On the numerical algorithm for isotropic kinematic hardening with the Armstrong Frederick evolution of the back stress

Transcription

1 Comut. Methods Al. Mech. Engrg. 191 (2002) On the numerical algorithm for isotroic kinematic hardening with the Armstrong Frederick evolution of the back stress Vlado A. Lubarda *, David J. Benson Deartment of Mechanical and Aerosace Engineering, University of California, 9500 Gilman Drive, San Diego, La Jolla, CA , USA Received 20 Setember 2001 Abstract The algorithmic consistency conditions are derived which yield the lastic loading index for the combined isotroic kinematic hardening lasticity with the Armstrong Frederick evolution equation for the back stress. Rate-indeendent elastic lastic, and rate-deendent viscoelastic lastic and elastic viscolastic material models are all encomassed by the analysis. The directions of the lastic and viscolastic increments of strain are determined from a develoed numerical algorithm, in conjunction with the radial return method. Numerical evaluations are resented to comare the redictions of various models under cyclic loading in simle shear. The effects of different material arameters on stress resonse are discussed. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Algorithmic consistency; Radial return; Viscolasticity; Anisotroic hardening 1. Introduction Numerical algorithms for treating the governing equations of lastic resonse have been well develoed for various rate-indeendent and rate-deendent constitutive models. Particular attention was given to both continuum henomenological models and hysical models based on the single crystal lasticity. The recent books by Simo and Hughes [1], and Belytschko et al. [2] can be consulted for the review and assessment of the roosed methods and their origin. The urose of this aer is to elaborate on the algorithmic consistency conditions and numerical algorithms for the combined isotroic kinematic hardening lasticity with the Armstrong Frederick evolution equation for the back stress. The rate-indeendent elastic lastic, and the rate-deendent viscoelastic lastic, and elastic viscolastic constitutive models are encomassed by the analysis. The directions of the lastic and viscolastic increments of strain are determined from a develoed algorithm using the method of radial return and a numerical recie based * Corresonding author. Tel.: ; fax: address: vlubarda@ucsd.edu (V.A. Lubarda) /02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S (02)

2 3584 V.A. Lubarda, D.J. Benson / Comut. Methods Al. Mech. Engrg. 191 (2002) on the generalized midoint rule. The obtained results extend the analysis of Simo and Hughes [1], which is restricted to combined isotroic kinematic hardening with the Prager s tye evolution equation for the back stress, and of Zienkiewicz and Cormeau [3] and Hornberger and Stamm [4]. Numerical evaluations are resented to comare the redictions of various models under cyclic forward and backward loading in simle shear. The effects of hardening and viscous arameters on stress resonse are then discussed. 2. Analytical background In the model of combined isotroic kinematic hardening J 2 lasticity the yield condition at an arbitrary state of deformation is secified by 1 ffiffiffi ks ak ¼Kð#Þ: ð1þ 2 The norm kk acts on a second-order tensor A by the square root trace oeration kak ¼ðA : AÞ 1=2 : ð2þ The deviatoric art of the Cauchy stress r in Eq. (1) is S ¼ r ðtrrþi=3. The rectangular comonents of the second-order unit tensor I are the Kronecker deltas d ij. The center of the yield circle in the deviatoric lane trr ¼ 0 is the back stress a. This is a traceless tensor, whose counterart in the lane trr ¼ constant is b ¼ a þ 1trr 3 I: ð3þ The current radius of the yield circle is K, which is assumed to deend on the equivalent lastic strain # ¼ ffiffiffi Z t 2 kde k: ð4þ 0 The integral is taken along the deformation ath from the initial to current instant of time on some quasistatic scale t. The lastic art of the strain increment de is denoted by de. The rate-indeendent materials are considered which comly with the normality rule, so that de is codirectional with the outward normal to the current yield surface in stress sace, i.e., de ¼ dc S a ; dc > 0: ð5þ ks ak The loading index dc is related to the increment of equivalent lastic strain by d# ¼ ffiffi 2 dc; ð6þ since kde k¼dc. According to the Armstrong Frederick [5] model, the evolution equation for the back stress a is da ¼ c 1 de c 2 akde k: ð7þ The arameters c 1 and c 2 can be either constants or functions of #. Ifc 2 ¼ 0, the Prager [6,7] hardening model is obtained in which the instantaneous translation of the yield surface is in the direction of the lastic increment of strain de, thus in the direction of the stress difference S a. The addition of the second term (the so-called recall term) in Eq. (7), roortional to a, modifies the direction of this translation and gives rise to hardening moduli for the reversed lastic loading that are in better agreement with exerimental data. Indeed, from the consistency condition the loading index can be exressed as dc ¼ 1 ðs aþ : ds ; ð8þ H ks ak

