This paper concentrates on the study of the three-dimensional free vibrations in a homogenous isotropic, viscothermoelastic hollow sphere whose surfaces are subjected to stress free, thermally insulated or isothermal boundary conditions. The use of governing partial differential equations is solved into a coupled system of ordinary differential equations. The equation for toroidal motion gets decoupled from rest of the motion and remains unaffected due to thermal variations. Matrix Fröbenious method of extended power series is employed to obtain the solution. The secular equations for the existence of various types of possible modes of vibrations in the considered hollow sphere are derived in the compact form. The special cases of spheroidal and toroidal modes of vibrations of a hollow sphere have also been deduced and discussed. In order to explore the characteristics of vibrations the secular equations are further solved by using fixed point iteration numerical technique with the help of MATLAB software tools. The computer simulated results have been presented graphically for copper material.
The nonlocal elasticity together with electro‐magneto thermal dual‐phase‐lag (DPL) model to a transversely isotropic generalized thermoelastic hollow sphere with voids is addressed in radial direction. The governing equations and stress–strain‐displacement relations have been resolved by applying time harmonics technique. The elimination technique is employed to resolve the homogeneous equations to find the unknowns. The unknowns such as dilatation, displacement, voids volume fraction, temperature change, and radial/de‐hoop stresses have been calculated analytically. In order to discover the vibration analysis, the numerical Iteration method has been applied to thermally insulated/isothermal boundary conditions. For computation purpose the software like MATLAB tool has been used. The real parts of generated data obtained from the frequency equations have also been shown graphically to frequency shift and natural frequencies. The effects of DPL model of the theory of generalized transversely isotropic thermoelasticity with voids have been represented with and without the effects of magnetic field for nonlocal/local elastic materials and validated with existing literature.
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