Connection Between Liquid Crystal Theory and Plate Theory

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

Chapter 21 in Integral Methods in Science and Engineering, 2002, pp 137-142 from Springer

Abstract: 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] 1 $${{\nabla }^{4}}\upsilon - \frac{{{{\chi }_{a}}}}{{{{K}_{1}}}}{{H}^{2}}{{\nabla }^{2}}\upsilon + {{\left( {\frac{\pi }{{d{{\lambda }_{0}}}}} \right)}^{2}}\upsilon = \frac{{4P}}{{\pi {{K}_{1}}{{c}_{0}}}} in S,$$ 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 1 ≤ a, 0 ≤ x 2 ≤ b} 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 (search for similar items in EconPapers)
Date: 2002
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DOI: 10.1007/978-1-4612-0111-3_21

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