On anisotropic polynomial relations for the elasticity tensor

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On anisotropic polynomial relations for the elasticity tensor

Nicolas Auffray, Boris Kolev, Michel Petitot

To cite this version:

Nicolas Auffray, Boris Kolev, Michel Petitot. On anisotropic polynomial relations for the elasticity tensor. Journal of Elasticity, Springer Verlag, 2014, 115 (1), pp.77-103. �10.1007/s10659-013-9448-z�.

�hal-00827948�

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On anisotropic polynomial relations for the elasticity tensor

N. Auffray · B. Kolev · M. Petitot

Received: date / Accepted: date

Abstract In this paper, we explore new conditions for an elasticity tensor to be- long to a given symmetry class. Our goal is to propose an alternative approach to the identification problem of the symmetry class, based on polynomial invariants and covariants of the elasticity tensorC, rather than on spectral properties of the Kelvin representation. We compute a set of algebraic relations which describe precisely the orthotropic ([D2]), trigonal ([D3]), tetragonal ([D4]), transverse isotropic ([SO(2)]) and cubic ([O]) symmetry classes in H⁴, the higher irreducible component in the decomposition ofEla. We provide a bifurcation diagram which describes how one

“travel” inH⁴from a given isotropy class to another. Finally, we study the link be- tween these polynomial invariants and those obtained as the coefficients of the char- acteristic or the Betten polynomials. We show, in particular, that the Betten invariants do not separate the orbits of the elasticity tensors.

Keywords Symmetry classes·Invariants·Anisotropy PACS 46.05.+b·46.25.-y·46.35.+z

Mathematics Subject Classification (2000) 74B05·15A72

N. Auffray

LMSME, Universit´e Paris-Est, Laboratoire Mod´elisation et Simulation Multi Echelle, MSME UMR 8208 CNRS, 5 bd Descartes, 77454 Marne-la-Vall´ee, France

E-mail: [email protected] B. Kolev

LATP, CNRS & Universit´e d’Aix-Marseille, 39 Rue F. Joliot-Curie, 13453 Marseille Cedex 13, France E-mail: [email protected]

M. Petitot

LIFL, Universit´e des Sciences et Technologies de Lille I, 59655 Villeneuve d’Ascq Cedex, France E-mail: [email protected]

1 Introduction

In this paper we consider the problem of identifying symmetry classes of elastic media symmetries governed by Hooke’s law, i.e. identifying the symmetry classes of the elasticity tensor. Different definitions of symmetry classes can be found in the literature, in our approach the definition of Forte and Vianello [17] is retained. In this frameworkEla, the vector space of elasticity tensors, is divided into eight conjugacy classes [17] which group tensors having conjugate symmetry groups.

The problem we study here is motivated by experimental needs. When a sample is tested, usually both the symmetries of the material and its orientation are unknown [18]. Expressed in a reference frame (in the laboratory), the measured elasticity ten- sors, generally, does not exhibit evident symmetry properties, therefore how can the symmetry class of a sample be identified ? This problem is invisible for isotropic materials, but becomes more prominent as the anisotropy of the material increases.

For a totally anisotropic (triclinic) material, how can we decide whether two sets of components represent the same material?

This question has already been addressed in the literature. The different approaches are usually based on the spectral decomposition of the elasticity tensor [29,11,10].

We propose here an alternative approach using polynomial invariants(andcovari- ants) of the elasticity tensor. Instead of resorting on the spectral decomposition of the elasticity tensor, we use the harmonic decomposition, which is a higher-dimensional analogue of the Fourier decomposition. As established in [17],Ela can be decom- posed as follow

Ela'2H⁰⊕2H²⊕H⁴. (1) in whichH^kis space ofk-th order harmonic tensors, the analogue of Fourier modes.

It is interesting to note thatH⁴andEla have the same symmetry classes.

In this paper, we develop an invariant-based approach to provide necessary and sufficient conditions for a tensor inH⁴to belong to a given isotropy class. This pro- cedure applied to the componentD∈H⁴of an elasticity tensor, givesinvariant nec- essary conditionsfor an elasticity tensor to belong to a given symmetry class. Under certain hypothesis, that will be investigated in a forthcoming paper, these conditions can further be generically sufficient.

This approach, investigated in the present contribution, follows a long series of papers [24,25,8]. It is interesting to notice that some of the materials introduced here are well-known by the high energy physics community [1,2,30], and has stimulated mathematical researches [31,27,28]. However, these methods do not seem to have yet been applied in elasticity.

The objectives of this paper are twofold:

1. Introduce a rigorous (and general) geometric framework to describe anisotropic features of tensor spaces;

2. Parameterize the symmetry classes ofH⁴.

The paper is organized as follows. We begin by recalling the harmonic decom- position of Ela. In section3, we introduce the geometric framework used for the analysis of a tensor representation, and in section4 the geometric parametrization

of orbit spaces. Although materials introduced in this section are not new, and are probably well-known by most mathematicians and the high-energy physics commu- nity, it does not seem to have been yet exploited by the mechanical community. We clarify the geometry behind the so called “normal forms” of elasticity tensors (lin- ear slices) and their ambiguity (monodromy groups). This permits us to justify, on a rigorous mathematical basis, the results given in table1and in table2. Calculations of the monodromy groups for closed subgroups of SO(3)are given in appendix A.

