The paper addresses the solution existence of equilibrium problem for a layered composite containing cracks and asymptotic behavior of solutions. Boundary conditions considered at the crack faces have an inequality type and describe a mutual nonpenetration and adhesion. At the external boundary of the composite, we impose the Neumann condition, which implies a non-coercivity of the boundary value problem. A solution existence of the problem is proved, and conditions imposed on the external forces guaranteeing the solution existence are found. An asymptotic analysis with respect to the parameter characterizing the elasticity tensor is provided, and limit models are analysed. In particular, the limit models describe the composite with a volume rigid inclusion and with a cavity.
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