A consistent finite elastoplasticity theory combining ...

International Journal of Plasticity 16 (2000) 143±177

A consistent ®nite elastoplasticity theory combining additive and multiplicative decomposition of the stretching and the deformation gradient H. Xiao, O.T. Bruhns*, A. Meyers Institute of Mechanics, Ruhr University Bochum, D-44780 Bochum, Germany Received in final revised form 16 August 1999 Dedicated to Professor Franz G. Kollmann on the occasion of his 65th birthday

Abstract A phenomenological ®nite deformation elastoplasticity theory is proposed by consistently combining the additive decomposition of the stretching D and the multiplicative decomposition of the deformation gradient F. The proposed theory is Eulerian type and suitable for both isotropic and anisotropic elastoplastic materials with general isotropic and kinematical hardening behaviour. Within the context of the proposed theory, the Eulerian rate type constitutive formulation based on the decomposition D ˆ D e ‡ D ep determines the total stress, the total kinematical quantities as well as the elastic part D e and the coupled elastic±plastic part D ep etc. Then, the two rate quantities D e and D ep are related to the elastic part F e and the plastic part F p in the decomposition F ˆ F e F p in a direct and natural manner. It is found that the just-mentioned relationship between the two widely used decompositions, together with a suitable elastic p relation de®ning the elastic stretch Ve ˆ F e F e T , consistently and uniquely determines the elastic deformation F e and the plastic deformation F p and all their related kinematical quantities, without recourse to the widely used ad hoc assumption about a special form of F e . Moreover, it is shown that for each process of purely elastic deformation the incorporated Eulerian rate type formulation intended for elastic response, which is based on the newly discovered logarithmic rate, is exactly-integrable to deliver a general hyperelastic relation with any given type of initial material symmetry, and thus the suggested theory is subjected to no self-inconsistency diculty in the rate form characterization of elastic response, as encountered by other existing Eulerian rate type theories. In particular, it is proved that, to achieve the just-mentioned goal, the logarithmic rate is the only choice among all possible (in®nitely many) objective corotational rates. Further, * Corresponding author. Tel.: +49-234-32-23080; fax:+49-234-32-14229 E-mail address: [email protected] (O.T. Bruhns). 0749-6419/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S0749-6419(99)00045-5

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the proposed theory is shown to ful®ll, in a full sense, the invariance requirement under the change of frame or the superposed rigid body motion. Accordingly, with the suggested theory the main fundamental discrepancies involving the decompositions of D and F disappear. # 2000 Elsevier Science Ltd. All rights reserved. Keywords: Finite elastoplasticity; Additive and multiplicative decomposition; Logarithmic rate; Uniqueness

1. Introduction A general theory of elastoplastic materials in the presence of ®nite strain and large rotation is usually formulated by decomposing the total deformation or the total deformation rate into an elastic and a plastic part and then establishing a separate constitutive relation for each part. Within this context, the decomposition composed of an elastic and a plastic part, among other things, is fundamental. Several decompositions have been used and advocated by various researchers, which have led to several schools of ®nite deformation elastoplasticity. Green and Naghdi (1965, 1966) postulated a formal additivity of the total right Cauchy-Green strain, in which the plastic strain is regarded as a primitive variable with certain speci®ed properties. Lee and Liu (1967) and Lee (1969) adopted and developed the multiplicative decomposition of the total deformation gradient. Nemat-Nasser (1979, 1982) proposed that the total stretching should be additive. Although the above-mentioned decompositions are motivated and justi®ed by their respective reasonable physical considerations, the several existing elastoplastic formulations based on them are somewhat controversial. A number of fundamental issues between them have been debated rigorously and extensively in the literature (see e.g. Green and Naghdi, 1971; Nemat-Nasser, 1979, 1982; Casey and Naghdi, 1980, 1981, 1983, 1992; Lee, 1981; Lubarda and Lee, 1981). Details can be found in a recent comprehensive critical review by Naghdi (1990). Certain main relevant issues are indicated below. First, we mention the main issue concerning the multiplicative decomposition F ˆ F e F p . It is noted that an arbitrary rigid body rotation superposed on a related intermediate con®guration has no e€ect on this decomposition, which renders F e and F p determinable only to within an arbitrary rotation. To eliminate this indeterminacy, one has to introduce an additional ad hoc special assumption that F e can be chosen to be a symmetric positive de®nite tensor, i.e. F e ˆ Ve is a pure stretch, with the rotational part of F e completely ignored. This widely used assumption has been shown to be inconsistent with the invariance requirement under the change of frame or under the superposed rigid body rotation in a general sense, see Green and Naghdi (1971), Casey and Naghdi (1980, 1981, 1983, 1992)1 and Naghdi (1990). Next, the other issues are related to the decomposition D ˆ D e ‡ D p . It is known that there is no natural, direct relationship between the pairs (D e ; D p ) and (F e ; F p ) 1

We note that Casey and Naghdi (1992) also have introduced a multiplicative decomposition without making the restrictive assumption that the intermediate con®guration is stress-free.

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according to the multiplicative decomposition of F with the aforementioned special assumption about F e , and hence it is thought that the foregoing decomposition of D could hold only for certain restrictive cases of deformations, such as small elastic strain etc., see Lee (1969, 1981). Moreover, to ful®ll the objectivity requirement, each rate type model based on the foregoing additive decomposition does not involve the usual material time derivative but an objective rate, such as the wellknown Zaremba-Jaumann rate and polar rate etc. Various researchers use di€erent objective rates and they do not all agree on the same rate. More essentially, a rate type model with an objective rate, if special care is not taken, may predict aberrant, spurious phenomena such as the so-called shear oscillation etc. (see e.g. Lehmann, 1972a,b; Dienes, 1979; and Nagtegaal and de Jong, 1982; see also Khan and Huang, 1995, for detail.) It seems that a model in this case is questionable and can not be regarded to be fully reliable and reasonable. Indeed, Simo and Pister's work (1984) reveals that several widely used Eulerian rate type equations intended for characterizing elastic response turn out to be self-inconsistent in the sense that none of them is exactly integrable to really deliver an elastic relation. With some recent development in kinematics of ®nite deformations and in rate type models by these authors (Bruhns et al., 1999; Xiao, et al., 1996, 1997a,b, 1998a, b,c, 1999) and other researchers (see Lehmann, Guo and Liang, 1991; Reinhardt and Dubey, 1995, 1996), in this article we attempt to develop a new ®nite deformation elastoplasticity theory by consistently combining the additive and the multiplicative decompositions of the stretching and the deformation gradient. The theory that will be proposed is Eulerian type and suitable for both isotropic and anisotropic elastoplastic materials with general isotropic and kinematical hardening behaviour. It will be shown that with the suggested theory the aforementioned main fundamental discrepancies concerning the two widely used decompositions disappear. This article is organized as follows. In Section 2, for later use we outline some relevant basic facts and results for kinematics of ®nite deformations of continua. In particular, we shall establish and demonstrate the pure kinematical fact (Lemma A in Section 2): Let t pertain to a time interval I. Then a deformation gradient F  …t† over I can be determined uniquely by its related left stretch and stretching V  …t† and D  …t† given over I, together with an initial value F  …0†. This result will prove to be essential to the subsequent development. In Section 3, postulating the additive decomposition D ˆ D e ‡ D ep and adopting the newly discovered logarithmic rate (see Bruhns et al., 1999; Xiao et al., 1996, 1997a,b, 1998a, b,c, 1999; Lehmann et al., 1991; Reinhardt and Dubey, 1995, 1996), we propose a self-consistent Eulerian rate type constitutive formulation of isotropic and anisotropic elastoplastic materials with general isotropic and kinematical hardening behaviour. In Section 4, in a direct and natural manner we relate the two rate quantities D e and D ep to the elastic deformation F e and the plastic deformation F p in the multiplicative decomposition A in Section 2) and a F ˆ F e F p . Utilizing the foregoing kinematical fact (Lemma p suitable elastic relation de®ning the elastic stretch V e ˆ F e F e T , we show that the just-mentioned relationship between the two widely used decompositions consistently and uniquely determines the elastic deformation F e and the plastic deformation F p , as well as all their related kinematical quantities. In Section 5, we prove

