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Size-dependent vibration and bending analyses of the piezomagnetic three-layer nanobeams

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Abstract

Vibration and electro-magneto-elastic bending analysis of a three-layer nanobeam with a nanocore and two piezomagnetic face sheets are studied in this paper. Timoshenko model of beam as well as nonlocal magneto-electro-elastic relations are used for analysis of this problem. The nanoface sheets are subjected to applied electric and magnetic potentials. The nanobeam rests on Winkler–Pasternak foundation. Electric and magnetic potentials are assumed as combination of linear function along the thickness direction that reflects applied electric and magnetic potentials and a cosine function that satisfies boundary conditions. Numerical results of this problem investigate the effect of some important parameters of nanobeam, such as nonlocal parameter, applied electric and magnetic potentials, and parameters of foundation on the vibration and magneto-electro-mechanical bending behaviors of the problem.

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Abbreviations

\({{u}},{{w}}\) :

Displacement of mid-surface

\(\theta\) :

Rotation component

\(\mu ,\lambda\) :

Lamé’s constants

\(\xi\) :

Nonlocal parameter

\({{\nabla }^{2}}\) :

Laplacian operator

\(\psi ,\phi\) :

Electric and magnetic potentials

\({{\sigma }_{xx}},{{\tau }_{xz}}\) :

Stress components

\({{\varepsilon }_{xx}},{{\gamma }_{xz}}\) :

Strain components

\({{C}_{ijkl}}\) :

Stiffness coefficients

\({{e}_{ijk}}\) :

Piezoelectric constants

\({{q}_{ijk}}\) :

Piezomagnetic coefficients

\({{E}_{k}}\) :

Electric field

\({{H}_{k}}\) :

Magnetic field

\({{N}_{ij}},{{M}_{ij}}\) :

Force and moment resultants

\({{\bar{D}}_{i}},{{\bar{B}}_{i}}\) :

Electric displacement and magnetic induction resultants

T:

Kinetic energy

U:

Strain energy

\({{k}_{1}},{{k}_{2}}\) :

Parameters of Pasternak’s foundation

W:

External works

\(K,J,P\) :

Stiffness, damping, and mass matrices

F :

Force matrix

L :

Length of beam

\(\omega\) :

Natural frequencies

q :

Transverse loads

\(U,\Theta ,W,\Psi ,\Phi\) :

Amplitude of unknown functions

\({{\psi }_{0}},{{\phi }_{0}}\) :

Applied electric and magnetic potentials

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Acknowledgements

The research described in this paper was financially supported by the University of Kashan. (Grant No.: 36346/5). The first author would also like to thank the Iranian Nanotechnology Development Committee for their financial support.

Author information

Authors and Affiliations

  1. Faculty of Mechanical Engineering, Department of Solid Mechanics, University of Kashan, Kashan, 87317-51167, Iran

    Mohammed Arefi

  2. Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

    Ashraf M. Zenkour

  3. Department of Mathematics, Faculty of Science, Kafrelsheikh University, Kafrelsheikh, 33516, Egypt

    Ashraf M. Zenkour

Authors
  1. Mohammed Arefi
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  2. Ashraf M. Zenkour
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Corresponding author

Correspondence to Mohammed Arefi.

