A Numerical Approach to Analyze Detail Mechanical Characteristic at the Crack Tip of SCC in Dissimilar Metal Welded Joints

+e mechanical characteristic at the crack tip is one of the main factors affecting the stress corrosion cracking (SCC) in dissimilar metal welded joints (DMWJs). In this research, to evaluate the effect of heterogeneous material properties on the mechanical characteristic at the crack tip of DMWJs accurately, a heterogeneous material model of the SA508 Cl.3-Alloy 52M DMWJ was established based on USDFLD subroutine in ABAQUS. +e comparison of the traditional “Sandwich” material model with the heterogeneous material properties, stress-strain conditions, and the plastic zone around the crack tip at the interference zone has been analyzed by the finite elementmethod (FEM).+e results indicated that the heterogeneousmaterial model could characterize the mechanical properties of the SA508 Cl.3-Alloy 52M DMWJs accurately. In addition, the crack at the interface zone between materials will deflect along with the weld metal in two material models.


Introduction
Nickel base alloys, such as alloy 152/52M, alloy 182/82, and alloy 309L/308L, have been widely used as weld metals to connect the pipe nozzle and safety end of the primary circuit of the nuclear pressure vessel [1][2][3][4][5]. However, several studies have demonstrated that the distribution of microstructure and material mechanical properties in the heat-affected zone (HAZ) and dilution zone (DZ) of the welded joint has become much complex during the welding progress [6][7][8][9][10][11]. Meanwhile, the complicated mechanical characteristic at the crack tip caused by the heterogeneous material properties is one of the main factors for stress corrosion cracking (SCC) of dissimilar metal welded joints (DMWJs) [12]. erefore, there is a great significance to characterize the material mechanical properties of DMWJs in detail for the safety evaluation and life prediction of the DMWJ in service environments.
Considerable research efforts have been devoted to the influence of the material constraint effect of welded joints on the mechanical characteristic at the crack tip by finite element method (FEM). Some scholars studied the mechanical characteristic at the crack tip of DMWJs based on the "Sandwich" material model, which ignores the existence of the HAZ and DZ and endows corresponding material properties to the base metal and weld metal [13,14], respectively. erefore, the traditional "Sandwich" material model can only roughly evaluate the influence of mismatch of the material mechanical properties between the base metal and weld metal on the safety of welded joints. However, the material mechanical properties of the DMWJ are not distinguished strictly at the interface zones between materials. Recent experiments in this field have obtained the material mechanical properties of the SA508 Cl.3-Alloy 52M DMWJ by microhardness test or minisized tensile test [15,16]. e experimental results show dramatic changes in the strength and work hardening exponent at the interface zones. erefore, the mechanical characteristic at the crack tip in the interface zones of welded joints is analyzed by the traditional "Sandwich" material model, which cannot meet the requirements. In previous studies, some scholars used refined zoning of the "Sandwich" material model to improve the solution accuracy [17], but that can only realize the gradient change of material in the model, and it increases the workload. Recently, many researchers have attempted to solve this problem by using the user subroutine in ABAQUS. Chi et al. [18] complied the stress-strain curve of the unsaturated soil based on the user-defined field (USDFLD) subroutine variables at a material point in the ABAQUS. e accuracy and reliability of the subroutine are verified by numerical simulation and experiment. Xue et al. [19] have established the continuous transition model of the welded joint based on the User-Defined Material Mechanical Behavior (UMAT) subroutine in the ABAQUS and found that the stress field at the crack tip of the continuous transition model deflects to the low yield strength zone. However, the writing progress of UMAT subroutines is complicated.
In this research, for further study of the mechanical characteristic at the crack tip of the SA508 Cl.3-Alloy 52M DMWJ, a heterogeneous material model has been established based on the USDFLD subroutine, and it has been compared with the traditional "Sandwich" material model. Furthermore, the stress-strain condition and plastic zone at the crack tips in interface zones are analyzed based on FEM, which provided a new method for accurately analyzing the mechanical field at the crack tip of DMWJs.

