Constitutive equations for elastoplastic bodies at finite ... - Springer Link

Acta Mechanica 100, 155-170 (1993)

ACTA MECHANICA ~D Springer-Verlag 1993

Constitutive equations for elastoplastic bodies at finite strain: thermodynamic implementation K. Ch. Le and H. Stumpf, Bochum, Federal Republic of Germany

Dedicated to the memory of Professor Theodor Lehmann (Received June 6, 1992; revised September 28, 1992)

Summary. Within the fi'amework of classical mechanics and field theories, equivalent sets of balance equations and entropy production inequality are formulated in spatial, referential and intermediate description for an elastoplastic body at finite strain The entropy production inequality formulated with respect to the intermediate configuration together with an assumption about the functional form of the free energy are used to substantiate new constitutive equations. Special attention is focused to the cases of small elastic/large plastic strain deformation (metal plasticity) and rigid-plastic deformation.

l

Introduction

When dealing with elastoplasticity at finite strain, one should take into account the following two phenomena. The first one is the irreversibility of the plastic deformation leading to the dissipation of energy (Farren and Taylor [1]) and the increase of entropy (Ziegler [2]). The second one concerns the microscopical mechanism of slip and the crucial idea of the relaxed intermediate configuration used for expressing the explicit independence of the free energy on the preceding plastic strain (Lee and Liu [3], Lee [4]). Therefore, thermodynamic requirements formulated with respect to the relaxed intermediate configuration should play a key role in establishing the constitutive equations for elastoplastic bodies. The present paper is devoted to this problem. The concept of the relaxed intermediate configuration is perhaps the most controversial one in finite strain elastoplasticity. According to Lee [4], to unstress the body undergoing nonhomogeneous plastic flow it must be considered cut into infinitesimal elements. Locally, one can map each element to IR3 (embedding) and use this map as a coordinate system (or a chart) to express the plastic deformation. The change of coordinate systems in overlapping domains must be forced to be diffeomorphic, otherwise one gets into trouble with the tensorial properties of the plastic deformation. This leads necessarily to the idea of using manifolds to describe globally the relaxed state of the body, if we want to maintain our consideration within the continuum mechanics. We can profit by the remaining freedom in choosing the geometrical structure of this manifold to ensure the incompatibility of the plastic strain. Namely, we introduce the intermediate configuration by decomposing the motion of an elastoplastic body into a plastic motion in some relaxed space considered as Riemannian manifold (Eckart [5], Sedov [6]), followed by an elastic motion. As consequence, one obtains the multiplicative decomposition of the deformation gradient, used by many modern authors to justify the existence of the local relaxed state (see [3], [4] and the review paper by Cleja-Tigoiu and Soos [7]). But in contrary to [3],

156

K. Ch. Le and H. Stumpf

[4] we view the relaxed space rather as original concept not implied by the multiplicative decomposition law. The kinematics of strain is developed, with strain measures defined on tangent spaces to the reference, spatial and intermediate configuration, respectively. Of special interest is the definition of objective rates for tensors referred to the intermediate and spatial configuration. This problem becomes very actual in our case, because the time-dependent intermediate configuration is chosen as reference configuration for the thermodynamic requirements. In this paper we use the intermediate Lie derivatives as objective rates for the intermediate strain tensors (Simo and Ortiz [8], Stumpf [9]). Within the framework of classical mechanics and field theories, we formulate a set of balance equations and entropy production inequality in integral form with respect to the spatial configuration for an arbitrary volume. Equivalent sets of equations in referential and intermediate description for an elastoplastic body at finite strain can be obtained by changing variables in the former equations and passing to their localized forms. The entropy production inequality formulated with respect to the intermediate configuration together with an assumption about the functional form of the free energy are used to substantiate new constitutive equations. It is shown that the associated intermediate energymomentum tensor and the intermediate plastic strain rate should enter the yield condition and the associated flow rule to ensure the positiveness of the plastic dissipation. Equivalent sets of constitutive equations in referential and spatial description are presented. Special attention is focused to the cases of small elastic/large plastic strain deformation (metal plasticity) and rigid-plastic deformation. In these cases the plastic stress tensor can approximately be replaced by the Cauchy stress tensor. Finally, a comparison with known results of the literature (Lee [4], Simo [10]) is carried out, which shows the advantage of our proposition.

