Archive of SID. An Exact Solution for Classic Coupled Magneto-Thermo-Elasticity in Cylindrical Coordinates.

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1 Joual of Solid Mechaics Vol. 4, No. (0) pp A Exact Solutio fo Classic Coupled Maeto-Themo-Elasticity i Cylidical Coodiates M. Jabbai,*, H. Dehbai Postaduate School, South Teha Bach, Islamic Azad Uivesity, Teha, Ia Sama techical ad vocatioal taii collee, Islamic Azad Uivesity, Vaami Bach, Vaami, Ia Received 8 Novembe 0; accepted 7 Jauay 0 INTRODUCTION I ABSTRACT I this pape, the classic coupled Maeto-themo-elasticity model of hollow ad solid cylides ude adial-symmetic loadi coditio (, t) is cosideed. A full aalytical ad the diect method based o Fouie Hakel seies ad Laplace tasfom is used, ad a exact uique solutio of the classic coupled equatios is peseted. The themal ad mechaical bouday coditios, the body foce, the heat souce ad maetic field vecto ae cosideed i the most eeal foms, whee o limiti assumptio is used. This eeality allows to simulate a vaiety of applicable poblems. The esults ae peseted fo themal ad mechaical shock, sepaately, ad compae the effect of maetic field o tempeatue ad displacemet. 0 IAU, Aak Bach. All ihts eseved. Keywods : Coupled maeto themo-elasticity; Hollow cylide; Exact solutio N ecet yeas, coupled themoelasticity has developed eomously because thee ae may applicatios of it i idusties elated to advaced desi such as aeospace, tubies, jet motos, uclea eactos, etc. Theefoe it is cucial to ead the defomatio ad tempeatue distibutios i themal shock cases. I coupled poblems of themoelasticity take ito accout the time ate of chae of the fist vaiatio ivaiat of stai teso i the fist law of themodyamics causes the coupli betwee elastic ad themal field. The mathematical teatmet of coupled themoelasticity poblems by aalytical methods is athe complicate so that oly vey basic poblems have bee teated i the liteatue ad may woks have bee doe by umeical methods of solutio fo coupled Maeto-themo-elasticity poblems. Thee ae some few papes that peseted the closed-fom o aalytical solutio fo coupled cases. Hetaski [] foud the solutio of coupled themoelasticity i the fom of seies fuctio. Hetaski ad Iaczak [] peseted a study of the oe-dimesioal themoelastic waves poduced by a istataeous plae souce of heat i homoeeous isotopic ifiite ad semi-ifiite bodies of the Gee-Lidsay type. Also these authos peseted a aalysis of lase-iduced waves popaati i a absobi themoelastic semi-space of the Gee-Lidsay type [3]. Geoiadis ad Lykotafitis obtaied a thee-dimesioal tasiet themoelastic solutio fo Rayleih-type distubaces popaati o the suface of a halfspace [4]. Wae [5] 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 used to pedict tempeatue distibutio ad the stais by a istataeous poit heat poit souce of heat o by a suddely applied delat foce. Achive of SID * Coespodi autho. Tel.: ; Fax: addess: mohse.jabbai@mail.com (M.Jabbai). 0 IAU, Aak Bach. All ihts eseved.

2 34 M. Jabbai ad H. Dehbai Majoity of papes focused o umeical methods i coupled poblems.lee [6] cosideed the poblem of theedimesioal axisymmetic quasi-static coupled maetothemoelasticity fo the lamiated cicula coical shells subjected to maetic ad tempeatue fields. Laplace tasfom ad fiite diffeece methods ae used to aalyze the poblem. He obtaied solutios fo the tempeatue ad themal defomatio distibutios i a tasiet ad steady state. Dai ad Fu [7] cosideed the maetothemoelastic poblem of fuctioally aded mateial (FGM) hollow stuctues subjected to mechaical loads. The mateial stiffess, the themal expasio coefficiet ad the maetic pemeability ae assumed to obey the simple powe-law vaiatio thouh the stuctues wall thickess. The aim of thei eseach was to udestad the effect of compositio o maeto- themoelastic stesses ad to desi optimum FGM hollow cylides ad hollow sphees. Wa ad Do [8] peseted theoetical methods fo aalyzi maetothemoelastic esposes ad petubatio of the maetic field vecto i a coducti ohomoeeous themoelastic cylide subjected to themal shock. By maki use of fiite Hakle iteal tasfoms, the aalytical expessios fo maetothemodyamic stess ad petubatio espose of a axial maetic field vecto i a o-homoeeous cylide ae obtaied. Dai ad Wa [9] peseted a aalytical method to solve the poblem fo the dyamic stess-focusi ad ceted-effect of petubatio of the maetic field vecto i othotopic cylides ude themal ad mechaical shock loads. Aalytical expessios fo the dyamic stesses ad the petubatio of the maetic field vecto ae obtaied by meas of fiite Hakel tasfoms ad Laplace tasfoms. Misa et al. [0] cosideed the poblem of a half-space ude the ifluece of a exteal pimay maetic field ad a elevated tempeatue field aisi out of a amp-type heati of the suface. It is foud that the stess distibutio ad the secoday maetic field ae almost idepedet of themal elaxatio time, but ae siificatly depedet o the mechaical elaxatio time. Massalas [] cosideed the pheomeoloical desciptio of the maetothemoelastic iteactios i feomaetic mateial i the famewok of the eealized theoy of themoelasticity poposed by Gee ad Laws. The mateial is supposed to be homoeeous, aisotopic ad elastic udeoi lae defomatios. The aalysis is based o the themodyamic laws of quasi-maetostatics. Misa et al. [] peseted a solutio fo the iduced tempeatue ad stess fields i a ifiite tasvesely isotopic solid cotiuum with a cylidical hole usi the iteal tasfom. The solid medium is cosideed to be exposed to a maetic field ad the cavity suface is assumed to be subjected to a amp-type heati. Gee ad Lidsay model is used to accout fo fiite velocity of heat coductio. Roy Choudhui ad Roy Chattejee [3] peseted the popaatio of maetothemoelastic distubaces poduced by a themal shock i a fiitely coducti elastic half-space i cotact with a vacuum, they used Laplace tasfom. The tempeatue-ate depedet theoy of themoelasticity poposed by Gee ad Lidsay is employed. It is foud that the tempeatue ad defomatio ae discotiuous at the wave fots wheeas the petubed maetic fields suffe delta fuctio siulaity at these locatios. Paul ad Naasimha [4] studied the poblem of axisymmetic axial stess wave eeatio i a themoelastic cicula cylidical ba i the pesece of a applied maetic field. It is assumed that the suface of the cylide is fee fom mechaical loadis ad themal adiatio. A eeal solutio is obtaied by petubatio techique ad is aihilated to a paticula case whee i the applied maetic field is costat i space ad time. Roy Chattejee ad Roychoudhui [5] peseted the maetothemoelastic distubaces i a pefectly electically coducti elastic half-space, i cotact with a vacuum, due to a themal shock applied o the plae bouday of the half-space. The theoy of themoelasticity poposed by Gee ad Lidsay is used to accout fo the iteactio betwee the elastic ad themal fields. Mauszewsk [6] peseted the oliea maetothemoelastic equatios i soft feomaetic ad elastic bodies. The symmety of couplis i these equatios is also ivestiated. Mady et al. [7] peseted the equatios of maeto-themoelasticity with two elaxatio times ad with vaiable electical ad themal coductivity fo oedimesioal poblems icludi heat souces that ae cast ito matix fom, usi the state space ad Laplace tasfom techiques. The esulti fomulatio is applied to a poblem fo the whole coducti space with a plae distibutio of heat souces. Che ad Lee [8] woked o maetothemoelasticity by itoduci two displacemet fuctios ad two stess fuctios. The ovei equatios of the liea theoy of maeto-electothemo-elasticity with tasvese isotopy ae simplified. The mateial ihomoeeity alo the axis of symmety ca be take ito accout ad a appoximate lamiate model is employed to facilitate deivi aalytical solutios. Tiahu et al. [9] epoted the theoy of eealized themoelasticity, based o the theoy of Lod ad Shulma with oe elaxatio time, used to study the electomaeto-themoelastic iteactios i a semi-ifiite pefectly coducti solid subjected to a themal shock o its suface whe the solid ad its adjoii vacuum is subjected to a uifom axial maetic field. He used Laplace tasfom. The Maxwell ' s equatios ae fomulated ad the eealized electomaeto-themoelastic coupled ovei equatios ae established. Shama ad pal [0] ivestiated the popaatio of maetic-themoelastic plae wave i a iitially ustessed, homoeeous isotopic coducti plate ude uifom static maetic field. The eealized theoy of themoelasticity is employed by Achive of SID 0 IAU, Aak Bach

3 A Exact Solutio fo Classic Coupled Maeto-Themo-Elasticity i Cylidical Coodiates 35 assumi the electical behaviou as quasi-static ad the mechaical behaviou as dyamic. At shot waveleth limits, the secula equatios fo symmetic ad skew-symmetic modes educe, to Rayleih suface wave fequecy equatio, because a fiite thickess plate i such a situatio behaves like a semi-ifiite medium. Abd-Alla et al. [] have ivestiated the stess, tempeatue, ad maetic field i a isotopic homoeeous viscoelastic medium with a spheical cavity i a pimay maetic field, whe the cuved suface of the spheical cavity subjected to peiodic loadi. The eealized theoy of themoelasticity poposed by Gee ad Lidsay elasticity is applied to accout fo fiite velocity of heat popaatio. I this wok, a full aalytical method is poposed to obtai the espose of the ovei equatios of the classical coupled maeto themoelasticity i cylidical coodiates, whee a exact solutio is peseted. The method of solutio is based o the Fouie expasio ad eiefuctio methods, which is a taditioal ad outie method 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. Fo the paticula solutio, that is, the espose to mechaical ad themal shocks ad maetic field vecto, the eiefuctio method ad Laplace tasfomatio ae used. This wok is the extesio of the pevious pape of []. GOVERNING EQATIONS A hollow cylide with ie ad oute adius i ad o made of isotopic mateial subjected to the adial-symmetic mechaical, themal shock loads ad maetic field vecto is cosideed. The classical theoy of coupled maeto themoelasticity fo the wave popaatio is cosideed. If u is the displacemet compoet i the adial diectio, the stai-displacemet elatios fo the adial-symmetic loadi coditios ae [30]: u u (), whee (,) deotes patial deivative. The stess-stai elatios fo plae stai coditios ae[30]: E Eα σ ν ε νεθθ T,t ν ν ν E Eα νυ ν σ νε ν ε T,t θθ θθ whee ij ad ij (, i j, ) ae the adial ad hoop stesses ad stai teso, espectively; Ttis (,) the tempeatue distibutio, is the coefficiet of themal expasio, E is the modulus of elasticity ad is the Poisso's atio. Assumi that the maetic pemeability,, hollow cylide is equal to the maetic pemeability of the medium aoud it, ad also the medium is o-feomaetic ad o-feoelectic ad ioi the Thompso effect, the simplified Maxwell's equatios of electodyamics fo a pefectly coducti elastic medium ae [7]: Achive of SID () J h h ( UH) divh 0 ( JH ) z (3) Applyi a iitial maetic field vecto H(0,0, H z ) i cylidical coodiates to Eq. (3), yields [7]: 0 IAU, Aak Bach

4 36 M. Jabbai ad H. Dehbai U ( u,0) h (0,0, hz ) u u (4) hz Hz( u ) z ( JH) Hz( u ) whee z,, H, J ad h ae electo-maetic stess (N/m ), maetic pemeability (H/m), maetic itesity vecto, electic cuet desity vecto ad petubatio of maetic field vecto, espectively. The equatio of motio i the adial diect is [30]: z σ, σ σθθ F(, t) ρu (5) whee Ft (,) is body foce i the adial diectio Fi. Fi. The eomety of a cylide ude a maetic field, Mechaical ad themal shock Usi the elatios () to (5), the Navie equatio i tem of the displacemet compoets is obtaied as: E E u, u, u T,, t Hz u, u, u ρu F(, t) The coupled heat coductio equatio fo the adial-symmetic loadi coditio is [30] : ρc Eα Q(,t) T T T T u u,, k k υ, K (7) Achive of SID whee k,, c, T ad Q(, t) ae the themal coductio coefficiet, the desity, the specific heat, iitial tempeatue ad heat eeatio souce. Mechaical ad themal bouday coditios ae [9]: (6) C u(, t) C u, (, t) C T(, t) f () i i i C u(, t) C u, (, t) C T(, t) f () 3 C T(, t) C T, (, t) f () 3 i 3 i 3 C T(, t) C T, (, t) f () (8) whee C ij ae the mechaical ad themal coefficiets, i ad o ae ie ad oute adius espectively. By assii diffeet values fo them, types of mechaical ad themal bouday coditios may be obtaied. These bouday coditios iclude the displacemet, stai, stess, specified tempeatue, covectio, ad heat flux. The iitial bouday coditios ae assumed i eeal fom [-8]: 0 IAU, Aak Bach

5 A Exact Solutio fo Classic Coupled Maeto-Themo-Elasticity i Cylidical Coodiates 37 U(,0) f 5() U(, 0) f () (9) 6 T(, 0) f () 7 3 SOLUTION Eqs. (6) ad (7) ae the system of ohomoeeous patial diffeetial equatios with ocostat coefficiets (fuctios of adius vaiable ) ad with eeal ad paticula solutios. 3. Geeal solutio with homoeeous bouday coditios Sice the coefficiets of these equatios ae idepedet of time vaiable (t), the expoetial fuctio fom of the time vaiable may be assumed fo the eeal solutio as [-8]: λt u(, t) U() e T(, t) () e λt (0) whee is eievalue ad show the atual fequecy. Substituti Eq. (0) ito homoeeous paes of Eqs. (6) ad (7), yields: u u ud dλ ud5 u u u 0 '' ' d3λ d4λ u u 0 Eqs.() ae system of odiay diffeetial equatios, whee the pime symbol (') shows diffeetiatio with espect to the adius vaiable () ad d -d 5 ae costat paametes ive i appedix. The fist solutios of U () ad () ae cosideed as [-8]: U () AJ () BJ 0 () whee shows odal lies i atual vibati modes ad A ad B ae the maximum amplitudes. Substituti Eq. () i Eq. (), ad usi the fomulas fo deivatives of the Bessel fuctio, such asdj( )/ d J( ) ( J( )/ ), yields: Achive of SID d d λ A Bd J ( ) 0 5 λd βa (-β λd)b J 0 () (3) Eq. (3) shows U ad ca be the solutio of Eqs. (), if oly ad if: d 5 d A λ dβ 0 λd B 4β β λd3 0 (4) The otivial solutio of Eq. (4) is obtaied by setti the detemiat of this equatio equal to zeo as: ( d5) d d3 dd4 0 (5) 0 IAU, Aak Bach

