D. H. BIlA"SD-UUond ,J. T. OVY."S: General ~lixl'd Finite Elullwnt ~[ethod" of Xonlinl'ar C~lI1tilllln 4-41

Il~

:r.,UDI :.:1. 111-·1:01 (1\173)

D. H. BHx~mARI !.J. T. ODE~

fh·Il(,l'allUhJ·tl Finit.e EIC'lllellt l\}('(hotls uf NOlllilH'Hl' ('ontillua.

III tier t'orlieyenden Arbeit werdell .llodelle tier /Iichtlillearell thermomechflllischen Reaktioll Jllr eine allgemeineyemi.schte Methode der Jiniten Eleme/lte koll.struiert. (;elCisse beuits existierelltle .1lodelle diesel' Art werden au.s del'Cll1gemeillen Formulierung abgeleitct u11d darges/dlt.

r:cmmli /llixed Jinite elemellt models oj the 1I01l1il1cClrtlrermoll/f:t;/I11/liatl reSp01l8/l oj dissipath'c /IIedia are fOlls/rut/ed.A nil mber oj e:ri.ytillY specialized finite eleme11t models are derived from the gweral for/llulatioll presented in th is paper.

B npe;l.~araellOii paGoTe npOJl31l0;rIlTen JlUnpOCllllc llOIlc.:ICii lIealllleiilloii TCpllOllCXaHIIQen;oii pcan-111111il.~n oGmero elicmalllloro lleToila J;OIH"lJlbIX :J;IClleIlToR. Ha oOllleil ITIO\llly.;lIIpOBlili BhIBC,ll'llhl IIoxapm,Tep"3unallhl Hel:OTophlC eymCCTIlYIOlllllC Yil>C lW:ll','1II ;)TOrO llOila.

\

1. Introduction

)Tost. of t.he early applications of t.he eoneept. of finite elt'll1cnts 10 problcms in solid ml'chanies dealt, withapproximations of the displacement. field which Wl're design cd so as to minimize t.he total potent.ial energyof It discrete model of an elaf!tic body. Such approximnl ions generally lead to all upper hound for t.he pokntialcnergy of the body. and t.Iwir construct.ion involves idcns that., in large measure. have been available foJ' overa century. In the early 1960's, anum hcr of alternnt.i-ves to what. may be termed the "st.andard approach"to finite t'lement formulat·iolls began to appear. Thest' involved t.he usc of models based on complenH'nt.aryenergy ideas and the approximat.ion of st.resses rat.hl·J' than displncements [I]. [21, [:3). Here lower bounds toenergy may be obtained, and the gelwl:at.ion of st.iffncss lIlat.ricc\s required t.he inversion of local flexibilitymat.rices t.hat arise naturally in the anulYHis. If one IIhandons the ,'cquirement. Hmt the finit<· ell'luent npproxi-mation provide hounds of one form or unot.her on the energy of t.he exact. Iiolution. then the usc of socalledmixed variat.ional principles for (·onstruet.ing finite clement mod<'lKsuggest.s itself, Ht'fl\ simultaneolls approxi-mation of bol.h displucclIlcnls and 8t rl'HSrS can be gcnel'lItl'd ViII, Iiay, thc II gr.LT~OEH· H 1~(SSNmt prineiple(see, for example, [4J, [5], [iiI). Thc entire area of so-called mixcd and hybrid finite ('Icment. formulations 1'01'

linenrly elnsl·ic hodies has been expllll'ed in some del.nil hy PfA~ IIllcl TONG [7J-[10): lUI up.to.date summal'Yof l.he ideas wil,h additional references to related work ('1m be found in [10).

If we look beyond linenI' elasticity 10 t.he general t.hermolllcchaniel.ll theories of nonlinear eont.inua, wesec many othl'I' field vnriahles in addit.ion to I.he displacements nncl sl·l'l'sses. Tn IHIl'ely t.hcrmal phenomcna,for l'xample, the equat.ions governing 1ll'lIt. l'onduct,ion and \.II(' (·ollllel'\'llt.ioll of energy involve t.he absolutetempernture, 1.empcrnturc gradients, t.lw specific ent.ropy, and eOlllponcnts of heat. flux. In 1ll0l'e generalprocesses, these quantit.ies nppcal' couplncl with th()lil~ variables related to mechanical l·l'sponse. 'l'11(\ mostnatural choice of dependent variables in t.he general enst' is the local displacement. field and temi>eratme fieldin each finite element. (e. g. [II]). since t.he usc of lllensmes of mot.ion and of changes of t.he degree of hotnessare suggested by t.he so· called axiom of euusalit.y [121. So far as \\'l' know. no IIttempts appcllr to have beenmnde at. constructing any sort of mixed model appropriate for t he general thm'11lodynamieal Iheory of con·tinuous media; indeed. att.empts at tll(\ dcvclopnH'nt. of relat.ively modest hybrid models fol' large quasi-st.aticdeformations of linearly elast.ic bodies have been confined to linearized incremental formulat.ions and arelimited to infinitesimal st.rains [13).

In the prC'sent paper, we develop geneI'll] mixed finite element. models of t.I1enonlineal' t.hermomechauiealresponse of dissipat.ive media, in which independent. approxilllat ions of t.lle displacement.s. st.rains, strcRses.int.ernal dissipnt.ion, heat. flux, temperature, temperatm'c gradit'nts, entropy, lind, in t.ho ('nse of continllumplasticity. the inelast.ic st.rain tensor or t.he int.ernal state (hiddl'n variable) tensor nre conf.\tnlCted. The result'" a complete discrete analogue of the entire collection of local field cquations for t.hermomcchanical belHl\Tiorof continuous media: e. g., linear moment.um, t.he stndn-displael'Illcnt equations. eonstitut.i,·c equationI'. heat.cond Ilction, E'tl'. "'hill' t.he l'esult.il1g collection of eqllll t.ions appelll' to be too com plex fOl' i1lI1llediate applieation,their generalit.y makes it pOlisible to view the idea of mixed modl'ls forlll in rat hel' unifying manner. Indeed,by assuming that cert.ain of the local field cquations arc identically fllll.isfied, a prim·i. we int.roduce correspondingdl'pendt'nl'es allIong various fields and t.hereby reduce the list of dependent. variables. Tn this way. we areallle' to pl'Odll(,(, a variet.y of "pecialized mixed models, For example, NlCKgl.l:S variatiollal model of linea.rt.hcI'llIo,·isl·opln"tieity pl'Ohlc-m [14] is shown t.o be del'iva hie as II Rpecia! ease from 0111' gelll'l'almodel. Cprt.ainof t.he mixed and hybrid models of PIAN llnd TO!\Cl[7J -[iO] art', of course. a.lso encOIllpnKIil'd by om model.

The vehil'lt' for om approximation is t.he variat.ional principle developed by ODEN [15] and ODE~ andBHANDART [16], [Ii] togethcl' with t.heir t.heory of "thermoplastil'ally simple" materials 1I8). These idcas are

*) SenioI' Hcsearch Assistant.**) Professor and Chairman, Depnrtnll'nt of Enginecring ~rechnnics,

D. H. BIIANIlARllIll1i ,J. T. OOt:N: r:I'Jl('ral :\lixl'd ~Fillite Eleml'nt :\11'111011.. of Nonlilwar eont,jnlla

reviewed hriefly in ~eetions 2 and :3 of t.he papl'r. In Sel,tion ..j. we de,,/dop 1!l'Ill'rnllllixed finite element modelsfor nOlllinear l'ollt.inuil and WI' provide interprptal,ions of certain result.s. ~'illall.v. in Hl'dioH 5 we pxalllinea mUll bpI' of spl'l'ial l'ases obl,a.ined from tIll' geHeral fOl'lllulat.ion.

