A collocation mixed finite element method for the analysis of flexoelectric solids

Tian, Xinpeng; Sládek, J.; Sládek, V.; Deng, Qian; Qun, LI

doi:10.1016/j.ijsolstr.2021.01.031

Cited by 41 publications

(14 citation statements)

References 38 publications

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“…In addition, one can observe that the induced electric intensity increases with the increasing flexoelectric coefficients. It is in agreement with the results recently presented by the authors in [36]. The largest values of the electric intensity are at the clamped end, where the bending moment and the strain-gradients reach their maximum values.…”

Section: Numerical Resultssupporting

confidence: 93%

“…In the following sections, we will use the finite elements with a length of 10 nm, when the convergence is guaranteed. Moreover, the variation of the beam deflection along x 1 for various flexoelectric coefficients is obtained with the 2D analysis by the mixed FEM program [36], where 2000 rectangular elements are used. An opposite verification of the new computer code for the 2D problems by the Euler-Bernoulli theory was applied in [36] for a body force load, where the analytical solution is available.…”

Section: Numerical Resultsmentioning

confidence: 99%

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Geometrical Nonlinearity for a Timoshenko Beam with Flexoelectricity

Repka

Sládek

2021

Nanomaterials

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The Timoshenko beam model is applied to the analysis of the flexoelectric effect for a cantilever beam under large deformations. The geometric nonlinearity with von Kármán strains is considered. The nonlinear system of ordinary differential equations (ODE) for beam deflection and rotation are derived. Moreover, this nonlinear system is linearized for each load increment, where it is solved iteratively. For the vanishing flexoelectric coefficient, the governing equations lead to the classical Timoshenko beam model. Furthermore, the influence of the flexoelectricity coefficient and the microstructural length-scale parameter on the beam deflection and the induced electric intensity is investigated.

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Section: Numerical Resultssupporting

confidence: 93%

Section: Numerical Resultsmentioning

confidence: 99%

Geometrical Nonlinearity for a Timoshenko Beam with Flexoelectricity

Repka

Sládek

2021

Nanomaterials

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“…Using our self-developed finite element code, 34 we conducted finite element simulations on the flexoelectric effect around the crack tip (for details, see SI. note7).…”

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confidence: 99%

Directly Observing the Evolution of Flexoelectricity at the Tip of Nanocracks

Tian

Deng

et al. 2022

Nano Lett.

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As an electromechanical coupling between strain gradients and polarization, flexoelectricity is largely enhanced at the nanoscale. However, directly observing the evolution of flexoelectric fields at the nanoscale usually suffers from the difficulty of producing strain gradients and probing electrical responses simultaneously. Here, we introduce nanocracks in SrTiO 3, Ba 0.67 Sr 0.33 TiO 3, and TiO 2 samples and apply continuously varying mechanical loading to them, and as a result, huge strain gradients appear at the crack tip and result in a significant flexoelectric effect. Then, using atomic force microscopy, we successfully measure the evolution of flexoelectricity around the crack tips. For the case of SrTiO 3, the maximum induced electric field reaches 11 kV/m due to the tensile load increasing. The proposed method provides a reliable way to identify the significance of the flexoelectric effect. It may also open a new avenue for the study of flexoelectricity involving multiple physics phenomena including flexoelectronics, the flexo-photovoltaic effect, and others.

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“…or pyramid [5][6][7], and introducing nano-cracks, dislocations, lattice mismatches, or nano-inclusions [8][9][10][11][12][13][14].…”

Section: Introductionmentioning

confidence: 99%

Modeling mechanical waves propagation in flexoelectric solids

Zhou,

Tian,

Deng

et al. 2024

Smart Mater. Struct.

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In this paper, the propagation of mechanical waves in flexoelectric solids with the consideration of both the direct and converse flexoelectric effects is studied via a collocation mixed finite element method (MFEM). The dynamic effects associated with mechanical waves propagation are accounted by introducing the kinetic energy in the Hamilton's principle. In the proposed collocation MFEM, a quadratic polynomial is independently assumed for each component of the mechanical strain and electric field. The independently assumed mechanical strain and electric field are collocated with their counterparts computed from the displacement and electric potential at 9 Gaussian quadrature points. Thus, except for the fundamental field variables, no additional degrees of freedom (DOFs) are introduced. By performing the numerical experiments using the collocation MFEM, it is found that due to the direct flexoelectric effect, the propagation of mechanical waves can result in electric polarization in materials. Besides, the converse flexoelectric effect can induce mechanical waves when there are non-uniform transient electric field applied to the material. Numerical results indicate that by increasing the loading speed of the time varying mechanical displacement load, the direct flexoelectric effect associated with the mechanical strain gradient could be significantly enhanced.

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A collocation mixed finite element method for the analysis of flexoelectric solids

Cited by 41 publications

References 38 publications

Geometrical Nonlinearity for a Timoshenko Beam with Flexoelectricity

Geometrical Nonlinearity for a Timoshenko Beam with Flexoelectricity

Directly Observing the Evolution of Flexoelectricity at the Tip of Nanocracks

Modeling mechanical waves propagation in flexoelectric solids

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