3-D Exact Vibration Analysis of a Generalized Thermoelastic Hollow Sphere with Matrix Frobenius Method

This paper presents exact free vibration analysis of stress free (or rigidly fixed), thermally insulated (or isothermal), transradially isotropic thermoelastic hollow sphere in context of generalized (non-classical) theory of thermoelasticity. The basic governing equations of linear generalized thermoelastic transradially isotropic hollow sphere have been uncoupled and simplified with the help of potential functions by using the Helmholtz decomposition theorem. Upon using it the coupled system of equations reduced to ordinary differential equations in radial coordinate. Matrix Frobenius method of extended series has been used to investigate the motion along the radial coordinate. The secular equations for the existence of possible modes of vibrations in the considered sphere are derived. The special cases of spheroidal (S-mode) and toroidal (T-mode) vibrations of a hollow sphere have also been deduced and discussed. The toroidal motion gets decoupled from the spheroidal one and remains independent of the both, thermal variations and thermal relaxation time. In order to illustrate the analytic results, the numerical solution of the secular equation which governs spheroidal motion (S-modes) is carried out to compute lowest frequencies of vibrational modes in case of classical (CT) and non-classical (LS, GL) theories of thermoelasticity with the help of MATLAB programming for the generalized hollow sphere of helium and magnesium materials. The computer simulated results have been presented graphically showing lowest frequency and dissipation factor. The analysis may find applications in engineering industries where spherical structures are in frequent use.


Introduction
The theory of thermoelasticity is well established, Nowacki [1].The governing field equations in classical dynamic coupled thermoelasticity (CT) are wave-type (hyperbolic) equations of motion and a diffusion-type (parabolic) equation of heat conduction.Therefore, it is seen that part of the solution of energy equation extends to infinity, implying that if a homogeneous isotropic elastic medium is subjected to thermal or mechanical disturbances, the effect of temperature and displacement fields is felt at an infinite distance from the source of disturbance.This shows that part of disturbance has an infinite velocity of propagation, which is physically impossible.With this drawback in mind, Lord and Shulman [2], Green and Lindsay [3], modified the Fourier law of heat conduction and constitutive relations so as to get a hyperbolic equation for heat conduction.These works include the time needed for the acceleration of heat flow and take into account the coupling between temperature and strain fields for isotropic materials.Dhaliwal and Sherief [4] extended the generalized thermoelasticity [2] to anisotropic elastic bodies.A wave-like thermal disturbance is referred as "second sound" by Chandrasekharaiah [5].These theories are also supported by experiments of Ackerman et al. [6] that exhibiting the actual occurrence of second sound at low temperatures and small intervals of time.The investigators Singh and Sharma [7] studied the propagation of plane harmonic waves in homogeneous anisotropic heat-conducting elastic materials.Sharma [8], Sharma and Sharma [9] presented an exact analysis of the free vibrations of simply supported, homogeneous, transversely isotropic cylindrical panel based on the three-dimensional generalized thermoelasticity.
The free vibrations of solid and hollow spheres have been the subject of study for a long period, frequently associated with interest in the oscillations of the earth.In the late nineteenth century, Lamb [10] showed that two basic types of free vibrations namely, 1) the vibrations with zero volume change and zero radial displacement; and 2) the vibrations with zero radial components of the curl of the displacement, exist in an isotropic sphere.These vibrations are referred as "vibrations of the first and second classes" respectively.Lapwood and Usami [11] named the first class of vibrations as "torsional or toroidal" and the second class as "spheroidal or poloidal".Lapwood and Usami [11] presented an excellent treatment of the vibration of a hollow sphere surrounding a liquid core having finite normal and shear rigidity which serves as an approximate model of the Earth.Lamb [10] derived the equations governing the free vibration of a solid sphere and subsequently Chree [12] obtained the secular equations of free vibrations of a sphere in the more convenient form.Much later, Sâto and Usami [13,14] computed and tabulated the natural frequency parameters for an extensive set of modes of vibration for the solid sphere.They provided equations and a comprehensive set of results for the distribution of displacement within the vibrating sphere.Shah et al. [15,16] studied the vibrations of hollow spheres by using twodimensional theory of elasticity to obtain natural frequency parameters.Gupta and Singh [17] investigated the problems of wave propagation in transradially isotropic elastic sphere.They showed that, for a transradially isotropic sphere, the toroidal and spheroidal modes of vibrations are independent of each other.
Bargi and Eslami [18] used Green-Lindsay theory of thermoelasticity to study the thermo-elastic response of functionally graded hollow sphere and investigated the material distribution effects on temperature, displacement and stresses.Sharma and Sharma [19], studied the generalized transradially thermoelastic solid sphere.
We have not come across any systematic and exact study on the effect of temperature variations on three dimensional vibration of heat conducting elastic generalized hollow spherical structures.Therefore, the purpose of this paper is to present the exact three dimensional vibration analysis of transradially isotropic, thermoelastic generalized hollow sphere subjected to stress free (or rigidly fixed), thermally insulated (or isothermal) boundary conditions.The secular equations governing three dimensional vibrations in a generalized hollow sphere have been derived by using Frobenius series method.The derived secular equations for spheroidal (S) modes of vibrations which are dependent on thermal variation, have been solved numerically for zinc and cobalt materials in order to compute lowest frequency and dissipation factor.The obtained results in case of toroidal vibrations are found to be in agreement with those of Cohen et al. [20].

