Weber Lushg Zavaliangos Anand 1990

International Journal o f Plasticity. Vol. 6, pp. 701-744. 1990 0749-6419/90 $3.00 + .00 Printed in the U,S.A. Copyright © 1991 Pergamon Press pie

A N O B J E C T I V E T I M E - I N T E G R A T I O N P R O C E D U R E F O R

I S O T R O P I C R A T E - I N D E P E N D E N T A N D R A T E - D E P E N D E N T

E L A S T I C - P L A S T I C C O N S T I T U T I V E E Q U A T I O N S

G. G. WEBER,* A. M. LUSH,'[" A. ZAVALIANGOS, and L. ANAND~

Massachusetts Institute of Technology

Abstract-In a large class of rate-independent and rate-dependent elastic-plastic constitutive equations the elasticity is modeled in hypoelastic form, with the stress rate being taken as the Jau- m a n n derivative, so as to make the constitutive model properly frame-indifferent or objective. Here, we present a fully-implicit, stable time-integration procedure for implementing such constitutive models in displacement-based finite element procedures. The numerical procedure preserves the very desirable feature of incremental objectivity. The overall procedure is a gen- eralization of the well known "radial-return" algorithm of classical rate-independent plasticity, and it is therefore well suited for implementation in large-scale finite element codes. As an example, we have implemented the time-integration procedure in the finite element code ABAQUS. To check the incremental objectivity, accuracy, and stability of the algorithm some representative problems are solved.

i. INTRODUCTION

The simplest, and most commonly used phenomenological theory for infinitesimal elastic-plastic deformations of polycrystalline metals at absolute temperatures 0 less than approximately (l/3)Om, where Om is the melting temperature of a material in degrees absolute, is the classical, rate-independent, flow theory with isotropic hardening. This theory has the following main ingredients:

1. The state of a material element at any given time is characterized by the applied Cauchy stress T, and a scalar internal variable called the isotropic deformation resistance or theflow strength. Here, we denote this scalar state variable by s; it is positive valued, and it has dimensions of stress.

The constitutive model consists essentially o f a coupled set of differential evolution equations for the state variables (T,s) .

2. The stress rate is given by a linear isotropic function of the elastic strain rate, which in turn is defined as the difference between the total strain rate and the plastic strain rate.

The plastic strain rate is given by a flow rule. No plastic flow is possible as long as a scalar equivalent tensile stress is less than a critical value--the isotropic deformation resistance. Plastic flow may occur when the value of the equivalent tensile stress equals the isotropic deformation resistance. Whether or not plastic flow

*Now with the Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, PA. "['Now with Alliant Computer Corporation, Littleton, MA. SAuthor to whom all correspondence should be sent.

701

702 G.G. WEsEgetaL

.

4.

actually occurs when this yield condition is satisfied depends on a loading/unloading criterion. When plastic flow does occur, it produces no volume change, and its tensorial direction is parallel to the outward unit normal to the yield surface at the current stress point. The rate of change in the deformation resistance is linearly related to an equivalent tensile plastic strain rate through a hardening function. The equivalent tensile plastic strain rate is defined by a consistency condition, which ensures that the evolution equation for the stress and the evolution equation for the deformation resistance are consistent with each other.

This classical rate-independent model has been generalized to a frame-indifferent form by Hn~L [1958,1959]. For a discussion of an appropriate loading/unloading criterion for non-hardening and softening materials see Hn.L [1967] and HUGHES [1984].

It has long been recognized that the notion of rate-independence of plastic response is only a convenient approximation at low homologous temperatures. Even at low temperatures plastic flow due to dislocation motion is inherently rate-dependent (e.g., Gn-

[1966]). Various extensions of the classical theory to model rate-dependent behavior have been proposed in the literature. We find the state variable formulation of RICE [1970,1971,1975] most attractive because of its strong physical basis. (For a recent large deformation, state variable formulation of a rate-dependent model see AN,~D [1985] and BROWN, IOM & ANAND [1987].) This rate-dependent model differs from the rate- independent model in that there is no yield condition, and no loading/unloading criterion is used. Instead, plastic flow is assumed to occur at all nonzero values of stress, although at low stresses the rate of plastic flow may be immeasurably small.t Further, the equivalent plastic strain rate, which is determined by the consistency condition in the rate-independent model, needs to be prescribed by an appropriate constitutive function in the rate-dependent model.

These finite strain, rate-independent, and rate-dependent state-variable constitutive models are expected to be useful for describing the deformation behavior of initially isotropic materials up to deformation levels where significant polycrystal texturing of the metal has not developed. These models possess an elegant mathematical structure and attractive numerical features that make them suitable for obtaining numerical solutions to initial/boundary value problems of engineering importance.

While the numerical implementation of the infinitesimal strain version of the rate- independent model into displacement based finite element procedures dates back to

1964 (Wn.Kncs [1964]), progress in the numerical implementation of the finite strain, frame-indifferent version of the rate-independent model has been slow. The early papers in this field were those by HmBrrT, M~c,~z. and RIcE [1970], NEEDLE~ [1972a,b], and OstAs and SW~DLOW [1974]. The state of affairs up to -1975 has been reviewed by McMEEK~G and RIcE [1975]. In their paper, McMeeking and Rice also presented their now widely used, "up-dated Lagrangian Formulation" based on Hill's rate equation o f virtual work [1959]. It has been recognized for some time now that a numerical scheme based on a rate-variational statement can lead to the drifting of the updated configurations of the body away from equilibrium, and various "corrective procedures" have been suggested in the literature (e.g., NEEDLEM.~,~ [1985]) tO minimize such tendencies.

tThe theory considered here differs from a variety of theories in the literature which retain an on/off switch for viscoplastic flow; cf., for example, PmzZYNA 11963].

Objective time-integration procedure 703

Numerical implementations based on the total equation of virtual work exist, and the most widely used are those in the commercial finite-element programs ABAQUS [1988], ADINA [1988], and MARC [1988].

In contrast to the large number of publications on the numerical implementation of rate-independent plasticity, there is a scarcity of published papers on the numerical implementation of the finite strain rate-dependent models. The purpose of this paper is to present a unified treatment of the time-integration problem for both the rate-independent and rate-dependent constitutive models. The plan of this paper is as follows. After introducing our notation in Section II, we review the basic structure of the simple isothermal rate-independent and the rate-dependent constitutive models in Section III. The evolution equation for stress in both models employs the Jaumann derivative, which renders these models properly frame indifferent. This entails the use of a hypoelastic relation to model elasticity. It is well known that in the absence of plastic flow, hypoelastic equations for stress lead to dissipation. However, it is easy to show (e.g., A s ~ [1985]; NFF-DLEMA~ [1985]) that the hypoelastic form of the constitutive equation for stress is a good approximation to a proper hyperelastic equation under situations where elastic stretches remain infinitesimal. This is a good approximation for most metals.

We start our discussion of the time-integration procedures for the constitutive models considered here, in Section IV. An important characteristic of a time-integration algorithm for formulations which use a hypoelastic relation to model elasticity is that it should be "incrementaily objective." The formulation of numerical time-integration schemes which ensure objectivity during computations has been bothersome. Important contributions to the formulation of objective time-integration algorithms have been made, for example, by HUGHES and WING•T [1980], NAOT~AAL and VELDPAUS [1982, 1984], RUnINSTEIN and ATLURI [1983], REED and ATLURI [1983,1985], HUCHES [1984], and NAGTE~AAL and REnELO [1986]. A detailed examination of this issue has recently been made by WEnER [1988], and the discussion of incremental objectivity presented here is taken from his thesis.

In Section V we detail the algorithm for updating the state variables (T,s) for the rate- independent constitutive model, and in Section VI we detail the corresponding algorithm for the rate-dependent model. We limit our discussion in this paper to the consistent and stable Euler backward time-integration procedure. For the rate-independent model this integration procedure corresponds to the well-known "radial-return" algorithm of WH.K~S [1964]; also, see IO~G and IC, tmo [1977] and SCHRE',rER, KULAK, and ~ [1979]. For a recent review and discussion of the generalized trapezoidal and the generalized midpoint integration algorithms, of which the Euler backward method is a particular case, see ORTIZ and PoPov [1985]. These authors indicate that for large strain increments (many times the yield strain, as would be desirable for metal working type applications) the Euler backward method is to be preferred because of the stability characteristics of the algorithm.

In Section VII the principle of virtual work is linearized to obtain a consistent, closed- form elasto-viscoplastic tangent operator (the "Jacobian") for use in solving for global balance of linear momentum in implicit, two-point, deformation-driven finite element algorithms.

In Section VIII we briefly discuss certain aspects of the implementation of our constitutive equations and integration algorithm in the finite element program ABAQUS [1988], and we present a few examples which illustrate the excellent performance of our algorithms. We conclude the paper in Section IX, with some closing remarks.

704 G. G. WEBER et al.

Ii. NOTATION

For the most part we shall use notat ion which is standard in modern continuum mechanics (cf., e.g., GORTII~ [19811):

x = i (p , t ) tBt = i ( ~ o , t ) p = $ ( x , t ) F(x,t) ffi (O/cgp)£(p,t), det F(x,t) > 0 .~(p,t) • (a /Ot )x (p , t ) v = ~(~(x,t),t) L(x,t) ~ (O/ax)v(x , t ) D(x,t) - sym L = (I/2)(L + L r) W(x,t) E skw L = (1/2)(L - L r) T(x,t) T' ~ T - (l /3)(tr T)I T v I= 1" - WT + TW b(x,t) z = £ ( p , r) z : i (p (x , t ) , r ) ~ i , (x , r ) Ft(x,~') = (0/0x)i t (x,r) , det F~(x,t) > 0 • , = i ( ~ o , r ) ut(x,r) ~ z - x Sl(x,r) -- [det F t (x , r ) )T( i t (x , r ) , r ){Ft (x , r ) ) - r b~(x,r) ~- (det F t (x , r ) lb( i t (x , r ) , r )

material point of a body in a reference configuration tB 0 at time 0; motion; configuration of the body at t; reference map; deformation gradient; velocity; spatial description of velocity; velocity gradient; stretching; spin; Cauchy stress; Cauchy stress deviator; Jaumann derivative of Cauchy stress; body force per unit volume; place occupied by p at time r > t; relative motion; relative deformation gradient; configuration of the body at time r; incremental displacement at time r; relative first Piola-Kirchhoff stress; relative reference body force.

