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Postbuckling analysis of hydrostatic pressurized FGM microsized shells including strain gradient and stress-driven nonlocal effects

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Abstract

Herein, with the aid of the newly proposed theory of nonlocal strain gradient elasticity, the size-dependent nonlinear buckling and postbuckling behavior of microsized shells made of functionally graded material (FGM) and subjected to hydrostatic pressure is examined. As a consequence, the both nonlocality and strain gradient micro-size dependency are incorporated to an exponential shear deformation shell theory to construct a more comprehensive size-dependent shell model with a refined distribution of shear deformation. The Mori–Tanaka homogenization scheme is utilized to estimate the effective material properties of FGM nanoshells. After deduction of the non-classical governing differential equations via boundary layer theory of shell buckling, a perturbation-based solving process is employed to extract explicit expressions for nonlocal strain gradient stability paths of hydrostatic pressurized FGM microsized shells. It is observed that the nonlocality size effect causes to decrease the critical hydrostatic pressure and associated end-shortening of microsized shells, while the strain gradient size dependency leads to increase them. In addition, it is found that the influence of the internal strain gradient length scale parameter on the nonlinear instability characteristics of hydrostatic pressurized FGM microsized shells is a bit more than that of the nonlocal one.

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Author information

Authors and Affiliations

  1. School of Mechanical Engineering, Xi’an Jiaotong University, 28 West Xianning Road, Xi’an, 710049, Shaanxi, China

    Xiaohui Yang

  2. School of Science and Technology, The University of Georgia, Tbilisi, 0171, Georgia

    Saeid Sahmani

  3. School of Mechanical Engineering, Eastern Mediterranean University, G. Magosa, TRNC Mersin 10, Turkey

    Babak Safaei

Authors
  1. Xiaohui Yang
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  2. Saeid Sahmani
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  3. Babak Safaei
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Correspondence to Xiaohui Yang.

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Appendices

Appendix A

$$\varphi_{1} = \frac{{A_{11}^{*} }}{{\left( {A_{11}^{*} } \right)^{2} - \left( {A_{12}^{*} } \right)^{2} }},\quad \varphi_{2} = \frac{{A_{12}^{*} }}{{\left( {A_{11}^{*} } \right)^{2} - \left( {A_{12}^{*} } \right)^{2} }},\quad \varphi_{3} = \frac{{A_{11}^{*} B_{11}^{*} - A_{12}^{*} B_{12}^{*} }}{{\left( {A_{11}^{*} } \right)^{2} - \left( {A_{12}^{*} } \right)^{2} }}$$
$$\varphi_{4} = \frac{{A_{11}^{*} B_{12}^{*} - A_{12}^{*} B_{11}^{*} }}{{\left( {A_{11}^{*} } \right)^{2} - \left( {A_{12}^{*} } \right)^{2} }},\quad \varphi_{5} = \frac{{A_{11}^{*} B_{11}^{**} - A_{12}^{*} B_{12}^{**} }}{{\left( {A_{11}^{*} } \right)^{2} - \left( {A_{12}^{*} } \right)^{2} }},\quad \varphi_{6} = \frac{{A_{11}^{*} B_{12}^{**} - A_{12}^{*} B_{11}^{**} }}{{\left( {A_{11}^{*} } \right)^{2} - \left( {A_{12}^{*} } \right)^{2} }}$$
$$\varphi_{7} = \frac{1}{{A_{66}^{*} }},\quad \varphi_{8} = \frac{{B_{66}^{*} }}{{A_{66}^{*} }},\quad \varphi_{9} = \frac{{B_{66}^{**} }}{{A_{66}^{*} }}$$
$$\varphi_{10} = D_{11}^{*} - B_{11}^{*} \varphi_{3} - B_{12}^{*} \varphi_{4} ,\quad \varphi_{11} = D_{12}^{*} - B_{11}^{*} \varphi_{3} - B_{12}^{*} \varphi_{4}$$
$$\varphi_{12} = D_{66}^{*} - B_{66}^{*} \varphi_{8} ,\quad \varphi_{13} = B_{11}^{*} \varphi_{5} + B_{12}^{*} \varphi_{6} - D_{11}^{**}$$
(38)
$$\varphi_{14} = B_{12}^{*} \varphi_{5} + B_{11}^{*} \varphi_{6} - D_{12}^{**} ,\quad \varphi_{15} = B_{66}^{**} \varphi_{9} - D_{66}^{**}$$
$$\varphi_{16} = B_{12}^{**} \varphi_{1} - B_{11}^{**} \varphi_{2} , \varphi_{17} = B_{11}^{**} \varphi_{1} - B_{12}^{**} \varphi_{2} - B_{66}^{**} \varphi_{7}$$
$$\varphi_{18} = B_{11}^{**} \varphi_{3} + B_{12}^{**} \varphi_{4} - D_{11}^{**} ,\quad \varphi_{19} = B_{12}^{**} \varphi_{3} + B_{11}^{**} \varphi_{4} - D_{12}^{**}$$
$$\varphi_{20} = B_{66}^{**} \varphi_{8} - D_{66}^{** } ,\quad \varphi_{21} = G_{11}^{*} - B_{11}^{**} \varphi_{5} - B_{12}^{**} \varphi_{6}$$
$$\varphi_{22} = G_{66}^{*} - B_{66}^{**} \varphi_{9} ,\quad \varphi_{23} = G_{12}^{*} - B_{11}^{**} \varphi_{6} - B_{12}^{**} \varphi_{5} .$$

