Generalized Continua as Models for Materials pp 1-15

Part of the Advanced Structured Materials book series (STRUCTMAT, volume 22) | Cite as

Shells and Plates with Surface Effects

Chapter

Abstract

The through-the-thickness integration procedure applied to a three-dimensional (3D) slender body leads to exact two-dimensional (2D) equations of plates and shells, see [36]. The procedure can be considered as a specific homogenization technique which results in a 2D generalized media—the non-linear theory of shells of Cosserat type. Within this theory the shell is described as a deformable surface each point of which has 3 translational and 3 rotational degrees of freedom similar to the 3D Cosserat continuum [15]. Below we discuss the through-the-thickness integration procedure applied to the non-classical problem of the theory of surface elasticity [21]. The theory can be applied to modeling of surface effects which are important in mechanics of nanostructured materials [11, 55]. Applying the through-the-thickness integration procedure we reduce 3D equations to 2D ones. The effective (apparent) stiffness properties of the shell are changed in comparison with the classical models of shells. Some examples of a plate bending are discussed taking into account surface effects.

References

  1. 1.
    Altenbach, H., Eremeyev, V.A.: On the shell theory on the nanoscale with surface stresses. Int. J. Eng. Sci. 49(12), 1294–1301 (2011)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Altenbach, H., Eremeyev, V.A., Lebedev, L.P.: On the existence of solution in the linear elasticity with surface stresses. ZAMM 90(7), 535–536 (2010)MathSciNetGoogle Scholar
  3. 3.
    Altenbach, H., Eremeyev, V.A., Morozov, N.F.: Linear theory of shells taking into account surface stresses. Dokl. Phys. 54(12), 531–535 (2009)CrossRefGoogle Scholar
  4. 4.
    Altenbach, H., Eremeyev, V.A., Morozov, N.F.: On equations of the linear theory of shells with surface stresses taken into account. Mech. Solid 45(3), 331–342 (2010)CrossRefGoogle Scholar
  5. 5.
    Altenbach, H., Eremeyev, V.A., Morozov, N.F.: Surface viscoelasticity and effective properties of thin-walled structures at the nanoscale. Int. J. Eng. Sci. 59, 83–89 (2012)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Altenbach, H., Morozov, N.F. (eds.): Surface Effects in Solid Mechanics—Models, Simulations, and Applications. Springer, Berlin (2013)Google Scholar
  7. 7.
    Chen, C.Q., Shi, Y., Zhang, Y.S., Zhu, J., Yan, Y.J.: Size dependence of Young’s modulus in ZnO nanowires. Phy. Rev. Lett. 96(7), 075505–4 (2006)Google Scholar
  8. 8.
    Chróścielewski, J., Makowski, J., Pietraszkiewicz, W.: Statyka i dynamika powłok wielopłatowych. Nieliniowa teoria i metoda elementów skończonych. Wydawnictwo IPPT PAN, Warszawa (2004)Google Scholar
  9. 9.
    Cuenot, S., Frétigny, C., Demoustier-Champagne, S., Nysten, B.: Surface tension effect on the mechanical properties of nanomaterials measured by atomic force microscopy. Phys. Rev. B 69(16), 165410–5 (2004)Google Scholar
  10. 10.
    Dahmen, K., Lehwald, S., Ibach, H.: Bending of crystalline plates under the influence of surface stress—a finite element analysis. Surf. Sci. 446(1–2), 161–173 (2000)CrossRefGoogle Scholar
  11. 11.
    Duan, H.L., Wang, J., Karihaloo, B.L.: Theory of elasticity at the nanoscale. In: Advances in Applied Mechanics, vol. 42, pp. 1–68. Elsevier, San Diego (2008)Google Scholar
  12. 12.
    Duan, H.L., Wang, J., Karihaloo, B.L., Huang, Z.P.: Nanoporous materials can be made stiffer than non-porous counterparts by surface modification. Acta Mater. 54, 2983–2990 (2006)CrossRefGoogle Scholar
  13. 13.
    Earnshaw, J.C., McGivern, R.C., McLaughlin, A.C., Winch, P.J.: Light-scattering-studies of surface viscoelasticity—direct data-analysis. Langmuir 6(3), 649–660 (1990)CrossRefGoogle Scholar
  14. 14.
    Eremeyev, V.A., Altenbach, H., Morozov, N.F.: The influence of surface tension on the effective stiffness of nanosize plates. Doklady Phys. 54(2), 98–100 (2009)MATHCrossRefGoogle Scholar
  15. 15.
    Eremeyev, V.A., Lebedev, L.P., Altenbach, H.: Foundations of Micropolar Mechanics. Springer, Heidelberg (2012)Google Scholar
  16. 16.
    Finn, R.: Equilibrium Capillary Surfaces. Springer, New York (1986)Google Scholar
  17. 17.