For brevity, and whenever there is no danger of confusion, we will omit the arguments (x, t) , (p, ~) and so forth, for the various field quantities listed above.

!11. CONSTITUTIVE MODELS

At each time, the variables governing the response of a material element are taken to be the pair (T , s ) , where T is the Cauchy stress, and s is a scalar internal state variable, with dimensions of stress, called the isotropic deformat ion resistance. It represents an averaged isotropic resistance to macroscopic plastic flow offered by the underlying "isotropic" strengthening mechanisms such as dislocation density, solid solution strengthening, subgrain and grain size effects, etc. The isothermal constitutive models that we shall consider here consist o f a pair o f coupled differential evolution equations for these two state variables. In the next subsection, we shall consider a rate-independent model (HmL [1958,1959,1967], MCMEEKIr~O and Rlcr~ [1975], HUGHES [1984]), and then we shall consider a rate-dependent model (Ar~AND [1982,1985], BROWN, KIM and A ~ [1989]). The rate-independent model employs the classical Mises yield surface, the associated normality flow rule, and a switching rule which differentiates between elastic and elastic-plastic processes. In contrast, in the rate-dependent model, al though the "direction" o f plastic flow may be thought to be directed normal to a Mises plastic po- tential surface, no yield condition and switching rules are employed. Plastic flow is assumed to occur for all nonzero values of stress, although at low stresses the rate of flow may be immeasurably small. This rate-dependent model belongs to the so-called class of "unified constitutive equations" in which "plasticity" and "creep" are unified, in that they are described by the same set o f flow and evolutionary equations. For a recent review of such models see LZSDnOLM [1987].


III. l. Rate-independent model

The evolution equation for stress T is taken as2:

T v = £ [ D - DP], (1)

where

,g - 2#0 + (r - (2/3)/~)I ® 1 (2)

is the fourth-order elastic stiffness tensor, with/z > 0 and r > 0 the elastic shear and bulk moduli . Also, 0 and 1 are the four th-order and second-order identity tensors, respectively.

D p is the plastic stretching tensor which is given by the flow rule:

D p = x3xl3"~PN p. (3)

Here, N p is a deviatoric (for plastic incompressibility) unit tensor defining the "direc- t ion" o f plastic flow, and

~P ~= ~](2/3)DP'D p >- O,

is the equivalent tensile plastic strain rate, which is (proportional to) the "magnitude" of the rate of plastic flow. It is a positive valued scalar parameter which is determined by the consistency condition to be considered below.

The parameter X is a switching parameter. When X = 0 we call the deformation process elastic, and when X = 1 we call the process elastic-plastic. The conditions under which X has a value 0 or l are based on the following notions of a yield condition and an unloading/loading criterion. Let

# -- ~/(3/2)T' .T ' (4)

define an equivalent tensile stress. Then, in this model # _< s is the yield condition, = s defines the current yield surface in the stress space, and

N ~= (3-~(T'/8) (5)

is the outward unit normal to the yield surface at the current stress point. If the current stress is within the yield surface, or if it is on the yield surface and a trial stress rate ,g [D] points towards the interior side o f the tangent plane to the yield surface, that is, if N- ,g [D] _.< 0, then the process is defined to be elastic and X = 0. Otherwise, X = 1, and the process is elastic-plastic (HtrcnEs [1984]):

"Hnx [1958] and McMEsgtr~G and Rice [1975] have recommended the use of the stress rate (T r + T tr D) instead of T v. Indeed, such a stress rate is also suggested by a finearization of a proper hyperelastic constitutive equation under the assumption of small elastic stretches. However, if the linearization is carried out in a consistent fashion, then it is the Jaumann derivative of the Canchy stress that appears in the rate constitutive equation for the stress. Further, since plastic deformation is assumed to be incompressible, the difference between the use of (1 "v + T tr D) and T ~ will be barely detectable computationally for typical situations where elastic stretches are small.

706 G.G. WEaEa et al.

[ ~ i f 6 < s , o r i f 6 = s a n d N . ~ [ D ] _ < ~ , l X = i f 6 s a n d N . ~ [ D ] > " (6)

Note that because ~ is isotropic,

N . ~ [ D ] = 2#N.D,

and, thus, the sign of (N.D) is the same as that of (N.-e.[D]). In the classical theory, the direction of plastic flow is taken to be in the direction of

the outward unit normal to the yield surface at the current stress point ("associated/normality flow rule"):

N p = N. (7)

Next, the evolution equation for the deformation resistance s is taken as

g -- h~ p, (8)

where

h = h ( s ) (9)

is a hardening function. The material is said to be strain-hardening or perfectly plastic according as h > 0 or h = 0, respectively.

During a plastic process the pair (T,s) must continue to satisfy the yield condition O = s. This is feasible only if the consistency condition

~ = ~ (10)

is satisfied. Using the result ~ = ~/3/2N.1" = 3q~-~N-T v, together with the evolution equation (8), we obtain the following condition for the evolution equation for the stress and the evolution equation for s to be "consistent":

~'P = 3,~13-~h-IN.T v. (I I)

Next, using (I I), (7), (3) and (I), we may rewrite ~P as

~P = 3~-~g- IN-d~[D] , (12)

where

g E h + (3/2)N.,g[N] = (31~) (l + ~ ). (13)

Finally, from the evolution equation (8) and the consistency condition (I0), we have that

d6 h---

d~p,


and thus h is equivalent to the (in general varying) slope of the true stress versus logarithmic plastic strain curve obtained from a uniaxial tension or compression test.

In summary, the classical rate-independent constitutive model for isotropic elastic- plastic solids with a Mises yield condition, an associated flow rule, and isotropic hardening consists of the following set of coupled evolution equations for the state variables (T,s):

Here,

Ill. 1 .a. Evolution equation for the stress T:

T v = ,g[D - DP]. (14)

• g ffi 2tL~t + (x - (2/3)#)1 ® 1 (15)

is the elasticity tensor, with Iz > 0 and x > 0 the elastic shear and bulk moduli. The plastic stretching tensor D p is given by the flow rule:

D p = X3x['a-~PN, (16)

where

N = , ~ ~ ( T ' / ~ ) , (17)

is the unit outward normal to the yield surface at the current stress point,

a ,= ",/(3/2)T'-T', (18)

is the equivalent tensile stress, and ~P is the equivalent tensile plastic strain rate given by

t p = 3vr3-~g-~N.~[D], (19)

where

g = h + (3/2)N.,g[N] = (3~)( l + 3~ ) . (20)

The parameter X is a switching parameter with values

X = [ ~ i f 0 < s , orif0=sandN.~[Dlif#=sandN.~[D]_><0,10 " (21)

For later use, note that by using (16), (17), and (19) the evolution equation for the stress (14) may alternatively be expressed as

T v = ~ [ D ] , (22)

where

e - ~. - X(3/2)g-~M ® M, (23)

708 G . G . WEBER et al.

is the fourth order elastic-plastic stiffness tensor, with

M E . g [ N ] .

III. 1 .b. Evolution equation for the deformation resistance s:

where

is a hardening function.

(24)

(25)

is the direction of plastic flow,

where

III.2. Rate-dependent model

The rate-dependent constitutive model for isotropic elastic-plastic solids with isotropic hardening consists of the following set of coupled evolution equations for the state variable (T,s):

III.2.a. Evolution equation for the stress T:

T v = P-[D - DP]. (27)

The plastic stretching tensor D p is given by the flow rule:

O p = 3x]~6PN, (28)

N = 34]-~(T'/8), (29)

8 - x/(3/2)T'.T ', (30)

is the equivalent tensile stress, and ~P is the equivalent tensile plastic strain rate prescribed by a constitutive function

tp = f(O,s). (31)

Ill.2.b. Evolution equation for the deformation resistance s:

= g(Ü,s) = h~ p - ~, (32)

To complete this rate-independent constitutive model for a particular material, the material properties/functions that need to be specified are the elastic shear and bulk moduli, # and r, respectively; the initial value of the deformation resistance s; and the hardening function/~ (s).

h = h (s) (26)


where

h = h(O,s) (33)

is a hardening function, and

P=~(s) (34)

is a static recovery function. To complete this rate-dependent constitutive model for a particular material the

material properties/functions that need to be specified are the elastic shear and bulk moduli, t~ and K, respectively; the constitutive function (31) for the equivalent tensile plastic strain rate; and the initial value and evolution function (32) for the deformation resistance. Even for a particular material it is not expected that the same constitutive functions for ~P and $ will be appropriate for all values of strain-rates and temperatures) Indeed, different particular forms for these functions will, in general, be necessary for different regimes of strain-rate and temperature. These forms should reflect the dominant features of the underlying microstructural mechanisms which govern the material response in the regime under consideration. For high temperature deformation of metals at moderate strain rates, specific forms for the flow function f and the evolution equation g have been given by AUAUD [19821 and BRows, KIM and ANAUD [1989].

Note that in the rate-dependent model there is no switching parameter which turns plastic flow off or on; plastic flow is assumed to occur at all nonzero values of stress. Further, the equivalent plastic strain rate, which is determined by the consistency condition in the rate-independent model, needs to be prescribed by a constitutive function. Since there is no yield condition to be satisfied in the rate-dependent model, there is also no consistency condition which needs to be satisfied in this model. The overall mathematical structure of the rate-dependent model is simpler because the plastic flow rule is a smooth function, although the particular form of the constitutive function for ~P may be mathematically very stiff in certain regions of plastic flow, requiring special care in formulating numerical algorithms.

IV. COMPUTATIONAL ASPECTS. INCREMENTAL OBJECTIVITY

In typical "implicit" finite element procedures which use nonlinear constitutive models, the discretized principle of virtual work, which enforces equilibrium and boundary conditions in a weak sense, generates an estimated incremental displacement field which is used to calculate the integration point values of the stress T and other field variables at the end of a time increment. If these stresses do not satisfy the principle of virtual work at the end of the increment, then the estimate of the incremental displacement field is revised, and new end of increment stresses are calculated; iteration continues until the principle of virtual work is satisfied to within acceptable tolerances.

Accordingly, we assume (i) that we are given Ixlx ~ ~,1 and the pair of variables

[T(t),s(t)], (35)

3Note that the rate-dependent model is not restricted to low homologous temperatures, as the rate- independent model is.

