TWO TEMPERATURE ELECTRO-MAGNETO THERMO-VISCO- ELASTIC RESPONSE WITH RHEOLOGICAL PROPERTIES AND TEMPERATURE DEPENDENT ELASTIC MODULI

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1 Inernaional Journal of Physics and Mahemaical Sciences ISSN: 77- (Online) 4 Vol. 4 () April-June, pp /Mondal and Mukhopadhyay TWO TEMPERATURE ELECTRO-MAGNETO THERMO-VISCO- ELASTIC RESPONSE WITH RHEOLOGICAL PROPERTIES AND TEMPERATURE DEPENDENT ELASTIC MODULI * Manoj K. Mondal and B. Mukhopadhyay Deparmen of Mahemaics, Seacom Engineering College, Howrah, India Deparmen of Mahemaics, Bengal Engineering and Science Universiy, Howrah, India * Auhor for Correspondence ABSTRACT The presen paper is concerned wih a hermo-visco-elasic problem of an isoropic maerial in cylindrical hole wih wo-emperaure due o he presence of a uniform magneic field. Rheological properies of volume and densiy of maerial are considered here. The problem is based on he concep of emperaure dependen mechanical properies. Generalized hea conducion equaion due o Lord-Shulman and Green- Lindsay are uilized. Eigen value approach is used o solve he problem. Keywords: Thermo-Visco-Elasiciy, Rheological Propery, Vecor-Marix-Differenial Equaion, Relaxaion Funcion, Two-Temperaure INTRODUCTION In classical heory of hermo-elasiciy, he hea conducion equaion is a parabolic ype differenial equaion which predics ha he effec of a hermal disurbance will insananeously manifes iself a infiniely large disance from he source. This predicion is unrealisic from a physical poin of view. During las four decades, non-classical heories which predic finie speed of hermal signal in elasic solids have been developed o remove his drawback. In he heory of hermoelasic diffusion he coupled hermoelasic model which is used implies infinie speeds of propagaion of hermoelasic waves. Lord and Shulman (967) obained a wave-ype hea equaion by consrucing a new law of hea conducion o replace he classical Fourier s law which ensure finie speeds of propagaion for hea and elasic waves which is known as he firs generalizaion of he coupled hermo-elasiciy heory. Green and Lindsay (97) incorporae emperaure rae erm ino he consiuive equaions of hermo-elasiciy and aained an explici version of he consiuive equaions. This heory depends on wo relaxaion imes. Linear visco-elasiciy including hermal sresses and classical hermo-visco-elasiciy increase he field area of elasic heory o he research workers. Drozdov (996) derived a consiuive model in hermovisco-elasiciy which accouns for changes in elasic moduli and relaxaion imes. Using generalized heory proposed by Lord-Shulman and Green-Lindsay, he problem on visco-elasic maerials has been discussed by Giorgi and Naso (6). Explanaion of elecro-magneo-hermo-visco-elasic plane waves in roaing media wih hermal relaxaion was discussed by Choudhuri and Chaapadhyay (7). The hermo-elasic problem wih he effec of magneic field and hermal relaxaion under diffusion was discussed by Ohman e al. (3). A high emperaure he mechanical properies of he maerial are emperaure-dependen. Mos invesigaion in hermo-visco-elasiciy was done by ignoring he emperaure-dependen mechanical properies. Thermo-visco-elasiciy including he emperauredependen mechanical properies increases he field area of elasic heory o he research workers. Problem on emperaure-dependen mechanical properies was analyzed by Aouadi and El-Karamany (4) in heir paper. Ezza e al., () expend heir valuable effors o recognize he effecs of modified Ohm s and Fourier s laws on generalized magneo-hermo-visco-elasiciy wih relaxaion volume properies. Kundu and Mukhopadhyay (5) invesigaed a hermo-visco-elasic problem of an infinie medium conaining a spherical caviy considering he rheological properies of volume, using he generalized heory of hermo-elasiciy. Mondal and Mukhopadhyay (3) discussed he effecs of rheological volume and densiy properies on heir problem having emperaure dependen mechanical properies. The sudy of hermo-visco-elasiciy wih wo-emperaure is of ineres in some branches of Copyrigh 4 Cenre for Info Bio Technology (CIBTech) 74

