Distributed Branch Points and the Shape of Elastic Surfaces with Constant Negative Curvature

Abstract

We develop a theory for distributed branch points and investigate their role in determining the shape and influencing the mechanics of thin hyperbolic objects. We show that branch points are the natural topological defects in hyperbolic sheets, they carry a topological index which gives them a degree of robustness, and they can influence the overall morphology of a hyperbolic surface without concentrating energy. We develop a discrete differential geometric approach to study the deformations of hyperbolic objects with distributed branch points. We present evidence that the maximum curvature of surfaces with geodesic radius R containing branch points grow sub-exponentially, $O(e^{c\sqrt{R}})$ in contrast to the exponential growth $O(e^{c' R})$ for surfaces without branch points. We argue that, to optimize norms of the curvature, i.e., the bending energy, distributed branch points are energetically preferred in sufficiently large pseudospherical surfaces. Further, they are distributed so that they lead to fractal-like recursive buckling patterns.

Appendix: Asymptotics of Painlevé III

We can get more accurate estimates than implied by the bounds in (5.6). For $\varphi \ll 1,$ the Painlevé III equation (3.18) and the associated boundary conditions reduce to

$$\begin{aligned} \varphi ''(z) + \frac{\varphi '(z)}{z} - \varphi (z) = 0, \quad \varphi (0) = \varphi _0, \quad \varphi '(0) = 0. \end{aligned}$$

The solution is given by $\varphi (z) = \varphi _0 I_0(z)$, where $I_0$ is the modified Bessel function of the first kind (Abramowitz and Stegun 1992, §9.6). From the small and large z asymptotics of $I_0$ (Abramowitz and Stegun 1992, §9.7), we get

$$\begin{aligned} \varphi _{\text {inner}}(z)&= \varphi _0\left( 1 + \frac{z^2}{4} + O\left( z^4 \right) \right) , \text { for } z\ll 1, \\ \varphi _{\text {outer}}(z)&= \varphi _0 \frac{e^{z}}{\sqrt{2\pi z}}\left( 1 + \frac{1}{8z} + O\left( \frac{1}{z^2}\right) \right) , \text {for} z \gg 1. \end{aligned}$$

For the regime $z \gg 1, \varphi \approx \pi$, we have the weakly damped pendulum equation:

$$\begin{aligned} \varphi ''(z) - \sin \varphi (z) = - \frac{\varphi '(z)}{z} \approx 0, \end{aligned}$$

(A.1)

with asymptotic solutions of the form

$$\begin{aligned} \varphi _{\text {pend}}(z) \approx \pi - A\sin (z^* - z), \end{aligned}$$

(A.2)

for a slowly varying amplitude A that changes over many cycles of the pendulum. We are only interested in the first crossing $\phi (z^*) = \pi$, so we can assume that A is constant and determine A by matching the large z asymptotics of the Bessel solution with the pendulum solution. From the Bessel solution, we derive initial data for the pendulum equation, fixing the energy level for this conservative system:

$$\begin{aligned} (\varphi _{\text {pend}}(0), \varphi _{\text {pend}}'(0)) \approx \left( \frac{\varphi _0 e^{z}}{\sqrt{2\pi z}}, \frac{\varphi _0 e^{z}}{\sqrt{2\pi z}}\right) = (\delta , \delta ), \end{aligned}$$

(A.3)

where we match at such a point z that $z \gg 1, \delta \ll 1$. The energy of the pendulum solution is given by

$$\begin{aligned} E = \frac{\varphi '^2}{2} + \cos \varphi \approx 1 + \frac{\delta ^4}{24}, \end{aligned}$$

(A.4)

as $\cos \varphi$ is the potential and $\delta \ll 1$. Substituting the data into the energy, we find

$$\begin{aligned} 1 + \frac{\delta ^4}{24}&\approx \frac{1}{2}\left( A'\sin (z^*-z) + A\cos (z^*-z)\right) ^2 + \cos \varphi , \\&\approx \frac{1}{2}\left( A'\sin (z^*-z) + A\cos (z^*-z)\right) ^2 - 1 + \frac{(\pi - \varphi )^2}{2}, \\&\approx \frac{1}{2}\left( A'\sin (z^*-z) + A\cos (z^*-z)\right) ^2 - 1 +\frac{1}{2}A^2\sin ^2(z^*-z), \\&\approx -1 \frac{1}{2}\left[ A'^2\sin ^2(z^*-z) - 2A'A\sin (z^*-z)\cos (z^*-z) + A^2\right] , \end{aligned}$$

which in the case of slowing varying A simplifies to

$$\begin{aligned} A&\approx 2\sqrt{1 + \frac{\delta ^4}{48}}. \end{aligned}$$

Fig. 20

Asymptotics using the Pendulum and Bessel approximations in the $\varphi _0\rightarrow 0$ limit compared to the numerical solution of the Painlevé equation for $\varphi _0 = \frac{\pi }{100}$. Our interest is in approximating the exact solution well on an interval $[0,z^*]$ where $z = z^* \approx 9$ is the first instance where $\varphi (z) = \pi$, depicted by the dashed horizontal line in the figure

We are now equipped with a complete asymptotic description of the solutions to Painlevé III for an initial angle $\varphi _0$. The description is divided into three regimes: $z \ll 1$ and $\varphi _0 \lesssim \varphi \ll \pi$, $z \gg 1$ and $\varphi _0 \ll \varphi \lesssim \pi$, and finally $z\gg 1$ and $\varphi \approx \pi$:

$$\begin{aligned} \varphi (z) \approx \left\{ \begin{matrix} \varphi _0\left( 1 + \frac{z^2}{4}\right) ,\ z\ll 1\text {~and~} \varphi _0 \lesssim \varphi \ll \pi \\ \varphi _0 \frac{e^{z}}{\sqrt{2\pi z}}\left( 1 + \frac{1}{8z}\right) ,\ z\gg 1\text {~and~} \varphi _0 \ll \varphi \lesssim \pi \\ \pi - 2\sqrt{1 + \frac{e^{4 z}}{192\pi ^2 z^2}}\sin (z^* - z),\ \varphi \approx \pi , z \lesssim z^* \approx -\log (\varphi _0). \end{matrix}\right. \end{aligned}$$

(A.5)

A numerical validation of these asymptotic relations is illustrated in Fig. 20 (we consider $\varphi _0 = \frac{\pi }{100}$). Using the expressions in (A.5) instead of the bounds (5.6) gives the optimal constant $C(\phi ^*) =1$ in Lemma 5.1.