EFFECT OF A TEMPERATURE-DEPENDENT THERMAL CONDUCTIVITY ON A FIXED UNBOUNDED SOLID WITH A CYLINDRICAL CAVITY

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1 U.P.B. Sci. Bull., Seies A, Vol. 78, Iss. 4, 016 ISSN EFFECT OF A TEMPERATURE-DEPENDENT THERMAL CONDUCTIVITY ON A FIXED UNBOUNDED SOLID WITH A CYLINDRICAL CAVITY Ashaf M. ZENKOUR 1 This aticle investigates the themoelastic inteactions in an othotopic unbounded solid containing a cylindical cavity with vaiable themal conductivity. A genealized solution is developed in the context of the one elaxation time themoelasticity theoy. The suface of the cylinde is constained and subjected to a hamonically vaying heat. The govening equations ae teated to be timeless dependence by using the Laplace tansfom. Finally, the tansfomed equations ae inveted by the numeical invesion of the Laplace tansfom. A numeical example has been calculated to illustate the effects of the vaiability themal conductivity paamete and the angula fequency of the themal vibation on all fields. Keywods: Themoelasticity, Constained cylindical hole, Othotopic medium, Vaiable themal conductivity, Hamonically vaying heat. MSC010: 74Dxx, 74Fxx, 74Kxx. 1. Intoduction The classical coupled theoy of themoelasticity is insufficient to deal with themoelasticity poblems. One pat of its solution to the heat equation is extended to infinity. This matte contay to the physical phenomenon since that a pat of mechanical o themal distubance should include an infinite velocity of popagation. This paadox may be teated afte using one of the genealized themoelasticity theoies [1 3]. Lod and Shulman [1] and Geen and Lindsay [] intoduced new theoies of genealized themoelasticity that pedict a finite speed fo heat popagation. Tzou [3] fomulated a new genealized themoelasticity theoy called dual-phase-lag (DPL) heat conduction model. Most investigations in themoelasticity ae based on the assumption of the tempeatue-independent mateial popeties [4 1]. The applicability of the solutions of such poblems is limited to cetain anges of tempeatue. Geneally, the themal conductivity should be tempeatue-dependent at high tempeatue, which definitely altes the themoelastic behavios. The effect of tempeatuedependent themal conductivity is investigated by many authos [13 ]. Most of 1 Pof., Depatment of Mathematics, Faculty of Science, King Abdulaziz Univesity, P.O. Box 8003, Jeddah 1589, Saudi Aabia; Depatment of Mathematics, Faculty of Science, Kafelsheikh Univesity, Kaf El-Sheikh 33516, Egypt. zenkou@kau.edu.sa, zenkou@sci.kfs.edu.eg

2 3 Ashaf M. Zenkou these authos ae taken into account the vaiable themal conductivity fo poblem concened with genealized themoelastic solids subjected to vaious types of heating souces. The aim of the pesent pape is to investigate the themoelastic inteactions in an othotopic unbounded body containing a cylindical cavity with vaiable themal conductivity. The suface of the pesent cylinde is constained and subjected to a time-dependent themal shock. The poblem is solved in the context of genealized themoelasticity with one elaxation time, developed by Lod and Shulman [1]. A diect appoach of the Laplace tansfom is used to obtain the solution of the pesent poblem in the Laplace domain. In addition, a numeical technique is employed to obtain the solution in the physical domain. The effect of the angula fequency of themal vibation and the vaiability of themal conductivity paametes is investigated gaphically and discussed.. Fomulation of the poblem The unbounded othotopic body with cylindical cavity and constained suface is consideed to be subjected to a hamonically vaying heat. The cylindical coodinates system (, θ, z) with z-axis as the axial axis of the cylinde. The pesent poblem is consideed as a 1D poblem due to symmety and all the functions ae depending on the adial distance and the time t. Fo axially symmetic poblem, the adial, hoop, and axial displacement components ae educed to be u = u(, t), u θ (, t) = u z (, t) = 0, (1) with adial ε and hoop ε θ stains given by ε = u(,t), ε θ = u(,t). () So, the stess-displacement elations may be witten as σ c 11 c 1 u β 11 { σ θ } = [ c 1 c ] { u } { β } Θ, (3) σ z c 13 c 3 β 33 whee σ, σ θ, and σ z ae the adial, hoop, and axial stess components, espectively, c ij ae the isothemal elastic constants, β ij ae the themal elastic coupling components, and Θ = T T 0 is the dynamical tempeatue incement of the esonato, in which T 0 is the envionmental tempeatue. The dynamic equation of motion of the cylindical cavity, without consideing the body foces o the heat souces acting in the medium, is expessed as σ + σ σ θ = ρ u t, (4) whee ρ is the mateial density of the medium. Fom Eq. (3), the above equation of motion will be in the fom

