Finite Element Procedures for Large Strain Elastic Plastic Structures

8/3/2019 Finite Element Procedures for Large Strain Elastic Plastic Structures

1/12

Capurrrrs d Srrucfurrs Vol. 28 NO. 3, pp. 39S-406. 1988R i n d In Gmt Briuin. My5 -7919/88 s3 00+ 0 00C 1% Pngmon Journslr Ltd

FINITE ELEMENT PROCEDURES FOR LARGESTRAIN ELASTIC-PLASTIC THEORIESJAMES D. LEE

Robot Systems Division, National Bureau of Standards, Gaithersburg. MD 20899, USA .(Received 21 March 1987)

Abstract-In a unified approach, the vrtua work equations in ra te form and in incremental forms arederived rigorously for elastic-plastic continuum subjected to large strains. T h e finite element p r d u r e sfor the analyses of clastic-plastic solid b a d on Lees theory and the Green-Naghdi theory arc presented.Also, i t i s shown that transformations can be performed among the Eulerian, the Total-Lagrangian, andth e Updated -Lagrangian formulations, and among different forms o f constitutive relations, without anyapproximation.

1. INTRODUCTIONThe variational principle, or the principle of virtualwork, for the analyses o f solid mecahnics with largestrains involved has been treated extensively byWashizu [I],nd by, for example, Hill [2], Eringen [3],Bathe [4], Malvern [SI, Oden [6], Hibbitt et al. [7],McMeeking and Rice [8], Schamhorst and Plan [9],Lubarda and Lee [ I O ] , etc. T h e virtual work equa -tions can be expressed in ra te form and in threeincremental forms. In Sec. 2, i t i s shown that,by starting from any one of the three forms ofCauchys law of motion-the equilibrium equation, auniversal virtual work equation in ra te form can bederived rigorously. In Sec. 3, in a straightforward andrigorous way, the Eulerian, Total-Lagrangian, andUpdated -Lagrangian forms of incrementa! virtualwork equations are derived, and i t i s shown that thethree forms of incremental virtual work equationscan be transformed from one to the other withoutany approximation. I t is shown also that the in-cremental virtual work equations are exactly the sameas the virtual work equation in rate form in thelimiting case-he size of the incremental step a pproaching zero. In Sec. 4, the finite element formu -lations based on the Total-Lagrangian and theUpdated-Lagrangian incremental virtual work equa -tions are made in detail for elastic-plastic continuumdivided into general, three-dimensional, solid ele -ments. The emphasis is put on the calculation ofnodal forces, which i s an exact treatment. The calcu-lated nodal forces are then taken as the basis to checkwhether the equilibrium i s reached pointwise. Theseformulations can then be applied to any theory ofelastic-plastic solid with large strains.Among numerous theories of elastic-plastic solid,

Lees theory[10 -15] i s the most unique one in theSense that the decomposition of total deformationinto the elastic and the plastic parts i s made at thedeformation gradient level, while Green-Naghdistheory[1621] i s the most general one-t allows

material anisotropy and various kinds of hardeningru les to be incorporated into the formulation. In Secs5 and 6, the detailed i terative procedures are outlinedfor Lee s theory and Green-Naghdis theory,respectively.Throughout this paper, the standard tensor sum-

mation convention i s adopted: the rectangularEulerian coordinates, x,(k = 1,2,3), and Lagrangiancoordinates, X,(K = I,2,3), are employed; an index,k or K, after a comma indicates a partialdifferentiation with respect to the coordinate, x, orXK; a superposed dot indicates the material timederivative; and some standard notations appearing inEringens book [3] are utilized.

2. VIRTUAL WORK EQUATIONIN RATE FORM

The equilibrium equation may be written in one ofthe following forms [3]:

Cy,, + PL = 0

where: uq i s the Cauchy stress tensor; p andpo are themass density in the deformed and the undeformedstate respectively; L i s the body force per unit mass;

i s the deformation gradient; and the firs1 orderand the second order Fiola-Kirchhoff stress tensorsare defined respectively as

In eqns (4) and (S), J i s the Jacobian of the trans -formation between the deformed configuration and

395


2/12

396the undeformed configuration, i.e.

