A Continuum-Mechanical Theory for Nematic Elastomers

Abstract

We develop a continuum theory for the mechanical behavior of rubber-like solids that are formed by the cross-linking of polymeric fluids that include nematic molecules as elements of their main-chains and/or as pendant side-groups. The basic kinematic ingredients of this theory are identical to those arising in continuum-level theories for nematic fluids: in addition to the deformation, which describes the trajectories of material particles, an orientation, which delineates the evolution of the nematic microstructure, is introduced. The kinetic structure of our theory relies on the precept that a complete reckoning of the power expended during the evolution of a continuum requires the introduction of forces that act conjugate to each operative kinematic variable and that to each such force system there should correspond a distinct momentum balance. In addition to conventional deformational forces, which expend power over the time-rate of the deformation and enter the deformational (or linear) momentum balance, we, therefore, introduce a system of orientational forces, which expend power over the time-rate of the orientation and enter an additional orientational momentum balance. We restrict our attention to a purely mechanical setting, so that the thermodynamic structure of our theory rests on an energy imbalance that serves in lieu of the first and second laws of thermodynamics. We consider only nematic elastomers that are incompressible and microstructurally inextensible, and a novel aspect of our approach concerns our treatment of these material constraints. We refrain both from an a priori decomposition of fields into active and reactive components and an introduction of Lagrange multipliers; rather, we start with a mathematical decomposition of the dependent fields such as the deformational stress based on the geometry of the constraint manifold. This naturally gives rise to active and reactive components, where only the former enter into the energy imbalance because the latter automatically expend zero power in processes consistent with the constraints. The reactive components are scaled by multipliers which we take to be constitutively indeterminate. We assume constitutive equations for the active components, and the requirement that these equations be consistent with the energy imbalance in all processes leads to the active components being determined by an energy density response function of the deformation gradient, the orientation, and the orientation gradient. We formulate the requirements of observer independence and material symmetry for such a function and provide, as a specialization, an expression that encompasses the energy densities used in the Mooney-Rivlin description of rubber and the Oseen-Zöcher-Frank description of nematic fluids.