## 2 Problem Description  
A model of a spherical shell is considered. It is affected by the external pressure \(p_{\rm o}\), and its inner surface undergoes mechano-chemical corrosion, as shown in Figure 1. Assuming that the rate of internal corrosion is \(v_{\rm o}\), the corrosion process causes the thickness \(h\) of the shell to change with time \(t\). We adopt the effective stress definition to characterize stress here, which is commonly used when corrosion is present. The problem in this paper is to predict (estimate) the critical thickness, critical stress, and service life of the spherical shell based on non-destructive testing data in a temperature and corrosive environment. This is essentially a non-destructive estimation problem involving variable boundary conditions. For clarity, we describe this complex and combined problem in two steps. Before All, It Is Necessary to Solve the Estimation of the Critical Thickness, Critical Stress, and Service Life of the Shell Based on NDT Data in a Non-Corrosion and Temperature-Independent Environment  
It involves two sub-problems. First, theoretically, we need to obtain the second-order buckling critical stress for the spherical shell model (see Appendix A), which is the accurate solution. Then, based on the format of this second-order solution, a test data-based estimation method for the shell critical stress will be established by introducing an existing non-destructive method (see [33]). The mathematical problem involved in obtaining the second-order buckling critical stress is related to the solution of the following Equations (1)-(4):  
\[\frac{dN_{x}}{d\theta}+\big{(}N_{x}-N_{y}\big{)}{\rm cot}\,\theta-Q_{x}+N_{y} \bigg{(}\frac{u}{R}+\frac{dw}{Rd\theta}\bigg{)}-Q_{x}\bigg{(}\frac{d^{2}w}{Rd \theta^{2}}+\frac{w}{R}\bigg{)}=0, \tag{1}\]  
\[\frac{dQ_{x}}{d\theta}+Q_{x}{\rm cot}\,\theta+N_{x}+N_{y}+pR+N_{x}\bigg{(} \frac{d^{2}w}{Rd\theta^{2}}+\frac{du}{Rd\theta}\bigg{)}+N_{y}\bigg{(}\frac{u} {R}+\frac{dw}{Rd\theta}\bigg{)}{\rm cot}\,\theta=0, \tag{2}\]  
\[\frac{dM_{x}}{d\theta}+\big{(}M_{x}-M_{y}\big{)}\,{\rm cot}\,\theta-Q_{x}R+M_{ y}\bigg{(}\frac{u}{R}+\frac{dw}{Rd\theta}\bigg{)}=0, \tag{3}\]  
\[\sigma=\frac{pR}{2h_{0}} \tag{4}\]  
where \(u\) is the displacement of the shell element in \(x\) direction, \(v\) is the displacement in \(y\) direction, \(w\) is the displacement in \(z\) direction, \(t_{0}\) is the shell thickness, \(p_{cr}\) is the classical  
Figure 1: A spherical shell subjected to both external pressure and internal corrosion. 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. When introducing the non-destructive method, the corresponding mathematical problem involved is how to express the relationship between non-destructively measurable quantities (such as: \(w\) and \(w/p\) at any point on the outer surface of the shell) as an equation of a straight line. The physical quantity to be estimated (e.g., the critical load \(p_{cr}\) of the shell) needs to appear exactly in the expression of the line slope. Assuming that we can obtain this equation line by non-destructive testing in the elastic stage, the quantity to be estimated can be obtained immediately. For example, let us take Equation (12) in which the equation slope is \(\frac{p_{cr}^{2}}{E\pi\beta^{2}}\). Then suppose we get the slope by testing and drawing; therefore, the only unknown quality in it, \(p_{cr}\), can be expressed (obtained) by the slope easily. Moreover, the assumptions within this non-destructive method are that the shell deformation is axisymmetric and the compression in the shell is uniform.