Fractional heat conduction with f inite wave speed in a thermo-visco-elast ic spherical shel l

This problem deals with the thermo-elastic interaction due to step input of temperature on the stress free boundaries of a homogeneous visco-elastic orthotropic spherical shell in the context of a new consideration of heat conduction with fractional order generalized thermoelasticity. Using the Laplace transformation, the fundamental equations have been expressed in the form of a vector-matrix differential equation which is then solved by eigen value approach and operator theory analysis. The inversion of the transformed solution is carried out by applying a method of Bellman et al (1966). Numerical estimates for thermophysical quantities are obtained for copper like material for weak, normal and strong conductivity and have been depicted graphically to estimate the effects of the fractional order parameter. Comparisons of the results for different theories (TEWED (GN-III), three-phase-lag model) have also been presented and the effect of viscosity is also shown. When the material is isotropic and outer radius of the hollow sphere tends to infinity, the corresponding results agree with that of existing literature. Keywords Generalized thermo-visco-elasticity, Three-phase-lag model, Fractional order heat equation, Eigen value approach, Vector-matrix differential equation, Step input temperatures. Fractional heat conduction with f inite wave speed in a thermo-visco-elast ic spherical shel l 1 INTRODUCTION Linear viscoelasticity has been an important area of research since the period of Maxwell, Boltzman, Voigt and Kelvin. Valuable information regarding linear viscoelasticity theory may be obtained in the books of Gross (1953), Staverman and Schwrzl, Alfery and Gurnee, Ferry, Bland and Lakes. Many researchers like Biot (1954, 1955), Gurtin and Sternberg, Liioushin and Pobedria, Tanner, Huilgol and Phan-Thein have contributed notably on thermoviscoelasticity. Freudenthal has pointed out that most of the solids, when subjected to dynamic loading, exhibit viscous effects. The Kelvin-Voigt model is one of the macroscopic mechanical models often used to describe the viscoelastic behavior of a material. The model represents the delayed elastic response subjected to A. Sur M. Kanoria * Department of Applied Mathematics, University of Calcutta, India b Department of Applied Mathematics, University of Calcutta, India Author e-mail: k_mri@yahoo.com M. Kanoria et al./ Fractional heat conduction with finite wave speed in a thermo-visco-elastic spherical shell 1133 Latin American Journal of Solids and Structures 11 (2014) 1132-1162 stress when the deformation is time dependent but recoverable. The dynamic interaction of thermal and mechanical fields in solids has great practical applications in modern aeronautics, astronautics, Nuclear reactors and high-energy particle accelerators. Several researchers are working in this field. A problem involving Two-Temperature Magneto-Viscoelasticity with thermal Relaxation time in Perfect conducting medium have been solved by Ezzat and El-Karamany (2009). A two temperature thermo-electro-viscoelastic problem subjected to modified Ohm's and Fourier's Laws have been solved by Ezzat et al. (2012). The classical theories of thermoelasticity involving infinite speed of propagation of thermal signals, contradict physical facts. During the last five decades, non-classical theories involving finite speed of heat transportation in elastic solids have been developed to remove the paradox. In contrast with the conventional coupled thermoelasticity theory, which involves a parabolic-type heat transport equation, these generalized theories involving a hyperbolic-type heat-transport equation are supported by experiments exhibiting the actual occurrence of wave-type heat transport in solids, called second sound effect. The first generalization to this theory is due to Lord and Shulman (1967) who formulated the generalized thermoelasticity theory involving one thermal relaxation time, which is known as extended thermo-elasticity theory (ETE). The second generalization to the coupled thermoelasticity theory due to Green and Lindsay (1972), involves two relaxation times. The third generalization to the coupled thermoelasticity theory is known as low-temperature thermoelasticity introduced by Hetnarski and Ignaczak called the H-I theory. This model is characterized by a system of non-linear field equations. The fourth generalization in concerned with the thermo-elasticity without energy dissipation (TEWOED) and thermoelasticity with energy dissipation (TEWED) introduced by Green and Naghdi (1991, 1992, 1993) and provide sufficient basic modifications in the constitutive equations that permit treatment of a much wider class of heat flow problems, labeled as types I, II, III. The natures of these three types of constitutive equations are such that when the respective theories are linearized, type-I is same as the classical heat equation (based on Fourier’s law) whereas types II and III permit propagation of thermal signals at a finite speed. When Fourier conductivity is dominant the temperature equation reduces to classical Fourier’s law of heat conduction and when the effect of conductivity is negligible, the equation has undamped thermal wave