Frictionless and adhesive nanoindentation: Asymptotic modeling of size effects

Widely used the Bulychev–Alekhin–Shorshorov relation for analyzing nanoindentation load–displacement data to determine elastic modulus of a thin specimen does not account for the size of specimen since the BASh relation is based on analytical solutions of the contact problems for an elastic half-space. In order to model the substrate effect, the unilateral contact problem for a spherical indenter pressed against an elastic layer on an elastic half-space is analyzed for different types of boundary conditions imposed at the interface between the specimen and the substrate. Approximate (asymptotically exact) solutions are obtained in explicit form. The influence of the substrate effect on the incremental contact stiffness is described in terms of the asymptotic constants possessing information about the thickness of the specimen and depending on the relative stiffness of the substrate.