We study the radial motion of an incompressible viscoelastic spherical shell with a nonconvex strain energy function that models a material that can undergo a phase transition. In addition to the classical Newtonian viscosity for viscoelastic materials, we consider a material with two microstructural coefficients that are supposed to sense local configurational changes that take place during a deformation. Conditions necessary to show the effect of the nonconvexity of the strain energy function during a phase transition of the material, are determined, and the resulting dynamics is analyzed. It is shown that, though small periodic vibrations are possible, the system can easily revert into a mode of large amplitude motion as a result of small external excitation. Such motion may be transient to periodic motion or to chaotic motion. Boundaries in parameter space for the occurrence of this type of motion are determined and examples are given.