3 where H is the hardening modulus given by H ¼ 2 dk d# þ c ðs aþ : a 1 c 2 ffiffi : ð9þ 2 K This clearly deends not only on # and Kð#Þ, but also exlicitly on the current state of stress S and a. For examle, in uniaxial tension r along the direction 1 we have r ij ¼ rd i1 d j1 ; b ij ¼ b 1 d i1 d j1 þ b 2 ðd i2 d j2 þ d i3 d j3 Þ; ð10þ and S ij ¼ rd i1 d j1 1d 3 ij ; ð11þ a ij ¼ ad i1 d j1 1aðd 2 i2d j2 þ d i3 d j3 Þ¼ 3ad 2 i1d j1 1d 3 ij ; ð12þ with the connections b 1 ¼ a þ 1r; 3 b 2 ¼ 1a þ 1 r: 2 3 ð13þ The stress deendent term in Eq. (9) therefore becomes ðs aþ : a ¼ r 3a a ¼ ffiffiffi 3 2 Ka: ð14þ The minus sign corresonds to lastic loading in the reversed direction (at the stress level r ¼ 3a=2 ffiffi 3 K). In the numerical evaluations it will be assumed that the radius of the yield surface, initially equal to K 0, saturates to the value K 1 at large # (saturation kinematic hardening), such that Kð#Þ ¼K 0 þðk 1 K 0 Þ 1 ex b 0# : ð15þ K 1 K 0 The arameter b 0 is an aroriate constant whose hysical interretation is available from an easily verified roerty dk b 0 ¼ lim #!0 d# : ð16þ The total increment of strain is the sum of elastic and lastic arts. For infinitesimal elastic comonent of deformation, and for an isotroic material, we write de ¼ 1 2l ds þ de : ð17þ The deviatoric art of the strain increment de is denoted by de, and l is the elastic shear modulus. Thus, ds ¼ 2lde 2lde : ð18þ Here, we identify ds e ¼ 2lde V.A. Lubarda, D.J. Benson / Comut. Methods Al. Mech. Engrg. 191 (2002) as the elastic art of the deviatoric stress increment ds, while ð19þ ds ¼ 2l de ¼ 2ldc S a ks ak ð20þ reresents its lastic art. The lastic loading condition is ds : de < 0; i:e:; ðs aþ : de > 0: ð21þ

4 3586 V.A. Lubarda, D.J. Benson / Comut. Methods Al. Mech. Engrg. 191 (2002) Viscoelastic lastic resonse A simle viscoelastic lastic model of material resonse is obtained by adding a viscous art of the strain increment to its elastic and lastic contributions [8]. If viscous resonse is governed by the Newton s viscosity law, we can write de ¼ 1 Sdt þ 1 S a ds þ dc g e 2l ks ak ; ð22þ where dt now reresents the actual time increment during which ds and de take lace. The viscosity coefficient is denoted by g e. The consistency condition for the continuing lastic deformation gives the lastic loading index dc according to Eq. (8), and the hardening modulus H according to Eq. (9). It can be readily shown that the stress increment ds is exressed in terms of the increments de and Sdt by " 1 ds ¼ 2l J 1 þ H=2l ðs aþðs aþ ks ak 2 # : de 1 Sdt : ð23þ g e The outer tensor roduct is denoted by, and J stands for the symmetric fourth-order unit tensor with the rectangular comonents ðd ik d jl þ d il d jk Þ=2. The lastic loading condition is ðs aþ : de 1 Sdt > 0: ð24þ g e The material time arameter s e will be conveniently introduced as s e ¼ g e 2l ; such that dt ¼ 1 dt : g e 2l s e ð25þ ð26þ 2.2. Elastic viscolastic resonse In an elastic viscolastic material model the strain increment is the sum of elastic and viscolastic arts, such that de ¼ 1 ds þ dev ð27þ 2l Using a viscolasticity model based on the concet of overstress [9 12], with anisotroic hardening accounted for via the back stress a, we may write de v ¼ dt D ks ak ffiffiffi E S a 2 Kð#Þ g ks ak : ð28þ The viscosity arameter is g, and the Macauley brackets hi are used, such that hwi ¼ðwþjwjÞ=2, for any scalar w. Consequently, the stress increment is ds ¼ 2l de dt s D ks ak ffiffiffi E S a 2 Kð#Þ ks ak : ð29þ

