# On questions of existence in shell theory

Philippe G. Ciarlet

► To cite this version:

Philippe G. Ciarlet. On questions of existence in shell theory. [Research Report] IRIA-RR-170, IRIA. 1976, pp.14. hal-04716355

HAL Id: hal-04716355

https://inria.hal.science/hal-04716355v1

Submitted on 1 Oct 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
IRIA

laboria

Institut de Recherche
d'Informatique
et d'Automatique

Domaine de Voluceau
Rocquencourt
B.P. 5 78150 - Le Chesnay
France
Tél.: 954 90 20

laboratoire de recherche
en informatique
et automatique

<figure><img src="image_2.png" /></figure>

ON QUESTIONS OF EXISTENCE
IN SHELL THEORY

<figure><img src="image_3.png" /></figure>

P. G. CIARLET

Rapport de Recherche N° 170

Juillet 1976
ON QUESTIONS OF EXISTENCE IN SHELL THEORY

P.G. Ciarlet*

Résumé :

On considère le modèle de Koiter des problèmes de coques, pour lequel un problème important consiste naturellement à montrer l'existence d'une solution. Ce problème est résolu dans un article à paraître de M. Bernadou et de l'auteur. Comme la démonstration dans le cas général est très technique et longue, on se contente ici de donner une démonstration simplifiée dans le cas d'un problème d'arc, qui est un bon problème « modèle » pour les problèmes de coques.

Abstract :

We consider Koiter's model of the shell problem, for which one basic problem is of course to prove the existence of a solution. This is done in a forthcoming paper of M. Bernadou and the author. Since the proof in the general case is highly technical and lengthy, it is the purpose of this paper to give a simplified proof in the case of the arch problem, which is a good « model problem » for the shell problem.

<footnote>* Analyse Numérique - Tour 55-65, Université Pierr et Marie Curie, 4 place Jussieu - 75230 PARIS CÉDEX 05
A paraître dans : Journal of the Indian Mathematical Society.</footnote>
1. **INTRODUCTION** - In this paper, we consider a modelization of the shell problem, known as Koiter's model. The description of this model is found at various stages of its development in Koiter (1966,1970), Koiter & Simmonds (1972). This model is based on certain physical hypotheses (essentially about the stress distribution across the thickness of the shell), which John (1965) has theoretically justified.

A different model has been proposed by Naghdi (1963, 1972). A simplified theory of the so-called "shallow" shells is presented in Timoshenko & Woinowsky-Krieger (1959) and Washizu (1968).

One basic problem is of course to prove the existence of a solution, which in turn amounts to showing that the *corresponding strain energy is elliptic*. This is done in Bernadou & Ciarlet (1976).

Since however the proof in the general case is highly technical and lengthy, we content ourselves here with the simpler question of the ellipticity of the strain energy of an arch. This simplification is justified, inasmuch as the arch problem is a "model problem" for the shell problem. In fact the proof found in Bernadou & Ciarlet (1976) follows basically a method set up in Ciarlet (1976 a) for circular arches and extended in the present paper for the case of an arch of varying curvature.

As regards the questions of ellipticity for various shell models, we mention the works of Rougée (1969) for "cylindrical" shells, Coutris (1973) for Naghdi's model, Gordeziani (1974) for the model of Vekua (1965), Shoiket (1974) for the model of Novozhilov (1970). For cylindrical shells, see Kołakowski & Dryja (1970), Miyoshi (1973). Nevertheless, it seems that this problem was not as yet solved in a completely satisfactory way (cf. Remark 1, at the end of this paper).
We finally mention that the ellipticity of the "continuous" model plays a crucial role in the convergence analysis of finite element approximations of shells, as shown in Ciarlet (1976a, 1976 b) (for introductions to the finite element method, see Babuška & Aziz (1972), Ciarlet (1975, 1976a), Oden & Reddy (1976), Strang & Fix (1973)).

2. **THE SHELL PROBLEM.** - Let $\Omega$ be a bounded subset in a plane $\varepsilon^2$, with boundary $\Gamma$. Then a **shell** $S$ is the image of the set $\bar{\Omega}$ by a mapping $\phi: \bar{\Omega} \subset \varepsilon^2 \to \varepsilon^3$, where $\varepsilon^3$ is the usual Euclidean space. In fact, the surface $S$ is the **middle surface** of the shell, but since we are only considering "thin" shells, **we shall constantly identify the shell with its middle surface**. The data $\Gamma$ and $\phi$ are assumed to be sufficiently smooth for all subsequent purposes.

