Generating complex fold patterns through stress-free deformation induced by growth

The 2D and 3D axisymmetric models for stress-free deformations induced by growth have been established for curvilinear coordinates in arbitrary geometries in the current paper. In the derivation of the models, the deformation gradient is decomposed into the product of an elastic tensor and a growth tensor, and the growth tensor is further polar decomposed, based on which the basis vectors are appropriately chosen in each configuration/state. The constraints for the growth functions are derived together with the governing equations for the position vector and the rotational angle, which constitute coupled nonlinear partial differential equations. Under a special case, the analytical solutions to this system are obtained, based on which two mechanisms for generating complex fold patterns are identified. One is the introduction of geometric imperfection to the initial configuration, and the other is the creation of new growth tensors through the switching of curvilinear coordinates. Applying these two mechanisms, we manage to find a proper initial geometry and growth tensor to generate a special type of fold pattern having convex or/and concave cusps at the inner and outer surfaces of an initial annulus, having relevance with the morphology of a brain organoid and a bitter gourd. It seems this type of pattern has not been reported before by elastic instabilities. Many other complex fold patterns are also generated, like the morphologies of moss sakura, a red maple leaf, a rose, celosia, a gastrula, single clove garlic, a tumor spheroid, morning glory and two types of mushrooms. This model provides possible explanations for the morphogenesis of some biological tissues with zero or negligible residual stress. The results can also be incorporated with other models yielding residual stress to give a comprehensive description of the morphologies and stress distributions of the organisms. And the deformation principles obtained can be used in the design of soft devices through 4D printing of soft material.