CENTRO INTERNACIONAL DE MÉTODOS NUMÉRICOS EN INGENIERÍA Modelo Constitutivo Elastoplástico Anisótropo en Deformaciones Finitas para Análisis de Materiales Compuestos Reforzados con Fibras E. Car S. Oller E. 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MODEL PHASE 3 COMPOUNDING 3 CONSTITUTIVE MODEL PHASE 2 COMPOUNDING 2 CONSTITUTIVE MODEL PHASE 1 COMPOUNDING 1 (σ ) (σ ) (σ ) i * 2 (σ ) i (σ ) i i 2 CONSTITUTIVE MODEL PHASE n (σ ) i * 1 COMPOUNDING n (σ ) Updated Configuration CONSTITUTIVE MODELS FOR EACH COMPOUND 3 n RECOMPOSITION σ i = ∑ k c (σ i )c c = NO CONVERGENCE CHECK RESIDUAL FORCE AND CONVERGENCE Reference Configuration S = φ (σ i ) NEXT INCREMENT: n+1 Iljxud 4= Hvtxhpd gh vroxflöq gh xq sureohpd qr olqhdo pxowlidvh ghvduuroodgr hq od frqjxudflöq uhihuhqfldo | dfwxdol}dgd 46 Od ghvfulsflöq gho frpsruwdplhqwr hoävwlfr dqlvöwursr gh xq vöolgr qr suhvhqwd judqghv glfxowdghv1 Hv srvleoh xwlol}du sdud wdo q odv irupdv jhqhudohv gh od whruðd gh od hodvwlflgdg +Kxoo/ 4<;:, +Pdwwkhzv dqg Udzolqjv/ 4<<7, +Shqgohwrq dqg Wxwwoh/ 4<;<,1 Od irupxodflöq gh xqd oh| frqvwlwxwlyd dghfxdgd sdud vöolgrv qr sursruflrqdohv ru0 wöwursrv r dqlvöwursrv frqvwlwx|h xq sureohpd gh hohydgd frpsohmlgdg1 Orv sulphurv lqwhqwrv gh irupxodu ixqflrqhv gh 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glvlsdflöq hq xq surfhvr hodvwrsoävwlfr lvrwìuplfr hq ho hvsdflr uhdo dqlvöwursr1 Od h{suhvlöq gh od glvlsdflöq uhvxowd gh frqvlghudu ho sulphu | vhjxqgr sulqflslr gh od whuprglqäplfd1 Ho sulphu sulqflslr srvwxod ho edodqfh gh od hqhujðd/ h{ljlhqgr od frqvhuydflöq gh od hqhujðd wrwdo lqwhuqd Z gho vlvwhpd1 Hvwh sulqflslr uhodflrqd od srwhqfld lqwurgxflgd do vlvwhpd | od fdqwlgdg gh fdoru T h{lvwhqwh hq ho plvpr/ frq ho fdpelr gh hqhujðd lqwhuqd joredo Z txh hvwh h{shulphqwd +Pdoyhuq/ 4<9<, +Oxeolqhu/ 4<<3, +Roohu/ 4<;<,1 Od irupd orfdo hxohuldqd vh h{suhvd frpr= p $@ j = g . pu gly +t, +5;, grqgh p hv od ghqvlgdg r pdvd yroýplfd hq od frqjxudflöq hvsdfldo/ $ hv od ghqvlgdg gh hqhujðd lqwhuqd hvshfðfd/ j hv ho whqvru gh whqvlrqhv gh Fdxfk|/ g ho judglhqwh gh yhorflgdghv/ u hv od ixhqwh lqwhuqd gh fdoru hvshfðfd | t hv ho xmr gh fdoru sru frqgxfflöq1 Od irupd orfdo odjudqjhdqd vh h{suhvd frpr= pf $@ V =H .pf u gly +Mt, +5<, pf vlhqgr od pdvd hq od frqjxudflöq uhihuhqfldo/ $ hv od hqhujðd lqwhuqd hvshfðfd/ V hv ho vhjxqgr whqvru gh whqvlrqhv gh Slrod0Nlufkr/ H ho whqvru gh ghirupdflrqhv gh Fdxfk|0Juhhq/ u hv od ixhqwh lqwhuqd gh fdoru hvshfðfd | t hv ho xmr gh fdoru sru frqgxfflöq1 Ghqlhqgr od ghqvlgdg gh hqhujðd oleuh gh Khopkrow} # frpr od sduwh gh od hqhujðd dsuryhfkdeoh sdud uhdol}du wudedmr d whpshudwxud frqvwdqwh uhvxowd= _es # @ $ +63, vlhqgr od hqwursðd hvshfðfd | xqd phglgd gh od whpshudwxud1 Vx yduldflöq whpsrudo uhvxowd= #@$ +64, Uhhpsod}dqgr od hf1+64, hq od hf1+5<, uhvxowd od h{suhvlöq gho sulphu sulqflslr gh od whuprglqäplfd hq ixqflöq gh od hqhujðd oleuh gh Khopkrow}1 pf # . . @ V =H .pf u gly +Mt, 4; pf # @ V =H .pf u gly +t, pf pf +65, Od irupd hxohuldqd orfdo gho sulphu sulqflslr uhvxowd gh frqvlghudu od wudqvirupdflöq 3A 3 j = g @ a V =H/ vlhqgr g @ I ho judglhqwh gh yhorflgdghv/ pf @ Mp od pdvd HI _T | t @ Mt= hq od frqjxudflöq uhihuhqfldo/ M @ ghw I @ _T f p #@ j = g . pu gly +t, p p +66, h{suhvlöq orfdo Hxohuldqd gho sulphu sulqflslr gh od whuprglqäplfd hq ixqflöq gh od hqhujðd oleuh gh Khopkrow}1 Ho vhjxqgr sulqflslr hvwdeohfh txh sdud xq surfhvr luuhyhuvleoh vh gheh fxpsolu txh ho fdpelr hq od surgxfflöq lqwhuqd gh hqwursðd gheh vhu pd|ru r ljxdo txh ho fdpelr gh hqwursðd lqwurgxflgd +Pdoyhuq/ 4<9<, +Roohu/ 4<;<,/ hvwr hv= g +67, @ @? gw U U 6o vlhqgr K ^ od hqwursðd lqwhuqd @ @ T pgY | od hqwursðd lqwurgxflgd @? @ T w gY . 7 w gV1 Wudqvirupdqgr hq od ghvljxdogdg dqwhulru odv lqwhjudohv gh vxshuflh hq lqwhjudohv yroxpìwulfdv | frpelqdqgr frq od irupd orfdo hxohuldqd +hf15;, gho sulphu sulqflslr gh od whuprglqäplfd uhvxowd od glvlsdflöq orfdo sru xqlgdg gh pdvd frpr +Roohu/ 4<;<,= 4 j=g @$. t=u +68, p p Whqlhqgr hq fxhqwd od ghqlflöq gh od hqhujðd oleuh gh Khopkrow} +hf1+63,, uhvxowd od h{suhvlöq orfdo Hxohuldqd gh od glvlsdflöq= 4 j=g t=u +69, @ # . p p 71615 Ixqflöq gh hqhujðd oleuh gh Khopkrow} Hq od ghqlflöq gh od hqhujðd oleuh gh Khopkrow} qr hv fruuhfwr hpsohdu frpr ydul0 deoh oleuh ho whqvru hoävwlfr gh Juhhq 0 Odjudqjh hq od frqjxudflöq uhihuhqfldo |d txh ho plvpr vh yh dihfwdgr sru ho xmr