An obstacle problem for Koiter’s shells

Abstract

In this paper, we define, a priori, a natural two-dimensional Koiter’s model of a ‘general’ linearly elastic shell subject to a confinement condition. As expected, this model takes the form of variational inequalities posed over a non-empty closed convex subset of the function space used for the ‘unconstrained’ Koiter’s model. We then perform a rigorous asymptotic analysis as the thickness of the shell, considered a ‘small’ parameter, approaches zero, when the shell belongs to one of the three main classes of linearly elastic shells, namely elliptic membrane shells, generalized membrane shells and flexural shells. To illustrate the soundness of this model, we consider elliptic membrane shells to fix ideas. We then show that, in this case, the ‘limit’ model obtained in this fashion coincides with the two-dimensional ‘limit’ model obtained by means of another rigorous asymptotic analysis, but this time with the three-dimensional model of a ‘general’ linearly elastic shell subject to a confinement condition as a point of departure. In this fashion, our proposed Koiter’s model of a linearly elastic shell subject to a confinement condition is fully justified in this case, even though it is not itself a ‘limit’ model.

Keywords

Linearly elastic shells Koiter’s model obstacle problems asymptotic analysis

1. Geometrical preliminaries

For details about the classical notions of differential geometry recalled in this section, see, e.g. Ciarlet [1, 2].

Greek indices, except $ε$ and $ν$ , 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 $E^{3}$ designates the three-dimensional Euclidean space; the Euclidean inner product and the vector product of $u, v \in E^{3}$ are denoted $u \cdot v$ and $u \land v$ ; the Euclidean norm of $u \in E^{3}$ is denoted $| u |$ . The notation $δ_{i}^{j}$ designates the Kronecker symbol.

Given an open subset $Ω$ of $ℝ^{n}$ , notations such as $L^{2} (Ω)$ , $H^{m} (Ω)$ or $H_{0}^{m} (Ω)$ , $m \geq 1$ , designate the usual Lebesgue and Sobolev spaces, and the notation $D (Ω)$ designates the space of all functions that are infinitely differentiable over $Ω$ and have compact supports in $Ω$ . The notation ${∥ \cdot ∥}_{X}$ designates the norm in a normed vector space X. Spaces of vector-valued functions are denoted with boldface letters. The notation ${∥ \cdot ∥}_{0, Ω}$ designates the norm of the Lebesgue space $L^{2} (Ω)$ , and the notation ${∥ \cdot ∥}_{m, Ω}$ , designates the norm of the Sobolev space $H^{m} (Ω)$ , $m \geq 1$ .

A domain in $ℝ^{n}$ is a bounded and connected open subset $Ω$ of $ℝ^{n}$ , whose boundary $∂Ω$ is Lipschitz-continuous, the set $Ω$ being locally on a single side of $∂Ω$ .

Let $ω$ be a domain in $ℝ^{2}$ , let $y = (y_{α})$ denote a generic point in $ω$ and let $\partial_{α} : = \partial ∕ \partial y_{α}$ and $\partial_{α β} : = \partial^{2} ∕ \partial y_{α} \partial y_{β}$ . A mapping $θ \in C^{1} (\bar{ω}; E^{3})$ is an immersion if the two vectors

a_{α} (y) : = \partial_{α} θ (y)

are linearly independent at each point $y \in \bar{ω}$ . Then the image $θ (\bar{ω})$ of the set $\bar{ω}$ under the mapping $θ$ is a surface in $E^{3}$ , equipped with $y_{1}, y_{2}$ as its curvilinear coordinates. Given any point $y \in \bar{ω}$ , the vectors $a_{α} (y)$ span the tangent plane to the surface $θ (\bar{ω})$ at the point $θ (y)$ , the unit vector

a_{3} (y) : = \frac{a_{1} (y) \land a_{2} (y)}{| a_{1} (y) \land a_{2} (y) |}

is normal to $θ (\bar{ω})$ at $θ (y)$ , the three vectors $a_{i} (y)$ form the covariant basis at $θ (y)$ and the three vectors $a^{j} (y)$ defined by the relations

a^{j} (y) \cdot a_{i} (y) = δ_{i}^{j}

form the contravariant basis at $θ (y)$ ; note that the vectors $a^{β} (y)$ also span the tangent plane to $θ (\bar{ω})$ at $θ (y)$ and that $a^{3} (y) = a_{3} (y)$ .

The first fundamental form of the surface $θ (\bar{ω})$ is defined by means of its covariant components

a_{α β} : = a_{α} \cdot a_{β} = a_{β α} \in C^{0} (\bar{ω}),

or by means of its contravariant components

a^{α β} : = a^{α} \cdot a^{β} = a^{β α} \in C^{0} (\bar{ω}) .

Note that the symmetric matrix field $(a^{α β})$ is the inverse of the matrix field $(a_{α β})$ , that $a^{β} = a^{α β} a_{α}$ and $a_{α} = a_{α β} a^{β}$ , and that the area element along $θ (\bar{ω})$ is given at each point $θ (y), y \in \bar{ω}$ , by $\sqrt{a (y)} dy$ , where

a : = \det (a_{α β}) \in C^{0} (\bar{ω}) .

Given an immersion $θ \in C^{2} (\bar{ω}; E^{3})$ , the second fundamental form of the surface $θ (\bar{ω})$ is defined by means of its covariant components

b_{α β} : = \partial_{α} a_{β} \cdot a_{3} = - a_{β} \cdot \partial_{α} a_{3} = b_{β α} \in C^{0} (\bar{ω}),

or by means of its mixed components

b_{α}^{β} : = a^{βσ} b_{ασ} \in C^{0} (\bar{ω}),

and the Christoffel symbols associated with the immersion $θ$ are defined by

Γ_{α β}^{σ} : = \partial_{α} a_{β} \cdot a^{σ} = Γ_{β α}^{σ} \in C^{0} (\bar{ω}) .

The Gaussian curvature at each point $θ (y), y \in \bar{ω}$ , of the surface $θ (\bar{ω})$ is defined by

κ (y) : = \frac{det (b_{α β} (y))}{det (a_{α β} (y))} = \det (b_{α}^{β} (y))

(the denominator in this relation does not vanish, since $θ$ is assumed to be an immersion). Note that the Gaussian curvature $κ (y)$ at the point $θ (y)$ is also equal to the product of the two principal curvatures at this point.

A surface $θ (\bar{ω})$ defined by means of an immersion $θ \in C^{2} (\bar{ω}; E^{3})$ is said to be elliptic if its Gaussian curvature is everywhere $> 0$ in $\bar{ω}$ or, equivalently, if there exists a constant $κ_{0}$ such that

0 < κ_{0} \leq κ (y) for all y \in \bar{ω} .

Given an immersion $θ \in C^{2} (\bar{ω}; E^{3})$ and a vector field $η = (η_{i}) \in C^{1} (\bar{ω}; ℝ^{3})$ , the vector field

\tilde{η} : = η_{i} a^{i}

can be viewed as a displacement field of the surface $θ (\bar{ω})$ , and is thus defined by means of its covariant components $η_{i}$ over the vectors $a^{i}$ of the contravariant bases along the surface. If the norms ${∥ η_{i} ∥}_{C^{1} (\bar{ω})}$ are small enough, the mapping $(θ + η_{i} a^{i}) \in C^{1} (\bar{ω}; E^{3})$ is also an immersion, so that the set $(θ + η_{i} a^{i}) (\bar{ω})$ is also a surface in $E^{3}$ , equipped with the same curvilinear coordinates as those of the surface $θ (\bar{ω})$ , called the deformed surface corresponding to the displacement field $\tilde{η} = η_{i} a^{i}$ . One can then define the first fundamental form of the deformed surface by means of its covariant components

a_{α β} (η) : = (a_{α} + \partial_{α} \tilde{η}) \cdot (a_{β} + \partial_{β} \tilde{η}),

and the second fundamental form of the deformed surface by means of its covariant components

b_{α β} (η) : = \partial_{α} (a_{β} + \partial_{β} \tilde{η}) \cdot \frac{(a_{1} + \partial_{1} \tilde{η}) \land (a_{2} + \partial_{2} \tilde{η})}{| (a_{1} + \partial_{1} \tilde{η}) \land (a_{2} + \partial_{2} \tilde{η}) |}

The linear part with respect to $\tilde{η}$ in the difference

\frac{1}{2} (a_{α β} (η) - a_{α β})

is called the linearized change of metric, or strain, tensor associated with the displacement field $η_{i} a^{i}$ , the covariant components of which are then given by

\begin{matrix} γ_{α β} (η) = \frac{1}{2} (a_{α} \cdot \partial_{β} \tilde{η} + \partial_{α} \tilde{η} \cdot a_{β}) \\ = \frac{1}{2} (\partial_{β} η_{α} + \partial_{α} η_{β}) - Γ_{α β}^{σ} η_{σ} - b_{α β} η_{3} \\ = γ_{β α} (η) . \end{matrix}

The linear part with respect to $\tilde{η}$ in the difference

\frac{1}{2} (b_{α β} (η) - b_{α β})

is called the linearized change of curvature tensor associated with the displacement field $η_{i} a^{i}$ , the covariant components of which are then given by

\begin{matrix} ρ_{α β} (η) = (\partial_{α β} \tilde{η} - Γ_{α β}^{σ} \partial_{σ} \tilde{η}) \cdot a_{3} \\ = \partial_{α β} η_{3} - Γ_{α β}^{σ} \partial_{σ} η_{3} - b_{α}^{σ} b_{σβ} η_{3} + b_{α}^{σ} (\partial_{β} η_{σ} - Γ_{βσ}^{τ} η_{τ}) + b_{β}^{τ} (\partial_{α} η_{τ} - Γ_{ατ}^{σ} η_{σ}) + (\partial_{α} b_{β}^{τ} + Γ_{ασ}^{τ} b_{β}^{σ} - Γ_{α β}^{σ} b_{σ}^{τ}) η_{τ} \\ = ρ_{β α} (η) . \end{matrix}

2. A natural Koiter’s model for a general shell subject to a confinement condition

Let $ω$ be a domain in $ℝ^{2}$ with boundary $γ$ , and let $γ_{0}$ be a non-empty relatively open subset of $γ$ . For each $ε > 0$ , we define the sets

Ω^{ε} : = ω \times] - ε, ε [and Γ_{0}^{ε} : = γ_{0} \times [- ε, ε],

we let $x^{ε} = (x_{i}^{ε})$ designate a generic point in the set $\bar{Ω^{ε}}$ , and we let $\partial_{i}^{ε} : = \partial ∕ \partial x_{i}^{ε}$ . Hence, we also have $x_{α}^{ε} = y_{α}$ and $\partial_{α}^{ε} = \partial_{α}$ . The set $Γ_{0}^{ε}$ is thus a subset of the lateral face of the undeformed reference configuration.