2

Geometry and kinematics of elastoplastic bodies

In order to describe finite plastic deformations one should necessarily use the concept of manifolds. We begin by identifying a reference configuration of an elastoplastic body with a manifold N which can be embedded into the three-dimensional Euclidean space g3. The reference configuration N is assumed to be the initial undeformed configuration. Material points in ~ are denoted X. A motion of M is a one-parameter family of deformations or mappings 05~: ~ g3 specified by x = 05(X, 0

Xc~,

x ~ g 3.

(2.1)

The function 05 is supposed to be as many times continuously differentiable as required. The tangent of 05 is denoted F and is called the deformation gradient of 05; thus F = T05: TxN ~ Txg 3 is the linear transformation, where TxN and T J a are the tangent spaces to N' and 4 3 at X and x, respectively. Let X A and x a denote coordinate systems on N and C 3, respectively. Then the components of the two-point tensor F are given by

F ~ ( X , t) -- 005a (X, 0,

a, A = 1, 2, 3

(2.2)

Constitutive equations for elastoplastic bodies

157

We suppose that ~btis an invertible and orientation preserving mapping at each moment of time, so that det F ~ > 0,

VX c ~ .

(2.3)

Eckart [5], Eglit [11] and Sedov [6] were the first to introduce the concept of a local, current, relaxed state of the elastoplastic body (with stresses removed and changes of temperature reduced to zero). While the relaxed configurations cannot be defined globally in the Euclidean space g3, one could introduce mentally a motion of the body in some relaxed space r which can be treated geometrically as Riemannian manifold (cf. [6]). This motion of the body will be called the plastic flow. The space N3 need not necessarily be Euclidean, meaning that in general its curvature tensor does not vanish. Because the plastic deformation tensor will be defined as the pull-back of the metric o f ~ 3 by the plastic flow, the plastic strain does not in general satisfy the homogeneous compatibility condition. As we shall see later, the statement that ~3 is non-Euclidean is equivalent to the statement that plastic strain occurs in the body. Let us denote the motion in N3 by qSP: .~ --+ N3 or, equivalently

r = ~ ( x , t),

r = ~ ( x ~, O.

(2.4)

Here ~ denotes points in ~3, ~ are coordinates of ~. We use Greek indices to refer to the relaxed space N 3 With (2.1), (2.4) one can decompose the total deformation q~ as follows: = 4, ~o 4}",

x ~ = ~ea(+P~(XA, t), t),

(2.5)

where q~P and q~e: qSP(~) ~ d~ are called the plastic and elastic deformations, respectively (see Fig. 1). The functions ~bp and q~ are supposed to be as many times continuously differentiable as required. If q~Pand ~b~were mappings from g3 to d~ the assumption of their regularity would lead to vanishing curvature tensors obtained by using the plastic and elastic deformation tensors. This would be in conflict with the incompatibility of the plastic and elastic strain. Therefore, to maintain the regularity and the incompatibility consistent, it is necessary to introduce N3 as non-Euclidean manifold. As consequence of (2.5) we have the following multiplicative decomposition of the deformation gradient: r = FeF,

(2.6)

F~ = F%FP~,

F e

~

) q5pI~) c 2 a

Fig. 1. Schematic sketch of reference, intermediate and spatial configurations of their tangent spaces and the associated multiplicative decomposition of the deformation gradient

158

K. Ch. Le and H. Sturnpf

where F p : T x ~ ~ T ~ 3 and F ~: T~N3 ~ Txif3 are tangents of ~P and ~b~, respectively, r ~ = T4",

0qbp~ F~5 = 5 ~ ,

F e = TO e ,

F ca-

(2.7)

~.

(2.8)

