An educational perspective on sphere–elastic half-space contact: Linking Hertz and Sneddon models

We have attempted to apply the computer-based finite element analysis (FEA) method to accurately measure the mechanical properties (e.g., hardness and elasticity) of a soft material by an indentation test. First, an axisymmetric model has been developed using commercially available FEA code ANSYS. The FEA model consisted of a thin Al-film resting on Si-substrate. A spherical indenter has been used to indent the Al-film, which traveled a predefined depth during the loading and unloading cycles. First, numerical simulations were conducted to get the force vs. displacement plot, which was later used to determine the modulus of elasticity and hardness of Al-film. The effects of substrate modulus and indentation depth were thoroughly investigated to determine the modulus and hardness of Al-film. The effect of friction, considered at the interface of indenter and Al-film, was found to offer minimum impact for relatively small indentation depth. The induced force on the Al-film by the indenter has been found to be higher with increasing indentation depth when friction was considered. However the contact stiffness, represented by the slope of the unloading curve, has been found almost the same with and without considering friction. The variation of substrate modulus has been found to be ineffective to capture the Al-film modulus for relatively small indentation depth. However for higher indentation depth, the substrate modulus has been found to offer profound effect to capture the film modulus. The hardness of the Al-film has also been found to be relatively unaffected with variation of substrate modulus. However, the hardness of the Al-film has been found to be higher with friction for relatively high indentation depth. Results obtained from this preliminary research are important to continue further investigation and to characterize the mechanical properties of other soft-materials, e.g., biofilms to minimize its detrimental effects and utilize its favorable aspects in industrial and biomedical applications.

Colloidal probes have been increasingly demanded for the characterization of cellular modulus in atomic force microscope because of their well-defined geometry and large contact area with cell. In this work, submicron colloidal probes are prepared by scanning electron microscope/focused ion beam and compared with sharp tip and micron colloidal probe, in conjunction with loading velocity and indentation depth on the apparent elastic modulus. NIM and cartilage cells are used as specimens. The results show that modulus value measured by sharp tip changes significantly with loading velocity while remains almost stable by colloidal probes. Also, submicron colloidal probe is superior in characterizing the modulus with increasing indentation depth, which could help reveal the mechanical details of cellular membrane and the modulus of the whole cell. To test the submicron colloidal probe further, the modulus distribution map of cell is scanned with submicron colloidal probe of 50 nm radius during small and large indentation depths with high spatial resolution. The outcome of this work will provide the effective submicron colloidal probe according to the effect of loading velocity and indentation depth, characterizing the mechanical properties of the cells.

A study of deformation behavior and phase transformation in 4H-SiC during nanoindentation process via molecular dynamics simulation

It is well known that plastic deformation is a highly nonlinear dissipative irreversible phenomenon of considerable complexity. As a consequence, little progress has been made in modeling some well-known size-dependent properties of plastic deformation, for instance, calculating hardness as a function of indentation depth independently. Here, we devise a method of calculating hardness by calculating the residual indentation depth and then calculate the hardness as the ratio of the load to the residual imprint area. Recognizing the fact that dislocations are the basic defects controlling the plastic component of the indentation depth, we set up a system of coupled nonlinear time evolution equations for the mobile, forest, and geometrically necessary dislocation densities. Within our approach, we consider the geometrically necessary dislocations to be immobile since they contribute to additional hardness. The model includes dislocation multiplication, storage, and recovery mechanisms. The growth of the geometrically necessary dislocation density is controlled by the number of loops that can be activated under the contact area and the mean strain gradient. The equations are then coupled to the load rate equation. Our approach has the ability to adopt experimental parameters such as the indentation rates, the geometrical parameters defining the Berkovich indenter, including the nominal tip radius. The residual indentation depth is obtained by integrating the Orowan expression for the plastic strain rate, which is then used to calculate the hardness. Consistent with the experimental observations, the increasing hardness with decreasing indentation depth in our model arises from limited dislocation sources at small indentation depths and therefore avoids divergence in the limit of small depths reported in the Nix-Gao model. We demonstrate that for a range of parameter values that physically represent different materials, the model predicts the three characteristic features of hardness, namely, increase in the hardness with decreasing indentation depth, and the linear relation between the square of the hardness and the inverse of the indentation depth, for all but 150 nm, deviating for smaller depths. In addition, we also show that it is straightforward to obtain optimized parameter values that give good fit to the hardness data for polycrystalline cold worked copper and single crystals of silver.

The cytoskeleton comprises microtubules, actin microfilaments, and intermediate filaments. The cytoskeleton plays a role in the mechanical function of cell adhesion, motility, and migration. Currently, cancer cell mechanics is an important topic of understanding cancer progression. Atomic force microscopy (AFM) is one of the techniques used to determine the mechanical properties of cells. In this study, AFM was used to measure the force curves of U937 cancer cells treated with two types of anticancer drugs (colchicine and taxol), and alterations in the Young’s modulus were subsequently investigated. The study results indicate that the force curve trend was dissimilar among the three types of cells (cells in the control group and in two types of anticancer drugs). In addition, when using the AFM to analyze the influence of drugs on the mechanical properties of cells, applying a small loading force from the AFM tip (small cell indentation depth) produced more accurate results. At a small cell indentation depth, the Young’s modulus order was Ecolchicine < Econtrol < Etaxol because colchicine caused the cell microtubules to disassemble, thereby reducing cell strength, whereas taxol caused the microtubules to assemble, which increased cell strength. At a large cell indentation depth, the Ecolchicine value was the largest because colchicine reduced the strength and height of the cells and the effect of the underlying substrate was reflected in and influenced the force curve.