5 V.A. Lubarda, D.J. Benson / Comut. Methods Al. Mech. Engrg. 191 (2002) The material time arameter s is introduced as s ¼ g 2l : ð30þ More involved constitutive equations of viscolastic resonse could also be used, such as those considered by Chaboche [13,14], Bammann [15], McDowell [16], M uller-hoee and Wriggers [17], Freed and Walker [18], and Lubarda and Benson [19]. 3. Algorithmic consistency condition for rate-indeendent resonse Fig. 1 shows a trace of the yield surface in the deviatoric lane at the two nearby instants t n and t nþ1. The center of the first yield surface is a n, and its radius is Kð# n Þ. The index n designates the number of deformation increments through which the current state has been reached. The center of the subsequent yield surface, after an increment of strain de nþ1 has been alied along the lastic loading branch, is denoted by a nþ1, and its radius by Kð# nþ1 Þ. We have # nþ1 ¼ # n þ d# nþ1 ; ð31þ and a nþ1 ¼ a n þ da nþ1 : ð32þ The radial return method with the backward Eulerian scheme will be emloyed. The udated stress state on the new yield surface is defined by S nþ1 ¼ S nþ1 2lde nþ1 : ð33þ The elastic redictor state of stress is S nþ1 ¼ S n þ ds e ¼ S n þ 2lde; while the remaining term on the right-hand side of Eq. (33) is the lastic corrector term [1,20], ð34þ ds nþ1 ¼ 2lde nþ1 ¼ 2ldc nþ1 k k : ð35þ Fig. 1. The yield surface with the center at the back stress a n, and the subsequent yield surface with the center at a nþ1 ¼ a n þ da nþ1. The elastic redictor state of stress S nþ1 is brought to the stress state S nþ1 on the yield surface by the lastic corrector 2lde nþ1.

6 3588 V.A. Lubarda, D.J. Benson / Comut. Methods Al. Mech. Engrg. 191 (2002) Consequently, and S nþ1 ¼ S n þ 2lde 2l dc nþ1 k k ð36þ ¼ S n a n þ 2lde 2l dc nþ1 k k da nþ1: ð37þ The increment of the back stress will be comuted from the Armstrong Frederick evolution equation, and the algorithmic recie da nþ1 ¼ c 1 de nþ1 c 2kde nþ1 k½ha n þð1 hþa nþ1 Š; 0 6 h 6 1: ð38þ The generalized midoint rule is used for the recall term (the value h ¼ 0 and 1 corresonding to the forward and backward Euler rules, resectively, and h ¼ 1=2 to the midoint rule). Substitution of Eq. (5) into Eq. (38) gives da nþ1 ¼ c 1 dc nþ1 k k c 2 dc nþ1 ½ha n þð1 hþa nþ1 Š; ð39þ having in mind that kde nþ1 k¼dc nþ1. Eq. (39) can be rewritten as da nþ1 ¼ a nþ1 dc nþ1 c 2 a n ; ð40þ k k c 1 where c 1 a nþ1 ¼ : ð41þ 1 þ c 2 ð1 hþdc nþ1 When Eq. (40) is inserted into Eq. (37), we obtain þ ð2l þa nþ1 Þdc nþ1 k k ¼ B n; ð42þ with B n ¼ S n a n þ 2l de þ b nþ1 dc nþ1 a n ð43þ and b nþ1 ¼ c 2 c 2 a nþ1 ¼ : ð44þ c 1 1 þ c 2 ð1 hþdc nþ1 To derive an equation for the loading arameter dc nþ1 we take a trace roduct of Eq. (42) with itself, which gives ks nþ1 a nþ1 kþð2l þ a nþ1 Þdc nþ1 2 ¼kS n a n k 2 þk2lde þ b nþ1 dc nþ1 a n k 2 þ 2ðS n a n Þ : ð2lde þ b nþ1 dc nþ1 a n Þ: ð45þ In view of Eq. (1), alied at the instants t n and t nþ1, Eq. (45) is rewritten as ffiffiffi 2 Kð#n þ ffiffi h 2 dcnþ1 Þþð2lþa nþ1 Þdc nþ1 ¼ 2K 2 ð# n Þ þk2ldeþb nþ1 dc nþ1 a n k 2 þ 2ðS n a n Þ : ð2l de þ b nþ1 dc nþ1 a n Þi 1=2: ð46þ

