Time-Independent Constitutive Theories For Cyclic Plasticity

Imernattonal Journal of Plus~tctty. Vol. 2, No. 2. ~0. 1-19-188. 1986 0749-6-t.19,86 $3.00 - .
0~3
Primed in ~he U.S.A. -" 1986 Pergamon Journals Ltd.
TIME-INDEPENDENT CONSTITUTIVE THEORIES

FOR CYCLIC PLASTICITY
J. L. CHABOCHE
Office National d'Etudes et de Recherches A6rospatiales

(Communicated by Erhard Krempl, Rensselaer Polytechnic Institute)
Abstract-The article is mainly concerned with time-independent plasticity in the range of cyclic
loadings. Three different approaches are considered for the description of kinematic behaviour:
(i) the use of independent multiyield surfaces, (ii) models with two surfaces only, (iii) the so-
called "nonlinear-kinematic hardening rule" defined by a differential equation. The thermo-
dynamic framework from which the third approach is derived and the conditions of varying
temperature are considered. Connections between the three kinds of models are pointed out.
Also, some specific rules to describe cyclic hardening or cyclic softening of the material are pro-
posed. Finally, the limits of the models considered, and the difficulties associated with their
practical use and their implementation in computer codes are discussed in detail.
I. INTRODUCTION
The concepts o f classical time-independent plasticity are very old. For m o n o t o n i c load-
ing isotropic or linear-kinematic rules are generally considered g o o d enough to predict
stresses and strains in structures.
During the past 15 years, the development o f computers, the increased knowledge of
the actual behaviour o f materials (low-cycle-fatigue), the emergence o f several indus-
trial applications involving the prediction o f stresses and strains under cyclic loadings,
led to the development and the use o f different constitutive equations in cyclic plastic-
ity. At high temperature, maybe eventually at room temperature, the problem becomes
more complex due to the effects of time (creep, viscoplasticity, recovery, aging, etc.).
In the present report we limit this study to the time-independent structure o f constitu-
tive equations. The case o f viscoplasticity and recovery is evaluated in previous papers
(CHABOCHE [1983a,b], [1984]).
W h e n considering cyclic plasticity, that is the development of constitutive equations
describing the behaviour o f materials under cyclic plastic strains, different kinds o f for-
mulations can be adopted. Two classes emerge from the literature, based on one o f the
following t h e r m o d y n a m i c a l concepts:
(i) The present state o f the material depends on the present values and the past his-
tory o f observable variables only (total strain, temperature, etc.) giving rise to
hereditary theories.
(ii) The present state o f the material depends only on the present values o f observ-
able variables and a set o f internal-state variables.
The first concept was used for example by VALANIS [1971-1980] in the development
o f the endochronic theory, by KREMPL [1975] in viscoplasticity, by GUELIN et al. [1977]
in the hereditary theory with discrete m e m o r y events.
The second a p p r o a c h is studied under m a n y different ways in order to generalize the
classical isotropic and linear kinematic theories:
149
150 L C>:.:,~ocm:
(a) using multilayer concepts (BEssELING [1958] a: MEIJERS [1980]),

(b) by means of muhiyield surfaces (MRoz [1967]),
(c) with two surfaces only (DAFALIAS et al. [1976] & KRIEG [1975]),
(d) in terms of differential equations (ARMSTRO.','G ,~ FREDERICK [1966], NI.~,*.I.~I.~
KHADJINSKY [19721.
In the present report the three last formulations are considered and compared. Atten-
tion is focused on the kinematic hardening in two examples: the Mroz Model, in Sec-
tion II (category b) and the so-called nonlinear-kinematic rule, in Section III, initially
introduced by Armstrong and Frederick, developed further and intensively used at
ENSET (MARQUIS [1979a,b]), and O N E R A (CHA~OCHE [1977], CHABOCHE eta/. [1979],
C.~ILLErAb'D et al. [1985]) in France. Some connections are pointed out with the two
surface models of DAFALIAS & POPOV [1976] and K~IEG [1975], and the range of appli-
cability of the different models is studied in detail. One of the main problems concerns
the description of random-type loadings and ratchetting effects: At the present time,
only updating rules lead to correct simulations of such effects, but some inconsisten-
cies of such updating procedures can be pointed out.
The superposition of isotropic hardening is studied in a separate section. It is gen-
erally used to describe cyclic hardening or softening of the materials. Discussion is made
about the range of applicability of the models (Section V), the methods used to deter-
mine the material dependent coefficients and what concerns the implementation in struc-
ture computer codes (Section VI).
All the above theories consider the partitioning of strain into elastic and plastic parts
which, in the small strain hypothesis, writes as
= % + e;, . (1)
Another c o m m o n feature is the use of a yield surface concept, in the stress space:
f=f(o, hardening) _< 0 . (2)
f < 0 indicates the elastic domain. Plastic flow takes place if f = 0. Figure l(a) illustrates
the concept together with the classical normality hypothesis for the plastic strain rate:
/:
( !
,,o
t
//
/ ,o,oio,
Vi =0
(a)
V (b)
C
Fig. 1. Schematic of the elastic domain, normality hypothesis, load-unload criterion.

Constitutive theories for cyclic plasticity 151
d~p=dA~ . (3)
Oo
In the following, only the associated flow rules are considered, in which the yield sur-
face and the loading surface (or plastic potential surface) coincide. A third common
hypothesis is relative to the loading-unloading criterion. The subspace f > 0 is excluded.
At constant temperature plastic flow only occurs if the loading index Of/Oo : do is posi-
tive.t The opposite case implies elastic unloading [see Fig. l(b)].
All the models considered are a generalization in some way of the linear kinematic
rule introduced by PRAGER [19491, where the yield surface is described by
f=¢(o-c~p)-k=O . (4)
Here the translation of the yield surface is denoted by the kinematic stress tensor (or
back stress or rest stress):
X=c~p , dX=c d~p , (5)
and the plastic multiplier d/X is determined through the consistency condition of time-
independent plasticity: f = d f = 0. By using eqns (3) and (4) one finds (at constant tem-
perature)
df=Of Of Of Of. Of 0
~:do- ~:dX = ~:do-cdA 0-~" 0"-~= " (6)
Taking account for the yield condition and the load-unload criterion leads to
(Of/Oo: do)
dA = H ( f ) c(Of/O¢) : Of/Oo ' (7)
where H is the Heaviside function, H ( u ) = 0 if U < 0, H ( u ) = 1 if u > 0, and the

brackets are defined by ( . ) = u H ( u ) . Equation (3) can be written as
d~p = ( 1 / c ) H ( f ) ( n : d o ) n , (8)
where n is the unit outward normal to the yield surface
0f/0o
n = [af/ao: af/ao] z/2 • (9)
The hardening moduli is the constant c,
do" : d~p
c- dtp:d~ (10)
~'The symbol : indicates the contracted tensorial product.

152 I.k.C.q.~,BOCHE
11. T H E M U L T I Y I E L D S U R F A C E MODELS
[[.1 The M r o z m o d e l
The main criticism of the Prager's kinematic hardening rule relates to the linearity
of the stress-strain behavior derived from it. Though the model accounts for some
Bauschinger effect, difficulties appear when it is applied to complex loading pro-
grammes involving unloading and subsequent loadings in reverse directions. Figure 2
illustrates the point by comparing the model to the experimental trends. Generalization
by considering c as a plastic strain-dependent positive parameter (so that a varying tan-
gent modulus is obtained) leads to inconsistencies for reversed plastic flow due to the
unique relationship:
X = c(<~,)¢~, . (t 1)
Improvements of the linear kinematic rule are obtained by means of the multilayer
model, based on the physical idea of nonhomogeneous grains and subgrains. Such an
approach is usually written in terms of scalars (BESSELING [1958]). Its multiaxial gen-
eralization does not consider explicitly the notion of surfaces. A different generalization
was proposed by MRoz [1967], introducing the concept of a "field of work-hardening
moduli" instead of the single modulus c given in (5).
In the case of tension-compression the model uses the approximation of the stress-
strain curve by linear segments with different hardening moduli. The multilayer model
reproduces the MASING' rule [1927] under cyclic loading. After close of the cycle at point
E [Fig. 3(a)] the branch K L M N P E is deduced homothetically by a factor 2 from the
initial tension curve O A B C D E (here the model is considered in its cyclically stable form
with no isotropic hardening). In the stress space the multiaxial generalization is obtained
by using several hypersurfaces f0, ./1 . . . . f,,,, where f0 is the initial yield surface and fL,
f2 . . . . f,,, are separate regions of constant hardening moduli. During the plastic load-
ing between surface J} and surface f~+~ the loading surface or active surface is ./i = 0,
and the inner surfaces f0 . . . . . f~_~ are tangent at the loading point, so that
--F
J !
/
7/
J Cl
b
It /
(a) (b)
Fig. 2. Cyclic tension-compression. (a) linear-kinematic hardening and experimental tendencies ( . . . . . ) (b)
inadequacy of an univocal nonlinear stress-strain relation.
I
I"'/
K
(a)
{b)
Fig. 3. (a) Approximation of the stress strain curve by portions of constant tangent moduti, (b) representa-
tion in the stress space.
fo = f l . . . . . J~-I =J~ = 0 , (12)
afo afl .. = d~l aft

(13)
d,Ao~ =dAl~ =" Oa "
Each surface can be described by equation similar to (4):
j~ = ¢#(a - x i ) - k i = 0 , (14)
where Xi and ki correspond to its center and its width, respectively.

