Mechanics of Solids

, Volume 50, Issue 5, pp 495–507 | Cite as

Natural vibrations of nanodimensional piezoelectric bodies with contact-type boundary conditions

  • V. A. Eremeev
  • A. V. NasedkinEmail author


Homogeneous problems of vibrations of nanodimensional piezoelectric bodies with surface stresses and electric charges taken into account are studied. The boundary conditions modeling the frictionless contact of a body with rigid massive punches and the body coverings by a system of open and grounded electrodes are considered. The weak statements of these problems are given. It is proved that the spectrum is real and discrete and the system of eigenfunctions is complete. Several theorems on the eigenfrequency variations are stated with the surface effects taken into account and in the case of variations in the mechanical and electrical boundary conditions and the material characteristics. The finite element approximations are used to state finite-dimensional generalized eigenvalue problems. The results of finite-element calculations of the model problem, which illustrate the influence of surface effects, are given.


electro-elasticity piezoelectricity nanomechanics eigenfrequency spectral property surface effect 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    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).zbMATHMathSciNetCrossRefADSGoogle Scholar
  2. 2.
    H. L. Duan, J. Wang, and B. L. Karihaloo, “Theory of Elasticity at the Nanoscale,” Adv. Appl. Mech. 42, 1–63 (2008).CrossRefGoogle Scholar
  3. 3.
    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), 618–620 (2009) [Dokl. Phys. (Engl. Transl.) 54 (2), 98–100 (2009)].MathSciNetGoogle Scholar
  4. 4.
    R. V. Goldstein, E. A. Kasparova, and P. S. Shushpannikov, “Role of Surface Effects in Deformation of Two-Layer Plates,” Vestnik Tambov. Univ. 15 (3–2), 1182–1185 (2010).Google Scholar
  5. 5.
    R. V. Goldstein, V. A. Gorodtsov, and K. V. Ustinov, “Influence of Surface Residual Stresses and Surface Elasticity on Deformation of Spherical Inclusions of Nano-Meter Dimensions in and Elastic Matrix,” Fiz. Mezomekh. 13 (5), 127–138 (2010).Google Scholar
  6. 6.
    J. Wang, Z. P. Huang, H. L. Duan, et al., “Surface Stress Effect inMechanics of Nanostructured Materials,” Acta Mech. Solida Sinica 24 (1), 52–82 (2011).CrossRefGoogle Scholar
  7. 7.
    A. A. Javili, A. A. McBride, and P. Steinmann, “Thermomechanics of Solids with Lower-Dimensional Energetics: On the Importance of Surface, Interface, and Curve Structures at the Nanoscale. A Unifying Review,” Appl. Mech. Rev. 65 (1), 010802–31 (2013).CrossRefADSGoogle Scholar
  8. 8.
    M. E. Gurtin and A. I.Murdoch, “A Continuum Theory of ElasticMaterial Surfaces,” Arch. Rat. Mech. Anal. 57 (4), 291–323 (1975).zbMATHCrossRefGoogle Scholar
  9. 9.
    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
  10. 10.
    H. Altenbach, V. A. Eremeev, and N. F. Morozov, “On Equations of the Linear Theory of Shells with Surface Stresses Taken into Account,” Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 3, 30–44 (2010) [Mech. Solids (Engl. Transl.) 45 (3), 331–342 (2010)].Google Scholar
  11. 11.
    H. Altenbach, V. A. Eremeyev, and L. P. Lebedev, “On the Existence of Solutions in the Linear Elasticity with Surface Stresses,” ZAMM 90 (3), 231–240 (2010).zbMATHMathSciNetCrossRefADSGoogle Scholar
  12. 12.
    V. A. Eremeyev and L. P. Lebedev, “Existence of Weak Solutions in Elasticity,” Math. Mech. Solids 18 (2), 204–217 (2013).MathSciNetCrossRefGoogle Scholar
  13. 13.
    H. L. Duan, J. Wang, B. L. Karihaloo, and Z. P. Huang, “Nanoporous Materials Can Be Made Stiffer Than Non-Porous Counterparts by SurfaceModification,” Acta Mater. 54 (11), 2983–2990 (2006).CrossRefGoogle Scholar
