Choosing an optimal shape of thin rigid inclusions in elastic bodies

Abstract

The optimal control problem for a three-dimensional elastic body containing a thin rigid inclusion as a surface is studied. It is assumed that the inclusion delaminates, which is why there is a crack between the elastic domain and the inclusion. The boundary conditions on the crack faces that exclude mutual penetration of the points of the body and inclusion are considered. The cost functional that characterizes the deviation of the surface force vector from the function prescribed on the external boundary is used; in this case, the inclusion shape is considered as a control function. It is proven that a solution of the described problem exists.

The Junction Problem for Two Weakly Curved Inclusions in an Elastic Body

“…The last few years have brought in the extensive studies of boundary value problems for the equilibria of elastic and inelastic bodies with cracks in the framework of models with the nonlinear boundary conditions on faces [1][2][3][4][5][6][7]. Similar results are obtained in the case of problems of equilibria for elastic bodies with thin inclusions in the presence of crack delamination [8][9][10][11][12][13][14]. Among the articles dealing with junction we note [15][16][17][18][19][20][21][22] which consider the junction problem of elastic objects.…”

On a Limiting Passage as the Thickness of a Rigid Inclusions in an Equilibrium Problem for a Kirchhoff-Love Plate with a Crack

“…The density of C ∞ 0 (Ω γ ) in H n 0 (Ω γ ) (n ∈ N) allows us to obtain from (17) the equality η − η = 0 in H 1 0 (Ω γ ) 2 × H 2 0 (Ω γ ). It remains to observe that η = η on γ ± by the construction.…”

“…At present, active studies of problems for various models of elastic bodies containing rigid inclusions and cracks with both linear and nonlinear boundary conditions are being carried out, and we refer the reader, for example, to [1][2][3][4][5][6][7][8][9][10][11]. Using the universality of the methods of the calculus of variations, various problems for bodies with rigid inclusions have been successfully formulated and investigated, see, for example, [1,[12][13][14][15][16][17][18][19][20][21]. In particular, within the framework of two-dimensional problems of elasticity theory, the first mathematical model on the equilibrium of a body with nonlinear Signorini-type conditions on a part of the boundary of a thin delaminated rigid inclusion was proposed in [1].…”

On hyperelastic solid with thin rigid inclusion and crack subjected to global injectivity condition

“…Applying such conditions, papers [5][6][7][8][9] study volumetric rigid inclusions with cracks on their boundaries. Later papers [10][11][12][13][14][15] consider the problems of elastic bodies with thin rigid inclusions and cracks in the same manner. In [16][17][18][19][20][21][22][23], for the linearly elastic solids with inclusions and cracks, shape sensitivity analysis was performed.…”