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 (CM) model, 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 set-up. We show that the evolution of wrinkles comes in tandem with collapse of transverse compressive stress, rather than vanishing transverse strain, 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.