6 38 M. Jabbai ad H. Dehbai Eq.(5) is the fist chaacteistic equatio. Thus, it is cocluded that the fist solutio fo U ad satisfy the system of Eq.(), ad they ae the fist solutios of this system. The secod solutio of the system of odiay diffeetial equatios with ocostat coefficiets Eq. () must be cosideed as [-8]: U () A J A J 3 () B J0 B3J (6) Substituti Eq. (6) to Eq. () yields: (( d 5)β λ d)a 3 dβb 3J0 ( (d 5)β λ d)a λ d A3 dβbj 0 β Aλd4β ( β λd)b 3 βb3j0 A3λd4β( β λd)b 3 3J 0 The expessios fo U ad ca be the solutio of Eqs. (), if oly ad if: ( d5)β λ d dβ A3 0 λd4β β λd3b3 0 ( ( d5)β λ d)a λ d A3 dβb0 β λd4βa ( β λd)b 3 βb3 0 (0) The oivial solutio of Eqs. (8) is obtaied by setti the detemiat equal to zeo as: ( d5) d d3 dd4 0 () The chaacteistic Eq.() is the same as the chaacteistic Eq. (5). This equality is iteesti as it pevets mathematical dilemma ad complexity, ad a sile value fo the eievalue β simultaeously satisfies both chaacteistic Eqs. (5) ad (). Eqs.(9) ad (0) ive the elatio betwee A, A 3, B, ad B 3, ad they play as the balaci atios that help Eq. (6) to be the secod solutio of the system of Eq. (). Eqs. (9) ad (0) ae two alebaic equatios with six ukow A, A 3, B, B 3, β ad. Sice the umbe of ukows is moe tha the equatios, thee is o estictio to bi ocompatibility betwee Eqs. (8) (0). The complete eeal solutios fo the solid cylide ae [-8]: Achive of SID (7) (8) (9) u () AJ A J ζj () Aζ J0 A ζ3j0 ζ4j Those fo hollow cylide ae: U() AJ AJ ζja 3Y A4 Y ζy () A ζ J A ζ J ζ J A ζ Y A ζ Y ζ Y ) () (3) 0 IAU, Aak Bach

7 A Exact Solutio fo Classic Coupled Maeto-Themo-Elasticity i Cylidical Coodiates 39 whee ζ - ζ 4 ae atios obtaied fom Eqs. (4), (9) ad (0) ad ae ive i appedix. Substitute U ad i homoeeous fom of bouday coditios Eq.(8), fou liea alebic equatios ae obtaied as: μ μ μ3 μ A 4 0 μ μ μ μ A μ A 3 μ3 μ33 μ μ A 4 μ4 μ43 μ (4) whee μ ij ae the coefficiets depedi o β ad ad ae ive i appedix. Setti the detemiat of coefficiets of Eq. (4) equal to zeo, the secod chaacteistic equatio is obtaied. A simultaeous solutio of this equatio ad Eq. (5), esult i a ifiite umbe of tow eievalues, β ad λ. Theefoe U ad ae ewitte as [-8]: U() A J ζ5j ζ6j ζ7y ζ8y ζ9y () A ζ J ζ J ζ J ζ Y ζ Y ζ Y ) whee ζ 5 - ζ 5 ae eievectos ca obtaied fom Eqs (4). Let us show the fuctios i the backets of Eqs. (5) by fuctios H ad H 0 as: H( ) J( ) ζ5j( ) ζ6j ( ) ζ7y( ) ζ8y( ) ζ9y ( ) H( ) ζ J( ) ζ J( ) ζ J ( ) ζ Y( ) ζ Y( ) ζ Y ( ) Accodi to the Stum-Liouville theoies, these fuctios ae othooal with espect to the weiht fuctio p() = as [-8]: 0 m o H(β)H(βm)d H(β i ) m (7) whee H( ) is om of the H fuctio ad equals [-8]: H(β ) o H (β)d i (8) Achive of SID Due to the othooality of fuctio H, evey piecewise cotiuous fuctio, such as f() ca be expaded i tems of the fuctio H (eithe fo H o H 0 ), ad is called the H-Fouie seies as [,3]: f() e H(β ) (9) (5) (6) whee e equlas: e o f() H() d (30) H(β) i Thee ae thee oups fo eievalues, whee the fist is eal ad eative, ad the secod ad thid oes, 0 IAU, Aak Bach

