Free Vibration Analysis of Graphene Platelets–Reinforced Composites Plates in Thermal Environment Based on Higher-Order Shear Deformation Plate Theory

Abstract

As a first endeavor, this article presents the free vibration of composite plates reinforced with graphene platelets (GPLs) based on the higher-order shear deformation plate theory. Moreover, it is assumed that the material properties are temperature dependent and are graded in the thickness direction. It is assumed that GPLs randomly spread out in each individual composite layer reinforced with graphene platelets. The theoretical formulation is derived based on higher-order shear deformation plate theory and the initial thermal stresses are evaluated by solving the thermo-elastic equilibrium equations. The Halpin–Tsai micromechanical model is used to evaluate the effective material properties of every layer of composite plates reinforced GPLs. Further, the Navier solution has been used to derive the governing equations of motion and evaluate the natural frequencies and dynamic response of simply supported graphene platelet reinforced composite plates. Four different GPL distribution pattern is modeled to find out its effect on the frequency of the plate and the other parameters. The result asserted that subjoining GPL to composite plates has a significant reinforcing effect on the free vibration of Graphene platelet reinforced composite (GPLRC) plates.

Appendix 1

$$ \begin{aligned} T & = \int\limits_{V} {\frac{1}{2}\rho \left[ {\left ( {\frac{\partial u}{\partial t}} \right)^{2} + \left( {\frac{\partial v}{\partial t}} \right)^{2} + \left( {\frac{\partial w}{\partial t}} \right)^{2} } \right]} {\text{d}}V: \hfill \\ \delta T & = \int\limits_{0}^{a} {\int\limits_{0}^{b} {\left[ \begin{aligned} & \left( { - I_{0} \frac{{\partial^{2} u_{0} }}{{\partial t^{2} }} - I_{1} \frac{{\partial^{2} \phi_{x} }}{{\partial t^{2} }} + I_{3} c_{1} \left( {\frac{{\partial^{2} \phi_{x} }}{{\partial t^{2} }} + \frac{{\partial^{3} w_{0} }}{{\partial t^{2} \partial x}}} \right)} \right)\delta u_{0} \hfill \\ & + \left( { - I_{1} \frac{{\partial^{2} u_{0} }}{{\partial t^{2} }} - I_{2} \frac{{\partial^{2} \phi_{x} }}{{\partial t^{2} }} + I_{4} c_{1} \left( {\frac{{\partial^{2} \phi_{x} }}{{\partial t^{2} }} + \frac{{\partial^{3} w}}{{\partial t^{2} \partial x}}} \right)} \right)\delta \phi_{x} \hfill \\ & + \left( {c_{1} I_{3} \frac{{\partial^{2} u_{0} }}{{\partial t^{2} }} + c_{1} I_{4} \frac{{\partial^{2} \phi_{x} }}{{\partial t^{2} }} - I_{6} c_{1}^{2} \left( {\frac{{\partial^{2} \phi_{x} }}{{\partial t^{2} }} + \frac{{\partial^{3} w}}{{\partial t^{2} \partial x}}} \right)} \right)\delta \phi_{x} \hfill \\ & + \left( { - c_{1} I_{3} \frac{{\partial^{3} u_{0} }}{{\partial x\partial t^{2} }} - c_{1} I_{4} \frac{{\partial^{3} \phi_{x} }}{{\partial x\partial t^{2} }} + I_{6} c_{1}^{2} \left( {\frac{{\partial^{3} \phi_{x} }}{{\partial x\partial t^{2} }} + \frac{{\partial^{4} w}}{{\partial t^{2} \partial x^{2} }}} \right)} \right)\delta w_{0} \hfill \\ & \left( { - I_{0} \frac{{\partial^{2} v_{0} }}{{\partial t^{2} }} - I_{1} \frac{{\partial^{2} \phi_{y} }}{{\partial t^{2} }} + I_{3} c_{1} \left( {\frac{{\partial^{2} \phi_{y} }}{{\partial t^{2} }} + \frac{{\partial^{3} w_{0} }}{{\partial t^{2} \partial y}}} \right)} \right)\delta v_{0} \hfill \\ & + \left( { - I_{1} \frac{{\partial^{2} v_{0} }}{{\partial t^{2} }} - I_{2} \frac{{\partial^{2} \phi_{y} }}{{\partial t^{2} }} + I_{4} c_{1} \left( {\frac{{\partial^{2} \phi_{y} }}{{\partial t^{2} }} + \frac{{\partial^{3} w}}{{\partial t^{2} \partial y}}} \right)} \right)\delta \phi_{y} \hfill \\ & + \left( {c_{1} I_{3} \frac{{\partial^{2} v_{0} }}{{\partial t^{2} }} + c_{1} I_{4} \frac{{\partial^{2} \phi_{y} }}{{\partial t^{2} }} - I_{6} c_{1}^{2} \left( {\frac{{\partial^{2} \phi_{y} }}{{\partial t^{2} }} + \frac{{\partial^{3} w}}{{\partial t^{2} \partial y}}} \right)} \right)\delta \phi_{y} \hfill \\ & + \left( { - c_{1} I_{3} \frac{{\partial^{3} v_{0} }}{{\partial y\partial t^{2} }} - c_{1} I_{4} \frac{{\partial^{3} \phi_{y} }}{{\partial y\partial t^{2} }} + I_{6} c_{1}^{2} \left( {\frac{{\partial^{3} \phi_{y} }}{{\partial y\partial t^{2} }} + \frac{{\partial^{4} w}}{{\partial t^{2} \partial y^{2} }}} \right)} \right)\delta w_{0} \hfill \\ & + \left( { - I_{0} \frac{{\partial^{2} w_{0} }}{{\partial t^{2} }}} \right)\delta w_{0} \hfill \\ \end{aligned} \right]} } \,{\text{d}}x\,{\text{d}}y, \hfill \\ \end{aligned} $$