7 V.A. Lubarda, D.J. Benson / Comut. Methods Al. Mech. Engrg. 191 (2002) This is the key equation of the numerical method. It reresents an algorithmic consistency condition for the considered hardening model with the Armstrong Frederick evolution of the back stress. Being in the form of a nonlinear equation for the loading arameter dc nþ1, its solution is sought by numerical means. If the Newton s iterative method is emloyed, the function F ¼ F ðdc nþ1 Þ is defined by F ¼ ffiffi 2 Kð#n þ ffiffiffi 2 dcnþ1 Þþð2lþa nþ1 Þdc nþ1 h 1=2: 2K 2 ð# n Þ þk2ldeþb nþ1 dc nþ1 a n k 2 þ 2ðS n a n Þ : ð2lde þ b nþ1 dc nþ1 a n Þi ð47þ whose gradient with resect to dc nþ1 is F 0 ¼ 2 dk d# ð# n þ ffiffiffi 2 dcnþ1 Þþa 0 nþ1 dc nþ1 þ 2l þ a nþ1 h 2K 2 ð# n Þ þk2ldeþb nþ1 dc nþ1 a n k 2 þ 2ðS n a n Þ : ð2lde þ b nþ1 dc nþ1 a n Þ ðb 0 nþ1 dc nþ1 þ b nþ1 ÞðS n a n þ 2lde þ b nþ1 dc nþ1 a n Þ : a n : i 1=2 ð48þ The iterative rocedure to find the zero value of F is then based on the recie dc iþ1 nþ1 ¼ dci nþ1 F ðdci nþ1 Þ F 0 ðdc i nþ1 Þ : ð49þ The iterations are ended when a desired accuracy has been reached, i.e., when jf ðdc iþ1 nþ1þj falls to within a rescribed error tolerance. The convergence is guaranteed because F is a convex function of dc nþ1. In the iterative rocess we use # i nþ1 ¼ # n þ ffiffi 2 dc i ð50þ nþ1 : It is assumed in Eq. (38) that c 1 and c 2 are either constants or deendent on # n only. If the generalization is made so that c i ¼ w i c i ð# n Þþð1 w i Þc i ð# nþ1 Þði ¼ 1; 2Þ, only the rates a 0 nþ1 and b0 nþ1 are affected. If c 2 ¼ 0, the numerical algorithm corresonding to Prager s evolution of the back stress is obtained. In this case, Eq. (46) simlifies to ffiffi 2 Kð#n þ ffiffi 2 dcnþ1 Þþð2lþc 1 Þdc nþ1 ¼kS nþ1 a nk; in accord with the result from Simo and Hughes [1]. It remains to secify the direction of de nþ1, i.e., to determine the normalized tensor ð51þ k k : To that goal, we rewrite Eq. (42) as ½k kþð2lþa nþ1 Þdc nþ1 Š k k ¼ B n: By taking a norm there follows: k kþð2lþa nþ1 Þdc nþ1 ¼kB n k: Consequently, Eq. (54) gives k k ¼ B n kb n k : ð52þ ð53þ ð54þ ð55þ

