Hertzian contact mechanics

Wenning and Miiser [74] extended the considerations made above for athermal, flat walls to the interaction between a curved tip and a flat substrate by including Hertzian contact mechanics. Since the Hertzian contact area A increases proportionally to they concluded that for a dry, nonadhesive, commensurate tip substrate system, Fg should scale linearly with L, since is independent of A. This has now been confirmed experimentally by Miura and Kamiya for M0S2 flakes on M0S2 surfaces [74a]. For a dry, nonadhesive, disordered tip pressed on a crystalline substrate, they obtained Fg oc which was obtained by inserting A oc into Fg oc Lfs/A. The predictions were confirmed by molecular dynamics simulations, in which special care was taken to obtain the proper contact mechanics. The results of the friction force curve are shown in Fig. 6. 

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. 

The highly anisotropic nature of PDA films allows us to show that in-plane properties of materials can be observed using IC-AFM. This is due to the tilt of the AFM cantilever which produces a small but significant in-plane component to the tip s motion. In the case of PDA monolayers, in-plane Action and shear deformation anisotropy leads to contrast in the IC-AFM phase image. The results can be explained using a sinq)le model that incorporates Hertzian contact mechanics with in-plane dissipation, and may be generalized to the study of other anisotropic materials. 

The first theoretical description of elastohydrodynamic lubrication for the fine contact problem relevant to gears that combined the Reynolds equation. Bams law, and a Hertzian contact mechanics was developed by Ertel [961] and pubhshed 10 years later by Gmbin and Vinogradova [962]. The approach was to assume a Hertzian contact to calculate the pressure in the gap and let the pressure in the lubricant before the entrance increase exponentially to match the Hertzian contact pressure curve. The pressure distribution for such a line contact between parallel cylinders is shown in Figure 9.12. With this approximation, Ertel derived an expression for the average thickness ho of the lubricant film within the gap, which was later refined by Dowson [963] ... 

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]. 

Fig. 4. Schematic of the JKR treatment of contact mechanics calculations. The point (, a, P) corresponds to the actual state under the action of interfacial forces and applied load P. P is the equivalent Hertzian load corresponding to contact radius a between the two surfaces, ( o, P) and (S], a, P ) are the Hertzian contact points. The net stored elastic energy and displacement S are calculated as the difference of steps 1 and 2.

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]... 

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... 

Fig. 17 Contact mechanics analysis of Herztian cracks within brittle materials.a Schematic description of a Hertzian cone crack induced under normal indentation by a rigid sphere, b Reduced plot of JC-field as function of cone crack length and for increasing loads pf < p// < pm during sphere-on-flat normal indentation of brittle materials. Arrowed segments denote stage of stable ring crack extension from Cf to cc (initiation), then unstable to ci at P = P,n (cone-crack pop-in) (From [67]). Branches (1) and (3) correspond to unstable crack propagation (dK/dc > 0), branches (2) and (4) to stable crack propagation (dK/dc < 0)...

For the Hertzian contact, no force is needed to pull away the contacting sphere from the flat plane in excess of the weight of the sphere. However, for the JKR contact, due to adhesion forces, this does not hold. The value of the nonzero pull-off force represents the adhesion of the contacting sphere with the flat plane. Strictly speaking, this force corresponds to adherence of the surfaces as energy dissipation, surface relaxation, etc. also influence its value. It should be stressed that the value of the JKR pull-off force only depends on the sphere (lens) radius and the work of adhesion in the medium in which the JKR experiment is conducted. Thus, the contact area and mechanical properties for true JKR contacts do not play a role for its value. All the above considerations for contact mechanics were based on pairwise additivity of molecular forces. 

---