A comprehensive continuum model for graphene in the framework of first strain gradient theory

Abstract

Capture of discrete nature of nanomaterials is central to predict their behavior in high precision. To this end, various augmented continuum theories have been presented. However, their accuracy is strongly dependent on the recognition of accurate values of associated characteristic length scale parameter, and many ambiguities and uncertainties still remain about them. As main purpose of this study, the accurate values of elasticity tensor elements in the framework of first strain gradient theory are calculated for graphene with trigonal crystal system. A promising way to achieve them could be through molecular mechanics model. So that, the energy, including strain and kinetic energy, based on first strain gradient theory can be equivalent to that ones on the basis of molecular mechanics model. This successful relation results in addressing the accurate values of elasticity tensor element. In this way, the dynamic and static behaviors of graphene can be accurately investigated.

Appendices

Appendix A

$$\begin{aligned} \pmb {\mathrm {A^{(11)}}}=\begin{bmatrix} a_{11} &{}\quad a_{12} &{}\quad a_{13} &{}\quad a_{14} &{}\quad a_{15}\\ &{}\quad a_{22} &{}\quad -a_{13}+\sqrt{2}\,a_3 &{}\quad a_1 &{}\quad a_2\\ &{}\quad &{}\quad -a_{12}+a^*_3 &{} \quad a_{34} &{}\quad a_{35}\\ &{}\quad &{}\quad &{}\quad a_{44} &{}\quad a_{45}\\ &{}\quad &{}\quad &{}\quad &{}\quad a_{55} \end{bmatrix} \end{aligned}$$

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with

$a_1=a_{14}-\sqrt{2}\,a_{34},\, a_2=a_{15}-\sqrt{2}\,a_{35}, \, a_3=\frac{a_{11}-a_{22}}{2}, \, a^*_3=\frac{a_{11}+a_{22}}{2}$

$$\begin{aligned} \pmb {\mathrm {A_c}}= & {} \begin{bmatrix} 1 &{}\quad -1 &{}\quad -\sqrt{2} &{}\quad 0 &{}\quad 0\\ &{}\quad 1 &{}\quad \sqrt{2} &{}\quad 0 &{}\quad 0\\ &{} \quad &{}\quad 2 &{}\quad 0 &{}\quad 0\\ &{} \quad &{}\quad &{}\quad 0 &{}\quad 0\\ &{}\quad &{} \quad &{}\quad 0 &{}\quad 0 \end{bmatrix} \end{aligned}$$

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$$\begin{aligned} \pmb {\mathrm {D^{(4)}}}= & {} \begin{bmatrix} d_{11} &{} \quad d_{12} &{}\quad -d_{12}\\ d_{11} &{}\quad -d_{12} &{}\quad d_{12}\\ 0 &{}\quad -\sqrt{2}\,d_{12} &{}\quad \sqrt{2}\,d_{12}\\ d_{41} &{}\quad 0 &{}\quad 0\\ d_{51} &{}\quad 0 &{}\quad 0 \end{bmatrix} \end{aligned}$$

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$$\begin{aligned} \pmb {\mathrm {F^{(8)}}}= & {} \begin{bmatrix} f_{11}&{} f_{12} &{} f_{13} &{} f_{14} &{} f_{15}\\ -f_{11}&{} -f_{12}+\beta _1 &{} f_{23} &{} -f_{12}+\beta _1 &{} -f_{15}-2\beta _2\\ -\sqrt{2}\,f_{11}&{} -\sqrt{2}\,\left( f_{12}-\frac{3\beta _1}{2}\right) &{} -\sqrt{2}\,(f_{15}+\beta _2) &{} -\sqrt{2}\,\left( f_{12}-\frac{\beta _1}{2}\right) &{} -\sqrt{2}\,(f_{13}-\beta _2)\\ 0 &{} 0 &{} f_{43} &{} 0 &{} -f_{43}\\ 0 &{} 0 &{} f_{53} &{} 0 &{} -f_{53} \end{bmatrix} \nonumber \\\end{aligned}$$

