( ) Abstract. 2 FEDSS method basic relationships. 1 Introduction. 2.1 Tensorial formulation

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1 Displacmnt basd continuous strss rcovry procdur, Mijuca D, Brkoviæ M. and Draškoviæ Z., Advancs in Finit Elmnt Tchnology, ISBN , Ed. B.H.V.Topping, Civil Comp Prss, 7-34 (996). Abstract In this papr th problm of a finit lmnt strss rcovry is considrd. First, a nw global coordinat indpndnt approximation of a continuous strss fild is prsntd. It has bn shown that th proposd, FEDSS (Finit Elmnt Displacmnt typ Strss Smoothing) mthod is computationally mor fficint, at last compard with th classical strss avraging procdur. Scond, thr is a numrical vidnc that, at last for four nodd isoparamtric lmnts, any strss rcovry procdur is lss accurat in strain nrgy than dirct FEA (Finit Elmnt Analysis). It has also bn shown that, for bilinar isoparamtric lmnts, th rlativ nrgy rror norm with rspct to th xact solution, computd by Gaussian intgration is smallst for th raw FEA, compard with any othr strss rcovry procdur and any typ of th numrical quadratur. Hnc, on can rcommnd (undr)intgration in th midpoint of an lmnt, i.. in th drivativ (strss) suprconvrgnt point, whn rror indicators of Z Z (Zinkiwicz Zhu) typ [] ar calculatd. Introduction Accurat strss prdiction is of crucial importanc in th analysis and dsign procdurs of ral physical bodis. Th quality of a solution in th finit lmnt analysis can b improvd by th msh rfinmnt. Howvr, incras of th msh dnsity drastically raiss th solution tim. Also, classical displacmnt typ finit lmnt analysis gnrats continuous displacmnt fild, but discontinuous strss fild ovr th modl. Obviously, an accptabl strss rcovry procdur should dlivr accurat nough continuous strsss ovr th modl in a rasonabl tim. In addition, drivativ rcovry tchniqus ar usd for rror stimats in adaptiv finit lmnt procdurs. Thr ar two gnral classs of th strss smoothing procdurs [,, 3]. If carrid out ovr a whol finit lmnt domain th procdur is known as a global smoothing. Local smoothing is prformd at ach nod or small group of nods. On of th simplst local smoothing procdurs is th avraging of strsss from nighbouring lmnts at a particular common nod. For th purpos of th prsnt papr th mthod is namd FEAavrg. Howvr, all ths approachs including also innovativ [4] ons, ar basd on th convntionally calculatd strsss. Hnc, all known approachs ar scalar, which mans that thy approximat only on strss componnt at a tim. It has bn shown in [] that to maintain invarianc of th finit lmnt approximations undr th coordinat transformations, tnsorial charactr of ths approximations should b strictly rspctd. A significant novlty of th proposd procdur should b pointd out it is both global and tnsorial. Ths two faturs, along with high accuracy and computational fficincy, ar combind in th prsnt FEDSS mthod. Th scond novlty introducd, which is albit applicabl in any Z Z typ rror stimator is aformntiond undrintgration of rror indicator. FEDSS mthod basic rlationships Proposd FEDSS mthod can b usd as a strss smoothing procdur as wll as an rror stimator. At varianc with th known approachs, it is basd not on th convntionally calculatd strsss, but on th displacmnts. This tchniqu is inspird with som rsults in [6], rlatd to th two-fild finit lmnt mthod. Nvrthlss, it has b shown that similar idas can b applid in th strss rcovry problms.. Tnsorial formulation In [7] starting point was th strss displacmnt rlationship: T ( ) A: t = u + u, whr t is th strss tnsor, u th displacmnt vctor, and A th lastic complianc tnsor. A wak solution of () on can gt by th Galrkin procdur using strss variations as th tst functions. Finit lmnt approximations ar introducd similarly as in [7]. Aftr rcasting of all quantitis in th coordinat form, on obtains th rlationship: () A t = B u () ΛstΓuv Γuv whr, for an lmnt : Γq Λst Γq ( ) ( ) ( ) L M i k m o AΛstΓuv = Λ Γ ( ) s ( ) t ( ) u ( ) Λ Λ Γ Γ v i k m o (3) fgh PL PM A dv, f g h ξ ξ ξ ξ

