., B in Fig. 16b), and this increase is more pronounced at tr/G = 4.0 x 10 -4, as is evident through a comparison of subgrains A and B in Fig. 16c. These elongated subgrains exhibit a ladder-like morphology, with their long boundaries oriented along the (110) primary slip direction. Similar microstructures of elongated and equiaxed subgrains, often distributed in alternate bands of narrow and wide subgrains, have been observed in many materials (Gupta and Strutt, 1967; Clauer et al., 1970; Hasegawa et al., 1970, 1971; Feltham and Sinclair, 1972; Poirier, 1972b; Orlov~i et al., 1972a; Htither and Reppich, 1973; Kestenbach et al., 1978). Significantly, there is also a variation in the internal microstructures within these two types of subgrains. The elongated subgrains (e.g., A in Fig. 16c) have a higher dislocation density, a larger number of cells with coarser boundaries, and fewer secondary subboundaries than their equiaxed neighbors (e.g., B in Fig. 16c). These observations suggest an apparent correlation between the aspect ratio of a subgrain and its internal microstructure. Hasegawa et al. (1970, 1971) have reported similar observations on Cu single crystals, where the elongated subgrains appeared to creep at slower rates than the equiaxed ones. These results suggest that the elongated subgrains are harder than their equiaxed neighbors due to differences in their internal dislocation microstructure. Earlier observations have shown that the elongated subgrains tend to become equiaxed by subboundary migration (Hasegawa et al., 1970, 1971; Clauer et al., 1970). At a value of or/G --- 7.5 x 10 - 4 and 473 K--well within the exponential creep region (Fig. 10)--no cells and subgrains are observed below e < 0.2 (Fig. 16d). Instead, the microstructure consists of uniform distribution of dislocations interspersed with light patches of lower etch pit density and long subboundaries oriented approximately along the (100) direction. These light patches of low dislocation density are probably the first indications of cell formation through dislocation annihilation, since they were not observed at lower temperatures and higher stresses. However, well developed subgrains were observed at e ~ 1.0, although their formation did not result in steady-state creep. At very high values of normalized stress corresponding to or/G = 1.4 x 10-3, the microstructure consists primarily of a uniform distribution of dislocations and few or no subboundaries (Fig. 16e). The substructure in the exponential creep regime is strongly dependent on strain and temperature, where an increase in both of these parameters tended to promote subgrain formation (Raj and Pharr, 1989).

Chapter 8 Modelingthe Role of Substructure during Class M and Exponential Creep 379 The general picture that emerges from these results is that the transition from class M to exponential creep is associated with gradual variations in the microstructure which involve a complex and subtle interaction between different elements of the substructure. These substructural changes occur both at the coarser level of the primary subgrains and at the finer levels of the cells, secondary subboundaries, and dislocations. Steady-state creep occurs when there is a dynamic equilibrium in the evolution of all these microstructural features. Figure 17 schematically summarizes the microstructural observations reported on NaC1 single crystals (Raj et al., 1991). The lower portion of the figure illustrates the idealized microstructures formed at different values of normalized stress above and below (~/G)vLB while the upper portion correlates the major substructural processes occurring at each microstructure with a corresponding simplified geometric representation. Figure 17 also lists the increasing probability of the recovery mechanisms likely to dominate at different values of normalized stress. Typical values of k k T / D 1 G b are also included in Fig. 17. The effect of strain or temperature would be essentially to increase the probability of attaining this equilibrium substructure (i.e., from right to left in Fig. 17). At high stresses and low temperatures corresponding to the exponential creep regime, the substructure consists predominantly of forest dislocations and a few subboundaries lying along secondary slip planes at low values of strain, and mainly of subgrains when e > 1 (Fig. 17a). Glide polygonization can lead to the formation of the light patches with lower dislocation density. As mentioned earlier, these light patches appear to be the first evidence of cell formation. Other recovery processes, such as cross-slip and dynamic recrystallization, can be important, depending on strain, temperature, and stacking fault energy. At lower stresses and higher temperatures close to the power-law breakdown criterion, but still within the exponential creep regime, the substructure consists of a mixture of elongated (hard) and equiaxed (soft) subgrains (Fig. 17b). The elongated subgrains contain a high density of dislocations and coarse-walled cells. In contrast, the equiaxed subgrains have an internal microstructure of thin-walled cells, secondary subboundaries, and fewer "free" dislocations. This difference in the internal microstructures of these two types of subgrains is expected to provide the driving force for subboundary migration, which ultimately leads to the formation of equiaxed primary subgrains (Fig. 17c). Considerable evidence now suggests that subboundary migration can contribute typically between 3 and 25% to the total creep strain, although contributions as much as 100% have also been reported (Exell and Warrington, 1972; Vollertsen et al., 1984; Caillard and Martin, 1987; Biberger and Blum, 1988, 1992a, b). Careful experiments suggest that subboundary migration occurs after large stress reductions, thereby allowing the microstructure to attain the equilibrium subgrain size characteristic of the reduced stress (Ferreira and Stang, 1979, 1983; Eggeler and Blum, 1981; Goel et al., 1983; Soliman et al., 1983; Mohamed et al., 1985).

380

S.V. Raj, I. S. Iskovitz, and A. D. Freed

~ l l l T l ~ | l , ' l l l ~ Schematic idealized creep substructures, and their simplified geometric representation, formed in NaCI single crystals in the exponential and class M creep regions. The different recovery mechanisms likely to dominate under various creep conditions are also indicated. (a) A high density of dislocations interspersed with light patches; a few subboundaries form when e < 0.2 but are transformed to well-formed subgrains when e > 1.0 if recovery mechanisms are sufficiently rapid. Some materials may recrystallize if cross-slip and glide polygonization do not occur rapidly enough to bring about significant recovery. (b) Elongated and equiaxed subgrains with different internal cellular and dislocation microstructures result in subboundary migration. Cell boundary refinement probably occurs by cross-slip and dislocation climb. (c) Equiaxed subgrains, cell boundary refinement, and secondary subboundary formation occur in the class M region. Steady-state creep is not observed. (d) An equilibrium substructure of equiaxed primary subgrains, secondary subboundaries, and narrow-walled cells is observed under conditions where steady-state creep is observed.

Other recovery mechanisms, such as dislocation climb, cross-slip, and subgrain rotation, are also expected to play an important role. The shape of the primary subgrains is essentially equiaxed and stable in the class M creep region, but there are differences in their internal microstructures from one subgrain to another. Some contain a large number of thick-walled cells while others contain mainly

Chapter 8 Modelingthe Role of Substructure during Class M and ExponentialCreep 381 thin-walled cells. Cell boundary refinement is dominant in this region, leading to a narrowing of the cell walls. There is also a corresponding decrease in the dislocation density within the cells and an increase in the density of secondary subboundaries (Fig. 17c). Steady-state creep does not occur until cell boundary refinement is completed and a dynamic equilibrium exists between different elements of the microstructure (Fig. 17d). Under these conditions, the microstructure consists of equiaxed primary subgrains containing cells and a network of secondary subboundaries. The cell boundaries appear to be the major sources of dislocations since they are likely to break up more easily than a subboundary to accommodate strain inhomogeneity in the material (Fig. 18) (Raj and Langdon, 1991b). Evidence that subboundaries, in contrast to cell boundaries, are mechanically stable at high stresses is presented in Section IV. Similar interpretations were advanced by Gupta and Strutt (1967), who also suggested that cell boundary disintegration during creep provided mobile dislocations for deformation.

3. Quantitative Descriptions of the Stress Dependence of the Dislocation Substructure Figure 19 shows the variation of dc/b and ds/b with or/G for NaC1 single crystals, where the data have been obtained at different temperatures and various values of strain (Raj et al., 1989; Raj and Pharr, 1989). The magnitude of (or/G)pL~ is also indicated on the figure. Three points may be noted from Fig. 19. First, cells and subgrains, once formed, attain an equilibrium size inversely proportional to and primarily determined by the normalized stress. The cell and subgrain sizes do not vary significantly with strain and temperature, which is consistent with previous observations (Bird et al., 1969). Second, subgrains and cells are stable even when ~r/G > (cr/G)pLB, thus suggesting that the class M to exponential creep transition does not involve any catastrophic break-up of cells and subgrains as postulated by Pharr (1981). Third, the two plots appear to converge at a value of or/G > 2 x 10 -3. This convergence in the two plots is expected to occur when both dc and ds approach the average spacing between the forest dislocations; i.e., dc ~ ds "~ (p)O.5. It is evident from Fig. 19 that both de and d~ decrease with increasing stress in accordance with the experimentally determined equations dc/b = 930(G/~) ~

(15a)

ds/b = 1 5 ( G / a ) 11

(15b)

Equations (15a) and (15b) suggest that the cell size exhibits a weaker dependence on the normalized stress than the subgrain size. Although cells have been observed in several materials (Gupta and Strutt, 1967; Hasegawa et al., 1970,

382

S.V. Raj, I. S. Iskovitz, and A. D. Freed

Schematic showing (a) cells and dislocations within a primary subgrain and (b) emmission of expended dislocations from a cell wall and its glide toward the opposite cell wall; bl and b2 are the Burgers vectors of the partial dislocations. (Reprinted from Acta Metall., Volume 39, S. V. Raj and T. G. Langdon, Creep behavior of copper at intermediate tempratures. III. A Comprison with theory, Pages 1823-1832, Copyright (1991), with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK.)

/l[llll,'|ll|tl

1971; Feltham and Sinclair, 1972; Streb and Reppich, 1973; Goel et al., 1983; Mohamed et al., 1985; Lee and Nam, 1988), quantitative information on the stress dependence of dc was not always reported. Previous measurements reported by Goel et al. (1983) on an AI-Zn alloy also suggest that dc is weakly dependent on the applied stress.

Chapter 8 Modelingthe Role of Substructure during Class M and Exponential Creep 383

~l[lll,ll~lllillll Variationsin the normalized cell and subgrain sizes with normalized stress for sodium chloride single crystals. (Reprinted with permission from Raj and Pharr, Mater. Sci. Eng., A122, 1989, 233-242.)

The stress dependence of ds (or de) can be expressed in a general form as ds/b = K(G/a)

m

(16)

where m and K are constants. A comparison of Eqs. (15b) and (16) suggests that K -- 15 and m = 1.1. The magnitude of these constants fall within the range of values reported for other ionic crystals (Raj and Pharr, 1986a). A compilation of the experimental values of m and K for several materials suggests that 0 < m < 2 and 0.01 < K < 105 (Raj and Pharr, 1986a). Although several values of m and K have been reported in the literature, it was demonstrated in an earlier study that these variations in the magnitudes of m and K are due to random error in the experimental measurements of subgrain size for most materials (Raj and Pharr, 1986a). As a result of this statistical scatter, the experimental values of K were

384

s.v. Raj, I. S. Iskovitz, and A. D. Freed

shown to be dependent on m through log K = - 2 . 8 m + 4.2

(17)

so that K ~ 23 when m = 1. These values were shown to fit most experimental data reasonably well, so Eq. (16) can be expressed as the universal relation

ds/b-- 23(G/o')

(18a)

Owing to the unique relation between m and K expressed by Eq. (17), several other combinations of these constants exist which would predict almost identical values of ds/b for a constant magnitude of or/G. Nevertheless, a value of m = 1 appears to be justified since experimental values of m approach unity whenever the number of measurements of the subgrain size exceeds 15 (Raj and Pharr, 1986a). The data relating dc/b with or/G for high-temperature creep are insufficient in comparison to the vast amount of similar data on ds. Thus, a universal relation, such as Eq. (18a), is unavailable. However, Eq. (15a) can be reformulated, assuming m = 0.5, to give

dc/b .~, 600(G/or) ~

(18b)

One reason for this apparent lack of data may be due to a misidentification of cells as subgrains in the literature. In addition, the terms "cells" and "subgrains" have been used interchangeably in the older literature (e.g., Feltham and Sinclair, 1962-1963), confounding the proper identification of the microstructure. The other possibility is that in some materials the random dislocations within the primary subgrains may prefer to cluster into three-dimensional dislocation networks rather than cells. Thus, the clustering of random dislocations may be envisioned as shown schematically in Fig. 20. In this case, the formation of primary subgrains is always favored, but additional dislocation rearrangement may sequentially favor the formation of either cells and secondary subboundaries or three-dimensional dislocation networks as the steady-state microstructure. Other microstructural features, such as microbands and recrystallized grains (Hansen and Jensen, 1991), can also form under certain deformation conditions, so the microstructural picture can be more complicated than illustrated by Fig. 20. Ultimately, the type of microstructure formed will be dictated by the need to lower the free energy of the deforming solid. As discussed in Section II. D, the magnitude of (cr/G)PLB is approximately constant for many materials (Figs. 9-11), thus suggesting that the transition from class M to exponential creep is due to some definite microstructural changes occurring in the material. In an earlier study, Pharr (1981) proposed that this transition involves a catastrophic breakdown in the subgrain microstructure so that (o'/G)PLB then represents a measure of the average strength of subboundaries in the material. However, the description of the microstructural changes occurring in the class M and exponential creep regions in Section III.A.2 clearly suggests that there is no sudden breakdown in the subgrain microstructure, which in fact can extend

Chapter 8 Modelingthe Role of Substructure during Class M and Exponential Creep 385 Uniform dislocation substructure

Primary subgrain formation

I

Threedimensional networks

Ces

I

Secondary subboundaries .-I Steady-state v I microstructure / l [ l l l l ' | | ~ [ l l Schematic showing the differentcreep substructures that may develop from a clustering of an initial uniform distribution of dislocations.

to values of cr/G > (cr/G)PLB. Instead, these observations suggest a gradual increase in the width of cell boundaries which appear to result in an elongated subgrain microstructure at the point of transition between class M and exponential creep. Measurements of the average dimensions of the cell walls, L h, and the cell interiors, L s, obtained over a wide range of strain, stress, and temperature reveal a remarkable correlation between the ratio Lh/L s and (or/G)PLB, despite a large amount of scatter (Fig. 21) (Raj and Freed, 1992). This scatter is largely influenced by the effect of strain and temperature, both of which affect the degree of refinement of the cell boundaries (S. V. Raj and G. M. Pharr, unpublished research, Rice University, Houston, TX 1986; Raj and Pharr, 1992; Raj and Freed, 1992). Therefore, an increased tendency toward a steady-state microstructure appears to result as Lh/L s tends toward a constant value of 0.2, assuming other elements of the substructure also attain dynamic equilibrium. Noting that the volume fraction of the cell boundaries, feb, is related to L h / L s through (Raj and Freed, 1992). /cb=

1-

[1/(1 + Lh/LS)] 3

(19)

we see that a value of Lh/L s -- 0.2 corresponds to feb = 0.42.2 2Equation (19) is a general derivation of fcb, in contrast to an earlier definition (Raj and Pharr, 1992), since it is independent of the geometry of the cell.

386

S.V. Raj, I. S. Iskovitz, and A. D. Freed

Ratio of the average dimensions of the cell width to the cell interior vs. normalized stress for NaCI single crystals deformed in the class M and exponential creep regimes. This ratio tends toward a constant value of about 0.2 under conditions when steady-state creep is attained. (Reprinted from Scr. Metall. Mater., Volume 27, S. V. Raj and A. D. Freed, Effect of strain sand stress on the relative dimensions of the 'hard' and 'soft' regions in crept NaC1 single crystals, Pages 1741-1746, Copyright (1992), with kind permission from Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK.)

The dislocation density within the subgrains is d e p e n d e n t on the normalized applied stress as

p = (1/otMb)2(cr/G) 2

(20)

where 0.1 < c~ _< 1, although typically c~ ,~ 0.5 for M -- 2 (Fig. 22). (Bird et al., 1969; Takeuchi and Argon, 1976). Equations (12) and (20) are similar in form when the terms are rearranged. In other words, Eq. (12) specifically applies when the obstacles to dislocation motion are other dislocations in the network within the primary subgrains. However, when cell boundaries are the obstacles, Eq. (18b) suggests that O'ob s is better defined as Crobs ~ 4 x 1 0 5 G ( b / d c ) 2

(21)

especially since the steady-state microstructures show that the cell interiors contain few or no dislocations (Figs. 13 and 15).

Chapter8 Modelingthe Role of Substructure during Class M and Exponential Creep 387 10--6

O

I

AI 9 Cu 9 Cu

'

I ' I'I' 346 ~ 550 ~ 745 ~

I

I

~ I ~ I ~I I

,oooc

600 ~ 800 ~ 900 ~ Mo 1650 ~ MgO 1700~

..~.~ Fe 10-7

a [] ~1

..Q Q,.

LiF

t ~

800 650 ~~ 500 ~ 400 o~ f . , p ~ ,

10-8

m

m

m

10_ 9 10-5

I~1

Illlll 10-4 "riG

I

,

I , I,I,

10--3

l|[l~ll~tll#ll Stress dependence of the dislocation density within the subgrain interiors in several crept materials. (Reprinted from S. Takeuchi and A. S. Argon, J. Mater. Sci., 11, 1976, 1542-1566, with the permission of Chapman & Hall.)

B. Effect of Creep Strain on Substructure 1. Class M Creep Regime Although stress plays an important role in determining the nature, size, and morphology of the substructure, strain and temperature can also influence the creep microstructure. An increase in both these parameters generally tends to favor cell and subgrain formation with a corresponding reduction in the dislocation density. It is fairly well established that the substructure morphology changes with creep strain for many materials, where most of these studies have been conducted in the class M creep region (Takeuchi and Argon, 1976). These investigations reveal a certain general trend in creep substructure formation, which has been summarized by Takeuchi and Argon (1976) (Fig. 23). First, a uniform dislocation substructure forms on loading. As creep progresses, recovery begins and the initial substructure transforms itself into a cellular or subgrain microstructure

388

S.V. Raj, I. S. Iskovitz, and A. D. Freed

Cb

r

t .~

"r

-.r

j~r

,,r162 r

; ~_

~'.

0r

r

.r ~

,,'.,-,G~).'Y

..I

~-

~1

44j , r %.r "~..b 4 _.~.._ *,I ~ . , r +" ,~I

r

>;- ~

>>r

r

,'2//",. ?r162

f""

/ . /.,../.

/

- >,~,

r r cf .~,r r r r _r > r ~->,,r ~r162r >,r162162 "r r ,i"r f

g" 1r

(a)/,r

>~

>:'_r

j

"_

.r

,~.

r162 r :>

r162

"-~

,~_ ~ r

r162 ,

Co) &'/

,r % .

,,,- -~,<,,r

' " ~ ' r 1 6 2

,." ~ ~.,

.r

~-

~-

(c)

r

/

"-"

T.

r

,,,,c

>

b.

," J,"

_b / / ~ " j

~> ."

/ ~.4~,,r /

,~

.

< >

I:r'

I

".'.,,"

r

>.

,,>r

r

r",,

-%

#': ,/I

,r

~r

r

".--.-~- / " ,I

(d)

/ *,~ ", 2.. ~ " " ;

Schematic showing the development of the creep substructure during class M creep at different stages of the primary and secondary creep regions: (a) On loading; (b) early stage of primary creep; (c) later stage of primary creep; and (d) secondary creep. (Reprinted from S. Takeuchi and A. S. Argon, J. Mater. Sci., 11, 1976, 1542-1566, with the permission of Chapman & Hall.)

Chapter 8 Modelingthe Role of Substructure during Class M and Exponential Creep 389 resembling that formed during stage III deformation at low-temperature deformation (Fig. 23a). Additional straining results in further clustering of the dislocations into a heterogeneous substructure consisting of some regions with a high density of parallel dislocation walls and others with low dislocation density (Fig. 23b). Continued straining produced a banded microstructure of alternate regions of elongated and equiaxed subgrains (Fig. 23c). The elongated subgrains tend to become equiaxed as the banded microstructure transforms to a homogeneous distribution of equiaxed subgrains with a cellular or a three-dimensional network within them (Fig. 23d). Experimental observations by Hasegawa et al. ( 1 9 7 1 ) on a Cu single crystal suggest that the local creep rate is higher in the equiaxed (i.e., coarse) than the elongated (i.e., fine) subgrains (Fig. 24a). However, the local creep rates in

14

I

I

12

Cu

T = 1018 K (r=2.0MPa

10 8

----(3-Am

O

-9 6

.tO

-

Fine region C o a r s e region

4

0

0

60 I I

I 100

I 200 t (h)

I 300

I

I

I

100

I 200 t (h)

I 300

400

50 40 E30 E, "~ 20 10 0

b

0

400

/I[Illl~|||U Variationsin the (a) local creep rate and (b) subgrain size for coarse and fine subgrains during primary and secondary stages of class M creep of copper single crystals (Hasegawa et al., 1971). (Reprinted from S. Takeuchi and A. S. Argon, J. Mater. Sci., 11, 1976, 1542-1566, with the permission of Chapman & Hall.)

