Universal Scaling Laws for Deep Indentation Beyond the Hertzian Regime

TL;DR

This paper develops a universal scaling law for deep indentation of soft materials, accurately predicting contact behavior beyond traditional limits using geometric mapping, validated by simulations and experiments across various materials.

Contribution

It introduces a universal framework for extreme indentation regimes, extending Hertzian contact theory to large deformations with validated closed-form solutions.

Findings

Hertz-type pressure distributions in highly deformed contact

Closed-form solutions match simulations up to $rac{ ext{delta}}{R} = 2.5$

Experimental validation across diverse soft materials

Abstract

Deep indentation of soft materials is ubiquitous across scales in nature and engineering, yet accurate predictions of contact behaviors under extreme deformations ($\delta/R > 1$) remain elusive due to geometric and material nonlinearities. Here, we investigate the indentation of rigid spheres into soft elastic substrates, resolving the highly nonlinear regime where the sphere becomes fully submerged. A universal geometric mapping approach reveals Hertz-type pressure distributions in the deformed configuration, validated by FEA. Closed-form solutions for contact force and radius agree with simulations up to $\delta/R = 2.5$. Experiments spanning soft polymers (Ecoflex, PDMS), food substrates (tofu), and biological tissues (octopus) validate the derived scaling law for hyperelastic materials. Our results establish a universal framework for extreme mechanical interactions, with applications in soft robotics, bioengineered systems, and tissue mechanics.