The materials in these sections are rather general and, as the methodology can be applied in other situations, we believe that it was important to properly introduce this framework. Once introduced, these notions are applied to the spaceH⁴, in order to provide necessary and sufficient invariant conditions for a tensor inH⁴to belong to the following symmetry classes : orthotropic ([D2]), trigonal ([D3]), tetragonal ([D4]), transverse isotropic ([SO(2)]) and cubic ([O]). For each of these classes, we provide a parametrization of the corresponding stratum by rational expressions involving up to six polynomial invariants. Bifurcation relations between related classes are also com- puted and summed-up in a diagram (c.f. figure2in section5.6). Finally, in section6, we prove, as a collateral observation, that the invariants defined by the coefficients of the Betten polynomial [7] do not separate the orbits ofEla. This paper is concluded in section7where extension of the method to a broader class of situations is discussed.

2 Decomposition of the elasticity tensor

In the theory oflinear elasticity, thestress tensorσand thestrain tensorεare related, at a fixed temperature, by theHooke’s law

σ^{i j}=C^{i jkl}ε_kl.

Thestress tensorσ belongs toS₂(R³), the vector space ofcontravariantsymmetric tensors onR³, while ε belongs toS²(R³), the vector space ofcovariant symmet- ric tensors onR³(both of dimension 6). As a consequence, the elasticity tensor is endowed withminor symmetries

C^{i jkl}=C^jikl=C^{i jlk}.

Forhyperelasticmaterials, the stress-strain relation is furthermore assumed toderive from an elastic potentialand we must add themajor symmetry

C^{i jkl}=C^{kli j}.

The space of (hyper) elasticity tensors is therefore the 21D vector space Ela=S₂S₂(R³).

Remark2.1. An elasticity tensorCcan also be viewed as a second-rank symmetric tensorCinS²(R³)'R⁶. The associate matrix representation [15] is given by

C=

















c₁₁ c₁₂ c₁₃ √ 2c₁₄√

2c₁₅√ 2c₁₆ c12 c22 c23

√2c₂₄√ 2c₂₅√

2c₂₆ c13 c23 c33

√ 2c34

√ 2c₃₅√

2c₃₆

√ 2c14

√ 2c24

√

2c34 2c44 2c₄₅ 2c₄₆

√ 2c₁₅ √

2c₂₅ √

2c₃₅ 2c₄₅ 2c₅₅ 2c₅₆

√ 2c₁₆ √

2c₂₆ √

2c₃₆ 2c₄₆ 2c₅₆ 2c₆₆















 ,

wherec_mnare the components of the elasticity tensor in an orthonormal frame. We use the conventional rule to recode a pair of indices(i,j)(i,j=1,2,3) by a integer m=α(i,j)(m=1,2, . . .6), whereα(i,i):=ifor 1≤i≤3 andα(i,j):=9−(i+j) fori6=j.

To each hyperelastic material corresponds an elasticity tensor Cbut this asso- ciation is not unique, it depends of the choice of afixed orientationof the material.

Hence, there is agauge group. As the material is rotated by an elementg= (g_i^j)in the rotation group SO(3), the elasticity tensorC= (C^{i jkl})is submitted to the following transformation

C^{i jkl}7→gⁱ_pg_q^jg^k_rg^l_sC^pqrs.

From the point of view of linear elasticity, the classification of elastic materials can be assimilated to the description of theorbits of this SO(3)-representation on the space of elasticity tensorsEla. Since throughout this paperR³is endowed with its traditional euclidean metric, no distinction will be made between covariant and con- travariant components. As a consequence, since now, only lower subscripts will be considered.

This representation can be decomposed into irreducible pieces, known as har- monic tensors(higher dimensional analogue of Fourier modes), and denoted byHⁿ(R³) (or simply byHⁿif there is no ambiguity). EachHⁿis the space of totally symmetric, traceless tensors of ordern. It is a vector space of dimension 2n+1. This decompo- sition, known as theharmonic decompositionis the following forEla

Ela'2H⁰⊕2H²⊕H⁴. (2) It has already been used in the study of anisotropic elasticity tensors [5,6,17] and we refer to these references for a deeper insight into this topic.

EachC∈Ela can therefore be written asC= (λ,µ,a,b,D)whereλ,µ∈H⁰, a,b∈H²andD∈H⁴. The explicit harmonic decomposition is well-known. Given an orthonormal frame(e₁,e2,e₃)of the Euclidean space, we get [8]

C_{i jkl}=λ δ_{i j}δ_kl+µ(δ_ikδ_jl+δ_ilδ_jk) +δi ja_kl+δkla_{i j}

+δ_ikb_jl+δ_jlb_ik+δ_ilb_jk+δ_jkb_il +D_{i jkl},

(3)

where(δ_{i j})are the components of the Euclidean metricqin the orthonormal frame (e_i). This formula can be inverted to obtain the 5 harmonic components(λ,µ,a,b,D) ofC. OnEla=S²S²(R³), there are only two different contractions¹(or traces)

d_{i j}= (tr₁₂C)_{i j}:=

3 k=1

∑

C_{kki j}, v_{i j}= (tr₁₃C)_{i j}:=

3 k=1

∑

C_{kik j},

wheredandv are known respectively, as thedilatationtensor and theVoigttensor [14]. Starting with (3) we get

d= (3λ+2µ)q+3a+4b, v= (λ+4µ)q+2a+5b. (4) Taking the traces of each equation, one obtains

tr(d) =9λ+6µ, tr(v) =3λ+12µ, and, finally:

λ= 1

15(2 tr(d)−tr(v)), µ= 1

30(−tr(d) +3 tr(v)), a=1

7(5 dev(d)−4 dev(v)), b=1

7(−2 dev(d) +3 dev(v)),

where dev(a):=a−¹₃tr(a)qis thedeviatoricpart (or traceless part) of the 2nd-order tensora.

