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On the size-dependent bending and buckling of the partially covered laminated microplate

Abstract

The bending and buckling of the microcomponents show size dependency. The strain gradient elasticity theory is proposed to explain the size dependency. In this paper, we derive the theoretical relations among the modified strain gradient elasticity theory, the modified couple stress theory and the general strain gradient elasticity theory, and clarify the degradation relation. The general theory includes all strain gradients while the modified strain gradient elasticity theory and the modified couple stress theory only contain part of strain gradients. By ignoring the deviatoric part of the strain gradients \(\eta _{ijk}^{'(2)}\) or the symmetric part of the strain gradients \(\eta _{ijk}^{s}\), the general theory is simplified as the modified couple stress theory or the modified strain gradient elasticity theory, respectively. The ability of the general theory and the reduced theories in describing the bending and buckling response of the partially covered laminated microplate is subsequently compared. Results reveal that the general theory predicts smaller bending deflection and axial displacement while larger buckling load than that of the reduced theories. The general theory is more effective in reflecting the size effects. In addition, it is found that the increase of the thickness or radius of the upper elastic layer makes the buckling load increase while the deflection increase firstly and then decrease. There exists the specific radius ratio and thickness ratio to make the clamped-clamped microplate achieve the maximum deflection.

Graphical abstract

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Acknowledgements

This work was supported by the Natural Science Foundation of Shandong Province of China (ZR2021QA078), Taishan Scholars Program of Shandong Province (tsqn20190401, tsqn201909108), the Natural Science Foundation of Shandong Province of China (ZR2020ME164, ZR2021MF042), the Key Research and Development Project of Zibo City (2020SNPT0088), the Open Fund of State Key Laboratory of Applied Optics (SKLAO2020001A16) and the Shandong Provincial Key Laboratory of Precision Manufacturing and Non-traditional Machining.

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Correspondence to Hongyu Zheng, Qianjian Guo or Xuye Zhuang.

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Appendix A

Appendix A

The matrix \(\left[ M(P) \right]\) in Eq. (74) is derived as

$$\begin{aligned} \left[ M(P) \right] = \left[ \begin{array}{ccccccccccc} 1 &{}R^2 &{}m_{1} &{}m_{2} &{} m_{3} &{} m_{4} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} m_{5} &{} m_{6} &{} m_{7} &{} m_{8} &{} m_{9} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} m_{10} &{} m_{11} &{} m_{12} &{} m_{13} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ m_{14} &{} m_{15} &{} m_{16} &{} m_{17} &{}m_{18} &{} m_{19} &{} -1 &{} m_{20} &{} m_{21} &{} m_{22} &{} 0 \\ 0 &{} m_{23} &{} m_{24} &{} m_{25} &{} m_{26} &{} m_{27} &{} 0 &{} m_{28} &{} m_{29} &{} m_{30} &{} 0\\ 0 &{} m_{31} &{} m_{32} &{} m_{33} &{} m_{34} &{} m_{35} &{} 0 &{} -2 &{} m_{36} &{} m_{37} &{} 0\\ 0 &{} 0 &{} 0 &{} m_{38} &{} m_{39} &{} m_{40} &{} 0 &{} 0 &{} m_{41} &{} m_{42} &{} 0 \\ 0 &{} m_{43} &{} m_{44} &{} m_{45} &{} m_{46} &{} m_{47} &{} 0 &{} m_{48} &{} m_{49} &{} m_{50} &{} m_{51} \\ 0 &{} 0 &{} m_{52} &{} m_{53} &{} m_{54} &{} m_{55} &{} 0 &{} 0 &{} m_{56} &{} m_{57} &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} m_{58} &{} m_{59} &{} m_{60} &{} m_{61} \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} m_{62} &{} m_{63}&{} 0 \\ \end{array} \right] \end{aligned}$$

