Paolo Piersanti
email: ppiersan@iu.edu

AN OBSTACLE PROBLEM FOR LINEARLY ELASTIC FLEXURAL SHELLS

In this paper we identify a set of two-dimensional variational inequalities that model an obstacle problem for a linearly elastic flexural shell subjected to a confinement condition expressing that all the points of the admissible deformed configurations remain in a given half-space.

Introduction

In this paper we recover, via a rigorous asymptotic analysis as the thickness approaches zero over a ad hoc three-dimensional model, a set of two-dimensional variational inequalities governing the displacement of a linearly elastic flexural shell subject to remain confined in a half-space. The problem under consideration is an obstacle problem.

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 [START_REF] Ciarlet | A justification of the two-dimensional linear plate model[END_REF][START_REF] Ciarlet | On the ellipticity of linear membrane shell equations[END_REF][START_REF] Ciarlet | Asymptotic analysis of linearly elastic shells. I. justification of membrane shell equations[END_REF][START_REF] Ciarlet | Asymptotic analysis of linearly elastic shells. III. justification of koiter's shell equations[END_REF][START_REF] Ciarlet | Asymptotic analysis of linearly elastic shells: "generalized membrane shells[END_REF][START_REF] Ciarlet | Asymptotic analysis of linearly elastic shells. II. Justification of flexural shell equations[END_REF] for the purpose of justifying Koiter's model. For the justification of Koiter's model in the time-dependent case where the action of temperature is considered, we refer the reader to [START_REF] Piersanti | On the justification of the frictionless time-dependent Koiter's model for thermoelastic shells[END_REF]. In these papers no confinement conditions are imposed.

In the recent papers [START_REF] Ciarlet | Obstacle problems for Koiter's shells[END_REF][START_REF] Ciarlet | A confinement problem for a linearly elastic Koiter's shell[END_REF][START_REF] Ciarlet | Un problème de confinement pour une coque membranaire linéairement élastique de type elliptique[END_REF][START_REF] Ciarlet | An obstacle problem for elliptic membrane shells[END_REF], 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, where only the "lower face" of the shell is required to remain above the "horizontal" plane. 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 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 in the case where the linearly elastic shell under consideration is a flexural shell appears not to have been addressed in the literature yet. The purpose of this paper is exactly to remedy this situation.

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 specialise 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 penalised 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 desired set of two-dimensional variational inequalities is recovered.

Geometrical preliminaries

For details about the classical notions of differential geometry used in this section and the next one, see, e.g. [START_REF] Ciarlet | Mathematical Elasticity[END_REF] or [START_REF] Ciarlet | An Introduction to Differential Geometry with Applications to Elasticity[END_REF].

Greek indices, except ε, 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 whose origin is denoted by O; the Euclidean inner product and the vector product of u, v ∈ E 3 are denoted u • v and u × v; the Euclidean norm of u ∈ E 3 is denoted |u|. The notation δ j i designates the Kronecker symbol.

Given an open subset Ω of R n , notations such as L 2 (Ω), H 1 (Ω), or H 2 (Ω), 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 • 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 : Ω → R are respectively denoted by: f + (x) := max{f (x), 0} and f -(x) := -min{f (x), 0} x ∈ Ω. 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 ∂ α := ∂/∂y α and ∂ αβ := ∂ 2 /∂y α ∂y β . A mapping θ ∈ C 1 (ω; E 3 ) is an immersion if the two vectors a α (y) := ∂ α θ(y) are linearly independent at each point y ∈ ω. Then the image θ(ω) of the set 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 ∈ ω, the vectors a α (y) span the tangent plane to the surface θ(ω) at the point θ(y), the unit vector

a 3 (y) := a 1 (y) ∧ a 2 (y) |a 1 (y) ∧ a 2 (y)|
is normal to θ(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) • a i (y) = δ j i
form the contravariant basis at θ(y); note that the vectors a β (y) also span the tangent plane to θ(ω) at θ(y) and that a 3 (y) = a 3 (y).

The first fundamental form of the surface θ(ω) is then defined by means of its covariant components

a αβ := a α • a β = a βα ∈ C 0 (ω),
or by means of its contravariant components

a αβ := a α • a β = a βα ∈ C 0 (ω).
Note that the symmetric matrix field (a αβ ) is then the inverse of the matrix field (a αβ ), that a β = a αβ a α and a α = a αβ a β , and that the area element along θ(ω) is given at each point θ(y), y ∈ ω, by a(y) dy, where

a := det(a αβ ) ∈ C 0 (ω).
Given an immersion θ ∈ C 2 (ω; E 3 ), the second fundamental form of the surface θ(ω) is defined by means of its covariant components

b αβ := ∂ α a β • a 3 = -a β • ∂ α a 3 = b βα ∈ C 0 (ω),
or by means of its mixed components b β α := a βσ b ασ ∈ C 0 (ω), and the Christoffel symbols associated with the immersion θ are defined by

Γ σ αβ := ∂ α a β • a σ = Γ σ βα ∈ C 0 (ω).
The Gaussian curvature at each point θ(y), y ∈ ω, of the surface θ(ω) is defined by

κ(y) := det(b αβ (y)) det(a αβ (y)) = det b β α (y)
(the denominator in the above 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 inverse of the product of the two principal radii of curvature at this point. Given an immersion θ ∈ C 2 (ω; E 3 ) and a vector field η = (η i ) ∈ C 1 (ω; R 3 ), the vector field η := η i a i can be viewed as a displacement field of the surface θ(ω), 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 (ω) are small enough, the mapping (θ + η i a i ) ∈ C 1 (ω; E 3 ) is also an immersion, so that the set (θ + η i a i )(ω) is also a surface in E 3 , equipped with the same curvilinear coordinates as those of the surface θ(ω), called the deformed surface corresponding to the displacement field η = η i a i . One can then define the first fundamental form of the deformed surface by means of its covariant components

a αβ (η) :=(a α + ∂ α η) • (a β + ∂ β η) =a αβ + a α • ∂ β η + ∂ α η • a β + ∂ α η • ∂ β η.
The linear part with respect to η in the difference 1 2 (a αβ (η) -a αβ ) is called the linearized change of metric tensor associated with the displacement field η i a i , the covariant components of which are thus defined by

γ αβ (η) := 1 2 (a α • ∂ β η + ∂ α η • a β ) = 1 2 (∂ β η α + ∂ α η β ) -Γ σ αβ η σ -b αβ η 3 = γ βα (η).
The linear part with respect to η in the difference 1 2 (b αβ (η) -b αβ ) is called the linearized change of curvature tensor associated with the displacement field η i a i , the covariant components of which are thus defined by

ρ αβ (η) := (∂ αβ η -Γ σ αβ ∂ σ η) • a 3 = ρ βα (η) = ∂ αβ η 3 -Γ σ αβ ∂ σ η 3 -b σ α b σβ η 3 + b σ α (∂ β η σ -Γ τ βσ η τ ) + b τ β (∂ α η τ -Γ σ ατ η σ ) + (∂ α b τ β + Γ τ ασ b σ β -Γ σ αβ b τ σ )η τ .
It turns out that, when a generic surface is subjected to a displacement field η i a i whose tangential covariant components η α vanish on a nonzero length portion of boundary of the domain ω, denoted γ 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 ∂ ν denotes the outer unit normal derivative operator along γ.

Theorem 2.1. Let ω be a domain in R 2 and let an immersion θ ∈ C 3 (ω; E 3 ) be given. Define the space

V K (ω) := {η = (η i ) ∈ H 1 (ω) × H 1 (ω) × H 2 (ω); η i = ∂ ν η 3 = 0 on γ 0 }.
Then there exists a constant c 0 such that

α η α 2 H 1 (ω) + η 3 2 H 2 (ω) 1/2 ≤ c 0 α,β γ αβ (η) 2 L 2 (ω) + α,β ρ αβ (η) 2 L 2 (ω) 1/2
for all η = (η i ) ∈ V K (ω).

The above inequality, which is due to [START_REF] Bernadou | Sur l'ellipticité du modèle linéaire de coques de W. T. Koiter[END_REF] and was later on improved by [START_REF] Bernadou | Existence theorems for twodimensional linear shell theories[END_REF] (see also [START_REF] Ciarlet | Mathematical Elasticity[END_REF]), 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.

The three-dimensional obstacle problem for a "general" linearly elastic shell

Let ω be a domain in R 2 , let γ := ∂ω, and let γ 0 be a non-empty relatively open subset of γ. For each ε > 0, we define the sets

Ω ε = ω × ]-ε, ε[ and Γ ε 0 := γ 0 × [-ε, ε] , we let x ε = (x ε i
) designate a generic point in the set Ω ε , and we let ∂ ε i := ∂/∂x ε i . Hence we also have x ε α = y α and ∂ ε α = ∂ α . Given an immersion θ ∈ C 3 (ω; E 3 ) and ε > 0, consider a shell with middle surface θ(ω) and with constant thickness 2ε. This means that the reference configuration of the shell is the set Θ(Ω ε ), where the mapping Θ : Ω ε → E 3 is defined by

Θ(x ε ) := θ(y) + x ε 3 a 3 (y) at each point x ε = (y, x ε 3 ) ∈ Ω ε .
One can then show (cf., e.g., Theorem 3.1-1 of [START_REF] Ciarlet | Mathematical Elasticity[END_REF]) that, if ε > 0 is small enough, such a mapping Θ ∈ C 2 (Ω ε ; E 3 ) is an immersion, in the sense that the three vectors

g ε i (x ε ) := ∂ ε i Θ(x ε
), are linearly independent at each point x ε ∈ 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 ) • g ε i (x ε ) = δ j i , constitute the contravariant basis at the same point. It will be implicitly assumed in the sequel that ε > 0 is small enough so that Θ : Ω ε → E 3 is an immersion.

