Validation of contact mechanics models for atomic force microscopy via finite elements analysis and nanoindentation experiments

In this work, we have validated the application of Hertzian contact mechanics models and corrections for the analysis of force vs. indentation curves, acquired using spherical indenters on linearly elastic samples, by means of finite elements simulations and AFM nanomechanical measurements of polyacrylamide gels possessing a thickness gradient. We have systematically investigated the impact of both large indentations and vertical spatial confinement (bottom effect) on the accuracy of the nanomechanical analysis performed with the Hertz model for the parabolic indenter compared to the Sneddon model for the spherical indenter. We demonstrated the accuracy of the combined corrections of large indentations and bottom effect for the Hertz model proposed in the literature in the framework of linearized force vs. indentation curves acquired using spherical indenters, as well as a validation of a new linearized form of the Sneddon model. Our results show theoretically and numerically, as well as through nanomechanical measurements on gels, that the corrected Hertz model allows to accurately quantify the Young's modulus of elasticity of linearly elastic samples using spherical tips in a wide range of indentation and thickness values that are relevant for the study of cellular systems.