390

S.V. Raj, I. S. Iskovitz, and A. D. Freed

I

Eel 1014

I

T = 873 K = 75 M P a

9 PT I r

O

Psb

A

Psi

1013 A

2x1012 [ 0.0

I

0.1

I

0.2 E

I

0.3

1 0.4

l l l [ l l l l | t l l l l l Variations in the total dislocation density, and the dislocation densities in the subboundaries and within the subgrains with creep strain in or-iron (Orlov~i et al., 1972; Orlovfi and (~adek, 1973). (Reprinted from T. Takeuchi and A. S. Argon, Mater Sci., 11, 1976, 1542-1566, with the permission of Chapman & Hall.)

both regions approach a common strain rate as the subgrain size attains a uniform value throughout the specimen (Fig. 24b). Figure 25 shows the variation of the total dislocation density, Pr, the dislocation density in the subboundaries, Psb, and that within the subgrains, Psi, with creep strain (Orlov~i et al., 1972a; Orlov~i and Cadek, 1973; Takeuchi and Argon, 1976), where Pr = Psb + Psi. The total dislocation density and the dislocation density within the subboundaries generally increase monotonically to steady-state values, whereas Psi first increases steeply and then decreases to a constant value with increasing creep strain.

2. Exponential Creep Regime A similar progressive variation in substructure with creep strain is observed in NaC1 single crystals deformed at 473 K and 10.0 MPa (i.e., cr/G = 7.5 x 10 -4) in the exponential creep regime (S. V. Raj and G. M. Pharr, unpublished research, Rice University, Houston, TX, 1986). The microstructure consists predominantly of a mixture of regions comprising a fairly uniform distribution of dislocations interspersed with light patches and subboundaries after a creep strain of about 10% (Fig. 26a). Long straight subboundaries, oriented approximately parallel to the (100) direction, form at e ~ 0.2 (Fig. 16d). The first signs of the formation of rudimentary cells and subgrains become evident at e ~ 0.25 as the initial dislocation substructure begins to recover (Fig. 26b). Continued deformation to e ~ 0.5

Chapter 8

Modeling the Role of Substructure during Class M and Exponential Creep

391

/I1[1t11~t|,[11 Strain dependence of the creep substructure in the exponential creep region in NaC1 single crystals crept at 473 K under a normalized stress of 7.5 x 10 -4. (a) e = 0.10; (b) e = 0.25; (c) E = 0.50.

392

S.V. Raj, I. S. Iskovitz, and A. D. Freed

results in the formation of elongated subgrains with higher dislocation density and equiaxed subgrains with lower dislocation density (Fig. 26c). Recent observations suggest that L h / L s decreases inversely with e. The empirical relation )(. = L h / L s ,m 0.2(1 + ( l / e ) )

(22a)

was observed to fit the experimental data fairly well (Raj and Freed, 1992). The smallest value of e is e0 at the start of a creep test, where e0 is the instantaneous creep strain on loading. Thus, Eq. (22a) suggests that L h / L s ~ 4 for a typical value of e0 ~ 0.05 at t = 0. However, an alternative mathematical fit derived from the viscoplasticity literature (Armstrong and Frederick, 1966) leads to a more attractive formulation for the functional form of X, in part because it contains hardening and dynamic softening terms and in part because it is more amenable to numerical analysis than Eq. (22a). Therefore, expressing the time derivative of ,~ as j(. = -B2[)(.o - (A2/B2)] e x p ( - B 2 e ) k

(22b)

)(. = (A2/B2) .-I-.[X0 - (A2/B2)] exp(-B2e)

(22c)

it can be shown that

where A2 and B2 are constants and X0 is the initial value of X. The first term in the parentheses in Eq. (22b) represents the hardening coefficient while the second term is the softening component. The ratio A2/B2 represents the steady-state value of X, so that from Fig. 21, A2/B2 ~, 0.2. Equation (22c) fits the experimental data fairly well for values of A2/B2 = 0.2, X0 = 4.0, and B2 = 5.4 (Fig. 27). The value of X0 = 4.0 corresponds to an initial value of fcb = 0.99.3 The above description of the creep microstructure reveals two important resuits. First, there are large similarities between substructure formation during the transient stages of the class M creep regime (Fig. 23) (Takeuchi and Argon, 1976) and that existing in the exponential creep region (Fig. 26). In other words, these observations suggest that the constitutive law for primary creep is the exponential creep law, which reduces to a power-law relation in the limit of steady-state deformation. This is shown schematically in Fig. 28 for constant strain rate, constant stress, and constant load conditions involving compression and tension tests. Figure 28a shows the different stages, A, B, C and D, in the primary and secondary creep regions. The structure parameters, Si, corresponding to the points A, B, C and D are also marked on the creep curve. The thin curves in Fig. 28b represent the constitutive k e x p ( Q c / R T ) - c r / G relation, for a constant value of 3It should be noted that cells have not formed during this early stage of deformation. Therefore, fcb has no physical meaning in terms of describing the volume fraction of the cell boundaries. Instead, a high value of fcb __ 0.99 signifies a homogeneous distribution of dislocations formed in the material soon after loading.

Chapter 8 Modelingthe Role of Substructure during Class M and Exponential Creep 393 3.0

I

I

I

2.5

I

T=473K o- = 1 0 . 0 M P a B 2 =5.4 X o = 4.0

2.0

,.-~ 1.5 1.0

NaCI <001 >

A2~= 0.2

m

B2

/"

-

_

///

= 0.0

0.0

92)

0.1

+ X o - A2 I 0.2

exp(-B2~ ) I 0.3 tE

I 0.4

I 0.5

0.6

/ l [ l ~ l l ~ t l l l ~ l l Changesin the ratio of the averagedimensions of the cell width to the cell interior with

creep strain in the exponential creep region in NaC1 single crystals crept at 473 K under a normalized stress of 7.5 x 10 -4. the structure parameters. The bold curve in Fig. 28b represents the experimentally observed class M and exponential creep relations with n > 3. The transition points A, B, C and D shown in Fig. 28a are also marked in Fig. 28b. Thus, based on the microstructural evidence presented earlier, primary creep can be interpreted as transition stages along several exponential creep curves. Although the same steady-state point, D, can be reached either by constant strain rate or constant stress deformation techniques (Weertman, 1956), Fig. 28b reveals that the development of the steady-state microstructures does not follow the same thermodynamic path. An outcome from Fig. 28 is a rationalization for the deviations of the experimental values of the stress exponent from the universal creep law. The broken line with n = 3 shown in Fig. 28 represents the universal creep law predicted by all climb-controlled models (Weertman, 1975). These models ignore contributions from other mechanisms, so this line is a limiting case. Therefore, based on Fig. 28, the higher values of n ~ 4.5 observed for many materials (Bird et al., 1969) suggest that the contributions from nondiffusional processes are sufficient to increase n from 3 to 4.5. Additionally, the experimental values of V* can be explained by including contributions from nondiffusional mechanisms. If class M behavior is solely due to dislocation climb, theory predicts that V* ~ lb 3 (Conrad, 1964; Evans and Rawlings, 1969). However, in actuality, V* ~ 5 x 102-104b 3 for many materials in the class M creep region (Balasubramanian and Li, 1970; S. V. Raj and G. M. Pharr, unpublished research, Rice University, Houston, TX, 1986; Raj, 1989), which is several orders of magnitude higher than that expected for dislocation climb. The implication of these observations is that experimen-

394

S.V. Raj, I. S. Iskovitz, and A. D. Freed

s6J Constant (riG

S5

Si = structure parameter

a S1 i

~3 31"~;51 $7

8 ~10 I S9 I

/ ,,t4,',1,/, / '" " , i Ill

;

I//I/~~/B i /I I ,////I

oEo It. X (9

Constant\

strain rate test ~

/ z___z_/~

/ / / /

.

I

I /--Observed curve

,/ i/~-.,~/S~~''~:/" /

n- 1.o ~ . ;'-7S-./T'~"~--~ . .!/~.;./..1 ~ ~ ~ . ~ - "I~

1

n=3

I / - - Constant

I [/~.

Constant load /I compression test --/

I 11~ ] ~ r-.-- Constant load

I ] I

b

stress test

(riG

[,, tensi~

=

N__ (er/G)pLB

/l[tll|l||l|

Schematic plot demonstrating that the transient primary creep stages A, B and C of the creep curve in (a) correspond to the structure-dependent (i.e., Si) exponential curves in the normalized creep rate-normalized stress relationship shown in (b); D is at or close to the steady-state point. The deformation paths followed in constant strain rate, constant stress, and constant load tests are indicated in (b). The broken line with n - 3 represents the natural creep law.

tal values of V* >> b 3 cannot be rationalized if contributions from nondiffusional creep mechanisms are ignored in the class M region. Second, Figs. 26a-c reveal that a homogeneous dislocation microstructure is essentially unstable during deformation, as it tends to transform to a heterogeneous substructure comprising hard regions with high dislocation densities (e.g., elongated subgrains, and cell and subgrain boundaries) and soft regions with low dislocation densities (e.g., equiaxed subgrains, and cell and subgrain interiors). These observations imply that the one-parameter modeling approach based solely

Chapter 8

Modeling the Role of Substructure during Class M and Exponential Creep

395

on the stress dependence of the dislocation density is incomplete (Mecking and Kocks, 1981; Estrin and Mecking, 1984). Nix and Gibeling (1985) and Nix et al., (1985) proposed using a two-parameter approach based on subgrain interior (soft regions) and subboundaries (hard regions), but in view of the actual nature of the deformation microstructures, even this method is unrealistic. In this regard, the DSV model proposed by Freed et al. (1992) provides a more realistic framework for developing a microstructure-based creep model.

C. Effect of Temperature on Substructure The literature offers relatively few creep data in which creep curves have been generated at a single value of normalized applied stress over a wide range of temperatures. Such information is useful not only in evaluating the specific form of Eq. (1) but also in isolating the effect of temperature on the development of the creep substructure. Figure 29 shows an example of the creep curve for NaC1 generated over a temperature range of 573-873 K at a nominal value of o-/G = 4 x 10 -4 (Raj and Pharr, 1992). It is seen that, while the creep rates decrease monotonically with increasing strain, the tendency toward steady-state behavior increases

1o-1

I

I

I

NaCI

<001 > 0-4--~ er/G = 4xl

10-2

873 K K

10 -3

A

'T -~1,/

= -

773 K ~ . _

10--4

723 K

-

673 K 10 -5

~

~

~

~

-

--

\ ~

623 K

10 .-6 _~-

10-7 _

573 K

_ m m

lO-8

0

I

0.1

I

I

0.2

0.3

I

0.4

0.5

t5

/|[l~lJ~|B~]ili Plot of creep rate vs. true strain for NaC1 single crystals deformed between 573 and 873 K under a normalized stress of about 4 x 10 -4. (Effect of temperature on the formation of creep substructure in sodium chloride single crystals, S. V. Raj and G. M. Pharr, 75, 1992, 347-352. Reprinted by permission of the American Ceramic Society.)

396

S.V. Raj, I. S. Iskovitz, and A. D. Freed

with increasing temperature. Detailed microstructural observations revealed that temperature variations affect the microstructure in a subtle, but significant manner, where microstructural refinement is greater at higher temperatures (Raj and Pharr, 1992). The net effect of this increased tendency toward microstructural refinement with increasing temperature results in a decrease in L h / L s (Fig. 30a) and fcb (Fig. 30b) with increasing temperature for the cells and cell boundaries. Similar results are expected to be valid for the subgrains. These temperature dependencies

l l l [ l l l l , ' t | t l J l Changes in (a) the ratio of the average dimensions of the cell width to the cell interior and (b) volume fraction of the cell boundaries with absolute temperature in NaCI single crystals crept in the class M and exponential creep regimes under a normalized stress of 4.0 • 10 -4. (Data from Raj and Pharr, 1992.)

Chapter 8 Modelingthe Role of Substructure during Class M and Exponential Creep 397 are given by the regression equations X = Lh/LS=

13.8exp(-0.005T)

f c b = 3.74 exp(-0.003T)

(23a) (23b)

As detailed in Section III.A, the stress dependence of the substructure, especially its effect on the subgrain size, is well characterized for a number of materials (Bird et al., 1969; Takeuchi and Argon, 1976; Thompson, 1977; Bendersky et al., 1985; Orlovfi and 0adek, 1986; Raj and Pharr, 1986a). In contrast, relatively little information is available on the effect of temperature on the cell or subgrain size. Early investigations on the subject suggested that temperature affects both the morphology and dimensions of the substructure (Myshlyaev, 1967; Orlovfi et al., 1972; Belkin et al., 1973; Michel et al., 1973; Streb and Reppich, 1973; Kestenbach et al., 1976, 1978). For example, it has been suggested that the substructure changes from a configuration of dislocation tangles and cells to well-formed subgrains with increasing temperature (Michel et al., 1973; Kestenbach et al., 1976, 1978). Although an increase in dc or ds with increasing temperature was reported in some instances (Myshlyaev, 1967; Orlovfi et al., 1972b; Belkin et al., 1973; Michel et al., 1973; Streb and Reppich, 1973), other observations suggest that dc and ds are relatively independent of temperature (Orlovfi et al., 1972; Young and Sherby, 1973; Blum et al., 1980). However, little direct evidence confirms this lack of temperature dependence of the cell and subgrain size since this conclusion was drawn primarily from studies in which the effects of temperature were of secondary importance to those of stress. For example, plots of d s / b against or/G show that the data are usually clustered around the mean line, thus suggesting that the effects of temperature are small (Raj and Pharr, 1986a). Although these indirect observations rule out strong temperature dependences, they do not permit the separation of weak temperature-dependent variations in dc and ds from experimental scatter. These small temperature effects, if present, can be better identified from direct measurements of dc and ds as a function of temperature at constant values of ~r/G. Unfortunately, the number of investigations in which such direct measurements have been made is small and the results are ambiguous. Direct measurements of ds as a function of temperature at constant values of cr (Orlov~i et al., 1972b; Blum et al., 1980) revealed that ds is essentially independent of temperature within experimental scatter. Recent results on NaC1 suggest that ds increases by a factor of about 2 with increasing temperature, while dc is almost independent of temperature for a variation in temperature by 300 K (Fig. 31) (Raj and Pharr, 1992). Since errors in measuring dc and ds are within a factor of 2, the small temperature dependence of ds shown in Fig. 31 cannot be considered significant. The limited direct measurements of the temperature dependence of the cell and subgrain sizes suggest that the final dimensions of the creep substructure are influenced primarily by the applied stress and strain, and only secondarily by

398

S.V. Raj, I. S. Iskovitz, and A. D. Freed 80

I

70

60

0.25 0.35

I

NaCI <001 > Cells Subgrains 9

I

9

V

A A

0.60

50 E m 40 "10 t,. o 0

"o

30

20 V

10

0

500

I

600

I

700

I

800

900

T (K)

IN[Ill;till Plot of the cell and subgrain size against absolute temperature for NaC1 single crystals deformed under a normalized stress of about 4.0 x 10 -4. (Effect of temperature on the formation of creep substructure in sodium chloride single crystals, S. V. Raj and G. M. Pharr, 75, 1992, 347-352. Reprinted by permission of the American Ceramic Society.)

temperature. However, since diffusion-controlled recovery mechanisms, such as dislocation climb, become significant at high temperatures, the kinetics of formation of the steady-state substructure increases with increasing temperature. As a result, excess dislocations are annihilated and the width of the cell and subgrain walls decreases with a corresponding increase in the dimensions of the cell interior (Fig. 30a).

D. Internal Stresses Associated with the Formation of Substructure The concept of a long-range internal back stress has a sound thermodynamic basis in the theory of deformation (Hirth and Nix, 1969; Kocks et al., 1975; Gibeling and Nix, 1980; Nadgornyi, 1988). Considerable experimental evidence now suggests

Chapter 8

Modeling the Role of Substructure during Class M and Exponential Creep

399

that the evolution of a steady-state creep substructure leads to a corresponding build-up of the internal stresses within the material. This in turn results in a decrease in the creep rate during primary creep to a constant value during steady-state creep. This close correlation between creep substructure and ab is clearly revealed in Fig. 32 in the classic study by Hasegawa et al. (1972) on a Cu-16 (at.%) A1 alloy. This alloy exhibits sigmoidal creep, where the creep rate first increases due to a corresponding rise in the total dislocation density and then decreases to a steady-state value with the advent of cell formation. The microstructure during the initial period of creep consists of isolated dislocations, and a b / a ~ 0.3 remains virtually unchanged during this time. As creep progresses, the isolated dislocated dislocations cluster to form cells, so that the dislocation density within the cells decreases with creep time or creep strain. Thus, cell formation is associated with a steep increase in a b / a and a corresponding decrease in the creep rate. The development of internal stresses within the material due to cell or subgrain formation is not unique to creep. The build-up of internal stresses can occur under other deformation modes, such as fatigue and uniaxial constant strain rate deformation. For example, Fig. 33a shows the distribution of the average global and local shear stresses present in the channels and walls of persistent slip bands in a fatigued copper specimen, where these stresses were determined from measurements of the radii of curvature of bowed-out dislocations (Mughrabi, 1983). Figure 33b is a schematic of the idealized illustration of this stress distribution showing the relative magnitudes of the local tensorial back, B, and forward, F, stresses with respect to the tensorial global stress, a. 4 Indirect evidence for the increase in ab with cell or subgrain formation is reflected in the decrease in L h / L s with increasing e (Fig. 27). The existence of an internal back stress acting on the dislocation within a cell or subgrain interior due to the presence of the cell or subgrain boundaries suggests that the local effective stress ae(X, t) -- a - ab(X, t), where the spatial and time dependence of ae and ab arise due to the dynamic nature of the substructure. Thus, the average creep rate can be expressed as k cx ( } ] N ( T e ( X , t ) / N ) n' cx (a - ~ N ~ b (x, t ) / N ) n', where N is the number of cell or subgrain boundaries and n' is the effective stress exponent. Since ab develops in response to microstructural changes occuring under the action of an applied stress during the course of an uninterrupted creep test, the latter type of experiment does not permit an effective separation of ab from a. However, it is clear that ae < 0, and hence k < 0, when a < ab. Therefore, reducing the applied stress from a > ab to a < Orb should lead to a decoupling of ab from a, thus providing a method for measuring ab. 4The nomenclature B, F, a, and r has been used in order to be consistent with the tensorial representations described in Section VI.B. The subscripts have been omitted for the unidirectional stresses, so that B = ab and F = af describe the special case of unidirectional local back and forward stresses, respectively.

400

S.V. Raj, I. S. Iskovitz, and A. D. Freed

10--6

u,l,l,l,I

A

U,Ip~xl,=,l,l,I

I,l,=~ !

I I I nllini

I n I nlnlnJ

I n I nln'lTI

l i I~l~lq

I I l Ullll

/

.~

10-8

I t I tlnlti

i ~ I X

1014 ~ - I J I IItlUi I J I Il~i~I _-- Isolated ! - dislocations ~ u - - ~ 1013 ~~E ~" ,2,o

pT_~ ~

t

~

CU-16%AI..-~ 723K

Cell

formation ~ - -/_p__d_ - - ' - 'd~ - Pd

-- 1.0 0.8 m

1012 ~.--

-- 0.6

_

"~

_0,4

_

1011 ~ 1010

- 0.2 I = l llllnJ

1.0

I = I~l~h,I

I , I~l,ld

-

0.6

-

0.4 -

0.0

BI(IIII'tIll]PI

I o I,IJl 91

0.0

I ~ l Ul~iq I ~ I~111~i I I I~1~1~I I I l UlUlu

0.8

0.2

~

--

-

10 2

-

I i l JlilnJ

10 3

I n I nlnJlJ

10 4 t (s)

I = I olnlnJ

105

-

I t I tltln

10 6

Effect of cell formation on the creep rate and internal back stress in Cu-16% AI. (Reprinted with permission from Hasegawa et al., 1972.)