One then defines the metric tensor associated with the immersion Θ by means of its covariant components

g ε ij := g ε i • g ε j ∈ C 1 (Ω ε ), or by means of its contravariant components g ij,ε := g i,ε • g i,ε ∈ C 1 (Ω ε ).
Note that the symmetric matrix field (g ij,ε ) is then the inverse of the matrix field (g ε ij ), that g j,ε = g ij,ε g ε i and g ε i = g ε ij g j,ε , and that the volume element in Θ(Ω ε ) is given at each point Θ(x ε ), x ε ∈ Ω ε , by g ε (x ε ) dx ε , where

g ε := det(g ε ij ) ∈ C 1 (Ω ε
). One also defines the Christoffel symbols associated with the immersion Θ by

Γ p,ε ij := ∂ i g ε j • g p,ε = Γ p,ε ji ∈ C 0 (Ω ε ). Note that Γ 3,ε α3 = Γ p,ε 33 = 0. Given a vector field v ε = (v ε i ) ∈ C 1 (Ω ε ; R 3
), the associated vector field ṽε := v ε i g i,ε , can be viewed as a displacement field of the reference configuration Θ(Ω ε ) of the shell, thus defined by means of its covariant components v ε i over the vectors g i,ε of the contravariant bases in the reference configuration.

If the norms v ε i C 1 (Ω ε ) are small enough, the mapping (Θ+v ε i g i,ε ) is also an immersion, so that one can also define the metric tensor of the deformed configuration

(Θ + v ε i g i,ε )(Ω ε ) by means of its covariant components g ε ij (v ε ) :=(g ε i + ∂ ε i ṽε ) • (g ε j + ∂ ε j ṽε ) =g ε ij + g ε i • ∂ j ṽε + ∂ ε i ṽε • g ε j + ∂ i ṽε • ∂ j ṽε .
The linear part with respect to ṽε in the difference

1 2 (g ε ij (v ε ) -g ε ij )
is then called the linearized strain tensor associated with the displacement field v ε i g i,ε , the covariant components of which are thus defined by

e ε i j (v ε ) := 1 2 g ε i • ∂ ε j ṽε + ∂ ε i ṽε • g ε j = 1 2 (∂ ε j v ε i + ∂ ε i v ε j ) -Γ p,ε ij v ε p = e ε j i (v ε ).
The functions e ε i j (v ε ) are called the linearized strains in curvilinear coordinates associated with the displacement field v ε i g i,ε . We assume throughout this paper that, for each ε > 0, the reference configuration Θ(Ω ε ) 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 λ ≥ 0 and µ > 0 (for details, see, e.g., [START_REF] Ciarlet | Mathematical Elasticity. Vol. I: Three-Dimensional Elasticity[END_REF]Section 3.8]).

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,ε ∈ L 2 (Ω ε ), and to a homogeneous boundary condition of place along the portion Γ ε 0 of its lateral face (i.e., the displacement vanishes on Γ ε 0 ). 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 ε i g i,ε must be such that all the points of the corresponding deformed configuration remain in a half-space of the form

H := {x ∈ E 3 ; Ox • q ≥ 0},
where q is a nonzero vector given once and for all. In other words, any admissible displacement field must satisfy

Θ(x ε ) + v ε i (x ε )g i,ε (x ε ) • q ≥ 0, for all x ε ∈ Ω ε ,
or possibly only for almost all (a.a. in what follows) x ε ∈ Ω ε when the covariant components v ε i are required to belong to the Sobolev space H 1 (Ω ε ) as in Theorem 3.1 below.

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

Θ(Ω ε ) ⊂ 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" ω × {-ε} of the reference configuration satisfy the confinement condition (see, e.g., [START_REF] Léger | Mathematical justification of the obstacle problem in the case of a shallow shell[END_REF], [START_REF] Léger | A linearly elastic shell over an obstacle: The flexural case[END_REF], [START_REF] Rodríguez-Arós | Mathematical justification of the obstacle problem for elastic elliptic membrane shells[END_REF]). Clearly, the confinement condition considered in the present paper 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 ∈ E 3 ; Ox • q = 0} and then to end up on the "other side" of this plane.

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 ε , is classical (see, e.g. [START_REF] Ciarlet | Mathematical Elasticity[END_REF]). More specifically, let

A ijk ,ε := λg ij,ε g k ,ε + µ g ik,ε g j ,ε + g i ,ε g jk,ε = A jik ,ε = A k ij,ε ,
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 u ε = (u ε i ) where the functions u ε i : Ω ε → R are the three covariant components of the unknown "three-dimensional" displacement vector field u ε i g i,ε of the reference configuration of the shell, should minimize the energy

J ε : H 1 (Ω ε ) → R defined by J ε (v ε ) := 1 2 Ω ε A ijk ,ε e ε k (v ε )e ε i j (v ε ) √ g ε dx ε - Ω ε f i,ε v ε i √ g ε dx ε , for each v ε = (v ε i ) ∈ H 1 (Ω ε
) over the set of admissible displacements defined by:

U (Ω ε ) :={v ε = (v ε i ) ∈ H 1 (Ω ε ); v ε = 0 on Γ ε 0 , (Θ(x ε ) + v ε i (x ε )g i,ε (x ε )) • q ≥ 0 for a.a. x ε ∈ Ω ε }.
The solution to this minimization problem exists and is unique, and it can be also characterized as the unique solution of the following problem:

Problem P(Ω ε ). Find u ε ∈ U (Ω ε
) that satisfies the following variational inequalities:

Ω ε A ijk ,ε e ε k (u ε ) e ε i j (v ε ) -e ε i j (u ε ) √ g ε dx ε ≥ Ω ε f i,ε (v ε i -u ε i ) √ g ε dx ε , for all v ε = (v ε i ) ∈ U (Ω ε
). The following result can be thus straightforwardly proved.

Theorem 3.1. The quadratic minimization problem: Find a vector field

u ε ∈ U (Ω ε ) such that J ε (u ε ) = inf v ε ∈U (Ω ε ) J ε (v ε ),
has one and only one solution. Besides, u ε is also the unique solution of Problem P(Ω ε ).

Proof. Define the space

V (Ω ε ) := {v ε = (v ε i ) ∈ H 1 (Ω ε ); v ε = 0 on Γ ε 0 }.
Then, thanks to the uniform positive-definiteness of the elasticity tensor (A ijk ,ε ) and to the boundary condition of place satisfied on Γ ε 0 = γ 0 × [-ε, ε] (recall that λ ≥ 0, µ > 0, and that γ 0 is a non-empty relatively open subset of γ = ∂ω), it can be shown (see [6, Theorems 3.8-3 and 3.9-1]) that the continuous and symmetric bilinear form

(v ε , w ε ) ∈ H 1 (Ω ε ) × H 1 (Ω ε ) → Ω ε A ijk ,ε e ε k (v ε )e ε i j (w ε ) √ g ε dx ε , is V (Ω ε )-elliptic; besides, the linear form v ε ∈ H 1 (Ω ε ) → Ω ε f i,ε v ε i √ g ε dx ε , is clearly continuous. Finally, the set U (Ω ε ) is nonempty (by assumption), closed in H 1 (Ω ε ) (any convergent sequence in V (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 classical (see, e.g., [START_REF] Duvaut | Inequalities in Mechanics and Physics[END_REF], [START_REF] Glowinski | Numerical Methods for Nonlinear Variational Problems[END_REF], or [START_REF] Ciarlet | Linear and Nonlinear Functional Analysis with Applications[END_REF]).

Since θ(ω) ⊂ Θ(Ω ε ), it evidently follows that θ(y) • q ≥ 0 for all y ∈ ω. But in fact, a stronger property holds: Lemma 3.1. Let ω be a domain in R 2 , let θ ∈ C 1 (ω; E 3 ) be an immersion, let q ∈ E 3 be a nonzero vector, and let ε > 0. Then the inclusion

Θ(Ω ε ) ⊂ H = {x ∈ E 3 ; Ox • q ≥ 0} implies that inf y∈ω (θ(y) • q) > 0.
Proof. Without loss of generality, assume that |q| = 1. Assume that there exists y 0 ∈ ω such that θ(y 0 ) • q = 0. It is then easily seen that the assumed relation θ(y) • q ≥ 0 for all y ∈ ω implies that a α (y 0 ) • q = 0, which means that the plane {x ∈ E 3 ; ox • q = 0} is tangent to the surface θ(ω) at the point θ(y 0 ). Consequently, either a 3 (y 0 ) = q and then (θ(y 0 ) -εa 3 (y 0 )) • q = -ε < 0, which contradicts (θ(y 0 ) -εa 3 (y 0 )) ∈ Θ(Ω ε ) ⊂ H; or a 3 (y 0 ) = -q and then (θ(y 0 ) + εa 3 (y 0 )) • q = -ε < 0, which contradicts θ((y 0 ) + εa 3 (y 0 )) ∈ Θ(Ω ε ) ⊂ H.

We must therefore have θ(y) • q > 0 for all y ∈ ω; hence inf y∈ω (θ(y)

• q) > 0 since ω is a compact subset of R 2 .
We now consider the "penalized" version of Problem P(Ω ε ). One such penalization transforms the set of variational inequalities in Problem P(Ω ε ) into a set of nonlinear variational equations, where the nonlinearity is defined in terms of the measure of the violation of the constraint. Let κ > 0 denote a "penalty parameter". The "penalized" variational formulation corresponding to Problem P(Ω ε ) takes the following form:

Problem P κ (Ω ε ). Find u ε κ ∈ V (Ω ε
) that satisfies the following variational equations:

Ω ε A ijk ,ε e ε k (u ε )e ε i j (v ε ) √ g ε dx ε - 1 κ Ω ε [Θ + u ε i,κ g i,ε ] • q -(v ε i g i,ε • q) √ g ε dx ε = Ω ε f i,ε v ε i √ g ε dx ε , for all v ε = (v ε i ) ∈ V (Ω ε
). The existence and uniqueness of the solution for Problem P κ (Ω ε ) is classical too, and resorts to the Browder fixed point theorem (cf., e.g., [START_REF] Kikuchi | Contact problems in elasticity: a study of variational inequalities and finite element methods[END_REF] and [START_REF] Lions | Quelques méthodes de résolution des problèmes aux limites non linéaires[END_REF]).

It can also be shown that, in the same spirit of [START_REF] Scholz | Numerical solution of the obstacle problem by the penalty method[END_REF], for each δ > 0 and for each ε > 0, we can find a number κ 0 = κ 0 (δ, ε) > 0 such that, for each 0 < κ < κ 0 , it results

(1)

u ε -u ε κ H 1 (Ω ε ) < δ 2 ,
where u ε and u ε κ respectively denote the solutions of Problem P(Ω ε ) and Problem P κ (Ω ε ).

