Phase transitions in one-dimensional nonlinear viscoelasticity: admissibility and stability

Abstract

For the motion of a one-dimensional viscoelastic material of rate type with a non-monotonic stress-strain relation, a mixed initial boundary value problem is considered. A simple existence theory is outlined, based on a novel transformation of the equation into the form of a degenerate reaction-diffusion system. This leads to new results characterizing the regularity of weak solutions. It is shown that each solution tends strongly to a stationary state asymptotically in time. Stable stationary states are characterized. Stable states may contain coexistent phases, i.e. they may have discontinuous strain. They need not be minimizers of energy in the strong sense of the calculus of variations; “metastable” and “absolutely stable” phases may coexist in a stable state. Furthermore, such states do arise as long-time limits of smooth solutions.Beyond the above, “hysteresis” and “creep” phenomena are exhibited in a model of a loaded viscoelastic bar. Also, a viscosity criterion is proposed for the admissibility of propagating waves in the associated purely elastic model. This criterion is then applied to describe the formation of some propagating phase boundaries in a loaded elastic bar.