Modelling Simul. Mater. Sci. h g . 2 (1994) 483-504. Printed in the UK

Finite-strain anisotropic plasticity and the plastic spin

N Aravas Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104, USA

Received 4 November 1993, accepted for publication 21 November 1993

Abstract. A detailed analysis of the k i n e M c s of anisotropic elastoplastic solids under finite isothermal deformations is presented. The formulation is based on a multiplicative decomposition of the deformatio? @adient temor into elastic and plastic p m . Emphasis is placed on the proper definition and the physical meaning of the so-called 'plastic spin'. which is the spin of the continuum relative to the material subshucmre. Constitutive equations for the plastic spin are derived for three different material systems: (i) a fibre-reinforced metal-matrix composite, in which the local axis of transverse isotropy is defined by the local orientation of the fibres, (ii) polymeric materials, in which the anisompy is deformation induced, and (E) an elastoplastic material which yields accarding to a yield condition of the kinematic hardenin% we, The numerical implementation of the elasroplastic equations in a finite element programme, as well as an algorithm for their numerical integration, are briefly discussed. The choice of the intermediate unstressed configuration and the proper definition of the plastic spin in 'non- isoclinic' configurations are also discussed.

1. Introduction

The mechanics of finite elastoplastic deformations has been well developed in recent years and the problem of the appropriate generalization of the classical laws of elastoplasticity to the case of finite deformations has been addressed in numerous publications. During finite plastic deformations, the kinematics of the material substructure, which defines the material symmetries, is not necessarily identical to that of the continuum. The quantity that emerges from such a distinction in kinematics is the plastic spin, which is the spin of the continuum relative to the material substructure (Dafalias 1983, 1984). Mandel (1971, 1973) and Kratochvil(l971, 1973) were the first to suggest that a complete macroscopic plasticity theory must include constitutive relations not only for the plastic part of the deformation rate but for the plastic spin as well. Using the representation theorems for isotropic functions, Kratochvil (1973) concluded that the plastic spin vanishes identically in isotropic materials. In anisotropic materials, however, the plastic spin is of major importance and has been the focus of a series of papers by Dafalias (1983, 1984, 1985, 1987, 1988) and Loret (1983) where constitutive equations are formulated for different anisotropies using tensorial structure variables.

The plastic spin and issues related to the evolution of anisotropy in finitestrain elastoplasticity has been the subject of several publications; amongst these, we mention the works of Hahn (1974). Nagtegaal and Wertheimer (1984), Zbib and Aifantis (1988), van der Giessen (19&9), Nemat-Nasser (1990). and Tigoiu and Soos (1990). It appears, however, that there still exists some confusion in the literature about the proper definition and the physical meaning of the plastic spin. A detailed discussion of the kinematics and the general form of the constitutive equations of anisolmpic elastoplastic solids under

0965-M93/94~A0483t22$1950 @ 1994 IOP Publishing Ltd 483

484 N Aruvus

finite isothermal deformations is presented in this paper. The formulation is based on the multiplicative decomposition of the deformation gradient F into an elastic and a plastic part, i.e. F = Fe . Fp (Lee 1969). In section 2, we discuss the physical meaning of the continuum and subsmctural spins, and introduce the definition of the plastic spin. The details of the elastic-plastic kinematics for finite deformations are discussed in section 3. Several examples of continua with microstructures are discussed and constitutive equations for the corresponding plastic spins are derived in section 4. The problem of finite simple shear for a kinematic hardening plasticity model is analysed in section 5. The numerical implementation of the elastoplastic constitutive equations in a finite element programme and an algorithm for their numerical integration are presented in section 6. The paper closes with a detailed discussion of the choice of the intermediate unstressed configuration and the proper definition of the plastic spin in 'non-isoclinic' intermediate configurations.

Standard notation is used throughout. Boldface symbols denote tensors, the order of which are indicated by the context. All tensor components are written with respect to a fixed Cartesian coordinate system, and the summation convention is used for repeated indices, unless otherwise indicated. The prefices tr and det indicate the trace and the determinant respectively, a superscript T the transpose of a second-order tensor, a superposed dot the material time derivative, and the subscripts s and a the symmetric and anti-symmetric parts of a second-order tensor. Let a and b be vectors, and A and B second-order tensors; the following products are used in the text: (ab)jj = uibj, (A * a)i = Aijuj, ( a . A)i = ujAji,

and (A . B)ij = Ai&.

2. Continuum versus microstructural spin

Consider a continuum which occupies a certain region BO of the physical space at a certain time b. The position of a material particle at this time is described by its position vector X. Let the position vector of the particle be x at time t . Then an equation of the form

I = I(X, t ) (1)

defines the motion of the continuum. Let B denote the new (deformed) configuration of the continuum.

The deformation gradient F of the motion is defined as

F ( X , t ) = axcx, wax (2)

and the corresponding velocity gradient L at B is given by

L ( ~ , t ) = a+, t)/az =: F . F - l = D + w (3)

where z1 is the velocity field, and (D, W) are the deformation rate and spin tensors, defined as the symmetric and anti-symmetric parts of L, respectively.

2.1. The continuum spin

It is a well-known result in continuum mechanics that the value of the spin tensor W(x) at a material point x defines the spin of the material line elements which, in B, are situated at I and directed along the principal axes of the deformation rate tensor D(x), i.e.

(4) ei = w. rj = w x ri

Finite-strain anisompic plasticity and the plastic spin 485

Figure 1. Orientation of a unit vector m in the deformed configuration B.

Figure 2. Fibrereinforced material subjected to simple shear

where ri is a unit vector attached to a material fibre which is instantaneously aligned with one of the eigenvectors of D, and w is the axial vector of W .