In order to describe mathematically the translation motion of surfaces of constant
hardening moduli it is assumed that they cannot intersect, but can consecutively act in
contact and move together. In view of the decreasing values of the corresponding
moduli, that point is essential to obtain consistent results under complex multiaxial load-
ing programs. The rule for the translation o f the active surface f~ = 0 is chosen by con-
154 J, k. CHABOCHE
sidering the stress point o:~a on the subsequent surface f/_f = 0, corresponding to the
same direction of the o u t w a r d normal. Figure 4 indicates this geometrical definition.
This rule imposes coincident outward normals when the two surfaces contact each other
dX, : d/~/(e;.t - ~) • (t5)
Suppose first the case o f nonexpanding surfaces, that is constant values for param-
eters k;. The consistency condition df / = 0 leads to
of,. do af~ a/, aj) (16)
and multiplier dt¢; is determined by
(Of/Oo : do} (n:do)

d~, : H(/,)affg~ 77 _ 7 - o) = H ( f / ) n: (o~+l - o)
(17)
The chosen rule imposes the same translation for the inner surfaces
dXo = dXt . . . . . dXt-i = dX; • (18)
In the original Mroz model the normality hypothesis for the plastic strain rate is con-
sidered independently, the plastic multiplier being independent o f d/~,
d~p = ( 1 / c ; ) H ( f j (n : d a ) n , (19)
with n being the o u t w a r d unit normal,
Fig. 4. The Prager and Mroz kinematic hardening.

n= 0f~/0a (20)
[Of#&r" Of#&r] 1/2 ,
and q the hardening modulus associated with the active surface ft.
If expansion of the surface is considered, that is if isotropic hardening is superposed
on kinematic hardening, the parameter ki depends for example on the accumulated
plastic strain defined by
=f (dep:dep) 1/2 . (21)
Each surface is now written as
f , = O(cr - X / ) - k ~ ( A ) . (22)
The translation rule (15) applies, but consistency condition leads now to
d#i = H ( f i ) (Of~./Oa : da) - dki for i _< 1

Ofi/c3Cr: (¢/+1 -- It)
or after rearranging with (19)-(21).
OkilOA ] (n : do) (23)

d/z, = H ( ~ ) 1- qtO~/-~;-~j3,],/2 n: (";,+;~o')
Thus all surfaces which are not reached by the stress point (i >__l + 1) expand (or con-
tact) uniformly, whereas the remaining surfaces, which are in mutual contact at the
stress point, undergo both translation and expansion (or contraction).
Let us study now the application of the Mroz model in the case of a Von Mises mate-
rial. Each surface can be written as
fi = J ( a - Xi) - ki(p) , (24)
where J corresponds to the octahedral shear stress,
J ( o - Xi) = [ ] ( o ' - X ~ ) : ( o ' - X~)] 1/'- (25)
where ~r' and X~ are the deviators of a and Xi, and p is the accumulated plastic strain,
with
dp = [2d~p: d~p] 1/2 (26)
Coefficients 3/2 and 2/3 in (25) and (26) respectively are chosen in order to easily iden-
tify the multiaxial equations and the pure tensile case. Then the rate equations (19) and
(23) become
3 H(ft)
d E p - 2 ctk z ( ( a ' - X ~ ) : d a ) ( a ' - X ~ ) , (27)
[ 56 ]. [- CHABOCHE
dXi=d/x,(crt, l - ~ ) , i_<l , (28)
(1-1 x/3 Ok,) <i,~'-X;):d,7> (29)
Let us note the possibility to write all equations in the deviatoric space: The term
(a' - Xl) : (~+~ - a) in eqn (29) for example is identical to ( a ' - XI) :(aL~ - a'), and
the translation rule (28) can be written independently for the deviatoric and the hydro-
static parts:
d X l = d/xi(e/+~ - o') (30)
dTr(Xi) = d~i[Tr(~t.1) - Tr(e)] . (31)
In fact, in that case the hydrostatic part of the rest stress plays no rote. In the stress
space, this surface is a circular cylinder, orthogonal to the deviatoric plane, and trans-
lation in the hydrostatic direction does not change the surfaces. It is permissible to con-
sider X as a deviator.
The Mroz model is a good way to generalize the linear-kinematic rule and it is pos-
sible to describe
• the correct nonlinearity of the stress-strain loops, even under cyclically stable con-
ditions,
• the Bauschinger effect,
• cyclic hardening or softening of the material with asymptotic plastic shakedown.
Under asymmetric loading conditions, for example alternating tension-compression

superposed on either a constant mean tension stress or a constant mean shear stress, the
Mroz model gives no ratchetting at all, exactly as does Prager's linear kinematic rule.
Figure 5 illustrates this fact for pure tension-compression: For cyclically stable con-
ditions (all k~ constants), shakedown takes place after one cycle only, for any mean
stress.
II.2 Two-surface models

One of the main difficulties with the Mroz model is the large number of surfaces nec-
essary to describe an actual material. Each surface needs the storage of a tensor vari-
able (usually six components) and a scalar one. Several models were then developed in
order to obtain the same specific properties by using only two surfaces, the yield sur-
face and the bounding surface (or limit surface).
II.2.1 The Dafalias-Popov model. The two-surface model of DAFALIAS & POPOV
[1976] is one of the best known. Only the main properties are pointed out in the pres-
ent paper, for comparison with the multisurface or Mroz model.
The field of hardening moduli c, is replaced by a generalized plastic modulus K:
K = K(a,a~n) , (32)
D __ Omo x
_ _ Ornin
O
Fig. 5. Immediate accommodation of the Mroz model.
depending upon the distance 6 = [(# - a) : (# - a)] ~/2 between the present stress state
(on the yield surface) and the stress state a on the bounding surface with the same out-
ward normal. The yield and bounding surfaces play the role of the active (J~ = 0) and
subsequent (J~+t = 0) surfaces in the Mroz model, respectively. Now the plastic modu-
lus is continuously varying between
K(6i~,6i~) = co and K(0,6in) =Ko • (33)
The quantity 6~n represents the value of 6 at the initiation of a new loading process
and measures how far the material state is f r o m the state represented by the bounds
(Fig. 6). The diminishing distance ~ gives a measure of the proximity of this limit state
during plastic deformation. 5~n changes at each reversal, but is constant during plastic
flow; thus it is associated with the most recent event of unloading-reloading. Let us note
here the use of an updating procedure in the model. The difficulties associated with
updating will be discussed in Section VI.2.3, with regard to the generality of the rule
under complex loading programs.
The flow equation obeys the normality rule:
1
d~p = ~ ( n : d a > n , (34)
with n defined by (9) as the unit outward normal to the yield surface. The accumulated
plastic strain is defined as in (21):
dA = (d~a : dEp) I/2 = ( 1 / K ) ( n : da) . (35)
In order to describe internal changes associated with plastic flow, DAFAI.U~S~ POPOV
use the concept of plastic internal variables. We restrict here the presentation to the use
158 J.L. CH.,,SO(rHE
(7 /" "~
/1
X
J
x' ]
/ \//_2
o 1 ! ~'
] (a)
Fig. 6. (a) Schematic illustration of 6 and &~, (b) representation of" the yield and bounding surfaces and illus-
tration of their motions.
of A as the internal variable describing isotropic changes, that is expansion or contrac-

tion of the surfaces. Moreover, we restrict ourselves to homogeneous functions of degree
one in stress to define the yield and b o u n d i n g surfaces:
f= O(a - X ) - k(A ) = 0 , (36)
f= 0(#- X) - / ~ ( A ) = 0 , (37)
where X and Y~ denote their respective centers.

As in the M r o z model, at contact (in an asymptotic situation) the two surfaces must
have the same outward normals. This can be specified by the following equation govern-
ing the translation rule for the b o u n d i n g surface [see Fig. 6(b)]:
dX=dX-dg(#-~r) . (38)
The translation o f the yield surface can be specified freely, using a unit direction v :
dX -
K~ -<n- • d~) d,~
r = K ~ - - ~ ' . (39)
K n:~ iI:~
K~, is determined t h r o u g h the consistency condition (d f = 0) for the yield surface:
1 ak
-- (40)
K~, = K - (Of/O~r: O f / O ~ ) ~/2 02t
In the same way the multiplier d~ in eqn (38) is determined by the consistency con-
dition for the b o u n d i n g surface. By using the notation
(n:dii) =KOdA=Ko K (n:da) , (41)
one finds
K -KS (n:da) _
dg=dX-* (a- a) I (42)
K n : (ii - a)
where KO is given by
1 aE
KJ=Ko-- (43)
(af/au
~af/a~)~/*
Zi *
KO is determined by K(O,&,)and represents the limiting hardening for saturated con-

ditions (k and I? constants).
In the particular case where the translation direction for the yield surface is chosen
as in the Mroz model, eqns (23) and (24) reduce respectively to
dX=K, (n:du)
K n:(+u)(a-u) ’
(44)
dX = “K,r”(a”l)(a -
2 a) . (45)
The above hardening rules are written for the case of plastic flow, that is whenf= 0.
The three main differences as compared to the Mroz model are
(9 the use of two surfaces only,