2 Inernaional Journal of Physics and Mahemaical Sciences ISSN: 77- (Online) 4 Vol. 4 () April-June, pp /Mondal and Mukhopadhyay maerial science, meallurgy, applied mahemaics ec. Now a day he effec of wo-emperaure has become an imporan area of research. According o Gurin and Williams (967) he second law of hermodynamics for coninuous bodies may involve wih win emperaures. In he heory of hermodynamics he emperaure caused by he hermal process is known as conducive emperaure φ and he emperaure due o mechanical process in he maerial is known as hermodynamic emperaure T. The heory of hea conducion depending on he above wo emperaures was originaed by Chen and Gurin (968). The propagaion of harmonic plane waves in he heory of wo-emperaure hermoelasiciy ware invesigaed by Puri and Jordan (6). Quinanilla (4) analyzed he exisence, srucural sabiliy, convergence and spaial behavior for he heory wo-emperaure hermo-elasiciy. By means of wo-emperaure generalized hermo-elasiciy Youssef and Al-Harby (7) explained he sae-space approach on an infinie body wih spherical caviy. Ailawilia e al. (9) invesigaed he deformaion of a roaing generalized hermoelasic medium wih wo emperaures under he influence of graviy subjeced o differen ype of sources. Banik and Kanoria () invesigaed he effecs of woemperaure on generalized hermo-elasiciy for infinie medium wih spherical caviy. The analysis of effecs of wo-emperaure in he maerial having emperaure dependen mechanical properies on generalized hermo-visco-elasic problem was discussed by Mondal and Mukhopadhyay (3). Shaw and Mukhopadhyay (3) discussed he moving hea source response in micropolar half-space wih woemperaure heory. Ezza and El-Karamany () esablished he model of one-dimensional equaions of he wo-emperaure generalized magneo-hermo-elasiciy heory wih wo relaxaion imes. Youssef () analyzed he effec of wo-emperaure in infinie medium under generalized hermoelasiciy wih cylindrical caviy. Various problems relaed o visco-elasiciy, done wihou aking he rheological densiy propery and also he mechanical properies which are aken are no emperaure dependen. The rheological properies of volume as well as densiy having emperaure dependen mechanical properies are being considered here for an infinie visco-elasic solid wih a cylindrical hole in he conex of he heory of generalized woemperaure magneo-hermo-elasiciy. The inverse of Laplace ransform for he differen expression is done numerically using he mehod adoped by Honig and Hirdes (984). Soluion for sresses, displacemen and emperaure are presened graphically wih respec o space and ime variables separaely. FORMULATION OF PROBLEM AND SOLUTION We consider a homogeneous, isoropic, perfecly conducing long hollow visco-elasic cylinder wih z- axis as he axis of he cylinder. We assume ha he whole body is siuaed a a consan applied magneic field H =,, H o aced along z-axis. I produce an induce magneic field h =,, h and a perurbed elecric field. For perfecly conducing slowly moving visco-elasic medium, he variaion of magneic field and elecric field are given by Maxwell equaions: curl H = J + D, () curl E = B, () D = ε E, (3) B = μ H, (4) div D =, (5) div B = (6) ogeher wih he generalized Ohm s law Copyrigh 4 Cenre for Info Bio Technology (CIBTech) 75

