-
Previous Article
A Landau--de Gennes theory of liquid crystal elastomers
- DCDS-S Home
- This Issue
-
Next Article
Existence of solutions to boundary value problems for smectic liquid crystals
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 |
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. |
[1] |
Mauro Fabrizio, Claudio Giorgi, Angelo Morro. Isotropic-nematic phase transitions in liquid crystals. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 565-579. doi: 10.3934/dcdss.2011.4.565 |
[2] |
Xiaoming Wang. Upper semi-continuity of stationary statistical properties of dissipative systems. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 521-540. doi: 10.3934/dcds.2009.23.521 |
[3] |
Marco Cicalese, Antonio DeSimone, Caterina Ida Zeppieri. Discrete-to-continuum limits for strain-alignment-coupled systems: Magnetostrictive solids, ferroelectric crystals and nematic elastomers. Networks and Heterogeneous Media, 2009, 4 (4) : 667-708. doi: 10.3934/nhm.2009.4.667 |
[4] |
M. Silhavý. Ideally soft nematic elastomers. Networks and Heterogeneous Media, 2007, 2 (2) : 279-311. doi: 10.3934/nhm.2007.2.279 |
[5] |
Boling Guo, Yongqian Han, Guoli Zhou. Random attractor for the 2D stochastic nematic liquid crystals flows. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2349-2376. doi: 10.3934/cpaa.2019106 |
[6] |
Geng Chen, Ping Zhang, Yuxi Zheng. Energy conservative solutions to a nonlinear wave system of nematic liquid crystals. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1445-1468. doi: 10.3934/cpaa.2013.12.1445 |
[7] |
Zdzisław Brzeźniak, Erika Hausenblas, Paul André Razafimandimby. A note on the stochastic Ericksen-Leslie equations for nematic liquid crystals. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 5785-5802. doi: 10.3934/dcdsb.2019106 |
[8] |
Tomás Caraballo, Cecilia Cavaterra. A 3D isothermal model for nematic liquid crystals with delay terms. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022097 |
[9] |
Pengyu Chen, Xuping Zhang. Upper semi-continuity of attractors for non-autonomous fractional stochastic parabolic equations with delay. Discrete and Continuous Dynamical Systems - B, 2021, 26 (8) : 4325-4357. doi: 10.3934/dcdsb.2020290 |
[10] |
Apala Majumdar. The Landau-de Gennes theory of nematic liquid crystals: Uniaxiality versus Biaxiality. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1303-1337. doi: 10.3934/cpaa.2012.11.1303 |
[11] |
Jihong Zhao, Qiao Liu, Shangbin Cui. Global existence and stability for a hydrodynamic system in the nematic liquid crystal flows. Communications on Pure and Applied Analysis, 2013, 12 (1) : 341-357. doi: 10.3934/cpaa.2013.12.341 |
[12] |
Chun Liu. Dynamic theory for incompressible Smectic-A liquid crystals: Existence and regularity. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 591-608. doi: 10.3934/dcds.2000.6.591 |
[13] |
Patricia Bauman, Daniel Phillips, Jinhae Park. Existence of solutions to boundary value problems for smectic liquid crystals. Discrete and Continuous Dynamical Systems - S, 2015, 8 (2) : 243-257. doi: 10.3934/dcdss.2015.8.243 |
[14] |
Ioan Bucataru, Matias F. Dahl. Semi-basic 1-forms and Helmholtz conditions for the inverse problem of the calculus of variations. Journal of Geometric Mechanics, 2009, 1 (2) : 159-180. doi: 10.3934/jgm.2009.1.159 |
[15] |
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 and Applied Analysis, 2015, 14 (2) : 637-655. doi: 10.3934/cpaa.2015.14.637 |
[16] |
Elisa Gorla, Maike Massierer. Index calculus in the trace zero variety. Advances in Mathematics of Communications, 2015, 9 (4) : 515-539. doi: 10.3934/amc.2015.9.515 |
[17] |
Bernard Dacorogna, Giovanni Pisante, Ana Margarida Ribeiro. On non quasiconvex problems of the calculus of variations. Discrete and Continuous Dynamical Systems, 2005, 13 (4) : 961-983. doi: 10.3934/dcds.2005.13.961 |
[18] |
Daniel Faraco, Jan Kristensen. Compactness versus regularity in the calculus of variations. Discrete and Continuous Dynamical Systems - B, 2012, 17 (2) : 473-485. doi: 10.3934/dcdsb.2012.17.473 |
[19] |
Xiaohui Zhang, Xuping Zhang. Upper semi-continuity of non-autonomous fractional stochastic $ p $-Laplacian equation driven by additive noise on $ \mathbb{R}^n $. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022081 |
[20] |
Yi-hang Hao, Xian-Gao Liu. The existence and blow-up criterion of liquid crystals system in critical Besov space. Communications on Pure and Applied Analysis, 2014, 13 (1) : 225-236. doi: 10.3934/cpaa.2014.13.225 |
2020 Impact Factor: 2.425
Tools
Metrics
Other articles
by authors
[Back to Top]