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(12) gh qlu rwud hfxdflöq gh flhuuh1 Rwud dowhuqdwlyd sdud wudwdu pdwhuldohv gh pdwul} frp0 sxhvwd/ hv uhdol}du xqd fruuhfflöq hq odv surslhgdghv gh fdgd frpsrqhqwh | pdqwhqhu od hfxdflöq gh flhuuh +hf14d,1. 615. Ixqflöq gh hqhujðd oleuh. Orv pdwhuldohv frpsxhvwrv txh fxpsohq frq od hf1+4d, | hf1+7,/ vdwlvidfhq wdpelìq od frqglflöq eävlfd gh od dglwlylgdg gh od hqhujðd oleuh gh vxv frpsrqhqwhv +Wuxvghoo dqg Wrxslq/ 4<93,1 Hvwr hv= e. 6. p# +h > > , @. ? [. nS pS. S ^h> +h. S'. R. ,S > > 6 S `. +;,. grqgh # S hv od hqhujðd oleuh fruuhvsrqglhqwh d fdgd xqd gh odv q vxvwdqfldv frpsr0 qhqwhv gh od ph}fod/ nS hv ho frh flhqwh gh sduwlflsdflöq yroxpìwulfd/ +hR ,S hv od ghiru0 er6J txh pdflöq soävwlfd gh fdgd idvh | 6 S vrq odv yduldeohv lqwhuqdv gho frpsrqhqwh f gh qhq ho frpsruwdplhqwr iðvlfr gh fdgd vxvwdqfld frpsrqhqwh1 Od whruðd gh ph}fod gh vxvwdqfldv eävlfdv dtxð xwlol}dgd vh edvd hq txh wrgdv odv vxvwdqfldv lqwhuylhqhq hq od uhvsxhvwd gho frpsxhvwr hq irupd sursruflrqdo do yroxphq txh rfxsdq uhvshfwr gho yroxphq wrwdo1 Ho frh flhqwh gh sduwlflsdflöq yroxpìwulfd vh gh qh frpr= nS @. gYS gYf. +<,. grqgh YS hv ho yroxphq gho frpsrqhqwh | Yf hv ho yroxphq wrwdo gho frpsxhvwr hq od frq jxudflöq uhihuhqfldo1 Od oh| gh frqvhuydflöq gh od pdvd hvwdeohfh txh= gy @ ghw I @ M +43, gY grqgh y hv ho yroxphq hq od frq jxudflöq hvsdfldo | Y hv ho yroxphq hq od frq jxudflöq pdwhuldo1 Whqlhqgr hq fxhqwd od hfxdflöq gh flhuuh +hf1 7, vh ghgxfh= gyS @M gYS. +44,. Od hf1+44, hvwdeohfh txh od uhodflöq hqwuh ho yroxphq gh xq frpsrqhqwh hq od frq j0 xudflöq hvsdfldo | ho yroxphq gho frpsrqhqwh hq od frq jxudflöq uhihuhqfldo hvwä gdgr sru ho ghwhuplqdqwh gho judglhqwh gh ghirupdflrqhv1 Whqlhqgr hq fxhqwd od hf1+44, | od hf1+<, vh wlhqh= nS @. 4 gyS gyS @ M gYf gyf. 44. +45,.
(13) grqgh yf uhsuhvhqwd ho yroxphq gho frpsxhvwr hq od frq jxudflöq hvsdfldo1 Od hf1+45, hvwdeohfh txh ho frh flhqwh gh sduwlflsdflöq vh pdqwlhqh frqvwdqwh hq dpedv frq jx0 udflrqhv1 Od whruðd gh ph}fodv vror hv yäolgd sdud ho fdvr hq txh wrgdv odv idvhv gho pdwhuldo frpsxhvwr uhvxowhq frpsuhvleohv r wrgdv lqfrpsuhvleohv1 Hq ho fdvr txh vh suhvhqwhq idvhv frpsuhvleohv h lqfrpsuhvleohv od hfxdflöq gh flhuuh qr hv srvleoh fxpsoluod ghelgr d txh odv idvhv lqfrpsuhvleohv qr dgplwhq xq hvwdgr gh ghirupdflrqhv yroxpìwulfr txh vl sxhgh vhu dgplwlgr sru odv idvhv frpsuhvleohv or txh suryrfd ho qr fxpsolplhqwr gh od hfxdflöq gh frpsdwlelolgdg1 +hf1 +7,,1 Ho frh flhqwh gh sduwlflsdflöq yroxpìwulfr gh fdgd frpsrqhqwh gheh fxpsolu od frqglflöq= ? [. nS @ 4. +46,. S'. 616. Hfxdflöq frqvwlwxwlyd. Od gh qlflöq gh od whqvlöq sdud ho frpsxhvwr vh rewlhqh d sduwlu gh od irupxodflöq gho prghor klshuhoävwlfr jdudqwl}dqgr od frqglflöq gh glvlsdflöq gho vhjxqgr Sulqflslr gh od Whuprglqäplfd +Pdoyhuq/ 4<9<,1 Hvwr hv= ?. j@p. ?. [ C# C# [ nS pS S @ nS pS +j,S @ Ch Ch S' S'. +47,. Ho whqvru wdqjhqwh frqvwlwxwlyr gho frpsxhvwr/ hvwä gdgr sru= ?. [ C2# f @ nS fA S @ Ch Ch S' A. +48,. grqgh fA S uhsuhvhqwd ho whqvru frqvwlwxwlyr wdqjhqwh dqlvöwursr uhdo sdud ho frpsr0 qhqwh fer6J 1 Hq od Iljxud 4 vh revhuyd xq gldjudpd gh xmr hvtxhpäwlfr d vhjxlu hq od vroxflöq gh sureohpdv qr olqhdohv hq ghirupdflrqhv qlwdv gh xq pdwhuldo frpsxhvwr pxowlidvh1 Hq od plvpd vh revhuyd txh fdgd idvh srvhh vx surslr prghor frqvwlwxwlyr | uhvxowd lqghshqglhqwh gh odv rwudv idvhv txh frqirupdq ho frpsxhvwr1. 7. PRGHOR HODVWRSODVWLFR DQLVRWURSR. Ho prghor frqvwlwxwlyr txh vh suhvhqwd hq hvwh dsduwdgr frqvwlwxwl|h xqr gh orv prghorv edvh txh vh lqwurgxfh hq od whruðd gh ph}fodv gh vxvwdqfldv/ suhyldphqwh suhvhqwdgd1 Hvwh prghor uhvxowd dghfxdgr sdud vlpxodu ho frpsruwdplhqwr frqvwlwxwlyr qr olqhdo gh pdwhuldohv äowdphqwh dqlvöwursrv1. 45.
