Mechanics of pressurized cellular sheets

Abstract

Everyday experience shows that cellular sheets are stiffened by the presence of a pressurized gas: from bicycle inner tubes to bubble wrap, the presence of an internal pressure increases the stiffness of otherwise floppy structures. The same is true of plants, with turgor pressure (due to the presence of water) taking the place of gas pressure; indeed, in the absence of water, many plants wilt. However, the mechanical basis of this stiffening is somewhat opaque: simple attempts to rationalize it suggest that the stiffness should be independent of the pressure, at odds with everyday experience. Here, we study the mechanics of sheets that are a single-cell thick and show how a pressure-dependent bending stiffness may arise. Our model rationalizes observations of turgor-driven shrinkage in plant cells and also suggests that turgor is unlikely to provide significant structural support in many monolayer leaves, such as those found in mosses. However, for such systems, turgor does provide a way to control leaf shape, in accordance with observations of curling upon drying of moss leaves. Guided by our results, we also present a biomimetic actuator that uncurls upon pressurization.

Keywords: bryophytes; cellular solids; turgor.

Figure 1.

(a,b) Gravity-driven deflections of a chain of inflated plastic ‘air pillows’ clamped vertically at one end. In (a), the air pillows are connected by a thin flexible strip; while in (b), they have been glued directly together (as indicated by the white arrows). (c) Photographs of the canopy, leaf and mid-leaf cross-section of the ground-dwelling moss Distichophyllum freycinettii. (Figure 1c was originally published in Waite & Sack [20] (2010; $\copyright$ The Authors) and is reproduced under license with permission of John Wiley and Sons, all rights reserved). (d) Sketch of the symmetric pressurized cell model considered in §2. The monolayer cell sheet is bent by the application of a bending moment, $\mu$. The structure's resistance to bending is described by the effective bending modulus of the cell, $B_{\mathrm{eff}}$, which we determine in terms of the cell geometry, the Young’s modulus of the cell walls, $E$, and the internal pressure, $p$.

Figure 2.

Plots of (a) the enclosed area of the cell, ${\hat{A}}_0$, and (b) the width of the cell, ${\hat{w}}_0$, as a function of pressure, $\mathcal{P}$. These plots were numerically determined by solving the Kirchhoff equations with parameters $\mathcal{A}=1/2$, $\mathcal{B}={10}^{-4}$ and $\mathrm{\Theta }=\pi /2$, $3\pi /8$, $\pi /4$, $\pi /8$ and $0$ (yellow to blue dots). The greyscale background corresponds to the maximum strain in the top and bottom beams, ${ϵ}_{\max }={\max }_S|{\alpha }_{\pm }-1|$, as delineated by the colourbar below these plots. In the inset of (a), the area of the flaccid cell (i.e. ${\hat{A}}_0$ at $\mathcal{P}=0$) is plotted as a function of the clamping angle, $\mathrm{\Theta }$; this is maximized at $\mathrm{\Theta }={\mathrm{\Theta }}_{\max }\approx 0.54$, for $\mathcal{A}=1/2$. In the main figures, the solution for $\mathrm{\Theta }={\mathrm{\Theta }}_{\max }$ is plotted as a black dashed curve. In this critical case, the cell is circular in shape, which qualitatively separates the different behaviours: for $\mathrm{\Theta }>{\mathrm{\Theta }}_{\max }$, the cell resembles a horizontal ellipse and an increase in pressure initially decreases the cell width, ${\hat{w}}_0$; while for $\mathrm{\Theta }<{\mathrm{\Theta }}_{\max }$, the cell resembles a vertical ellipse and an increase in pressure initially increases the cell width, ${\hat{w}}_0$. These regimes are delineated by ①–③. Note that all the curves (areas and widths) converge when the dimensionless pressure becomes close to one.

Figure 3.

Plot of the effective bending modulus, ${\hat{B}}_{\mathrm{eff}}$, as a function of the internal pressure, $\mathcal{P}$, for the configuration parameters $\mathcal{A}=1/2$, $\mathcal{B}={10}^{-4}$ and $\mathrm{\Theta }=\pi /2$, $3\pi /8$, $\pi /4$, $\pi /8$ and $0$ (yellow to blue). The effective moduli found by numerically solving the Kirchhoff equations are plotted as coloured dots, while asymptotic results are plotted as solid curves: ${\hat{B}}_{\mathrm{eff}}\sim {\hat{B}}_{\mathrm{eff}}^0$, valid for small pressures, is shown with the corresponding colour, and (2.8), valid for large pressures, is shown in black. The black dashed curve is the numerical solution with maximum encapsulated area, $\mathrm{\Theta }={\mathrm{\Theta }}_{\max }\approx 0.54$. The greyscale background corresponds to the maximum strain in the top and bottom beams, ${ϵ}_{\max }={\max }_S|{\alpha }_{\pm }-1|$, as delineated by the colourbar; the contour lines of ${ϵ}_{\max }={10}^{-4}$, ${10}^{-3}$, ${10}^{-2}$ and ${10}^{-1}$ are shown as vertical dashed lines. ①–③ correspond to the three regimes sketched in figure 2. Although ${\hat{B}}_{\mathrm{eff}}$ may initially decrease with $\mathcal{P}$ (i.e. for large values of $\mathrm{\Theta }$), it ultimately increases with $\mathcal{P}$. Note that all the curves (bending stiffnesses) converge when the dimensionless pressure becomes close to one.

