M Olive
email: molive@cmi.univ-mrs.fr

N Auffray

SYMMETRY CLASSES FOR EVEN-ORDER TENSORS

Keywords: Mathematics Subject Classification. 20C35, 74B99, 15A72 Anisotropy, Symmetry classes, Higher order elasticity

The purpose of this article is to give a complete and general answer to the recurrent problem in continuum mechanics of the determination of the number and the type of symmetry classes of an even-order tensor space. This kind of investigation was initiated for the space of elasticity tensors. Since then, different authors solved this problem for other kinds of physics such as photoelectricity, piezoelectricity, flexoelectricity, and strain-gradient elasticity. All the aforementioned problems were treated by the same computational method. Although being effective, this method suffers the drawback not to provide general results. And, furthermore, its complexity increases with the tensorial order. In the present contribution, we provide general theorems that directly give the sought results for any even-order constitutive tensor. As an illustration of this method, and for the first time, the symmetry classes of all even-order tensors of Mindlin second strain-gradient elasticity are provided.

1. Introduction 1.1. Physical motivation. Within these last years, an increasing interest in generalized continuum theories [START_REF] Forest | Mechanics of generalized continua: Construction by homogenization[END_REF][START_REF] Dell'isola | Generalized hooke's law for isotropic second gradient materials[END_REF][START_REF] Lebée | A bending gradient model for thick plates, part i : Theory[END_REF][START_REF] Dell' Šisola | How contact interactions may depend on the shape of cauchy cuts in n-th gradient continua: approach "à la d'alembert[END_REF] has been observed. These works based on the pioneering works of [START_REF] Mindlin | Micro-structure in linear elasticity[END_REF][START_REF] Mindlin | Second gradient of strain and surface-tension in linear elasticity[END_REF][START_REF] Toupin | Elastic materials with couple stresses[END_REF] propose extended kinematic formulation in order to take into account size effects within the continuum. The price to pay is the appearance in the constitutive relations of tensors of order greater than 4. These higher order objects are difficult to handle and extracting physical meaningful information is not straightforward. The aim of this paper is to provide general results concerning the type and number of anisotropic systems an even-order tensor can have.

Such results have important applications, at least, for the modeling and the numerical implementation of non classical linear constitutive laws:

Modeling: The stakes of modeling is given a material, and a set of physical variables of interest to construct the more general (linear, in the present context) constitute law that describes the behavior of that material. An example of such a method is provided by Thionnet and Martin [START_REF] Thionnet | A new constructive method using the theory of invariants to obtain material behavior laws[END_REF] where, given a set of variables V and the material symmetry group S, they derive mechanical behavior laws using the Theory of Invariants and Continuum Thermodynamics. In such perspective our results will say, without making any computation whether or not S is contained in the set of symmetry classes of L(v, v ′ ) the space of linear applications from v ∈ V to v ′ ∈ V . Numerical implementation: To implement a new linear constitutive law in a finite-element code, one has to known the complete set of matrices needed to model the associated anisotropic behavior. In that perspective, our result in a precious guideline as it tells you the number of the sought matrices, and the way to construct them. Such a way to proceed is illustrated, for example, in the case of 3D strain gradient elasticity in [START_REF] Auffray | Matrix representations for 3d strain-gradient elasticity[END_REF].

Constitutive tensors symmetry classes. In the field of mechanics, constitutive laws are usually expressed in terms of tensorial relations between the gradient of primary variables and their fluxes [START_REF] Gu | Interfacial discontinuity relations for coupled multifield phenomena and their application to the modeling of thin interphases as imperfect interfaces[END_REF]. As it is well-known, this feature is not restricted to linear behaviors since tensorial relations appear in the tangential formulation of non-linear ones [START_REF] Triantafyllidis | The influence of scale size on the stability of periodic solids and the role of associated higher order gradient continuum models[END_REF]. It is also known that a general tensorial relation can be divided into classes according to its symmetry properties. Such classes are referred to as symmetry classes in the field of mechanics [START_REF] Forte | Symmetry classes for elasticity tensors[END_REF], or as isotropic classes (or strata) in the field of mathematical physics [START_REF] Abud | The geometry of spontaneous symmetry breaking[END_REF][START_REF] Auffray | Invariant-based approach to symmetry class detection[END_REF].

In the case of second order tensors, the determination of symmetry classes is rather simple. According to a spectral analysis it can be concluded that any second-order symmetric tensor 1 can either be orthotropic ([D 2 ]), transverse isotropic ([O(2)]), or isotropic ([SO(3)]). Such kind of tensors are commonly used to describe, e.g., heat conduction, electric permittivity. For higher order tensors, the determination of the set of symmetry classes is more involved, and is mostly based on an approach introduced by Forte and Vianello [START_REF] Forte | Symmetry classes for elasticity tensors[END_REF] in the case of elasticity. Let us briefly detail this case.

The vector space of elasticity tensors, which will be noted Ela throughout this paper, is the subspace of 4th-order tensors endowed with the following index symmetries:

Minor symmetries: : E ijkl = E jikl = E jilk Major symmetry: : E ijkl = E klij The symmetries will be denoted using the following notation: E (ij) (kl) , where (..) indicates the invariance under the in-parenthesis indices permutations, and .. .. the invariance with respect to the underlined blocks permutations. Index symmetries encode the physics described by the mathematical operator. On one hand, the minor symmetries stem from the fact that rigid body motions do not induce deformation (symmetry of ε), and that the material is not subjected to volumic couple (symmetry of σ). And, on the other hand, the major symmetry is the consequence of the existence of a free energy. An elasticity tensor, E, can be viewed as a symmetric linear operator on T (ij) , the space of symmetric second order tensors. According to Forte and Vianello [START_REF] Forte | Symmetry classes for elasticity tensors[END_REF], for the classical action of SO(3), Ela is divided into the following 8 symmetry classes:

[Ela] = {[1], [Z 2 ], [D 2 ], [D 3 ], [D 4 ], [O(2)], [O], [SO(3)]} which correspond, respectively, to the following physical classes 2 : triclinic ([1]), monoclinic ([Z 2 ]), orthotropic ([D 2 ]), trigonal ([D 3 ]), tetragonal ([D 4 ]), transverse isotropic ([O(2)]), cu- bic ([O]
) and isotropic ([SO(3)]). The mathematical notations used above will be detailed in subsection 2.2. Besides this fundamental result, the interest of the Forte and Vianello paper was to provide a general method to determine the symmetry classes of any tensor space [START_REF] Auffray | Invariant-based approach to symmetry class detection[END_REF]. Since then, using this method, other results have been obtained: The limitations of the Forte-Vianello approach. The method introduced by Forte and Vianello is actually the more general one 3 . But, in the same time, it suffers from at least two limitations:

(1) The computation of the harmonic decomposition;

(2) The specificity of the study for each kind of tensor. In its original setting, the method requires the computation of the explicit harmonic decomposition of the studied tensor, i.e. its decomposition into the sum of its SO(3)-irreducible components, also known as harmonic tensors 4 . Its explicit computation, which is generally based 1 Such a tensor is related to a symmetric matrix, which can be diagonalize in an orthogonal basis. The stated result is related to this diagonalization.

2 These symmetry classes are subgroups of the spatial group of rotations SO(3). This is due to the fact that the elasticity tensor is even-order. To treat odd-order tensors, the full orthogonal group O(3) has to be considered. 3 Some other methods can be found in the literature, such as counting the symmetry planes [START_REF] Chadwick | A new proof that the number of linear elastic symmetries is eight[END_REF], or studying the SU(2)-action on Ela [8], . . . but these methods are difficult to generalize to any kind of vector space. 4 Harmonic tensors are completely symmetric and traceless, they inherit this name because of a well-known isomorphism in R 3 between these tensors and harmonic polynomials [START_REF] Backus | A geometrical picture of anisotropic elastic tensors[END_REF].

on an algorithm introduced by Spencer [START_REF] Spencer | A note on the decomposition of tensors into traceless symmetric tensors[END_REF], turns out to be intractable in practice as the tensorial order increases. But, this is not a real problem, since the only information needed is the number of different harmonic tensors of each order appearing in the decomposition, i.e. the isotypic decomposition. Based on arguments presented in [START_REF] Jerphagnon | The description of the physical properties of condensed matter using irreducible tensors[END_REF], there exists a direct procedure to obtain this isotypic decomposition from the tensor index symmetries [START_REF] Auffray | Décomposition harmonique des tenseurs -Méthode spectrale[END_REF]. Such an approach has been used in [START_REF] Le Quang | Symmetry groups and symmetry classes for sixth-order tensors[END_REF] to obtain the symmetry classes of 6th-order tensors.

As each kind of tensor space requires a specific study, this specificity constitutes the other limitation of the method. This remark has to be considered together with the observation that, for even-order tensors it seems that there exists, indeed, only two possibilities. Precisely, a tensor space has as many classes as:

• the full symmetric tensor space: e.g. Ela is divided into 8 classes as the full symmetric one [START_REF] Forte | Symmetry classes for elasticity tensors[END_REF]; • the generic tensor space5 : any other 4th-order tensor space (photoelasticity [START_REF] Forte | Symmetry classes and harmonic decomposition for photoelasticity tensors[END_REF], flexoelectricity [START_REF] Quang | The number and types of all possible rotational symmetries for flexoelectric tensors[END_REF],...) is divided into 12 classes such as the generic one. The same observation can also be made for 2nd-and 6th-order tensors [START_REF] Le Quang | Symmetry groups and symmetry classes for sixth-order tensors[END_REF]. Understanding what is the general rule behind this observation would be an important result in mechanics. Its practical implication is the direct determination of the number and the type of symmetry classes for any constitutive laws no matter their orders. This result is of valuable importance to understand the feature of generalized continuum theories, in which higher order tensors are involved in constitutive laws.

1.2. Organization of the paper. In section 2, the mains results of this paper: Theorems I, II and III are stated. As an application, the symmetry classes of the even-order constitutive tensor spaces of Mindlin second strain-gradient elasticity are determined. Results concerning the 6th-order coupling tensor and the 8th-order second strain-gradient tensor are given for the first time. Obtaining the same results with the Forte-Vianello approach would have been much more difficult. Other sections are dedicated to the construction of our proofs. In section 3, the mathematical framework used to obtain our result is introduced. Thereafter, we study the symmetry classes of a couple of harmonic tensors, which is the main purpose of the tool named clips operator. We then give the associated results for couples of SO(3)-closed subgroups (theorem 4.6 and table 2). Thanks to these results, and with the help of some previous works on that topic done by Golubistky and al. [START_REF] Ihrig | Pattern selection with O(3) symmetry[END_REF], we obtain in section 5 some general results concerning symmetry classes for general even-order tensors. In section 6 our main results are finally proved. The appendix is devoted to proofs and calculus of clips operations.

Main results

In this section, our main results are stated. In the first subsection, the construction of Constitutive Tensor Spaces (CTS in the following) is discussed. This construction allows us to formulate our main results in the next subsection. Finally, application of these results to Mindlin Second Strain-Gradient Elasticity (SSGE in the following) is considered. It worth noting that precise mathematical definitions of the symmetry classes are given in section 3.

2.1. Construction of CTS. Linear constitutive laws are linear applications between the gradient of primary physical quantities and theirs fluxes. Each of these physical quantities (see table 1) are in fact related to subspaces6 of tensors spaces: theses subspaces will be called State Tensor Spaces (STS in the following). These STS will be the primitive notion from which the CTS will be constructed. In the following, L(F, G) will indicate the vector space of linear application from F to G.

Let consider now two STS : E 1 = T G and E 2 = T f , which are pth-and qth-order STS with possibly index symmetries. As a consequence, they belong to subspaces of ⊗ p R 3 and ⊗ q R 3 . A constitutive tensor C is a linear application between E 1 and E 2 , i.e. an element of the space

Physical notions

Mathematical object Mathematical space Gradient Tensor state T 1 ∈ ⊗ p R 3 T G : tensor space with index symmetries Fluxes of gradient Tensor state T 2 ∈ ⊗ q R 3 T f : tensor space with index symmetries Linear constitutive law 1. Physical and mathematical links L(E 1 , E 2 ). This space is isomorphic, modulo the use of an euclidean metric, to E 1 ⊗ E 2 . Physical properties leads to some index symmetries on T ∈ E 1 ⊗ E 2 ; thus the vector space of such T is some vector subspace

C ∈ L(T G , T f ) T C ⊂ L(T G , T f ) Table
T C of E 1 ⊗ E 2 .
Now, each of the spaces E 1 , E 2 and E 1 ⊗ E 2 has a natural O(3) action. In this paper, we are concerned with cases in which p + q = 2n. In such a situation, it is known that the O(3)-action on E 1 ⊗ E 2 reduces to the one of SO(3) [START_REF] Forte | Symmetry classes for elasticity tensors[END_REF]. We therefore have

L(E 1 , E 2 ) ≃ E 1 ⊗ E 2 ⊂ T p ⊗ T q = T p+q=2n
Examples of such constructions are provided in the following table:

Property E 1 E 2 Tensor product for CTS Number of classes Elasticity T (ij) T (ij) Symmetric 8 Photoelelasticity T (ij) T (ij) Standard 12 Flexoelectricity T (ij)k T i Standard 12 First-gradient elasticity T (ij)k T (ij)k Symmetric 17
In the former table, two kinds of constitutive tensor spaces appeared whether they describe • Coupled physics : such tensors encode the coupling between two different physics, such as photoelasticity and flexoelectricity; • Proper physics : such tensors describe solely one physical phenomenon, such as classical and first-gradient elasticities.

On a mathematical side this implies :

• Coupled physics : the spaces E 1 and E 2 may differ, and when E 1 = E 2 linear applications are not self-adjoint; • Proper physics : we have E 1 = E 2 and linear applications are self-adjoint 7 .

Therefore, the elasticity tensor is a self-adjoint linear application between the vector space of deformation tensors and the vector space of stress tensors. These two spaces are modeled on T (ij) . The vector space of elasticity tensors is therefore completely determined by T (ij) and the symmetric nature of the tensor product, i.e. Ela = T (ij) ⊗ S T (kl) where ⊗ S denotes the symmetric tensor product. On the side of coupling tensors, the flexoelectricity is a linear application between E 1 = T (ij)k , the space of deformation gradient, and E 2 = T l the electric polarization, therefore

Flex = T (ij)k ⊗ T l .
2.2. Symmetry classes of even order tensor spaces. Let consider an even order constitutive tensor space T 2n , it is known [START_REF] Jerphagnon | The description of the physical properties of condensed matter using irreducible tensors[END_REF] that this space can be decomposed orthogonally8 into a full symmetric space and a complementary one which is isomorphic to a tensor space of order 2n -1, i.e. :

T 2n = S 2n ⊕ C 2n-1
Let us introduce : S 2n : the vector space of 2nth-order completely symmetric tensors; G 2n : the vector space of 2nth-order tensors with no index symmetries 9 .

The following observation is obvious :

S 2n ⊆ T 2n ⊆ G 2n
and therefore, if we note I the operator which to a tensor space gives the set of its symmetry classes, we obtain:

I(S 2n ) ⊆ I(T 2n ) ⊆ I(G 2n
) Symmetry group of even-order tensors are conjugate to SO(3)-closed subgroups [START_REF] Zheng | The description, classification, and reality of material and physical symmetries[END_REF][START_REF] Forte | Symmetry classes for elasticity tensors[END_REF]. The classification of SO(3)-closed subgroups is a classical result that can be found in many references [START_REF] Ihrig | Pattern selection with O(3) symmetry[END_REF][START_REF] Sternberg | Group theory and physics[END_REF]. These subgroups are, up to conjugacy,: Lemma 2.1. Every closed subgroup of SO( 3) is conjugate to precisely one group of the following list:

{1, Z n , D n , T , O, I, SO(2), O(2), SO (3)
} Among these groups, we can distinguish: planar groups: • Z n (n ≥ 2) the cyclic group of order n, generated by the n-fold rotation Q(k; θ = 2π n ), is the symmetry group of a chiral polygon; • D n (n ≥ 2) the dihedral group of order 2n generated by Z n and Q(i; π), is the symmetry group of a regular polygon; 2) the subgroup generated by SO(2) and Q(i; π). Classes of exceptional subgroups are: T the tetrahedral group of order 12 which fixes a tetrahedron, O the octahedral group of order 24 which fixes an octahedron (or a cube), and I the subgroup of order 60 which fixes an icosahedron (or a dodecahedron). In section 6, the symmetry classes of S 2n and G 2n are obtained: Lemma 2.2. The symmetry classes of S 2n are :

: {1, Z n , D n , SO(2), O (2)
• SO(2) the subgroup of rotations Q(k; θ) with θ ∈ [0; 2π[; • O(
I(S 2 ) = {[D 2 ], [O(2)], [SO(3)]} I(S 4 ) = {[1], [Z 2 ], [D 2 ], [D 3 ], [D 4 ], [O(2)], [O], [SO(3)]} n ≥ 3, I(S 2n ) = {[1], [Z 2 ], • • • , [Z 2(n-1) ], [D 2 ], • • • , [D 2n ], [O(2)], [T ], [O], [I], [SO(3)]} Lemma 2.3. The symmetry classes of G 2n are : I(G 2 ) = {[1], [Z 2 ], [D 2 ], [SO(2)], [O(2)], [SO(3)]} I(G 4 ) = {[1], [Z 2 ], • • • , [Z 4 ], [D 2 ], • • • , [D 4 ], [SO(2)], [O(2)], [T ], [O], [SO(3)]} n ≥ 3, I(G 2n ) = {[1], [Z 2 ], • • • , [Z 2n ], [D 2 ], • • • , [D 2n ], [SO(2)], [O(2)], [T ], [O], [I], [SO(3)]}
With the following cardinal properties:

n 1 2 ≥ 3 #I(S 2n ) 3 8 2(2n + 1) #I(G 2n ) 6 12 4n + 5
The symmetry classes of T 2n are clarified by the following theorem :

Theorem I. Let T 2n be a tensor space then either I(T 2n ) = I(S 2n ) or I(T 2n ) = I(G 2n ).

In other terms,the number and the type of classes are the same as • either S 2n the space of 2n-order completely symmetric tensors. In this case, the number of classes is minimal;

• or G 2n the space of 2n-order generic tensors. In this case the number of classes is maximal. In fact, as specified by the following theorems, in most situations the number of class is indeed maximal:

• For coupling tensors:

Theorem II. Let consider T 2p the space of coupling tensors between two physics described respectively by two tensors vector spaces E 1 and E 2 . If these tensor spaces are of orders greater or equal to 1, then I(T 2p ) = I(G 2p ).

• For proper tensors:

Theorem III. Let consider T 2p , the space of tensors of a proper physics described by the tensor vector space E. If this tensor space is of order p ≥ 3, and is solely defined in terms of its index symmetries, then

I(T 2p ) = I(G 2p ).
Remark 2.4. Exception occurs for: p = 1: the space of symmetric second order tensors is obtained; p = 2: in the case of T (ij) , the space of elasticity tensors is obtained. In each of the aforementioned situations the number of classes is minimal. There is no other situation where this case occurs. It should therefore be concluded that the space of elasticity tensors is exceptional.

2.3.

Second strain-gradient elasticity (SSGE). Application of the former theorems will be made on the even order tensors of SSGE. In first time the constitutive equations will be summed-up, then results will be stated. It worth noting, that obtaining the same results with the Forte-Vianello approach would have been far more complicated.

Constitutive laws. In the second strain-gradient theory of linear elasticity [START_REF] Mindlin | Second gradient of strain and surface-tension in linear elasticity[END_REF][START_REF] Forest | First vs. second gradient of strain theory for capillarity effects in an elastic fluid at small length scales[END_REF], the constitutive law gives the symmetric Cauchy stress tensor 10 σ (2) and the hyperstress tensors τ (3) and ω (4) in terms of the infinitesimal strain tensor ε (2) and its gradients η (3) = ε (2) ⊗ ∇ and κ (4) = ε (2) ⊗ ∇ ⊗ ∇ through the three linear relations: 4) : ε (2) + M (5) ∴ η (3) + N (6) :: κ (4) , τ

     σ (2) = E (
= M T (5) : ε + A (6) ∴ η (3) + O (7) :: κ (4) , ω (4) = N T (6) : ε (2) + O T (7) ∴ η (3) + B (8) :: κ (4)

(2.1)

where :, ∴, :: denote, respectively, the double, third and fourth contracted product. Above 11 ,

σ (ij) , ε (ij) , τ (ij)k , η (ij)k = ε (ij),k , ω (ij)(kl) and κ (ij)(kl) = ε (ij), (
kl) are, respectively, the matrix components of σ (2) , ε (2) , τ (3) , η (3) , ω (4) and κ (4) relative to an orthonormal basis (i; j; k) of R 3 .

And

E (ij) (lm) , M (ij)(lm)n , N (ij)(kl)(mn) , A (ij)k (lm)n , O (ij)k(lm)(no)
and B (ij)(kl) (mn)(op) are the matrix components of the related elastic stiffness tensors.

Symmetry classes. The symmetry classes of the elasticity tensors and of first strain-gradient elasticity tensors has been studied in [START_REF] Forte | Symmetry classes for elasticity tensors[END_REF] and [START_REF] Le Quang | Symmetry groups and symmetry classes for sixth-order tensors[END_REF]. Hence, here solely considered the spaces of coupling tensors N (6) and of second strain-gradient elasticity tensors B (8) will be considered.

• Let define Ces to be the space of coupling tensors between classical elasticity and second strain-gradient elasticity:

Ces = {N (6) ∈ G 6 |N (ij)(kl)(mn) } A direct application of theorem 2.2 leads to the following results: 10 Exceptionally, in this subsection, tensor orders will be indicated by in-parenthesis exponents. 11 The comma classically indicates the partial derivative with respect to spatial coordinates. The T exponent denotes transposition. The transposition is defined by permuting the p first indices with the q last, where p is the tensorial order of the image of a q-order tensor.

I(Ces) = {[1], [Z 2 ], • • • , [Z 6 ], [D 2 ], • • • , [D 6 ], [SO(2)], [O(2)], [T ], [O], [I], [SO(3)]}
Therefore Ces is divided into 17 symmetry classes.

• Let define Sgr to be the space of second strain-gradient elasticity tensors:

Sgr = {O (8) ∈ G 8 |O (ij)(kl) (mn)(op) }
A direct application of theorem 2.2 leads to the following results:

I(Sgr) = {[1], [Z 2 ], • • • , [Z 8 ], [D 2 ], • • • , [D 8 ], [SO(2)], [O(2)], [T ], [O], [I], [SO(3)]}
Therefore Sgr is divided into 21 symmetry classes.

Mathematical framework

In this section the mathematical framework of symmetry analysis is introduced. In the first two subsections the notions of symmetry group and class are introduced, meanwhile the last is devoted to the introduction of irreducible spaces. The presentation is rather general, and will be specialized to tensor spaces only at the end of the section.

3.1. Isotropy/symmetry groups. Let ρ be a representation of a compact real Lie group 12 G ρ : G → GL(E) on a finite dimensional R-linear space E. This action will be noted

∀(g, x) ∈ G × E, g • x = ρ(g)(x)
For any element of E, the set of operations g in G letting this element invariant is defined as

Σ x := {g ∈ G | g • x = x}
This set, for physicists, is called the symmetry group of x, and for mathematicians the stabilizer or the isotropy subgroup of x. It is worth noting that, owning to G-compactness, every isotropy subgroup is a closed subgroup of G. Conversely, a dual notion can be defined for G-elements. For any subgroup K of G, the set of K-invariant elements in E is defined as

E K := {x ∈ E | k • x = x ∀k ∈ K}
Such a set is referred to as a fixed point set and is a linear subspace of E. In this context we will note d(K) = dim E K . It has to be observed that fixed-point sets are group inclusion reversing, i.e. for subgroups K 1 and K 2 of G, we have the following property

K 1 ⊂ K 2 ⇒ E K 2 ⊂ E K 1
For a given isotropy group K, the former sets are linked by the following property:

x ∈ E K ⇒ K ⊂ Σ x 3.2. Isotropy/symmetry classes. We aim at describing objects that have the same symmetry properties but may differ by their orientations in space. The first point is to define the set of all the positions an object can have. To that aim we consider the G-orbit of an element x of E:

Orb(x) := {g • x | g ∈ G} ⊂ E
Due to G-compactness this set is a submanifold of E. Elements of Orb(x) will be said to be Grelated. A fundamental observation is that G-related vectors have conjugate symmetry groups. More precisely 13 Orb(x) = Orb(y)

⇒ ∃g ∈ G | Σ x = gΣ y g -1 (3.1) Let us define the conjugacy class of a subgroup K ⊂ G by [K] = {K ′ ⊂ G|∃g ∈ G, K ′ = gKg -1 } (3.
2) 12 In the following G will solely represent a compact real Lie group, therefore this precision will mostly be omitted. 13 With the classical coset notation if H is a subgroup of G and g ∈ G that is not in the subgroup H, then a left coset of H in G is defined gH = {gh : h ∈ H} and symmetricaly for a right coset.

An isotropy class (or symmetry class) [Σ] is defined as the conjugacy class of an isotropy subgroup Σ. This definition implies that there exists a vector x ∈ E such that Σ = Σ x and Σ ′ ∈ [Σ]; furthermore Σ ′ = gΣg -1 for some g ∈ G. The notion of isotropy class is the good notion to define the symmetry property of an object modulo its orientation: symmetric group is related to a specific vector, but we deal with orbits, which are related to symmetric classes because of 3.1. Due to G-compactness there is only a finite number of isotropy classes [START_REF] Bredon | Introduction to compact transformation groups[END_REF], and we note

I(E) := {[1]; [Σ 1 ]; • • • ; [Σ l ]}
the set of all isotropy classes. In the case G = SO(3) this result is known as the Hermann theorem [START_REF] Herman | Some theorems of the theory of anisotropic media[END_REF][START_REF] Auffray | Décomposition harmonique des tenseurs -Méthode spectrale[END_REF]. Elements of I(E) are conjugate to SO(3)-closed subgroups and this collection was introduced in subsection 2.2.

Irreducible spaces.

For every linear subspace F of E, we note

g • F := {g.x | ∀g ∈ G ; ∀x ∈ F} and we say that F is G-stable if g • F ⊂ F for every g ∈ G.
It is clear that, for every representation, the subspaces {0} and E are always G-stable. If, for a representation ρ on E, the only G-invariant spaces are the proper ones, the representation will be said irreducible. For a compact Lie group, the Peter-Weyl theorem [START_REF] Sternberg | Group theory and physics[END_REF] ensures that every representation can be split into a direct sum of irreducible ones. Furthermore, in the case G = SO(3), those irreducible representations are explicitly known.

There is a natural action of SO(3) on the space of R 3 -harmonic polynomials. If p denotes such a harmonic polynomial, and x ∈ R 3 , then, for every g ∈ SO(3) we note

g • p(x) = p(g -1 • x)
Harmonic polynomials form a graded vector space, and to each subspace of a given degree a SO(3)-irreducible representation is associated. H k will be the vector space of harmonic polynomials of degree k, with dim H k = 2k+1. If we take a vector space V to be a SO(3)-representation, it can be decomposed into SO(3)-irreducible spaces:

V = H k i
Grouping the same order irreducible spaces, the SO(3)-isotypic decomposition is obtained:

V = n i=0 α i H i
where α i is the multiplicity of the irreducible space H i in the decomposition, and n the order of the higher order irreducible space of the decomposition.

Application to tensor spaces. In mechanics, V is a vector subspace of ⊗ p R 3 . In R 3 there exists an isomorphism,φ, between harmonic polynomial spaces and harmonic tensor spaces [START_REF] Backus | A geometrical picture of anisotropic elastic tensors[END_REF][START_REF] Forte | Symmetry classes for elasticity tensors[END_REF]. Therefore all that have been previously said in term of harmonic polynomials can be translated in terms of harmonic tensors. A detailed discussion on this isomorphism can be found in [START_REF] Backus | A geometrical picture of anisotropic elastic tensors[END_REF]. Therefore H k = ϕ(H k ), is the space of harmonic tensors, that is the space of completely symmetric and traceless tensors. According to this isomorphism, any tensor space T n can be decomposed into SO(3)-irreducible tensors:

T n = n i=0 α i H i
The symmetry group of T ∈ T n is the intersection of the symmetry groups of all its harmonic components 14

Σ T = n i=0   α i j=0 Σ H i,j  
In the same way, I(T n ) will be obtained as a function of the symmetry classes of the irreducible representations involved in the harmonic decomposition of T n . The symmetry classes of SO(3)irreducible representations are explicitly known [START_REF] Ihrig | Pattern selection with O(3) symmetry[END_REF][START_REF] Golubitsky | Singularities and groups in bifurcation theory[END_REF], what is unknown is how to combine these results to determine the symmetry classes of V (or T n ).

Clips operations

The aim of this section is to construct symmetry classes of a reducible representation from the irreducible ones. To that aim a new class-operator, named clips operator, will be defined. The main result of this section is given in table 2, which contains all clips operations between SO(3)-closed subgroups. It is worth noting that this table contains more results than strictly needed for the proofs of our theorems. Nevertheless, we believe that these results are interesting in their own and may find applications for other problems. The explicit proofs of these results can be found in Appendix A.

In this section the intersection of solely two symmetry classes, is considered. Extensions to more general reducible representations will be treated in section 5. Let us start with the following lemma: Lemma 4.1. Let E be a representation of a compact Lie group G that split into a direct sum of two G-stable subspaces

E = E 1 ⊕ E 2 where g • E 1 ⊂ E 1 and g • E 2 ⊂ E 2 ∀g ∈ G

If we note by I the set of all isotropy classes associated to E, I i the set of all isotropy classes associated to E

i (i = 1, 2), then [Σ] ∈ I if and only if there exist [Σ 1 ] ∈ I 1 and [Σ 2 ] ∈ I 2 such as Σ = Σ 1 ∩ Σ 2 .
Proof. If we take [Σ 1 ] ∈ I 1 and [Σ 2 ] ∈ I 2 , we know there exists two vectors x 1 ∈ E 1 and x 2 ∈ E 2 such that Σ i = Σ x i (i = 1, 2). Then, let x := x 1 + x 2 .

For every g ∈ Σ 1 ∩ Σ 2 we have g

• x 1 + g • x 2 = x 1 + x 2 = x; thus Σ 1 ∩ Σ 2 ⊂ Σ x . Conversely for every g ∈ Σ x we have g • x = x = g • x 1 + g • x 2
But, as E i are G-stable and are in direct sum, we conclude that g • x i = x i (i = 1, 2). The reverse inclusion is proved.

The other implication is similar: if we take [Σ] ∈ I then we have Σ = Σ x for some x ∈ E. And x can be decomposed into x 1 + x 2 . The same proof as above shows that Σ = Σ x 1 ∩ Σ x 2 . Lemma 4.1 shows that the isotropy classes of a direct sum are related to intersections of isotropy subgroups. But as intersection of classes is meaningless, the results cannot be directly extended. To solve this problem, one tool named clips operator will be introduced. To that aim, let first consider the following lemma: Lemma 4.2. For every two G-classes [Σ i ] (i = 1, 2), and for every g 1 , g 2 in G, there exists

g = g -1 1 g 2 in G such that g 1 Σ 1 g -1 1 ∩ g 2 Σ 2 g -1 2 = [Σ 1 ∩ gΣ 2 g -1 ]
14 In the notation H i,j the first exponent refers to the order of the harmonic tensor, meanwhile the second indexes the multiplicity of H i in the decomposition.

Proof.

Let g = g -1 1 g 2 and Σ = g 1 Σ 1 g -1 1 ∩ g 2 Σ 2 g -1 2
For every γ ∈ Σ we have γ = g 1 γ 1 g -1 1 = g 2 γ 2 g -1 2 for some γ i ∈ Σ i (i = 1, 2); then

g 1 γg -1 1 = γ 1 ∈ Σ 1 and g 1 γg -1 1 = gγ 2 g -1 ∈ gΣ 2 g -1 Thus we have g 1 Σg -1 1 ⊂ Σ 1 ∩ gΣ 2 g -1
, and conversely. As g 1 Σg -1 1 is conjugate to Σ, we have proved the lemma.

Σ 2 ], noted [Σ 1 ] ⊚ [Σ 2 ], to be [Σ 1 ] ⊚ [Σ 2 ] := {[Σ 1 ∩ gΣ 2 g -1 ] for all g ∈ G} which is a subset of G-classes.
If we note 1 the identity subgroup, one can observe some immediate properties:

Proposition 4.4. For every G-class [Σ] we have [1] ⊚ [Σ] = {[1]} and [G] ⊚ [Σ] = {[Σ]}
Then, if we have two G-representations E 1 and E 2 , and if we note I i the set of all isotropy classes of E i , the clips operator can be extended to these sets and noted

I 1 ⊚ I 2 := Σ 1 ∈I 1 ,Σ 2 ∈I 2 [Σ 1 ] ⊚ [Σ 2 ]
Then, by lemma 4.1, we obtain the corollary:

⊚ [Z n ] [D n ] [T ] [O] [I] [SO(2)] [O(2)] [Z m ] [Z d ] [D m ] [Z d 2 ] [Z d ] [Z d 2 ] Z d ′ 2 , [Z dz ] [Z d ] , [D d ] [T ] [Z d 2 ] [Z d 3 ] [Z 2 ] [Z d 3 ] , [D d 2 ] [Z 2 ] [Z 3 ] [T ] [O] [Z d 2 ] [Z d 3 ] [Z d 4 ] [Z 2 ] [Z d 3 ] , [Z d 4 ] [D d 2 ] , [D d 3 ] [D d 4 ] [Z 2 ] [Z 3 ] [T ] [Z 2 ] [D 2 ] , [Z 3 ] [D 3 ] , [Z 4 ] [D 4 ] , [O] [I] [Z d 2 ] [Z d 3 ] [Z d 5 ] [Z 2 ] [Z d 3 ] , [Z d 5 ] [D d 2 ] [D d 3 ] , [D d 5 ] [Z 2 ] [Z 3 ] [T ] [Z 2 ] [Z 3 ] , [D 3 ] [T ] [Z 2 ] [Z 3 ] , [D 3 ] [Z 5 ] , [D 5 ] [I] [SO(2)] [Z n ] [Z 2 ] [Z n ] [Z 2 ] [Z 3 ] [Z 2 ] [Z 3 ] , [Z 4 ] [Z 2 ] [Z 3 ] , [Z 5 ] [SO(2)] [O(2)] [Z d 2 ] [Z n ] [Z 2 ] [D n ] [D 2 ] [Z 3 ] [D 2 ] [D 3 ] , [D 4 ] [D 2 ] [D 3 ] , [D 5 ] [Z 2 ] [SO(2)] [Z 2 ] [O(2)] Notations Z 1 := D 1 := 1 d 2 := gcd(n, 2) d 3 := gcd(n, 3) d 5 := gcd(n, 5) d ′ 2 := gcd(m, 2) dz := 2 if d = 1, dz = 1 otherwise d 4 := 4 if 4 | n 1 otherwise

Isotropy classes of harmonic tensors

In this section, the construction of the symmetry classes of a reducible representation from its irreducible components will be studied. To that aim, in subsection 5.1 the main results concerning the symmetry classes of irreducible representations are summed-up. In subsection 5.2, and using the results of the previous section, basic properties of reducible representations are obtained. These results will be used in section 6 to prove the theorems stated in section 2. From now on, all the results will be expressed in terms of tensor spaces. For forthcoming purposes, let us introduce the following notations. For each integer k, we note:

Γ T (k) := T if T ∈ I k ∅ otherwise ; Γ O(k) := O if O ∈ I k ∅ otherwise and Γ I(k) := I if I ∈ I k ∅ otherwise Σ(k) := SO(2) if SO(2) ∈ I k ∅ otherwise ; Ω(k) := O(2) if O(2) ∈ I k ∅ otherwise
where I k is the set of symmetry classes of H k .

