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.

Some 1-and 2-D dynamic problems of coupled thermoelasticity in an unbounded solid, half-space and slab are treated. The problems are basic, and have been addressed in the literature. This treatment considers (possibly finite) pre-stress and both Fourier and thermal relaxation models, and generates analytical results from exact solutions or inversions of asymptotic expressions for exact transform solutions. The results illustrate the role of wave speeds in solution behavior, how pre-stress affects these speeds, and how the models themselves define solution behavior, especially near wave fronts.

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