Asymptotic analysis of linearly elastic flexural shells subjected to an obstacle in absence of friction

Unilateral contact problems arise in many fields such as medicine, engineering, biology and material science. For instance, the description of the motion inside the human heart of the three Aorta valves, which can be regarded as linearly elastic shells, is governed by a mathematical model built up in a way such that each valve remains confined in a certain portion of space without penetrating or being penetrated by the other two valves. In this direction we cite the recent references [40], [41] and [51].

The displacement of a linearly elastic shell is modeled, in general, via the classical equations of three-dimensional linearized elasticity (cf., e.g., [9]). The intrinsic complexity of this model, however, prevents certain situations from being amenably studied like, for instance, when the shell is non-homogeneous and anisotropic (cf., e.g., [7] and [8]), or its thickness varies periodically (cf., e.g., [48] and [49]). It might thus be useful to perform a dimension reduction in order to obtain approximate models which are more amenable to analyze and to implement numerically.

The identification of two-dimensional limit models for time-independent linearly elastic shells was extensively treated by Ciarlet and his associates in the seminal papers [14, 15, 16, 17, 18, 21] for the purpose of justifying Koiter’s model in the case where no obstacles are taken into account. We also mention the papers [44, 45, 47], which are about the numerical computation of the solution of the obstacle-free linear Koiter’s model in the time-dependent case. For the justification of Koiter’s model in the time-dependent case where the action of temperature is considered, we refer the reader to [33]. In the aforementioned papers no confinement conditions were imposed. The recent papers [38] and [39] provide examples of variational-inequalities-based models arising in biology. The paper [37], instead, treats the modeling and analysis of the melting of a shallow ice sheet by means of a set of doubly nonlinear parabolic variational inequalities.

In the recent papers [19, 20, 22, 23], Ciarlet and his associates fully justified Koiter’s model in the case where the linearly elastic shell under consideration is an elliptic membrane shell subject to the aforementioned confinement condition.

The confinement condition we are here considering considerably departs from the Signorini condition usually considered in the existing literature. Indeed, the Signorini contact condition states that only the “lower face” of the shell is required to remain above the “horizontal” plane (cf., e.g. Chapter 63 of [50]). Such a confinement condition renders the asymptotic analysis considerably more difficult as the constraint now bears on a vector field, the displacement vector field of the reference configuration, instead of on only a single component of this field.

The recovery of a set of two-dimensional variational inequalities as a result of a rigorous asymptotic analysis conducted on a ad hoc three-dimensional model based on the classical equations of three-dimensional linearized elasticity in the case where the linearly elastic shell under consideration is a flexural shell has only been addressed, to our best knowledge, by Léger and Miara in the paper [28] in a very amenable geometrical framework. In the paper [28] the authors do not consider the action of friction. The method proposed in the paper [28], which makes use of a very lengthy argument exploiting the properties of cones, is suitable for treating the case where only one linearly elastic flexural shell is considered. In the context of multi-physics multi-scale problems, like for instance those considered by Quarteroni and his associates in [40], [41] and [51], it is not however clear whether the approach proposed by Léger and Miara is suitable. The purpose of this paper is to remedy this situation by presenting a new method, based on the rigorous asymptotic analysis technique developed by Ciarlet, Lods and Miara [21] and the properties of the penalty method for constrained optimization problem (cf., e.g., [10]), for recovering the same set of two-dimensional variational inequalities for linearly elastic flexural shells that were recovered in the recent papers [19, 20] in a more general geometrical framework. It is also worth mentioning that the method we present in this paper makes use of considerably less lengthy computations than those in [28]. In particular, the approach based on the penalty method we are going to implement will save us the effort of constructing a suitable vector field in the instance of recovering the sought two-dimensional limit model. We refer the reader to the references [26, 32, 43, 46].

More precisely, in this paper we recover, via a rigorous asymptotic analysis as the thickness approaches zero over a ad hoc three-dimensional model (three-dimensional in the sense that it is defined over a three-dimensional subset of $\mathbb{R}^{3}$), a set of two-dimensional (two-dimensional in the sense that it is defined over a two-dimensional subset of $\mathbb{R}^{2}$) variational inequalities governing the displacement of a linearly elastic flexural shell subject to remain confined in a half-space in absence of friction. The problem under consideration is an obstacle problem.

The paper is divided into five sections (including this one). In section 2 we recall some background and notation. In section 3 we recall the formulation and the properties of a three-dimensional obstacle problem for “general” linearly elastic shell. In section 4 we specialize the formulation presented in the previous section to the case where the linearly elastic shell under consideration is a flexural shell, and then we scale the three-dimensional obstacle problem in a way such that the integration domain becomes independent of the thickness parameter. The penalized version of the three-dimensional obstacle problem is introduced at the end of this section. Finally, in section 5, a rigorous asymptotic analysis is carried out and the sought set of two-dimensional variational inequalities is recovered.

For details about the classical notions of differential geometry used in this section and the next one, see, e.g. [11] or [12].

Greek indices, except $\varepsilon$, take their values in the set $\{1,2\}$, while Latin indices, except when they are used for indexing sequences, take their values in the set $\{1,2,3\}$, and the summation convention with respect to repeated indices is systematically used in conjunction with these two rules. The notation $\mathbb{E}^{3}$ designates the three-dimensional Euclidean space whose origin is denoted by $O$; the Euclidean inner product and the vector product of $\bm{u},\bm{v}\in\mathbb{E}^{3}$ are denoted $\bm{u}\cdot\bm{v}$ and $\bm{u}\times\bm{v}$; the Euclidean norm of $\bm{u}\in\mathbb{E}^{3}$ is denoted $\left|\bm{u}\right|$. The notation $\delta^{j}_{i}$ designates the Kronecker symbol.

