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Existence of an optimal size of a rigid inclusion for an equilibrium problem of a Timoshenko plate with Signorini-type boundary condition
- Nyurgun Lazarev^{1}Email author,
- Tatiana Popova^{1} and
- Galina Semenova^{1}
https://doi.org/10.1186/s13660-015-0954-3
© Lazarev et al. 2016
Received: 13 July 2015
Accepted: 23 December 2015
Published: 16 January 2016
Abstract
We study the contact problems for elastic plates with a rigid inclusion. We consider the case of frictionless contact between the rigid part of the plate and a rigid obstacle. The contact is modeled with the Signorini-type nonpenetration condition. The deformation of the transversely isotropic elastic part of the plate is described by the Timoshenko model. We analyze the dependence of solutions to the contact problems on the size of rigid inclusion. The existence of a solution to the optimal control problem is proved. For that problem, the cost functional characterizes the deviation of the displacement vector from a given function, whereas the size parameter of rigid inclusion is chosen as the control function.
Keywords
MSC
1 Introduction
Applications of composite materials are growing vastly along with the development of research interests concerning material behavior. A large variety of new materials represents a challenge in mathematical modeling. In practice, the strengthening of the body is often achieved by reinforcement constructions on the outer edge, so it is important to study the mathematical models concerning the elastic bodies with rigid inclusions on the outer boundary. In this regard, it is of interest to investigate the contact problems for plates that are reinforced by rigid inclusions.
There are a number of works related to the modeling of contact problems for composites (see, e.g., [1–4]. It is known that the classical approach to contact problems is characterized by a given contact area [5, 6]. In contrast to this, for the mathematical models with unilateral boundary conditions of Signorini type, the contact area is not known a priori [7–14]. The power and generality of variational methods make it possible to solve various problems for elastic bodies and plates with inclusions; see, for example, [15–21]. In particular, a framework for two-dimensional elasticity problems with a thin delaminated rigid inclusion and nonlinear Signorini-type conditions on a part of boundary is proposed in [15]. The three-dimensional case is considered in [22]. The paper [23] is devoted to the analysis of the shapes of cracks and thin rigid inclusions in elastic bodies. The formula for the shape derivative of the energy functional is obtained for the equilibrium problem for an elastic body with a delaminated thin rigid inclusion [19]. For a Kirchhoff-Love plate containing a thin rigid inclusion, the cases both with and without delamination of inclusion are considered [24]. In that work, for the plate without delamination of inclusions, it is established that by passing to the limit in the equilibrium problems for volume inclusions embedded in an elastic plate as the size of the inclusions tends to zero we obtain the equilibrium problem for the plate with a thin inclusion.
In this paper, we study the nonlinear equilibrium problem for a plate subject to the Signorini condition on a part of the boundary. We consider volume inclusions defined by three-dimensional domains and thin inclusions defined by cylindrical surfaces. The present study investigates the effect of varying the inclusion size. We formulate an optimal control problem with the cost functional characterizing the deviation of the displacement vector from a given function. The control functions depend on the size parameter of the rigid inclusion. We prove the existence of an optimal inclusion size.
Additionally, we establish a qualitative connection between the contact problems for plates with rigid inclusions of varying size. In particular, we prove the strong convergence of the solutions for problems with volume inclusions to the solution of the problem for thin inclusion as the size parameter of the volume inclusion tends to zero.
2 Equilibrium problems for an elastic plate containing a rigid inclusion
Let us formulate the family of contact problems for an elastic inhomogeneous plate containing a volume rigid inclusion. We suppose that the inhomogeneous clamped plate may come into contact with a rigid obstacle. Let \(\Omega\subset\mathbf{R}^{2}\) be a bounded domain with a smooth boundary Γ. Suppose that the smooth unclosed curves γ and \(\gamma_{0}\) lie on Γ such that \(\operatorname{meas}(\gamma)>0\), \(\operatorname{meas}(\gamma_{0})>0\), and \(\bar{\gamma}\cap\bar{\gamma}_{0}=\emptyset\).
- (a)
the boundaries \(\partial{\omega}_{t}\) are smooth such that \(\partial\omega_{t}\in C^{0,1}\);
- (b)
\(\gamma=\partial\omega_{t}\cap\Gamma\) for all \(t\in(0,t_{0}]\);
- (c)
\(\omega_{t}\subset\omega_{t'}\) for all \(t,t'\in(0,t_{0}]\), \(t< t'\);
- (d)
for any fixed \(\hat{t}\in(0,t_{0})\) and any neighborhood \(\mathcal{O}\) of the domain \(\overline{\omega}_{t}\), there exists \(t_{\mathcal{O}}>\hat{t}\) such that \(\omega_{t}\subset \mathcal{O}\) for all \(t\in[\hat{t},t_{\mathcal{O}}]\);
- (e)
for any neighborhood \(\mathcal{O}\) of the curve γ, there exists \(t_{\mathcal{O}}>0\) such that \(\omega_{t}\subset{\mathcal{O}}\) for all \(t\in (0,t_{\mathcal{O}}]\);
- (f)
\(\bigcup_{t< t'}{\omega_{t}}=\omega_{t'}\) for all \(t'\in(0,t_{0}]\);
- (g)
the sets \(\Omega\backslash\bar{\omega}_{t}\) are Lipschitzian domains for all \(t\in(0,t_{0}]\).
