Continuum Mechanics and Thermodynamics

, Volume 30, Issue 3, pp 675–688 | Cite as

Plane micropolar elasticity with surface flexural resistance

  • Alireza Gharahi
  • Peter Schiavone
Original Article


We propose a linear surface/interface model for plane deformations of a micropolar elastic solid based on a higher-order surface elasticity theory capable of incorporating bending and twisting effects. The surface/interface is modeled as a bending-resistant Kirchhoff micropolar thin shell perfectly bonded to the boundary of the solid. It is anticipated that by combining micropolar bulk and surface effects in this way, the enhanced model will most accurately capture the essential characteristics (in particular, size dependency) required in the modeling of materials with significant microstructure as well as in the modeling of classes of nanomaterials. The corresponding boundary value problems are particularly interesting in that they involve boundary conditions of order higher than that of the governing field equations. We illustrate our theory by analyzing the simple problem of a circular hole in a micropolar sheet noting, in particular, the extent to which surface effects and micropolar properties each contribute to the deformation of the sheet.


Micropolar elasticity Surface effects Bending resistance Plane problem 


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  1. 1.
    Anderson, W., Lakes, R.: Size effects due to cosserat elasticity and surface damage in closed-cell polymethacrylimide foam. J. Mater. Sci. 29, 6413–6419 (1994)ADSCrossRefGoogle Scholar
  2. 2.
    Ariman, T.: On the stresses around a circular hole in micropolar elasticity. Acta Mech. 4(3), 216–229 (1967)CrossRefzbMATHGoogle Scholar
  3. 3.
    Chen, H., Hu, G., Huang, Z.: Effective moduli for micropolar composite with interface effect. Int. J. Solids Struct. 44, 8106–8118 (2007)CrossRefzbMATHGoogle Scholar
  4. 4.
    Chen, T., Chiu, M.: Effects of higher-order interface stresses on the elastic states of two-dimensional composites. Mech. Mater. 43(4), 212–221 (2011)CrossRefGoogle Scholar
  5. 5.
    Cheng, Z., He, L.: Micropolar elastic fields due to a circular cylindrical inclusion. Int. J. Eng. Sci. 35(7), 659–668 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chhapadia, P., Mohammadi, P., Sharma, P.: Curvature-dependent surface energy and implications for nanostructures. J. Mech. Phys. Solids 59(10), 2103–2115 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chiu, B., Lee, D.: On the plane problem in micropolar elasticity. Int. J. Eng. Sci. 11, 997–1012 (1973)CrossRefzbMATHGoogle Scholar
  8. 8.
    Dai, M., Gharahi, A., Schiavone, P.: Analytic solution for a circular nano-inhomogeneity with interface stretching and bending resistance in plane strain deformations. Appl. Math. Modell. 55, 160–170 (2018)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Eremeyev, V., Altenbach, H.: Basics of Mechanics of Micropolar Shells, pp. 63–111. Springer, Berlin (2016)Google Scholar
  10. 10.
    Eremeyev, V., Lebedev, L., Altenbach, H.: Foundations of Micropolar Mechanics. Springer, Berlin (2013)CrossRefzbMATHGoogle Scholar
  11. 11.
    Eringen, A.: Theory of micropolar plates. ZAMP 18(1), 12–30 (1967)ADSCrossRefGoogle Scholar
  12. 12.
    Eringen, C.: Linear theory of micropolar elasticity. J. Math. Mech. 15(6), 909–923 (1966)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Gauthier, R., Jahsman, W.: A quest for micropolar elastic constants. J. Appl. Mech. 42(2), 369–374 (1975)ADSCrossRefzbMATHGoogle Scholar
  14. 14.
    Gurtin, M., Murdoch, A.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57, 291–323 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kaloni, P., Ariman, T.: Stress concentration effects in micropolar elasticity. Acta Mech. 4(3), 216–229 (1967)CrossRefzbMATHGoogle Scholar
  16. 16.
    Kreyszig, E.: Differential Geometry. University of Toronto Press, Toronto (1964)zbMATHGoogle Scholar
  17. 17.
    Lakes, R.: Physical meaning of elastic constants in cosserat, void, and microstretch elasticity. J. Mech. Mater. Struct. 11(3), 217–229 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Lebedev, L.P., Cloud, M.J., Eremeyev, V.A.: Tensor Analysis with Applications in Mechanics. World Scientifc (2010).
  19. 19.
    Miller, R., Shenoy, V.: Size-dependent elastic properties of nanosized structural elements. Nanotechnology 11, 139–147 (2000)ADSCrossRefGoogle Scholar
  20. 20.
    Mindlin, R.: Influence of couple-stresses on stress concentrations. Exp. Mech. 3(1), 1–7 (1963)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Nowacki, W.: Theory of Asymmetric Elasticity. Pergamon Press, Oxford (1986)zbMATHGoogle Scholar
  22. 22.
    Sargsyan, S.: General theory of micropolar elastic thin shells. Phys. Mesomech. 15, 69–72 (2012)CrossRefzbMATHGoogle Scholar
  23. 23.
    Schiavone, P., Ru, C.: Integral equation methods in plane-strain elasticity with boundary reinforcement. Proc. R. Soc. A 454, 2223–2242 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Sharma, P., Dasgupta, A.: Average elastic fields and scale-dependent overall properties of heterogeneous micropolar materials containing spherical and cylindrical inhomogeneities. Phys. Rev. B 66(224110), 1–10 (2002)Google Scholar
  25. 25.
    Sharma, P., Ganti, S., Bhate, N.: Effect of surfaces on the size-dependent elastic state of nano-inhomogeneities. Appl. Phys. Lett. 82, 535–537 (2003)ADSCrossRefGoogle Scholar
  26. 26.
    Steigmann, D., Ogden, R.: Plane deformations of elastic solids with intrinsic boundary elasticity. Proc. R. Soc. A 453(1959), 853–877 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Steigmann, D., Ogden, R.: Elastic surface-substrate interactions. Proc. R. Soc. A 455(1982), 437–474 (1999)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Warren, W., Byskov, E.: A general solution to some plane problems of micropolar elasticity. Eur. J. Mech. A Solids 27, 18–27 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Xun, F., Hu, G., Huang, Z.: Effective in plane moduli of composites with a micropolar matrix and coated fibers. Int. J. Solids Struct. 41, 247–265 (2004)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of AlbertaEdmontonCanada

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