3 Geometry of orbit spaces

In this section, we setup the geometric framework which is required to study rigor- ously a linear group action on a vector space. Our starting point is alinear represen- tation

ρ:G→GL(V)

of a real, compact Lie groupGon a finite dimensionalR-vector spaceV. The action onV will be denoted by

g·v:=ρ(g)v, g∈G,v∈V.

3.1 Orbits, symmetry groups and fixed point sets

The set of all vectorsv∈V which are related tovby an elementg∈Gis called the G-orbitofvand is denoted by

G·v:={g·v|g∈G}. TheG-orbits are compact submanifolds ofV[2,12].

1 The notation tri jindicates that the contraction should be done on thei-th andj-th indices.

The set of transformations of Gwhich leave a given vectorvfixed is a closed subgroup ofG, which is called theisotropy subgroup(or symmetry group) ofv. It will be denoted by

G_v:={g∈G;g·v=v}. The symmetry groups of vectors in a same orbit are conjugate

Gg·v=g Gvg⁻¹, v∈V,g∈G. (5) LetHbe any subgroup ofG. The set of vectorsv∈V which are fixed byH

V^H:={v∈V|h.v=vfor allh∈H},

is called the fixed point set of H. It it is alinear subspace of V. Notice that, for eachv∈V^H,H⊂G_vand that ifH₂=gH₁g⁻¹thenV^H2 =g·V^H1. IfH₁⊂H₂then V^H2⊂V^H1. However, it may happen thatV^H2=V^H1 butH₁6=H₂.

Given a subgroupHofG, thenormalizerofH N(H):=

g∈G|gHg⁻¹=H

is the maximal subgroup of G, in which H is a normal subgroup. It can also be defined as theisotropy groupofHfor the action ofGon the set of its subgroup (by conjugacy). The proof of the following important lemma can be found in [19].

Lemma 3.1. For each subgroup H of G, the fixed point set V^His N(H)-invariant. If H is moreover theisotropy subgroupof some vector v∈V , then N(H)is the maximal subgroup of G which leaves invariant V^H.

3.2 Symmetry classes and strata

Two vectorsv₁ andv₂ are in the same isotropy class(or symmetry class) if their isotropy subgroups areconjugateinG, that is if there existsg∈Gsuch that

G_v2=g G_v1g⁻¹.

Due to (5), two vectors v₁ andv₂ which are in the same G-orbit are in the same isotropy class but the converse is generally false.

Theconjugacy classof a subgroupH(i.e. the subset ofP(G)of all subgroups ofGwhich are conjugate toH) will be denoted by[H]. The conjugacy class of an isotropy subgroup will be called anisotropy class. In general, a compact groupG has an infinite number of different conjugacy classes, but for a given action, there is only afinitenumber of different isotropy classes (see [12]). In the particular case of SO(3)-representation on tensors, this result is known to the mechanical community as a consequence ofHermann’s theorem[20,3].

The set of all vectorsv∈V in the same isotropy class defined by[H]is denoted Σ_[H] and called a stratum. It is the union of allG-orbits of points which isotropy groups are conjugate toH(it is not a vector subspace in general). There is only afinite

number of (non empty) strata and each of them is a (G-invariant) smooth submanifold ofV [2,12]. The partition

V=Σ_[H0]∪Σ_[H1]∪ · · · ∪Σ_[Hn] is called theisotropy stratificationof(V,ρ).

On the set of conjugacy classes ofclosedsubgroups of a compact groupG, there is a partial order induced by inclusion. It is defined as follows:

[H₁][H₂] iffH₁is conjugate to a subgroup ofH₂.

Endowed with this partial order, the set of isotropy classes²has aleast elementand a greatest element. This partial order induces a (reverse) partial order relation on strata

Σ_[H2]Σ_[H1] iff [H₁][H₂].

The set of strata inherits therefore the structure of a finiteposet(partially ordered set). The topological closure of a stratumΣ_[H], denoted byΣ_[H], and called shortly theclosed stratum associated to[H], corresponds to vectors v∈V such thatvhas at leastisotropy[H]. Notice thatΣ_[H] corresponds to vectorsv∈V such thatvhas exactlythe isotropy[H]. A “closed stratum” is therefore a union of strata. A stratum Σ_[H2]Σ_[H1]means thatΣ_[H2]is contained in the boundary ofΣ_[H1]. A stratumΣ_[H2] is said to beadjacentto the stratumΣ_[H1]ifΣ_[H2]Σ_[H1]and there is no other stratum Σ_[H]betweenΣ_[H2]andΣ_[H1](i.e.Σ_[H2]Σ_[H]Σ_[H1]).

Remark3.2. AG-orbit is said to begenericif it belongs to the least isotropy class.

Thegeneric stratumΣ_[H0], which is the union of generic orbits can be shown to be a dense and openset inV. Moreover, for each other stratumΣ_[H], we have

dimΣ_[H]<dimΣ_[H0].

On the opposite side, the stratum which corresponds to the greatest isotropy subgroup is called theminimal stratum³.

3.3 Normal forms and monodromy groups

It is important to notice that a vectorv∈V belongs to the closed strataΣ_[H](i.e. has at least isotropy[H]), iff itsG-orbit intersects the fixed point setV^H. This observation leads to the possibility to define anormal form⁴for eachG-orbit, or at least to reduce the complexity of the problem of describingG-orbits.

2 The partially ordered set of conjugacy classes of all closed subgroups of SO(3)is described in sec- tionA.

3 Since the isotropy group of the null-vector isGitself, this minimal stratum is always represented by the isotropy class[G] ={G}. The stratumΣ_[G]is always reduced to{0}ifρis anon-trivial irreducible representation but it may not be otherwise.

4 In elasticity this is known as the possibility to choose a “natural coordinate system” in which the matrix representation of a given elasticity tensorChas a lot of zeros.