Given an open subset $\Omega$ of $\mathbb{R}^{n}$, notations such as $L^{2}(\Omega)$, $H^{1}(\Omega)$, or $H^{2}(\Omega)$, designate the usual Lebesgue and Sobolev spaces, and the notation $\mathcal{D}(\Omega)$ designates the space of all functions that are infinitely differentiable over $\Omega$ and have compact supports in $\Omega$. The notation $\left\|\cdot\right\|_{X}$ designates the norm in a normed vector space $X$. Spaces of vector-valued functions are denoted with boldface letters.

The positive and negative parts of a function $f:\Omega\to\mathbb{R}$ are respectively denoted by:

$$f^{+}(x):=\max\{f(x),0\}\quad\textup{ and }\quad f^{-}(x):=-\min\{f(x),0\}\quad x\in\Omega.$$

The boundary $\Gamma$ of an open subset $\Omega$ in $\mathbb{R}^{n}$ is said to be Lipschitz-continuous if the following conditions are satisfied (cf., e.g., Section 1.18 of [13]): Given an integer $s\geq 1$, there exist constants $\alpha_{1}>0$ and $L>0$, and a finite number of local coordinate systems, with coordinates $\bm{\phi}^{\prime}_{r}=(\phi_{1}^{r},\dots,\phi_{n-1}^{r})\in\mathbb{R}^{n-1}$ and $\phi_{r}=\phi_{n}^{r}$, sets $\tilde{\omega}_{r}:=\{\bm{\phi}_{r}\in\mathbb{R}^{n-1};|\bm{\phi}_{r}|<\alpha_{1}\}$, $1\leq r\leq s$, and corresponding functions

$$\tilde{\theta}_{r}:\tilde{\omega}_{r}\to\mathbb{R},$$

such that

$$\Gamma=\bigcup_{r=1}^{s}\{(\bm{\phi}^{\prime}_{r},\phi_{r});\bm{\phi}^{\prime}_{r}\in\tilde{\omega}_{r}\textup{ and }\phi_{r}=\tilde{\theta}_{r}(\bm{\phi}^{\prime}_{r})\},$$

and

$$|\tilde{\theta}_{r}(\bm{\phi}^{\prime}_{r})-\tilde{\theta}_{r}(\bm{\upsilon}^{\prime}_{r})|\leq L|\bm{\phi}^{\prime}_{r}-\bm{\upsilon}^{\prime}_{r}|,\textup{ for all }\bm{\phi}^{\prime}_{r},\bm{\upsilon}^{\prime}_{r}\in\tilde{\omega}_{r},\textup{ and all }1\leq r\leq s.$$

We observe that the second last formula takes into account overlapping local charts, while the last set of inequalities express the Lipschitz continuity of the mappings $\tilde{\theta}_{r}$.

An open set $\Omega$ is said to be locally on the same side of its boundary $\Gamma$ if, in addition, there exists a constant $\alpha_{2}>0$ such that

	$\displaystyle\{(\bm{\phi}^{\prime}_{r},\phi_{r});\bm{\phi}^{\prime}_{r}\in\tilde{\omega}_{r}\textup{ and }\tilde{\theta}_{r}(\bm{\phi}^{\prime}_{r})<\phi_{r}<\tilde{\theta}_{r}(\bm{\phi}^{\prime}_{r})+\alpha_{2}\}$	$\displaystyle\subset\Omega,\quad\textup{ for all }1\leq r\leq s,$
	$\displaystyle\{(\bm{\phi}^{\prime}_{r},\phi_{r});\bm{\phi}^{\prime}_{r}\in\tilde{\omega}_{r}\textup{ and }\tilde{\theta}_{r}(\bm{\phi}^{\prime}_{r})-\alpha_{2}<\phi_{r}<\tilde{\theta}_{r}(\bm{\phi}^{\prime}_{r})\}$	$\displaystyle\subset\mathbb{R}^{n}\setminus\overline{\Omega},\quad\textup{ for all }1\leq r\leq s.$

Following [13], a domain in $\mathbb{R}^{n}$ is considered as a bounded Lipschitz domain, namely, a bounded and connected open subset $\Omega$ of $\mathbb{R}^{n}$, whose boundary $\partial\Omega$ is Lipschitz-continuous, the set $\Omega$ being locally on a single side of $\partial\Omega$.

Let $\omega$ be a domain in $\mathbb{R}^{2}$, let $y=(y_{\alpha})$ denote a generic point in $\overline{\omega}$, and let $\partial_{\alpha}:=\partial/\partial y_{\alpha}$ and $\partial_{\alpha\beta}:=\partial^{2}/\partial y_{\alpha}\partial y_{\beta}$. A mapping $\bm{\theta}\in\mathcal{C}^{1}(\overline{\omega};\mathbb{E}^{3})$ is an immersion if the two vectors

$$\bm{a}_{\alpha}(y):=\partial_{\alpha}\bm{\theta}(y)$$

are linearly independent at each point $y\in\overline{\omega}$. Then the image $\bm{\theta}(\overline{\omega})$ of the set $\overline{\omega}$ under the mapping $\bm{\theta}$ is a surface in $\mathbb{E}^{3}$, equipped with $y_{1},y_{2}$ as its curvilinear coordinates. Given any point $y\in\overline{\omega}$, the vectors $\bm{a}_{\alpha}(y)$ span the tangent plane to the surface $\bm{\theta}(\overline{\omega})$ at the point $\bm{\theta}(y)$, the unit vector

