Evaluation of stress concentration around micro-sized holes within simplified models of strain gradient elasticity


DOI: 10.34759/trd-2021-121-04

Аuthors

Korolenko V. A.*, Soliaev J. O.**

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

*e-mail: korolenko.vmir@gmail.com
**e-mail: yos@iam.ras.ru

Abstract

This paper presents the results for the modeling of the deformed state and the level of stress concentration around the micro-sized holes. Analytical solutions are derived for the deformations of an infinite plate containing a cylindrical hole within several simplified one-parametric models of the strain gradient elasticity theory (SGET). Namely, we used the simplified strain gradient elasticity theory, the couple stress theory, the dilatation gradient elasticity theory and the fully symmetric Gusev-Lurie theory, which all are the special cases of the general Mindlin-Tupin SGET. Linear elastic isotropic behavior of the material is assumed. New variant of the Papkovich-Neuber solution of SGET equations in terms of displacements is involved for analytical derivations. It is shown that this solution can be reduced to the standard Helmholtz decomposition for the gradient part of the displacement field and to the standard representation of its classical part. Based on the derived solutions we investigate the changes of the stress and strain state around the holes of different diameter. It is shown that the choice of a suitable variant of a simplified SGET model and identification of the length scale parameter for the certain materials can be performed based on the experimental data for the failure loads for the samples containing holes of different diameters (with a minimum size of ~100 µm). It is also shown that identification can be also carried out on the basis of direct methods of strain measurements around the small-sized holes, for example, by using digital image correlation methods, which requires the use of microscopy or, at least, macro photography techniques at the micro-scale level.

Keywords:

stress concentration, Kirsch problem, strain gradient elasticity theory, identification of length scale parameters