The scaled three-dimensional problem for a family of flexural shells

In Section 3, we considered an obstacle problem for "general" linearly elastic shells. From now on, we will restrict ourselves to a specific class of shells, according to the following definition (proposed in [START_REF] Ciarlet | Asymptotic analysis of linearly elastic shells. II. Justification of flexural shell equations[END_REF]; see also [START_REF] Ciarlet | Mathematical Elasticity[END_REF]).

Consider a linearly elastic shell, subjected to the various assumptions set forth in Section 3. Such a shell is said to be a linearly elastic flexural shell if the following two additional assumptions are satisfied: first, ∅ = γ 0 ⊂ γ, i.e., the homogeneous boundary condition of place is imposed over a nonzero area portion of the entire lateral face γ 0 × [-ε, ε] of the shell, and second, the space

V F (ω) := {η = (η i ) ∈ H 1 (ω) × H 1 (ω) × H 2 (ω); γ αβ (η) = 0 in ω and η i = ∂ ν η 3 = 0 on γ 0 } contains nonzero functions, i.e., V F (ω) = {0}.
In this paper, we consider the obstacle problem (as defined in Section 3) for a family of linearly elastic flexural shells, all sharing the same middle surface and whose thickness 2ε > 0 is considered as a "small" parameter approaching zero. Our basic objective then consists in performing an asymptotic analysis as ε → 0, so as to seek whether we can identify a limit two-dimensional problem as ε → 0. To this end, we shall resort to a (by now standard) methodology first proposed by Ciarlet, Lods and Miara (cf. Theorem 5.1 of [START_REF] Ciarlet | Asymptotic analysis of linearly elastic shells. II. Justification of flexural shell equations[END_REF] and Theorem 6.2-1 of [START_REF] Ciarlet | Mathematical Elasticity[END_REF]): To begin with, we "scale" each problem P(Ω ε ), ε > 0, over a fixed domain Ω, using appropriate scalings on the unknowns and assumptions on the data. Note that these scalings and assumptions definitely depend on the type of shells that are considered; for instance, those used for the linearly elastic elliptic membrane shells considered elsewhere (cf. [START_REF] Ciarlet | An obstacle problem for elliptic membrane shells[END_REF] and also [START_REF] Ciarlet | Un problème de confinement pour une coque membranaire linéairement élastique de type elliptique[END_REF]) are different.

More specifically, let

Ω := ω × ]-1, 1[ , let x = (x i
) denote a generic point in the set Ω, and let

∂ i := ∂/∂x i . With each point x = (x i ) ∈ Ω, we associate the point x ε = (x ε i ) defined by x ε α := x α = y α and x ε 3 := εx 3 , so that ∂ ε α = ∂ α and ∂ ε 3 = 1 ε ∂ 3 .
To the unknown u ε = (u ε i ) and to the vector fields v ε = (v ε i ) appearing in the formulation of the problem P(Ω ε ) corresponding to a linearly elastic flexural shell, we then associate the scaled unknown u(ε) = (u i (ε)) and the scaled vector fields v = (v i ) by letting

u i (ε)(x) := u ε i (x ε
) and v i (x) := v ε i (x ε ), at each x ∈ Ω. Finally, we assume that there exist functions f i ∈ L 2 (Ω) independent on ε such that the following assumptions on the data hold

f i,ε (x ε ) = ε 2 f i (x) at each x ∈ Ω.
Note that the independence on ε of the Lamé constants assumed in Section 3 in the formulation of problem P(Ω ε ) implicitly constituted another assumption on the data.

In view of the proposed scaling, we define the "scaled" version of the geometrical entities introduced in section 2:

g i (ε)(x) := g i,ε (x ε ) at each x ∈ Ω, g(ε)(x) := g ε (x ε ) and A ijk (ε)(x) := A ijk ,ε (x ε ) at each x ∈ Ω, e α β (ε; v) := 1 2 (∂ β v α + ∂ α v β ) -Γ k αβ (ε)v k = e β α (ε; v), e α 3 (ε; v) = e 3 α (ε; v) := 1 2 1 ε ∂ 3 v α + ∂ α v 3 -Γ σ α3 (ε)v σ , e 3 3 (ε; v) := 1 ε ∂ 3 v 3 , where Γ p ij (ε)(x) := Γ p,ε ij (x ε ) at each x ∈ Ω. Define the space V (Ω) := {v = (v i ) ∈ H 1 (Ω); v = 0 on γ 0 × [-1, 1]},
and define, for each ε > 0, the set

U (ε; Ω) := {v = (v i ) ∈ V (Ω); θ(y) + εx 3 a 3 (y) + v i (x)g i (ε)(x) • q ≥ 0 for a.a. x = (y, x 3 ) ∈ Ω}.
We are thus in a position to introduced the "scaled" version of Problem P(Ω ε ), that will be denoted in what follows by P(ε; Ω). Problem P(ε; Ω). Find u(ε) ∈ U (ε; Ω) that satisfies the following variational inequalities:

Ω A ijk (ε)e k (ε; u(ε)) e i j (ε; v) -e i j (ε; u(ε)) g(ε) dx ≥ ε 2 Ω f i (v i -u i (ε)) g(ε) dx,
for all v ∈ U (ε; Ω). Proof. The variational Problem P(ε; Ω) simply constitutes a re-writing of the variational Problem P(Ω ε ), this time in terms of the scaled unknown u(ε), of the vector fields v, and of the functions f i , which are now all defined over the domain Ω. Then the assertion follows from this observation.

The functions e i j (ε; v) appearing in Problem P(ε; Ω) are called the scaled linearized strains in curvilinear coordinates associated with the scaled displacement vector field v i g i (ε).

For later purposes (like in Lemma 4.1 below), we also let

g i (ε)(x) := g ε i (x ε ) at each x ∈ Ω.
Likewise, one can introduce the "scaled" version of Problem P κ (Ω ε ), that will be denoted in what follows by P κ (ε; Ω).

Problem P κ (ε; Ω). Find u κ (ε) ∈ V (Ω) that satisfies the following variational equations:

Ω A ijk (ε)e k (ε; u κ (ε))e i j (ε; v) g(ε) dx - 1 κ Ω [Θ + u i,κ (ε)g i (ε)] • q -(v i g i (ε) • q) g(ε) dx = ε 2 Ω f i v i g(ε) dx, for all v ∈ V (Ω).
The following existence and uniqueness result can be thus easily proved.

Theorem 4.2. Then the scaled unknown u κ (ε) is the unique solution of the variational Problem P κ (ε; Ω).

Proof. The variational Problem P κ (ε; Ω) simply constitutes a re-writing of the variational Problem P κ (Ω ε ), this time in terms of the scaled unknown u κ (ε), of the vector fields v, and of the functions f i , which are now all defined over the domain Ω. Then the assertion follows from this observation.

A condition similar to (1) can be derived, i.e., for each δ > 0 and for each ε > 0, we can find a number κ 0 = κ 0 (δ, ε) > 0 such that, for each 0 < κ < κ 0 , it results

(2) u(ε) -u κ (ε) H 1 (Ω) < δ 2 ,
where u(ε) and u κ (ε) respectively denote the solutions of Problem P(ε; Ω) and Problem P κ (ε; Ω). Without loss of generality, we restrict ourselves to considering penalty parameters with the following property:

(3) 0 < κ ≤ ε, so that κ → 0 as ε → 0.

Remarks.

(1) Of course, the variational Problem P(ε; Ω) could have been equivalently written as a minimization problem, thus mimicking that found in Theorem 3.1. It turns out, however, that its formulation in Theorem 4.1 as a set of variational inequalities is more convenient for the asymptotic analysis undertaken in Section 5.

(2) It is immediately verified (cf., e.g., [START_REF] Ciarlet | Mathematical Elasticity[END_REF]) that other assumptions on the data are possible that would give rise to the same problem over the fixed domain Ω. For instance, should the Lamé constants (now denoted) λ ε and µ ε appearing in Problem P(Ω ε ) be of the form λ ε = ε t λ and µ ε = ε t µ, where λ ≥ 0 and µ are constants independent of ε and t is an arbitrary real number, the same Problem P(ε; Ω) arises if we assume that the components of the applied body force density are now of the form

f i,ε (x ε ) = ε 2+t f i (x) at each x ∈ Ω,
where the functions f i ∈ L 2 (Ω) are independent of ε.

The next lemma assembles various asymptotic properties as ε → 0 of functions and vector fields appearing in the formulation of Problem P(ε; Ω); these properties will be repeatedly used in the proof of the convergence theorem (Theorem 4.3):

In the next statement, the notation "O(ε)", or "O(ε 2 )", stands for a remainder that is of order ε, or ε 2 , with respect to the sup-norm over the set Ω, and any function, or vector-valued function, of the variable y ∈ ω, such as a αβ , b αβ , a i , etc. (all these are defined in Section 2) is identified with the function, or vector-valued function, of x = (y, x 3 ) ∈ Ω = ω × [-1, 1] that takes the same value at x 3 = 0 and is independent of x 3 ∈ [-1, 1]; for brevity, this extension from ω to Ω is designated with the same notation.

Recall that ε > 0 is implicitly assumed to be small enough so that Θ : Ω ε → E 3 is an immersion. Lemma 4.1. Let ε 0 be defined as in Theorem 3.1-1 of [START_REF] Ciarlet | Mathematical Elasticity[END_REF]. The functions A ijk (ε) = A jik (ε) = A k ij (ε) have the following properties:

A ijk (ε) = A ijk (0) + O(ε), A αβσ3 (ε) = A α333 (ε) = 0,
for all 0 < ε ≤ ε 0 , where A αβστ (0) = λa αβ a στ + µ(a ασ a βτ + a ατ a βσ ),

A αβ33 (0) = λa αβ , A α3σ3 (0) = µa ασ , A 3333 (0) = λ + 2µ,
and there exists a constant C 0 > 0 such that i,j

|t ij | 2 ≤ C 0 A ijk (ε)(x)t k t ij
for all 0 < ε ≤ ε 0 , all x ∈ Ω, and all symmetric matrices (t ij ).