solutions without energy dissipation. Applying the above theories of generalized thermoelasticity, several problems have been solved by Mallik and Kanoria (2008), Kar and Kanoria (2009), Islam and Kanoria (2011), Ghosh and Kanoria (2010), Banik and Kanoria (2011). Recently Roychoudhury (2007) has established a generalized mathematical model of a coupled thermoelasticity theory that includes three-phase lags in the heat flux vector, the temperature gradient and in the thermal displacement gradient. The more general model established reduces to the previous models as special cases. According to this model ! q = − K ! ∇T (P,t +τT )+ K ★ ! ∇ν(P,t +τν ) ⎡⎣ ⎤⎦ , where ! ∇ν ( " ν = T ) is the thermal displacement gradient and K ★ is the additional material constant. To study some practical relevant problems, particularly in heat transfer problems involving very short time intervals and in the problems of very high heat fluxes, the hyperbolic equation gives significantly different results than the parabolic equation. According to this phenomenon the lagging behavior in the heat conduction in solid should not be ignored particularly when the elapsed times during a transient process are very small, say about 7 10 s or the heat flux is very much high. 1134 M. Kanoria et al./Fractional heat conduction with finite wave speed in a thermo-visco-elastic spherical shell Latin American Journal of Solids and Structures 11 (2014) 1132-1162 Three-phase-lag model is very useful in the problems of nuclear boiling, exothermic catalytic reactions, phonon-electron interactions, phonon-scattering etc., where the delay time q τ captures the thermal wave behavior (a small scalar response in time), the phase-lag T τ captures the effect of phonon-electron interactions (a microscopic response in space), the other delay time ν τ is effective since, in the three-phase-lag model, the thermal displacement gradient is considered as a constitutive variable whereas in the conventional thermoelasticity theory temperature gradient is considered as a constitutive variable. Banik and Kanoria (2012) have solved the effect of three-phase-lag in an infinite medium with a spherical cavity. The magneto-thermo-elastic responses in a perfectly conducting medium under three-phase-lag model have been studied by Das and Kanoria (2012). However, over the last few decades, anisotropic materials have been increasingly used. There are materials which have natural anisotropy such as zinc, magnesium, sapphire, wood, some rocks and crystals, and also there are artificially manufactured materials such as fiber-reinforced composite materials which exhibit anisotropic character. The advantage of composite materials over the traditional materials lies on their valuable strength, elastic and other properties (1980). A reinforced material may be regarded to some order of approximation, as homogeneous and anisotropic elastic medium having a certain kind of elastic symmetry depending on the symmetry of reinforcement. Some glass fibre reinforced plastics may be regarded as transversely isotropic. Thus, problems of solid mechanics should not be restricted to the isotropic medium only. Increasing use of an anisotropic media demand that the study of elastic problems should be extended to anisotropic medium also. Differential equations of fractional order have been the focus of many studies due to their frequent appearance in various applications in fluid mechanics, viscoelasticity, biology, physics and engineering. The most important advantage of using fractional differential equations in these and other applications is their non-local property. It is well known that the integer order differential operator is a local operator but the fractional order differential operator is non-local. This means that the next state of a system depends not only upon its current state but also upon all of its historical states. This is more realistic, and this is one reason why fractional calculus has become more and more popular (1967, 1997, 1999). Fractional calculus has been used successfully to modify many existing models of physical processes. One can state that the whole theory of fractional derivatives and integrals was established in the second half of the nineteenth century. The first application of fractional derivatives was given by Abel who applied fractional calculus in the solution of an integral equation that arises in the formulation of the Tautochrone problem. The generalization of the concept of derivative and integral to a non-integer order has been subjected to several approaches, and some various alternative definitions of fractional derivatives appeared in Refs. (1974, 1997, 2000). In the last few years, fractional calculus was appli


INTRODUCTION
Linear viscoelasticity has been an important area of research since the period of Maxwell, Boltzman, Voigt and Kelvin.Valuable information regarding linear viscoelasticity theory may be obtained in the books of Gross (1953), Staverman and Schwrzl, Alfery and Gurnee, Ferry, Bland and Lakes.Many researchers like Biot (1954Biot ( , 1955)), Gurtin and Sternberg, Liioushin and Pobedria, Tanner, Huilgol and Phan-Thein have contributed notably on thermoviscoelasticity.Freudenthal has pointed out that most of the solids, when subjected to dynamic loading, exhibit viscous effects.
