domain: math spi

text:
Arno Roland

Ngatcha Ndengna

A nonlinear mathematical analysis of laminated composite shell models: an existence and uniqueness theorem

Keywords: Well-posed 2D/3D linear models, Laminated composite shells(LCS), Laminate Constitutive Equations(LCE), motion equation, Variational formulation, Existence and uniqueness results AMS Subject Classification: 22E46, 53C35, 57S20 1

This paper is devoted to nonlinear analysis of a three-dimensional(3D) shell model with its associated generalized ABCDE-matrix that accounts the warping and twisting effects. The proposed constitutive relation from this model is original and does not use any empirical consideration as in some 3D shell theories recently developed in the literature(without rigorous mathematical analysis). Here, we establish a Green formula and an existence and uniqueness theorem of weak solution to the first time for a 3D laminated shells and even 2D laminated using ABCDE-matrix. The proposed results can be used to prove the existence and uniqueness of a weak solution of shell models available in literature.

Introduction

This work is devoted to present 2D/3D linear laminate constitutive equations (LCE) and variational formulations that can be used to modeling the mechanical behavior of any laminated composite shells (LCS) and boundary conditions and which we propose existence and uniqueness theorems. These new LCE have been recently obtained in [START_REF] Ngatcha | A Two-Dimensional Model to Analyze the Static and Dynamic Mechanical Behavior of Multilayered shell Structures[END_REF] and 12 (see also [START_REF] Ndengna | Two dimensional static mechanic analysis of laminated composite tube using ABCDE matriix with no correction factor[END_REF] ) using 2D and 3D kinematic shell equations "of Kirchhoff type" proposed by Nzengwa and Tagne [START_REF] Nzengwa | A two-dimensional model for linear elastic thick shells[END_REF] and Nzengwa [START_REF] Nzengwa | A 2D model for dynamics of linear elastic thick shell with transversal strains variation proceeding of 8th conference on dynamical systems and applications[END_REF] respectively. The 3D kinematic shell equations (KSE) introduce a particular admissible displacement that accounts the stretching-through-the-thickness of the shell. This admissible displacement is solution of a 3D torsion problem (resolved here in original way) and allow to consider the section warping and the variation of the thickness during the deformation. When the laminate shell become thicker a simple variation of the thickness during the deformation leads to distortion of the membrane and creates the transverse shear stress. When the LCS is plane (2D shell) it necessary to maintain the plane deformation by applied transverse force σ i3 , i = 1, 2, 3 (seen as solution PDE proposed in [START_REF] Nzengwa | A two-dimensional model for linear elastic thick shells[END_REF] . For 2D linear isotropic shell this force have been computed rigorously by Feumo et al [START_REF] Feumo | Finite element model for elasic thick shells using gradient recorvery method[END_REF] and for 2D LCS by Ngatcha et al [START_REF] Ngatcha | A Two-Dimensional Model to Analyze the Static and Dynamic Mechanical Behavior of Multilayered shell Structures[END_REF] . For 3D LCS it's not necessary to maintain the deformation in plane direction.Therefore, transverse stress components appear directly in the 3D LCE. With that, it's possible to analyze exactly the LCS using directly these LCE. Some static and dynamic exact analysis for laminated composite plates and shells have been proposed in literature. Some analysis using numerical method as finite element method (see Bernadou and Ciarlet [START_REF]Sur l'ellipticit du modle linaire des coques de W.T Koitier[END_REF] , Chapelle and Bathe 8 ) or finite difference based method as that developed by Youssoufa et al ? . The exact or numerical analysis can use either the LCE or the elasticity equation directly in spite of the fact that it is yet to be justified mathematical by an existence and uniqueness theorem. This situation has been partially remedied by Ngatcha et al [START_REF] Ngatcha | A Two-Dimensional Model to Analyze the Static and Dynamic Mechanical Behavior of Multilayered shell Structures[END_REF] in the particular case of 2D static and dynamic exact analysis of LCS without section warping and stretching-through-the-thickness. There are few static and dynamic analysis of LCS that integrate any existence and uniqueness theorem. For some linear and nonlinear shell a few existence theorems have been established. We cite linear and nonlinear elastic thin and thick shells 5 , 6 , 7 , 4 , [START_REF]Sur l'ellipticit du modle linaire des coques de W.T Koitier[END_REF] . All these results have neglected the influence of third fundamental form for the case 2D.

Existence and uniqueness theorems have been established for 2D linear elastic isotropic shells accounting the Gauss tensor effect by [START_REF] Nzengwa | A two-dimensional model for linear elastic thick shells[END_REF] . This results has been demonstrated in a different fashion in [START_REF] Ngatcha | A Two-Dimensional Model to Analyze the Static and Dynamic Mechanical Behavior of Multilayered shell Structures[END_REF] . Accounting the section warping, Nzengwa [START_REF] Nzengwa | A 2D model for dynamics of linear elastic thick shell with transversal strains variation proceeding of 8th conference on dynamical systems and applications[END_REF] (see also ? ) has proposed an existence and uniqueness of a admissible displacement. There few existence and uniqueness theorems for three-dimensional LCS model in the literature. This work address this issue by a rigorous non linear analysis in appropriate Hilbert space.

The first objective of this paper is to further extend these existence to the case where the strain deformations matrices are in L 2 (Ω) = L 2 (Ω; M) and even in H -m (Ω) = H -m (Ω; M), m > 0 by means of different and to large extend simpler, more elegant and more natural. Let Ω be a domain in R 3 and let a strain vector X ∈ Ω. The Hooke's law can be given by:

σ(X) = C (X) in Ω, (1.1)
where ∈ L 2 (Ω, M) is the strain tensor and σ ∈ L 2 (Ω, M) is the stress tensor, C ∈ L ∞ (Ω, M) is the material matrix that verify the ellipticity property. Here, M is the vectorial space of symmetric of second order tensor. Let Λ ad = {σ ∈ L 2 (Ω; M), -div(σ(X)) = f in Ω and Γ n ∂Ω (σ, p) = 0 on ∂Ω}. For the problem of non divergence-free stress tensor in a smooth domain defined in Λ ad and given by: -div(σ(X)) = f, in Ω Γ n ∂Ω (σ(X), p) = 0 where the linear mapping σ : → Γ n ∂Ω (σ, p) = σ.n| ∂Ω -p is continuous from Λ in (H 1/2 (∂Ω)) [START_REF]Sur l'ellipticit du modle linaire des coques de W.T Koitier[END_REF] and where Λ = {σ ∈ L 2 (Ω; M), div(σ) ∈ (L 2 (Ω)) [START_REF]Sur l'ellipticit du modle linaire des coques de W.T Koitier[END_REF] }.

The variational formulation in term of strain reads:

Find X ∈ D (Ω) such that Ω (X) : (Y )dΩ = L(Y ), ∀Y ∈ D (Ω). (1.3)
Here D (Ω) denotes the space of distribution defined over Ω seen as the dual of D(Ω) (the space of function infinitely differentiable and have compact supports in Ω).

The second objective is to show that some of our proposed results (existence and uniqueness)obtained via an mathematical analysis of a class of problem given above allow to reformulate in a new way a 3D torsional problem to improve the prediction of the mechanical behavior of laminated composite shells. In this work, rigorous mathematical analysis of laminated shell models are proposed for the first time in the modern literature to justify some kinematic shell equations. A such analysis will help to improve kinematic formulations for further investigation or for improve some well-known existence and uniqueness theorems which neglect several considerations. The goals of this work is first, to expose two general constitutive equations, second present some mathematical relations and new results important in the mathematical elasticity theory applied to shell theories. The rest of paper is organized as follows.

In section 2 we present some notations, mathematical tools and geometry of shell. Section 3 expose some new results for the 3D shell model. We prove here an existence of admissible displacement by solving a 3D torsional problem. Section 4, we present two new constitutive relations for laminated composite shells. The advantages of these relations are clearly exposed. In section 5, we propose variational formulations for laminated shell models using these constitutive relations. We prove an existence and uniqueness of a variational solution. We establish existence and uniqueness theorems to well justify the developed constitutive relations. Finally in section 6, we propose Green formula for laminated thick shell models. We conclude in section 7.