In the following, we show that W ( z ) is also the average spin of all material fibres which, in B, are situated at I. Let m be a unit vector attached to a material fibre at S.

Referring to figure 1, we write

m = cos03 coshel + s ing coshez + sin43e3 (5)

where el, e2 and e3 are unit vectors along the coordinate axes. Using t!e components wi of the axial vector w, we can write the following representation for the spin tensor W :

Then, using the well-known result

m = (W f D . mm - mm . D ) .m (7)

486 N Aravas

together with (5) and (6) one can readily show that (Aravas and Aifantis 1991)

(4) = tu3 = -WIZ = w,, (8)

where

is the average spin about the coordinate axis x3 of all material fibres which, in 8, are situated at x. Similarly, if we consider the rotation rates about the x1 and xz axes, we find

( d i ) = w i i = l , 2 , 3 (10)

i.e. the components of W(x) define the average spin of all material fibres which. in B, are situated at x. In view of this interpretation, the spin tensor W will be referred to in the following as the ‘continuum spin’.

A two-dimensional version of this interpretation has been given by Cauchy (1841) (as referenced in Truesdell and Toupin 1960, p 355).

2.2. The substructural and the plastic spins

When an anisotropic material is subjected to finite deformations, the spin w of the ‘substructure’, which defines the material symmefries (or asymmetries), is not necessarily the same as that of the continuum (W).

For simplicity, we focus our attention on rigid plastic materials in this section and start with the example of a rigid plastic continuum reinforced by aligned fibres. A block of such a material is subjected to a finite simple shear y parallel to the fibres as shown in figure 2. Let X I and x z be the coordinate axes on the plane of the motion as shown in figure 2; then it can be readily shown that the continuum spin W is given by

Figure 2 makes it clear that, whereas the continuum spin W is non-zero, the orientation of the fibres (substructure) does not change during the shear deformation and, therefore, the substructural spin w vanishes at all times.

Returning to the general case of finite-strain anisotropic rigid plasticity, we note that a complete constitutive model, in addition to the usual constitutive equations for the elastic and plastic parts of the deformation rate, must specify how the material anisotropy evolves during plastic flow. Dafalias (1984, 1985) was the 6rst to suggest the use of the plastic spin WP as a convenient means of distinguishing between the kinematics of the continuum and the kinematics of the substructure. He defined W as the spin of the continuum relative to the substructure, i.e.

wp=w-w (12)

and argued that constitutive equations must be written not only for the plastic part of the deformation rate but for the plastic spin as well. At the risk of becoming redundant, we mention that the last equation can be also written as

w=w-wp (13)

Finite-strain anisotropic plasticity and the plastic spin 487

so that, if the motion of the continuum, and thus W , is known and a constitutive equation for W is given, then equation (13) defines the spin of the substructure, i.e. the plastic spin can be used to define indirectly the spin of the material substructure which, in turn, will define the evolution of anisotropy. Since Dafalias’ original work was published, several arguments and counterarguments have appeared in the literature debating wshether a constitutive equation for WQ is indeed necessary (e.g. Nemat-Nasser 1990). It appears that there still exists some confusion in the literature about the plastic spin and its meaning, mainly because the quantities referred to as ‘the plastic spin’ by various authors are not the same and do not always relate to the spin of the substructure. A detailed discussion of the various ‘plastic spins’ used by different authors is presented in section 6 below.

An alternative, and perhaps more natural, statement for Dafalias’ suggestion could be that a constitutive equation for the substructural spin w must be written, in addition to the usual equations for the plastic part of the deformation rate. The value of WP is then determined as the difference between W and U , according to equation (12). In the present paper, we follow this alternative approach and derive constitutive equations for the substructural spin w based on micromechanical considerations; whether the corresponding expression for the plastic spin WP, resulting from equation (12), should be called a ‘constitutive’ equation or not is simply a question of semantics.

We conclude this section by mentioning that, in elastic-plastic (as opposed to rigid- plastic) materials, the plastic spin is again the spin of the continuum relative to the substructure in the intermediate (elastically unloaded) configuration as discussed in the following section.

3. Elastic-plastic kinematics

The kinematics of finite elastic plastic deformation is described by the multiplicative decomposition of the deformation gradient F Gee 1969):

According to this decomposition, the neighbourhood of a material point is mapped first from the undeformed configuration Bo to the intermediate unstressed configuration Bi by the plastic part FP, and then carried to the current configuration B by the elastic part Fe. The intermediate unstressed configuration B,, that is the configuration of the continuum after removal of Fe, is not uniquely defined, since an arbitrary rigid-body rotation can be superposed to it and still leave the configuration unstressed. Motivated by the kinematics of the single crystal and in order to remove the aforementioned ambiguity, Mandel (1971) introduced a triad of local ‘director vectors’, say 4, embedded in the material substructure, and used the orientation of the 4s with respect to a fixed Cartesian coordinate system to define the orientation of the intermediate configuration Bi. The director vectors used in the definition of Bi can be associated with the direction of certain material fibres and follow the deformation of the continuum, or with the principal directions of tensorial state variables, in which case their transformation is not necessarily identical to that of the continuum. We choose to work in terms of the so-called ‘isoclinic configuration’ which is defined in such a way that the director vectors at Bi have, at all times, a fixed orientation in reference to a global coordinate system (Mandel 1971, Nagtegaal and Wertheimer 1984).

