BOUNDARY ELEMENT METHOD FOR SHEAR DEFORMABLE PLATE WITH MATERIAL NONLINEARIRTY

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1 OL., O. 8, APRIL 6 I ARP Jornl of Engnrng Ald cncs 6-6 Asn Rsrch Pblshng tor ARP). All rghts rsrvd..rnornls.com BOUDARY ELEET ETHOD FOR HEAR DEFORABLE PLATE WITH ATERIAL OLIEARIRTY ryono Drtmnt of chncl Engnrng, Fclty of Engnrng, Unvrsts hmmdyh rrt, Indons Eml: sryono@ms.c.d ABTRACT In ths r formlton of bondry lmnt mthod for shr dformbl lt thory th mtrl nonlnrty s rsntd. Th mtrl s ssmd to ndrgo smll strns. Th von ss crtron s sd to vlt th lstc zon lstc rfctly lstc mtrl bhvor s ssmd. An ntl strss formlton s sd to formlt th bondry ntgrl qtons. ot only th lstc strn d to bndng bt lso th lstc strns d to mmbrn r consdrd. Th domn ntgrl d to mtrl nonlnrty s vltd sng cll dscrtzton tchnq. A totl ncrmntl mthod s ld nstd of n ncrmntl trtv rocdr, to solv th nonlnr bondry ntgrl qtons. mrcl mls r rsntd to dmonstrt th vldty th ccrcy of th formlton. Kyords: shr dformbl lt, mtrl nonlnrty, totl ncrmnt mthod, bondry lmnt mthod. ITRODUCTIO Plt strctrs r dly sd n ngnrng lctons, for ml rcrft, crs, bolr drms, rssr vssls, bldng slbs, shs, tc. Drng ts srvcs, th lt my b sbctd to vrty of lods, sch s tnson, bndng or combnd tnson bndng. In th cs of tnson, th lt s consdrd s todmnsonl roblm t cn b solvd sng th ln strss thory of lstcty Tmoshno, 97). On th othr h, th bndng roblms r lnd by lt bndng thory Rssnr, 947, Tmoshno, 959) th combnd tnson bndng css cn b rrsntd by sroston of lt bndng thory to-dmnsonl ln strss thory. Thr r to dly sd lt thors. Th frst on s dvlod by Krchhof 85)] s commonly rfrrd to s th clsscl thory. Th othr s dvlod by Rssnr 945), s non s th shr dformbl thory. Th clsscl thory s dqt for nlyzng crtn lctons, hovr, for roblms nvolvng strss concntrtons crcs th thory hs bn shon not to b n grmnt th rmntl msrmnts Albd, 998). Unl th clsscl thory, th Rssnr thory ts nto ccont th shr dformton. In th shr dformbl lt thory, th roblm s rrsntd n trms of thr dgrs of frdom, nvolvng gnrlzd dslcmnts.. to rottons dflcton) gnrlzd trctons.. momnts trnsvrs shr forcs). Th nlyss of lt bndng roblms by th bondry lmnt mthod BE) hs bn rortd by mny rsrchrs. Hovr, ts lcton to lstolstc lt bndng nlyss s vry lmtd. Th lcton of th BE to lstolstc nlyss of Rssnr lts cn b fond n th ors by Krm Tlls 988), Krm Tlls 99), Krm Tlls 998) Rbro ntrn 998). Krm Tlls 99) rsntd formlton of lstolstc nlyss of Rssnr lts s n tnson of thr rvos or 988). Ltr, thy lso formltd th bondry ntgrl qtons for th sm lcton sng n ntl strss roch th cll dscrtzton tchnq s ld to vlt th domn ntgrl n hch th trnglr clls r sd 998). To solv th lstolstc roblm n ncrmntl trtv rocdr s dotd togthr th von ss Trsc yld crtr. Th clsscl lstcty thory s sd n hch lstc strns r tm ndndnt. Th lstc strn s consdrd only d to bndng lstc-rfctly lstc mtrl s consdrd. An ltrntv roch for dlng th lstolstc nlyss of Rssnr lts by BE s rsntd by Rbro ntrn 998). In thr or, n ncrmntl trtv lgorthm bsd on th ntl stffnss mthod s mlmntd. Ths r rsnts th bondry lmnt formlton of lstolstc nlyss of Rssnr lts. Th formlton follos closly th or by Krm Tlls 998); hovr not only th lstc strn d to bndng bt lso th lstc strn d to mmbrn r consdrd. Th totl ncrmntl tchnq s ld n dlng th nonlnr systm of qton. Th cll dscrtzton mthod sng 9-nods qdrltrl cll s mloyd to vlt th domn ntgrls rng n th formlton. Elstc-rfctly lstc mtrl s consdrd. Throghot ths r, th crtsn tnsor notton s sd, th Gr ndcs vryng from to th Ltn ndcs vryng from to. GOERIG EQUATIO In ordr to dfn gnrl formlton for mtrl nonlnrty of lt bndng, t s consdrd tht lstc strns r only d to bndng mmbrn, hnc totl strn rts cn b dfnd s: ; ) ; ) 59

