On the Continuum Mechanics Approach in Modeling Nanosized Structural Elements

  • Holm AltenbachEmail author
  • Victor A. Eremeyev
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 4)


During the last 50 years the nanotechnology is established as one of the advanced technologies manipulating matter on an atomic and molecular scale. By this approach new materials, devices or other structures possessing at least one dimension sized from 1 to 100  nm are developed. The question arises how structures composed of nanomaterials should be modeled. Two approaches are suggested—theories which take into account quantum mechanical effects since they are important at the quantum-realm scale or theories which are based on the classical continuum mechanics adapted to nanoscale problems. Here the second approach will be discussed in detail. It will be shown that the classical continuum mechanics (kinematics, stress states analysis, balances and constitutive equations) with some improvements is enough for a sufficient description of the mechanical behavior of nanomaterials and -structures in many situations. After a brief recall of the basics of Continuum Mechanics a theory with surface effects will be discussed.


Strain Tensor Material Point Surface Stress Reference Configuration Stress Vector 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Aifantis, E.: Update on a class of gradient theories. Mech. Mater. 35(3–6), 259–280 (2003)CrossRefGoogle Scholar
  2. 2.
    Alizada, A.N., Sofiyev, A.H.: Modified Young’s moduli of nano-materials taking into account the scale effects and vacancies. Meccanica 46, 915–920 (2011)CrossRefGoogle Scholar
  3. 3.
    Altenbach, H.: An alternative determination of transverse shear stiffnesses for sandwich and laminated plates. Int. J. Solids Struct. 37(25), 3503–3520 (2000)CrossRefGoogle Scholar
  4. 4.
    Altenbach, H.: On the determination of transverse shear stiffnesses of orthotropic plates. ZAMP 51, 629–649 (2000)CrossRefGoogle Scholar
  5. 5.
    Altenbach, H. (ed.): Holzmann Meyer Schumpich Technische Mechanik Festigkeitslehre, 10th edn. Vieweg + Teubner, Wiesbaden (2012)Google Scholar
  6. 6.
    Altenbach, H.: Kontinuumsmechanik—Eine elementare Einführung in die materialunabhängigen und materialabhängigen Gleichungen. Springer, Berlin (2012)Google Scholar
  7. 7.
    Altenbach, H., Eremeyev, V.A., Morozov, N.F.: On the equations of the linear theory of shells with surface stresse taken into account. Mech. Solids 45(3), 331–342 (2010)CrossRefGoogle Scholar
  8. 8.
    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)CrossRefGoogle Scholar
  9. 9.
    Altenbach, H., Eremeyev, V.A., Morozov, N.F.: On the influence of residual surface stresses on the properties of structures at the nanoscale. In: Altenbach, H., Morozov, N.F. (eds.): Surface Effects in Solid Mechanics—Models, Simulations, and Applications, Advanced Structured Materials, vol. 19, pp. 21–32. Springer, Berlin (2013)Google Scholar
  10. 10.
    Altenbach, H., Zhilin, P.A.: The theory of simple elastic shells. In: Kienzler, R., Altenbach, H., Ott I. (eds.): Critical Review of the Theories of Plates and Shells, Lecture Notes in Applied and Computational Mechanics, vol. 16, pp. 1–12. Springer, Berlin (2004)Google Scholar
  11. 11.
    Asghari, M.: Geometrically nonlinear micro-plate formulation based on the modified couple stress theory. Int. J. Eng. Sci. 51(18), 292–309 (2012)CrossRefGoogle Scholar
  12. 12.
    Asghari, M., Ahmadian, M.T., Kahrobaiyan, M.H., Rahaeifard, M.: On the size-dependent behavior of functionally graded micro-beams. Mater. Des. 31, 2324–2329 (2010)CrossRefGoogle Scholar
  13. 13.
