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Dynamic thermal buckling and postbuckling of clamped–clamped imperfect functionally graded annular plates

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Abstract

Dynamic thermal buckling and postbuckling of imperfect functionally graded material (FGM) annular plates are investigated based on the nonlinear plate theory. The transient temperature fields of the FGM annular plates under dynamic thermal loadings are obtained by Laplace transform in combination of power series method according to the theory of Fourier heat conduction. The nonlinear dynamic equations of large axisymmetric deformations are numerically solved by series expansions and Runge–Kutta method. The critical bucking and dynamic postbuckling responses are predicted by the maximum deflections of the FGM annular plates with positive or negative initial geometric imperfections. The effects of the loads, the material gradient and the initial geometric imperfections on the dynamic responses and the buckling critical temperatures of the FGM annular plates are analyzed in detail.

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Acknowledgements

This work was financially supported by the National Natural Science Foundation of China (Nos. 11662008, 11262010) and abroad exchange funding for young backbone teachers of Lanzhou University of Technology. The authors gratefully acknowledge all of the support.

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Authors and Affiliations

  1. School of Science, Lanzhou University of Technology, Lanzhou, 730050, China

    Jinghua Zhang & Like Chen

  2. CAEC, Zhejiang Fuchunjiang Hydropower Equipment Co., Ltd, Hangzhou, 311121, China

    Shuangchao Pan

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  1. Jinghua Zhang
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Correspondence to Jinghua Zhang.