Mathematical tools and geometry of shells

Shells are three-dimensional bodies bounded by two, relatively close, curved surfaces. In general, the distance between those surfaces is small compared with other shell parameters. This subsection presents geometry of laminated shell necessary to understand the rest of paper.

Mathematical tools and preliminaries

Let Ω be an open subset of R N and let X = (x i ) designate a generic point in Ω; the α is a multi-index (an n-tuple of non negative integers, α i . The length of α is given by |α| := n i=1 α i . We identify the three-dimensional Euclidean space with R 3 and | • | its Euclidean norm. Let Ω ⊂ R 3 denotes a shell and S = ϕ(ω) ∈ R 3 its mid-surface; where ω ∈ R 2 is a connected bounded open set and ϕ ∈ C 2 (ω, R 3 ) is an immersion. The space of all indefinitely derivable functions ϕ : Ω → R with compact support contained in Ω is denoted D(Ω) and the space of all the distributions overs Ω is denoted D (Ω). We denote L p (Ω) and W m,p (Ω), ∀m ≥ 1, p ≥ 1 the Lebesgue and Sobolev spaces. The Sobolev spaces also noted H m (Ω), is defined to be the dual space (H -m (ω)) with norm given by the dual norm. Note that different dual spaces can be used to define negative index Sobolev spaces, in particular, it is frequently useful to use the dual of a subspace of

∂
H -m (Ω). If the boundary of Ω is lipschitpz-continuous and if Γ 0 ⊂ ∂Ω, is a non-zero subset of the border of Ω such that Γ 0 = γ 0 ∪] -h 2 , h 2 [. W 1,p γ0 (S) := {v ∈ W 1,p (S), v = 0onγ 0 }, W 2,p γ0 (S) := {v ∈ W 2,p (S), v = ∂ ν v = 0onγ 0 } (2.
1) where ∂ ν denote the outer normal derivative operator along ∂Ω and

∂ ν = ν i ∂ ∂xi . If p = 2, we adopt the following notations H m (S) = W m,2 (S), H 1 γ0 (S) = W 1,2 γ0 (S); H 2 γ0 (S) = W 2,2
γ0 (S). In the same sense let us:

H m (Ω) = W m,2 (Ω), H 1 Γ0 (Ω) = W 1,2 Γ0 (S), H 2 Γ0 (Ω) = W 2,2 Γ0 (Ω) (2.2)
where

W 1,p Γ0 (ΩΩ) := {v ∈ W 1,p (S), v = 0onΓ 0 }, W 2,p Γ0 (Ω) := {v ∈ W 2,p (S), v = ∂ ν v = 0onΓ 0 }. (2.
3) The vector fields and symmetric matrix fields defined over A = Ω, ω are represented by italic capitals, Boldface italic capitals and special Roman capitals. The notations C m (Ω) ,0 ≤ m ≤ ∞, denote the usual spaces of continuous differentiable functions. The notations C m (Ω), m ≥ 0 and C ∞ (Ω) denote the space formed by the restrictions the closure Ω of all the function in the space C m (R n ) and C ∞ (R n ). The notation H m (Ω) for negative integers m, with H 0 (Ω) = L 2 (Ω), designate the usual Sobolev spaces. The notation in special roman capital L 2 (w), L 2 (Ω), H 1 0 (Ω), H -1 (Ω), D (Ω) denotes the spaces of symmetric matrix fields defined over Ω. Combining the above rules, we shall thus encounter spaces such as H 1 0 (ω), H 2 0 (ω), H m (ω), with H 0 (ω) = L 2 (ω). In the same sense,for any integer n ≥ 0 the symbol M n , S n , M -1 respectively designate the space of all square matrices of order n, the space of all symmetric matrix, the set of all matrices A ∈ M n with det(A) = 0. We have M n ⊂ S n . The Lebesgue integral of a dΩ-measurable nonnegative function f :

Ω → R is defined in R +∞ by Ω f dΩ = h 2 -h 2 S f dSdz. ( 2

Compact inclusions

The following results are essential to prove some estimate errors and convergence results in the next. We have the following compact inclusions:

H s+1 (Ω) ⊂ H s (Ω) ⊂ H -1 (Ω), s ∈ [0, 1].
(2.5)

Moreover, ∀s ∈ [0, 1] we have the following embedding following:

H s+1,1 (Ω) → H s,1 (Ω); L 2 (0, T ; H 1,1 (Ω)) → L 2 (0, T ; L 2 (Ω)) (2.6)

Geometry of shells: concept and preliminary results

Greek indices and exponents vary in the set {1, 2}, Latin indices and exponents vary in the set {1, 2, 3}, and the summation convention for repeated indices and exponents is used in conjunction with these rules. • is the scalar product and f ,α = ∂f ∂α (∂ is partial differential operator); the cross-product symbol × denotes the vector product. A generic point in R 3 is denoted X = (α, β, z). A generic point in R 2 is denoted x = (α, β). The tangent space T x R 3 to R 3 at any given X ∈ R 3 is identified with R 3 . The tangent space T x S to the surface S at any given x ∈ R 2 is identified with the plan parallel to T x S and passing through the origin of R 3 . The position of an arbitrary point within the shell medium is defined by curvilinear principal co-ordinates α and β upon the middle surface S or reference surface r(α, β), and z directed along the out-ward normal n(α, β). The mapping R : Ω → R 3 is defined by:

R(X) = r(x) + h(x) 2 ζn(x) at each X = (α, β, z) ∈ Ω (2.7) is a C 1 -diffeomorphism from Ω onto its image R(Ω) ⊂ R 3 . In Eq. (2.7), ζ = 2 h(x)
z with ζ ∈ [-1, 1] and h(x) is the total thickness of the shell.

Generally, r(x) is defined on fixe system as:

r(x) = r 1 (x)e 1 + r 2 (x)e 2 + r 3 (x)e 3 in ω (2.8)
where e 1 , e 2 , e 3 are the unit vectors of the global reference system (O, x 1 , x 2 , x 3 ). We note C m (ω, Y), m > 0 is the space of Y-valued fields with components in C m (ω). We can now defined the normal vector and the tangential vector a α , a β by using an immersion from ω into R 3 . We recall that an immersion from ω into R 3 is a mapping Ψ ∈ C 2 (ω, R 3 ) such that ∂ α Ψ : ω → R 3 are two tangential vector fields linearly independent at each point of ω. The image of ω by Ψ is noted Ψ(ω) and we have Ψ(ω) ⊂ R 3 . Given any curve γ ∈ C 2 (ω, R 3 ) of S, we defined the vectors fields a α = a α (Ψ) := ∂ α Ψ : S → R 3 .

We define the first fundamental form a αβ (Ψ) of the reference surface S. At each point x ∈ S, the bilinear form a αβ : T x S × T x S → R is given by:

a αβ (Ψ) = ∂ α Ψ.∂ β Ψ (2.9)
with a α .a β = δ α β in ω. We defined also the field vector normal to S noted a 3 : S → R 3 given by:

a 3 = a 3 (Ψ) := ∂ 1 Ψ × ∂ 2 Ψ |∂ 1 Ψ × ∂ 2 Ψ| ∈ C 1 (ω, R 3 ). (2.10) (a α (Ψ), a β (Ψ), a 3 (Ψ)) is a covariant basis.
Note that a αβ (Ψ) denotes the covariant component of the metric tensor of the deformed reference surface. In the same sense, we defined at each point x ∈ S the curvature tensor b .. :

T x S × T x S → R, also called the second fundamental form. We let b ∈ C 1 (ω): b = b αβ (Ψ) := ∂ αβ Ψ.a 3 (Ψ) (2.11)
We can deduce the normal vector n(x) by:

n(α, β) = r ,α × r ,β a α a β ∈ C 2 (ω, R 3 ) (2.12)
Finally using the second fundamental form of r(x), the principle radii of curvature can be evaluated as:

R α (x) = - r ,α .r ,α r ,αα .n R β (α, β) = - r ,β .r ,β r ,ββ .n (2.13)
We denote the mean curvature and Gauss curvature, respectively by:

H = 1 R α + 1 R β ∈ C 0 (ω); K = 1 R α 1 R β ∈ C 0 (ω) (2.14)
The third fundamental form of the surface c αβ (Ψ) is defined at each point x ∈ S as the bilinear form c .. : T x S × T x S → R defined by:

c αβ (Ψ) = a ργ (Ψ)b αγ (Ψ)b ρβ (Ψ) (2.15)
Previously, we have defined covariant components of the first fundamental form, second fundamental form and those of third fundamental form along the surface Ψ(ω). Given an immersion Ψ considered as fixed, we let (for brevity).

a 3 := a 3 (Ψ), c αβ = c αβ (Ψ), b αβ = b αβ (Ψ), c αβ = c αβ (Ψ) (2.16)
and a := a(Ψ) and, we let

e αβ = 1 2 (a αβ (Ψ) -a αβ ) (2.17) K αβ = b αβ (Ψ) -b αβ (2.18) Q αβ = 1 2 (c αβ (Ψ) -c αβ ) (2.19) (2.20)
respectively the change of first fundamental form, change of second fundamental and third fundamental form between Ψ(ω) and Ψ(ω).

Further, we have the Weingarten-Gauss relations

∂n ∂α = a α R α (no sum on α) (2.21)
we recall the Codazzi-Menarddi condition:

( a α R α ) ,β = a α,β R β , and ( a β R β ) ,α = a β,α R α , (2.22)
where α, β = 1, 2. By differentiation Eq. (2.7) we have the vector fields

g i ∈ C 2 defined at each X = (α, β, z) ∈ Ω g α = R ,α = r ,α +z ∂n ∂α = a ,α +z a α R α = (1+ z R α )a α , g 3α = g 3α = 0 in Ω, (2.23) (a α , a β , a 3
) and (a α , a β , a 3 ) denote the covariant and contravariant basis of midsurface and (g α , g β , g 3 ) and (g α , g β , g 3 ) denote the covariant and contravariant basis of the shell and det(g α , g β , g 3 ) > 0. generally, we have the Three-dimensional metric tensor g ij :

g αβ = µ λ α µ ρ β a λρ , g αβ = (µ -1 ) α λ (µ -1 ) β ρ a λρ , g 3α = g 3α = 0, g 33 = g 33 = 0, in Ω.
(2.24) Here the concept of mixed tensor of surface µ α β which allows the passage between the base of vectors defined on the middle plane and that defined at any point of the shell by µ α β = δ α β -zb α β , where δ α β is Kronecker's symbol; b α β ∈ C 0 (ω) is mixed curvature tensor of un-deformed reference surface defined by b α β = a αρ b ρβ with b ρβ ∈ C 0 (ω) denote curvature tensor components. µ α β is nonsingular. Its unique inverse (µ -1 ) α β satisfying (µ -1 ) ρ β µ α λ = δ α β . We can also expressed g αβ by

g αβ = a αλ ∞ p=0 (1 + p)(b p ) β λ z p , (b 0 ) β λ = δ α β , (b n ) β λ = 0, for n < 0. (2.25)
We have also the Lam parameters

A α = R ,α R ,α = (1 + z R α )a α ; A β = R ,β R ,β = (1 + z R β )a β A 3 = 1 (2.26)
In next, we will assume that the shell is at first order i.e. (1

+ z R α ) ≈ 1 and (1 + z R β ) ≈ 1 and in this case we set A α ≈ a α and A β ≈ a β .
A general differential line-element length dS (z) given by:

(dS (z) ) = (dS (z) α ) 2 + (dS (z) β ) 2 + (dS (z) z ) 2 = (A α dα) 2 + (A β dβ) 2 + (A 3 dz) 2 (2.27)
We define also elementary volume of the shell as follows:

dV = A α A β dαdβdz = (1 -2zb α α + z 2 det(b α β ))dSdz = ρ(α, β, z)dSdz (2.28)
where we set ρ(α, β, z)

= (1 -2zb α α + z 2 det(b α β )).

Space vectors covariant derivatives

A vector field can be expressed component wise indifferently in g i -basis or a i -basis as follows:

U = u i (X)g i = u i (X)a i ,
(2.29)

u α = (µ τ α ) u τ , u α = (µ τ α ) -1 u τ , where u i (X)a i : ω → R 3 , u i (X) : ω → R.
The calculation of partial derivative of space vector U leads to

U, i = U i | i g j with U i | j = U i,j -Γ p ij U p (2.30)
where U i | j is the 3D covariant derivative of contravariant components and where Γ p ij is the 3D Christoffel's symbol given by: Γ p ij = g p .g i,j andΓ pij = g p .g i,j .

(2.31)

Here, g p = (µ -1 ) p λ a λ and g α,β = µ ν α a ν,β -zb ν α,β a ν . So we have Γ γ αβ = g α,β .g γ = (µ -1 ) γ ν [Γ ν αβ -z(∇ β b ν α + Γ ν αβ b ν λ )]. (2.32)
We can define the relation between the 3D and 2D Christoffel's symbols by:

Γ γ αβ = Γ γ αβ + (µ -1 ) γ ν ∇ β µ ν α ∈ C 0 (w, R 3 ) ∇ γ Γ α αβ = ∇ β Γ α αγ .
(2.33)

We have also the Christoffel's symbols 3D follows:

Γ α β3 = -(µ -1 ) α λ b λ β , Γ β β3 = (µ -1 ) β λ b λ β , Γ 3 αρ = µ γ α b γρ , Γ α α3 = Γ β β3 .
(2.34)

In local base (g α , g 3 ), we have for each layer k = ( ) k ij (g i ⊗ g j ). The covariant derivation on the domain of laminated Ω denoted by | j .

Strain tensor

By definition the strain tensor or deformation tensor = ( ) ij defined as the small perturbations hypothesis apply on displacements field vector yields (

) ij = 1 2 (U i | j + U j | i )
; the strain tensor for each layer k (using the curvilinear coordinates α and β) shall be of great importance ? :

αα = µ λ α ( u λ,α + A λ,β A β u β -b λα u 3 ) where µ λ α =    1 - z R α 0 0 1 - z R β    (2.35)
The shear strain components α = β is:

2 αβ = (U α/β + U β/α ) = [µ λ α (∇ β u λ -b λβ u 3 ) + µ λ β (∇ α u λ -b λα u 3 )] (2.36) April 20, 2023 16:59 WSPC/INSTRUCTION FILE Shell˙Nonlinear˙M3AS˙2023
The transverse shear deformation α3 with α = 1, 2, are :

2 α3 = µ λ α u λ,3 + u 3,α + b λ α u λ = µ λ α (A λ u λ ) ,3 + ( u 3,α + b λ α A λ u λ ) (2.37)
the normal transverse strain component is

33 = U 3/3 = u 3,3 ; for each u 3 ∈ T x S (2.38)
So the equations are affected by the choice of the coordinate system, the material and curvature The stress tensor σ ∈ M written as

σ = σ αβ g α ⊗ g β = σ αβ g α ⊗ g β (2.39)
Using Christoffel symbol we have:

σ αβ | λ = ∂ αβ σ + Γ α λγ σ γβ + Γ β λγ σ αγ (2.40) σ αβ | αβ = ∂ α (σ αβ | β ) + Γ α αλ (σ αβ | β )

Shell models versus laminated composite formulation

The shell kinematic used here is formulated with the hypotheses of composite materials.

