## Abstract

A generalized transport theorem for convecting irregular domains is presented in the setting of Federer’s geometric measure theory. A prototypical r-dimensional domain is viewed as a flat r-chain of finite mass in an open set of an n-dimensional Euclidean space. The evolution of such a generalized domain in time is assumed to follow a continuous succession of Lipschitz embedding so that the spatial gradient may be nonexistent in a subset of the domain with zero measure. The induced curve is shown to be continuous with respect to the flat norm and differential with respect to the sharp norm on currents in $${\mathbb{R}^{n}}$$Rn. A time-dependent property is naturally assigned to the evolving region via the action of an r-cochain on the current associated with the domain. Applying a representation theorem for cochains, the properties are shown to be locally represented by an r-form. Using these notions, a generalized transport theorem is presented.

Original language | English |
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Pages (from-to) | 539-559 |

Number of pages | 21 |

Journal | Continuum Mechanics and Thermodynamics |

Volume | 28 |

Issue number | 1-2 |

DOIs | |

State | Published - 1 Mar 2016 |

## Keywords

- Homological integration theory
- Lipschitz embeddings
- Lipschitz maps
- Sharp chains
- Transport theorem

## ASJC Scopus subject areas

- Materials Science (all)
- Mechanics of Materials
- Physics and Astronomy (all)