8 40 M. Jabbai ad H. Dehbai ad 3, ae cojuate complex with a eative eal pat, -, ad a imaiay pat, ae the damped ad odamped themal-mechaical atual fequecies, ad d. Tems d ad is the dampi atio fo the th atual mode. 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. Usi Eqs. (0), (5), ad (6), the displacemet ad tempeatue distibutios due to the eeal solutio become [-8]: 3 m 3 m λmt m u (, t) { a e }H ( ) λmt m 0 T (, t) { N e }H ( ) Usi iitial coditios Eq.(9) ad with the help of Eqs. (6)ad (7)ad(9)to(3), ukow costats, a m, N m ae obtaied. 3. Paticula solutio with o-homoeeous bouday coditios The eeal solutios may be used as pope fuctios to uess the paticula solutio adopted to the ohomoeeous pats of Eqs. (6) ad (7) ad the ohomoeeous bouday coditios Eq. (8) as [-8]: p 5 p u (, t) { G (t)j ( ) G (t)j ( ) G (t)} T (, t) { G (t)j ( ) G (t)j ( ) G (t)} Fo solid cylide, the secod type of Bessel fuctio Y is excluded. It is ecessay ad suitable to expad the body foce F(,t) ad heat souce Q(,t) i the H-Fouie expasio fom as [-8]: F(, t) F (t)h ( ) Q(, t) Q (t)h ( ) 0 whee F (t) ad Q (t) ae o F() t F(,) t H( ) d H ( ) 0 i o Q() t Q(,) t H0( ) d H ( ) i Achive of SID (3) (3) (33) (34) Substituti Eqs.(3) ad (33) ito ohomoeeous fom of Eq. (6) ad (7) yields: 0 IAU, Aak Bach

9 A Exact Solutio fo Classic Coupled Maeto-Themo-Elasticity i Cylidical Coodiates 4 d5g dg dg dg3 d9d6g5 () t dd6g 5() t dd7 G 5 () t d9d7 G5 () t J( ) d0d7 G6 () t d0d6g6 () t d6d8d6f(,) t d6d8d7f(,) t d5 G dg dg4 () t d9d7g5 () t dd7g 5() t J0 ( ) 0 d0d7g6 () t d6d8d7f(,) t (35) d4g () t G3() t d3g 3() t G4() t d4d8g 5() t J0 ( ) d3d8g6 () t d5d8g 6 () t d7dd8q(,) t d4g () t G4() t dg 3 4() t dd 7 d9qt (,) J( ) 0 d4d9g 5() t d3d9g6 () t d5d9g 6() t whee d7 d9 ae coefficiet of H Expasios ad peseted i appedix. The uessed fuctios Eq.(3) ca satisfy ohomoeeous pat of Eqs.(6) ad (7) if oly ad if: d5g dg dg dg3 d9d6g5 dd6g 5 dd7 G 5 d9d7 G5 () t d0d7 G6 () t d0d6g6 () t d6d8d6f(,) t d6d8d7 F(,) t 0 d G d G d G d d G d d7g 5() t d0d7g6 () t d6d8d7f(,) t 0 d G () t G () t d G () t G () t d d G () t d d G () t d d G () t d d d Q(,) t 0 d G t G t d G t d d G t d d G t d d G t d d d () 4() 3 4() 4 9 5() 3 9 6() 5 9 6() 7 9Qt (,) 0 Taki the Laplace tasfom of Eqs. (36) ad usi tow bouday coditios of Eqs. (8) (fo solid cylide oly the secod ad foth bouday coditios ae applicable), ive: dd 9 6 dd6s d5 ds ds d 0 d0d7 d0d 6 dd7 s d9d7 0 d5 ds 0 d d9d7 dd7s d0d7 d4s 0 d3s d4d8s d3d8 d5d8s 0 d4s 0 d3s d4d9s d3d9 d5d9s CJ ( ) C J0( ) J( ) ) C3J 0 ( ) s s C4J 0 ( ) C4J ( ) s G () s ddd 6 8 6Fs (,) ddd 6 8 7Fs (,) G () s ddd 6 8 7Fs (,) G3 () s dd 7 d8qs (,) G4 () s dd 7 d9qs (,) G5 () s f() s G6 () s f4 () s Achive of SID (36) (37) 0 IAU, Aak Bach

