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Applied Physics A

, 124:800 | Cite as

On the electro-thermo-mechanical vibration characteristics of elastically restrained functionally graded nanobeams using differential transformation method

  • Salman Ebrahiminejad
  • Javad MarzbanradEmail author
  • Mahya Boreiry
  • Gholam Reza Shaghaghi
Article
  • 72 Downloads

Abstract

In this investigation, free vibration of piezoelectric functionally graded nanobeams in the presence of thermal field effects for various elastic boundary conditions is studied. Formulations are established in the framework of nonlocal elasticity theory of Eringen accompanied by Euler–Bernoulli beam theory. Further, the material characteristics are supposed to change through the thickness based on the power law. Discretization of the equations of motion and the relative boundary conditions are carried out by utilizing the differential transformation method. The presented model is validated through obtaining numerical results for conventional boundary conditions in comparison with the corresponding benchmark results. Finally, numerical results are given in details to demonstrate the impact of material gradient, nonlocal effect, piezoelectric voltage, along with temperature changes on vibration characteristic of piezoelectric functionally graded nanobeams, with various elastic boundary conditions.

List of symbols

\({A_{{\text{xx}}}},{B_{{\text{xx}}}},{C_{{\text{xx}}}}\)

Cross-sectional rigidities

b

Width of beam

D

Electric field

\({E_{\text{c}}} \cdot {E_{\text{m}}}\)

Young’s modulus (C ceramic, m metal)

\({e_{31}}\)

Piezoelectric coefficient

h

Height of beam

H

Moisture effect

\({k_{\text{p}}}\)

Pasternak parameter

L

Length of beam

M

Bending moment

N

Axial force

\({N^{\text{T}}}\)

Thermal load

\({N^{\text{P}}}\)

Preload

P

Power-law index

\({P_{{\text{electric}}}}\)

Piezoelectric load

\({P_0},{P_{ - 1}},{P_1},{P_2},{P_3}\)

Temperature-dependent coefficient

t

Time

T

Kinetic energy

u

Axial displacement

U

Strain energy

V

Volume fraction

\({W_{{\text{ext}}}}\)

Work done

x

x coordinate

z

z coordinate

\(\eta\)

Magnetic property

\({\lambda _{33}}\)

Dielectric constants

\(\beta\)

Moisture expansion coefficient

\(\rho\)

Density

\(\omega\)

Non-dimensional natural frequency

\(\mu\)

Nonlocal parameter

\(\nu\)

Poisson’s ratio

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Vehicle Dynamical Systems Research Laboratory, School of Automotive EngineeringIran University of Science and TechnologyTehranIran
  2. 2.Young Researchers and Elites club, Science and Research BranchIslamic Azad UniversityTehranIran

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