Mechanics of Solids

, Volume 45, Issue 3, pp 331–342 | Cite as

On equations of the linear theory of shells with surface stresses taken into account

  • H. Altenbach
  • V. A. Eremeev
  • N. F. Morozov


We construct equations of equilibrium and constitutive relations of linear theory of plates and shells with transverse shear strain taken into account, which are based on reducing the spatial elasticity relations with surface stresses taken into account to two-dimensional equations given on the shell median surface. We analyze the influence of surface elasticity moduli on the effective stiffness of plates and shells.

Key words

surface stresses shells plates effective stiffnesses surface tension 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    H. L. Duan, J. Wang, and B. L. Karihaloo, “Theory of Elasticity at the Nanoscale,” Adv. Appl. Mech. 42, 1–63 (2008).CrossRefGoogle Scholar
  2. 2.
    L. D. Landau and E.M. Lifshitz, Course of Theoretical Physics, Vol. 6: Hydrodynamics (Nauka, Moscow, 1986) [in Russian].Google Scholar
  3. 3.
    J. Wang, H. L. Duan, Z. P. Huang, and B. L. Karihaloo, “A Scaling Law for Properties of Nano-Structured Materials,” Proc. Roy. Soc. London. Ser. A 462(2069), 1355–1363 (2006).zbMATHCrossRefADSGoogle Scholar
  4. 4.
    H. L. Duan, J. Wang, Z. P. Huang, and B. L. Karihaloo, “Size-Dependent Effective Elastic Constants of Solids Containing Nano-Inhomogeneities with Interface Stress,” J. Mech. Phys. Solids 53(7), 1574–1596 (2005).zbMATHCrossRefMathSciNetADSGoogle Scholar
  5. 5.
    S. Cuenot, C. Frétigny, S. Demoustier-Champagne, and B. Nysten, “Surface TensionEffect on the Mechanical Properties of Nanomaterials Measured by Atomic Force Microscopy,” Phys. Rev. B 69(16), 165410–5 (2004).CrossRefADSGoogle Scholar
  6. 6.
    G. Y. Jing, H. L. Duan, X.M. Sun, et al., “Surface Effects on Elastic Properties of Silver Nanowires: Contact Atomic-Force Microscopy,” Phys. Rev. B 73(23), 235409–6 (2006).CrossRefADSGoogle Scholar
  7. 7.
    C. Q. Chen, Y. Shi, Y. S. Zhang, et al., “Size Dependence of Young’s Modulus in ZnO Nanowires,” Phys. Rev. Lett. 96(7), 075505–4 (2006).CrossRefADSGoogle Scholar
  8. 8.
    P. S. Laplace, Sur L’action Capillaire in Supplément au Traitéde Mécanique Céleste, T. 4, Livre X (Gauthier-Villars, Paris, 1805).Google Scholar
  9. 9.
    P. S. Laplace, À la Théorie de L’action Capillaire in Supplément au TraitédeMécanique Céleste, T. 4, Livre X (Gauthier-Villars, Paris, 1805).Google Scholar
  10. 10.
    T. Young, “An Essay on the Cohesion of Fluids,” Phil. Trans. Roy. Soc. London 95, 65–87 (1805).CrossRefGoogle Scholar
  11. 11.
    J. W. Gibbs, “On the Equilibrium of Heterogeneous Substances,” in The Collected Works of J. Willard Gibbs (Longmans, Green, New York, 1928), pp. 55–353.Google Scholar
  12. 12.
    E. Orowan, “Surface Energy and Surface Tension in Solids and Fluids,” Proc. Roy. Soc. London. Ser. A 316(1527), 473–491 (1970).CrossRefADSGoogle Scholar
  13. 13.
    Ya. S. Podstrigach and Yu. Z. Povstenko, Introduction to Mechanics of Surface Phenomena in Strained Rigid Bodies (Naukova Dumka, Kiev, 1985) [in Russian].Google Scholar
  14. 14.
    R. Finn, Equilibrium Capillary Surfaces (Springer, New York, 1986; Mir,Moscow, 1989).zbMATHGoogle Scholar
  15. 15.
    A. I. Rusanov, “Thermodynamics of Solid Surfaces,” Surf. Sci. Rep. 23(6–8), 173–247 (1996).CrossRefADSGoogle Scholar
  16. 16.
    A. I. Rusanov, “Surface Thermodynamics Revisited,” Surf. Sci. Rep. 58(5–8), 111–239 (2005).CrossRefADSGoogle Scholar
