Acta Mechanica

, Volume 230, Issue 10, pp 3741–3758 | Cite as

Contact of faces of a rectilinear crack under complex loading and various contact conditions

  • V. I. OstrikEmail author
Original Paper


The problem of partial contact of faces of a rectilinear crack in a uniform elastic plane under given tensile and shear stresses at infinity as well as two compressive concentrated forces applied near the crack is investigated. Three variants of problem statement are considered: smooth contact, sliding contact, and contact with slip and adhesion between the faces of the crack. Using Kolosov–Muskhelishvili complex potentials, the problem is reduced to a conjugation problem, and a solution of the last one is obtained in analytic form. Transcendental equations to determine the boundaries of contact area and adhesion zone are derived. Conditions are found for cases when the contact area of crack faces degenerates into a point or takes an entire crack. It is shown that an adhesion zone disappears and a mutual slip of crack faces occurs along an entire contact area, if a ratio of tangential and normal stresses given at infinity is larger than a certain value. The results of calculation of boundaries of contact area and adhesion zone, stress intensity factors, and distributions of normal and tangential contact stresses are presented.



  1. 1.
    Mossakovsky, V.I., Zagubizhenko, P.A.: On a mixed problem of elasticity theory for a plane, weakened by a rectilinear gap. Rep. USSR Acad. Sci. 94(3), 409–412 (1954). (in Russian)Google Scholar
  2. 2.
    Mossakovsky, V.I., Zagubizhenko, P.A.: On a compression of an elastic isotropic plane, weakened by a rectilinear crack. Rep. Ukr. SSR Acad. Sci. 5, 385–390 (1954). (in Ukrainian)Google Scholar
  3. 3.
    Aksogan, O.: Partial closure of a Griffith crack under a general loading. Int. J. Fract. 11(4), 659–670 (1975)CrossRefGoogle Scholar
  4. 4.
    Bowie, O.L., Freese, C.E.: On the “overlapping” problem in crack analysis. Eng. Fract. Mech. 8(2), 373–379 (1976)CrossRefGoogle Scholar
  5. 5.
    Grilytsky, N.D., Kit, G.S.: On a stress–strain state in the vicinity of a crack with partially contacting faces. Math. Methods Physicomech. Fields 8, 35–39 (1978). (in Ukrainian)Google Scholar
  6. 6.
    Zozulya, V.V.: Bending of a plate weakened by a crack with contacting edges under dynamic loading. Rep. USSR Acad. Sci. 4, 56–60 (1991). (in Ukrainian)MathSciNetGoogle Scholar
  7. 7.
    Shatsky, I.P.: Developing a model of a contact of crack faces in bending plate. Theor. Appl. Mech. (31), 91–97 (2000). (in Russian)Google Scholar
  8. 8.
    Zozulya, V.M., Men’shikov, A.V.: Contact interaction of the faces of a rectangular crack under normally incident tension–compression waves. Int. Appl. Mech. 38(3), 302–307 (2002)CrossRefGoogle Scholar
  9. 9.
    Omidvar, B., Rahimian, M., DorMohammadi, A.A.: Simultaneous analysis of dynamic crack growth and contact of crack faces in single-region boundary element method. Am. Eurasian J. Agric. Environ. Sci. 5(2), 273–283 (2009)Google Scholar
  10. 10.
    Opanasovych, V.K., Yatsyk, I.M., Sulym, H.T.: Bending of Reissner’s plate containing a through-the-thickness crack by concentrated moments taking into account the width of a contact zone of its faces. J. Math. Sci. 187(5), 620–634 (2012)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Dhondt, G.: Effect of contact between the crack faces on crack propagation. Key Eng. Mater. 577–578, 61–64 (2014)Google Scholar
  12. 12.
    Kaminsky, A.O., Selivanov, M.F., Chornoivan, YuO: Determination of contact stresses between faces of a crack of normal separation. Rep. Natl. Acad. Sci. Ukr. 5, 36–42 (2016). (in Ukrainian)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Muskhelishvili, N.I.: Some Basic Problems of Mathematical Theory of Elasticity. Noordhoff, Groningen (1953)zbMATHGoogle Scholar
  14. 14.
    Muskhelishvili, N.I.: Singular Integral Equations. Nauka, Moscow (1968). (in Russian)zbMATHGoogle Scholar
  15. 15.
    Bateman, H., Erdelyi, A.: Higher Transcendental Functions, vol. 3. McGraw-Hill, New York (1955)zbMATHGoogle Scholar
  16. 16.
    Fromm, H.: Berechnung des Schlupfes beim Rollen deformierbarer Scheiben. Z. Angew. Math. Mech. 7(1), 27–58 (1927). (in German)CrossRefGoogle Scholar
  17. 17.
    Spence, D.A.: An eigenvalue problem for elastic contact with finite fraction. Proc. Camb. Philos. Soc. 73, 249–268 (1973)CrossRefGoogle Scholar
  18. 18.
    Ostrik, V.I.: Indentation of a punch into an elastic strip with friction and adhesion. Mech. Solids 46(5), 755–765 (2011)CrossRefGoogle Scholar
  19. 19.
    Ostryk, V.I.: Axisymmetric contact of a punch of polynomial profile with an elastic half-space when there is friction and adhesion. J. Appl. Math. Mech. 77(4), 433–444 (2013)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Theoretical and Applied MechanicsTaras Shevchenko National University of KyivKyivUkraine

Personalised recommendations