Wrinkling in Sheets with Nonuniform Growth and Bending Rigidity

Thin elastic sheets bend easily, leading to mechanical instabilities such as wrinkling. Here, we investigate wrinkles at edges of bi-strips, which consist of two thin sheets, one that swells and one that does not, joined side-by-side. It is well known that when bending rigidity is uniform across an isolated bi-strip, swelling results in axisymmetric shapes like a wine bottle: two cylinders of different radii are joined by a smooth transition zone. However, when the bending rigidity of the swollen sheet differs from that of the non-swollen sheet, purely axisymmetric shapes are no longer energetically favorable, and wrinkles arise. When the bending rigidity of the non-swollen sheet is essentially infinite, the wrinkles coarsen with distance from the transition zone such that dimensionless wavelengths and widths are related by $\tilde{\lambda} \propto \tilde{w}^{2/3}$. If the bending rigidity of the non-swollen sheet is non-infinite (but~still significantly larger than that of the swollen sheet), then the non-swollen sheet assumes a non-infinite radius of curvature, $R_0$. We find that the wrinkles in this system extend a critical distance, $w_C$, beyond the junction of the two strips and that $w_C \propto R_0$. Local undulations of wrinkles are favorable in this system because they decrease the overall bending energy by allowing the non-swollen sheet to have a larger radius of curvature than would otherwise be dictated by its reference geometry. Our results are relevant to a wide range of sheets that experience non-uniform growth, whether in natural systems such as plants or in synthetic systems such as designed, responsive materials.