Stretching Hookean ribbons part II: from buckling instability to far-from-threshold wrinkle pattern

Abstract

We address the fully developed wrinkle pattern formed upon stretching a Hookean, rectangular-shaped sheet, when the longitudinal tensile load induces transverse compression that far exceeds the stability threshold of a purely planar deformation. At this “far-from-threshold” parameter regime, which has been the subject of the celebrated Cerda–Mahadevan model (Cerda and Mahadevan in Phys Rev Lett 90:074302, 2003), the wrinkle pattern expands throughout the length of the sheet and the characteristic wavelength of undulations is much smaller than its width. Employing Surface Evolver simulations over a range of sheet thicknesses and tensile loads, we elucidate the theoretical underpinnings of the far-from-threshold framework in this setup. We show that the evolution of wrinkles comes in tandem with collapse of transverse compressive stress, rather than vanishing transverse strain (which was hypothesized by Cerda and Mahadevan in Phys Rev Lett 90:074302, 2003), such that the stress field approaches asymptotically a compression-free limit, describable by tension field theory. We compute the compression-free stress field by simulating a Hookean sheet that has finite stretching modulus but no bending rigidity, and show that this singular limit encapsulates the geometrical nonlinearity underlying the amplitude–wavelength ratio of wrinkle patterns in physical, highly bendable sheets, even though the actual strains may be so small that the local mechanics is perfectly Hookean. Finally, we revisit the balance of bending and stretching energies that gives rise to a favorable wrinkle wavelength, and study the consequent dependence of the wavelength on the tensile load as well as the thickness and length of the sheet.

Graphic abstract

Appendix: TFT simulations

In simulating a sheet with no bending rigidity, any compression gives rise to an infinitely corrugated shape, limited only by the mesh size. In order to check that these simulations provide the TFT solution reliably, we performed simulations with a sequence of mesh densities, starting with the “base” density $\rho _n^{(0)} = 6.95 \times 10^5$, used in most of our simulations, then increasing the density to $4 \rho _n^{(0)}$ and to $16 \rho _n^{(0)}$. Figure 9 shows the numerical values of several macroscale features, which are predictable by TFT, for these mesh densities values. The variation among these different meshes is a tiny fraction ($\lesssim 10^{-3}$) of the characteristic differences between the TFT value and the finite-$\epsilon$ simulations, from which we extract the scaling laws in Figs. 3 and 4.