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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show all references

References:
[1]

Arch. for Rat. Mech. and Anal., 86 (1984), 125-145. doi: 10.1007/BF00275731.  Google Scholar

[2]

J. Elasticity, 56 (1999), 33-58. doi: 10.1023/A:1007647913363.  Google Scholar

[3]

J. Functional Analysis, 58 (1984), 225-253. doi: 10.1016/0022-1236(84)90041-7.  Google Scholar

[4]

Phys. Rev. E, (Rapid Comm.), 47 (1993), 3838. Google Scholar

[5]

Journal de Physique II, 4 (1993), 93-102. Google Scholar

[6]

J. Phys. II France, 4 (1994), 75-91. Google Scholar

[7]

European J. Appl. Math, 23 (2012), 121-154. doi: 10.1017/S0956792511000313.  Google Scholar

[8]

Math. Models Methods Appl. Sci., 19 (2009), 601-630. doi: 10.1142/S0218202509003541.  Google Scholar

[9]

J. Mech. Phys., 59 (2011), 787-803. doi: 10.1016/j.jmps.2011.01.007.  Google Scholar

[10]

Physical Review E., 66 (2002), 061710. Google Scholar

[11]

Journal of Mechanics and Physics of Solids, 50 (2002), 1431-1451. doi: 10.1016/S0022-5096(01)00120-X.  Google Scholar

[12]

Arch. Rational Mech. Anal., 161 (2002), 181-204. doi: 10.1007/s002050100174.  Google Scholar

[13]

Phys. D, 136 (2000), 175-191. doi: 10.1016/S0167-2789(99)00153-0.  Google Scholar

[14]

in Modeling of Soft Matter, IMA Vol. Math. Appl., 141, Springer, New York, 2005, 189-203. doi: 10.1007/0-387-32153-5_8.  Google Scholar

[15]

J. Phys. II France, (1997), 1059-1069. Google Scholar

[16]

Macromolecular Rapid Communications, 16 (1995), 679-686. Google Scholar

[17]

Annales de l'I.H.P., Section C, 11 (1994), 217-243.  Google Scholar

[18]

Arch. Rat. Mech. Anal., 100 (1988), 105-127. doi: 10.1007/BF00282200.  Google Scholar

[19]

J. Phys. II France, 6 (1996), 1049-1060. Google Scholar

[20]

J. Phys. II France, 6 (1996), 1273-1290. Google Scholar

[21]

Siberian math. J., 12 (1977), 515-531. Google Scholar

[22]

Oxford University Press, 2003. Google Scholar

[23]

Liquid Crystals, 26 (1999), 1531-1540. Google Scholar

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