Abstract
The contact of an elastic quarter- or eighth-space is studied under the condition that the movement of the side surface of the quarter-space is constrained: It can slide freely along the plane of the side surface but its normal movement is blocked (for example, by a rigid wall). The solution of this contact problem can be easily achieved by additionally applying a mirrored load to an elastic half-space. Non-adhesive contact and the Johnson-Kendall-Roberts (JKR)-type adhesive contact between a rigid sphere and an elastic quarter-space under such a boundary condition is numerically simulated using the fast Fourier transform (FFT)-assisted boundary element method (BEM). Contacts of an elastic eighth-space are investigated using the same idea. Depending on the position of the sphere relative to the side edge, different contact behavior is observed. In the case of adhesive contact, the force of adhesion first increases with increasing the distance from the edge of the quarter-space, achieves a maximum, and decreases further to the JKR-value in large distance from the edge. The enhancement of the force of adhesion compared to the half-space-contact is associated with the pinning of the contact area at the edge. We provide the maps of the force of adhesion and their analytical approximations, as well as pressure distributions in the contact plane and inside the quarter-/eighth-space.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Popov V L. Contact Mechanics and Friction. 2nd edn. Berlin: Springer, 2017.
Barber J R. Contact Mechanics. Cham (Switzerland): Springer International Publishing AG, 2018.
Hanson M T, Keer L M. Analysis of edge effects on rail-wheel contact. Wear 144(1–2): 39–55 (1991)
Guo L, Wang W, Zhang Z M, Wong P L. Study on the free edge effect on finite line contact elastohydrodynamic lubrication. Tribol Int 116: 482–490 (2017)
Ma H, Wang D, Tai X Y, Wen B C. Vibration response analysis of blade-disk dovetail structure under blade tip rubbing condition. J Vib Control 23(2): 252–271 (2017)
Benad J. Numerical methods for the simulation of deformations and stresses in turbine blade fir-tree connections. FU Mech Eng 17(1): 1–15 (2019)
Hetényi M. A method of solution for the elastic quarter-plane. JAppl Mech 27(2): 289–296 (1960)
Hetényi M. A general solution for the elastic quarter space. J Appl Mech 37(1): 70–76 (1970)
Keer L M, Lee J C, Mura T. Hetényi’s elastic quarter space problem revisited. Int J Solids Struct 19(6): 497–508 (1983)
Keer L M, Lee J C, Mura T. A contact problem for the elastic quarter space. Int J Solids Struct 20(5): 513–524 (1984)
Hanson M T, Keer L M. A simplified analysis for an elastic quarter-space. Q J Mech Appl Math 43(4): 561–587 (1990)
Guilbault R. A fast correction for traction-free surface of elastic quarter-space. WIT Trans Eng Sci 66: 37–48 (2010).
Zhang Z M, Wang W, Wong P L. An explicit solution for the elastic quarter-space problem in matrix formulation. Int J Solids Struct 50(6): 976–980 (2013)
Zhang H B, Wang W Z, Zhang S G, Zhao Z Q. Modeling of elastic finite-length space rolling-sliding contact problem. Tribol Int 113: 224–237 (2017)
Stan G. The effect of edge compliance on the adhesive contact between a spherical indenter and a quarter-space. Int J Solids Struct 158: 165–175 (2019).
Pohrt R, Li Q. Complete boundary element formulation for normal and tangential contact problems. Phys Mesomech 17(4): 334–340 (2014)
Liu S B, Wang Q, Liu G. A versatile method of discrete convolution and FFT (DC-FFT) for contact analyses. Wear 243(1–2): 101–111 (2000)
Pohrt R, Popov V L. Adhesive contact simulation of elastic solids using local mesh-dependent detachment criterion in boundary elements method. FU Mech Eng 13(1): 3–10 (2015)
Johnson K L, Kendall K, Roberts A D. Surface energy and the contact of elastic solids. P Roy Soc A 324(1558): 301–313 (1971)
Popov V L, Li Q, Lyashenko I A, Pohrt R. Adhesion and friction in hard and soft contacts: Theory and experiment. Friction 9(6): 1688–1706 (2021)
Wang Q J, Sun L L, Zhang X, Liu S B, Zhu D. FFT-based methods for computational contact mechanics. Front Mech Eng 6: 1–20 (2020)
Li Q. Edge effect and indentation depth-dependent contact behavior in contact of an elastic quarter-space. Int J Solids Struct 285: 112552 (2023)
Li Q. Simulation of a single third-body particle in fictional contact. FU Mech Eng 18(4):537–544 (2020)
Acknowledgements
This work has been conducted under partial financial support from DFG (Grant Nos. PO 810/55-3 and LI 3064/2-1).
Author information
Authors and Affiliations
Contributions
Valentin L. POPOV designed the concept of the paper. Qiang LI executed numerical simulations.
Corresponding authors
Ethics declarations
The authors have no competing interests to declare that are relevant to the content of this article. The author Valentin L. POPOV is the Editorial Board Member of this journal.
Additional information
Qiang LI. He is a postdoctoral researcher at the Berlin University of Technology. He studied mechanical engineering in East China University of Science and Technology. He obtained his doctorate at the Berlin University of Technology in 2014 and now works as a scientific researcher at the Department of System Dynamics and the Physics of Friction headed by Prof. V. L. POPOV. He has published over 70 papers in international journals, including Physical Review Letters. His scientific interests include tribology, elastomer friction, hydrodynamic lubricated contact, numerical simulation of frictional behaviors, fast numerical method based on boundary element method, and adhesion.
Valentin L. POPOV. He is full professor at the Technische Universität Berlin. He studied physics and obtained his doctorate in 1985 from the Moscow State Lomonosov University. In 1985–1998, he worked at the Institute of Strength Physics and Materials Science of the Russian Academy of Sciences and was a guest professor in the field of theoretical physics at the University of Paderborn (Germany) from 1999 to 2002. Since 2002, he is the head of the Department of System Dynamics and the Physics of Friction at the Berlin University of Technology. He has published over 300 papers in leading international journals and is the author of the book Contact Mechanics and Friction: Physical Principles and Applications which appeared in ten editions in German, English, Chinese, Russian, Spanish, and Japanese. He is the editor-in-chief of the Journal Forntiers in Mechanical Engineering/Tribology, member of editorial boards of many international journals and is organizer of more than 20 international conferences and workshops over diverse tribological themes. Prof. POPOV is an Honorary Professor of the Tomsk Polytechnic University, of the East China University of Science and Technology, and of the Changchun University of Science and Technology, and the Distinguished Guest Professor of the Tsinghua University. His areas of interest include tribology, nanotribology, tribology at low temperatures, biotribology, the influence of friction through ultrasound, numerical simulation of contact and friction, research regarding earthquakes, and synovial joints regenerative rehabilitation.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Li, Q., Popov, V.L. Non-adhesive and adhesive contacts of an elastic quarter-or eighth-space with freely sliding sides. Friction 12, 2052–2063 (2024). https://doi.org/10.1007/s40544-024-0866-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40544-024-0866-7