with

$$\begin{aligned} \begin{aligned} m_{1}&=lnR \quad m_{2}=R^2lnR \\ m_{3}&=I_{0}(dR) \quad m_{4}=k_{0}(dR)\\ m_{5}&=-2 c_{3(1)} I_{1} + 2g_{7(1)} h_{2}+4 A_{2(1)} h_{2}\\ m_{6}&=(-2A_{1(1)} +2A_{7(1)} )/R^4\\&\quad +(-c_{3(1)}I_{1}+g_{7(1)}h_{1}-2g_{2(1)}h_{1})/R^2\\ m_{7}&=(6A_{1(1)}-2A_{7(1)})/R^2-c_{3(1)}I_{1}(2lnR+1)\\&\quad +g_{7(1)}h_{1}(2lnR+1)+4A_{2(1)}(lnR+1)\\&\quad +2g_{2(1)}h_{1}(2lnR+3)\\ m_{8}&=-\frac{1}{4}d^4 A_{1(1)}+\frac{3}{2}I_{0}(dR) +2I_{2}(dR) \\&\quad +\frac{1}{2}I_{4}(dR) -\frac{A_{1(1)}d^3}{4R}(I_{3}(dR) +3I_{1}(dR))\\&\quad +\left( \frac{1.5A_{1(1)}d^2}{R^2} -\frac{ 0.5 A_{7(1)} d^2}{R^2}+0.5 A_{2(1)} d^2+g_{2(1)}h_{1} d^2\right) \\&\quad (I_{0}(dR)+I_{2}(dR)) \\&\quad +\left( -\frac{3A_{1(1)}d}{R^3}+\frac{A_{7(1)}d}{R^3}-\frac{c_{3(1)}I_{1}d}{R}\right. \\&\quad \left. +\frac{ g_{7(1)}h_{1}d}{R}+\frac{dA_{2(1)}}{R} \right) I_{1}(dR)\\ m_{9}&=-\frac{d^4A_{1(1)}}{4} \left( \frac{3}{2}k_{0}(dR) +2k_{2}(dR) + \frac{1}{2}k_{4}(dR) \right) \\&\quad + \frac{A_{1(1)}d^3}{4R} (3k_{1}(dR) +k_{3}(dR) )\\&\quad +\left( \frac{1.5A_{1(1)} d^2}{R^2}-\frac{0.5 A_{7(1)} d^2}{R^2} +0.5A_{2(1)} d^2 \right. \\&\quad \left. +g_{2(1)} h_{1} d^2\right) (k_{0}(dR)+k_{2}(dR))\\&\quad \left( \frac{3 A_{1(1)}d}{R^3}-\frac{A_{7(1)} d}{R^3} + \frac{c_{3(1)} I_{(1)} d}{R} \right. \\&\quad \left. -\frac{g_{7(1)} h_{(1)} d}{R} -\frac{d A_{2(1)} }{R} \right) k_{1}(dR) \end{aligned} \end{aligned}$$
(100)
$$\begin{aligned} \begin{aligned} m_{10}&=\frac{2A_{1(1)}-2A_{7(1)}}{R^3} \\ m_{11}&=\frac{2A_{1(1)}+2A_{7(1)}}{R}\\ m_{12}&=\frac{A_{1(1)} d^3}{4} (3I_{1}(dR)+I_{3}(dR))\\&\quad + \frac{A_{7(1)} d^2}{2R} (I_{0}(dR)+I_{2}(dR))\\&\quad +\frac{-A_{7(1)} d}{R^2} I_{1}(dR)\\ m_{13}&=\frac{A_{1(1)} d^3}{4} (-3k_{1}(dR)-k_{3}(dR))\\&\quad + \frac{A_{7(1)} d^2}{2R} (k_{0}(dR)+k_{2}(dR)) +\frac{A_{7(1)} d}{R^2} k_{1}(dR)\\ m_{14}&=1 \quad m_{15}=R_{1}^2 \quad m_{16}=lnR_{1} \\ m_{17}&=R_{1}^2 lnR_{1} \quad m_{18}=I_{0}(dR_{1}) \quad m_{19}=k_{0}(dR_{1})\\ m_{20}&=-R_{1}^2 \quad m_{21}=-J_{0}(\sqrt{s_{1}}R_{1})\\ m_{22}&=-J_{0}(\sqrt{s_{2}}R_{1}) \quad m_{23}=2R_{1} \quad m_{24}=\frac{1}{R_{1} }\\ m_{25}&=2R_{1}lnR_{1}+R_{1} \quad m_{26}=dI_{1}(dR_{1}) \\ m_{27}&=-dk_{1}(dR_{1}) \quad m_{28}=-2R_{1}\\ m_{29}&=\sqrt{s_{1}} J_{1}(\sqrt{s_{1}}R_{1}) \\ m_{30}&=\sqrt{s_{2}} J_{1}(\sqrt{s_{2}}R_{1}) \quad m_{31}=2 \quad m_{32}=-\frac{1}{R_{1}^2}\\ m_{33}&=2lnR_{1}+3 \quad m_{34}=\frac{d^2}{2}(I_{0}(dR_{1})+I_{2}(dR_{1}) )\\ m_{35}&=\frac{d^2}{2}(k_{0}(dR_{1})+k_{2}(dR_{1}) )\\ m_{36}&=\frac{s_{1}}{2} J_{0}(\sqrt{s_{1}}R_{1})-J_{2}(\sqrt{s_{1}}R_{1})\\ m_{37}&=\frac{s_{2}}{2} J_{0}(\sqrt{s_{2}}R_{1})-J_{2}(\sqrt{s_{2}}R_{1})\\ m_{38}&= \frac{4A_{2(1)}}{R_{1}} \\ m_{39}&=\frac{d^5A_{1(1)}}{4} (\frac{5}{2} I_{1}(dR_{1})+\frac{5}{4} I_{3}(dR_{1})+\frac{1}{4} I_{5}(dR_{1}))\\&\quad +\left( \frac{-3A_{1(1)}}{R_{1}^{4}} + \frac{d A_{2(1)}}{R_{1}^2} \right) I_{1}(dR_{1})\\&\quad +\left( \frac{3A_{1(1)}d^2}{2R_{1}^{3}}-\frac{ A_{2(1)}d^2}{2R_{1}}\right) (I_{0}(dR_{1})+I_{2}(dR_{1}) )\\&\quad +\left( \frac{-3A_{1(1)}d^3}{4R_{1}^{2}} + \frac{-A_{2(1)}d^3}{4} \right) ( 3 I_{1}(dR_{1}) +I_{3}(dR_{1}) )\\&\quad \frac{A_{1(1)}d^4}{2R_{1}} \left( \frac{3}{2}I_{0}(dR_{1}) + 2I_{2}(dR_{1})+\frac{1}{2}I_{4}(dR_{1}) \right) \\ m_{40}&=\frac{A_{1(1)}d^5}{4} (-\frac{5}{2}k_{1}(dR_{1})-\frac{5}{4}k_{3}(dR_{1})\\&\quad -\frac{1}{4}k_{5}(dR_{1}) +\left( \frac{3A_{1(1)}d}{R_{1}^4}-\frac{A_{2(1)}d}{R_{1}^2} \right) \\&\quad k_{1}(dR_{1}) +\left( \frac{3A_{1(1)}d^2}{2R_{1}^3}-\frac{A_{2(1)}d^2}{2R_{1}}\right) \\&\quad ( k_{0}(dR_{1}) +k_{2}(dR_{1}) ) +\left( \frac{3A_{1(1)}d^3}{4R_{1}^2}+\frac{A_{2(1)}d^3}{4}\right) \\&\quad ( 3k_{1}(dR_{1}) +k_{3}(dR_{1}) ) +\frac{A_{1(1)}d^4}{2R_{1}}\\&\quad \left( \frac{3}{2}k_{0}(dR_{1}) +2k_{2}(dR_{1})+\frac{1}{2}k_{4}(dR_{1})\right) \end{aligned} \end{aligned}$$
(101)
$$\begin{aligned} \begin{aligned} m_{41}&=-0.5(A_{3}s_{1}^{2.5}m-A_{1}s_{1}^{2.5}) \left( \frac{5}{4}J_{1}(\sqrt{s_{1}}R_{1})\right. \\&\quad \left. -\frac{5}{8}J_{3}(\sqrt{s_{1}}R_{1})+\frac{1}{8}J_{5}(\sqrt{s_{1}}R_{1})\right) \\&\quad -\left( \frac{3A_{1} \sqrt{s_{1}}}{R_{1}^4} -\frac{3A_{3}\sqrt{s_{1}}m}{R_{1}^4} -\frac{A_{2}\sqrt{s_{1}}}{R_{1}^2} +\frac{A_{4}\sqrt{s_{1}}m}{R_{1}^2} \right) \\&\quad J_{1}(\sqrt{s_{1}}R_{1}) -\left( \frac{-3A_{1}s_{1}}{2R_{1}^3} +\frac{3A_{3}s_{1}m}{2R_{1}^3} + \frac{A_{2}s_{1}}{2R_{1}} -\frac{A_{4}s_{1}m}{2R_{1}} \right) \\&\quad (J_{0}(\sqrt{s_{1}}R_{1})-J_{2}(\sqrt{s_{1}}R_{1})) \\&\quad -\left( \frac{3A_{1}s_{1}^{0.5}}{2R_{1}^2} -\frac{3A_{3}s_{1}^{1.5}m}{2R_{1}^2} + \frac{A_{2}s_{1}^{1.5}}{2} -\frac{A_{4}s_{1}^{1.5}m}{2} \right) \\&\quad (-1.5J_{1}(\sqrt{s_{1}}R_{1})+0.5J_{3}(\sqrt{s_{1}}R_{1}))\\&\quad -\left( -\frac{A_{1}s_{1}^{2}m}{R1}+\frac{A_{3}s_{1}^{2}m}{R1} \right) \\&\quad \left( -\frac{3}{4} J_{0}(\sqrt{s_{1}}R_{1})+J_{2}(\sqrt{s_{1}}R_{1}) - \frac{1}{4} J_{4}(\sqrt{s_{1}}R_{1})\right) \end{aligned} \end{aligned}$$
