Untitled

Abstract

The concept of internal constraints is extended to gradient materials. Here, interesting constraints can be introduced, such as pseudorigid ones. The stresses and the hyperstresses will be given by constitutive equations only up to reactive parts, which do no work during any compatible motion of the body. For the inclusion of thermodynamical effects, the theory is generalized to the case of thermomechanical constraints. Here one obtains reactive parts of the stresses, heat flux, entropy, and energy, which do not contribute to the dissipation. Some critical remarks on the classical concept of internal constraints are finally given. A method to introduce internal constraints in a natural way is described to overcome some conceptual deficiencies of the classical concept.Communicated by Francesco dell'Isola. MSC2010: 74A30. 1 2 ALBRECHT BERTRAM AND RAINER GLÜGE format. It turns out, and will be shown in the sequel, that such an extension is in fact straightforward once a theory of gradient materials has been constructedat least within the mechanical context.The extension to a thermomechanical format is more complicated. This is, however, necessary not only in order to investigate the compatibility of such constraints with the second law of thermodynamics, but also to study the temperature dependence of mechanical constraints.There has been some discussion about a sound format for the inclusion of the thermodynamical variables into such a theory of internal constraints; see [Green et al. ].The starting point for the present approach is a suggestion by [Trapp 1971; Bertram 2005], where a rate form of a thermomechanical constraint is assumed and the possibility for reactive parts of the stresses, heat fluxes, and energies is given that are not dissipative during any process that is compatible with the constraint. Again, the extension of this theory to gradient materials is straightforward.At the end of this contribution, some critical remarks on the standard theory of mechanical constraints are given and a procedure to avoid these shortcomings is suggested.Dedicated to Gérard Maugin, in gratitude for many years of inspiration and friendshipThe differential-geometric underpinnings of a unified theory of material uniformity and evolution are exposed in terms of the language of groupoids subordinate to geometric distributions. Both the standard theory of material uniformity and the extended theory of functionally graded materials are included in the formulation as well as their temporal counterparts in anelastic and aging processes.In this paper we reinvestigate the structure of the solution of a well-known Love's problem, related to the electrostatic field generated by two circular coaxial conducting disks, in terms of orthogonal polynomial expansions, enlightening the role of the recently introduced class of the Lucas-Lehmer polynomials. Moreover we show that the solution can be expanded more conveniently with respect to a Riesz basis obtained starting from Chebyshev polynomials.This paper deals with the modeling of cy...