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The mechanical response of nanobeams considering the flexoelectric phenomenon in the temperature environment

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Abstract

This article employs the novel shear strain theory for the first time to investigate the static bending and static buckling responses of nanobeams affected by the flexoelectric effect in a temperature environment. Consideration is given to the effect of applied voltage and rotation on the beam's fixed axis. The calculation formulas are based on the finite element method; the equilibrium equation is derived from Hamilton's principle, where charge polarization is considered via the strain gradient; and the voltage applied to the beam makes the electric field calculation expression significantly more complex than previously reported. This increases the realism of the research results. This study also investigated the effect of a series of rotational parameters, applied voltage, temperature, and boundary conditions on the nanobeams' mechanical response. The paper demonstrates through calculation results that the influence of the applied voltage increases the beam's load capacity. This parameter can be used to adjust the beam's bending response, which is the foundation of practical application.

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Author information

Authors and Affiliations

  1. Laboratory of Applied Physics, Science and Technology Advanced Institute, Van Lang University, Ho Chi Minh City, Vietnam

    Gia Thien Luu

  2. Faculty of Applied Technology, School of Technology, Van Lang University, Ho Chi Minh City, Vietnam

    Gia Thien Luu

  3. Division of Mechanics, Civil Engineering Department, Akdeniz University, Antalya, Turkey

    Ömer Civalek

  4. China Medical University Hospital, China Medical University, Taichung, 404327, Taiwan

    Ömer Civalek

  5. Thuyloi University, 175 Tay Son, Dong Da, Hanoi, Vietnam

    Bui Van Tuyen

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  1. Gia Thien Luu
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  2. Ömer Civalek
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Dr Gia Thien Luu was involved in methodology and supervision and wrote the manuscript. Prof. Ömer Civalek prepared all figures, solved the result number, and contributed to visualization Dr. Bui Van Tuyen contributed to methodology, solved the result number, and wrote the manuscript All authors reviewed the manuscript

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Correspondence to Bui Van Tuyen.

