Stretching Hookean ribbons part I: relative edge extension underlies transverse compression and buckling instability

The wrinkle pattern exhibited upon stretching a rectangular sheet has attracted considerable interest in the "extreme mechanics" community. Nevertheless, key aspects of this notable phenomenon remain elusive. Specifically-what is the origin of the compressive stress underlying the instability of the planar state? what is the nature of the ensuing bifurcation? how does the shape evolve from a critical, near-threshold regime to a fully developed pattern of parallel wrinkles that permeate most of the sheet? In this paper we address some of these questions through numerical simulations and analytic study of the planar state in Hookean sheets. We show that transverse compression is a boundary effect, which originates from the relative extension of the clamped edges with respect to the transversely contracted, compression-free bulk of the sheet, and draw analogy between this edge-induced compression and Moffatt vortices in viscous, cavity-driven flow. Next, we address the instability of the planar state and show that it gives rise to a buckling pattern, localized near the clamped edges, which evolves-upon increasing the tensile load-to wrinkles that invade the uncompressed portion of the sheet. Crucially, we show that the key aspects of the process-from the formation of transversely compressed zones, to the consequent instability of the planar state and the emergence of a wrinkle pattern-can be understood within a Hookean framework, where the only origin of nonlinear response is geometric, rather than a non-Hookean stress-strain relation.