Self-consistent approximations for the frictionless paraboloidal and conical depth-sensing indentation: The generalized bottom effect

Abstract

The axisymmetric frictionless indentation problem for a linearly-elastic layered medium is considered in the special cases of paraboloidal and conical indenters. A particular focus is placed on incorporating the general substrate effect into the force–displacement relation that is used in the inverse analysis of the depth-sensing indentation curves. A unified analytical modeling approach is developed by further exploring the concept of the indentation scaling factor. The asymptotically exact self-consistent approximations are derived for the dependences of the contact force and the indenter displacement on the contact radius, and their accuracy are tested against the numerically based analytical approximations available in the literature along with the asymptotic approximations. The substrate effect in the Johnson–Kendall–Roberts (JKR)-type adhesion for a paraboloidal or conical indenter is studied in detail.