Connection Between Liquid Crystal Theory and Plate Theory

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 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].