The Influence of Diffusion on Generalized Magneto- Thermo-Viscoelastic Problem of a Homogenous. Isotropic Material

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1 dv. Tho.. Mh. Vo. no Th Infn of Dffson on Gnzd Mgno- Tho-Vsos Pob of Hoognos Isoo M F. S. Byons Mhs Dn Fy of Sn U -Q Unvsy P. O. Box 9 Mh Sd b F.S.Byons@ho.o bs Th sn s d sdyng h ffs of vsosy nd dffson on gnzd gno- hos nons n n soo wh sh vy. hon fnon hnqs sd. Th h ssss nd dsn hv bn obnd. Ths xssons d ny fo o nd dd ghy. ywods: Vsos; Gnzd gno ho; Isoo; Dffson; Sh -Inodon Th dffson n b dfnd s h sonnos gon of sbsns fo gons of hgh onnon o gons of ow onnon. Thodffson n h ods s on of h nso osss h hs g on. Thodffson n n s sod s d o h ong of h fds of ss dffson nd h of sn. Now[-] dvod h hoy of hos dffson. In hs hoy h od hos od s sd whh s nfn sds of ogon of hos wvs. Shf.[] dvod h hoy of gnzd hos dffson h ds fn sds of ogon fo hos nd dffsv wvs. Shf nd Sh [] wod on ob of hos hf-s wh ng sbsn n on wh h bondng n n h onx of h hoy of gnzd hos dffson. Rny X.[6] sdd h dyn sons of n nfn body wh ynd vy whos sf

2 7 F. S. Byons sffs h sho sng fn n hod. Eff of oon n gnzd gno - ho - vsos d dsssd by bd-. [7]. Lod nd Shn [89] Gn nd Lndsy [] xnd wo on gnzd hos of hosy h hv bo h n of h ns of n sh n hs. So obs of hosy o gnzd hosy nvsgd by bd- [] bd-.[] nd Eng nd bd- [] svy.. Royhodh nd Mhodhyy [6] sdd h ff of oon nd xon s on n wvs n gnzd ho-vsosy. Royhodh nd Bn [7] nvsgd h gno-hos nons n n nfn vsos ynd of dndn sbd o od odng. Shy sy ho-vsos wvs n vsos d wh sh vy dsssd by Bn. [8]. - Foon of h ob L s onsd h d s f ondo nd h nzd Mxw qons govnng h ogn fd n h bsn of h dsn n SI n h fo s n Royhodh nd Mhodhyy [6] nd s [9] : J h h μ E dv h dv E U E μ H. wh h s h bd gn fd ov h y gn fd E s h nnsy J s h n dnsy μ s h gn by H s h onsn y gn fd nd U s h dsn vo. yng n n gn fd vo H H φ n sh oodn φ o Eq.. w hv U U H H H E H φ φ μ.

3 Infn of dffson 7 J hφ h H.b φ φ E E.. Th s d s ong nfoy wh n ng voy Ω Ωn wh n s n vo snng h don of h xs of oon. Th dsn qon of oon n h ong f hs wo ddon n on Ω Ω s h n on d o vyng oon ony nd Ω s h oos on nd Ω Ω. Foowng s [] h govnng qons fo n soo hoognos s sod wh gnzd gno-ho- vsos dffson nd ff of oon : Th Mxw's o-gn sss nso s gvn by μ Hh H h Hh. δ.. Th qon of oon: { Ω Ω } Ω ] μ μ T f ρ[. wh f s dfnd s Lonz's fo s [9] whh y b wn s f μ J H μ H φ h qon of h ondon : T ρ T T T. v v h qon of ss dffson: D DT Db.6 v h onsv qons: μ δ T T.7 [ ] P b T T.8 wh ρ s h ss dnsy of h s h ondvy μ μ nd svy h nd offns of n h nd dffson xnson v s sf h onsn sn μ L' s onsns T s h bso of h d T s h fn nfo of h body hosn sh h T T / T << T T s h hn xon d o h vsosy s h h xon w ns h

4 7 F. S. Byons h h ondon qon w d fn sd of h ogon nd s h dffson xon whh w ns h h qon ssfd by h onnon w so d fn sd of ogon of fo on d o h oh nd D s h dffsv offn. Th onsns nd b h ss of hodffson ffs nd dffsv ffs svy h oonns of h sss nso h oonns of h sn nso nd s h h on. Th non-vnshng dsn oonn s h d on s wh Δ..9 Thn h sn nso hs h foowng oonns.. Th b don s hs gvn by:.. D o sh syy Eqs..-.8 h fo ρ Ω Ω. T ρ T T. v D T Db. μ. μ.6 P b.7 D f H μ φ μ μ H..8 φ - Dnsonss qns Now w nod h.-.7: foowng non dnson vbs n Eqs.

