Linear Algebra Provides a Basis for Elasticity without Stress or Strain

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1 Soft, 05, 4, 5-4 Publshd Onln Sptmbr 05 n ScRs. Lnar Algbra Provds a Bass for Elastcty wthout Strss or Stran H. H. Hardy Math/Physcs Dpartmnt, Pdmont Collg, Dmorst, USA Rcvd 6 Novmbr 05; accptd 4 Dcmbr 05; publshd 7 Dcmbr 05 Copyrght 05 by author and Scntfc Rsarch Publshng Inc. Ths work s lcnsd undr th Cratv Commons Attrbuton Intrnatonal Lcns (CC BY). Abstract Lnar algbra provds nsghts nto th dscrpton of lastcty wthout strss or stran. Classcal dscrptons of lastcty usually bgn wth dfnng strss and stran and th consttutv quatons of th matral that rlat ths to ach othr. Elastcty wthout strss or stran bgns wth th postons of th ponts and th nrgy of dformaton. Th nrgy of dformaton as a functon of th postons of th ponts wthn th matral provds th matral proprts for th modl. A dscrt or contnuous modl of th dformaton can b constructd by mnmzng th total nrgy of dformaton. As prsntd, ths approach s lmtd to hypr-lastc matrals, but s approprat for nfntsmal and fnt dformatons, sotropc and ansotropc matrals, as wll as quas-statc and dynamc rsponss. Kywords Elastcty, Strss, Stran, Fnt Elastcty. Introducton Soft matrals lk rubbr, foam, and many bologcal matrals can strtch far byond th lmts of nfntsmal lastcty and yt rturn to thr orgnal shap whn forcs ar rmovd. It s usful to b abl to modl ths dformatons, but nfntsmal lastcty basd on strss and stran tnsors cannot b usd for ths larg dformatons. Th classcal quatons of fnt lastcty ar qut dffcult, rqurng 0 or mor tnsors to xplan th thory. Th dscrpton of lastcty prsntd hr prsnts th sam quatons for both nfntsmal and fnt lastcty, and rqurs only two tnsors to dfn th thory. Th dscrpton of lastcty prsntd hr wll b for hypr-lastc matrals. A hypr-lastc matral stors nrgy whn t s dformd and rturns ths nrgy to ts surroundngs whn t s rturnd to ts orgnal stat. Rubbr s th most common xampl. Th nrgy stord n th matral can b xprssd as a functon of th How to ct ths papr: Hardy, H.H. (05) Lnar Algbra Provds a Bass for Elastcty wthout Strss or Stran. Soft, 4,

2 postons of th ponts wthn th matral. By mnmzng th stord nrgy, dffrntal quatons of lastcty and forcs can b found. Th da of dscrbng dformatons n trms of ponts and forcs was frst usd by Eulr, Lagrang, and Posson and prdats Cauchy s ntroducton of strss and stran []. Th arlr rsarchrs, howvr, dd not complt th dscrpton of th gnral dffrntal quatons for fnt lastcty. I wll do that hr. To do ths, I wll follow th notaton of Spncr [] to dscrb th postons of th ponts wthn th body bfor and aftr dformaton.. Dformaton as a Mappng Dfn th ntal locaton of ach pont wthn a matral as th vctor, X, wth componnts X, =,,. Aftr th dformaton, ach pont n th matral wll b at som nw poston, x, wth componnts x, =,,. Th componnts of ach pont aftr th dformaton ar functons of th componnts of th poston of ach pont bfor dformaton. That s ( ) ( ) x = f X, X, X = x X, X, X. () To match th physcal ralty of th dformaton of a matral, nthr nvrsons nor a chang of dmnson of th matral wll b allowd (.. a thr-dmnsonal matral cannot b turnd nsd out or prssd nto a plan or a ln). As a rsult of ths rstrcton, vry pont n th matral aftr th dformaton wll corrspond to xactly on pont n th matral bfor