References

  1. Taylor D. The theory of critical distances, Engineering Fracture Mechanics, 2008, vol. 75, no. 7, pp. 1696-1705. DOI:10.1016/j.engfracmech.2007.04.007
  2. Kulesh M.A., Matveenko V.P., Shardakov I.N. Prikladnaya mekhanika i tekhnicheskaya fizika, 2001, vol. 42, no. 4, pp. 145-154.
  3. Bochkarev A.O., Grekov M.A. Fizicheskaya mezomekhanika, 2017, vol. 20, no. 6, pp. 62-76.
  4. Eshel N.N., Rosenfeld G. Effects of strain-gradient on the stress-concentration at a cylindrical hole in a field of uniaxial tension, Journal of Engineering Mathematics, 1970, vol. 4, no. 2, pp. 97-111.
  5. Khakalo S., Niiranen J. Gradient-elastic stress analysis near cylindrical holes in a plane under bi-axial tension fields, International Journal of Solids and Structures, 2017, vol. 110, pp. 351-366. DOI:10.1016/j.ijsolstr.2016.10.025
  6. Kaloni P.N., Ariman T. Stress concentration effects in micropolar elasticity, Zeitschrift für angewandte Mathematik und Physik ZAMP, 1967, vol. 18, no. 1, pp. 136-141. DOI:10.1007/BF01593904
  7. Dai M., Yang H.B., Schiavone P. Stress concentration around an elliptical hole with surface tension based on the original Gurtin-Murdoch model, Mechanics of Materials, 2019, vol. 135, pp. 144-148. DOI:10.1016/j.mechmat.2019.05.009
  8. Mindlin R.D. Microstructure in linear elasticity, Columbia Univ New York Dept of Civil Engineering and Engineering Mechanics, 1963. DOI:10.1007/BF00248490
  9. Polizzotto C. A hierarchy of simplified constitutive models within isotropic strain gradient elasticity, European Journal of Mechanics-A/Solids, 2017, vol. 61, pp. 92-109. DOI:10.1016/j.euromechso1.2016.09.006
  10. Askes H., Aifantis E. C. Gradient elasticity in statics and dynamics: an overview of formulations, length scale identification procedures, finite element implementations and new results, International Journal of Solids and Structures, 2011, vol. 48, no. 13, pp. 1962 −1990. DOI:10.1016/j.ijsolstr.2011.03.006
  11. Mindlin R. D., Tiersten H.F. Effects of couple-stresses in linear elasticity, Columbia Univ New York, 1962, no. 11, pp. 415-448.
  12. Lurie S.A. et al. Dilatation gradient elasticity theory, European Journal of Mechanics-A/Solids, 2021, vol. 88, pp. 104258. DOI:10.1016/j.euromechsol.2021.104258
  13. Gusev A.A., Lurie S.A. Symmetry conditions in strain gradient elasticity, Mathematics and Mechanics of Solids, 2017, vol. 22, no. 4, pp. 683-691. DOI:10.1177/1081286515606960
  14. Toupin R. Elastic materials with couple-stresses, Archive for rational mechanics and analysis, 1962, vol. 11, no.1, pp. 385-414. DOI:10.1007/BF00253945
  15. Askes H., Susmel L. Understanding cracked materials: is linear elastic fracture mechanics obsolete? Fatigue & Fracture of Engineering Materials & Structures, 2015, vol. 38, no. 2, pp. 154-160. DOI:10.1111/ffe.12183
  16. Vasil’ev V.V., Lur’e S.A., Salov V.A. Fizicheskaya mezomekhanika, 2018, vol. 21, no. 4, pp. 5-12. DOI: 10.24411/1683-805X-2018-14001
  17. Vasil’ev V.V., Lur’e S.A. Deformatsiya i razrushenie materialov, 2019, no. 9, pp. 12-19.
  18. Sciarra G., Vidoli S. Asymptotic fracture modes in strain-gradient elasticity: Size effects and characteristic lengths for isotropic materials, Journal of Elasticity, 2013, vol. 113, no. 1, pp. 27-53. DOI:10.1007/s10659-012-9409-y
  19. Morel S., Dourado N. Size effect in quasibrittle failure: Analytical model and numerical simulations using cohesive zone model, International Journal of Solids and Structures, 2011, vol. 48, no. 10, pp. 1403-1412. DOI:10.1016/j.ijsolstr.2011.01.014
  20. Bažant Z.P., Yu Q. Universal size effect law and effect of crack depth on quasi-brittle structure strength, Journal of engineering mechanics, 2009, vol. 135, no. 2, pp. 78-84. DOI:10.1061/(ASCE)0733-9399(2009)135:2(78)
  21. Lurie S. et al. Eshelby’s inclusion problem in the gradient theory of elasticity: applications to composite materials, International Journal of Engineering Science, 2011, vol. 49, no. 12, pp. 1517-1525. DOI:10.1016/j.ijengsci.2011.05.001
  22. Solyaev Y., Lurie S., Korolenko V. Three-phase model of particulate composites in second gradient elasticity, European Journal of Mechanics-A/Solids, 2019, vol. 78, pp. 103853. DOI:10.1016/j.euromechsol.2019.103853
  23. Morse P.M., Feshbach H. Methods of theoretical physics, American Journal of Physics, 1954, vol. 22, no. 6, pp. 410-413.
  24. Zieliński A.P. On trial functions applied in the generalized Trefftz method, Advances in Engineering Software, 1995, vol. 24(1—3), pp.147-155. DOI:10.1016/0965-9978(95)00066-6
  25. Lomakin E.V. et al. Doklady Rossiiskoi akademii nauk. Fizika, tekhnicheskie nauki, 2020, vol. 495, no. 1, pp. 50-56. DOI: 10.31857/S2686740020060139
  26. Vasil’ev V.V., Lur’e S.A., Salov V.A. Izvestiya Rossiiskoi akademii nauk. Mekhanika tverdogo tela, 2020, no. 4, pp. 43-49. DOI: 10.31857/S0572329920040133
  27. Solyaev Y., Babaytsev A. Direct observation of plastic shear strain concentration in the thick GLARE laminates under bending loading, Composites Part B: Engineering, 2021, vol. 224, pp. 109145. DOI:10.1016/j.compositesb.2021.109145
  28. Lur’e S.A., Dudchenko A.A., Nguen D.K. Trudy MAI, 2014, no. 75. URL: http://trudymai.ru/eng/published.php?ID=49674
  29. Endogur A.I., Kravtsov V.A. Trudy MAI, 2013, no. 64. URL: http://trudymai.ru/eng/published.php?ID=36558


Download

mai.ru — informational site MAI

Copyright © 2000-2024 by MAI

 Вход