On Thermal Stresses in an Infinite Body Heated by Continuous Point Source inside it

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Ariel Melton
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1 JOUNAL OF HMOLASICIY VOL. NO. Dcmb ISSN 8- (n) ISSN 8-X (Onln) h:// On hml Ssss n n Infn Body Hd by Connuous Pon Souc nsd S.P. Pw K.C. Dshmuh nd G.D. Kd* Absc In hs w consd n nfn body consn nl mu hd by by m dndn connuous on h souc nsd. h mu dsbuon dslcmn nd hml ssss obnd nd suls comd usng h dffn foms of soucs. h Llc nsfom s usd o obn h mu dsbuon funcon. h numcl suls llusd fo mssv lumnum ml. Kywods hml ssss nfn body on h souc Llc nsfom. I. INODUCION H sudy of oblms of dmnon of mu nd sss dsbuon n h nfn mdum und dffn hml nd mchncl condons bcoms subjc of xnsv sch n h fld of hmolscy. Such n och wh shcl objcs s of g monc n ngnng cc. h oblms of h clculon of mu fld nd s hml ssss whn mssv bods hd by connuous souc nsd h body ncound n vous hyscl oblms l souc of h hddn nsd h lg o xc; h s gnd nsd h mssv ml n h fbcon ocss. H my b oducd n solds du o vous sons-by nducon hng chmcl con docv dcy nd ssg of n lcc cun c. h conducon of h s vy mon hnomnon n h ngnng scncs. Snc Fou s wo L ho Anlyqu d l Chlu mny mhmcs mhods hv bn dvlod o focus on dffn ys of h nsf nd hml sss oblms. hough h fld of hmolscy s dvlod n mny wys sll nds o wo on som chllngng fundmnl oblms. Som xmls of hs S.P. Pw S.N. Mo AsComm nd Sm. G.D. Sf Scnc Collg ums Ind. K.C. Dshmuh D of Mhmcs M Ngu Unvsy Ngu Ind. * G.D. Kd D of Mhmcs M Ngu Unvsy Ngu Ind. (ml: gdd@dffml.co.n). ys blongng o goms wh nfn nd sm-nfn n shcl coodns o b sudd wh ll lu s vlbl. l lu on hs found o b on h homognous nfn ml wh consn nd unfom hmo hyscl os n h monoghs of Cslw [] M.N. Ozs [] Nod [] Yn nd Kc []. hy hv dscussd vy of oblms nvolvng sdy nd unsdy h conducon. Nod [] hs dscussd h sdy nd nsn oblms of hml ssss n shcl coodns n uncould hmolscy. In h vw of cn lu K.C. Dshmuh l [5] dmnd h unnown mu nd hml ssss on h cuvd sufc of sm-nfn ccul cylnd by ngl nsfom mhod G.D. Kd l [6] sudd h hml ssss n sm nfn sold ccul cylnd subjcd o by nl h suly nd suls obnd n ms of Bssl s funcons Yng Zhou [7] sudd h h conducon n sm-nfn body wh Pow y nl nd boundy condon. N. S. Soln [8] sudd h h conducon n sm-nfn ccul cylnd wh flow of h suls fom h nl dsbuon of mu hough h cylnd G.D. Kd l [] sudd h dmnon of hml dflcon n sm nfn hollow ccul cylnd subjcd o m y hng on h low fc. In h sn on dmnsonl qus-sc oblm of mu dsbuon dslcmn nd hml ssss n nfn body hd wh by m dndn h souc lcd s consdd. h m of h wo s o obn h mhmcl modl fo dcng h mu nd sss fld wh h ys of on soucs wh nl consn mu. hs s usful n hvng h nowldg of hml ssss n mssv bods n h fld of ngnng. II. FOMULAION OF POBLM MPAU DISIBUION Consd n nfn body consn nl mu. I s hd by on h souc n h body whch lss h connuously W fo m. W ssum consn hmohyscl os of ml. h mhmcl fomulon fo h on dmnsonl unsdy s dsbuon of mu

2 JOUNAL OF HMOLASICIY VOL. NO. Dcmb ISSN 8- (n) ISSN 8-X (Onln) h:// sd n shcl coodns s [] Subjcd o h followng boundy condon lm () s () Inl condon Wh () () (5) hml dffusvy c hml conducvy c Scfc h Dnsy HMOLASICIY UAIONS Fo on dmnsonl oblm n h shcl coodn sysm whch mns shclly symmc oblm h dslcmn chnqu s xnsvly usd. h os n shcl coodnnss nd dcon dncl nd u dnos h dslcmn n h dl dcon h sn dslcmn lons [] du u (6) d h cosondng hmolsc sss sn lon o Hoo s lons (7) (8) Wh nd h ssss n h dl nd ngnl dcon nd nd sns n dl nd ngnl dcon. s h mu chng obnd fom h h conducon quon() s h coffcn of hml xnson s h sn dlon nd nd h Lm consns ld o h modulus of lscy nd h Posson s o s (9) λ = h qulbum quon n dl dcon xcludng h body focs nd n m s d () d h dslcmn nd hml ssss nducd du o on dmnsonl h conducon fld n n nfn body whou cvy gvn by followng ngl quon [ g5] u d F () d F () F Wh F d Wh d () nd dl nd ngnl ssss (MP) s h coffcn of hml xnson s h modulus of lscy of sh s h Posson s o. h boundy condon fo con f sufc n hs cs bcoms [] s () h quons (-) consu h mhmcl fomulon of h oblm. III. ANALYICAL SOLUIONS W ulz h mhod of Llc nsfom chnqu o solv h h conducon oblm (-) ccodngly h Llc nsfom of wll ssfy followng boundy vlu oblm d d d d d d d d d lm d quon (5) wn s d d (5) (6) h soluon o hs dffnl quon s obnd s A B A (7) (8) s fn