Example3.3. For the standard action of the rotation group SO(3)on the space of 3×3 symmetric matrices, the least isotropy class is[D2]. In particular, each symmetric matrix hasat leastthis symmetry. Therefore, the SO(3)-orbit of each matrix meets the fixed point setV^D2 which corresponds to diagonal matrices. This gives a (global) normal formrepresentative for each orbit.

This general reduction procedure can be described as a consequence of lemma3.1.

For each subgroupH, the linear representationρ:G→GL(V)induces a linear rep- resentation ofN(H)onV^H, obtained by the restriction

ρ_/N(H):N(H)−→GL(V^H).

This induced representation is not faithful⁵because its kernel containsH. However, when H is an isotropy group, its kernel isexactly H and we get a faithful linear representation

ρ_ΓH :Γ^H−→GL(V^H)whereΓ^H:=N(H)/H.

Furthermore, two vectorsv₁,v₂inV^H∩Σ_[H]are in the sameG-orbit iff there are in the sameN(H)-orbit. We have therefore reduced (locally, for orbits inΣ_[H]) the problem of describing the orbit spaceV/Gto the the orbit spaceV^H/Γ^H. This is especially meaningful when the groupΓ^Hisfinite, in which case, we say thatV^His a linear slice. Then, eachG-orbit intersectsV^Hat most in a finite number of points and V^H/Γ^H is anorbifold⁶. The cardinal ofΓ^His equal to the index[N(H),H]ofHin its normalizerN(H). It is equal to the number of times, theG-orbit of a point inΣ_[H]

intersects the fixed point setV^H. Each of this point defines anormal formof the orbit (like the diagonal form of a symmetric matrix). But this normal form is not unique in general, these different normal forms are permuted by the groupΓ^H which is called themonodromy groupofV^Hand described thisambiguity(in the case of symmetric matrices this monodromy group corresponds to the permutations of the eigenvalues on the diagonal). It is quite remarkable that this ambiguity (the monodromy group) can be computeda priori, for each isotopy class (see AppendixA).

3.4 Strata dimensions

The main formula, which is the basis for all other equations relating dimensions of strata, fixed-point sets and isotropy subgroups is a consequence of the reduction pro- cedure explained in section3.3. It is summarized by the following lemma.

Lemma 3.4.

dimΣ_[H]=dimV^H+dimG−dimN(H) (6)

5 A representationρ:G→GL(V)is faithful if the only elementg∈Gsuch thatρ(g)is the identity in GL(V)is the unit element ofG.

6 Orbifolds have been introduced by Thurston [33]. They generalize manifolds to admit quotients of manifolds by finite groups.

Proof. The main observation is that Σ_[H]/G is a differentiable manifold, contrary to the whole the orbit spaceV/Gwhich is not in general. This can be justified as follows (see [12] for a more rigourous proof). EachG-orbit inΣ_[H]meetsV^H, and two vectors inV^H∩Σ_[H]are in the sameG-orbit iff they are in the sameΓ^H-orbit.

Therefore, the orbit spacesΣ_[H]/Gand(V^H∩Σ_[H])/Γ^Hare the same. Notice that the setV^H∩Σ_[H]is the subspace of vectors inV^Hwhose isotropy classes are exactly[H]

(generic elements inV^H). It is an open and dense set inV^H. ButΓ^H has no fixed point onV^H∩Σ_[H], from which it can be deduced that(V^H∩Σ_[H])/Γ^His a manifold of dimension

dim(V^H∩Σ_[H])/Γ^H=dimV^H−dimΓ^H.

Moreover, the restriction of the projection mapπ:V→V/Gto the stratumΣ_[H]

π_/Σ

[H] :Σ_[H]→Σ_[H]/G

can be shown to be afiber bundlewith fiberG/H, i.e. a space that locally looks like Σ_[H]/G×G/H, and therefore

dimΣ_[H]=dimG/H+dimΣ_[H]/G, where

dimG/H=dimG−dimH, and

dimΣ_[H]/G=dimV^H−dimΓ^H=dimV^H−(dimN(H)−dimH).

Finally, we get

dimΣ_[H]=dimV^H+dimG−dimN(H).

Remark3.5. The dimension ofV^H can be computed using thetrace formulafor the Reynolds operator(see [19]). Lettingχ_ρbe the character of the representation(V,ρ), we have

dimV^H= 1

|H|

∑

h∈H

χρ(h), (7)

if H is a finite group (and the preceding formula has to be replaced by the Haar integral overHfor an infinite compact Lie group). Explicit analytical formulas based on (7) were obtained in [19] and [4].

Fig. 1 Adjacency diagram for isotropy classes ofEla.

3.5 Application to the elasticity tensor

The isotropy classes for the elasticity tensor have been computed in [17,9] (a general algorithm to compute all the isotropy classes of any SO(3)-representation has been proposed in [23]). There are exactly eight classes: the isotropic class[SO(3)], the cubicclass[O], thetransversely isotropicclass[O(2)], thetetragonalclass[D4], the trigonalclass[D3], theorthotropicclass[D2], themonoclinicclass[Z2]and the the triclinicclass [1](see AppendixAfor the definitions of these classes). The corre- spondingadjacency diagramis illustrated in figure1.

The geometric characterization of Ela and its stratification are summed-up in table1, whereS_n is the symmetric group (the group of permutations acting on n elements) and the notations used for the closed subgroup of SO(3)are given in Ap- pendix A. The results appearing in table 1 are well-known in elasticity, but their constructions are usually ad-hoc and do not stand on rigorous mathematical founda- tions. The material provided in this section was primary exposed to fix and clarify the geometric framework underneeth⁷. The following remarks should be of importance to relate this framework with popular and historical considerations.

