#### Abstract

The stress and strain status associated with the material properties is one of the main factors affecting stress corrosion cracking (SCC) of structural components in nuclear power plants (NPPs). In many SCC prediction models, the stress intensity factor calculated with homogeneous materials is used to characterize the crack tip stress state. However, the mechanical and material properties in weld joints are heterogeneous, which will produce the discontinuous distribution of stress and strain nearby crack tip and affect the crack propagation. To understand the material mechanical heterogeneity effects on interface crack propagation, the specimens with ultimate tensile strength mismatch and elastic modulus mismatch were studied by using an extended finite element method (XFEM). The results indicate that the interface crack extension is easy to occur in the specimens with larger ultimate tensile strength mismatch, while the elastic modulus mismatch has little effects on crack extension. The different interface cracks in the dissimilar metal weld joints of the reactor pressure vessel used in NPPs tend to deviate from the initial direction into alloy 182, and the interface crack propagation path fluctuation is small.

#### 1. Introduction

It is widely known that welding defects such as inclusions, pores, and incomplete penetration are common in welding zone, which will affect the integrity of weld structure. However, a weld structure exists in subcomponents of pressure vessels, pressurizers, steam generators, piping, and deaerators in primary water systems of pressurized water reactors (PWRs) [1]. A typical application of weld joints in PWRs is the dissimilar metal weld (DMW) joints connecting the low alloy steel (LAS) reactor pressure vessel (RPV) nozzles to austenitic stainless steel (SS) pipes, as shown in Figure 1; initially, the Ni-base alloy buttering is predeposited on the RPV nozzle face, then welding is carried out between the buttering layer and the pipe with Ni-base alloy [2]. The operating experience of nuclear power plants (NPPs) shows that the weld zone is more susceptible to stress corrosion cracking (SCC), which seriously threatens the safety of NPPs [3–5]. In order to avoid the sudden failure of components and structures caused by SCC in NPPs, efforts have been made to understand the mechanism of SCC [6–8], and many SCC crack prediction models have been developed to predict the remaining life of main structures during the last three decades [9, 10]. Because the mechanical and material properties are key factors affecting SCC, in many SCC prediction models, the linear fracture mechanics are widely used to acquire the relationship between the crack propagation rate (*da*/*dt*) and the stress intensity factor (*K*) [11–14].

**(a)**

**(b)**

Since the materials are different at different positions in weld joints, especially in DMW joints, the mechanical and material heterogeneities of base metal, weld metal, and heat-affected zone (HAZ) are different; the different constraints induced by mechanical heterogeneity will lead to the discontinuity of stress and strain distribution nearby crack tip and affect the integrity of welded structures [15, 16]. Even the wide use of crack tip opening displacement (CTOD) and the *J*-integral as elastic-plastic fracture parameters in structural integrity assessment, this is relevant to the plastic constraint and the yield strength of material in the homogeneous case, there are no standard procedures for fracture mechanics testing of specimens with welds due to the difficulty to determine the CTOD from the crack mouth opening displacement (CMOD) with a heterogeneity welded joint. Thus, the relationship between the *J*-integral and the CTOD was investigated with different mismatch materials and crack depth by using an elastic-plastic finite element method, which supports the use of the *J*-integral instead of the CTOD [17]. The *J*-*Q*-*M* approach, where *Q* quanties the geometry effect and *M* the material mismatch effects, was produced to estimate the influence of fracture toughness on the effect of mismatch without the discussion of competing failure criteria related to ductile crack growth and plastic collapse [18]. By the *J*-*Q*-*M* approach, it was concluded that the weld-metal strength overmatch results in higher constraint than evenmatch, and the critical stress level for initiating brittle fracture in the HAZ is reached at the lowest loads in the case of overmatching.

Even the mismatch of weldments has been much discussed, most of them focus on the effects of yield strength mismatch [15–19], and little concerns the ultimate tensile strength and elastic modulus differences of the welded metals. Thus, the ultimate tensile strength and elastic modulus differences of welded metals and their effects on the interface crack propagation were focused, and the propagation characteristic of interface crack in dissimilar weld joints was summarized by adopting an extended finite element method (XFEM) in this study.

