Computer methods in applied

mechanics and englneerlng

ELSEWIER Comput. Methods Appl. Mech. Engrg. 128 (1995) 315-323

Consistent tangent operator for plasticity models based on the plastic strain rate potential

Lfiszl6 Szab6*, John J. Jonas Department of Metallurgical Engineering, McGill University, 3450 University Street, Montreal, Que., H3A 2A7, Canada

Received 24 August 1994; revised 2 May 1995

Abstract

A stress updating integration algorithm is derived for use with plastic strain rate potentials and with the backward implicit Euler method. An explicit expression is also deduced for the consistent tangent modulus. This analysis extends an earlier treatment developed for elastic-perfectly plastic materials in that it applies to materials which harden isotropically. The method is intended for use in finite element calculations that employ plasticity models based on the plastic strain rate potential. It is particularly suitable for cases where the form of the plastic dual potential has been obtained from texture measurements or from crystallographic models that describe texture evolution during metal forming.

1. Introduction

Most rate-independent plastic models are formulated in terms of rate-type constitutive equations. In these cases, the analysis of elastoplastic problems requires integration of the constitutive relation. Thus, the choice of integration algorithm has considerable influence on the efficiency, accuracy and convergence of the elastic-plastic incremental solution. Several integration methods have already been proposed, such as those of Hughes [l], Simo and Taylor [2], Ortiz and Simo [3], Simo and Taylor [4] and Matzenmiller and Taylor [5], which are based on the use of plastic potentials or yield functions defined in stress space. The accuracy and stability of integration algorithms have also been discussed by Simo and Taylor [4], Krieg and Krieg [6], Ortiz and Popov [7], de Borst and Feenstra [S] and Hartmann and Haupt [9].

Much less development has occurred, however, in the derivation of methods that apply to potential functions defined in plastic strain rate space, i.e. that are applicable to the so-called ‘dual potential’ approach. Examples of such treatments include those of Van Houtte et al. [lo], Martin and Nappi [ll], Barlat and Chung [12], Martin and Caddemi [13] and Bacroix and Gilormini [14].

The mathematical foundations of the dual potential analysis have been fairly extensively worked out by Ziegler [15] and Hill [16, 171, and its relation to the method of convex analysis was discussed by Eve et al. [18]. Nevertheless, unresolved questions remain with regard to the numerical implementations that must be prepared for use in finite element calculations. For example, when the displacement-based finite element method is used, integration methods are required for stress updating. In what follows, a return mapping algorithm is described for employment with the plastic strain rate potential. Although a somewhat similar approach has recently been proposed by Martin and Caddemi [13] and Bird and

* Corresponding author. On leave from Department of Applied Mechanics, Technical University of Budapest, H-1111 Budapest, Miiegyetem rkp. 5, Hungary.

0045-7825/95/$09.50 0 1995 Elsevier Science S.A. All rights reserved SSDZ 0045-7825(95)00884-5

316 L. Szab6, J.J. Jonas I Comput. Methods Appl. Mech. Engrg. 128 (1995) 315-323

Martin [19], the present treatment differs from theirs in that it applies to hardening (as opposed to perfectly plastic) materials and is designed for use with the finite element method based on the principle of virtual displacement (as opposed to the one that employs non-linear mathematical programming).

The numerical implementation described here is based on the use of the consistent tangent modulus. The solution of global equilibrium equations is frequently based on the standard Newton-Raphson iterative procedure. It is well known that to maintain the quadratic rate of asymptotic convergence for Newton’s method, the tangent operator must be consistent with the numerical algorithm employed to integrate the plasticity rate equations (see e.g. [2, 4, 20, 211). In the present paper, we show how to define a consistent tangent modulus for employment with the plastic strain rate potential.

