Asymptotic Justiﬁcation of Models of Plates Containing Inside Hard Thin Inclusions

: 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 ) .


Introduction
An equilibrium problem of a Kirchhoff-Love plate containing a nonhomogeneous inclusion is considered. It is assumed that the elastic properties of the inclusion depend on a small parameter characterizing width of the inclusion ε as ε N with N < 1. 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 (pure bending, see, e.g., [1][2][3][4]).
The aim of the present work is to justify passing to the limit as ε → 0. To do this, we apply a method that was originally introduced in [5,6] for problems of gluing plates. 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. It is shown that there exist two types of hard inclusions in dependence of N: thin rigid inclusion (N < −1) and thin elastic inclusion (N = −1). In case N ∈ (−1, 1), the influence of the inhomogeneity disappears in the limit. We get limit problems in a variational form, which is convenient, for example, for numerical analysis by the finite element method.
Let us give a short survey of works that are close to the present investigation. Note that there are not so many works devoted to study of models of thin inclusions in plates. We mention [7][8][9], in which thin elastic inclusions in pates were studied. Papers [10][11][12][13] are devoted investigations of thin rigid inclusions. We refer to [14][15][16][17][18][19][20][21] for asymptotic analyses for different models of bonded structures in Elasticity. We indicate also paper [22], where a geometry-dependent state problem for a heterogeneous medium with defects is investigated in framework of anti-plane elasticity.
Finally, 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. Wherein, the plate size does not vary and remains constant.

Statement of Problem
Let us fix a small parameter ε ∈ (0, 1) and consider an inhomogeneous rectangular plate Ω ⊂ R 2 with a thin rectangular inclusion Ω ε inc ⊂ Ω of width 2εd, where d is diameter of Ω. Let us specify some notations: Note that, for all small enough ε > 0 a family of subdomains Ω ε inc lies strictly inside Ω. Besides, let us define the following notations: We assume that S inc is divided into three subsets S α ⊂ S inc , where each S α is an union of finite number of segments or empty set, α = 1, 2, 3.
In our consideration, Ω is a composite plate, consisting of the elastic matrix Ω ε mat and the inhomogeneous inclusion where Moreover, in the sequel, we will use the following notations: Denote, by E 0 , E ε α and k 0 , k α , Young's modules and Poisson's ratios of parts Ω mat and Ω ε α of the composite plate Ω, respectively, α = 1, 2, 3. The compound character of the structure is expressed by the fact that E 0 , k 0 , and k α are constants, while Young's modulus E ε α depends on ε, as follows: where N 1 , N 2 , N 3 are real numbers, such that Parameters N 1 and N 2 correspond to hard inclusions in the plate Ω (see [6,24,25]). Moreover, put N 0 = 0.
Denote, by w, deflections of the composite plate Ω. Then the bending moments are defined by formulae (see, e.g., [26,27]) where the positive definite and symmetric tensor {d ijkl } is orthotropic with the following components: h is a thickness of the plate Ω that is constant. Note paper [28], where it was shown non-standard behaviour in the asymptotic two-dimensional reduction from three-dimensional elasticity, when the thickness and size of inclusions depend on the same parameter.
The potential energy functional of the plate has the following representation (see [27]): where f ∈ L 2 (Ω) is a bulk force acting on the plate Ω. Subsequently, the equilibrium problem of nonhomogeneous plate clamped on the external boundary ∂Ω can be formulated as the minimization problem: find a function w ε ∈ H 2 0 (Ω) such that Problem (2) is known to have a unique solution w ε (see, e.g., [26,29]), which satisfies the variational equality: Moreover, the function w ε is a unique solution the following boundary value problem: where ν is a unit normal vector ∂Ω.

Limit Problem
To justify passing to the limit as ε → 0, we need some auxiliary lemma proved in [5,6].

Lemma 1 (Poincare-typé inequalities). For any triplet
Our main result is the following theorem. (8); let w 0 ∈ K 0 be a solution to the following variational equality: Denote, by w ± , a restriction of w to subdomain Ω ± and, moreover, put Then, the following convergences w ε ± w ± weakly in H 2 (Ω ± ), w ε m w m weakly in L 2 (Ω m ), take place as ε → 0.
From definition of the set K ε , after passing to the limit as ε → 0, we obtain Because w m,1 = 0 in Ω m (see (16)), w m does not depend on z 2 . Therefore, taking into account (17), we conclude that there exists a function β(z 2 ) ∈ L 2 (Ω m ) such that Condition (18) means that the function w m is affine in the domain Ω m with respect to z 2 , i.e., there exists δ, γ ∈ R, such that Because of (19), we have Now, let us show that w ± satisfy the following equality: Indeed, from the relation Due to estimate (13) and the equalities w ε m,1 (±d, z 2 ) = εw ε ± (0, z 2 ) for z 2 ∈ (a, b) (see the definition of the set K ε ), we obtain as ε → 0. From (15) (the first line) and the compactness of trace operator, it follows w ε ±,1 → w ±,1 strongly in L 2 (S) as ε → 0, and (22) holds. At last, using the same arguments as in [6], we can prove additionally that and, moreover, 22 in Ω 2 , Now, let us define a function Conditions (19)- (23) imply that the function w 0 belongs to the set K 0 .

Concluding Remarks
We proposed a method of asymptotic derivation of plate models containing hard thin inclusions lying strictly inside the plate. The method is based on the variational properties of the solution of the equilibrium problem and allows for one to simultaneously construct all possible cases of hard thin inclusions. It is shown that there exist two type of thin inclusions in the Kirchhoff-Love plate, namely, the rigid inclusion S 1 for N < −1 and the elastic inclusion S 2 for N = −1. The inhomogeneity disappears in the case of N ∈ (−1, 1). The last means that we have no any peculiarity along the set S 3 .
In the conclusion, we note that the proposed method does not allow considering the case of the exponent N ≥ 1 simultaneously with the case of the exponent N < 1, because, for the first case, we need to use other type of test functions (see [6]), which cannot be substituted in variational equality for the second case of the exponent.

Conflicts of Interest:
The authors have not conflict of interest.