#### 2. Theory Model

##### 2.1. Solution of Interface Crack

As shown in Figure 2, a finite length crack 2*a* along the interface of an infinite body composed of two different isotropic, homogeneous materials; a tensile stress *σ*_{22} normal to the crack faces and an in-plane shear stress *τ*_{12} are applied remote from the crack; a complex stress intensity factor *K* at the crack tip could be calculated as [20]where *i* denotes the imaginary term, . *K*_{1} and *K*_{2} are real and denote the stress intensity factor of modes 1 and 2, respectively. *ε* is an oscillatory parameter given bywherewhere *μ*_{k} (*k* = 1, 2) are the shear modulus of the two different materials, respectively, and *ν*_{k} are Poisson’s ratios.

By means of Irwin’s crack closure integral, the interface energy release rate *G* of an interface crack is given as [21]wherewhere *k* = 1, 2 denotes the two different materials and *E*_{k} is the elastic modulus.

##### 2.2. Damage Model in XFEM

When dealing with the numerical simulation of crack propagation, a damage model contains crack initiation criterion and crack propagation law should be involved; once the crack initiation criterion is met, the crack can occur according to the defined crack propagation law.

###### 2.2.1. Crack Initiation Criterion and Crack Extension Direction

The stress intensity factor *K*, which is mainly determined by the maximum principal stress of crack tip, has an important effect in the SCC process; the crack starts to propagate if the crack-tip stress intensity factor reaches the threshold value *K*_{ISCC}. Thus, the maximum principal stress criterion built in the XFEM models was selected to describe the initiation of crack [22]:where is the maximum allowable principal stress; is the maximum principal stress; the Macaulay brackets are used to signify that the crack does not initiate with a purely compressive stress state; *f* is the maximum principal stress ratio, and the crack is assumed to initiate when *f* reaches a value of one.

When the maximum principal stress is specified, the newly introduced crack is always orthogonal to the maximum principal stress direction when the fracture criterion is satisfied.

###### 2.2.2. Crack Propagation Law

By defining the equivalent fracture energy release rate *G*_{equiv}, the power law based on energy was involved in this study to describe the crack propagation after the crack initiates [23]:where *G*_{equiv} is the equivalent fracture energy release rate and *G*_{equivC} is the critical equivalent fracture energy release rate; *G*_{I}, *G*_{II}, and *G*_{III} are critical energy release rates in Mode I, Mode II, and Mode III cracks, respectively; *a*_{m}, *a*_{n}, and *a*_{o} are damage exponents. In this study, the crack was Mode I, so only the first item in (7) was adopted.

The crack will propagate if *G*_{equiv} reaches *G*_{equivC}. According to Irwin, the fracture energy release rate in a two-dimensional crack is the energy that must be provided in order to break the material and create a cracked surface area Δ*A* [21]:where *G* is the fracture energy release rate; Π = *U* − *W* is potential energy, *W* is external work, and *U* is strain energy of the crack; *a* is the crack length, and *B* is the crack width.

According to (8), the calculation of *G* requires the crack increments Δ*a* approaching zero, and it obviously cannot be reached in numerical simulation; thus the Virtual Crack Closure Technique (VCCT) with two steps is adopted. The crack length equals to *a* in step 1, and the potential energy could be calculated as Π_{1} = *U*_{1} − *W*_{1}; the crack extends to *a* + Δ*a* in step 2, and the potential energy is Π_{2} = *U*_{2} – *W*_{2}; then, the fracture energy release rate *G* could be approximately calculated with a small enough increment of Δ*a* as

###### 2.2.3. The Description of Crack Status

The crack status in the simulation is described by a variable STATUSXFEM; the element is completely cracked if STATUSXFEM equals to 1, and the element contains no crack if STATUSXFEM equals to 0. If the element is partially cracked, the value of STATUSXFEM is in the range of 0 and 1.

