Hertz Model Research Articles

A comprehensive study of AFM stiffness measurements on inclined surfaces: theoretical, numerical, and experimental evaluation using a Hertz approach

Atomic Force Microscopy (AFM) is a leading nanoscale technique known for its significant advantages in the analysis of soft materials and biological samples. Traditional AFM data analysis is often based on the Hertz model, which assumes perpendicular indentation of a planar sample. However, this assumption is not always valid due to the varying geometries of soft materials, whether natural, synthetic or biological. In this study, we present a new theoretical model that incorporates correction coefficients into Hertz’s model to account for cone-like and spherical probes, and to consider local tilt at the probe-sample interface. We validate our model using finite element analysis (FEA) simulations and experimental AFM measurements on tilted polyacrylamide gels. Our results highlight the need to include local tilt at the probe-sample contact to ensure accurate AFM measurements. This represents a step forward in our understanding of the elastic properties at the surface of soft materials in the broadest sense.

Falling weight impact acceleration-time signals analysis for road modulus detection: Theoretical and experimental investigations

Structural modulus detection is essential for construction and maintenance in road engineering. Falling weight impact loading method has significant strengths in large-scale, continuous, and rapid detection. To provide theoretical support for this, theoretical analysis and falling weight impact experiments were employed in this study to investigate the influence of road structure parameters and falling weight impact parameters on impact acceleration-time signals. First, the applicability of the Hertz model in road detection was analyzed, and theoretical formulas for peak acceleration amax and impact duration t2 were established, considering the perfectly elastic collision between falling weight and half-space. Second, the influence of layered structures’ properties on impact acceleration-time curves was studied through model box impact experiments. Third, field tests were conducted to study engineering applications. The results showed that the influence patterns were similar in both theoretical and experimental studies, with differences reflected in the time of compression and restitution phases due to the road materials’ viscosity and plasticity. Additionally, in indoor experiments, amax showed an excellent power function relationship with structural layer modulus, with R2 more than 0.95, and the contribution of each layer’s modulus to amax decreased from top to bottom. Finally, amax and resilient modulus of an old road also followed a good power function relationship, with R2 around 0.76. This study revealed that amax was recommended for road surface modulus detection based on solid theoretical and experimental support.

Application of Flores model considering variable recovery coefficient in dynamics analysis of ball bearings

PurposeThis paper aims to select an appropriate contact force model and apply it to the interaction model between the balls and the cage in the rolling bearings to describe the elastic–plastic collision phenomena between the two.Design/methodology/approachTaking the ball–disk collision mode as an example, several main contact force models were compared and analyzed through simulation and experiment. In addition, based on the consideration of yield strength of materials and initial collision velocity, a variable recovery coefficient model was proposed, and its validity and accuracy were verified by the ball–disk collision experiments. Then, respectively, the Flores model and the Hertz model were applied to the interaction between the balls and the cage, and the dynamics simulation results were compared.FindingsThe results indicate that the Flores model has good regression of recovery coefficient, indicating good applicability for both elastic and elastic–plastic contacts and can be applied to the contact collision situations of various materials. Under certain working conditions, there are significant differences in the dynamics results of rolling bearings simulated using the Flores model and Hertz model, respectively.Originality/valueThis paper applies the Flores model with variable recovery coefficients to the dynamics simulation analysis of ball bearings to solve the elastic–plastic collision problem between the rolling elements and the cage that cannot be reasonably handled by the Hertz model.Peer reviewThe peer review history for this article is available at: https://publons.com/publon/10.1108/ILT-04-2024-0138/

Role of adhesion in the mechanics of pathogenic Listeria monocytogenes EGDe as a function of the pH of growth.

