The collocation mixed FEM is developed for 3-d crack analyses in solids with the direct flexoelectricity. In the direct flexoelectricity the polarization vector is proportional to the strain gradients. In contrast to piezoelectric polarization, this phenomenon is occurring in all dielectric materials. Since the large strain gradients arise at the crack tip vicinity, the large flexoelectric effect is localized around the crack tip. A standard domain discretization method with the C 0 continuous approximations is insufficient for solution of boundary value problems, since strain gradients are occurring in constitutive equations. To deal with higher order derivatives in governing equations of the gradient theory, it is developed the collocation mixed FEM (CMFEM). In CMFEM the C 0 continuous approximations are applied independently to displacements and strains. The constraint between the independent mechanical strains and displacement is satisfied by a collocation method at considered collocation points on each finite element. The J-integral expression for 3D crack problems in strain-gradient piezoelectricity is derived in this paper. It is shown that 3D J-integral expression is supplied with some additional surface integrals in comparison with 2D case.
Tian et al. (Thu,) studied this question.