Incremental Kinematics for Finite Element

8/12/2019 Incremental Kinematics for Finite Element

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INTERNATIONAL J O U R N A L FOR N U M E R I C A L METHODS I N E N G I N E E R I N G , V OL . 36, 3937-3956 (1993)

INCREMENTAL KINEMATICS FOR FINITE ELEMENTAPPLICATIONSM . M . RASHID

Engineering Mechanics and Material Modeling Department, Sandia National Laboratorre?, Alhicqueryue. NeN M exico ,87/85, U S A

S U M M A R YA kinematical algori thm is proposed in which the input is the incremental deformation grad ient correspond-ing to a n increment in motion for a t ime step, and the output is a constant rate-of-st retching tensor an da rotat ion tensor. The concept of algorithmic objectivity is discussed, with a stronger s tatcmcnt of objectivitythan those advanced in previous works being set forth. The im portance of strong objecticity is il lustrated forapplications in which moderately large rotation increments are to be expected. Accuracy of the proposedalgori thm is discussed and i l lustrated with example calculat ions: and i t is shown that the accuracy ofthe algori thm may be extended to arbi t rary order. Also, the computational effort associated with theproposed algori thm is seen to compare favourably to tha t of other algori thms.

1. INTRODUCTIONThis paper presents a means for processing the incremental motions generated by finite elementsolution procedures for use in constitutive integration algorithms. The basic premise of the paperis that, while the global solution procedures found in all updated-Lagrangian finite elementformulations generate only discrete deformation increments, the evolution equations defining thematerial state generally depend on the complete history of the deformation. It is thereforenecessary to assume a deformation path over each time step for the purpose of updating thematerial state.To be useful, the assumed deformation path over each time step must satisfy certain require-ments. Among these requirements is the need for the assumed path to be of such a form that it isconvenient for use in the constitutive integration algorithm. Although the precise implications ofthis requirement will depend on the particular constitutive relations of interest, the algorithmpresented here produces a constant stretching motion such that D = constant and W = 0 (whereD is the stretching rate, W is the vorticity, and D + W is the spatial velocity gradient), followed bya stepwise rotation. This representation is appropriate for a wide variety of inelastic constitutivemodels. The deformation path is constructed for each time step given the incremental deforma-tion gradient for the step, which in turn is trivial t o compute based on quantities normallyavailable in the finite element environment.Another requirement of any incremental algorithm is that it be objective, in some sense. Herethe notions of weak and strong incremental objectivity are introduced. Essentially, a weaklyobjective kinematical algorithm is objective for special input motions, namely, pure rotation andpure stretch. The notion of strong objectivity is related to the algorithmic behaviour in thepresence of both stretching and rotation. Tt is shown that the algorithm of Hughes and Winget,as well as other, related algorithms (e.g. that of Flanagan and Taylor), are only weakly objective.This characteristic can cause unintended coupling between the rotational part of the motion and0029-598 1/93/233937-20$15.00993 by John Wiley Sons, Ltd. Received 21 July 1992Revised 13 April 1993


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3938 M . M . R A SHIDthe stress update whenever moderately large rotation increments are present, as illustrated inSection 4. However, lack of strong objectivity is of little consequence when both the rotationincrement and the stretching increment are small, as is often the case in finite element applica-tions.Section 2 of the pa per dispenses with som e preliminary kinematics, and provides a statementof the problem to be addressed. The algorithm itself is presented in Section 3, including astep-by-step summary for purposes of implementation. A synopsis of the computationaleffort required by the algorithm is also given in Section 3. Example calculations are presented inSection 4, wherein the effects of the s tro ng objectivity of the prop osed algorithm are illustrated formotions involving moderately large rotation increments. The relationship of the present algo-rithm to previously proposcd ones is also discussed in Section 4. Finally, some kinematicalaspects of the computation of tangent moduli are addressed in Section 5. and a geometricalinterpretation of the rotation part of the algorithm is given in the appendix.Notat ion is stan dar d throughou t, w ith Cartesian tensors bcing employed. The switch betweenco-ordinate-free notation and indicia1 notation is made freely; where tensorial componentsappear, they should be understood to represent Cartesian components. An interposed dotindicates double contraction, and the summation convention is in force. Finally, the expressionx = O(c) is taken t o have the standard meaning lin~,-,ox/t: 0 and constant.

2. KI NEM ATI CAL PRELI M I NARI ESIn finite element applications involving inelastic material behaviour, it is generally necessary tointegrate rate-type constitutive equations over each time increment to obtain the updatedmaterial state. In the constitutive rate equations, qu antities relating to the rate of deformationand rotation app ear as forcing functions. Th e global solution procedures used in all finite elementcodes require th at the constitutive update be performed given only the deformation increment,but n ot the defo rmation history, over each time step. How ever, except in thc case of elasticity, theupdated material state clearly depends on the deformation history over the step, since thekinematical forcing functions that appear in the constitutive rate equations are pointwisefunctions of the deformation and spin rates. It is therefore clear that, for the purposes ofintegrating the constitutive rate e quations, an assum ption m ust be mad e regarding the variationof the imposed deform ation o ver each time increm ent, given the total deformation increment. Informulating this assumption, the following three considerations should be borne in mind:

1. The assumed deformation history over each time step should reproduce the given totaldeform ation increment w hen evaluated a t the end of the step, as accurately as possible if notexactly.2. The assumed deformation history should be reasonable in the sense that it should notinvolve a ny wild excursions enroute to the total deformation increment. Such excursionscould prod uce unrealistic results when the constitutive equations are integrated.3. The assumed deformation history should be such that the Constitutive rate equations arerendered a s easy to integrate a s possible.The issue of incrementaE ohjectiviry as defined by Hughes and Winget falls within the domainof Item 1 above. Essentially, Hughes and Winget define an incrementally objective kinematicalalgorithm as one in which the incremental motion tha t is returned by the algorithm is identical tothe input incremental motion, whenever the input motion is a pure rotation. An a lgorithm will becalled strongly objective if, in addition to the requirements for weak objectivity, it computesa stretching part that is independent of the input rotation, when the input incremental motion


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INCREMENTAL KINEMATICS 3939involves both stretch and rotation. These definitions will be made more precise in the next section,after the necessary notation has been introduced.The implications of Item 3 will depend, to some extent, on the form of the rate constitutiveequations. For the purposes of this development, it will be assumed that the constitutive rateequations are most easily integrated when the imposed deformation history is one of constantstretching rate (D = constant) and zero spin (W = 0), where

