Hertz Model Research Articles

Mapping Nanomechanical Properties near Internal Interfaces in Biological Materials

ABSTRACTModulus mapping using nanoDMA (Dynamic Mechanical Analysis) is a recently developed technique based on a nanoindentation instrument equipped with an AFM-like piezoscanner and dynamic force modulation system. The surface properties, storage and loss moduli are quantified based on the Hertz model for the contact mechanics of the sample-tip configuration. In this approach, the applied load, topography features, and their size may have a pronounced effect on the obtained results. In order to demonstrate that, internal interfaces of deep sea sponge (Monorhaphis chuni), which comprises alternating layers of relatively thick (4 μm in average) biosilica and thin (60 nm) organic material, were characterized using the nanoDMA modulus mapping technique. Experimental data were analyzed in tight interrelation with finite element simulations. This combination allowed us to evaluate elastic modulus of a 60 nm wide organic layers in M. chuni.

Size effects on the nanomechanical properties of cellulose I nanocrystals

Abstract

Verification of two minimally invasive methods for the estimation of the contact pressure in human vocal folds during phonation

The contact pressure on the vocal fold surface during high pitch or amplitude voice production is believed to be one major source of phonotrauma. Models for the quantitative estimate of the contact pressure may be valuable for prevention and treatment. Various indirect and minimally invasive approaches have been purported to estimate contact pressure. But the accuracy of these methods has not yet been objectively verified in controlled laboratory settings. In the present study, two indirect approaches for the estimation of the contact pressure were investigated. One is based on a Hertzian impact model, and the other on a finite element model. A probe microphone was used for direct measurements of the contact pressure and verifications of the indirect approaches. A silicone replica of human vocal folds was used as a test bed. Consistent contact pressure estimations were obtained using all three methods. The advantages and disadvantages of each approach for eventual clinical applications are described.

교량 상부구조물의 탄성받침 설치에 따른 충돌특성 분석

Seismic structure pounding between adjacent superstructures may induce the destruction of pier and bridge superstructures and cause local damage that leads to the collapse of the whole bridge system. The pounding problem is related to the expansion of joints, gap distance and seismic response of the abutments. In this research, methods of the contact element approach, the linear spring model, the Kelvin-Voigt model and the Hertz model were studied to analyse the pounding characteristics. The shaking table test for a model specimen such as a bridge superstructure with elastomeric bearings was performed to evaluate the contact element approach methods. Relationships between the time history response from the numerical analysis results and the measured response from the shaking table test are compared. The experimental results were not well matched with the numerical analysis results using the existing pounding stiffness models. Therefore, in this study, coefficients are proposed to calculate the appropriate pounding stiffness ratio.

Space Robot Self-Assembling Parameter Analysis

With the depth of space exploration and applications, the space robot self-assembling has become more and more important for space on-orbit service. Parameter analysis is an indispensable work of assembling process design. The dynamic model of the space robot with target is established firstly, then the Hertz model is introduced as the collision model between the docking link and the capturing cone, and the operational space control is applied to achieve the assembling operation control under the Cartesian space. Then the paper focuses on the discussion and analysis of the influences of the parameters such as the stiffness coefficient, friction coefficient, leading speed coefficient and vertical speed damping coefficient on the assembling characteristics. The simulation results would provide a basis for assembling control and parameter design and have an important academic value and engineering significance on space service.

Magnetic and mechanical properties of rolled-up Au/Co/Au nanomembranes with multiple windings

Rolled-up Au/Co/Au microtubes with up to three windings were fabricated by the combination of strain engineering, conventional photolithography, and electron beam deposition. First, magnetization properties of the initial 2D film arrays and the corresponding tube arrays were studied and strong influences of magnetostrictive and shape anisotropy are observed. Second, the mechanical deformation was examined by an instrumented indentation technique at the nanoscale and analyzed by contact mechanics theory. The loading curve fitting in the elastic regime by the Hertz model provides a first approximation of the nanomembrane radial elastic modulus of about 135 GPa.

