Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-26T11:11:48.211Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical study of liquid crystal elastomers by a mixed finite element method

Published online by Cambridge University Press:  22 August 2011

C. LUO  and
M. C. CALDERER
C. LUO
Affiliation:
Oxford Center for Collaborative Applied Mathematics, Mathematical Institute, University of Oxford, Oxford OX1 3LB, UK email: [email protected]
M. C. CALDERER
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Liquid crystal elastomers present features not found in ordinary elastic materials, such as semi-soft elasticity and the related stripe domain phenomenon. In this paper, the two-dimensional Bladon–Terentjev–Warner model and the one-constant Oseen–Frank energy expression are combined to study the liquid crystal elastomer. We also impose two material constraints, the incompressibility of the elastomer and the unit director norm of the liquid crystal. We prove existence of minimiser of the energy for the proposed model. Next we formulate the discrete model, and also prove that it possesses a minimiser of the energy. The inf-sup values of the discrete linearised system are then related to the smallest singular values of certain matrices. Next the existence and uniqueness of the Lagrange multipliers associated with the two material constraints are proved under the assumption that the inf-sup conditions hold. Finally numerical simulations of the clamped-pulling experiment are presented for elastomer samples with aspect ratio 1 or 3. The semi-soft elasticity is successfully recovered in both cases. The stripe domain phenomenon, however, is not observed, which might be due to the relative coarse mesh employed in the numerical experiment. Possible improvements are discussed that might lead to the recovery of the stripe domain phenomenon.

