Load, applied

Carpick et al [M] used AFM, with a Pt-coated tip on a mica substrate in ultraliigh vacuum, to show that if the defonnation of the substrate and the tip-substrate adhesion are taken into account (the so-called JKR model [175] of elastic adliesive contact), then the frictional force is indeed proportional to the contact area between tip and sample. Flowever, under these smgle-asperity conditions, Amontons law does not hold, since the statistical effect of more asperities coming into play no longer occurs, and the contact area is not simply proportional to the applied load. 

Salmeron M, Liu G-Y and Ogletree D F 1995 Molecular arrangement and mechanical stability of self-assembled monolayers on Au(111) under applied load Force in Scanning Probe Methods ed H-J Guntherodt et al (Amsterdam Kluwer)... 

Figure Bl.20.12. Measured frietion foree, F, and real eontaet area, A, against externally applied load, L, for two moleeularly smooth miea surfaees sliding in 0.5 M KCl solution, i.e. in the absenee of adhesion, with pemrission from [25],...

The use of fatigue data and crack length measurements to predict the remaining service life of a stmcture under cyclic loading is possibly the most common application of fracture mechanics for performance prediction. In complex stmctures the growth of cracks is routinely monitored at intervals, and from data about crack growth rates and the applied loadings at that point in the stmcture, a decision is made about whether the stmcmre can continue to operate safely until the next scheduled inspection. 

For a single-value toughness material, dT/dc = 0. Accordingly, if the applied stress intensity factor is always increasing with crack length, equation 4 is always satisfied. Thus, the condition for fracture is equation 5, where is given by the applied loading conditions. 

The calculated loading stress, L, on a component is not only a function of applied load, but also the stress analysis technique used to find the stress, the geometry, and the failure theory used (Ullman, 1992). Using the variance equation, the parameters for the dimensional variation estimates and the applied load distribution, a statistical failure theory can then be formulated to determine the stress distribution, f L). This is then used in the SSI analysis to determine the probability of failure together with material strength distribution f S). 

The formulations for the failure governing stress for most stress systems can be found in Young (1989). Using the variance equation and the parameters for the dimensional variation estimates and applied load, a statistical failure theory can be formulated for a probabilistic analysis of stress rupture. 

Crack extension is often observed to vary significantly at the same nominal value of AK (= Y AOpos Tta) for different values of R-ratio. Elber [26] was the first to explain this observation for metals in terms of the crack closure phenomenon. He determined, by measuring specimen compliance, that fatigue cracks open and close at the crack tip at positive values of stress due to contact between crack surfaces behind the crack tip. For elastic fatigue conditions it is generally found that P p = P, and Kop = K, where P is the applied load. 

A sphere in contact with a flat surface under the action of an applied load P (P > 0 for compression and P < 0 for tension) deforms as shown in Fig. 3. Let a be the radius of the contact zone. The center of the sphere is displaced by a... 

Fig. 3. Schematic of a sphere in contact with a flat surface, (a) The deformation when surfaces are in contact. The radius of the deformed zone is a, and the separation profile is given by D versus r. The central displacement, S, is shown as the distance between the center of the deformed zone and the tip of the undeformed sphere, represented by the bold line. S characterizes the displacement of the applied load, (b) When the applied load is —/ s, the pull-off force, the surfaces jump out of contact, and the undeformed shape of the surfaces is attained.

It can be seen from Eq. 11 that the contact radius is finite even at no applied load. The zero load contact radius Aq is given by... 

The surfaces jump out of contact when the applied load is equal to the pull-off force, i.e. P — —P. The contact radius at the point of separation, a, is given by... 

In their analysis, Johnson et al. [6] note that, under an applied load P, the aetual eontaet radius a is higher than the radius predieted by the Hertzian theory. Johnson et al. eaieuiate an equivalent Hertzian load P (> P) using Eq. 7, and the corresponding energy U ) stored in the system due to elastic deformation of the sphere under Hertzian conditions. Ui and P are 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.

In the JKR experiments, a macroscopic spherical cap of a soft, elastic material is in contact with a planar surface. In these experiments, the contact radius is measured as a function of the applied load (a versus P) using an optical microscope, and the interfacial adhesion (W) is determined using Eqs. 11 and 16. In their original work, Johnson et al. [6] measured a versus P between a rubber-rubber interface, and the interface between crosslinked silicone rubber sphere and poly(methyl methacrylate) flat. The apparatus used for these measurements was fairly simple. The contact radius was measured using a simple optical microscope. This type of measurement is particularly suitable for soft elastic materials. 

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