Low-velocity impact response of geometrically nonlinear functionally graded graphene platelet-reinforced nanocomposite plates

Abstract

This paper investigates the low-velocity impact response of functionally graded multilayer nanocomposite plates reinforced with a low content of graphene nanoplatelets (GPLs) in which GPLs are randomly oriented and uniformly dispersed in the polymer matrix within each individual layer with GPL weight fraction following a layer-wise variation along the plate thickness. The micromechanics-based Halpin–Tsai model is used to evaluate the effective material properties of the GPL-reinforced composite (GPLRC), and the modified nonlinear Hertz contact theory is utilized to define the contact force between the spherical impactor and the GPLRC target plate. The equations of motion of the plate are derived within the framework of the first-order shear deformation plate theory and von Kármán-type nonlinear kinematics and are solved by a two-step perturbation technique. The present analysis is validated through a direct comparison with those in the open literature. A parametric study is then performed to study the effects of GPL distribution pattern, weight fraction, geometry and size, temperature variation as well as the radius and initial velocity of the impactor on the low-velocity impact response of functionally graded GPLRC plates.

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Acknowledgements

This work is fully funded by two research grants from the Australian Research Council under Discovery Project scheme (DP140102132, DP160101978). The authors are grateful for the financial support. Dr. Mitao Song and Prof. Qinsheng Bi are also grateful for the support from the National Natural Science Foundation of China (Grant Nos. 11302087 and 11632008).

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Correspondence to Jie Yang.

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Appendices

Appendix A

The differential operators in Eqs. (25a)–(25d) are

$$\begin{aligned} L_{11}= & {} \gamma _{110} \frac{\partial ^{2}}{\partial x^{2}}+\gamma _{112} \frac{\partial ^{2}}{\partial y^{2}}, \quad L_{12} =\gamma _{120} \frac{\partial }{\partial x}, \\ L_{13}= & {} \gamma _{131} \frac{\partial }{\partial y}, \\ L_{21}= & {} \gamma _{210} \frac{\partial ^{3}}{\partial x^{3}}+\gamma _{212} \frac{\partial ^{3}}{\partial x\partial y^{2}}, \\ L_{22}= & {} \gamma _{221} \frac{\partial ^{3}}{\partial x^{2}\partial y}+\gamma _{223} \frac{\partial ^{3}}{\partial y^{3}}, \\ L_{23}= & {} \frac{\partial ^{4}}{\partial x^{4}}+\gamma _{232} \frac{\partial ^{4}}{\partial x^{2}\partial y^{2}}+\gamma _{234} \frac{\partial ^{4}}{\partial y^{4}}, \\ L_{31}= & {} \gamma _{310} \frac{\partial }{\partial x}, \quad L_{32} =\frac{\partial ^{2}}{\partial x^{2}}+\gamma _{322} \frac{\partial ^{2}}{\partial y^{2}}+\gamma _{32} , \\ L_{33}= & {} \gamma _{331} \frac{\partial ^{2}}{\partial x\partial y}, \\ L_{34}= & {} \gamma _{340} \frac{\partial ^{3}}{\partial x^{3}}+\gamma _{342} \frac{\partial ^{3}}{\partial x\partial y^{2}}, \\ L_{41}= & {} \gamma _{411} \frac{\partial }{\partial y}, \quad L_{42} =\gamma _{421} \frac{\partial ^{2}}{\partial x\partial y}, \\ L_{43}= & {} \gamma _{430} \frac{\partial ^{2}}{\partial x^{2}}+\gamma _{432} \frac{\partial ^{2}}{\partial y^{2}}+\gamma _{43}, \\ L_{44}= & {} \gamma _{441} \frac{\partial ^{3}}{\partial x^{2}\partial y}+\gamma _{443} \frac{\partial ^{3}}{\partial y^{3}} \end{aligned}$$

