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Journal of Elasticity

Erschienen in:

01.03.2014

On Anisotropic Polynomial Relations for the Elasticity Tensor

verfasst von: N. Auffray, B. Kolev, M. Petitot

Erschienen in: Journal of Elasticity | Ausgabe 1/2014

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Abstract

In this paper, we explore new conditions for an elasticity tensor to belong 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 tensor C, rather than on spectral properties of the Kelvin representation. We compute a set of algebraic relations which describe precisely the orthotropic (\([\mathbb {D}_{2}]\)), trigonal (\([\mathbb {D}_{3}]\)), tetragonal (\([\mathbb {D}_{4}]\)), transverse isotropic ([SO(2)]) and cubic (\([\mathbb {O}]\)) symmetry classes in \(\mathbb {H}^{4}\), the highest-order irreducible component in the decomposition of \(\mathbb {E}\mathrm {la}\). We provide a bifurcation diagram which describes how one “travels” in \(\mathbb {H}^{4}\) from a given isotropy class to another. Finally, we study the link between these polynomial invariants and those obtained as the coefficients of the characteristic or the Betten polynomials. We show, in particular, that the Betten invariants do not separate the orbits of the elasticity tensors.

Vorheriger Artikel Rayleigh Surface Waves on a Kelvin-Voigt Viscoelastic Half-Space

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The notation \(\operatorname {tr}_{ij}\) indicates that the contraction should be done on the i-th and j-th indices.

The partially ordered set of conjugacy classes of all closed subgroups of SO(3) is described in Appendix.

Since the isotropy group of the null-vector is G itself, this minimal stratum is always represented by the isotropy class [G]={G}. The stratum Σ _[G] is always reduced to {0} if ρ is a non-trivial irreducible representation.

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

A representation ρ:G→GL(V) is faithful if the only element g∈G such that ρ(g) is the identity in GL(V) is the unit element of G.

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

The dimensions of Σ _[H] and Σ _[H]/G are obtained using formula (6) of Sect. 3.3, dimV ^H is computed using the trace formula (7) and the explicit formulas provided [4].

Notice that G/N(H) is in bijection with the set of distinct subgroups of G which are conjugate to H, that is [H].

Polynomials P ₁,…,P _N are said to be algebraically independent over \(\mathbb {R}\) if the only polynomial in N variables which satisfies Q(P ₁,…,P _N)=0 is the zero polynomial.

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

This integrity basis has been computed in [32] and brings to the knowledge of the mechanic community in [8].

A linear map Φ:V ₁→V ₂, between two representations (V ₁,ρ ₁) and (V ₂,ρ ₂) of a same group G is equivariant if Φ(ρ ₁(g)⋅v ₁)=ρ ₂(g)⋅Φ(v ₁), for all v ₁∈V ₁ and g∈G. In that case, we say that v ₂:=Φ(v ₁) is covariant to v ₁.

I.e., for all symmetry classes, except the monoclinic and the triclinic classes.

Abud, M., Sartori, G.: The geometry of orbit-space and natural minima of Higgs potentials. Phys. Lett. B 104(2), 147–152 (1981) ADSCrossRefMathSciNet

Abud, M., Sartori, G.: The geometry of spontaneous symmetry breaking. Ann. Phys. 150(2), 307–372 (1983) ADSCrossRefMATHMathSciNet

Auffray, N.: Démonstration du théorème de Hermann à partir de la méthode Forte-Vianello. C. R., Méc. 336(5), 458–463 (2008) MATH

Auffray, N.: Analytical expressions for anisotropic tensor dimension. C. R., Méc. 338(5), 260–265 (2010) MATHMathSciNet

Backus, G.: A geometrical picture of anisotropic elastic tensors. Rev. Geophys. 8(3), 633–671 (1970) ADS

Baerheim, R.: Harmonic decomposition of the anisotropic elasticity tensor. Q. J. Mech. Appl. Math. 46(3), 391–418 (1993) MATHMathSciNet

Betten, J.: Irreducible invariants of fourth-order tensors. Math. Model. 8, 29–33 (1987) MATHMathSciNet

Boehler, J.P., Kirillov, A.A. Jr., Onat, E.T.: On the polynomial invariants of the elasticity tensor. J. Elast. 34(2), 97–110 (1994) MATHMathSciNet

Bóna, A., Bucataru, I., Slawinski, M.: Characterization of elasticity-tensor symmetries using SU(2). J. Elast. 75(3), 267–289 (2004) MATH

10.

Bóna, A., Bucataru, I., Slawinski, M.: Coordinate-free characterization of the symmetry classes of elasticity tensors. J. Elast. 87(2), 109–132 (2007) MATH

11.

