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A certain type of defectivity applicable to binary mixtures of uniform solids and liquid crystals is identified as arising from the possible lack of uniformity of the composite as a result of a misalignment between the structures of the underlying constituents. Various local measures of misalignment are derived for the case of a smectic-A combined with solids with discrete or continuous symmetry groups. [ABSTRACT FROM AUTHOR]

For any body-time manifold R × B there exists a groupoid, called the material groupoid, encoding all the material properties of the material evolution. A smooth distribution, the material distribution, is constructed to deal with the case in which the material groupoid is not a Lie groupoid. This new tool provides a unified framework to deal with general non-uniform material evolution. [ABSTRACT FROM AUTHOR]

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]

Inspired by the seminal works of Eshelby (Philos Trans R Soc A 244A:87–112, 1951, J Elast 5:321–335, 1975) on configurational forces and of Noll (Arch Ration Mech Anal 27:1–32, 1967) on material uniformity, we study a thermoelastic continuum undergoing volumetric growth and in a dynamical setting, in which we call the divergence of the Eshelby stress the Eshelby force. In the classical statical case, the Eshelby force coincides with the negative of the configurational force. We obtain a differential identity for the modified Eshelby stress, involving the torsion of the connection induced by the material isomorphism of a uniform body, which includes, as a particular case, that found by Epstein and Maugin (Acta Mech 83:127–133, 1990). In this identity, the divergence of the modified Eshelby stress with respect to this connection of the material isomorphism takes the name of modified Eshelby force. Moreover, we show that Eshelby's variational approach (1975) can be used to formulate not only the balance of material momentum, but also the balance of energy. In this case, we find that what we call Eshelby power is the temporal analogue of the Eshelby force, and we obtain a differential identity for the modified Eshelby power. This leads to concluding that the driving force for the process of growth–remodelling is the Mandel stress. Eventually, we find that the relation between the differential identities for the modified Eshelby force and modified Eshelby power represents the mechanical power expended in a uniform body to make the inhomogeneities evolve. [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]

• Background on differential geometric concepts is provided in the short space available. • The theory of G-structures is shown to be relevant to both approaches mentioned in title. • A non-trivial example is presented to illustrate the theory. • Some epistemological questions are addressed, if very briefly. An attempt is made to bring into harmony two of the paradigms commonly used in the theory of continuous distributions of defects. It is shown that the common differential geometric apparatus is provided neatly by the theory of G-structures. In the case of a structural model, based on putative experimental observations at the microscopic level, a G-structure can be shown to emerge from the group of linear transformations that preserve a tensorial quantity. For the phenomenological (macroscopic) constitutive model, the G-structure arises from the notion of material isomorphism and the underlying local symmetry group of the constitutive law. A comparative example is presented in the framework of certain smectic liquid crystals. [ABSTRACT FROM AUTHOR]

• Description of fibre recruitment by means of a material implant (theory of uniformity). • Description of the evolution of the recruitment parameters and the fibre orientation by use of the theory of uniformity. • Benchmark problem describing an idealised blood vessel, representing, e.g., a case of hypertension. This study focusses on the remodelling of biological tissues in the framework of the theory of material uniformity. A constitutive evolution model is introduced, including fibre recruitment and reorientation, and subjected to the entropy inequality, which enforces the Second Principle of Thermodynamics. The model is applied to a numerical example describing a pressurised fibre-reinforced cylinder, roughly representing an artery, and is able to capture the major characteristics of remodelling in arteries, as reported in the literature. [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]