Archive of Applied Mechanics

, Volume 76, Issue 11–12, pp 725–745 | Cite as

Finite element analyses of cracks in piezoelectric structures: a survey



Piezoelectric materials have widespread applications in modern technical areas such as mechatronics, smart structures or microsystem technology, where they serve as sensors or actuators. For the assessment of strength and reliability of piezoelectric structures under combined electrical and mechanical loading, the existence of cracklike defects plays an important role. Meanwhile, piezoelectric fracture mechanics has been established quite well, but its application to realistic crack configurations and loading situations in piezoelectric structures requires the use of numerical techniques as finite element methods (FEM) or boundary element methods (BEM). The aim of this paper is to review the state of the art of FEM to compute the coupled electromechanical boundary value problem of cracks in 2D and 3D piezoelectric structures under static and dynamic loading. In order to calculate the relevant fracture parameters very precisely and efficiently, the numerical treatment must account for the singularity of the mechanical and electrical fields at crack tips. The following specialized techniques are presented in detail (1) special singular crack tip elements, (2) determination of intensity factors K I K IV from near tip fields, (3) modified crack closure integral, (4) computation of the electromechanical J-integral, and (5) exploitation of interaction integrals. Special emphasis is devoted to a realistic modeling of the dielectric medium inside the crack, leading to specific electric crack face boundary conditions. The accuracy, efficiency, and applicability of these techniques are examined by various example problems and discussed with respect to their advantages and drawbacks for practical applications.


