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October 8, 2025Open Access

Universal Scaling Laws for Deep Indentation Beyond the Hertzian Regime

TMTong MuArmy Medical University CLChanghong LinghuCity University of Hong Kong YLYanju LiuHarbin Institute of Technology

Key Points

A universal scaling law for contact force applies to deep indentation in various materials, including polymers and biological tissues.
Experiments validate the framework with soft substrates like Ecoflex and tofu, confirming predictions up to a deformation ratio of 2.5.
Finite element analysis confirms that Hertz-type pressure distributions can be observed in the deformed state of soft materials.
These findings enhance understanding of mechanical interactions in fields like soft robotics and tissue mechanics.

Abstract

Deep indentation of soft materials is ubiquitous across scales in nature and engineering, yet accurate predictions of contact behaviors under extreme deformations (/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 /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.

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