dformaton, so that th mappng from X to x s on-to-on. Consdr now a pont nar th pont X,.. X = X + dx. Th pont X s mappd nto th pont x = x + dx. W can fnd th rlatonshp btwn dx and dx by dffrntatng Equaton () and usng th chan rul to gv x dx = d X, () X hr and throughout th rst of ths papr, th Enstn summaton notaton s usd so that rpatd ndcs ar x summd ovr. In gnral wll vary from pont to pont wthn th matral (s Fgur ). Th matrx formd by ths valus, X x = () X s calld th dformaton gradnt tnsor by Spncr []. Th rsults found so far ar approprat for any coordnat systm, but I wll us two spcfc nrtal coordnat systms (.. fxd coordnat systms whr Nw- Fgur. In ths fgur, ˆ llustrats th obsrvr coordnat systm. s th mappng dfnd at ach pont n spac from th X pont locaton bfor th dformaton to th sam pont x aftr th dformaton. Any local coordnat systm bfor th dformaton, ê, s mappd nto a nw local coordnat systm aftr th dformaton,. 6

3 ton s laws apply). On I wll call th obsrvr coordnat systm. Ths coordnat systm s th on chosn to solv som problm n (.g. a smulaton or an ngnrng problm). Th scond nrtal coordnat systm I wll call th xprmntal coordnat systm. Ths coordnat systm wll b th on chosn by an xprmntalst who wshs to masur th nrgy assocat wth a partcular dformaton of a partcular sampl. In th obsrvr coordnat systm, I wll dnot th componnts of o as. In th xprmntal coordnat systm I wll dnot th componnts as. I wll frst dscrb how th xprmntr should masur th matral proprts of th matral n hs coordnat systm.. Masurng th Enrgy of Dformaton It s suffcnt to lmt xprmntal dformatons to homognous dformatons of a homognous porton of th matral n ordr to dfn th nrgy of dformaton. If th body s ansotropc, t s ncssary to dfn th orntaton of ths ansotropy. Ths could b dtrmnd by a vsual nspcton (.g. wood gran) or by a knowldg of how th matral was mad (.g. rbar n concrt). Ornt th ansotropy to algn wth th xprmntal coordnat systm so that n th xprmntal coordnat systm th ansotropc coordnat systm of th matr- ( ) ( ) ( ) al s ˆ for =,,, and ˆ = (,0,0 ), ˆ = ( 0,,0 ), ˆ = ( 0,0,), or ˆ ( k ) = δk. (Not that hr th suprscrpt, k, dnots whch coordnat vctor and th subscrpt,, dnots whch componnt.) Usng th notaton of Equaton (), Equaton () n th xprmntal coordnats systm bcoms dx = dx (4) A homognous dformaton s a dformaton n whch vry pont n th body undrgos an affn transformaton. Lnar algbra dfns an affn transformaton as a mappng that transforms any pont n spac X to anothr pont n spac x by a matrx transformaton F, followd by a translaton d. Applyng ths to our matral body, vry pont n th body aftr th dformaton, x, corrsponds to a pont bfor th dformaton, X, as x = FX + d (5) whr X = coordnats of th poston of any pont bfor th dformaton, x = coordnats of th poston of ths sam pont aftr th dformaton, F = matrx of valus that ar th sam throughout th body, and d = componnts of a vctor that ar th sam throughout th body. Takng th drvatv of Equaton (5) and comparng th rsult wth Equaton (4), w fnd that n our xprmntal coordnat systm F =. Equaton (5) ndcats that to compltly dfn any dformaton n our xprmntal coordnat systm, w nd th nn componnts of and th thr componnts d. Ths valus can b found by rcordng th locaton of any four non-coplanar ponts n th xprmntal matral bfor and aftr th homognous dformaton. Th coordnats of ach of