~@ea

The decomposition (2.6) was postulated and actively used by Lee and Liu [3] to obtain adequate constitutive equations for elastoplastic bodies. However, it is difficult to imagine the deformation gradients F e and F p as linear mappings in Lee and Liu's approach, while the tangent spaces are not defined at all. With the relaxed space introduced above, this problem seems to be solved from the theoretical point of view. It should be emphasized that the relaxed space as well as the intermediate configuration are considered here as original concepts, not implied by the decomposition law (2.6). Of course, within the framework of the mechanics of Cauchy continua, one cannot determine the relaxed space. The situation here is similar to that of the infinitesimal theory, in which the intermediate configuration is undetermined up to an arbitrary infinitesimal rotation. In our opinion, constitutive equations for the plastic spin proposed in the literature cannot be used to determine the intermediate configuration (see e. g. Stumpf and Badur [13] and Lehmann [14]). A possible way to determine uniquely the intermediate configuration is to formulate finite elastoplasticity within the framework of oriented media and to consider the intermediate configuration as non-Riemannian manifold with torsion. This will be shown in a forthcoming paper. Fortunately, in the theory of this paper the intermediate configuration plays the role only as far as the thermodynamic requirement is concerned, and we shall see that the plastic flow q~P does not enter explicitly the entropy inequality. Moreover, by using push-forward/pull-back operations, one makes the plastic flow disappeared in the spatial and referential description. Also contrary to Simo [10], the relaxed space and the intermediate configuration are not endowed here with the metric tensor of the reference configuration, because this would mean that incompatibility due to the plastic deformation is impossible. Besides, Simo's assumption is also in contradiction to the covariance principle (see Section 5). Throughout this paper we shall use upper case letters to denote tensors defined on the reference configuration ~ c d~3, lower case ones for those on the current spatial configuration ~bt(N) ~ 8 3, and lower case letters with a superposed bar for those on the current relaxed intermediate configuration qS~(N) c ~3. Superscripts (.)e and (.)P denote "elastic" and "plastic" quantities, respectively, while an additional mark (.). indicates that the quantity does not consist of purely elastic or purely plastic deformations. Let W1, Wz be tangent vectors on T x ~ , which are transformed by push-forward operation (Marsden and Hughes [12]) to wl, w2 on Txo~s, respectively W1 = F W 1 ,

w2 = FW2,

w l a = FaAW1 A,

W2b = F b n W f .

(2.9)

To measure the lengths L1, L2 of W1, W2 and the angle O between them we need a metric tensor G: T x N x T x N --+ IR defined on the reference configuration. Thus L 1 L 2 c o s O = G ( X ) ( W I , W2),

L 1 L 2 c o s O = GABW1AW2 B.

(2.10)

The lengths 11, 12 of wl, w2 and the angle 0 between them in the deformed state can be measured with the help of a metric tensor g : T~d~ x T J 3 ~ IR on the spatial configuration as follows: 1112 cos 0 = g(x)(wl, W2),

11l2 COS 0 = gabWl"W2 b.

(2.11)


!59

When W,, W2 are known, these same lengths and angle can be calculated with the help of a Green deformation tensor C: T~N x T ~ ~ IR defined by C(X)(W1, I4/2) = g(x)(wl, w2),

C A B W 1 A W 2 B = gabwl~w2 b.

(2.12)

Thus, we say that C is the pull-back o f g to the reference configuration and denote this operation by qS*(g)([12]) c = ~*(g) = ergr,

c~.

= V"~goF.,

(2.13)

with F r : T~*g a ~ T x * ~ the transpose of F(a linear transformation of the dual cotangent spaces) I defined by (Fr/~)(lr =/~(FW) for fle T~*E 3 and W e TxN. Analogously, let us denote a metric tensor on ~3 by ~ and the pull-back and push-forward operations by qS*, q5e*, and ~b,, ~bg, respectively. We can then define total deformation and strain tensors on ~ , ~bF(~) and qS~(N) as follows:

1

c = 4),(63 = F-rGF-1,

e = -2 (g - c),

gv = qSP,(G) = F v - r G F P - 1 ,

1

e e = 4 e * ( g ) = F e T g F e,

(2.14)

1

= ~ (e " - e , ) ,

e = 5 ( c - 63.

By using pull-back q5P ~ and push-forward qS; operations we can also define the following elastic and plastic strain tensors: 1 E p = -2 (C p - 63,

C p = 40p*(~,) = Fpr,~Fp, 1

1

e e = ~ (e ~ - g ) ,

ep = ~ ~, - ep),

~e = O~(g) = F - ~ g F

-~,

(2.15)

1

~ = 5 (g -- ~e).

