Hertz model

Hertz [27] solved the problem of the contact between two elastic elliptical bodies by modeling each body as an infinite half plane which is loaded over a contact area that is small in comparison to the body itself. The requirement of small areas of contact further allowed Hertz to use a parabola to represent the shape of the profile of the ellipses. In essence. Hertz modeled the interaction of elliptical asperities in contact. Fundamental in his solution is the assumption that, when two elliptical objects are compressed against one another, the shape of the deformed mating surface lies between the shape of the two undeformed surfaces but more closely resembles the shape of the surface with the higher elastic modulus. This means the deformed shape after two spheres are pressed against one another is a spherical shape. 

FIGURE 21.18 Force-deformation curves of local points indicated by open circles in Figure 21.15c. The curve fitting against Hertz model are superimposed on each curve, (a) Carbon black (CB) region (upper circle), 1.01 0.03 GPa, (b) interfacial region (middle circle), 57.3 0.8 MPa, and (c) rubber region (lower circle), 7.4 0.1 MPa. 

The elasticity was quantitatively determined by analyzing the recorded force curves with the help of the Hertz model. The Hertz model describes the elastic deformation of two spherical surfaces touching imder the load, which was calculated theoretically in 1882 by Hertz. Other effects, such as adhesion or plastic deformation, were not included in this model. Sneddon extended the calculation to other geometries. For a cone pushing onto a flat sample, the relation between the indentation 5 and the loading force F can be expressed as ... 

Mapping of the elastic modulus of the glassy and rubbery blocks and clay regions was probed by employing Hertzian and Johnson-Kendall-Roberts (JKR) models from both approaching and retracting parts of the force-distance curves. In order to determine the elastic properties of SEBS nanocomposites in its different constituting zones, the corrected force-distance curve was fitted to the Hertz model ... 

Block and clay regions of SEBS nanocomposite Modulus from Hertz model ( Sample) MPa Localized sample deformation (d), nm Modulus from JKR model ( Sample) MP Bulk modulus3 of SEBS/clay nanocomposite, MPa... 

Compressive measurements provide a means to determine specimen stiffness, Young s modulus of elasticity, strength at failure, stress at yield, and strain at yield. These measurements can be performed on samples such as soy milk gels (Kampf and Nussi-novitch, 1997) and apples (Lurie and Nussi-novitch, 1996). In the case of convex bodies, where Poisson s ratio is known, the Hertz model should be applied to the data in order to determine Young s modulus of elasticity (Mohsenin, 1970). It should also be noted that for biological materials, Young s modulus or the apparent elastic modulus is dependent on the rate at which a specimen is deformed. 

In the absence of a load (F = 0) the indentation is zero and the contact radius is also zero. Since no attractive surface forces were considered there is also no adhesion in the Hertz model. 

Elastic deformation. For small loads we can use the Hertz model as a simple approximation. The microcontacts are thereby assumed to be spherical. Hertz theory predicts an actual contact surface for an individual sphere on a plane (see Eq. 6.64) ... 

JKR model In contrast to the Hertz model where the minimal load is zero, here we can apply negative loads, that is we can even pull on the particles. The greatest negative load is equal to the adhesion force 3 7SR = 0.471 p,N. 

There are two major sources of the deformation in contact-mode SFM the elasticity of the cantilever and the adhesion between the tip and sample surface. For purely elastic deformation, a variety of models have been developed to calculate the contact area and sample indentation. The lower limit for the contact diameter and sample indentation can be determined based on the Hertz model without taking into account the surface interactions [79]. For two bodies, i.e. a spherical tip and an elastic half-space, pressed together by an external force F the contact radius a and the indentation depth 8 are given by the following equations ... 

Mathematical modelling of the compression of single particles 2.5.2.7 Hertz model. The mechanics of a sphere made of a linear elastic material compressed between two flat rigid surfaces have been modelled for the case of small deformations, normally less then 10% strain (Hertz, 1882). Hertz theory provides a relationship between the force F and displacement hp as follows ... 