JE det (xI.K).and XK., i s related to as

JAMES D. LEEwhere the Kirchhoff stress tensor i s defined as

(6) T = Ju,,, (14)

I t i s noticed that, in this work, attention is focused onthe static problems, although the formulations couldbe extended to the dynamic cases without majordifficulty.I f eqn (1) i s differentiated with respect to time andthen multiplied by the virtual velocity 60, and thenintegrated over the deformed volume, the following i sobtained:

Similarly, from eqns (2) and (3), one may have thefollowing:

the Jaumann stress rate tensor is defined as [ I O ]

and wV i s the spin tensor.I t should be noted that eqn (13) i s the same resultas that obtainable from the Hills variational prin -ciple [2,8, lo]; also, the second te rm on the left-handside of eqn (13) indicates that the follower typeloading i s automatically incorporated in the formu -lation. I t should be emphasized that the virtual workequation in rate form, eqn (11) or eqn (13), i s asgeneral and exact as the equilibrium equation, eqns(l),2), or (3). However, the rate form may not bepractical at the problem -solving level, but i t can andwill be used as a basis to check the val idi ty of thevirtual work equation in incremental forms, whichwill be formulated in the next section.

3. EULERIAN AND LACRAICIANFORMULATIONSBy multiplying eqn (1) by the virtual displacement

may obtain the virtual work equation in the followingdr [(TKLx,.L)], K + p,,j, 6t:dV = O . (IO) 6u, and integrating over the deformed volume, oneSIform:After some mathematical manipulations, eqns (8)-(10) are all reduced to the same virtual work equation

in rate form as follows:

where: s + i s part o f surface surrounding 1 over whichthe surface traction, defined as T,= uunJ(n, i s the unitoutward normal of the surface), i s specified; daJ i s thedifferential area vector; d i s the deformation ratetensor; and the Truesdell s tr es s r a t e tensori,,sdefined as 122,231

where the infinitesimal strain tensor, e i s defined aset] E (#,Ik u J.C) /2 (1 7)

I f either eq n (2) or eqn (3) i s multiplied by the virtualdisplacement bu, and integrated over the undeformedvolume, the virtual work equation i s obtained asi.u (d , vL+ U N . L ) ~ U N ~ A K

+ pJK6URdV = TKLBEKLdV, (18)s swhere dA, is the differential area vector in theundeformed state; the Green-Lagrangian straintensor, EKL, i s defined as

Also, one may readily show that eqn (1 1) can berewritten as

uK andj, are respectively the displacement vector and(13) the body force vector expressed in the Lagrangian


3/12

Finite element procedures for large strain elastic -plastic theories 397coordinates, i.e

where b i s the direction cosine between the Euleriancoordinates, x,, and the Lagrangian coordinates, XSuppose the solutions a! state 1 are known; the

solutions at state 2 can be expressed as the sums ofthe solutions a t state I and the incremental solutions,e.g. Ta(2)= Ta ( l ) +ATKL . (21)Then, eqns (16) and (18) can be rewritten as

+ P(2)Lf,(l) + A m A 4 do (2)J= J[u,](l) + Au,]bAe,,dc(2)

6. + ATKL1[6NL + uN.L(l)+ AU N,L ]~AU NdAK+ POVK(l) + A/K1bAuKdVJ

=J[T,(l) +AT,]bAE,dV.I t i s worthwhile to mention that, in deriving eqns (22)and (23), in other words, in the process o f seeking theincremental solutions, the variations of the solutionsat state I are vanishing.

The incremental Truesdell stresses and the in -cremental Washizu strains are defined as [l]:

AE: ~X K , ; ( I )zY~, j ( l )Aa . (25)Combining eqn (24) with cqn (9 , it can be readilyshown thatu,.(l) + Aa;

uJ(2) all( 1) + Au o

where

Eringen [3] has shown that

(29)TKLE lim-TKL = JX,,,X,.~ t,,.AI-O A ICombining q n (24) with eqn (29), i t is seen that

AaI:6, = lim-AI- o A I which indicates the meaning of the incrementalTruesdell stresses. The incremental Washizu strainsare the incremental Green-Lagrangian strains takingstate 1 as t he reference state, and i t has been shownthat [11

