# American Institute of Mathematical Sciences

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
##### References:
 [1] E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations,, Arch. for Rat. Mech. and Anal., 86 (1984), 125. doi: 10.1007/BF00275731. Google Scholar [2] D. Anderson, D. Carlson and E. Fried, A continuum-mechanical theory for nematic elastomers,, J. Elasticity, 56 (1999), 33. doi: 10.1023/A:1007647913363. Google Scholar [3] J. M. Ball and F. Murat, $W^{1,p}$-quasiconvexity and variational problems for multiple integrals,, J. Functional Analysis, 58 (1984), 225. doi: 10.1016/0022-1236(84)90041-7. Google Scholar [4] P. Bladon, E. Terentjev and M. Warner, Transitions and Instabilities in liquid-crystal elastomers,, Phys. Rev. E, 47 (1993). Google Scholar [5] P. Bladon, E. Terentjev and M. Warner, Soft elasticity- deformation without resistance in liquid crystal elastomers,, Journal de Physique II, 4 (1993), 93. Google Scholar [6] P. Bladon, E. Terentjev and M. Warner, Deformation-induced orientational transitions in liquid crystal elastomer,, J. Phys. II France, 4 (1994), 75. Google Scholar [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. doi: 10.1017/S0956792511000313. Google Scholar [8] P. Cesana and A. DeSimone, Strain-order coupling in nematic elastomers: Equilibrium configurations,, Math. Models Methods Appl. Sci., 19 (2009), 601. doi: 10.1142/S0218202509003541. Google Scholar [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. doi: 10.1016/j.jmps.2011.01.007. Google Scholar [10] S. Conti, A. DeSimone and G. Dolzmann, Semisoft elasticity and director reorientation in stretched sheets of nematic elastomers,, Physical Review E., 66 (2002). Google Scholar [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. doi: 10.1016/S0022-5096(01)00120-X. Google Scholar [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. doi: 10.1007/s002050100174. Google Scholar [13] A. DeSimone and G. Dolzmann, Material instabilities in nematic elastomers,, Phys. D, 136 (2000), 175. doi: 10.1016/S0167-2789(99)00153-0. Google Scholar [14] A. DeSimone and G. Dolzmann, Stripe-domains in nematic elastomers: Old and new,, in Modeling of Soft Matter, (2005), 189. doi: 10.1007/0-387-32153-5_8. Google Scholar [15] H. Finkelmann, I. Kundler, E. M. Terejtev and M. Warner, Critical Stripe-Domain Instability of Nematic Elastomers,, J. Phys. II France, (1997), 1059. Google Scholar [16] I. Kundler and H. Finkelmann, Strain-induced director reorientation in nematic liquid single crystal elastomers,, Macromolecular Rapid Communications, 16 (1995), 679. Google Scholar [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., 11 (1994), 217. Google Scholar [18] V. Sverak, Regularity properties of deformations with finite energy,, Arch. Rat. Mech. Anal., 100 (1988), 105. doi: 10.1007/BF00282200. Google Scholar [19] Terentjev, M. Warner and G. C. Verwey, Non-uniform deformations in liquid crystal elastomers,, J. Phys. II France, 6 (1996), 1049. Google Scholar [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. Google Scholar [21] S. K. Vodopyanov and V. M. Goldstein, Quasiconformal mappings and spaces of functions with generalized first derivatives,, Siberian math. J., 12 (1977), 515. Google Scholar [22] M. Warner and E. M. Terenjtev, Liquid Crystal Elastomers,, Oxford University Press, (2003). Google Scholar [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. Google Scholar

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. doi: 10.1007/BF00275731. Google Scholar [2] D. Anderson, D. Carlson and E. Fried, A continuum-mechanical theory for nematic elastomers,, J. Elasticity, 56 (1999), 33. doi: 10.1023/A:1007647913363. Google Scholar [3] J. M. Ball and F. Murat, $W^{1,p}$-quasiconvexity and variational problems for multiple integrals,, J. Functional Analysis, 58 (1984), 225. doi: 10.1016/0022-1236(84)90041-7. Google Scholar [4] P. Bladon, E. Terentjev and M. Warner, Transitions and Instabilities in liquid-crystal elastomers,, Phys. Rev. E, 47 (1993). Google Scholar [5] P. Bladon, E. Terentjev and M. Warner, Soft elasticity- deformation without resistance in liquid crystal elastomers,, Journal de Physique II, 4 (1993), 93. Google Scholar [6] P. Bladon, E. Terentjev and M. Warner, Deformation-induced orientational transitions in liquid crystal elastomer,, J. Phys. II France, 4 (1994), 75. Google Scholar [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. doi: 10.1017/S0956792511000313. Google Scholar [8] P. Cesana and A. DeSimone, Strain-order coupling in nematic elastomers: Equilibrium configurations,, Math. Models Methods Appl. Sci., 19 (2009), 601. doi: 10.1142/S0218202509003541. Google Scholar [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. doi: 10.1016/j.jmps.2011.01.007. Google Scholar [10] S. Conti, A. DeSimone and G. Dolzmann, Semisoft elasticity and director reorientation in stretched sheets of nematic elastomers,, Physical Review E., 66 (2002). Google Scholar [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. doi: 10.1016/S0022-5096(01)00120-X. Google Scholar [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. doi: 10.1007/s002050100174. Google Scholar [13] A. DeSimone and G. Dolzmann, Material instabilities in nematic elastomers,, Phys. D, 136 (2000), 175. doi: 10.1016/S0167-2789(99)00153-0. Google Scholar [14] A. DeSimone and G. Dolzmann, Stripe-domains in nematic elastomers: Old and new,, in Modeling of Soft Matter, (2005), 189. doi: 10.1007/0-387-32153-5_8. Google Scholar [15] H. Finkelmann, I. Kundler, E. M. Terejtev and M. Warner, Critical Stripe-Domain Instability of Nematic Elastomers,, J. Phys. II France, (1997), 1059. Google Scholar [16] I. Kundler and H. Finkelmann, Strain-induced director reorientation in nematic liquid single crystal elastomers,, Macromolecular Rapid Communications, 16 (1995), 679. Google Scholar [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., 11 (1994), 217. Google Scholar [18] V. Sverak, Regularity properties of deformations with finite energy,, Arch. Rat. Mech. Anal., 100 (1988), 105. doi: 10.1007/BF00282200. Google Scholar [19] Terentjev, M. Warner and G. C. Verwey, Non-uniform deformations in liquid crystal elastomers,, J. Phys. II France, 6 (1996), 1049. Google Scholar [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. Google Scholar [21] S. K. Vodopyanov and V. M. Goldstein, Quasiconformal mappings and spaces of functions with generalized first derivatives,, Siberian math. J., 12 (1977), 515. Google Scholar [22] M. Warner and E. M. Terenjtev, Liquid Crystal Elastomers,, Oxford University Press, (2003). Google Scholar [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. Google Scholar
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