Optimal location of a rigid inclusion in equilibrium problems for inhomogeneous Kirchhoff–Love plates with a crack

Abstract

A non-linear model describing the equilibrium of a cracked plate with a volume rigid inclusion is studied. We consider a variational statement for the Kirchhoff–Love plate satisfying the Signorini-type non-penetration condition on the crack faces. For a family of problems, we study the dependence of their solutions on the location of the inclusion. We formulate an optimal control problem with a cost functional defined by an arbitrary continuous functional on a suitable Sobolev space. For this problem, the location parameter of the inclusion serves as a control parameter. We prove continuous dependence of the solutions with respect to the location parameter and the existence of a solution of the optimal control problem.

1. Introduction

The creation and use of high-performance engineered composites with unique properties is one of the foundations of technological progress. The development and implementation of composites involve a preliminary stage of simulation experiments based on suitable mathematical models. In this stage, optimal geometrical and mechanical properties of composites can be predicted and optimized. The accuracy of precalculations depends on the quality of selected mathematical models. In this regard, justification and examination of the most accurate mathematical models describing the stress–strain state of composites is a promising direction of scientific research. It is well known that under both exploitation loads as well as specific operating conditions, inhomogeneous properties of composite structures can lead to the nucleation of cracks. For cracked solids, it should be noted that the adequacy of a mathematical model to represent the actual physical process depends to a considerable extent on the choice of the boundary conditions on cracks.

From a mathematical point of view, the presence of cracks implies the formulation of a problem in a domain with non-regular boundary components. This circumstance represents one of the principal difficulties of crack problems [1–5]. For composite solids, the analysis of the influence of geometric and mechanical properties of inclusions on the sensitivity of the stress at a crack-tip is a challenging mathematical problem [6–10]. Moreover, it should be noted that imposing boundary conditions of an equality type on a crack may lead to physical inconsistency because a mutual penetration of the crack faces may happen [1, 11].

An alternative approach to the formulation of crack problems that excludes mutual penetration of the crack faces has been investigated since the 1990s. This approach is characterized by the Signorini-type boundary conditions at the crack faces [1, 2, 12–21]. In the last 10 years, within the framework of crack models subject to non-penetration (contact) conditions, a number of papers have been published concerning shape optimization problems for delaminated rigid inclusions; see, for example, [2, 22–27]. For a heterogeneous two-dimensional body with a micro-object (defect) and a macro-object (crack), the anti-plane strain energy release rate is expressed by means of the stress intensity factor that is examined with respect to small defects such as microcracks, holes, and inclusions [7]. The ability of velocity methods to describe changes of topology by creating defects such as holes was investigated in [28]. Employing the shape-topological sensitivity analysis [29], the existence of a solution of an optimal control problem related to the best choice of the location and shape of elastic inclusions was shown in [6]. In [30, 31] explicit formulae of derivatives of the energy functionals with respect to the rigid inclusion shapes were established.

In the present paper, we deal with a non-linear model describing the equilibrium of an elastic plate containing a volume rigid inclusion. We consider a family of variational problems that depends on the location parameter of the rigid inclusion. Here we suppose that the inclusion can shift its location along the graph of a given continuous function. It is important that the function of a graph may not satisfy the Lipschitz condition. Along with this family of problems, we formulate an optimal control problem with the cost functional defined by an arbitrary continuous functional on a suitable Sobolev space. We prove the continuous dependence of the solutions with respect to the location parameter of the rigid inclusion which plays the role of a control parameter. In contrast to the previous results related to the optimal size of rigid inclusions [24, 26, 32], to justify a passage to the limit in variational inequalities, we construct a suitable strongly converging sequence of test functions. The result concerning the optimal location of a rigid inclusion for a two-dimensional non-linear model describing the equilibrium of a cracked composite solid was obtained in [19].

2. Family of equilibrium problems

We start with a description of admissible geometries (see Figure 1). Let $\mathrm{\Omega }\subset {\mathbb{R}}^2$ be a bounded domain with the smooth boundary $\mathrm{\Gamma }\in C^{2,1}$ . Suppose that a Lipschitz curve $\gamma \subset \mathrm{\Omega }$ without self-intersections has the following properties. We assume that $\gamma$ can be extended in such a way that this extension crosses $\mathrm{\Gamma }$ at two points, and $\mathrm{\Omega }$ is divided into two subdomains ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$ with Lipschitz boundaries $\partial {\mathrm{\Omega }}_1$ , $\partial {\mathrm{\Omega }}_2$ , $\mathrm{meas}(\mathrm{\Gamma }\cap \partial {\mathrm{\Omega }}_i)>0$ , $i=1,2$ . Let us denote ${\mathrm{\Omega }}_{\gamma }=\mathrm{\Omega }\backslash \overline{\gamma }$ . Let ${\omega }_0$ be a domain of class $C^{2,1}$ such that $(0,x_2^0)\in {\omega }_0$ , ${\overline{\omega }}_0\subset {\mathrm{\Omega }}_{\gamma }$ . We consider the following family of domains $\{{\omega }_t\}$ , $t\in [0,T]$ defined by the relation

$${\omega }_t=\{x+(t,\psi (t))|x\in {\omega }_0\},\psi (t)\in C[0,T],\psi (0)=x_2^0.$$

We suppose the following assumptions.

