Constitutive equations for cyclic plasticity of Waspaloy

Pergamon International Journal of Plasticity, Vol. 12, No. 8, pp. 967-985, 1996

Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved

0749-6419/95 $15.00+.00

Plh S0749-6419(96)00037-X

CONSTITUTIVE EQUATIONS FOR CYCLIC PLASTICITY OF WASPALOY

A. Abdul-Latif*

LG2MS-mrcanique, URA 1505 du CNRS, Universit6 de Technologie de Compirgne, B.P. 649, 60206 Compi+gne cedex, France

(Received in final revised form 5 June 1996)

Abstract--The isotropic and kinematic hardening rules of the ONERA/LMT model suffer certain deficiencie,; in describing the cyclic behaviour of nickel-based Waspaloy where the second phase is sheared as in the under-aged state. An approach is suggested which takes into account the actual evolution of such hardening during cyclic loading. An internal variable is introduced to describe the effect of the starting of the second phase shearing processes during the first cycles. The ONERA/LMT kinematic hardening rule is extended in order to appropriately describe the softening phenomenon observed i:a almost all cases of applied loadings after cyclic stabilization, The resulting model is employed 1o simulate the mechanical behaviour of such materials under a proportional cyclic loading path. Furthermore, after adopting a supplementary assumption, these constitutive equations are coupled with the non-proportionality effect to characterize the cyclic behavior of this alloy under such conditions. Predictions of the proposed model are compared with experimental results. The constitutive equations describe fairly well the experiments of this alloy under simple and complex cyclic loadings. Copyright © 1996 Elsevier Science Ltd

I. INTRODUCTION

Consti tut ive model ing o f cyclic plasticity and cyclic viscoplasticity with internal state variables has achieved notable success in correlating simple and complex deformat ion histories. Because o f the increasing development and application o f computer-based analysis techniques, the need for appropr ia te constitutive equat ions has also grown con- currently. Knowledge o f the mechanical behavior o f materials under complex loadings is necessary to appropria te ly design mechanical structural systems.

Recently, a number o f literature surveys have represented the state o f the art in this field (for example, Krempl [1987]; Chaboche [1989]; Ohno [1990]) giving detailed discussions o f those theories, o f mechanical cyclic behavior o f metallic materials. The constitutive equations proposed to describe the uniaxial behavior are usually capable o f giving a very good correlat ion with the experimental data. However, modeling o f multiaxial deformat ion features becomes a difficult task due to the complexity o f the responses to non-propor - tional loading. Under condit ions o f highly non-propor t iona l cyclic loading, the kinematic and isotropic hardening rules must be complicated due to the effects o f changing plastic

*Present address: Universit~ Paris 8, IUT de Tremblay, 93290 Tremblay-en-France, France.

967

968 A. Abdul-Latif

strain rate direction on activating slip systems, etc. Many cyclic plasticity and viscoplasticity models have a complex mathematical form and possess a large number of material constants which need a wide experimental data-base to be identified.

For many metallic materials, and especially those with a low stacking fault energy, cyclic hardening under non-proportional loading becomes stronger than that under proportional loading. According to the experimental microstructural data in non-proportional cyclic tests, slip system multiplicity can be clearly observed, particularly in highly deformed regions. This multiplicity is strongly dependent on the loading path (for example, McDowell et al. [1988]; Clavel et al. [1989]; Doquet [1989]; Ferney et al. [1991]; Abdul-Latif et al. [1994]). This multiplicity leads consequently to different cyclic hardening behavior under non-proportional loading compared to proportional loading.

Partial recovery of the overhardening is experimentally observed during the decreasing of the non-proportionality, for example, in the case of stainless steel (BenaUal & Marquis [1987]; Doong & Socie [1989]) and in the case of Waspaloy (Clavel et al.. [1989]; Abdul- Latif et al. [1994]). In Waspaloy, this partial recovery originates mainly from variation in the isotropic component (Clavel et al. [1989]; Abdul-Latif et al. [1994]). It has been found that this is related to the material anisotropy occurring in many metallic materials after plastic straining (Nishino et al. [1986]). From a modeling point of view, many models are unable to describe this phenomenon, whereas the description of such a phenomenon is possible for others (McDowell [1985a]; Benallal & Marquis [1987]; Doong & Socie [1991]; Abdul-Latif et al. [1994], amongst others).

Concerning the non-proportionality, some models propose that this extra-hardening is completely based on isotropic hardening, like Benallal and Marquis's model [1987]. On the other hand, some other constitutive models considered the influence of the non-proportionality to be represented in the saturation amplitude due either to kinematic or both isotropic and kinematic hardenings (Krempl & Lu [1984]; McDowell [1985a, 1985b]; Krempl & Yao [1987]; McDowell [1987]; Moosbrugger & McDoweU [1989, 1990]; Moos- brugger [1991]; Abdul-Latif et al. [1994]).

