## 2 Theory and Formulation
### The Homogenization Process of MHC  
An overview of MHC is presented in Figure 2. Accordingly, the homogenization approach was based on Halpin-Tsai model and a micromechanical theory and consisted of two main steps. First, we computed the effective properties of the composite, i.e.,  
\[E_{11}=V_{F}E_{11}^{F}+V_{NCM}E^{NCM} \tag{1}\]  
\[\frac{1}{E_{22}}=\frac{1}{E_{22}^{F}}+\frac{V_{NCM}}{E^{NCM}}-V_{F}V_{NCM}- \frac{\frac{(v^{f})^{2}E^{NCM}}{E_{22}^{F}}+\frac{(v^{NCM})^{2}E_{22}^{F}}{E^{M }}-2v^{F}v^{NCM}}{V_{F}E_{22}^{F}+V_{NCM}E^{NCM}} \tag{2}\]  
\[\frac{1}{G_{12}}=\frac{V_{F}}{G_{12}^{F}}+\frac{V_{NCM}}{G^{NCM}} \tag{3}\]  
\[\rho=V_{F}\rho^{F}+V_{NCM}\rho^{NCM} \tag{4}\]  
\[\
u_{12}=V_{F}v^{F}+V_{NCM}v^{NCM} \tag{5}\]  
where \(\rho,~{}E,~{}G\) and \(\
u\) represent the mass density, elasticity modulus, shear modulus, and Poisson's ratio, respectively; the superscripts \(NCM\) and \(F\) indicate 'nanocomposite matrix' and 'fiber', respectively. The sum of volume fraction of the carbon fiber \(V_{F}\) and the volume fraction of the nanocomposite matrix \(V_{NCM}\) is equal to 1. \[V_{F}+V_{NCM}=1 \tag{6}\]  
Second, we obtained the effective properties of the nanocomposite using the extended Halpin-Tsai micromechanics theory:  
\[E^{NCM}=E^{M}(\frac{5}{8}(\frac{1+2\beta_{dd}V_{CNT}}{1-\beta_{dd}V_{CNT}})+ \frac{3}{8}(\frac{1+2(l^{CNT}/d^{CNT})\beta_{dl}V_{CNT}}{1-\beta_{dl}V_{CNT}})) \tag{7}\]  
in which \(\beta_{dd}\) and \(\beta_{dl}\) are given by  
\[\beta_{dl}=\frac{(E_{11}^{CNT}/E^{M})-(d^{CNT}/4t^{CNT})}{(E_{11}^{CNT}/E^{ M})+(l^{CNT}/2t^{CNT})},\quad\beta_{dd}=\frac{(E_{11}^{CNT}/E^{M})-(d^{CNT}/4t^{CNT})}{(E_{11}^{ CNT}/E^{M})+(d^{CNT}/2t^{CNT})} \tag{8}\]  
\(l^{CNT},t^{CNT},d^{CNT},E^{CNT},\) and \(V_{CNT}\) denoting the length, thickness, diameter, elasticity modulus, and volume fraction of CNTs, respectively. In addition, \(E^{M}\) and \(V_{M}\) are the elasticity modulus and volume fraction of the matrix, respectively. The volume fraction of CNTs can be calculated explicitly based on its weight fraction \(W_{CNT}\) according to following relation  
\[V_{CNT}^{*}=\frac{W_{CNT}}{W_{CNT}+(\frac{\rho^{CNT}}{\rho^{M}})(1-W_{CNT})}. \tag{9}\]  
\[E_{11}=V_{F}E_{11}^{F}+V_{NCM}E^{NCM} \tag{1}\]0 Schematic view of the multiscale hybrid annular plate, related parameters, and coordination. According to subsequent relations and Figure 3, three functionally graded and one uniform pattern were considered for the distribution of CNTs through the thickness of MHCAP. \[ \begin{array}{ll}V_{CNT}=V_{CNT}^{\star}\frac{|\xi_{j}|}{h}&\text{FG-X}\\ V_{CNT}=V_{CNT}^{\star}\Big{(}1+\frac{2\xi_{j}}{h}\Big{)}&\text{FG-V}\\ V_{CNT}=V_{CNT}^{\star}\Big{(}1-\frac{2\xi_{j}}{h}\Big{)}&\text{FG-A}\\ V_{CNT}=V_{CNT}^{\star}&\text{FG-UD}\end{array}  \tag{10}\]  
where \(\xi_{j}=\Big{(}\frac{1}{2}+\frac{1}{2N_{t}}-\frac{j}{N_{t}}\Big{)}h\quad j=1,2,\dots,Nt\). The sum of \(V_{M}\) and \(V_{CNT}\) as the two constituents of the nanocomposite matrix is equal to 1. \[V_{CNT}+V_{M}=1 \tag{11}\]  
The corresponding mass density, Poisson's ratio, and shear modulus of the nanocomposite matrix can be determined as follows  
\[\rho^{NCM}=V_{CNT}\rho^{CNT}+V_{M}\rho^{M} \tag{12}\]  
\[\
u^{NCM}=\
u^{M} \tag{13}\]  
\[G^{NCM}=\frac{E^{NCM}}{2(1+\
u^{NCM})} \tag{14}\]  
Figure 2: Overview of multiscale hybrid composite.