This point should be noted that the parameters of \(\vartheta_{i} \left( {i = 1, \ldots ,23} \right)\) are the dimensionless form of \(\varphi_{i}\).

The solutions in asymptotic forms corresponding to each of independent variables are extracted as below:

$$\begin{aligned} W = {\mathcal{A}}_{00}^{\left( 0 \right)} & + \epsilon^{3/2} \left[ {{\mathcal{A}}_{00}^{{\left( {3/2} \right)}} - {\mathcal{A}}_{00}^{{\left( {3/2} \right)}} \left( {\sin \left( {\frac{{\varGamma_{1} X}}{\sqrt \epsilon }} \right) + { \cos }\left( {\frac{{\varGamma_{1} X}}{\sqrt \epsilon }} \right)} \right)e^{{ - \frac{{\varGamma_{2} X}}{\sqrt \epsilon }}} - {\mathcal{A}}_{00}^{{\left( {3/2} \right)}} \left( {\sin \left( {\frac{{\varGamma_{1} \left( {\pi - X} \right)}}{\sqrt \epsilon }} \right) + { \cos }\left( {\frac{{\varGamma_{1} \left( {\pi - X} \right)}}{\sqrt \epsilon }} \right)} \right)e^{{ - \frac{{\varGamma_{2} \left( {\pi - X} \right)}}{\sqrt \epsilon }}} } \right] \\ & + \epsilon^{2} \left[ {{\mathcal{A}}_{00}^{\left( 2 \right)} + {\mathcal{A}}_{11}^{\left( 2 \right)} \sin \left( {mX} \right)\sin \left( {nY} \right) - {\mathcal{A}}_{00}^{\left( 2 \right)} \left( {\sin \left( {\frac{{\varGamma_{1} X}}{\sqrt \epsilon }} \right) + { \cos }\left( {\frac{{\varGamma_{1} X}}{\sqrt \epsilon }} \right)} \right)e^{{ - \frac{{\varGamma_{2} X}}{\sqrt \epsilon }}} - {\mathcal{A}}_{00}^{\left( 2 \right)} \left( {\sin \left( {\frac{{\varGamma_{1} \left( {\pi - X} \right)}}{\sqrt \epsilon }} \right) + { \cos }\left( {\frac{{\varGamma_{1} \left( {\pi - X} \right)}}{\sqrt \epsilon }} \right)} \right)e^{{ - \frac{{\varGamma_{2} \left( {\pi - X} \right)}}{\sqrt \epsilon }}} } \right] \\ & + \epsilon^{3} \left[ {{\mathcal{A}}_{00}^{\left( 3 \right)} + {\mathcal{A}}_{11}^{\left( 3 \right)} \sin \left( {mX} \right)\sin \left( {nY} \right)} \right] \\ & + \epsilon^{4} \left[ {{\mathcal{A}}_{00}^{\left( 4 \right)} + {\mathcal{A}}_{11}^{\left( 4 \right)} \sin \left( {mX} \right)\sin \left( {nY} \right) + {\mathcal{A}}_{20}^{\left( 4 \right)} \cos \left( {2mX} \right) + {\mathcal{A}}_{02}^{\left( 4 \right)} { \cos }\left( {2nY} \right)} \right] + O\left( {\epsilon^{5} } \right) \\ \end{aligned}$$