The functions Γ p ij (ε) and g(ε) have the following properties:

Γ σ αβ (ε) = Γ σ αβ -εx 3 (∂ α b σ β + Γ σ ατ b τ β -Γ τ αβ b σ τ ) + O(ε 2 ), Γ 3 αβ (ε) = b αβ -εx 3 b σ α b σβ , ∂ 3 Γ p αβ (ε) = O(ε), Γ σ α3 (ε) = -b σ α -εx 3 b τ α b σ τ + O(ε 2 ), Γ 3 α3 (ε) = Γ p 33 (ε) = 0, g(ε) = a + O(ε),
for all 0 < ε ≤ ε 0 and all x ∈ Ω. In particular then, there exist constants g 0 and g 1 such that 0 < g 0 ≤ g(ε)(x) ≤ g 1 for all 0 < ε ≤ ε 0 and all x ∈ Ω.

The vector fields g i (ε) and g j (ε) have the following properties:

g α (ε) = a α -εx 3 b σ α a σ , g 3 (ε) = a 3 , g α (ε) = a α + εx 3 b α σ a σ + O(ε 2 ), g 3 (ε) = a 3 . Proof.
See the proofs of Theorems 3.3-1 and 3.3-2 in [START_REF] Ciarlet | Mathematical Elasticity[END_REF].

Remark. The various relations and estimates in Lemma 4.1 hold in fact for any family of linearly elastic shells, i.e., irrespective of whether these shells are flexural ones or not.

When one considers a family of linearly elastic flexural shells whose thickness 2ε approaches zero, a specific Korn's inequality in curvilinear coordinates holds over the fixed domain Ω = ω × ]-1, 1[, according to the following theorem. That the constant C 1 that appears in this inequality is independent of ε > 0 plays a key role in the asymptotic analysis of such a family (see part (i) of the proof of Theorem 5.1). Theorem 4.3. Let there be given a family of linearly elastic flexural shells with the same middle surface θ(ω) and thickness 2ε > 0. Define the space

V (Ω) := {v = (v i ) ∈ H 1 (Ω); v = 0 on γ 0 × [-1, 1]}.
Then there exist constants ε 1 > 0 and

C 1 > 0 such that i v i 2 H 1 (Ω) 1/2 ≤ C ε i,j e i j (ε; v) 2 L 2 (Ω) 1/2
for all 0 < ε ≤ ε 1 and all v ∈ V (Ω).

Proof. See Ciarlet, Lods and Miara Theorem 4.1 of [START_REF] Ciarlet | Asymptotic analysis of linearly elastic shells. II. Justification of flexural shell equations[END_REF] or Theorem 5.3-1 of [START_REF] Ciarlet | Mathematical Elasticity[END_REF].

Rigorous asymptotic analysis

The ultimate goal of this paper is to show, in the same spirit as [START_REF] Ciarlet | Asymptotic analysis of linearly elastic shells. II. Justification of flexural shell equations[END_REF] (see also Theorem 6.2-1 of [START_REF] Ciarlet | Mathematical Elasticity[END_REF]), that the solutions u(ε) of the (scaled) three-dimensional problems P(Ω ε ) converge -as ε approaches zero -to the solution of a two-dimensional problem, denoted P F (ω) in what follows.

The proof proposed in [START_REF] Ciarlet | Asymptotic analysis of linearly elastic shells. II. Justification of flexural shell equations[END_REF] (see also Theorem 6.2-1 of [START_REF] Ciarlet | Mathematical Elasticity[END_REF]) resorts, however, to the usage of a specific vector field that was first introduced by Miara and Sanchez-Palencia [START_REF] Miara | Asymptotic analysis of linearly elastic shells[END_REF]. This argument is, in general, not easy to generalize to the context of variational inequalities, for which the test functions are in general chosen in a nonempty, closed and convex subset of a certain space.

In order to overcome this difficulty, we first prove that, under the assumption that κ ≤ ε, the solutions u κ (ε) of Problem P κ (ε; Ω) converge -as ε approaches zero -to the solution of the variational problem P F (ω).

Define the set

U F (ω) := {η = (η i ) ∈ V F (ω); θ(y) + η i (y)a i (y) • q ≥ 0 for a.a. y ∈ ω},
where the space V F (ω) has been defined in section 4. We observe that the set U F (ω) is nonempty, closed and convex in the space V F (ω). The vector fields a i and the functions Γ σ αβ , b αβ , a αβ , a, and γ αβ (η), have been defined in Section 2. We are thus in a position to define the two-dimensional problem P F (ω) as follows:

Problem P F (ω). Find ζ ∈ U F (ω) that satisfies the following variational inequal- ities: 1 3 ω a αβστ ρ στ (ζ)ρ αβ (η -ζ) √ a dy ≥ ω p i (η i -ζ i ) √ a dy, for all η = (η i ) ∈ U F (ω)
, where

a αβστ := 4λµ λ + 2µ
a αβ a στ + 2µ a ασ a βτ + a ατ a βσ and p i :=

1 -1 f i dx 3 .
In the same spirit as Theorem 3.1, it can be show that Problem P F (ω) admits one and only one solution.

We are now ready to show that, under the assumption that κ ≤ ε, the solutions u κ (ε) of Problem P κ (ε; Ω) converge -as ε approaches zero -to the solution of Problem P F (ω).

Theorem 5.1. Let ω be a domain in R 2 , let θ ∈ C 3 (ω; E 3 ) be the middle surface of a flexural shell, let γ 0 be a non-zero length portion of the boundary γ (cf. Section 4) and let q ∈ E 3 be a non-zero vector given once and for all. Let us consider the non-trivial space (cf. Section 4)

V F (ω) := {η = (η i ) ∈ H 1 (ω) × H 1 (ω) × H 2 (ω); γ αβ (η) = 0 in ω and η i = ∂ ν η 3 = 0 on γ 0 },
and let us define the set

U F (ω) := {η = (η i ) ∈ V F (ω); θ(y) + η i (y)a i (y) • q ≥ 0 for a.a. y ∈ ω}.
Let there be given a family of linearly elastic flexural shells with the same middle surface θ(ω) and thickness 2ε > 0, and let u κ (ε) ∈ V (Ω) denote for each ε > 0 the unique solution of Problem P κ (ε; Ω), where the penalty parameter κ is assumed to be as in [START_REF] Bernard | Density results on Sobolev spaces whose elements vanish on a part of the boundary[END_REF].

Then there exists u ∈ H 1 (Ω) independent of the variable x 3 and satisfying

u = 0 on Γ 0 = γ 0 × [-1, 1] , u(ε) → u in H 1 (Ω) as ε → 0.
Define the average

u = (u i ) := 1 2 1 -1 u dx 3 . Then u = ζ,
where ζ is the unique solution to the two-dimensional variational Problem P F (ω).

Proof. Strong and weak convergences as ε → 0 are respectively denoted by → and . For brevity, we let

e i j (ε) := e i j (ε; u κ (ε)).
The outline of the proof is to a large extent inspired by the proof of Theorem 6.2-1 of [START_REF] Ciarlet | Mathematical Elasticity[END_REF] (itself adapted from Theorem 5.1 in Ciarlet, Lods and Miara [START_REF] Ciarlet | Asymptotic analysis of linearly elastic shells. II. Justification of flexural shell equations[END_REF]), where no confinement condition was imposed. This is why some parts of the proof are reminiscent of those in [START_REF] Ciarlet | Mathematical Elasticity[END_REF]; otherwise, considering the confinement condition requires extra care.

(i) There exists a subsequence, still denoted (u(ε)) ε>0 , and there exists u ∈ H 1 (Ω) and there exist e 1 i j ∈ L 2 (Ω) satisfying

u = 0 on Γ 0 = γ 0 × [-1, 1]
and such that u(ε) u in H 1 (Ω) and thus u(ε) → u in L 2 (Ω), (θ(y) + u i (y, x 3 )a i (y)) • q ≥ 0 for a.a. x = (y, x 3 ) ∈ Ω, 1 ε e i j (ε) e 1 i j in L 2 (Ω).

Letting v = u κ (ε) in the variational equations of Problem P κ (ε; Ω). Combining the uniform positive-definiteness of the tensor (A ijk (ε)), the Korn inequality of Theorem 4.3, the asymptotic behavior of the function g(ε) (Lemma 4.1), and the fact that

- 1 κ Ω [Θ + u i,κ (ε)g i (ε)] • q -(u i,κ (ε)g i (ε) • q) g(ε) dx = - 1 κ Ω [Θ + u i,κ (ε)g i (ε)] • q -(Θ + u i,κ (ε)g i (ε) • q) g(ε) dx + 1 κ Ω [Θ + u i,κ (ε)g i (ε)] • q -(Θ • q) g(ε) dx ≥ 0,
for all κ > 0 and all ε > 0, we obtain for ε > 0 sufficiently small:

C -2 1 ε 2 i u i (ε) 2 H 1 (Ω) ≤ i,j e i j (ε) 2 L 2 (Ω) ≤ C 0 √ g 0 Ω A ijk (ε)e k (ε)e i j (ε) g(ε) dx ≤ ε 2 C 0 √ g 0 Ω f i u i (ε) g(ε) dx ≤ ε 2 C 0 g 1 g 0 i f i 2 L 2 (Ω) 1/2 i u i (ε) 2 L 2 (Ω) 1/2
. This chain of inequalities first shows that the norms u i (ε) H 1 (Ω) are bounded independently of ε, secondly, that the terms ε -1 e i j (ε) L 2 (Ω) are bounded uniformly with respect to ε and, finally, that the terms (4)

1 κε 2 [Θ + u i,κ (ε)g i (ε)] • q - L 2 (Ω)
are uniformly bounded with respect to ε as well (recall that κ = O(ε) by assumption).

Hence there exist a subsequence, a vector field u ∈ H 1 (Ω), and functions e 1 i j ∈ L 2 (Ω), such that:

(5)

u(ε) u in H 1 (Ω), 1 ε e i j (ε) e 1 i j in L 2 (Ω), 1 κ [Θ + u i,κ (ε)g i (ε)] • q -→ 0 in L 2 (Ω), [Θ + u i,κ (ε)g i (ε)] • q -→ 0 in L 2 (Ω).
The fact that u(ε) → u in L 2 (Ω) is a consequence of the Rellich-Kondrašov Theorem (Theorem 6.6-3 of [START_REF] Ciarlet | Linear and Nonlinear Functional Analysis with Applications[END_REF]). Combining the latter with the last convergence in [START_REF] Ciarlet | Mathematical Elasticity[END_REF] gives:

(θ(y) + u i (y, x 3 )a i (y)) • q ≥ 0, for a.a. x = (y, x 3 ) ∈ Ω.
That u = 0 on γ 0 × [-1, 1] follows from the continuity of the trace operator tr :

H 1 (Ω) → L 2 (γ × [-1, 1]). Indeed, for all i, we have that for all v ∈ H 1 (Ω) such that v = 0 on Γ -Γ 0 , 0 = Γ0 u i (ε)v dΓ → Γ0 u i v dΓ,
so that a density result proved by Bernard [START_REF] Bernard | Density results on Sobolev spaces whose elements vanish on a part of the boundary[END_REF] (see also Theorem 6.7-3 of [START_REF] Ciarlet | Linear and Nonlinear Functional Analysis with Applications[END_REF]) gives u i = 0 on Γ 0 .

