American Institute of Mathematical Sciences

Advanced
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

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 & Continuous Dynamical Systems - S, 2015, 8 (2) : 259-282. doi: 10.3934/dcdss.2015.8.259
References:
[1]

Arch. for Rat. Mech. and Anal., 86 (1984), 125-145. doi: 10.1007/BF00275731.  Google Scholar

[2]

J. Elasticity, 56 (1999), 33-58. doi: 10.1023/A:1007647913363.  Google Scholar

[3]

J. Functional Analysis, 58 (1984), 225-253. doi: 10.1016/0022-1236(84)90041-7.  Google Scholar

[4]

Phys. Rev. E, (Rapid Comm.), 47 (1993), 3838. Google Scholar

[5]

Journal de Physique II, 4 (1993), 93-102. Google Scholar

[6]

J. Phys. II France, 4 (1994), 75-91. Google Scholar

[7]

European J. Appl. Math, 23 (2012), 121-154. doi: 10.1017/S0956792511000313.  Google Scholar

[8]

Math. Models Methods Appl. Sci., 19 (2009), 601-630. doi: 10.1142/S0218202509003541.  Google Scholar

[9]

J. Mech. Phys., 59 (2011), 787-803. doi: 10.1016/j.jmps.2011.01.007.  Google Scholar

[10]

Physical Review E., 66 (2002), 061710. Google Scholar

[11]

Journal of Mechanics and Physics of Solids, 50 (2002), 1431-1451. doi: 10.1016/S0022-5096(01)00120-X.  Google Scholar

[12]

Arch. Rational Mech. Anal., 161 (2002), 181-204. doi: 10.1007/s002050100174.  Google Scholar

[13]

Phys. D, 136 (2000), 175-191. doi: 10.1016/S0167-2789(99)00153-0.  Google Scholar

[14]

in Modeling of Soft Matter, IMA Vol. Math. Appl., 141, Springer, New York, 2005, 189-203. doi: 10.1007/0-387-32153-5_8.  Google Scholar

[15]

J. Phys. II France, (1997), 1059-1069. Google Scholar

[16]

Macromolecular Rapid Communications, 16 (1995), 679-686. Google Scholar

[17]

Annales de l'I.H.P., Section C, 11 (1994), 217-243.  Google Scholar

[18]

Arch. Rat. Mech. Anal., 100 (1988), 105-127. doi: 10.1007/BF00282200.  Google Scholar

[19]

J. Phys. II France, 6 (1996), 1049-1060. Google Scholar

[20]

J. Phys. II France, 6 (1996), 1273-1290. Google Scholar

[21]

Siberian math. J., 12 (1977), 515-531. Google Scholar

[22]

Oxford University Press, 2003. Google Scholar

[23]

Liquid Crystals, 26 (1999), 1531-1540. Google Scholar

show all references

References:
[1]

Arch. for Rat. Mech. and Anal., 86 (1984), 125-145. doi: 10.1007/BF00275731.  Google Scholar

[2]

J. Elasticity, 56 (1999), 33-58. doi: 10.1023/A:1007647913363.  Google Scholar

[3]

J. Functional Analysis, 58 (1984), 225-253. doi: 10.1016/0022-1236(84)90041-7.  Google Scholar

[4]

Phys. Rev. E, (Rapid Comm.), 47 (1993), 3838. Google Scholar

[5]

Journal de Physique II, 4 (1993), 93-102. Google Scholar

[6]

J. Phys. II France, 4 (1994), 75-91. Google Scholar

[7]

European J. Appl. Math, 23 (2012), 121-154. doi: 10.1017/S0956792511000313.  Google Scholar

[8]

Math. Models Methods Appl. Sci., 19 (2009), 601-630. doi: 10.1142/S0218202509003541.  Google Scholar

[9]

J. Mech. Phys., 59 (2011), 787-803. doi: 10.1016/j.jmps.2011.01.007.  Google Scholar

[10]

Physical Review E., 66 (2002), 061710. Google Scholar

[11]

Journal of Mechanics and Physics of Solids, 50 (2002), 1431-1451. doi: 10.1016/S0022-5096(01)00120-X.  Google Scholar

[12]

Arch. Rational Mech. Anal., 161 (2002), 181-204. doi: 10.1007/s002050100174.  Google Scholar

[13]

Phys. D, 136 (2000), 175-191. doi: 10.1016/S0167-2789(99)00153-0.  Google Scholar

[14]

in Modeling of Soft Matter, IMA Vol. Math. Appl., 141, Springer, New York, 2005, 189-203. doi: 10.1007/0-387-32153-5_8.  Google Scholar

[15]

J. Phys. II France, (1997), 1059-1069. Google Scholar

[16]