5 Infn of dffson 7 η η η η η η η T T Ω Ω. wh ρ nd ρ v / η w hv Ω Ω... μ. μ.6 b P.7 wh. η ρ Db D T T v.8 - Bondy ondons Th hoognos n ondons snd by h foowng bondy ondons : Th vy sf s on f.. Th vy sf s sbd o h sho. H.b Th h on s so ssd o b nown fnon of h vy sf. H. wh nd onsns nd H s h Hvsd n s fnon. Th dsn fnon

6 7 F. S. Byons..d - Soon of h ob ssng s hon dndn fo fo h qns nd ong h fo s foowng:. G U Ε Θ. Eqs..-. bo U G Θ Ε. G Ε Θ Θ. G G Θ Ε.. By yng h oo o Eq.. w obn Θ Ε G. Fo Eqs..-. w obn 6 Θ Ε G.6 wh { } ] [.7 { }.8.9. Ω Ω μ μ φ H. Th bov sys of qons n b fozd s Θ Ε G. wh nd h oos of h hs qon. 6.

7 Infn of dffson 7 Th soon of Eq.. whh s bondd nfny s gvn by / Θ. / Ε. / G. wh nd s dndng ony on nd / s h odfd Bss fnon of h sond nd of od /. oby bwn Eqs..-. ong wh. nd. w gv s o.6..7 Sbsng fo Eqs..-. no. w obn /.8 /.9. /. Ingng boh sds of Eqs..9 n h on o h w obn /.. so fo.-.7 w g / /.

8 76 F. S. Byons / /. / P. wh. μ b By sng h bondy ondons w g ] [ ] [ ] [ ] [ ] [ ] [ / / / / / / M W N W N M W N W N M W N W N. / / H N φ μ.6 / W..7

9 Infn of dffson 77 M N [ W N [ W / / W W / / ] ] N [ W / W / ].8 Th dssson n hos dffson d s fo Fgs. - 6 nd n hos d s fo Fgs. 7-:.E.E.E.E.E.....E.E.E.E-.E.E-.E -.E E -.E.E -.E-.. -.E -.E -.E... Ω _ Ω. Ω..6E.E.E.E 8.E- 6.E-.E-.E-.E -.E E- -6.E E.E.E.E.E -.E... -.E Fg.: Th dsn n ho vsos dffson d

10 78 F. S. Byons.E... -.E- -.E- -6.E- -8.E- Ω Ω.... Ω E -.E.E -.E E- -.E- -.E- -.E- -.E- -.E- -.E- -.E-.E... -.E-. -.E- -.E E- -.E- -.E- -.E- -.E- -.E-.....E -.E E- -6.E- -8.E- -.E -.E -.E -.6E Fg.: Th n ho vsos dffson d

11 Infn of dffson 79.E-7.E-7.E-7 Ω Ω.... Ω..E-7.E-7.E-8.E E-6.E-6.E-6.E-6.E-6.E-7.E... -.E-7.E-6.E-6..E-6.E-6.E-6.E-6.E E -.E-7....E-6.E-6.E-6.E-6.E-6.E-6.E-6.E-6.E-6.E-7.E... Fg.: Th onnon n ho vsos dffson d

12 8 F. S. Byons.E 8.E 6.E.... Ω Ω.... Ω..E.E.E.E... -.E.E.E.E.E -.E... -.E -.E -.E.E.E.E -.E... -.E -6.E -8.E -.E -.E -.E.E.E E... -.E -.E -.E -.E -.E Fg.: Th sss n ho vsos dffson d

13 Infn of dffson 8.E 8.E E- Ω Ω.... Ω..E-.E-.E... -.E- -.E-.E.E.E.E-.E... -.E- -.E -.E.6E.E..E..E... 8.E-. 6.E-.E-.E-.E E.E.E.E.E.E.E.E.E-.E -.E-... Fg.: Th sss n ho vsos dffson d

