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Elliptical crack and punch problems solved by integral equation method for piezoelectric media

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Abstract

The problem of an elliptical crack embedded in an unbounded transversely isotropic piezoelectric media with the crack-plane parallel to the plane of isotropy of the media and subjected to remote normal mechanical as well as electric loading is considered first. The problem has been successfully reduced to a pair of coupled integral equations that are suitable for the application of an integral equation method developed earlier for three-dimensional problems of LEFM. Solution to the mechanical displacement and electric potentials are obtained for prescribed uniform loadings and expressions for corresponding intensity factors and crack opening displacement are deduced. The above method has further been applied to solve the problem of a rigid flat-ended elliptical punch indenting a transversely isotropic piezoelectric half-space surface with the plane of isotropy parallel to the surface. Solutions to mechanical stress and electric displacement are obtained for prescribed constant normal displacement and constant electric potential interior to the elliptical region and expression for the total force required to maintain a prescribed indentation is deduced.

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References

  1. Chao L.P., Huang J.H.: On a piezoelectric material containing a permeable elliptic crack. Int. J. Solids Struct. 37, 5161–5176 (2000)

    Article  MATH  Google Scholar 

  2. Chen W.Q., Shioya T.: Complete and exact solution of a penny shaped crack in a piezoelectric solid: antisymmetric shear loadings. Int. J. Solids Struct. 37, 2603–2619 (2000)

    Article  MATH  Google Scholar 

  3. Chen W.Q., Shioya T., Ding H.J.: Integral equations for mixed boundary value problem of a piezoelectric half-space and the applications. Mech. Res. Commun. 26, 583–590 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Gradshteyn, I.S., Ryzhik, I.M.: Tables of Integrals, Series and Products. [Corrected and enlarged edition. English Translation: A. Jeffery (Editor)]. Academic Press, Orlando Fl (2000)

  5. Huang J.H.: A fracture criterion of a penny-shaped crack in a transversely isotropic piezoelectric media. Int. J. Solids Struct. 34, 2631–2644 (1997)

    Article  MATH  Google Scholar 

  6. Kogan L., Hui C.-Y., Molkov V.: Stress and induction field of a spheroidal inclusion or a penny-shaped crack in a transversely isotropic piezoelectric material. Int. J. Solids Struct. 33, 2719–2737 (1996)

    Article  MATH  Google Scholar 

  7. Kuna M.: Finite element analyses of cracks in piezoelectric structures: a survey. Arch. Appl. Mech. 76, 725–745 (2006)

    Article  MATH  Google Scholar 

  8. Kuna M.: Fracture mechanics of piezoelectric materials—Where we are right now?. Eng. Frac. Mech. 77, 309–326 (2010)

    Article  Google Scholar 

  9. Li X.-F., Lee K.Y.: Effects of electric field on crack growth for a penny-shaped dielectric crack in a piezoelectric layer. J. Mech. Phy. Solids. 52, 2079–2100 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Li X.Y., Wang M.Z.: On the anisotropic piezoelastic contact problem for an elliptical punch. Acta Mech. 186, 87–98 (2006)

    Article  MATH  Google Scholar 

  11. Mukhopadhyay, M.: Some problems related to elliptic cracks and punches. PhD Thesis, University of Calcutta (1990)

  12. Parton V.Z.: Fracture mechanics of piezoelectric materials. Acta Astro. 3, 671–683 (1976)

    Article  MATH  Google Scholar 

  13. Qin T.Y., Yu Y.S., Noda N.A.: Finite-part integral and boundary element method to solve three-dimensional crack problems in piezoelectric materials. Int. J. Solids Struct. 44, 4770–4783 (2007)

    Article  MATH  Google Scholar 

  14. Roy A., Chatterjee M.: An elliptic crack in an elastic half-space. Int. J. Eng. Sci. 30, 879–890 (1992)

    Article  MATH  Google Scholar 

  15. Roy A., Saha T.K.: Weight function for an elliptic crack in an infinite medium. Part-I. Normal loading. Int. J. Frac. 103, 227–241 (2000)

    Article  Google Scholar 

  16. Shang F., Kuna M., Abendroth M.: Finite element analyses of three-dimensional crack problems in piezoelectric structures. Eng. Frac. Mech. 70, 143–160 (2003)

    Article  Google Scholar 

  17. Sneddon I.N.: The stress intensity factor for a flat elliptical crack in an elastic solid under uniform tension. Int. J. Eng. Sci. 17, 185–191 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  18. Wang B.: Three-dimensional analysis of a flat elliptical crack in a piezoelectric material. Int. J. Eng. Sci. 30, 781–791 (1992)

    Article  MATH  Google Scholar 

  19. Wang Z.K., Huang S.H.: Fields near elliptical crack tip in piezoelectric ceramics. Eng. Frac. Mech. 51, 447–456 (1995)

    Article  Google Scholar 

  20. Wang Z.K., Huang S.H.: Stress intensification near an elliptical border. Theor. Appl. Frac. Mech. 22, 229–237 (1995)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Surendranath College, 24/2 M.G. Road, Kolkata, 700009, India

    T. K. Saha

  2. Department of Mathematics, S.R.S. Vidyamahapitha, Kamarpukur, Hooghly, West Bengal, India

    S. Saha

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  1. T. K. Saha
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  2. S. Saha
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Correspondence to T. K. Saha.

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Saha, T.K., Saha, S. Elliptical crack and punch problems solved by integral equation method for piezoelectric media. Arch Appl Mech 82, 79–95 (2012). https://doi.org/10.1007/s00419-011-0540-1

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