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the uniqueness of the logarithmic rate in the sense of achieving an exactly-integrable rate type formulation of elastic response. In Section 6, we derive simpli®ed results for the particular yet important case of small elastic strain elastoplasticity with a quadratic yield function of von Mises type and an associated ¯ow rule. Finally, we summarize the main features of the suggested theory in Section 7. At the end of this introduction, we list some notations for later use. Let A; B and H be, respectively, two second order tensors and a fourth order tensor. Throughout, we shall use the notations A : B; AB, H: A and A:H to designate, respectively, the scalar and the second order tensors de®ned by A : B ˆ Aij Bij ; …AB†ij ˆ Aik Bkj ; …H : A†ij ˆ Hijkl Akl ; …A : H†ij ˆ Akl Hklij : Moreover, let Q be an orthogonal tensor. Then we shall use Q ? A and Q ? H to represent the second order and fourth order tensors given by …Q ? A†ij ˆ Qik Qjl Akl ;

i:e:

Q ? A ˆ QAQT ;

…Q ? H†ijkl ˆ Qip Qjq Qkr Qls Hpqrs : Throughout, the symbol AT is used to represent the transpose of the second order tensor A. For orthogonal tensors Q and R, the following identities hold. Q ? …H : A† ˆ …Q ? H† : …Q ? A†; Q ? …R ? A† ˆ …QR† ? A: …Q ? A† : …Q ? B† ˆ A : B: 2. Preliminaries in kinematics and a lemma Consider a body B with particles and identify each particle with its position vector X in a ®xed reference con®guration C0 in a 3-dimensional Euclidean space R3 . We assume that the body B has an initial natural stress-free state and always identify C0 with the latter. Let x be the position vector in R3 occupied by a typical particle X in

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the current con®guration C of B. A motion of the body B is a mapping  de®ned by x ˆ …X; t†, where t belongs to a time interval ‰0; aŠ with t=0 corresponding to the initial state, i.e. the reference state. The deformation gradient and its determinant are: F ˆ Grad ˆ

@ ; @X

detF > 0:

…1†

The particle velocity and the velocity gradient are denoted, respectively, by : v ˆ x;

L ˆ gradv ˆ

: @v ˆ FF ÿ1 : @x

…2†

Throughout, the superposed dot designates the material time derivative with respect to the time t holding X ®xed, and Aÿ1 represents the inverse of a second order tensor A. 2.1. Some basic facts According to the well-known polar decomposition theorem, the deformation gradient F or, equivalently, each second order tensor F with Eq. (1)2 has the unique left and right multiplicative decompositions: F ˆ VR ˆ RU;

RT ˆ Rÿ1 ;

detR ˆ 1;

…3†

where V and U are known as the left and right stretch tensors and R as the rotation tensor. Both V and U are positive de®nite symmetric tensors and determined uniquely by B ˆ V2 ˆ FFT ;

C ˆ U2 ˆ F T F;

…4†

where B and C are usually called the left and right Cauchy±Green tensors. Moreover, the rotation R is also uniquely determined by F. Direct formulas for the stretches and the rotation in terms of F are available in, e.g. Hoger and Carlson (1984) and Ting (1985). On the other hand, the velocity gradient L has the unique additive decomposition ( L ˆ D ‡ W; 1 1 …5† D ˆ …L ‡ LT †; W ˆ …L ÿ LT †; 2 2 where D and W are known as the stretching and the vorticity tensor, respectively. Under the change of frame speci®ed by the transformation of motion2 2 In general, the time variable in the transformed frame with + may have a translation relative to that in the original frame. This consideration is irrelevant to the subsequent development and hence is not pursued here.

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‡ …X; t† ˆ x0 …t† ‡ Q…X; t†;

^ …t†; QˆQ

…6†

the material particle X moves to the place x‡ ˆ ‡ …X; t† in the current con®guration C+ at the time t. Note that in the original frame without +, C0 serves as both an initial and a reference con®guration, while in the transformed frame with + the reference con®guration is C0 and the initial con®guration is C+ 0 , which is obtained from C0 through Eq. (6) at the initial instant t=0. Generally, C0 is di€erent from C+ 0 except for the case when x0 …0† ˆ 0 and Qjtˆo ˆ I. Here and henceforth, I is used to denote the second order identity tensor. For all quantities associated with the transformed frame, here and henceforth we use the same symbols as those for the original frame but with an attached plus ``+'' sign. Accordingly, we have the transformation formulas:  ‡ R‡ ˆ QR; F ˆ QF; …7† V‡ ˆ Q ? V; U‡ ˆ U; 

: L‡ ˆ Q ? L ‡ Q Q T ; : W‡ ˆ Q ? W ‡ QQT : D ‡ ˆ Q ? D;

…8†

2.2. Logarithmic strain and its work-conjugate stress According to Hill (1978) (see also Ogden, 1984), a general class of Eulerian and Lagrangean strain measures may be de®ned through one single scale function. Their forms can be given by (see Xiao et al. 1998b) E ˆ g…C† ˆ

m X g…w †C ;

…9†

ˆ1

e ˆ g…B† ˆ

m X g…w †B ;

…10†

ˆ1

where the scale function g… † is a smooth increasing function with the normalized property g(1)=g0 (1)ÿ12=0. Here and henceforth, w ;  ˆ 1;


; m, are used to denote the distinct eigenvalues of the Cauchy±Green tensors C and B, and C and B the corresponding subordinate eigenprojections of C and B. In particular, the natural logarithmic scale function g…w†=12lnw yields Hencky's logarithmic strain measures: Hˆ

m 1 1X ln C ˆ …lnw †C ; 2 2 ˆ1

…11†

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hˆ

m 1 1X …lnw †B : ln B ˆ 2 2 ˆ1

149

…12†

The following relations hold h ˆ R ? H;

H ˆ RT ? h:

…13†

Following Hill (1978) (see also Ogden, 1984), the symmetric second order tensor S determined through the work-conjugacy relation : S:Eˆ :D is called the work-conjugate stress measure of the Lagrangean strain measure E. In the above,  is used to designate the Kirchho€ stress, i.e.  ˆ …det F†; where  is the Cauchy stress. In particular, we denote the work-conjugate stress of the Lagrangean logarithmic strain measure H by P , and the Eulerian counterpart of P by , i.e.  ˆ R ? P;

P ˆ RT ? :

…14†

An explicit expression of P in terms of the right stretch tensor U and the rotated Cauchy stress RT ?  was ®rst derived by Hoger (1987). An alternative expression of P in terms of C and RT ?  can be derived by setting g…w† ˆ 12 lnw in the formula (5.8) given in (Xiao et al., 1998c). From this fact and Eq. (14)1 we obtain ˆ

m q X w  ÿ w ÿ1 wÿ1 B  B :  w lnw ÿ lnw ;ˆ1

…15†

2.3. Polar rate and logarithmic rate Let A be a time-di€erentiable objective symmetric second order Eulerian tensor and X a time-dependent skewsymmetric second order tensor. The latter, called a spin, can determine a proper orthogonal tensor Q to within a constant proper orthogonal tensor through the linear tensorial di€erential equation : Q ˆ ÿQ X  : Then we have : : : :  …Q ? A† ˆ Q AQT ‡ Q AQ T ‡ Q AQT ˆ Q ? A ;

…16†

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where the symmetric second order tensor :  A ˆ A ‡ AX ÿ X A is said to be a corotational rate of the Eulerian tensor A de®ned by the spin tensor X . It represents the rate-of-change of the Eulerian tensor A observed in a rotating frame with the spin X . A corotational rate of an objective Eulerian tensor need not be objective. The objectivity of a corotational rate depends on its de®ning spin. There are in®nitely many objective corotational rates. The well-known ZarembaJaumann rate de®ned by the spin X ˆ W, i.e. the vorticity tensor, provides an example of objective corotational rates. A general discussion of objective corotational rates and their de®ning spins can be found in Xiao et al., (1998a,b). For our purpose, the polar rate (see, e.g. Dienes, 1979, 1986) and the newly discovered logarithmic rate (see Lehmann et al., 1991; Reinhardt and Dubey, 1995, 1996; Xiao et al., 1996, 1997a,b, 1998a,b, 1999; Bruhns et al., 1999) will be of particular interest. They are de®ned by, respectively, the polar spin (see, e.g. Dienes, 1979, 1986; and Guo, 1984, for detail) p m X : 1 ÿ w =w p B DB XR ˆ RRT ˆ W ‡ 1 ‡ w =w 6ˆr