The plastic strain C p is the pull-back of ~ to the reference configuration. One can compute the "curvature t e n s o r " / d ' defined on N by using the tensor C p in place of G. Due to the fundamental property of the plastic strain, C p need not satisfy- any compatibility condition, or equivalently, C p cannot be derived from a displacement field within the Euclidean space g3. This means that in general the curvature K ~ does not vanish, Because the intermediate curvature tensor f of ~3 is the push-forward of K e [12], F also does not vanish. The same is true for k e, the curvature tensor on qSt(N) obtained from c e. Therefore, the assumption that the relaxed space ,~3 is in general non-Euclidean when plastic strain occurs is justified. F r o m (2.13)-(2.15) one can easily prove the following decomposition laws: E = E~ + E~,

E~ = 4~*(e ~) = F r a ~

= ~e + ~p, e = e ~ + e~,

(2.16.1-5) e~ = q~(6P) = F~-rFPF e-1

i Note the difference between our definition and that of Marsden and Hughes [12].

160


As we can see from (2.16) decompositions of the Green and Almansi strain tensors E and e into purely elastic and purely plastic parts are in general not possible. The deformation tensors E~ = E - E p and e~ = e - e e are not natural measures of the elastic and plastic deformations. Nevertheless, they can be used formally in uncoupling constitutive equations with obligation of push-forward/pull-back operations to return to the pure measures (Sedov [6], Green and Naghdi [15]). In contrary, the intermediate strain tensor i can be decomposed additively according to (2.16.3). Therefore, it seems to be comfortable to choose ~e and ~P as primary variables in the constitutive relations. However, the intermediate tensor ~ is defined on the unknown relaxed Riemannian manifold ~ 3 so to obtain basic equations in the observable spatial or the referential picture with known reference configuration one should always use push-forward/pull-back operations. F r o m (2.13)- (2.16) it follows also that for the special case of small plastic deformations with F v g 1 the approximate additive decomposition E ~ E ~ + E p is possible. Correspondingly, we obtain for small elastic deformations with F ~ g 1 the approximation e ~ e e + e p. Let us define the velocities of the total motion, the elastic motion, and the plastic flow by v--

Vo~ -1 ,

V~

ve = o~~ 5~-1,

oP = ~/70o ~)P - 1,

= ~ ~b"(X, t)x . . . . . t,

oe" = ~

~be"({, t)~ . . . . . ,,

~,pc~--__-& ~e=(X ' r)x . . . . . t.

(2.17)

(2.18)

(2.19)

F r o m the decomposition (2.5) it follows that = ~e + F ~

= v~ + ~;,(~,).

(2.20)

Note that the vector F~6p depends also on the elastic deformation, therefore it can be seen from (2.20) that the spatial velocity v cannot be decomposed additively into purely elastic and purely plastic parts. Let us consider now a decomposition of the rates of deformation. If the strain tensors are given on the reference configuration ~ , one can easily define their rates by taking the partial time derivative with X kept constant 1

1~

= ~ C "~- ~ -~ C ( X , t)l x . . . . . 1

t,

(2.21)

10

]E~ = -2 CP = 2 ~t CP(X' t)[x . . . . ~,.

(2.22)

Here and in what follows the dot is used to denote the partial time derivative with X kept constant. For those tensors defined on the current and current relaxed intermediate configurations the choice of their spatial rates is connected with some kind of arbitrariness due to the fact that the principle of objectivity alone does not uniquely determine objective rates. In this paper we shall use Lie derivatives (see e.g. [12]) as objective rates, which can be obtained by pull-back of the spatial object to the reference configuration, taking the partial time derivative with X kept constant and push-forward the result to the spatial configuration. The Lie derivative will also be formulated for tensors defined on the intermediate configuration ([8], [9]).


161

Let tensors t and i be defined on q~(~') and qStv(~), respectively. Then we have P~t = 4 ,

(2.23)

3 t ff)*(t)lx . . . . . ~ ,

~ / = q~ (~t qbP*(t-),x= oonst).

(2.24)

be

oo

ompon n

in s o m e

o

velocity vector, with components va. Then it is easy to show that the Lie derivative of t is the following ( : ) - t e n s o r (see, e.g. [12]) 3

~? ...... l

1.... Ova

(~t)}.2f, = ~ t~'".~,[. . . . . . t + ~?x~ rf...kv - U...k ~ x ~

(all upper indices)

~vt + tz".'.'.'l~x f + (all lower indices).

(2.25)

Analogously, for a (~)-tensor Z defined on qSf(N') we introduce the Lie derivative of f (~t-)4' .... .... = ~~ _

~

/~4'..... ....