Effect of Atomic-Scale Roughness on Contact Behavior

Large-scale molecular dynamics simulations are performed to study the effect of atomic-scale surface roughness on nano-contact. The modeling system consists of rigid spherical tips with different surface roughness and elastic flat substrate. Our results show that atomic-scale multi-asperity can change the contact behavior from consistent with the Hertz model to the Persson model. However, adhesion will reduce the influence of surface roughness, to the extent that the two tips with different roughness show similar variations of real contact area with applied load. The maximum compression and tensile stress of the rough tip is about 2 times and 1.5 times that of the smooth one, respectively. Moreover, the rough tip exhibits larger repulsive force and attractive force in the entire range of simulated load. Our simulations suggest that pull-off force cannot characterize the extent of the influence of adhesion on contact behavior at the nanoscale.

Effect of aminoglycoside on mechanical properties of zwitterionic phospholipid vesicles

The effect of the aminoglycoside (streptomycin) incorporation on the nanomechanical properties of pure dipalmitoylphosphatidylcholine (DPPC) vesicles was studied using atomic force microscope (AFM) on mica surface. The vesicles were prepared by extrusion and adsorbed on the mica surface. The forces, measured between an AFM tip and the vesicle, presented that the breakthrough of the tip into the vesicles occurred two times. Each breakthrough represented each penetration of the tip into each bilayer. Force data prior to the first breakthrough were fitted well with the Hertzian model to estimate Young's modulus and bending modulus of the vesicles. It was found that the Young's modulus and bending modulus were not varied with the incorporation of AGs (streptomycins) up to the 1:1 AG/DPPC vesicle system. This result may suggest that the AGs do not lead to the disruption of DPPC packing.

Mechanical Phenotyping of Mouse Embryonic Stem Cells: Increase in Stiffness with Differentiation

Atomic force microscopy (AFM) has emerged as a promising tool to characterize the mechanical properties of biological materials and cells. In our studies, undifferentiated and early differentiating mouse embryonic stem cells (mESCs) were assessed individually using an AFM system to determine if we could detect changes in their mechanical properties by surface probing. Probes with pyramidal and spherical tips were assessed, as were different analytical models for evaluating the data. The combination of AFM probing with a spherical tip and analysis using the Hertz model provided the best fit to the experimental data obtained and thus provided the best approximation of the elastic modulus. Our results showed that after only 6 days of differentiation, individual cell stiffness increased significantly with early differentiating mESCs having an elastic modulus two- to threefold higher than undifferentiated mESCs, regardless of cell line (R1 or D3 mESCs) or treatment. Single-touch (indentation) probing of individual cells is minimally invasive compared to other techniques. Therefore, this method of mechanical phenotyping should prove to be a valuable tool in the development of improved methods of identification and targeted cellular differentiation of embryonic, adult, and induced-pluripotent stem cells for therapeutic and diagnostic purposes.

Mechanical Characterization of Hepatoma Cells Using Atomic Force Microscope

In order to reveal variation of mechanical properties of hepatoma cells with nanometer resolution, atomic force microscopy (AFM)-based nanoindentation experiments are performed on hepatoma cell to derive Young’s modulus employing a corrected Hertz model. Under load conditions of nanoindentation force as 0.43809-0.73015nN and penetration rate as 0.4 Hz, the calculated value of Young’s modulus of hepatoma cells is 34.137±0.67kPa with a 95% confidence interval. The results demonstrate the Young’s modulus varies with the measurement position, and the center of cell possesses lower value than peripheral region. Variation of Young’s modulus is resulted from external reaction, which supports well the theory of cytoskeleton structure. Furthermore, the difference of Young’s modulus between normal cells and cancerous ones are also discussed, and it will provide possibility of a new route for early diagnosis of hepatoma.