14 8 F. S. Byons 6.E.E.E.... Ω Ω.... Ω..E.E.E.E....E.E.E.E.E.E.E.E.E-.E....E.E.E.E.E E... 6.E.E.E.E.E.E.E... Fg.6: Th h on n ho vsos dffson d

15 Infn of dffson 8 6- P s: - If w ng h dffson ff by nng Eqs..6 nd.8 nd by ng n Eqs... nd.8 w hv B 6. wh [ ] η η η η η ± 6. B B 6. By sng h bondy ondons w hv h h h h h h h h h B h h h 6. H h H h φ φ μ μ 6.6. h h 6.7 B

16 8 F. S. Byons 6.E.E.E.....E.E.E Ω Ω.... Ω..E.E.E.E.E.E... -.E.E... -.E 9.E 8.E 7.E 6.E..E.E.E.E.E.E.E E 6.E.E.....E.E.E -.E....E... Fg.7: Th dsn n ho vsos d

17 Infn of dffson 8.E.E... -.E -.E -.E -.E.... Ω Ω.... Ω..E.E... -.E -.E -.E -.E -.E.E.E -.E... -.E -6.E -8.E -.E -.E -.E -.6E -.8E E.E... -.E -.E -.E -.E -.E -.E Fg.8: Th n ho vsos d

18 86 F. S. Byons.E.E... -.E -.E -6.E -8.E -.E -.E -.E.... Ω Ω.... Ω..E.E... -.E -.E -6.E -8.E -.E -.E.E.E -.E... -.E -.E -.E -.E -6.E -7.E -8.E -9.E E 9.E- 8.E- 7.E- 6.E-.E-.E-.E-.E-.E-.E... Fg.9: Th sss n ho vsos d

19 Infn of dffson 87.8E.6E.E.E.....E 8.E 6.E.E.E.E -.E... Ω Ω.... Ω..6E.E.E.E 8.E 6.E.E.E.E -.E....E.E.E 8.E 6.E E.E.E... -.E.E... -.E- -.E- -.E- -.E- -.E- -6.E- -7.E- Fg.: Th sss n ho vsos d

20 88 F. S. Byons 7- N ss nd dssson Th o ws sd hosn fo oss of n vons. Th onsns of h ob gvn by [] nd [] μ.86 v 8. J / g 86 W /. g / s / s D b.9 g / s / s g s / g.98 T 9 ρ 89 g / η s / / g Usng hs vs ws fond h: μ. 9. Hφ Th n hnq ond bov ws sd o obn h d dsn d sss nd onnon s w s h h on dsbon nsd h sh. Ths dsbons shown n Fgs. - svy. Fo h s of bvy so oon ss no bng snd h. -In ho - vsos dffson d Fg. shows h von of h d dsn nd ffs h hng of oon Ω hn xon nd h ds. I ws fond h h vs of dsd wh n nsd of oon nd ds wh h vs of h s vs sgh hng dffn vs of hn xon. Fg. shows h dsbon nd ffs h hng of oon Ω hn xon nd h ds. I ws fond h h vs of dsd nd nsd wh n nsd of nd ds svy wh h vs of h s vs sgh hng dffn vs of hn xon nd oon. Fg. shows h onnon dsbon nd ffs h hng of oon Ω hn xon nd h ds. I ws fond h h vs of dsd wh n nsd of ds nd hn xon wh h vs of h s vs sgh hng dffn vs of oon. Fg. shows h d sss sss nd ffs h hng of oon Ω hn xon nd h ds. I ws fond h h vs of dsd nd nsd wh n nsd of nd oon nd ds svy wh h vs of h s vs sgh hng dffn vs of hn xon nd oon. Fg. shows h ngn sss nd ffs h hng of oon Ω hn xon nd h ds. I ws fond h