The Kelvin-Voigt model is one of the macroscopic mechanical models often used to describe the viscoelastic behavior of a material.The model represents the delayed elastic response subjected to Latin American Journal of Solids and Structures 11 (2014) 1132-1162 stress when the deformation is time dependent but recoverable.The dynamic interaction of thermal and mechanical fields in solids has great practical applications in modern aeronautics, astronautics, Nuclear reactors and high-energy particle accelerators.Several researchers are working in this field.A problem involving Two-Temperature Magneto-Viscoelasticity with thermal Relaxation time in Perfect conducting medium have been solved by Ezzat and El-Karamany (2009).A two temperature thermo-electro-viscoelastic problem subjected to modified Ohm's and Fourier's Laws have been solved by Ezzat et al. (2012).
The classical theories of thermoelasticity involving infinite speed of propagation of thermal signals, contradict physical facts.During the last five decades, non-classical theories involving finite speed of heat transportation in elastic solids have been developed to remove the paradox.In contrast with the conventional coupled thermoelasticity theory, which involves a parabolic-type heat transport equation, these generalized theories involving a hyperbolic-type heat-transport equation are supported by experiments exhibiting the actual occurrence of wave-type heat transport in solids, called second sound effect.The first generalization to this theory is due to Lord and Shulman (1967) who formulated the generalized thermoelasticity theory involving one thermal relaxation time, which is known as extended thermo-elasticity theory (ETE).The second generalization to the coupled thermoelasticity theory due to Green and Lindsay (1972), involves two relaxation times.
The third generalization to the coupled thermoelasticity theory is known as low-temperature thermoelasticity introduced by Hetnarski and Ignaczak called the H-I theory.This model is characterized by a system of non-linear field equations.The fourth generalization in concerned with the thermo-elasticity without energy dissipation (TE-WOED) and thermoelasticity with energy dissipation (TEWED) introduced by Green and Naghdi (1991, 1992, 1993) and provide sufficient basic modifications in the constitutive equations that permit treatment of a much wider class of heat flow problems, labeled as types I, II, III.The natures of these three types of constitutive equations are such that when the respective theories are linearized, type-I is same as the classical heat equation (based on Fourier's law) whereas types II and III permit propagation of thermal signals at a finite speed.When Fourier conductivity is dominant the temperature equation reduces to classical Fourier's law of heat conduction and when the effect of conductivity is negligible, the equation has undamped thermal wave solutions without energy dissipation.Applying the above theories of generalized thermoelasticity, several problems have been solved by Mallik and Kanoria (2008), Kar and Kanoria (2009), Islam and Kanoria (2011), Ghosh and Kanoria (2010), Banik and Kanoria (2011).
Recently Roychoudhury (2007) has established a generalized mathematical model of a coupled thermoelasticity theory that includes three-phase lags in the heat flux vector, the temperature gradient and in the thermal displacement gradient.The more general model established reduces to the previous models as special cases.According to this model where !∇ν ( " ν = T ) is the thermal displacement gradient and K ★ is the additional material constant.
