Transverse edge extension underlies compression, buckling and wrinkling in thin solids

When a rectangular sheet whose short edges are clamped is stretched, elongated wrinkles appear, indicating the presence of transverse compression. So far, the mechanism by which longitudinal tension and edge clamping act jointly to generate transverse compression has not been clarified. Here we employ analytic tools and numerical simulations to compare this problem with a new variant, where instead of stretching and edge clamping one only imposes edge-localized transverse strain, pulling the corners outward. We find that, despite the absence of longitudinal tension in the new variant, both models exhibit similar transverse stress profile in planar state, thus revealing that the generic origin of compression is the transverse extension of edges relative to the bulk. This similarity in planar stress profiles underlies similarity in the near-threshold buckling patterns exhibited by the two models. In contrast, the two models are sharply distinct in their respective far-from-threshold regime: the deflection of the stretched sheet consists of small wavelength wrinkles whereas the edge-extended sheet does not develop wrinkles. This indicates the role of longitudinal tension in providing an effective substrate resistance, which is crucial for the emergence of wrinkle patterns.