-.

" !Ill '" 1:'\ :' _I P .. ': "\ I'

11 ' " ",' ~ \

,III. ''1h

>t"c' _lfl,,1l

[ II~ ',\1':

'\ I 'III "'11\,' _ 1I1l:".Il.lli'1IIJ. ,Ill '1':[1 <'~ .. '.I:

"II,.:: ,Lllk,I":'1

( ,m:'II:.:lh'l1

S, N, ATLURI '* ".. , ','ec-..... '

"TRODL G IO\

" :":"': ~J ' " , ."::,.

" . , '': . I)

-

Computational Solid Mechanics I' (Finite Elements & Boundary Elements): Present Status and Future Directions '

In Ih.:: firs! p:H1 ,J ) this plP'::L .;ert:1I1l developments Jnd trel1d~ In .:ompuIJ· 1l!JIlJ1 m<.'thl)J~ themselvt's Jre t.hscussed. These mclude IOPI":S In fum .... ~Iemem Jnu 'luunuJry eio.'menl le~-hllologles (or solid mechanl":s: Ii) LBB ..:onditlon<; lor ~l'Il('rJI IlIl n.:: dl'rncl1I method s: Iii) ka ~l·order. s!lblc. 1I1\'HllrH. l>uparJmetrl":. ,lr~~,-t>J\<."d. h~bT1lJ IllLx.::d <!i<.'1l1ems : (I ii) use u! symholh: m:lI11pulJIIOIl (1\1 JJJr'Il\':> 'l1('sil re l lrll!menl. (VI tr:lIlSlent dynamic respnn~: Jnd In) bound..lr:-· <,kr~l<''11 :ll<,!htld~ lor lineJr dJ~Il<.:il\. JS well J5 tor linn.:: -ITJ1Il ?whl<'m) 11 1J1(',J,t:, :n.U':T1Jh. In th,: ~,:~ono part u r the papn. ~ertal:; tOPt~S m Ihl! rnl!~hJn: '(' "t' ,,)liJs. wi1<.!relll (U!llputJl1u nal methods h;.lve pbyed. Jno jr..: exp~.::eJ :0 "IJ\ '1!!lllfi(J1l! rok" Jre JI~..:usr.ed. These mdudl! (n cUmtllUll\e :TI!)Jl!i· Inc ,\ IneIJ)II'; mJtertJI I)du\"lor. (I i ) mechanICS ~) I lIleiJ)tl( Jnd JynJrr.lc t"fJdUrl!. tlli) n\)niln<'Jr contmuum me~hanl~sJnd )hellthem~. Jnd It\)strtLdUral ,ulltr"l. tll\'olvll1g ~olllr()1 ,lI JynJlmc response ot' brge 'PJ(~ Jnd ':Jrlh )tTU~' IUf,"

~t l ~ H 1J ~ ( ~~H ~ 7[,& i W ~ 7[ I

lJLJ? ,& * * ~~ Ii jJ rtiJ

•

" "

•

I.n , " ," ·<.'1\'

-, ' i

.. - __ ,', lit _,!:.! LBP ;-~ ..:~'f'

,',ttL':' ~ 6.:£ f. "Ii

'-

!! 1;1, '. , "

,[:.1:"111'\ ,'IIIJI1I<1I1, lur -;cn<'rJlll1l11,:.:k:1\"1I1 "h'[h,),h,IU!

','J', '1,1," '\Jol.:, lIl \ 'afl.l lll. h~omJ 1!1!'(<'u hOt"JfJP·,'t::,

'i..:"'I'!lI" 'J,.:<1 .)1) J~,llm,'d 'Ir.:\'i~'~, j, Jilar~~ 1-' 'h,' hllJI !ht"i.k.-mcllt hJst'u IS(\p.IIJmc'!r;~ ~1<'111<,1l! r",1

,I \111"nll( IllJtlIPUIJll'lIl til) JJJro11\1! :1\<,,:1 ~!·Tl·

':",t , '11. ~,', , II J 'UII~lent Jynal11k fl"r'U\'-l' ;nJ 1\11 "'''Ln\.!. \. "

.1 j JI':.'

,"

:,>:~,0111 :ll<.'tlhHh 1m 11Il11<' ~trJm In,':J'!k pr,'hknh hi

"c',';·>l1 ~,\<' JI',.:u" ,'enJIll IOPI,', [11 th,' 'lh',:I.m:.

'I ." I,h .. ~llI~h 11:1\<.' h.:,·n. Jnd Jr ',' 11\..(:\ 1(1 :,,' ,'Ill"I' • ..! :":1:;,111,\ llruugh the J<.'\'.:ltJplll<'l:t II dlmpIIIJ;. tlLl1

"LIJ lh',h.ITH,·,. ~.Ii!1e-J ;H::nJr!j~ III "kll <1, Th.::>1.' mduo<, (I) ~OIh,:tull\l' Ill"J,: :!,~ I]

-ilJI1'Il.lII,'Tl ,\IIi! h',' 11:\1 'I\hkl~h ilL.!

,II :h<, (\111..:: ','r tlh' \ ,11',111,,,"1\,'1'1 _:

\ 1",:\'111

. I ,)

Ih,"J_;:, IIJI<'nJI n.:hJ\[o r. (11) 1Tl<',:IJIlI" ,)1 :1;~[J"i,

IIiLI \\tlJlllk I rJ.:ture liii) nUnlineJf ~')Tlltllllllm '1\,'.1:.1111., .1 ,..:

"tl",I>.:l.l,', ,11 h -·d'Il1I"I'.j:.IlJJ :1'~.IIJ" "un:"" c m lPUT \TIONAL \lETHOIlS II .lIt"U'I'I'" "i"ll.Ii '<',',,·II':,.n [it,: 1 1I11~.\ \IJL'

nIl.' .,lIlt,',\ tI h,,· ::'IJIIIo.1.', " h..: rJN" If.'

t"ll'.I. In "" 'lUll ~ ,1<' .h,:h, .en.lIll ·..:nu, ,Ill,: ',::"

0.1"\"]"[,111':111 .!l ,'IlH"llIJII"llJl 'Ul,th"<J 'hcf1\'<'!\'" II I

'J :, 'nJi Con!," ~k~ ,11 fI'~ :-:::,J: ;Ih. \ [':Jllt:..; \1c ,:~,J!I;, JI LlIlJIl. I{

• • 1,1<' \1c: Jl" ,I ... i \ '.1

'''-<'''' , I, ',;111,' " 'J .. ,'. "11" .. . 'J,,..:

lie .. ·.• ,\,' Ji::..:u~~ rt'ct'1l1 Jd\311~~'i JnJ 'r,'II.j, n '::,:1<, 'i<'~~,":1 .11l,j nnundJ r\ <,I<,mclll t<,dlll(\iul!l<'~,

.. Fitlll(' - Elemenl T!!chnology

,n 'he rnii"v.mc 'P\'(j]~, ' :'1,

'Ii ! L.JJ"/hl'I"",I\J BJh'''''J· lk,·1!1 \ ' It.., , \

methods: Oi) element development based on non·standard (hybrid/mixed) formulations. to avoid klfiematic deforma· tlon modes. etc.. (iii) use of symbolic mampulation; (iv) adaptive mesh refinement: and (v) transient dynamic analysis.

1. 1. lB B Conditi ons Consider. for Instance. a linear elastic solid undergomg mfimtesima! deformation. Let the wlid. n. be discretlZed. for purposes of generatIng an :lpprmumaling solullon. mto a finlle number of elements Urn 1m '" L.;·n. [U" ~ n J. and a nm be the boundary

m en of nm In general. anm = Pm+Sum + Stm. where Pm is

the inter element boundary. and 5tlll and ~lIn are those segments 01 a nm v. hlch :Ire oommon With the external

boundary segments 5t (where tractions are prescTlbed) and Su (where displacements are prescTlbed). respectively. As stated m [I J. the govermng equations :lnd boundary condi­tions fo r the limte·element am'mbl~ are as follows:

FUIU (a. E. u)ET x E x V .. uch that

( I .a)

(I.b)

(I.e)

o iJ nJ .: II ,,\ I JI Stm ( I.J)

a IJ .J .j. I , • 0 m nm (I.e)

a, ) • a ) ' m 11m (I J)

• ", • ", " Pm (I.,)

( a I ) n J) ~ +W I ) II 1)- = 0 JI P m ( l.h)

where Ihe 1I0men.:iJlure IS JS tollo\\~ a iJ (stress tensor); Elj htrJIIl L u l (dlspiJ~emellll. (+) Jnd (-) denote. ar tlllTartl~. the two ~ I ue~ vI Pm Eq . (l.g) IS the tntere!· emem dlsplacemenl wmp:lllbdtty ~ondlt1on_ while (l.h) IS lhe Intere1emelll .. tracllon-reClproCllY" condl1ton_

A~ ~h()\ ... n III [I. :: \. Eqs. (I .J to I.h) may be treated as the Euler - Lagrange equations and natural boundary (.:onuJllons of geneT:lI \arlatlonal prtnClples in four different wa~ S H summarIZed below

CJ:.C Let the lTlal fun~llon u l obey (I.g, a prion. The variatio na l prtnclple governtng the remalllder of lhe equallons may be ~llted JS'

li[Llu.E,al]" 0, where

LiUE.a) = ~~J [1-:E1JkIiEklEIJ 11m

+0 IJ 1U l l .J) E IJ ) IIUllu\

+ J S urn a i j n j (ii i - ui) ds

(' )

Case 1: The Irial functions ui, ai j . and E i j are 111

arbitrary. We m~roduce Lagr.ange multiplier s f IP at

Pm. such that (Tip)+ + (Tip)-=Oat,vm' apnoti.

The variallonal prinCiple governing (I.a to I.h) may be

5tatedasOL(u.E.a.T p) =O.where:

+ Js 0 " n-- (ii - u)ds um IJ IJ I I

- f5tm

ti ui ds - JPm Ui Tip ds (3)

Case 3: Here also Ihe tnal funcllons ui. ai j. and €jJ

are all arbitrary. We Introduce :In Independent. ullique. displacement field uip at Pm and enforce the lnterelement

displacement mmpallbility condillons, ( u+i = U ip ) and

(ul = U ip ) at Pm. through arbllTary Lagrange multi'

pliers lip and Tip on either side Pm' The V'JT1a1!OIIal

prtnciple in this case becomes:

liUU,E.O.U p, Tp)=O, where

(4)

Case 4' Here again the trial functions uL OJ rand €i J

Jre arbitrary. In this case. which is a SimplificatiOn of Case 3 above. Inasmuch as TiP and Tfp are arbitrary on ei ther

Side of Pm. we set liP = (oi j nj)+ and lip = (oi Jnj)-Jt

Pm' The vanationai lprinciple in this case becomes:

li L (u. E. 0. U p) = O. where

( 5)

Note that in Eqs. (3), (4), and (5), the integrals over the e!ement~volume, n m' are identical to that in Eq. (2) and hence are not repeated.