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Appendix

Appendix

$$\begin{aligned} F_1= & {} \frac{\partial ^{3}W}{\partial x^{3}}+\frac{1}{x}\frac{\partial ^{2}W}{\partial x^{2}}-\frac{1}{x^{2}}\frac{\partial W}{\partial x},\\ F_2= & {} -\left( {\frac{\partial ^{2}W}{\partial x^{2}}-\frac{\partial ^{2}W_0 }{\partial x^{2}}+\frac{1-\nu }{2x}\frac{\partial W}{\partial x}} \right) \frac{\partial W}{\partial x}\\&-\,\left( {\frac{\partial ^{2}W}{\partial x^{2}}+\frac{1-\nu }{x}\frac{\partial W}{\partial x}} \right) \frac{\partial W_0 }{\partial x},\\ F_3= & {} \frac{\partial ^{2}W}{\partial x^{2}}+\frac{\partial ^{2}W_0 }{\partial x^{2}}+\frac{1}{x}\left( {\frac{\partial W}{\partial x}+\frac{\partial W_0 }{\partial x}} \right) , \\ F_4= & {} \frac{1}{x}\frac{\partial }{\partial x}x\frac{\partial }{\partial x}\frac{1}{x}\frac{\partial }{\partial x}\left( {xU} \right) ,\\ F_5= & {} \frac{1-3\nu }{x}\frac{\partial W}{\partial x}\frac{\partial ^{2}W}{\partial x^{2}}+\frac{1-2\nu }{x}\frac{\partial ^{2}W}{\partial x^{2}}\frac{\partial W_0 }{\partial x}\\&+\,\left[ \frac{2\left( {1-\nu } \right) }{x}\frac{\partial W}{\partial x} +\frac{\partial ^{2}W}{\partial x^{2}} \right] \frac{\partial ^{2}W_0 }{\partial x^{2}}+\frac{\partial W}{\partial x}\frac{\partial ^{3}W_0 }{\partial x^{3}},\\ F_6= & {} \frac{1}{x}\frac{\partial }{\partial x}\left[ {x\left( {\frac{\partial U}{\partial x}+\frac{\nu }{x}U} \right) \frac{\partial W}{\partial x}} \right] \\&+\,\left[ {\frac{\partial U}{\partial x}+\frac{\nu }{x}U} \right] \frac{\partial ^{2}W_0 }{\partial x^{2}}+\left( {\frac{\partial ^{2}U}{\partial x^{2}}+\frac{1+\nu }{x}\frac{\partial U}{\partial x}} \right) \frac{\partial W_0 }{\partial x},\\ F_7= & {} \frac{1}{x}\frac{\partial }{\partial x}\left[ {\frac{x}{2}\left( {\frac{\partial W}{\partial x}} \right) ^{2}+\frac{\partial W}{\partial x}} \right] \\&+\,\left( {2\frac{\partial W_0 }{\partial x}+\frac{3}{2}\frac{\partial W}{\partial x}} \right) \frac{\partial W}{\partial x}\frac{\partial ^{2}W_0 }{\partial x^{2}} \\&+\,\left[ \left( {3\frac{\partial W}{\partial x}+\frac{\partial W_0 }{\partial x}} \right) \frac{\partial ^{2}W}{\partial x^{2}}\right. \\&\left. +\left( {\frac{3}{2x}\frac{\partial W}{\partial x}+\frac{1}{x}\frac{\partial W_0 }{\partial x}} \right) \frac{\partial W}{\partial x} \right] \frac{\partial W_0 }{\partial x} \\ \eta _1= & {} \left( {\nu -15} \right) \frac{32A_8 ^{2}x^{13}}{\lambda ^{2}}+\left( {\nu -13} \right) \frac{48A_6 A_8 x^{11}}{\lambda ^{2}}\\&+\,\left( {\nu -11} \right) \left( {18A_6 ^{2}+32A_4 A_8 } \right) \frac{x^{9}}{\lambda ^{2}}\\&+\,\left( {\nu -9} \right) 16A_4 A_6 \frac{x^{7}}{\lambda ^{2}} \\&+\,\left( {\nu -9} \right) 24A_2 A_8 \frac{x^{7}}{\lambda ^{2}}+\left( {\nu -7} \right) \\&\times \,\left( {12A_2 A_6 +8A_4 ^{2}} \right) \frac{x^{5}}{\lambda ^{2}}+\left( {\nu -5} \right) \frac{8A_2 A_4 x^{3}}{\lambda ^{2}}\\&+\,\left( {\nu -3} \right) \frac{2A_2 ^{2}x}{\lambda ^{2}} \end{aligned}$$
$$\begin{aligned} \eta _2= & {} \left( {\nu -15} \right) \frac{64\delta _0 A_8 ^{2}x^{13}}{\lambda ^{2}}+\left( {\nu -13} \right) \frac{96\delta _0 A_8 A_6 x^{11}}{\lambda ^{2}}\\&+\,\left( {\nu -11} \right) \left( {36\delta _0 A_6 ^{2}+64\delta _0 A_4 A_8 } \right) \frac{x^{9}}{\lambda ^{2}} \\&+\,\left( {\nu -9} \right) \left( {48\delta _0 A_4 A_6 +32\delta _0 A_2 A_8 } \right) \frac{x^{7}}{\lambda ^{2}}\\&+\,\left[ \left( {24\delta _0 A_2 A_6 +16\delta _0 A_4 ^{2}} \right) \left( {\nu -7} \right) +\,\frac{384D_2 A_8 }{D_1 } \right] \frac{x^{5}}{\lambda ^{2}}\\&+\left[ \frac{144D_2 A_6 }{D_1 }+16\delta _0 A_2 A_4 \times \,\left( {\nu -5} \right) -12\delta _0 A_2 ^{2} \right] \frac{x^{3}}{\lambda ^{2}}\\&+\,\left( {\delta _0 A_2 ^{2}\nu +\frac{8D_2 A_4 }{D_1 }} \right) \frac{4x}{\lambda ^{2}} \\ \lambda _1= & {} \frac{5632D_1 D_4 ^{4}A_8 ^{3}x^{20}}{\lambda ^{3}}+\frac{11520D_1 D_4 ^{4}A_6 A_8 ^{2}x^{18}}{\lambda ^{3}}\\&\quad +\,\left( 7776A_6 ^{2}A_8 {+}6912A_4 A_8 ^{2}\right. \\&\quad \left. +\,3072A_2 A_8 ^{2} \right) \frac{D_1 D_4 ^{4}x^{16}}{\lambda ^{3}}\\&\quad +\,\left( {9216A_4 A_6 A_8 {+}1728A_6 ^{3}} \right) \frac{D_1 D_4 ^{4}x^{14}}{\lambda ^{3}}\\&\quad +\,\left( 4032A_2 A_6 A_8 {+}2688A_4 ^{2}A_8 \right. \\&\quad \left. +\,3024A_4 A_6 ^{2} \right) \frac{D_1 D_4 ^{4}x^{12}}{\lambda ^{3}}\\&\quad +\,\left( 2304A_2 A_4 A_8 {+}1296A_2 A_{6}^{2}\right. \\&\quad \left. +\,1728A_{4}^{2} A_6 \right) \frac{D_1 D_{4}^{4} x^{10}}{\lambda ^{3}}\\&\quad +\,\left( {320A_{4}^{3} {+}1440A_2 A_4 A_6 {+}480A_{2}^{2} A_8 } \right) \frac{320A_{4}^{3} x^{8}}{\lambda ^{3}} \\&\quad +\,\left( {288A_2 ^{2}A_6 {+}384A_2 A_4 ^{2}} \right) \frac{D_1 D_4 ^{4}x^{6}}{\lambda ^{3}}\\&\quad +\,\frac{144D_1 D_4 ^{4}A_2 ^{2}A_4 x^{4}}{\lambda ^{3}}+\frac{16D_1 D_4 ^{4}A_2 ^{3}x^{2}}{\lambda ^{3}}\\ \end{aligned}$$
