Alloy 600, a nickel-chromium alloy, has an outstanding corrosion resistance with excellent fabricability and is used in light water reactors at elevated temperatures. The alloy is also being considered for an advanced reactor concept because of its high allowable design strength at the elevated temperature. Alloy 600 is a power hardening material and basic plastic properties of the alloy are changed in the welded zone due to inhomogeneity in weld joints. The extended finite element method (XFEM) is used when the problem of variations invariably in the stress intensity factors (K) at a different instant rate exists. This paper focuses on the effect of variations in macrostructural properties of the alloy on stress corrosion cracking plastic zone ahead of the crack-tip using XFEM. To control the variations in the K, a new technique is also introduced in this research. The results show that the plastic zone is affected by K (increases with the increase of K), yield strength (plastic zone decreases with the increase in yield strength), and hardening exponent “n” (core region increases with the increase of exponent) of the materials. Simulations were performed and results are compared with experimental data.

## INTRODUCTION

Stress corrosion cracking (SCC) is a slow and stable crack propagation process in welded joints. Most authors^{1 } define SCC as a conjoint contribution of stresses, environment, and chemical action to the failure of metals. The necessary characteristics conditions for SCC are: tensile stresses, a specific environment, an environmental susceptibility of the material (metallurgical aspect), and time.^{2-3 } SCC is a cause of failure in the welded zones of different materials in light water reactors (LWRs).^{4-5 } Normal plastic strain rate around the crack-tip zone (ε_{ct}) is the main force acting on oxide film rupture around the crack-tip in the quantitative model of SCC crack growth rate.^{6 } The plastic region/zone around the crack-tip, in which the crack propagates, is also called the core region,^{7 } whose range is affected by the variations in the macrostructural properties of the material. The material properties are changed in the welded zone due to inhomogeneity.

### The Stress Corrosion Cracking Quantitative Model

The SCC crack growth rate F-A model based on the slip dissolving film and oxidation was established by Ford and Anderson,^{8-9 } which can be expressed by Equation (1). where da/dt the SCC crack growth rate in mm/s; m is exponent of the current decay curve which is related to the corrosion potential, solution conductivity, and chromium depletion. is the strain rate at crack-tip; κ_{a} is the oxidation rate constant, which is determined by the environment and material in the vicinity of the crack-tip, and is given by: where M is the metal molecular weight; ρ is the metal density; F is Faraday’s constant; z is the change in charge during the oxidation process; i_{0} is the oxidation current density of the bare surface; t_{0} is the time before the onset of the current decay; and is the oxide film degradation strain.

Considering that it is difficult to obtain the strain rate at the crack-tip in the F-A model, a crack-tip strain rate model based on the theory of crack-tip strain gradient was deduced by Shoji and shown in Equation (3). The FRI model establishes the relationship between the crack-tip strain rate and macro-mechanical parameters,^{10} and has been adopted by many laboratories around the world. Meanwhile, engineering applications have also been achieved to some extent, but the crack-tip characteristic distance r_{0} in the model has not been given a clear definition.

To obtain the strain rate at the crack-tip, Xue, et al.,^{11 } proposed an alternative algorithm of the strain rate at the crack-tip, that is the plastic strain at r_{0} in front of the crack-tip, and is used to replace the strain at the crack-tip, as shown in Equation (4), where r_{0} is the distance from the front of the crack-tip.

The variation of the distance r will cause an increase of crack-tip strain when the crack is growing. where dε_{p}/da is the strain rate at a characteristic distance r_{0} in front of the growing crack-tip.

Substituting Equation (5) into Equation (1), the SCC growth rate at the crack-tip in a high-temperature water environment can be expressed as: where:

For a stable propagation crack-tip, Equation (5) can be expressed as Equation (8), and the calculation method of the tensile plastic strain rate at the crack-tip is shown in Figure 1.

Substituting Equation (8) into Equation (6) to obtain the expression of the SCC crack growth rate is shown in Equation (9).

It can be seen from the model that plastic zone (r) in front of the crack-tip is an important region for the calculations of the SCC crack growth rate.

