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

A method is developed allowing to establish irreducible representations for anisotropic non‐polynomial constitutive equations. The case when a constitutive equation takes the form of an explicit relation between two symmetric second order tensoers is considered in detail. Transitions from the most general anisotropy to particular cases of anisotropy are established. As an example the transition from the general non linear forms to the case of classical linear elasticity is given. It appears that for the considered case of tensor functions the irreducible representations for the non‐polynomial case are similar to those concerning a polynomial function. This similarity disappears for functions involving a larger number of arguments.

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Representations for isotropic scalar‐valued functions, not necessarily polynomials, derived by Wang and by Smith through different procedures are not identical. In the paper both procedures are analysed and the number of invariants entering the functional bases is examined. After suitable amendments of the previously established bases a minimal basis for the chosen invariants is obtained. The technique of proving sufficiency of this set of invariants is given in Appendix.