Compu/. Ma/h. Applic. Vol. 15, No.3, pp. 169-174. 1988Printed in Great Britain. All rights reserved

0097-4943/88 $3.00+ 0.00Copyright © 1988 Pergamon Press pic

'I,

A NOTE ON THE NUMERICAL ANALYSIS OFMATERIAL DAMAGE BASED ON THE THEORY

OF MATERIALS OF TYPE N

S. J. KIM' and J. T. ODEN~

'Seoul National University, Department of Aerospace Engineering, Seoul. Korea2Texas Institute for Computational Mechanics, The University of Texas at Austin, Austin.

TX 78712-1085. U.S.A.

(Received 3 September 1987)

Abstract-A generalization of the theory of materials of type N to continuum damage mechanics ispresented. Then algorithms developed earlier by the authors for plasticity problems are extended andapplied to the analysis of progressive damage of materials under large elastoplastic deformation. Resultsobtained by applying these methods to representative problems of metal forming are presented.

I. INTRODUCTION

The success of continuum damage theories in modeling the loss of stiffness of highly stressedengineering materials has led to increased interest in this subject over the last decade. Typically.damage theories are characterized by evolution equations for one or more damage variables thatdepict Ihe growth of microcrack density. delamination or debonding of composite components.nuclealion of voids. etc. The damage of the material may characterize a loss of stiffness or strengthand can model the prelude to fracture and failure of the material specimen.

The earliest version of such continuum damage theories is often credited to Kachanov (I]. andnumerous generalizations have been proposed. We mention as important and representative thework of Lemaitre and Chaboche (2]. Lemaitre [3]. and Taylor and Flannagen [4]. Higher-orderdamage theories. in which the damage variable is vector-valued or tensor-valued have also beenproposed: see. e.g. Allen et ai, [5]. A detailed history and survey of the literature was given in 1984by Krajcinovic [6]. The bulk of these damage theories deal with intinitesimal deformations of metalsor rocks.

In the present note. we outline a simple damage theory that tits naturally into the frameworkof our theory for generalized potentials in tinite elastoplasticity which we refer to as a theory ofmaterials of "type N" (signifying a generalized notion of normality) [7-9]. In these earlier paperswe describe numerical procedures for solving the equations in tinite elastic plasticity. A straight-forward generalization of the methods in [14] leads to algorithms for calculating progressivedamage in materials undergoing very large elastoplastic deformations. We cite applications of theseprocedures to the calculation of damage in elastoplastic metals undergoing very large strains ina simulation of a metal forming process.

2. CONTINUUM DAMAGE MODEL

For details on kinematics and notations. we refer to [7-9]. We consider finite deformations ofa material body characterized by a collection of constitutive equations of the form

<I> = cP (E, e. g. 11,d)

C1 = r. (E. e, g, II, d)

tl = N (E. e. g. II. d)

q = Q (E. e. g. 11.d)

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t70 S. J. KIM and J. T. ODEN

and a generalized potential P. which may only be smooth enough to possess a generalizedsubdifferential in the sense of Clark (see [II]), such that

CPo -d)e81J'(CT,A) (2)

for arbitrary (CT. A). Here <I> is the Helmholtz free energy, CT is the Cauchy stress, tl is the entropydensity, and q is the heat flux. The "independent" constitutive variables are defined as follows:

E =L UC U-11,ym dt,

P =LUP U-Ilsym dr.

U = iJc + Up.{J = the absolute temperature.g = the temperature gradient.h = a hardness variable conjugate to an internal state variable z.d = the scalar-valued damage measure,ot = the state-damage pair = (h. d),A = the thermodynamic force conjugate to IX.

Here. U is the right stretch tensor. U is the right stretch-rate. and UC and UP are "elastic" and"plastic" parts of U; for an interpretation of these tensors, see [9].

Following arguments standard. in continuum thermodynamics, we are able to show that (if IJ'is differentiable and convex),

T 8<1> 8¢ a<l>S ::;:T CT R = p a E' t1 = - ao' a g = 0

a<l> afA = - Oot. 0 = - p aa' s: P + A: IX - 0 -I q .vo ~ °

alJ' alJ' alJ'P=- i=--- d=--as' ah' as·Here, R is the rotation tensor and D is the thermodynamic variable conjugate to d.

3. EXAMPLE CONSTITUTIVE EQUATIONS

As an example, we cite some possible forms of the constitutive functionals suggested by earlierwork. In [9], we presented some simpler generalizations of rules proposed by Bodner and Partom[10, I], drawn from the classical Prandtl-Reuss flow rule which simulates isotropic hardening ofmetals. We amend the constitutive functionals for free energy to account for loss of stiffness dueto damage in a manner suggested by work of Lemaitre [3]. The resulting potentials assume theforms:

(I - d)", 2<I> = -2 [A.(trE)- + 2µ tr E] - hi A - Ilm(hl - ho)exp( -mA).

