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Size-dependent vibration and instability of fluid-conveying functionally graded microshells based on the modified couple stress theory

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Abstract

In this article, the vibration and dynamic instability of cylindrical microshells made of functionally graded materials (FGMs) and containing flowing fluid are studied. In order to take the size effects into account, the modified couple stress elasticity theory is used in conjunction with the classical first-order shear deformation shell theory. The material properties of FGM microshells are considered to be graded in the thickness direction on the basis of the power-law function. By using Hamilton’s principle, the non-classical governing differential equations of motion and related boundary conditions are derived. Subsequently, a Navier-type exact solution method is carried out to obtain the imaginary and real parts of natural frequencies of different modes for various values of fluid velocity, length scale parameter, material property gradient index, compressive axial load, and length-to-radius ratio. It is found that for microshells with lower length-to-radius ratios, the system diverges at lower values of fluid velocity. Also, it is demonstrated that by increasing the value of material property gradient index of FGM microshell, the natural frequency of the first mode and the critical flow velocity of the system increase.

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, Iran

    R. Ansari, A. Norouzzadeh & S. Sahmani

  2. Department of Mechanical Engineering, Lahijan Branch, Islamic Azad University, P.O. Box 1616, Lahijan, Iran

    R. Gholami

Authors
  1. R. Ansari
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  2. R. Gholami
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  3. A. Norouzzadeh
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  4. S. Sahmani
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Corresponding authors

Correspondence to R. Ansari or R. Gholami.

Appendix 1

Appendix 1

$$\begin{aligned} M_{11} = &\,I_{1} ,M_{14} = I_{2} ,M_{22} = I_{1} ,M_{25} = I_{2} ,M_{33} = I_{1} + F_{mn} , \\ M_{41} = &\, I_{2} ,M_{44} = I_{3} ,M_{52} = I_{2} ,M_{55} = I_{3} \\ C_{33} = &\, \frac{{8F_{mn} mU}}{L} \\ \end{aligned}$$

The classical components:

$$\begin{aligned} K_{11}^{{\prime }} = &\, A_{11} \alpha_{m}^{2} + A_{55} \beta_{n}^{2} ,K_{12}^{{\prime }} = \left( {A_{12} + A_{55} } \right)\alpha_{m} \beta_{n} ,K_{13}^{{\prime }} = - \frac{{A_{12} \alpha_{m} }}{R}, \\ K_{14}^{{\prime }} = &\, B_{11} \alpha_{m}^{2} + B_{55} \beta_{n}^{2} ,K_{15}^{{\prime }} = (B_{12} + B_{55} )\alpha_{m} \beta_{n} , \\ K_{21}^{{\prime }} = &\, (A_{12} + A_{55} )\alpha_{m} \beta_{n} ,K_{22}^{{\prime }} = A_{11} \beta_{n}^{2} + A_{55} \alpha_{m}^{2} + \frac{{k_{s} A_{55} }}{{R^{2} }},K_{23}^{'} = - \frac{{k_{s} A_{55} + A_{11} }}{R}\beta_{n} , \\ K_{24}^{{\prime }} = &\, (B_{12} + B_{55} )\alpha_{m} \beta_{n} ,K_{25}^{{\prime }} = B_{11} \beta_{n}^{2} + B_{55} \alpha_{m}^{2} - \frac{{k_{s} A_{55} }}{R}, \\ K_{31}^{{\prime }} = &\, - \frac{{A_{12} \alpha_{m} }}{R},K_{32}^{{\prime }} = - \frac{{k_{s} A_{55} + A_{11} }}{R}\beta_{n} , \\ K_{33}^{{\prime }} = &\, k_{s} A_{55} \left( {\alpha_{m}^{2} + \beta_{n}^{2} } \right) + N_{xx}^{0} \alpha_{m}^{2} + \frac{{A_{11} }}{{R^{2} }} - F_{mn} \alpha_{m}^{2} U^{2} , \\ K_{34}^{{\prime }} = &\, k_{s} A_{55} \alpha_{m} - \frac{{B_{12} \alpha_{m} }}{R},K_{35}^{{\prime }} = k_{s} A_{55} \beta_{n} - \frac{{B_{11} \beta_{n} }}{R}, \\ K_{41}^{{\prime }} = &\, B_{11} \alpha_{m}^{2} + B_{55} \beta_{n}^{2} ,K_{42}^{{\prime }} = (B_{12} + B_{55} )\alpha_{m} \beta_{n} ,K_{43}^{{\prime }} = k_{s} A_{55} \alpha_{m} - \frac{{B_{12} \alpha_{m} }}{R}, \\ K_{44}^{{\prime }} = &\, D_{11} \alpha_{m}^{2} + D_{55} \beta_{n}^{2} + k_{s} A_{55} ,K_{45}^{{\prime }} = (D_{12} + D_{55} )\alpha_{m} \beta_{n} , \\ K_{51}^{{\prime }} = &\,\, (B_{12} + B_{55} )\alpha_{m} \beta_{n} ,K_{52}^{{\prime }} = B_{11} \beta_{n}^{2} + B_{55} \alpha_{m}^{2} - \frac{{k_{s} A_{55} }}{R},K_{53}^{{\prime }} = k_{s} A_{55} \beta_{n} - \frac{{B_{11} \beta_{n} }}{R}, \\ K_{54}^{{\prime }} = &\, (D_{12} + D_{55} )\alpha_{m} \beta_{n} ,K_{55}^{{\prime }} = D_{11} \beta_{n}^{2} + D_{55} \alpha_{m}^{2} + k_{s} A_{55} \\ \end{aligned}$$

