Ericksen's bar revisited: Energy wiggles

This paper addresses the non-uniqueness pointed out by Ericksen in his classical analysis of the equilibrium of a one-dimensional elastic bar with non-convex energy. According to Ericksen, for the bar in a hard device, the piecewise constant functions delivering the global minimum of the energy can have an arbitrary number N of discontinuities in strain (phase-boundaries). Following some previous work in this area, we regularize the problem in order to resolve this degeneracy. We add two non-local terms to the energy density: one depends on the high (second) derivatives of the displacement, the other contains low (zero) derivatives. The low-derivative term (scaled with a constant beta) introduces a strong non-locality, and simulates a three-dimensional interaction with the loading device, forcing the formation of layered microstructures in the process of energy minimization. The high-derivative (strain-gradient) term (scaled with a different constant alpha), represents a surface energy contribution which penalizes the formation of phase interfaces and prevents the infinite refinement of microstructures. In our description we consider the positions of interfaces as variables. This singles out in a natural way an infinite number of finite-dimensional subspaces, where all the essential nonlinearity is concentrated. In this way we can calculate explicitly the local minimizers (metastable states) and their energy, which turns out to be a multi-valued function of the interface positions and the imposed overall strain d. Our approach thus gives an explicit framework for the study of the rich variety of finite-scale equilibrium microstructures for the bar and their stability properties, This allows for the study of a number of properties of phase transitions in solids; in particular their hysteretic behavior. Among our goals is the investigation of the phase diagram of the system, described by the function N(d, alpha, beta) giving the number of phase-boundaries in the absolute minimizer. We observe the somewhat counterintuitive effect that the energy at the global minimum, as a function of the overall strain, generically develops non-smooth oscillations (wiggles). Copyright (C) 1996 Elsevier Science Ltd