The purpose of this paper is to describe new phenomena concerning the isotropic and kinematic cyclic hardening behaviors recorded experimentally in Under-Aged (UA) Was- paloy. During uniaxial cyclic loading, the evolution of the isotropic hardening of this material is different in comparison with other typical evolutions (exponential form). On the other hand, under uniaxial and multiaxial loadings, the softening phenomenon becomes effective only for the transient period after stabilization. This softening is governed by both hardening components (Ferney [1994]). Therefore, the classical constitutive hardening rules of an ONERA/LMT (Lemaitre & Chaboche [1985]) model for example, are inadequate to describe the behavior of the UA Waspaloy. This is due to the effect of the Second Phase Shearing Process (SPSP). The physical bases of these phenomena are not yet clearly understood. Generally, the mechanical behavior of the UA state can be characterized by shearing of 3 '~ precipitates. During proportional loading paths, the plastic strain localizes tightly in intense slip bands in the (111) family planes (Abdul-Latif et al. [1994]; Ferney [1994]). Moreover, in-between intense slip bands, very few dislocations were observed (Ferney et al. [1991]; Abdul-Latif et al. [1994]) in accordance with the localization of the deformation process. Because of small (but numerous) stacking faults scattered in-between the bands, therefore, the main part of the extra-hardening comes from the isotropic component. According to Ferney [1994], this is due to the interaction between the stacking faults which exists on different slip systems (latent hardening). It is

Constitutive equations for cyclic plasticity 969

shown that slip system multiplication enhances isotropic hardening. In this UA state, two microstructural mechanisms take place: the plastic strain localization in the form of slip bands as well as the cyclic softening which follows this plastic strain localization (Pineau [1979]). These two mechanisms provoke a local degradation of the precipitate structure. It is worth emphasizing that the repeated shearing phenomenon of the precipitates during cyclic loading is considered as a basis to interpret the existence of these two mechanisms (Pineau [19"79]). According to Pineau [1979], the shearing process of precipitates (often orderly) probably leads to the loss of order. In other words, softening can result from the decreasing of the contribution of order hardening of the alloy. This explanation has been clearly adopted by Calabrrse & Laird [1974]. In spite of the wide study performed by Ferney [1994] on this UA Waspaloy, a fundamental question arises concerning the remarkable ,cyclic hardenings: the isotropic hardening evolution during the first cycles and the softening phenomenon recorded after the steady state in both loadings (simple and complex). Undoubtedly, these physical phenomena are governed by a mechanism (or mechanisms) of the SPSP being coupled with the dislocation motion mechanisms.

In this work, based on the experimental study performed by Ferney [1994] and taking the related ~xficrostructural mechanisms into consideration, a new extension of the so- called ONERA/LMT model is proposed considering the isotropic and kinematic hardening rules. Using the framework of irreversible thermodynamics with internal state variables, a pair of internal variables is proposed to describe mainly the beginning evolution (first few cycles) of isotropic hardening. In fact, these variables can describe the effect of the starting of the SPSP during a few cycles, supposing here that the coupling between the non-proportionality and this effect can be governed by the non-proportionality parameter. The softening phenomenon observed experimentally is also taken into account by an extension of the kinematic hardening rule.

II. PHENOMENOLOGICAL MODEL

For characterizing the cyclic mechanical behavior of metallic materials in time-independent plasticity in the form of constitutive equations, internal variable theory in the framework of thermodynamics of irreversible processes is used. In this theory, two potentials which depend on the internal variables are sufficient to deduce the constitutive equations. The choice of these variables is based on physical considerations and experimental resuks. The present phenomenological model takes into account the decomposi- tion of the hardening into non-linear isotropic and non-linear kinematic components adopting isol:hermal and small strain conditions (Chaboche [1977]).

II. 1. Internal variables

In addition to the observable variables (total strain e and temperature), the classical internal hardLening variables are introduced. The variables (r, R) represent the isotropic hardening; the force R describes the expansion of the radius of the yield surface in the stress space. The couple of variables (tx, x) represents the kinematic hardening tensor and the back stress tensor, respectively, the latter represents the translation of the yield surface center in the stress space. In order to take into account the effect of the starting of the SPSP, a scalar internal variable "7" is introduced with the force "1 ~'' as an associated variable. Throughout this paper, the total strain tensor ~ = ~;e q- f-p is supposed as the

970 A. Abdul-Latif

unique "external" state variable with Cauchy stress tensor o- as an associated variable under isothermal conditions.

II.2. Free energy and dissipation potential

In the case of small strain and for an isothermal process, the state of the system can be described by its free energy (state potential) (Lemaitre & Chaboche [1985]). This free energy represents the sum of reversible and stored energies per unit volume, i.e.

p0 -- P0e(ee) + POp( r, o~, 3') (1)

Concerning the reversible energy, 0e (elastic part) a classical isotropic quadratic form is adopted, so that the Cauchy stress tensor er becomes:

o0 tr = P~ee = k tr gel + 2/-te-e (2)

where 1 denotes the unit second order tensor, p is the density of the material, and A and # are the classical Lame's constants.