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Appendices

Appendix 1

$$ E_{z1} = - \frac{1}{{k_{2}^{2} h}}\left( \begin{gathered} - 2\cosh \left( \frac{z}{h} \right)\varepsilon_{\rm{s}} f_{14} h^{3} k_{1} k_{2}^{2} - \sinh \left( \frac{z}{h} \right)e_{31} \varepsilon_{\rm{s}} h^{4} k_{1} k_{2}^{2} \hfill \\ + k_{2}^{2} \left\{ {k_{3} e_{31} z\left\{ {\varepsilon_{\rm{s}} .\left( {k_{0} + 1} \right) + \varepsilon_{\rm{b}} } \right\}h + \psi } \right\} \hfill \\ \end{gathered} \right); $$
$$ E_{z2} = - \frac{{\sinh \left( {\frac{{k_{2} h}}{2}} \right)}}{{k_{2}^{2} b_{33} h\left( {e^{{2k_{2} h}} - 1} \right)}}\left( \begin{gathered} \left( \begin{gathered} \varepsilon_{\rm{s}} hf_{14} \cosh \left( \frac{1}{2} \right) \hfill \\ - 4\varepsilon_{\rm{s}} h\left( {b_{33} k_{1} + \frac{1}{2}} \right)f_{14} \sinh \left( \frac{1}{2} \right) + \left\{ {\varepsilon_{\rm{s}} + \varepsilon_{\rm{b}} } \right\}f_{14} h \hfill \\ \end{gathered} \right)e^{{\frac{1}{2}k_{2} h}} \hfill \\ + 2\left( \begin{gathered} \frac{1}{2}\varepsilon_{\rm{s}} hf_{14} \cosh \left( \frac{1}{2} \right) \hfill \\ - 2\varepsilon_{\rm{s}} h\left( {b_{33} k_{1} + \frac{1}{2}} \right)f_{14} \sinh \left( \frac{1}{2} \right) + \frac{1}{2}\left\{ {\varepsilon_{\rm{s}} + \varepsilon_{\rm{b}} } \right\}f_{14} h \hfill \\ \end{gathered} \right)e^{{\frac{3}{2}k_{2} h}} \hfill \\ \end{gathered} \right); $$
$$ E_{z3} = \frac{{e^{{ - k_{2} z}} }}{{2\left( {e^{{2k_{2} h}} - 1} \right)k_{2} b_{33} }}\left( \begin{gathered} \left( \begin{gathered} - 2h\varepsilon_{\rm{s}} \left( {b_{33} e_{31} hk_{1} - \frac{1}{2}f_{14} } \right)\cosh \left( \frac{1}{2} \right) \hfill \\ - 4\varepsilon_{\rm{s}} h\left( {b_{33} k_{1} + \frac{1}{2}} \right)f_{14} \sinh \left( \frac{1}{2} \right) \hfill \\ + 2k_{3} e_{31} \left\{ {\varepsilon_{sx} .\left( {k_{0} + 1} \right) + \varepsilon_{\rm{b}} } \right\}b_{33} + \left\{ {\varepsilon_{\rm{s}} + \varepsilon_{\rm{b}} } \right\}f_{14} h \hfill \\ \end{gathered} \right)e^{{\frac{1}{2}k_{2} h}} \hfill \\ + 2\left( \begin{gathered} h\varepsilon_{\rm{s}} \left( {b_{33} e_{31} hk_{1} + \frac{1}{2}f_{14} } \right)\cosh \left( \frac{1}{2} \right) \hfill \\ - 2\varepsilon_{\rm{s}} h\left( {b_{33} k_{1} + \frac{1}{2}} \right)f_{14} \sinh \left( \frac{1}{2} \right) \hfill \\ - k_{3} e_{31} \left\{ {\varepsilon_{\rm{s}} .\left( {k_{0} + 1} \right) + \varepsilon_{\rm{b}} } \right\}b_{33} + \frac{1}{2}\left\{ {\varepsilon_{\rm{s}} + \varepsilon_{\rm{b}} } \right\}f_{14} h \hfill \\ \end{gathered} \right)e^{{\frac{3}{2}k_{2} h}} \hfill \\ \end{gathered} \right) $$
$$ E_{z4} = - \frac{{e^{{k_{2} z}} }}{{2\left( {e^{{2k_{2} h}} - 1} \right)k_{2} b_{33} }}\left( \begin{gathered} \left( \begin{gathered} - 2h\varepsilon_{\rm{s}} \left( {b_{33} e_{31} hk_{1} + \frac{1}{2}f_{14} } \right)\cosh \left( \frac{1}{2} \right) \hfill \\ + 4\varepsilon_{\rm{s}} h\left( {b_{33} k_{1} + \frac{1}{2}} \right)f_{14} \sinh \left( \frac{1}{2} \right) \hfill \\ + 2k_{3} e_{31} \left\{ {\varepsilon_{\rm{s}} .\left( {k_{0} + 1} \right) + \varepsilon_{\rm{b}} } \right\}b_{33} - \left\{ {\varepsilon_{\rm{s}} + \varepsilon_{\rm{b}} } \right\}f_{14} h \hfill \\ \end{gathered} \right)e^{{\frac{1}{2}k_{2} h}} \hfill \\ + 2\left( \begin{gathered} h\varepsilon_{\rm{s}} \left( {b_{33} e_{31} hk_{1} - \frac{1}{2}f_{14} } \right)\cosh \left( \frac{1}{2} \right) \hfill \\ + 2\varepsilon_{\rm{s}} h\left( {b_{33} k_{1} + \frac{1}{2}} \right)f_{14} \sinh \left( \frac{1}{2} \right) \hfill \\ - k_{3} e_{31} \left\{ {\varepsilon_{\rm{s}} .\left( {k_{0} + 1} \right) + \varepsilon_{\rm{b}} } \right\}b_{33} - \frac{1}{2}\left\{ {\varepsilon_{\rm{s}} + \varepsilon_{\rm{b}} } \right\}f_{14} h \hfill \\ \end{gathered} \right)e^{{\frac{3}{2}k_{2} h}} \hfill \\ \end{gathered} \right) $$

with

$$ k_{0} = \cosh \left( \frac{1}{2} \right);\;\;k_{1} = \frac{1}{{\kappa_{33} h^{2} - b_{33} }};\;\;k_{2} = \sqrt {\frac{{k_{33} }}{{b_{33} }}} ;\,\;k_{3} = \frac{1}{{\kappa_{33} }} $$

