A thermodynamic framework is presented that can be used to describe the solidification of polymer melts, both the solidification of atactic polymers into an amorphous elastic solid and the crystallization of other types of polymer melts to semi-crystalline elastic solids. This framework fits into a general structure that has been developed to describe the response of a large class of dissipative bodies. The framework takes into account the fact that the natural configuration of the viscoelastic melt and the solid evolve during the process and that the symmetries of these natural configurations also evolve. Different choices are made as to how the material stores energy, produces entropy, and for its latent heat, latent energy, etc., that lead to models for different classes of materials. The evolution of the natural configuration is dictated by the manner in which entropy is produced, how the energy is stored etc., and it is assumed that the constitutive choices are such that the rate entropy production is maximized, from an allowable class of constitutive models. Such an assumption also determines the crystallization kinetics, i.e., provides equations such as the Avrami equation. Using the framework, a model is developed within which the problem of fiber spinning is studied and we find that the model is able to predict observed experimental results quite well.