$$\bm{a}_{3}(y):=\frac{\bm{a}_{1}(y)\times\bm{a}_{2}(y)}{|\bm{a}_{1}(y)\times\bm{a}_{2}(y)|}$$

is normal to $\bm{\theta}(\overline{\omega})$ at $\bm{\theta}(y)$, the three vectors $\bm{a}_{i}(y)$ form the covariant basis at $\bm{\theta}(y)$, and the three vectors $\bm{a}^{j}(y)$ defined by the relations

$$\bm{a}^{j}(y)\cdot\bm{a}_{i}(y)=\delta^{j}_{i}$$

form the contravariant basis at $\bm{\theta}(y)$; note that the vectors $\bm{a}^{\beta}(y)$ also span the tangent plane to $\bm{\theta}(\overline{\omega})$ at $\bm{\theta}(y)$ and that $\bm{a}^{3}(y)=\bm{a}_{3}(y)$.

The first fundamental form of the surface $\bm{\theta}(\overline{\omega})$ is then defined by means of its covariant components

$$a_{\alpha\beta}:=\bm{a}_{\alpha}\cdot\bm{a}_{\beta}=a_{\beta\alpha}\in\mathcal{C}^{0}(\overline{\omega}),$$

or by means of its contravariant components

$$a^{\alpha\beta}:=\bm{a}^{\alpha}\cdot\bm{a}^{\beta}=a^{\beta\alpha}\in\mathcal{C}^{0}(\overline{\omega}).$$

Note that the symmetric matrix field $(a^{\alpha\beta})$ is then the inverse of the matrix field $(a_{\alpha\beta})$, that $\bm{a}^{\beta}=a^{\alpha\beta}\bm{a}_{\alpha}$ and $\bm{a}_{\alpha}=a_{\alpha\beta}\bm{a}^{\beta}$, and that the area element along $\bm{\theta}(\overline{\omega})$ is given at each point $\bm{\theta}(y),\,y\in\overline{\omega}$, by $\sqrt{a(y)}\,\mathrm{d}y$, where

$$a:=\det(a_{\alpha\beta})\in\mathcal{C}^{0}(\overline{\omega}).$$

Given an immersion $\bm{\theta}\in\mathcal{C}^{2}(\overline{\omega};\mathbb{E}^{3})$, the second fundamental form of the surface $\bm{\theta}(\overline{\omega})$ is defined by means of its covariant components

$$b_{\alpha\beta}:=\partial_{\alpha}\bm{a}_{\beta}\cdot\bm{a}_{3}=-\bm{a}_{\beta}\cdot\partial_{\alpha}\bm{a}_{3}=b_{\beta\alpha}\in\mathcal{C}^{0}(\overline{\omega}),$$

or by means of its mixed components

$$b^{\beta}_{\alpha}:=a^{\beta\sigma}b_{\alpha\sigma}\in\mathcal{C}^{0}(\overline{\omega}),$$

and the Christoffel symbols associated with the immersion $\bm{\theta}$ are defined by

$$\Gamma^{\sigma}_{\alpha\beta}:=\partial_{\alpha}\bm{a}_{\beta}\cdot\bm{a}^{\sigma}=\Gamma^{\sigma}_{\beta\alpha}\in\mathcal{C}^{0}(\overline{\omega}).$$

Given an immersion $\bm{\theta}\in\mathcal{C}^{2}(\overline{\omega};\mathbb{E}^{3})$ and a vector field $\bm{\eta}=(\eta_{i})\in\mathcal{C}^{1}(\overline{\omega};\mathbb{R}^{3})$, the vector field

$$\tilde{\bm{\eta}}:=\eta_{i}\bm{a}^{i}$$

can be viewed as a displacement field of the surface $\bm{\theta}(\overline{\omega})$, thus defined by means of its covariant components $\eta_{i}$ over the vectors $\bm{a}^{i}$ of the contravariant bases along the surface. If the norms $\left\|\eta_{i}\right\|_{\mathcal{C}^{1}(\overline{\omega})}$ are small enough, the mapping $(\bm{\theta}+\eta_{i}\bm{a}^{i})\in\mathcal{C}^{1}(\overline{\omega};\mathbb{E}^{3})$ is also an immersion, so that the set $(\bm{\theta}+\eta_{i}\bm{a}^{i})(\overline{\omega})$ is also a surface in $\mathbb{E}^{3}$, equipped with the same curvilinear coordinates as those of the surface $\bm{\theta}(\overline{\omega})$, called the deformed surface corresponding to the displacement field $\tilde{\bm{\eta}}=\eta_{i}\bm{a}^{i}$. One can then define the first fundamental form of the deformed surface by means of its covariant components

	$\displaystyle a_{\alpha\beta}(\bm{\eta}):=$	$\displaystyle(\bm{a}_{\alpha}+\partial_{\alpha}\tilde{\bm{\eta}})\cdot(\bm{a}_{\beta}+\partial_{\beta}\tilde{\bm{\eta}})$
	$\displaystyle=$	$\displaystyle a_{\alpha\beta}+\bm{a}_{\alpha}\cdot\partial_{\beta}\tilde{\bm{\eta}}+\partial_{\alpha}\tilde{\bm{\eta}}\cdot\bm{a}_{\beta}+\partial_{\alpha}\tilde{\bm{\eta}}\cdot\partial_{\beta}\tilde{\bm{\eta}}.$