In the current configuration B, the velocity gradient L can be written as

L = F . F - 1 - - E“. p-’+ ~ e . FP. ~ p - 1 .FC-’ = D + w (15)

D = D C t D Q and W = W * t W Q (16)

488 N Araws

where

De = (p * Ij"-l)s W' = (P . (17) DP = ( F e . @P . FP-' . (18) Wp = (F . @P . P - ' . In the isoclinic configuration we also define the 'purely plastic' quantities

Lp = fi. FP-' Di p - - ( &P. FP-'), wp ( F P . (19)

We show next that Wp is indeed the spin of the continuum relative to the substructure in the isoclinic configuration Bi. Since Bi is defined in such a way that the direction of the director vectors (substructure) does not vary with time, the substructural spin w is always zero at Bi. Therefore, the quantity W'' = (@. FP-l)=, which is the average spin of the continuum at Bj, is also the spin of the continuum relative to the non-rotating substructure at Bi.

It should be noted that the choice of the intermediate configuration can be made arbitrarily provided that the corresponding kinematical quantities, such as De, DP, etc, are properly defined (Dafalias 1984, 1985); a detailed discussion of the proper definition of the plastic spin when the intermediate configuration is not isoclinic is given in section 6 below.

4. Continua with microstructures

In all cases discussed in the following, we define the material symmetries and write the elastic and plastic constitutive equations in the intermediate unstressed configuration; this is, in a sense, a natural choice, since the intermediate configuration can be viewed as the 'deformed' configuration for the plastic part of the motion, and the 'undeformed' configuration for the elastic part. For simplicity, isotropic hyperelasticity is assumed in all cases and the elastic part of the constitutive equations is written as

se = poa\y(Ee)/aEe (20)

where Se = det(F)Fe-' . cr. Fe-T is the elastic second Piola-Kirchhoff stress, U is the true (Cauchy) stress, po is the mass density in the undeformed configuration BO, W is the elastic strain energy density function, and E' = O.S(FeT -Fe - r) is the elastic Green strain, I being the second-order identity tensor.

In the isoclinic configuration, we also define the stress tensor

z1 = det(F) Fe-' * U . F (21)

which is plastic work conjugate to Lp at Bi, i.e. the rate of plastic work per unit undeformed volume is given by

W P = tr(u. DP) = tr(Z * L;). (22)

When the elastic response is isotropic, Z is symmetric and equal to (kavas 1992)

z1 = d e t ( F ) R T -u. Re (23)

Finite-strain anisotropic plasticity and the plastic spin 489

where Re is the rotational part of Fe. The constitutive equations for the plastic part of the motion are of the form

where f is a scalar-valued yield function, A is a scalar plastic multiplier, s is a collection of scalar- and tensor-valued state variables, and (f, NF, np) are isotropic functions of their arguments (Dafalias 1987, 1988).

The constitutive model is completed by srecifying a constitutive equation for the evolution of the state variables s at Bi:

d = A?(X,S) (27)

where j is an isotropic function of its arguments. Several constitutive models for the yield function f and the plastic deformation date

Dp have been presented in the literature for both isotropic and anisotropic materials. In the following, we focus on the plastic spin Wp and derive constitutive equations for some commonly used anisotropic plastic constitutive models.

4.1. Fibre-reinforced metals

Consider a metal-matrix composite reinforced by short aligned fibres. In our continuum approach, each material point is characterized by persistent transversely isotropic symmetries in the isoclinic configuration. At each material point we define a local ‘director vector’ in a such a way that its orientation in the undeformed configuration 0, is the same as that of the aligned reinforcing fibres. The ‘director vectors’ are embedded in the continuum, follow its deformation, and their orientation in the intermediate configuration Bi defines the local axis of transverse isotropy. The multiplicative decomposition (14) is written at each material point, and the isoclinic configuration for that point is defined in such a way that the local ‘director vector’ (say n) at Bi has, at all times, a fixed orientation with reference to a global coordinate system, conveniently chosen to be the same as that of the corresponding orientation in the undefonned configuration 6, The isoclinic configuration is now uniquely defined to within a rigid rotation about the local axis of symmetry specified by n.

For definiteness, we let n be a unit director vector. The rate of change of n is

n = w . n (28)

where

w = W:+ Dp .nn - n n . Dp

is the substructural spin at Bi. Since the director vector n does not vary with time in the isoclinic configuration, we have that

490 N Aravas

The above equation shows that n is the axial vector of the antisymmetric tensor w = Wf + 0: . nn - nn . Dp and, therefore, we have the representation (e.g. Ogden 1984, P 30)

w = Wf + 0:. nn - nn = a(nzn3 - n3n2) (31)

where nz, n3, and n form an orthonormal basis, and (Y is an arbitrary constant. The right-hand side of (31) is a spin about n in the isoclinic configuration. Transverse isotropy implies rotational symmetry about n, so that a spin about n is inconsequential, therefore, 01 can be set to zero in (31), which reduces to

wp A .D; - D; . A (32)

where the orientation tensor A is defined by the dyadic product A = nn. The above equation is the constitutive equation for the plastic spin in the isoclinic

configuration. It can be readily shown that equation (32) is indeed of the form of (26) and consistent with the general expression given by Dafalias (1984) for transversely isotropic materials (Aravas 1992). A complete elastieplastic model for fibrereinforced metal-matrix composites has been presented recently by Aravas (1992).

4.2. Anisotropic polymers

Battennan and Bassani (1990) have developed a constitutive model for polymers under finite strain. Their model assumes that the principal directions of the plastic stretch comelate with the molecular orientation or ‘texture’ of the polymer, and, therefore, the principal axes of plastic stretch are taken as the principal (orthogonal) axes of orthotropy. Under arbitrary histories of deformation, each material point of the deforming polymer will possess orthofropic symmetries about the principal axes of the plastic stretch tensor BP = Fp FPT in the intermediate configuration. In Mandel’s terminology, the eigenvectors of BP play the role of the director vectors at each material point. We denote by A; (AY > A; > A;) the eigenvalues of (BP)”* and by q the corresponding unit eigenvectors. Plastic incompressibility is assumed, so that = 1. At every instant, the arbitrary rotation of the intermediate configuration at each point is chosen in such a way that the orientation of the mis at that point does not vary with time and equals the orientation of the corresponding m i s when the material point yields for the first time (Pereda et a1 1993).