3 Inernaional Journal of Physics and Mahemaical Sciences ISSN: 77- (Online) 4 Vol. 4 () April-June, pp /Mondal and Mukhopadhyay u J = ς E + μ H (7) where H = h + H is magneic inensiy vecor, J is curren densiy vecor. ε, μ and ς are he elecric permiiviy, he magneic permeabiliy and he elecric conduciviy respecively. u is he displacemen vecor and B, D respecively are he magneic inducion and elecric displacemen vecor. The cylindrical polar co-ordinaes suppose o be aken as r, θ, z. Due o radial symmery he displacemen can be aken as u = u r,,, and he emperaure be funcions of r and. Therefore he srain componens ake he form as ε rr = u, ε θθ = u, ε r zz = and dilaaion is = u + u. The basic r equaions for wo-emperaure elecro-magneo-hermo-visco-elasic solid in he conex of generalized heory, where rheological properies of volume as well as densiy are considered, may be aken as: The equaion of moion ς rr + ς rr ς θθ + J B r r = ρ u. (8) The generalized equaion of hea conducion k φ = C E R 3 τ τ The consiuive equaions e ij T τ + τ T τ dτ + 3T α T R τ S ij = R τ dτ, () τ ς = R τ τ e 3α T T T + τ T dτ. () Relaion beween conducive emperaure φ and hermodynamic emperaure T φ T = a φ () τ e τ + τ e 3 dτ. (9) τ where a is wo-emperaure parameer (Youssf 6). Here e ij, S ij are he deviaoric pars of srain ensor and sress ensor respecively; T is he reference emperaure; e, k, C E are dilaaion, hermal conduciviy, specific hea a consan srain respecively; ρ, α T are he densiy and co-efficien of linear hermal expansion a absolue emperaure respecively; τ, τ, τ 3 are hermal relaxaion imes; R, R, R 3 are non-negaive relaxaion funcion, relaxaion funcion characerized by rheological properies of volume and densiy respecively. is Laplacian operaor. Depending upon he values of hermal relaxaion imes τ, τ, τ 3 he above problem can be reduced according as bellow. When τ = τ = τ 3 = he above one corresponds he problem wih classical heory. When τ =, τ = τ 3 he above one corresponds he problem wih generalized Lord-Shulman heory. When τ, τ, τ 3 = he above one corresponds he problem wih generalized Green-Lindsay heory. The relaxaion funcions are aken in he form Copyrigh 4 Cenre for Info Bio Technology (CIBTech) 76

4 Inernaional Journal of Physics and Mahemaical Sciences ISSN: 77- (Online) 4 Vol. 4 () April-June, pp /Mondal and Mukhopadhyay R = μ M R = K M R 3 = ρ M 3 g d g d g d. (3) The funcion g generally aken in he form g = e β α (Kolunov 976) where < α <, β >, M M 3 M < Γ α, <. Using equaions (4), (7) and (), () we have J B r = μ H u + u r μ ε H u, ς ij = R τ e ij τ dτ + δ ij R τ τ e 3α T T T + τ T dτ. (4) Afer eliminaing J wih he help of equaion (7), we have from equaion () and () h = ς u E μ H ε E, (5) r re = μ h. (6) In he above we consider he maerial having emperaure dependen mechanical properies which are in he form μ = μ ψ T, K = K ψ T, ρ = ρ ψ T, α T = α T ψ T (Lomakin 976) where ψ i T = α i T T r, i =,, and α i >, i =, and α < in which μ, K, ρ and α T are he Lame consan, bulk modulus, densiy, Co-efficien of linear hermal expansion a room emperaure T r. For more convenien form we use he following non-dimensional variables r = r, u = u, = c, τ a a a = c τ a, τ = c τ a, τ 3 = c τ a 3, h h =, a ς H μ c E = ς ij where E a ς H μ c, R = R 3K, R = R K, R 3 = R 3 ρ, T = γ T T ρ c, φ = γ φ T ρ c, = ς ij K c = λ + μ ρ, a = k c c E ρ, γ = 3K α T Using non-dimensional erms and aking Laplace ransform wih respec o he equaions (8), (9) and ()-(6), dropping he primes, ake he following forms as p R + R + ε L u r p m ψ R + τ p φ ω φ L r Copyrigh 4 Cenre for Info Bio Technology (CIBTech) 77