Figure 1.

Geometry of the problem.

Assumption 1. The union of all domains ${\omega }_0^T$ defined in accordance with the following relation

$${\omega }_{\lambda }^{\beta }=\bigcup _{t\in [\lambda ,\beta ]}{\omega }_t,0\le \lambda <\beta \le T,$$

satisfies ${\overline{\omega }}_0^T\cap \overline{\gamma }={\overline{\omega }}_0^T\cap \mathrm{\Gamma }=\varnothing$ .

Remark 1. Let us observe that due to Assumption 1, there exist some domains $O$ , $O_{\omega }$ with the smooth boundaries $\partial O$ , $\partial O_{\omega }$ of class $C^{2,1}$ such that ${\overline{\omega }}_0^T\subset O_{\omega }$ , ${\overline{O}}_{\omega }\subset O$ , $\overline{O}\subset {\mathrm{\Omega }}_{\gamma }$ .

This assumption gives us an opportunity to use the well-known results concerning the spaces $H^2(S)$ defined on some domain $S$ with a boundary $\partial S$ of class $C^{2,1}$ , namely, the trace theorem [33] and the following characterization

$$v\in H_0^2(S)\iff v\in H^2(S),v=\frac{\partial v}{\partial m}=0\mathrm{on}\partial S,$$

where $m$ is a unit normal to $\partial S$ .

Next, we should write the following assumption for an arbitrary domain ${\omega }_{t\prime }$ corresponding to the fixed parameter $t\prime \in [0,T]$ .

Assumption 2. For an arbitrary strictly inner subdomain $D\subset {\omega }_{t\prime }$ there exists a positive sufficiently small number $\delta >0$ such that $D\subset {\omega }_{t\prime }\cap {\omega }_t$ for all $t\in (t\prime -\delta ,t\prime +\delta )\cap [0,T]$ .

In order to formulate the mathematical model, we consider a three-dimensional Cartesian space $\{x_1,x_2,z\}$ such that the set $\{{\mathrm{\Omega }}_{\gamma }\}\times \{0\}\subset {\mathbb{R}}^3$ corresponds to the middle plane of a plate. We assume that the thickness of the plate is constant and is equal to two. The curve $\gamma$ defines a crack (a cut) in the plate. This means that the cylindrical surface of the through crack specified by the relations $x=(x_1,x_2)\in \gamma$ , $-1\le z\le 1$ where $|z|$ is the distance to the middle plane. Next, we fix the parameter $t\in [0,T]$ and suppose that the domain ${\omega }_t$ refers to a rigid inclusion whereas the domain ${\mathrm{\Omega }}_{\gamma }\backslash {\overline{\omega }}_t$ corresponds to an elastic part of the plate.

Denote by $\chi =\chi (x)=(W,w)$ the displacement vector of the mid-surface points ( $x\in {\mathrm{\Omega }}_{\gamma }$ ), by $W=(w_1,w_2)$ the displacements in the plane $\{x_1,x_2\}$ , and by $w$ the displacements along the axis $z$ . The strain and integrated stress tensors are denoted by ${\epsilon }_{\mathrm{ij}}={\epsilon }_{\mathrm{ij}}(W)$ , ${\sigma }_{\mathrm{ij}}={\sigma }_{\mathrm{ij}}(W)$ , respectively [1]:

$${\epsilon }_{\mathrm{ij}}(W)=\frac{1}{2}(\frac{\partial w_j}{\partial x_i}+\frac{\partial w_i}{\partial x_j}),{\sigma }_{\mathrm{ij}}(W)=c_{\mathrm{ijkl}}{\epsilon }_{\mathrm{kl}}(W),i,j=1,2,$$

where $\{c_{\mathrm{ijkl}}\}$ is the given elasticity tensor, assumed as usual to be symmetric and positive definite:

$$\begin{array}{c}{c}_{\mathrm{ijkl}}=c_{\mathrm{klij}}=c_{\mathrm{jikl}},i,j,k,l=1,2,c_{\mathrm{ijkl}}\in L^{\infty }({\mathrm{\Omega }}_{\gamma }),\hfill \\ c_{\mathrm{ijkl}}{\xi }_{\mathrm{ij}}{\xi }_{\mathrm{kl}}\ge c_0|\xi {|}^2,\forall \xi ,{\xi }_{\mathrm{ij}}={\xi }_{\mathrm{ji}},i,j=1,2,c_0=\mathrm{const}.>0.\hfill \end{array}$$