ths ponts provds x and X n Equaton (5). Snc thr ar four ponts wth thr componnts ach, ths gvs a total of quatons and unknowns of th form of Equaton (5). From ths quatons, all componnts of and d can b found for any xprmntal dformaton. To masur th nrgy durng xprmnts, rcord th appld forcs, F m, and th corrspondng dsplacmnts of th ponts whr ths forcs ar appld, dx m wth m = to M, whr M s th numbr of appld forcs. Th nrgy of dformaton s th total work don by ths forcs,.. th sum of th ntgrals from th ntal to th fnal poston of th ponts whr th forcs ar appld: tot fnal = F dx (6) ntal Th nrgy pr orgnal volum, E, s found by dvdng th total nrgy by th ntal volum of th sampl, V. tot E = (7) V Wth ths masurmnts, an xprmntr can construct a tabl of th stord nrgy pr unt ntal volum as m m 7

4 a functon of th componnts of and d. To complt th dscrpton, us a lnar ntrpolaton or an quaton ft to th tabl of data to dfn E as a contnuous functon of and d,.. (, ) E = f d (8) Ths compltly dfns th nrgy pr unt orgnal volum for homognous dformatons of th matral sampl. 4. Spcal Exprmntal Cass W hav found that n gnral th xprmntally masurd nrgy pr unt volum, E, can b a functon of as many as componnt valus (nn n and thr n d ). It s a bt much to xpct an xprmntr to map out a dmnsonal nrgy spac for vry matral for vry applcaton, so t s usful to fnd som smplfcatons dpndng upon th partcular matral and applcaton w ar ntrstd n. For xampl, for most applcatons, th only xtrnal body forc that nds to b consdrd s gravty. In that cas, w only nd to xprss th nrgy of translaton as a functon of û d, whr û corrsponds to th drcton of th gravtatonal forc and d th dsplacmnt of th cntr of mass of th matral. If th xprmntal coordnat systm s algnd so that th thrd componnt of d s paralll wth th gravtatonal forc, w only nd to nclud nrgy changs du to d and can omt th two varabls d and d n our nrgy functon, bcaus ths dsplacmnts wll rsult n no chang n th nrgy of th body. A furthr rducton of paramtrs for rotatons and dformatons can b found f w tak a Sngular Valu Dcomposton (SVD) of. Th SVD of a matrx unquly dvds th matrx nto thr matrcs, two rotatonal ( R and R ), and on dagonal, Λ. In partcular, SVD = R ΛR (9) ( ) Ths form wll b usful n dscrbng both xprmntal procsss to masur th nrgy of dformaton and th quatons of lastcty. Snc any dformaton,, can b xprssd unquly as RΛR, thn vry possbl dformaton can b consdrd a combnaton of a rotaton, R followd by a strtch or comprsson along th thr fxd orthogonal coordnat axs, Λ, followd by a fnal rotaton, R. In th most gnral cas th nrgy of th matral can dpnd upon body forcs from lctrc or magntc forcs n addton to gravty. For xampl, a matral wth an lctrc dpol ( p ) or a magntc dpol ( µ ) n an lctrc (ξ ) or magntc ( ) fld, wll hav nrgy p ξ or µ, rspctvly so that th nrgy wll b a functon of R, Λ, and R. If th matral has a charg, q, nrgy must nclud qξ d. In ths cass th nrgy of dformaton may b a functon of all nn varabls n F as wll as th thr varabls n d. If th lctrc and magntc body forcs ar not sgnfcant n a partcular applcaton, th thr componnts of R nd not b ncludd n calculatng E snc n that cas th nrgy assocatd wth dformatons ar ndpndnt of body rotatons aftr th dformaton. If th body s sotropc, thn th nrgy wll b ndpndnt of both R and R snc rotatng th body bfor or aftr th