where

$$\begin{aligned}&\left[ {\gamma _{110} ,\gamma _{112} } \right] ={\kappa a^{2}\left[ {K_{22} ,\beta ^{2}K_{11} } \right] }/{\left( {\pi ^{2}D_{11}^{*} } \right) }, \\&\gamma _{120} ={\kappa a^{2}K_{22} }/{\left( {\pi ^{2}D_{11}^{*} } \right) }, \\&\gamma _{131} ={\kappa \beta a^{2}K_{11} }/{\left( {\pi ^{2}D_{11}^{*} } \right) }, \\&\left[ {\gamma _{210} ,\gamma _{212} } \right] ={\gamma _6 \left[ {B_{21}^{*} ,\beta ^{2}\left( {B_{11}^{*} -B_{33}^{*} } \right) } \right] }/{A_{22}^{*} }, \\&\left[ {\gamma _{221} ,\gamma _{223} } \right] ={\gamma _6 \left[ {\beta \left( {B_{22}^{*} -B_{33}^{*} } \right) ,\beta ^{3}B_{12}^{*} } \right] }/{A_{22}^{*} }, \\&\left[ {\gamma _{232} ,\gamma _{234} } \right] ={\beta ^{2}\left[ {2A_{12}^{*} +A_{33}^{*} ,\beta ^{2}A_{11}^{*} } \right] }/{A_{22}^{*} }, \\&\gamma _{310} ={-\kappa a^{2}K_{22} }/{\left( {\pi ^{2}D_{11}^{*} } \right) }, \\&\left[ {\gamma _{322} ,\gamma _{32} } \right] ={\left[ {\pi ^{2}\beta ^{2}D_{33}^{*} ,-\kappa a^{2}K_{22} } \right] }/{\left( {\pi ^{2}D_{11}^{*} } \right) }, \\&\gamma _{331} ={\beta \left( {D_{12}^{*} +D_{33}^{*} } \right) }/{D_{11}^{*} }, \\&\left[ {\gamma _{340} ,\gamma _{342} } \right] ={\left[ {-B_{21}^{*} ,\beta ^{2}\left( {B_{33}^{*} -B_{11}^{*} } \right) } \right] }/{\left( {\gamma _6 D_{11}^{*} } \right) }, \\&\gamma _{411} ={-\kappa \beta a^{2}K_{11} }/{\left( {\pi ^{2}D_{11}^{*} } \right) }, \\&\gamma _{421} ={\beta \left( {D_{21}^{*} +D_{33}^{*} } \right) }/{D_{11}^{*} }, \\&\left[ {\gamma _{430} ,\gamma _{432} ,\gamma _{43} } \right] \\&\quad ={\left[ {\pi ^{2}D_{33}^{*} ,\pi ^{2}\beta ^{2}D_{22}^{*} ,-\kappa a^{2}K_{11} } \right] }/{\pi ^{2}D_{11}^{*} }, \\&\left[ {\gamma _{441} ,\gamma _{443} } \right] ={\left[ {\beta \left( {B_{33}^{*} -B_{22}^{*} } \right) ,-\beta ^{3}B_{12}^{*} } \right] }/{\left( {\gamma _6 D_{11}^{*} } \right) } \end{aligned}$$

in which \(\gamma _6 = \left( {\frac{A_{11}^{*} A_{22}^{*} }{D_{11}^{*} D_{22}^{*} }} \right) ^{\frac{1}{4}}\).

Appendix B

Coefficients \(g_{\xi \zeta \eta }^{\mathrm{w}}\), \(g_{\xi \zeta \eta }^{\mathrm{f}}\), \(g_{\xi \zeta \eta }^{\mathrm{x}}\), \(g_{\xi \zeta \eta }^{\mathrm{y}}\), \(g_{\xi \zeta \eta }^{\mathrm{q}}\), \(\hat{{g}}_{113}^{\mathrm{f}}\), \(\hat{{g}}_{113}^{\mathrm{x}}\), \(\hat{{g}}_{113}^{\mathrm{y}}\), and \(\hat{{g}}_{113}^{\mathrm{q}}\) are