Chapter 8

Modeling the Role of Substructure during Class M and Exponential Creep

401

Spatial distribution of the effective internal stresses acting in the cell interior (B) and on the cell boundary (F), respectively. (a) Experimental data measured in copper in the persistent slip bands (data from Mughrabi, 1983); (b) schematic representation of the data in (a).

i|[llilJ[,lltll

402

S.V. Raj, I. S. Iskovitz, and A. D. Freed

I

I

I

AI single crystal T = 673 K

-200

(r = 0 . 4 M P a

~.. 0

_

((r o = 4 MPa)

t o = 0.22

-400

~u

~

-600

-8oo

Istrain I 0

500

I 1000

I

-

1500

I

2000

-

I

2500

I

3000

3500

t (s)

Anelastic creep strain observed in an AI single crystal deformed to a strain of e ~ 0.22 at 673 K under an initial stress of 4 MPa after the stress was reduced to 0.4 MPa. (From Gibeling (1979), cited by Nix and Ilschner (1980), with permission from the author.) The magnitude of ab is determined experimentally from either stress decrease ("stress dip") or stress relaxation tests. Several methods based on one or the other technique have been reported in the literature (e.g., Davies et al., 1973; Gibeling and Nix, 1977; Blum and Finkel, 1982; Pahutov~i e-t al., 1977, 1993; Yamada and Li, 1975; Toma et al., 1976; Yoshinaga, 1993); however, the interpretation of the results has been controversial and confusing, primarily because it is not always clear which specific microstructural event (e.g., dislocation bowing, grain boundary relaxation, dislocation run-off into the cell or subgrain boundaries, and subboundary migration) should be identified with ab. Significantly, anelastic recovery strain associated with negative creep rates when a < ab has been observed in single crystals where grain boundary effects can be ruled out as the primary cause of ab (Fig. 34) (Gibeling and Nix, 1981). These observations confirm that back stresses arise as a consequence of the heterogeneous nature of the creep substructure (Nix and Ilschner, 1980). Recent calculations also suggest that the local effective stresses acting in the cell interior, aci, and on the cell boundaries, acb, change with increasing temperature as the dislocation substructure becomes increasingly heterogeneous (Fig. 35) (Raj and Pharr, 1992). Assuming a particular form for the temperature dependence of ab, these calculations show that as the temperature increases, there is a steep increase in the forward stress, af, acting on the cell boundaries, which also results in a corresponding increase in Ocb (-- o ~- of) and a decrease in O'ci ( = O" - - O"b ) . These changes in the effective stresses arise due to a decrease in the dislocation density within the cells, Pci, and increase in that cell boundaries, Pcb, with increasing temperature. Similar variations are expected for subboundaries.

Chapter8

Modeling the Role of Substructure during Class M and Exponential Creep

403

T (~ 325 400 500 600 700 800 9001000

1.o

I

0.8-

I

I

I

~b~

o.6_

IJ.L...-~

12

/

10

~

8 6

0.4

~

4

2

o

I

0

0.8

;~

0.4

b

o 1015

10

9

~ / - -

"

o I

I

1014 t E ..Q

o

4 r

"

1013

I

I

Pcb~

p = 4x1012m -2 for er/G - 4x16z4 . . . . . . . __

012

t_

o 1011 ._

0

101o 10 9

0.3

I

I

I

I

I

I

0.4

0.5

0.6

0.7

0.8

0.9

1.0

T/T m ||[141J'||5"Jg Calculated values of the internal stresses and dislocation densities in the cell boundaries

and cell interiors as a function of temperature. (Effect of temperature on the formation of creep substructure in sodium chloride single crystals, S. V. Raj and G. M. Pharr, 75, 1992, 347-352. Reprinted by permission of the American Ceramic Society.)

MICROSTRUCTURAL STABILITY Although the microstructural observations reported in Section III provide important information on the qualitative and quantitative aspects of creep substructure, they do not address the problem of microstructural stability involving the morphological and dimensional variations in the substructure when the testing conditions are changed. Investigations on microstructural stability generally involve

404

S.V.Raj, I. S. Iskovitz, and A. D. Freed

producing an initial microstructure, which is then subjected to a different set of deformation conditions in order to understand its transformation to the final substructure. There has been a considerable amount of discussion in the literature (Robinson et al., 1974; Pontikis and Poirier, 1975; Parker and Wilshire, 1976; Miller et al., 1977; Sherby et al., 1977; Ferreira and Stang, 1979, 1983; Blum et al., 1980; Langdon et al., 1980; Eggeler and Blum, 1981; Goel et al., 1983; Soliman et al., 1983; Ginter et al., 1984; Mohamed et al., 1985; Raj et al., 1989) regarding the stability of subgrains after a stress reduction. Equation (16) implies that the subgrain size should increase if the stress is decreased. However, several experimental observations have resulted in contradictory results. The first set of experiments conducted on AgCI (Pontikis and Poirier, 1975), Cu (Parker and Wilshire, 1976), A1-5%Zn (Langdon et al., 1980) revealed that the subgrains formed at the higher stress are relatively stable after a stress reduction and do not increase to a size consistent with the reduced stress. These experiments appeared to cast doubt on the universality of Eq. (16). Miller et al. (1977) criticized some of these early observations, pointing out that the creep strain following the stress reduction was insufficient to permit the amount of subgrain coarsening necessary to allow the subgrain size to attain its equilibrium value at the lower stress. Langdon et al. (1980) addressed this criticism by measuring the subgrain size just after steady-state creep was reestablished, but failed to observe any change in the subgrain size. The second set of experimental observations on A1 (Ferreira and Stang, 1979, 1983; Soliman et al., 1983; Mohamed et al., 1985), A1-5%Zn (Blum et al., 1980; Goel et al., 1983), and NaCI (Eggeler and Blum, 1981) clearly demonstrated that substantial subgrain coarsening occurs after a stress reduction with a corresponding increase in the creep rate. The new subgrain size, similar to that predicted by Eq. (18) for the reduced stress, was typically attained when the creep strain following the stress reduction was greater than 3% (Raj et al., 1989). It was observed that subgrain coarsening after a stress reduction involves both the dissolution of some subboundaries and the migration of others (Eggeler and Blum, 1981; Goel et al., 1983; Soliman et al., 1983; Ginter et al., 1984; Mohamed et al., 1985). Soliman et al. (1983) suggested that Langdon et al. (1980) may have erroneously measured the cell size instead of the subgrain size, which could account for the discrepancy in the two sets of observations. The above stress change experiments have largely been conducted in the class M regime. However, they do not address two interesting and related questions regarding microstructural stability and its relation to the constitutive creep law: (1) Is the path along the class M to exponential creep plot, such as those shown in Figs. 9 and 10, reversible? (2) Is the microstructure formed in the exponential creep region stable in the class M regime and vice versa? The observations from experiments designed to address these questions are discussed below.

Chapter 8

Modeling the Role of Substructure during Class M and Exponential Creep

1~

I

_,,,.

10-q

I

I

I

I

--

0

I,,

0.1

l

0.2

,ac__J

I-

<001)

i~~~~.,~,~~ --, 'T 10-5

lo-71

I

405

T = 873 K

---

o = 1.0 RPa

I

0.3

,I

0.q E

I

0.5

I

0.6

I

0.7

"0.8

I|[~llltm{J Comparison of the creep curves for as-received and prestrained NaC1 single crystal specimens deformed at 873 K under a normalized stress of 1.1 x 10 -4. The prestrained specimen was initially crept to e ~ 0.2 in the exponential creep region at 473 K under a normalized stress of 7.5 x 10 -4. The prestrained microstructure was similar to Fig. 16d. (Reprinted from Raj, Pharr, and Whittenberger, Mater. Sci. Eng., A l l 3 , 1989, 161-175.)

A. Effect of Prior Deformation on Creep Substructure Two types of experiments were conducted on NaC1 single crystals in order to study the path dependence of the constitutive equations for creep and microstructural stability (Raj et al., 1989). In the first series of experiments, a specimen was deformed in the exponential creep region (Fig. 10). The specimen was cleaved parallel to the stress axis, and both halves were polished and etched. The substructure after creep under these conditions consisted of uniform distribution of etch pits and a few ill-formed subboundaries similar to that shown in Fig. 16d. One half of the cleaved specimen was then deformed further to a strain of s ~ 0.43 at 873 K and at a stress of 1.0 MPa (i.e., cr/G -- 1.1 x 10 - 4 ) corresponding to the class M creep region (Fig. 10). The creep curve obtained under these conditions is shown in Fig. 36 along with that for an as-received specimen. In order to ascertain the effect of static recovery on the initial prestrained microstructure, the other half of the cleaved specimen was also placed in the creep machine close to the test specimen in such a manner that no applied stress was on it. The initial creep rate of the prestrained specimen was less than 10 -8 s -1, which was below the detection limit of the strain-measuring equipment; measurable creep was detectable only after a period of 1 h. However, the creep rate rises very rapidly to about 2 x 10 -5 s -1 within the first few percent of strain before decreasing gradually to values close to that observed for the as-received specimen when s > 0.3 (Fig. 36). The microstructure of the prestrained specimen after subsequent creep to a true strain of about 0.43 is shown in Fig. 37a, where it is seen that the initial

406

S.V. Raj, I. S. Iskovitz, and A. D. Freed

l l [ l l l i , ' t l H I Microstructure of a NaC1 single crystal specimen prestrained in the exponential creep regime and subsequently deformed to e -~ 0.43 at 873 K under a normalized stress of 1.1 x 10 - 4 corresponding to the class M creep regime. Equiaxed primary subgrains, secondary subboundaries, and cells have formed from the initial microstructure shown in Fig. 16d. (b) Microstructure of an as-received specimen deformed to e ~ 0.40 at 873 K under a normalized stress of 1.1 x 10-4. (c) Effect of static annealing at 873 K on the prestrained microstructure shown in Fig. 16d (i.e., cr -- 0). Comparison of Figs. 37a and 37c reveals that the applied stress is the major driving force for substructure formation during creep.

s u b s t r u c t u r e (Fig. 16d) has t r a n s f o r m e d to a m i c r o s t r u c t u r e o f w e l l d e v e l o p e d p r i m a r y s u b g r a i n s , s e c o n d a r y s u b b o u n d a r i e s , a n d cells. This m i c r o s t r u c t u r e is s i m i l a r to that o b s e r v e d on an a s - r e c e i v e d s p e c i m e n d e f o r m e d to an a l m o s t identical v a l u e o f strain (Fig. 37b). T h e cell and s u b g r a i n sizes o f the p r e s t r a i n e d and the a s - r e c e i v e d m a t e r i a l s are in a g r e e m e n t to w i t h i n a f a c t o r o f 2, w h i c h is the n o r m a l

Chapter 8

Modeling the Role of Substructure during Class M and Exponential Creep

407

Comparison of the Average Sizes of Cells and Subgrains Formed in NaC1 Single Crystals a Test parameters

de (#m)

ds (/zm)

As received

T = 873 K o- -- 1.0 M P a e = 0.40

19.5 -4- 1.8

193.5 + 30.5

Prior creep at T -- 473 K, cr = 10.0 MPa, and e ~ 0.2

T -- 873 K, cr -- 1.0 M P a

19.1 + 2.1

126.3 -t- 35.3

Material condition

=0.43

Note: The error limits were determined at the 95% confidence level. a F r o m Raj et al. (1989).

level of scatter associated with these measurements (Table 3). For comparison, the predicted values of the cell and subgrain sizes from Eqs. (15a) and (15b) are dc ~ 22.5 and ds ~ 135/zm, respectively. Figure 37c shows the effect of static annealing on the initial predeformed microstructure (Fig. 16d). Although the extent of recovery due to static annealing is not as extensive as that during deformation, it is not negligible; and as shown in Fig. 37c, a few large subgrains, subdivided by cells, have already formed during this period. A comparison of Figs. 37a and 37c clearly demonstrates that stress-assisted recovery is much faster than static recovery. The sigmoidal primary creep observed in the prestrained material (Fig. 36) is analogous to similar behavior reported for Cu-16 at.% A1 (Fig. 32) (Hasegawa et al., 1972). Therefore, a similar rationale can be adopted to explain the nature of the creep curve for the prestrained specimen. The high dislocation density representative of that in the exponential creep region (Fig. 16d) suggests that few mobile dislocations are available initially for measurable creep to occur. However, as the dislocation substructure begins to recover, the density of mobile dislocations increases, which leads to the initial rise in the creep rate observed in Fig. 36. The subsequent decrease in the creep rate beyond the peak value can be attributed to an increase in the long-range internal stress in the specimen as cells and subgrains evolve during deformation. In the second series of experiments, a specimen was first deformed in the class M creep regime in Fig. 10. The microstructure after this deformation consisted of large equiaxed primary subgrains and cells (Fig. 38). Next, the specimen was retested at 473 K under a stress of 10.0 MPa (cr/G = 7.5 x 10-4), where the stress and temperature conditions correspond to the exponential creep regime in Fig. 10. The creep rates for the prestrained specimen were significantly lower than those for an as-received specimen deformed under similar stress and temperature conditions (Fig. 39). The prestrained specimen was removed for microstructural examination at points X and Y marked on the lower curve in Fig. 39.

408

S.V. Raj, I. S. Iskovitz, and A. D. Freed

~ll[llll't|tli Microstructure of a NaCI single crystal specimen prestrained to e ~ 0.20 at 923 K under a normalized stress of 7.5 • 10 -5 in the class M regime.

Comparison of the creep curves for as-received and prestrained NaC1 specimens deformed at 473 K under a normalized stress of 7.5 • 10 -4 corresponding to the exponential creep regime. The prestrained specimen was initially crept to e ~ 0.2 at 923 K under a normalized stress of 7.5 • 10 -5 corresponding to the class M regime. Microstructural observations were conducted on the prestrained specimen after deformation to points marked X and Y.

Chapter 8 Modelingthe Role of Substructure during Class M and Exponential Creep 409 Soon after loading (i.e., X in Fig. 39), the surface microstructure of the prestrained specimen perpendicular to the stress axis showed evidence of slip bands traversing the original subboundaries (Fig. 40a). On repolishing and re-etching the surface, it was found that the original subboundaries were still intact and there was no evidence of any catastrophic rupture of these boundaries anywhere on the surface of observation (Fig. 40b). However, localized migration of the subboundaries had occurred in some regions (e.g., A in Fig. 40b). The observation that the subboundaries are mechanically stable directly contradicts theoretical predictions that exponential creep is due to the mechanical instability of subboundaries (Pharr, 1981). Instead, the cellular microstructure, initially present within the primary subgrains (Fig. 38), was more or less replaced by a uniform distribution of etch pits, presumably due to the rupture of the original cell boundaries. However, the generation of fresh dislocations from newly activated sources at the higher stress can also account for some of this increase in the dislocation density. Continued deformation to e ~ 0.18 (i.e., Y in Fig. 39) resulted in the formation of new subboundaries and in the localized migration of the original subgrain boundaries (e.g., arrows in Fig. 41a). It was observed that the migration of the old subboundaries was partly responsible for nucleating new subgrains. For this to happen, the subboundary segment must bow beyond a critical radius, dsb/2 ~ fll Fsb/z', where dsb is the subgrain diameter of the new subgrains, Fsb is the surface energy of a subboundary, and [31 is a constant. In many areas of the specimen, the random dislocation microstructure shown in Fig. 40b was found to have been completely replaced by well developed subgrains (Fig. 4 l b). It should be noted that this microstructure is vastly different from the one observed in an as-received specimen deformed to e ~ 0.2 under similar stress and temperature conditions (Fig. 16d). However, a mixture of elongated and equiaxed subgrains was observed in an as-received specimen after deformation to e ~ 0.5 (Fig. 26c). Therefore, the creep curve for the prestrained specimen in Fig. 39 can be viewed as a lateral shift of the k-e plot to lower values of strain relative to the unstrained material. In effect, the ready evolution of a subgrain microstructure in the predeformed specimen also results in the development of a larger internal back stress in comparison to the as-received material. The above observations on the effect of prior deformation lead to some significant conclusions. First, microstructures formed in the exponential creep regime are inherently unstable and they will continue to transform toward a steady-state substructure. The first series of experiments described in this section supports this conclusion, which also provides an experimental basis for Fig. 28. Second, subboundaries exhibit considerable mechanical stability and do not break down catastrophically into individual dislocations under high stress. Instead, a stress increase is accommodated by subboundary migration, the probable break-up of cell boundaries, and the activation of new dislocation sources. Third, this irreversibility

Chapter 8 Modelingthe Role of Substructure during Class M and Exponential Creep 411 of the subgrain microstructure also leads to a path-dependent flow behavior. In other words, the transition from the exponential region to class M creep is irreversible and path-dependent.

NIX-GIBELING ONE-DIMENSIONAL TWO-PHASE CREEPMODEL As discussed in Sections III and IV, the deformation behavior of materials is influenced by the microstructure, and the microstructure in turn develops in response to external variables such as stress, temperature, and the initial microstructure. Thus, there is a tendency toward steady-state flow behavior as the deformation substructure tends toward an equilibrium configuration. It was also demonstrated that the deformation substructure is heterogeneous in nature, consisting of hard and soft regions with different deformation responses to the applied stress. The hard regions consist of elongated subgrains having a high dislocation density (e.g., A in Fig. 16b), cell and subgrain boundaries, and highly clustered regions with large dislocation densities. In contrast, the soft regions consist of equiaxed subgrains having a low dislocation density (e.g., B in Fig. 16b), cell and subgrain interiors, and regions of the crystal with low dislocation densities. Although it has long been recognized that creep is influenced by the microstructure, initial attempts to model creep deformation failed to recognize the heterogeneous nature of the creep substructure. Thus, these models have largely ignored contributions from processes other than dislocation climb, so that the ensuing rate equation always resulted in n = 3 if no special assumptions are made, no matter which microstructural feature is considered to be important in the model (Weertman, 1975). However, as mentioned earlier, experimental observations generally result in a value of n > 3 (Bird et al., 1969). The first serious attempt to develop a creep model based on the heterogeneous nature of the dislocation substructure was proposed by Nix and Ilschner (1980). Simultaneously, Mughrabi (1980) and Mughrabi and Essmann (1980) also considered the heterogeneity in the microstructure in understanding low-temperature and cyclic deformation. These approaches largely recognize the different deformation characteristics of the hard and soft regions. Subsequently, several other models have been developed for uniaxial and cyclic deformation using hard and soft ~|[l],lltlS[ll Microstructures at the point marked 'X' in Fig. 39; e + 0.034, T = 473 K, a/G = 7.5 • 10-4. (a) Unpolished and etched surface showing slip lines crossing a subboundary (Sb) originally formed after prior creep in the class M regime. (b) Polished and etched surface demonstrating that the original subboundaries after prestraining are mechanically intact. Local subboundary migration has occurred at some regions (e.g., at A) and the dislocation density in the subgrain interior has increased considerably compared to the prestrained microstructure shown in Fig. 38.

