Energy-minimizing nematic elastomers
2014
We prove weak lower semi-continuity and existence of energy-minimizers for a free energy describing stable deformations and the corresponding director
configuration of an incompressible nematic liquid-crystal elastomer subject to physically realistic boundary conditions. The energy is a sum of
the trace formula developed by Warner, Terentjev and Bladon (coupling the deformation gradient and the director field) and the Landau-de Gennes
energy in terms of the gradient of the director field and the bulk term for the director with coefficients depending on temperature. A key step in our analysis is to prove that the energy density has a convex extension to non-unit length director fields.
Our results apply to the setting of physical experiments
in which a thin incompressible elastomer in $\mathbb{R}^3$ is clamped on its sides
and
stretched perpendicular to its initial director field, resulting in
shape-changes and director re-orientation.
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