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

Handle URI:
http://hdl.handle.net/10754/599023
Title:
Numerical study of liquid crystal elastomers by a mixed finite element method
Authors:
LUO, C.; CALDERER, M. C.
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. © Copyright Cambridge University Press 2011.
Citation:
LUO C, CALDERER MC (2011) Numerical study of liquid crystal elastomers by a mixed finite element method. European Journal of Applied Mathematics 23: 121–154. Available: http://dx.doi.org/10.1017/S0956792511000313.
Publisher:
Cambridge University Press (CUP)
Journal:
European Journal of Applied Mathematics
KAUST Grant Number:
KUK-C1-013-04
Issue Date:
22-Aug-2011
DOI:
10.1017/S0956792511000313
Type:
Article
ISSN:
0956-7925; 1469-4425
Sponsors:
This publication was based on work supported in part by Award No. KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST). This publication was partially supported by the National Science Foundation, Grant numbers: DMS-FRG-0456232 and DMS 1009181.
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Publications Acknowledging KAUST Support

Full metadata record

DC FieldValue Language
dc.contributor.authorLUO, C.en
dc.contributor.authorCALDERER, M. C.en
dc.date.accessioned2016-02-25T13:51:22Zen
dc.date.available2016-02-25T13:51:22Zen
dc.date.issued2011-08-22en
dc.identifier.citationLUO C, CALDERER MC (2011) Numerical study of liquid crystal elastomers by a mixed finite element method. European Journal of Applied Mathematics 23: 121–154. Available: http://dx.doi.org/10.1017/S0956792511000313.en
dc.identifier.issn0956-7925en
dc.identifier.issn1469-4425en
dc.identifier.doi10.1017/S0956792511000313en
dc.identifier.urihttp://hdl.handle.net/10754/599023en
dc.description.abstractLiquid 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. © Copyright Cambridge University Press 2011.en
dc.description.sponsorshipThis publication was based on work supported in part by Award No. KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST). This publication was partially supported by the National Science Foundation, Grant numbers: DMS-FRG-0456232 and DMS 1009181.en
dc.publisherCambridge University Press (CUP)en
dc.subjectInf-sup conditionen
dc.subjectLiquid crystal elastomeren
dc.subjectMixed finite element methoden
dc.subjectSemi-soft elasticityen
dc.subjectVariational methodsen
dc.titleNumerical study of liquid crystal elastomers by a mixed finite element methoden
dc.typeArticleen
dc.identifier.journalEuropean Journal of Applied Mathematicsen
dc.contributor.institutionUniversity of Oxford, Oxford, United Kingdomen
dc.contributor.institutionUniversity of Minnesota Twin Cities, Minneapolis, United Statesen
kaust.grant.numberKUK-C1-013-04en
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