Application of Geometric measure Theory in Continuum Mechanics: The Configuration Space, Principle of Virtual Power and Cauchy's Stress Theory for Rough Bodies

TL;DR

This paper applies geometric measure theory to continuum mechanics, enabling the inclusion of generalized bodies with finite perimeter and formulating a corresponding stress theory and principle of virtual power.

Contribution

It introduces a new framework using homological integration theory to extend continuum mechanics to rough bodies with finite perimeter.

Findings

Framework allows for generalized bodies beyond smooth sets

A proper stress theory is formulated for these bodies

A generalized principle of virtual power is established

Abstract

In this work, the principles of Homological Integration Theory are applied to the mathematical formulation of continuum mechanics. A central guideline in the currently acceptable formulation of continuum mechanics is that an admissible body is represented by a set of finite perimeter. The proposed framework is shown to enable the inclusion of a class of generalized bodies for which a corresponding stress theory is properly formulated and a generalized principle of virtual power is presented.