On the polynomial invariants of the elasticity tensor

Abstract

We consider the old problem of finding a basis of polynomial invariants of the fourth rank tensor C of elastic moduli of an anisotropic material. Decomposing C into its irreducible components we reduce this problem to finding joint invariants of a triplet (a, b, D), where a and b are traceless symmetric second rank tensors, and D is completely symmetric and traceless fourth rank tensor (D e T~). We obtain by reinterpreting the results of classical invariant theory a polynomial basis of invariants for D which consists of 9 invariants of degrees 2 to 10 in components of D. Finally we use this result together with a well-known description of joint invariants of a number of second-rank symmetric tensors to obtain joint invariants of the triplet (a, b, D) for a generic D. O. Introduction. Motivation for the workThe main purpose of the work is to construct a classification of linear anisotropic elastic materials. In colloquial terms we ask the question: How do we give distinct names to distinct anisotropic elastic materials? Clearly a designation based on the 21 components of the tensor C of elastic moduli in a fixed reference frame is not good for this purpose because it provides, generically, different names for different orientations of a given material. As the material is rotated the tensor C moves on its orbit in the space T~, of elasticity tensors. Thus what is needed is a parametrization of distinct orbits. The set O c of the distinct orbits of elasticity moduli is a manifold of (21-3) = 18 dimensions that has a fairly complicated boundary. The problem of naming the distinct orbits would be solved if Or, the manifold of distinct orbits, can be mapped in a one-to-one and continuous manner into the linear space Rn; the coordinates of an image point would then serve as the name of the associated orbit. It is of interest to know what minimal dimension n is needed for this purpose.The following examples may be of help in thinking about this idea:If the manifold of interest is a circle which is one dimensional then the minimal dimension n = 2. If the manifold of interest is the group SO(3) of rigid

Isotropic invariants of a completely symmetric third-order tensor

“…As noted by Boehler et al 7 , the algebra of SO(3)-invariant polynomials on the real vector space H 3 ⊕H 1 is isomorphic to the algebra of SL(2, C)-invariant 54 polynomials on the complex vector space S 6 ⊕ S 2 ; that is…”

“…And, indeed, some material symmetry classes are described by higher-order structural tensors. 7 . Expressed in a generic basis, it is difficult to identity the symmetry class of a linear operator, and to determine one of its optimal basis or representation.…”

“…This distinction is important because, although the algebra of invariant polynomials separates the orbits, this set might be very large. As a consequence, an integrity basis is a functional basis, but the converse is generally false 7,40 . Hence, the cardinal of a minimal integrity basis is generally greater than that of a functional basis.…”

Symmetry classes for odd‐order tensors

“…In the present situation, tensor spaces will be decomposed into O(3)-invariant spaces. Such a decomposition has been widely used in the mechanical community for studying anisotropic elasticity [5,6,14,15] and is often referred to as the harmonic decomposition. In order to present our results in self-contained way, basic definitions of the harmonic decomposition are summed up here.…”

“…the tensorial product of two vector spaces generates a second-order tensor space which decompose as scalar (H 0 ), a vector (H ♯1 ) and a deviator (H 2 ). Spaces indicated with ♯ contain pseudo-tensors, 6 In order to introduce the harmonic decomposition as an operative tool, we do not mention in this subsection the 2 different representations of O(3) on a vector space. Nevertheless, and in order to be rigorous, this point is discussed in subsection 3.3.…”