…17†

and the logarithmic spin (see Xiao et al., 1996, 1997b, 1998a,b) X

Log

ˆW‡

 m  X 1 ‡ …w =w † 2 ‡ B DB : 1 ÿ …w =w † ln…w =w † 6ˆ

…18†

Pm Here and henceforth, the notation 6ˆ means the summation for all ;  ˆ 1;


; m except  ˆ . When m ˆ 1, such a summation is assumed to vanish. The corotational rates de®ned by the polar spin XR and the logarithmic spin X Log are accordingly called the polar rate and the logarithmic rate and denoted by :  AR ˆ A ‡ AXR ÿ XR A; :  ALog ˆ A ‡ AXLog ÿ XLog A: We call the proper orthogonal tensor RLog determined by the linear tensorial differential equation : RLog ˆ ÿRLog X Log ;

RLog jtˆ0 ˆ I

the logarithmic rotation. The following formula will be useful :  …RLog ? A† ˆ RLog ? A Log :

…19†

…20†

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In particular, by using the latter and D ˆ hLog (see, e.g. Xiao et al., 1997b) we have :  …RLog ? h† ˆ RLog ? h Log ˆ RLog ? D:

…21†

2.4. A lemma It may readily be understood that a time-di€erentiable deformation gradient F given over a time interval I uniquely determines all its related kinematical quantities over I. In many cases, however, the inverse may be concerned with: Some of : the kinematical quantities related to the deformation gradient F and its rate F are known, while the deformation gradient F itself is left to be determined. A relevant question of such kind is treated below, which will prove to be essential to the subsequent development. Lemma A. Let V  and D  be, respectively, a time-di€erentiable positive de®nite symmetric second order tensor and a symmetric second order tensor given over the time interval ‰0; aŠ. Then a non-symmetric second order tensor F  over ‰0; aŠ with det F  > 0 can be determined uniquely by V  and D  and an initial value of F  through the system of nonlinear tensor equations in F  : F  F  T ˆ V2 ;

: : F F ÿ1 ‡ …F F  ÿ1 †T ˆ 2D ;

…22†

if and only if the following consistency condition is ful®lled: m X ˆ1

V  D  V  ˆ

m X :    lÿ1  V V V ;

…23†

ˆ1

where l and V  ;  ˆ 1;


; m, are the distinct eigenvalues of V  and the corresponding subordinate eigenprojections of V  , respectively. Under the above condition, the unique solution F  of the foregoing system is given by F  ˆ V  R , where R is the proper orthogonal tensor determined by : R  ˆ X R  ;

 R  jtˆ0 ˆ Vÿ1 0 F0;

…24†

where V 0 ˆ V  jtˆ0 ; F 0 ˆ F  jtˆ0 and the spin X is given by 

X ˆ

m X 6ˆ

! : 2l l   ÿ1       V D V  ÿ …l ÿ l † V V V  : 2  l2  ÿ l

…25†

Proof. Applying the polar decomposition theorem and Eq. (22)1, we know that the second order tensor F  with det F  > 0 is of the form F  ˆ V  R  with a proper orthogonal tensor R  to be determined. Then, we have

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: : : F  ˆ V  R ‡ V  R  : Substituting the latter and F  ˆ V R into (22)2 and introducing the spin : X ˆ R  R T …ˆ ÿX T †; we derive the linear tensor equation for the spin X : : V  X V  ÿ1 ÿ V  ÿ1 X V ˆ 2D  ÿ V  V  ÿ1 ÿ V  ÿ1 V  By left- and right-multiplying the above equation by V  , we obtain : B  X ÿ X B  ˆ 2V D  V  ÿ B  ; : : : where B  ˆ V 2 and hence B  ˆ V  V  ‡ V  V . Then, utilizing a related result given in Xiao (1995) [see the expression following Eq. (23) therein], we infer that the above linear tensor equation for X has a solution if and only if the condition m X : V …2V  D  V  ÿ B  †V  ˆ O; ˆ1

or, equivalently, V ÿ1

! m X : V  …2V  D  V  ÿ B  †V  V ÿ1 ˆ O: ˆ1

Then, utilizing the equalities V  V  ˆ V  V  ˆ l V  ;

 V  V ÿ1 ˆ V ÿ1 V  ˆ lÿ1  V ;

one can easily derive the condition (23). Further, the spin X should be continuous with respect to B  . According to Guo et al. (1992), under a suitable consistency condition the latter tensor equation for the spin X has one and only one solution that is continuous with respect to B  . This unique solution may be derived by means of the eigenprojection method developed in (Xiao, 1995; Xiao et al., 1996, 1997a,b, 1998a, b,c, 1999; Bruhns et al., 1999) and is just given by Eq. (25). Finally, once the spin X is known, the rotation R  is uniquely determined by Eq. (24). Q.E.D. Lemma A indicates that a deformation gradient F  over a time interval I can uniquely be determined by its related left stretch and stretching given over I, together with its initial value. This pure kinematical result will play a crucial role in consistently combining the additive and the multiplicative decomposition of the total stretching and the total deformation gradient, as will be seen in Section 4.

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3. Self-consistent Eulerian rate type constitutive formulation We consider an elastoplastic solid with an initial stress-free natural state C0 and with initial material symmetry represented by an orthogonal subgroup G0. As mentioned earlier, the initial natural state is taken as the reference con®guration. Accordingly, we have the initial conditions Fjtˆ0 ˆ I;

 jtˆ0 ˆ O:

…26†

As commonly done, we assume the additive decomposition of the total stretching D, i.e. D ˆ D e ‡ D ep

…27†

for each process of elastic-plastic deformation. We call D e the elastic part of D and D ep the coupled elastic-plastic part of D. When the state of microstructures responsible for inelastic deformation, such as microcracks and dislocations etc., is held ®xed in the course of deformation, the deformation, by de®nition, is purely elastic and in this case the elastic part D e is just the total stretching D. As will be seen, in a natural and direct manner the elastic part D e may be interpreted as the deformation rate related to the elastic deformation F e in the decomposition F ˆ F e F p , while D ep is associated with both the elastic and the plastic deformation F e and F p .3 Based on the logarithmic strain measure and its work-conjugate stress measure as well as the logarithmic rate, in what follows we propose the constitutive formulations of the elastic part D e and the coupled elastic-plastic part D ep separately. Small and ®nite deformation elastoplasticity theories with the additive decomposition of the strain increment or the stretching D and their applications can be found in, e.g. the comprehensive and informative review articles by Drucker (1988), Neale (1981) and Nemat-Nasser (1983, 1992). 3.1. The complementary hyperelastic potential We assume that the purely elastic response of the elastoplastic body in question is hyperelastic. Then, in terms of the work-conjugate stress measure P of the Lagrangean logarithmic strain measure H there is a complementary hyperelastic potential  ˆ ~…P† that is invariant [see Eq. (A1) in Appendix] under the initial material symmetry group G0, such that the logarithmic strain H is derivable from this potential with respect to the work-conjugate stress measure P of H (see, e.g. Hill, 1978), i.e. Hˆ 3

1 @~ ln C ˆ 2 @P

That is why we here prefer the notation D ep rather than D p .

