Elastic body with thin nonhomogeneous inclusion in non-coercive case

Abstract

The paper addresses analysis of equilibrium state of an elastic body with a thin nonhomogeneous inclusion in a non-coercive case. We assume that a part of the inclusion is located outside the elastic body, and the traction free boundary condition at the external boundary of the elastic body does not provide a coercivity of the problem. The inclusion is delaminated from the surrounding elastic material that implies a presence of the interfacial crack. Constraint boundary conditions at the crack faces describe a mutual nonpenetration and have an inequality type form. We prove a solution existence of the equilibrium problem, analyze a passage to the limit as the rigidity parameter of the inclusion tends to zero, and prove a solution existence of an inverse problem. The inverse problem assumes that along with the displacement field we have to find an elasticity coefficient for the inclusion provided that an additional information is given which can be found from a measurement.

Thin inclusion at the junction of two elastic bodies: non-coercive case

“…General approaches to non-coercive boundary value problems for elliptic equations can be found in the book [27]. Various equilibrium problems for elastic bodies containing thin inclusions in non-coercive cases are investigated in the works [28][29][30].…”

The homogenized dynamical model of a thermoelastic composite stitched with reinforcing filaments

“…Owing to the similarity of such an asymptotic analysis, we do not perform it in the present paper. Also, let us mention some recent papers devoted to the studies of the so-called thin inclusion problems in elastic bodies [19][20][21][22][23][24]. Finally, we note papers [25][26][27][28][29] devoting investigations of homogenization problems of high-contrast periodic composites in the presence of transmission conditions between phases.…”