$$\begin{aligned} \lambda _2= & {} \frac{16896D_1 D_4 ^{4}\delta _0 A_8 ^{3}x^{20}}{\lambda ^{3}}+\frac{34560A_6 D_1 D_4 ^{4}\delta _0 A_8 ^{2}x^{18}}{\lambda ^{3}}\\&\quad +\,\left( {20736A_4 A_8 ^{2}{+}23328A_6 ^{2}A_8 } \right) \frac{D_1 D_4 ^{4}\delta _0 x^{16}}{\lambda ^{3}}\\&\quad +\,\frac{9216A_2 A_8 ^{2}D_1 D_4 ^{4}\delta _0 x^{14}}{\lambda ^{3}}\\&\quad +\,\left( {27648A_2 A_6 A_8 {+}5184A_6 ^{3}} \right) \\&\quad \times \,\frac{D_1 D_4 ^{4}\delta _0 x^{14}}{\lambda ^{3}}+\left( {1-3\nu } \right) \frac{448D_2 D_4 ^{4}A_8 ^{2}x^{12}}{\lambda ^{3}} \\&\quad +\,\left( {12096A_2 A_6 A_8 +8064A_4 ^{2}A_8 +9072A_4 A_6 ^{2}} \right) \\&\quad \times \,\frac{D_1 D_4 ^{4}\delta _0 x^{12}}{\lambda ^{3}}+3888A_2 A_6 ^{2}\frac{D_1 D_4 ^{4}\delta _0 x^{10}}{\lambda ^{3}} \\&\quad +\,\left( {6912A_2 A_4 A_8 +5184A_{4}^{2} A_6 } \right) \frac{D_1 D_4 ^{4}\delta _0 x^{10}}{\lambda ^{3}}\\&\quad +\,\left( {1-3\nu } \right) \frac{576D_2 D_{4}^{4} A_6 A_8 x^{10}}{\lambda ^{2}}\\&\quad +\,\left( {1-3\nu } \right) \frac{180D_2 D_{4}^{4} A_{6}^{2} x^{8}}{\lambda ^{2}} \\&\quad +\,\left( {1-3\nu } \right) \frac{320D_2 D_4 ^{4}A_4 A_8 x^{8}}{\lambda ^{2}}\\&\quad +\,\left( 960A_4 ^{3}+1440A_2 ^{2}A_8 \right. \\&\quad \left. +\,4320A_2 A_4 A_6 \right) \frac{D_1 D_4 ^{4}\delta _0 x^{8}}{\lambda ^{3}} \\&\quad +\,\left( {1-3\nu } \right) \frac{128D_2 D_4 ^{4}A_2 A_8 x^{6}}{\lambda ^{2}}+\left( {1-3\nu } \right) \\&\quad \times \,\frac{192D_2 D_4 ^{4}A_4 A_6 x^{6}}{\lambda ^{2}}+\left( 864A_2 ^{2}A_6\right. \\&\quad \left. +\,1152A_2 A_4 ^{2} \right) \frac{D_1 D_4 ^{4}\delta _0 x^{6}}{\lambda ^{3}}\\&\quad +\,\left( {1-3\nu } \right) \frac{48D_2 D_4 ^{4}A_4 ^{2}x^{4}}{\lambda ^{2}}+\left( {1-3\nu } \right) \\&\quad \times \,\frac{72D_2 D_4 ^{4}A_2 A_6 x^{4}}{\lambda ^{2}}+\frac{432D_1 D_4 ^{4}\delta _0 A_2 ^{2}A_4 x^{4}}{\lambda ^{3}}\\&\quad +\,\left( {1-3\nu } \right) \frac{32D_2 D_4 ^{4}A_2 A_4 x^{2}}{\lambda ^{2}}\\&\quad +\,\frac{48D_1 D_4 ^{4}\delta _0 A_2 ^{3}x^{2}}{\lambda ^{3}}+\left( {1-3\nu } \right) \frac{4D_2 D_4 ^{4}A_2 ^{2}}{\lambda ^{2}} \\ \end{aligned}$$
$$\begin{aligned} \lambda _3= & {} \frac{11264D_1 D_4 ^{4}\delta _0 ^{2}A_8 ^{3}x^{20}}{\lambda ^{3}}+\frac{23040D_1 D_4 ^{4}\delta _0 ^{2}A_6 A_8 ^{2}x^{18}}{\lambda ^{3}}\\&\quad +\,\left( {13824A_8 ^{2}{+}15552A_6 ^{2}A_8 } \right) \frac{D_1 D_4 ^{4}\delta _0 ^{2}x^{16}}{\lambda ^{3}} \\&\quad +\,\left( {6144A_2 A_8 ^{2}+18432A_4 A_6 A_8 +3456A_6 ^{3}} \right) \\&\quad \times \,\frac{D_1 D_4 ^{4}\delta _0 ^{2}x^{14}}{\lambda ^{3}}+\frac{\left( {4-\nu } \right) 1792D_2 D_4 ^{4}\delta _0 A_8 ^{2}x^{12}}{\lambda ^{2}} \\&\quad +\,\left( {6048A_4 A_6 ^{2}+5376A_4 ^{2}A_8 +8064A_2 A_6 A_8 } \right) \\&\quad \times \,\frac{D_1 D_4 ^{4}\delta _0 ^{2}x^{12}}{\lambda ^{3}}+\frac{\left( {7-2\nu } \right) 2304D_2 D_4 ^{4}\delta _0 A_6 A_8 x^{10}}{\lambda ^{2}} \\&\quad +\,\frac{\left( {3-\nu } \right) 720D_2 D_4 ^{4}\delta _0 A_6 ^{2}x^{8}}{\lambda ^{2}}+\left( 2592A_2 A_6 ^{2}\right. \\&\quad \left. +\,3456A_4 ^{2}A_6 +4608A_4 A_6 A_8 \right) \frac{D_1 D_4 ^{4}\delta _0 ^{2}x^{10}}{\lambda ^{3}} \\&\quad +\,\left( {640A_4 ^{3}+960A_2 ^{2}A_8 +2880A_2 A_4 A_6 } \right) \\&\quad \times \,\frac{D_1 D_4 ^{4}\delta _0 ^{2}x^{8}}{\lambda ^{3}}+\frac{\left( {3-\nu } \right) 1280D_2 D_4 ^{4}\delta _0 A_4 A_8 x^{8}}{\lambda ^{2}} \\&\quad +\,\frac{8D_1 D_4 ^{4}A_8 x^{7}}{\lambda }\frac{\partial ^{2}U\left( x \right) }{\partial x^{2}}+\left( 768A_2 A_4 ^{2}\right. \\&\quad \left. +\,576A_2 ^{2}A_6 \right) \frac{D_1 D_4 ^{4}\delta _0 ^{2}x^{6}}{\lambda ^{3}}\\&\quad +\,\left( {5-2\nu } \right) \frac{256D_2 D_4 ^{4}\delta _0 A_2 A_8 x^{6}}{\lambda ^{2}} \\&\quad +\,\left( {8+\nu } \right) \frac{8\nu D_1 D_4 ^{4}A_8 x^{6}}{\lambda }\frac{\partial U\left( x \right) }{\partial x}-\frac{64N_T D_4 ^{4}A_8 x^{6}}{\lambda }\\&\quad +\,\left( {5-2\nu } \right) \frac{384D_2 D_4 ^{4}\delta _0 A_4 A_6 x^{6}}{\lambda ^{2}}\\&\quad +\,\left( {56A_8 \nu U\left( x \right) +6A_6 \frac{\partial ^{2}U\left( x \right) }{\partial x^{2}}} \right) \frac{D_1 D_4 ^{4}x^{5}}{\lambda }\\&\quad +\,\left( {2-\nu } \right) \frac{192D_2 D_4 ^{4}\delta _0 A_4 ^{2}\nu x^{4}}{\lambda ^{2}}-\frac{36N_T D_4 ^{4}A_6 x^{4}}{\lambda } \\&\quad +\,\left( {2-\nu } \right) \frac{288D_2 D_4 ^{4}\delta _0 A_2 A_6 \nu x^{4}}{\lambda ^{2}}+\left( {6+\nu } \right) \\&\quad \times \,\frac{6D_1 D_4 ^{4}A_6 \nu x^{4}}{\lambda }\frac{\partial U\left( x \right) }{\partial x}+\frac{288D_1 D_4 ^{4}\delta _0 ^{2}A_2 ^{2}A_4 x^{4}}{\lambda ^{3}} \\&\quad +\,\left( {4A_4 \frac{\partial ^{2}U\left( x \right) }{\partial x^{2}}+30A_6 \nu U\left( x \right) } \right) \frac{D_1 D_4 ^{4}x^{3}}{\lambda }\\&\quad +\,\frac{32D_1 D_4 ^{4}\delta _0 ^{2}A_2 ^{3}x^{2}}{\lambda ^{3}}+\left( {4+\nu } \right) \frac{4D_1 D_4 ^{4}A_4 \nu x^{2}}{\lambda }\frac{\partial U\left( x \right) }{\partial x} \\ \end{aligned}$$
$$\begin{aligned}&\quad +\,\left( {3-2\nu } \right) \frac{128D_2 D_4 ^{4}\delta _0 A_2 A_4 \nu x^{2}}{\lambda ^{2}}-\frac{16N_T D_4 ^{4}A_4 x^{2}}{\lambda }\\&\quad +\,\left( {A_2 \frac{\partial ^{2}U\left( x \right) }{\partial x^{2}}+6A_4 \nu U\left( x \right) } \right) \frac{2D_1 D_4 ^{4}x}{\lambda } \\&\quad +\,\left( {2+\nu } \right) \frac{2D_1 D_4 ^{4}A_2 \nu }{\lambda }\frac{\partial U\left( x \right) }{\partial x}\\&\quad -\,\frac{4N_T D_4 ^{4}A_2 }{\lambda }+\frac{2D_1 D_4 ^{4}A_2 \nu U\left( x \right) }{x\lambda } \\ \lambda _4= & {} -\frac{D_0 A_8 x^{8}}{\lambda }-\frac{D_0 A_6 x^{6}}{\lambda }-\frac{D_0 A_4 x^{4}}{\lambda }\\&-\,\frac{D_0 A_2 