8 3590 V.A. Lubarda, D.J. Benson / Comut. Methods Al. Mech. Engrg. 191 (2002) This is a desired result, because, with the reviously determined loading arameter dc nþ1, the tensor B n is comletely secified. The back stress can now be udated by using Eq. (40). 4. Algorithmic consistency conditions for rate-deendent resonse 4.1. Viscoelastic lastic resonse The udated stress state on the new yield surface is given according to the backward Eulerian scheme by S nþ1 ¼ S nþ1 2l de nþ1 : ð56þ The viscoelastic redictor state of stress is from Eq. (22) S nþ1 ¼ S n þ 2lde ½uS n þð1 uþs nþ1 Š dt ; ð57þ s e where the generalized midoint rule with 0 6 u 6 1 is used for the viscous contribution, and s e ¼ g e =2l. The remaining term on the right-hand side of Eq. (56) is the lastic corrector term ds nþ1 ¼ 2lde nþ1 ¼ 2ldc nþ1 k k : ð58þ Introducing the arameter q ¼ 1 þð1 uþ dt 1 ; ð59þ s e it is readily found that dt S nþ1 ¼ qs n þ q 2lde us n 2lqdc s nþ1 ð60þ e k k and dt ¼ qðs n a n Þþq 2lde us n 2lqdc s nþ1 e k k þðq 1Þa n da nþ1 : ð61þ If the increment of back stress is comuted from the Armstrong Frederick evolution equation, and the algorithmic recie of Eqs. (38) and (40), there follows: þð2lq þ a nþ1 Þdc nþ1 k k ¼ B n; ð62þ where dt B n ¼ qðs n a n Þþq 2lde us n þ A n ð63þ s e and A n ¼ðb nþ1 dc nþ1 þ q 1Þa n : ð64þ The equation for the loading arameter dc nþ1 can be deduced by taking a trace roduct of Eq. (62) with itself. This gives

9 ks nþ1 V.A. Lubarda, D.J. Benson / Comut. Methods Al. Mech. Engrg. 191 (2002) a nþ1 kþð2lq þ a nþ1 Þdc nþ1 ¼ q 2 ks n a n k 2 þ dt 2 q 2l de us n þ A n s e dt þ 2qðS n a n Þ : q 2l de us n s e þ A n : ð65þ When k k¼ ffiffiffi 2 Kð#n þ ffiffiffi 2 dcnþ1 Þ; ks n a n k¼ ffiffi 2 Kð#n Þ; ð66þ are substituted into Eq. (65) we obtain the algorithmic consistency condition in the form of a nonlinear equation for the lastic loading index dc nþ1. The equation can be solved by the Newton s iterative method along the same lines as described in the revious section. If g e!1, i.e., dt=s e! 0, then q ¼ 1, and Eq. (65) reduces to Eq. (45). It can be readily verified that k k ¼ B n kb n k ; which secifies the direction of the lastic increment of strain de nþ1, i.e., B n de nþ1 ¼ dc nþ1 kb n k : ð67þ ð68þ The udated stress state follows from Eq. (60) Elastic viscolastic resonse The radial return method with the backward Eulerian scheme will again be emloyed. The udated stress state is defined by S nþ1 ¼ S nþ1 2ldev nþ1 : ð69þ The elastic redictor state of stress is (see Fig. 2) S nþ1 ¼ S n þ 2lde; ð70þ while the remaining term on the right-hand side of Eq. (69) is the viscolastic corrector term. The viscolastic art of the strain increment is defined by Eq. (28) as de v nþ1 ¼ dc nþ1 k k ; ð71þ where 2ldc nþ1 ¼ dt h k k ffiffi 2 Kð#nþ1 Þi ; ð72þ s rovided that the term within the square brackets is ositive. In the case of the Armstrong Frederick evolution of the back stress, we can write from Eq. (45) k k¼ ð2lþa nþ1 Þdc nþ1 h 1=2: þ ks n a n k 2 þk2lde þ b nþ1 dc nþ1 a n k 2 þ 2ðS n a n Þ : ð2lde þ b nþ1 dc nþ1 a n Þi ð73þ

