An analytical model to predict the response of fluid-filled shells to impact—a model for blunt head impacts

An approximate analytical model to predict the response of a fluid-filled shell of arbitrary thickness impacting with a solid elastic sphere is proposed and the limits of applicability of the equations developed are discussed. The model is based on combining the Hertzian contact stiffness and the effective local membrane and bending stiffness to derive implicit expressions for global impact characteristics including the duration of impact, the peak force transmitted, peak global acceleration of shell and sphere, and the resultant pressures induced in the fluid. Closed-form explicit expressions are also derived to predict whether the pressure response in the fluid will be hydrostatic or will exhibit large dynamic transients of pressure (and shear strain). It should be noted that the impact of hollow/empty shells with solid spheres, as well as the impact of shells with an elastic half-space, can be straightforwardly treated as limiting cases. The model is of obvious relevance to head impact modelling and selected parametric studies of the response of fluid-filled shells with geometric and material properties about those typical for the human head are given.