On a family of numerical models for couple stress based flexoelectricity for continua and beams

摘要

A family of numerical models for the phenomenological linear flexoelectric theory for continua and their particularisation to the case of three-dimensional beams based on a skew symmetric couple stress theory is presented. In contrast to the standard strain gradient flexoelectric models which assume coupling between electric polarisation and strain gradients, we postulate an electric enthalpy in terms of linear invariants of curvature and electric field. This is achieved by introducing the axial (mean) curvature vector as a strain gradient measure. The physical implication of this assumption is many-fold. Firstly, analogous to the standard strain gradient models, for isotropic (non-piezoelectric) materials it allows constructing flexoelectric energies without breaking material's centrosymmetry. Secondly, unlike the standard strain gradient models, nonuniform distribution of volumetric part of strains (volumetric strain gradients) do not generate electric polarisation, as also confirmed by experimental evidence to be the case for some important classes of flexoelectric materials. Thirdly, a state of plane strain generates out of plane deformation through strain gradient effects. Finally, under this theory, extension and shear coupling modes cannot be characterised individually as they contribute to the generation of electric polarisation as a whole. As a first step, a detailed comparison of the developed couple stress based flexoelectric model with the standard strain gradient flexoelectric models is performed for the case of Barium Titanate where a myriad of simple analytical solutions are assumed in order to quantitatively describe the similarities and dissimilarities in effective electromechanical coupling under these two theories. From a physical point of view, the most notable insight gained is that, if the same experimental flexoelectric constants are fitted in to both theories, the presented theory in general, reports up to 200% stronger electromechanical conversion efficiency. From the formulation point of a view, the presented flexoelectric model is also competitively simpler as it eliminates the need for high order strain gradient and coupling tensors and can be characterised by a single flexoelectric coefficient. In addition, three distinct mixed flexoelectric variational principles are presented for both continuum and beam models that facilitate incorporation of strain gradient measures in to a standard finite element scheme while maintaining the Co continuity. Consequently, a series of low and high order mixed finite element schemes for couple stress based flexoelectricity are presented and thoroughly benchmarked against available closed form solutions in regards to electromechanical coupling efficiency. Finally, nanocompression of a complex flexoelectric conical pyramid for which analytical solution cannot be established is numerically studied where curvature induced necking of the specimen and vorticity around the frustum generate moderate electric polarisation. (C) 2019 Elsevier Ltd. All rights reserved.