Hertzian theory

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 surfaces of all materials interact through van der Waals interactions and other interactions. These interfacial forces, which are attractive in most cases, result in the deformation of the solid bodies in contact. In practice, the radius of the contact zone is higher than the radius predicted by the Hertzian theory (Eq. 7). Johnson et al. [6] modified the Hertzian theory to account for the interfacial interactions, and developed a new theory of contact mechanics, widely known as the JKR theory. In the following section, we discuss the details of the JKR theory. The details of the derivation may be obtained elsewhere [6,20,21]. 

In their analysis, Johnson et al. [6] note that, under an applied load P, the aetual eontaet radius a is higher than the radius predieted by the Hertzian theory. Johnson et al. eaieuiate an equivalent Hertzian load P (> P) using Eq. 7, and the corresponding energy U ) stored in the system due to elastic deformation of the sphere under Hertzian conditions. Ui and P are given by... 

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... 

During a collision, the colliding solids undergo both elastic and inelastic (or plastic) deformations. These deformations are caused by the changes of stresses and strains, which depend on the material properties of the solids and the applied external forces. Theories on the elastic deformations of two elastic bodies in contact are introduced in the literature utilizing Hertzian theory for frictionless contact and Mindlin s approach for frictional contact. As for inelastic deformations, few theories have been developed and the available ones are usually based on elastic contact theories. Hence, an introduction to the theories on elastic contact of solids is essential. 

In this section, an introduction of the general relations of stresses in equilibrium in an infinitely large solid medium is presented, followed by a special application where a concentrated force is acting on a point inside the solid. Also presented is the case of forces on the boundary of a semiinfinite solid medium, which is of importance to the contact of two solid objects. As consequences of the boundary compression, displacements due to the changes of stresses and strains in the region of contact can be linked to the contact force by the Hertzian theory for frictionless contacts and by Mindlin s theory for frictional contacts. For more details on the Hertzian theory for contact, interested readers may refer to books on elasticity [Goldsmith, 1960 Timoshenko and Goodier, 1970 Landau and Lifshitz, 1970]. 

Assume that the normal pressure and displacements are not affected by the existence of the tangential traction and resulting displacements. Hence, the normal pressure and contact area can be determined by the Hertzian theory. For the sliding contact of spheres, substituting Eq. (2.69) into Eq. (2.77) gives rise to the tangential traction as... 

When the deviation from the elastic state of the material surface is small, the Hertzian theory can estimate the force of impact, contact area, and contact duration for collisions between spherical particles and a plane surface using Eqs. (2.132), (2.133), and (2.136), respectively. To account for inelastic collisions, we may introduce r as the ratio of the reflection speed to the incoming speed, V. Therefore, we may write... 

The basis for the characterization of the contact behaviour of curved bodies is given by the well known classical Hertzian theory (1). Because this theory basically applies only to perfectly elastic materials with ideal smooth surfaces, some additional factors must be considered in discussing the contact behaviour of materials like polymers. 

Nanomechanical mapping has been applied to several material systems to date, as introduced in Section 3.3. However, in these applications we adopted Hertzian theory and argued only elastic modulus, and therefore the analyses were subject to many restrictions. More seriously, practical measurements must be performed under appropriate conditions to avoid other complex interactions, such as adhesion and viscoelasticity, and to obtain precise and correct results. Measurement in an aqueous environment to avoid adhesion effects is a possible example, where we can suppress the water capillary effect, which is unavoidable, and the major contribution to the adhesion force under ambient conditions. 

For a water lubricated system we may however estimate the conditions quite differently. Considering that temperature rises because mechanical energy is transformed into thermal energy and assuming that water can be trapped in a confined space (between asperities, in superficial pores and/or cracks) the pressure would rise only due to the isochoric heating process. These pressures are in the order of several hundred Wa and thus still very high but much lower then those running into tens of GPa from Hertzian theory (Fig. I). 

Under dry conditions the contact between the rollers will consist of elongated areas of contact as shown in Figure 1. At very light loads these areas will be almost elliptical in shape and the contact areas and stresses will be close to those calculated from Hertzian theory based on the asperity tip curvatures. At loads sufficient to cause significant elastic deformation, however, the effects of a sinusoidal surface will become apparent and the contact areas are expected to depart from ah elliptical shape. 