Remark 3.6. The spaceV^H is defined for any closed subgroup H, not only for an isotropy subgroup. In elasticity, for instance, fixed-point sets have been considered

7 The dimensions ofΣ[H]andΣ[H]/Gare obtained using formula (6) of section3.3, dimV^His computed using thetrace formula(7) and the explicit formulas provided [4].

Table 1 Isotropy classes forEla.

Isotropy class H N(H) Γ^H cardΓ^H dimV^H dimΣ_[H]/G dimΣ_[H]

Triclinic 1 SO(3) SO(3) ∞ 21 18 21

Monoclinic Z2 O(2) O(2) ∞ 13 12 15

Orthotropic D2 O S3 6 9 9 12

Trigonal D3 D6 S2 2 6 6 9

Tetragonal D4 D8 S2 2 6 6 9

Transverse isotropic O(2) O(2) 1 1 5 5 7

Cubic O O 1 1 3 3 6

Isotropic SO(3) SO(3) 1 1 2 2 2

Table 2 Isotropy classes forH⁴.

Isotropy class H N(H) Γ^H cardΓ^H dimV^H dimΣ[H]/G dimΣ[H]

Triclinic 1 SO(3) SO(3) ∞ 9 6 9

Monoclinic Z2 O(2) O(2) ∞ 5 4 7

Orthotropic D2 O S3 6 3 3 6

Trigonal D3 D6 S2 2 2 2 5

Tetragonal D4 D8 S2 2 2 2 5

Transverse isotropic O(2) O(2) 1 1 1 1 3

Cubic O O 1 1 1 1 4

Isotropic SO(3) SO(3) 1 1 0 0 0

forZ3,Z4(which are notisotropy subgroups). This may be why it was believed, for such a long time, that there was ten types of elasticity classes rather than eight.

Remark3.7. The stratumΣ_[H]consists of tensors whose symmetry group is conju- gate toH. It corresponds to a symmetry class according to Forte and Vianello [17].

Formula (6) can be recast as

dimΣ_[H]=dimV^H+e_H

wheree_H=dimG/N(H)is theEuler number⁸. In elasticity, e_H is the number of Euler angles needed to define the appropriate coordinate system in which the elastic tensor has the minimum number of elastic constants [22]. The strata dimension cor- responds to what Norris, for example, calls the number ofindependent parameters characterizing a tensor.

Remark3.8. When themonodromy groupΓ^His finite,Σ_[H]/GandV^Hhave the same dimension, this is the situation for most of elasticity symmetry classes. At the op- posite, whenΓ^H is continuous, the dimension of the orbit spaceΣ_[H]/Gis strictly smaller than dimV^H. In elasticity, this case occurs only for classes[Z2]and[1].

The isotropy classes of the 9-D space H⁴, the higher order component of the harmonic decomposition ofEla, are the same as the 8 classes ofEla. The main char- acteristics of each symmetry classes are reported in the table2.

8 Notice thatG/N(H)is in bijection with the set of distinct subgroups ofGwhich are conjugate toH, that is[H]

4 Semialgebraic structure on orbit spaces

4.1 Polynomial invariants

The linear action of the groupGon the R-vector spaceV extends naturally to the vector space of polynomial functions defined onV with values inR. This extension is given by

(g·P)(v):=P(g⁻¹·v)

for every polynomial functionPonVand every vectorv∈V. The set of allinvariant polynomialsis a sub-algebra of the algebraR[V]of polynomial functions defined on V. Thisinvariant algebrais denoted byR[V]^G.

WhenGis acompactLie group, this algebraR[V]^Gis offinite type[21,26]. This means that it is possible to find a finite set of invariant polynomialsJ₁, . . . ,J_N which generatesR[V]^G(as an algebra). A minimal generating set is called anintegrity basis.

Notice that, in general, theinvariant algebrais notfree, which means that it is not always possible to find a minimal generator setJ₁, . . . ,J_N which are algebraically independent⁹. The polynomial relations amongJ₁, . . . ,J_N are calledsyzygies. The set of such relations is also finitely generated.

4.2 Inequalities defining orbit spaces

The orbit spaceV/Gof a linear representationV of a compact Lie groupGcan be given the structure of asemialgebraic set[13], that is a subset ofR^N defined by a boolean combination of polynomial equations and inequalities overR. A general ex- plicit construction has been proposed in [27] [27]. Since invariant polynomial func- tions onV separate the orbits[2, Appendix C], it is possible to find a finite set of polynomial invariants{J₁, . . . ,J_N}which defines a polynomial mappingJ:V→R^N such that

J(C) =J(C⁰) iffCandC⁰are in the same orbit.

This way, aG-orbit can be identified with a point inJ(V)⊂R^Nand the orbit space is described by the equations and inequalities which defined the subsetJ(V)ofR^N.

4.3 Equations for closed strata

Similarly, each closed stratum can be described by a finite set of polynomial equations and inequalities. Let[H]be a fixed isotropy class andΣ_[H]⊂V be the corresponding stratum. Let{J₁,J2, . . . ,J_N}be a finite set of invariant polynomials which separate the orbits ofG. To obtain the defining equations forΣ_[H], we fix a subgroupH in the conjugacy class [H] and choose linear coordinates(xⁱ)_1≤i≤q onV^H. Then, we

9 PolynomialsP₁, . . . ,PNare said to be algebraically independent overRif the only polynomial inN variables which satisfiesQ(P1, . . . ,PN) =0 is the zero polynomial.

evaluate{J₁,J₂, . . . ,J_N}onV^H(as polynomials in thexⁱ) and we try to obtainimplicit equationson theJ_kfor the parametric system

J_k=P_k(x¹, . . .x^q), k=1, . . . ,N, P_k∈R[X₁, . . . ,X_q], (8) satisfied by the restriction of theJ_ktoV^H.

