The configuration space and principle of virtual work for rough bodies

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.