A first-principle study of nonlinear large amplitude vibration and global optimization of 3D penta-graphene plates based on the Bees Algorithm


Penta-graphene, a new monolayer of carbon atoms, has been synthesized with ideal strength and temperature resistance. However, the mechanical behavior of penta-graphene has not been fully investigated yet. This paper presents an analytical investigation on the nonlinear large amplitude vibration of imperfect three-dimensional (3D) penta-graphene plates subjected to uniformly distributed external pressure with simply supported and immovable edges in thermal environments. The elastic constants and the thermal expansion coefficients of the 3D penta-graphene plate are determined using the density functional theory. The motion and compatibility equations are established based on the Reddy’s higher-order shear deformation plate theory in which the effect of von Karman nonlinear terms, the initial imperfection and the Pasternak elastic foundation are taken into consideration. The Galerkin method is applied to determine the closed-form expressions of linear frequency and nonlinear to linear frequency ratio while the dynamic response of the plate is obtained by using the fourth-order Runge–Kutta method. The Bees Algorithm is used to determine the optimum value of the natural frequency which depends on five variables including the thickness, the length and the width of penta-graphene plates and two stiffness coefficients of elastic foundations. The numerical results show the effects of width-to-thickness ratio, elastic foundations coefficients, initial imperfection parameter and temperature increment on the nonlinear vibration of the 3D penta-graphene plates.

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\(C_{ij} \,(ij=11,12,22,44,55,66)\) :

Elastic constants of 3D penta-graphene

\(\alpha _{11} ,\,\alpha _{22} \) :

Thermal expansion coefficients of 3D penta-graphene

uvw :

Displacement components parallel to the coordinates \(\left( {x,y,z} \right) \)

\(\phi _{x} ,\phi _{y} \) :

Rotations of the transverse normal about the y and x axes at \(z=0\)

\(\varepsilon _{x}^{0} ,\varepsilon _{y}^{0} ,\gamma _{xy}^{0} ,\gamma _{xz}^{0} ,\gamma _{yz}^{0} \) :

Strain components at the middle surface

\(\varepsilon _{x} ,\varepsilon _{y} ,\gamma _{xy} ,\gamma _{xz} ,\gamma _{yz} \) :

Strain components at the distance z from the mid-plane

\(\sigma _{xx} ,\sigma _{yy} ,\sigma _{xy} ,\sigma _{xz} ,\,\sigma _{yz}\) :

Stress components at the distance z from the mid-plane

\(\Delta T\) :

Temperature increment of the environment

\(k_{1} ,\,k_{2} \) :

Elastic foundations stiffness

a :

Length of the 3D penta-graphene plate

b :

Width of the 3D penta-graphene plate

h :

Thickness of the 3D penta-graphene plate

f :

Airy’s stress function

\(\varepsilon \) :

Viscous damping coefficient

q :

External pressure

mn :

Modes of vibration

\(W,\Phi _{x} ,\Phi _{y}\) :

Amplitude of the deflection and rotation angles

\(w^{*}\) :

Initial imperfection function of the 3D penta-graphene plate

\(\mu \) :

Imperfect function parameter

\(\omega _\mathrm{L}\) :

Linear frequency

\(\alpha \) :

Frequency ratio

\(\xi \) :

Amplitude deflection


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This work is funded through National Foundation for Science and Technology Development of Vietnam—NAFOSTED under Grant Number 107.02-2019.01.

