A dynamical expansion problem is examined for a spherical shell composed of a homogeneous isotropic incompressible modified Varga material, where the inner surface of the shell is subjected to a class of suddenly applied periodic step radial pressures. Under a constant pressure, the existence conditions of the periodic solutions of the differential equation that describes the motion of the shell are proposed, correspondingly, it is proved that the motion of the shell would present a nonlinearly periodic oscillation as the given pressure does not exceed a certain critical value and that the shell will be destroyed ultimately with the infinitely increasing time as the pressure exceeds the critical value. Under the periodic step pressures depending on time, all the controllability conditions for nonlinearly periodic oscillations of the spherical shell are presented, and numerical simulations are also given.
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