*2.3. Kinematic Relations*

This work aims to establish a new shear deformation shell theory. The main idea of the present theory comes from the four-variable refined theory [8,9,11,28,29]. According to assumptions of various four-variable refined theories, the transverse displacement *w* is partitioned into the bending component *wb* and shear component *ws*, the in-plane displacements *u* and *v* are also partitioned into the extension component *u*0, *v*0, the bending component *ub*, *vb*, and shear component *us*, *vs*. Therefore, the displacement field in the doubly curved shell space can be expressed as follows:

$$\begin{cases} u(x,y,z,t) = \left(1+\frac{z}{\mathbb{R}^x\_x}\right)u\_0(x,y,t) - z\frac{\partial w\_b(x,y,t)}{\partial x} - f(z)\frac{\partial w\_s(x,y,t)}{\partial x} \\\\ v(x,y,z,t) = \left(1+\frac{z}{\mathbb{R}^y\_y}\right)v\_0(x,y,t) - z\frac{\partial w\_b(x,y,t)}{\partial y} - f(z)\frac{\partial w\_b(x,y,t)}{\partial y} \\\\ w(x,y,z,t) = w\_b(x,y,t) + w\_s(x,y,t) \end{cases} \tag{4}$$

where *u*0, *v*<sup>0</sup> denote the displacements along *x* and *y* coordinate directions of the corresponding point on the reference surface; *wb* and *ws* are the bending and shear components of the transverse displacement, respectively; *f*(*z*) represents shape function determining the distribution of the transverse shear strains and stresses along the thickness. By the same methodology, in the previous study [29], we proposed a new shape function *f*(*z*) as follows:

$$f(z) = z \left[ -\frac{1}{8} + \frac{3}{2} \left( \frac{z}{h} \right)^2 \right] \tag{5}$$

Detail steps to construct this shape function for shell panels are listed in Appendix A. The strains associated with the displacement field in Equation (4) are:

$$\begin{array}{l} \boldsymbol{\kappa}\_{xx} = \frac{1}{1+z/R\_x} [\boldsymbol{\kappa}\_x^0 + z\boldsymbol{\kappa}\_x^b + f(z)\boldsymbol{\kappa}\_x^s];\\ \boldsymbol{\kappa}\_{yy} = \frac{1}{1+z/R\_y} [\boldsymbol{\kappa}\_y^0 + z\boldsymbol{\kappa}\_y^b + f(z)\boldsymbol{\kappa}\_y^s];\\ \boldsymbol{\gamma}\_{xy} = \frac{1}{1+z/R\_x} [\boldsymbol{\gamma}\_{xy}^0 + z\boldsymbol{\kappa}\_{xy}^b + f(z)\boldsymbol{\kappa}\_{xy}^s] + \frac{1}{1+z/R\_y} [\boldsymbol{\gamma}\_{yx}^0 + z\boldsymbol{\kappa}\_{yx}^b + f(z)\boldsymbol{\kappa}\_{yx}^s];\\ \boldsymbol{\gamma}\_{xz} = \frac{1}{1+z/R\_x} g(z)\boldsymbol{\gamma}\_{xz}^s;\\ \boldsymbol{\gamma}\_{yz} = \frac{1}{1+z/R\_y} g(z)\boldsymbol{\gamma}\_{yz}^s \end{array} \tag{6}$$

where:

ε0 *<sup>x</sup>* <sup>=</sup> <sup>∂</sup>*u*<sup>0</sup> <sup>∂</sup>*<sup>x</sup>* <sup>+</sup> *wb Rx* <sup>+</sup> *ws Rx* ; γ<sup>0</sup> *xy* <sup>=</sup> <sup>∂</sup>*v*<sup>0</sup> <sup>∂</sup>*<sup>x</sup>* ; ε0 *<sup>y</sup>* = - ∂*v*<sup>0</sup> <sup>∂</sup>*<sup>y</sup>* <sup>+</sup> *wb Ry* <sup>+</sup> *ws Ry* ; γ<sup>0</sup> *yx* <sup>=</sup> <sup>∂</sup>*u*<sup>0</sup> <sup>∂</sup>*<sup>y</sup>* ; *kb <sup>x</sup>* = - 1 *Rx* ∂*u*<sup>0</sup> <sup>∂</sup>*<sup>x</sup>* <sup>−</sup> <sup>∂</sup>*w*<sup>2</sup> *b* ∂*x*<sup>2</sup> ; κ*<sup>b</sup> <sup>y</sup>* = - 1 *Ry* ∂*v*<sup>0</sup> <sup>∂</sup>*<sup>y</sup>* <sup>−</sup> <sup>∂</sup>*w*<sup>2</sup> *b* ∂*y*<sup>2</sup> ; κ*b xy* = - 1 *Ry* ∂*v*<sup>0</sup> <sup>∂</sup>*<sup>x</sup>* <sup>−</sup> <sup>∂</sup>*w*<sup>2</sup> *b* ∂*x*∂*y* ; κ*<sup>b</sup> yx* = - 1 *Rx* ∂*u*<sup>0</sup> <sup>∂</sup>*<sup>y</sup>* <sup>−</sup> <sup>∂</sup>*w*<sup>2</sup> *b* ∂*x*∂*y* ; *ks <sup>x</sup>* <sup>=</sup> <sup>−</sup>∂2*ws* <sup>∂</sup>*x*<sup>2</sup> ; *<sup>k</sup><sup>s</sup> <sup>y</sup>* <sup>=</sup> <sup>−</sup>∂2*ws* <sup>∂</sup>*y*<sup>2</sup> ; <sup>κ</sup>*<sup>s</sup> xy* <sup>=</sup> <sup>∂</sup>2*ws* <sup>∂</sup>*x*∂*<sup>y</sup>* ; <sup>κ</sup>*<sup>s</sup> yx* <sup>=</sup> <sup>−</sup>∂2*ws* <sup>∂</sup>*x*∂*<sup>y</sup>* ; γ*xz* = <sup>1</sup> <sup>1</sup>+*z*/*Rx <sup>g</sup>*(*z*) <sup>∂</sup>*ws* <sup>∂</sup>*<sup>x</sup>* ; *g*(*z*) = (1 − *f* (*z*)); γ*yz* = <sup>1</sup> <sup>1</sup>+*z*/*Rx <sup>g</sup>*(*z*) <sup>∂</sup>*ws* ∂*y* (7)