$$ \left\{ {I_{{_{0} }} ,I_{1} ,I_{2} ,I_{3} ,I_{4} ,I_{5} ,I_{6} } \right\} = \sum\limits_{k = 1}^{{N_{L} }} {\int\limits_{ - h/2}^{h/2} {\rho^{k} } \left\{ {1,z,z^{2} ,z^{3} ,z^{4} ,z^{5} ,z^{6} } \right\}{\text{d}}z}, $$

$$ \begin{aligned} \delta U = & \frac{1}{2}\iiint\limits_{V} {\sigma _{{ij}} \delta \varepsilon _{{ij}} {\text{d}}V} \\ = & \iint\limits_{A} {\left[ {\begin{array}{*{20}l} {N_{{xx}} \frac{{\partial \delta u_{0} }}{{\partial x}} + M_{{xx}} \frac{{\partial \delta \phi _{x} }}{{\partial x}} - P_{{xx}} c_{1} \left( {\frac{{\partial \delta \phi _{x} }}{{\partial x}} + \frac{{\partial ^{2} \delta w_{0} }}{{\partial x^{2} }}} \right)} \hfill \\ {\;\; + N_{{yy}} \frac{{\partial \delta v_{0} }}{{\partial y}} + M_{{yy}} \frac{{\partial \delta \phi _{x} }}{{\partial y}} - P_{{yy}} c_{1} \left( {\frac{{\partial \delta \phi _{y} }}{{\partial y}} + \frac{{\partial ^{2} \delta w_{0} }}{{\partial y^{2} }}} \right)} \hfill \\ {\;\; + N_{{xy}} \frac{{\partial \delta u_{0} }}{{\partial y}} + N_{{xy}} \frac{{\partial \delta v_{0} }}{{\partial x}} + M_{{xy}} \left( {\frac{{\partial \delta \phi _{x} }}{{\partial y}} + \frac{{\partial \delta \phi _{y} }}{{\partial x}}} \right) - P_{{xy}} c_{1} \left( {\frac{{\partial \delta \phi _{x} }}{{\partial y}} + \frac{{\partial \delta \phi _{y} }}{{\partial x}} + 2\frac{{\partial ^{2} \delta w_{0} }}{{\partial x\partial y}}} \right)} \hfill \\ {\;\; + (Q_{{xz}} - 3S_{{xz}} c_{1} )\left( {\delta \phi _{x} + \frac{{\partial \delta w_{0} }}{{\partial x}}} \right) + (Q_{{yz}} - 3S_{{yz}} c_{1} )\left( {\delta \phi _{y} + \frac{{\partial \delta w_{0} }}{{\partial y}}} \right)} \hfill \\ \end{array} } \right]}{\text{d}}A \\ = & \iint\limits_{A} {\left[ {\begin{array}{*{20}l} { - \frac{\partial }{{\partial x}}N_{{xx}} \delta u_{0} - \frac{\partial }{{\partial x}}M_{{xx}} \delta \phi _{x} - c_{1} \left( { - \frac{\partial }{{\partial x}}P_{{xx}} \delta \phi _{x} + \frac{{\partial ^{2} }}{{\partial x^{2} }}P_{{xx}} \delta w_{0} } \right)} \hfill \\ { - \frac{\partial }{{\partial y}}N_{{yy}} \delta v_{0} - \frac{\partial }{{\partial y}}M_{{yy}} \delta \phi _{y} - c_{1} \left( { - \frac{\partial }{{\partial y}}P_{{yy}} \delta \phi _{y} + \frac{{\partial ^{2} }}{{\partial y^{2} }}P_{{yy}} \delta w_{0} } \right)} \hfill \\ { - \frac{\partial }{{\partial y}}N_{{xy}} \delta u_{0} - \frac{\partial }{{\partial x}}N_{{xy}} \delta v_{0} - \frac{\partial }{{\partial y}}M_{{xy}} \delta \phi _{x} - \frac{\partial }{{\partial x}}M_{{xy}} \delta \phi _{y} } \hfill \\ { - c_{1} \left( { - \frac{\partial }{{\partial y}}P_{{xy}} \delta \phi _{x} - \frac{\partial }{{\partial x}}P_{{xy}} \delta \phi _{y} + 2\frac{{\partial ^{2} }}{{\partial x\partial y}}P_{{xy}} \delta w_{0} } \right)} \hfill \\ {\;\; + Q_{{xz}} \delta \phi _{x} - \frac{\partial }{{\partial x}}Q_{{xz}} \delta w_{0} - 3c_{1} \left( {S_{{xz}} \delta \phi _{x} - \frac{\partial }{{\partial x}}S_{{xz}} \delta w_{0} } \right)} \hfill \\ {\;\; + Q_{{yz}} \delta \phi _{y} - \frac{\partial }{{\partial y}}Q_{{yz}} \delta w_{0} - 3c_{1} \left( {S_{{yz}} \delta \phi _{y} - \frac{\partial }{{\partial y}}S_{{yz}} \delta w_{0} } \right)} \hfill \\ \end{array} } \right]}{\text{d}}A \\ \end{aligned}. $$