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Appendix A

$$\begin{aligned} Z_{11}= & {} A_{44} -6c_{1} D_{44} +9c_{1}^{2} F_{44} ,\,\,Z_{12} =A_{55} -6c_{1} D_{55} +9c_{1}^{2} F_{55} , \\ Z_{13}= & {} -c_{1}^{2} (E_{11} I_{15} +E_{12} I_{25} +H_{11} ),\,\, \\ Z_{14}= & {} -c_{1}^{2} (4E_{66} I_{33} +4H_{66} +E_{11} I_{16} +E_{12} I_{26} +2H_{12} +E_{12} I_{15} +E_{22} I_{25} ),\,\, \\ \\ Z_{15}= & {} -c_{1}^{2} (E_{12} I_{16} +E_{22} I_{26} +H_{22} ), \\ Z_{16}= & {} c_{1} (E_{11} I_{13} -c_{1} E_{11} I_{15} +F_{11} -c_{1} H_{11} +E_{12} I_{23} -c_{1} E_{12} I_{25} ), \\ Z_{17}= & {} c_{1} (2E_{66} I_{32} -2c_{1} E_{66} I_{33} +2F_{66} -2c_{1} H_{66} +c_{1} E_{12} I_{13} -c_{1} E_{12} I_{15}\\&+F_{12} -c_{1} H_{12} +E_{22} I_{23} -c_{1} E_{22} I_{25} ), \\ Z_{18}= & {} c_{1} \left( {E_{12} I_{14} -c_{1} E_{12} I_{16} +E_{22} I_{24} -c_{1} E_{22} I_{26} +F_{22} -c_{1} H_{22} } \right) , \\ Z_{19}= & {} c_{1} (2E_{66} I_{32} -2c_{1} E_{66} I_{33} +2F_{66} \\&-2c_{1} H_{66} +E_{11} I_{14} -c_{1} E_{11} I_{16} +E_{12} I_{24} -c_{1} E_{12} I_{26} +F_{12} -c_{1} H_{12} ), \\ Z_{110}= & {} -c_{1} (E_{11} I_{12} -E_{12} I_{21} ), \\ Z_{111}= & {} -c_{1} (2E_{66} I_{31} -E_{11} I_{11} +2E_{12} I_{12} -E_{22} I_{21} ),\,\,Z_{112} =c_{1} (E_{12} I_{11} -E_{22} I_{12} ), \\ Z_{21}= & {} -A_{44} +6c_{1} D_{44} -9c_{1}^{2} F_{44} ,\,\,Z_{22} =-c_{1} (B_{11} I_{15} +F_{11} +B_{12} I_{25} -c_{1} E_{11} I_{15} -c_{1} H_{11} -c_{1} E_{12} I_{25} ), \\ Z_{23}= & {} -c_{1} (B{ }_{11}I_{16} +B_{12} I_{26} +F_{12} +2B_{66} I_{33} +2F_{66} -2c_{1} E_{66} I_{33}\\&-2c_{1} H_{66} -c_{1} E_{11} I_{16} -c_{1} E_{12} I_{26} -c_{1} H_{12} ), \\ Z_{24}= & {} B_{11} I_{13} -c_{1} B_{11} I_{15} +D_{11} -c_{1} F_{11} +B_{12} I_{23} -c_{1} B_{12} I_{25} -c_{1} E_{11} I_{13} +c_{1}^{2} E_{11} I_{15} -c_{1} F_{11} +c_{1}^{2} H_{11} \\&-c_{1} E_{12} I_{23} +c_{1}^{2} E_{12} I_{25} ,\,\,Z_{25} =B_{66} I_{32} -c_{1} B_{66} I_{33} \\&+D_{66} -c_{1} F_{66} -c_{1} E_{66} I_{32} +c_{1}^{2} E_{66} I_{33} -c_{1} F_{66} +c_{1}^{2} H_{66} , \\ Z_{26}= & {} B_{11} I_{14} -c_{1} B_{11} I_{16} +B_{12} I_{24} -c_{1} B_{12} I_{26} +D_{12} -c_{1} F_{12} \\&+B_{66} I_{32} -c_{1} B_{66} I_{33} +D_{66} -c_{1} F_{66} -c_{1} E_{66} I_{32} \\&+c_{1}^{2} E_{66} I_{33} -c_{1} F_{66} +c_{1}^{2} H_{66} -c_{1} E_{11} I_{14} +c_{1}^{2} E_{11} I_{16} -c_{1} E_{12} I_{24} +c_{1}^{2} E_{12} I_{26} -c_{1} F_{12} +c_{1}^{2} H_{12} , \\ Z_{27}= & {} -B_{11} I_{12} +B_{12} I_{21} +c_{1} E_{11} I_{12} -c_{1} E_{12} I_{21} , \\ Z_{28}= & {} B_{11} I_{11} -B_{12} I_{12} -B_{66} I_{31} -c_{1} E_{11} I_{11} +c_{1} E_{12} I_{12} \\&+c_{1} E_{66} I_{31} ,\,\,Z_{31} =-A_{55} +6c_{1} D_{55} -9c_{1}^{2} F_{55} , \\ Z_{32}= & {} -c_{1} (2B_{66} I_{33} +2F_{66} +B_{12} I_{15} +F_{12} +B_{22} I_{25} \\&-2c_{1} E_{66} I_{33} -2c_{1} H_{66} -c_{1} E_{12} I_{15} -c_{1} H_{12} -c_{1} E_{22} I_{25} ), \\ Z_{33}= & {} -c_{1} (B_{12} I_{16} +B_{22} I_{26} +F_{22} -c_{1} E_{12} I_{16} -c_{1} E_{22} I_{26} -c_{1} H_{22} ), \\ Z_{34}= & {} B_{66} I_{32} -c_{1} B_{66} I_{33} +D_{66} -c_{1} F_{66} +B_{12} I_{13} -c_{1} B_{12} I_{15} +D_{12} -c_{1} F_{12} +B_{22} I_{23} \\&-c_{1} B_{22} I_{25} -c_{1} E_{66} I_{32} +c_{1}^{2} E_{66} I_{33} -c_{1} F_{66} \\&+c_{1}^{2} H_{66} -c_{1} E_{12} I_{13} +c_{1}^{2} E_{12} I_{15} -c_{1} F_{12} +c_{1}^{2} H_{12} -c_{1} E_{22} I_{23} \\&+c_{1}^{2} E_{22} I_{25} ,\,\,Z_{35} =B_{66} I_{32} -c_{1} B_{66} I_{33} \\&+D_{66} -c_{1} F_{66} -c_{1} E_{66} I_{32} +c_{1}^{2} E_{66} I_{33} -c_{1} F_{66} +c_{1}^{2} H_{66} , \\ Z_{36}= & {} B_{12} I_{14} -c_{1} B_{12} I_{16} +B_{22} I_{24} -c_{1} B_{22} I_{26} +D_{22} -c_{1} F_{22} -c_{1} E_{12} I_{14} +c_{1}^{2} E_{12} I_{16} -c_{1} E_{22} I_{24} \\&+c_{1}^{2} E_{22} I_{26} -c_{1} F_{22} +c_{1}^{2} H_{22} ,\,\,Z_{37} =-B_{66} I_{31} -B_{12} I_{12} \\&+B_{22} I_{21} +c_{1} E_{66} I_{31} +c_{1} E_{12} I_{12} -c_{1} E_{22} I_{21} , \\ Z_{38}= & {} B_{12} I_{11} -B_{22} I_{12} -c_{1} E_{12} I_{11} +c_{1} E_{22} I_{12} . \end{aligned}$$