10 4 M. Jabbai ad H. Dehbai Eqs. (37) is a system of alebaic equatios ad is solved fo G (t) to G 6 (t) by the Came methods i the Laplace domai whee by the ivese Laplace tasfom the fuctio ae tasfomed ito the eal time domai. I the pocess of solutio, it is ecessay to coside the followi poits. -The iitial coditios Eq.(9) ae cosideed oly fo eeal solutios Eq.(3). The iitial coditios of G (t) - G 6 (t) fo the paticula solutios ae cosideed equal to zeo. -Eq. (37) is i polyomial fom fuctio of the Laplace paametes s (ot Bessel fuctios fom of s). Thefoe the exact ivese Laplace tasfoms is possible ad somehow simple. 3-Fo hollow cylide, it is eouh to iclude the secod type of Bessel fuctios Y() i the sequece of paticula solutio as [-8]: p u (,) t { G () t J ( ) G () t J ( ) G () t Y ( ) G () t Y ( )} G () t G () t p T (,) t { G () t J ( ) G () t J ( ) G () t Y ( ) G () t Y ( )} G () t G () t Substituti Eqs. (38) i Eqs. (6) ad (7), eiht equatios ae obtaied whee usi the fou bouday coditios Eq.(8), fuctios G (t) - G (t) ae obtaied fo the hollow cylide. 4 RESULTS AND DISCUSSIONS As a example, a solid cylide with o m made of alumiium is cosideed. The mateial popeties ae show i Table.[7,-3]: Table The iitial tempeatue T is cosideed 93º K. Mateial popeties of Alumium E=70 (G Pa) (/ K) 3 707( k / m ) K=04(W/m K) c=903(j/k K) 8 Hz 5 0 ( A/ m) Hz=0 & 3 Now, a istataeous hot outside suface tempeatue T(, t) 0 T whee () t is the uit Diac fuctio, is cosideed, ad the outside adius of the cylide is assumed to be fixed ( u(, t) 0).To daw the aphs, a odimesioal time ˆ vt t is cosideed, whee v E( ) / ( ) is the dilatatioal-wave velocity [3-35]. Fis. - 5 show the wave fots of displacemet, tempeatue, adial ad hoop stess with ad without maetic field at tˆ 0.. As a secod example, a mechaical shock wave of the fom u(, t) 0 u is applied to the outside suface of the cylide, whee the suface is assumed to be at zeo tempeatue ( T(, t) 0) ad ( u T) is a assumed iitial displacemet at tˆ 0.6. These fiues show the maetic field iceases the stiffess of body ad the deceases the tempeatue ad displacemet. Fis. 6-9 show the wave fots fo the displacemet, tempeatue, adial ad hoop stess with ad without maetic field distibutios alo the adial diectio. The coveece of the solutios fo these examples is achieved by cosideatio of 00 eievalues used fo the H-Fouie expasio. Moe tha this umbe of eievalues esults 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. The small oscillatios i Fis. ad 4 ae due to the coveece of solutios. Achive of SID (38) 0 IAU, Aak Bach

11 A Exact Solutio fo Classic Coupled Maeto-Themo-Elasticity i Cylidical Coodiates 43 Fi. No-dimesioal displacemet distibutio due to iput 3 T(, t) 0 T at o-dimesioal time tˆ 0. with ad without maetic field. Fi.3 No-dimesioal tempeatue distibutio due to iput 3 T(, t) 0 T at o-dimesioal time tˆ 0. with ad without maetic field. Fi.4 No-dimesioal adial stess distibutio due to iput 3 T(, t) 0 T at o-dimesioal time tˆ 0. with ad without maetic field. Achive of SID Fi.5 No-dimesioal hoop stess distibutio due to iput 3 T(, t) 0 T at o-dimesioal time tˆ 0. with ad without maetic field. Fi.6 No-dimesioal displacemet distibutio due to iput u(, t) 0 u at o-dimesioal time tˆ 0.6 with ad without maetic field. 0 IAU, Aak Bach

12 44 M. Jabbai ad H. Dehbai 5 CONCLUSIONS Fi.7 No-dimesioal tempeatue distibutio due to iput u(, t) 0 u at o-dimesioal time tˆ 0.6 with ad without maetic field. Fi.8 No-dimesioal adial stess distibutio due to iput u(, t) 0 u at o-dimesioal time tˆ 0.6 with ad without maetic field. Fi.9 No-dimesioal hoop stess distibutio due to iput u(, t) 0 u at o-dimesioal time tˆ 0.6 with ad without maetic field. Achive of SID I this pape, a aalytical solutio fo the coupled Maeto-themo-elasticity of thick cylides ude adial tempeatue o mechaical shock load with ad without maetic field is peseted. The method is based o the eiefuctio Fouie expasio, which is a classical ad taditioal method of solutio fo the typical iitial ad bouday value poblems. The steth of this method is its ability to eveal the fudametal mathematical ad physical popeties ad the itepetatios of the poblem ude study. I the coupled Maetothemoelastic poblem of the adial-symmetic cylide, the ovei equatios ae a system of patial diffeetial equatios with two idepedet vaiables, the adius () ad the time (t). The taditioal pocedue to solve this class of poblems is to elimiate the time vaiable by usi the Laplace tasfom. The esulti system is a set of odiay diffeetial equatios i tems of the adius vaiable, which falls ito the Bessel fuctio family. This method of aalysis bis the Laplace paamete (s) i the aumet of the Bessel fuctios, causi hadship o complicatios i cayi out the exact ivese of the Laplace tasfomatio. As a esult, the umeical ivese of the Laplase 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 the H-Fouie expasio. Thus, the Laplace paamete (s) appeas i polyomial fuctio foms, ad hece the exact Laplace ivesio tasfomatios ae possible. 0 IAU, Aak Bach