Piezoelectric fracture mechanics Finite element method Crack analyses Limited permeable cracks 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Gabbert, U., Tzou, H.: Smart structures and structonic systems. In: Proceedings of IUTAM-Symposium, Magdeburg, 2000. Kluwer, Dordrecht (2001)Google Scholar
  2. 2.
    Santini P., Marchetti M., Brebbia C.A. (1999): Computational methods for smart structures and materials. WIT, SouthamptonGoogle Scholar
  3. 3.
    Qin Q.-H. (2001): Fracture mechanics of piezoelectric materials. WIT, Southampton, BostonGoogle Scholar
  4. 4.
    Zhang T.-Y., Zhao M., Tong P. (2002): Fracture of piezoelectric ceramics. Adv. Appl. Mech. 38, 147–289CrossRefGoogle Scholar
  5. 5.
    Chen Y.H., Hashebe N. (2005): Current understanding on fracture behaviors of ferroelectric/piezoelectric materials. J. Intell. Mater. Syst. Struct. 16, 673–687CrossRefGoogle Scholar
  6. 6.
    Sosa H. (1991): Plane problems in piezoelectric media with defects. Int. J. Solids Struct. 28, 491–505zbMATHCrossRefGoogle Scholar
  7. 7.
    Pak Y. (1992): Linear electro-elastic fracture mechanics of piezoelectric materials. Int. J. Fract. 54, 79–100Google Scholar
  8. 8.
    Park S.B., Sun C.T. (1995): Effect of electric field on fracture of piezoelectric ceramics. Int. J. Fract. 70, 203–216CrossRefGoogle Scholar
  9. 9.
    Karapetian E., Kachanov M., Sevostianov I. (2002): The principle of correspondence between elastic and piezoelectric problems. Arch. Appl. Mech. 72(8): 564–587zbMATHCrossRefGoogle Scholar
  10. 10.
    Atluri S. (1986): Computational methods in the mechanics of fracture. North Holland, AmsterdamzbMATHGoogle Scholar
  11. 11.
    Anderson T. (1995): Fracture Mechanics. CRC, Boca Raton, FLzbMATHGoogle Scholar
  12. 12.
    Aliabadi M., Rooke D. (1991): Numerical Fracture Mechanics. Kluwer, DordrechtzbMATHGoogle Scholar
  13. 13.
    Kuna, M.: Entwicklung und Anwendung effizienter numerischer Verfahren zur bruchmechanischen Beanspruchungsanalyse am Beispiel hybrider finiter Elemente. Habilitation, Martin Luther University Halle (1990)Google Scholar
  14. 14.
    Allik H., Hughes T.J.R. (1970): Finite element method for piezoelectric vibration. Int. J. Numer. Methods Eng. 2, 151–157CrossRefGoogle Scholar
  15. 15.
    Görnandt A., Gabbert U. (2002): Finite element analysis of thermopiezoelectric smart structures. Acta Mech. 154, 129–140zbMATHCrossRefGoogle Scholar
  16. 16.
    Wriggers P. (2001): Nichtlineare Finite-Elemente-Methoden. Springer, Berlin Heidelberg New YorkGoogle Scholar
  17. 17.
    Taylor L.M., Flanagan D.P. (1989): Pronto-3D a three-dimensional transient solid dynamics program. SAND87-1912, Sanda National Laboraties, AlbuquerqueGoogle Scholar
  18. 18.
    Enderlein M., Ricoeur A., Kuna M. (2005): Finite element techniques for dynamic crack analysis in piezoelectrics. Int. J. Fract. 134, 191–208CrossRefGoogle Scholar
  19. 19.
    Wu C.C., Sze K.Y., Huang Y.Q. (2001): Numerical solutions on fracture of piezoelectric materials by hybrid element. Int. J. Solids Struct. 38, 4315–4329zbMATHCrossRefGoogle Scholar
  20. 20.
    Wu D., Wu C.C. (2006): Numerical analysis for piezoelectric crack under varied boundary conditions by optimized hybrid element method. Eng. Fract. Mech. 73, 649–670CrossRefGoogle Scholar
  21. 21.
    Wu, C.C., Liu, J.H., Yagawa, G.: Finite element dual analysis for piezoelectric crack. In: Computational Mechanics. WCCM VI in Conjunction with APCOM’04, pp. 746–751. Tsinghua Press, Beijing, & Springer, Berlin Heidelberg New York (2004)Google Scholar
  22. 22.
    Ricoeur A., Kuna M. (2003): Influence of electric fields on the fracture of ferroelectric ceramics. J. Eur. Ceram. Soc. 23(8): 1313–1328CrossRefGoogle Scholar
  23. 23.
    Sosa H.A., Pak Y.E. (1990): Three-dimensional eigenfunction analysis of a crack in a piezoelectric material. Int. J. Solids Struct. 26(1): 1–15zbMATHCrossRefGoogle Scholar
  24. 24.
    Eshelby J. (1970): Energy relations and the energy momentum tensor in continuum mechanics. In: Kanninen M.F., et al. (eds). Inelastic Behavior of Solids. McGraw-Hill, New York, pp. 77–114Google Scholar
  25. 25.
    Rice J. (1968): A path independent integral and the approximate analysis of strain concentration by notches and cracks. J. Appl. Mech. 35, 379–386Google Scholar
  26. 26.
    Cherepanov G. (1967): Rasprostranenie trechin v sploshnoi srede (About Crack Advance in the Continuum). Prikladnaja Matematika i Mekhanika 31, 478–488Google Scholar
  27. 27.
    Cherepanov G. (1977): Invariant G-integrals and some of their applications in mechanics. J. Appl. Math. Mech. (Translation of Prikladnaja Matematika i Mekhanika-PMM) 41(3): 397–410MathSciNetCrossRefGoogle Scholar
  28. 28.