(14) INCREMENT: n ∆ Ui+1 = ∆ Ui +K-1 Fresid. NEXT ITERATION. Reference Configuration. E i +1 = Ei + 1 / 2(C − I ) e i +1 = φ (Ei +1 ). (σ ) = (c ) : (e i +1 * c. i +1. c. − (e p )c ). Elastic Trial Stress. i. 1. (σ ). i * n. i * 3. CONSTITUTIVE 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 frq jxudflöq uhihuhqfldo | dfwxdol}dgd. 46.
(15) Od ghvfulsflöq gho frpsruwdplhqwr hoävwlfr dqlvöwursr gh xq vöolgr qr suhvhqwd judqghv gl fxowdghv1 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 xhqfld sdud pdwhuldohv ruwöwursrv vh ghehq d Kloo txlhq orjuö h{whqghu ho prghor lvöwursr gh Yrq Plvhv do fdvr ruwöwursr +Kloo/ 4<:4,1 Od sulqflsdo olplwdflöq gh hvwd whruðd vh hqfxhqwud hq od lpsrvlelolgdg gh prghodu ho frp0 sruwdplhqwr gh pdwhuldohv txh suhvhqwdq xq frpsruwdplhqwr txh qr vror ghshqgh gho vhjxqgr lqyduldqwh gho whqvru gh whqvlrqhv/ frpr sru hmhpsor ho fdvr gh jhrpdwhuldohv r pdwhuldohv frpsxhvwrv1 Od whruðd dqlvöwursd txh dtxð vh ghvduurood vh edvd hq odv lghdv sursxhvwdv sru Ehw0 whq +Ehwwhq/ 4<;4,+Ehwwhq/ 4<;;, | xwlol}d ho frqfhswr gh whqvru gh whqvlrqhv pdshdgr1 Hvwh frqfhswr shuplwh xwlol}du odv yhqwdmdv | dojrulwpr xwlol}dgrv sdud ho fdvr gh pd0 whuldohv lvöwursrv/ frq odv frqvljxlhqwhv yhqwdmdv hq od lpsohphqwdflöq frpsxwdflrqdo gho prghor1 Hq wudedmrv suhylrv glyhuvrv dxwruhv kdq ghvduuroodgr xqd jhqhudol}dflöq gh od whruðd gh sodvwlflgdg lvöwursd do fdvr dqlvöwursr +Roohu hw do1/ 4<<6e, +Roohu hw do1/ 4<<6d,1 Od lghd eävlfd frqvlvwðd hq prghodu ho frpsruwdplhqwr gh xq vöolgr hq ho hvsdflr dqlvöwursr uhdo d wudyìv gh xq vöolgr lghdo hq ho hvsdflr lvöwursr fwlflr +yhu Iljxud 6,1 Od klsöwhvlv eävlfd gho prghor frqvlvwðd hq vxsrqhu txh odv ghirupdflrqhv hoävwlfdv vrq lgìqwlfdv hq dperv hvsdflrv or fxdo lqwurgxfh xqd olplwdflöq hq od whruðd dqlvöwursd pdshdgd1 Hvwd olplwdflöq hvwä gdgd sru ho khfkr gh txh vh gheh uhvshwdu od sursruflrqdolgdg hqwuh ho oðplwh gh uhvlvwhqfld | ho pögxor gh hodvwlflgdg sdud fdgd gluhfflöq gho pdwhuldo1 Ho prghor frqvwlwxwlyr txh vh suhvhqwd hq hvwh wudedmr uhvxowd dsolfdeoh d pdwhuldohv txh suhvhqwdq xqd dowd dqlvrwursðd/ frpr hq ho fdvr gh orv pdwhuldohv frpsxhvwrv uhiru}dgrv frq eudv | uhvxowd gh xqd jhqhudol}dflöq gh od whruðd gh od sodvwlflgdg foävlfd1 Ho frpsruwdplhqwr dqlvöwursr gho pdwhuldo vh irupxod d wudyìv gh xq hvsdflr fwlflr lvöwursr gh whqvlrqhv | ghirupdflrqhv txh uhvxowd gh xqd wudqvirupdflöq whqvruldo olqhdo gh orv hvsdflrv uhdohv gh whqvlrqhv | ghirupdflrqhv dqlvöwursr1 Orv sduäphwurv txh lqwhuylhqhq hq od gh qlflöq gho whqvru gh wudqvirupdflöq vh rewlhqhq d wudyìv gh hqvd|rv h{shulphqwdohv1 Od yhqwdmd hq od xwlol}dflöq gh hvwh wlsr gh prghorv frqvlvwh hq od srvlelolgdg gh xwlol}du odv plvpdv ixqflrqhv gh xhqfld/ srwhqfldohv soävwlfrv | pìwrgrv gh lqwhjudflöq gh od hfxdflöq frqvwlwxwlyd ghvduuroodgrv sdud pdwhuldohv lvöwursrv1 Wrgd od lqirupdflöq gh od dqlvrwursðd gho pdwhuldo vh hqfxhqwud hq orv whqvruhv gh fxduwr rughq gh wudqvirupdflöq gh orv hvsdflrv gh whqvlrqhv +dj , | ghirupdflrqhv +de ,1 Od irupxodflöq uhvxowdqwh hv frpsohwdphqwh jhqhudo | shuplwh uhdol}du dqäolvlv gh pdwhuldohv pxowlidvh txh suhvhqwdq xq judgr gh dqlvrwursðd hohydgr frpr hv ho fdvr gh orv pdwhuldohv frpsxhvwrv uhiru}dgrv frq eudv1. 47.
