Fractional heat conduction with finite wave speed in a thermo-visco-elastic spherical shell

Fractional heat conduction with finite wave speed in a thermo-visco-elastic spherical shell

A. Sur; M. Kanoria^*

Department of Applied Mathematics, University of Calcutta, India

ABSTRACT

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.

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 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 , where is the thermal displacement gradient and 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 s or the heat flux is very much high. 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 captures the thermal wave behavior (a small scalar response in time), the phase-lag captures the effect of phonon-electron interactions (a microscopic response in space), the other delay time 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) as follows

where is the mass density, is the concentration, is the diffusion conductivity, the coordinate symbol, which takes the value 1, 2, 3. The notation is the Riemann-Liouville fractional integral, introduced as a natural generalization of the well-known n-fold repeated integral 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 non-viscous isotropic materials. The effect of the fractional order parameter is also discussed.

2 FORMULATION OF THE PROBLEM

We consider a homogeneous orthotropic thermo-visco-elastic spherical shell of inner radius and outer radius in an undisturbed state and initially at uniform temperature . We introduce spherical polar coordinates 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 and the temperature are assumed to be functions of and 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 are the stress tensor, is the temperature increase over the reference temperature , are the elastic constants, are the thermal moduli, is the coefficient of thermal conductivity along the radial direction, is the additional material constant along the radial direction, is the mass density, is the specific heat of the solid at constant strain, is the mechanical relaxation time, and are the phase-lag of temperature gradient and the phase-lag of the heat flux respectively. Also where is the phase-lag of thermal displacement gradient.

In the case and we arrive at the thermo-elasticity equations with energy dissipation (TEWED(GN-III)).

The stress equation of motion in spherical polar co-ordinate is given by

Introducing the following non-dimensional quantities

Equations (4)-(8) become

and

where

and

Also, and are dimensionless constants, being the thermoelastic coupling constant. Where is the non-dimensional thermal wave velocity and is the damping co-efficient.

The boundary conditions are given by

where and are dimensionless constants, and is the Heaviside unit step function. The above condition indicate that for time η P η₁⁰ there is no temperature on the inner boundary and for η # 0 there is no temperature on the outer boundary. Thermal shocks are given on the boundaries of the shell 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

3 METHOD OF SOLUTION

Let

with denote the Laplace transform of and respectively.

Since we have

On taking Laplace transform, equations (12) and (13) reduce to

and

where

and assuming

Differentiating equation (21) with respect to and using equation (22), we get

Equations (22) and (23) can be written in the form

and

where, we assume that

is the thermo-elastic coupling constant and

From equations (24) and (25), we have the vector-matrix differential equation as follows

where

and

4 EIGEN VALUE APPROACH

Let

where is a scalar, is a vector depending on and is a non-trivial solution of the scalar differential equation

Let . Therefore, from equation (30) we have

The solution of equation (30) is

Using equation (29) and (30) into equation (27) we get

where is the eigen vector corresponding to the eigen value .

The characteristic equation corresponding to can be written as

The roots of the characteristic equation (37) are of the form and , where

Equation (34) can be written as

Therefore, the positive roots of the equation (36) are

where

Therefore, and are real positive quantities.

The eigen vectors corresponding to the eigen values can be calculated as

Therefore, from equation (29) and using equation (28) we get

and

where and are the modified Bessel functions of order of first and second kind respectively. 's and 's are independent of but dependent of and are to be determined from the boundary conditions.

Using the recurrence relations of modified Bessel functions we obtain from equation (41)

since

where . Taking Laplace transform on the equations (9), (10) and (11) we get

where Using the boundary conditions on and on , where , on

Using the recurrence relations (Watson, 1980) from equations (42) and (44) we obtain

and

where for for and

From (44), the values of,,and are given as

5 SPECIAL CASES

For the homogeneous and transversely isotropic material

and Therefore from (9) and (10), Hence, from (45) and (46), we can write

Also for an isotropic material, and and for a non-viscous material, we have Hence, Thus, for an isotropic material, equations (45) and (46) reduce to

Moreover, for large value of i.e., for large value of , and tend to zero. Thus we have

Hence for large value of, the asymptotic expressions of and are given as

and

Therefore, for an infinitely extended body

where

The results agree with those of Kar and Kanoria (2007) for GN III model.

6 OPERATOR THEORY ANALYSIS

Equations (12) and (13) can be expressed in the following form

with and

where

and

Taking the Laplace transform, we have

and

Where and

Operating on (64) and using (65) we have

Similarly, operating on (65) and using (64) we have

where and are the two operators and and are the roots of the quadratic equation in given by

As the solution of equation (66) and (67) we have

and

Where and are the modified Bessel functions of order of first and second kind respectively; ,,and are independent of but dependent on

Therefore, substituting the expressions of and in equation (65), we get

and

Therefore

Equations (73) and (74) are the same as that of equations (40) and (42) (i.e., the solutions obtained by Eigen-value approach).

