Hertzian contact theory

Besides the oblique contact, tangential displacements may also be produced in the contact of two elastic spheres under the actions of a compressional twist, as shown in Fig. 2.13. Since the torsional couple does not give rise to a displacement in the z-direction, the pressure distribution is not influenced by the twist and is thus given by the Hertzian contact theory. 

Collisions between particles with smooth surfaces may be reasonably approximated as elastic impact of frictionless spheres. Assume that the deformation process during a collision is quasi-static so that the Hertzian contact theory can be applied to establish the relations among impact velocities, material properties, impact duration, elastic deformation, and impact force. 

As mentioned, the erosion of a solid surface depends on the collisional force, angle of incidence, and material properties of both surface and particles. Although abrasive erosion rates cannot be precisely predicted at this stage, some quantitative account of erosion modes which relates various impact parameters and properties is useful. In the following, a simple model for the ductile and brittle modes of erosion by dust or granular materials suspended in a gas medium moving at a moderate speed is discussed in light of the Hertzian contact theory [Soo, 1977]. 

Equivalent stress distribution which is shown in Figure 13 (loading below yield stress) is in good correlation with Hertzian contact theory. 

Hertzian contact theory [10] is applied to deduce the force-displacement relationshij). Spring coefficient is obtained as a function of the displacement as follows ... 

Examination of the coefflcient of friction of polymeric systems in the same manner as for metallic systems shows the functional differences. The primary difference between the two sliding systems is that even for polymeric systems that undergo initial plastic deformation, there will be a significant portion of the real area of contact that is in elastic contact. For elastomeric sliding systems virtually all of the real area of contact is loaded elastically. Thus, for the sake of discussion, assume that the contact is a sphere-on-flat configuration with completely elastic loading. One case where this occurs is for a soft elastomer on a hard, nominally flat counterface. In this situation, the real contact area is given by Hertzian contact theory radius calculation as... 

The number of cycles of disk rotation required to initiate the wear track correlated positively with the weight percent of the siloxane modifier in the epoxy. However, the initiation times for the ATBN- and CTBN-modified epoxies showed no significant correlation with the percentage of the incorporated modifier. The initiation of the wear track is assumed to result from the fatigue of the epoxy hence initiation time is related to the surface stresses. Because the surface stresses are inversely related to the elastic modulus as predicted by the Hertzian elastic contact theory 52), the initiation time data at ION load were compared to the elastic moduli of the materials in Fig. 16. The initiation times for the siloxane-modified epoxies were negatively correlated with their elastic moduli while samples modified with ATBN and CTBN showed positive correlations with their moduli. At lower loads the initiation times for the siloxane-modified epoxies increased. The effect of load on the CTBN- and ATBN-modified epoxies was too erratic to show any significant trends. 

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. 

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] ... 

The simplest theory for the analysis of surface elasticity based on AFM force-distance curve measurement is Hertzian contact mechanics [Landau and Lifshitz, 1967], As shown schematically in Figure 3.1, a force-distance curve is a plot of the displacement, z, of the piezoelectric scanner normal to the specimen s surface as the horizontal axis and the cantilever deflection, A, as the vertical axis. Hertzian contact mechanics cannot treat adhesive force in principle. We need to make some effort to minimize the adhesive force in a practical experiment. Measurement in aqueous conditions is effective for polymeric materials with low water absorbability. A cantilever with a large spring constant also hides weak van der Waals forces. Figure 3.1a shows the... 

The S-F plot (Figure 3.1c), derived from the z-A plot, is now fitted with the theory of Hertzian contact mechanics to provide an estimation of Young s modulus. 

There are two main pathways which a simulation of such conditions can follow. The first one is to come from the static point-of-view and to estimate the pressure from the theory of Hertzian contact pressures using simplified geometries for measured roughness profiles. The temperatures are then estimated from the boundary conditions such as shape, velocity and... 

The material removal mechanism of the CMP process was relatively well explained by the previous scientists. The material removal mechanism of dielectric CMP is further well explained by Cook in his paper pubhshed in 1990 [7]. It was explained that the rate of mass transportation during glass pohshing is determined by five factors the rate of water diffusion into the glass surface, the dissolution of the glass under the applied load, the adsorption rate of the dissolved material onto the abrasive surface, the redeposition of the dissolved material onto the surface of the work piece, and the aqueous corrosion between particle impacts. Water diffuses into sUoxane bonding (Si—O—Si) and the diffusion rate is controlled by multiple process conditions such as pressure or temperature. This hydrated oxide surface is removed by an abrasion process. The indentation process by each abrasive was modeled by Hertzian contact and their contact stress was calculated from the theory of elasticity. 