We will denote by $(a,b) \to a \cdot b$, $|\cdot|$, and $e^i$, $1 \le i \le 3$, the Euclidean scalar product, Euclidean norm, and an orthonormal basis of the space $\varepsilon^3$, respectively.

We shall assume that all points of the shell $S = \phi(\bar{\Omega})$ are regular, in the sense that the two vectors

$$ (1) \qquad a_{\alpha} = \partial_{\alpha} \phi, \quad \alpha = 1,2, $$

are linearly independent, for all points $\xi = (\xi^1, \xi^2) \in \bar{\Omega}$.

As a rule, we shall use Greek letters : $\alpha, \beta, \tau, \dots$, for indices which take their values in the set $\{1,2\}$, while Latin letters : $i, j, k, \dots$, will be used for indices which take their values in the set $\{1,2,3\}$. For these indices, we shall use Einstein's convention for summation. Finally, the usual symbols, such as $\partial_\alpha, \partial_{\alpha\beta}$, etc..., shall be also used for partial derivatives of **vector-valued** functions of the form $\theta=\theta_i e^i : \bar{\Omega} \subset \varepsilon^2 \to \varepsilon^3$. Thus, for instance, one has $\partial_\alpha \theta = \partial_\alpha \theta_i e^i$, $\partial_{\alpha\beta} \theta = \partial_{\alpha\beta} \theta_i e^i$, etc...

The vectors $a_\alpha$ are tangent to the **curvilinear coordinate lines** $\phi(\xi^\beta = \text{constant})$, $\beta \ne \alpha$, and they define the tangent plane at the point $\phi(\xi)$.
We introduce the vector

$$
(2) \qquad a_3 = a^3 = \frac{a_1 \times a_2}{\|a_1 \times a_2\|}.
$$

The first fundamental form (a<sub>αβ</sub>) of the surface is defined by

$$
(3) \qquad a_{\alpha\beta} = a_{\beta\alpha} = a_{\alpha} \cdot a_{\beta} = \partial_{\alpha} \phi \cdot \partial_{\beta} \phi.
$$

With the **covariant basis** (a_α) is associated the **contravariant basis**
(a^α) of the tangent plane, which is defined through the relations

$$
(4) \qquad a^{\alpha} \cdot a_{\beta} = \delta^{\alpha}_{\beta} \quad (\text{no summation if } \alpha = \beta),
$$

where $\delta^\alpha_\beta$ is the Kronecker's symbol. Then we have

$$
(5) \quad a_{\alpha} = a_{\alpha\beta} a^{\beta}, \quad a^{\alpha} = a^{\alpha\beta} a_{\beta}, \quad a^{\alpha\beta} = a^{\alpha} \cdot a^{\beta} = a^{\beta\alpha},
$$

where the matrix (a^α) is the inverse of the matrix (a_α), which is always
invertible since all points are regular, by assumption.

We recall that the <u>area measure</u> dS along the surface S is given by

$$
(6) \qquad dS = \sqrt{a} \, d\xi,
$$

where

$$
(7) \qquad a = \det(a_{\alpha\beta}) = a_{11}a_{22} - (a_{12})^2.
$$

We now come to the shell model, which is another example of a familiar
problem : the solution u, which will be defined below, minimizes the shell
energy

$$
(8) \qquad J(v) = \frac{1}{2} a(v, v) - f(v),
$$

when the functions v vary in an appropriate space V. We shall therefore
successively define the bilinear form a(·,·), the linear form f, and the
space V.
The unknowns are the three functions

$$u_i : \xi \in \bar{\Omega} \rightarrow u_i(\xi) \in \mathbb{R},$$

which represent the *covariant components* of the displacement $u=u(\xi)$
of the point $\phi(\xi)$. In other words we have

$$ (9) \qquad \vec{u} = u_i \vec{a}^i . $$

of course, it should be remembered that the vectors $\vec{a}^i$ are also functions
of $\xi$. The strain energy $\frac{1}{2} a(\mathbf{v},\mathbf{v})$ of the shell is a surface integral:

$$ (10) \qquad \frac{1}{2} a(\mathbf{v},\mathbf{v}) = \int_S A(\dots) \, dS = \int_{\Omega} A(\dots) \sqrt{a} \, d\xi. $$

