Marco Picchi Scardaoni
email: marco.picchiscardaoni@ing.unipi.it

Marco Montemurro
email: marco.montemurro@ensam.eu

Convex or non-convex? On the nature of the feasible domain of laminates

Keywords: Anisotropy, Plane elasticity, Polar method, Composite materials, Laminates, Quasi-triviality, Convexity

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Two still-open problems in anisotropic plane elasticity for composite laminates are: (a) the definition of the general expression of the feasible domain; (b) the definition of sufficient and necessary conditions to obtain membrane/bending uncoupled laminates (uncoupling property), and/or the same group of symmetry for membrane and bending stiffness tensors (homogeneity property).

One can deal with the two problems in the general framework of the First-Order Shear Deformation Theory (FSDT) [START_REF] Reddy | Mechanics of Laminated Composite Plates and Shells: Theory and Analysis[END_REF]). The FSDT is the generalization of the Classical Laminate Theory (CLT) [START_REF] Jones | Mechanics Of Composite Materials[END_REF]) and constitutes a theoretical framework to describe the behaviour of composite laminates.

In the FSDT [START_REF] Reddy | Mechanics of Laminated Composite Plates and Shells: Theory and Analysis[END_REF]), the analytic expression of the laminate stiffness tensor is

K lam =    A B O D O sym H    , ( 1
)
where A is the membrane stiffness tensor of the laminate, D the bending stiffness tensor, H the out-of-plane shear stiffness tensor, B the membrane/bending coupling stiffness tensor. It is convenient to introduce also the following normalised tensors:

A * := 1 h A, B * := 2 h 2 B, D * := 12 h 3 D, C * := A * -D * , H * := 1 h H, (2)
where h is the total thickness of the laminate.

In the special case of laminates made of identical plies, i.e. same material and thickness for the elementary layer, the expressions of the above tensors read:

A * = 1 N N k=1 a k Q(θ k ), B * = 1 N 2 N k=1 b k Q(θ k ), D * = 1 N 3 N k=1 d k Q(θ k ), H * = 1 N N k=1 h k Q(θ k ), C * = 1 N 3 N k=1 c k Q(θ k ). (3)
In Eq. (3), N is the number of plies of the laminate and θ k is the orientation angle of the k-th ply. Q(θ k ) is the in-plane reduced stiffness tensor of the k-th ply whose material frame is turned by an angle θ k with respect to the global reference frame of the laminate. Analogously, Q(θ k ) is the out-of-plane reduced stiffness tensor of the k-th ply. Furthermore, coefficients a k , b k , d k , h k and c k read:

a k = 1, b k = 2k -N -1, d k = 12k(k -N -1) + 4 + 3N (N + 2), h k = 1, c k = -2N 2 -12k(k -N -1) -4 -6N.
(4)

The ordered sequence of orientations, from the bottom to the top of the laminate, is called stacking sequence (SS), or stack.

It is well known that there is no bijective relationship between arbitrary macroscopic elastic properties of the laminate (in terms of A * , B * , D * and H * components) and SSs. This means that, given an arbitrary SS, a set of elastic properties is always uniquely determined, whilst the converse is generally false. This aspect is particularly important in practical applications, such as laminates design and SSs recovery. Without entering in the details, after the determination of the optimal stiffness distribution within a structure, with respect to a given merit function, SSs matching the target optimal elastic properties must be found out [START_REF] Catapano | A multi-scale approach for the optimum design of sandwich plates with honeycomb core. part II: the optimisation strategy[END_REF]; [START_REF] Montemurro | Least-weight composite plates with unconventional stacking sequences: Design, analysis and experiments[END_REF][START_REF] Montemurro | A two-level procedure for the global optimization of the damping behavior of composite laminated plates with elastomer patches[END_REF][START_REF] Montemurro | Design of the elastic properties of laminates with a minimum number of plies[END_REF]; Picchi Scardaoni and [START_REF] Picchi Scardaoni | A General Global-Local Modelling Framework for the Deterministic Optimisation of Composite Structures[END_REF]). It is then of primary importance to determine the expressions of the feasible domain of the aforementioned elasticity tensors. However, given a SS, the feasible domains of A * , B * , D * must be correlated, since these tensors are associated to the same SS.

A first kind of bounds derives directly from thermodynamic considerations, since the elasticity tensor is positive-definite. Therefore, the elastic constants live in a feasible domain whose boundary constitutes the so-called elastic-bound.

However, a second kind of bounds can be introduced, which are often referred as geometrical bounds. To this purpose, it is important to adopt some mathematical descriptors of the anisotropy. The most common approach makes use of the wellknown lamination parameters (LPs) [START_REF] Jones | Mechanics Of Composite Materials[END_REF]; M. (1982); [START_REF] Tsai | Introduction to composite materials[END_REF]; [START_REF] Tsai | Invariant properties of composite materials[END_REF]). LPs unquestionably provide a compact representation of the stiffness tensor of the laminate; although, they are not all tensor invariants. A sound alternative for describing plane anisotropy is represented by the polar formalism introduced in [START_REF] Verchery | Les invariants des tenseurs d'ordre 4 du type de l'élasticité[END_REF]). Thanks to the polar formalism, it is possible to represent any plane tensor by means of tensor invariants, referred as polar parameters (PPs), which are related to the symmetries of the tensor. In particular, for a fourth-order elasticity-like plane tensor (i.e. a tensor having both major and minor symmetries), all possible elastic symmetries can be easily expressed in terms of conditions on the PPs. Moreover, the polar formalism offers a frame-invariant description of any plane tensor [START_REF] Verchery | Les invariants des tenseurs d'ordre 4 du type de l'élasticité[END_REF]).

In [START_REF] Hammer | Parametrization in laminate design for optimal compliance[END_REF]), geometrical bounds were derived as a consequence of the nature of the trigonometric functions involved in the definition of the laminate stiffness tensors. In [START_REF] Vannucci | A note on the elastic and geometric bounds for composite laminates[END_REF]), Vannucci derives the expressions of the geometrical bounds in the framework of the polar method, showing that these bounds are always stricter than the elastic counterpart: hence these bounds must be considered in practical design problems.

In [START_REF] Grenestedt | Layup optimization of composite material structures[END_REF]) Grenestedt and Gudmundson claimed to provide a proof of the convexity of the feasible domain of general anisotropic laminates in the LPs space. However, as it will be discussed later in this paper, the proof provided in [START_REF] Grenestedt | Layup optimization of composite material structures[END_REF]) proves only that the projection of the LPs on each axis (in the LPs space, which is of dimension 12) is convex. Indeed, this proof does not respect a fundamental lemma of convex analysis [START_REF] Boyd | Convex Optimization[END_REF]), which states that if a set is convex then the projection over some of its coordinates is convex too, while the converse is not generally true. Based on this incorrect result, many works, dealing with the optimisation of laminates and making use of the formalism based on LPs, claim that this description endows the composite design problems with nice properties [START_REF] Bloomfield | On feasible regions of lamination parameters for lay-up optimization of laminated composites[END_REF]; [START_REF] Diaconu | Feasible region in general design space of lamination parameters for laminated composites[END_REF]; [START_REF] Hammer | Parametrization in laminate design for optimal compliance[END_REF]; [START_REF] Macquart | Optimisation of composite structures -enforcing the feasibility of lamination parameter constraints with computationally-efficient maps[END_REF]; [START_REF] Raju | On further developments of feasible region of lamination parameters for symmetric composite laminates[END_REF]; [START_REF] Setoodeh | Approximate feasible regions for lamination parameters[END_REF]). It is noteworthy that this is one of the main reasons at the basis of the wider diffusion of the approach based on LPs with respect to that based on PPs, which are intrinsic and frame-independent quantities of the stiffness tensor. However, several works, based on LPs description, seem to provide evidences, both mathematical and graphical, of the convexity of the feasible domain. How is it possible? The answer is probably due to the overlapping of two concepts: the geometrical-admissible region (i.e. the region delimited through the geometrical bounds) and the actual feasible region, i.e. the set of all the mapped SSs, through LPS (or PPs), in the corresponding space. In fact, although the geometrical bounds define a region, possibly convex, it is not generally true that the actual feasible domain, which is a subset of the geometrical-admissible region, is a convex set. It is evident the conceptual overlapping between the geometrical-admissible region (which can be convex or not) and the actual feasible region, which should be the true object of investigation. In [START_REF] Diaconu | Feasible region in general design space of lamination parameters for laminated composites[END_REF]) the authors formulate a variational problem to find the actual shape of the feasible domain in the LPs space. Moreover, they provide some plots of some 2D projections of the domain, which are convex set. Similar approaches are provided in [START_REF] Bloomfield | On feasible regions of lamination parameters for lay-up optimization of laminated composites[END_REF]; [START_REF] Setoodeh | Approximate feasible regions for lamination parameters[END_REF]). All of these studies, as stated by the authors themselves, rely on the fact that the feasible region, in the LPs space, is a convex sets. In the light of the results of this work, it is possible to consider the findings of the existing literature as convex hulls (regardless of the number of layers) of the actual feasible region, which can be narrower by far.

The derivation of the laminate actual feasible domain in a closed form is of paramount importance when formulating the laminate design problem as a constrained non-linear programming problem. Keeping this in mind, a simple investi-gation led to the discovery of the analytic expression of a further bound, valid for any kind of symmetry of the laminate, which makes the feasible region a stricter non-convex region. These new bounds introduce the dependence from the number of plies. Indeed, up to now, it is odd that the feasible region of laminates having even very different number of plies is provided by the same equations, which in turn do not depend on the number of plies! Starting from these considerations, and from some recent advances in the development of a global/local modelling approach for the multi-scale optimization of composite structures discussed in (Picchi Scardaoni and Montemurro (2020)), the research has been extended to the feasible domain determination problem for composite laminates. In particular, in this paper, the rigorous proof of the non-convexity of the laminate feasible domain is provided, regardless of the formalism used to describe the anisotropy (i.e. LPs or PPs).