Model of nanoindentation size effect incorporating the role of elastic deformation

When testing biological samples with atomic force microscopy (AFM) nanoindentation using pyramidal indenters, Sneddon's equation is commonly used for data processing, approximating the indenter as a perfect cone. While more accurate models treat the AFM tip as a blunted cone or pyramid, these are complex and lack a direct relationship between applied force and indentation depth, complicating data analysis. This paper proposes a new equation derived from simple mathematical processes and physics-based criteria. It is accurate for small indentation depths and serves as a viable alternative to complex classical approaches. The proposed equation has been validated for ℎ < 3R (where h is the indentation depth and R is the tip radius) and confirmed through simulations with blunted conical and pyramidal indenters, as well as experiments on prostate cancer cells. It is a reliable method for experiments where the tip radius cannot be ignored, such as in shallow indentations on thin samples to avoid substrate effects.

The Hertz equation is the most widely used equation for data processing in AFM nanoindentation experiments on soft samples when using spherical indenters. Although valid only for small indentation depths relative to the tip radius, it is usually preferred because it directly relates applied force to indentation depth. Sneddon derived accurate equations relating force and contact radius to indentation depth for shallow and deep indentations, but they are rarely used in practice. This paper presents analytical approaches to solving Sneddon’s nonlinear system. Using Taylor series expansions and a simple equation linking applied force, average contact radius, and indentation depth, we derive a two-term equation that directly relates force to indentation depth. This expression is accurate for h ≤ 1.5 R, where h is the indentation depth and R is the indenter radius, making it applicable to most practical AFM measurements on soft materials. It should be used instead of the Hertzian model for extracting Young’s modulus, thereby enhancing measurement accuracy without increasing the complexity of data processing. In addition, the results are generalized to produce a series solution that is valid for large indentation depths. The newly derived equations proposed in this paper are tested on both simulated and experimental data from cells, demonstrating excellent accuracy.

Indentation hardness is found to be related to indentation depth when indentation test is applied on homogeneous materials under small indentation depth, which shows strong size effect in the indentation. While in contrast, indentation hardness has a very limited relationship with indentation depth when it is large, showing distinct scaling relationships between hardness and material properties. Previous studies on scaling relationships under deep indentation condition of elastic-perfectly plastic homogeneous materials have been carried out systematically by finite element analysis. In this paper, a heterogeneous material, particle-reinforced matrix composite is detailed studied to investigate its scaling relationships under deep indentation with different particle positions and material properties by finite element analysis.

Abstract

The indentation stress characteristics of thin film/substrate systems by the flat cylindrical indenters have been simulated by means of the finite element method (FEM). The emphasis was put on the stress distribution ahead of the indenters. The influences of the friction coefficient between the indenter and the thin film, the thickness and hardening modulus of the thin film have been considered. It is found that the stress distribution was not affected by the friction coefficient. But the influence of the thickness and hardening modulus of the thin film on the stress distribution was obvious. At small indentation depth, the plastic deformation occurs at the edge of the indenter only, and the zone will propagation both vertically and laterally with the indentation depth increasing. When the indentation depth reaches a certain value, the thin film at the interface will occur the deformation plastic zone for the case studied in this paper. At lager depths, the two plastic zones will connect, and then the plastic zone propagates along the lateral direction. Beside, it is also found that the maximum of the Mises stress and the shearing stress on the interface occur at 0.8r and r(r is the radius of the indenter), respectively.

Indentation-based assessment of the dependence of geometrically necessary dislocations upon depth and strain rate in FCC materials

Plastic strain and strain gradients at very small indentation depths

Indentation has several advantages as a loading mode for determining constitutive behavior of soft, biological tissues. However, indentation induces a complex, spatially heterogeneous deformation field that creates analytical challenges for the calculation of constitutive parameters. As a result, investigators commonly assume small indentation depths and large sample thicknesses to simplify analysis and then restrict indentation depth and sample geometry to satisfy these assumptions. These restrictions limit experimental resolution in some fields, such as brain biomechanics. However, recent experimental evidence suggests that conventionally applied limits are in fact excessively conservative. We conducted a parametric study of indentation loading with various indenter geometries, surface interface conditions, sample compressibility, sample geometry and indentation depth to quantitatively describe the deviation from previous treatments that results from violation of the assumptions of small indentation depth and large sample thickness. We found that the classical solution was surprisingly robust to violation of the assumption of small strain but highly sensitive to violation of the assumption of large sample thickness, particularly if the indenter was cylindrical. The ramifications of these findings for design of indentation experiments are discussed and correction factors are presented to allow future investigators to account for these effects without recreating our finite element models.

A numerical fracture analysis of indentation into thin hard films on soft substrates