Figure 4.

Sketch of the asymmetric pressurized cell model considered in this section. The top and bottom surface walls have differing values of Young's moduli $E_{\pm }$, thicknesses $h_{\pm }$, relaxed lengths $L_{\pm }$ and/or clamping angles ${\mathrm{\Theta }}_{\pm }$. These asymmetries induce an effective curvature, ${\kappa }_{\mathrm{eff}}$, which varies with the internal pressure, $p$.

Figure 5.

Plots of the numerically determined effective cell curvature, ${\hat{\kappa }}_{\mathrm{eff}}$, as a function of the internal pressure for an asymmetry in either the Young's modulus (${\displaystyle \circ }$ in panel (a)), thickness (${\displaystyle \bullet }$ in panel (a)), relaxed length (${\displaystyle \mathbf{+}}$ in panel (b)), or clamping angle (${\displaystyle \mathbf{\times }}$ in panel (b)) of the surface walls, where the corresponding values of ${\mathcal{R}}_E$, ${\mathcal{R}}_h$, ${\mathcal{R}}_L$ or ${\mathcal{R}}_{\mathrm{\Theta }}$ (asymmetries in the Young's modulus, thickness, length and clamping angle, respectively) are given in the keys. (Note that the ratio values, and hence the sign of curvature, are chosen for presentation purposes only). The parameters are chosen such that their averages recover those used in figure 3, that is $(1/{\mathcal{Y}}_{+}+1/{\mathcal{Y}}_{-})/2=1$, $(1/{\mathcal{A}}_{+}+1/{\mathcal{A}}_{-})/2=1/2$, $({\mathcal{B}}_{+}+{\mathcal{B}}_{-})/2={10}^{-4}$ and $({\mathrm{\Theta }}_{+}+{\mathrm{\Theta }}_{-})/2={\mathrm{\Theta }}_{\mathrm{avg}}$, where in (a), ${\mathrm{\Theta }}_{\mathrm{avg}}=3\pi /8$ (blues) and $\pi /8$ (yellow and green), and in (b), ${\mathrm{\Theta }}_{\mathrm{avg}}=\pi /4$. Also plotted are the asymptotic results for small pressures (as dashed lines) and large pressures (as solid curves), as derived in ection II.D of [25], and the cell profiles for the asymmetry with the corresponding colour at $\mathcal{P}=0$, ${10}^{-5/2}$ and ${10}^{-1/2}$ (left to right).

Figure 6.

Plot of the extended lamina widths, ${\ell }_{\mathrm{width}}/2$, against estimated elastogravitational length, ${\ell }_{\mathrm{bg}}=[B_{\mathrm{eff}}/(\rho g{\ell }_{\mathrm{thick}}){]}^{1/3}$, for nine species of single-cell thick moss; here, colour denotes the species (see key), while point shape denotes the type as follows: trunk-dwelling mosses, ${\displaystyle ◻}$, (Pyrrhobryum pungens, Acroporium fuscoflavum, Campylopus hawaiicus); ground-dwelling mosses, ${\displaystyle ◯}$, (Distichophyllum freycinetii, Fissidens pacificus, Hookeria acutifolia); branch-dwelling mosses, ${\displaystyle \mathrm{△}}$, (Holomitrium seticalycinum, Macromitrium microstomum, Macromitrium piliferum). Here, we have used the cell width, cell height, cell wall thicknesses, lamina thickness and maximum leaf width reported by [20] for each species to estimate $L$, $\mathcal{A}L$, $h$, ${\ell }_{\mathrm{thick}}$ and ${\ell }_{\mathrm{width}}$, respectively. We further assume a density ${\displaystyle \rho =1000\text{k}\mathrm{g}{\mathrm{m}}^{-3}}$; clamping angle $\mathrm{\Theta }=\pi /4$ (based on the cross-sections shown in [20]); Young's moduli $E=100\mathrm{MPa}$, which is chosen as a lower bound to the typically accepted range in plant cells [41]; and turgor pressures (a) $p=0\mathrm{MPa}$ (i.e. a flaccid cell) and (b) $0.8\mathrm{MPa}\le p\le 2.1\mathrm{MPa}$ (the typical range found in moss [32], with variation shown by error bars in ${\ell }_{\mathrm{bg}}$). The lines ${\ell }_{\mathrm{width}}/2=2{\mathcal{D}}^{1/3}{\ell }_{\mathrm{bg}}$ for $\mathcal{D}={10}^{-2}$ (dash-dotted) and $\mathcal{D}={10}^{-3}$ (dotted) correspond to the deflection, $\mathcal{D}$, of an Euler–Bernoulli beam, (4.1); thus, the shaded region corresponds to where appreciable deformation under gravity would be expected.

