Stress State of an Orthotropic Piezoelectric Body with a Triaxial Ellipsoidal Inclusion Subject to Tension

  • V. S. KirilyukEmail author
  • O. I. Levchuk

The problem of the stress state in an orthotropic piezoelectric body with a triaxial ellipsoidal inclusion under homogeneous force and electric loads is considered. The problem is solved by the Eshelby method of equivalent inclusion generalized to the case of a piezoelectric orthotropic space. The approach is validated against the example of a spheroidal cavity in a transversely isotropic material (the axis of revolution coincides with the symmetry axis) for which the exact solution is known. The stress distribution over the surface of the ellipsoidal cavity subject to tension is analyzed numerically.


orthotropic piezoelectric body triaxial ellipsoidal inclusion generalized equivalent inclusion method stress distribution 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    V. T. Grinchenko, A. F. Ulitko, and N. A. Shul’ga, Electroelasticity, Vol. 1 of the five-volume series Mechanics of Coupled Fields in Structural Members [in Russian], Naukova Dumka, Kyiv (1989).Google Scholar
  2. 2.
    L. P. Khoroshun and O. I. Levchuk, “Revisiting the fundamentals of the nonlinear theory of electroelasticity,” Dop. NANU, No. 3, 43–52 (2018).Google Scholar
  3. 3.
    M. O. Shul’ga and V. L. Karlash, Resonant Electromechanical Vibrations of Piezoelectric Plates [in Ukrainian], Naukova Dumka, Kyiv (2008).Google Scholar
  4. 4.
    Y. Benveniste, “The determination of the elastic and electric fields in a piezoelectric inhomogeneity,” J. Appl. Phys., 72, No. 3, 1086–1095 (1992).ADSCrossRefGoogle Scholar
  5. 5.
    W. Q. Chen and C. W. Lim, “3D point force solution for a permeable penny-shaped crack embedded in an infinite transversely isotropic piezoelectric medium,” Int. J. Fract, 131, No. 3, 231–246 (2005).CrossRefzbMATHGoogle Scholar
  6. 6.
    W. Q. Chen, C. W. Lim, and H. J. Ding, “Point temperature solution for penny–shaped crack in an infinite transversely isotropic thermo-piezo-elastic medium,” Eng. Anal. Boundary Elem., 29, No. 6, 524–532 (2005).CrossRefzbMATHGoogle Scholar
  7. 7.
    C. R. Chiang and G. J. Weng, “The nature of stress and electric-displacement concentrations around a strongly oblate cavity in a transversely isotropic piezoelectric material,” Int. J. Fract, 134, No. 3–4, 319–337 (2005).CrossRefzbMATHGoogle Scholar
  8. 8.
    L. Dai, W. Guo, and X. Wang, “Stress concentration at an elliptic hole in transversely isotropic piezoelectric solids,” Int. J. Solids Struct., 43, No. 6, 1818–1831 (2006).CrossRefzbMATHGoogle Scholar
  9. 9.
    M. L. Dunn and M. Taya, “Electroelastic field concentrations in and around inhomogeneities in piezoelectric solids,” J. Appl. Mech., 61, No. 3, 474–475 (1994).ADSCrossRefGoogle Scholar
  10. 10.
    A. Y. Hodes and V. V. Loboda, “A contact zone approach for an arc crack at the interface between two electrostrictive materials,” Int. J. Solids Struct., 128, No. 1, 262–271 (2017).CrossRefGoogle Scholar
  11. 11.
    V. L. Karlash, “Conductance- and susceptance-frequency responses of piezoceramic vibrators,” Int. Appl. Mech., 53, No. 4, 464–471 (2017).ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    V. G. Karnaukhov, I. F. Kirichok, and V. I. Kozlov, “Thermomechanics of inelastic thin-walled structural members with piezoelectric sensors and actuators under harmonic loading (review),” Int. Appl. Mech., 53, No. 1, 6–58 (2017).ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    I. Yu. Khoma and T. M. Proshchenko, “Tension and shear of a transversely isotropic piezoceramic plate with a circular hole with mixed conditions on flat sides,” Int. Appl. Mech., 53, No. 6, 704–715 (2017).ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    L. P. Khoroshun, “Two-continuum mechanics of dielectrics as the basis of the theory of piezoelectricity and electrostriction,” Int. Appl. Mech., 54, No 2, 143–154 (2018).ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    V. S. Kirilyuk and O. I. Levchuk, “Electrostressed state of a piezoceramic body with a paraboloidal cavity,” Int. Appl. Mech., 42, No. 9, 1011–1020 (2006).ADSCrossRefzbMATHGoogle Scholar
  16. 16.
    V. S. Kirilyuk and O. I. Levchuk, “Stress state of an orthotropic piezoelectric material with an elliptic crack,” Int. Appl. Mech., 53, No. 3, 305–312 (2017).ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    L. V. Mol’chenko, “Influence of an extraneous electric current on the stress state of an annular plate of variable rigidity,” Int. Appl. Mech., 37, No. 12, 1607–1611 (2001).CrossRefGoogle Scholar
  18. 18.
    Yu. N. Podil’chuk, “Exact analytical solutions of static electroelastic and thermoelectroelastic problems for a transversely isotropic body in curvilinear coordinate systems,” Int. Appl. Mech., 39, No. 2, 132–170 (2003).CrossRefzbMATHGoogle Scholar
  19. 19.
    Yu. N. Podil’chuk, “Representation of the general solution of statics equations of the electroelasticity of a transversally isotropic piezoceramic body in terms of harmonic functions,” Int. Appl. Mech., 34, No. 7, 623–628 (1998).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Yu. N. Podil’chuk and I. G. Myasoedova, “Stress state of a transversely isotropic piezoceramic body with spheroidal cavity,” Int. Appl. Mech., 40, No. 11, 1269–1280 (2004).CrossRefGoogle Scholar
  21. 21.
    A. K. Soh and J. L. Liu, “Interfacial debonding of a circular inhomogeneity in piezoelectric–piezomagnetic composites under antiplane mechanical and in-plane electromagnetic loading,” Compos. Sci. Technol., 65, No. 9, 1347–1353 (2005).CrossRefGoogle Scholar
  22. 22.
    Z. K. Wang and B. L. Zheng, “The general solution of three-dimension problems in piezoelectric media,” Int. J. Solids Struct., 32, No. 1, 105–115 (1995).CrossRefzbMATHGoogle Scholar
  23. 23.
    T. Y. Zhang and C. F. Gao, “Fracture behaviors of piezoelectric materials,” Theor. Appl. Fract. Mech., 41, No. 1–3, 339–379 (2004).CrossRefGoogle Scholar
  24. 24.
    Y. Zhou, W. Q. Chen, and C. F. Lu, “Semi-analytical solution for orthotropic piezoelectric laminates in cylindrical bending with interfacial imperfections,” Compos. Struct., 92, No. 4, 1009–1018 (2010).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.S. P. Timoshenko Institute of MechanicsNational Academy of Sciences of UkraineKyivUkraine

Personalised recommendations