Singularity removal in the elasticity theory solution based on a non-euclidean model of a continuous medium

Cover Page

Cite item

Full Text

Open Access Open Access
Restricted Access Access granted
Restricted Access Subscription or Fee Access

Abstract

A representation for singularities of the classical elastic stress field was obtained using the Airy stress function for a plane-strained state of a continuous medium. For a non-Euclidean model of a continuous medium, the structure of the internal stress field of a plane-strained state was shown to consist of a classical elastic stress field and a non-classical stress field determined through the incompatibility function of deformations. The requirement for the absence of singularities in the internal stress field allowed to compensate for the singularity in the elasticity theory solution for the zero harmonic by choosing a singularity of the non-classical stress field.

Full Text

Restricted Access

About the authors

M. A. Guzev

Institute for Applied Mathematics FEB RAS

Author for correspondence.
Email: guzev@iam.dvo.ru
Russian Federation, Vladivostok

E. V. Chernysh

Institute for Applied Mathematics FEB RAS

Email: guzev@iam.dvo.ru
Russian Federation, Vladivostok

References

  1. Sinclar G.B. Stress singularities in classical elasticity I: Removal, interpretation and analysis // Appl. Mech. Rev., 2004, vol. 57(4), pp. 251–297. https://doi.org/10.1115/1.1762503
  2. Sinclair G.B. On ensuring structural integrity for configurations with stress singularities // A Review. Fatigue&Fracture of Engng. Mater.&Struct., 2016, vol. 39(5), pp. 523–535. https://doi.org/10.1111/ffe.12425
  3. Cherepanov G.P. Mechanics of Brittle Fracture. Moscow: Nauka, 1974. 640 p. (in Russian)
  4. Vasiliev V.V. Singular solutions in problems of mechanics and mathematical physics // Izv. RAN. MTT, 2018, no. 4, pp. 48–65. https://doi.org/10.31857/S057232990000702-2
  5. Vasiliev V.V., Lurie S.A. On the solution singularity in the plane elasticity problem for a cantilever strip // Izv. RAN. MTT, 2013, no. 4, pp. 40–49.
  6. Vasiliev V.V., Lurie S.A. Nonlocal solutions to singular problems of mathematical physics and mechanics // JAMM, 2018, vol. 82, no. 4, pp. 459–471.
  7. Vasiliev V.V., Lurie S.A. Differential equations and the problem of singularity of solutions in applied mechanics and mathematics // J. Appl. Mech.&Tech. Phys., 2023, vol. 64, no. 1, pp. 98–109.
  8. Lazar M. Non-singular dislocation loops in gradient elasticity // Phys. Lett. A, 2012, vol. 376(21), pp. 1757–1758.
  9. Lazar M. The fundamentals of non-singular dislocations in the theory of gradient elasticity: Dislocation loops and straight dislocations // Int. J. of Solids&Struct., 2013, vol. 50(2), pp. 352–362. https://doi.org/10.1016/j.ijsolstr.2012.09.017
  10. Lazar M., Po G. The non-singular Green tensor of Mindlin’s anisotropic gradient elasticity with separable weak non-locality // Phys. Lett. A, 2015, vol. 379(24–25), pp. 538–1543. https://doi.org/10.1016/j.physleta.2015.03.027
  11. Po G., Lazar M., Admal N.C., Ghoniem N. A non-singular theory of dislocations in anisotropic crystals // Int. J. of Plasticity, 2018, vol. 103, pp. 1–22. https://doi.org/10.1016/j.ijplas.2017.10.003
  12. Kioseoglou J., Konstantopoulos I., Ribarik G. et al. Nonsingular dislocation and crack fields: implications to small volumes // Microsyst. Technol., 2009, vol. 15, pp. 117–121. https://doi.org/10.1007/s00542-008-0700-6
  13. Aifantis E.C. A note on gradient elasticity and nonsingular crack fields // J. Mech. Behav. Mater., 2011, vol. 20, pp. 103–105.
  14. Konstantopoulos I., Aifantis E.C. Gradient elasticity applied to a crack // J. Mech. Behav. Mater., 2013, vol. 22, pp. 193–201.
  15. Parisis K., Konstantopoulos I., Aifantis E.C. Nonsingular solutions of GradEla models for dislocations: An extension to fractional GradEla // J. of Micromech.&Molec. Phys., 2018, vol. 03, no. 03n04, a. 1840013. https://doi.org/10.1142/s2424913018400131
  16. Guzev M., Liu W., Qi C. Non-Euclidean model for description of residual stresses in planar deformations // Appl. Math. Model., 2021, vol. 90, pp. 615–623.
  17. Godunov S.K. Elements of Continuum Mechanics. Moscow: Nauka, 1978. 304 p. (in Russian)
  18. Gurtin M.E. A generalization of the Beltrami stress functions in continuum mechanics // Arch. for Rat. Mech.&Anal., 1963, vol. 13, no. 1, pp. 321–329. https://doi.org/10.1007/BSF01262700
  19. Myasnikov V.P., Guzev M.A., Ushakov A.A. Structure of the self-equilibrated stress field in a continuous medium // Far Eastern Math. J., 2002, no. 2, pp. 231–241.
  20. Kondo K. On the geometrical and physical foundations of the theory of yielding // Proc. Jap. Nat. Congr. Apll. Mech., 1952, vol. 2, pp. 41–47.
  21. Bilby B.A., Bullough R., Smith E. Continuous distributions of dislocations: a new application of the methods of non-Reimannian geometry // Proc. Roy. Soc., 1955, vol. 231(1185), pp. 263–273. https://doi.org/10.1098/rspa.1955.0171
  22. Novikov S.P., Taimanov I.A. Modern Geometric Structures and Fields. Moscow: MCCME, 2005. 584 p. (in Russian)
  23. Guzev M.A. The structure of the kinematic and force fields in the Riemannian model of a continuous medium // J. of Appl. Mech.&Tech. Phys., 2011, vol. 52, no. 5, pp. 39–48.
  24. Gradshteyn I.S., Ryzhik I.M. Table of Integrals, Series, and Products. Moscow: Nauka, 1971. 1108 p. (in Russian)
  25. Williams M.L. Stress singularities resulting from various boundary conditions in angular corners of plates in extension // J. of Appl. Mech., 1952, vol. 19, no. 4, pp. 526–528.

Supplementary files

Supplementary Files
Action
1. JATS XML
Download

Copyright (c) 2025 Russian Academy of Sciences