International Applied Mechanics

, Volume 42, Issue 9, pp 1011–1020 | Cite as

Electroelastic stress state of a piezoceramic body with a paraboloidal cavity

  • V. S. Kirilyuk
  • O. I. Levchuk


The static equilibrium of an electroelastic transversely isotropic space with a paraboloidal cavity under axisymmetric mechanical and electric loads is analyzed. Paraboloidal coordinates and special harmonic functions are used to obtain an exact solution. The distribution of stresses and electric-flux density over the surface of the cavity subject to internal pressure is analyzed as an example


piezoelectricity three-dimensional problem paraboloidal cavity force and electric fields 


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  1. 1.
    E. I. Grigolyuk, L. A. Fil’shtinskii, and Yu. D. Kovalev, “Tension of a piezoceramic layer weakened by through tunnel cavities,” Dokl. RAN, 385, No. 1, 61–63 (2002).Google Scholar
  2. 2.
    V. T. Grinchenko, A. F. Ulitko, and N. A. Shul’ga, Electroelasticity, Vol. 1 of the six-volume series Mechanics of Coupled Fields in Structural Members [in Russian], Naukova Dumka, Kiev (1989).Google Scholar
  3. 3.
    S. A. Kaloerov, A. I. Baeva, and Yu. A. Glushchenko, “Two-dimensional problem of electroelasticity for a multiply connected piezoelectric body with cavities and flat cracks,” Teor. Prikl. Mekh., 32, 64–79 (2001).Google Scholar
  4. 4.
    P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Vol. 2, McGraw-Hill, New York (1953).MATHGoogle Scholar
  5. 5.
    V. Z. Parton and B. A. Kudryavtsev, Electromagnetoelasticity of Piezoelectric and Electroconductive Bodies [in Russian], Nauka, Moscow (1988).Google Scholar
  6. 6.
    Yu. N. Podil’chuk, “On the stress-strain state of a transversally isotropic body with a paraboloidal inclusion,” Int. Appl. Mech., 34, No. 11, 1079–1085 (1998).Google Scholar
  7. 7.
    Yu. N. Podil’chuk, “Electroelastic equilibrium of transversally isotropic, piezoceramic media containing cavities, inclusions, and cracks,” Int. Appl. Mech., 34, No. 10, 1023–1034 (1998).MathSciNetGoogle Scholar
  8. 8.
    K. V. Solyanik-Krassa, Axisymmetric Problem of Elasticity [in Russian], Stroitekhizdat, Moscow (1987).Google Scholar
  9. 9.
    W. Chen and T. Shioya, “Fundamental solution for a penny-shaped crack in a piezoelectric medium,” J. Mech. Phys. Solids, 47, 1459–1475 (1999).CrossRefMathSciNetGoogle Scholar
  10. 10.
    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, 231–246 (2005).CrossRefGoogle Scholar
  11. 11.
    M. L. Dunn and M. Taya, “Electroelastic field concentrations in and around inhomogeneties in piezoelectric solids,” Trans. ASME, Ser. E, J. Appl. Mech., 61, 474–475 (1994).Google Scholar
  12. 12.
    S. A. Kaloerov, A. I. Baeva, and Yu. A. Glushchenko, “Two-dimensional electroelastic problem for a multiply connected piezoelectric body,” Int. Appl. Mech., 39, No. 1, 77–84 (2003).CrossRefGoogle Scholar
  13. 13.
    I. F. Kirichok, “Flexural vibrations and heating of a circular bimorph piezoplate under electric excitation applied to nonuniformly electroded surfaces,” Int. Appl. Mech., 41, No. 9, 1037–1042 (2005).CrossRefGoogle Scholar
  14. 14.
    V. S. Kirilyuk, “Stress state of an elastic orthotropic medium with an elliptic crack under tension and shear,” Int. Appl. Mech., 41, No. 4, 358–366 (2005).CrossRefGoogle Scholar
  15. 15.
    V. S. Kirilyuk, “On the stress of a piezoceramic body with a flat crack under symmetric loads,” Int. Appl. Mech., 41, No. 11, 1263–1271 (2005).CrossRefGoogle Scholar
  16. 16.
    Yu. N. Podil’chuk and I. G. Kovalenko, “Thermoelectroelastic state of a piezoceramic body with a spheroidal cavity in a uniform heat flow,” Int. Appl. Mech., 41, No. 11, 1254–1262 (2005).CrossRefGoogle Scholar
  17. 17.
    F. Shang, M. Kuna, and M. Abendroth, “Finite element analysis of a three-dimensional crack problem in piezoelectric structures,” Eng. Fract. Mech., 70, 143–160 (2003).CrossRefGoogle Scholar
  18. 18.
    H. Sosa and N. Khutoryansky, “New developments concerning piezoelectric materials with defects,” Int. J. Solids Struct., 33, No. 23, 3399–3414 (1996).CrossRefMathSciNetGoogle Scholar
  19. 19.
    Z. K. Wang and S. H. Huang, “Stress intensification near an elliptical border,” Theor. Appl. Fract. Mech., 22, 229–237 (1995).CrossRefMathSciNetGoogle Scholar
  20. 20.
    Z. K. Wang and B. L. Zheng, “The general solution of three-dimensional problems in piezoelectric media,” Int. J. Solids Struct., 32, 105–115 (1995).CrossRefGoogle Scholar
  21. 21.
    J. Z. Zuo and G. C. Sih, “Energy density theory formulation and interpretation of cracking behavior for piezoelectric ceramics,” Theor. Appl. Fract. Mech., 34, 17–33 (2000).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • V. S. Kirilyuk
    • 1
  • O. I. Levchuk
    • 1
  1. 1.S. P. Timoshenko Institute of MechanicsNational Academy of Sciences of UkraineKiev

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