Skip to main content
    • Aa
    • Aa

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

  • C. LUO (a1) and M. C. CALDERER (a2)

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.

Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[2] J. Ball (1976) Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63 (4), 337403.

[3] P. Bladon , E. Terentjev & M. Warner (1993) Transitions and instabilities in liquid crystal elastomers. Phys. Rev. E 47 (6), 38383840.

[5] F. Brezzi & M. Fortin (1991) Mixed and Hybrid Finite Element Methods, Springer, Berlin.

[7] P. Cesana & A. DeSimone (2009) Strain-order coupling in nematic elastomers: equilibrium configurations. Math. Models Methods Appl. Sci. 19, 601630.

[8] D. Chapelle & K. Bathe (1993) The inf-sup test. Comput. Struct. 47, 537537.

[10] F. Clarke (1975) Generalized gradients and applications. Trans. Am. Math. Soc. 205, 247262.

[11] F. Clarke (1976) A new approach to Lagrange multipliers. Math. Oper. Res. 1 (2), 165174.

[12] S. Clarke , A. Hotta , A. Tajbakhsh & E. Terentjev (2001) Effect of crosslinker geometry on equilibrium thermal and mechanical properties of nematic elastomers. Phys. Rev. E 64 (6), 61702.

[13] S. Conti , A. DeSimone & G. Dolzmann (2002) Semisoft elasticity and director reorientation in stretched sheets of nematic elastomers. Phys. Rev. E 66 (6), 061710.

[14] S. Conti , A. DeSimone & G. Dolzmann (2002) Soft elastic response of stretched sheets of nematic elastomers: aA numerical study. J. the Mech. Phys. Solids 50 (7), 14311451.

[16] A. DeSimone (1999) Energetics of fine domain structures. Ferroelectrics 222 (1–4), 533542.

[17] A. DeSimone & G. Dolzmann (2000) Material instabilities in nematic elastomers. Phys. D: Nonlinear Phenom. 136 (1–2), 175191.

[18] A. DeSimone & L. Teresi (2009) Elastic energies for nematic elastomers. Eur. Phys. J. E: Soft Matter and Biol. Phys. 29 (2), 191204.

[19] J. Ericksen (1991) Liquid crystals with variable degree of orientation. Arch. Ration. Mech. Anal. 113 (2), 97120.

[21] F. Frank (1958) I. Liquid crystals. On the theory of liquid crystals. Discuss. Faraday Soc. 25, 1928.

[22] R. Hardt , D. Kinderlehrer & F. Lin (1986) Existence and partial regularity of static liquid crystal configurations. Commun. Math. Phys. 105 (4), 547570.

[24] Q. Hu , X. Tai & R. Winther (2009) A saddle point approach to the computation of harmonic maps. SIAM J. Numer. Anal. 47 (2), 15001523.

[25] I. Kundler & H. Finkelmann (1995) Strain-induced director reorientation in nematic liquid single crystal elastomers. Macromol. Rapid Commun. 16 (9), 679686.

[26] J. Küpfer & H. Finkelmann (1994) Liquid crystal elastomers: Influence of the orientational distribution of the crosslinks on the phase behaviour and reorientation processes. Macromol. Chem. Phys. 195 (4), 13531367.

[27] P. Le Tallec (1981) Compatibility condition and existence results in discrete finite incompressible elasticity. Comput. Methods Appl. Mech. Eng. 27 (2), 239259.

[28] A Logg & G. Wells (2010) DOLFIN: Automated finite element computing. ACM Trans. Math. Softw. 37 (2), 128. URL:

[30] A. Majumdar & A. Zarnescu (2010) Landau–De Gennes theory of nematic liquid crystals: The Oseen–Frank limit and beyond. Arch. Ration. Mech. Anal. 196 (1), 227280.

[33] G. Verwey , M. Warner & E. Terentjev (1996) Elastic instability and stripe domains in liquid crystalline elastomers. J. Phys. II France 6, 12731290.

[35] E. Zubarev , S. Kuptsov , T. Yuranova , R. Talroze & H. Finkelmann (1999) Monodomain liquid crystalline networks: Reorientation mechanism from uniform to stripe domains. Liq. Cryst. 26 (10), 15311540.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

European Journal of Applied Mathematics
  • ISSN: 0956-7925
  • EISSN: 1469-4425
  • URL: /core/journals/european-journal-of-applied-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 76 *
Loading metrics...

Abstract views

Total abstract views: 109 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 29th May 2017. This data will be updated every 24 hours.