dformaton wll produc no chang n th nrgy of dformaton. In that cas only th thr dagonal valus of Λ, Λ, ar ndd to dscrb th nrgy assocatd wth dformaton. If n addton, th matral can b consdrd ncomprssbl, thn th volum of th matral, V, s constant V = = Λ Λ Λ = constant, (0) and only two ndpndnt lmnts of Λ nd to b usd to dscrb th nrgy durng a dformaton. In addton to ths smplfcatons, t s suffcnt to masur th nrgy as a functon of only thos changs that ar xpctd n a partcular applcaton. So for xampl, f nfntsmal dformatons ar suffcnt to modl th problm at hand, only on small dsplacmnt masurmnt n ach drcton s rqurd. Altrnatvly, f th body s gong to b usd only n xtnson, thr s no nd to masur th nrgy assocatd wth comprssonal forcs. Rvln [] usd ths approach for rubbr. For hs applcaton th rubbr could b consdrd sotropc and ncomprssbl, so h only dformd th matral sampl n xtnson along two prpndcular drctons. Ths s suffcnt to fnd th nrgy as a functon of Λ and Λ. Fnally, f a dformaton contans only rgd body motons, matrals ar nthr comprssd nor xtnd, so that Λ = for =,,. Thus a rgd body rotaton, Equaton (9) gvs = RR. Snc rgd body rotatons can b xprssd n trms of a sngl rotaton matrx, th sx paramtrs n R and R rduc to only 8

5 thr. What w hav found s that w can rduc th numbr of varabls that th nrgy s a functon of from to as fw as n th cas of an ncomprssbl, sotropc matral whr w can gnor xtrnal body forcs lk gravty n our applcaton. In gnral, howvr, all varabls may b rqurd and t s hlpful to fnd th most computatonally ffcnt way to rprsnt th nrgy for ach applcaton. 5. Som Applcaton Issus W hav found that SVD( ) can b usd to smplfy th xprmntal masurmnts for partcular cass; howvr, SVD s a rathr computatonally havy calculaton to b usd durng smulatons. It s thrfor usful to rprsnt th dformatons n a mor computatonally ffcnt mannr for applcatons. For xampl, f w consdr th cas whr th matral s sotropc and thr ar no body forcs, th nrgy pr unt volum s a functon of only Λ. Thus any thr ndpndnt varabls spannng th sam spac as Λ may b usd to charactrz th nrgy functon. Th valus Λ, Λ, and Λ can b rwrttn [4] as In addton I = Λ + Λ + Λ I I = ΛΛ + ΛΛ + ΛΛ = ΛΛ Λ. I = a a+ b b+ c c I = + + I = ( a b) ( a b) ( a c) ( a c) ( b c) ( b c) a ( b c), whr a, b, and c ar th column vctors of. Thus n smulatons of sotropc bods t s not ncssary to comput SVD as th smulaton progrsss. All that s rqurd ar th componnts of to comput th ndd thr ndpndnt valus. As a rsult t s bst n ths cas to rdfn th xprmntally dfnd nrgy, E f Λ E = f I abc,,, I abc,,, I abc,, aftr th xprmnts ar compltd. Onc E has bn = ( ) as ( ( ) ( ) ( )) convrtd from a functon of Λ to a functon of th column vctors of, t s only ncssary to fnd th componnts of durng smulatons. Th SVD( ) s no longr rqurd. If th matral s ansotropc, th nrgy pr unt volum s a functon of th sx ndpndnt valus of Λand R, Thr from th dagonal lmnts of Λ and thr angls from QRD, provds a mor ffcnt vnu for calculatons than dos SVD( ) R. QR dcomposton, ( ). (For ths applcaton w must us th Gram- Schmdt QRD algorthm nstad of th mor common Housholdr QRD, bcaus th Housholdr algorthm prmts nvrsons.) QRD( ) producs R, whr s an uppr trangular matrx and R s a rotaton matrx. Snc th nrgy of dformaton s ndpndnt of any rotaton aftr th dformaton, th sx componnts of, lk th thr valus of Λ for th