Tài liệu Unified Plastic Constitutive Laws of Deformation doc

Unified
Plastic
Constitutive
Laws of
Deformation
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I
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 Tokyo 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.
Academic Press, Inc.
A Division of Harcourt Brace & Company
525 B Street, Suite 1900, San Diego, California 92101-4495
United Kingdom Edition published 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 ix
Preface xi
~l Unified Cyclic Viscoplastic Constitutive Equations:
Development, Capabilities, and Thermodynamic
Framework
J. L. Chaboche
List of Symbols 1
I. Introduction 2
II. A Cyclic Viscoplastic Constitutive Law 4
III. Capabilities of the Constitutive Model 20
IV. Thermoviscoplasticity 33
V. Conclusion 61
References 63
Dislocation-Density-Related Constitutive Modeling
Yuri Estrin
I. Introduction 69

II. One-Internal-Variable Model 72
III. Two-Internal-Variable Model 91
IV. Conclusion 103
References 104
~1 Constitutive Laws for High-Temperature Creep 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
143
~]1 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. Introduction 229
II. The Kinetics Equation 234
III. The State Equations 247
IV. Measurement and Analysis of the Charac-
teristic 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
III. Discussion 294
References 316
Contents
~
VII
Anisotropic and Inhomogeneous Plastic
Deformation of Polycrystalline Solids
J. Ning and E. C. Aifantis
I. Introduction 319
II. Constitutive Relations for a Single Crystallite
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
VI. Inhomogeneous Plastic Deformation 332
References 339

321
327
~~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 Single-
Phase 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
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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 Uni-
versity of Western Australia, Nedlands, Western Australia 6907, Australia
R. W. Evans (107), Interdisciplinary Research Centre in Materials for High Per-
formance Applications, Department of Materials Engineering, University of
Wales, Swansea SA2 8PP, United Kingdom
A. D. Freed (343), Lewis Research Center, National Aeronautics and Space Ad-
ministration, 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 Tech-
nological University, Houghton, Michigan 49931
S. V. Raj (343), Lewis Research Center, National Aeronautics and Space Admin-
istration, Cleveland, Ohio 44135
B. Wilshire (107), Interdisciplinary Research Centre in Materials for High Per-
formance 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 tech-
nology 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 constitu-
tive 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
X
X'
|174174
IIxlIM
ci
O/Ox
d/dx
6
I
Idev
scalar functions, parameters, or variables
second- or fourth-rank tensors
deviator of the second-rank tensor X
scalar product of vectors or contracted tensor product
tensorial product contracted twice
tensorial products
generalized second invariant ofX : IIXIIM = (X : M
: 1) 1/2

time derivative of a
partial derivative
total derivative
Kroenecker delta
fourth-rank unity tensor
fourth-rank deviatoric operator
Specific variables or functions
u "back strain" or state variable associated with back stress
a aging state variable
isotropic "drag" state variable
D 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 strain tensor
s elastic strain tensor
s
plastic strain or viscoplastic strain tensor
thermodynamic state potential
q~ dissipation potential (rates)
4~* dual dissipation potential (forces)
dissipation
f yield function
h hardening modulus
J, JT second invariant overstress and back stress
plastic multiplier
A elastic stiffness tensor
n direction of plastic flow
v direction of the back-stress rate
f2, ~p viscoplastic potential

~r
static recovery potential
p accumulated plastic strain
q heat flux vector
r isotropic "yield" state variable
R yield stress increase
S entropy
or, or' stress tensor, stress deviator
T temperature
V T temperature gradient
u internal energy
Wp plastic work
Ws stored energy
X, X' back stress tensor and its deviator
Y yield stress
Z thermodynamic force associated with aging variable a
~11 INTRODUCTION
The constitutive equations considered here, mainly those devoted to metallic ma-
terials, are essentially developed with the objective of the inelastic analysis of
structural components. Initially, they are based on the concepts of continuum
mechanics, where a particular representative volume element of material can be
considered as submitted to a macroscopically uniform stress, neglecting the micro-
stress/microstrain inhomogeneities at the microscale (but not their effects).
The physical facts, the precise role of the dislocations, their arrangements,
and their evolution are considered more in detail in several other chapters of
this book. Here, we concentrate on a macroscopic description of the various
processes, making reference to the microstructural events as often as possible (at
least qualitatively).
The application domains are limited to the quasistatic deformation of metallic
materials (strain rate between 10 -l~ and 10-1), especially under cyclic loading

