Skip to main content
SpringerLink
Log in

Modified couple stress theory in orthogonal curvilinear coordinates

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

Abstract

The formulations for the modified couple stress theory (MCST) are consistently derived in general orthogonal curvilinear coordinate systems. In particular, the expressions for the rotation vector, higher-order strain, and stress tensors, i.e., the rotation gradient tensor and the deviatoric part of the symmetric couple stress tensor, and the classical strain and stress tensors are derived for an arbitrary orthogonal curvilinear coordinate system. Additionally, using the theory of surfaces, the formulations for the MCST are derived for general doubly curved coordinates, which are more convenient to use for shells of arbitrary curvature. The expressions for special cases, i.e., cylindrical and spherical shells, are obtained. The MCST expressions derived in this study are comprehensive and generally and can be used for consistent utilisation of the MCST in any orthogonal curvilinear coordinate system.

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

Similar content being viewed by others

References

  1. Ghayesh, M.H., Amabili, M., Farokhi, H.: Nonlinear forced vibrations of a microbeam based on the strain gradient elasticity theory. Int. J. Eng. Sci. 63, 52–60 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Toupin, R.A.: Elastic materials with couple-stresses. Arch. Ration. Mech. Anal. 11, 385–414 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  3. Mindlin, R., Tiersten, H.: Effects of couple-stresses in linear elasticity. Arch. Ration. Mech. Anal. 11, 415–448 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  4. Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16, 51–78 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  5. Mindlin, R.D., Eshel, N.N.: On first strain-gradient theories in linear elasticity. Int. J. Eng. Sci. 4, 109–124 (1968)

    MATH  Google Scholar 

  6. Ghayesh, M.H., Farokhi, H.: Chaotic motion of a parametrically excited microbeam. Int. J. Eng. Sci. 96, 34–45 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Yazdani Sarvestani, H., Akbarzadeh, A.H., Mirabolghasemi, A.: Structural analysis of size-dependent functionally graded doubly-curved panels with engineered microarchitectures. Acta Mech. 1–27 (2018)

  8. Ghayesh, M.H., Farokhi, H., Alici, G.: Size-dependent performance of microgyroscopes. Int. J. Eng. Sci. 100, 99–111 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fernandes, R., Mousavi, S.M., El-Borgi, S.: Free and forced vibration nonlinear analysis of a microbeam using finite strain and velocity gradients theory. Acta Mech. 227, 2657–2670 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gholipour, A., Farokhi, H., Ghayesh, M.H.: In-plane and out-of-plane nonlinear size-dependent dynamics of microplates. Nonlinear Dyn. 79, 1771–1785 (2015)

    Article  Google Scholar 

  11. Ghayesh, M.H., Amabili, M., Farokhi, H.: Three-dimensional nonlinear size-dependent behaviour of Timoshenko microbeams. Int. J. Eng. Sci. 71, 1–14 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Taati, E., Molaei Najafabadi, M., Basirat Tabrizi, H.: Size-dependent generalized thermoelasticity model for Timoshenko microbeams. Acta Mech. 225, 1823–1842 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ghayesh, M.H., Farokhi, H., Amabili, M.: Nonlinear dynamics of a microscale beam based on the modified couple stress theory. Compos. Part B Eng. 50, 318–324 (2013)

    Article  MATH  Google Scholar 

  14. Asghari, M., Kahrobaiyan, M., Nikfar, M., Ahmadian, M.: A size-dependent nonlinear Timoshenko microbeam model based on the strain gradient theory. Acta Mech. 223, 1233–1249 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ghayesh, M.H., Farokhi, H., Amabili, M.: In-plane and out-of-plane motion characteristics of microbeams with modal interactions. Compos. Part B Eng. 60, 423–439 (2014)

    Article  Google Scholar 

  16. Ghayesh, M.H., Farokhi, H.: Coupled size-dependent behavior of shear deformable microplates. Acta Mech. 227, 757–775 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ghayesh, M.H., Farokhi, H.: Nonlinear dynamics of microplates. Int. J. Eng. Sci. 86, 60–73 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Farokhi, H., Ghayesh, M.H.: Thermo-mechanical dynamics of perfect and imperfect Timoshenko microbeams. Int. J. Eng. Sci. 91, 12–33 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Farokhi, H., Ghayesh, M.H.: Nonlinear dynamical behaviour of geometrically imperfect microplates based on modified couple stress theory. Int. J. Mech. Sci. 90, 133–144 (2015)

    Article  Google Scholar 

  20. Ghayesh, M.H., Farokhi, H., Amabili, M.: Nonlinear behaviour of electrically actuated MEMS resonators. Int. J. Eng. Sci. 71, 137–155 (2013)

    Article  MATH  Google Scholar 

  21. Ghayesh, M.H.: Nonlinear vibration analysis of axially functionally graded shear-deformable tapered beams. Appl. Math. Model. 59, 583–596 (2018)

