Abstract-The cyclic constitutive equations developed and used at ONERA and LMT-Cachan are presented in detail in terms of a hierarchy of various models.
International Journal of Plasticity, Vol. 5, pp. 247-302. 1989
0749-6419/89 $3.00 + .00 Copyright '© 1989 Maxwell Pergamon Macmillan pie
Printed in the U.S.A.
CONSTITUTIVE EQUATIONS FOR CYCLIC PLASTICITY AND CYCLIC VISCOPLASTICITY
J.L. CItxnOC~IE Office National d'Etudes et de Recherches A6rospatiales (Communicated by David Rees, Brunel, The University of West London) Abstract-The cyclic constitutive equations developed and used at ONERA and LMT-Cachan are presented in detail in terms of a hierarchy of various models. Both the time-independent and the viscoplasticity versions of the equations are discussed, as well as their ability to describe correctly most of the experimentally observed effects under monotonic or cyclic loading, constant or variable temperature, including strain hardening and time recovery effects. The reported experimental data concerns stainless steels and have been published previously. Four other theories are then presented and compared in a systematic way. They include the Ohno-Kachi time-independent plasticity theory, two unified viscoplastic models by Walker and by Krempl and Yao, the new developments of the endochronic theory by Watanabe and Atluri. All these approaches show some similarities with the first one, especially in that concerns the non.linearity of kinematic hardening, which represents the key for describing the cyclic behaviour of metallic materials.
I. INTRODUCTION
The modern methods for life prediction in structures need inelastic analyses, which lead to large computing times, especially under cyclic loadings. Corresponding to the larger level of accuracy involved in the computing methods, it is necessary to use better constitutive equations to describe the material behaviour. During the past decade much progress has been made in the development of constitutive equations to represent the behaviour of materials, under cyclic loading conditions at high temperatures. At the very beginning, two classes of constitutive equations emerge from the literature, based on one of the following thermodynamical concepts: I. The present state of the material depends on the present values and the past history of observable variables only (total strain, temperature, etc . . . . ) giving rise to hereditary theories. 2. The present state of the material depends only on the present values of observable variables and a set of internal state variables. The fn'st concept was used for example by VALANIS[1971-1980] in the development of the endrochronlc theory, by ~ L [1975] in viscoplasticity, by GtnzLn~et al. [1977] in the hereditary theory with discrete memory events. The second approach has been developed in various ways, using the concept of yield surface in the case of time-independent plasticity, and multilayer (BEss~mo [1958]) or multiyield surface models (MRoz [1967], DAF~IAS ~ Popov [19761, Kltmo [1975]). The time-dependency was introduced either by separating plastic and creep strains (RoBn~247
248
J.L. CHABOCHE
SON et al. [1976]), taking into account the coupling effects through the hardening rules (KAwAz-OH.ASHI[1987], CONTESTt-CAtU.ETAUD[19871), or in the framework of unified constitutive equations, incorporating only one inelastic strain (CHABOCHE-RoussELmR [19831). This paper deals with the last approach of constitutive modeling, especially through unified viscoplastic models based on the superposition of kinematic hardening, isotropic hardening (to describe cyclic hardening or cyclic softening) and time-recovery (thermal recovery) of the hardening (LAGNEBORG [1972], PONTER-LECKIE [1976]). The model developed in France, especially at ONERA (CHABOCHE [1975]) and LMTCachan (MARQUIS[1979]) is recalled in detail together with some new developments. It is presented in a hierarchical form, similar to the one used recently for soil modeling (DEsAX [1987]). The time-independent version of the approach (section II) and the more general viscoplastic framework (section III) are successively considered. Both versions include the same hierarchy of hardening rules (see Fig. 1): The so-called "Non-LinearKinematic" rule initially proposed by ARMSTRONG& FREDERICK [1966], the superposition of an isotropic rule to take into account cyclic hardening (or softening) of materials (MARQtltS [1979]), a strain memory variable introduced by CHABOCHEet al. [1979] to describe the complex cyclic hardening effects in materials such as stainless steels. The time-dependency is described in terms of the unified viscoplasticity (CHAROCr~ [1975]), incorporating time recovery of hardening and several additional effects. Equations used in the ONERA model are indicated in Table 1.