#### 3. Finite Element Model of Interface Crack

##### 3.1. Geometry Model and Mesh Model

A typical DMW joint in RPV has a pipe inner diameter of 735 mm and a wall thickness of 74 mm. When the inner diameter of pipe is far greater than the crack length, the welded joint could be simplified as a two-dimensional plane strain problem; a three-point bending specimen model was built according to ASTM E1820-05a [24], as shown in Figure 3. The initial crack length is 15 mm, and the precrack length is 3 mm.

The calculation of crack propagation is on the mesh model; even an accurate solution can be obtained with a coarser mesh with XFEM; the element size near the crack tip still affects the accuracy and stability of the numerical simulation. The element size in the crack propagation zone should satisfy the demands ofwhere *h* is the element size along the crack propagation direction; *G*_{c} is the critical fracture energy release rate at crack tip; *E* is the elastic modulus; is Poisson’s ratio; and *σ*_{0} is yield strength.

The element size in a crack propagation zone is modeled as 0.05 mm × 0.05 mm, which satisfied the demands of (10), the element size far away from the crack propagation zone is 1 mm × 1 mm, and the element with a size of 0.05 mm∼0.2 mm is used in the transition zone. The finite element model was meshed to 38735 elements with 4-node biquadratic plane strain quadrilateral (CPE4), as shown in Figure 4. A small size blunt notch was designed at the crack tip with a radius of 0.05 mm to eliminate the calculation singularity. The geometric nonlinearity should be used to satisfy the computing demands when larger displacement and larger deformation appear in the simulation.

##### 3.2. Material Model

The fracture mode of welded joints in nuclear power plants mainly manifests as ductile damage, and the materials of welded joints generally belong to a power hardening materials; thus the nonlinear relationship between stress and strain of power hardening materials can be described by the Ramberg-Osgood equation:where *ε* is strain, *σ* is stress, *σ*_{0} is the yield strength of the material, *α* is the yield offset, *n* is the hardening exponent for the plastic, and *n* can be calculated by [17]where *κ* = 0.163 and *σ*_{0} is the yield strength.

The materials are different at the two sides of interface crack, in order to quantitate the difference of material properties on both sides of the crack, definedwhere *γ* is the material mismatch rate; the larger the value, the greater the differences of the two materials; it will be represented as *γ*_{u} and *γ*_{E} subsequently to denote the ultimate tensile strength mismatch rate and elastic modulus mismatch rate, respectively; *P* represents elastic modulus or ultimate tensile strength; subscripts *l* and *r* represent left and right, respectively.

Assuming that the specimen is made of A508 at the beginning, in order to obtain the effects of elastic modulus mismatch and ultimate tensile strength mismatch on the crack propagation, the elastic modulus and ultimate tensile strength of the material at the left side of interface are arbitrarily changed. The mechanical properties of A508 and other materials used in DMW joints of RPV are listed in Table 1, and the three-point bending specimens built are listed in Table 2.

#### 4. Effects of Material Properties Mismatch on Interface Crack Propagation

##### 4.1. Ultimate Tensile Strength Mismatch

Under the condition of ultimate tensile strength mismatch, the load-displacement curves and displacement-crack extension curves are shown in Figures 5 and 6, respectively. It can be seen that the loads increase with the increase in applied displacement, followed by the crack extension during the first stage; this indicates that the short crack tends to propagation than the long crack under the same load condition. As the applied displacement increases, the loads increase to the maximum value and then decrease. With the increase in the ultimate tensile strength mismatch rate, the maximum load and the corresponding applied displacement decrease; this indicates that the increase in the ultimate tensile strength mismatch rate will reduce the load crack propagation needed; cracks are more likely to extend at the interface with large mismatch. With a certain crack extension length, the applied displacement also decreases with the increase in the ultimate tensile strength mismatch rate, which also indicates that the cracks are more likely to extend at the interface with large mismatch.

As shown in Figure 7, at the initiation of crack propagation, *J*-integral increases with the mismatch rate, which means that the crack initiation resistance is greater in interface crack with a larger mismatch rate. As the crack extends, the *J*-integral reduces, and the largest reduction gradient appears in the specimen with a larger mismatch rate. When the crack propagates to 1 mm, the *J*-integral starts to increase with the extension of crack. In the specimens with different mismatch rates, the *J*-integral decreases with the increase of mismatch rate, and this also indicates that the crack tends to extend easily in the specimen with a larger mismatch rate.

