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Low-velocity impact response of nanotube-reinforced composite sandwich curved panels

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Abstract

The low-velocity impact response of the sandwich curved panels with functionally graded carbon-nanotube-reinforced composite (FG-CNTRC) surface and isotropic foam core is discussed in this paper. Five types of stacking arrangements including uniform distribution of FG-CNTRC considering thermal environment were analysed. Using the Hertz contact law and rule of mixture model as well as the Kármán-type equations, the nonlinear formulations were built and solved by the two-step perturbation method. The carbon nanotubes’ volume fraction, the structure size, the original impact velocity, temperature, relative thickness and the influence of gradient forms on the panels’ impact behaviours were analysed. The outcomes show that the stiffness of the non-contact surface has large influence on contact response and various types of FG-CNTRC can be used for different operating conditions providing stiffness or cushion performance.

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Acknowledgements

The authors would gratefully acknowledge the financial support from the Natural Science Funds for Young Scholar of Jiangsu Province (No. BK 20170837) and the fundamental research funds for the central universities (No. 309181B8807).

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Authors and Affiliations

  1. Department of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing, China

    Wuning Ma, Chunhao Yang, Dawei Ma & Jianlin Zhong

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  1. Wuning Ma
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  2. Chunhao Yang
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  3. Dawei Ma
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  4. Jianlin Zhong
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Corresponding author

Correspondence to Chunhao Yang.

Appendix I.

Appendix I.