Po

Here we have denoted

I (11 + I)B=- - , lieN.3" 11

J~= 1/2 tr (sy S' = S - 1/3 tr (S).

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A note on the numerical analysis of material damage

Then, from relations presented earlier. we have

0<1> p), PS=p ::>E=-(trE)I+-2µE.

u Po Po

4. A NUMERICAL RESULT

t7t

In [9], we described a finite element model of the quasistatic behavior of materials characterizedby constitutive equations similar to those given above. In addition, an incremental total Lagrangianalgorithm was presented for the solution of problems with large plastic deformation and isotropichardening.

A straightforward extension of the algorithms and discrete formulation of[9] permits us 10 nowinclude continuum damage effects. One merely adds to the evolution equations for the internal statevariable that for damage d having made appropriate modifications in the form of the constitutivefunctionals of <I> and IJ'.

Numerical results obtained by following such procedures using the constitutive equations listedin the preceding section are illustrated in Figs I and 2. In these calculations. the following valuesof the material parameter were used:

;. = 93667 N/mm2,

µ = 44000 N/mm2•

11 = I,

Do = 1.3 x to's-I.

m = 5.75 x 104•

III = 1450 N/mm2,

110 = 1150 N/mm2•

C1=0.001 mm2fN·s.

The results shown correspond to a 20% crushing of a metallic billet in finite plane strain. Thedimensions of the specimens are 4 x 5 units and the finite element model consists of 80 quadrilateralelements composed of a composite of four constant-strain triangles, as shown. Damage is measuredon a non-dimensionalized scale, 0 ~ d ~ 10, with d = 10 signaling fracture of the material. Theresults in this particular calculation allow one to trace the evolution of damage during the metalforming process and to estimate residual damage in the final produce for a 10 s loading time. Whileno fracture is suggested in the formed product in the present example. one does see a significantloss of stiffness in some portions of the billet after completion of the loading.

172 S. J. KIM and J. T. ODEN

Fig. I. Progressive damage in a dynamically deformed billet at finite strain. Progressing upward:(a) 0 < d < 5; (b) 5 ~ d < 6: (c) 6 ~ d < 7; (d) 7 ~ d < 8; (e) 8 ~ d. In (a) # - d > 8; elsewhere 7 ~ d < 8.

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A note on the numerical analysis of material damage

Fig. 2. A second example of a progressively damaged billet: (a) 0 < d < 4.5; (b) 4.5 ~ d < 5.5;(c) 5.5 ~ d < 6.5; (d) 6.5 ~ d < 7.5; and (e) d ~ 7.5.

CAMWA.ISl-B

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174 S. J. KIM and J, T, ODEN

Acknoll'ledgement- The support of this work by the Army Research Office under Contract DAAL03-86-K-0043 is gratefullyacknowledged.

REFERENCES

l. M. L. Kachanov, On the creep fracture time. Izv. AklJd. Nallk. SSR 8, 26--31 (1958).2. J. Lemailre and J, L. Chaboche. Aspects phcnomenologiques de la rupture par endommagemenl. J. Mech. Appl. 2.

o. 3 (/978).3. J. Lemaitre. Coupled elasto-plasticity and damage con.stitutive equations. Compo Meth. appl. Mech. Engng 51,31-49

(1985).4. D. p, Flanagan and L. M. Taylor, An accurate numerical algorithm for stress integration with finite rotations. Compo

Merh. appl. Mech. Engng. In press.5. D, H. Allen. W. E. Haislcr and C. H. Harris. Research on damage modcls for conlinuous fiber composites.

A FOSR-84-0067 (1985).6. D. Krajcinovic, Continuous damage mechanics revisited: basic concepts and definitions. J. appl. Mech. 85-WA/APM-5

(1985).7. S. J. Kim and J. T. Oden, Generalized potentials in finite elastoplasticity. Int. J. Engng Sci. 22, 1235-1257 (1984).8. S. J. Kim and J. T. Oden, Gencralized potcntials in finite elastoplasticity, II. Int. J. Engng Sci. 23, 515-530 (1985).9. S, J. Kim and J. T. Oden, Finite element analysis of a class of problems in finite elastoplasticity based on thc

thermodynamical theory of materials of type N. Compo Meth. appl. Mech. ElIglIg 53, 277-302 (1985).10, S. R. Bodner and y, Partom, A large deformation elastic-viscoplastic analysis of a thick-walled spherical shell. J. appl.

Mech. TrailS. ASME 39, 75t-757 (1972).II. S. R. Bodner and Y. Partom, Constitutive equations for elastic-viscoplastic strain-hardening materials. J. appl. Mech.

Trans. ASME 42(2), 385-389 (1975).

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