The variational problems involved in (2) to (5) are multiple S.lddle~ point problems and represent the most general bases for formulating a wide variety of finite element methods such as the mIXed, hybfld. and mIXed~hybrid as well as the compatible displacement methods. The question naturally arises, if the multifield variational pnnclples as in (2 to 5 ) are used, are there anv crite·T1a that one must use 10 Jpproximale each 01

th~ several vaTlJbles (ui' €ij, Oij. Uip,TiP,Tip"et c. )

In each element. such that the finite element solution eXists and is stable? Even though a wide variety of hybnd and mIXed elements have been developed over the last 20 years [see Rei. 3 for J comprehensive summary], several of these were based on t rial and error and were plagued at tLmes by lack of stability and appe:nance of zero~energy (kmematic) modes (other than the flgld­body modes) al the elemem - leveL due 10 the lack of mathematical answers 10 the quemon posed above . In this context. BrezZl [41 prescmed what appears 10 be a ploneermg study on the existence. uniqueness. and stabi­

lity of SOlUll0ns to a variaIJonal problem with a single constraint. i.e. a vartational problem with two vaTLable fields. Later, Ying Jnd Atlun [5J extended Brezzi·s work to slUdy the problem of J hybnd - finne - element solullon of Stoke·s flow vf In incompresSible flow - a problem which Involves three vanable fields and two Lagrange multipliers. Recently. the eXIstence and stability condi· tlons for discrete solutIons based on the most-generJi multiple saddle-pomt problems 01 solid mechanics. based on Eqs. (2 10 5), have been systematically explored [6 to 81. Here we present these wnditions for finne element soJullons based on the variatll.mal principle m Eq. (4) which. m J way. IS the more general of Ihe four cases presented Jbove. Thus. finne dement >alUllons based on Eq. (4) eXlSt. Jre stable . lnd converge. provided vnly the follOWing ..:o nditlonsare met

Sup

V v € \' rn J Pm vi tIP;;' a

\' ~ V

j3 II v il V V v € (ker (C) )'

Sup Ve f

It ·1 Vlp€Ker(D)(t'-l b ) P T(pi

(0.":)

where

(6J)

( Ker(C))' =!v € V ,~Jpm Vi tip ds = 0, V tp€ Ker(D)j

(6,g)

(K"(P))'= IrE T, ~ lIn r '(' ') d, m m IJ l.J

[Ker(L))' =j€ € E, ~ f €ij Tij = O. V T € (Ker(p))lf (6.i)

When the constant 11 in the above relations is positive and independent of the finite element mesh used. conver· gence of the solution is achieved. The above conditions are somel1rnes referred to as the Ladyzhenskaya [9J. Babuska [9), and Brezzi [4) conditions. The reduction of the above global condit ions 10 an element level is discuss· ed In [5.6.7,8j.Tt IS important to note that while Eqs. (6.a - 6.i) Jnd theu local counterparts. specify the conditions to be S.ll1sfied by the respective fields approximated in each nnlle element, they do no!. unfortunately. give any clues as 10 how one should go about choosing these fields to sal1sfy the stated cTlteria. ThiS question has been a subJ­ect of much recent study [10-14], and some frulllul results are summar IZed below.

1.2, Elemem Development Based on Hybrid-\lixed Formulations to Satisfy Element Stability Condi­tions ::md Avoidance of Kinematic Modes We mall consider two specific examples: (i) a mIXed finite element method. wheretn the fields Qjj and Uj. satisfying the strain­

stress relallon. and the Eqs. (I.e. l.f, l nd \.g). J prion are assumed tn each element: and (ij) a hybrid fintte element mel hod w herein an equ Ilibrated 0i j that satisfies Eqs.(Le::l nd

\ .0. J pnorl. is assumed tn eJch n m . and an tnterelemem·

..:ompa\lble displacement field iiw that satisfies (I . ..: Jnd

l.g,. a priori. is assumed at a nm

baSIS of the above mIXed method Eq. (2) \0 be:

only. TIle vaT1ational

can be deduced from

( J )

Likewise. t he vaTlallonal basis of the hybTld method mentiOned above can be deduced from Eq. (5) to be:

( " )

wherem U IP is assumed such Ihat U tp : ui at Sum' and

~ el) oil okQ == W I.; (0) IS the .;omplemenlary energy

density According to [he geneTJI theory for eXistence and

stability [6,- ,8,]l ndicJted in Eqs, (6 ), the finite element soluILons based on (7) eXIst and are stable If:

~fn 0 1J vlJdn Vo€T nl m ' ;.,3 vDVVv€V,,;o

'0 1) T 14,:1 I

Jnd\\'~,IOIJ,olJ';,a a TVo€kenGI

""h el<' 0., ~ Jre ?n~111\'e (Jnu mdependent 01 mesh I, Jnd

(9.d\

":OTTe,p\l!lJl!l~ ~(\ndIPons for bJ~('d "n 1 'II JT('

1111l1(, eicmcrll ;,olull()n~

'vo€

"her('

\- , t \ I .<'

T,"'jr1J c'H rr!m 1 r iJ,J""J

W IJ - 7'1· I)) Hl Q m V m I JnJ \'~() lSJ~J,'IIIll'J tnl'IJI

( IO,J I

i lO,bl

t IIh l

IIO.d\

In :he :\\":IJ 'nethoJ ~J~L'd 0111 \"1\. 11 11 is pos~lhle

to ':'I<' :~<.l ~:p 'cHm Pm UllIqU<'I\ ,nlll rlm. 11 Ii rm111edl'

J[e!\ ,e,'n 'h:H 'h .... .:ondllilln ,10JI t-e(omes ~~nonYIl1\JU~ "lIh r<l J 1 B,'ln ..:nnJalon:; lmpi\ thJI

Soo V r} E T(nrT,,)~ Oq\(L.)\>OV\ E V..:o !J 111

( II I

le! 'lie JU!\o.'mI\JI\ ,11 01 T" .11 1111 he HI Jnu Ih:l1 1\

\ '0 '"Ie':1. fil<'!l 1III :!lll"'l1es !lUI ill n JIlJ thJ! tho.' fJn" .)J 'I\e ':lJtr;\ [3 III

J

vl B o: r.J O"V(' ·)dv - - - m 11 I) I,)

m (1 ~ ., )

mould be 'n'. Since vi" ° at Sum for veY..:o , It can be

seen that Vi appearing in (II) do not mc1ude any global

rigid body modes, assummg that the original bou ndJr y conditions at Su are such t hat they preclude any rtgld

mOllon o f the solid as a whole, However, e\'en while global rigid mOllon may be precluded, such rigid motion may be ~o,nsidered at the element ,level. T~u.s. v0.Jl '" ° for 'r ngld modes (r '" 6 for .>- D and _, lor IWO dun<!ns' 10r:~ ) for each element displacement. v m, Furthermore,

a! J IS arbitrary and mdependent for each element. <,x~ept that wllhlll e'J ch element, am e To. Thus. (II) m3~ be wrmen as:

Sup ¥ a € To ~J un VI'!', dj.»0 '" V € Y '0

III nm I) " ... \ I :.bl

where ~d denote non'T1gld modes III each element. A suffiC ient cllnd Lllo n for (I ~,b) to hold is:

( 13 )

Lei )1 ~ be the number of stress modes 3ssumed III

<'3lh element nm, :md leI Nqbe the numot'r oJ! dispb .. :. n md

ement modes III e3ch Hm. Then Ihe dtm<!nslOn 01 Vi IS

(:\q-r). Thu s.

( 1.1 )

Thu s. lor (13) [0 hold. )1{3 :;.. (;\q_rl JIlJ ,he 'Jnk

01 !.J~ m :)/lOuld be (Nq -r ).

Rem3rk 1 If We (Q" , q) III e:a~h element. \-' OE T\).denol.

ed lS Wc Iq, ql . ..:an be wrmen as:

m W, (0, 0 1 '" a mt Hm

" am

( 1:' )

then 11 .::a n he ~hown i 15}lh31 the demelll >tl11ne~) I'!lJt!IX t..m ":3 n be wrLllen as:

k : Bm H \ Bmt _ m _ _ ( I b I

"here !.J m IS defined 'lhrou2h

~ ,- I

Here. ~m mcludes both rigid and non-rigid mOOes.

and Jim (-::m):; Nq . Note lhat the rank o f ~ IS the same

as thai 01 B*1n Smce Wo,;m (0. a) IS poSlilve delinue.

II follows IhJI Ihe rank of Ihe element suffness malnx

IS (Nqr I. provided Ihat 01 ~m i~ (Nq- r).

Remark ~ '\IOle that both 0IJ Jnd vi In \ I-n are compo·

nent s In the '::lTleSlln system '<I The momentum olilnce

o.:ondlllon Involves Jllferenuauon 01 0lj W.r.! X1' while

the ~lTlHIS v \LJ)abo Involve dllt'erenlla\lon W.f.t '<J' In

thc usual lsoparametm: dement formuiltlon. the geomel·

flCl1 transformation between the (no ndimenSlonal) ..

p;Hent" ekment Jnd that In th..: ph~slcal Jomatn IS '<J

"'Xl t E k) ..... here EJ... usually laken 10 be - I";;; Ek ..;; I. are

o:uf\llinear ..:oordtnate~. In J .. hsplacement iormulatlon.

one usuall~ aSsun1<.'S u\ = ul (~l. Jnd In an isoparam etTlc

-r eprt"se11lat Ion t h ... repre>entatton ror Xl J S well as u tCO nlal1l

all eq ua l number L11 bhlS fun..:tl\l!lS m ~k The stillness

matflX .)1 the d ... ment In Ihe Jlsp!Jcem ... nt !ormu!Juon.

whl,'h Jepends un W( th l aXk l ..:a n be IDo ..... n 10 be ob·

Ject l\t' o r ()b~ef\er HWar lJnt --ThJ I meJIIS. I t km IS the dement stlffn ... ss malflX In

I n ISUp;ITJlIletTl': dlspb.: ... ment tormubuon In th ... XI .:oor·

ditute ~y~t ... rn. then liS r ... pre~ntJIl\ll1 In any uther o:afte~lan

s~~I ... m ~. = q~ IS given h} ~'m = q~'1l ~T ..... here 9 IS

lllthll~onal

To malntJIn the Obj"'CIlVlI} .)1 the dement ~lLlfness 1l1alrLX In J mlxed·hybrld formulallon. 11 has been ~hl)wn

[II-loll that the ,ITeh tensor. q. Jlou!d he Jssum ... d 111

In ... lem ... nt·I\)~al .:o\)rdlnate ~ystem Jnd .!!2.!. III J glob:.1i

.:oordln:at ... ~~~tem

Ilrlll... el ... m ... ms "t :;quJte Jnd ~unlc (or re..:t JngulJr Jnd r"'dJng uiJr pth1l1t ,hlp"'s. re~p"'~llvei\ Here. the Ih~'H\ ,I t ,ymrnetrt,' ~fllUpS ha., h ... en demonstrJled [11 .1 .. '1 it) 0 ... J u')<!tu! tooi trI _hll')'il1l~ i.:J)t-order ~trc)) !1 ... ld, ( \l iJ = " 4 /I that I"'Ju 10 Ih ... ·lllalrLX ~",m l see eq IJ)

tlr ran k ("4 rI. Jnd a ,ulfness malH" ~m. Whl..:h IS nb·

leC\l\'e and JI\\\ <I t Jnk ".:j r l In th\) ..:aSt'. J ..:anC,lan ":llll1dlllJt~ ,ntem lo.;ated Jt the ~entrtlld 1)1 Ih ... c'l ... m ... 1l1

Jnd Jlllng Ihl' J x ... ~.1\ ,ymmt"l1\ III t he eh~m ... nt. I~ u.,ed In

[11 1-lI.bolhOIJJnd\I.J[( 1.]lmplytngiH I i,,] IJre

J"'Cllmpo,ed 1IlII) l1lvarlant Irr ... du":lblc ,pJ..:e, uSing .;rllup IhellT\ In lerm'!! III IheS<.' Irr ... du":lbk repre:.entJIHln.,. the maITI'. '\.3\ I B-*) ' . .:orrespondln~ 10 Bl a. \" ) IlU e:a..:h element. b ... .:nme'i "4UJ,1 ·JIJl!.onar: nm s. group Ihellr~ ... nJol",., ,HI'" to pld, 0 1] 11l ~J..:h ... Il.'men!. for J gl\en \\

, u..:h thJt Ih ... rcsultmg elcmem lormulJuul1 IS tn\l,lrtJnt JIlO'IJbk. It ha,h ... en;huwnll: 1-l]Ih.:tt (I)IOIJIOUI ­nllu ... J :>quar ........ I\h :-':q - r '" ~. Ihere Jr ... 1 ..... \1 po)slbl ...

