The well-known solution for a rigid punch bonded to an elastic half plane has oscillatory singularities at the corners of the punch. It is shown here that the resulting displacement of the free surface of the half plane actually spirals in upon itself and is therefore physically inadmissible. The problem of a flat punch with sloped sides is then formulated and a solution obtained which does not have an oscillatory singularity. The limit is taken as the punch sides become perpendicular with its base. This leads to square root singularities in the shear stress and to logarithmic singularities in the normal stress along the bond.

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