spatial asymptotic directors, asymmetric theories of mechanics of continuous deformable bodies, stress tensor, auxetic.

In this paper new representations of three-dimensional asymmetric stress tensor and the corresponding form of the differential equilibrium equations are given. Asymmetric theories of solid mechanics continue to attract attention in connection with the necessity of mathematical modelling of the mechanical behaviour of the advanced materials (e.g., auxetics by means of the hemitropic micropolar theory of elasticity). The study is restricted to such asymmetric second rank tensors, for which it is still possible to retain the notion of real eigenvalues, but not to accept the mutual orthogonality of the directors of the principal trihedron. The exact algebraic formulation of these asymmetry conditions is discussed. The study extends the tensor representations of the symmetric stress tensor based on the notion of asymptotic directions. The obtained results are a clear evidence in favor of algebraic hyperbolicity both the symmetric and asymmetric second rank tensors in three-dimensional space.

Radayev Yuri Nickolaevich

e-mail: radayev@ipmnet.ru, y.radayev@gmail.com, Dr. Sci. Phys. & Math., Professor, Leading Researcher, Institute for Problems in Mechanics of RAS, Moscow, Russia.