A new term is added to the classical plastic free energy (Chaboche [1977]; Lemaitre & Chaboche [1985]) for taking into consideration the effect of the starting of the second phase shearing process exhibited by the UA state. This state should favor an isotropic hardening (Ferney et al. [1991]; Ferney [1994]). Thus, the plastic part (stored energy) can be defined as:

1 1 " pOp(r, tx, 7) = ~ Q r2 + ~ Z C i ° l i " O~i ~- (1 -- A)~stCsr 7

i : l (3)

where (s and Ks are material dependent parameters of the starting of the SPSP. The existence of the non-proportionality parameter A (see eqn (25)) in the third term in the right- hand side of eqn (3) represents a choice for describing the influence of the loading path on the starting of the SPSP. Note that this parameter is systematically equal to zero for proportional loading path and varies with the change of the non-proportionality. The effect of this change will be discussed later. Parameters Q and Ci represent the moduli of the isotropic and kinematic hardening, respectively. In the case of non-proportionality, these moduli become functions of the parameter A and the plastic strain rate modulus ,b given by:

~--- F..p " Ep (4)

where Ep is the plastic strain rate. The superposition of the kinematic hardening rules is possible in this framework according to Chaboche [1977] and Lemaitre & Chaboche [1985]. In this work, two kinematic hardening variables are used (n=2) (eqn (3)). The state laws are now defined as the derivatives of 0, for example:

00 2 gi ~--- P ~ i ~ i --'~ 3 CiO~i (5)


In UA Waspaloy, Ferney [1994] observed that the softening is more governed by the kinematic hardening than the isotropic one. Thus, for modeling this softening phenomenon, it is supposed here that, after the steady state, the second kinematic hardening modulus C2 becomes a function of the accumulated plastic strain expressed by:

Cin if p' < 0 C2 = Cin - C~ if p I > 0 (6a)

where Cin is the initial second classical kinematic hardening modulus. To start this softening, a threshold p' =P-Psat is introduced with Psat representing the accumulated plastic strain at the steady state. The rate of change of C a can be written as:

Coo = at(Cf- Coo)17'H(p -- Psat) (6b)

where Cf is the final asymptotic value of kinematic hardening modulus after the softening. Here a' is a parameter which controls the speed of saturation of Ca. H(p-Psat)=0 if P <Psat and H(p-Psat) = 1 ifp>_Psat. It is experimentally observed that the rate of softening differs with the employed loading path, i.e. the greater the non-proportionality, the greater the rate of softening is recorded to attain the stabilization. So, a' can be defined as:

a ' - af (6c) ( 1 - A ) + m f

where af ant] mf are coefficients of the material. According to the nature of the isotropic hardening evolution, a certain choice of the

internal variables is adopted. This choice is taken in a manner that the coupling is directly established in the state laws between the isotropic hardening variables and the internal variables of the shearing process (7, F) (eqns (7) and (8)).

o¢ R : P-~-r : Q r + (1 - A)(s ~s7 (7)

o¢ r : (1 - A)(st%r (8)

For isothermal deformation the second law of thermodynamics requires that the dissipation rate /} shall be positive (Clausius-Duhem inequality) (Lemaitre & Chaboche [1985]), i.e.

2 / ) = {7" ip -- Z X i " ~ i - R r - F'~ 2 0 (9)

i=1

After determining the force variables o-, xi, R and r , the formulation is completed by the rate equations of the internal variables kp, &i, i and -~. These evolution rates can be obtained by the introduction of an elastic domain in the stress space (von-Mises yield funct ion)f as well as a potential of dissipation F considering the non-associated plasticity case. It is worth noting that the non-linearity of the two sources of hardening can be introduced by the plastic potential F. Thus, the yield surface can be expressed by:

972 A. Abdul-Latif

f = J2 (o- - x) - R - k < 0 (1o)

where k is the initial size of the yield surface in the uniaxial tension condition and J2 the von Mises norm defined in the stress space by:

J2(o- - x) = V/~ (s - x) - s - x) ( l l )

where s = o--(1/3) tr o-1 is the deviatoric part of the Cauchy stress tensor o-. The purely deviatoric tensor x is given by:

2

x = Z x i (12) i=1

The plastic potential can be written as:

1 2'r~ai 2 lbR2 r(o-,x, R, F) = f ( o - , x , R ) + ~ l~. ,~/J2(xi ) + ~ + I'e -p~s+m (13)

where ai and b are the material dependent coefficients characterizing the nonlinearity of the kinematic and isotropic hardening, respectively; #s is a material dependent parameter characterizing the starting of the SPSP. It is worth noting that p in the third term in the right-hand side of eqn (13) represents a parameter and it is not a state variable. This choice is adopted based on two reasons: firstly, to simplify the constitutive equations (essentially for R) and secondly to describe the starting of the SPSP which is a function of the cycling, i.e. a function of the accumulated plastic strain.

The evolution laws (normality rule) of the plastic strain and internal variables are:

OF 3)~( s - x "~ ~.p = )i ~ = ~ \J2- ( ; - - -x)J

j, OF a i = - ~ = i.p - .Xai,xi

"OF= j~[1-b{r + (1- ~(s~sT}]

X OF

where )~ =/~

(14)

(15)

(16)

(17)

The existence of the additional terms in the plastic free energy (eqn (3)) and in the plastic potential (eqn (13)) as compared to Chaboche's model represents a means of describing the effect of SPSP on the isotropic hardening force evolution R. By supposing here that A remains constant for each type of loading, this evolution can be deduced as follows:

R = A[(Q - bR) - (1 - A)(st%e -p~s+~] (18)


