Analytical study of the damping vibration behavior of the metal foam nanocomposite plates reinforced with graphene oxide powders in thermal environments

Abstract

This article performs an analytical study on the damping vibration behavior of metal foam nanocomposite plate reinforced with graphene oxide powders (GOPs) in thermal environment. The GOPs are dispersed through the thickness of the structure according to three functionally graded (FG) and one uniform distribution patterns. The Halpin–Tsai micromechanical model is chosen for estimating the effective material properties of the structure having GOPs as reinforcement phase. Also, different porosity types are taken into account for the metal foam matrix. The plate is resting on a three-parameter viscoelastic medium containing Winkler and Pasternak layers in combination with viscous dampers which can dissipate the oscillation of the structure in some special cases. The Governing differential equations are derived via Hamilton’s principle on the basis of refined higher order shear deformation theory and then solved with employing Galerkin solution method to obtain the natural frequencies of the proposed structure. Moreover, various boundary conditions (B.Cs) including simply supported, fully clamped and different combinations of these B.Cs are considered in this study. The influences and confrontation of different significant parameters such as GOPs’ weight fraction, foundation parameters, aspect and side-to-thickness ratios, porosity coefficients, thermal environment, and FG patterns are investigated through various graphical and numerical results. Our findings declare that the dynamic behavior of the graphene oxide powder reinforced metal foam (GOPRMF) plate remarkably depends on these parameters.

Appendix

The components of the stiffness matrix are as follows:

$$\begin{aligned} k_{11} & = A_{11} r_{1} + A_{66} r_{2} , \, k_{12} = (A_{12} + A_{66} )r_{2} , \, \hfill \\ k_{13} &= - B_{11} r_{1} - (B_{11} + 2B_{66} )r_{2}, \,\\ k_{14} &= - B_{11}^{s} r_{1} - (B_{11}^{s} + 2B_{66}^{s} )r_{2} , \, \\ k_{21} &= (A_{12} + A_{66} )r_{3} , \, k_{22} = A_{22} r_{4} + A_{66} r_{3} \hfill \\ k_{23} &= - B_{22} r_{4} - (B_{12} + 2B_{66} )r_{3} , \,\\ k_{24} & = - B_{22}^{s} r_{4} - (B_{12}^{s} + 2B_{66}^{s} )r_{1} \hfill \\ k_{31} &= B_{11} r_{5} + (B_{12} + 2B_{66} )r_{6} , \,\\ k_{32} & = B_{22} r_{7} + (B_{12} + 2B_{66} )r_{6} \hfill \\ k_{33} &= - D_{11} r_{5} - 2(D_{12} + 2D_{66} )r_{6} \hfill \\&\quad- D_{22} r_{7} - k_{w} r_{8} + (k_{p} - N_{T} )(r_{10} + r_{9} ) \hfill \\ k_{34} & = - D_{11}^{s} r_{5} - 2(D_{12}^{s} + 2D_{66}^{s} )r_{6} \hfill \\&\quad - D_{22}^{s} r_{7} - k_{w} r_{8} + (k_{p} - N_{T} )(r_{10} + r_{9} ) \hfill \\ k_{41} & = B_{11}^{s} r_{5} + (B_{11}^{s} + 2B_{66}^{s} )r_{6} , \, \\ k_{42} &= B_{22}^{s} r_{7} + (B_{12}^{s} + 2B_{66}^{s} )r_{6} \hfill \\ k_{43} & = - D_{11}^{s} r_{5} - 2(D_{12}^{s} + 2D_{66}^{s} )r_{6} - D_{22}^{s} r_{7} \hfill \\&\quad - k_{w} r_{8} + (k_{p} - N_{T} )(r_{10} + r_{9} ) \hfill \\ k_{44} & = - H_{11}^{s} r_{5} - 2(H_{12}^{s} + 2H_{66}^{s} )r_{6} - H_{22}^{s} r_{7} \hfill \\&\quad - k_{w} r_{8} + (A^{s} + k_{p} - N_{T} )(r_{10} + r_{9} ). \hfill \\ \end{aligned}$$