In all that follows, we are given an injective immersion $θ \in C^{3} (\bar{ω}; E^{3})$ and $ε > 0$ , and we consider a shell with middle surface $θ (\bar{ω})$ and with constant thickness $2 ε$ . This means that the reference configuration of the shell is the set $Θ (\bar{Ω^{ε}})$ , where the mapping $Θ : \bar{Ω^{ε}} \to E^{3}$ is defined by

Θ (x^{ε}) : = θ (y) + x_{3}^{ε} a^{3} (y) at each point x^{ε} = (y, x_{3}^{ε}) \in \bar{Ω^{ε}} .

Note that the injectivity assumption is made here for physical reasons, but that it is otherwise not needed in the proofs. One can then show (compare with Theorem 3.1-1 of Ciarlet [1] or Theorem 4.1-1 of Ciarlet [2]) that, if the thickness $ε > 0$ is small enough, such a mapping $Θ \in C^{2} (\bar{Ω^{ε}}; E^{3})$ is a $C^{2}$ –diffeomorphism from $\bar{Ω^{ε}}$ onto $Θ (\bar{Ω^{ε}})$ , and hence is in particular an injective immersion, in the sense that the three vectors

g_{i}^{ε} (x^{ε}) : = \partial_{i}^{ε} Θ (x^{ε})

are linearly independent at each point $x^{ε} \in \bar{Ω^{ε}}$ ; these vectors then constitute the covariant basis at the point $Θ (x^{ε})$ , while the three vectors $g^{j, ε} (x^{ε})$ , defined by the relations

g^{j, ε} (x^{ε}) \cdot g_{i}^{ε} (x^{ε}) = δ_{i}^{j},

constitute the contravariant basis at the same point.

It will be implicitly assumed in what follows that the immersion $θ \in C^{3} (\bar{ω}; E^{3})$ is injective and that $ε > 0$ is small enough so that $Θ : \bar{Ω^{ε}} \to E^{3}$ is a $C^{2}$ -diffeomorphism onto its image.

We henceforth assume that the shell is made of a homogeneous and isotropic linearly elastic material and that its reference configuration $Θ (\bar{Ω^{ε}})$ is a natural state, i.e., is stress-free. As a result of these assumptions, the elastic behaviour of this elastic material is completely characterized by its two Lamé constants $λ \geq 0$ and $μ > 0$ (see, e.g., Section 3.8 in Ciarlet [3]).

We also assume that the shell is subjected to applied body forces whose density per unit volume is defined by means of its contravariant components $f^{i, ε} \in L^{2} (Ω^{ε})$ , i.e., over the vectors $g_{i}^{ε}$ of the covariant bases, and to a homogeneous boundary condition of place along the portion $Γ_{0}^{ε}$ of its lateral face, i.e., the admissible displacement fields vanish on $Γ_{0}^{ε}$ .

A commonly used two-dimensional set of equations for modelling such a shell (‘two-dimensional’ in the sense that it is posed over $\bar{ω}$ instead of $\bar{Ω^{ε}}$ ) was proposed in 1970 by Koiter [4, 5]. We now describe the modern formulation of this model.

First, define the space

V_{K} (ω) : = {η = (η_{i}) \in H^{1} (ω) \times H^{1} (ω) \times H^{2} (ω); η_{i} = \partial_{ν} η_{3} = 0 on γ_{0}},

where the symbol $\partial_{ν}$ denotes the outer unit normal derivative operator along $γ$ , and define the norm $∥ \cdot ∥_{V_{K} (ω)}$ by

∥ η ∥_{V_{K} (ω)} : = {\sum_{α} ∥ η_{α} ∥_{1, ω}^{2} + ∥ η_{3} ∥_{2, ω}^{2}}^{1 ∕ 2} for each η = (η_{i}) \in V_{K} (ω) .

Next, define the fourth-order two-dimensional elasticity tensor of the shell, viewed here as a two-dimensional linearly elastic body, by means of its contravariant components

a^{αβστ} : = \frac{4 λμ}{λ + 2 μ} a^{α β} a^{στ} + 2 μ (a^{ασ} a^{βτ} + a^{ατ} a^{βσ}) .

Finally, define the bilinear forms $B_{M} (\cdot, \cdot)$ and $B_{F} (\cdot, \cdot)$ by

\begin{matrix} B_{M} (ξ, η) : = \int_{ω} a^{αβστ} γ_{στ} (ξ) γ_{α β} (η) \sqrt{a} dy, \\ B_{F} (ξ, η) : = \frac{1}{3} \int_{ω} a^{αβστ} ρ_{στ} (ξ) ρ_{α β} (η) \sqrt{a} dy, \end{matrix}

for each $ξ = (ξ_{i}) \in V_{K} (ω)$ and each $η = (η_{i}) \in V_{K} (ω)$ , and define the linear form $ℓ^{ε}$ by

ℓ^{ε} (η) : = \int_{ω} p^{i, ε} η_{i} \sqrt{a} dy, for each η = (η_{i}) \in V_{K} (ω),

where $p^{i, ε} (y) : = \int_{- ε}^{ε} f^{i, ε} (y, x^{ε}) d x^{ε}$ at each $y \in ω$ .

Then the total energy of the shell is the quadratic functional $J : V_{K} (ω) \to ℝ$ defined by

J (η) : = \frac{ε}{2} B_{M} (η, η) + \frac{ε^{3}}{2} B_{F} (η, η) - ℓ^{ε} (η) for each η \in V_{K} (ω) .

The terms $ε B_{M} (\cdot, \cdot) ∕ 2$ and $ε^{3} 2 B_{F} (\cdot, \cdot) ∕ 2$ respectively represent the membrane part and the flexural part of the total energy, as aptly recalled by the subscripts ‘M’ and ‘F’.

In Koiter’s model, the unknown ‘two-dimensional’ displacement field $ζ_{i, K}^{ε} a^{i} : \bar{ω} \to E^{3}$ of the middle surface $θ (\bar{ω})$ of the shell is such that the vector field $ζ_{K}^{ε} : = (ζ_{i, K}^{ε})$ should be the solution of the following problem: find a vector field $ζ_{K}^{ε} : \bar{ω} \to ℝ^{3}$ that satisfies

ζ_{K}^{ε} \in V_{K} (ω) and J (ζ_{K}^{ε}) = inf_{η \in V_{K} (ω)} J (η),

or equivalently, find $ζ_{K}^{ε} \in V_{K} (ω)$ that satisfies the following variational equations:

ε B_{M} (ζ_{K}^{ε}, η) + ε^{3} B_{F} (ζ_{K}^{ε}, η) = ℓ^{ε} (η) for all η \in V_{K} (ω) .

As first shown by Bernadou and Ciarlet [6] (see also Bernadou et al. [7]), this problem has one and only one solution.

Assume now that the shell is subject to the following confinement condition: the unknown displacement field $ζ_{i, K}^{ε} a^{i}$ of the middle surface of the shell must be such that the corresponding ‘deformed’ middle surface remains in a given half-space, of the form

ℍ : = {x \in E^{3}; ox \cdot p \geq 0},

where $p \in E^{3}$ is a given non-zero vector. Equivalently, the deformed middle surface must remain ‘on one side’ of the plane ${x \in E^{3}; ox \cdot p \geq 0}$ , then viewed as an obstacle; it is this observation that motivates the use of the term ‘obstacle problem’ in the title of this paper. Of course, it will be henceforth assumed that the ‘undeformed’ middle surface satisfies this confinement condition, i.e., that

θ (\bar{ω}) \subset ℍ .

It is thus physically sound to assume that, while the total energy of the shell remains unchanged, the set over which the energy is to be minimized is now a strict subset of $V_{K} (ω)$ (denoted $U_{K} (ω)$ in the following) that takes into account the imposed confinement condition. These assumptions lead to the following definition of a problem, denoted $P_{K}^{ε} (ω)$ in the next theorem, which constitutes Koiter’s model for a shell subject to a confinement condition. In what follows, ‘a.e.’ stands for ‘almost everywhere’.

Theorem 1. The minimization problem: Find

ζ_{K}^{ε} \in U_{K} (ω) : = {η = (η_{i}) \in V_{K} (ω); (θ (y) + η_{i} (y) a^{i} (y)) \cdot p \geq 0 for a . y \in ω}

such that

J (ζ_{K}^{ε}) = inf_{η \in U_{K} (ω)} J (η)

has one and only one solution.

Besides, this solution is also the unique solution of problem $P_{K}^{ε} (ω)$ : find $ζ_{K}^{ε} \in U_{K} (ω)$ that satisfies the following variational inequalities:

ε B_{M} (ζ_{K}^{ε}, η - ζ_{K}^{ε}) + ε^{3} B_{F} (ζ_{K}^{ε}, η - ζ_{K}^{ε}) \geq ℓ^{ε} (η - ζ_{K}^{ε}) for all η = (η_{i}) \in U_{K} (ω) .

