Skip to main content
SpringerLink
Log in

On the Continuum Mechanics Approach in Modeling Nanosized Structural Elements

  • Chapter
  • First Online:
New Frontiers of Nanoparticles and Nanocomposite Materials

Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 4))

  • 2333 Accesses

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 119.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Aifantis, E.: Update on a class of gradient theories. Mech. Mater. 35(3–6), 259–280 (2003)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  3. Altenbach, H.: An alternative determination of transverse shear stiffnesses for sandwich and laminated plates. Int. J. Solids Struct. 37(25), 3503–3520 (2000)

    Article  Google Scholar 

  4. Altenbach, H.: On the determination of transverse shear stiffnesses of orthotropic plates. ZAMP 51, 629–649 (2000)

    Article  Google Scholar 

  5. Altenbach, H. (ed.): Holzmann Meyer Schumpich Technische Mechanik Festigkeitslehre, 10th edn. Vieweg + Teubner, Wiesbaden (2012)

    Google Scholar 

  6. Altenbach, H.: Kontinuumsmechanik—Eine elementare Einführung in die materialunabhängigen und materialabhängigen Gleichungen. Springer, Berlin (2012)

    Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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. 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. Asghari, M.: Geometrically nonlinear micro-plate formulation based on the modified couple stress theory. Int. J. Eng. Sci. 51(18), 292–309 (2012)

    Article  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  Google Scholar 

  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. Chen, S.H., Feng, B.: Size effect in micro-scale cantilever beam bending. Acta Mech. 219, 291–307 (2011)

    Article  Google Scholar 

  16. Cosserat, E., Cosserat, F.: Théorie des Corps Déformables. A. Herman et fils, Paris (1909)

    Google Scholar 

  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. 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. 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)

    Article  Google Scholar 

  20. Eremeyev, V.A., Zubov, L.M.: Mechanics of Elastic Shells (in Russian). Nauka, Moscow (2008)

    Google Scholar 

  21. Finn, R.: Equilibrium Capillary Surfaces. Springer, New York (1986)

    Book  Google Scholar 

  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. 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. 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)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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. Gross, D., Hauger, W., Schröder, J., Wall, W.A., Bonet, J.: Engineering Mechanics, vol. 2, Mechanics of Materials, Springer, Berlin (2011)

    Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  Google Scholar 

  30. Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57(4), 291–323 (1975)

    Article  Google Scholar 

  31. Guz, A.N., Rushchitsky, J.J.: Establishing foundations of the mechanics of nanocomposites(review). Int. Appl. Mech. 47(1), 2–44 (2011)

    Article  Google Scholar 

  32. Haupt, P.: Continuum Mechanics and Theory of Materials, 2nd edn.. Springer, Berlin (2002)

    Book  Google Scholar 

  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. 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. 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)

    Article  CAS  Google Scholar 

  36. Krivtsov, A.M., Morozov, N.F: On mechanical characteristics of nanocrystals. Phys. Solid State 44(12), 2260–2065 (2002)

    Article  CAS  Google Scholar 

  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)

    Article  Google Scholar 

  38. Landau, L.D., Lifshitz, E.M.: Course of Theoretical Physics, vol. 6, Fluid Mechanics. Butterworth-Heinemann, Oxfort (1987)

    Google Scholar 

  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. Lazopoulos, K.A.: On the gradient strain elasticity theory of plates. Europ. J. Mech. A/Solids 23(5), 843–852 (2004)

    Article  Google Scholar 

  41. Lebedev, L.P., Cloud, M.J., Eremeyev, V.A: Tensor Analysis with Applications in Mechanics. World Scientific, Singapore (2010)

    Book  Google Scholar 

  42. Libai, A., Simmonds, J.G: The Nonlinear Theory of Elastic Shells, 2nd edn. Cambridge University Press, Cambridge (1998)

    Book  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  46. Murdoch, A.I.: A thermodynamical theory of elastic material interfaces. Quart. J. Mech. Appl. Math. 29(3), 245–274 (1976)

