In this interesting paper, the authors address an anomaly which arises when a rigid, square-ended block is pressed against a linear elastic half plane and slid along. The authors note that, within the framework of linear elasticity, the singularity in the contact pressure, and hence shearing traction, produces, adjacent to the edges, regimes in which the implied local relative slip direction dominates the rigid-body sliding velocity, and hence produces a violation of the Coulomb friction law. They seek to resolve the paradox by appealing to a more sophisticated strain definition. All of this is within the context of a quasistatic formulation. The authors recognize, of course, that in any real problem the paradox is unlikely to arise because of (a) the finite strength of the contact giving rise to a yield zone, and (b) the absence of an atomically sharp corner at the contact edge where there is, in all probability, a finite edge radius. Here, we wish to address these issues quantitatively, and so demonstrate that it is unlikely that the paradox described, though interesting, will have any bearing in a real contact.
On the Navier–Stokes system with the Coulomb friction law boundary condition