soävwlfr d wudyìv gh od frpsrqhqwh soävwlfd gho judglhqwh gh ghirupdflrqhv IR / sru or wdqwr hv frqyhqlhqwh ghqlu od hqhujðd oleuh gh Khopkrow} frpr= # @ # +he > > , +6:, grqgh hv od whpshudwxud devroxwd/ he hv ho whqvru gh ghirupdflrqhv gh Dopdqvl | xq frqmxqwr gh yduldeohv gh hvwdgr soävwlfdv/ frpr sru hmhpsor od frkhvlöq hqwuh sduwðfxodv/ od yduldeoh gh hqgxuhflplhqwr/ ho äqjxor gh ur}dplhqwr lqwhuqr/ hwf1 4< 71616 Klsöwhvlv gh hodvwlflgdg ghvdfrsodgd Xq fdvr sduwlfxodu gh od ixqflöq gh hqhujðd oleuh or frqvwlwx|h od klsöwhvlv gh hodvwlflgdg ghvdfrsodgd h{suhvdgd phgldqwh od hfxdflöq= # +he > > , @ # e +he > , . # R +> , +6;, Hvwd klsöwhvlv vh edvd hq txh orv qlyhohv gh hqhujðd qhfhvdulrv sdud glvwruvlrqdu od uhg fulvwdolqd/ uhvsrqvdeoh gh od ghirupdflöq hoävwlfd/ vrq glihuhqwhv gh orv qlyhohv gh hqhujðd qhfhvdulrv sdud ho ghvol}dplhqwr lqwhufulvwdolqr1 Hvwd klsöwhvlv kd vlgr xwlol}dgd sru glyhuvrv dxwruhv | frqvwlwx|h od edvh gh prghorv frpsxwdflrqdohv sursxhvwrv/ hqwuh rwurv sru +Jdufðd Jdulqr dqg Rolyhu/ 4<<4, +Jdufðd Jdulqr dqg Rolyhu/ 4<<5, +Oxeolqhu/ 4<<3, +Roohu/ 4<;<, +Oxfflrql hw do1/ 4<<9,1 Odv yduldeohv oleuhv gho sureohpd vrq od whpshudwxud | od fxrwd hoävwlfd gh od ghirupdflöq gh Dopdqvl ghqlgd frpr= ] | 4 R e R 3 h @h h @ h gw +6<, Le 5 f grqgh hR hv od ghirupdflöq soävwlfd hq od frqjxudflöq hvsdfldo/ e hv ho whqvru l}txlhugr R gh Fdxfk|0Juhhq/ | h uhsuhvhqwd od yduldflöq whpsrudo gh od ghirupdflöq soävwlfd hq od frqjxudflöq hvsdfldo1 71617 H{suhvlöq gh od glvlsdflöq Od yduldflöq whpsrudo gh od hqhujðd oleuh vh h{suhvd frpr= C# e C# C# g . . +73, Che C C grqgh ge @ O +he , 7 uhsuhvhqwd od yduldflöq whpsrudo gh od fxrwd hoävwlfd gh od ghiru0 pdflöq1 Uhhpsod}dqgr od hf1+64, hq od hf1+69, vh rewlhqh od h{suhvlöq gh od glvlsdflöq= C# e C# C# 4 @ p g . . p . = g t=u 3 +74, Che C C #@ Rshudqgr od h{suhvlöq gh od glvlsdflöq txhgd= C# C# 4 C# e t=u 3 p @ = g eg p p Ch C C +75, Whqlhqgr hq fxhqwd od ghvfrpsrvlflöq dglwlyd gho whqvru yhorflgdg gh ghirupdflöq g @ ge . gR +Jdufðd Jdulqr dqg Rolyhu/ 4<<4, od hf1+75, txhgd= C# C# C# 4 e @ p e =g p t=u 3 . = gR p Ch C C 7 Ghulydgd gh Ohh1 Yhu Dsìqglfh L 53 +76, H{suhvlöq txh uhfleh ho qrpeuh gh ghvljxdogdg gh Fodvlxv0Gxkhp1 Hvwd lqhfxdflöq gheh vhu vdwlvihfkd sdud wrgrv orv srvleohv surfhvrv1 Sxhvwr txh wdqwr g frpr uhs0 uhvhqwdq yduldflrqhv whpsrudohv duelwuduldv gh odv yduldeohv oleuhv gho sureohpd/ sdud jdudqwl}du ho fxpsolplhqwr gh od hf1+76, sdud xq gdgr hvwdgr whuprglqäplfr/ vxv pxo0 wlsolfdgruhv ghehq vhu lgìqwlfdphqwh qxorv1 Gh dtxð vxujh od hfxdflöq frqvwlwxwlyd gho pdwhuldo= C# C# > @ +77, e Ch C Odv hfv1 +77, vrq odv oh|hv frqvwlwxwlydv jhqhudohv sdud wrgr wlsr gh pdwhuldohv1 Hvwdeohfhq txh ho hvwdgr whqvlrqdo | od hqwursðd vh rewlhqhq frpr od ghulydgd gh od ixqflöq gh hqhujðd oleuh uhvshfwr gh od fxrwd gh ghirupdflrqhv hoävwlfdv gh Dopdqvl | od whpshudwxud uhvshfwlydphqwh/ vlhqgr qhfhvdulr sru hoor ghqlu od h{suhvlöq gh od ixqflöq gh hqhujðd oleuh1 j@p 71618 H{suhvlöq gh od hqhujðd oleuh1 Sduwlfxodul}dflöq d shtxhôdv ghiru0 pdflrqhv hoävwlfdv | hvwdgrv lvrwìuplfrv Orv pdwhuldohv frpsxhvwrv/ d h{fhsflöq gh orv pdwhuldohv hodvwöphurv/ suhvhqwdq ghiru0 pdflrqhv hoävwlfdv shtxhôdv1 Hvwr frqgxfh d txh od sduwh hoävwlfd gho whqvru judglhqwh gh ghirupdflrqhv Ie vh dsur{lph d od xqlgdg | od sduwh hoävwlfd gho whqvru l}txlhugr gh Fdxfk|0Juhhq +ee ,3 wlhqgh do whqvru pìwulfr hvsdfldo j1 Hq hvwh fdvr od glvwlqflöq hqwuh frqjxudflöq lqwhuphgld | ghirupdgd fduhfh gh lpsruwdqfld1 Vlq hpedujr/ odv ghirupdflrqhv soävwlfdv frqwlqýdq vlhqgr qlwdv | hv qhfhvdulr pdqwhqhu od suhvhqfld gho whqvru ghuhfkr gh Fdxfk|0Juhhq F hq od h{suhvlöq pdwhuldo gho prghor frqvwlwxwlyr1 Vl hvwh whqvru qr vh wlhqh hq fxhqwd h{soðflwdphqwh vh hvwä ghqlhqgr xq pdwhuldo glihuhqwh/ pdwhpäwlfdphqwh yäolgr shur vh slhugh ydolgh} iðvlfd1 Sdud ho fdvr gh shtxhôdv ghirupdflrqhv hoävwlfdv uhvxowd vxflhqwh fdudfwhul}du od frpsrqhqwh hoävwlfd gh od hqhujðd oleuh phgldqwh xqd ixqflöq fxdguäwlfd gho whqvru hoävwlfr gh ghirupdflrqhv gh Dopdqvl he / hvwr hv= 4 e j e +78, h =f =h 5p Whqlhqgr hq fxhqwd od klsöwhvlv gh hodvwlflgdg ghvdfrsodgd +hf1 6;,/ od h{suhvlöq gh od sduwh hoävwlfd gh od hqhujðd oleuh +hf1 78, | od h{suhvlöq gho whqvru frqvwlwxwlyr hq ho hvsdflr dqlvöwursr uhdo hq ixqflöq gho whqvru lvöwursr fwlflr +hf1 58, sxhgh uhhvfuleluvh frpr= #e @ 4 +he = f = he , . # R 5p l 4 e k j 3 #@ h = +d , = f = +de , = he . # R +, 5p #@ 54 +79, +7:, Vxvwlwx|hqgr od hf1+7:, hq od hf1+77, vh rewlhqh od hfxdflöq frqvwlwxwlyd hq ho hvsd0 flr dqlvöwursr uhdo ghqlgd hq wìuplqrv gho whqvru frqvwlwxwlyr hq ho hvsdflr lvöwursr fwlflr/ hvwr hv= k l C# j 3 e @ +d , = f = +d , = he Che l k j @ +dj ,3 = f = he @ +dj ,3 = j j@p +7;, +7<, grqgh he uhsuhvhqwd od fxrwd gh ghirupdflöq hoävwlfd gh Dopdqvl hq ho hvsdflr lvöwursr fwlflr | j uhsuhvhqwd od whqvlöq hq ho hvsdflr lvöwursr fwlflr1 717 Uhjod gh xmr1 Oh| gh hyroxflöq gh odv yduldeohv lqwhuqdv Od oh| gh hyroxflöq hq ho wlhpsr gh od ghirupdflöq soävwlfd hvwä gdgd sru= Cj +83, Cj Whqlhqgr hq fxhqwd txh wrgd od dqlvrwursðd gho pdwhuldo hvwä frqwhqlgd hq ho whqvru gh wudqvirupdflöq dj / od ixqflöq gh srwhqfldo soävwlfr sdud ho vöolgr dqlvöwursr uhvxowd= R gR @h @ j +j> , @ j +j> dj > , @ j +j> , @ N +84, Uhhpsod}dqgr od hf1+84, hq od hf1+83, ho lqfuhphqwr gh od fxrwd soävwlfd gh od ghirupdflöq gh Dopdqvl uhvxowd= R h @ j Cj Cj Cj Cj @ = @ = dj @ h = dj Cj Cj Cj Cj +85, j grqgh h hv ho xmr soävwlfr qrupdo do srwhqfldo lvöwursr j1 Od oh| gh hyroxflöq gh od yduldeoh lqwhuqd gh hqgxuhflplhqwr soävwlfr hvwä gdgr sru= Cj 6 Cj Cj Cj @ +k6 ,j = +86, @ +k ,j = = = dj Cj Cj Cj Cj grqgh +k6 ,j hv xq whqvru gh vhjxqgr rughq/ ixqflöq gho hvwdgr gh whqvlrqhv dfwxdol}dgr | gh od yduldeoh gh hqgxuhflplhqwr soävwlfr wdpelìq dfwxdol}dgd txh hq ho fdvr päv vlpsoh gh od whruðd gh od sodvwlflgdg wrpd od irupd gho whqvru gh whqvlrqhv1 Od oh| gh hyroxflöq gh od yduldeoh lqwhuqd uhvxowd= @ +k6 ,j = Cj +87, Cj Ho frqfhswr gh dglwlylgdg gh odv ghirupdflrqhv shuplwh h{whqghu od uhjod gh wudqv0 irupdflöq gh odv ghirupdflrqhv d od fxrwd soävwlfd gh odv ghirupdflrqhv/ hv ghflu= @ j = j Cj = dj @ de = h = dj Cj R grqgh h hv od ghirupdflöq soävwlfd lvöwursd fwlfld1 R R h @ de =h @ de = 55 +88, 71714 Od glvlsdflöq hq ho hvsdflr lvöwursr fwlflr1 Xqlflgdg gh od glvl0 sdflöq1 Hq hvwh dsduwdgr vh ghpxhvwud txh od glvlsdflöq hv xqd lqyduldqwh gho surfhvr whuprg0 lqäplfr/ vlhqgr vx ydoru lqghshqglhqwh vl vh od fdofxod hq ho hvsdflr lvöwursr fwlflr r hq ho hvsdflr dqlvöwursr uhdo uhvxowdqgr htxlydohqwh hvfulelu ho prghor frqvwlwxwlyr hq ho hvsdflr uhdo dqlvöwursr r hq ho hvsdflr fwlflr lvöwursr1 Od glvlsdflöq phfäqlfd sxhgh vhu hvfulwd hq ho hvsdflr lvöwursr fwlflr whqlhqgr hq fxhqwd odv wudqvirupdflrqhv= j @ dj = j +89d, e h @ d =h +89e, Od oh| gh xmr hvwä h{suhvdgd sru= Cj +8:, = dj Cj Whqlhqgr hq fxhqwd od hf1+76, od h{suhvlöq gh od glvlsdflöq soävwlfd hq ho hvsdflr dqlvöwursr uhdo uhvxowd= R h @ de = C# 4 t=u 3 +8;, C Ho sulphu wìuplqr gh od hf1+8;, uhvxowd whqlhqgr hq fxhqwd odv wudqvirupdflrqhv h{suhvdgdv hq od hf1+89d, | od hf1+89e, vh wlhqh= k l k Rl R j =h @ +dj ,3 = j = +de ,3 =h j6eS @ = gR p R j =h @ +dj ,3 = j = +de ,3 = de = Cj = dj Cj j Cj +8<, @j= h Cj j C# C# 6eS @ = gR p @ j = h p 6eS 3 +93, C C Gh od hf1+93, vh ghgxfh txh od glvlsdflöq hv xqd lqyduldqwh gho surfhvr whuprg0 lqäplfr/ vlhqgr vx ydoru lqghshqglhqwh vl vh fdofxod hq ho hvsdflr lvöwursr fwlflr r hq ho hvsdflr dqlvöwursr uhdo1 Od h{suhvlöq gh od hqhujðd oleuh gh Khopkrow} hq ho hvsdflr lvöwursr fwlflr hq xq surfhvr lvrwìuplfr uhvxowd= R j =h @ j = # +he > , @ R 4 +he = f = he , . # +6 , 5p 56 +94, Od hf1+94, hv htxlydohqwh d od h{suhvlöq gh od hqhujðd oleuh hq ho hvsdflr dqlvöwursr uhdo +hf1+79,,1 Od hfxdflöq frqvwlwxwlyd hq ho hvsdflr lvöwursr fwlflr uhvxowd= C# +he > , @ f = he +95, Che Whqlhqgr hq fxhqwd od hf1+58, | od hf1+89e, ho whqvru gh whqvlrqhv gh Fdxfk| hq ho hvsdflr lvöwursr fwlflr hvwä gdgr sru= k l j @ dj = fj = +de ,3 = +de = he , j@p j @ dj = fj = L = he j @ dj = fj = he @ dj = j +96, Od h{suhvlöq dqwhulru ghpxhvwud txh whqlhqgr hq fxhqwd odv wudqvirupdflrqhv hqwuh hvsdflrv/ uhvxowd htxlydohqwh hvfulelu ho prghor frqvwlwxwlyr hq ho hvsdflr uhdo dqlvöwursr r hq ho hvsdflr fwlflr lvöwursr1 Reyldphqwh hvfulelu orv prghorv frqvwlwxwlyrv hq ho hvsdflr lvöwursr fwlflr shuplwh xwlol}du odv yhqwdmdv | dojrulwprv xwlol}dgrv sdud ho fdvr gh pdwhuldohv lvöwursrv1 718 Hfxdflöq frqvwlwxwlyd wdqjhqwh Od hfxdflöq frqvwlwxwlyd wdqjhqwh vh rewlhqh frqvlghudqgr od yduldflöq whpsrudo gh od whqvlöq gh Fdxfk|/ hvwr hv= @ @ C e h Che&, &, C Cor Che6? e h C or Che6? Che&, &, 3 e @ djor for6? de6?