To study some practical relevant problems, particularly in heat transfer problems involving very short time intervals and in the problems of very high heat fluxes, the hyperbolic equation gives significantly different results than the parabolic equation.According to this phenomenon the lagging behavior in the heat conduction in solid should not be ignored particularly when the elapsed times during a transient process are very small, say about 7 10 − s or the heat flux is very much high.
Latin American Journal of Solids and Structures 11 (2014) 1132-1162 Three-phase-lag model is very useful in the problems of nuclear boiling, exothermic catalytic reactions, phonon-electron interactions, phonon-scattering etc., where the delay time q τ captures the thermal wave behavior (a small scalar response in time), the phase-lag T τ captures the effect of pho- non-electron interactions (a microscopic response in space), the other delay time ν τ is effective since, in the three-phase-lag model, the thermal displacement gradient is considered as a constitutive variable whereas in the conventional thermoelasticity theory temperature gradient is considered as a constitutive variable.Banik and Kanoria (2012) have solved the effect of three-phase-lag in an infinite medium with a spherical cavity.The magneto-thermo-elastic responses in a perfectly conducting medium under three-phase-lag model have been studied by Das and Kanoria (2012).However, over the last few decades, anisotropic materials have been increasingly used.There are materials which have natural anisotropy such as zinc, magnesium, sapphire, wood, some rocks and crystals, and also there are artificially manufactured materials such as fiber-reinforced composite materials which exhibit anisotropic character.The advantage of composite materials over the traditional materials lies on their valuable strength, elastic and other properties (1980).A reinforced material may be regarded to some order of approximation, as homogeneous and anisotropic elastic medium having a certain kind of elastic symmetry depending on the symmetry of reinforcement.Some glass fibre reinforced plastics may be regarded as transversely isotropic.Thus, problems of solid mechanics should not be restricted to the isotropic medium only.Increasing use of an anisotropic media demand that the study of elastic problems should be extended to anisotropic medium also.
Differential equations of fractional order have been the focus of many studies due to their frequent appearance in various applications in fluid mechanics, viscoelasticity, biology, physics and engineering.The most important advantage of using fractional differential equations in these and other applications is their non-local property.It is well known that the integer order differential operator is a local operator but the fractional order differential operator is non-local.This means that the next state of a system depends not only upon its current state but also upon all of its historical states.This is more realistic, and this is one reason why fractional calculus has become more and more popular (1967,1997,1999).
Fractional calculus has been used successfully to modify many existing models of physical processes.One can state that the whole theory of fractional derivatives and integrals was established in the second half of the nineteenth century.The first application of fractional derivatives was given by Abel who applied fractional calculus in the solution of an integral equation that arises in the formulation of the Tautochrone problem.The generalization of the concept of derivative and integral to a non-integer order has been subjected to several approaches, and some various alternative definitions of fractional derivatives appeared in Refs.(1974,1997,2000).In the last few years, fractional calculus was applied successfully in various areas to modify many existing models of physical processes, e.g., chemistry, biology, modeling and identification, electronics, wave propagation and viscoelasticity (1971,1974,1983,1984,1997).One can refer to Padlubny (1999) for a survey of applications of fractional calculus.
Recently, a considerable research effort is expended to study anomalous diffusion, which is characterized by the time-fractional diffusion-wave equation by Kimmich (2002) where ρ is the mass density, c is the concentration, κ is the diffusion conductivity, i the coordinate symbol, which takes the value 1, 2, 3.The notation I ξ is the Riemann-Liouville fractional integral, introduced as a natural generalization of the well-known n-fold repeated integral ( ) n I f t written in a convolution-type form as in (2000).Youssef (2010) introduced another formula of heat conduction in the following form and a uniqueness theorem has also been proved.