Appendix B

$$\begin{aligned} l_{11}= & {} -k_{1} -k_{2} \left( {\lambda _{m}^{2} +\delta _{n}^{2} } \right) +Z_{13} \lambda _{m}^{4} +Z_{14} \lambda _{m}^{2} \delta _{n}^{2} +Z_{15} \delta _{n}^{4} +Z_{110} Q_{1} \lambda _{m}^{4} +Z_{111} Q_{1} \lambda _{m}^{2} \delta _{n}^{2} +Z_{112} Q_{1} \delta _{n}^{4} , \\ l_{12}= & {} -Z_{11} \lambda _{m} +Z_{16} \lambda _{m}^{3} +Z_{17} \lambda _{m} \delta _{n}^{2} +Z_{110} Q_{2} \lambda _{m}^{4} +Z_{111} Q_{2} \lambda _{m}^{2} \delta _{n}^{2} +Z_{112} Q_{2} \delta _{n}^{4} , \\ l_{13}= & {} -Z_{12} \delta _{n} +Z_{18} \delta _{n}^{3} +Z_{19} \lambda _{m}^{2} \delta _{n} +Z_{110} Q_{3} \lambda _{m}^{4} +Z_{111} Q_{3} \lambda _{m}^{2} \delta _{n}^{2} +Z_{112} Q_{3} \delta _{n}^{4} , \\ l_{14}= & {} \frac{32Q_{2} \lambda _{m} \delta _{n} }{3ab},\,\,l_{15} =\frac{32Q_{3} \lambda _{m} \delta _{n} }{3ab},\,\,n_{1} =-Z_{11} \lambda _{m}^{2} -Z_{12} \delta _{n}^{2} ,\,\,n_{2} =\frac{32Q_{1} \lambda _{m} \delta _{n} }{3ab}, \\ n_{3}= & {} -\frac{8Z_{110} \lambda _{m} \delta _{n} }{3abI_{21} }-\frac{8Z_{112} \lambda _{m} \delta _{n} }{3abI_{11} },\,\,n_{4} =-\frac{\lambda _{m}^{4} }{16I_{11} }-\frac{\delta _{n}^{4} }{16I_{21} },\,\,n_{5} =\frac{16}{mn\pi ^{2}}, \\ l_{21}= & {} -\lambda _{m}^{3} (Z_{22} +Q_{1} Z_{27} )-\lambda _{m} \delta _{n}^{2} (Z_{23} +Q_{1} Z_{28} ),\,\,l_{22} =Z_{21} -Z_{24} \lambda _{m}^{2} \\&-Z_{25} \delta _{n}^{2} -Z_{27} Q_{2} \lambda _{m}^{3} -Z_{28} Q_{2} \lambda _{m} \delta _{n}^{2} , \\ l_{23}= & {} -Z_{26} \lambda _{m} \delta _{n} -Z_{27} Q_{3} \lambda _{m}^{3} -Z_{28} Q_{3} \lambda _{m} \delta _{n}^{2} ,\,\,n_{6} =Z_{21} \lambda _{m} ,\,\,n_{7} =\frac{8Z_{27} \delta _{n} }{3abI_{21} },\,\,l_{31} =-\delta _{n}^{3} (Z_{33} +Q_{1} Z_{38} ) \\&-\lambda _{m}^{2} \delta _{n} (Z_{32} +Q_{1} Z_{37} ),\,\,l_{32} =-Z_{34} \lambda _{m} \delta _{n} -Z_{38} Q_{2} \delta _{n}^{3} -Z_{37} Q_{2} \lambda _{m}^{2} \delta _{n} ,\,\,l_{33} =Z_{31} -Z_{35} \lambda _{m}^{2} -Z_{36} \delta _{n}^{2} \\&-Z_{38} Q_{3} \delta _{n}^{3} -Z_{37} Q_{3} \lambda _{m}^{2} \delta _{n} ,\,\,n_{8} =Z_{31} \delta _{n} ,\,\,n_{9} =\frac{8Z_{38} \lambda _{m} }{3abI_{11} }. \end{aligned}$$

