Continuum Mechanics of Line Defects in Liquid Crystals and Liquid Crystal Elastomers

Amit Acharya and Kaushik Dayal

 (To appear in Quarterly of Applied Mathematics)

This paper presents a generalization of traditional continuum approaches to liquid crystals and

liquid crystal elastomers 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 materialframe

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 aproach to derive new, non-singular,

finite-energy planar solutions for a family of axial wedge disclinations.