An Alternative Route to the Localized Momentum Balance Relations in Classical Continua

Abstract This paper discusses an alternative to the classical arguments used to obtain the localized laws of momentum balance, i.e. Cauchy's stress theorem and the equations of motion. We show they can be obtained from the integral form of momentum balance using only the localization theorem without any asymptotic analysis of small parts. A key step in the process is to recast the integral momentum balance equation into an equivalent form before performing localization. This modified form leads to a direct deduction of the equations of motion using the localization theorem. Cauchy's stress theorem follows shortly afterward. Since the other localized balance laws of thermomechanics are commonly found via the localization theorem, it is hoped that the simple derivation provided herein could serve a pedagogical purpose, providing for a unified argument structure by which all standard balance laws in classical mechanics can be deduced. The derivation can also be generalized to produce an alternate proof of Noll's theorem, which shows that the traction on a surface at a point depends only on the outward normal of the surface and not any of its higher-order local descriptors such as curvature, also known as Cauchy's postulate.