Wrinkling in Sheets with Nonuniform Growth and Bending Rigidity

TL;DR

This study investigates wrinkles at the edges of bi-strips formed from two thin sheets with nonuniform growth and bending rigidity. By fabricating back-to-back latex bi-strips with a stiffened non-swollen region and inducing differential swelling via paraffin, the authors reveal two key scaling laws: the wrinkle wavelength obeys $\tilde{\lambda} \propto \tilde{w}^{2/3}$ in both flat and curved geometries, and a curvature-controlled transition exists where wrinkles extend only up to a critical width $w_C$ with $w_C \propto R_0$. The work shows that local undulations can reduce the total bending energy by allowing the non-swollen region to attain a larger radius of curvature, linking nonuniform growth, bending rigidity, and pattern formation. These findings enhance understanding of growth-induced morphologies in natural and synthetic responsive sheets and provide design principles for tuning wrinkle patterns via spatially varying bending rigidity.

Abstract

Thin elastic sheets bend easily, leading to mechanical instabilities such as wrinkling. Here, we investigate wrinkles at edges of bi-strips, which consist of two thin sheets, one that swells and one that does not, joined side-by-side. It is well known that when bending rigidity is uniform across an isolated bi-strip, swelling results in axisymmetric shapes like a wine bottle: two cylinders of different radii are joined by a smooth transition zone. However, when the bending rigidity of the swollen sheet differs from that of the non-swollen sheet, purely axisymmetric shapes are no longer energetically favorable, and wrinkles arise. When the bending rigidity of the non-swollen sheet is essentially infinite, the wrinkles coarsen with distance from the transition zone such that dimensionless wavelengths and widths are related by $\tildeλ \propto \tilde{w}^{2/3}$. If the bending rigidity of the non-swollen sheet is non-infinite (but~still significantly larger than that of the swollen sheet), then the non-swollen sheet assumes a non-infinite radius of curvature, $R_0$. We find that the wrinkles in this system extend a critical distance, $w_C$, beyond the junction of the two strips and that $w_C \propto R_0$. Local undulations of wrinkles are favorable in this system because they decrease the overall bending energy by allowing the non-swollen sheet to have a larger radius of curvature than would otherwise be dictated by its reference geometry. Our results are relevant to a wide range of sheets that experience non-uniform growth, whether in natural systems such as plants or in synthetic systems such as designed, responsive materials.

Figures (7)

• Figure 1: Configurations after swelling a bi-strip. (a) The reference geometry is a rectangular sheet with length ($L$), total width ($w_0+w$), and thickness ($h$) subjected to a uniaxial differential swelling field. Over most of the sheet, lengths increase by either a factor of 1 (no swelling) or a factor of $1+\alpha$ (maximum swelling). These two regions are joined by a sigmoidal transition over a distance $\delta$. (b) Swelling without stretching results in isometric embedding of the reference geometry: two concentric cylinders are joined by a transition over distance $\delta$. (c) When stretching is allowed, finite thickness effects increase the radii of curvature of both cylinders (now denoted $R_0$ and $R$) and the width of the transition region (now denoted $\Delta$). Stretching energy accumulates in this transition region. (d) The radius of curvature of the non-swollen region can increase further if the two regions have different bending rigidities, which increases the width of the transition region and introduces wrinkles of wavelength $\lambda$. (e) When $\Delta>w$, wrinkles cover the entire swollen region.
• Figure 2: Four back-to-back bi-strips. Each has a central, unswollen, flat region of length 9 cm. In swollen regions with a small width, $w$, a single 1st-generation wavelength usually emerges (labeled "1"). In wider swollen regions, wrinkles can merge into a 2nd generation (labeled "2").
• Figure 3: Measurement of the lengthening factor, $\alpha(t)$. (a) Photograph of latex sheets of different thicknesses submerged in molten paraffin. Each sheet began as an 8 × 35 mm rectangle. (b) Lengthening as a function of time for the sheets in panel a. Each swelling curve was fit to an exponential function with an asymptotic lengthening factor, $\alpha_0$, and a characteristic time, $\tau$. (c) Characteristic times from the exponential fits as a function of the thickness, $h$, of each latex sheet. The timescales follow $\tau\propto h^2$, as expected from the diffusion of a solvent into a porous material. (d) Asymptotic lengthening factors are independent of the thicknesses of the latex sheets. The mean lengthening factor is $\alpha_0 \approx 0.28$, with a standard deviation of $0.02$.
• Figure 4: Profiles of 1st-generation wrinkles that appear in a bi-strip as it swells in liquid paraffin. (a) Photographs from the side, through time, of one edge of a back-to-back bi-strip. The sheet is flat at $t =$ 0 s (not shown). Over time, wrinkles grow. The bottom image corresponds to $t =$ 80 s. The profile of the edge (blue) is fit with a sine wave (red) to extract the linear swelling factor, $\alpha(t)$. (b) The linear swelling factor extracted from Panel a (gray circles) is fit well by an exact prediction from the calibrated swelling factor in Fig. \ref{['fig:calibration']}, which means there is no significant stretching in 1st-generation wrinkles.
• Figure 5: Scaling of wrinkles in back-to-back bi-strips in which the unswollen central region is flat. (a) Wrinkles in latex sheets with thickness 0.31 mm and length 9 cm (blue sheet, top) and 0.51 mm (white sheet, bottom). The sheets have non-swollen central regions and swollen edge regions of widths $w_1$, $w_2$, $w_3$, and $w_4$. (b) Measured wavelength versus width of swollen regions. Color corresponds to the thickness of the sheet before swelling. Squares denote 1st-generation wrinkles fit by $\lambda = 3.255w$ (black line)MoraBoudaoud2006Bi-strip. Circles denote wrinkles that have undergone further coarsening. (c) Data from coarsened 2nd- and 3rd-generation wrinkles collapse to the predicted power law of $\tilde{\lambda} \propto \tilde{w}^{2/3}$ when replotted as dimensionless wavelength $\tilde{\lambda}\equiv\lambda/h$ versus dimensionless width $\tilde{w}\equiv\alpha^{-1/4}w/h$.