DISCRETIZATION AND HYSTERESIS

This paper presents a simple and explicit mathematical example of the effects of discretization on a nonconvex variational problem. We descr ibe 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 q uadratic coupling. In the framework of classical elasticity theory, th e model has a strong integral nonlocality. Under special constitutive hypotheses, one can construct a collection of stationary points with a n arbitrary number of interfaces between phases. We show that solution s with more than one interface are saddle points of the energy, unstab le with respect to motion of the interface. We then discretize the ene rgy and show that the saddle points of the continuum problem all corre spond 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.