Journal of Mathematical Sciences

, Volume 160, Issue 4, pp 470–477 | Cite as

Pressure of an elastic half space on a rigid base with rectangular hole in the case of a liquid bridge between them

  • R. M. Martynyak
  • B. S. Slobodyan
  • V. M. Zelenyak

A model of contact between an elastic half space and a rigid base with a shallow surface rectangular hole is proposed. The hole contains an incompressible liquid and gas. The liquid occupies the middle part of the hole and forms a capillary bridge between the opposite surfaces. The remaining volume of the hole is filled with gas under a constant pressure. The liquid completely wets the surfaces of the bodies. The pressure drop at the liquid–gas interface caused by the surface tension is defined by the Laplace formula. The corresponding plane contact problem for the elastic half space is essentially nonlinear because the pressure of the liquid and the length of the capillary in the contact-boundary conditions are not known in advance and depend on the external load. The problem is reduced to a system of three equations (a singular integral equation for the function of height of the hole and two transcendental equations for the length of the capillary and the height of the meniscus). An analytic-numerical procedure for the solution of these equations is proposed. Dependences of the length of the capillary and the pressure drop at the liquid–gas interface on the external load, volume of liquid, and its surface tension are analyzed.


Surface Tension Pressure Drop Contact Problem Elastic Body Liquid Bridge 
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  1. 1.
    S. A. Artsybyshev, Course in Physics. Part 1. Mechanics and Heat [in Russian], Ministerstvo Prosveshcheniya RSFSR, Moscow (1951).Google Scholar
  2. 2.
    I. G. Goryacheva and Yu. Yu. Makhovskaya, “Adhesive interaction of elastic bodies,” Prikl. Mat. Mekh., 65, No. 2, 279–289 (2001).zbMATHGoogle Scholar
  3. 3.
    I. G. Goryacheva and Yu. Yu. Makhovskaya, “Contact of elastic bodies in the presence of capillary adhesion,” Prikl. Mat. Mekh., 63, No. 1, 128–137 (1999).zbMATHGoogle Scholar
  4. 4.
    G. S. Kit, R. M. Martynyak, and I. M. Machishin, “Influence of a gas-liquid filler of an intercontact space on the stressed state of mating bodies,” Prikl. Mekh., 39, No. 3, 52–60 (2003).zbMATHGoogle Scholar
  5. 5.
    R. M. Martynyak, “Contact of a half space with not plane base in the case of an intercontact gap filled with ideal gas,” Mat. Met. Fiz.-Mekh. Polya, 41, No. 4, 144–149 (1998).zbMATHGoogle Scholar
  6. 6.
    R. M. Martynyak, “Contact interaction of two half spaces in the presence of a surface hole partially filled with incompressible liquid,” Fiz.-Khim. Mekh. Mater., 26, No. 2, 91–94 (1990).Google Scholar
  7. 7.
    R. M. Martynyak, “Mechanothermodiffusion interaction of bodies with regard for a filler of intercontact gaps,” Fiz.-Khim. Mekh. Mater., 36, No. 2, 124–126 (2000).Google Scholar
  8. 8.
    R. M. Martynyak, Thermoelastic Interaction in the Case of a Nonideal and Mechanical Contact [in Ukrainian], Candidate-Degree Thesis (Physics and Mathematics), Lviv (1987).Google Scholar
  9. 9.
    R. M. Martynyak and B. S. Slobodyan, “Interaction of two bodies in the presence of capillaries in an intercontact gap,” Mat. Met. Fiz.-Mekh. Polya, 49, No. 1, 164–173 (2006).zbMATHGoogle Scholar
  10. 10.
    B. E. Monastyrs’kyi, “Influence of a filler of an intersurface gap on the interaction of bodies under conditions of nonideal contact,” Prikl. Probl. Mekh. Mat., Issue 1, 78–82 (2003).Google Scholar
  11. 11.