5 Inernaional Journal of Physics and Mahemaical Sciences ISSN: 77- (Online) 4 Vol. 4 () April-June, pp /Mondal and Mukhopadhyay = r p m ψ + c μ ε ε u, (7) L φ = r a φ + a 3L u, (8) φ T = ω φ, (9) R = β ψ p M π p+β R = ψ p M π p+β, () R 3 = ψ p M 3 π p+β ς rr = pr u u r ς θθ = pr u r u + pr u + u r m ψ + τ p (φ ω φ), () + pr u + u r m ψ + τ p (φ ω φ), () ς zz = pr u + u r + pr u + u r m ψ + τ p (φ ω φ), (3) h = pu δ + v p E, (4) L E = p r h. (5) Where bar denoes he Laplace ransform wih respec o, p is he parameer of Laplace ransform and ω = a a, β = 4μ, m = ρ c, ε = T γ a, ε 3K K kρ c = μ H, δ K = ς μ c a, v = ε μ c, a = p R 3 + τ p + p ωr 3 + τ p, a 3 = ε ψ p R + τ 3 p + p ωr 3 + τ p, L r + r. Solving equaions (7) and (8) and combining equaions (4), (5) we have L u = r a u + r a φ, (6) L φ = r a u + r a φ, (7) L h where = r a 3 u + r a 3 φ + r a 33 h (8) a = ψ m + v ε p, a q = R ψ m p + τ p ωa q, a = a 3 a, Copyrigh 4 Cenre for Info Bio Technology (CIBTech) 78

6 Inernaional Journal of Physics and Mahemaical Sciences ISSN: 77- (Online) 4 Vol. 4 () April-June, pp /Mondal and Mukhopadhyay a = a + a 3 a, a 3 = pa, a 3 = p a, a 33 = p δ + v p, q = p R + R + ε + mωpr ψ a 3 + τ p. Now he equaions (6), (7) and (8) in he form of vecor-marix differenial equaion may be wrien as L V = r A V (9) where u φ h a a V = ; A = a a. a 3 a 3 a 33 Soluion of equaion (9) may be wrien as V = 3 i= A i X i w i r, (3) where A i are he arbirary consans, X i are he eigenvecors of he marix A corresponding o he eigenvalue of he marix A and w i r, saisfies he modified Bessel s differenial equaion r d w dw + r dr dr r λ + w =. (3) Soluion of equaion (3) is w i r, = K r (3) where K r is he modified Bessel s funcion of second kind of order, so ha X i = a a a 3 a a 3 a a 33 ; a 33, i =, and X 3 λ 3 =, for λ 3 = a 33. (33) Therefore from (3) we have u = A i i= a K r, 34 φ = A i a i= 3 K r, (35) h = A i a 3a a 3 a i= a 33 3 λ K λ ir A 3 3 i λ K λ 3r (36) 3 where K r is he modified Bessel s funcion of second kind of order. Using (34)-(36) we have from equaions (9) and ()-(4) Copyrigh 4 Cenre for Info Bio Technology (CIBTech) 79

7 Inernaional Journal of Physics and Mahemaical Sciences ISSN: 77- (Online) 4 Vol. 4 () April-June, pp /Mondal and Mukhopadhyay T = i= A i a ω 3 K r, (37) ς rr = A i a p R + R K λ r + 3pR i rλ K r i i= Q, (38) p R ς θθ = A i a R K λ r 3pR i rλ K r i i= Q, (39) ς zz = A i a p R R i= K r Q, (4) E = δ + v p i= A i pa + a 3a a 3 a a 33 K r + A 3 λ 3 K λ 3r (4) where Q = R ψ m p + τ p A i a i= ω 3 K r. In free space, where J, if E and h represen he componen of elecric and induced magneic field inensiies in he direcion of θ and z respecively, hey saisfy he following non-dimensional equaions in ransform domain r re = ph, (4) h = v p E. (43) Wih he help of soluion of (4), he soluion of (43) becomes h = p v I pvr (44) where I is he modified Bessel funcion of firs kind of order. Boundary Condiions The ransverse componens of he elecric and induced magneic field inensiies are coninuous across he surface of he cylinder and hence E r, = E r, ; h r, = h r, for > a he surface of he hole. (45) For mechanical and hermal boundary condiions we consider he following wo cases: Case I) ς rr = H, φ = a he surface of he hole; Copyrigh 4 Cenre for Info Bio Technology (CIBTech) 8