    Article  Google Scholar 

  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. Orowan, E.: Surface energy and surface tension in solids and fluids. Philos. Trans. Royal Soc. Lond. Ser. A 316, 473–491 (1970)

    CAS  Google Scholar 

  49. Palmov, V.A: Vibrations of Elasto-Plastic Bodies. Springer, Berlin (1998)

    Book  Google Scholar 

  50. Parisch, H.: Festkörper-Kontinuumsmechanik: Von den Grundgleichungen zur Lösung mit Finiten Elementen. Teubner, Stuttgart (2003)

    Book  Google Scholar 

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

    Article  Google Scholar 

  52. Podio-Guidugli, P., Caffarelli, G.V: Surface interaction potentials in elasticity. Arch. Ration. Mech. Anal. 109(4), 343–383 (1990)

    Article  Google Scholar 

  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. Povstenko, Y.Z: Theoretical investigation of phenomena caused by heterogeneous surface tension in solids. J. Mech. Phys. Solids 41(9), 1499–1514 (1993)

    Article  Google Scholar 

  55. Rusanov, A.I: Thermodynamics of solid surfaces. Surf. Sci. Rep. 23, 173–247 (1996)

    Article  CAS  Google Scholar 

  56. Rusanov, A.I.: Surface thermodynamics revisited. Surf. Sci. Rep. 58, 111–239 (2005)

    Article  CAS  Google Scholar 

  57. Salençon, J.: Handbbok of Continuum Mechanics. Berlin, Berlin (2001)

    Google Scholar 

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

    Article  Google Scholar 

  60. Stolken, J.S., Evans, A.G: Microbend test method for measuring the plasticity length scale. Acta Mater. 46(14), 5109–5115 (1998)

    Article  CAS  Google Scholar 

  61. Wang, Z.Q., Zhao, Y.P: Thermo-hyperelastic models for nanostructured materials. Sci. China Phys. Mech. Astron. 54(5), 948–956 (2011)

    Article  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  63. Willner, K.: Kontinuums- und Kontaktmechanik: Synthetische und analytische Darstellung. Springer, Berlin (2003)

    Book  Google Scholar 

  64. Wriggers, P.: Nichtlineare Finite-Element-Methoden. Springer, Berlin (2001)

    Book  Google Scholar 

  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)

    Article  Google Scholar 

  66. Young, T.: An essay on the cohesion of fluids. Philos. Trans. Royal Soc. Lond. 95, 65–87 (1805)

    Article  Google Scholar 

  67. Zhou, L.G., Huang, H.: Are surfaces elastically softer or stiffer. Appl. Phys. Lett. 84(11), 1940–1942 (2004)

    Article  CAS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Lehrstuhl für Technische Mechanik, Institut für Mechanik, Otto-von-Guericke-Universität Magdeburg, Universitätsplatz 2, 39106, Magdeburg, Germany

    Holm Altenbach & Victor A. Eremeyev

Authors
  1. Holm Altenbach
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Victor A. Eremeyev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Holm Altenbach .

Editor information

Editors and Affiliations

  1. , Faculty Biosciences and, Universiti Teknologi Malaysia, Block A, Building V01, Johor Bahru, 81310, Malaysia

    Andreas Öchsner

  2. of Technology, Fac. Mechanical Engineering, Khajeh Nasir Toosi University, Tehran, Iran

    Ali Shokuhfar

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Altenbach, H., Eremeyev, V.A. (2012). On the Continuum Mechanics Approach in Modeling Nanosized Structural Elements. In: Öchsner, A., Shokuhfar, A. (eds) New Frontiers of Nanoparticles and Nanocomposite Materials. Advanced Structured Materials, vol 4. Springer, Berlin, Heidelberg. https://doi.org/10.1007/8611_2012_67

Download citation

Publish with us

Policies and ethics

Search

Navigation