Peridynamics model for flexoelectricity and damage

A flexoelectric peridynamic (PD) theory is proposed. In the PD framework, the formulation introduces a nanoscale flexoelectric coupling that entails non-uniform strain in centrosymmetric dielectrics. This potentially enables PD modeling of a large class of phenomena in solid dielectrics involving cracks, discontinuities etc. wherein large strain gradients are present and the classical electromechanical theory based on partial differential equations do not directly apply. PD electromechanical equations, derived from Hamilton's principle, satisfy the global balance laws. Linear PD constitutive equations reflect the electromechanical coupling effect, with the mechanical force state affected by the polarization state and the electrical force state in turn by the displacement state. An analytical solution to the PD electromechanical equations is presented for the static case when a point mechanical force and a point electric force act in an infinite 3D solid dielectric. A parametric study on how different length scales influence the response is undertaken. In addition, the model is extended to incorporate damage using phase field – an order parameter, supplemented with a PD bond breaking criterion to study flexoelectric effects in damage and fracture problems. To demonstrate the performance of our proposal, we first simulate, considering small flexoelectricity effect and no damage, an externally pressured 2D flexoelectric disk subjected to a potential difference between the inner and outer surfaces and compare the results with existing solutions in the literature. Next, we simulate a plate with a central pre-crack under tension considering damage and flexoelectricity effects, and study the effect of various constitutive parameters on the damage evolution. We also furnish a classical derivation of phase field based flexoelectricity in Appendix I.