Appendix 2

$$\left\{ {{A}_{1}},{{A}_{2}},{{A}_{8}} \right\}=\underset{-\frac{h}{2}-{{h}_{p}}}{\overset{-\frac{h}{2}}{\mathop \int }}\,{{c}_{11}}\left\{ 1,{{x}_{3}},x_{3}^{2} \right\}\text{d}{{x}_{3}}+\underset{-\frac{h}{2}}{\overset{\frac{h}{2}}{\mathop \int }}\,E\left\{ 1,{{x}_{3}},x_{3}^{2} \right\}\text{d}{{x}_{3}}+\underset{\frac{h}{2}}{\overset{\frac{h}{2}+{{h}_{p}}}{\mathop \int }}\,{{c}_{11}}\left\{ 1,{{x}_{3}},x_{3}^{2} \right\}\text{d}{{x}_{3}},$$
$$\left\{ {{A}_{3}},{{A}_{9}} \right\}=\underset{-\frac{h}{2}-{{h}_{p}}}{\overset{-\frac{h}{2}}{\mathop \int }}\,{{e}_{113}}\frac{\pi }{{{h}_{p}}}\sin \left( \frac{\pi {{{\tilde{x}}}_{3}}}{{{h}_{p}}} \right)\left\{ 1,{{x}_{3}} \right\}\text{d}{{x}_{3}}+\underset{\frac{h}{2}}{\overset{\frac{h}{2}+{{h}_{p}}}{\mathop \int }}\,{{e}_{113}}\frac{\pi }{{{h}_{p}}}\sin \left( \frac{\pi {{{\tilde{x}}}_{3}}}{{{h}_{p}}} \right)\left\{ 1,{{x}_{3}} \right\}\text{d}{{x}_{3}},$$
$$\left\{ {{A}_{4}},{{A}_{10}} \right\}=\underset{-\frac{h}{2}-{{h}_{p}}}{\overset{-\frac{h}{2}}{\mathop \int }}\,{{q}_{113}}\frac{\pi }{{{h}_{p}}}\sin \left( \frac{\pi {{{\tilde{x}}}_{3}}}{{{h}_{p}}} \right)\left\{ 1,{{x}_{3}} \right\}\text{d}{{x}_{3}}+\underset{\frac{h}{2}}{\overset{\frac{h}{2}+{{h}_{p}}}{\mathop \int }}\,{{q}_{113}}\frac{\pi }{{{h}_{p}}}\sin \left( \frac{\pi {{{\tilde{x}}}_{3}}}{{{h}_{p}}} \right)\left\{ 1,{{x}_{3}} \right\}\text{d}{{x}_{3}},$$
$${{A}_{5}}=\underset{-\frac{h}{2}-{{h}_{p}}}{\overset{-\frac{h}{2}}{\mathop \int }}\,{{c}_{12}}\text{d}{{x}_{3}}+\underset{-\frac{h}{2}}{\overset{\frac{h}{2}}{\mathop \int }}\,\mu \text{d}{{x}_{3}}+\underset{\frac{h}{2}}{\overset{\frac{h}{2}+{{h}_{p}}}{\mathop \int }}\,{{c}_{12}}\text{d}{{x}_{3}},$$
$$\left\{ {{A}_{6}},{{A}_{7}} \right\}=\underset{-\frac{h}{2}-{{h}_{p}}}{\overset{-\frac{h}{2}}{\mathop \int }}\,\left\{ {{e}_{113}},{{q}_{113}} \right\}\cos \left( \frac{\pi {{{\tilde{x}}}_{3}}}{{{h}_{p}}} \right)\text{d}{{x}_{3}}+\underset{\frac{h}{2}}{\overset{\frac{h}{2}+{{h}_{p}}}{\mathop \int }}\,\left\{ {{e}_{113}},{{q}_{113}} \right\}\cos \left( \frac{\pi {{{\tilde{x}}}_{3}}}{{{h}_{p}}} \right)\text{d}{{x}_{3}},$$
$$\left\{ {{A}_{11}},{{A}_{12}},{{A}_{13}} \right\}=\underset{-\frac{h}{2}-{{h}_{p}}}{\overset{-\frac{h}{2}}{\mathop \int }}\,\left\{ {{\eta }_{11}},{{g}_{11}},{{\mu }_{11}} \right\}{{\cos }^{2}}\left( \frac{\pi {{{\tilde{x}}}_{3}}}{{{h}_{p}}} \right)\text{d}{{x}_{3}}+\underset{\frac{h}{2}}{\overset{\frac{h}{2}+{{h}_{p}}}{\mathop \int }}\,\left\{ {{\eta }_{11}},{{g}_{11}},{{\mu }_{11}} \right\}\cos \left( \frac{\pi {{{\tilde{x}}}_{3}}}{{{h}_{p}}} \right)\text{d}{{x}_{3}},$$
$$\left\{ {{A}_{14}},{{A}_{15}},{{A}_{16}} \right\}=\underset{-\frac{h}{2}-{{h}_{p}}}{\overset{-\frac{h}{2}}{\mathop \int }}\,\left\{ {{\eta }_{33}},{{g}_{33}},{{\mu }_{33}} \right\}{{\left[ \frac{\pi }{{{h}_{p}}}\sin \left( \frac{\pi {{{\tilde{x}}}_{3}}}{{{h}_{p}}} \right) \right]}^{2}}\text{d}{{x}_{3}}+\underset{\frac{h}{2}}{\overset{\frac{h}{2}+{{h}_{p}}}{\mathop \int }}\,\left\{ {{\eta }_{33}},{{g}_{33}},{{\mu }_{33}} \right\}{{\left[ \frac{\pi }{{{h}_{p}}}\sin \left( \frac{\pi {{{\tilde{x}}}_{3}}}{{{h}_{p}}} \right) \right]}^{2}}\text{d}{{x}_{3}},$$