  17. 17.
    M. E. Gurtin and A. I. Murdoch, “A Continuum Theory of Elastic Material Surfaces,” Arch. Rational Mech. Anal. 57(4), 291–323 (1975).zbMATHMathSciNetGoogle Scholar
  18. 18.
    M. E. Gurtin and A. I. Murdoch, “Addenda to Our Paper: A Continuum Theory of ElasticMaterial Surfaces,” Arch. Rational Mech. Anal. 59(4), 389–390 (1975).zbMATHCrossRefMathSciNetADSGoogle Scholar
  19. 19.
    A. I. Murdoch, “A Thermodynamical Theory of Elastic Material Interfaces,” Quart. J. Mech. Appl. Math. 29(3), 245–274 (1976).zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    P. Podio-Guidugli and G. V. Caffarelli, “Surface Interaction Potentials in Elasticity,” Arch. Rational Mech. Anal. 109(4), 343–383 (1990).zbMATHCrossRefMathSciNetADSGoogle Scholar
  21. 21.
    Yu. Z. Povstenko, “Theoretical Investigation of Phenomena Caused by Heterogeneous Surface Tension in Solids,” J. Mech. Phys. Solids 41(9), 1499–1514 (1993).zbMATHCrossRefMathSciNetADSGoogle Scholar
  22. 22.
    D. J. Steigmann and R.W. Ogden, “Elastic Surface-Substrate Interactions,” Proc. Roy. Soc. London. Ser.A 455(1982), 437–474 (1999).zbMATHCrossRefMathSciNetADSGoogle Scholar
  23. 23.
    V. A. Eremeev and L.M. Zubov, Mechanics of Elastic Shells (Nauka, Moscow, 2008) [in Russian].Google Scholar
  24. 24.
    P. Lu, L. H. He, H. P. Lee, and C. Lu, “Thin Plate Theory Including Surface Effects,” Int. J. Solids Struct. 43(16), 4631–4647 (2006).zbMATHCrossRefGoogle Scholar
  25. 25.
    Y. J. Shi, W. L. Guo, and C. Q. Ru, “Relevance of Timoshenko-Beam Model to Microtubules of Low Shear Modulus,” Phys. E: Low-Dimens. Syst. Nanostruct. 41(2), 213–219 (2008).CrossRefADSGoogle Scholar
  26. 26.
    D. W. Huang, “Size-Dependent Response of Ultra-Thin Films with Surface Effects,” Int. J. Solids Struct. 45(2), 568–579 (2008).zbMATHCrossRefGoogle Scholar
  27. 27.
    J. Peng, J. Wu, K. C. Hwang, et al., “Can a Single-Wall Carbon Nanotube be Modeled as a Thin Shell?” J. Mech. Phys. Solids 56(6), 2213–2224 (2008).zbMATHCrossRefMathSciNetADSGoogle Scholar
  28. 28.
    K. Dahmen, S. Lehwald, and H. Ibach, “Bending of Crystalline Plates under the Influence of Surface Stress-a Finite Element Analysis,” Surf. Sci. 446(1–2), 161–173 (2000).CrossRefADSGoogle Scholar
  29. 29.
    R. E. Miller and V. B. Shenoy, “Size-Dependent Elastic Properties of Nanosized Structural Elements,” Nanotech. 11(3), 139–147 (2000).CrossRefADSGoogle Scholar
  30. 30.
    J. G. Guo and Y. P. Zhao, “The Size-Dependent Elastic Properties of Nanofilms with Surface Effects,” J. Appl. Phys. 98(7), 074306–11 (2005).CrossRefADSGoogle Scholar
  31. 31.
    C. F. Lu, C. W. Lim, and W. Q. Chen, “Size-Dependent Elastic Behavior of FGM Ultra-Thin Films Based on Generalized Refined Theory,” Int. J. Solids Struct. 46(5), 1176–1185 (2009).CrossRefGoogle Scholar
  32. 32.
    V. A. Eremeyev, H. Altenbach, and N. F. Morozov, “The Influence of Surface Tension on the Effective Stiffness of Nanosize Plates,” Dokl. Ross. Akad. Nauk 424(5), 618–621 (2009) [Dokl. Phys. (Engl. Transl.) 54 (2), 98–100 (2009)].Google Scholar
  33. 33.
    H. Altenbach and P. A. Zhilin, “General Theory of Elastic Simple Shells,” Uspekhi Mekh. 11(4), 107–148 (1988).MathSciNetGoogle Scholar
  34. 34.
    H. Altenbach, “An Alternative Determination of Transverse Shear Stiffness for Sandwich and Laminated Plates,” Int. J. Solids Struct. 37(25), 3503–3520 (2000).zbMATHCrossRefGoogle Scholar
  35. 35.
