Connection Between Liquid Crystal Theory and Plate Theory

  • Julie E. Kidd
  • Christian Constanda
  • John A. Mackenzie
  • Iain W. Stewart

Abstract

Layer deformations in a finite sample of smectic A liquid crystal, caused by the application of a pressure and a magnetic field, can be modeled by the equation [1]
$${{\nabla }^{4}}\upsilon - \frac{{{{\chi }_{a}}}}{{{{K}_{1}}}}{{H}^{2}}{{\nabla }^{2}}\upsilon + {{\left( {\frac{\pi }{{d{{\lambda }_{0}}}}} \right)}^{2}}\upsilon = \frac{{4P}}{{\pi {{K}_{1}}{{c}_{0}}}} in S,$$
(1)
subject to the “hinged boundary” conditions
$$\begin{array}{*{20}{c}} {\upsilon = 0 on \partial S,} \\ {{{\upsilon }_{{{{x}_{1}}{{x}_{1}}}}} = 0 for {{x}_{1}} = 0, a, 0 \leqslant {{x}_{2}} \leqslant b,} \\ {{{\upsilon }_{{{{x}_{2}}{{x}_{2}}}}} = 0 for {{x}_{2}} = 0, b, 0 \leqslant {{x}_{1}} \leqslant a,} \\ \end{array}$$
where S is the rectangle {(x 1, x 2) ∈ ℝ2 : 0 ≤ x 1a, 0 ≤ x 2b} of boundary ∂S, K 1, d, c 0, λ 0 and P are positive constants, χ a = const > 0, and H is the (constant) magnetic field. Plots of the layer deformations v(x 1, x 2) for increasing values of H can be found in [1].

Keywords

Tral Verse 

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References

  1. 1.
    I.W. Stewart, Layer undulations in finite samples of smectic-A liquid crystals subjected to uniform pressure and magnetic fields, Phys. Rev. E 58 (1998), 5926–5933.CrossRefGoogle Scholar
  2. 2.
    C. Constanda, A mathematical analysis of bending of plates with transverse shear deformation, Pitman Res. Notes Math. Ser. 215, Longman-Wiley, Harlow-New York, 1990.MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2002

Authors and Affiliations

  • Julie E. Kidd
  • Christian Constanda
  • John A. Mackenzie
  • Iain W. Stewart

There are no affiliations available

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