2 OL., O. 8, APRIL 6 I ARP Jornl of Engnrng Ald cncs 6-6 Asn Rsrch Pblshng tor ARP). All rghts rsrvd..rnornls.com ) hr, r th totl bndng strn rts, r th totl n-ln strn rts, shr strn rts rsctvly. Th totl bndng strn rts consst lnr rts nonlnr rts. mlrly totl n-ln strn rts consst of lnr rts nonlnr rts. Th nonlnr rts of qtons ) ) r d to lstcty ) thy cn b rssd s 4) 5) On th othr h, th rts of th strss rsltnts cn lso b sttd s ; 6) for momnt rsltnts Q Q ; 7) for shr rsltnts 8) for mmbrn strss rsltnts. Th strssdslcmnt rltonshs cn b rsntd s D,, ), Q C, ); ) B,,, ) ; ) hr, D Eh, B Eh C Eh. ) Th qlbrm qton cn b rssd s: Q ; ) ; 9) Q, q ; ), 4) DIPLACEET AD TRE EQUATIO Alctons BE n sold mchncs r bsd on th omgln s dntts. omgln s dntty for dslcmnts n lstolstcs shr dformbl lt bndng roblms stts tht th rt of th dslcmnts to rottons on dflcton) t ny onts X [ X )] tht blong to domn X Є ) to th bondry vls of dslcmnt rts [ ] trcton rts [ ] cn b rssd s Krm, 998): X') W X', d P X', d W X ', X ) q X ) d d 5) X ', X ) X ) for rotton dflctons X') U X', t d T X', d ', X ) X X ) d 6) for n-ln dslcmnt. Whr, W X,, P X,, χ X,X), U X,, T X,, X,X) r clld fndmntl soltons rrsntng dslcmnt, trcton strn n th drcton t ont X d to nt ont forc n th drcton t ont X cn b fond n ryono 7). Eqton 5) 6) r vld for ny sorc onts thn domn X Є ), n ordr to fnd soltons on th bondry onts, t s ncssry to consdr th lmtng rocss s X Є. Aftr lmtng rocss, bondry dslcmnt ntgrl qtons cn b rssd s C ') W ', d P ', d W ', X ) q X ) d ', X ) X ) d 7) C ') U ', t d T ', d ', X ) X ) d 8) 5