$$\left\{ {{N}^{\psi }},{{M}^{\psi }} \right\}=\underset{-\frac{h}{2}-{{h}_{p}}}{\overset{-\frac{h}{2}}{\mathop \int }}\,{{e}_{113}}\frac{2{{\psi }_{0}}}{{{h}_{p}}}\left\{ 1,{{x}_{3}} \right\}\text{d}{{x}_{3}}+\underset{\frac{h}{2}}{\overset{\frac{h}{2}+{{h}_{p}}}{\mathop \int }}\,{{e}_{113}}\frac{2{{\psi }_{0}}}{{{h}_{p}}}\left\{ 1,{{x}_{3}} \right\}\text{d}{{x}_{3}},$$
$$\left\{ {{N}^{\phi }},{{M}^{\phi }} \right\}=\underset{-\frac{h}{2}-{{h}_{p}}}{\overset{-\frac{h}{2}}{\mathop \int }}\,{{q}_{113}}\frac{2{{\phi }_{0}}}{{{h}_{p}}}\left\{ 1,{{x}_{3}} \right\}\text{d}{{x}_{3}}+\underset{\frac{h}{2}}{\overset{\frac{h}{2}+{{h}_{p}}}{\mathop \int }}\,{{q}_{113}}\frac{2{{\phi }_{0}}}{{{h}_{p}}}\left\{ 1,{{x}_{3}} \right\}\text{d}{{x}_{3}},$$
$$\left\{ {{{\bar{D}}}^{\psi }},{{{\bar{B}}}^{\psi }} \right\}=\underset{-\frac{h}{2}-{{h}_{p}}}{\overset{-\frac{h}{2}}{\mathop \int }}\,\left\{ {{\eta }_{33}},{{g}_{33}} \right\}\frac{2{{\psi }_{0}}}{{{h}_{p}}}\frac{\pi }{{{h}_{p}}}\sin \left( \frac{\pi {{{\tilde{x}}}_{3}}}{{{h}_{p}}} \right)\text{d}{{x}_{3}}+\underset{\frac{h}{2}}{\overset{\frac{h}{2}+{{h}_{p}}}{\mathop \int }}\,\left\{ {{\eta }_{33}},{{g}_{33}} \right\}\frac{2{{\psi }_{0}}}{{{h}_{p}}}\frac{\pi }{{{h}_{p}}}\sin \left( \frac{\pi {{{\tilde{x}}}_{3}}}{{{h}_{p}}} \right)\text{d}{{x}_{3}},$$
$$\left\{ {{{\bar{D}}}^{\phi }},{{{\bar{B}}}^{\phi }} \right\}=\underset{-\frac{h}{2}-{{h}_{p}}}{\overset{-\frac{h}{2}}{\mathop \int }}\,\left\{ {{g}_{33}},{{\mu }_{33}} \right\}\frac{2{{\phi }_{0}}}{{{h}_{p}}}\frac{\pi }{{{h}_{p}}}\sin \left( \frac{\pi {{{\tilde{x}}}_{3}}}{{{h}_{p}}} \right)\text{d}{{x}_{3}}+\underset{\frac{h}{2}}{\overset{\frac{h}{2}+{{h}_{p}}}{\mathop \int }}\,\left\{ {{g}_{33}},{{\mu }_{33}} \right\}\frac{2{{\phi }_{0}}}{{{h}_{p}}}\frac{\pi }{{{h}_{p}}}\sin \left( \frac{\pi {{{\tilde{x}}}_{3}}}{{{h}_{p}}} \right)\text{d}{{x}_{3}}.$$

Appendix 1

Appendix 1

$$\left\{ A,B,C \right\}=\underset{-\frac{h}{2}-{{h}_{p}}}{\overset{-\frac{h}{2}}{\mathop \int }}\,{{\rho }^{p}}\left\{ 1,{{x}_{3}},x_{3}^{2} \right\}\text{d}{{x}_{3}}+\underset{-\frac{h}{2}}{\overset{\frac{h}{2}}{\mathop \int }}\,{{\rho }^{c}}\left\{ 1,{{x}_{3}},x_{3}^{2} \right\}\text{d}{{x}_{3}}+\underset{\frac{h}{2}}{\overset{\frac{h}{2}+{{h}_{p}}}{\mathop \int }}\,{{\rho }^{p}}\left\{ 1,{{x}_{3}},x_{3}^{2} \right\}\text{d}{{x}_{3}}.$$

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Arefi, M., Zenkour, A.M. Size-dependent vibration and bending analyses of the piezomagnetic three-layer nanobeams. Appl. Phys. A 123, 202 (2017). https://doi.org/10.1007/s00339-017-0801-0

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