A Non-Destructive Method for Predicting Critical Load, Critical Thickness and Service Life for Corroded Spherical Shells under Uniform External Pressure Based on NDT Data

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This work can be potentially valuable to be used as a reference for existing estimating methods based on NDT.

Abstract

A pressurized spherical shell that is continuously corroded will likely buckle and lose its stability. There are many analytical and numerical methods to study this problem (critical load, critical thickness, and service life), but the friendliness (operability) in engineering test applications is still not ideal. Therefore, in this paper, we propose a new non-destructive method by combining the Southwell non-destructive procedure with the stable analysis method of corroded spherical thin shells. When used carefully, it can estimate the critical load (critical thickness) and service life of these thin shells. Furthermore, its procedure proved to be more practical than existing methods; it can be easily mastered, applied, and generalized in most engineering tests. When used properly, its accuracy is acceptable in the field of engineering estimations. In the context of the high demand for non-destructive analysis in industry, it may be of sufficient potential value to be used as a reference for existing estimating methods based on NDT data.

1. Introduction

Due to the increasing use of shell-type structures in spacecrafts, submarines, buildings, and storage tanks, there has been a corresponding increase in the interest of researchers and practical engineers in the stability of shells. Hemispherical shells are the most important structural element in engineering applications because they can resist higher pure internal pressure loads than any other geometric vessel with the same wall thickness and radius.

In practice, most pressure vessels experience external loads due to hydrostatic pressure or external shocks-. Therefore, they should be designed to withstand the worst load combinations without failure. Loads transmitted by cylindrical rigid actuators applied on top of the sphere are considered common external loads. Therefore, it is important to study its effect on the initial buckling behaviour of such shells. Meanwhile, corrosion is defined as the gradual destruction of a material due to chemical reactions within the environment. The most common type of corrosion is uniform corrosion or general corrosion, which is distributed almost uniformly over the entire exposed surface. General wear can occur both with the formation of a fully protective ultra-thin coating of corrosion products and without an oxide layer. The formation of a blocking passivation film, as well as changes in the concentration of one or the other reactants, may inhibit when the corrosion rate should decay exponentially (decline) with time [1,2,3].

On the other hand, as with other types of damage (e.g., [4]), corrosion can be enhanced by the applied load [5]. Experimental data suggests that there is a stress corrosion threshold, after which mechanical stress accelerates corrosion [5,6,7]. In this case, the stress changes due to the reduction in shell thickness, and the changed stress in turn enhances the corrosion process. In general, for the strength analysis of structural elements under mechano-chemical corrosion conditions, an initial boundary value problem with unknown evolutionary boundaries must be solved.

In addition to stress, there are many other effects that can affect the corrosion rate, such as temperature; it has a great influence on the rate of galvanic corrosion of metals. In the case of neutral-solution corrosion (oxygen depolarization), elevated temperature has a favourable effect on the overpotential and oxygen diffusion rate for oxygen depolarization but leads to a decrease in oxygen solubility. When corrosion (hydrogen depolarization) occurs in acidic media (such as sea water), the corrosion rate increases exponentially with increasing temperature due to the reduced hydrogen evolution overpotential. An Arrhenius-type experimental dependence was observed between corrosion rate and temperature [8]. The effect of temperature on acid corrosion, most commonly in hydrochloric and sulfuric acid, has been the subject of extensive research [9,10,11,12,13,14,15,16,17,18,19,20,21,22]. In hydrochloric acid, the effective activation energies for corrosion processes vary from 57.7 to 87.8 kJ mol/L, where most are concentrated around 60.7 kJ mol/L. In some cases, studies were performed at only three temperature values using a single experimental method, which increased the likelihood of erroneous determination of the corrosion activation energy. In this regard, further research is advisable, as it may provide a reliable comparative basis for discussing the obtained results.

In summary, we know that temperature has a great influence on corrosion rate, corrosion and stress can interact with each other, and finally, they can jointly affect the stability of the shell. In this regard, we should explore the relationships of temperature, corrosion, and stress to shell stability. Experimental [23], analytical [24,25,26,27,28], and numerical methods [29,30,31] were used to study the buckling of (uniformly compressed hemispherical of moderate thickness) metal shells using corrosion and temperature. A related study [28] also demonstrated the high accuracy of these methods. However, in practical engineering applications, they often lack operability and are less friendly to workers.