A laminate is uncoupled if and only if B * = O. This means that in-plane forces do not produce curvatures and, similarly, bending moments do not deform the laminate middle plane. A laminate is said homogeneous if and only if C * = O. The homogeneity property is linked to the design of the pure bending tensor D * . The design of D * is quite difficult because its properties depend not only on the plies orientation angles, but also on their position within the SS. In order to have the same properties both in membrane and bending in any direction, i.e. the same group of symmetry, the homogeneity property must be imposed.

The uncoupling property is often sought in many engineering applications. Since coefficients b k assume antisymmetric values with respect to the laminate middle plane, a simple way to obtain B * = O consists of using a symmetric SSs, as commonly done in several works [START_REF] Adams | Genetic algorithm optimization and blending of composite laminates by locally reducing laminate thickness[END_REF]; [START_REF] Bloomfield | On feasible regions of lamination parameters for lay-up optimization of laminated composites[END_REF]; [START_REF] Macquart | Derivation and application of blending constraints in lamination parameter space for composite optimisation[END_REF]; [START_REF] Raju | On further developments of feasible region of lamination parameters for symmetric composite laminates[END_REF]; [START_REF] Seresta | Optimal design of composite wing structures with blended laminates[END_REF]). Of course, this is only a sufficient condition, since asymmetric uncoupled SSs exist. In [START_REF] Caprino | A note on specially orthotropic laminates[END_REF]), the existence of uncoupled anti-symmetric stacks was proven, while in [START_REF] Verchery | Une méthode d'aide graphique à la conception des séquences d'empilement dans les stratifiés[END_REF]), the existence of completely asymmetric uncoupled SSs was shown. In [START_REF] Vannucci | A special class of uncoupled and quasihomogeneous laminates[END_REF]), a special class of uncoupled and possibly homogeneous laminates was found; the solutions belonging to this class are called quasi-trivial (QT) and represent a class of arithmetically-exact solutions. Furthermore, authors have shown that the number of independent QT solutions is by far larger than the number of symmetric ones. An efficient enumerating algorithm has been recently proposed in [START_REF] Garulli | Quasitrivial stacking sequences for the design of thick laminates[END_REF]). In this work, the enumeration of all independent QT SSs, up to N = 35, is presented.

Up to the present, it is not clear if quasi-triviality is a sufficient or even neces-sary condition to have uncoupled and/or homogeneous laminates. This work aims at clarifying this aspect too, by showing that quasi-triviality is, in general, only a sufficient condition. However, for a particular case, quasi-triviality becomes also a necessary condition. In other words, for that particular case, QT solutions constitute the full set of all possible uncoupled and/or homogeneous SSs. The paper is organized as follows: a short recall of the polar formalism in the FSDT framework and of the LPs in the CLT framework is presented in Section 2. Section 3 provides the proofs of the non-convexity of the feasible domain in the PPs and LPs spaces for general anisotropic and membrane-orthotropic laminates. Sections 4 and 5 present the new analytical bounds for multilayer plates, together with some exact solutions. Section 6 provides the proof that quasi-triviality is not a necessary condition for uncoupling and/or homogeneity in the most general case. Finally, Section 7 ends the paper with some meaningful conclusions and prospects.

Fundamentals of the Polar Method and of the Lamination Parameters

Polar Method

The Polar Method allows representing any n-th order plane tensor in terms of invariants: this method was introduced, for the first time, in [START_REF] Verchery | Les invariants des tenseurs d'ordre 4 du type de l'élasticité[END_REF]). For a deeper insight in the matter, the reader is addressed to [START_REF] Vannucci | Anisotropic Elasticity[END_REF]). In this context, a second-order symmetric plane tensor Z can be expressed in the local frame Γ = {O, x 1 , x 2 , x 3 } as:

Z 11 = T + R cos 2Φ, Z 12 = R sin 2Φ, Z 22 = T -R cos 2Φ, ( 5
)
where T is the isotropic modulus, R the deviatoric one and Φ the polar angle. Among them, only the moduli are invariants, whilst Φ is needed to set the reference frame. If L is a fourth-order plane elasticity-like tensor, i.e. with major and minors symmetries, its Cartesian components can be expressed by means of four moduli and two polars angles, namely T 0 , T 1 , R 0 , R 1 , Φ 0 , Φ 1 . The complete expression reads:

L 1111 = T 0 + 2T 1 + R 0 cos 4Φ 0 + 4R 1 cos 2Φ 1 , L 1112 = R 0 sin 4Φ 0 + 2R 1 sin 2Φ 1 , L 1122 = -T 0 + 2T 1 -R 0 cos 4Φ 0 , L 1212 = T 0 -R 0 cos 4Φ 0 , L 2212 = -R 0 sin 4Φ 0 + 2R 1 sin 2Φ 1 , L 2222 = T 0 + 2T 1 + R 0 cos 4Φ 0 -4R 1 cos 2Φ 1 . (6)
In Eq. ( 6), T 0 and T 1 are the isotropic moduli, R 0 and R 1 the anisotropic ones, Φ 0 and Φ 1 the polar angles. Among them, only the four moduli and the difference Φ 0 -Φ 1 are tensor invariants.

One of the advantages of the polar method is that for a fourth-order elasticity-like plane tensor, the polar invariants are related to the elastic symmetries of the tensor. Indeed the polar formalism offers an algebraic characterization of the elastic symmetries [START_REF] Catapano | Invariant formulation of phenomenological failure criteria for orthotropic sheets and optimisation of their strength[END_REF]; [START_REF] Catapano | On the correlation between stiffness and strength properties of anisotropic laminates[END_REF]). In particular, four different symmetries can be defined:

• Orthotropy: this symmetry corresponds to the condition Φ 0 -Φ 1 = K π 4 , K = 0, 1.
• R 0 -Orthotropy: the algebraic condition to obtain this special orthotropy is R 0 = 0. This case has been studied in [START_REF] Vannucci | A special planar orthotropic material[END_REF]).

• Square symmetry: it can be achieved by imposing R 1 = 0. This case represents the 2D counterpart of the well-known 3D cubic syngony.

• Isotropy: the condition to be satisfied is

R 0 = R 1 = 0.
In order to properly analyse the mechanical behaviour of a laminate, it is possible to express the stiffness tensors appearing in Eq. ( 3) in terms of their PPs. In particular, A * , B * , D * , and thus C * , are fourth-order elasticity-like plane tensor, while H * behaves like a second-order symmetric plane tensor (Montemurro (2015a,b)). The PPs of the laminate stiffness tensors can be expressed as functions of the PPs of the lamina reduced stiffness matrices and of the geometrical properties of the stack (i.e. the layers orientation, position and number). The polar representation of the normalised stiffness tensors of the laminate reads:

T A * 0 = T 0 , T A * 1 = T 1 , R A * 0 e i4Φ A * 0 = R 0 N e i4Φ 0 N k=1 a k e i4θ k , R A * 1 e i2Φ A * 1 = R 1 N e i2Φ 1 N k=1 a k e i2θ k , (7)
T B * 0 = T B * 1 = 0, R B * 0 e i4Φ B * 0 = R 0 N 2 e i4Φ 0 N k=1 b k e i4θ k , R B * 1 e i2Φ B * 1 = R 1 N 2 e i2Φ 1 N k=1 b k e i2θ k , (8)
T D * 0 = T 0 , T D * 1 = T 1 , R D * 0 e i4Φ D * 0 = R 0 N 3 e i4Φ 0 N k=1 d k e i4θ k , R D * 1 e i2Φ D * 1 = R 1 N 3 e i2Φ 1 N k=1 d k e i2θ k , (9)
T H * = T, R H * e i2Φ H * 1 = R N e i2Φ N k=1 h k e -i2θ k = R A * 1 R R 1 e i2(φ+φ 1 -φ A * 1 ) , (10)
T C * 0 = 0, T C * 1 = 0, R C * 0 e i4Φ C * 0 = R 0 N 3 e i4Φ 0 N k=1 c k e i4θ k , R C * 1 e i2Φ C * 1 = R 1 N 3 e i2Φ 1 N k=1 c k e i2θ k , (11)
In the above equations, T 0 , T 1 , R 0 , R 1 , Φ 0 and Φ 1 are the polar parameters of the in-plane reduced stiffness matrix Q of the lamina, while T , R, and Φ are those of the transverse shear stiffness matrix Q: all of these parameters solely depend upon the ply material properties. Since basic layers are actually orthotropic, without loss of generality it can be assumed that Φ 1 = 0 (Φ 0 = K π 4 with K ∈ {0, 1}) [START_REF] Vannucci | A note on the elastic and geometric bounds for composite laminates[END_REF]). For further details on the Polar Method in the FSDT framework, the reader is addressed to (Montemurro (2015a,b)).

Lamination Parameters

LPs are often used together with the parameters of Tsai and Pagano to describe the anisotropic behaviour of the laminate in the framework of the CLT (Grenestedt and Gudmundson (1993); [START_REF] Hammer | Parametrization in laminate design for optimal compliance[END_REF]; [START_REF] Jones | Mechanics Of Composite Materials[END_REF]; [START_REF] Reddy | Mechanics of Laminated Composite Plates and Shells: Theory and Analysis[END_REF]). LPs express the properties of a laminate in terms of moments, relative to the plate mid-plane, of the trigonometric functions entering in the frame rotation formulas [START_REF] Hammer | Parametrization in laminate design for optimal compliance[END_REF]). For the sake of brevity, the complete expressions of the LPs in the CLT framework are not here reported. The interested reader is addressed to [START_REF] Jones | Mechanics Of Composite Materials[END_REF]).

LPs are closely related to PPs: from a mathematical viewpoint, it is sufficient to report the following identities between PPs and LPs, using the same nomenclature of [START_REF] Vannucci | Anisotropic Elasticity[END_REF]).

ξ 1 + iξ 2 = 1 N N k=1 a k e i4θ k , ξ 3 + iξ 4 = 1 N N k=1 a k e i2θ k , ξ 5 + iξ 6 = 1 N 2 N k=1 b k e i4θ k , ξ 7 + iξ 8 = 1 N 2 N k=1 b k e i2θ k , ξ 9 + iξ 10 = 1 N 3 N k=1 d k e i4θ k , ξ 11 + iξ 12 = 1 N 3 N k=1 d k e i2θ k . (12)
It is evident that 12 parameters are necessary to completely define the behaviour of the laminate in the CLT framework.

The Non-Convexity of the Feasible Domain

For a generic anisotropic laminate, the geometrical bounds in the PPs space read [START_REF] Vannucci | A note on the elastic and geometric bounds for composite laminates[END_REF]):

           0 ≤ ρ 0 ≤ 1, 0 ≤ ρ 1 ≤ 1, 2ρ 2 1 ≤ 1 -ρ 2 0 1 -(-1) K ρ 0 cos 4Φ A * 0 , (13)
where

ρ 0 := R A * 0 R 0 = 1 N N j=1 cos 4θ j 2 + N j=1 sin 4θ j 2 , (14)
ρ 1 := R A * 1 R 1 = 1 N N j=1 cos 2θ j 2 + N j=1 sin 2θ j 2 . ( 15
)
In the LPs space, they read [START_REF] Hammer | Parametrization in laminate design for optimal compliance[END_REF])

         2ξ 2 3 (1 -ξ 1 ) + 2ξ 2 4 (1 + ξ 1 ) + ξ 2 1 + ξ 2 2 -4ξ 3 ξ 2 ξ 4 ≤ 1 ξ 2 3 + ξ 2 4 ≤ 1 -1 ≤ ξ 1 ≤ 1. (16)
For a membrane-orthotropic laminate, geometrical bounds in the PPs space simplify to

         -1 ≤ ρ 0K ≤ 1 0 ≤ ρ 1 ≤ 1 2ρ 2 1 -1 -(-1) -K ρ 0K ≤ 0 (17)
where

ρ 0K := (-1) K A * ρ 0 , K A * ∈ {0, 1}.
In the LPs space, they read (M. ( 1982))

   -1 ≤ ξ 3 ≤ 1 2ξ 2 3 -1 ≤ ξ 1 ≤ 1. ( 18
)
As discussed in (Montemurro (2015a,b); Vannucci (2017)), the design of a laminate lives in R 12 , since four PPs are needed to define the anisotropic part of tensors A * , B * and D * (the deviatoric part of tensor H * being directly related, in general, to the anisotropic one of tensors A * and D * ). Regarding LPs, the same remark holds since 12 LPs are needed to fully describe the laminate behaviour in the CLT framework, as stated above. For both representations, the feasible domain is then a subset of R 12 . The determination of the feasible domain, in the most general case, is still an open problem. As stated in the introduction, in the literature some misleading results can be found about the convexity of the laminate feasible domain. For example, in [START_REF] Grenestedt | Layup optimization of composite material structures[END_REF]), the feasible domain in the LPs space is claimed to be convex. Conversely, in [START_REF] Vannucci | Anisotropic Elasticity[END_REF]), the feasible domain in the PPs space is claimed to be non-convex for anisotropic laminates and convex for orthotropic ones.

The main result of this Section is a rigorous proof of the non-convexity of the feasible domain in the LPs and PPs spaces, both for anisotropic and orthotopic laminates. To this purpose, some nomenclature must be introduced.

Let z * ∈ [0, 1] be the dimensionless coordinate defined through the laminate thickness, from the bottom to the top. The layup function is a combination of piecewise functions defined as

θ(z * ) := N i=1 c i χ[∆z * i ], (19)
where

N ≥ 1, N ∈ N, c i ∈ [-π 2 , π 2 ] and χ[∆z * i ] is the indicator function assuming unit value in the interval ∆z * i . The function χ [∆z * i ] is defined as: χ [∆z * i ] :=    1 if z * ∈ ∆z * i , 0 otherwise . (20) Intervals ∆z * i are such that N i=1 ∆z * i = [0, 1], N i=1 ∆z * i = ∅ and meas(∆z * i ) = 1/N . It is clear that the range of θ(z * ) is [-π 2 , π 2 ].
The layup function can be seen as the SS counterpart when the integral description is used instead of summations.

Proposition 3.1. The set of layup functions

Θ N := θ(z * ) : θ(z * ) = N i=1 c i χ[∆z * i ], c i ∈ [-π 2 , π 2 ] is convex.
Proof. Let α ∈ [0, 1] and suppose that θ(z * ) and θ(z * ) belong to Θ N . Therefore, there exist some ĉi and či belonging to the interval

[-π 2 , π 2 ] such that θ(z * ) = N i=1 ĉi χ[∆z * i ] and θ(z * ) = N i=1 či χ[∆z * i ].
A convex combination of these two elements satisfies the following identity:

α θ + (1 -α) θ = α N i=1 ĉi χ[∆z * i ] + (1 -α) N i=1 či χ[∆z * i ] = N i=1 (α ĉi + (1 -α)č i ) χ[∆z * i ] =: N i=1 ci χ[∆z * i ], (21)
where ci are defined through the last equality. Therefore, the right-hand side member belongs to Θ N for every α and for every θ(z * ), θ(z * ). This claim is sufficient to conclude the proof.

The following lemma [START_REF] Boyd | Convex Optimization[END_REF]) is needed before introducing the main result of this Section.

Lemma 3.1. Let C ⊂ R m×n be a convex set. Then, the projection over some of its

coordinates P(C) = {x 1 ∈ R m | (x 1 , x 2 ) ∈ C for some x 2 ∈ R n } is convex.
It is then immediate the following Corollary 3.1. If a projection P(S) (in the sense of Lemma 3.1) of a set S is non-convex, then S is a non-convex set.

Of course, Lemma 3.1 has an important consequence: if a projection of the set S is convex, the set is not necessarily convex. Indeed, the proof provided in [START_REF] Grenestedt | Layup optimization of composite material structures[END_REF]) about the convexity of the LPs space does not take into account for this aspect.

Let p[θ] denote the vector consisting of twelve PPs (or LPs) obtained with the layup function θ(z * ). The following four Propositions express the non-convexity of the feasible domain regardless of the anisotropy representation. A more detailed proof is given only for full-anisotropic laminates in PPs space, the remaining four being conceptually identical.

Proposition 3.2. The feasible region in the PPs space of all anisotropic laminates composed of N plies

Π N := {p[θ(z * )] ∀ θ(z * ) ∈ Θ N } is a non-convex bounded subset of R 12 for every N > 1 (N ∈ N).
Proof. For the boundedness, it is sufficient to see that the twelve components of p are bounded. In fact, the six dimensionless anisotropic moduli (ρ 0 , ρ 1 for tensor A * , B * , D * ) take values in the set [0, 1], whilst the six dimensionless polar angles

(φ 0 := Φ 0 π/4 and φ 1 := Φ 1 π/2 for tensors A * , B * , D * ) take values in the set [-1, 1]. Therefore, Π N is a bounded subset of the 12D-parallelepiped [0, 1] 6 × [-1, 1] 6 ⊂ R 12 .
Let p := p[ θ(z * )] and p := p[ θ(z * )] be two points of the feasible domain for the layup functions θ(z * ) and θ(z * ) belonging to Θ N . Moreover, consider the interval [0, 1] subdivided into N disjoint intervals ∆z * i of equal length, and let α ∈ [0, 1]. A convex set, by definition, contains the whole line segment that joins any two points belonging to the set. For the characterisation of Π N , its convexity would imply the existence of a layup function θ ∈ Θ N such that p[ θ] = αp + (1 -α)p belongs to Π N for every α and for every p, p. Therefore, to prove the thesis of the proposition, it is sufficient to prove the violation of such definition at least for one case. Moreover, thanks to Proposition 3.1 and Corollary 3.1, it is sufficient to seek the violation in the projection of Π N onto some of its hyper-planes. To this purpose, consider the projection of Π N onto the plane (ρ 1 , ρ 0 ).

If layup functions are considered instead of SSs:

ρ 0 [θ(z * )] :=   1 0 cos 4θ(z * )dz *   2 +   1 0 sin 4θ(z * ) dz *   2 = 2 1 0 1 0 cos 2 2(θ(z * ) -θ(t * )) dz * dt * -1, (22)
and

ρ 1 [θ(z * )] :=   1 0 cos 2θ(z * )dz *   2 +   1 0 sin 2θ(z * ) dz *   2 = 1 0 1 0 cos 2(θ(z * ) -θ(t * )) dz * dt * , (23)
where the identities f (x) dx g(x) dx = f (x)g(t) dx dt, cos(α -β) = cos α cos β + sin α sin β and cos(4α) = 2 cos 2 2α -1 have been used. Consider two points of the (ρ 1 , ρ 0 ) plane having coordinates P = (1, 1) and P = (1 -2 N , 1). These two points belong to Π N . In fact, P corresponds to layup functions of the form

N i=1 cχ[∆z * i ] (c ∈ [-π 2 , π 2
]), whilst P corresponds to layup functions of the form

N -1 i=1 cχ[∆z * i ] + -π 2 + c χ[∆z * N ] (c ∈ [-π 2 , π 2
]), as it can be easily verified. Then, assuming that the convex combination of these two points belongs to Π N , one has:

α 1 -2 N 1 + (1 -α) 1 1 = ρ 1 [ θ(z * )] ρ 0 [ θ(z * )] , (24)
for some θ(z * ) ∈ Θ N . The second component of Eq. ( 24) is satisfied only if ρ 0 [ θ(z * )] = 1. Considering Eq. ( 22), it means that

1 0 1 0 cos 2 2(θ(z * ) -θ(t * )) dz * dt * =
1. This condition is achieved if θ(z * ) is the constant function (the line segment represented by Eq. ( 24) would collapse in a point, so this case is disregarded) or if it is of the form

θ(z * ) = M i=1 cχ[∆z * i ] + N i=M +1 - π 2 + c χ[∆z * i ], (25)
for some M < N , M ∈ N. Assuming θ(z * ) as in Eq. ( 25), the value of ρ 1 θ (z * ) can be calculated from Eq. ( 23) as:

ρ 2 1 [ θ(z * )] = 1 0 1 0 cos 2( θ(z * ) -θ(t * ))dt * dz * = 1 0 1 0 cos 2 M i=1 cχ[∆z * i ] + N i=M +1 - π 2 + c χ[∆z * i ]+ - M i=1 cχ[∆t * i ] - N i=M +1 - π 2 + c χ[∆t * i ] dt * dz * = M/N 0 M/N 0 cos 2 c M i=1 χ[∆z * i ] -χ[∆t * i ] dt * dz * + M/N 0 1 M/N cos 2 c M i=1 χ[∆z * i ] -- π 2 + c N i=M +1 χ[∆t * i ] dt * dz * + 1 M/N M/N 0 cos 2 -c M i=1 χ[∆t * i ] + - π 2 + c N i=M +1 χ[∆z * i ] dt * dz * + 1 M/N 1 M/N cos 2 - π 2 + c N i=M +1 χ[∆z * i ] -χ[∆t * i ] dt * dz * = M N 2 - M N 1 - M N - M N 1 - M N + 1 - M N 2 = 1 -2 M N 2 , (26)
and, hence,

ρ 1 [ θ(z * )] = 1 -2 M N 2 . ( 27
)
Finally, from Eqs. ( 24) and ( 27), one obtains

1 - 2α N = 1 -2 M N 2 =    1 -2 M N , if 1 > 2 M N , 2 M N -1, otherwise . (28)
From Eq. ( 28), either M = α or M = N -α. Both results are absurd, since M must be, by definition, an integer number (for all α). Therefore, no θ(z * ) exists satisfying Eq. ( 24), and the entire line segment between P and P does not belong to the projection of Π N onto the (ρ 1 , ρ 0 ) plane. This counter-example proves the statement of the Proposition, remembering also Corollary 3.1.

It is noteworthy that Proposition 3.2 considers only geometrical aspects of laminate layups. With the same argument used to prove Proposition 3.2, one can prove that the feasible domain of laminates with an orthotropic membrane behaviour is a non-convex bounded set.

Proposition 3.3. The feasible region in the PPs space of all membrane-orthotropic

laminates composed of N plies Π Ort N := {p[θ(z * )] | Φ A * 0 -Φ A * 1 = K A * π 4 , K A * ∈ {0, 1}, ∀ θ(z * ) ∈ Θ N } is a non-convex bounded subset of R 12 for every N > 1 (N ∈ N).
Proof. To show the boundedness, the argument is the same of the proof of Proposition 3.2. Adopting the integral description,

Φ 0 [θ(z * )] := 1 4 arctan      1 0 sin 4θ(z * )dz * 1 0 cos 4θ(z * )dz *      , Φ 1 [θ(z * )] := 1 2 arctan      1 0 sin 2θ(z * )dz * 1 0 cos 2θ(z * )dz *      . (29)
For the non-convexity, it is sufficient to notice that points P and P , used for the the proof of Proposition 3.2, actually belong to Π Ort N (with K A * = 0).

Proposition 3.4. The feasible region in the LPs space of all anisotropic laminates composed of N plies

Π N := {p[θ(z * )] ∀ θ(z * ) ∈ Θ N } is a non-convex bounded subset of R 12 for every N > 1 (N ∈ N).
Proof. To show the boundedness, the argument is the same as the proof of Proposition 3.2. For the non-convexity, the same argument of Proposition 3.2 is used, but settled in the (ξ 3 , ξ 1 ) plane. Adopting the integral description,

ξ 1 = 1 0 cos 4θ(z * )dz * , ξ 3 = 1 0 cos 2θ(z * )dz * . ( 30
)
Let P = (1, 1) and P = (1 -2 N , 1) be the same points used for Proposition 3.2. These two points belong to Π N . In fact, P corresponds to layup functions of the form

N i=1 0χ[∆z *
i ], whilst P corresponds to layup functions of the form

N -1 i=1 0χ[∆z * i ]+ -π 2 + 0 χ[∆z * N ]
, as it can be easily verified. The convexity condition, to be confuted, reads:

ξ 3 [ θ(z * )] ξ 1 [ θ(z * )] = α P + (1 -α) P = α 1 -2 N 1 + (1 -α) 1 1 (31)
for some θ(z * ) ∈ Θ N and for all α ∈ [0, 1]. Considering the second component of Eq. ( 31), θ(z * ) is necessarily of the form

θ(z * ) = M i=1 c 1 χ[∆z * i ] + N i=M +1 c 2 χ[∆z * i ] (32) with c 1 , c 2 ∈ {0, -π 2 , π 2 } and M < N , M ∈ N.
As a consequence, it is easy to see that, for all combinations of c 1 and c 2 ,

ξ 3 [ θ(z * )] ∈ {-1, 1, -1 + 2 M N , 1 -2 M N }. The first component of Eq. (31) reads {-1, 1, -1 + 2 M N , 1 -2 M N } = 1 - 2α N
, which either cannot be satisfied regardless of α or violates the condition M ∈ N. Considerations similar to those of the proof of Proposition 3.2 can also be repeated for this case, allowing to conclude the proof.

Proposition 3.5. The feasible region in the LPs space of all membrane-orthotropic

laminates composed of N plies Π Ort N := {p[θ(z * )] | Φ A * 0 -Φ A * 1 = K A * π 4 , K A * ∈ {0, 1}, ∀ θ(z * ) ∈ Θ N } is a non-convex bounded subset of R 12 for every N > 1 (N ∈ N).
Proof. The proof follows straightforwardly from the previous ones. Indeed, points P and P used in the proof of Proposition 3.4 correspond to membrane-orthotropic laminates (they are a particular case of the ones chosen in the proof of Proposition 3.2 and 3.3). Therefore, the proof can be considered concluded.

Propositions 3.2 and 3.4 claim that, for an anisotropic laminate, the feasible region Π N , in terms of PPs or PPs, is a non-convex set. Propositions 3.3 and 3.5 state that the non-convexity of the feasible domain holds also in the case of laminates having an orthotropic membrane stiffness tensor. Specifically, the non-convexity is maintained in the projection of the 12-D feasible domain onto either the (ρ 1 , ρ 0K ) or the (ξ 3 , ξ 1 ) plane1 .

A way to visualise the non-convexity of the feasible domain is, trivially, to plot all the possible points of Π N and Π Ort N , projected in the corresponding plane according to the anisotropy representation, for an arbitrary number of plies N . Inasmuch as the number of all the possible SSs grows exponentially with the number of layers N , a sampling step is assumed. It is convenient to introduce the notion of number of groups, denoted by m, which is is the number of distinct orientations within a SS. The number of plies associated to the i-th orientation is denoted by n i (obviously m i=1 n i = N ). Figure 1 shows the feasible domain projections for a laminate having N = 4 plies (K = 0). Indeed, the projected feasible domains are non-convex sets. In [START_REF] Grenestedt | Layup optimization of composite material structures[END_REF]), it was claimed that the feasible domain in LPs space is convex. The thesis and the proof of this claim are erroneous. The authors consider one of the twelve components of p at time. Each component of p is a continuous function whose range is a continuous bounded set: in R such a set (a segment) is necessary convex. Moreover, considering Lemma 3.1, one cannot conclude that a set is convex by knowing that some (not all) of its projections are convex sets. The point is that the components of p are not independent, since they rely on the same SS. This mutual influence is the actual reason of the shrinkage of the feasible domain from the 12D parallelepiped, as Fig. 1 clearly shows.

The feasible region of the limit infinite-ply laminate, i.e. Π ∞ , is, actually, the convex hull of the feasible region of laminates regardless of the number of plies. However, a laminate has always a finite number of plies, which makes the determination of Π N the true problem to address.

Some Exact Solutions for Geometrical Bounds

In this section, only anisotropic laminates, in the PPs space, will be considered. The non-convexity of the feasible region Π N is preserved in the projection onto the (ρ 1 , ρ 0 ) plane.

Fig. 2 shows some results for the cases N = 2, • • • , 7, equivalent to the cases m = 2, • • • , 7 with n i mutually equal. It is evident the non-convexity of the feasible domain, which may degenerate to a curve or to a point. In fact, it is easy to see that, if N = 1 or m = 1, i.e. the case of a laminate where all plies share a single orientation, ρ 0 = ρ 1 = 1.