Assumptions

Let assume in addition for a current modeling purpose that: (i) Each constituent layer has its own geometrical and physico-mechanical characteristics; (ii) the material of each constituent layer is linearly elastic and anisotropic;

(iii) the layers are in perfect bond, no slip between two adjacent laminae may occur;(iv) the displacement field function in the thickness direction is assumed to be constant or linearly varying the throughout the thickness; (v) the mid-surface of the laminated and the each laminae is bounded and sufficiently smooth for all subsequent computations. (vi) The perfect-bonding conditions for the layers are formulated as follows: the stresses σ(x, y, z) and displacements U i (x, y, z) are continuous at the interface between the (k -1)th and kth layers:

U k i | Γ k-1,k = U k-1 i | Γ k-1,k , [σ] k n i,k = -[σ] k n i,k-1 . (3.1)
where Γ k-1,k (see Fig. (??)) is the interface between the layer k -1 and k; n i,k (respectively n i,k-1 ) is the normal of layer k(respectively of layer k -1) along the direction i. We note that n i is the components of the normal n to the interface along the direction i where n is the normal of domain Ω k given by

n = (n α , n β ) = (cos(ϕ α ), cos(ϕ β )) (3.2)
where ϕ α and ϕ β are the angles between the normal n and the directions α and β, respectively.

In this work, we present a 3D model that introduces a particular from of admissible displacement obtained as a solution of a torsional loading problem in 3D solid. The model accounts all the components of transverse deformation and transverse stress without use any empiric consideration as in many 3D shell models available in the literature. Let us φ = (φ α , φ 3 ) the total field of transverse deformation where φ α = 2 α3 α = 1, 2; and we φ 3 = 33 deduce by limit analysis that rot(φ) = 0. So we have: φ α = ∇ α q = 2 α3 and φ 3 = ∇ 3 q = 33 .

Existence of a potential distortion and kinematic equations

Mid-surface displacement u(x, y, z) is computed in order to satisfy the plane strain state and the change of surface along the thickness U α (x, y, z) = µ λ α u λ where µ ρ γ = δ ρ γ -zb ρ γ . In this case 2 α3 = 0 and u 3 does not depend on z-parameter. The general solution of Equation 2 α3 = 0 is found in Nzengwa et al ? and consists to find u α in the base (a α , a 3 ) that satisfies Equations (3.3)

u γ (x, y, z) = µ ρ γ u ρ (α, β) -zu 3,γ ; for each(u ρ , u 3 ) ∈ (T x S) 2 u 3 (α, β, z) = u 3 (α, β) (3.3)
We note that the transverse strains is given by

α3 = 1 2 (µ ρ α u ρ,3 + ( u ρ,3 + B ρ α u ρ ))
Let us φ = (φ α , φ 3 ) the total field of transverse deformation where φ α = 2 α3 α = 1, 2 and where φ 3 = 33 .

Proof of existence of a admissible displacement q(x, y, z)

Admitting that rot(φ) = 0, there exist a function q(x, y, z) whose derive φ and in this case, we have:

φ α = ∇ α q = 2 α3 , and φ 3 = ∂ 3 q = 33 . (3.4)
We recall that

U α (x, y, z) = (µ) β α u α (x) -zu 3,α (x), in Ω, U 3 (x, z) = u 3 (x), (3.5)
We multiply the first equation of (3.5) by (µ -1 ) β α and we obtain:

(µ -1 ) α β U α (x, y, z) = u β (x) -z(µ -1 ) α β u 3,α (x). (3.6)
We derive with respect to z and we get:

(µ -1 ) α β U α (x, y, z) = -z(µ -1 ) α β U α,3 (x) = -(µ -1 ) ρ β (µ -1 ) α ρ ( u 3 ) ,z +(µ -1 ) ρ β (µ -1 ) α ρ q ,α , (3.7
(z(µ -1 ) α β u 3,α ) ,z = (µ -1 ) ρ β (µ -1 ) α ρ u 3,α -(µ -1 ) ρ β (µ -1 ) α ρ ( u 3,z ) ,α . (3.8)
Due to fact that u 3 = ∂ 3 q = ∂ z q, we have ( u 3,z ) ,α = q ,α . Therefore, q(x, y, z) satisfies the following additional system of equations:

(µ -1 ) ρ α u α (x, y, z) ,z + (µ -1 ) ρ β (µ -1 ) α ρ ∂ α ( u 3 -q) = 0 ( u 3 ) ,z = q ,z (3.9)
The last equation of (3.9) implies the existence of normal displacement depending only of x. Noting u 3 this displacement, we get:

(( u 3 ) -q) ,z = 0 ⇒ u 3 (x, y) = u 3 -q. (3.10)
With (3.9) and (3.10) we have proven the existence of admissible displacement q (stretching function) by using a 3D torsion problem. It's possible to write a 2D kinematic with through-the-thickness variable.

Kinematic equations

The stretching function introduce the transverse displacement and according to Nzengwa ? , the static kinematic equations read:

U α (x, z) = u α (x) -zθ α (x) + z 2 Φ α (x) U 3 (x, z) = u 3 (x) + q(x, z) (3.11)
Here, q(x, z)) is the stretching displacement through the thickness. The plan rotation

θ α = θ ρ (u α ) = (u 3,α + 2B ρ α u ρ ) :      θ 1 = 2u 1 R 1 - u 3,x A 1 θ 2 = 2u 2 R 2 - u 3,y A 2 (3.12)
and where

Φ α = (B ρ γ B γ α u ρ + B ρ α u 3,ρ
) is Gauss rotation given by:

     Φ 1 = 1 R 1 (b τ 1 u τ + θ 1 ) Φ 2 = 1 R 2 (b τ 2 u τ + θ 2 ) (3.13)
Explanation of the term q(x, z)

This term is a particular admissible displacement and can be expressed as a function (non polynomial or polynomial) in z:

q(x, z) = w(z)q(x). (3.14) where w(z) is the transverse distribution function. The term q(x, z) is the stretching-through-the-thickness and provides the transverse normal deformation/stress contributions in the laminated composite shells. More generally, we have:

q(x, z) = q(x, 0) + zq 1 (x) + z 2 q 2 (x) + ........ + z n q n (x). (3.15)
When z = 0, we find q(x, z) = 0 and we retrieve the model proposed in Nzengwa and Tagne ? . the K-L based kinematic or FSD based models. The order of this term determine the refinement of 3D exact solutions of the composite shell problems. In first order, we read q(x, z) = zq(x) = w(z)q(x) where w(z) = z.

(3.16)

In this paper, we take q(x, z) at the first order.

Strain-Displacement Field

The stress-strain relationship for a typical k -th lamina in a laminated composite shell made of N laminas (1 ≤ k ≤ N ) as shown in Fig. ?? on page ?? is written in local basis by Hooke equation:

σ k ij = [Q(θ k )] k ij (3.17)
where σ k ij , k ij , [Q(θ k )] represent the 3D stress tensor, 3D strain tensor, and 3D reduced stiffness matrix, respectively. The 3D strain tensor components are given by equation ((3.18) or (3.19)) see ? , [START_REF] Ndengna | Two dimensional static mechanic analysis of laminated composite tube using ABCDE matriix with no correction factor[END_REF] .