[
Time-independent Non-linear kinematic hardening
Time.dependent
i i
(2)
(3) linear --....
~,
-',-. I V,=oplast,c ) "'"" " "" " ~ [ potential I $i,, f "'~ ~" S , = " I I.tropic I I"" ""'x" / : : ~ I hardening I"~'~ /I "../ 1 '1 ~ . ](cyclic evolutions)I'"-. I " / "..~] Kinematic I
/
!
t
J
,
f
/
".
" ~
.,~"'"
""~,'i I "..
~3 II memorization Jp"
,,/
~ l time-recovery I 6 .....
""I-'. "1 v / " ' ' . _ [ Isotropic
-..., .'".
"/ /
/
I
Anisotropy cross-hardening out-of-phase distortion
I time.recovery I 6, .... . .,,,
Aging temperatu re history
/
t/ °,1
,-"
,,
~2+ vIh
,,.o
Fig. 1. A hierarchical model for metals in time-independent and time-dependent plasticity.
Equations for cyclic plasticity and cyclic viscoplasticity
249
Table !. Summary of the equations used in the model developed at ONERA Viscoplastic strain rate: Of/
3
w' - X '
Kinematic hardening: X=~Xi
2 X i = 3 ciep - 7 i c k ( P ) X i P
i
Plastic strain memorization: 2 F=~J(¢p-i')-pb
( I - ~)HIF)
¢' - X' J-'~# - X )
n =
(n : n* )n*#
n* =
J(E# - ~')
isotropic hardening: ~ ( p ) = # = + (1 -- ~ = ) e - o p
= 2?t(QM - Q)~ R = b(Q
-
R)p
K * = Ko + o t x R
R * = ctRR
Time-recovery for kinematic hardening: Xi = (Xi)hm-d -- 'Yi° [ J ( X i ) ] m , - I Xi
Time recovery for isotropic hardening: [~ = b ( ~ - R ) p + ~lr[Qr - R [ m sign(Qr - R)
\
Q.
/J
It can be shown from treating the irreversible processes with sets of internal variables that the form of this constitutive equation is compatible with a general thermodynamic framework ( G E ~ [1973], G E ~ et al. [1983]). In a second part, recently developed cyclic constitutive equations are presented and discussed in detail. They are the Om~o-I~cm [1985-1986] time-independent plasticity model, introduced in terms of a two-surfaces approach (section IV), the unified model used by W~,.rt~ [1981] (section V), the viscoplastic theory based on overstress proposed by gatmoL et al. [1984] (section VI), the endochronic theory of VAtAmS [1980], with the practical rules developed by W^T~,,.~ a A ~ t r ~ [1986a,b] (section VII). Each approach is compared systematically to the ONERA/LMT model. Many similarities are pointed out, especially in respect of the Non-Linear-Kinematic rule. This rule provides a common frame in those cyclic constitutive equations which do not continuously update the current material properties (Cn_Jmoc~ [1986]). Various relationships are studied and
250
J.k. CnAaocH~:
compared separately (section VIII) for describing the limiting case of high strain rates in terms of unified viscoplasticity. Also the domains of applicability of the models and the difficulties of their determination are briefly discussed. Let us begin with a brief historical background about Non-Linear-Kinematic hardening and the additional theories. • The first use of a recall term in classical Linear-Kinematic hardening (PRAGER [1949]) was employed by ARMSTRONG& FREDERICK[1966]. MALININ& KHADJINSKY [19721 developed a different modification within the framework of viscoplasticity, including time recovery. The combination of the two models was proposed by CHABOCHE [1975] and applied to the high temperature behaviour of a Nickel base superalloy. • The incorporation of cyclic hardening (or softening) was proposed by MARQUIS [1979a,b], who modified the kinematic rule and by CHABOCHEet al. [1979] who increased the yield surface size with a new concept of strain range memorization. • WALKER[1981] introduced the same kind of unified constitutive equations in an integral form. KREMPLet al. [19841 modified the Non-Linear-Kinematic rule, using a driving term proportional to the total strain rate. • At the same period, in the framework of time-independent plasticity, DAFALIAS [19831 & OHNO [1982] proposed two modifications to the strain range memorization concept. In particular, the OHNo-KAcHI [1985] model incorporates cyclic hardening in a slightly different way, using the concept of non-hardening surface. Later, TANAKA et al. [19851 further developed the concept with monotonic and cyclic nonhardening surfaces. • From a completely different approach, the new endochronic theory of plasticity and viscoplasticity introduced by VALANIS[1980] was specialized by decomposing the kernel in a series of exponentials, giving rise to a variant of the