(102)
$$\begin{aligned} \begin{aligned} m_{42}&= -(-0.5(A_{1}s_{2}^{2.5}m+A_{3}s_{2}^{2.5}n)\\&\quad \left( \frac{5}{4}J_{1}(\sqrt{s_{2}}R_{1}) -\frac{5}{8}J_{3}(\sqrt{s_{2}}R_{1}) +\frac{1}{8}J_{5}(\sqrt{s_{2}}R_{1})\right) \\&\quad - \left( \frac{3A_{1} \sqrt{s_{2}}}{R_{1}^4} -\frac{3A_{3} \sqrt{s_{2}}n}{R_{1}^4} -\frac{A_{2} \sqrt{s_{2}}n}{R_{1}^2} +\frac{A_{4} \sqrt{s_{2}}n}{R_{1}^2} \right) \\&\quad J_{1}(\sqrt{s_{2}}R_{1}) - \left( \frac{-3A_{1}s_{2}}{2R_{1}^3} +\frac{3A_{3}s_{2}n}{2R_{1}^3} + \frac{A_{2}s_{2}}{2R_{1}} -\frac{A_{4}s_{2}n}{2R_{1}} \right) \\&\quad (J_{0}(\sqrt{s_{2}}R_{1})-J_{2}(\sqrt{s_{2}}R_{1})) \\&\quad - \left( \frac{3A_{1}s_{2}^{1.5}}{2R_{1}^2} -\frac{3A_{3}s_{2}^{1.5}n}{2R_{1}^2} + \frac{A_{2}s_{2}^{1.5}}{2} -\frac{A_{4}s_{2}^{1.5}n}{2} \right) \\&\quad (1.5J_{1}(\sqrt{s_{2}}R_{1})+0.5J_{3}(\sqrt{s_{2}}R_{1}))\\&\quad - \left( -\frac{A_{1}s_{2}^{2}}{R_{1}}+\frac{A_{3}s_{2}^{2}n}{R_{1}} \right) \\&\quad \left( -\frac{3}{4} J_{0}(\sqrt{s_{2}}R_{1})+J_{2}(\sqrt{s_{2}}R_{1}) - \frac{1}{4} J_{4}(\sqrt{s_{2}}R_{1})\right) \end{aligned} \end{aligned}$$
(103)
$$\begin{aligned} \begin{aligned} m_{43}&=-2c_{3(1)}I_{(1)}+2g_{7(1)}h_{(1)}+4A_{2(1)}+4g_{2(1)}h_{(1)}\\ m_{44}&=\frac{-2A_{1(1)}+2A_{7(1)}}{R_{1}^4}\\&\quad +\frac{-c_{3(1)}I_{(1)}+g_{7(1)}h_{(1)}-2g_{2(1)}h_{(1)}}{R_{1}^2}\\ m_{45}&=\frac{6A_{1(1)}-2A_{7(1)}}{R_{1}^2} -c_{3(1)}I_{(1)}(2ln(R_{1})+1)\\&\quad +g_{7(1)}h_{(1)}(2ln(R_{1})+1)\\&\quad +4A_{2(1)}(2ln(R_{1})+1) +2g_{2(1)}h_{(1)}(2ln(R_{1})+3) \end{aligned} \end{aligned}$$
(104)
$$\begin{aligned} \begin{aligned} m_{46}&=\frac{-d^4A_{1(1)}}{4} \left( \frac{3}{2} I_{0}(dR_{1}) +2 I_{2}(dR_{1}\right) \\&\quad +\frac{1}{2} I_{4}(dR_{1}) ) +\frac{d^3A_{1(1)}}{4R_{1}} ( 3I_{1}(dR_{1}) +I_{3}(dR_{1}) ) \\&\quad +\left( \frac{1.5A_{1(1)}d^2}{R_{1}^2} -\frac{0.5A_{7(1)}d^2}{R_{1}^2} +0.5A_{2(1)}d^2 +g_{2(1)}h_{1}d^2 \right) \\&\quad ( I_{0}(dR_{1})+I_{2}(dR_{1} ) \\&\quad +\left( -\frac{3A_{1(1)}d}{R_{1}^3} +\frac{A_{7(1)}d}{R_{1}^3} -\frac{c_{3(1)}I_{(1)}d}{R_{1}}\right. \\&\quad \left. +\frac{g_{7(1)}h_{(1)}d}{R_{1}} +\frac{A_{2(1)}d}{R_{1}} \right) I_{1}(dR_{1}) \end{aligned} \end{aligned}$$
(105)
