Hertz contact theory

The contact half-width, b, can be calculated from the Hertz contact theory by using the actual values of physical property for the loaded gel and the gel sheet such as Poisson ratio, the Young modulus, etc. As a result of calculation from the above equation, the minimum inclination angle was 3° and it was the same as the angle resulted from the experiment. It was found experimentally and supported by the model that the sheer wave fl ont of the peristaltic motion was necessary to transport cylindrical gels. 

The typical force-displacement curve of a primary spherical particle (Y-AI2O3) in Fig. 7.9a shows clearly both elastic and plastic displacement ranges. The elastic displacement of the spherical particle up to the yield point F was described using the Hertz contact theory of spheres (Hertz, 1882 Antonyuk et al., 2011) according to... 

Section 8.6.2 within the framework of a particular model which can capture the zipping up process in a realistic way. The model is based on the theory of elastic contact of solids with rounded surfaces known as Hertz contact theory, with the added feature that the contacting surfaces cohere. 

The results of Hertz contact theory are recovered if the second term on the right sides of both (8.113) and (8.114) are ignored. It is evident in both of these expressions that the classical results are recovered whenever d 1 but that the effect of cohesion is significant otherwise. 

Figure 31 shows the schematic of a particle of diameter d attached to a flat surface. Here, P is the external force exerted on the particle, a is the contact radius, and Fad is the adhesion force. The classical Hertz contact theory provides for the elastic deformation of bodies in contact, but neglects the adhesion force. Several models for particle adhesion to flat surfaces were developed in the past that improves the Hertz model by including the effect of adhesion (van der Waals) force. 

According to Hertz contact theory, the normal load P which the pin is subjected to, is given by ... 

By combining Hertz s contact theory (Eq. 1) and with Hamaker s functional form for the attractive force (Eq. 17), the Derjaguin model takes the form... 

In this chapter, two simple cases of stereomechanical collision of spheres are analyzed. The fundamentals of contact mechanics of solids are introduced to illustrate the interrelationship between the collisional forces and deformations of solids. Specifically, the general theories of stresses and strains inside a solid medium under the application of an external force are described. The intrinsic relations between the contact force and the corresponding elastic deformations of contacting bodies are discussed. In this connection, it is assumed that the deformations are processed at an infinitely small impact velocity and for an infinitely long period of contact. The normal impact of elastic bodies is modeled by the Hertzian theory [Hertz, 1881], and the oblique impact is delineated by Mindlin s theory [Mindlin, 1949]. In order to link the contact theories to collisional mechanics, it is assumed that the process of a dynamic impact of two solids can be regarded as quasi-static. This quasi-static approach is valid when the impact velocity is small compared to the speed of the elastic... 

When the particle deformation is small compared to the size of the undeformed spheres, the contacts obey Hertzian contact mechanics. According to Hertz s theory, the elastic energy associated with a single contact is [118] ... 

Fig-6 Periodic box of concentrated dispersion of soft spherical particles. Each pair of particles at contact forms a facet, as shown in Fig. 4, that deforms according to Hertz s theory or similar law... 

If a circle with radius a (m) is regarded as the contact area between a stainless steel ball and substrate under a normal load P (0.49 N), Hertz s theory affords the following relationship using Young s modulus of stainless steel and silicon wafer, (1.96 x 10" Pa) and (1.30 x 10" Pa), and Poisson s ratio D (0.30) and Ub (0.28), respectively ... 

In 1885 Joseph Boussinesq (6), trying to extend the validity of these results to the case of axi-symetrical rigid convex punches indenting a flat semi-infinite elastic medium, demonstrates that, without an adequate boundary comlition, the size of the contact area is generally unknown. In ordo to overcome this difficulty, he imposes that normal stresses vanish on the border of the contact area. In other words, the profile of the distorted medium must be tangent to the surfiice of the punch on the border of the contact area. Note that this condition is the same as the condition presupposed by the Hertz s theory. With this assumption, the size oh of the contact area and the penetration depth 5h are completely defined (Figure 1). 

The subj t of adhesive contact mechanics may be said to have started when Kendall (//), solving the problem of the adhesive contact of a rigid flat cylinder punch indenting the smooth plane surface of an elastic medium, demonstrated that the border of the contact area can be considered as a crack tip. The more complex problem of a spherical punch was solved in 1971 by Johnson, KendaU and Roberts (72). The JKR theory predicts the existence of contact area greater then that ven by the elastic contact Hertz s theory. The molecular attractive forces are responsible for this increase and, even in the absence of external compressive loading, the contact area has a finite size. Separating the two solids requires the application of an adherence force despite the existence of infinite normal stresses in the border of the contact area. 

Figure 8. lUf-width of the equilibrium contact area between a rigid cylindw and the smooth surface of an elastic solid as a fimction of the normal plied load per unit axial length, in reduced coordinates. E q)erimental data M in the immediate vicinity of the theoretical curve (heavy line). The curve deduced from the classical Hertz s theory (non-adhesive elastic contact) is given for conq>arison. 

