Hertz model | ScienceGate

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.

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Adhesion between elastic cylinders based on the double-Hertz model

International Journal of Solids and Structures ◽

10.1016/j.ijsolstr.2014.04.003 ◽

2014 ◽

Vol 51 (14) ◽

pp. 2706-2712 ◽

Cited By ~ 9

Author(s):

Fan Jin ◽

Wei Zhang ◽

Sulin Zhang ◽

Xu Guo

Keyword(s):

Hertz Model ◽

Elastic Cylinders

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Hertz model for contact of water droplet with superhydrophobic surface

Physics Letters A ◽

10.1016/j.physleta.2009.01.001 ◽

2009 ◽

Vol 373 (8-9) ◽

pp. 804-806 ◽

Cited By ~ 16

Author(s):

Michel Molotskii ◽

Ilya Torchinsky ◽

Gil Rosenman

Keyword(s):

Water Droplet ◽

Superhydrophobic Surface ◽

Hertz Model

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Space Robot Self-Assembling Parameter Analysis

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.308-310.1975 ◽

2011 ◽

Vol 308-310 ◽

pp. 1975-1980

Author(s):

Tian Xi Liu ◽

Lei Liang ◽

Wei Cheng ◽

Yang Zhao

Keyword(s):

Process Design ◽

Space Robot ◽

Parameter Analysis ◽

Parameter Design ◽

Hertz Model ◽

Operational Space ◽

Self Assembling ◽

Vertical Speed ◽

Engineering Significance ◽

Operation Control

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.

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Generalized Hertz model for bimodal nanomechanical mapping

Beilstein Journal of Nanotechnology ◽

10.3762/bjnano.7.89 ◽

2016 ◽

Vol 7 ◽

pp. 970-982 ◽

Cited By ~ 35

Author(s):

Aleksander Labuda ◽

Marta Kocuń ◽

Waiman Meinhold ◽

Deron Walters ◽

Roger Proksch

Keyword(s):

Contact Mechanics ◽

Small Amplitude ◽

Sample Surface ◽

Hertzian Contact ◽

Modes Of Operation ◽

Hertz Model ◽

Force Microscopy ◽

Atomic Force ◽

Resonant Modes ◽

Shape And Size

Bimodal atomic force microscopy uses a cantilever that is simultaneously driven at two of its eigenmodes (resonant modes). Parameters associated with both resonances can be measured and used to extract quantitative nanomechanical information about the sample surface. Driving the first eigenmode at a large amplitude and a higher eigenmode at a small amplitude simultaneously provides four independent observables that are sensitive to the tip–sample nanomechanical interaction parameters. To demonstrate this, a generalized theoretical framework for extracting nanomechanical sample properties from bimodal experiments is presented based on Hertzian contact mechanics. Three modes of operation for measuring cantilever parameters are considered: amplitude, phase, and frequency modulation. The experimental equivalence of all three modes is demonstrated on measurements of the second eigenmode parameters. The contact mechanics theory is then extended to power-law tip shape geometries, which is applied to analyze the experimental data and extract a shape and size of the tip interacting with a polystyrene surface.

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Behavior of Soft Spheres During Impact by High-Speed Photography

Journal of Engineering Materials and Technology ◽

10.1115/1.3225621 ◽

1983 ◽

Vol 105 (1) ◽

pp. 67-73 ◽

Cited By ~ 5

Author(s):

Yoichi Tatara

Keyword(s):

Deformation Process ◽

High Speed ◽

High Impact ◽

High Speed Photography ◽

Tennis Ball ◽

Hertz Model ◽

Hertz Theory ◽

Soft Spheres ◽

Almost All ◽

The Impact

Previously, it has been verified experimentally for durations of impact that the Hertz theory (the quasi-statical theory) holds during impact of spheres without any exception. However, no measurement of duration of impact has been presented for spheres of materials other than metal. This study presents exceptional cases of impacts of spheres during which the Hertz model does not directly hold. By the use of a high-speed camera running at a speed of 5000 frames/s, durations of impact are measured directly for impacts of two solid rubber spheres of the same size and content and impacts of a soft ball (Japanese type-soft tennis ball) on a rigid foundation. As a result, the measured durations of impact in the two impacting cases are found to be decreased as the impact velocity is increased, similar in tendency to durations of impact of elastic metal spheres during which the Hertz theory holds. However, the measured durations of impact are found to be clearly shorter than results calculated according to the Hertz theory, approximately half in the former impacts at high impact velocities, and about 70 percent of the Hertzian results in the latter impacts at almost all impact velocities. Deformation process of the ball impacting on the foundation is also presented to indicate both durations in the compressive process and the restitution one to be shorter than those expected by the Hertz theory. The other results observed on the films are noted to investigate the origin of the great discrepancies between the measured and Hertzian durations (that is, the impacting mechanism of the rubber spheres or the rubber ball packed with air treated here).

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Erratum to: “Hertz model for contact of water droplet with superhydrophobic surface” [Phys. Lett. A 373 (2009) 804]

Physics Letters A ◽

10.1016/j.physleta.2009.02.039 ◽

2009 ◽

Vol 373 (16) ◽

pp. 1500

Author(s):

Michel Molotskii ◽

Ilya Torchinsky ◽

Gil Rosenman

Keyword(s):

Water Droplet ◽

Superhydrophobic Surface ◽

Hertz Model

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Dynamic Behavior Analysis of Vehicle Shimmy System With Consideration of Clearance in Steering Linkage Mechanism

Volume 1: 24th Conference on Mechanical Vibration and Noise, Parts A and B ◽

10.1115/detc2012-70215 ◽

2012 ◽

Author(s):

Jian-Wei Lu ◽

Jia-Yun Xin ◽

Jongkil Lee ◽

Alexander F. Vakakis ◽

Lawrence A. Bergman

Keyword(s):

Numerical Analysis ◽

Dynamic Response ◽

Dynamic Model ◽

Bifurcation Diagram ◽

Dynamic Behavior ◽

Nonlinear Dynamic ◽

Theoretical Basis ◽

Phase Plane ◽

Linkage Mechanism ◽

Hertz Model

Dynamic behavior of the vehicle shimmy system with consideration of steering linkage clearance was analyzed. Based on nonlinear dynamic theories, a 4-DOF dynamic model of vehicle shimmy system with consideration of steering linkage clearance was presented. In which, the revolute movement pair of steering linkage with clearance between the steering tie rod and tie rod arm was described based on Hertz model. Numerical analysis on the dynamic response of the vehicle shimmy system with clearance was carried out, and the results were presented with phase plane, Poincaré map, and bifurcation diagram. The dynamic behavior of the vehicle shimmy system with clearance is compared with that without clearance, and the influence of the clearance on the dynamic behavior of the system is discussed, which will provide theoretical basis for attenuation of vehicle shimmy.

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

Current Nanoscience ◽

10.2174/1573413715666190509112358 ◽

2019 ◽

Vol >15 (5) ◽

pp. 541-546

Author(s):

Ya Li ◽

Haidong Liu

Keyword(s):

Mechanical Properties ◽

Lung Cancer ◽

Elastic Modulus ◽

Cancer Cells ◽

Evaluation System ◽

Finite Thickness ◽

Lung Cancer Cells ◽

Hertz Model ◽

Technological Support ◽

Accurate Evaluation

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.

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