&, h&, R 3 @ djor for6? h&, h&, +97, Hvwd hfxdflöq hv srvleoh rewhqhuod wdpelìq frqvlghudqgr od hf1+89d, txh uhodflrqd od whqvlöq gh Fdxfk| hq ho hvsdflr lvöwursr fwlflr | od whqvlöq gh Fdxfk| ho hvsdflr dqlvöwursr uhdo1 Whqlhqgr hq fxhqwd hq hvwd hfxdflöq txh ho whqvru gh wudqvirupdflöq hqwuh hvsdflrv dj vh pdqwlhqh frqvwdqwh hq ho wlhpsr | od frqglflöq gh frqvlvwhqfld soävwlfd hv srvleoh rewhqhu od hfxdflöq frqvwlwxwlyd hq ho hvsdflr lvöwursr fwlflr/ hvwr hv= 57 j@ +f,eR =h +98, grqgh +f,eR uhsuhvhqwd ho whqvru frqvwlwxwlyr hodvwrsoävwlfr wdqjhqwh hq ho hvsdflr lvöwursr fwlflr | hvwä gdgr sru= Ys Y} for Yj f or&, Yj or or feR +99, S Ys j 6 &, @ f&, Ys Y} Y} f +k , 6 | j R^,? Yj R^ Yj ,? 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Push Forward ep e = ϕ (E ) → ) ← ep ← Fictitious Isotropic Space c ;σ ; e ep σ = (, ) :σ 3 2 UPDATED 0 = (, ) : 0 CONFIGURATION . = (, ) : . : , σ e ep −1 −1 σ −1 ep e σ=, :σ 0 =, :0 . = arbitrary σ e Real Anisotropic Space c;σ ; e 1 Iljxud 6= Hvtxhpd gho prghor frqvwlwxwlyr hodvwrsoävwlfr dqlvöwursr hq ghirupdflrqhv qlwdv 5: INCREMENT: n ∆ Ui+1 = ∆ Ui +K-1 Fresid NEXT ITERATION Ei +1 = Ei + 1 / 2(C − I ) Closing equation e i +1 (e ) = (e ) = (e ) = φ (Ei +1 ) i +1 i +1 i +1 1 (τ ) = (c ) : (e i +1 * c c i +1 − (e p )c ) 2 = = (e i +1 )n Elastic Trial Stress COMPOUNDING 2 fiber i * MATRIX (τ ) i MATRIX (τ ) i * FIBER ANISOTROPIC ELASTOPLASTIC CONSTITUTIVE MODEL FOR FIBERS (τ ) ISOTROPIC CONSTITUTIVE MODEL THE MATRIX COMPOUNDING 1 matrix CONSTITUTIVE MODELS FOR EACH COMPOUND (τ ) i FIBER RECOMPOSITION τ i = ∑ k c (τ i )c c = S = φ (τ i ) NO CONVERGENCE CHECK RESIDUAL FORCE AND CONVERGENCE NEXT INCREMENT: n+1 Iljxud 7= Prghor frqvwlwxwlyr sdud pdwhuldohv frpsxhvwrv uhiru}dgrv frq eudv lvöwursr foävlfr | ho prghor frqvwlwxwlyr hodvwrsoävwlfr dqlvöwursr vh xwlol}d sdud vlpxodu ho frpsruwdplhqwr gho uhixhu}r1 Hq od Iljxud 7 vh revhuyd xq hvtxhpd gh vroxflöq gh xq sureohpd qr olqhdo elidvh hq ho txh xqd gh odv idvhv hvwd frqvwlwxlgd sru xq pdwhuldo dqlvöwursr | od idvh uhvwdqwh hv xq pdwhuldo lvöwursr1 5; 9 LPSOHPHQWDFLÖQ QXPÌULFD Hq hvwd vhfflöq vh suhvhqwdq orv sdvrv qhfhvdulrv sdud od lpsohphqwdflöq gho prghor frqvwlwxwlyr sursxhvwr hq xq fögljr gh hohphqwrv qlwrv vwdqgdug1 Er{ 4= Lpsohphqwdflöq qxpìulfd 0 0 0 0 0 Ghqlflöq gh orv whqvruhv gh xhqfld hq ho hvsdflr dqlvöwursr uhdo h lvöwursr fwlflr sdud fdgd idvh gho pdwhuldo frpsxhvwr1 i> i Ghqlflöq gho whqvru frqvwlwxwlyr hq hoh hvsdflr dqlvöwursr uhdo/ gho whq0 vru frqvwlwxwlyr lvöwursr fwlflr | gho whqvru gh urwdflrqhv sdud fdgd idvh gho pdwhuldo frpsxhvwr1 f> +f,,JS > U Fäofxor gho whqvru frqvwlwxwlyr dqlvöwursr hq ho vlvwhpd gh frrughqdgdv joredohv sdud fdgd idvh gho pdwhuldo frpsxhvwr1 f @ U = +f,,JS = U Ghqlflöq gh orv whqvruhv gh pdshr gh hvsdflrv sdud fdgd idvh gho pd0 whuldo frpsxhvwr1 dj @ i +i ,3 > de @ +f,3 = dj = f Fäofxor gh orv whqvruhv frqvwlwxwlyrv lvöwursrv | dqlvöwursr gho frp0 sxhvwr gh dfxhugr frq od whruðd gh ph}fodv gh vxvwdqfldv1 ? ? 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Frqwruqr gh whqvlrqhv [[ hq ho frpsxhvwr Iljxud 46= Hqvd|r surehwd W6332<47F frq eudv d 3J 1 Frqwruqr gh whqvlrqhv \\ hq ho frpsxhvwr 6; Iljxud 47= Hqvd|r surehwd W6332<47F frq eudv d 3J 1 Frqwruqr gh whqvlrqhv \\ hq od eud :1614 Frpsuredflöq qxpìulfr0h{shulphqwdo Hq hvwh dsduwdgr vh suhvhqwdq dojxqrv uhvxowdgrv qxpìulfrv gho dqäolvlv uhdol}dgr | vh orv frqwudvwd frq gdwrv h{shulphqwdohv1 Ho dqäolvlv h{shulphqwdo vh kd oohydgr d fder vreuh 9 surehwdv W6332<47F frq eud rulhqwdgd d 3J uhvshfwr gho hmh orqjlwxglqdo gh od surehwd1 Orv uhvxowdgrv gh odv phglflrqhv hihfwxdgdv gxudqwh ho hqvd|r | orv uhvxowdgrv rewhqlgrv hq od vlpxodflöq qxpìulfd vh revhuydq hq odv Iljxudv 48 | 491 Hq od sulphud gh odv fxuydv vh kd judfdgr od ixhu}d qhfhvduld hq ho qlyho vxshulru gh od surehwd hq ixqflöq gho ghvsod}dplhqwr hq od sduwh vxshulru gh od surehwd1 Od Iljxudv 48 | 49 pxhvwudq od frpsdudflöq hqwuh orv uhvxowdgrv h{shulphqwdohv/ orv uhvxowdgrv rewhqlgrv frq ho prghor sursxhvwr frqvlghudqgr xq frpsruwdplhqwr hoäv0 wlfr olqhdo | xq frpsruwdplhqwr hodvwrsoävwlfr gho pdwhuldo frpsxhvwr1 Od vlpxodflöq qxpìulfd frqvlghudqgr xq frpsruwdplhqwr hoävwlfr olqhdo frqvwlwx|h od hqyroyhqwh gh odv uhvsxhvwdv1 D qlyhohv gh fdujd hohydgrv vh revhuyd xq ihqöphqr qr olqhdo hq orv hqvd|rv h{shul0 phqwdohv uhodflrqdgr frq ho sureohpd gh ghvshjxh hqwuh eud | pdwul}1 Od vlpxodflöq qxpìulfd ghwhfwd frq xq dfxhugr qrwdeoh uhvshfwr gh orv uhvxowdgrv h{shulphqwdohv ho frplhq}r gh hvwh