Archive of Applied Mechanics

, Volume 88, Issue 1–2, pp 97–110 | Cite as

Anti-plane loading of microstructured materials in the context of couple stress theory of elasticity: half-planes and layers

  • Thanasis Zisis


The problem of calculating the displacements and stresses in a layered system often arises in engineering analysis and design, ranging from the field of mechanical engineering, to the field of materials science and soil mechanics. The present work focuses on the anti-plane response of half-planes and layers of finite thickness bonded on rigid substrates, under a point load, in the context of couple stress elasticity. The theory of couple stress elasticity is used to model the material microstructure and incorporate the size effects into the macroscopic response. This problem in plane strain configuration is referred to as Burmister’s problem. The purpose is to derive the pertinent Green’s functions that can be effectively used for the formulation of anti-plane contact problems in the context of couple stress elasticity. Full-field solutions regarding the out-of-plane displacements, the strains and the equivalent stress are presented, and of special importance is the behavior of the new solutions near to the point of application of the force where pathological singularities and discontinuities exist in the classical solutions.


Anti-plane problems Couple stress elasticity Micromechanics Bonded layer 


  1. 1.
    Bardet, J.-P., Vardoulakis, I.: The asymmetry of stress in granular media. Int. J. Solids Struct. 38(2), 353–367 (2001)CrossRefMATHGoogle Scholar
  2. 2.
    Bogy, D.B., Sternberg, E.: The effect of couple-stresses on singularities due to discontinuous loadings. Int. J. Solids Struct. 3(5), 757–770 (1967)CrossRefMATHGoogle Scholar
  3. 3.
    Borrelli, A., Horgan, C.O., Patria, M.C.: Saint–Venant’s principle for anti-plane shear deformations of linear piezoelectric materials. SIAM J. Appl. Math. 62, 2027–2044 (2002)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bower, A.F.: Applied Mechanics of Solids. CRC Press, Boca Raton (2009)Google Scholar
  5. 5.
    Burmister, D.M.: Evaluation of pavement systems of the WASHO road test by layered system methods. Highw. Res. Board Bull. 177, 26–54 (1958)Google Scholar
  6. 6.
    Cosserat, E., Cosserat, F.: Theorie Des Corps Deformables. Herman, Paris (1909)MATHGoogle Scholar
  7. 7.
    de Borst, R.: A generalisation of J2-flow theory for polar continua. Comput. Methods Appl. Mech. Eng. 103(3), 347–362 (1993)CrossRefMATHGoogle Scholar
  8. 8.
    Eringen, A.C.: Nonlinear Theory of Continuous Media. McGraw-Hill, New York (1962)Google Scholar
  9. 9.
    Gavardinas, I.D., Giannakopoulos, A.E. Zisis, T.: A pre-stressed plate analogue for solving anti-plane problems in couple stress and dipolar elasticity. Int. J. Solids. Struct. (2017) (under review)Google Scholar
  10. 10.
    Georgiadis, H.G., Velgaki, E.G.: High-frequency Rayleigh waves in materials with micro-structure and couple-stress effects. Int. J. Solids Struct. 40(10), 2501–2520 (2003)CrossRefMATHGoogle Scholar
  11. 11.
    Gourgiotis, P.A., Zisis, T.: Two-dimensional indentation of microstructured solids characterized by couple-stress elasticity. J. Strain Anal. Eng. Des. 51(4), 318–331 (2016)CrossRefGoogle Scholar
  12. 12.
    Gourgiotis, P.A., Zisis, T., Baxevanakis, K.P.: Analysis of the tilted flat punch in couple-stress elasticity. Int. J. Solids Struct. 85, 34–43 (2016)CrossRefGoogle Scholar
  13. 13.
