WAVES IN AN INHOMOGENEOUS VISCOELASTIC HOLLOW SPHERE

The problem of propagation of nonstationary longitudinal waves in a sphere with a concentric cavity consisting of homogeneous viscoelastic spherical layers with continuity conditions for displacement and normal stresses at the boundaries between the contacting layers is solved. A uniformly distributed normal load acts on the surface of the sphere, the cavity remains free. The solution of the problem is constructed using the integral Laplace transform in time. The solution in originals is presented in a new form, which is especially convenient for numerical implementation with a large number of homogeneous layers for both regular relaxation kernels and singular Rzhanitsyn – Koltunov kernels. This new form, also suitable for other problems, made it possible to significantly simplify dynamic calculations and, with an increase in the number of layers, it is easy to proceed to the study of transients in a sphere of viscoelastic functionally graded material with continuously changing physical and mechanical properties in the radial direction. A method for approximating the continuous inhomogeneity of the sphere material by a layered medium is applied, which is often used in stationary dynamic problems for elastic, thermoelastic and piezoelectroelastic bodies. The validity of this approach for nonstationary problems was previously confirmed by the author's calculations for bodies with cylindrical and plane boundaries. For the sphere, the convergence of the results was also observed with an increase in the number of layers under a continuously time-varying load. Transient processes with exponential type of the sphere material inhomogeneity, including the inhomogeneity of the singular relaxation kernel, are investigated.