The rate of change of the q s can be written as

(33) m. , - w e m i -

where the substructural spin w is now the Eulerian spin of the plastic part of the motion and is given by (Biot 1965, Hill 1970)

(no sum over k) (34) Ai2 + A;* A:’ - AY‘

(D:), A i # A; k # 1 (no sum over k, I ) (35)

where all tensor components are with respect to the unit eigenvectors mi. When the intermediate configuration is isoclinic, the rate of change of the director vectors vanishes and, therefore, w = 0. Equations (34) and (35) now reduce to

( W p ) k k = 0 (no sum over k) (36)

Finite-strain anisotropic plasticiry and the plastic spin 491

which are constitutive equations for the plastic spin. When two of the plastic stretches are equal, say A; = A!, the material becomes locally

transversely isotropic about the m1 direction in the isoclinic configuration. In such a case, in view of the local rotational symmetry, a spin about the ml axis is inconsequential and can be set arbitrarily to zero. Therefore, we write

( W ~ ) X I = (W:)I~ = 0 when A{ = Ay k # 1. (38)

Finally, when all three plastic stretches are equal (A; = A: = $ = I), the material is locally isotropic and (Dafalias 1983, 1984, Loret 1983)

wp = 0. (39)

A detailed discussion of the complete elastoplastic constitutive equations for anisotropic polymers can be found in Pereda eta1 (1993).

4.3. Kinematic hardening

We begin with a brief review of the commonly used small-strain version of the kinematic hardening model. The yield condition is of the form

f ( r r - a, F) = 0

;p = ($P : € P ) 1 / 2

(40)

(41)

where a is the back stress, ?P is the equivalent plastic strain defined from

ep being the plastic part of the infinitesimal strain tensor. The plastic flow rule is of the form

€P ;\nP (42)

where A is a plastic multiplier that vanishes during elastic unloading or neutral plastic loading, and np is a second-order tensor that determines the direction of the plastic strain rate.

The collection s of state variables is now s = (a, P'). The simplest version of the evolution equation for the back stress a is due to F'rager and is of the form

c i = c 6 ' = A c n P or a = c @ (43)

where c is a material constant. Also, in view of (41) and (42) the evolution equation of E P

is

;p = &P g P = ($ nP : nP) ' /2 , (44)

In the following, we present a finite-strain version of the kinematic hardening model and discuss issues related to the plastic spin. The yield condition and the plastic flow rule are written in the isoclinic configuration in terms of the stress tensor E = det(F) ReT . cr. Re and the symmetric traceless back stress tensor A, which are both defined at ai:

f (E - A, i p , hjP) = 0 DP = AN: Wp = ASlp (45)

492 N Aravas

where f, Np and sly are isotropic functions of Z and s, where s is the collection (A, E'P, A!). The evolution equation of 3 is

2' = 3 ( Z D P . 1 ' Dp)l/'= E "P - - ( 3 ZN? I . . N/)l/z, (46)

The corresponding evolution equation for AY is (Pereda et a[ 1993)

i p J = Lip J i p J = h;mj . (NP + sl;) mj (no sum over j ) . (47)

The model is completed with the specification of the evolution equation of the back stress A, which will also define the specific form of 0; in (454. The eigenvectors 4 of the back stress A play the role of the director vectors and define the orientation of the substructure. We write

3 A = Ajddi

k l

where the djs are understood to be unit eigenvectors. At each material point, the intermediate (isoclinic) configuration is defined in such a

way that the orientation of the 4s does not vary with time and equals the orientation of the corresponding 4 s when the material point fist yields. A specific example of the definition of the isoclinic configuration is given in section 7.3 where the problem of finite simple shear is analysed in detail.

There are two possible forms for the evolution equation of A. In the first one, the back stress A depends only on the final plastic state of the continuum and is independent of the history of deformation. whems in the second the final value of A depends on the history of plastic deformation. The two cases are discussed separately in the following.

4.3.1. A history-independent equation for A. A natural extension of equation (43tr) to finite strains is to assume that the symmetric back stress A is co-axial (same principal directions) with the plastic stretch tensor BP = FP . F p T and that the principal values of A are functions of the plastic stretches Ay, i.e.

where mi are unit eigenvectors of BP, and the functions ki are such that

3 z k i = 0 i=l

ki = 0

k, = ki when Ay = A; (i # j ) .

when A: = 1

(50)

Parks el al (1984) and Boyce er a1 (1988) developed a model that falls in this category for the mechanical behaviour of anisotropic polymers at finite strains.

Using arguments similar to those used in section 4.2, we can readily show that the plastic spin Wf is again given by equations (36t(39).

Finite-strain anisotropic plasticity and the plastic spin 493

4.3.2. A history-dependent eqwrrion for A. In the most general case of a history-dependent back stress, one has to write a rate equation for the evolution of A. Starting with

we have to specify the evolution of both the principal values Ai and the eigenvectors 4. The general form of the equation for the rate of change of the Ais will he

For a rate-independent theory, the functions gi must he homogeneous of degree one in both @ and i,': Therefore, taking into account equations (46) and (47), we conclude that (54) can he wntten as

3

Ai = A &(E, A, ZP, E.,') Cii = 0 i = l

(55)

where the ijs are isotropic functions of their arguments. The rate of change of the unit eigenvectors 4 will be of the form

C i , = w . & (56)

where w is again the substructural spin. The specification of w is crucial in the development of the model: it defines the way the hack stress A evolves and leads to an expression for the plastic spin Wf. The general form of w will he

w = h(W/, Dp3 E, A, $,A,', i p , A;). (57)

Equations (29) and (34) and (35) for the fibre-reinforced metal and the anisotropic polymer respectively are special cases of the general expression for w given in equation (57).