The function $A(\dots)$ is given by (cf. Koiter (1970), equ. (3.16)) :

$$ (11) \qquad A(\dots) = \frac{Ee}{2(1-\sigma^2)} \left\{ (1-\sigma) \begin{pmatrix} \gamma_{\beta}^{\alpha} & \gamma_{\alpha}^{\beta} + \sigma & \gamma_{\alpha}^{\alpha} & \gamma_{\beta}^{\beta} \\ \gamma_{\beta}^{\alpha} & \gamma_{\alpha}^{\beta} + \sigma & \gamma_{\alpha}^{\alpha} & \gamma_{\beta}^{\beta} \end{pmatrix} \right. \\ \qquad \left. + \frac{Ee^3}{24(1-\sigma^2)} \left\{ (1-\sigma) \begin{pmatrix} \bar{\rho}_{\beta}^{\alpha} & \bar{\rho}_{\alpha}^{\beta} \\ \bar{\rho}_{\beta}^{\alpha} & \bar{\rho}_{\alpha}^{\beta} \end{pmatrix} + \sigma \begin{pmatrix} \bar{\rho}_{\alpha}^{\alpha} & \bar{\rho}_{\beta}^{\beta} \\ \bar{\rho}_{\alpha}^{\beta} & \bar{\rho}_{\beta}^{\alpha} \end{pmatrix} \right\}, $$

where $e$ is the thickness of the shell, $E$ is its Young's modulus, $\sigma$ is
its Poisson's coefficient, and the mixed tensors $(\gamma_{\beta}^{\alpha})$ and $(\bar{\rho}_{\beta}^{\alpha})$ are
obtained from the doubly covariant strain tensor $(\gamma_{\alpha\beta})$ and change of
curvature tensor $(\bar{\rho}_{\alpha\beta})$ through the tensorial operations

$$ (12) \qquad \gamma_{\beta}^{\alpha} = a^{\alpha\nu} \gamma_{\nu\beta}, \quad \bar{\rho}_{\beta}^{\alpha} = a^{\alpha\nu} \bar{\rho}_{\nu\beta}. $$

The tensors $(\gamma_{\alpha\beta})$ and $(\bar{\rho}_{\alpha\beta})$ are given by

$$ (13) \qquad \gamma_{\alpha\beta} = \gamma_{\beta\alpha} = \frac{1}{2} (v_{\alpha}|\beta + v_{\beta}|\alpha) - b_{\alpha\beta} v_{3}, $$

$$ (14) \qquad \bar{\rho}_{\alpha\beta} = \bar{\rho}_{\beta\alpha} = v_3|_{\alpha\beta} - c_{\alpha\beta} v_3 + b_{\alpha}^{\lambda} v_{\lambda}|_{\beta} + b_{\beta}^{\lambda} v_{\lambda}|_{\alpha} + b_{\alpha}^{\delta} v_{\delta}|_{\beta} $$

the various symbols occuring in these expressions being now defined.
The second fundamental form $(b_{\alpha\beta})$ of the surface is given by

$$ (15) \qquad b_{\alpha\beta} = b_{\beta\alpha} = -a_{\alpha} \cdot \partial_{\beta} a_{3} = a_{3} \cdot \partial_{\beta} a_{(4)} = \frac{1}{2} a_{3} \cdot (\partial_{\beta} a_{\alpha} + \partial_{\alpha} a_{\beta}), $$

from which the third fundamental form $(c_{\alpha\beta})$ of the surface is derived
by letting
$$ (16) \qquad c_{\alpha\beta} = c_{\beta\alpha} = b_{\alpha}^{\lambda} b_{\lambda\beta}, \text{ where } b_{\alpha}^{\lambda} = a^{\lambda\beta} b_{\alpha\beta}. $$

Then the Christoffel symbols ($\Gamma^\alpha_{\beta\gamma}$) of the surface are defined by the formulas

$$ (17) \qquad \Gamma^\alpha_{\beta\gamma} = a^\alpha \cdot \partial_\beta a_\gamma = a^{\alpha\nu} a_\nu \cdot \partial_\beta a_\gamma = a^{\alpha\nu} \partial_\nu \phi \cdot \partial_\gamma \phi. $$

These functions are symmetric with respect to the lower indices in the sense that

$$ (18) \qquad \Gamma^\alpha_{\beta\gamma} = \Gamma^\alpha_{\gamma\beta}, $$

and they satisfy

$$ (19) \qquad \Gamma^\alpha_{\beta\gamma} = a^{\alpha\lambda} \Gamma_{\lambda\beta\gamma}, \text{ with } \Gamma_{\alpha\beta\gamma} = \frac{1}{2} (\partial_\gamma a^\alpha + \partial_\alpha a^\gamma - \partial_\alpha a^\gamma). $$

