Skip to main content
SpringerLink
Log in

Analysis of a Griffith crack at the interface of two piezoelectric materials under anti-plane loading

  • Original Article
  • Published:
Continuum Mechanics and Thermodynamics Aims and scope Submit manuscript

Abstract

The main objective of this work is the contribution to the study of the piezoelectric structures which contain preexisting defect (crack). For that, we consider a Griffith crack located at the interface of two piezoelectric materials in a semi-infinite plane structure. The structure is subjected to an anti-plane shearing combined with an in-plane electric displacement. Using integral Fourier transforms, the equations of piezoelectricity are converted analytically to a system of singular integral equations. The singular integral equations are further reduced to a system of algebraic equations and solved numerically by using Chebyshev polynomials. The stress intensity factor and the electric displacement intensity factor are calculated and used for the determination of the energy release rate which will be taken as fracture criterion. At the end, numerical results are presented for various parameters of the problem; they are also presented for an infinite plane structure.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions. National Bureau of Standards. Appl. Math. Ser. 55, 773 (1964)

  2. Beom, H.G., Atluri, S.N.: Near-tip and intensity factors for interfacial cracks in dissimilar anisotropicpiezoelectric media. Int. J. Fract. 75, 163–183 (1996)

    Article  Google Scholar 

  3. Chen, H., Wei, W., Liu, J., Fang, D.: Propagation of a mode-III interfacial crack in a piezoelectric-piezomagnetic bi-material. Int. J. Solids Struct. 49, 2547–2558 (2012)

    Article  Google Scholar 

  4. Chen, S., Gao, H.: Dynamic behaviors of mode III interfacial crack under a constant loading rate. Contin. Mech. Thermodyn. 22, 515–530 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Chiang, C.R.: Mode-III crack problems in a cubic piezoelectric medium. Acta. Mech. 224, 2203–2217 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Deng, W., Meguid, S.A.: Analysis of conducting rigid inclusion at the interface of two dissimilar piezoelectric materials. ASME. J. Appl. Mech. 65, 76–84 (1998)

    Article  ADS  Google Scholar 

  7. Erdogan, F., Gupta, G.D., Cook, T.: Numerical solution of singular integral equations. In: Sih, G.C. (ed.) Methods of Analysis and Solutions of Crack Problems, pp. 368–425. Noordhooff. Int. Publ, Leyden (1973)

    Chapter  Google Scholar 

  8. Ferdjani, H., Abdelmoula, R., Marigo, J.J., El Borgi, S.: Study of size effects in the Dugdale model through the case of a crack in a semi-infinite plane under anti-plane shear loading. Contin. Mech. Thermodyn. 21, 41–55 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Gao, H., Zhang, T.Y., Tong, P.: Local and global energy release rates for an electrically yielded crack in a piezoelectric ceramic. J. Mech. Phys. Sol. 45, 491–510 (1997)

    Article  ADS  Google Scholar 

  10. Gao, C.F., Wang, M.Z.: Collinear permeable cracks between dissimilar piezoelectric materials. Int. J. Solids Struct. 37, 4969–4986 (2000)

    Article  MATH  Google Scholar 

  11. Gao, C.F., Zhao, M., Tong, P., Zhang, T.Y.: The energy release rate and the J integral of an electrically insulated crack in piezoelectric material. Int. J. Eng. Sci. 42, 2175–92 (2004)

    Article  Google Scholar 

  12. Guimard, J.-M., Allix, O., Pechnik, N., Thévenet, P.: Characterization and modeling of rate effects in the dynamic propagation of mode-II delamination in composite laminates. Int. J. Fract. 160, 55–71 (2009)

    Article  Google Scholar 

  13. Häusler, C., Jelitto, H., Neumeister, P., Balke, H., Schneider, G.A.: Interfacial fracture of piezoelectric multilayer actuators under mechanical and electrical loading. Int. J. Fract. 160, 43–54 (2009)

    Article  Google Scholar 

  14. Hu, K.Q., Kang, Y.L., Li, G.Q.: Moving crack at the interface between two dissimilar magnetoelectroelastic materials. Acta. Mech. 182, 1–16 (2006)

    Article  MATH  Google Scholar 

  15. Keqiang, Hu, Zengtao, Chen, Jiawei, Fu: Moving Dugdale crack along the interface of two dissimilar magnetoelectroelastic materials. Acta. Mech 226, 2065–2076 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kuo, C.M., Barnett, D.M.: Stress singularities of interfacial cracks in bounded piezoelectric half-spaces. In: Wu, J.J., Ting, T.C.T., Barnett, D.M. (eds). Modern Theory of Anisotropic Elasticity and Applications, SIAM Proceedings Series, Philadelphia. pp. 33–50. (1991)

  17. Li, X.F.: Transient response of a piezoelectric material with a semi-infinite mode-III crack under impact loads. Int. J. Fract. 111, 119–130 (2001)

    Article  Google Scholar 

  18. Li, X.: Electroelastic analysis of an internal interface crack in a half-plane consisting of two bonded dissimilar piezoelectric quarter-planes. Meccanica 38, 309–323 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. Li, X., Lee, K.Y.: Fracture analysis of cracked piezoelectric materials. Int. J. Solids Struct. 41, 4137–4161 (2004)

    Article  MATH  Google Scholar 

  20. Li, Y.D., Leeb, K.Y.: Anti-plane fracture analysis for the weak-discontinuous interface in a non-homogeneous piezoelectric bi-material structure. Euro. J. Mech. A/Solids. 28, 241–247 (2009)

