Optimal control of the thickness of a rigid inclusion in equilibrium problems for inhomogeneous two-dimensional bodies with a crack

Abstract

The equilibrium problems for two-dimensional elastic body with a rigid delaminated inclusion are considered. In this case, there is a crack between the rigid inclusion and the elastic body. Non-penetration conditions on the crack faces are given in the form of inequalities. We analyze the dependence of solutions and derivatives of the energy functionals on the thickness of rigid inclusion. The existence of the solution to the optimal control problem is proved. For that problem the cost functional is defined by derivatives of the energy functional with respect to a crack perturbation parameter while the thickness parameter of rigid inclusion is chosen as the control function.

Optimal control of the radius of a rigid circular inclusion in inhomogeneous two-dimensional bodies with cracks

“…This means that displacements on the crack's faces are not required to have a prescribed structure of infinitesimal rigid displacements. Another difference between the problems that have been considered in [31] is that in the present work a family of rigid inclusions have not a fixed common boundary curve. The optimal control problem analyzed in this paper consists in the best choice of the radius r * ∈ [r 0 , R] of the circular rigid inclusion.…”

“…This approach to solving crack problems is characterized by inequality type boundary conditions at the crack faces, is indeed what we employ in the present paper. Within this approach, various problems for bodies with rigid inclusions has been successfully formulated and investigated using variational methods, see for example [9,25,27,31,32,33,34]. In contrast to a previous study of an optimal control problem for a two-dimensional elastic body with a rigid delaminated inclusion, as considered in [31], we suppose that crack curve touches the inclusion's boundary only at the crack's tip.…”

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

“…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]. For a heterogeneous two-dimensional body with a micro-object (defect) and a macro-object (crack), the anti-plane strain energy release rate is expressed by means of the stress intensity factor that is examined with respect to small defects such as microcracks, holes, and inclusions [7].…”

“…We prove the continuous dependence of the solutions with respect to the location parameter of the rigid inclusion which plays the role of a control parameter. 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].…”

Optimal size of a rigid thin stiffener reinforcing an elastic plate on the outer edge

“…Such linear models allow the opposite crack faces to penetrate to each other, such state leads to inconsistency with practical situations . Since the beginning of 1990, the crack theory with non‐penetration conditions of inequality type is under active study . Using the variational methods, various problems for bodies with rigid inclusions have been successfully formulated and investigated; see for example .…”