3 Effect of a tempeatue-dependent themal conductivity [ ] solid with a cylindical cavity 33 c 11 ( u + 1 u ) c u = ρ u + β Θ t 11 Hee, the genealized heat conduction equation is given by [1] 1 (K Θ t ) [ρc E Θ t + T 0 + (β 11 β ) Θ. (5) t (β 11 u + β u )], (6) ) = (1 + τ 0 whee K is the themal conductivity, C E is the specific heat pe unit mass at constant stain, and τ 0 is the themal elaxation time paamete. The above heat equation is given accoding to the genealized dynamical theoy of themoelasticity of Lod and Shulman [1] that eliminate the paadox of the classical coupled theoy of themoelasticity. 3. Tempeatue-dependent themal conductivity Geneally, the assumption that the solid body is themosensitivity (the themal popeties of the mateial vay with tempeatue) leads to a nonlinea heat conduction poblem. The exact solution of such poblem can be found by assuming the themal conductivity K and the specific heat C E to be linealydepending on the tempeatue [3], but themal diffusivity is assumed be constant. That is K = K (Θ) = k 0 (1 + k 1 Θ), (7) whee k 0 is the themal conductivity at ambient tempeatue T 0 and k 1 is the slope of the themal conductivity-tempeatue cuve divided by the intecept k 0. Now, we will conside the Kichhoff tansfomation [3] ψ = 1 Θ K k (Θ)dΘ, (8) 0 0 whee ψ is a new function expessing the heat conduction. By substituting Eq. (7) in Eq. (8), one gets ψ = Θ(1 + 1 k 1Θ). (9) Fom Eq. (9), it follows that ψ = K (Θ) Θ. (10) t k 0 t ψ = K (Θ) Θ, k 0 Afte substituting Eq. (10) into Eq. (6), the new fom of the geneal heat equation of solids with tempeatue-dependent themal conductivity is obtained by ψ = (1 + τ 0 ) t [1 ψ + T 0 (β u k t k 0 t 11 + β u whee k = K /ρc E is themal diffusivity and )], (11) = + 1. (1) Then, the equation of motion, Eq. (5), will be in the fom c 11 ( u + 1 u ) c u = ρ u + β 11 ψ t 1+k 1 Θ + β 11 β [ 1 + k k 1 1 ψ 1], (13) o in an expanding fom

4 34 Ashaf M. Zenkou c 11 ( u + 1 u ) c ] + β 11 β k 1 u = ρ u + β ψ t 11 [1 (k 1Θ) + (k 1 Θ) [1 + 1 (k 1ψ) 1 (k 8 1ψ) + 1]. (14) Now, it is assumed that the tempeatue change Θ = T T 0 accompanying the defomation is small and does not esult in significant vaiations of the elastic and themal coefficients. So, one can conside these coefficient be egaded as independent of T. In addition to the assumption Θ/T 0 1 one can assume that second powes and poducts of the components of stain may be neglected in compaison with the stains themselves. Thus, the usual linea theoy of themoelasticity is obtained by consideing the case whee only tems linea in stain and tempeatue change. Then, Eqs. (14) and (3) take the foms c 11 ( u + 1 u ) c u = ρ u + β ψ t 11 + (β 11 β ) ψ, (15) σ c 11 c 1 u β 11 { σ θ } = [ c 1 c ] { u } { β } ψ. (16) σ z c 13 c 3 β 33 In what follows, the following dimensionless vaiables will be used {, u, R } = c 0 {, u, R}, k {t, τ 0 } = c 0 {t, τ k 0}, ψ = ψ, σ T j = σ j, k 0 c 1 = T 0 k 1, c 0 = c 11, (j =, θ, z). (17) 11 ρ Theefoe, the govening equations take the following foms afte dopping the pimes fo convenience, u + 1 u c u = u + ε ψ t 1 + ε ψ 6, (18) ψ = (1 + τ 0 ) t [ ψ + (ε u t t 4 + ε u 5 )], (19) σ 1 c u 1 ε 1 { σ θ } = [ c 1 c ] { u } { ε } ψ, (0) σ z c 3 c 4 ε 3 whee {ε 1, ε, ε 3, ε 6 } = T 0 {β c 11, β, β 33, β 11 β }, 11 {ε 4, ε 5 } = 1 {β ρc 11, β }, {c 1, c, c 3, c 4 } = 1 {c E c 1, c, c 13, c 3 }. (1) Initial and bounday conditions Both the initial and bounday conditions of the poblem should be consideed. The initial conditions ae assumed to be in the fom u(, 0) = u(,t) = 0, ψ(, 0) = ψ(,t) = 0. () t t=0 t t=0 The following bounday conditions hold since the bounday of the cylinde is constained and subjected to a to hamonically vaying heat