    Asghari, M., Kahrobaiyan, M.H., Ahmadian, M.T: A nonlinear Timoshenko beam formulation based on the modified couple stress theory. Int. J. Eng. Sci. 48(12), 1749–1761 (2010)CrossRefGoogle Scholar
  14. 14.
    Chen, C.Q., Shi, Y., Zhang, Y.S., Zhu, J., Yan, Y.J.: Size dependence of Young’s modulus in ZnO nanowires. Phys. Rev. Lett. 96, 075, 505, 1–4 (2006)Google Scholar
  15. 15.
    Chen, S.H., Feng, B.: Size effect in micro-scale cantilever beam bending. Acta Mech. 219, 291–307 (2011)CrossRefGoogle Scholar
  16. 16.
    Cosserat, E., Cosserat, F.: Théorie des Corps Déformables. A. Herman et fils, Paris (1909)Google Scholar
  17. 17.
    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, 165,410, 1–5 (2004)Google Scholar
  18. 18.
    Duan, H., Wang, J., Karihaloo, B.: Theory of elasticity at the nanoscale. In: Aref, H., van der Giessen, E. (eds.): Advances in Applied Mechanics, vol. 42, pp. 1–68. Elsevier, London (2008)Google Scholar
  19. 19.
    Duan, H.L., Wang, J., Huang, Z.P., Karihaloo, B.L: Size-dependent effective elastic constants of solids containing nano-inhomogeneities with interface stress. J. Mech. Phys. Solids 53(7), 1574–1596 (2005)CrossRefGoogle Scholar
  20. 20.
    Eremeyev, V.A., Zubov, L.M.: Mechanics of Elastic Shells (in Russian). Nauka, Moscow (2008)Google Scholar
  21. 21.
    Finn, R.: Equilibrium Capillary Surfaces. Springer, New York (1986)CrossRefGoogle Scholar
  22. 22.
    Fleck, N.A., Muller, G.M., Ashby, M.F., Hutchinson, J.W: Strain gradient plasticity: Theory and experiment. Acta Metall. Mater. 42(2), 475–487 (1992)Google Scholar
  23. 23.
    Gibbs, J.W.: On the equilibrium of heterogeneous substances. Transactions Connecticut Acadamy of Arts and Sciences III, pp. 343–524 (1874–1878)Google Scholar
  24. 24.
    Govindjee, S., Sackman, J.L: On the use of continuum mechanics to estimate the properties of nanotubes. Solid State Commun. 110(4), 227–230 (1999)CrossRefGoogle Scholar
  25. 25.
    Greer, J.R., Nix, W.D: Size dependence of mechanical properties of gold at the sub-micron scale. Appl. Phys. A 80, 1625–1629 (2005)CrossRefGoogle Scholar
  26. 26.
    Grigolyuk, E.I., Seleznev, I.T.: Nonclassical Theories of Vibration of Beams, Plates and Shells (in Russian), Itogi nauki i tekhniki. Mekhanika tverdogo deformiruemogo tela, vol. 5. VINITI, Moskva (1973)Google Scholar
  27. 27.
    Gross, D., Hauger, W., Schröder, J., Wall, W.A., Bonet, J.: Engineering Mechanics, vol. 2, Mechanics of Materials, Springer, Berlin (2011)Google Scholar
  28. 28.
    Gurtin, M.E., Markenscoff, X., Thurston, R.N.: Effect of surface stress on the natural frequency of thin crystals. Appl. Phys. Lett. 29(9), 529–530 (1976)CrossRefGoogle Scholar
  29. 29.
    Gurtin, M.E., Murdoch, A.I.: Addenda to our paper: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 59(4), 389–390 (1975)CrossRefGoogle Scholar
  30. 30.
    Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57(4), 291–323 (1975)CrossRefGoogle Scholar
  31. 31.
    Guz, A.N., Rushchitsky, J.J.: Establishing foundations of the mechanics of nanocomposites(review). Int. Appl. Mech. 47(1), 2–44 (2011)CrossRefGoogle Scholar
  32. 32.
