Three dimensional viscoelastic medium under thermal shock

Transcription

1 Engnrng Sol Mhans Conns lss aalabl a GrongSn Engnrng Sol Mhans hompag:.grongsn.om/sm hr mnsonal solas mm nr hrmal shok Abhk Sr * an M. Kanora Dparmn of Appl Mahmas Unrs of Cala 9 A.P.C. Roa Kolkaa-79 Ws Bngal Ina A R I C L EI N F O A B S R A C Arl hsor: R 6 Marh 6 Ap Jn 6 Aalabl onln Jn 6 Kors: Gnral hrmolas Dal-phas-lag hrmolas mol Kln-Vog-mol Fn a sp Normal mo analss hs arl als h h hrmolas nraon n a hr-mnsonal homognos an sorop solas mm nr h Dal-phas-lag mol of gnral hrmolas. h rslng non-mnsonal opl aons ar appl o a spf problm of a halfspa hos srfa s raon-fr an s sbj o a m-pnn hrmal shok. h analal prssons for h splamn omponns srss mprar an sran ar oban n h phsal oman b mplong normal mo analss. hs prssons ar also alla for a oppr-lk maral an ha bn p graphall. Dsssons ha bn ma o hghlgh h ff of sos on h s fl. h phnomnon of a fn sp of propagaon s obsr for ah fl. Also f h ff of sos s ngl h rsls agr h h sng lrar. 6 Grong Sn L. All rghs rsr.. Inroon Lnar solas has bn an mporan ara of rsarh sn h pro of Mall Bolman Vog an Kln. Valabl nformaon rgarng lnar solas hor ma b oban n h books of Gross 95 Alfr an Grn 956 Frr 97 Blan 96 an Laks 998. Man rsarhrs lk Bo Grn an Srnbrg 96 Ioshn an Pobra 97 annr 988 ha onrb noabl on hrmosolas. Frnhal 954 ha pon o ha mos of h sols hn sbj o nam loang hb sos ffs. h Kln-Vog mol s on of h marosop mhanal mol ofn s o srb h solas bhaor of a maral. h mol rprsns h la las rspons sbj o srss hn h formaon s m pnn b rorabl Sr an kanora 4a; Sr an Kanora 4b. h nam nraon of hrmal an mhanal fls n sols has gra praal applaons n morn aronas asronas nlar raors an hgh-nrg alraors for * Corrsponng ahor. E-mal arsss: abhksr4@gmal.om A. Sr 6 Grong Sn L. All rghs rsr. o:.567/j.sm.6.6.