Appendix C

$$\begin{aligned} a_{1}= & {} -\frac{\left( {l_{21} l_{33} -l_{23} l_{31} } \right) }{\left( {l_{22} l_{33} -l_{23} l_{32} } \right) },\,\,a_{2} =-\frac{\left( {n_{6} l_{33} -n_{8} l_{23} } \right) }{\left( {l_{22} l_{33} -l_{23} l_{32} } \right) }, \\ a_{3}= & {} -\frac{\left( {n_{7} l_{33} -n_{9} l_{23} } \right) }{\left( {l_{22} l_{33} -l_{23} l_{32} } \right) },\,\,a_{4} =\frac{\left( {-\lambda _{m} \overline{\overline{j_{5} }} l_{33} +\delta _{n} \overline{\overline{j_{5}^{*} }} l_{23} } \right) }{\left( {l_{22} l_{33} -l_{23} l_{32} } \right) }, \\ a_{5}= & {} \frac{\left( {l_{21} l_{32} -l_{22} l_{31} } \right) }{\left( {l_{22} l_{33} -l_{23} l_{32} } \right) },\,\,a_{6} =\frac{\left( {n_{6} l_{32} -n_{8} l_{22} } \right) }{\left( {l_{22} l_{33} -l_{23} l_{32} } \right) }, \\ a_{7}= & {} \frac{\left( {n_{7} l_{32} -n_{9} l_{22} } \right) }{\left( {l_{22} l_{33} -l_{23} l_{32} } \right) },\,\,a_{8} =-\frac{\left( {-\lambda _{m} \overline{\overline{j_{5} }} l_{32} +\delta _{n} \overline{\overline{j_{5}^{*} }} l_{22} } \right) }{\left( {l_{22} l_{33} -l_{23} l_{32} } \right) }, \\ b_{11}= & {} \left( {l_{11} +l_{12} a_{1} +l_{13} a_{5} } \right) ,\,\,b_{12} =\left( {n_{1}^{1} +l_{12} a_{2} +l_{13} a_{6} } \right) , \\ b_{13}= & {} \left( {n_{2}^{1} +l_{14}^{1} a_{1} +l_{15}^{1} a_{5} } \right) ,\,\,b_{14} =\left( {n_{3} +l_{12} a_{3} +l_{13} a_{7} } \right) , \\ b_{15}= & {} \left( {l_{14}^{1} a_{2} +l_{15}^{1} a_{6} } \right) ,\,\,b_{16} =\left( {n_{4}^{1} +l_{14}^{1} a_{3} +l_{15}^{1} a_{7} } \right) , \\ j_{0}= & {} j_{1} -\bar{{\bar{{j}}}}_{7} \lambda _{m}^{2} -\bar{{\bar{{j}}}}_{7}^{*} \delta _{n}^{2} ,\,\overline{j_{0} } =j_{0} -\left( {l_{12} a_{4} +l_{13} a_{8} } \right) , \\ j_{0}^{*}= & {} \left( {l_{14}^{1} a_{4} +l_{15}^{1} a_{8} } \right) . \end{aligned}$$

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Dat, N.D., Quan, T.Q., Tran, P. et al. A first-principle study of nonlinear large amplitude vibration and global optimization of 3D penta-graphene plates based on the Bees Algorithm. Acta Mech (2020). https://doi.org/10.1007/s00707-020-02706-7

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