Second-order surface potentials in finite elasticity

A study of elastic, shape-dependent body-environment interactions is presented where the contact loads the environment applies on the body at a given deformation are modelled by a surface potential whose density depends not only on the first deformation gradient but also on the second. The cases of tangential and simple surface potentials are given special attention, as well as the case of surface potentials admitting an energetically equivalent replacement by a volume potential being a second-order null lagrangian. Throughout the paper the body-environment boundary is assumed to be a smooth surface, except in the last section, where a body with a line discontinuity of the boundary normal is considered.