Yoffe-type moving crack in a functionally graded piezoelectric material

In this paper, the problem of a functionally graded piezoelectric material (FGPM) with a constant-velocity Yoffe-type moving crack is considered. The strip is assumed to be under an anti-plane mechanical loading and an in-plane electric loading and its material properties, such as the elastic stiffness, piezoelectric constant, dielectric permittivity and mass density, are assumed to vary continuously along the thickness of the strip. By using the Fourier transform, the problem is first reduced to two pairs of dual integral equations and then into Fredholm integral equations of the second kind. The closed forms of the singular stress, electric field and electric displacement are obtained from the asymptotic expansion of the stresses and electric fields around the crack tip. Different from the case of a stationary crack in an FGPM, it is found that at the tip of the crack the electric field also exhibits the singularity of the inverse square root, along with the stress and electric displacement singularities. It is also observed that increasing the gradient of the material properties can reduce the magnitudes of the stress and electric displacement intensity factors but it has little effect on the electric field intensity factor. When the crack moving velocity increases, both the stress and the electric displacement intensity factors decrease but the electric field intensity factor increases.