Failure behavior of single sand grains: Theory versus experiment

Grain-scale brittle fracture and grain rearrangement play an important role in controlling the compaction behavior of reservoir rocks during the early stages of burial. Therefore, the understanding of single-grain failure is important. We performed constant displacement rate crushing tests carried out on selected, wellrounded, single sand grains and on randomly sampled grains from different grain size (d) batches of pure quartz sand. Applying a Hertzian fracture mechanics model for grain crushing, the critical load at failure (Fc) data obtained for the selected grains were converted into an accurate estimate of the size of flaws associated with failure (cf). Similarly, the distributed Fc data obtained from the different batch samples were converted into distributions of grain failure stress. Weibull weakest link theory could not explain the observed grain failure behavior. On the contrary, the Hertzian grain failure criterion enabled the conversion of the distributed Fc data, for the batch samples, into distributions of cf, assuming spherical grains, or of effective radius of curvature (rg), characterizing contact surface asperities in the case of non-spherical

Microelasticity of red blood cells in sickle cell disease

Translation of cellular mechanics findings is crucial in many diseases, including Alzheimer’s disease, Parkinson’s disease, type II diabetes, malaria, sickle cell disease, and cancer. Atomic force microscopy (AFM) is appropriate for measuring mechanical properties of living and fixed cells due to its high force sensitivity and its ability to measure local and overall properties of individual cells under physiological conditions. A systemic force–displacement curve analysis is reported on the quantification of material stiffness via AFM using two theoretical models derived from the Hertz model. This analysis was applied to red blood cells from patients with sickle cell disease to determine the Young’s modulus of these cells in the oxygenated and deoxygenated state. Sickle cell disease pathophysiology is a consequence of the polymerization of sickle hemoglobin in red blood cells upon partial deoxygenation and the impaired flow of these cells in the microcirculation. A model is presented for a four-sided pyramidal indenter that is subsequently shown to have a better fit to the obtained data than that using a model of a parabolic indenter. It is concluded that deoxygenation and therapeutic treatment have a significant impact on the stiffness. This analysis presents a new approach to addressing medical disorders.

Extreme Hardening of PDMS Thin Films Due to High Compressive Strain and Confined Thickness

Polymers confined to small dimensions and that undergo high strains can show remarkable nonlinear mechanics, which must be understood to accurately predict the functioning of nanoscale polymer devices. In this paper we describe the determination of the mechanical properties of ultrathin polydimethylsiloxane (PDMS) films undergoing large strains, using atomic force microscope (AFM) indentation with a spherical tip. The PDMS was molded into extremely thin films of variable thickness and adhered to a hard substrate. We found that for films below 1 μm in thickness the Young's modulus increased with decreasing sample thickness with a power law exponent of 1.35. Furthermore, as the soft PDMS film was indented, significant strain hardening was observed as the indentation depth approached 45% of the sample thickness. To properly quantify the nonlinear mechanical measurements, we utilized a pointwise Hertzian model which assumes only piecewise linearity on the part of the probed material. This analysis revealed three regions within the material. A linear region with a constant Young's modulus was seen for compression up to 45% strain. At strains higher than 45%, a marked increase in Young's modulus was measured. The onset of strain induced stiffening is well modeled by finite element modeling and occurs as stress contours expanding from the probe and the substrate overlap. A third region of mechanical variation occurred at small indentations of less than 10 nm. The pointwise Young's modulus at small indentations was several orders of magnitude higher than that in the linear elasticity region; we studied and ruled out causes responsible for this phenomenon. In total, these effects can cause thin elastomer films to become extremely stiff such that the measured Young's modulus is over a 100-fold higher than the bulk PDMS. Therefore, the mechanics of a polymer can be changed by adjusting the geometry of a material, in addition to changing the material itself. In addition to understanding the mechanics of thin polymer films, this work provides an excellent test of experimental techniques to measure the mechanics of other nonlinear and heterogeneous materials such as biological cells.

