Buckling, wrinkling, and crumpling of simulated thin sheets

Abstract

Ubiquitous across nature and technology, thin films are subject to three primary out-of-plane of deformations: buckling, wrinkling, and crumpling. Plants' leaves gain structure and support from their buckled curvature, yet metalworkers are plagued by the same defects in steel production. Growing biofilms form patterns of radial wrinkles, reminiscent of the clever folds and pleats necessary to drape flat planes of cloth over the rounded human figure. Large scale movement of tectonic plates force crumples and folds into the Earth's relatively thin and fragile crust, while even the slightest thermal fluctuations drive crumpling in atomically thin materials. Thin sheets, in their infinite usefulness, surround us in time and space, and the current era of computational power promises new methods to study their deformations. Eventually we could use this understanding to squash manufacturing bugs and program novel design features. But we must begin by characterizing, predicting, and mastering the mechanics of buckling, wrinkling, and crumpling.

We first introduce an irregular lattice mass-spring-model (MSM) we have developed to simulate elastic thin sheets. Our mechanical MSM reliably maps bulk properties--Young's modulus, Poisson's ratio, shear modulus, and bending rigidity--onto a discrete network of randomly arranged mesh nodes. The MSM we propose is inspired by previously established discrete stretching and bending models, but altered to comply with analytical predictions for a mesh of equilateral triangles. To our knowledge, a combined stretching and bending MSM has not been quantitatively evaluated for numerical convergence and accuracy. We measure the error associated with benchmark MSMs from the literature and our proposed MSM, in both regular and random mesh networks. We find that in both mesh types our proposed MSM has a lower magnitude of error in Young's modulus, Poisson's ratio, and bending rigidity than the benchmark models. The proposed model, however, performs worse in the shear modulus tests than the benchmarks. We conclude that all errors in the simulated sheets are within the range of variation one could expect from physical samples of materials.

Next we simulate and study the deformation modes of a thin elastic ribbon as a function of applied end-to-end twist and tension. Our simulations reproduce all reported experimentally observed modes, including transitions from helicoids to longitudinal wrinkles, creased helicoids and loops with self-contact, and transverse wrinkles to accordion self-folds. Our simulations also show that the twist angles at which the primary longitudinal and transverse wrinkles appear are well described by various analyses of the F\"oppl-von K\'arm\'an (FvK) equations, but the characteristic wavelength of the longitudinal wrinkles has a more complex relationship to applied tension than previously estimated. The clamped edges are shown to suppress longitudinal wrinkling over a distance set by the applied tension and the ribbon width, but otherwise have no apparent effect on measured wavelength. Further, by analyzing the stress profile, we find that longitudinal wrinkling does not completely alleviate compressive stress, but caps the magnitude of the compression. Nonetheless, the width over which wrinkles form is observed to be wider than analytical predictions in the so-called near-threshold (NT) regime-- the width is more consistent with the predictions of the far-from-threshold (FT) analysis framework. However, the end-to-end contraction of the ribbon as a function of twist is found to more closely follow the corresponding NT prediction as tension in the ribbon is increased, in contrast to the expectations of FT analysis. These results point to the need for further theoretical analysis of this rich thin elastic system, guided by our physically robust and intuitive simulation model.

We then propose a dimensionless bendability parameter, $\epsilon^{-1} = [\left(\frac{h}{W}\right)^2 \frac{1}{T}]^{-1}$ to describe wrinkling of thin, twisted ribbons. Bendability has been identified as a useful parameter in several configurations of wrinkled thin sheets, and similarly proves useful in the context of twisted ribbons. Recasting predictions for the onset, wavelength, and stress distribution of longitudinal wrinkles in terms of bendability efficiently collapses data collected across a range of ribbon thicknesses, widths, and applied tensions. This demonstrates that key wrinkling quantities depend primarily on the ribbon's bendability, including the residual stress in the buckled ribbon, which saturates at the critical buckling stress. We identify scaling relations for onset, wavelength, critical stress, and residual stress in the wrinkled ribbon that are valid both in the highly bendable range ($\epsilon^{-1} > 20$), as well as a range of moderately bendable ribbons ($\epsilon^{-1} \in (0,20]$). When data are restricted to highly bendable sheets---the range in which NT methods are expected to be valid---the resultant scaling exponents for wrinkle onset and wavelength reproduce theoretical NT predictions. The critical stress scaling, however, is insensitive to this data exclusion.

Turning to sheets with plasticity, we determine that cylindrical sheets crumpled via twisting accumulate total ridge length $\ell$ with a logarithmic dependence on crumpling iteration $n$, in accordance with the model previously introduced to describe sheets crumpled via axial compression. This is the first systematic study of crease length accumulation in a crumpling configuration without external confinement. We introduce a process for extracting crease length through a fully automated image processing pipeline which returns a clean crease map and facet segmentation for the crumpled sheet. We emphasize the importance of ``noise'' in generating logarithmic growth for physical and simulated sheets, such as the inversion of some crumple facets in the sheet, introduced by physically handling the sheets or alternating twist directions in simulations. Our simulations unlock a new method of analyzing crumpling via plasticity, a primary output of the simulation. By examining the plastic deformations accumulated in edges of the simulation mesh, we justify crease length as an appropriate measure of damage in a sheet, confirm that an unfurled sheet's crease map is representative of damage in the compact configuration, and prove that increases in $\ell$ from $n$ to $n+1$ are overwhelmingly due to new damage conferred during the $n+1$ iteration. Finally we investigate the softening and sharpening of existing ridges, and the rate of new ridge growth in weakly, moderately, and strongly crumpled cylinders.

Finally we summarize several efforts toward determining a geometric predictor for crumpliness. Statistical models robustly couple ridge length distribution to the log-normal distribution in weakly confined sheets, and to the gamma distribution in strongly compressed systems with jamming. These successful descriptions of geometry's effects on the successive fragmentations of ridges clearly indicate a strong dependence of crease evolution on the geometry and degree of compaction. We introduce a set of simulations which radially compress square sheets to probe crease length growth in a more symmetric confinement geometry. Our simulations show early plateaus of cumulative crease length due to a near absence of mechanical noise in the unfurling and recrumpling processes. Visual inspection of the samples also reveals a strong geometric signature in the crumple patterns where damage at small confinements is mostly concentrated around the midpoints of each sheet edge. At higher compactions, however, this signature washes out and the density of crumple facets is homogeneous throughout the entire sheet. To more directly compare the axially compressed, twisted cylinder, and radially compressed crumpling experiments, we additionally introduce a confinement parameter $\rho = \left(V_f - V_\text{min}\right)/\left(V_i - V_\text{min}\right)$ where $V_f$ and $V_i$ are the final and initial confinement volumes, and $V_\text{min}$ is the smallest volume to which the sheet could reasonably be compressed. Thus $\rho$ quantifies how close a given system is to maximum compaction. We attempt to scale cumulative crease length $\ell$ of the crumpled sheets as a function of crumpling iteration $n$, using the compaction parameter $\rho$ as a normalizing factor, demonstrating reasonable collapse across the axially compressed data, and partial collapses for radially compressed and wrung cylinder datasets.