We define a three-dimensional Cartesian space \(\{x_{1},x_{2},z\}\) such that the set \(\{\Omega\}\times\{0\}\subset\mathbf{R}^{3}\) corresponds to the middle plane of the plate. Fix the parameter \(t\in(0,t_{0}]\). According to our arguments, the rigid inclusion is specified by the set \(\omega_{t}\times[-h,h]\), that is, the boundary of the rigid inclusion is given by the cylindrical surface \(\partial\omega_{t}\times[-h,h]\). The elastic part of the plate corresponds to the domain \(\Omega\backslash\overline{\omega}_{t}\). The thickness of the plate is considered to be constant and equal to 2h.
Denote by \(({W},w)\) the vector of mid-plane displacements (\(x\in\Omega\)), where \({W}=(w_{1},w_{2})\) are the displacements in the plane, and \(\{x_{1},x_{2}\}\) and w are the displacements along the axis z. We denote the angles of rotation of a normal fiber by \(\psi=\psi(x)=(\psi_{1},\psi_{2})\) (\(x\in\Omega\)).
Remark 1
3 An optimal control problem
Theorem 1
There exists a solution of the optimal control problem (13).
Proof
Before proceeding further, we first prove the following lemma.
Lemma 1
Let \(t^{*}\in[0,t_{0}]\) be a fixed real number, and let \(\{t_{n}\}\subset[t^{*},t_{0}]\) be a sequence of real numbers converging to \(t^{*}\) as \(n\to\infty\). Then for an arbitrary function \({\eta}\in K_{t^{*}}\), there exist a subsequence \(\{t_{k}\}=\{t_{n_{k}}\}\subset\{t_{n}\}\) and a sequence of functions \(\{{\eta}_{k}\}\) such that \({\eta}_{k}\in K_{t_{k}}\), \(k\in\mathbf{N}\), and \({\eta}_{k}\to{\eta}\) weakly in \(H(\Omega)\) as \(k\to\infty\).
Proof
Now we can prove the following statement.
Lemma 2
Let \(t^{*}\in[0,t_{0}]\) be a fixed real number. Then \(\xi_{t}\to\xi_{t^{*}}\) strongly in \(H(\Omega)\) as \(t\to t^{*}\), where \(\xi_{t}\) is the solution of (6) corresponding to \(t\in(0,t_{0}]\), whereas \(\xi_{t^{*}}\) is the solution corresponding to (6) for \(t^{*}>0\) and to problem (11) for \(t^{*}=0\).
Proof
We will prove it by contradiction. Let us assume that there exist a number \(\epsilon_{0}>0\) and a sequence \(\{t_{n}\}\subset(0,t_{0}]\) such that \(t_{n}\to t^{*}\), \(\|\xi_{n}-\xi_{t^{*}}\|\geq\epsilon_{0}\), where \(\xi_{n}=\xi_{t_{n}}\), \(n\in\mathbf{N}\), are the solutions of (6) corresponding to \(t_{n}\).
Observe that, as \(t_{n} \to t^{*}\), there must exist either a subsequence \(\{t_{n_{l}}\}\) such that \(t_{n_{l}}\leq t^{*}\) for all \(l\in\mathbf{N}\) or, if that is not the case, a subsequence \(\{t_{n_{m}}\}\), \(t_{n_{m}}>t^{*}\) for all \(m\in\mathbf{N}\).
By Lemma 1, for any \({\eta}\in K_{t^{*}}\) there exist a subsequence \(\{t_{k}\}=\{t_{n_{k}}\}\subset\{t_{n}\}\) and a sequence of functions \(\{{\eta}_{k}\}\) such that \({\eta}_{k}\in K_{t_{k}}\) and \({\eta}_{k}\to{\eta}\) weakly in \(H(\Omega)\) as \(k\to\infty\).
4 Conclusion
The existence of a solution to the optimal control problem (13) is proved. For that problem, the cost functional \(J(t)\) characterizes the deviation of the displacement vector from a given function ξ∗, whereas the size parameter t of the rigid inclusion is chosen as the control function.
Lemmas 1 and 2 establish a qualitative connection between the contact problems for plates with rigid inclusions of varying size. In particular, it is shown that as the size parameter of volume rigid inclusion tends to zero, the solutions of the contact problems converge to the solution of the contact problem for a plate containing a thin rigid inclusion.
In contrast to the classical Kirchhoff-Love theory of plates, the Timoshenko-type plate theories take into account the transversal shear deformation and rotation inertia. Therefore, this model is more accurate, particularly, for moderately thin plates and when transverse shear plays a significant role; see [33, 34]. The reference [35] is devoted to the construction of the Timoshenko-Reissner theories by the asymptotic method. In this work, for the dynamic case, a method is suggested for the extension of the range of applicability of the Timoshenko-Reissner theory.
Declarations
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided 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.
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