Thermo-mechanical wave dispersion analysis of nonlocal strain gradient single-layered graphene sheet rested on elastic medium

Abstract

Present research is mainly devoted to show the effects of temperature change on the mechanical properties of in-plane waves propagating in a single-layered graphene sheet. Graphene sheet is assumed to be rested on an elastic medium. Influences of elastic substrate on the behavior of graphene sheet is modeled by the means of a two parameter elastic foundation. In addition, a trigonometric refined plate theory is applied to derive the kinematic relations. Also, a nonlocal strain gradient theory is utilized to show the size-dependency of graphene sheet. In the framework of Hamilton’s principle, the nonlocal governing differential equations are derived. Moreover, an analytical approach is applied to find the unknowns of the final eigen value equation. At the end of the paper, it is tried to clarify the influences of each parameter by the means of some diagrams.

Appendix

In Eq. (31) $k_{ij}$ and $m_{ij}$, $(i,j = 1,2)$ are defined as follows:

$$ \left\{ \begin{aligned} k_{11} = & - \,(1 + \eta^{2} (\beta_{1}^{2} + \beta_{2}^{2} ))(D_{11} \beta_{1}^{4} + 2(D_{12} + 2D_{66} )\beta_{1}^{2} \beta_{2}^{2} + D_{22} \beta_{2}^{4} ) \\ & + \,(1 + \mu^{2} (\beta_{1}^{2} + \beta_{2}^{2} ))((N^{T} - k_{p} )(\beta_{1}^{2} + \beta_{2}^{2} ) - k_{w} ) \\ k_{12} = & - \,(1 + \eta^{2} (\beta_{1}^{2} + \beta_{2}^{2} ))(D_{11}^{s} \beta_{1}^{4} + 2(D_{12}^{s} + 2D_{66}^{s} )\beta_{1}^{2} \beta_{2}^{2} + D_{22}^{s} \beta_{2}^{4} ) \\ & + \,(1 + \mu^{2} (\beta_{1}^{2} + \beta_{2}^{2} ))((N^{T} - k_{p} )(\beta_{1}^{2} + \beta_{2}^{2} ) - k_{w} ) \\ k_{21} = & - \,(1 + \eta^{2} (\beta_{1}^{2} + \beta_{2}^{2} ))(D_{11}^{s} \beta_{1}^{4} + 2(D_{12}^{s} + 2D_{66}^{s} )\beta_{1}^{2} \beta_{2}^{2} + D_{22}^{s} \beta_{2}^{4} ) \\ & + \,(1 + \mu^{2} (\beta_{1}^{2} + \beta_{2}^{2} ))((N^{T} - k_{p} )(\beta_{1}^{2} + \beta_{2}^{2} ) - k_{w} ) \\ k_{22} = & - \,(1 + \eta^{2} (\beta_{1}^{2} + \beta_{2}^{2} ))\left( \begin{aligned} H_{11}^{s} \beta_{1}^{4} + 2(H_{12}^{s} + 2H_{66}^{s} )\beta_{1}^{2} \beta_{2}^{2} + H_{22}^{s} \beta_{2}^{4} \\ + A_{55}^{s} \beta_{1}^{2} + A_{44}^{s} \beta_{2}^{2} \\ \end{aligned} \right) \\ & + \,(1 + \mu^{2} (\beta_{1}^{2} + \beta_{2}^{2} ))((N^{T} - k_{p} )(\beta_{1}^{2} + \beta_{2}^{2} ) - k_{w} ) \\ \end{aligned} \right. $$

(36)

$$ \left\{ \begin{aligned} & m_{11} = - (1 + \mu^{2} (\beta_{1}^{2} + \beta_{2}^{2} ))(I_{0} + I_{2} (\beta_{1}^{2} + \beta_{2}^{2} )) \\ & m_{12} = - (1 + \mu^{2} (\beta_{1}^{2} + \beta_{2}^{2} ))(I_{0} + J_{2} (\beta_{1}^{2} + \beta_{2}^{2} )) \\ & m_{21} = - (1 + \mu^{2} (\beta_{1}^{2} + \beta_{2}^{2} ))(I_{0} + J_{2} (\beta_{1}^{2} + \beta_{2}^{2} )) \\ & m_{22} = - (1 + \mu^{2} (\beta_{1}^{2} + \beta_{2}^{2} ))(I_{0} + K_{2} (\beta_{1}^{2} + \beta_{2}^{2} )) \\ \end{aligned} \right.. $$

(37)