Asymptotic Justification of the Models of Thin Inclusions in an Elastic Body in the Antiplane Shear Problem

The equilibrium problem for an elastic body having an inhomogeneous inclusion with curvilinear boundaries is considered within the framework of antiplane shear. We assume that there is a power-law dependence of the shear modulus of the inclusion on a small parameter characterizing its width. We justify passage to the limit as the parameter vanishes and construct an asymptotic model of an elastic body containing a thin inclusion. We also show that, depending on the exponent of the parameter, there are the five types of thin inclusions: crack, rigid inclusion, ideal contact, elastic inclusion, and a crack with adhesive interaction of the faces. The strong convergence is established of the family of solutions of the original problem to the solution of the limiting one.


INTRODUCTION
The equilibrium problem for an elastic body having an inhomogeneous inclusion with a Lipschitz boundary is considered within the framework of the antiplane shear model. We assume that shear moduli of the inclusion and its width depend on a small parameter ε. In this case, the shear moduli of the material of the inclusion are proportional to ε N , where N is an arbitrary real number. Mixed boundary conditions are set on the outer boundary of the elastic body. The equilibrium problem is formulated as the problem of minimizing the energy functional in the Sobolev space H 1 .
The main goal of the paper is to investigate the asymptotics of the solution as ε vanishes. Basing on the variational properties of the solution to the minimization problem, we propose some method of asymptotic analysis that allows us to prove the strong convergence of the family of solutions of the original problem to the solution of the limiting one. This method made it possible to obtain the limiting model of an elastic body with a thin inclusion for all values of the parameter N . Moreover, we show that, depending on N , there are the five types of thin inclusions: a crack for N > 1; a rigid inclusion for N < −1; an ideal contact for −1 < N < 1 which means that the limiting problem is the equilibrium problem for a homogeneous body; an elastic inclusion for N = −1; and a crack with adhesive contact of the faces for N = 1. Also we established the strong convergence of the family of solutions of the original problem to the solution of the limiting one.
Let us present a brief overview of the works closest to the present study. In [1][2][3][4][5], the asymptotic analysis was carried out for the models of elastic bodies bonded together by a thin adhesive layer. Moreover, in these articles, unlike the present paper, the method of asymptotic justification of the limiting model (Γ-convergence, formal asymptotic expansion, etc.) depended on the elastic properties of the adhesive layer.
In the recent studies [6][7][8][9][10], some models were investigated of elastic bodies with thin inclusions and cracks, in particular, those accounting for the interaction of crack faces with each other. The results of numerical simulation of the behavior of elastic bodies with thin inclusions can be found in [11][12][13][14][15][16]. Here, we note the works [17][18][19] on the numerical solution of the problems of the theory of the cracks with possible contact of faces.

STATEMENT OF THE PROBLEM
In the space R 2 we define a domain Ω with a Lipschitz boundary ∂Ω. Let Γ N and Γ D be some parts of ∂Ω with nonzero Hausdorff measures such that Γ D ∩ Γ N = ∅ and Γ D ∪ Γ N = ∂Ω. Denote by S the intersection of Ω with the abscissa axis. Let us define two intervals I 1 and I 2 lying on the abscissa axis and included in Ω such that I 1 ⊂ I 2 and I 2 ⊂ S. In addition, we assume that there is a convex subdomain D ⊂ Ω such that its intersection with the abscissa axis is I 2 and, moreover, D ⊂ Ω.
Consider the two Lipschitz functions ψ ± defined on S such that ±ψ ± > 0 on I 1 and ψ ± = 0 on S \ I 1 . Let us fix a small parameter ε > 0 and introduce the notations: Finally, we assume that I 1 is subdivided into five subsets S i ⊂ I 1 , where S i is the union of finitely many intervals or an empty set, i = 1, 5; whereas Condition (1) means that S 4 and S 5 lie strictly inside I 1 . Finally, we define We assume that Ω is an elastic inhomogeneous body consisting of an elastic matrix Ω ε 0 that contains strictly inside it an inhomogeneous elastic inclusion Ω ε m . Wherein, the stress-strain state of the body Ω is described within the framework of the antiplane shear model. This state occurs when the displacement field has one nonzero component directed perpendicular to the plane in which the other two components of the displacement field are equal to zero (see [20]). Denote by k 0 , k ε i the shear moduli for the subdomains Ω ε 0 and Ω ε m i respectively, i = 1, 5. We assume that k 0 > 0 is constant, while k ε i depend on ε as follows: In the domain Ω, we define the following piecewise constant function: We assume that the body Ω is fixed on some part Γ D of the outer boundary ∂Ω, whereas the force g ∈ L 2 (Γ N ) is applied to the remaining part Γ N . In the case of small deformations, the formulation of the problem of an elastic body equilibrium is as follows: where Note that (2) is a weak variational formulation of the boundary value problem which was studied in [12,[21][22][23] within the framework of the antiplane shear model. Here ν is the unit outward normal to ∂Ω. The further goal is to investigate the behavior of the solution to problem (2) as ε vanishes.

