Three-dimensional Signorini-type problem for composite bodies contacting with sharp edges of rigid inclusions

A new type of non-classical three-dimensional contact problems formulated over non-convex admissible sets is proposed. Namely, we assume that a composite body in its undeformed state touches a wedge-shaped obstacle at a single point of contact. Investigated composite bodies consist of an elastic matrix and a rigid inclusion. In this case, displacements on a set corresponding to a rigid inclusion have a given structure that describes possible parallel translations and rotations of the inclusion. A rigid inclusion is located on the outer boundary of the body and has a special geometric shape in the form of a cone. A presence of a rigid inclusion makes it possible to write out a new type of a non-penetration condition for some geometrical configurations of an obstacle and a composite body near the contact point. In this case, sets of admissible displacements can be nonconvex. For the case of a thin rigid inclusion described by a cone, energy minimization problems are formulated. Based on the analysis of auxiliary minimization problems formulated over convex sets, the solvability of problems under study is proved. Under the assumption of a sufficient smoothness of the solution, equivalent differential statements are found. The most important result of this research is the justification of a new type of mathematical models for contact problems with respect to three-dimensional composite bodies.