Nonlinear post-buckling and vibration of 2D penta-graphene composite plates

Abstract

The newly developed penta-graphene is a two-dimensional (2D) carbon allotrope with promising mechanical properties. This paper investigates the nonlinear post-buckling and vibration of imperfect three-dimensional penta-graphene composite plates resting on elastic foundations and subjected to uniform external pressure and axial compressive load. The elastic constants of the single-layer penta-graphene are fully determined by the density functional theory by fitting the equation of strain energy to the density functional theory energy. Specifically, the elastic constant $C_{66}$ which has not been considered by other authors is also determined. The motion and compatibility equations are derived based on the classical plate theory taking into account von Karman geometrical nonlinearity, initial geometrical imperfection and Pasternak type elastic foundations. For nonlinear post-buckling, the Bubnov–Galerkin method is applied to obtain the load–deflection amplitude curves while the Runge–Kutta method and harmonic balance method are used to obtain the deflection amplitude–time curves and the amplitude–frequency curves for nonlinear vibration. Numerical results show the effects of geometrical parameters, initial imperfection and elastic foundations on the nonlinear post-buckling and vibration of the imperfect 2D penta-graphene plates. The present results are also compared to others to validate the accuracy of the applied method and approach.

Appendices

Appendix A

$$\begin{aligned} \begin{aligned} b_1^1&=\frac{abh\lambda _m \delta _n }{16}\left[ {\begin{array}{l} P_1 \frac{\left( {Q_2 Q_4 -Q_1 Q_3 } \right) }{Q_2^2 -Q_1^2 }\lambda _m^4 +P_2 \frac{\left( {Q_2 Q_4 -Q_1 Q_3 } \right) }{Q_2^2 -Q_1^2 }\delta _n^4 \\ +P_3 \frac{\left( {Q_2 Q_4 -Q_1 Q_3 } \right) }{Q_2^2 -Q_1^2 }\lambda _{m}^{2} \delta _n^2 -P_4 \frac{\left( {Q_2 Q_3 -Q_1 Q_4 } \right) }{Q_2^2 -Q_1^2 } \\ -P_5 \frac{\left( {Q_2 Q_3 -Q_1 Q_4 } \right) }{Q_2^2 -Q_1^2 }+P_6 \lambda _m^4 +P_7 \delta _n^4 +P_8 \lambda _{m}^{2} \delta _{n}^{2} -k_1 -k_2 \left( {\lambda _{m}^{2} +\delta _{n}^{2}} \right) \\ \end{array}} \right] , \\ b_2^1&=-\frac{\lambda _m^{2}\delta _n^{2}}{12}\left( {P_1 \frac{1}{A_{22}^*}+P_2 \frac{1}{A_{11}^*}} \right) h^{2}, \\ b_3^1&=\left( {\frac{ab\lambda _m \delta _n }{256}\left( {\frac{\delta _n^4 }{A_{22}^*}+\frac{\lambda _m^4 }{A_{11}^*}} \right) } \right) h^{3}, \\ b_4^1&=-\left( {\frac{8\lambda _m \delta _n }{12}\frac{\left( {Q_2 Q_4 -Q_1 Q_3 } \right) }{Q_2^2 -Q_1^2 }} \right) h^{2}. \\ \end{aligned} \end{aligned}$$

Appendix B

$$\begin{aligned} \begin{aligned} b_1^2&=-\frac{1}{h\lambda _m^2 }\left[ \begin{array}{l} P_1 \frac{\left( {Q_2 Q_4 -Q_1 Q_3 } \right) }{Q_2^2 -Q_1^2 }\lambda _m^4 +P_2 \frac{\left( {Q_2 Q_4 -Q_1 Q_3 } \right) }{Q_2^2 -Q_1^2 }\delta _n^4 \\ +P_3 \frac{\left( {Q_2 Q_4 -Q_1 Q_3 } \right) }{Q_2^2 -Q_1^2 }\lambda _{m}^2 \delta _n^2 -P_4 \frac{\left( {Q_2 Q_3 -Q_1 Q_4 } \right) }{Q_2^2 -Q_1^2 } \\ -P_5 \frac{\left( {Q_2 Q_3 -Q_1 Q_4 } \right) }{Q_2^2 -Q_1^2 }+P_6 \lambda _{m}^4 +P_7 \delta _n^4 +P_8 \lambda _{m}^2 \delta _n^2 -k_1 -k_2 \left( {\lambda _m^2 +\delta _n^2 } \right) \\ \end{array} \right] , \\ b_2^2&=-\frac{4\delta _n }{3ab\lambda _m }\left( {P_1 \frac{1}{A_{22}^*}+P_2 \frac{1}{A_{11}^*}} \right) , \\ b_3^2&=\frac{1}{16h\lambda _m^2 }\left( {\frac{\delta _n^4 }{A_{22}^*}+\frac{\lambda _m^4 }{A_{11}^*}} \right) , \\ b_4^2&=-\frac{32\delta _n }{3ab\lambda _m }\frac{\left( {Q_2 Q_4 -Q_1 Q_3 } \right) }{Q_2^2 -Q_1^2 }. \end{aligned} \end{aligned}$$

Appendix C

$$\begin{aligned} \begin{aligned} m_1&=-\frac{ab}{4}\left[ P_1 \frac{(F_2 F_4 -F_1 F_3 )}{F_2^2 -F_1^2 }\lambda _m^4 +P_2 \frac{(F_2 F_4 -F_1 F_3 )}{F_2^2 -F_1^2 }\delta _n^4 +P_3 \frac{(F_2 F_4 -F_1 F_3 )}{F_2^2 -F_1^2 }\lambda _m^2 \delta _n^2 \right. \\&\quad \left. -P_4 \frac{(F_2 F_3 -F_1 F_4 )}{F_2^2 -F_1^2 }-P_5 \frac{(F_2 F_3 -F_1 F_4 )}{F_2^2 -F_1^2 }+P_6 \lambda _m^4 +P_7 \delta _n^4 +P_8 \lambda _{m}^{2} \delta _{n}^{2} -k_1 -k_2 (\lambda _{m}^{2} +\delta _n^2 )\right] , \\ m_2&=-\left[ {\frac{8}{3}\frac{(F_2 F_4 -F_1 F_3 )}{F_2^2 -F_1^2 }\lambda _m \delta _n } \right] ,\quad m_3 =-\left[ {\frac{1}{3}\lambda _m \delta _n \left( {P_1 \frac{1}{A_{22}^*}+P_2 \frac{1}{A_{11}^*}} \right) } \right] , \\ m_4&=-\left[ {-\frac{ab}{64}\left( {\frac{1}{A_{22}^*}\delta _n^4 +\frac{1}{A_{11}^*}\lambda _m^4 } \right) } \right] ,\quad m_5 =-\frac{4}{\lambda _m \delta _n }, m_6 =\rho _1 \frac{ab}{4}. \end{aligned} \end{aligned}$$