Hertzian elastic contact

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. 

Change in the junction resistance can be related to change in contact area and film indentation using a Hertzian elastic contact between the probe and the sample as described earlier, in combination with the Simmons model for nonresonant tunneling. The tunneling current can be related to the contact area by... 

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... 

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. 

No significant difference is found for the predictions of the yield stresses from these three criteria. However, Tresca s criterion is more widely used than the other two because of its simplicity. When two solid spherical particles are in contact, the principal stresses along the normal axis through the contact point can be obtained from the Hertzian elastic... 

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... 

Scanning force microscopy (SFM) was used for probing micromechanical properties of polymeric materials. Classic models of elastic contacts, Sneddon s, Hertzian, and JKR, were tested for polyisoprene rubbers, polyurethanes, polystyrene, and polyvinylchloride. Applicability of commercial cantilevers is analyzed and presented as a convenient plot for quick evaluation of optimal spring constants. We demonstrate that both Sneddon s and Hertzian elastic models gave consistent and reliable results, which are close to JKR solution. For all polymeric materials studied, correlation is observed between absolute values of elastic moduli determined by SFM and measured for bulk materials. For rubber, we obtained similar elastic modulus from tensile and compression SFM measurements. 

In the present communication, we report the results from studies of micromechanical properties on polymeric materials interpreted using classic theories of elastic contacts, Sneddon s, Hertzian, and Johnson-Kendall-Roberts (JKR). These models are tested for a set of polymeric materials with known Young s modulus, E, from 1 MPa to 3 GPa. Special attention is paid to the elucidation of applicability of different contact models and optimization of experimental probing procedures. 

By combining optimal cantilever parameters and experimental conditions one can obtain reliable force distance data which is appropriate for further contact mechanics analysis for a wide selection of polymeric materials. Both Sneddon s and Hertzian models of elastic contact give consistent results in the range of indentation depth up to 100 nm. Close correlation is observed between elastic moduli determined by SFM in compression mode (approaching cycle) and measured values for bulk materials. As shown, for elastic materials force-distance curves can be used for evaluation of tensile elastic moduli from retracing cycle. For rubber material, the latest is in a good agreement with measuremente in compression mode. 

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 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]. 

Stresses at asperity or micro contacts within the overall Hertzian contact are key features, so that a thorough understanding of the stresses at asperity contacts is clearly needed. Useful insights into the behaviour of the contact of real surfaces can be obtained using dry elastic contact simulations [18,19] or elastic/plastic contact simulations [20]. However, it is necessary to consider the micro-EHL effects of the oil film since these influence the contact pressures and, through non-Newtonian and time-dependent behaviour, determine the traction forces at the micro contacts. The important subsurface stress field is therefore affected by the presence of a micro EHL film. In general when roughness is present the maximum sub-surface shear stress field occurs much closer to the surface as seen in the micro EHL simulations in section 4. 

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... 

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) ... 

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. 

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... 

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]. 

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