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

Abstract

A two-dimensional model describing the equilibrium state of a cracked inhomogeneous body with a rigid circular inclusion is investigated. The body is assumed to have a crack that reaches the boundary of the rigid inclusion. We assume that the Signorini condition, ensuring non-penetration of the crack faces, is satisfied. We analyze the dependence of solutions on the radius of 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, with the radius of inclusion chosen as the control parameter.

Fictitious Domain Method for Equilibrium Problems of the Kirchhoff-Love Plates with Nonpenetration Conditions for Known Configurations of Plate Edges

“…Suppose that the elastic coefficients depend on λ by formulas (10), moments m λ ij and stresses σ λ ij , i, j = 1, 2, are expressed by the formulas (12). Consider a following energy functional Π λ (Ω γ ; η) that has the same form as in (13) and defined on H(Ω γ ) ( the space H(Ω γ ) is defined in (11)). So, for any fixed λ ∈ (0; λ 0 ], we consider the following variational statement of an equilibrium problem for a plate:…”

“…The method of the fictitious domain has proven useful in establishing the solvability of problems that describe equilibrium of bodies with cracks crossing the external boundary at zero angles [2,4,8,9]. In the last years, within the framework of crack models subject to non-penetration boundary conditions, numerous works have been published, see, for example, [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25].…”

Equilibrium problem for an thermoelastic Kirchhoff–Love plate with a nonpenetration condition for known configurations of crack edges

“…Considering (19) with test functionsW = W +W ,W ∈ H 1 (Ω γ ), [W ]ν ≥ 0 on γ, we obtain by the Green's formulas (11), (12) the following boundary conditions (20) [σ ν (W ) − θ] = 0, σ τ (W ) = 0 on γ.…”

Equilibrium problem for a thermoelastic Kirchhoff–Love plate with a delaminated flat rigid inclusion

“…Among this type of nonlinear mathematical models, a wide range of various problems for Kirchhoff-Love plates in the framework of elastic constitutive relations has been studied [4,6,[13][14][15][16]. Problems for elastic plates with rigid inclusions are investigated in [17][18][19][20][21][22][23]. Thermoelastic models of plates with cracks have been studied, e.g.…”