Analysis of the Hertz Contact Model for Evaluating Mechanical Properties of Polymers Using the Finite Element Method

Atomic force microscopy (AFM) is widely used to quantify mechanical properties, typically Young's modulus, by fitting force-indentation data with various mathematical contact models. However, results across laboratories often diverge, and the causes remain unresolved. Here, we interrogate the methodology by which mechanical properties are defined in AFM indentation and identify key limitations of the Hertz model, the standard model for determining mechanical properties, notably that the contact radius is not directly determined, which limits the accuracy of the estimated Young's modulus. We hypothesize that this inference systematically overestimates the true tip-sample contact, which inflates the contact area and thereby underestimates Young's modulus. This bias is amplified under large indentation conditions, which are frequently used in soft-material studies. To isolate and clarify the issue, we focus on a well-characterized polymer, polyvinyl chloride (PVC), using it as a controlled testbed for contact radius overestimation. Our analysis is focused on the contact radius and Hertz-based extraction of Young's modulus. We determined the contact radius and Young's modulus using AFM with two different probes: a sphere with a 20 nm radius (SPHERE20) and a sphere with a 2 µm radius (SPHERE2000). The results were compared to macroscopic data obtained using a standard measurement (ISO 527-1:2019) of Young's modulus. The contact was modeled using finite element analysis (FEA). The dependence of the contact radius on the indentation was compared to the Hertz model. The results from FEA fit corrected contact radius values, and it is smaller by 15.46% (SPHERE20) and 57.9% (SPHERE2000) than those calculated by the Hertz model.