$\begin{gathered} m_{11} = I_{0} r_{11} , \, m_{12} = 0, \, m_{13} = - I_{1} r_{11} , \, m_{14} = - J_{1} r_{11} \hfill \\ m_{21} = 0, \, m_{22} = I_{0} r_{12} , \, m_{23} = - I_{1} r_{12} , \, m_{24} = - J_{1} r_{12} \hfill \\ m_{31} = - I_{1} r_{10} , \, m_{32} = I_{1} r_{9} , \, m_{33} = I_{0} r_{8} - I_{2} (r_{10} + r_{9} ) \hfill \\ m_{34} = I_{0} r_{8} + J_{2} (r_{10} + r_{9} ), \, m_{41} = - J_{1} r_{10} , \, m_{42} = J_{1} r_{9} \hfill \\ m_{43} = I_{0} r_{8} - J_{2} (r_{10} + r_{9} ), \, m_{44} = I_{0} r_{8} - K_{2} (r_{10} + r_{9} ) \hfill \\ \end{gathered}$The components of the mass matrix are defined as

$$\begin{gathered} m_{11} = I_{0} r_{11} , \, m_{12} = 0, \, m_{13} = - I_{1} r_{11} , \, m_{14} = - J_{1} r_{11} \hfill \\ m_{21} = 0, \, m_{22} = I_{0} r_{12} , \, m_{23} = - I_{1} r_{12} , \, m_{24} = - J_{1} r_{12} \hfill \\ m_{31} = - I_{1} r_{10} , \, m_{32} = I_{1} r_{9} , \, m_{33} = I_{0} r_{8} - I_{2} (r_{10} + r_{9} ) \hfill \\ m_{34} = I_{0} r_{8} + J_{2} (r_{10} + r_{9} ), \, m_{41} = - J_{1} r_{10} , \, m_{42} = J_{1} r_{9} \hfill \\ m_{43} = I_{0} r_{8} - J_{2} (r_{10} + r_{9} ), \, m_{44} = I_{0} r_{8} - K_{2} (r_{10} + r_{9} ). \hfill \\ \end{gathered}$$

Also the nonzero components of the damping matrix are stated as

$$c_{33} = c_{34} = c_{43} = c_{44} = - i\mu r_{8}$$

where

$$\begin{aligned} \left\{ {r_{1} ,r_{2} ,r_{{11}} } \right\} &= \int_{0}^{a} {\int_{0}^{b} {X_{m}^{\prime } } } (x)Y_{n} (y)\left\{ {X_{m}^{{\prime \prime \prime }} (x)Y_{n} (y),}\right.\\&\quad \left.{X_{m}^{\prime } (x)Y_{n}^{{\prime \prime }} (y),X_{m}^{\prime } (x)Y_{n} (y)} \right\}\hfill \\\left\{ {r_{3} ,r_{4} ,r_{{12}} } \right\} &= \int_{0}^{a} {\int_{0}^{b} {X_{m} } } (x)Y_{n}^{\prime } (y)\left\{ {X_{m}^{{\prime \prime }} (x)Y_{n}^{\prime } (y),}\right.\\&\quad \left.{X_{m} (x)Y_{n}^{{\prime \prime \prime }} (y),X_{m} (x)Y_{n}^{\prime } (y)} \right\} \hfill \\ \left\{ {r_{5} ,r_{6} ,r_{7} } \right\} & = \int_{0}^{a} {\int_{0}^{b} {X_{m} } } (x)Y_{n} (y)\left\{ {X_{m}^{{\prime \prime \prime \prime }} (x)Y_{n} (y),}\right.\\&\quad \left.{X_{m}^{{\prime \prime }} (x)Y_{n}^{{\prime \prime }} (y),X_{m} (x)Y_{n}^{{\prime \prime \prime \prime }} {\mkern 1mu} (y)} \right\} \hfill \\ \left\{ {r_{8} ,r_{9} ,r_{{10}} } \right\} &= \int_{0}^{a} {\int_{0}^{b} {X_{m} } } (x)Y_{n} (y)\left\{ {X_{m} (x)Y_{n} (y),}\right.\\&\quad \left.{X_{m} (x)Y_{n}^{{\prime \prime }} (y),X_{m}^{{\prime \prime }} (x)Y_{n} (y)} \right\}. \hfill \\ \end{aligned}$$