$$\begin{aligned} g_{111}^{\mathrm{f}}= & {} \frac{1}{g_{11} }\left| {{\begin{array}{ccc} 0&{} {m^{3}\gamma _{210} +mn^{2}\gamma _{212} }&{}\quad {m^{2}n\gamma _{221} +n^{3}\gamma _{223} } \\ {m\gamma _{310} }&{}\quad {m^{2}+n^{2}\gamma _{322} -\gamma _{32} }&{}\quad {mn\gamma _{331} } \\ {n\gamma _{411} }&{}\quad {mn\gamma _{421} }&{}\quad {m^{2}\gamma _{430} +n^{2}\gamma _{432} -\gamma _{43} } \\ \end{array} }} \right| , \\ g_{111}^{\mathrm{x}}= & {} \frac{1}{g_{11} }\left| {{\begin{array}{ccc} {m^{4}+m^{2}n^{2}\gamma _{232} +n^{4}\gamma _{234} }&{}\quad 0&{}\quad {m^{2}n\gamma _{221} +n^{3}\gamma _{223} } \\ {m^{3}\gamma _{340} +mn^{2}\gamma _{342} }&{}\quad {m\gamma _{310} }&{}\quad {mn\gamma _{331} } \\ {m^{2}n\gamma _{441} +n^{3}\gamma _{443} }&{}\quad {n\gamma _{411} }&{}\quad {m^{2}\gamma _{430} +n^{2}\gamma _{432} -\gamma _{43} } \\ \end{array} }} \right| , \\ g_{111}^{\mathrm{y}}= & {} \frac{1}{g_{11} }\left| {{\begin{array}{ccc} {m^{4}+m^{2}n^{2}\gamma _{232} +n^{4}\gamma _{234} }&{}\quad {m^{3}\gamma _{210} +mn^{2}\gamma _{212} }&{}\quad 0 \\ {m^{3}\gamma _{340} +mn^{2}\gamma _{342} }&{}\quad {m^{2}+n^{2}\gamma _{322} -\gamma _{32} }&{}\quad {m\gamma _{310} } \\ {m^{2}n\gamma _{441} +n^{3}\gamma _{443} }&{}\quad {mn\gamma _{421} }&{}\quad {n\gamma _{411} } \\ \end{array} }} \right| , \end{aligned}$$
$$\begin{aligned} g_{111}^{\mathrm{q}}= & {} m^{2}\gamma _{110} +n^{2}\gamma _{112} +m\gamma _{120} g_{111}^{\mathrm{x}} +n\gamma _{131} g_{111}^{\mathrm{y}} \\&-\beta ^{2}\gamma _{14} \left( {n^{2}J_{\mathrm{f}00}^{[0]} +m^{2}J_{\mathrm{F}00}^{[0]} } \right) , \\ g_{202}^{\mathrm{f}}= & {} \frac{1}{2}\frac{\beta ^{2}m^{2}n^{2}\left( {\gamma _{32} -4m^{2}} \right) }{16m^{4}\left( {\gamma _{32} -4m^{2}} \right) +64m^{6}\gamma _{210} \gamma _{340} }, \\ g_{022}^{\mathrm{f}}= & {} \frac{1}{2}\frac{\beta ^{2}m^{2}n^{2}\left( {\gamma _{43} -4n^{2}\gamma _{432} } \right) }{16n^{4}\gamma _{234} \left( {\gamma _{43} -4n^{2}\gamma _{432} } \right) +64n^{6}\gamma _{223} \gamma _{443} }, \end{aligned}$$
$$\begin{aligned} g_{202}^{\mathrm{x}}= & {} -\frac{4\beta ^{2}m^{5}n^{2}\gamma _{340} }{16m^{4}\left( {\gamma _{32} -4m^{2}} \right) +64m^{6}\gamma _{210} \gamma _{340} }, \\ g_{022}^{\mathrm{y}}= & {} -\frac{4\beta ^{2}m^{2}n^{5}\gamma _{443} }{16n^{4}\gamma _{234} \left( {\gamma _{43} -4n^{2}\gamma _{432} } \right) +64n^{6}\gamma _{223} \gamma _{443} }, \\ g_{202}^{\mathrm{q}}= & {} m^{2}n^{2}\beta ^{2}\gamma _{14} g_{111}^{\mathrm{f}} -2m\gamma _{120} g_{202}^{\mathrm{x}} , \\ g_{022}^{\mathrm{q}}= & {} m^{2}n^{2}\beta ^{2}\gamma _{14} g_{111}^{\mathrm{f}} -2n\gamma _{131} g_{022}^{\mathrm{y}} , \\ \end{aligned}$$