Regarding notation, tensors are denoted by bold-face characters, the order of which is indicated in the text. Tensor products are denoted by 8, and the following symbolic operations apply: g * n = gini, (A* n)i = Aijnj, (A *B)ij = A,$,, A : B = A,B, and (C : A)ij = CijklAkl, with the summation convention over repeated indices. The superposed dot denotes the material time derivative or rate, the superscripts t and -1 denote transpose and inverse, and the prefix tr indicates the trace. In addition, Z is the fourth-order and S the second-order identity tensor.

2. Formulation of the elastic-plastic constitutive model

In this section, the governing equations of the commonly used classical theory of plasticity are recapitulated. This outline follows the treatments of Simo and Taylor [2, 41 and Ortiz and Simo [3]; the plasticity model is then described in terms of a plastic potential expressed in strain-rate space.

The constitutive equations of a wide range of elastoplastic materials can be summarized as follows. The total strain is assumed to be the sum of the elastic and plastic strains

& = Ee + &p ) (1)

and, for the case of linear elasticity, the stress tensor is related to the elastic strains as

a=De:k=, (2)

where D” is the fourth-order elasticity tensor. In what follows, linear isotropic elasticity is assumed, so that

D”=2GT+K6@6, (3)

where T = Z - +S 696 is the deviatoric operator, and G and K are the elastic shear and bulk moduli, respectively.

In the classical theory of plasticity, the plastic strain rate tensor is proportional to the gradient of the plastic potential cp(a, r), which is defined in stress space. Here, 7 represents the hardening parameter. For associative plasticity, the plastic potential is equal to the yield function, cp(a, r) = 4(a, 7). In the present paper, we assume isotropic hardening and the yield condition takes the form

9(@,~)=f(u)-r(Y)~O, (4)

where the function y+ r(y) defines the hardening law in terms of a scalar plastic state variable y. The flow rule for the plastic strain rates, using Eq. (4), is given by

where i is a plastic multiplier. In Eq. (5) one can write, for simplicity of notation,

af V=aa.

Note that v is the normal to the yield surface in stress space. The loading/unloading conditions can be expressed in Kuhn-Trucker form as

fpco, /i>o, df#J=O. (7)

L. SzabcS, J.J. Jonas I Comput. Methods Appl. Mech. Engrg. 128 (1995) 315-323 317

The plastic parameter i in Eq. (5), using the plastic consistency condition 8, = 0 and the rate form of Eqs. (1) and (2), can be expressed as

e.. i= v:D .E

H+v:De:v’ (8)

where H is the generalized plastic hardening modulus. Note that when the scalar plastic state variable y is defined by

Y' d dt ,

the plastic hardening modulus is given by H = &lay. Finally, the rate form of the constitutive equation can be expressed as

&=D’P:g, Dep=De-_ De:v@v:De

H+v:D=:v’

(9)

where Dep is the so-called elastoplastic, or continuum, tangent modulus tensor. In the classical flow theory discussed above, the yield function f(a) = T, represents a convex potential

in stress space. As discussed by Ziegler [15] and Hill [17], there is a dual potential which can be defined in strain rate space, g(dp) = i. The stress tensor, using this potential, can be derived as

(11)

When the functions f(a) and g(tip) are made homogeneous of degree one with respect to positive multipliers, the rate of plastic work is given by

I@J = u : &P = Ti = Tg($). (12)

The function g(kp) acts as a work-equivalent measure of the plastic strain rate, since the rate of plastic work is the same for all ep with a common value of g(S”).

There are several suggestions regarding the form of the plastic strain rate potential (see e.g. [lo, 12, 14, 21, 221); this topic is considered further in the Appendix.

It is important to note that, for plastically incompressible materials, the yield condition does not depend on the hydrostatic component of the stress. The function f is then defined by the deviatoric stresses. It follows that the function g in Eq. (11) acts as a potential for the stress; however, only its deviator is so determined.

