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

Abstract

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

Simultaneous determination of the mechanical properties and turgor of a single bacterial cell using atomic force microscopy

“…However, the Sneddon model cannot be used when indenting a spherical sample with a pyramidal or conical probe, as this can lead to ∼20% underestimation of the maximum force at a relative deformation of 10% for a sample with a Poisson's ratio of 0.49. 38 Therefore, we have developed an extended Sneddon model based on dimensionless analysis and finite element modeling. For simplification, this new model adopts a quadratic polynomial equation for the empirical function f ( h / D , α ), where the coefficients can be correlated to the tip angle as described in eqn (2)–(6).…”

“…For simplification, this new model adopts a quadratic polynomial equation for the empirical function f ( h / D , α ), where the coefficients can be correlated to the tip angle as described in eqn (2)–(6). 38 a = 39.657 α + 0.101 b = −2.767 α − 2.614 c = −0.416 α + 1.710where D is the diameter of the sample; α (in radians) is the semi-included angle of the tip; and a , b , and c are the fitting coefficients determined by numerical fitting to FE results.…”

“…In this model, the sample is bonded to the rigid plate and the conical tip with a semi-included angle of 25° is assumed rigid since it is far stiffer than the samples. 38 The tip radius is 10 nm, the same to AFM probe. A frictionless contact between the indenter and cell is used as friction effects on the measured mechanical properties are not significant.…”

“…A frictionless contact between the indenter and cell is used as friction effects on the measured mechanical properties are not significant. 38,44 The diameter of the spherical shell sample is 1 μm based on the average diameter of S. epidermidis cell of 0.5–1.5 μm. 45 FE models with different cell wall thicknesses were compared and these showed that the S. epidermidis cells measured had a wall thickness in the range 20–40 nm.…”

Measurement of Liver Stiffness Using Atomic Force Microscopy Coupled with Polarization Microscopy

“…Young's modulus, as the change in the value of Poisson's ratio from 0.4 to 0.5 results only in a 0.893x decrease in Young's modulus values calculated according to the Sneddon's equation. Given the multi-fold differences in Young's modulus between the different durations of CCl 4 treatments, the errors produced by approximating Poisson's ratio are only marginal.We used withdraw curves to calculate stiffness values, as we were interested in the elastic response of the tissue to the load provided by the cantilever rather than in the plastic response to indentation68 . Due to the viscoelastic response of the soft tissue, fitting withdraw curves might overestimate Young's modulus, which should be kept in mind.…”