Chapter 8 Modelingthe Role of Substructure during Class M and Exponential Creep 413 microstructure-based deformation concepts (Mughrabi, 1981, 1983; Vogler and Blum, 1990; Argon and Haasen, 1993; Hofmann and Blum, 1993; Zhu and Blum, 1993). Typically, these models consider two microstructural parameters involving the dislocation density and the cell or subgrain size. Nix and Ilschner (1980) attempted to model class M and exponential creep behavior by assuming that the creep rate is given by ~- kg -nt- ~cl

(24)

where ~g is the creep rate for thermally activated glide given by Eq. (11) and kcl is the dislocation-climb-controlled creep rate given by Eq. (3). It was assumed that ~g is the creep rate in the soft regions of the subgrain interior while kcl is the creep rate in the hard regions of the subboundaries. Although Eq. (24) suggests that the class M to exponential creep transition will occur naturally as a consequence of the dominance of nondiffusional creep mechanisms, its formulation is fundamentally flawed since it treats the deformation of the hard and soft regions as independent of each other. Subsequently, while preserving the concept of hard and soft regions, Nix and Gibeling (1985) and Nix e t al. (1985) proposed a two-phase composite model for creep. This approach treats the microstructure as a composite of hard and soft regions, where the deformation occurring in the cell or subgrain interior is coupled with that taking place in the cell or subgrain boundaries (Fig. 42). The model assumes that screw dislocations glide, multiply, cross-slip, and annihilate in the soft regions, while simultaneously depositing two edge components at the cell or subgrain walls. Recovery is assumed to occur at elevated temperatures by the climb and annihilation of the dislocations in the cell or subgrain walls. The original model was strictly valid for high stacking fault energy materials. However, a modified derivation is presented here which should be applicable for low stacking fault energy materials as well.

A. Modified Nix-GibelingTwo-PhaseCompositeModel The modified approach is shown in Fig. 43a. The screw dislocations of the unit Burgers vector, b, are assumed to be separated into two partial dislocations of Burgers vectors, bl and b2, respectively, in the cell or subgrain interior. These dislocations are assumed to glide in the extended state, while depositing edge components at the boundaries by forming constrictions at the points marked A (Fig. 43b). Subsequently, these edge components are assumed to split into partial

~|(~lll'taL~ilO Microstructures at the point marked Y in Fig. 39; e ~ 0.18, T = 473 K, a / G = 7.5 x 10-4. (a) The original subboundaries show evidence of localized migration, as indicated by the arrows, and recovery within the subgrain interior. (b) New subgrains have formed within the original subgrains, due in part to subboundary migration.

41 4

S.V. Raj, I. S. Iskovitz, and A. D. Freed

Y

I•

T i_...

7

l

t.s

I/'

~--Cell wall

-~- 3t." t.

--1

~ | [ I L | l t t l l ~ l l Schematic illustration of a one-dimensional cell consisting of infinitely long cell walls. Screw dislocations glide in the cell interior and deposit edge dislocations in the cell walls (Nix and Gibeling, 1985; Nix et al., 1985). (Reprinted from Nix et al. (1989) with permission from Metall. Trans. and the author.)

dislocations in the cell or subgrain walls. It follows from Fig. 43a that recovery within the cell or subgrain interiors can occur by the cross-slip of the screw components after the two partials form a constriction. In comparison, recovery in the cell or subgrain boundaries can occur by climb of the extended dislocations in the walls or to a limited extent by cross-slip of the screw components at A (Fig. 43c). Dislocation glide in the cell walls is assumed to occur along the x direction. In order to maintain compatibility, and assuming the Voigt approximation, ~, _. ~,ps + f S / G = ~,ph + ~ h / G

(25)

where ~,ps and ~,ph are the plastic strain rates in the soft (s) and the hard (h) regions, respectively, and f s / G and f h / G are the elastic shear strain rates in the soft and hard phases, respectively. A second condition that must be satisfied in order to maintain mechanical equilibrium leads to (Mughrabi, 1983) r = f h r h + fSrS

(26)

where r s and r h a r e the effective shear stresses acting in the soft and hard regions, respectively, and fs and fh are the volume fractions of the soft and the hard regions, respectively, so that fh+fs

= 1

(27)

Chapter 8

Modeling the Role of Substructure during Class M and Exponential Creep

415

(a) Schematic illustration of the one-dimensional cell model shown in Fig. 42 for the case when the dislocations are split into partial dislocations. (b) Detailed view of the split dislocations of Burgers vectors bl and b2 showing their constriction at A and further separation in the cell walls.

Noting that r h = r + rf and r s = r - rb, where rf and rb are the forward and back shear stresses acting on the hard and soft regions, respectively, Eqs. (26) and (27) lead to Tf--

(fs/fh)rb

(28)

As shown in Fig. 35, the forward stress is several times greater than the back stress, implying that f s > f h . Assuming that dislocation glide in the soft and hard regions occurs by the intersection of rectangular obstacles, the strain rates in the two regions can be formulated in terms of Eq. (11) as (Nix and Gibeling, 1985) ~)p~b = 4 x 1012(-~4~/G)2

exp[-(AF/kT)(1

-

re/t0)]

(29)

where 4~ -- h or s and ~'e is the shear strength of the obstacle in each region related

416

s.v. Raj, I. S. Iskovitz, and A. D. Freed

to the appropriate dislocation density, pO, by (Taylor, 1934)

~ = a ~ Gb(p~) ~

(30)

As deformation progresses, the strength of the obstacles constantly evolves in the two regions, and it is necessary to establish the structure evolution laws. Screw dislocations glide in the cell interior until they are stopped by the elastic fields of other dislocations. Strain hardening occurs as these dislocations are statistically stored in the cell interior. The plastic strain increment associated with this hardening event is given by d y ps -- bASd ps

(31)

where AS, the average distance traveled by a dislocation in the cell interior before it is stopped, has a value of about 100/(pS)0.5. The rate of increase in the dislocation density is then (jbs) + _-[(~"fi-~)/100b]~,p s

(32)

The rate of decrease in the dislocation density, (tSs)-, in the cell interior due to cross-slip and annihilation of two screw dislocations is given by (Nix et al., 1985) 05s) - = - 20[(q/-~) / L*]vo e x p ( - AGcs/ k T)

(33)

where vo is the Debye frequency and L* is the activated length for nucleating a single cross-slip event. Nix and Gibeling (1985) and Nix et al. (1985) assumed that L* = 1000b, but this is unrealistically large. Noting that A* ~ 300b 2 for crossslip (Conrad, 1964; Evans and Rawlings, 1969; Bonneville and Escaig, 1979) and assuming a stacking fault width of 0.5 to 10b, it is more likely that L* ,~ 30 to 600b.Using Eqs. (32) and (33), it can be shown that the rate of change of the dislocation density in the soft region, t5s, is given by /5s = ( ~ / 1 0 0 b ) ~ , p s - 2 0 ( ~ - ~ / L * ) V D e x p ( - A G c s / k T )

(34)

The rate of change in the obstacle strength in the cell interior is given by (Nix and Gibeling, 1985; Nix et al., 1985)

d~S/dy ps -- G/200 - IO(b/L*)vD(G/~ 'ps) e x p ( - A G c s / k T )

(35)

The first term in Eq. (35) represents the linear work-hardening rate, while the second term accounts for recovery within the cell or subgrain interior due to the annihilation of screw dislocations by cross slip. The inclusion of the recovery term accounts for the finite probability that some of the screw dislocations can form a constriction according to the Friedel-Escaig mechanism (Friedel, 1959, 1964, 1977; Escaig, 1968a, b). The activation free energy for cross-slip is given by Eq. (10). It is noted that Q01 in Eq. (10) increases as the stacking fault energy decreases through its inverse dependence on the adjustable parameter cd, so that

Chapter 8

Modeling the Role of Substructure during Class M and Exponential Creep

417

the probability of forming constrictions and the rate of recovery by cross-slip also decrease with F/Gb. As noted earlier, it is expected that Q01 ~ 185 kJ mo1-1 and oe' ~ 500-950 for NaC1 single crystals based on the experimental data shown in Fig. 7. Alternatively, the activation energy can be expressed in terms of Eq. (8). The glide of a screw dislocation in the cell interior deposits two edge components at the cell walls. Strain hardening in the cell walls occurs as these edge dislocations knit into the walls with a corresponding increase in the stress in the cell walls. The strain increment due to this process is similar to that given by Eq. (31), except that the distance, A h, traversed by the dislocation in the cell walls is expected to be equal to the mesh spacing, h, in the walls. The dislocation spacing in the boundaries is given by A h ~ h = (K1 Gb/f h) and K1 is a constant equal to about 5. The rate of increase in the dislocation density in the cell walls is then (/~h)+-

[(r

(36)

where ph is the dislocation density in the cell walls. Assuming that the recovery in the walls occurs only by the climb and annihilation of the extended dislocations, the rate of decrease of redundant dislocations in the cell walls given by (Prinz et al., 1982) (/sh) - = - - 2 p h / t c

(37)

where tc is the characteristic time required for a pair of edge dislocations to climb a distance h/2 in the cell or subgrain boundary before annihilating each other. The characteristic time is given by tc = h/2vc, where vc is the climb velocity of the extended dislocations. Following Argon and Moffatt (1981),

VC -- f l z C j b ( F / G b ) 2 ( D 1 G f f 2 / k T b ) ( r h / G )

(38)

where Cj is the density of extended jogs per unit length along the dislocation line, and f12 is a constant equal to about 1000. Although recent studies suggest that Cj is dependent on F / G b through a complex power-law relation (Kong and Li, 1993), the magnitudes of several parameters used in the derivation of Cj are unknown. The magnitude of Cj is assumed to be independent of F / G b in the present paper for simplicity. Assuming f2 ~ 0.7b 3 and using Eq. (30) with ot~ -- o~h, the rate of decrease in the dislocation density in the cell walls is ( p h ) - ~,~ _(1.4flzCjb/K1)(oth)Z(F/Gb)Z(D1Gb3/kT)(ph)2

(39)

The total rate of change in the dislocation density in the cell walls is obtained by combining Eq. (37) and (39): (/bh) ~_ [(%//-~h)/Klb] ~ph _

(1.4flzCjb/K1)(~h)Z(F/Gb)Z(D~Gb3/kT)(ph)2 (40)

418

S.V. Raj, I. S. Iskovitz, and A. D. Freed

Assuming K1 ~ 5,/32 ~ 1000 (Argon and Moffatt, 1981), and Cjb ~ 0.01 (Prinz et al., 1982), Eq. (40) reduces to 05 h) = [(x/~--ff)/5b] ~ ph -(2.8)(t~h)Z(F/Gb)Z(D, G b 3 / k T ) ( p h ) 2

(41)

The rate of the change of the obstacle strength in the hard regions is given by

d ~ h / d y ph = (d~h/dyPh) + q- (d~h/dyPh) -

(42)

where (dfh/d}/Ph) + is the increase in the obstacle strength in the walls due to the deposition of the edge components and ( d f h / d y P h ) - is the rate of decrease in the obstacle strength in the walls due to recovery. Noting that (Haasen el al., 1986).

(dfh/dyPh) + = (d~'h/dph)(p)+(l/}/ph)

(43)

the second term in Eq. (42) can be expressed as (dfh/d},,Ph) - --- (d~h/dt)-(1/~/P h)

(44)

( d f h / d t ) - = [(othGb)2/2fh](ph )-

(45)

where

Combining Eqs. (36), (39), and (42)-(45), the rate of change in the obstacle strength in the boundaries is

( d ~ h / d y oh) = (c~h)(G/10) -- 1.4(F/Gb)2(DIGb/kT)(~h/G)3(G/~,Ph) (46) The second term in Eq. (46) exhibits the natural third-power-law dependence on stress associated with many creep theories (Weertman, 1975). The present treatment differs from that derived by Nix and Gibeling (1985) and Nix et al., (1985) in two important ways. First, the climb velocity is assumed to be dependent on the stacking fault energy so that F / G b enters the model in a natural manner. Second, Nix and Gibeling (1985) and Nix et al., (1985) assumed that dislocations in the cell wall glide a constant distance equal to L h ~, 100 nm. This is an unrealistic assumption since it implies that the edge components do not interact with the dislocations forming the cell walls. Thus, this assumption has been discarded in the present approach. Instead, the glide distance is assumed to be related to the mesh spacing in the cell walls in order to allow the glide distance to change constantly during deformation as the microstructure evolves. It is important to note that the recovery processes occurring in the cell interior as well at the boundaries are affected by the magnitude of the stacking fault energy through its influence on cross-slip and climb.

B. Limitationsof the One-Dimensional Composite Model Despite its usefulness is describing creep, the one-dimensional model is limited in a number of ways, some of which were discussed by Nix and Gibeling (1985) and

Chapter 8 Modelingthe Role of Substructure during Class M and Exponential Creep 419 Nix et al., (1985). First, the model unrealistically assumes that cell and subgrain boundaries extend infinitely in one direction; therefore, it fails to account for the three-dimensional nature of the substructure. Second, the model is strictly applicable for the special dislocation boundaries and processes illustrated in Figs. 42 and 43. Thus, recovery events due to dislocation climb of edge components are ignored in the soft regions, as are those due to cross-slip of screw dislocations occurring in other types of boundaries. Since both edge and screw dislocation components are expected to be present in the hard and soft regions, the model artificially isolates cross-slip events to the cell or subgrain interior and climb to the cell or subgrain walls. Third, the model implicitly assumes that ceils or subgrains are present from the beginning of a creep test, and it does not consider the time-dependent variation of fh and fs or the manner in which dislocations pattern themselves into dislocation clusters (Kubin, 1993). Fourth, in view of the discussion in Section III, the model assumes an idealized and oversimplified picture of the creep substructure. Fifth, the model is strictly valid for single-phase, coarse-grained materials and single crystals exhibiting class M behavior since all grain boundary effects are ignored. Sixth, the number of slip systems activated in the material is not explicitly considered although the model implicitly assumes that multiple slip results in high rates of dislocation storage. Despite these limitations, this simple model can explain the development of internal stresses within a creeping solid as a natural consequence of the development of a heterogeneous microstructure. As will be demonstrated in Section VI.A, the Nix-Gibeling model is the limiting case of the more realistic three-dimensional DSV analysis (Freed et al., 1992).

DEVELOPMENT OF A MULTIPHASE THREE-DIMENSIONAL CREEPMODEL As discussed in Section V. B, the one-dimensional, two-phase creep model has several limitations. Although several similar two-phase deformation models have been proposed (e.g., Mughrabi, 1981, 1983, 1987; Vogler and Blum, 1990), they, too, are limited in a similar manner. Recently, Qian and Fan (1991) developed a three-dimensional viscoplastic model based on dislocation substructure, but they did not provide for substructure strain compatibility.

A. The Dislocation SubstructureViscoplasticityModel The present three-dimensional model accounts for this strain compatibility by using the Budiansky and Wu (1962) self-consistent formalism with an Eshelby criterion (1957) for strain compatibility between the hard and soft regions. The model has been termed the dislocation substructure viscoplasticity model (DSV) (Freed et al.,

420

S.V. Raj, I. S. Iskovitz, and A. D. Freed

1992). The result is a rate-dependent viscoplastic theory that incorporates a selfconsistent effect of dislocation substructure on material response. The internal state variables of this theory are the dislocation densities of the hard and soft regions, the average size of the dislocation substructure, and the relative volume fractions of the two regions. The inclusion of variable geometric or size effects as well as the potential applicability to the deformation of several microstructural features (e.g., cells and subgrains) are unique features of the present model. Several assumptions have been made in the construction of the model. First, the self-consistent method of Budiansky and Wu (1962) adequately represents the hard and soft regions to a first approximation, with both regions having identical elastic moduli. Second, the cells and subgrains are treated as equiaxed, isotropic spherical inclusions in the Eshelby (1957) analysis for strain compatibility. Third, ~ r m a t i o n involves multiple slip systems so that the material is isotropic and the plastic strain rate of each region is coaxial with its deviatoric stress. Consequently, the constitutive equations of Prandtl (1924) and Reuss (1930) are taken to apply in each region (i.e, no back stress is considered in the macroscopic flow law). Fourth, the von Mises (1913) criterion is used to describe the topology of the nested set of flow surfaces. The von Mises equivalent stress and plastic strain rate are considered to correlate with the stress and plastic strain rate of a critically resolved slip system via Taylor's relation (1938). Fifth, the local plastic strain rates of the hard and soft regions are governed by dislocation mechanics, where the dislocation structure is assumed to evolve during deformation. Sixth, grain boundary effects are assumed to be negligible. For simplicity the model is developed for a two-phase cellular substructure, but the procedure can be extended to other substructures, including multiphase microstructures.

B. CompositeThree-DimensionalModel Through the volume averaging process, each global tensor field, s a y to its local fields X~ through the summation

"f(ij- ~ f~X~ q~--1

such that

1-- ~

f~

f(ij,is related (47)

~b--I

where f ~ is the relative volume fraction associated with field X/~ of phase ~ (i.e., each type of substructure, such as cells and subgrains). This is a rule of mixtures relation, where each local field of the continuum represents an integrated volume average of that field over its associated phase of a microcontinuum, which is designated by a unit cell. A bar placed over a variable identifies that it represents the volume average of its local variables over the entire unit cell. In principle, Eq. (47) should describe a microstructure of several cells and subgrains, such as those illustrated schematically in Figs. 17 and 18a. For simplicity, only a two-

Chapter 8

Modeling the Role of Substructure during Class M and Exponential Creep

421

phase composite structure consisting of cell walls and cell interiors is considered here, for which Eq. (47) reduces to Xij - (1 - f h ) X isj , _4j- f h x

(48)

h. ij,

It is cautioned that although the individual tensor fields are considered to add up according to the rule of mixtures, mappings from one field to another (e.g., constitutive equations) need not satisfy such a rule. The Prandtl-Reuss (Prandtl, 1924; Reuss, 1930) equations are used to relate the local stresses, crib, and strain rates, ~/~, in phase r so that k r 6 ~ -- 3to (~k~- OecJCg~k) + ~-crik

(49)

(so) where

s~- ~ - ~

* 6ij -50"l&

(51a)

E~ -- e~ - -Sekk6ij

(51b)

and

are the deviatoric stresses and strains, respectively, in phase 0; tc is the bulk modulus, Oecis the mean coefficient of thermal expansion, 6ij is the Kronecker delta, Jv~ describes the kinetics of plastic flow; k = (Otc/OT)T, and (~ = (OG/OT)T. These are the governing constitutive equations of a thermoelastoplastic isotropic Hookean material. In plasticity (i.e., rate-independent condition), Jv~ would be described through a consistency condition; whereas, in viscoplasticity (i.e., ratedependent condition), Jv~ is described through a kinetics equation (an equation of state). A significant, but realistic assumption in the current treatment is that T, Otc, G, and tc are assumed to be identical in both the hard and soft regions, which substantially simplifies the ensuing theory. Writing the flow equation for plastic straining as k/~ -- 3 [ikp~ II IIS* II

(52)

implies that the kinetics of Prandtl (1924) and Reuss (1930) be described by ~.4,_ 3 IlkP4' II 2118'11

(53)

where IlSoll- v / 3 s ~ s ~

and

]]~pq5]]_V/2~pjqS~iP~b

(54)

422

S.V. Raj, I. S. Iskovitz, and A. D. Freed

Equation (54) gives the von Mises criteria for stress and plastic strain rate when normalized for tension. The Prandtl-Reuss kinetic equation can be expressed in terms of the local shear stresses, r ~, and plastic shear strain rates, ~Pr acting on phase 4~ as ~4,

3~'P4~ 2M2rr

=

(55)

The Taylor factor, which has a value between 2.65 and 3.06 for cubic systems depending on the number of slip systems (Kocks, 1970), establishes the transformation r ~ = IIS~II/M

yPr = MlleP~ll

and

(56)

As shown in Section V.A, yPq~ is a function of r ~, T, p~, cell or subgrain size, L (-- L h + LS), and fh. Thus, Eq. (56) establishes a relationship between the Prandtl-Reuss kinetics and a set of microstructural variables, which naturally leads to a viscoplastic model based on dislocation physics. Since the elastic and thermal moduli in the hard and soft regions are the same, the volume-averaged stress, ~ij, and strain, if?ij, that one measures in a laboratory experiment on an isotropic Hookean material are described by #kk -- cr~ = 3X(gkk --C~c(T- T0)6kk)

(57)

Sij = 2G ( E, ij - EP )

