A nonlinear toroidal shell model for surface morphologies and morphogenesis

Biological tissues with core–shell structures usually exhibit non-uniform curvatures such as toroidal geometry presenting interesting features containing positive, zero, and negative Gaussian curvatures within one system, which give rise to intriguing instability patterns distinct from those observed on uniformly curved surfaces. Such varying curvatures would dramatically affect the growing morphogenesis. To understand the underlying morphoelastic mechanism and to quantitatively predict morphological instability patterns, we develop a nonlinear toroidal core–shell model and incorporate advanced numerical techniques for pattern prediction. Analytical solutions indicate that regions with positive Gaussian curvature (outer ring) require higher critical buckling stresses than those with negative Gaussian curvature (inner ring), with the critical threshold positively correlated to the key dimensionless parameters that are composed of curvature and stiffness of the system. Using the Asymptotic Numerical Method (ANM) as a robust path-following continuation approach, we continuously trace the post-buckling evolution and the associated wrinkling topography. We reveal that for donut-like toroidal core–shell structures, stripes initially form in the inner region with negative Gaussian curvature, and then evolve into a non-uniform hexagonal pattern in the post-buckling stage, while localized dimples may appear in core–shell tori with low stiffness. For cherry-like core–shell tori, the outer region with positive Gaussian curvature usually exhibits axisymmetric stripes or hexagonal patterns. A phase diagram on wrinkling topography at the critical buckling threshold is provided, in line with analytical predictions, offering fundamental insights into the complex interplay between curvature and material stiffness on multi-phase pattern selection in core–shell structures.