Another approach is to extend the classical contact theory for indenters on elastic half-spaces developed by Hertz [77] and Huber [78] to the case of layered materials. An example of such an approach is ref. [79], in which the authors modify the Hertz/Huber analysis by considering the coating material properties as a function of indentation depth. Mathematically, the authors treat the transition from coating to substrate as a discontinuity in Young s modulus and Poisson s ratio represented by a Heaviside step function, and re-derive the appropriate Hertzian equations. The results match FEA calculations well. 

The surface of an asperity of rubber is taken to be spherical. The increase in the area of contact with the time of contact may be expressed in terms of creep properties, on the assumption that normal load is borne by the asperity as soon as it makes contact with the rigid countersurface. It is convenient to relate the area of contact Ap (t) after a time t, reckoned from the instant of first contact to the area A corresponding to the rubber elastic state. The latter may be determined from considerations of surface statistics and Hertz s theory. 

Based on the presented theory, it is intended to construct the contact force model between the two spheres. The initial indentation velocity between the two spheres is = 0.3 m/s. The speed of deformation waves is 2.6xlO m/s, which provides a limiting value of 0.(126 m/s for the impact to be considered elastic. Hence, the Hertz contact force model with permanent indentation is a valid one. The generalized parameter K is calculated from equation (2), with v = 0.33, to be equal to 5.50x10 N/m -. The equivalent mass of the two spheres is obtained from equation (7) as m = 0.046 kg. From equations (12), (13), and (14), the unknown parameters in the contact force model are evaluated as... 

Dispersive Work of Adhesion (W°) and Adhesive Contact Theories (Hertz, DMT, JKR)... 

The exponent m is mostly in the order of m 1 /6 — 1 /4. An exponent m= /6 results from Hertz s theory (see Section 6.8.2) for spherical grain contacts under pressure thus, this type of equation is preferred for sediments with a granular structure (sand, sandstone). 

The factor (6.143) also covers the pressure influence on the elastic properties. Hertz s theory applied on the grain-grain contact results in a power law ... 

According to Hertz s theory of elastic collision, the change rate of the contact area Af during the collision is given by... 

It should be noted that the contact radius mentioned above is obtained from the DEM simulation conditions, based on the Hertz elastic contact theory. However, in DEM simulation, the Young s modulus is usually set to 1 100 MPa to reduce the computing effort, while the Young s modulus of real hard materials like glass beads would be much larger than this value range, e.g., around 50 GPa. Therefore, an additional correction coefficient c is introduced by Zhou et al. (2010b) ... 

Contact mechanics, in the classical sense, describes the behavior of solids in contact under the action of an external load. The first studies in the area of contact mechanics date back to the seminal publication "On the contact of elastic solids of Heinrich Hertz in 1882 [ 1 ]. The original Hertz theory was applied to frictionless non-adhering surfaces of perfectly elastic solids. Lee and Radok [2], Graham [3], and Yang [4] developed the theories of contact mechanics of viscoelastic solids. None of these treatments, however, accounted for the role of interfacial adhesive interactions. 

Dutrowski [5] in 1969, and Johnson and coworkers [6] in 1971, independently, observed that relatively small particles, when in contact with each other or with a flat surface, deform, and these deformations are larger than those predicted by the Hertz theory. Johnson and coworkers [6] recognized that the excess deformation was due to the interfacial attractive forces, and modified the original Hertz theory to account for these interfacial forces. This led to the development of a new theory of contact mechanics, widely referred to as the JKR theory. Over the past two decades or so, the contact mechanics principles and the JKR theory have been employed extensively to study the adhesion and friction behavior of a variety of materials. 

The Hertz theory of contact mechanics has been extended, as in the JKR theory, to describe the equilibrium contact of adhering elastic solids. The JKR formalism has been generalized and extended by Maugis and coworkers to describe certain dynamic elastic contacts. These theoretical developments in contact mechanics are reviewed and summarized in Section 3. Section 3.1 deals with the equilibrium theories of elastic contacts (e.g. Hertz theory, JKR theory, layered bodies, and so on), and the related developments. In Section 3.2, we review some of the work of Maugis and coworkers. 

The classical theory of contact mechanics, due to Hertz, treats the bodies in contact with a hard wall repulsive interaction, i.e. there is no attractive interaction whatsoever, and a steep repulsion comes into play when the surfaces of the bodies are in contact. The Hertzian theory assumes that only normal stresses exist, i.e. the shear stress in the contact region is zero. Under these conditions, the contact radius a), central displacement (3) and the distribution of normal stress (a) are given by the following expressions ... 

The JKR theory, much like the Hertz theory, assumes a parabolic approximation for the profile of sphere, which is valid for small ratios of contact radii to the sphere s radius. Maugis [34] has shown that for small particles on a soft substrate, this ratio could be so large that such parabolic approximation is no longer valid. Under such conditions, the use of exact expression for the sphere profile is necessary for the applicability of the JKR theory, which is expressed as... 

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