…28†

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for each process of purely elastic deformation. Applying Lemma B given in the appendix, the potential  may be formulated equivalently in terms of the Eulerian counterpart of P [see Eq. (A2) in Appendix] as follows:  ˆ ^…†  ~…RT ? †;

…29†

where the potential function ^ …† in terms of the Eulerian stress measure  ˆ R ? P is invariant [see Eq. (A3) in Appendix] under the R-rotated material symmetry group R ? G0 ˆ fR ? Q0 j Q0 2 G0 g:

…30†

From Lemma B, we obtain the equivalent Eulerian formulation of the Lagrangean formulation (28) as follows: hˆ

1 @^ ln B ˆ ; 2 @

…31†

It should be noted that the potential function ~…P† depends on the Lagrangean stress measure P and the initial material symmetry axes that characterize the structure of the group G0, while the potential function ^…† relies on the Eulerian stress measure  and the R-rotated material symmetry axes.4 This can be clearly seen by observing the fact: if the vectors fa10 ; a20 ;


g, which represent the initial material symmetry axes, characterize the structure of the group G0, then fRa10 ; Ra20


g characterize the structure of the R-rotated group R?G0. If a representation for ~ …P† [resp. ^…†] is available, then a representation for ^…† [resp. ~…P†] can be obtained merely by replacing P (resp. ) and the material symmetry axes fa10 ; a20 ;


g (resp. fRa10 ; Ra20 ;


g) with their Eulerian (resp. Lagrangean) counterparts. For instance, let G0 be the transverse isotropy group with a preferred axis represented by the unit vector a0 . Then we have ~ …P† ˆ …trP ; trP 2 ; trP 3 ; a0
P a0 ; a0
P 2 a0 †; ^ …† ˆ …tr; tr2 ; tr3 ; a
a; a
2 a†;

a ˆ Ra0 :

Generally, we have (29) and the relationship [see Eq. (A4) in Appendix] @^ @~ ˆR? : @ @P

4

…32†

We emphasize that here the material symmetry axes are ®xed with respect to the continuum. Thus the underlying anisotropy is not a general one.

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3.2. Exactly-integrable Eulerian rate type formulation of hyperelastic response Now we are concerned with the constitutive formulation of the elastic part D e . We assume 

D ˆ …@^=@†Log : e

…33†

It will be seen in the next section that the gradient @^=@ can be interpreted as the logarithm of an elastic stretch tensor. Accordingly, the above formulation simply means that the logarithmic rate of an elastic logarithmic strain measure furnishes the elastic part of the total deformation rate, i.e., the total stretching D. This relation, or equivalently, the rate Eq. (33), is motivated by and consistently based on the rigorous kinematical fact (see Xiao et al., 1996, 1997a,b, 1998a,b; Lehmann et al., 1991; Reinhardt and Dubey, 1995, 1996): the logarithmic rate of the Eulerian logarithmic strain measure h is identical with the the stretching D. In fact, the foregoing relation is reduced to the latter fact for each process of purely elastic deformation. Since a constitutive formulation of D e is intended for characterizing elastic response, for each process of purely elastic deformation it must be exactly-integrable to really yield an elastic, in particular, a hyperelastic relation between an elastic strain measure and a stress measure. If the just-stated self-consistency condition or requirement is not ful®lled, the resulted elastoplasticity theory will be self-inconsistent in the sense of characterizing elastic response. In this case, a theory may be questionable and some spurious behaviour may be predicted, such as shear oscillations etc., refer to the relevant remark in Section 1 for detail. It will be shown in Section 5 that, to ful®ll the foregoing self-consistency requirement, the Eulerian rate type formulation suggested is the unique choice among all formulations of its kind with all possible (in®nitely many) objective corotational rates to be chosen. Moreover, it will be seen in Section 5 that the suggested rate type formulation is exactlyintegrable to indeed yield a general hyperelastic relation with any given type of initial material symmetry. As mentioned before, the gradient @^=@ depends on the R-rotated material symmetry axes. Since the logarithmic rate is the rate-of-change observed in a rotating frame with the logarithmic spin :X Log and the latter is not coincident with a rotating frame with the polar spin RRT , for an initially anisotropic material the right-hand side of Eq. (33) must incorporate the contribution of the rates of the foregoing R-rotated material symmetry axes. This fact is consistent with the fact that has been correctly indicated by Brannon (1998) by examining an extreme case. Another form of Eq. (33) can be derived. In a rotating frame with the polar spin XR , each R-rotated material symmetry axis keeps unchanged. From this fact we deduce the equality @2 ^  : R : …@^=@†R ˆ @@ 

…34†

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Then, by using the latter we recast Eq. (33) in the form ! ! ^ @2 ^ @^  R e LR LR @ : ‡ X ÿX D ˆ @@ @ @

…35†

with XLR ˆ XLog ÿ XR ˆ

 m  p X 2 w =w 2 ‡ B DB : 1 ÿ …w =w † ln…w =w † 6ˆ

…36†

It is pointed out in passing that the rate equation (33) can not be recast in the form 

D e ˆ …@^=w † Log ˆ

@2 ^  : Log : @@

The reason lies in the fact that the chain rule 

…†Log ˆ

@  : Log @

cannot hold true for a symmetric second order tensor-valued function …† that is invariant under a R-rotated anisotropy group. However, the above chain rule is indeed true for the case of initial isotropy (see Xiao et al., 1999). A simpli®ed form of rate type formulation for isotropic elastic response is obtainable, refer to Xiao, et al. (1997a) and Bruhns et al. (1999) for detail. 3.3. Yield function, ¯ow potential and ¯ow rules We assume that the current yield surface in the stress space is de®ned by ~; k† ˆ f^…; ; k† ˆ 0; f ˆ f~…P; 

~ ˆ RT ? : 

…37†

Here, k is a scalar internal variable characterizing the isotropic hardening behaviour and , called the back stress, an Eulerian symmetric second order tensor internal variable characterizing the kinematic hardening behaviour5. Moreover, the yield functions f~ and f^ are, respectively, invariant [see Eqs. (A1) and (A3) in Appendix] under the initial material symmetry group G0 and the R-rotated material symmetry group R ? G0 . We further assume that in the stress space there exists another surface

5 Generally, more internal variables may be introduced to characterize the complicated inelastic behaviour in a more realistic manner (see, e.g. Bruhns and Diehl, 1989, for detail), but this will not change the structure of the theory developed.

H. Xiao et al. / International Journal of Plasticity 16 (2000) 143±177

~; k† ˆ g^…; ; k†; g ˆ g~…P; 

157

…38†

where also the scalar functions g~ and g^ are, respectively, invariant under the initial material symmetry group G0 and the R-rotated material symmetry group R ? G0 , such that the gradient of this surface is in the direction of the coupled elastic-plastic part D ep , i.e. the ¯ow rule is of the form : @g^ : D ep ˆ @

…39†

Hence, g is called the ¯ow potential. The relationship between the Lagrangean type and the Eulerian type scalar functions, f~ and f^; g~ and g^, is speci®ed by Lemma B in the Appendix. For a process of continued plastic ¯ow, the stress point :must remain on the current yield surface and hence we have the consistency condition f ˆ 0 for plastic ¯ow, i.e. : @f~ @f~ : @f~ : :P ‡ : ~ ‡ k ˆ 0: @P @~ @k

…40†

Utilizing the last identity in Section 1, we recast the latter in the form …R ?

: : @f~ @f~ @f~ : † : …R ? P † ‡ …R ? † : …R ? ~ † ‡ k ˆ 0; @P @~ @k

and then applying Eqs. (14)1, (37)2, (A4) and the following identities for the polar rate: :  R ? ~ ˆ R ;

:  R ? P ˆ R ;

we convert the Lagrangean formulation (40) to the Eulerian formulation @f^  R @f^  R @f^ : : ‡ :  ‡ k ˆ 0: @ @ @k

…41†

Moreover, we assume that the evolution equations for the internal variables  and k have the general forms : k ˆ k : D ep ;

 Log ˆ H : D ep ; 

i.e. : @g^ : ; k ˆ k : @

…42†

@g^ :  :  Log ˆ H : @

…43†

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In the above, k ˆ k^…; ; k† and H ˆ H^ …; ; k† are Eulerian type symmetric second order and fourth order tensor-valued functions, respectively, invariant under the R-rotated material symmetry group R?G0. The latter has the minor index symmetry property. Namely,   k^…Q ? ; Q ? ; k† ˆ Q ? k^…; ; k† ; 8Q 2 R ? G0 ; 

Hijkl ˆ Hjikl ˆ Hijlk ; H^ …Q ? ; Q ? ; k† ˆ Q ? …H^ …; ; k††;

8Q 2 R ? G0 :

From the consistency condition (41) for the plastic ¯ow and Eqs. (42)±(43), we : derive an expression for the plastic multiplier as follows: ! 8 ^  ^ > @f @f : > > : R ‡ : …XLR  ÿ XLR † ; < ˆ ÿ @ @ …44† > ^ ^ > ^ ^ @f @g @f @g > : ˆ ‡ …k : †: :H: @ @k @ @ In the above, the loading-unloading factor is speci®ed by (see Bruhns et al., 1999) 8 > @f^  Log > < 0 if f < 0 or f ˆ 0 and : < 0; @ …45† ˆ ^ > > : 1 if f ˆ 0 and @f :  Log 50: @ Combining Eqs. (33) and (39), or (35) and (39), we obtain Dˆ

 : @g^ ; Log ‡ ^ @ …@=@†

…46†

or, equivalently, ! ! ^ @2 ^ @^ : @g^  R LR LR @ : ‡ : X ÿX ‡ Dˆ @@ @ @ @

…47†

Left- and right-multiplying Eqs. (43) and (46) by the logarithmic rotation RLog and its transpose, respectively, then applying Eqs. (20)±(21) and (26) and integrating the obtained expressions over the interval ‰0; tŠ, we derive the integral forms of the rate Eqs. (42), (43) and (46) as follows:

H. Xiao et al. / International Journal of Plasticity 16 (2000) 143±177

kˆ

…t

@g^ : ds;

k : @ 0

 ˆ …RLog †T ?

