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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]

Highlights • Laminates and bundles are important cases of functionally graded materials. • Classical inhomogeneity theory is not able to treat defects in these materials. • A novel definition of inhomogeneity in laminates and bundles is introduced. Abstract Laminates and bundles are common particular cases of functionally graded materials (FGMs). Within this class of non-uniform media, laminates and bundles are distinguished by a property of partial uniformity along surfaces or lines. The question addressed in this paper is whether this property can be exploited to define rigorously a notion of homogeneity eventually leading to a quantitative measure of a density of continuous defects. [ABSTRACT FROM AUTHOR]

A material body with smoothly distributed microstructure can be seen geometrically as a fibration or, when the symmetry group is specified, as a fiber bundle. Within this very general framework, we present a geometric description of such material bodies in terms of fiber jets. We introduce the notion of fiber frame and construct the corresponding Lie groupoid and fiber G-structure. Then, physical properties of a material body with microstructure as uniformity and homogeneity can be translated in geometrical terms as transitivity for the Lie groupoid or integrability for the fiber G-structure. [Copyright &y& Elsevier]

The constitutive characterization of the uniformity and homogeneity of binary elastic composites is presented in terms of a combination of the material groupoids of the individual constituents. The incorporation of these two groupoids within a single double groupoid is proposed as a viable mathematical framework for a unified formulation of this and similar kinds of problems in continuum mechanics. [ABSTRACT FROM AUTHOR]