(ii) a more general translation rule for the yield surface, but v in eqn (39) has to
be specified, and the best possibility seems to be to use the Mroz’s formulation,
that is eqn (44).
(iii) The function K(6,6i”) defining a continuously varying plastic modulus with the
possibility to describe a smooth elastic-plastic transition, using for example
K=K(6,8i”)=Ko+h& .
I”
For each plastic loading process K varies between an infinite value (for 6 = Si,) and KO
for limit conditions. Recall that 6i” is the initial value of the distance 6 between u and
3 at the beginning of each plastic loading. Then the smooth elastic-plastic transition is
obtained by the way of an updating procedure.
A recent generalization of the model (DAFALLG[1980]) is the introduction of the new
concept of an elastic nucleus.
11.2.2 The Krieg’s model. It also defines two surfaces, the yield and the bounding sur-
faces, and each shows both kinematic and isotropic transformations. Krieg used homo-
geneous functions of order two for the definition of these surfaces, which does not
160 J.L. CH.~,8OCHE
change the main properties of the model. The two surfaces are written in the deviatoric
stress space (KRn~ [1975]):
f=(a'-X'):(o'-X')-R 2=0 , (47)
f= (o' - X') : (o' - X ' ) -/~-~ = 0 . (48)
Plastic flow occurs when f = 0 and Of/Oo:do>_ O. In that case the plastic strain
rate is
of
d~p = d/~ ~ = 2 d,~(o' - X') . (49)
The a c c u m u l a t e d plastic strain p is defined by
d p = (~ dl~p: d~p) I/2 = 2 4 ( 2 / 3 1 R d,~ . (50)
C o n t r a r y to the preceding formulation, Krieg defines directly the translation and

expansion rules in terms o f differential equations. The chosen forms are
dX = b(1 - / 3 ) ( 0 ' - o ' ) dp , (51)
(o' - o') : (o' - X ')

d R = bfl dp , (521
R
dX = (2~c(1 - 3') d~p , (53)
d/~ = c3, d p , (54)
where b, /3, c and ~/ are material dependent coefficients. As shown by eqn (51), the
translation o f the yield surface obeys the M r o z hypothesis. W h e n 0 is the stress state
on the b o u n d i n g surface with the same o u t w a r d normal, then
O' - - X ' O" - - X'

n - (55)
R
The plastic strain multiplier is determined by the consistency condition d f = 0. One

finds
dZ = (n : do) (56)
2 4 ( 2 / 3 ) b R n : (0' - 0')
Let us note two inconveniences with the f o r m u l a t i o n :

(1) There is no limit for the isotropic hardening o f the b o u n d i n g surface, eqn (54),
which gives no s h a k e d o w n under cyclic loading.
(2) Under uniaxial loading there are very few degrees o f freedom in the model, as can
be seen in the particular case where the bounding surface is .fixed (c = 0). Under pure
tension for example (51) and (52) reduce to
dtr = b ( 6 - o) dp , (57)
if e is constant the solution is
~r = ~ + (~o - e ) e - b P (58)
I!I. NONLINEAR KINEMATIC HARDENING RULE
In the preceding sections the cyclic plastic behaviour of materials was considered in
terms of kinematic hardening. Nonlinearity of the kinematic hardening was introduced
either by the field of hardening moduli associated with several concentric surfaces, or
by the concept of a continuously varying hardening modulus from which the transla-
tion rule of the yield surface is deduced.
Both approaches present difficulties. In the first one the number of variables neces-
sary to obtain a good description is very high, and in absence of isotropic hardening,
that is for cyclic stabilized conditions, no ratchetting (or mean stress relaxation) can
occur. In the two-surface model of DArALIAS*, POPOV the key to describe smooth elastic-
plastic transition and qualitatively correct ratchetting effects is the updating procedure
which, as will be discussed in Section VI.2.3, leads to inconsistencies under complex
loading conditions.
In the present section attention is focused on an intermediate approach, where
nonlinear-kinematic hardening is introduced directly, choosing the form of the differ-
ential equations governing the kinematic variables. The notion of bounding surface is
not necessary but results from the chosen model as seen in Section III.3.1. The vary-
ing hardening modulus is obtained as a consequence of these equations.
III. 1 F o r m u l a t i o n o f the m o d e l
III.1.1 Basic f o r m u l a t i o n . The concept of a yield surface is used, which obeys for
example the Von Mises criterion
f=J(o-X)-R-k=O , (59)
J(o- X) = [,3-(o'- X') : (o'- X ' ) ] 1/2 . (60)
The kinematic and isotropic variables are X and R, respectively, which define position
and size of the yield surface; k is the initial size of the surface, with R(O) = O.
The plastic flow follows the normality rule
Of 3 u' - X' (61)

d~P = dA a-~a = 2 dA R + k
The plastic multiplier d,~ is derived from the hardening rule through the consistency con-
dition f = d f = 0 when plastic flow occurs (no plastic flow when f < 0 or d f < 0).
The usual linear-kinematic rule can be easily modified in order to introduce some non-
162 J. k CH.~sOCH~.
i
i% ~'A
/ k
~/ k
/ ! ,
< %
Y
(a) (b)
Fig. 7. Behaviour of the nonlinear-kinematic rule in stress space (a) and in tension-compression (b), in the
case where R = 0.
linear evolution, with an acceptable description of cyclic loadings (concavity of the

stress-strain curves under tensile-compressive loading for example). This modification,
initially proposed by ARMSTROYC ~ FREDERIC~C [1966], in which an evanescent strain
m e m o r y effect is introduced (evanescent along the plastic strain path), can be written
in its simplest form (MARQUIS [1979a,b]) as
dX=~cd~,-TXdp . (62)
The key of this simple model is the recall term, the second term on the right-hand side,
proportional to X and the modulus of plastic strain rate
dp=(2dep:dep)l/2=dA . (63)
Application of the consistency condition (d f = 0) yields the plastic multiplier
dp=dA= ~ :da = ~ ~+~- :da , (64)
and the hardening modulus
h = -2 c Of : Of 3,x :
af(~af
:
of) ~/z (65)
3 Oo O,~ ~ aa
For the Von Mises criterion (61) one obtains
3 __°'-X' =
(66)
h=c- ~3'X: R+k c- 3'X:n
with n being the unit outward normal

f.-
af/Oa 13 ~r' - X '

(67)
n = ( O f / O a : O f / o a ) l/2 = ~2 R + k
Now the plastic strain rate is given by
3 1 (~a'-X' )a'-X' 3 l(n:da)n . (68)

d p= g R+k-2h
Let us note that the kinematic variable governed by eqns (61) and (62) is a deviator,
due to the plastic strain incompressibility. Assuming X = 0 in the initial conditions, then
d T r ( X ) = ~cTr(dEp) - 3' Tr(X) dp = -3' Tr(X) dp . (69)
Integration gives
Tr(X) = T r ( X 0 ) e -*. = 0 .
III.1.2 U n i a x i a l l o a d i n g . For the uniaxial case one finds from (60)-(63)
f =l,~-XI - R-k:O ,
dp = Id~pl = 1 (_(o- X ) da) h=c-3"X+_ (a-X) , (70)
d X = c dCp - 3'Xld~p[ .
Let us note the essential difference between the two plastic strain rate terms in
eqn (62), d~p and dp, giving rise to de u and Idepl, respectively. The nonlinearity intro-
duced by the recall term is thus not the same during a flow under tensile or under com-
pressive loading. The relation between X and e, is nonunique and the concavity of the
stress-strain curve is correctly reproduced. At each half cycle, beginning with Epo, X0,
the kinematic model (70) is integrated explicitly to
X(ep) = v-
c + - v e-7(~p-~po) (71)
3'
where v = +_1 gives the direction of the flow. The nonlinearity can also be deduced from
the expression for the hardening modulus.
During each plastic loading the stress is given by
a=v-+
° - v - c) e-V('p-~po)+k+R . (72)
3/ 3'
164 J.L. CHA~OC~!~-
The nonlinear-kinematic model gives not only the shape of the hysteresis loop, but also
the relations between the amplitudes at the stabilized cycle (here stabilization is very fast
for a periodic symmetrical loading). The cyclic curves are expressed as
_1o_ C tanh (' __'~_) + k + R . (73)

2 ~
Note that stabilization in the nonlinear-kinematic model occurs only if the loading
is symmetrical (zero mean stress). Qualitatively, the model thus describes the ratchet
effects found under nonzero c~mean. That point will be discussed in more detail in Sec-
tion V.3.
III. 1.3 G e n e r a l i z a t i o n . The range of validity of the model is widened, and the quan-
titative description of the ratchet effects is improved when we superimpose several mod-
els of the same type as follows (CHABOCHE e, ROUSSELIER [1981]):
t~t
x = ~ x, , (74)
i=I
dXi = ic, dep - ~{iXid p . (75)
Each of the variables Xi then works independently with the same kind of nonlinear
rule. One of them can be linear, e.g.
dXm = ~c,, dep . (76)
Under proportional loading the model is still explicitly integrable. Figure 8 illustrates
the modeling possibilities with three kinematic variables which, in practical situations,
is largely sufficient to cover strain ranges between 0.01°70 and 4%. A smooth elastic-
plastic transition is not exactly reproduced but the initial hardening modulus is high (see
Fig. 8).
Let us note the possibility of rewriting the above generalizations (74) and (75) in a
different form which will be used later. With three kinematic variables, denoting the
additional variables as
y= (1-72tX2+ (1-7-2)X3 , Z= (1-7-23)(1-'t-23)X3 , (77)

