The crack tip fields for anti-plane crack in functionally graded piezoelectric materials with arbitrarily gradient direction

The higher order crack tip fields for anti-plane crack in functionally graded piezoelectric materials(FGPMs) under mechanical and electrical loadings are investigated. To make the analysis tractable as usual, the elastic stiffness, piezoelectric parameter, and dielectric permittivity of FGPMs are assumed to be exponential functions with arbitrary gradient direction. The crack surfaces are assumed to be electrically impermeable. Similar to the Williams´ solution of homogeneous elastic materials, the high order crack tip stress and electric displacement fields are obtained by the eigen-expansion method.