Also, it should be noted that

which indicates the meaning of the incrementalWashizu strains.Starting from either eqn (22) or eqn (23), one may

prove that the virtual work equation can be expressedas

+ P(l)Lf , ( l ) + A L P 4 dr(1)f=J[o,(l)+Au:~bA~:d.(l). (33)

I t i s noticed that the integrations of eqns (22), (23)and (33) are to be performed, respectively, on theconfigurations of state 2, initial state, and state 1;therefore, Eulerian, Total-Lagrangian, and Updated -Lagrangian formulations are named for eqns (22),(23) and (33), respectively. I t should be emphasizedtha t these three equations are nothing but the samevirtual work equation expressed in different in -m e n t a l forms and, during the process of deri-vation, no approximation has ever been made.I t i s straightforward to show that eqn (33) can be

wr i t ten as

1A.t 666: do (1) + a 1)Au,.,(l Au k.,( 1) dr (11i= f ~ ( l )A / ;dAu, dv(l)


4/12

398where

Nm=

JAMES D. LEE

N, 0 0 N2 0 0 . .0 N, 0 0 N2 0 . '0 0 N, 0 0 N 2 . .

as

I n view of eqn (16) and by recalling that uJl) andp(1) are the solutions at state 1 subjected to the bodyforceJ(1) and t he sur face traction u,,(l)n,(l) on s* ,i t is concluded that R is vanishing. Then, with thefollowing equalities

it can be shown that

which means, in the limiting case, the virtual workequation in incremental form i s exactly the same asthe virtual work equation in rate form. Therefore, thevalidity of the incremental virtual work equations,eqns (22), (23) and (33) is established.

4. FINITE ELEMENT FORMULATIONSLet a solid body be devided into many elements;

each element consists o fN nodal points. Correspond -ingly, there are N shape functions, N,,y = 1,2,3, . . .,N, so that the Eulerian coordinates o fa gener ic point (x,y,z) within the element can belinked to the nodalpoint coordinates (Z7,1,. 2,) in thefollowing matr ix form:

x; = N,f,, (39)where:

. . .

. . . ? j N 1 .0 0 NN. . .Similarly, in Lagrangian coordinates, the counterpartof eqn (39) can be written as

where is the nodal point coordinate vector in theundeformed state expressed in Lagrangian coordi -nates, and N has the same form as N Also. thedisplaaments and incremental displacements of ageneric point within the element can be linked to thecounterparts of the nodal points as

Through a very standard procedure (4,24,25], thedisplacement and the incremental displacementgradients can be expressed as

where B i s a (3 l 3 l 3N)matrix than can be obtainedthrough the shape functions and the nodal pointcoordinates.In theUpdated -Lagrangianapproach, suppose the

solutions at state 1 are obtained; then eqn (16) writtenin the following form has to be satisfied:

{u,,(l 1 de,(l) dc (1 1- P(l)f,(l)u, dr (1)Ic

which can be rewritten as

6u, [ la,( l) BU do (1) - ~ ( 1f,() N dc (11I- I e IJl) N, da,(l)t 6u, F. = 0. (46)

I t is noticed that F i s the nodal force vector of ageneric element. Ifeqn (46) i s summed over all the


5/12

Finite element procedures for large strain clastic-plastic theories 399elements of the solid body, one may obtain thefollowing:

buiF;=O [=1,2,3 , . . . Nd, (47)where Nd stands for the number of degrees of free-dom of the entire solid body. Because eqn (47) should

- be valid for arbitrary 6u; except for those componentswhich are specified through displacement boundaryconditions, therefore, one may conclude that: (1) fora ll the components where the displacements arespecified, the reactive forces are obtained through thecalculations indicated in eqn (46); and (2) for all the

. other components, F i =O , which means the equi-librium i s reached pointwise. The calculation of nodalforces may now be expressed by

F; = a B dr. - pAN dt. - u vN da, . (49)s s 6.Similarly. in the Total-Lagrangian approach, recalleqn (19) and notice that the variation of theGreen-Lagrangian strains can be expressed as

For the calculation o f nodal forces, the counterpartof eqn (49) can be written as E F$?) dAus, (56)

where~7;~~pecifies the constitutive relation for theelastic -plastic solid, i.e.

