Emergence of Spontaneous Twist and Curvature in Non-Euclidean Rods: Application to Erodium Plant Cells

We present a limiting model for thin non-Euclidean elastic rods. Originating from the three-dimensional (3D) reference metric of the rod, which is determined by its internal material structure, we derive a 1D reduced rod theory. Specifically, we show how the spontaneous twist and curvature of a rod emerge from the reference metric derivatives. Thus, the model allows calculating the unconstrained equilibrium configuration of a thin rod directly from its internal structure. The model is applied to the study of cells from members of the Geraniaceae plant family and their configurational response to dehydration. We show how the geometrical arrangement of cellulose fibrils on the cell walls determines the helical shapes of isolated cells.

Figure 1

Thin elastic bodies in stork’s bill. A (a) wet and (b) dry awn (scale bar—1 cm). (c) Scanning electron microscopy image of a dry cell taken from the top of the coiling region of the awn (scale bar—$50\mu \mathrm{m}$).

Figure 2

(a) At every point on the cell wall, we define the angle $\psi$ between the orientation of the fibrils and the cell’s longitudinal axis. $\psi$ is approximately independent of the longitudinal coordinate $x^1$ but may depend on the azimuthal direction $\chi$ within the cross section plane. (b) A regular helix, in which $\psi (\chi )$ is constant and equals $\theta$. (c) A tilted helix at tilt angle $\varphi$, in which $\psi (\chi )$ ranges between $(\theta -\varphi )$ and $(\theta +\varphi )$, where $\theta$ is the mean fibril angle.

Figure 3

Predicted shapes for different cells with several values of the cell parameters $\theta$ and $\varphi$, for $\alpha =0.5$. The black lines on the sides of each rod represent the $\pm x^2$ directions in intrinsic coordinates.

Figure 4

(a) Raw images of a cell at different shrinkage ratios $\alpha$ (scale bar—$100\mu \mathrm{m}$). (b) The same images isotropically rescaled and superimposed onto one another to demonstrate collapse onto the same curve. Calculation of the internal structure parameters from these shapes shows a decrease in $\alpha$ from 1 to $\sim 0.7$, with approximately constant $\theta =69°\pm 5°$ and $\varphi =28°\pm 5°$.