Figure 8 illustrates the crack propagation path in the specimens with different mismatch rates; it can be seen that the crack propagation will deviate from the initiation direction. The crack extends to the left side of the specimens when mismatch rate exists, and the crack propagation path fluctuation is small in the specimen with a small mismatch rate, which indicates the larger the mismatch rate, the more unstable the interface crack propagation path is. Since the specimens all have a small ultimate tensile strength at the left side, the crack tends to propagate to the side with small ultimate tensile strength; the crack path fluctuation increases as the ultimate tensile strength mismatch increases. When the mismatch rate equals to 0, the crack direction will deviate from the initiation direction to the left or right side random.

##### 4.2. Elastic Modulus Mismatch

The load-displacement curves in Figure 9 show the maximum load required for crack extension decreases with the increase in the elastic modulus mismatch rate, which indicates that the crack tends to extend at the interface with a larger elastic modulus mismatch rate. As the displacement-crack extension curves in Figure 10 are coincident with each other, the differences of *J*-integral-Δ*a* curves shown in Figure 11 are also small; it could be concluded that the elastic modulus mismatch has little effects on the interface crack extension.

Figure 12 shows the crack propagation path in specimens with different elastic modulus mismatch rates; it can be seen that the crack propagation will deviate from the initiation direction to the side with larger elastic modulus. The crack extension direction seems to be the same if the crack extend length is smaller than 7 mm even if the elastic modulus mismatch exists. When the crack continues to extend, the crack path in the specimen with a small elastic modulus mismatch rate seems similar to the specimen of no mismatch, but the crack path has a big deflection in the specimen with a small elastic modulus mismatch rate; thus the crack extension path fluctuation increases as the elastic modulus mismatch rate increases.

#### 5. Interface Crack Propagation in the Safe-Ends Weld Joints

Two specimens with austenite stainless steel 316L, nickel alloy 182, and low-alloy steel A508 are modeled to simulate the crack propagation in DMW joints. Of the two specimens, left and right materials are modeled with 316L and alloy 182, respectively, in one specimen, and another one is modeled using alloy 182 and A508 as left and right materials, respectively. Figure 13 illustrates the load-displacement curves of the two different weld joint specimens; although the maximum load is needed in the 316L-Alloy 182 specimen, the two curves have little differences. The applied displacement-crack extension curve and *J*-integral-crack extension curves in Figures 14 and 15 also coincide with each other; this indicates that the interface crack propagation resistances have little differences in two specimens.

The crack extension directions of the two specimens are shown in Figure 16; it can be seen that the two interface cracks both deviate from the initiation direction and both deviate into alloy 182 specimen. When the specimens fracture, the crack path offset in the *X*-direction is in the range of 0.2∼0.3 mm, and the crack extend direction has little fluctuation.

#### 6. Conclusion

The effects of material properties’ mismatch on interface crack extension are calculated with XFEM. The crack extension resistance is larger in the specimens with a larger ultimate tensile strength mismatch rate initially, and the crack seems to easily initiate in specimens with small ultimate tensile strength mismatch. When the crack extends to a certain length, the crack extension resistance is small in the specimen with larger ultimate tensile strength mismatch; the crack extends easily in the specimens with larger ultimate tensile strength mismatch. The elastic modulus mismatch has little effects on crack extension resistance. The crack extension resistances have little differences in the DMW bimaterial specimens, and the crack tends to deviate from the initial direction into alloy 182, and the crack path fluctuation is small.

#### Data Availability

The data used to support the findings of this study were calculated according to the finite element method, and they are included within the article. The parameters used in the calculation model were cited from references listed.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This work was supported by the Research Fund of State Key Laboratory for Marine Corrosion and Protection of Luoyang Ship Material Research Institute (LSMRI) under Contract No. 61429010102, the National Natural Science Foundation of China (Grant Nos. 11502195 and 51775427), the Key Research and Development Program of Shaanxi Province (Grant No. 2017GY-034), and the Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2018JQ5193).