2 ( ) ( ) ( ) Γq L Γ K i k BΛst = Λ K PL Pa ( Λ ) s ( Λ ) t ξ i k a ( Γ ) pq ( ) g g dv. Γ p In ths xprssions a, b, c, d ar indics of local lmnt ( ) coordinat systms; Λ K is th incidnc matrix which maps global nods into th local nods of an lmnt; P K ar th ( Γ ) pq local valus of shap functions; symbols ar th componnts of th contravariant mtric tnsor at nod ( Λ ) p Γ, indics p, q, r, s, t rfr to th nodal coordinat systm x at nod Λ; i, j, k, l ar th indics of global Cartsian coordinat systm z i of a modl; symbols g ab ar th componnts of th contravariant mtric tnsor in th intrior of an lmnt and a, b, c, d ar th indics of a local, lmnt coordinat systm ξ a. Finally, A fgh ar th contravariant componnts of th lastic complianc tnsor. Omission of th indx in (3), (4), tc., mans global valus of th quantitis undr considration, i.., simpl summation of ths. In th Cartsian coordinats, applicabl for flat two-dimnsional and for thr dimnsional configurations, xprssion (3) has th following intrprtation: ( ) ( ) ( ) i j k L M AΛst Γuv = Λ Γ ( Λ ) s ( Λ ) t ( Γ) u g l ( ) P A P dv Γ v L ijkl M.. Matrix rprsntation of th proposd tchniqu Hrin on hav to solv th systm of quations (). Th first stp will b th matrix rprsntation of a problm. Matrix formulation will b mad in th sam mannr as it has bn don in [6]. Global systm of quations () has th following matrix rprsntation: Γuv Γq [ AΛstΓuv ] { t } { BΛstuΓq } (4) () =. (6) Matrix [ A ΛstΓuv ] is positiv dfinit, symmtric and spars. At varianc with undrintgration which will b rcommndd for th computation of rror indicators, this matrix (to avoid ill-conditioning) should b intgratd by full Gaussian, or vntually Lobatto typ [] formula.. Comparison of th FEDSS and som wll known strss rcovry mthods Th numrical rsults of th prsnt procdur ar, up to th ordr of th rounding rror, th sam as in global strss smoothing mthods proposd arlir by Odn [], Hinton and Campbll [3], Zinkiwicz and Zhu [], dspit th fact that in th krnl of th systm matrics (3) and () w hav lastic cofficints, whil in th aformntiond mthods thr ar only th shap functions, P L P M, in th krnl. Of cours, th right hand sid is also diffrnt. Anyhow, such idntical rsults ar rathr unxpctd. It should b howvr pointd out that, to gt idntical rsults, th sam st of th points of numrical intgration should b takn for both mthods. Th advantag of th proposd procdur is a fact that it can b usd also in th cas whn only th displacmnts ar availabl, ithr from thortical considrations, numrical analysis, or an xprimnt. Also, spcial attntion is paid on th tnsorial invarianc of th of corrsponding xprssions. Hnc th prsnt formulation allows us of diffrnt coordinat systm at ach obsrvd nod. Obviously, tnsorial invarianc can b kpt indpndntly on th kind of strss smoothing approach [8]. 3 Dtrmination of th strain nrgy and appropriat norms Traditionally, th strain nrgy T U = d t A t, (7) has bn usd [9] for stimation of accuracy and convrgnc of th finit lmnt solutions. In this xprssion t is an xact strss solution, A th lastic complianc tnsor and th domain of th modl. Th nrgy of finit lmnt solution at th lmnt lvl is obviously: Uh = th T A t h d, (8) whr is th domain of an lmnt. Not also that t h is a finit lmnt strss, dpndnt on th strss rcovry procdur. Th popularity of abov masur is partially du to a fact that it is, at a systm lvl, bcaus of th First Law of Thrmodynamics, qual to th work of th xtrnal forcs (at last for hyprlastic matrials) which can b asily calculatd [0]. Howvr, if a local discrtization rror stimat for a givn msh is ndd, it is practical to introduc so calld nrgy (rror) norm: T = ( t th ) A( t th ) d / It is usually takn as grantd that smoothd (continuous) strss fild is mor accurat than finit lmnt discontinuous strss pattrn. Howvr, th numrical tsts show that th strain nrgy (8) at global lvl, if raw FE strsss ar considrd, is closr to (7) than if w us any sophisticatd strss smoothing (s Figur 3). Hnc, for th purpos of th strss stimation, in th quation (8) w can rplac th xact strss t by t FEA raw finit lmnt rsults, and t h by t smooth th rsult of any smoothing procdur. Th corrsponding a postriori rror indicator of Z Z (Zinkiwicz Zhu) typ [] is, for an lmnt: T = ( t t ) A ( t fa t smooth ) d fa fa smooth / (9) (0) whr t FEA is a raw strss fild from th displacmnt FE analysis, whil t smooth is a strss fild obtaind by som smoothing procdur. Analogously, on can introduc th absolut nrgy norm:

3 u or mor spcifically: = U () h t = E x (6) u T = t A t fad fa fa / () It is asy now to dfin th rlativ prcntag rror [] or prcision [] as: η = 00 (3) u ( ) = 00 (4) u η fa fa fa Th lmnt rror indicator (4) is almost indntical to thos proposd in [] and []. Dnominator in ths rfrncs / is dfind as ( u + ), assuring η <00% for coars mshs, but approaching () for ralistic ons. Howvr, most important diffrnc btwn th prsnt and prvious approachs is undrintgration (intgration of in th drivativ (strss) suprconvrgnt points [3]) which dcisivly incrass rliability of Z Z typ rror stimators. 4 Numrical xampls In this sction two typs of numrical tsts ar prformd. Th first on is th study of quality of th proposd FEDSS strss rcovry procdur. Scond is an analysis of th problm of rror stimation of finit lmnt solution. In th availabl rfrncs, diffrnt strss rcovry procdurs ar gnrally prsntd as suprconvrgnt, mor accurat than undrlying finit lmnt analysis. Numrical xampls prsntd hr will show that such rsults ar probably obtaind with intgration formula of highr ordr than it is justifid and ncssary whn th nrgy rror norm (indicator) is calculatd. 4. On-dimnsional xampl Lt us considr on dimnsional xampl proposd in [4]. For th purpos of this papr it is modlld by on row of two-dimnsional lmnts. This modl is particularly usful for th dmonstration of diffrnt strss rcovry approachs, lik FEA, FEDSS and FEAavrg. Spcial attntion is paid on th dpndnc of rsults on th ordr of th Gaussian quadratur. Th xrcis is sktchd on Figur. Modulus of lasticity and Poisson s cofficint ar takn to b E = and ν = 0, rspctivly. Th govrning quations for this problm ar: t 4Ex = 0, u( 0) = 0, u( 3) = t = E u. Exact strss is:, x, () Figur. 3 On dimnsional xampl On Figur th xact strsss, strss solutions of classical finit lmnt analysis and two typs of strss smoothing procdurs, FEDSS and FEAavrg ar shown. Strss/E Figur. Exact FEA FEDSS FEA avrg x Effcts of th various strss rcovry procdurs Th finit lmnt strsss ar discontinuous ovr th boundaris of lmnts, whil xact and smoothd strsss ar continuous. In th cntrs of lmnts th diffrnc btwn EXACT and FEA strsss is smallst compard with th combinations EXACT FEDSS and EXACT FEAavrg. This is not th cas for othr positions of th Gaussian intgration points whr FEA solution is gnrally mor in rror than smoothd solutions. Consquntly, numrical xaminations show that for highr ordr Gaussian intgration rlativ nrgy rror norms ar largr than for on point Gaussian quadratur, in contrary to th full strain nrgis, which ar mor accurat (closr to th xact valus) whn intgratd by th highr ordr quadratur, Tabl. Finally, on can conclud that th strss rror, Figur, and th rrors in nrgy, Tabl, smallr for th global (FEDSS) than for th local (FEAavrg) strss smoothing, ar mor clarly rprsntd by on point than by two point Gaussian intgration. In fact "accurat" intgration of nrgy rror norm intgrats th squar of a local gap (s Figur ) btwn th xact and FE solution, and hnc th nrgy rror norm is inadquatly rprsntd by highr ordr intgration