10 3592 V.A. Lubarda, D.J. Benson / Comut. Methods Al. Mech. Engrg. 191 (2002) Fig. 2. The static yield surface with the center at the back stress a n. The static state of stress S 0 n is on the yield surface, while the corresonding dynamic state of stress S n is outside the yield surface. The subsequent static yield surface has the center at a nþ1 ¼ a n þ da nþ1, with the static and dynamic states of stress S 0 nþ1 and S nþ1. The elastic redictor state of stress S nþ1 is brought to the stress state S nþ1 by the viscolastic corrector 2lde v nþ1. As in Section 3, the arameters are introduced c 1 a nþ1 ¼ ; ð74þ 1 þ c 2 ð1 hþdc nþ1 b nþ1 ¼ c 2 c 2 a nþ1 ¼ : ð75þ c 1 1 þ c 2 ð1 hþdc nþ1 The substitution of Eq. (73) into Eq. (72) yields a nonlinear equation for the viscolastic loading index dc nþ1. This is 2l 1 þ a nþ1 2l þ s dc dt nþ1 þ ffiffi 2 Kð#n þ ffiffi 2 dcnþ1 Þ h 1=2: ¼ ks n a n k 2 þk2lde þ b nþ1 dc nþ1 a n k 2 þ 2ðS n a n Þ : ð2lde þ b nþ1 dc nþ1 a n Þi ð76þ With a determined dc nþ1, the direction of the viscolastic increment of strain de v nþ1 k k ¼ B n kb n k ; is secified by where B n ¼ S n a n þ 2l de þ b nþ1 dc nþ1 a n : ð78þ The viscolastic model corresonding to the Prager s evolution of the back stress is obtained by setting c 2 ¼ 0 and a nþ1 ¼ c 1. The resulting equation for dc nþ1 is 2l 1 þ c 1 2l þ s dc dt nþ1 þ ffiffiffi 2 Kð#n þ ffiffi 2 dcnþ1 Þ¼kS nþ1 a nk: ð79þ In the limit s =dt! 0, we recover the equations for the loading index of the rate-indeendent lasticity from Section 3. In articular, Eq. (76) reduces to Eq. (45), while Eq. (79) reduces to Eq. (51). ð77þ

11 V.A. Lubarda, D.J. Benson / Comut. Methods Al. Mech. Engrg. 191 (2002) Numerical results and conclusions To illustrate the alication of the derived formulas and comare the redictions of various constitutive models, we consider a strain history shown in Fig. 3. The shear strain of amount is first alied at constant strain-rate of 100 s 1, followed by a segment of constant strain during s, and the reversal of strain to zero value at the strain-rate of negative 100 s 1. The material is subsequently subjected to an oosite ulse of shear strain. The calculated time history of the corresonding shear stress in the case of viscoelastic lastic model is shown in Fig. 4. The solid curve corresonds to Prager hardening, and the dashed curve to Armstrong Frederick hardening. The utilized material arameters were: l ¼ 80 GPa, K 0 ¼ 100 MPa, K 1 ¼ 200 MPa, b 0 ¼ 2 GPa, and g e ¼ 4 MPa s. In the case of Prager hardening the constant c 1 ¼ 10 GPa, while for the Armstrong Frederick hardening we took c 1 ¼ 10 GPa and c 2 ¼ 100 MPa. The stress relaxations in Fig. 4 occur during the time intervals of constant strain. Fig. 5 shows the stress strain curves obtained in the two cases. The difference in the hardening rates between the two models is clear. The hardening rates can be further adjusted by varying the material arameters, as required by a articular exerimental data. For examle, the effect of the arameter c 2 is illustrated in Fig. 6, which shows that the increase of this arameter decreases the hardening rate and makes the stress increase more gradual. The stress time variation corresonding to shear strain inut from Fig. 3 in the case of elastic viscolastic model is shown in Fig. 7. The two curves again corresond to two different hardening models. Fig. 3. The time history of alied forward and backward cycles of shear strain. Fig. 4. The time history of the shear stress corresonding to shear strain cycles from Fig. 3, in the case of a viscoelastic lastic material model. The darker curve is for the Armstrong Frederick hardening, and the lighter curve for the Prager kinematic hardening.

12 3594 V.A. Lubarda, D.J. Benson / Comut. Methods Al. Mech. Engrg. 191 (2002) Fig. 5. The shear stress shear strain curves for a viscoelastic lastic material model corresonding to time variations of strain and stress deicted in Figs. 3 and 4. Fig. 6. The effect of the Armstrong Frederick arameter c 2 on the stress resonse of a viscoelastic lastic material. The darker curve is for c 2 ¼ 50 MPa, and the lighter curve for c 2 ¼ 100 MPa. Fig. 7. The time history of the shear stress corresonding to shear strain cycles from Fig. 3, in the case of an elastic viscolastic material model. The darker curve is for the Perzyna Armstrong Frederick model, and the lighter curve for the Perzyna Prager model. The utilized material arameters were as in the revious case, excet that the viscosity arameter was g ¼ 2 MPa s. Fig. 8 shows the corresonding stress strain curve and the hysteretic loos obtained in the two cases. The effect of the Armstrong Frederick arameter c 2 on the stress resonse is illustrated in Fig. 9.