Formulation and Solution
We consider the thermoelastic problem for homogeneous, transradially thermally conducting, elastic generalized hollow sphere of inner and outer radii R 1 and R 2 , re-spectively, initially maintained at uniform temperature 0 in the undisturbed state.For generalized spherically isotropic thermoelastic medium, in the spherical polar coordinates   , the basic governing equations of motion, heat conduction and constitutive relations can be expressed as follows Sharma and Sharma [19].
, r , r , , , ,   where rr  is stress along radial direction and r  , r  are along tangential direction.Here is the temperature change, 11 12 13 and 44 are five independent isothermal elasticity; and of course

Boundary Conditions
We consider the free vibrations of a generalized hollow sphere subjected to stress free (or rigidly fixed), thermally insulated (or isothermal) boundary conditions at the surfaces 1 (inner radius) and (outer radius).Mathematically, this leads to 1) For stress free, thermally insulated (or isothermal) boundary of the sphere.

at
(inner radius) and r (outer radius); 2) For rigidly fixed, thermally insulated (or isothermal) boundary of the sphere.

Solution of the Problem
We define the dimensionless quantities  are shear wave velocity and characteristic frequency of the generalized hollow sphere respectively.The primes have been suppressed for convenience.
It is advantageous to express the displacements , u u and r in terms of functions u , , G w  and defined by Sharma and Sharma [19] as.
) Using Equation (11) in Equations ( 1)-( 4), we find that where Assume spherical wave solution of the form where   Upon substituting solution (16) in Equations ( 12)-( 15), we obtain The quantities and 17) and ( 18) are defined as The uncoupling of equations for the displacement potential n from n and n indicates the existence of two distinct modes of vibrations.The solution of Equation ( 11) for corresponds to the Toroidal mode.There is no effect of temperature and generalization on toroidal mode.Toroidal frequencies will be same as obtained by Cohen et al. [20].
The solution of the spherical Bessel Equation ( 17) is given by where and J  and Y  is Bessel function of first kind and second kind.
1 n and 2 n are arbitrary constants determined from the boundary conditions.

Generalized Series Method
The system of Equations ( 18) has been solved with the help of matrix Frobenius method.Clearly the point 0 r  (i.e.0   ) is a regular singular point of Equations (18) and all the coefficients of the differential Equations (18) are finite, single valued and continuous in the interval The quantities satisfy all the necessary conditions to have series expansions and hence the Frobenius power series method is applicable to solve the coupled system of differential Equations (18).Thus, we have took the solution vector of the type 0 where , where s is a constant (real or complex) to be determined and , , k are unknown coefficients to be determined.We need solution in the domain 1 2 , 1 .The solution ( 22) is valid in some deleted interval   (about the origin) where is the radius of convergence.

