Hysteresis and interface dynamics in mathematical models of phase transitions

Materials undergoing stress-induced martensitic phase transformations (in particular, shape memory alloys) exhibit a markedly hysteretic behavior under cyclic loading. The hysteresis loops on the load-elongation diagrams are often serrated. The serrations are accompanied by a nonsmooth, ''jerky'' motion of the phase boundaries.

In this talk, we consider two mathematical models describing the dynamics of phase transitions. In both models, a bar is subjected to time-dependent displacement boundary conditions. The bar is assumed to be an elastic continuum that can deform only in the direction along its length. The local deformation of the bar is described by the displacement field and its spatial derivative, called strain. The elastic properties of the bar are determined by an elastic energy density which is a function of strain. We assume this function to be a nonconvex double-well potential. The wells in the elastic energy density represent two different material phases, austenite and martensite. The dynamic models take into account both inertia and dissipative viscous terms. The first model also includes the interfacial energy, modeled by a strain-gradient term. In the second model, this term is omitted. The models result in initial-boundary value problems for nonlinear parabolic PDE. Both models predict hysteresis which is primarily due to dynamic solution getting locked in local minimizers of the potential energy functional. The hysteresis loops persist even when the loading rate is very slow, and viscosity effects are minor. We find that in the model without the interfacial energy term, the hysteresis loops are serrated, and a stick-slip interface motion is observed. We show that for a given loading this solution behavior is a singular limit of the model with interfacial energy term as this term tends to zero. On the other hand, at fixed strain-gradient coefficient and slow enough (quasistatic) loading the model including the interfacial energy results in a smooth interface motion and smaller, non-serrated hysteresis loops.

Refreshments will be served in Math Annex Room 1115, 3:15 p.m.