5 Effect of a tempeatue-dependent themal conductivity [ ] solid with a cylindical cavity 35 The suface of the cylinde = R is subjected to a hamonically vaying heat Θ(R, t) = Θ 0 cos(ωt), ω > 0, (3) whee ω is the angula fequency of the themal vibation and Θ 0 is a constant. Using Eq. (9), then one gets ψ(r, t) = Θ 0 cos(ωt) + 1 k 1[Θ 0 cos(ωt)]. (4) It is to be noted that ω = 0 fo the themal shock poblem. The mechanical bounday condition is due to the displacement of the suface is constained. That is u(r, t) = 0. (5) 5. Solution of the poblem in the Laplace tansfom domain Using the Laplace tansfom of Eqs. (18)-(0) and taking into account the initial conditions given in Eq. (1) and assuming that β 11 = β (i.e., ε 4 = ε 5 = ε) and c 11 = c (i.e., c = 1), we obtain the following equations: d u + 1 du u dψ d d s u = ε 1, (6) d d ψ + 1 dψ = s(1 + τ d d 0s) [ψ + ε ( du + )], (7) d u ε 1 ε 1 σ 1 c du 1 d { σ θ } = [ c 1 1 ] { } { } ψ. (8) u σ z c 3 c 4 ε 3 Hee, any vaiable with an ove ba denotes the Laplace tansfom of this vaiable and s denotes the tansfom paamete. Equations (6) and (7) can be witten in the foms (DD 1 s )u = ε 1 Dψ, (9) εqd 1 u = (D 1 D q)ψ, (30) whee D = d d, D 1 = d d + 1, q = s(1 + τ 0s). (31) Now, let us define the adial displacement u in tems of the themoelastic potential function φ by the elation u = dφ d, (3) then, one can ewite Eqs. (9) and (30) as (D 1 D s )φ = ε 1 ψ, (33) εqd 1 Dφ = (D 1 D q)ψ. (34) Eliminating ψ fom Eqs. (33) and (34), one gets { 4 [s + q(1 + εε 1 )] + qs }φ = 0. (35) The above equation leads to the modified Bessel equation fo φ of zeo ode

6 36 Ashaf M. Zenkou ( d + 1 d d m d 1 ) ( d + 1 m d d ) φ = 0, (36) whee m 1 and m ae the oots m 1 = 1 (A + A 4B), m = 1 (A A 4B), (37) in which A = s + q(1 + εε 1 ), B = qs. (38) The solution of Eq. (36) unde the egulaity conditions that u, Θ, ψ 0 as can be witten as φ = i=1 A i K 0 (m i ), (39) whee K 0 ( ) is the modified Bessel s function of the fist kind of ode zeo and A i, i = 1, ae two paametes depending on s of the Laplace tansfom. Using Eqs. (33) and (39), we obtain ψ = 1 ε i=1 (m i s )A i K 0 (m i ). (40) 1 Substituting fom Eq. (41) into the Laplace tansfom of Eq. (3), we obtain u = i=1 A i K 1 (m i ). (41) whee K 1 ( ) is the modified Bessel function of the fist kind of ode one. So, the stesses can be witten as σ = [s K 0 (m i ) + m i (1 c 1 ) i=1 K 1 (m i )] A i, (4) σ θ = {[s + m i (c 1 1)]K 0 (m i ) + m i (c 1 1) i=1 K 1 (m i )} A i,(43) i=1 σ z = {[ m i c 3 ε 3 d (m ε i s )] K 0 (m i ) m ic 4 K 1 1 (m i ) + m i c 3 K (m i )} A i. (44) whee K ( ) is the modified Bessel function of the fist kind of ode two. In addition, the bounday conditions given in Eqs. (4) and (5), afte using Laplace tansfom, take the foms s + k 1(s +ω ) s +ω ψ (R, s) = Θ 0 [ ] = G (s), s(s +4ω ) (45) u (R, s) = 0. (46) Substituting Eqs. (40) and (41) into the above bounday conditions, one obtains two equations in the unknown paametes A i, as i=1 (m i s )A i K 0 (m i R) = ε 1 G (s), (47) i=1 m i A i K 1 (m i R) = 0. (48) The solution of the poblem will be completed in the Laplace tansfom domain afte obtaining the two constants A 1 and A. Solving the above two equations, one gets A 1 = A = ε 1 G (s)m K 1 (m R) m (m 1 s )K 0 (m 1 R)K 1 (m R) m 1 (m s )K 0 (m R)K 1 (m 1 R), (49) ε 1 G (s)m 1 K 1 (m 1 R). m 1 (m s )K 0 (m R)K 1 (m 1 R) m (m 1 s )K 0 (m 1 R)K 1 (m R)

