A numerical method for axisymmetric adhesive contact based on kalker’s variational principle

Authors

DOI:

https://doi.org/10.15587/1729-4061.2018.132076

Keywords:

adhesive contact, boundary element method, Kalker’s variational principle, wavy roughness, arc-length method

Abstract

A numerical method for axisymmetric adhesive contact of elastic bodies is proposed. It allows computing the size of the contact spot, the force of interaction as well as the contact pressure distribution unrestricted to any particular form of the initial gap between the bodies. Therefore, compared to the existing analytical theories, it is a more versatile research tool that can be used to study such phenomena as adhesive strength of conjugate bodies and stability loss induced energy dissipation in oscillating contact. A variational principle that can be used to construct an approximate solution is proposed. The derived nonlinear equations of the discretized mini-max problem determine the unknown radius of the circular contact spot and the nodal values of the thought-for contact pressure. Unlike other numerical methods where contact domain is updated by subtracting or adding separate boundary elements of finite size, the proposed approach enables gradual continuous variation of the contact area. The arc-length method was implemented in the numerical routine in order to solve for the unstable sections of the adhesive interaction process. Besides the distance and force variables, the increment of the contact area is included in the control for the sake of convergence. The numerical error of the approximate method with respect to the known analytical solutions is evaluated. Linear convergence with mesh refinement in computed force and contact area is observed. Extension of the proposed approach for arbitrary three-dimensional shape of the contacting bodies is planned for the future. This is required to study the impact of the random surface roughness on their adhesive properties.

Author Biography

Mykola Tkachuk, National Technical University «Kharkіv Polytechnic Institute» Kyrpychova str., 2, Kharkiv, Ukraine, 61002