A summation convention over repeated indices is used throughout the paper. Next we denote the bending moments by formulae [1]

$$m_{\mathrm{ij}}(w)=d_{\mathrm{ijkl}}w{,}_{\mathrm{kl}},i,j=1,2,(w{,}_{\mathrm{kl}}=\frac{{\partial }^{2}w}{\partial x_k\partial x_l})$$

where the tensor $\{d_{\mathrm{ijkl}}\}$ has the same properties as the tensor $\{c_{\mathrm{ijkl}}\}$ . Let $B(S,·,·)$ be a bilinear form defined by the equality

$$B(S,\chi ,\overline{\chi })={\int }_S\{{\sigma }_{\mathrm{ij}}(W){\epsilon }_{\mathrm{ij}}(\overline{W})+m_{\mathrm{ij}}(w)\overline{w}{,}_{\mathrm{ij}}\}\mathrm{dx},$$

(1)

where $\chi =(W,w)$ , $\overline{\chi }=(\overline{W},\overline{w})$ , and $S$ is some subdomain of ${\mathrm{\Omega }}_{\gamma }$ . The potential energy functional of the plate has the following representation [1]:

$$\mathrm{\Pi }({\mathrm{\Omega }}_{\gamma },\chi )=\frac{1}{2}B({\mathrm{\Omega }}_{\gamma },\chi ,\chi )-{\int }_{{\mathrm{\Omega }}_{\gamma }}F\chi \mathrm{dx},\chi =(W,w),$$

where vector $F=(f_1,f_2,f_3)\in L_2(\mathrm{\Omega }{)}^3$ describes the body forces [1]. We introduce the Sobolev spaces

$$H^{1,0}({\mathrm{\Omega }}_{\gamma })=\{v\in H^1({\mathrm{\Omega }}_{\gamma })|v=0\mathrm{on}\mathrm{\Gamma }\},$$

$$H^{2,0}({\mathrm{\Omega }}_{\gamma })=\{v\in H^2({\mathrm{\Omega }}_{\gamma })|v=\frac{\partial v}{\partial n}=0\mathrm{on}\mathrm{\Gamma }\},$$

$$H({\mathrm{\Omega }}_{\gamma })=H^{1,0}({\mathrm{\Omega }}_{\gamma }{)}^2\times H^{2,0}({\mathrm{\Omega }}_{\gamma }),$$

where $n=(n_1,n_2)$ is the unit external normal vector to $\mathrm{\Gamma }$ . We should note here that by the assumption concerning the domain ${\mathrm{\Omega }}_{\gamma }$ , Friedrich’s and Korn’s inequalities are valid in the non-smooth domain ${\mathrm{\Omega }}_{\gamma }$ (see [1, 34]). They provide the following inequality

$$B({\mathrm{\Omega }}_{\gamma },\chi ,\chi )\ge c\parallel \chi {\parallel }^2\forall \chi \in H({\mathrm{\Omega }}_{\gamma }),(\parallel \chi \parallel =\parallel \chi {\parallel }_{H({\mathrm{\Omega }}_{\gamma })})$$

(2)

with a constant $c>0$ independent of $\chi$ (see [1]).

Remark 2. The inequality (2) yields the equivalence of the standard norm in $H({\mathrm{\Omega }}_{\gamma })$ and the semi-norm determined by the left-hand side of (2).

For the precise formulation, we have in mind that the notion of a rigid inclusion allows only displacement $\chi {|}_{{\omega }_t}=\zeta$ in the space $R({\omega }_t)$ of infinitesimal rigid displacements on ${\omega }_t$ , where

$$\begin{array}{c}R({\omega }_t)=\{\zeta (x)=(\rho ,l)|\rho (x)=\hfill \\ =b(x_2,-x_1)+(c_1,c_2);l(x)=a_0+a_1{x}_1+a_2{x}_2,x\in {\omega }_t\},\hfill \end{array}$$

(3)

where $b,c_1,c_2,a_0,a_1,a_2\in \mathbb{R}$ , see [35]. The condition of mutual non-penetration of the crack faces is given by

$$[W\nu ]\ge \left|\left[\frac{\partial w}{\partial \nu }\right]\right|\mathrm{on}\gamma ,(W\nu =w_i{\nu }_i),$$

where $\nu =({\nu }_1,{\nu }_2)$ is a unit outward normal to $\gamma$ , $[v]=v^{+}-v^{-}$ is the jump of a $v$ , and the traces $v^{\pm }$ on ${\gamma }^{+}$ , ${\gamma }^{-}$ correspond to the positive and negative directions of $\nu$ , respectively, refer to [1, 35]. An equilibrium problem for a plate with a crack and a rigid inclusion can be formulated as the following minimization problem:

$$\mathrm{Find}{\xi }_t=(U_t,u_t)\in K_t\mathrm{such}\mathrm{that}\mathrm{\Pi }({\mathrm{\Omega }}_{\gamma },{\xi }_t)=\underset{\chi \in K_t}{inf}\mathrm{\Pi }({\mathrm{\Omega }}_{\gamma },\chi ),$$

(4)

where the set of admissible functions is in the form

$$K_t=\{\chi =(W,w)\in H({\mathrm{\Omega }}_{\gamma })|[W]\nu \ge |[\frac{\partial w}{\partial \nu }]|\mathrm{on}\gamma ;\chi {|}_{{\omega }_t}\in R({\omega }_t)\}.$$

