An Exact Solution for Kelvin-Voigt Model Classic Coupled Thermo Viscoelasticity in Spherical Coordinates

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1 Joual of Solid Mechaics Vol. 8, No. (6 pp A Exact Solutio fo Kelvi-Voit Model Classic Coupled Themo Viscoelasticity i Spheical Coodiates S. Bahei, M.Jabbai * Mechaical Eieei Depatmet, Islamic Azad Uivesity, South Teha Bach, Teha, Ia Received 5 Decembe 5; accepted 4 Febuay 6 ABSTRACT I this pape, the classic Kelvi-Voit model coupled themo-viscoelasticity model of hollow ad solid sphees ude adial symmetic loadi coditio is cosideed. A full aalytical method is used ad a exact uique solutio of the classic coupled equatios is peseted. The themal ad mechaical bouday coditios, the body foce, ad the heat souce ae cosideed i the most eeal foms ad whee o limiti assumptio is used. This eeality allows simulate vaieties of applicable poblems. At the ed, umeical esults ae peseted ad compaed with classic theoy of themoelasticity. 6 IAU, Aak Bach.All ihts eseved. Keywods : Coupled themo viscoelasticity; Hollow sphee; Exact solutio; Kelvi- Voit method. INTRODUCTION A T the classical ucoupled theoy of themo-elasticity pedicts, such as the heat equatio is of a paabolic type, pedicti ifiite speeds of popaatio fo heat waves ad the equatio of heat coductio of this theoy does ot cotai ay elastic tems cotay to the fact that the elastic chaes poduce heat effects, etc. which that ae ot compatible with physical obsevatios. Theefoe the theoy coupled themo-elasticity has eceived much attetio i the liteatue dui the past seveal decades []. The umbes of papes that peset the closed-fom o aalytical solutio fo the coupled themoelasticity fuctio. Hetaski [] foud the solutio of the themo elasticity i the fom of a seies fuctio. Hetaski ad Iaczak peseted a study of the oe-dimesioal themo elastic waves poduced by a istataeous plae souce of heat i homoeeous isotopic ifiite ad semi-ifiite bodies of the Gee-Lidsay type []. These authos also peseted a aalysis fo lase-iduced waves popaati i a absobi themo elastic semi-space of the Gee- Lidsay type [4]. Geoiadis ad Lykotafitis obtaied a thee-dimesioal tasiet themo elastic solutio fo Rayleih-type distubaces popaati o the suface of the half-space [5]. Wae [6] peseted the fudametal matix of a system of patial diffeetial opeatos that oves the diffusio of heat ad the stais i elastic media. This method ca be used to pedict the tempeatue distibutio ad the stais by a istataeous poit heat, poit souce of heat, o by a suddely applied dilate foce. Bahtui ad Eslami [7] studied the coupled themoelastic espose of a fuctioally aded cicula cylidical shell, ad used a Galeki fiite elemet fomulatio i the space domai ad the Laplace tasfom i the time domai. Bai ad Eslami [8] peseted a solutio fo oedimesioal eealized themoelasticity of a disk. They employed the Laplace tasfom ad Galeki fiite elemet method to solve the ovei equatios. * Coespodi autho. Tel.: addess: mohse.jabbai@mail.com (M.Jabbai. 6 IAU, Aak Bach. All ihts eseved.

2 S. Bahei ad M.Jabbai 5 The Kelvi-Voit model at poblem of maeto-themo-viscoelasticity has extesive uses. This theoy uses i dives fields, such as eophysics fo udestadi the effect of the Eath`s maetic field o seismic wave, dampi of acoustic waves i a maetic field, developmet of a hihly sesitive supecoducti maetomete, electical powe eieei, optics, supesoic aiplaes, etc. [9]. Kopoff [], Chadwick [] ad Nowacki [] studied these types of poblems at the beii. Misa et al. [, 4] Abd-Alla et al. [5] ad Kaliski [6] studied these types of poblems cosidei viscoelastic solid of Kelvi-Voit type. Abd-Alla ad Mahmoud [7] peseted a aalytical solutio fo maeto-themo-viscoelastic o-homoeeous medium with a spheical cavity subjected to peiodic loadi. So et al. [8] studied the poblems of a plae hamoic wave at the iteface betwee to viscoelastic media ude eealized themo-viscoelastic theoy whe the media pemeate a uifom maetic field. S.M. Abo-Dahab [9] studied the effects of the themally iduced vibatio, maetic field ad viscoelasticity i a isotopic homoeeous ubouded body with a spheical cavity. Shama et al. [, ] employed kelvi-voit model of viscoelasticity to study Rayleih-Lamb waves i themoelastic plates i the cotext of eealized (GL ad LS ad coupled theoies of themoelasticity. Roy-Chudhui ad Mukhopdhyay [] studied the effect of otatio ad elaxatio time o plae waves i a ifiite eealized themoviscoelastic solid of Kelvi-Voit type with the etie medium otati with a uifom aula velocity. M. I. A. Othma ad I. A. Abbas [] peseted a ivestiatio of the tempeatue, displacemet, ad stess i a viscoelastic half space of Kelvi-Voit type which the o dimesioal ovei equatios ae solved by the fiite elemet method. Avijt Ka ad M. Kaoia [4] peseted a iteactio due to step iput of tempeatue o the stess fee boudaies of a homoeeous visco-elastic isotopic spheical shell i the cotext of eealized theoies of themo-elasticity. Ezzat et al. [5, 6] applied the state space appoach to oe-dimesioal poblems of eealized themo-visco- elasticity. I the peset wok a full aalytical method is used to obtai the espose of the ovei equatios, theefoe a exact solutio is peseted. The method of solutio is based o the Fouie s expasio ad Eie- fuctio methods, which ae taditioal ad outie methods i solvi the patial diffeetial equatios. Sice the coefficiets of equatios ae ot fuctios of the time vaiable (t, a expoetial fom is cosideed fo the eeal solutio matched with the physical wave popeties of themal ad mechaical waves. Fo the paticula solutio, that is the espose to mechaical ad themal shocks, the Eie-fuctio method ad Laplace tasfomatio is used. This wok is the extesio of the pevious pape that peseted a exact solutio i the spheical coodiates [7]. GOVERNING EQUATIONS A hollow sphee with ie ad oute adius ad i o, espectively, made of isotopic mateial subjected to adialsymmetic mechaical ad themal shock loads, is cosideed. If u is the displacemet compoet i the adial diectio, the stai-displacemet elatios i spheical coodiates ae as follow: u,,, u v cot si u v u,, u v v v,, v cot si u,, si v,, v cot si ( whee, deotes patial deivative. The o-vaishi displacemet compoet is u u,t, so that, 6 IAU, Aak Bach

3 54 A Exact Solutio fo Kelvi-Voit Model Classic Coupled. u u, ( The stess-stai-tempeatue elatio fo eealized themo-viscoelastic Kelvi-voit mateial type is T T t t ij kk ij ij ( whee i, j,,, ad ae Lame s costats,, is the themal expasio coefficiet, T is the absolute tempeatue, is the themal elaxatio, is the mechaical elaxatio time (sesitive pat of the tem of the viscosity. Fo a spheical adial-symmetic system the o-vaishi stesses compoets may witte as: t t u ( u, T T t t u ( u, T T t t (4 The equatio of motio i the adial diectio is, F (, t u (5 whee F, t is the body foce i the adial diectio. Substituti Eq. (4 ito Eq. (5, the Navie equatio i tems of the displacemet compoets is obtaied as: u, u u, T, T, u F (. t t t t ( ( ( (6 Heat coductio equatio i adial-symmetic diectio with the mechaical coupli tem is C T T, T, ( T T ( u, u Q(, t k k k (7 whee is desity of the mateial, k is themal coductivity, themal elaxatio paamete, themal bouday coditios ae T is efeece tempeatue solid, C v is specific heat of the mateial pe uit mass, is Q, t is heat eeatio souce. Mechaical ad i i i i, i, (,, ( C u, t C u, t C T, t f ( t, C u, t C u, t C T, t f ( t, C T t C T t f t, C T t C T t f t 4 4, 4 (8 whee C ij ae mechaical ad themal coefficiets ad by assii diffeet values fo them, diffeet type of mechaical ad themal bouday coditio may be obtaied. These bouday coditios iclude the displacemet, stai, stess (fo the fist ad secod bouday coditios, specified tempeatue, ad covectio, heat flux coditio (fo the thid ad fouth bouday coditios. The f to f 4 ae abitay fuctios which show 6 IAU, Aak Bach

