Extended displacement discontinuity method for an interface crack in a three-dimensional transversely isotropic piezothermoelastic bi-material. Part 2: Numerical method

The extended displacement discontinuity method is proposed to analyze the nonlinear electric and thermal effects on an interface crack in a three dimensional (3D), transversely isotropic, piezothermoelastic bi-material under combined mechanical-thermo-electrical loadings. The fundamental solutions for uniformly distributed, extended displacement discontinuities applied over a triangular element are obtained by integrating the fundamental solutions for the unit-point, extended displacement discontinuities given by Part 1 over the triangular area. In order to eliminate the oscillatory singularity of the stresses near the crack front, the Delta function in the fundamental solutions is approximated by the Gaussian distribution function, and accordingly, the Heaviside step function is replaced by the Error function. The extended, displacement discontinuity boundary element method with an iterative approach is proposed to determine the value of the fields in the crack interior for opening-crack model. As an example, an elliptical interface crack is studied under different electrical and thermal boundary conditions. The extended stress intensity factors (SIFs) without oscillatory singularities, the forms of the energy release rate (ERR) and local J-integral which can be both expressed in terms of intensity factors are obtained. The numerical method is validated and the influence of combined loadings and material-mismatch on the results is studied. The effects of different boundary conditions and ellipticity ratio on the results are also investigated.