$$\begin{aligned} \begin{aligned} m_{47}&=\frac{-d^4A_{1(1)}}{4} \left( \frac{3}{2} k_{0}(dR_{1}) +2 k_{2}(dR_{1}) +\frac{1}{2} k_{4}(dR_{1}) \right) \\&\quad +\frac{d^3A_{1(1)}}{4R_{1}} ( 3k_{1}(dR_{1}) +k_{3}(dR_{1}) ) \\&\quad +\left( \frac{1.5A_{1(1)}d^2}{R_{1}^2} -\frac{0.5A_{7(1)}d^2}{R_{1}^2} +0.5A_{2(1)}d^2 +g_{2(1)}h_{1}d^2 \right) \\&\quad ( k_{0}(dR_{1})+k_{2}(dR_{1} ) \\&\quad +\left( -\frac{3A_{1(1)}d}{R_{1}^3} -\frac{A_{7(1)}d}{R_{1}^3} +\frac{c_{3(1)}I_{1}d}{R_{1}} -\frac{g_{7(1)}h_{(1)}d}{R_{1}} -\frac{A_{2(1)}d}{R_{1}} \right) k_{1}(dR_{1}) \end{aligned} \end{aligned}$$
(106)
$$\begin{aligned} \begin{aligned} m_{48}&= 2(c_{3(1)}I_{(1)}+c_{3(2)}I_{(2)}) -2(g_{7(1)}h_{(1)}+g_{7(2)}h_{(2)}) \\&\quad -4(g_{2(1)}h_{(1)} +g_{2(2)}h_{(2)}) -4A_{2}\\ m_{49}&= -\frac{A1 s_{1}^2 m}{2} \left( -\frac{3}{4}J_{0}(\sqrt{s_{1}}R_{1}) +J_{2}(\sqrt{s_{1}}R_{1}) -\frac{1}{4}J_{4}(\sqrt{s_{1}}R_{1})\right) \\&\quad - \frac{A_{(1)} s_{1}^{1.5} -A_{(1)} s_{1}^{1.5} m }{2R_{1}} \left( -\frac{3}{2} J_{1}(\sqrt{s_{1}} R_{1}) +\frac{1}{2} J_{3}(\sqrt{s_{1}} R_{1}) \right) \\&\quad - \left( \frac{-3A_{(1)} s_{1} +3A_{(3)} s_{1} m }{2R_{1}^{2}} +\frac{A_{(7)} s_{1} -A_{(8)} s_{1} m }{2R_{1}^{2}}\right. \\&\quad \left. -(g_{2(1)}h_{(1)}+g_{2(2)}h_{(2)})s_{1} -\frac{A_{(2)}s_{1}}{2} +\frac{A_{(4)}s_{1}m}{2} \right) \\&\quad (J_{0}(\sqrt{s_{1}}R_{1})-J_{2}(\sqrt{s_{1}}R_{1} ) \\&\quad - \left( \frac{3A_{1} \sqrt{s_{1}} -3A_{3} \sqrt{s_{1}}m}{R_{1}^3} +\frac{-A_{7} \sqrt{s_{1}} +A_{8} \sqrt{s_{1}}m}{R_{1}^3}\right. \\&\quad +\frac{(c_{3(1)}I_{(1)}+c_{3(2)}I_{(2)}) \sqrt{s_{1}} }{R_{1}} -\frac{(g_{7(1)}h_{(1)}+g_{7(2)}h_{(2)}) \sqrt{s_{1}} }{R_{1}}\\&\quad \left. -\frac{(c_{3(1)}S_{(1)}+c_{3(2)}S_{(2)}) \sqrt{s_{1}} m}{R_{1}} -\frac{A_{2} \sqrt{s_{1}} }{R_{1}} +\frac{A_{4} \sqrt{s_{1}} m}{R_{1}}\right) J_{1}(\sqrt{s_{1}} R_{1}) \end{aligned} \end{aligned}$$
(107)
$$\begin{aligned} \begin{aligned} m_{50}&= -\frac{A1 s_{2}^2- A3 s_{2}^2 n}{2} \left( -\frac{3}{4}J_{0}(\sqrt{s_{2}}R_{1}) +J_{2}(\sqrt{s_{2}}R_{1})\right. \\&\quad -\frac{1}{4}J_{4}(\sqrt{s_{2}}R_{1})) - \frac{A_{(1)} s_{2}^{1.5} -A_{(3)} s_{2}^{1.5} n }{2R_{1}}\\&\quad \left( -\frac{3}{2} J_{1}(\sqrt{s_{2}} R_{1}) +\frac{1}{2} J_{3}(\sqrt{s_{2}} R_{1}) \right) \\&\quad - \left( \frac{-3A_{(1)} s_{2} +3A_{(3)} s_{2} m }{2R_{1}^{2}} +\frac{A_{(7)} s_{2} -A_{(8)} s_{2} n }{2R_{1}^{2}}\right. \\&\quad \left. -(g_{2(1)}h_{(1)}+g_{2(2)}h_{(2)})s_{2} -\frac{A_{(2)}s_{2}}{2} +\frac{A_{(4)}s_{2}n}{2} \right) \\&\quad (J_{0}(\sqrt{s_{2}}R_{1})-J_{2}(\sqrt{s_{2}}R_{1} ) \\&\quad - \left( \frac{3A_{1} \sqrt{s_{2}} -3A_{3} \sqrt{s_{2}}n}{R_{1}^3} +\frac{-A_{7} \sqrt{s_{2}} +A_{8} \sqrt{s_{2}}n}{R_{1}^3}\right. \\&\quad +\frac{(c_{3(1)}I_{(1)}+c_{3(2)}I_{(2)}) \sqrt{s_{2}} }{R_{1}} -\frac{(g_{7(1)}h_{(1)}+g_{7(2)}h_{(2)}) \sqrt{s_{2}} }{R_{1}}\\&\quad -\frac{(c_{3(1)}S_{(1)}+c_{3(2)}S_{(2)}) \sqrt{s_{2}} m}{R_{1}}\\&\quad \left. -\frac{A_{2} \sqrt{s_{2}} }{R_{1}} +\frac{A_{4} \sqrt{s_{2}} n}{R_{1}}\right) J_{1}(\sqrt{s_{2}} R_{1}) \end{aligned} \end{aligned}$$
