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On the concept of far points in Hertz contact problems.pdf

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On the concept of far points in Hertz contact problems.pdf

International Journal of Engineering Science 113 (2017) 20–36 Contents lists available at ScienceDirect International Journal of Engineering Science journal homepage: /locate/ijengsci On the concept of “far points” in Hertz contact problems Ivan Argatov a, Mark Kachanov b, Gennady Mishuris c,? a Institut für Mechanik, Technische Universit?t Berlin, 10623 Berlin, Germany b Department of Mechanical Engineering, Tufts University, Medford, MA 02155, USA c Department of Mathematics, Aberystwyth University, Ceredigion SY23 3BZ, Wales, UK article info Article history: Received 5 September 2016 Accepted 27 November 2016 Available online 4 January 2017 Keywords: Hertzian contact Far points Contact compliance Asymptotic model abstract 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 speci?c 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 identi?ed and, in the vast majority of cases of practical interest, the concept is indeed legitimate. Note that the mentioned 2-D and 3D problems are quite challenging and the presented solutions may be of interest of their own. ? 2016 Elsevier Ltd. All rights reserved. 1. Introduction A small contact area between two elastic bodies creates a highly compliant zone in the vicinity of the contact: when the bodies are pressed against one another, most of the overall deformation comes from this zone. The contact zone compliance is usually characterized by approach of two “remote points” – points on two s