It is clear that the second term in the right-hand side of eqn (18) gives the isotropic hardening evolution notably during the first few cycles. R evolves rapidly when the accumulated plastic strain p is small, whereas this evolution will be reduced as p increases. The mathematical role of the parameters concerning this phenomenon (G, n~, #s) will be depicted later (Section 1II). In the case where there are no SPSP, one can obtain: G = ns = #~ = 0 giving rise to a classical evolution of R:

= J~(Q - b R ) (19)

As far as the second kinematic hardening rate is concerned, such a rate can be deduced according to the proposed modification on C2 (eqn (6)):

2

with

x 2 ( 0 ) = 0 (20)

B = a ' ( C f - C ~ ) (20a)

The classical evolution can be retrieved w h e n B = 0

3 . ~ a 2 x 2 , x2=~C2~p - x2(0)--0 (21)

The classical Lagrange multiplier ~ is given by the consistency condition j" = 0 and must verify ~he following loading-unloading condition:

f < O , X > O , X f = O (22)

One can deduce:

/ 0

if f = 0 and j" = 0

otherwise

(23)

where < y > = y if y > 0 and < y > = 0 if y _< 0. H represents the scalar tangent hardening modulus which can be divided in two contributions:

n = niso + / / k i n (24a)

The isotropic hardening contribution (Hiso) can be expressed by:

Hiso = Q - b R - (1 - A)(se~se -p~s+m (24b)

while the kinematic hardening contribution has the following form:

H k i n = C l + C 2 - - ~ S---X)" : a l x l + x 2 a2+ (24c)

974 A. Abdul-Latif

I1.3. Effect o f non-proportionality

It is now well known that under non-proportional ioadings, cyclic hardening is stronger than in proportional loading for many metallic materials. Theoretically, it has been proposed (Benallal & Marquis [1987]) that the isotropic hardening modulus Q is considered as a new variable. They assumed that this variable is a strain hardening one. Its evolution depends on the non-proportional parameter A, whereas, the cyclic behavior of UA Was- paloy exhibits an extra-hardening affected by the two sources of hardening (isotropic and kinematic) in this loading path (Ferney et al. [1991]; Moosbrugger [1993]; Abdul-Latif et al. [1994]; Ferney [1994]). To describe this particular cyclic behavior of Waspaloy, a model was recently proposed by Abdul-Latif et al. [1994] supposing that C1 becomes a function ofp and A under non-proportionality.

As a matter of fact, the actual starting mechanisms of the SPSP have, until now, been undetermined in the case of proportional and non-proportional loading paths due to the absence of a fundamental physical study. This merits a future development for under- standing these mechanisms during simple and complex loadings. In any event, for the sake of simplicity, it is assumed here that the loading path plays an important part in this phenomenon, i.e. the greater the non-proportionality, the lower the number of cycles (under the same equivalent strain or stress) required to terminate this process. Therefore, a simple coupling is adopted between the non-proportionality and this phenomenon governed directly by the non-proportionality parameter A.

It is clear that the new couple of internal variables (7, 1-') is strongly related to the isotropic hardening but not to the kinematic one. Hence, to introduce the effect of such variables on the non-proportionality, only the equations of overhardening due to the isotropic hardening (Benallal & Marquis [1987]) will be affected, while the equation of overhardening due to the first kinematic component (Abdul-Latif et al. [1994]) remains unchanged. On the other hand, a simple coupling concerning the second kinematic hardening is introduced to describe the effect of the loading trajectory on the softening rate (eqn (6c)).

Generally, the non-proportionality can be characterized by an angle (/3) (between the back stress x and its rate x). According to BenaUal [1989] the parameter (0 _< A < 1) that describes this non-proportionality, is defined as:

A = sin2/3 = 1 (x" x)2 (25) (x:x)(x:x)

The limiting cases of this angle are:/3 = 0 giving A = 0 (proportional path), while/3 = 90 ° leading to A = 1.

According to Benallal & Marquis's model [1987], the isotropic hardening modulus Q has been considered as a new variable depending on the non-proportionality factor A and the plastic strain rate modulus/) :

O. = D(A)(QAs -- a ) p (26a)

where D(A) = (dr - f r )A + fr (26b)

gra Qr + (1 -- A)Qo and QAs - grA + (1 - A) (26c)


where dr, f~, gr, Qr and Qo are material dependent parameters. Note that the initial value of Q is Qo. In the proportional loading path (A = 0) Q remains always equal to Qo- This leads to the classical isotropic hardening rule (eqn (19)). On the other hand, for the second extreme case (90 ° out-of-phase), i.e. A = 1, eqn (26a) becomes Q = dr(Qr - Q)p.

Using the above method, Abdul-Latif et al. [1994] proposed a modification of the ONERA/LlVIT model to describe the remarkable mechanical behavior of the Waspaloy under non-proportionality. They considered that the first kinematic hardening modulus C1 becomes a function of A and p governed by the following differential equations.

Cl = I(A)(GAs -- C1)p (27a)

I(A) = (fbx -- Wx)A + Wx (27b)

,TxACx + (1 - A)Co CAs = (27c)

r&A + (1 - A)

where ~bx, w~, ~x, Cx and Co are material dependent parameters. The limiting cases (proportional and non-proportional 90 ° out-of-phase paths) are exactly the same for eqn (27a) as for eqn (26a). A basic question arises here concerning eqns (27a) and (26a). These equations are not very compatible with the thermodynamic (eqn (3)) due to the fact that Q and C~ haw~ to be consequently derived with respect to p (or r). However, the method adopted represents an approximated and simplified solution, otherwise, this solution becomes precise but quite complex.

