Self-consistent approximations for the tangential-displacement correction to the incremental indentation stiffness

An axisymmetric problem of unilateral frictionless contact between a paraboloidal or conical indenter and a transversely isotropic elastic half-space is considered in the refined formulation by accounting for the tangential (radial) displacements of the surface points of the elastic body. Using the idea of self-consistent approximation, a system of two coupled relations for the main contact variables (contact force, indenter displacement, and contact radius) is derived. The concept of the true contact radius is discussed, and it has been argued that the incremental stiffness relation should be expressed in its terms. Correction factors for the force–displacement relation and the incremental indentation stiffness (as a function of the true contact radius) are evaluated in explicit form.