  Nonsingular disclination dynamics in a uniaxial nematic liquid crystal is
modeled within a mathematical framework where the kinematics is a direct
extension of the classical way of identifying these line defects with
singularities of a unit vector field representing the nematic director. It is
well known that the universally accepted Oseen-Frank energy is infinite for
configurations that contain disclination line defects. We devise a natural
augmentation of the Oseen-Frank energy to account for physical situations
where, under certain conditions, infinite director gradients have zero
associated energy cost, as would be necessary for modeling half-integer
strength disclinations within the framework of the director theory. Equilibria
and dynamics (in the absence of flow) of line defects are studied within the
proposed model. Using appropriate initial/boundary data, the gradient-flow
dynamics of this energy leads to non-singular, line defect equilibrium
solutions, including those of half-integer strength. However, we demonstrate
that the gradient flow dynamics for this energy is not able to adequately
describe defect evolution. Motivated by similarity with dislocation dynamics in
solids, a novel 2D-model of disclination dynamics in nematics is proposed. The
model is based on the extended Oseen-Frank energy and takes into account
thermodynamics and the kinematics of conservation of defect topological charge.
We validate this model through computations of disclination equilibria,
annihilation, repulsion, and splitting. We show that the energy function we
devise, suitably interpreted, can serve as well for the modeling of equilibria
and dynamics of dislocation line defects in solids making the conclusions of
this paper relevant to mechanics of both solids and liquid crystals.
