Inhomogeneous Deformation Fields within a Double-Twist Cylinder

Nematic liquid-crystal elastomers combine heterogeneous molecular orientation fields with mechanically anisotropic cross-linking. One example is the cross-linked double-twist structure found in collagen fibrils and keratin macrofibrils, both important structural components of many animal tissues. Theoretical treatments of elastomers under strain typically impose a deformation field throughout the structure. However, in many experimental setups, the deformation is only imposed at the boundaries — for example at the ends of long fibrils. Since in real systems the molecular director and cross-link density fields can be heterogeneous, the strain field within the elastomer is generally heterogeneous as well. We present a general thermodynamic framework for modelling the deformation of a nematic liquid crystal elastomer. We derive coupled partial differential equations which fully characterize a spatially heterogeneous deformation gradient field for a given set of boundary conditions. To demonstrate the utility of this method we solve for the full deformation gradient field within an anisotropically cross-linked double-twist cylinder subjected to experimentally relevant boundary conditions.