AFM can also be used to probe local mechanical properties of thin films of food biopolymers, which are difficult to measure using traditional rheological methods. Several mechanical models have been developed to analyze the Young s modulus of food systems. One of the simplest models, the Hertz model, assumes that only the elastic deformation exists in a surface with spherical contacts, and the adhesion force can be neglected (Hugel and Seitz 2001). Equation (8.2) describes the relationship between the loading force, F and the penetration depth, d, where a is the radius of contact area, R the curvature of the tip radius, Vi and the Poisson s ratios of the two contact materials that have Young s modulus, Ei and E2. ... 

Figure 8.3 shows the typical force-distance curve for a K-carrageenan film. From the slope of the curve, where the AFM tip is in contact with the film surface. Young s modulus of the K-carrageenan film can be estimated by using the Hertz model with the proper measurement of the AFM tip radius using scanning electron microscopy (SEM) and estimated value of Poisson s ratio based on the characteristics of film surface, which is around 1.4 MPa. 

The resolution of the contact mode depends on a contact area at the tip apex. The contact diameter (a) can be estimated with the Hertz model in which the contact area increases with applied force (F) on the tip due to elastic deformation between tip and sample. 

The Hertz model, extended by Sneddon to other geometries provides a relation between load force F and the indentation 8 (with Young s modulus E, Poisson ration v, and the half opening angle of the indenting cone a). 

The average of the approach and retract curves determined experimentally are fitted with the Hertz model the combination of the two equations above yields... 

Interestingly, for thick gelatin films (>1 micron) the Hertz model matches the data well (here E > 20 kPa). For thinner films, a disagreement between the model and the experimental data has been observed and it is attributed to the effect of the underlying substrate (Fig. 4.21). 

QCM measurements, we also recorded elasticity maps of MDCK cells before and after fixation with PFA and GA by scanning force microscopy (SFM). Applying the commonly used Hertz model to the recorded raw data, we obtained a median Young s modulus of 2.5 0.3 kPa for native MDCK cells that was increased to 3.7 0.9 kPa after PFA fixation. The highest median values of 25 3 kPa were found after a 30 min fixation with GA. Thus, the well-established SFM measurements indicate that there is a graded and individual stiffening of the cells when different fixatives are used. Consistent with the QCM experiments, PFA was found to be less efficient in cell stiffening than GA. In SFM studies the cortical actin cytoskeleton is considered to be the dominant contributor to the mechanical properties of the cell membrane [46]. Since the QCM readout correlates with SFM measurements, the conclusion may apply that the cortical actin cytoskeleton is also predominantly responsible for the acoustic load of the resonator. 

In this context, Mazur has corrunented [32b] that for films dried at T > MFT, fire extent of deformation for an elastic particle should be predicted by the Hertz model, not the JKR model. Thus the reduced neck diameter (x/a) should be proportional to fl (Equation 14.3), and not rfirectly dependent on particle size... 

A scanning force instrument also allows for the acquisition of force-distance curves to characterize the local mechanical properties of the sample. Well-defined indentation experiments on soft surfaces like swollen hydrogels in aqueous media are possible with the colloidal probe technique. Raw data are assessed, for example, according to the Hertz model, with the assumption... 

Further information about the possible range of mechanical effects that occur within thin synthetic gel substrates has been provided in a study that used gel indentation with the cantilever of an atomic force microscope (AFM). Gels of varying thickness, H, were made at the same time with the same polyacrylamide gel solutions to maintain a constant E of tissue-like ( kPa) elasticities, and the gels were all indented by l-2 pm at forces that bend the cantilever in the nano-Newton (nN) range. An apparent elasticity E pp was obtained in this AFM experiment by fitting the force/versus indentation depth d with a generalized Hertz model ... 

Fig. 1 Single cell elasticity measurements with microindentation technique. (A) AFM allows both live cell imaging and mechanical testing (e.g., microindentation) in near physiological conditions. The AFM tip attached to a cantilever descents slowly to a surface and causes an indentation. The depth of indentation is detected by laser light diffraction pattern. (B) Typical force-distance curves are obtained for hard and soft surfaces and usually analyzed with the classical Hertz model that relates the applied force to the indentation depth.

The Young s modulus of a cell can be measured using an atomic force microscope (AFM) by indenting the cell with the AFM tip. Based on Hertz model, the Young s modulus, E, can be calculated as [7]... 

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