## 6 Conclusions
### Derive the Relationship between \(w\) and \(\varphi\) of the Spherical Shell
 \tag{102}\]  
\[B_{n}=-\frac{B_{n}^{\prime}\varphi\lambda_{n}[(\alpha+1)\lambda_{n}+2(v+ \varphi)]}{[-\alpha\lambda_{n}^{2}(1-\alpha)+(\varphi-2\alpha+\varphi\alpha) \lambda_{n}+v(3\varphi-2\alpha)+\varphi+v^{2}-\alpha-1]\lambda_{n}}, \tag{103}\]  
After canceling \(\lambda_{n}\), and neglecting the small quantities as \(\alpha\), \(\varphi\), and their products in comparison with unity, we obtain:  
\[B_{n}=-\frac{B_{n}^{\prime}\varphi[\lambda_{n}+2(v+\varphi)]}{-\alpha\lambda_ {n}^{2}+(\varphi-2\alpha+\varphi\alpha)\lambda_{n}+v^{2}-1}. \tag{104}\]  
Coming back to the definition of the displacement \(w\), one may write the equation \(w=\sum_{n=0}^{\infty}B_{n}P_{n}\)--or writing it in detail:  
\[ \begin{array}{ll}w=B_{0}P_{0}+B_{1}P_{1}+B_{2}P_{2}+\cdots\cdots\\ w=\big{[}B_{0}+0.25B_{2}+\frac{\varphi}{6t}B_{4}+\cdots\big{]}+\big{[}B_{1}+ \frac{3}{8}B_{3}+\cdots\big{]}\cos\theta+[B_{2}+\cdots]\cos 2\theta+\end{array}  \tag{105}\]  
here \(\lambda_{n}=n(n+1)-2\), which is the minimum for \(n=\frac{1}{2}\); therefore, it has the same values for \(n\) equals to minus one and zero. Since \(n\) must be an integer, it is chosen as zero, which yields \(\lambda=-2\), and corresponds to \(B_{0}\), which is a function of \(\lambda_{n}\), and gets smaller when \(\lambda_{n}\) becomes greater. Thus, it is possible to neglect all the terms and simply write \(w\cong B_{0}\) since the terms which contain \(\cos\theta\), \(\cos 2\theta\dots\) are much smaller--so, buckling is usually expected at the places where \(\theta\) is large [33]:  
\[w\cong\frac{B_{0}^{\prime}\varphi[-2+2(v+\varphi)]}{4\alpha+(\varphi-2\alpha+ \varphi\alpha)2+1-v^{2}}, \tag{106}\]  
\[w\cong\frac{2B_{0}^{\prime}\varphi(v+\varphi-1)}{2\varphi(1+\varphi)+1-v^{2}}, \tag{107}\]  
\[w\cong\frac{B_{0}^{\prime}(v-1)}{1+\frac{1-v^{2}}{2\varphi}}. \tag{108}\]  
Appendix B Deriving the Corrosion Rate of the Spherical Shell as a Function of Temperature and Stress, Based on the Arrhenius Type  
It is assumed that the shell is subjected to the simultaneous action of a constant external pressure \(Q\) and uniform internal corrosion. The rate of the thickness decrease at each moment of time \(t\) is equal to the corrosion rate [5]:  
\[\frac{dh}{dt}=-f(\sigma,T)\frac{dh}{dt}=-f(\sigma,T), \tag{109}\]where \(f(\sigma,T)\) is a sufficiently smooth function of compressive stress \(\sigma\), defined by Equation (A2), and the temperature \(T\). The corrosion rate dependence on the stress value, according to experimental data, can be approximated by exponential function suggested in [1], while the temperature effect is usually described by the Arrhenius-type law [11]. Combination of both functions yields the following relation Equation (A73). Temperature has a great effect on the rate of metal electrochemical corrosion. In the case of corrosion in a neutral solution (oxygen depolarization), the increase of the corrosion rate increases exponentially with temperature increase because the hydrogen evolution overpotential decreases. Experimental dependence of the Arrhenius type is observed between the corrosion rate and temperature. Using the current density \(j_{corr}\), we express the corrosion rate [1]:  
\[j_{corr}=\lambda\exp(E_{a}/R_{g}T).\] (A71)  
where \(E_{a}\) is the effective activation energy of the corrosion process in kJ mol\({}^{-1}\), \(R_{g}\) is the molar gas constant in J mol\({}^{-1}\) K\({}^{-1}\), T is the absolute temperature in \(K\), \(\lambda\) is the pre-exponential factor, and \(j_{corr}\) is the corrosion current density, A cm\({}^{-2}\). Equation (A71) provides the determination of the effective activation energy of the corrosion process. When \(\sigma=0\), \(E_{a}=E_{co}\), we have \(T=T_{0}\), and \(j_{corr}=\lambda\exp(E_{c0}/R_{g}T)\).