Energy-minimizing nematic elastomers

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

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, United States

Received May 2013 Revised November 2013 Published July 2014

Abstract
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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.

Keywords: Nematic elastomers, trace formula., lower semi-continuity, calculus of variations, liquid crystals, existence.

Mathematics Subject Classification: Primary: 35J57; Secondary: 35J50, 49J3.

Citation: Patricia Bauman, Andrea C. Rubiano. Energy-minimizing nematic elastomers. Discrete and Continuous Dynamical Systems - S, 2015, 8 (2) : 259-282. doi: 10.3934/dcdss.2015.8.259

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.

show all references

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.

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