4 S. Bahei ad M.Jabbai 55 mechaical ad themal shocks, espectively. The iitial bouday coditios ae assumed i the followi eeal fom u, f ( u, f ( T, f ( (9 5, t 6 7 whee f 5 to f 7 ae abitay fuctios which show iitial distibutios of displacemet ad tempeatue, espectively. SOLUTION The Eq. ( ad Eq. ( costitute a system of ohomoeeous patial diffeetial equatios with o-costat coefficiets (fuctios of the adius oly has eeal ad paticula solutio.. Geeal solutio with homoeeous bouday coditios A fom of solutio ca be suitable fo Eq. ( ad Eq. ( may be assumed fo the eeal solutio as: t u, t u ( e T, t ( e t ( By substituti Eq. ( ito the homoeeous pats of Eqs. (6 ad (7 yields, u u u u ( ( Cv T u u k k ( Eq. ( is a system of odiay diffeetial equatios, whee the pime symbol (' shows diffeetiatio with espect to adial vaiable ad if suppose: C T d d d d v,,, ( ( k k (. Chaes i depedet vaiables To obtai a solutio fo Eq. (, the depedet vaiables ae chaed as: u u( ( ( ( Substituti Eq. ( ito Eq. ( ives 9 u u u d d d u 4 d d 4u d 4 u 4 (4 6 IAU, Aak Bach

5 56 A Exact Solutio fo Kelvi-Voit Model Classic Coupled.. Solutio justificatio The fist solutio u ad ae cosideed fo the solid sphee as: u A J ( ( B J ( (5 Substituti Eq. (5 ito Eq. (4 ad usi the fomulas fo deivatives of the Bessel fuctio, such as: J J / J ad J J / J, yield d A d B J ( 4 d A d B J ( (6 Eq. (6 shows that u ad ca be the solutio of Eq. (4 if ad oly if d d A d B 4 d (7 The o-tivial solutio of Eq. (7 is obtaied by equati the detemiat of this equatio to zeo ad bis the fist chaacteistic equatio. The secod solutios of u ad ae cosideed as: B J B J u A J A J 5 (8 Substituti Eq. (8 ito Eq. (4 yields ( d A db J ( ( d A ( d A db db J ( d A d B B B J ( d A d B J ( 4 4 (9 The expessios fo u ad ca be the solutio of Eq. (4 if ad oly if d d A d B 4 d ( d A d A db db 4 d A d B B ( The o-tivial solutio of Eq. ( is obtaied by equati the detemiat to zeo as: d d dd 4 ( ( 6 IAU, Aak Bach

6 S. Bahei ad M.Jabbai 57 The equality of Eq. (7 with Eq. ( is iteesti as it pevets mathematical dilemma ad complexity, ad a sile value fo the eievalue simultaeously satisfies both chaacteistic equatios yielded by Eq. (7 ad Eq. (. Eqs. ( ad ( ive the elatio betwee A, A, B ad B, ad they play as the balaci atios that help Eq. (8 to be the secod solutio of the system of Eqs. (4. The complete uique eeal solutios fo the solid sphee ae 5 u A J ( A J ( J ( ( A J ( A J ( 4J ( (4 Those fo the hollow sphee ae u A J ( A J ( J ( A Y ( A Y ( Y ( ( A J ( A J ( 4J ( AY ( A Y ( 4Y ( (5 whee 4 ae atios obtaied fom Eqs.(7, (-( ad ae ive i Appedix A. Substituti the homoeeous fom of the bouday coditios (Eq.(8, fou liea alebaic equatios ae obtaied as: u ad i 4 A 4 A 4 A A (6 whee ij ae ive i the Appedix. Setti the detemiat of the coefficiets of Eq. (6 equal to zeo, the secod chaacteistic equatio is obtaied. A simultaeous solutio of this equatio ad o-tivial solutio of Eq. (7 esults i a ifiite umbe of two eievalues, ad. Theefoe, u ad ae ewitte as: u A J ( [ J ( J ( ] Y ( [ Y ( Y ( ] ( A J ( [ J ( J ( ] Y ( [ 4Y ( 5Y ( ] (7 whee 5 to 5 ae atios peseted i Appedix A ad ae obtaied fom Eq.(6. Let us show the fuctios i the backets of Eq. (7 by fuctios H ad H as: H J ( [ J ( J ( ] Y ( [ Y ( Y ( ] H ( J ( [ J ( J ( ] Y ( [ Y ( Y ( ] 4 5 (8 Accodi to the Stum Liouville theoies, these fuctios ae othooal with espect to the weiht fuctio as: m H ( H ( m d H ( m (9 whee H is the om of the H fuctio ad equals 6 IAU, Aak Bach