The linear part with respect to $\tilde{\bm{\eta}}$ in the difference $\dfrac{1}{2}(a_{\alpha\beta}(\bm{\eta})-a_{\alpha\beta})$ is called the linearized change of metric tensor associated with the displacement field $\eta_{i}\bm{a}^{i}$, the covariant components of which are thus defined by

$$\gamma_{\alpha\beta}(\bm{\eta}):=\dfrac{1}{2}\left(\bm{a}_{\alpha}\cdot\partial_{\beta}\tilde{\bm{\eta}}+\partial_{\alpha}\tilde{\bm{\eta}}\cdot\bm{a}_{\beta}\right)=\frac{1}{2}(\partial_{\beta}\eta_{\alpha}+\partial_{\alpha}\eta_{\beta})-\Gamma^{\sigma}_{\alpha\beta}\eta_{\sigma}-b_{\alpha\beta}\eta_{3}=\gamma_{\beta\alpha}(\bm{\eta}).$$

The linear part with respect to $\tilde{\bm{\eta}}$ in the difference $\dfrac{1}{2}(b_{\alpha\beta}(\bm{\eta})-b_{\alpha\beta})$ is called the linearized change of curvature tensor associated with the displacement field $\eta_{i}\bm{a}^{i}$, the covariant components of which are thus defined by

	$\displaystyle\rho_{\alpha\beta}(\bm{\eta})$	$\displaystyle:=(\partial_{\alpha\beta}\tilde{\bm{\eta}}-\Gamma_{\alpha\beta}^{\sigma}\partial_{\sigma}\tilde{\bm{\eta}})\cdot\bm{a}_{3}=\rho_{\beta\alpha}(\bm{\eta})$
		$\displaystyle=\partial_{\alpha\beta}\eta_{3}-\Gamma_{\alpha\beta}^{\sigma}\partial_{\sigma}\eta_{3}-b_{\alpha}^{\sigma}b_{\sigma\beta}\eta_{3}+b_{\alpha}^{\sigma}(\partial_{\beta}\eta_{\sigma}-\Gamma_{\beta\sigma}^{\tau}\eta_{\tau})+b_{\beta}^{\tau}(\partial_{\alpha}\eta_{\tau}-\Gamma_{\alpha\tau}^{\sigma}\eta_{\sigma})+(\partial_{\alpha}b_{\beta}^{\tau}+\Gamma_{\alpha\sigma}^{\tau}b_{\beta}^{\sigma}-\Gamma_{\alpha\beta}^{\sigma}b_{\sigma}^{\tau})\eta_{\tau}.$

It turns out that, when a generic surface is subjected to a displacement field $\eta_{i}\bm{a}^{i}$ whose tangential covariant components $\eta_{\alpha}$ vanish on a non-zero length portion of boundary of the domain $\omega$, denoted $\gamma_{0}$ in the statement of the next result, the following inequality holds (this inequality plays an essential role in our convergence analysis; cf. the proof of Theorem 5.1). Note that the components of the displacement fields, linearized change of metric tensor and linearized change of curvature tensor appearing in the next theorem are no longer assumed to be continuously differentiable functions; they are instead to be understood in a generalized sense, since they now belong to ad hoc Lebesgue or Sobolev spaces. Throughout the paper the symbol $\partial_{\nu}$ denotes the outer unit normal derivative operator along the boundary $\gamma$.

The above inequality, which is due to [3] and was later on improved by [4] (see also Theorem 2.6-4 of [11]), constitutes an example of a Korn inequality on a general surface, in the sense that it provides an estimate of an appropriate norm of a displacement field defined on a surface in terms of an appropriate norm of a specific “measure of strain” (here, the linearized change of metric tensor and the linearized change of curvature tensor) corresponding to the displacement field considered.

Let $\omega$ be a domain in $\mathbb{R}^{2}$, let $\gamma:=\partial\omega$, and let $\gamma_{0}$ be a non-empty relatively open subset of $\gamma$. For each $\varepsilon>0$, we define the sets

$$\Omega^{\varepsilon}=\omega\times\left]-\varepsilon,\varepsilon\right[\quad\textup{ and }\quad\Gamma^{\varepsilon}_{0}:=\gamma_{0}\times\left[-\varepsilon,\varepsilon\right],$$

we let $x^{\varepsilon}=(x^{\varepsilon}_{i})$ designate a generic point in the set $\overline{\Omega^{\varepsilon}}$, and we let $\partial^{\varepsilon}_{i}:=\partial/\partial x^{\varepsilon}_{i}$. Hence we also have $x^{\varepsilon}_{\alpha}=y_{\alpha}$ and $\partial^{\varepsilon}_{\alpha}=\partial_{\alpha}$.

Given an immersion $\bm{\theta}\in\mathcal{C}^{3}(\overline{\omega};\mathbb{E}^{3})$ and $\varepsilon>0$, consider a shell with middle surface $\bm{\theta}(\overline{\omega})$ and with constant thickness $2\varepsilon$. This means that the reference configuration of the shell is the set $\bm{\Theta}(\overline{\Omega^{\varepsilon}})$, where the mapping $\bm{\Theta}:\overline{\Omega^{\varepsilon}}\to\mathbb{E}^{3}$ is defined by

$$\bm{\Theta}(x^{\varepsilon}):=\bm{\theta}(y)+x^{\varepsilon}_{3}\bm{a}^{3}(y)\text{ at each point }x^{\varepsilon}=(y,x^{\varepsilon}_{3})\in\overline{\Omega^{\varepsilon}}.$$