2 88 ampl. Bo 956 formla h opl hrmolas hor o lmna h parao nhrn n h lassal nopl hor ha las formaon has no ff on h mprar. h fl aons for boh h hors ar of a m parabol-hprbol p hh pr nfn sps for hrmolas sgnals onrar o phsal obsraons. Hnarsk an Ignaak 999 amn f gnralaons o h opl hor of hrmolas. h frs gnralaon s o Lor an Shlman 967 ho formla h gnral hrmolas hor nolng on rlaaon m. hs hor s rfrr o as L-S hor or n hrmolas hor EE n hh h Mall-Caano la rplas h Forr la of ha onon b nrong a sngl paramr ha as as a rlaaon m. h son gnralaon o h opl hor of hrmolas s o Grn an Lnsa 97 all G-L hor or h mprar-ra pnn hor RDE hh nols o rlaaon ms. Problms onrnng hs gnral hors sh as EE an RDE ha bn s b Chanraskharaah 986 Ignaak 989. Mllr 97 propos an nrop proon nal ha l o rsrons on a lass of ons aons. A gnralaon of hs nal as lop b Grn an Las 97. Grn an Lnsa oban a mof rson of h ons aons. hs aons r npnnl oban b Shb 975. For a r orks of Ignaak 989 ma b mnon hr prsnaon of h o hors an som mporan rsls ar ah n hs fl. h hr gnralaon o h opl hor of hrmolas s knon as lo-mprar hrmolas nro b Hnarsk an Ignaak 996 all h H-I hor. hs mol s harar b a ssm of non-lnar fl aons. Lo-mprar non-lnar mols of ha onon ha pr a-lk hrmal sgnals an hh ar sppos o hol a lo mprars ha also bn propos an s n som orks b Kosnsk 989 an Kosnsk an Cmmmll 997. h forh gnralaon o h opl hor s onrn h h hrmolas hor ho nrg sspaon EWOED nro b Grn an Nagh rfrr o as G-N hor of p II n hh h lassal Forr la s rpla b a ha fl ra-mprar gran rlaon. h ha ranspor aon os no nol a mprar-ra rm an as sh hs mol ams namp hrmolas as n hrmolas maral. In h on of lnar rson of hs hor horm on nnss of solons has bn sablsh b Chanraskharaah 996a 996b. h forh gnralaon of h hrmolas hor lop b Grn an Nagh also nols a ha onon la hh nls h onnonal la an on ha nols h hrmal splamn gran among h ons arabls. hs mol s rfrr o as h GN mol III hh nols sspaon of nrg n gnral an ams amp hrmolas as. ahr al. 5 ha mplo Grn-Nagh hors of p II an p III o s h hrmal an mhanal as n an annls oman. Mallk an Kanora 7 ha s on mnsonal hrmolas srban n an sorop fnonall gra mm n h on of gnral hrmolas ho nrg sspaon. Problms onrnng hs hor ha bn s b man ahors sh as Banopaha an Rohohr 5 Rohohr an Banopaha 5 Kar an Kanora 9b Bank al. 7 9 Rohohr an Da 5 Mallk an Kanora 8 9. h ffh gnralaon o h hrmolas hor s knon as h al phas lag mol lop b o 995 an Chanraskharaah 998. o 995 onsr mro-srral ffs no h la rspons n m n h marosop formlaon b akng no aon ha h nras of h la mprar s la o phonon-lron nraons on h marosop ll. A marosop laggng or la rspons bn h mprar gran an h ha fl or sms o b a possbl oom o sh progrss nraons. o 995 nro o-phas lags o boh h ha fl or an h mprar gran an onsr a ons aon o srb h laggng bhaor n h ha onon n sols. Aorng o hs mol

3 A. Sr an M.Kanora / Engnrng Sol Mhans lassal Forr s la k has bn rpla b P K P hr h mprar gran a a pon P of h maral a m orrspons o h ha fl or a h sam pon a m. Hr K s h hrmal on of h maral. h la m s nrpr as ha as b h mrosrral nraons an s all h phas-lag of h mprar gran. h ohr la m s nrpr as h rlaaon m o h fas ransn ffs of hrmal nra an s all h phas-lag of h ha fl. h as hn orrspon o lassal Forr s la. If an o rfrs o h mol as sngl-phas-lag mol. Rohohr 7 has s on mnsonal hrmo-las a propagaon n an las half-spa n h on of al-phas-lag mol. Rnl sral rsarhrs ha amp o sol hr problms on h bass of h hor of al-phas-lag mol. Qnanlla has sol sral problms on h bass of hs mol. h ponnal sabl an onon of h la paramrs n h al-phas-lag hor nr hs mol ha bn s b Qnanlla. Kmar Prasa an Mkhopaha ha s h propagaon of fn hrmal a n h on of al-phas-lag mol. h problm o fn hrmal a propagaon n a half-spa nr arabl hrmal loang ha bn s b Sr an Kanora 4. Rnl Karmakar. al 6 ha s h hrmolas a propagaon n an nfn bo hang a sphral a nr hs hor. Also a rmarkabl hr-mnsonal problm mplng hs hor ha bn sol b Kalkal an Dsal 4. h obj of h prsn onrbon s o onsr a hr-mnsonal Kln-Vog hrmosolas mm n hh h bonng plan s sbj o prsrb mprar an bonar s fr of raons. h ha onon aon has bn formla nrong al-phas-lag ha mol of ha onon. Inrong h normal mo analss h gornng aons ha bn prss n arsan oorans an ar appl o a hrmal shok problm n a solas bo hh flls h half-spa. h nmral smas of h hrmophsal ans ha bn omp for a oppr-lk maral an ha bn p graphall o s h ff of sos on h phsal ans. h rsls orrsponng o h ass hn h ff of sos s nsn agr h h rsls of h sng lrar.. Bas Eaons h srss-sran-mprar rlaons ar * * j j [ * ] j j hr j s h srss nsor s h rfrn mprar s h mprar fl h bal laon an j s h sran nsor gn b j j j. Srss aon of moon n absn of bo for s j j j h ha aon for h nam opl gnral hrmosolas bas on h al-phaslag hrmolas mol s gn b K å 4