sotropc cas, can b usd to dfn th nrgy of dformaton. Th Gram-Schmdt QRD algorthm s not a partcularly havy numrcal calculaton and can b usd n applcatons, but an altrnatv s also possbl. As I hav notd prvously [5] th componnts of can b wrttn n trms of dot and cross products of th column vctors of. Thus th nrgy of dformaton can b wrttn n trms of dot and cross products of th column vctors of and onc th nrgy s xprssd n ths trms, t s not ncssary to calculat any othr tnsors than n carryng out applcatons; howvr, w do hav to consdr th orntaton of any ansotropy for ngnrng applcatons. 6. Coordnat Algnmnt Whn an ansotropc matral s placd n srvc t s ncssary to know th ntal orntaton of th ansotropy. Ths s bcaus th stord nrgy s a functon of th orntaton of th sotropc matral rlatv to th obsrvr s coordnat systm. For xampl consdr a lamnat. If th lamnat s orntd so that th lamna ar paralll to th x-y plan n th obsrvr s coordnat systm and xtndd n th x drcton a fxd amount thr wll b a chang n nrgy of th matral. Howvr, f th sam lamnat s ntally orntd so that th lamna () () 9

6 ar paralll to th y-z plan n th obsrvr s coordnat systm and xtndd n th x drcton th sam fxd amount thr wll b a dffrnt chang n nrgy of th matral. Thus th ntal orntaton of th lamna n th obsrvr s coordnat systm must b known n ordr to corrctly calculat th stord nrgy n th matral. Whn th matral s placd n srvc bfor any dformaton has occurrd, th ansotropc coordnat systm n whch th nrgy masurmnts wr mad, ˆ ( k ), and th obsrvr coordnat systm, ˆ ( k ), may not algn. Dfn a rotaton matrx, ˆ ( k ) = δk 0 R, whr 0 R maps th componnts of th obsrvr s coordnat systm,, nto th componnts of ansotropc coordnat systm bfor any dformaton,.. ˆ = 0ˆ. () ( k) ( k) R Th local dformaton whch s xprssd n th obsrvr s coordnat systm, xprmntal coordnat systm, Not that 0 R l s at most only a functon of, by a smpl chang of coordnats of ( R ) T 0 o 0 k kl l o, can b xprssd n th o [6], = R (4) X. It s not a functon of x snc t s dfnd bfor any d- formaton has takn plac. For an nhomognous matral, th matrx may vary from pont to pont, so x that th nrgy would b a functon of X as wll as. (Of cours any nhomognous matral mght also b mad up of dffrnt matrals whch hav dffrnt nrgy maps at dffrnt locatons n spac, X X.) Thus n th most gnral cas, So that n gnral, 7. Is Ths Enrgy a Scalar? x = f, X an d d = f ( x, X ) (5) X x E = f (, d ) = f, x, X (6) X It may sm strang that w must transfr th coordnat valus of th matral n th obsrvr s coordnat systm back nto th xprmntal coordnat systm n ordr to fnd th local chang n nrgy, but ths s xactly as t should b. Enrgy must b a scalar, whch s ndpndnt of th choc of th coordnat systm. That ths s th cas can b sn f w xprss th nrgy n trms dot products of vctors, whch ar ndpndnt of th ( ) coordnat choc. Not that th local ansotropy coordnat systm vctors, ˆ k, ar mappd nto a nw st of ( k ) vctors,, whch ar n gnral nthr orthogonal, nor unt vctors (.g. s Fgur ). In th obsrvr s coordnat systm ths mappng s ( k) o ( k) = ˆ (7) Th corrspondng mappng of ths sam vctors n th xprmntal coordnat systm s ( k) ( k) ˆ = (8) ( ) ( ) W can choos to xprss th nrgy as a functon of th nvarant, ˆ, whch must hav th sam valu n both coordnat systms. To s th conncton btwn th maps n th dffrnt coordnat systms, xpand th dot product n both systms and compar th rsult. In th xprmntal coordnat systm, ( ) ( ) ( ) ( ) ( ) ( ) ˆ = ˆ ˆ ˆ k k = k k = δkk δ = (9) ( ) In th obsrvr s coordnat systm, whr ˆ k ( ) ( ) = δk and ˆ k 0ˆ k = R, w hav 0