The effect of the new internal variables (7,F) on the tangent plastic modulus (H) as well as on the evolution rate of isotropic hardening rule k is now examined. It is clear that the equations of Hiso and k must be modified to take into account the evolution of Q and the effect of the internal variable governing the phenomenon of the starting of the SPSP. Besides, the kinematic hardening contribution Hk~n shall take the new extension on the second kinematic hardening into consideration, whereas the evolution rate of the kinematic hardening rule XI takes the same form as in Abdul-Latif et al. [1994]. Thus, one can obtain:

Hiso = Q - b R + D(A) ( ~ - l) [R- (1 - A)~s~s'y] - ( 1 - A)~s~se -'~+m (28)

3 - I(~t) ~--~- 1 }Xl + x2 (a2 ~-22)] : (j---~--x))) (29)

k = i ' Q + Q r - ( 1 - A ) ~ s a s ' ~ , R(0) = 0 (30a)

C'I 2 XI :~ I IX1 "-I-'~Cl~.p - ,~alx1 , Xl(0) = 0 (31)

Note that the second non-linear kinematic hardening rule xz has the same form as in eqn (20).

As far as 1the extreme case of the non-proportionality (A = 1) is concerned, i.e. out-of- phase 90 °, one can find theoretically that the cyclic behavior is not affected by the SPSP,

976 A. Abdul-Latif

supposing that, for the same of simplicity, this operation could take place in a very rapid manner• Thus, the evolution rate of the isotropic hardening of this limiting case and the equations of Hiso are the same as in Abdul-Latif et al. [1994], i.e.

[~ = J~(Q - bR) + ~ R (32)

In the case of a purely proportional loading path (A = O) giving Q = dl = 0 and without softening phenomenon (B = 0) the isotropic and classical kinematic hardening rules are recovered.

/~ = ~[(Q - bR) - (s~;se -p~+m] , R(0) = 0 (34)

2 xi = 5 Cigp - J~aixi, xi(0) = 0 (35)

Moreover, if there is neither the non-proportionality effect nor the second phase shearing process, the classical isotropic hardening rule is retrieved (eqn (19)).

IIL APPLICATIONS

Let us discuss now the different applications conducted on the UA state of Waspaloy. All the tests are carried out at room temperature using a servo-hydraulic INSTRON machine (type 1340). Note that the experimental procedures as well as the experimental results were discussed in Abdul-Latif et al. [1994] and Ferney [1994].

In general, under strain-controlled conditions, the loading paths in von Mises strain ~(_~1~ follows: space (e(t), v5 j are defined as

e(t) = eosinqcot (36)

7(0 = %sin(cot - ~) (37)

t5 = "Y-2-° = v~ (38) Eo

In the above, e(t) is the axial strain, 3'(0 represents the shear strain, eo and 70 are the amplitudes of the strains, w is the frequency of oscillation and qD is the phase angle between the two strains. In all employed paths q is equal to one except the "butterfly" test in which q is equal to two; 6 represents the in-phase strain ratio. On the other hand, in the "cross" test (tension-compression alternately with torsion-torsion) a zero equivalent von Mises strain was imposed at the beginning of each alteration. Note that all the tests are conducted where the maximum von Mises equivalent plastic strain is maintained constant at 0.5%. These employed loading paths are presented in Fig. 1.

III. 1. Identification

The identification process which determines the different material coefficients is presented. First of all, the proposed model is programmed into a small computer code. Then, this identification process is performed by using, of course, an appropriate experimental


data base. In order to satisfactorily terminate this operation, the following procedures are adopted:

(a) Identification of the coefficients concerning the uniaxial loading: this procedure is carried out on the experimental tension-compression cycles. For determining the evolution of the isotropic hardening, the first 10 cycles and a stabilized one are used. In general, there are 12 coefficients which require identification for describing this simple loading up to steady state. From the first cycle (precisely the first loading in tension) the Young's modulus (E) and the initial yield stress (k) are determined. The initial yield stress is defined using a plastic strain offset of 5 × 10 s. The experimental evolution of the isotropic and kinematic hardening components are measured according to method developed by Kulhman-Wilsdorf & Laird [1979], and further details about these measurements can be found in Abdul-Latif et al. [1994]. To start off the identification procedure, the initial values of hardening coefficients are chosen based on these experimental measurements. Moreover, for the sake of improving the correlated solutions, two kinematic hardening components are used. Thus, the classical parameters of the kinematic hardening (Cl, a~, fin and a2) as well as the isotropic hardening coefficients (classical coefficients: Q and b, new ones: (s, ns and #s) are specified by having the best fit between the theoretical and experimental results. As regards these new parameters of the starting of the second phase shearing process, it is instructive to demonstrate the mathematical role of each one. It is seen that (F:ig. 2) (s describes the rate of decreasing (or increasing) of R at the beginning of straining, whereas, ~s has a role of controlling the rate of increasing (or decreasing) of R just after the minimum (or maximum) value. The minimum (or maximum) value can be mainly governed by the third parameter (#~). By using the identified parameters (Table 1),

EMPLOYED LOADING

PATHS

TC, "f=0

UA3

TT, q~./6

UA6

TT, q~=~2

UA2

SCHEMATIC R E P R E S E N T A -

TIONS

• !7N3

7 N 3

( . ~ / -5

EMPLOYED LOADING

PATHS

T'r, q ~ / 2

Then TC

UA7

1W

q~-o, q=2

Butterfly

,,..J-.