.:h\)\~~, ttll J IlW-paIJmet ... ! c'qulhbrat~d ,tr ... ss lle1d (HI :m In ~u!.ht '1Od ... J "<lUJI'" v.uh "'q _ f '" l.t. there

Jre ':1 ..:hOl":"', ;,H J 1; -OJrJlllel"'l ,tress li~ld.lll1 ) I.l! In <!l\!hl lII.d ... d ..:tlh· th~!" Jr ... eu::ill ..:hol ~ il)T J "r ... " h ... ld With \1 3 = I .... Jnt.! 11\1 t'li J 20 l!uu ... J .uhe· dl'::lc'

are 384 choices for a stress field wllh M(J =54 ; all of which

lead to stable and objective elements. The 'best' selection among all these chOIce s may depend upon : (j) Ihe lowest

e lgenv:alue o f the matrIX ( B·m) ( B*ml) ( B*m betng dcflllcd tn Eq. ( 14) 1 and (ii) the capabililY of Ihe ca ndldale stress field to represelll the cardinal states o f stress of pure tensIOn. shear. bending. and lorsion in each eleme nt. A ..:omprehensive study at such tests is gIven in (I -:. 131·

Remark -l . Consider a mIXed hybrid element 01' a general curv Ll mear shape and tntroduce a geometric mappmg ot'

the type "i = xi( ~k). wah - I ~ ~k "I. Lei ~k (~m) and I!~ (~m) be the covana nt and ..:ont ravanant base vectors

. respecllvely. of the ..:urvtltnear ..:oorduut es en Let ~k represent the ,;;ovana nt baseveclOrs at the centrOid. I. ....

£k = .! k t ~m = 0), and lei ~k be a ..:arlCSlln system Jt

~m = O. then. II has been sho .... n [!~ 1.J1 that requlIe· menlS 01 in\lanance may be mel b} represenllng the str~~s tensor 11l t he alternall\'c !orms:

O=O Ij(Xk)~i~j' a I) e To ( I"> a)

=OlJt~k)~i~) . a Ij € T ( I~,b)

=0 t~k ) glg). IJ ~_ a'J , T ( 1 ~.c)

= olj ,~k) ~ 1 ~j , a' J , T ( 1 :-'.J I

=al]\~k)~ 2 1-::] . 01j e T

Olher possible representatio ns are discussed 111 [ loll. In l ib) To IS the spa ce 01 v.hue T IS that II! differentable ,tresses. It IS )<!en Ihal (18. J and 18.b ) can eaSily represent stat~~ ')1 .;onsta1ll ~'Hess tn the ..:art eslJn ,:oofdlnate S}Slem Jnd h ... ,ll· ... _In pass the so ·..:alled .. ..:onsI3111 stress" pah'h

(ulhIJ ... rmg J ~1ale III ..:onstalll stress. :>.I). Q = (II

~1 ~J . v.here (I) He o.:01lStJ1II S. it IS se ... !! \hal re preSt"!II·

Ilio n ( I~.:) .:In pass the pal(h lest If 0 1) (~ I tndude~ tun..:t IOns suo.:h thai

tJ9.a I

SI!1( .... til an lsoparam ... utc 10rmuiallOn. ,a'm ap, 1~ J )Implc pol}nomlJl til r . 11 is pOSSible. In genenL

thai J polynorlllal repr ese ntat IOn ~X15tS for ail l~kl tn

( 18 . ..: ) whICh passes the patc h tesl. However. th ... mess t icid ..... 111 nOI be. In general. nt the "least-order"

Ot! the u lher hand. t Ih.d) ..:an pJSS the pal..:h lest :i a I) I ~ k) tndudes fune! IOns such I mt

a 11 1 ~k ,= Cmn(~:n.~I) I ~n ~J t (~O.J I

a t' a ti ",e rnn --aXm aXn

For the usu:lI isoparame tric formul:l1ion. It is secn

that ali I ~k) of C~O.b) 3re no longer smlple polynormals. Hence. represemat lOn (IS.dl wilh po[ynot1llJi funCtions

a ij i~") \\lll nol. In gener:il . pass the patch test. However. (lb,el will pa~s Ihe pat~h lesl. ~mce . in this..:ase.

\ ~ I )

whi.'r~ limn Jr.: ~lllhtJIHS . Jnd hct1l:C J sunple / .. ,>,en least­order) pol~normJI repr ~~en[Juon \m duJing ":OIlSlao\

terrns l \\!llsut"ii..:e (or O IJ I~kl.

Rem:.H" :) To formulate In lsoparametric curvilinear n1l.xed.hybnd demem. o ne rna) use JltcrnatJve represent­

ations for )lr~S' JS III (l8.a 18.e) Jnd JSSUTne "i (~k) to be III ihl! ,,;.Ime (orm 3.S '(i(~k). NotethJI "iJrecartes­ian ,"omponcms or displacement. For the J!tern:Hlve repro:s~lltJll0 lb ,)! ~trc~S::lS In lIb), the bilineJr form B( o. \' 1. r'or .!Jch element . take s () n the respe(\!ve represent-31 10n

:J nmOmnl~k.hLk. Jk/J~:J~/(det J)d~l d~d~3 C.:!.:!·c)

J mn I:!-. : nor .. · \\lnJ Idet ~~11l . nn (~~.d)

v.lwe I I.m Jo!note~ 0 ( ) a ~ . J mJ = (a '\ m a ~J) Jnd

:1 m): JmJI~k.=O I.

In R<.'mJrk.3 ,:olh:ermn!! >qua res Jnd ~ubes. J group ­IheorellC:li method which .!nables a choice 01 a Ij (Xk). for

J gl\"('n vi / xk). tll:l! gives the ra nk tN q - r) to B"m was

descnbed For such squares and cubes. the bilinear (orm IS ~ompuled uSing (14). Compa rlllg (l .. Q Jnd 1.:!2l. It can be seen lhal there exists no sunple way of choosmg the stress 3S m (18 ) lor .:urvlline3r elements -;uch that the rank ot B"m is delermtned :l pnori. However. it has been demonstr:lted 1Il[1.3.14]that If 0i) (~k).LIr olj O~k) of

(I i'.>.b-I b.e ) IS chosen to be 01 the SJme polynomial form

(i.e. by "eplaclllg "(k b) ~):iS that 01 0ij (xk) which IS Jd!WU ~y uSllIg ~roup theory lor squ:Hes and cubes. th<'n the fanl-..: .)f B*1l1 IS mallllal!lo!d 10 he (Nq- r ) even ',n .en ,evereiy Jistorted eie!1le!lh. Funher. 1\

11J~ -ee~l .. i<'ari\ JeTll(llhlT:lteJ tlut :h~ leJ~t .)rder.

invar ia nt . isoparamet ric, curvilinear mixed-hybrid elements are less d istortion- sensitive and lead to more accurate results compared to the standa rd d isplacement elements III a variety of examples. Methods for suppreSSIon of zero energy modes m hybrid-st ress elements. based on heunst\c reasoning, have ;tlsa been mdependently suggested recently in{161·

1.3. Use of Symbolic Manipulatio n In recent YellS.

there has bee n a surge in research activity. likely 10 burgeon tn the (uture. in the area of co mpuler symbolic manipu. lanon: (i) in the evaluation of element stiffness coel"fiCients for HOlte elements to elimillJ t e errors introduced by numerical quadrature and to improve the el"ficiency 01 the generation of relevant element propertleso. (ij) to compare the performance of different elements and to synthesize desirable elements. and (iii) to capitalize on the symmetry and/or other properties of a panicu!ar elemem or group of elements 10 generate thelT chara.:ter­istic arrays in an efficient way.

The computer symbolic manipulation systems MACSYM A [17]. INTER. and FO RM AC have been used in several studies. Ref. {181 used MACSYMA 10 gener:lte the stiffness coefficients of finite elements as funcl ions of the specified material and geometric parameters while earlier studies (19] employed simple algebraic polynomial mampulators. The manipulator INTE R was used in [.:!O] while FORM AC was used in [:!l]. While most of these studies pertam 10 linear problems. the use of MACS't'MA

in nonlinear finite element analysis. wherein the geometnc nonline:H1ties lead 10 cubic and higher-order Sllffness terms. has been discussed III Ref. [2:!] which also presents a sununary of studies employing computerized symbolic manipulation up to 1980. More recent studies emploY lllg symbolic manipulatlon have been presented by Park and colleagues [23 - 25] who used 5lJch methods for J Founer analYSIS of spurious mecha nisms and locking III the (inlle element method as well as to construct a rank-5lJITicient. one-pomt integrated. four-noded plate- bending element based upon a discrete Founer analySIS techmque by which the uncoupled discrete o perator governing the transverse displacement is directly wmpared wnh Ihe corresponding contllluum operator.

1.4. Adaptive Mesh Refinement The economics 01 tinae element comput;ttions dictate that a given accura­cy be achieved with the mimmum man-hour as weI! as .:omputer costs. Convergence of fimte element resu lts is usually sought in several ways: (i) h-convergence.

i.e. retimng t he spatial mem while keeptng lhe order 01 interpolation in each element the same.:md (ii) p -.:onver· gence. i.e. increasmg the order o f function-mle rpolauon in each element while keeplllg the spatial mesh the same. or (iii) a co mbinatlon o f both. In any event. the IOta I number o f degrees of freedom is progressively increased.

Early studies [26] focused on the determlllatlon o( o ptunal finite element meshes (h-verslon) which mml' mize the error of the fimte elemem solulion ior a given number of degrees of freedom. The cost 01 this optlmlZ' ation itself is often prohibitive. :md pracllcal "gUidelines" (:!7] are often used to generate efficient meshes. An lVenue of research 01 greal promIse. that has been recently pursued and one tml is likely to rece tve much Jltenllon.