Non-destructive estimation methods, such as in the field of pressure vessels, have been a research hotspot due to their advantages of simple operation, widespread use, and low cost. As one of the non-destructive methods, the non-destructive approach of Southwell’s column analysis is now extended to spherical shells, which are subjected to uniform external pressure [32]. However, there are few non-destructive methods for estimating the buckling of pressure spherical shells of this type. The paper [33] established a non-destructive estimation method for a spherical shell under external pressure that can predict its critical load. It is based on an exact first-order solution of the critical stress and requires two assumptions: one is that the thickness of the shell does not change; and the second is that the temperature does not affect the critical buckling of the shell. In any case, when the shell is in a corrosive environment, the thickness of the shell varies and the effect of temperature on both the stress and the corrosion rate is unavoidable. No non-destructive method has been found as of yet to predict the critical loads (stresses) of the shell under this condition, including the service life. Therefore, in the industry, people still hope to obtain a new lossless method. This method can predict the critical load (or stress) and critical thickness of the shell under external pressure from corrosion on the basis of a higher-order (second-order) exact solution. At the same time, we are also oriented to predict its useful (remaining) life. For this reason, in this paper, we will analytically extend the Southwell process method for the non-destructive prediction of critical loads, critical stresses, and service life of hemispherical shells subjected to uniform external pressure considering corrosion and ambient temperature through rigorous mathematical derivation.

2. Problem Description

A model of a spherical shell is considered. It is affected by the external pressure p_o, and its inner surface undergoes mechano-chemical corrosion, as shown in Figure 1. Assuming that the rate of internal corrosion is v_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.

2.1. 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):

$$\frac{d{N}_x}{d\theta }+\left(N_x-N_y\right)\mathit{cot}\theta -Q_x+N_y\left(\frac{u}{R}+\frac{dw}{Rd\theta }\right)-Q_x\left(\frac{d^{2}w}{Rd{\theta }^2}+\frac{w}{R}\right)=0,$$

(1)

$$\frac{d{Q}_x}{d\theta }+Q_x\mathit{cot}\theta +N_x+N_y+pR+N_x\left(\frac{d^{2}w}{Rd{\theta }^2}+\frac{du}{Rd\theta }\right)+N_y\left(\frac{u}{R}+\frac{dw}{Rd\theta }\right)\mathit{cot}\theta =0,$$

(2)

$$\frac{d{M}_x}{d\theta }+\left(M_x-M_y\right)\cot \theta -Q_{x}R+M_y\left(\frac{u}{R}+\frac{dw}{Rd\theta }\right)=0,$$

(3)

$$\sigma =\frac{pR}{2h_0}$$

(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, t₀ is the shell thickness, p_cr 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.

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 $\frac{p_{cr}^2}{E{m}^3{\Delta }^2}$. 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. At the same time, it does not assume the specific location of buckling and the number of buckling waves, as shown in Figure 2.

2.2. Then, on the Basis of the Previous Step 2.1, We Can Further Solve the Problem of Critical Load (Stress) and Service Life in an Environment Where Corrosion and Temperature Coexist

The mathematical problem that needs to be solved is similar to that in Section 2.1, but an extra equation (corrosion kinetic equation) needs to be taken into consideration—the relationships between the corrosion rate (that is, the derivative of thickness), temperature, and stress:

$$\frac{dh}{dt}=-f\left(\sigma ,T\right)\frac{dh}{dt}=-f\left(\sigma ,T\right)$$

(5)

here, for f (σ, T) we adopt:

$$f\left(\sigma ,t\right)=v_0\mathit{exp}(\frac{V\sigma }{R{T}_g})$$

(6)

where h is the thickness of the shell, t is the corrosion time, $v_0$ is the initial corrosion rate, $\sigma$ is the stress, T is the temperature, $R_g$ is the molar gas constant, and V is the material molar volume.

It is noted that the derivation process of Equation (6) is shown in Appendix B, where we found that the relationship between corrosion rate and temperature conforms to the Arrhenius type [5]. Meanwhile, it should be observed that according to the physical definition of shell buckling, the critical thickness in the case described in this section will be identical to its expression obtained in Section 2.1.

