Gradient piezoelectricity for cracks under an impact load

The flexoelectric effect on elastic waves is investigated in nano-sized cracked structures. The strain gradients are considered in the constitutive equations of a piezoelectric solid for electric displacements and the higher-order stress tensor. The governing equations with the corresponding boundary conditions are derived from the variational principle. The finite element method (FEM) is developed from the principle of virtual work. It is equivalent to the weak-form of derived governing equations in gradient elasticity. The computational method can be applied to analyze general 2D boundary value problems in size-dependent piezoelectric elastic solids with cracks under a dynamic load. The FEM formulation is implemented for strain-gradient piezoelectricity under a dynamic load.