Nearly 200 years ago, Cauchy was able to formulate the question: do global equilibrium solutions exist for 3D elasticity? Still no one knows a rigorous answer under realistic physical assumptions.

In 1977, John Ball recognized the necessary criterion of polyconvexity for the existence of energy minimizers, and yet the calculus of variations fails to provide weak solutions to the equilibrium equations for these minimizers. We will discuss the difficulties of obtaining weak solutions with the ``simplifying'' assumption of incompressibility, and discuss a second-gradient model which provides weak solutions to higher order equilibrium equations.