$$ \begin{aligned} \left\{ {N_{xx} ,N_{yy} ,N_{xy} } \right\} & = \sum\limits_{k = 1}^{{N_{\text{L}} }} {\int_{z} {\left\{ {\sigma_{xx}^{k} ,\sigma_{yy}^{k} ,\sigma_{xy}^{k} } \right\}} } {\text{d}}z, \hfill \\ \left\{ {M_{xx} ,M_{yy} ,M_{xy} } \right\} & = \sum\limits_{k = 1}^{{N_{\text{L}} }} {\int_{z} {\left\{ {\sigma_{xx}^{k} ,\sigma_{yy}^{k} ,\sigma_{xy}^{k} } \right\}} } z{\text{d}}z, \hfill \\ \left\{ {P_{xx} ,P_{yy} ,P_{xy} } \right\} & = \sum\limits_{k = 1}^{{N_{\text{L}} }} {\int_{z} {\left\{ {\sigma_{xx}^{k} ,\sigma_{yy}^{k} ,\sigma_{xy}^{k} } \right\}} z^{3} } {\text{d}}z, \hfill \\ \left\{ {Q_{xz} ,Q_{yz} } \right\} & = \sum\limits_{k = 1}^{{N_{\text{L}} }} {\int_{z} {\left\{ {\sigma_{xz}^{k} ,\sigma_{xy}^{k} } \right\}} } {\text{d}}z, \hfill \\ \left\{ {S_{xz} ,S_{yz} } \right\} & = \sum\limits_{k = 1}^{{N_{\text{L}} }} {\int_{z} {\left\{ {\sigma_{xz}^{k} ,\sigma_{xy}^{k} } \right\}z^{2} } } {\text{d}}z, \hfill \\ \end{aligned} $$