    N. I. Muskhelishvili, Some Main Problems of the Mathematical Theory of Elasticity [in Russian], Academy of Sciences of the USSR, Moscow (1954).Google Scholar
  12. 12.
    B. S. Slobodyan and R. M. Martynyak, “Simulation of interaction of bodies with regard for the surface tension of liquid in an intercontact gap,” Fiz.-Mat. Model. Inform. Tekhnol., 6, 19–29 (2007).Google Scholar
  13. 13.
    O. G. Chekina, “On the friction of rough surfaces separated by a thin layer of liquid,” Tren. Iznos, 19, No. 3, 306–311 (1998).Google Scholar
  14. 14.
    S. A. Chizhik, “Capillary mechanism of adhesion and friction of rough surfaces separated by a thin layer of liquid,” Tren. Iznos, 15, No. 1, 11–26 (1994).Google Scholar
  15. 15.
    Y. Ando, “Effect of capillary formation on friction and pull-off forces measured on submicron-size asperities,” Tribology Lett., 19, No. 1, 29–36 (2005).CrossRefGoogle Scholar
  16. 16.
    Z. Dai, Y. Min, and S. Gorb, “Frictional characteristics of the beetle head-joint material,” Wear, 260, 168–174 (2006).CrossRefGoogle Scholar
  17. 17.
    T. Kato, S. Watanabe, and H. Matsuoka, “Dynamic characteristics of an in-contact headslider considering meniscus force. Part 1. Formulation and application to the disk with sinusoidal undulation,” J. Tribology, 122, 633–638 (2000).CrossRefGoogle Scholar
  18. 18.
    T. Kato, S. Watanabe, and H. Matsuoka, “Dynamic characteristics of an in-contact headslider considering meniscus force. Part 2. Application to the disk with random undulation and design conditions,” J. Tribology, 123, 168–174 (2001).CrossRefGoogle Scholar
  19. 19.
    S. Kobatake, Y. Kawakubo, and S. Suzuki, “Laplace pressure measurement on laser textured thin-film disk,” Tribology Int., 36, 329–333 (2003).CrossRefGoogle Scholar
  20. 20.
    P. Lambert and A. Delchambre, “Parameters ruling capillary force at the submillimetric scale,” Langmuir, 21, No. 21, 9537–9543 (2005).CrossRefGoogle Scholar
  21. 21.
    Li Shi and M. Arunava, “Thermal transport mechanisms at nanoscale point contacts,” J. Heat Transfer., 124, 329–337 (2002).CrossRefGoogle Scholar
  22. 22.
    C. Pailler-Mattei and H. Zahouani, “Analysis of adhesive behaviour of human skin in vivo by an indentation test,” Tribology Int., 39, No. 1, 12–21 (2006).CrossRefGoogle Scholar
  23. 23.
    W. Peng and B. Bhushan, “Sliding contact analysis of layered elastic/plastic solids with rough surfaces,” Trans. ASME. J. Tribology, 124, 46–61 (2002).CrossRefGoogle Scholar
  24. 24.
    Y. Rabinovich, M. Esayanur, and B. Moudgil, “Capillary forces between two spheres with a fixed volume liquid bridge: theory and experiment,” Langmuir, 21, 10992–10997 (2005).CrossRefGoogle Scholar
  25. 25.
    A. Rennie, P. Dickrell, and W. Sawyer, “Friction coefficient of soft contact lenses: measurements and modeling,” Tribology Lett., 18, No. 4, 499–504 (2005).CrossRefGoogle Scholar
  26. 26.
    F. Soulie, F. Cherblanc, M. El Youssoufi, and C. Saix, “Influence of liquid bridges on the mechanical behaviour of polydisperse granular materials,” Int. J. Numer. Anal. Meth. Geomech., 30, 123–228 (2006)CrossRefGoogle Scholar
  27. 27.
    J. Zheng and J. L. Streutor, “A liquid bridge between two elastic half-spaces: A theoretical study of interface instability,” Tribology Lett., 16, Nos. 1–2, 1–9 (2004).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2009

Authors and Affiliations

  • R. M. Martynyak
    • 1
  • B. S. Slobodyan
    • 1
  • V. M. Zelenyak
    • 1
  1. 1.Pidstryhach Institute of Applied Problems of Mechanics and MathematicsUkrainian Academy of SciencesLvivUkraine

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