The constitutive relation for an individual layer can be determined by the generalized Hooke's law, namely [30,31]:

$$
\begin{pmatrix}
\sigma\_{xx}^{k} \\
\sigma\_{yy}^{k} \\
\tau\_{yz}^{k} \\
\tau\_{zx}^{k} \\
\tau\_{xy}^{k}
\end{pmatrix} = \begin{bmatrix}
\overline{Q}\_{11}^{k} & \overline{Q}\_{11}^{k} & 0 & 0 & \overline{Q}\_{16}^{k} \\
\overline{Q}\_{12}^{k} & \overline{Q}\_{22}^{k} & 0 & 0 & \overline{Q}\_{26}^{k} \\
0 & 0 & \overline{Q}\_{44}^{k} & \overline{Q}\_{45}^{k} & 0 \\
0 & 0 & \overline{Q}\_{45}^{k} & \overline{Q}\_{55}^{k} & 0 \\
\overline{Q}\_{16}^{k} & \overline{Q}\_{26}^{k} & 0 & 0 & \overline{Q}\_{66}^{k}
\end{pmatrix} \begin{pmatrix}
\varepsilon\_{xx} \\
\varepsilon\_{yy} \\
\gamma\_{yz} \\
\gamma\_{xz} \\
\gamma\_{xy}
\end{pmatrix} \tag{8}
$$

where *Qk ij* are the transformed material constraints expressed in terms of material constants:

$$\begin{cases} \overline{Q}\_{11} = Q\_{11}\cos^{4}\theta^{k} + 2(Q\_{12} + 2Q\_{66})\sin^{2}\theta^{k}\cos^{2}\theta^{k} + Q\_{22}\sin^{4}\theta^{k};\\ \overline{Q}\_{12} = (Q\_{11} + Q\_{22} - 4Q\_{66})\sin^{2}\theta^{k}\cos^{2}\theta^{k} + Q\_{22}\{\sin^{4}\theta^{k} + \cos^{4}\theta^{k}\};\\ \overline{Q}\_{22} = Q\_{11}\sin^{4}\theta^{k} + 2(Q\_{12} + 2Q\_{66})\sin^{2}\theta^{k}\cos^{2}\theta^{k} + Q\_{22}\cos^{4}\theta^{k};\\ \overline{Q}\_{16} = (Q\_{11} - Q\_{12} - 2Q\_{66})\sin^{2}\theta^{k} + (Q\_{12} - Q\_{22} + 2Q\_{66})\sin^{3}\theta^{k}\cos\theta^{k};\\ \overline{Q}\_{26} = (Q\_{11} - Q\_{12} - 2Q\_{66})\sin^{3}\theta^{k}\cos\theta^{k} + (Q\_{12} - Q\_{22} + 2Q\_{66})\sin^{3}\theta^{k}\cos^{3}\theta^{k};\\ \overline{Q}\_{66}^{k} = (Q\_{11} + Q\_{22} - 2Q\_{12} - 2Q\_{66})\sin^{2}\theta^{k}\cos^{2}\theta^{k} + Q\_{66}(\sin^{4}\theta^{k} + \cos^{4}\theta^{k});\\ \overline{Q}\_{44}^{k} = Q\_{44}\cos^{2}\theta^{k} + Q\_{55}\sin^{2}\theta^{k};\\ \overline{Q}\_{55}^{k} = (Q\_{55} - Q\_{44})\cos\theta^{k}\sin\theta^{k};\\ \overline{Q}\_{$$

in which, *Qij* are the plane stress-reduced stiffnesses defined in terms of the engineering constants in the material axes of the layer. For each CNT layer:

$$\begin{aligned} Q\_{11} &= \frac{\underline{E}\_{11}(z)}{1 - \underline{\nu}\_{12}\underline{\nu}\_{21}};\ Q\_{12} = \frac{\underline{\nu}\_{12}\underline{E}\_{22}(z)}{1 - \underline{\nu}\_{12}\underline{\nu}\_{21}};\ Q\_{22} = \frac{\underline{E}\_{22}(z)}{1 - \underline{\nu}\_{12}\underline{\nu}\_{21}};\\ \underline{Q}\_{44} &= \mathcal{G}\_{23}(z);\ \mathcal{Q}\_{55} = \mathcal{G}\_{13}(z);\ \mathcal{Q}\_{66} = \mathcal{G}\_{12}(z) \end{aligned} \tag{10}$$

⎧

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