$$\begin{aligned} g_{133}^{\mathrm{w}}= & {} \frac{2\beta ^{2}m^{2}n^{2}\gamma _{14} g_{022}^{\mathrm{f}} }{g_{13} } \\&\left| {{\begin{array}{ccc} {m^{4}+9m^{2}n^{2}\gamma _{232} +81n^{4}\gamma _{234} }&{}\quad {m^{3}\gamma _{210} +9mn^{2}\gamma _{212} }&{}\quad {3m^{2}n\gamma _{221} +27n^{3}\gamma _{223} } \\ {-m^{3}\gamma _{340} -9mn^{2}\gamma _{342} }&{}\quad {-m^{2}-9n^{2}\gamma _{322} +\gamma _{32} }&{}\quad {-3mn\gamma _{331} } \\ {-3m^{2}n\gamma _{441} -27n^{3}\gamma _{443} }&{}\quad {-3mn\gamma _{421} }&{}\quad {-m^{2}\gamma _{430} -9n^{2}\gamma _{432} +\gamma _{43} } \\ \end{array} }} \right| ,\\ g_{133}^{\mathrm{f}}= & {} -\frac{2\beta ^{2}m^{2}n^{2}\gamma _{14} g_{022}^{\mathrm{f}} }{g_{13} } \\&\left| {{\begin{array}{ccc} 0&{} {m^{3}\gamma _{210} +9mn^{2}\gamma _{212} }&{}\quad {3m^{2}n\gamma _{221} +27n^{3}\gamma _{223} } \\ {m\gamma _{310} }&{}\quad {-m^{2}-9n^{2}\gamma _{322} +\gamma _{32} }&{}\quad {-3mn\gamma _{331} } \\ {3n\gamma _{411} }&{}\quad {-3mn\gamma _{421} }&{}\quad {-m^{2}\gamma _{430} -9n^{2}\gamma _{432} +\gamma _{43} } \\ \end{array} }} \right| , \\ g_{133}^{\mathrm{x}}= & {} \frac{2\beta ^{2}m^{2}n^{2}\gamma _{14} g_{022}^{\mathrm{f}} }{g_{13} } \\&\left| {{\begin{array}{ccc} 0&{}\quad {m^{4}+9m^{2}n^{2}\gamma _{232} +81n^{4}\gamma _{234} }&{}\quad {3m^{2}n\gamma _{221} +27n^{3}\gamma _{223} } \\ {m\gamma _{310} }&{}\quad {-m^{3}\gamma _{340} -9mn^{2}\gamma _{342} }&{}\quad {-3mn\gamma _{331} } \\ {3n\gamma _{411} }&{}\quad {-3m^{2}n\gamma _{441} -27n^{3}\gamma _{443} }&{}\quad {-m^{2}\gamma _{430} -9n^{2}\gamma _{432} +\gamma _{43} } \\ \end{array} }} \right| , \\ g_{133}^{\mathrm{y}}= & {} -\frac{2\beta ^{2}m^{2}n^{2}\gamma _{14} g_{022}^{\mathrm{f}} }{g_{13} } \\&\left| {{\begin{array}{ccc} 0&{} {m^{4}+9m^{2}n^{2}\gamma _{232} +81n^{4}\gamma _{234} }&{}\quad {m^{3}\gamma _{210} +9mn^{2}\gamma _{212} } \\ {m\gamma _{310} }&{}\quad {-m^{3}\gamma _{340} -9mn^{2}\gamma _{342} }&{}\quad {-m^{2}-9n^{2}\gamma _{322} +\gamma _{32} } \\ {3n\gamma _{411} }&{}\quad {-3m^{2}n\gamma _{441} -27n^{3}\gamma _{443} }&{}\quad {-3mn\gamma _{421} } \\ \end{array} }} \right| , \\ g_{313}^{\mathrm{w}}= & {} \frac{2\beta ^{2}m^{2}n^{2}\gamma _{14} g_{202}^{\mathrm{f}} }{g_{31} } \\&\left| {{\begin{array}{ccc} {81m^{4}+9m^{2}n^{2}\gamma _{232} +n^{4}\gamma _{234} }&{}\quad {27m^{3}\gamma _{210} +3mn^{2}\gamma _{212} }&{}\quad {9m^{2}n\gamma _{221} +n^{3}\gamma _{223} } \\ {-27m^{3}\gamma _{340} -3mn^{2}\gamma _{342} }&{}\quad {-9m^{2}-n^{2}\gamma _{322} +\gamma _{32} }&{}\quad {-3mn\gamma _{331} } \\ {-9m^{2}n\gamma _{441} -n^{3}\gamma _{443} }&{}\quad {-3mn\gamma _{421} }&{}\quad {-9m^{2}\gamma _{430} -n^{2}\gamma _{432} +\gamma _{43} } \\ \end{array} }} \right| ,\\ g_{313}^{\mathrm{f}}= & {} -\frac{2\beta ^{2}m^{2}n^{2}\gamma _{14} g_{202}^{\mathrm{f}} }{g_{31} } \\&\left| {{\begin{array}{ccc} 0&{} {27m^{3}\gamma _{210} +3mn^{2}\gamma _{212} }&{}\quad {9m^{2}n\gamma _{221} +n^{3}\gamma _{223} } \\ {3m\gamma _{310} }&{}\quad {-9m^{2}-n^{2}\gamma _{322} +\gamma _{32} }&{}\quad {-3mn\gamma _{331} } \\ {n\gamma _{411} }&{}\quad {-3mn\gamma _{421} }&{}\quad {-9m^{2}\gamma _{430} -n^{2}\gamma _{432} +\gamma _{43} } \\ \end{array} }} \right| , \\ g_{313}^{\mathrm{x}}= & {} \frac{2\beta ^{2}m^{2}n^{2}\gamma _{14} g_{202}^{\mathrm{f}} }{g_{31} } \\&\left| {{\begin{array}{ccc} 0&{} {81m^{4}+9m^{2}n^{2}\gamma _{232} +n^{4}\gamma _{234} }&{}\quad {9m^{2}n\gamma _{221} +n^{3}\gamma _{223} } \\ {3m\gamma _{310} }&{}\quad {-27m^{3}\gamma _{340} -3mn^{2}\gamma _{342} }&{}\quad {-3mn\gamma _{331} } \\ {n\gamma _{411} }&{}\quad {-9m^{2}n\gamma _{441} -n^{3}\gamma _{443} }&{}\quad {-9m^{2}\gamma _{430} -n^{2}\gamma _{432} +\gamma _{43} } \\ \end{array} }} \right| , \\ \end{aligned}$$
$$\begin{aligned} g_{313}^{\mathrm{y}}= & {} -\frac{2\beta ^{2}m^{2}n^{2}\gamma _{14} g_{202}^{\mathrm{f}} }{g_{31}}\\&\left| {{\begin{array}{ccc} 0&{} {81m^{4}+9m^{2}n^{2}\gamma _{232} +n^{4}\gamma _{234} }&{}\quad {27m^{3}\gamma _{210} +3mn^{2}\gamma _{212} } \\ {3m\gamma _{310} }&{}\quad {-27m^{3}\gamma _{340} -3mn^{2}\gamma _{342} }&{}\quad {-9m^{2}-n^{2}\gamma _{322} +\gamma _{32} } \\ {n\gamma _{411} }&{}\quad {-9m^{2}n\gamma _{441} -n^{3}\gamma _{443} }&{}\quad {-3mn\gamma _{421} } \\ \end{array} }} \right| , \\ \hat{{g}}_{113}^{\mathrm{f}}= & {} \frac{I}{g_{11} }\left| {{\begin{array}{ccc} 0&{}\quad {m^{3}\gamma _{210} +mn^{2}\gamma _{212} }&{}\quad {m^{2}n\gamma _{221} +n^{3}\gamma _{223} } \\ {g_{111}^{\mathrm{x}} }&{}\quad {-m^{2}-n^{2}\gamma _{322} +\gamma _{32} }&{}\quad {-mn\gamma _{331} } \\ {g_{111}^{\mathrm{y}} }&{}\quad {-mn\gamma _{421} }&{}\quad {-m^{2}\gamma _{430} -n^{2}\gamma _{432} +\gamma _{43} } \\ \end{array} }} \right| , \end{aligned}$$
$$\begin{aligned} \hat{{g}}_{113}^{\mathrm{x}}= & {} \frac{I}{g_{11} }\left| {{\begin{array}{ccc} {m^{4}+m^{2}n^{2}\gamma _{232} +n^{4}\gamma _{234} }&{}\quad 0&{}\quad {m^{2}n\gamma _{221} +n^{3}\gamma _{223} } \\ {-m^{3}\gamma _{340} -mn^{2}\gamma _{342} }&{}\quad {g_{111}^{\mathrm{x}} }&{}\quad {-mn\gamma _{331} } \\ {-m^{2}n\gamma _{441} -n^{3}\gamma _{443} }&{}\quad {g_{111}^{\mathrm{y}} }&{}\quad {-m^{2}\gamma _{430} -n^{2}\gamma _{432} +\gamma _{43} } \\ \end{array} }} \right| , \\ \hat{{g}}_{113}^{\mathrm{y}}= & {} \frac{I}{g_{11} }\left| {{\begin{array}{ccc} {m^{4}+m^{2}n^{2}\gamma _{232} +n^{4}\gamma _{234} }&{}\quad {m^{3}\gamma _{210} +mn^{2}\gamma _{212} }&{}\quad 0 \\ {-m^{3}\gamma _{340} -mn^{2}\gamma _{342} }&{}\quad {-m^{2}-n^{2}\gamma _{322} +\gamma _{32} }&{}\quad {g_{111}^{\mathrm{x}} } \\ {-m^{2}n\gamma _{441} -n^{3}\gamma _{443} }&{}\quad {-mn\gamma _{421} }&{}\quad {g_{111}^{\mathrm{y}} } \\ \end{array} }} \right| , \\ g_{113}^{\mathrm{q}}= & {} 2\beta ^{2}m^{2}n^{2}\gamma _{14} \left( {g_{022}^{\mathrm{f}} +g_{202}^{\mathrm{f}} } \right) -\beta ^{2}\gamma _{14} \left( {n^{2}g_{002}^{\mathrm{f}} +m^{2}g_{002}^{\mathrm{F}} } \right) , \\ \hat{{g}}_{113}^{\mathrm{q}}= & {} m\gamma _{120} \hat{{g}}_{113}^{\mathrm{x}} +n\gamma _{131} \hat{{g}}_{113}^{\mathrm{y}} +I_0 \end{aligned}$$

in which

$$\begin{aligned} g_{11}= & {} \left| {{\begin{array}{ccc} {m^{4}+m^{2}n^{2}\gamma _{232} +n^{4}\gamma _{234} }&{}\quad {m^{3}\gamma _{210} +mn^{2}\gamma _{212} }&{}\quad {m^{2}n\gamma _{221} +n^{3}\gamma _{223} } \\ {m^{3}\gamma _{340} +mn^{2}\gamma _{342} }&{}\quad {m^{2}+n^{2}\gamma _{322} -\gamma _{32} }&{}\quad {mn\gamma _{331} } \\ {m^{2}n\gamma _{441} +n^{3}\gamma _{443} }&{}\quad {mn\gamma _{421} }&{}\quad {m^{2}\gamma _{430} +n^{2}\gamma _{432} -\gamma _{43} } \\ \end{array} }} \right| , \\ g_{13}= & {} \left| {{\begin{array}{l} {m^{2}\gamma _{110} +9n^{2}\gamma _{112} -\beta ^{2}\gamma _{14} \left( {m^{2}J_{\mathrm{F}00}^{[0]} +9n^{2}J_{\mathrm{f}00}^{[0]} } \right) } \\ 0 \\ {m\gamma _{310} } \\ {3n\gamma _{411} } \\ \end{array} }} \right. \\&\left. { \, {\begin{array}{ccc} 0&{}\quad {m\gamma _{120} }&{}\quad {3n\gamma _{131} } \\ {m^{4}+9m^{2}n^{2}\gamma _{232} +81n^{4}\gamma _{234} }&{}\quad {m^{3}\gamma _{210} +9mn^{2}\gamma _{212} }&{}\quad {3m^{2}n\gamma _{221} +27n^{3}\gamma _{223} } \\ {-m^{3}\gamma _{340} -9mn^{2}\gamma _{342} }&{}\quad {-m^{2}-9n^{2}\gamma _{322} +\gamma _{32} }&{}\quad {-3mn\gamma _{331} } \\ {-3m^{2}n\gamma _{441} -27n^{3}\gamma _{443} }&{}\quad {-3mn\gamma _{421} }&{}\quad {-m^{2}\gamma _{430} -9n^{2}\gamma _{432} +\gamma _{43} } \\ \end{array} }} \right| , \\ g_{31}= & {} \left| {{\begin{array}{l} {9m^{2}\gamma _{110} +n^{2}\gamma _{112} -\beta ^{2}\gamma _{14} \left( {9m^{2}J_{\mathrm{F}00}^{[0]} +n^{2}J_{\mathrm{f}00}^{[0]} } \right) } \\ 0 \\ {3m\gamma _{310} } \\ {n\gamma _{411} } \\ \end{array} }} \right. \\&\, \left. {{\begin{array}{ccc} 0&{}\quad {3m\gamma _{120} }&{}\quad {n\gamma _{131} } \\ {81m^{4}+9m^{2}n^{2}\gamma _{232} +n^{4}\gamma _{234} }&{}\quad {27m^{3}\gamma _{210} +3mn^{2}\gamma _{212} }&{}\quad {9m^{2}n\gamma _{221} +n^{3}\gamma _{223} } \\ {-27m^{3}\gamma _{340} -3mn^{2}\gamma _{342} }&{}\quad {-9m^{2}-n^{2}\gamma _{322} +\gamma _{32} }&{}\quad {-3mn\gamma _{331} } \\ {-9m^{2}n\gamma _{441} -n^{3}\gamma _{443} }&{}\quad {-3mn\gamma _{421} }&{}\quad {-9m^{2}\gamma _{430} -n^{2}\gamma _{432} +\gamma _{43} } \\ \end{array} }} \right| \\ \end{aligned}$$