C("Inc.:,,~ ad~p\lh' letl;;ement of fimte element meshes [ ::!lS ~91 fhe mLsh ,efmement (h or p. or a combuutlon :hdfo\, IS .,d:lptive In ,he ~nse that each step depends UII :t.e 'nformatlOh con ... erlllng the error mdicators and "( torrl){ "stulI.tlors prov 'dt'd by the previous ones, The Nub:.:,[, hI! srlf adapllve (:!S .:!91 In the sellse thai ilL Usel n,cr .. ctloll l~ ne..:,;ssary to trigger the adaptive mesh re!:n<:ment process.

1.5. Tra nSien t Uynamic Analysis The standard approach 10 recent years for transient dynamic response Lit solids. and transient analYSIS of solid-tlUid IOleraCtlon problems. has been 10 first discrellze. in space. the govern· 109 partl:ll different III equatIons. In space and tllne ..:oor· duutes. to obtalll a system o! ..:oupled (nonlinear) ordinary diffcrenllal equations 10 lime. There have been Significant stndes made In Integrating these time- differential eqw· tlolIS uSlIIg impliclt. expliCit. and mIXed forms. ImpoTiant results have also JPpcared recently concerning operator­splitting and panllionmg methods. For an excellent sum· mary o! these Jnd related tOpiCS. the reader is referred to J recent monograph [30]

2. Boundary-Element Techn ology Since about the early 1970·s. the applic:Lllon of weighted reSidual methods. wherelll only boundar)' resldua.is enter IIItO ..;onSlderallon. ha.\e re":~lved Ihelr overdue allenlLon \ 31 33]' Contrary 10 ..:bllilS often made III ..:ommercI:LI

sort ware - vendor crrcies. the boundary element method is nell her "'different" nor "better" than the tinne element method-II IS. III J way. a (inl1e element method If the modern :Jc:Jdemlc vIew th:lt JII discrete methods :Jre somt (orm of weighted resldu:J1 methods IS taken. ThiS IS. III fact. Ihe VIew that led to the lII\ToduclJon of gradwu courses IIIled "Finlte Elements. Boundary Elements. and Other Computational \lei hods III ~teclunLcs" IIItO the currH.:ulum at G<!orgla [nslllute of Technoloogy in 1<)--

There Jre "everal ways HI which J so-..:alled bound:lTY <!lemerll method rna) be deVised \3:]. Two ot thes<! .HC ba~ed on (i) the use u t J "SlllEula.r" solution to wher th<! ~nltJe differential <!qu3tlon \)j- the problem or to the dL1fcrenual operator of Ihe problem th:J! ..:ol\taUlS the tughe~1 order deTlvatlves. as test !UIICILOns m the weIghted resldu31 approach. Jnd luI the use ot asymplOllc solullons to Ihe differentul cqualLon ot' the problem lor )Impi<! domams such as seml-mllllLle domaills. elc .. as \TuJ tunc· lions III Ihe weighted reSidual approa..:h. These are hrL<!tl~ dls..:ussed belo .....

Consider the prohlem ,II linear 'sotroPIC elastl(Ll}. Fllr an appmxlm:Jte )OlutLon. consider a global ITJallUllc· 11011 Uk (Le. valid over the entITe domaml Jnd J glohal te'St fUn(tloll vk' Let the ..:ompallbllity condition and Slress .. stram rei:Jtlon be S.lllslied a pTlon. i.e ..

0IJ(u\.,)=Eljkr u(k.\:)

"d

°IJ (\\.,):EIJk\\(1.; \\ '"

n

n

wht:r1;'111 a II

't ~ 11 E I Uk) Implle~ I he .tress a q JefLved from U \.;..

.. I ~:l' th~ "U;II ?r"!,en\ El\.;.I = E";"I

::;: E k 2 i j, etc" the stresses in (23) and (~4) are symmet ric tensors and hence 53l1Sfy the angular momentum balance conditions of the noo- polar contmuum. The only other condl1lons to be met by the trial function Uk are:

0ij(Uk).j ,

fi '" 0 '" n (:!S.a)

a 'J oJ ~

" " s, (25.b)

U ~ , Uj " Su (:5.c)

The .... elghled reSIdual forms of (:5a- ~ Sc) are

I ~6.a.)

I ~6,b)

and JS

(ui - uJtj{vk)ds'" 0 U

(';b.c)

where Ii (vk) '" OlJ (vk)nj' USing the divergence theorem

III C6.J) . one Cln "add" Eqs. (::!6.a-~6d to ob taLlI a

summed weighted residual equJtion:

SIII>:I! Ihe mJter]JI is linear. we ha\'e:

0IJIU\.,JVII.J) '" EljUU(k,~)v(i.j)

'" a U. (VI' U (k.~)

()SLIIg t ~8) III (:!7) Jnd uSing the divergence on the 'Iolume IIItegral. II IS eas)' to obtain:

-f Su

'"

(~8)

theorem

(l9)

frz !Oij (Vk).jUi + fiVild!2+JantiiUJ..,)v\JS

(30)

"'rlen' 'n Jetimtloo. an =

lhat the values II (uk) and u l 01 the trial solulion may be prescTlbec + at approprllte ~gments of a n.

The key step IS now to make a specifi..: o.::holce for the ~ iunLtton vk We WIll o.::hoose vk to be Ihe )mgular solution tor a po lOt load m an mfimte spa..:e. In the present 3-dimenslonal o.::Jse. let the po lOt load he UI the ~~

direction Jt the 10CatlOll '<m := ~rn' thus. 11 IS seen thai vk SJllsfies the equatlon

rO!J\Vk ll] ~ Oh' m ~mH5ll el = 0

lor tl"'I.: .. ~1 (31)

.... here 0 '''m - ~ml IS the DLTJ": (un":I]!Jn. 6 r l I) Ihe

Kwne..:ke r deltJ. Jnd H denoles simp ly Ihe direction Ot Ih ... load. U)1I11! 1.'11111 tJOI Jnd reo.:Jlling the property 01 DlrJo.: iUTl..:tloTls. 1I I:) )eeTl dut

The .;,oluIIlln {or a pomt ILIad In 1Illtnile :ipa~·e. when the mJ[ er III ~ lSO[ fOPlo.::. I. e wh ... n the [ensor E i j kQ, In \ ~3 :ltId :·H IS ISlHroPh: Jnd !TI\"\11\"esonl~ 1 .... 0 ..:onstants E Jnd v WJ, ~.I\"en b~ Lord Kelvlll more Ihan J ..:entury Jgo (3~]. In KdvUl') ,OIUlllll1, the dl)pIJ..:ernent m the lth dlTeo.::llon al 'm due to J um! I"Jd III the I.th dlTe..:tton JI ~n 15 given b} VII. t'<Ill' ~m). Jnd the tra..:ilon ill the lIh JlreCllon 011 Jnlflemeu >Urlao.:~ 'Wllh normal ,;osme) 11k at '<m. du ... to J unll IOJU 1fI the lth dtreo.:llon at ~m ~s given b}

tCit'<1l1.~rnl Thu~.irom ( 30 )JndI 3:) \.\ehJ\e

In 1;.;1. ~ml~Jn~ poml.\.\hd'::'mis Ihe

duw-fr.\ \anar-ie C'uher .n n "r ,)11 )!1 h .nuI,,:a led . ' ote thJ! lie Ke:\\n 'lliu[IIIIl, \11. Jnd III Jre 'lfI~ulJr JS Em - '\~II \\ h.:::: ~ III b IJken !IIlhe lirml to JPPfl1J..:h J bound· Jf\ ppml ,lne mt!'>1 ~nmld~r (Ju..:h~ prlll..:Jpal value~

ot the .nt~~rlb in (33) Jnd !hu~ obl all1 l31. 3:1 Ihe '0..).

.;:.olled hounOJf\ IIllegralequJIl()Il.

... I[ II1J\' )1 ~uur~<! I"h' lerllle>.l <!J~tl\. thJt e\'en II one had ~IJ rteJ ,Jut .\ Ilh tTlll lUIl..:Jlom Uk th;1t ;,a\Jsty the JIspIJ' ~"'J:l"'llI '111UllL1Jl\ dJIlUJIlon 1 ~~ ... l J rflorl. llle >1111 end,

111 ,nill:OI

wheretn the superscript b denotes a ··boundlry-value··. The so-called ''boundary-e lement method" seeks to satisfy Eq. (34) in a weighted residual sense. ~ote rust that while l volume integral does appelr 111 (34). 11 does not involve t~ function Uk. Eq. (34) may be satlsfied in Jny number of ways In a discrete sense l3~]. includ ing c01l0callon and Galerkln and "variallonal- techmques. When a Galerkin scheme IS used. on each boundary element. arbitrary-order Interpolallons may be used for

u~ (e b ) and ti (x! ). In terms of I heu respecllve nodal·

values It the element nodes. TIllS results In an equation 01 the form:

~9*=~9* (35)

where ~( is a vector 01 boundlry nodal Ji5pll~ements ( some 01 which lre speCIfied). and <;r is the vector ot" boun· dary nodaltraCltons (some of which Jre speCified)

[t 15 Important to note that a singular solullon [0 Eq . (31) IS tmposslble to obtain when Eijk( is stro ngly am· sOtr0PIC lS In modern compoSile mateflals. Theretn lies the essenual limttatlon 01 the boundary element melhod.

Ano[her simple 3nd more direct "boundary -element"' method may be developed when the trial iun..:tlons Uk

themselves satlsiy the Navier differentlal equallon. but !!2.! Ihe boundary condit iOn, i.e ..

(36)

lsymptotic solU1ions that satisfy (36) may be Jemed for seml-tnltmte strIpS. near holes and voids. ~rlC\... etc. [34.35]. When (36) IS used tn (~7). one.)btalfls the .. bound3ry - weighted -residual" equat ion.

The methodology Ifl1phed III Eqs. P6) l!1d (37) has been labeled "the edge-function method" m[34. 35].

Similar ~OllceplS may. of o.:ourse. be e,<t.!nded to either geometrtc311y or m~tertally nonlmeJr problems. Hcre we illustrale Ihe case Llf finne deiormalton mel3s, IIcilY. using the rate t'ormulallon [36.37]wherem [he rates Jre rele rred 10 the currently deiormed conrlguranon. Usmg a cartesian system for slfT1pliclly.let Vi be the velOCity oi a matertal panicle as a iuncl10n of ~urrent wordtoates '<k With Lij = Vi.] being the veloclt)' gradient. Dij '" II:.

(L I] .,. LJj ) the rl[e 01 deformation lensor. and WI]::

(LIJ - l;i ) bemg the Spin tensor. such that Lij "'( Dij

.,. W ij)' Let Si] be the rlte o r Second Plaia - Kirchhoff

stress tensor and TiJ be the CJu~hy stress In Ihe (ur rent ..:onllgur3 tion. The tield equallOIlS to Ihe rlle larm Jrc

I·; -.. " J

(L:-'IB):(Sij+ T1 kVj.k)j+ f)= O.

