International Journal of Solids and Structures

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1 Intenational Jounal of Solids and Stuctues 47 (1) Contents lists available at ScienceDiect Intenational Jounal of Solids and Stuctues jounal homepage: Magneto-themoelasticity fo an infinite body with a spheical cavity and vaiable mateial popeties without enegy dissipation Mohmed N. Allam a, Khaled A. Elsibai a, *, Ahmed E. Abouelegal a,b a Mathematics Depatment, Faculty of Science, Mansoua Univesity, Mansoua 35516, Egypt b Depatment of Mathematics, Faculty of Science, King Khaled Univesity, Abha, P.O. Box 94, Saudi Aabia aticle info abstact Aticle histoy: Received 1 May 8 Received in evised fom 1 Decembe 9 Available online 15 May 1 Keywods: Genealized themoelasticity Geen and Naghdi model Spheical cavity Mateial popeties Magneto-themoelasticity The model of genealized themoelasticity poposed by Geen and Naghdi, is applied to study the electomagneto themoelastic inteactions in an infinite pefectly conducting body with a spheical cavity. The modulus of elasticity ae taking as linea function of tempeatue. By means of the Laplace tansfom and Laplace invesion, the poblem is solved. The closed fom solutions fo displacement, tempeatue, and themal stesses ae epesented gaphically. A compaison is made with the esults in the case of tempeatue-independent. Ó 1 Elsevie Ltd. All ights eseved. 1. Intoduction In fact, the change of body tempeatue has an effect on the stain/stess fields and convesely, i.e. mechanical action, and coesponding stain poduce a tempeatue field. A pat of the mechanical enegy due to stain is conveted into heat. The change of tempeatue being small, the coesponding tems in field equations and coupling tems in heat conduction equations ae neglected in the classical uncoupled theoy of themoelasticity. This theoy has two shotcomings. The fist and obvious one is that the elastic defomation has no effect on the tempeatue field, and the othe is that the theoy pedicts an infinite speed of popagation fo heat waves as well as fo mechanical distubance. Biot (1956) deived the equations of the coupled theoy of themoelasticity by changing the equations of motion, thus taking cae of the fist shotcoming which is existing in the classical dynamical uncoupled theoy of themoelasticity. In this theoy, the heat equation, still emains paabolic. To take cae of this paadox, some theoies of genealized themoelasticity ae poposed which allow a finite speed fo the popagation of themal and mechanical distubances. The fist genealization, fo isotopic bodies, is due to Lod and Shulman (1967) who obtained a wave-type heat equation by postulating a new law of heat conduction to eplace the classical Fouie s law. The anisotopic case was late developed by Sheief and * Coesponding autho. addesses: dkhaledelsibai@yahoo.com (K.A. Elsibai), ahabogal@mans. edu.eg (A.E. Abouelegal). Dhaliwal (198). The second genealization is known as the theoy of themoelasticity with two elaxation times, o the theoy of tempeatue-ate-dependent themoelasticity, and was poposed by Geen and Lindsay (197). It does not violate Fouie s law of heat conduction when the body unde consideation has a cente of symmety, and it is valid fo both isotopic and anisotopic bodies. The theoy of themoelasticity without enegy dissipation is anothe genealized theoy and was fomulated by Geen and Naghdi (type II) (Geen and Naghdi, 1993). It includes the themal displacement gadient among its independent constitutive vaiables, and diffes fom the pevious theoies in that it does not accommodate dissipation of themal enegy. Sinha and Elsibai (1996) consideed the themoelastic inteactions caused in an infinite body with spheical cavity fo the diffeent theoies of genealized themoelasticity, that is (L-S) and (G-L) and classical dynamical coupled theoy. The solution of the poblem is obtained in the context of two diffeent cases, using the Laplace tansfom technique. Allam et al. () deduced the themal stess distibutions in a hamonic field fo a homogeneous, isotopic infinite body with a cicula cylindical hole based on Geen and Naghdi theoy. A numeical example was given and the esults wee pesented gaphically to illustate the effect of the paamete of Geen and Naghdi theoy. Abd-Alla et al. (4) investigated the magneto themo viscoelastic inteactions in an unbounded body with a spheical cavity subjected to a peiodic loading. They applied the genealized theoy poposed by Geen and Lindsay, when the cavity is placed in a constant pimay magnetic field. Youssef (5) solved a poblem of an infinite body with a cylindical -7683/$ - see font matte Ó 1 Elsevie Ltd. All ights eseved. doi:1.116/j.ijsolst.1.4.1