Atomic force microscopy was utilized to estimate the adhesion strengths to silicon nitride as well as the cellular elasticities of pathogenic Listeria monocytogenes EGDe cells cultured in media adjusted to five different pH conditions of growth (5, 6, 7, 8, and 9) under water with 0.0027 fixed ionic strength. Particularly, the role of adhesion on the bacterial elastic properties was investigated. The nonadhesive Hertz model of contact mechanics was used to extract Young's moduli of elasticity of bacterial cells from the approach force-indentation data. Additionally, the adhesive models of contact mechanics: Johnson-Kendall-Roberts (JKR) and Derjaguin-Muller-Toporov (DMT) were used to estimate Young's moduli of elasticity of bacterial cells from the retraction force-indentation data. Our results indicated that adhesion to silicon nitride was the highest for cells cultured at a pH of 7. Similarly, bacterial cells cultured at pH 7 were characterized by the highest Young's moduli of elasticities compared to the lower or higher pH conditions of growth. Young's moduli of elasticities estimated from the Hertz model were stiffer than those estimated using JKR or DMT models. As the adhesion between bacterial cells and indenters increased, the difference between the Hertz model and JKR or DMT models estimates of Young's moduli of elasticity increased as well. Contradicting the current norm of using the Hertz model to quantify bacterial elasticity in the literature, our results highlight the extreme importance of utilizing contact mechanics models with adhesion components in them such as the JKR and DMT models to estimate bacterial elasticity.

Mechanoluminescence of ZnS: Cu@Al2O3 enabled optical fiber microlens passive tactile sensor for hardness recognition.

We propose an optical fiber microlens (OFM) passive tactile sensor (OFMPTS) for hardness recognition. The sensor features a core-shell structure, with an OFM as the core and an elastic mechanoluminescence (ML) matrix shell, which is composed of ZnS: Cu@Al2O3 doped with 10 nm SiO2 particles and polydimethylsiloxane (PDMS). Utilizing the Hertz model, the ML intensity of the sensor is correlated to the elastic modulus of the sample, which enables precise hardness detection. The microlens fiber structure significantly enhances photon collection efficiency, thereby allowing for effective coupling of the ML signal. In press mode, OFMPTS differentiates between five PDMS hardness levels. It can also operate in scan or tap mode, identifying hidden foreign bodies and tissue masses through response curve analysis. The sensor, which requires no external light source, expands the capabilities of optical hardness measurement.

Research on Spatial Developable Mechanism Considering Revolute Clearance Joints with Irregular Rough Surfaces

Due to assembling, manufacturing errors, and wear, irregular rough surfaces inevitably exist in joints, which will increase the contact stiffness nonlinearity at the joint of the spatial developable mechanism, and the traditional contact force model is difficult to accurately predict the contact force change of irregular rough surface clearance. Aiming at the difficulty of accurately predicting the dynamic behavior of the spatial developable mechanism caused by rough joint clearance, an improved clearance contact force modeling method based on an uncoordinated contact model is proposed in this paper. The influence of rough peak distribution on the contact area is analyzed. The stiffness of the traditional Hertz model is modified based on the probability distribution density function of the rough peak, and an improved contact force model based on dynamic contact stiffness is established. Based on the Lagrange multiplier method, the dynamics model of spatial developable mechanism is constructed. Based on the model, the dynamic analysis of the spatial expansion mechanism is carried out, and the influence of the roughness of the contact surface and the size of the clearance on the dynamic characteristics of the system are explored; as well, the influence of different clearance parameters on the dynamic characteristics of the development mechanism is revealed. Through the analysis, this paper provides a theoretical basis for predicting the effect of a spatial revolute clearance joint on the dynamic characteristics of the mechanism and lays a good foundation for the manufacture and application of the mechanism.

Quantification of Young's modulus of kaolin, sodalite and nanocomposite based polycaprolactone/sodalite using atomic force microscopy