( U ) =        αβ (U ) = e αβ ( u) -zK αβ ( u) + z 2 Q αβ ( u) + qΥ αβ α3 (U ) = 1 2 (µ ρ α u ρ,3 + ( u ρ,3 + B ρ α u ρ )) 33 (U ) = U 3,3 (3.18)
( U ) =                  αβ (U ) = e αβ ( u) -zK αβ ( u) + z 2 Q αβ ( u) i3 (U (η)) = 0 = η i3 αβ (U (q)) = q(x, z)Υ αβ = q α3 (U (q)) = 1 2 ∂ α q(α, z) = 1 2 φ α 33 (q) = ∂ 3 q = φ 3 (3.19)
where Υ αβ = -(µ ν α b νβ + µ ν β b να ), and q(x, y, z) = w(z)q(x, y) Finally the total strain tensor deformation which accounts the transverse shear deformation is given by: For orthotropic material the stress-strain constitutive relations is given by Equation (3.21):

ij ( U ) = ij (U (η)) + ij (U (q)) = η + q . ( 3
σ =        σ αβ = [Q][e αβ ] -z[Q][K αβ ] + z 2 [Q][Q αβ ] + q[Q][Υ αβ ], σ α3 = 1 2 [Q][φ α ], σ 33 = [Q][φ 3 ], (3.21)
where e αβ ( u), K αβ ( u), Q αβ ( u) and qΥ αβ define respectively, the membrane deformation tensor, the change of curvature tensor, the change in third fundamental form (or gauss curvature tensor) and the stretching tensor. We have:

e αβ ( u) = 1 2 (∇ β u α + ∇ α u β -2u 3 b αβ ); (3.22) K αβ ( u) = ∇ α b ρ β u ρ + b ρ α ∇ β u ρ + b ρ β ∇ α u ρ + ∇ α ∇ β u 3 -b τ α b τ β u 3 ; 2Q αβ ( u) = b ρ α ∇ β b τ ρ u τ + b ρ α b τ ρ ∇ β u τ + b ρ α ∇ β ∇ ρ u 3 + b ρ β ∇ α b τ α u τ + b ρ β b τ ρ ∇ α u τ + b ρ β ∇ α ∇ ρ u 3 ; Υ αβ = - 1 2 µ τ α b τ β + µ τ β b τ α
The terms Q αβ ( u) (that is the product of the curvature tensor and the variation of the rotational vector) and Υ αβ disappear usually in some 3D laminated shell models recently developed in the literature. We remark that these tensors, referred respectively to hereafter as the Gauss deformation tensor and the warping section tensor.

Proposition 3.1. The following relation is hold:

e αβ , K αβ , Q αβ ∈ L 2 (ω) if (u α , u 3 , q) ∈ (H 1 (ω)) 2 × H 2 (ω) × H 1 (ω). Moreover, e αβ ∈ H -1 (ω), K αβ ∈ H -2 (ω), Q αβ ∈ H -2 (ω), [qΥ αβ ] ∈ H -1 (ω) if (u α , u 3 , q) ∈ (L 2 (ω)) 2 × L 2 (ω) × L 2 (ω).
Theorem 3.1. Let there be given a domain ω ⊂ R 2 with boundary γ 1 ⊂ ∂ω non-zeros measure and an immersion φ ∈ C 2 (ω, R 3 ); and let a vector field U = (u α , u 3 , q) ∈ H(ω

) := (H 1 (ω)) 2 × H 2 (ω) × H 1 (ω) such that e αβ ( u) = K αβ ( u) = Q αβ ( u) = 0 in ω and u i = q = θ α = Φ α = 0, then u α = u 3 = q = 0.
Theorem 3.2. Let there be given three tensors fields e = e αβ ∈ L 2 (ω);

K = K αβ ∈ L 2 (ω); Υ = Υ αβ ∈ L 2 (ω); Q = Q αβ ∈ L 2 (ω) then exists a vector u = (u α , u 3 ) and a function q, with x = (x 1 , x 2 ) → ( u(x), q(x)) ∈ R 2 × R 2 with (u α , u 3 , q) ∈ (H 1 0 (ω)) 2 × H 2 0 (ω) × H 1 0 (ω) such that e αβ ( u) = e αβ , K αβ ( u) = K αβ , Q αβ ( u) = Q αβ Υ αβ , φ α ∈ L 2 (ω) if and only if ω N αβ e αβ + M αβ K αβ + M * αβ Q αβ + S αβ [qΥ αβ ] + S i3 φ i ds = 0, i = 1, 2, 3.
that satisfies d α (N, M, M * , S, S i ) = 0 in H -1 (ω), d 3 (N, M, M * , S, S i ) = 0 in H -2 (ω).

New constitutive relations for anisotropic shells

In this section, we presents new general 2D and 3D laminate constitutive equations which most appropriate to analyze the mechanical behavior of laminate composite plates and shells. This novel constitutive relation take into account the effect of curvature and geometric parameters of shell. The proposed constitutive relations is one of more general existing in the literature and extend the previous constitutive relation used in some work without clear justifications. These LCE give the possibility to study some problems related at the static and dynamic behavior of composite structure.

Best Hooke's law for laminated composite shells

The following Equation is the most general(anisotropic) possible for each lamina

σ ij k = L ijkl k kl , (4.1)
where i, j = x, y, z are the problem physical coordinates. and where

L ijkl k = [Q(θ k )]G ij G kl
satisfying the ellipticity condition i.e. there exist a constant a > 0 such that for any symmetric tensor χ we have L ijkl χ ij χ kl ≥ aχ kl χ kl . For two-dimensional laminated composite shell, the Hooke's law reads:

σ αβ k = [Q(θ k )]A αβ A ργ ργ = L αβργ k (x, z) ργ (4.2)
σ k αβ , αβ , represent the stress tensor for each layer and the strain tensor respectively.

σ αβ = C αβγδ γδ + C αβγ3 γ3 + C αβ33 33 (4.3)
We will express the tensors γ3 and 33 in function of γδ and we get:

γ3 = H γα C α3γδ γδ + H γ3 C αβγ3 γδ (4.4) 33 = H 3α C α3γδ γδ + H 33 C 33γδ γδ +
where

H γα = (2C γ3α3 ) -1 , H γ3 = (2C γ333 ) -1 , H 33 = (C 3333 ) -1 .
Here γ3 and 33 are obtained by assuming that the plane stresses i.e. σ 33 = 0, σ α3 = 0. In plane stresses, associating Eq.( 4.3) and Eq.(), the stress tensor σ αβ given in (??) becomes:

σ αβ = L αβργ (x, z) γρ , (4.5)
where

L αβργ (x, z) = C αβγδ -2C αβγδ (2C γ3α3 ) -1 C ρ3γδ (2C γ333 ) -1 -C αβ33 (2C γ333 ) -1 C ρ3γδ + C 3333 ) -1 C 33γδ (4.6) L αβργ k (x, z) = [Q(θ k )
]a αβ a ργ ρ(x, z) the tangent modulii tensor(taking into account geometry parameter of shell) where [Q(θ k )] is the reduced stiffness matrix [START_REF] Ndengna | Two dimensional static mechanic analysis of laminated composite tube using ABCDE matriix with no correction factor[END_REF] . For the first order approximation L αβργ k (x, z) = L αβργ k (x, 0). According to Hooke's Law, we can take:

L αβργ (x, 0) = a αβ a γρ [Q] - I Q 66 (4.7)

2D laminate constitutive equations (2D LCE): ABCDE-matrix

The 2D strain tensor can be presented further as follows ( αβ ) = ( 11 , 22 , 2 12 ) t the components are given in Eq. (??) or Eq. (??). To combine the lamina stiffness, it is necessary to invoke the definition of forces and moment resultants N, M, andM * for each layer as integral of stress through the thickness of the lamina.

(N αα , M αα , M * αα ) = 1 a β N L k=1 h k h k-1 A 2 ρ(X)(1, -z, z 2 )σ αα dz (4.8) (N ββ , M ββ , M * ββ ) = 1 a α N L k=1 h k h k-1 A 1 ρ(X)(1, -z, z 2 )σ ββ dz (N αβ , M αβ , M * αβ ) = 1 a β N L k=1 h k h k-1 A 2 ρ(X)(1, -z, z 2 )σ αβ dz; (N βα , M βα , M * βα ) = 1 a α N L k=1 h k h k-1 A 1 ρ(X)(1, -z, z 2 )σ αβ dz
More generally, we have:

(N nn , M nn , M * nn ) = 1 a n N L k=1 h k h k-1 A n (1, -z, z 2 )σ nn dz, n = α, β. (4.9)
where N = N αα , N ββ , N αβ , N βα t is the vector resultant forces; M = M αα , M ββ , M αβ , M βα t is the vector of resultant moment; M * = M * αα , M * ββ , M * αβ , M * βα t is the vector of resultant Gauss moment; a α , a β , A 1 , A 2 are defined in Eq. (2.9) and Eq. (2.26) respectively. In Eq. (4.8), we consider the effect of curvatures and geometric parameters in the evaluation of the stress resultants. The classical R-M approach neglect these effects. In deriving the equations of motion, we have assumed that where

N αβ = N βα , M αβ = M βα , M * αβ = M * βα
Ā = Āij = A ij + K α B ij , B = Bij = B ij + K α C ij , (4.11)
C = Cij = C ij + K α D ij , D = Dij = D ij + K α E ij , Ē = Ēij = E ij + K α T ij and where K α = C
Rα with C the tracer and R α the radii of composite shells.