4 mthod strain nrgy xact= η 3 l η l analytical analytical FEA FEA FEAavrg FEA avrg FEDSS FEDSS Tabl. Strain nrgy and rlativ nrgy of on-dimnsional modl 4. Rctangular in plan loadd plat with th prscribd displacmnt Th problm is borrowd from [4]. Aim of this subsction is to show how th ordr of Gaussian quadratur influncs th quality of strss rcovry procdurs in a two dimnsional msh. Rctangular domain dtrmind by th points (0,0), (,0), (,) and (0,) is considrd. Modulus of lasticity is E = and Poisson s cofficint ν = 0. Analytical displacmnts ar givn by th rlationship: 3 u = v = x x x ( x )( + y) y y y ( y )( + x). 3 4 (7) Exact strains and strsss ar calculatd from th quations of lasticity. For th plat thicknss, th total strain nrgy is U = Finit lmnt modls consist of bilinar rctangular four nodd isoparamtric lmnts. Th squnc of mshs, 4, 4 4, 8 4, 8 8, 6 6, 3 3 is considrd, with dgrs of frdom 8, 30, 0, 90, 6, 78, 78, rspctivly. Matrics in (6) wr formd using Gaussian quadratur. Th quantitis of primary intrst wr th convrgnc of th strain nrgy and th rlativ nrgy rror norm according to quations (7) (4). Exact strsss ar compard to strsss obtaind with strss rcovry mthods FEA, FEDSS and FEAavrg. Strsss and xprssions (7) (4) wr valuatd using, and 3 3 Gaussian quadratur. Strain nrgy convrgnc of th xamind modls is prsntd on Figur 3. From this Figur th following facts ar vidnt: th xact strain nrgy, as a high ordr polynomial (), is vry much dpndnt on th ordr of th Gaussian intgration rul, th strain nrgy of th raw finit lmnt solution (FEA) is practically indpndnt on intgration rul, bcaus it bhavs mor or lss as a constant in th intrior of ach lmnt, for th rasonably dns mshs FEA strain nrgy practically coincids with th thortical valu intgratd at midpoints of lmnts, total strain nrgy for both FEDSS and FEAavrg modls is lss accurat than for raw FEA solution, both modls ar mor snsitiv to th ordr of intgration rul than raw FE solution. Howvr, th rsulting diffrncs ar qualitativly insignificant, FEDSS approach is clarly mor accurat than FEAavrg. Strain Enrgy mthod - Gaussian points: EXACT (3x3) EXACT (x) EXACT (x) FEA (x, x, 3x3) FEDSS (x) FEDSS (x) FEA avrg (x) FEA avrg (x) Figur 3. Strain nrgy convrgnc Explanation is analogous to that for on dimnsional xampl, Figur, whr at th lmnt midpoint (suprconvrgnt point [, 3]) thr is a smallst diffrnc btwn FEA and xact strsss. Conclusion is that if a Gaussian quadratur is usd th xact solution can b rplacd with FEA solution for linar or bilinar lmnts. It is xpctd that analogous conclusions can b mad for th lmnts of highr ordr. Figur 4 shows th rlativ nrgy rror norm (with rspct to th xact solution) (3) of diffrnt strss rcovry solutions, vrsus th numbr of dgrs of frdom. From this figur on can conclud that: th rror norm of FEA solution is smallst whn undrintgratd (by Gaussian quadratur). This is in th accordanc with th fact that FEA and xact strain nrgy ar vry clos (Figur 3) whn intgration is mployd, th rror norms of FEAavrg and FEDSS ar also smallr whn intgratd by schm than by schm, opposing formally th situation on Figur 3, whr th intgration of full strain nrgis is mor accurat if schm is usd, than. Th rason for this discrpancy is a fact than smoothd solutions (s Figur ) ar similarly to th gnric FEA solution, closst to th xact solution at th lmnt midpoints (suprconvrgnt points).