    Haupt, P.: Continuum Mechanics and Theory of Materials, 2nd edn.. Springer, Berlin (2002)CrossRefGoogle Scholar
  33. 33.
    Huang, Z.P., Wang, J.: Micromechanics of nanocomposites with interface energy effect. In: Bai, Y.L., Zheng, Q.S., Wei, Y.G. (eds.): IUTAM Symposium on Mechanical Behavior and Micro-Mechanics of Nanostructured Materials, Solid Mechanics and its Applications, vol. 144, pp. 51–59. Springer, Dordrecht (2007)Google Scholar
  34. 34.
    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), 235,409–235,416 (2006)Google Scholar
  35. 35.
    Kong, S., Zhou, S., Nie, Z., Wang, K.: The size-dependent natural frequancy of Bernoulli-Euler micro-beams. Int. J. Eng. Sci. 46, 427–437 (2008)CrossRefGoogle Scholar
  36. 36.
    Krivtsov, A.M., Morozov, N.F: On mechanical characteristics of nanocrystals. Phys. Solid State 44(12), 2260–2065 (2002)CrossRefGoogle Scholar
  37. 37.
    Lam, D.C.C., Yang, F., Chonga, A.C.M., Wang, J., Tong, P.: Experiments and theory in strain gradient elasticity. J. Mech. Phys. Solids 51, 1477–1508 (2003)CrossRefGoogle Scholar
  38. 38.
    Landau, L.D., Lifshitz, E.M.: Course of Theoretical Physics, vol. 6, Fluid Mechanics. Butterworth-Heinemann, Oxfort (1987)Google Scholar
  39. 39.
    Laplace, P.S.: Traité de Mécanique Céleste. Livre X, vol. 4, Suppl. 1, chap. Sur l’action capillaire. Supplément à à la théorie de l’action capillaire, pp. 771–777. Gauthier-Villars et fils, Paris (1805)Google Scholar
  40. 40.
    Lazopoulos, K.A.: On the gradient strain elasticity theory of plates. Europ. J. Mech. A/Solids 23(5), 843–852 (2004)CrossRefGoogle Scholar
  41. 41.
    Lebedev, L.P., Cloud, M.J., Eremeyev, V.A: Tensor Analysis with Applications in Mechanics. World Scientific, Singapore (2010)CrossRefGoogle Scholar
  42. 42.
    Libai, A., Simmonds, J.G: The Nonlinear Theory of Elastic Shells, 2nd edn. Cambridge University Press, Cambridge (1998)CrossRefGoogle Scholar
  43. 43.
    Lü, C.F., Chen, W.Q., Lim, C.W: Elastic mechanical behavior of nano-scaled fgm films incorperation surface energies. Compos. Sci. Technol. 69, 1124–1130 (2009)CrossRefGoogle Scholar
  44. 44.
    Lü, C.F., Lim, C.W., Chen, W.Q.: Size-depencent elastic behavior of fgm ultra-thin films based on generalized refined theory. Int. J. Solids Struct. 46, 1176–1185 (2009)CrossRefGoogle Scholar
  45. 45.
    Ma, H.M., Gao, X.L., Reddy, J.N: A mucrostructure-dependent Timoshenko beam model based on a modified couple stress theory. J. Mecha. Phys. Solids 56, 3379–3391 (2008)CrossRefGoogle Scholar
  46. 46.
    Murdoch, A.I.: A thermodynamical theory of elastic material interfaces. Quart. J. Mech. Appl. Math. 29(3), 245–274 (1976)CrossRefGoogle Scholar
  47. 47.
    Naghdi, P.: The theory of plates and shells. In: Flügge, S. (ed.): Handbuch der Physik, vol. VIa/2, pp. 425–640. Springer, Heidelberg (1972)Google Scholar
  48. 48.
    Orowan, E.: Surface energy and surface tension in solids and fluids. Philos. Trans. Royal Soc. Lond. Ser. A 316, 473–491 (1970)Google Scholar
  49. 49.
