Variational setting of nonlinear equilibrium problem: wedge discontinuity lines

A nonlinear equilibrium problem in elasticity is here studied. In particular, the model adopted consists of two elastic bodies which interact with each other: the first one represents the elastic body of interest and the latter the environment which is surrounding it. Accordingly, they are said to form a body-environment pair. The equilibrium problem is considered in the case when the interaction body-environment is live, that is the energy functional depends not only on the position and on the deformation which takes place at that position, but also on deformation gradients. In particular, when a material is of grade 1, the energy functional depends on the first deformation gradient and, when a material is of grade 2 it depends on the first deformation gradient and, in addition, on the second deformation gradient. Here, the environment is assumed to be a simple material, namely of grade 1, while the body immersed into it, of grade 1 or 2. In the two different cases,, respectively, the equilibrium conditions are written under the assumption that the body boundary is not a regular one, but is obtained as the union of two regular surfaces which intersect each other on a regular line. The latter, represents a wedge discontinuity line of the body boundary; that is, a line on which no outer normal unit vector is defined. The two different models, termed also First-Order and Second-Order Surface Interaction Potentials, in turn, are analyzed under this assumption on the body boundary. Thus, it is shown that further conditions need to be imposed. A comparison between the conditions in the two different cases is provided.