B Desmorat
email: boris.desmorat@sorbonne-universite.fr

N Auffray

Space of 2D elastic materials: a geometric journey

Keywords: 2D elasticity, Invariants, Integrity basis, Eigenvalues

In this paper, we describe geometrically the elastic material domain in terms of invariants of the integrity basis (including the positive definiteness condition), prove a theoretical link between those polynomial invariants and the Kelvin invariants of the elasticity tensor, to finally introduce the concept of design transformation which leads to elastic materials subdomains with identical Kelvin invariants.

Introduction

In linear elasticity, it is customary to describe an elastic material by its tensor specified with respect to some basis. Such a way to describe elastic materials is not satisfactory since rotating the matter produces, with respect to the same basis, another elasticity tensor describing the same elastic material. Hence an elastic material is not defined by an unique tensor but rather by the collection of all elastic tensors related by orthogonal transformations. This idea can be condensed by describing elastic materials by a collection of quantities which are invariant with respect to orthogonal transformations. These quantities are often simply reffered to as the invariants of the elasticity tensor [START_REF] Boehler | On the polynomial invariants of the elasticity tensor[END_REF][START_REF] Vianello | An integrity basis for plane elasticity tensors[END_REF][START_REF] Auffray | Invariant-based reconstruction of bidimensionnal elasticity tensors[END_REF][START_REF] Olive | A minimal integrity basis for the elasticity tensor[END_REF]. A basis of invariants that are polynomial in terms of tensor components is called an Integrity Basis (ℐℬ). Elements of ℐℬ, also reffered to as Boehler invariants [START_REF] Boehler | On the polynomial invariants of the elasticity tensor[END_REF], satisfy polynomials relations defining the set of elastic materials as a domain within a higher dimensional space. Points of this domain uniquely defined elastic materials, and their symmetry classes is encoded in the the topology of the domain. In the case of planar elasticity the domain of elastic materials lives in R 5 (the situation is much more complicated for 3D elasticity and will not be discussed here).

On the other hand, the Kelvin representation, originaly due to Kelvin [START_REF] Thomson | Elements of a mathematical theory of elasticity[END_REF] and appearing again in the litterature of the 80's [START_REF] Rychlewsky | On Hooke's law[END_REF][START_REF] Mehrabadi | Eigentensors Of Linear Anisotropic Elastic Materials[END_REF], allows for the definition of 3 polynomial invariants (the elementary symmetric functions), denoted Kelvin invariants, which are uniquely related to the 3 eigenvalues of such tensorial representation. Such invariants are used in mechanics with different purposes : dissymmetric elastic behaviour in tension and in compression [START_REF] Desmorat | Compliance optimization with nonlinear elastic materials: Application to constitutive laws dissymmetric in tension-compression[END_REF], yield and strength criteria [START_REF] Arramon | A multidimensional anisotropic strength criterion based on Kelvin modes[END_REF][START_REF] Desmorat | Non-quadratic Kelvin modes based plasticity criteria for anisotropic materials[END_REF], continuum damage mechanics [START_REF] Biegler | An energy-based constitutive model for anisotropic solids subject to damage[END_REF][START_REF] Franois | Identification des symtries matrielles de matriaux anisotropes[END_REF][START_REF] Desmorat | Dissymtrie de comportement lastique anisotrope coupl ou non l'endommagement[END_REF][START_REF] Desmorat | Kelvin decomposition and multiple effective stresses concept in anisotropic materials[END_REF][START_REF] Franois | A damage model based on Kelvin eigentensors and Curie principle[END_REF][START_REF] Mattiello | Rate dependent ductility and damage threshold: Application to Nickel-based single crystal CMSX-4[END_REF] ... Moreover, the link between the eigenvalues of the Kelvin decomposition of 2D elasticity tensor and the polar formalism [START_REF] Verchery | Les invariants des tenseurs d'ordre 4 du type de l'lasticit[END_REF][START_REF] Vannucci | Plane anisotropy by the polar method[END_REF] was described in [START_REF] Desmorat | An alternative to the Kelvin decomposition for plane anisotropic elasticity[END_REF].

In this paper, devoted to the 2D case, we will describe geometrically the elastic material domain in terms of invariants of the integrity basis (including the positive definiteness condition), prove a theoretical link between the two non-equivalent sets of polynomial Kelvin and Boehler invariants, to finally introduce the concept of design transformation which leads to elastic materials subdomains with identical Kelvin invariants.

Organization of the paper

First, section 2 is devoted to a brief description of the space of 2D linear elastic material. Next, in section 3, the decomposition of second-order symmetric tensor space into deviatoric and spheric subspaces is introduced. In section 4, the Clebsh-Gordan harmonic decomposition of elasticity tensors is detaiiled and the integrity basis defined accordingly. The section 5 the definite positiveness conditions expressed in terms of the invariants of the integrity basis are provided, and geometrical description of the elastic material domain is proposed. In section 6, an explicit link between Boehler and Kelvin invariants, that makes use of the spheric direction introduced in section 3 is shown. And, finally, we introduce in section 7 the concept of design transformation which transforms elastic materials while preserving the eigenvalues of the Kelvin representation of an elasticity tensor. An explicitly characterization of geometric domains obtained by all possible design transformations for an initially isotropic material is studied, with an application to the specific case of pentamode materials.

Notations

Throughout this paper, the physical space is modelled on the Euclidean space ℰ 2 with E 2 its associated vector space. Once an arbitrary reference point chosen those spaces can be associated and 𝒫 = {e 1 , e 2 } will denote an orthonormal basis of E 2 . For forthcoming need, let also defined 𝒦 = {ê 1 , ê2 , ê3 } the orthonormal canonical basis of R 3 , 𝒦 will be referred to as the Kelvin basis.

Tensor spaces:

-T denote a tensor space; -𝑆 2 (R 𝑛 ) is the space of symmetric second-order tensors in 𝑛-D.

-K 𝑛 is the space of 𝑛-th order completely symmetric and traceless tensors on R 2 , called harmonic tensors.

Tensors of order ranking from 0 to 4 are denoted, respectively, by 𝛼, v, a

∼ , A ≈ .
The simple, double and fourth-order contractions are written •, :, :: respectively. In components, with respect to 𝒫, these notations correspond to

u • v = 𝑢 𝑖 𝑣 𝑖 , a ∼ : b ∼ = 𝑎 𝑖𝑗 𝑏 𝑖𝑗 , (A ≈ : B ≈ ) 𝑖𝑗𝑘𝑙 = 𝐴 𝑖𝑗𝑝𝑞 𝐵 𝑝𝑞𝑘𝑙 , (A ≈ :: B ≈ ) = 𝐴 𝑝𝑞𝑟𝑠 𝐵 𝑝𝑞𝑟𝑠
where the Einstein summation on repeated indices is understood. When needed, index symmetries of both spaces and their elements are expressed as follows: (..) indicates invariance under permutations of the indices in parentheses, .. .. indicate symmetry with respect to permutations of the underlined blocks.

Tensor products:

-⊗ stands for the classical product, and ⊗ 𝑛 indicates its 𝑛-th power; -𝑆 2 denotes its the completely symmetrized product, and 𝑆 𝑛 its extension to product of 𝑛 elements; -⊗ indicates the twisted tensor product defined by: (a

∼ ⊗ b ∼ ) 𝑖𝑗𝑘𝑙 = 1 2 (𝑎 𝑖𝑘 𝑏 𝑗𝑙 + 𝑎 𝑖𝑙 𝑏 𝑗𝑘 )
Special tensors:

-1

∼

the second-order identity tensor;

-I ≈ = 1 ∼ ⊗ 1 ∼ the fourth-order identity tensor of 𝑆 2 (R 𝑛 ); -K ≈ = 1 2 1 ∼ ⊗ 1 ∼ the spheric projector; -J ≈ = I ≈ -K ≈ the deviatoric projector.

Matrix spaces:

-ℳ 𝑛 is the space of 𝑛 × 𝑛 dimensional square matrices ; -ℳ 𝑆 𝑛 is the space of 𝑛 × 𝑛 dimensional symmetric square matrices ; -ℳ 𝑝,𝑞 is the space of 𝑝 × 𝑞 rectangular matrices.

Groups:

The following matrix groups are considered in the paper:

-GL(𝑑): the group of all linear invertible transformations of R 𝑑 , i.e. F ∈ GL(𝑑) iff det(F) ̸ = 0; -O(𝑑): the orthogonal group, that is the group of all isometries of -Id, the identity group; -Z 𝑘 , the cyclic group with 𝑘 elements generated by R(2𝜋/𝑘); -D 𝑘 , the dihedral group with 2𝑘 elements generated by R(2𝜋/𝑘) and P e 2 .

R 𝑑 i.e. Q ∈ O(𝑑) iff Q ∈ GL(𝑑) and Q -1 = Q 𝑇 ,

Miscellaneous notations:

-ℐℬ means Integrity Basis.

-≃ denotes hereafter an isomorphism.

-𝒮 𝑛 denotes he unit 𝑛-sphere defined as:

𝒮 𝑛 = { 𝑥 ∈ R 𝑛+1 : ‖𝑥‖ = 𝑟 } .
-ℒ(𝐸, 𝐹 ) indicates the space of linear applications from 𝐸 to 𝐹 ; -ℒ(𝐸) indicates the space of linear applications from 𝐸 to 𝐸; -ℒ 𝑠 (𝐸) indicates the space of self-adjoint linear applications on 𝐸;

2 Space of linear elastic materials

The space elasticity tensors

In the field of linear elasticity, the constitutive law is a local linear relation between the second-order symmetric Cauchy stress tensor 𝜎 ∼ and the second-order symmetric infinitesimal strain tensor 𝜀 ∼ :

𝜎 𝑖𝑗 = 𝐶 𝑖𝑗𝑘𝑙 𝜀 𝑘𝑙 (1) 𝜎 ∼ and 𝜀 ∼ belong to 𝑆 2 (R 2
), the space of bi-dimensional symmetric second-order tensors. As a consequence the elasticity tensor possesses minor index symmetries:

𝐶 𝑖𝑗𝑘𝑙 = 𝐶 𝑗𝑖𝑘𝑙 = 𝐶 𝑖𝑗𝑙𝑘
which are condensed in the notation: 𝐶 (𝑖𝑗)(𝑘𝑙) . Due to the potential energy associated to the elastic behavior another index symmetry has to be taken into account:

𝐶 𝑖𝑗𝑘𝑙 = 𝐶 𝑘𝑙𝑖𝑗
This so-called major symmetry is encoded in the notation: 𝐶 𝑖𝑗 𝑘𝑙 . Hence, combined with the minor ones, we obtain the elastic index symmetries: 𝐶 (𝑖𝑗) (𝑘𝑙) . The vector space of 2D elasticity tensors is defined as1 :

Ela := {C ≈ ∈ ⊗ 4 R 2 |𝐶 (𝑖𝑗) (𝑘𝑙) }, dim Ela = 6
and can also be viewed as

Ela = ℒ 𝑠 (𝑆 2 (R 2 ))
For being admissible, an elasticity tensor, considered as a quadratic form on

𝑆 2 (R
T ≈ = Q ⋆ T ≈
in which the star product ⋆ stands for the standard tensorial action. In components, with respect to 𝒫, this action can be detailed:

𝑇 𝑖𝑗𝑘𝑙 = 𝑄 𝑖𝑝 𝑄 𝑗𝑞 𝑄 𝑘𝑟 𝑄 𝑙𝑠 𝑇 𝑝𝑞𝑟𝑠
The nature of a material does not change when it is subjected to a rotation or a flip (mirror isometry though a line). On the contrary the elasticity tensor will change with respect to a given reference frame, for instance 𝒫, so that multiple tensors can be associated to one elastic material. From a physical point of view, the necessary and sufficient conditions for two elastic tensors T 1

≈ , T 2 ≈ ∈ Ela to represent the same elastic material, denoted T 1 ≈ ∼ T 2 ≈ , writes T 1 ≈ ∼ T 2 ≈ ⇔ ∃Q ∈ O(2) | T 1 ≈ = Q ⋆ T 2 ≈
The collection of all elasticity tensors that describe the same elastic material is a geometric object called the orbit of T 1 ≈ and defined by

𝒪(T 1 ≈ ) = {T ≈ ∈ Ela, ∃Q ∈ O(2) | T ≈ = Q ⋆ T 1 ≈ }
For some transformation, the resulting tensor is identical to the original one. The set of orthogonal transformations letting T 1 ≈ invariant constitutes its symmetry group:

G T 1 ≈ := {Q ∈ O(2), |T 1 ≈ = Q ⋆ T 1 ≈ }
Let consider the following equivalence relation among elements of Ela, [START_REF] Auffray | On anisotropic polynomial relations for the elasticity tensor[END_REF][START_REF] Auffray | Handbook of bidimensional tensors: Part I: Decomposition and symmetry classes[END_REF]. The space of 2D elasticity tensors is divided into 4 strata [START_REF] He | On the symmetries of 2d elastic and hyperelastic tensors[END_REF][START_REF] Vianello | An integrity basis for plane elasticity tensors[END_REF][START_REF] Auffray | Handbook of bidimensional tensors: Part I: Decomposition and symmetry classes[END_REF]:

T 1 ≈ ≈ T 2 ≈ ⇔ ∃Q ∈ O(2) | G T 1 ≈ = QG T 2 ≈ Q -1
Ela = 𝛴 [Z 2 ] ∪ 𝛴 [D 2 ] ∪ 𝛴 [D 4 ] ∪ 𝛴 [O(2)]
In mechanical terms, 𝛴 to isotropic materials [START_REF] Zheng | The description, classification, and reality of material and physical symmetries[END_REF].

3 Decomposition of second-order symmetric tensors space Since elasticity tensors are linear symmetric applications on 𝑆 2 (R 2 ), a first step to describe Ela is to understand the structure of 𝑆 2 (R 2 ). This is the aim of the present section. )

𝒫

It is possible to turn this second-order symmetric tensor into a genuine vector of R 3 by defining the following linear application 𝜑 :

𝑆 2 (R 2 ) → R 3 : ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ê1 = 𝜑(e 1 ⊗ e 1 ) ê2 = 𝜑(e 2 ⊗ e 2 ) ê3 = 𝜑( √ 2 2 (e 1 ⊗ e 2 + e 2 ⊗ e 1 )) (2)
The cannonical basis 𝒦 = {ê 𝑖 } of R ) and a spheric one (K 0 ). In formula:

𝑆 2 (R 2 ) ≃ K 2 ⊕ K 0 (3)
Hence any t ∼ ∈ 𝑆 2 (R 2 ) can be decomposed accordingly. This decomposition, which is unique, is given by the well-known formula:

t ∼ = d ∼ + 𝛼1 ∼ , with 𝛼 = 1 2 t ∼ : 1 ∼ , d ∼ = t ∼ -𝛼1 ∼ (4) with d
∼ ∈ K 2 and 𝛼 ∈ K 0 . Associated to this decomposition, we define a new basis

ℋ = { f1 , f2 , f3 } , with f1 = √ 2 2 (ê 1 -ê2 ) f2 = ê3 f3 = √ 2 2 (ê 1 + ê2 )
This new basis ℋ will be referred to as the harmonic basis. The passage matrix P from 𝒦 to ℋ is given by ) 𝑇 corresponds respectively to the spheric and deviatoric parts of t. The relations between these different bases are summed up in the following diagram:

𝑃 𝑖𝑗 = ê𝑖 • f𝑗 P = ⎛ ⎜ ⎝ √ 2 2 0 √ 2 2 - √ 2 2 0 √ 2 2 0 1 0 ⎞ ⎟ ⎠
Physical Space: R 2 , 𝒫 = {e 𝑖 } ⊗ / / 𝒫 ⊗ 𝑛 = {e 𝑖 1 ⊗ . . . ⊗ e 𝑖 𝑛 } 𝜑 Strain-Stress Space (Kelvin basis): R 3 , 𝒦 = {ê 𝑖 } ⊗ / / P 𝒦 ⊗ 𝑚 = {ê 𝑖 1 ⊗ . . . ⊗ ê𝑖 𝑚 } P Strain-Stress Space (Harmonic basis): R 3 , ℋ = { f𝑖 } ⊗ / / ℋ ⊗ 𝑚 = { f𝑖 1 ⊗ . . . ⊗ f𝑖 𝑚 }

Representation of physical rotations

The interest of the different bases introduced is revealed when studying how tensors transform with respect to active rotation, that is to element of the rotation subgroup SO( 2). An element of SO( 2) is parametrized by: R

∼ = ( cos 𝜃 -sin 𝜃 sin 𝜃 cos 𝜃 ) 𝒫 Its action on t ∼ of 𝑆 2 (R 2 ) gives a new element t ∼ of 𝑆 2 (R 2 )
, the components of which are related to those of t ∼ in the following manner

𝑠 𝑖𝑗 = 𝑅 𝑖𝑝 𝑅 𝑗𝑞 𝑠 𝑝𝑞 By defining R ∼ = 𝜑(R ∼ ⊗ R ∼
), the former action expressed in R 2 , can be reformulated

directly in R 3 : t = R ∼ t
In the same way, a rotated elasticity tensor expressed as a second-order tensor in

R 3 is T ∼ = R ∼ T ∼ R ∼ 𝑇 R ∼
has the following matrix expression in 𝒦: 3) and corresponds to a rotation of 2𝜃 around the axis k 𝑠 . Expressed in ℋ, the matrix of Q ∼ has the following expression:

R ∼ = 1 2 ⎛ ⎝ 1 + cos 2𝜃 1 -cos 2𝜃 - √ 2 sin 2𝜃 1 -cos 2𝜃 1 + cos 2𝜃 √ 2 sin 2𝜃 √ 2 sin 2𝜃 - √ 2 sin 2𝜃 2 cos 2𝜃 ⎞ ⎠ 𝒦 It can be checked that Q ∼ ∈ SO(
Q ∼ = ⎛ ⎝ cos(2𝜃) -sin(2𝜃) 0 sin(2𝜃) cos(2𝜃) 0 0 0 1 ⎞ ⎠ ℋ (5)
Physical rotation matrices are well-structured in the harmonic basis. The geometric content is clear, when an element of 𝑆 2 (R 2 ) is rotated by an angle 𝜃, its spherical part is invariant, while its deviatoric part is turned by an angle 2𝜃.

Invariant description of Ela

To construct a geometric description of the space of 2D elastic materials the use of invariant functions is required. For our needs, those functions will mostly be considered as polynomial. To construct the polynomial invariants a of tensor, the first step is to decomposed this tensor into elementary parts. This decomposition is the object of subsection 4.1. The construction of polynomial invariants and their geometric interpretation will be the object of subsection 4.2 and subsection 4.3.

Decompositions of fourth-order elasticity tensors

Let's go back to the study of the space Ela, this space is shown to have the following isotypic structure [START_REF] Auffray | Handbook of bidimensional tensors: Part I: Decomposition and symmetry classes[END_REF]:

Ela ≃ K 4 ⊕ K 2 ⊕ 2K 0
in which K 4 is the space of 2D fourth-order harmonic tensors, that is of complete symmetric and traceless fourth-order tensors. Since more than one copy of K 0 are involved there are multiple explicit harmonic decompositions [START_REF] Golubitsky | Singularities and groups in bifurcation theory[END_REF][START_REF] Auffray | Handbook of bidimensional tensors: Part I: Decomposition and symmetry classes[END_REF]. This leads to multiple couples of isotropic parameters. A specific choice is the Clebsch-Gordan interpretation of the harmonic decomposition [START_REF] Auffray | Gomtrie des espaces de tenseurs, application l'lasticit anisotrope classique et gnralise[END_REF]:

T ≈ = D ≈ + 1 2 (1 ∼ ⊗ d ∼ + d ∼ ⊗ 1 ∼ ) + 𝜅(1 ∼ ⊗ 1 ∼ ) + 𝛾 J ≈ (6)
where

D ≈ ∈ K 4 , d ∼ ∈ K 2 and {𝜅, 𝛾} ∈ K 0 .
Using the Kelvin representation a fourth-order elasticity tensor is represented as the following matrix in basis 𝒦:

T ∼ = ⎛ ⎝ 𝑇 1111 𝑇 1122 √ 2 𝑇 1112 𝑇 1122 𝑇 2222 √ 2 𝑇 1222 √ 2 𝑇 1112 √ 2 𝑇 1222 2 𝑇 1212 ⎞ ⎠ 𝒦 (7) d ∼ ∈ K 2 and D ≈ ∈ K 4 can be parametrized in the Kelvin basis by d ∼ = ⎛ ⎝ 𝑑 1 -𝑑 1 √ 2 𝑑 2 ⎞ ⎠ 𝒦 D ∼ = ⎛ ⎝ 𝐷 1 -𝐷 1 √ 2𝐷 2 -𝐷 1 𝐷 1 - √ 2𝐷 2 √ 2𝐷 2 - √ 2𝐷 2 -2𝐷 1 ⎞ ⎠ 𝒦 (8)
A structure is made apparent by writing the harmonic decomposition of Ela in the basis 2 ℋ:

T ∼ = ⎛ ⎝ 2𝐷 1 + 𝛾 2𝐷 2 𝑑 1 2𝐷 2 -2𝐷 1 + 𝛾 𝑑 2 𝑑 1 𝑑 2 2𝜅 ⎞ ⎠ ℋ (9)
The previous matrix is structured which means,

⎧ ⎨ ⎩ 𝜎 ∼ 𝑑 = T ≈ 𝑑𝑑 : 𝜀 ∼ 𝑑 + T ≈ 𝑑𝑠 : 𝜀 ∼ 𝑠 𝜎 ∼ 𝑠 = T ≈ 𝑠𝑑 : 𝜀 ∼ 𝑑 + T ≈ 𝑠𝑠 : 𝜀 ∼ 𝑠 (10) with T
≈ 𝑑𝑑 = D ≈ + 𝛾 J ≈ ; T ≈ 𝑑𝑠 = 1 2 d ∼ ⊗ 1 ∼ ; T ≈ 𝑠𝑠 = 𝜅(1 ∼ ⊗ 1 ∼ )
This particular block form for writing the elasticity tensor will be referred to as the Clebsch-Gordan representation. This form consists in writing the elasticity tensor as a symmetric linear application on K 2 ⊕ K 0 : This decomposition is induced on the linear operator and the elasticity tensor appears to be structured by blocks:

Ela ≃ ℒ 𝑠 (K 2 ⊕ K 0 ) = ℒ 𝑠 (K 2 ) ⊕ ℒ 𝑠 (K 0 ) ⊕ ℒ(K 0 , K 2 ).
T ≈ = [ T ≈ 𝑑𝑑 T ≈ 𝑑𝑠 T ≈ 𝑠𝑑 T ≈ 𝑠𝑠 ] (11)

Integrity basis for elasticity tensors

Integrity basis is, for a given space V and for a given group action 𝐺, a set of fundamental polynomial invariants such that any 𝐺-invariant polynomial on V is a polynomial in the elements of the integrity basis [START_REF] Weyl | The classical groups[END_REF]. Integrity basis will be denoted ℐℬ(V, 𝐺).

