Finite element methods for piezoelectricity and flexoelectricity with fracture mechanics applications

Abstract

Flexoelectricity is the coupling of polarization to strain gradients. This effect is size-dependent and is most prominent in nanoscale structures in which gradients are large. In our previous work, we have established a theoretical framework to study flexoelectricity and obtained analytic solutions for various problems. However, a rigorous and complete computational framework that accounts for the gradient effects in electromechanical problems is still absent. The challenges are that gradients bring in additional length scales and raise the order of the governing differential equations. In the present report, we overcome this difficulty by introducing a “mixed” formulation. In traditional finite element theory, displacement is the primary variable. Here, we treat displacement and displacement gradients as independent variables. This helps reduce the required smoothness of the displacement field and makes the formulation simple to implement numerically. Similarly, the electric potential and polarization are treated as two independent variables. This maintains the symmetry of the stiffness matrix. Based on this idea, we work out a weak formulation for flexoelectric solids. In accordance with the formulation, we develop a new plane-strain element, which is a 9-node quadrilateral element with 87 degrees of freedom. The proposed technique and element clears the patch tests and gives excellent agreement to benchmark problems with known analytic solutions. We study the size-effect and the flexoelectric reduction of the stress intensity factor. The shape effect could lead to ways of making piezoelectric structures from cen-tro-symmetric materials and even isotropic materials. © 2017 ICF 2017 - 14th International Conference on Fracture. All rights reserved.