Material Geometry

Resumen

In continuum physics the physical properties of a elastic body are characterized for all the constitutive relations. This measures the mechanical response produced at each particle by a deformation in a local neighbourhood of the particle. Di erential geometry provides a rigorous mathematical framework not only to present the constitutive properties but to discover and prove results. For applications, it is usual that the bodies are assumed uniform and homogeneous in the sense of that the body is made of a unique material and there is a con guration in such a way that the mechanical response is the same at all the points. The main purpose of this thesis is to follow the Noll's approach to present a mathematical framework based on groupoids, algebroids and distributions to deal with non-uniform and inhomogeneous simple bodies. For any simple body a unique groupoid, called material groupoid, may be naturally associated. The unifomity of the body coincides with the transitivity of the groupoid. If the material groupoid turns out to be a Lie groupoid the associated Lie algebroid, called material algebroid, is available. Then, the homogoneneity is characterized by the integrability of both (material groupoid and material algebroid). However, the property of being Lie groupoid is not guaranteed. In fact, smooth uniformity corresponds to that imposition of di erentiability on the material groupoid. Smooth distributions are now introduced to deal with this case. In fact, two smooth distributions, called material distributions, may be canonically de ned generalizing the notion of Lie algebroid. Thus, it is proved that we can cover the simple body by a material foliation whose leaves are (smoothly) uniform. These new tools are also used to present a \measure" of uniformity and an homogeneity for non-uniform bodies. The construction of the material distribution is generalized to a much more abstract framework in which the case of an arbitrary subgroupoid of a Lie groupoid is treated. We also study Cosserat media by imposing that the corresponding material groupoid is a Lie groupoid.