This is however a difficult task in general (one might consider [16] for a full discussion about theimplicitization problem). Moreover, it could happen that the al- gebraic varietyVI defined by such an implicit system is bigger than the varietyVP, defined by the parametric system (8). Besides, we are confronted to the difficulty that we work overRwhich is not algebraically closed.

To overcome these difficulties, we observe that the restrictions of{J₁,J₂, . . . ,J_N} toV^HareΓ^H-invariant and therefore can be expressed as polynomial expressions of some generatorsσ1, . . . ,σrof the invariant algebra of the monodromy groupΓ^H on V^H. This observation reduces the complexity of the problem. We consider first the implicitization problemfor the parametric system

J_k=p_k(σ₁, . . . ,σr), k=1, . . . ,N, p_k∈R[X₁, . . . ,Xr]. (9) To solve this problem, we use aGroebner basis[16]. The main observation is that, in each case we considered, the system (9) leads torational solutions. More precisely, we obtained a system of implicit equations (syzygies) which characterizesthe closed stratumΣ_[H]

S_j(J₁,J₂, . . . ,J_N) =0, j=1, . . .l S_j∈R[X₁, . . . ,X_N], (10) and a system

σ_i=R_i(J₁,J₂, . . . ,J_N), i=1, . . . ,r R_i∈R(X₁, . . . ,X_N), (11) which express theσias rational¹⁰ functions of{J₁,J₂, . . . ,J_N}onthe open stratum Σ_[H].

Remark 4.1. Beware, however, that these rational expressions may not be unique.

However if someσ_i can be written as P₁/Q₁ as well asP₂/Q₂ thenP₁Q₂−P₂Q₁ belongs to the ideal generated by theS_j.

Because the expressions of σi are rational, the fact that the field on which we work is real or complex does not matter at this level. Therefore, eachrealsolution (J₁,J₂, . . . ,J_N)of (10) corresponds to aunique real solution(σ₁, . . . ,σr)given by (11).

The second step is to obtain a real point (x¹, . . . ,x^q)inV^H from a real solution (σ₁, . . . ,σr)of (11). For this, we need to compute an additional system of inequalities on theσi, or equivalently on theJ_k, which permits to exclude complex solutions of

σi=Qi(x¹, . . .x^q), i=1, . . . ,r, Qi∈R[X₁, . . . ,Xq].

In section 5 where our results are presented, the only non-trivial monodromy group that we encountered were the standard action of the symmetric groupsS₂and

10 The fact that the solutions are rational will not be justified here. We just observe that this is the case for all classes we have treated in this article.

S₃(in appropriate coordinates). In these cases, and more generally when the mon- odromy group isS_nand the action is isomorphic to the standard one,r=q=n, the invariantsσ₁, . . . ,σ_nare algebraically independent and (in an appropriate coordinates system)x¹, . . . ,xⁿare the roots of the polynomial

p(z) =zⁿ−σ₁zⁿ⁻¹+· · ·+ (−1)ⁿσ_n.

Therefore, the problem reduces to finding conditions on theσithat ensure that all the roots ofpare real. The solution is due to Hermite, [13]. He has proved that the number of distinct real roots of a real polynomial pof degreen is equal to thesignatureof the Hankel matrixB(p):= Si+j−2

1≤i,j≤n, whereS_k:=∑ⁿ_i=1(xⁱ)^kis the power sum of the roots ofp. In particular,phas real roots if and only ifB(p)is non-negative.

5 Symmetry classes ofH⁴

Contrary toEla, an integrity basis forH⁴is known¹¹. The aim of this section is to describe the strata ofH⁴, applying the algebraic procedure detailed in section4.3and using this integrity basis. We obtain a characterization, in terms of polynomial re- lations between invariants, for isotropy classes havingfinitemonodromy groupΓ^H, namely thecubicclass[O], thetransversely isotropicclass[O(2)], thetrigonalclass [D3], thetetragonalclass[D4]and theorthotropicclass[D2]. For each of them, ex- plicit polynomial relations between the elementary invariant ofH⁴is given. Finally, a bifurcation diagram is provided to make explicit how we “travel” from a given isotropy class to another.

LetD∈H⁴, we introduce the following 2nd-order tensorsd₂, . . . ,d₁₀(which are covariant¹²toD)

d₂=tr₁₃(D²) d₃=tr₁₃(D³) d₄=d²₂ d₅=d₂Dd₂ d₆=d³₂ d₇=d²₂Dd₂ d₈=d²₂D²d₂ d₉=d²₂Dd²₂ d₁₀=d²₂D²d²₂

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Proposition 5.1. An integrity basis forH⁴is given by J_k:=tr(d_k), k=2, . . . ,10.

The first six invariantsJ₂, . . . ,J₇are algebraically independent. The last three ones J₈,J₉,J₁₀are linked to the formers by polynomial relations. Thesefundamental syzy- gieswere computed in [32]. For technical details and historical considerations con- cerning these fundamental invariants, we refer to the original publication of Boelher and al.[8] and the references therein.

Remark5.2. Notice that the first invariantJ₂(D) =hD,Diis the squared norm ofD, it corresponds to the squaredFrobenius normofD. In particular, J₂(D) =0 if and only ifD=0.

11 This integrity basis has been computed in [32] and brings to the knowledge of the mechanic commu- nity in [8].

12 A linear mapΦ:V1→V2, between two representations(V1,ρ1)and(V2,ρ2)of a same groupGis equivariantifΦ(ρ1(g)·v₁) =ρ2(g)·Φ(v₁), for allv₁∈V₁andg∈G. In that case, we say thatv₂:=Φ(v1) iscovarianttov₁.

5.1 Cubic symmetry ([O])

AnO-invariant tensorD∈H⁴has the following matrix representation:

D=

















8δ −4δ −4δ 0 0 0

−4δ 8δ −4δ 0 0 0

−4δ −4δ 8δ 0 0 0

0 0 0 −8δ 0 0

0 0 0 0 −8δ 0

0 0 0 0 0 −8δ

















(13)

whereδ ∈R.

As indicated in Table1, the monodromy group for the cubic system is1, therefore linear slice corresponding to the octahedral group is of degree 1. In other terms, there is no ambiguity in defining the normal form (13). The evaluation of the invariants (12) on this slice gives the following parametric system

J₂=480δ², J₃=1920δ³, J₄=76800δ⁴, J₅=0, J₆=12288000δ⁶, J₇=0,

J₈=0, J₉=0, J₁₀=0.

We notice that the parameter 4δ=J3/J₂is a rational invariant.

Proposition 5.3. A harmonic tensor D∈H⁴is in the closed stratumΣ_[O]if and only if the invariants J₂(D)· · ·J₁₀(D)satisfy the following polynomial relations

3J₄=J₂², J₅=0, 30J₃²=J₂³, 9J₆=J₂³,

J₇=0, J₈=0, J₉=0, J₁₀=0. (14) D is in the cubic class[O]if moreover J2(D)6=0. In that case, it admits the normal form(13)where4δ:=J₃(D)/J₂(D).

We observe that∀J₂,J₃∈R,δ∈R. Therefore we do not need to add any inequal- ity to the syzygy system (14) to ensure thatδis real. This fact is general to any linear slice of degree 1.

Remark5.4. All 2nd-order tensors inH²covariant toDvanish.

As a consequence, for the cubic case, the following characterization is possible.

Corollary 5.5. A harmonic tensor D∈H⁴is in the closed stratumΣ_[O]if and only if there exists c₂,c₃∈R^?such that

d₂=c₂q, d₃=c₃q, and 10c²₃−c³₂=0. (15) Proof. Sinced_kare covariant to anO-invariant tensor, they are multiple of the iden- tity, i.e.,d_k=c_kq, fork=2, . . . ,10. ThusJ_k=tr(d_k) =3c_kand the set of syzygies (14) can be expressed in terms of thec_k. The announced relations follow, using the additional fact thatDq=0 and the definitions of thed_k.

Conversely, suppose that these relations are verified. The fundamentalDcovari- ants, provided in proposition (5.1), are eitherd₃, powers ofd₂, or involved products of

Dandd₂. Ifd₂=c₂q, powers ofd₂are scalar multiples of the metricq, and because Dis harmonic all terms involvingDd₂vanish. ThereforeJ₅,J₇,J₈,J₉,J₁₀vanish and the relations betweenJ₂, J₄andJ₆are automatically satisfied. Because trq=3 and 10c²₃−c³₂=0, we have moreover 10J₃²−J₂³=0. Therefore the fundamental syzygies of the cubic class are satisfied.

5.2 Transversely isotropic symmetry ([O(2)])

An O(2)-invariant tensorD∈H⁴has the following matrix representation:

D=

















3δ δ −4δ 0 0 0

δ 3δ −4δ 0 0 0

−4δ −4δ 8δ 0 0 0

0 0 0 −8δ 0 0

0 0 0 0 −8δ 0

0 0 0 0 0 2δ

















(16)

whereδ ∈R.

As in the previous case, the monodromy group for the transversely isotropic sym- metry is1, and the linear slice is of degree 1. The evaluation of the invariants (12) on this slice gives

J₂=280δ², J₃=720δ³, J₄=32800δ⁴, J₅=80000δ⁵, J₆=4528000δ⁶, J₇=17600000δ⁷, J₈=211200000δ⁸, J₉=3872000000δ⁹, J₁₀=46464000000δ¹⁰ The parameterδ=7J₃/18J₂is a rational invariant. The linear slice corresponding to the group O(2)is of degree 1, which was already known (see sectionA).

Proposition 5.6. A harmonic tensor D∈H⁴is in the closed stratumΣ_[O(2)] if and only if the invariants J2(D)· · ·J₁₀(D)satisfy the following polynomial relations

98J₄=41J₂², 63J₅=25J₃J₂, 3430J₃²=81J₂³, 1372J₆=283J₂³, 882J₇=275J₂²J₃, 4802J₈=165J₂⁴, 12348J₉=3025J₂³J₃, 67228J₁₀=1815J₂⁵.

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D is in the transversely isotropic class[O(2)]if moreover J₂(D)6=0. In that case, it admits the normal form(16)whereδ=7J3(D)/18J₂(D).

As for[O]the expression ofδ ensures that, for allJ2,J3∈R, the solutionδ is real.

Remark5.7. All 2nd-order tensors inH²covariant toDare multiple of diag(1,1,−2) and are eigenvectors ofDcorresponding to the eigenvalue 12δ.

5.3 Trigonal symmetry ([D3])

AD3-invariant tensorD∈H⁴has the following matrix representation:

D=

















3δ δ −4δ −√

2σ 0 0

δ 3δ −4δ

√2σ 0 0

−4δ −4δ 8δ 0 0 0

−√ 2σ √

2σ 0 −8δ 0 0

0 0 0 0 −8δ −2σ

0 0 0 0 −2σ 2δ

















(18)

where(δ,σ)∈R².