TC and

Torsion-Torsion

Cross

SCHEMATIC REPRESENTA -

TIONS

,~'N3

i

"r~3

TC :Tension-Compression, Tr : Tension-Torsion

Fig. 1. Employed loading paths in the strain space.

978 A. Abdul-Latif

Plastic strain modulus (p)

0 / t I I , - . \,,,, 0.05 0.1 0.15 0.2

:~ -4o ,,

"~ -80

" ~ - - - Classical Model

-160 New Model

/ ¢ ( 3 )

Fig. 2. A plot of predicted isotropic hardening evolutions of Chaboche's model and the new model showing the influence of each coefficient during cycling.

Table 1. The value of the identified coefficients of UA state of Waspaloy

Chaboche's model [1977]

E(MPa) 215000 u 0.32 k(MPa) 804 Q(MPa) -16542 b 200 CI(MPa) 8280 al 89.5 Cin(MPa) 96959 a2 330

New extension

(s(MPa) 474 ns 35 #s 0.76 Cj(MPa) 31000 af 1.02 mf 0.032

Extension of Q,(MPa) 3287 Benallal and gr 4.1 Marquis [1987] d, 1.1

f , 2.83

Cx(MPa) 113772 Extension of r/x 2.07 Abdul-Latif et al. Ox 6.7 [1994] wx 1.67


1200

800

400

0

-400

-800

-1200

<

time (sec)

(a)

. . . . exp. Theo. ]

1200

Axial s t r e s s (MPa) 800.

, /

i ........................................ 1.2~

I

o. i, (b)

400 ...................................................................................................

300

e L

2oo

X

lOO

o

eL - 1 0 0

~,, -200

- - A A O ¢ f , . ~ O U g U m m ¢ C ¢ ¢ ~ - - A

Number of cycles I I I

5 I0 15 2~ O,0- b" b'O "0"0-0"° ° x~ ° U tr t) i

o..a" i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . !

l . . . . . . . Theo. Theo. O Exp. • Ex~

(c)

Fig. 3. For the UA Waspaloy state in tension-compression loading: (a) correlation of the correlated axial stress- time with axial stress-time experimental responses; (b) stabilized hysteresis loop (experimental and correlations results); (c) comparison between the experimental and predicted evolutions of the isotropic (R) and kinematic (X) hardenings.

980 A. Abdul-Latif

it is seen from Fig. 3 that the correlations of this model agree well with the experimental results. Furthermore, the constitutive equation of the isotropic hardening (eqn (18)) which is plotted in Fig. 3, shows a good correlation with the experimental measurement of this hardening evolution during this uniaxial loading. A comparison between the experimental results for a first half cycle and two theoretical solutions is exhibited by Fig. 4. At the beginning of the yielding, this alloy behaves as an elastic-perfectly plastic material (prac- tically without any hardening effect) (Fig. 4). The interpretation of this behavior is based on the fact that competition between rapid softening (governed by the isotropic hardening) and the hardening (controlled by the kinematic hardening component) is taking place. This abrupt softening of the isotropic hardening is induced by the SPSP. For this reason, the new model can describe the transient cyclic behavior.

(b) To correlate non-proportional experiments, the related parameters (dr, fr, g,, Qr, 4~x, cox, ~/x and Cx) must be determined. This task is achieved by using the experimental data of a 90 ° out-of-phase circular straining. It is worth noting that this step should be done by fixing all of the identified coefficients concerning the uniaxial loading path (previous step). Then, the first 10 cycles, like the tension-compression case, are used. Using the experimental circular loading results (UA2), the values of (dr, Qr, c~:, and Cx) can be identified. On the other hand, with the non-proportional loading of 30 ° out-of-phase (UA6), gr and r/x will be identified. Finally, the cyclic test UA7 (Fig. 1) permits determination offr and a; x. For identifying the three softening coefficients (Cf, af and my), many cycles after the steady state are chosen for various loading paths (TC, UA2). Table 1 shows a complete set of these identified coefficients. In the full non-proportionality case (UA2), comparisons between experimental and theoretical results in this complex cyclic path are plotted in Fig. 5. In fact, the experimental measurements of the isotropic hardening evolution during such types of loading are not possible, so that comparison between the experimental and theoretical evolutions of R becomes impossible. It is limited, for this reason, by the comparison of the maximum equivalent stress evolution (Figs 5 and 6) as a response to the

1200

~" 900

600

< 300

0 I I I I I

0 0.01

l 0 Exp. ' New Model I

. . . . . . . Classical Model ! . ° ° . - - - J ° . ° •

0

0

O °

0.002 0.004 0.006 0.008

Axial strain

Fig. 4. Plots showing the predictions of the classical and new proposed model with corresponding experimental responses for the UA Waspaloy state of a half first cycle in tension-compression loading (Chaboche's model: Ct = 13,660, C2 = 118,535, a~ = 103.8, a2 =489, Q-20,585, b = 145.5 as in Abdul-Latif et al. [1994]).


corresponding experimental results. It is concluded that this correlation describes sufficiently well the observed experimental results, especially the softening phenomenon after the steady state.