(1.)"'3()3x)

(A]\IB). 5 lJ :: 5) i

(CompatibIlity): DlJ = '-:tv iJ + vpl

(Boundar~ ("011(1Itlon51 IldSI)'" Tlk Vj.kJ

::, J at 51

(30)

(30)

(40 )

In linLle delmmatton problems. one needs to ..:onSlder the obJ~..:tlvlt} Jnd ffiJt;:n:!I Ir3memdirler;,~nce 01 the constitutive lJy, To IhlS end. one may ..:onslder an oblee· live fal e of the Klrchhorf stress 0 1) 1= J TI J' where J IS Ihe JawhlJn \) 1' deforffiJIlOn grJdlem In the ~urre!lt

conllguratl\ln) Lei the ub)~ctlve rate be 0i J. y, hlch ..:an be anyone ,)1 Jn tnlLntte!y many po:,slblhlles. For l!lstance

13Q J'

,

,

,

,

(~3.a )

a'i • Lf..IOkJ ·Olklkl tCu!ter-RI\Jinl

(43.b)

°'1 Llf..of..J· 0lf.. LkJ (mLxed)

(.n.e l

a IJ ... Lf..1 0Jf.. - a if.. lJk imlxed) ( 43.d)

OlJ \\ d, Of..] -O'k\\ Jf.. dJumJnn- 'all Z:HemDa)

(.1',~ 1

0lf.. nJf.. IGreen \I ~Glilnhl

t.1 ',n

y,h~r~ a II I) :he m:II~:lJj f.ltC 'Jl Klr~hhllif mes:.. Jnd

nil:: R. I ).. R,I" y,here RJk b the tool ~otat10n:n 'he

~urr~nt ~Omll!.UrJIi\ln. h Jelermmed Irom J pnl:ll ·de':lJm­po~lIlon nl the ~urrent d~lormaIK)n gladlem

l. n ,)blC(I1\<, ~()rmllull\e rCIJIHHl ,)1 lh~ tlte IOfm I~

pO~lulaled. 1II\"1f..1rl\! th~ pr1rlelpl~ ll[ 'lbJe..:tI\'lI~ 1401 JS.

where I h In IsotropIC temor lun~l!on j3q . .JO] .. c. under ob'>~r\'er Iran~l orm.1tLon 01 (l1gld rotalloni O mn.

II ~IIJH1 mdu.:ed .ll1lSOIHlP~ 1~ ~o he ,me rna\ ,ntr<luu~e ,11l~rnal \"artahle~ illch

t.1q

~onskicred.

.lS the had; ,Hess u~\ :11 .lfll;"l1rO(ll( h;:LTJ~nlt1!! ?I.1'Il":lI~ Jnd 1\Tlle

Ihe ;O!l~tlIUI1\'" ;.1\\, .1,

o

Oil ::: f(Dkf2. Ilk£ . oH ) (46)

"d o

Ilij ::: g(DU, O:kf2. 0ke) (47)

wherein both f and g are isotrOpic tensor functions . It has been demonstrated [39 . .11\ that. as long as the te nsor functions f and g o f (46) and (47) are general enough. one may use anyone of the mfinnely many stress-rates. such .l S (43t on the left hand side of (46) and (47).

The general rebtlons (46) and (47) produce physically pla­USIble stresses in homogeneous deformallon problems. L e. nono~lliJtory stresses In Simple filllte-shell.

In a iinne-deformalion melastic problem. one m:ty decompose Ihe melastlc SIT3ms as:

d. , 1 (48)

wherem )uperscnpn e and 1 denote ebsll<.: and melastL~. respectl\'el~ Henceiorth. we Will consider. wlIhoUI loss 01 gener3Iit~·. an lsotroplc-hardenmg eustoplasllc Tebllon 101 metals twnh small elastIC stlJlns) ]37. 3<)1 as.

where 0::: 1 If the matef\.31 undergoes pbsllc strain. p. and A 3rc Lame constants. OISlhe yield-mess m uniaXIal tenSIon. k IS the slope 0 1 the true stress versus logarithmIC plasllc )trall1 curve m unuxlJI tension. and O"k ~ IS the

de\lator of Kirchhoff stress. When elaStiC 'MJmS are I'en 'i1ll311. JS \I.e shall assume. oil m (49) maybeldentlfud

JS !he )Jumann late 01 Stress \3Q ], wuhout In} lI1.:on· ~ISh'rk\ "or slmplicll},.( 49 1 rna} bewrttten.1S:

I Q, , -ok~o lJ) Dkl: !

1501

t 5 1 )

The deltnll ion of BIlk( IS app3rem by ~ompaTlng

p4, ~O Jnd 51).

""' ,lte that E1)k( m (50) and (51) IS the Iilstalltaneous

tensor ,II IsotrOPIC eI35I1cII),. when the m31Cfl31 is ~onSLder·

eu 10 he tnlll3l1y IsotrOPIC.

TIle relation between Slj and the laurnann rate OIl IS

\3'. 3' I

( 52,a I

t 5':.b)

(52.d)

In wr!llOg (52.b), (51) has been used. The meamng of FiJk~ In t52.d) is apparent. Note that 5iJ In (52)

is symmetric. by defilHllon. ~o.".. let 'Ii the test functlon for veloclIY. Without

loss of generality, we :lS5um e thai the thesl funclLons saliS!} the geometric boundary co nditi£'O' (42) . . Let d 1J tv) be the veloell} strains der tveo Trom vi and 51) (v)

be the second P- K mess-rate denved from v. USing (52.d) Thu~. the onl~ tield ~quallons to be satisfied 3TC:

[SIJ {\"'+T tk"J,kl.l'" f) "' 0 In n

"d

( 5j)

(5.)

l':l \', I be the lest fun":llon. whICh. In general. may not valH~h Jl Su The summed weighted resldlLJl forms ot \ 5.; IJndt:'-ll Jlt.'

f n jIS1Jt \)+T1k\ J. kl.l'" :\fwJ

0; f Ii \\ Q '

( 56 )

,-,lte Ihlt E IJ k \. IS Ihe IsotrOPIC tensor ot hne:lT elas·

11(1\\

Thu~,

.1. \ k J n

(58)

Now. we may choose the test funCllon w I such that It is a solullon of the KelVin problem:

[E ij kQDJ I(W m)],Q +8(x m - Em} o..:.p ep = 0

(p • 1,:.3) (60)

It IS then immediately apparent that an Integul rel­ation for vk tn Sl and hence an tntegral equation ror vk on a n. results from (59) and (60) in just the same way as in Eqs. (31 to 34). In (59) it is understood that tj (v) would be spectfied at ~ and vk would be specified at Su. whue the test functX)n wk is non-zero at Su, and [Elj H Dj i (w) J are non-zero at~.

Now several conclusions may be drawn. In ISOtrOPIC Eq. 33). the unknown trial funClion uQ. appears only on the boundary.

Hence any discrete method to SOItisfy (33), such as the Galerkin method. involves only boundary- tnlerpolallon for uQ and. hence. the name "boundary - element method"

On the other hand. when lar e deformations or tnel· asticlly. are present (see Eq. 59 , the unknown trial [unclln (for partIcle velocity from the current configuration) vk

appears not only on the boundary. bu t also 10 the inteIlOr. Hence. in a direct deveJ0Jlment of a discrete method . the trial functnn vk must oe discrellzed not only on the boundary, but also In the mterior of the domam. Hence. \I IS no longer 3. "boundary·element'· method. but rather a finite element method of 3. specIal kind. If. on the Oter hand. an lIeratlve solullon IS used and the boundary· mlegral equatIon resultmg from (59) is VIewed JS l rei· ation for Ith Iterative value for vk al a n on the lelt-Iund side. tn terms of the known (i- I) Ih Iterallve \'alue lor Dk.£ (v) and vk.2 in non the right-hand Side. then one m3.Y consIder. in a way, a "boundary-element" method analogous to that of hnear elastlcay wllh a rather complex system 01 "body forces" due the effects of nnlle delor· mat Ions a nd melasllcuy (corresponding to the (i-I ) th mlerallve solullon). In thiS case. the methodOlogy IS similar to the finite element method wherem only t he linear-elastiC st iffness matrIX IS used throughOUt. and the solution for fume deformations and tnelasllclty IS obtained through a slow process of "modified" Ne Wlon· Raphson Iter:lIio ns. On the other hand. when finite ele· ment Interpolations are used In the mtenor of the domain. it is common practice to obtatn "tangent-stiffness" matr­ices to account for Hnlle deformations and inelasllclIY, 10 speed up convergence. All saId and done. Lt LS perhaps approprl:lle 10 label the solullon methodology based on a duect apphcanon of (59) and (60) as a "mIXed fmlte­element {boundary element method".

OccaSIOns. wherein a pure boundary element method (with bounda ry interoolatlOn only for tTial tun~l!ons)

,

and a pure finite element method (with intenor interpol· ations for trial funldons ) may be combmed JudicIOusly. are discussed in [23. 4::!J.

COMPUTATIONAL MECHANICS

Here we discuss certam topics in the mechanics of solids. wherein computJlIonal 10015 have played In the recent years. and are expected to play In the ooming years. a slgmficant. but secondary. role.

1. Constitutive Modeling While. until a fe w years back. simple constnutlye relations such as lSOUOPIC harden· ing or linear kmemallc hardening plasticity were the mam· stay of computer programs. currently there is a widespread interest m the constllullve modeling of expenmentally observed behaVior of materla !s Im'olving plastic and creep detormallons under monolOnic and cyclic loading.

The general theory of inter nal va riab les. as skctched

in (46) and (47) fo r instance, has played a key role in the development of more and more realistic constitutive models to characterize inelastic behavior. Such internal variables that are being widely employed include. for instance: 0) the tensor locating t he center of the yield surface in the stress space ("the back-stress"), Oi) the parameters that characterize the expansion of the yield su rface. (iii) the parameters that characleTIZe the "bound-109-su rface" in multi- yield-surface theones {43-45J oi plasllcity. and (iv) the back-stress and drag- stress used to characterize creep rurface. etc.

The multitude of conslltutive relations for melasticity. proposed in literature. appear on the surface to be unTelat· ed to each other and to be based on totally diverse concepts

II has been found recently {46,47Jthat such is not the case and that the "internal-time" (endochronic) theory {48.49J. the multi-yield-surface theories {43 10 451. and the internal variable theories {50. 51 J are essenllally the same. wllh only mmor variations. It is also >hown that lhe difierentla l forms of the suess-strain relations for

Endochrumc Theory: {( . ) denotes a derivative of ( ) with respect to Newtonian-time or a Newtonian·time-like extem:tl parameter such as cxtcrnalloadJ

tr ( g) = t~'"b + jAn) tr (V· where ~o. Ao are Lame contants

f(O = (I + J3r> (linear) or an = {a + t I - a)cxp ( - TnJ(expon)

S~ldfd\) , c=l

r = ~ r(D . h* = ~ .. I ~ - 1

KmemallC Hardenm~

Q. r(d . i l l) = ~Jl PI e p - -'--~ (eP· ~p) :-:( no ~um on 1) for i = l.~ ...... - - a 1~ f -

i= ~ ~(i)= ~J.loPl (o)~P _! , ,

IsotropIC Hardenmg:

o Sy = SyP (!p ~p) l-: (linear i)

= TIs: - S; {a "-il-a)cxp ( Tn!} (~p ~P)~(-;JtuT31ed t)

Jable I SUmm3l\ rJ! th.: Proe~! Imclnal·Timc Theory 'll PbSllClty

plastlclty lnd the differenllal forms of evolution equations for mternal variables given m [46, 47] mclude, as special cases, the multi-yield-surface theories of Mro z (43], Krieg [44], 3nd Defalias and Popov [45]: the nonlinear kinelllallc hardemng theories of Chaboche et al. [50,51]; and Ihe classiC31 Pr:lger - Melan hardenmg rule.