8 Inernaional Journal of Physics and Mahemaical Sciences ISSN: 77- (Online) 4 Vol. 4 () April-June, pp /Mondal and Mukhopadhyay Case II) ς rr = δ, φ = a he surface of he hole. When he boundary condiions are used, a sysem of hree linear equaions in A, A, A 3 can be obained from he equaions (35), (38) and combinaion of (36) wih (44). Solving hese equaions, we achieve he values of A i for wo disinc cases. Case I) A = X, A X p Y X X Y =, A p Y X X Y 3 = I Z X X Z vp Z 3 pz 3 Y X X Y Case II) X A = Y X X Y, A = where X i = N i K i X Y X X Y, A 3 = I vp Z 3 Z X X Z Z 3 Y X X Y, Y i = a pr K i + 3pR K i N N i ω 3 K i, Z i = a 3a a 3 N i N 3i 3 K i, K i = K, K i = K, N i = a, N 3i = a 33 ; i =, and Z 3 = λ 3 3 K λ 3, N = R ψ m p + τ p, R = R + R, I = I pv. Equaions (34)-(4) ogeher wih he above derived values of A, A and A 3 for disinc cases provide he evenual soluions in ransform domain. Now for graphical represenaion i required he numerical informaion s. RESULTS AND DISCUSSION The expressions for displacemen, sress, emperaure and he fields can found ou numerically wih he help of above consequen values of A i. For infiniesimal emperaure deviaions from reference emperaure we can ake ψ i T = α i T T r ; i =,, such ha α >, α > & α < (Nowacki 959). To sudy he behavior of he quaniies in deails and wih he inenion of demonsraing he oucomes obain in he above we aemp o achieve he numerical values of he differen characerisic parameers of he maerial. For execuion of he graphical represenaion we ake for graned he numerical values for a magnesium crysal-like maerial as (Ezza e al., ) ρ =.74 3 kg m 3, C E = J K kg, k = 56W K m, λ = N m, μ = 58 7 N m, λ + μ = N m, α T o = 5. 6 K, T o = 98K, K = N m, γ = 3λ + μ α T o = N m K. The above considered numerical values imply m =.4443, ε =.37, β = For he graphical evaluaion he oher consans in his paper may be aken as β =.5, α =, M =.6, M = M 3 =.8, ψ =.8, ψ =.9, ψ =.5, ε =.5, δ =.8, v = To ge he soluion for hermal displacemen, emperaure, sress and he fields we apply inverse Laplace ransform numerically using he mehod based on Honig and Hirdes (984) o he equaions (34)-(4). The resuls of he presen invesigaions are given in form of figures for differen values of ime, radial disance and wo-emperaure parameer (emperaure discrepancy). Moreover we draw he graphs considering emperaure independen mechanical properies (TIMP) and considering emperaure Copyrigh 4 Cenre for Info Bio Technology (CIBTech) 8

9 Inernaional Journal of Physics and Mahemaical Sciences ISSN: 77- (Online) 4 Vol. 4 () April-June, pp /Mondal and Mukhopadhyay dependen mechanical properies (TDMP) and in addiion comparison are also made considering wih rheological densiy propery (WDP) and considering wihou rheological densiy propery (WODP). In paricular cases when we choose, a = ha is ω =, our considered problem reduced o one, relaed o unique-emperaure. M 3 =, our considered problem reduced o one, relaed o null rheological densiy propery. M =, our considered problem reduced o one, relaed o null rheological volume propery. M 3 = and M =, our considered problem reduced o one, relaed o null rheological densiy propery as well as null rheological volume propery. ψ = ψ = ψ =, our considered problem reduced o one, relaed o emperaure independen mechanical properies. ψ = ψ = ψ =, ω =, our considered problem reduced o one, relaed o emperaure independen mechanical properies and unique-emperaure. ψ = ψ = ψ =, M 3 =, our considered problem reduced o one, relaed o emperaure independen mechanical properies and null rheological densiy propery. ψ = ψ = ψ =, M 3 =, M =, our considered problem reduced o one, relaed o emperaure independen mechanical properies and null rheological densiy propery as well as null rheological volume propery. ψ = ψ = ψ =, M 3 =, M =, ω =, our considered problem reduced o one, relaed o emperaure independen mechanical properies and null rheological densiy propery as well as null rheological volume propery wih unique-emperaure. Figure: Curves represenaion Figure: Plo hermodynamic emperaure vs ime Figure 3: Plo conducive emperaure vs ime Figure 4: Plo conducive emperaure vs radius Copyrigh 4 Cenre for Info Bio Technology (CIBTech) 8