710 G . G . WEBER et aL

at time t, with the Cauchy stress T(t) satisfying equilibrium; and (ii) that we are also given Izlz ~ a l l . With these given, we take the computational problems to be:

1. A stable, accurate and efficient computation of the pair {T(r),s(r)}. 2. The computation of a consistent Jacobian matrix to be used in a Newton-type iter-

ative method for revising the estimated displacements such that the updated stresses better satisfy the principle of virtual work at the end of the increment.

As emphasized by HtJOHES [1984], these are the main problems of computational plasticity, with item 1 above being the central problem, because Jacobian matrices are used only in the search for displacement fields that lead to satisfaction of the principle of virtual work, but in the end have no effect on the accuracy of the solution. Indeed, in "explicit" finite element procedures, which are widely used in large scale inelastic analysis, Jacobian matrices are not required, and item 1 above is the only function required of a "constitutive equation subroutine."

IV. 1. Quasi-static incremental boundary-value problem

Assume that the current configuration ~ t of a body which is in equilibrium is known, and suppose that the equilibrium stress field T(x, t) together with the field of state variable s(x, t) and the relevant material parameters are given. Let ~ t and St be complementary regular subsurfaces of the boundary OtBt of tBt. Denote the relative displacements ut(x, r) prescribed at points x of ~ t by fit, and denote the surface tractions St(x,7")fl(X,/) prescribed on points x of $t by it. (More complicated boundary conditions may be envisioned, but we do not go into that matter here.)

Given these data, a sufficiently smooth vector field ut(x, r) generates kinematical quantities which, through the evolution equations for T and s, allow us (via appropriate time-integration procedures) to calculate the stress field T(z, r) and the internal variable field s(z, r) on the configuration ~ , at time r. If this relative displacement field u/(x, r) is such that the stress field T(z, z) when expressed in terms of the relative first Piola-Kirchhoff stress gt (x, r) satisfies

St(x,'r) • ~x fit(x, r) d V - b,(x, ¢).fi,(x, r) d V I t

- fa i,(x,r).fit(x,r) d A = O, tBt

(36)

for every sufficiently smooth "variational" vector field fit (x, r) (which is not identically zero, and which vanishes on ~t) , then it is called a (weak) solution of the mixed incremental problem.

In order to obtain such a solution we first need to assume a deformation path between tBt and ~ , for purposes of discretization of the relevant kinematical quantities, and then to integrate the constitutive equations across a time-step, 4 4 t = r - t, to obtain T(z, r). To this end, in the next few subsections we first transform the constitutive equa-

4Since plasticity is path-dependent, we must restrict the time increment • t and the relative deformation gradient Ft ( r ) m F ( r ) F ( t ) - t to be (in some sense) "small ."


tions to a convenient form for performing such an integration, and then examine, in some detail, suitable schemes for interpolating relevant kinematical quantities.

IV.2. A transformed set of constitutive equations

In the rate-independent and rate-dependent elastic-plastic constitutive equations considered here, the elasticity is modeled in hypoelastic form, with the stress rate being taken as the Jaumann derivative, so as to make the constitutive model "frame-indifferent" or "objective." The formulation of numerical time-integration procedures which ensure objectivity during computations has been endeavored by many authors, and important contributions have been made, for example, by HUGHES and WXr~GET [1980], NAGTEG~ and VELDPAUS [1982,1984], RUmr~STEI~ and ATLtrRX [1983], REED and ATLURX [1983,1985], HUGm~S [1984], and NAGTEG.~a. and I~BELO [1986]. Here, we describe an incrementally objective procedure based on the work of WEBER [1988].

Let P(~') be a rotation tensor which is defined to be the solution of the initial-value problem

P(~') = W ( D P ( D , t _< ~'_ r, (37)

P( t ) = 1, (38)

where W(~') is the spin tensor. Using rotations P(~') so-defined, we define the bar-transformation of a symmetric

second-order spatial tensor A(~') by

P.(~) - Pr(~)A(~)P(~) . (39)

In particular,

-r(~-) = Pr(~')T(~')P(~-), with "f(t) = T(t). (40)

Further, on using (39) and (37) we have the important result

~(~') = Pr(~')TV(~-)P(~'). (41)

We shall use these definitions and results to obtain "bar-transformations" of the constitutive models of Section III. After transformation, the evolution equations for stress in both models involve only the material time derivative of T instead of the more complicated Jaumann derivative of T. Such a transformation has previously been used, for example, by NA6"r~G~L and VELDPAUS [1982,1984] and Hu~I~S [1984]. Nagtegaal and Veldpaus call the bar-transformed quantities "rotation-neutralized" quantities. Regard- less of terminology, after transformation the constitutive equations take on a simple form which greatly aids the numerical integration procedure.

IV.3. Bar form of the constitutive models

Using (39--41), the isotropy of ~ , and the isotropy of the other constitutive functions, we obtain the following bar forms of the constitutive equations:

712 G . G . WEBER el' al.

IV.3.a. Bar form o f the rate-independent model.

I. Evolution equation for the stress T:

= ,l~[D - DP],

with

where

and

Dp = x 3~;Y~tpS,

1N = 3 ,~-~(T' /#) ,

# = q (3 /2 ) 'T ' . t ' ,

tP = 343434343434343434~g-'lq.,e[D],

g = h + (3/2)1~. ~.[l~l],

[ ~ i f a < s , or if # = s and l~ . ,g [D] _< 0, ]

X = if # = s and N- ,g[[ ) ] > 0 "

2. Evolution equation for the deformation resistance s:

where

= h~ p,

h = h ( s ) .

IV.3.b. Bar form of the rate-dependent model.

1. Evolution equation for the stress T:

= .C[D - Dp] ,

with

where

Dp = ,~ '~ tPS ,

I~ = 3~(3-22(T'/8),

a = q(3/2)T'.T',

~'p =f(a,s).

2. Evolution equation for the deformation resistance s:

= g ( a , s ) .

(42)

(43)

(44)

(45)

(46)

(47)

(48)

(49)

(50)

(51)

(52)

(53)

(54)

(55)

(56)


and

In our numerical implementation we shall leave the evolution equation for s in this generic form. The hardening-recovery form of eqn (32) is a special case, and this can be accommodated easily during actual computations.

For later use, whenever convenient, we will write

b p = ~P(T , s ) ,

s = ~ ( T , s )

for the flow rule and the evolution equation for s, for both the rate-independent and the rate-dependent models in the bar form, with the understanding that the functions ~J' and ~ are the specific isotropic functions of their arguments considered above.

When the constitutive equations are expressed in the bar form, the response of the material to an imposed relative motion y = i t (x, ~'), ~" E. [t, r ] , is obtained by first inte- grating the constitutive problem expressed in terms of "r and ~. Note that B is the driving quantity in this integration problem. After integration of the constitutive problem, the stress tensor T( r ) is transformed to obtain

T( r ) = P ( r ) T ( r ) P r ( r ) . (57)

At first glance, it would appear that since one needs to interpolate for B(~') = Pr(~')D(~')P(~'), one needs also to interpolate for P(~') for all times ~'E [ t , r ] . How- ever, as we shall see shortly, in the scheme proposed below, this is not necessary. It will be seen, that to evaluate the Cauchy stress T at time r, one needs to know the value of P only at time ¢, and not for all times ~" E [t ,¢], and if an appropriate value of P( r ) can be found by suitable interpolation techniques, the integration of the kinematical initial value problem (37,38) need never be performed.

IV.4. An objective time-integration algorithm

Consider a relative motion

Y = i t ( x , ~ ' ) , t ~ ~'--<r, (58)

and let i~ be another motion of the body which is related to i t by a change in frame:

iT(x, ~') : q(~') + Q(~') l i t (x , ~') - o], (59)

where q(~') is a point in space, o is a fixed origin, and Q(~') is a rotation with Q(t) = 1.

Definition. Following REED and Art.trea [1983,1985], we define a time-integration algorithm for the evolution equations for [T,s} to be incrementally objective, 5 if for all

~The well-known criterion of incremental objectivity of HuoHr:s and Wr~ol~r [1980] is a restricted form o f this more general statement. The definit ion o f incremental objectivity set forth by these authors is that if z = t t ( x , r ) = e + R , ( r ) x , where • is a fixed point, then T ( r ) must transform according to T ( r ) = Rt(r)T(t)Rtr(r). It is stated only for rigid relative motions.

714 G.G. WEnER et al.

relative motions z = i,(x, r), the numerical approximations to T(r) and s(r) transform under a change in frame according to the same rules as the physical quantities them- selves; that is, if

T*(r) = Q ( r ) T ( r ) Q r ( r ) , (60)

s*(r) = s(r ) . (61)

Proposition. A numerical algorithm used to evaluate

"F(r) = "F(t) + ~ d - ,g P(T(~'),s(f)) d , (62)

s (r ) = s ( t ) + ~(T(~'),s(f)) dr, (63)

T(r) = P(r)~r(r )pr(r ) , (64)

will be absolutely incrementally objective if under a change in frame the interpolated quantities D(~') and P(r) , transform according to 6

B*(~-) = B(D, (65)

P*(r) = Q( r )P ( r ) ; (66)

that is, if they are objective.

Proof." Since D(~') is the only driving quantity for evaluation of (62), and also since the functions in (62) and (63) are isotropic, if (65) holds under a change in frame, then we will obtain

T ' ( r ) = T(r) . (67)

Using (39), that implies that

Next, if (66) holds, then

T*(r) = P * ( r ) p r ( r ) T ( r ) P ( r ) p * r ( r ) .

T*(r) = Q( r )T ( r )Qr( r).

Finally, from (63) and (67) it is clear that under a change in frame s(r ) transforms as s*(r) = s ( r ) .

6Recall that b(~') = Pr(~')D(DP(~'), and thus (65) and (66) require that D*(~') = Q ( D D ( D Q r ( D .


An objective interpolation o f D(~) and P (r). In a two-point, deformation-driven problem Ixlx E (Bt} and Izlz E @,} are the only data. Let

Ft(r) m F ( r ) F ( t ) - ' , (68)

be the known relative deformation gradient, and

F,(r) = Rt(r)Ut(r) (69)

its right polar decomposition, then we take the interpolated values of D(~') and P(r) to be expressible as

2(~') = f}(Ft(r),~'), with ~(1 , f ) = 0, (70)

P(r) = F'(F,(r)), with [~(U,(¢)) = 1. (71)

As a restriction on the interpolated quantities [}(~') and P(r) , we require that under a change in frame 2(~') and P(r) transform according to (65) and (667. Since FT(r) = Q(r )Ft ( r ) , this requires that the following hold for all rotations Q(r):

= D(Q(r)Ft( r ) , ~'),

F'(F,(r)) = Qr(r)[~(Q(r)Ft(r)).

