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1 Unified Cyclic Viscoplastic Constitutive Equations: Development, Capabilities, and Thermodynamic Framework scalar functions, parameters, or variables second- or fourth-rank tensors

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 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 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

Includes index

ISBN 0-12-425970-7 (alk paper)

1 Deformations (Mechanic) Mathematical models 2 Plasticity-

-Mathematical models 3 Dislocations in crystals Mathematical

Contents

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

II One-Internal-Variable Model 72

III Two-Internal-Variable Model 91

Contents

II Traditional Approaches to Creep and Creep

VI Creep under Nonsteady Loading Conditions

143

for Modeling Solute Effects and

Yield-Surface Distortion

Gregory A Henshall, Donald E Helling, and Alan K Miller

III Simulating Solute Effects through Short Range

The Constitutive Law of Deformation Kinetics

A S Krausz and K Krausz

IV Measurement and Analysis of the Charac-

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

References 430

~ ~ l Comments and Summary

K Krausz and A S Krausz

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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

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

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

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)

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

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

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

(1978), and Robinson (1978) Further refinements were introduced thoughout the 1980s, for example, by Walker (1981), Bruhns (1982), Lowe and Miller (1984),

(1990)

Comparative reviews of constitutive theories in cyclic plasticity or viscoplas-

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 C h a b o c h e

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:

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

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) = ~ ( J T ( X i ) X t i ) m (2.13)

JT(Xi) 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 Unified Cyclic Viscoplastic Constitutive Equations 7 (equal to or different from M):

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 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 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

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 S t ) ] 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:

X i : -~Ci~p - y i q b ( R ) ( J ( X i ) - X l i ) m J ( X i ) p - } " i S i ( X i ' R ) X i (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:

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):

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

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

(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

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 Sub- sequently, 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 More- over, 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 memoriza- tion: Kinematic hardening is evanescent in nature, and isotropic hardening sat- urates 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"

x/~/2(1 - ~ ) H ( F ) ( n 9 n*)n*~b (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 = A e p / 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 intro- duced by considering an asymptotic isotropic hardening Q in Eq (2.27) depending

on p For instance,

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-Independent Limiting Case

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 imple- mented 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 1 b for a power dependency such as

()n

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

,-+oc [ undetermined 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

of rate-independent plasticity are valid for cases where the plastic tangent modulus

is positive Softening materials need slightly different procedures

D The Limiting Creep 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:

Bi - - Yi 1 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

b(Q - ~ r J ( o ' ) ) P s - - y r ( ~ r J ( O ' ) Q r ) mr ( 2 4 3 )

[ g s = ( ( 1 - ~ - ' ~ J ~ J - ~ r ) J ( ~ n D

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 = mi = 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 asymptot- ically (Qr = 0)

Chapter I Unified Cyclic Viscoplastic Constitutive Equations 15 Under these particularizations, the above system of equations reduces to

PS (J(cr)),

where we have posed Ks K + }-]~j(Cj/?,sj) 1/" + (bQ/yr) 1/n We obtain ~bs -

have effectively proved that our viscoplastic constitutive model degenerates ex- plicitly 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 se- lection of the experimental data The viscous and static recovery effects are deter- mined in a second step

9 The response of the hardening models may be obtained analytically for uni- axial monotonic or cyclic conditions (in the rate-independent case)

1 6 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)

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 ap- proximation ~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 dura- tions (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 (relax- ation 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 constitu- tive equations have some similarities with many other models, either in terms of the linear/nonlinear kinematic hardening for the hardening/dynamic recovery/static re- covery 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:

C1 -~-C2

fx = J (X - X2) - < 0 (2.56)

)'1 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"

C1 -+-C2 j~ J(tr - X 2 ) - (kl + R ) < 0 kt - - k + (2.57)

yl

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

In the case of more than two back stresses, the generalization of the above demonstration leads to the geometric construction of "nested surfaces," as pro- posed by Wang and Ohno (1991), the (simultaneous) movement of each one being obtained by a similar "image point" concept We do not express this decomposi- tion 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 con- stitutive model is defined in terms of the two surface presentation The translation

Chapter 1 Unified Cyclic 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), Moosbrugger and McDowell (1990), or by Voyiadjis and Kattan (1990)

9 The distance between the stress state and the image point, defined as a

J (orl or), may be used explicitly as in several models, and (2.59) above can the replaced by

X - g

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

The increase in degrees of freedom is evident if we note that (2.53) corresponds

to the particular case K(a) - y16

Another interesting point concerns the generalization of the two-surface ap- proach to the anisotropic description introduced in Section II.A If we define the yield and the bounding surfaces by

f - [(or - X) 9 M " (or - X ) ] 1 / 2 - k < 0 (2.61)

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

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

Of Ofz M " (or - X) Mz " (ort - Xl)

Replacing orz and or in (2.63) as taken from (2.65), we obtain

X y'[(klM/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)

with N - g(kzMi -1 - kM-1) We immediately notice that the dynamic recovery

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 N L K 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

CAPABILITIES OF 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 Viscosity and Kinematic Hardening

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 Unified Cyclic 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

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 2 3

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 a change in the yield limit Y = k + R

9 As an evolution of the tangent modulus in the kinematic hardening, by the function r given by (2.25)

9 As 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 hard- ening 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 CrMs ~ O-Mo

previously (Chaboche et al., 1979)

24 J.L Chaboche

l Step cyclic test on 316 L stainless steel at room temperature, showing the strain range • • •

memorization effect

C Limiting Case 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 hard- ening equations (Kawai and Ohashi, 1987; Contesti and Cailletaud, 1989) This permits freedom for the modeling, but leads to increasing computational difficul- ties

(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 25

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 Temperature Viscosity and Static Recovery

At high temperature, all the thermally activated processes are effective The vis- cosity 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 load- ings, 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: