LAGRANGIAN EQUATIONS FOR THE MULTIBODIES HEREDITARY SYSTEMS 1 UDK: 534. O.A. Goroshko, N.P. Puchko

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1 UIVESIY O IŠ he sef ourl ACA UIVESIAIS Seres: Mehs Auo Corol obos Vol o pp 9 - Eor of seres: Sefovć Herh Aress: Uverzes rg 8 š YU el: : hp:// u/ LAGAGIA EUAIOS O HE MULIBODIES HEEDIAY SYSEMS UD: 54 OA Gorosho uho ol rs Shevheo Uvers Volors 64 v Ure Absr hs pper s oere wh he ulboes sses whh he ero bewee he boes re esrbe b he sr or wel sgulr herer oels Srg fro he geerl euo he Lgrg euos suppleee b he geerlze relo re osrue A lss of ll eere sses s sepre for whh he hree fors of oo euos re presee: fferel Lgrg euos of he r or hgher orer wo egro-fferel fors of euos wh rheologl relol erels he pl eples re gve IODUCIO he ehs of ou wh he herer properes obe we vee ue o usg of ew sruurl erls hvg well-efe rheologl properes he esg pre A he prese e he bss of hs ehs re wel usg he egeerg lulos Vre osruve ehos usg e Gler-pe ehos for reuo of he sses wh fe uber of freeo egrees hve bee evelope for vesgos he s of ulboes sree herer sses for gog fro ople eu o he sree oels I he prese wor he ehos of he ll s re evelopg for he sree ehl sses wh he herer eros bewee her prles Lgrg euos fors re eere for hese sses ELIMIAY IDIGS O HE MECHAICS O HEEDIAY MAEIALS I he urre lerure he ers vsoels herer els re euvle he herer heor of els s he er oe b V Volerr Yu he reserh esrbe hs publo ws suppore b he Ierol Soros See Euo rogr of Ierol essse ouo gr SU47 eeve Deeber 6 996

2 OA GOOSHO UCHO boov [] observe h hs er efes els fer-os ver el he oel of sr rheologl bo elv bo wel sgulr oel of herer bo obe he wes pplo for esrbg of he herer pheoe MODEL O SADAD HEEDIAY BODY Ul sress se of rheologl bo s esrbe b fferel relo σ σ E ε Eε where E E re seous els oulus prologe oe s e of relo I he ehs of herer sree sses fore of ero bewee he prles of hese sses b log wh E s esrbe b euo where re seous rg prologe oe of elee relze he ero; s eforo of hs elee Solvg he euo for he fore he elogo wo egrl euos re obe Here 4 ep s erel of relo ep s erel of rheolog he E uer he fg elogo esrbes eresg relo of eso wh e he herer elee; he E 4 esrbes he elogo of he rheologl elee If l oe of oo of he sse he rheologl he relg proesses use b he prelr hsor of he log oo ple he he epressos E 4 re egre over fro o hus he sr oel of herer eros be esrbe hree ws el he fferel euo of se he egrl euo of sress relo or he egrl euo of rheolog 4 I hs se he erels of he ls euos re epresse b he epoel fuos

3 Lgrg Euos for he Mulboes Herer Sses 4 WEALY-SIGULA MODEL O HEEDIAY BODY A furher evelope of he sr rheologl bo els he wel-sgulr oel whh he wel-sgulr rol-epoel fuos β β Ae V Ae U 5 α α re hose s he erels of relo rheolog Here U- V- re soe polols or he seres posve power -; β β re he oeffes of relo rheolog B pplo of wel-sgulr egrl euos he eforos re ore pproprel eere he begg pero of he log he ore ople wel-sgulr erels re use lso Yu boov [] propose spel fuo Ξ β α α β 6 Γ α where Γz s Euler fuo he fuo 6 geeres we lss of epoel fuos esrbe he properes of he herer boes However he wel-sgulr oel llows esrbg of he herer proesses ol he for of egrl E 4 o e soe eples of he wel-sgulr erels A zhs [] propose he hree-preers erel of rheolog: β e α < 7 α he four-preers erel ws ope b M A oluov []: α ep p α < β < 8 β Alogous erel for represeo of he properes of hghl els rubber ws propose b GY Slos [4]: e α ep 9 α 5 HE OMS O HE MOIO EUAIOS O A HEOLOGICAL OSCILLAO Le us oser he shee of he sple rheologl osllor whh osse of ss rese o he rheologl elee s subee o ve fore A shee of hs osllor s show g Is oo euo wh respe o 'Aleber's prple hs he e for where s reo of he rheologl elee eere b E or 4 Ele reo wh he use of E we ob he egro-fferel euo of