2, \lllllilll'al' 'l'hl·rlllllvisl'II)1last.il'ity

. \ dl't.a i led d iSl:ussioll of t,lll' l'onstitu t,iVI' t~heol'Y of t.hl'rllllwisl'o- plastil'it y is gi vuu ill [I RJ: ('onseq uen Ily. herewe shall only r('view briefly Ole fi{'ld equations govl'l'ning the belHlvior of tlwJ'/lIoplastil'ully t-;illlple materiait-; .

ConKider a mat,pl'inl body Jj oC'C'llpying a r{'ferelll'C {'onfigurut,ion Co in thr/'u·dilllensionlli euelidean Sp1lC'l'.Following tho not.at.ion in [II J. we est,ablish a fixed spatial frame of rd{,l'{,lll'l' Zi (i = 1,2, :3) whil'h. for simplicity,is r{'ctnnglliar l·artoHiall. 'Vhile ill Cu' a mat.erial pal't.ide .1' is lalleled :1', U = 1.2. :J) and. for convenience.we take :1', = z, at, till1l' T = O. The motion of J) is t.hcn the one-parameter family of eOllfiglll'lltions definedby the mapping ZI = XI(.I'. t) and 1/,(.1'./) = Xi(.I'. I) - :1', al'l' the eart.esian cOlllpOlwntH of displaeement reJat,ivet.o Co.

The st.min.displacement, and t,Jll'rll1al-gradient. relations an' I!iven hy t,l1l' following f'xpl'essions

IY'f = -.;(/Ii.; +'1;.; + 1I",.i 1l1Ii.;)

y, = O·"

(:?la)

(2.1 b)

wlll're ;,(.1', t). 11(.", I). f/(,r. t) and 0(,1', I) denote respeeti,·ely. the litl'ain tensor. the displaeem{'nt '-ector. thermalgrildif'/It and t.IH' I1hsolut{' tempernt'lue. The l'omnla denot{'s partial diff{'r{'ntiat.io/l with I·t'spe{'t to t.ht' materialcoordinatpli ,1', whidl wc assumed to he rectangular cartesian in t.he l'efel't'lwt' eonfigu\'lItion.

The loeal fOl'llls of t.lw laws of lineal' lIlonwntum, angulal' mOll1{'ntulI1 Ilnd tlw energy are

La;j(,).,,} -I- 11 .... ;)1.; +!! Pm = fJ ii",; aij = ail

a; i Y f / -I- '.! Ii + 1/; = fJ ;.;

(2.2a. b)

(2.2e)

wherein all fJuant,it,ills are I'dl>ITcd 10 tIll' rd{'rclll'e configurat,ion: a(.I'. I) Ill'ing t,hl' seeoml PJOL.~\-Krn(,IIJlOFFstl'ess 1(\nsor, F(.I'./) the hody forl'e \'I'dol', (1(.1', 1) till' mass densit.y of t.he solid. Ii (.1', l) is the illt,el'llal hent, supplied1'('1' unit; lI1ass JlI'1' 1I II it, time, q(,I'.I) is the Iwa\. flllx "cetol'. and 1'(,1', I) I.he int,l'l'nnlenel'gy dl'nsily. The Impel',posed dot indil'nt,l's t Iw partin.1 diffl,l'tmt.illt,ioll \\'it.h l'cslll'et, to t.ill1p and I)q dl'notcs KIWl'n;CKlm delt.a. 1':qul1-lion (2.2(') eall hI' \I'I'ittl'n in an alternate fOl'm hy int.J'()ducillg the II J.:l.lI1l01.'I'Z frPI' pnpl'lIY dpnsit,y If = e - 8 fJand t,lw int.PI'lud diHsipnlion f1 = aij YI/ - U(';' + s ril illto (2.2c). i. ('.

I/; + !! Ii -I- f1 = !! '1'0 .S. (2.2d)

1 .. (> > I) ,. f 1 ' I I I f) ~; I ", 2. I 60Sn Wl'lldng :'.:'1 , or IIl't leI' con"I'lIIeIWI'. \\'e }five rpp lweI n )YJ (J n W 1('1'(' ...' == T .

. uTo eompll'tp t.11lldeHeript,ion of the hehaviol' of a l,hl'I'nlOplllst.icl1l1y simpll' solid. lI'e need only ltdd to

t,he basic kilwlI1al.ical and physienl la\\'s tbe cOllst,it,ut.i\'l~ {'quatiolll'l whidl ehnTlwtl'dze I.he nmt{'rial of whi{'hthe solid is eompoRed. In [18]. a t,ht'1'IlIoplast.ieally simple Illat,crial is chal'llct.l'l'i:l.pd hy II set of five const.it.utiveequations depid,ing the hcr cnergy rp, the stress a. the ent.ropy S, t.he inP)llstie st,ra.in-rat.e '1 and t,lll' heat.flux (1 as func't ionalll of the histories of strain ", the illl'lastie st.rain '1. a hsolntc j,plllpel'llt.ure (J anel t.he current."alue of t,he lellll)('rllt.ur{' gradient. y.

This tlwory lll'esupposes a suffi{'icntlr smooth transition (rolllan ;'clast.ie stllt.e·' to an "inelastiestate"to allow FnEclIET different.iation of certain eonstituti,-c funetionals. Wit.hout, I,his assumption. which \\'t'lllaint.ain is aCl'l'pt.llhl{'. the elahorate machinery of OWE~S pU]. [20] rnusl. he employed. Happily, our smoot.htheory yields liS spcdal eIlH!'S COLE.\IAX·S thermodynamics of simple materials 121 J liS well as t.he gt'neral ttworyof visen-plastie mall'rials of VAL\~IS [221. [2:n

In [I8]. t.hermodynamical rest.riel ions, in the s{'nse of l'oLE~IA~ [21], show t.hat this class of materials00

can 1)(' ehal'1l1,t.l:I·i'l.l'd hy only t.hrel' eonst.it.uti,·{' funetionals, Ollt' r!('s('rihing t.lw free energy <J> which is indet-00 00 .c=1I

pendent. of y lind t.IlP otlwr t,wo being heat flux to and t.he ineillst.i{' st.min-rllte ')1 . It is also shown t.hatj=(l 3=1)

t,he cnnst,it.ut,ivl' l'qllat,ions fo\' st.ress and ent,ropy arc d{'rivable f!'Om the hl'l' 10 IWrg,\' alone and the jnternaJdissipation is o!lt.lliIH'd frol11 the eonst.itllt,i\'t~ equations of rp anll ;1. Then t·hl' 1ll'1 of ('onst,it,utivc equat,ions isaiven I,\," . (x:J (.'(;

(J = 0,. (/> [1';': I'J = f'j U'~: I'JH ,II N=-O

- 00 I/> [I'~; J'] = cJ' [r~: l'J,IJ =11 ~=0

(i.aa)

(2.:H)

D. H. 1311.1»;1),1111 ami ,J. T. ODE»;: Gcm·ral ~lixl'c1 Finite Elenll'ut ~l('thocls of Xonlilll'lll' (:ollt.illllll

and"

fJ = clr~: /'. f11H =~o

<X-

ii = '.)1 In: r. f11_~ It

(2.:J d)

(2.:Je)

wherein (2.:1) we han' uSl'd I,ll(' notal ion l' = (,"'}.0) and tilt' tot.al histories al'l) dl:colIl)los('d into I.hl' 'past,hist,orics' and t.hl' 'CUlTen!; \'/lIII('s' of I,he rcspeet.ive al'gllllwni fUIH't,jol1s, c. g. 1.11f't,olal history f"( ..) (0 ~ .. s00)can be rep"csentl'd hy a pnir In(.~). J'I(O)) where 1'~(.~)= r'(.~) [.~E (0, 00)].

The Ilymholism a.", Of) dOllot.es partial diffl'rent.illtion with resped. to YlJ and () \\'h('('eas f)., alld f)1' denot.f'spurt,inl FRECHF.T diffl'l'enl,iatiollllssoeint.ed wit.h variations of the past hislories of '} nnd /': Ihe \'crlicnl st,rokcin t,lle argunlt'nl of /}I'q) in (2.:1c) indicates lilwarity in the quantity \\'hiell follows (i. 0.. {':). Tlw int.el'lllllllissipation a(.r. I) of (2.:1<.;) is not. of course. a dep{'ndent. eonstit.uli,'e \'ilriable in t.lll' strict. S(,IISI': t.hl' funC'tional1)[. ;.] is. in gencl'lll. uniquely dd.ermined frOIl1 qJ and 9l \'ia \.Ill' Cl.At1SIUS·j)ITHE~1 inequ/llil.y.