7 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 's are computed from roots of the shifted Legendre polynomial of degree 7 (see ^Appendix) with The computations for the state variables are carried out for different values of and values of The materials chosen for numerical evaluation are copper material. The physical data for orthotropic material are (2009)

and the hypothetical values of the relaxation time parameters are taken as

Here, in this article we have considered three-phase-lag model. Now, for this model, the solution of heat conduction is stable if where 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, is an additional material constant depending on the material. For copper like material, we take

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 and small time for weak conductivity, normal conductivity and strong conductivity respectively 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 and outer boundary of the hollow sphere. Figures 1 and 2 depict the variation of the radial stress against the radial distance of the sphere. From the figures it is observed that the radial stress vanishes at the inner boundary and the outer boundaries 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 oscillation term in the heat equation of three-phase-lag model. For weak conductivity, the oscillatory nature is also seen for GN III model.

Figure 2 represents the variation of for and respectively. It is seen that 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, the effect of is very prominent inside the shell, whereas, for and, the radial stress vanishes for and respectively, which is physically plausible.

Figures 3 and 4 are plotted to show the variation of the stress along the radius of the sphere for different values of the non-local fractional parameter and for respectively. In figure 3, variation of is shown for larger time for same set of parameters. As seen from the figure, attains the maximum magnitude near the inner boundary of the shell. The magnitude of for is larger than that of which is again larger than that of when and the rate of decay in magnitude of for is faster than that of which is again faster than that obtained for .

In both cases, the stress corresponding to each model and for different nonlocal fractional parameter, the stress propagates for For small time, the stress is compressive in nature near the inner boundary of the shell. Here also, for the stress component almost disappears for and for , it vanishes for . For ,i.e., the region equidistant from the boundaries vanishes in earlier situations for normal conductivity and strong conductivity of the materials.

Figures 5 and 6 are plotted to show the variation of the stress component for different fractional parameter From the figure 5, it is seen that when the magnitude of is maximum near the inner boundary of the shell. Also it is observed that increase in the nonlocal fractional parameter also increases the magnitude of the stress component For and the decay in magnitude of is more rapid compared to that of As may seen from the figure, is compressive near the inner boundary of the shell and the similar qualitative behavior is seen in the variation of as that in figure 4.

Figures 7 and 8 depict the variation of the temperature along the radius of the sphere for different values of It is seen that whenever (i.e., for ) the inner boundary of the shell is kept at zero temperature whereas for the inner boundary is maintained the fixed temperature value while in both situations, outer boundary maintains the same step-input-temperature For larger time, attains the maximum magnitude near for GN III model. Whereas in the earlier situations, the magnitude of decays sharply near the inner boundary of the shell for compared to that of and The rise in magnitude near the outer boundary is rapid also. For and for , the magnitude of the temperature almost disappears for

Figures 9-16 are plotted to show the effect of viscosity for two set of times for weak conductive materials. Form figures 9-10 it is seen that satisfies our theoretical boundary conditions. As may seen from figure 10, it is seen that attains the maximum value for non-viscous material for both models near the inner boundary of the shell.

Figures 11-12 are plotted to show the variation of versus from these figures it is seen that the effect of viscosity is more prominent in GN III model compared to that of 3P lag model for a large time when Whereas, for earlier situations, the effect of viscosity for GN III model is very prominent near the boundaries of the shell compared to the interior of the shell.

From figures 13-14, the similar qualitative nature is seen in the variation of for both viscous and non-viscous materials.

Figures 15 and 16 are plotted to show the effect of viscosity on temperature 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 , for 3P lag model, the magnitude of is larger for viscous material compared to the non-viscous material. Whereas for , the magnitude is larger for non-viscous material compared to the viscous material.

Figures 17-19 are plotted to show the variations of ,and respectively against the time whenever and 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.

Figures 20-22 are plotted to draw the comparison between isotropic and orthotropic material for and for for viscous material. From figure 20, it is seen that for orthotropic material, the stress waves are reflected from either boundary whereas for isotropic material, the propagation of each of the waves are found to occur. Also, amplitude of decreases with the increase of the non-local fractional parameter.

Figure 22

Figure 21 depicts the variation of versus 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 is maximum near the outer boundary of the shell for an isotropic material.

8 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:

(i) When the time is small, , i.e., at early stage of wave propagation, both the models give close results, whereas for comparatively large time , significant differences are observed for weak, normal and strong conductivities respectively. Also, in the earlier situations, maximum magnitude occurs for weak conductivity whereas for large time, magnitudes are maximum when conductivity is high inside the body.

(ii) It is observed that maximum magnitude of stresses will occur for viscous material and for strong conductivity.

(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 is more prominent near the inner boundary for orthotropic material compared to that of an isotropic material.

Acknowledgements We are grateful to Professor S. C. Bose of the Department of Applied mathematics, University of Calcutta, for his kind help and guidance in preparation of the paper. We also express our sincere thanks to the reviewer for his valuable suggestions for the improvement of the paper.

www.matweb.com.

Appendix

Let the Laplace transform of be given by

We assume that is sufficiently smooth to permit the use of the approximate method to apply.

Substituting in equation (A.1) we obtain

where

Applying the Gaussian quadrature rule to (A.2) we obtain the approximate relation

where 's are the roots of the shifted Legendre polynomial and 's are the corresponding weights and

Thus, we have

Therefore

As the matrix is the product of multiplied by Vander monde matrix, it can be shown that the matrix is non-singular.

Hence, are known. For we have

From equations in (A.5) we can calculate the discrete values of i.e., and finally using interpolation, we obtain the stress components

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Appendix

Publication Dates

• Publication in this collection
  13 Mar 2014
• Date of issue
  Dec 2014