2 63 M.N. Allam et al. / Intenational Jounal of Solids and Stuctues 47 (1) cavity and vaiable mateials popeties subjected to themal mechanical shocks by using Laplace tansfom technique. Tianhu et al. (5) used the theoy of genealized themoelasticity, based on the theoy of Lod and Shulman with one elaxation time, to study the electomagneto themoelastic inteactions in an infinitely long pefectly conducting solid cylinde subjected to a themal shock on its suface when the cylinde and its adjoining vacuum is subjected to a unifom axial magnetic field. Themoelasticity has seen a apid development by vaious engineeing science. Most of the investigations wee done unde the assumption tempeatue-independent mateial popeties, which limit the applicability of the solution obtained to cetain anges of tempeatue. At a high tempeatue the mateial chaacteistics such as the modulus of elasticity, Poisson s atio, the coefficient of themal expansion, and the themal conductivity ae no longe constants (Lomakin, 1976). In ecent yeas, due to the pogess in vaious fields in science and technology, it has become necessay to take into account consideation the eal behavio of the mateial chaacteistics. The expeimental data show that the changes of Poisson s atio and the coefficient of linea themal expansion due to high tempeatue can be neglected (Manson, 1954). The modulus of elasticity and themal conductivity ae consideed as the tempeatue-dependent mateial paamete in this aticle. The idea of the pesent wok, is to investigate the displacement, tempeatue, and themal stesses in an unbounded body with a spheical cavity assuming that the suface of the cavity is subjected to themal shocks and zeo stess and the cavity is placed in a magnetic field with constant intensity. The moduli of elasticity ae taken to be functions of tempeatue. Finally, by taking an appopiate mateial, the esults ae plotted gaphically to illustate the poblem and compaed the esults in the case of tempeatue-independent mechanical popeties.. Fomulation of the poblem Fo a homogeneous, isotopic, and in the absence of extenal body foces and inne heat souces, the motion equation has the following fom ijj þ F i ¼ u whee ij is the components of stess tenso, u i is the components of displacement vecto, q is the mass density and F i is the components of Loentz foce, whose expession is F i ¼ l e ðj H i whee H is the magnetic field vecto, J is the cuent density vecto and l e is the magnetic pemeability. The heat conduction equation in the context of genealized themoelasticity poposed by Geen and Naghdi (type II) is T T ¼ qc þ whee e = e kk is the stain dilatation, T is the absolute tempeatue of the medium, T is the efeence tempeatue, K is the mateial constant chaacteistic of Geen and Naghdi theoy, c =(3k + l) a t, a t is the linea themal expansion coefficient, k and l ae Lame s constants, and C E is the specific heat at constant stain. We conside a homogeneous, isotopic infinite body with a spheical cavity of adius a. We take a spheical pola coodinates (,H,u) and take the oigin of the coodinate system at the cente of the spheical cavity. The sphee is placed in a magnetic field with constant intensity H = (,,H ). Due to the application of the initial magnetic field H, thee esult an induced magnetic field h which be small, so that, thei poducts with u i and thei deivatives ð1 ð ð3 can be neglected fo lineaization and an induced electic field E. The simplified linea equations of electodynamics fo a pefectly homogeneous conducting elastic solid h ¼ J E ¼l E ¼l H ð6 :h ¼ :E ¼ ð7 whee is the electic pemeability. The induced field components in the sphee ae obtained fom Eqs. (4) (7) in the foms E ¼ð E ¼ l h ¼ð h ¼ð H e: ð8 The components of Loentz foce can be obtained fom Eqs. (4) and (8) in the fom F ¼ l e H l e Due to the symmeties of the sphee, the only nonvanishing displacement component will be u = u(,t). So that, the components of stain tenso ae given by e HH ¼ u ¼ e uu e H ¼ e u ¼ e Hu ¼ : The cubic dilatation e will be e ¼ e þ e uu þ e þ u : ð1 The components of the stess tenso fo a spheical symmetic system, ae given by ¼ þ ke ð11 HH ¼ l u þ ke ct ¼ uu H ¼ u ¼ Hu ¼ : ð1 Fom Eqs. (9), (11) and (1), we obtain the equation of motion (1) k þ l þ l @ ¼ l e H þ : ð13 Applying the div opeato to both sides of Eq. (13), we obtain k þ l þ l e H e c T ¼ l e H þ ð14 whee is the Laplace s opeato in spheical coodinates, given @ : Conside a themoelastic body of mateial having tempeatuedependent popeties on the fom (Youssef, 5) k ¼ k f ðt l ¼ l f ðt c ¼ c f ðt k ¼ k f ðt ð15 whee k is the themal conductivity, k, l, c and k ae consideed to be constants f(t) is given in a non-dimensional function of tempeatue. In the case of tempeatue-independent modulus of elasticity, f(t) = 1. We will conside that (Youssef, 5) f ðt ¼1 a T whee a is called the empiical mateial constant.

