# Buckling Analysis of CNTRC Curved Sandwich Nanobeams in Thermal Environment
## 2 Theoretical Formulation  
### Geometric and Mechanical Properties  
A symmetric cross-ply single-walled carbon nanotube reinforced composite (CNTRC) curved sandwich beam of length \(L\), thickness \(h\), and radius of curvature \(R\) is considered, as shown in Figure 1. Different volume fraction distributions of CNTs are here assumed throughout the thickness (see Figure 2), in agreement with the following relations [22]:  
* UD (Uniformly-Distributed) CNTRC multilayered nanobeam: \[V_{cnt}=V_{cnt}^{*}\] (1)* FG-O CNTRC multilayered nanobeam: \[V_{cnt}=2\Bigg{(}1-\frac{\left|2|z|-\left|z_{(k-1)}+z_{(k)}\right|\right|}{z_{(k) }-z_{(k-1)}}\Bigg{)}V_{cnt}^{*}\] (2)
* FG-X CNTRC multilayered nanobeam: \[V_{cnt}=2\frac{\left|2|z|-\left|z_{(k-1)}+z_{(k)}\right|\right|}{z_{(k)}-z_{(k- 1)}}V_{cnt}^{*}\] (3)  
More specifically, UD CNTRC refers to a uniform distribution of CNTs, whereas FG-V CNTRC, FG-O CNTRC and FG-X CNTRC account for different non-uniform FG distributions. Moreover, \(z_{(k)}\) and \(z_{(k-1)}\) refer to the thickness coordinates at the bottom and top sides of the \(k\)th layer within the laminated nanobeam; \(V_{cnt}^{*}\) is the total volume fraction of CNTs, defined as  
\[V_{cnt}^{*}=\frac{W_{cnt}}{W_{cnt}+(\rho_{cnt}/\rho_{m})(1\;-\;W_{cnt})} \tag{4}\]  
Figure 1: Geometry of a carbon nanotube reinforced composite (CNTRC) curved sandwich beam: (**a**) geometric parameters of the curved beam, (**b**) cross-section of the curved beam.  
Figure 2: Cross-sections of various (CNTRC) carbon nanotube reinforced composite sandwich beams.  
where \(W_{cnt}\) is the CNTs mass fraction; \(\rho_{cnt}\), \(\rho_{m}\) refer to the CNTs and polymer mass density, respectively.  
The Mori-Tanaka scheme [31] is here applied together with the rule of mixtures and molecular dynamics, as suggested in [1,2]. Thus, the effective Young's modulus and shear modulus for each CNTRC sheet is described as  
\[\begin{array}{l}E_{11}^{k}=\eta_{1}V_{cnt}^{k}E_{11}^{cnt}+V_{p}^{k}E_{p}\\ \frac{\eta_{2}}{E_{22}}=\frac{V_{cnt}^{k}}{E_{22}^{cnt}}+\frac{V_{p}^{k}}{E_{p}} \\ \frac{\eta_{3}}{G_{12}^{k}}=\frac{V_{cnt}^{k}}{G_{12}^{cnt}}+\frac{V_{p}^{k}}{ G_{p}}\end{array} \tag{5}\]  
where \(E_{11}^{k}\) and \(E_{22}^{k}\) are the elasticity modulus along the in-plane directions \((x,z)\) for the \(k\)th layer and \(G_{12}^{k}\) is its shear modulus. The subscripts \(p\) and \(cnt\) refer to the polymer and SWCNT properties, respectively, assuming the CNT efficiency parameters \(\eta_{1},\ \eta_{2},\ \eta_{3}\) as proposed in [6] and summarized in Table 1.  
The Poisson's ratio \(\
u_{12}^{k}\), the density \(\rho^{k}\), and the thermal expansion coefficients in the longitudinal and transverse directions \(\alpha_{11}^{k}\), \(\alpha_{22}^{k}\), for each sheet are given as follows:  
\[\
u_{12}^{k}=V_{cnt}^{k}\
u_{12}^{cnt}+V_{p}^{k}\
u_{p} \tag{6}\]  
\[\rho^{k}=V_{cnt}^{k}\rho_{cnt}+V_{p}^{k}\rho_{p} \tag{7}\]  
\[\alpha_{11}^{k}=V_{cnt}^{k}\alpha_{11}^{cnt}+V_{p}^{k}\alpha_{p} \tag{8}\]  
\[\alpha_{22}^{k}=\big{(}1+\
u_{12}^{cnt}\big{)}V_{cnt}^{k}\alpha_{22}^{cnt}+ \big{(}1+\
u_{p}\big{)}V_{p}^{k}\alpha_{p}\ -\ \
u_{12}^{k}\alpha_{11}^{k} \tag{9}\]