Two-Point Rotations in Geometry of Finite Deformations

Abstract

The paper is devoted to a study of deformed states of continuous medium. The study is restricted to the case when a deformed state admits a comparison to the referential state. The latter are considered to be immersed in the three-dimensional Euclidean space. A derivation of the two-point tensor of finite rotation and its unconventional orthogonality are discussed. One-point rotation tensors are introduced. Both of the two one-point rotation tensors are orthogonal in the conventional sense thus allowing to determine all geometrical characteristics related to a rotation in a three-dimensional space. Priority in the paper is given to simple algorithmic procedures for obtaining natural components of measures and tensors of finite deformations, as well as transformations of the fundamental equations of continuum mechanics realized by the rotation tensors and corresponding vectors of finite rotations. The two pseudovectors of finite rotations are defined and are to be employed, along with the pseudovectors of the “extra” rotations, as the principal kinematic parameters in mathematical models of micropolar elastic continuum.