\\ e present here 1 synopsIs of the Iheory developed In

[-46.-P I. We restm:t ourselves to small strams and deform· atlons. Let ~ be tile small -str:lln tensor Jnd g liS rate. Let the dev\JlOr of ~ be ~. Let Q be the stress and Q Its rate. and the deviator of 0 is denmed as S. The rate fo rm 01

the ~tress--str:lm relatIOns tor .:omblned lsotr0plc-kmem· au.: hardening plaso":IIY denved In [-46. -47] are summaTlz· ed in the Jbove labk.

The re iJllon s given III Table 1 have been shown [-46} to be no more Jlffi..:ult to implement In a computatlonal algorithm than the .:l:mical plasticllY theory. However. In ..:onlTlst to the d:lsslcal theory. the relalions m Table I predici ..:ydic h:udnlllg. cross-hardemng. etc. as lccur­:lIely .IS desued. A umfied creep-pbsllclly-theory. arull)~OUS to Ihat In Table I. has also been presemed [-46. J-\.

-\ nmher IOpl": In ..:ons\ltutlve modeling that h:ls engen­dered mu..:h Jis,,:usslOn In the past three years concerns linue )train pIJSlI":lIY. The "controversy" surrounded th(' results tor ,he:1T mess. which were found to be osctll· Jlor~ 1!I !IIne[~:]. In fimte sunple she:lr. when Ihe Jaumann - \ \111 ::lte wa) u~ed for the emlUllon equallon tor back­Slre~) 2. III J Simple generJliZJl!on of Prager-~lebn linear klllenlJtl": hard(,lllnl! rulc to finite deformallons. This "Jnomal~" has prompted Lee el al. {53] to derive modifi­

ed humann rales Jnd \)(hers to "Ihrow Ihe bbme" on the JaumJnn rale Jnd IIlstead to use the Green-~kGinms rale

[I has hetn dernonstr:lted In [3q l thaI none o f Ihe strt~, rates Jre 10 b~ "blamed". and in fa,t no o ne stress­rale ~s preferable 10 others. prOVIded lhat the isolToplC len",-IT funclions. sUlh as r :lnd g lS III 1-46) Jnd P7). :lTe e'\panded JPproprlllel~ un the TIght ·hand sides 01 (-46) Jnd ,,,-) lrrespe.:I1\·e ,)1 .... hlCh obJectlw rale IS used on :he kit hand -;lde \)1 Ph 1 Jnd (-n).

I 'll\g tluly geneTJI ~'\panslons fOI t' and g as In P6) Jllo.i (J-) Jnll i..I~l!lg. lor e'\lmple. the humlnn rales In the .elt hanJ ~Ide~ "1 1-46) Jnd tJ""l. modeling .)\ test JalJ !II l finne deformallon dastlc-plasl!": tenSlOn­!\IT'l\lll 'e5t has been pertormed successfully lor the lirst 11J1le ,n luerJture. Jnd Ihe ,;oc31led S .... \1t dfe<:t ha, been J~''''uTltely modeled

Slullle~ Jlong Ihc lbo\'e lines [ll. l6 . .17] Jre expect · cd "1 1'13\ J "ilglllllo.:Jm role III the ne3! tuture III undersl­J.nll!n~ materul beh:l.\lor m brge ~lTaIllS.

"'l COlllputJtional FrJCllI rc MecilJn ics The rliedlJrlh,;S \)1 lrJOUle (especially ot lIlelasli..: dynamic lnu 1 hret'-Junenslonal 1 rJ..:1 ure 1 IS one area where compUl· Jtl~lIlJI method,; ha\'e played an unquestionably tmportant 10k Jnd Jrt' likdy to contmue 10 do:;o. A ..:omprehenslve ,urnmary ,)t .:ompu t:llloIlJI mcthods In the met.:hamcs of tradure hasre.:entl~ he~n prepared {5-41

Here >\e "T1ell~ touch upon 1 .... 0 topiCS: modeling ,)1 tlvnaml': ..:ra.:),; propagallon Jnd three-dimenSIOnal ~ronkms 01 ~mnedJed ,IT 'illTTa..:e Tbv.s.

In rhe phI 'LX yeJrs ,lr )\). "l1IovlIlg ,mguIJr dement"

procedures of analyzing dynamiC crack propagation and arrest have been ext ensively developed {55-59 ]. In this procedure, the eigenfunctions for displacement. velocity, and acceleration near the tip of a dynamically propagating crack, as derived in (60], lre used 3S basis functlons of the "slngular element" which surrounds the crack-tip. The Singular element may move by an arbitrary amount of crack length 1Ilcrement .6.30 in each time increment ..It of the numerical time-integration procedure. The movmg Singular element. within which the crackttp always has a fixed locatIOn. retains its shape 3t all times: but the mesh of regular (isoparametTlc) finite elemenl5. surrounding the moving singular element. deforms accordingly. To Slffiulate large amountS of crack-propagallon. the mesh pattern of regular elements is readjusted perIodically.

The analYSIS of dynamic crack-propagation has. however. been greally simplified due to the recent develop· ments [60-641 concermng far-field path-mdependem Integrals governtng the fields neal the proplgltmg crack­tip. 3nalogous 10 the well- known J of elasto-statics. For instance. for elastodYllJnuc crack propagJllon. the integrll which IS eqUivalent to the ra le of energy-release IS given by

where \II IS Ihe stram energy densay, T the klnellc energy denSIty, ti the tractlons. ui the displacements. (.) denOies a nme-denvallve. r€ IS a near-field path. and r the 131-

field palh. The above far -field Inlegr:lis eruble In a~.:urJte :lnd eificlent analYSIS of dynamiC crJck propagaTion. uSing ordinary (non- smgulal) isoparametrl': finne ele· menlS [65. 66].

Another interesting and Impurtlnt development In

.:ompulatlOnal fracture mechamcs has been the enhance­ment of Ihe Schwarz-Newmann alterllJtlng method tor solvmg three-dImenSional problems of embedded 01

surface flaws In structural components. [n the alternatlng method [671. the analytlc3l solution

for an embedded elliptical ..:rack m an injimtc elamc medIUm. which IS a basic soluuon reqUired 111 the alternat· Ing techmque. has been limited to 1 .:ublt.: Varll110n 01

normal pressure on the crack surface [68]. This limitation IS thought to be one of the major reasons for the rebuve maccuracy of Ihe aiternallng mel hod as t.:ompared to hybrid finite element procedures or Ihe boundarY-integral equallon procedures.

Recently a general solullon procedure has been deTIVed in [69\ for the problem of an mlinlle daSllc medIUm with an embedded ellipllcal crack. the faces of which are subleCt to arbItrary 'larllIlOnS of normal lS well as shear tractions. Later a more detailed solullon. lS .... ell as a general procedu­re for the evaluallon of the requued eJhplI": Integrals. wJsobtalned m (""0\.

... I~- -

Since 1971 no work has appeared in literature to generalize the solution in [68] to an arbitrary pressure variation on the crack surface due to the seemingly insur­mountable mathematical and algebraic difficulties. While the analytical solution [69, 70J can be reduced to a closed­form solution for a relatively simple loading such as const­ant or linear variation of the tractions, for a high-order polynomial variation of the tractions_ the solution pro­cedure requires a digital computer. To obtain the stress components at a given pOint by uSing a computer. a general evaluallon procedure [70J for obtaining the partial denva· tives 01 the potential functions used In the formuhtion is also one of the key algebraic steps in the successful applicalion of the present analytical solulion.

Recently a major improvement of the alternating method has been made in [70. 71 J. In the new alternating method [70. 71 J, the complete . general analyl1cal solution [69. 70] for an elliptical crack explained e:ulier was im­p].;~mented in conjunction with the finite element method.

The major steps requITed in the nOite element alterna t· ing method are gIVen In the foHowmg: (i) Solve the uncra­cked body under the given external load by using the finlle element method. To save computation time in solvmg the nnlle element equation for multlple right· hand SIdes. J spet'ial solution techmque was implemented PO]. (ij) Using the finite element :.olution. compute the residual ~tresses at the locallon uf the origmal crack 10 the uncracked solid: (iii) Compare the reSIdual messes calculated m Step (Ii) With J permissible stress magn itude: (iv) To SJtislY the suess houndJry condition on the crack surface. reverse the reSidual stresses. Then determine the analytical :;olutlon for the crack subjected 10 these revers· ed reslduJI stresses: (v) Evaluate the slress lOtenslty factors in the JnalYllcal solution for the current iteration: (VI) Calculate the residual stresses 011 external surt"aces of lhe body due to the applied stresses on the crack surfJce In

Step (iv). To sallsfy the stress boundary condlllOn o n the externJl surfaces 01 the body. reverse the reSidual stre)ses Jnd cakubte the equIvalent nodal forces: and (Vii) Cunslder these nodal furces JS external applied loads acting on the uflcrJcked body. Repeat all steps 10 the Iterallon pro..:edure unlll the reSidual stresses on the .:rack surtJce become tleghglble (Step iii). Tu obtam the rinal solUllon. Jdd the s\T('ss lOtensity lactors of alillerallons.

In the Jbove. ,everal novel wmputJtJOnal techniques were al:;o Implemented to save the computJtton ume Jnd to Impruve the convergency Jnd accuracy ur the present linlle demem .lltemaling method [70- "' 3]. Sma.l very W;Jr~e mc~h ~an be used to analyze the uncracked body. the JiteTtiallng method becomes :.I very lnexpenSI\e pro· cedure for rounne evaluation ot Jccurate ,;tress mtenslty factors tor Ibws In structures. It was fo und [70] IhJt thiS new Jlterm.lIng method IS .It least an order 01 mag· tlnude lOexpeoslve compared 10 the eJrlier hybTld element prllceduf~ ]-"'] .

Several )tudies have .1150 been completed !-5. 7h] m USltlg the alternating method for muillple sem l-ellipncal cr Jck s 10 pressure vessels Jnd for thermJI "hock analysis 01 sunace tlaws In pressure ve~sel~.

lna~n uch .:!..:' no tlllmertcal modelin!! .)1' crac'k from Sll1gUiar!lll"S IS pellvrmed. the present tmne-eiement JlternJtln~ .t1ethuti hJ~ hec n IOU110 to he h\ f:H the ka~t· expenSive .IS ,~ell I~ th .. ' ,nll~t .iL..:urJte m'~thlld J~ ~(lm·

pared to th ree-dimensional hybrid-crack element pro­cedu res [77J or three-dimensional boundary element approaches [31. 781.

