Analysis of thermoelastic response in functionally graded hollow sphere without load

In this article, a model concerning free vibrations of spherically symmetric, thermoelastic, isotropic, and functionally graded sphere has been developed and analyzed in the context of linear theory of generalized thermoelasticity with one relaxation time. Laplace transform has been used to solve the problem which yields natural frequencies of free vibrations without performing inversion of the transform. The analytical results for coupled, uncoupled, and homogeneous spheres have been deduced as special cases of the general case. Natural frequencies of first 10 modes of vibrations have been obtained for different values of grading index of cobalt material regarding coupled thermoelastic, generalized thermoelastic, and elastic functionally graded spheres. The frequency shift and thermoelastic damping for Fourier and non-Fourier processes of heat propagation, temperature change, radial, and hoop stresses have been presented graphically. It has been analyzed here that grading index parameter helps in detecting the strength of signals in such material devices and the thermal relaxation time contributes in improving the quality of signals. The analysis also leads to the fact that grading index parameter is useful from design point of view and it can be tailored to specific applications for controlling the stress.