Corrections to the stiffness relationship in 3-sided and conical indentation problems

Abstract

One key relationship in the depth-sensing indentation technique is the proportionality between the contact stiffness and the contact size, as can be proved from the Sneddon's solution of axisymmetric frictionless contact. However, Sneddon's solution is only accurate when the indenter approaches a half-space (e.g., for conical indenter, the half-apex angle approaches 90º) and the interface is frictionless. As Hay et al. (J. Mater. Res., 1999) pointed out, sharp indenters lead to a radial inward displacement on the sample surface, thus leading to extra indentation force needed to push the surface back to conform with the conical indenter. In this paper, we argue that the physical origin arises from the incorrect use of reference and deformed coordinates in the boundary conditions that define Sneddon's problem. This yields two correction factors for both load and depth solutions, which are needed for sharp pyramidal indenters and frictional contact. Approximate solutions are derived which compare favorably well with the finite element simulations. We also find that the stiffness correction factor of three-sided indenter is about 11∼15% times higher than that of conical indenter, and this multiplicative factor is only a weak function of the indenter angle but does not depend on the friction coefficient and Poisson's ratio.