A Simple Sufficient Condition for the Quasiconvexity of Elastic Stored-energy Functions in Spaces Which Allow for Cavitation

In this note we formulate a sufficient condition for the quasiconvexity at x 7→ λx of certain functionals I(u) which model the stored-energy of elastic materials subject to a deformation u. The materials we consider may cavitate, and so we impose the well-known technical condition (INV), due to Müller and Spector, on admissible deformations. Deformations obey the condition u(x) = λx whenever x belongs to the boundary of the domain initially occupied by the material. In terms of the parameters of the models, our analysis provides an explicit λ0 > 0 such that for every λ ∈ (0, λ0] it holds that I(u) ≥ I(uλ) for all admissible u, where uλ is the linear map x 7→ λx applied across the entire domain. This is the quasiconvexity condition referred to above.