71 / 71 , ~t'i 72
and using the coefficients
b, ~Cl -I-c2+c3 , b2 : c,(1 - ~f~l)-I-C3 ('{-~1)

1-- , b3=c3 ("Y~)(
1- 1- 7-~2) ,
(78)
one can write eqns (74) and (75) as

l
I( = 1 6 0
2OO
X 2 = 100 (1 - e,m (-200 ep))
/ X 3 = 2 5 0 0 ep
x ! =40 (1-rip (-2000 ep))
I i J'
o 1 2 % (~p
Fig. 8. Superposition of two nonlinear and a linear kinematic variables in tension.
dX = ~bl d ~ p - 7 1 ( X - Y) dp ,
dy=2bzdEp-Tz(Y-Z) dp , (79)
dZ = 2b 3 d~p - "y3Z dp .
Then, if X is considered as the primary kinematic variable (appearing in the yield sur-
face expression), Y and Z play the roles of secondary and tertiary kinematic variables
(see Section III.3.3).
III.2 Use under varying temperature

Several practical applications of cyclic constitutive equations involve varying temper-
ature together with cyclic loadings. In nuclear components, cyclic temperature is often
considered as the secondary loading process. Temperature can affect plastic flow in two
ways:
• The size of the yield surface, for example k in eqn (59), depends on temperature.
• The evolution equation for the center of the yield surface, for example coefficients
c and 3' in eqn (62), depend on temperature.
166 J. L CHABOCHE
The thermodynamic framework developed below shows that the influence or' temper-
ature must be introduced on "strain variables" associated with X and R defining the
yield surface (59).
III.2.1 T h e r m o d y n a r n i c f r a r n e w o r k . The above developed kinematic hardening, as

well as isotropic hardening considered in Section IV. 1, can be incorporated in a gen-
eral thermodynamic framework (GERMAIN [1973], SIDOROFF [1975], GERM.adN e t al.
[1983]), which makes it possible to express the irreversible processes with internal vari-
ables. Let us summarize this theory briefly as it applies to small quasistatic transfor-
mations, using T for the temperature, ~ for the thermoelastic strain tensor and % for
the family of internal variables that describe the current state of the volume element.
Two potentials are used, a thermodynamic potential, e.g. the free energy
',~ = ¢(e~, T,c~;) . (80)
The total strain is divided into thermoelastic strain % and plastic strain ~p such that
= e~ + ep. The second potential controls the dissipation
~=¢(i~,~j ; ~e,%,Y) . (81)
The variables ~e, T,c~j appearing in this potential are considered as parameters. The
Clausius-Duhem inequality, expressing the second law of thermodynamics, leads to
a¢ a¢ a¢
= p ~ , S=---aT ' A j : P --a~j ' (82)
where a is the stress tensor, -S the specific entropy, p the density, and A s are the ther-
modynamic forces associated with the internal variables %. The dissipation is then
expressed as
If) : ~7 : i p -- A j ~ j -- ( 1/T)q-gradT . (83)
This must be positive because of the second principle. This is done automatically if we
adopt the hypothesis of generalized normality, using the potential ~* obtained from
by a Legendre-Fenschel transformation on the variables de, &j (GERMAIN [t973,1983]:
~* = ¢*(o, Aj;~e, T,c~ s) . (84)
For a theory with time (or rate) effects the generalized normality is expressed by
&," , d; 0¢* (85)

ip = 0a = - OAj
We see that the intrinsic dissipation is necessarily positive if ¢* is convex, positive and
cancels out at the origin (a = Aj = 0):
O~* A.0¢* >_., > 0 . (86)

Di=a:ip-Ai&j=a:-~-. + I OA j
In the limiting case of time-independent plasticity, (85) will be replaced by
Og Og
dep = dA ~ , d~j = -dA OAj ' (87)
where dA is the plastic multiplier, and g represents a plastic potential for instantaneous
plasticity. If g = 0 is chosen identical to the yield surface f = 0, the theory reduces to
the associated flow rule in terms of the generalized variables (a, A j ) and (~p, - d g ) .
Materials which obey such an hypothesis are often called "standard generalized mate-
rials" (HAtvr~N & NGUYEN [1975]). In general g is not identical to f , and one can speak
about nonassociated flow rule in the generalized space. If g is a convex function of
(a, - A j ) , is non-negative and contains the origin, i.e.
g = g(a, Aj;~e, T, aj) -> 0 , g(0,0; Ce, T, c~j) = 0 , (88)
then the intrinsic dissipation is necessarily positive:
dDi = ~r : dcp - A j d a j = d.~ ~r : -~g + A j _>gd,~__O . (89)
III.2.2 Application to the nonlinear-kinematic rule. The nonlinear kinematic model

presented in Section III. 1 fits into this thermodynamic framework with the two expres-
sions
p~.b= ~A:ee:e~+ ~cot:ot+ W ( r , T ) , (90)
33' 33' X : X , (91)

g=J(a-X)-R-k+~cX:X=f+ 4---c
where ot is the internal "strain variable" associated with the center X of the elastic
domain, r is the isotropic variable, associated with R. The coefficients c, 3', k, depend
on the temperature. One get the associated variables from the first potential by apply-
ing relations (82):
a=A:ee , X=]cot , R= W'(r) , (92)
the generalized normality hypothesis (87) gives
Og 3 a' - X'
d~p=dA~=~dAj(a_X) ' (93)
dot='d~=d~p-~c
Og 3 3' X d,~ (94)
168 J.L. CffABOC~,~i
Og )12
dr= - d/~-- =dA=dp= id~:d~p 195)
OR . '
For ~ = O one recovers the linear kinematic rule [eqn (94) shows that ~ is the plastic
strain itself]. Equation (95) demonstrates that the isotropic variable is the accumulated
plastic strain p. Combination of eqns (92) and (94) leads to the nonlinear-kinematic rule
2
dX= xcd~p-yXdp . (96)
o
Let us note that the generalization (74) and (75) is easily obtained within the same
framework by making the function ~ and g the sum of several similar terms.
III.2.3 I n f l u e n c e o f t e m p e r a t u r e . The influence of temperature on the plastic flow

is now defined, the independent state variables being clearly selected in the thermody-
namic potential: ~, a , p , T. Under varying temperature, eqn (94) must be used for the
evolution of kinematic hardening and not eqn (96). p is the isotropic hardening vari-
able. The associated variables X and R are obtained thereafter by eqns (92). The pres-
ent choice of the state variables is supported by physical concepts: Hardening is the
result of dislocation substructure and intergranular strain incompatibilities. If one sup-
poses a uniform thermal dilatation, which is coherent with the present level of con-
tinuum mechanics, no change can be expected in the substructure during a rapid
temperature change. Material changes such as precipitation have to be treated through
specific equations as in CAILLETAUD et al. [1972].
Taking as an example the tensile monotonic loading, integrating eqn (70) under
isothermal conditions with X ( 0 ) = 0 gives
X = (c/./)( 1 - e '~,') .
The coefficient c, for example, with the dimension of a stress, strongly depends on tem-
perature. Using (92) and (94) leads to
1 Oc
k = - vxf, + - x (97)
cO-T "
The present choice, rarely discussed though mentioned in CHABOCHE [1977] and
WALKER [1981], is very important in practical applications with cyclic temperature.
WA~.KER [1981] has shown a typical example, illustrating the importance of this choice.
III.3 H o w to d e s c r i b e classical m o d e l s
III.3.1 A t w o - s u r f a c e m o d e l . As mentioned already in MARQuIs [1979a] the above
internal variable approach of plasticity through a nonlinear kinematic rule can be con-
sidered as a two-surfaces theory. These two surfaces are
° the yield surface, which is defined by f = O,

