Abstract
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 εN 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).
Highlights
An equilibrium problem of a Kirchhoff–Love plate containing a nonhomogeneous inclusion is considered
The problem is formulated as a variational one; namely, as a minimization problem of the energy functional over a set of admissible deflections in the Sobolev space H 2. This implies that the deflections function is a solution of a boundary value problem for bi-harmonic operator
The method is based on variational properties of the solution to the corresponding minimization problem and allows for finding a limit problem for any N < 1 simultaneously
Summary
An equilibrium problem of a Kirchhoff–Love plate containing a nonhomogeneous inclusion is considered. The problem is formulated as a variational one; namely, as a minimization problem of the energy functional over a set of admissible deflections in the Sobolev space H 2. Note that there are not so many works devoted to study of models of thin inclusions in plates. We mention paper [23], where the mechanical behavior of an anisotropic nonhomogeneous linearly elastic three-layer plate with soft adhesive, including the inertia forces, was studied, and the various limiting models in the dependence of the size and the stiffness of the adhesive was derived. The problem under consideration in the present paper is different from the mentioned paper because we consider the hard inhomogeneity lying strictly inside the plate and derive limiting problem depending on the size and stiffness of the inclusion.
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