III.2. Vali&Ttion

As for the validation, simulation tests are achieved by using the proposed model for characterizing the cyclic hardening behavior especially in the uniaxial tests. Therefore, three uniaxial tests (tension-compression) were conducted under plastic strain-controlled condition with amplitudes: Aepl 1 = 1%, 1.5% and 2.2%. Figure 7 represents the experimental hardening and maximum equivalent stresses evolution during this cycling in comparison with the predicted solutions for these employed loadings. It is clear that this proposed model is able to characterize well such evolutions in the first cycles as well as the softening phenomenon after the steady state (Fig. 7).

~-~ 1400

~ 1300

~ 1200

~ 1100

~ looo

N 9oo

Theo.- WU2

] • Exp.- WU2

I I I i 0.2 0.4 0.6 0.8


Fig. 5. Comparison between experimental and correlated responses of the maximum equivalent stress for UA state under non-proportional loading 90 ° out-of-phase (UA2).

~" 1500

Y

1300

~, 1100

o 900

700 i I I , M

I 0 0.25 0.5 0.75 1 1.25 I

/ Plastic strain modulus (p)

Fig. 6. Correlation of the correlation with experimental results for UA Waspaloy state under UA7 test.

982 A. Abdul-Latif

(b)

(a)

1100

500

400

300 e~

X 200

100 ¸

0

~ -100 •

-200

ooeo°e°e°°°°°°°°o°°oooeooeo [ '~ . . . . . . . . . . . . . _ . . o . . . . .

Oo I

• Exp.-2.2%

O Exp.-1.5%

A E, xp.-l%

(R)Theo.

(X)Theo.- 1% . . . . (X)Theo.-2.2% . . . . . . . (X)l'heo.-l.5%

I I Number of cycles

I 10 20 30 40

{ 5".0

~. 1050

1000

950 / O ,O

. . . . Theo.-I%

900 { ~ O Exp.- 1%

7 - - T h e o . - l . 5 %

850 T A Exp.- 1.5%

! 800 4 - { { {

0 10 20 30

Number of cycles

(c) i ~ : 5 ~

i < o

40

1200 '

~ 6~

.~ 200

o o

Axial Stral,

~ O Exp.- 2.2% I - Theo.- 2.2°/,

40 80 120 160

Number of cycles

i .............................. 1-500 ............................................

[ o E pl

Fig. 7. Comparison between the experimental and predicted evolutions for UA Waspaloy under tension-compression loading with different axial plastic strains-controlled (Aep,, = 1%, 1.5% and 2.2%) for: (a) isotropic (R) and kinematic (X) hardenings, (b) max imum equivalent stress, (c) hysteresis loop number 30 for Aep,, = 2.2%.


1500

1200

8 900

. , = q

o

~ 3oo

Theo.-butterfly

O Exp.-butterfly [ Theo.-cross

[] Exp.-cross

0 I I I 0 0.3 0.6 0.9 1.2


Fig. 8. Plots showing comparisons between the experimental responses of the maximum equivalent stress and corresponding predictions for UA Waspaloy state during different complex cyclic Ioadings: butterfly and cross tests.

Other comparisons (prediction-experimental results) are performed using complex straining paths (butterfly and cross tests (Fig. 1)). In spite of an exceptional behavior of this alloy under these tests which needs further comment (See Abdul-Latif et al. [1994]), the developed model tries to describe quantitatively the cyclic behavior of UA Waspaloy as shown in Fig. 8. The difference between the theoretical and experimental results (Fig. 8) is based on the fact that the maximum value of the extra-hardening comes theoretically from 90 ° out-of-phase where .4 = 1, while experimentally (for this alloy) the extra-hardening is more important in the case of butterfly than 90 ° out-of-phase loading. So, the mathematical definition of the parameter ,4 needs future development.

IV. CONCLUSION

In the uniaxial loading conditions, it has been demonstrated that the proposed model is capable of reasonable prediction of the cyclic hardening behavior of the UA Waspaloy, i.e. the isotropic hardening evolution during the first cycles and the softening phenomenon recorded after the steady-state behavior.

The manner of coupling between these two phenomena concerning the hardening components and the non-proportionality by the parameter A seems to be reasonable but it requires justification in future work. Due to the lack of an experimental direct method to measure the isotropic hardening evolution during this type of loading, one cannot per- form a direct comparison (prediction-experimental results). However, according to the different comparisons between the predictions and experimental observations, it can be concluded that the presented model describes sufficiently well the actual mechanical behavior of UA state under such conditions, especially the softening phenomenon.

984 A. Abdul-Latif

From a physical point of view, up to now the actual mechanism of the SPSP has not been determined, neither under proportional nor under non-proportional loading. Therefore, it will be necessary to understand the actual mechanism (or mechanisms) of the precipitate shearing process in the course of cycling by performing a fundamental physical study. This will undoubtedly be coupled with the dislocation motion mechanisms.

Acknowledgements--The author is grateful to Dr V. Ferney for supplying all the experimental results. Dr K. Saanouni is thanked for his useful discussion.

