A modified Sneddon model for the contact between conical indenters and spherical samples

Indentation techniques have proven to be effective to characterize the mechanical properties of materials. For the elastic deformation, the commonly used models are Hertz model and Sneddon model. However, neither of them works for indenting the spherical samples using the pyramid or conical indenter. Therefore, one modified Sneddon model has been developed to determine the Young’s modulus of spherical samples from indentation results. In this study, the effects of sample diameter and indenter angles on indentation tests were investigated by finite element method (FEM). The empirical correction parameters in the new mathematical model were introduced based on dimensional analysis and determined by the numerical fitting to FEM results. Experimental tests with different conical indenters have demonstrated that the new model is capable to reliably determine the Young’s modulus of the spherical samples. The new model can fill the gap of the contact mechanics and enrich the experimental solid mechanics for the interpretation of indentation results. Graphic abstract

Is it necessary to calculate Young’s modulus in AFM nanoindentation experiments regarding biological samples?

Background: The determination of the mechanical properties of biological samples using Atomic Force Microscopy (AFM) at the nanoscale is usually performed using basic models arising from the contact mechanics theory. In particular, the Hertz model is the most frequently used theoretical tool for data processing. However, the Hertz model requires several assumptions such as homogeneous and isotropic samples and indenters with perfectly spherical or conical shapes. As it is widely known, none of these requirements are 100 % fulfilled for the case of indentation experiments at the nanoscale. As a result, significant errors arise in the Young’s modulus calculation. At the same time, an analytical model that could account complexities of soft biomaterials, such as nonlinear behavior, anisotropy, and heterogeneity, may be far-reaching. In addition, this hypothetical model would be ‘too difficult’ to be applied in real clinical activities since it would require very heavy workload and highly specialized personnel. Objective: In this paper a simple solution is provided to the aforementioned dead-end. A new approach is introduced in order to provide a simple and accurate method for the mechanical characterization at the nanoscale. Method: The ratio of the work done by the indenter on the sample of interest to the work done by the indenter on a reference sample is introduced as a new physical quantity that does not require homogeneous, isotropic samples or perfect indenters. Results: The proposed approach, not only provides an accurate solution from a physical perspective but also a simpler solution which does not require activities such as the determination of the cantilever’s spring constant and the dimensions of the AFM tip. Conclusion: The proposed, by this opinion paper, solution aims to provide a significant opportunity to overcome the existing limitations provided by Hertzian mechanics and apply AFM techniques in real clinical activities.

Contact Model for Incompressible Neo-Hookean Materials Under Finite Spherical Indentation

In this paper, a contact model is proposed to predict the contact response of an incompressible neo-Hookean half-space under finite spherical indentations. The axisymmetric finite element (FE) model is created to simulate the contact behaviors. Inspired by the numerical results, the radius of the contact circle is derived. The contact force is then obtained by modifying the radius of the contact circle of the Hertz model. The format of the distribution of the contact pressure is also developed according to the Hertz model. A parameter, determined by fitting the numerical results, is introduced to characterize the effect of the indentation depth on the shape of the distribution function of the contact pressure. The newly proposed contact model is numerically validated to predict well the contact behaviors, including the contact force, the radius of the contact circle, and the distribution of the contact pressure, for the incompressible neo-Hookean half-space under spherical indentation up to the indenter radius. However, the Hertz model is verified to offer acceptable predictions of the contact behaviors for the incompressible neo-Hookean materials within the indentation depth of 0.1 times of the indenter radius.

A discussion regarding the application of the Hertz contact theory on biological samples in AFM nanoindentation experiments

: Atomic Force Microscopy (AFM) Nanoindentation procedure regarding biological samples poses significant challenges with respect to the accuracy of the provided results. These challenges are related to the inhomogeneity of biological samples, various uncertainties in experimental methods and certain approximations regarding the theoretical analysis. The most commonly used theoretical model for data processing at the linear elastic regime regarding biological samples is the Hertz model. This paper focuses on the investigation of the resulting errors of the basic equation of the Hertz theory that depend on the ratio, indentation depth/indenter’s radius regarding the Young’s modulus calculation. Several examples in the literature that do not take into account the value of the ratio indentation depth/indenter’s radius are reported and the related errors are presented and discussed. In addition, an extended new equation is derived which takes into account the influence of the aforementioned ratio on the calculation of the Young’s modulus and can be easily used for calculations. Moreover, a rational explanation, regarding the extended differences of the Young’s modulus calculations using the same experimental results when these are processed using the Hertz model and the Oliver & Pharr analysis (which is the general model that applies for any axisymmetric indenter) is provided. In conclusion, the derived equation is further combined with equations which take into account the shape of the sample in order to provide a complete and reliable theoretical tool which can be generally applied in order to reduce the errors produced by the current methodology.

