Topological aspects of linear elastisity theory (methodological notes)

Abstract

A comparative discussion is given of the deformation properties of three-dimensional (3D) and two-dimensional (2D) solids, which are considered in the approximation of continuum mechanics as elastic continua with three and two spatial dimensions.
Attention is drawn to the effectiveness of attracting concepts and methods of geometry to establish the general patterns of deformation of such systems without taking into account the physicochemical properties of atoms and the interatomic forces. In a geometric description, these continua are elastic spaces with different topological properties, which leads to significant differences in the relationships between the characteristics of their elasticity: Young's moduli, shear, bulk moduli, and Poisson's ratio. Deformation characteristics that can be considered as unique topological invariants of 3D and 2D elastic continua are established