    Article  MathSciNet  Google Scholar 

  22. Ghayesh, M.H.: Dynamics of functionally graded viscoelastic microbeams. Int. J. Eng. Sci. 124, 115–131 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  23. Farokhi, H., Ghayesh, M.H., Gholipour, A., Hussain, Sh: Motion characteristics of bilayered extensible Timoshenko microbeams. Int. J. Eng. Sci. 112, 1–17 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ghayesh, M.H., Farokhi, H., Gholipour, A.: Oscillations of functionally graded microbeams. Int. J. Eng. Sci. 110, 35–53 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  25. Ghayesh, M.H., Farokhi, H., Gholipour, A.: Vibration analysis of geometrically imperfect three-layered shear-deformable microbeams. Int. J. Mech. Sci. 122, 370–383 (2017)

    Article  Google Scholar 

  26. Farokhi, H., Ghayesh, M.H.: Supercritical nonlinear parametric dynamics of Timoshenko microbeams. Commun. Nonlinear Sci. Numer. Simul. 59, 592–605 (2018)

    Article  MathSciNet  Google Scholar 

  27. Farokhi, H., Ghayesh, M.H.: Nonlinear mechanics of electrically actuated microplates. Int. J. Eng. Sci. 123, 197–213 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ghayesh, M.H., Farokhi, H., Gholipour, A., Tavallaeinejad, M.: Nonlinear oscillations of functionally graded microplates. Int. J. Eng. Sci. 122, 56–72 (2018)

    Article  MATH  Google Scholar 

  29. Ghayesh, M.H.: Functionally graded microbeams: Simultaneous presence of imperfection and viscoelasticity. Int. J. Mech. Sci. 140, 339–350 (2018)

    Article  Google Scholar 

  30. Park, S.K., Gao, X.L.: Bernoulli-Euler beam model based on a modified couple stress theory. J. Micromech. Microeng. 16, 2355 (2006)

    Article  Google Scholar 

  31. Yang, F., Chong, A.C.M., Lam, D.C.C., Tong, P.: Couple stress based strain gradient theory for elasticity. Int. J. Solids Struct. 39, 2731–2743 (2002)

    Article  MATH  Google Scholar 

  32. Eringen, A.C.: Mechanics of continua. Wiley, (1967)

  33. Ghayesh, M.H., Paidoussis, M.P., Amabili, M.: Nonlinear dynamics of cantilevered extensible pipes conveying fluid. J. Sound Vib. 332, 6405–6418 (2013)

    Article  Google Scholar 

  34. Ghayesh, M.H., Farokhi, H., Hussain, Sh: Viscoelastically coupled size-dependent dynamics of microbeams. Int. J. Eng. Sci. 109, 243–255 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  35. Ghayesh, M.H., Amabili, M.: Nonlinear dynamics of axially moving viscoelastic beams over the buckled state. Comput. Struct. 112–113, 406–421 (2012)

    Article  Google Scholar 

  36. Ghayesh, M.H., Amabili, M., Paidoussis, M.P.: Nonlinear vibrations and stability of an axially moving beam with an intermediate spring support: two-dimensional analysis. Nonlinear Dyn. 70(1), 335–354 (2012)

    Article  MathSciNet  Google Scholar 

  37. Farokhi, H., Ghayesh, M.H.: Size-dependent parametric dynamics of imperfect microbeams. Int. J. Eng. Sci. 99, 39–55 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  38. Ghayesh, M.H., Amabili, M.: Coupled longitudinal-transverse behaviour of a geometrically imperfect microbeam. Compos. Part B: Eng. 60, 371–377 (2014)

    Article  Google Scholar 

  39. Ghayesh, M.H.: Stability characteristics of an axially accelerating string supported by an elastic foundation. Mech. Mach. Theory 44, 1964–1979 (2009)

    Article  MATH  Google Scholar 

  40. Ghayesh, M.H.: On the natural frequencies, complex mode functions, and critical speeds of axially traveling laminated beams: parametric study. Acta Mech. Solida Sinica 24(4), 373–382 (2011)

    Article  Google Scholar 

  41. Haque, M.A., Saif, M.T.A.: Strain gradient effect in nanoscale thin films. Acta Mater. 51, 3053–3061 (2003)

    Article  Google Scholar 

  42. Chen, S.H., Feng, B.: Size effect in micro-scale cantilever beam bending. Acta Mech. 219, 291–307 (2011)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical and Construction Engineering, Northumbria University, Newcastle upon Tyne, NE1 8ST, UK

    Hamed Farokhi

  2. School of Mechanical Engineering, University of Adelaide, Adelaide, SA, 5005, Australia

    Mergen H. Ghayesh

Authors
  1. Hamed Farokhi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mergen H. Ghayesh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mergen H. Ghayesh.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Farokhi, H., Ghayesh, M.H. Modified couple stress theory in orthogonal curvilinear coordinates. Acta Mech 230, 851–869 (2019). https://doi.org/10.1007/s00707-018-2331-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-018-2331-z

Search

Navigation