Contact Hertzian

Modelling of the tme contact area between surfaces requires consideration of the defonnation that occurs at the peaks of asperities as they come into contact with mating surfaces. Purely elastic contact between two solids was first described by H Hertz [7], The Hertzian contact area (A ) between a sphere of radius r and a flat surface compressed under nonnal force N is given by... 

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.

Via an ad hoc extension of the viscoelastic Hertzian contact problem, Falsafi et al. [38] incorporated linear viscoelastic effects into the JKR formalism by replacing the elastic modulus with a viscoelastic memory function accounting for time and deformation, K t) ... 

Figure 22 shows variation of the him thickness with velocities. The three curves in the hgure are results from the EHL solution, experimental data, and TFL solution, respectively. The maximum Hertzian contact pressure is 0.125 GPa and the atmosphere viscosity of oil is 0.062 Pa s. While the velocity is higher than 100 mm s, i.e., the him is thicker than 50 nm, all the results from EHL, TFL, and experimental data are very close to each other, which indicate that when in the EHL lubrication regime, bulk viscosity plays the main role and the results of three types are close to each other. When... 

NR with standard recipe with 10 phr CB (NR 10) was prepared as the sample. The compound recipe is shown in Table 21.2. The sectioned surface by cryo-microtome was observed by AFM. The cantilever used in this smdy was made of Si3N4. The adhesion between probe tip and sample makes the situation complicated and it becomes impossible to apply mathematical analysis with the assumption of Hertzian contact in order to estimate Young s modulus from force-distance curve. Thus, aU the experiments were performed in distilled water. The selection of cantilever is another important factor to discuss the quantitative value of Young s modulus. The spring constant of 0.12 N m (nominal) was used, which was appropriate to deform at rubbery regions. The FV technique was employed as explained in Section 21.3.3. The maximum load was defined as the load corresponding to the set-point deflection. 

When two elastic and frictionless spheres are brought into contact under compressional forces or pressures, deformation occurs. The maximum displacement and contact area depend not only on the compressional force but also on the elastic material properties and radii of the spheres. The contact between two elastic and frictionless spherical bodies under compression was first investigated by Hertz (1881) and is known as the Hertzian contact. 

Therefore, for Hertzian contact, using Eq. (2.75) and Eq. (2.76), the maximum radius of contact rc, the maximum approaching distance a, and the corresponding maximum pressure can be calculated on the basis of the contact force, the elastic material properties of the spheres, and the radii of the spheres. 

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. 

Furthermore, assume that Eq. (2.74), obtained for Hertzian contact, is valid with rc being the actual radius of contact area so that Ar F ... 

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]. 

Calculate the Hertzian contact stress, cr, and assess its acceptability by comparing it with the... 

With respect to Hertzian contact stress, it is recommended that in the absence of actual data, a should be no greater than twice the yield stress of the ear material. Since the yield stress is given as 250 N/mm2, that criterion is satisfied in this example. 

Related Calculations. The expression for calculating the Hertzian contact stress can be simplifies to... 

Fig. 1.3 Comparison of elastic Hertzian contact (left) and adhesive JKR contact (right), (a) Hertzian contact Dashed line (sphere) shape of contacting spherical lens prior to pressing to the flat surface by force L. Hertzian contact profile shown by solid line, with radius under external load L aH (b) JKR contact Schematic of adhesion force (adhesive zone model, forces schematically indicated by vectors) further deforming a spherical lens from Hertzian contact (solid line) to JKR contact (dotted line) with radius aJKR. Reproduced from [7] with permission copyright Springer Verlag...

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. 

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] ... 

To model the elastic properties of dispersions of soft particles, we consider a dispersion of N spheres in a periodic box, as shown in Fig. 6. The particles are either monodisperse with radius R or polydisperscd with a Gaussian distribution around a mean radius R. The concentration of particles is above the random close-packed volume fraction of 0c = 0.64 so that the particles are jammed together and form facets at contact. The contacts are assumed to be purely repulsive and frictionless and hence exert only a normal repulsive force at contact. The total elastic energy stored in the structure is the summation of the pairwise contact energies. Even at the highest volume fraction at near-equilibrium conditions, i.e., without flow, deformation of a particle is no more than 10% of its radius. Thus, the particle deformation is small compared to the size of the undeformed sphere and the contacts obey the Hertzian contact potential given by (1). 

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