REFERENCES

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1985 Lemaitre, J., and Chaboche, J.L., Mrcanique des Matbriaux Solides. Dunod, Bordas, Paris. 1985a McDowell, D.L., "A Two Surface Model for Transient Nonproportional Cyclic Plasticity: Part l - -

Development of Appropriate Equations," ASME J. Appl. Mech., 52, 298. 1985b McDowell, D.L., "A Two Surface Model for Transient Nonproportional Cyclic Plasticity: Part 2

Comparison of Theory with Experiments," ASME J. Appl. Mech. 52, 303. 1986 Nishino, S., Hamada, N., Sakane, M., Ohnami, M., Matsumura, N. and Tokizane, M., "Micro-

structural Study of Cyclic Strain Hardening Behavior in Biaxial Stress States at Elevated Temperature", Fatigue and Fracture of Engng. Mater. Struct., 9, 65.

1987 Benallal, A. and Marquis, D., "Constitutive Equations for Nonproportional Cyclic Elasto-Viscoplasti- city," ASME J. Engng. Mater. Technol., 109, 326.

1987 Krempl, E., "Models of Viscoplasticity - - Some Comments on Equilibrium (Back) Stress and Drag Stress," Acta Mech., 69, 25.

1987 Krempl, E. and Yao, D., "The Viscoplasticity Theory Based on Overstress Applied to Ratchetting and Cyclic Hardening," in Rie, K.T. (ed). Low Cycle Fatigue and Elasto-Plastic Behavior of Materials. Elsevier, London, pp. 137-148.

1987 McDowell, D.L., "An Evaluation of Recent Developments in Hardening and Flow Rules for Rate- Independent Nonproportional Cyclic Plasticity," ASME J. Appl. Mech., 54 323.

1988 McDowell, D.L., Stahl, D.R., Stock, S.R. and Antolovitch, S.D., "Biaxial Path Dependence of Defor- mation Substructure of Type 304 Stainless Steel," Metall. Trans., 19A, 1977.

1989 Benallal, A., "Thermoviscoplasticit6 et Endommagement des Structures," Thrse de Doctorat d'Etat, Universit6 Pierre et Marie Curie, Paris 6.

1989 Chaboche, J.L., "Constitutive Equations for Cyclic Plasticity and Cyclic Viscoplasticity." Int. J. Plasti- city, 5, 247.

1989 Clavel, M., Pilvin, P. and Rahouadj, R., "Analyse Microstructurale de la D~formation Plastique Sous Sollicitations Non Proportionnelles dans un Alliage Base Nickel," C. R. Acad. Sci. Paris, 309, Srrie II, 689.

1989 Doong, S. H. and Socie, D. F., "Deformation Mechanisms of Metals Under Complex Nonproportional Cyclic Loading," Proc., Third Int. Conf. on Biaxial/Multiaxial Fatigue, 3-6 April, Stuggart, Germany.

1989 Doquet, V., "Comportement et endommagement de deux aciers fi structure cubique centrre et cubique faces centr~es, en fatigue oligocyclique, sous chargement multiaxial non proportionnel," Thrse de Doc- torat, Ecole Nationale Suprrieure des Mines de Paris.

1989 Moosbrugger, J.C. and McDowell, D.L., "On a Class of Kinematic Hardening Rules for Nonpropor- tional Cyclic Plasticity," ASME J. Engng. Mater. Technol., 111, 87.

1990 Moosbrugger, J.C. and McDowell, D.L., "A Rate-Dependent Bounding Surface Model with a Gen- eralized Image Point for Nonproportional Cyclic Plasticity,," J. Mech. Phys. Solids, 38, 627.

1990 Ohno, N., "Recent Topics in Constitutive Modeling of Cyclic Plasticity and Viscoplasticity," Appl. Mech. Rev., 43, 283.


1991 Doong, S.H. and Socie, D.F., "Constitutive Modeling of Metals Under Nonproportional Cyclic Load- ing," ASME J. Engng. Mater. Technol., 113, 23.

1991 Ferney, V., Hautefeuille, L. and Clavel, M., "Influence de la Microstructure sur l'Ecrouissage Cyclique d'Alliages fi Durcissement Structural en Sollicitations Multiaxiales: Partie II--Comportement de l'Alli- age Waspaloy," M~moires et Etudes Scientifiques Revue de M6tallurgie, 10, 679.

1991 Moosbrugger, J.C., "Some Developments in the Characterization of Material Hardening and Rate Sensitivity for Cyclic Viscoplasticity Models," Int. J. Plasticity, 7, 405.

1993 Moosbrugger, J.C., "Experimental Parameter Estimation for Nonproportional Cyclic Viscoplasticity: Nonlinear Kinematic Hardening Rules for Two Waspaloy Microstructures at 650°C, ' ' Int. J. Plasticity, 9, 345.

1994 Abdu]-Latif, A., Clavel, M., Ferney, V. and Saanouni, K., "On the Modeling of Nonproportional Cyclic Plasticity of Waspaloy," ASME J. Engng. Mater. Technol., 116, 35.

1994 Ferney, V., "Etude de l'Ecrouissage Cyclique sous Sollicitations Complexes," Th6se de Doctorat, Universit6 de Technologie de Compi6gne.

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Constitutive equations for cyclic plasticity of Waspaloy