7 ˆ R ˆ R ( ˆ ) ( ) ( ) ( ) ( ) ( ) o ( ) 0 ( ) o 0 ( ) = ˆ ˆ ˆ k k = k kl l = kmm kl lnn ( ) T o o o 0 kmδmkl lnδ n = kkl l k kl l = R R R R = R R. H. H. Hardy ( ) ( ) Comparng Equaton (9) to Equaton (0) rturns us to Equaton (4) snc ˆ s nvarant and must b th sam valu n both coordnat systms. Ths s th sam typ of xplanaton that must b usd n xprssng th nrgy n trms of th componnts of dsplacmnt vctors, although w usually do not dscuss t n ths trms. Usually th nrgy assocatd wth th dsplacmnt can b calculatd from a formula nstad of havng to carry out ndvdual xprmnts for ach matral. For xampl, a dsplacmnt n th prsnc of gravty changs th nrgy stord n th matral, but t s asly xprssd as mg d whr m s th mass and g s th acclraton du to gravty and no xprmnts ar ncssary. Howvr, f w dd carry out th xprmnts, w would masur th nrgy of dformaton n trms of th componnts of d n th xprmntal coordnat systm. In that cas w would nd to tak nto account any chang n th componnts of d n th obsrvrs coordnats. For xampl, assum th nrgy stord du to gravty s xprssd n th xprmntal coordnat systm as mgd, whr th forc of gravty n th xprmntal coordnat systm has bn chosn to algn wth X. If w placd ths matral nto an obsrvr coordnat systm whr gravty s n th X drcton, w must frst rotat th coordnat systm to algn th X drcton wth th X drcton bfor lookng up th corrspondng nrgy that w stord n our nrgy map w compld by masurng th nrgy n th xprmntal coordnat systm. Of cours f w choos to xprss th nrgy n trms of th dot product, mg d all of ths would tak car of tslf. ( ) A word of cauton s ncssary hr. Th vctors ˆ ( ) and ar th sam vctors n both th obsrvr and th xprmntal coordnat systms. On th othr hand, th column vctors, a, b, and c, ar only dfnd n th xprmntal coordnat systm, and thrfor th transformaton n Equaton (4) must b carrd out bfor xtractng th column vctors of to calculat th nrgy of dformaton for an ansotropc body. Ths s not ncssary for sotropc matrals bcaus n that cas th nrgy s ndpndnt of all rotatons and applyng th transformaton Equaton (4) has no ffct on th fnal nrgy calculaton,.. th Λ s ar th sam for and. 8. Smulatons o I hav now compltly dfnd a mthod to masur th nrgy of dformaton of a homognous body xprmntally and how to plac th matral n a gvn applcaton. Bcaus ngnrng applcatons ar oftn complx, t s usually ncssary to put ths nformaton nto a computr smulaton. Th smplst approach s to ust pastng small pcs of th matral togthr to dfn th complt matral. Ths can b don by randomly postonng ponts n th matral and us ths to dvd th matral to b smulatd nto small ttrahdrons of volum ΔV k, ach boundd by four ponts. Th ntal and fnal locatons of ths four ponts put nto Equaton (5) can b usd to dfn th paramtrs ndd to calculat th nrgy pr unt orgnal volum for ach ttrahdron. Th total nrgy of th systm s ust th wghtd sum of th nrgy pr unt volum of ach ttrahdron, x Etot = Ek, x, X ΔVk () X whr k s summd ovr all th ttrahdra n th matral. Apply th boundary condtons and mov th ntrnal ponts to produc mnmum total nrgy, E tot, and w hav a soluton. I hav calld ths ths dscrt rgon modl [4]. An altrnatv mthod s to us a contnuous Eulr-Lagrang tchnqu to mnmz th functonal, x Etot = E, x, d matral X V X () Th rsult of ths approach s a st of partal dffrntal quatons whch can b solvd by any numrcal tchnqu (.g. fnt dffrnc, fnt lmnt, Raylgh-Rtz, tc.). I hav calld ths Eulr-Lagrang lastcty [5]. (0)