Non-LinearKinematic rule (WATANABE& ATLURI [1986]). • Since 1981, several additional rules were superposed to the ONERA-LMT constitutive model in order to incorporate more adequately the complex behaviour of stainless steels at high temperature (CI-IAaOCHE~z ROtISSELmR[1983], NOUAILHASet al. [1983a], NouArLHAS et al. [1983b1, NOUAILHASet al. [19841, BENALLALet al. [1985], BENALLAL& MARQUIS [1987], NOUAILHAS[1987]). • Other modifications to the model have been proposed, introducing the separation between micro-strains and macro-strains (DELoBEtLE [1985], DELOBELLE-OYTANA [1986]) and the separation between plastic strains and creep strains (KAWAI & Om~.Sm [1987], CONTESTI-CAILLETAUD[1987]). During the past ten years, several other sets of constitutive equations were proposed and developed to treat the monotonic and cyclic behaviour of metals and alloys, especially at high temperature. Among them, let us mention the MATMOD equations developed by MILLER [19761, SCmaDT 8: MILLER [19811, LOWE • MILLER[1984] which incorporate several strengthening mechanisms, the LEE • ZAVI~RL [19781 approach, which includes the treatment of initially anisotropic materials, the models developed by BODNI~R [1975], by PONTER & L E C ~ [1976], by BRUItNS [1982], by ROBINSON [1983]. There are some similarities between the above theories, especially in the unified character of the viscoplastic framework. However, with the exception of some later versions of these models, the kinematic hardening is mainly introduced in a linear form for large strain rate conditions, with a superimposed thermal recovery term.
Equations for cyclicplasticityand cyclicviscoplasticity
251
!!. A HIERARCHICAL MODEL FOR METALS: TIME-INDEPENDENT PART
Inelastic cyclic straining of metals induces many complex phenomena which have to be described separately or simultaneously. The observed experimental facts and the corresponding microstructural processes can be presented in a hierarchical way, similar to the one used in some soil modeling (DEsaz [1987]). The model developed at ONERA and LMT-Cachan is easy to incorporate in that hierarchy. With reference to Fig. 1, this is defined as follows: The first layers describe the most important facts for cyclic loading in a decreasing order of importance and an increasing order of complexity (6t,6z . . . . ). Each layer can be used as required, depending on the considered application (material, temperature, loading c o n d i t i o n s . . . ). The left part of Figure 1 shows the time-independent set of layers. On the right-hand side are given the time-dependent aspects which may be written in a unified framework: Each hardening rule of the left-hand side may be incorporated as required in the viscoplastic (v) constitutive equations, giving rise to a gi+~ level. Similarly the time recovery (r) effects may be taken into account with a 8i+~+r formulation (in some case, it could be possible to consider time-dependent plasticity without viscosity as a 6i+r model). The last level, not considered here, is the incorporation of continuum damage processes (d) and their coupling effects with plasticity, viscoplasticity and hardening (LE~LArrRE CHABOCHE [1985]). The equations employed with the ONERA model are summarized in Table 1. They incorporate the various possibilities discussed in sections II. 1 to II.4, III.l to III.3 and III.5. II. 1 Time-independent plasticity The present approach uses the decomposition of hardening into kinematic and isotropic parts, each obeying respective differential equations. In the framework of timeindependent plasticity, using the yield surface concept and taking into account the plastic incompressibility though a von-Mises yield function, leads to:
f = J ( ¢ - X) - R - k < 0
(1)
where X is the back stress (or rest stress), k the initial size of the yield surface, R its evolution (increase or decrease). J denotes the Von-Mises distance in the deviatoric stress space:
J(¢ - X) = ~/3/2(¢' - X ' ) ( ¢ ' - X')
(2)
where ¢' and X' are the deviators of ¢ and X. Plastic flow occurs under the conditions f = 0 and 0f/00: ¢ > 0. Under the classical normality hypothesis, the plastic strain rate becomes:
Of
3
¢' - X'
(3)
R+---T
A is the plastic multiplier, here identical to the modulus of the strain rate: = p = ~2/3~p : ~p.