In the isoclinic configuration, the orientation of the director vectors 4 does not v a q with time; therefore w = 0 or, equivalently,

h(W,'. OF, E, A, Zp, A;, ip, A,') = 0 (58)

which eventually leads to an equation of the form

W/ = A flp(E, A, 9'. A,") (59)

where Cl! is an isotropic function of its arguments (Dafdias 1987). For example, one could use a model similar to that proposed by Lee et al (1983) and

assume that the spin of the principal directions of the hack stress A coincides with the spin of the material fibre which, in Bi, is instantaneously aligned with the eigenvector, say flax, of A that corresponds to the eigenvalue of A with maximum absolute value. In that case,

(60) w = W / + Dp . P d m -d'"""d'"" * Dp.

494 N Aravm

The requirement that w = 0 leads to

W: = P d - . 0,' - 0,' .PP (61)

which is the constitxtive equation for the plastic spin. Note that in such a model, when a material point yields for the first time, A = 0 (isotropy) and the vectors P and cl j are not uniquely defined. One possible way to overcome this difficulty would be to assume that, when A = 0, P = c y and d, = q, where q are the eigenvectors of Dp and c y is the eigenvector of 0; that corresponds to the eigenvalue of @ with maximum absolute value. In that case, in view of equation (61), Wf = 0 when A = 0, which is consistent with the well-known result that the'plastic spin vanishes in isotropic materials (Kratochvil 1973, Hahn 1974, Dafalias 1983, 1984).

Returning to the general case of a history-dependent equation for A, we note that, since the orientation of the 4s does not vary with time in the isoclinic configuration, the evolution equation of A can be written as

where G is an isotropic function of its arguments. Note that, in the isoclinic configuration, all material time derivatives are also co-rotational with the substructure (since w = 0). Therefore, the explicit use of any co-rotational rates is avoided when the intermediate configuration is chosen to be isoclinic, thus simplifying greatly the numerical implementation of the elastoplastic equations in a finite element programme. However, when the intermediate configuration is not isoclinic, rates co-rotational with the substructure must be used in the constitutive equations; a detailed discussion of this topic is presented in section 6 below.

We conclude this section by mentioning that Dafalias (1983, 1984, 1985) and Loret (1983) use the representation theorems for isotropic functions F a n g 1970a, b, Smith 1971, Liu 1982) to write a constitutive equation for the plastic spin Ti' directly, without going through the specification of the substructural spin w first; in that case, the expression for w is calculated from the difference between the continuum and the plastic spin. It should be emphasized that the two approaches, i.e. the one presented herein and that of Dafalias and Loret, are completely equivalent. However, it is the author's opinion that it is preferable to start with the specification of the substructural spin w based on microstructural considerations directly, instead of specifying the plastic spin first through the use of the aforementioned representation theorems.

5. Numerical implementation in a finite element programme

In the following, we summarize first the most general form of the elastoplastic constitutive equations and discuss next their numerical implementation in a finite element programme.

The intermediate configuration Bj is chosen to be isoclinic. We define the material symmetries and write the elastoplastic constitutive equations in the isoclinic configuration. The elastic part of the constitutive law is of the form

Se = 8Y(EC)/8Ee. (63)

Finite-strain anisotropic plasticily and the plastic spin 495

The yield condition and the plastic flow rule are written as

f(X, s) = 0 Dp = AA'[(X,s) W: = An!@, s). (64)

The evolution equation of the state variables is of the form

s = AS(& s). (65)

We mention again that, since the intermediate configuration is isoclinic, the evolution equation of the state variables is written in terms of the usual material time derivatives even for tensor-valued state variables.

In a finite element environment, the solution of the elastoplastic problem is developed incrementally and the constitutive equations are integrated at the element Gauss points. In a displacement-based finite element formulation the solution is deformation driven. At a material point, the solution (EL, s., F f , Fn) at time t. as well as the deformation gradient F.+I at time tn+l = t,, + At are supposed to be known and one has to determine the solution (%+I. sn+1, F,+i).

The starting point is equation (19) which, after using (&I), can be written as

@ = ~p .qp = A(N; + np). F P . (66)

The direction of plastic flow is assumed to be constant over the increment and equal to (Np + CZp)" = X,. Integration of the above equation yields

FL.' = Fi-' . exp(-AAX,) = Fl-' . [ I - AAX" + iAA2X: + O(AA3)]

which is truncated to (67)

FLG' = F,"-' ' ( I - AAX, + 4AA'X;). (68) The evolution equations of the state variables are integrated using a forward Euler technique. A summary of the algorithm for the integration of the constitutive equations is given in the following:

F:+, = F ~ ~ . ( I - AAX, + $AA*x~) Fe-'

F:+i = Fn+l . n+l

G + I = F,cTl . F,c,l

(6%

(70)

(71)

E,+, = ;cc;+, -1) (72) = ,m*/aEe)n+l (73)

%+I = %+I . c;+, (74)

sn+i = sn + AAS(Xn,sn) (75) f(%+l> & + I ) = 0 (76)

where F&, = Fn+l . Fi-' . We choose AA as the primary unknown, treating the yield condition (76) as the basic equation for its determination.

