On rigid inclusions and cavities in elastic body with a crack: non-coercive case

Аннотация

In the paper, we consider an equilibrium problem for an elastic body with a crack
in a case of Neumann boundary conditions at the external boundary. The Neumann boundary conditions imply a non-coercivity of the problem. Inequality constraints are imposed on the solution providing a mutual non-penetration between the crack faces.
Various passages to limit with respect to the parameter characterizing a rigidity of the body are analyzed, and
limit models are investigated. In particular,
an existence of solutions
is proved for all cases considered; necessary and sufficient conditions imposed on the external forces are found.
The limit models describe the elastic body with a volume rigid inclusion and the body with a cavity.
These results are obtained both for the case when the crack is located inside the elastic body and for the case when it extends to the outer boundary.