(4)
252
J.L. CHABOCHE
Its determination involves the definition of the hardening rules together with the use of" the consistency condition f = 0 during plastic flow. Let us note from the experimental determination of the yield surface (MoRETON e t aL [1981]) that there is evidence o f a large kinematic translation together with a small isotropic dilatation and a distortion of the surface. The last transformation is not considered in the first layers o f the model (see section II-5). II.2 T h e N o n - L i n e a r - K i n e m a t i c
r u l e (61)
Widely used in France ( C n . ~ a o c ~ [1975], MARQUIS [19791, etc . . . . ), these kinematic equations were initially introduced by ARMSTRONG & FREDERZCK [1966]. Non-linearities are given as a recall term in the Prager rule:
(5)
X = 2/3 C~.p - y X p
where C and 3' are two material dependent coefficients and 3' = 0 stands for the LinearKinematic rule. The key o f the model is the plastic strain rate (tensor) in the first term and the plastic strain rate modulus (scalar) in the second term. Using the consistency condition f = f = 0 leads to the following equations to express the modulus o f plastic strain rate and the hardening modulus h: I
(6)
A = f9 = H ( f ) h
3
X : (a' - X')
h=C-~y
(7)
k+R
Application to the case o f tension-compression leads to:
f= Io-xJ -R-k-0
_r'= 0 if g < 0 or n* :ep > 0
(45)
where n* = (og/aep)/llag/cgep)ll and l[ipll = 4~p: ip. One can readily check the complete identitybetween relations(42) to (45) and the memory concept of eqns (18) to (21). The bounding surface and the yield surface are respectivelytaken as: f=
2 ~. (#' - X , ' ) : ( q ' - X ' ) - ~2 = 0
f=
~ (o' -- X ' ) : (¢' - X') - K2 = 0
2
(46)
(47)
where #' and g' are the deviators of # and ¢. # represents the stress state on the bounding surface with the same outward normal as for the actual stress state on the yield surface (direction of plastic flow). X' and X' are respectively the centers of the bounding surface and the yield surface. The motion of the bounding surface is considered from superposing isotropic and kinematic hardening:
= ~(q)
2[
.~ -" "~ K + ( I -
I")
q =/'p
(48)
O:]
(49)
ip -- KrXp.
The transformation of the yield surface is assumed to be purely kinematic, using the same hypothesis as in the M n o z model [1967]: =
a(a' - ¢')p
(50)
the normality hypothesis governs the plastic flow:
a/ ip = A a-99" Using the consistency condition d f = 0 during plastic flow leads to:
(51)
272
J. L° CHABOCHE
2 1 2 o'-X' ~p = ~ ~ ( n : o ) n = ~ p ~
(52)
K
h = ~
A(O - a ) : n
(53)
2° equivalent to (21). where n is defined as ~JIl~,,ll = ~cp/p, O~No a KAcm [1986] make the following particular choice for the function (48):
(54)
g(q) = go + Lq
where q expresses linearly as a function of p by using (43) and (48): 1 q = - (P - Po)-
(55)
IV.2 Particular case o f tension-compression Under tension-compression, the three-dimensional model reduces to (OHr,~o-KAcm [1985]): g= (ep-~')2-p2_ o
4 - WATANABE-ATLUR I (127) 5 - M I L L E R relation (128) 6 - MILLER modified (n -* 1.5)
O~
o .J
2.3
3 ~
5
7-DELOBELLErelation(129~-~ 2.1 -
1-2-6-7
~
,
,
J~2 ......'.._ ~ - ~ .