In the present situation, that is for V = Ela and 𝐺 = O(2), the integrity basis 1. is finitely generated, i.e. ♯ℐℬ(Ela, O(2)) < +∞; 2. separates the orbits

ℐℬ(Ela, O(2))(T 1 ≈ ) = ℐℬ(Ela, O(2))(T 2 ≈ ) ⇔ T 1 ≈ ∼ T 2 ≈ .
In other terms, the invariants of the integrity basis define an application from the space of elasticity tensors to Ela/O(2), the space of elastic materials [START_REF] Auffray | Invariant-based reconstruction of bidimensionnal elasticity tensors[END_REF]. From now on, to save space, the notation we will shortened to ℐℬ. Integrity bases for O(2)action on the space of plane elasticity tensors are known since the second-half of the 90 [START_REF] Blinowski | Two-dimensional Hooke's tensors Isotropic decomposition, effective symmetry criteria[END_REF][START_REF] Vianello | An integrity basis for plane elasticity tensors[END_REF] and are constructed from the harmonic decomposition of Ela [START_REF] Boehler | On the polynomial invariants of the elasticity tensor[END_REF][START_REF] Olive | Isotropic invariants of completely symmetric third-order tensors[END_REF][START_REF] Auffray | Invariant-based reconstruction of bidimensionnal elasticity tensors[END_REF]. Let consider the following quantities:

𝐼 1 = 𝜅 𝐽 1 = 𝛾 𝐼 2 = d ∼ : d ∼ 𝐽 2 = D ≈ :: D ≈ 𝐼 3 = d ∼ : D ≈ : d ∼ (12)
Those elements are O(2)-invariant and, in the chosen notation, the subscript indicates the degree of the poynomial invariants in the elasticity tensor.

We have the following result [START_REF] Vianello | An integrity basis for plane elasticity tensors[END_REF]:

Theorem 4.1. A minimal integrity basis for O(2)-action on Ela is

ℐℬ = (𝐼 1 , 𝐽 1 , 𝐼 2 , 𝐽 2 , 𝐼 3 ) .
Those elements are free meaning that they are not related by any polynomial relation. They satisfy, however, the following inequality:

𝐼 2 2 𝐽 2 -2𝐼 2 3 ≥ 0 ( 13
)
This Cauchy-Schwarz type inequality comes from the fact we are dealing with real valued tensors. This inequality is fundamental and its geometric meaning elucidated in the next subsection. We define the following application from Ela to Ela/O(2) that associate to a tensor its uniquely defined elastic material.

ℐℬ(T

≈ ) := ( 𝐼 1 (T ≈ ), 𝐽 1 (T ≈ ), 𝐼 2 (T ≈ ), 𝐽 2 (T ≈ ), 𝐼 3 (T ≈ ) )
This set of polynomial invariants will be referred to as Boehler invariants.

Remark

𝐼 2 = 2(𝑑 2 1 + 𝑑 2 2 ) 𝐽 2 = 8(𝐷 2 1 + 𝐷 2 2 ) 𝐼 3 = 4𝐷 1 (𝑑 2 1 -𝑑 2 2 ) + 8𝐷 2 𝑑 1 𝑑 2 (15)
4.3 A geometric representation of the space of elastic materials (vector space Ela)

As introduced in section 2, Ela is divided into 4 strata:

Ela = 𝛴 [Z 2 ] ∪ 𝛴 [D 2 ] ∪ 𝛴 [D 4 ] ∪ 𝛴 [O(2)]
For the least symmetric class, that is for biclinic elastic materials, the functions of ℐℬ are algebraically independent. A biclinic material is described by five independent quantities, that is by a point in R 5 . The location of this point is not any, since constrained by the relations ( 13) and [START_REF] Desmorat | Non-quadratic Kelvin modes based plasticity criteria for anisotropic materials[END_REF].

In Figure 1 are summed-up the different transitions from a symmetry class to another expressed in terms of polynomial relations between invariants. While are special orthotropic tensors, they are not ordinary orthotropic since extra restriction, i.e. other than invariance properties, are needed to define them (those tensors correspond to 𝑅 0 -orthotropic tensors [START_REF] Vannucci | A special planar orthotropic material[END_REF]).

𝛴 [D 2 ] = 𝛴 𝑜𝑟𝑑 [D 2 ] ∪ 𝛴 𝑠𝑝𝑒𝑐 [D 2 ] ,
𝛴 [Z 2 ] 𝐽 2 =0 { { 𝐼 2 2 𝐽 2 -2𝐼 2 3 =0, 𝐽 2 ̸ =0 # # 𝛴 𝑑 [D 2 ] 𝐼 2 =0 𝛴 𝑔 [D 2 ] 𝐼 2 =0 𝛴 [D 4 ] 𝐽 2 =0 { { 𝛴 [O(2)]
Fig. 1: Breaking symmetry conditions of Ela

In Figure 2 is represented the space of elastic materials (vector space Ela) with respect to (𝐼 2 , 𝐽 2 , 𝐼 3 ), without taking the positive definiteness property into account.

The surface, which corresponds to the polynomial equation 𝐼 2 2 𝐽 2 -2𝐼 2 3 = 0 contains all the at-least-orthotropic materials (stratum

𝛴 [D 2 ] = 𝛴 [D 2 ] ∪𝛴 [D 4 ] ∪𝛴 [O(2)]
). The condition 𝐼 2 2 𝐽 2 -2𝐼 2 3 > 0 indicates on which side of the orthotropic surface are the biclinic materials located (strata 𝛴 [Z 2 ] ). Finally, we get that, independently of the values of the isotropic invariants 𝐼 1 and 𝐽 1 :

biclinic materials (stratum 𝛴 It can be checked by a direct calculation that

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ 𝜎 1 = 2(𝐼 1 + 𝐽 1 ) 𝜎 2 = 4𝐼 1 𝐽 1 + 𝐽 2 1 - 1 2 (𝐼 2 + 𝐽 2 ) 𝜎 3 = 1 2 (𝐼 3 -𝐼 2 𝐽 1 + 4𝐼 1 𝐽 2 1 -2𝐼 1 𝐽 2 ) ( 16
)
so that the the positivity conditions can be expressed in terms of elements of the integrity basis:

T ≈ is positive definite ⇔ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 𝐼 1 + 𝐽 1 > 0 4𝐼 1 𝐽 1 + 𝐽 2 1 - 1 2 (𝐼 2 + 𝐽 2 ) > 0 𝐼 3 -𝐼 2 𝐽 1 + 4𝐼 1 𝐽 2 1 -2𝐼 1 𝐽 2 > 0 (17)
To obtain simpler expressions, the Clebsch-Gordan form of the elasticity tensor can be exploited. To that aim consider the following lemma [START_REF] Petersen | The matrix cookbook[END_REF]: This approach applied respectively to T ≈ 𝑑𝑑 and to T ≈ 𝑠𝑠 (defined in [START_REF] Desmorat | Compliance optimization with nonlinear elastic materials: Application to constitutive laws dissymmetric in tension-compression[END_REF]) provides the following equivalent sets of positiveness conditions expressed in terms of the elements of the integrity basis:

T ≈ is positive definite ⇔ ⎧ ⎪ ⎨ ⎪ ⎩ 𝐽 1 > 0 2𝐽 2 1 -𝐽 2 > 0 𝐼 3 -2𝐽 2 𝐼 1 -𝐼 2 𝐽 1 + 4𝐽 2 1 𝐼 1 > 0 ⇔ ⎧ ⎪ ⎨ ⎪ ⎩ 𝐼 1 > 0 8𝐼 1 𝐽 1 -𝐼 2 > 0 𝐼 3 -2𝐽 2 𝐼 1 -𝐼 2 𝐽 1 + 4𝐽 2 1 𝐼 1 > 0
The positiveness conditions show that for any elastic material, the isotropic invariants 𝐼 1 and 𝐽 1 are strictly positive quantities. A look at the two sets of conditions suggests to introduce the following adimensional anisotropic invariants:

𝑖 2 = 𝐼 2 8𝐼 1 𝐽 1 𝑗 2 = 𝐽 2 2𝐽 2 1 𝑖 3 = 𝐼 3 8𝐼 1 𝐽 2 1 (18)
The positive definiteness conditions read

T ≈ is positive definite ⇔ { 0 ≤ 𝑗 2 < 1 2𝑖 3 -𝑗 2 -2𝑖 2 + 1 > 0 ⇔ { 0 ≤ 𝑖 2 < 1 2𝑖 3 -𝑗 2 -2𝑖 2 + 1 > 0
The adimensionalization of the inequality [START_REF] Desmorat | Kelvin decomposition and multiple effective stresses concept in anisotropic materials[END_REF] shows that

-1 < 𝑖 3 < +1
Together with condition (13), the complete set4 of conditions for positive definiteness of a real valued fourth-order elasticity tensor read

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0 ≤ 𝑖 2 < 1 0 ≤ 𝑗 2 < 1 -1 ≤ 𝑖 3 ≤ +1 1 2 (-1 + 2𝑖 2 + 𝑗 2 ) < 𝑖 3 𝑖 2 3 ≤ 𝑖 2 2 𝑗 2 (19)
Such a domain is presented in figure 3. Note that the quantities on the three axes are different from figure 2. The geometric domain of positive definite elasticity tensors is strictly located above the plane and inside the volume defined by the surface. For 𝑗 2 = 1, the plane and the surface are coincident.

An explicit link between Boehler and Kelvin invariants

The invariants considered in this paper are polynomial functions of the tensor considered as a geometric object in the physical space R 2 , the group of invariance being O(2). Hereafter these quantities will be referred to as the Boehler invariants [START_REF] Boehler | On the polynomial invariants of the elasticity tensor[END_REF][START_REF] Auffray | Invariant-based reconstruction of bidimensionnal elasticity tensors[END_REF]. The eigenvalues of symmetric tensors constitute another system of invariant functions. Since eigenvalues are algebraic invariants, we will rather consider the coefficients of the characteristic polynomial which are given by the elementary symmetric polynomials. Those quantities will be referred to as (polynomial) Kelvin invariants. The two sets (with Boehler and Kelvin invariants) coincide only for second-order tensors. For the space of fourth-order elasticity tensors Ela, the Boelher set comprises 5 invariants (𝐼 1 , 𝐽 1 , 𝐼 2 , 𝐽 2 , 𝐼 3 ), while Kelvin set comprises only 3 quantities (𝜎 1 , 𝜎 2 , 𝜎 3 ). To build a bridge between these sets for elasticity tensors, let consider the following classical result [START_REF] Rivlin | Further Remarks on the Stress-Deformation Relation for Isotropic Materials[END_REF][START_REF] Smith | On isotropic functions of symmetric tensors, skew-symmetric tensors and vectors[END_REF][START_REF] Boehler | Lois de comportement anisotrope des milieux continus[END_REF][START_REF] Zheng | Theory of representations for tensor functions -A unified invariant approach to constitutive equations[END_REF] :

Lemma 6.1. Let (a ∼ , u) ∈ V = 𝑆 2 (R 3 ) × R 3 and set: 𝑖 1,0 := tr a ∼ , 𝑖 2,0 := tr a ∼ 2 , 𝑖 3,0 := tr a ∼ 3 , 𝑖 0,2 := u • u, 𝑖 1,2 := u • (a ∼ • u), 𝑖 2,2 := u • (a ∼ 2 • u), (20)
A minimal integrity basis ℐℬ for V with respect to standard O(3)-action, is given by the collection:

ℐℬ(V) = {𝑖 1,0 , 𝑖 2,0 , 𝑖 3,0 , 𝑖 0,2 , 𝑖 1,2 , 𝑖 2,2 }.
In the notation 𝑖 𝑝,𝑞 , 𝑝 is the degree in a ∼ and 𝑞 the degree in u. 5 The matrices H

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ tr a ∼ = 𝜎 1 tr a ∼ 2 = 𝜎 2 1 -2𝜎 2 tr a ∼ 3 = 𝜎 3 1 -3𝜎 1 𝜎 2 + 3𝜎 3 ⇔ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ 𝜎 1 = tr a ∼ 𝜎 2 = 1 2 ( (tr a ∼ ) 2 -tr a ∼ 2 ) 𝜎 3 = 1 6 ( (tr a ∼ ) 3 -3 tr a ∼ tr a ∼ 2 + 2 tr a ∼ 3 )
This allows to exchange the first 3 invariants 𝑖 1,0 , 𝑖 2,0 , 𝑖 3,0 in (20) of lemma 6.1 with the elementary symmetric functions. Proof. The vector ŝ is unitary, hence 𝑖 0,2 = 1 and no specific information is given by this quantity. The following relations can be verified by a direct calculation:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 𝜎 1 = 2(𝐼 1 + 𝐽 1 ) 𝜎 2 = 4𝐼 1 𝐽 1 + 𝐽 2 1 -1 2 (𝐼 2 + 𝐽 2 ) 𝜎 3 = 1 2 (𝐼 3 -𝐼 2 𝐽 1 + 4𝐼 1 𝐽 2 1 -2𝐼 1 𝐽 2 ) 𝑖 1,2 = 2𝐼 1 𝑖 2,2 = 1 2 𝐼 2 + 4𝐼 2 1
This non-linear system is easily uniquely inverted by substitution in the following:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 𝐼 1 = 1 2 𝑖 1,2 𝐽 1 = 1 2 (𝜎 1 -𝑖 1,2 ) 𝐼 2 = 2(𝑖 2,2 -𝑖 2 1,2 ) 𝐽 2 = 1 2 (𝜎 1 + 𝑖 1,2 ) 2 -2(𝜎 2 + 𝑖 2,2 ) 𝐼 3 = 2𝜎 3 -2𝜎 2 𝑖 1,2 + 𝑖 2 1,2 (𝜎 1 + 𝑖 1,2 ) + 𝑖 2,2 (𝜎 1 -3𝑖 1,2 )
Again, this system is uniquely invertible (by substitution). This ends the proof.

This result shows that Kelvin invariants is the subset of Boehler invariants obtained when the spheric direction ŝ is forgot. As a consequence the eigenvalues of the elasticity tensor do not define uniquely an elastic material, but rather a family of elastic materials.

Using the spheric direction ŝ, it is possible to give the following alternative definition of Boehler invariants. Consider ŝ the vector direction of the spheric space in R 3 and construct the projectors The Boehler invariants of the integrity basis are:

𝐼 1 = 1 2 tr T ∼ 𝑠𝑠 𝐼 2 = T ∼ 𝑠𝑑 : T ∼ 𝑠𝑑 𝐼 3 = 2 tr( T ∼ 𝑠𝑑 T ∼ 𝑑𝑑⋆ T ∼ 𝑠𝑑 ) 𝐽 1 = 1 2 tr T ∼ 𝑑𝑑 𝐽 2 = T ∼ 𝑑𝑑⋆ : T ∼ 𝑑𝑑⋆
in which the notation ⋆ is defined by

T ∼ ⋆ := T ∼ - 1 2 (tr T ∼ )P ∼ 𝑑

Design transformations

Having two different actions acting on the space of elasticity tensors motivates the introduction of a new concept that we called design transformation.

First, the design transformation is defined in section 7.1. Then, we will prove in section 7.2 that any elastic material can be design-transformed into a tetragonal elastic material. Finally, we will study in details in section 7.3 the design transformations of initially isotropic elastic materials.

Definition and fundamental properties of a design transformation

Definition 7.1. Applications between elastic materials preserving eigenvalues will be called design transformations.

The design transformations have the following fundamental properties:

1. the set of design transformations of Ela is SO(3); 2. any initially positive definite tensor remains positive definite after any designtransformation; 3. trivial design transformations are the physical rotations.

Tetragonal equivalent elastic material

We have the following remarkable property: ] ℋ allows for defining a rotation which ends up with a diagonal matrix in basis ℋ. Let denote {𝜆 𝑖 } 1≤𝑖≤3 the set of eigenvalues. Suppose first that the {𝜆 𝑖 } 1≤𝑖≤3 are all distinct (this is the standard situation for a initial tensor that belongs to

𝛴 [D 4 ] , 𝛴 [D 2 ] or 𝛴 [Z 2 ]
). In this case there are 6 seemingly different permutations of this set. It can be observed that a 𝜋 2 physical rotation will permute the first two eigenvalues. Hence we end up with 3 distinct 𝛴 [D 4 ] materials. Suppose now that among {𝜆 𝑖 } 1≤𝑖≤3 two eigenvalues are equal. Let say that we have {𝜆 1 , 𝜆 1 , 𝜆 2 } (this is the standard situation for a initial tensor that belongs to 𝛴 [O(2)] and the exceptional situation for tensor in

𝛴 [D 4 ] or 𝛴 [D 2 ]
). In this case there are 3 seemingly different permutations of this set. It can be observed that a

Study of design-transformed initially isotropic material

Let consider an isotropic elastic material:

T ∼ = ⎛ ⎝ 𝛾 0 0 0 0 𝛾 0 0 0 0 2𝜅 0 ⎞ ⎠ ℋ (21)
Using a unit quaternion, any rotation in SO(3) can be encoded as (see appendix B)

Q ∼ = ⎛ ⎝ 𝜔 2 + 𝑣 2 𝑥 -𝑣 2 𝑦 -𝑣 2 𝑧 2(𝑣 𝑥 𝑣 𝑦 -𝜔𝑣 𝑧 ) 2(𝜔𝑣 𝑦 + 𝑣 𝑥 𝑣 𝑧 ) 2(𝑣 𝑥 𝑣 𝑦 + 𝜔𝑣 𝑧 ) 𝜔 2 -𝑣 2 𝑥 + 𝑣 2 𝑦 -𝑣 2 𝑧 2(𝑣 𝑦 𝑣 𝑧 -𝜔𝑣 𝑥 ) 2(𝑣 𝑥 𝑣 𝑧 -𝜔𝑣 𝑦 ) 2(𝜔𝑣 𝑥 + 𝑣 𝑦 𝑣 𝑧 ) (𝜔 2 -𝑣 2 𝑥 -𝑣 2 𝑦 + 𝑣 2 𝑧 ) ⎞ ⎠ with 𝜔 2 + 𝑣 2 𝑥 + 𝑣 2 𝑦 + 𝑣 2 𝑧 = 1 Let introduce the parameter 𝛼 ∈ [0, 1] defined by 𝛼 := 𝑣 2 𝑥 + 𝑣 2 𝑦 . (22)
Using equation [START_REF] Franois | Identification des symtries matrielles de matriaux anisotropes[END_REF] and property 𝛼 = 1 -𝑣 2 𝑧 -𝜔 2 , the family of elastic materials obtained by carrying out design transformations on the initially isotropic elastic material ( 21) is obtained as a parametric set of the following form, with 𝛼 ∈ [0, 1]:

ℐℬ(𝛼) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 𝐼 1 (𝛼) = 𝜅 0 -2𝛼(1 -𝛼)(2𝜅 0 -𝛾 0 ) 𝐽 1 (𝛼) = 𝛾 0 + 2𝛼(1 -𝛼)(2𝜅 0 -𝛾 0 ) 𝐼 2 (𝛼) = 8𝛼(1 -𝛼)(1 -2𝛼) 2 (2𝜅 0 -𝛾 0 ) 2 𝐽 2 (𝛼) = 8𝛼 2 (1 -𝛼) 2 (2𝜅 0 -𝛾 0 ) 2 𝐼 3 (𝛼) = 16𝛼 2 (1 -𝛼) 2 (1 -2𝛼) 2 (2𝜅 0 -𝛾 0 ) 3 (23)
It can be noticed that:

-ℐℬ(𝛼) = ℐℬ(1 -𝛼) ∀𝛼 ∈ [0, 1] so that the variation of parametric parameter 𝛼 can be reduced to the interval [0, 1 2 ]; for 𝛼 = 0 the initial isotropic material is retrieved; for 𝛼 = 1 2 a tetragonal material is obtained; -𝐼 1 (𝛼) + 𝐽 1 (𝛼) = 𝜅 0 + 𝛾 0 , ∀𝛼 ∈ [0, 1 2 ] so that the curve in the (𝐼 1 , 𝐽 1 ) plane is a line segment; -2𝐼 3 (𝛼) 3 -𝐼 2 (𝛼) 2 𝐽 2 (𝛼) = 0, ∀𝛼 ∈ [0, 1 2 ] so that the curve is drawn on the surface of orthotropic elastic materials.

In other terms, the last item indicates that no biclinic material can be reached by the design transformation of an initially isotropic material.

The parametric set of equations ( 23) describe a curve in the space of elastic materials. This curve is represented in figure 4 for specific values of 𝜅 0 and 𝛾 0 (as the material remains positive definite, the positive definiteness condition is not plotted on the graph). The variable 𝛼 can be eliminated from ( 23) leading to the following equivalent system of equations, where 𝐼 1 ∈ [ 𝛾 0 2 , 𝜅 0 ]:

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 𝐽 1 = 𝛾 0 + 𝜅 0 -𝐼 1 𝐼 2 = 4(2𝐼 1 -𝛾 0 )(𝜅 0 -𝐼 1 ) 𝐽 2 = 2(𝜅 0 -𝐼 1 ) 2 𝐼 3 = 4(2𝐼 1 -𝛾 0 )(𝜅 0 -𝐼 1 ) 2
Moreover, the variables (𝛾 0 , 𝜅 0 ) can be eliminated from the previous system, leading to the following implicit algebraic system of equations, where 𝐼 1 > 0 and

𝐽 1 > 0: ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 𝐼 2 2 -32𝐼 2 1 𝐽 2 -4𝐼 2 𝐽 2 + 32𝐼 1 𝐽 1 𝐽 2 -8𝐽 2 1 𝐽 2 + 4𝐽 2 2 = 0 𝐼 2 2 -8𝐼 1 𝐼 3 + 4𝐼 3 𝐽 1 -2𝐼 2 𝐽 2 = 0 -𝐼 2 𝐼 3 + 4𝐼 1 𝐼 2 𝐽 2 + 2𝐼 3 𝐽 2 -2𝐼 2 𝐽 1 𝐽 2 = 0 2𝐼 2 3 -𝐼 2 2 𝐽 2 = 0 (24)
The algebraic system of equations ( 24) characterizes the set of at-least-orthotropic elastic materials that can be design-transformed into an isotropic material. Moreover, we have the following property (proof given in Appendix C):

Proposition 7.4. The algebraic system (24) corresponds exactly to the subset of at-least-orthotropic materials with two identical eigenvalues.

Particular case of pentamode materials

Let consider an isotropic elastic material:

T ∼ 0 = ⎛ ⎝ 0 0 0 0 0 0 0 0 2𝜅 0 ⎞ ⎠ ℋ
The integrity basis reads then

ℐℬ(𝛼) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 𝐼 1 (𝛼) = (1 -2𝛼) 2 𝜅 0 𝐽 1 (𝛼) = 4𝛼(1 -𝛼)𝜅 0 𝐼 2 (𝛼) = 32𝛼(1 -𝛼)(1 -2𝛼) 2 𝜅 2 0 𝐽 2 (𝛼) = 32𝛼 2 (1 -𝛼) 2 𝜅 2 0 𝐼 3 (𝛼) = 128𝛼 2 (1 -𝛼) 2 (1 -2𝛼) 2 𝜅 3 0 , 𝛼 ∈ [0, 1 2 ]. (25)
When the variable 𝛼 is eliminated, this leads to the following equivalent system of equations, where 𝐼 1 ∈ [0, 𝜅 0 ]:

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 𝐽 1 = 𝜅 0 -𝐼 1 𝐼 2 = 8𝐼 1 (𝜅 0 -𝐼 1 ) 𝐽 2 = 2(𝜅 0 -𝐼 1 ) 2 𝐼 3 = 8𝐼 1 (𝜅 0 -𝐼 1 ) 2
The parameter 𝜅 0 can be eliminated from the previous system of equation, where

𝐼 1 ≥ 0 and 𝐽 1 ≥ 0: ⎧ ⎪ ⎨ ⎪ ⎩ 𝐼 2 = 8𝐼 1 𝐽 1 𝐽 2 = 2𝐽 2 1 𝐼 3 = 8𝐼 1 𝐽 2 1 (26)
(26) is an algebraic system of equations that defines the entire set of at-leastorthotropic elastic materials that can be design-transformed into an isotropic material with two eigenvalues equal to zero. Moreover, we have the following property (proof given in Appendix C):

Proposition 7.5. The algebraic system (26) corresponds to the set of at-leastorthotropic materials with 2 eigenvalues equal to zero.

A Normal forms

In this appendix, normal forms for tensors of different symmetry classes are provided with respect to the harmonic basis6 ℋ. Normal forms are provided for generic tensors within each symmetry class. In the following, • corresponds to independent coefficients while × indicates equal coefficients.

-

𝛴 [Z 2 ] ⎛ ⎝ • 0 • • • • ⎞ ⎠ ℋ ⎛ ⎝ • • 0 • • • ⎞ ⎠ ℋ ⎛ ⎝ • • • • 0 • ⎞ ⎠ ℋ Remark: A physical rotation of 𝜋 2 switches normal form 2 to normal form 3. -𝛴 [D 2 ] ⎛ ⎝ • 0 0 • • • ⎞ ⎠ ℋ ⎛ ⎝ • 0 • • 0 • ⎞ ⎠ ℋ
Remark: A physical rotation of 𝜋 2 switches the normal form one to another.

-𝛴 [D 4 ] ⎛ ⎝ • 0 0 • 0 • ⎞ ⎠ ℋ (27)
-

𝛴 [O(2)] ⎛ ⎝ × 0 0 × 0 • ⎞ ⎠ ℋ (28)

B Quaternionic parametrization of rotations

Let denote 𝒮 𝑑 the unit sphere in dimension 𝑑.

Consider 𝑞 a pure quaternion:

𝑞 ∈ Im(H) ⇔ 𝑞 = 𝑞𝑥i + 𝑞𝑦j + 𝑞𝑧k with the Hamilton relations i 2 = j 2 = k 2 = ijk = -1
As 𝑞 ∈ Im(H), 𝑞 will be denoted by q from now on. SO(3) action on Im(H) is q ⋆ = ℎqℎ -1 with ℎ ∈ 𝒮 3 the space of unit quaternions.

ℎ ∈ 𝒮 3 ⇔ ℎ = 𝜔 + 𝑣𝑥i + 𝑣𝑦j + 𝑣𝑧k with 𝜔 2 + 𝑣 2 𝑥 + 𝑣 2 𝑦 + 𝑣 2 𝑧 = 1
As a quaternion the expression of h can be shortened to

ℎ = 𝜔 + v
in which 𝜔 is the scalar/real part of ℎ and v is the vector/imaginary part of ℎ. Unit quaternions can also be expressed under the so called polar form:

ℎ = cos ( 𝜃 2
) + n sin ( 𝜃 2
) with n ∈ 𝒮 2
By identification

𝜔 = cos ( 𝜃 2
) 𝑣 𝑖 = sin ( 𝜃 2
) 𝑛 𝑖 (29
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 𝑥𝑡 = 𝑅 cos ( 𝜃 2 ) 𝑥𝑧 = 𝑅 sin ( 𝜃 2
) cos(𝜑)

𝑥𝑥 = 𝑅 sin ( 𝜃 2
) sin(𝜑) cos(𝜓)

𝑥𝑦 = 𝑅 sin ( 𝜃 2
) sin(𝜑) sin(𝜓)

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
with 𝑅 the hyper radius. We can define the radius of the projected sphere by

𝑟 = 𝑅 sin ( 𝜃 2 )
and the radius of the projected circle

𝜌 = 𝑟 sin(𝜑) = 𝑅 sin ( 𝜃 2 ) sin(𝜑) in such a way that ⎛ ⎜ ⎜ ⎝ 𝑥𝑡 = 𝑅 cos ( 𝜃 2 ) 𝑥𝑧 = 𝑟 cos(𝜑) 𝑥𝑥 = 𝜌 cos(𝜓) 𝑥𝑦 = 𝜌 sin(𝜓) ⎞ ⎟ ⎟ ⎠
For points on the unit hypersphere:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 𝑅 = 1 𝑟 = sin ( 𝜃 2 ) 𝜌 = 𝑟 sin(𝜑) = sin ( 𝜃 2
) sin(𝜑)

Let's take back to the rotation formula q ⋆ = ℎqℎ -1 = (𝜔 + v)q(𝜔 -v) = 𝜔 2 q + 𝜔(vq -qv) -vqv

The multiplication of two pure quaternions gives:

vq = -v • q + v ∧ q Hence q ⋆ = ℎqℎ -1 = (𝜔 2 + v • v)q + 2𝜔(v ∧ q) + 2v ∧ (v ∧ q) = q + 2𝜔(v ∧ q) + 2v ∧ (v ∧ q)
In the R 3 canonical basis

𝑄 𝑖𝑗 = q ⋆ 𝑗 • e 𝑖 = 𝛿 𝑖𝑗 + 2𝜔(v ∧ e 𝑗 ) • e 𝑖 + 2v ∧ (v ∧ e 𝑗 ) • e 𝑖
Using vectorial identities it finally can be established that:

Q ∼ = (𝜔 2 -‖v‖ 2 ) I ∼ -2𝜔 𝜖 ≃ • v + 2v ⊗ v (30)
In matrix form

Q ∼ = ⎛ ⎝ 𝜔 2 + 𝑣 2 𝑥 -𝑣 2 𝑦 -𝑣 2 𝑧 2(𝑣𝑥𝑣𝑦 -𝜔𝑣𝑧) 2(𝜔𝑣𝑦 + 𝑣𝑥𝑣𝑧) 2(𝑣𝑥𝑣𝑦 + 𝜔𝑣𝑧) 𝜔 2 -𝑣 2 𝑥 + 𝑣 2 𝑦 -𝑣 2 𝑧 2(𝑣𝑦𝑣𝑧 -𝜔𝑣𝑥) 2(𝑣𝑥𝑣𝑧 -𝜔𝑣𝑦) 2(𝜔𝑣𝑥 + 𝑣𝑦𝑣𝑧) (𝜔 2 -𝑣 2 𝑥 -𝑣 2 𝑦 + 𝑣 2 𝑧 ) ⎞ ⎠ (31)
If we introduce the polar parametrization of quaternion in [START_REF] Verchery | Les invariants des tenseurs d'ordre 4 du type de l'lasticit[END_REF] we obtain:

Q ∼ = cos(𝜃) I ∼ -sin(𝜃) 𝜖 ≃ • n + (1 -cos(𝜃))n ⊗ n
which is the well-known Rodrigues formula for rotations.

Using the parametrization (29), equation ( 22) reads, with 𝑛 2 𝑥 + 𝑛 2 𝑦 = sin 2 (𝜑) and 𝜌 2 = 𝛼:

𝜌 2 = sin 2 ( 𝜃 2 ) sin 2 (𝜑) (32)
Moreover, as n ∈ 𝒮 2 , we get

𝑛 2 𝑧 = cos 2 (𝜑) (33)
Consider first the case 𝜌 ∈]0, 1[ and suppose that 𝜑 ̸ = 0. In this case the system (32)-( 33) can be rewritten as [START_REF] Zheng | The description, classification, and reality of material and physical symmetries[END_REF] and the set of rotations associated to the parameter 𝜌 2 is ) = 0. The case sin(𝜑) 2 = 0 corresponds to a physical rotation:

{ 𝜃 = ±2 arcsin ( 𝜌 sin(𝜑) ) 𝑛𝑧 = ± cos(𝜑)
𝛷 -1 (𝜌 2 ) = ⎧ ⎨ ⎩ n ± = ⎛ ⎝ sin(𝜑) cos(𝜓) sin(𝜑) sin(𝜓) ± cos(𝜑) ⎞ ⎠ , 𝜃 = ±2 arcsin ( 𝜌 sin(𝜑) ) , ∀(𝜑, 𝜓) ∈]0, 𝜋] × [0, 2𝜋] ⎫ ⎬ ⎭ Consider now the case 𝜌 = 1. In such a case, system (32)-(33) implies sin 2 ( 𝜃 2 ) = 1 ; sin(𝜑) 2 = 1 Hence 𝛷 -1 (1) = ⎧ ⎨ ⎩ n ± = ⎛ ⎝ cos (
𝛷 -1 (0) = ⎧ ⎨ ⎩ n ± = ⎛ ⎝ 0 0 ±1 ⎞ ⎠ , 𝜃 ∈ [0, 2𝜋] ⎫ ⎬ ⎭ while the case sin 2 ( 𝜃 2
) = 0 corresponds to a "null" rotation around any direction:

𝛷 -1 (0) = { n ± ∈ 𝑆 2 , 𝜃 = 2𝑘𝜋, ∀𝑘 ∈ Z }
C Proofs of propositions 7.4 and 7.5

First, recall that:

the tensors considered are real valued, so that the Boehler invariants ∈ R; -𝐼 2 ≥ 0 and 𝐽 2 ≥ 0 by definition; the elementary symmetric functions can be expressed in terms of Boehler invariants, see [START_REF] Franois | A damage model based on Kelvin eigentensors and Curie principle[END_REF].

Proof. Proposition 7.4

The condition for an elastic material to have at least 2 equal eigenvalues reads in terms of elementary symmetric functions:

-𝜎 2 1 𝜎 2 2 + 4𝜎 3 2 + 4𝜎 3 1 𝜎 3 -18𝜎 1 𝜎 2 𝜎 3 + 27𝜎 2 3 = 0 (35)
Using ( 16), the subset of at-least-orthotropic materials with two identical eigenvalues is defined by the algebraic system:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 2𝐼 2 3 -𝐼 2 2 𝐽 2 = 0 -(𝐼 1 + 𝐽 1 ) 2 (𝐼 2 -2𝐽 1 (4𝐼 1 + 𝐽 1 ) + 𝐽 2 ) 2 - 1 2 (𝐼 2 -2𝐽 1 (4𝐼 1 + 𝐽 1 ) + 𝐽 2 ) 3 + ... + 16(𝐼 1 + 𝐽 1 ) 3 (𝐼 3 -𝐼2𝐽 1 + 4𝐼 1 𝐽 2 1 -2𝐼 1 𝐽 2 ) -... -9(𝐼 1 + 𝐽 1 )(-𝐼 2 + 2𝐽 1 (4𝐼 1 + 𝐽 1 ) -𝐽 2 )(𝐼 3 -𝐼 2 𝐽 1 + 4𝐼 1 𝐽 2 1 -2𝐼 1 𝐽2) + ... + 27 4 (𝐼 3 -𝐼 2 𝐽 1 + 4𝐼 1 𝐽 2 1 -2𝐼 1 𝐽 2 ) 2 = 0 (36)
1. For 𝐼 2 = 0, [START_REF] Rivlin | Further Remarks on the Stress-Deformation Relation for Isotropic Materials[END_REF] and (36) reduce to the same polynomial conditions:

{ 𝐼 3 = 0 𝐽 2 (-8𝐼 2 1 + 8𝐼 1 𝐽 1 -2𝐽 2 1 + 𝐽 2 ) 2 = 0
2. The case 𝐼 2 ̸ = 0 and 𝐽 2 = 0 cannot occur as in this case the material is 𝑅 0 -orthotropic and have three distinct eigenvalues: 𝐼 2 √ 𝐽 2 . Solutions 𝐽 𝑎,𝑏,𝑐 1 of (36) are

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 𝜆 1 = 𝐽 1 𝜆 2 = 1 2
( 2𝐼 1 + 𝐽 1 - √ 2𝐼 2 + (2𝐼 1 -𝐽 1 ) 2 ) 𝜆 3 = 1 2 ( 2𝐼 1 + 𝐽 1 + √ 2𝐼 2 + (2𝐼 1 -𝐽 1 ) 2 ) 3
𝐽 𝑎 1 = 2𝐼 1 - 𝐼 2 -2𝐽 2 2 √ 2 √ 𝐽 2 𝐽 𝑏,𝑐 1 = 2𝐼 1 - √ 𝐽 2 √ 2 ± 2𝑖 √ 𝐼 2 √ 2
so that 𝐽 𝑎 1 is the only real valued solution. On the other hand, the Groebner basis of (24) reads in this case: 𝐼 2 √ 𝐽 2 is similar to the previous case up to some sign changes.

-𝐼2 √ 𝐽 2 √ 2 + 4𝐼 1 𝐽 2 -2𝐽 1 𝐽 2 + √ 2𝐽
We have thus proved that the real valued solution of (36) are identical to the (real valued) solution of [START_REF] Rivlin | Further Remarks on the Stress-Deformation Relation for Isotropic Materials[END_REF]. This ends the proof.

Proof. Proposition 7.5

The necessary and sufficient conditions for an elastic material to have at least 2 eigenvalues equal to zero reads in terms of elementary symmetric functions:

𝜎 2 = 0 and 𝜎 3 = 0 (38) Using ( 16), the subset of at-least-orthotropic material with two zero eigenvalues is defined by the algebraic system:

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 2𝐼 3 -𝐼 2 2 𝐽 2 = 0 4𝐼 1 𝐽 1 + 𝐽 2 1 - 1 2 (𝐼 2 + 𝐽 2 ) = 0 𝐼 3 -𝐼 2 𝐽 1 + 4𝐼 1 𝐽 2 1 -2𝐼 1 𝐽 2 = 0 (39)
The Groebner basis of the previous algebraic system reads (with lexicographic order (𝐼 3 , 𝐽 2 , 𝐼 2 , 𝐽 1 , 𝐼 1 )):

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ (8𝐼 2 1 + 𝐼 2 )(𝐼 2 -8𝐼 1 𝐽 1 ) 2 = 0 𝐼 2 -8𝐼 1 𝐽 1 -2𝐽 2 1 + 𝐽 2 = 0 2𝐼 1 𝐼 2 + 𝐼 3 -16𝐼 2 1 𝐽 1 -𝐼 2 𝐽 1 = 0
The case 𝐼 1 = 0 implies that 𝐼 2 = 0, so that 𝐽 2 = 2𝐽 2 1 and 𝐼 3 = 0. The case 𝐼 1 ̸ = 0 implies that 8𝐼 2 1 + 𝐼 2 ̸ = 0 because 𝐼 2 ≥ 0, so that 𝐼 2 = 8𝐼 1 𝐽 1 , and finally 𝐽 2 = 2𝐽 2 1 and 𝐼 3 = 8𝐼 1 𝐽 2 1 . We have thus prooved that (39) implies [START_REF] Smith | On isotropic functions of symmetric tensors, skew-symmetric tensors and vectors[END_REF]. It is trivial to check that (26) implies (39). This ends the proof.

D On reduced anisotropic invariants

}

The equivalence class can also be defined as the (non-compact) orbit of T 1 ≈ with respect to Dil. As well-known the space Ela can be partitioned into disjoint sets of equivalence classes. It can be observed that the adimensional anisotropic invariants:

𝑖 2 = 𝐼 2 8𝐼 1 𝐽 1 𝑗 2 = 𝐽 2 2𝐽 2 1 𝑖 3 = 𝐼 3 8𝐼 1 𝐽 2 1 (40)
introduced in section 5 are constant for the equivalence classes introduced above. For example, let consider a tensor T 1 ≈ having (𝑖 2 , 𝑗 2 , 𝑖 3 ) as adimensional invariants and that the associated elastic material is defined by (𝐼 1 , 𝐽 2 , 𝐼 2 , 𝐽 2 , 𝐼 3 ). A tensor T 2 ≈ in the same equivalence class is associated to the elastic material (𝛽 2 𝐼 1 , 𝛼 2 𝐽 1 , 𝛼 2 𝛽 2 𝐼 2 , 𝛼 4 𝐽 2 , 𝛼 4 𝛽 2 𝐼 3 ), a direct computation shows that the associated adimensional invariants will remain unchanged. The converse is not true, since different tensors (and not elastic materials) can have the same set of reduced adimensional invariants without being related by the equivalence relation. For example, consider T 1 ≈ and T 2 ≈ two tensors which are conjugated module SO [START_REF] Auffray | Gomtrie des espaces de tenseurs, application l'lasticit anisotrope classique et gnralise[END_REF]. Those tensors possess the same set of adimensional invariants but are not related by any element of Dil.

[Z 2 ]

 2 corresponds to the set of biclinic materials, 𝛴 [D 2 ] to the set of orthotropic materials, 𝛴 [D 4 ] to the set of tetragonal materials and 𝛴 [O(2)]

𝒦

  and can be shown to be an element of O(3)∖SO[START_REF] Auffray | Handbook of bidimensional tensors: Part I: Decomposition and symmetry classes[END_REF]

2

  The matrix normal forms of elasticity tensor T ≈ associated to the different symmetry classes are given in the basis ℋ in Appendix A.

  [Z 2 ] ) are strictly inside the volume defined by the surface, point 𝑂 corresponds to isotropic materials (stratum 𝛴 [O(2)] ), open ray ]𝑂𝐴) corresponds to tetragonal materials (stratum 𝛴 [D 4 ] ), open ray ]𝑂𝐵) corresponds to 𝑅 0 -orthotropic materials (stratum 𝛴 𝑠𝑝𝑒𝑐 [D 2 ] ), surface without {𝑂} ∪ ]𝑂𝐴) ∪ ]𝑂𝐵) corresponds to ordinary orthotropic materials (stratum 𝛴 𝑜𝑟𝑑 [D 2 ]).

Fig. 2 :

 2 Fig. 2: Algebraic variety of elastic material with respect to (𝐼 2 , 𝐽 2 , 𝐼 3 ) (without taking the positive definiteness property into account).

Lemma 5 . 2 .

 52 Consider 𝑀 ∈ ℳ 𝑆 𝑝+𝑞 a real symmetric matrix having the following block shape 𝐴 ∈ ℳ 𝑆 𝑝 , 𝐶 ∈ ℳ 𝑆 𝑞 , 𝐵 ∈ ℳ 𝑝,𝑞 . Let 𝑀/𝐴 be the Schur complement of 𝐴 in 𝑀 : 𝑀/𝐴 = 𝐶 -𝐵 𝑇 𝐴 -1 𝐵 and 𝑀/𝐶 be the Schur complement of 𝐶 in 𝑀 : 𝑀/𝐶 = 𝐴 -𝐵𝐶 -1 𝐵 𝑇 Then the following conditions are equivalent: 1. 𝑀 is positive definite if and only if 𝐴 and 𝑀/𝐴 are positive definite, 2. 𝑀 is positive definite if and only if 𝐶 and 𝐶/𝐴 are positive definite.

Fig. 3 : 1 ≈ , T 2 ≈∈

 312 Fig. 3: Positive definiteness elasticity domain in terms of adimensional anisotropic invariants (front and back views).

  𝛼, 𝛽 ∈ R , (𝛼, 𝛽) ̸ = (0, 0) } possess identical adimensional invariants. More details can be found in Appendix D.

∼ 3

 3 are related to the elementary symmetric functions 𝜎 1 , 𝜎 2 , 𝜎 3 by the uniquely invertible following relations:

Proposition 6 . 3 . 1 ⊗ e 1 + e 2 ⊗ e 2 )

 631122 Consider T ≈ ∈ Ela and T ∼ its representation as a second-order tensor in R 3 . Consider also the tensor s and ŝ its vector representation in R 3 . The set of Boehler invariants of T ≈ for the O(2)-action is equivalent to the set of invariants of the pair ( T ∼ , ŝ) for the O(3)action.

Proposition 7 . 2 .∈

 72 Except for isotropic elasticity tensors with 3 identical eigenvalues, any elasticity tensor T ≈ Ela is equivalent, up to a design transformation, to at least one tetragonal elasticity tensor. Proof. Consider an elasticity tensor T ≈ ∈ Ela and T ∼ its second-order tensor representation in 𝑆 2 (R 3 ). Consider [ T ∼ ] ℋ , the matrix representation of T ∼ in the harmonic basis ℋ. Diagonalization of [ T ∼

𝜋 2

 2 physical rotation will permute the first two eigenvalues. Hence we end up with 2 distinct situations : {𝜆 1 , 𝜆 1 , 𝜆 2 } and {𝜆 1 , 𝜆 2 , 𝜆 1 }. The first one correspond to a tensor in 𝛴 [O(2)] tensor, while the other to a tensor in 𝛴 [D 4 ] .

5 J1Fig. 4 : 2 .

 542 Fig. 4: An iso-eigenvalues curve in the space of elastic material. The initial material is isotropic (point O , = 0), with the design transformation it becomes orthotropic (curved line, 0 < 𝛼 < 1 2 ) and becomes tetragonal for 𝛼 = 1 2 (point A). Figure plotted with 𝛾 0 = 1 and 𝜅 0 = 3 2 .

⎠

  ) with the following spherical coordinate for n (𝜑, 𝜓) ∈]0, 𝜋] × [0, 2𝜋], n = ⎛ ⎝ 𝑛𝑥 = sin(𝜑) cos(𝜓) 𝑛𝑦 = sin(𝜑) sin(𝜓) 𝑛𝑧 = cos(𝜑) ⎞ In fact we have the following hypersherical coordinates

⎭

  = (2𝑘 + 1)𝜋, ∀𝑘 ∈ Z, 𝜓 ∈ [0, 2𝜋] ⎫ ⎬Consider now the case 𝜌 = 0. The system (32)-(either sin(𝜑) 2 = 0 or sin2 ( 𝜃 2

1 √ 2 𝐼 2 √ 𝐽 2 (

 1222 . With 𝐼 2 ̸ = 0 and 𝐽 2 ̸ = 0, 𝐼 3 = ± recall that by definition 𝐽 2 ≥ 0). (a) Let consider the case 𝐼 3 = + 1 √ 2

  unique solution with respect to 𝐽 1 is 𝐽 𝑎 1 again. (b) The case 𝐼 3 = -1 √ 2

1 ≈∈ 1 ≈

 11 Let consider the harmonic basis ℋ = { f1 , f2 , f3 } introduced subsection 3.1, and define the following second order tensors:d ∼ := f1 ⊗ f1 + f2 ⊗ f2 ; s ∼ := f3 ⊗ f3We can specify the following group of transformations Dil This relation can be shown to be : reflexive, symmetric and transitive. In other words, this is an equivalence relation. The equivalence class of an element T ≈ Ela is defined as [T

  where the superscript 𝑇 denotes the transposition.-SO(𝑑): the special orthogonal group, i.e. the subgroup of O(𝑑) consisting of transformations satisfying det(Q) = 1.

	(	1 0 0 -1	)	,

Let detail the case 𝑑 = 2. As a matrix group, O(2) can be generated by: R(𝜃) = ( cos 𝜃 -sin 𝜃 sin 𝜃 cos 𝜃 ) , 0 ≤ 𝜃 < 2𝜋, and P(e 2 ) = in which R(𝜃) is a rotation by angle 𝜃 and P(e 2 ) is the reflection across the 𝑥 axis. SO(2) corresponds to the group of plane rotations generated by R(𝜃). The following finite subgroups of O(2) will be used :

  2 ), should further be positive definite, meaning that its eigenvalues 𝜆 𝑖 should verify ∃ 𝑀 ∈ R *+ , 0 < 𝜆 𝑖 ≤ 𝑀

	2.2 From active physical rotations to elastic materials
	Consider O(2) the set of 2D isometric transformations. Its action on an element
	T

≈ of Ela gives a new element T ≈ of Ela,

  Through this relation two tensors are equivalent if their symmetry group are conjugate. The equivalence classes for this relation are called strata. More specifically, in what follows 𝛴 [𝐺] will denote the equivalence class of elasticity tensors having their symmetry group conjugate to 𝐺. In other words, [𝐺] is the symmetry class of the elements of the stratum 𝛴 [𝐺]

  3 will be referred to as the Kelvin basis. With respect to 𝒦, we can define the vector t image of t

					∼	by 𝜑
		⎛	𝑡 11	⎞
	t =	⎝	𝑡 22 √ 2 𝑡 12	⎠ 𝒦
	These two representations are isometric since t ∼	: t ∼	= t • t. A third representation
	is possible. This representation is associated to the decomposition of 𝑆 2 (R 2 ) into
	a deviatoric space (K 2

  4.2. The invariants 𝐼 2 and 𝐽 2 are, by definition, such that Remark 4.3. Using the parametrization (8), the invariants 𝐼 2 , 𝐽 2 , 𝐼 3 of the integrity basis read:

	𝐼 2 ≥ 0	𝐽 2 ≥ 0	(14)

  the set 𝛴 𝑜𝑟𝑑 [D 2 ] and 𝛴 𝑠𝑝𝑒𝑐 [D 2 ] have been considered in this figure. Elements of the set 𝛴 𝑜𝑟𝑑 [D 2 ] are ordinary orthotropic elasticity tensors. It only contains elements obtained just by imposing orthotropic invariance to generic anisotropic tensors. Elements of the set 𝛴 𝑠𝑝𝑒𝑐 [D 2 ]

Even if obvious, it is worth mentioning that both the stiffness tensor and its inverse, the compliance tensor, belong to the same vector space. Hence in the following no physical interpretation (stiffness or compliance) will be given to elements of Ela.

A subset 𝒞 of a vector space V is a cone if for each 𝑝 in 𝒞 and positive scalars 𝜆, 𝜆𝑝 ∈ 𝒞. The cone is said convex provided 𝜆𝑝 + 𝜇𝑞 ∈ 𝒞, for any positive scalars 𝜆, 𝜇 and any 𝑝, 𝑞 ∈ 𝒞

The number of inequations in the presented set is not minimal.

It is worth emphasizing that usually normal forms are provided with respect to the Voigt or the Kelvin representation.


Jacqueline Bourdeau
email: bourdeau@licef.teluq.uqam.ca

France Henri
email: henri@licef.teluq.uqam.ca

Aude Dufresne
email: aude.dufresne@licef.teluq.uqam.ca

Josephine Tchetagni
email: chetagni@licef.teluq.uqam.ca

Racha Ben Ali

R Ben

Collaborative Learning and Research Training: Towards a Doctoral Training Environment

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

Introduction

What is doctoral training made of? This learning situation has not been studied thoroughly, nor have the problems of learning environments to support this situation in the field of Technology-Enhanced Learning (TEL).

We claim that immersion in scientific communities and participating in collaborative learning activities are two essentials of doctoral education. A research training environment should support these activities through interactions among several actors using toplevel instrumentation. An analysis of actors, activities, resources, and competencies has been conducted in order to make explicit and to model the dynamics of research training through immersion in scientific communities and participating in collaborative learning activities.

An authoring environment has been prototyped to support such activities inside specific PhD programs. One Use Case consists of two or three research distributed teams sharing observations and discussions, a research training situation involving immersion and collaborative learning. A series of tests and co-experimentations involving Inquiry Learning Environments as a topic of study in the field of Technology-Enhanced Learning was conducted. An international collaboration happened through Kaleidoscope and the co-experimentations were made possible by an optical network infrastructure providing high quality interactions in terms of sharing and telepresence.

This paper describes the problem tackled, the methodology that was selected, and the results obtained.

The problem

In their "Recommendations from National Studies on Doctoral Education", Nyquist and Wulff [9] reviewed recent national studies on doctoral education in the US: "three themes that strongly emerge are: Current graduate education does not adequately match the needs and demands of the changing academy and broader society; there is a lack of systematic, developmentally appropriate supervision for many who are seeking careers that require or benefit from the attainment of a Ph.D.; and there exists a growing concern about the high level of attrition among doctoral students". Problems brought up in the literature on doctoral training [1], [2], [3], [4], [5] include the lack of definition of expected competencies, the inadequation between program requirements and available research training activities.

We claim that the main reason for those problems is the lack of explicit knowledge about the nature and the characteristics of a doctoral training in general and of specific doctoral programs in particular. We believe that most programs share many common elements although they have different cultures, depending on the disciplines, countries and institutions.

The first issues triggered in this project deal with the process of making explicit and formalizing this knowledge which is required to design a DTE on a scientific basis.

Second, a doctoral training environment should be aware of current evolution in the scientific world, and be ready to support what is now called e-Science. The term was coined by John Taylor who stated: 'e-Science is about global collaboration in key areas of science and the next generation of infrastructure that will enable it ' [6]. In the United Kingdom, e-Science projects span a range of disciplines from particle physics and astronomy to engineering and healthcare, and include international collaborative activities. Access to top-level instrumentation is a key to conducting e-Science, and optical networks are pivotal to this access.

In the spirit of a 'Collaboratory', the DTE needs to be equipped with a network infrastructure that can support immersion in a community and access to scientific instrumentation.

Methodology

First, a general characterization was carried out to draw the mandatory DTE orientations for the training of future generations of scientists, to prepare them for what is called 'E-science ' [6]. An inspiring model, the Collaboratory [7], was selected and provided the foundations for practicing collaborative research. Two features were identified as essential for doctoral training: immersion in scientific activities (such as coexperimentation) and in communities, and access to top-level instrumentation based on performance, safety, and privacy.

Then, a systematic analysis of processes, actors and resources was conducted, a fundamental step in designing a learning environment. Defining research competencies proved to be quite an ordeal. A substantial challenge concerns the support to developing research competencies, beginning with their explicitness.

Concerned with reusability, the team emphasized the fact that common grounds should be found for doctoral processes, and that generic learning objects or resources should be shareable as is recommended and now supported by recent technologies [8].

As a result, the team's objectives was to design an authoring system that allows DTE designers to instantiate their own DTE which reflects their own views, to pilot and instantiate a DTE in order to test and validate the ideas with authentic users placed in specific situations.

How can a doctoral training environment be sufficiently sophisticated to support the development of high-level skills through interaction among several actors using top-level instrumentation? An analysis of actors, activities, resources, and competencies has been conducted in order to make explicit and to model the dynamics of research training, and to obtain a design rationale.

Making Knowledge about PhD Training Explicit

Can the knowledge about PhD Training be made explicit? If so, why should this be? Traditionally, PhD supervisors resist the idea that it is possible and potentially beneficial to students for reasons such as: training methods are too peculiar, fuzzy, and interpersonal; it is better for students to discover by themselves, to make up their own mind and to pave their own way; it is the prerogative of each supervisor to say what they want to their students, etc.

Other reasons for this resistance could be the belief that doctoral training is so intertwined with research, and research is so competitive, that this knowledge ought to remain secret; or that doctoral education consists of initiating rather than training, and should therefore be transmitted 'by word of mouth'.

On the other hand, students often indicate they wish they had known in advance (and all along the process) what they should expect and what is expected from them [9], [10]. They also regret lacking a priori knowledge about the doctoral training process: "Had I only known…." Such feelings might contribute to the high dropout rates.

Such a situation appears paradoxical for researchers in our field. Specializing in computer and cognitive sciences, and experienced in principled design of learning environments, a team such as ours, could not remain indifferent to this challenge.

Below is a short description of our analysis of the doctoral training process, and of research training competencies.

The next section introduces the building and testing of a ligthpath network, and the design of an econtrolled experimental laboratory. This is followed by a short description of co-experimentations, showing the inclusion of an international dimension in a private scientific network.

The Doctoral Training Process

From the very beginning, the team tackled the analysis of the main processes and sub-processes of doctoral training, as well as the actors and resources associated with them [11]. It soon became clear that it was impossible to strictly draw the line between processes common to most PhD programs and those which are specific to a discipline or an institution. A reasonable approximation was made: the doctoral training process can be split up into five main subprocesses. These sub-processes are: registration, academic program, student support, scientific immersion, student life, and career preparation (Fig. 1). All of these sub-processes imply the participation of at least two of the following actors: student (enrolled in a program), professor, research staff, manager, and public (none of the above). These actors interact with each other and with the environment. A single user (human being) can play the roles of one or several actors.

Research Training Competencies

Similarly to the analysis of the doctoral process, it was impossible to strictly distinguish between competencies common to most PhD programs from those which are specific to a discipline or an institution [12]. Another challenge consisted of reconciling competencies with skills and also sometimes with objectives, since these terms do not always share the same semantics, both in the literature [13], and in the actual everyday life of a PhD program [14]. The notion of competency itself is currently under scrutiny in order to orient the future design of an agent capable to diagnose these competencies. The list of skills proposed by the British Research Councils and the Arts and Humanities Research Board [15] was selected as a reasonable common basis for most PhD programs. These 'Skills training requirements for research students' are organized into five categories:

A) 'Research skills and techniques -to be able to demonstrate skills such as: The ability to recognize and validate problems; Original, independent and critical thinking… B) Research Environment -to be able to: show a broad understanding of the context, at the national and international level; understand the processes for funding and evaluation of research… C) Research Management -to be able to: apply effective project management; design and execute systems for the acquisition and collation of information… D) Personal Effectiveness -to be creative, innovative and original; demonstrate self-discipline… E) Communication Skills -to write clearly and in a style appropriate to purpose, e.g. progress reports, published documents, thesis; to construct coherent arguments … F) Networking and Teamworking -to develop and maintain co-operative networks and working relationships … G) Career Management -to appreciate the need for and show commitment to continued professional development; to take ownership for and manage one's career progression…'

Results

This section introduces the results obtained sofar: a prototype of a doctoral training environment with graph-based navigation and support to emergent activities within communities, the building and testing of a network infrastructure, the access to an econtrolled research laboratory, and the conduct of coexperimentations at the international level.

1. Prototype of a Doctoral Training Environment

After extracting the design rationale, a prototype of a Doctoral Training Environment (DTE) was developed with its processes, actors, and resources. This section introduces the DTE Architecture and the implementation of the DTE authoring tool, called DTE-Author.

At the heart of the DTE is a set of links between conceptual entities: actors and their roles, activities and actions, competencies and resources. These links form the conceptual architecture [16], as illustrated in Fig. 2. The DTE Authoring Tool allows users to design and generate Doctoral Training Environments (DTE), hence its name: DTE-Author. It was built upon CONCEPT@, Télé-université's authoring environment, in connection with PALOMA, its learning objects repository. The rationale underlying the DTE-Author results from the cognitive analysis and includes three main features: 1) a doctoral process composed of six sub-processes: registration, program support, program activities, scientific immersion, student life and career preparation, 2) a competency model for future researchers, 3) interactions amongst various actors.

The goal of the DTE-Author is to allow users (authors) to create a process-based multi-actor environment specific to a given program (hereafter named DTE-X) that supports the development of competencies needed by future researchers. Thus, the authoring tool proposes the following method to design a DTE-X: 1) clarify each basic process in terms of doctoral activities, 2) specify each doctoral activity by linking it with the acquisition of a research competency, 3) assign specific resources and actors to each doctoral activity. The activity is considered as a set of actions, each action being carried out by an actor and supported by a resource.

