Discretization and hysteresis

This paper presents a simple and explicit mathematical example of the effects of discretization on a nonconvex variational problem. We describe a one-dimensional model which we call the Ericksen-Timoshenko bar. The energy includes a term that is nonconvex in the strain, quadratic terms in an internal variable and its derivatives, and the simplest quadratic coupling. In the framework of classical elasticity theory, the model has a strong integral nonlocality. Under special constitutive hypotheses, one can construct a collection of stationary points with an arbitrary number of interfaces between phases. We show that solutions with more than one interface are saddle points of the energy, unstable with respect to motion of the interface. We then discretize the energy and show that the saddle points of the continuum problem all correspond to local minimizers of the discrete problem. Thus, the ''energy landscape'' of the continuum problem is essentially smooth, while the landscape of the discretization is bumpy. This result, which is due to the constraints imposed by the discretization, is independent of mesh size.