10 Inernaional Journal of Physics and Mahemaical Sciences ISSN: 77- (Online) 4 Vol. 4 () April-June, pp /Mondal and Mukhopadhyay Figure 5: Plo displacemen vs ime Figure 6: Plo displacemen vs radius Figure 7: Plo elecric field vs ime Figure 8: Plo elecric field vs radius Figure 9: Plo perurbed magneic field vs ime Figure : Plo perurbed magneic field vs radius Figure : Plo sress componen vs ime Figure : Plo sress componen vs ime Copyrigh 4 Cenre for Info Bio Technology (CIBTech) 83

11 Inernaional Journal of Physics and Mahemaical Sciences ISSN: 77- (Online) 4 Vol. 4 () April-June, pp /Mondal and Mukhopadhyay Figure 3: Plo sress componen vs ime Here he figures represen he variaion of hermodynamic emperaure, conducive emperaure, displacemen, elecric field, perurbed magneic field and sresses componens when coninuous radial sress is applied on he hermally insulaed boundary and all he graphs are shown in view of wo relaxaion imes. Figure sands for represenaion of curves for remaining all graphs. The variaions of hermodynamic emperaure verses ime in he presence of rheological densiy propery, as well as wihou i, are ploed in he figure and he assessmen is made beween TDMP and TIMP. The effec of wo-emperaure parameer is clearly seen from his figure. A a disance from he verical axis all ype of curves produce he noable variaions. I is observed from his figure ha for increasingly fixed values of r, he deviaion gradually decreases. Figure 3 and figure 4 describe he variaions of conducive emperaure wih respec of ime variable and radial disance respecively. I is observed from boh hese figures ha he conducive emperaure akes differen values in he presence of densiy propery and woemperaure parameer and wihou hem. From figure 4 i is found ha he curves for TDMP always provide lower values han ha of he curves for TIMP and seadily converge o zero. Figure 5 depics he variaion of displacemen versus ime. For fixed value of r, a lile change in displacemen is observed a he iniial sage and afer a cerain ime-pass i gradually decreases and hen converges o a cerain value. From his figure i is seen ha when ime increases he curves for TDMP and TIMP converge nearly parallel o each oher and he naure of he curves jus change for WDP and WODP. I is also observed ha he iniial ime-pass increases wih he increasing value of r. Figure 6 shows he variaion of displacemen wih respec o radial disance. I is found from he figure ha for fixed he magniude of displacemen decreases wih he increasing value of he radial disance and ha i approaches zero a a disance far from he boundary of he hole. Figures 7-8 represen he disribuions of elecric field and figures 9- represen he disribuions of perurbed magneic field when coninues radial sress is applied a he boundary of he hole. I is seen from boh he figures 7 and 9 ha as ime increase boh he fields decreases and he magniude of he fields are inversely proporional wih radial disance. Figures 8 and show ha for fixed values of, elecric field and perurbed magneic field increase wih he radial disance and afer he raverse of cerain disance he seady sae condiion arrives and gradually converges o zero. Figures -3 illusrae he sress disribuions wih respec o ime for WDP and WODP as well as aking TDMP and TIMP. Afer a cerain span of ime i is found from he figures ha he sress componens expose a sudden incremens and hen decrease wih ime. I is noiced ha he magniude of he sudden incremens decreases wih he increasing value of he radial disance. Conclusions In he presence of uniform magneic field and in he conex of classical heory and generalized heory, he governing equaions of hermo-visco-elasiciy wih wo-emperaure have been invesigaed wih he effec of emperaure dependen mechanical properies and rheological densiy propery. The leading equaions have been derived for he cases when coninuous radial sress and insananeous radial sress applied on he hermally insulaed boundary. Eigen value approach has been used o solve he vecormarix-differenial equaion. We conclude ha he magniude of he displacemen for he curve TIMP is Copyrigh 4 Cenre for Info Bio Technology (CIBTech) 84