3D Laminate Constitutive Equations (3D-LCE): generalized ABCDE-matrix

The force and moment resultants are obtained by integrating the stresses over the shell thickness as in [START_REF] Ngatcha | A Two-Dimensional Model to Analyze the Static and Dynamic Mechanical Behavior of Multilayered shell Structures[END_REF] . The stress resultant equations extend those obtained in previous work of [START_REF] Ngatcha | A Two-Dimensional Model to Analyze the Static and Dynamic Mechanical Behavior of Multilayered shell Structures[END_REF] (see also [START_REF] Ndengna | Two dimensional static mechanic analysis of laminated composite tube using ABCDE matriix with no correction factor[END_REF] ) and read:

           N M M * S S 1z S 2z S 3z            = B =            A -B C S 1 [0] 6×6 [0] 6×6 [0] 6×6 -B C -D S 2 [0] 6×6 [0] 6×6 [0] 6×6 C -D E S 3 [0] 6×6 [0] 6×6 [0] 6×6 S 1 S 2 S 3 S 4 [0] 6×6 [0] 6×6 [0] 6×6 [0] 6×6 [0] 6×6 [0] 6×6 [0] 6×6 S 4 [0] 6×6 [0] 6×6 [0] 6×6 [0] 6×6 [0] 6×6 [0] 6×6 [0] 6×6 S 4 [0] 6×6 [0] 6×6 [0] 6×6 [0] 6×6 [0] 6×6 [0] 6×6 [0] 6×6 S 5                       e K Q γq ∂ 1 q ∂ 2 q q            (4.12)
The above equations given by (4.15) is the 3D laminate constitutive relations or Laminated Fundamental Law for three-dimensional thick elastic shells in first approximation. Here the effect of geometric of shell do not take into account. This methodology is also use for laminated plates in ? and in more recent book and papers available in the literature. The laminate stiffness or composite-shell stiffness coefficients A, B, C, D, E, S 1 , S 2 , S 3 , S 4 , S 5 are defined by:

H l = 1 l N b k=1 [Q ij ] k (h l k -h l k-1 ), 1 ≤ i, j ≤ 6. (4.13)
More generally each composite-shell stiffness coefficients can be written in first approximation:

Z = N b k=1 [Q ij ] k h k h k-1 z l w (t) (z)dz ; l = 0, 1, 2, 3, 4; t = 0, 1, 2, ...., n; (4.14)
where the terms [Q ij ] are the elastic stiffness coefficients for the material. The strain have been obtained by a complex inverse calculation and we get [START_REF] Arno | New threedimensional Mathematical Modellings to Analyse of Mechanical behavior of Laminated Anisotropic Elastic Shell (LCES) according to First order Shear Deformation Theory[END_REF] :

           e K Q qΥ ∂ 1 q ∂ 2 q q            = B =            A 11 B 11 C 11 S 1 1 [0] 6×6 [0] 6×6 [0] 6×6 E 11 F 11 G 11 S 1 2 [0] 6×6 [0] 6×6 [0] 6×6 I 11 J 11 K 11 S 1 3 [0] 6×6 [0] 6×6 [0] 6×6 S 1 4 S 1 5 S 1 6 S 1 7 [0] 6×6 [0] 6×6 [0] 6×6 [0] 6×6 [0] 6×6 [0] 6×6 [0] 6×6 S 1 8 [0] 6×6 [0] 6×6 [0] 6×6 [0] 6×6 [0] 6×6 [0] 6×6 [0] 6×6 S 1 9 [0] 6×6 [0] 6×6 [0] 6×6 [0] 6×6 [0] 6×6 [0] 6×6 [0] 6×6 S 1 10                       N M M * S S 1z S 2z S 3z            (4.15)
All the coefficients of the above matrix are available in [START_REF] Arno | New threedimensional Mathematical Modellings to Analyse of Mechanical behavior of Laminated Anisotropic Elastic Shell (LCES) according to First order Shear Deformation Theory[END_REF] .

Simple exact analysis

Simple 2D and 3D exact analysis of the mechanical behavior of the laminated shells can consist to solve a problem of the form:

U = A 2D,3D F, (4.16)
where U is strain components vector, F is the vector components force, A 2D are the classical ABCDE-matrix (for 2D materials) and where A 3D is the generalized ABCDE-matrix (for 3D materials). The matrices A 2D,3D will be the essential key to prove the existence and uniqueness of a variational solution. We will establish an existence and uniqueness theorem in term of strain components. This is different of the existence theorem presented in 9 in term of displacements.

Variational Problem for static case

We defined the following space:

V = {(V, φ) ∈ (H 1 (Ω k , R 3 )) 2 , ∂ αβ V.a 3 ∈ L 2 (Ω k ), ∂ i3 V ∈ L 2 (Ω k ), ∂ α φ ∈ L 2 (Ω k ), ∂ α V.a 3 ∈ L 2 (Ω k )} (5.
1) and the associated norm defined by:

(V, φ) V = { (V, φ) (H 1 ) 2 + α,β ∂ αβ V.a 3 L 2 + i,3 ∂ i3 V L 2 + α ∂ α φ L 2 + α ∂ α V.a 3 L 2 } (5.2) We have the following injection V ⊂ H where H = H 2 (S)×H 1 (S)×H 2 (S)×H 1 (S).
The space M associated to V V is a Hilbert space. Let us u = (u 1 , u 2 , w = u 3 , q) and X = (θ α , Φ α ); α = 1, 2. We define the admissible displacement space denoted by M γ0 reads:

M γ0 = { ∈ M, = 0onΓ 0 } (5.3) M γ0 = {(u, q) ∈ (W 1,2 γ0 (S)) 2 × (W 2,2 γ0 (S)) × W 1,2 γ0 (S) } (5.4)
in the same sense we defined the admissible rotation space by W r par:

W r = {X ∈ (H 2 γ0 (S) × H 2 γ0 (S)) 2 } (5.5)
Finally we defined the admissible solutions space as follows:

V ad = M γ0 × W r (5.6)
V ad is a Hilbert space.

Statement problem

Let the border of S, ∂S = γ 0 ∪ γ 1 , be partitioned in two parts and the border of

the shell ∂Ω = Γ 0 ∪ Γ 1 with Γ 0 = γ 0 × - h 2 , h 2
, and

Γ 1 = γ 1 × - h 2 , h 2 ∪ Γ -∪ Γ + we denote Γ -= S × {- h 2 } and Γ + = S × {+ h 2 }.
We consider a laminated composite shell of thickness h, clamped on part of its border Γ 0 , and subject to volume forces f α and f 3 and to surface forces gα and g3 on the rest of its border Γ 1 . To study the mechanical behavior of laminated composite shell we consider the following problem:

                     div[σ] k = f in Ω k [σ].n = P on Γ + [σ] k = [Q(α k )] k [e αβ ] -z[Q(α k )] k [K αβ ] + z 2 [Q(α k )] k [Q αβ ] +[Q(α k )] k [qΥ αβ ] + 1/2[Q(α k )] k [φ α ] + [Q(α k )] k [φ 3 ] [σ]n k i,k-1 = [σ]n k-1 i,k
on Γ k-1,k Stress continuity between two layers

u k Γ k,k-1 = u k-1 Γ k,k-1 on Γ k-1,k Displacement continuity between two layers (5.7)
where σ k and k are the stress and strain fields for the layer k defined respectively by: σ k : Ω → S 3 , k : Ω → S 3 such that σ k is given by the Hooke's law for anisotropic material defined in Eq.(4.5). The source terms f ∈ L 2 (Ω); and the surface forces g ∈ L 2 (Γ + ). The stress field components for each layer k satisfy:

σ 33 k ∈ H 2 (- h k 2 , h k-1 2 ; H 2 (ω k )), (5.8)
σ α3 k ∈ H 1 (- h k 2 , h k-1 2 ; H -1 (ω k )), (5.9)
σ αβ k , ∂σ αβ ∈ L 2 (- h k 2 , h k-1 2 ; L 2 (ω k )).
(5.10)

Best variational formulations for laminated composite shells

Let us φ = w(z) φ and H(S) = L 2 (S)×L 2 (S)×L 2 (S)×L 2 (S)×L 2 (S)×L 2 (S)×L 2 (S).

