Space symmetries draw elasticity theory

TL;DR

This paper demonstrates that for high-symmetry crystals, the fundamental equations of elasticity can be derived solely from spatial symmetry considerations, without relying on traditional physical principles like the Cauchy principle or stress symmetry.

Contribution

It shows that classical elasticity equations can be obtained from symmetry arguments alone in high-symmetry cases, challenging traditional derivations.

Findings

Elasticity equations derived from symmetry for high-symmetry crystals.

Traditional principles are not necessary for deriving elasticity in these cases.

Highlights the power of symmetry in fundamental physical theories.

Abstract

The foundation of continuum elasticity theory is based on two general principles: (i) the force felt by a small volume element from its surrounding acts only through its surface (the Cauchy principle, justified by the fact that interactions are of short range and are therefore localized at the boundary); (ii) the stress tensor must be symmetric in order to prevent spontaneous rotation of the material points. These two requirements are presented to be necessary in classical textbooks on elasticity theory. By using only basic spatial symmetries it is shown that elastodynamics equations can be derived, for high symmetry crystals (the typical case considered in most textbooks), without evoking any of the two above physical principles.