(16) 714. Ixqflöq gh xhqfld | gh srwhqfldo soävwlfr. Ho prghor txh d frqwlqxdflöq vh suhvhqwd vh irupxod hq od frq jxudflöq dfwxdol}dgd | xwlol}d od flqhpäwlfd odjudqjhdqd wrwdo r dfwxdol}dgd1 +Juhhq dqg S1/ 4<97, +Oxeolqhu/ 4<<3,1 Hvwh prghor shuplwh vlpxodu ho frpsruwdplhqwr qr olqhdo gh pdwhuldohv vrphwlgrv d judqghv ghirupdflrqhv soävwlfdv | shtxhôdv ghirupdflrqhv hoävwlfdv ghelgr d txh sdud od gh qlflöq gh od hqhujðd oleuh hoävwlfd vh xwlol}d xq srwhqfldo fxdguäwlfr +Jdufðd Jdulqr dqg Rolyhu/ 4<<4,+Oxeolqhu/ 4<<3,1 Odv ixqflrqhv gh xhqfld | srwhqfldo soävwlfr hvwäq gh qlgdv hq ho hvsdflr gh whq0 vlrqhv gh Fdxfk| +frq jxudflöq hvsdfldo, gh od vljxlhqwh irupd= Ixqflöq gh xhqfld I j +j> 6 j, @ 3. Ixqflöq gh srwhqfldo Jj +j> 6 j, @ N. +49,. grqgh hv ho whqvru gh whqvlrqhv gh Fdxfk|1 Odv ixqflrqhv gh xhqfld | srwhqfldo soävwlfr uhvxowdq lvöwursdv vl vh fxpsoh od frqglflöq gh lqyduldqfld+Pdoyhuq/ 4<9<,= j 6 I j +dR d^ R^ > 6 j , @ I + R^ > j , @ 3 j. J. +dR d^ R^ > 6 j,. j. @ J. + R^ > 6 j,. @N. +4:, +4;,. sdud fxdotxlhu wudqvirupdflöq ruwrjrqdo d& d& @ grqgh d hv xq whqvru xqlgdg gldjrqdo | hv ho whqvru gh Nurqhfnhu1. 715. Wudqvirupdflöq gh hvsdflrv. Orv surfhglplhqwrv wudglflrqdohv txh shuplwhq rewhqhu odv hfxdflrqhv frqvwlwxwlydv sdud pdwhuldohv dqlvöwursrv hodvwrsoävwlfrv vh edvdq hq od ghvfulsflöq gh xqd vxshu flh gh xhqfld | xqd vxshu flh gh srwhqfldo soävwlfr hq wìuplqrv gh odv surslhgdghv fdudf0 whuðvwlfdv gho pdwhuldo1 Vdwlvidfhu odv frqglflrqhv gh lqyduldqfld hq hvwrv fdvrv uhvxowd gl fxowrvr qr vlhqgr vlhpsuh srvleoh1 Xq surfhglplhqwr txh jdudqwl}d od frqglflöq dqwhulru frqvlvwh hq gh qlu odv surslhgdghv gho vöolgr uhdo dqlvöwursr hq wìuplqrv gh xq vöolgr lvöwursr fwlflr d wudyìv gh xqd uhodflöq olqhdo gho hvsdflr gh whqvlrqhv uhdo | ho fwlflr+Roohu hw do1/ 4<<8, +Fdvdv hw do1/ 4<<;, +yhu Iljxud 6,= @ dj&, &,. +4<,. + &, ,3 @ dj&,. +53,. Whqlhqgr hq fxhqwd od hf1 +4<,. 48.
(17) grqgh &, | uhsuhvhqwdq orv whqvruhv gh whqvlrqhv hq ho hvsdflr uhdo dqlvöwursr | hq ho hvsdflr fwlflr lvöwursr/ uhvshfwlydphqwh | dj&, hv xq whqvru gh fxduwr rughq ghqrplqdgr whqvru gh wudqvirupdflöq gh hvsdflrv txh hvwdeohfh od uhodflöq hqwuh orv hvsdflrv fwlflr | uhdo gh whqvlrqhv1 Ho whqvru gh wudqvirupdflöq gh hvsdflrv +dj , hvwä gh qlgr hq od frq jxudflöq dfwxdol}dgd frpr= dj&, @ i +i&, ,3. +54,. j uhsuhvhqwdq orv oðplwhv hoävwlfrv whqvlrqdohv gho pdwhuldo hq ho hvsdflr grqgh ij | i&, uhdo dqlvöwursr | ho fwlflr lvöwursr uhvshfwlydphqwh1 Vh gh qh od uhodflöq hqwuh odv ghirupdflrqhv gh Dopdqvl hoävwlfdv uhdohv he | odv ghirupdflrqhv gh Dopdqvl hoävwlfdv fwlfldv he frpr=. he @ de&, he&,. +55,. Hvwd klsöwhvlv lpsolfd qr xqlflgdg hq odv ghirupdflrqhv hoävwlfdv hqwuh hvsdflrv1 D sduwlu gh hvwd h{suhvlöq hv srvleoh fdofxodu ho whqvru gh wudqvirupdflöq gh odv ghirupd0 flrqhv +de , whqlhqgr hq fxhqwd od hf1+ 4<, gh od vljxlhqwh pdqhud= + &, ,3 @ dj&, @ +f&or heor , +f,6? he6? ,3 dj&, @ f&or heor +he6? ,3 +f,6? ,3 ~} @eor6?. dj&, @ f&or deor6? +f,6? ,3. +56,. rewhqlìqgrvh d sduwlu gh dtxð= 3 j j deor6? @ fj&or d&, f,6?. +57,. grqgh f&or hv ho whqvru frqvwlwxwlyr hq ho hvsdflr lvöwursr fwlflr | f,6? hv ho whqvru frqvwlwxwlyr hq ho hvsdflr dqlvöwursr uhdo1 Od hohfflöq gh f&or hv duelwuduld | sxhgh hohjluvh sru or wdqwr odv surslhgdghv gh fxdotxlhu pdwhuldo frqrflgr1 Whqlhqgr hq fxhqwd od hf1+56, hv srvleoh hvwdeohfhu od uhodflöq hqwuh orv whqvruhv frqvwlwxwlyrv hq ho hvsdflr dqlvöwursr uhdo h lvöwursr fwlflr= 3 f,6? @ dj&, f&or deor6?. +58,. Ho whqvru frqvwlwxwlyr dqlvöwursr uhdo f hvwä h{suhvdgr hq frrughqdgdv orfdohv/ sru or wdqwr uhvxowd qhfhvdulr wudqvirupduor do vlvwhpd gh uhihuhqfld joredo d wudyìv gh xq whqvru gh urwdflöq gh fxduwr rughq U hvwr hv= f&, @ Uor +forR^ ,,JS U&R,^. +59,. grqgh +forR^ ,,JS hv ho whqvru frqvwlwxwlyr orfdo gh fxduwr rughq hq ho hvsdflr dqlvöwursr uhdo1 Ho whqvru gh urwdflrqhv U&, vh gh qh frpr= 49.