D = $(L + LT), W = $(L LT) (1)where L is the spatial velocity gradient. The implications of this specification are discussed below.A few definitions are now introduced. The generic time increment under consideration isassumed to be such that t E I t , , t ,+ 1.The configurations of the material element under consid-eration at I t, and t = t,+ are denoted by K, and K, + 1 , respectively. The incremental motionover the time increment is assumed to be given in the form of the inverse of the deformationgradient p of ti,,+ with respect to K , , which may be written as

Here ;(x) is the incremental displacement field for the time step, and x is the position vector ofmaterial points in t i , + l . I t is remarked that E - is readily obtained within a finite elementenvironment, since for a typical element e, dG/ax is given byc;BOXj u: B;,,, (no sum on e ) (3)

In (3), u: (i = 1 , 2 , 3 ; A = 1,2,. . . , number of nodes per element) are the components ofincremental nodal displacements for element e and for the current time step, whereas BTA are thecomponents of the shape function gradient for node A and element r , evaluated at time t = t,+and a t the quadrature point o f interest. It is emphasized that (3) was written assuming the B;,,, thatare available in the finite element code are the components of the shape function gradient withrespect to K, 1 . This is the case in many large-deformation finite element formulations. If insteadthe BT,, are the components of thc shape function gradient with respect to K, (i.e. evaluated att = t,), then 6 is given by

where X is the position vector of material points in ti,,, and where now( 5 )dBi- _ ufAB;,, (no sum on e)OX,

within element e . With regard to the kinematical algorithm described in the next section, theprocedure will be developed using 6 as input. while it is noted that 6 may be used in place of@ - with only minor modification. Also, it bears repeating that fi is not the deformation gradientwith respect to some global reference configuration, but rather the incremental deformationgradient of ti, with respect to K , . Hence, fi = F,+ F,-l , where F, is the total deformationgradient at time t,.

must bedeveloped subject toIn line with the comments made above equation (l) , a deformation history is sought such that

As mentioned previously, a suitable time-varying deformation F(t), t E [ t , , t,+F t,) = 1 F(t,+l)= E (6)


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3940 M. M. RASHIDFF-I =D (constant and symmetric), t,, < t < t,+, ( 7 4

F(t;+ ) =0 symmetric positive definite) (7b)F(t,,+ ) = R f i = 6 (R proper orthogonal) (7 4

Equation (7b) is a consequence of (7a). According to (7a)47c), the deformation is assumed to varyover the time step in such a way that stretching occurs along fixed principal directions andwithout rotation in 1 E ( t , , , t ,+, ), with all of the rotation occurring impulsively at t = r , , , with noadditional stretch.

For future use, it is noted that any rotation tensor R may be expressed in the formR i j= d j j + (1 - os @ ( p i p j i j) in O s i j k p k (8)

in which p is the unit vector along the rotation axis, 0 is the rotation angle, and E i j k = f 1 or 0 isthe alternator symbol. In deriving an approximation to R in the next section, use will be madeof the distance function d ( R 1 ,R 2 ) between two rotations R 1 nd R 2 .The metric on the manifoldSO (3) of three-dimensional rotations is usually defined so that d(R, ,R,) s equal to the minimumangle of rotation required to take R1 nto R, , i.e. d ( R , , R,) is the angle of rotation of R z R raround a fixed axis. Using (8) and letting pl, p, and 0, ,8, be the rotation axes and rota tion anglesof R , and R 2 , respectively, it can be shown that

4 sin2 [d(R , , R 2 ) ] = 2[S?(l + C,) + S$(1 + C, ] SfS?+ 4aS, S2(1 c1 cz c1C,)+ a2[4 1 C 1 C2 + C1C , ) 6 S : S $ ]4a3S1s2(1 c 1- c2+ c1 ,)

4(1 c1 c , + c ,c2)z (9)In (9), a = p1 p 2 ,S, = sin 0, and C, = cos O , , where a = 1,2. Equation (9) serves to define thedistance function on S O ( 3 ) when d(R', R 2 ) < n/2, which is not restrictive for the purposesconsidered herein. Finally, it is noted that (9) reduces to the expected expression

[d(R', R2)]' = 0: + Q nfflB2 + c4 (10)It is here remarked that the deformation history given by (7aH7c) is convenient for thewhen 1O,1 E < 1.integration of the usual rate form of isotropic plasticity, i.e.

. F = C * ( D -DP) 1 1)In (1 l),C is the rank-four isotropic elastic modulus tensor, DP s a function of Cauchy stress T andany hardening parameters that may exist, and f = T + TW W T represents the Jaumann stressrate. With the deformation history (7aH7c), 1 1) becomes

T = C .(D DP), D constant (12)where now only the material rate of Cauchy stress, and not a corotational rate, appears on theleft-hand side. The solution to (12) is evaluated at t = t n + l to obtain Ti;, from which the updatedCauchy stress is obtained as R'fR'.Some authors have replaced the Jaumann rate of Cauchy stress in (11) with other co-rota-tional stress rates, e.g. that involving C2 = RRT instead of W (see discussions by Johnson andB a m m a n ~ ~ , ~t l ~ r i , ~ee et al.,' Dafalias6 and Dienes'). (Here,R arises from the polar decomposi-


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INCREMENTAL K I N E M A H C S 3941tion F = RU of the total deformation gradient.) The justification for this replacement is oftenbased on the aphysical oscillations in stress components observed when (1 1) is used to determinethe stress response in large shearing motions. However, use of the so-called Green-McInnis rate(i.e. that based on Q instead of W) introduces an awkward dependence of the stress on the chosenglobal reference configuration that may be no more physical than the stress oscillations in simpleshear predictcd by the Jaumann-rate form of plasticity and hypoelasticity. In the context ofincremental kinematics, this dependence manifests itself as the need to carry descriptors of thetotal deformation from step to step. In this paper, the view is taken that any history dependence ofthe constitutive response, including dependence on a particular, favoured reference configuration,should be explicitly accounted for through suitable state variables.' The criteria (6) and (7aH7c)were formulated consistent with this point of view, and in particular such that the desireddescription of the deformation history over each time step [i.e. (6) and (7aH7c)l could bedetermined based on the incremental motion only (as represented by k ' or @), ndependent ofany global reference configuration.

