A constitutive and numerical model for P-formulation of flexoelectricity with type II strain gradient elasticity

As the size of sensors, actuators, and electronic components is driven to the micro- and nanoscale ranges, flexoelectricity, a higher order electromechanical coupling effect, becomes more prominent. Conventional finite element framework requires elements of at least $C^1$-continuity to resolve strain gradient. This requirement can be circumvented by using mixed finite element method, which considers strain components as independent degrees of freedom. Most previous studies have employed the $E$-formulation because it is relatively straightforward to implement. However, developing a $P$-formulation is necessary, as it enables phase-field modeling of ferroelectric domain evolution (with electric polarization $P$ serving as an order parameter) that is additionally influenced by strain gradients. Current research based on type II strain gradient elasticity presents a mixed finite element formulation capable of capturing the flexoelectric coupling effect using a kinematic constraint between relaxed strain and mechanical strain. The proposed mixed finite element is verified with a patch test and a thick cylinder problem, whose results are extensively discussed.