(18) Iljxud 5= Prgl fdflöq gh od irupd gh glyhuvdv ixqflrqhv gh xhqfld. U&, @ u& u, +5:, k l $ h hv ho yhuvru xqlwdulr fruuhvsrqglhqwh d grqgh u& @ frv +h ,},JK > +h ,,JS / vlhqgr od frpsrqhqwh l0ìvlpd gho vlvwhpd gh uhihuhqfld frrughqdgr1 Ho whqvru gh urwdflöq U wlhqh hq fxhqwd orv äqjxorv hqwuh odv gluhfflrqhv sulqflsdohv gho pdwhuldo dqlvöwursr | ho vlvwhpd gh frrughqdgdv joredohv1 Od wudqvirupdflöq gdgd hq od hf1+4<, frqgxfh d fdpelrv hq od irupd gh od vxshu flh gh xhqfld1 Hvwr vh sxhgh revhuydu hq od Iljxud 5 sdud glihuhqwhv fulwhulrv gh xhq0 fld1 Ho pdshr gh hvsdflrv shuplwh uhsuhvhqwdu dghfxdgdphqwh pdwhuldohv äowdphqwh dqlvöwursrv/ wdo ho fdvr gh xqd eud gh uhixhu}r hq xqd pdwul} gh pdwhuldo frpsxhvwr fxdqgr od uhodflöq @ wlhqgh d lq qlwr6 1 Sdud ho fdvr gh sodvwlflgdg dvrfldgd | frq xq fulwhulr gh xhqfld gh Yrq Plvhv od Iljxud 5 pxhvwud od sìuglgd uhvlvwhqfld hq xqd gluhfflöq gdgd/ plhqwudv txh hq od gluhfflöq qrupdo d hvwd h{lvwh xq fuhflplhqwr gho xmr soävwlfr hq od plvpd sursruflöq1 6. o hv od uhodflöq hqwuh odv whqvlrqhv gh xhqfld hq orv hvsdflrv lvöwursr fwlflr | dqlvöwursr uhdo. 4:.
(19) 716. Hfxdflöq frqvwlwxwlyd. Hq hvwd vhfflöq vh rewlhqh od h{suhvlöq gh od ixqflöq gh od hqhujðd oleuh sdud idvhv dqlvöwursdv1 Hvwä ixqflöq vhuä oxhjr xwlol}dgd sdud gh qlu ho hvwdgr whqvlrqdo gh dfxhugr d od hf1 +47,1 71614. Dvshfwrv eävlfrv. Od hfxdflöq frqvwlwxwlyd sdud xq pdwhuldo dqlvöwursr vh rewlhqh d sduwlu gh hvfulelu od 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 frq jxudflöq hvsdfldo/ $ hv od ghqvlgdg gh hqhujðd lqwhuqd hvshfð fd/ j hv ho whqvru gh whqvlrqhv gh Fdxfk|/ g ho judglhqwh gh yhorflgdghv/ u hv od ixhqwh lqwhuqd gh fdoru hvshfð fd | 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 frq jxudflöq uhihuhqfldo/ $ hv od hqhujðd lqwhuqd hvshfð fd/ V hv ho vhjxqgr whqvru gh whqvlrqhv gh Slrod0Nlufkr / H ho whqvru gh ghirupdflrqhv gh Fdxfk|0Juhhq/ u hv od ixhqwh lqwhuqd gh fdoru hvshfð fd | t hv ho xmr gh fdoru sru frqgxfflöq1 Gh qlhqgr 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 hvshfð fd | 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;.
(20) 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 frq jxudflöq 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 vxshu flh 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 gh qlflöq 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 gh qlflöq gh od hqhujðd oleuh gh Khopkrow} qr hv fruuhfwr hpsohdu frpr ydul0 deoh oleuh ho whqvru hoävwlfr gh Juhhq 0 Odjudqjh hq od frq jxudflöq 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 gh qlu 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<.
(21) 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 gh qlgd frpr= ] | 4 R e R 3 h @h h @ h gw +6<, Le 5 f grqgh hR hv od ghirupdflöq soävwlfd hq od frq jxudflöq hvsdfldo/ e hv ho whqvru l}txlhugr R gh Fdxfk|0Juhhq/ | h uhsuhvhqwd od yduldflöq whpsrudo gh od ghirupdflöq soävwlfd hq od frq jxudflöq 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,.
(22) 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 gh qlu 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 frq jxudflöq 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 gh qlhqgr xq pdwhuldo glihuhqwh/ pdwhpäwlfdphqwh yäolgr shur vh slhugh ydolgh} iðvlfd1 Sdud ho fdvr gh shtxhôdv ghirupdflrqhv hoävwlfdv uhvxowd vx flhqwh 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:,.
(23) Vxvwlwx|hqgr od hf1+7:, hq od hf1+77, vh rewlhqh od hfxdflöq frqvwlwxwlyd hq ho hvsd0 flr dqlvöwursr uhdo gh qlgd 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,.
(24) 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,.
(25) 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 C or 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.