R
Upon substituting solution (22) along with its derivatives in Equations ( 18) and simplifying, we get where The elements, 2 Copyright © 2012 SciRes.WJM Equating to zero the coefficients of lowest powers of in Equation ( 23), we obtain: where where the coefficients A and C are given by The roots of indicial Equations (28) are given as Clearly the roots j are related through the relation 4 s is real but the roots 1 s and 2 s may be, in general, complex.In case the parameter, s, is complex, then leading terms in the complex series solution (22) are of the type: In order to obtain two independent real solutions, according to Neuringer [21], it is sufficient to use any one of the complex root in a part and taking its real and imaginary parts.Also, the treatment of complex case is unlike that of the real root case with the advantage that the differential equation is required to be solved only once in the former case rather than twice as in latter one.For the choice of roots of the indicial equations, the system of Equations ( 27) leads to following eigen vectors: where A is a constant.Thus we have as the corresponding eigen vectors.Again equating to zero the coefficients of next lowest degree term 1 s   in Equation ( 23) and noting that the matrix   H s  is nonsingular for each j, we obtain: where The matrices   equal to zero, we obtain following recurrence relation: where the matrices 1 2 , H H and H are defined in Equation (23).This implies that Now putting 0, 1, 2, 3 k   in Equation (36) successively and simplifying, we get It can be easily shown that the matrix has similar form to that of for odd values of k .
Thus we have: Here the elements are given by Equations (A2)-(A3) as defined in the Appendix.Moreover, it can be shown that in Equation ( 22) is analytic and hence can be differentiated term by term.Moreover, the derived series are also analytic functions.
Thus the general solution of the system of Equations ( 18) has the form where , , are eigenvectors corresponding to the eigen-values j s and integer .The quantities k jk are arbitrary constants to be evaluated.Consequently, the potential functions and T are written from Equations ( 12) by using (42) as under: Noting that both the matrices and , as and using the fact that k k  , if each component sequence converges (Cullen [22]), we can conclude that the series (22) are absolutely and uniformly convergent with infinite radius of convergence.Therefore, the considered series

Stress-Free Generalized Hollow Sphere
The unknowns ,

Toroi
The secular dispersion Eq where t*(=h/R) is thickness to mean radius ratio, where thickness of the sphere defined as h R R   and mean radius as Clearly and the secular Equation (61) reduces to    It can be shown that Equation ( 63) is identical to the one obtained by Love [24], [page 284, Equation (38)].These modes have also been discussed in detail by Cohen et al. [20] and Ding et al. [23] and the corresponding frequencies of such Toroidal modes are same as in case of elastokinetics.The analysis in case of coupled thermoelasticity (CT) can be obtained by setting  4) and ( th ly fixed and thermal boundary conditions (9b) at the surface 5) then using e resulting values of these parameters in different relations at various stages.

Rigidly Fixed Sphere
Taking the rigid and of the sphere on displacements we get following dete antal equations 0 , 1,2,3,4,5,6 , for thermally insulated 2 , for isothermal The elements  

Toroidal Vibrations
Equation (65) corresponds to first class vibrations (Toroidal mode).Clearly these modes do not depend on thermal and relaxation time variations as expected.

Spheroidal Vibrations
The secular Equations (64) govern the Spheroidal vibrations (S-modes) in case of and respectively in a transrad Gene moelastic hollow sphere subjected to stress fr conditions.These relations contain complete information  , n ralized theree boundary regardin cy a g frequen nd other characteristics of the Spheroidal modes of vibrations in a transradially isotropic Generalized thermoelastic hollow sphere.

Numerical Results and Discussion
We consider the case of free vibrations of a transradially, isotropic generalized thermoelastic hollow spheres made up of solid helium and magnesium materials whose physical data is given in Table 1.As given by Sh an d Dhaliwal and Sin general plex transcen s co plex values of the fre quency (ω) giv frequency arma d Sharma [19] an gh [26].Due to the presence of dissipation term in heat conduction Equation ( 4 Figures 1 and 3 show the variations of lowest frequency (Ω) with thickness to mean radius ratio (t*) for different values of degree of spherical harmonics (n) for solid helium and magnesium respectively in case of stress free generalized hollow sphere.From Figure 3, it is observed that the profile of lowest frequency increases parabolically with the increase of t* and the order with respect to generalized theories of thermoelasticity is Ω(CT) > Ω(GL) >Ω(LS) for n = 1 and n = 2. Figure 3 reveals that the lowest frequency vari mains dispersionless at all values of the degree rical harmonics (n) in the context of LS, GL a ge s of therm the order for lowest frequency interlaces is T) > Ω(GL) for n = 1 and n = 2.It can be concluded that with the increase of degr of spherical harmonics (n), est frequency increases.From both the Figures we conclude that with the increase of t* lowest frequency increases.
Figures 2 and 4 show the variations of damping factor (D) with thickness to mean radius ratio (t*) for different values of degree of spherical harmonics (n) for solid he- Wm deg Wm deg  An examination is made on the variations of dimensionless lowest frequency with respect to thickness to mean radius ratio (t*) ranging from thin spherical shell (t* = 0.05) to the thick spherical shell (t* = 0.1) of isotropic materials.The findings confirm that the variation of the Spheroidal frequencies increases with t* and for degree of spherical harmonics (n), the same trend of pro ickness to mean radius ratio (t*) for ifferent values of degree of spherical harmonics (n) for two materials solid helium and magnesium respectively in case of rigidly fixed boundary condition.For both the materials similar behaviour has been observed i.e. the lowest frequency increases with the increase of t*.In     with time ratio for different values of thickness to mean radius ratio (t*) for solid helium and magnesium respectively.The profiles of lowest frequency are observed to thermal relaxation time in both the materials with increasing values of thickness to mean radius ratio (t*).From both the Figures it is observed that with the increase of time ratio lowest frequency is nearly constant value, also lowest frequency has more value for thick shell as compared to thin shells.with the increase of time ratio the magnitude of damping factor is nearly constant value, also lowest frequency has more value for thick shells as compared to thin shells.[24], Lamb [10] and Cohen et al. [20] elastokinetics.The lowest frequency of spheroidal vibrations is noticed to be significantly affected due to both temperature variations and relaxation time, in hollow spheres of both helium and magnesium materials.

