## 6 Conclusions
The authors ([PERSON] & [PERSON]) thank [PERSON] in USA and [PERSON] in Aberystwyth University for fruitful discussions. The author ([PERSON]) also thanks [PERSON] in the Department of Mathematics of Hefei University of Technology, [PERSON] and [PERSON] in the Department of Civil and Structural Engineering of the University of Sheffield for discussing in its earlier stage. **Conflicts of Interest:** The authors declare no conflict of interest. **Notation List** \begin{tabular}{l l} \(S\) & slope of \(w\) vs. \(w/p\) line \\ \(u\) & displacement of the shell element in x direction \\ \(v\) & displacement of the shell element in y direction \\ \(w\) & displacement of the shell element in z direction \\ \(U_{0}\) & effect of initial imperfections \\ \(V\) & shearing force in straight members in y direction (buckling coefficient to be \\  & determined experimentally) \\ \(\varepsilon_{x}\) & unit elongation or strain in \(x\)-direction \\ \(\varepsilon_{y}\) & unit elongation or strain in \(y\)-direction \\ \(\varepsilon_{y}\) & unit elongation of middle surface in \(x\)-direction \\ \(\varepsilon_{2}\) & unit elongation of middle surface in \(y\)-direction \\ \(\mu\) & Poisson's ratio \\ \(\chi_{y}\) & change of curvature in x-direction \\ \(\chi_{x}\) & change of curvature in y-direction \\ \(t_{0}\) & thickness of the shell \\ \(p_{cr}\) & buckling pressure \\ \(H()\) & mathematical operator \\ \(N_{x},N_{y}\) & resultant forces \\ \(Q_{x},Q_{y}\) & shear forces \\ \(M_{x}\) & My bending moments \\ \(p_{o}\) & outer pressure \\ \(v_{i}\) & inner mechano-chemical corrosion rate \\ \(t\) & time \\ \(h\) & thickness \\ \(\sigma\) & principal stress \\ \(\sigma_{e}\) & effective stress \\ \(r\) & distance between a point in the shell material and the origin of the coordinate \\  & system/radius of two concentric spheres \\ \(r_{o}\) & distance between the point in the outer shell surface and the origin of the coordinate \\  & system \\ \(b\) & corrosion inhibition effect \\ \(\mathrm{d}r\) & radius of two concentric spheres \\ \(\mathrm{d}\theta,\mathrm{d}\varphi\) & top angles of four wedge-shaped sections \\ \(r_{c}\) & midsurface radius \\ \end{tabular} \begin{tabular}{l l} \(x\) & thickness to midsurface radius ratio \\ \(t^{*}\) & time required for a corroded pressure shell to fail for the first time due to buckling \\  & or yielding \\ \(h\) & corresponding thickness of the shell under the critical failure state \\ \end{tabular}   
Appendix A Buckling Formulas for Spherical Shells in the Case of Section 2.1 (without Corrosion and Temperature)  
_Appendix A.1. Derivation of the Second-Order Critical Buckling Load (Stress)_  
Using the above derived angles instead of the initial ones, \(d\theta\), \(\sin\theta d\psi\), and \(\cos\theta d\psi\), the equations of equilibrium of the element \(ABCD\) become [33]:  
\[w+7.27385\;w/p=16.59268,\] (A1)  
\[\frac{dQ_{x}}{d\theta}+Q_{x}\cot\theta+N_{x}+N_{y}+pR+N_{x}\bigg{(}\frac{d^{2 }w}{Rd\theta^{2}}+\frac{du}{Rd\theta}\bigg{)}+N_{y}\bigg{(}\frac{u}{R}+\frac{ dw}{Rd\theta}\bigg{)}\cot\theta=0,\] (A2)  
\[\frac{dM_{x}}{d\theta}+\big{(}M_{x}-M_{y}\big{)}\cot\theta-Q_{x}R+M_{y}\bigg{(} \frac{u}{R}+\frac{dw}{Rd\theta}\bigg{)}=0.\] (A3)  
Figure 11: Hemispherical shells and one-half analysis model. Figure 12: Meridian of a spherical shell before and after buckling. Here, \(u\) is the displacement of the shell element in \(x\) direction, \(v\) is the displacement of the shell element in \(y\) direction, \(w\) is the displacement of the shell element in \(z\) direction, \(t_{0}\) is the thickness of the shell, \(p_{\sigma}\) is the classical buckling pressure, \(N_{x}\), \(N_{y}\) are the resultant forces, \(Q_{x}\), \(Q_{y}\) are the shear forces, \(M_{x}\), \(M_{y}\) are the bending moments, and \(\theta\), \(\psi\) are the angles of the shell element.