Phase field fracture modelling of flexible piezoelectric materials considering different electrical boundary conditions

Flexible piezoelectric materials have gained considerable attention due to their remarkable properties, including electromechanical coupling and high stretchability. These characteristics make them valuable in the realm of flexible electronic devices. However, the issue of fracture in these materials cannot be ignored. In general, these flexible/stretchable materials experience fractures when subjected to significant deformation, unlike brittle piezoelectric materials with low failure strain which have been extensively studied. There is a pressing need to investigate the fracture behavior of flexible piezoelectrics under finite deformation conditions. Within the framework of the phase field method, this work addresses the fracture of flexible piezoelectrics utilizing a nonlinear electromechanical material model. To investigate the influence of electrical boundary conditions on fracture behavior, a function related to the electric permittivity ratio and phase field variable is employed to degrade the electric energy density. By adjusting the electric permittivity ratio, the analysis encompasses the fracture behavior of flexible piezoelectric materials under the assumptions of electrically impermeable, semi-permeable, and permeable conditions, respectively. In order to solve the coupled governing equations, a residual controlled staggered algorithm (RCSA) is employed in the user element subroutine of commercial software ABAQUS. The simulation results indicate that fracture behavior in flexible piezoelectric materials is influenced by several factors, including material parameters, geometry, polarization direction, and the external electric field. Notably, when the poling direction is perpendicular to the electric field direction, variations in the external electric field have a minimal impact on fracture behavior. In contrast, when the poling direction is parallel to the electric field direction, the influence on fracture behavior is pronounced. These findings provide valuable insights for developing strategies to enhance the fracture resistance and durability of flexible piezoelectric materials in practical applications.