• the bounding surface (or limit surface), which defines the limiting states of stress
and can be obtained from the two inequalities: J ( a - X) _ R + k and J ( X ) _< c / 7 ,

the last one being easily verified from the rule (62). That gives for the bounding
surface
f = J(ff) - R - k - c/~ ~ 0 . (98)
Figure 7(a) illustrates this property for R = 0. It is also evident from the uniaxial par-
ticular case (72). These two surfaces can expand isotropically through the function
R ( p ) . The generalization with superposition of several kinematic variables (74) also
obeys this two-surface property. With several nonlinear-kinematic variables, one finds
(99)
f= J(a) - R - k - ~ -ci
-<0 .
i "Yi
In that case and in (98) the bounding surface does not translate. Translation of the
bounding surface can be accomplished if one of the kinematic variables is linear (76)
m--I ) m-I
J(o" - Kin) __. J o" - X + E Xi <- J(o" - X ) + ~ J(Xi) ,
i=1 i=1
(lOO)
m-I
f=e(,,-x,.)-R-k- E c_~ .
i= 1 ~i
III.3.2 T h e M r o z m o d e l . A second analogy has to be pointed out. The nonlinear-

kinematic rule (62) corresponds to a two-surface form of Mroz model. Let us note
the stress state on the hounding surface with the same outward normal as the direction
of plastic flow. The two surfaces are given as [Fig. 7(a)]
f=J0r-X)-R-k=0 ,
(101)
f= J ( e ) - R - k - c/'y = 0 .
For the Von Mises criterion one gets
(102)
n = R ~ k - R + k + c/',/
The plastic strain rate is given by
Of 3 a' - X' (103)

dep=dA ~= ~ dp R + k
The nonlinear-kinematic rule yields, using (102) and (103) and the fact that X is a
deviator,
170 J . L . C~{ABOCH:2
dX= c R+k -7X' dp= R+k R+k

c)(o R + k + c/~/ # dp ,
after simplification
dX=-~(o'-a' dp , (lO4)
which is very similar to Mroz's hypothesis (15). The only differences are the use of stress
deviator in place of stress and of plastic strain increment dp in place of the multiplier
d~. The first difference (deviators) does not change anything due to the plastic uncom-
pressibility hypothesis.
Then, using the Mroz's hypothesis, together with the Von Mises criterion and con-
sidering proportionality between the multiplier d~ and the plastic strain rate dp auto-
matically results in nonlinear kinematic hardening. Let us note that the subsequent
surface (here the bounding surface) is attained only asymptotically.
III.3.3 O t h e r m o d e l s . We have seen that the two surfaces are implicitly introduced
by nonlinear-kinematic rule. This fact can be used to compare it to the models devel-
oped in Section II.2. The present approach corresponds to a particular choice of the
model of DAFALIAS & POPOV. For example, for the case of three kinematic variables
(74)-(76), one of them being linear, the following expressions are obtained for the dif-
ferent tangent moduli (isotropic hardening is neglected)
K3=c3 , K~,=K=ct+c2+c3-q-{3/2(yiXt+y:X2):n , (I05)
The kinematic part of the Krieg's model is a particular case of the present approach.
Selecting ~ =3' = 0 (pure kinematic hardening) in eqns (51)-(54) leads to
dX = b ( # ' - o') dp , (106)
dX = q(2/3)cde n . (107)
Due to the similar expression for the yield and loading surfaces [compare eqns. (47)-
(48) and (59), (100)] eqn (106) is identical to the nonlinear-kinematic rule. Using
o ' = "X' + I R / R ) ( a ' - X ' ) , one can get with (49) and (50)
dX=b (a'-X')-(X'-X')
I dp=b
I%]~(/~-R)dep-(X'-'X')dP
(108)
Equations (107) and (108) are identical to (79) with ~/2 = b3 = Y3 = 0 and with X replac-
ing Y.
Other models are similar to the presently developed kinematic rule. Let us mention
the work of ZAV~RL & LEE [1978], using what they call primary and secondary inter-
nal variables. Concerning the kinematic part the model yields
dX = q ( X L - X) dp , (109)
where X z is a limiting state for X, defined through the use of a secondary state vari-
able a s
X L = ca' + a ' , (110)
da' = ga'dp . (111)
a' is the stress deviator, q,c,g are coefficients. After some manipulation the above equa-
tions can be rewritten as
dX=q(1-c) ~d~p- (X--- 1-c dp , (112)
da'=g(~kd,p+Xdp) , (113)
where k is the initial elastic limit. Equation (112) clearly shows similarities with the non-
linear-kinematic rule; compare, for example, eqns (79) with Z = 0. However, eqn (113)
gives an important difference as compared to the present approach. In (113) the sec-
ond term on the right-hand side is positive and can give rise to an unlimited growth of
a" [compare to (79) where these terms are negative]. Integration under tension for
example shows evidence of an exponential term in + el, for X, then for o.
IV. SUPERPOSITION OF ISOTROPIC HARDENING
The two preceding sections were concerned with the description of kinematic harden-
ing, which represents rapid changes in the dislocation structure. During each half-cycle
(in tension-compression for example) dislocations are remobilized when unloading and
reverse loading take place. Then kinematic hardening gives a description of monotonic
rapid evolutions during each branch in a cyclic loading.
Independently of the kinematic effect, accumulation of dislocations can be repre-
sented by the accumulated plastic strain. The corresponding strength modification can
be introduced in the modeling through a change in the width of the elastic domain: This
change is given by the isotropic internal stress in equation (59). It is directly related to
the increase in the dislocation density, but may also depend on the dislocation config-
uration, e.g. creation of dislocation cells, size and fineness of the cells, etc.
IV. 1 Classical isotropic hardening

To introduce the evolution of R one can use any function R ( p ) , given by W'(p) in
(92). One can also restrict the evolution, with the use of some differential equations.
In the isothermal case a simple form,, used also by ZAVERI. & LEE [1978], is
dR = b(Rs - R) dp , (114)
where b and Rs are two constants.

The internal stress R varies as a function of the accumulated plastic strain p. After
a certain number of cycles (which decreases as the strain amplitude increases) it stabi-
[72 J . L . CH.~,BOCH~
lizes at the value R s . This is necessary, or else the only possible stabilized cycle would
be elastic. We integrate to get
R(p) = Rs( I - e-°r') . {i15)
In tension-compression the stress in each half-cycle is given by
cr = X ( e p ) + vk + vR(p) . (116)
where u = +1 or - 1 for positive or negative plastic strain rate, and where X(e~,) is
given by integrating the kinematic rule, eqn (71). Under symmetric cyclic loading, one
gets at each cycle
o vf = X , v ( d e n ) + k + R(p) . (117)
J~p is the plastic strain range, c~t, and Xvs the maximum stress and back stress. Xts
depends on the plastic strain range and is little different from c / y tanh(-y ,len/2)
(equality stands for stabilized conditions). Applying relation (117) to each cycle, to the
stabilized cycle and to the first cycle, we get approximately
GAI -- O'~fO ~ R
-- 1 -- e-bp = 1- e -2baep''' (liS)
~A, ls - d,'~lo Rs
where aMs and o:w0 are the stress peaks in the stabilized cycle and in the first cycle,
respectively. Taking the example of stainless steel 316 (GOODALL e t al. [1980]), Fig, 9
shows that the variation in the above quantity depends on the accumulated plastic strain,
independent of the strain amplitude, and that relation (118) gives a good approximation.
Under stabilized conditions the only hardening variable that is not constant is the
kinematic variable.
~M-~Mo
O'M~_aMo
10
/,o% ~
12 xlO -2 cf° '~

08- a 0,8 x I0 -2 n;
q7 x!O -2 o~/I
06.
zx q5 x 10 .2
• •
./ &
04
•a~'//~& ~-0"o : I _ exp (_ 3j85p)
eo Z
02 0 •
~,--00 - - • i 9
!0-3 lo- lO-, 16
p -'z_ 2N ~&p
Fig. 9. Variation of hardening with accumulated plastic strain p.

Introduction of isotropic hardening is also possible through the equation of evolu-

tion of the kinematic variable. Equation (62) can be replaced by
dX = ~cde e - 3'(p)X dp , (119)
where 3'(p) is a suitable function, usually taken as (MARQUIS [1979a1, WAZKER [19811):
3'(p) = ~,, + (3"0- 3 ' , ) e - ~ . (120)
Evidence of such effect of isotropic hardening can be given in a strain controlled

cyclic test by comparing the plastic modulus at maximum strain for the first or the sec-
ond cycle with the one in the stabilized cycle. Neglecting d R / d p , which is small enough
when stabilization takes place after twenty cycles or more, the two plastic moduli can
be approximated in the uniaxial case by c - "r0X,vt and c - 3"sXM. In order to reduce
the number of unknown coefficients one can suppose parallelism between the evolution
of R and the evolution of 3' by writing
7(P) = 3'0 + (Ts - 3"o)(R/Rs) , (121)
which is equivalent to (115) and (120) with ~ = b. Figure 10 illustrates the case of 316 L
with a strain controlled test [10(a) and (b)] and a stress controlled test [10(c) and (d)]
(MARQUIS [1979a]).
IV.2 M a x i m u m plastic strain m e m o r y

