Analytical models for the impact of a solid sphere on a fluid-filled spherical shell incorporating the stress wave propagation effect and their applications to blunt head impacts

A new analytical (non-linear) model for the impact of a solid sphere on a fluid-filled spherical shell is developed by including the stress wave propagation effect in addition to the Hertzian contact deformations and the shell membrane and bending actions. The expressions for determining the maximum mutual approach and impact duration are analytically derived using the principle of energy conservation. A simplified (linearized) model incorporating the elastic energy loss due to the stress wave propagation is then formulated by using a linear force-deflection relation, which leads to a closed-form expression for the impact duration. It is shown that the new non-linear model reduces to that of Young (2003) and the linearized model recovers that of Mansoor-Baghaei and Sadegh (2011) when the stress wave propagation effect is not considered. By directly applying the newly obtained non-linear and linearized models, three representative problems simulating blunt head impacts are analyzed. The first problem characterizes the blunt impact of a human head on the ground or on an automobile, the second one simulates the blunt impact of a non-lethal projectile on a human head, and the third one represents two football players' head collision. Numerical results for the maximum deflection, maximum impact force and impact duration are provided to quantitatively show their variations with the impact velocity and shell thickness. The values predicted by the current new models are also compared with those given by the two existing models and with available finite element simulation results, and a good agreement is observed.