## 3 The Analytical Solution for Orthotropic Materials Adopting Lame's Approach  
Material composite stiffness properties depend on the reinforcement orientation and reinforcement volume fraction [31]. For a unidirectional (UD) composite ply, a scale factor \(\lambda\) between the Young's modulus in the principal reinforcement direction \(\mathrm{E}_{1}\) and in the transversal one \(\mathrm{E}_{2}\) can be defined as  
\[\lambda^{2}=\frac{\mathrm{E}_{1}}{\mathrm{E}_{2}} \tag{1}\]  
The solution of the equilibrium and compatibility equations for the thick-walled composite vessel under plane stress assumptions returns the radial displacement \(\mathrm{u}(\mathrm{r})\) as the following function of the radial position \(\mathrm{r}\),  
\[\mathrm{u}(\mathrm{r})=\mathrm{C}_{1}\mathrm{r}^{-\frac{1}{\lambda}}+\mathrm{ C}_{2}\mathrm{r}^{\frac{1}{\lambda}} \tag{2}\]  
where the \(\mathrm{C}_{1}\) and \(\mathrm{C}_{2}\) constants may be calculated with the aid of the boundary values of the cylinder radial stress (\(\sigma_{\mathrm{r}}\)) at the inner radius (\(\mathrm{r}_{\mathrm{i}}\)) and at the outer radius (\(\mathrm{r}_{\mathrm{e}}\)) of the cylinder, see Equation (3). \[ \begin{array}{l}\sigma_{\mathrm{r}}|_{\mathrm{r}=\mathrm{r}_{1}}=-\mathrm{ p}_{1}\\ \sigma_{\mathrm{r}}|_{\mathrm{r}=\mathrm{r}_{\mathrm{e}}}=0\end{array}  \tag{3}\]  
Thus, obtaining  
\[ \begin{array}{l}\mathrm{C}_{1}=-\frac{\mathrm{r}_{1}\mathrm{p}_{1}\mathrm{r }_{\mathrm{e}}^{\frac{1}{\lambda}}-\left(\mathrm{v}_{12}\lambda+1\right)}{ \lambda\cdot\mathrm{E}_{1}\left(\mathrm{r}_{\mathrm{e}}^{-\frac{1}{\lambda}} \mathrm{r}_{\mathrm{i}}^{\frac{1}{\lambda}}-\mathrm{r}_{\mathrm{i}}^{-\frac{1} {\lambda}}\mathrm{r}_{\mathrm{e}}^{\frac{1}{\lambda}}\right)}\\ \mathrm{C}_{2}=\frac{\mathrm{r}_{1}\mathrm{p}_{1}\mathrm{r}_{\mathrm{e}}^{- \frac{1}{\lambda}}\cdot\left(\mathrm{v}_{12}\lambda-1\right)}{\lambda\cdot \mathrm{E}_{1}\left(\mathrm{r}_{\mathrm{e}}^{-\frac{1}{\lambda}}\mathrm{r}_{ \mathrm{i}}^{\frac{1}{\lambda}}-\mathrm{r}_{\mathrm{i}}^{-\frac{1}{\lambda}} \mathrm{r}_{\mathrm{e}}^{\frac{1}{\lambda}}\right)}\end{array}  \tag{4}\]  
where \(\mathrm{v}_{12}\) is the major Poisson's ratio of the composite lamina. The circumferential stress (5) and the radial displacement (6), as sampled at the inner cylinder tube diameter where both the quantities peak, may be expressed as follows,  
\[\sigma_{\mathrm{\Earth|r=\mathrm{r}_{1}}}=\frac{1}{\lambda}\frac{\mathrm{r}_{ \mathrm{e}}^{\frac{2}{\lambda}}+\mathrm{r}_{\mathrm{i}}^{\frac{2}{\lambda}}}{ \mathrm{r}_{\mathrm{e}}^{\frac{2}{\lambda}}-\mathrm{r}_{\mathrm{i}}^{\frac{2}{ \lambda}}}\mathrm{P}_{1} \tag{5}\]  
\[\mathrm{u}_{\mathrm{r}}|_{\mathrm{r}=\mathrm{r}_{1}}=\frac{\mathrm{P}_{1}\cdot \mathrm{r}_{\mathrm{i}}}{\mathrm{E}_{1}}\cdot\left(\mathrm{v}_{12}+\frac{1}{ \lambda}\cdot\frac{\mathrm{r}_{\mathrm{e}}^{\frac{2}{\lambda}}+\mathrm{r}_{ \mathrm{i}}^{\frac{2}{\lambda}}}{\mathrm{r}_{\mathrm{e}}^{\frac{2}{\lambda}}- \mathrm{r}_{\mathrm{i}}^{\frac{2}{\lambda}}}\right)=\varepsilon_{\mathrm{ \Earth|r=\mathrm{r}_{1}}}\cdot\mathrm{r}_{\mathrm{i}} \tag{6}\]  
Formula (6) also recalls the relation between the circumferential strain (\(\varepsilon\)) component at the inner radius and the local radial displacement. \begin{table}
\begin{tabular}{c c} \hline \hline
**Reference Cylinder Tube** & **Structural Properties** \\ \hline Pi & 67 MPa \\ \(\sigma_{\mathrm{\Earth|r=\mathrm{r}i}}\) & 483 MPa \\ \(\mathrm{u}_{\mathrm{r}}|_{\mathrm{r}}=\mathrm{r}\) & 0.359 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Reference cylinder tube stress and radial deformation at the first-yield. In the problem under scrutiny, the inner radius and the operating pressure are imposed as design constraints, whereas the material and the outer radius are free variables and they are selected in order to ensure admissible values for (a) the peak stresses and (b) the radial compliance at the inner radius. In the context of a technical feasibility study, a limited set of materials was selected, see Section 4 below.