61
1.9
1.7
l .....
,
1.5 --14
,
-10
-6
,
LT'°I°)
-2
2
6
(a)
200
--4.-7
6 r
160
i
,
120
80
!
2
40
5 3
0
~ 80
I 92
~
I
I
'
I
104
116
128
140
ov
(b) Fig. 18. Comparison between seven asymptotic relationships: (a) viscous stress/plastic strain ra(e; (b) equivalent exponent/viscous stress. Continued on facin 8 page.
Equations for cyclic plasticity and cyclic viscoplasticity
295
Figure 18 is constructed for the following values:
n=No=24, Ovo=60, p o = l O -=° or* = 120, p* -- 10 -2. From these comparisons, it appears that the asymptotic effect is more or less rapid and pronounced. In fact, only the relation (127) gives a stress completely limited by /(2. In relations (124) and (129) the limit has an exponential character but in (125) and (126) the limit corresponds to a second power function, with a larger exponent. The Krempl model (126) and the Miller expression (128) do not describe correctly the asymptotic behaviour which can be met with the other theories. In the Miller theory (128) the asymptotic character can be obtained by changing exponent 1.5 into a larger value, for instance n2 as in the expression used by Dclobclle (in Figs. 18a,b,c a second simulation is given by interchanging exponents 1.5 and n in eqn (128)). Figure 18c shows the asymptotic effects relating the stress and the ratio of actual strain rate to the strain rate calculated by the normal power relationship (130). In practical applications, the relation (125), giving rise to a limited asymptotic effect (in the regime 10 -3 < p < 1) with a high power exponent, can be used with simple (explicit) numerical integration schemes of viscoplasticity. This constitutes a good corn-
3
140
5
Ov
2 128
1-6 7 4
116
104
[-
92
b/(ov/K) n 80
t
0
!
80
.
I
,
160
I
240
l
I
320
m
400
(c) Fig. 18 continued. (c) viscous stress/increasing factor for strain rate.
296
J . L . CHABOCHE
promise when the exponential and hyperbolic functions present some difficulties at very high strain rates where the equivalent exponent takes values larger than 100 (Fig. 18b). The completely asymptotic term of the endochronic theory (127) is only tractable with similar iterative techniques to those used in time-independent plasticity.
IX. CONCLUSION
The cyclic constitutive equations developed in France at ONERA and LMT-Cachan have been presented in detail within a hierarchical framework. They incorporate many possibilities to describe complex cyclic behaviour, including multiaxial and out-of-phase loading conditions, high temperatures, time recovery and aging effects. The basis of the hierarchy is to use the so-called Non-Linear-Kinematic hardening rule, initially proposed by ARMSTRONG& F~.~DEmCK [1966]. Its generalization superimposes several N.L.K. rules and eventually incorporates a linear kinematic hardening term. A particular part of the model may be selected depending on the material and temperature, but also depending on the considered application (loading conditions) and on the available set of experimental data. One feature of these constitutive equations is the containment, within the same hardening/softening rules, of the three following approaches: • Classical viscoplasticity (unified framework), • Time-independent plasticity, which is considered as a limiting case of viscoplasticity, • Combined effects, within the framework of unified viscoplasticity, incorporating some asymptotic effects for high strain rates (with a quasi time-independent behaviour). This consistency between these different frameworks is particularly interesting for quantifying the different parts of the model. For example, tests with constant strain rates may be used to determine the hardening rules within the time-independent version. The viscosity effect may then be obtained as was indicated in section III. 1 and finally the recovery effects are determined as required. Several other constitutive theories have been studied in this paper and compared systematically to ONERA model: • The theory of OHNO & ~ACHI [1985] applies to the time-independent framework. It employs the concept of strain range memorization to describe isotropic hardening (cyclic hardening). Then, contrary to the first approach, it cannot describe materials with cyclic hardening in the absence of memorization effects. Kinematic hardening is introduced in terms of a two surface theory. Under stabilized cyclic conditions, the hardening rule becomes identical to the first approach, through superimposing a non-linear kinematic and a linear kinematic rule. • The WAL~R model [1981] appears within the unified viscoplastic framework. It does not include a yield surface and the isotropic hardening rule acts as a drag term in the viscoplastic equation. The kinematic hardening is identical to the ONERA approach, with the superposition of one NLK and one LK rules. Also the kinematic time recovery effects have a similar form and the influence of temperature changes