Chapter
1 Unified Cyclic Viscoplastic Constitutive Equations 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.
t
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 mod-
els 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 viscoplas-

ticity 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 pro-
cedure 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
4 J.L. Chaboche
polycrystalline materials. Various complicated processes can be modeled, includ-
ing Bauschinger effects, creep, relaxation, strain rate effects, monotonic hardening,
cyclic hardening or softening, static recovery effects at high temperature, creep-
plasticity 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 thermody-
namic 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 condi-
tions, based on the thermodynamic theory and also the modeling of metallurgical
effects, such as aging, that are induced by temperature changes.
A CYCLIC VISCOPLASTIC CONSTITUTIVE LAW
This section is devoted to the presentation and development of a set of consti-
tutive equations based on the combination of kinematic hardening and isotropic
hardening with both "yield" and "drag" effects. The unified viscoplastic frame-
work is chosen, but the limiting cases of rate-independent plasticity and station-
ary 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 consid-
ered. In a Cartesian reference configuration, the strain e is taken to be com-
posed 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 back-
stress 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-
Hd~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:
X EXi
i
Xi

Ni
" F.p ~i(Xi,
R)Qi
" XiP Si
(Xi,
R)Qi
" Xi
hyp- ry(R)Rp- Sy(R)R
b = hap- rd(D)Dp- Sd(D)D
(2.9)
(2.10)
(2.11)
(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 temperature-
dependent!) 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) = ~(JT(Xi)
Xti) m
(2.13)
JT(Xi)
where the MacCauley bracket ( ) is zero when
JT(Xi) ~
Xli.
For the definition of
the distance, we may also use an anisotropy effect with the fourth-order tensor T
Chapter
1 Unified Cyclic 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 situa-

tions.
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
is
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 tension-
torsion, 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 (2.16)
2
Ni sCiI Qi = yil
T = M (2.17)
where I is the fourth-rank symmetric identity tensor, expressed from the Kronecker
delta symbol:
1 (~ik~j l -Jr- ~il~jk)
(2.18)
Ii j kl ~
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
3 (O.t __ X t) . (o.t __ St)] 1/2
(2.19)
y(,~ - x) =

[~
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):
~P " 00" 2
J(o" - X) ~ 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 = - (2.22)
2
J(o" - X)
2
with kp /)n and ~n 9 n 1. The back-stress rate equation reduces to
2 Xi
Xi : -~Ci~p - yiqb(R)(J(Xi) - Xli) m
J(Xi) p -
}"iSi(Xi' R)Xi
(2.23)
and Xi is identical to its deviator X~, provided Tr(Xi) = 0 for some initial condi-
tions.
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 + (2.24)
n2 -+- 1
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 kine-
matic 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):
1
~b(R) =
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 -~Ci~p- ~'i(~(g)(J(Xi) - Xli) j(Xi ) ~3
- Ys; [ J (Xi) ]mi- 1Xi
k b(Q- R)p-
yr]R-
Qr[mrsign(R-

Qr)
D - b'(Q'- D)p- diD- Q'rlm'rsign(D-
Q'r)
(2.26)
(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'
Y r(-Dl-mr,
Q' - coQ + K, and
Q'r
coQr
-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
P
~ I
Ov=O-g
l
/
y/
r
(a) Stress decomposition in uniaxial tension compression; (b) schematics of the lim-

iting 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