Prediction of in-plane elastic properties of graphene in the framework of first strain gradient theory

Abstract

In the present study, the in-plane elastic stiffness coefficients of graphene within the framework of first strain gradient theory are calculated on the basis of an accurate molecular mechanics model. To this end, a Wigner–Seitz primitive cell is adopted. Additionally, the first strain gradient theory for graphene with trigonal crystal system is formulated and the relation between elastic stiffness coefficients and molecular mechanics parameters are calculated. Thus, the ongoing research challenge on providing the accurate mechanical properties of graphene is addressed herein. Using results obtained, the in-plane free vibration of graphene is studied and a detailed numerical investigation is implemented.

Appendix

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

(54)

with

$$\begin{aligned}&a_1=a_{14}-\sqrt{2}\,a_{34},\, a_2=a_{15}-\sqrt{2}\,a_{35}, \, a_3=\frac{a_{11}-a_{22}}{2}, \, a^*_3=\frac{a_{11}+a_{22}}{2}\nonumber \\&\pmb {\mathrm {A_c}}=\left[ \begin{array}{ccccc} 1 &{} -1 &{} -\sqrt{2} &{} 0 &{} 0\\ &{} 1 &{} \sqrt{2} &{} 0 &{} 0\\ &{} &{} 2 &{} 0 &{} 0\\ &{} &{} &{} 0 &{} 0\\ &{} &{} &{} 0 &{} 0 \end{array}\right] \end{aligned}$$

(55)

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

(56)

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

(57)

$$\begin{aligned}&\pmb {\mathrm {H^{(6)}}}=\left[ \begin{array}{ccccc} h_{11}&{} h_{12} &{} h_{13} &{} h_{12} &{} h_{13} \\ &{} h_{22} &{} h_{23} &{} h_{22} &{} h_{23}\\ &{} &{} h_{33} &{} h_{23} &{} h_{33}\\ &{} &{} &{} h_{22} &{} h_{23}\\ &{} &{} &{} &{} h_{33} \end{array}\right] \end{aligned}$$

(58)

$$\begin{aligned}&\pmb {\mathrm {J^{(4)}}}=\left[ \begin{array}{ccccc} j_{11} &{} j_{12} &{} j_{12}\\ &{} j_{22} &{} j_{23}\\ &{} &{} j_{22} \end{array}\right] \end{aligned}$$

(59)

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

(60)

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

(61)

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

(62)

with

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