Then the covariant derivatives $v_\alpha|\beta$, $v_3|\alpha\beta$, $b_\alpha^\delta|\beta$ are given by

$$ (20) \qquad v_{\alpha|\beta} = \partial_{\beta} v_{\alpha} - \Gamma^{\gamma}_{\alpha\beta} v_{\gamma}, $$

$$ (21) \qquad v_3|\alpha\beta = \partial_{\alpha\beta} v_3 - \Gamma^{\gamma}_{\alpha\beta} \partial_{\gamma} v_3, $$

$$ (22) \qquad b_{\alpha|\beta}^{\delta} = \partial_{\beta} b_{\alpha}^{\delta} + \Gamma_{\beta\lambda}^{\delta} b_{\alpha}^{\lambda} - \Gamma_{\alpha\beta}^{\lambda} b_{\lambda}^{\delta}. $$

Using the Mainardi-Codazzi and Ricci identities, one can derive the equalities

$$ (23) \qquad b_{\alpha|\beta}^{\delta} = b_{\beta|\alpha}^{\delta}, $$

which in turn imply the symmetry of the tensor $\bar{\rho}_{\alpha\beta}$ of (14).

For references in Tensor Calculus and Differential Geometry, the reader may consult Goujon (1963), Lelong-Ferrand (1963), Lichnérowicz (1967), Valiron (1950, Chapters 12, 13, 14).
From all the previous formulas, it follows that in the integrand (11) appearing in the strain energy (10) of the shell, one finds the three functions $v_i$ and some of their partial derivatives, which we shall sometimes record as the following twelve functions $V_I$, $1 \le I \le 12$:

$$ (24) \qquad (v_I)^{12}_{I=1} = \begin{cases} v_1, & \partial_1 v_1, \partial_2 v_1, \\ v_2, & \partial_1 v_2, \partial_2 v_2, \\ v_3, & \partial_1 v_3, \partial_2 v_3, \partial_1 v_3, \partial_2 v_3, \partial_2 v_3, \end{cases} $$

while the notation $v$ is reserved for the triple $(v_1, v_2, v_3)$.

Associating as in (24) twelve functions $U_I$, $1 \le I \le 12$, with another generic function $u=(u_1, u_2, u_3)$, we are able to state the main properties of the bilinear form (10). The proof, which is essentially a matter of lengthy checkings, is left to the reader.

**THEOREM 1.** The bilinear form which occurs in the definition of the strain energy of the shell is of the followins form:

$$ (25) \qquad a(u,v) = \int_{\Omega} \sum_{I,J=1}^{12} A_{IJ}(\xi) U_I V_J d\xi. $$

Denoting by $\phi_i$ the components of the mapping $\phi = \phi_i e^i$, we have, for each $(I,J)$:

$$ (26) \qquad A_{IJ}(\xi) = f_{IJ}(\partial_{\alpha} \phi_i(\xi), \partial_{\alpha\beta}\phi_i(\xi), \partial_{\alpha\beta\gamma}\phi_i(\xi)), $$

where the function $f_{IJ}$ is a quotient of a polynomial in its arguments by a denominator which is an integer power of the expression

$$ (27) \qquad \sqrt{a} = \sqrt{\det(a_{\alpha\beta})} = \left(\sum_{i=1}^{3} (\partial_1 \phi_i)^2\right) \left(\sum_{j=1}^{3} (\partial_2 \phi_j)^2\right) - \left(\sum_{i=1}^{3} (\partial_1 \phi_i \partial_2 \phi_i)^2\right)^{1/2} $$

The bilinear form is symmetric in its arguments $u$ and $v$ and, finally, it is defined and continuous over the space $H^1(\Omega) \times H^1(\Omega) \times H^2(\Omega)$. ■
The potential energy of the exterior forces is another surface integral, of the form

$$ (28) \qquad f(v) = \int_S f \cdot v dS = \int_{\Omega} f_i^i v_i \sqrt{ad\xi}, $$

where the functions $f^i$ represent the contravariant components, i.e., over the basis $(a_i)$, of the reduced density per unit surface of the exterior forces. Clearly, such a linear form is also continuous over the space $H^1(\Omega) \times H^1(\Omega) \times H^2(\Omega)$.

Let then $V$ be a space such as

$$ (29) \qquad V = H_0^1(\Omega) \times H_0^1(\Omega) \times H_0^2(\Omega), \quad \text{resp.} \quad V = H_0^1(\Omega) \times H_0^1(\Omega) \times (H_0^2(\Omega) \cap H_0^1(\Omega)), $$

which corresponds to the case of a _clamped shell_, resp. _a simply supported shell_. Then the problem of showing the existence of a displacement $u \in V$ such that $J(u) = \inf_{v \in V} J(v)$, or equivalently such that $a(u,v) = f(v)$ for all $v \in V$, with $J$, $a(\cdot, \cdot)$ and $f(\cdot)$ as in (8), (25) and (28) respectively, reduces to the problem of showing the $V$-ellipticity of the bilinear form.