As indicated in the table1for this symmetry class is no more1butS₂, and the linear slice is of degree 2. Its action is given by

δ7→δ, σ7→ −σ.

Therefore, we will have to make a choice betweenσ and−σ to define our normal form. The evaluation of the invariants (12) on this slice gives

J₂=280δ²+16σ² J₃=144δ 5δ²−σ²

J₄=32800δ⁴+2720σ²δ²+88σ⁴ J₅=32δ −σ²+50δ²²

J₆=4528000δ⁶+436800σ²δ⁴+20640σ⁴δ²+496σ⁶ J₇=320δ 22δ²+σ²

−σ²+50δ²2

J₈=3840δ² 22δ²+σ²

−σ²+50δ²2

J₉=3200δ −σ²+50δ²2

22δ²+σ²2

J10=38400δ² −σ²+50δ²2

22δ²+σ²2

We notice that the parameter:

δ=−1 4

J5

J₂²−3J₄

is a rational invariant, therefore for all J₂,J₄,J₅∈R, the parameter δ is real (c.f.

section4.3). However, the minimal equation satisfied byσis of degree 2

J₂=280δ²+16σ². (19)

A condition is needed forσto be real. This condition, which isσ²≥0, is given by the following inequality onJ₂,J₄,J₅:

2J₂(J₂²−3J₄)²−35J₅²≥0 (20) Having computedδ, the equation (19) has two roots with opposite signs.

Proposition 5.8. A harmonic tensor D∈H⁴is in the closed stratumΣ_[D3]if and only if the invariants J₂(D)· · ·J₁₀(D)satisfy the following polynomial relations

192J₆=−51J₂³+216J₂J₄+10J₃² 36J₇=−2J₂²J₃+6J₃J₄+27J₂J₅

768J₄²=−99J₂⁴+552J₂²J₄+10J₂J₃²+240J₃J₅ 240J₈=−33J₂⁴+96J₂²J₄+30J₂J₃²+40J₃J₅ 576J4J₅=−41J23J3+120J2J3J4+216J22J₅+30J33

1152J9=−99J23J3+296J2J3J4+648J22J₅+10J33

1440J₅²=−11J25+32J23J4−70J22J32+240J2J3J₅+240J32J4

8640J₁₀=−891J₂⁵+2592J₂³J₄+730J₂²J₃²+2160J₂J₃J₅+240J₃²J₄ (21)

together with inequality(20). D is in the trigonal class[D3]if moreover3J4−J₂²6=0 and98J4−41J₂²6=0. In that case, it admits the normal form(18)where

δ=−1 4

J₅ J₂²−3J₄ and whereσis the positive root of J2=280δ²+16σ².

Remark5.9. All 2nd-order tensors inH²covariant toDare multiple of diag(1,1,−2) and are eigenvectors ofDcorresponding to the eigenvalue 12δ.

5.4 Tetragonal symmetry ([D4])

AD4-invariant tensorD∈H⁴has the following matrix representation:

D=

















−σ+3δ σ+δ −4δ 0 0 0 σ+δ −σ+3δ −4δ 0 0 0

−4δ −4δ 8δ 0 0 0

0 0 0 −8δ 0 0

0 0 0 0 −8δ 0

0 0 0 0 0 2σ+2δ

















(22)

where(δ,σ)∈R². As for the[D3]class, the monodromy group isS₂, and the linear slice is of degree 2. Therefore a choice has to be made betweenσ and−σ to define the normal form.

The evaluation of the invariants (12) on this slice gives J₂=8σ²+280δ²

J3=48δ σ²+15δ²

J4=32σ⁴+960δ²σ²+32800δ⁴ J₅=128δ (5δ−σ)²(5δ+σ)²

J₆=128σ⁶+5760σ⁴δ²+86400σ²δ⁴+4528000δ⁶ J₇=512δ 55δ²+σ²

(5δ−σ)²(5δ+σ)² J₈=6144δ² 55δ²+σ²

(5δ−σ)²(5δ+σ)² J₉=2048δ(5δ−σ)²(5δ+σ)² 55δ²+σ²2

J₁₀=24576δ²(5δ−σ)²(5δ+σ)² 55δ²+σ²2

We notice that the parameter

δ=−1 4

J₅ J₂²−3J₄

is a rational invariant, therefore for all(J₂,J₄,J₅)∈R³,δ∈R. However, the minimal equation satisfied byσis of degree 2

J₂=8σ²+280δ² (23)

As for[D3], the condition

2J₂(J₂²−3J₄)²−35J₅²≥0 (24) has to be added in order to ensureσto be real.

Having computed δ, the equation (23) has two roots with opposite signs. The linear slice corresponding to the groupD4is of degree 2, which was already known (see sectionA).

Proposition 5.10. A harmonic tensor D∈H⁴is in the closed stratumΣ_[D4] if and only if the invariants J₂(D)· · ·J₁₀(D)satisfy the following polynomial relations

6J₆=−3J₂³+9J₂J₄+20J₃² 3J₇=J₂²J₃−3J₃J₄+3J₂J₅

6J₄²=−3J₂⁴+9J₂²J₄+20J₂J₃²−20J₃J₅ 5J₈=−3J₂⁴+6J₂²J₄+30J₂J₃²−5J₃J₅ 3J₄J₅=7J₂³J₃−15J₂J₃J₄+3J₂²J₅−60J₃³

6J₉=5J₂³J₃−13J₂J₃J₄+6J₂²J₅−20J₃³

5J₅²=−2J₂⁵+4J₂³J₄+10J₂²J₃²−20J₂J₃J₅+30J₃²J₄ 15J₁₀=−9J₂⁵+18J₂³J₄+85J₂²J₃²−30J₂J₃J₅+15J₃²J₄

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