Specimen and Material
Model. Compact tension (CT) specimens are usually used in the SCC experiments in hightemperature of nuclear power plants materials, and their geometric dimensions are shown in Figure 1 [20]. Due to the limitation in the size of the SA508 Cl.3-Alloy 52M DMWJ, the 0.5T-CT specimen with prefabricated crack was performed in FEM analysis and guided by the American Society for Testing and Materials Standards [21]. e schematic diagram of the 0.5T-CT specimen in the SA508 Cl.3-Alloy 52M DMWJ is shown in Figure 2, and the length of the prefabricated crack is 2 mm. During the welding progress, the uneven distribution of the microstructure at the interface zones makes the mechanical properties complicated and unevenly distributed [22,23]. erefore, the interface regions between materials of the DMWJ are the focus of the SCC. e materials composed of the "Sandwich" material model and heterogeneous material model are shown in Figure 3. In the heterogeneous material model, the weld transition zone consists of HAZ in base metal (SA508), while the high hardness zone consists of DZ in weld metal (Alloy 52M) with widths of 2 mm, 0.1 mm, and 1.7 mm, respectively. In this study, the interface crack is located at the interface between materials in the "Sandwich" material model and at the fusion boundary line in the heterogeneous material model. To study the effect of the mechanical field at the noninterface crack caused by the material constraint effect of DMWJs and the noninterface crack position in the "Sandwich" material model, the heterogeneous material model of DMWJs is shown in Figure 3 and summarized in Table 1. e Ramberg-Osgood equation can be used to express the true stress-strain (σ − ε) relation of the DMWJ [3], which is written as follows: where σ y is the yield strength, ε y is the yield strain, and α is the Ramberg-Osgood coefficient. e work hardening exponent n of the welded joint can be simply obtained as follows [24]: where k � 0.163. Some scholars have obtained the microhardness distribution of the DMWJ on the surface of SA508 Cl.3-Alloy 52M, which is shown in Figure 4. e plateau values of the hardness for the base metal and weld metal are 186 and 175, respectively. Peng et al. [25] have summarized the linear relationship between Vickers hardness HV 0.1 and yield strength σ y . e yield strength of the HAZ of DMWJs can be obtained according to the following equation: Due to the difference between the chemical composition and strength of the DZ and HAZ, the yield strength σ y can be obtained by using the following equation proposed by Pargeter [26]: (4) Figure 5 shows the yield strength σ y distribution near the SA508 Cl.3-Alloy 52M DMWJ fusion line based on (3) and (4). e yield strength σ y of the SA508 Cl.3-Alloy 52M DMWJ at 340°C has been used in the finite element simulation and is shown in Figure 6. e yield strength of base metal and weld metal is 420 MPa and 385 MPa in the "Sandwich" material model, respectively. e black lines shown in Figure 6 represent the yield strength distribution of the heterogeneous material model.

FEM Model.
In the SCC experiments, the crack front along the thickness of the specimen of the SA508 Cl.3-Alloy 52M DMWJ is mainly dominated by plane strain conditions. erefore, the elastic-plastic finite element analysis is carried out under the condition of plane strain. e global grid for a specimen containing cracks is shown in Figure 7(a), where the x-direction is the direction of the crack growth and the y-direction is normal to the direction of the crack at the coordinate system. e finite element model has 12,756 nodes of 8-node biquadratic plane strain quadrilateral (CPE8). e meshes around the crack tip are refined to investigate the mechanical field distribution near the crack tip, where 8400 CPE8 are adopted around the crack tip, as shown in Figure 7 e material constraint effect of DMWJs on the mechanical characteristic at the SCC crack tip has been investigated under the constant load. During the numerical simulation process, the specimen was loaded by concentrated force in the center point of two loaded holes in the vertical direction. All other motions of the center point were restrained except the vertical direction. e coupling constraint was defined between the center point and the loaded hole. e stress intensity factor (K I ) was used as the mechanical parameter, and the constant load was taken as the loading condition of the finite element. In this research, the stress intensity factor was set to be a constant value of 30 MPa·m 1/2 , and the load P can be obtained by where P is load and B is sample thickness. As shown in Figure 1, if a/W � 0.5, then the f (a/w) � 9.66. ree observation paths were established in the finite element model to characterize the influence of the heterogeneous material mechanical properties on the mechanical characteristic around the crack tip in detail. Path 1 and path 2 were established along 45°and −45°in the Φ � 0.02 mm circular region around the crack tip, and path 3 was established along the front of the crack tip. Zhang et al. [27] proposed that the region of strain field is 1 < r/(J/σ y ) < 5, where J is J-integral. According to the obtained J-integral of the specimen, the distance from the start point to the end point of each path from the crack tip is 0.008∼0.02 mm in this study, which is shown in Figure 8.