Ezzat established a new model of fractional heat conduction equation by using the new Taylor series expansion of time-fractional order, developed by Jumarie (2010) as El-Karamany and Ezzat (2011) introduced two general models of fractional heat conduction law for a non-homogeneous anisotropic elastic solid.Uniqueness and reciprocal theorems are proved, and the convolutional variational principle is established and used to prove a uniqueness theorem with no restriction on the elasticity or thermal conductivity tensors except symmetry conditions.For fractional thermoelasticity not involving two-temperatures, El-Karamany and Ezzat (2011) established the uniqueness, reciprocal theorems and convolution variational principle.The dynamic coupled and Green-Naghdi thermoelasticity theories result as limit cases.The reciprocity relation in case of quiescent initial state is found to be independent of the order of differintegration.Fractional order theory of a perfect conducting thermoelastic medium not involving two temperatures was investigated by Ezzat and El-Karamany (2011).Thermal wave propagation in an infinite half-space under fractional order Green-Naghdi theory was studied by Sur and Kanoria (2012).
To the authors' knowledge, under three-phase-lag effect, no solution of visco-elastic orthotropic materials for fractional heat conduction equation has been reported.With this motivation in mind the present analysis is to study the thermoelastic stresses, displacement and temperature distribution in a orthotropic hollow sphere in the context of GN-III and three-phase-lag model of generalized thermoelasticity where the heat equation consists of some non-local fractional operator signifying not only the present state, but also the previous states due to sudden temperature change on the stress-free boundaries.The governing equations are formed in Laplace transform domain which is then solved by eigen-value approach and operator theory analysis.The inversion of the transformed solution are carried out numerically applying the method of Bellman et al.A comprehensive analysis of the result have been presented for 3P model and GN-III model for both viscous and nonviscous isotropic materials.The effect of the fractional order parameter is also discussed.

FORMULATION OF THE PROBLEM
We consider a homogeneous orthotropic thermo-visco-elastic spherical shell of inner radius a and outer radius b in an undisturbed state and initially at uniform temperature T 0 .We introduce spherical polar coordinates (r,θ,φ) with the center of the cavity at the origin as shown in Figure a.
We consider spherically symmetric thermal problem so that the displacement component !u = [u(r,t),0,0] and the temperature T are assumed to be functions of r and t only.The stress-strain-temperature relations in the present problem are (Kelvin-Voigt type) and the generalized heat conduction equation for fractional order three-phase-lag model is where τ ij (i, j = r,θ,φ) are the stress tensor, T is the temperature increase over the reference tem- perature T 0 , C ij (i, j = 1,2,3) are the elastic constants, β i (i = r,θ,φ) are the thermal moduli, K r is the coefficient of thermal conductivity along the radial direction, K r ★ is the additional material constant along the radial direction, ρ is the mass density, C e is the specific heat of the solid at constant strain, 0 t is the mechanical relaxation time, τ T and τ q are the phase-lag of temperature gra- dient and the phase-lag of the heat flux respectively.Also τ ν ★ where τ ν is the phase-lag of thermal displacement gradient.
The stress equation of motion in spherical polar co-ordinate is given by Introducing the following non-dimensional quantities Equations ( 4)-( 8) become Latin American Journal of Solids and Structures 11 (2014) 1132-1162 where and are dimensionless constants, ε being the thermoe- lastic coupling constant.Where C T is the non-dimensional thermal wave velocity and C K is the damping co-efficient.
The boundary conditions are given by where χ 1 and χ 2 are dimensionless constants, and H (η) is the Heaviside unit step function.The above condition indicate that for time η P η 1 0 there is no temperature (Θ = 0) on the inner boundary and for η # 0 there is no temperature (Θ = 0) on the outer boundary.Thermal shocks are given on the boundaries of the shell ( R = 1, S ).Thermal stresses in the elastic medium due to the application of these thermal shocks are calculated.We assume that the medium is at rest and undisturbed initially.
The initial and the regularity conditions can be written as Latin American Journal of Solids and Structures 11 (2014) 1132-1162 with Re( p) > 0 denote the Laplace transform of U and Θ respectively.