    P. A. Zhilin, Applied Mechanics. Foundations of Shell Theory (Izd-vo Polytechn. Univ., St. Petersburg, 2006) [in Russian].Google Scholar
  36. 36.
    H. Altenbach and V. A. Eremeyev, “Direct Approach Based Analysis of Plates Composed of Functionally Graded Materials,” Arch. Appl. Mech. 78(10), 775–794 (2008).zbMATHCrossRefGoogle Scholar
  37. 37.
    L. M. Zubov, Methods of Nonlinear Elasticity in Shell Theory (Izd-vo RGU, Rostov-on-Don, 1982) [in Russian].Google Scholar
  38. 38.
    S. P. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw Hill, New York, 1959; Fizmatgiz, Moscow, 1963).Google Scholar
  39. 39.
    A. L. Goldenveizer, Theory of Thin Elastic Shells (Nauka, Moscow, 1976) [in Russian].Google Scholar
  40. 40.
    V. V. Novozhilov, K. F. Chernykh, and E. I. Mikhailovskii, The Linear Theory of Thin Shells (Politekhnika, Leningrad, 1991) [in Russian].Google Scholar
  41. 41.
    L. P. Lebedev, M. J. Cloud, and V. A. Eremeyev, Tensor Analysis with Applications in Mechanics (World Scientific, Singapore, 2010).zbMATHGoogle Scholar
  42. 42.
    E. B. Wilson, Vector Analysis, Founded upon the Lectures of J. W. Gibbs (Yale Univ. Press, New Haven, 1901).Google Scholar
  43. 43.
    P. A. Zhilin, Vector and Second-Rank Tensors in Three-Dimensional Space (Nestor, St. Petersburg, 2001) [in Russian].Google Scholar
  44. 44.
    L. M. Zubov and M. I. Karyakin, Tensor Calculus. Foundations of the Theory (Vuzovskaya Kniga, Moscow, 2006) [in Russian].Google Scholar
  45. 45.
    E. I. Grigolyuk and I. T. Selezov, Nonclassical Theories of Rod, Plate, and Shell Vibrations in Results in Science and Technology. Mechanics of Deformable Solids, Vol. 5 (VINITI, Moscow, 1973) [in Russian].Google Scholar
  46. 46.
    E. I. Grigolyuk and P. P. Chulkov, Stability and Vibrations of Three-Layer Shells (Mashinostroenie, Moscow, 1973) [in Russian].Google Scholar
  47. 47.
    H. Altenbach, J. Altenbach, and W. Kissing, Mechanics of Composite Structural Elements (Springer, Berlin, 2004).Google Scholar
  48. 48.
    R. Szilard, Theories and Applications of Plate Analysis. Classical, Numerical, and Engineering Methods (Wiley, Hoboken, New Jersey, 2004).CrossRefGoogle Scholar
  49. 49.
    C. M. Wang, J. N. Reddy, and K. H. Lee, Shear Deformable Beams and Shells (Elsevier, Amsterdam, 2000).Google Scholar
  50. 50.
    H. Altenbach and V. A. Eremeyev, “Eigen-Vibrations of Plates Made of Functionally Graded Material,” CMC 9(2), 153–178 (2009).Google Scholar
  51. 51.
    J. Chróścielewski, J. Makowski, and W. Pietraszkiewicz, Statics and Dynamics of Multifold Shells: Nonlinear Theory and Finite Element Method (Wydawnictwo IPPT PAN, Warsaw, 2004) [in Polish].Google Scholar
  52. 52.
    A. Libai and J. G. Simmonds, The Nonlinear Theory of Elastic Shells (Cambridge Univ. Press, Cambridge, 1998).zbMATHCrossRefGoogle Scholar
  53. 53.
    W. Pietraszkiewicz, Finite Rotations and Lagrangian Description in the Nonlinear Theory of Shells (Polish Sci. Publ., Warsaw-Poznan, 1979).Google Scholar
  54. 54.
    P.M. Naghdi, “The Theory of Shells and Plates,” in Handbuch der Physik, Vol. 2, Ed. by S. Flügge (Berlin, 1972), pp. 425–640.Google Scholar
  55. 55.
    M. B. Rubin, Cosserat Theories: Shells, Rods, and Points (Kluwer, Dordrecht, 2000).zbMATHGoogle Scholar

Copyright information

© Allerton Press, Inc. 2010

Authors and Affiliations

  1. 1.Martin Luther University Halle-WittenbergHalle (Saale)Germany
  2. 2.South Federal UniversityRostov-on-DonRussia
  3. 3.St. Petersburg State UniversitySt. PetersburgRussia

Personalised recommendations