3 M.N. Allam et al. / Intenational Jounal of Solids and Stuctues 47 (1) Fo lineaity of the govening patial diffeential equations of the poblem, we have to take into account the condition jtt j T 1, which give us the appoximating function of f(t) tobe in the fom. f ðt 1 a T : Then heat equation will take the fom MK T þ Mc e qc : The equation of motion is MC 1 þ C e Mc T ¼ C c þ 1 and the constitutive equations ¼ M þ k e c T HH ¼ M l u þ k e c T whee C 1 ¼ ðk þ l c q ¼ l e M ¼ f ðt ¼ 1 a T C ¼ l e H q and K ¼ K is Geen and Naghi paamete: qc E ð16 In ode to solve the poblem, we assumed that the initial conditions of the poblem ae taken to be homogeneous, while the bounday conditions ae taken as follows: (1) The suface of the cavity ae taction fee (zeo stess) i.e. ð t ¼ ¼ a: ð17 () The themal bounday condition is that the suface of the cavity subjected to a themal shock i.e. Tð t ¼T Hðt ¼ a ð18 whee H(t) is the Heaviside unit step function. To tansfom the above equations to dimensionless foms, we define the following non-dimensional quantities 9 u ¼ðqC =c 1 T au ¼ =a >= T ¼ T=T t ¼ðk =a t > ð19 ¼ =ðc T HH ¼ HH=ðc T : Using these non-dimensional vaiables the govening Eqs. (11), (1), (14) and (3) educe to (dopping pimes fo convenience) ¼ þ g u T HH ¼ þðg þ 1 u T e g e T T T ¼ g þ e ð1 and the bounday conditions (17) and (18) take the foms ¼ at ¼ 1 T ¼ Hðt at ¼ 1 ð ð3 ð4 whee g ¼ k =ðk þ l C 3 ¼ C 1 þ C C 4 ¼ C þ c =ðc M c ¼ 1 l e g 1 ¼ C 1 C 3 g ¼ k C 4 a C 3 g 3 ¼ k Ma K g 4 ¼ Meg 3 e ¼ c T : q C E C 1 3. Solution of the poblem in the Laplace domain Applying the Laplace tansfom with paamete s (denoted by a ba) which is defined as Fðs ¼ ff ðtg ¼ Z 1 f ðte st dt into both sides of Eqs. () and (3) unde the initial conditions, we obtain the following equations g s e ¼ g 1 T ð5 g 4 s e ¼ g 3 s T: ð6 Eliminating T, between Eqs. (5) and (6), we get fouth-ode equation satisfied by e in the fom ½ 4 A þ CŠe ¼ whee M A ¼ðg 3 þ g þ g 1 g 4 s ¼ as C ¼ g g 3 s 4 ¼ bs 4 : In a simila manne, we can show that T satisfies the equation ½ 4 A þ CŠT ¼ : Intoducing m i (i = 1,) into Eq. (7), we find m 1 m 1 e ¼ ð7 ð8 ð9 whee m 1 and m ae the oots of the following chaacteistic equation m 4 Am þ C ¼ p m i ¼ a ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 4b s = ¼ x i s p x i ¼ a ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 4b = i ¼ 1 : ð3 By taking e ¼ð1=w and substituting into Eq. (9), we get the solution of e in the fom e ¼ 1 ð A 1 expðm 1 þa expðm þa 3 expðm 1 þa 4 expðm ð31 whee A i,(i = 1,,3,4) is some paamete depending on s, and at infinity (? 1), fo egulaity condition, the displacement and tempeatue must be continuous, so we take A 3 and A 4 equal to zeo. By the same technique, we get T ¼ 1 A 1 expðm 1þA expðm ð3 whee A i ði ¼ 1 ae paametes depending on s and substituting fom Eqs. (31) and (3) into (6), we get the following elation A i ¼ s g 4 m i g 3 s A i ¼ g 4 x i g 3 A i ¼ X i A i X i ¼ g 4 x i g 3 : ð33