$$\begin{aligned} N_{xx} & = A_{11} \frac{{\partial u_{0} }}{\partial x} + B_{11} \frac{{\partial \phi_{x} }}{\partial x} - D_{11} c_{1} \left( {\frac{{\partial \phi_{x} }}{\partial x} + \frac{{\partial^{2} w_{0} }}{{\partial x^{2} }}} \right)\\& \quad + A_{12} \frac{{\partial v_{0} }}{\partial y} + B_{12} \frac{{\partial \phi_{y} }}{\partial y} - D_{12} c_{1} \left( {\frac{{\partial \phi_{y} }}{\partial y} + \frac{{\partial^{2} w_{0} }}{{\partial y^{2} }}} \right), \\ N_{yy} & = A_{22} \frac{{\partial v_{0} }}{\partial y} + B_{22} \frac{{\partial \phi_{y} }}{\partial y} - D_{22} c_{1} \left( {\frac{{\partial \phi_{y} }}{\partial y} + \frac{{\partial^{2} w_{0} }}{{\partial y^{2} }}} \right)\\& \quad + A_{12} \frac{{\partial u_{0} }}{\partial x} + B_{12} \frac{{\partial \phi_{x} }}{\partial x} - D_{12} c_{1} \left( {\frac{{\partial \phi_{x} }}{\partial x} + \frac{{\partial^{2} w_{0} }}{{\partial x^{2} }}} \right), \\ N_{xy} & = A_{44} \frac{{\partial u_{0} }}{\partial y} + A_{44} \frac{{\partial v_{0} }}{\partial x} + B_{44} \left( {\frac{{\partial \phi_{x} }}{\partial y} + \frac{{\partial \phi_{y} }}{\partial x}} \right)\\& \quad - D_{44} c_{1} \left( {\frac{{\partial \phi_{x} }}{\partial y} + \frac{{\partial \phi_{y} }}{\partial x} + 2\frac{{\partial^{2} w_{0} }}{\partial x\partial y}} \right), \\ M_{xx} & = B_{11} \frac{{\partial u_{0} }}{\partial x} + C_{11} \frac{{\partial \phi_{x} }}{\partial x} - E_{11} c_{1} \left( {\frac{{\partial \phi_{x} }}{\partial x} + \frac{{\partial^{2} w_{0} }}{{\partial x^{2} }}} \right)\\& \quad + B_{12} \frac{{\partial v_{0} }}{\partial y} + C_{12} \frac{{\partial \phi_{y} }}{\partial y} - E_{12} c_{1} \left( {\frac{{\partial \phi_{y} }}{\partial y} + \frac{{\partial^{2} w_{0} }}{{\partial y^{2} }}} \right), \\ M_{yy} & = B_{22} \frac{{\partial v_{0} }}{\partial y} + C_{22} \frac{{\partial \phi_{y} }}{\partial y} - E_{22} c_{1} \left( {\frac{{\partial \phi_{y} }}{\partial y} + \frac{{\partial^{2} w_{0} }}{{\partial y^{2} }}} \right) \\& \quad + B_{12} \frac{{\partial u_{0} }}{\partial x} + C_{12} \frac{{\partial \phi_{x} }}{\partial x} - E_{12} c_{1} \left( {\frac{{\partial \phi_{x} }}{\partial x} + \frac{{\partial^{2} w_{0} }}{{\partial x^{2} }}} \right), \\ M_{xy} & = B_{44} \frac{{\partial u_{0} }}{\partial y} + B_{44} \frac{{\partial v_{0} }}{\partial x} + C_{44} \left( {\frac{{\partial \phi_{x} }}{\partial y} + \frac{{\partial \phi_{y} }}{\partial x}} \right)\\& \quad - E_{44} c_{1} \left( {\frac{{\partial \phi_{x} }}{\partial y} + \frac{{\partial \phi_{y} }}{\partial x} + 2\frac{{\partial^{2} w_{0} }}{\partial x\partial y}} \right), \\ p_{xx} & = D_{11} \frac{{\partial u_{0} }}{\partial x} + E_{11} \frac{{\partial \phi_{x} }}{\partial x} - G_{11} c_{1} \left( {\frac{{\partial \phi_{x} }}{\partial x} + \frac{{\partial^{2} w_{0} }}{{\partial x^{2} }}} \right)\\& \quad + D_{12} \frac{{\partial v_{0} }}{\partial y} + E_{12} \frac{{\partial \phi_{y} }}{\partial y} - G_{12} c_{1} \left( {\frac{{\partial \phi_{y} }}{\partial y} + \frac{{\partial^{2} w_{0} }}{{\partial y^{2} }}} \right), \\ p_{yy} & = D_{22} \frac{{\partial v_{0} }}{\partial y} + E_{22} \frac{{\partial \phi_{y} }}{\partial y} - G_{22} c_{1} \left( {\frac{{\partial \phi_{y} }}{\partial y} + \frac{{\partial^{2} w_{0} }}{{\partial y^{2} }}} \right) \\& \quad + D_{12} \frac{{\partial u_{0} }}{\partial x} + E_{12} \frac{{\partial \phi_{x} }}{\partial x} - G_{12} c_{1} \left( {\frac{{\partial \phi_{x} }}{\partial x} + \frac{{\partial^{2} w_{0} }}{{\partial x^{2} }}} \right), \\ p_{xy} & = D_{44} \frac{{\partial u_{0} }}{\partial y} + D_{44} \frac{{\partial v_{0} }}{\partial x} + E_{44} \left( {\frac{{\partial \phi_{x} }}{\partial y} + \frac{{\partial \phi_{y} }}{\partial x}} \right)\\& \quad - G_{44} c_{1} \left( {\frac{{\partial \phi_{x} }}{\partial y} + \frac{{\partial \phi_{y} }}{\partial x} + 2\frac{{\partial^{2} w_{0} }}{\partial x\partial y}} \right), \\ Q_{xz} & = (A_{55} - 3C_{55} c_{1} )\left( {\phi_{x} + \frac{{\partial w_{0} }}{\partial x}} \right), \\ S_{xz} & = (C_{55} - 3E_{55} c_{1} )\left( {\phi_{x} + \frac{{\partial w_{0} }}{\partial x}} \right), \\ Q_{yz} & = (A_{66} - 3C_{66} c_{1} )\left( {\phi_{y} + \frac{{\partial w_{0} }}{\partial y}} \right), \\ S_{yz} & = (C_{66} - 3E_{66} c_{1} )\left( {\phi_{y} + \frac{{\partial w_{0} }}{\partial y}} \right). \\ \end{aligned} $$