4 9 hr ar h splamn omponn an s h ns s h spf ha a onsan sran an h paramrs an ar fn as * * * 5 hr s h solas rlaaon m bng Lam s onsan ; bng h offn of lnar hrmal panson. If an nglng E. 4 rs o h ha onon aon orrsponng o Lor-Shlman mol; bng h rlaaon m for Lor-Shlman LS mol.. Formlaon of h Problm W no onsr an sorop homognos an hrmolas mm h mprar pnn mhanal proprs n hr-mnsonal spa hh flls h rgon hr s fn b } < < < < : { sbj o a hrmal shok on h bonng plan o h srfa. h bo s nall a rs an h srfa s assm o b fr of raon. Inrong h ranglar arsan ssm hang an orgn on h srfa h h as ral no h mm an h omponns of h splamn or as. h ons rlaons ar h aons of moon n absn of bo fors ar

5 A. Sr an M.Kanora / Engnrng Sol Mhans h ha onon aon for al-phas-lag mol s gn b. K 5 Es. 7-9 an b ras n h follong form Inrong h follong nonmnsonal arabls ' j j hr an K an afr rmong prms h abo aons an b rn n nonmnsonal form as follos 9 hr. an In a smlar mannr an ransform h ons rlaons n nonmnsonal forms. h mnsonlss prssons for h ons ar oban b ang Es. 9- as follos

6 9. W shall onsr h n aran srss o b h man al of h prnpal srsss as follos. Sbsng h prssons of an no h abo prssons oban 4 hr Normal mo analss In hs mho h solons of h phsal arabls an b ompos n rms of normal mos n h follong form [ ] [ ] p[ m n] 5 j j hr an ar h ampls of h fnons s h anglar frn m an n ar h a nmbrs n an rons rspl. Usng h normal mos o Es. -4 an hn lmnang from h rslng prssons oban h ssm of ornar ffrnal aons * * * D m n 6 D m n a a 7. hr a an a. Elmnaon of from E. 6 an E. l h follong forh-orr ffrnal aon 4 D LD M 8 hr L M a m n a m n. Sn h nn s ha h solons ansh a nfn so as o sasf h rglar onon a nfn no anl h ponnal par ha has pos por. hrfor ha j R 9

7 A. Sr an M.Kanora / Engnrng Sol Mhans R Usng E. 9 an E. arr from E. 4 R hr. an a R R 5. Applaon In orr o rmn h onsans R an R n o onsr h follong bonar onons on h srfa as follos hrmal bonar onon h bonng plan s sbj o m-pnn hrmal shok n h form f hrfor. ] [ n m f Mhanal bonar onon h bonng plan s fr of raon. So ha. on Usng E. ha. on 4 Applng h bonar onons o h abo ssm of aons an solng h rslng ssm ha. f R f R No prformng h normal mo analss or E. 4 an sng E. an E. arr h solon of h splamn as follos l l R 5 hr. R l n m