The extended finite element method (XFEM) is a unique tool, and while the literature is full of crack simulations of SCC growth rate, XFEM has not been used much with SCC crack growth rate. The reason is XFEM cracks have constantly changing stress intensity factors (SIFs) at different instant rates. Zhong, et al.,^{12 } studied the intergranular SCC. Congleton and Shoji^{13-14 } studied reactors pressure vessel SCC at high temperatures and presented different theories about the SCC crack growth rate. Xue, et al.,^{15 } have studied the effect of single tensile overload on the SCC crack growth rate. None of the aforementioned researchers has used XFEM. In this paper, the XFEM is used with the new technique of controlling the SIF described in *Methods* section.

### The Extended Finite Element Method

The XFEM formulation is a powerful tool in numerical modeling and now widely used for studying crack propagation under different loading and environmental conditions without re-meshing.^{16-17 } Before the extensive use of the XFEM, the finite element method (FEM) had also been used in solving engineering structural problems. Meanwhile, the issues related to the discontinuity or singularity around the crack-tip were quite problematic.^{18 } Major problems with FEM were the mesh regeneration around the crack-tip to align the mesh with crack boundaries and stress concentration around the crack-tip required mesh refinement for accurate representation.^{19 } The accuracy of prediction and computational efficiency is negatively affected by mesh refinement and regeneration.^{20 } Belytschko and Black^{21 } introduced a new method for crack simulations with minimal mesh refinement named XFEM. Their methodology was developed based on the research of Melenk and Babuska.^{22 } Farukh, et al.,^{23 } proved the capability of the XFEM for predicting the crack growth rate. The methodology was based on the partition of unity theorem of conventional FEM.^{24 } The XFEM method enriched the conventional FEM mesh with a special shape function to take into account the displacement and discontinuity.

This method was considered well suited for crack propagation^{25 } and was implemented in commercial finite element analysis (FEA) codes such as ABAQUS^{†} and ANSYS^{†}.^{26-27 }

The partition of unity and enrichment function around the crack-tip in case of elastic-plastic analysis is important to discuss here. In XFEM, during the analysis, the enrichment function consists of the near-tip asymptotic functions that capture the singularity around the crack-tip and a discontinuous function that represents the jumps in the displacement across crack surfaces (see Figure 2).

The approximation for a displacement vector surface with the partition of unity enrichment^{28 } is: where N_{i}(X) is the usual nodal shape function; μ_{i} is the usual nodal displacement vector associated with the continuous part of the finite element solution; H(X) is the discontinuous jump functions across the crack surface, also named the Heaviside function, which is associated with the product of the nodal enriched degree of freedom vector (a_{i}); and the third term is the nodal enriched degree of freedom vector () and the associated elastic asymptotic crack-tip function (F_{a}[X]). The first term on the right-hand side applies to all of the nodes in the model, the second term applies to all of the nodes whose shape function support is cut by the crack interior, and the third term is only for those nodes whose shape function support is cut by crack-tip.^{29 } Where: where δ(x) is the normal distance from the crack or discontinuity function. The crack-tip singularity incorporating the elastic-plastic enrichment function is given^{30 } by:

## MATERIALS

Nickel-based Alloy 600 has been used in the LWRs at high temperatures in welded zones due to its excellent properties.^{31 } In the welded joints (dissimilar metal region) of LWRs, where the temperature is high, different defects may occur due to the particularity of the welding technology such as pores, cracks, inclusions, impermeability, composition gradients, inevitable residual stresses, and variations in the mechanical properties.^{32-33 } Cold bending is used mainly to manufacture the pipe used in LWRs. Different degrees of work hardening occurs due to an increase in the thickness of the pipe. The cold working can promote and accelerate the SCC crack growth rate in the LWRs.^{34 }

Alloy 600 is a power hardening material and its relationship with the stress and strain can be described by the Ramberg-Osgood model shown in Equation (13).

where ε is the total plastic strain; ε_{0} is the plastic strain at yield point; σ is the total stress; σ_{0} is the yield stress; α is the offset coefficient; and n is the strain hardening exponent.

The elastic properties of the Alloy 600 are shown in Table 1. For plastic properties, see Figure 3. The tensile true stress-strain curve is drawn by experimenting on the “Slow Strain Rate Stress Corrosion Testing Machine” as shown in Figure 4. The engineering values are converted into true values.

As the material properties are changed in the welded zone, to understand the effect of variations on the core region, yield strength was set to 300 MPa, 500 MPa, and 700 MPa, and the hardening exponent was set to 2, 4, and 6.