Macromolecular Rapid Communications, 16 (1995), 679-686. Google Scholar

[17]

Annales de l'I.H.P., Section C, 11 (1994), 217-243.  Google Scholar

[18]

Arch. Rat. Mech. Anal., 100 (1988), 105-127. doi: 10.1007/BF00282200.  Google Scholar

[19]

J. Phys. II France, 6 (1996), 1049-1060. Google Scholar

[20]

J. Phys. II France, 6 (1996), 1273-1290. Google Scholar

[21]

Siberian math. J., 12 (1977), 515-531. Google Scholar

[22]

Oxford University Press, 2003. Google Scholar

[23]

Liquid Crystals, 26 (1999), 1531-1540. Google Scholar

[1]

Luigi C. Berselli, Jishan Fan. Logarithmic and improved regularity criteria for the 3D nematic liquid crystals models, Boussinesq system, and MHD equations in a bounded domain. Communications on Pure & Applied Analysis, 2015, 14 (2) : 637-655. doi: 10.3934/cpaa.2015.14.637
[2]

Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Existence results and stability analysis for a nonlinear fractional boundary value problem on a circular ring with an attached edge : A study of fractional calculus on metric graph. Networks & Heterogeneous Media, 2021, 16 (2) : 155-185. doi: 10.3934/nhm.2021003
[3]

Frank Sottile. The special Schubert calculus is real. Electronic Research Announcements, 1999, 5: 35-39.

[4]

Giovanni Cimatti. Forced periodic solutions for piezoelectric crystals. Communications on Pure & Applied Analysis, 2005, 4 (2) : 475-485. doi: 10.3934/cpaa.2005.4.475
[5]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2739-2776. doi: 10.3934/dcds.2020384
[6]

Hala Ghazi, François James, Hélène Mathis. A nonisothermal thermodynamical model of liquid-vapor interaction with metastability. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2371-2409. doi: 10.3934/dcdsb.2020183
[7]

Huy Dinh, Harbir Antil, Yanlai Chen, Elena Cherkaev, Akil Narayan. Model reduction for fractional elliptic problems using Kato's formula. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021004
[8]

Changjun Yu, Lei Yuan, Shuxuan Su. A new gradient computational formula for optimal control problems with time-delay. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021076
[9]

Wei Liu, Pavel Krejčí, Guoju Ye. Continuity properties of Prandtl-Ishlinskii operators in the space of regulated functions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3783-3795. doi: 10.3934/dcdsb.2017190
[10]

Rama Ayoub, Aziz Hamdouni, Dina Razafindralandy. A new Hodge operator in discrete exterior calculus. Application to fluid mechanics. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021062
[11]

Mengjie Zhang. Extremal functions for a class of trace Trudinger-Moser inequalities on a compact Riemann surface with smooth boundary. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021038
[12]

Qian Liu. The lower bounds on the second-order nonlinearity of three classes of Boolean functions. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020136
[13]

Alessandro Fonda, Rodica Toader. A dynamical approach to lower and upper solutions for planar systems "To the memory of Massimo Tarallo". Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3683-3708. doi: 10.3934/dcds.2021012
[14]

Matheus C. Bortolan, José Manuel Uzal. Upper and weak-lower semicontinuity of pullback attractors to impulsive evolution processes. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3667-3692. doi: 10.3934/dcdsb.2020252
[15]

Yumi Yahagi. Construction of unique mild solution and continuity of solution for the small initial data to 1-D Keller-Segel system. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021099
[16]

Masahiro Ikeda, Ziheng Tu, Kyouhei Wakasa. Small data blow-up of semi-linear wave equation with scattering dissipation and time-dependent mass. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021011
[17]

Hao Li, Honglin Chen, Matt Haberland, Andrea L. Bertozzi, P. Jeffrey Brantingham. PDEs on graphs for semi-supervised learning applied to first-person activity recognition in body-worn video. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021039
[18]

Linyao Ge, Baoxiang Huang, Weibo Wei, Zhenkuan Pan. Semi-Supervised classification of hyperspectral images using discrete nonlocal variation Potts Model. Mathematical Foundations of Computing, 2021  doi: 10.3934/mfc.2021003
[19]

Yangrong Li, Fengling Wang, Shuang Yang. Part-convergent cocycles and semi-convergent attractors of stochastic 2D-Ginzburg-Landau delay equations toward zero-memory. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3643-3665. doi: 10.3934/dcdsb.2020250
[20]

Haiyan Wang. Existence and nonexistence of positive radial solutions for quasilinear systems. Conference Publications, 2009, 2009 (Special) : 810-817. doi: 10.3934/proc.2009.2009.810

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (38)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences