# American Institute of Mathematical Sciences

April  2015, 8(2): 259-282. doi: 10.3934/dcdss.2015.8.259

## Energy-minimizing nematic elastomers

 1 Department of Mathematics, Purdue University, West Lafayette, IN 47906 2 Franklin W. Olin College of Engineering, Needham, MA 12492, United States

Received  May 2013 Revised  November 2013 Published  July 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.
Citation: Patricia Bauman, Andrea C. Rubiano. Energy-minimizing nematic elastomers. Discrete & Continuous Dynamical Systems - S, 2015, 8 (2) : 259-282. doi: 10.3934/dcdss.2015.8.259
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