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.

3.1. 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:

$$\frac{2Em}{1-v^2}=\frac{p}{\phi }$$

(7)

Here, m is the ratio of the thickness to the radius of the sphere.

We rewrite Equation (A51) in Appendix A.1 as:

$$p_{cr}^2=Em.Em.m^2{\left(\frac{2}{\sqrt{3\left(1-v^2\right)}}-\frac{m\nu }{1-v^2}\right)}^2,$$

(8)

here, $m=\frac{h_0}{R}$.

Let $\Delta =\frac{2}{\sqrt{3\left(1-v^2\right)}}-\frac{m\nu }{1-v^2}$, from Equations (7) and (8), we can obtain:

$$\frac{p}{\varphi }=\frac{2Em}{1-v^2}=\frac{2p_{cr}^2}{E{m}^3\left(1-v^2\right){\Delta }^2}$$

(9)

and then

$$\frac{1}{\varphi }=\frac{2Em}{1-v^2}=\frac{2p_{cr}^2}{pE{m}^3\left(1-v^2\right){\Delta }^2}$$

(10)

By substituting Equation (10) into Equation (A69) in Appendix A.2, we get:

$$w\cong \frac{B_0^{\prime }\left(v-1\right)}{1+\frac{1-v^2}{2}.\frac{2p_{cr}^2}{pE{m}^3\left(1-v^2\right){\Delta }^2}}$$

(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 φ shown in Equation (5).

By cross-multiplying in Equation (11), we can achieve:

$$w+\frac{p_{cr}^2}{E{m}^3{\Delta }^2}\frac{w}{p}=B_0^{\prime }\left(v-1\right).$$

(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 $S=\frac{p_{cr}^2}{E{m}^3{\Delta }^2}$; Equation (12) can be rewritten as:

$$w+S\frac{w}{p}=B_0^{\prime }\left(v-1\right)$$

(13)

one can get:

$$p_{cr}=m\Delta \sqrt{SEm}$$

(14)

By substituting Equation (A53) in Appendix A.1 into Equation (7), we get:

$$\varphi =\frac{{\left(h^{*}\right)}^2\sqrt{\frac{1-v^2}{3}}}{Rh}.$$

(15)

Substituting Equation (15) into (A69) in Appendix A, we get:

$$w+J\left(wh\right)=B_0^{\prime }\left(v-1\right),$$

(16)

where:

$$J=\frac{R\sqrt{3\left(1-v^2\right)}}{2{\left(h^{*}\right)}^2},$$

(17)

then we obtain:

$$h^{*}=\frac{\sqrt{R}\sqrt[4]{3\left(1-v^2\right)}}{\sqrt{2J}}.$$

(18)

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)).

3.2. 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:

$$\frac{d\sigma }{dt}=\frac{2{\sigma }^2}{pR}{v}_0\exp \left({\overline{E}}_{c0}\left(1-\frac{{\overline{E}}_c}{\overline{T}}\right)\right)\exp \left(\frac{V\sigma }{R_{g}T}\right).$$

(19)

Through performing variable separation on Equation (19) in $\left[t_0,t^{*}\right], \left[{\sigma }_0,{\sigma }^{*}\right],$ we get:

$${\sigma }^{-2}\exp \left(-\frac{V\sigma }{R_{g}T}\right)d\sigma =\frac{2}{pR}{v}_0\exp \left({\overline{E}}_{c0}\left(1-\frac{{\overline{E}}_c}{\overline{T}}\right)\right)dt,$$

(20)

where, t₀ refers to the initial time, t* to the service life, σ₀ to the stress at time t₀, and σ^* to the critical stress.

Integrating Equation (20), we get:

$${{\displaystyle \int }}_{{\sigma }_0}^{{\sigma }^{*}}\exp \left(-\frac{V\sigma }{R_{g}T}\right){\sigma }^{-2}d\sigma =\frac{2}{pR}{v}_0\exp \left({\overline{E}}_{c0}\left(1-\frac{{\overline{E}}_c}{\overline{T}}\right)\right){{\displaystyle \int }}_{t_0}^{t^{*}}dt.$$

(21)

Considering that Equation (10) establishes consistency, through $\sigma \equiv \frac{pR}{2h}$, the service life t^* can be obtained:

$$t^{*}=\frac{h_0{\sigma }_0}{{\upsilon }_0}\left[-{\overline{E}}_{co}(1-{\overline{E}}_c/\overline{T}\right]{{\displaystyle \int }}_{{\sigma }^0}^{{\sigma }^{*}}{\sigma }^{-2}\mathit{exp}(\frac{-V\sigma }{R_{g}T})d\sigma .$$

(22)

If we still consider Equation (10), Equation (A77) can be transformed into:

$$\frac{dh}{dt}=v_0\exp \left({\overline{E}}_{c0}\left(1-\frac{{\overline{E}}_c}{\overline{T}}\right)\right)\exp \left(\frac{VpR}{2R_{g}Th}\right).$$

(23)

In order to separate the variables t and h in [t₀, t*] and [h₀, h*], first we get:

$$\exp \left(-\frac{VpR}{2R_{g}Th}\right)dh=v_0\exp \left({\overline{E}}_{c0}\left(1-\frac{{\overline{E}}_c}{\overline{T}}\right)\right)dt,$$

(24)

then:

$${{\displaystyle \int }}_{h_0}^{h^{*}}\exp \left(-\frac{VpR}{2R_{g}Th}\right)dh=v_0\exp \left({\overline{E}}_{c0}\left(1-\frac{{\overline{E}}_c}{\overline{T}}\right)\right){{\displaystyle \int }}_{t_0}^{t^{*}}dt,$$

(25)

and finally:

$$t^{*}=\frac{1}{v_0\exp \left({\overline{E}}_{c0}\left(1-\frac{{\overline{E}}_c}{\overline{T}}\right)\right)}{{\displaystyle \int }}_{h_0}^{h^{*}}\exp \left(-\frac{VpR}{2R_{g}Th}\right)dh+t_0.$$

(26)

Note that in this section we still have $p_{cr}=m\Delta \sqrt{SEm}$, ${\sigma }^{*}=\frac{mR\Delta \sqrt{SEm}}{2h_0}$, and $h^{*}=\frac{\sqrt{R}\sqrt[4]{3\left(1-v^2\right)}}{\sqrt{2J}}$.

4. Example Analysis

The whole process described above (including those in Appendix A and Appendix B), showed that the non-destructive method of this paper used mathematical logic rigorously. To further validate it, we compare its results with those of other methods on this section. Due to the limitation of NDT data found in literature on these kinds of shells, we compare our method within a special case in Section 2.1 (without corrosion and temperature) with another existing method.

The NDT data adopted here is taken from an experiment described in [33]. To maintain consistency with the experimental model of [33], we also apply internal suction to our FE model to simulate equivalently the external pressure. The geometric parameters for the FE model (meeting thin wall hypothesis) are: the radius of shell r = 0.05 m and the wall thickness h₀ = 0.0005 m, yielding a h₀/r ratio of 0.01. The mechanical properties are: the Young modulus E = 650 MPa, Poisson ratio ν = 0.4, and density ρ = 1150 kg/m³.

A 1/8 symmetrical spherical shell model built by the ANSYS software is shown in Figure 3. The first two buckling modes obtained from the eigenvalue buckling analysis are shown in Figure 4.

First, we calculate the value of $p^{*}$ using the described method. The results for w and w/p are plotted in Figure 5.

Within the plot, it is shown that the fitting line equation for the w and w/p points is:

w + 7.27385 w/p = 16.59268,

(27)

therefore, we get S = 7.27385 by Equation (13), and then obtain $p^{*}$ = 0.08593 through Equation (14).

In order to verify the practicality and accuracy of this method, we also calculate the $p^{*}$ value adopting the other five existing methods, including one-order analytical value, numerical value, two-order analytical value, etc., which are shown in

Table 1. Critical load of our method comparing with other methods.