8 94 Also an R s a onsan o b rmn. From E. an E. 4 on sng E. 5 arr a D. In of h bonar onons h abo aon ransforms o 6 D f on. 7 Wh h hlp of E. 5 ha l l R. 8 Hn h fnal solons for h mnsonlss srss mprar sran an splamn an b from Es. 9- an E. 5 as follos os m n[ R R ] 9 os m n 4 [ R R ] a os m n [ 4 a R a R ] a l l 4 os m n R. 6. Nmral rsls an sssons Wh an am o llsra h rsls oban n h prng son no prsn h analal rsls nmrall. In h nmral ompaon ha onsr a oppr-lk maral. Sn s h ompl m onsan ha hn os sn. h als of h maral onsans ar akn o b Kalkal an Dsal kgm K 86 Wm K. 9 K Jkg K.5 s... s. K.. Frhr h als of h ohr non-mnsonal paramrs arsng n h prsn analss ar akn o b f m. n.. In orr o s h ff of sos on h hrmophsal ans for LS an DP lag mols Fgs. -4 ha bn plo. In hs fgs. h onnos lns rprsn h graphs orrsponng o nonsos NVIS mol an h o lns orrspons o ha for a sos VIS mm.

9 4 A. Sr an M.Kanora / Engnrng Sol Mhans θ DPVIS DPNVIS LSVIS LSNVIS Fg.. araon of rss for.4 Fg. ps h araon of h mprar agans h san hn. an h m.4. As sn from h fgr aans s mamm magn on h plan.. nar h plan of applaon of h hrmal loang hh ompls h h ral saon an as for largr al of. D o h prsn of sos rm h magn of s lssr han ha orrsponng o a nonsos mm n < <.5. Frhr h ra of a of s fasr n LS mol han ha of DP lag mol. hs phnomnon s obsr for boh sos an nonsos mm Fg.. Varaon of rss for.4 DPVIS DPNVIS LSVIS LSNVIS Fg. ps h araon of h splamn omponn agans h san hn. an.4 for boh sos an nonsos mm. I s obsr ha h splamn fl sars h a mamm al n all h fgrs on h plan an almos mnshs for a larg al of. h a of h splamn omponn s fasr for nonsos mm han ha of h sos mm. Also for boh sos an nonsos mm h magn of s largr for DP mol han ha of LS mol an fnall mnshs o ro a far from h pon of applaon of h hrmal loang. Fg. ps h araon of hrmal srss agans h san for h sam s of paramrs. I s obsr ha on h plan anshs sasfng h mhanal bonar onon. Frhr h srss fl nrass sharpl n < <.5 o aan s mamm al a.5 an hn mnshs appromal o ro al for a larg al of. h fgr also ral h fa ha o h prsn of h sos rm h magn of as han ha for nonsos mm. Frhr for boh sos an nonsos mm h magn of srss a s largr for DP mol han ha of LS mol.

10 σ Fg.. Varaon of rss for.4 DPVIS DPNVIS LSVIS LSNVIS Fg. 4 has bn plo o s h araon of h longaon agans h san. For boh sos an nonsos mm h srss fl follos a smlar rn onl hang ffrns n magn. I s obsr ha h sran fl has mamm magn nar h n of h applaon of hrmal loang hh s phsall plasbl an hn mnshs o ro as rgs from h pon of h sor applaon. h sran also shos sgnfan sns on h ff of sos. For boh h mols h magn of h longaon mnshs o h prsn of h sos rm DPVIS DPNVIS LSVIS LSNVIS 4 6 Fg. 4. Varaon of rss for.4 m prar srbon Fg. 5. Profl of rss an for sos mm

11 A. Sr an M.Kanora / Engnrng Sol Mhans Fgs. 5-8 ha bn plo o s h profl of h hrmophsal ans for a sos mm hn. an for ffrn lngh an m rspl. D splam n srbon Fg. 6. Profl of rss an for sos mm Sharng Srss srbon Fg. 7. Profl of rss an for sos mm 5 Sran srbon Fg. 8. Profl of rss an for sos mm As sn from hs fgs. h h nras of h m h magn of h hrmophsal ans h also nras hh s plasbl.