  14. 14.
    H. Ibach, “The Role of Surface Stress in Reconstruction, Epitaxial Growth and Stabilization of Mesoscopic Structures,” Surf. Sci. Rep. 29 (5–6), 195–263 (1997).CrossRefADSGoogle Scholar
  15. 15.
    K. B. Ustinov, R. V. Goldstein, and V. A. Gorodtsov, “On the Modeling of Surface and Interface Elastic Effects in the Case of Eigenstrains,” in Advanced Structural Materials, Vol. 30: Surface Effects in Solid Mechanics–Models, Simulations and Applications, Ed. by H. Altenbach and N. F. Morozov (Springer, Berlin, 2013), pp. 167–180.Google Scholar
  16. 16.
    V. B. Shenoy and R. E. Miller, “Size-Dependent Elastic Properties of Nanosized Structural Elements,” Nanotechnol. 11 (3), 139–147 (2000).CrossRefADSGoogle Scholar
  17. 17.
    V. B. Shenoy, “Atomistic Calculations of Elastic Properites of Metallic Fee Crystal Surfaces,” Phys. Rev. 71 (9), 094104–11 (2005).CrossRefADSGoogle Scholar
  18. 18.
    G. Y. Huang and S. W. Yu, “Effect of Surface Piezoelectricity on the Electromechanical Behaviour of a Piezoelectric Ring,” Phys. Status Solidi B 243 (4), R22–R24 (2006).MathSciNetCrossRefADSGoogle Scholar
  19. 19.
    A. V. Nasedkin and V. A. Eremeyev, “Spectral Properties of Piezoelectric Bodies with Surface Effects,” in Advanced Structural Materials, Vol. 30: Surface Effects in Solid Mechanics–Models, Simulations and Applications, Ed. by H. Altenbach and N. F.Morozov (Springer, Berlin, 2013), pp. 105–121.Google Scholar
  20. 20.
    A. V. Belokon’ and I. I. Vorovich, “Several Mathematical Problems of the Theory of Electro-Elastic Bodies,” in Actual Problems ofMechanics of DeformableMedia (DGU, Dnepropetrovsk, 1979), pp. 52–67 [in Russian].Google Scholar
  21. 21.
    H. Altenbach, V. A. Eremeyev, and L. P. Lebedev, “On the Spectrum and Stiffness of an Elastic Body with Surface Stresses,” ZAMM 91 (9), 699–710 (2011).zbMATHMathSciNetCrossRefADSGoogle Scholar
  22. 22.
    A. V. Belokon’ and A. V. Nasedkin, “Some Properties of the Natural Frequencies of Electroelastic Bodies of Bounded Dimensions,” Prikl. Mat. Mekh. 60 (1), 151–158 (1996) [ J. Appl. Math. Mech. (Engl. Transl.) 60 (1), 145–152 (1996)].MathSciNetGoogle Scholar
  23. 23.
    F. Riesz and B. Sz.-Nagy, Lectures in Functional Analysis (Budapest, 1953; Mir, Moscow, 1979).Google Scholar
  24. 24.
    S. G. Mikhlin, Variational Methods in Mathematical Physics (Pergamon, New York, 1964; Nauka, Moscow, 1970).Google Scholar
  25. 25.
    K. J. Bathe, Finite Element Procedures (Prentice Hall, Englewood Cliffs, HJ, 1996; Fizmatlit, Moscow, 2010).Google Scholar
  26. 26.
    O. Zienkiewicz and K. Morgan, Finite Elements and Approximation (Wiley, NewYork, 1983; Mir, Moscow, 1986).Google Scholar
  27. 27.
    G. Iovane and A. V. Nasedkin, “Some Theorems about Spectrum and Finite Element Approach for Eigenvalue Problems for Elastic Bodies with Voids,” Comput. Math. Applicat. 53 (5), 789–802 (2007).zbMATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    G. Iovane and A. V. Nasedkin, “Modal Analysis of Piezoelectric Bodies with Voids. I I. Finite Element Simulation,” Appl.Math. Model. 34 (1), 47–59 (2010).zbMATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    E. Dieulesaint and D. Royer, Elastic Waves in Solids. Application to Signal Processing (Masson, Paris, 1974; Wiley New York 1980; Nauka, Moscow, 1982).Google Scholar

Copyright information

© Allerton Press, Inc. 2015

Authors and Affiliations

  1. 1.Otto-von-Guericke Universität MagdeburgMagdeburgGermany
  2. 2.South Scientific CenterRussian Academy of SciencesRostov-on-DonRussia
  3. 3.South Federal UniversityRostov-on-DonRussia

Personalised recommendations