Deep Indentation of Hyperelastic Materials Reveals Tip Independent Parabolic Force Depth Response via Strain Energy Delocalization

Abstract

Indentation is a practical route for probing soft materials when standard tests are difficult, destructive, or cannot be performed in situ. Conventional indentation is usually interpreted in the shallow-depth regime, where the indentation depth D is small compared with the indenter radius R. In this limit, the response is controlled by local contact geometry and primarily identifies the small-strain Young's modulus E. Here, we show that at deep indentation, D >> R, flat and spherical indenters converge to the same parabolic force-depth law, F = beta E D^2. The coefficient beta is independent of indenter radius and tip shape, only mildly affected by interfacial friction, and controlled by the hyperelastic strain-stiffening response. Finite-element simulations show that this scaling arises from strain-energy delocalization: the region where SED/mu > 0.01 expands into a spheroidal domain whose size scales with D. The activated volume therefore scales as D^3, giving stored elastic energy U ~ E D^3 and force F = dU/dD ~ E D^2. Far from contact, the strain-energy-density fields collapse toward the Boussinesq far-field solution when distances are normalized by a = sqrt(F/E), which scales as D in the deep-indentation regime. These results provide a mechanistic basis for tip-shape independence and link beta to the Ogden strain-stiffening parameter alpha, enabling hyperelastic parameter extraction from deep-indentation data.