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Research Papers

Insights Into Flexoelectric Solids From Strain-Gradient Elasticity

[+] Author and Article Information
Sheng Mao

Department of Mechanical Engineering
and Applied Mechanics,
University of Pennsylvania,
Philadelphia, PA 19104
e-mail: maosheng@seas.upenn.edu

Prashant K. Purohit

Department of Mechanical Engineering
and Applied Mechanics,
University of Pennsylvania,
Philadelphia, PA 19104
e-mail: purohit@seas.upenn.edu

1Corresponding author.

Manuscript received March 27, 2014; final manuscript received April 2, 2014; accepted manuscript posted April 18, 2014; published online May 5, 2014. Assoc. Editor: Pradeep Sharma.

J. Appl. Mech 81(8), 081004 (May 05, 2014) (10 pages) Paper No: JAM-14-1138; doi: 10.1115/1.4027451 History: Received March 27, 2014; Revised April 02, 2014; Accepted April 18, 2014

A material is said to be flexoelectric when it polarizes in response to strain gradients. The phenomenon is well known in liquid crystals and biomembranes but has received less attention in hard materials such as ceramics. Here we derive the governing equations for a flexoelectric solid under small deformation. We assume a linear constitutive relation and use it to prove a reciprocal theorem for flexoelectric materials as well as to obtain a higher-order Navier equation in the isotropic case. The Navier equation is similar to that in Mindlin's theory of strain-gradient elasticity. We also provide analytical solutions to several boundary value problems. We predict size-dependent electromechanical properties and flexoelectric modulation of material behavior. Our results can be used to interpret experiments on flexoelectric materials which are becoming increasingly sophisticated due to the advent of nanoscale probes.

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Figures

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Fig. 1

Size-dependent stiffening of flexoelectric beams. χ in the legend is the susceptibility constant. The bending rigidity plotted on the y-axis is normalized against EI where E is the Young's modulus and I is the moment of inertia of the cross section. The thickness of the beam is normalized against 3l. As the beam gets thinner the strain gradients increase, so the effects of flexoelectricity become more prominent.

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Fig. 2

In (a), a point load Q is applied, while in (b) there is a potential difference between the upper and lower surface over a portion of the beam

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Fig. 3

A disk/cylinder, with inner and outer radius ri and ro. It is subject to a potential difference V0 between the surfaces as well as internal and external pressure pi and po.

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Fig. 6

Stresses in the disk loaded by internal/external pressure. (a) Compares the profiles of the θθ component of the true stress and Cauchy stress, they differ by little. (b) and (c) plot the radial and circumferential normal true stress respectively.

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Fig. 7

Stress concentration factor in the disk loaded by internal/external pressure. (a) plots the asymptotic behavior of SCF with ro/ri. (b) plots the flexoelectric reduction of SCF with increasing f. f2 is normalized against fmax2 whose value is determined by requiring the energy to be positive definite.

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Fig. 8

Modulation of mechanical/electrical quantities in disk loaded by internal/external pressure. (a) plots SCF as a function of potential Vo holding po and pi fixed. (b) plots polarization at the inner surface as a function of pressure pi, holding Vo = 0. Nondimensionalization is carried out in the same manner as in Fig. 4.

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Fig. 4

Variation of the electric quantities along the radial direction in the disk loaded by internal/external pressure. We choose ro/ri = 2, po/pi = 2, ν = 0.3, χ = 1, l = 0.2ri, and lf = 0.5l, as in the legend. All quantities are normalized to be nondimensional, in the unit system where length, force, and charge are measured by ri,Eri2,and ri2ɛ0E respectively.

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Fig. 5

Variation of displacement and strains in the disk loaded by internal/external pressure. (a) Normalized magnitude of displacement. (b) and (c) plot the radial and circumferential normal strain respectively.

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Fig. 9

A solution for the in-plane shear of a disk is obtained with the following parameters: ro/ri = 2, τ0/E = 1, ν = 0.3, χ = 1, l = 0.2ri, and lf = 0.5l. (a) plots normalized shear strain and (b) the normalized displacement as functions of the radial coordinate. (c) plots the distribution of normalized azimuthal polarization. It reaches a minimum around the middle of the disk. Note that all quantities are nondimensionalized, in the same manner as in Fig. 4.

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