5 Rlativ Enrgy Error Norm (%) 0 FEA (x Gaussian points) FEAavrg (x Gaussian points) FEAavrg (x Gaussian points) FEDSS (x Gaussian points) FEDSS (x Gaussian points) FEA (x Gaussian points) Rlativ Enrgy Error Norm (%) 0 FEAavrg (x Gauss Point) FEDSS (x Gauss Point) Figur 4. Rlativ nrgy rror norm of th strsss with rspct to th xact valus Figur. Rlativ nrgy rror norm of th strsss rlatd to th xact valus Howvr, on both Figurs 3 and 4, FEDSS and FEAavrg curvs ar qualitativly similar, inrspctivly on th intgration schm usd. What is mor important from both figurs on can conclud, without any doubt, that FEDSS is mor accurat smoothing schm than classical FEAavrg. In th Figur FEAavrg and FEDSS rrors with rspct to th thortical valus, intgratd by Gaussian quadratur, xtractd from th Figur 4, ar prsntd. In Figur 6 th sam valus intgratd by quadratur ar givn. Similarly in Figur 7 rrors with rspct to FEA valus (i.. Z-Z typ rror indicators) intgratd by rul ar givn. Finally, in Figur 8 rror indicators intgratd by full, rul ar shown. It is vidnt that for rasonably dns mshs th charactr of th curvs on Figur and Figur 7 is vry similar, both qualitativly and quantitativly. In contrary, comparison of Figurs 6 and 8 show that usual, full, intgration of th rror indicator, masks th diffrnc btwn FEAavrg and FEDSS procdurs. Shortly spaking, th strain nrgy of th FEDSS (and any othr quivalnt or similar procdur) is mor accurat compard with, say, that of simpl strss avraging. Howvr, this fact is dubious if for th calculation of Z-Z typ rror indicator on uss xact, in th cas considrd, intgration. At varianc, th advantag of mor advancd procdurs is clar if th drivativ suprconvrgnt points (in th prsnt cas ) ar usd, s Tabls and 3. Rlativ Enrgy Error Norm (%) FEAavrg (x Gaussian Points) FEDSS (x Gaussian Points) Figur 6. Rlativ nrgy rror norm of th strsss rlatd to th xact valus

6 Error Indicator η fa (%) 00 0 FEAavrg (x Gaussian points) FEDSS (x Gaussian points) Tabl. η FEA η FEDSS η FEAavrg Rlativ nrgy norms (%), intgration xact η FEA FEDSS η FEA FEAavrg η FEA Tabl 3. Error indicators (%), in tgration Figur Error indicators intgratd at th lmnt midpoint 4.3 Squar plat with a circular hol A problm of th squar plat with a circular hol [] is considrd in Figur 9. Only a quartr of it is analyzd du to th symmtry of that systm. Isotropic, homognous matrial proprtis and th plan strss bhavior ar assumd. Modulus of lasticity is and Poisson ratio 0.3. Plain isoparamtric four-nodd quadrilatral lmnts and Gauss quadratur ar usd. q= Error Indicator η fa (%) 0 Y 0. X FEAavrg (x Gaussian points) FEDSS (x Gaussian points) Figur 8. Error indicators intgratd by th full Gaussian quadratur Figur 9. Plat with a circular hol Rcovrd FEDSS strsss ar compard with simpl nodal avragd strsss FEAavrg. To study th ffctivnss of FEDSS mthod as a strss rcovry and smoothing procdur two typs of th analysis wr prformd. First, Z Z typ rror indicator vrsus numbr of lmnts, and scond, th sam indicator as a function of th xcution tim, is xamind. Th numrical studis wr mad for th squnc of

7 mshs from to On of thm ( ) is prsntd in Figur FEA avrg (x Gaussian points) FEDSS (x Gaussian points) Y C 0. Error indicator η fa (%) 0 0 X A B Figur 0. Finit lmnt msh On possibl usr dfind st of coordinat systms, usful for th dtrmination of hoop strsss at global nods is shown on Figur numbr of lmnts along AB Figur. Error indicators vrsus msh dnsity FEA avrg (x Gaussian points) FEDSS (x Gaussian points) Error indicator η fa (%) 0 0 Figur. Usr dfind coordinat systms. Figur shows a comparison of th rror indicators vrsus th numbr of dgrs of frdom. As it can b sn, FEDSS formulation, prsntd hr, is significantly mor accurat than convntional smoothing FEAavrg. Evn mor important is a quantitativ masur, i.., th xcution tim for th accuracy rquird. In Figur 3 rror indicators rlatd to th tim of xcution ar compard. It can b said that FEDSS strss rcovry procdur vidntly incrass th quality of FE solution and rducs th solution tim. Th rror of % masurd in th nrgy rror norm by th us of intgration rul is achivd twic fastr by th mthod prsntd in this papr, compard with classical avraging procdur, dspit th additional tim ncssary for th solution of a systm (6) for strsss Excution Tim (s) Figur 3. Error indicators vrsus xcution tim Not that quantitativ rsults (accuracy in th function of th solution tim) can sldom b found in th availabl rfrncs on th postprocssing procdurs.