3 OL., O. 8, APRIL 6 I ARP Jornl of Engnrng Ald cncs 6-6 Asn Rsrch Pblshng tor ARP). All rghts rsrvd..rnornls.com hr, C ) r fr trm tht r C )=δ )+α ), for smooth bondry th fr trm s.5. Th omgln s dntty for strsss cn b rssd rsctvly s: X ') W X ', d P X ', d W X ', X ) q X ) ) ) 8 d l l X ', X ) X ) d 9) for momnt rsltnts, Q X ') W X ', d P X ', d W X ', ) q X ) d l X ', X ) X ) d ) for shr rsltnts, X') U X', t d T X', d ', X ) X X ) d ) ) 8 ) for mmbrn rsltnts. DICRETIZATIO AD YTE OF EQUATIO In ordr to solv qton 7), 8), 9), ) ), nmrcl mthod s mlmntd. Th bondry s dscrtzd sng qdrtc sormtrc lmnts. Th domn s dvdd nto nmbr of clls of 9 nods qdrltrl cll s shon n Fgr-). Cll nod Bondry nod ) ) ) ) Th sh fnctons α r dfnd s ) ) ) ) ) Aftr dscrtzton ont collocton on th bondry s ll s n th domn, th qtons 7) 8) cn b rttn n th mtr form s H G b T 4) H G t T hr [H] [G] r th ll-non bondry lmnt nflnc mtrcs, [T] s th nflnc mtr d to lstcty. Th srscrt sho th lt th n-ln mod rsctvly {.}, {}, {.} {t} r th dslcmnt th trcton rt vctors on th bondry. {b} s th lod rt vctors on th domn { } { } r th nonlnr trm d to lstcty. Aftr mosng bondry condton, qtons 4) cn b rttn s A T f 5) T hr, [A] s th systm mtr, {} s th nnon vctor {f } s th vctor of rscrbd bondry vls. mlrly, th strss ntgrl qtons of qtons 9), ) 4) cn b rsntd n mtr form s G Q G H H G t b T b H E T T 6) h Fgr-. Dscrtzton. In ths formlton, bondry rmtr, th nnon bondry vls of dslcmnts trctons r romtd sng ntrolton fncton, n follong mnnr: OLUTIO ALGORITH Th totl ncrmntl mthod solvs th nonlnr systm of qtons of qton 5) bsd on th ncrmntl rocdr. It hs n lgorthm s:. olv th qton 5), ssm tht th nonlnr trms, =) r ql to zro for th frst lod ncrmnt. It mns tht th lnr systm qtons 5

4 OL., O. 8, APRIL 6 I ARP Jornl of Engnrng Ald cncs 6-6 Asn Rsrch Pblshng tor ARP). All rghts rsrvd..rnornls.com r solvd. For th +)th lod ncrmnt t s ssmd tht ) +)th= ) th. olv qton 6) bsd on th bondry vls obtnd from nmbr. Th sm cs s nmbr s mlmntd for th nonlnr trm.. Evlt of th lstc zon bsd on th strss obtnd from th nmbr. In ths stg th von ss crtron s sd. 4. If th lstcty hs tn lc thn, obtn th nonlnr trm othrs go to th nmbr 5. Th clr lnton of th dtrmnton of th lstcs trm cn b fond n th or by ryono 7). 5. If th lod s lss thn th fnl lod thn go to nmbr rt ntl th lod s ql fnl lod UERICAL EXAPLE In ordr to sho th vldty th ccrcy of th BE formlton totl ncrmnt mthod n dlng th nonlnr trm, n ml s rsntd. A ln strss roblm of rctnglr lt th cntr hol s shon n Fgr- s sbctd to tnson lod =5 P) s ll s nformly trnvrs lod q=.84 P). Th tnson lod s ctng on th to sd of mm lngth, hrs th nformly trnvrs lod s ctng on th hol srfc of th lt. Th mtrl hs rorts of E=7 G, ν=. σ Y=4. σ B 5 mm 6 mm Fgr-. Hol-Plt th tnson of srfc rsr of q. BE modl s Fgr-) hch hs 7 lmnts on th bondry th 64 domn clls fond to gv convrgd soltons s sd n ths ml. A mm σ Fgr-4 shos th dflctons t ont B drng lodng hch r comrd to th FE rslts. Th FE rslts r obtnd sng AY th 76 domn clls of lstc 4-nod 4 shll lmnt hch s lso fond to gv convrgd rslts. Th nondmnsonl rmtrs r dfnd s n th fgr. It cn b sn from th fgr tht BE FE rslts r n good grmnt. Q = q Fgr-4. Dflcton t ont B of th ml lt th bndng lod q tnson lod. Fgr-5 rsnts th dslcmnt contor n th drcton Fgrs b) th dflcton contor Fgrs c d) of th ml lt n mllmtr. It cn b sn tht th contors loo smlr, hovr FE dflcton contor hs slghtly bggr rgon th mmm dflcton comrd to th BE. Th vls for both dslcmnt dflcton dffr slgthy, bt r consdrd cctbl s th to mthods r bsd on nmrcl solton. Plstc zon von ss strss) t th fnl lod s rsntd n Fgr-6 for both FE BE nlyss n P. It cn b sn tht th contors s ll s th vls loo slghtly dffrnt; hovr thy r consdrd cctbl s th to mthods r bsd on nmrcl solton. W = r FE BE y BE msh B Bondry nod Cll nod Fgr-. BE modl. 5

OL., O. 8, APRIL 6 I 89-668 ARP Jornl of Engnrng Ald cncs 6-6 Asn Rsrch Pblshng tor ARP). All rghts rsrvd..rnornls.