7 Effect of a tempeatue-dependent themal conductivity [ ] solid with a cylindical cavity 37 Hence, one can easily obtain the displacement and stesses as well as othe physical quantities of the medium. The tempeatue Θ can be obtained by solving Eq. (9) afte applying the Laplace tansfom as Θ (, s) = 1+ 1+k 1ψ k 1. (50) 6. Numeical esults and discussion In this section, the tempeatue Θ, adial displacement u, and stesses σ, σ θ and σ z distibutions will be obtained inside the medium in thei inveted foms. To invet the Laplace tansfom in Eqs. (39)-(44), a numeical invesion method based on a Fouie seies expansion [4] shouuld be adopted. Any function in Laplace domain can be inveted in this method to the time domain as f(t) = ect t {1 f (c) + Re [ N ( 1) n n=1 f (c + inπ )] }. (53) t In most numeical expeiments, the eliable value of c should satisfies the elation ct 4.7 [5]. So, the numeical calculations ae faste convegence fo the same value of c. Numeical evaluations ae made by choosing an othotopic mateial such as the cobalt. The popeties of such mateial ae thus given in SI units [6] as c 11 = c = (N/m), c 1 = (N/m), β 11 = β = (N/m K), β 33 = (N/m (54) K), C E = 47 (J/kg K), K = 69 (W/m K s), ρ = 8836 (kg/m 3 ). The cylindical cavity of adius R = 1 with its cente at the oigin is consideed. The esults ae illustated gaphically in Figues 1 5 fo diffeent values of R, (R 1). It is assumed in all cases studied, except othewise stated, that T 0 = 98 K, k 1 = 0.5, ω = 5, and t = The vaiations of the field quantities along the adial diection ae plotted in Figues 1 5 fo vaious paametes: (a) themal conductivity k 1, (b) angula fequency ω, and (c) time t. Figues 1a, a, 3a, 4a and 5a epesent the fist case in which thee diffeent values the themal conductivity paamete k 1 ae consideed to discuss the effect of themal conductivity. It is assumed that k 1 = 1 and 0.5 fo tempeatuedependent themal conductivity while k 1 = 0 othewise. The second case of esults is illustated in Figues 1b, b, 3b, 4b and 5b fo diffeent values of the angula fequency paamete of themal vibation ω. Fo themal shock poblem, we put ω = 0 and fo hamonically heat ω is set to be eithe 5 o 10. In the last case of esults, Figues 1c, c, 3c, 4c and 5c display the values of the consideed

8 38 Ashaf M. Zenkou physical vaiables in the diection of wave popagation fo diffeent values of dimensionless time t which is taken to be 0.05, 0.07, and 0.1. Fig. 1 shows that the tempeatue Θ is inceasing as k 1 inceases and as both ω and t decease. The distibution of tempeatue may be found as a wave type of heat popagation in the medium. The heat wave font moves fowad with a finite speed in the medium with the passage of time. The tempeatue Θ maybe have a local maximum values at the position = Fig. 1. Vaiation of tempeatue Θ along the adial diection fo vaious paametes: (a) themal conductivity k 1, (b) angula fequency ω, and (c) time t. Fig.. Vaiation of adial displacement u along the adial diection fo vaious paametes: (a) themal conductivity k 1, (b) angula fequency ω, and (c) time t.