Conclusions
The lowest frequency and dissipation factor have shown strong dependency on the degree of spherical harmonics (n) and hence the importance of the degree of spherical harmonics must be taken into consideration while designing a spherical structure.
The effect of relaxation time ratio on lowest frequency and dissipation factor of vibration under consideration is also observed in hollow spheres of helium and m ies where spherical structures are in frequent use.

Appendix
The quantities used in Equations (34) are defined as   , , 1,2,3 The quantities

1 3 
respectively, the coefficients of linear thermal expansion and thermal conductivities along and perpendicular to the axis of symmetry,  and e C are the mass density and specific heat at constant strain and and 1 are thermal relaxation times respectively 1l 0 t t  is Kronecker's delta in which for Lord-Shulman (LS) theory and for Green-Lindsay (GL) theory of thermoelasticity.The comma notation is used for spatial derivatives and the superposed dot denotes time differentiation.It can be proved thermodynamically Sharma and Sharma[9] that1 3 that and the isothermal elasticity are components of a positive definite fourth-order tensor.The necessary and sufficient conditions for the satisfaction of latter requirement are

2 H
are defined as

1 0
) For the existence of non-trivial solution of Equations (26) we must have   H s  , which results in the fol- the Appendix A1.Now equating the coefficients of powers of s k   by using boundary conditions(9) at the inner and outer boundaries of the generalized hollow sphere.Upon employing stress free and thermal boundary conditions (9a) at the surface 1

D
), the secular equations are in com dental equations which provide u mquency (ω).The real part of frevalues of n and k.The computer simulated profiles of lowest frequency and dissiin spheres of solid helium and magnesium materials have been in Figu 2 for different values of thickness to mean radius ratio (t*) in the context of LS, GL, and CT theories of coupled thermoelasticity.The values of the thermal relaxation time has been estimated from Equation (2.5) of Chanas given by Sharma and Sharma [ t num

Figure 1 .Figure 2 .Figure 3 .
Figure 1.Variation of lowest frequenc with thickness to mean radius ratio for different values of degree of spherical harmo ics (n) for lium material in e of stress free boundary condition.y n he cas

Figure 4 .
Figure 4. Variation of damping factor with thickness to mean radius ratio for different values of degree of spherica

Figure 5 .Figure 6 .Figure 7 .
Figure 5. Variation of lowest frequency with thickness to mean radius ratio for different values of degree of spherical harmonics (n) for helium material in case of rigidly fixed oundary condition.b

Figure 5 -In Figure 7
Figure 5 the trend of profiles of lowest frequency inter laces is - Figure 10  shows t magnitude of dissipation is constant in the region

Figure 8
Figure 8. Variation m harmonics (n) for magnesium ma

Figure 9 .Figure 10 .Figure 11 .Figure 12 .
Figure 9. Variation of lowest frequency with time ratio for various value of thickness to mean radius ratio (t*) for helium material.
© 2012 SciRes.WJM The detail of Spheroidal vibrations on neglecting the thermal effects and considering only the radial vibrations have been discussed by Ding et al.