- Research
- Open Access
Existence of an optimal size of a delaminated rigid inclusion embedded in the Kirchhoff-Love plate
- Nyurgun Lazarev^{1}Email author
- Received: 26 June 2015
- Accepted: 10 September 2015
- Published: 6 October 2015
Abstract
We consider equilibrium problems for an inhomogeneous plate with a crack situated at the inclusion-matrix interface. The matrix of the plate is assumed to be elastic. The boundary condition on the crack curve are given in the form of inequalities and describes the mutual nonpenetration of the crack faces. We analyze the dependence of solutions on the size of rigid inclusion. The existence of the 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, while the size parameter of rigid inclusion is chosen as the control function.
Keywords
- crack
- plate
- rigid inclusion
- nonpenetration condition
- variational inequality
MSC
- 74G55
- 49J40
- 49J30
1 Introduction
It is well known that the presence of inclusions as well as of cracks in an elastic body can cause a high stress concentration. The mechanical and geometric properties of inclusions are responsible for crack initiation and propagation. Problems for different models of elastic bodies containing rigid inclusions and cracks with both linear and nonlinear boundary conditions have been under active study; see [1–9]. Using the variational methods, various problems for bodies with rigid inclusions have been successfully formulated and investigated; see for example [1, 10–14]. In particular, a framework for two-dimensional elasticity problems with nonlinear Signorini-type conditions on a part of boundary of a thin delaminated rigid inclusion is proposed in [1]. The three-dimensional case is considered in [15]. Reference [16] 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 [12]. For a Kirchhoff-Love plate containing a thin rigid inclusion the cases both with and without delamination of inclusion are considered [17]. 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 investigate equilibrium problems for the Kirchhoff-Love plate with a rigid inclusion. We consider volume inclusions defined by three-dimensional domains and thin inclusions defined by cylindrical surfaces. For all cases, we suppose that the crack is situated at the inclusion interface. 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 equilibrium problems for the Kirchhoff-Love plate with delaminated thin rigid inclusions and delaminated volume inclusions. 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. It should be noted that we impose boundary conditions of inequality type on the crack. Investigations on mathematical modeling of the crack theory with nonlinear conditions for nonpenetration of the opposite crack faces presented as a system of equalities and inequalities, with relevant bibliography can be found in [16, 18–24]. Other models of deformable solids can be found in [25, 26].
2 Equilibrium problems for an elastic plate containing a rigid inclusion
- (a)
the boundaries \(\partial{\omega}_{t}\) are smooth such that \(\partial\omega_{t}\in C^{1,1}\);
- (b)
\(\omega_{t}\subset\omega_{t'}\), \(\overline{\omega}_{t'}\subset\Omega\) for all \(t,t'\in(0,t_{0}]\), \(t< t'\);
- (c)
for any fixed \(\hat{t}\in(0,t_{0})\) and any neighborhood \(\mathcal{O}\) of the set \(\overline{\omega}_{t}\) there exists \(t_{\mathcal{O}}>\hat{t}\) such that \(\omega_{t}\subset \mathcal{O}\) for all \(t\in[\hat{t},t_{\mathcal{O}}]\);
- (d)
for any neighborhood \(\mathcal{O}\) of the curve γ there exists \(t_{\mathcal{O}}>0\) such that \(\omega_{t}\subset{\mathcal{O}}\) for all \(t\in (0,t_{\mathcal{O}}]\);
- (e)
\(\gamma\subset\partial\omega_{t}\) for all \(t\in(0,t_{0}]\);
- (f)
\(\bigcup_{t< t'}{\omega_{t}}=\omega_{t'}\) for all \(t'\in(0,t_{0}]\).
For simplicity, suppose the plate has a uniform thickness \(2h=2\). Let us assign a three-dimensional Cartesian space \(\{x_{1},x_{2},z\}\) with the set \(\{\Omega_{\gamma}\}\times\{0\}\subset\mathbf{R}^{3}\) corresponding to the middle plane of the plate. The curve γ defines a crack (a cut) in the plate. This means that the cylindrical surface of the through crack may be defined by the relations \({x}=(x_{1},x_{2})\in\gamma\), \(-1\le z\le1\) where \(|z|\) is the distance to the middle plane. For fixed \(t\in(0,t_{0}]\) the volume rigid inclusion is specified by the set \(\omega_{t}\times[-1,1]\), i.e. the boundary of the rigid inclusion is defined by the cylindrical surface \({\partial\omega_{t}}\times[-1,1]\). The elastic part of the plate corresponds to the domain \(\Omega\backslash\overline{\omega}_{t}\). Depending on the direction of the normal \(\nu=(\nu_{1},\nu_{2})\) to γ we will speak about a positive face \(\gamma^{+}\) or a negative face \(\gamma^{-}\) of the curve γ. The jump \([q]\) of the function q on the curve γ is found by the formula \([q]=q|_{ {\gamma}^{+}}-q|_{ {\gamma}^{-}}\).
Remark 1
The inequality (2) yields the equivalence of the standard norm and the semi-norm determined by the bilinear form \(B( \cdot, \cdot)\) in the space \({H(\Omega_{\gamma})}\).
3 An optimal control problem
Theorem 1
There exists a solution of the optimal control problem (9).
Proof
Before proceeding 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 \({\eta }=(V,v)\in K_{t^{*}}\) there exist a subsequence \(\{t_{k}\}=\{t_{n_{k}}\}\subset\{t_{n}\}\) and a sequence of functions \(\{{\eta}_{k}\}\) such that \({\eta}_{k}=(V_{k},v_{k})\in K_{t_{k}}\), \(k\in \mathbf{N}\) and \({\eta}_{k}\to{\eta}\) weakly in \(H(\Omega_{\gamma})\) 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 \(H(\Omega_{\gamma})\) as \(t\to t^{*}\), where \(\xi_{t}\) is the solution of (4) corresponding to \(t\in(0,t_{0}]\), while \(\xi_{t^{*}}\) is the solution corresponding to (4) for \(t^{*}>0\) and to the problem (7) 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 (4) corresponding to \(t_{n}\).
It can be proved analogously that \(\tilde{U}|_{\omega_{t^{*}}}= b(x_{2},-x_{1})+(c_{1},c_{2})\) a.e. in \(\omega_{t^{*}}\). Thus, we conclude that \(\tilde{\xi}|_{\omega_{t^{*}}}\in R(\omega_{t^{*}})\). Therefore, in all possible cases we have \(\tilde{\xi}|_{\omega_{t^{*}}}\in R(\omega_{t^{*}})\).
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 \(m\in\mathbf{N}\).
From 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 \(H(\Omega_{\gamma})\) as \(k\to\infty\).
4 Conclusion
The existence of the solution to the optimal control problem (9) is proved. For that problem the cost functional \(J(t)\) characterizes the deviation of the displacement vector from a given function \(\xi^{*}\), while the size parameter t of the rigid inclusion is chosen as the control function.
Lemmas 1 and 2 establish a qualitative connection between the equilibrium problems for plates with rigid delaminated inclusions of varying thickness. In particular it is shown that as the thickness of volume rigid inclusion tends to zero, the solutions of the equilibrium problems converge to the solution of the equilibrium problem for a plate containing a thin rigid delaminated inclusion.
The obtained results can be used to investigate some mathematical and mechanical problems concerning inhomogeneous solids with rigid inclusions.
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.
Authors’ Affiliations
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