By Marcelo Epstein;Marek Elzanowski

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**Additional resources for Material Inhomogeneities and their Evolution: A Geometric Approach (Interaction of Mechanics and Mathematics)**

**Sample text**

Any simply connected chunk of this body can be “straightened” and thus brought back locally into a homogeneous conﬁguration, but not so the whole body. In other words, B is only locally homogeneous. A uniform body which is not locally homogeneous is said to be inhomogeneous or to contain a distribution of inhomogeneities. Historically, the theory of continuous distributions of inhomogeneities in material bodies stemmed from various attempts at generalizing the geometric ideas of the theory of isolated dislocations arising in crystalline solids.

In other words, no (even local) homogeneous conﬁguration can exist. 2 Homogeneity in terms of a material connection Given a uniformity ﬁeld on an open set U, how can we ascertain whether or not it is integrable? In other words, given a uniformity ﬁeld expressed in terms of some coordinate system X I , how can we know whether or not there exists a local coordinate system such that the given uniformity bases are adapted to it? To obtain a necessary condition for this question of integrability, let us 32 1 An overview of inhomogeneity theory assume that such a coordinate system, say Y M , does exist and let us write the change of coordinates to our system X I by means of three smooth and smoothly invertible functions: X I = X I (Y M ) (I, M = 1, 2, 3).

We will, accordingly, call F (Rn ) the standard frame bundle (in n dimensions). Given the frame bundle of an n-dimensional manifold and a subgroup G of the general linear group GL(n; R), it is not necessarily true that the frame bundle is reducible to this subgroup. The standard frame bundle F (Rn ), however, is reducible to any subgroup G of GL(n; R). To see why this is the case, we need only consider at each point of Rn the standard natural basis and construct the set of all bases obtained from it by the right action of all elements of G.