Two-Scale Geometric Modelling for Defective Media

Abstract: A new geometrically exact micro-structured model is constructed using a generalisation of the notion of Riemann-Cartan manifolds and fibre bundle theory of rank 3. This model is based around the concept of two different length scales: a macroscopic scale -- of dimensions 1, 2, or 3 -- and a microscopic one -- of dimension 3. As they interact with each other, they produce emergent behaviours such as dislocations (torsion) and disclinations (curvature). A first-order placement map F : TB --> TE between a micro-structured body B and the micro-structured ambient space E is constructed, allowing to pull the ambient Riemann-Cartan geometry back onto the body. I norder to allow for curvature to arise, F is, in general, not required to be a gradient. Central to this model is the new notion of pseudo-metric, providing, in addition to a macroscopic metric (the usual Cauchy-Green tensor) and a microscopic metric, a notion of coupling between the microscopic and macroscopic realms. A notion of frame indifference is formalised and invariants are computed. In the case of a micro-linear structure, it is shown that the data of these invariants is equivalent to the data of the pseudo-metric.