A rigid ellipsoidal die slides on the surfaces of transversely isotropic half-spaces. In one case the material symmetry axis coincides with the half-space surface normal. In the other, the axis lies in the plane of the surface. In both cases sliding proceeds with constant sub-critical speed along a straight path at an arbitrary angle to the principal material axes. A three-dimensional dynamic steady state is considered, i.e., the contact zone surface must conform to the die profile and contact zone traction remains constant in the frame of the die. Exact solutions for contact zone traction are derived in analytic form, as well as formulas for contact zone geometry. Symmetry need not be assumed in the solution process. Anisotropy is found to largely determine zone shape at low sliding speed, but the direction of sliding can become a major influence at higher sliding speeds. Cartesian coordinates are used in the analysis, but introduction of quasi-polar coordinates allows problem reduction to a Cauchy singular integral equation.

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