## METHODS

A simple plate of 100 mm × 50 mm with an embedded (XFEM) crack of 20 mm length was used as shown in Figure 5, along with the boundary conditions and loading direction.

The simulations for static loading were performed using XFEM. Three different static loads of 200 MPa, 300 MPa, and 350 MPa were applied to check the influence of the SIF on the core region.

### Control of Stress Intensity Factor

Researchers^{36-37 } have applied the SCC load in terms of constant values of the SIF while dealing with techniques other than XFEM. In the case of XFEM, SIFs are always changing at different instant rates.^{38-39 } However, the values of the SIFs can be controlled by introducing predefined stresses in the materials, see Figure 6, and can be estimated by using displacement extrapolation and stress extrapolation techniques.^{40-41 }

The predefined stresses are introduced in step I of the simulations and for the next step, the (*.odb) file is used as part 1 of the 2 required for the XFEM crack simulations, and the (*.odb) file also acts as the load case in the next step.

### Finite Element Model and Crack Simulations

A typical four-node finite element mesh (structured mesh) model with 111,848 elements and 227,700 nodes is shown in Figure 7(a), where the X-coordinate is parallel to the direction of the crack and the Y-coordinate is normal to the direction of the crack. A small subset in circular form with a 15 mm radius is generated around the crack-tip to visualize the crack growth rate clearly, see Figure 7(b). The crack simulation consists of three steps: crack initiation, crack propagation, and failure.^{42 } All of these steps are simulated in the ABAQUS 6.14^{†} software without any re-meshing near the crack-tip. The maximum principal stress criterion is used which is represented as follows: where represents the maximum allowable principal stress and the damage starts when the ratio becomes unity or greater than unity;^{26 } see Figures 8, 9, and 10. The damage evolution technique based on power law has been used to simulate crack propagation. The step time is 1 s with 0.001 increment size. The total time for the simulations is more than 10 h.

## RESULTS AND DISCUSSION

### Effect of Macrostructural Properties on Plastic Zone Size

The preliminary condition for the SCC crack growth rate is to keep the K almost constant. In the case of debonding crack, it is convenient to apply load in terms of constant K. That is the reason many authors have used the debonding crack mechanism while dealing with the SCC crack growth rate. Since the last decade, the fatigue crack simulations are normally performed using XFEM in the FEA software due to less computational load as compared to that in case of debonding. The current need is to shift SCC to the XFEM.^{43 }

Figure 11 represents the SIF variations at different applied loads concerning the distance from the crack-tip. It can be seen that the values of the SIFs in the case of XFEM are constantly changing. The necessary condition for the SCC crack growth rate is to keep K constant. To keep K an exact constant, a triangular load is required. Exact constant values of the K are hard to achieve practically. Slight variations in the K for SCC crack growth rate can be neglected. These values have been estimated by the stress extrapolation method. Constant values of the SIFs in the case of XFEM cannot be achieved. There are an immense increase and abnormality in the values of K in the case of XFEM. However, the values of the K can be controlled by using the technique in this paper, see section 3.1 *Control of Stress Intensity Factor* (almost constant as compared to the abnormal values of K in the case of XFEM, see Figure 6). The values of the SIF can be varied by varying the applied load. The larger the values of the applied load, the larger the overall values of the SIF. However, the SIF increases for the distance from the initial crack-tip (core region) which is quite understandable.

The values of K have been controlled and made constant by using the technique described in section 3.1 *Control of Stress Intensity Factor*. The slope of the controlled plot of K in Figures 6 and 11 (exaggerated view) are extracted directly from the FEA software. In view, these two figures (6 and 11) may appear to have a different slope, but not at all. To give a clear view, Figure 11 starts from 0.5 of the plastic zone, while Figure 6 has 0 as a starting point (see the starting point of controlled K in the figures). At the start, the slope is slightly different than that when the crack starts propagating.

Figure 12 represents the effect of the hardening exponent on the plastic core region. The plastic core region increases with the increase of the hardening exponent. For a comparatively larger value of the hardening exponent, the plastic core region at a constant value of the SIF is comparatively larger.

### Effect of Macrostructural Parameters on Normal Plastic Strain

Figure 13 represents the distribution of plastic strain from the distance of the crack-tip in the core region on different yield strength on the Alloy 600.