12 98 7. Conlsons In hs prsn analss a mahmaal ramn has bn prsn o plor h ff of sos on a propagaon n a hr-mnsonal gnral Kln-Vog hrmolas mol nr al-phas lags. h problm has bn sol horall an mplf hrogh spf mols LS an DP mols. All h fgrs plo ar slf-planaor n hbng h ffrn plars hh or n h propagaon of as h follong rmarks ma b a.. h prsn of h hrmo-solas rlaaon paramrs ll ras h magn of h hrmophsal ans an h a of h phsal ans boms slor o h prsn of h sos rm. So h paramr of sos ha a sgnfan ff on all h hrmophsal ans.. From h srbon of h mprar s ha h hrmal a s propagang h a fn sp hh alas h hprbol hrmolas hor.. Hr all h rsls for sos paramr an for Dal-phas lag mol agr h h sng lrar. Aknolgmns W ar grafl o Prof. S. C. Bos of h Dparmn of Appl Mahmas Unrs of Cala for hs alabl sggsons an gan n prparaon of h papr. Rfrns Alfr. & Grn E. F In: Erh F.R. E. In Rholog: hor an Applaons ol.. Aam Prss N York. Banopaha N. & Rohohr S. K. 5. hrmolas a propagaon ho nrg sspaon n an las half spa. Blln of Cala Mahmaal So Bank S. Mallk S. H. & Kanora M. 7. hrmolas nraon h nrg sspaon n an nfn sol h srb proall arng ha sors. Inrnaonal Jornal of Pr an Appl Mahmas Bank S. Mallk S. H. & Kanora M. 9. A nf gnral hrmolas formlaon: applaon o an nfn bo h a lnral a an arabl maral proprs. Inrnaonal Jornal of Appl Mhans an Engnrng 4-6. Blan D. R. 96. h hor of Lnar Vsolas. Prgamon Prss Ofor. Bo M. A hor of srss-sran rlaons n an sorop solas an rlaaon phnomna. Jornal of Appl Phss Bo M. A Varaonal prnpal n rrrsbl hrmonams h applaon o solas. Phss R Bo M. A hrmolas an rrrsbl hrmonams. Jornal of Appl Phss Chanraskharaah D. S hrmolas h son son; A r. Appl Mhans R Chanraskharaah D. S. 996a. A nnss horm n h hor of hrmolas ho nrg sspaon. Jornal of hrmal Srsss Chanraskharaah D. S. 996b. A no on h nnss of solon n h lnar hor of hrmolas ho nrg sspaon. Jornal of Elas Chanraskharaah D. S Hprbol hrmolas: A r of rn lrar. Appl Mhans R Frr J. D. 97. Vsolas Proprs of Polmrs. J. Wl an Sons N York.