When the constitutive equations are properly normalized, AA can be interpreted as the magnitude of the plastic strain increment. In such a case, if the material is plastically incompressible (i.e. N; + Slp = 0), it can be readily shown that

de@+,) = 1 + O(AA2) (77)

i.e. the above algorithm preserves plastic incompressibility to within terms of O(AA2) in comparison to unity (kavas 1992).

496 N Aravas

6. The choice of the intermediate configuration

In this section, we use the subscripts i and U to distinguish between quantities dcfined in the isoclinic configuration Bi and an arbitrary intermediate unstressed configuration a,, e.g. (q, I?') and (Pi , Ff) will be the elastic and plastic parts of the deformation gradient at Bi and & respectively (F = F: . qP = F t . Ff). A thorough analysis of the continuum and substructural kinematics for a non-isoclinic intermediate configuration has been given by Dafalias (1987, 1988).

The kinematics of the elastoplastic deformations are written as follows (Dafalias 1987):

L = F . F-! (78) - - q . Fe-' + q .e. Kp-1 . qe-1

= $" . F:-l + Fi . Ft. e-' . F,-' = i;. q-1 + F.' . @. Ft-1 . pi-' (79)

(80)

and

D = D e + DP W = W" f WP (81)

where

De = (e. = (@:. F,'-')s (82)

(83)

(84)

(85)

In the above equations, a superposed o denotes time derivatives co-rotational with the substructure at a., i.e.

$ =e + F e . U, fi: = Fl- U*. (86)

where w, is the spin of the substructure in the arbitrary intermediate configuration a, defined by Ff.

In the arbitrary intermediate configuration we also define the 'purely plastic' quantities

(87)

W" = (e . q-'), = ($" e . F e - 1 " DP = (q . eP . e-' . q-l)s = (e. &$. F:-' . q-')*

)a

WP = (T . eP. qp-' . Fe-' i )a = (Pi. I?:. q - 1 .

DE = (k$. FP-'), = (F:. F:-')$,

and

W,P = (Fj . Ff-')a = (e . -U,. (88)

1 Recalling that (E".p. F f - )a is the average continuum spin at a,,, we readily conclude from equation (88) that Wf is again the spin of the continuum relative to the substructure in the intermediate configuration B.. In this connection, we note that several authors choose to call the quantity (I?!. 'the plastic spin', even when the intermediate configuration is not isoclinic (e.g. Nemat-Nasser 1990). Of course, there is nothing wrong with using that name, provided one realizes that (I??. F:-l)a is just the continuum spin at E,, it is different

Finitestrain anisotropic plasticiry and the plastic spin 491

from what is called 'the plastic spin' here (and in Dafalias's work), and that (I? . Ff-'), alone cannot be used to differentiate between the continuum and the substmctural spin at Bu.

The isoclinic and the arbitrary intermediate configurations differ by a time-dependent rigid-body rotation R,, i.e

The director vectors dr at Bu are related to the corresponding 4s at Bi through

d r = R , . & . (91)

Taking into account that the 4s do not vary with time at Bj, we readily conclude that

dr = W. . d: (92)

where

w . = % . l q (93)

is the substructural spin at B,,. We also have the following relationships between quantities at B, and B,:

Recall that the plastic constitutive equations in the isoclinic Configuration are of the form

where f , Np, Cl! and ? are isotropic functions of their arguments, In the following, we discuss the corresponding form of the above equations when the intermediate configuration is not isoclinic. We define

E" = R, . E. q T (100)

and

su = &[SI (101)

where

%[SI = s i f s is a scalar (102)

R,[s l=R, . s i f s is a vector (103)

&[SI = R, . s. if s is a second-order tensor. ( 104)

498 N Aravas

Then, using (94) and (95) and taking into account that f , NP, SlP and 5 are isotropic functions, we readily conclude that

p", S U I = 0

where

(109)

(110)

(1 11) Note that the functional form of the plastic constitutive equations at U, is identical to that at Ui, i.e. equations (105)-(108) at U, can be obtained from the corresponding equations (96)-(99) at U;, if one replaces (E, s, DP, W:, S) by (E", s,, DE, W!, in) (Dafalias 1988).

e .

S" = S"

S" = s. - U" . S"

s, = s. - w. . s, + s. . w,,

ifs, is a scalar

ifs, is a vector

ifs. is a second-order tensor.

* .

0 .

7. An example: kinematic hardening and simple shear

7.1. The yield condition and the plasticJlow rule We consider a rigid plastic material which yields according to a kinematic-hardening von Mises yield criterion. The yield condition is written in the isoclinic configuration and is of the form

f(E, A, Fp) = [$(E' - A) : (E' - A)]''* - oY(EP) = 0 (1 12) where a prime denotes the deviatoric part of a second-order tensor. A normality rule is used in the isoclinic configuration, so that

Using the definition of @, the flow rule (113). and the yield condition we can readily show that @ = A, so that (113) can be written as

3 ELP

2 U Y Df --(E'- A).

Since the material is rigid plastic, the elastic part of the deformation gradient is a rigid- body rotation, i.e. Fe = Re. The yield condition and the flow rule can be written in the deformed configuration U as follows:

f(o, a, ZP) = [;(U'- a) : (U' - CY)]''* - UY(ZP) = 0

where u =Re. E . Rer, a = R e . A . Rer and DP = E. 0,'. Rer. The corresponding uniaxial stress-strain relationship of the material is

0 1 1 = ay(';) + +I1 (116)

where XI is the axis of tension and E: = In ,ky (A; = ,k!, = l / f i ) .

