A morphoelastic stability framework for post-critical pattern formation in growing thin biomaterials

Morphological instabilities play a key role in the evolution of form and function in growing biomaterials. Spatially varying differential growth, in particular, leads to residual stresses that for soft materials or slender geometries are favourably relieved by out-of-plane buckling or wrinkling. While the onset of instability in growing biomaterials has been studied extensively, the post-critical regime remains poorly understood. To this end, this paper presents a robust computational modelling framework for morphoelastic instabilities and associated post-critical pattern formation. The seven-parameter shell element—commonly used for elasto-plastic analysis of engineering materials—is implemented alongside the multiplicative decomposition of the deformation gradient tensor into a growth and an elastic part. The governing nonlinear equations are solved using a generalised path-following/numerical continuation solver that facilitates comprehensive exploration of the stability landscape through pinpointing of critical points and branch switching at bifurcations. The utility and power of the computational framework is demonstrated by unveiling complex pattern formation phenomena that arise as a result of sequentially occurring morphological instabilities. In particular, we highlight the central role that exponential edge growth plays in fractal wrinkling patterns, the blooming of doubly-curved flower petals, and wavy daffodil trumpets. In addition, the ability of the solver to track critical points through parameter space enables efficient sensitivity studies into the role of material parameters on pattern formation. The presented computational framework is thus a versatile tool for modelling the evolution of form and function in growing biological systems and the design of biotechnology applications.