Morphomechanics of growing curled petals and leaves

Petals and leaves are usually curled and exhibit intriguing morphology evolution upon growth, which contributes to their important biological functions. To understand the underlying morphoelastic mechanism and to determine the crucial factors that govern the growth-induced instability patterning in curved petals and leaves, we develop an active thin shell model that can describe variable curvatures and spontaneous growth, within the framework of general differential geometry based on curvilinear coordinates and hyperelastic deformation theory. Analytical solutions of distinguished growing shapes such as saddle surface and cylindrical mode are then derived. We reveal distinct morphological evolutions of doubly curved leaves/petals with different curvatures κx (along the main vein) and κy (perpendicular to the main vein) upon differential growth. Compared to the flat (zero curvature) configuration, leaves/petals with longitudinal curvature κx experience a global bending deformation. With the increase of growth strain, the leaf/petal undergoes a coupling behavior of edge wrinkling and global bending deformation, associated with a pitchfork bifurcation. Conversely, the transverse curvature κy does not lead to significant bending behavior, but results in delayed critical buckling threshold and reduced wrinkling amplitude. Physical insights into curvature effects on morphology evolutions are further provided by the analysis of nonlinear competition between bending and membrane energies. Moreover, we explore the effect of vein constraint on pattern formation, showing that, unlike the edge wrinkling observed in leaves with strong vein constraint, those with weak vein constraint are prone to grow into a saddle shape, consistent with analytical solutions. The results uncover the intricate interplay between configurational curvature and vein confinement on plant morphogenesis, providing fundamental insights into a variety of growing shapes of curled petals and leaves.