13 V.A. Lubarda, D.J. Benson / Comut. Methods Al. Mech. Engrg. 191 (2002) Fig. 8. The shear stress shear strain curves for an elastic viscolastic material model corresonding to time variations of strain and stress deicted in Figs. 3 and 7. Fig. 9. The effect of the Armstrong Frederick arameter c 2 on the stress resonse of an elastic viscolastic material. The darker curve is for c 2 ¼ 50 MPa, and the lighter curve for c 2 ¼ 100 MPa. Fig. 10. The effect of the viscosity arameter g on the stress resonse of an elastic viscolastic material. The darker curve is for g ¼ 4 MPa s, and the lighter curve for g ¼ 2 MPa s. More gradual change of stress is achieved by higher values of this arameter. Fig. 10 shows the effect of the viscosity arameter g. The resonse in the straining under constant rate of strain is stiffer, and the stress relaxation under constant strain less raid for higher values of g.

14 3596 V.A. Lubarda, D.J. Benson / Comut. Methods Al. Mech. Engrg. 191 (2002) Acknowledgements Research funding rovided by the Los Alamos National Laboratories is kindly acknowledged. References [1] J.C. Simo, T.J.R. Hughes, Comutational Inelasticity, Sringer, New York, [2] T. Belytschko, W.K. Liu, B. Moran, Nonlinear Finite Elements for Continua and Structures, Wiley, New York, [3] O.C. Zienkiewicz, I.C. Cormeau, Visco-lasticity lasticity and cree in elastic solids a unified numerical solution aroach, Int. J. Numer. Meth. Engrg. 8 (1974) [4] K. Hornberger, H. Stamm, An imlicit integration algorithm with a rojection method for viscolastic constitutive equations, Int. J. Numer. Meth. Engrg. 28 (1989) [5] P.J. Armstrong, C.O. Frederick, A mathematical reresentation of the multiaxial Bauschinger effect, GEGB Re. RD/B/N 731, [6] W. Prager, The theory of lasticity: a survey of recent achievements (James Clayton Lecture), Proc. Inst. Mech. Engrg. 169 (1955) [7] W. Prager, A new method of analyzing stresses and strains in work-hardening lastic solids, J. Al. Mech. 23 (1956) [8] F.J. Zerilli, R.W. Armstrong, Thermal activation based constitutive equations for olymers, J. Phys. IV 10 (2000) [9] V.V. Sokolovskii, Proagation of elastic viscolastic waves in bars, Prikl. Mat. Mekh. 12 (1948) , in Russian. [10] L.E. Malvern, The roagation of longitudinal waves of lastic deformation in a bar of material exhibiting a strain-rate effect, J. Al. Mech. 18 (1951) [11] P. Perzyna, The constitutive equations for rate sensitive lastic materials, Q. Al. Math. 20 (1963) [12] P. Perzyna, Fundamental roblems in viscolasticity, Adv. Al. Mech. 9 (1966) [13] J.-L. Chaboche, Constitutive equations for cyclic lasticity and cyclic viscolasticity, Int. J. Plasticity 5 (1989) [14] J.-L. Chaboche, Unified cyclic viscolastic constitutive equations develoment, caabilities, and thermodynamic framework, in: A.S. Krausz, K. Krausz (Eds.), Unified Constitutive Laws for Plastic Deformation, Academic Press, San Diego, 1996, [15] D.J. Bammann, Modeling temerature and strain rate deendent large deformations of metals, Al. Mech. Rev. 43 (5/2) (1990) S312 S319. [16] D.L. McDowell, A nonlinear kinematic hardening theory for cyclic thermolasticity and thermoviscolasticity, Int. J. Plasticity 8 (1992) [17] N. M uller-hoee, P. Wriggers, Numerical integration of viscolastic material laws at large strains with alication to imact roblems, Comut. Struct. 45 (1992) [18] A.D. Freed, K.P. Walker, Viscolasticity with cree and lasticity bounds, Int. J. Plasticity 9 (1993) [19] V.A. Lubarda, D.J. Benson, On the evolution equation for the rest stress in rate-deendent lasticity, Int. J. Plasticity 18 (2002) [20] R.D. Krieg, D.B. Krieg, Accuracies of numerical solution methods for the elastic-erfectly lastic model, J. Pressure Vessel Techn. 99 (1977)