The variation of indentation load as a function of penetration depth for a sphere of radius R indenting the flat surface of an elastic solid is given by the Hertzian theory of elastic contact as... 

Figure 2.16 presents results obtained for several contact pressures 0.66,0.83,1.12 and 1.42 GPa (pressures calculated using Hertzian theory) with an optimal concentration of 1 wt%. A decrease in the friction coefficient is observed when the contact pressure is increased. Moreover, the low friction coefficient is obtained from the very beginning of the tests. This characteristic is very interesting for potential industrial applications since it is known that traditional... 

The sliding contact used here is composed of a sphere on a plane. Under normal loading, the solids are reversibly deformed (purely elastic and isotropic behaviour of the substrates). Thanks to Hertzian theory, the contact area is determined to be circular with a radius equal to a (Figure 2.29). The calculation of the distance between the ball and the flat where the crown appears would lead to a better understanding of the formation mechanism of the crown. The formula of the deformation of the ball determined by Cameron gives the distance between the pin and flat at a given distance (r) [54] ... 

Raman spectroscopy is very efficient in measuring pressure inside the contact area. Mansot used it to measure the pressure in a sphere/flat contact containing a thin polymer film [90]. Pressure curves obtained are in agreement with Hertzian theory. Similar works determined the disttibution of the pressure in a contact in an elastohydrodynamic lubrication regime [91]. 

The position of the peaks on spectra obtained at various normal loads enables the pressure applied on the particles inside the contact to be determined. Calculations are carried out using the straight fine of Raman displacement obtained from hydrostatic pressure experiments (Figure 2.81). Figure 2.86 presents pressures calculated for 2H-WS2 and IF-WS2 compared with those determined by Hertzian theory (contact without particles, contact pressures determined as above). Differences between the experimental and theoretical curves can be observed. Several hypotheses can explain these differences. First, it was found that the analysed point does not correspond to the centre of the contact area. This would explain the low values obtained. The most probable hypothesis is that the pressure is not uniformly distributed in the contact but is more important on the fuUerene clusters (Figure 2.87). 

Figure 2.86 Contact pressures determined from Raman shift observed on the Raman spectra obtained at different normal loads with (a) 2H-WS2 and (b) IF-WS2. These pressures are compared to those calculated from Hertzian theory...

Hertzian Theory (Repulsion between Elastic Bodies)... 

The working principles of an existing computational valve train model developed by Dickenson [12] and originally constructed by Ball [21] are summarised in the following section. The kinematics are calculated based on the geometry present in Figure 1 prior to the Hertzian theory of elastic contacts being used. 

The extraordinary temperature rise in the dimple zone would be caused by high pressure. It can be predicted easily that the pressure would be higher than the pressure predicted by the Hertzian theory at the dimple zone according to the deformation. As shear force is added in the high pressure zone, heat is generated in the oil film. Furthermore, the heat generated in such thick film as the dimple is hard to diffuse into the sinfaces. Therefore, the temperature would increase in the dimple zone markedly and sometimes reaches above 300 K. 

The deformability of a purely elastic contact can be characterised by the strain to elastic shakedown limit, which depends on the actual yield stress. The elastic deformability is often related to a ratio of some power of the hardness and elastic modulus of the surface layer [6]. This approach is based on the Hertzian theory. Furthermore, it assumes that the average Hertzian pressure is equal to the hardness at the onset of plastic flow. 

The relationship between the real contact area and applied load is shown in Fig. 6. For a comparison, the ideal smooth case is also plotted. It can be seen that the rough surface contact area is linearly proportional to the applied load and represents only a small percentage of the smooth contact result. It is consistent with Greenwood and Trip [18] extension of the Hertzian theory for the case of the elastic contact of rough spheres. Figure 7 shows the variation of real contact area with applied load for four different spatial resolutions. For each spatial resolution, a nearly linear behaviour is observed. The slope of the lines decreases with increasing spatial resolution. Therefore at the constant load the contact pressure increases and real contact area decreases with spatial resolution. This is in accordance with previously published works [19], [20]. 

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