Equations for cyclic plasticity and cyclic viscoplasticity
297
is treated in the same way. However, there is no time recovery for isotropic hardening and an initial, non recoverable back stress is used, The viscoplasticity theory developed by Kau~a,t et al. [1984,1985] is written in terms of total strain, without the use of a yield surface. The kinematic hardening describes the evolution of the equilibrium stress which, in the ONERA theory, is the projection of the stress state on the present elastic domain (the equilibrium stress is the sum of the back stress and half the size of the elastic domain). In the limiting case of time-independent plasticity, or for materials with very low viscosity, the kinematic model is still identical to the superposition of an NLK model and an LK rule (under proportional loading conditions), The particular form of the Endochronic Theory ( V ~ x s [19801) developed by W^r,~a,~,~ ~, Arttnu [1986], also reduces to the superposition of several NLK hardening rules for both the time-independent and viscoplastic cases. The only slight difference is due to the particular choice of the influence of isotropic hardening in the back stress evolution equation. This choice is a consequence of the internal time concept in the endochronic theory. Within the unified viscoplastic framework, this theory incorporates an asymptotic time-independent term. That could create difficulties for viscoplastic numerical integration schemes. From these comparisons, it appears that recent theories employ or superimpose several Non-Linear-Kinematic rules and a Linear Kinematic rule with several other features already incorporated in the hierarchical model developed at ONERA and LMT-Cachan. The use of the NLK-LK hardening theory is one of the only ways to describe correctly the cyclic response of materials without the use of updating rules (see ~ H ~ [1986]). The only alternative would be to use multilayer theories, with a finite number of layers (MEm~RS [1980], KuJ^wsKx • Mgoz[1980]) or an infinite number as in the hereditary theory of GU~LIN [1977]. Let us remark here the clear connections which have been demonstrated (see W^TANASE a ATLURI [1986]) between the endochronic theory, one hereditary theory, and the more classical plasticity and viscoplasticity theories, based on internal variables. The two fundamentally different approaches of thermodynamics, hereditary and internal variables, are then conciliated. Several other unified constitutive equations may be mentioned that employ kinematic hardening: the MATMOD equations developed by Mn.LER [1976], which are based on linear kinematic hardening (recently incorporating some non-linearity in Low~ ~ Mnlg~ [1984]), the model of RoBn~soN [1983], which separates the stress space into different domains in a non-classical way, the new models of W,~J~a~R a, BOD~ER presented in LINDHOLMet al. [1984], the model of DELOBELLE[1985], (DELOBELLE & OYTANA [1986]) which incorporates two different hardening rules when separating micro-plasticity and macro-plasticity. Also generalizations have been made within a non-unified plasticitycreep framework, as was mentioned in the introduction. To conclude, the past ten years of research into constitutive equations have shown great developments for cyclic plasticity and cyclic viscoplasticity. Several almost identical approaches describe correctly a wide range of behaviour. These constitutive equations have already been used to solve practical problems in structural components, as reviewed and discussed in ~ 1 ~ [1987b] and in C~n-T.F-TAuD-BULLET[1987]. HOwever, one effect not yet satisfactorily represented is that of cyclic ratchetting, as was discussed by CHAnOCH~ [1987a]. Further research must be conducted here to modify the kinematic rules (which govern this property) or to introduce additional rules.
298
J.L. C~OCHE
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