The Hertz Contact Problem and Its Volumetric Reduction with Computational Applications
A method of computational reduction of an elastic contact model for rigid bodies in frame of the Hertz contact model is considered. An algorithm to transform the outer surfaces geometric properties to the local contact coordinates system is described. It tracks the surfaces of the bodies which are able to contact permanently in time.
An approach to compute the normal elastic force is represented. That one deals with the reduction to one transcendental scalar equation that includes the complete elliptic integrals of the fist and second kinds. Simulation of the Hertz model was accelerated essentially due to use of the differential technique to compute the complete elliptic integrals and due to the replacement of the implicit transcendental equation by the differential one.
Based on the Hertz contact problem classic solution an invariant form for the force function which depends on the geometric properties of an intersection for the undeformed rigid bodies volumes, so-called volumetric model, is proposed then. The resulting reduced expression for the force function supposed to be in use in cases of the classic contact theory hypotheses are broken. The expression derived has been applied to several cases of the elastic bodies contacting, and in particular back to the source Hertz model itself.
The volumetric model showed a reliable behavior and an acceptable accuracy. Finally an implementation of the ball bearing dynamics computer model as an example of the contact models application is under consideration.