A Simple Approximation for Indentation of Finite Elastic Layers by Parabolic, Conical, and Flat-ended Cylindrical Indenters

In this work, we develop simple approximation formulas for the indentation of compressible elastic layers of arbitrary thickness by parabolic, conical, and flat-ended cylindrical indenters. Most surprisingly, a compact "Pythagoras-like" interpolation between the half-space and thin-layer limits provides an excellent solution to the problem. The proposed empirical expressions are validated against extensive Boundary Element Method simulations, showing remarkable accuracy across a wide range of elastic moduli, Poisson's ratios (up to 1/3), indenter geometries, penetration depths, and layer thicknesses. The approach is further extended to adhesive contacts of flat-ended punches, yielding compact estimates of adhesion forces in layered systems. Owing to their simplicity and robustness, the presented formulas provide practical correction rules for interpreting experimental indentation data, particularly in nanoindentation and atomic force microscopy of thin films and biological materials.