By Marcelo Epstein;Marek Elzanowski
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Extra info for Material Inhomogeneities and their Evolution: A Geometric Approach (Interaction of Mechanics and Mathematics)
3, in each case the change in the uniformity ﬁeld consists on a multiplication of P to the right by a member of the archetypal symmetry group (for the degree of freedom permitted by the material symmetry) or a member of the general linear group (for the degree of freedom permitted by an arbitrary change of archetype). Since we are only considering natural-state archetypes, the latter degree of freedom boils down to a multiplication by an arbitrary member of the orthogonal group. Moreover, since the body is an elastic solid, the former degree of freedom is a subgroup of the orthogonal group.
10. A (not necessarily smooth) global section ❊❊ ❊ ❇ ❇ ❇ ☎ ☎ ❊ ❊ ❇ ❊ ❊ ❇ ❇ ❇ ❊ ☎ ☎ ☎ ☎ ✂ ✂ ✂ ☎ ✂ ✂ ✂ ✂ B Fig. 11. Smooth local sections The principal bundle of frames Assume now that we were to look at the body B as a manifold devoid of any constitutive connotation. In that case, we could consider the ﬁbre bundle F (B) obtained by attaching to each point the collection of all frames at that point. This is the so-called bundle of linear frames, or frame bundle, of the base manifold B. It is also denoted by F B or L(B).
We recall that a groupoid is said to be transitive if for each ordered pair of points of the base manifold there exists at least one groupoid element with one of the points as the source and the other as the target. In the case of the material groupoid, transitivity means, therefore, that each body point is materially isomorphic to every other body point. In other words, the uniformity of a body corresponds exactly to the notion of a transitive groupoid, namely, a groupoid whereby none of the sets Pij is empty.