Cite item


This article is aimed at theoretical study of the stress-strain state of an infinite plate with two semi-infinite symmetrical edge notches. The analytical solution is obtained by means of decomposition in the M. Williams series expansion and subsequent calculation of the amplitude coefficients of the expansion using the complex representation of stresses. An analysis of the multiparametric expansion of the stress field and a computational experiment with different number of terms are carried out. A comparison of the complex representation of the stress field with the asymptotic series of M. Williams
obtained shows the need for an accurate estimate of the number of terms keeping in the expansion series depending on the distance from the crack tip.

About the authors

L. N. Kosygina

Samara National Research University

Author for correspondence.
Russian Federation


  1. Stepanova L.V., Roslyakov P.S. Multi-parameter description of the crack-tip stress field: Analytic determination of coefficients of crack-tip stress expansions in the vicinity of the crack tips of two finite cracks in an infinite plane medium // International Journal of Solids and Structures, 2016, Vol. 100–101, pp. 11–28. doi: 10.1016/j.ijsolstr.2016.06.032 .
  2. Stepanova L.V., Roslyakov P.S. Polnoe asimptoticheskoe razlozhenie M. Uil’iamsa u vershin dvukh kollinearnykh treshchin konechnoi dliny v beskonechnoi plastine . Vestnik Permskogo natsional’nogo tekhnicheskogo universiteta. Mekhanika , 2015, no. 4, pp. 188–225. DOI:
  3. 15593/perm.mech/2015.4.12 .
  4. Stepanova L.V., Adylina E.M. Napriazhenno-deformirovannoe sostoianie v okrestnosti vershiny treshchiny v usloviiakh smeshannogo nagruzheniia . Prikladnaia matematika i tekhnicheskaia fizika , 2014, Vol. 55, no. 5, pp. 885–895 .
  5. Igonin S.A., Stepanova L.V. Asimptotika polei napriazhenii i sploshnosti u vershiny ustalostnoi treshchiny v povrezhdennoi srede v usloviiakh ploskogo napriadennogo sostoianiia . Vestnik Samarskogo gosudarstvennogo universiteta , 2013, no. 9-2, pp. 97–108. Available at:
  6. .
  7. Stepanova L.V., Roslyakov P.S., Gerasimova T. Complete Williams Asymptotic expansion near the crack tips of collinear cracks of equal lengths in an infinite plane. Solid State Phenomena, 2017, Vol. 258, pp. 209–212. doi: 10.1016/j.prostr.2016.06.225 .
  8. Stepanova L.V., Roslyakov P.S., Lomakov P.N. A Photoelastic Study for Multiparametric Analysis of the Near Crack Tip Stress Field Under Mixed Mode Loading. Procedia Structural Integrity, 2016, Vol. 2, pp. 1797–1804. doi: 10.1016/j.prostr.2016.06.226 .
  9. Gupta M., Alderliesten R.C., Benedictus R. A review of T-stress and its effects in fracture mechanics. Engineering Fracture Mechanics, 2015, Vol. 134, pp. 218–241. DOI: .
  10. Berto F., Lazzarin P. On higher order terms in the crack tip stress field. International Journal of Fracture, 2010, Vol. 161, pp. 221–226. DOI: .
  11. Malikova L., Vesely V. Significance of Higher-order Terms of the Williams Expansion for Plastic Zone Extent Estimation Demonstrated on a Mixed-mode Geometry. Procedia Materials Science, 2014, Vol. 3, pp. 1383–1388. doi: 10.1016/j.mspro.2014.06.223
  12. Hello G., Tahar M.B., Roelandt J.M. Analytical determination of coefficients in crack-tip stress expansions for a finite crack in an infinite plane medium. International Journal of Solids and Structure, 2012, Vol. 49, pp. 556–566. DOI: .
  13. Hello G., Tahar M.B. On the exactness of truncated crack-tip stress expansions. Procedia Materials Science, 2014, Vol. 3, pp. 750–755. doi: 10.1016/j.mspro.2014.06.123 .
  14. Williams M.L. Stress singularities resulting from various boundary conditions in angular corners of plates in tension. Journal of Applied Mechanics, 1952, Vol. 19, pp. 109–114. DOI:
  15. .
  16. Williams M.L. On the stress distribution at the base of a stationary crack. Journal of Applied Mechanics, 1957, Vol. 24, pp. 109–114. Available at:
  17. .
  18. Williams M.L. The Stresses Around a Fault or Crack in Dissimilar Media. Bulletin of the Seismological Society of America, 1959, Vol. 49, no 2, pp. 199–204. Available at:
  20. Zak A.R., Williams M.L. Crack Point Stress Singularities at a Bi-Material Interface. Journal of Applied Mechanics, 1963, Vol. 30, pp. 142–143. Available at: .
  21. Stepanova L.V. Matematicheskie metody mekhaniki razrusheniia .
  22. М.: Fizmatlit, 2009, 336 p. . Available at: .
  23. Knott J.F. Osnovy mekhaniki razrusheniia . М.: Metallurgiia, 1978, 256 p. .
  24. Broek В. Osnovy mekhaniki razrusheniia . M.: Vyssh. shkola, 1980, 368 p. .
  25. Tada H., Paris P.C., Irwin G.R. The Stress Analysis of Cracks Handbook. NY: ASME Press, 2000, 678 p. Available at: .

Copyright (c) 2018 Л. Н. Косыгина

This website uses cookies

You consent to our cookies if you continue to use our website.

About Cookies