-~.... -)..... .... Ogw ~ . . . . . , -~ ~ ~? t4' .... ~p2 __ t4' ~ --

(all upper indices)

&spz + t)..... ~ - + (all lower indices).

(2.26)

Here 6 p is the velocity of the plastic flow defined on the relaxed space ~3: gp~ = 3~p~/&. The difference between (2.26) and the previous definition of the intermediate Lie derivative given in [9], [13] is that, due to the regularity of qSP,the velocity gradient i v = iw/~- ~ now can be expressed as true gradient: [P = grad gP. Note that the Lie derivatives of scalar functions coincide with their material time derivatives, U=~+v'Vf=D,L

Thus, according to (2.16) we have the following decompositions of the strain rates: +/~,

(2.27.1)

d = P ~ = d~ + flv = f3U + ~ v ,

(2.27.2)

d = ~e = d~ + dg = ~e ~ + ~eg.

(2.27.3)

D = F. = D ~ + O p

=/~

Note that the tensor d~ = ~g~ depends also on the velocity of the plastic flow and therefore is not a pure measure of the intermediate elastic strain rate. Directly from the definitions (2.23), (2.24) it can be seen that !2c = O,

.~Op = 0,

52g = 2 d ,

f2~ = 2d p,

s

= 0,

(2.28)

~e v-* = 0.

(2.29)

These properties will play an essential role in the subsequent study of the constitutive equations.

162

3


Balance principles in spatial, material and intermediate description

We shall now present the basic balance equations and the Clausius-Duhem inequality for the etastoplastic body in spatial, referential and intermediate description, Let O(x, t), v(x, t), and O(x, t) be the mass density, the spatial velocity, and the temperature field, respectively, which have to be determined in the spatial description. Let ~' be any open nice subset o f ~ , with a boundary ~3~, and let dv and d c~be the volume and surface elements in g 3 respectively. Within the classical non-polar mechanics, we formulate the following balance principles and the Clausius-Duhem inequality in the spatial description (see, e.g., Truesdell and Toupin [16]):

dt

0 dv = 0 (conservation of mass),

(3.1)

4,t (~)

dt

Ov dv = Ct (~)

f

f

ob dv +

Ct (o1~)

an da (balance of momentum),

OCt(qz)

d f o(xxv)a~= f o(x•

f xx(an)d~

dt

ckt(o~)

4~ (og)

ag,t (ou)

(balance of moment of momentum), d--t

0 e+-~

f

(3.2)

-~ d,~ -

4,t (~)

f

q ' n d~ --0-

(entropy production inequality).

(3.5)

OCt(~

Here a(x, t) is the (contravariant) Cauchy stress tensor, e the internal energy density, q(x, t) the entropy and q(x, t) the heat flux vector. We regard the mass force b(x, t) and the heat source r(x, t) as given externally. Symbols (., .) and x are used to denote the scalar and vector products, respectively, and n is the unit outward normal to ~bt(q/) (one-form). Standard procedures enable one to pass to the localized form of the basic field equations (c.f. [12]) Dto + ~ div v = 0

(mass),

(3.6)

(momentum),

(3.7)

(moment of momentum),

(3.8)

QDtv = •b + div a

a = ar

#Dte+ div q = Qr + a- d (energy), V0

(3.9)

Q(tlDtO + D,O) - a. d + --0- "q 0.

(4.13)

with

I ~ = lO~(2p, 0, g - ~) __>0,

The dissipation functions IOp and lOb are supposed to be positive definite, convex and lower-semi-continuous with respect to their arguments tlp and ~, respectively. The symbol ~3is used to denote the sub-differential of convex functions (Moreau [25]). An example of the dissipation functions can be given for the case of isotropic materials with a generalization of yon Mises' and Fourier's laws referred to the intermediate configuration ~r0(2/3) 1/2 [~P6d~/~d~6]1/2 if ~r DP(dp, 0, ~- 1) = ( + ov otherwise, 1

= 0

(4.14)

(4.15)

Here ao is the yield stress in uniaxial tension experiment and z the thermal conductivity. From (4.15) it follows that v = 1/2 and (4.12.2) reduces to Fourier's equation (cf. [2]) = - x grad 0.

(4.16)

Further, it is convenient (but not necessary) to suppose that no volume change occurs in plastic flow so that ~=0o,

div 6 v = , ~ - l ' d P = O .

