Geodesic
Problem of the equilibrium of a two-dimensional elastic body with two contacting thin rigid inclusions
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh international winter mathematical school "Modern methods of function theory and related problems", Voronezh, January 27 - February 1, 2023, Part 1, Tome 227 (2023), pp. 51-60.

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A new nonlinear mathematical model is proposed that describes the equilibrium of a two-dimensional elastic body with two thin rigid inclusions. The problem is formulated as a minimizing problem for the energy functional over a nonconvex set of possible displacements defined in a suitable Sobolev space. The existence of a variational solution to the problem is proved. Optimality conditions and differential relations are obtained that characterize the properties of the solution in the domain and on the inclusion; these conditions are satisfied for sufficiently smooth solutions.
Keywords: crack, rigid inclusion, nonpenetration condition, variational problem 
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N. P. Lazarev; V. A. Kovtunenko. Problem of the equilibrium of a two-dimensional elastic body with two contacting thin rigid inclusions. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh international winter mathematical school "Modern methods of function theory and related problems", Voronezh, January 27 - February 1, 2023, Part 1, Tome 227 (2023), pp. 51-60. http://geodesic.mathdoc.fr/item/INTO_2023_227_a3/

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