Reduced Membrane Model for Liquid Crystal Polymer Networks: Asymptotics and Computation

Abstract

We examine a reduced membrane model of liquid crystal polymer networks (LCNs) via asymptotics and computation. This model requires solving a minimization problem for a non-convex stretching energy. We show a formal asymptotic derivation of the 2D membrane model from 3D rubber elasticity. We construct approximate solutions with point defects. We design a finite element method with regularization, and propose a nonlinear gradient flow with Newton inner iteration to solve the non-convex discrete minimization problem. We present numerical simulations of practical interests to illustrate the ability of the model and our method to capture rich physical phenomena.

Figures (16)

• Figure 1: Computed solution with the blueprinted director field $\mathbf{m}_1$ that has degree $1$ defect, $\lambda<1$ and $\alpha_r=0\text{ (right)},\pi/2\text{ (left)}$. We refer to Section \ref{['sec:dir_defect']} for details of these numerical simulations.
• Figure 2: Director field $\mathbf{m}_{1/2}$ from \ref{['eq:degree-1half']} (left), lifted surface $\mathbf{y}_{1/2}$ from \ref{['eq:soln-degree-1half']}-\ref{['eq:lifted-degree-1half']} for $\lambda =2^{1/3}$ (middle), and computed solution in a unit disc domain with $\mathbf{m}=\mathbf{m}_{1/2}$ and a Dirichlet boundary condition that is compatible with \ref{['eq:soln-degree-1half']}-\ref{['eq:lifted-degree-1half']} (right). Note that the gradient of $\phi_{1/2}$ is parallel to $\mathbf{m}_{1/2}$ whereas $\mathbf{m}_{1/2}^\perp$ is the typical director field for a $1/2$ defect.
• Figure 3: Approximate lifted surface for degree 2 defect (left) and computed solution with the director field $\mathbf{m}_2$ in Section \ref{['sec:dir_defect']} (right). Our derivation requires $x_1>0$, but the solution should be symmetric across the $x_2x_3$ plane, which is why we plot a reflected solution for $x_1<0$. We recover two bumps, consistent with the simulation but at the cost of a singularity at the origin.
• Figure 4: Contour plot of approximate lifted surface for degree 3/2 defect for $a = .75$ and $x_1>0$ (left) and computational result for a degree 3/2 defect obtained in Section \ref{['sec:dir_defect']}. The profile matches the computed "bird beak" shape. To see this, notice that contour lines pinch off as $x_1\to0$. As a result, the lifted surface gets steeper near the origin. This helps explains the "bird beak" shape.
• Figure 5: Director fields with point defects of degree $n$. First row displays $n=2,3/2,-1$ and $\alpha=0$ (from left to right). Each panel shows experimental and expected configurations from mcconney2013topography as well as two views of the computed solution. Second row depicts experimental pictures from de2012engineering and our simulations of the cone structure $n=1, \alpha=\frac{\pi}{2}$ (left) and anti-cone structure $n=1, \alpha=0$ (right). The numerical model reproduces experimental observations well.

Theorems & Definitions (11)

• Remark 1: comparison with ozenda2020blend
• Lemma 1: stretching vs. 3D energy
• proof
• Proposition 1: nondegeneracy
• Proposition 2: target metric
• proof
• Corollary 1: immersions of $g$ are minimizers with vanishing energy
• Lemma 2: energy density gap
• proof
• Theorem 1: convergence of discrete minimizers