DECOMPOSITION OF THE PROBLEM AND TRANSFORMATION OF COORDINATES
Let us rewrite problem (2) in equivalent form decomposing it into several subproblems defined in the domains Ω ε ± and Ω ε m whose solutions are related on the common parts of the boundaries of the domains. To this end, we define the set Then problem (2) can be reformulated as follows: It is required to find a triple of functions (u ε− , u ε+ , u εm ) ∈ K ε satisfying the variational equality By analogy to [16], we can prove that problem (3) has the unique solution u ε = (u ε− , u ε+ , u εm ) ∈ K ε . Moreover, u ε± is the restriction of u ε to Ω ε ± , while u εm is the restriction of u ε to Ω ε m , where u ε is the solution of (2). Put We now define the coordinate transformations which map the domains Ω ε ± and Ω ε m to the domains that are fixed and independent of ε. To this end, we choose an arbitrary cut-off function θ ∈ C ∞ (R 2 ) so that In the domains Ω ± and Ω m , we consider the following coordinate transformations: It is easy that transformations (4) are one-to-one mappings of Ω ± to Ω ε ± . Moreover, due to the smoothness of the functions ψ ± , the mappings (4) and (5) also establish a one-to-one correspondence between the spaces H 1 (Ω ± ), H 1 (Ω m ), and H 1 (Ω ε ± ), H 1 (Ω ε m ), respectively. Note that the norms of the corresponding elements are related via the usual formula for changing variables in integrals (see [24,Lemma 3.2] or [25, p. 46]). Moreover, K ε is mapped to the set K, where We apply (4) and (5) to (3). In result, we conclude that the triple of functions u ε − , u ε + , u ε m ∈ K is the unique solution of the following variational equality: where Hereinafter, the subscripts after the comma indicate differentiation with respect to the corresponding coordinate. Note that in (7) all exponents of ε are positive.
In what follows, it is convenient to use the asymptotic representation for bilinear forms b ε ± up to o(1); namely, the following formulas hold:

THE LIMITING PROBLEM
In this section, we will justify the passage to the limit as ε → 0. The following auxiliary lemma is true (the proof can be found in [26,27]): We now state and prove the main (6), and let (u − , u + ) ∈ K l be a solution of the following variational inequality: where Then the following convergence takes place as ε → 0: where Proof. We insert u ε − , u ε + , u ε m into (6) as some test function. As a result, we arrive at the estimate where C is a certain constant independent of ε. From (11), Lemma 1, and the definition of K we have u ε m L 2 (Ωm i ) ≤ c, i = 1, 2, 4, 5.
Inequalities (11) and (12) imply the existence of such that for some subsequence ε n , n ∈ N, still denoted by ε, the following convergences hold as ε → 0: Next, we investigate the properties of the limit functions in (14). It is obvious that Let us show that Indeed, from the proof of the theorem (see Section 4.3, [28]) we can conclude that, for v m ∈ L 2 (Ω m ) such that v m,2 ∈ L 2 (Ω m ), the linear trace operators on S ± are defined. Whence it follows that At the same time, owing to u ε m | S ± = u ε ± | S (see the definition of K) and the convergence u ε ± | S → u ± | S strongly in L 2 (S), we obtain (17).
Let us now find a variational equality that is satisfied by the limit functions u − , u + , and u m in (14). To this end, we take the pair otherwise. (We remind that in this case v + (z 1 , 0) = v − (z 1 , 0).) Then, since ψ ± are Lipschitz functions, the triplet (v − , v + , v m ) belongs to the set K and can be inserted as a test function in (6). Taking (14) into account, after passage to the limit as ε → 0, we obtain the following variational equality for (u − , u + , u m ): Further, owing to (9) and (21), we have the chain of equalities whereas from (10) and (20) we obtain Thus, we proved that the limit functions u − and u + satisfy In this case, u m is determined by formulas (9) and (10).
From the Poincar´e inequality and the continuity of the trace operator, it follows that the set K l is the Banach space with the norm Due to the fact that (C 1 (Ω − ) × C 1 (Ω + )) ∩ K l is dense in K l with respect to the norm · K l , we obtain variational equality (8) for all pairs (v − , v + ) ∈ K l . Let us show that (8) has a unique solution. This will mean that the convergence in (14) is valid not only for some subsequence, but also for ε → 0. First, note that (1) implies that there is a constant (8) is equivalent to the following minimization problem: Find a pair (u − , u + ) ∈ K l such that where To prove the solvability of (23), it suffices to establish the coercivity of the functional Π, which follows from the Poincar´e and Cauchy-Bunyakovsky inequalities (see [8,9,29], where a similar situation was considered for models of two-dimensional theory of elasticity). The uniqueness of the solution to problem (23) is obvious.
Let us show that strong convergence takes place in (14). For this we insert u ε − , u ε + , and u ε m into (6) as test functions. After passing to the limit as ε → 0 and taking into account (14), weak semicontinuity of norms in the spaces H 1 Γ D (Ω ± ) and L 2 (Ω m i ), i = 1, 2, 3, 4, 5, and also the fact that the functions u ± and u m satisfy (22), we conclude that p i = 0 and q j = 0 fori = 1, 2, 3, 4 and j = 1, 2, 3, 5. Inserting again u ε − , u ε + , and u ε m into (6) and passing to the limit in the resultant identity, we obtain the convergence of the norms of the functions from (14) to the norms of the corresponding limits. This fact and the weak convergence of the corresponding sequences entail the strong convergence in (14). Finally, the strong convergence in (14) and the lemma imply (13).
The theorem is proved.
Let us rewrite problem (8) in equivalent form. To this end, define Then the pair (u − , u + ) is a solution to problem (8) if and only if the function u ∈ K 0 , where is a solution to the following variational equality: Hereinafter [·] denotes the jump of a function on the corresponding curve (in this case, S 5 ).
4. CONCLUDING REMARKS In this article we propose a method of asymptotic substantiation of the models of elastic bodies having thin inclusions. The method is based on the variational properties of the solution and allows us simultaneously and from a unified position to construct all possible cases of thin inclusions within the framework of the antiplane shear model. Namely, the set S 1 simulates a thin rigid inclusion on which the displacement jump is zero, while the jump of normal stresses can be nonzero; the set S 2 models a perfect contact for which the jumps of displacements and their normal derivatives equal zero; the set S 3 corresponds to a crack; the set S 4 models a thin elastic inclusion; and the set S 5 simulates a crack with adhesive contact of the faces; i.e., normal derivative of displacements on each of the faces is proportional to the displacement jump.
Below, for illustrative purposes, we consider each case separately and write the equivalent differential formulations with the complete set of boundary conditions on inclusions. All differential formulations are derived from the corresponding variational ones by using the generalized Green's formula and inserting some properly chosen test functions. We omit the derivation details and only note that these details can be found, for example, in [30]. 4.1. N < −1. A Model of a Thin Rigid Inclusion in an Elastic Body Let S = S 1 . Then the variational formulation of the equilibrium problem for an elastic body with a thin rigid inclusion is as follows: It is required to find a function u 1 ∈ K 1 which satisfies the variational equality If u 1 has additional smoothness then we can write for (25) the equivalent differential formulation: Note the papers [2,[32][33][34][35][36] in which the problems on thin rigid inclusions for boundary value problems with the Laplace operator were considered. In [11,14,37,38], the problems of the theory of elasticity for bodies with thin rigid inclusions were under study. (−1, 1). A Model of Antiplane Shear of a Homogeneous Body (without Inclusions) Let S = S 2 . Then the set K 0 in (24) will coincide with the entire space H 1 Γ D (Ω), and the variational formulation of the equilibrium problem for an elastic body is as follows:

