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This paper concerns an equilibrium problem for an an elastic body with a thin rigid inclusion crossing an external boundary of the body at zero angle. The inclusion is assumed to be exfoliated from the surrounding elastic material that provides an interfacial crack. To avoid nonphysical interpenetration of the opposite crack faces, we impose inequality type constraints. Moreover, boundary conditions at the crack faces depend on a positive parameter describing a cohesion. A solution existence of the problem with different conditions on the external boundary is proved. Passages to the limit are analyzed as the damage parameter tends to infinity and to zero. Finally, an optimal control problem with a suitable cost functional is investigated. In this case, a part of the rigid inclusion is located outside of the elastic body, and a control function is a shape of the inclusion. [ABSTRACT FROM AUTHOR]

In the framework of 2D-elasticity, an equilibrium problem for an inhomogeneous body with a curvilinear inclusion located strictly inside the body is considered. The elastic properties of the inclusion are assumed to depend on a small positive parameter δ characterizing its width and are assumed to be proportional to δ − 1 . Moreover, it is supposed that the inclusion has a curvilinear rough boundary. Relying on the variational formulation of the equilibrium problem, we perform the asymptotic analysis, as δ tends to zero. As a result, a variational model of an elastic body containing a thin curvilinear rod is constructed. Numerical calculations give a relative error between the initial and limit problems depending on δ. [ABSTRACT FROM AUTHOR]

The effect of a functional gradient in the cross-section material (FGM) of a thin ribbon-like interfacial deformable inclusion on the stress–strain state of a piecewise homogeneous linear–elastic matrix under longitudinal shear conditions is considered. Based on the equations of elasticity theory, a mathematical model of such an FGM inclusion is constructed. An analytic–numerical analysis of the stress fields for some typical cases of the continuous functional gradient dependence of the mechanical properties of the inclusion material is performed. It is proposed to apply the constructed solutions to select the functional gradient properties of the inclusion material to optimize the stress–strain state in its vicinity under the given stresses. The derived equations are suitable with minor modifications for the description of micro-, meso- and nanoscale inclusions. Moreover, the conclusions and calculation results are easily transferable to similar problems of thermal conductivity and thermoelasticity with possible frictional heat dissipation. [ABSTRACT FROM AUTHOR]

A numerical–analytical approach to the problem of determining the stress–strain state of bimaterial structures with interphase ribbon-like deformable inhomogeneities under combined force and dislocation loading has been proposed. The possibility of delamination along a part of the interface between the inclusion and the matrix, where sliding with dry friction occurs, is envisaged. A structurally modular method of jump functions is constructed to solve the problems arising when nonlinear geometrical or physical properties of a thin inclusion are taken into account. A complete system of equations is constructed to determine the unknowns of the problem. The condition for the appearance of slip zones at the inclusion–matrix interface is formulated. A convergent iterative algorithm for analytical and numerical determination of the friction-slip zones is developed. The influence of loading parameters and the friction coefficient on the development of these zones is investigated. [ABSTRACT FROM AUTHOR]

An equilibrium problem of the Kirchhoff–Love plate containing a nonhomogeneous inclusion is considered. It is assumed that elastic properties of the inclusion depend on a small parameter characterizing the width of the inclusion ε as ε N with N < 1 . The passage to the limit as the parameter ε tends to zero is justified, and an asymptotic model of a plate containing a thin inhomogeneous hard inclusion is constructed. It is shown that there exists two types of thin inclusions: rigid inclusion ( N < − 1 ) and elastic inclusion ( N = − 1 ). The inhomogeneity disappears in the case of N ∈ (− 1 , 1) . [ABSTRACT FROM AUTHOR]