13 A. Sr an M.Kanora / Engnrng Sol Mhans Frnhal A. M Eff of rhologal bhaor on hrmal srss. Jornal of Appl Phss Grn A. E. & Lnsa K. A. 97. hrmolas. Jornal of Elas -7. Grn A. E. & Las N. 97. On h nrop proon nal. Arh for Raonal Mhans an Analss Grn A. E. & Nagh P. M. 99. A r-amnaon of h bas poslas of hrmomhans. Prongs of Roal So Srs A Grn A. E. Nagh P. M. 99. hrmolas ho nrg sspaon. Jornal of Elas Gross B. 95. Mahmaal srr of h hors of solas. Hrmann: Pars. Grn M. E. & Srnbrg E. 96. On h lnar hor of solas. Arh for Raonal Mhans an Analss Hnarsk R. B. Ignaak J Solon-lk as n a lo-mprar nonlnar hrmolas sol. Inrnaonal Jornal of Engnrng Sn Hnarsk R. B. & Ignaak J Gnral hrmolas. Jornal of hrmal Srsss Ignaak J Gnral hrmolas an s applaons. hrmal Srsss III hap. 4. In: Hnarsk R.B. E. Mhanal an Mahmaal Mhos. Norh Hollan. Ioshn A. A. & Pobra B. E. 97. Mahmaal hor of hrmal Vsolas. Naka Moso. Kalkal K. K. & Dsal S. 4. Effs of phas-lags on hr-mnsonal a propagaon h mprar pnn proprs. Inrnaonal Jornal of hrmophss Kar A. & Kanora M. 9b. Gnral hrmo-so-las problm of a sphral shll h hr-phas-lag ff. Appl Mahmaal Mollng Karmakar R. Sr A. & Kanora M. 6. hrmolas nraon n an nfn bo nr al phas lags. Jornal of Appl Mhans an hnal Phss In Prss. Kosnsk W. & Cmmll V. A Gran gnralaon o nral sa arabls an a hor of spr fl. Jornal of horal an Appl Mhans Kosnsk W Elas Was n h Prsn of a N mprar Sal n Elas Wa Propagaon. Elsr N York Laks R. S Vsolas Sols. CRC Prss N York. Lor H. W. & Shlman Y. H A gnral namal hor of hrmolas. Jornal of Mhans an Phss of Sols Mallk S. H. & Kanora M. 7. Gnral hrmolas fnonall gra sol h a proall arng ha sor. Inrnaonal Jornal of Sols an Srrs Mallk S. H. & Kanora M. 8. A o mnsonal Problm for a ransrsl sorop gnral hrmolas hk pla h spaall arng ha sor. Eropan Jornal of Mhans A/Sols Mallk S. H. Kanora M. 9. A nf gnral hrmolas formlaon: applaon o pnn shap rak analss. Jornal of hrmal Srsss 9 -. Mllr I. M. 97. h Colnss a nrsal fnon n hrmolas bos. Arh for Raonal Mhans an Analss Prasa R. Kmar R. & Mkhopaha S.. Propagaon of harmon plan as nr hrmolas h al-phas-lags. Inrnaonal Jornal of Engnrng Sn Qnanlla R. & Horgan C. O. 5. Spaal bhaor of solons of h al-phas-lag ha aon. Mahmaal Mhos n Appl Sns Qnanlla R Rak R. 6 A no on sabl n al-phas-lag ha onon. Inrnaonal Jornal of Ha an Mass ransfr Qnanlla R. 9. A ll-pos problm for h hr-al-phas-lag ha onon. Jornal of hrmal Srsss 7-78.

14 Qnanlla R.. Eponnal sabl n h al-phas-lag ha onon hor. Jornal of Non-Elbrm hrmonams Qnanlla R.. A onon on h la paramrs n h on-mnsonal al-phas-lag hrmolas hor. Jornal of hrmal Srsss Rohohr S. K. & Banopaha N. 5. hrmolas a propagaon n roang las mm ho nrg sspaon. Inrnaonal Jornal of Mahmas an Mahmaal Sns Rohohr S. K. & Da P. S. 5. hrmolas nraon ho nrg sspaon n an nfn sol h srb proall arng ha sors. Inrnaonal Jornal of Sols an Srrs Rohohr S. K. 7. On-mnsonal hrmolas as n las half-spa h alphas-lag ff. Jornal of Mhans of Marals an Srrs Shb E. S hrmolas sols Chapr. In: Erngn A.C. E. Connm Phss Par ol.. Aam Prss N York. Sr A. & Kanora M. 4a. hrmolas nraon n a solas fnonall gra halfspa nr hr-phas-lag mol. Eropan Jornal of Compaonal Mhans Sr A. & Kanora M. 4b. Fraonal ha onon h fn a sp n a hrmo-solas sphral shll. Lan Amran Jornal of Sols an Srrs 7-6. Sr A. & Kanora M. 4. Fn hrmal a propagaon n a half-spa o arabl hrmal loang. Applaons an Appl Mahmas ahr H. Farbor S. J. & Eslam M. R. 5. hrmolas analss of an annls sng h Grn-Nagh mol. Jornal of hrmal Srsss annr R. I Engnrng Rholog. Ofor Unrs Prss. o D. Y A nf fl approah for ha onon from maro o mro sals. ASME Jornal of Ha ransfr