Proof. The bilinear forms $B_{M} : V_{K} (ω) \times V_{K} (ω) \to ℝ$ and $B_{F} : V_{K} (ω) \times V_{K} (ω) \to ℝ$ and the linear form $ℓ^{ε} : V_{K} (ω) \to ℝ$ are clearly continuous. Since the two-dimensional elasticity tensor $(a^{αβστ})$ of the shell is uniformly positive definite, in the sense that there exists a constant $c = c (ω, θ, μ) > 0$ such that

\sum_{α, β} | t_{α β} |^{2} \leq c a^{αβστ} (y) t_{στ} t_{α β}

for all $y \in \bar{ω}$ and all symmetric matrices $(t_{α β})$ , the bilinear form

(ξ, η) \in V_{K} (ω) \times V_{K} (ω) \to ε B_{M} (ξ, η) + ε^{3} B_{F} (ξ, η) \in ℝ

is $V_{K} (ω)$ -elliptic, on the one hand, thanks to the Korn’s inequality on a general surface due to [6] (see also [7]; this inequality is also recalled in Theorem 5 below; note that this inequality holds in particular because $dγ$ -meas $γ_{0} > 0$ ).

On the other hand, it is easily seen that $U_{K} (ω)$ is a non-empty (by assumption), closed (any convergent sequence in $U_{K} (ω)$ contains a subsequence that pointwise converges to the same limit), and convex, subset of the space $V_{K} (ω)$ .

Hence, by a classical result (see, e.g., Duvaut and Lions [8], Glowinski [9], or Theorems 6.1–1 and 6.1–2 of Ciarlet [10]), this minimization problem, or equivalently problem $P_{K}^{ε} (ω)$ , has one and only one solution. □

Problem $P_{K}^{ε} (ω)$ is meant to apply to any shell, i.e., regardless of the subset $γ_{0}$ of $γ$ , the asymptotic behaviour of the applied forces as $ε \to 0$ , and the geometry of the middle surface. It is in this sense that it is a model for a ‘general’ shell.

Conversely, it has been shown (in the series of papers by Ciarlet and Lods [11, 12], and Ciarlet et al. [13]; see also Chapters 4, 5 and 6 in Ciarlet [1]) that a rigorous asymptotic analysis of the equations of ‘unconstrained’ (i.e., not subject to any confinement condition) linearly elastic shells as their thickness approaches zero leads one to distinguish three types of ‘limit’ equations, corresponding to elliptic membrane shells, generalized membrane shells and flexural shells, depending on the subset $γ_{0}$ , the behaviour of the applied forces as $ε \to 0$ and the geometry of the middle surface. It has been furthermore shown (see Theorems 3.2, 4.1 and 5.2 in the references provided just before) that, remarkably, the equations of the linear two-dimensional ‘unconstrained’ Koiter’s model (i.e., that described at the beginning of this section) exhibit exactly the same ‘limit’ behaviour, thus fully justifying Koiter’s model in the ‘unconstrained’ case, i.e., when no confinement condition is imposed.

It is thus natural to investigate whether our proposed Koiter’s model for a shell subject to a confinement condition shares the same features. This is the primary objective of this paper.

To this end, we thus need to identify the possible limit behaviours of the solution to our model, i.e., of problem $P_{K}^{ε} (ω)$ , as $ε \to 0$ , depending on the type of shell under consideration: this is the objective of the next three sections.

3. Koiter’s model for an elliptic membrane shell subject to a confinement condition

Consider a linearly elastic shell, subject to the various assumptions set forth in Section 2. Following the terminology proposed in Section 4.1 of Ciarlet [1], such a shell is said to be a linearly elastic elliptic membrane shell if the following two additional assumptions are satisfied: first, $γ_{0} = γ$ , i.e., the homogeneous boundary condition of place is imposed over the entire lateral face $γ \times [- ε, ε]$ of the shell, and second, its middle surface $θ (\bar{ω})$ is elliptic, according to the definition given in Section 1. Note that the assumption $γ_{0} = γ$ implies that the space $V_{K} (ω)$ introduced in Section 2 now reduces to

V_{K} (ω) = H_{0}^{1} (ω) \times H_{0}^{1} (ω) \times H_{0}^{2} (ω) .

To begin with, we recall a crucial inequality that holds for elliptic surfaces.

Theorem 2. Let $ω$ be a domain in $ℝ^{2}$ and let $θ \in C^{3} (\bar{ω}; E^{3})$ be an immersion such that $θ (\bar{ω})$ is an elliptic surface. Define the space

V_{M} (ω) : = H_{0}^{1} (ω) \times H_{0}^{1} (ω) \times L^{2} (ω),

and the norm $∥ \cdot ∥_{V_{M} (ω)}$ by

∥ η ∥_{V_{M} (ω)} : = {\sum_{α} ∥ η_{α} ∥_{1, ω}^{2} + ∥ η_{3} ∥_{0, ω}^{2}}^{1 ∕ 2} for each η = (η_{i}) \in V_{M} (ω) .

Then there exists a constant $c = c (ω, θ) > 0$ such that

∥ η ∥_{V_{M} (ω)} \leq c {\sum_{α, β} ∥ γ_{α β} (η) ∥_{0, ω}^{2}}^{1 ∕ 2}

for all $η = (η_{i}) \in V_{M} (ω)$ .

This inequality, which was proves by Ciarlet and Lods [14] and Ciarlet and Sanchez-Palencia [15] (see also Ciarlet [1], Theorems 2.7–3), constitutes an example of a Korn inequality on an elliptic surface; it constitutes a ‘Korn inequality’ in the sense that it provides an estimate of an appropriate norm of a displacement field defined on an elliptic surface in terms of an appropriate norm of a specific ‘measure of strain’ (here, the linearized change of metric tensor) corresponding to the displacement field considered.

The forthcoming analysis resorts to an argument similar to the one in Theorem 7.2-1 (itself based on Destuynder [16], Sanchez-Palencia [17], Caillerie and Sanchez-Palencia [18] and, especially, on Theorem 2.1 in Ciarlet and Lods [19]) of Ciarlet [1], and constitutes the first convergence result of this paper. The set $U_{K} (ω)$ appearing in the next theorem is defined in Theorem 1.

Theorem 3. Let $ω$ be a domain in $ℝ^{2}$ and let $θ \in C^{3} (\bar{ω}; E^{3})$ be an immersion. Consider a family of elliptic membrane shells with thickness $2 ε$ approaching zero and with each having the same middle surface $θ (\bar{ω})$ , and assume that there exist functions $f^{i} \in L^{2} (Ω)$ independent of $ε$ such that

f^{i, ε} (x^{ε}) = f^{i} (x) at each x^{ε} \in Ω^{ε} for each ε > 0 .

Finally, assume that the following ‘density property’ holds:

U_{K} (ω) is dense in U_{M} (ω) with respect to the norm ∥ \cdot ∥_{V_{M} (ω)} .

For each $ε > 0$ , let $ζ_{K}^{ε}$ denote the solution to problem $P_{K}^{ε} (ω)$ (Theorem 1). Then the following convergences hold:

\begin{matrix} ζ_{α, K}^{ε} a^{α} \to ζ_{α} a^{α} in H^{1} (ω) as ε \to 0, \\ ζ_{3, K}^{ε} a^{3} \to ζ_{3} a^{3} in L^{2} (ω) as ε \to 0, \end{matrix}

where $ζ$ is the unique solution to the following problem $P (ω)$ : Find

ζ \in U_{M} (ω) : = {η = (η_{i}) \in V_{M} (ω); (θ (y) + η_{i} (y) a^{i} (y)) \cdot p \geq 0 for a . a . y \in ω}

that satisfies the following variational inequalities:

\int_{ω} a^{αβστ} γ_{στ} (ζ) γ_{α β} (η - ζ) \sqrt{a} dy \geq \int_{ω} p^{i} (η_{i} - ζ_{i}) \sqrt{a} dy for all η = (η_{i}) \in U_{M} (ω),

where

p^{i} : = \int_{- 1}^{1} f^{i} d x_{3} \in L^{2} (ω) .

Proof. (i) Problem $P_{M} (ω)$ has one and only one solution. To see this, we notice that the bilinear form defined by

B_{M} (ξ, η) = \int_{ω} a^{αβστ} γ_{στ} (ξ) γ_{α β} (η) \sqrt{a} dy for all ξ, η \in U_{M} (ω),

is continuous and $V_{M} (ω)$ -elliptic thanks to the Korn’s inequality of Theorem 2, the set $U_{M} (ω)$ is a non-empty closed and convex subset of $V_{M} (ω)$ , and the linear form $ℓ$ defined by

ℓ (η) : = \int_{ω} p^{i} η_{i} \sqrt{a} dy for each η = (η_{i}) \in V_{M} (ω)

is continuous over $V_{M} (ω)$ .

(ii) Uniform boundedness of the family ${(ζ_{K}^{ε})}_{ε > 0}$ . In what follows, strong and weak convergences are respectively denoted → and ⇀. By virtue of the assumption made on the applied body force density, the variational inequalities in problem $P_{K}^{ε} (ω)$ reduce to

B_{M} (ζ_{K}^{ε}, η - ζ_{K}^{ε}) + ε^{2} B_{F} (ζ_{K}^{ε}, η - ζ_{K}^{ε}) \geq ℓ (η - ζ_{K}^{ε}),

for all $η \in U_{K} (ω)$ ; hence,

B_{M} (ζ_{K}^{ε}, ζ_{K}^{ε}) + ε^{2} B_{F} (ζ_{K}^{ε}, ζ_{K}^{ε}) \leq B_{M} (ζ_{K}^{ε}, η) + ε^{2} B_{F} (ζ_{K}^{ε}, η) - ℓ (η - ζ_{K}^{ε})

for all $η \in U_{K} (ω)$ . Thanks to the uniform positive definiteness of the tensor $(a^{αβστ})$ and to Theorem 2, there exists a constant $C_{1} > 0$ such that

∥ ζ_{K}^{ε} ∥_{V_{M} (ω)}^{2} \leq C_{1} B_{M} (ζ_{K}^{ε}, ζ_{K}^{ε}) .

