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Research Papers

A Three-Dimensional Mixed Finite Element for Flexoelectricity

[+] Author and Article Information
Feng Deng

State Key Laboratory for Strength and Vibration
of Structures,
School of Aerospace Engineering,
Xi'an Jiaotong University,
Xi'an 710049, China

Qian Deng

State Key Laboratory for Strength and
Vibration of Structures,
School of Aerospace Engineering,
Xi'an Jiaotong University,
Xi'an 710049, China
e-mail: tonydqian@mail.xjtu.edu.cn

Shengping Shen

State Key Laboratory for Strength
and Vibration of Structures,
School of Aerospace Engineering,
Xi'an Jiaotong University,
Xi'an 710049, China
e-mail: sshen@mail.xjtu.edu.cn

1Corresponding authors.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received November 17, 2017; final manuscript received December 30, 2017; published online January 24, 2018. Editor: Yonggang Huang.

J. Appl. Mech 85(3), 031009 (Jan 24, 2018) (10 pages) Paper No: JAM-17-1640; doi: 10.1115/1.4038919 History: Received November 17, 2017; Revised December 30, 2017

Flexoelectric effect is a universal and size-dependent electromechanical coupling between the strain gradient and electric field. The mathematical framework for flexoelectricity, which involves higher-order gradients of field quantities, is difficult to handle using traditional finite element method (FEM). Thus, it is important to develop an effective numerical method for flexoelectricity. In this paper, we develop a three-dimensional (3D) mixed finite element considering both flexoelectricity and strain gradient elasticity. To validate the developed element, we simulate the electromechanical behavior of a flexoelectric spherical shell subjected to inner pressure and compare the numerical results to analytical results. Their excellent agreement shows the reliability of the proposed FEM. The developed finite element is also used to simulate the electromechanical behavior of a nanometer-sized flexoelectric truncated pyramid. By decreasing the sample size, we observed the increase of its effective piezoelectricity. However, due to the effects of strain gradient elasticity and the influence of flexoelectricity on stiffness, the dependency of effective piezoelectricity on the sample size is not trivial. Numerical results indicate that, when the sample size is smaller than a certain value, the increase of effective piezoelectricity slows down. This finding also shows the importance of a numerical tool for the study of flexoelectric problems.

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Figures

Grahic Jump Location
Fig. 1

Sketch of a hexahedral element. The element has 27 nodes, eight corner nodes (marked in square block) and 19 other nodes (marked in dot) which include 12 edge-centered nodes, six surface-centered nodes, and a volume-centered node.

Grahic Jump Location
Fig. 2

Model of spherical shell subjected to inner/outer pressure pi/po. Inner and outer radius of the sphere are ρi and ρo.

Grahic Jump Location
Fig. 3

Finite element model for the 1/8 spherical shell. There are 6000 hexahedral elements in total, 20 elements on radius, and 300 elements on spherical surface. Symmetric boundary conditions are used in three symmetric surface, (x = 0) plane, (y = 0) plane, and (z = 0) plane.

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Fig. 4

Comparison of FEM results with analytical solutions. (a) FEM result and analytical solution of radial displacement uρ versus radius. (b) FEM and analytical result of electric potential φ versus radius ρ. All the FEM results are adopted from the nodal DOF on the sphere's centered line (x = y = z).

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Fig. 5

(a) FEM result and analytical solution of radial displacement uρ versus radius ρ for different flexoelectric coefficients. (b) Analytical result of radial displacement uρ at inner surface versus Inner pressure pi for different flexoelectric coefficients.

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Fig. 6

Sketch of dielectric material with micropyramid structures

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Fig. 7

(a) Schematic of nanopyramid subjected to uniform pressure at top surface. At the bottom of the model, electric potential and displacement component at height direction are set to zero. (b) Grid information in simulation. There are 2800 elements and 151,357 DOFs in total. Symmetric boundary conditions are used in symmetric surface, (x = 0) plane, (y = 0) plane.

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Fig. 8

(a) Color map of electric potential generated by flexoelectricity and (b) variation of electric potential and strain (εzz) versus height

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Fig. 9

(a) Variation of effective piezoelectric modulus versus flexoelectric coefficients for different size of microstructure and (b) variation of effective piezoelectric modulus versus the size of microstructure for different flexoelectric coefficients

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