(ii) The weak limits u i ∈ H 1 (Ω) found in (i) are independent of the variable x 3 ∈ [-1, 1], in the sense that they satisfy

∂ 3 u 3 = 0 in L 2 (Ω) and ∂ 3 u 3 = 0 in D (Ω).
Besides, the average u satisfies u ∈ U F (ω), namely,

u = (u i ) ∈ H 1 (ω) × H 1 (ω)×H 2 (ω) and u i = ∂ ν u 3 = 0 on γ 0 , γ αβ (u) = 0 in ω, θ(y) + u i (y)a i (y) • q ≥ 0 for a.a. y ∈ ω.
Apart from the latter property, the proof is identical to that of part (ii) of the proof of Theorem 6.2-1 in [START_REF] Ciarlet | Mathematical Elasticity[END_REF]. Let us thus prove that θ(y) + u i (y)a i (y) • q ≥ 0 for a.a. y ∈ ω.

By part (i), we have that (θ(y) + u i (y, x 3 )a i (y)) • q ≥ 0 for a.a. x = (y, x 3 ) ∈ Ω. Since u = (u i ) is independent of x 3 , we have that (θ(y) + u i (y, x 3 )a i (y)) • q = θ(y) + 1 2

1 -1 u i (y, x 3 ) dx 3 a i (y) • q, so that u = (u i ) ∈ U F (ω).
(iii) The weak limits e 1 i j ∈ L 2 (Ω) and u ∈ H 1 (Ω) found in (i) satisfy a αβ e 1 α β in Ω.

-∂ 3 e 1 α β = ρ αβ (u) in L 2 (Ω),
The equality -∂ 3 e 1 α β = ρ αβ (u) in L 2 (Ω) follows from Theorem 5.2-2 of [START_REF] Ciarlet | Mathematical Elasticity[END_REF]. Let v = (v i ) ∈ V (Ω) be arbitrarily chosen. It is known (cf., e.g., part (iii) of Theorem 6.2-1 of [START_REF] Ciarlet | Mathematical Elasticity[END_REF]) that

εe α β (ε; v) → 0 in L 2 (Ω), εe α 3 (ε; v) → 1 2 ∂ 3 v α in L 2 (Ω), εe 3 3 (ε; = ∂ 3 v 3 for all ε > 0.
These relations, combined with the boundedness of the terms ε -1 e i j (ε) L 2 (Ω)

independently of ε > 0 (part (i)) and the asymptotic behavior of the functions A ijk (ε) and g(ε) as ε → 0 (Lemma 4.1), give 4) and ( 5)),

Ω A αβστ (ε) e σ τ (ε) ε + A αβ33 (ε) e 3 3 (ε) ε εe α β (ε; v) g(ε) dx → 0 as ε → 0, Ω 4A α3σ3 (ε) e σ 3 (ε) ε εe α 3 (ε; v) g(ε) dx → Ω 2µa ασ e 1 σ 3 v α √ a dx as ε → 0, Ω A 33στ (ε) e σ τ (ε ε ) + A 3333 (ε) e 3 3 (ε) ε εe 3 3 (ε; v) g(ε) dx → Ω λa στ e 1 σ τ + (λ + 2µ)e 1 3 3 v 3 √ a dx as ε → 0, - 1 κ Ω [Θ + u i,κ (ε)g i (ε)] • q -(v i g i (ε) • q) g(ε) dx → 0 as ε → 0 (cf. (
ε 2 Ω f i v i g(ε) dx → 0 as ε → 0. Consequently, Ω (2µa ασ e 1 σ 3 )v α + λa στ e 1 σ τ + (λ + 2µ)e 1 3 3 v 3 √ a dx = 0.
Since this inequality holds for any vector field v = (v i ) ∈ V (Ω), it follows that e 1 σ 3 = 0 and λa στ e 1 σ τ + (λ + 2µ)e 1 3 3 = 0 in L 2 (Ω). In particular, the latter gives:

e 1 3 3 = - λ λ + 2µ
a αβ e 1 α β in L 2 (Ω).

(iv) The weak limit u = (u i ) is the solution of Problem P F (ω).

Given any η ∈ U F (ω), define the vector field w(ε; η) = (w i (ε; η)) by:

w α (ε; η) := η α -εx 3 (∂ α η 3 + b σ α η σ ), w 3 (ε; η) := η 3 .
We have that w(ε; η) ∈ V (Ω). Following the same steps as in part (iv) of [START_REF] Ciarlet | Mathematical Elasticity[END_REF], observe that

w(ε; η) →η in H 1 (Ω), 1 ε e α β (ε; w(ε; η)) →{-x 3 ρ αβ (η)} in L 2 (Ω), 1 ε e α 3 (ε; w(ε; η)) ε>0 converges in L 2 (Ω).
For each ε > 0, define

Λ(ε) := Ω A ijk (ε) 1 ε e k (ε) -e 1 k 1 ε e i j (ε) -e 1 i j
g(ε) dx, so that, combining the uniform positive-definiteness of the three-dimensional elasticity tensor and the asymptotic behavior of the function g(ε) with the variational equations of Problem P(ε; Ω) specializing v = w(ε; η) -u κ (ε), we obtain:

0 ≤ i,j 1 ε e i j (ε) -e 1 i j 2 L 2 (Ω) ≤ C 0 √ g 0 Λ(ε).
This gives (6)

0 ≤ Λ(ε) = Ω A ijk (ε) 1 ε e k (ε) 1 ε e i j (ε) g(ε) dx -2 Ω A ijk (ε) 1 ε e k (ε) e 1 i j g(ε) dx + Ω A ijk (ε)e 2 k e 1 i j g(ε) dx ≤ Ω A ijk (ε) 1 ε e k (ε)} 1 ε e i j (ε; w(ε; η)) g(ε) dx - 1 κ Ω [Θ + u i,κ (ε)g i (ε)] • q -(w i (ε; η) -u i,κ (ε)g i (ε) • q) g(ε) dx -2 Ω A ijk (ε) 1 ε e k (ε) e 1 i j g(ε) dx + Ω A ijk (ε)e 1 k e 1 i j g(ε) dx - Ω f i (w i (ε; η) -u i (ε)) g(ε) dx.
The asymptotic behavior of the functions w(ε; η) and 1 ε e i j (ε; w(ε; η)), the asymptotic behavior of the three-dimensional elasticity tensor A ijk (ε) and g(ε)

(Lemma 4.1), the weak convergences 1 ε e i j (ε) e 1 i j in L 2 (Ω) established in part (i), and the relations satisfied by e 1 i 3 (part (iii)) together give:

Ω A ijk (ε) 1 ε e k (ε)} 1 ε e i j (ε; w(ε; η)) g(ε) dx → Ω (A αβστ (0)e 1 σ τ + A αβ33 (0)e 1 3 3 ){-x 3 ρ αβ (η)} √ a dx, 2 Ω A ijk (ε) 1 ε e k (ε) e 1 i j g(ε) dx - Ω A ijk (ε)e 1 k e 1 i j g(ε) dx → Ω A ijk (0)e 1 k e 1 i j √ a dx = Ω a αβστ e 1 σ τ e 1 α β √ a dx, Ω f i (w i (ε; η) -u i (ε)) g(ε) dx → Ω f i (η i -u i ) √ a dx = ω p i (η i -u i ) √ a dy.
Finally, we observe that the asymptotic behavior of the functions g(ε), w i (ε; η) and (4) gives that the limit as ε → 0 of

Ω [Θ + u i,κ (ε)g i (ε)] • q - ε 2 κ [(w i (ε; η) -u i,,κ (ε))g i (ε) • q] g(ε) dx = Ω [Θ + u i,κ (ε)g i (ε)] • q - ε 2 κ [(Θ + w i (ε; η)g i (ε)) • q] g(ε) dx - Ω [Θ + u i,κ (ε)g i (ε)] • q - ε 2 κ [(Θ + u i,κ (ε)g i (ε)) • q] g(ε) dx
exists and is nonnegative.

We have yet to take into account the relations ρ αβ (u) = -∂ 3 e 1 α β in L 2 (Ω) established in part (iv). Since u is independent of x 3 (part (ii)), these relations show that the functions e 1 α β are of the form e 1 α β = Υ αβ -x 3 ρ αβ (u) with Υ αβ ∈ L 2 (ω). The asymptotic behaviors observed in (6), Lemma 3.1, and the uniform positivedefiniteness of the fourth order two-dimensional elasticity tensor (a αβστ ) in turn imply

0 ≤ lim ε→0 Λ(ε) ≤ - ω p i (η i -u i ) √ a dy - Ω A ijk (0)e 1 k e 1 i j √ a dx + Ω A ijk (0)e 1 k {-x 3 ρ αβ (η)} √ a dx = - ω p i (η i -u i ) √ a dy - 1 2 Ω a αβστ Υ στ Υ αβ √ a dx + 1 2 ω a αβστ x 2 3 ρ στ (u)ρ αβ (η -u) √ a dx ≤ 1 3 ω a αβστ ρ στ (u)ρ αβ (η -u) dy - ω p i (η i -u i ) √ a dy.
In conclusion, the latter yields

0 ≤ ω a αβστ Υ στ Υ αβ √ a dy ≤ 1 3 ω a αβστ ρ στ (u)ρ αβ (η-u) dy- ω p i (η i -u i ) √ a dy,
which establishes that u is the unique solution for Problem P F (ω).