21 Infn of dffson 89 h vs of dsd wh n nsd of oon nd ds wh h vs of h s vs sgh hng dffn vs of hn xon nd oon. Fg. 6 shows h h on dsbon nd ffs h hng of oon Ω hn xon nd h ds. I ws fond h h vs of dsd wh n nsd of oon nd ds wh h vs of h s vs sgh hng dffn vs of hn xon nd oon. -In ho vsos d Fg. 7 shows h d dsn nd ffs h hng of oon Ω hn xon nd h ds. I ws fond h h vs of dsd wh n nsd of oon nd ds wh h vs of h s vs sgh hng dffn vs of hn xon nd oon. Fg. 8 shows h dsbon nd ffs h hng of oon Ω hn xon nd h ds. I ws fond h h vs of dsd nd nsd wh n nsd of oon nd ds svy wh h vs of h s vs sgh hng dffn vs of hn xon nd oon. Fg. 9 shows h d sss nd ffs h hng of oon Ω hn xon nd h ds. I ws fond h h vs of dsd nd nsd wh n nsd of nd ds nd oon svy wh h vs of h s vs sgh hng dffn vs of hn xon nd oon. Fg. shows h ngn sss nd ffs h hng of oon Ω hn xon nd h ds. I ws fond h h vs of dsd nd nsd wh n nsd of nd oon nd ds svy wh h vs of h s vs sgh hng dffn vs of hn xon nd oon. 8- onsons Fo h vos dssson n ho vsos dffson d w hv h vs of h dsn h onnon dsbon h sss nd h h on dsbon dsd wh nsd vs of oon Ω nd h ds nd h s vs sgh

22 9 F. S. Byons hng dffn vs of hn xon. Wh h vs of h dsbon h sss hngs vs nsd dsd o sgh hng wh nsd vs of oon Ω hn xon nd h ds. Fo h vos dssson n ho vsos d w hv h vs of h dsn dsd wh nsd vs of oon Ω nd h ds nd h s vs sgh hng dffn vs of hn xon. Wh h vs of h dsbon h sss nd h sss hngs vs nsd dsd o sgh hng wh nsd vs of oon Ω hn xon nd h ds. Fny h ss obnd n hs h shod ov sf fo sh n sn dsgns of nw s ow- hysss s w s fo hos wong on h dvon of hoy of hybo ho dffson. oss ffs of h gn fd oon nd ss dffson xhng wh h nvonn sng fo nd nsd n os nfn h dsgn nd oons. Sdy of h hnonon of dffson s so sd o ov h ondons of o xon. Rfns [] Now. W. 97 Dyn obs of hodffson n sods B. d. Po. S. S. S. Th. -6. [] Now. W. 97 Dyn obs of hodffson n sods II B. d. Po. S. S. S. Th.9-. [] Now. W. 97 Dyn obs of ho dffson n sod III B. d. Po.S. S. S. Th [] Shf. H. Hz. F. Sh. H. Th hoy of gnzd hos dffson In. J. Eng. S [] Shf. H. Sh. H. hf - s ob n h hoy of gnzd hos dffson In. J. Sods S [6] X. R. Tn. X. Shn.Y. 9Th nfn of dffson on gnzd hos obs of nfn body wh ynd vy In. J. Eng. S [7] bd-.. M. Byons. F. S. Eff of oon n gnzd gno-ho - vsos d dvn Tho nd d Mhn. no.. -.

23 Infn of dffson 9 [8] Lod H.W. Shn Y.967 gnzd dyn hoy of hosy J. Mh. Phys. Sods [9] Lod H. W. Shn. Y 967 gnzd hodyn hoy of ho-sy. Esy J. Mh. Phys. Sods [] Gn.E. Lndsy. 97 Thosy J. Esy. 7. [] bd-.m. 99 Th sss n nsvsy soo ynd d o n nsnnos h so. Mh. o [] bd-. M. 996 Tnsn sons of nhoognos nsvsy soo hos ynd o dyn n J.. Mh.o 7.. [] bd-. M. 999 Pogon of Rygh wvs n n s hf-s of ohoo d Mhs nd oon Vo [] bd-. M. bd-.n. Zdn N.. Th ssss n non-hoognos ohoo s yd ynd J.Th Ssss. 8. [] E-Ngg.M. bd-.m. 987 On gnzd hos ob n n nfn ynd nd n sss Ehs Moon Pns 7.. [6] Royhodh S.. Mhodhyy S. Eff of oon nd xon s on n wvs n gnzd ho vsosy IJMMS [7] Royhodh S.. Bn S. 998 Mgno-hos Inons n n nfn vsos ynd of dndn sbd o od odng In. J. Eng. S 6 / [8] Bn S. Mhodhyy Ryhodh S.. 99 Shy sy ho - vsos wvs n vsos d wh sh vy o. Mh [9] s J.D. 98 Eogns. M Gw-H Nw Yo. [] Sngh H. Sh J. N.98 Gnzd hos wvs n nsvsy soo d. J. os. So

24 9 F. S. Byons [] Thos. L.96 Fndns of H Tnsf. Pn -H In Engwood ffs Nw Jsy. Rvd: Novb

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