Skip to main content
SpringerLink
Log in

Finite element implementation of gradient elasticity theory on thin and thick plates

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

In this paper, static analysis of thin and thick plate bending problems with stress singularities is performed using gradient elasticity theory, and the obtained results are compared with each other and also for different boundary conditions. Two different rectangular finite elements having three degrees of freedom per node are used in the finite element implementation where the formulations of the finite elements are based on the Kirchhoff and Reissner–Mindlin plate theories. It is demonstrated through several examples that the stress singularities at sharp crack tips and under point loads of the plates are removed when using gradient elasticity. Convergence studies are also carried out to indicate the effectiveness of the implementations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27
Fig. 28

Similar content being viewed by others

References

  1. Toupin, R.A.: Elastic materials with couple stresses. Arch. Ration. Mech. Anal. 11, 385–414 (1962). https://doi.org/10.1007/BF00253945

    Article  MathSciNet  MATH  Google Scholar 

  2. Mindlin, R.D., Tiersten, H.F.: Effects of couple-stresses in linear elasticity. Arch. Ration. Mech. Anal. 11, 415–448 (1962). https://doi.org/10.1007/BF00253946

    Article  MathSciNet  MATH  Google Scholar 

  3. Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16, 51–78 (1964). https://doi.org/10.1007/BF00248490

    Article  MathSciNet  MATH  Google Scholar 

  4. Mindlin, R.D.: Second gradient of strain and surface tension in linear elasticity. Int. J. Solids Struct. 1, 417–438 (1965). https://doi.org/10.1016/0020-7683(65)90006-5

    Article  Google Scholar 

  5. Triantafyllidis, N., Aifantis, E.C.: A gradient approach to localization of deformation. I. Hyperelastic materials. J. Elast. 16, 225–237 (1986). https://doi.org/10.1007/BF00040814

    Article  MathSciNet  MATH  Google Scholar 

  6. Aifantis, E.C.: On the role of gradients in the localization of deformation and fracture. Int. J. Eng. Sci. 30, 1279–1299 (1992). https://doi.org/10.1016/0020-7225(92)90141-3

    Article  MATH  Google Scholar 

  7. Altan, S.B., Aifantis, E.C.: On the structure of the mode III crack-tip in gradient elasticity. Scr. Metall. Mater. 26, 319–324 (1992). https://doi.org/10.1016/0956-716X(92)90194-J

    Article  Google Scholar 

  8. Ru, C.Q., Aifantis, E.C.: A simple approach to solve boundary-value problems in gradient elasticity. Acta Mech. 101, 59–68 (1993). https://doi.org/10.1007/BF01175597

    Article  MathSciNet  MATH  Google Scholar 

  9. Altan, B.S., Aifantis, E.C.: On some aspects in the special theory of gradient elasticity. J. Mech. Behav. Mater. 8, 231–282 (1997). https://doi.org/10.1515/JMBM.1997.8.3.231

    Article  Google Scholar 

  10. Askes, H., Aifantis, E.C.: Gradient elasticity in statics and dynamics: an overview of formulations, length scale identification procedure, finite element implementations and new results. Int. J. Solids Struct. 48, 1962–1990 (2011). https://doi.org/10.1016/j.ijsolstr.2011.03.006

    Article  Google Scholar 

  11. Askes, H., Gutiérrez, M.A.: Implicit gradient elasticity. Int. J. Numer. Methods Eng. 67, 400–416 (2006). https://doi.org/10.1002/nme.1640

    Article  MathSciNet  MATH  Google Scholar 

  12. Askes, H., Morata, I., Aifantis, E.C.: Finite element analysis with staggered gradient elasticity. Comput. Struct. 86, 1266–1279 (2008). https://doi.org/10.1016/j.compstruc.2007.11.002

    Article  Google Scholar 

  13. Askes, H., Gitman, I.: Non-singular stresses in gradient elasticity at bi-material interface with transverse crack. Int. J. Fract. 156, 217–222 (2009). https://doi.org/10.1007/s10704-009-9357-0

    Article  MATH  Google Scholar 

  14. Askes, H.: Gradient elasticity theories and finite element implementations for static fracture. In: First IJFatigue and FFEMS Joint Workshop, Forni di Sopra (UD), Italy, March 7–9 (2011)

  15. Çalık-Karaköse, Ü.H., Askes, H.: A recovery-type a posteriori error estimator for gradient elasticity. Comput. Struct. 154, 204–209 (2015). https://doi.org/10.1016/j.compstruc.2015.04.003

