Cattaneo–Mindlin plane problem with Griffith friction

• Cattaneo–Mindlin problem is extended for Griffith condition.
• Solution using superposition of pressure distributions.
• The corrective distribution formally equivalent to a JKR solution.
• Geometry affects the apparent friction coefficient.
• Example solutions are given.

The classical Cattaneo–Mindlin problem for elastic halfplanes is extended for a Griffith condition for inception of slip, and otherwise following the standard Coulomb law in the sliding zone. A general solution is found using the idea of superposing normal contact pressure distributions for arbitrary 2D geometry. In particular, the full sliding component of shear is corrected with a distribution in the stick region which is formally equivalent to a JKR solution for the normal contact problem insisting on the stick area. We show that geometry affects the apparent friction coefficient (the maximum tangential load at the inception of slip), since a sudden transition to slip occurs when the stick region reaches a critical size which corresponds to the phenomenon of pull-off in the JKR solution. Example solutions are given for Hertzian geometry, power law punches and a sinusoidal profile.