The non-classical components:

$$\begin{aligned} K_{11}^{{\prime \prime }} = &\, \frac{1}{4}A_{55} l^{2} \left( {\alpha_{m}^{2} \beta_{n}^{2} + \beta_{n}^{4} + 4\frac{{\beta_{n}^{2} }}{{R^{2} }}} \right),K_{12}^{{\prime \prime }} = - \frac{1}{4}A_{55} l^{2} \left( {\alpha_{m}^{3} \beta_{n} + \alpha_{m} \beta_{n}^{3} + \frac{{\alpha_{m} \beta_{n} }}{{R^{2} }}} \right), \\ K_{13}^{{\prime \prime }} = &\, \frac{1}{2}A_{55} l^{2} \frac{{\alpha_{m} \beta_{n}^{2} }}{R},K_{14}^{{\prime \prime }} = - \frac{5}{4}A_{55} l^{2} \frac{{\beta_{n}^{2} }}{R} + \frac{1}{4}B_{55} l^{2} \left( {\alpha_{m}^{2} \beta_{n}^{2} + \beta_{n}^{4} } \right), \\ K_{15}^{{\prime \prime }} = &\, \frac{1}{4}A_{55} l^{2} \frac{{\alpha_{m} \beta_{n} }}{R} - \frac{1}{4}B_{55} l^{2} \left( {\alpha_{m}^{3} \beta_{n} + \alpha_{m} \beta_{n}^{3} } \right), \\ K_{21}^{{\prime \prime }} = &\, - \frac{1}{4}A_{55} l^{2} \left( {\alpha_{m}^{3} \beta_{n} + \alpha_{m} \beta_{n}^{3} + \frac{{\alpha_{m} \beta_{n} }}{{R^{2} }}} \right), \\ K_{22}^{{\prime \prime }} = &\, \frac{1}{4}A_{55} l^{2} \left( {\alpha_{m}^{4} + \alpha_{m}^{2} \beta_{n}^{2} + 2\frac{{\alpha_{m}^{2} }}{{R^{2} }} + \frac{{\beta_{n}^{2} }}{{R^{2} }} + \frac{1}{{R^{4} }}} \right), \\ K_{23}^{{\prime \prime }} = &\, - \frac{1}{4}A_{55} l^{2} \left( {3\frac{{\alpha_{m}^{2} \beta_{n} }}{R} + \frac{{\beta_{n}^{3} }}{R} + \frac{{\beta_{n} }}{{R^{3} }}} \right), \\ K_{24}^{{\prime \prime }} = &\, \frac{1}{2}A_{55} l^{2} \frac{{\alpha_{m} \beta_{n} }}{R} + \frac{1}{4}B_{55} l^{2} \left( { - \alpha_{m}^{3} \beta_{n} - \alpha_{m} \beta_{n}^{3} + \frac{{\alpha_{m} \beta_{n} }}{{R^{2} }}} \right), \\ K_{25}^{{\prime \prime }} = &\, \frac{1}{4}A_{55} l^{2} \left( {3\frac{{\alpha_{m}^{2} }}{R} + \frac{{\beta_{n}^{2} }}{R} - \frac{1}{{R^{3} }}} \right) + \frac{1}{4}B_{55} l^{2} \left( {\alpha_{m}^{4} + \alpha_{m}^{2} \beta_{n}^{2} - \frac{{\alpha_{m}^{2} }}{{R^{2} }}} \right), \\ K_{31}^{{\prime \prime }} = &\, \frac{1}{2}A_{55} l^{2} \frac{{\alpha_{m} \beta_{n}^{2} }}{R},K_{32}^{{\prime \prime }} = - \frac{1}{4}A_{55} l^{2} \left( { - \frac{{\alpha_{m}^{2} \beta_{n} }}{R} + \frac{{\beta_{n}^{3} }}{R} + \frac{{\beta_{n} }}{{R^{3} }}} \right), \\ K_{33}^{{\prime \prime }} = &\, \frac{1}{4}A_{55} l^{2} \left( {\alpha_{m}^{4} - 2\alpha_{m}^{2} \beta_{n}^{2} + \beta_{n}^{4} + \frac{{\alpha_{m}^{2} }}{{R^{2} }} + \frac{{\beta_{n}^{2} }}{{R^{2} }}} \right), \\ K_{34}^{{\prime \prime }} = &\, - \frac{1}{4}A_{55} l^{2} \left( {\alpha_{m}^{3} + \alpha_{m} \beta_{n}^{2} + \frac{{\alpha_{m} }}{{R^{2} }}} \right),K_{35}^{{\prime \prime }} = \frac{1}{4}A_{55} l^{2} \left( {3\alpha_{m}^{2} \beta_{n} - \beta_{n}^{3} + \frac{{\beta_{n} }}{{R^{2} }}} \right), \\ K_{41}^{{\prime \prime }} = &\, - \frac{5}{4}A_{55} l^{2} \frac{{\beta_{n}^{2} }}{R} + \frac{1}{4}B_{55} l^{2} \left( {\alpha_{m}^{2} \beta_{n}^{2} + \beta_{n}^{4} } \right), \\ K_{42}^{{\prime \prime }} = &\, \frac{1}{2}A_{55} l^{2} \frac{{\alpha_{m} \beta_{n} }}{R} + \frac{1}{4}B_{55} l^{2} \left( { - \alpha_{m}^{3} \beta_{n} - \alpha_{m} \beta_{n}^{3} + \frac{{\alpha_{m} \beta_{n} }}{{R^{2} }}} \right), \\ K_{44}^{{\prime \prime }} = &\, \frac{1}{4}A_{55} l^{2} \left( {\alpha_{m}^{2} + 4\beta_{n}^{2} + \frac{1}{{R^{2} }}} \right) - \frac{1}{2}B_{55} l^{2} \frac{{\beta_{n}^{2} }}{R} + \frac{1}{4}D_{55} l^{2} \left( {\alpha_{m}^{2} \beta_{n}^{2} + \beta_{n}^{4} } \right), \\ K_{45}^{{\prime \prime }} = &\, - \frac{3}{4}A_{55} l^{2} \alpha_{m} \beta_{n} - \frac{1}{4}D_{55} l^{2} \left( {\alpha_{m}^{3} \beta_{n} + \alpha_{m} \beta_{n}^{3} } \right), \\ K_{51}^{{\prime \prime }} = &\, \frac{1}{4}A_{55} l^{2} \frac{{\alpha_{m} \beta_{n} }}{R} - \frac{1}{4}B_{55} l^{2} \left( {\alpha_{m}^{3} \beta_{n} + \alpha_{m} \beta_{n}^{3} } \right), \\ K_{52}^{{\prime \prime }} = &\, \frac{1}{4}A_{55} l^{2} \left( {3\frac{{\alpha_{m}^{2} }}{R} + \frac{{\beta_{n}^{2} }}{R} - \frac{1}{{R^{3} }}} \right) + \frac{1}{4}B_{55} l^{2} \left( {\alpha_{m}^{4} + \alpha_{m}^{2} \beta_{n}^{2} - \frac{{\alpha_{m}^{2} }}{{R^{2} }}} \right), \\ K_{53}^{{\prime \prime }} = &\, \frac{1}{4}A_{55} l^{2} \left( { - \alpha_{m}^{2} \beta_{n} - \beta_{n}^{3} + \frac{{\beta_{n} }}{{R^{2} }}} \right), \\ K_{54}^{{\prime \prime }} = &\, - \frac{3}{4}A_{55} l^{2} \alpha_{m} \beta_{n} - \frac{1}{4}D_{55} l^{2} \left( {\alpha_{m}^{3} \beta_{n} + \alpha_{m} \beta_{n}^{3} } \right), \\ K_{55}^{{\prime \prime }} = &\, \frac{1}{4}A_{55} l^{2} \left( {4\alpha_{m}^{2} + \beta_{n}^{2} + \frac{1}{{R^{2} }}} \right) + \frac{1}{2}B_{55} l^{2} \frac{{\alpha_{m}^{2} }}{R} + \frac{1}{4}D_{55} l^{2} \left( {\alpha_{m}^{4} + \alpha_{m}^{2} \beta_{n}^{2} } \right) \\ \end{aligned}$$

where \(K_{ij} = K_{ij}^{{\prime }} + K_{ij}^{{\prime \prime }}\).

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Ansari, R., Gholami, R., Norouzzadeh, A. et al. Size-dependent vibration and instability of fluid-conveying functionally graded microshells based on the modified couple stress theory. Microfluid Nanofluid 19, 509–522 (2015). https://doi.org/10.1007/s10404-015-1577-1

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