Robust mixed FE for analyses of higher-order electromechanical coupling in piezoelectric solids

Abstract

Direct flexoelectricity is a size-dependent phenomenon, very prominent at smaller scales, that connects the strain gradients and the electric field. The very existence of strain gradients enhances noncentrosymmetry and heightens the interaction between piezoelectricity and flexoelectricity, demanding fully coupled higher-order electromechanical formulations. The numerical instability of the existing finite elements used to model flexoelectricity alone is revealed due to their reliance on the stabilization parameter. Thus, two new finite elements () and () are proposed for mixed FEM that are numerically robust without any need of such stabilization parameters. Additionally, the existing finite element [ in (Deng et al. in J Appl Mech 84:081004, 2017)], is implemented from scratch to replicate known results and benchmark the performance of newly proposed finite elements. To verify the robustness of these elements, various benchmark problems for flexoelectricity in dielectric solids, such as a thick cylinder and truncated pyramid are simulated. The great agreement of the numerical results with the existing ones reflects the foundational nature of the proposed elements. Furthermore, the proposed mixed finite elements were used to successfully analyze cantilever beam and truncated pyramid problems where piezoelectric effects were taken into account for the first time. Current results are intrumental in simulating flexoelectricity and piezoelectricity together to highlight their interactions using newly proposed numerically robust finite elements.

A novel 3D mixed finite element for flexoelectricity in piezoelectric materials

“…The three-dimensional 27-noded mixed FE used here is a 3D extension of QL1-16. 46 The DOFs are u 1 , u The schematic representation of a mixed finite element is shown in Figure 1. A thorough code verification process was conducted for QL1-16, 46 ensuring the stability and accuracy of the finite element.…”

“…46 The DOFs are u 1 , u The schematic representation of a mixed finite element is shown in Figure 1. A thorough code verification process was conducted for QL1-16, 46 ensuring the stability and accuracy of the finite element. Primarily, the focus was on the LBB (Ladyzhenskaya-Babuška-Brezzi) condition, addressing saddle point problems.…”

Evaluating non-intrinsic contribution in flexoelectric measurements

“…properties but also originate from the non-intrinsic flexoelectric contributions [13,14]. Surface piezoelectricity, one of the non-intrinsic flexoelectric contributions, may contribute to the flexoelectric response to a comparable extent to bulk flexoelectricity [15,16]. Additionally, the contribution of the surface-terminated layer to flexoelectricity can be highlighted based on first-principles [17].…”