Bóna, A., Bucataru, I., Slawinski, M.: Space of SO(3)-orbits of elasticity tensors. Arch. Mech. 60(2), 123–138 (2008) MATHMathSciNet

12.

Bredon, G.E.: Introduction to Compact Transformation Groups. Pure and Applied Mathematics, vol. 46. Academic Press, New York (1972) MATH

13.

Coste, M.: An Introduction to Semialgebraic Geometry. Université de Rennes (2002)

14.

Cowin, S.C.: Properties of the anisotropic elasticity tensor. Q. J. Mech. Appl. Math. 42, 249–266 (1989) MATHMathSciNet

15.

Cowin, S.C., Mehrabadi, M.M.: Eigentensors of linear anisotropic elastic materials. Q. J. Mech. Appl. Math. 43, 15–41 (1990) MATHMathSciNet

16.

Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms, 3rd edn. Undergraduate Texts in Mathematics. Springer, New York (2007) MATH

17.

Forte, S., Vianello, M.: Symmetry classes for elasticity tensors. J. Elast. 43(2), 81–108 (1996) MATHMathSciNet

18.

Francois, M., Geymonat, G., Berthaud, Y.: Determination of the symmetries of an experimentally determined stiffness tensor: application to acoustic measurements. Int. J. Solids Struct. 35(31), 4091–4106 (1998) MATH

19.

Golubitsky, M., Stewart, I., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory. Vol. II. Applied Mathematical Sciences, vol. 69. Springer, New York (1988)

20.

Herman, B.: Some theorems of the theory of anisotropic media. Dokl. Akad. Nauk SSSR 48(2), 89–92 (1945) MathSciNet

21.

Hilbert, D.: Theory of Algebraic Invariants. Cambridge University Press, Cambridge (1993) MATH

22.

Norris, A.N.: On the acoustic determination of the elastic moduli of anisotropic solids and acoustic conditions for the existence of symmetry planes. Q. J. Mech. Appl. Math. 42, 413–426 (1989) MATHMathSciNet

23.

Olive, M., Auffray, N.: Symmetry classes for even-order tensors. Math. Mech. Complex Syst. 1(2), 177–210 (2013). doi:10.2140/memocs.2013.1.177

24.

Olver, P.J.: Group theoretic classification of conservation laws in elasticity. In: Systems of Nonlinear Partial Differential Equations, Oxford, 1982. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 111, pp. 323–331. Reidel, Dordrecht (1983)

25.

Olver, P.J.: Canonical anisotropic elastic moduli. In: Modern Theory of Anisotropic Elasticity and Applications, Research Triangle Park, NC, 1990, pp. 325–339. SIAM, Philadelphia (1991)

26.

Olver, P.J.: Classical Invariant Theory. London Mathematical Society Student Texts, vol. 44. Cambridge University Press, Cambridge (1999) MATH

27.

Procesi, C., Schwarz, G.: Inequalities defining orbit spaces. Invent. Math. 81(3), 539–554 (1985) ADSMATHMathSciNet

28.

Procesi, C., Schwarz, G.W.: The geometry of orbit spaces and gauge symmetry breaking in supersymmetric gauge theories. Phys. Lett. B 161(1–3), 117–121 (1985) ADSMathSciNet

29.

Rychlewski, J.: On Hooke’s law. J. Appl. Math. Mech. 48(3), 303–314 (1984) MathSciNet

30.

Sartori, G., Valente, G.: Tools in the orbit space approach to the study of invariant functions: rational parametrization of strata. J. Phys. A 36(7), 1913–1929 (2003) ADSMATHMathSciNet

31.

Schwarz, G.W.: Lifting smooth homotopies of orbit spaces. Publ. Math. Inst. Hautes Études Sci. 51, 37–135 (1980) MATH

32.

Shioda, T.: On the graded ring of invariants of binary octavics. Am. J. Math. 89, 1022–1046 (1967) MATHMathSciNet

33.

Thurston, W.P.: Three-Dimensional Geometry and Topology. Vol. 1. Princeton Mathematical Series, vol. 35. Princeton University Press, Princeton (1997) MATH

Titel: On Anisotropic Polynomial Relations for the Elasticity Tensor
verfasst von: N. Auffray
B. Kolev
M. Petitot
Publikationsdatum: 01.03.2014
Verlag: Springer Netherlands
Erschienen in: Journal of Elasticity / Ausgabe 1/2014
Print ISSN: 0374-3535
Elektronische ISSN: 1573-2681
DOI: https://doi.org/10.1007/s10659-013-9448-z

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