Johnson, Kendall, and Roberts (JKR) calculated the Hertzian contact area between two spherical surfaces when the adhesion energy could not be disregarded [15]. They verified their theory by their own experiments using the material combination of rubber and glass. In the JKR theory, it is assumed that adhesion energy is proportional to the contact area and that the attractive force acting on the outside of the contact area can be ignored. The JKR theory is outlined below. 

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. 

Attention is confined to isotropic materials. Also, we deal only with halfplane problems and rigid indentors. However, the results are applicable to mildly curved surfaces and, with certain modifications, to the case of contact between two viscoelastic bodies. This is the familiar argument used in the theory of Hertzian contact. The modifications mentioned are not trivial in the viscoelastic case, as they are in the elastic case, involving as they do, the combining of viscoelastic... 

Various theories have been developed on the basis of these works in order to process cases of lateral creep or spin, conformal contact and multi- or non-Hertzian contact [7]. 

Theory of Sliding Contact. Based on Hertzian contact model, Hamilton and Goodman (11), which was later elaborated by Hamilton (12), presented the scratch process as a combination of an indentation process and a sliding process. Figure 3 shows the schematic of the scratch process in this model. At the contact surface, where z = 0,... 

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]. 

To account for some of the shortcomings of the JKR theory, Derjaguin and coworkers [19] developed an alternative theory, known as the DMT theory. According to the DMT theory, the attractive force between the surfaces has a finite range and acts outside the contact zone, where the surface shape is assumed to be Hertzian and not deformed by the effect of the interfacial forces. The predictions of the DMT theory are significantly different compared to the JKR theory. 

Whereas the JKR model approached the topic of particle adhesion from a contact mechanics viewpoint, the DMT theory simply assumes that the adhesion-induced contact has the same shape as a Hertzian indentor. The normal pressure distribution Ph(p) for the Hertzian indentor is related to the repulsive force and the distance from the center of the contact circle to the point represented by r according to the relationship [49]... 

Hertzian mechanics alone cannot be used to evaluate the force-distance curves, since adhesive contributions to the contact are not considered. Several theories, namely the JKR [4] model and the Derjaguin, Muller and Torporov (DMT) model [20], can be used to describe adhesion between a sphere and a flat. Briefly, the JKR model balances the elastic Hertzian pressure with attractive forces acting only within the contact area in the DMT theory attractive interactions are assumed to act outside the contact area. In both theories, the adhesive force is predicted to be a linear function of probe radius, R, and the work of adhesion, VFa, and is given by Eqs. 1 and 2 below. 

Also adhesion between the tip and sample can cause deformation of the sample. Several theories have been developed to include the effect of adhesive forces. In the JKR theory adhesion forces outside the contact area are neglected and elastic stresses at the contact line are infinite [80]. Even under zero load, the adhesion force results in a finite contact radius a=(9jtR2y/2 E)1/3 as obtained from Eq. 7 for F=0. For example, for a tip radius R=10 nm, E=lGPa, typical surface energy for polymers y=25 mN/m, and typical SFM load F=1 nN, the contact radius will be about a=9.5 nm and 8=9 nm, while under zero load the contact radius and the deformation become a=4.5 nm and 8=2 nm, respectively. The experiment shows that under zero load the contact radius for a 10 nm tungsten tip and an organic film in air is 2.4 nm [240]. The contact radius caused only by adhesion is almost five times larger than the Hertzian diameter calculated above. It means, that even at very small forces the surface deformation as well as the lateral resolution is determined by adhesion between the tip and sample. 

The JKR approximation works well for high adhesion, large radii of curvature and compliant materials but may underestimate surface forces. An alternative theory have been developed by Derjaguin, Muller, Toporov (DMT) to include noncontact adhesion forces acting in a ring-shaped zone around the contact area [81]. On the other hand, the DMT approximation constrains the tip-sample geometry to remain Hertzian, as if adhesion forces could not deform the surfaces. The DMT model applies to rigid systems with small adhesion and radius of curvature, but may underestimate the contact area. For many SFM s, the actual situation is likely to lie somewhere between these two models [116]. The transition between the models their applicability for SFM problems were analysed elsewhere [120,143]. 

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... 

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