$$\begin{aligned}L_{11}\left( \right) &=\frac{4}{3h^2}\left[ F_{11}^{*}\frac{\partial ^4}{\partial X^4}+2\left( F_{16}^{*}+F_{61}^{*} \right) \frac{\partial ^4}{\partial X^3\partial Y} \right. \\&\quad +\left( F_{12}^{*}+F_{21}^{*}+4F_{66}^{*} \right) \frac{\partial ^4}{\partial X^2\partial Y^2}\\&\quad \left.+2\left( F_{26}^{*}+F_{62}^{*} \right) \frac{\partial ^4}{\partial X\partial Y^3}+F_{22}^{*}\frac{\partial ^4}{\partial Y^4}\right],\\ L_{12}\left( \right) &= \left[ D_{11}^{*}-\frac{4}{3h^2}F_{11}^{*} \right] \frac{\partial ^3}{\partial X^3}\\&\quad +\left[ 3D_{16}^{*}-\frac{4}{3h^2}\left( F_{61}^{*}+2F_{16} \right) \right] \frac{\partial ^3}{\partial X^2\partial Y}\\&\quad +\left[ \left( D_{12}^{*}+2D_{66}^{*} \right) -\frac{4}{3h^2}\left( F_{12}^{*}+2F_{66}^{*} \right) \right] \frac{\partial ^3}{\partial X\partial Y^2}\\&\quad +\left[ D_{26}^{*}-\frac{4}{3h^2}F_{26} \right] \frac{\partial ^3}{\partial Y^3},\\ L_{13}\left( \right) &= \left[ D_{16}^{*}-\frac{4}{3h^2}F_{16}^{*} \right] \frac{\partial ^3}{\partial X^3}\\&\quad +\left[ \left( D_{12}^{*}+2D_{66}^{*} \right) -\frac{4}{3h^2}\left( F_{21}^{*}+2F_{66}^{*} \right) \right] \frac{\partial ^3}{\partial X^2\partial Y}\\&\quad +\left[ 3D_{26}^{*}-\frac{4}{3h^2}\left( F_{62}^{*}+2F_{26}^{*} \right) \right] \frac{\partial ^3}{\partial X\partial Y^2}\\&\quad +\left[ D_{22}^{*}-\frac{4}{3h^2}E_{22}^{*} \right] \frac{\partial ^3}{\partial Y^3},\\ L_{14}\left( \right) &= B_{21}^{*}\frac{\partial ^4}{\partial X^4}+\left( 2B_{26}^{*}-B_{61}^{*} \right) \frac{\partial ^4}{\partial X^3\partial Y}\\&\quad +\left( B_{11}^{*}+B_{22}^{*}-2B_{66}^{*} \right) \frac{\partial ^4}{\partial X^2\partial Y^2}\\&\quad +\left( 2B_{16}^{*}-B_{62}^{*} \right) \frac{\partial ^4}{\partial X\partial Y^3}+B_{12}^{*}\frac{\partial ^4}{\partial Y^4},\\ L_{15}\left( {\bar{N}}^T \right) &= \frac{\partial ^2}{\partial X^2}\left( B_{11}^{*}{\bar{N}}_{x}^{T}+B_{21}^{*}{\bar{N}}_{y}^{T} \right) +2\frac{\partial ^2}{\partial X\partial Y}\left( B_{b6}^{*}{\bar{N}}_{xy}^{T} \right) \\&\quad +\frac{\partial ^2}{\partial Y^2}\left( B_{12}^{*}{\bar{N}}_{x}^{T}+B_{22}^{*}{\bar{N}}_{y}^{T} \right) ,\\ L_{16}\left( {\bar{M}}^T \right) &= \frac{\partial ^2}{\partial X^2}\left( {\bar{M}}_{x}^{T} \right) +2\frac{\partial ^2}{\partial X\partial Y}\left( {\bar{M}}_{xy}^{T} \right) +\frac{\partial ^2}{\partial Y^2}\left( {\bar{M}}_{y}^{T} \right) ,\\ L_{21}\left( \right) &= A_{22}^{*}\frac{\partial ^4}{\partial X^4}-2A_{26}^{*}\frac{\partial ^4}{\partial X^3\partial Y}+\left( 2A_{12}^{*}+A_{66}^{*} \right) \frac{\partial ^4}{\partial X^2\partial Y^2}\\&\quad +2A_{16}^{*}\frac{\partial ^4}{\partial X\partial Y^3}+A_{11}^{*}\frac{\partial ^4}{\partial Y^4},\\ L_{22}\left( \right) &= \left[ B_{21}^{*}-\frac{4}{3h^2}E_{21}^{*} \right] \frac{\partial ^3}{\partial X^3}\\&\quad +\left[ B_{26}^{*}-B_{61}^{*}-\frac{4}{3h^2}\left( E_{26}^{*}-E_{61}^{*} \right) \right] \frac{\partial ^3}{\partial X^2\partial Y}\\&\quad +\left[ B_{11}^{*}-B_{66}^{*}-\frac{4}{3h^2}\left( E_{11}^{*}-E_{66}^{*} \right) \right] \frac{\partial ^3}{\partial X\partial Y^2}\\&\quad +\left[ B_{16}^{*}-\frac{4}{3h^2}E_{16}^{*} \right] \frac{\partial ^3}{\partial Y^3},\\ L_{23}\left( \right) &= \left[ B_{26}^{*}-\frac{4}{3h^2}E_{26}^{*} \right] \frac{\partial ^3}{\partial X^3}\\&\quad +\left[ B_{22}^{*}-B_{66}^{*}-\frac{4}{3h^2}\left( E_{22}^{*}-E_{66}^{*} \right) \right] \frac{\partial ^3}{\partial X^2\partial Y}\\&\quad +\left[ B_{16}^{*}-B_{62}^{*}-\frac{4}{3h^2}\left( E_{16}^{*}-E_{62}^{*} \right) \right] \frac{\partial ^3}{\partial X\partial Y^2}\\&\quad +\left[ B_{12}^{*}-\frac{4}{3h^2}E_{12}^{*} \right] \frac{\partial ^3}{\partial Y^3},\\ L_{24}\left( \right) &= \frac{4}{3h^2}\left[ E_{21}^{*}\frac{\partial ^4}{\partial X^4}+\left( 2E_{26}^{*}-E_{61}^{*} \right) \frac{\partial ^4}{\partial X^3\partial Y} \right. \\&\quad +\left( E_{11}^{*}+E_{22}^{*}-2E_{66}^{*} \right) \frac{\partial ^4}{\partial X^2\partial Y^2}\\&\quad +\left. \left( 2E_{16}^{*}-E_{62}^{*} \right) \frac{\partial ^4}{\partial X\partial Y^3}+E_{12}^{*}\frac{\partial ^4}{\partial Y^4}, \right] \\ L_{25}\left( {\bar{N}}^T \right) &= \frac{\partial ^2}{\partial X^2}\left( A_{12}^{*}{\bar{N}}_{x}^{T}+A_{22}^{*}{\bar{N}}_{y}^{T} \right) -\frac{\partial ^2}{\partial X\partial Y}\left( A_{66}^{*}{\bar{N}}_{xy}^{T} \right) \\&\quad +\frac{\partial ^2}{\partial Y^2}\left( A_{11}^{*}{\bar{N}}_{x}^{T}+A_{12}^{*}{\bar{N}}_{y}^{T} \right) ,\\ L_{31}\left( \right) &= \left[ A_{55}-\frac{8}{h^2}D_{55}+\frac{16}{h^4}F_{55} \right] \frac{\partial }{\partial X}\\&\quad +\left[ A_{45}-\frac{8}{h^2}D_{45}+\frac{16}{h^4}F_{45} \right] \frac{\partial }{\partial Y}\\&\quad +\frac{4}{3h^2}\left[ \left( F_{11}^{*}-\frac{4}{3h^2}H_{11}^{*} \right) \frac{\partial ^3}{\partial X^3} \right. \\&\quad +\left( F_{16}^{*}+2F_{61}^{*}-\frac{4}{h^2}H_{16}^{*} \right) \frac{\partial ^3}{\partial X^2\partial Y}\\&\quad +\left( F_{21}^{*}+2F_{66}^{*}-\frac{4}{3h^2}\left( H_{12}^{*}+2H_{66}^{*} \right) \right) \frac{\partial ^3}{\partial X\partial Y^2}\\&\left. +\left( F_{26}^{*}-\frac{4}{3h^2}H_{26}^{*} \right) \frac{\partial ^3}{\partial Y^3} \right] ,\\ L_{32}\left( \right) &= \left[ A_{55}-\frac{8}{h^2}D_{55}+\frac{16}{h^4}F_{55} \right] \\&\quad -\left[ D_{11}^{*}-\frac{8}{3h^2}F_{11}^{*}+\frac{16}{9h^4}H_{11}^{*} \right] \frac{\partial ^2}{\partial X^2}\\&\quad -2\left[ D_{16}^{*}-\frac{4}{3h^2}\left( F_{16}^{*}+F_{61}^{*} \right) +\frac{16}{9h^4}H_{16}^{x} \right] \frac{\partial ^2}{\partial X\partial Y}\\&\quad -\left[ D_{66}^{*}-\frac{4}{3h^2}F_{66}^{*}+\frac{16}{9h^4}H_{66}^{*} \right] \frac{\partial ^2}{\partial Y^2},\\ L_{33}\left( \right) &= \left[ A_{45}-\frac{8}{h^2}D_{45}+\frac{16}{h^4}F_{45} \right] \\&\quad -\left[ D_{16}^{*}-\frac{4}{3h^2}\left( F_{16}^{*}+F_{61}^{*} \right) +\frac{16}{9h^4}H_{16}^{*} \right] \frac{\partial ^2}{\partial X^2}\\&\quad -\left[ D_{12}^{*}+D_{66}^{*}-\frac{4}{3h^2}\left( F_{12}^{*}+F_{21}^{*}+2F_{66}^{*} \right) \right. \\&\quad +\left. \frac{16}{9h^4}\left( H_{12}^{*}+H_{66}^{x} \right) \right] \frac{\partial ^2}{\partial X\partial Y}\\&\quad -\left[ D_{26}^{*}-\frac{4}{3h^2}\left( F_{26}^{*}+F_{62}^{*} \right) +\frac{16}{9h^4}H_{26}^{*} \right] \frac{\partial ^2}{\partial Y^2},\\ L_{34}\left( \right) &= L_{22}\left( \right) , L_{35}\left( {\bar{N}}^T \right) \\&\quad = \frac{\partial }{\partial X}\left[ \left( B_{11}^{*}-\frac{4}{3h^2}E_{11}^{*} \right) {\bar{N}}_{x}^{T} \right. +\left. \left( B_{21}^{*}-\frac{4}{3h^2}E_{21}^{*} \right) {\bar{N}}_{y}^{T} \right] \\&\quad +\frac{\partial }{\partial Y}\left[ \left( B_{66}^{*}-\frac{4}{3h^2} \right) {\bar{N}}_{xy}^{T} \right] \\ L_{36}\left( {\bar{S}}^T \right) &= \frac{\partial }{\partial X}\left( {\bar{S}}_{x}^{T} \right) +\frac{\partial }{\partial Y}\left( {\bar{S}}_{xy}^{T} \right) ,\\ L_{41}\left( \right) &= \left[ A_{45}-\frac{8}{h^2}D_{45}+\frac{16}{h^4}F_{45} \right] \frac{\partial }{\partial X}\\&\quad +\left[ A_{44}-\frac{8}{h^2}D_{44}+\frac{16}{h^4}F_{44} \right] \frac{\partial }{\partial Y}\\&\quad +\frac{4}{3h^2}\left[ \left( F_{16}^{*}-\frac{4}{3h^2}H_{16}^{*} \right) \frac{\partial ^3}{\partial X^3} \right. \\&\quad +\left( F_{12}^{*}+2F_{66}^{*}-\frac{4}{3h^2}\left( H_{12}^{*}-2H_{66}^{*} \right) \right) \frac{\partial ^3}{\partial X^2\partial Y}\\&\quad +\left( F_{26}^{*}+2F_{62}^{*}-\frac{4}{h^2}H_{26}^{*} \right) \frac{\partial ^3}{\partial X\partial Y^2}\\&\left. +\left( F_{22}^{*}-\frac{4}{3h^2}H_{22}^{*} \right) \frac{\partial ^3}{\partial Y^3} \right] ,\\ L_{42}\left( \right) &= L_{33}\left( \right) ,\\ L_{43}\left( \right) &= \left[ A_{44}-\frac{8}{h^2}D_{44}+\frac{16}{h^4}F_{44} \right] \\&\quad -\left[ D_{66}^{*}-\frac{8}{3h^2}F_{66}^{*}+\frac{16}{9h^4}H_{66}^{*} \right] \frac{\partial ^2}{\partial X^2}\\&\quad -2\left[ D_{26}^{*}-\frac{4}{3h^2}\left( F_{26}^{*}+H_{62}^{*} \right) +\frac{16}{9h^4}H_{26}^{*} \right] \frac{\partial ^2}{\partial X\partial Y}\\&\quad -\left[ D_{22}^{*}-\frac{8}{3h^2}F_{22}^{*}+\frac{16}{9h^4}H_{22}^{*} \right] \frac{\partial ^2}{\partial Y^2},\\ L_{44}\left( \right) &= L_{23}\left( \right) ,\\ L_{45}\left( {\bar{N}}^T \right) &= \frac{\partial }{\partial X}\left[ \left( B_{16}^{*}-\frac{4}{3h^2}E_{16}^{*} \right) {\bar{N}}_{x}^{T} \right. +\left( B_{26}^{*}-\frac{4}{3h^2}E_{26}^{*} \right) {\bar{N}}_{y}^{T}\\&\quad +\left. \left( B_{66}^{*}-\frac{4}{3h^2}E_{66}^{*} \right) {\bar{N}}_{xy}^{T} \right] +\frac{\partial }{\partial Y}\left[ \left( B_{12}^{*}-\frac{4}{3h^2}E_{12}^{*} \right) {\bar{N}}_{x}^{T} \right. \\&\quad +\left( B_{22}^{*}-\frac{4}{3h^2}E_{22}^{*} \right) {\bar{N}}_{y}^{T}+\left. \left( B_{62}^{*}-\frac{4}{3h^2}E_{62}^{*} \right) {\bar{N}}_{xy}^{T} \right] \\ L_{46}\left( {\bar{S}}^T \right) &= \frac{\partial }{\partial X}\left( {\bar{S}}_{xy}^{T} \right) +\frac{\partial }{\partial Y}\left( {\bar{S}}_{y}^{T} \right) ,\\ L\left( \right) &= \frac{\partial ^2}{\partial X^2}\frac{\partial ^2}{\partial Y^2}-2\frac{\partial ^2}{\partial X\partial Y}\frac{\partial ^2}{\partial X\partial Y}+\frac{\partial ^2}{\partial Y^2}\frac{\partial ^2}{\partial X^2}. \end{aligned}$$

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Ma, W., Yang, C., Ma, D. et al. Low-velocity impact response of nanotube-reinforced composite sandwich curved panels. Sādhanā 44, 227 (2019). https://doi.org/10.1007/s12046-019-1214-x

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