Since we have On taking Laplace transform, equations ( 12) and ( 13) reduce to and where Differentiating equation ( 21) with respect to R and using equation ( 22), we get Equations ( 22) and ( 23) can be written in the form and Latin American Journal of Solids and Structures 11 (2014) 1132-1162 where, we assume that A = 2, ε is the thermo-elastic coupling constant and From equations ( 24) and ( 25), we have the vector-matrix differential equation as follows where and where m is a scalar, !X is a vector depending on R and ω (R,m) is a non-trivial solution of the scalar differential equation Let ω = R −1/2 ω 1 .Therefore, from equation (30) we have The solution of equation ( 30) is Latin American Journal of Solids and Structures 11 (2014) 1132-1162 Using equation ( 29) and (30) into equation ( 27) we get where !X (m) is the eigen vector corresponding to the eigen value m 2 .
The characteristic equation corresponding to !A can be written as The roots of the characteristic equation ( 37) are of the form Equation ( 34) can be written as Therefore, the positive roots of the equation ( 36) are where Therefore, m 1 and m 2 are real positive quantities.
The eigen vectors X (m j ), j = 1,2 corresponding to the eigen values m j 2 , j = 1,2 can be calculated as Therefore, from equation ( 29) and using equation ( 28) we get and where I 3/2 (m i R) and K 3/2 (m i R) are the modified Bessel functions of order 3/ 2 of first and second kind respectively.A i 's and B i 's (i = 1,2) are independent of R but dependent of p and are to be determined from the boundary conditions.Using the recurrence relations of modified Bessel functions we obtain from equation ( 41) where P = I, K .Taking Laplace transform on the equations ( 9), ( 10) and ( 11) we get where Using the recurrence relations (Watson, 1980) from equations ( 42) and ( 44) we obtain   Hence, from ( 45) and ( 46), we can write Also for an isotropic material, ( ) and for a non-viscous material, we have 0 0. t = Hence, 4 1. a = Thus, for an isotropic material, equations ( 45) and ( 46) reduce to Moreover, for large value of b i.e., for large value of S , 0 ( ) i K m S and 1 ( ) i K m S tend to zero.
Thus we have Therefore, for an infinitely extended body ) The results agree with those of Kar and Kanoria (2007) for GN III model.
Latin American Journal of Solids and Structures 11 (2014) 1132-1162 6 OPERATOR THEORY ANALYSIS Equations ( 12) and ( 13) can be expressed in the following form where Taking the Laplace transform, we have ( ) Where ( ) ( ) Operating 1 D D on (64) and using (65) we have Similarly, operating 1 DD on (65) and using (64) we have As the solution of equation ( 66) and ( 67) we have and Where ( ) K m R are the modified Bessel functions of order j of first and second kind respectively; i A , i B , i C and i D are independent of R but dependent on .
p Therefore, substituting the expressions of U and Θ in equation ( 65), we get Therefore Equations ( 73) and ( 74) are the same as that of equations ( 40) and (42) (i.e., the solutions obtained by Eigen-value approach).

NUMERICAL RESULTS AND DISCUSSIONS
To get the solutions for displacement, temperature distribution and stresses in the space-time domain we have to apply Laplace inversion formula to the equations ( 40), ( 42), ( 44) and (45) respectively, which have been done numerically using the method of Bellman et al. (1966) for fixed value of the space variable and for , where i η 's are computed from roots of the shifted and the hypothetical values of the relaxation time parameters are taken as 7 7 7 7 0 1.0 10 sec, 2.0 10 sec, 1.5 10 sec, 1.0 10 sec Here, in this article we have considered three-phase-lag model.Now, for this model, the solution i.e., the stability condition of Quintanilla and Racke is verified (2008).Also, for an isotropic material, the physical data are taken as (www.matweb.com).