4 634 M.N. Allam et al. / Intenational Jounal of Solids and Stuctues 47 (1) In the Laplace Tansfom domain, solving Eq. (1) using (31), and assuming that u vanishes at infinity, we obtain u ¼ 1 ðm 1 þ 1 m 1 A 1 expðm 1 þ ðm þ 1 A m expðm : ð34 Substituting the solutions fo u and T in the elations () and (1), we obtain the non-dimensional constitutive equations in the Laplace tansfom domain as follows! ¼ Me ðm ð1 gðm 1 1 þ 1 þ 1 X 1 A m þ Me ðm HH ¼ Me ðm 1 þ Me ðm ð1 gðm þ 1 þ 1 X m 3 ðg 1ðm 1 þ 1 þ g! X 1 3 m 1 ðg 1ðm þ 1 þ g X 3 m! A A 1! A : ð35 ð36 In the Laplace tansfom domain, the bounday conditions (4) become ¼ at ¼ 1 ð37 T ¼ 1=s at ¼ 1: By substituting Eq. (37) in Eqs. (3) and (35), we find the constants A 1 and A in the fom F A 1 ¼ sðf 1 L F L 1 F 1 A ¼ sðf 1 L F L 1 whee F i ¼ ð1 gðm i þ 1 þ 1 X m i expðm i i L 1 ¼ X i expðm i i ¼ 1 : ð38 Intoducing the constants A 1 and A in Eqs. (3), (34), (35) and (36), then we can wite the solutions in the Laplace tansfom domain, fo the dimensionless displacement, tempeatue, and themal stesses in the following foms u ¼ 1 k 11 x 1 s þ k 1 s þ k 13 s þ k 14 þ k 15 expðc 3 s s 1 s s 1 s 1 k 1 x s þ k s þ k 3 s þ k 4 þ k 5 expðc 3 s s 1 s s s ð39 T ¼ X 1 v v v 11 s þ 1 13 þ expðc s s 1 s s 1 s þ X v v v 1 s þ 3 þ expðc s s 1 s s s ð4 ¼ g M q 11 x 1 3 s þ q 1 s þ q 13 s þ q 14 þ q 15 expðc 3 s s 1 s s 1 s þ g M q 1 x 3 s þ q s þ q 3 s þ q 4 þ q 5 expðc 3 s s 1 s s s ð41 HH ¼ g M n 11 x 1 3 s þ n 1 s þ n 13 s þ n 14 þ n 15 expðc 3 s s 1 s s 1 s þ g M n 1 x 3 s þ n s þ n 3 s þ n 4 þ n 5 expðc 3 s s 1 s s s ð4 whee k mj, v mj, q mj, n mj, c m,(m =1,, j =1,,...,5) ae listed in the Appendix A. 4. Invesion of the Laplace tansfoms To obtain the solutions of the displacement, tempeatue and themal stesses in the physical domain, it is necessay to pefom Laplace invesion fo the consideed vaiables obtained in Laplace tansfom domain. We can take the invese Laplace tansfom of Eqs. (39) (4), and find expessions fo the dimensionless displacement, tempeatue, and themal stesses in the foms u ¼ 1 X T ¼ 1 m¼1 ð1 m Hðt c m ðk x m1 þ k m ðt c m m þ k m3 ðt c m þ k m4 e s 1ðtc m þ k m5 e s ðtc m X m¼1 ¼ g M 3 X X m ð1 m Hðt c m v m1 þ v m e s 1ðtc m þ v m3 e s ðtc m m¼1 ð1 m Hðt c m x m þq m4 e s 1ðtc m þ q m5 e s ðtc m HH ¼ g M 3 X m¼1 q m1 þ q m ðt c m þ q m3 ðt c m ð1 m Hðt c m ðn x m1 þ n m ðt c m m þ n