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$$\begin{aligned} \pmb {\mathrm {H^{(6)}}}= & {} \begin{bmatrix} h_{11}&{}\quad h_{12} &{}\quad h_{13} &{}\quad h_{12} &{} \quad h_{13} \\ &{}\quad h_{22} &{}\quad h_{23} &{}\quad h_{22} &{}\quad h_{23}\\ &{} \quad &{}\quad h_{33} &{}\quad h_{23} &{}\quad h_{33}\\ &{}\quad &{}\quad &{}\quad h_{22} &{}\quad h_{23}\\ &{}\quad &{}\quad &{}\quad &{}\quad h_{33} \end{bmatrix} \end{aligned}$$

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$$\begin{aligned} \pmb {\mathrm {J^{(4)}}}= & {} \begin{bmatrix} j_{11} &{}\quad j_{12} &{} \quad j_{12}\\ &{}\quad j_{22} &{}\quad j_{23}\\ &{}\quad &{}\quad j_{22} \end{bmatrix} \end{aligned}$$

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$$\begin{aligned} \pmb {\mathrm {f\left( F^{(8)}\right) }}= & {} \begin{bmatrix} \sqrt{2}\,\alpha &{}\quad \beta _2 &{}\quad -2\beta _3-\beta _2\\ 0 &{}\quad \beta _2 &{}\quad 3\beta _2-2\beta _3\\ \alpha &{}\quad -2\sqrt{2}\,(\beta _2-\beta _3) &{}\quad 0 \\ 0 &{}\quad f_{43} &{}\quad f_{43} \\ 0 &{}\quad f_{53} &{}\quad f_{53} \end{bmatrix} \end{aligned}$$

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$$\begin{aligned} \pmb {\mathrm {f\left( D^{(4)}\right) }}= & {} \begin{bmatrix} 0 &{}\quad \frac{\sqrt{2}}{2}\,d_{11} &{}\quad 0 &{}\quad -\frac{\sqrt{2}}{2}\,d_{11} &{} \quad 0\\ 0 &{}\quad \frac{\sqrt{2}}{2}\,d_{11} &{} \quad 0 &{}\quad -\frac{\sqrt{2}}{2}\,d_{11} &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{} \quad \frac{\sqrt{2}}{2}\,d_{41} &{}\quad 0 &{}\quad -\frac{\sqrt{2}}{2}\,d_{41} &{}\quad 0\\ 0 &{}\quad \frac{\sqrt{2}}{2}\,d_{51} &{} \quad 0 &{}\quad -\frac{\sqrt{2}}{2}\,d_{51} &{}\quad 0 \end{bmatrix} \end{aligned}$$

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$$\begin{aligned} \pmb {\mathrm {f\left( J^{(4)}\right) }}= & {} \begin{bmatrix} 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ &{}\quad 0 &{}\quad 0 &{} -j_{11} &{}\quad -\sqrt{2}\,j_{12}\\ &{} \quad &{} \quad 0 &{} \quad -\sqrt{2}\,j_{12} &{}\quad -(j_{22}+j_{23})\\ &{}\quad &{}\quad &{}\quad 0 &{} \quad 0\\ &{} \quad &{}\quad &{} \quad &{}\quad 0 \end{bmatrix} \end{aligned}$$

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with $\beta _1=\frac{f_{12}-f_{14}}{2},\,\beta _2=\frac{f_{13}+f_{23}}{2},\,\beta _3=\frac{f_{13}-f_{15}}{2}$

Appendix B

$$\begin{aligned} u= & {} -zw_{,x},\,\,\,v = -zw_{,y},\,\,\,w = w \end{aligned}$$

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$$\begin{aligned} \varepsilon _{xx}= & {} -zw_{,xx},\,\,\,\varepsilon _{yy}=-zw_{,yy},\,\,\,\varepsilon _{xy}=-zw_{,xy} \end{aligned}$$

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$$\begin{aligned} \eta _{xxx}= & {} \varepsilon _{xx,x}=-zw_{,xxx},\,\,\,\eta _{xxy}=\varepsilon _{xx,y}=-zw_{,xxy},\,\,\,\eta _{xyy}=\varepsilon _{xy,y}= -zw_{,xyy}\nonumber \\ \eta _{yyy}= & {} \varepsilon _{yy,y}= -zw_{,yyy},\,\,\,\eta _{yxx}=\varepsilon _{xy,x}=-zw_{,xxy},\,\,\,\eta _{yyx}=\varepsilon _{yy,x}=-zw_{,xyy}\nonumber \\ \eta _{xxz}= & {} -w_{,xx},\,\,\,\eta _{yyz}=-w_{,yy},\,\,\,\eta _{xyz}=-w_{,xy} \end{aligned}$$