One can then show (cf., e.g., Theorem 3.1-1 of [11]) that, if $\varepsilon>0$ is small enough, such a mapping $\bm{\Theta}\in\mathcal{C}^{2}(\overline{\Omega^{\varepsilon}};\mathbb{E}^{3})$ is an immersion, in the sense that the three vectors

$$\bm{g}^{\varepsilon}_{i}(x^{\varepsilon}):=\partial^{\varepsilon}_{i}\bm{\Theta}(x^{\varepsilon}),$$

are linearly independent at each point $x^{\varepsilon}\in\overline{\Omega^{\varepsilon}}$; these vectors then constitute the covariant basis at the point $\bm{\Theta}(x^{\varepsilon})$, while the three vectors $\bm{g}^{j,\varepsilon}(x^{\varepsilon})$ defined by the relations

$$\bm{g}^{j,\varepsilon}(x^{\varepsilon})\cdot\bm{g}^{\varepsilon}_{i}(x^{\varepsilon})=\delta^{j}_{i},$$

constitute the contravariant basis at the same point. It will be implicitly assumed in the sequel that $\varepsilon>0$ is small enough so that $\bm{\Theta}:\overline{\Omega^{\varepsilon}}\to\mathbb{E}^{3}$ is an immersion.

One then defines the metric tensor associated with the immersion $\bm{\Theta}$ by means of its covariant components

$$g^{\varepsilon}_{ij}:=\bm{g}^{\varepsilon}_{i}\cdot\bm{g}^{\varepsilon}_{j}\in\mathcal{C}^{1}(\overline{\Omega^{\varepsilon}}),$$

or by means of its contravariant components

$$g^{ij,\varepsilon}:=\bm{g}^{i,\varepsilon}\cdot\bm{g}^{i,\varepsilon}\in\mathcal{C}^{1}(\overline{\Omega^{\varepsilon}}).$$

Note that the symmetric matrix field $(g^{ij,\varepsilon})$ is then the inverse of the matrix field $(g^{\varepsilon}_{ij})$, that $\bm{g}^{j,\varepsilon}=g^{ij,\varepsilon}\bm{g}^{\varepsilon}_{i}$ and $g^{\varepsilon}_{i}=g^{\varepsilon}_{ij}\bm{g}^{j,\varepsilon}$, and that the volume element in $\bm{\Theta}(\overline{\Omega^{\varepsilon}})$ is given at each point $\bm{\Theta}(x^{\varepsilon})$, $x^{\varepsilon}\in\overline{\Omega^{\varepsilon}}$, by $\sqrt{g^{\varepsilon}(x^{\varepsilon})}\,\mathrm{d}x^{\varepsilon}$, where

$$g^{\varepsilon}:=\det(g^{\varepsilon}_{ij})\in\mathcal{C}^{1}(\overline{\Omega^{\varepsilon}}).$$

One also defines the Christoffel symbols associated with the immersion $\bm{\Theta}$ by

$$\Gamma^{p,\varepsilon}_{ij}:=\partial_{i}\bm{g}^{\varepsilon}_{j}\cdot\bm{g}^{p,\varepsilon}=\Gamma^{p,\varepsilon}_{ji}\in\mathcal{C}^{0}(\overline{\Omega^{\varepsilon}}).$$

Note that $\Gamma^{3,\varepsilon}_{\alpha 3}=\Gamma^{p,\varepsilon}_{33}=0$.

Given a vector field $\bm{v}^{\varepsilon}=(v^{\varepsilon}_{i})\in\mathcal{C}^{1}(\overline{\Omega^{\varepsilon}};\mathbb{R}^{3})$, the associated vector field

$$\tilde{\bm{v}}^{\varepsilon}:=v^{\varepsilon}_{i}\bm{g}^{i,\varepsilon},$$

can be viewed as a displacement field of the reference configuration $\bm{\Theta}(\overline{\Omega^{\varepsilon}})$ of the shell, thus defined by means of its covariant components $v^{\varepsilon}_{i}$ over the vectors $\bm{g}^{i,\varepsilon}$ of the contravariant bases in the reference configuration.

If the norms $\left\|v^{\varepsilon}_{i}\right\|_{\mathcal{C}^{1}(\overline{\Omega^{\varepsilon}})}$ are small enough, the mapping $(\bm{\Theta}+v^{\varepsilon}_{i}\bm{g}^{i,\varepsilon})$ is also an immersion, so that one can also define the metric tensor of the deformed configuration $(\bm{\Theta}+v^{\varepsilon}_{i}\bm{g}^{i,\varepsilon})(\overline{\Omega^{\varepsilon}})$ by means of its covariant components

$\displaystyle g^{\varepsilon}_{ij}(v^{\varepsilon}):=(\bm{g}^{\varepsilon}_{i}+\partial^{\varepsilon}_{i}\tilde{\bm{v}}^{\varepsilon})\cdot(\bm{g}^{\varepsilon}_{j}+\partial^{\varepsilon}_{j}\tilde{\bm{v}}^{\varepsilon})=g^{\varepsilon}_{ij}+\bm{g}^{\varepsilon}_{i}\cdot\partial_{j}\tilde{\bm{v}}^{\varepsilon}+\partial^{\varepsilon}_{i}\tilde{\bm{v}}^{\varepsilon}\cdot\bm{g}^{\varepsilon}_{j}+\partial_{i}\tilde{\bm{v}}^{\varepsilon}\cdot\partial_{j}\tilde{\bm{v}}^{\varepsilon}.$