In the remainder of this section, we summarize the basic relations of the plasticity model based on the plastic strain rate potential. The rate form of the constitutive equations is given by

&=D’:(k-S’), (13)

and the deviatoric stress is calculated using

s=r)u, (14)

where p is the normal to the plastic potential in plastic strain rate space, as defined by

Since + = i = g(tip), the rate equation for the function r( y ) can be written as

(15)

It is clear that for this model, a linear tangent relation between the strain rate and stress rate similar to that of Eq. (10) cannot be defined. However, it will be shown below that an explicit expression for the consistent tangent operator can be derived.

318 L. Szab6, J.J. Jonas 1 Comput. Methods Appl. Mech. Engrg. 128 (1995) 315-323

3. Stress updating integration algorithm

In computational plasticity, the basic problem is the integration of the constitutive equations over a finite time step. In the displacement-based finite element method, the constitutive quantities are known at time t, and the global solution is advanced from t, to t, + 1 using an iterative Newton method. In this work, the implicit backward Euler method is employed to determine the updated stress and plastic state variables.

Let [t, , t, + 1] be the time interval of interest, and let us denote equilibrium states by n, n + 1, . . . and intermediate, not equilibrated, states during the iteration by i - 1, i, . . . . The total strains are known at time ti according to the formula

ci = E, + AE~ , (17)

where Aci is the total strain increment between the preceding equilibrium state and the current configuration at time ti. This total strain increment is additively decomposed into elastic and plastic parts, as follows

AE~ = AE~ + AE~’ . (18)

The elastic predictor or so-called trial stresses are defined by

u;=u,,+D~:AE~, (19)

and the final stresses at ti, using (X3), become

u~=u,,+D”:AE~=u,+D”:(AE~-AE~). (20)

From (19) and (20), we finally obtain

ui=uT-De:Ae~. (21)

This equation, using the deviatoric quantities, can be rewritten in the following form

si = s; - 2G A$ . (22)

For the constitutive model in which the plastic potential is defined in plastic strain rate space, the deviatoric stresses at time tj can be calculated from

MA&P 1 ‘i ='i aAsP = ~∾) , (23)

where 7i is calculated using

To = ~[x + g(A&;)] = T~(AE;) . (24)

It should be noted that the plastic strain rate at time ti, corresponding to the fully implicit integration scheme, is calculated from the plastic strain increment using the relation dp = A&p/At, where At = tj - t,. Moreover, the increment of the plastic state variable can be calculated from

Ax = Agi = g(AEp) . (25)

It is now clear that the problem is reduced to finding the unknown plastic strain increments A&P from Eqs. (22) and (23). The combination of these equations, leads to

T~(AE;)&AEP) + 2G A&P - s; = 0. (26)

The problem of integrating the elastoplastic equations reduces to the solution of the non-linear algebraic equation system (26) for A$.

REMARK 1. In the classical theory of plasticity described in the first part of Section 2, calculation of the plastic strain increment vector, using the implicit backward Euler method, yields only one non-linear equation (see e.g. [l, 2, 4, 23-251). In this case, the unknown variable AA is defined along

L. Szabb, J.J. Jonas I Comput. Methods Appl. Mech. Engrg. 128 (1995) 315-323 319

the lines of the incremental form of Eq. (5): AeP = Ahv; here the gradient of the yield surface v is calculated at time ti. 0

It then follows from Eq. (26) that the updated stresses are given by

ai = si + i tr u,,S + K tr(AEi)S , (27)

where si is calculated from (23), using the value of AEP obtained from (26). Finally, we note that the yield function is not explicitly known for the plasticity model discussed

above. Thus, there is no direct way to check whether plastic deformation occurs from the trial stress of relation (19), as in the case of the classical models using the stress form of the yield function. However, following the work of Bacroix and Gilormini [14], a trial function I can be defined from the expression for the rate of plastic work, Eq. (12), as follows

Z(s, EP) = 7g(&p) - s : EP . (28)

By substituting the trial deviatoric stress s: in this function and minimizing for the plastic strain rate, the following conditions are obtained

Min{l(sT, sP)} < 0 if ST lies outside the yield surface

= 0 if sT lies on the yield surface

> 0 if ST lies inside the yield surface . (29)