More interesting is the case m = 2. After few calculations, considering that n 2 = N -n 1 , one achieves the following relation:

ρ 0 = λ 2 + (1 -λ) 2 + 2λ(1 -λ) ρ 2 1 -λ 2 -(1 -λ) 2 2λ(1 -λ) 2 -1 , (33)
or, in the implicit form:

F 2 (ρ 0 , ρ 1 , λ) := ρ 0 -λ 2 + (1 -λ) 2 + 2λ(1 -λ) ρ 2 1 -λ 2 -(1 -λ) 2 2λ(1 -λ) 2 -1 , (34)
where λ := n 1 /N . Eq. ( 33) means that, for m = 2, the locus of lamination points, i.e. the pairs (ρ 1 , ρ 0 ), is represented by a family of curves, parametrised with the relative number of plies of each orientation within the SS. Eq. ( 33) intersects the axis ρ 0 = 1 for the following values of ρ 1 :

         1 -2λ, if λ ≤ 1/2, 2λ -1, if λ ≥ 1/2, 1, ( 35
) (a) N = 2 (discretisation step = 1 • ) (b) N = 3 (discretisation step = 1 • ) (c) N = 4 (discretisation step = 2 • ) (d) N = 5 (discretisation step = 3 • ) (e) N = 6 (discretisation step = 5 • ) (f) N = 7 (discretisation step = 6 • )
Figure 2: Feasible domain projection onto the plane (ρ 1 , ρ 0 ) for anisotropic laminates of N plies (equivalent to consider laminates with m distinct orientations, all of them appearing the same number of times within the SS)

in perfect agreement with the proof of Proposition 3.2. The family of implicit curves, expressed by Eq.( 34) in the form of F 2 (ρ 0 , ρ 1 , λ) = 0, admits envelope, represented by the well-known geometrical-bound ρ 0 = 2ρ 2 1 -1 [START_REF] Vannucci | A note on the elastic and geometric bounds for composite laminates[END_REF]). As an example, Fig. 3 shows the lamination points locus for a laminate having N = 20, m = 2, when n 1 varies in the range [0, N ] as a subset of N. The analytic solution of Eq. ( 33) is represented by the red curves. It is noteworthy the discrete nature of the locus, due to the discrete nature of the stack. The process may be conceptually extended to a higher number of groups. However, computations become intricate. For the sake of simplicity, consider the case m = 3. Eqs. ( 14) and ( 15) can be rearranged as (see Appendix A)

ρ 0 = 1 N 3 j=1 n 2 i -2 2 i=1 3 j=i+1 n i n j + 4 2 i=1 3 j=i+1 n i n j cos 2 (2∆θ ij ), (36)
ρ 1 = 1 N 3 j=1 n 2 i + 2 2 i=1 3 j=i+1 n i n j cos(2∆θ ij ). (37)
Injecting, for example, the expression of cos (2∆θ 13 ) from Eq. (37) into Eq. ( 36), one obtains

F 3 (ρ 0 , ρ 1 ; ∆θ 12 , ∆θ 23 ) := ρ 0 - 1 N 3 j=1 n 2 i -2 2 i=1 3 j=i+1 n i n j + 4 2 i=1 n i n (i+1) cos 2 (2∆θ i(i+1) ) + 1 n 1 n 3 N 2 ρ 2 1 - 3 j=1 n 2 i -2 2 i=1 n i n (i+1) cos(2∆θ i(i+1) ) 2   1/2 . ( 38
)
20

For the envelope, the following system must be solved:

               F 3 (ρ 0 , ρ 1 ; ∆θ 12 , ∆θ 23 ) = 0, ∂F 3 ∂∆θ 12 (ρ 0 , ρ 1 ; ∆θ 12 , ∆θ 23 ) = 0, ∂F 3 ∂∆θ 23 (ρ 0 , ρ 1 ; ∆θ 12 , ∆θ 23 ) = 0, (39)
which simplifies to

                   F 3 (ρ 0 , ρ 1 ; ∆θ 12 , ∆θ 23 ) = 0, sin(2∆θ 12 ) N 2 ρ 2 1 - 3 j=1 n 2 i -2n 1 (n 2 + n 3 ) cos(2∆θ 12 ) -2n 2 n 3 cos(2∆θ 23 ) = 0, sin(2∆θ 23 ) N 2 ρ 2 1 - 3 j=1 n 2 i -2n 3 (n 1 + n 2 ) cos(2∆θ 23 ) -2n 1 n 2 cos(2∆θ 12 ) = 0. (40)
Four are the possibilities to annihilate the last two expressions of Eq. ( 40).

Case A: i+1) , i = 1, 2. This case corresponds to a SS composed by parallel laminae. This condition corresponds to the point ρ 0 = ρ 1 = 1.

2∆θ 12 = K 12 π, 2∆θ 23 = K 23 π, K 12 , K 23 ∈ Z.. Therefore, cos(2∆θ i(i+1) ) = (-1) K i(
Case B: 2∆θ 12 = K 12 π, 2∆θ 23 = K 23 π, K 12 , K 23 ∈ Z.. From Eq.(40) 3 , one obtains the expression of cos(2∆θ 23 ), and thus

               cos(2∆θ 23 ) = N 2 ρ 2 1 - 3 i=1 n 2 i -2n 1 n 2 (-1) K 12 2n 3 (n 1 + n 2 ) , ρ 0 = 1 N 3 j=1 n 2 i -2 2 i=1 3 j=i+1 n i n j + 4n 1 n 2 + 4n 3 (n 1 + n 2 ) cos 2 (2∆θ 23 ) 1/2 . ( 41
)
Only the case K 12 = 0 is effective, corresponding to the case m = 2, because orientations 1 and 2 coincide.

Case C:

2∆θ 12 = K 12 π, 2∆θ 23 = K 23 π, K 12 , K 23 ∈ Z.
. This case is analogous to case B, exchanging indices 1, 2 with 2, 3, respectively. Fig. 4 depicts the curves of cases B or C for a laminate having N = 6, m = 3 and n 1 = 1, n 2 = 2, n 3 = 3, obtained by permuting indices 1, 2, 3 modulo 3. This is due to the arbitrary substitution of cos (2∆θ 13 ) in the derivation of Eq. ( 38). Because of the cyclic permutation, cases B and C coincide. A further case, arising from a different change of variables, is worthy to be considered.

Case E: 2∆θ 12 = K 12 π, 2∆θ 23 = K 23 π, K 12 , K 23 ∈ Z.. Subtracting term by term the last two formulae of Eq.( 40), one gets the condition cos(2∆θ 12 ) = cos(2∆θ 23 ). Note that cos(2∆θ 13 ) = cos(2∆θ 12 + 2∆θ 23 ). From Eq. (37) one obtains the value of parameter cos(2∆θ 12 ).

                     cos(2∆θ 12 ) = cos(2∆θ 23 ) = N 2 ρ 2 1 - 3 i=1 n 2 i -2n 1 n 3 2n 2 (n 1 + n 3 ) , if 2∆θ 12 = -2∆θ 23 , cos(2∆θ 12 ) = cos(2∆θ 23 ) = -b ± √ b 2 -4c 2 , if 2∆θ 12 = 2∆θ 23 , F 3 (ρ 0 , ρ 1 ; ∆θ 12 , ∆θ 23 ) = 0, (43)
where

b = n 2 (n 1 + n 3 ) 2n 1 n 3 , c = - N 2 ρ 2 1 - 3 i=1 n 2 i + 2n 1 n 3 4n 1 n 3 . ( 44
)
Fig. 6 depicts the bound. The area it encloses is always inside the actual feasible region.

4.1. Exact analytic expression of the feasible domain for the case n 1 = n 2 = n 3 = n

In the case n 1 = n 2 = n 3 = n, for every N = 3n, the aforementioned relationships can be further simplified. It is noteworthy that for this case, Eqs. ( 41) and (43) provide the analytic expression of the actual projection of the feasible domain, i.e.

                       0 ≤ ρ 0 ≤ 1, 0 ≤ ρ 1 ≤ 1, ρ 0 ≤ ρ 1 |3ρ 1 + 2| , ρ 0 ≥ ρ 1 |3ρ 1 -2| , ρ 0 ≤ 1 3 1 + (9ρ 2 1 -5) 2 2 , (45)
which, of course, does not depend on N . Fig. 7 shows the result for this case. To the best of the authors' knowledge, it is the first time that an exact closed-form expression for the projection of the feasible domain of a laminate is given.

New Geometrical Bounds

It is possible to infer that Eq. ( 34) evaluated for λ = 1/N , i.e. F 2 (ρ 0 , ρ 1 , 1/N ) = 0, is, in any case, a geometrical bound for anisotropic laminates. Similarly, F 2 ((-1) K ρ 0K , ρ 1 , 1/N ) = 0 is a geometrical bound for membrane-orthotropic laminates.

From the proofs of Propositions 3.2 and 3.3, there exists a region in the (ρ 1 , ρ 0 ) plane that is excluded from the feasible domain projection. This results has a physical interpretation. For a SS made of N plies, the case m = 2, n 1 = 1, n 2 = N -1 is the minimal condition to make the feasible domain not to degenerate into a point (case n 1 = 0). Therefore, no lamination points can lie in the epigraph of F 2 (ρ 0 , ρ 1 , 1/N ) = 0. If so, the amount of plies would be a non-integer number, leading, thus, to a contradiction. The excluded area is more important for relative small N , whilst it can be negligible as N increases.

Therefore, new geometrical bounds can be proposed for a general anisotropic laminate:

                 0 ≤ ρ 0 ≤ 1, 0 ≤ ρ 1 ≤ 1, 2ρ 2 1 ≤ 1 -ρ 2 0 1 -(-1) K ρ 0 cos 4Φ A * 0 , F 2 (ρ 0 , ρ 1 , 1/N ) ≤ 0. (46)
Moreover, for a laminate having an orthotropic membrane stiffness tensor, the above bounds simplifies to

               -1 ≤ ρ 0K ≤ 1, 0 ≤ ρ 1 ≤ 1, 2ρ 2 1 -1 -(-1) K ρ 0K ≤ 0 F 2 ((-1) K ρ 0K , ρ 1 , 1/N ) ≤ 0. (47)
It is noteworthy that, for a quasi-homogeneous laminate, Eqs. ( 46) and (47) offer a description of the laminate feasible domain richer than those available in the literature (M. (1982); Vannucci (2012)).

Figure 8 shows the Vannucci's geometrical bounds [START_REF] Vannucci | A note on the elastic and geometric bounds for composite laminates[END_REF]) and and the proposed bound for anisotropic (Eq. ( 46)) and membrane-orthotropic (Eq. ( 47)) laminates with N = 4.

It is remarkable that the proposed bounds formulation depends also on the number of plies of the laminate.

On the Necessary and Sufficient Conditions for Uncoupling and Homogeneity

QT SSs are characterised by an interesting and very useful property: membrane/bending uncoupling and/or homogeneity requirements can be exactly met regardless of the values of the orientation angles. In particular, these requirements can be fulfilled by acting only on the position of the layers into the stack [START_REF] Garulli | Quasitrivial stacking sequences for the design of thick laminates[END_REF]; [START_REF] Vannucci | A special class of uncoupled and quasihomogeneous laminates[END_REF]). QT SSs have been efficiently used in many practical problems [START_REF] Montemurro | On the effective integration of manufacturability constraints within the multi-scale methodology for designing variable angletow laminates[END_REF]; [START_REF] Montemurro | A multi-scale approach for the simultaneous shape and material optimisation of sandwich panels with cellular core[END_REF][START_REF] Montemurro | Least-weight composite plates with unconventional stacking sequences: Design, analysis and experiments[END_REF][START_REF] Montemurro | A general multi-scale two-level optimisation strategy for designing composite stiffened panels[END_REF]). Specifically, a QT stack represents an equivalence class for all possible orientations that each group of plies can assume. As an example, {90

• , -26 • , 90 • , 90 • , 90 • , -26 • , 90 • } and {1 • , 42 • , 1 • , 1 • , 1 • , 42 • , 1 • }
are elements of the same equivalence class [{0, 1, 0, 0, 0, 1, 0}], where 0 and 1 are just labels identifying two possibly distinct orientations. Of course, the choice of the orientations depends upon the desired elastic behaviour of the laminate.

To explain clearly the concept of QT solutions, consider a laminate with N plies and m ≤ N different orientations and define

G j := {k : θ k = θ j }, (48)
the set of indices within the SS sharing the same orientation θ j . Conditions for uncoupling and homogeneity can be than split as multiple sums over the different sets G j , j = 1, . . . , m [START_REF] Garulli | Quasitrivial stacking sequences for the design of thick laminates[END_REF]). Therefore, the uncoupling condition reads:

N k=1 b k e iβθ k = m j=1 e iβθ j k∈G j b k = 0, β = 2, 4, (49)
while the homogeneity requirement can be expressed as:

N k=1 c k e iβθ k = m j=1 e iβθ j k∈G j c k = 0, β = 2, 4. (50)
In this context, a group of plies oriented at θ j , for which

k∈G j b k = 0, k∈G j c k = 0, (51)
is called saturated group with respect to coefficient b k and c k , respectively. QT SSs are entirely composed of saturated groups. For more details on this topic, the reader is addressed to [START_REF] Garulli | Quasitrivial stacking sequences for the design of thick laminates[END_REF][START_REF] Vannucci | Anisotropic Elasticity[END_REF]).

In this Section, some theorems of linear algebra are used to prove that quasitriviality is only a sufficient condition for uncoupling and/or homogeneity for m ≥ 3. For m = 2, quasi-triviality is also a necessary condition. The results are based on the following two theorems discussed in [START_REF] Green | The linear dependence of functions of several variables, and completely integrable systems of homogeneous linear partial differential equations[END_REF]), which are reported here for the sake of completeness. Theorem 6.1. Consider the p-dimensional cube A := (-π/2, π/2) p . Let y 1 and y 2 be functions of the p independent variables u 1 , u 2 ,. . . ,u p for which all partial derivatives of the first order, ∂y 1 /∂u k , ∂y 2 /∂u k , (k = 1, 2, . . . , p) exist throughout the region A. Furthermore, suppose that one of the functions, i.e. y i , does not vanish in A. Then, if all the two-rowed determinants in the matrix

              y 1 y 2 ∂y 1 u 1 ∂y 2 u 1 ∂y 1 u 2 ∂y 2 u 2 . . . . . . ∂y 1 u p ∂y 2 u p               , (52)
Eq. ( 52) reads

   e iβθ 1 e iβθ 2 iβe iβθ 1 0 0 iβe iβθ 2    . (55)
One can easily check that none of the two-rowed determinants of Eq.( 55) is singular in A = (-π/2, π/2) 2 . Therefore, it follows that, for m = 2, the combination of functions e iβθ j , j = 1, 2, β = 2, 4 is linearly independent. Accordingly, the proof can be considered concluded.

Remark 6.1. Proposition 6.1 means that uncoupled and/or homogeneous laminates imply the annealing of the sum of coefficients b k and/or c k associated to each orientation. In other words, when m = 2, uncoupled and/or homogeneous laminates can be composed only of QT stacks and vice versa.

Remark 6.2. In the light of Theorem 6.1 and Proposition 6.1, it is not possible to obtain membrane-isotropic laminates with only two different orientations, since the sum of coefficients a k cannot be zero. This result is in agreement with the evidence that no isotropic SS is known having only two distinct orientation angles [START_REF] Vannucci | Anisotropic Elasticity[END_REF]; [START_REF] Warren | Mechanical properties of laminate design to be isotropic[END_REF]; [START_REF] Wu | Fully Isotropic Laminates and Quasi-Homogeneous Anisotropic Laminates[END_REF]).

Remark 6.3. The application of Theorem 6.2 for m ≥ 3 is cumbersome, and it is not reported here for the sake of simplicity. The application of such a Theorem proves that functions e iβθ j , j = 1, . . . , m, β = 2, 4 are linearly dependent for m ≥ 3. Therefore, for m ≥ 3, one can find uncoupled and/or homogeneous laminates which are not QT solutions. Moreover, in this case, membrane-isotropic laminates exist. For example, the laminate [0/ -60/60/0/60 2 / -60 3 /0 2 /60], from [START_REF] Vannucci | Anisotropic Elasticity[END_REF]), is membrane-isotropic (the isotropic behaviour in membrane has been obtained by using the well-known Warren and Norris rule [START_REF] Warren | Mechanical properties of laminate design to be isotropic[END_REF])). For this laminate, the isotropy condition is achieved because of the linearly dependence of the complex exponential functions appearing in the definition of the anisotropic moduli, since a k = 1, ∀k.

Conclusions

In this paper two aspects linked to the nature of the feasible domain of composite laminates have been studied. Proofs of the non-convexity of the feasible domain in polar parameters and lamination parameters spaces have been provided for both anisotropic and orthotropic-membrane laminates. The results correct the erroneous common belief of the convexity of the feasible domain in lamination parameters space. In the light of this finding, the old feasible domain can be interpreted as the convex hull of the projected feasible domain, regardless of the number of plies composing the laminate.

This work clarifies some preliminary aspects of the feasible domain determination problem. Although tackling the problem in full generality is probably impractical, a closed form of stricter feasibility bounds has been derived in terms of the membrane stiffness tensor polar parameters. This is an aspect of paramount importance in the optimisation of composite laminates, where the anisotropy is tailored to satisfy some merit functions and constraints.

Furthermore, the problem of retrieving sufficient and necessary conditions for membrane / bending uncoupling and homogeneity has been addressed in this study. It has been shown that, in general, quasi-triviality is only a sufficient condition. It becomes also a necessary one if the stacking sequence has only two distinct orientations.

As prospects of this study are concerned, the derivation of the analytic expressions of the feasible domain of laminates (in both lamination parameters and polar parameters spaces), at least for the membrane stiffness tensor, should be derived in more general cases, maintaining the dependence from the number of plies. These expressions must be included in the formulation of the optimisation problem of the composite in order to get true feasible optimal solutions. Moreover, to support the composite design, necessary and sufficient conditions to achieve uncoupled and/or homogenous and/or isotropic laminates should be derived. This could represent a step-forward to the definition of the feasible domain considering membrane and bending stiffness tensors at once.

Of course, the main (and most difficult) problem to address still remains the derivation of the actual feasible domain of a laminate when the full stiffness matrix is considered either in the fully anisotropic case or when introducing some hypotheses on the laminate stiffness tensors elastic symmetries.

Figure 1 :

 1 Figure 1: Feasible domain projections for anisotropic (on the left) and membrane-orthotropic (on the right) laminates (K = 0) of N = 4 plies, on the top in terms of PPs, on the bottom in terms of LPs (discretisation step = 5 • ). In red the points P and P used in the proofs of Section 3.

Figure 3 :

 3 Figure 3: Lamination points for N = 20, m = 2 (discretisation step = 3 • ).

Figure 4 :

 4 Figure 4: Cases B or C for n 1 = 1, n 2 = 2, n 3 = 3, (discretisation step = 3 • ) by permuting indices 1, 2, 3 modulo 3.