159

…48†

… t

 @g^ :

RLog ? …H : †ds ; @ 0

… t  1 @^ : Log @g^ Log T ‡ …R † ? ds :

R ? h ˆ …lnB† ˆ 2 @ @ 0

…49†

…50†

The above expressions clearly show the e€ects of the ®nite deformation and rotation history on the current strain and the hardening behaviour via the logarithmic rotation RLog . When the yield function f and the ¯ow potential g are identical, the plasticity is said to be governed by an associated ¯ow rule, otherwise the ¯ow rule is said to be non-associated. Simpli®ed results for some particular yet important cases will be derived in Section 6. 4. Kinematical quantities related to the decomposition F ˆ F e F p The constitutive formulation based on the additive decomposition (27), i.e. Eqs. (33), (39) and (42)±(46), together with Cauchy's equations of motion and well-posed initial and boundary conditions, determine the total stress, the total kinematical quantities F and L, as well as the elastic part D e and the coupled elastic-plastic part D ep , etc. On the other hand, for a process of elastoplastic deformation, it is required to de®ne and specify elastic and plastic deformations and their related kinematical quantities. It should be noted that if there is no a priori de®nition for elastic and plastic deformations, no de®nite information about the latter can be drawn from the rate quantities D e and D ep . On the contrary, D e and D ep need to be related to ``elastic deformation'' and ``plastic deformation'' in an appropriate sense. To introduce and separate elastic and plastic deformations, the physically-motivated multiplicative decomposition of the total deformation gradient is widely used, which was ®rst introduced by KroÈner (1960) with reference to linearized theory, subsequently utilized by Backman (1964) and Willis (1969), and systematically and extensively used and developed by Lee and other researchers, see, e.g. Lee and Liu (1967), Lee (1969), Lubarda and Lee (1981), Agah-Tehrani et al. (1987) and Dafalias (1987, 1988). According to this decomposition, the total deformation gradient F has the multiplicative decomposition F ˆ FeFp;

det F e > 0;

det F p > 0;

…51†

for any process of elastic-plastic deformation. Usually, F e and F p are called, respectively, the elastic and the plastic part of F. For each process of purely elastic deformation, F e is just F.

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At each instant t, the above decomposition is associated with a stress-free intermediate con®guration C , which is realized by elastic unloading from the current con®guration C or by another approach. Accordingly, the elastic part F e is related with a motion from the intermediate con®guration C to the current con®guration C, while the plastic part Fp from the initial con®guration C0 to the intermediate con®guration C . In accordance with the initial condition (26)1, it is evident that at the initial instant t=0 the three con®gurations C0, C and C coincide with each other.6 Hence, we have F e jtˆ0 ˆ F p jtˆ0 ˆ I:

…52†

From the decomposition (51) we derive the expressions : : L ˆ F e F eÿ1 ‡ F e F p F pÿ1 F eÿ1 ;

…53†

: : D ˆ sym…F e F eÿ1 † ‡ sym…F e F p F pÿ1 F eÿ1 †;

…54†

: : W ˆ skw…F e F eÿ1 † ‡ skw…F e F p F pÿ1 F eÿ1 †:

…55†

Here and henceforth we use the notations symA and skwA to designate the symmetric and the skewsymmetric part of a second order tensor A, i.e. 1 symA ˆ …A ‡ AT †; 2

1 skwA …A ÿ AT †: 2

Now we proceed to establish the relationship between the two decompositions (27) and (51). Towards this goal, let us compare Eqs. (27) and (54). Clearly, the ®rst term of the right-hand side of Eq. (54) relies on the elastic part F e only, while the second term depends on both the elastic and the plastic part F e and F p . Thus, a natural, direct relationship between the two decompositions (27) and (51) should be7 : D e ˆ sym…F e F eÿ1 †;

: D ep ˆ sym…F e F p F pÿ1 F eÿ1 †:

…56†

In the above two relations, the former implies the latter and vice versa. The righthand side of the latter explains why D ep has been termed the coupled elastic-plastic part of D before. 6

Generally, the initial intermediate con®guration may be chosen as one that has a rigid body rotation relative to the initial con®guration. 7 Such direct relationship could not be possible, if the elastic part F e is restricted to the special case when F e is a positive de®nite symmetric tensor, as usually done. In the just mentioned restricted special case, this relationship fails to obey the objectivity requirement, and hence other kind of relationship has to be introduced, see the aforementioned relevant literature.

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161

Consider the constitutive formulation (33) for the elastic part D e of the stretching D. As pointed out before, the rate equation (33) should be exactly integrable to produce an elastic relation. Since B e ˆ F e F eT

…57†

de®nes an elastic strain measure of Cauchy±Green type, generally we may assume that the foregoing elastic relation is of the form …B e † ˆ

@^ : @

For each process of purely elastic deformation, we have B e =B and the above relation should coincide with the hyperelastic relation (31), i.e. …B† ˆ h ˆ

1 ln B: 2

Therefore, it is straightforward to propose the elastic relation he ˆ

1 @^ ln B e ˆ : 2 @

…58†

Then, from the latter and Eq. (33) we derive the relationship8 De ˆ

1  Log : 2 …ln B e †

…59†

This indicates that the elastic part D e of D is just the logarithmic rate of the elastic logarithmic strain measure h e . Now, with Lemma A a crucial observation could be made: The above established relationship between the two widely used decompositions (27) and (51) can consistently and uniquely determine the elastic part F e and the plastic part F p in the decomposition (51), as well as all their related kinematical quantities, with no ad hoc assumption about restricted special forms of F e and/or F p . Indeed, from the outset of this section we know that the total stress, the total deformation gradient F and the total velocity gradient L and the two parts D e and D ep , etc., can be obtained by integrating the constitutive Eqs. (33), (39), (42)-(46) and Cauchy's equations of motion with well-posed initial p and boundary conditions. Then, Eq. (58) determines  the elastic stretch tensor Ve ˆ F e F eT . Utilizing Eqs. (59), (25) in Xiao et al. (1997b) and the equalities V e …ln V e † ˆ …ln V e †V e ˆ …ln le †V e ; 8 Note that through Eq. (56)1 D e is consistently and uniquely related with the purely elastic part F e although the logarithmic rate through Eq. (18) seems to be a function of the total deformation. However, with this respect we refer to Eq. (64) from which X Log can be expressed as function of purely elastic quantities.