In case of GN theory, K ★ is an additional material constant depending on the material.For copper like material, we take . 4 tively for 3P and GN III models.In these figures, the magnitudes of the variation of stresses and temperature are observed for viscous material when the step-input temperatures are applied on the inner boundary 1 R = and outer boundary 4 S = of the hollow sphere.Figures 1 and 2 depict the variation of the radial stress against the radial distance R of the sphere.From the figures it is observed that the radial stress ( ) R σ vanishes at the inner boundary ( 1) R = and the outer boundaries ( 4) R = of the shell which satisfy our theoretical boundary conditions.The magnitude of the radial stress is maximum near for a strong conductive material and for GN III model.Also, for three-phase-lag model, the oscillatory nature is observed.This is due to the presence of the oscilla-= 0.0257750, 0.138382, 0.352509, 0.693147, 1.21376,  σ vanishes at the boundaries of the shell where there are thermal sources which agree with our theoretical boundary conditions.As may seen from the figure, the stress wave is compressive in nature near both the boundaries.Also, at earlier stage of wave propagation, both the models give close results, whereas with advancement of time time, the stress wave is propagating with different speeds.For        Figures 15 and 16 are plotted to show the effect of viscosity on temperature Θ for two sets of time.For both viscous and non-viscous material, the temperature satisfies our thermal boundary conditions.Also, the effect of viscosity is very prominent in earlier situations than latter.As may seen from the figures, when 1.21 η = , for 3P lag model, the magnitude of Θ is larger for viscous material compared to the non-viscous material.Whereas for 0.026 η = , the magnitude is larger for nonviscous material compared to the viscous material.α = From these figures, it is seen that at the beginning of time, oscillatory natures are seen in the propagation of the stress components.Finally they reach to a steady state which supports the physical fact.Figure 21 depicts the variation of θ σ versus R for isotropic and orthotropic materials.As may be seen from the figure that for an isotropic material, the oscillatory nature is observed due to the reflection as mentioned earlier.However, the magnitude of θ σ is maximum near the outer boundary of the shell for an isotropic material.than that of 1.0 α = for an orthotropic material.As may seen from the figure, oscillatory behavior is seen near the boundaries for an isotropic material.It is seen that for isotropic material, when 2.5 R = , i.e., at the surface equidistant from the boundaries, Θ almost disappears at the primary stage of thermal load application.

CONCUSIONS
The problem of investigating the radial stress, hoop stress, temperature in a homogeneous isotropic viscoelastic spherical shell is studied in the light of three-phase-lag model and GN-III model in the context of space-fractional heat conduction equation.The method of Laplace Transform is used to write the basic equations in the form of a vector-matrix differential equation which is then solved by eigen-value approach.The numerical inversion of Laplace Transform is computed by the method of Bellmen.The analysis of the result permits some concluding remarks: (ii) It is observed that maximum magnitude of stresses will occur for viscous material and for strong conductivity ( 1.2) α = .
(iii) For an isotropic material, the maximum temperature occurs near the boundaries of the shell and it almost disappears in the interior of the shell.(iv) The effect of R σ is more prominent near the inner boundary for orthotropic material compared to that of an isotropic material.
( , ) ( , ) As the matrix is the product of { } i diag W multiplied by Vander monde matrix, it can be shown that the matrix is non-singular.

★
The results of the numerical evaluation of the thermo-elastic stress variations and temperature distribution are illustrated in figures 1-8 for both large time ( 1.21) ofSolids and Structures 11 (2014) 1132-1162    tion term in the heat equation of three-phase-lag model.For weak conductivity ( 0also seen for GN III model.

Figures 5
Figure 2 Figure 5 Figure 6 φ σ versus R for Figure 8 Θ versus R for Figure 13 Figure14 Figures 17-19 are plotted to show the variations of R σ , Figure17 Figure 20

Figure 22 Θ
Figure 22 Θ versus R for 0.026 η = and 0.5, 1.0.α = e., at early stage of wave propagation, both the models give close results, whereas for comparatively large time ( 1.21) , in the earlier situations, maximum magnitude occurs for weak conductivity whereas for large time, magnitudes are maximum when conductivity is high inside the body.
as follows From equations in (A.5) we can calculate the discrete values of