Establishment of Heterogeneous Material Model Based on USDFLD.
ere are a lot of material constitutive models provided in the ABAQUS for the users. However, the mechanical properties at the DMWJs are heterogeneous. In this research, a heterogeneous material model of the SA508 Cl.3-Alloy 52M DMWJ has been established based on the USDFLD subroutine. e subroutine is used to redefine the field variables at the element integration point, which is programmed in the FORTRAN language [28]. e flowchart of the USDFLD subroutine is shown in Figure 9. e change of material mechanical properties associated with the functional relationship between longitudinal coordinate Uy and the field variable fx in FEM is established. At the same, the predefined material "MAT" in the FEM is invoked by the state variable STATE V (x) for the field variable fx. e yield strength σ y distribution is fitted according to Figure 6. e relationship between the field edited function fx and the y-direction of the CT specimen is established in the following equation:

e Stress Field
Feature around the Crack Tip. e stress and strain conditions at the crack tip of DMWJs can significantly affect the SCC behavior. ey are usually used as mechanical parameters to predict the SCC crack initiation. Figure 10(a) shows the Mises stress σ m contour of crack 1 in the range of crack tip at Φ � 0.8 mm, where the Mises stress contour around crack 1 is suddenly interrupted at the interface between the base metal and weld metal. erefore, the abrupt change of material mechanical properties at the interface in the "Sandwich" material model will cause discontinuity of the stress filed around the crack tip. e Mises stress σ m contour of crack-FB is shown in Figure 10(b). As can be seen from Figure 10(b), the Mises stress contour around the crack tip is continuously distributed in the heterogeneous material model, and the contour near the high hardness zones becomes wrinkled and deformed.
e Mises stress curves along path 1 and path 2 of crack 1 and crack-FB are shown in Figure 11(a), where the Mises stress value near r � 0 is the largest, and the Mises stress curve value of crack-FB is greater than that of crack 1. Combined with the asymmetric Mises stress contour around the crack tip of crack 1 and crack-FB, the stress curve is symmetrically distributed along the front of the crack in a small region around the crack tip. As shown in Figure 6, the yield strength of the heterogeneous material model at the fusion line is significantly higher than that at the interface of the "Sandwich" material model. erefore, the Mises stress value at the interface obtained by the traditional "Sandwich" material model is small. Figure 11(b) shows the Mises stress curves of crack 1, crack 2, and crack 3 in the "Sandwich" model, in which it can be seen that the three curves are varied closely from one another. erefore, the material constraint effect caused by the material properties of the base metal and weld metal given to the "Sandwich" material model has a little effect on the Mises stress value around crack tip in interface zones. e Mises stress curves at the crack tip in the heterogeneous material model are shown in Figure 12(a), where the Mises stress curve value at the crack tip is larger than other curves. It is indicated that under the same external load, the Mises stress distribution around the crack tip in the fusion line is larger than that in DZ and HAZ due to the material constraint effect caused by the welding. Figure 12(   properties of the SA508 Cl.3-Alloy 52M DMWJ. It is indicated that larger tensile stress will be provided for the _-type crack when it is located at the FB line. Figure 13 shows the equivalent plastic strain contour around the crack tip in the range of crack tip at Φ � 0.8 mm. Similar to Mises stress contour around the crack tip in the "Sandwich" material model, the discontinuity of equivalent plastic strain around the crack tip is suddenly interrupted when the crack is present at the interface between the materials, as shown in Figure 13(a). e equivalent plastic strain of crack-FB is shown in Figure 13(b). Contrary to the Mises stress contour around the crack tip in the heterogeneous material model, the equivalent plastic strain around the crack tip is contracted, and it is continuously distributed.