m3 ðt c m þ n m4 e s 1ðtc m þ n m5 e s ðtc m 5. Numeical esults and discussion Now, we conside a numeical example to illustate the analytical pocedue pesented ealie,. The coppe mateial was chosen fo pupose of numeical evolutions, which we take the following mateial values of the diffeent physical paametes (Youssef, 5) k ¼ 7: N=m l ¼ 3: N=m e ¼ :168 a ¼ :5 K 1 H ¼ 1 7 =4p Am 1 q ¼ 8: kg=m 3 C E ¼ 383:1 m K 1 s a t ¼ 1: K 1 l e ¼ 4p 1 7 Hm 1 : The numeical technique outlined above was used to obtain the tempeatue, displacement and themal stesses distibutions inside the body. Figs. 1 4 pesent the dimensionless values of displacement, tempeatue and adial and hoop stesses distibutions fo Tempeatue Fig. 1. Tempeatue distibution with. : t =.1 t =.5

5 M.N. Allam et al. / Intenational Jounal of Solids and Stuctues 47 (1) diffeent values of adial distance and fo two values of time, namely fo t =.5 and.1. In Fig. 1, we obseve that the tempeatue distibution T has a maximum value at the bounday of the cavity but, inside of this egion the value vanishes apidly. Displacement t =.1 t =.5 Fig.. Displacement distibution with. Fig. epesent the vaiations of the displacement distibution u, which we obseved that at the displacement eaches a maximum value at nea the suface of the cavity and then stat to deceases with the incease of adial distance. In Fig. 3, the adial stess at the suface cavity is zeo. This is coincides with the mechanical bounday condition that the cavity suface is taction fee. It is also obseved that the adial stess attains it minimum value at some distance fom the bounday of the cavity, then inceases with the incease of adial distance. In Fig. 4, the hoop stess have thei minimum values at the suface of the cavity, then inceases with the incease of adial distance. Figs. 5 8 pesent the effect of efeence tempeatue on displacement, tempeatue and adial and hoop stesses distibutions when the modulus of elasticity is a linea function of efeence tempeatue ðm ¼ð1 a T ¼:5 1:5 and in the case of a tempeatue-independent modulus (M = 1). The effect of Geen and Naghdi paamete K is obseved in Figs. 9 1, which we see that displacement and tempeatue is inceasing with the incease of G-N paamete, while the themal stesses ae deceasing. 6. Concluding emaks In this wok, we have discussed the model of magneto-themoelasticity fo an infinite body with a spheical cavity depending on Radial stess -.1 t =.1 -. t =.5 Tempeatue M =.8 M = 1 M = Fig. 3. Radial stess distibution with. Fig. 5. Tempeatue distibution with M =.8 M = 1 M = 1.5 Hoop stess -.1 t =.1 Displacement t = Fig. 4. Hoop stess distibution with. Fig. 6. Displacement distibution with.

