Non-linear relations of anisotropic elasticity and a particular postulate of isotropy

A general approach to the construction of six-dimensional images of strain processes is proposed with the introduction of a vector basis which, in special cases, is identical to the well-known bases of A. A. Il’yushin, V. V. Novozhilov and Ye. I. Shemyakin and S. A. Khristianovich. The analysis of the properties of materials is based on the use of the concept of characteristic elastic states, which was introduced in the papers of J. Rychlewski. In the case of an isotropic material and four types of anisotropic materials belonging to the cubic, hexagonal, trigonal and tetragonal systems, characteristic subspaces, corresponding to the multiple eigenvalues of the elasticity tensor are defined in a six-dimensional space. In accordance with Hooke's law, the components of the stress and strain vectors in these subspaces preserve their axial alignment for any of their orthogonal transformations. The particular postulate of isotropy, formulated by Il’yushin, is therefore satisfied by definition within the framework of isotropic characteristic subspaces for linear elastic materials. An extension of the particular postulate to strain processes in non-linear anisotropic materials is proposed, on the basis of which a general form of constitutive relations is obtained containing a minimum number of experimentally determinable material functions.