Skip to main content
SpringerLink
Log in

Weak solutions of the problem of torsion of micropolar elastic beams

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

In this paper, we consider torsion of a micropolar beam whose cross-section is bounded by an irregular curve. Using the boundary integral equation method we find the (weak) solution for the warping function in a Sobolev space setting and illustrate the effectiveness of the method using an example of torsion of a micropolar graphite square beam. The example demonstrates the effect of material microstructure.

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. Brebbia C.A.: The Boundary Element Method for Engineers. Pentech Press, London (1978)

    Google Scholar 

  2. Chudinovich I., Constanda C.: Variational and Potential Methods in the Theory of Bending of Plates with Transverse Shear Deformation. Chapman&Hall/CRC, London/Boca Raton (2000)

    MATH  Google Scholar 

  3. Eringen A.C.: Linear theory of micropolar elasticity. J. Math. Mech. 15, 909–923 (1966)

    MATH  MathSciNet  Google Scholar 

  4. Horgan C.O.: Anti-plane shear deformations in linear and nonlinear solid mechanics. SIAM Rev. 37, 53–81 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  5. Iesan D.: Torsion of micropolar elastic beams. Int. J. Eng. Sci. 9, 1047–1060 (1971)

    Article  MATH  Google Scholar 

  6. Lakes R.: Experimental methods for study of Cosserat elastic solids and other generalized elastic continua. In: Muhlhaus, H.-B. (eds) Continuum Models for Materials with Microstructure, pp. 1–22. Wiley, New York (1995)

    Google Scholar 

  7. Love A.E.H.: A Treatise on the Mathematical Theory of Elasticity. Dover, New York (1944)

    MATH  Google Scholar 

  8. Miranda C.: Partial Differential Equations of Elliptic Type. Springer, Berlin (1970)

    MATH  Google Scholar 

  9. Nowacki W.: Theory of Asymmetric Elasticity. Polish Scientific Publishers, Warsaw (1986)

    MATH  Google Scholar 

  10. Park H.C., Lakes R.S.: Cosserat micromechanics of human bone: strain redistribution by a hydration-sensitive constituent. J. Biomech. 19, 385–397 (1986)

    Article  Google Scholar 

  11. Potapenko S., Schiavone P., Mioduchowski A.: Generalized Fourier series solution of torsion of an elliptic beam with microstructure. Appl. Math. Lett. 17, 189–192 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  12. Potapenko S., Schiavone P., Mioduchowski A.: Anti-plane shear deformations in a linear theory of elasticity with microstructure. J. Appl. Math. Phys. ZAMP 56, 516–528 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  13. Potapenko S., Schiavone P., Mioduchowski A.: On the solution of torsion of cylindrical beams with microstructure. Math. Mech. Solids 11, 181–195 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  14. Potapenko, S., Shmoylova, E.: Solvability of weak solutions to anti-plane Cosserat elasticity by means of boundary integral equations. Math. Mech. Solids. doi:10.1177/1081286508095810 (2008)

  15. Schiavone P.: On existence theorems in the theory of extensional motion of thin micropolar plates. Int. J. Eng. Sci. 27, 1129–1133 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  16. Shmoylova E., Potapenko S., Rothenburg L.: Weak solutions of the interior boundary value problems of plane Cosserat elasticity. J. Appl. Math. Phys. ZAMP 57, 506–522 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  17. Shmoylova E., Potapenko S., Dorfmann A.: Weak solutions to fundamental boundary value problems in anti-plane elasticity with microstructure. Arch. Mech. 59, 519–539 (2007a)

    MATH  MathSciNet  Google Scholar 

  18. Shmoylova E., Potapenko S., Rothenburg L.: Boundary element analysis of stress distribution around a crack in plane micropolar elasticity. Int. J. Eng. Sci. 45, 199–209 (2007b)

    Article  MathSciNet  Google Scholar 

  19. Smith, A.C.: Torsion and vibrations of cylinders of a micro-polar elastic solid. In: Eringen, A.C. (ed.) Recent Advances in Engineering Science, vol. 5, pp. 129–137 (1970)

  20. Sokolnikoff I.S.: Mathematical Theory of Elasticity. McGraw-Hill, New York (1946)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Civil and Environmental Engineering, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Stanislav Potapenko

  2. Research and Development Division, Maplesoft, Waterloo, ON, N2E 1K8, Canada

    E. Shmoylova

Authors
  1. Stanislav Potapenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. Shmoylova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stanislav Potapenko.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Potapenko, S., Shmoylova, E. Weak solutions of the problem of torsion of micropolar elastic beams. Z. Angew. Math. Phys. 61, 529–536 (2010). https://doi.org/10.1007/s00033-009-0018-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00033-009-0018-y

Mathematics Subject Classification (2000)

Keywords

Search

Navigation