A Necessary Condition for Energy-Minimizing Plane Deformations of Elastic Solids with Intrinsic Boundary Elasticity

In the plane-strain theory of the finite deformation of a hyperelastic solid that has a thin elastic coating on all or part of its boundary, the properties of the Hessian matrix of the coating energy, regarded as a function of the boundary stretch and a curvature-like variable, have been shown recently to play an important role in connection with the analysis of stability. Here it is proved that the Hessian is positive semidefinite in an energy-minimizing configuration. The proof is given for the case in which the bulk material to which the coating is bonded is not subject to any internal kinematic constraints and is then extended to allow for the constraint of incompressibility.