According to [3, 35], the problem (4) is known to have a unique solution ${\xi }_t\in K_t$ , which satisfies the variational inequality

$${\xi }_t\in K_t,B({\mathrm{\Omega }}_{\gamma },{\xi }_t,\chi -{\xi }_t)\ge {\int }_{{\mathrm{\Omega }}_{\gamma }}F(\chi -{\xi }_t)\mathrm{dx}\forall \chi \in K_t.$$

(5)

3. Optimal control problem

Let $G:H({\mathrm{\Omega }}_{\gamma })\to \mathbb{R}$ be an arbitrary continuous functional, and consider the cost functional

$$J(t)=G({\xi }_t)$$

defined on an interval $[0,T]$ .

As an example of such functionals having physical sense, we can give the functional $G_1(\chi )=\parallel \chi -{\chi }_0{\parallel }_{H({\mathrm{\Omega }}_{\gamma })}$ , which characterizes the deviation of the displacement vector from a given function ${\chi }_0$ . We mention that under the appropriate additional conditions on $F$ , $\mathrm{\Omega }$ , and $\gamma$ , there exists the first derivative

$$G_2(\chi )=\frac{d\mathrm{\Pi }({\mathrm{\Omega }}_{\epsilon };{\chi }^{\epsilon })}{d\epsilon }{|}_{\epsilon =0}$$

of the energy functional $\mathrm{\Pi }({\mathrm{\Omega }}_{\epsilon };{\chi }^{\epsilon })$ with respect to the domain perturbation parameter $\epsilon$ at $\epsilon =0$ . Here ${\chi }^{\epsilon }$ denotes solutions of equilibrium problems for perturbed domains ${\mathrm{\Omega }}_{\epsilon }$ . We refer the reader to [36] for the exact expression of the formula of the first derivative and corresponding explanations. It follows from [36] that the functional that expresses the mentioned first derivative is continuous in the space $H({\mathrm{\Omega }}_{\gamma })$ . Consider the optimal control problem:

$$\mathrm{Find}{t}^{*}\in [0,T]\mathrm{such}\mathrm{that}J(t^{*})=\underset{t\in [0,T]}{sup}J(t).$$

(6)

This means that we want to find the best location of the inclusion along the graph of the function $\psi$ that provides the maximal value for the cost functional. Our main existence result is as follows.

Theorem 1. There exists a solution of the optimal control problem (6).

Proof. Let $\{t_n\}$ be a maximizing sequence. By the boundedness of the interval $[0,T]$ , there exists a convergent subsequence $\{t_{n_k}\}\subset \{t_n\}$ such that

$$t_{n_k}\to t^{*}\mathrm{as}k\to \infty ,t^{*}\in [0,T].$$

It is easy to see that if there exists a subsequence $\{t_{n_l}\}\subset \{t_{n_k}\}$ with the following properties $t_{n_l}\equiv t^{*}$ , $l\in \mathbb{N}$ , we establish that $J(t^{*})$ is a solution of (6). Therefore, we assume that $t_{n_k}\ne t^{*}$ for sufficiently large $k$ .

In view of Lemma 2, the solutions ${\xi }_k$ of (4) corresponding to the parameters $t_{n_k}$ converge to the solution ${\xi }_{t^{*}}$ strongly in $H({\mathrm{\Omega }}_{\gamma })$ as $k\to \infty$ . This allows us to obtain the convergence

$$J(t_{n_k})\to J(t^{*}),$$

which yields

$$J(t^{*})=\underset{t\in [0,T]}{sup}J(t).$$

The theorem is proved.  □

4. Auxiliary lemmas

Now we have to establish auxiliary results that were used to prove Theorem 1. For convenience, we formulate these results as the following two consistent lemmas.

Lemma 1. Let $t^{*}\in [0,T]$ be a fixed real number and let $\{t_n\}\subset [0,T]$ 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 ${\eta }_k\in K_{t_k}$ , $k\in \mathbb{N}$ and ${\eta }_k\to \eta$ strongly in $H({\mathrm{\Omega }}_{\gamma })$ as $k\to \infty$ .

Proof. Fix $\eta \in K_{t^{*}}$ . First note that if there exists a subsequence $\{t_{n_k}\}$ such that $t_{n_k}=t^{*}$ , then the assertion of the lemma holds for ${\eta }_k\equiv \eta$ , $\{t_n\}$ . Without loss of generality, we can assume that is a strictly monotonic decreasing sequence such that

$$t^{*}=\underset{n\to \infty }{lim}{t}_n.$$

Denote by ${\zeta }^{*}=({\rho }^{*},l^{*})$ with ${\rho }^{*}=b^{*}(x_2,-x_1)+(c_1^{*},c_2^{*})$ , $l^{*}(x)=a_0^{*}+a_1^{*}{x}_1+a_2^{*}{x}_2$ the function describing the structure of $\eta$ in ${\omega }_{t^{*}}$ . We extend the function ${\zeta }^{*}=({\rho }^{*},l^{*})$ to the whole domain $\mathrm{\Omega }$ by the equalities:

$${\zeta }^{*}=({\rho }^{*},l^{*}),\mathrm{where}\{\begin{array}{cc}{\rho }^{*}=b(x_2,-x_1)+(c_1^{*},c_2^{*}),\hfill & x\in \mathrm{\Omega },\hfill \\ \multicolumn{1}{c}{l^{*}(x)=a_0^{*}+a_1^{*}{x}_1+a_2^{*}{x}_2,}& x\in \mathrm{\Omega }.\hfill \end{array}$$