x^{2}}{\lambda }-\frac{D_0 }{\lambda }\\ \lambda _5= & {} \frac{8D_1 D_4 ^{4}\delta _0 A_8 x^{7}}{\lambda }\frac{\partial ^{2}U}{\partial x^{2}}+\left( {8\frac{\partial U}{\partial x}-\frac{8N_T }{D_1 }+\nu \frac{\partial U}{\partial x}} \right) \\&\times \frac{8D_1 D_4 ^{4}\delta _0 A_8 x^{6}}{\lambda }{+}\left( {56A_8 \nu U+6A_6 \frac{\partial ^{2}U}{\partial x^{2}}} \right) \frac{D_1 D_4 ^{4}\delta _0 x^{5}}{\lambda } \\&+\,\left( {6\frac{\partial U}{\partial x}+\nu \frac{\partial U}{\partial x}-\frac{6N_T }{D_1 }} \right) \frac{6D_1 D_4 ^{4}\delta _0 A_6 x^{4}}{\lambda }\\&+\,\left( {30A_6 \nu U+4A_4 \frac{\partial ^{2}U}{\partial x^{2}}} \right) \frac{D_1 D_4 ^{4}\delta _0 x^{3}}{\lambda }\\&-\,\frac{16N_T D_4 ^{4}\delta _0 A_4 x^{2}}{\lambda }\\&+\,\left( {4+\nu } \right) \frac{4D_1 D_4 ^{4}\delta _0 A_4 x^{2}}{\lambda }\frac{\partial U}{\partial x}+\left( {6\nu A_4 U+A_2 \frac{\partial ^{2}U}{\partial x^{2}}} \right) \\&\times \,\frac{2D_1 D_4 ^{4}\delta _0 x}{\lambda }+\frac{D_2 D_4 ^{4}U}{x^{3}}-\frac{D_2 D_4 ^{4}}{x^{2}}\frac{\partial U}{\partial x} \\&+\,\frac{2D_2 D_4 ^{4}}{x}\frac{\partial ^{2}U}{\partial x^{2}}+\frac{2D_1 D_4 ^{4}\delta _0 A_2 \nu U}{\lambda x}+\left( {\nu +2} \right) \\&\times \,\frac{2D_1 D_4 ^{4}\delta _0 A_2 }{\lambda }\frac{\partial U}{\partial x}+D_2 D_4 ^{4}\frac{\partial U}{\partial x}-\frac{4N_T D_4 ^{4}\delta _0 A_2 }{\lambda } \\ {\eta }'_1= & {} \left( {\nu -15} \right) \frac{A_8 ^{2}x^{15}}{7\lambda ^{2}}+\left( {\nu -13} \right) \frac{2A_6 A_8 x^{13}}{7\lambda ^{2}}+\left( {\nu -11} \right) \\&\times \left( {9A_6 ^{2}+16A_4 A_8 } \right) \frac{x^{11}}{60\lambda ^{2}}+\left( {\nu -9} \right) 3A_4 A_6 \frac{x^{9}}{10\lambda ^{2}} \\&+\,\left( {\nu -9} \right) 2A_2 A_8 \frac{x^{9}}{10\lambda ^{2}}+\left( {\nu -7} \right) \left( {3A_2 A_6 +2A_4 ^{2}} \right) \\&\times \,\frac{x^{7}}{12\lambda ^{2}}+\left( {\nu -5} \right) \frac{A_2 A_4 x^{5}}{3\lambda ^{2}}+\left( {\nu -3} \right) \frac{A_2 ^{2}x^{3}}{4\lambda ^{2}} \\ \end{aligned}$$
$$\begin{aligned} {\eta }'_2= & {} \left( {\nu -15} \right) \frac{2\delta _0 A_8 ^{2}x^{15}}{7\lambda ^{2}}+\left( {\nu -13} \right) \frac{4\delta _0 A_8 A_6 x^{13}}{7\lambda ^{2}}\\&+\,\left( {\nu -11} \right) \left( {9\delta _0 A_6 ^{2}+16\delta _0 A_4 A_8 } \right) \frac{x^{11}}{30\lambda ^{2}} \\&+\,\left( {\nu -9} \right) \left( {12\delta _0 A_4 A_6 +4\delta _0 A_2 A_8 } \right) \frac{x^{9}}{10\lambda ^{2}}\\&\quad +\,\left[ \left( {3\delta _0 A_2 A_6 +2\delta _0 A_4 ^{2}} \right) \left( {\nu -7} \right) \right. \\&\quad \left. +\,\frac{48D_2 A_8 }{D_1 } \right] \frac{x^{7}}{6\lambda ^{2}} +\left[ \frac{36D_2 A_6 }{D_1 }\right. \\&\quad \left. +\,4\delta _0 A_2 A_4 \left( {\nu -5} \right) -3\delta _0 A_2 ^{2} \right] \\&\quad \times \,\frac{x^{5}}{6\lambda ^{2}}+\left( {\delta _0 A_2 ^{2}\nu +\frac{8D_2 A_4 }{D_1 }} \right) \frac{x^{3}}{2\lambda ^{2}} \\ \eta _1 ^{\prime \prime }= & {} -\left( {\nu -15} \right) \frac{A_8 ^{2}\left( {x_1 ^{16}-1} \right) }{7\lambda ^{2}\left( {x_1 ^{2}-1} \right) }\\&-\,\left( {\nu -13} \right) \frac{2A_6 A_8 \left( {x_1 ^{14}-1} \right) }{7\lambda ^{2}\left( {x_1 ^{2}-1} \right) }\\&-\,\left( {\nu -11} \right) \left( {9A_6 ^{2}+16A_4 A_8 } \right) \frac{\left( {x_1 ^{12}-1} \right) }{60\lambda ^{2}\left( {x_1 ^{2}-1} \right) } \\&-\,\left( {\nu -9} \right) \left( {3A_4 A_6 +2A_2 A_8 } \right) \frac{\left( {x_1 ^{10}-1} \right) }{10\lambda ^{2}\left( {x_1 ^{2}-1} \right) }\\&-\,\left( {\nu -7} \right) \left( {3A_2 A_6 +2A_4 ^{2}} \right) \frac{\left( {x_1 ^{8}-1} \right) }{12\lambda ^{2}\left( {x_1 ^{2}-1} \right) } \\&-\,\left( {\nu -5} \right) \frac{A_2 A_4 \left( {x_1 ^{6}-1} \right) }{3\lambda ^{2}\left( {x_1 ^{2}-1} \right) }-\left( {\nu -3} \right) \frac{A_2 ^{2}\left( {x_1 ^{4}-1} \right) }{4\lambda ^{2}\left( {x_1 ^{2}-1} \right) } \\ \eta _2 ^{\prime \prime }= & {} -\left( {\nu -15} \right) \frac{2\delta _0 A_8 ^{2}\left( {x_1 ^{16}-1} \right) }{7\lambda ^{2}\left( {x_1 ^{2}-1} \right) }\\&-\,\left( {\nu -13} \right) \frac{4\delta _0 A_8 A_6 \left( {x_1 ^{14}-1} \right) }{7\lambda ^{2}\left( {x_1 ^{2}-1} \right) }\\&-\,\left( {\nu -11} \right) \left( {9A_6 ^{2}+16A_4 A_8 } \right) \frac{\delta _0 \left( {x_1 ^{12}-1} \right) }{30\lambda ^{2}\left( {x_1 ^{2}-1} \right) } \\&-\,\left( {\nu -9} \right) \left( {3A_4 A_6 +A_2 A_8 } \right) \frac{4\delta _0 \left( {x_1 ^{10}-1} \right) }{10\lambda ^{2}\left( {x_1 ^{2}-1} \right) }\\&-\,\left[ \left( {3\delta _0 A_2 A_6 +2\delta _0 A_4 ^{2}} \right) \left( {\nu -7} \right) \right. \\&\left. +\,\frac{48D_2 A_8 }{D_1 } \right] \frac{\left( {x_1 ^{8}-1} \right) }{6\lambda ^{2}\left( {x_1 ^{2}-1} \right) } \\&-\,\left[ {\frac{36D_2 A_6 }{D_1 }+4\delta _0 A_2 A_4 \left( {\nu -5} \right) -3\delta _0 A_2 ^{2}} \right] \\ \end{aligned}$$
$$\begin{aligned}&\times \,\frac{\left( {x_1 ^{6}-1} \right) }{6\lambda ^{2}\left( {x_1 ^{2}-1} \right) }-\left( {\delta _0 A_2 ^{2}\nu +\frac{8D_2 A_4 }{D_1 }} \right) \frac{\left( {x_1 ^{4}-1} \right) }{2\lambda ^{2}\left( {x_1 ^{2}-1} \right) } \\ \eta _3 ^{\prime \prime }= & {} \left( {\nu -15} \right) \frac{A_8 ^{2}\left( {x_1 ^{16}-x_1 ^{2}} \right) }{7\lambda ^{2}\left( {x_1 ^{2}-1} \right) }+\left( {\nu -13} \right) \frac{2A_6 A_8 \left( {x_1 ^{14}-x_1 ^{2}} \right) }{7\lambda ^{2}\left( {x_1 ^{2}-1} \right) }\\&+\,\left( {\nu -11} \right) \left( {9A_6 ^{2}+16A_4 A_8 } \right) \frac{\left( {x_1 ^{12}-x_1 ^{2}} \right) }{60\lambda ^{2}\left( {x_1 ^{2}-1} \right) }\\&+\,\left( {\nu -9} \right) \left( {3A_4 A_6 +2A_2 A_8 } \right) \frac{\left( {x_1 ^{10}-x_1 ^{2}} \right) }{10\lambda ^{2}\left( {x_1 ^{2}-1} \right) }\\&+\,\left( {\nu -7} \right) \left( {3A_2 A_6 +2A_4 ^{2}} \right) \frac{\left( {x_1 ^{8}-x_1 ^{2}} \right) }{12\lambda ^{2}\left( {x_1 ^{2}-1} \right) } \\&+\,\left( {\nu -5} \right) \frac{A_2 A_4 \left( {x_1 ^{6}-x_1 ^{2}} \right) }{3\lambda ^{2}\left( {x_1 ^{2}-1} \right) }+\left( {\nu -3} \right) \frac{A_2 ^{2}\left( {x_1 ^{4}-x_1 ^{2}} \right) }{4\lambda ^{2}\left( {x_1 ^{2}-1} \right) } \\ \eta _4 ^{\prime \prime }= & {} \left( {\nu -15} \right) \frac{2\delta _0 A_8 ^{2}\left( {x_1 ^{16}-x_1 ^{2}} \right) }{7\lambda ^{2}\left( {x_1 ^{2}-1} \right) }\\&+\,\left( {\nu -13} \right) \frac{4\delta _0 A_8 A_6 \left( {x_1 ^{14}-x_1 ^{2}} \right) }{7\lambda ^{2}\left( {x_1 ^{2}-1} \right) }\\&+\,\left( {\nu -11} \right) 9\delta _0 A_6 ^{2}\frac{\left( {x_1 ^{12}-x_1 ^{2}} \right) }{30\lambda ^{2}\left( {x_1 ^{2}-1} \right) } \\&+\,\left( {\nu -11} \right) 16\delta _0 A_4 A_8 \frac{\left( {x_1 ^{12}-x_1 ^{2}} \right) }{30\lambda ^{2}\left( {x_1 ^{2}-1} \right) }+\left( {\nu -9} \right) \\&\times \left( {12\delta _0 A_4 A_6 +4\delta _0 A_2 A_8 } \right) \frac{\left( {x_1 ^{10}-x_1 ^{2}} \right) }{10\lambda ^{2}\left( {x_1 ^{2}-1} \right) } \\&+\,\left[ {\delta _0 \left( {3A_2 A_6 +2A_4 ^{2}} \right) \left( {\nu -7} \right) +\frac{48D_2 A_8 }{D_1 }} \right] \\&\times \frac{\left( {x_1 ^{8}-x_1 ^{2}} \right) }{6\lambda ^{2}\left( {x_1 ^{2}-1} \right) }+4\delta _0 A_2 A_4 \left( {\nu -5} \right) \frac{\left( {x_1 ^{6}-x_1 ^{2}} \right) }{6\lambda ^{2}\left( {x_1 ^{2}-1} \right) } \\&+\,\left( {\frac{36D_2 A_6 }{D_1 }-3\delta _0 A_2 ^{2}} \right) \frac{\left( {x_1 ^{6}-x_1 ^{2}} \right) }{6\lambda ^{2}\left( {x_1 ^{2}-1} \right) }\\&+\,\left( {\delta _0 A_2 ^{2}\nu +\frac{8D_2 A_4 }{D_1 }} \right) \frac{\left( {x_1 ^{4}-x_1 ^{2}} \right) }{2\lambda ^{2}\left( {x_1 ^{2}-1} \right) } \\ \end{aligned}$$

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Zhang, J., Pan, S. & Chen, L. Dynamic thermal buckling and postbuckling of clamped–clamped imperfect functionally graded annular plates. Nonlinear Dyn 95, 565–577 (2019). https://doi.org/10.1007/s11071-018-4583-5

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