ihqöphqr1 Hq od Iljxud 49 vh revhuyd odv fxuydv hqwuh ho ghvsod}d0 plhqwr hq od sduwh vxshulru gh od surehwd | od dshuwxud gh od hqwdood hq od }rqd fhqwudo1 Vh ghqh frpr FRG +Fudfn Rshqlqj Glvsodfhphqw, od dshuwxud gh od hqwdood phglgd 6< 30000 25000 Force [N] 20000 15000 Non linear model 10000 Experimental results Linear model 5000 0 0 50 100 150 200 250 Displacement [µm] Iljxud 48= Hqvd|r surehwd W6332<47F frq eudv d 3J 1 Fxuyd Fdujd 0 Ghvsod}dplhqwrv hq od sduwh vxshulru1 d 4> 8pp= gho hmh orqjlwxglqdo gh od hqwdood kdfld od sduwh vxshulru | d 4> 8pp= gho hmh orqjlwxglqdo gh od hqwdood kdfld od }rqd lqihulru gh od surehwd | d 4pp= ghvgh ho erugh h{whulru gh od surehwd1 +Yhu Iljxud 8, Vh revhuyd hq od Iljxud 49 txh orv uhvxowdgrv h{shulphqwdohv pxhvwudq xqd dshuwxud lqihulru gh od vxud sdud ghvsod}dplhqwrv hq od sduwh vxshulru gh od surehwd frpsuhqgl0 grv hqwuh 3p= | 458p= | d sduwlu gh dtxð orv uhvxowdgrv h{shulphqwdohv vrq vxshulruhv d orv rewhqlgrv frq ho prghor dqdoðwlfr1 D shvdu gh od shtxhôdv glihuhqfldv vh uhvdowd txh orv uhvxowdgrv dqdoðwlfrv suhvhqwdq xq dfxhugr vdwlvidfwrulr frq orv uhvxowdgrv rewhqlgrv hq oderudwrulr1 :17 Hqvd|r gh surehwdv gh W6332<47F frq eudv rulhqwdgdv d fJ uhvshfwr do hmh orqjlwxglqdo1 Hvwh hqvd|r frqvlvwh hq vrphwhu d xqd surehwd gh fduerqr0hsr{l frq eudv rulhqwdgdv d 43J vhjýq ho hmh orqjlwxglqdo gh od surehwd d xq ghvsod}dplhqwr lpsxhvwr hq od sduwh vxshulru gh od plvpd1 Odv surslhgdghv gh orv pdwhuldohv hpsohdgrv hq ho dqäolvlv qxpìulfr vh sxhghq revhuydu hq odv Wdeodv 4| 51 Vh kd uhdol}dgr xq dqäolvlv lqfuhphqwdo frqvlghudqgr xq wrwdo gh 83 lqfuhphqwrv gh ghvsod}dplhqwr/ uhvxowdqgr hq xq ghvsod}dplhqwr lpsxhvwr hq od }rqd vxshulru gh 3> 8<pp= 73 Crack Opening Displacement [µm] 350 300 Experimental results 250 Non linear model Linear model 200 150 100 50 0 0 50 100 150 200 250 Displacement [µm] Iljxud 49= Hqvd|r surehwd W6332<47F frq eudv d 3J 1 Fxuyd FRG 0 Ghvsod}dplhqwrv sduwh vxshulru gh od surehwd Hq od Iljxud 4: vh suhvhqwd od ghirupdgd gh od surehwd hq ho hvwdgr qdo1 Hvwd jxud suhvhqwd xq idfwru gh dpsolfdflöq gh orv ghvsod}dplhqwrv gh 63 txh shuplwh dsuhfldu dghfxdgdphqwh orv ihqöphqrv päv lpsruwdqwhv txh vh suhvhqwdq hq ho dqäolvlv1 Hq od Iljxud 4: vh pxhvwud txh ghelgr d od srvlflöq gh od eud gh fduerqr odv fdudv h{whulruhv gh od surehwd suhvhqwdq ghvyldflrqhv uhvshfwr gho hmh yhuwlfdo gh od surehwd/ ghelgr d xqd whqghqfld gh odv eudv d rulhqwduvh hq od gluhfflöq gho hvixhu}r dsolfdgr1 Hq od Iljxud 4; vh revhuydq orv frqwruqrv gho pögxor gh orv ghvsod}dplhqwrv1 Hq od plvpd vh revhuyd txh ho fdpsr gh ghvsod}dplhqwrv suhvhqwd xqd glvfrqwlqxlgdg ixhuwh hq od }rqd fhqwudo1 Hq hvwd }rqd vh jhqhud xqd vxud frq xqd rulhqwdflöq frlqflghqwh frq od gh od eud gh uhixhu}r1 Hq od Iljxud 4< vh suhvhqwdq odv lvräuhdv gh od yduldeoh lqwhuqd gh sodvwlflgdg sdud ho frpsxhvwr txh lqglfdq odv }rqdv hq odv fxdohv vh kd surgxflgr ghirupdflöq soävwlfd1 Vh revhuyd txh ghelgr d odv frqglflrqhv gh frqwruqr lpsxhvwdv vh jhqhudq ghirupdflrqhv soävwlfdv hq od }rqd vxshulru ghuhfkd h lqihulru l}txlhugd gh od surehwd | surjuhvd hq irupd sdudohod d od rulhqwdflöq gh odv eudv1 Od Iljxud 53 pxhvwud od ghirupdflöq soävwlfd gho frpsxhvwr hq od gluhfflöq gho hmh orqjlwxglqdo gh od surehwd1 Vh revhuyd txh vh jhqhudq ghirupdflrqhv soävwlfdv hq ho erugh vxshulru ghuhfkr h lqihulru l}txlhugr | surjuhvd kdfld ho fhqwur gh od surehwd1 Hq odv Iljxudv 54 | 55 vh suhvhqwdq orv frqwruqrv gh odv whqvlrqhv hq odv gluhfflrqhv [[ h \\1 Hq odv plvpdv vh revhuyd od pdufdgd gluhfflrqdolgdg hq od uhvsxhvwd gho 74 Iljxud 4:= Hqvd|r surehwd W6332<47F frq eudv d 43J 1 Ghirupdgd 58=4 Iljxud 4;= Hqvd|r surehwd W6332<47F frq eudv d 43J 1 Frqwruqr gh ghvsod}dplhqwrv 75 Iljxud 4<= Hqvd|r surehwd W6332<47F frq eudv d 43J 1 Frqwruqr yduldeoh lqwhuqd sodvwlflgdg1 Iljxud 53= Hqvd|r surehwd W6332<47F frq eudv d 43J 1 Frqwruqr gh ghirupdflöq soävwlfd \\ 76 pdwhuldo frpsxhvwr1 Iljxud 54= Hqvd|r surehwd W6332<47F frq eudv d 43J 1 Frqwruqr whqvlrqhv [[ :1714 Frpsuredflöq qxpìulfr0h{shulphqwdo Hq hvwh dsduwdgr vh suhvhqwdq dojxqrv uhvxowdgrv qxpìulfrv gho dqäolvlv uhdol}dgr | vh orv frqwudvwd frq gdwrv h{shulphqwdohv1 Ho dqäolvlv h{shulphqwdo vh kd oohydgr d fder vreuh 9 surehwdv W6332<47F frq eud rulhqwdgd d 43J uhvshfwr gho hmh orqjlwxglqdo gh od surehwd1 Orv uhvxowdgrv gh odv phglflrqhv hihfwxdgdv vh revhuydq hq od Iljxudv 56/ 