(39)
$$\begin{aligned} F = & - {\mathcal{B}}_{00}^{\left( 0 \right)} \left( {\beta^{2} X^{2} + \frac{{Y^{2} }}{2}} \right) + \epsilon \left[ { - {\mathcal{B}}_{00}^{\left( 1 \right)} \left( {\beta^{2} X^{2} + \frac{{Y^{2} }}{2}} \right)} \right] \\ & + \epsilon^{2} \left[ { - {\mathcal{B}}_{00}^{\left( 2 \right)} \left( {\beta^{2} X^{2} + \frac{{Y^{2} }}{2}} \right) + {\mathcal{B}}_{11}^{\left( 2 \right)} \sin \left( {mX} \right)\sin \left( {nY} \right)} \right] \\ & + \epsilon^{5/2} \left[ {{\mathcal{A}}_{00}^{{\left( {3/2} \right)}} \left( {b_{10}^{\left( 2 \right)} \sin \left( {\frac{{\varGamma_{1} X}}{\sqrt \epsilon }} \right) + b_{01}^{\left( 2 \right)} { \cos }\left( {\frac{{\varGamma_{1} X}}{\sqrt \epsilon }} \right)} \right)e^{{ - \frac{{\varGamma_{2} X}}{\sqrt \epsilon }}} + {\mathcal{A}}_{00}^{{\left( {3/2} \right)}} \left( {b_{10}^{\left( 2 \right)} \sin \left( {\frac{{\varGamma_{1} \left( {\pi - X} \right)}}{\sqrt \epsilon }} \right) + b_{01}^{\left( 2 \right)} { \cos }\left( {\frac{{\varGamma_{1} \left( {\pi - X} \right)}}{\sqrt \epsilon }} \right)} \right)e^{{ - \frac{{\varGamma_{2} \left( {\pi - X} \right)}}{\sqrt \epsilon }}} } \right] \\ & + \epsilon^{3} \left[ { - {\mathcal{B}}_{00}^{\left( 3 \right)} \left( {\beta^{2} X^{2} + \frac{{Y^{2} }}{2}} \right) + {\mathcal{A}}_{00}^{\left( 2 \right)} \left( {b_{10}^{\left( 3 \right)} \sin \left( {\frac{{\varGamma_{1} X}}{\sqrt \epsilon }} \right) + b_{01}^{\left( 3 \right)} { \cos }\left( {\frac{{\varGamma_{1} X}}{\sqrt \epsilon }} \right)} \right)e^{{ - \frac{{\varGamma_{2} X}}{\sqrt \epsilon }}} + {\mathcal{A}}_{00}^{\left( 2 \right)} \left( {b_{10}^{\left( 3 \right)} \sin \left( {\frac{{\varGamma_{1} \left( {\pi - X} \right)}}{\sqrt \epsilon }} \right) + b_{01}^{\left( 3 \right)} { \cos }\left( {\frac{{\varGamma_{1} \left( {\pi - X} \right)}}{\sqrt \epsilon }} \right)} \right)e^{{ - \frac{{\varGamma_{2} \left( {\pi - X} \right)}}{\sqrt \epsilon }}} } \right] \\ & + \epsilon^{4} \left[ { - {\mathcal{B}}_{00}^{\left( 4 \right)} \left( {\beta^{2} X^{2} + \frac{{Y^{2} }}{2}} \right) + {\mathcal{B}}_{20}^{\left( 4 \right)} \cos \left( {2mX} \right) + {\mathcal{B}}_{02}^{\left( 4 \right)} \cos \left( {2nY} \right)} \right] + O\left( {\epsilon^{5} } \right) \\ \end{aligned}$$
(40)
$$\begin{aligned} \varPsi_{X} & = \epsilon^{2} \begin{array}{*{20}c} {\left[ {{\mathcal{C}}_{11}^{\left( 2 \right)} \cos \left( {mX} \right){ \sin }\left( {nY} \right) + \left( {c_{10}^{\left( 2 \right)} \sin \left( {\frac{{\varGamma_{1} X}}{\sqrt \epsilon }} \right) + c_{01}^{\left( 2 \right)} \cos \left( {\frac{{\varGamma_{1} X}}{\sqrt \epsilon }} \right)} \right)e^{{ - \frac{{\varGamma_{2} X}}{\sqrt \epsilon }}} } \right.} \\ {\left. { + \left( {c_{10}^{\left( 2 \right)} \sin \left( {\frac{{\varGamma_{1} \left( {\pi - X} \right)}}{\sqrt \epsilon }} \right) + c_{01}^{\left( 2 \right)} \cos \left( {\frac{{\varGamma_{1} \left( {\pi - X} \right)}}{\sqrt \epsilon }} \right)} \right)e^{{ - \frac{{\varGamma_{2} \left( {\pi - X} \right)}}{\sqrt \epsilon }}} } \right]} \\ \end{array} \\ & \quad + \epsilon^{3} \left[ {{\mathcal{C}}_{11}^{\left( 3 \right)} \cos \left( {mX} \right){ \sin }\left( {nY} \right)} \right] + \epsilon^{4} \left[ {{\mathcal{C}}_{11}^{\left( 4 \right)} \cos \left( {mX} \right){ \sin }\left( {nY} \right) + {\mathcal{C}}_{20}^{\left( 4 \right)} { \sin }\left( {2mX} \right)} \right] + O\left( {\epsilon^{5} } \right) \\ \end{aligned}$$
(41)
$$\begin{aligned} \varPsi_{Y} & = \epsilon^{2} \left[ {{\mathcal{D}}_{11}^{\left( 2 \right)} \sin \left( {mX} \right){ \cos }\left( {nY} \right)} \right] + \epsilon^{3} \left[ {{\mathcal{D}}_{11}^{\left( 3 \right)} \sin \left( {mX} \right){ \cos }\left( {nY} \right)} \right] \\ & \quad + \epsilon^{4} \left[ {{\mathcal{D}}_{11}^{\left( 4 \right)} \sin \left( {mX} \right){ \cos }\left( {nY} \right) + {\mathcal{D}}_{02}^{\left( 4 \right)} \sin \left( {2nY} \right)} \right] + O\left( {\epsilon^{5} } \right) , \\ \end{aligned}$$
(42)