12 Inernaional Journal of Physics and Mahemaical Sciences ISSN: 77- (Online) 4 Vol. 4 () April-June, pp /Mondal and Mukhopadhyay lesser hen ha of he curve for TDMP. In he presence of emperaure discrepancy (ω =.7) and rheological densiy propery he magniude of he displacemen increases. The conducive emperaure and he hermodynamic emperaure exhibi he remarkable changes on behalf of various siuaions. The elecric field and perurbed magneic field ake he values wih very small dispariy for differen condiions. Sress componens provide he similar naure as i was for null emperaure discrepancy and for void rheological densiy propery. REFERENCES Ailawalia P, Khurana G and Kumar S (9). Effec of roaion in a generalized hermoelasic medium wih wo emperaure under he influence of graviy. Inernaional Journal of Applied Mahemaics and Mechanics 5(5) Aouadi M and El-Karamany AS (4). The relaxaion effecs of volume properies in wodimensional generalized hermoviscoelasic problem. Applied Mahemaics and Compuaion Banik S and Kanoria M (). Two-Temperaure Generalized Thermoelasic Ineracions in an Infinie Body wih a Spherical Caviy. Inernaional Journal of Thermophysics Chen PJ and Gurin ME (968). On a heory of hea conducion involving wo emperaures. Zeischrif für Angewande Mahemaik und Physik 9(4) Drozdov AD (996). A consiuive model in hermo-viscoelasiciy. Mechanics Research Communicaions Ezza MA and El-Karamany AS (). Two-emperaure heory in generalized magneohermoelasiciy wih wo relaxaion imes. Meccanica Ezza MA, Zakaria M and El-Karamany AS (). Effecs of modified Ohm s and Fourier s laws on generalized magneoviscoelasic hermoelasiciy wih relaxaion volume properies. Inernaional Journal of Engineering Science Giorgi C and Naso MG (6). Mahemaical models of Reissner Mindlin hermoviscoelasic plaes. Journal of Thermal Sresses Green AE and Lindsay KA (97). Thermoelasiciy. Journal of Elasiciy -7. Gurin ME and Williams WO (967). An axiom foundaion for coninuum hermodynamics. Archive for Raional Mechanics and Analysis Honig G and Hirdes U (984). A mehod for he numerical inversion of Laplace ransforms. Journal of Compuaional and Applied Mahemaics 3-3. Kolunov MA (976). Creeping and Relaxaion (Izd. Vishaya, Shkola, Moscow). Kundu MR and Mukhopadhyay B (5). A Thermoviscoelasic Problem of an Infinie Medium wih a Spherical Caviy Using Generalized Theory of Thermoelasiciy. Mahemaical and Compuer Modelling Lomakin VA (976). The Theory of Elasiciy of Non-Homogeneous Bodies (Moscow Sae Universiy Press, Moscow). Lord H and Shulman Y (967). A generalized dynamical heory of hermoelasiciy. Journal of he Mechanics and Physics of Solids Mondal MK and Mukhopadhyay B (3). A cylindrical problem wih rheological volume, densiy propery on hermo-visco-elasic medium in magneic field. Inernaional Journal of Applied Mahemaics and Mechanics 9() -38. Mondal MK and Mukhopadhyay B (3). Effec of wo-emperaure on hermoviscoelasiciy problem wih rheological properies. Inernaional Journal of Physics and Mahemaical Sciences 3() Nowacki W (959). Some Dynamic Problems of Thermoviscoelasiciy. Archiwum Mechaniki Sosowane 59. Copyrigh 4 Cenre for Info Bio Technology (CIBTech) 85

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