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References
N.I. Muskhelishvili, Singular Integral Equations, (English translation by J.R.M. Radok, Noordhoff, Groningen, (1953)). For applications to elasticity problems, including the above contact problem, see N.I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity, (English translation by J.R.M. Radok, Noordhoff, Groningen, (1953)). For a simpler discussion of the solution of contact problems involving singular integral equations, see K.L.Johnson, loc. cit., §2.7.
C. Cattaneo, Sul contatto di corpi elastici, Acad. dei Lincei, Rediconti, Ser 6, Vol. 27 (1938), 342–348, 433–436, 474–478.
R.D. Mindlin, Compliance of elastic bodies in contact, ASME J.Appl.Mech., Vol. 17 (1949), 259–268.
Notice that the assumption is that all points in −c>x>c are in a state of stick throughout the loading process. It is therefore possible to integrate (12.77) and write the boundary condition in terms of h(x) instead of ḣ(x) In general, this is possible as long as no point passes from a state of slip to one of stickduring the loading. In particular, the stick zone must not advance into the slip zone during loading. For an exhaustive study of the effect of loading history in frictional contact problems, see J. Dundurs and M. Comninou, An educational elasticity problem with friction: Part 1, Loading and unloading paths for weak friction, ASME J. Appl. Mech., Vol. 48 (1981), 841–845; Part 2: Unloading for strong friction and reloading, ibid., Vol. 49 (1982), 47–51; Part 3: General load paths, ibid, Vol. 50 (1983), 77–84.
The effect of non-monotonic loading in the related problem of two contacting spheres was considered by R.D. Mindlin and H. Deresiewicz, Elastic spheres in contact under varying oblique forces, ASME J. Appl. Mech., Vol. 21 (1953), 327–344. The history-dependence of the friction law leads to quite complex arrangements of slip and stick zones and consequent variation in the load-compliance relation. These results also find application in the analysis of oblique impact, where neither normal nor tangential loading is monotonic (see N. Maw, J.R. Barber and J.N. Fawcett, The oblique impact of elastic spheres, Wear, Vol. 38 (1976), 101–114).
K.L. Johnson, Energy dissipation at spherical surfaces in contact transmitting oscillating forces, J. Mech. Eng. Sci., Vol. 3 (1961), 362.
Jäger, J. (1997) Half-planes without coupling under contact loading. Arch. Appl. Mech. Vol. 67 (1997), 247–259.
Ciavarella, M. The generalized Cattaneo partial slip plane contact problem. I-Theory, II-Examples. Int. J. Solids Structures. Vol. 35 (1998), 2349–2378.
Hills, D.A. and Nowell, D. (1994) Mechanics of fretting fatigue. Kluwer, Dordrecht, Szolwinski, M.P. and Farris, T.N. (1996) Mechanics of fretting fatigue crack formation. Wear 198, 93–107.
Ciavarella M. and Hills, D.A. (1998) Some observations on the oscillating tangential forces and wear in general plane contacts, Europ. J. Mech. A-Solids. Vol. 18 (1999), 491–497.
Note that this applies to the uncoupled problem (β=0) only. With dissimilar materials, there is generally a leading slip zone and there can also be an additional slip zone contained within the stick zone. This problem is treated by R.H. Bentall and K.L. Johnson, Slip in the rolling contact of dissimilar rollers, Int. J. Mech. Sci., Vol. 9 (1967), 389–404.
F.W. Carter, On the action of a locomotive driving wheel, Proc. Roy. Soc. (London), Vol. A112 (1926), 151–157.
For a more extensive discussion of frictional problems of this type, see K.L. Johnson, loc. cit., Chapters 5,7,8.
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(2004). Plane Contact Problems. In: Elasticity. Solid Mechanics and Its Applications, vol 107. Springer, Dordrecht. https://doi.org/10.1007/0-306-48395-5_12
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