## 3 Problem Solving  
In order to solve the problems highlighted in this paper (see Section 2), we run the following two steps to explain our methods. First Step: Establish a Non-Destructive Method for Predicting Spherical Shell Life Regardless of Corrosion and Temperature  
From (A50) in Appendix A.1, we rewrite the it as:  
\[\frac{2 Em}{1-v^{2}}=\frac{p}{\varphi} \tag{7}\]  
Here, \(m\) is the ratio of the thickness to the radius of the sphere. We rewrite Equation (A51) in Appendix A.1 as:  
\[p_{cr}^{2}=Em.Em.m^{2}\Bigg{(}\frac{2}{\sqrt{3(1-v^{2})}}-\frac{mv}{1-v^{2}} \Bigg{)}^{2}, \tag{8}\]  
here, \(m=\frac{h_{0}}{R}\). Let \(\Delta=\frac{2}{\sqrt{3(1-v^{2})}}-\frac{mv}{1-v^{2}}\), from Equations (7) and (8), we can obtain:  
\[\frac{p}{\phi}=\frac{2 Em}{1-v^{2}}=\frac{2p_{cr}^{2}}{Em^{3}(1-v^{2})\Delta^{2}} \tag{9}\]  
and then  
\[\frac{1}{\phi}=\frac{2 Em}{1-v^{2}}=\frac{2p_{cr}^{2}}{pEm^{3}(1-v^{2})\Delta^ {2}} \tag{10}\]  
By substituting Equation (10) into Equation (A69) in Appendix A.2, we get:  
\[w\cong\frac{B_{0}^{\prime}(v-1)}{1+\frac{1-v^{2}}{2}.\frac{2p_{cr}^{2}}{pEm^{ 2}(1-v^{2})\Delta^{2}}} \tag{11}\]  
As can be seen from the equations above, we have successfully established the relationship between the displacement and the pressure of the shell by appropriately rewriting the form of the second-order critical load and combining the relationship between the displacement \(w\) and the rotation angle \(\varphi\) shown in Equation (5). By cross-multiplying in Equation (11), we can achieve:  
\[w+\frac{p_{cr}^{2}}{Em^{3}\Delta^{2}}\frac{w}{p}=B_{0}^{\prime}(v-1). \tag{12}\]  
Formally, Equation (12) is the equation of a straight line following the Southwell procedure described in Section 2.1. It has \(w\) as one axis and \(w/p\) as another. Furthermore, the expression of the slope of this line contains the unknown critical load \(p_{cr}\). Therefore, the slope of this line can be gained experimentally, and the critical load can then be obtained. Let \(S=\frac{p_{cr}^{2}}{Em^{3}\Delta^{2}}\); Equation (12) can be rewritten as:  
\[w+S\frac{w}{p}=B_{0}^{\prime}(v-1) \tag{13}\]one can get:  
\[p_{cr}=m\Delta\sqrt{SEm} \tag{14}\]  
By substituting Equation (A53) in Appendix A.1 into Equation (7), we get:  
\[\phi=\frac{{{({h^{*}})}^{2}}\sqrt{\frac{{1-{v}^{2}}}{3}}}{Rh}. \tag{15}\]  
Substituting Equation (15) into (A69) in Appendix A, we get:  
\[w+J(wh)=B_{0}^{\prime}(v-1), \tag{16}\]  
where:  
\[p_{cr}^{2}=Em.Em.m^{2}\Bigg{(}\frac{2}{\sqrt{3(1-v^{2})}}-\frac{mv}{1-v^{2}} \Bigg{)}^{2}, \tag{8}\]0  
then we obtain:  
\[p_{cr}^{2}=Em.Em.m^{2}\Bigg{(}\frac{2}{\sqrt{3(1-v^{2})}}-\frac{mv}{1-v^{2}} \Bigg{)}^{2}, \tag{8}\]1  
With the previous steps, we have obtained a non-destructive estimation of the critical thickness (see Equation (18)) in the same way that we have used to solve the critical load above (see Equation (14)). Second Step: Establish a Non-Destructive Method for Predicting Critical Load, Critical Thickness, and Service Life of Spherical Shells in the Presence of Corrosion and Temperature  
According to the description in Section 2.2, this case requires an additional corrosion-rate equation than in Section 2.1--see (A77) in Appendix B. Considering Equation (10), from Equation (A77) we obtain:  
\[p_{cr}^{2}=Em.Em.m^{2}\Bigg{(}\frac{2}{\sqrt{3(1-v^{2})}}-\frac{mv}{1-v^{2}} \Bigg{)}^{2}, \tag{8}\]2  
then:  
\[p_{cr}^{2}=Em.Em.m^{2}\Bigg{(}\frac{2}{\sqrt{3(1-v^{2})}}-\frac{mv}{1-v^{2}} \Bigg{)}^{2}, \tag{8}\]3  
and finally:  
\[p_{cr}^{2}=Em.Em.m^{2}\Bigg{(}\frac{2}{\sqrt{3(1-v^{2})}}-\frac{mv}{1-v^{2}} \Bigg{)}^{2}, \tag{8}\]4  
Note that in this section we still have \(p_{cr}=m\Delta\sqrt{SEm}\), \(\sigma^{\star}=\frac{mR\lambda\sqrt{SEm}}{2h_{0}}\), and \(h^{\star}=\frac{\sqrt{R}\frac{4}{3}\sqrt{3(1-v^{2})}}{\sqrt{2f}}\).