Isotropic hardening thus describes the cyclic hardening, but only one value Rs of the
variable R is obtained under stabilized conditions, regardless of the strain history.
Experiments show that this is not true in reality, at least for certain materials. For exam-
ple, for 316 L steel at room temperature, the following is found:
(i) After cyclic hardening saturation (R = Rst) at a given strain level (say +_1%), an
increase in the strain amplitude (_+1.5%) causes the cyclic hardening to commence again
[Fig. 1 l(a)], until a new saturated condition is reached. The model can predict this only
if two values for Rs are assumed.
(ii) After cycling at a higher level (+3%), the corresponding hardening is at least par-
tially memorized, and the subsequent stabilized cycle at the lower level has a much
higher stress amplitude than when there is no precycling [compare Figs. I l(a) and (b)].
The cyclic curves (relating Act to de) are very different, depending on whether a single
test piece is used for each level or an incremental method is used (Fig. 12). The incre-
mental test is realized by increasing strain range at each cycle until a maximum value,
then decreasing it until 0. The sequence is repeated until stabilization of the stress-strain
response. The corresponding cyclic curve is given by the set o f maximum values for
stress and strain in the intermediate cycles during the increasing sequence. Figure 12
shows the difference between the incremental tests and the normal cyclic curve (different
specimens or increasing levels only). Also the influence of the applied maximum strain
range in the sequence is marked.
A model has been proposed to express the observed dependence between the satu-
rated value of R and the maximum plastic strain amplitude (CHAaOCrm et al. [1979]).
It consists of introducing a new hardening variable q that memorizes the maximum plas-
tic strain amplitude. On the physical level, microstructure analysis shows this variable
I74 d.L.. CHABOCHE
t.
/ 0001 //
¢/
(a) (b)
5"-
t 0 0 PlPo.
/ QP:
E I/g E
(c} (d)
Fig. 10. Cyclic behavior of 316 L at room temperature: (a) the first cycle, (b) the stabilized cycle: - - test,
. . . . simulation, (c) stress controlled test, (d) simulation of test (c). Equation (119) is used in the simulation.
m a y be associated with the presence of dislocation cells, expressing both their size a n d
the thickness of their walls. The effect can also be partly a t t r i b u t e d to the creation of
twins (NouAILHAS et al. [1983]).
The general f o r m u l a t i o n (CHABOCHE et al. [t979]) consists of i n t r o d u c i n g a n o n -
h a r d e n i n g surface in the plastic strain space (Fig. 13):
F=ZJ(ep-f)-q<O . (122)
As long as the strain state is inside the surface, we assume that d F = 0. C h a n g e in the
m e m o r y state takes place only if F = 0 a n d ( O F / & p ) : dep > 0 :
dq = ~ H ( F ) ( n : n*)dp , (123)
500
25o
c('l.I
-2~0
i
• - - T~t with itw:nli~ ~ .

i -500
b - E x ~ t i m e n t a J c y c t ~ e f ~ f Dfw.-Vcling of z 3 ~ .
(a) (b)
///;°
c -- C J c u l m ~ of ¢VclN " l " , d -- CaJculatian of cvclts " b " .
(c) (d)
Fig. 11. Controlled strain testing on 316 L steel at 20°C, showing the hardening memoryeffect.
where n and n* are respectively the unit outward normals to the surface f = 0 in the
stress space and to the surface F = 0 in the strain space. The consistency condition
( d F = O) associated with this rule determines the evolution o f the center
d~=l ( 3 ~ H ( F ) ( n :n*)n*dp . (124)
The cyclic hardening law (isotropic hardening) is then modified to take the value of
q into account. For example, we can say
dR = b(Q - R) dp , (125)
1"76 J.L. CHABOCHE
Maximum ~train (%)

~, range
t a°/2CMp'l " G ~
i +
400" i ¢ , o n o t ~ a
4,-I, 1 speCimen per level ]

~00' I / o, Increment,I re,t, [
/ Q Telts with Increming levels ]
A • Monotonic loading !
__ Calcuhltton : model 2 (
i
A&/2(%)
Fig. 12. Prediction of cyclic curves obtained by different testing procedures.
ram path
Fig. 13. Definition of variables describing the memory of maximum strain range.
d Q = 2/~(Qs - Q) d q = I~H(F)(Qs - Q) dp , (126)
w h e r e / z a n d Qs are two coefficients. U n d e r t e n s i l e - c o m p r e s s i v e cycling, q - A e p ~ / . ,

-- , f")
where A%.ax represents the m a x i m u m plastic strain range. R e l a t i o n (126) is integrated

a n d the s a t u r a t i o n value o f the cyclic h a r d e n i n g b e c o m e s
Q(q)=Q(~) =Qs + (Qo-Qs)e -"a'p~, (127)

This model, with complete memory, expresses rather well the effects observed on the
316 stainless steel. Combining this with the nonlinear-kinematic hardening eqn (62) the
model can reconstruct at the same time the monotonic tensile test curve (Fig. 12), the
successive cycles with several levels of saturation [Figs. 1 l(a) and (c)], the memory at
the ___1070 level of hardening induced by cycling at +3070 [Figs. l l(b) and (d)], the nor-
mal cyclic curve obtained by a single test at each level, and the difference between the
cyclic curve obtained by the incremental methods (Fig. 12), in particular the effect of
the maximum strain range of these particular cyclic loadings (see 2% and 4070 strain
range).
A modification proposed by OHr~o [1982], and used also by DAr~IAS [1983]
introduces a certain gradual effect into the memory, thus distinguishing a single cycle
from several successive cycles. This is done by replacing (123) by
dq=~H(F)(n : n*)dp . (128)
The value of ~ is between 0 and 0.5 07 = 0.08, for example). Modelings of certain cyclic
loadings are improved, in particular when the mean strain is modified at the same time
as the strain amplitude (OrtNo [1982]).
This expression for the memory is thus perfect, insofar as it cannot diminish. After
monotonic or cyclic hardening, testing at a lower strain level shows a slow cyclic soft-
ening spread out over a large number of cycles. The case is well known for cold-worked
materials that show cyclic softening effects. Research is currently under way to include
such effects in a single model (NouAILHhS et al. [1983]).
V. RANGE OF APPLICABILITY FOR STAINLESS STEEL
V. 1 Time dependency
The presently developed hardening models are presented in the framework of time-
independent plasticity. The generalization of nonlinear-kinematic rule to the viscoplastic-
ity is very easy through the following assumptions:
(i) Consider the yield surface f = 0 as the surface for zero plastic strain rate and do
not use the load-unload and consistency conditions.
(ii) Stress state is allowed to be in the space f > 0 (Fig. 14). The viscous stress, or
overstress (K~t,a,L [1975]), is then defined as the minimum distance between the
stress state and the surface f = 0.
(iii) The normality rule applies to the equipotential surface and the generalized nor-
mality hypothesis (85) can also be used. The hardening rules apply in exactly the
same way, eqns (62), (74), (75) and (114), for example.
(iv) In addition it is possible to introduce time-independent plasticity as a limiting
case for infinite strain rate, with two inelastic strains (plastic and viscoplastic)
and a unique set of hardening variables, obeying the same rules ( ~ E L [1971],
HALpar~r~ ~, SoN [1975]).
The constitutive framework deduced from assumptions (i), (ii) and (iii) is often con-
sidered as "unified constitutive equations," only one inelastic strain and one set of
hardening rules are being introduced to describe plastic flow. For elevated temperature
applications such unified viscoplastic equations need to take into account time recov-
17S 3 [.. CHABOCHE
VISCOPLAST1CIT~
ST1CIT~'
Fig. 14. Viscoplastic equipotential surfaces and limiting cases of time-independent plasticity.
ery, eventually time hardening or aging effects (CHABOCHE & ROUSSELIER [1983]). The
corresponding developments cannot be considered in the present paper.
The limits of use of the time-independent plasticity models depend on the material,
the temperature and the considered loading conditions. For stainless steels they consti-
tute acceptable modelization for temperature below 450°C, even if viscoplastic effects
take place at room temperature (KREMPL [1979]), However, the apparent time indepen-
dence at temperatures above ambient could correspond to the combination of two oppo-
site time effects (CHABOCHE & ROUSSELIER [1983]). For higher temperatures
time-independent plasticity can also be used to consider extreme loading rates (very low
or very high); this scheme corresponds then to a limiting case of viscoplasticity.
More generally, if the loading conditions are such as to induce in the structure
approximately constant strain rates, the time-independent plasticity can still be an
acceptable scheme. In that case the above mentioned viscous stress is incorporated into
the width of the elastic domain, the equipotential which corresponds to the actual strain
rate being taken as the yield surface f = 0.
V.2 Limits for cyclic hardening or softening

In Section IV the evolution of the strength of the material during a cyclic plastic flow
was introduced through isotropic hardening, considering changes in the width of the
yield surface a n d / o r in the plastic modulus (119). The proposed rule was shown to give
an adequate description of the cyclic hardening of 316 stainless steel.
The influence of the plastic strain range upon the level of cyclic hardening, together
with its memorization, is described by means of the plastic strain m e m o r y introduced
in Section IV.2. The possibilities are then very large. For analysis of structures one can
use the kinematic hardening and the normal isotropic hardening [either on R(p) or on
function 3'(P)] if only low strain ranges (Aep < 1°70) are considered, or the plastic
strain m e m o r y for larger strain ranges or for varying cyclic loadings.
Let use note that the present description concerns monotonous evolutions (cyclic
hardening for example), valid for the first portion of the life in low-cycle-fatigue tests.
The stabilized conditions correspond to the maximum stress range in the life. For stain-
less steels initially in quenched conditions this state is attained after 20 to 200 cycles.
The partial evanescence of hardening for larger numbers of cycles (low strain ranges)
or for prehardened materials (cold worked, roiled, etc.) can be described by introduc-
ing more sophisticated models (NoUAILHAS et al. [1983]).
V.3 Limits for random loading or ratchetting under tension-compression

