Acta Mechanica 114, 217-224 (1996) ACTA MECHANICA 9 Springer-Verlag 1996 The geometry of uniformity in second-grade elasticity M. de Le6n, Madrid, Spain, and M. Epstein, Calgary, Alberta (Received April 6, 1994) Summary. The underlying structure of the theory of continuous distributions of defects, such as dislocations and disclinations, in second-grade elastic bodies is presented in terms of second-order G-structures. 1 Introduction Since the early work of Kondo [10], Bilby [7], Kr6ner [9], and others, it was understood that the theory of continuous distributions of defects in crystalline media finds its natural expression in the language of modern differential geometry. Later work by Noll [5] and Wang [6] on the theory ofinhomogeneities in continuous media, while making no mention, heuristic or otherwise, of the underlying crystalline structure, led to similar conclusions. More recently, the realization [3] that the basic geometric object involved in the formulation is a G-structure, has permitted the derivation of specific results. In extending this last approach to second-grade elastic materials, we bear in mind that the precise connection between some physical quantities, such as disclinations and couple-stresses, and their putative geometric counterparts, remains still an open question, and that it is within the realm of second-grade elasticity that the natural answer is likely to be found. The main aim of this paper, however, is to introduce the basic idea of second-grade homogeneity as the physical counterpart of the integrability of a certain, precisely defined geometric object: a second-order G-structure. This object, although well established, is still in need of further mathematical study. We have, nevertheless, refrained from presenting new mathematical results, some of which have appeared elsewhere [2], and we have, rather, focussed on the most elementary presentation possible, given the complexity of the subject. In the final Section, an attempt to foster an intuitive picture of the rather complicated situation at hand is presented. We hope that the interest of mechanicists in this fertile area of research will be aroused or renewed. 2 Review of the first-grade case A material body B is a three-dimensional differentiable manifold that can be covered with just one chart ~:B ~P,~ 3 , which may be interpreted physically as a configuration in a Euclidian space referred to a fixed basis. By composition, it is possible (and convenient) to refer all configurations 9 to a fixed