The linear part with respect to $\tilde{\bm{v}}^{\varepsilon}$ in the difference $\dfrac{1}{2}(g^{\varepsilon}_{ij}(\bm{v}^{\varepsilon})-g^{\varepsilon}_{ij})$ is then called the linearized strain tensor associated with the displacement field $v^{\varepsilon}_{i}\bm{g}^{i,\varepsilon}$, the covariant components of which are thus defined by

$$e^{\varepsilon}_{i\|j}(\bm{v}^{\varepsilon}):=\frac{1}{2}\left(\bm{g}^{\varepsilon}_{i}\cdot\partial_{j}^{\varepsilon}\tilde{\bm{v}}^{\varepsilon}+\partial^{\varepsilon}_{i}\tilde{\bm{v}}^{\varepsilon}\cdot\bm{g}^{\varepsilon}_{j}\right)=\frac{1}{2}(\partial^{\varepsilon}_{j}v^{\varepsilon}_{i}+\partial^{\varepsilon}_{i}v^{\varepsilon}_{j})-\Gamma^{p,\varepsilon}_{ij}v^{\varepsilon}_{p}=e_{j\|i}^{\varepsilon}(\bm{v}^{\varepsilon}).$$

The functions $e^{\varepsilon}_{i\|j}(\bm{v}^{\varepsilon})$ are called the linearized strains in curvilinear coordinates associated with the displacement field $v^{\varepsilon}_{i}\bm{g}^{i,\varepsilon}$.

We assume throughout this paper that, for each $\varepsilon>0$, the reference configuration $\bm{\Theta}(\overline{\Omega^{\varepsilon}})$ of the shell is a natural state (i.e., stress-free) and that the material constituting the shell is homogeneous, isotropic, and linearly elastic. The behavior of such an elastic material is thus entirely governed by its two Lamé constants $\lambda\geq 0$ and $\mu>0$ (for details, see, e.g., Section 3.8 of [9]).

We will also assume that the shell is subjected to applied body forces whose density per unit volume is defined by means of its covariant components $f^{i,\varepsilon}\in L^{2}(\Omega^{\varepsilon})$, and to a homogeneous boundary condition of place along the portion $\Gamma^{\varepsilon}_{0}$ of its lateral face (i.e., the displacement vanishes on $\Gamma^{\varepsilon}_{0}$).

For what concerns surface traction forces, the mathematical models characterized by the confinement condition considered in this paper (confinement condition which is also considered in [27] in a more amenable geometrical framework) do not take any surface traction forces into account. Indeed, there could be no surface traction forces applied to the portion of the three-dimensional shell boundary that engages contact with the obstacle.

The confinement condition considered in this paper is more suitable in the context of multi-scales multi-bodies problems like, for instance, the study of the motion of the human heart valves, conducted by Quarteroni and his associates in [40, 41, 51] and the references therein.

In this paper we consider a specific obstacle problem for such a shell, in the sense that the shell is also subjected to a confinement condition, expressing that any admissible displacement vector field $v^{\varepsilon}_{i}\bm{g}^{i,\varepsilon}$ must be such that all the points of the corresponding deformed configuration remain in a half-space of the form

$$\mathbb{H}:=\{x\in\mathbb{E}^{3};\,\bm{Ox}\cdot\bm{q}\geq 0\},$$

where $\bm{q}$ is a nonzero vector given once and for all. In other words, any admissible displacement field must satisfy

$$\left(\bm{\Theta}(x^{\varepsilon})+v^{\varepsilon}_{i}(x^{\varepsilon})\bm{g}^{i,\varepsilon}(x^{\varepsilon})\right)\cdot\bm{q}\geq 0,$$

for all $x^{\varepsilon}\in\overline{\Omega^{\varepsilon}}$, or possibly only for almost all (a.a. in what follows) $x^{\varepsilon}\in\Omega^{\varepsilon}$ when the covariant components $v^{\varepsilon}_{i}$ are required to belong to the Sobolev space $H^{1}(\Omega^{\varepsilon})$ as in Theorem 3.1 below.

We will of course assume that the reference configuration satisfies the confinement condition, i.e., that

$$\bm{\Theta}(\overline{\Omega^{\varepsilon}})\subset\mathbb{H}.$$

It is to be emphasized that the above confinement condition considerably departs from the usual Signorini condition favored by most authors, who usually require that only the points of the undeformed and deformed “lower face” $\omega\times\{-\varepsilon\}$ of the reference configuration satisfy the confinement condition (see, e.g., [27], [28], [42]). Clearly, the confinement condition considered in the present paper, which is inspired by the formulation proposed by Brézis & Stampacchia [6], is more physically realistic, since a Signorini condition imposed only on the lower face of the reference configuration does not prevent – at least “mathematically” – other points of the deformed reference configuration to “cross” the plane $\{x\in\mathbb{E}^{3};\;\bm{Ox}\cdot\bm{q}=0\}$ and then to end up on the “other side” of this plane (cf., e.g., Chapter 63 in [50]). It is evident that the vector $\bm{q}$ is thus orthogonal to the plane associated with the half-space where the linearly elastic shell is required to remain confined.

Such a confinement condition renders the asymptotic analysis considerably more difficult, however, as the constraint now bears on a vector field, the displacement vector field of the reference configuration, instead of on only a single component of this field.