    Pak Y.E., Herrmann G. (1986): Conservation laws and the material momentum tensor for the elastic dielectric. Int. J. Eng. Sci. 24, 1365–1374zbMATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Pak Y.E. (1990): Crack extension force in a piezoelectric material. J. Appl. Mech. 57, 647–653zbMATHGoogle Scholar
  30. 30.
    Kuna M. (1995): Energiebilanzintegrale für Risse in piezoelektrischen Werkstoffen unter elektrischen und mechanischen Beanspruchungen. Tech. Mech. 15, 195–204Google Scholar
  31. 31.
    McMeeking R.M. (1989): Electrostrictive stresses near crack–like flaws. J. Appl. Math. Phys. (ZAMP) 40, 615–627zbMATHCrossRefGoogle Scholar
  32. 32.
    Zhang T.Y., Gao C.F. (2004): Fracture behaviors of piezoelectric materials. Theor. Appl. Fract. Mech. 41, 339–379MathSciNetCrossRefGoogle Scholar
  33. 33.
    Barsoum R.S. (1976): On the use of isoparametric finite elements in linear fracture mechanics. J. Numer. Methods Eng. 10, 25–37zbMATHCrossRefGoogle Scholar
  34. 34.
    Kumar S., Singh R.N. (1996): Crack propagation in piezoelectric materials under combined mechanical and electrical loadings. Acta Mater. 44(1), 173–200zbMATHGoogle Scholar
  35. 35.
    Kuna, M.: FEM-Techniken zur Analyse von Rissen unter elektrischen und mechanischen Beanspruchungen, Berichte 29. Tagung DVM Arbeitskreis Bruchvorgänge, pp.~369–379 (1997)Google Scholar
  36. 36.
    Kuna M. (1998): Finite element analyses of crack problems in piezoelectric structures. Comput. Mater. Sci. 13, 67–80CrossRefGoogle Scholar
  37. 37.
    Kuna, M., Ricoeur, A.: Theoretical investigations on the cracking of ferroelectric ceramics. In: Smart Structures and Materials 2000: Active Materials Behavior and Mechanics, Proceedings of SPIE, Vol. 3992, pp. 185–196 (2000)Google Scholar
  38. 38.
    Kemmer, G.: Berechnung von elektromechanischen Intensitätsparametern bei Rissen in Piezokeramiken, VDI, Düsseldorf, Reihe 18, Nr. 261 (2000)Google Scholar
  39. 39.
    Buchholz, F.G.: Improved formulae for the finite element calculation of the strain energy release rate by the modified crack closure integral method. In: Robinson, J. (ed.) Accuracy, Reliability and Training in FEM Technology. Robinson and Associates, Dorset, pp.~650–659 (1984)Google Scholar
  40. 40.
    Park S., Sun C.-T. (1995): Fracture criteria for piezoelectric ceramics. J. Am. Ceram. Soc. 78(6): 1475–1480CrossRefGoogle Scholar
  41. 41.
    Shang F., Kuna M., Abendroth M. (2003): Finite element analyses of three-dimensional crack problems in piezoelectric structures. Eng. Fract. Mech. 70(2): 143–160CrossRefGoogle Scholar
  42. 42.
    Shih C.F., Moran B., Nakamura T. (1986): Energy release rate along a three-dimensional crack front in a thermally stressed body. Int. J. Fract. 30, 79–102Google Scholar
  43. 43.
    Abendroth M., Groh U., Kuna M., Ricoeur A. (2002): Finite element-computation of the electromechanical J-integral for 2-D and 3-D crack analysis. Int. J. Fract. 114, 359–378CrossRefGoogle Scholar
  44. 44.
    Stern M., Becker E.B., Dunham R.S. (1976): Contour integral computation of mixed-mode stress intensity factors. Int. J. Fract. 12(3): 359–368Google Scholar
  45. 45.
    Yau J., Wang S., Corton H. (1980): A mixed-mode crack analysis of isotropic solids using conservation laws of elasticity. J. Appl. Mech. 47, 335–341zbMATHCrossRefGoogle Scholar
  46. 46.
    Hao T., Shen Z. (1994): A new electric boundary condition of electric fracture mechanics and its applications. Eng. Fract. Mech. 47, 793–802CrossRefGoogle Scholar
  47. 47.
    Balke, H., Kemmer, G., Drescher, J.: Some remarks on fracture mechanics of piezoelectric solids. In: Proceedings of Micro Materials 1997, Berliner Congress Center (BCC), 16–18 April 1997, pp. 398–401. Deutscher Verband für Materialforschung und -Prüfung e.V., Berlin (1997)Google Scholar
  48. 48.
    McMeeking R. (1999): Crack tip energy release rate for a piezoelectric compact tension specimen. Eng. Fract. Mech. 64, 217–244CrossRefGoogle Scholar
  49. 49.
    Gruebner, O.: FE-Analyse von Rissspitzenfeldern in Piezokeramiken unter Berücksichtigung der elektrischen Permeabilität der Risse und des nichtlinearen Materialverhaltens. Ph.D. thesis, Universität Karlsruhe (2001)Google Scholar
  50. 50.
    Wippler K., Ricoeur A., Kuna M. (2004): Towards the computation of electrically permeable cracks in piezoelectrics. Eng. Fract. Mech. 71, 2567–2587CrossRefGoogle Scholar
  51. 51.
    Wang Z.K., Huang S.H. (1995): Stress intensification near an elliptical border. Theor. Appl. Fract. Mech. 22, 229–237MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Institute of Mechanics and Fluid DynamicsTechnische Universität Bergakademie FreibergFreibergGermany

Personalised recommendations