T he detailed expression o f a" will be indicated in thenext section. Now, the governing matr ix equationsfor a generic element become

[K$ + K$]Au a = F!," +FZ')+F:". (58)In order to solve for the incremental solution usingthe Total -Lagrangian approach, first, the incrementalGreen-Lagrangian strains can be found, by using eqn(191, as

where

In order to solve for the incremental solutions usingthe Updated -Lagrangian approach, the te rms ineqn (33) are treated as follows:

Def ine AGKL and AH as


6/12

400 JAMES D. LaNow, the te rms in eqns (23) are treated as

where AUMW specifies the following constitutiverelation of the elastic-plastic solid.

T h e detailed expression of A will be indicated inSec. 6. Now, the governing matrix equations for ageneric element assume the same form as eqn (58).When the matr ix equations for al l the elements in th e

solid body are summed up, the grand matr ix equa -tions of the entire mechanical system are obtained.After th e displacement -specified boundary conditionsarc imposed, onemay solve for the incremental nodalpoint displacements. I t should -be emphasized thatwhether a set of solutions i s acceptable ornot sependson whether the equilibrium of nodal forces, thecalculation of which i s an exact treatment, is reachedor not; there fore , the approximations made in deriv -ing eqns (53), (56), (62) and (65) in order to obtainthe incremental solutions in an i terative process donot imply that the f inal accepted solutions are ap-proximated ones. The point will be seen clearly in thefollowing sections.

5. ITERATIVE PROCEDURESFOR LEES THEORYLee and h is co -workers[10 -15 ] have formulated a

theory of plasticity based on the exact nonlinearkinematics of elastic and plastic deformations.Among all the existing theories of plasticity, Leestheory i s a very unique one. Instead o f assuming thatthe total strain is the summation of the elastic strainand the plastic strain, Lees theory begins with thefact that the deformation gradient i s the product ofthe elastic and the plastic defonnation gradients.Therefore, i t i s worthwhile to demonstrate how thefinite element formulations derived in the previoussections can be applied for Lee s theory.To begin with, l e t Lees theory be briefly describedin the following [IO].First, le t the deformation gr a -

dient x , ,~be expressed as

where F and FP may be named, respectively, as theelastic part and plastic part of the deformationgradient and F specifies the mapping from the un-stressed plastically deformed configuration to theelastically -plastically deformed configuration. How -ever, t he decomposition as indicated in eqn (67)i s not unique because any arbitrary loca rotationin the unstressed state gives another unstressedconfiguration. Therefore, further restrictions have tobe imposed on F, namely, F i s symmetric and hasthe same principal directions as the stress tensor andthe plastic part of the deformation r a t e tensor. Theserestrictions imply that the material under consid -eration i s isotropic and obeys the isotropic hardeningru le . Define FI and F P to be the inverse of P and FP,respectively, i.e.

T h e elastic and the p las ti c pa rt s of the deformation


7/12

Finite element procedures for large strain clastic-plastic theories 4 0 1one can show that

I t can be shown that

a,, + dP, (74)CO ,,= CO~+/t(r,oRF;-Fs~oa,FF:,), (75)

where eqn (74) indicates that the deformation ratetensor i s not just the sum of the elastic and the plasticparts-a difference between Lee's theory and othertheories of plasticity. Then, Lubarda and Lee [ IO ]obtained the constitutive relation between theKirchhoff stress tensor and the elastic Cauchy-Greentensor as

where Z is the Helmholtz free energy density andc EFfkFjk . (77) I t i s noticed that

Since thematerial i s isotropic, Zmust be the function b tp , = bMiJ= bJUW7= b t pmn 9 (90)and, also, b i s a function of e. In view of eqn @I),can be transformed to be a function o f the stresses.

