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arXiv · 1711.09022

Characteristic distribution: An application to material bodies

V. M. Jiménez·M. de León·M. Epstein
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Abstract

Associated to each material body $\mathcal{B}$ there exists a groupoid $\Omega \left( \mathcal{B} \right)$ consisting of all the material isomorphisms connecting the points of $\mathcal{B}$. The uniformity character of $\mathcal{B}$ is reflected in the properties of $\Omega \left( \mathcal{B} \right)$: $\mathcal{B}$ is uniform if, and only if, $\Omega \left( \mathcal{B} \right)$ is transitive. Smooth uniformity corresponds to a Lie groupoid and, specifically, to a Lie subgroupoid of the groupoid $\Pi^1\left({\mathcal B}, {\mathcal B} \right)$ of 1-jets of $\mathcal B$. We consider a general situation when $\Omega \left( \mathcal{B} \right)$ is only an algebraic subgroupoid. Even in this case, we can cover $\mathcal{B}$ by a material foliation whose leaves are transitive. The same happens with $\Omega \left( \mathcal{B} \right)$ and the corresponding leaves generate transitive Lie groupoids (roughly speaking, the leaves covering $\mathcal B$). This result opens the possibility to study the homogeneity of general material bodies using geometric instruments.

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BibTeXRIS

V. M. Jiménez, M. de León, M. Epstein. 2017-11-24. Characteristic distribution: An application to material bodies. https://doi.org/10.1016/j.geomphys.2018.01.021

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