This is done in Bernadou & Ciarlet (1976), under the sole assumption that the mapping $\phi$ is of class $C^3(\bar{\Omega})$. Rather than giving the lengthy and technical proof here, we shall instead focus our attention on a simpler problem, which nevertheless displays all the essential features of the general shell problem : _the arch problem_, where a single variable (instead of two) is needed.

### 3. THE ARCH PROBLEM

The arch $\mathcal{A}$ is the image of an interval $I=[0,L]$, $L > 0$, by a mapping

$$ \phi : s \in I \to \phi(s) = \phi_i(s) e^i \in \mathcal{E}^2, $$

where $\mathcal{E}^2$ denotes an Euclidean plane equipped with an orthonormal basis $e^1, i = 1,2$. We shall assume that the parameter $s$ is the curvilinear abscissa along the arch, so that the vector $a_i = \phi'$ is a unit vector. We let $a_2$ denote the unit normal vector oriented in the usual way.
Then the energy of the arch has the following form (cf. for example
Moan (1974)) :

$$
(30) \quad J(v) = \frac{1}{2} \int_I \{ EA \left(v_1' - \frac{v_2}{R}\right)^2 + EI \left((v_2' + \frac{v_1}{R})'\right)^2 \} ds - \int_I f \cdot v ds.
$$

In this expression, the functions v₁ and v₂ : 1 → ℝ are the
tangential and normal components of the admissible displacements
v = v₁a¹ + v₂a², the function R: ℝ → ℝ is the (algebraically counted)
radius of curvature, so that the function 1/R : ℝ → ℝ is the curvature
of the arch. Finally, the constant E is the Young's modulus of the
constitutive material of the arch, the constant A is the area of a
cross-section of the arch and the constant I is the moment of inertia
of a cross-section of the arch. Since these three constants are strictly
positive, there is no loss of generality in assuming that EA = EI = 1,
as will be henceforth assumed.

In the next theorem, the proof of the ellipticity of the bilinear
form is given in such a form that it can be extended to a shell, as in
Bernadou & Ciarlet (1976).

In what follows, we shall use the following standard notations for
norms and semi-norms in Sobolev spaces :

$$
|v|_{m, I} = \left( \int_I |v^{(m)}(s)|^2 ds \right)^{1/2},
$$

$$
\|v\|_{m, I} = \left( \sum_{i=0}^{m} |v|_{1, I}^2 \right)^{1/2},
$$

$$
|v|_{0,\infty, I} = \operatorname*{ess\,sup}_{s \in I} |v(s)|.
$$

THEOREM 2. If the function 1/R is continuously differentiable over the interval I, the bilinear form defined by

$$
(31) \quad a(u,v) = \int_I \left\{ \left( (u')^2 - \frac{u_2}{R} \right) \left( (v')^2 - \frac{v_2}{R} \right) + \left( (u_2')^2 + \frac{u_1}{R} \right)' \left( (v_2')^2 + \frac{v_1}{R} \right)' \right\} ds
$$

is $H_o^1(I) \times (H^2(I) \cap H_o^1(I))$-elliptic, and thus it is a fortiori
$H_o^1(I) \times H_o^2(I)$-elliptic.
Proof. We shall equip the space

$$
(32) \qquad V = H_o^1(I) \times (H^2(I) \cap H_o^1(I))
$$

with the norm

$$
(33) \qquad |v| = (|v_1|^2_{1,I} + |v_2|^2_{2,I})^{1/2}
$$

(it is easily verified that over the space H<sup>2</sup>(I) ∩ H<sub>O</sub><sup>1</sup>(I), the semi-norm
|·|<sub>2,I</sub> is a norm, equivalent to the norm ||·||<sub>2,I</sub>).

The proof consists in three steps.

(i) There exists a constant $\lambda > 0$ and a constant $\mu$ such that

$$
(34) \quad \forall v \in V, a(v,v) \ge \lambda |v|^2 + \mu (|v_1|^2_{0,I} + ||v_2||^2_{1,I}).
$$

Let $\beta = \left| \frac{1}{R} \right|_{0,\infty,I}$. For all $\varepsilon > 0$, we have

$$
\int_I \left(v_1' - \frac{v_2}{R}\right)^2 ds \ge |v_1'|_{1,I}^2 - 2\beta |v_1|_{1,I} |v_2|_{0,I} \\
\ge (1-\varepsilon\beta) |v_1|_{1,I}^2 - \frac{\beta}{\varepsilon} |v_2|_{0,I}^2
$$

and thus if we choose $\epsilon \in ]0, \frac{1}{\beta}[$, we have found a constant $\lambda_1 > 0$ and
a constant $\mu_1$ such that

$$
\forall v_1 \in H_1^1(I), \forall v_2 \in L_2^2(I),
$$

$$
(35) \quad \int_I \left( v_1' - \frac{v_2}{R} \right)^2 ds \ge \lambda_1 |v_1|^2_{1,I} + \mu_1 |v_2|^2_{0,I}.
$$

