Unloading properties based elastoplastic continuous contact force model: Restitution and stiffness characterization

A nonlinear contact force model is developed for spherical elastoplastic impacts. The nonlinear contact force model is constructed based on an existing piecewise model and the other based on the force–displacement diagram. The energetic impact law of the coefficient of restitution is used to solve for the unloading restitution stiffness. An integration of the dynamic equation of motion and proper use of initial conditions is performed to solve for the maximum deformation. Furthermore, the contact force law from the force–displacement diagram is used at maximum deformation to obtain a physical expression of the final indentation in terms of the maximum deformation and compression and restitution stiffness. In addition, the principle of the unloading linear stiffness is used and two equivalent closed form solutions are derived for the unloading linear stiffness. Moreover, a closed form solution of the maximum force is derived based on the unloading stiffness closed form solutions. The final and unloading indentation properties of the new model are investigated by performing numerical simulations for all coefficient of restitution values ranging from elastic to plastic impacts.