3. Nonlinear Continuum Mechanics and SheUTheory The works of contemporary mechanicians such as T ruesdeU, Noll , Hill. and othe rs have done much in plaCing the theories of nonlinear behavior of continua on a fi rm. rational basis. The advent of nonlinear computational mechanics in the past decade or so has not only brought these aesthellcaUy appealing theoretical develop· ments to the pract ical level of a technologist but also has enriched. and has the enormous potential for enrichmg. these theones through the search for aJternallve formula· tions that may lead to computationally more-efficient algorithms. On a broad philosophical level. Gne may observe that most theories of modern conunuum mech· anics IOvolve differenllal or integral operators. whereas computatIOnal mechanics relies on discrete operators. Thus. it is fruitful to think of formulallng theones of mechanics from a discrete viewpoint. A case 10 pOint is the question of objectivity of stress-rates used in con­tinuum mechanics. whereas computational mechamc s involves the employment of stress-increments whICh remam objectlve over finite time-steps [79, SO].

A recent monograph [Sll exemplifies the diversity of work In nonlinear compulallonal mechanics. The future work will undoubtedly involve development of techniques for large-strain large·rotatlon analySIS of :.hells ustng alternate theories [S2J and the development of Jlternate finite element methodologies to treat large­stram behaVIOr of metallic mat enals.

TIle use of Bubnov-Galerkin techniques. using glObal· local approXimatIOns. for studying the global nonlmeJr response of structures [S3J and development of :.olullon ::ligOTlthms. such as [84], are topics that deserve a Wider attenuon.

.t . Stru c tu ral Cont rol The subject 01 "aeroelasllcllY "- a ,;tudy of the interaction of fleXible elastic bodies with the surrounding Iluid medium-has blossomed 10 the iJte 1940's and early 1950'5 and played an unponant role In the design 01 modern aITcraf!. In aeroebS\lCllY the Jerodynamic forces actmg on the elastic body have an eqUIValent intluence of "negallve dampIng". I.e .. the surrounding flUId medium acts as an energy supplier to the Vibrating elastlc body. A problem of the opposite vanety arises in the deSign of large space structures( LSS) Intended for a vanety of operJ\lons 10 outerspace. These structures are enVisaged 10 be as large as Manhattan Island. for m· stance. and 10 be very tlexlble. Other examples of LSS lre the large space antennae. The ce ntral problems 1Il

the design of these LSS are vibratlon suppression and shape control. when the LSS are subject 10 disturbances such as due to unbalanced rotating machinery on board. thruster finngs, sleWing/pointing maneuvers. thermal transIents. etc. Vibrallon suppressIon and shape control 01 LSS are sought through either active or passive (or a combination ot" the two) types of control mechan!sms. Thus. in parallel With the ,;ohJect of aeroelasllcny. we have the emergmg 5uhJect o! ser\,oelaS1lcny - thai of (ontrol ,)1' dynamiC mOllon ' It detormabk '!Tuctures. In Jt.lditlOn to LSS. slll1!hr problems aT!se in the deSign ,)1 tall hudd1Ol!S nn

earth, wherein It is required to co ntrol t he dynamic motion, say under se IsmIc loads. to ampli tudes within the bounds of human comfort levels.

The tOpiCS germane to the Issue of LSS contro llability are lS follows: (i) Wlule for the usual free or forced vlbra· tlon response analysIs of structures. efficient algorithms based on many thousands of degrees of freedom eXIst. the algornhms for optimal control are current ly limIted to bUl a few degrees of freedom. In the traditional finite element modelling of a Slructure. several hundreds or thousands of element-nodal degrees of freedom may have to be used to obtam even the (irst few (sat. 10 or ~O) fundamental frequencies and global mode shapes. Thus. there IS J need tor Jiternallve :lpproaches for reduced· order modelling ut the suucture-1.e .. its stiffness and mertll. (11) Design of :algortthms for irnplememallon of optunJI -:ont rol of systems of moderate dimension of. sa~ 50-100 (I.e .. 50-100 global modes); (iii) The ml)fC prevalent concepts (85. 861 have been to either Ignore dampmg or conSider it bemg proportional to mass or sllffness Jnd obtain the "normal modes" of the (linear) struo.:ture. Based on the orthogonality of these normal mode s. Ihe system of (linear) ordinary differenCtal equa· lions are com pletely decoupled. and conlIol of response of ~ach decoupled equation IS attempted mdividuaJly. Tlu s -:oncep! is labelled the so-o.:alled " Independent Modal Splo.:e Control'" Tlus approach. whtle mathematically sImple. depends un the use of as many ,ontrol-iorce actuators as the numb..:r of decoupled modes bemg COntIOl!· ed On the other hand. If damping exists due to deliberate deSIgn oJf pass\\'e dampers or due to deliber:Jle deSIgn 01 JOIntS. 1\ may be ot "nonproporllonal'" type; and thus. the (linear ordinary differential) equations of motIon cannot be decoupled. Thus. one needs 10 implement control based on the system ot coupled equatiOns of mOllon Jnd wl\h an JrbllTary number of active control· force actuators that IS perlups much smaller than the Older of the ,ystem bemg o.:ontrolled: (iv) The effect of no· n!ine:mlles In the system. such 11u! the discrellZed equa· 110m 0\ mOllon are nonlinear ordinary differential equa· oons m Hme

D"pendlng un the spatlJl and temporal vari:l!10ns 01 the dlSlurbJncrs. Ihe LSS may be modeled as a thlre· dlmenslonJI nctwork 01 beams and bars [87. 881. or. aiternatlvel) . an eqUivalent continuum model such as a piatt' ma\ be used. Recently [891 a methodology was presented wherein (i) an eqUIvalent contmuum mod~l III the form 0\ J flee-tree plate is used. (ii) the linear trans· lent JYLUmh': response 01 the plate is modeled by a boun·

dan -dement technique based on the singular soiullon :)1 J blharmOnLC operator. (iii) a nodal-control. I.e .. control 01 nodal response based on the completely coupled nodal system eli equallons IS used. (Jv) non proportlonal dampIng is consldrred. Jnd (v ~ the tina! lime III the control algoTtthm

IS set to be miLnuy so that a neady-state Rlccati eqUlllon IS solved.

In another recent study [901, a singular-solution approach was used to drnve un" discrete coupled ordinary dlfferenllal equauons governmg {he tranSient dynamiC response 01 3n IIllllally stressed !lat plate that is assumed to be l continuum model of:1 large space Structure. ~on­propoIllonal dampmg IS assumed. A reducedorder model ot \I (" n ) coupled urdinary Ji(fercntlal equations lre

derived in terms of t he amplitudes of psuedo- modes of the nominally unda mped system. Optimal co ntrol is implemented using m « N ) cont ro l-force actuators. in addilion to a possible number p « m) paSSive VISCOUS dampers. Algorithms fo r efficient solution of Riccati equatio ns are implemented. The problem of control spillover is discussed. Several example problems involving sup pression of Vibratio n of "free-free" and "simply· supported" pla les were presented and discussed.

Structura l control. Lnyolving Ihe active or paSSive control of nonlinear dynamic response of deformable bodies. is likely to be a subject of intense research actiVity of the next decade.

ACKNOWLEDGEMENTS

The studies which are bmfly discussed herein, were made possible through past and present research support from the Office of Nayal Research. the Air force Office of Scientific Research. the National Science Foundation. and NASA. The encouragement of Drs. Y. Rajapakse. A. Ku shner, A. K. Amos, C. C. Charms, l. Be rke. and C. J. Astill, oyer the past several years. is thankfully acknowledg. ed. II is a pleasure to thank Ms. 1. Webb for her assIstance in the preparallon of this paper.

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29. de S.R. Gago. J.P ., Kelly, OW., Zienkiewicz, O.c.. and Babuska. L , A posteriori error analySIS and adapHve processes in the finite element method: part lI ·adaptive mesh refinement, IntI. Irl. Num. Meth. Eng. 19. 1657·1669 (1983).

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32. Atluri. S.N. and Crannell. 1.1.. Boundary element methods (BEM) and combination of BEM~FHL Report No. GIT- ESM - SA- 78-16. Center for the Adv. of Compo Mech., Georgia Tech. Atlanta. GA (1 978).

33. Brebbia. C.A" The Boundary Element ~ethod ror Engineers (pentech Press 1978).

34. QUillian, PM .. Fitzgerald. 1,E., and Atlurt, S.N .. TIle edge, function method, Proc. IntI. Symp. on Innova· tive Num. Methods in App!. Engg, Science, Versailles, France (1977).

35. Qumlan, P,M .. Grannell. 1.1 .. Atluri. S.N .. and Fltz, gerald. 1.E" The edgefunction method for three' dimensional stress analysis. including embedded elliptical cracks and surface flaws, in Brebbia. C.A. (ed.), Boundary Element Methods in Engineenng (Springer-Verlag 1982) 457·471.

36. Atlurl, S.N. and Murakawa. H .. On hybrid finite element methods in nonlinear solid mechamcs. in Bergan, P.C. et al. (eds.), Fimte Elements in Nonlinear Mechanics. I (Tapir Press. Trondheim. Norway 1977) 3 '44.

37. Atlur!. S.N., On some new general and complementary energy theorems for the rate problems in !lnlle stralll. claSSIcal elastoplasticity , JmJ. Structur:1l Mechantcs, 8.1 . 6 1·92 (1980).

38. Atluri. S.N .. On rate principles for finite stra in analy sis of elastic and melastic nonlinear solids. Recent Re· search on Mechanical Beh .. viour of Solids (professor H. Miyamoto's 60th Anniversary Volume). (Umv. of Tokyo Press 1979) 79·107.

39. At luri. S.N., On constitutive relations 3t finne stram: hypo,elasticity and elasto·plasticity with isotropic or kinematic hardemng, Comp, Me th. App!. Mech. & Engg ... B, 137·171 (1984),

40. Truesdell. C, and Noll. W .. The nonlinear field theoTle s of mechanics. in Flugge, S, (ed,), Encyclopedia Df PhYSICS (Springer. Berlin 1965).

41. Reed, KW . and Atluri. S.N., Constitutive modelmg and computational implementation in [imte strain plasticllY. Inti. lrnl. of Plasticity (Special issue m honor of Professor I .F. Bell's 70th Anmversary) (m press).

J"2 Atlun. S.N. and Nishioka, T., Hybrid methods of Jnaly sls. m Kardestuncer. H. (ed.). Umfication o t Finll e Element Methods (j .H. Argyris' 70th An· nlversa ry Volume) (North·llolbnd 19H-l) 05,06.

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~5. Oafalias. Y.F and Popov. EP .. PI:lSIic internal variables formalism of cyclic plastlcLty. J rn!. of App!. Mech .. 43, 645-651 (1976).

