Axisymmetric Wave Propagation in a Solid Viscoelastic Sphere, 1967

(1967) Axisymmetric Wave Propagation in a Solid Viscoelastic Sphere, 1967. Iowa State University

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Abstract

In this paper we solve the more complex problem of the viscoelastic sphere under axisymmetric loading; the material of the sphere may have general relaxation characteristics but its Poisson's ratio is restricted to having a constant value for all time. Evidently the solution of the elastic problem may be obtained as a special case. The method of solution is based on a superposition principle proposed by Valanis; this principle was discussed at length in previous papers<2,3). However, for the sake of completeness, we give here an outline of its essential features. Two problems of linear wave propagation in a viscoelastic solid sphere are solved. The waves are generated by two types of impact on the surface of the sphere. The deformation has symmetry with respect to an axis through the center of the sphere. The solution is based on a superposition principle which reduces the general solution to a static elastic solution, an elastic solution of an eigenvalue problem and an integral equation of the Volterra type involving time only. The solutions are given in double infinite series involving spherical Bessel functions, Legendre polynomials and Legendre functions of the first kind and order one.