In particular, choosing Q(r) = Rr(r) , we obtain the following reduced forms of the interpolation functions for D(~-) and P(r):

2(~') = [)(Ut(r),~'), with [1(1,~') = 0, (72)

P(r) = Rt(r). (73)

Finally, since U~(r) = U t ( T ) and RT(r) = Q(r )Rt ( r ) , the interpolation functions (72) and (73) satisfy the requirements (65) and (66).

Thus, to obtain an absolutely incrementally objective time integration algorithm, one needs to take P(r) = R/(¢), and to assure (eqn (72)) that the assumed deformation path between the initial and final configurations be invariant under superposed rigid body motions, that is, the interpolation for D(~') be a function of only the right relative stretch tensor Ut (r) and ~" E [t, r] . Various interpolation functions for D(~') satisfying the above requirement may be envisaged. Here, following WEBER [1988], we propose the use of the constant interpolant

2(~') = Di-- .constant, with (74)

1 D t = - - l n U t ( r ) , A t = r - - t . (75)

At

We adopt (73-75) in all that follows. Based on considerations different than those presented here, these interpolation formulae have also been recently suggested independently by NAGrEG~X and REBELO [1986].

716 G.G. WEaEg et al.

Before closing this section, it is important to note that the considerations above are tantamount to assuming the following objective incremental constitutive equation for the stress:

"F(r) = "F(t) + .g[Et(r) - EP(~')], (76)

where

~£(r) - R r ( r ) T ( r ) R t ( r ) with "F(t) = T(t) , (77)

F,t(r) - In Ut(r) , (78)

f ; -

E~(r) = BP(T(J'),s(~')) dL (79)

It is well known (cf., e.g., RICE [1975], Hxu. [1978]) that the stress measure which is "work-conjugate" to the logarithmic strain increment Et (r) for "moderate" stretches is the stress measure T(r) defined above. 7 Thus, in the spirit of Biot's work [1965], the objective incremental constitutive equation (76) above is the natural analog to the objective rate constitutive equation (1). Indeed, the instantaneous time-derivative (that is, (O/Ot)f(r)l ,=t) of (76) is exactly equal to eqn (1).

V. TIME-INTEGRATION PROCEDURE. RATE-INDEPENDENT MODEL

Previously, we have defined the current configuration of the body as the configuration at time t, and the subsequent configuration as the configuration at time z. For most of this section it is more convenient to make the following identifications

/ n m t ,

/n+l ~-~ T,

to emphasize the incremental nature of the time-integration procedure. The solution being supposed to have been obtained up to time G, and that at time t.+t being sought. Further, subscripts 'n' and 'n + I' on variables will indicate that the variables are evaluated at times t. and t.+=, respectively.

Assume that the state (T.,s.) at time t. is known. Note that T. = T.. Then the problem is to integrate the evolution equations for T and s across a time increment A t = t.+l -- t. and thereby calculate (T~+t,s.+~), transform T,,+t to T.+~, and march for- ward in time.

An algorithm to integrate the rate evolution equations should in general be (a) consistent with the constitutive equations, Co) numerically stable, and (c) incrementally objective. We have considered the problem of incremental objectivity in the previous section. Regarding the other two characteristics, we will use the consistent and stable Euler-backward method of integration. Thus, from eqns (42), (43), and (49), together with the use of the definition (2) of .g, we obtain

T.+~ = f ; + , - ×(~4t)~g+zY~+~, (80)

s~+l = s. + Ath(sn+l)~ff+,, (81)

7Actually, "~(r) = {det Ut(r)lRT(r)T(r)Rt(r).


where

"r~,+~ = T. + £ [ d E ] (82)

is a trial value for the stress at the end of the increment, with

z i g . In{U,,(t~+~)} (83)

the incremental logarithmic strain. In a displacement-based finite element program, :I [: can be computed from known kinematic information. Thus, T~+~ is known at the beginning of the increment. To update the state variables using (80) and (81), we need to calculate ~[.+j, ~P e.+t and s.+,. We show next that 1~+1 is known in terms of T~,+,, and, hence, it is also known at the beginning of the solution process.

Taking the deviatoric part of (80), using T~,+s = x/2-~0~+~l~.+~ and rearranging, we obtain

(2~-73~.+, + XV~pzit~P.+, )lgl.+ I = T*~.,, (84)

which shows that lgI.+l is parallel in direction to "r*~.l. Hence.

R.+l = T~,;,/nT~,~.z~ = 3 V 3 7 ~ ( T ~ ; ~ / o . + ~ ) , (85)

where

~'+~ = V(3/2)T~,;I .T ,~.~. (86)

Equation (85) may be rearranged to read

T:,+l = ~.+iT~%, where ~.+l = (8~+l/8,~+I). (87)

We call ~,,+~ the "radial return factor." As an alternate to evaluating T.+~ from (80), we can first evaluate its deviatoric part T:,+~ using (87), and then use the fact that the mean normal pressure P,,+I " (-I/3)trT.+~ is equal to the mean normal pressure p~,+~ - (-I/3)trT~,+t. Thus, the problem reduces to finding 8.+i and s~+z.

From (80), during plastic flow (X = I ), we obtain

0n+l = O*+t - 3/~At~+t. (88)

Further, since 0 = s during plastic flow, from (81) we obtain

On+l = Sn+! = Sn "~" dth(sn+t)~+t, (89)

and from (88) and (89),

1 s~+1 - s~ - ~(s~+t) ~ { ~,+t - s~+tl = 0. (9o)

Hence, the problem of determining O.+t = s.+~ reduces to solving eqn (90). For a constant hardening rate h the solution of (90) is trivial; however, if h is a function of s (as

718 G.G. WEBER et al.

is usually the case), then we need to resort to some numerical method to solve (90) for an+ l -~ 'Sn+l .

Assume that ~+~ = s~+~ has been obtained. Then the final step is to use (87) to calculate

T.+~ = ,1 .+,I7 , ; - , - P ; + , 1, (91)

and to rotate it according to (see eqns (39) and (73))

T~+l = IRt.(tn+l)lT.+l {R,.(tn+l)l r, (92)

to obtain the Cauchy stress at the end of the increment.

V.I. Summary of numerical algorithm for the rate-independent model:

Step 1. Calculate the relative deformation gradient

Ftn(tn+l) = F( t ,+l ) (F(t,)) -I •

Step 2. Perform the polar decomposition

Ft.( t .+l) = [Rt.(t .+l)l [U,.(t.+~)l.

Step 3. Compute the incremental logarithmic strain A ~:. To do this, first compute the spectral decomposition of U,.(t ,+l):

3

U, . ( t .+ , ) = Y, a")e "~ ® e "), i=l

where A ci~ are the eigenvalues and e ti) are the eigenvectors of Ut.(tn+~ ), and then

3 z ig = ~ (lnA(i))e (i) ® e (i).

j= l

For later use, we note that z ig is well approximated by:

z ig --- 2[U,n(t,+,) - II [U,.(tn+l) + 11-' ; (93)

As shown by WEBER [1988], the rational expansion (1 st Pade approximation) above of the tensor logarithm is an excellent approximation to the true value, when the principal stretches of Utn(tn+l ) are in the range 0.7 </~(i) < 1.3. Since in a typical time step the incremental stretches should, in general, be "small," this approximation may be used with impunity.

Step 4. Calculate the trial stress:

Y?,+, = T . + a e [ a t ] .


Step 5. Calculate the mean normal pressure corresponding to "rY,+l:

P*+ t = - ( l/3)tr'F~,+l.

Step 6. Calculate the deviatoric part of T*+l:

T,~-i = "r*+l + p*+ll .

Step 7. Calculate the trial equivalent tensile stress 6,~+l:

--st O*+l = x/(3/2)T~+l.T~'+l.

Step 8. If

0"+, < Sn,

T.+I = "iT,+t - (4g~4t) tC+~l~.+, ,

T*+, = T. + ~.[AE]

1~.+, = 3 , /~¢T%,/0~+,) .

(94)

(95)

(96)

then the process is elastic, and

"rn+l = ~?,+,,

Sn+I = Sn,

T~+l = (Rt.(G+l)}Tn+l {R,.(tn+l)} r.

Otherwise, the process is elastic-plastic. Continue. Calculate sn+l = 0,+z by solving

1 s . + , - s . - ~ ( s . + , ) ~ {o7,+, - s .+~ } = o.

Step 9. Calculate the radial-return factor 7/,+l:

~n+l ~'~ ( a n + l / 0 n + l ) .

Step 10. Update the stress:

T n + ~ = , h + ~ T?,~, ~ - P ~ + t 1 .

T~+l = {Rt.(t~+l)}Tn+, {R,.(t~+l)} r.

VI. TIME-INTEGRATION PROCEDURE. RATE-DEPENDENT MODEL

F r o m eqns (51)-(56), by paralleling the arguments of Section V we obtain

720 G . G . W~nER et al.

~/(3/2)Tn+ t. 'T~,~.~, (97) 8 " + ! = ' -*'

T:,+t = ~n+lT*~l, (98)

~ + l = 8~+t/O~+l, ( 9 9 )

~+l --- f(On+l,S~+l), (I00)

8~+1 = 8~+1 - 3#At~+t , (101)

s~+l = s~ + Atg(O~+l,Sn+l). (102)

Thus, unlike the procedure for the rate-independent model, s here we have to solve for sn+j and O,,+t from the pair of equations 9

sn+t - sn - Atg(On+l,S~+l) = 0, (103)

8~+t -- 8~+~ + At3#f(#~+i,S~+l) = 0, (104)

which are obtained from (100-102). LusH, WrBm~ and AN~V [19891 have recently detailed a robust and efficient iterative method for obtaining the solution for a large class of functions f and g.

VI.I. Summary o f numerical algorithm f o r the rate-dependent model

The algorithm is identical to that for the rate-independent model detailed in the previous Section, except, replace Step 8 in that algorithm by:

Step 8. Calculate 8~+t and s.+~ by solving

Sn+! -- Sn -- A t g ( O n + l , S n + l ) = O,

0.+1 - O*+t + At3#f(#n+~,s.+j) = O.