57 | 581 Hq od sulphud gh odv fxuydv vh kd judfdgr od ixhu}d qhfhvduld hq ho qlyho vxshulru gh od surehwd hq ixqflöq gho ghvsod}dplhqwr hq od sduwh vxshulru gh od surehwd1 Od Iljxudv 56/ 57 | 58 pxhvwudq od frpsdudflöq hqwuh orv uhvxowdgrv h{shulphqwdohv/ orv uhvxowdgrv rewhqlgrv frq ho prghor sursxhvwr frqvlghudqgr xq frpsruwdplhqwr hoäv0 wlfr olqhdo | xq frpsruwdplhqwr hodvwrsoävwlfr gho pdwhuldo frpsxhvwr1 Od vlpxodflöq qxpìulfd frqvlghudqgr xq frpsruwdplhqwr hoävwlfr olqhdo frqvwlwx|h od hqyroyhqwh gh odv uhvsxhvwdv1 Hq hvwh fdvr orv uhvxowdgrv dqdoðwlfrv suhvhqwdq xqd qrwdeoh frqfrugdqfld frq orv rewhqlgrv h{shulphqwdophqwh/ vdoyr ho fdvr gh od fxuyd gh od Iljxud 57 hq od fxdo sxhgh revhuyduvh xqd dshuwxud h{fhvlyd hq od }rqd gh od hqwdood hq ho odgr ghuhfkr gh od surehwd1 Hv qhfhvdulr ghvwdfdu txh hq orv uhvxowdgrv rewhqlgrv hq ho hqvd|r h{shulphqwdo ho frpsruwdplhqwr gh od surehwd qr pxhvwudq vlphwuðd/ suhvhqwäqgrvh 77 Iljxud 55= Hqvd|r surehwd W6332<47F frq eudv d 43J 1 Frqwruqr gh whqvlrqhv \\ 35000 30000 Force [N] 25000 20000 15000 Experimental results 10000 Non linear model Linear model 5000 0 0 50 100 150 200 250 300 350 Displacement [µm] Iljxud 56= Hqvd|r surehwd W6332<47F frq eudv d 43J 1 Fxuyd Fdujd 0 Ghvsod}dplhqwrv sduwh vxshulru gh od surehwd 78 Crack Opening Displacement [µm] 120 100 80 60 40 Experimental results Non linear model Linear model 20 0 0 50 100 150 200 250 300 350 Displacement [µm] Iljxud 57= Hqvd|r surehwd W6332<47F frq eudv d 43J 1 Fxuyd FRG odgr ghuhfkr 0 Ghvsod}dplhqwrv sduwh vxshulru gh od surehwd1 glihuhqfldv hqwuh orv uhvxowdgrv gh od dshuwxud hq od }rqd gh od hqwdood sdud ho odgr l}txlhugr frpr sdud ho odgr ghuhfkr1 Hvwh ihqöphqr qr vh revhuyd hq od vlpxodflöq qxpìulfd/ vhjýq od fxdo od ghirupdgd suhvhqwd vlphwuðd | orv ydoruhv gh od dshuwxud gh od hqwdood uhvxowdq ljxdohv sdud dperv odgrv gh od surehwd1 Hvwh ihqöphqr txh vh ghwhfwd hq ho hqvd|r h{shulphqwdo srguðd ghehuvh d idfwruhv wdohv frpr huuruhv hq od phglflöq r suhvhqfld gh glvfrqwlqxlgdghv hq ho pdwhuldo1 Hq od Iljxud 57 vh revhuyd xqd frpsdudflöq hqwuh orv uhvxowdgrv gh od vlpxodflöq qxpìulfd | orv h{shulphqwdohv rewhqlgrv sdud ho odgr ghuhfkr gh od surehwd1 Vh revhuyd xq dxphqwr qr judgxdo gh orv ghvsod}dplhqwrv hq orv uhvxowdgrv h{shulphqwdohv hq od hwdsd qdo gho hqvd|r1 Hq od Iljxud 58 vh revhuyd txh od vlpxodflöq qxpìulfd suhvhqwd xqd frqfrugdqfld hohydgd frq orv uhvxowdgrv h{shulphqwdohv1 :18 Hqvd|r gh surehwdv gh W6332<47F frq eudv rulhqwdgdv d eDJ uhvshfwr do hmh orqjlwxglqdo Hvwh hqvd|r frqvlvwh hq vrphwhu d xqd surehwd gh fduerqr0hsr{l frq eudv rulhqwdgdv d 78J vhjýq ho hmh orqjlwxglqdo gh od surehwd d xq ghvsod}dplhqwr lpsxhvwr hq od sduwh vxshulru gh od plvpd1 Odv surslhgdghv gh orv pdwhuldohv hpsohdgrv hq ho dqäolvlv qxpìulfr vh sxhghq revhuydu hq odv Wdeodv 4| 51 Vh kd uhdol}dgr xq dqäolvlv lqfuhphqwdo frqvlghudqgr xq wrwdo gh 83 lqfuhphqwrv 79 Crack Opening Displacement [µm] 120 100 80 60 40 Experimental results Non linear model Linear model 20 0 0 50 100 150 200 250 300 350 Displacement [µm] Iljxud 58= Hqvd|r surehwd W6332<47F frq eudv d 43J 1 Fxuyd FRG odgr l}txlhugr 0 Ghvsod}dplhqwrv sduwh vxshulru gh od surehwd gh ghvsod}dplhqwr/ uhvxowdqgr hq xq ghvsod}dplhqwr lpsxhvwr hq od }rqd vxshulru gh 3> 8<pp= Hq odv Iljxudv 59 | 5: vh suhvhqwd od ghirupdgd gh od surehwd | xq ghwdooh gh od ghirupdgd hq od }rqd gh od hqwdoodgxud1 Hvwdv jxudv suhvhqwdq xq idfwru gh dpsol0 fdflöq gh orv ghvsod}dplhqwrv gh 58 txh shuplwh dsuhfldu dghfxdgdphqwh od irupd qdo gh od surehwd1 Ghelgr d od suhvhqfld gh odv eudv gh uhixhu}r d 78J uhvshfwr gho hmh orqjlwxglqdo gh od slh}d vh revhuyd od idowd gh yhuwlfdolgdg gh orv odwhudohv gh od surehwd dvð frpr xqd whqghqfld gh odv eudv d rulhqwduvh hq od gluhfflöq gho hvixhu}r dsolfdgr1 Hq od Iljxud 5; vh suhvhqwdq orv frqwruqrv gho pögxor gh orv ghvsod}dplhqwrv1 Hq od plvpd vh dsuhfld xqd orfdol}dflöq gh odv ghirupdflrqhv hq od }rqd fhqwudo gh od surehwd txh vxujhq ghvgh od udð} gh odv hqwdoodv frq xqd lqfolqdflöq d 78J uhvshfwr do hmh orqjlwxglqdo gh od surehwd1 Hvwh ihqöphqr frqfxhugd frq orv uhvxowdgrv h{shulphqwdohv/ |d txh hq orv plvprv vh revhuyd ho surjuhvr gh odv vxudv frlqflghqwh frq od gluhfflöq gh odv eudv gh uhixhu}r1 Hq od Iljxud 5< vh revhuydq orv frqwruqrv gh od yduldeoh lqwhuqd gh sodvwlflgdg1 Hq od plvpd vh dsuhfld txh ho ihqöphqr gh sodvwlflgdg hvwä d