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48. V;ilJnts. K. C. Fundamental ..:onsequence of a new mtnnSI':: (Jme measure plamcuy as J ILmlt of the ~ndo..:hrontc theory. Ar .::h. of Mech. J2. 2.171-191 ( 19bO)

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50. Chaboche. J.L. and Rou.seher. G .. On the plastic and vLscopbstic .::onStltU!LVe equations. part I: rules devel· oped WIth mlernal \'Jrtable concept. part II : applica· \lon 0 1 mternal varuble .::oncepts to 316 stamless steel. m B:.Iyla..:. G. (cd .), InelastIC Analysis and Life Pr edIctIon III Elevated Temp Design. AS~I E -PVP­

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n01HsotroPlc workhardenmg til finlle stram plaslLcLty. III Lee. E Hand \h llc::( R.L. (eds.). PlastlcLty of \\e(als J( F11Iue Stram' Theon. Expenment. and (ompUIJILOll (O\'n. 01 .-\ppl. \ I<,th .. Stanford. lnd Dc::pt \1 <'..:h. Engg .. RPL IQ8:) o5·10:!.

53 L<'<'. E.I\.. ~IJ llett. R .L.. and W<'Tthelmer. T.B .. Stress JnalY~Ls lor JllLsotr0PLC hardenmg 111 finne deformatton plaslI..:Lty. lrnl. of App!. '\1<'ch .. 50. 554-560 0983).

::;4 Atlur1. S.:'\' (ed.). (omputalLonal .\1ethods III the \1<,..:hJIl1":s 01 Fracture (~orth.Holbnd) (to appe:.lr).

~, At lurL SS . 'Jr shlOkJ. T.. Jnd :"Jakagakl . .\1 .. Numencal \todehnl! 01 dynamLc and nonhnear cr:ick propagatron m tinae bodLes. by movll1g stngular elements. 111

Perrone. 'J . :.Ind Atlun. S 'J.leds.\. AS1'-tE- AMO- Vol. . UI:\.S~IE [9 -9) 37·06

50. 'JrmlOkJ. T . .lnd AtiuTl. S.;-": .. Numerrcal modehng o i Jynarm..: ~ra.::k propagatlon 111 t'\nne bodies by movmg ~1I1gular dements. part [ formulation: part 11: results. 1rn!. 01 App!. .\i ech .. 47 i 1980) 570·576 and 577-582 . respe..:trvely

5- .... LshLoka. T .. Stonesrler. R.B .. Jnd AtluTl. S.N .. An <,va]uJllon 01 :,everJI movmg s1I1gu]arny [lilLIe element procedures tor Jnalysl~ 01 last Ira..:ture. Engg. Fr:.lcture \I e.::h" IS. ~05·~1 b t Iqhl)'

511. \1"I)LokJ. T. Jnd .-\tlun. S.:"\ -";umert..:al JnalY~ls 01

J\n:.lllll.:: crJ..:k PlOp:.lCJ\ml1 ~ell<,r31l0n Jnd predLctron ,1:1...ile· Em:; F'J~'Jr'.' \k~:\Jnt..:,. Itl .. :O.~·_~_~:::: r [lhe).

59. Nishioka, T . and Atluri, S.N., Finite element sLfTlula· tion of fast fracture in steel OCB specImen. Engg. Fracture Mechanics. 16. 157·17 5 (1982).

60. Nishioka, T. and Atluri, S.N., Path· iade?endem in· tegrals. energy release rates, and general solutions of near·tip fields in mixed mode dynamic fracture mech· all1cs. Engg. Fracture Mechanic~. 18, !. 1·12 (1983).

61. Atlun. S.N" "P ath·independent integrals 111 finite elasticity and inei.:tsticity with body forces and ar· bltrary crack-face conditions, Engg. Fracture Mechan· ics. 16. J. 341·364 (1982).

62. Atlurl. S.N .. Nishioka. T .. and Nakagaki . .\1.. Recent studies m energy Integrals and therr applicallons.

Plenary Paper .6th IntI. Congress on Fracture (Perg. amon Press. New Delhi. 1984) (10 appear).

63. Atlun. S.N. and Nishioka. 1.. On path'lndependent Lntegrals in the mechanics of elamc. melastLc. and dynamiC fracture. Jrnl. of the AeronaulLcal Society of India (Special issue in honor of Professor G.R. lrwLl,) (l984)(in press).

64. Atlurr. S.N .. Nishioka. T., and Nakagaki. ~1.. Incre­mental path·independent integrals In melastlc and dynamic fracture mechanics. Engg. Fracture ~I e{.:h·

antcs (in press). 65. Nishioka, T. and Atluri. S.N .. DynamIC crack propaga·

t lon USLng a new path·independent lnlegral and moving isoparametric elements. A IAA hnl .. 2~. 3. -l()9415

(1984). 66. NLshloka. T. and Atiuri. S.N., A numerJcal study of

the use of path-independent integrals In elaslO·dynamL{.: crack propagation, Engg. Fracture Me..:hantcs. 18. I. ~3·33 (1983).

67. Shah. R.C. and Kobayashi. A.S., On the surface Oaw

problem. 111 Swedlow. J.L. (ed.). The Suriace Cra{.:k. PhYSical Problems and Computational Studies (AS~\E 1972) 79·124.

68. Shah. R. C. and KobayashI. A.S .. Stress intensLty fa.:tor for elliptical crack under arbitrary normal loadmg, Engg. Fracture Mechanics. 3. 71-96 (1971).

69. Vijayakumar. K. and AtluTL. S.N .. An embedded ellip(ical tlaw in an infinite solid. subJe~t 10 arbiITar~ crack'Iace tracttons, Jrnl. of Applied .\\echanl":s . .l8. 88·06(1981).

70. Nishioka. T. and AtiurL. S.N., Analyu .. :al ,;oIUILon fOI embedded elliptical cracks. and finue element aiternatLng method for elliptical surface t13 ..... s. ,ubJe..:t· ed to arbnrary loadings. Engg. Fra cture Mechanr..:s. I" ~47·268(1983).

., I. :'-lishtoka. T. and AtiuTl. S.N .. A maJO! de\'elopmelll towards a cost·effectlve alternating method for Ira..:ture analySIS of st eel reactor pressure vessels. Paper G·I ~. Trans. 6th SM IRT .Pans(1981) .

n. NLshlOka. T. and AtlurL. S.N .. Ana.lysls ot SUf(J(e t1m. 111 pressure vessels by a new 3·dimenSlonal 3lternJILng method. J rn!. of Plessure Vessel Technology. 104. 299- 307 (1982).

73. NishLoka. T .and Atluri. S.N .. AnrnexpenslVe 3·D 111lne element alternallng method for the analym of SUr!·J..:e nawed 3Ifcralt structural components. AlA:\. lrni.. ~l. 5,749.758(1983).

7 .. t At luTL. S.N. lnd Kathiresan. K. 3·0 anal\\LS o{ mrlJ.::e I1lws III thICk·wailed rel.::tor pf('ssure' ves..els u~mg drspllcement·hybnd filllte element method. :,\udear Em.!!! & D':slgn. 51.~. 163')-01 Iq-qf

75. O'Donoghue , P.E., Nishioka . T., and Atluri. S.N. , Multlple coplanar embedded elliptical cracks in an mfinlte solid subject 10 arbitrary crack face tractions, Imi. Jm!. Num . Meth. in Engg. (1984) (in press).

76. O'Donoghue. P.E.. Nishioka. T., and At luri. S.N .. ~I ultlple surface cracks In pressure vessels, Engg. Fr:u;ture Mech. (1984) (in press).

77. Allun, S.j'J lnd Kathiresan. K .. 3·D analyses of surface !laws tn thlck·walled reactor pressure vessels UStng a dlsplacement·hybrld finne element method. Xuclear Engg. & DeSign. 51. t 63·176 (1979).

-So HelioL J Labbcns. R.C. and Pellissler-Tallon. A .. SemL-elhptL..:a.1 ..:ra.cks tn a cylinder subjected to stress gradients, Fracture ~lecharlLCS ASTM 5TP 677, 34 1· 364 (l97Q I.

-Q RubtnSletn. R. and AIlur\. S.N., ObJectlvllY of mcre­mental ~onSILtutlve relalio ns over finlle time steps tn .:omputltlonal nnlle deformalLon analySIS. Compo \I elh. tn Appl. ~I ech. & Engg .. 36,:!77·:90 t1983)'

SO. Reed. K.W. lnd AlluT!, S.N., AnalYSIS of large quasi· ~1atlc de(ormatLons of inelastic bodies by a new hybrid­stress (jnne element algomhm. Compo Meth. 1O Appl. Mech. & Engg .. 39. ~45·295 (1 983), and "applications ·'.-10.171·198(1983).

81 AtluT!. S.N. and Perrone. N._ {eds.}. Computer ~Ielhods tor SlonlineJr Sohd and Structural Mechanics. ASr..IE· .\MD. 54 IAS\lE 1'183) 5~4 pp.

~~ -\tluTl. SOS. Al lerrutc stress measures. conjugate )H:lln meJ~urrs. :Ind mL'(ed vaTLalLonal 10rmulaILons tnvolvlOg rigid rotatIons. (or .:omput:nion:l! analyses vI IlOnely delOrmed sohds. wllh applicalLon to plates Jnd ~ell~·part L theory. Computers & Structures, IS.1.93·llbC]Qb4).

S3 \;ovr. .-\ K. Re..:ent Jdvances III reduction methods ror nonltne:1T problems. Comput ers & Structures. 13,

31-44 (1981) , 84. Hughes, TJ .R., Winget. J .. levit . I.. and Tezduyar.

T.E .. New allernalLng direClion procedures III finlle element analysis based on EBE approximate factoriza­tions. III Atlun, S.N. and Perrone . N, (eds.). Computer Methods fo r Nonlinear Solids and Suuclural Mechanics ,ASME-AMD, 54 (1983),

85. Melrovltch. L. . Baruh, H., and Oz. H., A comparison of control techntques for large fleXIble systems. Jm!. GUidance. Control. and DynamiCs, 6. 4, 30::!·3 l 0 tl983 ),

86. \Ie\!ovllch. L.. Baruh . H., Montgomery. R.L and Williams. 1 p " Nonlinea.r natu ral cont rol of an experlm· ental beam. Jrn!. GUidance. Control. and Dynamics. 7.4. 43-'--+4 3 (1984).

87. Kondoh , K. and Atlur\' S.N., Influence of local buckl­ing on global instability : simplified fi n1le deformation. post-buck ling analyses of trusses. Computers & Stru· ctures (in press).

88. Kondon. K. and Alluri. S.N ., A simplified finite ele­ment method for large defOlmatlon post·buckling ana lysis of large frame structures. uSillg explicitly denved tangent stiffness matrices. In tI. 1m!. Num. Meth. in Eogg. (in Te'o'lew).

89. O'Donoghue , P.E. and Atlu rt. S.N .. Connol of tran­SIent dynamic response of structures. ~UMET A85. Swansea. UK (1985) (in press) .

QO. O'Donoghue. P.E . and AliuT!. S.N .. Servo-elastlc osc~la t1ons: control of tranSient dynam1c mOllon of a plate. Advances in Aerospace StructureS-bolo ASME WAM (December I 984)(tn press).

Manus.:npt recel\'ed December ~l. 1984. accepted for publicalLo n January ~4. 1985.