VII. JACOBIAN MATRIX

Recall that the quasi-static incremental boundary value problem was to find a relative displacement field ut(x, ~') such that

S t ( X , T ) • ~ f i t (X,~ ' ) d V - - b t ( x , r ) ' f i t ( x , ~ ' ) d V - i t ( x , r ) . f i t ( x , r ) d A = O, t t ¢Bt

(105)

for every variational vector field fit(x, ~'). In order to solve the nonlinear set of equations generated from (105) by the finite element discretization, iterative techniques must be used. The most commonly used iterative method is Newton's method. This method

SRecall that in the rate-independent model 0n+m --sn+t during plastic flow.

9For the class of func t ions f which are invertible and in which 0 can be written as a function of (~P,s), or s as a function of ( to ,0) , the pair o f nonlinear equations can be collapsed into a single nonlinear equation which needs to be solved.


requires the linearization of (105) around the last estimate for the solution ut(x, T). Let d~ denote a small change in a quantity ~, then the linearized form of (105) is simply

f $ 0 dS,(x,r) • ~x i~(x,7) dV t

(106)

and the identities

the result

St(v) = (det Ft(~))T(r)Ft(~) - r ,

T(7") = RtO')T(T)Rt(7) T,

where

dFt( r) -r = -( dFt( ¢)Ft( 7.)-i ) rFt( r) -T,

d(det Ft(T)) = (det Ft(1"))tr(dFt(¢)Ft(¢)-l),

we obtain

f. o f.[ dS,(r) • -~x f i t ( x , r )dV= IR , ( r )~ ( r ) [d4E l R , ( r ) r l t f

+ [(dRt(r)R,(r)r)T(r) - T(r)(dR,(r)Rt(r)r)}

- {T(r)(dFt(r)Fr(r) -1)}

+ [T(r)tr(dFt(r)Ft(r) -I)}]

• [ I 0-~ fi,(x, r)l Ft(~)-'(det F,(T))] dV,

~(¢) -- aagT(T), (108)

are the linearization moduli which depend on the constitutive equations and the time integration algorithm used to evaluate "i'O').

In order to complete the linearization procedure, it remains to derive the differentials d4~,dR,(~-)RtO-) r, and the linearization moduli ~ (r). We proceed first with the evai-

(1o7)

In order to achieve the quadratic convergence which is characteristic of Newton's method, it is important to evaluate this linearized form accurately (NAGTEGX~J. ~ VELD- PÛS [1984], S~O • TA~IOR [1985]). In this paper we concentrate on evaluating only the first term in the linearized expression (106) above; evaluation of the second term depends on the particular body forces and surface tractions applied.

Using the definition of the relative first Piola-Kirchhoff stress,

722 G . G . WEnER et al.

uation of dAE,,dRt(r)Rt(r) r. In what follows, we describe an approximate, ~° but computationally economic, evaluation of these differential quantities.

Differentiating eqn (93) and rearranging terms we get

dAF, = 4(Ut(r) + 1 ) - JdUt( r ) (Ut ( r ) + 1) -~. (109)

Next, differentiating Ut (r) 2 ( = Ft ( r ) TF t ( "r )) and premultiplying by Ut ( r ) - ~, we obtain

Ut( r ) - I d (Ut ( 7") 2) = Ut( r)-I dU,( r)Ut( r) + d U t ( r) -- 2Ut(r) -I sym(Ft( r)rdFt( r)).

( l l0)

Introducing the Biot incremental strain measure E~(r) -ffi Ut(r) - l , and expanding Ut(r) -t in terms of E~(r), we arrive at

Ut(r) - l = 1 - E~(r) + O(E~(r)2). ( I l l )

Substituting (l I l) in (110), and rearranging terms we obtain the expression

dUt ( r ) = sym[Ut(r) -I sym(Ft(r)rdFt(¢))] + O(E~(r)2). ( l l2)

Finally, substituting (112) in (109) we obtain

dAE - 4(Ut(r) + l) -I sym[Ut(r)-2sym(Ft(r)rdFt(z))] (Ut(r) + l) - j . ( l l3)

Also, differentiating Ft(r) = R,(~')Ut(r) and using ( l l2) we get

dRt(¢)Rt(r) r - dFt(r)Ft(~)-I

- R/ ( r ) sym[Ut( r ) -I sym(Ft ( r ) rdFt (r))]Ut(r) -t Rt(r ) r, (114)

which is of the same accuracy as the expression (112) for dUt ( r ) above.

VII. I. The linearization moduli

From (80) and (94) for the rate-independent case during loading and the rate-dependent case in general,

~,( r ) = Oae.T*(¢) - (~r6/zzlt) [ l~l(r) (~) Oa£~P(r) + ~qr)a~t~(r)]. (115)

Using (82), (85), and (86), straightforward calculations give

04~.T*(r) = P-, (116)

0a~.#*(r) = 3 ~ M ( r ) , (117)

'°Note that- this approximation might affect the rate of convergence of the global iteration scheme but not the accuracy of the algorithm.


where

/ . m/TT5 \ o , ~ s ( r ) = I ~ t2,,(~ - ( i / 3 ) , ® ~) - s , r ) ® ~ ( r ) l ,

\0 (r) / (1{8)

where

.~,(r) ~= 2~( r ) (9 - (1/3)1 @ 1) + xl @ 1, (121)

/2(r) *= ~/(r)#, (122)

[o(r) 1 ,1(r) ~ O*(r----)J" (123)

From (120) we note that the only difference in ~ between the rate-independent model and the rate-dependent model arises from the difference between the values for the term

~p aa{E (r). In what follows, we evaluate this term for the two models.

V11.2. Linearization moduli for the rate-independent model

Linearization of (88) and (89) gives

d0( r ) = dO*(r) - 3#Atd~P(r), (124)

d0( r ) = ds ( r ) = Ath( r ) d~P(r), where (125)

h(r) ~(r ) == , (126)

1 - A t ~s ~a(r) T

Substituting for dO(r) from (125) into (124) and solving for d~P(~ -) we obtain

dO*(r) (127) d~a(r) = At(h(r) + 3#)"

Next, using (117) and (127) we obtain

,p 3 3 ~ ( r ) Oat~ (r) = At{3p(l -- c~(r)) -~ } ' (128)

M( r ) - .g[iq(r)] = 2Oil(r) . (119)

Substituting (116) and (118) into (115), and rearranging gives

O'(r)

(120)

724 G . G . WEBER e t al.

where

~(r) c,i(r) ffi 3# + h( r )" (129)

Substitution of (128) into (120) gives

where

~,(r) = . g ( r ) - (3/2)~(r)-I~i(r) ® M ( r ) .

~(r)-~ = ( ~ ) In(r) - cri(r)}.

(130)

(131)

Crd(r) = lal + a2(~2) l -I,

a I m 1 + 3lz4td~fl, , (135)

a2 E 3#AtOsfl~, (136)

bl - 4tOag],, (137)

b2 =- 1 - ztOsgl,. (138)

Substituting for d#(r) from (133) into (132), solving for d~P(r) and using (117) we obtain

,v/3-~ M(r) d,~£~P(r) = (139)

Atl3#(l -- c~a(r))-t } '

substitution which into (120) gives

~,(r) - .l~(r) - (3 /2 )~( r ) - iM(r ) ® M(r) . (140)

(134)

with

where

VII.3. Linearization moduli for the rate-dependent model

Linearization of (101) gives

dS(r) = d0*(r) - 3#Atd~P(r). (132)

The quantity dS(r) is obtained by linearizing (103) and (104) and solving the resulting pair of equations. This gives

dO(r) = Cra(r) dS*(r), (133)


where

~ ( r ) - ~ = ( ~ - ~ ) { ~ ( r ) - C r a ( r ) l . (141)

From (131) and (141) it is clear that the difference in the contribution ~ to the Jaco- bian from the rate-independent and rate-dependent models is only the difference between cri of eqn (129) and cra of eqn (134). It is important to note from (129) that for the rate- independent model when h = 0 we have c,~ = 0, whereas, from (134), in the nonhardening version of the rate-dependent model Cra = a~ -I * O.

VII.4. Summary o f Jacobian

As an alternate to (107) we may write

0 f m d S t ( r ) ' - ~ x f i t ( x , r ) d V = / m , [ ~ ( r ) [ d A E ]

+ { (dRt ( r )R , ( r ) r )T(r ) - T ( r ) (dRt ( r )R t ( r ) r ) }

- {T(1.)(dFt(r)Ft(r)- i) l

+ {T(r ) t r (dFt (r )Ft (r ) - I ) l ]

• / I O " , (x , r ) l F t ( r ) - ' (de t Ft(r))] dV,

where

with

d A E "~ R,(~') IdAF, IR,(r) r,

and

where

dAE, "- 4(Ut(r) + 1)-Zsym[Ut(r)- tsym(Ft(r)rdFt(r))] (U,(r) + 1)-I;

dRt ( r )R t ( r ) r - dFt(r )Ft(r ) - I

- R t (r )sym[Ut(r ) - i sym(Ft(r )rdFt(r ) )]U,(r ) -I R,(r)r;

~ ( r ) = ~ ( r ) - (3 /2 )~( r ) - iM(r ) ® M(r) ,

~.(r) -- 2Mr) (a - (1/3)1 (~) 1) + K1 @ 1,

#(r) - ~(r)#,

1 ~(r) "- O.(r ) ) ,

(142)

(143)

726 G . G . WEBER el a[.

and

g(r)-t=(~'~)l~(r)-Cr~(r)l,

with c,~ given by (129), for the rate-independent model, and

~(r)-t=(~'~)l'7(r)-c,a(r)l,

with c,a given by (134), for the rate-dependent model. Also

M(r) ~ ~ [N(r ) ] = 2t~N(r). (144)

For programming purposes it is useful to rewrite ~ (r) as

~ ( r ) = 2/2(r)~1 + {K - (2/3)/~(r)11 ® 1 - 2/~11/(r) - Cri/ra}N(r) ~ N(7). (145)

Note that the linearization moduli ~ for both the rate-independent and the rate-dependent models have exactly the same form as the constitutive modulus e of eqn (23) for rate-independent plasticity. However, the incremental shear modulus/2, and the parameter ~ in the linearization moduli are different from the corresponding quantities in e . Also, the direction tensor N in the linearization moduli is evaluated at the end of the increment instead of at the beginning of the increment. The use of e for the rate- independent model or .g for the rate-dependent model, in the Newton procedure, instead of the appropriate linearization moduli ~ , can lead to very slow convergence.

