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

Abstract

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 C 0 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.

Modeling flexoelectricity in soft dielectrics at finite deformation

“…To address this, several numerical alternatives have been proposed, such as mesh-free approximations (Abdollahi et al, 2014, 2015a,b, Abdollahi and Arias, 2015, isogeometric analysis (Ghasemi et al, 2017, Nanthakumar et al, 2017, Thai et al, 2018, Hamdia et al, 2018, Nguyen et al, 2019, C 1 Argyris triangular element approximation (Yvonnet and Liu, 2017) and the B-spline-based immersed boundary method (Codony et al, 2019). Another family of numerical methods are those circumventing the C 1 continuity requirement by introducing additional variables, such as mixed formulations (Mao et al, 2016, Deng et al, 2017, or those based on micromorphic theories of continua (Poya et al, 2019, McBride et al, 2019. Recently, a few works report the application of these methods to large deformation flexoelectricity (Thai et al, 2018, Poya et al, 2019, McBride et al, 2019, Yvonnet and Liu, 2017, Nguyen et al, 2019 but the continuum formulation at finite deformation is still open, see previous paragraph, and there is a need for validation of the computational results.…”

“…Another family of numerical methods are those circumventing the C 1 continuity requirement by introducing additional variables, such as mixed formulations (Mao et al, 2016, Deng et al, 2017, or those based on micromorphic theories of continua (Poya et al, 2019, McBride et al, 2019. Recently, a few works report the application of these methods to large deformation flexoelectricity (Thai et al, 2018, Poya et al, 2019, McBride et al, 2019, Yvonnet and Liu, 2017, Nguyen et al, 2019 but the continuum formulation at finite deformation is still open, see previous paragraph, and there is a need for validation of the computational results.…”

“…However, most of these works assume infinitesimal deformations, and are therefore suitable to model crystalline ceramics only. Efforts have been re-cently made to extend the theory to polymers or elastomers undergoing large deformations, but the literature is still scarce (Liu, 2014, Yvonnet and Liu, 2017, Thai et al, 2018, Poya et al, 2019, McBride et al, 2019, Nguyen et al, 2019. Some of these works model flexoelectricity as a linear coupling between strain gradients and the electric displacement (Poya et al, 2019) or the electric field (McBride et al, 2019, Nguyen et al, 2019 instead of the electric polarization, which however is the most natural choice according to experiments and first-principle calculations (Ma and Cross, 2001a, Zubko et al, 2007, Resta, 2010, Hong and Vanderbilt, 2011.…”

“…Efforts have been re-cently made to extend the theory to polymers or elastomers undergoing large deformations, but the literature is still scarce (Liu, 2014, Yvonnet and Liu, 2017, Thai et al, 2018, Poya et al, 2019, McBride et al, 2019, Nguyen et al, 2019. Some of these works model flexoelectricity as a linear coupling between strain gradients and the electric displacement (Poya et al, 2019) or the electric field (McBride et al, 2019, Nguyen et al, 2019 instead of the electric polarization, which however is the most natural choice according to experiments and first-principle calculations (Ma and Cross, 2001a, Zubko et al, 2007, Resta, 2010, Hong and Vanderbilt, 2011. Furthermore, works modeling flexoelectricity as a coupling between strain gradients and electric polarization consider a coupling tensor of mixed material-spatial character (Liu, 2014, Yvonnet and Liu, 2017, Thai et al, 2018, leading in general to a lack of objectivity in the resulting polarization as argued in Section 2.3.…”

An analytical micromechanics‐based multiscale model for the analysis of functionally graded nanocomposites

“…Higher‐order theories have attracted researchers' interest in their ability to include the size effect—both at the microscopic level and the structure's overall geometry level. Modified couple stress theory introduced by Yang et al 1 has gained popularity in the study of micro/nano‐beams, 2 plates 3,4 and shells 5 …”