From Assumptions 1 and 2, there exists $\delta$ such that for all $t\in [0,T]\cap (t^{*},t^{*}+\delta )$ the domain ${\omega }_{t^{*}}^t$ is a simply connected domain with the Lipschitz boundary. It is now necessary to fix an arbitrary value $t\in [0,T]\cap (t^{*},t^{*}+\delta )$ and consider the following family of auxiliary problems

$$\mathrm{Find}\mathrm{an}\mathrm{element}{Q}_t\in K_t\prime \mathrm{such}\mathrm{that}p(Q_t)=\underset{\chi \in K_{t\prime }}{inf}p(\chi ),$$

(7)

where $p(\chi )=B({\mathrm{\Omega }}_{\gamma },\chi -\eta ,\chi -\eta )$ ,

$$\begin{array}{c}{K}_t\prime =\{\chi =(W,w)\in H({\mathrm{\Omega }}_{\gamma })|\chi =\eta ,\frac{\partial w}{\partial \mu }=\frac{\partial v}{\partial \mu }\mathrm{on}\partial O,\hfill \\ \chi {|}_{{\omega }_{t^{*}}^t}={\zeta }^{*},\chi =\eta \mathrm{in}\mathrm{\Omega }\backslash \overline{O}\},\hfill \end{array}$$

(8)

where $\mu$ is a unit normal to $\partial O$ . It is obvious that the functional $p(\chi )$ is coercive and weakly lower semicontinuous on the space $H({\mathrm{\Omega }}_{\gamma })$ . In addition, one can verify that the set $K_t\prime$ is convex and closed in $H({\mathrm{\Omega }}_{\gamma })$ . This properties provide the existence of a unique solution ${\eta }_t$ of the problem (7), see [1]. This solution is characterized equivalently by a variational inequality

$${\eta }_t\in K_t\prime ,B({\mathrm{\Omega }}_{\gamma },Q_t-\eta ,\chi -Q_t)\ge 0\forall \chi \in K_t\prime .$$

(9)

Note that applying a lifting operator for the smooth domains $\mathrm{\Omega }\backslash \overline{O}$ and $O\backslash {\overline{O}}_{\omega }$ (see Remark 1), we can construct a function $\hat{\chi }\in H({\mathrm{\Omega }}_{\gamma })$ such that $\hat{\chi }={\zeta }^{*}$ in $O_{\omega }$ and $\hat{\chi }=\eta$ , $\frac{\partial w}{\partial \mu }=\frac{\partial v}{\partial \mu }$ on $\partial O$ . As $\hat{\chi }\in K_t\prime$ for all $t\in [0,T]\cap (t^{*},t^{*}+\delta )$ , we can substitute $\hat{\chi }$ in (9) as the test function, which yields the inequality

$$B({\mathrm{\Omega }}_{\gamma },Q_t-\eta ,\hat{\chi })+B({\mathrm{\Omega }}_{\gamma },\eta ,Q_t)\ge B({\mathrm{\Omega }}_{\gamma },Q_t,Q_t)\forall t\in (0,T].$$

From this relation and Korn’s inequality, we obtain the following uniform upper bound:

$$\parallel Q_t\parallel \le c\forall t\in [0,T]\cap (t^{*},t^{*}+\delta ).$$

Therefore, we can extract from the sequence $\{Q_{t_n}\}$ a subsequence $\{Q_l\}$ , which is defined by equalities $Q_l=Q_{t_{n_l}}$ , $l\in \mathbb{N}$ (henceforth, we define a sequence $\{t_l\}$ by the equality $t_l=t_{n_l}$ ) and $\{Q_l\}$ weakly converges to some function $\tilde{\eta }$ in $H({\mathrm{\Omega }}_{\gamma })$ . It is now necessary to show that $\tilde{\eta }=\eta$ . By construction, and the smoothness of the domains $O\backslash {\overline{\omega }}_{t^{*}}$ , the inclusion $(Q_l-\eta )\in H_0^1(O\backslash {\overline{\omega }}_{t^{*}}{)}^2\times H_0^2(O\backslash {\overline{\omega }}_{t^{*}})$ holds. Consequently, bearing in mind the weak closedness of $H_0^1(O\backslash {\overline{\omega }}_{t^{*}}{)}^2\times H_0^2(O\backslash {\overline{\omega }}_{t^{*}})$ we have $(\tilde{\eta }-\eta )\in H_0^1(O\backslash {\overline{\omega }}_{t^{*}}{)}^2\times H_0^2(O\backslash {\overline{\omega }}_{t^{*}})$ . We consider now the functions of the form ${\chi }_l^{\pm }=Q_l\pm \alpha$ , where $\alpha$ is the function defined by zero extension of an arbitrary function $\tilde{\alpha }\in C_0^{\infty }(O\backslash {\overline{\omega }}_{t^{*}}{)}^3$ into ${\mathrm{\Omega }}_{\gamma }$ . It is observed that for sufficiently large $l$ the functions ${\chi }_l^{\pm }$ belong to $K{\prime }_{t_l}$ . We may therefore substitute ${\chi }_l^{+}$ and ${\chi }_l^{-}$ into (9), and then combining the resulting inequalities, we find

