The Dynamic solution of a nonlinear viscoelastic spherical membrane subject to internal pressure is studied. The constitutive equation is of an integral form with the relaxation function generated by the constitutive function of a nonlinear elastic material. The instantaneous response of the viscoelastic material is the same as the generating elastic material. For spherically symmetric motions, the equation of motion is a nonlinear integro-differential equation. This equation is converted to a system of nonlinear differential equations. A phase space analysis is carried out to study the properties of the solutions (trajectories). For the constitutive model generated from the Mooney-Rivlin materials, a numerical analysis is performed, with special attention paid to the transition from bounded solutions to unbounded solutions. Such a transition corresponds to, for equilibrium solutions, the bifurcation from the stable equilibrium deformations to unstable equilibrium deformations.
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