The nonlinear-kinematic rule can be used for any kind of loading conditions. For
approximately reversed conditions it gives good results. Under nonreversed loading the
rule reproduces qualitatively the ratchetting effect (under stress control), and the mean
stress relaxation (under strain control). In fact, the two effects are generally overesti-
mated by the model when only reversed conditions are used for the identification of the
coefficients. This overestimate is due to the rapid evanescence of the kinematic harden-
ing, rapid evanescence which is necessary to describe the nonlinearity of stress-strain
loops and a correct Bauschinger effect. Figure 15 illustrates that point in the case of
one kinematic variable (DAFAr[AS [1983]). However, the description improved when
several kinematic variables are superposed.
On the other hand multilayer surface models, such as Besseling's model or the Mroz
model, cannot describe ratchetting or mean stress relaxation. As with the linear-
kinematic rule, the accommodation (or elastic adaptation) takes place immediately after
the first cycle. Superposition of isotropic hardening improves the situation for transient
conditions but does not modify this conclusion for stabilized conditions.
The model of Dafalias & Popov permits a good description of random loadings, due
to the smooth elastic-plastic transition. At the beginning of a new stress-strain loop,
the plastic modulus is infinite, which eliminates the above mentioned inadequacy of the
nonlinear-kinematic rule. However, the improvement is obtained through the use of an
updating procedure, which leads to difficulties under complex loadings. In fact the prop-
at RaTche~'incj strain : ~(p

C-"[ x L° l-'--q
./-ii /
/
/.J A/~ ~p
/
I
o " /'I !
/ I
/ ~.'t~-( c
i/
i /
/ O'min ....
I
t
C' L.- •~k~ Accomodotion
(a) (b)
Fig. 15. (a) Behaviourof one nonlinear-kinematicvariable under nonsymmetricalloading conditions. (b)
Ratchetting effect under tension-compression.
I80 J, L. CHABOCHE
erty seems general. Kinematic rules without updating always Nve rise to an elastic-plastic
transition with a discontinuity in the stress-strain slope. Research could be conducted
on kinematic concepts using total strain rate instead of plastic strain rate as in KRE~PL
et al. [1984], but the thermodynamic framework would then have to be reconsidered.
V.4 Limits f o r multiaxiality
The above formulations can be used under multiaxial loading conditions. They gen-
erally give good results under proportional loadings. Some experimental works on stain-
less steels, clearly demonstrate the importance of kinematic effect. After a monotonic
plastic flow one can observe (MORETON et al. [1981]):
• a very small isotropic change,

• a large translation of the elastic domain,
• a small rotation of the surface.
With a very precise strain offset, distortion of the surface can be measured, but this
can be considered as a secondary effect. Inadequacy of the models is observed under
two situations:
(i) Superposition of a constant primary loading and a cyclic secondary loading, for
example a low constant tensile stress and a cyclic shear stress: In that case the descrip-
tion of ratchetting shows the same difficulties as in tension-compression. The linear-
hardening rule or the Mroz model gives no ratchetting under asymptotic conditions, but
nonlinear-kinematic hardening presents a too rapid evanescence and predicted ratch-
erring is too large (Fig. 16). An adequate superposition of several kinematic variables
can improve the modeling.
(ii) Out-of-phase plastic loadings often introduces an overhardening effect (LA_uSA
& SIDEBOTTOM [1978], McDOWELL [1983], NOUAILHAS et al. [1983b], KREMPL & Lu
[1984], CAILLETAUDet al. [1985]). Comparison of in-phase and out-of-phase strain con-
trolled tests, with the same equivalent strain range, shows a larger equivalent stress range
for out-of-phase than for in-phase conditions. Classical models cannot describe such
an effect and researches are conducted under two ways. Introduce a parametrization
of the cycle (McDoWELL [1983]) use anisotropic hardening (Nou.MLHAS et al. [1984]) or
more complex rules (TANAKA[1985]).
Let us note also that, at higher temperature, in the creep range, evidence of kinematic
hardening is clearly experimentally observed on several materials, including 316 stain-
less steel (DELOBELLE et al. [1984]).
Vl. DIFFICULTIES W I T H I M P L E M E N T A T I O N
The practical use of constitutive equations in the range of cyclic plasticity shows two
kinds of difficulties (the same is true for viscoplasticity). The first one concerns the
determination of the model, the second one the implementation in computer codes.
VI. 1 Identification o f the m o d e l

Two cases have to be considered, depending on whether the strain memory effect
exists or not. In the case without memory effect, the kinematic and isotropic rules can
easily be identified separately. The possibility of explicit integration of the models facil-
,/Tr
v ~ Trnox
~....
d¢ p
v~-T rni
Fig. 16. Ratchetting under tension-torsion for the nonlinear-kinematic rule.
itates identification under tension-compression. Let us note the possibility to measure

directly the isotropic and kinematic variables (TuEGEL [1985]).
VI. 1.1 Cyclic stabilized behaviour. Let us consider materials where no strain mem-
ory effect is present. This property can be checked through the following test results:
(i) The cyclic curve can approximately be deduced from the monotonic one by trans-
lation (in stress direction).
(ii) The cyclic curve is uniquely defined (for a given strain rate). Incremental test,
multilevel test (increasing or decreasing), one specimen per level, all methods give
the same results.
(iii) The Masing rule is approximately true, that is the form of each stabilized stress-
strain loop is deduced homothetically from the cyclic curve. The apparent elas-
tic domain in each loop is the same.
In that simple case, the coefficients of the nonlinear kinematic rule can be immedi-
ately identified from the cyclic stress-strain curve, choosing first the width k of the elas-
tic domain, then coefficients c,3, from the nonlinearity of the curve. Figure 17 shows
some examples. One of them is a superposition of a linear and a nonlinear-kinematic
variable. The isotropic hardening is identified separately using for example relation
(118). As mentioned in Section IV. 1 one can also introduce isotropic hardening by var-
iations in the coefficient 3' of the nonlinear-kinematic rule. The need for this is mea-
sured from the evolution of hardening modulus at maximum strain during cyclic strain
controlled tests.
182 J . L . CE,-~BOCHE
/-
15G0-
PT]~ 3,:30
ICi : 224000MP"~
a/
!'f ~ : 4 0 0
~2 : 200013
IOO0
j~----o-- NhMONIC
o/"° / ~ TA6V
/"
500-
5TF£" L
~.~-- _ _ ~ COBALT
1
0,5 Z
Fig. 17. Cyclic stress-strain curves of different materials interpreted by the nonlinear-kinematic hardening.
VI. 1.2 Influence of strain memorv. In stainless steels none of the above three prop-
erties is observed. In Section IV.2 the five-level strain controlled test was shown to be
a good test to observe the influence of the maximum plastic strain range. Moreover,
it is easy to see that the Masing rule does not apply. Figure 18(a) shows five stabilized
stress-strain loops (tensile branch), translated to their minimum. Superposition of the
E B C D E
A !
SJJ-
(al (b)
Fig, 18. (a) Inadequacy of the Masing rule for 316 L stainless steel (room temperature). (b) Translations defin-
ing the increase in saturated isotropic hardening. Data from C~tABOCHEet al. [1979].
loops is clearly impossible. Consequently the form of the cyclic curve is very different
from that predicted by the Masing rule.
Identification of the kinematic model is in the saturated state possible by superpos-
ing all loops by the translations shown in Fig. 18(b). The differences in the translational
values give the differences in the saturated values of twice the isotropic variable R. Sub-
tracting these values from the stress range and plotting the difference as a function of
plastic strain range leads to the determination of the pure kinematic effect. One can pro-
ceed as in Section VI.I.1 and find coefficients ci,~/;.
The measured differences in the saturated values of R easily give the function
Q(zlep) of relation (127) and the coefficients Qo, Qs,lz. The rapidity of saturation of
the isotropic hardening is then obtained [coefficient b of (114) or (118)], using for exam-
ple results shown in Fig. 9.
VI.2 Implementation in computer codes

Elastoplastic stress analysis for large structures presents some difficulties related to
the number of variables (storage), the incrementation procedures (CPU Time), the treat-
ment of the results (input-output, mass storage, etc.). This is particularly true for cyclic
analysis, where a number of successive cycles have to be calculated. Three typical aspects
are considered here.
VI.2.1 Cyclic stabilization o f the solutions. Usually, in cyclic elastoplasticity one can
consider only the stabilized cyclic behaviour of the material, since the crack initiation
predictions are generally based on the stabilized state. In the case of approximately
reversed loadings, only a few successive cycles are sufficient to stabilize stress redistri-
butions. However, if loading of the structure can result in cyclic mean-stress relaxation
due to cyclic plastic flow (repeated loads for example), a larger number of calculated
cycles is needed and simplified procedures are not so easy to develop and to implement.
On the other hand it is often necessary to take into account the process of cyclic
hardening or cyclic softening by itself. Considering directly the stabilized behaviour for
each element of the structure can lead to large errors. In fact only the cyclically plasti-
fied elements have to be considered as hardened or softened. This needs at least two
steps in the cyclic elastoplastic analysis: a first one using the initial monotonic
behaviour, taking into account the Bauschinger effect and the beginning of the cyclic
change; a second one with an asymptotic saturation of the constitutive equations for
the cyclically plastified elements (elements with no elastic shakedown).
The importance of the problem is shown in CAttLETAtrDet al. [1985] and CHa~BOCa~
a CAXLLETAUD[1985]. In the case of hardening material (316 L), for an axisymmetric
notched specimen, stabilization with the complete constitutive equations (describing ini-
tial behaviour, cyclic hardening, stabilized hardened state) needs more than 50 cycles
to obtain the good stabilized solution. In that case the solution coincides approximately
with the one obtained using the stabilized constitutive equations. In the opposite case
of cyclic softening materials, the example of a turbine disk (CJdLLETAUDet al. [1985])
shows that the two solutions do not coincide. With the initially hard material accom-
modation takes place only in a small plastic zone and the elastic zone is sufficient to
retain the structure; with the softened material (directly stabilized) the elastic zone is
not sufficient to retain plastic flow.
Let us note that the above considerations about cyclic stabilization are valid for any
184 J . L . CH.~,BOCHE
kind of constitutive equations. They illustrate the difficulties of calculations in cyclic