7 58 A Exact Solutio fo Kelvi-Voit Model Classic Coupled. H ( H ( d ( Due to the othooality of fuctio H, evey piecewise cotiuous fuctio, such as tems of the fuctio H (eithe fo H o H ad is called the H-Fouie seies as: f, ca be expaded i f ( e H ( ( whee e equals e f H d H ( i ( ( ( Accodi to the umeical esults, thee ae thee oups fo eievalues, whee the fist is eal ad eative, ad the secod ad thid oes, ad, ae cojuate complex with a eative eal pat,, ad a imaiay pat, d. Tems d ad ae the damped ad odamped themal-mechaical atual fequecies, ad is the dampi atio fo the -th atual mode. The o-tivial solutio of Eq. (5 is a alebaic equatio i polyomial fom, ad the detemiat of Eq. (4 is a alebaic equatio i the Bessel fuctio fom. The exact aalytical solutio fo this system of oliea alebaic equatios is complicated, ad the umeical method of solutio is employed i this pape. Sice the Bessel fuctios ae peiodic, the system has a ifiite umbe of oots. The umeical esults of ad fo 5 oots ae peseted i Sec. 4. Usi Eqs. (, (, (7 ad (8, the displacemet ad tempeatue distibutios due to the eeal solutio become p t t u, t a e e b cos t c si t H ( p d d t t T, t a e e b cos t c si t H ( d d ( Usi the iitial coditios (Eq. (9 ad with the help of Eqs. (-(, fou ukow costats, a, b, c ad d ae obtaied..4 Paticula solutio with o-homoeeous bouday coditios The eeal solutios may be used as pope fuctios fo uessi the paticula solutio suitable to the ohomoeeous pats of the Eqs. (6 to (7 ad the o-homoeeous bouday coditios (8 as: p p, ( ( ( 5 ( ] 5 ( u t G t J G t J G t, ( ( 4 ( ( ] 6 ( T t G t J G t J G t (4 Fo solid sphee, the secod type of Bessel fuctio Y is excluded. It is ecessay ad suitable to expad the, Q, t i the H-Fouie expasio fom as: body foce F t ad heat souce 6 IAU, Aak Bach

8 S. Bahei ad M.Jabbai 59 F (, t F ( t H ( Q (, t Q ( t H ( (5 whee F t ad Q t ae F( t F (, t H( d H ( i Q( t Q (, t H ( d H ( i (6 Substituti Eqs. (4 ad (5 ito the ohomoeeous fom of Eqs.(6 ad (7 yields G t G t G t G t b G t G t ( ( ( ( ( ( [ ( ( ( ( ( ( b G t G t b G t b G t b G t G t b G ( t G ( t b G ( t G ( t b G ( t G b G ( t b G ( t b F ( t b F ( t ] J ( [ G ( t G ( t b G ( t G ( t b G ( t 4 4 b G ( t G ( t b G ( t G ( t b G ( t b F ( t ] J ( [ b G ( t G ( t b G ( t G ( t G ( t b ( G ( t b ( G ( t b ( G ( t G ( t b ( Q ( t ] J ( [ b G ( t G ( t b G ( t G ( t b G ( t b G ( t b G ( t G ( t b Q ( t ] J ( ( t (7 whee b6 b ae the coefficiets of H-expasio ad b b5 ae ive i Appedix A. The iitial bouday coditios fo the paticula solutios ae assumed i the followi eeal fom u, u, T, (8 Theefoe, t G ( G ( G ( G ( G ( G ( G ( G ( G ( 5 (9 The uessed fuctios (Eq. (4 ca satisfy the ohomoeeous pat of avie equatio ad heat equatio Eq. (7 if ad oly if 6 IAU, Aak Bach

