Citation: Kontomaris, S.V.; Stylianou, A.; Chliveros, G.; Malamou, A. AFM Indentation on Highly Heterogeneous Materials Using Different Indenter Geometries. Appl. Mech. 2023, 4, 460–475. https://doi.org/10.3390/ applmech4020026 Received: 7 March 2023 Revised: 11 April 2023 Accepted: 14 April 2023 Published: 18 April 2023 Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Article AFM Indentation on Highly Heterogeneous Materials Using Different Indenter Geometries Stylianos Vasileios Kontomaris 1,2, * , Andreas Stylianou 3 , Georgios Chliveros 2 and Anna Malamou 4 1 BioNanoTec Ltd., 2043 Nicosia, Cyprus 2 Faculty of Engineering and Architecture, Metropolitan College, 15125 Athens, Greece; gchliveros@mitropolitiko.edu.gr 3 School of Sciences, European University Cyprus, 2404 Nicosia, Cyprus; an.stylianou@euc.ac.cy 4 Independent Power Transmission Operator S.A. (IPTO), 10443 Athens, Greece * Correspondence: rnd@bionanotec.eu or skontomaris@mitropolitiko.edu.gr Abstract: Hertzian mechanics is the most frequently used theory for data processing in Atomic Force Microscopy (AFM) indentation experiments on soft biological samples, due to its simplicity and significant scientific results previously published. For instance, using the Hertz model, it has been proven that there are significant differences in the mechanical properties of normal and cancerous tissues and that cancer cells’ invasive properties are correlated with their nanomechanical properties. However, many scientists are skeptical regarding the applicability of the Hertz theory to biological materials, as they are highly heterogeneous. The main critical question to be addressed is “what do we calculate” when fitting the force-indentation data to Hertz equations. Previous studies have shown that when using cylindrical, parabolic, or conical indenters, the fitting parameter is the average Young’s modulus. In this paper, it is demonstrated that it is also valid to fit equations derived from Hertzian mechanics to force-indentation data when testing soft, heterogeneous samples for any indenter geometry. The fitting factor calculated through this approach always represents the average Young’s modulus for a specific indentation depth. Therefore, Hertzian mechanics can be extended to soft heterogeneous materials, regardless of the indenter’s shape. Keywords: hertz model; heterogeneous samples; fitting; depth-dependent behavior; biological materials; deep spherical indentations; axisymmetric indenters 1. Introduction Atomic Force Microscopy (AFM) nanoindentation is a powerful method for the char- acterization of biological samples at the nanoscale [1–3]. Significant scientific results have been published during the last two decades, indicating the possibility of early diagnosis of various diseases such as cancer or osteoarthritis using AFM [2,4–11]. The force-indentation data in a typical experiment is usually processed using equations arising from Hertzian mechanics. However, these equations are valid only for materials that can be approximated as elastic half spaces [3]. In contrast, biological materials at the nanoscale are highly het- erogeneous [12]. This is a major limitation of AFM indentation since the results strongly depend on the indentation depth when using Hertzian equations for data processing [12]. As a result, the stiffness in terms of Young’s modulus (often referred to as the ‘apparent Young’s modulus’) will differ when using two different indentation depths on the same sample. Therefore, the results are user-dependent and can be difficult to reproduce. Thus, a critical question arises: “What are we calculating” when fitting the force-indentation data to classic Hertzian equations? Is this approach valid for determining the mechanical properties of biological samples at the nanoscale? Despite the mathematical limitations, it has been experimentally proven that classic fitting procedures can provide important information regarding the mechanical properties. For example, Hertzian mechanics can be used to characterize cancer cells since they are softer than healthy ones [1,2]. Furthermore, Appl. Mech. 2023, 4, 460–475. https://doi.org/10.3390/applmech4020026 https://www.mdpi.com/journal/applmech