plasticity and the necessary cost of such calculations. Then, if calculation has to be
done, it is better to use the best constitutive equation, without changing significantly
the cost, but changing certainly the range of validity of the results.
VI.2.2 Convergence of the incremental algorithms. Incremental plasticity needs a step-

by-step decomposition of the external loading and an iterative procedure, in order to
obtain equilibrium of the structure, to check the yield criterion at each step and to fol-
low correctly the hardening rule. Let us remark that stability of the numerical scheme
is never sufficient. One has to check also the correctness of the solution with regard to
the hardening state. Moreover, the convergence through the solution in terms of the
equilibrium can also be unsufficient. An illustration of that point can be given as
follows.
For isotropic hardening, the convergence criterion based on the equilibrium between
stresses and plastic strains (considered as initial stresses), automatically induces conver-
gence in terms of the hardening variable p.
For linear-kinematic hardening, convergence as regards equilibrium also induces con-
vergence in terms of the hardening variable which is identical to the plastic strain.
For nonlinear-kinematic hardening or all other nonlinear rules the same convergence
criterion is not sufficient. The actual state of hardening not only depends upon the
actual state of plastic strain but also on the whole history.
Therefore, contrary to the cases of isotropic hardening or linear-kinematic harden-
ing, calculations in the nonlinear case must use a sufficiently fine discretization of the
loading, especially under cyclic conditions. That point is also common for the differ-
ent models considered in this article.
VI.2.3 Comparative difficulties. Three kinds of constitutive equations were consid-

ered. Each of them presents specific difficulties for implementation in computer codes.
The nonlinear-kinematic hardening rule, based on one to three tensorial variables,
each of them being governed by an independent constitutive equation is easy to imple-
ment, presents a median level of mass storage and is general as concerns loading con-
ditions. The main difficulty comes from the nonlinearity of the differential equations
and the nonuniqueness of the hardening state as function of the plastic strain. How-
ever, it integrates explicitly for proportional loadings, which facilitates identification.
The same difficulty arises for the multisurface models but, in order to describe cor-
rectly the actual behaviour, a large number of surfaces are needed, resulting in a large
number of scalar and tensorial internal variables. Here the storage requirements are
higher than for the first model. One advantage is the step-by-step linearity of the
hardening rule, but this advantage decreases when many elements undergo plastic defor-
mation successively.
The two-surface models need only two tensorial variables (median storage require-
ment), but nonlinearity of the equation is a difficulty. A second general difficulty is the
need for an updating rule, which increases in fact the storage level (it is necessary to
memorize discrete events). Moreover, updating rules generally presents several incon-
sistencies when general random multiaxial loadings are considered. Two examples are
illustrated in Fig. 19. The first one corresponds to a uniaxial load-unload-reload situ-
ation where plastic flow is very small during unloading. The reloading is computed with
updated quantities. In the case of the Dafalias approach the distance ~in is reinitialized,
o6
Predicted with ~
uDdating ~,,,.~ ~ ~'.
\ ~ .. ~ ActuaJ
~ / .- t (schematic)
/// ~in (3)
6'n (I)
~,, Cp
(a)
,/5-r
Out-of-phase
loading
(7
(b)
Fig. 19. Inconsistencies of updating theories: (a) overshooting of the tensile curve after unloading, (b) out-
of-phase tension-torsion loading.
which permits an initial infinite plastic modulus but leads to an overshooting of the new
tensile curve. The second typical example is the special case of out-of-phase multiax-
ial loading, where no unloading takes place although cycling is evident.
VII. CONCLUDING REMARKS
Different kinds of constitutive models were evaluated here, based either
(i) on a family of surfaces, successively attained during a typical loading, the last
attained being the loading surface (active surface),
(ii) on two surfaces only, the yield surface and the bounding surface, attained
186 d, L. C'HaBOCHE
asymptotically. The example chosen was the model of DAFALL~S & POPOV
which uses an updating procedure in order to describe a smooth elastic-plastic
transition,
(iii) on two surfaces, with center and width considered to be internal variables
governed by differential equations giving rise to a nonlinear-kinematic hardening.
The specific properties of the three approaches were discussed on the basis of cyclic
loading and can be summarized as follows:
Each of them is able to describe correctly the approximately reversed loading con-
ditions.
The two last ones can describe ratchetting effects but the nonlinear-kinematic rule
overestimates this effect.
The second one is better for random loading conditions, clue to the updating proce-
dure, but plastic flow must be sufficiently reversed in each stress-strain loop.
None of the three approaches is able to describe the anisotropy and the over-
hardening effect induced by out-of-phase loadings.
Two domains of research are open, in order to develop better models:
(i) the problem of ratchetting, or mean-stress relaxation, which needs specific rules
and specific sets of coefficients,
(ii) the last-mentioned problem of out-of-phase loadings, which can be treated by
introducing some nonkinematic anisotropy of hardening.
Several researchers introduce updating procedures in order to improve random Ioad-

ings and the elastic-plastic transition (KuJAWSKI & MROZ [1980], BRUHNS [1982], GOOD-
MAN [1983]). The possible inconsistencies of such approaches for complex loadings have
to be studied carefully. Their general writing must probably use more complex rules than
presently proposed. Such an example of rules is given in MRoz [1981].
Acknowledgements-Part of this work was realized in the "Groupement d'Int6r& Scientifique Rupture :a
Chaud" with the support of the French Ministry of Industry and Research, which is gratefully acknowledged.
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(Received 3 April 1985; In final revised form 16 November 1985)

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Imernattonal Journal of Plus~tctty. Vol. 2, No. 2. ~0. 1-19-188. 1986 0749-6-t.19,86 $3.00 - .

TIME-INDEPENDENT CONSTITUTIVE THEORIES

Office National d'Etudes et de Recherches A6rospatiales

(a) using multilayer concepts (BEssELING [1958] a: MEIJERS [1980]),

f=f(o, hardening) _< 0 . (2)

Fig. 1. Schematic of the elastic domain, normality hypothesis, load-unload criterion.

X=c~p , dX=c d~p , (5)

where H is the Heaviside function, H ( u ) = 0 if U < 0, H ( u ) = 1 if u > 0, and the

where n is the unit outward normal to the yield surface

The hardening moduli is the constant c,

~'The symbol : indicates the contracted tensorial product.

fo = f l . . . . . J~-I =J~ = 0 , (12)

afo afl .. = d~l aft

Each surface can be described by equation similar to (4):

where Xi and ki correspond to its center and its width, respectively.

dX, : d/~/(e;.t - ~) • (t5)

of,. do af~ a/, aj) (16)

and multiplier dt¢; is determined by

(Of/Oo : do} (n:do)

dXo = dXt . . . . . dXt-i = dX; • (18)

with n being the o u t w a r d unit normal,

Fig. 4. The Prager and Mroz kinematic hardening.

=f (dep:dep) 1/2 . (21)

Each surface is now written as

d#i = H ( f i ) (Of~./Oa : da) - dki for i _< 1

or after rearranging with (19)-(21).

OkilOA ] (n : do) (23)

fi = J ( a - Xi) - ki(p) , (24)

where J corresponds to the octahedral shear stress,

J ( o - Xi) = [ ] ( o ' - X ~ ) : ( o ' - X~)] 1/'- (25)

dp = [2d~p: d~p] 1/2 (26)

dXi=d/x,(crt, l - ~ ) , i_<l , (28)

(1-1 x/3 Ok,) <i,~'-X;):d,7> (29)

d X l = d/xi(e/+~ - o') (30)

dTr(Xi) = d~i[Tr(~t.1) - Tr(e)] . (31)

Under asymmetric loading conditions, for example alternating tension-compression

II.2 Two-surface models

Fig. 5. Immediate accommodation of the Mroz model.

K(6i~,6i~) = co and K(0,6in) =Ko • (33)

dA = (d~a : dEp) I/2 = ( 1 / K ) ( n : da) . (35)

of A as the internal variable describing isotropic changes, that is expansion or contrac-

f= O(a - X ) - k(A ) = 0 , (36)

where X and Y~ denote their respective centers.

K~, is determined t h r o u g h the consistency condition (d f = 0) for the yield surface:

(n:dii) =KOdA=Ko K (n:da) , (41)

KO is determined by K(O,&,)and represents the limiting hardening for saturated con-

(9 the use of two surfaces only,

f=(a'-X'):(o'-X')-R 2=0 , (47)

f= (o' - X') : (o' - X ' ) -/~-~ = 0 . (48)

The a c c u m u l a t e d plastic strain p is defined by

d p = (~ dl~p: d~p) I/2 = 2 4 ( 2 / 3 1 R d,~ . (50)

C o n t r a r y to the preceding formulation, Krieg defines directly the translation and

dX = b(1 - / 3 ) ( 0 ' - o ' ) dp , (51)

(o' - o') : (o' - X ')

dX = (2~c(1 - 3') d~p , (53)

d/~ = c3, d p , (54)

O' - - X ' O" - - X'

The plastic strain multiplier is determined by the consistency condition d f = 0. One