4. Consistent elastoplastic tangent modulus

The explicit expression for the consistent tangent modulus will now be given for the plasticity model presented in Section 2. To derive this quantity, Eq. (20) is first differentiated to give

da, = D" : (da, - dds;) . (30)

Then, by differentiating the dual form of the flow rule, (23), we obtain

(31)

where aT,/ay, is the plastic hardening modulus Hi defined at time t,. Replacing the term within parentheses on the right-hand side by the fourth-order plastic modulus

tensor Dp, Eq. (31) can be simplified to read

dsi = D' : dA&P , (32)

where

Dp =r,M,+ HipiC3pi. (33)

Here, the fourth-order tensor iVi denotes the second-order term in Eq. (31), namely

Eq. (32) can be inverted, which then leads to

dhs; = C; : dsi ,

where Cp = Dp-' 1s the plastic compliance tensor.

(34)

(35)

REMARK 2. It can readily be shown that the tensor M, in Eq. (32) is not invertible. Since the function

320 L. Szabd, J.J. Jonas I Comput. Methods Appl. Mech. Engrg. 128 (1995) 315-323

g(Aep) is homogeneous of degree one, the matrix array of its second derivative annihilate the respective column A&P (iWi : A&P = 0, see [26]). In other words, the matrix of the tensor Mi represented by (6 X 6) matrix is singular. Thus, in the inversion of the tensor Df’, the well-known Sherman-Morrison formula cannot be applied. 0

In Eq. (35), dsi can be replaced by da,, using the fourth-order deviatoric operator T, so that

dA$‘=Cy:dsi=C;:T:dai=C‘P:dai. (36)

It should be noted that multiplication by T does not effect the symmetry of Cp in the above relation and that Cp cannot be inverted.

On substituting (36) into (30), after some manipulation, we obtain

du,=tiep:dei, (37)

where the consistent tangent modulus tensor ayp, defined by

aep = {C” + CP,-l, (38)

is symmetric. In Eq. (38), the tensor C” = De-’ is the elastic compliance defined by the well-known expression

1 1 C’=~GT++@S.

For numerical implementation, an equivalent but more useful form can be derived. First, the plastic strain increment is calculated from the expression

dA$’ = 2G(2GZ + Dp)-’ : de, , (40)

where de, is the deviatoric strain increment and the deviatoric forms of Eqs. (30) and (32) are employed to derive this relation. By substituting Eq. (40) into the deviatoric form of Eq. (30) the deviatoric stress increment can be expressed as

dsi = [2GZ - 4G2(2GZ + Df'-'1 : dei . (41)

It is of interest that the above relation was derived by Bird and Martin [19] for elastic-perfectly plastic materials. Under the latter conditions, Z-Z = 0 by definition and Dp in Eq. (41) only contains the first term in relation (33).

Eq. (41), making use of the relation tr da,. = 3K tr dei, can be rewritten as

dq=[D”-D”:T:(2GZ+Df’-‘:T:D”]:dq. (42)

In Eq. (42), the inverse of the term 2GZ + Dp can be evaluated by substituting (33) into the above expression and using the Sherman-Morrison formula; in this way, we obtain

(2GZ + TIM, + Hipi @pi)-’ = (Ai + Hipi @cc,)-’

=A,:’ - Hi

1+ Hipi :A-’ : pi A-' : pi@pi :A-’ . (43)

where Ai = 2GZ + riM.

REMARK 3. The fourth-order tensor Ai can also be inverted, using the work of Betten [27]. However, this calculation is more complex and is not discussed here. More details about the calculation may be found in [28]. In addition, it is well known that when the time step At-O, in the case of stress space plasticity, the consistent tangent operator reduces to the classical elastoplastic (or continuum) moduli given by (10) ( see e.g. (2, 41). Because of a continuum tangent moduli for the plasticity model discussed in the present paper, cannot be derived, the question of the limit value of the consistent (or algorithmic)

L. Szab6, J.J. Jonas I Comput. Methods Appl. Mech. Engrg. 128 (1995) 315-323 321

tangent operator (Eqs. (38) or (42)) for decreasing time step still needs to be addressed for the dual plasticity model (see also [28]). 0

5. Concluding remarks

In this paper, a stress updating integration algorithm has been derived for use with plastic strain rate potentials and with the backward implicit Euler method. An explicit expression has also been derived for the consistent tangent modulus. This analysis extends an earlier treatment developed for elastic- perfectly plastic materials in that it applies to materials which harden isotropically.

