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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

Plate Theory Finite Sample Layer Potential Mixed Derivative Liquid Crystal Layer 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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