3. AN ALGORITHM FOR INCREMENTAL KINEMATICSCalculation f the stretching r a t e

Equations (7aH7c) indicate that the rotation tensor R (from the polar decomposition F = R6)must be extracted from F-'. An approximate method for obtaining R that is both accurate andefficient will be presented shortly. First, however, an approximate D is obtained from p - '.To thisend, it is noted that(13)f j - l ) 2 = F - l @ - T = e 1

where c = @ @. From (7a), (7b) and the assumed constancy of D in (r,, r,+ ,), 0, and D sharecommon (fixed) principal directions; one is therefore justified in writingU = exp(At D), At t, + 1 , (14)

Equation (14) may be inverted to yield D = ( l / A t ) log 0,or, using (13),1

AtD = og { [(e 1 1 1 2 ] ' }The indicated operations in (1 5) are justified by the co-axiality of D and the argument of the logfunction, and by the positive-definiteness of c. [In principal, the functions appearing in (15) areevaluated by writing in spectral form and then applying the indicated operations to theprincipal values.] Next, it is noted that 2. ' differs from 1 by only a small amount, since thestretches associated with the increment in motion over a single time step are expected to be small.Accordingly, the function of c-' appearing in (15) is expanded in a Taylor series around 1:Retaining only a finite number of terms in the series expansion (16) results in some truncationerror. Specifically, integrating U = DU exactly subject to the initial condition U(l,) = 1, andwhere D is obtained from the first few terms of (16), results in a value of U(t,,+,) that differsslightly from 6.n the next section, it is demonstrated that the first two terms provide a highlyaccurate approximation to D for finite element applications.Before proceeding to the ro tation part of the algorithm, a few observations are made regarding


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M. . ASHID942(16).First, U is written as

i j = l + Y (17)where Y is symmetric and small in magnitude compared to unity. Using (17j, e-' may beexpressed as

c-1= 1 2 y + 3 y 2 - 4y3 + . . . (18)from which it is evident that the kth term in the expansion (16) is of O(Yk). ccordingly, (16) isrewritten as

where D s an approximation to I .One possible scheme by which to compute D s to simplycalculate F - ' as given by (2), then form c - ' using (13), and finally use in (19). However, ifthe computations are carried out in single precision, this procedure can lead to undesirableround-off error since e-' differs from 1 by terms that are small in magnitude. To remedy thisproblem, it is noted that

(20)a;c 1 1 = A A T - A - A T A 1 @ - I = - dXBy using (20) in (l9 ), problems caused by round-off erro r are avoided since all compone nts ofA (and not just the off-diagonal ones) are typically small compared to unity.Fo r applications in which the bulk response is much stiffer tha n the d eviatoric response, it maybe desirable to represent the d ilatational part of the incremental motion exactly, while allowingfor some error in the deviatoric part. T his is easily accomplished by comp uting the deviatoric p artof D according to (19), but replacing the spherical part by

1A tt r D = ~ - 1 o g , J [det @ ' Iulculation of the rotution tensorTurning now to the determination of R, it is emphasized t hat the app roximation to the rotationtensor R returned by the algorithm must be s tr ic t ly proper ortho gonal. Using (17)and expanding

0 - I in a Taylor series around 1, @ - I becomes(22)- I = f (y 2 + y3 . . .)R'.

Next, the vector a is defined by

and Q by


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INCREMENTAL KINEMATICS 3943Using (8)?(22) and (23), it may be shown that

a= sin 0 [1 + p . Y p r Y + O(Y2)]'12 and p + O(Y)2 J Qwhere 0 and p are the rotation angle and rotation axis of R, respectively. In writing (25) , t hasbeen assumed that 0 E [0, n), ince rotation angles in the range 0 E [n,2 n ) are accommodated byreversing the direction of p. Next, the trace of F-' is taken to produce

from which one obtainsC O S O = f ( t r F p l 1) + O(Y)

Approximate values of sin2 0 and cos2 0, here named sin20 and cos2 Q , are now defined by(27)

3P2[1 P + Q ) ] 2P3[l (f' + Q)1( p + Q Iosz 0 = P + ~ _ _ _p+ Q)

sin28 = 1 cos28 (29)(30)

A rotation tensor 8 which is an approximation to R, but which is nonetheless exactly properin which P is defined by

P = + [ t r P ' 12orthogonal: is now defined using the form (8) together with (28) and (29):

The reason for defining cos2 0 and sin20 s in (28) and ( 2 9 , rather than simply settingsin2 0 = Q, cos28 = 1 Q (321is related to the conditioning of (32) for near n/2. Specifically, it is evident from ( 2 5 ) hat Q isclose to unity when 0 = n / 2 . However, when 0 = nj2 exactly, Q differs from unity slightly due tothe terms of O(Y) appearing in (25). This error in Q results in a relatively large error in cos 0 if

cos 0 is obtain ed by extracting the square root of ( 3 2 ) 2 , ue to the behaviour of the square rootfunction near zero. Consequently, the function of P and Q on the right-hand side of (28) waschosen so that cos2B = P when 0 is near n/2,whereas cos' 0 ;z 1 Q when 0 is near zero, witha smo oth transition from one case to the other. Mo re precisely, using (28) l im p~ .ocos2Od/P)=for any Q , whereas limQ,o(sin2 8 / Q )= 1 for any P. Also, it is noted that ( 2 8 ) results in0 d cos 20 1 always, since P , Q 0.In extracting cos 0 from (28), the choice of sign is based on the range of the rotation angle 0 ;i.e. cos 0 r 0 if 8 E [O, n/2)a n d < 0 if 0 E ixj2, n). It should be noted that B can always be takento lie in the range [0, n), since the sensc of the rotat ion is automatically determined by thedirection of a,)ccordingly, the sign of cos 8 is set by examining the sign of (tr F 1).Finally, t w o observations are mad e regarding the evaluation of (31). First, from (28) and (29),the term siniY/2JQ may be written as

(33)


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3944 M . M . RASHIDwhich is easily evaluated when Q is close to zero. Secondly, the evaluation of the term( I cosOa)/4Q in (31) is problematic when Q is close to zero; in this case, the truncated Taylorseries

1 costr 1 P2 1 2 ( P - 1 ) (P 2 j ( P 1OP + 32)64P31104 992P + 376P2 2 P 3 + 5P4

512 P4

= - + Q 3 2 p 2 + Q 24Q 8

+ 0 Q4) (34)Q 3 _____ __should be used. The approximation given in (34) is within 10 of the exact functional value whenQ is 0.01 or smaller.It can be shown that d(R ,8 ) s of O(Y).As will be demonstrated in Section 4, this level ofaccuracy is virtually always adequate for large-deformation finite element computations. How-ever, it i s possible to increase the order of accuracy of R a arbitrarily by preconditioning @ ~'before using it in (23) and (30). Specifically, 6 ' = 0 R' can be premultiplied by an approx-imation to U, so that the result differs from fi by terms of O(Ykj. k > 1. Again expanding ina Taylor series around C ~ = 1,