The mechanical properties of clays, zeolite, and nanocomposite reinforced by these minerals are unique regarding their states and functions. Nevertheless, it is difficult to acknowledge the degrees of mechanical properties because of the nano-size of these materials and the broad distribution of these properties. This study aims to the quantification of Young’s modulus of three materials, namely kaolin, sodalite, and nanocomposite PCL/sodalite using the atomic force microscope (AFM). The Oliver-Pharr and Hertz models are considered for this quantification. The values of these modulus ranging from 11.19 to 69.15 GPa for the Oliver-Pharr model and 10.89 to 127.09 GPa for Hertz model (kaolin). On the other side, the Young’s modulus of sodalite varies between 69.02 and 12.01 GPa for the Hertz and the Oliver-Pharr models (sodalite). This modulus for nanocomposite ranges from 1.75 to 6.66 GPa and 1.63 to 45.36 GPa for the cited models, This research presents a simple and easy way to calculate this modulus without resorting to Nano-indentation. Moreover, this technique was also used to investigate the morphology of these materials. Therefore, it was deduced that the mechanical and morphological properties of the studied materials are completely different, especially nanocomposites, even though they have the same origin, which is kaolinite.

Effects of Vitamins E and D on Mechanical Properties of Breast Cancerous Cells

Background: Several investigations have demonstrated that vitamins can treat or prevent cancer by altering actin filaments and inhibiting cell migration and proliferation. Vitamins D and E are fat-soluble. This research aims to determine the short-term impact of vitamin D and E on the mechanical properties of breast cancer cells before comparing them with normal breast cells. Methods: Atomic force microscopy (AFM) was used to examine the deformation of MCF-10 normal breast cells, MCF-7 breast cancer cells, and MCF-7 breast cancer cells treated with 0.03 μM vitamin D and 16 μM vitamin E solutions. Young's modulus was calculated employing the Hertz model to determine cell stiffness. Results: The Young's modulus of vitamin D-treated cancer cells (585.8 Pa) was substantially similar to that of normal cells (455.6 Pa). Nevertheless, vitamin E treatment did not affect Young's modulus of cancer cells, which remained remarkably similar to that of untreated cancer cells (216.6 and 203.4 Pa, respectively). Conclusion: Unlike vitamin E, vitamin D enhances the stiffness of tumor cells and makes their mechanical properties similar to normal cells by interfering with actin filaments and cell skeletons, which may inhibit tumor cell migration. Based on these findings, vitamin D appears to be an effective drug for cancer treatment.

Size Matters: Rethinking Hertz Model Interpretation for Cell Mechanics Using AFM.

Cell mechanics are a biophysical indicator of cell state, such as cancer metastasis, leukocyte activation, and cell cycle progression. Atomic force microscopy (AFM) is a widely used technique to measure cell mechanics, where the Young modulus of a cell is usually derived from the Hertz contact model. However, the Hertz model assumes that the cell is an elastic, isotropic, and homogeneous material and that the indentation is small compared to the cell size. These assumptions neglect the effects of the cytoskeleton, cell size and shape, and cell environment on cell deformation. In this study, we investigated the influence of cell size on the estimated Young's modulus using liposomes as cell models. Liposomes were prepared with different sizes and filled with phosphate buffered saline (PBS) or hyaluronic acid (HA) to mimic the cytoplasm. AFM was used to obtain the force indentation curves and fit them to the Hertz model. We found that the larger the liposome, the lower the estimated Young's modulus for both PBS-filled and HA-filled liposomes. This suggests that the Young modulus obtained from the Hertz model is not only a property of the cell material but also depends on the cell dimensions. Therefore, when comparing or interpreting cell mechanics using the Hertz model, it is essential to account for cell size.

Dynamics of a bouncing capsule: An impulse model vs a Hertzian model

A capsule-shaped object, unlike a sphere, can rebound to a higher height than it is dropped from if it has an initial spin. In this paper, we compare two quantitative models of this phenomenon: an impulse-based analytic model and a force-based Hertzian model. We experimentally studied the effects of impact angular velocity and contact angle on the capsule's rebounding velocity and angular velocity. Our experimental findings showed successful agreement with our Hertzian model but not with our impulse model. We attribute the comparative success of the Hertzian model to its intricate treatment of the slip–stick behavior of friction during the collision.