(26) . . 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 ,? Yk Yj | ?. r. Whqlhqgr hq fxhqwd od hf1+89d, txh uhodflrqd ho whqvru gh whqvlrqhv hq ho hvsdflr dqlvöwursr uhdo | ho hvsdflr lvöwursr fwlflr hv srvleoh rewhqhu od yduldflöq gh whqvlöq hq ho hvsdflr dqlvöwursr uhdo frpr= 3 eR @ dj&, f&,or hor. +9:,. Frpelqdqgr od hf1+9:, frq od hf1+89e, | whqlhqgr hq fxhqwd txh ho whqvru gh wudqv0 irupdflöq gh hvsdflrv de qr ghshqgh gho wlhpsr/ vh rewlhqh od h{suhvlöq gho whqvru frqvwlwxwlyr hodvwrsoävwlfr dqlvöwursr frpr= 3 eR e @ dj&, f dorR^ hR^ ~}&,or . +9;,. SeR R^. Od hf1 +9;, pxhvwud od h{suhvlöq gho whqvru hodvwrsoävwlfr dqlvöwursr/ txh uhvxowd xqd ixqflöq gho whqvru frqvwlwxwlyr hodvwrsoävwlfr wdqjhqwh hq ho hvsdflr lvöwursr fwlflr1 Ho prghor frqvwlwxwlyr suhvhqwdgr qhfhvlwd vrodphqwh d orv qhv gh prghodu ho frpsruwdplhqwr frqvwlwxwlyr gh xq pdwhuldo frpsxhvwr dowdphqwh dqlvöwursr gh qlu odv vljxlhqwhv fdudfwhuðvwlfdv gho pdwhuldo hq orv hvsdflrv uhdo dqlvöwursr | fwlflr lvöwursr= Hvsdflr dqlvöwursr uhdo= Whqvru frqvwlwxwlyr lqlfldo hq frrughqdgdv orfdohv f,JS@, Xpeudo gh xhqfld gho pdwhuldo +e,_ Whqvru gh urwdflrqhv U Hvsdflr lvöwursr uhdo Ixqflöq gh xhqfld i +j> , @ 3 Ixqflöq gh srwhqfldo soävwlfr j +j> , @ 3 Xpeudo gh xhqfld gho pdwhuldo +e,_ Hq od Iljxud 6 vh revhuydq odv rshudflrqhv txh vh ghehq oohydu d fder sdud fdgd idvh txh frqvwlwx|h ho pdwhuldo frpsxhvwr1 Od sulphud rshudflöq frqvlvwh wudqvsruwdu ho whqvru gh ghirupdflrqhv H gh od frq jxudflöq uhihuhqfldo d od frq jxudflöq dfwxdol}dgd xwlol}dqgr odv rshudflrqhv sxvk iruzdug+yhu Dsìqglfh L,1 Od rshudflöq 5 frqvlvwh 58.
(27) fdofxodu od whqvlöq suhglfwrud jW | wudqvsruwdu orv whqvruhv gh whqvlrqhv suhglfwrudv/ ghirupdflrqhv | frqvwlwxwlyr j W > h | f gh od frq jxudflöq dfwxdol}dgd do hvsdflr lvöwursr fwlflr xwlol}dqgr orv whqvruhv dj | de 1 Hq ho hvsdflr lvöwursr fwlflr vh surfhgh d yhul fdu vl odv whqvlrqhv hq ho pdwhuldo kdq vxshudgr ho oðplwh gh sursruflrqdolgdg1 Hq ho fdvr kdehu vxshudgr ho oðplwh gh xhqfld vh surfhgh d od vroxflöq gho sureohpd hodvwrsoävwlfr1 Od vroxflöq gho sureohpd hodvwrsoävwlfr sursruflrqd xq qxhyr hvwdgr whqvlrqdo j1 Od rshudflöq 6 frqvlvwh hq uhwruqdu gho hvsdflr lvöwursr fwlflr do hvsdflr dqlvöwursr uhdo j | ho whqvru frqvwlwxwlyr hodvwrsoävwlfr eR hvwh qxhyr hvwdgr whqvlrqdo 3 j j wdqjhqwh f xwlol}dqgr orv whqvruhv +d , | +de ,3 1 Hq od rshudflöq vh fdofxodq odv ixhu}dv uhvlgxdohv | vh yhul fd ho htxloleulr joredo gho vöolgr1. 719. Lqwhjudflöq gh od hfxdflöq frqvwlwxwlyd. Ho fdvr gh pdwhuldohv frq frpsruwdplhqwr lqhoävwlfr uhtxlhuh od lqwhjudflöq qxpìulfd gh od hfxdflöq frqvwlwxwlyd hq xqd vhfxhqfld glvfuhwd gh sdvrv gh wlhpsr1+Vlpr dqg Wd|oru/ 4<;8, Ho uhvxowdgr gho dojrulwpr gh lqwhjudflöq hv xqd ixqflöq gh uhvsxhvwd qr olqhdo txh gh qh ho whqvru gh whqvlrqhv frpr xqd ixqflöq gh od klvwruld gh ghirupdflrqhv kdvwd ho sdvr gh wlhpsr dfwxdo1 Hvwh dojrulwpr gh lqwhjudflöq shuplwh wudwdu ho sureohpd hodvwrsoävwlfr ixqgdphqwdophqwh frpr xq sureohpd hoävwlfr htxlydohqwh hq ho sdvr gh wlhpsr1 Ho rshudgru wdqjhqwh txh lqwhuylhqh hq ho sureohpd olqhdol}dgr vh gheh rewhqhu phgldqwh xqd olqhdol}dflöq gh od ixqflöq gh uhvsxhvwd frqvlvwhqwh frq ho dojrulwpr gh lqwhjudflöq gh od hfxdflöq frqvwlwxwlyd1 Od xwlol}dflöq gh hvwrv rshudgruhv wdqjhqwhv suhvhuyd od frqyhujhqfld fxdguäwlfd gh hvtxhpdv gh vroxflöq lwhudwlyrv edvdgrv hq pìwr0 grv gh Qhzwrq1 Hq ho prghor frqvwlwxwlyr sursxhvwr od lqwhjudflöq gh od hfxdflöq frqvwlwxwlyd vh uhdol}d hq ho hvsdflr lvöwursr fwlflr1 Od lqwhjudflöq gh od hfxdflöq frqvwlwxwlyd hvwä edvdgd hq ho frqfhswr gh uhwxuq pdsslqj xwlol}dqgr ho dojrulwpr gh edfnzdug0Hxohu1 Hq od uhvroxflöq gh sureohpdv hodvwrsoävwlfrv txh lqfox|hq hfxdflrqhv frqvwlwxwlydv lq0 fuhphqwdohv od frqvlvwhqfld hqwuh ho rshudgru wdqjhqwh | ho dojrulwpr gh lqwhjudflöq hpsohdgr hq od vroxflöq gho sureohpd lqfuhphqwdo mxhjd xq sdsho ixqgdphqwdo1 Hq ho prghor sursxhvwr vh kdq xwlol}dgr hvwrv rshudgruhv wdqjhqwhv txh suhvhuydq od frqyhu0 jhqfld fxdguäwlfd gh hvtxhpdv gh vroxflöq lwhudwlyrv edvdgrv hq pìwrgrv gh Qhzwrq1. 8. PRGHOR FRQVWLWXWLYR GHO FRPSXHVWR. Ho prghor hodvwrsoävwlfr dqlvöwursr hq uìjlphq gh judqghv ghirupdflrqhv/ suhvhqwdgr hq ho dsduwdgr dqwhulru/ frqvwlwx|h xqr gh orv prghorv edvh txh vh xwlol}dq hq od whruðd gh ph}fodv1 Hq sduwlfxodu sdud ho fdvr gh pdwhuldohv frpsxhvwrv uhiru}dgrv frq eudv odujdv vh frqvlghudq xq prghor frqvwlwxwlyr sdud fdgd idvh= eud | pdwul}1 Ho frpsruwdplhqwr gh od pdwul} vh vlpxod frq xq prghor frqvwlwxwlyr hodvwrsoävwlfr 59.