    Garcia, R., Gómez, C.J., Martinez, N.F., Patil, S., Dietz, C., Magerle, R.: Identification of nanoscale dissipation processes by dynamic atomic force microscopy. Phys. Rev. Lett. 97(1), 1–4 (2006). doi:10.1103/PhysRevLett.97.016103 Google Scholar
  18. 18.
    Gibbs, J.W.: On the equilibrium of heterogeneous substances. In: Willard Gibbs, J. (ed.) The Collected Works, pp. 55–353. Longmans, Green & Co, New York (1928)Google Scholar
  19. 19.
    Guo, J.G., Zhao, Y.P.: The size-dependent elastic properties of nanofilms with surface effects. J. Appl. Phys. 98(7), 074306–11 (2005)Google Scholar
  20. 20.
    Gurtin, M.E., Markenscoff, X., Thurston, R.N.: Effect of surface stress on natural frequency of thin crystalS. Appl. Phys. Lett. 29(9), 529–530 (1976)CrossRefGoogle Scholar
  21. 21.
    Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57(4), 291–323 (1975)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    He, L.H., Lim, C.W., Wu, B.S.: A continuum model for size-dependent deformation of elastic films of nano-scale thickness. Int. J. Solid Struct. 41(3–4), 847–857 (2004)MATHCrossRefGoogle Scholar
  23. 23.
    Huang, D.W.: Size-dependent response of ultra-thin films with surface effects. Int. J. Solid Struct. 45(2), 568–579 (2008)MATHCrossRefGoogle Scholar
  24. 24.
    Huang, Z., Sun, L.: Size-dependent effective properties of a heterogeneous material with interface energy effect: from finite deformation theory to infinitesimal strain analysis. Acta Mech. 190, 151–163 (2007)MATHCrossRefGoogle Scholar
  25. 25.
    Huang, Z., Wang, J.: A theory of hyperelasticity of multi-phase media with surface/interface energy effect. Acta Mech. 182, 195–210 (2006)MATHCrossRefGoogle Scholar
  26. 26.
    Huang, Z., Wang, J.: Micromechanics of nanocomposites with interface energy effect. In: Handbook on Micromechanics and Nanomechanics, p. 48 (in print). Pan Stanford Publishing (2012)Google Scholar
  27. 27.
    Javili, A., McBride, A., Steinmann, P.: Numerical modelling of thermomechanical solids with mechanically energetic (generalised) Kapitza interfaces. Comput. Mater. Sci. 65, 542–551 (2012)Google Scholar
  28. 28.
    Javili, A., McBride, A., Steinmann, P., Reddy, B.D.: Relationships between the admissible range of surface material parameters and stability of linearly elastic bodies. Phil. Magazine 92, 3540–3563 (2012)Google Scholar
  29. 29.
    Javili, A., Steinmann, P.: A finite element framework for continua with boundary energies. Part I: the two-dimensional case. Comput. Method Appl. Mech. Eng. 198, 2198–2208 (2009)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Javili, A., Steinmann, P.: A finite element framework for continua with boundary energies. Part II: the three-dimensional case. Comput. Method Appl. Mech. Eng. 199, 755–765 (2010)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Javili, A., Steinmann, P.: On thermomechanical solids with boundary structures. Comput. Method Appl. Mech. Eng. 47, 3245–3253 (2010)MathSciNetMATHGoogle Scholar
  32. 32.
    Jing, G.Y., Duan, H.L., Sun, X.M., Zhang, Z.S., Xu, J., Li, Y.D., Wang, J.X., Yu, D.P.: Surface effects on elastic properties of silver nanowires: contact atomic-force microscopy. Phys. Rev. B 73(23), 235409–6 (2005)Google Scholar
  33. 33.
    Kajiyama, T., Tanaka, K., Ge, S.R., Takahara, A.: Morphology and mechanical properties of polymer surfaces via scanning force microscopy. Prog. Surf. Sci. 52(1), 1–52 (1996)CrossRefGoogle Scholar
  34. 34.
    Lagowski, J., Gatos, H.C., Sproles, E.S.: Surface stress and normal mode of vibration of thin crystals: GaAs. Appl. Phys. Lett. 26(9), 493–495 (1975)CrossRefGoogle Scholar
  35. 35.
    Laplace, P.S.: Supplément à la théorie de l’action capillaire. In: Traité de mécanique céleste, vol. X, pp. 1–68. Gauthier-Villars et fils, Paris (1805)Google Scholar
  36. 36.
    Libai, A., Simmonds, J.G.: The Nonlinear Theory of Elastic Shells, 2nd edn. Cambridge University Press, Cambridge (1998)Google Scholar
  37. 37.
    Lu, C.F., Lim, C.W., Chen, W.Q.: Size-dependent elastic behavior of FGM ultra-thin films based on generalized refined theory. Int. J. Solid Struct. 46(5), 1176–1185 (2009)CrossRefGoogle Scholar
  38. 38.
    Lu, P., He, L.H., Lee, H.P., Lu, C.: Thin plate theory including surface effects. Int. J. Solid Struct. 43(16), 4631–4647 (2006)MATHCrossRefGoogle Scholar