To the syst.{,111 of field (·quat.ions, we adjoin initial and houndary condit,ions. TIll' displacement andtraction boundary {'onditions nrC'

...lim - lim on cJJu X (- 00. 00)

on cJJT X (- 00. 00).

(2Aa)

(2.-1b)

The t{'mpC'rature and heat flux boundary eondilions an'

0=0 ...q = 1/( f/ = fJ

on aJJa X (- 00, 00)

on o:Ji" X (- 00. (0) .(2Ae)(2Ad)

For simpli{'ity and wit,JlOUt. loss of gelll'ralit~·. we take the associat·ed initial ('onditions to ill'

n(,r. 0) = "(.1;. 0) = '}P', 0) = O.

11(;",0) = ",(.1'.0) = 0: 0(.1'.0) = '1'0 .(2.-le)

(2AL g)

Siner the solut ion of t.hl' initial-houndary value "robl{'ms requi1'{''' t.hal, WB inl'orpol'llIt, the illit ial con-ditions t'xplieitly into t.he finld equations. it. is eon\'l'ni{'nt to take t.he LAPLACE t.ransforlll of (2.211). (2.td)and (t.3e) and t,hen, I,at-ing I.lll' ilwerse t,,·ansforll1. \\'1' writl'

fI*raii(f)",/ + '/I/II.i)]". -I- '.! //11 = (! /I.",. ...

r/* f/ i + /I + ,2.' = '.! '1'0 8('0

y'* '1(1 = A'ij rl'~:/'. y]"=0

wherein y(t) = I. y'(/) = I. I\nd

1m = l!l * Fill] (.r. t)

II + E = fI' * Ig II + a] (.1', I) + Q(.I') '1'08(,1'. 0)

Tht' symbol (*) appl'lIring in (2.5) denotes t.he eon\'olution integral which is defined nsI

Ii * V = f u(1 - T) V(T) tiTII

(2.fJa)

(2.:3 h)

(:Ui('

(t.Ha)

(t.£ih)

(2.i)

wher{' t.he dependcn{'c of /I and" on .1' is understood. Thus the systcm of equal ions (2.1). (2.:{a)-(2.3d).(2.5). t.oget.hCl' wit.h UIP houndar,\' eondit,ions (2A) ('omJlletel~' ehara{'terize a t,hennoplastil'all~' simpl!' lIl11t.el·ial.

:l.. \ (:"III'I'al "nrilll iOllal ])rilll'illll' for ~oJ\IiJ\('nr TIH'rlllo\'is('olllastitity

A generalizat.ion of VAD1BJ.:HO·S I,hl'orem 1.2-1Jt.o nonlinear initialllnd boundary value prohlellls in eont.inuumme{'hanies has heen prescnt.l'd alld used by ODES and BHA1\IlARI in st'\'el'1I1 papers [151-1171. An out./ineof t.he rnel,llOd with its lIpplil'llt.ion t.o nonlinear t.hermo\'iscOlllast.ieit.y is given helow; for dl'l,ails of the IIwt.hodsee /,pferellees [1;3J-[liJ.

OUI' principal aim her'I' is 1,0 eonsl,l'u('t a. general vllriat.ional 11I'inC'iple assoeiat.cd with t.Ill' nonlineal' t,h!'oryof 1,lwrmoYisl'opluI4icit.y d(·filwd hy (2.1). (2.:{a)-(2.:Jd). (2A) and (2.5). The uoundul'Y-\'lIhw pl'ohl('m de-seribed by (2.1 )-(2.5) ean bo writ.t.en in t.lll' f01'll1

(:J.I)

where ~ is a nonlinear operator dcfined on some dense set fl ~ 71, into 'V',V and 'Ii' !lPillg !'l'ul BMC\l:1I spaces.A = .1(,1'. I) is an elen1l'nt. of D. 1,1ll' dOlllain of~, lind r = r(,I'. t) E I;. The dOIllain of ~ l;llllsist.s of 1ll'de\'l'dninp-tupll's /1 = {II, :'. n. fT. '1. y. O. S. Ij} who,.,e eomponents are flllH'tions of posit,ion ;1' IIl1d t.illH' t.

444 D. IL III1ANIlARI lind J. T. OVES: General :\lixed J.'inite Element .Methods of i'lonlineor Cont.inua

The operator '.p(it) dpfined hy t.he formula

-- l -('.p(A). /1) = Lilli ~ [K(/I + (\If) - 1\"(/1)]

~ ... f1 :\(3.2)

(3.3)

where ]{(il) is a flln<'l.ional on D, :.- E 'ii, Ilnd (' , -) is an uppropriate hilinear mup from V' ® V into E isealled the gradiellt of 1\ at .t und is denoted '.p(A) = gmd ]{(A). The opcrat,ol' '.p(A) is potent.ial on D, ifthere exists a fllnetionnl K(/t) su{'h that. g1'lld K(A) = '.p(.!) for "I E D. It is shown by VAINBERG (24) that. if'.p(A) is a potcnl,ial opem1.m' then tl1l're exists II funcl.ional K(.1) whose gradient, is '.p(.t) which is given by thedircet. int.egral

IK(.1) = f <'.p(.'lo -I- s(.'t - itO»)' (.1 - ,to» ds -I- ](0

o

where ]((.10) = Ko is a constant and 8 a real parameter.Following t,he Jlrocedure adopted in [15)-[17]. we introduce the bilinear map of GUR'I'DI (25)

<11, v) = f [11 * v) d8iJJ

where 11 * v is definNI in (2.7). Then, subst.illlt.jng (3.1) int.o (3.:3), we ohtain t.he fllnetional

(3.4)

]{(.t) = ~J{Um *!! lim - 2 g * "m * t1~ii- g * 11m * (aijIIm,j),; - 2 lI,n *!! 1m - 2 g * Yli * aii +

.'1100 00

+ 2 (1:1< Yli * /fii [J':. rI- (I * a * a -I- 2 g * 2) [l':: 1'] * a - (I * 8 * 8 -I-,=(1 .~n00 00

'r 2 (/ * 8 * c'f [r~: J'] - 2 (J * qi * (II + 2 (/ * (2i [r~; J'. fJ I * (Ii - 2 (/ * (g' * q~i + If + E -.-0 •~f1

~ 00

- f! 7'" 8) * 0 - y * y' ... I/i} * III/ + 2 (/ * Jlij [r:; 1', fJ) * tJf}} rlJJ +,=0

-I- ~ I {g * [111 aii(·lIm.i + 21).111) - 2 'I'm] * 11m} deY -I- I {f/ * It", * 7''''} d1 -I-8.J/.I' 8'11"

-I- I {r/ * g' * (j * Q} rtf + I {g * g' * (Q - (d) * O} d c'f' .

D,'lIo D.1Jq

(3.5)

In arriving (3.5) from (3.3), we have set Ko = 0, .to = O. It. is now a simple mlltt.l'l' to ,·erify that the functional1\(./1) assnmes a stat.ionary ,'alue whenever (3.1) is Rat.isfied. In other words, equations (2.1) -(2.5) are theEULER equat.ions of the functiollal K(,t) of (:~.5). All nine fields II, ~'. G, cr, q, fI, O. ::; and '1 are varied inde-pendent.ly.