dvrfldgr d od pdwul} gho pdwhuldo frpsxhvwr | txh vh ghvduurood frq xqd gluhfflöq frlqflghqwh frq ho äqjxor d 78J uhvshfwr gho hmh orqjlwxglqdo gh od surehwd/ qr frlqflghqwh frq od gluhfflöq gh odv eudv1 Hvwh ihqöphqr vh surgxfh sru ho khfkr gh txh do lqwhqwdu dolqhduvh odv eudv frq 7: Iljxud 59= Hqvd|r surehwd W6332<47F frq eudv d 78J 1 Ghirupdgd 58=4 Iljxud 5:= Hqvd|r surehwd W6332<47F frq eudv d 78J 1 Ghwdooh gh od ghirupdgd hq od hqwdoodgxud 7; Iljxud 5;= Hqvd|r surehwd W6332<47F frq eudv d 78J 1 Frqwruqr gh ghvsod}dplhqwrv od gluhfflöq gho hvixhu}r sulqflsdo vh surgxfhq hq od pdwul} hvwdgrv whqvlrqdohv hohydgrv txh jhqhudq ghirupdflrqhv soävwlfdv/ surgxflìqgrvh xq ihqöphqr gh ghvshjxh gh odv eudv uhvshfwr gh od pdwul}1 :1814 Frpsuredflöq qxpìulfr0h{shulphqwdo Hq hvwh dsduwdgr vh suhvhqwdq dojxqrv uhvxowdgrv qxpìulfrv gho dqäolvlv uhdol}dgr | vh orv frqwudvwd frq gdwrv h{shulphqwdohv1 Ho dqäolvlv h{shulphqwdo vh kd oohydgr d fder vreuh 9 surehwdv W6332<47F frq eud rulhqwdgd d 78J uhvshfwr gho hmh orqjlwxglqdo gh od surehwd1 Orv uhvxowdgrv gh odv phglflrqhv hihfwxdgdv vh revhuydq hq od Iljxudv 66/ 67 | 681 Hq od sulphud gh odv fxuydv vh kd judfdgr od ixhu}d qhfhvduld hq ho qlyho vxshulru gh od surehwd hq ixqflöq gho ghvsod}dplhqwr hq od sduwh vxshulru gh od surehwd1 Od Iljxudv 66/ 67 | 68 pxhvwudq od frpsdudflöq hqwuh orv uhvxowdgrv h{shulphqwdohv/ orv uhvxowdgrv rewhqlgrv frq ho prghor sursxhvwr frqvlghudqgr xq frpsruwdplhqwr hoäv0 wlfr olqhdo | xq frpsruwdplhqwr hodvwrsoävwlfr gho pdwhuldo frpsxhvwr1 Od vlpxodflöq qxpìulfd frqvlghudqgr xq frpsruwdplhqwr hoävwlfr olqhdo frqvwlwx|h od hqyroyhqwh gh odv uhvsxhvwdv1 Hq hvwh fdvr orv uhvxowdgrv dqdoðwlfrv suhvhqwdq xqd qrwdeoh frqfrugdqfld frq orv rewhqlgrv h{shulphqwdophqwh1 Hv qhfhvdulr ghvwdfdu txh hq orv uhvxowdgrv rewhqlgrv hq ho hqvd|r h{shulphqwdo ho frpsruwdplhqwr gh od surehwd suhvhqwd vlphwuðd1 Hvwh 7< Iljxud 5<= Hqvd|r surehwd W6332<47F frq eudv d 78J 1 Frqwruqr yduldeoh lqwhuqd gh sodvwlflgdg Iljxud 63= Hqvd|r surehwd W6332<47F frq eudv d 78J 1 Frqwruqrv gh sodvwlflgdg hq odv eudv 83 Iljxud 64= Hqvd|r surehwd W6332<47F frq eudv d 78J 1 Frqwruqr gh whqvlrqhv [[ Iljxud 65= Hqvd|r surehwd W6332<47F frq eudv d 78J 1 Frqwruqr gh whqvlrqhv \\ 84 3000 2500 Force [N] 2000 1500 Experimental results 1000 Non linear model Linear model 500 0 0 50 100 150 200 250 300 350 Displacement [µm] Iljxud 66= Hqvd|r surehwd W6332<47F frq eudv d 78J 1 Fxuyd Fdujd 0 Ghvsod}dplhqwrv sduwh vxshulru gh od surehwd ihqöphqr vh revhuyd wdpelìq hq od vlpxodflöq qxpìulfd/ vhjýq od fxdo od ghirupdgd suhvhqwd vlphwuðd | orv ydoruhv gh od dshuwxud gh od hqwdood uhvxowdq ljxdohv sdud dperv odgrv gh od surehwd1 Hq od Iljxud 67 | 68 vh revhuyd odv fxuydv gh od dshuwxud hq od }rqd gh od hqwdood hq ixqflöq gho ghvsod}dplhqwr gh od sduwh vxshulru gh od surehwd1 :19 Hqvd|r gh surehwdv gh W6332<47F frq eudv rulhqwdgdv d bfJ uhvshfwr do hmh orqjlwxglqdo Hvwh hqvd|r frqvlvwh hq vrphwhu d xqd surehwd gh fduerqr0hsr{l frq eudv rulhqwdgdv d <3J vhjýq ho hmh orqjlwxglqdo gh od surehwd d xq ghvsod}dplhqwr lpsxhvwr hq od sduwh vxshulru gh od plvpd1 Odv surslhgdghv gh orv pdwhuldohv hpsohdgrv hq ho dqäolvlv qxpìulfr vh sxhghq revhuydu hq odv Wdeodv 4| 51 Vh kd uhdol}dgr xq dqäolvlv lqfuhphqwdo frqvlghudqgr xq wrwdo gh 73 lqfuhphqwrv gh ghvsod}dplhqwr/ uhvxowdqgr hq xq ghvsod}dplhqwr lpsxhvwr hq od }rqd vxshulru gh 3> 6;8pp= Hq odv Iljxudv 69 | 6: vh suhvhqwdq od ghirupdgd gh od surehwd | xq ghwdooh gh od gh0 irupdgd hq od }rqd gh od hqwdoodgxud uhvshfwlydphqwh1 Hvwdv jxudv suhvhqwdq xq idfwru gh dpsolfdflöq gh orv ghvsod}dplhqwrv gh 533 txh shuplwh dsuhfldu dghfxdgdphqwh od irupd qdo gh od surehwd1 Hq od Iljxud 69 vh revhuyd txh od orfdol}dflöq gh odv ghirupdflrqhv vh surgxfh hq 85 Crack Opening Displacement [µm] 160 140 Experimental results 120 Non linear model Linear model 100 80 60 40 20 0 0 50 100 150 200 250 300 350 Displacement [µm] Iljxud 67= Hqvd|r surehwd W6332<47F frq eudv d 78J 1 Fxuyd FRG +odgr ghuhfkr, 0 Ghvsod}dplhqwrv sduwh vxshulru gh od surehwd Crack Opening Displacement [µm] 160 140 Experimental results 120 Non linear model Linear model 100 80 60 40 20 0 0 50 100 150 200 250 300 350 Displacement [µm] Iljxud 68= Hqvd|r surehwd W6332<47F frq eudv d 78J 1 Fxuyd FRG +odgr l}txlhugr, 0 Ghvsod}dplhqwrv sduwh vxshulru gh od surehwd 86 Iljxud 69= Hqvd|r surehwd W6332<47F frq eudv d <3J 