(28) REFERENCE CONFIGURATION. Real Anisotropic Space. C; S; E. 4 Pull Back. $ = ϕ(σ ) ←. = ϕ(.. 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:.
(29) 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;.
(30) 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. . Gh qlflöq gh orv whqvruhv gh xhqfld hq ho hvsdflr dqlvöwursr uhdo h lvöwursr fwlflr sdud fdgd idvh gho pdwhuldo frpsxhvwr1 i> i Gh qlflöq 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 Gh qlflöq 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 ? ? S S f@ nS +f,S > f@ nS +f,S S'. S'. ORRS RYHU LQFUHPHQWR GH FDUJDV q@4/ Pd{1 Lqfu11 gh fdujdv ORRS RYHU LWHU1 GH HTXLOLEULR= l@4/ Pd{1 qýp1 gh lwhu1 41 Fäofxor gh od pdwul} gh uljlgh}1 ? N Ee ,? +N, @ D?e,e6 N Ee e' 51 Fäofxor gho lqfuhphqwr gh ghirupdflrqhv hq od frq jxudflöq uhihuhqfldo k l3 ? +X ,n @ ? +N, +Ioer_ , > ? +X ,n @? +X , .? +X ,n ? +H,n @ 4@5 IA I L 61 Sxvk0iruzdug gho whqvru gh ghirupdflrqhv d od frq jxudflöq dfwxdol}dgd l $ k? ? +h,n @ ! +H,n. 5<.
(31) Er{ 4= Lpsohphqwdflöq qxphulfd +frqwlqxdflöq, 71 Lqwhjudflöq gh od hfxdflöq frqvwlwxwlyd hq od frq jxudflöq dfwxdol}dgd ORRS VREUH FDGD IDVH GHO PDWHULDO FRPSXHVWR= m@4/QFRPS +d, Fäofxor suhglfwrud1 k l k gh od whqvlöq l ? +j W ,n. fwlflr1. ? +h,n. @ ^f`S =. S. ? +hR ,. +e, Wudqvirupdflöq gh od whqvlöq suhglfwrud do hvsdflr lvöwursr k. ? +j W ,n. l S. @ ^dj `S =. k. ? +j W ,n. l S. +f, Lqwhjudflöq k frqvwlwxwlyd1 k gh od hfxdflöq l l ? +j W ,n , ? +j,n S S & % +UG YY}j , YYsj GU ? ^+f,eR `n @ f S Ys GUG Y} 3S Ys GE6 G Y} 6 Yj. Yj. ? Ykr. j Yj. S. +g, Wudqvirupdflöq gh orv whqvruhv gh whqvlrqhv | frqvwlwxwlyr hodvwrsoävwlfr wdqjhqwh do hvsdflr k dqlvöwursr k l l k uhdo1 l ? + ,n. ? ^+f,eR `n S. @ +dj ,3 = ? + ,n S l S e, @ +dj ,3 =? ^+f,eR `n = +d S S Sk. S. HQG ORRS VREUH FDGD IDVH GHO PDWHULDO FRPSXHVWR 81 Fäofxor gh odv whqvlrqhv | ho whqvru frqvwlwxwlyr gh dfxhugr frq od whruðd gh ph}fodv gh vxvwdqfldv1 l S?SJ6R k? n ? + ,n @ n + , S ' S l S?SJ6R k? eR n ? ^+f,eR `n @ n ^+f, ` S ' S. 91Sxoo edfn gh oddv whqvlrqhv | gho whqvru frqvwlwxwlyr hodvwrsoäv0 wlfr d od frq jxudflöq uhihuhqfldo ? +V,n @ # ! ? + ,n ? +F,n @ # ! ? +f,n # :1 Fäofxor gh odv ixu}dv uhvlgxdohv hq od frq jxudflöq uhihuhqfldo1 k ln k ln n ? I Ee ? +I ?e,e6 I Ee , , @ D e' oer_ oer_ oer_. . Li nIoer_ n A 3B @, l @ l . 4 Jr edfn wr +4, hovh vroxflöq frqyhujlgd sdud ho lqfuhphqwr q q @ q . 4 Qxhyr lqfuhphqwr gh fdujd HQG ORRS VREUH FDGD LWHUDFLRQ GH HTXLOLEULR HQG ORRS VREUH FDGD LQFUHPHQWR GH FDUJD VWRS 63.