4 OA GOOSHO UCHO osllor oo wh he relo erel he followg for g sple osllor wh herer elee b shee of he fores g o he osllor ss Elug reo fro E wh he help of E 4 we ob he egrofferel euo of oo wh he erel of rheolog he boh fors of E were use b A s [7] he reserh of vbro of he bes he sple osllors I he se whe he properes of herer elee re esrbg b he oel of sr rheologl bo he erels of E 4 re he epoel fuos B ffereg E g he erve epressos o he l oes boh euos hve bee brough o he pure fferel euos whou egrl ebers: hs euo be obe b elug reo fro E wh he help of E I usg he wel-sgulr oel of herer bo orer o esrbe he oo of osllor E oes o res he pure fferel for I hs se he oo s esrbe ol wo egro-fferel fors hus he esrpo of oo of he sr herer osllor be obe b hree ws: he for of fferel euo wo fors of egro-fferel euos he esrpo of oo of he wel-sgulr osllor be proue wo egro-fferel fors ol

5 Lgrg Euos for he Mulboes Herer Sses 6 LAGAGIA EUAIOS O HE DISCEE HEEDIAY SYSEMS HE GEEAL CASE Esure h he geerl se for he esrpo of he oo of herer sses be erve Lgrg euos he egro-fferel for wh he erels of relo he geerl euo of he s of herer sse oe erl pos s of he followg for e X 4 where e e re reo oses of he reos re he eforos of rheologl elees he reos of herer elees re efe b he relo euos 5 A u of rheologl elees he sse s rbrr Le us he oo of he prles of sse re boue b s- holoo osrs he he oo of sse re efe b geerlze oores Cres oores re represee he fuos of he geerlze oores ssg o he geerlze oores elg he reos fro E 4 wh he help of E 5 we ob e X 6 Usg Lgrg ees ore wh proeure ssue he ll ehs [6] E 6 re rsfore o he followg for: Π 7 where re he e eerg he geerlze fores Π e 8 s poel eerg of herer elees erve wh respe o seous rges he epressos

6 4 OA GOOSHO UCHO b 9 wll be e he geerlze relo of he reos of rheologl elees Here b re eere s follows e b O ou of he epeee of he vros of geerlze oores fro he geerl euo of s 7 we ob Lgrg euos of he seo pe for herer sree sse he egro-fferel for Π If he fores X re poel he s bul up he geerl epresso of he poel eerg lug poel eerg of herer elees poel eerg of he fores X B log wh E 8 s ppropre o roue fuo of he reos relo of he rheologl elees Π b he forul Π e I hs se he geerlze relo of reos 9 re lule wh he forule Π he geerlze fores relo be efe wh he help of he epresso of vrul wor of he ve fores relo lso see below Epple Iegro-fferel euos relo re uversl I s es o e sure h geerl se he osruo of Lgrg euos for he herer sses he rheologl for b log wh E preses o possbles Elug he reos fro he geerl euo of s 4 wh he help of se euos he rheologl for we obe he e euo e X 4 he furher elug of he geerl se s possble I he se er we esure h he geerl se s possble o osru Lgrg euos for he sr rheologl sses he pure fferel for usg rheologl relos s

7 Lgrg Euos for he Mulboes Herer Sses 5 7 DYAMICALLY DEEMIAE HEEDIAY DISCEE SYSEMS Ou of ll he se of herer sses le us se off he sses llowg he osruo of Lgrg euos he pure fferel for wo egrl fors hs sses wll be e ll eere hs lss of sses s por solvg he pple probles Wrg ow he geerl euo of s he for b 5 where he oeffes b re eere b forul B epeee of he vros ou of E 5 we ob euos of Lgrge oe he reos of rheologl eros b 6 Defo: Herer sree sse s ere ll eere whe he u of he rheologl elees volve hs sse s eul or less he he uber of s egrees of freeo for whh s possble o sepre sse of euos of orer fro Lgrg E 6 b 7 h he eer forg b he elees b s o eul zero: e b 8 I vew of E8 he subsse 7 s solvble for he reos for whh he e relos re obe 9 he reos of eros he sr herer sse re eere b he followg euos Elug he reos fro E 7 we ob fferel euos of he hr orer

8 6 OA GOOSHO UCHO he oher - euos of he sse 6 re prese wh he help of E 9 he for b As sr boh wel sgulr herer ll eere sses llows he osruo of wo egro-fferel fors of he oo euos el: Elug he reos fro E 6 we ob egro-fferel euos oe he erels of relo he reg euos of hs sse hve he for Elug he reos fro E wh he help of 9 we ob he egrofferel euos wh he erels of relo he for 4 he oher - euos of sse re wre he for So h b log wh he oo of he ll eere sses be esrbe b hree fors of he ofe Lgrg euos 8 HE SIMLE DYAMICALLY DEEMIAE SYSEMS Whe he sruure of hese sses llows o sele he eforos of herer elees s he geerlze oores 5 he fors of Lgrg euos re splfe I hs se he rheologl E 4 for hese sses re represee s 6

