Friction interaction of two half planes in the presence of a surface groove
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FRICTION INTERACTION OF TWO HALF PLANES IN THE PRESENCE OF A SURFACE GROOVE N. I. Malanchuk1 and A. Kaczyński2
UDC 539.3
We study the problem of contact interaction of two elastic isotropic half planes (one of which contains a shallow groove) made of identical materials with regard for the local friction slip. First, the bodies are pressed to each other by normal forces up to their full contact and then monotonically increasing shear forces are applied to these bodies which leads to their partial slip. The problem is reduced to a singular integral equation with Cauchy kernel for the relative tangential shift of the boundaries of half planes in the slip zone. The sizes of this zone are found from the condition of boundedness of tangential stresses on its edges. We also obtain analytic solutions of the problem for some profiles of the groove and analyze the dependences of the length of slip zone and contact stresses on the applied loads. Keywords: stick–slip contact, groove, singular integral equation, relative tangential shift.
In various branches of science, in particular, in tribology, geophysics, machine building, and biomechanics, much attention is given to the investigation of the laws of friction contact. The presence of friction in the formulation of contact problems significantly complicates their solution. In the theoretical investigations of friction contact, the researchers most often use the Coulomb–Amonton law [1] according to which the relationship between the tangential stresses s caused by the friction forces and the normal pressure p is linear, i.e., s = ± fp , where f is the friction coefficient. The sign “+” or “–” of the parameter f is chosen from physical considerations with regard for the fact that the friction forces withstand the mutual slip of the boundaries of the bodies. According to this law, every point of the contact surface is either in the state of sticking when the inequality s < fp holds or in the state of sliding for s = ± fp . In the case where the last equality is true over the entire contact surface, we get the sliding contact of the bodies. These problems are characterized by the following specific feature: the zones of contact, sticking, and slip are unknown in advance. In addition, the final stress-strain state depends on the history of loading. The phenomena of friction contact were studied in detail in the past. The results of fundamental investigations and their applications can be found in the monographs and works [2–10]. In the present work, we focus our attention on the contact interaction of bodies with compatible surfaces (according to Johnson’s terminology [2]). These surfaces are often encountered in the nature and engineering but, nevertheless, are poorly studied. The main results in this direction have been obtained for the case of frictionless contact of bodies of this sort (see, e.g., [11–15] and bibliography therein). In [16, 17], plane contact problems were studied with regard for the sliding friction contact of bodies with compatible shapes. The contact inte
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