U = e, K, Q, Υq, ∂ 1 q, ∂ 2 q, q ∈ H(S) (5.11) The space H(S) is equipped with the norm: where we have voluntarily omitted the subscript L 2 (S) and L 2 (S) on the norm.

V H(S) = e 2 + K 2 + Q 2 + Υq 2 + ∂ α q 2 + q 2
The variational formulation in term of strain is proposed here for the problem given by Eq. (5.7). Next, we will study the existence and uniqueness of a variational solution.

Ω k -divσ( U). VdΩ k = Ω k f VdΩ ∀ V ∈ H(S), ⇒ Ω k t σ( U) : ∇ VdΩ k = Ω k f VdΩ + ∂Ω k (σ( U).n)∇ Vds ∀ V ∈ M Γ0 ⊆ V ad , ⇒ Ω k t σ( U) : ( V)dΩ k = Ω k f VdΩ + Γ + k P Vds ∀ V ∈ V ad , ⇒ Ω k t σ αβ (u) : αβ (v)dΩ k + 2 Ω k t σ α3 (φ) : α3 ( φ)dΩ k + Ω k t σ 33 (φ) : 33 ( φ)dΩ k = Ω k f VdΩ + Γ + k P Vds ∀ V ∈ V ad , S N : eds + S M : Kds + S M * : Q ds + S S : Υ φds + S S 1 : ∂ 1 φ + S S 2 : ∂ 2 φ + S S 3 : φ = Ω k f VdΩ + Γ + k P Vds ∀ V ∈ V ad . for all (N, M, M * , S) = (N αβ ), (M αβ ), (M * αβ ), (S αβ ) ∈ (L 2 (ω)) 4 , (S n ) ∈ L 2 (ω), n = 1, 2, 3,
where N, M, M * , S, S 1 , S 2 , S 3 are the force, moment, Gauss moment, twisting, and the transverse force respectively.

The final variational formulation reads: (5.13) Using Eq.(4.15), equation (5.13) can be reformulated as follows:

       Find U ∈ V ad such that ∀ V ∈ V
A( U, V) = S e, K, Q, Υφ, ∂ 1 φ, ∂ 2 φ, φ A 3D             e K Q Υ φ ∂ α φ ∂ β φ φ             = L( V), V ∈ H(S). (5.14)
The 3D matrix A 3D is given by 3D LCE above. We prove the existence and uniqueness of variational solution in term of unknowns associated to strains.

The matrix A 3D contains some mechanical coupling If we regroup all the terms that appear in the variational formulation (in term of displacements) some contributions in term of energy respectively the membrane energy, the flexional energy, the plane shear energy, the stretching plane energy and the transverse deformation energy noted respectively: The 3D variational formulation with all these contribution:

A m (u, φ, v φ), A f (u, φ, v, φ), A cis (u, φ, v, φ), A s (u, φ, v, φ), A t (u, φ, v, φ).
A(u, φ, v, φ) = A m (u, φ, v, φ)+ A f (u, φ, v, φ)+ A cis (u, φ, v, φ)+ A s (u, φ, v, φ)+ A dt (u, φ, v, φ) = L(v, φ).
(5.15) We let A = S U t A 3D U. The mapping A(., .) defined by A : H(S) × H(S) → R n is bilinear positive definiteness and coercive. Moreover, the mapping L(.) defined by: L : H(S) → R n is linear and continuous. We will show that A(., .) is coercive i.e. there exist m > 0 such that:

m U 2 ≤ | A( U, U)| = |A 3D U. U| in H(S).
(5.16)

We have according to the 3D Generalized ABCDE-matrix formulation the following relation holds

| A( U, U)| ≥ A e 2 + C K 2 + E Q 2 + S 4 qΥ 2 + S 5 q 2 (5.17) + S 4 ∂ 1 q 2 + S 4 ∂ 2 q 2 ≥ m U 2 ,
where U is defined above and where m

= min ( A L ∞ , C L ∞ , E L ∞ , S 4 L ∞ , S 5 L ∞ ) > 0 with L ∞ = L ∞ - h 2 , h 2
, L 2 (S) . The existence and uniqueness is proved using the Lax-Milgram lemma.

Variational formulation for 2D case: existence and uniqueness of a solution

We present a variational formulation in term of strain tensors e αβ , K αβ , Q αβ using the 2D ABCDE-matrix developed in a previous works (see [START_REF] Ngatcha | A Two-Dimensional Model to Analyze the Static and Dynamic Mechanical Behavior of Multilayered shell Structures[END_REF] ). Let there be given three tensors fields (e, K, Q) ∈ (L 2 (S)) [START_REF]Sur l'ellipticit du modle linaire des coques de W.T Koitier[END_REF] . We can find a unique vector

(v) = (v α , v 3 ) ∈ M Γ0 ⊆ V ad such that e αβ (v) = e αβ , K αβ (v) = K αβ , Q αβ (v) = Q αβ ∈ L 2 (ω).
The variational formulation for 2D case laminated composite shell can also be obtained easily:

find ū ∈ M γ0 such that S e, K, Q A 2D e, K, Q t dS = L(v), ∀v ∈ M γ0 , (5.18)
where () t is the transpose of vector and where A 2D ∈ M is the composite-shell stiffness matrix given in [START_REF] Ndengna | Two dimensional static mechanic analysis of laminated composite tube using ABCDE matriix with no correction factor[END_REF][START_REF] Ngatcha | A Two-Dimensional Model to Analyze the Static and Dynamic Mechanical Behavior of Multilayered shell Structures[END_REF] . In the same way as in the above demonstration, we prove that there a unique solution Y = (e, K, Q) ∈ (L 2 (S)) 3 such that:

A(Y, Y ) = L(Y ), Y ∈ H(S) = (L 2 (S)) 3 ⊆ V ad (5.19)
As for the 3D case, we show easily A(., .) is bilinear definite positive. Moreover, A(., .) is coercive i.e. there exist m > 0 such that

m Y 2 ≤ | A(Y, Y )| = |A 2D Y.Y | in H(S).
(5.20)

We have according to the 2D ABCDE-matrix the following relation holds

| A(Y, Y )| ≥ A e 2 + C K 2 + E Q 2 ≥ m Y 2 (5.21) where Y = e 2 + K 2 + Q 2 1 2 and where m = min( A L ∞ , C L ∞ , E L ∞ ) > 0 with L ∞ = L ∞ - h 2 , h 2 , L 2 (S) .
The existence and uniqueness is proved using the Lax-Milgram lemma.

Remark 5.1. We can see that when the Gauss disappears we retrieve the wellknown variational formulation of Kirchhoff-Love shell model proposed by [START_REF] Reddy | Mechanics of laminated composite plates and shells.Theory and analysis[END_REF] . The 2D variational formulation proposed here extend the 2D variational formulation developed in 15 , ? , 10 for isotropic material. The presence of several mechanical couplings in the laminated constitutive equation can allow to study a large class of 2D and 3D problem related to the analysis of the laminated plates and shells structures without major modifications.

