On the concept of "far points" in Hertz contact problems

Relative displacement of “far points” is used in the Hertzian contact mechanics as a measure of contact compliance. However, to be legitimate, it should be almost insensitive to the exact choice of the “far points”, and this is not always the case. The present work aims at examination of legitimacy of this concept, on specific examples of one-dimensional problem of a long rod, 2-D problem of heavy disk and 3-D problem of a sphere resting on a smooth rigid foundation. It is found that, whereas in the 1-D problem this concept may well become inadequate, in the considered 2-D and 3-D problems, the parameter controlling the legitimacy of this concept are identified and, in the vast majority of cases of practical interest, the concept is indeed legitimate. Note that the mentioned 2-D and 3-D problems are quite challenging and the presented solutions may be of interest of their own