Obvious from Figure 13 that plastic zone size decreases by increasing the yield strength of the material. The trends are following the literature.^{44-}^{45 } At a constant distance from crack-tip, the larger the value of the yield strength, the smaller the value of the normal tensile plastic strain. The yield strength of the material is a function of the temperature and welding variables. In the weld joints due to inhomogeneity, the yield strength of the material decreases,^{46 } moreover, the temperature in the LWRs is too high, hence the plastic strain region increases rendering Alloy 600 a susceptible material to SCC.^{47 }

The tensile normal strain distribution concerning the crack-tip distance is shown in Figure 14. With an increase in the value of hardening exponent the values of plastic strain increase.

### Effect of Macrostructural Parameters on Quantitative Stress Corrosion Cracking Crack Growth Rate

The strain rate around the crack-tip (ε_{ct}) is the main parameter to the oxide film rapture around the crack-tip in the SCC crack growth rate.^{6 } The values of the strain gradients are calculated from Figures 11,12, and 13 and are used in Equation (9) to estimate the SCC crack growth rate. The value of the constant is 7.478 × 10^{−7}. The hydrochemical properties used for nickel-based Alloy 600 are shown in Table 2.

Figure 15 represents the SCC crack growth rate in the core region with different yield strengths. The larger the yield strength, the smaller the plastic core region (see Figure 13). The smaller the plastic zone size, the smaller the crack growth rate (see Figure 15).

Figure 16 shows the effect of the applied load on the SCC crack growth rate in the core region. Here the important point to understand is that the applied loads are merely the representation of the SIFs. By changing the values of the applied loads, the SIFs are varied, along with the size of the plastic core zone. Figure 16 represents the effect of SIFs. The variations in the SCC crack growth rate at the start is due to the direction that the XFEM tends to take due to abnormal values of the K. These variations can also be seen in the normal plastic strain. In other words, the variations in the normal plastic strain are the cause of variations in the SCC crack growth rate at the start. These variations are removed as the crack propagates further in the core region.

Figure 17 is the representation of the SCC crack growth rate variations due to a change in hardening exponent values. It can be seen that the SCC crack growth rate increases with an increase in the hardening exponent values. This is because of the normal plastic strain increase in the plastic core region with an increase in exponent values.

Figure 18 gives the comparison between experimental and computational SCC crack growth rates. To further improve the prediction ability of the SCC growth rate prediction model based on elastic plastic finite element method, the SCC experimental results are compared with the prediction model of Equation (9) under different K.

Some of the curves are not smooth (see Figures 11,13, 14, 16, and 18), and there is also a gap between the experimental and the simulated values. This is all because the direction is not specified in the case of XFEM crack. The XFEM crack takes its direction. Most of the researchers have performed SCC with the debonding technique. This is probably for the first time that SCC is being performed with the XFEM technique. The variations in the K values are more unpredictable in the case of XFEM than of debonding technique. The values of K have been made almost constant in this technique as the other authors have done in case of the debonding technique.

The quantitative effect of the macrostructural parameters on the core region is shown in Figure 19.

## CONCLUSIONS

- •
A new technique to control (make variations small) stress intensity factor in the case of XFEM is introduced in this paper.

- •
The effect of stress intensity factors (in this case applied load) and hardening exponents are directly related to each other and their trend on the size of the quantitative SCC core region/zone is the same, i.e., the size of plastic core region increases with an increase in the values of stress intensity factors and hardening exponents. However, the length of the size of the core region is different in both cases. The effect of the yield strength on the size of the plastic core zone has an inverse relation. As the values of the yield strength decrease, which usually happens in weld joints in LWRs at high temperature for Alloy 600, the size of the plastic core region increases and the SCC crack growth increases. Among all of the microstructural parameters of the Alloy 600, the stress intensity factor (applied load) has the greatest impact on the size of the plastic zone.

## CONFLICTS OF INTEREST

The authors declare that there are no conflicts of interest regarding the publication of this paper. The mentioned funding in the Acknowledgments section does not lead to any conflicts of interest regarding the publication of this manuscript, there are no other possible conflicts of interest as well.

^{†}

Trade name.

## ACKNOWLEDGMENTS

This work was financially supported by the International Exchanges program scheme, a project by the National Natural Science Foundation of China and Royal Society (51811530311), and Shaanxi Province Science and Technology (Department) Key R & D Technology Project (2107GY-034).