Exact and numerical elastodynamic solutions for spherically symmetric problems of functionally graded thick-walled spheres subjected to pressure shocks

In the present paper, analytical and numerical elastodynamic solutions are developed for spherically symmetric problems of functionally graded thick-walled spheres subjected to arbitrary dynamic and shock loads. Both transient dynamic response and elastic wave propagation characteristics are studied in the mentioned nonhomogeneous structures. Variations of the material properties across the thickness are described according to both polynomial and power law functions. The numerical consistent transfinite element formulation is presented for both functions whereas the exact solution is presented for the power law function. The functionally graded material sphere is not divided into isotropic sub-spheres. An approach associated with dividing the dynamic radial displacement expression into quasi-static and dynamic parts and expansion of the transient wave functions in terms of a series of eigenfunctions is employed to propose the exact solution. Results are obtained for various exponents of the functions of the material properties distributions, various radius ratios, and variety of dynamic and shock loads.