(4.17)

This condition excludes automatically the case, in which the dissipation function of (4.14) is equal to infinity. Note that while lob is a quadratic form of ~ (Onsager's principle), lop is only a homogeneous function of first degree with respect to dv. This can be explained by comparing the mechanism of heat exchange with the nature of plastic flow. In the latter case a certain amount of the plastic (dissipative) stress should be reached to initiate the plastic flow (similar to the mechanism of dry friction). From the other side it can be shown that Eq. (4.12.1) is equivalent to the conventional yield condition in the plastic stress space (with convex yield surface) and the associated flow rule if and only if the dissipation function is homogeneous of first degree with respect to d p (cf. Ivlev [26]). The Eq. (4.12.1) then establishes the (nonsingle-valued) connection between/i and alP. However, it is worthwhile to emphasize that the dissipation itself is uniquely determined by dr. To rewrite (4.12.1) in the more usual form, we apply the Fenchel transformation [25] to the dissipation function lop to obtain the so-called plastic potential ~(ff, 0, ~). Then the


167

constitutive equation (4.12.1) is equivalent to the flow rule d p = 0e@(/i, 0, R).

(4.18)

For the dissipation function (4.14) the plastic potential has the form

qS(fi, 0,~) =

if oo

~,~p~ _ 3 g~pg,~

f ~ fTa

< Oo(2/3) ~/2

otherwise.

(4.19) (4.20)

We present also the above constitutive equations in referential description by pulling them back with F p and those in spatial description by pushing forward with F e, in full agreement with the covariance principle. Constitutive equations in referential description 1

tp = det F p - - aS(C, O, BP), 0o 0~' S = 20o ~ , P = BPM,

OF' N = -- a--~, M

=

(4.23)

CS,

[BpAC BvBV j ~ E ~ o ] */2

( + oo

(4.21)

(4.22)

- Oo ~PI +

DP(E p, O, B p) = ~ 0"~

p = O~ID p,

B p = (C p)- * = (G + 2E p)- *,

if

BPABE~B -=- 0

otherwise,

00 QA = _ zBpA~ ~OX "

(4.24)

(4.25)

Constitutive equations in spatial description

= (det F e) - , _1 a3(g, 0, be), Q a = 2 0 a~ = = bep,

(p(p, O, Ce) =

b e = (c e)-I = Fe~- 1 F r '

0~ 00'

(4.27) (4.28)

It = - - o ~ i + g a ,

if

ceccge _

oo

otherwise

5d(e -- e e) = O,,~o(r~, O, ee),

(4.26)

3

C~cCc~a pabpCa

aO

q" = - - x b e"~ - -

Oxb "

< ao(2/3)i/z

"~"

(4.29)

(4.30)

Here n is related to p by = f / J p = j / J ~;(ff) = j / J FeffFer .

(4.31)

As we can see from (4.23), (4.24), the constitutive equations in the referential description depend also on the plastic deformation B v. The same is valid for the constitutive equations (4.28), (4.29) in the spatial description, which depend also on the elastic deformation b e. Thus, within the spatial

168


(referential) description, anisotropy appears at large elastic (plastic) strain. The constitutive equations (4.26)- (4.31)(Eqs. (4.21)- (4.25)) together with Eqs. (3.6)- (3.11) (Eqs. (3.12)-(3.17)) compose the closed system of equations for elastoplastic bodies in the spatial (referential) description.

5

Special cases and comparison with known results

The constitutive equations proposed in the preceding Section can be simplified in the case of large plastic/small elastic strain deformation and small deviations of the temperature from a value 00 (uniform base temperature). This is an adequate assumption for many metal forming problems. The free energy per unit intermediate volume is then given by (in the infinitesimal case a similar caloric equation is proposed by Ziegler [27]) ~

= ~ 0 o - 0~o(0 - 0o) + ~1 ~.(g~e~)2 + ..,,~'~,v~e ~ ~ ~ # ae _ (32 + 2y) Ct~P~p(0 0c

~ o (0 - 0o) z .

00) (5.1)

Here ~Po and rio are the specific free energy and the specific entropy in the state (:e = 0, 0 = 0o), respectively, 2 and ~ are Lamb's moduli, ~ is the coefficient of thermal expansion and c is the specific heat capacity. Applying (4.6.1) to (5.1), we get the stresses U~ = i~P~:Y~a + 2kt~:~'a~a - (32 + 2#) ~ : ' ( 0 - 00).

