On J. Ball’s Fundamental Existence Theory and Regularity of Weak Equilibria in Nonlinear Radial Hyperelasticity

Abstract

In 1982, J. Ball formulated a pioneering theory on the existence and uniqueness of weak radial equilibria to the pure displacement boundary value problem associated with isotropic, frame-invariant strain-energy functions in nonlinear hyperelasticity. In the theory [Bal82], he posed the following question: “Does strong ellipticity (‘of the stored energy’) imply that all solutions to the equilibrium equations which pass through the origin and have finite energy are trivial?” J. Ball’s work depended critically on the number of elasticity dimensions.

In this chapter, we will present models in n-dimensional elasticity that establish that the answer to J. Ball’s question is negative. This work extends to higher dimensional elasticity the approach and results we presented, for the first time, on this question in [Ha07]. These models also provide further insight into another central, (very) difficult problem of nonlinear elasticity, namely, that of regularity of weak equilibria, which would be hard to gain by other methods such as the common, but delicate, phase plane analysis.