Likewise there exists a constant $\lambda_2 > 0$ and a constant $\mu_2'$ such that

$$
\int_I \left( \left(v_2' + \frac{v_1}{R}\right)' \right)^2 ds \ge \lambda_2 |v_2|^2_{2,I} + \mu_2' \left|\left(\frac{v_1}{R}\right)'\right|^2_{0,I}.
$$

Since

$$
|\left(\frac{v_1}{R}\right)'|_{0,I} \le \beta |v_1|_{1,I} + \beta^2 |R'|_{0,\infty,I} |v_1|_{0,I'}
$$
we have found a constant $\lambda_2 > 0$ and two constants $\mu_2$ and $\nu_2$ such that

$$ \forall v_1 \in H^1(I), \forall v_2 \in H^2(I), $$

$$ (36) \qquad \int_I \left( \left(v_2' + \frac{v_1}{R}\right)' \right)^2 ds \geq \lambda_2 |v_2|^2_{2,1} + \mu_2 |v_1|^2_{1,1} + \nu_2 |v_1|^2_{0,1}. $$

If the constant $\mu_2$ is positive, then inequality (34) is a direct consequence of inequalities (35) and (36). If $\mu_2 < 0$, let $\varepsilon$ be so chosen that $0 < \varepsilon < \min\left\{\frac{|\lambda_1|}{|\mu_2|}, 1\right\}$. Then

$$ \begin{aligned} a(v,v) &\ge \int_I \left(v_1' - \frac{v_2}{R}\right)^2 ds + \varepsilon \int_I \left(\left(v_2' + \frac{v_1}{R}\right)'\right)^2 ds \\ &\ge (\lambda_1 - \varepsilon|\mu_2|) |v_1|^2_{1,1} + \varepsilon|v_2| |v_1|^2_{0,1} + \mu_1 |v_2|^2_{0,1} + \varepsilon\lambda_2 |v_2|^2_{2,1}, \end{aligned} $$

and inequality (34) is proved in all cases.

### (ii) The mapping

$$ v \to \sqrt{a(v,v)} $$

is a norm over the space V. Clearly it is a semi-norm, so it remains to prove that $a(v,v)=0$ implies $v=0$, i.e., we are facing the problem of solving, in the sense of distributions, the coupled system of differential equations :

$$ (37) \qquad \left\{ \begin{array}{l} v_1' - \frac{v_2}{R} = 0 \quad \text{on } I, \\ (v_2' + \frac{v_1}{R})' = 0 \quad \text{on } I, \end{array} \right. $$

along with the boundary conditions :

$$ (38) \qquad v_1(0) = v_2(0) = v_1(L) = v_2(L) = 0. $$

As suggested by the geometry of the problem, let us introduce the angle $\theta$ between the vectors $e_1$ and $a^1$, so that the following relations hold :
$$ (39) \qquad \theta' = \frac{1}{R}, $$

$$ (40) \qquad \phi_1' = \cos \theta, \quad \phi_2' = \sin \theta. $$

We also introduce the Cartesian components $V_1$ and $V_2$ of the displacement. From the relations

$\forall s \in I, v(s) = v_1(s)a^1(s) + v_2(s)a^2(s) = V_1(s)e^1 + V_2(s)e^2,$
we deduce that

$$ \begin{cases} V_1 = v_1 \cos \theta - v_2 \sin \theta, \\ V_2 = v_1 \sin \theta + v_2 \cos \theta. \end{cases} $$

Since the functions $v_1$ and $v_2$ are both in the space $H^1(I)$ and since the function $\theta$ is in the space $C^1(\bar{I})$, both functions $V_1$ and $V_2$ are in the space $H^1(I)$, and

$$ (41) \qquad \begin{cases} V_1' = (v_1'-v_2\theta')\cos \theta - (v_2'+v_1\theta')\sin \theta, \\ V_2' = (v_2'+v_1\theta')\cos \theta + (v_1'-v_2\theta')\sin \theta. \end{cases} $$