Besides, the continuity of the bilinear forms $B_{M} (\cdot, \cdot)$ and $B_{F} (\cdot, \cdot)$ and the continuity of the linear form $ℓ$ imply that there exists a constant $C_{2} > 0$ such that

\begin{matrix} B_{M} (ζ_{K}^{ε}, η) + ε^{2} B_{F} (ζ_{K}^{ε}, η) - ℓ (η - ζ_{K}^{ε}) \\ \leq C_{2} (∥ ζ_{K}^{ε} ∥_{V_{M} (ω)} ∥ η ∥_{V_{M} (ω)} + ε^{2} ∥ ζ_{K}^{ε} ∥_{V_{M} (ω)} ∥ η ∥_{V_{M} (ω)} + ∥ ζ_{K}^{ε} ∥_{V_{M} (ω)} + ∥ η ∥_{V_{M} (ω)}) for all η = (η_{i}) \in U_{K} (ω) . \end{matrix}

Letting $η = 0$ thus gives

∥ ζ_{K}^{ε} ∥_{V_{M} (ω)} \leq C_{1} C_{2} for all ε > 0 .

(iii) Weak convergence of the family ${(ζ_{K}^{ε})}_{ε > 0}$ . By (ii), the family ${(ζ_{K}^{ε})}_{ε > 0}$ is bounded in $V_{M} (ω)$ . Therefore, there exists a subsequence, still denoted ${(ζ_{K}^{ε})}_{ε > 0}$ , a vector field $ζ^{*} \in V_{M} (ω)$ and functions $ρ_{α β}^{- 1} \in L^{2} (ω)$ such that:

\begin{matrix} ζ_{K}^{ε} ⇀ ζ^{*} in V_{M} (ω), \\ ε ρ_{α β} (ζ_{K}^{ε}) ⇀ ρ_{α β}^{- 1} in L^{2} (ω), \end{matrix}

the second convergence being also a consequence of the uniform positive definiteness of the tensor $(a^{αβστ})$ . Then $ζ^{*} \in U_{M} (ω)$ , since the set $U_{M} (ω)$ is non-empty, closed and convex (compare with, e.g., Theorem 3.7 in Brezis [20] or Theorem 5.13–1 in Ciarlet [10]). Fix $η \in U_{K} (ω)$ and observe that, since the vector field $ζ_{K}^{ε}$ solves problem $P_{K}^{ε} (ω)$ , the following variational inequalities hold:

B_{M} (ζ_{K}^{ε}, ζ_{K}^{ε}) \leq B_{M} (ζ_{K}^{ε}, η) + ε^{2} B_{F} (ζ_{K}^{ε}, η) - ℓ (η - ζ_{K}^{ε}),

so that letting $ε \to 0$ gives

\underset{ε \to 0}{limsup} B_{M} (ζ_{K}^{ε}, ζ_{K}^{ε}) \leq B_{M} (ζ^{*}, η) - ℓ (η - ζ^{*}),

on the one hand. On the other hand,

0 \leq B_{M} (ζ_{K}^{ε} - ζ^{*}, ζ_{K}^{ε} - ζ^{*}) = B_{M} (ζ_{K}^{ε}, ζ_{K}^{ε}) - 2 B_{M} (ζ_{K}^{ε}, ζ^{*}) + B_{M} (ζ^{*}, ζ^{*}),

which in turn implies that

2 B_{M} (ζ_{K}^{ε}, ζ^{*}) - B_{M} (ζ^{*}, ζ^{*}) \leq B_{M} (ζ_{K}^{ε}, ζ_{K}^{ε}) .

Hence, letting $ε \to 0$ gives

B_{M} (ζ^{*}, ζ^{*}) \leq \underset{ε \to 0}{liminf} B_{M} (ζ_{K}^{ε}, ζ_{K}^{ε}) .

In conclusion,

B_{M} (ζ^{*}, η - ζ^{*}) \geq ℓ (η - ζ^{*}) for all η = (η_{i}) \in U_{K} (ω) .

Furthermore, the assumed ‘density property’ gives

B_{M} (ζ^{*}, η - ζ^{*}) \geq ℓ (η - ζ^{*}) for all η = (η_{i}) \in U_{M} (ω),

which shows that $ζ^{*}$ is a solution to problem $P_{M} (ω)$ . Since problem $P_{M} (ω)$ admits a unique solution, we conclude that $ζ = ζ^{*}$ . Hence, the whole family ${(ζ_{K}^{ε})}_{ε > 0}$ weakly converges to $ζ$ in $V_{M} (ω)$ as $ε \to 0$ .

(iv) Strong convergence of the family ${(ζ_{K}^{ε})}_{ε > 0}$ . The $V_{K} (ω)$ -ellipticity of the bilinear form $B_{M} (\cdot, \cdot)$ and the assumed ‘density property’ together give

\begin{matrix} 0 \leq \frac{1}{C_{1}} ∥ ζ_{K}^{ε} - ζ ∥_{V_{M} (ω)}^{2} \\ \leq B_{M} (ζ_{K}^{ε} - ζ, ζ_{K}^{ε} - ζ) \\ = B_{M} (ζ_{K}^{ε}, ζ_{K}^{ε}) - 2 B_{M} (ζ_{K}^{ε}, ζ) + B_{M} (ζ, ζ) \\ \leq B_{M} (ζ_{K}^{ε}, η) + ε^{2} B_{F} (ζ_{K}^{ε}, η) - ℓ (η - ζ_{K}^{ε}) - 2 B_{M} (ζ_{K}^{ε}, ζ) + B_{M} (ζ, ζ), \end{matrix}

for all $η = (η_{i}) \in U_{M} (ω)$ . Hence, letting $η = ζ$ and letting $ε \to 0$ gives

\underset{ε \to 0}{limsup} ∥ ζ_{K}^{ε} - ζ ∥_{V_{M} (ω)}^{2} \leq 0,

which shows that

ζ_{K}^{ε} \to ζ in V_{M} (ω),

as was to be proved. □

Note that realistic sufficient conditions insuring that the ‘density property’ holds are given by Ciarlet et al. Ciarlet et al. [21] (see also Ciarlet et al. [22]).

4. Koiter’s model for a generalized membrane of the ‘first kind’ subjectto a confinement condition

Consider a linearly elastic shell subject to the various assumptions set forth in Section 2. Following the terminology proposed in Section 5.1 of Ciarlet [1], such a shell is said to be a linearly elastic generalized membrane shell if the following two additional assumptions are simultaneously satisfied. First, $length γ_{0} > 0$ (an assumption that is satisfied if $γ_{0}$ is a non-empty relatively open subset of $γ$ , as assumed here). Second, the space of admissible linearized inextensional displacements defined by

V_{F} (ω) : = {η = (η_{i}) \in H^{1} (ω) \times H^{1} (ω) \times H^{2} (ω); η_{i} = \partial_{ν} η_{3} = 0 on γ_{0} and γ_{α β} (η) = 0 in ω},

does not contain any non-zero vectors, i.e.,

V_{F} (ω) = {0},

but the shell is not an elliptic membrane shell in the sense of Section 3 (note in this respect that, indeed, the space $V_{F} (ω)$ also reduces to ${0}$ if the shell is an elliptic membrane one; compare with Theorem 2). The second condition in the definition of a generalized membrane shell, namely

V_{F} (ω) = {0},

is equivalent to stating that the semi-norm ${| \cdot |}_{ω}^{M}$ defined by

| η |_{ω}^{M} : = {\sum_{α, β} | γ_{α β} (η) |_{0, ω}^{2}}^{1 ∕ 2},

for each $η = (η_{i}) \in H^{1} (ω) \times H^{1} (ω) \times L^{2} (ω)$ becomes a norm over the space

V_{K} (ω) = {η = (η_{i}) \in H^{1} (ω) \times H^{1} (ω) \times H^{2} (ω); η_{i} = \partial_{ν} η_{3} = 0 on γ_{0}} .

Generalized membranes are themselves subdivided into two categories described in terms of the spaces

\begin{matrix} V (ω) : = {η = (η_{i}) \in H^{1} (ω); η = 0 on γ_{0}}, \\ V_{0} (ω) : = {η = (η_{i}) \in V (ω); γ_{α β} (η) = 0 in ω} . \end{matrix}

A generalized membrane shell is ‘of the first kind’ if

V_{0} (ω) = {0},

or, equivalently, if the semi-norm ${| \cdot |}_{ω}^{M}$ is already a norm over the space $V (ω)$ (hence, a fortiori, over the space $V_{K} (ω) \subset V (ω)$ ).

Otherwise, i.e., if

V_{F} (ω) = {0} but V_{0} (ω) \neq {0},

or, equivalently, if the semi-norm ${| \cdot |}_{ω}^{M}$ is a norm over $V_{K} (ω)$ but not over $V (ω)$ , the linearly elastic shell is a generalized membrane shell ‘of the second kind’. In this paper, we shall only consider generalized membranes of the ‘first kind’ (which are most frequently encountered in practice).

The forthcoming analysis resorts to an argument similar to that of Caillerie and Sanchez-Palencia [18] used in Theorem 7.2–2 of Ciarlet [1] (itself based on Caillerie and Sanchez-Palencia [18] and, especially, on Ciarlet and Lods [12]) and constitutes the second convergence result of this paper.

In addition to a simple assumption regarding the asymptotic behaviour of the applied forces (in effect, the same as for elliptic membrane shells; compare with Theorem 3), we will need (as in the previous references) to assume that the applied body forces are admissible, in the sense that there exist functions $φ^{α β} = φ^{β α} \in L^{2} (ω)$ independent of $ε$ such that, for each $ε > 0$ , the linear form $ℓ^{ε}$ appearing in problem $P_{K}^{ε} (ω)$ can also be written as

ℓ^{ε} (η) = \int_{ω} p^{i, ε} η_{i} \sqrt{a} dy = ε \int_{ω} φ^{α β} γ_{α β} (η) \sqrt{a} dy for all η = (η_{i}) \in V_{K} (ω) and for each ε > 0 .

Note that this assumption is stronger than the assumption that there exist functions $f^{i} \in L^{2} (Ω)$ independent of $ε$ such that

f^{i, ε} (x^{ε}) = f^{i} (x) at each x^{ε} \in Ω^{ε} for each ε > 0

made for elliptic membrane shells, since it implies that the linear forms $ℓ^{ε}$ are now continuous with respect to ${| \cdot |}_{ω}^{M}$ .