Modeling of L(v)

Now, we give a formulation of δP . We start by give the following transformation:

U α = u α (α, β) -z(∇ α u 3 + 2b τ α u τ ) + z 2 (b λ λ b τ α u τ + b τ α ∇ α u 3 ), (5.22) = u α (α, β) -zβ α (α, β) + z 2 Ψ α (α, β), = w ρ α (z)u ρ + wρ α (z)∇ ρ u 3 -zb ρ α u ρ = u α -z(θ α + b ρ α u ρ ) -z 2 θ α b ρ α , (5.23)
U 3 = u 3 (α, β) + q(x, z), where w ρ α (z) = δ ρ α -zb ρ α + z 2 b λ λ b τ α ; wρ α (z) = -zδ ρ α + z 2 b ρ τ ; θ α = -(∇ α u 3 + b ρ α u ρ ); β α (α, β) = ∇ α u 3 + 2b τ α u τ ; and the thickness coordinate z ∈ - h 2 , h 2 .
Let us v = (v α , v3 ) = (η ρ , η 3 ). One has: [q α η α + q 3 δη 3 + q

L(v) = Ω f vdX + Γ gvdΓ (
q α = h 2 -h 2 gτ -zb α ρ gρ dz q 3 = h 2 -h 2 g3 dz q 4 = h 2 -h 2 w(z)g 3 dz m α = h 2 -h 2 (zg τ -z 2 b α ρ gρ )dz .
In Eqs.(5.24)-(5.25), q α , q 3 are the line load. m = m α A α are the linear density moment obtained on the free contour γ that contains the forces on Γ 1 take on S. In fact, we have m = m α A α = m ν ν + m t t, where m t and m ν are the bending density moments and the twisting density moments applied on γ 1 . The coupled moment

m α θ α = m t θ ν -m ν θ t = m t θ ν + m ν n.∂ t v, ( 5

Boundary conditions

The effect of stretching-through-the-thickness modifies the boundary condition given for 2D case in [START_REF] Ngatcha | A Two-Dimensional Model to Analyze the Static and Dynamic Mechanical Behavior of Multilayered shell Structures[END_REF] . Here the equations on the boundaries are given by:

          
N αν + M νν b α ν + (2M νt + M * νt -m ν )b α t -q α = 0, Q ν + Q * ν -∂ s (m ν -M νt -M * νt ) -q 3 = 0, S ν3 -q 4 = 0, M νν + M * νν + m s = 0.

(5.28)

We choice a direct landmark(t, ν, a n ). Note that in many applications of laminated shell structure, we have m ν = 0(moment normal axe on border) where m s = m.ν and m ν = m.t are the density moments. On the clamped border γ 1 we have ∂ ν u α = 0, ∂ ν w = 0, u α = 0, w = 0; ∂ ν q = 0; q = 0.

(5.29)

Green formula for thick shell models

In this section, we establish Green formula for thick laminated shells. The derived relation can be used to model some composite structures without any and-hoc assumptions. We recall the following relations:

Given a domain w ⊂ R 2 with a smooth enough boundary γ 1 , let (ν α ) Let t α be the unit tangent to be curve (C) the surface a then n β = βγ t γ is the unit to (C) in the surface (S). Let us θ α = θ s t α + θ n n α . We have θ s = -dw ds + b γ α u γ t α . In the same sense, we have Φ s = Φ α t α . Theorem 6.1. Let there be given a domain ω ⊂ R 2 with a boundary γ 1 of class C 1,1 and an immersion ϕ ∈ C 2 (ω, R 3 ). We assume that e αβ , K αβ , Q αβ ∈ L 2 (ω) for all N = N αβ ∈ H 1 (ω), M = M αβ ∈ H 1 (ω), M * = M * αβ ∈ H 1 (ω), S = S i3 ∈ H 1 (ω) i = 1, 2, 3 and for all vector (u α , u 3 , q) ∈ (H 1 (ω)) 2 × H 2 (ω) × H 1 (ω). where for any smooth symmetric tensor field A αβ , we have:

∇ β A αβ := ∂ β (∇ β A αβ ) + Γ α βρ A αβ + Γ τ ρτ A αβ ,

( 1

 1

R 2

 2 ∂x i is the partial derivatives operators of the first order. Let ω be an open subset of and x = (x α ) a generic point in ω the n-order derivative operator ∂ ∂x α1 ......∂x αn .

( 3 .

 3 23)for all (N, M, M * , S) = (N αβ ), (M αβ ), (M * αβ ), (S αβ ) ∈ (L 2 (ω)) 4 , (S n3 ) ∈ L 2 (ω), n = 1, 2

  . The 2D LCE writes: N = Āe -BK + CQ; M = -Be + CK -DQ; M * = Ce -DK + ĒQ; (4.10)

  ad S N : eds + S M : Kds + S M * : Q ds + S S : Υ φds + S S 1 : ∂ 1 φ + S S 2 : ∂ 2 φ + S S 3 : φ = Ω k f VdΩ + Γ + k P Vds.

2 f α vα dSdz + S h 2 -h 2 f 3 g 3

 22233 δv 3 dSdz + Γ1 g α δv α dΓ + Γ1 v3 dΓ = S [p α η α + p 3 δη 3 + p 4 δy]dS + γ1

  .26) where m t = m. t, m ν = m. ν work on the angle due to the variation of ν while m ν work on the angle to variation of t. Using the fact that∂ t = t α ∂ α , ∂ n v α = B ρ α A ρ , n.v = v 3 and m ν n.∂ t v = -∂ t m ν n.v -m ν ∂ t n.v.We can show easily that the term γ1 m α δθ α dγ in (5.24):γ1 m α δθ α dγ = γ1 m t δθ ν -∂ t m ν δv 3 + m ν t α B ρα δv ρ dγ (5.27)

S

  ∇ α M αβ dS = ∂S M αβ ν α dγ (6.1) or S ∇ α M αβ ν β dS = -S M αβ ∇ α ν β dS + ∂S M αβ ν α ν β dγ(6.2)We shall denote byb ν τ = b τ α ν α , b τ τ = b τ α t α , M ντ = M αβ ν β t β , M ντ = M ατ ν α .

SN

  αβ e αβ (v) + M αβ K αβ (v) + M * αβ Q αβ (v) + S αβ Υ αβ (v) + φ i S i3 ds (6.4) + w d i (N, M, M * ) viα ds = γ1 b i (N, M, M * )u i + b ν (N, M, M * )∂ ν u 3 ds, where d i (N, M, M * ) ∈ L 2 (ω), b i (N, M, M * ) ∈ L 2 (γ) are defined by: b α (N, M, M * ) = N αν + M νν b α ν + (2M νt + M * νt -m ν )b α t , b 3 (N, M, M * ) = Q ν + Q * ν -∂ s (m ν -M νt -M * νt ) b ν (N, M, M * ) = M νν + M * νν + m s ,

Acknowledgements

This work is under the project sanctioned by the laboratory of energy modeling materials and methods (E3M), National Advanced School Polytechnic of Douala of University of Douala-Cameroon. April 20, 2023 16:59 WSPC/INSTRUCTION FILE Shell˙Nonlinear˙M3AS˙2023

and

Lemma 6.1. Moreover, for N = N αβ ∈ H 1 0 (S), M = M αβ ∈ H 2 0 (S), M * = M * αβ ∈ H 2 0 (S) and u i ∈ (L 2 (S)) [START_REF]Sur l'ellipticit du modle linaire des coques de W.T Koitier[END_REF] . The following Green formula with little regularity holds:

where (e(ũ), K(ũ), Q(ũ)) ∈ H -1 (S)×H -2 (S)×H -2 (S) and where

The Green's formula given by equation (6.4) is also satisfied for all tensors fields (N αβ , M αβ , M * αβ ) ∈ (N(S), M(S), M * (S)) where N(S), M(S), M * (S) are the Hilbert space defined by:

Concluding remarks

In this paper, we have proposed an nonlinear analysis of Kirchhoff type shell models. The first model accounts the effect of third fundamental form and the second account the stretching-through-the-thickness. We have used ABCDE-matrix for 2D shells and generalized ABCDE-matrix for 3D shell models to prove the existence and uniqueness of a variational solution. We have also proposed some mathematical results related to existence and uniqueness of studied shell models. These results have proved that the constitutive relation developed in our previous works and presented here, are well-posed mathematically and physically and can help to improve prediction of mechanical behavior of laminate composite shells.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.