## 3 Fundamental Equations  
Based on the TSDT [24], the displacement components \(u,v,w\) of the shell along the \(x,y,z\)-axes (see Figure 1), can be expressed as  
\[ \begin{array}{l}u(x,y,z)=u_{0}(x,y)\,\,+\,\,z\psi(x,y)\,\,+\,\,z^{2}\theta_{ x}(x,y)\,\,+\,\,z^{3}\lambda_{x}(x,y)\\ v(x,y,z)=v_{0}(x,y)\,\,+\,\,z\phi(x,y)\,\,+\,\,z^{2}\theta_{y}(x,y)\,\,+\,\,z^{3 }\lambda_{y}(x,y)\\ w(x,y,z)=w_{0}(x,y)\end{array}  \tag{8}\]  
\(u_{0},v_{0},w_{0}\) being the displacement components of a point belonging to the reference surface of the shell at \(z=0\), while \(\psi\) and \(\phi\) represent its rotations about the \(x\)- and \(y\)-axes, respectively. Figure 2: Sigmoidal distributions of the \(V_{CNT}\) through the thickness for different exponents \(s\). CNT: Carbon NanoTubes. By considering the stress-free conditions on the inner and outer surfaces of the shell, i.e., for \(\sigma_{xz}(x,y,\pm h/2)=\sigma_{yz}(x,y,\pm h/2)=0\), the relation (8) can be rewritten as  
\[ \begin{array}{l}u(x,y,z)=u_{0}(x,y)\ +\ zy(x,y)\ -\ c_{1}z^{3}(\psi\ +\ w_{0,x})\\ v(x,y,z)=v_{0}(x,y)\ +\ z\phi(x,y)\ -\ c_{1}z^{3}\big{(}\phi\ +\ w_{0,y}\big{)}\\ w(x,y,z)=w_{0}(x,y)\end{array}  \tag{9}\]  
where \(c_{1}=-4/3\hbar^{2}\). The following nonlinear strain-displacement expressions can be determined as  
\[\left\{ \begin{array}{l}\varepsilon_{xx}\\ \varepsilon_{yy}\\ \gamma_{xy}\end{array} \right\}\ =\ \left\{ \begin{array}{l}\varepsilon_{xx}^{(0)}\\ \varepsilon_{yy}^{(0)}\\ \gamma_{xy}^{(0)}\end{array} \right\}\ +\ z\left\{ \begin{array}{l} \varepsilon_{xx}^{(1)}\\ \varepsilon_{yy}^{(1)}\\ \gamma_{xy}^{(1)}\end{array} \right\}\ +\ z^{3}\left\{ \begin{array}{l} \varepsilon_{xx}^{(3)}\\ \varepsilon_{yy}^{(3)}\\ \gamma_{xy}^{(3)}\end{array} \right\} \tag{10}\] \[\left\{ \begin{array}{l}\gamma_{xz}\\ \gamma_{yz}\end{array} \right\}\ =\ \left\{ \begin{array}{l}\gamma_{xz}^{(0)}\\ \gamma_{yz}^{(0)}\end{array} \right\}\ +\ z^{2}\left\{ \begin{array}{l} \gamma_{xz}^{(2)}\\ \gamma_{yz}^{(2)}\end{array} \right\}\]  
with  
\[\left\{ \begin{array}{l}\varepsilon_{xx}^{(0)}\\ \varepsilon_{yy}^{(0)}\\ \gamma_{xy}^{(0)}\end{array} \right\} =\left\{ \begin{array}{l}u_{0,x}+w_{0,x}^{2}/2\\ v_{0,y}+w/R+w_{0,y}^{2}/2\\ u_{0,y}+v_{0,x}+w_{0,x}w_{0,y}\end{array} \right\} \tag{11}\] \[\left\{ \begin{array}{l}\varepsilon_{xx}^{(1)}\\ \varepsilon_{yy}^{(1)}\\ \gamma_{xy}^{(1)}\end{array} \right\} =\left\{ \begin{array}{l}\psi_{,x}\\ \phi_{,y}\\ \psi_{,y}+\phi_{,x}\end{array} \right\}\] \[\left\{ \begin{array}{l}\varepsilon_{xx}^{(3)}\\ \varepsilon_{yy}^{(3)}\\ \gamma_{xy}^{(3)}\end{array} \right\} =-c_{1}\left\{ \begin{array}{l}\psi_{,x}+w_{0,xx}\\ \phi_{,y}+w_{0,yy}\\ \psi_{,y}+\phi_{,x}+2w_{0,xy}\end{array} \right\}\] \[\left\{ \begin{array}{l}\gamma_{xz}^{(2)}\\ \gamma_{yz}^{(2)}\end{array} \right\} =-3c_{1}\left\{ \begin{array}{l}\psi+w_{0,x}\\ \phi+w_{0,y}\end{array} \right\}\] \[\left\{ \begin{array}{l}\gamma_{xz}^{(0)}\\ \gamma_{yz}^{(0)}\end{array} \right\} =\left\{ \begin{array}{l}\psi+w_{0,x}\\ \phi+w_{0,y}\end{array} \right\}\]  
In the above relations (10) and (11), \(\varepsilon_{xx}\) and \(\varepsilon_{yy}\) refer to the normal strains, whereas \(\gamma_{xy}\), \(\gamma_{xz}\) and \(\gamma_{yz}\) refer to the shear strains. The constitutive relations defining a linear thermos-elastic material can be expressed as a linear function of the strain field and the temperature variation, namely  
\[\left\{ \begin{array}{l}\sigma_{xx}\\ \sigma_{yy}\\ \tau_{yz}\\ \tau_{xz}\\ \tau_{xy}\end{array} \right\} =\left[ \begin{array}{cccc}Q_{11}&Q_{12}&0&0&0\\ Q_{21}&Q_{22}&0&0&0\\ 0&0&Q_{44}&0&0\\ 0&0&0&Q_{55}&0\\ 0&0&0&0&Q_{66}\end{array} \right]\left\{ \begin{array}{l}\varepsilon_{xx}- \alpha_{11}\Delta T\\ \varepsilon_{yy}-\alpha_{22}\Delta T\\ \gamma_{yz}\\ \gamma_{xz}\\ \gamma_{xy}\end{array} \right\} \tag{12}\]where \(\Delta T=T-T_{0}\) and \(T\) and \(T_{0}\) refer to a varied temperature and a reference temperature, respectively.