# Nonlinear Vibrations of Carbon Nanotubes with Thermal-Electro-Mechanical Coupling
## 1 Introduction  
In 1991, carbon nanotubes (CNTs) were discovered by [PERSON] [1], and their excellent mechanical, chemical, electrical properties have since, extensively attracted the attention of researchers [2; 3; 4; 5; 6]. CNTs are formed by the crimping of graphene sheets in a particular direction. Moreover, researchers observed the flexoelectric effect in CNTs [7]. The lattice of graphene is symmetrical, and the uniform deformation of graphene will not lead to polarization. When graphene is curling into CNTs, the symmetry of the lattice may break and may induce the piezoelectric effect. The second-order flexible electric effect in graphene can be caused by the strain gradient. In fact, the strain gradient changes the position of ions, leading to the asymmetric redistribution of electron density [8; 9]. [PERSON] has described in detail the zigzag and armchair \(\gamma\)-graphene-acetylene nanotubes using the Density functional theory function and non-equilibrium Green's function methods. He also reported the application prospect of these devices in nano transistors and switching memory circuits [10; 11; 12]. At the same time, CNTs also have technical potential in field emission, nano-electronic devices, micro-electro-mechanical system, biosensing, high-frequency nanoelectronics, and so on [13; 14; 15; 16; 17]. These applications are used in many physicalfields, such as temperature and electrostatic fields. [PERSON]'s results show that, when CNTs are deformed under a transverse concentrated load, their conductivity decreases greatly [18]. This experiment is of great significance, showing that the coupling effect should be specially considered [19]. CNTs can be employed in a variety of technical applications when combined with a polymer composite material to create carbon nanotube-reinforced composites [20]. In these applications, the influence of the temperature field is usually accompanied. However, it is necessary to comprehensively research the thermal-electro-mechanical coupling of CNTs [21]. Because CNTs are curled by graphene and the thickness of graphene is uncertain, the stiffness of CNTs cannot be determined by the classical beam model. This means that the bending and stretching stiffness must be considered separately [22; 23; 24]. For example, [PERSON] established a thermal-electro-mechanical coupling nanobeam model of CNTs with independent bending and stretching stiffness [23]. This nanobeam model describes the axial load and radial load of the beam. Generally, when the beam is subjected to axial load and radial load, curvature and axial strain will occur [25]. From the above discussion, one can find that CNTs have extensive applications. However, their mechanical properties under the multi-field coupling conditions have received little attention. For example, the deformations of CNTs when the ambient temperature and the applied voltage at both ends of the beam change have not been studied in depth thus far. However, this is common in nanoelectromechanical systems. In fact, [PERSON] [23] proposed a beam model for the first time to describe the mechanical properties of a CNT under thermal and electric fields. This paper will further analyze the model proposed by [PERSON] [23]. Because it is difficult to solve partial differential equations, the Galerkin method [26] is used to discretize the partial differential equation into ordinary differential equations. The accuracy of the Galerkin truncation depends on the order of truncation. Therefore, the first-order and second-order mode vibration equations of the partial differential equation are used in this paper. We will solve the two vibration equations using the multi-scale method [27; 28]. The steady-state solutions response to the primary resonance and the subharmonic resonance are obtained. The effect of temperature and electric fields on steady-state motion can be discovered by the analytical solutions. The accuracy of the solutions is checked by numerical calculations.