(58)

where To is a reference temperature and gP is the volume-averaged plastic strain. It should be noted that a ~ = #kk is a hydrostatic form of the Reuss (1929) comr posite approximation and consequently, that ekk = gkk is a hydrostatic form of the Voigt (1889) composite approximation. Furthermore, since the Voigt and Reuss approximations are upper and lower bounds, respectively, it then follows that the hydrostatic response is exact. However, this is not the case for the deviatoric stress response. The self-consistent formulation that is employed here is a realistic approximation that lies between the extremes of the Voigt-Reuss bounds. There are two long-range internal stress states that arise from this composite substructure; these are the backward, Bij, and forward, Fij, stress tensors, which were introduced in one-dimensional form in Section V. The back stress is a derived property in DSV; however, it is a phenomenological variable in the classical theories of plasticity and viscoplasticity. It is defined as the difference between the averaged applied stress and the stress of the soft region: (o'ihj -- O'ijs )

(59a)

nij -- Sij - Sij = fh ( s h _ SiSj )

(59b)

Bij = o'ij - o'ijs

_

fh

Similarly, the forward stress is defined as the difference between the stress in the

Chapter 8 Modeling the Role of Substructure during Class M and Exponential Creep

423

hard region and the averaged applied stress: F i j =-- cr h -- ~ i j - Fij -- sh

(1 - f h ) ( a h _ o.ij )

-- S i j - - ( l -

fh)(S h _ Sij)

(60a) (60b)

Therefore, fh

(61)

Bij = 1 - f h Fij

which is the three-dimensional analog of Eq. (28). These long-range internal stresses are deviatoric as a consequence of the fact that the hydrostatic contributions for the stresses of the hard and soft regions are equivalent. As expected, the rule of mixtures (Eq. (48)) is satisfied for the data shown in Fig. 33a since it is the requirement for equilibrium of the local stress fields. Compatibility between the strains in the hard and soft regions, as derived by the self-consistent method, produces the local strain fields (Freed et al., 1992) s

Eij -- Eij +

fh

--

E ' i j "3V

(

fl

2G

(62)

1 - fl

2-G

(63)

1-

where fi = [2(4 - 5v)/15(1 - v)] ,~ 0.5 is the shape factor for a spherical inclusion and v is Poisson's ratio. Equations (62) and (63) also satisfy the rule of mixtures given by Eq. (48). Since the cells are assumed to be equiaxed, and hence isotropic in their properties, it is reasonable to approximate them as spheres in the Eshelby analysis, although in reality they are not spherical. Since cells are not spherical, it may be necessary to treat fl as a variable parameter rather than a constant as determined by the Eshelby analysis given above. The last terms in Eqs. (62) and (63) are the self-consistent correction to the Voigt (1889) approximation for strain compatibility; i.e., E~ - - E'ij. These correction terms are a consequence of the Eshelby (1957) approximation for strain compatibility when implemented into the self-consistent framework of Budiansky and Wu (1962), as derived in the Appendix. The Voigt strain compatibility is achieved when fl = 0 (i.e., v = 0.8) in Eqs. (62) and (63). This compatibility condition has been used by several investigators (Mughrabi, 1983, 1987; Nix and Gibeling, 1985; Nix et al., 1985; Vogler and Blum, 1990; Lan et al., 1992) to analyze the deformation behavior of hard and soft regions in the one-dimensional substructural model discussed in Section V (Figs. 40 and 41). Equations (62) and (63) make apparent the additional contributions brought into the theory by Eshelby strain compatibility and, as a consequence, the importance of the role that the internal stresses play in ensuring compatibility of the local strain fields. The order of magnitude for

424

s.v. Raj, I. S. Iskovitz, and A. D. Freed

these corrections is that of the elastic strain, and therefore they are second-order corrections in most inelastic applications. Nevertheless, they could significantly impact the overall response, for example, under nonproportional loading histories.

C. Solution Algorithm for the DSV Model At the current time, t, all variables are assumed to be known. In other words, the set of global variables, aij, gij, gP, and T, and the set of local variables, a h, a/~, e h, esj, e pjh, eijps . ph . . pS . fh and L have known values. At some future time, t + 6t, values are assigned to T and el j, while values for all remaining global and local variables are left to be calculated. The theory contains four microstructural or internal state variables: ph, ps, fh, and L. The algorithm presented below is the solution for this problem statement resulting from the theoretical construction given above. The first-order, ordinary differential equations describing DSV that must be solved are 1 + 1 -

Sij +

= 2ckij

+

-G

Sisj

(64)

2c

2Gk/j +

-

1 -

1-t -

2Gj~S + 1 - fl"

+l-t

fl ((1 - fh)SiSj _ ~rhSiSj) 1 --fl

(65)

S ' = 00[r ~~ p~0, fh, L, T]

(66)

)-h __ .r

p~O,fh, L, T]

(67)

L = / ~ [ r ~~ p~o, fh, L, T]

(68)

whose kinetics are described by ~4,

=

3~,p4,[r~0 p~0 fh L T] ' ' ' ' 2MZr4 '

(69)

Equations (64) and (65) are obtained by combining the Prandtl-Reuss equation (50) with the time rate of change of the local deviatoric strains given in Eqs. (62) and (63). Representative relationship describing ~,P4~ and/5 ~ for each phase are given by Eqs. (29), (34), and (41) for the simple one-dimensional substructure model (i.e., when/3 = 0). The equations are likely to be more complex when the dislocation mechanics are formulated for the more realistic three-dimensional cell or subgrain model. This formulation has not been completed, so the present

Chapter 8

Modeling the Role of Substructure during Class M and Exponential Creep

425

analysis is restricted to the one-dimensional case. The rate of change in the volume fraction of the hard regions is detailed in Section VI.D. The form of Eq. (68) describing the rate of change in the cell or subgrain size, {, is poorly understood at present. This change in the subgrain size influences the creep rate through its effect on Eqs. (64)-(66). The importance of including { comes from the fact that considerable evidence now suggests that the subgrain size changes during primary creep (Fig. 18) (Hasegawa et al., 1971) and that it grows to a new equilibrium value after the stress is reduced from an initial creep stress (Eggeler and Blum, 1981; Ferreira and Stang, 1979, 1983; Goel et al., 1983; Soliman et al., 1983). However, as discussed in Section IV, measurements of the subgrain size after a reduction in the creep stress value resulted in contradictory results. It is possible that the changes in the cell size with strain or time are much smaller than the subgrain size (Goel et al., 1983; Soliman et al., 1983). It is clear that Eq. (68) should lead to Eq. (18) when a stead state is established after the stress drop. As discussed in Section VI.D, the present analysis assumes that L cx 1/f, where f is the average strength of the obstacles in the hard and soft regions. After integrating the local deviatoric stresses and the internal state variables, one updates the hydrostatic response

6kk = 3 K ( g k k - Otc(T- To)3kk)

(70)

and the remaining global and local variables

Sij -- (1 - f h )Sijs + fh S h

Sij

gP -- Eij

1 - fl

eh

1 --

--

ms

(72)

2G

Eij -- Eij + --

(71)

2G

(73) (74)

_

%-eP-

2c

(75)

ePjh -- ~P - ( ) F 1i J fll .

2G'

(76)

whose derivations are given in the Appendix. As expected, Eqs. (71)-(76) satisfy the rule of mixtures, Eq. (48). It should be noted that Bij and Fij a r e related to SSj and S/~ through Eqs. (59b) and (60b). The differential equations (64) and (65) for the evolution of the local stresses are strongly coupled and highly nonlinear; therefore, they are best solved using either semi-implicit or implicit integration methods. The nonlinearity is caused, for the most part, by the kinetics of ~.r The right-hand side of these equations

426

S.V. Raj, I. S. Iskovitz, and A. D. Freed

(i.e., the nonhomogeneous contribution) is where the coupling is strongest. In particular, a change in the stress in one region affects the change in stress of the other region, and vice versa. This coupling (in both the stress and stress rate) can lead to numerical instabilities for unreasonable initial guesses of the stress and stress rate in the two regions. For some integration algorithms, therefore, it may be necessary to first implicitly integrate the uncoupled differential equations for the local stresses obtained by using the Voigt approximation S/~ + (2G~ r - CJ/G)S~ = 2 G k i j

(77)

to acquire a reasonable initial guess for S~ and S/O/. These quantities should then be used to initiate and implicit integration of the more accurate differential equations (64) and (65) where, because of the coupling, the nonhomogeneous contributions of the right-hand sides are evaluated one iteration in arrears. Equation (77) is equivalent to setting the shape factor fl = 0 in Eqs. (64) and (65). As stated earlier, the model is applied to the simpler case of fl = 0 corresponding to the substructure configuration described in Section V and illustrated by Figs. 42 and 43. The case fl = 0 is also examined using the one-dimensional dislocation model, although it is not strictly valid, in order to study the effect of fl on the stresses and strains developed within the material. ,J

D. Rateof Changeof Microstructural Parameters Equations (64)-(68) require some knowledge of the rates of change of the microstructural parameters in order to evaluate the validity of the DSV model for situations where fl -- 0. The rates of change in the dislocation densities are given by Eqs. (34) and (41) for the one-dimensional case, but these formulations are likely to change for a three-dimensional substructure model. The dislocation physics for the latter geometry is still to be formulated. As mentioned in Section VI.C, the functional form of Eq. (68) is unknown at present. Defining f as .~ = fh.~h + (1 -- fh).~s

(78)

L can be expressed as a function of f through

L = (K/Mm)b(G/f) m

(79)

L = - ( m K / M m ) b ( G / f ) (m+l)f

(80)

Thus,

Equations (18a) and (18b) represent the steady-state forms of Eq. (79) when f is proportional to the applied stress. In defining L in terms of f, it is clear that L will change in a natural way after a stress change due to corresponding changes in f h , ~h, .l~s, and f.

Chapter 8 Modelingthe Role of Substructure during Class M and Exponential Creep 427 The experimental data discussed in Section III can be used to formulate an empirical relation for fh in terms of strain. Noting that L = L h -}- L s

(81)

and

(82)

it is found that L s = L[1/(1 + Z)]

L h = L[X/(1 + Z)]

Thus,

fh = 1 - [1/(1 --+-X)] 3

(83)

/h__ 311/(1 q- X)]4)(

(84)

and

Using Eqs. (22b), (22c), (83), and (84), both fh and / h can be obtained as a function of strain. The parameters fh and X can be used interchangeably as the independent variable. Equation (83) is similar in form to that derived by Dobes and Orlovfi (1990), but differs from that used by other investigators (Mughrabi, 1981, 1983, 1987; Nix and Gibeling, 1985; Nix et al., 1985; Qian and Fan, 1991) who defined fh as a linear fraction and assumed that the latter was equal to the volume fraction.

E. Coupling of DSV and Dislocation Physics One of the objectives of this paper outlined in Section I.B was to develop the theoretical basis for interlinking dislocation mechanisms occurring at the level of the substructure with the macroscopic stress state. The scale-up from the local to the global state is achieved through DSV (Fig. 2). The treatment presented in Sections V.A, V.B, V.C, and V.D demonstrates that this coupling is fairly complex, even for the relatively simple one-dimensional dislocation model. Equations (64) and (65), together with Eqs. (69)-(76), form the basis for this interlinkage between the local and the global variables, provided that the functional forms for Eqs. (66)(68), j~h and ~,~, are known. The local strain rates are described by Eq. (29) for the hard and soft regions, where the evolution of f~ during deformation can be obtained form Eqs. (35) and (46). Equations (34) and (41) give the rate of change in the dislocation densities in the hard and soft phases for the one-dimensional dislocation model. These equations can be used along with Eq. (30) to evaluate the change in f-~ with strain. The evolution of f'~ during deformation also influences L through Eq. (80), so the change in the cell or subgrain size can be evaluated in a natural manner. An examination of Eqs. (64), (65), and (71) shows that their solution requires some knowledge of fh and jch, where the latter are related to X and X through Eqs. (83) and (84), respectively. Equations (22b) and (22c) give

428

S.V. Raj, I. S. Iskovitz, and A. D. Freed

the specific functional forms for )~ and X, respectively, based on the experimental data shown in Figs. 21 and 27. Since the global strain rate evolves with ~?0, the quantities )r and )~, and hence fh and )eh, also evolve with f0. Equations (22b) and (22c) are empirical representations for the evolution of )r and )~. In principle, the exact forms could be derived from theories dealing with dislocation patterning (Kubin, 1993), although their computations are likely to be quite involved.

SUMMARY A detailed review of the current understanding of the effect of microstructural parameters on class M creep behavior is presented. Microstructural observations conducted in the power-law and exponential creep regimes suggest that creep is influenced by a complex interaction between several elements of the microstructure, such as dislocations, cells, and subgrains, and steady-state behavior is attained when the microstructure reaches a dynamic equilibrium between these different substructural features. Thus, the formation of equiaxed subgrains and the observation of class M creep (i.e., power-law creep) need not be a sufficient condition for steady-state behavior. Instead, recent observations reveal that the refinement of cell boundaries and the formation of secondary subboundaries are equally important components of steady-state creep. Quantitative measurements of the ratio of the dimensions of the cell boundary to the cell interior suggest that steady-state behavior is likely when this ratio is about 0.2. It is demonstrated that the microstructures formed in the exponential creep region are similar to those formed during the early stages of normal primary creep in the class M region. These experimental observations are used to formulate a phenomenological approach to understanding transient and steady-state deformation behavior in terms of the strain rate-stress deformation laws. A three-dimensional dislocation substructure creep model is developed and coupled with a modified one-dimensional dislocation model. Although the present dislocation analysis is still limited in scope, it nevertheless, extends previous analyses by taking into account the dynamic evolution of the microstructure during deformation.

APPENDIX A Derivation of the Coupled Differential Equations Used in the DSV Model For a general anisotropic Hookean material, the averaged response is governed by tYij -- Dijke (g?ke -

-

e-P ke

-

o~:t(T

-

TO))

(A.1)

Chapter 8

Modeling the Role of Substructure during Class M and Exponential Creep

429

while the local responses of the soft and hard regions are governed by P4'

(T

To))

(A.2)

where Dijkl is the elastic modulus tensor and oe~l is the coefficient of thermal expansion tensor. They are assumed to be identical for the hard and soft regions, and for the averaged material. Similarly, T is taken to be uniform over both the phases. Taking into account the fact that the elastic and thermal moduli of both regions are the same, one obtains the following strain compatibility relation from the generalized self-consistent method (Gramoll et al., 1991)" 4

-

+

-p - She )

(A.3)

where Sij~l is the Eshelby tensor (Eshelby, 1957), whose components are constantvalued for the family of ellipsoidal inclusions. Consequently, there is no transformation strain unless there is plastic strain. When substituted into Eq. (A.2), this condition for strain compatibility can be written in terms of the stresses as follows" 4,

-

O'ij -- (Yij -- Dijkg(Ikemn

-- Skgmn) \(sP4'mn -- 8mn-P )

(1.4)

where Iklmn is the identity tensor for second-rank symmetric tensors. This result is analogous to the one derived by Kr6ner (1961). When the material is isotropic, the averaged stress-strain response described by Eq. (A.1) becomes Eqs. (57) and (58), while the local response given by Eq. (A.2) leads to Eqs. (49) and (50) when differentiated. In addition, the criterion for compatibility described by Eq. (A.4) becomes S~

Sij -- 2G(1 - fl)(s p4' .j - gP)

(A.5)

where, the fourth-rank t e n s o r s Dijkl , Iklmn , and Sklmn reduce to the scalars 2G, 1, and fl, respectively. Equation (A.5) was first derived by Budiansky and Wu (1962) using another approach. Solving for the deviatoric response, where the applied strain, Eij, is given and the local stresses, S/~, are known from integration, six tensor equations with six tensor unknowns are obtained. They are the constitutive equations

Sij -- 2G ( Eij - ~?P) s

Sij - 2G(Eij S~ - 2G(E~

s

_ps

(A.6)

- sij )

(1.7)

- g~)

(A.8)

the compatibility equations derived from Eq. (A.5) Sij -Jr- 2G(1 - i~)8i ps -

S h + 2G(1 - fl)sPi~

(A.9)

430

S.V. Raj, I. S. Iskovitz, and A. D. Freed

and the volume averages Sij - - (1 -

f h ) Sijs + f h S h

~P = (1 - fh)eps + fhePhij

(A. 10)

(A.11)

ps

where {Sij, e p, E~, ESj, ePiC,8ij} the set of unknowns. Inverting this linear system of six equations to solve for these six unknown variables in terms of the known variables {Eij, S~, SSj, fh} leads to Eqs. (71)-(76). The fact that the coefficients of the tensors in Eqs. (A.6)-(A. 11) are all scalar quantities instead of fourth-rank tensors greatly simplifies the inversion of these equations. -

-

a r e

ACKNOWLEDGMENTS The authors thank Drs. Rob Dickerson and Mike Nathal for their comments and suggestions. The permission granted by various publishers and authors for reproducing some of the figures used in the text is also gratefully acknowledged.

REFERENCES Argon, A. S., and Haasen, P. (1993). A new mechanism of work hardening in the late stages of large strain plastic flow in F. C. C. and diamond cubic crystals. Acta Metall. Mater. 41, 3289-3306. Argon, A. S., and Moffatt, W. C. (1981). Climb of extended edge dislocations.Acta Metall. 29, 293-299. Argon, A. S., and Takeuchi, S. ( 1981 ). Internal stresses in power-law creep.Acta Metall. 29, 1877-1884. Armstrong, P. J., and Frederick, C. O. (1966). "A Mathematical Representation of the Multiaxial Bauschinger Effect," Rep. RD/B/N731, pp. 1-16, Berkeley Nuclear Laboratories, Central Electric Generating Board, Berkeley, UK. Balasubramanian, N., and Li, J. C. M. (1970). The activation areas for creep deformation. J. Mater. Sci. 5, 434-444. Barnby, J. T. (1966). Effect of strain aging on creep of an AISI 316 austenitic stainless steel. J. Iron Steel Inst. 204, 23-27. Barr, L. W., Hoodless, I. M., Morrison, J. A., and Rudham, R. (1960). Effects of gross imperfections on chloride ion diffusion in crystals of sodium chloride and potassium chloride. Trans. Faraday Soc. 56, 697-708. Barr, L. W., Morrison, J. A., and Schroeder, P. A. (1965). Anion diffusion in crystals of NaC1. J. Appl. Phys. 36, 624-631. Barrett, C. R, and Sherby, O. D. (1964). Steady-state-creep characteristics of polycrystalline copper in the temperature range 400 ~ to 950~ Trans. Metall. Soc. AIME 230, 1322-1327. Barrett, C. R., and Sherby, O. D. (1965). Influence of stacking-fault energy on high-temperature creep of pure meals. Trans. Metall. Soc. AIME 233, 1116-1119. Belkin, E. G., Demikhovskaya, N. N., Kurov, I. E. and Myshlyaev, M. M. (1973). Kinetics of substructure formation during creep in aluminum. Phys. Status Solidi A 16, 425-431. Bendersky, L. Rosen, A., and Mukherjee, A. K. (1985). Creep and dislocation substructure. Int. Met. Rev. 30, 1-15.

