# Nonlinear Vibrations of Carbon Nanotubes with Thermal-Electro-Mechanical Coupling
## 2 Materials and Methods
\begin{split}& P\frac{\partial^{2}\widetilde{w}}{\partial x^{2}}+Q \frac{\partial^{4}\widetilde{w}}{\partial x^{4}}+C\frac{\partial^{5}\widetilde {w}}{\partial t\partial x^{4}}+R\frac{\partial^{2}\widetilde{w}}{\partial x^{ 2}}\int\limits_{0}^{l}\left(\frac{\partial\widetilde{w}}{\partial x}\right)^{2 }dx\\ &-S\Big{\{}12\frac{\partial\widetilde{w}}{\partial x}\frac{ \partial^{2}\widetilde{w}}{\partial x^{2}}\frac{\partial^{3}\widetilde{w}}{ \partial x^{3}}+3\left(\frac{\partial\widetilde{w}}{\partial x}\right)^{2} \frac{\partial^{4}\widetilde{w}}{\partial x^{4}}+3\left(\frac{\partial^{2} \widetilde{w}}{\partial x^{2}}\right)^{3}\Big{\}}=\widetilde{f}(x,t),\end{split}  \tag{4}\]  
Here,  
\[ \begin{split}& P=\frac{k_{S}(k_{U}U+k_{T}T)}{mu\zeta_{0}^{2}l^{2}},\ R=\frac{k_{S}(1+k_{U}U+k_{T}T)}{2 mu\zeta_{0}^{2}l^{2}},\\ & Q=\frac{k_{B}}{mu\zeta_{0}^{2}l^{4}},\ S=\frac{k_{B}}{mu\zeta_{0} ^{2}l^{6}},\ C=\frac{\overline{C}}{mu\zeta_{0}^{2}l^{4}}.\end{split}  \tag{5}\]  
To reveal the load's influence on the second-order mode, we consider a linear load \(F(l-x)\), as shown in Figure 1:  
In this paper, we will focus on the nonlinear mechanical properties of \((10,10)\) SWCNT with thermal-electro-mechanical coupling. Dimensionless Equation (4) is a partial differential equation. It is difficult to find an accurate analytical solution for the equation. Therefore,  
Figure 1: Load on a CNT. we use the Galerkin method [26] to obtain an approximate solution. We assume that the solution of the equation is:  
\[\widetilde{w}=\sum_{j=1}^{n}\eta_{j}(t)\sin(j\pi x). \tag{6}\]  
For the analysis of the first and second order mode, we take \(n=2\) in Equation (6) and substitute it into Equation (4). Then, we multiply \(\sin(\pi x)\) on the two sides of the equation and the interval integral in \([0,1]\) (the Galerkin integral [26; 32]) to obtain the vibrational equation of the first mode. That of the second mode can be obtained by multiplying \(\sin(2\pi x)\) on the two sides, then, the integral in \([0,1]\). Therefore, we obtain the following equations:  
\[ \begin{array}{l}\ddot{\eta}_{1}+c_{1}\dot{\eta}_{1}+k_{11}\eta_{1}+k_{12} \eta_{1}\eta_{2}^{2}+k_{13}\eta_{1}^{3}=F_{1},\\ \dot{\eta}_{2}+c_{2}\dot{\eta}_{2}+k_{21}\eta_{2}+k_{22}\eta_{1}^{2}\eta_{2}+k_ {23}\eta_{2}^{3}=F_{2}.\end{array}  \tag{7}\]  
In Equation (7), the dot on the letter indicates the derivative in time. The parameters of Equation (7) are:  
\[ \begin{array}{l}k_{11}=1-\frac{\pi^{2}k_{5}(k_{1}l(l+k_{T}T)}{m\omega_{0}^{2 }t^{4}},\ k_{12}=\frac{\pi^{4}k_{5}(1+k_{12}l(l+k_{T}T)}{m\omega_{0}^{2}t^{4}} -\frac{30\pi^{6}k_{h}}{m\omega_{0}^{2}t^{6}},\\ k_{13}=\frac{\pi^{4}k_{5}(1+k_{12}l+k_{T}T)}{4m\omega_{0}^{2}t^{4}},\ k_{1}= \frac{4}{m\omega_{0}^{2}t^{6}}f,\\ k_{21}=16-\frac{4\pi^{2}k_{5}(k_{12}l(l+k_{T}T)}{m\omega_{0}^{2}t^{4}},\ k_{2}= \frac{\pi^{4}k_{5}(1+k_{12}l(l+k_{T}T)}{m\omega_{0}^{2}t^{4}}-\frac{30\pi^{6} k_{h}}{m\omega_{0}^{2}t^{6}},\\ k_{23}=\frac{4\pi^{4}k_{5}(1+k_{12}l+k_{T}T)}{m\omega_{0}^{2}t^{4}}-\frac{96\pi ^{2}k_{5}}{m\omega_{0}^{2}t^{6}},\ F_{2}=\frac{2}{m\omega_{0}^{2}t^{2}}f,\\ c_{1}=\frac{\overline{c}}{m\omega_{0}t^{4}},\ c_{2}=\frac{16\overline{c}}{m \omega_{0}t^{4}},\ c_{2}=16c_{1}.\end{array}  \tag{8}\]  
From the buckling theory of elastic columns, one can know that the CNTs will lose stability if \(k_{11}=0\). This case means that the critical temperature is a function of the critical electric field and critical length, as shown in Figure 2.  
We can find from Figure 2 that the electric field has a greater impact on the thermal-electro-mechanical coupling model of CNTs than the temperature field due to \(k_{U}\gg k_{T}\). This means that the electric field has an outstanding influence on the stiffness of CNTs. From the data of quantum calculations, the CEE is always bigger than the CTE. For example, using Hartree-Fock and density functional quantum mechanics simulations [30], [PERSON] obtained a CEE of about \(k_{U}=0.025\ \mathrm{nm/V}\). However, [PERSON] obtained a CTE of about \(k_{U}=0.025\ \mathrm{nm/V}\).