Finite-strain anisotropic plasticiv and the plastic spin 499

7.2. The evolution of the back stress and the plastic spin

7.2.1. History-independent back stress. The value of the back stress tensor A in the isoclinic configuration Bj is assumed to be co-axial with BP:

i = l

where the kis satisfy the conditions (50)<52). The plastic spin W/ is given in that case by equations (36)-(39).

The corresponding equation for the back stress in the current configuration is

where = Re. mj are the eigenvectors of the total stretch tensor B = F . FT. Also note that the eigenvalues is given by equations (36)<39), where now W/ and 0,' are replaced by WP and Dp, and all tensor components are with respect to the unit eigenvectors nj.

7.2.2. History-dependent back stress. We consider next the case in which the spin of the principal directions of the back stress A in the isoclinic configuration Bi coincides with the spin of the material fibre which, in Bj, is instantaneously aligned with the eigenvector P of A that corresponds to the eigenvalue of A with maximum absolute value, i.e.

of Bp are the same as those of B. The plastic spin wp at

We assume that the rate of change of the principal value of A in the i-principal direction is proportional to the value of the corresponding normal component of Dp in that direction, i.e.

(121)

When A = 0, the 4s are assumed to coincide with the eigenvectors of 0: and their spin equals the spin of the material fibre that is instantaneously aligned with the eigenvector of Dp that corresponds to the eigenvalue of Dp with maximum absolute value.

A j = c(E'P)dj . Dp . dj (no sum over j ) .

The corresponding equations in the current configuration B are

w = w - w p

where li = Re .d+. Again, when (Y = 0, the lis are defined by the eigenvectors of Dp.

500 N Aravas

7.3. Simple shear

A block of material is subjected to plane strain sim;lle shear y along the x I direction as shown in figure 2. The deformation gradient F is written as

F = I + y e l e z (126)

and the equivalent plastic strain is found to be 3' = y/&. The eigenvalues and eigenvectors of 9''' are

hl = 1 1 + + Y 2 + Y ( l + ~ Y ) 1 2 112 1 112 hz = l/hl h3 = 1 ( 127)

and

nl = coseel + singe2 n 2 = -sinBel +cosBel n3 = e3 (128)

where

B = ;tan4(2/y). (129)

In order to simplify the calculations, we choose the elastic rotation to be equal to the identity matrix, i.e. Fe = Rc = I. In this case, the intermediate configuration B, is non-isoclinic and coincides with the current configuration U. Therefore,

D = Dp = iP(ele2 + e2e1)

w = w + Wp = +,'(elez - ezel).

Using the flow rule (1 15), we readily conclude that

(130)

(131)

Note that, because of incompressibility, the stresses can be determined only to within an arbitrary pressure.

Finite-strain anisotropic plasticity and the plastic spin so1

7.3.1. History-independent back stress. Using (50)-(52), (118), (127) and (128), and we find the back seess to be

Q =k~(ntn~ - - z ~ z ) = kl[cos28(elel -e~e~)+sin2B(elez+ezei)l.

The stresses are determined using (137):

aiI = -ui2 = kt cos26

where B is given in terms of y by (129).

(138)

(139) CY 012 = - + k, sin28 a;, = (123 = U N = 0 v5

We consider next the particular case in which

(140) k. - - N-1 P , - :bay<, ei

where b and N are constants, 6: = lni;, and E, = pep)'/'. Figure 3 shows the variation of q2/aY and u;,/a, with y for N = 1 and b = 1.

Y

0 0.5 1 1.5 2 Y

Figure 3. Evolution of swsses in simple shear

Y

0 0.5 1 1.5 2

Y

Figure 4. Evolution of smsses in simple shear.

7.3.2. History-dependent back stress. The back stress is written as

The general form of the eigenvectors li is

11 =cos@el+sin@ez 12=-sin@et+cos@ez h = e s (142)

with 1'"" = Z1 , At y = Of, the. lis are aligned with the eigenvectors of DP, i.e. @ = 45" when y = 0. Using the above expressions for the lis, we find that the most general form of 01 is

a =allelel +azzezez+orse3e3+cuit(eiezfezei) (143)

502 N Aravas

where

=at cos2@ + rvzsin2 @ (YZ = azcos2@ + azsin' @

ff33 = ff3 a12 = ;(U1 -ff2)sin%$. (144)

Using (123), (125). (130) and (142), we can readily show that

(145) 2 w = y sin @(elez - S e , ) .

The spin of the eigenvectors of a is defined by w. so that

li = w. l j . (146)

Using (142) for li and (145) for w in the above equation, we can easily show that

(147) = - y sin 2 4 or @ = tan-I(l+ y ) - 1 . The evolution equation (124) for a can be written as

3 d: = A - w + W . a = c z ( l j . DP. ZJliZi -CY * w + w. a. (148)

i=l

Using equations (142). (130), (143) and (145) for lj, DP, a and w, respectively, we find

ci = y(kIzsin*@ + acsin4+)(elel - ezez)

+ ~ ( U Z Z -a11 +2ccosZ~)(e tez+eze l ) . (149)

Comparing (149) to (144), we conclude that

dall/dy = -druzz/dy = k l z s i n Z @ + $csin4@

daiz/dy = (an - ( Y I I + ~ c c o s ~ + ) s ~ ~ ~ +

(150)

(151)

da3s/dy = 0. (152)

When c = constant, integration of the above system of equations yields

where @ is given in terms of y by (147). We consider next the case in which c(P) = co?pN, where CO and N are constants. The

system of equations (150)-('I52) is integrated numerically using a forward Euler technique and the stresses are found using equations (137). The variation of UIZ and ui1 with y is shown in figure 4 for 0; = CO = uo =constant and N = 1.