  39. 39.
    Miller, R.E., Shenoy, V.B.: Size-dependent elastic properties of nanosized structural elements. Nanotechnology 11(3), 139–147 (2000)CrossRefGoogle Scholar
  40. 40.
    Mindlin, R.D.: Influence of rotatory inertia and shear on flexural motions of isotropic elastic plates. Transactions of ASME. J. Appl. Mech. 18, 31–38 (1951)MATHGoogle Scholar
  41. 41.
    Orowan, E.: Surface energy and surface tension in solids and fluids. Proc. Roy. Soc. London A 316(1527), 473–491 (1970)CrossRefGoogle Scholar
  42. 42.
    Pietraszkiewicz, W.: Finite Rotations and Langrangian Description in the Non-linear Theory of Shells. Polish Sci. Publ, Warszawa-Poznań (1979)Google Scholar
  43. 43.
    Podio-Guidugli, P., Caffarelli, G.V.: Surface interaction potentials in elasticity. Arch. Ration. Mech. Anal. 109(4), 343–383 (1990)MATHCrossRefGoogle Scholar
  44. 44.
    Podstrigach, Y.S., Povstenko, Y.Z.: Introduction to Mechanics of Surface Phenomena in Deformable Solids (in Russian). Naukova Dumka, Kiev (1985)Google Scholar
  45. 45.
    Reissner, E.: On the theory of bending of elastic plates. J. Math. Phys. 23, 184–194 (1944)MathSciNetMATHGoogle Scholar
  46. 46.
    Ru, C.Q.: Size effect of dissipative surface stress on quality factor of microbeams. Appl. Phys. Lett. 94, 051905–1-051905–3 (2009)Google Scholar
  47. 47.
    Rusanov, A.I.: Surface thermodynamics revisited. Surf. Sci. Rep. 58(5–8), 111–239 (2005)CrossRefGoogle Scholar
  48. 48.
    Rusanov, A.I.: Thermodynamics of solid surfaces. Surf. Sci. Rep. 23(6–8), 173–247 (1996)CrossRefGoogle Scholar
  49. 49.
    Sahoo, N., Thakur, S., Senthilkumar, M., Das, N.C.: Surface viscoelasticity studies of Gd$_2$O$_3$, SiO$_2$ optical thin films and multilayers using force modulation and force-distance scanning probe microscopy. Appl. Surf. Sci. 206(1–4), 271–293 (2003)CrossRefGoogle Scholar
  50. 50.
    Seoánez, C., Guinea, F., Castro Neto, A.H.: Surface dissipation in nanoelectromechanical systems: Unified description with the standard tunneling model and effects of metallic electrodes. Phys. Rev. B 77(12), 195409 (2008)Google Scholar
  51. 51.
    Steigmann, D.J., Ogden, R.W.: Elastic surface-substrate interactions. Proc. Roy. Soc. London Ser. A Math. Phys. Eng. Sci. 455(1982), 437–474 (1999)MathSciNetMATHCrossRefGoogle Scholar
  52. 52.
    Tranchida, D., Kiflie, Z., Acierno, S., Piccarolo, S.: Nanoscale mechanical characterization of polymers by atomic force microscopy (AFM) nanoindentations: viscoelastic characterization of a model material. Measur. Sci. Technol. 20(9), 9 (2009)CrossRefGoogle Scholar
  53. 53.
    Wang, G.F., Feng, X.Q.: Effects of surface elasticity and residual surface tension on the natural frequency of microbeams. Appl. Phys. Lett. 90(23), 231,904 (2007)Google Scholar
  54. 54.
    Wang, J., Huang, Q.A., Yu, H.: Young’s modulus of silicon nanoplates at finite temperature. Appl. Surf. Sci. 255(5), 2449–2455 (2008)CrossRefGoogle Scholar
  55. 55.
    Wang, J., Huang, Z., Duan, H., Yu, S., Feng, X., Wang, G., Zhang, W., Wang, T.: Surface stress effect in mechanics of nanostructured materials. Acta Mech. Solida Sin. 24, 52–82 (2011)Google Scholar
  56. 56.
    Wang, X.P., Xiao, X.D., Tsui, O.K.C.: Surface viscoelasticity studies of ultrathin polymer films using atomic force microscopic adhesion measurements. Macromolecules 34(12), 4180–4185 (2001)CrossRefGoogle Scholar
  57. 57.
    Wang, Z.Q., Zhao, Y.P.: Self-instabilityand bending behaviors of nano plates. Acta Mech. Solida Sin. 22(6), 630–643 (2009)Google Scholar
  58. 58.
    Young, T.: An essay on the cohesion of fluids. Philos. Trans. Roy. Soc. London 95, 65–87 (1805)CrossRefGoogle Scholar
  59. 59.
    Zhu, H.X., Wang, J.X., Karihaloo, B.L.: Effects of surface and initial stresses on the bending stiffness of trilayer plates and nanofilms. J. Mech. Mater. Struct. 4(3), 589–604 (2009)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Lehrstuhl für Technische Mechanik, Institut für Mechanik, Fakultät für MaschinenbauOtto-von-Guericke-Universtät MagdeburgMagdeburgGermany
  2. 2.Faculty of Mechanical EngineeringOtto-von-Guericke UniversityMagdeburgGermany
  3. 3.South Scientific Center of RASci and South Federal UniversityRostov on DonRussia

Personalised recommendations