(78) For the plastic part o f the constitutive relations, le tthe yield function be(79)

of the three invariants of c which are defined as1,= c,, ,1 2 = (1: - cyc,,)/2,

-

d-P,

-f =3/2z,;z;-s2,I,5 I / 6 e , ~err, E C Ck, = det (C). (80)

Then, eqn (76) can be written as where S is the current yield strength and 7 i s theKirchhoff stress deviator, defined as

where The loading rate, L, defined as(93)

can be shown to beI t i s noticed that cqn (81) does imply that theKirchhoff stress tensor and the elastic Cauchy -Greentensor, defined in eqn (77), have the same principaldirections. With the equality

L = 37:/?,,.Upon loading, defined as

(94)

' E. 'kJ +',&'kJ f = O . L > O , (95)


8/12

40 2-

JAMES D. LEEi t i s assumed that the following equality:

which can be rcwritten asdP,=94~:17 - f~, (97)

where qcan be a function of the invariants of r,,. Dueto the requirement of continuity in plasticloading [161,

3xS' = 3/27; 5 (98) = 2F, Fh70C.J

dZolds for the entire loading process. In unloading,defined as f < O or L


9/12

Finite element procedures for large strain elastic -plastic theories 40 3Step 4

cqn (1051, byApproximate the constitutive relation in ra te form,

u +u + AuaUi +-Uij(2).

Then, the B,n matrix should be u@ated also and allthe other quantities, i f needed, can be updated.- where StepI

The nodal force vector is now calculated accordingto cqns (48) and (49). i f the calculated nodal forcevector indicates that the equilibrium i s not reachedpointwise, then, in view of qns (49), (M), (55) and(56). the nodal force vector, F, indicated in eqn (48),actually serves as the forcing te rm for further iter -ation, namely, one should go back to Step 1.I f theequilibrium i s reached pointwise, or, at least,the error i s within certain given error tolerance, onemay change the applied loading, or even thedisplacement -specified boundary conditions, and goback to Step 1 to seek the solutions at the nextincremental level.

This i s actually the f i rst approximation made in thispaper. Also, in view of eqns (30) and (32), i t i s noticedthat, in the limiting case eqn (I10) approaches eqn(105). Then, recall eqn (27) and rewri te i t as

I t i s seen that eqns (1 10)-(112) involve iterations.Le t SI and S be defined as

6 I TERAWE PROCEDURES FORTHE GREEN-NAGHDI THEORYS:E 3/2rb(l)rb(l)S iE 3/2 132) rk(2). Green and Naghdi [16,17] formulated a very gen-eral theory for elastic-plastic continuum. Later,Casey and Naghdi [18-2IJ addressed further issues

related to that general theory. Since theGreen-Naghdi theory has been referenced by manyresearchers in the field of plasticity, i t has beendecided to demonstrate how the finite element fonnu -lations derived previously can be applied for thattheory.The basic constitutive relations, with the thenno -mechanical coupling being neglected, of theGreen-Naghdi theory are listed in the following [161:

I f SIS or SI= S and S 2 S, le t

e I f SI< S and S > S, le t

-'

+~~~ ( 8 ( 2 ) ) ( S , - S ) ~ / ( S , - ~ , ) ,115)where P and i*re the inverse of e and a*, re -spectively.Step 5For those elements which never experience plastic

deformation, Step 3 and 4 should be bypassed.Instead, the following will be calculated.

Eu = EL -+ EP,Z= Z(E')

- =1 L (loading)aTKLI t i s noticed that this step does not involve theapproximation made in eqn ( 1 10).Step 6Update the geometry, the displacement field, and

the stress distributions as

" '( '+A%= af )L (loading) (127)as

k&= K = 0 (neutral loading or unloading), (128)where E' i s the elastic Green-Lagrangian strain; E P i sthe plastic Green-Lagrangian strain; F = 0 specifiesa+ f, + Au,