The mathematical modeling of such an obstacle problem for a linearly elastic shell is then clear; since, apart from the confinement condition, the rest, i.e., the function space and the expression of the quadratic energy $J^{\varepsilon}$, is classical (see, e.g. [11]). More specifically, let

$$A^{ijk\ell,\varepsilon}:=\lambda g^{ij,\varepsilon}g^{k\ell,\varepsilon}+\mu\left(g^{ik,\varepsilon}g^{j\ell,\varepsilon}+g^{i\ell,\varepsilon}g^{jk,\varepsilon}\right)=A^{jik\ell,\varepsilon}=A^{k\ell ij,\varepsilon},$$

denote the contravariant components of the elasticity tensor of the elastic material constituting the shell. Then the unknown of the problem, which is the vector field $\bm{u}^{\varepsilon}=(u^{\varepsilon}_{i})$ where the functions $u^{\varepsilon}_{i}:\overline{\Omega^{\varepsilon}}\to\mathbb{R}$ are the three covariant components of the unknown “three-dimensional” displacement vector field $u^{\varepsilon}_{i}\bm{g}^{i,\varepsilon}$ of the reference configuration of the shell, should minimize the energy $J^{\varepsilon}:\bm{H}^{1}(\Omega^{\varepsilon})\to\mathbb{R}$ defined by

$$J^{\varepsilon}(\bm{v}^{\varepsilon}):=\frac{1}{2}\int_{\Omega^{\varepsilon}}A^{ijk\ell,\varepsilon}e^{\varepsilon}_{k\|\ell}(\bm{v}^{\varepsilon})e^{\varepsilon}_{i\|j}(\bm{v}^{\varepsilon})\sqrt{g^{\varepsilon}}\,\mathrm{d}x^{\varepsilon}-\int_{\Omega^{\varepsilon}}f^{i,\varepsilon}v^{\varepsilon}_{i}\sqrt{g^{\varepsilon}}\,\mathrm{d}x^{\varepsilon},$$

for each $\bm{v}^{\varepsilon}=(v^{\varepsilon}_{i})\in\bm{H}^{1}(\Omega^{\varepsilon})$ over the set of admissible displacements defined by:

$$\bm{U}(\Omega^{\varepsilon}):=\{\bm{v}^{\varepsilon}=(v^{\varepsilon}_{i})\in\bm{H}^{1}(\Omega^{\varepsilon});\;\bm{v}^{\varepsilon}=\textbf{0}\text{ on }\Gamma^{\varepsilon}_{0},(\bm{\Theta}(x^{\varepsilon})+v^{\varepsilon}_{i}(x^{\varepsilon})\bm{g}^{i,\varepsilon}(x^{\varepsilon}))\cdot\bm{q}\geq 0\textup{ for a.a. }x^{\varepsilon}\in\Omega^{\varepsilon}\}.$$

The solution to this minimization problem exists and is unique, and it can be also characterized as the unique solution of the following problem:

The following result can be thus straightforwardly proved.

Since $\bm{\theta}(\overline{\omega})\subset\bm{\Theta}(\overline{\Omega^{\varepsilon}})$, it evidently follows that $\bm{\theta}(y)\cdot\bm{q}\geq 0$ for all $y\in\overline{\omega}$. But in fact, a stronger property holds (cf. Lemma 2.1 of [23]).

We now consider the “penalized” version of Problem $\mathcal{P}(\Omega^{\varepsilon})$. One such penalization transforms the set of variational inequalities in Problem $\mathcal{P}(\Omega^{\varepsilon})$ into a set of nonlinear variational equations, where the nonlinearity is defined in terms of the measure of the “violation” of the constraint. Let $\kappa>0$ denote a penalty parameter. The “penalized” variational formulation corresponding to Problem $\mathcal{P}(\Omega^{\varepsilon})$ takes the following form:

Note that the penalty term corresponds to the operator $\bm{\beta}^{\varepsilon}:\bm{L}^{2}(\Omega^{\varepsilon})\to\bm{L}^{2}(\Omega^{\varepsilon})$ defined by:

(1)

$$\bm{\beta}^{\varepsilon}(\bm{v}):=\left(-\{(\bm{\Theta}+v_{j}\bm{g}^{j,\varepsilon})\cdot\bm{q}\}^{-}\dfrac{\bm{g}^{i,\varepsilon}\cdot\bm{q}}{\sqrt{\sum_{\ell=1}^{3}|\bm{g}^{\ell,\varepsilon}\cdot\bm{q}|^{2}}}\right)_{i=1}^{3},\quad\textup{ for all }\bm{v}=(v_{i})\in\bm{L}^{2}(\Omega^{\varepsilon}).$$

Note that the denominator appearing in the definition of $\bm{\beta}^{\varepsilon}$ is always positive, since the three vector fields $\bm{g}^{i,\varepsilon}$ are linearly independent at each $x^{\varepsilon}\in\overline{\Omega^{\varepsilon}}$.

In order to show the existence and uniqueness of solution for Problem $\mathcal{P}_{\kappa}(\Omega^{\varepsilon})$, we have to show that the operator $\bm{\beta}^{\varepsilon}$ is monotone. The following lemma, whose proof can be found, for instance, in [37] serves for this purpose.

The existence and uniqueness of the solution for Problem $\mathcal{P}_{\kappa}(\Omega^{\varepsilon})$ is classical too (cf., e.g., Theorem 3.15 in [26], or [29]).

