Date of Original Version
Published by the American Mathematical Society
Abstract or Description
This paper generalizes the Ericksen-Leslie continuum model of liquid crystals to allow for dynamically evolving line defect distributions. In analogy with recent mesoscale models of dislocations, we introduce fields that represent defects in orientational and positional order through the incompatibility of the director and deformation `gradient' fields. These fields have several practical implications: first, they enable a clear separation between energetics and kinetics; second, they bypass the need to explicitly track defect motion; third, they allow easy prescription of complex defect kinetics in contrast to usual regularization approaches; and finally, the conservation form of the dynamics of the defect fields has advantages for numerical schemes.
We present a dynamics of the defect fields, motivating the choice physically and geometrically. This dynamics is shown to satisfy the constraints, in this case quite restrictive, imposed by material-frame indifference. The phenomenon of permeation appears as a natural consequence of our kinematic approach. We outline the specialization of the theory to specific material classes such as nematics, cholesterics, smectics and liquid crystal elastomers. We use our approach to derive new, non-singular, finite-energy planar solutions for a family of axial wedge disclinations.
Quarterly of Applied Mathematics, 72, 1, 33-64.