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Modeling the Role of Substructure during Class M and Exponential Creep

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Beniere, E, Beniere, M., and Chemla, M. (1970). Conductibilite, nombres de transport et autodiffusion des ions dans diff6rents monocristaux de chlorure de sodium. J. Phys. Chem. Solids 31, 1205-1220. Beniere, M., Beniere, E, Catlow, C. R. A., Shukla, A. K., and Rao, C. N. R. (1977). Energy of migration of monovalent ions in NaCI: An experimental and theoretical study. J. Phys. Chem. Solids 38, 521-527. Biberger, M., and Blum, W. (1988). Subgrain boundary migration during creep of LiE In "Strength of Metals and Alloys" (E O. Kettunen, T. K. Lepist6, and M. E. Lehtonen, eds.), Vol. 2, pp. 875-880. Pergamon, Oxford. Biberger, M., and Blum, W. (1992a). Subgrain boundary migration during creep of LiF: I. Recombination of subgrain boundaries. Philos. Mag. [8], 65A, 757-770. Biberger, M., and Blum, W. (1992b). Subgrain boundary migration during creep of LiF: II. Constantstress experiments. Philos. Mag. [8], 66A, 27-40. Bird, J. E., Mukherjee, A. K., and Dorn, J. E. (1969). Correlations between high-temperature creep behavior and structure. In "Quantitative Relation Between Properties and Microstructure" (D. G. Brandon and A. Rosen, eds.), pp. 255-342. Israel Universities Press, Jerusalem. Blum, W. (1977). Dislocation models of plastic deformation of metals at elevated temperature. Z. Metallkd. 68, 484-492. Blum, W. (1993). High-temperature deformation and creep of crystalline solids. In "Plastic Deformation and Fracture of Materials" (H. Mughrabi, ed.), Mater. Sci. Technol. Vol. 6, pp. 360-405. VCH Publishers, Weinheim, Germany. Blum, W., and Finkel, A. (1982). New technique for evaluating long range internal back stresses. Acta Metall. 30, 1705-1715. Blum, W., and Ilschner, B. (1967). Uber das Kriechverhalten von NaC1-Einkristallen. Phys. Status Solidi 20, 629-642. Blum, W., Absenger, A., and Feilhauer, R. (1980). Dislocation structure in polycrystalline A1Zn during transient and steady state creep. In "Strength of Metals and Alloys" (E Haasen, V. Gerold, and G. Kostorz, eds.), Vol. 1, pp. 265-270. Pergamon, Oxford. Blum, W., Straub, S., and Volger, S. (1991). Creep of pure materials and alloys. In "Strength of Metals and Alloys" (D. G. Brandon, R. Chaim, and A. Rosen, eds.), Vol. 1, pp. 111-126. Freund, London. Bonneville, J., and Escaig, B. (1979). Cross-slipping process and the stress-orientation dependence in pure copper. Acta Metall. 27, 1477-1486. Budiansky, B., and Wu, T. T. (1962). Theoretical prediction of plastic strains of polycrystals Proc. U.S. Natl. Congr. Appl. Mech. 4th, Vol. 2, pp. 1175-1185. Caillard, D., and Martin, J. L. (1987). New trends in creep microstructural models for pure metals. Rev. Phys. Appl. 22, 169-183. Carrard, M., and Martin, J. L. (1987). A study of (001) glide in [112] aluminum single crystals. I. Creep characteristics Philos. Mag. [8] 56, 391-405. Carrard, M., and Martin, J. L. (1988). A study of (001) glide in [112] aluminum single crystals. II. Microscopic Mechanism Philos. Mag. [8] 58, 491-505. Carter, N. L., and Hansen, E D. (1983). Creep of rocksalt. Tectonophysics 92, 275-333. Challenger, K. D., and Moteff, J. (1973). Quantitative characterization of the substructure of AISI 316 stainless steel resulting from creep. Metall. Trans. 4, 749-755. Clauer, A. H., Wilcox, B. A., and Hirth, J. E (1970). Dislocation substructure induced by creep in molybdenum single crystals. Acta Metall. 18, 381-397. Coble, R. L. (1963). A model for boundary-diffusion controlled creep in polycrystalline materials. J. Appl. Phys. 34, 1679-1682. Conrad, H. (1964). Thermally activated deformation of metals. J. Met. 16, 582-588. Cuddy, L. J. (1970). Internal stresses and structures developed during creep. Metall Trans. 1,395-401. Davies, C. K. L., Davies, E W., and Wilshire, B. (1965). The effect of variations in stacking-fault energy on the creep of nickel-cobalt alloys. Philos. Mag. [8] 12, 827-839.

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Stokes, R. J. (1966). Mechanical properties of polycrystalline sodium chloride, Proc. Br. Ceram. Soc. 6, 189-207. Streb, G., and Reppich, B. (1973). Steady state deformation and dislocation structure of pure and Mg-doped LiF single crystals. Phys. Status Solidi A 16, 493-505. Takeuchi, S., and Argon, A. S. (1976). Steady-state creep of single-phase crystalline matter at high temperature. J. Mater. Sci. 11, 1542-1566. Taylor, G. I. (1934). The mechanism of plastic deformation of crystals. Proc. R. Soc. London, Ser. A 145, 362-387. Taylor, G. I. (1938). Plastic strain in metals. J. Inst. Met. 62, 307-324. Thompson, A. W. (1977). Substructure strengthening mechanisms. Metall. Trans., A 8A, 833-842. Toma, K., Yoshinaga, H., and Morozumi, S. (1976). Internal stresses during high-temperature deformation of pure aluminum and an A1-Mg alloy. Trans. Jpn. Inst. Met. 17, 102-110. Vandervoort, R. R. (1969). On the possible influence of stacking fault energy on the creep of pure BCC metals. Trans. Metall. Soc. AIME 245, 2269-2075. Vogler, S., and Blum, W. (1990). Micromechanical modeling of creep in terms of the composite model. In "Creep and Fracture of Engineering Materials and Structures" (B. Wilshire and R. W. Evans, eds.), pp. 65-79. Institute of Metals, London. Vogler, S., Messner, A., and Blum, W. (1991). Does the subgrain size influence the rate of creep? Mater. Sci. Eng. 145A, 13-19. Voigt, W. (1889). Uber die Beziehung zwischen den beiden Elatizit~itskonstanten isotroper K6rper, Wied. Ann. 38, 573-589. Vollersten, F., Hofbeck, R., and Blum, W. (1984). Double etchingnA simple method of investigating subboundary migration during creep. Mater. Sci. Eng. 67, L9-L 14. von Mises, R. (1913). Mechanik der festen K6rper im Plastisch-Deformablen Zustand. Nachr K. Ges. Wiss. G6ttingen Math. Phy. Kla. pp. 582-592. Wang, J., Horita, H., Furukawa, M., Nemoto, M., Tsenev, N. K., Valiev, R. Z., Ma, Y., and Langdon, T. G. (1993). An investigation of ductility and microstructural evolution in an AI-3% Mg alloy with submicron grain size. J. Mater. Res. 8, 2810-2818. Wawersik, W. R. (1984). Alternatives to a class M creep model for rock salt at temperatures below 160~ In "Mechanical Behavior of Salt" (H. R. Hardy, Jr. and M. Langer, eds.), pp. 1-26. Gulf, Houston, TX. Wawersik, W. R. (1985). Determination of steady creep rates and activation parameters for rock salt. ASTM Spec. Tech. PubL STP 869, 72-92. Weertman, J. (1955). Theory of steady-state creep based on dislocation climb. J. Appl. Phys. 26, 1213-1217. Weertman, J. (1956). Creep of polycrystalline aluminum as determined from strain rate tests. J. Mech. Phys. Solids 4, 230-234. Weertman, J. (1957). Steady-state creep of crystals. J. Appl. Phys. 10, 1185-1189. Weertman, J. (1965). Theory of the influence of stacking-fault width of split dislocations on high-temperature creep rate. Trans. Metall. Soc. AIME 233, 2069-2075. Weertman, J. (1975). High temperature creep produced by dislocation motion. In "Rate Processes in Plastic Deformation of Materials" (J. C. M. Li and A. K. Mukherjee, eds.), pp. 315-336. American Society for Metals, Metals Park, OH. Wolf, H. (1960). Die Aktivierungsenergie fur die Quergleitung aufgespaltener Schraubenversetzungen. Z. Naturforsch., A 15A, 180-193. Yamada, H., and Li, C. Y. (1975). Stress relaxation and mechanical equation of state in nickel and TD nickel, In "Rate Processes in Plastic Deformation of Materials" (J. C. M. Li and A. K. Mukherjee, eds.), pp. 298-314. American Society for Metals, Metals Park, OH. Yang, Z., Xiao, Y., and Shih, C. (1987). High temperature creep of Ni-Cr-Co alloys and the effect of stacking fault energy. Z. Metallkd. 78, 339-343.

Chapter 8

Modeling the Role of Substructure during Class M and Exponential Creep

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Yavari, R, Mohamed, E A., and Langdon, T. G. (1981). Creep and substructure formation in an A1-5% Mg solid solution alloy. Acta Metall. 29, 1495-1507. Yoshinaga, H. (1993). High-temperature deformation mechanisms in metals and alloys. Mater. Trans. JIM 34, 635-645. Young, C. M., and Sherby, O. D. (1973). Subgrain formation and subgrain-boundary strengthening in iron-based materials. J. Iron Steel Inst. 211, 640-647. Zhu, Q., and Blum, W. (1993). Explanation of the transition from Class M to Class A deformation behavior in terms of the composite model. In "Aspects of High Temperature Deformation and Fracture in Crystalline Materials" (Y. Hosoi, H. Yoshinaga, H. Oikawa, and K. Maruyama, eds.), pp. 649-656. Japan Institute of Metals, Sendai.

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Commentsand Summary K. Krausz and A. S. Krausz University of Ottawa Ontario, Canada

One may take the view that there is only one constitutive law of plastic deformation. But if so, it is an extremely complicated one and cumbersome to apply: in fact, it cannot be practical. A useful one should give the optimum application for the specific purpose. It is in this context that the contributions to this volume are presented. There are many aspects on which the contributors to this volume agree. The constitutive law of plastic deformation is now taking shape. Much, however, remains to be established; and the reader should bear in mind that the different views expressed by the contributors are not contradictions, but considerations among which the practitioner must select for the specific purpose of design. The following summaries assist the reader in this task. In his chapter J. L. Chaboehe ("Unified Cyclic Viscoplastic Constitutive Equations: Development, Capabilities, and Thermodynamic Framework") considers the development of the unified cyclic viscoplastic constitutive equations within the framework of continuum mechanics for the description of the macroscopic behavior of metallic materials. He attributes their great development during the past twenty years largely to the improvement in computing and the increasing use of the full inelastic analysis of structures in many industrial applications. He introduces the classical framework for the development of unified constitution equations under the small strain assumption and quasistatic conditions. The theory uses kinematic and isotropic state variables, and separates the yield and drag effects. In the first part of the article the general framework for the initially anisotropic material is introduced, both for the yield surface concept and the hardening effects. The equations are then expressed for the initially isotropic materials. Explicit forms are presented for the various rate equations used currently at ONERA and in several Unified Constitutive Laws of Plastic Deformation Copyright (~) 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

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industrial applications. In the hardening rules, three terms are introduced systematically: hardening, dynamic recovery, and static recovery terms. Additional effects are also discussed: 9 strain range memorization, 9 the two limiting cases of rate-independent plasticity, and 9 secondary creep. The correspondence with and difference from the multisurfaces approaches of plasticity are considered. In the second part, the capabilities of the proposed unified constitutive framework are demonstrated. The experimental results obtained on several classes of metallic materials, including nickel-base superalloys (used for disk or blade components) and stainless steels are discussed. Various aspects are presented, including 9 kinematic hardening, 9 cyclic hardening or softening, 9 high strain rates, 9 creep-plasticity interactions, 9 high-temperature static recovery effects, and 9 several forms of ratcheting effects. In the third part of the article, Chaboche discusses the thermodynamic aspects of the constitutive equations in detail using the classical framework of continuum thermodynamics with internal state variables. A restricted standard treatment is presented. It is based on the thermodynamic state potential and the dissipation potential, which facilitate the checking of the second thermodynamic principle in an a priori manner. The author explains why he does not consider the rates of the observable state variables in the rate equations in the proposed framework for internal state variables. He considers a more general treatment without using the standard form of a dissipation potential. Chaboche discusses several techniques that lead to investigation of the validity of the second principle with slightly different assumptions. The thermodynamic aspects are also examined by considering the stored and heat-dissipated energies during a (visco)plastic flow. It is shown that only the nonlinear kinematic hardening model (with a dynamic recovery term) is able to recover the decreasing ratio of the stored energy to the plastic work as a function of the plastic strain. Several analyses of experimental results demonstrate the validity of the constitutive models, both for their mechanical response and for the energy dissipation. The last subsection is devoted to varying temperature effects in terms of 9 its influence on the constitutive equations parameters, 9 the necessary presence of temperature rate terms in the back-stress rate equations, and

Chapter 9 Commentsand Summary 443 ~ the description of temperature history effects associated with metallurgical changes. Finally, Chaboche presents as example for the time aging (at room temperature) of some aluminum alloys. A simple annealing/aging model is proposed, using only one additional state variable; this model is able to describe the main facts within the thermodynamic framework. The chapter by Y. Estrin ("Dislocation-Density-Related Constitutive Modeling") deals with a unified elastic-viscoplastic constitutive model proposed ten years ago by Kocks and Mecking. It represents the microstructural state of the material with the dislocation density as the internal variable. The author emphasizes that the model is a versatile tool for the description of the mechanical responses of metallic materials. Taking this approach, the study accounts for the evolution of the dislocation density by introducing in the constitutive equations such microstructural quantities as grain size, particle spacing, and the characteristics of lamellar or composite structure. It is shown that, depending on the complexity of the system, one or two dislocation-density-related internal variables may be used. Further sophistication is achieved by taking solute effects into account. Special attention is paid to dynamically strain aging materialsl Estrin emphasizes recent developments and the predictive capabilities of this type of microstructure-related modeling. The chapter opens with a brief historical review on the dislocation-based approach to constitutive modeling, followed by an outline of the structure of a model. In succeeding sections, the concept is developed from a prototype "structureless" reference material (e.g., a course-grained single-phase metal). The effects on nonshearable and shearable second-phase particles are considered with the further development of the progressively more complex system. This is followed by the consideration of the effects of solutes on strain-hardening behavior. Two cases are considered: (1) immobile solutes, which affect the rate of recovery through their influence on the stacking-fault energy, and (2) mobile solutes, which inhibit cross-slip by a dynamic interaction with dislocations. The paper by R. W. Evans and B. Wilshire ("Constitutive Laws for HighTemperature Creep and Creep Fracture") argues that conventional theoretical and practical approaches to creep and creep fracture should be abandoned in favor of the Projection Concept proposed by the authors. Their arguments are based on the view that the normal creep curves recorded for metals and alloys at high temperatures cannot be described adequately by reporting just a few standard parameters, such as the minimum creep rate, the rupture life and the creep ductility. The authors make the point that simple reliance on measurements of the minimum creep rate

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rupture life and the creep ductility disregard the primary and tertiary stages, so that a major proportion of the information available from a creep curve is ignored totally. The deficiencies of conventional approaches are illustrated by the fact that, when standard plots are made of the variations in creep ductility and the rupture life with stress and temperature, differences in the behavior patterns observed under different test conditions must be interpreted by assuming that different mechanisms become dominant in different stress/temperature regimes. Evans and Wilshire emphasize that, despite decades of study, no general agreement has been reached on the precise mechanisms involved, and that particular difficulties have been encountered in attempting to account for the anomalous properties apparently exhibited by commercial particle-strengthened alloys. Moreover, the limitations of traditional theoretical ideas lead inevitably to practical problems. For instance, Evans and Wilshire consider that conventional procedures do not allow extrapolation of short-term results, so that costly and protracted test programs must be completed to supply long-term materials data for engineering design. The authors suggest that the Projection Concept, in contrast, introduces equations capable of quantifying the detailed shape of full creep curves and the variations in curve shape with stress and temperature. As developed by Evans and Wilshire, these equations provide a theoretically sound description of creep behavior based on micromodeling of the dislocation processes that govern primary creep and the damage processes that can cause the creep rate to accelerate during the tertiary state. Using the Projection Concept relationships, differences in behavior patterns observed in different stress and temperature regimes can be predicted and explained without invoking conventional multimechanism ideas, simultaneously rationalizing the apparently anomalous property characteristics reported for commercial creep-resistant alloys. Evans and Wilshire show that by providing unified constitutive laws relating stress-strain-time-temperature, the Projection Concept methodology is of direct practical applicability, as illustrated by extensive data sets observed for 0.5 CrO.SMo0.25V ferritic steel, a material widely used for construction of power and petrochemical plants. Evans and Wilshire demonstrate that the full range of long-term data needed for engineering design can be predicted accurately by Projection Concept analysis of short-term high-precision creep curves. Furthermore, they show that the extrapolation procedure also allows short-term results for service-exposed material to be used to estimate the remaining life components, an essential requirement for safe and cost-effective continued operation of plant approaching the end of its original design life. New results are then quoted by the authors to show that the Projection Concept relationships introduced originally to quantify uniaxial creep properties can be extended to analyze and interpret materials behavior even under multiaxial stress states. The authors indicate that in little more than a decade, the Projection Concept has progressed from an iconoclastic

Chapter 9 Commentsand Summary 445 idea to a unified theoretical and practical approach to creep and creep fracture, thus offering a radical alternative to traditional views. The article by G. A. Henshall, D. E. Helling, and A. K. Miller ("Improvements in the MATMOD Equations for Modeling Solute Effects and Yield-Surface Distortion") presents the MATMOD (MATerials MODel) family of unified constitutive equations. These equations were developed with the goal of predicting nonelastic deformation in engineering material subjected to complex loadings. To achieve this goal, the MATMOD family was designed to simulate a broad range of phenomena, including low-temperature "plasticity," high-temperature "creep," cyclic deformation, strain softening, solute strengthening, and multiaxial deformation. This was accomplished using the "physical-phenomenological" approach, in which internal state variables represent, in an approximate but physically meaningful manner, the controlling deformation processes. The overall structure of each model and the functional interrelationship among its variables follow from an understanding of these physical causes and effects. Adoption of the "unified" approach, in which one set of equations treats all of the above phenomena, follows logically from the fact that they are all controlled by the same set of physical mechanisms. To provide the required accuracy, the specific equations in the models were derived by fitting the quantitative behaviors observed across entire classes of materials. The focus of effort on a variety of deformation phenomena has led to several versions of MATMOD, which are briefly reviewed. This chapter focuses on two later versions of the equations that offer improvements beyond the earlier models. Specifically, additional effort was undertaken to simulate multiaxial deformation, particularly yield-surface distortions, and to combine the capability of modeling both solute effects and complex strain-softening behaviors within a single model. This latter model was kept as simple as possible so that the difficult job of determining the material-dependent constants for specific alloys would not become impossible. The key modeling concept that links these two new versions is the use of two back stress variables, which interconnect a wide range of deformation behavior. These authors first describe how they have modified the multiaxial MATMOD4V equations to permit prediction of distortions in the yield surface that are typically observed at small strain offsets following prestraining. These data indicate the presence of both long-range and short-range back stresses in several alloys. Distortion of the yield surface has been modeled using a Hill-type anisotropy tensor to modify the yield function in the model. Allowing the coefficients in the Hill approach to be a function of both the short- and long-range back stresses (the kinematic hardening variables in the model) results in an evolution of anisotropy during deformation. This improved model, MATMOD-4V-DISTORTION, retains the capability to predict expansion and translation of the yield surface, as well as a wide variety of mechanical behaviors simulated by MATMOD-4V. Verification of the model was achieved both through simulations and by independent predictions of

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yield locus distortions under proportional and complex nonproportional loadings for 1100 A1, 70:30 brass, and 2024-T7 A1. Having demonstrated (from multiaxial experiments) the existence of separate short-range and long-range back stresses, the authors utilize the short-range back stresses to treat solute effects in a simple manner in the MATMOD-BSSOL equations. Henshall and co-workers argue that this approach is consistent with the experimentally observed behavior and with most accepted theories and physical models of solute strengthening. They claim further that it provides, in addition, a physically plausible link between the low- and high-temperature behavior of solution-strengthened alloys while retaining strain softening and most of the previous MATMOD capabilities. Other improvements, unrelated to solute effects, simplify the equations without significantly compromising their physical basis or capabilities. MATMOD-BSSOL has been verified through simulations of peaks and plateaus in the low-stress versus temperature curves, steady-state creep behavior, and through independent predictions of a variety of nonelastic deformation behavior for pure A1 and dilute binary A1-Mg alloys. Finally, the methods used for numerical integration of the equations and evaluation of the material constants are briefly described. Numerical integration of MATMOD-4V-DISTORTION is achieved through a Fortran program employing the Gear Method for integrating systems of stiff differential equations. MATMODBSSOL is integrated using the NONSS program, which was designed to interface efficiently with finite-element solid mechanics codes. Evaluation of the material constants for each model involves trial-and-error procedures through which model simulations are compared with a variety of mechanical test data. The authors provide the values of these temperature-independent constants for several materials. A. S. Krausz and K. Krausz ("The Constitutive Law of Deformation Kinetics") develop the law from basic physical principles and express it in terms of engineering quantities appropriate for design and maintenance practice. These include typical design quantities: 9 stress, 9 strain, 9 strain rate, and 9 temperature and physical quantities that are readily measurable with long-established standard methods: 9 activation energy, 9 mobile dislocation and vacancy density, and 9 stress to work conversion. The theory distinguishes two components: the kinetics equation and the state variables. The combination of these two results in the unified evolutionary constitutive