Keywords

Liquid crystal elastomerSemi-soft elasticityVariational methodsMixed finite element methodInf-sup condition
Type
Papers
Information
European Journal of Applied Mathematics , Volume 23 , Issue 1: Liquid Crystals , February 2012 , pp. 121 - 154
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Adams, R. & Fournier, J. (1975) Sobolev Spaces, Vol. 65, Academic Press, New York.Google Scholar
[2]Ball, J. (1976) Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63 (4), 337–403.CrossRefGoogle Scholar
[3]Bladon, P., Terentjev, E. & Warner, M. (1993) Transitions and instabilities in liquid crystal elastomers. Phys. Rev. E 47 (6), 3838–3840.CrossRefGoogle ScholarPubMed
[4]Bochkanov, S. & Bystritsky, V. ALGLIB [online]. URL: http://www.alglib.net/Google Scholar
[5]Brezzi, F. & Fortin, M. (1991) Mixed and Hybrid Finite Element Methods, Springer, Berlin.CrossRefGoogle Scholar
[6]Calderer, M., Liu, C. & Yan, B. (2006) A mathematical theory for nematic elastomers with non-uniform prolate spheroids. Advances in applied and computational mathematics, 245–259.Google Scholar
[7]Cesana, P. & DeSimone, A. (2009) Strain-order coupling in nematic elastomers: equilibrium configurations. Math. Models Methods Appl. Sci. 19, 601–630.CrossRefGoogle Scholar
[8]Chapelle, D. & Bathe, K. (1993) The inf-sup test. Comput. Struct. 47, 537–537.CrossRefGoogle Scholar
[9]Ciarlet, P. (1988) Mathematical Elasticity, Vol 1, North-Holland, The Netherlands.Google Scholar
[10]Clarke, F. (1975) Generalized gradients and applications. Trans. Am. Math. Soc. 205, 247–262.CrossRefGoogle Scholar
[11]Clarke, F. (1976) A new approach to Lagrange multipliers. Math. Oper. Res. 1 (2), 165–174.CrossRefGoogle Scholar
[12]Clarke, S., Hotta, A., Tajbakhsh, A. & Terentjev, E. (2001) Effect of crosslinker geometry on equilibrium thermal and mechanical properties of nematic elastomers. Phys. Rev. E 64 (6), 61702.CrossRefGoogle ScholarPubMed
[13]Conti, S., DeSimone, A. & Dolzmann, G. (2002) Semisoft elasticity and director reorientation in stretched sheets of nematic elastomers. Phys. Rev. E 66 (6), 061710.CrossRefGoogle ScholarPubMed
[14]Conti, S., DeSimone, A. & Dolzmann, G. (2002) Soft elastic response of stretched sheets of nematic elastomers: aA numerical study. J. the Mech. Phys. Solids 50 (7), 1431–1451.CrossRefGoogle Scholar
[15]de Gennes, P. & Prost, J. (1995) The Physics of Liquid Crystals, Oxford University Press, USA.Google Scholar
[16]DeSimone, A. (1999) Energetics of fine domain structures. Ferroelectrics 222 (1–4), 533–542.CrossRefGoogle Scholar
[17]DeSimone, A. & Dolzmann, G. (2000) Material instabilities in nematic elastomers. Phys. D: Nonlinear Phenom. 136 (1–2), 175–191.CrossRefGoogle Scholar
[18]DeSimone, A. & Teresi, L. (2009) Elastic energies for nematic elastomers. Eur. Phys. J. E: Soft Matter and Biol. Phys. 29 (2), 191–204.CrossRefGoogle ScholarPubMed
[19]Ericksen, J. (1991) Liquid crystals with variable degree of orientation. Arch. Ration. Mech. Anal. 113 (2), 97–120.CrossRefGoogle Scholar
[20]Evans, L. (1998) Partial Differential Equations. American Mathematical Society, USA.Google Scholar
[21]Frank, F. (1958) I. Liquid crystals. On the theory of liquid crystals. Discuss. Faraday Soc. 25, 19–28.CrossRefGoogle Scholar
[22]Hardt, R., Kinderlehrer, D. & Lin, F. (1986) Existence and partial regularity of static liquid crystal configurations. Commun. Math. Phys. 105 (4), 547–570.CrossRefGoogle Scholar
[23]Hébert, M., Kant, R. & De Gennes, P. (1997) Dynamics and thermodynamics of artificial muscles based on nematic gels. J. Physique 7 (7), 909–919.Google Scholar
[24]Hu, Q., Tai, X. & Winther, R. (2009) A saddle point approach to the computation of harmonic maps. SIAM J. Numer. Anal. 47 (2), 1500–1523.CrossRefGoogle Scholar
[25]Kundler, I. & Finkelmann, H. (1995) Strain-induced director reorientation in nematic liquid single crystal elastomers. Macromol. Rapid Commun. 16 (9), 679–686.CrossRefGoogle Scholar
[26]Küpfer, J. & Finkelmann, H. (1994) Liquid crystal elastomers: Influence of the orientational distribution of the crosslinks on the phase behaviour and reorientation processes. Macromol. Chem. Phys. 195 (4), 1353–1367.CrossRefGoogle Scholar
[27]Le Tallec, P. (1981) Compatibility condition and existence results in discrete finite incompressible elasticity. Comput. Methods Appl. Mech. Eng. 27 (2), 239–259.CrossRefGoogle Scholar
[28]Logg, A & Wells, G. (2010) DOLFIN: Automated finite element computing. ACM Trans. Math. Softw. 37 (2), 1–28. URL: https://launchpad.net/fenicsCrossRefGoogle Scholar
[29]Luo, C. (2010) Modeling, Analysis and Numerical Simulation of Liquid Crystal Elastomer, PhD Thesis, University of Minnesota, USA.Google Scholar
[30]Majumdar, A. & Zarnescu, A. (2010) Landau–De Gennes theory of nematic liquid crystals: The Oseen–Frank limit and beyond. Arch. Ration. Mech. Anal. 196 (1), 227–280.CrossRefGoogle Scholar
[31]Rudin, W. (1991) Functional analysis. In: International Series in Pure and Applied Mathematics, McGraw-Hill, USA.Google Scholar
[32]Tallec, P. L. (1994) Numerical methods for nonlinear three-dimensional elasticity. In: Handbook of Numerical Analysis, Vol. 3, pp. 465–622. North-Holland, The Netherlands.Google Scholar
[33]Verwey, G., Warner, M. & Terentjev, E. (1996) Elastic instability and stripe domains in liquid crystalline elastomers. J. Phys. II France 6, 1273–1290.Google Scholar
[34]Warner, M. & Terentjev, E. (2007) Liquid Crystal Elastomers, Oxford University Press, USA.Google Scholar
[35]Zubarev, E., Kuptsov, S., Yuranova, T., Talroze, R. & Finkelmann, H. (1999) Monodomain liquid crystalline networks: Reorientation mechanism from uniform to stripe domains. Liq. Cryst. 26 (10), 1531–1540.CrossRefGoogle Scholar