e Strain Field Feature around the Crack Tip.
is indicates that the yield strength of the crack tip is smaller than that of the base metal and weld metal, and the equivalent plastic strain of the crack tip shrinks. Combined with Figures 10(b) and 11(b), the continuous deformation distribution of Mises stress and equivalent plastic strain contour can verify the yield strength distribution in the SA508 Cl.3-Alloy 52M DMWJ. As shown in Figure 6, the yield strength increases in the high hardness zone, and the corresponding equivalent plastic strain decreases, while the stress value increases. e equivalent plastic strain ε eq curves around the crack tip of crack 1 and crack-FB are shown in Figure 14(a). It can be seen that the equivalent plastic strain curves are asymmetrically distributed along with them ahead of the crack tip. Contrary to the Mises stress curve around the crack tip, the equivalent plastic strain curve of crack-FB is smaller than crack 1. e equivalent plastic distribution curves around crack tip in the "Sandwich" material model are shown in Figure 14(b). It is obvious from Figure 14     Advances in Materials Science and Engineering interface crack between materials is symmetrical. erefore, the complex strain field will be generated on both sides of the interface crack caused by the material constraint effect in the "Sandwich" material model. e equivalent plastic curves at the crack tip in the heterogeneous material model are shown in Figure 15(a). It is evident that the equivalent plastic curve value of crack-FB is smaller than crack-DZ and crack-HAZ, and the curves of crack-DZ and crack-HAZ are very close to each other. Contrary to the stress field at the crack tip in the heterogeneous material model, the strain field at the crack tip at the fusion line is smaller than that in DZ and HAZ. e normal strain curves ahead of the crack tip along path 3 in the heterogeneous material model are shown in Figure 15(b). It is obvious from Figure 15(b) that the tensile strain on the front of the crack-DZ is the largest. is shows that the DZ is Define the mechanical properties of materials and named"MAT" Advances in Materials Science and Engineering easily yielded under the same load. From this perspective, the SA508 Cl.3-Alloy 52M dissimilar metal welded joint is more prone to damage in the DZ.

e Plastic Zone around the Crack Tip.
e plastic zone around the crack tip is defined as the yield strain, that is, 0.2% equivalent plastic strain. e larger plastic zone can store more energy around the crack tip. erefore, the plastic zone size is an important mechanical parameter for SCC crack initiation. Figure 16(a) shows the distribution of plastic zones at the crack tip at Φ � 0.8 mm with interface crack 1 and subinterface crack 2 and crack 3 in the "Sandwich" model, where the plastic zone of crack 1 is asymmetrically distributed along the crack front. e crack tip plastic zone in the weld metal is larger than that of the base metal. e plastic zones of noninterface crack 2 and crack 3 are symmetrically distributed along the crack front.
ese results indicate that more energy is stored in the plastic zone near the weld metal. From this point of view, interface crack 1 will be deflected from the weld metal. e plastic zones around the crack tip at the transition zone of the heterogeneous material model are shown in Figure 16(b), where the plastic zones of the crack are asymmetrical. Moreover, the plastic zone area at the crack-DZ is larger than crack-FB and crack-HAZ. By   Figure 6, it is obvious that the crack tip plastic zone will decrease as the yield strength increases under the same load. erefore, the plastic zone around the crack tip at the interface zone between materials based on the traditional "Sandwich" material model will be larger.

Conclusions
In this research, the heterogeneous material model of the SA508 Cl.3-Alloy 52M DMWJ has been established based on the USDFLD subroutine. e comparison of the traditional "Sandwich" model with the heterogeneous material properties, the stress and strain condition, and the plastic zone around the crack tip at interface zones is analyzed by the finite element method. e main acquired results are summarized as follows: (1) By comparing the stress and strain contour around the crack tip in two material models, it is concluded that the stress and strain contour is symmetrical at the interface zones in the heterogeneous material model based on the USDFLD subroutine, which eliminates the sudden changes of the contour caused by the material properties given by the "Sandwich" material model. (2) When the stress intensity factor K I is constant, the crack at the fusion line obtains greater tensile stress in the heterogeneous material model, which provides a greater driving force for SCC crack to initiate. However, the variations in the tensile stress around the crack tip difference between interface crack and noninterface crack in the "Sandwich" material model can be ignored. (3) By considering the effect of the plastic zone and the strain curves around the crack tip carefully, it is concluded that both material models show that the crack at the interface zones between materials is more likely to deflect along with the weld metal.
Data Availability e article includes the numerical simulation data of the stress and strain curves and the plastic zone around the crack-tip used to support this study's findings.

Conflicts of Interest
e authors declare that they have no conflicts of interest.