$$ \begin{aligned} &\left\{ {A_{11} ,B_{11} ,C_{11} ,D_{11} ,E_{11} ,F_{11} ,G_{11} } \right\} \\& \quad = \sum\limits_{k = 1}^{{N_{\text{L}} }} {\int\limits_{ - h/2}^{h/2} {Q_{11}^{k} \left\{ {1,z,z^{2} ,z^{3} ,z^{4} ,z^{5} ,z^{6} } \right\}} } {\text{d}}z, \hfill \\& \left\{ {A_{12} ,B_{12} ,C_{12} ,D_{12} ,E_{12} ,F_{12} ,G_{12} } \right\} \\& \quad = \sum\limits_{k = 1}^{{N_{\text{L}} }} {\int\limits_{ - h/2}^{h/2} {Q_{12}^{k} \left\{ {1,z,z^{2} ,z^{3} ,z^{4} ,z^{5} ,z^{6} } \right\}} } {\text{d}}z, \hfill \\& \left\{ {A_{22} ,B_{22} ,C_{22} ,D_{22} ,E_{22} ,F_{22} ,G_{22} } \right\} \\& \quad = \sum\limits_{k = 1}^{{N_{\text{L}} }} {\int\limits_{ - h/2}^{h/2} {Q_{22}^{k} \left\{ {1,z,z^{2} ,z^{3} ,z^{4} ,z^{5} ,z^{6} } \right\}} } {\text{d}}z, \hfill \\& \left\{ {A_{55} ,B_{55} ,C_{55} ,D_{55} ,E_{55} } \right\}\\& \quad = \sum\limits_{k = 1}^{{N_{\text{L}} }} {\int\limits_{ - h/2}^{h/2} {Q_{55}^{k} \left\{ {1,z,z^{2} ,z^{3} ,z^{4} } \right\}} } {\text{d}}z, \hfill \\ &\left\{ {A_{66} ,B_{66} ,C_{66} ,D_{66} ,E_{66} } \right\}\\& \quad = \sum\limits_{k = 1}^{{N_{\text{L}} }} {\int\limits_{ - h/2}^{h/2} {Q_{66}^{k} \left\{ {1,z,z^{2} ,z^{3} ,z^{4} } \right\}} } {\text{d}}z. \hfill \\\end{aligned}$$

$$ \delta W_{1} = \iint\limits_{A} {\left[ {N_{1}^{T} \frac{\partial w}{\partial x}\frac{\partial \delta w}{\partial x} + N_{2}^{T} \frac{\partial w}{\partial y}\frac{\partial \delta w}{\partial y}} \right]}\,{\text{d}}x{\text{d}}y, $$

$$ N_{1}^{\text{T}} = N_{2}^{\text{T}} = \int_{ - h/2}^{h/2} {E\,\alpha (T - T_{0} ){\text{d}}z}. $$

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