The M-integral for calculating intensity factors of an impermeable crack in a piezoelectric material

Leslie Banks-Sills*, Yael Motola, Lucy Shemesh

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


In this study, a conservative integral is derived for calculating the intensity factors associated with piezoelectric material for an impermeable crack. This is an extension of the M-integral or interaction energy integral for mode separation in mechanical problems. In addition, the method of displacement extrapolation is extended for this application as a check on results obtained with the conservative integral. Poling is assumed parallel, perpendicular and at an arbitrary angle with respect to the crack plane, as well as parallel to the crack front. In the latter case, a three-dimensional treatment is required for the conservative integral which is beyond the scope of this investigation. The asymptotic fields are obtained; these include stress, electric, displacement and electric flux density fields which are used as auxiliary solutions for the M-integral. Several benchmark problems are examined to demonstrate the accuracy of the methods. Numerical difficulties encountered resulting from multiplication of large and small numbers were solved by normalizing the variables. Since an analytical solution exists, a finite length crack in an infinite body is also considered. Finally, a four point bend specimen subjected to both an applied load and an electric field is presented for a crack parallel, perpendicular and at an angle to the poling direction. It is seen that neglecting the piezoelectric effect in calculating stress intensity factors may lead to errors.

Original languageEnglish
Pages (from-to)901-925
Number of pages25
JournalEngineering Fracture Mechanics
Issue number5
StatePublished - Mar 2008


  • Finite element method
  • Interaction energy integral
  • M-integral
  • Piezoelectric material
  • Stress and electric flux density intensity factors


Dive into the research topics of 'The M-integral for calculating intensity factors of an impermeable crack in a piezoelectric material'. Together they form a unique fingerprint.

Cite this