Percolation to jamming in polymethylvinylsiloxane/silica nanocomposites

Researchers have conducted extensive experimental and theoretical studies on the jamming transition of colloidal and granular suspensions, but the existence and characteristics of jamming transition in nanoparticle-filled polymers remain elusive. In this work, two kinds of polymethylvinylsiloxane/silica nanocomposites were studied. A liquid-solid transition was found at the silica volume fraction of about 5 vol%, corresponding to the percolation of particle agglomerates. At higher silica content (about 10 vol%), a solid-solid transition was found in yield stress, equilibrium modulus, and the contact stress between agglomerates. The characteristic length of the particle network and inter-molecular distance also showed a transition at this transition. This transition corresponds to the simultaneous jamming of particles inside agglomerates and between agglomerates. This bi-level jamming behavior is distinct from the well-known jamming behavior in soft or hard particulate suspensions. Moreover, the dependence of the yield stress and equilibrium modulus on the distance to the jamming concentration differs from the simulations using simple interaction models like harmonic and Hertzian models. It implies that the interfacial viscoelastic layer may offer additional attractive and adhesive interactions between nanoparticles, which is critical in the jamming of polymer nanocomposites.

A Linear Fit for Atomic Force Microscopy Nanoindentation Experiments on Soft Samples

Atomic Force Microscopy (AFM) nanoindentation is a powerful technique for determining the mechanical properties of soft samples at the nanoscale. The Hertz model is typically used for data processing when employing spherical indenters for small indentation depths (h) compared to the radius of the tip (R). When dealing with larger indentation depths, Sneddon’s equations can be used instead. In such cases, the fitting procedure becomes more intricate. Nevertheless, as the h/R ratio increases, the force–indentation curves tend to become linear. In this paper the potential of using the linear segment of the curve (for h > R) to determine Young’s modulus is explored. Force–indentation data from mouse and human lung tissues were utilized, and Young’s modulus was calculated using both conventional and linear approximation methods. The linear approximation proved to be accurate in all cases. Gaussian functions were applied to the results obtained from both classic Sneddon’s equations and the simplified approach, resulting in identical distribution means. Moreover, the simplified approach was notably unaffected by contact point determination. The linear segment of the force–indentation curve in deep spherical indentations can accurately determine the Young’s modulus of soft materials at the nanoscale.

An electro-mechanical contact model for particulate systems

This study presents an electro-mechanical contact model to investigate the electrical response within a particulate system under mechanical loading. The model considers the electrical resistance across particle-to-particle and particle-to-wall contacts by formulating bulk and contact resistances, separately. Hertz and Holm's models are utilised for the contact resistance in the overlapping region. The bulk resistance for a particle is estimated considering particle geometry transformations during the compression. The proposed electro-mechanical contact model is verified and validated against published analytical solutions and experimental data. Finally, the contact model is applied to simulate a high-pressure torsion test to investigate the influence of three different materials on the electrical conduction properties of the system after particle fragmentation. The intensity distributions of electric potential, current and resistance of the granular system are analysed at both particle and system scales. An electroactivity index is proposed for evaluating the contribution of each fragment sieve cut on the bulk electrical conduction behaviour.

A thick wall concept for robust treatment of contacts in DEM simulation of highly polydisperse particulate systems

Modelling particulate systems with the Discrete Element Method (DEM) is an established practice, both in the representation and analysis of natural phenomena and in scale-up and optimization of industrial processes. Since the method allows tracking individual particles, each element can possess geometrical, physical, mechanical or chemical surface properties different from those of the other particles. One example is a polydisperse particulate system, i.e., characterized by a size distribution, opposed to the idealized monodisperse case. In conventional DEM, a softer particle stiffness is commonly adopted to reduce the computational time. It might happen that artificially soft particles, when colliding against a wall boundary, exhibit such large, unrealistic overlap that they “pass through” the wall and exit the domain. In the case of highly polydisperse systems, this often occurs when fine particles are pushed against the wall by coarse particles with masses several orders of magnitude larger. In the manuscript, a novel method is proposed, named thick wall, to allow the particles in contact with the walls to experience relatively large overlaps without ending up ejected out the domain. In particular, a careful way to calculate the particle-wall overlap and force unit vector can accommodate normal displacements larger than the maximum usually allowed, i.e., typically the particle radius, thereby preventing particles from being expelled from the domain. First, critical velocities for which single particles and pairs of fine/coarse particle escape the domain are analytically characterized using the linear and the Hertz models. The thick wall concept is then introduced and its effect on the maximum critical velocity is demonstrated with both contact models. Finally, application to pharmaceutical powder composed of carrier (coarse) and active pharmaceutical ingredient (API) (fine) particles in a shaken capsule prove this to be an example of vulnerability to the phenomenon of fine particle ejection and to significantly benefit from the thick wall modification.

