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A Lie groupoid, called material Lie groupoid, is associated in a natural way to any elastic material. The corresponding Lie algebroid, called material algebroid, is used to characterize the uniformity and the homogeneity properties of the material. The relation to previous results in terms of G−structures is discussed in detail. An illustrative example is presented as an application of the theory. [ABSTRACT FROM AUTHOR]

Lie groupoids and their associated algebroids arise naturally in the study of the constitutive properties of continuous media. Thus, continuum mechanics and differential geometry illuminate each other in a mutual entanglement of theory and applications. Given any material property, such as the elastic energy or an index of refraction, affected by the state of deformation of the material body, one can automatically associate to it a groupoid. Under conditions of differentiability, this material groupoid is a Lie groupoid. Its associated Lie algebroid plays an important role in the determination of the existence of material defects, such as dislocations. This paper presents a rather intuitive treatment of these ideas. [ABSTRACT FROM AUTHOR]

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 1-jets 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. [ABSTRACT FROM AUTHOR]