    Article  Google Scholar 

  16. Bagni, C., Askes, H.: Unified finite element methodology for gradient elasticity. Comput. Struct. 160, 100–110 (2015). https://doi.org/10.1016/j.compstruc.2015.08.008

    Article  Google Scholar 

  17. Bagni, C., Askes, H., Aifantis, E.C.: Gradient-enriched finite element methodology for axisymmetric problems. Acta Mech. 228, 1423–1444 (2017). https://doi.org/10.1007/s00707-016-1762-7

    Article  MathSciNet  MATH  Google Scholar 

  18. Tsiatas, G.C.: A new Kirchhoff plate model based on a modified couple stress theory. Int. J. Solids Struct. 46(13), 2757–2764 (2009). https://doi.org/10.1016/j.ijsolstr.2009.03.004

    Article  MATH  Google Scholar 

  19. Thai, H.T., Vo, T.P., Nguyen, T.K., Kim, S.E.: A review of continuum mechanics models for size-dependent analysis of beams and plates. Compos. Struct. 177, 196–219 (2017). https://doi.org/10.1016/j.compstruct.2017.06.040

    Article  Google Scholar 

  20. Repka, M., Sladek, V., Sladek, J.: Gradient elasticity theory enrichment of plate bending theories. Compos. Struct. 202, 447–457 (2018). https://doi.org/10.1016/j.compstruct.2018.02.065

    Article  Google Scholar 

  21. Aghazadeh, R., Dag, S., Cigeroglu, E.: Modelling of graded rectangular micro-plates with variable length scale parameters. Struct. Eng. Mech. 65(5), 573–585 (2018). https://doi.org/10.12989/sem.2018.65.5.573

    Article  Google Scholar 

  22. Repka, M., Sladek, V., Sladek, J.: Numerical study of size effects in micro/nano plates by moving finite elements. Compos. Struct. 212, 291–303 (2019). https://doi.org/10.1016/j.compstruct.2019.01.010

    Article  MATH  Google Scholar 

  23. Tenek, L.T., Aifantis, E.C.: A two-dimensional finite element implementation of a special form of gradient elasticity. Comput. Model. Eng. Sci. 3, 731–741 (2002). https://doi.org/10.3970/CMES.2002.003.731

    Article  MathSciNet  MATH  Google Scholar 

  24. Oñate, E.: Structural analysis with the finite element method linear statics, Volume 2. In: Beams, Plates and Shells, International Center for Numerical Methods in Engineering (CIMNE), Barcelona, Spain (2013)

  25. Kirchhoff, G.: “Über das Gleichqewicht und die Bewegung einer elastichen Scheibe. J. Reine Angew. Math. 40, 51–88 (1850). https://doi.org/10.1515/crll.1850.40.51

    Article  MathSciNet  MATH  Google Scholar 

  26. Reissner, E.: The effect of transverse shear deformation on the bending of elastic plates. J. Appl. Mech. 67, A67–A77 (1945). https://doi.org/10.1115/1.4009435

    Article  MathSciNet  Google Scholar 

  27. Mindlin, R.D.: Influence of rotary inertia and shear on flexural motion of isotropic elastic plates. J. Appl. Mech. 18, 31–38 (1951). https://doi.org/10.1115/1.4010217

    Article  MATH  Google Scholar 

  28. Ihlenburg, F., Babuška, I.: Finite element solution of the Helmholtz equation with high wave number. Part I: The h-version of the FEM. Comput. Math. Appl. 30(9), 9–37 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  29. Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method, Vol. 1-The Basis, 5th edn. Butterworth-Heinemann, Oxford (2000)

    MATH  Google Scholar 

Download references

Acknowledgements

Prof. Harm Askes is gratefully acknowledged for his fruitful and valuable comments and suggestions on this study.

Author information

Authors and Affiliations

  1. Department of Civil Engineering, Istanbul Technical University, 34469, Istanbul, Turkey

    Ülkü H. Çalık-Karaköse

Authors
  1. Ülkü H. Çalık-Karaköse
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ülkü H. Çalık-Karaköse.

Ethics declarations

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Çalık-Karaköse, Ü.H. Finite element implementation of gradient elasticity theory on thin and thick plates. Acta Mech 234, 511–531 (2023). https://doi.org/10.1007/s00707-022-03410-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-022-03410-4

Search

Navigation