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Theoretical analysis of thermoelastic damping of silver nanobeam resonators based on Green–Naghdi via nonlocal elasticity with surface energy effects

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Abstract

The vibration of a silver nanobeam resonator, considering the surface effect as well as the thermal effect has been investigated in this paper. The emerged bending in nanobeam resonator causes the surface effects to appear in nanobeam. The governing equations for nanobeam are obtained considering the surface and thermal effects and using the nonlocal elasticity theory. The temperature effects based on the Green–Naghdi thermoelasticity theory, and considering the thermoelastic damping, are taken into account. The vibration governing equations are derived by the coupled Green–Naghdi thermoelastic, nonlocal elasticity theory, and surface effect for Euler–Bernoulli beam model. The dynamic and temperature responses of the nanobeam are obtained in the Laplace domain using the Laplace method. The technique of inverse Laplace, called a Talbot method, is utilized to calculate the dynamic and thermal responses of the nanobeam in the time domain. To investigate the results, the effects of the various parameters, such as the surface effects, nonlocal parameter, and the initial temperature conditions, on the dynamic and temperature responses of the microbeam are scrutinized.

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Correspondence to Seyyed Amirhosein Hosseini.

Appendix

Appendix

$$ Q_{1} = \frac{{\sqrt {\frac{{ - 2\left( { - 2} \right)^{1/3} \hat{A}^{2} + 6\left( { - 2} \right)^{1/3} \hat{A}\,\hat{C} - 2\hat{B}\beta^{1/3} + \left( { - 2} \right)^{2/3} \beta^{2/3} }}{{\hat{A}\beta^{1/3} }}} }}{\sqrt 6 } $$
$$ Q_{2} = \frac{1}{2\sqrt 3 }\left( {\sqrt {\frac{1}{{\hat{A}\beta^{1/3} }}\left( {2 \times 2^{1/3} {\text{i}}\left( {{\text{i}} + \sqrt 3 } \right)\hat{B}^{2} + 2^{1/3} \left( {6\hat{A}\left( {\hat{C} - {\text{i}}\sqrt 3 \hat{C}} \right) + 2^{1/3} \left( { - 1 - {\text{i}}\sqrt 3 } \right)\beta^{2/3} } \right) - 4\hat{B}\beta^{1/3} } \right)} } \right) $$
$$ Q_{3} = \frac{{\sqrt {\frac{{2 \times 2^{1/3} \hat{B}^{2} - 6 \times 2^{1/3} \hat{A}\,\hat{C} - 2\hat{B}\beta^{1/3} + 2^{2/3} \beta^{2/3} }}{{\hat{A}\beta^{1/3} }}} }}{\sqrt 6 } $$
$$ \beta = - 2\hat{B}^{3} + 9\hat{A}\hat{B}\hat{C} - 27\hat{A}^{2} \hat{D} + \sqrt {4\left( { - \hat{B}^{2} + 3\hat{A}\hat{C}} \right)^{3} + \left( { - 2\hat{B}^{3} + 9\hat{A}\hat{B}\hat{C} - 27\hat{A}^{2} \hat{D}} \right)^{2} } $$
$$ \varepsilon_{1} = \frac{{ - \mu_{1} }}{{\mu_{4} \chi_{1} }}\,\,\,\,\,\,\,\varepsilon_{2} = \frac{{ - \mu_{2} + \mu_{4} \chi_{2} }}{{\mu_{4} \chi_{1} }}\,\,\,\,\,\,\varepsilon_{3} = \frac{{ - \mu_{3} }}{{\mu_{4} \chi_{1} }} $$
$$ \begin{aligned} \Delta_{1} = \varepsilon_{1} Q_{1}^{4} + \varepsilon_{2} Q_{1}^{2} + \varepsilon_{3} \hfill \\ \Delta_{2} = \varepsilon_{1} Q_{2}^{4} + \varepsilon_{2} Q_{2}^{2} + \varepsilon_{3} \hfill \\ \Delta_{3} = \varepsilon_{1} Q_{3}^{4} + \varepsilon_{2} Q_{3}^{2} + \varepsilon_{3} \hfill \\ \end{aligned} $$
$$ \begin{aligned} V_{1} = \Delta_{1} A_{1} \,\,\,\,V_{2} = \Delta_{1} A_{2} \hfill \\ V_{3} = \Delta_{2} A_{3} \,\,\,\,V_{4} = \Delta_{2} A_{4} \hfill \\ V_{5} = \Delta_{3} A_{5} \,\,\,\,\,V_{6} = \Delta_{3} A_{6} \hfill \\ \end{aligned} $$

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Hamidi, B.A., Hosseini, S.A., Hassannejad, R. et al. Theoretical analysis of thermoelastic damping of silver nanobeam resonators based on Green–Naghdi via nonlocal elasticity with surface energy effects. Eur. Phys. J. Plus 135, 35 (2020). https://doi.org/10.1140/epjp/s13360-019-00037-8

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