Investigation of nanoparticles on cell elasticity using the nanoindentation method

Nowadays, the biomechanics of living cells has been the focus of attention, as high-resolution techniques such as atomic force microscopy (AFM) allow the properties of living cells to be studied with nanoscale and precision. There are a lot of results showing that the mechanical changes in the cell or its extracellular matrix (ECM) are used as a biomarker of pathological changes. Therefore, cell stiffness is an index that evaluates the status of cell cultures. Elastic properties of cells can be deduced by nanoindentation experiments with AFM. The quantitative parameter that determines cellular deformation in the nanoindentation experiment is Young’s modulus. In recent studies, it has also been shown that gold nanostructures (Au-NP) can play an active role in the detection-diagnosis of diseases and/or targeting tumor and inflammatory cells for therapeutic reasons. In this study, first of all, 2D axial biomechanical models of keratinocytes and glioma cells were made with the Finite Element Method (FEM modeling) and their mechanical properties were calculated. The deformation and mechanical changes that occur in the cell during the uptake and distribution of gold nanostructures were also investigated through the modeling. Keratinocyte cells modeled in the study are epidermal cells that make keratin, and Glioma is a group of tumor cells formed in glial cell cells, which are the supporting tissue of the brain. The Young Modulus values ​​calculated using the Hertz model and elasticity values obtained from different techniques in the literature were compared with the simulation outputs.

Shock-Induced Microstructural Evolution, Phase Transformation, Sintering of Al-Ni Dissimilar Nanoparticles: A Molecular Dynamics Study.

Molecular dynamic simulations have been performed to explore contact behavior, microstructure evolution and sintering mechanism of Al-Ni dissimilar nanoparticles under high-velocity impact. We confirmed that the simulated contact stress, contact radius, and contact force under low-velocity impact are in good agreement with the predicted results of the Hertz model. However, with increasing the impact velocity, the simulated results gradually deviate from the predicted results of the Hertz model due to the elastic-plastic transition and atomic discrete structure. The normalized contact radius versus strain exhibits a weak dependence on nanosphere diameter. Below a critical velocity, there are very few HCP atoms in the nanospheres after thermal equilibrium. There are two different sintering mechanisms: under low-velocity impact, the sintering process relies mainly on the dislocation slip of Al nanospheres, while the dislocation slip of Ni nanospheres and the atomic diffusion of Al nanospheres predominate under high-velocity impact.

Rationally designed chitosan-interpenetrated cryobeads functionalized with polyacrylamide chains: Comparative analysis by Hertzian model and rubber elasticity

Optimization of the synthesis of polymer microspheres and millimeter-sized gel beads has gained importance due to efficiency and design advantages in applications. A systematic study is presented to allow for a molecular-based understanding of elasticity of crosslinked-chitosan (CS) beads. Unique results were obtained examining the effect of polymerization temperature and gel-preparation form on physico-mechanical properties of CS-incorporated poly (N-isopropylacrylamide‑sodium acrylate)/polyacrylamide, PNIPA/PAAm-CS, beads. ATR-FTIR, and thermogravimetric analysis results confirmed the successful preparation and enhanced thermal stability of CS-based gel beads in the form of semi-IPN. The structural changes of semi-IPN gels were studied based on powder X-ray diffraction analysis. After being incorporated with CS, the cryopolymerization was carried out under cryo-conditions, and PNIPA/PAAm structure became much more resistant to mechanical load. Addition of CS to semi-IPN structure caused a 2-fold increase in compressive elastic modulus, while the gel preparation under cryoconditions also improved the mechanical properties considerably by lowering the polymerization temperature. The scaling parameter calculations estimated by Hertz model for PNIPA/PAAm-CS semi-IPN cryobeads are related to displacement of compression force with an exponent of 1.63 ± 0.19. As cryobead diameter increased, swelling degree tendency increased, while a decrease in modulus was observed with increasing swelling. The presence of CS in semi-IPNs improved pH-response in an acidic environment, but stiffness of CS reduced the shrinkage ability of cryobeads upon increasing swelling temperature. Based on the interaction between semi-IPN structure and salt solutions, an improvement in elastic modulus was observed in various ammonium salts and sodium tripolyphosphate solution. On-off switching of cryobeads was a reversible process that was consistent with changes in ammonium salt concentration. Qualitative comparisons with experimental results showed that the prepared cryobeads can be designed as drug release carriers by ionic strength-switching modulation.