8 9. Dffrntal Equatons Th dscrt rgon mthod collapss nto Eulr-Lagrang lastcty f th sz of ach ttrahdron approachs zro as th numbr of ttrahdra, N, ncras wthout bound,.. N x x Etot = lm Ek, x, X Vk = E, x, X d V. N V 0 k = X X () To fnd th dffrntal quatons of lastcty w nd to mnmz (or fnd th xtrma) of E tot,.. x δetot = δ E, x, X dv = 0. X (4) Wth dv = dxdxdx. Ths s a classc Calculus of Varatons problm wth multpl varabls [7]. Th rsults of ths mnmzaton ar th followng thr Eulr quatons: E d E = 0. x dx ( x X ) Ths thr quatons ar qut gnral, bng approprat for both nfntsmal and fnt dformatons, for sotropc and ansotropc matrals, and can nclud surfac forcs, gravty, and lctrcal and magntc forcs. 0. Spcal Applcaton Cass If th matral s homognous, E df wll not b a functon of X. If gravty s th only xtrnal body forc, E can b sparatd nto th nrgy of dformaton, E df, and nrgy of body forcs, E body : x x E, x, X = Edf + Ebody ( x, X) (6) X X If only nfntsmal dformatons ar ndd, thn E can b xpandd n a Taylor s xpanson whch ylds th sam dffrntal quatons Landau drvd for nfntsmal dformatons [8] usng classcal strss and stran tchnqus. If tm dpndnc s rqurd, dfn th Lagrangan, = T E, (7) x x x ρv ρ whr T = = + + t t t, wth ρ th mass pr orgnal volum. Fndng th xtrma of (5) dx dx dx dt (8) = rsults agan n thr Eulr quatons, now of th followng form: d d = 0 x dx ( x ) dt X ( x t) Ths thr quatons ar th tm dpndnt dffrntal quatons for hypr-lastcty [9]. All that s ndd now s to nclud boundary condtons and forc. Boundary condtons consst of Numann and Drchlt boundary condtons. Drchlt boundary condtons ust st th postons of boundary ponts of th matral. Numann boundary condtons can b xprssd n trms of appld forcs on th surfacs of th matral [5], surfac forc E df = da (0) x X ( ) whr d A ar th componnts of th orgnal surfac ara whr th forcs ar appld. Equatons (0) provd dffrntal quatons to st th boundary condtons f th appld surfac forcs ar known. (9)

9 . Comparson to Othr Elastcty Thors Th most obvous dffrnc of ths approach and classcal lastcty s that n ths approach thr s no dfnton of strss or stran. Hr dsplacmnts and forcs ar th altrnatvs to strss and stran. Ths approach also rqurs th dfnton of only on scond ordr tnsor, th dformaton gradnt tnsor. In lastcty wth strss and stran mor than 0 tnsors hav bn usd to dscrb fnt lastcty [0] []. ( ) ( ) ( ) ( ) For classcal lastcty wth strss and stran, th nvarants ˆ ˆ, or, ar usd to dscrb stran n T T trms of or. In that cas stran s scond ordr n th dsplacmnts, dx In th approach pr- ( ) ( ) sntd n ths papr, th nvarants usd ar ˆ, and th nrgy of dformaton s calculatd from th matrx tslf, whch s only frst ordr n dx. Infntsmal lastcty also rqurs compatblty quatons rlatng strss and stran whch xprsss matral proprts as fourth ordr tnsors. Th approach gvn hr xprsss matral proprts for all hypr-lastc matrals as a scalar, th nrgy pr orgnal volum, E. Som dscrptons of lastcty dfn dx and dx to b rprsntd n dffrnt coordnat bass []. In ths dscrptons, s thn calld a two-pont tnsor. In ths prsntaton, all