Theorem 4. Let $ω$ be a domain in $ℝ^{2}$ and let $θ \in C^{3} (\bar{ω}; E^{3})$ be an immersion. Consider a family of generalized membrane shells ‘of the first kind’ with thickness $2 ε$ approaching zero and with each having the same middle surface $θ (\bar{ω})$ , and assume that each shell is subject to a boundary condition of place along a portion of its lateral face, whose middle curve is the set $θ (γ_{0})$ . Define the spaces

\begin{matrix} V (ω) : = {η = (η_{i}) \in H^{1} (ω); η = 0 on γ_{0}}, \\ V_{M}^{♯} (ω) : = completion of V (ω) with respect to {| \cdot |}_{ω}^{M} . \end{matrix}

For each $ε > 0$ , let $ζ_{K}^{ε}$ denote the solution to problem $P_{K}^{ε} (ω)$ (Theorem 1). Then the following convergence holds:

ζ_{K}^{ε} \to ζ in V_{M}^{♯} (ω) as ε \to 0,

where $ζ$ denotes the unique solution to problem $P_{M}^{♯} (ω)$ . Find

ζ \in U^{♯} (ω) : = closure of U_{K} (ω) with respect to {| \cdot |}_{ω}^{M}

that satisfies the following variational inequalities:

B_{M}^{♯} (ζ, η - ζ) \geq L_{M}^{♯} (η - ζ) for all η = (η_{i}) \in U^{♯} (ω),

where $B_{M}^{♯} (\cdot, \cdot)$ and $L_{M}^{♯}$ designate the unique continuous linear extensions from $V (ω)$ to $V_{M}^{♯} (ω)$ of the bilinear form $B_{M} (\cdot, \cdot)$ , and of the linear form $L_{M}$ defined by

L_{M} (η) : = \int_{ω} φ^{α β} γ_{α β} (η) \sqrt{a} dy for all η = (η_{i}) \in V (ω) .

Proof. (i) Problem $P_{M}^{♯} (ω)$ has one and only one solution. We first observe that the space $V_{M}^{♯} (ω)$ is also the completion of the space $V_{K} (ω)$ with respect to the norm ${| \cdot |}_{ω}^{M}$ . Clearly, problem $P_{M}^{♯} (ω)$ admits a unique solution, since the bilinear form $B_{M}^{♯} (\cdot, \cdot)$ is continuous and $V_{M}^{♯} (ω)$ -elliptic (recall that the tensor $(a^{αβστ})$ is uniformly positive definite), the set $U^{♯} (ω)$ is non-empty, closed with respect to ${| \cdot |}_{ω}^{M}$ , and convex, and the linear form $L_{M}^{♯}$ is continuous.

(ii) Uniform boundedness of the family ${(ζ_{K}^{ε})}_{ε > 0}$ . Because the applied body forces are admissible, the variational inequalities appearing in the problem $P_{K}^{ε} (ω)$ read

B_{M} (ζ_{K}^{ε}, η - ζ_{K}^{ε}) + ε^{2} B_{F} (ζ_{K}^{ε}, η - ζ_{K}^{ε}) \geq L_{M} (η - ζ_{K}^{ε})

for all $η = (η_{i}) \in U_{K} (ω)$ . By virtue of the continuity of the linear form $L_{M}$ with respect to the norm ${| \cdot |}_{ω}^{M}$ , there then exists a constant $C_{1} > 0$ such that

\begin{matrix} B_{M} (ζ_{K}^{ε}, ζ_{K}^{ε}) + ε^{2} B_{F} (ζ_{K}^{ε}, ζ_{K}^{ε}) \leq B_{M} (ζ_{K}^{ε}, η) + ε^{2} B_{F} (ζ_{K}^{ε}, η) - L_{M} (η - ζ_{K}^{ε}) \\ \leq B_{M} (ζ_{K}^{ε}, η) + ε^{2} B_{F} (ζ_{K}^{ε}, η) + C_{1} | η - ζ_{K}^{ε} |_{ω}^{M}, \end{matrix}

for all $η \in U_{K} (ω)$ . Thanks to the uniform positive definiteness of the tensor $(a^{αβστ})$ , there exists a constant $C_{2} > 0$ such that

{(| ζ_{K}^{ε} |_{ω}^{M})}^{2} \leq C_{2} B_{M} (ζ_{K}^{ε}, ζ_{K}^{ε}) and \sum_{α, β} | ε ρ_{α β} (ζ_{K}^{ε}) |_{0, ω}^{2} \leq C_{2} ε^{2} B_{F} (ζ_{K}^{ε}, ζ_{K}^{ε}) .

Hence, letting $η = 0$ gives

{(| ζ_{K}^{ε} |_{ω}^{M})}^{2} + \frac{1}{3} \sum_{α, β} | ε ρ_{α β} (ζ_{K}^{ε}) |_{0, ω}^{2} \leq C_{1} C_{2} | ζ_{K}^{ε} |_{ω}^{M} .

This inequality shows that the family ${(ζ_{K}^{ε})}_{ε > 0}$ is bounded in $V_{M}^{♯} (ω)$ and that each family ${(ε ρ_{α β} (ζ_{K}^{ε}))}_{ε > 0}$ is bounded in $L^{2} (ω)$ .

(iii) Weak convergence of the family ${(ζ_{K}^{ε})}_{ε > 0}$ . As a consequence of (ii), there exist a subsequence, still denoted ${(ζ_{K}^{ε})}_{ε > 0}$ , a vector field $ζ^{♯} \in V_{M}^{♯} (ω)$ and functions $ρ_{α β}^{- 1} \in L^{2} (ω)$ , such that

\begin{matrix} ζ_{K}^{ε} ⇀ ζ^{♯} in V_{M}^{♯} (ω) as ε \to 0, \\ ε ρ_{α β} (ζ_{K}^{ε}) ⇀ ρ_{α β}^{- 1} in L^{2} (ω) as ε \to 0 . \end{matrix}

The vector field $ζ^{♯}$ belongs to the set $U^{♯} (ω)$ , which is a non-empty, closed and convex subset of $V_{M}^{♯} (ω)$ .

Letting $ε \to 0$ in these variational inequalities gives

\underset{ε \to 0}{limsup} B_{M} (ζ_{K}^{ε}, ζ_{K}^{ε}) \leq B_{M}^{♯} (ζ^{♯}, η) - L_{M}^{♯} (η - ζ^{♯}) for all η \in U_{K} (ω),

and, therefore, by definition of $U^{♯} (ω)$ ,

\underset{ε \to 0}{limsup} B_{M} (ζ_{K}^{ε}, ζ_{K}^{ε}) \leq B_{M}^{♯} (ζ^{♯}, η) - L_{M}^{♯} (η - ζ^{♯}) for all η \in U^{♯} (ω),

on the one hand. On the other hand, we have

0 \leq B_{M}^{♯} (ζ_{K}^{ε} - ζ^{♯}, ζ_{K}^{ε} - ζ^{♯}) = B_{M} (ζ_{K}^{ε}, ζ_{K}^{ε}) - 2 B_{M}^{♯} (ζ_{K}^{ε}, ζ^{♯}) + B_{M}^{♯} (ζ^{♯}, ζ^{♯}),

which in turn implies

B_{M}^{♯} (ζ^{♯}, ζ^{♯}) \leq \underset{ε \to 0}{liminf} B_{M} (ζ_{K}^{ε}, ζ_{K}^{ε}) .

Therefore,

B_{M}^{♯} (ζ^{♯}, η - ζ^{♯}) \geq L_{M}^{♯} (η - ζ^{♯}) for all η = (η_{i}) \in U^{♯} (ω),

which shows that $ζ^{♯}$ is a solution to problem $P_{M}^{♯} (ω)$ . Since the solution to problem $P_{M}^{♯} (ω)$ is unique by (i), we conclude that

ζ^{♯} = ζ

and that the whole family ${(ζ_{K}^{ε})}_{ε > 0}$ weakly converges to $ζ$ in $V_{M}^{♯} (ω)$ as $ε \to 0$ .

(iv) Strong convergence of the family ${(ζ_{K}^{ε})}_{ε > 0}$ . The positive definiteness of the two-dimensional fourth-order elasticity tensor of the shell, together with the definition of the norm ${| \cdot |}_{ω}^{M}$ and of the bilinear form $B_{M} (\cdot, \cdot)$ and its extension $B_{M}^{♯} (\cdot, \cdot)$ , show that establishing the announced strong convergence is equivalent to establishing the convergence

B_{M}^{♯} (ζ_{K}^{ε} - ζ, ζ_{K}^{ε} - ζ) \to 0 as ε \to 0 .

Since $ζ = ζ^{♯} \in U^{♯} (ω)$ , we have, in particular,

\underset{ε \to 0}{limsup} B_{M} (ζ_{K}^{ε}, ζ_{K}^{ε}) \leq B_{M}^{♯} (ζ, ζ) .

Noting that

\begin{matrix} 0 \leq B_{M}^{♯} (ζ_{K}^{ε} - ζ, ζ_{K}^{ε} - ζ) = B_{M} (ζ_{K}^{ε}, ζ_{K}^{ε}) - 2 B_{M}^{♯} (ζ_{K}^{ε}, ζ) + B_{M}^{♯} (ζ, ζ) \end{matrix}

and noting that the weak convergence $ζ_{K}^{ε} ⇀ ζ$ in $V_{M}^{♯} (ω)$ as $ε \to 0$ , established in (iii), implies that

B_{M}^{♯} (ζ_{K}^{ε}, ζ) \to B_{M}^{♯} (ζ, ζ) as ε \to 0,

we infer from these relations that

\begin{matrix} 0 \leq \underset{ε \to 0}{limsup} B_{M}^{♯} (ζ_{K}^{ε} - ζ, ζ_{K}^{ε} - ζ) \\ = \underset{ε \to 0}{limsup} (B_{M} (ζ_{K}^{ε}, ζ_{K}^{ε}) - 2 B_{M}^{♯} (ζ_{K}^{ε}, ζ) + B_{M}^{♯} (ζ, ζ)) \\ = \underset{ε \to 0}{limsup} B_{M} (ζ_{K}^{ε}, ζ_{K}^{ε}) - B_{M}^{♯} (ζ, ζ) \leq 0 . \end{matrix}

Hence, the strong convergence

ζ_{K}^{ε} \to ζ in V_{M}^{♯} (ω)

holds, as announced. □

5. Koiter’s model for a flexural shell subjected to a confinement condition

Consider a linearly elastic shell, subject to the various assumptions set forth in Section 2. Following the terminology proposed in Section 6.1 of Ciarlet [1], such a shell is said to be a linearly elastic flexural shell if the following two additional assumptions are satisfied: first, $length γ_{0} > 0$ (an assumption that is satisfied if $γ_{0}$ is a non-empty relatively open subset of $γ$ , as here), and second, the following space of admissible linearized inextensional displacements:

V_{F} (ω) : = {η = (η_{i}) \in H^{1} (ω) \times H^{1} (ω) \times H^{2} (ω); η_{i} = \partial_{ν} η_{3} = 0 on γ_{0} and γ_{α β} (η) = 0 in ω}

contains non-zero functions, i.e.,

V_{F} (ω) \neq {0} .