6 636 M.N. Allam et al. / Intenational Jounal of Solids and Stuctues 47 (1) K = K = 4 Radial stess M =.8 M = 1 M = 1.5 Displacement Fig. 7. Radial stess distibution with Fig. 1. Displacement distibution with. Hoop stess -.1 M =.8 M = 1 M = 1.5 Radial stess K = K = Fig. 8. Hoop stess distibution with. Fig. 11. Radial stess distibution with. 1.1 Tempeatue.7.3 K = K = 4 Hoop stess K = K = Fig. 9. Tempeatue distibution with. a efeence tempeatue in the context of Geen and Naghdi theoy without enegy dissipation. The poblem is solved by means of the Laplace tansfom and Laplace invesion. We concluded that: (1) The basic equations of the poblem can be descibed by a fouth-ode chaacteistic equation. -. Fig. 1. Hoop stess distibution with. () The mechanical distibutions indicate that the wave popagates as a wave with finite velocity in medium. It is completely diffeent fom the case fo the classical theoy of

7 M.N. Allam et al. / Intenational Jounal of Solids and Stuctues 47 (1) themoelasticity whee an infinite speed of popagation is inheent and hence all the consideed functions have a non-zeo value fo any point in the medium. (3) The fact that in themoelasticity without enegy dissipation, the waves popagate with finite speeds is evident in all these figues. Appendix A a 1 ¼ ð1 g x 1 a 11 ¼ ð1 gx 1 a x 1 ¼ 1 X 1 c ¼ x x 1 g 4 ¼ x 1 ðg X 1 g g 5 ¼ x ðg X c g 1 ¼ x 1 x ð1 g a ¼ a 1 ¼ ð1 gx a ¼ 1 X g ¼ 1 g x 1 x b 1 ¼ X a 1 X 1 a b 11 ¼ X a 11 X 1 a 1 b 1 ¼ X a 1 X 1 a b ¼ X 1 a X a 1 b 1 ¼ X 1 a 1 X a 11 b ¼ X 1 a X a 1 c 1 ¼ a 1 x 1 a c ¼ a þ x 1 a 1 c 3 ¼ x 1 a c 11 ¼ a 11 x a 1 c 1 ¼ a 1 þ x a 11 c 13 ¼ x a 1 d 1 ¼ a 1 þ a x 1 d ¼ a þ a 1 x 1 þ a 1 a x 1 =ðg d 3 ¼ a x 1 þ a 1 a 1 x 1 =ðg d 4 ¼ a 1 a x 1 =ðg d 11 ¼ a 11 þ a 1 x d 1 ¼ a 1 þ a 11 x þ a a 1 x =ðg d 13 ¼ a 1 x þ a a 11 x =ðg d 14 ¼ a a 1 x =ðg h 1 ¼ a 1 þ a x 1 h ¼ a þ a 1 x 1 þ g 4 a h 3 ¼ a x 1 þ a 1 g 4 h 4 ¼ a g 4 h 11 ¼ a 11 þ a 1 x h 1 ¼ a 1 þ a 11 x þ g 5 a 1 h 13 ¼ a 1 x þ a 11 g 5 h 14 ¼ a 1 g 5 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s 1 ¼ b 11 þ b 11 4b 1b 1 s ¼ b 11 b 11 4b 1b 1 b 1 b 1 v 11 ¼ a s 1 s v 1 ¼ ða þ a 1 s 1 þ a s 1 s 1 ðs 1 s v 13 ¼ ða a 1 s a s s ðs 1 s v 1 ¼ a 1 v ¼ a þ a 11 s þ a 1 s s 1 s s 1 ðs 1 s v 3 ¼ a 1 a 11 s 1 a 1 s 1 s ðs 1 s k 11 ¼ s c s 1 þ s c 1s 1 þ s a þ s c 1 s 1 þ s 1s a þ a s 1 k 13 ¼ a s 1 s s 3 1 s3 k 1 ¼ s c 1s 1 þ s 1 c 11s þ s 1 a 1 þ s 1 c 11 s þ s 1s a 1 þ a 1 s k 3 ¼ a 1 s 1 s s 3 1 s3 k 1 ¼ ðs c 1 s 1 þ s a þ a s 1 s s 1 k 14 ¼ a þ c 1 s 1 þ c s 1 þ c 3s 3 1 s 3 1 ðs 1 s k ¼ ðs 1c 11 s 1 þ s 1 a 1 þ a 1 s s s 1 k 4 ¼ a 1 þ c 11 s