A unified modified couple stress model for size-dependent free vibrations of FG cylindrical microshells based on high-order shear deformation theory

Abstract

The aim of present work is to propose a unified modified couple stress-based shear deformable shell model that has capability of analyzing size-dependent free vibrations of functionally graded (FG) cylindrical microshells. In the current model, the classical strain tensors and curvature tensor components are given within the framework of a unified high-order shear deformation theory. The trapezoidal shape factor is taken into account for achieving reasonable stress resultants over thickness and makes the proposed model more accurate for thick shell structures. Various classical boundary conditions of the cylindrical microshells are considered using the admissible functions of the displacements containing a beam modal function. The governing equations associated with free vibrations of microshells are derived using a standard Lagrange method. A few comparisons are conducted and verify that the results from the current model coincide favorably to the ones from three-dimensional elasticity solutions. The comparisons also prove that the proposed model has a sufficient capability of dealing with free vibration problems of the thick functionally graded cylindrical microshells. To the end, the size-dependent free vibrations of an FG cylindrical microshell are investigated. As new results, natural frequencies of the FG microshells with various boundary conditions are tabulated, and the effects of some important parameters on vibration characteristics of FG microshells are discussed.

Appendices

Appendix 1

The matrixes [C] and [A] in Eq. (20):

$$ \left[ {\mathbf{C}} \right] = \left[ {\begin{array}{*{20}c} {\frac{{\partial \xi_{u} }}{\partial x}} & {f\frac{{\partial \xi_{u} }}{\partial x}} & 0 & 0 & { - z\frac{{\partial^{2} \xi_{w} }}{{\partial x^{2} }}} \\ 0 & 0 & {\frac{1}{R}\frac{{\partial \xi_{v} }}{\partial \theta }} & {\frac{f}{R}\frac{{\partial \xi_{v} }}{\partial \theta }} & { - \frac{z}{{R\left( {R + z} \right)}}\frac{{\partial^{2} \xi_{w} }}{{\partial \theta^{2} }} + \frac{{\xi_{w} }}{R + z}} \\ {\frac{1}{R + z}\frac{{\partial \xi_{u} }}{\partial \theta }} & {\frac{f}{R + z}\frac{{\partial \xi_{u} }}{\partial \theta }} & {\left( {\frac{R + z}{R}} \right)\frac{{\partial \xi_{v} }}{\partial x}} & {\left( {\frac{R + z}{R}} \right)f\frac{{\partial \xi_{v} }}{\partial x}} & { - \frac{z}{R}\left( {\frac{2R + z}{R + z}} \right)\frac{{\partial^{2} \xi_{w} }}{\partial x\partial \theta }} \\ 0 & 0 & 0 & {\frac{R + z}{R}f'\xi_{v} } & 0 \\ 0 & {f^{{\prime }} \xi_{u} } & 0 & 0 & 0 \\ \end{array} } \right] $$

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$$ \left[ {\mathbf{A}} \right] = \left[ {\begin{array}{*{20}c} {A_{1} \left( t \right)} & {A_{1} \left( t \right)} & {A_{1} \left( t \right)} & {A_{1} \left( t \right)} & {A_{1} \left( t \right)} \\ \end{array} } \right]^{T} $$

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$$ \left[ {\varvec{\Sigma}} \right] = \left[ {\begin{array}{*{20}c} {Q_{11} } & {Q_{12} } & 0 & 0 & 0 \\ {Q_{12} } & {Q_{22} } & 0 & 0 & 0 \\ 0 & 0 & {Q_{66} } & 0 & 0 \\ 0 & 0 & 0 & {Q_{44} } & 0 \\ 0 & 0 & 0 & 0 & {Q_{55} } \\ \end{array} } \right]. $$

