The configuration space and principle of virtual work for rough bodies

TL;DR

This paper develops a generalized theory of Cauchy fluxes and virtual work for rough bodies in Euclidean space, using geometric measure theory to handle irregular boundaries and extending classical concepts to a broader class of bodies.

Contribution

It introduces a new framework for analyzing bodies with irregular boundaries using flat chains and cochains, extending classical virtual work principles.

Findings

Generalized Cauchy flux theory for irregular bodies.

Extension of admissible bodies to flat chains.

Representation of stress as flat forms via Wolfe's theorem.

Abstract

In the setting of an $n$-dimensional Euclidean space, the duality between velocity fields on the class of admissible bodies and Cauchy fluxes is studied using tools from geometric measure theory. A generalized Cauchy flux theory is obtained for sets whose measure theoretic boundaries may be as irregular as flat $(n-1)$-chains. Initially, bodies are modeled as normal $n$-currents induced by sets of finite perimeter. A configuration space comprising Lipschitz embeddings induces virtual velocities given by locally Lipschitz mappings. A Cauchy flux is defined as a real valued function on the Cartesian product of $(n-1)$-currents and locally Lipschitz mappings. A version of Cauchy's postulates implies that a Cauchy flux may be uniquely extended to an $n$-tuple of flat $(n-1)$-cochains. Thus, the class of admissible bodies is extended to include flat $n$-chains and a generalized form of the principle of virtual power is presented. Wolfe's representation theorem for flat cochains enables the identification of stress as an $n$-tuple of flat $(n-1)$-forms.