To begin with, we state a crucial inequality that holds for ‘general’ surfaces (this inequality was already needed in the proof of Theorem 1).

Theorem 5. Let $ω$ be a domain in $ℝ^{2}$ and let $θ \in C^{3} (\bar{ω}; E^{3})$ be an immersion. Let $γ_{0}$ be a non-empty relatively open subset of $γ = ∂ω$ . Define the space

V_{K} (ω) : = {η = (η_{i}) \in H^{1} (ω) \times H^{1} (ω) \times H^{2} (ω); η_{i} = \partial_{ν} η_{3} = 0 on γ_{0}} .

Then there exists a constant $c = c (ω, γ_{0}, θ) > 0$ such that

{\sum_{α} ∥ η_{α} ∥_{1, ω}^{2} + ∥ η_{3} ∥_{2, ω}^{2}}^{1 ∕ 2} \leq c {\sum_{α, β} ∥ γ_{α β} (η) ∥_{0, ω}^{2} + \sum_{α, β} ∥ ρ_{α β} (η) ∥_{0, ω}^{2}}^{1 ∕ 2}

for all $η = (η_{i}) \in V_{K} (ω)$ .

This inequality, was first proved by Bernadou and Ciarlet [6] (see also Bernadou, Ciarlet and Miara [7] and Ciarlet [1], Theorems 2.7–3), constitutes an example of a Korn inequality on a general surface; it constitutes a ‘Korn inequality’ in the sense that it provides a basic 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.

The forthcoming analysis resorts to an argument similar to the one used in Theorem 7. 2–3 of Ciarlet [1] (itself based on Sanchez- Palencia [21] and, especially, on Ciarlet and Lods [17]) and constitutes the third convergence result of this paper.

Theorem 6. Let $ω$ be a domain in $ℝ^{2}$ and let $θ \in C^{3} (\bar{ω}; E^{3})$ be an immersion. Consider a family of flexural shells with thickness $2 ε$ approaching zero and with each having the same middle surface $θ (\bar{ω})$ . Let $γ_{0}$ be a non-empty relatively open subset of $γ$ , and assume that each shell is subject to a boundary condition of place along a portion of its lateral face, whose middle curve is the set $θ (γ_{0})$ . Finally, assume that there exist functions $f^{i} \in L^{2} (Ω)$ independent of $ε$ such that:

f^{i, ε} (x^{ε}) = ε^{2} f^{i} (x) at each x^{ε} \in Ω^{ε} for each ε > 0 .

For each $ε > 0$ , let $ζ_{K}^{ε}$ denote the solution to problem $P_{K}^{ε} (ω)$ (Theorem 1). Then the following convergence holds:

ζ_{K}^{ε} \to ζ in V_{K} (ω) as ε \to 0,

where $ζ$ denotes the solution to problem $P_{F} (ω)$ : Find

ζ \in U_{F} (ω) : = {η = (η_{i}) \in V_{F} (ω); (θ (y) + η_{i} (y) a^{i} (y)) \cdot p \geq 0 for a . a . y \in ω}

that satisfies the following variational inequality:

\frac{1}{3} \int_{ω} a^{αβστ} ρ_{στ} (ζ) ρ_{α β} (η - ζ) \sqrt{a} dy \geq \int_{ω} p^{i} (η_{i} - ζ_{i}) \sqrt{a} dy for all η = (η_{i}) \in U_{F} (ω),

where

p^{i} : = \int_{- 1}^{1} f^{i} d x_{3} \in L^{2} (ω) .

Proof. (i) Problem $P_{F} (ω)$ admits a unique solution. To see this, observe that the bilinear form $B_{F} (\cdot, \cdot)$ , which is defined by

B_{F} (ξ, η) = \frac{1}{3} \int_{ω} a^{αβστ} ρ_{στ} (ξ) ρ_{α β} (η) \sqrt{a} dy for each ξ, η \in V_{F} (ω),

is continuous and $V_{F} (ω)$ -elliptic, $U_{F} (ω)$ is a non-empty closed and convex subset of $V_{F} (ω)$ , and the linear form $ℓ$ defined by

ℓ (η) : = \int_{ω} p^{i} η_{i} \sqrt{a} dy for all η = (η_{i}) \in V_{F} (ω),

is continuous over $V_{F} (ω)$ . Hence the conclusion follows.

(ii) Uniform boundedness of the family ${(ζ_{K}^{ε})}_{ε > 0}$ . By virtue of the assumption on the applied body forces, the variational inequalities in problem $P_{K}^{ε} (ω)$ read

\frac{1}{ε^{2}} B_{M} (ζ_{K}^{ε}, η - ζ_{K}^{ε}) + B_{F} (ζ_{K}^{ε}, η - ζ_{K}^{ε}) \geq ℓ (η - ζ_{K}^{ε}),

for all $η = (η_{i}) \in U_{K} (ω)$ , and by the continuity of the linear form $ℓ$ , there exists a constant $C_{1} > 0$ such that

\begin{matrix} \frac{1}{ε^{2}} B_{M} (ζ_{K}^{ε}, ζ_{K}^{ε}) + B_{F} (ζ_{K}^{ε}, ζ_{K}^{ε}) \leq \frac{1}{ε^{2}} B_{M} (ζ_{K}^{ε}, η) + B_{F} (ζ_{K}^{ε}, η) - ℓ (η - ζ_{K}^{ε}) \\ \leq \frac{1}{ε^{2}} B_{M} (ζ_{K}^{ε}, η) + B_{F} (ζ_{K}^{ε}, η) + C_{1} ∥ η - ζ_{K}^{ε} ∥_{V_{K} (ω)}, \end{matrix}

for all $η = (η_{i}) \in U_{K} (ω)$ . Thanks to the uniform positive definiteness of the tensor $(a^{αβστ})$ and Theorem 5, there exists a constant $C_{2} > 0$ such that

∥ ζ_{K}^{ε} ∥_{V_{K} (ω)}^{2} \leq C_{2} (\frac{1}{ε^{2}} B_{M} (ζ_{K}^{ε}, ζ_{K}^{ε}) + B_{F} (ζ_{K}^{ε}, ζ_{K}^{ε})) for all 0 < ε \leq 1 .

Hence, combining these two inequalities and letting $η = 0$ gives

∥ ζ_{K}^{ε} ∥_{V_{K} (ω)} \leq C_{1} C_{2} .

(iii) Weak convergence of the family ${(ζ_{K}^{ε})}_{ε > 0}$ . By (ii), the family ${(ζ_{K}^{ε})}_{ε > 0}$ is bounded in $V_{K} (ω)$ . Therefore, there exists a subfamily, still denoted ${(ζ_{K}^{ε})}_{ε > 0}$ , a vector field $ζ^{*} \in V_{K} (ω)$ and functions $γ_{α β}^{- 1} \in L^{2} (ω)$ , such that

\begin{matrix} ζ_{K}^{ε} ⇀ ζ^{*} in V_{K} (ω), \\ \frac{1}{ε} γ_{α β} (ζ_{K}^{ε}) ⇀ γ_{α β}^{- 1} in L^{2} (ω), \end{matrix}

the second weak convergence being a consequence of the uniform positive definiteness of the tensor $(a^{αβστ})$ . The same weak convergence in turn implies that

γ_{α β} (ζ_{K}^{ε}) \to 0 in L^{2} (ω),

on the one hand. On the other hand, the weak convergence $ζ_{K}^{ε} ⇀ ζ^{*}$ in $V_{K} (ω)$ clearly implies that

γ_{α β} (ζ_{K}^{ε}) ⇀ γ_{α β} (ζ^{*}) .

Hence, the uniqueness of the weak limit shows that

γ_{α β} (ζ^{*}) = 0,

i.e., that $ζ^{*} \in V_{F} (ω)$ . Besides, $ζ^{*}$ belongs to $U_{K} (ω)$ , because $U_{K} (ω)$ is a non-empty, closed and convex subset of $V_{K} (ω)$ . In conclusion, $ζ^{*}$ belongs to $U_{F} (ω)$ .

For any given $η \in U_{F} (ω)$ , the following inequality holds:

\begin{matrix} B_{F} (ζ_{K}^{ε}, ζ_{K}^{ε}) \leq \frac{1}{ε^{2}} B_{M} (ζ_{K}^{ε}, η) + B_{F} (ζ_{K}^{ε}, η) - ℓ (η - ζ_{K}^{ε}) \\ = B_{F} (ζ_{K}^{ε}, η) - ℓ (η - ζ_{K}^{ε}), \end{matrix}

so that, letting $ε \to 0$ in this inequality, we obtain

\underset{ε \to 0}{limsup} B_{F} (ζ_{K}^{ε}, ζ_{K}^{ε}) \leq B_{F} (ζ^{*}, η) - ℓ (η - ζ^{*}) .

We also have

0 \leq B_{F} (ζ_{K}^{ε} - ζ^{*}, ζ_{K}^{ε} - ζ^{*}) = B_{F} (ζ_{K}^{ε}, ζ_{K}^{ε}) - 2 B_{F} (ζ_{K}^{ε}, ζ^{*}) + B_{F} (ζ^{*}, ζ^{*}),

which in turn implies that

2 B_{F} (ζ_{K}^{ε}, ζ^{*}) - B_{F} (ζ^{*}, ζ^{*}) \leq B_{F} (ζ_{K}^{ε}, ζ_{K}^{ε}) .

Hence, letting $ε \to 0$ , we obtain

B_{F} (ζ^{*}, ζ^{*}) \leq \underset{ε \to 0}{liminf} B_{F} (ζ_{K}^{ε}, ζ_{K}^{ε}) .