Characterization of Ocular Tissues Using Microindentation and Hertzian Viscoelastic Models

The authors applied a novel microindentation technique to characterize biomechanical properties of small ocular and orbital tissue specimens using the hertzian viscoelastic formulation, which defines material viscoelasticity in terms of the contact pressure required to maintain deformation by a harder body. They used a hard spherical indenter having 100 nm displacement and 100 μg force precision to impose small deformations on fresh bovine sclera, iris, crystalline lens, kidney fat, orbital pulley tissue, and orbital fatty tissue; normal human orbital fat, eyelid fat, and dermal fat; and orbital fat associated with thyroid eye disease. For each tissue, stress relaxation testing was performed using a range of ramp displacements. Results for single displacements were used to build quantitative hertzian models that were, in turn, compared with behavior for other displacements. Findings in orbital tissues were correlated with quantitative histology. Viscoelastic properties of small specimens of orbital and ocular tissues were reliably characterized over a wide range of rates and displacements by microindentation using the hertzian formulation. Bovine and human orbital fatty tissues exhibited highly similar elastic and viscous behaviors, but all other orbital tissues exhibited a wide range of biomechanical properties. Stiffness of fatty tissues tissue depended strongly on the connective tissue content. Relaxation testing by microindentation is a powerful method for characterization of time-dependent behaviors of a wide range of ocular and orbital tissues using small specimens, and provides data suitable to define finite element models of a wide range of tissue interactions.

An Improved Contact Model for Pounding Simulation of Base-isolated Highway Bridge

A modified Hertz model with nonlinear damping (Hertz damp model) is proposed for capturing the seismic pounding response of adjacent structures. Relevant parameters in the model are theoretically derived and numerically verified. Then, this model is used to simulate pounding response of a base-isolated highway bridge subjected to near fault ground motions. At the same time, nonlinear viscous dampers are installed between bridge decks for pounding prevention. The appropriate damping coefficient of the dampers can be found by parametric studies. It is shown that nonlinear viscous dampers are effective in reducing the relative displacement between bridge decks. At last, the hysteresis curve, the maximal damper force and stroke are used to demonstrate the behavior of nonlinear viscous dampers. The results indicate that satisfied viscous dampers can be produced to eliminate pounding according to the current manufacturing skills.

Spatial and temporal dependence of the cerebral endothelial cells elasticity

The reliable determination of the mechanical properties of a living cell is one of the most important challenges of the atomic force microscopic measurements. In the present study the spatial and temporal dependency of the force measurements on cerebral endothelial cells was investigated. Besides imaging the cells, two different sequences of force measurements were applied: Acquisition of force curves in short time at several points across the cell surface investigating spatial dependence of the elasticity. Acquisition of force curves for long time at a previously determined place, over the cell nucleus, which provides the temporal stability/variation of the measured forces/values. Three different stages of endothelial cell cultures of the hCMEC/D3 cells were used: sub-confluent living, confluent living, and confluent fixed cells. The Young's modulus was calculated from the force curves using the Hertz model and the results were plotted against time or location correspondingly. The rational of using the three stage of culture was to clarify whether the observed effect belongs to the individual cell, to the ensemble of cells or just to some, not living cell component. In case of sub-confluent cells the results revealed a softer nuclear region compared to the periphery, while an attenuated oscillation like fluctuation in time, with a period of about 10-30 min, was observed. Confluent living cells showed similar tendencies to the sub-confluent cells, but the changes were larger and the temporal oscillations had longer period. The spatial dependency of the elasticity on confluent cells was confirmed by force-volume measurement too. In case of fixed cells neither spatial nor temporal differences were observed between the nuclear and peripheral region, however the Young's modulus and the error of the measurement was larger, compared to the sub-confluent living cells.

