Skip to main content
SpringerLink
Log in

A new Bernoulli–Euler beam model incorporating microstructure and surface energy effects

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

Abstract

A new Bernoulli–Euler beam model is developed using a modified couple stress theory and a surface elasticity theory. A variational formulation based on the principle of minimum total potential energy is employed, which leads to the simultaneous determination of the equilibrium equation and complete boundary conditions for a Bernoulli–Euler beam. The new model contains a material length scale parameter accounting for the microstructure effect in the bulk of the beam and three surface elasticity constants describing the mechanical behavior of the beam surface layer. The inclusion of these additional material constants enables the new model to capture the microstructure- and surface energy-dependent size effect. In addition, Poisson’s effect is incorporated in the current model, unlike existing beam models. The new beam model includes the models considering only the microstructure dependence or the surface energy effect as special cases. The current model reduces to the classical Bernoulli–Euler beam model when the microstructure dependence, surface energy, and Poisson’s effect are all suppressed. To demonstrate the new model, a cantilever beam problem is solved by directly applying the general formulas derived. Numerical results reveal that the beam deflection predicted by the new model is smaller than that by the classical beam model. Also, it is found that the difference between the deflections predicted by the two models is very significant when the beam thickness is small but is diminishing with the increase of the beam thickness.

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. Cammarata R.C.: Surface and interface stress effects in thin films. Prog. Surf. Sci. 46, 1–38 (1994)

    Article  Google Scholar 

  2. Chen Y., Lee J.D., Eskandarian A.: Atomistic viewpoint of the applicability of microcontinuum theories. Int. J. Solids Struct. 41, 2085–2097 (2004)

    Article  MATH  Google Scholar 

  3. Ellis R.W., Smith C.W.: A thin-plate analysis and experimental evaluation of couple-stress effects. Exp. Mech. 7, 372–380 (1967)

    Article  Google Scholar 

  4. Eringen A.C.: On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54, 4703–4710 (1983)

    Article  Google Scholar 

  5. Eringen A.C., Edelen D.G.B.: On nonlocal elasticity. Int. J. Eng. Sci. 10, 233–248 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  6. Gao X.-L.: An expanding cavity model incorporating strain-hardening and indentation size effects. Int. J. Solids Struct. 43, 6615–6629 (2006)

    Article  MATH  Google Scholar 

  7. Gao, X.-L., Huang, J.X., Reddy, J.N.: A non-classical third-order shear deformation plate model based on a modified couple stress theory. Acta Mech. (published on-line on May. 30, 2013) (2013). doi:10.1007/s00707-013-0880-8

  8. Gao X.-L., Ma H.M.: Solution of Eshelby’s inclusion problem with a bounded domain and Eshelby’s tensor for a spherical inclusion in a finite spherical matrix based on a simplified strain gradient elasticity theory. J. Mech. Phys. Solids 58, 779–797 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  9. Gao X.-L., Mall S.: Variational solution for a cracked mosaic model of woven fabric composites. Int. J. Solids Struct. 38, 855–874 (2001)

    Article  MATH  Google Scholar 

  10. Gao, X.-L., Zhou, S.-S.: Strain gradient solutions of half-space and half-plane contact problems. Z. Angew. Math. Phys. (published on-line on Nov. 7, 2012) (2012). doi:10.1007/s00033-012-0273-1

  11. Gurtin M.E., Murdoch A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57, 291–323 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  12. Gurtin M.E., Murdoch A.I.: Surface stress in solids. Int. J. Solids Struct. 14, 431–440 (1978)

    Article  MATH  Google Scholar 

  13. Hutchinson J.W.: Plasticity at the micron scale. Int. J. Solids Struct. 37, 225–238 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  14. Lam D.C.C., Yang F., Chong A.C.M., Wang J., Tong P.: Experiments and theory in strain gradient elasticity. J. Mech. Phys. Solids 51, 1477–1508 (2003)

    Article  MATH  Google Scholar 

  15. Lazopoulos K.A., Lazopoulos A.K.: Bending and buckling of thin strain gradient elastic beams. Eur. J. Mech. A/Solids 29, 837–843 (2010)

    Article  Google Scholar 

  16. Lim C.W., He L.H.: Size-dependent nonlinear response of thin elastic films with nano-scale thickness. Int. J. Mech. Sci. 46, 1715–1726 (2004)

    Article  MATH  Google Scholar 

  17. Liu C., Rajapakse R.K.N.D.: Continuum models incorporating surface energy for static and dynamic response of nanoscale beams. IEEE Trans. Nanotech. 9, 422–431 (2010)

    Article  Google Scholar 

  18. Liu, C., Rajapakse, R.K.N.D., Phani, A.S.: Finite element modeling of beams with surface energy effects. ASME J. Appl. Mech. 78, 031014-1–031014-10 (2011)

    Google Scholar 

  19. Lu P., He L.H., Lee H.P., Lu C.: Thin plate theory including surface effects. Int. J. Solids Struct. 43, 4631–4647 (2006)