$$Q_l\in K{\prime }_{t_l},B({\mathrm{\Omega }}_{\gamma },Q_l-\eta ,\alpha )=0.$$

(10)

The function $\alpha$ is now fixed and by passing to the limit in (10) it is established that

$$B({\mathrm{\Omega }}_{\gamma },\tilde{\eta }-\eta ,\alpha )=B(O\backslash {\overline{\omega }}_{t^{*}},\tilde{\eta }-\eta ,\alpha )=0\forall \alpha \in C_0^{\infty }({\mathrm{\Omega }}_{\gamma }\backslash {\overline{\omega }}_{t^{*}}{)}^3.$$

(11)

Next, since by Assumption 1 the domain $O\backslash {\overline{\omega }}_{t^{*}}$ belongs to the class $C^{2,1}$ , we have the density of $C_0^{\infty }(O\backslash {\overline{\omega }}_{t^{*}})$ both in $H_0^1(O\backslash {\overline{\omega }}_{t^{*}})$ and $H_0^2(O\backslash {\overline{\omega }}_{t^{*}})$ . Therefore, (11) yields that $\tilde{\eta }-\eta =0$ in $H_0^1(O\backslash {\overline{\omega }}_{t^{*}}{)}^2\times H_0^2(O\backslash {\overline{\omega }}_{t^{*}})$ . Finally, by construction, the equality $\tilde{\eta }=\eta$ is satisfied in the domains ${\omega }_{t^{*}}$ and $\mathrm{\Omega }\backslash \overline{O}$ . Consequently, $\tilde{\eta }=\eta$ in $H({\mathrm{\Omega }}_{\gamma })$ . As a result, there is a sequence $\{Q_l\}$ such that $Q_l\in K{\prime }_{t_l}$ , $l\in \mathbb{N}$ and $Q_l\to W$ weakly in $H({\mathrm{\Omega }}_{\gamma })$ as $l\to \infty$ . Now we are in a position to prove the strong convergence. By virtue of the Mazur theorem there exist a function $N:\mathbb{N}\to \mathbb{N}$ and a sequence of sets of real numbers $\{\alpha (n{)}_i|i=n,\dots ,N(n)\}$ satisfying $a(n{)}_i\ge 0$ and $\sum _{i=n}^{N(n)}\alpha (n{)}_i=1$ such that the sequence $\{{\hat{Q}}_n\}$ defined by the convex combination

$${\hat{Q}}_n=\sum _{i=n}^{N(n)}\alpha (n{)}_i{Q}_i$$

converges to $\eta$ strongly in $H({\mathrm{\Omega }}_{\gamma })$ . We can now determine the required sequence $\{{\eta }_{n_k}\}$ by the equations

$${\eta }_{n_1}={\hat{Q}}_{N(1)},{\eta }_{n_2}={\hat{Q}}_{N(N(1))},\dots {\eta }_{n_k}={\hat{Q}}_{N^k(1)},\dots$$

The constructed functions ${\eta }_{n_k}$ , $k=1,2,\dots$ belong to the set $K_{t_{N^k(1)}}$ corresponding to the subsequence $\{t_{N^k(1)}\}$ of the sequence $\{t_l\}\subset \{t_n\}$ .

In the other case, namely when the sequence $\{t_n\}$ is a strictly monotonic increasing sequence, we should consider the sets $K{\prime }_t$ which are defined with the help of domains ${\omega }_t^{t^{*}}$ , for fixed $t\in [0,T]\cap (t^{*}-\delta ,t^{*})$ . Lemma 1 is proved.□

Now, we are in a position to prove an auxiliary statement that was used in the proof of Theorem 1.

Lemma 2. Let $t^{*}\in [0,T]$ be a fixed real number. Then ${\xi }_t\to {\xi }_{t^{*}}$ strongly in $H({\mathrm{\Omega }}_{\gamma })$ as $t\to t^{*}$ , where ${\xi }_t$ , ${\xi }_{t^{*}}$ are the solutions of (4) corresponding to parameters $t\in [0,T]$ , $t^{*}\in [0,T]$ , respectively.

Proof. We will prove it by contradiction. Let us assume that there exist a number ${ϵ}_0>0$ and a sequence $\{t_n\}\subset (0,T]$ such that $t_n\to t^{*}$ , $\parallel {\xi }_n-{\xi }_{t^{*}}\parallel \ge {ϵ}_0$ , where ${\xi }_n={\xi }_{t_n}$ , $n\in \mathbb{N}$ are the solutions of (4), corresponding to $t_n$ . Because of ${\chi }^0\equiv 0\in K_t$ for all $t\in [0,T]$ , we can insert $\chi ={\chi }^0$ in (5) for fixed $t\in (0,T]$ . This provides

$${\xi }_t\in K_t,B({\mathrm{\Omega }}_{\gamma },{\xi }_t,{\xi }_t)\le {\int }_{{\mathrm{\Omega }}_{\gamma }}F{\xi }_t\mathrm{dx}\forall t\in [0,T].$$

From here, the following estimate follows being uniform in $t\in [0,T]$

$$\parallel {\xi }_t\parallel \le c.$$

Consequently, there exists a subsequence of $\{{\xi }_n\}$ , still indexed by $n$ , such that ${\xi }_n$ converges to some function $\tilde{\xi }$ weakly in $H({\mathrm{\Omega }}_{\gamma })$ .

Now we show that $\tilde{\xi }\in K_{t^{*}}$ . Indeed, we have ${\xi }_n{|}_{{\omega }_{t_n}}={\zeta }_n\in R({\omega }_{t_n})$ . In accordance with the Sobolev embedding theorem [33], we deduce that

$${\xi }_n{|}_{{\omega }_{t^{*}}}\to \tilde{\xi }{|}_{{\omega }_{t^{*}}}\mathrm{strongly}\mathrm{in}{L}_2({\omega }_{t^{*}}{)}^3\mathrm{as}n\to \infty ,$$

(12)

$$U_n{|}_{\gamma }\to \tilde{U}{|}_{\gamma }\mathrm{strongly}\mathrm{in}{L}_2(\gamma {)}^2\mathrm{as}n\to \infty ,$$

(13)

$$\frac{\partial u_n}{\partial \nu }{|}_{\gamma }\to \frac{\partial \tilde{u}}{\partial \nu }{|}_{\gamma }\mathrm{strongly}\mathrm{in}{L}_2(\gamma )\mathrm{as}n\to \infty .$$

(14)

Choosing a subsequence, if necessary, we assume as $n\to \infty$ that ${\xi }_n\to \tilde{\xi }$ almost everywhere on ${\omega }_{t^{*}}$ .

To proceed further with our proof, we fix an arbitrary strictly inner subdomain $D\subset {\omega }_{t^{*}}$ . There exists a sufficiently large $N$ such that if $n\ge N$ , then $D\subset {\omega }_{t^{*}}\cap {\omega }_n$ , see Assumption 2. Therefore, the sequence $\{{\rho }_n\}$ converges to $\tilde{\xi }$ almost everywhere on $D$ as $n$ tends to infinity. This allows us to conclude that each of the numerical sequences $\{b^n\}$ , $\{c_1^n\}$ , $\{c_2^n\}$ , $\{a_{0n}\}$ , $\{a_{1n}\}$ , $\{a_{2n}\}$ defining the structure of ${\zeta }_n$ , $n=1,2,\dots$ in $D$ is bounded in $\mathbb{R}$ . Thus, we can extract subsequences (retain notation) such that

$$b_n\to b,a_{0n}\to a_0,c_{\mathrm{in}}\to c_i,a_{\mathrm{in}}\to a_i,i=1,2,\mathrm{as}n\to \infty .$$

Therefore, for the fixed domain $D$ , we have

$$\begin{array}{c}{U}_k\to (b{x}_2+c_1,-b{x}_1+c_2)\mathrm{a}.\mathrm{e}.\mathrm{in}D\mathrm{as}k\to \infty ,\hfill \\ u_k\to a_0+a_1{x}_1+a_2{x}_2\mathrm{a}.\mathrm{e}.\mathrm{in}D\mathrm{as}k\to \infty .\hfill \end{array}$$

Consequently, we obtain that

$$\begin{array}{c}\tilde{U}=(b{x}_2+c_1,-b{x}_1+c_2)\mathrm{a}.\mathrm{e}.\mathrm{in}D,\hfill \\ \tilde{u}=a_0+a_1{x}_1+a_2{x}_2\mathrm{a}.\mathrm{e}.\mathrm{in}D.\hfill \end{array}$$

Owing to arbitrariness of the domain $D\subset {\omega }_{t^{*}}$ we get that

$$\begin{array}{c}\tilde{U}=(b{x}_2+c_1,-b{x}_1+c_2)\mathrm{a}.\mathrm{e}.\mathrm{in}{\omega }_{t^{*}},\hfill \\ \tilde{u}=a_0+a_1{x}_1+a_2{x}_2\mathrm{a}.\mathrm{e}.\mathrm{in}{\omega }_{t^{*}}.\hfill \end{array}$$

Whence the inclusion $\tilde{\xi }{|}_{{\omega }_{t^{*}}}\in R({\omega }_{t^{*}})$ holds.