By means of a different classical approach based on energy estimates [29, 43] (see also part (iv) of Theorem 9.14-1 of  [13]), one can prove Theorem 3.2 by establishing that for each $\varepsilon>0$

(2)

$$\bm{u}^{\varepsilon}_{\kappa}\rightharpoonup\bm{u}^{\varepsilon},\quad\textup{ in }\bm{V}(\Omega^{\varepsilon}),$$

and that, thanks to the monotonicity of the penalty term established in Lemma 3.2, the weak limit satisfies Problem $\mathcal{P}(\Omega^{\varepsilon})$. As a result of the convergence (2) and the continuity of the linear operator $e^{\varepsilon}_{i\|j}:\bm{H}^{1}(\Omega^{\varepsilon})\to L^{2}(\Omega^{\varepsilon})$, we have that

$$e^{\varepsilon}_{i\|j}(\bm{u}^{\varepsilon}_{\kappa})\rightharpoonup e^{\varepsilon}_{i\|j}(\bm{u}^{\varepsilon}),\quad\textup{ in }L^{2}(\Omega^{\varepsilon}).$$

Actually, a stronger conclusion holds: Given $\varepsilon>0$, the following strong convergence holds

$$\bm{u}^{\varepsilon}_{\kappa}\to\bm{u}^{\varepsilon},\quad\textup{ in }\bm{V}(\Omega^{\varepsilon}).$$

To see this, observe that the uniform positive-definiteness of the elasticity tensor $(A^{ijk\ell,\varepsilon})$ (cf., e.g., [9]), Korn’s inequality (viz. Theorem 1.7-4 of [11]), the monotonicity of the penalty term (Lemma 3.2), the continuity of the operators $e_{i\|j}^{\varepsilon}$, and the weak convergence (2) give

	$\displaystyle\|\bm{u}^{\varepsilon}_{\kappa}-\bm{u}^{\varepsilon}\|_{\bm{H}^{1}(\Omega^{\varepsilon})}^{2}$	$\displaystyle\leq\dfrac{c_{0}c_{e}}{\sqrt{g_{0}}}\int_{\Omega^{\varepsilon}}A^{ijk\ell,\varepsilon}e^{\varepsilon}_{k\|\ell}(\bm{u}^{\varepsilon}_{\kappa}-\bm{u}^{\varepsilon})e^{\varepsilon}_{i\|j}(\bm{u}^{\varepsilon}_{\kappa}-\bm{u}^{\varepsilon})\sqrt{g^{\varepsilon}}\,\mathrm{d}x^{\varepsilon}$
		$\displaystyle=\dfrac{\varepsilon c_{0}c_{e}}{\sqrt{g_{0}}\kappa}\int_{\Omega^{\varepsilon}}\dfrac{\left\{[\bm{\Theta}+u_{j,\kappa}^{\varepsilon}\bm{g}^{j,\varepsilon}]\cdot\bm{q}\right\}^{-}}{\sqrt{\sum_{\ell=1}^{3}|\bm{g}^{\ell,\varepsilon}\cdot\bm{q}|^{2}}}((u^{\varepsilon}_{i,\kappa}-u^{\varepsilon}_{i})\bm{g}^{i,\varepsilon}\cdot\bm{q})\sqrt{g^{\varepsilon}}\,\mathrm{d}x^{\varepsilon}$
		$\displaystyle\quad+\dfrac{c_{0}c_{e}}{\sqrt{g_{0}}}\int_{\Omega^{\varepsilon}}f^{i,\varepsilon}(u^{\varepsilon}_{i,\kappa}-u^{\varepsilon}_{i})\sqrt{g^{\varepsilon}}\,\mathrm{d}x^{\varepsilon}$
		$\displaystyle\quad-\dfrac{c_{0}c_{e}}{\sqrt{g_{0}}}\int_{\Omega^{\varepsilon}}A^{ijk\ell,\varepsilon}e^{\varepsilon}_{k\|\ell}(\bm{u}^{\varepsilon})e^{\varepsilon}_{i\|j}(\bm{u}^{\varepsilon}_{\kappa}-\bm{u}^{\varepsilon})\sqrt{g^{\varepsilon}}\,\mathrm{d}x^{\varepsilon}$
		$\displaystyle\leq\dfrac{c_{0}c_{e}}{\sqrt{g_{0}}}\int_{\Omega^{\varepsilon}}f^{i,\varepsilon}(u^{\varepsilon}_{i,\kappa}-u^{\varepsilon}_{i})\sqrt{g^{\varepsilon}}\,\mathrm{d}x^{\varepsilon}$
		$\displaystyle\quad-\dfrac{c_{0}c_{e}}{\sqrt{g_{0}}}\int_{\Omega^{\varepsilon}}A^{ijk\ell,\varepsilon}e^{\varepsilon}_{k\|\ell}(\bm{u}^{\varepsilon})e^{\varepsilon}_{i\|j}(\bm{u}^{\varepsilon}_{\kappa}-\bm{u}^{\varepsilon})\sqrt{g^{\varepsilon}}\,\mathrm{d}x^{\varepsilon}\to 0,$

as $\kappa\to 0$. In particular, we have that for each $\varepsilon>0$ and for each $\delta>0$, we can find a number $\kappa_{0}=\kappa_{0}(\delta,\varepsilon)>0$ such that, for each $0<\kappa<\kappa_{0}$, it results

(3)

$$\|\bm{u}^{\varepsilon}-\bm{u}^{\varepsilon}_{\kappa}\|_{\bm{H}^{1}(\Omega^{\varepsilon})}<\dfrac{\delta}{2},$$

for all $\varepsilon>0$, where $\bm{u}^{\varepsilon}$ and $\bm{u}^{\varepsilon}_{\kappa}$ respectively denote the solutions of Problem $\mathcal{P}(\Omega^{\varepsilon})$ and Problem $\mathcal{P}_{\kappa}(\Omega^{\varepsilon})$.