Abstract
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.
Highlights
The theory of thermoelasticity is well established, Nowacki [1]
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
Due to the presence of dissipation term in heat conduction Equation (4), the secular equations are in general complex transcendental equations which provide us complex values of the frequency (ω)
Summary
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. 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. Sharma [8], Sharma and Sharma [9] presented an exact analysis of the free vibrations of supported, homogeneous, transversely isotropic cylindrical panel based on the three-dimensional generalized thermoelasticity
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