9 6 A Exact Solutio fo Kelvi-Voit Model Classic Coupled. 4 4 G ( t G ( t G ( t G ( t b G ( t G ( t b G ( t G ( t b G ( t b G ( t b G ( t G ( t b G ( t G ( t b G ( t G ( t b G ( t G b 5 F( t b6 F( t b5g ( t G bg 6 ( t b G ( t G ( t b G ( t G ( t b G ( t ( t b G ( t b G ( t b G ( t G ( t b G ( t G ( t b G ( t b F ( t ( t b G ( t G ( t G ( t b ( G ( t b ( G ( t b ( G ( t G ( t b ( Q ( t b G ( t G ( t b G ( t G ( t b G ( t G ( t G ( t b Q ( t (4 Taki the Laplace tasfom of Eq. (8 ad usi two bouday coditios of Eq. (8 (fo solid cylides, oly the secod ad fouth bouday coditios ae applicable ive s bs s b s b ( s b( s s bs b ( s b5s b4 ( s s b s b ( s s 5 4 b b b b b7 b76 ( s b9 ( 5 b96 s b8( 5 b86 ( s b76 ( s b96s b86 ( s b( s b ( b ( ( s s bs b b ( s s b7 b 8 b G ( s bf ( s G ( s b6f ( s G ( s b6 ( Q ( s G4 ( s b6q ( st G 5 ( s F ( s G6 ( s F4 ( s b b (4 6 IAU, Aak Bach

10 S. Bahei ad M.Jabbai 6 whee b b G s G6 s by the Came methods i the Laplace domai, whee by the ivese Laplace tasfom the fuctios ae tasfomed ito the eal time domai. I the pocess of solutio, it is ecessay to coside the followi poits:. Eq. (4 is i polyomial fom fuctio of the Laplace Paamete S (ot the Bessel fuctio fom of S.Theefoe, the exact ivese Laplace tasfom is possible ad somehow simple. Y i the sequece ae ive i Appedix A. Eq. (4 is solved fo. Fo the hollow sphee, it is eouh to iclude the secod type of the Bessel fuctio of paticula solutio as: u p, t { G ( t J ( G ( t J ( ] [ G ( t Y ( G ( t Y ( ] G ( t G ( t } T p, t G ( t J ( G ( t J ( ] [ G ( t Y ( G ( t Y ( ] G ( t G ( t (4 Substituti Eq. (4 i Eqs. (6 ad (7, eiht equatios ae obtaied, whee usi the fou bouday coditios (Eq. (8, fuctios ae obtaied fo the hollow sphee. 4 RESULTS AND DISCUSSIONS As a example, a solid sphee with adius oe mete made of Alumium is cosideed. The mateial popeties ae: 6 E 7 GFa ; v.; / K ; 77 k / m ; K 4 W / mk ; ; ; c 9 J / kk. The iitial tempeatue T is cosideed to be 9K. Now, a istataeous hot outside suface tempeatue T, t t t is a uit Diac fuctio, is cosideed ad the outside adius of the sphee is, whee assumed to be fixed ( u, t. Fis. -4 show the wave fots fo the displacemet ad tempeatue distibutios alo the adial diectio, whee the compaiso is well justified betwee the elastic theoy ad viscoelastic theoy. u, t t is applied to the outside suface of Fo the secod example, a mechaical shock wave of the fom the sphee, whee suface is assumed to be at zeo tempeatue ( T, t. Fis. 5-8 show the wave fots fo the displacemet ad tempeatue, whee the compaiso is well justified betwee the elastic theoy ad viscoelastic theoy. The coveece of the solutios fo these examples is achieved by cosideatio of eievalues used fo the H-Fouie expasio. Moe tha these umbes of eievalues esult i the iceased oud-off ad tucatio eos, which affect the quality of the aphs. The coveece of solutio is faste fo displacemet i compaiso with the tempeatue. Fi. Tempeatue distibutio due to iput T, t t at 5 5 s. 6 IAU, Aak Bach

11 6 A Exact Solutio fo Kelvi-Voit Model Classic Coupled. Fi. Displacemet distibutio due to iput T, t t at 5 5 s. Fi. Tempeatue distibutio due to iput T, t t at 4 s. Fi.4 Displacemet distibutio due to iput T, t t at 4 s. Fi.5 Tempeatue distibutio due to iput u, t t at 5 5 s. 6 IAU, Aak Bach