    Gourgiotis, P.A., Bigoni, D.: Stress channelling in extreme couple-stress materials part I: strong ellipticity, wave propagation, ellipticity, and discontinuity relations. J. Mech. Phys. Solids 88, 150–168 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gourgiotis, P.A., Bigoni, D.: Stress channelling in extreme couple-stress materials part II: localized folding vs faulting of a continuum in single and cross geometries. J. Mech. Phys. Solids 88, 169–185 (2016)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Grentzelou, C.G., Georgiadis, H.G.: Uniqueness for plane crack problems in dipolar gradient elasticity and in couple-stress elasticity. Int. J. Solids Struct. 42(24), 6226–6244 (2005)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Horgan, C.O.: Anti-plane shear deformation in linear and nonlinear solid mechanics. SIAM Rev. 37, 53–81 (1995)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Horgan, C.O., Miller, K.L.: Anti-plane shear deformation for homogeneous and inhomogenuous anisotropic linearly elastic solids. J. Appl. Mech. 61, 23–29 (1994)CrossRefMATHGoogle Scholar
  18. 18.
    Johnson, K.L.: Contact Mechanics. Cambridge University Press, Cambridge (1987)MATHGoogle Scholar
  19. 19.
    Koiter, W.T.: Couple-stresses in the theory of elasticity, I & II. Philos. Trans. R. Soc. Lond. B. 67, 17–44 (1969)MATHGoogle Scholar
  20. 20.
    Lubarda, V.A., Markenscoff, X.: Conservation integrals in couple stress elasticity. J. Mech. Phys. Solids 48(3), 553–564 (2000)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Mindlin, R.D., Tiersten, H.F.: Effects of couple-stresses in linear elasticity. Arch. Ration. Mech. Anal. 11, 415–448 (1962)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16(1), 51–78 (1964)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Mindlin, R.D.: Influence of couple-stresses on stress concentrations. Exp. Mech. 3(1), 1–7 (1963)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Radi, E.: On the effects of characteristic lengths in bending and torsion on mode III crack in couple stress elasticity. Int. J. Solids Struct. 45(10), 3033–3058 (2008)CrossRefMATHGoogle Scholar
  25. 25.
    Roos, B.W.: Analytic Functions and Distributions in Physics and Engineering. Wiley, NewYork (1969)MATHGoogle Scholar
  26. 26.
    Sofonea, M., Dalah, M., Ayadi, A.: Analysis of an anti-plane electro-elastic contact problem. Adv. Math. Sci. Appl. 17, 385–400 (2007)MathSciNetMATHGoogle Scholar
  27. 27.
    Shu, J.Y., Fleck, N.A.: The prediction of a size effect in microindentation. Int. J. Solids Struct. 35(13), 1363–1383 (1998)CrossRefMATHGoogle Scholar
  28. 28.
    Takeuti, Y., Noda, N.: Thermal stresses and couple-stresses in square cylinder with a circular hole. Int. J. Eng. Sci. 11(5), 519–530 (1973)CrossRefMATHGoogle Scholar
  29. 29.
    Toupin, R.A.: Elastic materials with couple-stresses. Arch. Ration. Mech. Anal. 11(1), 385–414 (1962)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Toupin, R.A.: Theories of elasticity with couple-stress. Arch. Ration. Mech. Anal. 17(2), 85–112 (1964)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Weitsman, Y.: Couple-stress effects on stress concentration around a cylindrical inclusion in a field of uniaxial tension. J. Appl. Mech. 32(2), 424–428 (1965)CrossRefGoogle Scholar
  32. 32.
    Zhou, Z.-G., Wang, B., Du, S.-Y.: Investigation of anti-plane shear behavior of two collinear permeable cracks in a piezoelectric material by using the nonlocal theory. J. Appl. Mech. 69, 1–3 (2002)CrossRefGoogle Scholar
  33. 33.
    Zisis, T., Gourgiotis, P.A., Baxevanakis, K.P., Georgiadis, H.G.: Some basic contact problems in couple stress elasticity. Int. J. Solids Struct. 51(11), 2084–2095 (2014)CrossRefGoogle Scholar
  34. 34.
    Zisis, T., Gourgiotis, P.A., Dal Corso, F.: A contact problem in couple stress thermoelasticity: the indentation by a hot flat punch. Int. J. Solids Struct. 63, 226–239 (2015)CrossRefGoogle Scholar

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© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Mechanics Division, School of Applied Mathematics and Physical ScienceNational Technical University of AthensAthensGreece

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