Continuum mechanics of defective media: an approach using fiber bundles
Résumé
The kinematics of a micro-structured material is geometrically modeled through the framework of fiber bundle geometry. The material continuum is a fiber bundle M B where B is compact and orientable. It is commonly agreed that connections with curvature and torsion can describe defect densities in micro-structured materials. The aim of this work is to introduce a method to derive these objects from the kinematics in an intrinsic way. The material bundle M is therefore placed in the Euclidean fiber bundle E ≡ TE E using a placement map φ : M E. A first-order transformation F : TM TE generalizing Tφ is then introduced. Finally, using F, a metric on B, a connection on M and a solder form on M are inferred from the Euclidean structure on E. These new objects are grouped into a single one, called a pseudo-metric, which allows us to describe the current state of matter through, among other things, the curvature (disclinations) and torsion (dislocations) tensors. On one hand, we see that the torsion tensor can be non-zero even in the holonomic F = Tφ case. On the other-hand, in order for the material to have a non-zero curvature tensor, we see that one must have a non-holonomic first-order transformation: that is, F ̸ = Tφ.
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