Two-Scale Geometric Modelling for Defective Media
Résumé
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 models 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. In
order 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.
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