PhD, Senior Researcher

Department of Theory and Systems of Automated Design of Mechanisms and Machines

References

  1. Johnson, K. L., Kendall, K., Roberts, A. D. (1971). Surface energy and the contact of elastic solids. In Proceedings of the Royal Society of London A: a thematical, Physical and Engineering Sciences, 324 (1558), 301–313.
  2. Guduru, P. R. (2007). Detachment of a rigid solid from an elastic wavy surface: Theory. Journal of the Mechanics and Physics of Solids, 55 (3), 445–472. doi: 10.1016/j.jmps.2006.09.004
  3. Derjaguin, B. V., Muller, V. M., Toporov, Yu. P. (1975). Effect of contact deformations on the adhesion of particles. Journal of Colloid and Interface Science, 53 (2), 314–326. doi: 10.1016/0021-9797(75)90018-1
  4. Pastewka, L., Robbins, M. O. (2016). Contact area of rough spheres: Large scale simulations and simple scaling laws. Applied Physics Letters, 108 (22), 221601. doi: 10.1063/1.4950802
  5. Sauer, R. A., Li, S. (2007). An atomic interaction-based continuum model for adhesive contact mechanics. Finite Elements in Analysis and Design, 43 (5), 384–396. doi: 10.1016/j.finel.2006.11.009
  6. Sauer, R. A. (2015). A Survey of Computational Models for Adhesion. The Journal of Adhesion, 92 (2), 81–120. doi: 10.1080/00218464.2014.1003210
  7. Feng, J. Q. (2000). Contact behavior of spherical elastic particles: a computational study of particle adhesion and deformations. Colloids and Surfaces A: Physicochemical and Engineering Aspects, 172 (1-3), 175–198. doi: 10.1016/s0927-7757(00)00580-x
  8. Greenwood, J. A. (2009). Adhesion of small spheres. Philosophical Magazine, 89 (11), 945–965. doi: 10.1080/14786430902832765
  9. Medina, S., Dini, D. (2014). A numerical model for the deterministic analysis of adhesive rough contacts down to the nano-scale. International Journal of Solids and Structures, 51 (14), 2620–2632. doi: 10.1016/j.ijsolstr.2014.03.033
  10. Pohrt, R., Popov, V. L. (2015). Adhesive contact simulation of elastic solids using local mesh-dependent detachment criterion in boundary elements method. Facta Universitatis, Series: Mechanical Engineering, 13 (1), 3–10.
  11. Popov, V. L., Pohrt, R., Li, Q. (2017). Strength of adhesive contacts: Influence of contact geometry and material gradients. Friction, 5 (3), 308–325. doi: 10.1007/s40544-017-0177-3
  12. Papangelo, A., Hoffmann, N., Ciavarella, M. (2017). Load-separation curves for the contact of self-affine rough surfaces. Scientific Reports, 7 (1). doi: 10.1038/s41598-017-07234-4
  13. Ciavarella, M., Papangelo, A. (2017). A random process asperity model for adhesion between rough surfaces. Journal of Adhesion Science and Technology, 31 (22), 2445–2467. doi: 10.1080/01694243.2017.1304856
  14. Prokopovich, P., Starov, V. (2011). Adhesion models: From single to multiple asperity contacts. Advances in Colloid and Interface Science, 168 (1-2), 210–222. doi: 10.1016/j.cis.2011.03.004
  15. Kalker, J. J. (1987). Variational and non-variational theory of frictionless adhesive contact between elastic bodies. Wear, 119 (1), 63–76. doi: 10.1016/0043-1648(87)90098-6
  16. Kesari, H., Lew, A. J. (2011). Adhesive Frictionless Contact Between an Elastic Isotropic Half-Space and a Rigid Axi-Symmetric Punch. Journal of Elasticity, 106 (2), 203–224. doi: 10.1007/s10659-011-9323-8
  17. Kalker, J. J. (1977). Variational Principles of Contact Elastostatics. IMA Journal of Applied Mathematics, 20 (2), 199–219. doi: 10.1093/imamat/20.2.199
  18. Johnson, K. L. (1985). Contact Mechanics. Cambridge, UK: Cambridge University Press. doi: 10.1017/cbo9781139171731
  19. Fuller, K. N. G., Tabor, D. (1975). The Effect of Surface Roughness on the Adhesion of Elastic Solids. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 345 (1642), 327–342. doi: 10.1098/rspa.1975.0138
  20. Briggs, G. A. D., Briscoe, B. J. (1977). The effect of surface topography on the adhesion of elastic solids. Journal of Physics D: Applied Physics, 10 (18), 2453–2466. doi: 10.1088/0022-3727/10/18/010
  21. Kim, H.-C., Russell, T. P. (2001). Contact of elastic solids with rough surfaces. Journal of Polymer Science Part B: Polymer Physics, 39 (16), 1848–1854. doi: 10.1002/polb.1159
  22. Fuller, K. N. G., Roberts, A. D. (1981). Rubber rolling on rough surfaces. Journal of Physics D: Applied Physics, 14 (2), 221–239. doi: 10.1088/0022-3727/14/2/015
  23. Guduru, P. R., Bull, C. (2007). Detachment of a rigid solid from an elastic wavy surface: Experiments. Journal of the Mechanics and Physics of Solids, 55 (3), 473–488. doi: 10.1016/j.jmps.2006.09.007
  24. Graveleau, M., Chevaugeon, N., Moës, N. (2015). The inequality level-set approach to handle contact: membrane case. Advanced Modeling and Simulation in Engineering Sciences, 2 (1). doi: 10.1186/s40323-015-0034-8
  25. Martynyak, R. M., Slobodyan, B. S. (2009). Contact of elastic half spaces in the presence of an elliptic gap filled with liquid. Materials Science, 45 (1), 66–71. doi: 10.1007/s11003-009-9156-9
  26. Kozachok, O. P., Slobodian, B. S., Martynyak, R. M. (2017). Interaction of Two Elastic Bodies in the Presence of Periodically Located Gaps Filled with a Real Gas. Journal of Mathematical Sciences, 222 (2), 131–142. doi: 10.1007/s10958-017-3287-6

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Published

2018-05-24

How to Cite

Tkachuk, M. (2018). A numerical method for axisymmetric adhesive contact based on kalker’s variational principle. Eastern-European Journal of Enterprise Technologies, 3(7 (93), 34–41. https://doi.org/10.15587/1729-4061.2018.132076

Issue

Vol. 3 No. 7 (93) (2018): Applied mechanics

Section

Applied mechanics

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Copyright (c) 2018 Mykola Tkachuk

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