Classes de symétrie des solides piézoélectriques

We apply here the harmonic and Cartan decomposition techniques to piezoelectric material symmetries classification. We show in particular that we shall reduce from 19 to 17 the number of symmetry classes corresponding to the piezoelectric phenomenon. To cite this article: G. Geymonat, T. Weller, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 847–852.  2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS Abridged English version We extend to linearly piezoelectric solids the physical symmetries classification of hyperelastic solids derived in [2]. According to constitutive equations (1), we shall consider that the behaviour of a piezoelectric solid is given by a triplet (A,P,S) ∈ Ela × Piez × Sym, where Ela, Piez and Sym are respectively the sets of hyperelastic tensors, piezoelectric tensors and symmetric tensors of order 2. The action of O 3 which follows from definition (2) of a piezoelectric solid symmetry group G(A,P,S) is defined by (Q M)...ijk... = · · ·QipQjqQkr · · ·M...pqr... where Q is an element of O3 and M a tensor of any order. Letting g(M)= {Q ∈O3; Q M = M}, we then have G(A,P,S)= g(A)∩ g(P)∩ g(S). The harmonic decomposition (cf. [8]) maps each P ∈ Piez onto a unique quadruplet (H,C, ν,v) ∈ Hrm×Dev×R3×R3, Hrm and Dev being the spaces of harmonic tensors (totally symmetric and traceless) of third and second orders respectively. Via the canonical isomorphism (3) between harmonic tensors and harmonic polynomials of corresponding degree, we transfer the action onto Hn, the space of homogeneous harmonic polynomials of degree n. Adresse e-mail : [email protected] (G. Geymonat).  2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés S1631-073X(02)02573-6/FLA 847 G. Geymonat, T. Weller / C. R. Acad. Sci. Paris, Ser. I 335 (2002) 847–852 Figure 1. – Characteristic tree of Piez symmetry classes. The Cartan decomposition (cf. [4]), which is a way to see Hn as the direct sum of n− 1 subspaces of dimension 2 and one subspace of dimension 1 is the final step to derive the symmetry classes of any P ∈ Piez. We then show that symmetry classes of piezoelectric tensors are found to be 14 and ranged as shown in Fig. 1. These physical (or constitutive) symmetries correspond to piezoelectric behaviour of materials whose crystallographic system (or material symmetry classe, cf. Section 5 for further explanations about these terms) is either triclinic (Id), monoclinic (Z2 and Z − 2 ), orthorhombic (D2 and D z 2), trigonal (Z3, D3 and D z 3), tetragonal (D d 4 ), hexagonal (D d 6 ) or tetrahedral (T ). Moreover, the transverse hemitropy (SO(2)) and transverse isotropy (O(2), O(2)−) physical symmetries correspond to the tetragonal or hexagonal crystallographic system. Finally we show that crystalline symmetries respectively written 4̄, (6, 6̄,622,6mm) and 23 cannot be seen as constitutive ones but as the trace on the tetrahedral, hexagonal and cubic crystallographic systems respectively of superior physical symmetries as shown in the order relation derived in [4]. This fact allows us to reduce to 17 the number of the piezoelectric solids physical symmetries up to now considered to be 19. 1. Définitions et notations On note Lin l’ensemble des tenseurs du deuxième ordre, I l’identité dans Lin et LT le transposé d’un élément L ∈ Lin. On définit alors les ensembles Sym = {L ∈ Lin; L = LT} et O3 = {L ∈ Lin; L LT = LTL = I} des tenseurs du deuxième ordre respectivement symétriques et orthogonaux. Le sous-groupe des rotations est l’ensemble des éléments de O3 dont le déterminant vaut 1, il est noté O+. On définit les espaces Ela et Piez des tenseurs hyperélastiques A et piézoélectriques P vérifiant respectivement Aijkl = Ajikl = Aijlk = Aklij et Ppqr = Pqpr , les composantes des tenseurs étant prises dans une base orthonormée (i, j,k) fixée. On considère ici un solide linéairement piézoélectrique. La loi de comportement d’un tel matériau lie le tenseur des contraintes T ∈ Sym et le vecteur déplacement électrique d ∈ R3 au tenseur des déformations E ∈ Sym et au vecteur gradient du potentiel électrique g ∈R3. Elle est donnée par : { T(E,g)= A[E] − P[g], d(E,g)= PT[E] + S[g], (1)