Finite-strain anisotropic piasticify and the plastic spin 503

Acknowledgments

Fruitful discussions with Professor Y F Dafalias are gratefully acknowledged. This work was canid out while the author was supported by the NSF MRL program at the University of Pennsylvania under Grant No DMR-9120668.

References

Aravas N 1992 Finite elastoplastic t r ans fodons of transversely isotropic metals Inf. I. Solidr Stmcf. 29 2137-57 Aravas N and Aifantis E C 1991 On the geometry of slip and spin in finite plastic deformation id. I. Plasriciry 7

141-60 Bassani I L and Banerman S D 1987 Deformation induced anisotropy and neck propagation in polymers Consrimrive

Modeling for Nonfradiflonnl Materials AMD-vol85, ed V K Stokes and D Kmjcinovic (New York ASME)

Banerman S D and Bassani I L 1990 Yielding, anisotropy, and deformtion processing of polymers Polymer Eng.

Biot M A 1965 Mechanics of Incremental Deformations mew York: Wiley) Boyce M C, Parks D M and Argon A S I988 Large inelastic deformation of glassy polymers. Part I &-dependent

constiMive model Meck Marer, 7 15-33 Cauchy A-L 1841 M6moire sur les dilatations, les condensations et les rotations produits par un changement de

forme dans un systeme de paints mat6riels Er. Con. Phys. Mafh. 2 302130 Dafalias Y F 1983 On the evolution of structure variables in anisotropic yield criteria at large plastic vansformations

Colloq. Int. du CNRS no 3.51 on Crir>res de rupture des mafiriaza a sfwcfure infeme orient& (villnrd-de-hs, France. June 1983) ed J P Bcehler (Paris: Editions du CNRS), in press

- 1984 The plastic spin concept and a simple illustration of its role in finite plastic transformations Mech. Mafer. 3 2U-33

- 1985 The plastic spin J. Appl. Mech. 52 865-71 - 1987 Issues in constitutive formulation at large elwtoplastic deformation. Part 1: k i n e d c s Acta Meeh 69

- 1988 Issues in constitutive formulation al large elastoplastic deformation. Part 2 kinetics Acfa Mech. 73

Hahn H T 1974 A finite-deformation theory of plasticity Inf. I. Solids Strucf. 10 111-21 W R 1970 Constitutive inequalities for isotropic elastic solids under finite strain Pmc. R. Soc. A 314 457-72 Kratochvil J 1971 Finite-strah theory of crystalline elastic-inelastic materials I. AppL Phys. 42 IlO&8 - 1973 On a finite s w a b theory of elastic-inelastic materials A m Mech. 16 12742 Lee E H 1969 Elastic-plastic deformations al finite strains 3, Appl. Mech 36 1-5 Lee E 4 Mallett R L and Wertheimer T B 1983 Stress analysis for anisotropic hardening in finite-deformation

Liu I-S 1982 On representations of anisotropic invariants Inr. I. Eng. Scl. 20 1099-1109 Loret B 1983 On the effects of plastic rotation in the finite deformation of anisotropic elastoplastic materials Mech.

Mandel I 1971 Plasticif6 Classique et Wscoplarlicilp. Courses and Lectures, No 97, International Center for

- 1973 Equations constitutives et directeurs dans les milieux plastiques et vismplastiques Inl. J. Solids Sfmcr.

Naglegad I D and T B Wertheimer 1984 Constitutive equations for anisotropic large sVain plasticity Constitutive

Nemat-Nasser S 1990 Certain basic issues in finite-deformation continuum plasticity Menmica 25 223-9 Ogden R W 1984 Non-linear E h f i e Deformtiom (Chichester: Ellis Honvood) Parks D M, Argon A S and Bagepalli B 1984 Large elastic-plastic deformation of glassy polymers. Part I:

constitutive modeling M17 Program in Polymer Science Reporl Massachusetts InstiNte of Technology Pereda 1 J, Aravas N and Bassani J L 1993 r i t e deformation of anisotropic polymers Mech. Mafer. 15 3-20 Smith G F 1971 On isotropic functions of symmetric tensors, skew-symmetric tensors and vectors Int. I. Eng. Sci.

ligoiu S C and Soos E I990 Elastoviscoplastic models with relaxed configurations and internal state variables

pp 3 3 4 5

Sei. 30 1281-7

119-38

12146

plasticity I. Appl. Mech. 50 554-50

Mater. 2 287-304

Mechanical Sciences, Udine (Berlin: Springer)

9 7ZAO

Equafions: Macro and Compufafioml Aspem ed K J Willam (New York ASivlQ pp 73-86

9 899-1916

Appl. Mech. Rev. 43 131-51

504 N Aravas

Truesdell C and Toupin R A 1960 The classical field theories Hnndbuck der Phyjk vol IIM, ed S Flugge (Berlin:Springer) pp 226793

Van Der Giessen E 1989 Continuum models for large defarmatlon plasticity. Part I: large deformation phsticity and the concept of a natural reference stnte Eur. 3. Mech NSolids 8 15-34

Wang C-C 1970a A new representation theorem for isotropic functions: an answer to Professor 0 F Smith's criticism of my papen on representations of isotropic functions. p;Ur 1: scalar-valued isotropic functions Anh. Ration Mech Anal 36 166-91

- 1970b A new representation theorem for isotropic functions: an answer to Professor G F Smith's criticism of my papers on representations of isotropic functions. Part 2: vector-valued isotropic functions. symmtric tensor-valued isotropic functions, and skew-symmetric tensor-valued isotropic functions Arch. Ration. Mech. A n d 36 198-223

Zbib H M and Aifantis E C 1988 On the concept of relative and plastic spin and its implications to large deformation theories. Part I: hypwlasticity and vertex-type plasticity Acta Meeh I 5 15-33