9 Lgrg Euos for he Mulboes Herer Sses 7 he geerl euo of s he geerlze oores re wrg he for 7 espeve Lgrg euos oe he reos hve he for 8 Elug he reos of herer elees fro 8 he frs euo 6 we ob he sse of fferel Lgrg euos he for 9 hs s sse of he r orer If uber of he herer elees s less he he uber of he egrees of freeo < us he soe euos of he sse 9 wll hve lssl for 4 Elug he reos fro E 8 b he relo euos 6 we ge egro-fferel Lgrg euos wh he erels of relo 4 A fll elg he reos of osrs fro E 8 rheologl E 6 we ge he sse of egro-fferel Lgrg euos rheologl for 4 hus he ll eere herer sses llows he osruo of Lgrg euos he pure fferel for for sr sses wo for of egro-fferel euos s for wel-sgulr boh for sr herer sses

10 8 OA GOOSHO UCHO 9 HE EXAMLES O HE COSUCIO O MOIO EUAIOS O HEEDIAY SYSEMS g Vbroproeol sse wh per Eple Vbroproeol sse wh per he vbroproeo of obe s relze b per wh rg he rheologl eergbsorber See g he per s represee b ss wh he rheologl elee hs s he ll eere rheologl sse he bsolue splee of ss he relve splee of ss be hose s he geerlze oores he eforos of rheologl elees re eere b hese splees he herer elees of hs sse re sr her se s esrbe b he euos 4 he e poel eerges re of he followg for Π C g g 44 he geerlze ve fores re 45 he euos of oo org o he forule 9 hve he e for: Π Π hus he oo of he rheologl vbroproeol sse wh per s esrbe b wo euos of he hr orer If he wel-sgulr oel s use for esrpo of he oo of sse show g he he reos of he rheologl elees re eere b euos 46

11 Lgrg Euos for he Mulboes Herer Sses 9 47 where - - re wel-sgulr erels of relo respeve wh E 4 euos of oo re represee s g - g - C 48 If he se of herer elees re esrbe b he rheologl euos wh wel-sgulr or sr erels he for 49 he epeg o he forule 4 he egro-fferel euos rheologl for re wre s follows [ ] g g g g C ] [ 5 A hoosg of oe hree presee bove fors of E for vesgo of he properes of vbroproeo s orolle b he pe of l foro bou he properes of herer elees Eple he osruo of he euos of oo of rheologl peulu g Sr rheologl oel he herer properes of peulu hre re efe b euo 5

12 OA GOOSHO UCHO g he rheologl peulu Whe gle of splee elogo re hose s he geerlze oores he hs sse s sple ll eere b efo e poel eerges of peulu re wre s g g υ υ s os 5 I ore wh 9 he euos of oo of rheologl peulu hve he for g g s ] [ ] os [ 5 Wel-sgulr oel he properes of herer peulu hre b euos 54 Correspog euos he ers of relo 4 hve he e for s ] [ os ] [ g g 55 he rheologl for s obe f he frs euo 55 s reple b he followg oe os } os ] [ { ] [ g g 56

13 Lgrg Euos for he Mulboes Herer Sses Eple Dll eere herer sse hs sse s pure g 4 he u of s herer elees s ore he he uber of he egrees of freeo he properes of hese elees re efe b he euos he ers of relo 57 e poel eerges wh respe o he forule 8- re esrbe b he epressos Π 58 g4 Dll eere herer sse or efo of he geerlze fores we voe he epresso of vrul wor of ve fores relo A 59 ro hs epresso follows h

14 OA GOOSHO UCHO 6 he he euos of oo wll loo le 6 I s p h he le sze of hs rle oes o llow o llusre opleel he ehue of pplo of ll vre of oo euos fors of sree herer sses EEECES boov Yu 977 Elees of Herer Mehs of Sols Mosow u uss zhs A 949 Soe uesos of he Mehs of Deforg e Sses Mosow GI uss oluov MA 968 o he ueso of he hoosg of erels soluo of probles oug reep relo Mehs of olers o 4 uss 4 Slos GL 96 O he lw of eforg of hgh els poler boes ep A S USS Vol 4 o uss 5 Sv G ushs Y Y 976 Elees of Herer Me Mehs v Vsh shol Ur 6 rs LA 964 A rese o All Ds Hee Loo LAGAGE-OVE JEDAČIE ZA ASLEDE SISEME SA VIŠE ELA OA Gorosho uho Ov r se bv sse vše l o oh e uzo esvo zeđu el opso obč sr l slb sgulr sle oel očevš o opše če eče ssvlee su Lgrge-ove eče opuee geerlzovo uopšeo relso Izvoe e ls č oređeh sse z oe su prze eče r obl re: ferele Lgrge-ove eče rećeg l všeg re ve egro-ferele fore eč s reološ relso ezgr D su pč ogovruć prer