The coefficients \(J_{\mathrm{f}00}^{[0]}\) and \(J_{\mathrm{F}00}^{[0]}\) can be obtained by substituting Eqs. (34b)–(34d) into Eqs. (26c) and (27c) or Eqs. (26d) and (27d).

Appendix C

Coefficients \(\chi _1\), \(\chi _2\), \(\chi _3\) and \(\chi _{\mathrm{f}}\) in Eq. (37) are

$$\begin{aligned} \chi _1= & {} \left[ m^{2}\gamma _{110} +n^{2}\gamma _{112} +m\gamma _{120} g_{111}^{\mathrm{x}} +n\gamma _{131} g_{111}^{\mathrm{y}}\right. \\&\left. -4\beta ^{2}\gamma _{14} \left( {m^{2}\lambda _x +n^{2}\lambda _y } \right) \right] \Big /{\hat{{g}}_{113}^{\mathrm{q}} }, \\ \chi _2= & {} {4\varGamma \left( {g_{202}^{\mathrm{q}} +g_{022}^{\mathrm{q}} } \right) }/{\left( {\pi ^{2}\hat{{g}}_{113}^{\mathrm{q}} } \right) }, \\ \chi _3= & {} {2\beta ^{2}m^{2}n^{2}\gamma _{14} \left( {g_{022}^{\mathrm{f}} +g_{202}^{\mathrm{f}} } \right) }/{\hat{{g}}_{113}^{\mathrm{q}} }, \\ \chi _{\mathrm{f}}= & {} {4\sin (mx_{\mathrm{c}} )\sin (ny_{\mathrm{c}} )}/{\left( {\pi ^{2}\hat{{g}}_{113}^{\mathrm{q}} } \right) } \end{aligned}$$

in which

$$\begin{aligned} \varGamma =\left\{ {{\begin{array}{ll} -\frac{4}{3mn},&{} m\hbox { and }n\hbox { are both odd numbers} \\ 0,&{} m\hbox { or }n\hbox { is an even number} \\ \end{array} }} \right. \end{aligned}$$

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Song, M., Li, X., Kitipornchai, S. et al. Low-velocity impact response of geometrically nonlinear functionally graded graphene platelet-reinforced nanocomposite plates. Nonlinear Dyn 95, 2333–2352 (2019). https://doi.org/10.1007/s11071-018-4695-y

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Keywords

  • Graphene nanoplatelets
  • Functionally graded materials
  • Low-velocity impact
  • The first-order shear deformation plate theory