Energy-minimizing nematic elastomers

Patricia Bauman; Andrea C. Rubiano

doi:10.3934/dcdss.2015.8.259

Article Contents

2015, Volume 8, Issue 2: 259-282. Doi: 10.3934/dcdss.2015.8.259

This issue Previous Article Existence of solutions to boundary value problems for smectic liquid crystals Next Article A Landau--de Gennes theory of liquid crystal elastomers

Energy-minimizing nematic elastomers

Patricia Bauman^1, and
Andrea C. Rubiano^2,

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

Received: April 30, 2013

Revised: October 31, 2013

Published: March 2015

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Abstract

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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Mathematics Subject Classification: Primary: 35J57; Secondary: 35J50, 49J30.

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References

[1]	E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations, Arch. for Rat. Mech. and Anal., 86 (1984), 125-145.doi: 10.1007/BF00275731.
[2]	D. Anderson, D. Carlson and E. Fried, A continuum-mechanical theory for nematic elastomers, J. Elasticity, 56 (1999), 33-58.doi: 10.1023/A:1007647913363.
[3]	J. M. Ball and F. Murat, $W^{1,p}$-quasiconvexity and variational problems for multiple integrals, J. Functional Analysis, 58 (1984), 225-253.doi: 10.1016/0022-1236(84)90041-7.
[4]	P. Bladon, E. Terentjev and M. Warner, Transitions and Instabilities in liquid-crystal elastomers, Phys. Rev. E, (Rapid Comm.), 47 (1993), 3838.
[5]	P. Bladon, E. Terentjev and M. Warner, Soft elasticity- deformation without resistance in liquid crystal elastomers, Journal de Physique II, 4 (1993), 93-102.
[6]	P. Bladon, E. Terentjev and M. Warner, Deformation-induced orientational transitions in liquid crystal elastomer, J. Phys. II France, 4 (1994), 75-91.
[7]	M. C. Calderer and C. Luo, Numerical study of liquid crystal elastomers in a mixed finite element method, European J. Appl. Math, 23 (2012), 121-154.doi: 10.1017/S0956792511000313.
[8]	P. Cesana and A. DeSimone, Strain-order coupling in nematic elastomers: Equilibrium configurations, Math. Models Methods Appl. Sci., 19 (2009), 601-630.doi: 10.1142/S0218202509003541.
[9]	p. Cesana and A. DeSimone, Quasiconvex envelopes of energies for the nematic elastomers in the small strain regime and applications, J. Mech. Phys., 59 (2011), 787-803.doi: 10.1016/j.jmps.2011.01.007.
[10]	S. Conti, A. DeSimone and G. Dolzmann, Semisoft elasticity and director reorientation in stretched sheets of nematic elastomers, Physical Review E., 66 (2002), 061710.
[11]	S. Conti, A. DeSimone and G. Dolzmann, Soft elastic response of stretched sheets of nematic elastomers: A numerical study, Journal of Mechanics and Physics of Solids, 50 (2002), 1431-1451.doi: 10.1016/S0022-5096(01)00120-X.
[12]	A. DeSimone and G. Dolzmann, Macroscopic response of nematic elastomers via relaxation of a class of SO(3)-invariant energies, Arch. Rational Mech. Anal., 161 (2002), 181-204.doi: 10.1007/s002050100174.
[13]	A. DeSimone and G. Dolzmann, Material instabilities in nematic elastomers, Phys. D, 136 (2000), 175-191.doi: 10.1016/S0167-2789(99)00153-0.
[14]	A. DeSimone and G. Dolzmann, Stripe-domains in nematic elastomers: Old and new, in Modeling of Soft Matter, IMA Vol. Math. Appl., 141, Springer, New York, 2005, 189-203.doi: 10.1007/0-387-32153-5_8.
[15]	H. Finkelmann, I. Kundler, E. M. Terejtev and M. Warner, Critical Stripe-Domain Instability of Nematic Elastomers, J. Phys. II France, (1997), 1059-1069.
[16]	I. Kundler and H. Finkelmann, Strain-induced director reorientation in nematic liquid single crystal elastomers, Macromolecular Rapid Communications, 16 (1995), 679-686.
[17]	S. Müller, T. Qi and B. S. Yan, On a new class of elastic deformations not allowing for cavitation, Annales de l'I.H.P., Section C, 11 (1994), 217-243.
[18]	V. Sverak, Regularity properties of deformations with finite energy, Arch. Rat. Mech. Anal., 100 (1988), 105-127.doi: 10.1007/BF00282200.
[19]	Terentjev, M. Warner and G. C. Verwey, Non-uniform deformations in liquid crystal elastomers, J. Phys. II France, 6 (1996), 1049-1060.
[20]	M. Verwey, M. Warner and E. M. Terenjtev, Elastic instability and stripe domains in liquid crystalline elastomers, J. Phys. II France, 6 (1996), 1273-1290.
[21]	S. K. Vodopyanov and V. M. Goldstein, Quasiconformal mappings and spaces of functions with generalized first derivatives, Siberian math. J., 12 (1977), 515-531.
[22]	M. Warner and E. M. Terenjtev, Liquid Crystal Elastomers, Oxford University Press, 2003.
[23]	E. Zubarev, S. Kuptsov, T. Yuranova, R. Talroze and H. Finkelmann, Monodomain liquid crystalline networks: Reorientation mechanism from uniform to stripe domains, Liquid Crystals, 26 (1999), 1531-1540.