N ∈
It is required to find such a function u 2 ∈ H 1 Γ D (Ω) that satisfies the variational equality Problem (26) describes the equilibrium of a homogeneous elastic body without any inclusions within the framework of the antiplane shear model. In this case, the differential formulation has the known form (for instance, see [20]):

N > 1.
A Model of an Elastic Body with a Crack Let S = S 3 . Then the variational formulation of the equilibrium problem for an elastic body with a crack is as follows: It is required to find such a function u 3 ∈ K 3 that satisfies the variational equality The differential formulation of problem (27) has the following form: Due to condition (1), we cannot formally put S = S 4 . Nonetheless, extending the Lipschitz functions ψ ± till the intersection with the abscissa axis so that the assumptions of Section 1, in particular, the conditions on the intervals I 1 and I 2 are fulfilled, we arrive at the problem in which S 2 = I 1 \ S 4 and k ε 2 = k 0 . Then, after passing to the limit as ε → 0, we obtain the following variational problem on the equilibrium of an elastic body containing a thin elastic inclusion: Applying the generalized Green's formula, we derive the differential formulation of problem (28) that looks as follows: Such a model arises when describing elastic bodies containing nanofibers which have surface stresses (for example, see [6,12,15,39,40]). We also note [7,10,16,29,41] where the nonlinear models were under study for thin elastic inclusions in elastic bodies in the presence of delamination cracks.

N = 1. A Model of an Elastic Body Having a Crack with Adhesive Interaction of the Faces
As in Section 4.3, we continue the Lipschitz functions ψ ± defined on S 5 so that S 5 ⊂ I 1 and all conditions of Section 1 are fulfilled. Putting S 2 = I 1 \ S 5 and k ε 2 = k 0 and passing to the limit as ε → 0, we arrive at the following variational problem on the equilibrium of an elastic body with a crack in the presence of adhesive contact of the crack faces: The differential formulation of problem (29) is as follows: where S n 5 and S p 5 are positive and negative edges of the crack S 5 respectively, ν is the unit outward normal to ∂Ω ∪ S 3 .
Problems of this kind were widely studied in [9,10,13,42,43] for the models of elastic bodies with cracks; in particular, within the framework of the antiplane shear model [12,44,45], and also in [1,[3][4][5]46] for the problems of gluing of elastic bodies. OPEN ACCESS This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.