The method is intended for use in finite element calculations that employ plasticity models based on the plastic strain rate potential. It is particularly suitable for cases where the form of the plastic dual potential has been derived from texture measurements (see the Appendix that follows) or from crystallographic models that describe texture evolution during metal forming. Details of numerical implementation of the method presented in this paper will be given a forthcoming publication [28].

Acknowledgments

The authors are indebted to the Natural Sciences and Engineering Research Council of Canada (NSERC) for financial support. This research has also been funded by the National Development and Research Foundation, Hungary (under contract no. OTKA 5-721 and 5-821). The authors thank Prof. Pierre Gilormini (Laboratoire de Mecanique et Technologie, Ecole Normale Superieure, Cachan, France) for helpful discussions during the course of this work.

Appendix A: Some suggestions for the plastic strain rate potential

Van Houtte et al. [lo], using texture data, proposed an analytical expression for the plastic potential in the following form

where ki, i E [l, 51 is a plastic strain rate vector defined as

(A.21

and tj = ($&iji)1’2 is the von Mises equivalent plastic strain rate. In this model, the deviatoric stress is calculated using

(A.3)

where Si, i E [l, 51 is a deviatoric stress vector defined so as to be work conjugate with b in Eq. (A.2), and 7’ is the critical resolved shear stress.

The ai,... coefficients in Eq. (A.l) are calculated by a least-squares fitting to the average Taylor factors. Note that many of these are equal to zero because of the various symmetry properties of these matrices, as has been discussed by van Houtte et al. [lo].

A fourth-order potential, using the texture-adjustment method (see e.g. [21]), has also been proposed by Bacroix and Gilormini [14], namely

(A.4)

322 L. Szabd, J.J. Jonas I Comput. Methods Appl. Mech. Engrg. 128 (199.5) 315-323

where the functions XK(kp), are defined by

X, = (%)4 2 X6 = (6g3d;, ) x,, = (43’(~~2)’ 3 Xl, = (~;,)*(~L)*

x2 = @g4 9 x, = (62p2)3ql ) Xl, = (@;,)*(g,)* 9 xl, = (L-:3)2(k~2)2

x3 = (e3)4 7 x&3 = (~:l)*(&)* 9 xl3 = (g;2)2(d:3)2 > Xl, = 6~1~;2(tt;3)2

x4 = (e3)4 9 x, = (67, )*N,)* 9 Xl, = (g,)*(~:,)* , Xl, = s;, g*( t’y3)*

x5 = (q4 7 Xl, = (~~l)‘(~~3)’ 7 x,, = (~g,)‘(~~,)’ 7 x2, = 6~1~;*(~;*)*

x 21

= 8P &P gP dP 11 23 13 12 9

x 22

= &P gP &P &P 22 23 13 12 2

and

~~~P~~ = ti{(k:,)2 + (k;*)* + s+;* + (8:*)* + (k’2p3)* + (6~3)2}1’2.

They introduced the following plastic strain rate vector

ET = [.+, it*, i;*, i2p3, iE] )

together with the appropriate dual deviatoric stress vector

(A.9

(A-6)

In terms of this notation, and using the strain rate potential, Eq. (AS), the deviatoric stress vector can be calculated from

(A.71

It should be noted that when the above potentials (A.l) and (A.4) are employed in finite element calculations in conjunction with the present algorithms, the relevant five-dimensional stress and strain vectors (A.2), (A.5) and (A.6) must be transformed into their six-dimensional forms.

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