Flexoelectric effect for cracks in piezoelectric solids

The finite element method (FEM) is developed to analyse 2-D crack problems in piezoelectric solids that exhibit the size-effect. This phenomenon, which typically occurs in micro/nano electronic structures, is described by the strain- and electric field-gradients in the constitutive equations. The variational principle with the appropriate boundary conditions is employed to derive the governing equations, from which the FEM formulation with C1-continuous elements is developed and implemented. An example is presented to demonstrate the effects of the strain- and electric intensity-gradients on the electro-mechanical behavior of the cracked solid