Inhomogeneous deformation in hyperspace

Abstract

Geometrical approach to continuum mechanics allowing an atomistic treatment is developed. It suggests a body as a surface immersed into ambient space. Four kinds of surfaces inherent to basic deformations determine classical dynamics of the atoms alongside their spatial coordinates. For uniaxial stretch, this surface is isomorphic to the cylinder. For a simple shear, it is the twisted cylinder, for a bending — a cone, and for a torsion — a helicoid. Scalar parameters of the metric of these surfaces that are stretch ratios, shear, cone, and torsion angles used instead of the strain tensor constitute extra coordinates of the particles in hypersurface. Correspondingly, vector-valued tensions are used instead of stress. In the case of pure deformation, the normal and extra coordinates of atoms obey the classical equation of motion admitting periodic boundary conditions. It is shown how the employing of artificial orders of freedom leads to a minimum of elastic energy. In boundary problems, the system of governing equations unambiguously reproduce the stress–strain–displacement relations with the help of using the tension boundary theorem instead of the Cauchy traction principle. The derived direct method of elastostatic makes the system of governing equations closed and the compatibility condition unnecessary. Commonest examples of bending problem normally following from the Airy stress function illustrate the new concept.