Chapter 9 Commentsand Summary 447 law. The kinetics equation is rigorously derived and its establishment for any specific application is described for a step-by-step procedure that lends itself well to computer processing. The state variables are considered for the representation of the microstructure as rigorous expressions, and as empirical relations when a physically based relation is lacking. Materials-testing concepts and methods are described for the representation of the state and development of the microstructure. Examples given for the analysis of the kinetics equation demonstrate the good agreement of the deformation kinetics method with the measured behavior. The kinetics law was derived as one of the thermally activated processes; this recognition led to the representation of plastic deformation within one framework for 9 diffusion, the mechanism of high-temperature creep deformation, 9 chemical reactions, the mechanism of corrosion and other environmental effects, 9 radiation reactions, which are essential components of nuclear reactor environments, and 9 light radiation, which affects polymeric materials. Importantly, the framework of thermal activation resulted in the exact description of the effect of temperature. The kinetics method led to a constitutive law that is valid for any material: metals, alloys, polymers, ceramics, and their composites. This is a considerable advantage in the present industrial environment where new materials are introduced at an impressive rate. The contribution by E. K r e m p l ("A Small-Strain Viscoplasticity Theory Based on Overstress") delineates the properties of the viscoplasticity theory based on overstress which is given for isotropy and small strain in three-dimensional form. This unified theory was developed by the author and his students during the last twenty years. As in many unified theories, the concept of a yield surface is not used and all inelastic deformations are considered rate-dependent. The concept of an effective stress, termed overstress by the author, plays a major role. The effective stress or overstress is the difference between the applied stress and the back stress (called equilibrium stress) and the inelastic strain rate is made to depend on this quantity. For the materials modeled by the author (stainless steel, a fully age-hardened A1 alloy, and a titanium alloy at room temperature: a modified cram steel at 538~ and an alloy 800 H at homologous temperatures exceeding 0.7) no need to introduce a drag stress was evident. Consequently, the paper considers only models whose sole effective stress (overstress) depends on the inelastic strain rates. Whether or not the drag stress needs to be included in a deformation model depends on the outcome of specially conceived mechanical tests which are described in the text. The growth of the equilibrium or back stress depends on two other state variables: the tensorial kinematic stress, which is intended to model work hardening

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(softening) in monotonic loading, and the rate-independent or isotropic stress, which models cyclic hardening or softening. When this quantity is constant, cyclic neutral behavior is reproduced by Krempl's model, which he calls VBO (viscoplasticity theory based on overstress). In the growth law for the kinematic stress given in the paper, work hardening (softening) is assumed to be linear after a nonlinear transition. The final slope of the kinematic stress, which also determines the slope of the stress, is equal to the tangent modulus Et; and this quantity can be positive (work hardening), zero (ultimately constant stress), or negative (ultimately work-softening). The growth law for the equilibrium (back) stress deviates from the normal pattern of hardening and dynamic recovery. Although it contains these terms, it also includes a term multiplied by the stress rate (or the elastic strain rate in the strain formulation), which effects a change of the equilibrium stress in the quasielastic regions. The author justifies this unusual feature by the need to properly represent elastic regions. According to conventional assumptions, state variables should grow only during inelastic deformation since they represent the internal structure of the material. This argument does not consider the modeling of elastic regions, which are important for engineering applications. Two other researchers have claimed that the elastic growth violates thermodynamic principles. Krempl discusses their papers and refutes their conclusions. Krempl also discusses asymptotic solutions of the coupled differential equations for constant strain rate loading, suggesting that the asymptotic solutions correspond to established inelastic flow, i.e., the point at which flow stress is reached. The asymptotic solutions permit the identification of material properties suitable for the model. Creep and relaxation behavior is discussed in detail, and qualitative arguments are used to show that the model can sustain a nonzero stress at rest. For a monotonically work-hardening material, only primary creep can be modeled. This property is called "cold creep." For the modeling of "hot creep," a static recovery term must be included in the growth law for the equilibrium stress. While this addition permits the modeling of secondary creep at stress levels in the quasi-linear region of the stress-strain diagram, it cannot model cyclic softening. As a consequence, a growth law for the isotropic stress is added to model cyclic softening and strain rate history effects. Other extensions include cyclic hardening (softening) and anisotropy. The relation of the Krempl's theory to those of Perzyna and Chaboche is discussed. Of special interest is the significant influence of the direction of the dynamic recovery term in the growth law for the equilibrium stress. The author gives two versions of the growth law for the equilibrium stress, which differ only by the direction of the dynamic recovery term. Numerical experiments show that the behavior in 90 ~ out-of-phase loading of the two is substantially different. Such a significant effect was not expected. The paper gives a review of the continuummechanics-based VBO model. Krempl claims that the model was derived using

Chapter 9 Commentsand Summary 449 macroscopic experiments as a guide. It therefore contains microstructural mechanisms which influence the outcome of experiments. The chapter by J. Ning and E. C. Aifantis ("Anisotropic and Inhomogeneous Plastic Deformation of Polycrystalline Solids") contains some of the recent contributions of plasticity theory by the second author (and his co-workers). It describes deformation-induced anisotropy, texture, and heterogeneity. For the description of anisotropy and texture effects, the so-called "scale-invariance" approach is adopted, thus allowing information and constitutive relations pertaining to single slip to be cast in the form of macroscopic constitutive equations. An orientationdistribution function and a texture tensor are introduced into the earlier version of the scale-invariance framework to describe texture effects in plastically deformed polycrystalline aggregates. The authors demonstrate that these effects are conveniently represented by the proposed theory; they present examples of different deformation modes, such as tension, rolling, and tension-torsion. For the description of heterogeneity and the associated problem of size effects, the strain-gradient approach is adopted. Various aspects of this approach, including the physical interpretation of the gradient coefficients, are considered. In particular, theoretical estimates for the strain gradient coefficient are provided based on "self-consistent" arguments. The applicability of the gradient approach for the interpretation of size effects in torsion is illustrated by comparing the theoretical predictions with experimental data measured in twisting experiments on high-purity (99.99%) copper wires. A detailed review of the current understanding of the effect of microstructural parameters on class M creep (i.e., power-law creep) behavior is presented by S. u Raj, I. S. Iskowitz, and A. D. Freed ("Modeling the Role of Dislocation Substructure during Class M and Exponential Creep"). Microstructural observations conducted in the power-law and exponential creep regimes suggest that creep is influenced by a complex interaction between several elements of the microstructure, such as dislocations, cells, and subgrains; steady-state behavior is attained when the microstructure reaches a dynamic equilibrium among these different substructural features. Thus, the formation of equiaxed subgrains and the occurrence of class M creep need not be a sufficient condition for steady-state behavior. Instead, recent observations reveal that the refinement of cell boundaries and the formation of secondary subboundaries are equally important components of steady-state creep. Quantitative measurements of the ratio of the dimensions of the cell boundary to the cell interior suggest that steady-state behavior is likely when this ratio is about 0.2. It is demonstrated by Raj, Iskowitz, and Freed that the microstructures formed in the exponential creep region are similar to those formed during the early stages of normal primary creep in the class M region. These experimental observations are used to formulate a phenomenological approach to understanding transient and steady-state deformation behavior in terms of the strain rate-stress deformation laws. A three-dimensional dislocation substructure creep model is

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developed by the authors and coupled with a modified one-dimensional dislocation model. Although the dislocation analysis presented is still limited in scope, it nevertheless extends previous analyses by taking into account the dynamic evolution of the microstructure during deformation. In closing the Editors draw the reader's attention to the purpose for which the constitutive law was developed and to which its application is directed: The model has to be economical. That is, its processing has to be optimal in the conglomeration of 9 available expertise, 9 material cost, 9 cost of material testing, including the time cost, 9 modifications for specific applications, 9 cost of stress analysis, including the time cost, 9 design of manufacturing process, 9 product design process, 9 product performance, including maintenance and reliability, 9 company policy, and 9 marketing decisions. This is only a short list of the considerations that are involved in the selection of the appropriate constitutive law. The choice of the constitutive law must be made so that optimum economic benefits are derived from the product. This decision is made in a dynamic marketplace. A tall order indeed. The authors of this volume offer guidance and help in the service of this need.

Index

Activation energy changes, 247-249 evaluation, 262-263 in kinetic equation, 235-236, 238-240 for lattice diffusion, 356 for primary and tertiary rate constants, 124-126 reverse rate constant, 255-256 variation for creep, 354 Activations forward, over single energy barrier, 250-251 frequency, 245 reverse effect on strain rate, 252 rate, 241-242 Activation volume, evaluation, 262-263 Aftereffect, creep or relaxation test, 293-294 AGICE program, developed at ONERA, 17 Aging, strain, dynamic, 88-90, 194 Aging effects, modeling, 57-61 Algorithm, solution, for DSV model, 420-423 Alloys Class I, 208-209 Class II, 197-199 creep-resistant, 444 particle-hardened power-law creep, 112-114 transition period, 147 Aluminum alloy 1100, 95-96 alloy 2024, 60-61 alloy with Mg MATMOD-BS SOL material constants, 222 MATMOD-BSSOL simulation, 205-210

gradient coefficient c, 335 1100-O, 185-187 1100-O and 2024-T7, 164-167, 220 pure, 206-213 stationary creep behavior, 52-53 superpurity, grain boundary cavities, 146-147 wavy slip, 360 Amplitude, vibrational, fluctuation, 238 Anisotropy and fourth-order tensors, 6-7 plastic deformation of polycrystalline solids, 319-339 tensor, in MATMOD equations, 168-178 in viscoplasticity theory, 306-307 Annihilation extended dislocations, 414 reactions, dislocation, 73 Ansatz representing decline in glide resistance, 84 for static recovery coefficient, 74 Apparatus, for determining MATMOD-BSSOL material constants, 221 Asymptotic properties, in stress space, 288-290 Atoms diffusion from grain boundaries, 348 and elementary rate constant, 236-243

Backstress due to solutes, 158 effective, 172-179 internal, 396-399 in MATMOD models, 445-446 with plastic strain, 99-100 rate equation, 6-8, 14, 54

451

452

Index

Backstress (continued) short- and long-range, 162-168, 223 short-range, solute effect simulation through, 189-214 two components, 188 Barrier energy, combinations, 244-246 strength, dependence on dislocation density, 119 Behavior, see also Creep behavior cyclic neutral, 298, 301 deformation, particle effects, 82-86 flow, types, 346-352 MATMOD variables, 180, 184 mechanical, materials, 276 in plastic deformation, 236 plastic flow, texture effect, 327-332 power-law, prediction, 126-129 single-energy-barrier mechanism, 253 stabilized cyclic, 20-34 stress-relaxation, 258-262 in lead, 266-267 stress-strain, 48-50 threshold-like, 88 Bond breaking, energy requirement, 237-240, 273-275 Boundaries cell, refinement, 377 grain cavities, 121, 146-147 effects, 352 primary and secondary, 371-373 Boundary structure, subgrain, 73 Bounding surface, image point on, 18-19 Brass, 70:30, 164, 220

C Calculation anisotropy tensor Mij, 171-179 rupture life, 130-132 Cartesian reference configuration, 4 Cauchy stress deviator, 326 Cavities grain boundary, 121, 146-147 strain-controlled development, 126 Cells boundaries etch pits in, 371-373 refinement, 377

distinction from subgrains, 368-370 increasing temperature, 393-395 walls, strain hardening, 413--414 Chaboche's analysis, in viscoplasticity theory, 308-310 Chaboche theory, 284, 295 Changes activation energy, 247-249 creep behavior, with grain refining, 80 in cyclic behavior, 302 flow unit density, 248-249 internal variables, 308 microstructure during deformation, 236 in state equations, 247-256 stress experiments during creep, 146-148 time-dependence, 259-262 subgrain size, 421--422 Channels, related dislocations, 96-100 Class M creep dislocation substructure role, 343-430 effect of stacking fault energy, 363-365 regime, 384-387 Climb, dislocation in cell boundary refinement, 377 class M behavior due to, 390-391 high- and low-temperature, 353-357 Clustering, random dislocations, 381 Coarsening, subgrain, 401 Coleman-Gurtin formalism, 307 Composite models one-dimensional, 415-4 16 three-dimensional, 417-420 two-phase, 411-415 Constants additional, in modified growth laws, 285 creep multiaxial, 140 uniaxial, 130 material, temperature independence, 306 0, primary, 138-139 Constitutive equations in case of cyclic deformation, 96-100 MATMOD family, 154-159 unified cyclic viscoplastic, 1-63 viscoplasticity theory based on overstress, 282-284 Constitutive laws cyclic viscoplastic, 4-20

Index deformation kinetics, 229-276 for high-temperature creep and creep fracture, 107-151 for primary creep, 389-390 Constitutive model, capabilities, 20-34, 62 Constitutive relations, for single crystallite, 321-322 Constraints, imposed constant applied strain as, 258 conversion, 249-250 Constrictions, preexisting, cross-slip at, 358 Conversion, of imposed constraint, 249-250 Copper activation energies, 362 creep, 355-356 gradient coefficient c, 335 polycrystalline, creep data, 359-360 single crystal, 49-50, 386-387 tough pitch, 261-262 Coupling DSV and dislocation physics, 424 kinematic and isotropic hardening, 9-10 microscopic kinetic equations and macroscopic stresses, 351 undesirable, 39 Creep cold and hot, 448 at constant stress, 76-78 exponential and class M, dislocation substructure role, 343-430 high-temperature constitutive laws, 107-151 in context of MATMOD-BSSOL, 193-194 interaction with plasticity, 29-32 life, predictions, 131 modeling, 290-291 under multiaxial stress states, 132-142 nondiffusional mechanisms, 357-368 under nonsteady loading conditions, 143-150 stationary, 14-15, 52-53 steady- state, MATMOD-B SSOL, 205-209 substructure normalized stress effect, 368-383 prior deformation effect, 401-409 Creep behavior change with grain refining, 80 description, 444 high-temperature, 111 modeling, 118-122, 304-305 primary, secondary, and tertiary, 291-292

453

steady-state, 371-373 Creep curve for cycles, 29-32 predicted for stress-temperature conditions, 149-150 shape, 109-110, 124-126 stages, 347 strain/time, 136-137 in stress-strain plane, 313-314 structure parameters on, 389-390 Creep data alloy 800 H, 83-84 extrapolation, parametric procedures, 114-117 polycrystalline copper, 359-360 uniaxial, engineering designs based on, 108-109

Creep fracture, constitutive laws, 107-151 Creep law limiting, 13-15 universal, 390 Creep model multiphase three-dimensional, 416-424 Nix-Gibeling one-dimensional two-phase, 409-416 Creep rate decrease during primary creep, 396 normalized, 354-355,363-368 parameter prediction, 130 for prestrained specimen, 404-405 rapid rise, 402 secondary and tertiary, 109-110 steady-state, 83-84 Creep strain effect on substructure, 384-392 separation from plastic strain, 24-26 Cross-slip mechanisms, nondiffusional creep, 357-361 screw dislocations, 357-361,412-413, 415-416 solute effect, 89-90 stacking fault energy effect, 364-365 Crystallite, single, constitutive relations, 321-322 Crystallographic texture, anisotropy tensor due to, 171-172, 178 Cyclic hardening description, 21-24 in viscoplasticity theory, 301-302

454

Index

Cyclic softening, in high homologous temperature deformation, 303

D Damage deformation, accumulation, 140-142 intergranular, 121-123 Dashpot, viscosity coefficient, 312-313 Decomposition, stress, for uniaxial tension-compression, 10 Defects, microstructural, distribution change, 236 Deformation behavior, particle effects, 82-86 for constant microstructure, 345 curves, in two-internal-variable model, 95 cyclic, 96-103, 197 damage accumulation, 140-142 isotropic finite, 315 kinetics, constitutive law, 229-276 models, two-phase, 350 multiaxial, 223 nonelastic, prediction, 445 plastic, polycrystalline solids, 319-339 prior, effect on creep substructure, 401-409 progressive, 28-29 quasistatic, metallic materials, 3 substructure, 195 tension-torsion, 331-332 transient strains for, 78-79 uniaxial, 75 Derivation coupled differential equations in DSV model, 425-427 MATMOD-BSSOL, equations, 202 Design data, long-term, prediction, 129-132, 151 Differential equation coupled, derivation, 425-427 solution, 269 Diffusion along dislocation cores, 111-112 atoms from grain boundai'ies, 348 controlled creep mechanisms, 353-357 related creep concepts, 128-129 Direction dislocation, 146-147 two-directional movement, 240-241 dynamic recovery terms, 285 final stress path, 165-166

flow units, 274 shear, effective strain rate, 169-171 stress and overstress, 289-290 Dislocation annihilation reactions, 73 arrangement, true state defined by, 54 athermal breakaway, 203 channel-like, 96-100 cores diffusion along, 111-112 diffusion-controlled creep mechanisms, 353-357 creep processes, 128 density, 276 geometric necessary, 338 mobile, intermittent arrest, 88-89 movement along alloy matrix, 113-114 effective stress governing, 146-147 through soft region, 337 two-directional, 240-241 physics, coupling with DSV, 424 related obstacles, 79-82 screw-oriented, 357-361, 412-413, 415-416 substructure during class M and exponential creep, 343-430 development, 191 stress dependence, 378-383 three-dimensional, 449-450 Dislocation density dependence of barrier strength, 119 evolution, 443 related constitutive modeling, 69-104 within subgrain, 383-389 Dislocation glide independent of vacancy migration, 244 kinetics controlled by, 71-72 obstacle-controlled, 361-362 resistance models, 72-73 in soft and hard regions, 412 viscous resistance, 287-288 Dislocation substructure viscoplasticity, 350, 352, 392 Dislocation substructure viscoplasticity model, 416-417, 420-423,425-427 Dissipation inequality, intrinsic and thermal, 36 Dissipative potential different from yield surface, 43-44 standard generalized materials, 37, 40-42

Index Distance glide, 415 interatomic, increase, 237-240 Distortions, yield surface, modeling, 160-188 Distribution elongated subgrains, 373-375 microstructural defects, change, 236 Double-logarithmic plot, 252-253 Drag stress decrease to zero, 12 evolution, 42 independent of yield stress, 6 DSV, see Dislocation substructure viscoplasticity

Economical model, 450 testing methods, 249-250 Elastic limit evolution, 60-61 temperature dependency, 57-58 Elastic-plastic interaction law, 333-334 Elastic region, exactly linear, 296 Energy for bond breaking, 237-240, 273-275 stacking fault, effect on class M creep, 363-365 stored and heat-dissipated, 47-50 Energy barrier combinations, 244-246 single and forward activation, 250-251 model, 264 symmetric, 267-270 Engineering applications, typical operating temperatures, 349 Engineering designs based on uniaxial creep data, 108-109 long-term data prediction, 151 Equations Fokker-Planck, 323 kinetic, 235-236, 238-240 L6vy-von Mises, 70 MATMOD, improvements, 153-224 MATMOD-BSSOL, development, 194-202 Prandtl-Reuss, 417-419 state, 230-233,247-256 unified cyclic viscoplastic constitutive, 1-63

455

Equilibrium stress, growth law, 283-284, 298-301,447-448 Etch pits in cell boundaries, 371-373 low density, 375-376 uniform distribution, 402, 407 Evolution drag stress, 42 effect on constitutive response, 21-24 elastic limit, 60-61 kinematic stress, 309-310 MATMOD-4V-DISTORTION variables, 184 nonmonotonic, ratio F = Ws / Wp, 48-49 strategy, 93-95 yield strength plateaus, 190 Evolution equations for aging variable, 58-59 for hardening variables (isothermal case), 6-7 hybrid, 79-82 for internal variables, 38-40 in two-internal-variable model, 91-92 Evolution laws, kinetic and microstructure, 344-345 Exponential creep dislocation substructure role, 343-430 regime, 387-392 Extrapolation, creep data, 114-117