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Appendix 2

The matrix [S] in Eq. (21):

$$ \left[ {\mathbf{S}} \right] = \frac{1}{2}\left[ {\begin{array}{*{20}c} 0 & 0 & { - \frac{2}{R}\frac{{\partial \xi_{v} }}{\partial x}} & { - \left( {\frac{2f}{R} + \left( {1 + \frac{z}{R}} \right)f'} \right)\frac{{\partial \xi_{v} }}{\partial x}} & {\frac{2}{R}\frac{{\partial^{2} \xi_{w} }}{\partial x\partial \theta }} \\ { - \frac{1}{{\left( {R + z} \right)^{2} }}\frac{{\partial \xi_{u} }}{\partial \theta }} & {\left( {\frac{{f^{{\prime }} }}{R + z} - \frac{f}{{\left( {R + z} \right)^{2} }}} \right)\frac{{\partial \xi_{u} }}{\partial \theta }} & {\frac{1}{R}\frac{{\partial \xi_{v} }}{\partial x}} & {\frac{f}{R}\frac{{\partial \xi_{v} }}{\partial x}} & { - \left( {\frac{1}{R} + \frac{R}{{\left( {R + z} \right)^{2} }}} \right)\frac{{\partial^{2} \xi_{w} }}{\partial x\partial \theta }} \\ {\frac{1}{{\left( {R + z} \right)^{2} }}\frac{{\partial \xi_{u} }}{\partial \theta }} & {\left( {\frac{f}{{\left( {R + z} \right)^{2} }} - \frac{{f^{{\prime }} }}{R + z}} \right)\frac{{\partial \xi_{u} }}{\partial \theta }} & {\frac{1}{R}\frac{{\partial \xi_{v} }}{\partial x}} & {\left( {\frac{R + z}{R}f^{{\prime }} + \frac{1}{R}f} \right)\frac{{\partial \xi_{v} }}{\partial x}} & { - \frac{{z\left( {2R + z} \right)}}{{R\left( {R + z} \right)^{2} }}\frac{{\partial^{2} \xi_{w} }}{\partial x\partial \theta }} \\ 0 & {\frac{{f^{'} }}{2}\frac{{\partial \xi_{u} }}{\partial x}} & { - \frac{1}{{R\left( {R + z} \right)}}\frac{{\partial \xi_{v} }}{\partial \theta }} & { - \left( {\frac{1}{{R\left( {R + z} \right)}}f + \frac{1}{2R}f^{{\prime }} } \right)\frac{{\partial \xi_{v} }}{\partial \theta }} & { - \frac{{\partial^{2} \xi_{w} }}{{\partial x^{2} }} + \frac{1}{{R\left( {R + z} \right)}}\frac{{\partial^{2} \xi_{w} }}{{\partial \theta^{2} }}} \\ { - \frac{1}{{2\left( {R + z} \right)}}\frac{{\partial^{2} \xi_{u} }}{\partial x\partial \theta }} & { - \frac{f}{{2\left( {R + z} \right)}}\frac{{\partial^{2} \xi_{u} }}{\partial x\partial \theta }} & {\frac{R + z}{2R}\frac{{\partial^{2} \xi_{v} }}{{\partial x^{2} }}} & {\frac{R + z}{2R}f\frac{{\partial^{2} \xi_{v} }}{{\partial x^{2} }} - \frac{1}{2}\left( {f^{{\prime \prime }} + \frac{z}{R}f^{{\prime \prime }} + \frac{3}{R}f^{{\prime }} } \right)\xi_{v} } & { - \frac{{z^{2} }}{{2R\left( {R + z} \right)}}\frac{{\partial^{3} \xi_{w} }}{{\partial x^{2} \partial \theta }}} \\ { - \frac{1}{{2\left( {R + z} \right)^{2} }}\frac{{\partial^{2} \xi_{u} }}{{\partial \theta^{2} }}} & {\frac{1}{2}\left( {f'' - \frac{{f^{{\prime }} }}{R + z}} \right)\xi_{u} - \frac{f}{{2\left( {R + z} \right)^{2} }}\frac{{\partial^{2} \xi_{u} }}{{\partial \theta^{2} }}} & {\frac{1}{2R}\frac{{\partial^{2} \xi_{v} }}{\partial x\partial \theta }} & {\frac{f}{2R}\frac{{\partial^{2} \xi_{v} }}{\partial x\partial \theta }} & {\frac{1}{R + z}\frac{{\partial \xi_{w} }}{\partial x} - \frac{{z^{2} }}{{2R\left( {R + z} \right)^{2} }}\frac{{\partial^{3} \xi_{w} }}{{\partial x\partial \theta^{2} }}} \\ \end{array} } \right] $$