A (35)= 1 - + ( c - ' 1) + &c-l I)* - (e - ' 113 + . . .where ~ ' -- 1 is obtained from (20).It can be shown that premultiplying F - ' by N terms of (35)and then using the reFult in ( 2 3 )and (30) to compute R a results in d(R ,R ) = O (Y N ) or arbitraryrotation angles. However, it is emphasized that the additional accuracy realized by precondi-tioning is rarely worth the associated computational expense (see below for an estimate of thecomputational expense involved in this and other kinematical algorithms).Finally, it is mentioned that the direction of a departs significantly from that of p as therotation angle 8 approaches 180 . This behaviour is ultimately related to the fact that a mustchange sign at 0 = 180 . For this reason, the algorithm should be restricted to rotations of notmore than, say, 178 in practice. This restriction can be removed by replacing the direction ofa with that of E , , ~ F ~ , , , F ; ~cjkF;kF; when 0 is near 180 (see the discussion on geometricalinterpretation in the appendix). Alternatively, in the event that the rotation angle is found to benear 180 , the input F - ' may be postmultiplied by a rotation of 180 around the direction ofa before entering the kinematical algorithm.* In this way the rotation that is returned by thealgorithm is rendered small, and the incremental rotation for the step is found by composing thissmall rotation with the transpose of the one used to postmultiply @ - I . In any case, theseprocedures are invoked rarely if at all, since a 180 rotation in a single increment is so large that itis unlikely to be encountered in practice.

- - A

Outline of the algorithmThe computational steps in the proposed algorithm are outlined below. The number of floatingpoint operations involved in each step are indicated. The input to the algorithm is A = 1 6 '

as defined in (20), and is computed according to (3).Step I . (42 operations):Compute C - ' 1 = AAT A A'

*This idea was advanced by Dr. Sam Key.


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LNCREMENTAL KINEMATICS 3945St ep 2. (50 operations):Approximate D by D = -- - { $(e ' 1) + i(c ' 1 2Step 3. (6 operations):Compute a , = c l l k F l i l= -- EllkAlkStep 4 . (1 1 operations):C o m p u t e Q = i x , a , , t r @ - ' - l and P = ( t r E - ' - 1 j 2S tep 5 . (24 operations, plus two evaluations of square root):Compute cos20 from (28) and take the square root, and compute sin Oa/2& from (33).Step 6. (21 operations using first three terms of equation (34)):

1At

-

Compute (1 cos P)/4Q using the sign of (tr6 1) for the sign of cos 8 ; use (34) when Q isclose to zero.S t e p 7. (24 operations):

The total computational effort involved in the algorithm is 178 floating point operations, plustwo evaluations of square root. This level of effort compares favourably with that associated withother kinematical algorithms, as discussed in the next section. Preconditioning E ' with the firsttwo terms of (35) requires 54 additional operations, whereas preconditioning with the first threeterms, yielding d(R , R) = O(Y3), requires 66 additional operations. After the stress integrationhas been performed using D as given in Step 2 and with W = 0, the final updated stress isobtained by rotating the stress forward by R'. This rota tion requires 75 floating point operations.It bears mention that all of the operations involved in the forgoing algorithm vectorize fully onvector processors.Objectivity of the algorithm

In order to discuss objectivity of kinematical algorithms, it is convenient to introduce thefollowing definition: the output ofa kinematical algorithm is the deformation gradient Pd fi.6.that results from exact integration over [t,, rT+ '1 of the description of the motion produced bythe algorithm, subject to the initial condition F ( t , , )= 1. For example, in the algorithm describedabove, 6 =Raca,here c a exp [Ar D ] and R is as given in (31). An algorithm will be calledweakly objectiue if1. 6 = @ whenever fi is proper orthogonal and2 . R a = 1 whenever 6 is positive definite symmetric.

Here @ is the input incremental deformation gradient, as before. 'Incremental objectivity' asdefined by Hughes and Winget' corresponds to Item 1 above, whereas Pinsky et al.9 include bothCriteria 1 and 2 in their definition of incremental objectivity.A kinematical algorithm will be called strongly objective if Criteria 1 and 2 above for weakobjectivity are satisfied, and in addition if3. 6 emains unchanged when is replaced by Q@,where Q is an arbitrary rotation tensor.


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3946 M . M . RASHIDCo mp arin g the two definitions, it is clear that the essential difference lies in the be haviou r of thestretch in the presence of rotation: weak objectivity requires that there be no output rotation fora pure stretch input, and no o utp ut stretch for a pu re rotation input, whereas strong objectivityrequires in addition that the output stretch be unaffected by the input rotation for non-vanishinginput stretch. This stronger statement is perhaps closer to the general concept of objectivityencou ntered in mechanics; namely, that two m otions th at differ only by a rotation should induc eresponses that differ only by the same rotation. Strong objectivity obviously implies weakobjectivity, but strong objectivity does n o t require that the algorithm in question be exact(i.e. Pa= F always). Indeed, an algorithm may return approximate values of R a and f ix nd stillbe strongly objective, so long as 0. s independent of the input rotation.Th at the propo sed algorithm is strongly objective (and therefore weakly objective) follows fromthe facts that u = 0 and cos 6' = 1when f i = 1, implying that fia= 1 in this case, and that D sindependent of R trivially (see equations (23) , (24), ( 2 8 ) (31) and (19)).

4. P E R F O R M A N C E A N D C O M P A K l SQ N W I T H O T H E R A L G O R I T H M SThe algorithm of Hugh es and Winget

The proposed algorithm may be directly compared to the algorithm of Hughes and Winget'(see also Reference lo), since the inp ut an d ou tpu t of the two algorithm s arc essentially the same.In the procedure of Hughes and Winget (hereafter referred to as HW), the spatial gradient G ofthe displacement field evaluated at the mid-poinr of the time step is used to form both R and Dfor the step. To obtain G, t is necessary to mak e some assumption regarding the man ner in whichthe no dal velocities vary over th e step; in Reference 1 , they are taken to be constant. Mak ing thisassumption, G may be obtained by evaluating the shape function g radient B:,, at the time-stepmid-point. Alternatively, G may be expressed in terms of the deform ation g radient for the step as(36)

Using not atio n consistent with that ofG = 2 ( @ ) ( @ + I )

An expression similar to (36) is also given by Pinsky etSection 3, Hughes and Winget approximate D (constant over the step) by

and R byR a = (1 + *o) l w ) ~ , o )(G GT) (38)

The origin of the H W algorithm is as follows. First, the assumption is made that the muterialvelocity field is co nst ant in time over the step, i.e. the velocity of each m aterial point is assumed tobe independent of time. Then, the spatial velocity gradient correspo nding to this velocity field isevaluated at the mid-point of the time step, yielding 1/At G . The approxim ate stretching ratetensor D s set equal t o the symm etric part of this mid-point velocity gradient, and o s set equalto the antisymmetric part. The approximate rotation tensor R is obtained by integratingR = WR using the modified mid-point rule with a single step. This approximate integrationobviously requires knowledge of the mid-point spin o only. Rubinstein an d Atluri haveproposed a similar procedure for the determination of ria rom the mid-point spin o, xcept thatthey employ the Cayley-Hamilton theorem to obta in an exact expression for exp(Ato nstead ofusing the modified mid-point rule. Pinsky et al.,' o n the other hand, prove that, if R = WR is


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IKCREMENTAL KINEMATICS 3947integrated approximately using a generalized linear single-step scheme, then the result is strictlyproper orthogonal only when the modified mid-point rule is used.Hughes an d Winget show that h a s given above is strictly proper ortho gon al for arbitrary F,and that D = 0 and R a = fr when P = , where k s a rotation of less than 180'-. I n theterminology of this paper, then, the HW algorithm is at least weakly objective. However, it turn sout that the H W algorithm is not strongly objective as defined in the previous section. Indeed,using ( 3 7 ) and (16),one can show that

A t D = Y Yz+ by3+ . . . + fI( (UZ ZY) + $(ZY' Y'Z) + . . . }+ 82(+(Z2Y + YZ2) + tzvz -+- . . . 1 + (39)for small Y aizd sinall rotation angle 8 , where Z , = 6 i j k p k and p is the r otation axis of R. NOW, ifthe deformation K, -+ x,+ is a pure stretch (i s . if 8 = O), then substitution of (18) into (16) andcomparison of the result with (39) reveals t hat D =D + O(Y ,) In this case. This is the sa me or derof accuracy a s that obtained by using (19) to calculate Da n the algorithm of Section 3. However,if 8 0, and in particular if the rotation angle is fairly large. then (39) indicates thatD = I) + O(Y), i.e. the error in D s first order in Y.

Th e consequences of this characteristic can be illustrated by considering the following problem.imagine a block of isotropic, linearly elastic material subjected to a motion whose deformationgradient is given by

0 1 C 0 , t 0 (40)0 1 Ip t

cos cut in wt I t b t 0[ F i j ]= [Rik] [ . Ik j ]= sin t cos tot 0L 0 1 O

where v is the Poisson's ratio of the m aterial. Th e motion given by (40) gives rise to a uniaxialstress state along the material direction that is aligned with the s , -axis a t t = 0, along witha simultaneously occurring rotatio n arou nd the x,-axis. Let 1) = n a n d p = IO4/E ( E the Young'smodulus), so tha t at t = 1.0, the tensile axis will again be aligned with the x,-axis an d the stresswill have reached a value of lo4 . Now, imagine partitioning the interval 0 < 1.0 in to 6 equaltime steps, so that Ar = and t,*= n At, n = 0, 1 . . . , 6. The incremental deformation gradient@ of FC, with respect to K , , is given by

P, = F,F;-ll 41)where F,, rz = 0. 1 , . . , 6, is computed according to (49)with replaced by 1, . The P, obtainedfrom (41) is used in bo th the algorithm of Section 3 (with no preconditioning) as well as in the H Walgorithm as represented by (361438). In bo th cases, the increment in stress is compu tedaccording to

(42)which. is exact within the approximation of small total stretch. In (42), and p are the usual Lami:constants corresponding to E = 10' and = 0.3. After incrementing the stress, the final updatedstress tensor is obtained by rotating forward using k .The computed solutions using both the algorithm of Section 3 and the HW algorithm arecompared to the exact solution in Figure 1. It is emphasized that. although the rotation incrementin each step is 30 : the stretches are very small, with a total stretch on the order of Th esignificant error exhibited by the H W algorithm i s related to the f-i-term in (39), which in turn isultimately due to the assumption of constant nodal velocities within each time step . However, it is

AT = Al(A1 t r D + 2pD )


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3948 M M R A SHID10000.0

exact solution andproposed algorithin7500.0

5000.0

2500.0

0.0

-2500.0

-5000.00.0 0.2 0.4 0.6 0.8 1 oTIME (SEC)

Figure 1. Comparison of the exact solution to those generated by the proposed algorithm (circles) and thcHughes-Winget algorithm (squares) for a linearly elastic material element simultaneously undergoing stretching androtation. Total rotation is 180 , whereas total stretch in the direction of nonzero stress is 0.001

further emphask ed t h a t the erro r illustrated in Figu re 1 is a direct consequence of the fairly largerotation increments, which are only occasionally encountcred in finite element practice.In writing (39), it has been assumed tha t th e H W algorithm is used in the same mode as thealgorithm proposed in Section 3, in the sense tha t the stress update is first performed subject to Dand W = 0, after which the incremented stress is rotated by R . In Reference I it is suggested thatthese steps be performed in the reverse order [see their equation s ( 1 1)and (12)]. If the rotat ion isperformed first, followed by a pure stretch along fixed directions, then the expression for thecorresponding stretching ra te D is(43)

where = 1 + y is the left stretch tensor in the pola r decom position @ = qk, nd y is symmetrican d small. Using y instead of Y , the expression for I) [as defined by ( 3 7 ) ] that replaces (39) isAtD = y y 2 + b y 3 + ' . + 8{:(yZ Zy) + $(Zy2 y2%)+ . . . $

+ UZ{i(ZZy + y z q + 2 Z y Z + . . } + . . . (44)Compar ison of (39) with (44) reveals tha t D as defined by ( 3 7 )can be used equally well to updatethe stress (an d other state variables) either before or after the rota tion has been applied. The erro rincurred by using the same D in bo th cases is of the same ord er as the #-terms in (39) and (44),which are present in any case.Turning now to the rotation part of the HW algorithm, it can be shown that the error in R agiven by (38) is second orde r in Y, i.e. that d(R ,k) O(Yz). To achieve this order of accuracyusing the algorithm of Section 3, it is necessary t o precondition @-'ith the first two terms of