Cortical cell stiffness is independent of substrate mechanics

Cortical stiffness is an important cellular property that changes during migration, adhesion, and growth. Previous atomic force microscopy (AFM) indentation measurements of cells cultured on deformable substrates suggested that cells adapt their stiffness to that of their surroundings. Here we show that the force applied by AFM onto cells results in a significant deformation of the underlying substrate if it is softer than the cells. This ‘soft substrate effect’ leads to an underestimation of a cell’s elastic modulus when analyzing data using a standard Hertz model, as confirmed by finite element modelling (FEM) and AFM measurements of calibrated polyacrylamide beads, microglial cells, and fibroblasts. To account for this substrate deformation, we developed the ‘composite cell-substrate model’ (CoCS model). Correcting for the substrate indentation revealed that cortical cell stiffness is largely independent of substrate mechanics, which has significant implications for our interpretation of many physiological and pathological processes.

Evaluation Method of Mechanical Properties of Living NSCLC Cells Based on Nano-indentation

Background:In AFM study of cell mechanical properties, the apparent elastic modulus of a cell is affected by many factors, especially the AFM tip geometry, indentation site of the cell, the application of the mathematical model and testing conditions.Background:In AFM study of cell mechanical properties, the apparent elastic modulus of a cell is affected by many factors, especially the AFM tip geometry, indentation site of the cell, the application of the mathematical model and testing conditions.Methods:In this study, indentation experiments of living cells under different conditions were performed aiming to build an accurate evaluation system of mechanical properties of lung cancer cells based on AFM. Comparisons of the effects of spherical and pyramid AFM tips, Hertz model of semiinfinite and finite thickness, cell nuclear and cytoplasmic indentation regions on the cell apparent elastic modulus were accomplished.Methods:In this study, indentation experiments of living cells under different conditions were performed aiming to build an accurate evaluation system of mechanical properties of lung cancer cells based on AFM. Comparisons of the effects of spherical and pyramid AFM tips, Hertz model of semiinfinite and finite thickness, cell nuclear and cytoplasmic indentation regions on the cell apparent elastic modulus were accomplished.Results:Compared with the calculated results by spherical tip, the elastic modulus distribution of non-small lung cancer cells (NSCLC) by pyramid tip was observed to be similar while the absolute values increased obviously, which were more than twice the numerical values by the spherical tip (p<0.05). The apparent elastic modulus values were the overvalued cause of the underestimation of the contact region in pyramidal tip measurement. Two different indentations over nucleus or lamellipodium of NCI-H520 cell and NCI-H1299 cell were analyzed. Consequently, the exact elastic modulus over the nucleus area can be calculated accurately using the semi-infinite Hertz model while the finite thickness Hertz model should be used for elasticity assessment of cell lamellipodium with a small thickness.Results:Compared with the calculated results by spherical tip, the elastic modulus distribution of non-small lung cancer cells (NSCLC) by pyramid tip was observed to be similar while the absolute values increased obviously, which were more than twice the numerical values by the spherical tip (p<0.05). The apparent elastic modulus values were the overvalued cause of the underestimation of the contact region in pyramidal tip measurement. Two different indentations over nucleus or lamellipodium of NCI-H520 cell and NCI-H1299 cell were analyzed. Consequently, the exact elastic modulus over the nucleus area can be calculated accurately using the semi-infinite Hertz model while the finite thickness Hertz model should be used for elasticity assessment of cell lamellipodium with a small thickness.Conclusion:This evaluation system provides technological support for accurate evaluation of viscoelastic properties of living cancer cells.Conclusion:This evaluation system provides technological support for accurate evaluation of viscoelastic properties of living cancer cells.