    Article  MATH  Google Scholar 

  20. Lü C.F., Wu D.Z, Chen W.Q.: Nonlinear responses of nanoscale FGM films including the effects of surface energies. IEEE Trans. Nanotech. 10, 1321–1327 (2011)

    Article  Google Scholar 

  21. Ma H.M., Gao X.-L., Reddy J.N.: A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. J. Mech. Phys. Solids 56, 3379–3391 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  22. Ma H.M., Gao X.-L., Reddy J.N.: A non-classical Reddy–Levinson beam model based on a modified couple stress theory. Int. J. Multiscale Comput. Eng. 8, 167–180 (2010)

    Article  Google Scholar 

  23. Ma H.M., Gao X.-L., Reddy J.N.: A non-classical Mindlin plate model based on a modified couple stress theory. Acta Mech. 220, 217–235 (2011)

    Article  MATH  Google Scholar 

  24. Mahmoud F.F., Eltaher M.A., Alshorbagy A.E., Meletis E.I.: Static analysis of nanobeams including surface effects by nonlocal finite element. J. Mech. Sci. Tech. 26, 3555–3563 (2012)

    Article  Google Scholar 

  25. McFarland A.W., Colton J.S.: Role of material microstructure in plate stiffness with relevance to microcantilever sensors. J. Micromech. Microeng. 15, 1060–1067 (2005)

    Article  Google Scholar 

  26. Miller R.E., Shenoy V.B.: Size-dependent elastic properties of nanosized structural elements. Nanotechnology 11, 139–147 (2000)

    Article  Google Scholar 

  27. Mindlin R.D.: Influence of couple-stresses on stress concentrations. Exp. Mech. 3, 1–7 (1963)

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  29. Nix W.D., Gao H.: Indentation size effects in crystalline materials: a law for strain gradient plasticity. J. Mech. Phys. Solids 46, 411–425 (1998)

    Article  MATH  Google Scholar 

  30. Nix W.D., Gao H.: An atomistic interpretation of interface stress. Scripta Mater. 39, 1653–1661 (1998)

    Article  Google Scholar 

  31. Papargyri-Beskou S., Tsepoura K.G., Polyzos D., Beskos D.E.: Bending and stability analysis of gradient elastic beams. Int. J. Solids Struct. 40, 385–400 (2003)

    Article  MATH  Google Scholar 

  32. Park S.K., Gao X.-L.: Bernoulli–Euler beam model based on a modified couple stress theory. J. Micromech. Microeng. 16, 2355–2359 (2006)

    Article  Google Scholar 

  33. Park S.K., Gao X.-L.: Variational formulation of a modified couple stress theory and its application to a simple shear problem. Z. Angew. Math. Phys. 59, 904–917 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  34. Shenoy V.B.: Size-dependent rigidities of nanosized torsional elements. Int. J. Solids Struct. 39, 4039–4052 (2002)

    Article  MATH  Google Scholar 

  35. Shenoy, V.B.: Atomistic calculations of elastic properties of metallic fcc crystal surfaces. Phys. Rev. B 71, 094104-1–094104-11 (2005)

  36. Steigmann D.J., Ogden R.W.: Plane deformation of elastic solids with intrinsic boundary elasticity. Proc. R. Soc. A 453, 853–877 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  37. Steigmann D.J., Ogden R.W.: Elastic surface–substrate interactions. Proc. R. Soc. A 455, 437–474 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  38. Timoshenko, S.P., Goodier, J.N.: Theory of Elasticity, 3rd edn. McGraw-Hill, New York (1970)

  39. Wang L.: Size-dependent vibration characteristics of fluid-conveying microtubes. J. Fluids Struct. 26, 675–684 (2010)

    Article  Google Scholar 

  40. 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 

  41. Yang, F.Q.: Size dependent effective modulus of elastic composite materials: spherical nanocavities at dilute concentrations. J. Appl. Phys. 95, 3516–3520 (2004)

    Google Scholar 

  42. Zhou S.-S., Gao X.-L.: Solutions of half-space and half-plane contact problems based on surface elasticity. Z. Angew. Math. Phys. 64, 145–166 (2013)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, University of Texas at Dallas, 800 West Campbell Road, Richardson, TX, 75080-3021, USA

    X.-L. Gao

  2. Department of Mechanical Design and Production Engineering, Zagazig University, Zagazig, 44511, Egypt

    F. F. Mahmoud

Authors
  1. X.-L. Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. F. F. Mahmoud
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to X.-L. Gao.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gao, XL., Mahmoud, F.F. A new Bernoulli–Euler beam model incorporating microstructure and surface energy effects. Z. Angew. Math. Phys. 65, 393–404 (2014). https://doi.org/10.1007/s00033-013-0343-z

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00033-013-0343-z

Mathematics Subject Classification (2000)

Keywords

Search

Navigation