(v) The weak convergences 1 ε e i j (ε)

e 1 i j in L 2 (Ω) established in part (i) are in fact strong, i.e., 1 ε e i j (ε) → e 1 i j in L 2 (Ω).
Besides, the limits e 1 i j are unique; hence these convergences hold for the whole family 1 ε e i j (ε) ε>0

.

Let Λ(ε) be as in part (v). Since u ∈ U F (ω), we define a vector field w(ε) ∈ V (Ω) as follows:

wα (ε) := u α -εx 3 (∂ α u 3 + b σ α u σ ), w3 (ε) := u 3 .
Specializing v = w(ε) -u κ (ε) in the variational equations of Problem P(Ω ε ) and repeating the same computations as in part (iv) gives:

0 ≤ lim sup ε→0 Λ(ε) ≤ - ω a αβστ Υ στ Υ αβ √ a dy ≤ 0.
In conclusion, by the uniform positive-definiteness of the two-dimensional fourthorder elasticity tensor (Lemma 4.1), we obtain Υ αβ = 0 and lim ε→0 Λ(ε) = 0. These relations in turn imply that the strong convergence 1 ε e i j (ε) → e 1 i j in L 2 (Ω) holds. The functions e 1 α β are uniquely determined, since they are given by e 1 α β = -x 3 ρ αβ (u) and the vector field u is uniquely determined as well. That the functions e 1 i 3 are uniquely determined then follows from the relations established in part (iv).

(vi) The weak convergence u

(ε) u in H 1 (Ω) established in part (i) is in fact strong, i.e., u(ε) → u in H 1 (Ω).
The proof is identical to that of part (vi) of the proof of Theorem 6.2-1 of [START_REF] Ciarlet | Mathematical Elasticity[END_REF] and for this reason is omitted.

The theorem is thus completely proven.

The convergence as ε → 0 of the solutions (u(ε)) of Problem P(ε; Ω) to the solution ζ of Problem P F (ω) can thus be established as a corollary of Theorem 5.1.

Corollary 5.2. Let ω be a domain in R 2 , let θ ∈ C 3 (ω; E 3 ) be the middle surface of a flexural shell, let γ 0 be a non-zero length portion of the boundary γ (cf. Section 4) and let q ∈ E 3 be a non-zero vector given once and for all. Let us consider the non-trivial space (cf. Section 4)

V F (ω) := {η = (η i ) ∈ H 1 (ω) × H 1 (ω) × H 2 (ω); γ αβ (η) = 0 in ω and η i = ∂ ν η 3 = 0 on γ 0 },
and let us define the set U F (ω) := {η = (η i ) ∈ V F (ω); θ(y) + η i (y)a i (y) • q ≥ 0 for a.a. y ∈ ω}.

Let there be given a family of linearly elastic flexural shells with the same middle surface θ(ω) and thickness 2ε > 0, and let u(ε) ∈ U (ε; Ω) denote, for each ε > 0, the unique solution of Problem P(ε; Ω).

Then, we have that

1 2 1 -1 u(ε) dx 3 → ζ, in H 1 (ω) as ε → 0,
where ζ is the unique solution to the two-dimensional variational Problem P F (ω).

Proof. The conclusion of Theorem 5.1 reads as follows: For each δ > 0 there exists a number ε 2 > 0 such that for all 0 < ε < ε 2 it results (7) 1 2

1 -1 u i,κ (ε)(•, x 3 ) dx 3 -ζ i H 1 (ω) < δ 2 ,
where u κ (ε) = (u i,κ (ε)) is the unique solution of Problem P κ (ε; Ω), and ζ = (ζ i ) is the unique solution of Problem P F (ω), and κ is as in [START_REF] Bernard | Density results on Sobolev spaces whose elements vanish on a part of the boundary[END_REF]. As a result of ( 2) and ( 7), it can be shown that for each δ > 0 there exists a number ε > 0 such that for all 0 < ε < ε it results In order to prove [START_REF] Ciarlet | A justification of the two-dimensional linear plate model[END_REF], fix δ > 0 and take ε := ε 2 . For each 0 < ε < ε 2 , we have that an application of the triangle inequality and ( 7) gives ( 9)

1 2 1 -1 u i (ε) dx 3 -ζ i H 1 (ω) ≤ 1 2 1 -1 u i,κ (ε)(•, x 3 ) dx 3 -ζ i H 1 (ω) + u i (ε) -u i,κ (ε) H 1 (Ω) < δ 2 + δ 2 = δ,
for all 0 < κ ≤ min{κ 0 (δ, ε), ε}. The estimate [START_REF] Ciarlet | On the ellipticity of linear membrane shell equations[END_REF] means that the solutions u(ε) of Problem P(ε; Ω) converge to the solution of Problem P F (ω) as ε → 0, as it had to be proved. It remains to "de-scale" the results of Theorem 5.1 and corollary 5.2, which apply to the solutions u(ε) of the scaled problem P(ε; Ω). This means that we need to convert these results into ones about the unknown u ε i g i,ε : Ω ε → E 3 , which represents the physical three-dimensional vector field of the actual reference configuration of the shell. As shown in the next theorem, this conversion is most conveniently achieved through the introduction of the averages 1 2ε ε -ε u ε i g i,ε dx ε 3 across the thickness of the shell, a procedure which also clearly highlights the striking difference (in terms of function spaces) between the asymptotic behaviors of the tangential and normal components of the displacement field of the middle surface of the shell.

Theorem 5.3. Let the assumptions on the data be as in Section 4 and let the assumptions on the immersion θ ∈ C 3 (ω; E 3 ) be as in Theorem 4.3. Let u ε = (u ε i ) ∈ U (Ω ε ) denote for each ε > 0 the unique solution of the variational Problem P(Ω ε ) and let ζ ∈ U F (ω) denote the unique solution to the variational inequalities in Problem P F (ω). Then

1 2ε ε -ε u ε α g α,ε dx ε 3 → ζ α a α , in H 1 (ω) as ε → 0, 1 2ε ε -ε u ε 3 g 3,ε dx ε 3 → ζ 3 a 3 , in H 1 (ω) as ε → 0.
Proof. The proof is analogous to that of Theorem 6.4-1 in [START_REF] Ciarlet | Mathematical Elasticity[END_REF] and for this reason is omitted.

In view of the scalings, viz., u ε i (x ε ) = u i (ε)(x) at each x ε ∈ Ω ε , and of the assumption on the data, viz., f i,ε (x ε ) = f i (x) at each x ε ∈ Ω ε , made in Section 4, it is natural to also "de-scale" the unknown appearing in the limit two-dimensional problem found in Theorem

f i dx 3 = ε -ε f i,ε dx ε 3
instead of their scaled counterparts p i = 1 -1 f i dx 3 . In this fashion, it is immediately found that ζ ε = (ζ ε i ) ∈ U F (ω) is the unique solution to the variational inequalities

ε 3 3 ω a αβστ ρ στ (ζ ε )ρ αβ (η -ζ ε ) √ a dy ≥ ω p i,ε (η i -ζ ε i ) √ a dy,
for all η = (η i ) ∈ U F (ω). These inequalities now display the factor ε 3 3 , which always appears in the left-hand sides of equations modelling flexural shells.

Theorem 4 . 1 .

 41 Then the scaled unknown u(ε) is the unique solution of the variational Problem P(ε; Ω).

1 u

 1 i (ε)(•, x 3 ) dx 3 -ζ i H 1 (ω) < δ,where u(ε) = (u i (ε)) is the unique solution of Problem P(ε; Ω) and ζ = (ζ i ) is the unique solution of Problem P F (ω).

Figure 1 .

 1 Figure 1. Geometrical interpretation of Corollary 5.2. Let δ > 0 be fixed and let 0 < ε < ε 2 , where ε 2 is the parameter found in Corollary 4.2. Let ζ denote the solution to Problem P F (ω). If κ > κ 0 (δ, ε), it might happen (Theorem 4.2) that the distance between the averaged solution of Problem P κ (ε; Ω) (i.e., the vector field ζ κ(ε)) and the averaged solution of Problem P(ε; Ω) (i.e., the vector field ζ(ε)) is, with respect to the H 1 (ω) norm, strictly greater than the fixed number δ. If, however, we replace the penalty parameter κ by a new parameter κ ≤ min{ε, κ 0 (δ, ε)}, then we can find an element ζ κ (ε) whose distance from both ζ κ(ε) and ζ(ε) is strictly less than δ/2.

1 - 1

 11 4.3, by letting ζ ε i (y) := ζ i (y) at each y ∈ ω, and by using in its formulation the contravariant components p i,ε := ε 3

Acknowledgements

The author is greatly indebted to Professor Philippe G. Ciarlet for his encouragement and guidance.


The collection of Cesar Franck, commonly encountered under the misleading title "L'organiste," consists of a series of compositions in groups of seven pieces, bound together in a progressive ordering by semitone in major and minor tonalities. The series was composed, according to an original annotation on the autograph manuscript, for the harmonium. The importance of these "small" compositions is great, especially because they reveal a neat image of what music education and music composition could have meant for C. Franck during the last years of his life. They constitute a summa of his art of composing. Inside the ordering progress of these groups the composer exploits, in a methodical way, some of the most characteristic elements of music composition. He, therefore, presents techniques and rudiments related to the form and the content of music composition, and to performance practices. An illustrative presentation is the purpose of this article which will try to expose, in an analytical manner, the construction of the ordering of these pedagogical rudiments, aiming to shed new light on the importance and authenticity of this collection.

Introduction

The series of compositions by César Franck, which we often find under the erroneous title "L'organiste," is probably among the most misunderstood works of the composer from the point of view of artistic quality. The first edition of César Franck's "L'organiste" appeared in 1892 and was published by Enoch editions in Paris. Following this first edition, others appeared after the composer's death in 1890: the second volume of the Enoch editions in 1905 annotated with the fingering of Charles Tournemire, Franck's successor at the St. Clotilde church (Flor 1972, p. 395), the edition was still under the responsibility of Charles Tournemire in 1934 and 1935; the Schott edition in 1997 in which Gunther Kauzinger presented all the existing compositions of the collection (63 pieces), adding the last four compositions of the last completed series; and more recently the Leduc edition in 1998 by Joël-Marie Fauquet and Joris Verdin, which proposed an important supplement of critical notes in several languages (Near 2002, p. 459). The year of 1892, date of the publication of the first edition, was most probably the time when, historically, the pieces for harmonium acquired the erroneous title "L'organiste" since Franck himself had left a handwritten note in the autograph regarding this issue: "Pièces pour Harmonium" 1 (Tchorek 2007, pp. 337-397).

