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Numerical study of liquid crystal elastomers by a mixed finite element method
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| Zeitschriftentitel: | European Journal of Applied Mathematics |
|---|---|
| Personen und Körperschaften: | , |
| In: | European Journal of Applied Mathematics, 23, 2012, 1, S. 121-154 |
| Format: | E-Article |
| Sprache: | Englisch |
| veröffentlicht: |
Cambridge University Press (CUP)
|
| Schlagwörter: |
| author_facet |
LUO, C. CALDERER, M. C. LUO, C. CALDERER, M. C. |
|---|---|
| author |
LUO, C. CALDERER, M. C. |
| spellingShingle |
LUO, C. CALDERER, M. C. European Journal of Applied Mathematics Numerical study of liquid crystal elastomers by a mixed finite element method Applied Mathematics |
| author_sort |
luo, c. |
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LUO, C. CALDERER, M. C. 0956-7925 1469-4425 Cambridge University Press (CUP) Applied Mathematics http://dx.doi.org/10.1017/s0956792511000313 <jats:p>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.</jats:p> Numerical study of liquid crystal elastomers by a mixed finite element method European Journal of Applied Mathematics |
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European Journal of Applied Mathematics |
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| title |
Numerical study of liquid crystal elastomers by a mixed finite element method |
| title_unstemmed |
Numerical study of liquid crystal elastomers by a mixed finite element method |
| title_full |
Numerical study of liquid crystal elastomers by a mixed finite element method |
| title_fullStr |
Numerical study of liquid crystal elastomers by a mixed finite element method |
| title_full_unstemmed |
Numerical study of liquid crystal elastomers by a mixed finite element method |
| title_short |
Numerical study of liquid crystal elastomers by a mixed finite element method |
| title_sort |
numerical study of liquid crystal elastomers by a mixed finite element method |
| topic |
Applied Mathematics |
| url |
http://dx.doi.org/10.1017/s0956792511000313 |
| publishDate |
2012 |
| physical |
121-154 |
| description |
<jats:p>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.</jats:p> |
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| author | LUO, C., CALDERER, M. C. |
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| description | <jats:p>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.</jats:p> |
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| physical | 121-154 |
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| series | European Journal of Applied Mathematics |
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| spelling | LUO, C. CALDERER, M. C. 0956-7925 1469-4425 Cambridge University Press (CUP) Applied Mathematics http://dx.doi.org/10.1017/s0956792511000313 <jats:p>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.</jats:p> Numerical study of liquid crystal elastomers by a mixed finite element method European Journal of Applied Mathematics |
| spellingShingle | LUO, C., CALDERER, M. C., European Journal of Applied Mathematics, Numerical study of liquid crystal elastomers by a mixed finite element method, Applied Mathematics |
| title | Numerical study of liquid crystal elastomers by a mixed finite element method |
| title_full | Numerical study of liquid crystal elastomers by a mixed finite element method |
| title_fullStr | Numerical study of liquid crystal elastomers by a mixed finite element method |
| title_full_unstemmed | Numerical study of liquid crystal elastomers by a mixed finite element method |
| title_short | Numerical study of liquid crystal elastomers by a mixed finite element method |
| title_sort | numerical study of liquid crystal elastomers by a mixed finite element method |
| title_unstemmed | Numerical study of liquid crystal elastomers by a mixed finite element method |
| topic | Applied Mathematics |
| url | http://dx.doi.org/10.1017/s0956792511000313 |