Two cases of rapid contact on an elastic half-space: Sliding ellipsoidal die, rolling sphere

In one case a rigid ellipsoidal die translates over the surface of a half-space.Because of friction, both compression and shear force are required.In the other, a rigid sphere rolls on the surface under a compressive force.Both motions occur along a straight path at constant subcritical speed.A dynamic steady state is treated, that is, the contact zone and its traction remain constant in the frame of the die or sphere.Exact solutions for contact zone traction are derived in analytic form, as well as formulas for the contact zone shape.Axial symmetry is not required in the solution process.Cartesian coordinates are used, but a system of quasipolar coordinates is introduced that allows problem reduction to singular integral equations similar in form to those found in 2D contact.