Figure 7.

Optical coherence tomography images of a cross-section of a (a) dehydrated and (b) hydrated Physcomitrium patens moss leaf. The two laminar halves protruding from the midrib are approximated by a circle (shown as dashed curves), the radius of which yields the dry and wet signed curvatures, ${\kappa }_{\mathrm{dry}}$ and ${\kappa }_{\mathrm{wet}}$, respectively. (c) Plot of ${\kappa }_{\mathrm{wet}}$ against ${\kappa }_{\mathrm{dry}}$ for 11 Physcomitrium patens leaves at various cross-sections along the leaves' lengths (points with the same colour correspond to measurements at different points along the same leaf). The majority of the points lie below $y=x$ (black dashed line), showing the leaves tend to curl when dehydrated.

Figure 8.

A 3D-printed realization of the asymmetric pressurized structure of figure 4 uncurves upon pressurization. Panel (a) shows a three-dimensional render of the full computer aided design (CAD), while panel (b) shows a half section. To enable inflation, the device has been designed with a ‘lid’ and a ‘base’ in the third-dimension (separated by $30\mathrm{mm}$). The other geometrical parameters are $L_{\mathrm{L}}=10\mathrm{mm}$, $L_{\pm }=20\mathrm{mm}$, $h_{\mathrm{L}}=1\mathrm{mm}$, $h_{\pm }=1\mathrm{mm}$, ${\mathrm{\Theta }}_{+}=0$ and ${\mathrm{\Theta }}_{-}=\pi /2$. Equal pressurization of each cell is ensured by elliptical holes (of width $6\mathrm{mm}$ and length $12\mathrm{mm}$) that connect neighbouring cells—as seen in (b). The structure was 3D printed using the PolyJet elastomer Agilus-30 [50]. Panels (c)–(e) show increasing pressurizations ($\mathrm{\Delta }p=0$, $5.87\mathrm{kPa}$ and $8.13\mathrm{kPa}$, respectively) of the 3D-printed structure with the effective curvature, ${\kappa }_{\mathrm{eff}}$, decreasing correspondingly. Quantitative measures of the macroscopic curvature as a function of measured air pressure are presented in the main figure of (f); different colours indicate different experimental runs and different shapes indicate different Young's modulus, $E$, as given in the key. The predicted effective curvature from the two-dimensional model developed here is plotted as the black dashed curve.

Figure 9.

Sketch of the bending of a pressurized two-dimensional channel. In the left sketch, an interior pressure, $p$, induces a tension, $T_p$, in the two bounding surface walls. In the right sketch, the pressurized channel is bent under a moment, $\mu$, which induces a differential strain across the channel.

Figure 10.

Plot of the extended lamina widths, ${\ell }_{\mathrm{width}}/2$, against estimated strain length, ${\ell }_{\mathrm{bend}}=[B_{\mathrm{eff}}/(\rho g{\ell }_{\mathrm{thick}}^2){]}^{1/2}$, for nine species of single-cell thick moss, defined by colour, and point shape denoting the type: trunk-dwelling mosses, ${\displaystyle ◻}$, (Pyrrhobryum pungens, Acroporium fuscoflavum, Campylopus hawaiicus); ground-dwelling mosses, ${\displaystyle ◯}$, (Distichophyllum freycinetii, Fissidens pacificus, Hookeria acutifolia); branch-dwelling mosses, ${\displaystyle \mathrm{△}}$, (Holomitrium seticalycinum, Macromitrium microstomum, Macromitrium piliferum). Here, we have used the cell width, cell height, interior and surface cell wall thicknesses, lamina thickness and maximum leaf width from [21] to estimate $L$, $L_{\mathrm{L}}$, $h_{\mathrm{L}}$, $h_{\mathrm{S}}$, ${\ell }_{\mathrm{thick}}$ and ${\ell }_{\mathrm{width}}$, respectively. We further assume a density ${\displaystyle \rho =1000}$ kg m⁻³; clamping angle $\mathrm{\Theta }=\pi /4$ (an approximation based on cell photographs [21]); Young's moduli $E_{\mathrm{L}}=E_{\mathrm{S}}=100\mathrm{MPa}$, which are chosen as a lower bound to the typically accepted range in plant cells [45]; and turgor pressures (a) $p=0\mathrm{MPa}$ (i.e. a flaccid cell) and (b) $0.8\mathrm{MPa}\le p\le 2.1\mathrm{MPa}$ (the typical range found in moss [33], which gives the range in ${\ell }_{\mathrm{bend}}$). The lines ${\ell }_{\mathrm{width}}/2=2{\mathcal{E}}^{1/2}{\ell }_{\mathrm{bend}}$ for $\mathcal{E}={10}^{-3}$ (dash-dotted) and $\mathcal{E}={10}^{-4}$ (dotted), correspond to the maximal differential strain, $\mathcal{E}$, caused by bending a Euler–Bernoulli beam, and the shaded region corresponding to significant deformations under gravity.