J, (I('IIl'rlll :\Iixrd Finite- Element. 11odl,Is

'Ve shall now examine general mixed finite-element. models of nonlinear continua obtained by the obviousRITz-procedure suggested by (3.5). Following the philosophy of finite-elements, we begin by representing

Ethe body 8i as a colledion U (j}. of material finite-t'lement.s (J}., {'onnectt'd cont.inuously t.oget.her at finite

•-1number G of nodal point,s XLI. J = 1,2 .... ,G. In finite-element analysis each clement. is t.emporarily con-sidered to he disconnect.ed from t,he entire collection and once the properties of individual elements are estab-lished, t.hen the process of conncct.ing elements t.ogether involvcs only the topology of model and is accomplishedby simple t.ransformations. Since t.his has been discussed at great. length elsewhere [II]. we shall confine Ourattent.ion to It t.ypieal element (J}. of (J} .

Consider, then, a t.ypical element. (J}" in ",hidl Nt nodal points a'~, .,.~, .•. , ;r{e have been ident.ified,Let ,A. = {Ile' " •• a•. C1•• qt, y" 0., S •. 'I.} denote the restl'iction of A = {Il, ", a. a, '1, y, 0, S, 'I} to (J}, andlet Al\'(t) denntn t Iwir vnlues at node ;J'. = 'l·~·.Then we int.rodllee the followin/o{ local apPI·oximat.ions of Ae:

'11.(.1', I) = I/.(:)(t) tpW(;") .

".(.1', I) = "(;>(1) tp~)(.~1'),

a,(.I', t) = al~)(t) ~W(·1') ,

a.(;", I) = aft)(I) VW(;I') ,

q,(J.', t) = qft)(I) P~)(;r) ,

y,(,r, t) = g{:)(1) G~~)(,1'),

8,(a', I) = To + T{!)(f) TW(;1') ,

S,(J.', I) = 8(:)(1) JI~~)(.l') ,

11,("', I) = '1(:)(1) l.~~)(,I') ,

(4.1 )

D. R. BHANDABI and J. T. OlH,N: Gent'rnll\lix('d .Finite EIt'nll'lIt ~Iethods of i\onlinear Continua 445

wlll're till' local interpolation functions tI'p(,r), ip~(J')' VI~I(.r),etc. each set sat.isfies the properties

1J'W(.r.11) = o:~!;lp~I(.z:) = 0 ". It 81.

1J'~)(,r) > 0 .1' E cJJ,(4.2)

1I11dt.he rcpeated index N in (4.1) iHsummed from I to X •.\Ve coulcl, of course. IItll' higher order repr{'sentatillns for II,. ~/., a•. ctc. in which nodal valuetl of various

spulial derivat.ives of these functions also appeal' [11]. For example, let Z~ denot.e the space of ordered triplellof non.negative integers; i. e. if 0( E Z;, tht'n oc = ("1' lX2, (\3)' Pilch (\c heing a non-negative integer. Usingthe mult.i index notl1l.ion

(4.3)

(Vi)

(4.!l )if .,' It .J}, .

WI' define a local iutl'l'polation fuud ion 1J'~:(.r) of dcgrcl' If al'l~ording t.o

D~ 1J';}'I(".JI) = b:~ Op: ()p; ()~: ' locI. I/~I ~ I) , (4.4)

where JlI, N = ]. 2 .....• Y,.. oc./~ E Z.; lind V)~(r)(.r) == 0 if .1: It ell,. l'roperticlS (4.4) l'nlSure that the loeal repre-tlcntation of a funetion II, over an clement YJ. is a Iilwnr eombinahol1 of the valllcs of all d('rivatin's of II, oforder loci :S q at. t'aeh node ,1'." E (j)•. That is; if

J 27 a.~(·)(/) 1J';Jtl(",)11.(.1', I) = 1",,::;;q o

then (4.4) implies thatN . N

([.~('l(l) = 1J~ 11,,(."'\. I) == 112 (,,>(/) .

Thus a higher mdcl' reprelSl'ntahol1 of degree q of t.he local displacement field 11.(.1'. t) t.akes I,he form\.- • . li

".(.1', I) = 1: 2: /l.~ ('>(/) V'i'I(;r) (4.i).\'~llsl:;:·/

and ill a I;illlilar manner higher order I'cprcsent;\tionl; fO!' :'.(.1', i), a.(,r, i), etc. can be conl;ll'uded. The nodalCjull11litics 1I~(tl, a~(">, ... , 1/~(·) lire now funcl ions of t;iml', so t.hat we clIn IIlso eonsh'uct; a finite elt'lllcnt appro·ximation of t.hcse «llllntitieR over t.lw "time element" (lice [II, p, 153]).

Let f(T(.I) dcnote t.l1C value of J{.(.·t.) at time i = ']', Wc }lartit.ion Ihe illll~I'\'al [0, '1'] into partitionsII = [1.,1'+1] and represent, cneh llodlll fllnet.ion It~(')(i), :,~'('I(I),... , '/~(tl(t) by sOllie 1IJ1J1I'Opl'iutcdiffel'cncl'lidlClllC in Ihe t,illlC domain; P.g.

.+1ll~'(t) = :E lI!n(ip) lp/Ht) (4.8)

p-.where, fol' convellience, we !luVI' d rllJlped the element la Ill'l 'e' ; I E Ii., tnIl and rp;W) arc the i11terpoln tion fune-tions in time of degl'ee k, ddinell so that

dill II s.' In, n = O. ]. :! l, - I ,(I) - Ull () .

dilll rp/l " - '" ~/I' R - I "1\, I' - . - .

and

j.V = 1,:!' ..... 1'/ •. (4.10)

Then. lIlSillg the l'cprest'ntationH of the furm (4.i) llnd (4.10), loeal approximat.ions of A. = {II., :' .. a•. a,. (I.,Y., 0., S •. 'I.} al'e discretized illt.O t.he form

k-l ~ J.\..~

( i) " " ~ '1 NP 11(1) • ( )It •• ", =..;;.. __ "- .;. It~ II (f" 1J'N.1':I~I;j;,/ II~ 0 /1=1 ,\' = 1

k-l ::! S,( I) ~' " ,. ....' IfPII(/) -~( )? .", =..... .:.. ~ .;..;"~ II q"/I 1J'/I',1' •

I~I:>:'I 1/'-0 P=I .\'=1k-I ~ s, (4.11)

y ....' ,..., ~ NP "'.11 ( ) ~ ( )a,(.I'. I) = .-;.. _ .;:., _ a~ II q" I 1J''N.r .1.I<q ,,~O jl~l .\'~1

k-- J :! .Na "

(, I) - ....... " ~ ",,' N.8 '''(t) •• ( ,)'1. ,1 , -.::. .... _ ~ 'Is II " /I . Xli.1 .!.j:;Oq 11-0 P=l -"·~1

In order to obtain t.lw discretizcd f01'111 of the functional 1\(.'1) of (3.5). we need to substitute (4.11) into(:1.5). This substit ulion obviously leads to nn ext.relllcly lengt hy expression. However, to indicate the forlllof the resulting t'quations describing the diseret.e lllorleland. at t.he same t.ime, to keep the lllgebra within the

·Uti I). H. RIUXlHRJ and ,J. T. OIH;X: Gelll'ral ~lixl'd Finite Elellll'nt ~Iet.hod!; of Xonlinl'ar Continua

reaSOllll ble limil.,.;, we shall cOllxidcr here reprcxcnta.t.iolls only of the fin,t·ordel' (i.e. the fi.'st. ordl'r represrmt.atiolIof tIll' local displacement. field ".(,1', I) = u.\"' q~(1) tp.\'(.r)).

