A perturbation method used for static contact and low velocity impact

An approximate method for the solution of static and dynamic contact problems between bodies with non-linear material behaviour is described. The method is a perturbation technique based on the linear elastic quasi-static solution. Here the method is applied to the problem of a sphere in contact with a half-space which means that the Hertz solution is used. The governing equations are rewritten so that the problem for the perturbed variables is one with surface forces in the contact region and volume forces inside the bodies. The latter are due to accelerations and strain gradients calculated from the quasi-static solution and the equation of motion. The contact condition results in an integral equation for the surface forces. Results are compared with FEM calculations, which show very good agreement for the dynamic case, both with linear elastic and non-linear (plastic) material behaviour. For the static case with non-linear material behaviour the results are good approximately up to the point where the inelastic zone reaches the surface of the bodies.