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I N C K k M k N T A L KI NbM AT lCS 3949(35). A comparison between the rotation part of the proposed scheme and that of the H Walgorithm may be illustrated by considering the following example problem. Let g l ,= l k u k , ,where

[fi ] = diagj l L Y ,+ *a , 1 + 4x1 (45)Furthcr, let R be specified by the rotation axis and rotation angle

174[/4]=--(1,1,1),3 B = -3 ( 4 4 )

Using (45), he magnitude of Y is I Y = (Y .Y) = a. Both the proposed algorithm and the HWalgorithm were used to com pute the approximate rotation R d orresponding to the incrementalmotion given by (45) an d (46). The results are plotted in F igure 2 for 0 < CI < 0.04, with thedistance d(k,6 ) being plotted versus a for both algorithms. From Figure 2, it is clear that therotation part of the H W algorithm is extremely accurate. In the context of the proposedalgorithm, the accuracy gains to be realized by preconditioning are evident from Figure 2.However, CI = 0.04 represents a fairly large incremental stretch in finite element practice; the erro rin fiawithout preconditioning still seem5 to be acceptable for most applications. In any case, an ydegree of accuracy can be obtained by preconditioning w ithout incurring undue com putationalexpense (see the comments at the end of Section 3).Finally, it is observed that the H W algorithm as expressed by (36 ( 3 8 ) requires approximately216 floating point op erations to return R a and D.This may be compared with the 232 floatingpoint operations plus two evaluations of square root required by the algorithm of Section 3 usingfirst-order preconditioning (1 78 plus two square roots withou t preco nditioning). The 216 figure isbased on the assumption that (or k erves as the input to the algorithm. In most finiteelement formulations, it is more computationally efficient to form G using (36) than to computeG directly by recomputing the shape function gradients at the time step m id-point.

Cornparison to other proceduresFlanagan and Taylor have presented a kinematical algorithm tha t is similar in motivation t oboth the algorithm of Section 3 and the H W method , in that the constitutive update procedure isprovided with a deform ation p ath w hich simplifies the o bjective stress rate t o simply a materialrate. In their algorrthm (hereafter referred to as FT), Flanagan and Taylor consider theGreen-McInnis objective stress rate rather than the Jau ma nn rate. Accordingly, the FT algo-rithm generates an approxim ate deformation path in which D = constant and R = constant overt E t n , n f l ) , after which R changes stepwise to its updated value at t = t,+ I . Here R is therotation tensor associated with the gradient of the total deformation. This deformation path maybe contrasted w ith th at given by (7a H 7c) , in which the rotation associated with the incrementaldeformation is held at R = 1 over the step.In the F T algorithm, the mid-point values of D and W are obtained as in the HW algorithm.Then, a result of Dienes, in which 0 = RRT s related to D, W and V (the tota l left stretch tenso r),is applied. Th e updated total rotation tensor is then ob tained by applying the modified mid-pointrule as in Reference 1, but using i2 instead of W. Since the total R s available in the FT scheme,the constitutive integration is performed in the so-called unrotated configuration. That is, D isback-rotated to RZD R, for the purposes of performing the stress update, after which the update dstress is forward-rotated to R, T R;f for use in the statement of momentum balance. The leftstretch tensor V is updated using a single forward Euler step, and is stored for use in Dienesrelation for 0


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3950 M. M. ASHID1.500

_ -0.000 L j

0.050

0.040Fzw[rb 0.030a3g 0.0202a:w

0.010

STRAIN MAGNITUDE (ALPHA) iPROPOSED ALGORITHM,SECOND-ORDERPRECONDITIONING /\ /

HUGHES-WINGET ALGORITHM f l 8,\ROPOSED ALGORITHM,FIRST-ORDER

0.000STRAIN MAGNITUDE (ALPHA)

Figure 2. Error in the computed rotation for a dcfurmation increment that corresponds to a 45' rotation-about the(1, I , 1) oxis, with a stretch as given b y equat ion 145). I n this figure, thr: ro ta t ion error i s defined as d(R R)/(z/4)

The behaviour of the FT algorithm is similar to that of the WW algorithm. In particular, thecoupling between the stretching and ro:ation as exhibited by equa lions (39) and (44) s present forlarge rotation increments, d u e to tht: LIFC of the mid-point velocity gradient and assumed constantmaterial velocity field. The I T algori thm requires approximately 500 floating point operations,


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INCREMENTAL KINEMATICS 395 1including the determin ation of the mid-po int velocity gradient from [using ( 3 6 ) ] and theback-rotation of D, but not including the forward rotatio n of the up dated stress. It bears mentiontha t the F T algorithm has been in successful use in the PRONTO an d J AC I3 family of finiteelement codes for several years.Attention is now focused on the proced ure described by Pinsky et ~ 1 . ~lthough this algorithmis not truly a kinematical algorithm in that it does not provide a deformation path for theconstitutive integration, it has some features relevant to the present discussion. The essentialaspect in Pinsky et a/. is the observation tha t any generalized rate of stress (such as the Jaum annco-rotational rate or the Truesdell rate) may be written as a malerial rate of an appropriatelytransformed stress followed by the reverse transformation. This conclusion arises naturally intheir formulation, as they define all tensor fields over the manifold of body points, rather thanusing Cartesian tensors defined over a manifold of spatial points as is done here. Within thiscontext, all generalized rates may be tho ug ht of as special forms of the Lie derivative, in which th etensor quantity in question is pulled back to some other co nfiguration, the material rate is taken,an d the result is pushedj imvard to the cu rrent configuration. Pinsky et aL9consider rate forms ofhyperelasticity an d hypoelasticity, in which the stress rate is appro ximated by a weighted averageof the end-point stress tensors pulled back to a common configuration. An equation for theupdated stress results from equating this weighted average to the pull-back of the stress rateevaluated at an intermediate time point. This latter quantity generally involves the rate-of-stretching D and the stress evaluated at th e intermediate point.Pinsky et nL9 assume th at th e material velocity field is co nst an t over the step, which leads to therestriction that the intermediate point must be chosen to be the time mid-point if the overallprocedure is to be (weakly) objective. However, due to the constant-material-velocity-fieldassumption, the stress update procedure of Pinsky et aL9 is not strongly objective for the samereasons given in connection with the HW algorithm. They give a numerical example problem(their Section 6.3) that is similar to the problem of Figure 1 above, but which involves a totalstretch of 2.0 and a total rotation of 360. A departure of the numerical solution from the exactone is not seen in their calculation, since both the stretch and the rotatio n increments are of thesame orde r of magnitude, rendering the error in the stress small [see equa tion (39)l. Rerunningtheir example problem using their algorithm, but with a total stretch ratio of 1.01 instead of 2,resuits in a comparison between the computed and exact solutions similar to that shown inFigure 1.A completely different class of constitutive update procedures has recently emerged in whichkinematical algorithms of the type considered here play no role. Examples of such procedures,here referred to as non-rate-form update methods, have been advanced by, e.g., Moran et aI.I4and Weber and A nand; and , in the context of the so-called operator-split methodology, bySimo and Ortiz and Ortiz er Th e essential idea is to first integrate a rate equation for theplastic part of the deformation, after which the elastic deformation and the stress are obtained byfunction evaluation. Perhaps chief among the attractions of the non-rate-form approach is t h a tstron g objectivity is trivially satisfied. However, this ap pro ach is limited by two essential aspects:i t only applies to constitutive formulations for which there exists some form of kinematicaldecomposition of the m otion into elastic and plastic parts (e.g. the multiplicative decompositionF = FTPj, nd it is useful only when the plastic evolution eq uat ion ca n be adequately integratedwith a singlc-step integration rule, so that the imposed deformation is sampled only at thetime-step end-points. These limitations imply that there are many situations of practical interestfo r which it is necessary or desirable to appea l to a treatmen t of the incremental kinematics of thetype described herein.


