Radial oscillations of highly stressed, nonhomogeneous, thick-walled cylindrical and spherical shells

This work investigates the infinitesimal breathing motions of spherical and long cylindrical shells of arbitrary wall thickness subjected to a finite, axisymmetric deformation field caused by uniform internal and/or external pressures. A neo-Hookean material with a material constant varying continuously along the radial direction is used. The shell is first subjected to finite, axisymmetric, static deformations and is then exposed to a secondary, axisymmetric, dynamic displacement field. Based on the theory of small deformations superposed on large deformations, closed form expressions are obtained for the frequency of small oscillations about the highly prestressed state. Frequency versus initial deformation parameter curves are given for several nonhomogeneity functions and for various wall thicknesses. The softening or the hardening behaviors of the shells for varying prestress values are observed from these curves. When the frequency of breathing motions cease to be real valued, the superposed secondary motions are no longer periodic. Thus, the critical value of the prestress causing instability is defined as the one which corresponds to zero frequency. It is seen that when the nonhomogeneity parameters are taken zero, the known results of the homogeneous case are obtained. (Auth.)