    Palmov, V.A: Vibrations of Elasto-Plastic Bodies. Springer, Berlin (1998)CrossRefGoogle Scholar
  50. 50.
    Parisch, H.: Festkörper-Kontinuumsmechanik: Von den Grundgleichungen zur Lösung mit Finiten Elementen. Teubner, Stuttgart (2003)CrossRefGoogle Scholar
  51. 51.
    Park, S.K., Gao, X.: Bernoulli-Euler beam model based on a modified couple stress theory. J. Micromech. Microeng. 16, 2355–2359 (2006)CrossRefGoogle Scholar
  52. 52.
    Podio-Guidugli, P., Caffarelli, G.V: Surface interaction potentials in elasticity. Arch. Ration. Mech. Anal. 109(4), 343–383 (1990)CrossRefGoogle Scholar
  53. 53.
    Podstrigach, Y.S., Povstenko, Y.Z.: Introduction to Mechanics of Surface Phenomena in Deformable Solids (in Russian). Naukova Dumka, Kiev (1985)Google Scholar
  54. 54.
    Povstenko, Y.Z: Theoretical investigation of phenomena caused by heterogeneous surface tension in solids. J. Mech. Phys. Solids 41(9), 1499–1514 (1993)CrossRefGoogle Scholar
  55. 55.
    Rusanov, A.I: Thermodynamics of solid surfaces. Surf. Sci. Rep. 23, 173–247 (1996)CrossRefGoogle Scholar
  56. 56.
    Rusanov, A.I.: Surface thermodynamics revisited. Surf. Sci. Rep. 58, 111–239 (2005)CrossRefGoogle Scholar
  57. 57.
    Salençon, J.: Handbbok of Continuum Mechanics. Berlin, Berlin (2001)Google Scholar
  58. 58.
    Şimşek, M.: Dynamic analysis of an embedded microbeam carrying a moving microparticle based on a modified couple stress theory. Int. J. Eng. Sci. 48, 1721–1732 (2010)Google Scholar
  59. 59.
    Steigmann, D.J., Ogden, R.W: Elastic surface-substrate interactions. Proc. Roy. Soc. London. Ser. A 455, 437–474 (1999)CrossRefGoogle Scholar
  60. 60.
    Stolken, J.S., Evans, A.G: Microbend test method for measuring the plasticity length scale. Acta Mater. 46(14), 5109–5115 (1998)CrossRefGoogle Scholar
  61. 61.
    Wang, Z.Q., Zhao, Y.P: Thermo-hyperelastic models for nanostructured materials. Sci. China Phys. Mech. Astron. 54(5), 948–956 (2011)CrossRefGoogle Scholar
  62. 62.
    Wang, Z.Q., Zhao, Y.P., Huang, Z.: The effects of surface tension on the elastic properties of nano structures. Int. J. Eng. Sci. 48, 140–150 (2010)CrossRefGoogle Scholar
  63. 63.
    Willner, K.: Kontinuums- und Kontaktmechanik: Synthetische und analytische Darstellung. Springer, Berlin (2003)CrossRefGoogle Scholar
  64. 64.
    Wriggers, P.: Nichtlineare Finite-Element-Methoden. Springer, Berlin (2001)CrossRefGoogle Scholar
  65. 65.
    Yang, F., Chong, A., Lam, D., Tong, P.: Couple stress based strain gradient theory for elasticity. Int. J. Solids Struct. 39, 2731–2743 (2002)CrossRefGoogle Scholar
  66. 66.
    Young, T.: An essay on the cohesion of fluids. Philos. Trans. Royal Soc. Lond. 95, 65–87 (1805)CrossRefGoogle Scholar
  67. 67.
    Zhou, L.G., Huang, H.: Are surfaces elastically softer or stiffer. Appl. Phys. Lett. 84(11), 1940–1942 (2004)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Lehrstuhl für Technische MechanikInstitut für MechanikMagdeburgGermany

Personalised recommendations