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we deduce m m m : X X X : e e e V e D e Ve ˆ V e …ln Ve †V e ˆ leÿ1  V V V ; ˆ1

ˆ1

ˆ1

where le and V e ;  ˆ 1;


; m; are the distinct eigenvalues of V e and the corresponding subordinate eigenprojections of V e . The latter relation indicates that the positive de®nite symmetric tensor V e and the symmetric tensor D e obey the consistency condition (23). Thus, applying Lemma A and the initial condition (52) we infer that the elastic deformation F e ˆ V e Re over a time interval ‰0; aŠ is consistently and uniquely determined by V e and D e given over ‰0; aŠ, where the elastic stretch tensor V e is determined by Eq. (58), i.e. ! ^ @ ; …60† V e ˆ exp @ and the elastic rotation Re is obtained by integrating the linear tensorial di€erential equation : R e ˆ X e Re ;

Re jtˆ0 ˆ I;

…61†

with [see Eq. (25)] ! e e : 2l l Xe ˆ  e2  e2 Ve D e Ve ÿ …le ÿ le †ÿ1 Ve V e Ve : 6ˆ l ÿ l   m

…62†

A more convenient form for the spin Xe is available. In fact, by setting E ˆ Ve and E ˆ ln Ve in Eq. (3.5) given in (Xiao et al., 1998b; note that Be and Ve have the same eigenprojections) and using Eq. (59) we gain : le ÿ le e V …ln V e †V e e e  ln l ÿ ln l   ;ˆ1

m X : Ve ˆ

ˆ

m X

le ÿ le e Log e e ‡ XLog he †V e e e V  …D ÿ h X ln l ÿ ln l   ;ˆ1

ˆ XLog V e ÿ V e XLog ‡

m X

le ÿ le e e e e e V D V : ln l ÿ ln l   ;ˆ1

Hence, substituting the latter into Eq. (62) we obtain ! m X 2le le 1 Log e e e e ÿ ‡ X ˆX e e V D V : e2 e2 ÿ ln l ln l l ÿ l   6ˆ  

…63†

…64†

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163

Note that for a process of purely elastic deformation, the spin Xe becomes the polar spin Eq. (17) and accordingly the elastic stretch tensor V e and the elastic rotation R e become the total stretch tensor V and the total rotation R. Once the elastic deformation F e is available, one can immediately obtain the plastic deformation F p by F p ˆ F eÿ1 F:

…65†

Now we consider the rate quantities related to F e and Fp . Let : Le ˆ F e F eÿ1 ;

…66†

: Lp ˆ F p F pÿ1 :

…67†

Then, we have : : Le ˆ sym…F e F eÿ1 † ‡ skw…F e F eÿ1 † ˆ D e ‡ W e ;

…68†

where D e is given by Eqs. (33) or (35) and W e by : : W e ˆ skw…F e F eÿ1 † ˆ skw…V e V eÿ1 ‡ V e Xe V eÿ1 †;

…69†

: where V e , V e and Xe are, respectively, given by Eqs. (60), (63) and (64). Moreover, from Eqs. (53) and (66)±(68) we derive Lp ˆ F eÿ1 …L ÿ Le †F e ˆ F eÿ1 …L ÿ D e ÿ W e †F e :

…70†

Hence, we have ÿ
D p ˆ symLp ˆ sym F eÿ1 …L ÿ D e ÿ W e F e †;

…71†

ÿ
W p ˆ skwLp ˆ skw F eÿ1 …L ÿ D e ÿ W e F e †:

…72†

From the above analysis, we conclude that, within the context of the ®nite deformation elastoplasticity theory suggested in this and last sections, the elastic deformation F e and the plastic deformation F p and all their related kinematical quantities such as the spins W e and W p etc., can be consistently and uniquely determined. A similar fact concerning the latter two spins and others was ®rst observed and demonstrated by Nemat-Nasser (1990, 1992) in a di€erent context. Its implication in constitutive formulations can be found in the just-mentioned references. We conclude this section with a discussion concerning the objectivity requirement under the change of frame or the invariance requirement under the superposed rigid body motion in a full sense. Towards this goal, consider the change of frame speci®ed

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by Eq. (6). In a full sense,9 in the transformed frame with + we allow the initial intermediate con®guration at t ˆ 0, denoted by C + 0 , to be one that has any given 10 rotation Q relative to the initial con®guration C+ 0 . Accordingly, we have F e‡ tˆ0 ˆ Q tˆ0 Q T ;

F p‡ tˆ0 ˆ Q :

…73†

and hence the system (61) governing the rotation Re‡ should be changed to : R e‡ ˆ Xe‡ R e‡ ;

R e‡ tˆ0 ˆ Q tˆ0 Q T :

…74†

From Lemma B given in the appendix we know that the gradient @^=@ is objective and hence its logarithmic rate is also. From these facts and Eqs. (33) and (60) we derive D e‡ ˆ Q ? D e ;

V e‡ ˆ Q ? V e :

…75†

Moreover, from Eqs. (7)3, (8)2, (8)3 and (18) we deduce : XLog‡ ˆ Q ? XLog ‡ QQ T :

…76†

Then, from Eqs. (62) and (75)±(76) we infer : Xe‡ ˆ Q ? Xe ‡ QQ T :

…77†

Comparing Eqs. (74) and (77) with (61), we arrive at R e‡ ˆ QR e Q T :

…78†

Thus, Eqs. (75)2 and (78) produce F e‡ ˆ V e‡ Re‡ ˆ QF e Q T ;

…79†

and then the latter and Eq. (65) yield F p‡ ˆ …F e‡ †ÿ1 F ‡ ˆ Q F p :

…80†

Eqs. (79) and (80) are just the transformation formulas concerning the multiplicative decomposition (51), derived by Green and Naghdi (1971), Casey and Naghdi (1980, 1981) and Naghdi (1990). This indicates that the suggested theory obeys, in a full sense, the invariance requirement stated in the just-mentioned references.

9

See the preceding sixth footnote. Note that the initial con®guration C0 is transformed to C+ 0 by Qjtˆ0 , while the reference con®guration in the transformed frame remains C0. 10

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165

5. Uniqueness of the logarithmic rate in rate type formulation of elastic response In the Eulerian rate type Eq. (33) for elastic response, a particular objective corotational rate, i.e. the logarithmic rate, is adopted and a particular form of tensor function, i.e. the gradient @^ =@ of a scalar potential ^…† with respect to the stress measure , is used. There are, however, in®nitely many objective corotational rates to be chosen. This fact suggests that any other objective corotational rate may be used to replace the logarithmic rate in Eq. (33) and other relevant places. This nonuniqueness, if any, may result in a puzzling situation about whether there is an objective corotational rate that is preferable to any other one or, conversely, every objective corotational rate can well serve for our purpose. Moreover, it may be ^ …†. possible to replace the gradient @^=@ by a general form of tensor function U Accordingly, in general we are concerned with the Eulerian rate type equation 

D ˆ U^ …† ; e

…81†

where the right-hand side is the corotational rate de®ned by a spin X , i.e. :  ^ …†X ÿ X U ^ …†: ^ …† ‡ U U^ …† ˆ U The main objective of this section is to demonstrate that among all Eulerian rate type formulations of the form (81), Eq. (33) is the only choice in the self-consistency sense that for each process of purely elastic deformation the adopted rate equation must be exactly integrable to really deliver a general hyperelastic relation with any given type of initial material symmetry. Speci®cally, we shall prove the following result. ^ …† be a di€erentiable symTheorem A. Let X be any given spin tensor and let U ^ metric second order tensor-valued function with U …O† ˆ O. Then, for each process of purely elastic deformation the rate equation (81) is exactly integrable to yield a hyper^ …† is the gradient of a scalar potential elastic relation, if and only if X ˆ XLog and U ^…† with respect to , i.e., if and only if the rate equation (81) is identical with the rate equation (33). Moreover, for each process of purely elastic deformation the integration of the rate equation (33) exactly yields the general hyperelastic relation (31). Proof: First, we verify the suciency. For each process of purely elastic deformation, we have D e ˆ D. Eq. (33) becomes 

D ˆ …@^=@†Log : It is shown in Xiao et al., (1996, 1997b, 1998b) that the stretching D can be written as the logarithmic rate of Hencky's logarithmic strain measure h. Then, left- and right-multiplying the foregoing equation by the logarithmic rotation RLog and its transpose, respectively, and then utilizing the equality (21) we deduce

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…R

:

Log

: ? h† ˆ R

Log

! @^ : ? @

Integrating the above equality over the interval ‰0; tŠ and using the initial condition (26) and …@^ =@† jtˆ0 ˆ O, we derive RLog ? h ˆ RLog ?