8 Conclusions Nw global, coordinat indpndnt approximation of continuous displacmnt fild has bn prsntd and applid to two-dimnsional lasticity problms. Proposd procdur dlivrs th ndd accuracy in a significantly shortr tim than undrlying classical finit lmnt analysis. Howvr, as it has bn notd in Subsction., many of th global intrpolation procdurs ar of th similar accuracy. But, th advantag of th prsnt algorithm is its flxibility (i.. possibility to us arbirtrary local coordinat systms). In addition, it is implmntd as an univrsal postprocssing routin for which th input data (displacmnts) can b takn from any finit lmnt analysis packag, thory, or xprimnt. Howvr, th most important nw dtail in th abov study is prhaps a fact that undrintgration (intgration in th drivativ (strss) suprconvrgnt points) givs much sharpr pictur of th rror indicator. This fact rcommnds th us of th proposd approach not only for th quality assssmnt of th postprocssing procdurs, but also as a prcis tool in th adaptiv msh rfinmnt algorithms. Rfrncs [] Zinkiwicz, O.C., Zhu, J.Z., A simpl rror stimator and adaptiv procdur for practical nginring analysis, Int. J. Num. Mths. Eng. 4, , 987. [] Odn, J. T., Brauchli, H. J., On th calculation of consistnt strss distributions in finit lmnt approximations, Int.J.Num.Mth.Engng, 3, 37 3, 97. [3] Hinton, E., Campbll, J.S., Local and global smoothing of discontinuous finit lmnt functions using a last squars mthod, Int.J.Num.Mths. Eng. 8, , 974. [4] Riggs, H.R., Tsslr, A., Continuous vrsus wirfram variational smoothing mthods for finit lmnt strss rcovry, Advancs in Post and Postprocssing for FET, CIVIL-COMP Ltd, Edinburgh, Scotland, 37 44, 994. [] Draš koviæ, Z., On invarianc of finit lmnt approximations, Mchanika Tortyczna i Stosowana, 6, 97-60, 988. [6] Brkoviæ, M., Draš koviæ, Z., On th ssntial mchanical boundary conditions in two-fild finit lmnt approximations, Comput. Mth. in Appl. Mch. Engrg. 9, 339 3, 99. [7] Odn, J. T., Finit lmnts of nonlinar continua, McGraw-Hill, Nw York 97. [8] Mijuca, D., Brkoviæ, M., Som strss rcovry procdurs in th classical finit lmnt analysis, Procdings of th YUCTAM Niš, 99. [9] Sandr, G., Application of dual analysis principl, IUTAM Colloquium on High Spd Computing of Elastic Structurs, Lig, Blgium, 970. [0] Odn, J., T., Cary, G. F., Backr, E., B., Finit lmnts, An Introduction, Volum I, Prntic Hall, 98. [] Babuš ka, I., Validation of a postriori rror stimators by numrical approach, Int.J.Num.Mth.Engng, 37, 073-3, 994. [] Bckrs, P., Zhong, H. G., Msh adaption for two dimnsional strss analysis, Advancs in Pr and Postprocssing for FET, CIVIL-COMP Ltd, Edinburgh, Scotland, 47 9, 994. [3] Wibrg, N E., Abdulwahab, F., Error stimation with post procssd FEsolution, Advancs in Post and Postprocssing for FET, CIVIL-COMP Ltd, Edinburgh, Scotland,, 994. [4] Blackr, T., Blytschko, Suprconvrgnt patch rcovry with quilibrium and conjoint intrpolant nhancmnts, Int.J.Num.Mth.Engng, 37, 7-36, 994. [] Rao, A. K., Raju, I. S., Krishna Murty, A. V., A powrful hybrid mthod in finit lmnt analysis, Int. J.Num. Mth. Engng, 3, , 97.

AS 5850 Finite Element Analysis

AS 5850 Finite Element Analysis AS 5850 Finit Elmnt Analysis Two-Dimnsional Linar Elasticity Instructor Prof. IIT Madras Equations of Plan Elasticity - 1 displacmnt fild strain- displacmnt rlations (infinitsimal strain) in matrix form