Method No.	1st	2nd	3rd	4th	5th	6th
Method name	The lowest Eigenvalue method	Traditional one-order analytical method	Non-destruction	Analytical two-order method [33]	Numerical method [33]	Experimental method [33]
			Our method
Critical load value $p^{*}$	0.081574	0.08127	0.08593	0.08189	0.08790	0.07800
Formulas	$\left[K\right]+{\lambda }_i\left[S\right])\left\{{\psi }_i\right\}=0.$ Here [K],[S] are the constants, ${\lambda }_i$ is the buckling load multiplier, $\left\{{\psi }_i\right\}$ is the buckling mode.	$p_{cr}=\frac{2E{m}^2}{\sqrt{3\left(1-v^2\right)}}$	$p_{cr}=m\Delta \sqrt{SEm}$	$p_{cr}=\frac{2E{m}^2}{\sqrt{3\left(1-v^2\right)}}-\frac{E{m}^{3}v}{\left(1-v^2\right)}$	Ansys FEM software.	Non-destructive Testing
Relative Error	0.03718	0.04193	0.10167	0.04988	0.12692	Reference

. The highest error is produced from the numerical method of [33], while the lowest error is produced from the lowest eigenvalue method.

Because the error values from our method, when used correctly, are within the range of the other approaches, its accuracy can be considered acceptable in the field of engineering. Besides that, compared with other methods, there is no doubt that our method is the most practical and easiest one to operate in engineering.

From Table 1, we also note that the experimental values are the smallest relative to any other method (numerical methods of various orders, analytical methods, etc.). There may be several reasons to explain this behavior: for example, the roundness of the actual shell (not ideal), but with geometric imperfections; also, the realistic boundary support is not a 100%-hinged support or a 100%-fixed support either. However, we use the 100%-hinged support in the numerical model for simplification.

5. Practical Implementation of This Method

This method is friendly to inspectors working for practical projects and easy to implement. Its general steps are as follows: basically, we should apply an external uniform pressure (or equivalent uniform pressure) p by n times to the shell. Here, $p$ = $p_0$ + $\Delta p_i$, i = 1, 2, 3, …, n. At the same time, we record the deformation $w_i$ after each loading. Here, the parameter S can be obtained through Equation (13). Then the $p_{cr}$ is obtained by Equation (14). Then by the same way, we can obtain parameter J by Equation (17), and accordingly obtain $h^{*}$ by Equation (18), and finally obtain $t^{*}$ by Equation (26).

6. Conclusions

The highlight of this study is that it proposes a non-destructive method based on the Southwell procedure to estimate the critical load, critical thickness, and service life of internally corroded shells under external pressure while considering the temperature’s effect. Of course, it is also applicable for the shell in special cases, such as industrial environments without corrosion or temperature.

Based on rigorous theories (shell stability theory, the Southwell plot method, and corrosion dynamic theory), and following a scientific route from simple to complex, we derived and acquired our new method step-by-step. We first derived the critical load and critical thickness of the spherical shell in the absence of corrosion and temperature using the Southwell procedure method. Second, we derived non-destructive methods for critical load, critical thickness, and service life prediction under corrosion and temperature conditions. Third, we compared our method with other methods.

The results show that, if used properly, engineering precision requirements can be met. Furthermore, when the results are carefully interpreted, this technique provides useful estimates of elastic buckling loads (critical thickness and service life). The utility of this approach lies in the fact that it is versatile, simple, and non-destructive. Furthermore, it does not require any assumptions about the buckling wave number or the precise location of buckling.

It has to be pointed out that since the service life of a spherical shell is usually very long (and so is its corrosion process), NDT data for the shells that are regularly detected, especially those collected regularly during the shell corrosion process, and fully recorded and published data, has not been found. A very small part of it was found in some data from experiments, which is employed in this paper in Section 4. This is why the method in this paper cannot be successfully verified in more working conditions currently. However, the authors will try to obtain (retrieve data from peer industries) NDT data under more operating conditions to fully evaluate the non-destructive prediction ability of this method for the shell service life (such as serving in corrosive and temperature environments) in the near future.

In summary, the highlight of this method in evaluating critical loads (thickness, lifetime) for shells is that it is non-destructive. Additionally, it is the first time introducing the Southwell plot method into the stabilization analysis of shells of time-varying thickness. Its precision meets engineering requirements, and more importantly, it is practice- and implementation-friendly.