On a contact problem for a homogeneous plane with a finite crack under friction

Cover Page

Cite item

Full Text

Open Access Open Access
Restricted Access Access granted
Restricted Access Subscription Access

Abstract

An exact solution to the contact problem of indentation of an absolutely rigid punch with a straight base, taking into account friction, into one of the edges of a finite crack located in a homogeneous elastic plane was derived. It is assumed that shear contact stresses are directly proportional to normal contact pressure. In this case, it is assumed that the friction coefficient is directly proportional to the coordinates of the contacting points of the contacting surfaces. The governing system of equations for the problem was derived in the form of the heterogeneous Riemann problem for two functions with variable coefficients and its closed solution is constructed in quadratures. Simple formulas for contact stresses and the normal dislocation component of displacements of crack edge points were obtained. The patterns of changes in contact stresses and crack opening depending on the maximum value of the friction coefficient have been studied.

About the authors

V. Hakobyan

Institute of Mechanics of the National Academy of Sciences of the Republic of Armenia

Author for correspondence.
Email: vhakobyan@sci.am
Armenia, Yerevan

H. Amirjanyan

Institute of Mechanics of the National Academy of Sciences of the Republic of Armenia

Email: amirjanyan@gmail.com
Armenia, Yerevan

L. Hakobyan

Institute of Mechanics of the National Academy of Sciences of the Republic of Armenia

Email: lhakobyan@gmail.com
Armenia, Yerevan

References

  1. Galin L.A., 1980. Contact Problems in the Theory of Elasticity and Viscoelasticity. Nauka, 1980. 304 p.
  2. Shtaerman I.Ya. Contact Problem of Elasticity Theory. Gostekhteorizdat, 1949. 270 p.
  3. Panasyuk V.V., Savruk M.P., Datsyshin A.P. The Stress Distribution near Cracks in Plates and Shells. Kiev: Naukova Dumka, 1976. 443 p.
  4. Muskhelishvili N.I. Some Problems of Mathematical Theory of Elasticity. M.: Nauka, 1966. 708 р.
  5. Berezhnitsky L., Panasiuk V., Staschuk N. 1983. Interaction of Rigid Linear Inclusions and Cracks in Deformable Body. Kiev: Naukova Dumka, 1983. 288 p.
  6. Popov G.Ya. About Concentration of the Elastic Stresses Near Thin Detached Inclusion. “Contemporary Problems of Mechanics and Aviation”, Dedicated to I. Abraztsov. 1980. P. 156–162.
  7. Hakobyan V.N., Mirzoyan S.E., Dashtoyan L.L. Axisymmetric Mixed Boundary Value Problem for Composite Space with Coin-Shaped Crack // Herald of the Bauman Moscow State Technical University, Series Natural Sciences. 2015. № 3 (60). P. 31–46; https://doi.org/10.18698/1812-3368-2015-3-31-46
  8. Hakobyan V.N. 1995. On One Mixed Problem for Composite Plate, Weakened by Crack // Mechanics. Proceedings of National Academy of Sciences of Armenia. 1995. V. 48. № 4. P. 57–65.
  9. Hakobyan V.N. Stress Concentrators in Continuous Deformable Bodies, Advanced Structured Materials, V. 181. Springer, 2022. 397 p.
  10. Il’ina I.I., Silvestrov V.V. The Problem of a Thin Interfacial Inclusion Detached from the Medium Along One Side // Mech. Solids. 2005. V. 40. № 3. P. 153–166.
  11. Cherepanov G.P. Solution of a Linear Boundary Value Problem of Riemann for Two Functions and its Application to Certain Mixed Problems in the Plane Theory of Elasticity // J. Appl. Math. Mechanics. 1962. V. 26. № 5. P. 907–312.
  12. Mkhitaryan S.M. On the Stress-Strain State of an Elastic Infinite Plate with a Crack Expanding by Means of Smooth Thin Inclusion Indentation // Mechanics. Proceedings of National Academy of Sciences of Armenia. 2019. V. 72. № 4. P. 38–64; https://doi.org/10.33018/72.4.4
  13. Hakobyan V.N., Hakobyan L.V. Contact problem for a homogeneous plane with crack // Proc. NAS RA. Mechanics. 2020. V. 73. № 4. P. 3–12.
  14. Hakobyan V.N., Amirjanyan H.A., Dashtoyan L.L., Sahakyan A.V. Indentation of an Absolutely Rigid Thin Inclusion into One of the Crack Faces in an Elastic Plane Under Slippage at the Ends. In Book: Altenbach H., Bauer S., Belyaev K. et al. (eds), Advances in Solid and Fracture Mechanics, A Liber Amicorum to Celebrate the Birthday of N. Morozov. 2022. P. 187–197.
  15. Hakobyan V.N., Hakobyan L.V. On a model of friction for contact problems of the theory of elasticity // Proc. NAS RA. Mechanics. 2023. V. 76. № 2. P. 20–31; https://doi.org/10.54503/0002-3051-2023.76.2-20.
  16. Muskhelishvili N.I. Singular integral equations. Moscow: Nauka, 1968. 511 p.
  17. Prudnikov A.P., Brychkov Yu.A., Marichev O.I. Integrals and Series. Moscow: Nauka, 1981. 738 p.
  18. Sahakyan A.V., Amirjanyan H.A. Method of mechanical quadratures for solving singular integral equations of various types // IOP Conf. Series: J. Physics: Conf. Series. 2018. V. 991. 012070; https://doi.org/10.1088/1742-6596/991/1/012070

Supplementary files

Supplementary Files
Action
1. JATS XML
Download

Copyright (c) 2024 Russian Academy of Sciences