# Nonlinear Vibrations of Carbon Nanotubes with Thermal-Electro-Mechanical Coupling
## 2 Materials and Methods  
_Thermal-Electro-Mechanical Coupling Beam Model of CNTs_  
In Reference [23], [PERSON] expanded the deformation energy in extensional stiffness, bending stiffness, the CEE, and the CTE. Then, using the Hamiltonian principle, he obtained a partial differential equation that models the dynamical properties of CNTs with thermal-electro-mechanical coupling. The equation can be shown as:  
\[ \begin{array}{l}m\frac{\partial^{2}w}{\partial t^{2}}+k_{S}(k_{l}U+k_{T}T) \frac{\partial^{2}w}{\partial x^{2}}+k_{B}\frac{\partial^{4}w}{\partial x^{2}} -\frac{k_{S}(1+k_{l}U+k_{T}T)}{2l^{2}}\frac{\partial^{2}w}{\partial x^{2}} \frac{l}{\partial}\left(\frac{\partial w}{\partial x}\right)^{2}dx\\ -k_{B}\left\{12\frac{\partial w}{\partial x}\frac{\partial^{2}w}{\partial x^{2} }\frac{\partial^{3}w}{\partial x^{3}}+3\left(\frac{\partial w}{\partial x} \right)^{2}\frac{\partial^{4}w}{\partial x^{4}}+3\left(\frac{\partial^{2}w}{ \partial x^{2}}\right)^{3}\right\}=f(x,t).\end{array}  \tag{1}\]  
Here \(m\) is the mass per unit length of CNTs; \(l\) is the length of CNTs; \(k_{S}\) is the extensional stiffness; \(k_{B}\) is the bending stiffness; \(k_{l\bar{l}}\) is the CEE; and \(k_{T}\) is the CTE. Normally, there is no piezoelectric effect in isotropic materials [29]. However, if graphene is curled into CNTs, its hexagonal symmetry will be distorted and may cause the piezoelectric effect of tubes. This causes strain due to the external electric field, and the strain is proportional to the field strength [30]. Equation (1) is the plane motion equation of CNTs. If a CNT is treated as a beam with two hinged ends, its boundary conditions are:  
\[w=\frac{\partial^{2}w}{\partial x^{2}}=0,\ at\ x=0,l. \tag{2}\]  
We can add a linear damping term \(\overline{C}\partial^{5}w/\partial t\partial x^{4}\) to Equation (1) to obtain the equation of motion with damping dissipation as:  
\[ \begin{split}& m\frac{\partial^{2}w}{\partial t^{2}}+\overline{C} \frac{\partial^{5}w}{\partial t\partial x^{4}}+k_{S}(k_{U}U+k_{T}T)\frac{ \partial^{2}w}{\partial x^{2}}\\ &+k_{B}\frac{\partial^{4}w}{\partial x^{4}}-\frac{k_{S}(1+k_{U}U+ k_{T}T)}{2l^{2}}\frac{\partial^{2}w}{\partial x^{2}}\int\limits_{0}^{l}\left(\frac{ \partial w}{\partial x}\right)\ dx\\ &-k_{B}\Big{\{}12\frac{\partial w}{\partial x}\frac{\partial^{2} w}{\partial x^{2}}\frac{\partial^{2}w}{\partial x^{3}}+3\left(\frac{\partial w }{\partial x}\right)^{2}\frac{\partial^{4}w}{\partial x^{4}}+3\left(\frac{ \partial^{2}w}{\partial x^{2}}\right)^{3}\Big{\}}=f(x,t).\end{split}  \tag{3}\]  
[PERSON] used molecular dynamics calculations to obtain the following parameters: extensional stiffness \(k_{s}=\alpha(d-d_{0})\) and bending stiffness \(k_{B}=\beta(d-d_{0})^{3}\), where \(\alpha=1128.15\)nN/nm, and \(\beta=142.54\)nN/nm [2]. In this paper, we will take \((10,10)\) single walled carbon nanotubes (SWCNTs) as an example. Therefore, the tube's diameter is \(d=1.356\)nm, and the parameter \(d_{0}=2.7\times 10^{-7}\)nm. Because the \(d_{0}\) is much smaller than tube's diameter \(d\), we use \(d_{0}=0\). This makes \(k_{S}=1.53\times 10^{3}\)nN and \(k_{B}=0.355\times 10^{3}\)nN \(\cdot\)nm\({}^{2}\). The CTE is \(k_{T}=6.0\times 10^{-6}\)K\({}^{-1}\) from Reference [31], and the CEE is \(k_{U}=0.025\)nm/V, which is obtained through the linear fitting FDT calculations [30]. For convenience, we reduce Equation (3) to a dimensionless form. Letting \(\widetilde{w}=w/l,\widetilde{x}=x/l,\widetilde{t}=\omega_{0}t\), we obtain:  
\[