Deep Indentation of Soft Thin Films Beyond the 10 % Rule

Indentation is one of the simplest ways to feel the stiffness of matter. Interpreting indentation, however, requires care, especially for thin films-such as biological cells or soft coatings supported by rigid substrates-because the substrate effect can make a film appear orders of magnitude stiffer than it actually is. To minimize this artifact, the empirical "10% rule of thumb," which limits indentation depth to one-tenth of the film thickness, has often been adopted, though it becomes unreliable and impractical for ultrathin films. Here, we explore the traditionally forbidden regime: deep indentation, where the indentation depth is comparable to the film thickness. We develop an asymptotic theory that incorporates nonlinear geometry and elasticity, elucidating a simple indentation force-depth relation in this deep regime. We further show that the pull-off force in such indentation tests is remarkably independent of both the material law and geometric nonlinearity. The results provide a unified framework for interpreting thin-film mechanics under deep indentation, a regime that is generally much more accessible in experiments on very thin films.