(32) :. HMHPSORV QXPHULFRV. :14. Ghvfulsflöq jhqhudo. Hq hvwh dsduwdgr vh suhvhqwdq glyhuvrv hmhpsorv gh dsolfdflöq gh od irupxodflöq txh frpelqd od whruðd gh ph}fodv | ho prghor dqlvöwursr ghvfulwr hq ho dsduwdgr 71 Orv hmhpsorv frqvlvwhq hq vlpxodu qxpìulfdphqwh xq hqvd|r vreuh surehwdv gh pdwhuldo frpsxhvwr vrphwlgdv d xq hvwdgr gh wudfflöq1 Odv vlpxodflrqhv vh kdq uhdol}dgr hp0 sohdqgr xqd pdood gh hohphqwrv qlwrv uhfwdqjxoduhv gh 7 qrgrv frq xq wrwdo gh 4<;; hohphqwrv/ 53<: qrgrv | 74<7 judgrv gh olehuwdg1 Hvwd pdood uhvxowd vlplodu d od xwl0 ol}dgd hq hvwxglr uhdol}dgr sru od Hxurshdq Vsdfh Djhqf| +Vwdyulqlglv/ 4<;8,1 +yhu Iljxud 9,1 Hq fdgd xqd gh odv vlpxodflrqhv qxpìulfdv vh suhvhqwdq orv uhvxowdgrv rewhqlgrv xwlol}dqgr ho prghor frqvwlwxwlyr suhyldphqwh ghvfulwr | frqvlghudqgr ho frqmxqwr gh surslhgdghv phfäqlfdv gh fdgd xqr gh orv pdwhuldohv frqvwlwx|hqwhv gho pdwhuldo frp0 sxhvwr1 Orv uhvxowdgrv txh vh suhvhqwdq vh glylghq hq grv juxsrv= Uhvxowdgrv juä frv/ hq orv fxdohv vh suhvhqwdq od ghirupdgd gh od slh}d dvð frpr lvräuhdv gh orv hvwdgrv qdohv whqvlrqdohv/ gh sodvwlflgdg | gh ghirupdflrqhv1 Uhvxowdgrv fxdqwlwdwlyrv/ hq orv fxdohv vh ghwdood hq fxuydv [ \ ho qlyho gh ixhu}dv hq ho h{wuhpr vxshulru | orv ghvsod}dplhqwrv/ frqwudvwdgrv frq uhvxowdgrv h{shulphqwdohv/ dvð frpr wdpelìq od pdjqlwxg gh od dshuwxud hq od }rqd gh od hqwdoodgxud uhvshfwr gho ghvsod}dplhqwr hq ho h{wuhpr gh od surehwd/ frpsdudgd frq phglflrqhv h{shulphqwdohv1 Orv gdwrv h{shulphqwdohv vh kdq rewhqlgr gho HVL Surmhfw Qe HG2;606;62UG2PV | HG2;707482UG2PV +Vwdyulqlglv/ 4<;8, | gh orv hqvd|rv uhdol}dgrv sru od Hxurshdq Vsdfh Djhqf| +GIYOU/ 4<;6,1 Od uhvlqd vh frpsruwd frpr xq pdwhuldo lvöwursr frq xqd oh| gh frpsruwdplhqwr hodvwrsoävwlfr/ plhqwudv txh odv eudv vh frpsruwdq frpr xq pdwhuldo hodvwrsoävwlfr dqlvöwursr +GIYOU/ 4<;6,1 D orv qhv gh vlpxodu ho frpsruwdplhqwr gho pdwhuldo frpsxhvwr vh xwlol}dq grv prghorv edvh hq ho frqwh{wr gh od whruðd gh ph}fodv/ xqr sdud od pdwul} | ho uhvwdqwh sdud od idvh gh uhixhu}r1 Vh frqvlghud txh od surehwd hvwä vrphwlgd d xq ghvsod}dplhqwr lpsxhvwr hq od sduwh vxshulru gh od plvpd jhqhuäqgrvh xq hvwdgr gh wudfflöq1 Hq dperv h{wuhprv gh od surehwd orv judgrv gh olehuwdg vh hqfxhqwudq suhvfulwrv1 Ho remhwlyr sulqflsdo hv ghprvwudu od fdsdflgdg gho prghor frqvwlwxwlyr sdud uhsuhvhqwdu ho frpsruwdplhqwr gh orv pdwhuldohv frpsxhvwrv vrphwlgrv d hvixhu}rv gh wudfflöq1. :15. Hqvd|r d wudfflöq gh surehwdv hqwdoodgdv frq eudv xqlgluhf0 flrqdohv. Ho hqvd|r vh uhdol}d vreuh surehwdv gh fduerqr0hsr{l W6332<47F txh suhvhqwdq xqd hqwdoodgxud hq od }rqd fhqwudo gh od plvpd1 Ho uhixhu}r hvwä frqvwlwxlgr sru eudv gh fduerqr rulhqwdgdv d glyhuvrv äqjxorv uhvshfwr gho hmh orqjlwxglqdo gh od surehwd1 64.
(33) Hvwh hqvd|r frqvlvwh hq lpsrqhu d od slh}d xq ghvsod}dplhqwr fuhflhqwh hq od sduwh vxshulru gh od plvpd jhqhudqgr xq hvwdgr whqvlrqdo gh wudfflöq1 Hq od Iljxud 8 vh revhuydq odv glphqvlrqhv gh od surehwd | xq ghwdooh gh od }rqd fhqwudo hq od txh vh ghvwdfd orv sxqwrv hq orv fxdohv vh uhdol}durq odv phglflrqhv txh ghwhuplqdq od dshuwxud gh od hqwdood uhvshfwr gho ghvsod}dplhqwr gho erugh vxshulru1. Iljxud 8= Glphqvlrqhv gh od surehwd1 Ghwdooh gh od }rqd fhqwudo Hq od }rqd fhqwudo grqgh vh hqfxhqwudq odv hqwdoodv vh jhqhudq }rqdv gh frqfhq0 wudflöq gh whqvlrqhv/ fuhäqgrvh shuwxuedflrqhv orfdohv hq od pdwul} | jhqhuäqgrvh hvwd0 grv whqvlrqdohv txh vxshudq ho oðplwh hoävwlfr gh od pdwul}1 Hq ho fdvr gh surehwdv frq uhixhu}r xqlgluhfflrqdo od vxud frplhq}d | vh sursdjd vlhpsuh hq od pdwul} |/ d glihu0 hqfld gh rwurv hqvd|rv uhdol}dgrv vreuh surehwdv gh xq pdwhuldo lvöwursr krprjìqhr hvwd vxud wlhqgh d sursdjduvh sdudohod d od gluhfflöq gh odv eudv1 Odv rulhqwdflrqhv gh od eudv gh fduerqr hq odv glvwlqwdv surehwdv hqvd|dgdv vrq 3J / 43J / 78J | <3J uhvshfwr gho hmh orqjlwxglqdo gh od surehwd1 Orv uhvxowdgrv h{shulphqwdohv vh hqfxhqwudq uhvxplgrv hq ho HVL Surmhfw Qe HG2;60 6;62UG2PV | HG2;707482UG2PV Yroxphq := Ilqdo V|qwkhvlv Uhsruw +Vwdyulqlglv/ 4<;8,1. 65.
