Transient Thermo-elastic Analysis Of Functionally Graded Spheres Based On State Space Theory

In this paper,by the heat conductivity equation,the thermoelasticity equation and the equilibrium equation,and introducing the heat flux,the governing equations are constituted for the spherically symmetric transient heat conductivity and thermoelasticity problems of functionally graded spheres.By discretizing the governing equations with the division of the sphere into subshells in space domain and finite difference scheme in time domain,the state equations of the transient heat conductivity and thermoelasticity problems are established for the sphere with material properties varying arbitrarily in the radial direction.By employing the continuity of the state vector across all interfaces between the adjacent subshells and the initial and boundary conditions,the state space solutions are obtained for the transient heat conductivity and thermoelasticity of functionally graded spheres.For the two cases with material properties being continuous and discontinuous functions,the relevant analyses are performed in the numerical examples below:First,the influences of the number of subshells and the time step length on convergence and stability of the temperature,displacement and stress fields are considered for functionally graded spheres under transient condition.Next,by selecting the proper number of subshells and time step,the temperature,displacement and stress within the sphere are studied and plotted as functions of the radial coordinate and time,respectively.Finally,the effects of the exponential distribution factor on the transient temperature,displacement and stress fields are analyzed for the sphere with the exponential material properties.The numerical examples show that the present solutions are not only correct and efficient,but also applied to the analyses of spherically symmetric transient heat conductivity and thermoelasticity of spheres with arbitrarily graded material properties.