(108)
$$\begin{aligned} \begin{aligned} m_{51}&= - 2 \left( c_{3(1)}S_{(1)}+c_{3(2)}S_{(2)}\right) -4A_{4}\\ m_{52}&= \frac{2A_{1(1)}-2A_{7(1)}}{R_{1}^3}\\ m_{53}&= \frac{2A_{1(1)}+2A_{7(1)}}{R_{1}}\\ m_{54}&= \frac{A_{1(1)}d^3}{4} (3I_{1}(dR_{1})+I_{3}(dR_{1}))\\&\quad +\frac{A_{7(1)}d^2}{2R_{1}} (I_{0}(dR_{1})+I_{2}(dR_{1}))\\&\quad -\frac{A_{7(1)}d}{R_{1}^{2}} I_{1}(dR_{1})\\ m_{55}&= \frac{A_{1(1)}d^3}{4} (-3k_{1}(dR_{1})-k_{3}(dR_{1}))\\&\quad +\frac{A_{7(1)}d^2}{2R_{1}} (k_{0}(dR_{1})+k_{2}(dR_{1}))\\&\quad +\frac{A_{7(1)}d}{R_{1}^{2}} k_{1}(dR_{1})\\ m_{56}&=-(\frac{-A_{(1)} s_{1}^{1.5} +A_{(3)} s_{1}^{1.5} m }{2}\\&\quad \left( -\frac{3}{2} J_{1}(\sqrt{s_{1}}R_{1})+\frac{1}{2} J_{3}(\sqrt{s_{1}}R_{1}) \right) \\&\quad -(\frac{A_{(8)} s_{1}m-A_{(7)} s_{1} }{2R_{1}} (J_{0}(\sqrt{s_{1}}R_{1})- J_{2}(\sqrt{s_{1}}R_{1}))\\&\quad - \frac{A_{7} \sqrt{s_{1}}-A_{8} \sqrt{s_{1}} m}{R_{1}^2}\\&\quad J_{1}(\sqrt{s_{1}}R_{1}) \end{aligned} \end{aligned}$$
(109)
$$\begin{aligned} \begin{aligned} m_{57}&=-(\frac{-A_{(1)} s_{2}^{1.5} +A_{(3)} s_{2}^{1.5} n }{2}\\&\quad \left( -\frac{3}{2} J_{1}(\sqrt{s_{2}}R_{1})+\frac{1}{2} J_{3}(\sqrt{s_{2}}R_{1}) \right) \\&\quad -\left( \frac{A_{(8)} s_{2}n-A_{(7)} s_{2} }{2R_{1}} ( J_{0}(\sqrt{s_{2}}R_{1})- J_{2}(\sqrt{s_{2}}R_{1}) \right) \\&\quad - \frac{A_{7} \sqrt{s_{2}}-A_{8} \sqrt{s_{2}} n}{R_{1}^2}\\&\quad J_{1}(\sqrt{s_{2}}R_{1}) \end{aligned} \end{aligned}$$
(110)
$$\begin{aligned} \begin{aligned} m_{58}&= \frac{-A_{3} s_{1}^{2} +A_{5} s_{1}^{2} m }{2} \\&\quad \left( -\frac{3}{4} J_{0}(\sqrt{s_{1}}R_{1})+ J_{2}(\sqrt{s_{1}}R_{1}) -\frac{1}{4} J_{4}(\sqrt{s_{1}}R_{1}) \right) \\&\quad + \frac{-A_{3} s_{1}^{1.5} +A_{5} s_{1}^{1.5} m }{2R_{1}} \\&\quad \left( -\frac{3}{2} J_{1}(\sqrt{s_{1}}R_{1})+\frac{1}{2} J_{3}(\sqrt{s_{1}}R_{1}) \right) \\&\quad + \left( \frac{3A_{(3)} s_{1}-3A_{5} s_{1} m }{2R_{1}^{2}}\right. \\&\quad \left. +\frac{-A_{8} s_{1}+A_{9} s_{1} m }{2R_{1}^{2}} +\frac{A_{4} s_{1}}{2} -\frac{A_{6} s_{1} m}{2}\right) \\&\quad ( J_{0}(\sqrt{s_{1}}R_{1})- J_{2}(\sqrt{s_{1}}R_{1}) ) \\&\quad +\left( \frac{-3A_{3} s_{1}^{0.5}+3A_{5} s_{1}^{0.5} m }{R_{1}^{3}} + \frac{A_{8} s_{1}^{0.5}-A_{9} s_{1}^{0.5} m }{R_{1}^{3}} \right. \\&\quad -\frac{(c_{3(1)}S_{(1)}+c_{3(2)}S_{(2)}) \sqrt{s_{1}} }{R_{1}} +\frac{(c_{3(1)}h_{(1)}+c_{3(2)}h_{(2)}) \sqrt{s_{1}}m }{R_{1}} \\&\quad \left. +\frac{A_{4} \sqrt{s_{1}} }{R_{1}} -\frac{A_{6} \sqrt{s_{1}}m }{R_{1}}\right) J_{1}(\sqrt{s_{1}}R_{1}) \end{aligned} \end{aligned}$$
(111)
$$\begin{aligned} \begin{aligned} m_{59}&= \frac{-A_{3} s_{2}^{2} +A_{5} s_{2}^{2} n }{2}\\&\quad \left( -\frac{3}{4} J_{0}(\sqrt{s_{2}}R_{1})+ J_{2}(\sqrt{s_{2}}R_{1}) -\frac{1}{4} J_{4}(\sqrt{s_{2}}R_{1}) \right) \\&\quad + \frac{-A_{3} s_{2}^{1.5} +A_{5} s_{2}^{1.5} n }{2R_{1}} \left( -\frac{3}{2} J_{1}(\sqrt{s_{2}}R_{1})+\frac{1}{2} J_{3}(\sqrt{s_{2}}R_{1})\right) \\&\quad + \left( \frac{3A_{(3)} s_{2}-3A_{5} s_{2} m }{2R_{1}^{2}} +\frac{-A_{8} s_{2}+A_{9} s_{2} n }{2R_{1}^{2}}\right. \\&\quad \left. +\frac{A_{4} s_{2}}{2} -\frac{A_{6} s_{2} n}{2} \right) ( J_{0}(\sqrt{s_{2}}R_{1})- J_{2}(\sqrt{s_{2}}R_{1}) ) \\&\quad + \left( \frac{-3A_{3} s_{2}^{0.5}+3A_{5} s_{2}^{0.5} m }{R_{1}^{3}} + \frac{A_{8} s_{2}^{0.5}-A_{9} s_{2}^{0.5} m }{R_{1}^{3}} \right. \\&\quad -\frac{(c_{3(1)}S_{(1)}+c_{3(2)}S_{(2)}) \sqrt{s_{2}} }{R_{1}} +\frac{(c_{3(1)}h_{(1)}+c_{3(2)}h_{(2)}) \sqrt{s_{2}}n }{R_{1}}\\&\quad \left. +\frac{A_{4} \sqrt{s_{2}} }{R_{1}} -\frac{A_{6} \sqrt{s_{2}}n }{R_{1}}\right) J_{1}(\sqrt{s_{2}}R_{1}) \end{aligned} \end{aligned}$$
(112)
$$\begin{aligned} \begin{aligned} m_{60}&=2(c_{3(1)}S_{(1)}+c_{3(2)}S_{(2)})-4A_{4}\\ m_{61}&=4A_{6}-2(c_{3(1)}h_{(1)}+c_{3(2)}h_{(2)})\\ m_{62}&=\frac{A_{(3)} s_{1}^{1.5}-A_{5} s_{1}^{1.5} m }{2}\\&\quad (-\frac{3}{2} J_{1}(\sqrt{s_{1}}R_{1})+ \frac{1}{2}J_{3}(\sqrt{s_{1}}R_{1}) +\frac{A_{8} s_{1}-A_{9} s_{1} m }{2R_{1}}\\&\quad (J_{0}(\sqrt{s_{1}}R_{1})-J_{2}(\sqrt{s_{1}}R_{1} ) \\&\quad + \left( \frac{-A_{8} \sqrt{s_{1}}+A_{9} \sqrt{s_{1}} m }{R_{1}^{2}} \right) J_{1}(\sqrt{s_{1}}R_{1}) ) \end{aligned} \end{aligned}$$
(113)
$$\begin{aligned} \begin{aligned} m_{63}&=\frac{A_{(3)} s_{2}^{1.5}-A_{5} s_{2}^{1.5} n }{2}\\&\quad \left( -\frac{3}{2} J_{1}(\sqrt{s_{2}}R_{1})+ \frac{1}{2}J_{3}(\sqrt{s_{2}}R_{1}) +\frac{A_{8} s_{2}-A_{9} s_{2} n }{2R_{1}}\right. \\&\quad (J_{0}(\sqrt{s_{2}}R_{1})-J_{2}(\sqrt{s_{2}}R_{1} )\\&\quad + \left( \frac{-A_{8} \sqrt{s_{2}}+A_{9} \sqrt{s_{2}} n }{R_{1}^{2}} \right) J_{1}(\sqrt{s_{2}}R_{1}) ) \end{aligned} \end{aligned}$$
(114)

