An adaptive phase-field model for flexoelectric materials based on THB-splines

This work presents a thermodynamically consistent model for flexoelectric fracture analysis utilizing truncated hierarchical B-splines (THB-splines) and an adaptive phase-field method. The model integrates elastic, gradient elastic, piezoelectric and flexoelectric effects, solving the problem as a coupled system, while crack propagation is addressed within a phase-field formulation. By developing a novel adaptive THB discretization strategy, the requirements of both C 1 continuity and local mesh refinement are automatically fulfilled. To overcome the challenge of reproducing the flexoelectric crack-tip behavior, three different degradation functions are assigned to the elastic, gradient elastic and dielectric energies. The model is validated through representative numerical examples, providing insight into the interaction mechanisms between flexoelectricity and fracture. Fracture gives rise to a giant flexoelectric tip effect while simultaneously restricting the peak electrical response. In turn, flexoelectricity not only strengthens the fracture resistance of materials but also alters crack paths through the application of an electric field. The study shows that the presence of cracks substantially enhances flexoelectric sensitivity as well as energy conversion efficiency, providing theoretical support for the development of novel sensing and energy harvesting devices.