The symmetry principle in continuum mechanics

Abstract

For problems of the mechanics of an anisotropic inhomogeneous continuum, theorems are given concerning the uninterrupted symmetrical and antisymmetrical analytical continuation of the solution through the plane part of the boundary surface of the medium. Theorems are given for two types of mechanics problem; in the first of these both symmetrical and antisymmetrical continuations of the solution are allowed, while in the second only symmetrical continuation of the solution is allowed. Problems of the first type include problems which are reduced to linear thermoelastic dynamic differential equations of motion of an inhomogeneous anisotropic medium possessing a plane of elastic symmetry, to linear thermoelastic dynamic differential equations of motion of an inhomogeneous Cosserat medium, to non-linear differential equations describing the static elastoplastic stress state of a plate, etc. The second type includes problems which are reduced to non-linear differential equations describing geometrically non-linear strains of shells, to Navier–Stokes equations, etc. These theorems extend the principle of mirror reflection (the Riemann–Schwartz principle of symmetry) to linear and non-linear equations of continuum mechanics. The uninterrupted continuation of the solutions is used to solve problems of the equilibrium state of bodies of complex shape.