Failure acceleration, 150 during design life, 143 tensile creep, 122-123 F)~ equation, MATMOD-BSSOL, 199-200 J~p equation, MATMOD-BSSOL, 199-200 Ferritic steel, 0.5Cr0.5Mo0.25V, 113-116, 124-131,135-137 Finite-element analysis, 350, 352 Flattening, yield surface loci, 165-166, 216-217 Flow behavior, types, 346-352 plastic kinetics, 71-72 texture effect, 327-332 three-dimensional, 323-325,330-332 Flow law constitutive equations, 282-283 deviatoric, 304

456

Index

Flow stress asymptotic, 286-287 plateaus, 203-205 Flow unit density, change in, 248-249 Flow units, direction, 274 Fokker-Planck equation, 323 Forest density, 91-92 dislocations, 156, 159, 376 Formalism Coleman-Gurtin, 307 multicriterion, 46-47 ODF, 320-324 Formation jog, thermally activated, 98 substructure, NaC1 single crystals, 368-399 uniform dislocation substructure on loading, 384-385 Forward-reverse mechanism, in constitutive law, 266-267 Fracture, creep, constitutive laws, 107-151 Frank-Read source, 358 Friedel-Escaig mechanism, 413 Friedel model, 358 Fsot equation, MATMOD-BSSOL, 200-202

G Geometric necessary dislocation, 338 Geometric obstacles and dislocation-related obstacles, 79-82 high density, 77-79 Glide polygonization, 376 Gradient approach, relation to self-consistent approach, 333-335 Gradient coefficient c, estimates, 335-337 Grain boundary cavities, 121, 146-147 effects, 352 Grains, see also Subgrains orientation effects, 325,330 recrystallized, 381 size effects, 77-82 Growth law, for eqilibrium stress, 283-285, 298-301

Hard and soft regions, of deformation substructure, 409-4 11 Hardening, see also Work hardening

cyclic description, 21-24 in viscoplasticity theory, 301-302 and dynamic recovery terms, 196-197 grain boundary, 78 kinematic associated internal variable, 40 and isotropic hardening, 4-20 nonlinear, 49, 442 rules, temperature rate in, 53-56 Heat, stored in material, 47-50 H-H model, 335-336 History MATMOD family of constitutive equations, 157-159 simple and complex loading, 209-211 thermal, 153-154 History effects, temperature, 56-57 Hookean material isotropic, 418-419 thermoelastic response, 38-39 Hybrid model, 79-82 Hysteresis, modeled, 310-311

I Image point, distance from stress state, 19 Improvements, in MATMOD equations, 153-224 Incompressibility deviatoric property implied by, 325 in Mij, 173-174 off-diagonal components calculated from, 178 Inconel 738 LC, deformation data, 100-103 Industrial activity, constitutive law in, 271 Inhomogeneity, plastic deformation of polycrystalline solids, 319-339 Integration implicit, 422-423 numerical MATMOD equations, 214 MATMOD-4V-DISTORTION, 224, 446 operational equation, 265 Invertibility, in Malmberg's analysis, 311 Isothermal case, evolution equations for hardening variables, 6-7 Isotropy, materials, restriction to, 8-9

I Jog formation, thermally activated, 98 Jump test, strain rate, 94-95, 101-102

Index

457

Logarithmic law, as empirical relation, 232 Low-energy dislocation substructure, 345 K-B-W model, 335,339 Kinematic hardening associated internal variable, 40 and isotropic hardening, 4-20 nonlinear, 49 and normal viscosity, 21 Kinematic stress, growth law, 284 Kinetic equation MATMOD-BSSOL, 194-195 plastic deformation, 234-246 in power law form, 70-71 in unified evolutionary constitutive law, 446-447 Kinetics deformation, constitutive law, 229-276 dislocation glide-controlled, 71-72 first-order reaction-rate, 120-122 Kocks-Mecking model, 72, 76, 80, 84-85, 88, 103

Lead, stress-relaxation behavior, 266-267 Legendre-Fenchel transform, 37-38 L6vy-von Mises equation, 70 Limitation, one-dimensional composite model, 415-416 Limiting case high strain rates, 24-26 rate-independent, plasticity theory, 12-13 Linear relation, in deformation kinetics, 261-262 Loading cyclic, under small strains, 61-62 formation of uniform dislocation substructure on, 384-385 histories, simple and complex, 209-211 monotonic standard linear solid in, 314 work hardening (softening) in, 448 nonproportional, 298-299, 301-302 nonproportional multiaxial, 12 nonsteady, creep under, 143-150 out-of-phase cyclic, 300 proportional and nonproportional, 179-185 proportional multiaxial, 16 tension-compression, 26-29 for testing 0.5Cr0.5Mo0.25V steel, 135 thermomechanical, 34

M Magnesium, alloy with A1 MATMOD-BSSOL material constants, 222 MATMOD-B SSOL simulation, 205-210 Malmberg's analysis, in viscoplasticity theory, 310-311 Material constants, MATMOD, evaluation, 215-221 Materials coarse-grained single-phase, 74-77 constitutive law expressed for, 231 deformation kinetics theory, 274 granular, 332-333 Hookean isotropic, 418-419 thermoelastic response, 38-39 industrial, constitutive law, 272 initially isotropic, 8-12 metallic, quasielastic deformation, 2-3 parameters, determination procedure, 15-17 single-phase, class M and exponential creep, 352-368 standard generalized, 37 MATMOD-BSSOL material constants, evaluation, 217-221 modeling solute effects with, 189-194 simulations and predictions, 202-211 specific equations, development, 194-202 verification, 446 MATMOD-4V-DISTORTION material constants, evaluation, 215-217 in modeling yield-surface distortions, 162-169 numerical integration, 446 predictive capabilities, 179-188 Mechanistic features, yield-locus distortions, 167-168 Memorization plastic strain, 11-12 strain-range, 24, 442 Metal-forming operations, yield-surface distortions during, 161-162 Metals polycrystalline, creep curves, 121 pure, power-law creep, 111-112 Microstructure evolution laws, 344-345

4511

Index

Microstructure (continued) formed in class M and exponential creep regimes, 370-378 modification, 145 parameters, rate of change, 423-424 quantities, analysis, 256-270 stability, 400-409 state equations, 231-232 Migration grain boundary, 127 solute, toward dislocations, 89 subboundaries, 407 vacancy, 240-242 Minimization, quality function, 93 Misorientation angle, subboundary, 369 Modeling aging effects, 57-61 constitutive dislocation density related, 69-104 MATMOD philosophy, 154-157 creep, 290-291 creep behavior, 118-122, 304-305 nonelastic deformation by MATMOD-BSSOL, 213-214 relaxation, 291-293 stored energy, 48 tensile curves, 51 Models composite one-dimensional, 415-416 three-dimensional, 417-420 two-phase, 411-4 15 constitutive, capabilities, 20-34 DSV, 416-4 17, 420--423 hybrid, 79-82 Nix-Gibeling, 409-416 Nix-Ilschner, 349 NLK, 55 one-internal-variable, 72-90 radial return, 7 two-internal-variable, 91-104 Modification microstructures, 145 Nix-Gibeling two-phase composite model, 411-415 yield function, with anisotropy tensor, 169-171 Monkman-Grant relationship, 115 Mroz's translation rule, 18-19 Multiaxiality factor, loadings, 12

Multisurface approaches, rate-independent limit case, 17-20 Mutation, in evolution strategy, 94

Nabarro-Herring creep, 112, 348 Newtonian viscous creep, 347-348 NLK rule, 19-20 Nonlinear system solver method, MATMOD equations, 214 Nonshearable particles, dispersion, 82-84 Normality rule, generalized, 43-46 Notation dislocation substructure during creep, 427-430 unified cyclic viscoplastic constitutive equations, 1 Notch designs, testing for triaxial stress, 133 Numerical integration MATMOD equations, 214 MATMOD-4V-DISTORTION, 224 operational equation, 265

O Obstacle dislocation glide limited by, 361-362 geometric, 77-82 strength, rate of change, 414--415 ODF, see Orientation distribution function Odqvist's law, 13-15 ONERA AGICE program developed at, 17 rate equations used at, 441-442 Operational equation deformation kinetics, 249-256 stress relaxation example, 257-270 Orientation distribution function in scale invariance approach, 320-321 and texture effects, 322-324 Overstress in viscoelasticity, 312-315 viscoplasticity theory based on, 282-294, 447-448

Palm-Voce equation, 75, 84-85, 88 Parameters for creep data extrapolation, 114-117

Index materials, determination procedure, 15-17 microstructural, rate of change, 423-424 model, identification techniques, 93-95 Particle effects, on deformation behavior, 82-86 Particle-hardened alloys power-law creep, 112-114 transition period, 147 Particularization, material functions, 9-10 Peierls-Nabarro mechanism, 246 Peierls stress, negligibility, 90 Perzyna theory, 294 Phenomena, simulated by MATMOD equations, 155 Physical processes, at atomic level: elementary rate constant, 236-243 Pile-ups dislocation, 159, 180 in RA equation, 196 Pinning effect, solute, 89 7r-Plane, first rotation in, 175-177 Plant life extension, traditional approaches, 144-145 Plastic deformation, anisotropic and inhomogeneous, 319-339 Plasticity interaction with creep, 29-32, 211-214 low-temperature, 189-193 rate-independent, 45-46, 62 strain gradient, 338-339 Plastic strain flow equation, 418 separation from creep strain, 24-26 Plastic strain rate direction, 8-10 temperature dependence, 71 and viscoplastic potential, 5-6 Plateaus, flow stress, 203-205 Polycrystalline metals, creep curves, 121 solids, plastic deformation, 319-339 Potential dissipative different from yield surface, 43-44 standard generalized materials, 37, 40-42 state variables as parameters in, 44-45 viscoplastic, and hardening variables, 5-6 Power law behavior, prediction, 126-129 breakdown creep behavior, 193-194

459

criterion, 365-368 stress, 348 creep particle-hardened alloys, 112-114 pure metals, 111-112 as empirical relation, 232 form for kinetic equation, 70-71 Prandtl-Reuss equation, 417-419 Prandtl-Reuss flow rule, 162 Prediction capabilities of MATMOD-4V-DISTORTION, 179-188 independent, MATMOD-BSSOL, 209-211 long-term design data, 129-132 power-law behavior, 126-129 Prestrain increments, 216 proportional, 180-182 Principal stress, and 0 variation, 138-142 0 Projection Concept, 108-109, 117-123, 443-445 Properties asymptotic, in stress space, 288-290 long-term stress-rupture, scatter in, 117 viscoplasticity theory, 295-301

Ratcheting in rate-independent limiting case, 33-34 viscous, 26-29 Rate constant elementary, 236-243 kinetics combination, 244-246 reverse, activation energy, 255-256 Rate dependence, relation to temperature, 287-288 Rate equations, for thermodynamic forces, 42 Rate-independent limiting case, plasticity theory, 12-13 Rate of change microstructural parameters, 423-424 obstacle strength, 414-415 Ratio F = Ws/ Wp, nonmonotonic evolution, 48-49 RA equation, MATMOD-BSSOL, 196-199 R~ equation, MATMOD-BSSOL, 199-200 Recovery dislocation density, 98-99 due to static annealing, 403-404

460

Index

Recovery (continued) dynamic, 33-34, 73-74 at elevated temperatures, 411 mechanical, 191 mechanisms, 377-378 static, and high-temperature viscosity, 26 strain, anelastic, 399 Recovery terms dynamic direction, 285 and hardening, 196-197 identification, 43-44 static and dynamic, 6-7, 41 thermal, 197-199 Relaxation modeling, 291-293 stress, 26-29, 101,257-270 termination, 297 Remanent life assessment, 0 approach, 145, 149-150 Resistance creep, enhancement, 82-84 crystal lattice, 90 Robinson Life Fraction Rule, 144 Rolling, and plastic flow behavior, 328-329 Rupture life calculation, 130-132 minimum creep rate, 443-444

Scale invariance approach, incorporation of ODE 320-321 Scatter experimental, 394 in long-term stress-rupture properties, 117 statistical, 380-382 Scatter bands minimum property values, 143 wide, 131 Schwartz inequality, 18 Screw dislocations, cross-slip, 357-361, 412-413,415-416 Screw segments, dislocation, mobile, 97 Self-adaptive forward Euler method, MATMOD equations, 214 Self-consistent approach, relation to gradient approach, 333-335 Separation, plastic and creep strains, 24-26 Service performance, constitutive laws for, 233

Shear, simple, plastic flow behavior in, 327-328 Shearable particles, associated strain localization, 84-86 Sherby-Burke criterion, 366-368 Simulations MATMOD-BSSOL, 202-209 solute effects through short-range backstresses, 189-214 yield loci during loading, 179-185 Single crystals copper, 49-50, 386-387 decaying creep curves, 120 NaC1, 351, 3'68-399 Size effects grain, 77-82 in plastic deformation of polycrystalline solids, 337-339 subgrain, 421-422 in torsion, 449 Slip, see also Cross-slip crystallographic, 321-322 wavy, in A1, 360 Sodium chloride, single crystals, 351 substructure formation, 368-399 Solute drag, peaks, 205-209 Solute effects flow stress, 86-90 modeling, MATMOD equations for, 153-224 simulation, 189-214 Solute strengthening, at low temperatures, 200-202 Solutions algorithm, for DSV model, 420-423 asymptotic, 286-288 closed-form, 16 differential equation, 269 Spin, plastic, 326 Stability, microstructural, 400-409 Stacking fault energy, effects class M creep, 363-365 cross-slip and climb, 413-415 Stainless steel 304, viscous ratcheting, 29-30 asymptotic stress value, 11-12 316L creep tests and cyclic relaxation, 27-28 cyclic hardening, 23-24 Standard generalized normality, 45-46, 59 Standard linear solid, relaxation and creep in, 312-314

Index State equations microstructure changes, 247-256 plus kinetics equation, 230-233 State law, in thermodynamics with internal variables, 35-36, 38-40 State variables for aging, 57-61 kinematic and isotropic, 441-443 as parameters in potential, 44-45 in unified evolutionary constitutive law, 446-447 Static annealing, effect on predeformed microstructure, 403-404 Static recovery, and high-temperature viscosity, 26 Stationary solutes, affecting flow stress, 86-88 Strain, s e e a l s o Prestrain aging, dynamic, 88-90, 194 anelastic, 180 compatibility, 426 creep, accumulation, 117-118 elastic and plastic, 258 inelastic, in unified viscoplastic constitutive equations, 3-20 localization, associated with shearable particles, 84-86 path, locus distortions related to, 166-167 stress-dependent, 251 Strain hardening, in cell walls, 413-414 Strain-hardening rule, 141,150 Strain rate decreased by work hardening, 248 described by deformation kinetics, 235 v e r s u s driving stress, 252-253 effective, in shear direction, 169-171 inelastic, 286-287, 294 jump test, 94-95, 101-102 and overstress, linear relation, 313-314 sensitivity of flow stress, 77 v e r s u s stress coordinate system, 273 temperature-compensated, 208-211 Stress, s e e a l s o Overstress change, experiments during creep, 146-148 constant, creep at, 76-78, 85 decomposition, for uniaxial tension-compression, 10 dependence dislocation substructure, 378-383 strain, 251 deviatoric, 418,422

461

equilibrium, 283-284 internal associated with formation of substructure, 395-399 polycrystalline aggregate, 333 kinematic, 284 normalized effect on creep substructure, 368-383 v e r s u s normalized creep rate, 366 Peierls, 90 power-law breakdown, 348 relaxation, 26-29, 101,257-270 triaxial, testing with notch designs, 133 von Mises, 135, 138 work as linear function, 243,275 Stress path, final, direction, 165-166 Stress rupture post-exposure testing, 144-145 properties, long-term, 115-117, 123 Stress space, asymptotic properties, 288-290 Stress states, multiaxial, creep under, 132-142 Stress-strain curve, viscoplasticity theory, 296 Structural steel, gradient coefficient c, 335 Subboundaries intact and mechanically stable, 404-409 well defined, 368-369 Subgrains boundary structure, 73 distinction from cells, 368-370 exiaxed formation, 425 microstructure, 370-378 interior, 410-412 size change, 421-422 stability after stress reduction, 401 Substructure creep strain effect, 384-392 deformation, 195 dislocation role in class M and exponential creep, 343-430 stress dependence, 378-383 formation in NaC1 single crystals, 368-399 temperature effect, 392-395 Symbols, unified cyclic viscoplastic constitutive equations, 1-2

T Taylor factor, 361,419 Taylor series expansion, 332-334

462

Index

Temperature dependence material constants, 157-158 nonelastic deformation, 195 effect on substructure, 392-395 high creep at, 107-151 region for deformation, 265-270 high homologous, 303-304 high and low, climb at, 353-357 history effects, 56-57 operating, in engineering applications, 349 region, low and high, 242-243 stress relaxation as function, 259-270 variable, 304-306 varying, constitutive equations under, 50-61 Tensile creep data analysis, 123-132 failure, 122-123 Tensile curve, modeling, 51 Tension, uniaxial, 330 Tension-compression test, under strain control, 23-24 uniaxial rate-independent plasticity under, 55 stress decomposition, 10 Tension-torsion deformation, 331-332 Tension-torsion test, 134-135 Tensors anisotropy, in MATMOD equations, 168-178 fourth-order, and anisotropy, 6-7 texture, and average procedures, 324-326 Testing methods creep under multiaxial stress, 133-135 economical, 249-250 microstructural quantities, 256-270 post-exposure stress rupture, 144-145 Texture effects and ODF, 322-324 on plastic flow and yield, 327-332 Texture tensor, and average procedures, 324-326 Thermodynamic acceptability, 57-59 Thermodynamics with internal variables, 35-40, 62-63 introduction of constitutive equations, 40-47 in viscoplasticity theory, 307-311 Thermoviscoplasticity, related equations, 34-61 Threshold, in dynamic recovery term for backstresses, 7 neglected, 16, 18

Threshold stress and dislocation mechanisms, 114 mechanical, 86-87 Time, dependence of stress change, 259-262 Time-temperature parameter, 116 Transient, prior to stress change, prediction, 211 Transition, from class M to exponential creep, 376 Transition temperature, effect of stacking fault energy, 366-368 Translation rule, Mroz, 18-19 Two-surface theory, 20

Vacancy diffusion mechanism, 239 Vacancy migration, 240-242 Validation, two-internal-variable model, 100-103 Variables hardening, and viscoplastic potential, 5-6 internal reversible changes, 308 thermodynamics with, 3540, 62-63 MATMOD, behavior, 180, 184 MATMOD-BSSOL structure, 156 one-internal, models, 72-90 state for aging, 57-61 kinematic and isotropic, 441-443 as parameters in potential, 44-45 in unified evolutionary constitutive law, 446-447 strain-like, 53-56 two-internal, models, 91-104 Vibrational amplitude, fluctuation, 238 Viscoelasticity, overstress concept in, 312-315 Viscoplasticity dislocation substructure, 350, 352 small-strain, overstress-based theory, 281-315 theoretical development, 3-4 Viscoplastic potential, and hardening variables, 5-6 Viscosity high-temperature, and static recovery, 26 normal, and kinematic hardening, 21 Voigt-Reuss bounds, 419 von Mises stress, 135, 138

Index von Mises tensor, !78 von Mises yield criteria, 160-164, 418

W Walls adjoining, linked by channels, 97-98 dislocation, parallel, 386 recovery in, 414 Work, linear function of stress, 243,275 Work hardening decrease of strain rate, 248 in monotonic loading, 448 Work softening, in monotonic loading, 448

Yield behavior, texture effect, 327-332

463

Yield loci measurement, 164-167 simulations during loading, 179-185 Yield strength, in MATMOD-BSSOL equations, 191 Yield stress, and isotropic hardening, 5-6 Yield surface dissipative potential different from, 43-44 distortion, MATMOD equations for, 153-224 identical to boundary of elastic domain, 13 in simple shear, 329 in tension-torsion deformation, 331 and translation rule, 18-19

Zener-Hollomon function, 52-53

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