Abstract:
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.
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