Th BE formltons rsntd n ths r hv good grmnt th FE t hs bn shon tht BE modl smlr comrd to th FE modl.

, Plt Bndng Anlyss th Bondry Elmnt, Comttonl chncs Pblctons, othmton 998). Krm,.J. Tlls, J.C.F. On bondry lmnts for Rssnr s lt thory, Engnrng Anlyss, 5, -7 988).

d) BE modl dflcton contor Fgr-5. Dslcmnt n drcton dflcton contor for both FE BE nlyss. Krchhoff, G., Ubr ds glchgcht nd d bgng nr lstschn schb, J. Rn Ang. th., 4,.

) FE von ss strss contor b) BE von ss strss contor O IE 45. 5.976 86.69 57. 7.965 98.678 69.99 9.959.67 Fgr-6. Plstc zon t th fnl lod for both FE BE nlyss.

from th rslts obtnd t cn b concldd tht: Rssnr, E., On Bndng of Elstc Plts, Qrtrly of Ald thmtcs 5, 55-68 947). ryono, Albd,.H.

5 OL., O. 8, APRIL 6 I ARP Jornl of Engnrng Ald cncs 6-6 Asn Rsrch Pblshng tor ARP). All rghts rsrvd..rnornls.com ) FE modl contor b) BE modl contor c) FE modl dflcton contor U E W Th BE formltons rsntd n ths r hv good grmnt th FE t hs bn shon tht BE modl smlr comrd to th FE modl.. Th totl ncrmntl mthod s shon to b n ffcnt roch for ths roblm s rtd solton of systm of qtons s not rqrd th nonlnr trms r dtd by bc sbsttton. REFERECE Albd,.H., Plt Bndng Anlyss th Bondry Elmnt, Comttonl chncs Pblctons, othmton 998). Krm,.J. Tlls, J.C.F. On bondry lmnts for Rssnr s lt thory, Engnrng Anlyss, 5, ). Krm,. J. Tlls, J. C. F., Th BE ld to lt bndng lstolstc nlyss sng Rssnr s thory, Engnrng Anlyss th Bondry Elmnts, 9,. 5-57, 99. Krm,.J. Tlls, J.C.F., onlnr mtrl nlyss of Rssnr s lts, Plt Bndng Anlyss th Bondry Elmnt, 7-6, Comttonl chncs Pblctons, othmton 998). d) BE modl dflcton contor Fgr-5. Dslcmnt n drcton dflcton contor for both FE BE nlyss. Krchhoff, G., Ubr ds glchgcht nd d bgng nr lstschn schb, J. Rn Ang. th., 4,. 5-88, 85). Rssnr, E., Th Effct of Trnsvrs hr Dformton on th Bndng of Elstc Plts, Jornl of Ald chncs,, A ). ) FE von ss strss contor b) BE von ss strss contor O IE Fgr-6. Plstc zon t th fnl lod for both FE BE nlyss. COCLUIO Th lcton of BE to mtrl nonlnrty for shr dformbl lt bndng nlyss s rsntd th totl ncrmntl mthod s mlmntd to solv th nonlnr systm of qton,.from th rslts obtnd t cn b concldd tht: Rssnr, E., On Bndng of Elstc Plts, Qrtrly of Ald thmtcs 5, ). ryono, Albd,.H., Anlyss of hr Dformbl Plts th Combnd Gomtrc trl onlnrts by Bondry Elmnt thod, Intrntonl Jornl of old trctr 44, ). Tmoshno,., Wonosy-Krgr,., Thory of Plts hlls, nd dton, cgr-hll Intrntonl Edtons, ngor, 959). Tmoshno,., Goodr, J.., Thory of Elstcty, rd dton, cgr-hll Intrntonl Edtons, ngor, 97. 5

Fundamentals of Continuum Mechanics. Seoul National University Graphics & Media Lab

Fundamentals of Continuum Mechanics. Seoul National University Graphics & Media Lab Fndmntls of Contnm Mchncs Sol Ntonl Unvrsty Grphcs & Md Lb Th Rodmp of Contnm Mchncs Strss Trnsformton Strn Trnsformton Strss Tnsor Strn T + T ++ T Strss-Strn Rltonshp Strn Enrgy FEM Formlton Lt s Stdy