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3952 M. . ASHID5. THE TANGENT MODULUS TENSOR

In quasi-static finite element codes, and in dynamic codes in which the equations of motion areintegrated using an implicit operator, it is generally necessary to provide the global solutionalgorithm with a tangent modulus tensor. The tangent modulus appears in the expression for rateof change of internal nodal forces with respect to incremental nodal displacements, which in turnis used in the global Newton-Raphson (or other suitable) iterative solution procedure. Since thederivative of the incremental deformation gradient F with respect to incremental nodal displace-ments is element-formulation dependent, the focus in this section will be on the derivative ofupdated stress with respect to 6-l.As with the kinematical algorithm itself, the correspondingexpressions taking F as given are obtained b y slightly modifying those given below.

respectively. Also, letthe Cauchy stress obtained by integrating the constitutive equations over t E ( t n , t,+ subject toD =D [as given by 19)] and W = 0, be ?.?'hen,

Let the initial and updated Cauchy stress for time step n be T, and T, +

T,+ = RaTRdT (47)where Ra is given by (31). The objective is to compute the rank-four tensor

It will be assumed that the rank-four tensor- 39C = -dUis available from the constitutive update routine. In connection with (49), it is noted that

a?e = At~ O(Y)dD49)

For economy of presentation, R' and F - ' will hereafter be replaced with R and A, respectively.Combining (47)-(49),it is seen that (in component form)

The terms i?R/BA and dU/BA are not trivial to calculate, and are the essential results of thissection.

The approach taken here is to look for linear relations between A and R, nd between A and U .To this end, consider the polar decomposition F = R U , from which one obtains

A = A(RU + R U ) A ( 5 2 )Taking the rate of the expression U = AA', using the result in (52),and rearranging leads to

j = FikAImRmj4 4 UikAmlRjmk + & R j k )Recalling that U differs from I by terms of O(Y), one arrives at

( 5 3 )

A similar procedure leads to


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INCREMENTAL KINEMATICS 3953The O(1) terms given in (54) nd ( 5 5 )are expected to be entirely adequate for use in computing thetangent modulus tensor C. Combining (55) , (54) and (51), the final expression IS

C i j ~ f m n { R j n [ $ d k m 6 , 1 n , l R t k + i R I k R m l ]+ R c r n [ : 6 k n d j l n l R j k f $ R ) k R n l l } l rn Rw R j vC nr rn k (56)

6. CONCLUSIONThe purpose of the kinematical algorithm proposed herein is to express the deformationincrement for a time step in a form that is convenient for use in constitutive integrationalgorithms. The kinematical algorithm takes as input the inverse 8 of the incrementaldeformation gradient for the step. which is easily obtained from the incremental displacementsand the shape-function gradients. The output is a rate-of-stretching tensor D and a rotationtensor R , such that the gradient of the deformation corresponding to D = Dd constant), W = 0for t E [ t n , n + followed by stepwise application of the rotation R a , is very nearly $ at t = t n+ .A feature of the algorithm is that it can be made arbitrarily accurate simply by incurring theadditional expense o f retaining more terms. That is, D can be made arbitrarily close to the exactvalue ( l /At) log o, and R a to the exact value R, where 6 = Rc. or the basic form of thealgorithm presented in Section 3, it is shown that D = D + O(Y3), and that d(R , R ) = O(Y),where Y =6 1 and where d( , ) represents the distance function on the three-dimensionalrotation manifold SO(3) .This level of accuracy is generally more than adequate for finite elementapplications.

The important issue of objectivity is discussed in relation to both the present algorithm as wellas to previously-proposed algorithms. Previous treatments have interpreted algorithmic objectiv-ity in terms o f pure input rotation and pure input stretch. Here, the notion of strong objectivity isintroduced, which, in addition to the criteria for weak objectivity, requires that the output stretch(i.e. f l a= exp [At D ] in the present case) remains unchanged when an arhitrury input deforma-tion is altered by a rotation. While this stronger statement of algorithmic objectivity is perhapscloser to the usual concept o f objectivity encountered in mechanics, it is not the strongeststatement that might be made. Indeed, the most severe requirement would be to simply demandthat the output and input deformations be exact ly equal for arbitrary input deformations.

One of the basic differences between the present algorithm and those proposed by Hughes andWinget,' Flanagan and Taylor,' and Pinsky e t aL9 is that the latter three proposals all computethe stretching rate based on the assumption that the material velocity field is constant in timeover the time step. This assumption leads to a lack o f strong objectivity, i.c. to a coupling betweenthe stretching and rotation parts of the incremental motion. This coupling can lead to significanterrors in the stress update whenever the rotation increment is moderately large, even when theincremental stretch is infinitesimal. In the present approach, care is taken to avoid sucha coupling between the rotational and stretching parts o f the incremental deformation. The resultis a strongly objective kinematical algorithm for which the computational effort involved (about200 floating point operations) is comparable to that of other kinematical algorithms that havebeen proposed.

