Maugis

The Hertz theory of contact mechanics has been extended, as in the JKR theory, to describe the equilibrium contact of adhering elastic solids. The JKR formalism has been generalized and extended by Maugis and coworkers to describe certain dynamic elastic contacts. These theoretical developments in contact mechanics are reviewed and summarized in Section 3. Section 3.1 deals with the equilibrium theories of elastic contacts (e.g. Hertz theory, JKR theory, layered bodies, and so on), and the related developments. In Section 3.2, we review some of the work of Maugis and coworkers. 

The JKR theory, much like the Hertz theory, assumes a parabolic approximation for the profile of sphere, which is valid for small ratios of contact radii to the sphere s radius. Maugis [34] has shown that for small particles on a soft substrate, this ratio could be so large that such parabolic approximation is no longer valid. Under such conditions, the use of exact expression for the sphere profile is necessary for the applicability of the JKR theory, which is expressed as... 

Fig. 18. Adhesive contact of elastic spheres. pH(r) and pa(r) are the Hertz pressure and adhesive tension distributions, (a) JKR model uses a Griffith crack with a stress singularity at the edge of contact (r = a) (b) Maugis model uses a Dugdale crack with a constant tension aa in a < r < c [1111.

It is somewhat disconcerting that the MYD analysis seems to present a sharp transition between the JKR and DMT regimes. Specifieally, in light of the vastly different response predicted by these two theories, one must ponder if there would be a sharp demarcation around /x = 1. This topic was recently explored by Maugis and Gauthier-Manuel [46-48]. Basing their analysis on the Dugdale fracture mechanics model [49], they concluded that the JKR-DMT transition is smooth and continuous. 

There have been several theories proposed to explain the anomalous 3/4 power-law dependence of the contact radius on particle radius in what should be simple JKR systems. Maugis [60], proposed that the problem with using the JKR model, per se, is that the JKR model assumes small deformations in order to approximate the shape of the contact as a parabola. In his model, Maugis re-solved the JKR problem using the exact shape of the contact. According to his calculations, o should vary as / , where 2/3 < y < 1, depending on the ratio a/R. 

Up to this point, the discussion has been limited to cases where the adhesion-induced stresses result in elastic deformation. However, as previously discussed, this need not be the case. Indeed, these stresses can be, in many instances, comparable to the Young s moduli of the contacting materials, often resulting in plastic deformations. This area has been most notably explored by Maugis and Pollock [63]. 

According to the theory proposed by Maugis and Pollock, hereafter referred to as the MP model, if the adhesion induced stresses cause at least one of the contacting materials to yield and undergo a totally plastic response, the contact region formed will increase in size until the force causing the yielding is balanced... 

As also discussed by Maugis and Pollock, the hardness of the material is related to its yield strength Y by H = 2>Y. The factor of 3 is a consequence of the deformation constraints of the indentor geometry used in hardness measurements. In the absence of an applied load, the MP theory predicts that... 

An example of a Maugis-Pollock system is polystyrene particles having radii between about 1 and 6 p.m on a polished silicon substrate, as studied by Rimai et al. [64]. As shown in Fig. 4, the contact radius was found to vary as the square root of the particle radius. Similar results were reported for crosslinked polystyrene spheres on Si02/silicon substrates [65] and micrometer-size glass particles on silicon substrates [66]. 

Additional suggested resources for the reader include introductory articles on scanning probe techniques for materials properties measurement [82,83J. A comprehensive manual describing various surface preparation techniques, experimental procedures and instrumentation is also a good resource [84J, although the more recent modulation based techniques are not covered. Key textbooks include Johnson s on contact mechanics [51J and Israelachvili s on surface forces [18J, as well as a treatment of JKR/DMT issues by Maugis [85J. 

Barquins, M. and Maugis, D., Tackiness of elastomers. J. Adhes., 13, 53-65 (1981). Creton, C. and Lakrout, H., Micromechanics of flat-probe adhesion tests of soft vi.scoelas-tic polymer films. J. Polym. Sci. B Polym. Phys., 38(7), 965-979 (2000). 

Maugis, D., Contact, Adhesion and Rupture of Elastic Solids. Solid State Sciences. Springer, Berlin, 2000. 

An issue, at present unresolved, is that Derjaguin, Muller and Toporov [24,25] have put forward a different analysis of the contact mechanics from JKR. Maugis has described a theory which comprehends both the theories as special cases [26]. 

However, use of the loss tangent in Eq. (13) is an oversimplification. Instead, we shall make use of an empirical dissipation law obtained in the study of rate-dependent adhesion of elastomers (19). Maugis and Barquins [19], in order to explain the dynamic adhesion of elastomers in a variety of test configurations (peel, flat punch, etc), proposed the following relation ... 

Electrical current via mechanical contact Electrical tunnelling current Scan velocity Maugis parameter Thermal conductivity... 

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