It would be possible to say that the reason for this lack of recognition was also due to the small size of the compositions included in this long series 2 (Graces 1922, p. 840), as will be shown in the following pages. With this, we can perhaps associate the notoriety of the instrument for which they were composed, the harmonium (compared to the organ or the piano), though the same instrument "enjoyed considerable popularity in the salons of French high society" (Near 2002, p. 459). Nevertheless, the musical reality of these salons apparently did not represent the main stream of composers of this era. Thus, the use of the harmonium was not particularly frequent outside 1 Today we know it more commonly with the title "The Organist" but apparently and as Fauquet and Verdin inform us, in page xiii of the autograph manuscript we find an annotation by the hand of the composer "Pieces for harmonium". I thus think that no misunderstanding can be perpetuated concerning the supposed title and in particular the essence of this collection: the 63 compositions were written for the harmonium. From the other hand, it is possible to argue, that Franck was able to create confusion on the score by providing the term "organ" even if the harmonium was concerned. It is the case of the arrangement of his prelude, variation and fugue. 2 As H. Graces wrote in a clear way: "Every year we sit through dozens of orchestral works and scores of violin and pianoforte solos that are longer, duller, and of infinitely less value than the pick of Franck's twelve organ pieces". The situation does not seem to have changed greatly over the years". these salons, nor did it have "a high profile among music instruments," as Near points out 3 . On the other hand, Franck was keen on composing music of larger dimensions and formal structures, fully integrating himself through his music language into the French Romantic period. Consequently, this series of small compositions may have appeared as minor, compared with the others on his list.

These are perhaps the most important causes that, although they permitted this collection of compositions to acquire some fame, did not allow their full appreciation vis-à-vis Franck's other works. I may be permitted to argue that even today, the lack of an in-depth and focused music analysis continues to prevent recognition of this collection, though the same appears to change for Franck's other works (Schneider 2008, pp. 241-244), which essentially remains hidden or out of reach of auditors, researchers and musicians. At the most, it is marginally taught inside an organ conservatory class to undergraduate students, but almost never outside the ranks of organists and their congregations.

Thus, I would like to make a modest contribution to a re-evaluation of this series of small compositions by César Franck. It is my conviction that these pieces for harmonium constitute a summa (a sort of manual) of compositional techniques and, in general, of the art of composing as perceived, professed and practiced by Franck. It should also be remembered that Franck composed this work during the last years of his life; and so I am of the opinion that they should be given the most appropriate attention because they represent the final style of Franck as a composer 4 (Roe 1980, pp. 690-691). I cannot elude the fact that a comparative analysis with Franck's other works in his last days could significantly influence the results of research on this Harmonium collection.

It would, therefore, be appropriate to conduct a parallel search with works such as Fantasia, Cantabile, Pièce Héroïque, the transcriptions for the organ, and the three Chorals. Having acknowledged this deficiency, however, may I be permitted to focus on the score without providing other comparative elements. 3 Near gives the names of some, few, well known composers that have written music for this instrument, or have arranged music for it: Hector Berlioz, Louis Vierne, Charles-Marie Widor, Camille Saint-Saens, René Vierne, Alexandre Guilmant as well as Antonín Dvořák, 4 Franck has been active until the last days of his life. The three organ chorales and their later sketches dated 30 September 1890 are a consistent proof of this fact.

A manual of composition.

The importance of these small compositions is inestimable from various points of view, but above all, they give a clear idea of what music teaching and music composition could have meant for Franck. In his progression with work he methodically presented all those elements of teaching that were important to him, in a cyclical way. Elements that concern both form and content (techniques) of composition as well as performance practice are systematically and methodically exposed. It may seem inappropriate to point out that every composer, in a personal way, always associates himself or herself intrinsically with the compositions he or she brings to light. This personal link lends different hues to one's works that are difficult to define precisely and to comprehend from the results of a music analysis. In the case of this collection, however, the attention of the composer is focused on the presentation of compositional techniques, having at his disposal elementary and reduced music forms, so as to make their didactic aspect even more evident. Franck did not want to create a music-riddle for the reader. This is why he methodically arranged different kinds of compositions as well as compositional techniques in a clear and progressive way. He intended to deliver a clear message.

It should not be necessary to thoroughly analyze the compositions in order to acknowledge that their writing style is not precisely meant for the organ (though they can be played extremely well on it and perhaps they did). Several elements are able to demonstrate this fact, for example, the absence of the pedal and of harmonically rich and consistent backgrounds, which are present on the contrary in his organ works. The instrument, Gabriel-Joseph Grenié's "expressive organ" as perfected by the Mustels (Tchorek 2007, p. 338) (August Victor during the years of César Franck and later Alphonse) was already expressive (notably because of its free reeds) and Franck did not need to enhance it through his writing. Nevertheless, despite the clear instrumental destination of this collection (as stated by the composer), we find in the score elements that concern specific performance practices-piano accompaniment, choral writing or organ invention-appropriate for other keyboard instruments (Figure 1). This assertion may be considered as an element that demonstrates the multi-faceted objective of this collection. Franck intentionally composed in different writing styles. Figure 1 presents some of the most characteristic writing styles that are exploited in the series; and among others, it is possible to find a clear pianistic writing style (accompanied melodies), contrapuntal writing style (fugato and canons) and an organ-choral style. In the later years of Franck's life, the harmonium was able to perform melodic lines (thus replacing monophonic solo instruments) and served as an accompanying instrument as well. Present in the salons and also in some of the parishes that could not afford an organ, its usefulness doubled.

Nevertheless, the questions remains, for which specific harmonium did Franck compose his works and for what purpose? It is rather impossible to answer this question as it could have given us elements to better understand the purposes of this collection which undeniably constitutes an exemplary style of composing of César Franck. A pedagogical work? A work for posterity? A work for the ordinarium? A summa of César Franck as a composer? I think that we can with some hesitation accept the synthesis of these four hypotheses as the ultimate goal of this collection. Yet, it should not be forgotten that Franck was primarily an organist and church musician and, therefore, the religious context of this collection should not be lost sight of (Grace 1922, p. 842).

The constitution of the series

The series consists of nine groups of seven compositions each, with a supplement of four manuscript compositions that were first included by Kauzinger in the 1997 edition. These groups are arranged chromatically from the tonality of C to that of Ab (Figure 2), although originally

Franck planned to compose one hundred pieces. The collection is made up of 63 compositions, which today are listed in four existing sources, and apparently Franck did not have the time to complete it. Thus, the collection remains incomplete. The series is composed of short pieces which, even if short, cannot be considered humble as stated recently (Near 2002, p. 460). Apparently, the initial drawing would have foreseen the completion of the cycle with at least all the tones of an octave, from C to B. Thus, those in A, Bb and B should be considered at least as the missing part of the collection. The construction of each group progresses in a similar way and is finalized in the last piece of each one of them (often called "offertoire" or "sortie") in which Franck uses a technique that he greatly appreciated: that of the addition of preceding elements often called "cyclisme" (Figure 3). It is true that each series takes into consideration a specific element in order to exploit it, which I would characterize as "personal." It is thus impossible for us to have a clear idea about how Franck interpreted his works, on the organ or on the harmonium (Archbold 1988, p. 59 ;Flor 1972, pp. 499-500). Although a musicological term could be attributed to this personal element (provided by music analysis), it would not always be appropriate because the evolution and development of the music discourse in the series often acquires more subjective connotations such as general mood, use of stops, intrinsic rhythmic progression, or other characteristic "piece titles" commonly in use during this period (Tchorek 2007, p. 331). However, in trying to decipher a global drawing, it is possible to acknowledge that Franck's method of composition is based mainly on three points:

form, compositional techniques and execution practice. His vision on composition is consequently global and embraces the totality of the series. That is, these three axes are taken under consideration for the entire cycle of compositions.

Each group is composed of seven compositions which find their apex in the last one, which always serves as a stand where part of what has been exposed so far (inside the series) can be reproduced, whether elaborated or not. Also, this in-group procedure does not preclude an exchange of material between different groups. Furthermore, the construction of every group is based on a different element (a specific technique, a rhythm, a melody, or a harmonic background), although within three parameters that seem to remain fixed for them: form, technique and performance. This means that Franck had a clear idea of the progression of the 63 compositions based on these three fundamental aspects. Consequently, in this arrangement of compositions we find different genres and forms (Figure 4). The method of approach for these three parameters remains unaltered, but the content (the specific technique) may differ substantially. In Figure 4, we see a synoptic array of the most recurrent elements used by Franck: the form (preestablished or invented), the compositional and elaboration techniques (on harmony, rhythm and writing style) and the performance practice (piano, organ or harmonium styles).

The compositions present simple structural forms (ABA, AB or AA) which often are applied on pre-established compositional outlines. Thus we find a constellation generated by chorals, canons, fugati, preludes, marches, and variations as well as compositions written on pre-existing melodies 5 . At the same time, the compositional techniques that are employed demonstrate an intention to create a summa, a kind of composition manual in miniature. Among the most important techniques, we find elaborations on rhythm (syncopations, off-beats) as well as harmony (double counterpoint, inversions, reversions) and, more generally, an effort towards rhetorical progression of the musical discourse through continuous transformations and combinations of elements.

Regarding the overall structure of the series, I would like to bring in as an example the second group as it is rather characteristic of Franck and represents his method of composition (Figure 5). In this group, Franck decided to work mainly the rhythmic aspect of the compositions particularly by making use of the technique of inversion and reversion, the latter being the exchange of the positioning of the voices (flipping). The seven pieces included in this group are divided into three sections (1-3, 4-6 and 7) and each one of them re-elaborates material especially from its own section. As has already been said, exception is made to the seventh and last piece which reelaborates eventually elements from all the other pieces of the group. In order to give some vivid examples, I present some elaborations as they can be found inside the b7 which represents the apex of the entire group. These examples show only some parts of the relocated material inside the b7 (Figure 6). Generally, as far the whole series is concerned, the elaboration of a common element inside a group can bring to the fore different characteristics that are not always easy to individuate as they can be subject to interpretations. Nevertheless, every group presents a clear evolution of the music discourse which in the case of the third one (in D) is based on a melodic elaboration. That is, a characteristic melodic pattern is taken up and exploited in several pieces of the group. In this way, Franck succeeds in creating an internal global unity, at the same time by elaborating a single technique: melodic invention. The examples in Figure 6 show the method of elaboration of a basic melodic pattern, which is examined through its rhythmic effects (the anapest in red signs in Figure 7) and the direction of its intervals (especially at the end of a phrase). The green signs indicate an arpeggio effect (as direction) common in c2 and c5, which could be found with a looser interpretation in c1 and c3 as well. A similar effect takes place in the group in G (h), in which the melodic evolution of its fragments creates the same result as in the third one (Figure 8), that is an evolution of the melodic fragment based on development and elaboration. Figure 8 shows in colored signs the relation among the basic melodic segments (especially the h1) inside different compositions of the same group (h).