Appendix 2

$$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\varvec{E}}_{z1} = \left\{ {\frac{1}{{k_{2}^{2} h}}\left( \begin{gathered} 2\cosh \left( \frac{z}{h} \right)f_{14} h^{3} k_{1} k_{2}^{2} {\varvec{\varXi }}_{s2x} \hfill \\ + \sinh \left( \frac{z}{h} \right)e_{31} h^{4} k_{1} k_{2}^{2} {\varvec{\varXi }}_{s2x} \hfill \\ - k_{2}^{2} \left\{ {k_{3} e_{31} z\left\{ {{\varvec{\varXi }}_{s2x} .\left( {k_{0} + 1} \right) + {\varvec{\varXi }}_{b2x} } \right\}h} \right\} \hfill \\ \end{gathered} \right)} \right\}; $$
$$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\varvec{E}}_{z2} = \frac{{\sinh \left( {\frac{{k_{2} h}}{2}} \right)}}{{k_{2}^{2} b_{33} h\left( {e^{{2k_{2} h}} - 1} \right)}}\left( \begin{gathered} \left( \begin{gathered} - hf_{14} \cosh \left( \frac{1}{2} \right){\varvec{\varXi }}_{s2x} \hfill \\ + 4h\left( {b_{33} k_{1} + \frac{1}{2}} \right)f_{14} \sinh \left( \frac{1}{2} \right){\varvec{\varXi }}_{s2x} \hfill \\ - \left\{ {{\varvec{\varXi }}_{b2x} + {\varvec{\varXi }}_{s2x} } \right\}f_{14} h \hfill \\ \end{gathered} \right)e^{{\frac{1}{2}k_{2} h}} \hfill \\ + \left( \begin{gathered} - \varepsilon_{\rm{s}} hf_{14} \cosh \left( \frac{1}{2} \right){\varvec{\varXi }}_{s2x} \hfill \\ + 4h\left( {b_{33} k_{1} + \frac{1}{2}} \right)f_{14} \sinh \left( \frac{1}{2} \right){\varvec{\varXi }}_{s2x} \hfill \\ - \left\{ {{\varvec{\varXi }}_{b2x} + {\varvec{\varXi }}_{s2x} } \right\}f_{14} h \hfill \\ \end{gathered} \right)e^{{\frac{3}{2}k_{2} h}} \hfill \\ \end{gathered} \right); $$
$$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\varvec{E}}_{z3} = \frac{{e^{{ - k_{2} z}} }}{{2\left( {e^{{2k_{2} h}} - 1} \right)k_{2} b_{33} }}\left( \begin{gathered} \left( \begin{gathered} 2h\left( {b_{33} e_{31} hk_{1} - \frac{1}{2}f_{14} } \right)\cosh \left( \frac{1}{2} \right){\varvec{\varXi }}_{s2x} \hfill \\ + 4h\left( {b_{33} k_{1} + \frac{1}{2}} \right)f_{14} \sinh \left( \frac{1}{2} \right){\varvec{\varXi }}_{s2x} \hfill \\ - 2k_{3} e_{31} \left\{ {{\varvec{\varXi }}_{s2x} .\left( {k_{0} + 1} \right) + {\varvec{\varXi }}_{b2x} } \right\}b_{33} \hfill \\ - \left\{ {{\varvec{\varXi }}_{b2x} + {\varvec{\varXi }}_{s2x} } \right\}f_{14} h \hfill \\ \end{gathered} \right)e^{{\frac{1}{2}k_{2} h}} \hfill \\ + 2\left( \begin{gathered} - h\left( {b_{33} e_{31} hk_{1} + \frac{1}{2}f_{14} } \right)\cosh \left( \frac{1}{2} \right){\varvec{\varXi }}_{s2x} \hfill \\ + 2h\left( {b_{33} k_{1} + \frac{1}{2}} \right)f_{14} \sinh \left( \frac{1}{2} \right){\varvec{\varXi }}_{s2x} \hfill \\ + k_{3} e_{31} \left\{ {{\varvec{\varXi }}_{s2x} .\left( {k_{0} + 1} \right) + {\varvec{\varXi }}_{b2x} } \right\}b_{33} \hfill \\ - \frac{1}{2}\left\{ {{\varvec{\varXi }}_{b2x} + {\varvec{\varXi }}_{s2x} } \right\}f_{14} h \hfill \\ \end{gathered} \right)e^{{\frac{3}{2}k_{2} h}} \hfill \\ \end{gathered} \right); $$
$$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\varvec{E}}_{z4} = \frac{{f_{14} e^{{k_{2} z}} }}{{2\left( {e^{{2k_{2} h}} - 1} \right)k_{2} b_{33} }}\left( \begin{gathered} \left( \begin{gathered} 2h\left( {b_{33} e_{31} hk_{1} + \frac{1}{2}f_{14} } \right)\cosh \left( \frac{1}{2} \right){\varvec{\varXi }}_{s2x} \hfill \\ - 4h\left( {b_{33} k_{1} + \frac{1}{2}} \right)f_{14} \sinh \left( \frac{1}{2} \right){\varvec{\varXi }}_{s2x} \hfill \\ - 2k_{3} e_{31} \left\{ {{\varvec{\varXi }}_{s2x} .