Hyperelastic characterization via deep indentation

Abstract

Hyperelastic material characterization is crucial for understanding the behavior of soft materials -- such as tissues, rubbers, hydrogels, and polymers -- under quasi-static loading before failure. Traditional methods typically rely on uniaxial tensile tests, which require the cumbersome preparation of dumbbell-shaped samples for clamping in a uniaxial testing machine. In contrast, indentation-based methods, which can be conducted \textit{in-situ} without sample preparation, have been underexplored. To characterize the hyperelastic behavior of soft materials, deep indentation is required, where the material response extends beyond linear elasticity. In this study, we perform finite element analysis to link the force ($F$) versus indentation depth ($D$) curve with the hyperelastic behavior of a soft incompressible material, using a one-term Ogden model for simplicity. We identify three indentation regimes based on the ratio between indentation depth and the radius ($R$) of the spherical-tipped cylindrical indenter: the Hertzian regime ($D < 0.1R$), where $F = \frac{16}{9} E R^{0.5} D^{1.5}$; the parabolic regime ($D > 10R$), where $F = E D^2 \beta$ and the indenter radius becomes irrelevant; and an intermediate regime ($0.1R < D < 10R$) bridging the two extremes. We find that the Ogden strain-stiffening coefficient ($\alpha$) increases the parabolic indentation coefficient ($\beta$), allowing for the estimation of $\alpha$ from $\beta$. Furthermore, we observe that Coulomb friction increases $\beta$, potentially masking the effect of strain-stiffening for small $\alpha$. However, for $\alpha > 3$, friction has a negligible effect. Finally, our results show good agreement with experimental data, demonstrating that deep indentation can be an effective method for extracting hyperelastic properties from soft materials through \textit{in-situ} testing.