This segment appears inverted (h1 to h4), elaborated (h1 to h2 and h3), and elaborated as well as inverted (h1 to h5). Besides the use of a specific technique in each piece or group, Franck has also elaborated the overall morphological structure of each series as a whole. These composite and sometimes complex outlines indicate the premeditation level of the composer, as they present a calculated scheme of relocation of the same material inside a single group. By these means, Figure 8 shows a simple use of this artifact, that is, an almost mere relocation (whether slightly elaborated or not) of several parts of other compositions inside the last one of the group in F (f). Apparently, the form that better allows Franck to use this relocation process is the three-part form in ABA. In the above figure ( 9), we can see the relocation of elements of preceding compositions of the group, which have been used inside the last one (the 7 th ). In this particular case, the relocation has been made to specific and predetermined parts of the last composition (ABA) as well. That is,

Franck decided what to relocate and where to relocate it. The seventh piece of the group uses an ABA form and every part of this form uses elements that can be found in preceding pieces of the same group (f4, f2, f1 in A, f3 and f2 in B and f4, f2, f1 in the final A). A thoroughly detailed representative table or figure would be needed to show the interrelation of all the elements (or patterns in a broad sense) that have been relocated inside f7, mainly because they can be found everywhere. Franck does not only relocate a specific element from a preceding composition to the final one in order to elaborate it. He also elaborates it while he advances from the first composition to the last one. By these means, a basic element (pattern), which has its origin in the first composition of the group, can be found in any other (elaborated or not) before being relocated to the last one. Thus, he succeeds in creating a continuous procedure of elaboration (evolution) inside the whole group, which gives unity to the group and the evolution of the musical discourse.

In the f group (and in order to present some characteristic examples, figure 10) the anapest rhythm accompanied by syncopation (bb. 24-25, f1) is being reintroduced in an elaborated form in f2 (the syncopation in b. 1 and the anapest in b. 2). The anapest continues as the most relevant rhythmic pattern in f3 (bb. 2-17). The syncopation and the repeated notes (found in b.1-end in f1 and b. 1-end in f3) constitute the main characteristic of all the f4. It should be observed that a more detailed analysis will certainly reveal stricter relations among the compositions. In this way, a pattern, before reaching its final destination in the seventh composition, has already been significantly elaborated during the progression of the group. That is why, even if a pattern has not yet been relocated inside the final composition of a group, some of its structural characteristics can already be found inside it (as a natural evolution of their elaboration). The actual method of relocation stands as a clear sign of repetition of a pattern; but on the other side, the continuous elaboration of a pattern can be acknowledged as more concealed. Most frequently, this kind of relocation considers harmonic modulations, melodic elaborations and interval modifications of the original patterns, though it is not uncommon for Franck to relocate a specific structure (an original pattern) without modifying it. Although it could be difficult to define precisely the terms "pattern" or "structure" that are being relocated, usually they can be designated as polyphonic environments, characteristic harmonic successions, characteristic melodic and rhythmic patterns and segments or a combination of all of them. The example of the f group (in F+/F-) constitutes a rather simple model that exploits the technique of relocation, because in other groups this same technique (of relocation) and elaboration creates structures of major complexity and premeditation.

It is the case of the g group in F#/Gb (figure 11) which stands as one of the most elaborated in the whole collection. The overall structure of the seventh composition of the group is again an ABa form, though more elaborated with the addition of a Coda. Franck creates a miniature threepart form which is based on the grand ABA' one. I believe that the didactic subjacent objective of the exploitation of the ABA' form should be considered as certain in this case and generally in the whole collection. with the addition of a final Coda. Thus, the construction of each one of its sections (ABa Coda) contains elements of the previous compositions of the group (A of g5, g2, g4, B of g4, g6, etc.).

The composer is not, however, limited to a simple relocation of elements of the previous compositions (g1-6) inside the last one, but he elaborates (transforms) them in a very significant way. More precisely in A, the elements taken from g5 and g2 are partially augmented. In B, the elements taken from g6 are transformed into a binary tempo and those used for the coda (notably of the g6) are diminished and inverted by employing also a reversion of voices. In this way, Franck not only relocated previous material in the last composition of the group, but he also focused on the following actions:

Creation of the Aba + Coda, form.

Distribution of previous elements inside the new form in a coordinated manner (5, 2, 4 -4, 6 -5, 2, 4 & 6, 4).

Choice of a specific technique for each section (Aba + Coda): Augmentation (A) -tempo modification (B) -elaboration (a) -diminution/inversion/reversion (Coda).

It should be noted that due to the small size of the pieces, the application of these compositional techniques may seem elementary. In other words, Franck does not have the space to expand these processing phenomena. They are, therefore, applied on segments and on very small elements, and the interpretation of which can sometimes be debated. Nevertheless, the highly elaborated structure of these small compositions cannot in any way be considered a coincidence even if it is subject to interpretations.

Conclusion

With this analytical approach, I hope that I have been able to give a clearer idea of the complexity and premeditation with which Franck created this work. It is worth mentioning that the entire collection of compositions is homogeneous and a factor of this unity is always a specific element in each of the groups. It is therefore difficult to define in precise terms the substance of this progression (its manifestation) without the risk of making a subjective interpretation. This unity becomes evident especially when one plays the compositions. Often, Franck examines an element (a rhythm, a phrase, a technique, or a melody) which becomes a vehicle of unity and connection within each group. In this way the compositions, although different in form and in genre, acquire a relation within the same group so as to create the impression of one composition divided into parts. Moreover, the continuous interchange of elements taken from the different compositions of each group enhances their already strengthened relation.

In order to obtain an image of this collection as a whole, I believe that attention should be given to Franck's improvisation skills. It is my opinion that many of the compositional techniques employed by Franck are essentially related to his mastery of improvisation. By this I do not mean to exclude or even minimize his premeditated compositional strokes in the series. What in musicology is defined as an elaboration of a pattern may have been for Franck a free inventiveness during improvisations (literally during playing time). The rhetorical continuity of his musical discourse (the evolution and development of all the musical elements) betrays a natural way of improvisation that sometimes may seem abrupt (especially in regard to harmony), but it always has a rhetorical direction to an end. The rhythmic evolution and development comes naturally and passes from one composition to another without insinuating a premeditated process of composition, though it always exists.

Today, we do not have sufficient information on these novel compositions for an appreciation and evaluation of this collection; and understandably, many questions about it have not yet been answered (or even posed). Among them I may mention: Which is the ultimate purpose of the collection and for which precise instrument were they composed? I believe we can admit the factas already mentioned by others-that these compositions have a place in the religious sphere, as is evident from their characteristic titles and as can be found in their sources (offertoire, sortie, or known melodies). Thus, we are not in a position to argue about the religious background of the collection. It has been documented that Franck as a musician had a religious background despite having a relative lack of knowledge of religious and polyphonic music as a composer (Schneider 2008, p. 244).

Unfortunately, I cannot affirmatively react to this statement. For Franck, that was a late composition and represented his mature compositional style. It must be mentioned that Franck remained active and a prolific composer until the last years of his life (quartet in D, sonata for violin in A, and various pieces for organ); so the hypothesis that these small compositions might represent the style of a composer in his declining years must be discarded.

The Pièces pour Harmonium is a collection of free forms based on elements of different nature, where progression and relation is manifested by the use of a precise element for each group.

Inside the collection we find a constellation of techniques and practices that follow one another, ordered by semitone. The outline of the groups inside the series and the relation among the compositions inside each group are highly premeditated; so are the techniques that are employed each time. We must not forget that among the qualities of Franck, we must also recognize those of a music teacher (having had many illustrious disciples). In my opinion, more than pedagogical, the purpose of this collection is perhaps that of a personal summa of the art of composing, left to us by the composer. I may be permitted to translate it into French as "L'Art de Toucher la Composition." However, the way the material is presented (in short, defined pieces) also adds to the collection a strong didactic connotation that cannot in any way be neglected. It seems to me highly improbable for a composer (especially an organist), after Bach, to compose a keyboard collection of pieces by semitone without necessarily alluding to an educational purpose.

Figure 1 .

 1 Figure 1. Different types of performance and composition practices.

Figure 2 .

 2 Figure 2. The structure of the series.

Figure 3 .

 3 Figure 3. Concentration of material inside the last composition of each group.

Figure 4 .

 4 Figure 4. Techniques used in the series and the three basic parameters.

Figure 5 .

 5 Figure 5. Overall structure of the second series (b).

  of material of b2 in red signs and of b6 in green signs (bb. 28-29, b7). d. Presentation of material of b4 (b. 40, b7).

Figure 6 .

 6 Figure 6. Elaboration techniques inside the b7.

Figure 7 .

 7 Figure 7. Melodic relation among different pieces of the same group (c, 1, 2, 3 and 5).

Figure 8 .

 8 Figure 8. Melodic evolution and relation among different compositions of the same group (h, 1-5).

Figure 9 .

 9 Figure 9. Elaboration of material inside a final (7 th ) ABA form, f7.

Figure 10 .

 10 Figure 10. Elaboration of material inside the f group, rhythmic evolution. Anapest and syncopation (red), repeated notes (yellow), three notes pattern (green).

Figure 11 .

 11 Figure 11. Structure of the seventh (last) composition of the g group in F#-/Gb+.

These tables provide information about the general morphological structure of each group, together with related material in regard to the tonality, main characteristics and techniques as well as the form. They do not constitute an exhaustive source and are subject to interpretation. They should therefore be considered as an analytical guide.