On Second-Order Surface Interaction Potentials in Elasticity

We here study a class of body/environment interactions that depend on the deformed shape of the body's surface. To concentrate on the issue, we let the stress response of the body under study be standard, that is, described by a stored energy mapping that depends on the first gradient of deformation. The surface interaction, however, is described by a potential whose surface density depends also on the second deformation gradient; this assumption allows for substantial generalization of some previous work by P.Podio-Guidugli and G.Vergara Caffarelli. With the use of a variational argument, we derive a set of conditions characterizing the tangentiality of the interaction operator at the boundary. A fuller treatment of this and other related issues will be given.