To ]JI'oel'cd furl.her, \1'1' 1I0te thai t.lll' gl'lIl'l'alizl'd force vedor nt. node N is p' = {"(f). Thus, if \\'1' ap-proximate 1""(/) in t.il\l(· by

p'(t) = ip,(tJlx" (4.1t)then

1'(.1'. t) = r'" ifJ.,(f) 1.,,(.1') . (4.1:l)8u bst.it ution of (4.1 :~) toget/ll'1' wit.h t./ll' first onfer refll'{'scnta t ions £01' .tI, into (:l.5). leads to the followi lIg 10cIII11pproxi mlltioIH; of 1\(,1):

I J (I) (~)1\ ( .) = -- m ,- II w"" /l.1I JI (/"" - :! l~· II /l'Y' a;j.ll JI c("'" - 1;'!.l1 I' '1/,\'" a; j.ll fl/lR8 a"', -

~ 4. ~ 1 ~,. m hi ~\.,... .' .) ",... III 111 (\1 '

-.) Ii .j.1IIIu.\" ii' . - ·1 -1. y"', aijJI/1 7fooJ -I- -) 1',,> ;ii f.} II' - B· aX' (j.1J(l b-"J \-.... .\I.' '" m .\ I' -....'\.". J ,( - I J ../L. N t·\ .\ .\1 ,\ IJ. "0.::) '\'- A.

-I- :! as' II'.\' {- } b,; - C.".II S"" S.l1 fl C~/I -I- :! SX, (-) .d· } I'~- :! /)". M y{., {1Jµ I~',I+_t::::(j 1I ,-0oX 00

-I- " gX,' ~.;. {.} b - J. 1/"'" 11,11(/ G'~ + ,)II'\" 1.'; j {.} C' --; '.\ ~\' ~.\I I J 'J t\{1 - I J .\ N ..\"~,,,O ~ .. II

[ ((I) (I) (~) (:I) (:I) (I) )

- 2 '1'0 g~1I '1;.1/11 dp + A',II liJJtl e;, + I~·.lla.llµ d'iJ - 1'u EM S·l/tlc;; -I-(1). (I) (~) (;1) (3) (4) )]}

-I- '1'-"" 1/\'.11 qi .11P d~"1 + L S.II 1i·1I I' C""I -I- L .,..11 a .1111d~<fl - 'I'll L .,.'" S .111'c:ofl +1 1(:1»). II' . II I ' .- (,I); ,.. II J ~" ~ B(I) ,. T~ II II ~ .• ]+-- [', 11 'a'}' In"" -:! "1/"' 'a"'" "1I ,,-~ ' II'" m. "e"" +2 .\ ." n It. .\ P If ., III \\ JI .\ .\( PIt .\ p

(~l (II (I) (~) ~ ~) (:I) ~ _

-I- Bx,lI /I;~T"o.IIIJe.~'11 + lIs.uQ·II(/'/'-"'d:/i -l/s.uQ·up TS'd,:(/ + H.,·.u T-"·'Q.lI{Ja"'tI,(4.14)

where (.he 111111J'ix 1l1'1'1IYl'l and Ih!' I·CI'Ill!! "~.'J'V'"I" dc. al'l' I!in~n ill La hie I anel 2, llnd the first. 01'<1('1' 1'l'llI'esentnt,ioll1I!!{'d in (4.14) nrc gin'll in Appl'lldix .-I.

(~) -E.\! = 'f!! liN(.,') dJJ:

./I.

(~)

1~/.Ilu = f 'I'.d·'·) (Vl.II(·I') 'IJI1,j(:")).i tlJJ ;.11.

BS.II = f p.d,") 1',,,(.") dJJ: (\ . .11 = f ILs(,,') µ.,,(,r) doil;J1. J1,

(~) -£.,'.1/ = f '! Ts(·I') Ii.II(.r) dJj:

.JJ.(~I

I~/III: = f 1/, 'Ps("') $.,,(.1') 'I'I1,i(·I') rlf:~.JI./,

(I)

1:".11 = f'p,v(,") $.1/.;(,1:) I/:JJ:.II,

'I'll hll' I~llIt)'ix AI'I'ays

iii.,·.11 = f !! '1}s(·I') 11'.11("') tlJJ:./I,

l's." = f I! V'.II(·") J.d;") dJl;J1,

(I).

Ds.II = f U;I'('I') P.I/(.1:) tl:Jl: ,'."." = f 1.s(·") 1..1/(;") dJi: E:II = f tp.l/.;("·) rljl:../1. .II, ,il.

(:I) (I) (II .EN = f I'.d·") I~]J: I~',y = J!! 11.\'('1') lUI; L\.I1 = J T.d,") "'.I/.i(·'·) dJ/:

~ ~ h(;I) (4)

LX.lI = J Ts("') ".1/(.1') dJJ: L"·.I1 = f '..! T.d·") 1'.1/(''') d·li:$. J1.

(:I).

1,\- JJ = J IIi tp,..-(.I') ~.,,(.r) dci':8$.,·

(I)

lls.ll = f 1\/(;,') TS(·") rI.·'{:i.JJq

(I) ,BNM = f 'PsI,") 1.,,(:,') df:

8J1T(~) ~/ISM = f F.I/("·) Tx(:':) dJ:

b 11"

(~)

liS.11 = f Vis(.I') /.,,(.,.) elf:f:JJ ..

(:1)

11,,-.1/ = f TS(:") "'.,,(.,.) ll(j'/J.II'1

Tn blc 2l'oeffiei{,11 Is

e~'{J = a * !I' ... 11:(1) * ~(t) .e.~= g * !I' ... l~(I);

(f~"J = qW) * I/W): '1::/1 = y * fl'~(1) * ;;,~(t); u.;'J. = y ... '1~(t) * %(1) ?~(t);

a: = !/'" qW): ii~µ = til,(I) * 'pij(I); Ii: = !J ... r~(t): f,;~>11= f/ *lj;;(1) * q'pU):~h;'fl = IJ ... p:(1) JiW): b~ =!J'" p:'(t): 17;. = Y * p:(t):

1:.:" = 1/ .. !I' 7,:;(1) * ';;1(/); r~ = !I * ';:(1): 2~ = (I '" j~;(I):1~1 (:1)

d, = y ... y' ... ".:(1): ":·tJ = !I ... (I' ... ,/~(I) ... pW);

b~" = Y ... fJ~(1) ... fJµ(f):

1;;'" = !I * h~(I) >I< hW):(I)

el~µ = (I ... !I' ... 1t~(/) ... p~(t):(4) i!.)d~ = !1 * !I' ... p:(I): d.: = !I * II' ... Ji:(t):

1>. B. BIIA~lI,'lti IIIllI .1. T. ODEX: C:I'lwrlll ~Jixcd Fillil!' Eh'llIt'lIt Method .. lIf )\onlilll'lll' Continua -1-47

The eonst.itutive functionals of st.I'css. ent.I'O)lY. heat flux, internal dissipation and th{' inelast.ic I:ltraill-I'llh' fort,he element, $: aPlwlIl'ing in (4.14) HI'(' obt.ained h~' using thl' (Il'oe{'(Iun' of ODE); (see fl]. p. :WOj) and al'edufined as:

~. 00

'!~~1-] = f ¥iN(·r) Jii [1';."(1); /,x IdJl .A~-U ./1, -,,,--0

~) 00

(-J..- 1-] = f !lS(·I') rJ r 1';."(1); r·'·] dJi ,.. ~-.o 11. ,,=H

"" 00<S'\, [ -] = f 0",(.1') Ci [1';\'(1): rs. y"'] del) .!( .• 0 ..1}, .~:- n00 ocIPN L -] = f 1'.\'(.1') :/) r /';.\'(1); r·'·] dJJ .

~ _-0 n .11, $=()

0.) "":.

J~ [-] = f ;(t..(.r) .IVij [/,s(t): /''''. rr"] tid) .'-0 (JJ. 1t~O

(4..10)

'Ye then ge1ll'rale the complete syst{,lIl of equal,ions gun·rning th{' dis(,I'e!(' I\\odel b....sdtin!! tIll' fil'"t"lIl'jation of t.he fune!ional of eq. (.4-,14.) cqulll to zero. i.e. flA,Il. = O. ThC'1l

0/\. = 0:Oll;~ .'

ol\~=O:a.~x.

ob:. = U:OYi'/

'01\'_=0:(hn'/

_a/I,oris.' = 0;

0/\ eoqi5,-ji = 0 .

