## 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 representing the flat (n - 1)-cochains associated with the Cauchy flux.

Original language | English |
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Pages (from-to) | 1049-1072 |

Number of pages | 24 |

Journal | Mathematics and Mechanics of Solids |

Volume | 20 |

Issue number | 9 |

DOIs | |

State | Published - 1 Oct 2015 |

## Keywords

- Continuum mechanics
- Lipschitz configurations
- flat chains and cochains
- geometric measure theory
- stress theory