VIII. NUMERICAL EXAMPLES

The constitutive equations and time-integration procedures described in this paper have been implemented in the implicit finite element code ABAQUS (Version 4.7) [1988], by writing a "user material" subroutine, UMAT. The generality of the other features of ABAQUS, combined with the provision for adding a separate "user material" subroutine, makes it an extremely useful tool for the development and implementation of new constitutive equations.

The subroutine UMAT is called once for each integration point in the model for every global iteration. Part of the input provided to UMAT consists of the stress "r, Q~,wT,,Qrw and the internal state variable sn at the beginning of the increment. The relative rotation Qxw, between the local configuration of a material neighborhood at time tn and that at time t,+~, calculated by using the Hoorms and W~O~:T [1980] algorithm, is also supplied as input. In order to use our time-integration algorithm, we have to first calculate "In from "rn and Qttw. Next, from ABAQUS it is possible to obtain the deformation gradients F, and F,+t at the beginning and the end of an increment. 1~ With these inputs, the output expected from UMAT consists of the values of the stress

t 'This information is not currently routinely supplied as input to UMAT. Instead, ABAQUS supplies a strain in~ement calculated by using the H u o l ~ a Wn~cE'r 11980] algorithm. Appropriate subroutines that need to be called to obtain Fn and Pn+, were kindly supplied to us by Dr. H. D. Hibbitt of HKS, Inc.


T,,+~ and the internal state variable s~+~ at the end of the increment, plus the constitutive contribution ~, eqn (145), to the Jacobian matrix used in the Newton scheme for global equilibrium.

As in the paper by LusH, WEBER and ASA/qD [1989], all ABAQUS calculations with variable time-stepping presented in this Section were performed with a slightly modi- fied version of its STATIC analysis procedure. The most important modification was to enhance the automatic time-stepping procedure in ABAQUS to control the accuracy of the constitutive time-integrations. This was done by using as a control measure the maximum equivalent plastic tensile strain increment AdmP~ occurring at any integration point in the model during an increment. Although it is not a direct measure of the constitutive time-integration error, this measure was found to be very effective for control- ling the accuracy. Efficient calculations were obtained by keeping 4dmPa~ close to a specified nominal value Ad~ Accordingly, the automatic time-stepping algorithm op- erated to keep the ratio

close to 1.0 by adjusting the size of the time increments. After an equilibrium solution for a time increment Atn = t~+j - t~ was found, the value of R was checked to deter- mine whether this solution would be accepted. If R was greater than 1.25, then the solution was rejected and a new time increment was done that was smaller by the factor (0.85/R). If R _< 1.25, then the solution was accepted and the value of R was used to determined the first trial size of the next time increment. The following algorithm was used:

If 0 . 8 < R - < 1.25 then AG+~=AG/R;

if 0.5 < R_< 0.8 then Atn+~ = 1.25Atn;

if R < 0.5 then At~+~ = 1.504t~.

Note that the measure 4 ~ was allowed to exceed the user specified value by up to 25°7o. This was done to avoid having to recalculate increments that came out just slightly above the specified nominal value but were otherwise essentially acceptable. In the example problem of upset forging presented below, a considerable improvement in effi- ciency was obtained by doing this.

VIII.I. Simple shear o f a rate-independent, elastic-perfectly-plastic material

With respect to a rectangular cartesian coordinate system with origin o and orthonor- real base vectors [ei[ i = 1,3], a simple shearing motion is described by

x = p + (~t)p2el, (146)

or, in component form,

x i = p l + ('~t)p2, x2=P2, x3=P3, (147)

with -~ __ 0 a constant.

728 G . G . WEBER et aL

The analytical solution for the case of a rate-independent, nonhardening, elastic-plastic material subjected to this motion has been obtained by SPEr~CER and FERRtER [1973] (also, see Moss [1984]). For sufficiently small values of t the behavior is elastic, and the solution for the components of the stress T is

TI2 - - / z sin('i,t),

T22 = --Tii = --/x(l - cos('~t)), (148)

7"13 = T23 = T33 = 0,

where # is the elastic shear modulus. With So the deformation resistance in tension, let k ~ So/X~3 denote the deformation

resistance in shear. Then, the time to at which yielding commences is given by

t o = - C O S -I 1 - (149) "t 2 '

and the values of the nonzero stress components at yield are

[ I (TI2)o =/~s in cos - ' 1 - ~ , (T22)o = --(Tl ,)o = - -~ (150)

We confine our attention to materials with infinitesimal values of (k /#) . For such materials we expect that the amount of shear % -~ (6/to) at onset of yield is also infinitesimal. For small values (~/to) the two term series expansion of cos('~to) is

cos('~to) -=- 1 - ~ (f/to) 2.

Thus, using (149) and the series approximation above we have that the amount of shear at yield is

70--- (k /# ) . (151)

Hence, for materials with (kl~) ~ I the elastic solution (148) applies only for small values of 7 -< (k/~), and therefore the apparent periodic nature of this elastic solution is not real. Also, from (149-15 I) the values of the nonzero stress components at yield, to order (k/~) 2, are

(TI2)o -- /z sin(k//z) ffi k , (T22) o = - - ( T l l ) o = - - ] •

The elastic-plastic solution for t > to may be expressed in terms of a parameter 0 by

TI2 = k cos ¢~, T22 --- - T i l --- - k since, (153)

with

,-- + (°°- (154)


where

~o = sin-I (~ ~ ) (155)

is the value of ~ at time t = to. To examine the asymptotic behavior of the elastic-plastic solution, eqn (154) for may be expressed, using the approximations 3'o = 7to = (k/#) and sin~o = ¢'o =

( l /2)(k/~) , as

,,:2,=(, 7/] ..6, Because the argument of the exponential function in (156) contains the large factor (tL3"/k), the exponential term tends rapidly to zero as 3' increases. Indeed, T~2 decreases ~2 slightly from its value (T~2)o at initial yield (Moss [1984]). As 3' increases, ¢~ --, (k/tO, and the corresponding limiting values of the nonzero stress components are

(Tl2)o. = k cos(k/l~) = k, (Tzz)** = -(Tll)o, = - k s i n ( k / p ) = - k t ( ~ ) z. (157)

Note that normal stress effects occur in the elastic-plastic solution, but they are always of second order.

The numerical solution using our UMAT in ABAQUS for this problem was obtained by using a single four-noded plane strain element 13 (ABAQUS type CPE4). For the material properties we used a value of the Young's modulus E E 2~(1 + ~) = 25,000 MPa, a value of the Poisson's ratio J, = 0.3, and a value of the tensile deformation resistance of so = 50 MPa. This corresponds to (k/p) = 0.003. The calculations were performed for constant strain increments of A3, equal to 0.01 and 0.02. In Fig. 1 we show the comparison between the results of our numerical calculations and the analytical solution for the normalized stress components (T~z/k) and (T22/k) versus the shearing strain 3'. The agreement is very good.

VIII.2. Simple shear with superposed rigid rotation. Rate-independent, elastic-perfectly-plastic material

To verify the objectivity of our numerical algorithm we have carried out the numerical calculation for a rate-independent, elastic-perfectly-plastic material subjected to sim-

12Moss [1984] has discussed at great length this "instability" characteristic of the solution for materials with large values of (k/t~). Since the normal stresses are increasing in magnitude, in order to satisfy the yield condition with a constant value of So, the value of Ti2 has to decrease from its value (TI2)o at yield. The amount of decrease increases as (k/l~) increases. We emphasize again that the hypoelastic form for the constitutive equation for the stress is used here only as an approximation to a proper hyperelastic equation under situations where elastic stretches remain infinitesimal; this is a good approximation for most metals. The "instability" characteristic of the constitutive response to simple shear for large values of (k/p) emphasized by Moss is not important for most metals.

S3These elements in ABAQUS address the problem of mesh-locking in (near) incompressible situations by using the method of NAOT~OAAL, P.~q, xS, and Rtc~: [1974].

730 G . G . WEaER et aL

A

t"q t"q

v

t"q r--I

1.5

1.0

0.5

0.0

i . . . . I . . . . ! . . . . I . . . . I

-- T 12/k Analytical .... T--22/k Analytical

• Delta Gamma = 0.01 Q Delta Gamma = 0.02

Tl2/k

T22/k

- , - ~ - , . - ~ - , - ~ - .,, - i q - ~ - ~ - ~ - ~m- -~ , - . ~ - - , - ..1~- - , , - - ~ , - - , - - ~

I . . . . I . . . . I , . - • ! . . . . I

0.00 0.05 0.10 0.15 0.20

AMOUNT OF SHEAR, %'

Fig. 1. Comparison of results from a numerical calculation against the analytical solution for simple shear of a rate-independent, elastic-perfectly-plastic material.

pie shear with superposed rigid rotation. The homogeneous motion under consideration is given by

x = Q( t ) [p + (5't)p2e:] , (158)

with

Q( t ) = (el ® el + e2 ® e2)cos(0t) + (el ~ e2 - e2 ~ et)s in(0t) + e3 ® e3. (159)

The calculation was performed with 0 = 2~" radians per second and 5' = 0.2 per second, for t E [0,1], so that 0 -, 0t E [0,2x] and 3, - 5't E [1,0.2]. For fixed time-steps of A t = 0.1, the motion is shown in Fig. 2. Here A'y = 0.02 and A O = 36 °.

In performing our numerical calculations we used the same material parameters as those in the previous subsection. Let the solution for the stress corresponding to this motion be denoted by T*(t) . The results for the normalized components (T:a/k) and

Objective t ime-integration procedure 731

G.,0.0, T-0 G-0.02, T-36

I I / ' , , I

G-0.06, %.108 G-0.08, T.144

G=O.04, T=72

I" I I I I I I I

G-0.10, T-180

G=0.12, T-216 I I I I

! I

/ G-0.14, T-252

~ .16, T-288 o 2 8 . T.3 , G=0.20, T=360

Fig. 2. Simple shear with superposed rigid rotation. The total amoun t o f shear in this motion is G m 7 = 0.2, and the total rotat ion is T m 0 = 360 °. The mot ion is imposed in ten equal increments of A ~ = 0.02 and AO = 36 °.