þ c 1 s þ c 13s 3 s 3 1 ðs 1 s k 15 ¼ a c 1 s c s c 3s 3 s 3 ðs 1 s k 5 ¼ a 1 c 11 s 1 c 1 s 1 c 13s 3 1 s 3 ðs 1 s q 11 ¼ s d s 1 þ s d 1s 1 þ s a þ s d 1 s 1 þ s 1s a þ a s 1 q 13 ¼ a s 1 s s 3 1 s3 q 1 ¼ s 1 d 1s þ s 1 d 11s þ s 1 a 1 þ s 1 d 11 s þ s s 1 a 1 þ a 1 s q 3 ¼ a s 1 s q 1 ¼ ðs d 1 s 1 þ s a þ a s 1 s s 1 s 3 1 s3 q 14 ¼ a þ d 1 s 1 þ d s 1 þ d 3s 3 1 þ d 4s 4 1 s 3 1 ðs 1 s q 4 ¼ a 1 þ d 11 s þ d 1 s þ d 13s 3 þ d 14s 4 s 3 1 ðs 1 s q 15 ¼ a 1 þ d 11 s þ d 1 s þ d 13s 3 þ d 14s 4 s 3 ðs 1 s q 5 ¼ a þ d 1 s 1 þ d s 1 þ d 3s 3 1 þ d 4s 4 1 s 3 ðs 1 s q ¼ ðs d 11 s 1 þ s 1 a 1 þ a 1 s s s 1 n 11 ¼ s h s 1 þ s h 1s 1 þ s a þ s h 1 s 1 þ s 1s a þ a s 1 n 13 ¼ a s 1 s s 3 1 s3 n 1 ¼ s 1 h 1s þ s 1 h 11s þ s 1 a 1 þ s 1 h 11 s þ s s 1 a 1 þ a 1 s n 3 ¼ a s 1 s n 1 ¼ ðs h 1 s 1 þ s a þ a s 1 s s 1 s 3 1 s3 n 14 ¼ a þ h 1 s 1 þ h s 1 þ h 3s 3 1 þ hd 4s 4 1 s 3 1 ðs 1 s n 4 ¼ a 1 þ h 11 s þ h 1 s þ h 13s 3 þ h 14s 4 s 3 1 ðs 1 s n 15 ¼ a 1 þ h 11 s þ h 1 s þ h 13s 3 þ h 14s 4 s 3 ðs 1 s n 5 ¼ a þ h 1 s 1 þ h s 1 þ d 3s 3 1 þ h 4s 4 1 s 3 ðs : 1 s Refeences n ¼ ðs h 11 s 1 þ s 1 a 1 þ a 1 s s s 1 Biot, M.A., Themoelasticity and ievesible themodynamics. J. Appl. Phys. 7, Lod, H.W., Shulman, Y., A genealized dynamical theoy of themoelasticity. J. Mech. Phys. Solids 15,

8 638 M.N. Allam et al. / Intenational Jounal of Solids and Stuctues 47 (1) Sheief, H., Dhaliwal, R., 198. A uniqueness theoem and a vaiational pinciple fo genealized themoelasticity. J. Them. Stesses 3, Geen, A.E., Lindsay, K.A., 197. Themoelasticity. J. Elast., 1 7. Geen, A.E., Naghdi, P.M., Themoelasticity without enegy dissipation. J. Elast. 31, Sinha, S.B., Elsibai, K.A., Themal stesses fo an infinite body with spheical cavity with tow elaxation times. J. Them. Stesses 19, Allam, M.N., Elsibai, K.A., Abouelegal, A.E.,. Themal stesses in a hamonic field fo an infinite body with a cicula cylindical hole without enegy dissipation. J. Them. Stesses 5, Abd-Alla, A.M., Hammad, H.A., Abo-Dahab, S.M., 4. Magneto themo viscoelastic inteactions in an unbounded body with a spheical cavity subjected to a peiodic loading. Appl. Math. Comput. 155, Youssef, H.M., 5. Genealized themoelasticity of an infinite body with a cylindical cavity and vaiable mateial popeties. J. Them. Stesses 8, Tianhu, H., Xiaogeng, T., Yapeng, S., 5. A genealized electomagnetothemoelastic poblem fo an infinitely long solid cylinde. Eu. J. Mech. A/ Solids 4, Lomakin, V.A., The Theoy of Elasticity of Non-homogeneous Bodies, Moscow. Manson, S.S., Behavio of mateial unde conditions of themal stess, NACA Repot,, P. 117.