9 Effect of a tempeatue-dependent themal conductivity [ ] solid with a cylindical cavity 39 Fig. shows that the adial displacement u is inceasing as k 1 and t incease and as ω deceases. In Figue, u is no longe inceasing along the adial diection and has its maximum value at the location = 1.11 and this iespective to the values of k 1, ω and t. Fig. 3 shows that the adial stess σ is inceasing as k 1 deceases and as ω and t incease. The adial stess is inceasing though the adial diection accoding to all cases. It stats with negative values at = 1 and it is continuously inceasing to diminish to zeo value. Fig. 3. Vaiation of adial stess σ along the adial diection fo vaious paametes: (a) themal conductivity k 1, (b) angula fequency ω, and (c) time t. Fig. 4. Vaiation of hoop stess σ θ along the adial diection fo vaious paametes: (a) themal conductivity k 1, (b) angula fequency ω, and (c) time t.

10 40 Ashaf M. Zenkou Fig. 5. Vaiation of axial stess σ z along the adial diection fo vaious paametes: (a) themal conductivity k 1, (b) angula fequency ω, and (c) time t. Fig. 4 shows that the hoop stess σ θ is inceasing as k 1 deceases, and as ω and t incease. The hoop stess is inceasing though the adial diection. Also, it stats with negative values at = 1 and it is inceasing to diminish to zeo value. It maybe have a local maximum value at = 1.15 fo some values of k 1, ω and t. Finally, Figs. 5a and 5b show that the axial stess σ z is inceasing as k 1 deceases ω inceases. The axial stess σ z is no longe deceasing along the adial diection and has its minimum value at the location = 1.11 and this iespective to the values of k 1 and ω. In Figue 5c, the minimum values the axial stess σ z ae occuing at diffeent positions in the neighbohoods of = 1.1 and this depending on the value of t. In geneal, it is to be noted that the vaiability themal conductivity paamete k 1 has a significant effect on all the fields which add an impotance to ou consideation about the themal conductivity to be vaiable. The behavio of the thee cases of the angula fequency paamete ω is geneally quite simila and ω has a significant effect on all fields. The influence of time paamete is vey ponounced on all the studied field vaiables. It is to be noticed that all the vaiables behave the same manne due to the change in the values of time paamete with some diffeence in thei magnitudes. 7. Conclusions In this wok, we constuct the equations of genealized themoelasticity fo a homogeneous othotopic infinite unbounded body containing a cylindical cavity with a vaiable themal conductivity based on the Lod-Shulman s model.

11 Effect of a tempeatue-dependent themal conductivity [ ] solid with a cylindical cavity 41 The oute suface is taken to be fixed and subjected to a time-dependent tempeatue. The poblem has been solved numeically using the Laplace tansfom technique. Numeical esults fo adial displacement, tempeatue, and themal stesses ae illustated gaphically. Compaisons ae made between the esults pedicted by the theoy of genealized themoelasticity with one elaxation time. It is concluded, fom the numeical esults, that the vaiability themal conductivity paamete has significant effects on the speed of the wave popagation of all the studied fields. The themoelastic tempeatue, displacement, and stesses have stong dependencies on the angula fequency paamete. The heat popagates as a wave with finite velocity instead of infinite velocity in medium since the genealized themoelasticity theoy with one elaxation time is used. The theoy of coupled themoelasticity can extacted fom ou model as special case. Finally, the esults pesented hee should pove useful fo eseaches in scientific and engineeing, as well as fo those woking on the development of mechanics of solids. R E F E R E N C E S [1]. H. W. Lod, Y. Shulman, A genealized dynamical theoy of themoelasticity, J. Mech. Phys. Solids, vol. 15, no. 5, 1967, pp []. A. E. Geen, K. A. Lindsay, Themoelasticity, J. Elast., vol., no. 1, 197, pp [3]. D. Y. Tzou, A unified field appoach fo heat conduction fom maco- to mico-scales, J. Heat Tansfe, vol. 117, no. 1, 1995, pp [4]. M. Aouadi, Genealized themo-piezoelectic poblems with tempeatue-dependent popeties, Int. J. Solids Stuct., vol. 43, no. 1, 006, pp [5]. S. Mukhopadhyay, R. Kuma, Solution of a poblem of genealized themoelasticity of an annula cylinde with vaiable mateial popeties by finite diffeence method, Comput. Meth. Sci. Tech., vol. 15, no., 009, pp [6]. M. N. Allam, K. A. Elsibai and A. E. Abouelegal, Magneto-themoelasticity fo an infinite body with a spheical cavity and vaiable mateial popeties without enegy dissipation, Int. J. Solids Stuct., vol. 47, no. 0, 010, pp [7]. A. M. Zenkou, D. S. Mashat and A. E. Abouelegal, Genealized themodiffusion fo an unbounded body with a spheical cavity subjected to peiodic loading, J. Mech. Sci. Tech., vol. 6, no. 3, 01, pp [8]. A. M. Zenkou, D. S. Mashat and A. E. Abouelegal, The effect of dual-phase-lag model on eflection of themoelastic waves in a solid half space with vaiable mateial popeties, Acta Mech. Solida Sinca, vol. 6, no. 6, 013, pp [9]. A. M. Zenkou, A. E. Abouelegal, State-space appoach fo an infinite medium with a spheical cavity based upon two-tempeatue genealized themoelasticity theoy and factional heat conduction, Z. angew. Math. Phys. ZAMP, vol. 65, no. 1, 014, pp [10]. A. Su, M. Kanoia, Factional ode genealized themoelastic functionally gaded solid with vaiable mateial popeties, J. Solid Mech., vol. 6, no. 1, 014, pp [11]. A. M. Zenkou, A. E. Abouelegal, Effect of tempeatue dependency on constained othotopic unbounded body with a cylindical cavity due to pulse heat flux, J. Them. Sci. Tech., vol. 10, no. 1, 015, pp. 1-1.

