Fully explicit Agmon's condition for general states of a special incompressible elastic material

Abstract

Agmon's condition arises as a necessary condition at the boundary for minimizers in compressible and incompressible elasticity. It is commonly formulated as a statement concerning the solution set of a family of ODEs with constant coefficients. As such, it is algebraic “in principle”. In both the compressible and incompressible cases, Agmon's condition may be recast in a more overtly algebraic form, namely the requirement that a certain family of algebraic Riccati equations (parametrized over the tangent plane) should possess positive solutions. In order to reduce Agmon's condition to a fully explicit set of inequalities involving the components of the incremental elasticity tensor, one must be able to solve the algebraic Riccati equation explicitly. Known situations where this can be done tend to involve highly symmetric states of isotropic materials. It is therefore noteworthy that Agmon's condition may be rendered explicit for any boundary-point of an arbitrarily deformed incompressible neo-Hookean body.