The effects of composition and microstructure on compressive strength of Ag-Au nanoparticles

Hemispherical nanoparticles of Ag-33 at.% Au and Ag-60 at.% Au alloys were fabricated by solid state dewetting of Ag-Au bilayers deposited on a sapphire substrate. The in-situ microcompression tests of the particles in the scanning electron microscope have revealed the plastic deformation behavior typical of plasticity which is controlled by the nucleation of new dislocations: high elastic strain reaching about 5% of the particle diameter, followed by a catastrophic particle collapse. The modulus-normalized average contact pressure at the onset of plasticity calculated employing the isotropic Hertz model was concentration-independent for pure Ag and two studied alloys. The atomistic molecular dynamics simulations of shear and compressive deformation of bulk Ag-Au alloys revealed a weak solid solution hardening effect for dislocation nucleation stress and its absence for the nucleation strain. The particles strength and critical strain obtained in simulations were in excellent quantitative agreement with the experimental results. The lack of solid solution hardening or weakening effects in the Ag-Au has been discussed in terms of small atomic size mismatch between Ag and Au. Finally, the effect of the twin boundaries observed in the Ag-Au particles on their strength was discussed.

Analysis of tyre-pavement contact behaviour of heavy truck in full-scale test

ABSTRACT Tyre-pavement contact behaviour is one of the core issues in pavement engineering. In this study, the tyre-pavement contact behaviour was investigated in full-scale tests. The pavement texture was characterised by a 3D scanner and the sand patch method. The data of tyre-pavement contact stress were collected by a pressure distribution measurement system. The characteristics of the tyre-pavement contact stress distribution were explored by introducing the Boltzmann model. The feasibility of the Hertz model in assessing the tyre-pavement contact behaviour was evaluated. The results showed that the tyre-pavement carries a larger load by extending the contact area in longitudinal direction and increasing the contact stress mainly at the lateral edges. The pavement texture has significant effects on the contact behavior. Under the regular tyre vertical force range, the value of contact stress is within 1000 kPa for about 80% of the tyre-pavement contact area, but the maximum contact stress can be up to 3000 kPa. The Boltzmann model can be used to fit the contact stress distribution, and some parameters of the model have linear relationships with pavement surface macrotexture. The Hertz model can reflect the average state of tyre-pavement contact behaviour.

A Theoretical Contact Model for Rough Elastic Spheres

Abstract Investigating the contact of rough spheres is constructive to wide engineering applications and scientific research. However, the coexistence of global curvature and surface roughness brings difficulty in the analysis of this problem. In this work, we present a theoretical model for the contact between a rough elastic sphere and a rigid plane. As a fundament, the whole-range contact of nominally flat Gaussian rough surfaces is dealt with using a finite element method. With the load–area relation for the contact of nominally flat rough surfaces, the local real contact area of rough spherical contact is determined, in which the local pressure is given by the Hertzian model. Then, the total real contact area of rough spherical contact is obtained by integrating over the Hertzian contact area. It is found that the load–area relation for the contact of rough spheres is linear at light loads and converges to the result of smooth spherical contact as the load increases. This rough spherical contact model is validated through direct finite element simulations.