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Recent experiments on hydrogels subjected to large elongations have shown elastic instabilities resulting in the formation of geometrically intricate fringe and fingering deformation patterns. In this paper, we present a robust numerical framework addressing the challenges that emerge in the simulation of this complex material response from the onset of instability to the post-bifurcation behavior. We observe that the numerical difficulties stem from the non-convexity of the strain energy density in the near-incompressible, large-deformation regime, which is responsible for the coexistence of multiple equilibrium paths with vastly-different, sinuous deformation patterns immediately after bifurcation. We show that these numerical challenges can be overcome by using sufficiently-high order of interpolation in the finite element approximation, an arc-length-based nonlinear solution procedure that follows the entire equilibrium path of the system, and an implementation enabling parallel, large-scale simulations. The resulting computational approach provides the ability to conduct highly-resolved, truly quasi-static simulations of complex elastic instabilities. We present numerical results illustrating the ability of the path-following approach to describe the full evolution of fringe and fingering instabilities observed experimentally in recent experiments of confined cylindrical specimens of soft hydrogels subject to tension. Importantly, we observe that the robustness of the static solution procedure enables complete access to the multiplicity of solutions occurring immediately after the onset of bifurcation, as well as to the settled post-bifurcation states. [ABSTRACT FROM AUTHOR]

Modeling the spontaneous evolution of morphology in natural systems and its preservation by proportionate growth remains a major scientific challenge. Yet, it is conceivable that if the basic mechanisms of growth and the coupled kinetic laws that orchestrate their function are accounted for, a minimal theoretical model may exhibit similar growth behaviors. The ubiquity of surface growth, a mechanism by which material is added or removed on the boundaries of the body, has motivated the development of theoretical models, which can capture the diffusion-coupled kinetics that govern it. However, due to their complexity, application of these models has been limited to simplified geometries. In this paper, we tackle these complexities by developing a finite element framework to study the diffusion-coupled growth and morphogenesis of finite bodies formed on uniform and flat substrates. We find that in this simplified growth setting, the evolving body exhibits a sequence of distinct growth stages that are reminiscent of natural systems, and appear spontaneously without any externally imposed regulation or coordination. The computational framework developed in this work can serve as the basis for future models that are able to account for growth in arbitrary geometrical settings, and can shed light on the basic physical laws that orchestrate growth and morphogenesis in the natural world. [ABSTRACT FROM AUTHOR]