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Research Papers

Exact Solutions for Flexoelectric Response in Nanostructures

[+] Author and Article Information
M. C. Ray

Department of Mechanical Engineering,
Indian Institute of Technology,
Kharagpur 721302, India

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received May 6, 2014; final manuscript received June 2, 2014; accepted manuscript posted June 5, 2014; published online June 19, 2014. Assoc. Editor: Pradeep Sharma.

J. Appl. Mech 81(9), 091002 (Jun 19, 2014) (7 pages) Paper No: JAM-14-1198; doi: 10.1115/1.4027806 History: Received May 06, 2014; Revised June 02, 2014; Accepted June 05, 2014

This paper is concerned with the derivation of the exact solutions for the static responses of the simply supported flexoelectric nanobeams subjected to the applied mechanical load and applied voltage difference across the thickness of the beams. Considering both the direct and the converse flexoelectric effects, the governing equations and the associated boundary conditions of the beams are derived to obtain the exact solutions for the displacements and the electric potential in the beams. Due to the converse flexoelectric effect, the active beams significantly counteract the applied mechanical load. The normal and the transverse shear deformations in the beams are affected by the converse flexoelectric effect in the beams resulting in the coupling of bending and stretching deformations in the beams. For the particular values of the length of the beam and the applied voltage, the deflection of the nanobeam due to the converse flexoelectric effect significantly increases with the decrease in the thickness of the beam. But the deflection of the beam remains invariant with the change in length of the beam for the particular values of the thickness of the beam and the applied voltage. Also, for the particular values of the thickness of the beam and the applied mechanical load, the induced transverse electric polarization on the surface of the beam is independent of the variation of the length of the beam and the value of the polarization increases with the decrease in the thickness of the beam. The benchmark results presented here may be useful for verifying further research and the present study suggests that the flexoelectric nanobeams may be effectively exploited for advanced applications as smart sensors and actuators at nanoscale.

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References

Figures

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Fig. 1

Schematic diagram of a simply supported flexoelectric nanobeam

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Fig. 2

Variation of the center deflection of the thick nanobeam along the length (h = L/10, q0 = 106 N/m2, V1 = 0)

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Fig. 3

Variation of the center deflection of the thin nanobeam along the length (h = L/100, q0 = 104 N/m2, V1 = 0)

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Fig. 4

Distribution of the axial normal stress across the thickness of the thick nanobeam (h = L/10, σ = 106 N/m2, V1 = 0)

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Fig. 5

Distribution of the axial normal stress across the thickness of the thin nanobeam (h = L/100, σ = 104 N/m2, V1 = 0)

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Fig. 6

Distribution of the transverse normal stress across the thickness of the thick nanobeam (h = L/10, σ = 106 N/m2, V1 = 0)

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Fig. 7

Distribution of the transverse shear stress across the thickness of the thick nanobeam (h = L/10, σ = 106 N/m2, V1 = 0)

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Fig. 8

Effect of the thickness variation on the center deflection of the nanobeams along the length (L = 200 nm, q0 = 0, V1 = 0, V2=0.02 V)

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Fig. 9

Effect of variation of the length on the center deflection of the nanobeams for different values of thickness (q0 = 0, V1 = 0, V2=0.02 V)

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Fig. 10

Distribution of the gradient of the transverse electric field across the thickness of the thick nanobeam (h = L/10, q0 = 106 N/m2, V1 = 0)

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Fig. 11

Distribution of the electric polarization on the surface of the thick nanobeam (L = 200 nm, σ = 106 N/m2, V1 = V2 = 0)

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Fig. 12

Effect of the variation of length on the electric polarization in the nanobeams for different values of thickness (q0 = 106 N/m2, V1 = V2= 0)

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