The vector D in Eq. (74) is given as

$$\begin{aligned} \left[ D \right] = \left[ \begin{array}{ccccccccccc} 0 &{}0 &{}0 &{}m_{64} &{} m_{65} &{} m_{66} &{} m_{67} &{} m_{68} &{} m_{69} &{} m_{70} &{} m_{71} \\ \end{array} \right] \end{aligned}$$

with

$$\begin{aligned} \begin{aligned} m_{64}&=t_{0} R_{1}^4 \quad m_{65}=4t_{0} R_{1}^3 \\ m_{66}&=12t_{0} R_{1}^2\quad m_{67}=-32A_{2}t_{0}R_{1}+8A_{4}t_{1}R_{1}\\ m_{68}&=-24A_{1}t_{0}+6A_{3}t_{1}+2A_{8}t_{1}-8A_{7}t_{0}-(c_{3(1)}I_{(1)}\\&\quad +c_{3(2)}I_{(2)}) 4 R_{1}^2 t_{0}\\&\quad +(g_{7(2)}h_{2}+g_{7(1)}h_{1})4 R_{1}^2 t_{0}+(c_{3(2)}S_{2}+c_{3(1)}S_{1}) R_{1}^2 t_{1}\\&\quad +24 t_{0}(g_{2(2)}h_{2}+g_{2(1)}h_{1}) R_{1}^2\\&\quad +16t_{0} R_{1}^2 A2-16t_{3} R_{1}^2 A_{4}\\ m_{69}&=24t_{0}R_{1}A_{1}+8t_{0}R_{1}A_{7}-6t_{1}R_{1}A_{3}-2t_{1}R_{1}A_{8}\\ m_{70}&=-24A_{3}t_{0}+6A_{5}t_{1}-8A_{8}t_{0}+2A_{9}t_{1}\\&\quad -(c_{3(2)}S_{(2)}+c_{3(1)}S_{(1)}) 4 R_{1}^2 t_{0}\\&\quad +(c_{3(2)}h_{2}+c_{3(1)}h_{1}) R_{1}^2 t_{1} +16t_{0} R_{1}^2 A_{4}-16t_{3} R_{1}^2 A_{6}\\ m_{71}&=24t_{0}R_{1}A_{3}+8t_{0}R_{1}A_{8}-6t_{1}R_{1}A_{5}-2t_{1}R_{1}A_{9}\\ t_{0}&=\frac{q A_{6}}{64(A_{2}A_{6}-A_{4}^{2} )} \\ t_{1}&=\frac{q A_{4}}{16(A_{2}A_{6}-A_{4}^{2} )}\\ t_{3}&=\frac{q A_{4}}{64(A_{2}A_{6}-A_{4}^{2} )} \end{aligned} \end{aligned}$$
(115)

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Fu, G., Zhang, Z., Ma, Y. et al. On the size-dependent bending and buckling of the partially covered laminated microplate. Engineering with Computers (2022). https://doi.org/10.1007/s00366-022-01658-x

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  • DOI: https://doi.org/10.1007/s00366-022-01658-x

Keywords

  • Size dependency
  • Bending
  • Buckling
  • Laminated microplate