12 4 Ashaf M. Zenkou [1]. A. M. Zenkou, Thee-dimensional themal shock plate poblem within the famewok of diffeent themoelasticity theoies, Compos. Stuct., 13(015), [13]. A. M. Zenkou, I. A. Abbas, A genealized themoelasticity poblem of an annula cylinde with tempeatue-dependent density and mateial popeties, Int. J. Mech. Sci., vol. 84, 014, pp [14]. A. M. Zenkou, I. A. Abbas, Nonlinea tansient themal stess analysis of tempeatuedependent hollow cylindes using a finite element model, Int. J. Stuct. Stab. Dynam., vol. 14, no. 6, 014, (17 pages). [15]. A. M. Zenkou, A. E. Abouelegal, K. A. Alnefaie, X. Zhang and E.C. Aifantis, Nonlocal themoelasticity theoy fo themal-shock nanobeams with tempeatue-dependent themal conductivity, J. Them. Stesses, vol. 38, no. 8, 015, pp [16]. A. M. Zenkou, A.E. Abouelegal, Nonlocal themoelastic nanobeam subjected to a sinusoidal pulse heating and tempeatue-dependent physical popeties, Micosys. Techno., vol. 1, no. 8, 015, pp [17]. H. M. Youssef, I. A. Abbas, Themal shock poblem of genealized themoelasticity fo an infinitely long annula cylinde with vaiable themal conductivity, Comput. Meth. Sci. Tech., vol. 13, 007, pp [18] H. M. Youssef, A. A. El-Bay, Two-tempeatue genealized themoelasticity with vaiable themal conductivity, J. Them. Stesses, vol. 33, no. 3, 010, pp [19]. S. M. Abo-Dahab, I. A. Abbas, LS model on themal shock poblem of genealized magnetothemoelasticity fo an infinitely long annula cylinde with vaiable themal conductivity, Appl. Math. Model., vol. 35, no. 8, 011, pp [0]. H. H. Sheief, A. M. Abd El-Latief, Effect of vaiable themal conductivity on a half-space unde the factional ode theoy of themoelasticity, Int. J. Mech. Sci., vol. 74, 013, pp [1]. A. M. Zenkou, A. E. Abouelegal, Nonlocal themoelastic vibations fo vaiable themal conductivity nanobeams due to hamonically vaying heat, J. Viboeng., vol. 16, no. 8, 014, pp []. A. M. Zenkou, A. E. Abouelegal, Non-simple magnetothemoelastic solid cylinde with vaiable themal conductivity due to hamonically vaying heat, Eathq. Stuct., vol. 10, no. 3, 016, pp [3]. N. Noda, Themal Stesses in Mateials with Tempeatue-dependent Popeties, Themal Stesses I, R.B. Hetnaski (Edito), Noth-Holland, Amstedam, [4]. G. Honig, U. Hides, A method fo the numeical invesion of Laplace tansfom, J. Comp. Appl. Math., vol. 10, no. 1, 1984, pp [5]. D. Y. Tzou, Maco to Mico-scale Heat Tansfe: The Lagging Behavio, Taylo and Fancis, Washington DC, [6]. J. C. Misa, N. C. Chattopadhyay and S. C. Samanta, Study of the themoelastic inteactions in an elastic half space subjected to a amp-type heating a state-space appoach, Int. J. Eng. Sci., vol. 34, no. 5, 1996, pp