Material Geometry

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.…”

On the homogeneity of non-uniform material bodies

“…In addition, homogeneity may be generalized in such a way that any simple body can be tested to be homogeneous. A first step in this direction was done in [12] where the authors give a homogeneity condition for bundle and laminated bodies.…”

“…Finally, using the body-material distribution, we will be able to define a more general notion of smooth uniformity. This notion was introduced in [12]. We will end up using the foliation by uniform subbodies to interpret it over the material groupoid.…”

“…So, it makes sense to develop some kind of homogeneity for non-uniform material which measures the absence of defects and generalizes the known one. In the literature we can already find some partial answer of this question ([2, 9] for FGM's and [8,12] for laminated and bundle materials).…”