(5.2)

Let us calculate the plastic stresses according to (4.9.2)

p~ = - 0 ~ ~e + g~(~, + 2 ~ ) ~ .

(5.3)

Because ~e ~ 1 and 0 - 0o ~ 1, one can replace (5.3) by g=e to give p ~ s.

(5.4)

Thus, we can use the stress tensor g instead of/7. Let us find the plastic potential in the spatial configuration. Because c~=2e ~ + g ~ g ,

f/J~l

for

Fe~l,

(5.5)

we can replace (4,29) by q)(~, 0, g) =

if

g,~gbe- ~ ga~g~d a ~a~a

=< a0(2/3)~/2

(5.6)

OO otherwise. This is exactly yon Mises' plastic potential, leading to the corresponding associated flow rule ~(e -- e ~) = $~q)(~, 0, g).

(5.7)

We consider now the limiting case, when 2, #, a --+ co,

0 - - 0o'-+ 0,

g~ --+ 0,

P-+I,


169

so that according to (5.2)

(5.8)

g~P = "~g~Pg~6G6+ 2/~g~g~6G6 - (32 + 2~L)c~a(0 - 0o) ~ some finite value.

This corresponds to purely plastic deformation and we call such a body rigid-plastic. One can easily prove, that the free energy of the rigid-plastic body tends to zero, and all power of the external forces is dissipated into heat. The plastic stress tensor g is equal to the Cauchy stress tensor a, J = 1, and the flow rule has the form

(5.9)

,~e ~ = ~e = d = 8,~q~(a, 0, g),

with the plastic potential q) given by (5.6). Let us compare our constitutive equations (4.6), (4.14), (4.18), (4.21)-(4.30) with those proposed by Lee [4]. The Eqs. (18) and (19) in [4] for stress and entropy rewritten in our notation coincide with (4.6). At the same time, the flow rule proposed by Lee ([4, Eq. (40)]) differs from (4.18) if the elastic strain is not small. When the hardening is neglected, Lee's flow rule can be written in our notation as follows: d~ =

a.e(x,

0),

~ = J/fir.

(5.10)

Inconsistency arises in (5.10) concerning the appearance of the spatial stress tensor x on the right hand side of (5.10). This is due to Lee's assumption about the special form of F~:F ~ = Ve (the elastic stretch tensor) and R e = 1 (no elastic rotation). Our Eq. (4.18) is free from such special choice and can therefore be applied for a broader class of materials and problems. Let us consider the model of elastoplastic bodies proposed by Simo [10] (see also Mfiller-Hoeppe [28]). As it was noticed above, Simo has identified the metric tensor of the intermediate configuration ~ with G. But the equation ~p6A=CSB~ = GA~ is in contradiction to the covariance principle. In fact, every coordinate transformation in N can destroy this equation. With this assumption, Simo has introduced the following measures of elastic and plastic deformation [10, Eqs. (1.2a), (1.4)] Cg = F p T G F , b~ = F e G - 1 F eT,

C~AB = 6~CFP~GcD6aDFP~, b eab• = Feac~A~GABFeOp(~B~.

(5.11) (5.12)

It is easy to show that C~_(b$) is not a tensor defined on the reference (spatial) configuration. Considering for simplicity the isochoric motion, we can present Simo's free energy in the form [10, Eqs. (2.7 a)] 1

"beab

(5.13)

This is not a scalar equation, because every coordinate transformation in N leads to a change of energy. As consequence, applying Doyle-Ericksen formula one obtains from (5.13) a stress measure which is not a spatial tensor at all.

Acknowledgement The financial support of this work by DFG under contract ST 135/1-- 1 is gratefully acknowledged.