We now turn to the inequality condition in the definition of the set $K_{t^{*}}$ . We have to show that $\tilde{\xi }$ satisfies the inequality $[\tilde{U}]\nu \ge |[\frac{\partial \tilde{u}}{\partial \nu }]|$ on $\gamma$ . Bearing in mind the convergence (13), if necessary, we can once again extract a subsequences satisfying $U_n{|}_{\gamma }\to \tilde{U}{|}_{\gamma }$ , $\frac{\partial u_n}{\partial \nu }{|}_{\gamma }\to \frac{\partial \tilde{u}}{\partial \nu }{|}_{\gamma }$ almost everywhere on both ${\gamma }^{+}$ and ${\gamma }^{-}$ . This fact allows us to pass to the limit in the following inequality

$$[U_n]\nu \ge \left|\left[\frac{\partial u_n}{\partial \nu }\right]\right|\mathrm{on}\gamma .$$

This leads to $[\tilde{U}]\nu \ge |[\frac{\partial \tilde{u}}{\partial \nu }]|$ on $\gamma$ . Therefore, we get the inclusion $\tilde{\xi }\in K_{t^{*}}$ .

For the sequel, we have to prove the equality $\tilde{\xi }={\xi }_{t^{*}}$ and to establish the existence of a sequence ${\xi }_n={\xi }_{t_n}$ , $n=1,2,\dots$ of solutions strongly converging in $H({\mathrm{\Omega }}_{\gamma })$ to ${\xi }_{t^{*}}$ . Now, let us prove that $\tilde{\xi }={\xi }_{t^{*}}$ . For this purpose, we analyze the variational inequality (5) and its limiting case. 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 ${\eta }_k\to \eta$ strongly in $H({\mathrm{\Omega }}_{\gamma })$ as $k\to \infty$ .

The properties established above for the convergent sequences $\{{\eta }_k\}$ and $\{{\xi }_n\}$ allow us to pass to the limit as $k\to \infty$ in following inequalities derived from (5) for $t_k$ and with the test functions ${\eta }_k$

$$B({\mathrm{\Omega }}_{\gamma },{\xi }_{n_k},{\eta }_{n_k}-{\xi }_{n_k})\ge {\int }_{{\mathrm{\Omega }}_{\gamma }}F({\eta }_{n_k}-{\xi }_{n_k})\mathrm{dx}.$$

(15)

As a result, we have

$$B({\mathrm{\Omega }}_{\gamma },\tilde{\xi },\eta -\tilde{\xi })\mathrm{dx}\ge {\int }_{{\mathrm{\Omega }}_{\gamma }}F(\eta -\tilde{\xi })\mathrm{dx}\forall \eta \in K_{t^{*}}.$$

As $\eta$ was arbitrary, we see that the last inequality is variational. The unique solvability of this variational inequality yields that $\tilde{\xi }={\xi }_{t^{*}}$ .

To complete the proof we should establish the strong convergence ${\xi }_n\to {\xi }_{t^{*}}$ . By substituting $\chi =2{\xi }_t$ and $\chi =0$ into the variational inequalities (5) for $t\in [0,T]$ , we get

$${\xi }_t\in K_t,B({\mathrm{\Omega }}_{\gamma },{\xi }_t,{\xi }_t)={\int }_{{\mathrm{\Omega }}_{\gamma }}F{\xi }_t\mathrm{dx}\forall t\in [0,T].$$

(16)

The equalities (16) together with the weak convergence ${\xi }_n\to {\xi }_{t^{*}}$ in $H({\mathrm{\Omega }}_{\gamma })$ as $n\to \infty$ imply

$$\underset{n\to \infty }{lim}B({\mathrm{\Omega }}_{\gamma },{\xi }_n,{\xi }_n)=\underset{n\to \infty }{lim}{\int }_{{\mathrm{\Omega }}_{\gamma }}F{\xi }_n\mathrm{dx}={\int }_{{\mathrm{\Omega }}_{\gamma }}F{\xi }_{t^{*}}\mathrm{dx}=B({\mathrm{\Omega }}_{\gamma },{\xi }_{t^{*}},{\xi }_{t^{*}}).$$

As we have the equivalence of norms (see Remark 2), one can see that ${\xi }_n\to {\xi }_{t^{*}}$ strongly in $H({\mathrm{\Omega }}_{\gamma })$ as $n\to \infty$ .

Thus, we have a contradiction to the initial assumption. Lemma 2 is thus proved.  □

5. Conclusion

In this paper, we have analyzed a family of variational problems describing the equilibrium of cracked plates with inclusions having different location parameters $t\in [0,T]$ . The existence of the solution to the optimal control problem (6) has been proved. For that problem, the cost functional $J(t)$ is defined by an arbitrary continuous functional, whereas the location parameter $t$ of the rigid inclusion serves as a control. Lemmas 1 and 2 show a connection between the equilibrium problems for bodies with rigid inclusions of varying locations. Note that this approach can be applied for equilibrium problems related to the two-dimensional solids with classical linear conditions as well as for cases of reinforced solids without cracks.