12 S. Bahei ad M.Jabbai 6 Fi.6 Displacemet distibutio due to iput u, t t at 5 5 s. Fi.7 Tempeatue distibutio due to iput u, t t at 4 s. Fi.8 Displacemet distibutio due to iput u, t t at 4 s. 5 CONCLUSIONS I the peset pape, a aalytical solutio fo the coupled themoviscoelasticity of thick sphee ude adial tempeatue is peseted. Fis. to 8 show elaxatio time effect o vaiatio of displacemet ad tempeatue. It is obseved that the peak value of Classic coupled themoelastic theoy fo displacemet ad tempeatue iceases. The method is based o the eiefuctios Fouie expasio, which is a classical ad taditioal method of solutio of the typical iitial ad bouday value poblems. The o-competetive steth of this method is its ability to eveal the fudametal mathematical ad physical popeties ad itepetatios of the poblem ude studyi. I the coupled themoviscoelastic poblem of adial-symmetic sphee, the ovei equatios costitute a system of patial diffeetial equatios with two idepedet vaiables, adius ad time (t. The taditioal pocedue to solve this class of poblems is to elimiate the time vaiable usi the Laplace tasfom. The esulti system is a set of odiay diffeetial equatios i tems of the adius vaiable, which solutio falls i the Bessel fuctios family. This method of the aalysis bis the Laplace paamete (s i the aumet of the Bessel 6 IAU, Aak Bach

13 64 A Exact Solutio fo Kelvi-Voit Model Classic Coupled. fuctios, causi hadship o difficulties i cayi out the exact ivese of the Laplace tasfomatio. As a esult, the umeical ivesio of the Laplace tasfomatio is used i the papes deali with this type of poblems i liteatue. I the peset pape, to pevet this poblem, whe the Laplace tasfom is applied to the paticula solutios, it is postpoed afte elimiati the adius vaiable by H-Fouie Expasio. Thus, the Laplace paamete (s appeas i polyomial fuctio foms ad hece the exact Laplace ivesio tasfomatio is possible. APPENDIX A Cv T b, b, b, b4, b5, b4 ( ( ( k k k 5 b H ( d, b b H ( d 7 8 H( H( 5 9 H 4 ( H ( b b H ( d, b H ( d 7 b5h (, 4 H ( ( H ( b b C J ( C [ J ( J ( ] d b b H d 6 b4 C J ( J ( C J ( J ( J ( b5 C J (, b6 C J (, b7 C C, b8 C b C J ( C J (, b C J ( C J ( C J ( b C C 4 4 d d4 d d, d d d d d d d, d d d d,,,, ,,,,, IAU, Aak Bach

14 S. Bahei ad M.Jabbai , CJ ( i C J ( i J ( i CJ ( i i C J ( J i i 5 i i 4 i ( C J ( i J ( i J 5 ( i i J ( i J 5 ( i i C J ( J ( CY ( i C Y ( i Y ( i CY ( i i C Y ( Y 4 i i 5 i i 4 i 5 i ( i C Y ( i Y ( i Y 5 ( i i Y ( i Y 5 ( i i C Y ( Y ( C J ( C J ( J ( C J ( C J ( J ( 4 5 C J ( J ( J 5 ( J ( J 5 ( C J ( J ( C Y ( C Y ( Y ( C Y ( C Y ( Y ( 4 i i i i 4 i i 4 i C Y ( Y ( Y 5 ( Y ( Y 5 ( C Y ( Y ( C J ( C J ( C J ( J ( C J ( J ( J ( J ( C Y ( C Y ( i i C Y ( Y ( i i 4 i i 4 i 4 i C Y ( Y ( Y ( Y ( i 4 i i 4 i 4 i 6 IAU, Aak Bach

