J. Mech. Phys. Solids 212 (2026) 106572

Keywords:  Contact mechanics, Deep indentation, Soft materials, Large deformation, Geometric nonlinearity

Deep indentation of soft materials is common in biological and engineering systems, yet accurate predictions of contact behaviors at large indentation depths remain elusive. Classical Hertz theory and its extensions break down when the indentation depth becomes comparable to or exceeds the indenter radius, because these models are formulated in the undeformed configuration and cannot capture the dominant geometric nonlinearity. Here we develop a geometric mapping approach that establishes a Hertz-type normal pressure distribution in the deformed configuration, enabling analytical solutions for contact pressure, force, and radius under extremely large deformations. Closed-form expressions predict the full nonlinear evolution of the contact region, including the depth at which the sphere becomes fully submerged. Finite element simulations using neo-Hookean and Arruda Boyce materials confirm that geometric effects dominate over material nonlinearity for a broad class of elastomers. Experiments on polymers, food substrates, and biological tissues further validate a universal scaling law that collapses all measurements onto a single master curve. The proposed framework unifies deep indentation across soft materials and provides a fundamental basis for nonlinear contact mechanics with direct implications for soft robotics, tactile sensors, medical diagnostics, and tissue mechanics.

Tong Mu, Changhong Linghu, Yanju Liu, Jinsong Leng,  Huajian Gao, K. Jimmy Hsia

Universal scaling laws and a geometric mapping framework for deep indentation of soft materials.pdf