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Appendix C

$$\begin{aligned} M_{xx}&= -\int _{-\frac{h}{2}}^{\frac{h}{2}} (z\sigma _{xx}+\tau _{xxz})\,dz= \frac{h^3}{12}\left( C_{11}w_{,xx}+C_{12}w_{,yy})+h(h_{22}w_{,xx}+(h_{22}-J_{11})w_{,yy}\right) \end{aligned}$$

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$$\begin{aligned} M_{xy}&= -\int _{-\frac{h}{2}}^{\frac{h}{2}} 2(z\sigma _{xy}+\tau _{xyz})\,dz= \frac{h^3}{6}(C_{11}-C_{12})w_{,xy}+2hJ_{11}w_{,xy} \end{aligned}$$

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$$\begin{aligned} M_{yy}&= -\int _{-\frac{h}{2}}^{\frac{h}{2}} (z\sigma _{yy}+\tau _{yyz})\,dz= \frac{h^3}{12}(C_{12}w_{,xx}+C_{11}w_{,yy})+h((h_{22}-J_{11})w_{,xx}+h_{22}w_{,yy})\end{aligned}$$

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$$\begin{aligned} Y_{xxx}&= -\int _{-\frac{h}{2}}^{\frac{h}{2}} (\tau _{xxx})\,zdz= \frac{h^3}{12}\left( (\eta +a_{11})w_{,xxx}+\left( -3\eta +a_{12}+\sqrt{2}a_{13}\right) w_{,xyy}\right) \end{aligned}$$

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$$\begin{aligned} Y_{xxy}&= -\int _{-\frac{h}{2}}^{\frac{h}{2}} (2\tau _{xyx}+\tau _{xxy})\,zdz= \frac{h^3}{12}\left( -\left( 2\sqrt{2}a_{13}-3a_{11}+2a_{12}\right) w_{,xxy}\nonumber \right. \\&\left. \quad +\left( a_{12}+\sqrt{2}a_{13}\right) w_{,yyy}\right) \end{aligned}$$

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$$\begin{aligned} Y_{xyy}&= -\int _{-\frac{h}{2}}^{\frac{h}{2}} (2\tau _{xyy}+\tau _{yyx})\,zdz\nonumber \\&= -\frac{h^3}{12}\left( \left( 3\eta -a_{12}-\sqrt{2}a_{13}\right) w_{,xxx}+\left( -9\eta -3a_{11}+2a_{12}+2\sqrt{2}a_{13}\right) w_{,xyy}\right) \end{aligned}$$

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$$\begin{aligned} Y_{yyy}&= -\int _{-\frac{h}{2}}^{\frac{h}{2}} \tau _{yyy}\,zdz= \frac{h^3}{12}\left( \left( \sqrt{2}a_{13}+a_{12}\right) w_{,xxy}+a_{11}w_{,yyy}\right) \end{aligned}$$

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Appendix D

$$\begin{aligned}&M_{xx,xx}+M_{xy,xy}+M_{yy,yy}-Y_{xxx,xxx}-Y_{xxy,xxy}-Y_{xyy,xyy}-Y_{yyy,yyy}\nonumber \\&\quad -\rho h\, \ddot{w}=0 \end{aligned}$$

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$$\begin{aligned}&(M_{xx,x}+M_{xy,y}-Y_{xxx,xx}-Y_{xxy,xy}-Y_{xyy,yy})\,\delta w\mid _{x=0,{\bar{a}}}=0 \nonumber \\&(M_{xx}-Y_{xxx,x}-Y_{xxy,y})\,\delta w_{,x}\mid _{x=0,{\bar{a}}}=0,\,\,\,\,(Y_{xxx})\,\delta w_{,xx}\mid _{x=0,{\bar{a}}}=0 \nonumber \\&(M_{xy,x}+M_{yy,y}-Y_{yyy,yy}-Y_{xyy,xy}-Y_{xxy,xx})\,\delta w\mid _{y=0,{\bar{b}}}=0 \nonumber \\&(M_{yy}-Y_{yyy,y}-Y_{xyy,x})\,\delta w_{,y}\mid _{y=0,{\bar{b}}}=0,\,\,\,\,(Y_{yyy})\,\delta w_{,yy}\mid _{y=0,{\bar{b}}}=0 \end{aligned}$$

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