170

K. Ch. Le and H. Stumpf: Constitutive equations for elastoplastic bodies

References [1] Farren, W. S, Taylor, G. I.: The heat developed during plastic extension of metals. Proc. Roy. Soc. London A107, 422-451 (1925). [2] Ziegler, H.: An attempt to generalize Onsager's principle, and its significance for rheological problems. ZAMP 9, 748-763 (1958). [3] Lee, E. H., Liu, D. T.: Finite strain elastic-plastic theory with application to plane-wave analysis. J. Appl. Phys. 38, 19--27 (1967). [4] Lee, E. H.: Elasto-plastic deformation at finite strains. J. Appl. Mech. 36, 1 - 6 (1969). [5] Eckart, C.: The thermodynamics of irreversible processes. 4. The theory of elasticity and anelasticity. Phys. Rev. 73, 373-382 (1948). [6] Sedov, L. I.: Foundations of the non-linear mechanics of continua. Oxford: Pergamon Press 1966. [7] Cleja-Tigoiu, S., Soos, E.: Elastoviscoplastic models with relaxed configurations and internal state variables. Appl. Mech. Rev. 43, 131-151 (1990). [8] Simo, J. C., Ortiz, M.: A unified approach to finite deformation plasticity based on the use ofhyperelastic constitutive equations. Comp. Meth. Appl. Mech. Eng. 49, 2 2 1 - 2 4 8 (1985). [9] Stumpf, H.: Theoretical and computational aspects in the shakedown analysis of finite elastoplasticity. Int. J. Plasticity (in press). [10] Simo, J. C.: A framework for finite strain elastoplasticity based on maximum plastic dissipation and multiplicative decomposition. Part I. Continuum formulation. Comp. Meth. Appl. Eng. 66, 199-219 (1988). [11] Eg•it• M. E.: Tens•ria• characteristi•s •f finite def•rmati•ns (in Russian). Prik•• Mat. Mekh. 24, 94 7 - 95• (1960). [12] Marsden, J. E., Hughes, T. J. R.: Mathematical foundations ofelasticity. New Jersey: Prentice-Hall 1983. [13] Stumpf, H., Badur, J.: On missing links of rate-independent elastoplasticity at finite strains. Mech. Res. Comm., 17(5), 353-364 (1990). [14J Lehmann, T.: Some remarks on so-called plastic spin. ZAMM 71, 273--275 (1991). [15] Green, A. E., Naghdi, P. M.: A general theory of an elastic-plasticcontinuum. Arch. Rat. Mech. Anal. 18, 251-281 (1965). [16] Truesdell, C., Toupin, R.: The classical field theories. In: Handbuch der Physik Bd. III/3 (Truesdell, C., Noll, W., eds.) pp. 226-793. Berlin Heidelberg: Springer 1960. [17] Eshelby, J. D.: The force on an elastic singularity. Phil. Trans. Royal Soc. London A 224, 87 - 112 (1951). [18] Stumpf, H., Le, K. Ch.: Variational principles of nonlinear fracture mechanics. Acta Mech. 83, 2 5 - 3 7 (1990). [19] Epstein, M., Maugin, G. A.: The energy-momentum tensor and material uniformity in finite elasticity. Acta Mech. 83, 1 2 7 - i 3 3 (1990). [20] Coleman, B. D., Noll, W.: The thermodynamics of elastic materials with heat conduction and viscosity. Arch. Rat. Mech. Anal. 13, 167-178 (1963). [21] Miiller, I.: Thermodynamik. Diisseldorf: Bertelsmann 1973. [22] yon Mises, R.: Mechanik der plastischen Form/inderungen yon Kristallen. ZAMM 8, 161--185 (1928). [23] Hill, R.: A variational principle of maximum plastic work in classical plasticity. Q. J. Mech. Appl. Math. 1, 1 8 - 2 8 (1948). [24] Drucker, D. C.: A more fundamental approach to plastic stress-strain relations. In: Proceeding of the first US national congress of applied mechanics, pp. 487-491. New York: ASME 1951. [25] Moreau, J. J.: Sur les lois de frottement, de viscosit~ et plasticit& C. R. Acad. Sci. Paris S&. 1 Mat. 271, 608-- 611 (1970). [26] Ivlev, D. D.: Dissipative function in the theory of plastic media. Dokl. Akad. Naak SSSR 176 N5, 1037-1039 (1967). [27] Ziegler, H.: An introduction to thermomechanics. Amsterdam: North-Holland 1977. [28] M/iller-Hoeppe, N.: Beitr/ige zur Theorie und Numerik finiter inelastischer Deformationen. Dissertation, Bericht-Nr. F 90/4, Universitfit Hannover 1991. Authors' address: Prof. Dr.-Ing. H. Stumpf and Dr. Phys. Math. Sc. K. Ch. Le, Lehrstuhl fiir Allgemeine Mechanik, Ruhr-Universit/it Bochum, D-44780 Bochum, Federal Republic of Germany