@^ : @

Thus, the general hyperelastic relation (31) is recovered. Next, we verify the necessity. We suppose that for each process of purely elastic deformation, the rate equation (81) is exactly-integrable to yield an elastic relation, i.e.  ˆ  …F†; and hence we have ^ …† ˆ U

…F†

…82†

^ ‰  …F†]. We ®rst prove that under the above condition the corotawith …F†  U tional rate in Eq. (81) must be the logarithmic rate. In fact, since De is objective, the right-hand side of Eq. (81) must be objective. Hence, the spin X must de®ne an objective corotational rate. According to Xiao et al. (1998a,b), a general form of spin tensors de®ning objective corotational rates is given by11 X ˆ W ‡

m X h…w ; w ; trB†B DB ;

…83†

6ˆ

where h…x; y; z† de®nes the spin X and is called the spin function. As has been shown in Xiao et al.: (1998a,b), several well-known spins, such as the vorticity tensor W, the polar spin RR T and the logarithmic spin XLog etc., are incorporated as particular cases of the above general formula with several particular forms of the spin function h…x; y; z†. Now, let Q be the proper orthogonal tensor determined by the linear tensorial di€erential equation12 : Q ˆ ÿQ X ; 11

Q tˆ0 ˆ RT tˆ0 :

…84†

This form is more general than that given by Theorem 2 in Xiao et al. (1998a), which can be derived by dropping out the third requirement in Section 3 in Xiao et al. (1998a), refer to the proof of Theorem 2 preceding Eq. (34) therein. 12 Generally, the initial value Q jtˆ0 may be prescribed as any proper orthogonal tensor. Although here a particular Q jtˆ0 is chosen, the subsequent argument essentially applies to any other Q jtˆ0 . This follows from the fact that the right-hand side of Eq. (85) is independent of Q jtˆ0

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167

Then, left- and right-multiplying Eq. (81) with D e ˆ D by Q and its transpose, respectively, and then using the equality (16) we deduce :  ^ …†† ˆ Q ? D: …Q ? U Then, by integrating the latter over the interval ‰0; tŠ and using the initial condi^ …O† ˆ O, we derive tion (26) and U …t ^ …† ˆ QT ? … Q ? Dds†: U 0

Hence, comparing the latter with Eq. (82), we obtain …t …F† ˆ QT ? … Q ? Dds†: 0

…85†

We are in a position to demonstrate that the symmetric second order tensorvalued function …F† obeys the invariance requirement …QFQ0 † ˆ Q ? … …F††

…86†

for any time-dependent proper orthogonal tensor Q and any constant proper tensor Q0 . In fact, over any given time interval [0,a] the spin of the form (83) is a function of the deformation gradient F and the velocity gradient L. Hence, from this fact and Eq. (84) we know that the proper orthogonal tensor Q is a functional of F and L over [0, a]. Consider the motion with the deformation gradient F ˆ QFQ0 , where Q is any time-dependent proper orthogonal tensor and Q0 any constant proper orthogonal tensor. For the kinematical quantities associated with the motion with the deformation gradient F , we use the same symbols as those for the motion with the deformation gradient F but with a bar. Thus, we have R ˆ QRQ0 ;

B ˆ Q ? B;

D ˆ Q ? D;

:  ˆ QQT ‡ Q ? W: W

Moreover, from Eqs. (83) and (84) we have  ˆQ?X Q



: ‡ Q QT

and :   ; Q  Q  ˆ ÿQ  X

tˆ0

ˆ R T

tˆ0

ˆ QT0 …QR†T tˆ0 :

Then, from the last two equations we infer

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Q  ˆ QT0 Q QT : Thus, using Eq. (85) and the related results derived above and the penultimate identity in Section 1 we infer …t …QFQ0 † ˆ …F † ˆ Q T ? … Q  ? D ds† 0 … t  …QT0 Q QT † ? …Q ? D†ds ˆ …QQT Q0 † ? 0   …t T ˆ Q ? Q ? … Q ? Dds† ˆ Q ? … …F††; 0

i.e. Eq. (86) holds. From Eq. (86) we deduce that …F† is a symmetric second order tensor-valued isotropic function of the left stretch tensor V, or, equivalently, the logarithmic strain measure h, i.e. …F† ˆ …h† with …Q ? h† ˆ Q ? ……h†† for each orthogonal tensor Q. From these and Eq. (82) we derive ^ …†: …h† ˆ U Then, the latter and Eq. (81) with D e ˆ D produce 



D ˆ ^…† ˆ …h† : Applying the chain rule for corotational derivatives of symmetric second order tensor-valued isotropic functions, derived in Xiao et al. (1999), we obtain @   : h ˆ D: @h Further, utilizing the latter and the relation [see Eq. (22) in Xiao et al. 1999] 

h  ˆ L : D; where L=L …V† is given by Eqs. (21) or (30) in Xiao et al. (1999), we derive

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169

@ : …L : D† ˆ D; @h i.e.

@ : L ˆ I: @h

Here I is the fourth order identity tensor given by 1 Iijkl ˆ …il jk ‡ ik jl †: 2 Thus, we infer that L must be invertable and @ ˆ Lÿ1 : @h The latter holds if and only if the gradient @Lÿ1 =@h, which is a sixth order tensor, ful®lls the index symmetry properties (see Simo and Pister, 1984) 

@Lÿ1 @h

 ijklrs

ˆ

 ÿ1  @L ; @h ijrskl

i.e. @L~ @h

! ijklrs

@L~ ˆ @h

! ;

L~ ˆ Lÿ1 ÿ I;

…87†

ijrskl

which is just the condition (58) in Xiao et al., (1999). Then, from the relevant argument given in Xiao et al. (1999) it follows that Eq. (87) holds if and only if the spin X is the logarithmic spin XLog , i.e. the corotational rate used in Eq. (81) is the logarithmic rate. Further, when X ˆ XLog , by virtue of the same procedure used before we infer that the integration of the rate equation (81) produces the elastic relation ^ …† hˆU for each process of purely elastic deformation. If this relation is hyperelastic, then ^ …† ˆ @^=@ i.e. Eq. (31) holds. there is a scalar potential  ˆ ^…† such that U Q.E.D. In Xiao et al. (1999) and Bruhns et al. (1999), the uniqueness of the logarithmic rate in rate type formulation of elastic response is shown to be true for the particular case of initial isotropy. Theorem A indicates that this fact can be extended to a general case of any given type of initial material symmetry.

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6. Small elastic strains and quadratic yield function In this section, we shall derive simpli®ed results for some particular yet important cases. We shall consider ®nite elastoplastic deformation with small elastic strains. Many problems involving the inelastic behaviour of metals and alloys may fall within the scope of this particular case. In our consideration, the initial material symmetry group G0 is allowed to cover all kinds of material symmetry, incorporating the important cases such as isotropy, transverse isotropy and orthotropy etc. as q particular cases. By small elastic strains we mean that the norm kVe ÿ Ik ˆ tr…Ve ÿ I†2 is small. Hence the logarithm of Ve may be approximated by Ve ÿ I, i.e. he ˆ ln Ve ˆ …Ve ÿ I† ‡ O…kVe ÿ Ik†:

…88†

Moreover, we may assume the linear relationship @~ ˆ C0 : P ; @P

…89†

where C0 is a Lagrangean type fourth order tensor that is invariant under the initial material symmetry group G0 and possesses the minor and major index symmetry properties, i.e. Q0 ? C0 ˆ C0 ;

Q0 2 G0 ;

C0ijkl ˆ C0jikl ˆ C0ijlk ˆ C0klij :

…90† …91†

Thus, Eqs. (14)1, (32) and (89), together with the ®rst of the last three identities in Section 1, yield @^ ˆ C : ; @

C ˆ R ? C0 :

…92†

From Eqs. (92)2 and (90)-(91) the fourth order Eulerian tensor C is invariant under the R-rotated material symmetry group R ? G0 and enjoys the minor and major index symmetry properties as shown by Eq. (91). We call the fourth order tensors C0 and C the initial and current tangential elastic compliance tensors, respectively. Although the initial tangential elastic compliance tensor C0 may be a constant tensor related to the initial material symmetry axes, the current tangential elastic compliance tensor C depends on the total rotation R through the R-rotated material symmetry axes, except for the case of isotropy, as can be seen from Eq. (92)2. From Eqs. (88) and (92), for small elastic strains the elastic relation (58) may be approximated by

H. Xiao et al. / International Journal of Plasticity 16 (2000) 143±177

Ve ÿ I ˆ C :  ‡ O…kVe ÿ Ik†;

171

…93†

and the rate relation (33) or (35) for the elastic part D e of the total stretching D is given by 

D e ˆ …C : †Log ˆ C :  R ‡ …C : †XLR ÿ XLR …C : †: 