    Article  ADS  Google Scholar 

  21. Loboda, V., Lapusta, Y., Govorukha, V.: Mechanical and electrical yielding for an electrically insulated crack in an interlayer between piezoelectric materials. Int. J. Eng. Sci. 46, 260–272 (2008)

    Article  MATH  Google Scholar 

  22. Nishioka, T., Shen, S.: Higher order asymptotic solution for an interfacial crack in piezoelectric bi-material under impact. Mater. Sci. Res. Int. 7, 157–165 (2001)

    Google Scholar 

  23. Ou, Z.C., Chen, Y.H.: Interface crack problem in elastic dielectric/piezoelectric biomaterials. Int. J. Fract. 130, 427–454 (2004)

    Article  Google Scholar 

  24. Pak, Y.E., Hermann, G.: Conservations laws and the material momentum tensor for the electric dielectrics. Int. J. Eng. Sci. 24, 1365–1374 (1986)

    Article  Google Scholar 

  25. Pak, Y.E.: Crack extension force in a piezoelectric material. J. Appl. Mech. 57, 647–653 (1990)

    Article  ADS  MATH  Google Scholar 

  26. Park, S.B., Sun, C.T.: Fracture criteria for piezoelectric ceramics. J. Am. Ceram Soc. 78, 1475–1480 (1995)

    Article  Google Scholar 

  27. Qin, Q.H., Mai, Y.W.: Crack branch in piezoelectric biomaterial system. Int. J. Eng. Sci. 38, 673–693 (2000)

    Article  Google Scholar 

  28. Qin, Q.H., Yu, S.W.: An arbitrarily-oriented plane crack terminating at interface between dissimilar piezoelectric materials. Int. J. Solids Struct. 34, 581–590 (1997)

    Article  MATH  Google Scholar 

  29. Ricoeur, A., Kuna, M.: Influence of electric fields on the fracture of ferroelectric ceramics. J. Eur. Ceram Soc. 23, 1313–1328 (2003)

    Article  Google Scholar 

  30. Ryvkin, M.: Steady-state mode III delamination crack in a periodically layered medium. Contin. Mech. Thermodyn. 22, 635–646 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Scherzer, M., Kuna, M.: Combined analytical and numerical solution of 2D interface corner configurations between dissimilar piezoelectric materials. Int. J. Fract. 127, 61–99 (2004)

    Article  MATH  Google Scholar 

  32. Shen, S., Nishioka, T.: Fracture of piezoelectric material: energy density criterion. Theor. Appl. Fract. Mech. 33, 57–65 (2000)

    Article  Google Scholar 

  33. Shin, J.W., Lee, Y.S.: Anti-plane moving crack in a functionally graded piezoelectric layer between two dissimilar piezoelectric strips. J. Mech. Sci. Tech. 26(4), 1017–1025 (2012)

    Article  Google Scholar 

  34. Singh, B.M., Rokne, J.R., Dhaliwal, R.S., Vrbik, J.: Scattering of anti-plane shear waves by an interface crack between two bonded dissimilar functionally graded piezoelectric materials. Proc. R. Soc. A 465, 1249–1269 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. Soh, A.K., Fang, D.-N., Lee, K.L.: Fracture analysis of piezoelectric materials with defect using Energy density theory. Int. J. Solids. Struct. 38, 8331–8344 (2001)

    Article  MATH  Google Scholar 

  36. Song, H.P., Gao, C.F.: Interaction of a piezoelectric screw dislocation with a blunt mode III crack in a piezoelectric material. Eng. Fract. Mech. 96, 687–700 (2012)

    Article  Google Scholar 

  37. Sosa, H.A., Pak, Y.E.: Three-dimensional eigenfunction analysis of a crack in a piezoelectric material. Int. J. Solids Struct. 26, 1–15 (1990)

    Article  MATH  Google Scholar 

  38. Suo, Z., Kuo, C.M., Barnett, D.M., Willis, J.R.: Fracture mechanics for piezoelectric ceramics. J. Mech. Phys. Sol. 40, 739–765 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  39. Tian, W.Y., Rajapakse, R.K.N.D.: Fracture parameters of a penny-shaped crack at the interface of a piezoelectric bi-material system. Int. J. Fract. 141, 37–48 (2006)

    Article  MATH  Google Scholar 

  40. To, A.C., Li, S., Glaser, S.D.: Propagation of a mode-III interfacial conductive crack along a conductive interface between two piezoelectric materials. Wave Motion 43, 368–386 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  41. Wang, T.C., Han, X.L.: Fracture mechanics of piezoelectric materials. Int. J. Fract. 98, 15–35 (1999)

    Article  Google Scholar 

  42. Zhou, Z.G., Du, S.Y., WANG, B.: Investigation of anti-plane shear behavior of a Griffith crack in a piezoelectric material by using the non-local theory. Int. J. Fract. 111, 105–117 (2001)

    Article  Google Scholar 

  43. Zuo, J.Z., Sih, G.C.: Energy density theory formulation and interpretation of cracking behavior for piezoelectric ceramics. Theor. Appl. Fract. Mech. 34, 17–33 (2000)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Structural Mechanics Research Laboratory, Mechanical Engineering Department, Blida University, Blida, Algeria

    M. Gherrous & H. Ferdjani

Authors
  1. M. Gherrous
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H. Ferdjani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Gherrous.

Additional information

Communicated by Andreas Öchsner.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gherrous, M., Ferdjani, H. Analysis of a Griffith crack at the interface of two piezoelectric materials under anti-plane loading. Continuum Mech. Thermodyn. 28, 1683–1704 (2016). https://doi.org/10.1007/s00161-016-0501-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00161-016-0501-6

Keywords

Search

Navigation