An Approximate Force–Indentation Equation for n-Sided Blunt Pyramidal Indenters

Accurate AFM nanoindentation analysis requires models that account for the rounded apex of real pyramidal indenters. Although exact force-indentation equations for n-sided blunt pyramids exist, their numerical complexity limits routine use. In this work, a simple closed-form analytical approximation is developed that directly relates force to indentation depth for blunt pyramidal indenters. The method employs first-order Maclaurin series expansions of the geometric terms and the generic indentation differential equation, yielding a closed-form second-degree polynomial expression that is readily implemented in AFM data analysis. Comparison with the exact solutions showed that the approximation error decreases with indentation depth and is governed by the pyramid geometry rather than the tip radius. Simulated and experimental AFM data confirmed accurate Young’s modulus estimation above a geometry-dependent validity threshold. For a four-sided blunt pyramidal indenter, the proposed criterion predicts minimum indentation depths ranging from approximately 10.4 Rc for θ = 15° to 2.4 Rc for θ = 45° where Rc is the tip radius and θ is the pyramid’s semi-included angle. Application of the model to simulated AFM datasets yielded Young’s modulus values between 18.5 and 19.8 kPa for a true modulus of 20 kPa, corresponding to errors below 8% in all examined cases. Furthermore, the closed-form equation provided very good agreement with AFM nanoindentation data obtained from human prostate cancer cells. It is also shown that the generic derived equation includes the case of a spheroconical indenter as a limiting case. Young’s modulus is obtained directly from the quadratic coefficient, eliminating the need for tip-radius calibration. In addition, the formulation is applicable to heterogeneous materials, providing an effective local modulus through the weighted mean value theorem for integrals. The approach offers a practical and computationally efficient alternative for AFM data processing, improving the robustness of modulus estimation for soft biological materials.