Isotropy classes of direct sum.

We have this obvious lemma, directly deduced from theorem 5.1 :

Lemma 5.2. We have Γ T (k) = ∅ ⇒ {[D 2 ], [D 3 ]} ⊂ I k ) ; Γ O(k) = ∅ ⇒ {[D 2 ], [D 3 ], [D 4 ]} ⊂ I k ; Γ I(k) = ∅ ⇒ {[D 2 ], • • • , [D 5 ]} ⊂ I k .
We note I(k, n), the (n -1) self clips operations of I k ,which is the set of isotropy classes of a n-uple of k-th harmonic tensors [START_REF] Auffray | Invariant-based approach to symmetry class detection[END_REF], i.e. nH k . The basic operations are, for every integers k ≥ 1 and n ≥ 2 I(k, n) := I k ⊚ I(k, n -1) and I(k, 1) := I k On the simple example of H 2 , the following fact can be observed

I(2, n) := I 2 ⊚ I(2, n -1) = I 2 ⊚ I 2 = {[1] , [Z 2 ] , [D 2 ] , [O(2)] , [SO(3)]}
This result can be generalized: Corollary 5.3. For every integers n ≥ 2 and k ≥ 2, the isotropy classes of nH k are

I(k, n) = I k ⊚I k = {[1], [Z 2 ], • • • , [Z k ], [D 2 ], • • • , [D k ], [Γ T (k) ], [Γ O(k) ], [Γ I(k) ], [Σ(k)], [Ω(k)], [SO(3)]}
Proof. From theorem 5.1 we know that [D l ] ∈ I k for 2 ≤ l ≤ k; furthermore [SO(3)] ∈ I k then from lemma 4.4 we know that, for all integers 2 ≤ l ≤ k we will have (by induction), for all n ≥ 2 [D l ] ∈ I(k, n). Then, when we compute

I(k, n) ⊚ I k we will have [D l ] ⊚ [D l ] = {[1], [Z l ], [D l ]} Neither [O(2)
] and [SO(2)], with cyclic or dihedral conjugacy classes, generate other cases. The same occurs for clips operation of cyclic groups. Now, because of the lemma 5.2 we also see that no exceptional conjugacy class generates other cases. We have then prove our corollary.

Corollary 5.4. For every integers 2 ≤ 2p < 2q, we have

I(2p, 2q) := I 2p ⊚ I 2q = {[1], [Z 2 ], • • • , [Z max(q;2p) ], [D 2 ], • • • , [D 2q ], [Γ T (2p) ∪ Γ T (2q)], [Γ O (2p) ∪ Γ O (2q)], [Γ I (2p) ∪ Γ I (2q)], [O(2)], [SO(3)]} Proof. First of all, it is clear that because [SO(3)] ∈ I k i (i = 1, 2) we will have all [D l ] for 2 ≤ l ≤ 2q. Now we will have all [Z i ] ⊚ [SO(3)] for all 1 ≤ i ≤ p. We also have [Z j ] ∈ [D j ] ⊚ [D j ]
for all 1 ≤ j ≤ 2q; this show that

{[1], [Z 2 ], • • • , [Z max(q;2p) ]} ⊂ I[2p, 2q]
Now, we can observe that clips operation of dihedral groups and [O(2)] does not generate cyclic groups; and lemma 5.2 shows that no other cases can be generated with exceptional subgroups.

6. Isotropy classes of constitutive tensors 6.1. The symmetry classes of even-order tensor space. Let consider the constitutive tensor space T 2n , it is known that this space can be decomposed orthogonally into a full symmetric space and a complementary one which is isomorphic to a tensor space of order 2n -1 [START_REF] Jerphagnon | The description of the physical properties of condensed matter using irreducible tensors[END_REF], i.e. :

T 2n = S 2n ⊕ C 2n-1
Let consider the SO(3)-isotypic decomposition of T 2n

T 2n = 2n k=0 α k H k , with α 2n = 1
The part related to S 2n solely contains even order harmonic tensors with multiplicity one [START_REF] Jerphagnon | The description of the physical properties of condensed matter using irreducible tensors[END_REF], i.e.

S 2n = n k=0 H 2k and C 2n-1 = 2n-1 k=0 α ′ k H k with α ′ k = α k for k odd α k -1 for k even
Using the clips operator, the symmetry classes of T 2n can be expressed:

I(T 2n ) := I(S 2n ) ⊚ I(C 2n-1 )
Let us first determined the symmetry classes of S 2n . Using the results of the previous section, we have:

Lemma 6.1. n = 1, I(S 2 ) = {[D 2 ], [O(2)], [SO(3)]} n = 2, I(S 4 ) = {[1], [Z 2 ], [D 2 ], [D 3 ], [D 4 ], [O(2)], [O], [SO(3)]} ∀ n ≥ 3, I(S 2n ) = {[1], [Z 2 ], • • • , [Z 2(n-1) ], [D 2 ], • • • , [D 2n ], [O(2)], [T ], [O], [I], [SO(3)]}
With the following cardinal properties:

#I(S 2 ) = 3 ; #I(S 4 ) = 8 ; #I(S 2n ) = 2(2n + 1)
Proof. The case n = 1 is obtained as a direct application of the theorem 5.1 and the proposition 4.4. For n ≥ 2, let consider the corollary 5.4 in the case k 1 = 2(n -1) and k 2 = 2n.

I(2(n -1), 2n) := {[1], [Z 2 ], • • • , [Z 2(n-1) ], [D 2 ], • • • , [D 2n ], [Γ T (2(n -1)) ∪ Γ T (2n)], [Γ O (2(n -1)) ∪ Γ O (2n)], [Γ I (2(n -1)) ∪ Γ I (2n)], [O(2)], [SO(3)]}
In the collection of planar isotropy classes, [Z 2n-1 ] and [Z 2n ] are missing. It should be observed that the clips operation I[2(n -1), 2n] ⊚ I 2(n-2) can never complete the sequence. For exceptional groups it can be observed that for any n ≥ 3 the SO(3)-irreducible decomposition will contain H 6 . As {[T ], [O], [I]} are isotropy classes for H 6 , it would be the same for any space that contains H 6 .

• Therefore, for n ≥ 3

I(S 2n ) = {[1], [Z 2 ], • • • , [Z 2(n-1) ], [D 2 ], • • • , [D 2n ], [O(2)], [T ], [O], [I], [SO(3)]}
and, #I(T 2n s ) = 2(2n -1). • For the case n = 2, we obtain the same result but without the class T and I and, in such a case, #I(T 4 s ) = 8. Definition 6.2. For a given SO(3) representation on the tensor space T 2n (n ≥ 3), we define

C(2n) = {[1], [Z 2 ], • • • , [Z 2n ], [D 2 ], • • • , [D 2n ], [SO(2)], [O(2)], [T ], [O], [I], [SO(3)]} We also define n = 1, C(2) = {[1], [Z 2 ], [D 2 ], [SO(2)], [O(2)], [SO(3)]} n = 2, C(4) = {[1], [Z 2 ], • • • , [Z 4 ], [D 2 ], • • • , [D 4 ], [SO(2)], [O(2)], [T ], [O], [SO(3)]}
One can observe that these sets are in fact all the isotropy classes allowed by the Hermann theorem, and we clearly have the following cardinal properties: #C(2) = 6 ; #C(4) = 12 ; #C(2n, n ≥ 3) = 4n + 5 Definition 6.3. Let T 2n be a tensor space which SO(3)-irreducible decomposition is T 2n ≃ 2n k=0 α k H k . T 2n is said to be even-harmonic (EH) if α 2p+1 = 0 for each 0 ≤ p ≤ (n -1). Lemma 6.4. Let G 2n be the vector space of 2n-th order tensors with no index symmetries, G 2n is not EH.

Proof. For n ≥ 1, the induced reducible SO(3)-representation on G 2n = ⊗ 2n R 3 is construct by tensorial products of the vectorial one. Such a construction implies odd-order tensors in the harmonic decomposition of G 2n . Now, the following proposition can be proved Theorem I. Let T 2n be a tensor space, for any n ≥ 3, if T 2n is EH then I(T 2n ) = I(S 2n ), otherwise I(T 2n ) = I(G 2n ).

Proof. Let consider the SO(3)-irreducible decomposition of T 2n , this decomposition can be written T 2n ≃ S 2n ⊕ C 2n-1 . The following inclusion is always true

I(S 2n ) ⊆ I(T 2n ) ⊆ I(G 2n ) ⊆ C(2n)
If T 2n is not EH, there exists at least one k ∈ N such as α 2k+1 = 0 I(S 2n ) ⊚ [SO(2)] ⊆ I(S 2n ) ⊚ I(C 2n-1 ) = I(T 2n ) since, from theorem 5.1, any odd-order harmonic tensor contains [SO [START_REF] Auffray | Décomposition harmonique des tenseurs -Méthode spectrale[END_REF]] in its symmetry classes. As, from lemma 6.1, dihedral groups are contained up to 2n in I(S 2n ), the missing cyclic groups of I(S 2n ) are obtained by clips products with [SO [START_REF] Auffray | Décomposition harmonique des tenseurs -Méthode spectrale[END_REF]]. Therefore

I(S 2n ) ⊚ SO(2) = C(2n) hence I(T 2n ) = C(2n)
As G 2n is not EH, I(G 2n ) = C(2n). Therefore we reach the sought conclusion, that if T 2n is not EH I(T 2n ) = I(G 2n ).

Conversely, if T 2n is EH, C 2n-1 solely contains even order irreducible spaces and its leading harmonic spaces are, at most, of order 2(n -1). If the leading harmonic spaces order are strictly less than 2(n -1), the same analysis as for lemma.6.1 lead to the same conclusion. Suppose now, that α ′ 2(n-1) ≥ 1, using all the previous results,we have

I(T 2n ) = I(S 2n ) ⊚ I(C 2n-1 ) = (I 2n ⊚ I 2(n-1) ) ⊚ (I(2(n -1), α ′ 2(n-1) ) ⊚ I(2(n -2), α ′ 2(n-2) )) = (I 2n ⊚ I(2(n -1), 2)) = I(S 2n ) ⊚ I 2(n-1)
As I 2(n-1) does not contain [SO [START_REF] Auffray | Décomposition harmonique des tenseurs -Méthode spectrale[END_REF]] the missing classes cannot be generated, therefore I(T 2n ) = I(S 2n ). Therefore it should be concluded that for any 2n-order tensor space, symmetry classes are either the same as S 2n or the same as G 2n . Let us now investigate, in the next subsection, under which conditions a constitutive tensor space falls into one of the identified two possibilities. 6.2. Construction of a constitutive tensor space. This last subsection will be devoted to the proof of our main result. The space of constitutive tensors is a subspace of linear applications from E 1 to E 2 . As seen in section 2

L(E 1 , E 2 ) ≃ E 1 ⊗ E 2 ⊂ T p ⊗ T q ≃ T 2n=p+q
These vector spaces described the physical quantities involved in the problem understudy. We know, from the previous section, that any CTS has as many symmetry classes as either the complete symmetric tensor spaces, or the generic tensor space. Here we are interested in obtaining the conditions both on E 1 to E 2 and on the tensor product (symmetric or not) under which T 2n is even order, and therefore has a minimal number of symmetry classes. To that aim, distinction will be made between coupling and proper tensor spaces, in the sense previously defined in section 2.

Coupling tensor spaces. We consider here two STS given by their SO(3) isotypic decomposition:

E 1 = T p = p i=0 β i H i and E 2 = T q = q i=0 γ j H j with β p = γ q = 1
Now results are given by the two following lemmas: Lemma 6.5. If E 1 = E 2 , p > q and if T p ⊗ T q is EH, then T p is EH and T q = H 0 ; Proof. It is sufficient to consider the tensor product of the two leading irreducible spaces, and to use the Clebsch-Gordan product for SO(3) [START_REF] Jerphagnon | The description of the physical properties of condensed matter using irreducible tensors[END_REF][START_REF] Auffray | Décomposition harmonique des tenseurs -Méthode spectrale[END_REF]. We obtain

H p ⊗ H q = p+q i=p-q H i
Therefore p must be even, and q = 0, therefore T q = γ 0 H 0 , and by hypothesis γ 0 = 1. Thus T p has to be EH. Lemma 6. [START_REF] Backus | A geometrical picture of anisotropic elastic tensors[END_REF]. If E 1 = E 2 (and then p = q) and if T p ⊗ T p is EH then

β i = γ i . Furthermore, if L(E 1 ) is not self-adjoint then T p = H 0 ;
Proof. The demonstration is the same as the preceding proof.

These results can be summed-up in the following theorem:

Theorem II. Let consider T 2p the space of coupling tensors between two physics described respectively by two tensor vector spaces E 1 and E 2 . If these tensor spaces are of orders greater or equal to 1, then I(T 2n ) = I(G 2n ) Proof. This is a direct application of the two former lemmas.

Proper tensor spaces. In this case we have the following lemma:

Lemma 6.7. If E = E 1 = E 2 , T p ⊗ S T p is EH and if L(E) is self-adjoint then • if p = 2m + 1 then T 2m+1 = H 2m+1 ; • if p = 2m then T 2m = H 2m ⊕ β 0 H 0 ;
Proof. Because L(E) is self-adjoint the tensor product is replaced by the symmetric tensor product, and if T p = p i=0 β i H i , the symmetric tensor product T p ⊗ S T p can be decomposed into a direct sum of

β 2 i H i ⊗ S H i and β i β j H i ⊗ H j with i < j ∈ {0, • • • , p} with
the following Clebsch-Gordan rule for the symmetric product :

H k ⊗ S H k = k i=0 H 2i
Therefore, we cannot have the tensor product H i ⊗ H j for 1 ≤ i ≤ p -1 and i = j; thus we deduce that T p = β 0 H 0 ⊕ H p and T p ⊗ S T p = β 0 H 0 ⊕ (H p ⊗ S H p ) ⊕ 2β 0 H p Then : either p is odd, and in such case β 0 = 0; either p is even and then T p = β 0 H 0 ⊕ H p .

We therefore obtain the following result Theorem III. Let consider T 2p , the space of tensors of a proper physics described by the tensor vector space E. If this tensor space is of order p ≥ 3, and is solely defined in terms of its index symmetries, then I(T 2n ) = I(G 2n ).

Proof. Any tensor subspace defined in terms of its index symmetries contains, as a subspace, the space of full symmetric tensors. Since p = 3, the harmonic decomposition of S 3 does not satisfied the condition of lemma 6.7. Direct application of this lemma leads to the conclusion. Remark 6.8. It can be observed, that CTS having a minimum number of classes can nevertheless be constructed. They consists in spaces of autoadjoint linear applications between harmonic spaces, which are defined both from complete symmetry under index permutations and traceless property.

Conclusion

In this paper the symmetry classes determination of even order tensors has been studied. Based on a new geometric approach, a complete and general answer to this recurrent problem in continuum mechanics has been given. Application of our results are direct and solve directly problems that would have been difficult to manage with the Forte-Vianello method. As an example, and for the first time, the symmetry classes of the even-order tensors involved in Mindlin second strain-gradient elasticity were given. To reach this goal a geometric tools, named the clips operator, has been introduced. Its main properties and all the products for SO(3)closed-subgroups were also provided. We believe that these results may find applications in others context. Using the geometrical framework introduced in this paper, some extensions of the current method can be considered:

• To extend this approach to odd-order tensors;

• To take into account the coexistence of different symmetry properties for the physical properties of architectured multimaterials; These extensions will be the objects of forthcoming papers. A.2. Axes and subgroup classes. For every SO(3)-subgroups, we defined its g conjugate in the following way K g = gK 0 g t where the g exponent indicates the transformation, and 0 the initial configuration. To proceed toward our analysis we need to introduce the following group decomposition [START_REF] Ihrig | Pattern selection with O(3) symmetry[END_REF][START_REF] Golubitsky | Singularities and groups in bifurcation theory[END_REF] Definition A.

= Q k; π 3 A 1 A 2 A 7 A 6 A 5 A 8 A 3 A 4 O i j k 1
1. Let K 1 , K 2 , • • • , K s be subgroups of Σ. Then Σ is the direct union of the K i 's if a) Σ = s i=1 K i ; b) K i ∩ K j = {e} ∀i = j.
This decomposition is noted K = s i=1 K i when we have a direct union of subgroups. We give some important details about geometric structure of SO(3)-subgroup. Indeed:

• Z 0
n is characterized by the Oz axis, generated by k. For every rotation g ∈ SO(3), we note a the axis generated by gk and note Z a n = Z g n to indicate the rotation axis. • D 0 n is characterized by its primary axis Oz and several secondary axis b l . Therefore

D 0 n = Z 0 n n-1 l=0 Z b l 2 (A.1)
Each b l is perpendicular to Oz and are related by the Z 0 n generator. D 0 n is chosen such as one b l is generated by i. For every rotation g ∈ SO(3) we define a -generated by gk -to be the primary axis and b -generated by gi -the secondary one, this is noted D a,b n = D g n

• The subgroup T 0 can be split into a direct union of cyclic subgroups [START_REF] Ihrig | Pattern selection with O(3) symmetry[END_REF].

T 0 = 4 i=1 Z vt i 3 3 j=1 Z et j 2 (A.2)
where the vertex axes of the tetrahedron are noted vt i and his edge axes are noted et j , the details of these axis appear on figure 2. Each conjugate subgroup T g will be characterized by the set of its axes (gvt i , get i ), g ∈ SO(3). • The octahedral subgroup O 0 split into Now, for the study of Γ = T 0 ∩ T g the arguments will be based on T subgroups as well as on the axes.

O 0 = 3 i=1 Z f c i
• First, we can find a g such as all the axes are modified; in this case Γ = 1;

• A rotation around a face or an edge axis can be found such as only this axis is leave fixed. Then Γ = Z 2 or Γ = Z 3 depending on the fixed axis. • If we have Γ ⊃ D 2 then we can deduce that g carry two edge axes onto two edge axes.

After a given permutation of axes (which leave fix T 0 ) we can suppose that g leaves fix axes vt 1 and vt 2 ; we then conclude that g fixes also the axis vt 3 and then T g = T 0 . Thus we have here Γ = T 0 We deduce here the lemma: To study the clips operation with dihedral groups, we proceed in the same way as for the tetrahedron subgroup. The purpose is to examine axes of the cube and dihedral group. The arguments are about the same in each case. Therefore we will only detail the cases where 4 ∤ n, 3 | n and n is odd.

• Either we have a = vc j , then if b = f c j , Γ = D 3 or Γ = Z 3 otherwise.

• Or a = f c j = 0, then if b = f c j , Γ = Z 2 or Γ = 1 otherwise.

• Else a = ec j , then if b = ec j or b = f c j , Γ = Z 2 . All these arguments leads us to the lemma:

  } which are O(2)-closed subgroups; exceptional groups: : {T , O, I, SO(3)} which are the symmetry groups of platonician polyhedrons. Let us detail first the set of planar subgroups. If we fix a base (i; j; k) of R 3 , and note Q(v; θ) ∈ SO(3) the rotation about v ∈ R 3 and of angle θ ∈ [0; 2π[ we have • 1 the identity;

Definition 4 . 3 (

 43 Clips Operator). For every G-classes [Σ 1 ] and [Σ 2 ], we define the clips operator of [Σ 1 ] and [

Corollary 4 . 5 .Theorem 4 . 6 .

 4546 For every two G-representations E 1 and E 2 , if I 1 denotes the isotropy classes of E 1 and I 2 the isotropy classes of E 2 , then I 1 ⊚ I 2 are all the isotropy classes of E 1 ⊕ E 2 . For every two SO(3)-closed subgroups Σ 1 and Σ 2 , we have 1 ∈ [Σ 1 ] ⊚ [Σ 2 ]. The other classes of the clips operation [Σ 1 ] ⊚ [Σ 2 ] are given in the table 2

5. 1 .n ≤ k when k is odd n ≤ k 2

 12 Isotropy classes of irreducibles. The following result was obtained by Golubitsky and al. [21, 18]: Theorem 5.1. Let SO(3) acts on H k . The following groups are symmetry classes of H k : (a) 1 for k ≥ 3; (b) Z n (n ≥ 2) for   when k is even (c) D n (n ≥ 2) for n ≤ k; (d) T for k = 3, 6, 7 or k ≥ 9; (e) O for k = 1, 2, 3, 5, 7, 11; (f) I for k = 6, 10, 12, 15, 18 or k ≥ 20 and k = 23, 29; (g) SO(2) for k odd; (h) O(2) for k even; (i) SO(3) for any k.

1 2Figure 1 .

 11 Figure 1. Composite figure associated to D 4 ∩ rD 4 r t where r = Q k; π 3

Figure 2 .Figure 3 .•

 23 Figure 2. Cube C 0

Figure 4 .A. 3 . 1 •= 2απ d ∈ Z 0 dFigure 5 .

 43105 Figure 4. Symmetry axis of C 0 transformed axes (gf c i , gec j , gvc l ).• The icosahedral group I 0 split intoI 0 =

Lemma A. 8 .

 8 We have [T ] ⊚ [T ] = {[1], [Z 2 ], [Z 3 ], [T ]Octahedral subgroup. We begin by taking back the decomposition A.3 in the case of the tetrahedron, we directly get the lemma: Lemma A.9. For every integer n, we noted 2 (n) = gcd(n, 2) ; d 3 (n) = gcd(n, 3) ; d 4 (n) = 4 if 4 | n 1 otherwise Then we have [Z n ] ⊚ [O] = {[1], [Z d 2 (n) ], [Z d 3 (n) ], [Z d 4 (n) ]

Table 2 .

 2 Clips operations on SO(3)-subgroups

The nth-order generic tensor is a nth-order tensor with no index symmetry.

because of some symmetries

This is a consequence of the assumption of the existence of a free energy.

The related dot product is constructed by 2n products of the R 3 canonical one.

Formally this space is constructed as G 2n = ⊗ 2n R 3 .

The arrows of figure5are to be understood in terms of partial order, that is inclusion of conjugates.

Appendix A. Clips operation on SO(3)-subgroups In this section results concerning the clips operations on SO(3)-subgroups will be established. The geometric idea to study the intersection of symmetry classes relies on the symmetry determination of composite figures which symmetry groups is the intersection of two elementary figures. As an example let consider the rotation r = Q k; π 3 , determining D 4 ∩ rD 4 r t is amount to establish the set of transformations letting the composite figure 1 invariant.

A.1. Parameterization of subgroups. We will define geometric elements for each SO(3) closed subgroup:

• The cyclic group Z n is characterized by the Oz axis; it will be noted Z 0 n := Z n ; • The same convention is retained for the dihedral group D n , i.e. D 0 n := D n ; • For the cube C 0 (c.f fig. 2) we defined its vertex collection

• For the tetrahedron we consider the figure 2 and define T 0 to be the tetrahedron

T 0 is T 0 -invariant. • For the dodecahedron (c.f. fig. 3), we note D 0 the figure which vertices are 15 • Twelve vertices of type ± a 2 , ±φ 2 a 2 , 0 circularly permuted;

The converse inclusion is obvious, so we can conclude the lemma.

A direct application of lemma A.2 to the intersection Z 0 n ∩ Z g n leads to the result: Lemma A.3. For every integers n and m, we note d = gcd(n, m); we have

2 the following cases has to be considered:

• When Oz = a, the intersection Γ = Z 0 n ∩ Z a m and one can apply lemma A.2; • When, for some l, a = b l , Z 0 n ∩ Z a m = 1 and one has to considered Z b 2 ∩ Z a m , which equals the identity as soon as m is odd.

Lemma A.4. For every two integers n and m, we note d := gcd(n, m) and d 2 (m) := gcd(m, 2); then we have

Now consider the second kind of intersection

The following cases have to be considered • When Oz = a, and Ox = gb l 0 for some l : if d = 1 then Γ = Z b 0 2 and D 0 m otherwise; • When Oz = a, and Ox = gb l : Γ = Z 0 d ; • When Oz = gb l for some l : if n is even then Γ = Z 2 and 1 otherwise. Results are the same when the primary axis of D g m coincides to a secondary axis of D 0 n . Lemma A.5. For every integers n and m, we note d := gcd(n, m) and

A.4. Clips operations on exceptional and maximum subgroups. Here we are concern with the subgroups T 0 , O 0 , I 0 , SO(2) 0 , and O(2) 0 . For these studies, we will use results concerning their proper subgroups [START_REF] Ihrig | Pattern selection with O(3) symmetry[END_REF]. This information is summed-up in the following diagram 16 [4]:

Tetrahedral subgroup. First of all we take back the decomposition A.2:

We begin by the study of T 0 ∩ Z a n . As a consequence of the lemma A.2, the primary axis of Z a n must be an edge axis or a face axis of the tetrahedron. We therefore directly obain the lemma:

Lemma A.6. For every integer n, we note

We will use primary axes and secondary axes of dihedral subgroup:

We recall that the vertex axes of the tetrahedron are noted vt i and his edge axes are noted et j . It is clear that Γ is a subgroup of T 0 . Furthermore:

• In any case, when we only have b = et j , then Γ = Z 2 . Finally we can deduce the lemma:

Lemma A.10. For every integer n, we note

Now, we take Γ = O 0 ∩ T g and one can observe that Γ is necessarily a common subgroup of O 0 and T g , then its class must contain (in the sense of the partial order) [

or [T ]. After that:

• There exists a rotation g around an edge axis of T 0 (that is a common face axis of the cube) such that only this axis is fixed; and then Γ = Z 2 ; • There exists a rotation g around a vertex axis of T 0 (that is a common vertex axis of the cube) such that only this axis is fixed; and then Γ = Z 3 ; • As soon as Γ ⊃ D 2 , as in tetrahedral case, we necessarily have Γ = T 0 . We conclude here the lemma:

For the study of Γ = O 0 ∩ O g we will also use arguments based on subgroups. Some results are nevertheless more subtle:

• First, there exists a rotation g that fix only one edge axis, and in that case Γ = Z 2 ;

• Then there exists a rotation that leaves fix only one vertex axis, and in that case Γ = Z 3 ;

• There exists also a rotation that leaves fix only one face axis, and no other axis is fixed. See for example figure 6(b): in that case Γ = Z 4 ; • We can also find a rotation that leaves fix a face axis and which bring an edge axis onto a face axis. Indeed when we take g

We can exactly compute that gf c 3 = ec 6 , gec 1 = f c 1 and gec 2 = ec 5 and no other axes corresponds (c.f. figure 6(a)).

• If we take g = Q(vc 1 , π) we will find Γ = D 3 with vc 1 as primary axis and ec5 a secondary axis. • If Γ ⊃ T then, necessarily, g leaves fix the three edge axes of the tetrahedron, and then g will fix the cube C 0 ; thus Γ = O 0 . Finally we get the lemma: Lemma A.12. We have

Icosahedral subgroup. We take the decomposition A.4

Now, for the study of I 0 ∩ D g n we again use arguments about axes:

When we argue on the secondary axis of D g n , we see that we can always have Z 2 . Finally we get the lemma: Lemma A.14. For every integer n, we note

For the intersection I 0 ∩ T g it is clear, because of the inclusion T 0 ⊂ I 0 that we can obtain all the classes of [T ] ⊚ [T ]. If now this intersection contains a subgroup D 2 , we will necessarily have

where get i are the three edge axis of the tetrahedron gT 0 . These three axes will then have to correspond to three perpendicular axes of the dodecahedron. After permutation of the axes, which leaves fix the dodecahedron, we can suppose that these three axes are generated by the three vectors of the basis. But, then, the vertex axes of the tetrahedron will correspond to vertex axes of the embedded cube in the dodecahedron. We can then deduce that the intersection will be the whole T subgroup. Now we have to study Γ = I 0 ∩ O g . For that, we refer to the common subgroups of [O] and [I]. Such subgroups can clearly be taken from the poset 5. First, it is clear that, when the cube related to O g is the embedded cube in the dodecahedron, we will have Γ = T .

We also can find a rotation g such as Γ contains D 3 : indeed, g has to bring the vertex axis of the cube vc 1 onto the vertex axis of the dodecahedron vd 5 and the edge axis of the cube ec 5 onto the edge axis of the dodecahedron ed 7 . With maximality argument, we can deduce that Γ = D 3 . We now have to examine the case of D 2 , Z 3 and Z 2 :

• When Γ ⊃ D 2 , then, after permutation of axis of the dodecahedron, we can suppose that g leaves fix the three axis of the basis vector. But these three axis are two order rotations of the dodecahedron. Thus, g will fix the cube C 0 and we can deduce that O g = O 0 . We then have Γ = T . • We can find a rotation g around a vertex axis, for example vd 5 such that Γ = Z 3 ;

• As above, we can find a rotation around an edge axis, such that Γ = Z 2 . We finally conclude to the formula: • When Γ ⊃ T or Γ ⊃ D 2 , it then contains all the three two order rotations around each base axes, which will be three edge axes of the dodecahedron. Thus we can deduce that g, after permutation of these axes, leaves fix three perpendicular axes, and then g leaves fix I 0 ; finally Γ = I; • There exist a rotation g around an edge axis so that Γ = Z 2 . The same argument leads us to Z 3 and Z 5 . • If we take g to be the two order rotation around the axis vd 3 , we can compute that this rotation only leaves fix the axes vd 3 , ed 6 , ed 8 and ed 15 , and then Γ = D 3 . • If we take g to be the two order rotation around the face axis f d 1 we can also compute that it only leaves fix the axes f d 1 , ed 7 , ed 11 , ed 12 and ed 14 , thus Γ = D 5 .


D Cédat

O Fandeur

C Rey

D Raabe

Keywords: metal-ceramic composite, numerical simulation, crystal plasticity, polycrystal modelling, damage

The mechanical behavior of a Mo-TiC 30 vol.% ceramic-metal composite was investigated over a large temperature range (25°C to 700°C). High-energy X-ray tomography was used to reveal the percolation of the hard titanium carbide phase through the composite. Using a polycrystal approach for a two-phase material, finite element simulations were performed on a real 3D aggregate of the material. The 3D microstructure, used as starting configuration for the predictions, was obtained by serial-sectioning in a dual beam Focused Ion Beam (FIB)-Scanning Electron Microscope (SEM) coupled to an Electron Back Scattering Diffraction system (3D EBSD, EBSD tomography). The 3D aggregate consists of a molybdenum matrix and a percolating TiC skeleton. As most BCC metals, the molybdenum matrix phase is characterized by a change in the plasticity mechanisms with temperature. We used a polycrystal model for the BCC material, which was extended to two phases (TiC and Mo). The model parameters of the matrix were determined from experiments on pure molydenum. For all temperatures investigated, the TiC particles were considered as brittle. Gradual damage of the TiC particles was treated, based on an accumulative failure law that is approximated by an evolution of the apparent particle elastic stiffness. The model enabled us to determine the evolution of the local mechanical fields with deformation and temperature. We showed that a 3D aggregate representing the actual microstructure of the composite is required to understand the local and global mechanical properties of the studied composite.

Introduction

The aim of this study is to understand the mechanical behaviour and the damage evolutions of a metal-ceramic composite (Mo-TiC 30vol.% ) as a function of deformation and temperatures ranging from 800°C up to 1000°C.

For compression tests performed at high temperature, damage observations cannot be monitored and simulations are needed at grain scale, to get a better understanding of damage initiation.. In a preceding study, Cédat et al. [4] simulated the local mechanical fields using a 3D aggregate that was obtained by a random stack of layers of columnar grains. These grains were characterized in terms of crystal orientations that were determined by EBSD. When using this aggregate, the polycrystal model failed to describe the mechanical behavior of the composite. Different effects might be responsible for the discrepancy between experiment and predictions, namely, microstructural size effects, higher complexity of the required constitutive laws of the involved phases, and the spatial distribution (connectivity, and topology) of the TiC brittle phase. The current paper addresses this problem of properly predicting the micromechanical behavior of complex metal-ceramic composites in the 5 subsequent sections. In section 2, investigations (using tomography experiments) reveal a percolation of the TiC phase. To take into account such phase percolation within the computation of the local mechanical fields, a new representative aggregate has been built, using tomographic microstructure data obtained by a dual beam FIB-SEM experiment. The used polycrystal model, implemented in a Finite Element code, is described in section 3. Damage is represented by a decrease of TiC effective elastic modulus, implemented in the model. Section 4 presents the meshing and the boundary conditions applied to the 3D EBSD aggregate. New model parameters for Mo are determined by an inverse method. Section 5 deals with the results of the mechanical predictions for the composite. Section 6 draws conclusions.

Material and experimental procedures

The investigated material was obtained by powder metallurgy synthesis [START_REF] Ettmayer | 12th Plansee Seminar[END_REF], [START_REF] Flem | [END_REF] (Hot Isostatic Pressing (HIP) at 1600°C). No homogenization heat treatment was performed after HIP processing. The composite microstructure was characterized in a previous paper [4] using different chemical and physical methods. According to Cédat [1], the composite exhibited three phases: Molybdenum (Mo), Titanium Carbide (TiC) and a (Mo,Ti)C phase which is a face-centered cubic (FCC) structure with lattice parameters close to those of TiC (Fig. 1). The hard particles (TiC) showed a core/shell (or core/rim) structure, with molybdenum as binder phase. The third phase, identified as TiC-Mo 15at.% had the same structure as TiC. In this paper, the mechanical behavior of the third phase is assumed to be equivalent to that of TiC.

Chemical composition

The chemical composition of the powders is given in tables 1 and 2.

X-ray tomography results

Using high-energy X-ray tomography, a 3D investigation of the material was performed at the European Synchrotron Radiation Facility (ESRF, Grenoble, France). This characterization technique proceeds via taking a sequence of recordings of a sample at different angular positions. Then, an appropriate tomography reconstruction algorithm builds a 3D map of the microstructure based on the inversion of the X-ray attenuation coefficient projection paths that penetrated the material upon synchrotron illumination. Fig. 2 shows a 3D image of the microstructure of the Mo-TiC material obtained with the highest spatial resolution (0.28 µm) available on the ID19 beam line (X-ray energy: 65 keV) of the ESRF equipment. Fig. 2 represents the set of all points in the carbide phase (purple) that can be linked together to the bottom and top faces of the parallelepiped specimen (30 X 30 X 30µm 3 ), by paths entirely contained in the carbide phase. The microstructure of the composite revealed a percolating skeleton of carbides, embedded in a molybdenum metallic matrix, with a high 3D connectivity of the carbide phase.

3D aggregate mapping via 3D EBSD

To accurately compute the local mechanical fields, a new representative 3D aggregate has to be designed, taking into account the observed percolation of the TiC phase. Our approach to 3D EBSD was inspired by the works of Uchic et al. [5], Zaefferer et al. [6], Bastos et al. [7] and Konrad et al. [8], as well as by the 3D texture measurements, via synchrotron radiation published by Larson et al. [9] and Yang et al. [START_REF] Yang | MRS Symposium proceedings[END_REF]. The 3D EBSD experiments were conducted at the Max-Planck Institute (Düsseldorf), using a joint high-resolution field emission SEM/EBSD set-up together with a FIB system in the form of a Zeiss cross-beam 3D crystal orientation microscope. Details are given in [9] [10]. The analysis method applied in this study, involves highly precise and fully automated serial sectioning via a Ga + FIB and the subsequent mapping of the crystallographic texture in each of those layers using high-resolution EBSD. For more details on this method, see Zaafarani et al [START_REF] Zaafarani | [END_REF] and Zaefferer et al [12].

The spatial resolution of the 3D orientation and phase pixel information retrieved amounts to 50 x 50 x 50 nm 3 . The ion beam did not create noticeable damage, i.e. no serious deterioration of the EBSD pattern quality could be observed after milling. This stability of the material, when exposed to an ion beam, is attributed to its high melting point and the absence of phase transformations. The EBSD measurements were carried out sequentially in each layer after FIB serial sectioning at a 300nm step size between the abutting layers. The entire process of alternating FIB sectioning and EBSD measurements was carried out for a set of 22 subsequent layers for the reconstruction of the whole aggregate. Fig. 3 shows SEM images of two different serial sections and the entire 3D microstructure reconstructed as a 3D map. Fig. 3: 30 X 30 X 6.6 µm 3 aggregate obtained by 3D electron orientation microscopy (3D EBSD). (a) TiC, (b) aggregate. The colour code is given by the crystallographic standard triangle. For simulations, the loading direction is parallel to 2  axis.

Experimental analysis of the mechanical behavior of the composite and of the Molybdenum phase

Composite: The composite compression curves are given in Fig. 4 for temperatures between 25°C and 700°C. The specimen are cylinders with a diameter D=12mm and cylinder length L=18mm. The initial strain rate upon loading was about The compression curves show a linear elastic stage, followed by a linear hardening stage, and finally a weak strain hardening regime. The two strain hardening regimes change as a function of the temperature and deformation. The extension of the linear stage decreases with increasing total deformation. The strain hardening decays with increasing temperature. Fracture surface observations reveal that at 25°C, the two phases are characterized by a brittle behavior and that at 300°C, the molybdenum matrix is ductile while the TiC phase fails in a brittle manner. The mechanical properties of pure molybdenum had been previously studied by Cédat et al [1] using two kinds of tensile tests, namely, strain rate change tests (5.10 -4 and 5.10 -2 s -1 ) over a temperature ranging from 25 to 700°C and temperature change tests. A transition in the polycrystal mechanical mechanisms was observed above 150°C. For T>300°C, the viscoplastic flow weakly varied with temperature. These results, confirmed by TEM observations, suggested that molybdenum (like most BCC metals) reveal different plasticity properties at low and high temperatures. In BCC transition metals, screw dislocations are less mobile than edge dislocations. Screw dislocation motion proceeds via thermally activated kink pair formation and expansion. Optical microscopy (OM) and SEM observations [13] performed on molybdenum surfaces as well as on fracture surfaces after tensile testing, confirmed these results. For T= 25°C, molybdenum rupture occurs by grain boundary decohesion. The studied fracture surfaces reveal features of brittle failure. For T=200°C, grain boundary decohesion and plastic glide is observed. For T=300°C, deformation occurs by plastic glide and the fracture surface is characterized by ductile rupture features.

According to these observations, we concluded in our earlier work that the plasticity mechanisms of this material change in the temperature range around 200°C. One should note that this transition temperature T 0 is different from the brittle-to-ductile transition temperature of molybdenum (which is close to 25°C).

Polycrystal model

In order to determine the local mechanical fields in the different phases and especially the location and origin of damage in the TiC phase, a previous model [14] was upgraded and numerical simulations were conducted. The used crystal plasticity model was developed in the framework of large transformations [15,[START_REF] Teodosiu | Large Plastic Deformation[END_REF]17,178,[START_REF] Libert | Etudes expérimentale et numérique de l'effet des mécanismes de plasticité sur la rupture fragile par clivage dans les aciers faiblement alliés[END_REF][START_REF] Roters | [END_REF] (small elastic distortion but large lattice rotation) and was implemented in the Abaqus finite element code, using an UMAT User Subroutine.

The polycrystal is composed of different grains, each one having its own size and crystallographic orientation. Each grain is considered as a single crystal that is characterized by a constitutive law that is presented below. This polycrystal model was developed for BCC metals or alloys, taking into account temperature effects as will be outlined in more detail below. The two plasticity mechanisms controlling plastic strain hardening of molybdenum are double kink formation (low temperature) and interactions between mobile and forest dislocations (high temperature). In this paper, we summarize the polycrystal model extended to multi-phase composites. For more details, see Libert [18], Libert et al. [14].

Single crystal constitutive law for the molybdenum matrix

For the BCC crystal structure of the molybdenum matrix, plastic glide occurs on the 24 slip systems {110} <111> and {112} <111>. The chosen viscoplastic law can describe the competition between lattice friction and forest hardening. According to Kubin et al. [21], Rauch [22] and Tabourot et al. [23], the slip rate on a given slip system (hereafter referred to by the index (s)), is given by:

) sgn( kT ) ( G exp s s eff 0 s                  (1) Where ) ( s eff G  
is the energy of activation for crystallographic slip of lattice dislocations, which is a function of the effective shear stress s eff  . s  and 0  are the resolved shear stress on system (s) and the reference shear rate, respectively. When T<T 0, the shear rate 0  depends on the geometry of the segments of the screw dislocations. The energy of activation is then associated with the double kink mechanism and it is given by a phenomenological description, as proposed by Kocks et al. [24]:

  q p R s eff 0 s eff τ τ 1 ΔG τ ΔG                         (2)
Where p and q are parameters describing the energy associated with the Peierls barrier. In Eq.2, R  is the effective stress that is required to create a double kink at T=0 K, when the contribution of the thermal activation is equal to zero. This description assumes that the glide of dislocations on a system (s) is activated when s eff τ is larger than R  .

3.2 Single crystal strain hardening law for the molybdenum matrix The strain hardening evolution as a function of temperature depends on the interactions between mobile dislocations with lattice friction and/or forest dislocations. Lattice friction generates short range stresses described by the effective stress s eff τ . The contributions described by s int τ and 0

 are shear stresses and stand for the interaction between mobile and latent dislocations. Thus, the yield stress required to activate the glide system (s) is equal to the sum of the three contributions:

s int s eff 0 s τ τ τ τ    (3)
By considering a segment of a screw dislocation pinned by two obstacles, submitted to the lattice friction ( 0  ), Rauch [22] proposed the following expression of the internal stress:

  0 s u 24 , 1 u su 2 s int τ τ ρ a b μ τ     (4)
a su is a component of the dislocation interaction matrix as proposed by Franciosi [25], that is due to the interaction strength between the slip systems (s) and (u) (self and latent hardening). Latent hardening experiments [25], performed on single crystals under tensile testings, revealed that the a su parameters depend on the strain values: they increase up to 0.5% -1% straining, then reach asymptotic values. In this paper, we use the asymptotic values. u  is the dislocation density on the glide system (u). When merging expressions (3) and ( 4) and solving the resulting equation, one obtains:

               u u su 2 2 s eff s eff 0 s a b 4 2 1 2       (5)
The general expression of the hardening law ( 5) is appropriate to describe a continuous evolution of the shear stress as a function of the temperature: In the case of low temperature behavior (T<T 0 ), plasticity is governed by the reduced mobility of the screw dislocations and the double kink mechanism hence, int  is negligible compared to eff  . Eq.5 leads to:

  u eff 24 , 1 u u su 2 s eff s a b          (6)
In the case of high temperature, (T>T 0 ), plasticity mainly depends on forest hardening and eff  is negligible compared the two other terms. Eq.5 leads to:

    24 , 1 u u su s 0 s a b     (7)
At low temperature, plastic deformation occurs according to two non equivalent slip system families <111>{110} and <111>{112}.

Dislocation density evolution law

The dislocation density evolution law is a generalization of the relation proposed by Estrin and Mecking [26]. For each slip system, Eq. 8 describes the evolution of the 24 dislocation densities with the strain:

                 s c s u u grain s s ) T ( g ) T ( K D 1 b       (8)
This expression is derived from the balance between dislocation accumulation (Orowan's relationship) and dislocation annihilation which induces softening.

  s u u / ) T ( K 
is the dislocations mean free path which increases with decreasing temperature. As the evolution of the dislocation density is weak for low temperature, we assume that K(T) increases with decreasing temperature. The annihilation of dislocations is controlled by the mean dislocation-dislocation spacing g c , the temperature dependence of which is expressed here in terms of an Arrhenius law:

        ) T ( k E exp g ) T ( g B gc 0 c c (9)
According to Eq.9, two dislocations of the same system (s) may undergo annihilation, as soon as they are separated by a spacing below g c . Similar dislocation-based constitutive laws for crystal plasticity finite element models were applied, as suggested by Ma et al. [17,27].

Damage law for titanium carbide particles

Damage, in the compound, starts by initiation of cracks within the brittle carbide phase. A simple criterion that rests on the local stress in the brittle phase is introduced in the polycrystal model. The failure strength of ceramics under compression being approximately 10 to 15 times larger than the tensile strength [28], the critical stress leading to damage initiation is given by the following approximation:

15 e compressiv tensile c      with a K IC tensile    (10) C 1 K
is the toughness factor and a the mean size of the crack.

The results of the simulations show that the carbide phase is mainly under tension when the composite is mechanically loaded, while the molybdenum phase is under compression. For the carbide phase, the chosen damage criterion is formulated as a measure of the gradual accumulation of damage. It is based on the evolution of the elastic law: ( is the local equivalent stress) is verified within an element, its effective modulus is decreased as given by Eq.11, 12. For D(t) =1 (C(t)=0), the element is equivalent to an "hole". Changes of some elements stiffness of the TiC phase, contribute to that of the overall compound.

  ) t ( D 1 C ) t ( C ~0   (11) with   ) t t ( exp( 1 ) t ( D 0        (12)

Boundary conditions, meshing and model parameters

The polycrystal model was implemented in the Abaqus TM finite element code using an UMAT subroutine. Most of the molybdenum parameters are determined on the basis of the experimental curves, by using an inverse fitting method using a coupling ( [18,[START_REF] Roters | [END_REF]) between the Sidolo TM and Abaqus TM software packages. The initial density of the dislocations before straining was determined by TEM. The parameters used for the molybdenum are given in tables 3 and 4.

Aggregate. The polycrystalline aggregate is shown in Fig. 3. The Finite Element meshing is derived from the grid corresponding to the 3D EBSD (0.3 µm step) measurements. It is built by 220,000 elements (C3D8R). Each element corresponds to an original material square pixel that is 0.3 µm wide. The crystallographic microstructure properties of the two phases are described in terms of the Euler angles for each pixel of the two phases in the mesh, the TiC elastic properties, and the molybdenum constitutive laws parameters. The connectivity among the two phase sets and the percolation of the TiC are thus automatically taken into account.

Boundary conditions:

The considered aggregate corresponds to 1/8 of the whole computed structure. For symmetry reasons, the lower face of the crystalline aggregate, shown in Fig. 3, is loaded in compression mode normal to direction 2  . The displacement is u 2 =0 on the lower face. The other boundary conditions are u 3 =0 on the face normal to 3. Set of parameters identified for molybdenum by Cédat et al [1] In addition to this set of parameters, this new investigation has identified the parameters depending on temperature by inverse method on the base of experimental tensile curves performed at 25°C to 700°C. 4. Set of temperature dependent parameters identified for molybdenum 0  : shear stress due to lattice friction (Eq.3) a su : component of the interaction matrix between dislocation (s) and (u) (Eq.5,6,7) 0  : initial dislocation density g c0: material parameter of the dislocation-dislocation annihilation law (Eq. 9) Eg c : material parameter of the dislocation -dislocation annihilation law (Eq.9) D grain : grain size R  : effective stress required to create de double kink (Eq.2) 0   : shear rate reference (Eq.1) 0 G  : material parameter associated to the energy of activation of slip system (Eq.2) p and q :parameters describing the energy associated with the Peierls barrier (Eq.2)

Molybdenum parameters depending on temperature T

) T ( G  (eV) 0  (MPa) K g c (
) eff ( G ) T ( G    
: slip systems energy of activation (Eq.1) K(T): material parameter associated to the dislocation mean free path (Eq.8) g c : dislocation-dislocation annihilation distance (Eq.8,9)

The titanium carbide elasticity is assumed isotropic. Its features are given in table 5.  the rupture critical parameter in tension. According to our observations of fracture surfaces, we assume a perfect cohesion between TiC and Mo matrix.

E (GPa)    c  (

Results and discussion

For a pure molybdenum metal representing the polycrystal matrix, the simulation results delivered by the current polycrystal model were already published in [4]. The compression test simulations were in good agreement with the experiments results.

Mo-TiC composite results for T=25°C

The numerical and experimental curves are compared in Fig. 5. At room temperature, a very good agreement is observed between the numerical (Fig. 5) and experimental curves (Fig. 4), up to the maximum stress. Compared to our first simulations [1] that were carried out on crystalline aggregates (constructed as superposition of random layers of columnar grains), a large improvement is observed in the current full field 3D predictions. This improvement is attributed to the more realistic 3D mapping of the incipient microstructure where both, percolation and damage of the TiC phase are taken into account.

For each phase, the computed local stress averages 22  versus the applied strain 22 E are represented by dotted lines. The molydenum curve presents a perfect plastic behaviour, whereas TiC shows a non linear behaviour for The computed maps (Fig. 6) represent the evolution of distribution of local strain 22  in the grains of Mo and TiC and of damage within the TiC particles. The local strains ( 22 ) distribution is homogeneous in TiC grains and heterogeneous in large molybdenum grains. In TiC the damage parameter D(t) (introduced in eq. 10-12), increases with applied deformation and locally reach 1 for an applied strain

% 10 E 22 
. The value D=1 corresponds to rupture of the particles. According to our numerical results, we can propose for damage the following scheme: during stage I (elastic stage), the TiC skeleton supports most of the deformation. At the end of stage I and up to 3% compression strain rate (stage II), some bridges of TiC (normal to the compression axes) are submitted to local tensile stress. As soon as the critical stress criterion (eq.10) is verified, some bridges between TiC particles are damaged. During stage II, the damage parameters D increases and reaches 1 (rupture of the bridge) for

% 10 E 22 
. In stage III, the composite behaves as a matrix with large disconnected particles.

Composite results for high temperature

The numerical simulation and the corresponding experimental curves for high temperatures are given in Fig. 7. The comparison between the composite (Fig. 7a) and the pure molydenum curves (Fig. 7b) reveals, for an applied strain

09 . 0 E 22  
, a large increase of the macroscopic yield stress form 600MPa to 1450MPa. Fig. 7a shows, for

09 . 0 E 22  
, a good agreement between numerical and experimental results. The difference 22  between macroscopic experimental stress is less than 10MPa. For small strains, the agreement is only qualitative. For the composite, the small differences between experiment and simulation observed at the very beginning of plasticity stage come from the material parameters determined for pure molybdenum. For pure molybdenum, the simulated yield stresses are lower than the experimental ones (Fig. 7b). The material parameters of pure molydenum (tables 3, 4) were identified, for

% 2 . 0 E 22 
, on simplified stress-strain curves. An accurate identification would need a higher numbers of material parameters. In addition, the damage may be considered as too much simple. In stage II, the average computed effective elastic modulus of TiC bridges decreases less than actually observed. ). Damage evolution is then slowed down in stage III: the skeleton of TiC presents a lot of micro-cracks and the composite is reduced to a Mo matrix with large TiC inclusions. As a result, the composite hardening slopes become constant (plateau) before presenting a negative slope. For high temperatures, the evolution with applied deformation of damage parameter, is slowed down compared to low temperature, leading to an increase of the size of the plateau Damage is represented by a decay of the effective elastic modulus in some elements of the TiC mesh as soon as the criterion of the critical stress is satisfied. The square elements of the meshing are more representative of small holes, than actual thin microcracks. The increasing number of "holes" in TiC is sufficient to explain the weak negative slope of simulated stress-strain curves for large strain.

Conclusions

Understanding the mechanical behaviour of two-phase Mo-TiC30vol.% metal-ceramic composite, in a wide temperature range, requires simultaneous experimental and numerical studies. The first difficulty is to work on a realistic representation of the material, then to take into account of the damage of the brittle phase and its effect on the ductile one. The studied composite presents small grains and a brittle phase presenting percolation which crystalline orientation is not easily observable. Mechanical polishing and EBSD measurement cannot be used in this case and aggregate built by random layers of grains lead to wrong results. Consequently, a dual beam FIB-SEM was used to characterize and reconstruct layer by layer a real 3D polycrystalline two-phase Mo-TiC30vol.% metal-ceramic composite aggregate.

Implementation of damage of the brittle phase at the scale of the grains is required and provides good results on the microscopic stress-strain fields when applied to this 3D aggregate. The TiC damage criterion is described in terms of an accumulative decay in the elastic stiffness of this phase as a function of straining. A very good agreement with experimental curves is observed at 25°C. For any temperature, the stage III behavior is accurately described. The proposed 3D aggregate and the corresponding two-phase polycrystal model can be used to forecast the behavior of any related metal-ceramic or metal-metal composites at different temperatures, just by a change of the matrix and particles parameters.

Figure captions

Fig. 1. SEM picture showing the microstructure of the Mo-TiC 30%vol composite revealing three types of phases, namely, Molybdenum (in light grey), TiC (in black), and (Mo,Ti)C (in grey) [1]. Fig. 2: 3D representation of the phases obtained by X-ray tomography. (a) TiC , (b) Mo, (c) composite. Specimen size: (30 X 30 X 30 µm 3 ). Fig. 3: 30 X 30 X 6.6 µm 3 aggregate obtained by 3D electron orientation microscopy (3D EBSD). TiC and TiC-Mo 15at.% phases (top row), whole aggregate (bottom row). The colour code is given by the crystallographic standard triangle. For simulations, the loading direction is parallel to 2  axis.

Fig. 1 .

 1 Fig.1. SEM picture showing the microstructure of the Mo-TiC 30%vol composite revealing three types of phases, namely, Molybdenum (in light grey), TiC (in black), and (Mo,Ti)C (in grey) [1].

Fig. 2 :

 2 Fig.2: 3D representation of the phases obtained by X-ray tomography. (a) TiC , (b) composite. Specimen size: (30 X 30 X 30 µm 3 )

Fig. 4 :

 4 Fig.4: Composite behavior compression experimental tests; sigma: tensile stress; F: force; S(t): cross section at each time interval ; 22 E : strain

   and   are the local strain and local strain rate at a given point, respectively. effective modulus" that gradually decays as a function of the strain rate and of the strain.  and  are parameters that are arbitrarily chosen in order to create an abrupt decrease of the Young modulus due to micro-cracks formation. A positive critical stress value C  is determined from Eq.10. At each increment of the simulation, the criterion (Eq.10) is tested for all elements of the TiC meshing. As soon as the criterion c   

the 3 

 3 direction, the parallel surface is free. The u 2 applied displacement corresponds to a strain rate of about 5.10 -4 s -1 .

Fig. 4 :

 4 Fig.4: Composite behavior compression experimental tests; sigma: tensile stress; F: force; S(t): cross section at each time interval ; 22 E : strain.

Fig. 5 :

 5 Fig.5: Mo, TiC and Mo-TiC 30% numerical and experimental compression curves (T=25°C); sigma22: compression stress; epsilon22: strain. Scheme of TiC damage.

  maximum, then a decrease. The perfect plastic behaviour of the the Mo pure metal behaviour). The matrix yield stress is about 550MPa and the constant stress corresponding to the plateau is equal to22  =620MPa up to E22=0.15. The non linear TiC curve corresponds to a decrease of the effective elastic modulus induced by damage.

Fig. 6 :strain 22  2 

 6222 Fig.6: Local strain 22 

Fig. 7 :

 7 Fig.7: Experimental and numerical stressstrain curves on a 3D numerical and a real polycrystalline aggregate of the composite for five temperatures. Experimental and numerical stressstrain curves on molybdenum are given in the insight.

The computed distributions of internal stress 22  22 Fig. 8 : 22  22  22 E 22  22  22  22 

 2222822222222222222 Fig.8: Distribution of the uniaxial stress Sigma=

Fig. 4 :Fig. 5 :Fig. 6 . 2 

 4562 Fig.4: Composite behavior compression experimental tests; sigma: tensile stress; F: force; S(t): cross section at each time interval ; 22 E : strain.

Fig. 7 .

 7 Fig.7. Experimental and numerical stressstrain curves on a 3D numerical and a real polycrystalline aggregate of the composite for five temperatures. Experimental and numerical stressstrain curves on molybdenum are given in the insight.

Fig. 8 . 21 FIG1 22 FIG2a 23 FIG2b 24 FIG3a 25 FIG3b 26 FIG4 27 FIG5 28 FIG6

 82122232425262728 Fig.8. Distribution of the uniaxial stress Sigma= 22 

Table 1 .

 1 TiC composition (%mass).

	C	O	N	Ca	Co	W	Ni	Al	Fe	S
	19.23	0.6126 0.0279 0.002 0.032 0.39	<4E-4 0.0014 0.0061 0.0019
			Mo		O	Fe	K
			99.98	0.0620 9ppm	29ppm

Table 2 .

 2 Molybdenum composition (% mass)

Table 5 :

 5 Parameters of titanium carbide (TiC) for the proposed model E is the Young modulus,  the Poisson coefficient, c

	MPa)

Table 1 .

 1 TiC composition (%mass).

	C	O	N	Ca	Co	W	Ni	Al	Fe	S
	19.23 0.6126 0.0279 0.002 0.032 0.39	<4E-4 0.0014 0.0061 0.0019

Table 2 .

 2 Molybdenum composition (% mass)

	Mo	O	Fe	K
	99.98	0.0620 9ppm	29ppm

Table 3 .

 3 Set of parameters identified for molybdenum by Cédat et al[1]

		0	a uu =a su		0		g c0	E gc	D grain
	125 MPa 0.01	10 -11 m -2 14nm 2.17x10 -2 eV 3 µm
		R	 	0		G	0	p	q
	498 MPa 10 -1	1.1 eV	0.2	1.5

Table 4 .

 4 Set of temperature dependant parameters identified for molybdenum

	Molybdenum parameters depending on temperature
	T		G	(	T	)	(eV) 0  (MPa) K	g c (nm)
	25°C	0.35		125	440	20
	150°C	0.5				70	80	12
	300°C	0.97		70	50	26
	500°C	1.18		70	50	30
	700°C	-					70	30	36

Table 5 :

 5 Parameters of titanium carbide (TiC) for the proposed model

	E (GPa) 			c  (MPa)
	440	0.19	0.013	0.013	250

Table 2 .

 2 Molybdenum composition (% mass)

	Mo	O	Fe	K
	99.98	0.0620 9ppm	29ppm

Table 3 .

 3 Set of parameters identified for molybdenum by Cédat et al[1]

		0	a uu =a su		0		g c0	E gc	D grain
	125 MPa 0.01	10 -11 m -2 14nm 2.17x10 -2 eV 3 µm
		R	 	0		G	0	p	q
	498 MPa 10 -1	1.1 eV	0.2	1.5

Table 4 .

 4 Set of temperature dependant parameters identified for molybdenum

	Molybdenum parameters depending on temperature
	T		G	(	T	)	(eV) 0  (MPa) K	g c (nm)
	25°C	0.35		125	440	20
	150°C	0.5				70	80	12
	300°C	0.97		70	50	26
	500°C	1.18		70	50	30
	700°C	-					70	30	36

Table 5 :

 5 Parameters of titanium carbide (TiC) for the proposed model

	E (GPa) 			c  (MPa)
	440	0.19	0.013	0.013	250

Acknowledgements

The microstructure of the starting composite was characterized at the Max Planck Institute Düsseldorf. The authors are very grateful to Prof. S. Zaefferer for his kind and helpful discussions. The studied composite MoTiC was elaborated at Comissariat à l'Energie Atomique. The authors thank Dr. M. Le Flem and Dr. J.L. Bechade to give us the material and for financial help for tomography experiments. The authors are grateful to Prof. J.H. Schmitt (ECP) for stimulating and helpful discussions.