The second differential equation of (37) implies the existence of a constant $a$ such that

$$ (42) \qquad v_2' + \frac{v_1}{R} = a, $$

so that we obtain, upon combining relations (37), (39), (40), (41) and (42) :

$$ \begin{cases} V_1' + a\phi_2' = 0, \\ V_2' + a\phi_1' = 0. \end{cases} $$

Therefore, there exists constants $b_1$ and $b_2$ such that

$$ \begin{cases} V_1 = -a\phi_2 + b_1, \\ V_2 = a\phi_1 + b_2. \end{cases} $$

Let then $e^3 = e^1 \times e^2$. We have proved that the general solution of the differential system (37) is of the form
$$
(43) \qquad v = a \times \phi + b
$$

where the constant vectors *a* and *b* are given by

$$
(44) \qquad a = ae^3, \quad b = b_1 e^1 + b_2 e^2
$$

(see Remark 1 for the interpretation of such a solution).

Finally, it is an easy matter to show that any solution of the
form (43) necessarily vanishes when it is subjected to the boundary conditions
(38), since $\phi(0) \neq \phi(L)$ by assumption.

(iii) Using steps (i) and (ii), we are in a position to show the
V-ellipticity of the bilinear form. If it were not V-elliptic, there would
exist a sequence $v^k = (v_1^k, v_2^k) \in V$ such that

$$
(45) \quad \lim_{k \to \infty} a(v^k, v^k) = 0, \quad |v^k| = 1 \text{ for all } k.
$$

Since the sequence $(v_1^k)$ is bounded in $H_o^1(I)$, there exists a
subsequence, which we shall still denote by $(v_1^k)$ for convenience, which
weakly converges in $H_o^1(\Omega)$ and strongly converges in $L^2(I)$ to the same
limit $v_1$.

Likewise, the boundedness of the sequence $(v_2^k)$ in $H^2(I) \cap H_o^1(\Omega)$
implies that there exists a subsequence, still denoted by $(v_2^k)$, which
weakly converges in $H^2(I) \cap H_o^1(I)$ and strongly converges in $H^1(I)$ to the
same limit $v_2$.

The function $v \in V \to a(v,v)$ is continuous for the strong topology of the space $V$ and it is a convex function (it is even a strictly convex function since its second derivative is positive definite, as was shown in step (ii)). Therefore, it is weakly lower semi-continuous. As a consequence, we have, setting $v = (v_1,v_2)$,

$$
a(v,v) = \liminf_{k \to \infty} a(v^k, v^k) = \lim_{k \to \infty} a(v^k, v^k) = 0,
$$
and thus $v=0$ by step (ii). By step (i), we have for all $k$,

$$a(v^k, v^k) \geq \lambda |v^k|^2 + \mu (|v_1^k|^2_{0,I} + ||v_2^k||^2_{1,I})$$

(cf. (34)). We have therefore reached a contradiction, since

$$\lim_{k \to \infty} a(v^k, v^k) = 0 \quad (\text{cf. (45)}),$$

$$\lim_{k \to \infty} \{\lambda |v^k|^2 + \mu (|v_1^k|^2_{0,I} + |v_2^k|^2_{1,I})\} = \lambda > 0.$$

**REMARK 1.** - By step (ii) in the preceding proof, any displacement $v \in H^1(\Omega) \times H^2(\Omega)$ which satisfies $a(v,v) = 0$ is of the form (43), i.e., it corresponds to a **rigid body motion** in the plane of the arch. This condition is an instance of the **rigid displacement condition** that a mathematical model for an elastic system should be such that **the vanishing of the strain energy corresponds to rigid body motions.**

In addition, this interpretation provides an approach for integrating in a simple way the differential system (37) by suggesting the introduction of the function $\theta$, $V_1$ and $V_2$. Otherwise, one has to use more sophisticated tools from distribution theory, such as the fact that all solutions in the sense of distributions of a system of differential equations with smooth coefficients coincide with the classical solutions whereas, with this approach, it suffices to know that a distribution whose derivative is zero is a constant.

The same observation is true for shells : in this case, one has to solve, again in the sense of distributions, a homogeneous system of six **partial differential** equations in the three unknown functions $v_1, v_2, v_3$, for which no unicity result seems to be applicable (even though it is an overdetermined system) : in particular it is not possible to extract system of three equations which would be elliptic in the sense of Hörmander (1964, Definition 10.6-1). Fortunately, just as in the present case, the geometrical and physical origin of the problem suggests the proper changes of functions, which, in spite of their complicated forms, eventually lead
to partial differential equations which can be very simply integrated. This
is precisely the point that seemed to have been overlooked in the other
publications on the same subject.

REFERENCES

BABUSKA, I. ; AZIZ, A.K. (1972) : Survey Lectures on the Mathematical Foundations of the Finite Element Method, in *The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations*, pp. 3-359, Academic Press, New York.