(T22/k) of T(t) = Q(t)rT*(t)Q(t) versus the amount of shear % obtained from the numerical calculation are shown as solid points in Fig. 3. In this figure we also show as solid and chained lines, respectively, the corresponding analytical solutions for (Tj2/k) and (T~/k) in the absence of a superposed rotation. The excellent agreement between the numerical calculation and the analytical result verifies the objectivity of our algorithm.

For comparison, in Fig. 3 we also show as open points the results for the same problem obtained by using the capability built in ABAQUS for carrying out rate-independent plasticity calculations. Presently, ABAQUS (V4.7) employs the widely used HUGHr~S and W~GET [1980] algorithm for numerical objectivity. As is clear from Fig. 3, the performance of the Hughes-Winget algorithm for such large rotation increments is poor.

For 4 t = 0.05, in which case 43, = 0.01 and zl0 = 18 °, the numerical results from our algorithm and the Hughes-Winget algorithm are compared in Fig. 4. In this case

0 0

0 0

0 0

0 0

0 0

-T-12/k

Analytical

---- T 22/k Analytical

.

T-12/k Present Algorithm

.

T-22/k Present Algorithm

0

T-12/k Hughes-Winget Algorithm

0

TZ22/k Hughes-Winget Algorithm

T22/k

__

_‘_

__

~_

_‘,_

__

‘)_

__

~_

__

~_

__

~_

__

~_

__

_

l __

_

_.

I .

. ..l

....l.

.

..I..

I

0.00

0.

05

0.10

0.

15

AMOUNT OF SHEAR, y

0.20

Fig

. 3. Comparison

of

resu

lts from numerical

calc

ula

tio

n a

gai

nst

th

e an

alyt

ical

F

ig.

4.

Co

mp

aris

on

of

resu

lts

fro

m n

um

eric

al c

alcu

lati

on

ag

ain

st t

he

anal

ytic

al

solu

tio

n f

or

sim

ple

sh

ear

wit

h s

up

erp

ose

d r

igid

ro

tati

on

o

f a

rate

-in

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end

ent,

so

luti

on

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r si

mp

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hea

r w

ith

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rig

id r

ota

tio

n

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a ra

te-i

nd

epen

den

t,

elas

tic-

per

fect

ly-p

last

ic

mat

eria

l. In

th

e m

oti

on

a s

hea

r st

rain

of

y =

0.2

and

a

elas

tic-

per

fect

ly-p

last

ic

mat

eria

l. In

th

e m

oti

on

a s

hea

r st

rain

of

y =

0.2

and

a

rota

tio

n o

f 0

= 36

0” i

s im

po

sed

in I

O e

qu

al i

ncr

emen

ts o

f d

y =

0.02

an

d d

0 =

rota

tio

n

of

0 =

360”

is

imp

ose

d i

n 2

0 eq

ual

in

crem

ents

of

d y

= 0.

01 a

nd

38

=

36O

. Th

e so

lid p

oin

ts a

re t

he

resu

lts

of

the

calc

ula

tio

ns

usi

ng

th

e in

crem

enta

lly

18”.

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e so

lid p

oin

ts a

re t

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the

calc

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ng

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e in

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enta

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ject

ive

alg

ori

thm

dev

elo

ped

in t

his

pap

er,

and

th

e o

pen

po

ints

are

th

e re

st&

s o

bje

ctiv

e al

go

rith

m

dev

elo

ped

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th

is p

aper

, an

d t

he

oft

en p

oin

ts a

re t

he

resu

lts

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the

calc

ula

tio

n u

sin

g th

e cu

rren

t cap

abili

ties

in A

IIAQ

US

(V

4.7)

w

hic

h e

mp

loys

o

f th

e ca

lcu

lati

on

usi

ng

th

e cu

rren

t ca

pab

iliti

es

in A

BA

QU

S

(V4.

7)

wh

ich

em

plo

ys

the

Hu

gh

es a

nd

Win

get

alg

ori

thm

. 1h

c H

ug

hes

an

d W

in@

al

go

rith

m.

I.5

1.0

0.5

0.C

I ‘.

. .

1 1.

‘.

I ‘.

. .

1.

‘. .

1

TI22/k Hughes-Winget Algorithm

D.0

0 0.

05

0.10

0.

15

AMOUNT OF SHEAR, ')'


although the Hughes-Winget algorithm performs a little better, the agreement between the numerical results from this algorithm and the analytical results is still not good. The results start to improve only for relatively small (_<2 ° ) rotation increments. We shall come back to this point later when we discuss our solution for an axisymmetric upsetting problem. As we shall see, when the time-increment and therefore the relative deformation gradients are limited to small values in order to satisfy global equilibrium in complex problems, then the performance of the Hughes-Winget algorithm is acceptable.

VIII.3. Thick-wailed cylinder subjected to a constant radial displacement rate at its inner wall

A thick-walled cylinder, as shown schematically in Fig. 5, was subjected to a constant radial displacement rate at the inner wall of the cylinder. Let the radii of the inner and outer walls of the cylinder be denoted by a and b, respectively, and let U denote the constant expansion rate prescribed at the inner radius r = a. The initial values of a and b are denoted by ao and bo. The finite element model was built using eight noded axisymmetric elements CAX8H from ABAQUS. This is a hybrid element with independent pressure and displacement interpolation functions designed for use with incompressible

l

/

\

Z

lllllllllll -I

r

(b)

Fig. 5. Schematic of a thick-walled cylinder under prescribed velocity conditions at its inner wall.

(a)

734 G.G. WEnER et al.

materials. Ten elements in the radial direction were employed. The mesh is also shown in Fig. 5.

In this subsection, we compare our numerical solutions against the well-known analytical solutions for the reaction pressure versus the displacement of the inner wail, for incompressible, nonhardening, rigid-plastic (neglect of elasticity) rate-independent and rate-dependent materials.

VIlI.3.a. Rate-independent case. Here, the stress component T,r in a cylindrical coordinate system is given by PRAGER and HODG,~ [1951]:

[ r' ] Tr,(r,t) = k i n b# + a ( t ) z - a # " (160)

._~ 1.4

~3, 1.2

1.0

to 0.8 to

0.6

0.4

~ 0.2 0

0.0

! ! I ! l

@

Analytical • Numerical

@ I , I , I , I , I ,

0 2 4 6 8 10 NORM. DISPLACEMENT, U

! . . . . i . . . . I . . . . . . . . I . . . .

,.~ 1.4 ~ ' Analytical

. 1.2

to 1.0 to

0.6

0.8

I , , • , I , i " " I . . . . I . . . . l . . . . I

0 100 2 0 0 3 0 0 4 0 0 5 0 0

NORM. DISPLACEMENT, U

Fig. 6. Comparison of results for the normalized pressure (p/k) versus the normalized displacement U - [( U/oe)/(so/E)] from numerical calculations against the analytical solutions for a thick-walled cylinder sub- jeered to a constant radial displacement rate at its inner wall. The cylinder is made from a rigid-perfectly- plastic, rate-independent material. (a) For small normalized displacements. (b) For large normalized displacements.


~. !.2

i.0

0.8

m 0.6

0.4

0.2 0 ~ 0.0

! I I ! I I

0 • w w o

| i

2

NORM.

-- Analytical } • Numerical

I , , I , I

4 6 8 10

DISPLACEMENT, U

I I I I I I

1.2 ~ l ' Analytical

~ 0.8

~ 0.6 0 Z

I , I , I , I , I , I

0 100 200 300 400 500


Fig. 7. Comparison of results for the normalized pressure (p/k) versus the normalized displacement [7 m [(U/ao)/(so/E)] from numerical calculations against the analytical solutions for a thick-walled cylinder subjected to a constant radial displacement rate at its inner wall. The cylinder is made from a rigid-perfectly- plastic, rate-dependent material (m = 0.2). (a) For small normalized displacements. Co) For large normalized displacements.

With p = -T , , [ , f f i a denoting the pressure at the inner wall of the cylinder, eqn (160) gives

{p( t ) / k } = l n [ l + (b°2 - a°2) (ao ~ U-~J' with U = Ut, (161)

where, as before, k -- So/~. Note that the pressures p decreases as the flow progresses. In our numerical calculation, we take E -- 25,000 MPa, p -- 0.499, and So -- 50 MPa.

The selection o f a Poisson's ratio close to 0.5 makes the material almost incompressible. Also, since the "yield strain" eo - so/E is only 0.002, the elastic strains become negligible compared to plastic strains as deformation progresses and the normalized displacement 0 m ((U/ao)/eo) increases to value o f 5 and beyond.

736 G.G. Weaex et al.

.s~ !.4

D, 1.2

1.0

m 0.8

0.6 a~ t~

0.4

~ 0.2 0 Z 0.0

I D ! I I I

[] m-- N -J

[]

[]

-- Rate-ind., Analytical. Q Rate-ind., Numerical • Rate-dep., m=0.001, Num.

[] f , I , I , I I , I

0 2 4 6 8 10


I I | ! I

,4 \ I • Rate-dep., m=0.001, Num. N ~ ( Rate-ind., Analytical

].2 -" " ' "

m 1.o o~

t~ t~ 0.8

0 Z 0.6 I , I , I , I , I , I

0 100 200 300 400 500

NORM. DISPLACEMENT,

Fig. 8. Comparison of results for the normalized pressure (p/k) versus the normalized displacement /7 m [(U/ao)/(So/E)] from numerical calculations for a cylinder made from a rigid-perfectly-plastic, rate- dependent material with a very low strain-rate sensitivity (m = 0.001), against the analytical and numerical solutions for a rate-independent material model. (a) For small normalized displacements. (b) For large normalized displacements.

A comparison between the analytical and numerical results for the normalized pressure ( p / k ) versus the normalized displacement Uis shown in Figs. 6a and 6b for small and large values o f U, respectively. Note the large displacement increments 4 0 in Fig. 6b. The numerical results are in excellent agreement with the analytical solution.

VllI.3.b. Rate-dependent case. ~4 For a flow function (see eqn (31)) in the power-law form

/ 8 ~t/m f =~O~o} . (162)

'4This example has been previously considered by P~mce, Sum and NEEDL~t~t [1984] in the context of evaluating the accuracy of their semi-implicit time-integration scheme for elasto-viscoptastic materials.


with ~o, m and So w ~r3k material constants, the normalized pressure is given by

'P( l) /k ' = I ~ I 2 a ( l ) ( i l m l [ ( b ( ' ) ~ 2 m - |] ) J L\~- -~ ]

b(t) = 4(bg - a g) + a(ti 2, (163)

a( t ) = ao + (It.

