Characteristic distribution: An application to material bodies

Abstract

Associated to each material body B there exists a groupoid Ω (B) consisting of all the material isomorphisms connecting the points of B. The uniformity character of B is reflected in the properties of Ω (B): B is uniform if, and only if, Ω (B) is transitive.Smooth uniformity corresponds to a Lie groupoid and, specifically, to a Lie subgroupoid of the groupoid Π 1 (B, B) of 1jets of B. We consider a general situation when Ω (B) is only an algebraic subgroupoid. Even in this case, we can cover B by a material foliation whose leaves are transitive. The same happens with Ω (B) and the corresponding leaves generate transitive Lie groupoids (roughly speaking, the leaves covering B). This result opens the possibility to study the homogeneity of general material bodies using geometric instruments. the ICMAT Severo Ochoa projects SEV-2011-0087 and SEV-2015-0554. V.M. Jiménez wishes to thank MINECO for a FPI-PhD Position and the referee for the suggestions.

Lie groupoids and algebroids applied to the study of uniformity and homogeneity of material bodies

“…Notice that, at general, we cannot ensure that Ω (B) ⊆ Π 1 (B, B) is a Lie subgroupoid (see for instance [11,14,15]). Our assumption is that Ω (B) is in fact a Lie subgroupoid and, in this case, Ω (B) is said to be the material groupoid of B.…”

“…These issues are just another proof of how mathematical concepts are behind physical theories, providing new ways to get a better knowledge of them. Some of the new results provided by the theory of groupoids are contained in [11,14,15], where the use of groupoids has permitted to introduce the notion of material distribution and extend the concepts of uniformity and homogeneity. So the notion of groupoids is being more and more interesting to get new insights in material science.…”

The evolution equation: An application of groupoids to material evolution

“…However, there are materials for which the material groupoid is not differentiable, and therefore is not a Lie groupoid. In [14,17] we have developed a construction that generalizes the notion of Lie algebroid and yields the so-called characteristic distribution. This is a generalized distribution (in the sense of Stefan and Sussmann [24,25]) but involutive, thus determining generalized foliations that divide the body in a differentiable way.…”

“…It is important to say that both distributions are singular (their leaves may have different dimension). In fact, the material distributions of the material evolution may be seen as a particular example of the so-called characteristic distributions [4,14]. Theorem 4.3.…”

On the homogeneity of non-uniform material bodies

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A groupoid Ω (B) called material groupoid is naturally associated to any simple body B (see [11,9,10]). The material distribution is introduced due to the (possible) lack of differentiability of the material groupoid (see [13,15]). Thus, the inclusion of these new objects in the theory of material bodies opens the possibility of studying non-uniform bodies.

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