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$$ \left[ {\varvec{\Xi}} \right] = 2\mu \left( z \right)l^{2} {\mathbf{diag}}\left( {\begin{array}{*{20}c} {1,} & {1,} & {1,} & {2,} & {2,} & 2 \\ \end{array} } \right). $$

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Appendix 3

The matrix [M₀] in Eq. (24):

$$ \left[ {{\mathbf{M}}_{{\mathbf{0}}} } \right] = \left[ {\begin{array}{*{20}c} {\xi_{u}^{2} } & {f\xi_{u}^{2} } & 0 & 0 & { - z\xi_{u} \frac{{\partial \xi_{w} }}{\partial x}} \\ {f\xi_{u}^{2} } & {\left( {f\xi_{u} } \right)^{2} } & 0 & 0 & { - zf\xi_{u} \frac{{\partial \xi_{w} }}{\partial x}} \\ 0 & 0 & {\left( {\frac{R + z}{R}\xi_{v} } \right)^{2} } & {f\left( {\frac{R + z}{R}\xi_{v} } \right)^{2} } & { - \frac{{\left( {R + z} \right)z}}{{R^{2} }}\xi_{v} \frac{{\partial \xi_{w} }}{\partial \theta }} \\ 0 & 0 & {f\left( {\frac{R + z}{R}\xi_{v} } \right)^{2} } & {\left( {\frac{R + z}{R}f\xi_{v} } \right)^{2} } & { - \frac{{\left( {R + z} \right)z}}{{R^{2} }}f\xi_{v} \frac{{\partial \xi_{w} }}{\partial \theta }} \\ { - z\xi_{u} \frac{{\partial \xi_{w} }}{\partial x}} & { - zf\xi_{u} \frac{{\partial \xi_{w} }}{\partial x}} & { - \frac{{\left( {R + z} \right)z}}{{R^{2} }}\xi_{v} \frac{{\partial \xi_{w} }}{\partial \theta }} & { - \frac{{\left( {R + z} \right)z}}{{R^{2} }}f\xi_{v} \frac{{\partial \xi_{w} }}{\partial \theta }} & {\left( {z\frac{{\partial \xi_{w} }}{\partial x}} \right)^{2} + \left( {\frac{z}{R}\frac{{\partial \xi_{w} }}{\partial \theta }} \right)^{2} + \left( {\xi_{w} } \right)^{2} } \\ \end{array} } \right]. $$

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Appendix 4

The matrix [K] and [M] in Eq. (28):

$$ \left[ {\mathbf{K}} \right] = \int\limits_{0}^{L} {\int\limits_{0}^{2\pi } {\int\limits_{ - h/2}^{h/2} {\left\{ {\left[ {\mathbf{C}} \right]^{\text{T}} \left[ {\varvec{\Sigma}} \right]\left[ {\mathbf{C}} \right] + \left[ {\mathbf{S}} \right]^{\text{T}} \left[ {\varvec{\Xi}} \right]\left[ {\mathbf{S}} \right]} \right\}\left( {1 + \frac{z}{R}} \right)R{\text{d}}z{\text{d}}\theta {\text{d}}x} } } $$

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$$ \left[ {\mathbf{M}} \right] = \int\limits_{0}^{L} {\int\limits_{0}^{2\pi } {\int\limits_{ - h/2}^{h/2} {\left\{ {\rho \left( z \right)\left[ {{\mathbf{M}}_{{\mathbf{0}}} } \right]} \right\}\left( {1 + \frac{z}{R}} \right)R{\text{d}}z{\text{d}}\theta {\text{d}}x} } } . $$

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