The ''author'' is the only actor who uses the DTE-Author to design a DTE-X. The DTE design process is composed of four steps: identify the DTE-X, develop, configure, and generate the DTE-X. Each step is briefly described below:

1. Identify the DTE-X. This first step includes the following: name, select an abbreviation, and save. The

Link

Actor Action/Activity Link Action/Activity and system launches the six doctoral sub-processes and its corresponding activities, illustrated by files in a tree diagram whose specific nodes represent learning activities.

2. Developing the DTE-X. The second step consists in defining its structure in detail: add/modify a sub-process, delete a doctoral sub-process, add/modify an activity within a process, delete an activity within a process.

3. Configuring the DTE-X. The third step consists in configuring the activities:

-Associate a research competency to an activity by selecting from a set of competencies or by adding another one -Associate a resource to an activity. The DTE-Author includes four types of resources: application resources, NetNuke resources, generic resources and default resources. Application Resources consist of software which can be used for an activity. DNN Resources are resources used to design a DNN: newsgroups, e-filing, Weblist, etc. Generic Resources are documents: MS Office files, HTML, URL, pictures, videos, etc. For certain activities, the DTE-Author offers default resources that can belong to any of the three previous categories. Designers can view and select those considered most appropriate. Activity and resources deemed less relevant can be replaced by resources considered more appropriate. Designers can add their own applications.

-Associate an action to a specific activity if needed. The DTE-Author provides designers with two possibilities for each activity: select an action amongst a set of actions proposed for each activity; add a new action.

-Associate a resource with each action. To do so, authors consult ors send a request to a Learning Objects Repository. Furthermore, they can select or add a new resource of their choice.

-Associate an actor with each resource involved in the activity. In the framework of this action, users' rights must be taken into consideration.

At this point, once all links have been created as illustrated in Fig 4, the instance, i.e. a DTE-X, can be generated. -Configure a generic/permanent Website menu that provides generic services (a CONCEPT@ functionality): select a template, test, and generate the DTE-X.

As a result, the user-author obtains a learning environment organized as a tree structure. However, she can also select an alternate mode to visualize this structure and navigate: the graph-based mode.

Graph-Based Navigation

The doctoral process can be viewed as an institutional accreditation process: institutional steps are well defined with rules for registration, academic program, thesis defense, etc. However, it is also a research and development process, where creativity, innovation and adaptation for individual research take an important part. For such areas, a more open and emergent coordination framework must be drawn. Although it is less organized and uniform, some kind of structure and support can be provided to coordinate activities amongst the academic participants, for example for a series of seminars, or to formalize emergent coordination between professors and their students, to ensure follow up, support and progress.

For those more open and emergent structures of activities, we decided to integrate Explor@Graph. This flexible navigation environment is based on conceptual maps, where different structures of activities, resources, concepts or others entities can be freely described and navigated according to users' rights. Explor@Graph first presents a metaphor appropriate for the DTE: space exploration, brain neurons, connection amongst elements. Second, the system respects ergonomic principles for improved usability, offering high visibility, flexible navigation, progress feedback, collaborative features and adaptive interfaces to support progression. For example, it is possible to introduce dynamic help features, deadline reminders or simply remind users of the conference of the day.

Explor@Graph was developed from a standalone application in VB to a Web-based VB.Net version, called Explora@GraphNet. It was then linked to the DTE. Therefore, it offers graphic interfaces as an alternate browser within the DTE, exporting the process and sub-process structures, and allowing the creation of other more flexible structures with the Explor@Graph Editor. As a result, the system can display the same structures either as a tree diagram or a graph, and can launch the same resources in either mode. In order to integrate the processes defined in one mode or the other, we developed a framework of ontologically defined data exchange using a SESAME database. In this generic solution, (sub)process structures are defined as OWL structures, and an exportation module is specified from the DTE and from Explor@Graph to communicate process structures to the SESAME database. For flexible navigation within the DTE, Explora@GraphNet displays such structures (Fig. 5). This generic system could be used to communicate with other applications developed in the context of LORNET, such as resource or participant structures.

Emergent activities within communities

The need to 'leave room for emergence' was clearly expressed by the focus groups organized by the pilot team. The pilot study consists of instantiating a DTE-X for the PhD Program in Cognitive Informatics at TELUQ-UQAM, hereafter called DCI. As a result of two focus groups, one with professors and the other with students, the team decided to complement the DTE activity structure and to innovate by designing for emergent activities within communities -scientific community, program community, student community, etc. A conceptual and functional architecture is currently being designed to achieve this goal.

Our main challenge in designing DCI's DTE is to materialize our conceptualisation of doctoral studies in terms of human communities. In our view, immersion into a scientific community is a key factor for a doctoral program because it permits doctoral students to get involved in authentic research situations allowing them to develop their research competencies and to get acquainted with the collaboration process within such communities. Collaboration among researchers at local, regional or international levels is fundamental to improve research productivity. Getting exposed to the activity of a research community is crucial for the training of doctoral students. Collaboration constitutes the locus for immersion into the scientific community. Since immersion is an emergent process that cannot be systematically planned, we had to find a way to induce and support it. Immersion in a scientific community banks on communication, sharing and networking and on people's adhesion; it calls for influence, conviction and autonomy. Three main principles have been adopted to guide the design of the DCI's DTE in terms of immersion in a scientific community. Firstly, doctoral processes and research competencies associated with them have to be made explicit to the students in order for them to be able to manage their own competency development. Secondly, the various resources that could support competency acquisition and immersion processes should be easy to access and to manage. Thirdly, communication and information flow should be highly transparent for students as well as for all members of the DIC program using the DCI's DTE. These three principles are implemented in the DCI's DTE mainly through the design of the interfaces, the creation of a personalized administration space, as well as by letting the members of the DCI's community free to use their own tools (email, chat, blog, etc.), and by the use of RSS feed. Using an RSS aggregator allows for the visualization and handling of consolidated information. It is an effective way to support information flow and the sharing of information. The home page is personalized and divided in three main sections. In the upper part, a TreeMap represents the doctoral processes, and indicates to students their own progress. In the lower part, the students access the various RSS feeds referenced and an interactive calendar that gives access to the feeds' history. There are two types of feeds: the ones created and selected by students and the mandatory ones given by the DCI's DTE manager (for example, News from the program, Changes in the program site, etc.). The heading, common to all pages of the environment, gives users access to their own administration space as well as to their favorite URLs and other RSS feeds. This home page has been designed to be easily integrated by the user in an external site. Giving the users the possibility to implement this page on their own site should stimulate their participation. Another section of the DCI's DTE gives access to a TreeMap representing the research competencies. It allows the students to manage their progress throughout the process of acquiring competencies.

The personal administration space allows users to manage their personal email addresses and syndication channels (RSS feeds), and to add their favorite RSS feeds and references. This space also identifies the resources and the information the user wants to access, and the resources and information she wants other members of the DCI's community to have access to.

As of instantiating the structured activities, such as the academic program, we started with the detailed specification of two use cases: registration and defense. These use cases involve all actors, require many interactions amongst them, and require services that can operate optimally over an optical network. The registration process involves the actor 'public' (potential student) who needs to share and present her intentions and portfolio to a potential supervisor; the supervisor introduces her research lab, team, and facilities, as well as expectations and requirements. The use case 'defense' involves the program 'manager' in the organization and ruling, and requires students to present and demonstrate their results (demos); members of the jury need a high performing network connection to share their views and discuss relevant issues. Obviously, many of these services are common with the services needed for immersion in scientific activities and communities.

4. A Network Infrastructure

A ligthpath provides a scientific community with a network which is highly efficient, secure, and private. A range of services is needed to support immersion and access to instrumentation. Integration of these services -videoconferencing for example-onto the network is a challenge in terms of quality of service (QoS) and of security. will involve Vancouver and use the multipoint Access Grid mode. The multicast IP technology has been configured in order to improve the quality of service by reducing the latency in transmitting an audio and video flow duplicated to distributed participants. We expect this solution to provide an improved quality over the MCU Polycom since instead of sending a mosaic of low resolution images of participants, Access Grid will send these images with their original resolution.

Further development and testing include 3D applications, models and simulations, and high resolution visualisation.

An E-controlled Experimental Lab

How can remote teams access experimental research laboratories? This is the challenge faced by the LORIT@D team in the BEST project. LORIT* is a Télé-université experimental research lab that allows the observation and capture of multimedia data from multiple pre-synchronized sources [17] In order for distributed teams to be able to conduct remote experiments, E-control is currently being implemented to allow observation, manipulation and data collection from remote or distributed locations, therefore called LORIT at a distance, or LORIT@D (Fig. 7). Further tests with users will assess the LORIT@D technical performance and usability both with and without the ligthpath connection. The ligthpath connection reduces latency to its absolute minimum. Propagation delay of the optical signal (2*45 = 90 msec for an approximate lightpath length of 4800km between Montréal and Vancouver) and a minimum transmission delay (2*8kb/1Gbps = 16 µsec for a 1Gbps optical route) provides this minimal latency. Note that this minimal latency is uncompressible and corresponds to the speed of light. Most E-control applications and remote manipulation of instruments require this reduced latency, as LORIT@D does. Another advantage of using the 1Gpbs bandwidth over a private ligthpath is to transmit high resolution videoconferencing without reducing the quality of service. Observations can be improved by an order of magnitude, and scientific discussions can be more efficient, as is needed in co-experimentation settings. 5 ollaboratory', and they require a thorough preparation in terms of research protocol as well as of technical support. Aside the LORIT@D work reported in the previous section, other innovative ways of coexperimenting have been organized and investigated, at the national and international levels.

In conjunction with our partne LEIDOSCOPE European research network in the field of Technology-Enhanced Learning, we prepared a co-experimentation where two research teams composed of PhD students and researchers discussed by holding a videoconference and running application sharing software [18]. Both teams used called ' to share the manipulation, observation and argumentation protocol (Fig. 9). This activity is also part of the CIEL project in Kaleidoscope. For doctoral students, this activity can

Fig. 1 .

 1 Fig. 1. The doctoral process

Fig. 2 .

 2 Fig. 2. The conceptual architecture of the DTEThis conceptual architecture provides the foundation for the functional architecture (Fig.3) and consequently the development of the DTE authoring tool.

Fig. 3 .

 3 Fig. 3. The Functional Architecture of the DTE

Fig. 4 .

 4 Fig. 4. Configuring the DTE-X 4. Generating the DTE-X. The fourth and last step consists of generating a Website or portal which supports this learning environment:-Configure a generic/permanent Website menu that provides generic services (a CONCEPT@ functionality): select a template, test, and generate the DTE-X.

Fig. 5 .

 5 Fig. 5. Graph-based Navigation with Explora@GraphNet

  Such a network is currently being constructed between several universities in Canada and European partners in the field of Technology-Enhanced Learning. Called SCORE*, this program aims to support and test innovative ideas in the area of research training. The network architecture is illustrated in Fig 6. The SCORE ligthpath network is currently being implemented and tested in the BEST project to ensure it supports the DTE and the services it offers, such as videoconferencing, 3D models and simulations, and high resolution vizualisation. CANARIE provides the ligthpath connectivity within Canada over CA*net4 (Canadian backbone) and with the European partners, in conjunction with the SVL action of the Kaelidoscope network of excellence . The first measures proved to be close to theoretical values, including when testing with the University of Twente in the Netherlands.

Fig. 6 .

 6 Fig. 6. The SCORE network architecture

Fig. 7 .

 7 Fig. 7. The LORIT@D in action User interface design raises specific challenges ranging from latency to cognitive load and perceptual accuracy. Such topics are currently being studied by two teams, in Montreal and in Vancouver.The objectives of the BEST project for the LORIT@D are the following: 1) to install and program the control software for a new Web-based controller interface, 2) to study possible usage scenarios, 3) to program the controller interface for each of those scenarios, 4) to design a reservation and technical