BERNADOU, M. ; CIARLET, P.G. (1976) : Sur l'ellipticité du modèle linéaire de coques de W.T. Koiter, to appear in the Proceedings of the Second International Symposium on Computing Methods in Applied Sciences and Engineering, I.R.I.A., Versailles, 15-19 December, 1975.

CIARLET, P.G. (1975) : *Lectures on the Finite Element Method*, Tata Institute of Fundamental Research, Bombay.

CIARLET, P.G. (1976a) : *Numerical Analysis of the Finite Element Method*, Séminaires de Mathématiques Supérieures, Presses de l'Université de Montréal.

CIARLET, P.G. (1976b) : Conforming finite element methods for the shell problem, to appear in the Proceedings of the Conference on the Mathematics of Finite Elements and Applications, Brunel University, 07-10 April, 1975.

COUTRIS, N. (1973) : Flexions élastique et élastoplastique d'une coque mince, J. Mécanique 12, 463-475.

GORDEZIANI, D.G. (1974) : On the solvability of some boundary value problems for a variant of the theory of thin shells, Dokl. Akad. Nauk SSSR 215, N° 6, 677-680.

GOUYON, R. (1963) : *Calcul Tensoriel*, Vuibert, Paris.

HÖRMANDER, L. (1964) : Linear Partial Differential Operators, Springer-Verlag, Berlin.

JOHN, F. (1965) : Estimates for the derivatives of the stresses in a thin shell and interior shell equations, Comm. Pure and Appl. Math. 18, 235-267.

KOITER, W.T. (1966) : On the nonlinear theory of thin elastic shells, Proc. Kon. Ned. Akad. Wet. B69, 1-54.
KOITER, W.T. (1970) : On the foundations of the linear theory of thin elastic shells. I, II, Proc. Kon. Ned. Akad. Wet. B73, 169-195.

KOITER, W.T. ; SIMMONDS, J.C. (1972) : Foundations of shell theory, Proceedings of the 13th International Congress of Theoretical and Applied Mechanics, Moscow, pp. 150-176.

KOŁAKOWSKI, K. ; DRYJA, M. (1974) : A boundary value problem for equations of elastic cylindrical shell, Bulletin de l'Académie Polonaise des Sciences, Série des Sciences Techniques XXII, N° 11, 37-42.

LELONG-FERRAND, J. (1963) : *Géométrie Différentielle*, Masson, Paris.

LICHNEROWICZ, A. (1967) : *Eléments de Calcul Tensoriel*, Armand Colin, Paris (8th edition).

MIYOSHI, T. (1973) : Finite element method of mixed type and its convergence in linear shell problems, Kumamoto J. Sci. (Math) 10, 35-58.

MOAN, T. (1974) : A note on the convergence of finite element approximations for problems formulated in curvilinear coordinate systems, Computer Methods in Applied Mechanics and Engineering 3, 209-235.

NAGHDI, P.M. (1963) : Foundations of elastic shell theory, in *Progress in Solid Mechanics*, Vol. 4, pp. 1-90, North-Holland, Amsterdam.

NAGHDI, P.M. (1972) : *The Theory of Shells and Plates*, Handbuch der Physik, Springer-Verlag, Berlin.

NOVOZHILOV, V.V. (1970) : *Thin Shell Theory*, Wolters-Noordhoff.

ODEN, J.T. ; REDDY, J.N. (1976) : *An Introduction to the Mathematical Theory of Finite Elements*, Wiley Interscience, New York.

ROUGEE, P. (1969) : *Equilibre des Coques Elastiques Minces Inhomogènes en Théorie non Linéaire*, Thesis, University of Paris.

SHOIKET, B.A. (1974) : On existence theorems in linear shell theory, PMM 38, 567-571 [English translation : Journal of Applied Mathematics and Mechanics 38 (1974), 527-531].

STRANG, G. ; FIX, G.J. (1973) : *An Analysis of the Finite Element Method*, Prentice-Hall, Englewood Cliffs.

TIMOSHENKO, S. ; WOINOWSKY-KRIEGER, S. (1959) : *Theory of Plates and Shells*, McGraw-Hill.

VALIRON, G. (1950) : *Equations Fonctionnelles. Applications*, Masson, Paris (2nd edition).

VEKUA, I.N. (1965) : Theory of thin shallow shells of variable thickness, Akad. Nauk. Gruzin. SSR Trudy Tbilissi Mat. Inst. Razmadze 30, 3-103.

WASHIZU, K. (1968) : *Variational Methods in Elasticity and Plasticity*, Pergamon.