\left( {k_{0} + 1} \right) + {\varvec{\varXi }}_{b2x} } \right\}b_{33} \hfill \\ + \left\{ {{\varvec{\varXi }}_{s2x} + {\varvec{\varXi }}_{b2x} } \right\}f_{14} h \hfill \\ \end{gathered} \right)e^{{\frac{1}{2}k_{2} h}} \hfill \\ - 2\left( \begin{gathered} h\left( {b_{33} e_{31} hk_{1} - \frac{1}{2}f_{14} } \right)\cosh \left( \frac{1}{2} \right){\varvec{\varXi }}_{s2x} \hfill \\ 2h\left( {b_{33} k_{1} + \frac{1}{2}} \right)f_{14} \sinh \left( \frac{1}{2} \right){\varvec{\varXi }}_{s2x} \hfill \\ - k_{3} e_{31} \left\{ {{\varvec{\varXi }}_{s2x} .\left( {k_{0} + 1} \right) + {\varvec{\varXi }}_{b2x} } \right\}b_{33} \hfill \\ - \frac{1}{2}\left\{ {{\varvec{\varXi }}_{s2x} + {\varvec{\varXi }}_{b2x} } \right\}f_{14} h \hfill \\ \end{gathered} \right)e^{{\frac{3}{2}k_{2} h}} \hfill \\ \end{gathered} \right) $$
$$ {\varvec{K}}_{\rm{e}}^{\rm{Beam}} =\int\limits_{\Omega } {\left( {{\varvec{\varXi }}_{b2x}^{\rm T} z^{2} C_{11} {\varvec{\varXi }}_{b2x} + {\varvec{\varXi }}_{b2x}^{\rm T} zfC_{11} {\varvec{\varXi }}_{s2x} + {\varvec{\varXi }}_{s2x}^{\rm T} zfC_{11} {\varvec{\varXi }}_{b2x} + {\varvec{\varXi }}_{s2x}^{\rm T} f^{2} C_{11} {\varvec{\varXi }}_{s2x} + {\varvec{\varXi }}_{sx}^{\rm T} g_{z} c_{66} {\varvec{\varXi }}_{sx} } \right.} $$
$$ - \left( {z{\varvec{\varXi }}_{b2x} + f(z){\varvec{\varXi }}_{s2x} } \right)^{\rm T} e_{31} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\varvec{E}}_{z1} - \left( {z{\varvec{\varXi }}_{b2x} + f(z){\varvec{\varXi }}_{s2x} } \right)^{\rm T} e_{31} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\varvec{E}}_{z2} $$
$$ + \left( {z{\varvec{\varXi }}_{b2x} + f(z){\varvec{\varXi }}_{s2x} } \right)^{\rm T} e_{31} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\varvec{E}}_{z3} - \left( {z{\varvec{\varXi }}_{b2x} + f(z){\varvec{\varXi }}_{s2x} } \right)^{\rm T} e_{31} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\varvec{E}}_{z4} $$
$$ - \left( {{\varvec{\varXi }}_{b2x} + \frac{\partial f(z)}{{\partial z}}{\varvec{\varXi }}_{s2x} } \right)^{\rm T} f_{14} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\varvec{E}}_{z1} - \left( {{\varvec{\varXi }}_{b2x} + \frac{\partial f(z)}{{\partial z}}{\varvec{\varXi }}_{s2x} } \right)^{\rm T} f_{14} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\varvec{E}}_{z2} $$
$$ + \left( {{\varvec{\varXi }}_{b2x} + \frac{\partial f(z)}{{\partial z}}{\varvec{\varXi }}_{s2x} } \right)^{\rm T} f_{14} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\varvec{E}}_{z3} - \left( {{\varvec{\varXi }}_{b2x} + \frac{\partial f(z)}{{\partial z}}{\varvec{\varXi }}_{s2x} } \right)^{\rm T} \left. {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\varvec{E}}_{z4} } \right)\text{d}\Omega ; $$
$$ {\varvec{F}}_{\rm{e}}^{\psi } = \frac{1}{2}\int\limits_{\Omega } {\left( {\left( {z{\varvec{H}}_{b2x} + f(z){\varvec{H}}_{s2x} } \right)^{\rm T} e_{31} \left\{ {\frac{\psi }{h}} \right\} + \left( {{\varvec{\varXi }}_{b2x} + \frac{\partial f(z)}{{\partial z}}{\varvec{\varXi }}_{s2x} } \right)^{\rm T} f_{14} \left\{ {\frac{\psi }{h}} \right\}} \right)} \text{d}\Omega $$

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Luu, G.T., Civalek, Ö. & Van Tuyen, B. The mechanical response of nanobeams considering the flexoelectric phenomenon in the temperature environment. Arch Appl Mech 94, 493–514 (2024). https://doi.org/10.1007/s00419-023-02532-y

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