in which

$$\varGamma_{1} = \sqrt {\frac{{\sqrt {\frac{1}{{\vartheta_{4}^{2} }}} + \frac{{\vartheta_{4} }}{{\vartheta_{1} \vartheta_{2} + \vartheta_{4}^{2} }}}}{2}} ,\quad \varGamma_{2} = \sqrt {\frac{{\sqrt {\frac{1}{{\vartheta_{4}^{2} }}} - \frac{{\vartheta_{4} }}{{\vartheta_{1} \vartheta_{2} + \vartheta_{4}^{2} }}}}{2}} .$$
(43)

Appendix B

$${\mathcal{P}}_{q}^{\left( 0 \right)} = {\mathcal{U}}_{0} {\mathcal{U}}_{1} {\mathcal{U}}_{8} + {\mathcal{U}}_{2} {\mathcal{U}}_{8} \epsilon^{2}$$
(44)
$$\begin{aligned} {\mathcal{P}}_{q}^{\left( 2 \right)} & = 8{\mathcal{U}}_{1} {\mathcal{U}}_{3} {\mathcal{U}}_{7} {\mathcal{U}}_{8} + \frac{{8{\mathcal{U}}_{1} {\mathcal{U}}_{3} {\mathcal{U}}_{8} \left( {{\mathcal{U}}_{0} {\mathcal{U}}_{1} {\mathcal{U}}_{6} {\mathcal{U}}_{8} H_{20} + {\mathcal{U}}_{0} {\mathcal{U}}_{3} {\mathcal{U}}_{5} H_{20} } \right)}}{{{\mathcal{U}}_{0} {\mathcal{U}}_{1} {\mathcal{U}}_{8} H_{20} - {\mathcal{U}}_{5} }} + \frac{{8{\mathcal{U}}_{1} {\mathcal{U}}_{3} \left( {{\mathcal{U}}_{0} {\mathcal{U}}_{6} + {\mathcal{U}}_{0}^{2} {\mathcal{U}}_{3} H_{20} } \right)}}{{{\mathcal{U}}_{0} {\mathcal{U}}_{1} {\mathcal{U}}_{8} H_{20} - {\mathcal{U}}_{5} }} \\ & \quad + \frac{{8{\mathcal{U}}_{0} {\mathcal{U}}_{1} {\mathcal{U}}_{3} \left( {{\mathcal{U}}_{6} + {\mathcal{U}}_{0} {\mathcal{U}}_{3} } \right)}}{{{\mathcal{U}}_{0} {\mathcal{U}}_{1} {\mathcal{U}}_{8} - {\mathcal{U}}_{5} }} + 16{\mathcal{U}}_{0} {\mathcal{U}}_{3} {\mathcal{U}}_{4} {\mathcal{U}}_{8} \\ \end{aligned}$$
(45)
$$\delta_{q}^{\left( 0 \right)} = \left[ {\frac{{\vartheta_{1} }}{2} - \vartheta_{2} + \left( {\frac{{\left( {2\vartheta_{1} \vartheta_{2} - \vartheta_{2}^{2} } \right)\varGamma_{2} }}{{\pi \vartheta_{1} \left( {\varGamma_{1}^{2} + \varGamma_{2}^{2} } \right)}}} \right)\epsilon^{1/2} } \right]{\mathcal{P}}_{q} + \left[ {\left( {\frac{{3^{1/4} \left( {\varGamma_{1}^{2} + \varGamma_{2}^{2} } \right)\left( {2\vartheta_{1} - \vartheta_{2} } \right)^{2} }}{{6\pi \varGamma_{2} }}} \right)\epsilon } \right]{\mathcal{P}}_{q}^{2}$$
(46)
$$\delta_{q}^{\left( 2 \right)} = \left[ {\frac{{3^{3/4} m^{2} }}{32}} \right]\epsilon^{ - 3/2} ,$$
(47)

where

$$H_{11} = 1 + \pi^{2} {\mathcal{G}}_{1}^{2} \left( {m^{2} + \beta^{2} n^{2} } \right),\quad H_{20} = 1 + 4\pi^{2} {\mathcal{G}}_{1}^{2} m^{2}$$
(48)
$$G_{11} = 1 + \pi^{2} {\mathcal{G}}_{2}^{2} \left( {m^{2} + \beta^{2} n^{2} } \right),$$

where \({\mathcal{U}}_{i} \left( {i = 0, \ldots ,8} \right)\) are constant parameters extracted via the perturbation sets of equations:

$${\mathcal{S}}_{1} = - \left[ {\left( {2\vartheta_{1} - \vartheta_{2} } \right)\left( {{\mathcal{P}}_{q}^{\left( 2 \right)} } \right)} \right]$$
(49)
$${\mathcal{S}}_{2} = - \left( {2\vartheta_{1} - \vartheta_{2} } \right)\left( {{\mathcal{P}}_{q}^{\left( 0 \right)} } \right).$$
(50)

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Yang, X., Sahmani, S. & Safaei, B. Postbuckling analysis of hydrostatic pressurized FGM microsized shells including strain gradient and stress-driven nonlocal effects. Engineering with Computers 37, 1549–1564 (2021). https://doi.org/10.1007/s00366-019-00901-2

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