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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 unified theory of material defects, incorporating both the smooth and the singular descriptions, is presented based upon the theory of currents of Georges de Rham. The fundamental geometric entity of discourse is assumed to be represented by a single differential form or current, whose boundary is identified with the defect itself. The possibility of defining a less restrictive dislocation structure is explored in terms of a plausible weak formulation of the theorem of Frobenius. Several examples are presented and discussed. [ABSTRACT FROM AUTHOR]

The theory of singular dislocations is placed within the framework of the theory of continuous dislocations using de Rham currents. For a general n-dimensional manifold, an (n − 1)-current describes a local layering structure and its boundary in the sense of currents represents the structure of the dislocations. Frank’s rules for dislocations follow naturally from the nilpotency of the boundary operator. [ABSTRACT FROM PUBLISHER]

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]