3 JOUNAL OF HMOLASICIY VOL. NO. Dcmb ISSN 8- (n) ISSN 8-X (Onln) h:// B (9) Wh B s nonzo consn usng (6) on gs B () hus () L () Usng h convoluon on h bc of () d d 8 () W consd followng h css of connuous on souc. Consn on h souc. m dndn souc. m dndn souc n h fom fc Fo bov gvn css usng ()nd () n () h mu dsbuon dslcmn nd ssss obnd s CAS fc () f fc u (5) f fc (6) fc f fc (7) CAS fc (8) fc u (9) fc ()

4 JOUNAL OF HMOLASICIY VOL. NO. Dcmb ISSN 8- (n) ISSN 8-X (Onln) h:// fc fc () CAS fc () fc f fc u () fc f fc () fc fc f fc (5) III. SULS AND ANALYSIS h xc nlycl soluons obnd fo mu dsbuon dslcmn nd hml ssss n h vous. h mhmcl sofw MALAB s usd fo fuh numcl clculons. In numcl w sn snv suls fo lumnum (u) wh hmo mchncl os Dnsy / 69 f lb Scfc h F f h Bu c.. /.8 hml dffusvy h. f / Posson s o 5. Coffcn of ln xnson F /.8 6 S h vlu of (Fo cs only m n on souc funcon) Bu Inl mu F In fg. nd h mu dsbuon s shown long dus fo h dffn css of on souc funcons. In cs whn h mssv body s hd wh consn connuous on souc fo. h mu dly ncss long h dus nd hn n h sm wy coms down shly nd chs h sdy s nd ns nl vlu.fo nd lg ms h cuvs obnd show h h mu s mxmum vlus n h souc nd gdully dcss.fo cs nd whn h souc s m dndn mu chs o g vlus s comd o cs. In cs of souc fo smll ms mu ncs

5 m m m JOUNAL OF HMOLASICIY VOL. NO. Dcmb ISSN 8- (n) ISSN 8-X (Onln) h:// s nglgbl h body s bou nl mu.fo cs h mu s s lss bu smoohly dcss long h dus nd h ffc of h souc dcss whn w go wy fom h souc. Fg 5 nd 6 show h von n h dl dslcmn. I s obsvd fom h los h h dslcmn s vy smll fo cs nd lg fo cs.in l cs dslcmn covs lg sc whl fo l h dslcmn s hgh n h vsny of souc.h nu of dslcmn lso vs wh souc funcon.fo cs nly ncss nd gduly dcss fo dffn ms. Fo cs ncss long h dus nd fo cs h sm dcs long h dus s obsvd. h dffn dslcmn ofls xs fo hs css. In h sudy s nsng o obsvd h h s hnomnl dffnc fo ll h h dscussd css whch could succssfully ld n h fld of ngnng. Fg. 7 8 nd 9 sn von of h dl sss long h dl dcon. Fo cs nd h sss funcon dvlos nsl sss whl fo cs h ssss comssv n h nu.h dl sss funcon nds o zo s h nducd mchncl condon. Fg. nd shows h von of ngnl sss long h dus. Fo cs nd ssss nsl n nu whl fo cs s comssv.h ffc dcss wh ncs n vlu of. IV. CONCLUSSIONS In hs sudy h xc nlycl soluons obnd fo mu dsbuon dslcmn nd hml ssss fo sooc nd homognous nfn body hd wh connuous h souc nsd. h oblm s nlyzd ndndnly wh h css of consn nd m dndn souc funcon. h mu von long h dus fo cs s obsvd sgnfcn fo m. whl sm s nglgbl fo cs.h mu ofl s vsd fo h souc funcon of y h mu gdn s lg fo smll m nd dmnshs s m sss fuh. hs bcus of nu of m dndn souc funcon usd. h concluson dwn fom h sul s h h mu gdn vs wh consn nd m dndn connuous on souc wh h body wh m s wll s nu of souc funcon usd. h dslcmn lso shows von ccodng o mu gdn nd found o b chngd olly wh souc funcon. h dl sss funcon dvlos nsl ssss long h dus fo vy smll m fo cs nd bu oducs comssv ssss fo h souc usd n cs.a von s obsvd fo comvly g ms. h ngnl ssss nsl fo css nd nd comssv fo cs. h ffc of dslcmn dl nd ngnl ssss bcom null s w go wy fom h souc whch s n gmn h h body s nfn. h suls obnd fo mssv lumnum ml nd hs s usful nd hs mhmcl modl cn b modfd by chngng h fom of m dndn souc funcon n h fld of ngnng fo dffn mssv ml Fgu : mu von long dus (cs ) Fgu : mu von long dus (cs ) =. = = = Fgu : mu von long dus (cs ) = = =.5 = =5

