Hertz equation

Because the indentation varies with time, the modulus must be specified for a certain indentation time, eg, a 10-s modulus. The Hertz equation holds only for purely elastic materials. However, it has been appHed to viscoelastic materials, including polymers and coatings, with excellent results (249—256). Indentation hardness vs temperature curves are shown in Figure 40 (249,251). 

There are several distinctive features worth noting about the JKR equation. The first is in the limit of no adhesion (or, equivalently, large applied loads, as commonly occurs with macroscopic particles), Eq. 24 reduces to the Hertz equation... 

For the case where the substrate is rigid compared to the particle, substitution of Eqs. 40-44 into the Hertz equation (Eq. 1) gives... 

Upon substituting this relationship into the Hertz equation and solving numerically, Quesnel found that a varied as For situations of intermediate... 

Herschel-Bulkley model Herschel effect Hertz equation... 

Plasticizer diffusion, deposition, and accumulation on a material surface are controlled by the degree of the compatibihty between the plasticizer and the matrix, the surface energy of the liquid/solid interface, and the volatility of plasticizer. Volatihty and migration are controlled by different properties of the plasticizers therefore they are not related (Figure 7.8). Plasticizers have high boiling points so evaporation is typically slow. The rate can be calculated from the Hertz equation ... 

Such behavior is important to the understanding of frictional phenomena. It has been known since Amontons ° in the 17th century that friction, the force required to slide one solid over another, increases in proportion to the load pressing the solids together. This is clearly inconsistent with the Hertz equation for contact area. In the Hertz case, the contact diameter increases with thus the contact area increases with and therefore the friction force should also increase with, at odds with the Amontons observations. However, the behavior shown in Fig. 9.9(b), after sliding causes scratching of the surfaces, is consistent with Amonton s Law. The contact size increases with so that the contact area, and therefore friction, is then proportional to load... 

Only small particles, less than 1 pm in diameter, would show this effect. Krupp explained this in terms of the equations for London-van der Waals attractive forces between rigid spheres, together with the Hertz equations of contacL Because the attraction is proportional to particle diameter, the force at the particle contact decreases with D. However, the elastic area of the contact spot decreases faster, from the Hertz Equation (9.1), with Thus, as the particle gets smaller, the contact pressure must rise to the point at which plastic deformation occurs. 

Other measurements to assess film quahty are also possible using AFM. The mechanical properties of films can be quantified by applying a force to the surface through the tip and measuring the depth of the indent created on the surface. Experimental data of indentation depth (5) versus force (F) can be applied to the Hertz equation (Eq. [2.4]) using the fitting parameter a, which is related to Young s modulus ( ), as shown in Eq. [2.5] (Choukourov et al., 2012). 

Figure 7. (a) AFM phase image at the surface of a CoCrMo disc after sliding against UHMWPE. (b) Force-distance curve measured on one spot shown in Figure 6(b) ( ), the line was calculated from the Hertz equation for a sphere with an effective modulus ( ) of 0.9 GPa. 

Other penetrometer-indentometers include transducers to sense the position and movement of the probe and microprocessors for temperature control and data collection and reduction. These instruments are used mainly to measure softening points, which are not glass transitions but are usually close to those values. Because a softening point is indicative of behavior under load, it is often more useful for predicting performance than the Tg. Penetrometer-indentometers can also be used to measure indentation hardness, creep, creep recovery, and modulus. Examples of such instruments include the TA Instruments, Mettler, Perkin-Elmer, Seiko, and Shimadzu thermomechanical analyzers (TMAs). They can be used to generate modulus and modulus-temperature data from indentation-time plots by applying the Hertz equation (eq. 36) (170,296), where E is the elastic or Young s modulus, jx the Poisson s ratio, r the radius of the hemispherical indentor, P the force on the indentor (mass load x g), h the indentation, and ifk the indentation hardness. 

Johnson, Kendall and Roberts [47] measured d for some natural rubber spheres and found deviations from the Hertz equation at low loads, but conformity at high loads. Data are shown in Fig. 10. At low loads the zones of contact were greater than predicted by Hertz. This was due to the forces of attraction between the surfaces of the two spheres, and it was shown that the diameter of the zone of contact was now given by Eq. (12), where W is the work of adhesion. 

In the case of solids that undergo only elastic deformation in the contact zone, the dependence of 4on/is different the size of the active zone can in this case be estimated from the Hertz equation (see Section 5.2). 

O Figure 22.9 shows the schematics for the indentation test of a flat specimen with a spherical indenter. The depth of penetration of a rigid spherical indenter beneath the specimen is ht at full load max When the load is removed, assuming no reverse plasticity, the unloading is elastic and at complete unload, there is a residual impression of depth hr. If the load P ax is reapphed, then the reloading is elastic through a distance he = h,— hr according to the Hertz equation ... 

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