Optimal location of a rigid inclusion in equilibrium problems for inhomogeneous two‐dimensional bodies with a crack

Abstract

A two‐dimensional model describing equilibrium of a cracked inhomogeneous body with a rigid inclusion is studied. We assume that the Signorini condition, ensuring non‐penetration of the crack faces, is satisfied. For a family of corresponding variational problems, we analyze the dependence of their solutions on the location of the rigid inclusion. The existence of a solution of the optimal control problem is proven. For this problem, a cost functional is defined by an arbitrary continuous functional on a suitable Sobolev space, with the location parameter of the inclusion is chosen as a control parameter.

Optimal location of a rigid inclusion in equilibrium problems for inhomogeneous Kirchhoff–Love plates with a crack

“…An alternative approach to the formulation of crack problems that excludes mutual penetration of the crack faces has been investigated since the 1990s. This approach is characterized by the Signorinitype boundary conditions at the crack faces [1,2,[12][13][14][15][16][17][18][19][20][21]. In the last 10 years, within the framework of crack models subject to non-penetration (contact) conditions, a number of papers have been published concerning shape optimization problems for delaminated rigid inclusions; see, for example, [2,[22][23][24][25][26][27].…”

“…In contrast to the previous results related to the optimal size of rigid inclusions [24,26,32], to justify a passage to the limit in variational inequalities, we construct a suitable strongly converging sequence of test functions. The result concerning the optimal location of a rigid inclusion for a two-dimensional non-linear model describing the equilibrium of a cracked composite solid was obtained in [19].…”

On the connection between two equilibrium problems for cracked bodies in the cases of thin and volume rigid inclusions

“…The use of such boundary conditions, in contrast to the classical formulations of the problems of the crack theory [23,24], does not impose an a priori known zone of contact for the crack faces. The wide range of applicability of variational methods enables successful formulation and investigation of various problems for solids with rigid or elastic inclusions, see, for example [25][26][27][28][29][30][31][32][33][34][35][36][37]. In particular, a foundational reference for two-dimensional elasticity problems with Signorini-type conditions on a part of the boundary of a thin delaminated rigid inclusion is [4].…”