10/12

404 JAUES D. Lnthe yield surface; K may be caed th e hardeningparameter; L i s the loading rate; the case of loadingis defined as F = 0 and L > 0; the case of neutralloading i s defined as F = 0 and L = 0; and the caseof unloading i s defined as F < 0 or L < 0. I t i s seenthat, first, the decomposition of elastic and plasticparts i s made at the strain level, instead of at thedeformation gradient level as in the Lee ' s theory.Second, the yield function, f,and the hardening rule,eqn (127), are more general than the counterparts inthe Lee theory-this means the size and the locationof the yield surface in the stress space can be changedduring the loading process. Third, the Green-Naghditheory i s not restricted to isotropic material withisotropic hardening rule. Also, i t i s noticed that thecontinuity condition i s imposed by eqn (127). i.e. inloading,

and the Drucker's normali ty condition i s imposed byeqn (126), in which 1can be a function of Tu andEL.Equations (123) and (126) can also be written as

a2x .fKL = ELN 5 BuMNEL, (130)aE aE

Combining eqns (130) and (1 31) obtains

Step IBased on the current stresses, plastic strains, and

displacements, the element stiffness matrices K " )andK'2) are calculated according to eqns (62) and (63) andth e forcing terms, F"), P 2) and F") are calculatedaccording to oqns (63)-(65).Step 2T h e grand matrix equations can be formed as

KAU = F. (136)After the displacement -specified boundary conditionsare imposed, one may solve for AU, the nodal pointincrement displacement vector.Step 3The incremental Green-Lagrangian strains are cal -

culated, according to eqn (59), asAEU= 1/2(BG + 6AU

+ 1/2 B BMLp AU, AU, . (137)Step 4Calculate E'(1) according to eqn (123) as

Let E'(2) be E'(1) + AE and then calculate the follow -ing:

IfF SO, then Tu is updated and go to Step 6;otherwise, go to Step 5.where A i s the inverse of Step 5

(A) Make an educated guess for Tu(2), and calcu -(B) Calculate 6&(2) according to

E BUM + A--' '' (I33) late E according to eqn (I39).aTKL arMNand B i s the inverse of BH" In case of neutralloading or unloading, A = BUMN. In finite ele- k.pKL(2)= EL(1)+E&(l)+ AE -Et(2) . (141)ment analysis, the following incremental constitutiverelation i s needed: (C) Calculate the following:

A ~ M NT(2), EP(2)1J. (135)Now, the finite element procedures for large strainelastic-plastic solid based on the Green-Naghditheory can be outlined as follows.

@) I f fI = K and f2 > K, le tA = 1/2[A(1) + A(2)]


11/12

Finite element procedures for large strain elastic-plastic theories 405and if be expressed by any one of the following:

let

and then they can be transformed from one to the otherby the following rues.

where A stands for the inverse of A.(E) Calculate TL(2) as+ 5 u + a a + a a) ( 1 56)

The linearizations made in eqns (53) and (56) or ineqns (67) and (65), in order to evaluate the stiffnessmatr ix and the forcing term, do not present approx -imations for the overall formulation because thevalidity of the solutions i s based on the equilibriumso f the nodal forces which are calculated exactly. Also,i t is noticed that the follower -type forces are incorpo -rated automatically into this formulation. On theother hand, it should be mentioned again that thefollowing approximations,

I f TK L i s approximately equal to Tu(2), within agiven error tolerance, then update the following:

and go to Step 6; otherwise, go back to (A).Step 6Now the nodal point displacements are updated as-

and, also, 8 should be updated according to eqn(52). Calculate the nodal force vector according toeqns (48) and (51). I f the equilibrium i s not reachedpointwise, or the error i s not within a given errortolerance, then in view of eqns (S I ) , (63), (64) and(as), the calculated nodal force vector actually servesas the forcing te rm for further iteration, namely, oneshould go back to Step 1; otherwise, one may changethe force and/or displacement speced boundaryconditions and go back to Step 1 to seek the solutionsat the next incremental level.

have been made for Lees theory and the Green-Naghdi theory, respectively. However, this kind ofapproximation perhaps cannot be avoided if theincremental finite element procedures ar e employed.I t i s believed that, in view of eqns (I14) and (I15) oreqns (144) and (145), this approximation i s reason -able and accurate. Besides, this approximation i s notemployed for al l those elements which have neverexperienced plastic deformation, in Lees theory, andfor all those elements in the process of neutral loadingor unloading (including elastic theory), in the Green-Naghdi theory. Actually, for realistic problems, themajority of the elements are in that category. Onthe other hand, i t i s recognized that the solutions forelastic-plastic solid are path-dependent, i f the in-cremental step i s too large, and even if the stablesolutions are obtained, the path-dependency intro -duced numerically may not be acceptable. On theother hand, if the size of the incremental step i sreduced, say, to half, and no appreciable difference i sobserved, then the doubt can be eliminated.

