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


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


Cross-sectional rigidities


Width of beam


Electric field

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

Young’s modulus (C ceramic, m metal)


Piezoelectric coefficient


Height of beam


Moisture effect


Pasternak parameter


Length of beam


Bending moment


Axial force


Thermal load




Power-law index


Piezoelectric load

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

Temperature-dependent coefficient




Kinetic energy


Axial displacement


Strain energy


Volume fraction


Work done


x coordinate


z coordinate


Magnetic property

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

Dielectric constants


Moisture expansion coefficient




Non-dimensional natural frequency


Nonlocal parameter


Poisson’s ratio


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© 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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