15 66 A Exact Solutio fo Kelvi-Voit Model Classic Coupled. C J ( C J ( C J ( J ( C J ( J ( J ( J ( C Y ( C Y ( C Y ( Y ( C 4 Y ( Y ( Y ( Y ( REFERENCES [] Lahii A., Ka T. K., 7, Eievalue appoach to eealized themoviscoelasticity with oe elaxatio time paamete, Tamsui Oxfod Joual of Mathematical Scieces (: [] Hetaski R. B., 964, Solutio of the coupled poblem of themoelasticity i the fom of seies of fuctios, Achiwum Mechaiki Stosowaej 6: [] Hetaski R. B., Iaczak J., 99, Geealized themoelasticity closed-fom solutios, Joual of Themal Stesses 6: [4] Hetaski R. B., Iaczak J., 994, Geealized themoelasticity: e-spose of semi-space to a shot lase Pulse, Joual of Themal Stesses 7: [5] Geoiadis H. G., Lykotafitis G., 5, Rayleih waves eeated by a themal souce: a thee-dimesioal tasiet themoelasticity solutio, Joual of Applied Mechaics 7: 9-8. [6] Wae P., 994, Fudametal matix of the system of dyamic liea themoelasticity, Joual of Themal Stesses 7: [7] Bahtui A., Eslami M. R., 7, Coupled themoelasticity of fuctioally aded cylidical shells, Mechaics Reseach Commuicatios 4: -8. [8] Bai A., Eslami M. R., 4, Geealized coupled themoelasticity of disks based o the lod-shulma model, Joual of Themal Stesses 7: [9] Abd-Alla A.M., Hammad H. A. H., Abo-Dahab S.M., 4, Maeto-themo-viscoelastic iteactios i a ubouded body with a spheical cavity subjected to a peiodic loadi, Applied Mathematics ad Computatio 55: [] Kopoff L., 955, The iteactio betwee elastic wave motios ad a maetic field i electical coductos, Joual of Geophysical Reseach 6: [] Chadwick P., 957, Elastic waves popaatio i a maetic field, Poceedi of the Iteatioal Coess of Applied Mechaics, Busseles, Belium. [] Nowacki W., Facis P.H., Hetaski R.B., 975, Dyamic Poblems of Themoelasticity, Noodhoff, Leyde. [] Misa J. C., Samata S. C., Chakabati A. K., 99, Maeto-themomechaical iteactio i a aeolotopic viscoelastic cylide pemeated by a maetic field subjected to a peiodic loadi, Iteatioal Joual of Eieei Sciece 9 (: 9-6. [4] Misa J. C., Chatopadhyay N. C., Samata S. C., 994, Themo-viscoelastic waves i a ifiite aeolotopic body with a cylidical cavity-a study ude the eview of eealized theoy of themoelasticity, Composite Stuctues 5 (4: [5] Abd-alla A. N., Yahia A.A., Abo-Dahab S. M.,, O the eflectio of the eealized maeto-themoviscoelastic plae waves, Chaos, Solitos & Factal 6: -. [6] Kaleski S., 96, Abopatio of maeto-viscoelastic suface waves i a eal coducto i a maetic field, Poceedis of Vibatio Poblems 4 : 9-9. [7] Abd-Alla A. M., Mahmoud S. R.,, Maeto-themo-viscoelastic iteactios i a ubouded o-homoeeous body with a spheical cavity subjected to a peiodic loadi, Applied Mathematical Scieces 5(9: [8] So Y. C., Zha Y. Q., Xu H. Y., Lu B. H., 6, Maeto-themo-viscoelastic wave popaatio at the iteface betwee two micopola viscoelastic media, Applied Mathematics ad Computatio 76: [9] Abo-Dahab S.M.,, Effect of maeto-themo-viscoelasticity i a ubouded body with a spheical cavity subjected to a hamoically vayi tempeatue without eey dissipatio, Meccaica 47:6-6. [] Shama J. N., 5, Some cosideatios o the ayleih-lamb wave popaatio i visco-themoelastic plate, Joual of Vibatio ad Cotol : - 5. [] Shama J. N., Sih D., Kuma R., 4, Popaatio of eealized visco-themoelastic Rayleih-Lamb waves i homoeeous isotopic plates, Joual of Themal Stesses 7: [] Roy-Chudhui S. K., Mukhopdhyay S.,, Effect of otatio ad elaxatio o plae waves i eealized themoviscoelasticity, Iteatioal Joual of Mathematics ad Mathematical Scieces : IAU, Aak Bach

16 S. Bahei ad M.Jabbai 67 [] Othma M. I. A., Abbas I. A.,, Fudametal solutio of eealized themo-viscoelasticity usi the fiite elemet method, Computatioal Mathematics ad Modeli (: [4] Ka A., Kaoia M., 9, Geealized themo-visco-elastic poblem of a spheical shell with thee-phase-la effect,. Applied Mathematical Modelli : [5] Ezzat M. A., Othma M. I., El Kaamay A.S.,, State space appoach to eealized themo-viscoelasticity with two elaxatio times, Iteatioal Joual of Eieei Sciece 4: 8-. [6] Ezzat M. A., El Kaamay A. S., Smaa A. A.,, State space fomulatio to eealized themo-viscoelasticity with themal elaxatio, Joual of Themal Stesses 4: [7] Jabbai M., Dehbai H., Eslami M. R.,, A exact solutio fo classic coupled themoelasticity i spheical coodiates, Joual of Pessue Vessel Techoloy (:. 6 IAU, Aak Bach