  (a) Lamination points and envelope, n 1 = 1, n 2 = 2, n 3 = 3 (b) Lamination points and envelope, n 1 = 1, n 2 = 2, n 3 = 6 (c) Lamination points and envelope, n 1 = 1, n 2 = 2, n 3 = 9 (d) Lamination points and envelope, n 1 = 1, n 2 = 2, n 3 = 12

Figure 5 :

 5 Figure 5: Case D for four different laminates (discretisation step = 3 • ) .

  (a) Lamination points and envelope, n 1 = 1, n 2 = 2, n 3 = 3 (b) Lamination points and envelope, n 1 = 1, n 2 = 2, n 3 = 6 (c) Lamination points and envelope, n 1 = 1, n 2 = 2, n 3 = 9 (d) Lamination points and envelope, n 1 = 1, n 2 = 2, n 3 = 12

Figure 6 :

 6 Figure 6: Case E for four different laminates.

Figure 7 :

 7 Figure 7: Lamination points and envelope for the case n 1 = n 2 = n 3 = n (discretisation step = 3 • ).

Figure 8 :

 8 Figure 8: Vannucci's geometrical bound (Vannucci (2012)) and proposed bounds for a laminate having N = 4 (discretisation step = 3 • )

In the proof of Proposition 3.3, the non-convexity appears in the (ρ 1 , ρ 0 ) plane. However, since the points P and P correspond to orthotropic laminates with K A * = 0, the line segment delimited by the two points is still not included in the projection of the feasible domain onto the (ρ 1 , ρ 0K ) plane.

Acknowledgements

This paper presents part of the activities carried out within the research project PARSIFAL (PrandtlPlane ARchitecture for the Sustainable Improvement of Future AirpLanes), which has been funded by the European Union under the Horizon 2020 Research and Innovation Program (Grant Agreement n.723149). The authors would like to thank Prof. Massimiliano Lucchesi (University of Florence) for constructive criticism of Section 3.

vanish identically in A, y 1 and y 2 are linearly dependent in A. Theorem 6.2. Consider the p-dimensional cube A := (-π/2, π/2) p . Moreover, consider the matrix M s (y 1 , y 2 , . . . , y r ) in which the first row consists of the functions y 1 , y 2 up to y r (for some r) and the other s rows of derivatives of these functions:

Let the set of n functions y 1 , y 2 ,. . . ,y n of the p independent variables u 1 , u 2 ,. . . ,u p possess enough partial derivatives, of any orders whatever, to form a matrix M = M (n-2) (y 1 , y 2 , . . . , y n ), of n columns and n -1 rows, in which at least one of the (n -1)-rowed determinants, i.e.

vanishes nowhere in A. Moreover, suppose that all of the first derivatives of each of the elements of the above matrix M exist, and adjoin to the matrix M such of these derivatives as do not already appear in M, to form the new matrix M = M q (y 1 , y 2 , . . . , y n ), which has n columns and at least n rows, so that q ≥ n -1.

Then if all the n-rowed determinants of the matrix M in which the determinant W n is a first minor vanish identically in A, the functions y 1 , y 2 ,. . . ,y n are linearly dependent in A.

Thanks to the above Theorems, the following Proposition can be introduced.

Proposition 6.1. Quasi-trivial stacking sequences represent a necessary and sufficient condition to satisfy membrane/bending uncoupling and/or homogeneity requirements when the number of saturated groups m is equal to two.

Proof. Consider a laminate with a SS composed of m different orientations and apply Theorem 6.1 to the set of functions e iβθ j , j = 1, . . . , m, β = 2, 4. When m = 2, A. A note on the square of the sum of n terms Consider the square of the sum of terms g 1 , g 2 , . . . , g n , n ∈ N. Hence, it easy to see that

where the distributivity property of sum and product operators has been used. Therefore, for a laminate having N plies and m distinct orientations, Eq. ( 14) reads where the identity cos(2α) = 2 cos 2 α -1 has been used. In a similar manner, the expression of ρ 1 of Eq. ( 15) can be rearranged as Eq. ( 37).


Côme Bommier

Le suicide assisté : quand le législatif ne peut se substituer à l'existentiel Assisted suicide: when the legislative cannot

Keywords: Éthique, Euthanasie, Santé publique, Sens, Suicide assisté, Ethics, Euthanasia, Public health, Sense, Assisted suicide

take the place of the existential

Introduction

La question du sens de la vie se pose autant au jeune qu'à l'ancien, dès lors qu'il expérimente sa finitude par une quelconque expérience de mort. Le regard d'un jeune médecin est celui d'un témoin régulier des sévices de la mort, même si elle ne l'a pas encore couvert de son ombre menaçante.

À 30 ans généralement, on ne peut encore prétendre avoir la sagesse nécessaire qui se bâtit en fréquentant la mort au quotidien, en expérimentant sa présence impalpable dans la douleur de vivre ou en sentant sa brise dans l'altération existentielle que peut être une maladie grave. Mais à 30 ans, on a de nombreux chemins devant soi, creusés par tant d'ancêtres. Il nous revient de regarder l'horizon, de relier certains chemins et d'en éviter d'autres, de jeter un regard franc sur nous-mêmes et de faire des choix au service de ce que nous poursuivons sans cesse : le sens.

Le chemin biologique

Les cellules de notre corps meurent principalement par apoptose ou par nécrose. L'apoptose est la mort cellulaire programmée, ou mort cellulaire naturelle. Son étymologie grecque se traduit par la chute au loin, comme si on parlait de la chute des feuilles d'un arbre. Cette mort est soit induite par un signal extérieur venant de l'environnement cellulaire (voie extrinsèque), soit par des signaux internes à la cellule (voie intrinsèque). Cette mort, décrite comme naturelle, participe à la fois à l'homéostasie de nos tissus et elle est aussi nécessaire à de nombreux mécanismes du développement (par exemple, sans apoptose, nous aurions une palme au lieu de 5 doigts). De nombreuses maladies reposent en partie sur la défaillance de l'apoptose, de maladies du développement aux cancers. À l'inverse de l'apoptose, la nécrose est la mort non naturelle, non programmée, anticipée de la cellule.

Cette mort a une origine exogène contre laquelle la cellule n'a pas de mécanisme de défense suffisant. Enfin, ces morts cellulaires se distinguent de l'autophagie (autolyse, autophagocytose) qui permet à la cellule de survivre en « se mangeant soi-même », en grignotant une grande partie de ses composants afin de les recycler elle-même.

Si nous devions comparer le suicide à une de ces morts, peut-être que nous le comparerions à l'apoptose en tant qu'elle est réalisée par la cellule elle-même. Nous pourrions même dire que l'apoptose par voie extrinsèque serait en quelque sorte un suicide assisté. La cellule, trop vieille ou trop défaillante, s'autodétruit. À l'échelle cellulaire, le suicide -qu'il soit assisté ou non -est donc la règle commune.

Le chemin phylogénétique

Les espèces non-humaines habitant la Terre partagent avec nous, les Hommes, une ancestralité phylogénique. Nous partageons des caractères, qu'ils soient moléculaires, cellulaires, organiques, comportementaux ou sociétaux. Après que Charles Darwin l'ait suspecté, nous avons compris la manière dont nous partageons cette ancestralité avec les autres habitants terrestres et nous avons élaboré une théorie de l'évolution, faisant de celui qui est le plus adapté à son milieu de vie le meilleur candidat à la survie. Bien que des comportements d'autodestruction aient été supposés chez des animaux vivant au milieu des Hommes, nous pouvons avancer trivialement que la mort dans le monde non-humain est directement liée à la loi du plus fort ou du plus intelligent : « la loi de la nature », dira-t-on.

Le chemin anthropologique

Animal lui aussi, Homo sapiens est un mammifère primate hominoïde omnivore bipède au cerveau très développé, à l'origine de civilisations qui le dépassent individuellement. Ces à la fois perçu comme un acte de sagesse (« j'ai conscience de ce qui m'attend »), de fugue libertaire (« contre toute attente, je vais partir ailleurs, là où on ne pourra plus me trouver »), voire même de générosité (« je ne veux pas peser sur mes proches ou sur la société, je vous soulage de mon poids »). Cette vision s'appuie sur les chemins biologiques et phylogénétiques évoqués plus haut : l'homme doit mourir pour permettre aux autres de mieux se développer, surtout lorsqu'il n'est plus adapté à son milieu de vie. Le corps devient un déchet en puissance qu'il s'agira de traiter et de recycler ; l'esprit s'éteindra sur un ultime coup d'éclat ; quant à l'âme, qui peut savoir ? Quoiqu'il en soit, la fascination pour le suicide s'appuie sur une exploration intellectuelle, une recherche de réponse absolue en opposition à tout relativisme, qui paradoxalement va prendre corps dans l'anéantissement.

Si la Révolution française est à l'origine de la dépénalisation du suicide, elle a aussi repris à son compte la pensée stoïcienne attribuant à chaque vie humaine une dignité. Sur cette dignité se fondent une multitude de droits de l'Homme en tant qu'Homme. Et puisque nous sommes Hommes et que nous respectons la dignité de tous, alors une vie sociale est possible de même que l'élaboration d'une culture commune. Vouloir régler la question du suicide, ce serait vouloir que nous ayons tous les mêmes concepts de dignité humaine : revient alors la question de l'âme. Si nous regardons de nouveau attentivement les motifs de suicide, nous constatons qu'en dehors de toute croyance religieuse assumée, nous retrouvons les notions suivantes : regard rétrospectif sur sa vie, résolution d'un problème intérieur, perte de sens de la vie à venir. Ces notions témoignent largement de valeurs morales, que celles-ci se rapportent finalement à la déesse-philosophie, la déesse-civilisation ou à la déesse-jeunesse : dans toutes les situations, elles témoignent d'une croyance religieuse (« qui relie ») dont les structures symboliques sont changeantes. L'Homme renverse les codes religieux traditionnels : pétri par la culpabilité de par son conflit avec le réel, il prétend par l'hyperpuissance se donner son propre « jugement dernier », et refuse l'espérance en s'offrant le néant. Se suicider, ce serait donc un non-sens, croire en l'illusion de l'indépendance alors que l'acte se fonderait sur des valeurs morales par essence collectives et donc religieuses. Tout comme la mort est un non-sens absolu [START_REF] Jankélévitch | [END_REF].