Quasi-linear viscoelastic properties of PC-12 neuron-like cells measured using atomic force microscopy

Atomic force microscopy has been widely used to measure the mechanical properties of living cells. Models based on Hertz theory are often utilized to estimate the Young's modulus of cells. In practice, the nonlinear viscoelastic behavior of cells can often be observed in indentation experiments. In this article, the quasi-linear viscoelasticity (QLV) theory, which has been successfully applied to many biological tissues, was employed to construct the constitutive equation of neuron-like PC-12 cells. The values of Young's modulus obtained via our proposed method were one or two orders of magnitude, depending on the indenting rate, less than those obtained using methods based on the modified Hertzian model. The Kelvin model was also introduced to describe the solid and liquid behavior of cells. The prediction error of the Kelvin model was larger than that of the proposed QLV model, especially at the peak force and the toe portion of the force response. The results revealed that the Young's modulus estimated by the QLV model was less affected by indenting rates than that estimated by the modified Hertzian model or the Kelvin model. The proposed QLV model accounting for the indenting conditions was capable of describing both the nonlinear elastic response and relaxation behavior and, therefore, was more appropriate for modeling the biomechanical behaviors of PC-12 cells than the nonlinear elastic model or the linear viscoelastic model.

Indentation measurements of the subendothelial matrix in bovine carotid arteries

Artery biomechanics are an important factor in cardiovascular function and atherosclerosis development; as such, the macro-mechanics of whole arteries are well-characterized. However, much less is known about the mechanical properties of individual layers in the blood vessel wall. Since there is significant evidence to show that cells can sense the mechanical properties of their matrix, it is critical to characterize the mechanical properties of these individual layers at the scale sensed by cells. Here, we measured subendothelium mechanics in bovine carotid arteries using atomic force microscopy (AFM) indentation. To specifically indent the subendothelium, we evaluated three potential de-endothelialization methods: scraping, paper imprinting, and saponin incubation. Using scanning electron microscopy, histology stains, immunohistochemistry, and multiphoton microscopy, we found that scraping was the only effective de-endothelialization method capable of removing endothelial cells and leaving the subendothelial matrix largely intact. To determine the indentation modulus of the subendothelial matrix, both untreated and scraped (de-endothelialized) bovine carotid arteries were indented with a spherical AFM probe and the data were fit using the Hertz model. Both the endothelium on the untreated artery and the en face subendothelium had similar indentation moduli: E=2.5±1.9 and 2.7±1.1kPa, respectively. These measurements are the first to quantify the micro-scale mechanics of the subendothelial layer, and constitute a critical step in understanding the relationship between altered subendothelial micromechanics and disease progression.

Space Robot Self-Assembling Dynamics and Control

In the space robot self-assembling process, the assembling control becomes quite complicated and important for the nonlinear characteristics, which are caused by the dynamic coupling between the arm and the base, and the collision between the target and the assembling hole on the base. The dynamic model of the space robot with target is established firstly, then the Hertz model is introduced as the collision model between the docking link and the capturing cone, and the operational space control is applied to achieve the assembling operation control under the Cartesian space. Through simulating and analyzing the influence of the stiffness coefficient, the soft and hard behaviors during assembling are discovered, which would provide a basis for assembling control and parameter design. The results would have important academic value and engineering significance on space robot achieving on-orbit self-assembling.

Finite Element Analysis of Elastic-Plastic Contact Mechanic Considering the Effect of Contact Geometry and Material Properties

Each surface of roughness has different shape of asperity which is modeled with various shapes of analytical models. In this paper, the differences among various models of shape of asperity investigate using the Finite Element Method (FEM) and various analytical models. The contact stresses in rough surfaces are calculated analytically using various asperity shape models. Finite element analysis is also carried out assuming three types of material properties namely, the linear, the elastic-perfect plastic and the elastic-nonlinear hardening. The analytical results are compared with the results obtained by the finite element method. The results illustrate for using a deterministic approach which the numerical models are suitable. In hertz model, the result of force is very big in interface of causing deformation plastic, while Model Zhao has almost same result with FEM nonlinear property model. It is observed that the results obtained from Zhao’s model are generally in a better agreement with the results obtained from various finite element models especially in elastic-plastic and plastic zones, hence it may be concluded that Zhao’s model can be used for analyzing the rough surfaces in contact mechanics.