The number and types of all possible rotational symmetries for flexoelectric tensors

Quang, H. Le; He, Qi‐Chang

doi:10.1098/rspa.2010.0521

Cited by 87 publications

(77 citation statements)

References 31 publications

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“…We refer to [21] for an extensive analysis of other possible symmetries for the flexoelectric tensor.…”

Section: Appendix B Materials Tensorsmentioning

confidence: 99%

“…Within the continuum flexoelectric theory, the symmetry of the flexoelectric tensor is well understood [20,21], although its full characterization is still lacking for most materials [19]. The equations are a coupled system of 4th-order partial differential equations, which renders analytical solutions difficult to obtain and precludes the use of conventional C 0 finite elements.…”

Section: Introductionmentioning

confidence: 99%

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An immersed boundary hierarchical B-spline method for flexoelectricity

Codony

Marco

Fernández‐Méndez

et al. 2019

Computer Methods in Applied Mechanics and Engineering

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This paper develops a computational framework with unfitted meshes to solve linear piezoelectricity and flexoelectricity electromechanical boundary value problems including strain gradient elasticity at infinitesimal strains. The high-order nature of the coupled PDE system is addressed by a sufficiently smooth hierarchical B-spline approximation on a background Cartesian mesh. The domain of interest is embedded into the background mesh and discretized in an unfitted fashion. The immersed boundary approach allows us to use B-splines on arbitrary domain shapes, regardless of their geometrical complexity, and could be directly extended, for instance, to shape and topology optimization. The domain boundary is represented by NURBS, and exactly integrated by means of the NEFEM mapping. Local adaptivity is achieved by hierarchical refinement of B-spline basis, which are efficiently evaluated and integrated thanks to their piecewise polynomial definition. Nitsche's formulation is derived to weakly enforce essential boundary conditions, accounting also for the non-local conditions on the non-smooth portions of the domain boundary (i.e. edges in 3D or corners in 2D) arising from Mindlin's strain gradient elasticity theory. Boundary conditions modeling sensing electrodes are formulated and enforced following the same approach. Optimal error convergence rates are reported using high-order B-spline approximations. The method is verified against available analytical solutions and well-known benchmarks from the literature.

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“…We refer to [21] for an extensive analysis of other possible symmetries for the flexoelectric tensor.…”

Section: Appendix B Materials Tensorsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An immersed boundary hierarchical B-spline method for flexoelectricity

Codony

Marco

Fernández‐Méndez

et al. 2019

Computer Methods in Applied Mechanics and Engineering

View full text Add to dashboard Cite

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“…6 are employed. For flexoelectric coefficients, Le Quang and He [24] represented all the possible rotational symmetries for flexoelectric tensors, Shu et al [25] discussed the symmetry of flexoelectric coefficient in crystalline medium. We assume the flexoelectric coefficient as follows for sake of simplicity [25,26] …”

Section: Timoshenko Beam Model With Flexoelectricitymentioning

confidence: 99%

A Timoshenko dielectric beam model with flexoelectric effect

Zhang

Shen

2015

Meccanica

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In this paper, a Timoshenko dielectric beam model with the consideration of the direct flexoelectric effect is developed by using the variational principle. The governing equations and the corresponding boundary conditions are naturally derived from the variational principle. The effect of flexoelectricity on the Timoshenko dielectric beams is analytically investigated. The developed beam model recovers to the classical Timoshenko beam model when the flexoelectricity is not taken into account. To illustrate this model, the deflection and rotation of Timoshenko dielectric nanobeam under two different boundary conditions are calculated. The numerical results reveal that both the deflection and rotation predicted by the current model are smaller than those predicted by the classical Timoshenko beam model. Moreover, the discrepancies of the deflection and rotation between the values predicted by the two models are very large when the beam thickness is small. The current model can also reduce to the Bernoulli-Euler dielectric beam model wherein the shear deflection is not considered. This work may be helpful in understanding the electromechanical response at nanoscale and designing dielectric devices.

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“…The static flexoelectric effect, firstly introduced by Mashkevich, Tolpygo [29], and Kogan [30], manifesting itself in the appearance of electric polarization variation linearly proportional to the strain gradient Considering the importance of the flexoelectric coupling (shortly "flexocoupling") for the physical understanding of the gradient-driven couplings in mesoscale and nanoscale solids, one has to determine its symmetry and numerical values. Unfortunately, the available experimental and theoretical data about the flexocoupling tensor symmetry, specifically the amount of independent components allowing for the point group symmetry [35,36] and "hidden" permutation symmetry [37], and numerical values are contradictory [38]. Namely, the upper limits for the values ijkl f established by Yudin et al [39], as well as the values calculated from the first principles for bulk ferroics [40,41,42,43,44], the can be several orders of magnitude smaller than those measured experimentally in ferroelectric ceramics [45,46,47] and thin films [48], ferroelectric relaxor polymers [49] and electrets [50], incipient ferroelectrics [51,52] and biological membranes [53,54].…”

Section: Introductionmentioning

confidence: 99%

Size effect of soft phonon dispersion in nanosized ferroics

Morozovska

Eliseev

2019

Phys. Rev. B

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Using Landau-Ginsburg-Devonshire theory, we derive and analyze analytical expressions for the frequency dispersion of soft phonon modes in nanosized ferroics and perform numerical calculations for a thin SrTiO 3 film. We revealed the pronounced "true" size effect in the dependence of soft phonon spatial dispersion on the film thickness and predict that it can lead to the "apparent" or "false" size effect of dynamic flexoelectric coupling constants. Also we derived analytical expressions describing the influence of finite size effect on the appearance and properties of the incommensurate spatial modulation induced by the static flexoelectric effect in ferroic thin films.To verify the theoretical predictions experimental measurements of the soft phonon dispersion in nanosized ferroics seem urgent.

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The number and types of all possible rotational symmetries for flexoelectric tensors

Cited by 87 publications

References 31 publications

An immersed boundary hierarchical B-spline method for flexoelectricity

An immersed boundary hierarchical B-spline method for flexoelectricity

A Timoshenko dielectric beam model with flexoelectric effect

Size effect of soft phonon dispersion in nanosized ferroics

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