Face à la mort, le courage d'être

En période de pandémie et de confinement, l'humanité fait l'expérience de sa finitude. Le confinement a réinstallé un sentiment de solitude dans une société qui vit d'une hypercommunication permanente. Confiner des paysans du XIX° siècle n'aurait pas eu le même impact psychologique que de confiner des citadins du XXI°. En effet, notre monde utopique a globalement oublié la fréquentation de la solitude, de la souffrance, de la mort. Même seuls,

  civilisations obéissent aussi à une loi macro-évolutionniste puisque certaines civilisations se sont éteintes au profit d'autres plus vigoureuses. Ces civilisations s'organisent sur la base d'une culture toujours en évolution. Nous pouvons reprendre L-V. Thomas en disant que « toute civilisation se construit sur l'illusion de son immortalité »[START_REF] Thomas | Anthropologie de la Mort[END_REF]. Sans imaginer de lendemain, une civilisation n'a pas de raison à sa survie. Souvenons-nous avec tristesse des cas de suicides collectifs chez les indiens Guarani en Amazonie. À l'échelle individuelle, Freud prétend que « la mort n'existe pas pour l'inconscient »[START_REF] Sfreud | Deuil et mélancolie[END_REF]. Il est donc difficile de concevoir le suicide d'une civilisation, sauf si tous les hommes la composant décidaient simultanément de rompre avec leurs croyances ou de se suicider. Le suicide est la treizième cause de mortalité dans le monde et la première chez les jeunes. Tant qu'il ne dérange personne, il est accepté. S'il interfère avec l'illusion d'immortalité civilisationnelle, il devient réprimé. On comprend alors qu'il soit découragé dans une société théiste faisant du corps un don de Dieu, qu'il soit regretté dans n'importe quelle société communautaire ou encore qu'il soit banalisé dans une société individualiste et athée. Paradoxalement d'ailleurs, quand la mort est désirée, c'est certainement qu'on lui attribue une valeur positive, un apaisement, un soulagement, un certain bonheur quel que soit sa forme. Et dès lors qu'est suggérée une opposition entre un bien et un mal, c'est peut-être finalement un aveu de théisme. Regard sur le suicide L'absence de mention d'une quelconque répression du suicide dans les textes législatifs datant de la Révolution française font de cette période le début de la dépénalisation de celui-ci en France. Il est certain que le suicide est un acte des plus intimes et peut, à cet égard, être considéré comme un acte de liberté puisqu'il soulage la perception du corps de toute loi physique, la perception étant abolie par le décès. D'autres actes sont tout aussi intimes : penser, prier. À écouter Descartes (« Cogito ergo sum » [3]) puis Pascal, la raison classique nous apprend que cette intimité est ce qui nous plonge dans le réel. Logiquement, il semble donc que la question du suicide soit directement liée à l'appréhension du réel. Si le réel est une étendue neutre dont la valence est la même que celle de la mort, la mort n'est ni crainte ni souhaitée et advient quand elle doit advenir : dans cette perspective, le suicide n'a pas de raison particulière et s'il est réalisé, il est en fait un acte d'autonomie, une manière pour l'individu de se prouver qu'il se détermine en dernier recours, à compter que cette détermination ne soit influencée par une quelconque pensée extérieure. Si le réel est une prison des pensées, un enferment des possibilités, une restriction de l'Être, alors la mort devient désirable : quelle que soit la souffrance associée à l'agonie, le suicide est vu comme une libération ou une fin, selon que le commettant espère ou pas une vie après la mort. Il est toujours délicat d'envisager le suicide pour une personne religieuse : celle-ci sera plutôt attirée par le martyr que par l'annihilation. Si le réel est perçu positivement, le désir de suicide est alors la conséquence d'une perte de sens : l'individu ne se sentant plus digne de vivre, il a trahi ses idéaux, a perdu la personne qui donnait sens à son existence ou voit sa vision du sens de la vie complètement perturbée par un événement particulier (rupture amoureuse, perte d'un emploi, etc.). Dans le cas des personnes âgées (ou plutôt « se sentant âgées ») qui désirent mourir, on pourrait utiliser l'image d'une fleur qui se fane : le réel était une étendue positive puis la restriction de possibilité de pouvoir en profiter et l'angoisse d'en subir des souffrances poussent l'individu à raccourcir la fin de sa vie. Le suicide est alors

  nous avons le téléphone, internet et les réseaux sociaux. Même malades, nos sociétés feignent l'apaisement en remplaçant la sagesse par l'hédonisme. Il aura fallu qu'une cathédrale brûle pour se rappeler que nous avions une culture, et que surgisse une pandémie inédite pour nous rappeler à nos destins mortels. Nous avons même envisagé une horrible mort, par étouffement en l'absence de ventilateurs et dans une douleur indicible en l'absence de sédatifs suffisants. Nous avons imaginé la faucheuse rugir dans nos foyers, glacer notre sang d'un frisson soudain et tétaniser tous ceux qu'on aime. Mais l'Homme est. Il a le coeur et la rage d'être, et en cela il se différencie de ses congénères phylogéniques. Certes, comme les autres primates et les autres animaux, il élève sa progéniture et veille à la survie de son espèce dans la mesure du possible. Mais plus que cela, l'Homme aime et est capable de s'oublier pour un ou des autres. Cela est une part essentielle de son mode de vie, et le sens qu'il en tire le rend capable d'aller encore plus au-delà. Ce comportement est tout sauf naturel : à l'encontre de sa satisfaction immédiate et de son instinct, il prend un risque pour lui afin d'aider les autres. Au cours des dernières semaines, nous avons pu observer toutes les strates de notre société se mobiliser : face au spectre mortel du COVID-19, les soignants ont reconfigurés leurs hôpitaux, les industriels ont reconfigurés leurs outils de production, les grandes surfaces alimentaires ont assuré l'approvisionnement nécessaire, les distributeurs de tout secteur se sont mis à contribution, etc. Et alors que les mobilisés ne se sont jamais autant senti Hommes, des confinés débattent de la réglementation du suicide. Le suicide assisté, ou le non-sens par l'absurde Les promoteurs du suicide disent sans arrêt qu'ils désirent la mort avec sérénité, car il n'est pas d'angoisse mortelle sans angoisse d'éternité. De nombreux poisons existent et ont des effets certains : la toxine botulique (paralysie rapide et décès par asphyxie), le cyanure (insuffisance respiratoire aiguë), la ricine (décès plus lent en quelques jours), la strychnine (convulsions et état de mal allant jusqu'à l'engagement cérébral), les anatoxines (insuffisances hépatique et rénale aiguës) et d'autres encore. Pourtant, ces promoteurs manifestent afin d'être aidés dans le dernier geste et surtout pour ne pas être conscient lors de leur agonie. Qui plus est, des propositions de lois sont écrites pour « encadrer » ce geste, comme si le mourant invoquait la complicité de la déesse-république, entrainant avec lui les « bourreaux » (sous couvert du nom de « soignants » ?) qui devraient lui administrer le produit létal. Nous n'épiloguerons pas non plus sur les enjeux économiques qui participent à la décision du suicide ou encore sur le rôle des proches dans cette décision. Lorsqu'on est Homme bercé par un rêve prométhéen (tout-faire) ou par un désir brûlant d'amour (toutaimer), le suicide doit être évité parce qu'il est un non-sens autant pour l'individu que pour la collectivité. Au-delà de la médicalisation du suicide, une prise de conscience politique des causes menant au suicide La souffrance psychique ne saurait être dénoncée, elle est inhérente à la nature humaine. Loin d'une pathologisation du suicide, la question de reconnaître juridiquement le droit au suicide (non assisté) pour les personnes le réussissant doit être posée. Cependant, puisque la majorité n'y arrive pas et pour que nos existences gardent un sens, la souffrance doit être combattue, ainsi que les facteurs qui amènent au désir de mort. Aux Etats-Unis où la santé est payante et donc chère, le cancer est la première cause d'endettement et la réflexion sur l'euthanasie est plus promue qu'en France par exemple. Toutes les causes de misère sociale aboutissant à un sentiment de dégradation de la dignité et de l'honneur doivent être combattues. Pour que le suicide régresse, vivre le handicap tout autant que la vieillesse devrait être plus valorisé : une société où le validisme règne en maître souverain est une société de l'aujourd'hui sans lendemain, de la performance sans transmission, soit une société à l'agonie. De nombreux contemporains vantent la liberté d'expression acquise avec la sécularisation de notre société, et ils ont raison : cependant, notre religion-république n'offre pas la possibilité d'un pardon et d'une rédemption, autant qu'elle est sur le point -avec la suggestion de la légalisation de l'euthanasie -de mettre la mort à la table de l'amour. C'est pour cette raison anthropologique, cette question de

survie, que la liberté de culte et le respect de la vie privée sont désormais des enjeux de tout premier plan.

Ensuite, concernant la souffrance physique induite par des maladies graves, ceci est du domaine médical. Un arsenal législatif (terme guerrier) existe déjà et la loi Léonetti est bien celle dont l'autorité est la plus acceptée par les soignants tout autant que les patients. Si celle-ci traite avec beaucoup de justesse de la fin de vie des personnes atteintes de maladies graves et sans espoir thérapeutique, elle ne semble pas s'appliquer avec autant de mesure pour les patients atteints de troubles neurodégénératifs, de patients en état végétatif ou même d'enfants à naître ou tout juste nés et atteints de pathologies très graves. Ces lacunes sont un champ de recherche majeur en éthique dans les années à venir et dont les soignants doivent s'emparer.

Bibliographie