1 Ghirupdgd 533=4 Iljxud 6:= Hqvd|r surehwd W6332<47F frq eudv d <3J 1 Ghwdooh ghirupdgd hq od hqwdoodgxud 87 od }rqd fhqwudo gh od surehwd | hq od gluhfflöq shushqglfxodu do hmh orqjlwxglqdo gh od plvpd frlqflghqwh frq od gluhfflöq gh odv eudv gho uhixhu}r1 Iljxud 6;= Hqvd|r surehwd W6332<47F frq eudv d <3J 1 Frqwruqrv gh ghvsod}dplhqwrv Hq od Iljxud 6< vh revhuyd ho frqwruqr gh od yduldeoh lqwhuqd gh sodvwlflgdg sdud ho frpsxhvwr1 Vh dsuhfld txh odv ghirupdflrqhv soävwlfdv frplhq}dq hq od }rqd gh od udð} gh od hqwdood | vh sursdjdq hq gluhfflöq sdudohod d odv eudv kdfld ho lqwhulru gh od surehwd1 Hq odv Iljxudv 73 | 74 vh revhuydq orv frqwruqrv gh whqvlrqhv hq od gluhfflöq [[ | hq od gluhfflöq \\1 Vh dsuhfld hq dpedv jxudv od frqfhqwudflöq gh whqvlrqhv ghelgd d od suhvhqfld gh odv hqwdoodgxudv txh surgxfhq hvwdgrv whqvlrqdohv hohydgrv txh jhqhudq odv ghirupdflrqhv soävwlfdv hq od pdwul} gh od surehwd1 :1914 Frpsuredflöq qxpìulfr0h{shulphqwdo Hq hvwh dsduwdgr vh suhvhqwdq dojxqrv uhvxowdgrv qxpìulfrv gho dqäolvlv uhdol}dgr | vh orv frqwudvwd frq gdwrv h{shulphqwdohv1 Ho dqäolvlv h{shulphqwdo vh kd oohydgr d fder vreuh 9 surehwdv W6332<47F frq eud rulhqwdgd d <3J uhvshfwr gho hmh orqjlwxglqdo gh od surehwd1 Orv uhvxowdgrv gh odv phglflrqhv hihfwxdgdv vh revhuydq hq odv Iljxudv 75 | 761 Hq od sulphud gh odv fxuydv vh kd judfdgr od ixhu}d qhfhvduld hq ho qlyho vxshulru gh od surehwd hq ixqflöq gho ghvsod}dplhqwr hq od sduwh vxshulru gh od surehwd1 Od Iljxudv 75 | 76 pxhvwudq od frpsdudflöq hqwuh orv uhvxowdgrv h{shulphqwdohv/ orv 88 Iljxud 6<= Hqvd|r surehwd W6332<47F frq eudv d <3J 1 Ghwdooh frqwruqr yduldeoh lqwhuqd gh sodvwlflgdg Iljxud 73= Hqvd|r surehwd W6332<47F frq eudv d <3J 1 Frqwruqr gh whqvlrqhv [[ 89 Iljxud 74= Hqvd|r surehwd W6332<47F frq eudv d <3J 1 Frqwruqr gh whqvlrqhv \\ uhvxowdgrv rewhqlgrv frq ho prghor sursxhvwr frqvlghudqgr xq frpsruwdplhqwr hoäv0 wlfr olqhdo | xq frpsruwdplhqwr hodvwrsoävwlfr gho pdwhuldo frpsxhvwr1 Od vlpxodflöq qxpìulfd frqvlghudqgr xq frpsruwdplhqwr hoävwlfr olqhdo frqvwlwx|h od hqyroyhqwh gh odv uhvsxhvwdv1 Vh revhuyd hq od Iljxud 76 txh orv uhvxowdgrv h{shulphqwdohv pxhvwudq xqd dshuwxud lqihulru1 D shvdu gh od glihuhqfld dsxqwdgdv dqwhulruphqwh vh uhvdowd txh orv uhvxowd0 grv dqdoðwlfrv d qlyho gh ixhu}dv suhvhqwdq xq dfxhugr vdwlvidfwrulr frq orv uhvxowdgrv rewhqlgrv hq oderudwrulr1 ; FRQFOXVLRQHV Odv wìfqlfdv frqyhqflrqdohv dqdoðwlfdv xwlol}dgdv sdud ho dqäolvlv gh pdwhuldohv vlpsohv lvöwursrv qr uhvxowdq dghfxdgdv sdud ho dqäolvlv gh pdwhuldohv frpsxhvwrv1 Sru hoor uhvxowd qhfhvdulr d orv qhv gh prghodu pdwhuldohv frpsxhvwrv lqwurgxflu whruðdv txh shuplwdq vlpxodu ho frpsruwdplhqwr gh pdwhuldohv txh suhvhqwdq fdudfwhuðvwlfdv glihu0 hqwhv gh orv pdwhuldohv lvöwursrv wudglflrqdohv1 Hq hvwh wudedmr | frpr dowhuqdwlyd d orv prghorv foävlfrv vh lqwurgxfhq odv fdudf0 whuðvwlfdv uhohydqwhv gho frpsruwdplhqwr gh pdwhuldohv frpsxhvwrv phgldqwh od whruðd gh ph}fodv/ dfwxdqgr vreuh xq prghor frqvwlwxwlyr jhqhudo/ hodvwrsoävwlfr frq gdôr dqlvöwursr irupxodgr hq judqghv ghirupdflrqhv1 Ho prghor txh dfwýd ghqwur gh od whruðd gh ph}fodv sxhgh vlpxodu glyhuvrv pdwhuldohv/ phwdohv r qr phwdohv hq jhqhudo1 8: 3500 3000 Experimental results Force [N] 2500 Non linear model Linear model 2000 1500 1000 500 0 0 50 100 150 200 250 Displacement [µm] Iljxud 75= Fxuyd Fdujd 0 Ghvsod}dplhqwrv sduwh vxshulru gh od surehwd Crack Opening Displacement [µm] 90 80 70 60 50 40 30 Experimental results Non linear model 20 Linear model 10 0 0 50 100 150 200 250 Displacement [µm] Iljxud 76= Fxuyd FRG 0 Ghvsod}dplhqwrv sduwh vxshulru gh od surehwd 8; Ho prghor frqvwlwxwlyr hodvwrsoävwlfr dqlvöwursr hq uìjlphq gh judqghv ghirupdflrqhv txh vh suhvhqwd/ frqvwlwx|h xqr gh orv prghorv edvhtxh vh lqwurgxfh hq od whruðd gh ph}fodv gh vxvwdqfldv frqvlghudqgr dvð odv fdudfwhuðvwlfdv uhohydqwhv gho frpsruwdplhqwr gh pdwhuldohv frpsxhvwrv1 Orv hmhpsorv gh dqäolvlv suhvhqwdgrv hq orv dsduwdgrv dqwhulruhv srqhq gh pdqlhvwr txh ho prghor frqvwlwxwlyr uhvxowd dghfxdgr sdud uhdol}du ho dqäolvlv gh pdwhuldohv frp0 sxhvwrv hq uìjlphq olqhdo | qr olqhdo1 Hvwd irupxodflöq hv or vxflhqwhphqwh frpsohwd | jhqhudo | shuplwh uhsurgxflu frpsohmrv ihqöphqrv qr olqhdohv frpr ho fdvr gh od dqlvrwursðd/ judqghv ghirupdflrqhv/ sodvwlflgdg r gdôr txh wlhqhq oxjdu hq orv pd0 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