(34) Pögxor gh \rxqj Frh flhqwh gh Srlvvrq Whqvlöq gh xhqfld Oh| gh frpsruwdplhqwr srvwhulru d od xhqfld Hqhujðd gh iudfwxud Y6. 46333 P sd 3> 658 76> 656 P sd h{srqhqfldo frq deodqgdplhqwr 8Q@p 85> 8(. Wdeod 4= Surslhgdghv uhvlqd hsr{l Pögxor gh \rxqj Frh flhqwh gh Srlvvrq Whqvlöq gh xhqfld Oh| gh frpsruwdplhqwr srvwhulru d od xhqfld Ys. 56<=884 P sd 3> 3 633 P sd olqhdo frq hqgxuhflplhqwr 7:> 8(. Wdeod 5= Surslhgdghv eud gh fduerqr. :16. Hqvd|r gh surehwdv 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 vhjýq ho hmh orqjlwxglqdo gh od surehwd d xq ghvsod}dplhqwr lpsxhvwr hq od sduwh vx0 shulru gh od plvpd1 Odv surslhgdghv gho pdwhuldo hpsohdgdv hq ho dqäolvlv qxpìulfr vh uhvxphq hq odv Wdeodv 4| 51 Vh frqvlghud txh od uhvlqd vh frpsruwd frpr xq pdwhuldo lvöwursr frq xqd oh| gh frpsruwdplhqwr hodvwrsoävwlfr11 Odv surslhgdghv phfäqlfdv gh od uhvlqd hsr{l vh sxhghq revhuydu hq od Wdeod 41 Vh frqvlghud txh odv eudv vh frpsruwdq frpr xq pdwhuldo hodvwrsoävwlfr dqlvöwursr1 Odv surslhgdghv phfäqlfdv gh odv eudv xwlol}dgdv hq od vlpxodflöq qxpìulfd vh uhvxphq od Wdeod 51 Vh kd uhdol}dgr xq dqäolvlv lqfuhphqwdo frqvlghudqgr xq wrwdo gh 63 lqfuhphqwrv gh ghvsod}dplhqwr/ uhvxowdqgr hq xq ghvsod}dplhqwr lpsxhvwr hq od }rqd vxshulru gh 3> 6;8pp= Hq odv Iljxudv : | ; vh suhvhqwd od ghirupdgd gh od surehwd hq ho hvwdgr qdo | xq ghwdooh gh od ghirupdgd hq od }rqd fhqwudo txh suhvhqwd od hqwdoodgxud uhvshfwlydphqwh1 Hvwdv jxudv suhvhqwdq xq idfwru gh dpsol fdflöq gh orv ghvsod}dplhqwrv gh 83 txh shuplwh dsuhfldu dghfxdgdphqwh orv ihqöphqrv päv lpsruwdqwhv txh vh suhvhqwdq hq ho dqäolvlv1 Hq od Iljxud : vh revhuyd fodudphqwh txh ghelgr d od srvlflöq yhuwlfdo gh od eud gh fduerqr odv fdudv h{whulruhv gh od surehwd qr suhvhqwdq ghvyldflrqhv uhvshfwr gho hmh yhuwlfdo gh od surehwd/ vlwxdflöq txh vl vh surgxfh fxdqgr od eud suhvhqwd xq äqjxor uhvshfwr do hmh orqjlwxglqdo gh od surehwd + Yhu hqvd|rv frq eudv d 43J | 78J Iljxudv 4: | 59,1 Vh revhuydq txh vh surgxfhq fxdwur vxudv txh frplhq}dq hq od udð} gh odv 66.
(35) Iljxud 9= Pdood gh hohphqwrv qlwrv hpsohdgd hq ho dqäolvlv hqwdoodv | surjuhvdq vhjýq ho hmh orqjlwxglqdo gh od surehwd frlqflghqwh frq od gluhfflöq gh odv eudv gh fduerqr1 Hvwh ihqöphqr vh sxhgh dsuhfldu frq päv ghwdooh hq od Iljxud ;1 Hq od plvpd vh pxhvwud od }rqd gh ruljhq gh grv sduhv gh vxudv txh qdfhq hq od udð} gh fdgd hqwdood | vh sursdjdq hq od gluhfflöq gho hmh orqjlwxglqdo gh od surehwd1 Hq od Iljxud < vh suhvhqwdq orv frqwruqrv gho pögxor gh orv ghvsod}dplhqwrv1 Vh revhuyd txh hq od }rqd fhqwudo gh od surehwd frpsuhqglgd hqwuh dpedv hqwdoodv ho fdpsr gh ghvsod}dplhqwr suhvhqwd xqd yduldflöq vxdyh/ h{lvwlhqgr fxdwur }rqdv fodudphqwh glihuhqfldgdv hq odv fxdohv ho fdpsr gh ghvsod}dplhqwrv suhvhqwd xqd yduldflöq ixhuwh1 Hvwdv }rqdv frplhq}dq hq od udð} gh fdgd xqd gh odv hqwdoodv | surjuhvdq sdudohodphqwh do hmh orqjlwxglqdo gh od surehwd frlqflghqwh frq od gluhfflöq gh odv eudv1 Hq od Iljxud 43 vh revhuydq odv lvräuhdv gh od yduldeoh lqwhuqd gh sodvwlflgdg sdud ho frpsxhvwr txh lqglfdq odv }rqdv hq odv fxdohv vh kd surgxflgr ghirupdflöq soävwlfd1 Vh pxhvwud txh odv }rqdv päv dihfwdgdv vh hqfxhqwudq hq od udð} gh odv hqwdoodv ghelgr d od frqfhqwudflöq gh whqvlrqhv hq hvd }rqd | txh od sodvwlflgdg surjuhvd hq irupd sdudohod do hmh orqjlwxglqdo gh od surehwd1 Hq od Iljxud 44 vh suhvhqwd xq frqwruqr gh odv }rqdv grqgh vh kd surgxflgr ho ihqöphqr gh gherqglqj r ghvshjxh gh odv eudv uhvshfwr gh od pdwul}1 Xqr gh orv pr0 wlyrv gho frpsruwdplhqwr qr olqhdo gh orv pdwhuldohv frpsxhvwrv uhiru}dgrv frq eudv vh gheh do ihqöphqr gh irupdflöq gh julhwdv hq od pdwul} | ho frqvljxlhqwh ghvol}d0 plhqwr r prylplhqwr uhodwlyr hqwuh eud | pdwul}1 Hvwh ihqöphqr gh djulhwdplhqwr gh od pdwul} | ghvol}dplhqwr hqwuh eud | pdwul} uhgxfh od uljlgh} gho pdwhuldo h lqgxfh 67.
(36) Iljxud := Hqvd|r surehwd W6332<47F frq eudv d 3J 1 Ghirupdgd 83=4. Iljxud ;= Hqvd|r surehwd W6332<47F frq eudv d 3J 1 Ghwdooh ghirupdgd hq od hqwdo0 odgxud 68.
(37) Iljxud <= Hqvd|r surehwd W6332<47F frq eudv d 3J 1 Frqwruqr gh ghvsod}dplhqwrv. Iljxud 43= Hqvd|r surehwd W6332<47F frq eudv d 3J 1 Frqwruqr yduldeoh lqwhuqd gh sodvwlflgdg 69.
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