(4.16)

Equat.iuns (4..16) 111'8 tht' finite elelll{'nt. allalogllt!s of 1.11t'field equations (:!.l )-(:U»; l'.l!. equat.ion(4.] 0h is the finite el<,nwnt, analogue of t·he 101'al lill{'lll' momentulll l·lJlIat..ioll

and t.he t,I'l1Ct.iOIlboundal' ....condit.ion(:I) (1) (I) ~

/ ~. .II (11 "' .1111 (/~,o'l + J ~{1I11 /I:'~'(1; i .II/I 'iI~'#' = U .'·.If '1'''' .1111 e,:~

(4.l7a)

(4.n b)

IIlId equal.jolIs (4')())2-(4..16)G COlTI'S)lOlHIto the finite·(.(ement anlllogu(':; of tlw eonstituti"e (·qullt,ion,.; fol',the sln'ss,

'Xl

..l ,\' .If Oi i.ll II fi~'fl= .:r.~{-} It:;'=0

t.he int..<,rlHlldissipation,00

U'v.Il (T.lI{J b~"11 = 1]1N {.} h.~.=-o

the ent ropy,00

U:o/ 8.11 fJ r~'11= fJ x {-} i:~8=11

Ihe heat flux., 00

j) .".If qi .1111 7).~o'l = ;f~,.{-} Z:K.=.O

and the inelast.ie strain rate,00

J IIfJ ~.. 1.'" { }" "'.• "'.II 'Iij 1',,11 = ...'}; . I:"

H=(J

(4.18)

(4.10)

(··UO)

(-!.:! I)

Equation (4.]fjh is the discrete analogue of the energy equation lind the Iwat flux boundary eondition i.e.

(I). (II (~) (:I) Pl (II1,:,· .IllJi.l/ll tI~'fJ + L,\·.II ".1/11 (0.;/1 + /'.".11 fr·1I11 ti::'it = To /J,",II ...,.1/11(;';'/1,

(1) (I) (~) ~ _

11:0/Q.lJ tI It tI = 1111' .II Q.l1 tI d~tI .

(4.2:la)

(4.2:lb)

To deri\'!' the finit(>·element. I1nalnglwl'l of t.he ('olllpatihilit.y l'quat.ioll and thl' displaeellll'nt houndlll'~'conditions (4.10)8 and t.he thermal temperllture-gradi(·nt. J·elat.ioll to!!!'th!'I' wit.h tllP t.emperntnl'l· boundar,\'

448 D. H. llUAXIJAHI and J. T. OUES: General ~[ixl'fl Finite Element ~Il'thods of Xonlinear Continua

conditiol1li (4.16)9' we apply GHKEK-GAUSS I,heorcm to (:l,ii) 10 obtain t,he alternate form of the func-tionall\(. f):

- 1 (r I ....1\(/1) = -:!-, Ill", * [! "m - :! I(m *!! "' + y. (Il;.i + IIi.; + 11m.; 1I""j) * a'l -:! Y * {if * ai' +J/

00 00

+ 2 fI * )'1 I * pI [I'~; I'J -!/ * Cr * IT + 2 (J * a * JJ II'~; I'J - !/ * S * s +R-tl 8=0

oc 00

+ 2!/ * is * (f [r~: I'J - :! g • y' * 1/ * (Ii + 2 ff * II' * qi * O. i + 2 g * Q; [r~; r. yJ * 1/1-.=11 •• (J

- 2 (I * (l-f + 1.' - g '1'0,9) * Ii - y *!l .'1'1 * 1/'1 + 2 Y. 'iii * clViill'~; r. gJ} d(j} -.=0

.- I'fI * (II", - It".) * '1'''' dd;- (!/ * 'L'III• II". (t(/' - FrI.rI'* (0 -6hQdd +~~, tlT a,~u

(t.:!4)

(4-.2,5a)

(4.2011)

(-t·,20 b)

+ (y*y'*Q*OdJ.

8Ji'l011('1' again, following t.he proCC(hll'I' Ilsed in equat.ion (4-.14), we obtain t.he discrdo model of 1{(. I) of

(4.24-). \Vo t,l1l'n consider the '·lll'iation uf 1\,(/1,) wit.h respccl. to aii.lIp and qL\1{J in the interior of t.he body (j),as well with respect. to 'l''''x~ and (IS, 011 t.11t'boundaries oJi" and 0$0. respectively. This yields t.he disc-rete('ompatibility equat,ion

(1) (I) t!)-; _y~,""':!: I fl. "y, ......,: /-ii ,x" II. ""Q~O _.~ t ,\,", ~~')1S.lllIj (/~IIT .\.U1!i 0,\,1 t- .\311:"111 "", ((J.p.---"JS.lIYij a"p

and the displaeement hOllndary eonditionl-;U;~'\ = v;~'\ . (4.25b)

The disercte forms of thc t.emperatllre t.hcl'mal-gradi{'nt relat,ioll llnd I.he temperature boundary condit.ion!' arc<2) (I) ~

.J:\".It 'l's" d.~'p= Ds.ll y;"" b~11.

\\' hcl'l~

(4.27)

(!)

hilt 11 = f 'P.\'.,(,r) 'tp1I,i("') V!,\/(.I') clJJ ,.R

(1)

l.\'.1/ = f V'S.iU") Y,.I/(.l·) dJJ;.J}

(I)

.1\.1/ = f T,\·...(.'·) [0'.11(:") d:JJ(/J

and A,\.""lJl';.I/' a~'(I' u:',ictc.lIl'e defined in tables I and 2.Thus equations (.1.l7)-(·L2:3) and (4,25)-(4.27), lIrc t·he complclo fillit.e-element IIllalogues of t.he field

equat,ionN (2.1) -(2.5).

r,. Some Spl'l'illl faSl'S

There CxiMtsa number of important spec-illI eases that fall within the gene1'll1 framework of the theory dev(.lopedt.hus far, 1.11 this sectioll, W,l' shall disclisii Home of t.ltese ('aMeliwhich ean lJl' del'i,ed from the general for\l1ulat.ionIll'l-sented in Sectiull 4.

We note that by set.l,illg III = II and lldet.ing the const.itutive equation (:!.5c) from the collection of fieldequations (2.] )-(2.5). equations (4.W) reduce to t.he finite-element. analogues of the field equations go\'erningthc behavior of nonlinear t,heI'Illovlscoelast;ieit.y. Furthel'llIore, if only the approximations ill displac{,lIlent andtemperat.ure fields al'l' cOllsidcrt'd, t.hen we obt,ain discrt'le models of thel'll1ornechanical beha\~o)' of materialswith melllOl'y presented hy ODEX and HAlllUEZ [26)).

COlll'idcr now tho Ht,at-ic t,llCrmoelast.ic case whell t.el'llitl involving t.he t.iIlle rates !llld historics l:all bencgleet.ed. '1']\('n \.1ll' funl'liunal A'{.Il) of (:1.5) redlW('1l to

A't(A') = ~I {2 "i I Jii(,l, 0) - ~ I'f I (Ii; - f1~ + 2 IT :lJ(y, 0) - 2 II", a~'li- 2 u", '! Fm -,1/

- a",(aU IIIII.i),i - 82 + 28 cJ'(", 0) - 2 qi Yj + 2 YI ai(, •• 0, y) - 2 (CZ:i + Ij It + a) O} dJ} ++-~-r {[ni aii (a"',i -I- 2ilm/) - 2 1'111] Itm} iiI + {' «", '1'111 drf +

1/;./ a;;,+.r {to - O)Q}dJ + fQ(JdY.8,:AU B.:iJ,/

(5.1 )

0. R. BHAXDARI and ,J. T. OOE:-: U(·ncral )lixed .Finite Element MclhOllli of .xonlinear Cont.illua 4;4,9

wherein Jij( -), :lJ( -). J"( -) and ail -) are now functions of the current yalues of their respective arguments.Then introducing t.he approximations (4.1)1-(4.1)8 into (5.1) and noting t.hut. thc nodal values UN, "N .... , 8-'"are indcJll'ndt'nt uf time, WI' obtain. a,nalogolls to (4-.14). a function:

