A simple sufficient condition for the quasiconvexity of elastic stored-energy functions in spaces which allow for cavitation
Jonathan J. Bevan, Caterina Ida Zeppieri · Calculus of Variations and Partial Differential Equations · 2016
In this note we formulate a sufficient condition for the quasiconvexity at $$x \mapsto \lambda x$$ 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 $$u(x)= \lambda x$$ 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 $$\lambda _0>0$$ such that for every $$\lambda \in (0,\lambda _0]$$ it holds that $$I(u) \ge I(u_{\lambda })$$ for all admissible u, where $$u_{\lambda }$$ is the linear map $$x \mapsto \lambda x$$ applied across the entire domain. This is the quasiconvexity condition referred to above.