A C K N O W L E D G E M E N T SThe author would like to express appreciation to Dr. Sam Key for fruitful discussions inconnection with this work. The work reported herein was performed at Sandia NationalLaboratories in Albuquerque, New Mexico, and was supported by the United States Departmentof Energy under contract DE-AC04-76DP00789.


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3954 M . M . RASHIDAPPENDIX

The rotat ion part of the kinematical algorithm presented in Section 3 admits an interestinggeometrical interpretation, as follows. Consider two material p oints with position vectors X 1 andX2 in configuration ti,,. Suppose these two material points move to positions x 1 and x 2 in thehomogeneous deformation K, -+ K , , + ~ defined by E. Let the unit vector M that lies alongX2 XI be given by the Cartesian components (MI ,M 2 , M 3 )= (sin 4 cos i b sin $ sin $, cos $),where {$, ( 1 ) are polar angles. Now, the relative displacement (x2 x,) X2 XI of the twopoints lies along the vector FM M. This vector is pictured in Figure 3 for M in the { x l , c 2 }plane and for 6 corresponding to a pure stretch, a pure rotation, and a more general motioninvolving both stretch and rotation. Figure 3 suggests that the integral of the cross product( F M M ) x M ( = f?M x Mj over all orientations of M vanishes when F is a pure stretch, i.e.

On the other hand, it turns out that this integral lies along the ro tation axis of R when @ = R:- 8 n ,jo n. ( k M M ) x M sin 6 4 d$ =~ in O p3

where again 8 and p are the rotation angle and rotation axis of 6. More generally,(59)871 .1 M)x M s i n 4 d+ d$ = __-sin f l p t- Q ( Y )3

Carrying out this integral leads to an expression similar to (25),.Finally, and with reference to the comments made in Section 3 regarding the direction of

x2

Figure 3(a)


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INCREMENTAL. KfNEMATICS 3955

\ I

,///'-- \ f//I I /

Figure 3. Plot of the displacement vector (EM - M ) when P s (a)a pure stretch, (b) a pure totatton and (c) a generaldeformation

a when 0 180 ,it can be shown that

J o J oEvaluation of the integral in (60) yields a vector that is proportional to C , , ~F~ , , , F ; ~ ,hosedirection, it turns out, differs from that ofp by terms of O(Y), and is a good approximation t o thedirection of p when f? is near 180 .

REFEREN CES1. T. . R . Hughes and J. Wingct, 'Finite rotation effects in numerical intcgration of rate constitutive equat ions arising inlarge-deformation analyses', lnt. . numer. methods eng., 15, 1862-1867 (1980).


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3956 M. M. RASHID2. D. P. Flanagan and L. M. Taylor. An accurate numerical algorithm for stress integration with finite rotations,3. G. C. Johnson and D. J. Bammann, A discussion of stress rates in finite deform ation problems, Int. J. Solids Strucf. ,4 . S . N . Atluri, Onconstitutive re lations at finite strain: hypo-elasticity an d elasto-plasticity with isotropic or kinematic5 . E. H. Lee, R. L. Mallett and T. B. Wertheimer, Stress analysis for anisotropic hardening in finite-deformation6 . Y. F. Da hlias , Tor ota tio na l rates for kinematic hardening at large plastic deformations, J . Rppl . Mech. , 50, 56 1-5657. J. K. Dienes; On the analysis of rotation and stress ratc in deform ing bodies, Acta hfrch., 32, 217-232 (1979).8. M.M.Rashid, A class of constitutive m odels for rate-depen dent inelasticity in metals, in S.-C.Chou , (ed.),Proc. 12thArmy Symposium o n Solid Mechanics, Plymouth , MA, November 1991 pp. 521-537.9. P. M. Pinsky , M . Ortiz an d K. S. Pister, Numerical integration of rate consti tutive equa tions in finite deformationanalysis, Comput . Methods 4ppI. M e c h . Eng., 40, 137-1 58 ( 1 983).10. T. J. R. Hughes, Numerical implementation of constitutive models: rate-independent deviatoric plasticity. in S.Neniat-Nasser, R . J. Asaro and G . A. Hegemier, (eds.), Theoretical Foundation fo r Lar ge-S cak Computations .forNonlinear Material Rehatlior, Martinus Nijhoff, Dordrecht, 1984 pp. 29 57.11 . R. Rubinstein and S. N. Atluri, Objectivity of incremental consti tutive relations over finite t ime steps in com puta-tional finite deformation analyses, C om puf . Methods Appl. M e c h . Eny., 36, 277-290 (1983).12. L. M . Tay l o r an d D. P. Flanagan , PRONT03D a three-dimensional transient solid dynamics program. SandiaNational Laborntorirs Report SANDR6-0594, Albuquerque, New Mexico, 1989.

13. J. H . Biflle, JAC3D--a three-dim ensional finite element com pute r pro gra m for the nonlinear quasi-static response ofsolids with the conjugate gradient method, Sundia Nalionul Laboratories Report SANDX7-I .305, Albuquerque,New Mexico, 1993.14. B. Moran; M. Ort iz and C. F. Shih, Formulation of implicit f inite element methods for multiplicative finitedeform ation plasticity, Int . j . numrr. m e h o d s eng. , 29, 483-514 (1990).15. G. Weber and L. Ana nd, Finite deformation consti tutive equation s and a t ime integration procedure for isotropic,hypcrelastic-viscoplastic solids, Comput . Methods Appl. Mech. Eng., 79 , 173-202 (1990).16. J . C. Simo and M. Or tiz, A unified ap proa ch t o finite deform ation e lastoplastic analysis based on th e use ofhyperelastic constitutive equations, Comput . Methods Appl . Mech. Eng., 49, 221L245 (1985).17. M. Ortiz, P. M . Pinsky and R. L. Taylor, Operator split methods for the numerical solution of the elastoplasticdynamic problem. Comput . Methods A p p l . Mech. Eng., 39. 137-1 S7 (1983).

Coniput. Methods Appl . Mech. Eng., 6 2 , 305-320 (1987).20, 725- 737 (1984).hardening, Cornput. Mrthods Appl . M e c h . Eng., 43, 137~-1711984).plasticity, J. Appl. M e c h . , 50, 554-560 (1983).(1983).

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Incremental Kinematics for Finite Element