- - 1 { N ~.. .f ".. • U ~ \' ,., ]1 ,., \' (11 (1) i '~,. u1\1(.1) = 2 2 y,j :~.!~(-)- 2 ,,11\'.I/ YO a'l· - BX.II a· .a' + 2 rJ' ~. 1\'( -) - 21-'".1121;" aim. -

(~!. - --1:(111l;~;aii.lt '11;:! - ~ IJ.\'.II /;~ ll;~; - C,'·.I1 SoY SM -I- 2 8?\' (-IN (-) - 2 DXMYi" qi.ll -+-

_. [(1). (:!) (3») ((I). (2)+ :!!Ii" J\ (-) -:! 'I'n WIJ qi.11 + I~.II It.ll -I- B.II fIJI -I- '1'-'" L~\'.11 qi.lI + L,,·.II !b.ll -I-(:1) ,., )j} I [(a). .. (.1).. . (I) ,.,]+ LS.II a·l1 + -:) I il1l1 lI~t~a'I.1111:'~ + ~/.".11 I(;~.a"" M - 2 HN JI u~ ,/,111.11 +(~) (1) (:!),., (3),.,

-;- B.v,lf ff;~ T,,·.11 + 1/ S M QJI 1'.\' - 11.\'.\1 QJI 1'-" + f/ N .1/ 1'-" Q'u . (5.2)

In (5.2), t.hl' functiuns 1~~(-). es( -), IPs( -) and J;d -) for t.he {'lemclII. cJl, arc defined in a similar manner ast.hn,.;e in (.t.loj) except. that. in thi!; casco t,hese arc funet.ions of t,he CU!Tent· values of i' and 0 rather than thefUllet.ionnls of their hi!;torics. Thus. once again. the nl11ishin~ of t he first; variation of 1{, in (5.2) yields thefinite-denwnt analogues of tl1l' field l'lJUIIt ions goyel'lling the beha viol' of nonlinear t.hermoelastic solid. 'Veomit. the detllils here.

Xow consider the case of Iinel1l'i;-:pdthcl'llIoviseoelastieity in whieh \.lw displacement, field u(x, t) and thekm I){'\'l\t Ul'e field 0(.1', t) arc kinematically and t,hel'mally admissible (i.e. the displaeements ll(X, t) and t.empera-tUl'e 0(;1', I) sat.isfy the strain-displaeemenl rpll1t.ion, tomp{,l'l1ture thermal-grndient. relation and the constitu-tive' ('((uations). Th(,ll t,he funct.iollal of (:3.5) yields

I .,J """{II, O} = -:) ( till * '! III + 2y >I< lIi,j'" Jij [,,:.O~: £'. 0] - 2 f! II, * I, +

- .=0;'

+ 2 g ... g' * Qi 1,,:,0:; 1'.0. y] * fJ,i-2 y ... H ... 0-21] ToY * 'J [,,:,0:, ". 0] * O}d.cJl-.=0 ,=0 \- I!I ...1'; * II, d rJ' - .J' ri' ...Y >I< Q>I< fJ dJ" . (5,3)

8,.'/1T 8$'1

In arl'i\'ing lit. (5.:1) from (:1.5) wc han! set· t,he illtf'1'llul dissipation functioll 0- equal 10 zero and",

.Fij [i':, O~; i'· 01 •A~IJ

0.'

q; = ()i r,,:. 0:: ", O,!J] ,8... t1

(Y"

and S = J" [,,~, 0:; ", 0].-0arc lilll'llI' in the strain-history "I and the t.empernture histol'Y 0', We note that the functional 11{,} of (;),3) isidentical with (IV.l) of [14].

Xuw suhst,itut.ing the first ordel' loelll IIpproximat;ion;; (4.11) fOl' displacement field 1I.(x, I) lind the I.cm·perat.ure fipld 0.(.1'. t) into the functional II {.} of (5.:l), we ohtain the diseret.i;-:ed form which is identical to thatof [14]:

",{II., O.} = ~ r'iiiNM 'II;"" lqltrJ1J.~'{1 + 2 l1·r' ~Vj t} a~ - 2 }jx.II j;.lttl Iti"" ii·~tl + 2 'l's" i:\'i r} (:l~-- . 41_Ct R=O

(~) (~) ~. (1) , (2)... Z]- .• ,/,S" L . d' _ ,) '/'.\', (-) . r.} /,0 _ :) If. 1/'\'" 'I'i .Ittl e 'jI' _ ') 1I. '1'.\'_, Q.ll/l dO' .- ., '" - . . .\ \ '., - .\.11 i ,~,\ .1/ . ·.. ,1

.=0(5.4)

H{'re the funetionals of stress, enlmp)' lind heat f1nx for the elements cJl. arc defined in a similar mannerliS t.hose in (4.15); i.e.

0(, 00

'i~\~j {- } = J 'Ps.Ar) Jii [11;\'(1), '1';\'(1): II''', p'] t/-.JJ ,••. n JJ, .=()

~ 00

(-)x r} = ']'0 f ~)T:...(.l') J" [11;\(/), '1';.\'(1): ",". TS] dc71..~() JJ, .=0

00 00

:ii\" {.} = f T\- '(.1.) ai [11·\'(1) '}'.\'(I). It.\' '/'.\' (}.\'] dSi• & ' ... ),' r' r . ~ ~• .".f) foR. A'=O

(5.aa)

(5.5b)

(5.50)

The quantities 111,\',1/'PS.II' a~'fJ' a:, ii:p et.e. up[ll'l\j'illg ill (5A) are defined ill (aules (1)-(2).AMa final comment, we note that. certaill of the mixed or hybrid fillite-element. models of PlAN lind TONG

[7]-[10], bnsed on llELLJNOER-RETSSNERprinciple can also be obtained from t.he general formulat.ions presentedin Section 4 of this paper. The HELLI!\GER-HEISSNEll principle for finite deformations of elast.ic bodies followsimmediately fwm (4.24) by ignoring all t.lJl'I'I11111 effects, assuming each of thl' remaining mechanic III eonstitu-

3')

450 1>. R. BHANDARI and .J. T. ODF.S: Gcneral )Iixed Fillite Element )lethods of :\onlilwal' Continua

tive equations arc satisfied. ignoring ratn !('nus in (2.1) -(:!.5) (and, hence, II \'oiding the lwccssit.y of the 0Jlem-t.ion g*), lind setting

2 ,'iI (c7;i - aii) = - 2 B(a;i) .

Here B(a;i} is the complementary energy del1sit,y: i.e. if A (YiI) is t.he strain {'nergy fund-ion,

B = aii 'Yo - A .Incorporating all of these rest;riet.ions inlo (4.24), we obtain

Iln =f [- B(a;;} + ~aii (lIi.i + IIi,; + urn,; Ilm.i) - III J\] d~ - J~(l/.I - 111) 'l'i dS - j'Ifr; "l cIS .

.:JJ 8;//" &JJ'j'

(;'J.{i)Note thai, IIY neglecting fill' second order tel'IllS in (5.u), unci Mien subst.itllt,ing the local approximatiolls for"c(x} and 0c('.c}, we obt.ain f,jJe mixed models which nrc essent.ially the sallie as those of PIAN and TONG.

A number of other alternate variational models can nlso be preeipitntcd from the geneml principles pre-sented herein. Since t.he g{'neral proeedurl' for {'onst.ruct.in~ Iltese has b{'en adequately demonstrated, we shallnot elnbo1'llte further.