…94†

Characterizing plastic ¯ow, we adopt the associated ¯ow rule and a quadratic yield function of von Mises type, namely, the plastic potential function g and the yield function f are coincident and given by f ˆ g ˆ … ÿ † : Y : … ÿ † ÿ k:

…95†

Here, the fourth order Eulerian tensor Y, called the yield tensor, is invariant under the R-rotated material symmetry group R?G0 and has the minor and major index symmetry properties as shown by Eq. (91). The isotropic hardening parameter k is a function of the plastic work , i.e. …t …96† k ˆ k^ …†;  ˆ  : D ep ds; 0

and hence [see Eq. (99) below] : dk^ : dk^ : dk^  ˆ … : D ep † ˆ 2  : Y : … ÿ †: kˆ d d d

…97†

In addition, we adopt the kinematic hardening rule of Prager±Ziegler type based on the logarithmic rate, i.e.  Log ˆ D ep 

…98†

with  a kinematic hardening parameter. Hence, for the particular case at issue, the ¯ow rule (39) assumes the simpli®ed form : D ep ˆ 2 Y : … ÿ †;

…99†

and Eq. (98) becomes :   Log ˆ 2 Y : … ÿ †:

…100†

From the relevant expressions given above and the consistency condition (41) we : derive an expression for the plastic multiplier as follows:

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:

ˆ

   … ÿ † : Y :  Log ÿ … ÿ †XLR ‡ XLR … ÿ † dk^  : Y : … ÿ † 2… ÿ † : Y : … ÿ † ‡ d

:

…101†

2

Moreover, Eq. (46) becomes  : D ˆ …C : †Log ‡ 2 Y : … ÿ †;

…102†

or, equivalently, Eq. (47) becomes  : D ˆ C : R ‡ …C : †XLR ÿ XLR …C : † ‡ 2 Y : … ÿ †:

…103†

For a simpler initial material symmetry G0, further simpli®cation is possible. In particular, for the commonly considered case, i.e. the initial isotropy, both the group G0 and R?G0 become the full orthogonal group. In this case, the current tangential elastic compliance tensor C and the yield tensor Y are of the forms C ˆ aI I ‡ b I;

…104†

Y ˆ cI I ‡ d I;

…105†

where a and b, c and d, are material constants characterizing the elastic behaviour and the yielding behaviour, respectively. Furthermore, the stress measure  may be replaced equivalently by the Kircho€ stress  . The complete results and their derivation can be found in a recent work by these authors (Bruhns et al., 1999). 7. Conclusion In the previous sections, a consistent ®nite deformation elastoplasticity theory is proposed by combining the additive decomposition of the stretching D and the multiplicative decomposition of the deformation gradient F. The result derived is suitable for any given type of initial material symmetry, for general combining isotropic-kinematic hardening behaviour and for unrestricted deformations and rotations. The constitutive formulation based on the additive decomposition of D [see Eq. (27)] supplies a complete system of Eulerian rate type constitutive equations governing the total stress and the total deformation quantities, as well as the internal variables characterizing the hardening behaviour. The just-mentioned quantities and the elastic part D e and the coupled elastic-plastic part D ep of D are thus determined. Then, the latter two rate quantities are, in a direct and natural manner, related to the elastic and the plastic part F e and F p of the deformation gradient F in the multiplicative decomposition (51). This relationship and a well-de®ned elastic relation [see

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173

p Eqs. (58) or (60)] for the elastic stretch tensor Ve ˆ F e F eT , together with the total kinematical quantities, are shown to consistently and uniquely determine the elastic and the plastic deformations Fe and F p and all their related kinematical quantities. The main novelty of the suggested theory is the self-consistent Eulerian rate type constitutive formulation based on the newly discovered logarithmic rate and the natural and consistent combination of the two kinds of widely used decompositions concerning the stretching D and the deformation gradient F. It is shown that the rate equation used to represent elastic response is exactly-integrable to indeed deliver a general hyperelastic relation with any given type of initial material symmetry, and the suggested theory is thus subjected to no self-inconsistency diculty in the rate form characterization of elastic response, as encountered by any other existing Eulerian rate type elastic-plastic models of the similar kind. In particular, it is proved that, to achieve the just-mentioned goal, the logarithmic rate is the only choice among all possible (in®nitely many) objective corotational rates. Further, it is demonstrated that not only the two kinds of widely used decompositions concerning the stretching D and the deformation gradient F can be combined in a natural and consistent manner, but also such a combination results in unique determination of the elastic and the plastic deformation F e and F p and all their related kinematical quantities, with no ad hoc assumption about a restricted special form of F e or F p , and thus the main fundamental discrepancies concerning the foregoing two decompositions disappear. The problem of ®nite simple shear is widely recognized as a test problem for new elastoplasticity models. In a recent work (Bruhns et al., 1999), the ®nite simple shear response of the suggested elastoplasticity model for the case of initial isotropy has been studied by means of numerical integration. The results obtained compare favourably with previous corresponding results. Acknowledgements This work was completed under the ®nancial support of the Deutsche Forschungsgemeinschaft (DFG) (contract no. Br 580/26-1). The authors wish to express their sincere gratitude to this support. Appendix Let G0 be a material symmetry group relative to an initial state and P and  a pair of Lagrangean and Eulerian symmetric second order tensors related to each other by the rotation tensor R in a manner as shown by Eq. (14). Moreover, let  ˆ ~ …P† be a di€erentiable scalar function in terms of the Lagrangean tensor P that is invariant under the group G0, i.e. ~ …Q0 ? P† ˆ ~ …P†;

8Q0 2 G0 :

…A1†

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In this appendix, we demonstrate the following fact. Lemma B. The Lagrangean formulation  ˆ ~ …P† with Eq. (A1) may equivalently be converted to the Eulerian formulation  ˆ ^ …† via ^ …†  ~ …RT ? †:

…A2†

The function ^ …† is invariant under the R-rotated group R ? G0 [see Eq. (30)], i.e. ^ …Q ? † ˆ ^ …†;

8Q 2 R ? G0 ;

…A3†

and the following relationship holds: @^ @~ ˆR? : @ @P

…A4†

The gradient @^ =@ furnishes an objective Eulerian symmetric second order tensor. Proof. First, we show that the Eulerian formulation ^ …† de®ned by Eq. (A2) is invariant under the R-rotated group R ? G0 , i.e. Eq. (A3) holds. In fact, by using the penultimate identity in Section 1 we deduce RT ? …Q ? † ˆ …RT ? Q† ? …RT ? † for each orthogonal tensor Q. Then, from the above equality and Eqs. (A2) and (14)2 we infer ÿ
^ …Q ? † ˆ ~ RT ? …Q ? † ÿ
ˆ ~ …RT ? Q† ? …RT ? † ÿ
ˆ ~ …RT ? Q† ? P for each orthogonal tensor Q. Hence, from Eqs. (14)2, (A1), (A2) and the penultimate identity in Section 1, as well as the fact: Q 2 R ? G0 ) RT ? Q 2 G0 ; we derive ÿ
^ …Q ? † ˆ ~ …RT ? Q† ? P ˆ ~ …P† ˆ ~ …RT ? † ˆ ^ …† for each orthogonal tensor Q 2 R ? G0 .

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175

Next, we show that Eq. (A4) holds. In fact, by the de®nition of the gradient of a tensor function we have ^ † ÿ ^ …† @^ ^ ^ … ‡ "A ; : A ˆ lim" ! 0 " @ ~ † ÿ ~ …P † @~ ~ ~ …P ‡ "A : A ˆ lim" ! 0 ; @P " ^ and A ~ . Applyfor any Eulerian and Lagrangean symmetric second order tensors A ing Eqs. (14)2, (A2) and the above two equalities and the last two identities in Section 1, we infer …R ?

~ @~ ~ † ˆ @ : A ~ † : …R ? A @P @P ˆ lim" ! 0

  ~ † ÿ ~ …RT ? † ~ RT ? … ‡ "R ? A

" ~ † ÿ ^ …† ^ … ‡ "RT ? A ˆ lim" ! 0 " @^ ~ †: : …RT ? A ˆ @ Thus, Eq. (A4) holds. Finally, from Eqs. (7)2, (7)4 and (A4) and the fact that the gradient @~ =@P is a Lagrangean symmetric tensor, we know that the gradient @^ =@ is an objective Eulerian symmetric tensor. Q.E.D.

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