-

1 DISCUSSIONT h e virtual work equation in rate form, eqn(I1) or

eqn (13), i s obtained exactly. The virtual work equa-tions in incremental forms, namely, the Eulerian, theTotal-Lagrangian, and the Updated -Lagrangian for-mulations, are rigorously derived to be eqns (22), (23)and (33), respectively. Actually, the three incrementalforms can be transformed from one to the otherwithout any approximation. Besides, in the limitingcase, the incremental forms and the rate forms areidentical. Also, i t i s noticed that i f the constitutiverelation, in rate form, of any theory of plasticity can

REFERENCES1. K. Washnu, VariationalMethods in Elasticity andPlas-ticity, 2nd Edn. Pergamon Press, Oxford (1975).


12/12

406 Jlw~sD. Ln2. R.Hill,1.Mech. Phys. Sol& 7, 209 (1959).3. A. C. Eringen, Mechanics o Continua. Krieger, Hunt-ington, NY (1980).4. K. J. Bathe, Finite Uement Procedures in EngineeringAnalysis. PrcntictHall, Englewood Cliffs, NJ (1982).5. L. E. Malvern, Introduction I the Mechanics of aContinuous Medium. Prcntice-Hall, Englewood Cliffs,N J (1969).6. J. T. Wen, Fini te Elements o Nonlinear Continua.McGraw-Hill, New York (1972).7. H.D.Hibbitt. P. V. Marcal and J. R.bce,Int. J. SolidsSINCI 6, 1069 (1970).8. R. M. McMbcking and J. R.Ria, Int. J. Soit& Struct.11, 60 1 (1975).

9. T. Scharnhorst and T.H.H. Pian, Int. J.Nm r . Mefh.Engng 12, 665 (1978).10. V. A. Lubarda and E. H. Lee1.appl. Mech. 48 35(1981).11 . E. H. Lee, J. appl. Mech. 36, 1 (1969).12. E. H.k,Int.1. Solidc Struct. 17, 859 (1981).13. E. H.Lee In Numerical Methods in Indu-trialFormingProcesses (Edited by Plttmann, Wood, Alexander andZienkiewicz). Pinendgc Press, Swansea, U.K. (1982).14. E. H. Lee and R. M. McMaking, Inr. J.Soli& Dru c r .16, 715 (1980).15. E. H. Lee Constitutive Equations: Macro and Com-

putational Aspects, p. 103. ASME, New York (1984).

16. A. E. Green and P. M. Naghdi, A r c h ration. Mech.Anulysi.5 18, 251 (1965).17. A. E. Green and P. M. Naghdi, in Proc. IUTAMSymposium on Irreversible Aspects of ContinuumMechanics and Transjer o Physical Characteristics inMooing Flu& (Ededby Parkus and Sedov). Springer,Berlin (1966).18. J. Casey and P. M. Naghdi, J. appl. Mech. 48 285(1981).19. J. Casey and P.M. Naghdi, QWIJ. Mech. appl. Math.37, 231 (1984).20. J. Casey and P. M. Naghdi, Consritutiue Equations:Macro and Computational Aspects, p. 53. ASME, NewYork (1984).21. J. Casey and P. M. Naghdi, J. appl. Mech. 50 350(I83).22 C. Truesdell, Comm. Pure appl. Math. 8, 123 (1955).23. C. Truesdell, 1.Rar. Mech. AM / . 4, 83 (1955).24. L. J. Segerlind, Applied Finite Element Analysis. John25. 0. C. Zienkiewicz, The Finite Elemnt Method, 3rdEdn.26. D. C. Drucker, Proc. 1st U.S. National Congr. Appl.27. A. C. Palmer, G. Maier and D. C. Drucker, J. appl.

Wiley, New York (1976).McGraw -Hill, London (1983).Mech.. p. 487. ASME, New York (1951).Mech. 34 464 (1967).

Documents

Finite Element Procedures for Large Strain Elastic Plastic Structures