vctors and tnsors ar xprssd n th sam coordnat bass, thr all n th obsrvr or all n th xprmnt coordnat systms. Thr ar no two-pont tnsors. In som dscrptons of lastcty, only obctv tnsors ar usd to formulat th consttutv quatons. Obctv tnsors ar rqurd to b ndpndnt of th moton of th matral that s bng dformd. That s, ths tnsors should b th sam n both a fxd rfrnc coordnat systm and n a coordnat systm that dforms wth th matral []. In ths papr all physcal quantts ar xprssd only n trms of on of two coordnat systms fxd n spac bfor any dformaton taks plac. In ths papr, nthr th obsrvr coordnats nor th xprmntal coordnats dform as th matral dforms. Both ar nrtal coordnat systms fxd n spac. (Th ansotropy coordnats ˆ ( k ) ( k ) do dform n spac nto aftr a dformaton, but ths ar not usd as bass to dscrb th componnts of vctors or tnsors.) Som dtals may b asly confusd btwn ths approach and th Classcal approach. Thr xampls follow: T Λ ar not th of nvarants of th rght Cauchy-Grn dformaton tnsor, whch s dscrbd as. Instad th Λ valus ar th dagonal lmnts of SVD( ). QRD s not th sam as Polar dcomposton that s usd n classcal lastcty thory. Polar dcomposton producs R, whr s a symmtrc matrx, whras QRD producs R, whr s an uppr trangular matrx. E Th trm n Equaton (0) s quvalnt to Cauchy strss only for nfntsmal dformatons. x X. Concluson ( ) A mthod of dscrbng hypr-lastcty usng lnar algbra has bn prsntd that uss ponts wthn th matral and forcs nstad of strss and stran. Th thory provds a straght forward way to masur matral proprts and allows th ncluson of magntc and lctrc flds as wll as gravty. Ths dscrpton uss th sam quatons for both fnt and nfntsmal dformatons. Numann boundary condtons ar xprssd n trms of masurd forcs nstad of computd strsss. Th rsult s a complt thory of hypr-lastcty whch ncluds nfntsmal and fnt dformatons, sotropc and ansotropc matrals, quas-statc and dynamc lastc rsponss. Acknowldgmnts I would lk to acknowldg Mr. Joshua Wood and Dr. Mchal Brglund for thr vry frutful dscussons and hlpful suggstons. Rfrncs [] Todhuntr, I. (886) A Hstory of th Thory of Elastcty and of th Strngth of Matrals from Gall to th Prsnt Tm. Cambrdg Unvrsty Prss, Cambrdg. [] Spncr, A.J. (980) Contnuum Mchancs. Dovr, Nw York.

10 [] Rvln, R.S. and Saundrs, D.W. (95) Larg Elastc Dformatons of Isotropc Matrals. VII. Exprmnts on th Dformaton of Rubbr. Phlosophcal Transactons of th Royal Socty of London. Srs A. Mathmatcal and Physcal Scncs, 4, [4] Hardy, H.H. and Shmdhsr, H. (0) A Dscrt Rgon Modl of Isotropc Elastcty. Mathmatcs and Mchancs of Solds, 6, [5] Hardy, H.H. (0) Eulr-Lagrang Elastcty: Dffrntal Equaton for Elastcty wthout Strss or Stran. Journal of Appld Mathmatcs and Physcs,, [6] Glbrt, J.D. (970) Elmnts of Lnar Algbra. Intrnatonal Txtbook Company, Scranton. [7] Glfand, I.M. and Fomn, S.V. (99) Calculus of Varatons. Dovr, Nw York. [8] Landau, L.D. and Lfshtz, E.M. (005) Thory of Elastcty, Cours of Thortcal Physcs. Volum 7, Elsvr, London. [9] Hardy, H.H. (04) Eulr-Lagrang Elastcty wth Dynamcs. Journal of Appld Mathmatcs and Physcs,, [0] Trusdal, C. and Noll, W. (004) Th Non-Lnar Fld Thors of Mchancs. Sprngr-Vrlag, Nw York. [] Wu, H.-C. (004) Contnuum Mchancs and Plastcty. CRC Prss, Nw York. [] Klly, P. (05) Mchancs Lctur Nots Part III: Foundatons of Contnuum Mchancs. 4