Lie groupoids and algebroids applied to the study of uniformity and homogeneity of Cosserat media

Abstract

A Lie groupoid, called second-order non-holonomic material Lie groupoid, is associated in a natural way to any Cosserat media. This groupoid is used to give a new definition of homogeneity which does not depend on a reference crystal. The corresponding Lie algebroid, called second-order non-holonomic material Lie algebroid, is used to characterize the homogeneity property of the material. We also relate these results with the previously ones in terms of non-holonomic second-order G-structures. ContentsKey words and phrases. Groupoid, Lie algebroid, constitutive theory, Cosserat media, frame bundle, G−structure.This work has been partially supported by MINECO Grants MTM2013-42-870-P and the ICMAT Severo Ochoa projects SEV-2011-0087 and SEV-2015-0554. V.M. Jimnez wishes to thank MINECO for a FPI-PhD Position.

Lie groupoids and algebroids applied to the study of uniformity and homogeneity of material bodies

“…Notice that the identities are material isomorphisms, and the composition and the inversion of 1−jets preserve Eq. (16). Hence, Ω (B) has structure of groupoid over B which is, indeed, a subgroupoid of the 1−jets groupoid Π 1 (B, B).…”

“…For each two points we will denote by G (X, Y ) the collection of all 1−jets j 1 X,Y ψ which satisfy Eq. (16). So, the set of all material ismorphisms can be written as follows,…”

“…As we have seen, a body is uniform is the function W depends on the point X precisely according to Equation (16). In addition, a body is said to be homogeneous if we can choose a global section of the material groupoid which is constant on the body, more precisely: Definition 3.2.…”

“…Given a simple material body B we identify an embbeding φ 0 : B → R 3 as a reference configuration. A Lie groupoid, called material groupoid, can be naturally associated to any material body (see for example [10], or even [8,16] for Cosserat media). In particular, given two different points X and Y of the material body B, a material isomorphism is a linear isomorphism P XY : T X B → T Y B such that the mechanical response at X and Y is the same; more precisely, if W is the response functional depending on the deformation gradient F at any point X of B, then, W (F P, X) = W (F, Y ) , for any F .…”

The evolution equation: An application of groupoids to material evolution

“…If the material groupoid is a Lie groupoid, then we can associate with it the corresponding Lie algebroid, which is the infinitesimal approximation in the same way that every Lie group has a Lie algebra associated with it. The integrability of that Lie groupoid characterizes the homogeneity of the material [16,15].…”