13 A Exact Solutio fo Classic Coupled Maeto-Themo-Elasticity i Cylidical Coodiates 45 The method descibed i this pape is a exact solutio of the coupled Maeto-themo-elasticity of thick cylides with two ive bouday coditios. The exact solutios foud i liteatue fo the coupled themoelasticity poblems ae limited to the ifiite spaces ad half-spaces. The aalytical method of solutio peseted i this pape is fo a fiite domai with specified ad ive bouday coditios. This is the ovelty of the pape, whee the solutio of a popula stuctual compoet (a thick cylide) with two specified bouday coditios ude the coupled Maeto-themo-elasticity assumptio is ive aalytically ad i tems of the seies solutio. Appedix E k k ( ) c E d d d3 d4 T d5 Hz d6 d7 d8 E E k d 3d d d d d d d d 3d d d d C d C d E d E ζ ζ ζ ζ b / a d / c/ b/ a b / a/ c b/ a b/ a bζ / / a d c b/ a bζ / / a 3 4 C J C C ζ J J ( ) J ( ) i 0 i i 3 0 i i i i C J ζ J i i i C ζ J ζ J C J ( ) J ( ) ζ J ζ J ( ) ζ J ( ) i 0 i i i i i i 3CY i C 0 C3ζY0 Y ( i) Y ( i) 4C Y iζ iy ic Y0 ( i) Y ( i) ζy iζ iy ( i) ζy ( i ) i i i i C ζ Y ζ Y ) i Achive of SID C J A C C ζ J J ( ) J ( ) o 0 o o 3 0 o o C J oζ oj oc J0 ( o) J( o) ζj oζoj ( o) ζj ( o) o o o o C ζ J ζ J IAU, Aak Bach

14 46 M. Jabbai ad H. Dehbai 3CY oc Y0 ( o) Y ( o) C3ζY0 o o 4C Y oζ oy oc Y0 ( o) Y ( o) ζy oζoy ( o) ζy ( o) o C3 ζ3y0 oζ 4oY o 3C3ζJ0 ic3ζj( i) 3C3 ζ3j0 i ζ 33C3ζY0 ic3ζy ( i) 4 C4ζJ0 C4ζ J( o ) A 4C4 ζ3j0 ζ4j 43C4ζY0 C4ζY ( o ) J C ζ J ( ) ζ J ζ J ( ) ζ J ( ) 4 i 3 3 i 4 i 4 i 0 i 4 i C ζ Y ζ Y C ζ Y ( ) ζ Y ζ Y ( ) ζ Y ( ) i 4 i i 3 3 i 4 i 4 i 0 i 4 i C ζ J ( ) ζ J ζ J ( ) ζ J ( ) 4 3 o 4 o 4 o 0 o 4 o C ζ Y ζ Y ) C ζ Y ( ) ζ Y ζ Y ( ) ζ Y ( ) o 4 o 4 o 0 o 4 o REFERENCES [] Hetaski R. B., 964, Solutio of the coupled poblem of themoelasticity i the fom of seies of fuctios, Achiwum Mechaiki Stosowaej 6: [] Hetask R.B., Iaczak J., 993, Geealized Themoelasticity: Closed-Fom Solutios, Joual of Themal Stesses 6: [3] Hetask R.B., Iaczak J., 994, Geealized Themoelasticity: Respose of Semi-Space to a Shot Lase Pulse, Joual of Themal Stesses 7: [4] Geoiadis H. G., Lykotafitis G., 005, Rayleih Waves Geeated by a Themal Souce: A Thee-dimesioal Tasiat themoelasticity Solutio, Joual of Applied Mechaics 7:9-38. [5] Wae P., 994, Fudametal Matix of the System of Dyamic Liea Themoelasticity, Joual of Themal Stesses 7: [6] Lee Z.Y., 009, Maeto themoelastic aalysis of multilayeed coical shells subjected to maetic ad vapo fields, Iteatioal Joual of Themal Scieces 48:50-7. [7] Dai H.L., Fu Y.M., 007, Maeto themoelastic iteactios i hollow stuctues of fuctioally aded mateial subjected to mechaical loads, Iteatioal Joual of Pessue Vessels ad Pipi 84:3-38. [8] Wa X., Do K., 006, Maeto themodyamic stess ad petubatio of maetic field vecto i a ohomoeeous themoelastic cylide, Euopea Joual of Mechaics - A/Solids 5: [9] Dai H.L., X. Wa, 006, The dyamic espose ad petubatio of maetic field vecto of othotopic cylides ude vaious shock loads, Iteatioal Joual of Pessue Vessels ad Pipi 83:55-6. [0] Misa S. C., Samata S. C., Chakabati A. K., 99, Tasiet maeto themoelastic waves i a viscoelastic halfspace poduced by amp-type heati of its suface, Computes & Stuctues 43: [] Massalas C. V., 99, A ote o maeto themoelastic iteactios, Iteatioal Joual Eieei Sciece 9:7-9. [] Misa J. C., Samata S. C., Chakabati A. K., Misa Subhas C., 99, Maeto themoelastic iteactio i a ifiite elastic cotiuum with a cylidical hole subjected to amp-type heati, Iteatioal Joual Eieei Sciece 9: [3] Roy Choudhui S. K., Chattejee Roy G., 990, Tempeatue-ate depedet maeto themoelastic waves i a fiitely coducti elastic half-space, Joual Computes & Mathematics with Applicatios 9: [4] H. S., Naasimha R., 987, Maeto themoelastic stess waves i a cicula cylide, Iteatioal Joual of Eieei Sciece, 5: [5] Gai Chattejee(Roy), Roychoudhui S. K., 985, The coupled maeto themoelastic poblem i elastic half-space with two elaxatio times, Iteatioal Joual Eieei Sciece 3: [6] Mauszewsk B., 98, Dyamical maeto themoelastic poblem i cicula cylides-i: Basic equatios, Iteatioal Joual Eieei Sciece 9: [7] Ezzat M. A., El-Kaamay A. S., 003, Maeto themoelasticity with two elaxatio times i coducti medium with vaiable electical ad themal coductivity, Joual Applied Mathematics ad Computatio 4: Achive of SID 0 IAU, Aak Bach