Therefore,

B_{F} (ζ^{*}, η - ζ^{*}) \geq ℓ (η - ζ^{*}) .

We have thus shown that

B_{F} (ζ^{*}, η - ζ^{*}) \geq ℓ (η - ζ^{*}) for all η = (η_{i}) \in U_{F} (ω),

i.e., that $ζ^{*}$ is a solution to problem $P_{F} (ω)$ . Since problem $P_{F} (ω)$ admits a unique solution, we conclude that $ζ = ζ^{*}$ . Hence, the whole family ${(ζ_{K}^{ε})}_{ε > 0}$ weakly converges to $ζ$ in $V_{K} (ω)$ as $ε \to 0$ .

(iv) Strong convergence of the family ${(ζ_{K}^{ε})}_{ε > 0}$ . Combining the Korn’s inequality on a general surface (Theorem 5), the strong convergence $γ_{α β} (ζ_{K}^{ε}) \to γ_{α β} (ζ) = 0$ in $L^{2} (ω)$ and the uniform positive definiteness of the tensor $(a^{αβστ})$ , establishing the strong convergence $ζ_{K}^{ε} \to ζ$ in $V_{K} (ω)$ amounts to establishing that $B_{F} (ζ_{K}^{ε} - ζ, ζ_{K}^{ε} - ζ) \to 0$ as $ε \to 0$ .

Noting that we have

\begin{matrix} 0 \leq B_{F} (ζ_{K}^{ε} - ζ, ζ_{K}^{ε} - ζ) \\ = B_{F} (ζ_{K}^{ε}, ζ_{K}^{ε}) - 2 B_{F} (ζ_{K}^{ε}, ζ) + B_{F} (ζ, ζ) \\ \leq B_{F} (ζ_{K}^{ε}, ζ) - ℓ (ζ - ζ_{K}^{ε}) - 2 B_{F} (ζ_{K}^{ε}, ζ) + B_{F} (ζ, ζ), \end{matrix}

since $ζ \in U_{F} (ω)$ , we obtain

\underset{ε \to 0}{limsup} ∥ ζ_{K}^{ε} - ζ ∥_{V_{K} (ω)}^{2} \leq 0 .

Hence,

ζ_{K}^{ε} \to ζ in V_{K} (ω),

which completes the proof. □

6. Justification of the proposed model

Given an immersion $θ \in C^{3} (\bar{ω}; E^{3})$ and $ε > 0$ small enough, let the sets $Ω^{ε}$ , $Γ_{0}^{ε}$ , the $C^{2}$ –diffeomorphism $Θ \in C^{2} (\bar{Ω^{ε}}; E^{3})$ and the vector fields $g_{i}^{ε} \in C^{2} (\bar{Ω^{ε}}; E^{3})$ and $g^{j, ε} \in C^{2} (\bar{Ω^{ε}}; E^{3})$ be defined as in Section 2.

One then defines the metric tensor by means of its covariant components

g_{ij}^{ε} : = g_{i}^{ε} \cdot g_{j}^{ε} \in C^{1} (\bar{Ω^{ε}}),

or by means of its contravariant components

g^{ij, ε} : = g^{i, ε} \cdot g^{j, ε} \in C^{1} (\bar{Ω^{ε}}) .

Note that the symmetric matrix field $(g^{ij, ε})$ is then the inverse of the matrix field $(g_{ij}^{ε})$ , that $g^{j, ε} = g^{ij, ε} g_{i}^{ε}$ and $g_{i}^{ε} = g_{ij}^{ε} g^{j, ε}$ and that the volume element in $Θ (\bar{Ω^{ε}})$ is given at each point $Θ (x^{ε})$ , $x^{ε} \in \bar{Ω^{ε}}$ , by $\sqrt{g^{ε} (x^{ε})} d x^{ε}$ , where

g^{ε} : = det (g_{ij}^{ε}) \in C^{1} (\bar{Ω^{ε}}) .

One also defines the Christoffel symbols associated with the immersion $Θ : \bar{Ω^{ε}} \to E^{3}$ by

Γ_{ij}^{p, ε} : = \partial_{i} g_{j}^{ε} \cdot g^{p, ε} = Γ_{ji}^{p, ε} \in C^{0} (\bar{Ω^{ε}}) .

Given a vector field $v^{ε} = (v_{i}^{ε}) \in C^{1} (\bar{Ω^{ε}}; ℝ^{3})$ , the associated vector field

{\tilde{v}}^{ε} : = v_{i}^{ε} g^{i, ε}

may be viewed as a displacement field of the reference configuration $Θ (\bar{Ω^{ε}})$ of the shell, thus defined by means of its covariant components $v_{i}^{ε}$ over the vectors $g^{i, ε}$ of the contravariant bases in the reference configuration.

If the norms ${∥ v_{i}^{ε} ∥}_{C^{1} (\bar{Ω^{ε}})}$ are small enough, the mapping $(Θ + v_{i}^{ε} g^{i, ε})$ is also an immersion, so that one can also define the metric tensor of the deformed configuration $(Θ + v_{i}^{ε} g^{i, ε}) (\bar{Ω^{ε}})$ by means of its covariant components

g_{ij}^{ε} (v^{ε}) : = (g_{i}^{ε} + \partial_{i}^{ε} {\tilde{v}}^{ε}) \cdot (g_{j}^{ε} + \partial_{j}^{ε} {\tilde{v}}^{ε}) .

The linear part with respect to ${\tilde{v}}^{ε}$ in the difference $(g_{ij}^{ε} (v^{ε}) - g_{ij}^{ε})$ is then called the linearized strain tensor associated with the displacement field $v_{i}^{ε} g^{i, ε}$ , the covariant components of which are then given by

\begin{matrix} e_{i ∥ j}^{ε} (v^{ε}) : = g_{i}^{ε} \cdot \partial_{j} {\tilde{v}}^{ε} + \partial_{i}^{ε} {\tilde{v}}^{ε} \cdot g_{j}^{ε} \\ = \frac{1}{2} (\partial_{j}^{ε} v_{i}^{ε} + \partial_{i}^{ε} v_{j}^{ε}) - Γ_{ij}^{p, ε} v_{p}^{ε} \\ = e_{j ∥ i}^{ε} (v^{ε}) . \end{matrix}

The functions $e_{i ∥ j}^{ε} (v^{ε})$ are called the linearized strains in curvilinear coordinates associated with the displacement field $v_{i}^{ε} g^{i, ε}$ .

As in Sections 2 to 5, we assume that, for each $ε > 0$ , the reference configuration $Θ (\bar{Ω^{ε}})$ of the shell is in its natural state (i.e., it is ‘stress-free’) and that the material constituting the shell is homogeneous, isotropic and linearly elastic, the behaviour of which is thus governed by its two Lamé constants $λ \geq 0$ and $μ > 0$ .

We then consider a specific obstacle problem for such a shell, in the sense that the shell is subject to a confinement condition, expressing that any admissible displacement vector field $v_{i}^{ε} g^{i, ε}$ must be such that the corresponding deformed configuration remains in the same half-space as in Section 2, i.e., of the form

ℍ : = {x \in E^{3}; ox \cdot p \geq 0},

where $p \in E^{3}$ is a non-zero vector given once and for all. In other words, every admissible displacement field must satisfy the ‘constraint’

(Θ (x^{ε}) + v_{i}^{ε} (x^{ε}) g^{i, ε} (x^{ε})) \cdot p \geq 0 for all x^{ε} \in \bar{Ω^{ε}},

or possibly only almost everywhere in $Ω^{ε}$ . Note that, unlike the obstacle-free case, we do not consider applied surface forces.

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

Θ (\bar{Ω^{ε}}) \subset ℍ .

It is worth pointing out that this confinement condition considerably departs from the usual Signorini condition preferred by most authors, who usually require that only the points of the undeformed and deformed ‘lower face’ $ω \times {- ε}$ of the reference configuration satisfy the confinement condition (see, e.g., Léger and Miara [23 –25] or Rodrguez-Ars [26]). The confinement condition considered in this investigation is more physically realistic, since a Signorini condition imposed only on the lower face of the reference configuration does not prevent – at least theoretically – other points of the deformed reference configuration from ‘crossing’ the plane ${x \in E^{3}; ox \cdot p = 0}$ and then ending up on the ‘other side’ of this plane.

The mathematical modelling 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 total energy $J^{ε}$ , is classical. More specifically, let

\begin{matrix} A^{ijkℓ, ε} : = λ g^{ij, ε} g^{kℓ, ε} + μ (g^{ik, ε} g^{jℓ, ε} + g^{iℓ, ε} g^{jk, ε}) \\ = A^{jikℓ, ε} \\ = A^{kℓij, ε} \end{matrix}

denote the contravariant components of the three-dimensional elasticity tensor of the shell, viewed here as a three-dimensional elastic body. Then the unknown of the problem, which is the vector field $u^{ε} = (u_{i}^{ε})$ , where the functions $u_{i}^{ε} : \bar{Ω^{ε}} \to ℝ$ are the three covariant components of the unknown ‘three-dimensional’ displacement vector field $u_{i}^{ε} g^{i, ε}$ of the reference configuration of the shell, should minimize the energy $J^{ε} : H^{1} (Ω^{ε}) \to ℝ$ defined by

J^{ε} (v^{ε}) : = \frac{1}{2} \int_{Ω^{ε}} A^{ijkℓ, ε} e_{k ∥ ℓ}^{ε} (v^{ε}) e_{i ∥ j}^{ε} (v^{ε}) \sqrt{g^{ε}} d x^{ε} - \int_{Ω^{ε}} f^{i, ε} v_{i}^{ε} \sqrt{g^{ε}} d x^{ε}

for each $v^{ε} = (v_{i}^{ε}) \in H^{1} (Ω^{ε})$ over the set of admissible displacements defined as follows:

U (Ω^{ε}) : = {v^{ε} = (v_{i}^{ε}) \in H^{1} (Ω^{ε}); v^{ε} = 0 on Γ_{0}^{ε}, (Θ (x^{ε}) + v_{i}^{ε} (x^{ε}) g^{i, ε} (x^{ε}))) \cdot p \geq 0 for a . e . x^{ε} \in Ω^{ε}} .

