General load-depth relations for spherical, conical, and flat-ended cylindrical indentations of soft elastic layers: From ultra-thin-film to half-space

The thickness dependency of the mechanical response of layered structures challenges the applicability of classical Hertz-Sneddon contact mechanics in the indentation measurements of soft thin specimens. Although various modified theories have been reported for the contact of very thin or relatively thick elastic layers placed on a rigid substrate, unified analytical solutions covering the complete spectrum of layer thickness are still missing. Here, we establish explicit expressions of the load-depth relations for spherical, conical, and flat-ended cylindrical indentations of soft layers of arbitrary thickness. Two fundamental boundary conditions between the elastic layer and the rigid support are considered: (1) connected without friction and (2) perfectly bonded. Crucially, the derived relations demonstrate mathematical continuity from the ultra-thin-film limit to the half-space limit, which offer accurate yet convenient-to-use formulae for determining elastic moduli of thin materials via indentations, particularly for soft layers exhibiting intermediate thickness ranges. The validity of our general relations is confirmed through excellent agreements with experimental data and existing solutions within certain ranges.