Ap}lendix A.The first ordel' local repl'esl'ntutionll of .'1. in (4.11) arc diseret.ized in the following forlll

IIc(.r, t) = liS ~ V'N(,l') 'P~(t): i'.(J·. t) = i's ~ Y'r;('l') ;P;(l)

o,(.r. t) = oS~ V;,v(.r) ~W); ac(d:, t) = [J.\', 1',V(ol') Jl~(t)

'1.('1', t) = fJs" FN(.,.) 1iW): !Ie(..r. t) = y.\' or. O..,.(.r) p~(t) (A·I)0.(:1', t) = 1'0 + 'l's" T,,,(..1') ".~(t): 8.(.r, t) = SS.' I' N("') h~(t)

1/,(,.1', t) = '/S" 1.",(.1') hW) .In addition to (A-I), we nlso approximate locally III" IJl'esl'fibed functions for displncenll'nts li(x. I). tradions 1'(,.1', I). tempera-ture 0(;1'. t) lWlIt flux 0(.1" I) of (2.4) and the heal, supply h(.r. t) by the followillg

il,(.l', I) = Its" iiiN(.r) q;,,(I); 'r.(.r,t) = -'ix., 1;(,.1') (~(f)

Oc(.,\ t) = ,rs., T .v(.l·) f~(t): (/(,.1', I) = 'i\'·, r.\,(.r) fi;(t) (A.2)

h(.". t) = 'I'\' ~ "N(,,·)~;(t) .

A rkuowletlgl'lIl1'ntThe support of this work by t.hl' V. S. Air Foret' UWee 01 Scicntifit· He~earch und.'r Contract .F 446:W,li9·C·0124 is gmtefull.racknowledged.

I BEST, n. C., A Forllluia for Certain Types of Stifness ~latriees "I' Structural Elelllents, AlAA J .. I. Jlp. 478-479 (lUli3).2 BEST. G. C.. and OIlE1\', ,). '1' .. Al1!1lysig of Shell-tYII<' Stru('\u ...'8, ~:ng. Hcs. Hcp. 157. General Dynamics, Fort Worth,

Texas, December (1963).:l PlAN, '1'. H. R., Derivntion uf Element Stiffness )Iatricl's hJ; Assumed Stress DiHlributions, AIAA .J .. 2, Pl'. 1333-1:336

(1904)..1 DUNliAll, H. S.. and Prsn;a, K. S.. A Finite EI!'llll'nt, Applil'1I1ion of Hellingl'r.lteissner Variationlll Theorem. Pro!'. 2nd

Confercnc'() ~latrix )lethods Stl'lIe. ;\lech .. AI,'I·'DL·TU.-6K-I;'O. pp.471-487, \\'l'ight-l'atte!'1;oll AFB, Ohio. Oet. Jil-17,1968.

5 PB~. '1', II. H .. Elemcnt St.iffne~s .\Iatriel's 1'''1' Boundary COIIIJllllillilit.j' and for I'rosC'rihcd Boundlll'Y Stresses, Proe. (;unf .on MatTix ~Jethods in Structul'lll )lechanil: .., AFFDL·TH-lili-HlI, pp . .157-·t77, Wright-Patterson AFB. Ohio. ] !lU5.

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7 FlAX, T. H, H .. and TONG, P., Basis of Finito Element ~[cthod for Solid Continua, lut. J. NUlllerieal ~lethods Eng., 1,pp. :1-28 (I!J69),

8 PUN, '1'. H. H .. llnd TONG, 1'.. HlItionalizat.ion in Deriving Ell'ment StH'fneHl>lIIat.rix by ~\ssumed Stress Approach, 1'l'Oe.2nd ConI'. Matrix Methods Struct. ~1eeh .. A 1·'FDL·TR·ti8-liill, pp. 441-470, Wdght-l'atterl>on AI·'B. Ohio, 01'1 .. 1\168.

!l TONG, 1'., and PIAN, '1'. H. II .. A Variational Principll' RlIll t·he ConvCI'gcnce of a Finite Element Method Bascd on AssumedStress Distribution, Int. J. Solid and Structures, ii, pp. 463-472 (1\J69).

10 Pux, T. H. H .. Formulat.ion of Finite Element ~Iethods for Solid Continua, PmI'. of the U. S.. Japan Seminar 011 ~Iatl'ix)1ethods in Structural Anlll.rsis and Design, Uni\'ersity of "\llIhal1la Press. University. Tokyo, 196\1.

11 ODES, J. T .. Finite Elcment./! of Nonlinear Cont.jnlla, '1cGraw·lIill, :\ew York (1H7J).12 ERIXOBN. A. C., Nonlinear Theory of Continllous :Media, :IIe(: l'/L 11"Hill, New York (19U2).l:l PUN, '1'. H. H., lind TONG, 1'., Val'iational FUl'llIulat.ion of Finile Displaeen1l'nt Allalysis, Syrup. II1TA)1 on High Speed

Compllting of Elastic Structures, Liege, I !J70.14 NlCKl>:LL, n.. K, Stress Wllve Anlllysis in Ll\ycred Therllloviscoelustie I\latcrillis by Extended Hitz Method, Tech. Rep.

S-175, II, Rohm and HUllS Co.. Huntsville, AlII. (1968).15 ODEN, J. '1'., Finite Element Formulation of Nonlinear BOlll\llary- Value Problems, NATO ~\dvllneed Study f nstitute on

Finit~ Element Methods in Continllum ~[echanics. Li3boll (1971).Hi OIlEN, J. T., and B~IANDAJlI,D. R.. Variational Principles in Xonlinear Viseoeillsticity. Int. J. of Solids and Stl'uetllres

8, pp. 1ll17-1026 (1972).

"

D. R. BHAKUAIU and .J. ']'. OUEN: GClll'ral lIlixed Fillite ElclIlI'lIt i.\ldhods of 1'\onlincar cont.inlla 451

17 HIU:\DARI. V. R .. and OUJo.::\. J. 1'.. A Variational Prillciplp for Xonlincllr Thprlllovis('oplasticity, lnt. ,). Xonlincar ~Iech ..7, pp. 480-501 (1972).

IS (lun.; .•J. '1' .. lind BUA"SDARI. D. R .. A Theory of Thcrmoplnstic ;lIatcrinl" with ~[em()I''y.. J. Engrg. ;lIech. Div., ASCE. 911.pp. 131-Hi5 (1973).

19 OWES. D. H .. Thermodynamics of ~llllcrjllis wit.h Elast·ic Hange, Arch. Bnlion. ;lIech. Anal.. 31. 1'(1.91-112 (1968).20 0\\'£:\, D. H .. ~[echanical TheolT of )[aterials with Elastic Hange, Arch. Ration. ;I[ech. Anal.. :Ii. lip. 8a-1I0 (1970).21 COLElIL\K. B. D., Thermodynamics of ~Iateril\hl with ~IclIlory. Arch. Ral·ion. ~Iech. AlIlIl., 17, PI" 1-4G (19G!).22 VALANIS. K. C.. A Theory of VisC'o(llasticity Wit hout a Yil'ld Surface-Part 1.. Arch. Mech. ArchiwlIlIl Mechllniki St080-

110.4, PI'. 517-533 (1971).23 VAl,Al\l!'. K. C., A Theory of Viscoplnsticit.y Wil,hollt. a Yiclrl Surfal'e·l'art II. Arch. ~Iech. Ardliwulll )Iechaniki Stoso·

wanej. 23, :'\0.4. pp. 535-5;')\ (HI711.24 \'AISBERG, ~1. ~1.. Varilltional Principll's for t.he Study of XOlllincar Opcrutors (Trans. from the 1956 Rllssian edition by

A. Feinstcin), Holden.Day. San Francisco (IU64).2;; (; eRTIN, ~l. \0:., Val'ialiolllli Principles in the Linl'III' Theory or Viscoelasticity. Ar·ch. Hation. ;I[cch. Anal .. 1:1, pp. 179- HJl

(Hl6a).26 OnEN. J. '1'., llnd R..ullHt;Z. A. G.. Formulation of General Discrete )lodel8 of Thel'lllomechanielli Behavior of ;I[aterials

with )lemory, Int·. J. Solids and Structures, ii, pp. 10ii-HlU:\ (19ti9).

Eingorl'icht am 27.4.1972

A.llschriJt: Prof .. J. T. OnEN, Heseal'ch Institute, The University of .-\Iabamn in Hunts\'ille. Hllntsville, Alabama :!580i, liSA

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