Nonlinear continuum physics: grand themes and pretty problems

The first part of this lecture begins with a brief description of the nonlinear equations of continuum physics emphasizing the fundamental role of constitutive equations and the difficulties in determining them. Here the connection between experimentation, inverse methods, and qualitative methods is discussed. Some pressing fundamental open problems for the differential equations are presented. The second and major part of the lecture treats the solutions of a collection of conceptually simple specific problems illustrating the general themes raised in the first part: studies of of thresholds in material response, inverse problems for constitutive behavior, design and control problems for smart materials, asymptotic justification of quasistaticity, admissibility conditions for shock and other discontinuities, etc.