15 A Exact Solutio fo Classic Coupled Maeto-Themo-Elasticity i Cylidical Coodiates 47 [8] Che W.Q., Lee K.Y., 003, Alteative state space fomulatios maetoelectic themoelasticity with tasvese isotopy ad the applicatio to bedi aalysis of ohomoeeous plates, Joual Solids & Stuctues 40: [9] Tiahu H., Yape S., Xiaoe T., 004, A two-dimesioal eealized themal shock poblem fo a half-space i electomaeto-themoelasticit, Iteatioal Joual Eieei. Sciece 4: [0] Shama J.N., Pal M., 004, Rayleih-Lamb waves i maeto themoelastic homoeeous isotopic plate, Iteatioal Joual Eieei Sciece 4: [] Abd-Alla A.M., Hammad H.A.H., Abo-Dahab S.M., 004, Maeto-themo-viscoelastic iteactios i a ubouded body with a spheical cavity subjected to a peiodic loadi, Joual Applied Mathematics ad Computatio 55: [] Jabbai M., Dehbai H., Eslami MR., 00, A Exact Solutio fo Classic Coupled Themo elasticity i Spheical Coodiates, Joual of Pessue Vessel, ASME, Tasactio 3: [3] Jabbai M., Dehbai H., Eslami MR., 0, A Exact Solutio fo Classic Coupled Themo elasticity i Cylidical Coodiates, Joual of Pessue Vessel, ASME, Tasactio 33: [4] Jabbai M., Dehbai H., 009, A Exact Solutio fo Classic Coupled Themopooelasticity i Cylidical Coodiates, Joual of Solid Mechaics : [5] Jabbai M., Dehbai H., 00, A Exact Solutio fo Classic Coupled Themopooelasticity i Ax symmetic Cylide, Joual of Solid Mechaics : [6] Jabbai M., Dehbai H., 00, A Exact Solutio fo Lod-Shulma Geealized Coupled Themo pooelasticity i Spheical Coodiates, Joual of Solid Mechaics : [7] Jabbai M., Dehbai H., 0, A Exact Solutio fo Lod-Shulma Geealized Coupled Themopooelasticity i Cylidical Coodiates, Published i 9th iteatioal coess o themal, Budapest, Huy. [8] Jabbai M., Dehbai H., 0, A Exact Solutio fo Quasi-Static Poo-Themoelasticity i Spheical Coodiates, Iaia joual of mechaical eieei tasactios of the ISME : [9] Necati Ozisik M., 980, Heat coductio, Wiley & Sos. [30] Hetaski R. B., Eslami M. R., 009, Themal Stesses Advaced Theoy ad Applicatios, Spie, New Yok.. [3] Beezovski A., Eelbecht J., Maui G. A., 003, Numeical Simulatio of Two-Dimesioal Wave Popaatio i Fuctioally Gaded Mateials, Eu. J. Mech. A/Solids : [3] Beezovski A., Maui G. A., 003, Simulatio of Wave ad Fot Popaatio i Themoelastic Mateials With Phase Tasfomatio, Comput. Mate. Sci 8: [33] Beezovski A., Maui G. A., 00, Simulatio of Themoelastic Wave Popaatio by Meas of a Composite Wave- Popaatio Aloithm, J.Comput. Phys 68: [34] Eelbecht J., Beezovski A., Salupeea A., 007, Noliea Defomatio Waves i Solids ad Dispesio, Wave Motio 44: [35] Ael Y. C., Achebach J. D., 985, Reflectio ad Tasmissio of Elastic Waves by a Peiodic Aay of Cacks: Oblique Icidece, Wave Motio 7: Achive of SID 0 IAU, Aak Bach