6 Dslcmn Sgm Dslcmn Sgm Dslcmn Sgm JOUNAL OF HMOLASICIY VOL. NO. Dcmb ISSN 8- (n) ISSN 8-X (Onln) h:// 6 x -5 =. =. =. =. =.5 - =. - =.6-6 = Fgu : Dslcmn long dus (Cs ) Fgu 7: dl sss long dus (Cs ) =. =. = =. =.6 = Fgu 5: Dslcmn long dus (Cs ) Fgu 8: dl sss long dus (Cs ).5 x 5.5 =. =. =. x - - = uh =6 uh =9 uh Fgu 6: Dslcmn long dus (Cs ) Fgu 9: dl sss long dus (Cs )

7 Sgmh Sgmh Sgmh JOUNAL OF HMOLASICIY VOL. NO. Dcmb ISSN 8- (n) ISSN 8-X (Onln) h:// =. =.6 = Fgu : ngnl sss long dus ( Cs ) =. =.6 = Fgu : ngnl sss long dus ( Cs ).5 x = uh =6 uh =9 uh [] Nod N Hns Bnd ngw Y hml Ssss ylo nd Fncs Nw Yo nd d:. [] Ymn Yn nd Sd Kc H Conducon ylo & Fncs h d:8. [5] Dshmuh KC Wbh SD Kd GD. nd Kuln VS. Invs h Conducon oblm n sm-nfn ccul l nd s hml dflcon by us-sc och. An nnonl jounl (AAM) ; 5():-7. [6] Kd GDWbh SDDshmuh KC nd Kuln VS. hml ssss n Sm nfn sold ccul cylnd. InJ. Of Mh nd Mch. ;8(): 8-6. [7] Yng Zhou H Conducon n Sm-nfn body wh Pow-y Inl nd boundy Condons.In.J.hmohysOc;Do7/s [8] Soln Nnd Sngh.on Coducon of H n sm-nfn ccul cylnd.advncs n Comuonl Mhmcs nd s lcons(acma);():5-58. [9] DshmuhKCWbh SDKd GD nd Kuln VS. Invs H Conducon Poblm n sm-nfn cylnd nd Is hml Ssss by us-sc och lcons Ald Mhmcs(AAM) ;6() [] Kd GDWbh SDDshmuh KC nd Kuln VS. hml ssss n Sm nfn sold ccul cylnd. InJ. Of Mh nd Mch. ;8(): 8-6. [] Pns VA Unsdy s mu fld n sm-nfn body Hd by Ds Sufc H souc. Hgh mu6; ():9-8. [] Josh Mush M. Dffusson of H n Infn cylnd Hvng Souc of H Whn. Advncs n Mchncl ngnng nd s Alcons(AMA) ;(): -6. [] Kd GDWbh SDDshmuh KC nd Kuln VS. hml ssss n Sm nfn sold ccul cylnd. InJ. Of Mh nd Mch. ;8(): 8-6. [] Kd GD nd Dshmuh KC.hml Dflcon n sm-nfn hollow ccul cylnd subjcd o m y hng. Asn Jounl of Cun ngnng nd Mhs;: 7-9. [5] Ozs MN H Conducon Wly nd sonsinc nd d:99 [6] dwd W.Ng nd Muy Gll. A bl of Ingls of h o FunconsJ. of sch of h Nonl Buu of Sndds-B.Mhmcl Scncs 969;Vol.7BNo.. [7] Pw SPDshmuh KC nd Kd GD.ffc of h gnon on qus-sc hml ssss n sold sh.iosjounl of Mhmcs(IOS-JM)Aug; 7():6-67. [8] Pw SPDshmuh KC nd Kd GD. ffc of Innl H Gnon nd mu Asymmy n Hollow Sh nd Is hml Ssss. Jounl of hmolscy(jo)s ;(): Fgu : ngnl sss long dus ( Cs ) V. Acnowldgmn h uhos hnful o Unvsy Gns Commsson Nw Dlh o ovd h l fnncl sssnc und mjo sch ojc schm. V. FNCS [] Cslw HS nd Jg JC Conducon of h n solds Clndon Pss nd d :959. [] Ozs MN Boundy Vlu Poblm of H Conducon Innonl x boo Comny Scnon Pnnsylvn: 968.

Appendix. In the absence of default risk, the benefit of the tax shield due to debt financing by the firm is 1 C E C

Appendix. In the absence of default risk, the benefit of the tax shield due to debt financing by the firm is 1 C E C nx. Dvon o h n wh In h sn o ul sk h n o h x shl u o nnng y h m s s h ol ouon s h num o ssus s h oo nom x s h sonl nom x n s h v x on quy whh s wgh vg o vn n l gns x s. In hs s h o sonl nom xs on h x shl