The solution to this minimization problem exists and is unique; it can also be characterized as the unique solution of a set of appropriate variational inequalities, as shown in the next theorem.

Theorem 7 The quadratic minimization problem: Find a vector field $u^{ε} \in U (Ω^{ε})$ such that

J^{ε} (u^{ε}) = inf_{v^{ε} \in U (Ω^{ε})} J^{ε} (v^{ε})

has one and only one solution, which is also the unique solution of the variational problem $P (Ω^{ε})$ : Find

u^{ε} \in U (Ω^{ε})

that satisfies the following variational inequalities:

\int_{Ω^{ε}} A^{ijkℓ, ε} e_{k ∥ ℓ}^{ε} (u^{ε}) (e_{i ∥ j}^{ε} (v^{ε}) - e_{i ∥ j}^{ε} (u^{ε})) \sqrt{g^{ε}} d x^{ε} \geq \int_{Ω^{ε}} f^{i, ε} (v_{i}^{ε} - u_{i}^{ε}) \sqrt{g^{ε}} d x^{ε}

for all $v^{ε} = (v_{i}^{ε}) \in U (Ω^{ε})$ .

Proof. Define the space

V (Ω^{ε}) : = {v^{ε} = (v_{i}^{ε}) \in H^{1} (Ω^{ε}); v^{ε} = 0 on Γ_{0}^{ε}} .

Then, thanks to the uniform positive definiteness of the elasticity tensor $(A^{ijkℓ, ε})$ and to the boundary condition of the place satisfied on $Γ_{0}^{ε} = γ_{0} \times [- ε, ε]$ (recall that $γ_{0}$ is a non-empty, relatively open, subset of $∂ω$ ), it can be shown (see Ciarlet [2], Theorems 3.8–3 and 3.9–1) that the continuous and symmetric bilinear form

(v^{ε}, w^{ε}) \in H^{1} (Ω^{ε}) \times H^{1} (Ω^{ε}) \to \int_{Ω^{ε}} A^{ijkℓ, ε} e_{k ∥ ℓ}^{ε} (v^{ε}) e_{i ∥ j}^{ε} (w^{ε}) \sqrt{g^{ε}} d x^{ε}

is $V (Ω^{ε})$ -elliptic; moreover, the linear form

v^{ε} = (v_{i}^{ε}) \in H^{1} (Ω^{ε}) \to \int_{Ω^{ε}} f^{i, ε} v_{i}^{ε} \sqrt{g^{ε}} d x^{ε}

is clearly continuous. Finally, the set $U (Ω^{ε})$ is non-empty (by assumption), closed in $H^{1} (Ω^{ε})$ (any convergent sequence in $U (Ω^{ε})$ contains a subsequence that pointwise converges almost everywhere to its limit) and convex (as is immediately verified).

The existence and uniqueness of the solution to the minimization problem and its characterization by means of variational inequalities is then a consequence of the projection theorem. □

We now recall the convergence result recently established by Ciarlet et al. [21] (see also Ciarlet et al. [22]) regarding the asymptotic behaviour as $ε \to 0$ of the solution to problem $P (Ω^{ε})$ when the shell is an elliptic membrane one.

Theorem 8. Let $ω$ be a domain in $ℝ^{2}$ and let $θ \in C^{3} (\bar{ω}; E^{3})$ be an immersion. Consider a family of elliptic membrane shells with thickness $2 ε$ approaching zero and with each having the same middle surface $θ (\bar{ω})$ and assume that there exist functions $f^{i} \in L^{2} (Ω)$ independent of $ε$ such that the following assumption on the applied body force density holds:

f^{i, ε} (x^{ε}) = f^{i} (x) at each x^{ε} \in Ω^{ε} for each ε > 0 .

Finally, assume that the following ‘density property’ holds:

U_{K} (ω) is dense in U_{M} (ω) with respect to the norm ∥ \cdot ∥_{H^{1} (ω) \times H^{1} (ω) \times L^{2} (ω)} .

Let $ζ \in U_{M} (ω)$ denote the solution to problem $P_{M} (ω)$ (Theorem 3) and, for each $ε > 0$ , let $u^{ε} = (u_{i}^{ε})$ denote the solution to problem $P (Ω^{ε})$ (Theorem 1). Then the following convergences hold:

\begin{matrix} \frac{1}{2 ε} \int_{- ε}^{ε} u_{α}^{ε} g^{α, ε} d x_{3}^{ε} \to ζ_{α} a^{α} in H^{1} (ω) as ε \to 0, \\ \frac{1}{2 ε} \int_{- ε}^{ε} u_{3}^{ε} g^{3, ε} d x_{3}^{ε} \to ζ_{3} a^{3} in L^{2} (ω) as ε \to 0 . \end{matrix}

A comparison with the convergences

\begin{matrix} ζ_{α, K}^{ε} a^{α} \to ζ_{α} a^{α} in H^{1} (ω) as ε \to 0, \\ ζ_{3, K}^{ε} a^{3} \to ζ_{3} a^{3} in L^{2} (ω) as ε \to 0, \end{matrix}

established in Theorem 3 thus shows that the solution to the three-dimensional obstacle problem $P (Ω^{ε})$ and to the two-dimensional obstacle problem $P_{K}^{ε} (ω)$ exhibit the same limit behaviour as $ε \to 0$ . This observation then fully justifies the formulation of our proposed Koiter’s model for an elliptic membrane shell subjected to a confinement condition.

Similar justifications of our proposed Koiter’s model for a generalized membrane shell or for a flexural shell subjected to a confinement condition will be provided in forthcoming works.

Footnotes

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Research Grants Council of the Hong Kong Special Administrative Region (grant numbers 9042536, CityU 11300317).

References

Ciarlet

PG.

Mathematical elasticity, vol. III: theory of shells. Amsterdam: North-Holland, 2000.

Ciarlet

PG.

An introduction to differential geometry with applications to elasticity. Dordrecht: Springer, 2005.

Ciarlet

PG.

Mathematical elasticity, vol. I: three-dimensional elasticity. Amsterdam: North-Holland, 1988.

Koiter

. On the foundations of the linear theory of thin elastic shells. I. P K Ned Akad Wet, Ser B 1970; 73: 169–182.

Koiter

. On the foundations of the linear theory of thin elastic shells. II. P K Ned Akad Wet, Ser B 1970; 73: 183–195.

Bernadou

Ciarlet

. Sur l’ellipticité du modèle linéaire de coques de WT Koiter. In: Glowinski

Lions

(eds) Computing methods in applied sciences and engineering (Lecture Notes in Economics and Mathematical Systems, vol. 134). Berlin: Springer, 1976, 89–136.

Bernadou

Ciarlet

Miara

Existence theorems for two-dimensional linear shell theories. J Elast 1994; 34: 111–138.

Duvaut

Lions

JL.

Inequalities in mechanics and physics. Berlin: Springer, 1976.

Glowinski

Numerical methods for nonlinear variational problems. New York: Springer-Verlag, 1984.

10.

Ciarlet

PG.

Linear and nonlinear functional analysis with applications. Philadelphia: Society for Industrial and Applied Mathematics, 2013.

11.

Ciarlet

Lods

. Asymptotic analysis of linearly elastic shells. I. Justification of membrane shell equations. Arch Rational Mech Anal 1996; 136(2): 119–161.

12.

Ciarlet

Lods

Asymptotic analysis of linearly elastic shells: ‘generalized membrane shells’. J Elast 1996; 43(2): 147–188.

13.

Ciarlet

Lods

Miara

. Asymptotic analysis of linearly elastic shells. II. Justification of flexural shell equations. Arch Rational Mech Anal 1996; 136(2): 163–190.

14.

Ciarlet

Lods

On the ellipticity of linear membrane shell equations. J Math Pures Appl 1996; 75: 107–124.

15.

Ciarlet

Sanchez-Palencia

An existence and uniqueness theorem for the two-dimensional linear membrane shell equations. J Math Pures Appl 1996; 75: 51–67.

16.

Destuynder

A classification of thin shell theories. Acta Appl Math 1985; 4: 15–63.

17.

Sanchez-Palencia

Statique et dynamique des coques minces. I. Cas de flexion pure non inhibée. C R Acad Sci Paris Sér I Math 1989; 309(6): 411–417.

18.

Caillerie

Sánchez-Palencia

A new kind of singular stiff problems and application to thin elastic shells. Math Models Methods Appl Sci 1995; 5: 47–66.

19.

Ciarlet

Lods

. Asymptotic analysis of linearly elastic shells. III. Justification of Koiter’s shell equations. Arch Rational Mech Anal 1996; 136(2): 191–200.

20.

Brezis

Functional analysis, Sobolev spaces and partial differential equations. New York: Springer, 2011.

21.

Ciarlet

Mardare

Piersanti

An obstacle problem for elliptic membrane shells. Math Mech Solids. Epub ahead of print 30 November 2018 DOI: 10.1177/1081286518800164.

22.

Ciarlet

Mardare

Piersanti

Un problème de confinement pour une coque membranaire linéairement élastique de type elliptique. CR Acad Sci Paris, Sér I 2018; 356(10): 1040–1051.

23.

Léger

Miara

Mathematical justification of the obstacle problem in the case of a shallow shell. J Elast 2008; 90: 241–257.

24.

Léger

Miara

Erratum to ‘Mathematical justification of the obstacle problem in the case of a shallow shell’. J Elast 2010; 98: 115–116.

25.

Léger

Miara

A linearly elastic shell over an obstacle: the flexural case. J Elastic 2018; 131: 19–38.

26.

Rodrguez-Ars

Mathematical justification of the obstacle problem for elastic elliptic membrane shells. Appl Anal 2018; 97: 1261–1280.