In our numerical calculation we take E = 25,000 MPa, J, = 0.499, So = 50 MPa, ~0 = 0.002 sec -~, and m = 0.2. The displacement is applied at constant normalized rate of ((i/ao~o) = 1.0.

A comparison between the analytical and numerical results for the normalized pressure, ( p / k ) versus the normalized displacement O - ((U/ao)/eo) is shown in Figs. 7a and 7b for small and large values of U, respectively. Again, note the large displacement increments A U in Fig. 7b; the capability of taking such large steps in the finite element solution is a direct consequence of the robustly stable, implicit time-integration procedure used here. The numerical results are in excellent agreement with the analytical solution

In Fig. 8 we show our numerical results for a very low value of the strain rate sensitivity parameter, m = 0.001. For such small values of m the material behavior is nearly rate-independent. For comparison in this figure we also show the corresponding numerical and analytical solutions of the problem obtained from the rate-independent theory (see Fig. 6). The agreement of the numerical solution for the rate-dependent model with m = 0.001 and the results from the purely rate-independent model is very good.

AXIS

DIE FACE

OUTER SURFACE

MIDDLE PLANE

Fig. 9. Finite element mesh for an axisymmetric upsetting problem.

738 G .G. WEBER et al.

IIIII111

• - - ~ ÷ ~ ~ - - ÷ - ~ ÷ - - ~ - l i l l l l l l i l l I I I I I I I I I I I ~--+--+--+--+--+--+--+--+--+--t DEFORMED MESH I I I I I I I I I I I I I I I I I I I I I I , - - , - - , - - , - - , - - , - - , - - , - - , - - , - - , I I I I I I I I I I I I I I I I I I I I I I , - - , - - , - - , - - , - - , - - , - - , - - , - - , - - , I l l l l l l l l l l I I I I I I I I I I I

~- -+- -+- -+- -+- -+- -+- -+- -+- -+- -4 I I I I I I I I I I I I I I I I I I I l l l

~--+--+--+-- +--+--+--+--+--+--~ ORIGINAL MESH I I I I I I I I I I I I I I I I I I I I I I

Fig. 10. Deformed mesh for the upsetting problem after a height reduction of 60°70.

VIII.4. Upset forging o f a cylindrical billet

As a simple metal-forming example, the prototypical problem of isothermal upset forging of a cylindrical billet was solved. To simplify the calculation while retaining the important features, the dies were modeled as being rigid, with sticking friction acting to prevent sliding between the billet and the die faces when they are in contact. This friction causes the billet to barrel, with the material near the corners folding over to come in contact with the dies. Consequently, this example problem exhibits the realistic features of inhomogeneous deformation, with variable rates of straining at material points and time- varying die contact geometry. In our calculations we took the billet to be 2 mm diame- ter and 3 mm high. Figure 9 shows the finite element mesh containing 107 four-noded axisymmetric elements 15 (ABAQUS type CAX4). Near the corner where roll-over is expected to occur, the elements are triangular in shape to accommodate this deformation mode. Symmetry in the problem allowed a 1/4 model o f the billet to be used. The die face was modeled as a rigid surface and the external surface of the model was covered with interface elements (ABAQUS type IRS21A) to model the contact conditions.

VIII.4.a. Rate-independent case. Here, we considered a nonhardening material with E = 25,000 MPa, J, = 0.3, and So = 50 MPa. Figure 10 shows the deformed finite element mesh superposed on the undeformed mesh after a height reduction o f 60°70. The billet is seen to have expanded radially by a considerable amount. Five elements have folded over and come in contact with the die. Figure 11 shows the history of total die force versus die displacement for this calculation. Note that jumps in die force occur in the calculated result whenever new nodes came in contact with the die. In this figure, we also show the results from an identical calculation using the capabilities built

I SThese elements in ABAQUS address the problem of mesh-locking in (near) incompressible situations by using the method of NAOTEOAAt, PA~UCS, and Rice [1974].


450

400

350

300

250

200

150

i00

50

f O

/ ,/

• RATE-IND., h=0.0, UMAT • RATE-IND., h=0.0, ABAQUS O RATE-DEP., h=0.0, m=0.001, UMAT

I I I I I I I I I

0.0 0.2 0.4 0.6 0.8

DISPLACEMENT, mm

Fig. 11. Load versus displacement curve for an axisymmetric upsetting of a cylindrical billet made from a rate-independent, elastic-perfectly-plastic material The calculation was performed using our UMAT (solid square points) and also the capabilities built in ABAQUS (crosses). For comparison the results of a calculation, using our UMAT, for a rate-dependent material with a very low rate-sensitivity (m = 0.001) (open cir- cles) are also shown.

in ABAQUS for rate-independent plasticity. Also, we solve the same problem with our time-integration procedure for a rate-dependent material with a low value of the rate- sensitivity parameter, m = 0.001, and ~0 = 0.002. In the calculations using our UMATs we used our automatic time-stepping algorithm with A ~ = 0.05. The load versus displacement results from the three calculations, as shown in Fig. 11, are indistinguishable.

For this problem, the use o f the HUOnEs-Wn,~G~T [1980] algori thm in s tandard ABAQUS 0/4.7) gives results which are not too different f rom those obtained f rom our numerically objective algorithm. As mentioned previously, this is because o f the small relative rotations encountered in each increment in this problem.

VII I .4 .b Rate-dependent case. The specific constitutive functions used for the viscoplastic part o f the model (see eqns (31) and (32)) used in this calculation were the isothermal version of the functions for high temperature deformat ions proposed recently by BROWN, Ku~ and AUAND [1989] for hot-working:

f ( 8 , s ) = A {sinh(~8/s)}l/m, g ( 8 , s ) = h ( O , s ) f ( O , s ) , (164)

740 G . G . WEaER et al.

Table 1. Material parameters for A I I I00-O at 673°K (BROWN, KIM a ANAND [1989])

Material Parameter Value

A 4.75 x 10 -7 sec - I 7.0

m 0.23348 -~o 29.7 MPa ho 1115.6 MPa a 1.3

18.92 MPa n 0.07049 # 20.2 MPa x 66.0 MPa

150

125

z i00

o 7s

50

25

I I I I

• FIXED TIME-STEP OF 0.I SECS. o AUTOMATIC TIME-STEPPING, DELTA e = 0.05

/ /

f f

0 | I I I I I I I

0.0 0.2 0.4 0.6 0.8

DISPLACE~NT,

Fig. 12. Load versus displacement curve for an axisymmetric upsetting o f a cylindrical billet made from a hyperbolic-sine rate-dependent, s train-hardening material. The height o f the billet was reduced by 60% in 90.0 s. The calculation was performed by using a fixed time step o f 0.1 s in a total o f 900 increments, and also with au tomat ic time-stepping with a # r = 0.05 which required only 72 increments.


l.D. VALUE l 0.0 2 0.2 3 0.4 4 0.6 5 0.8 6 1.0 7 1.2 8 1.4 9 1.6

I0 1.8 11 2.0

Fig. 13. Contours of the equivalent tensile plastic strain eP after upsetting of 60°70.

with the function h defined by

h(O,s ) = hol l - ( s / s* ) l~s ign[ l - ( s / s*)] , s* = g[~P/A]". 065)

The static recovery rate ~ in (32) is taken to be zero. The values of the viscoplastic material parameters A, ~, re, So, ho, a, i , n, and the elastic moduli/~ and r, for 1100-O alu- minum at 400°C given by Brown et al. are listed in Table I.

In this rate-dependent problem, the height of the billet was reduced by 60% in 90.0 seconds. This corresponds to a nominal strain rate of 0.01 sec -~. Figure 12 shows the load versus displacement curve obtained by using fixed time-increments of 0. l seconds, and also by using our automatic time-stepping procedure with zlg~ = 0.05. The results from the two calculations are in very close agreement with each other. The calculation with the automatic time-stepping parameter set at zlgs p = 0.05 required only 72 increments.

In Fig. 13 we show the contours of the equivalent tensile plastic strain after 60°70 upsetting. A "dead zone" with less equivalent plastic strain is seen in the upper left por- tion of the mesh. This is a result of the constrained radial flow due to frictional forces from the die. A zone of more intense plastic flow is seen in the upper right corner of the mesh. This is associated with the fold over of the elements at the corner. Figure 14 shows contours of the deformation resistance s after 60% upsetting. A region of softer material is seen at the upper left in the figure, corresponding to the dead zone.

A similar calculation with slightly different geometrical dimensions of the upsetting specimen has been previously carried out by LusH, WEnvX and ANAr~D [1989]. In that paper we did not use the objective algorithm presented here. ~ Instead, we used the Hughes-Winget algorithm which was then standard in ABAQUS (V4.5). The results of our present calculations, when properly scaled for geometric dimensions, are indistinguishable from those presented in our previous paper. As discussed for the rate- independent case above, this is because of the small relative rotations encountered in each increment in this problem.

IX. CONCLUDING REMARKS

We have formulated, implemented, and evaluated the performance of a numerical ly object ive and stable, implicit time-integration procedure for a generic class of finite de-

ISOther aspects of the time-integration algorithm of our previous calculation were identical to that used

here,

742 G .G. WEaER et al.

I.D. VALUE 1 31. 2 32. 3 33. 4 34. 5 35. 6 36. 7 37. 8 38. 9 39.

10 40. 11 41.

Fig. 14. Contours of the deformation resistance s (in MPa) after upsetting of 60°'/0.

formation, isotropic, rate-independent, and rate-dependent constitutive equations for elastic-plastic materials. The constitutive equations and the time-integration procedure are straightforward generalizations of the classical constitutive equations of small-strain, rate-independent plasticity, and the classical radial-return algorithm, respectively. These features of this work make it well suited for wide use in the computational analysis of problems which involve moderately large plastic deformations of initially isotropic materials.

Acknowledgements--Helpful technical discussions with D. M. Parks are gratefully acknowledged. The finan- cial support for this work was provided by the U. S. Office of Naval Research (Grant No. N00014-86-8-0308) and the U.S. National Science Foundation (Grant No. MEA-8315117).

1951 1958

1959

1963

1964

1965 1966

1967

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1971

1972a

1972b

1973

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Department of Mechanical Engineering Massachusetts Institute of Technology Cambridge, MA 02139

(Rece/ved 2 July, 1989)

Weber Lushg Zavaliangos Anand 1990

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