Reference gap technique in viscoelestic sphere impact modeling

Hydrogel spheres, which undergo large deformations under external force, are widely used in applications ranging from soil conservation to biomedicine, soft robotics, and wearable devices. These large deformations induce strong nonlinearities and emergent phenomena, such as collective granular jamming, which remain difficult to capture within a unified numerical framework. In this study, we propose a discrete-continuous model incorporating a reference gap to construct a continuum hydrogel sphere with tunable Young's modulus. When elastic and surface energies become comparable, surface tension is explicitly incorporated into the model. Then, the relationship between the macroscopic Young's modulus and the microscopic stiffness coefficient is established by quasi-static compression of hydrogel spheres and isotropic compression of rectangular hydrogel blocks. We further evaluate the dynamic behavior of hydrogel spheres with Young's modulus Y ranging from 100 Pa to 50 000 Pa and a Poisson's ratio of 0.5, impacting a rigid substrate at velocities ranging from 0.5 m s^-1 to 4.5 m s^-1, and validate the results against experimental observations. The evolution of morphology and spreading diameter of the superelastic hydrogel spheres, as well as the forces and stresses exerted on spheres, are investigated. The simulation results demonstrate a transition in contact morphology from point contact to Hertzian contact, and eventually to global deformation, accompanied by a corresponding evolution in stress distribution, from point like concentration to single-ring and multi-ring patterns. These phenomena are governed by the propagation of mechanical waves and are characterized by elastic-capillary Mach numbers corresponding to supersonic, critical, subcritical, and highly low subcritical regimes.