Generalization of the classical theory of symmetry of tensors

An important addition is made (1977) to the tensor algebra, namely the concept of oriented tensors, i.e. tensor objects which depend on orientation in both a three-dimensional space, and in its subspaces. The theory of symmetry [1 - 3] is formulated for oriented tensors, and it generalises the classical theory of symmetry, which applies to the Euclidean tensors only. It was shown that the application of the classical theory, for example, to axial tensors, i.e. objects dependent on orientation in a 3D space, leads to wrong conclusions. The proposed theory is needed to obtain the constitutive equations for shells and other multipolar media, as well as when studying ionic crystals.
  1. Zhilin P.A. General theory of constitutive equations in the linear theory of elastic shells // Izvestiya AN SSSR, Mekhanika tverdogo tela (Transactions of the Academy of Sciences of the USSR, Mechanics of Solids). 1978. N 3. P. 190. (In Russian.)
  2. Zhilin P.A. Basic equations of non-classical theory of shells // Dinamika i prochnost mashin (Dynamics and strength of machines.) Trudi LPI (Proceedings of Leningrad Polytechical Institute.) N 386. 1982. P. 29-46. (In Russian.)
  3. Zhilin P.A. Applied mechanics. Foundations of shells theory. Tutorial book. St. Petersburg State Polytechnical University. 2006. 167 p. (In Russian.)