# Characterization of Crack Tip Mechanical Field of Inhomogeneous Materials in Welded Joints by USDFLD Subroutine

Authors: Yueqi Bi; Xiaoming Yuan; MingRui Hao; Daming Li; Shuai Wang; He Xue
Journal: Advances in Materials Science and Engineering
 (2022)

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## Abstract

The unevenly distributed mechanical properties of welded joints are one of the important factors, which affect welded joints’ safety seriously. The user-defined field (USDFLD) subroutine in ABAQUS finite element software can introduce variable-dependent material properties. This method can define inhomogeneous materials whose mechanical parameters depend on spatial coordinates and can be used to characterize the uneven mechanical properties of heat-affected zone and fusion zone. By establishing the static crack finite element model of welded joint with continuous change in yield strength, the sudden change in mechanical field in the numerical simulation process of sandwich structure model is eliminated. Compared with the existing numerical simulation methods of local mechanical heterogeneity of welded joints, this method provides a method to realize the structural integrity evaluation of welded joints.


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## Body

## 1. Introduction

Because of the intensity of welding process, the metals near heat source are significantly affected by the temperature, resulting in uneven distribution of the material microstructure of the welded joint [1–3]; thus, under the interaction of residual stress, nonuniformity of mechanical properties of microstructure, and other complex mechanical and environmental factors, micro-cracks are easy to occur in welded joints, such as welding hot breaking, cold crack, reheat crack, lamellar tearing, and stress corrosion crack, which has become a potential safety hazard in the service process of welded joints. Therefore, the welded joints are the key areas of structural integrity analysis, and local mechanical heterogeneity needs to be properly expressed. Considering the mechanical heterogeneity of welded joints, the sandwich composite structure is usually used in engineering applications, in which a method simplifies the welded joint to analyze mechanical properties by dividing the weld area (such as base metal and weld material) and giving corresponding material properties to each part. Such a simplified method of the welded joint is called the sandwich structure finite element welded joint model [4–8]. Although this method described the mechanical characteristics of the welded structure totally, it cannot match the local geometric and mechanical properties of the welded joint. In particular, the mechanical properties of materials are the key factors affecting crack propagation resistance [9–11]. Therefore, it is necessary to calculate and predict the crack propagation direction and the strain field and stress field at the crack tip. In general, the mechanical property mismatch is defined by the strength mismatch as the ratio of the yield strength of the weld metal to the base metal [12–14]. In most studies, the welded joint is simplified as a sandwich structure. For example, Fan et al. [15–17] used the sandwich structure model to study the effects of metal hardening rate mismatch, as well as crack position on crack propagation path and fracture toughness. Xue et al. [18] also used the sandwich structure model to study the effect of yield strength mismatch on the crack tip mechanical field. Wang et al. [19, 20] established a sandwich structure model to study the effect of crack position on crack propagation resistance and crack propagation path of welded joints of nuclear equipment. However, in the process of finite element analysis, the sandwich structure model will cause the phenomenon of stress field mutation, which is due to the different material properties leading to the stress discontinuity at the boundary between the two materials under the action of external load [21]. The USDFLD subroutine in ABAQUS finite element software can be used to introduce solution-dependent material properties. This method can be easily defined as a function of field variables [22]. Using the USDFLD subroutine, the mechanical properties of materials can be introduced into the parameters dependent on spatial coordinate variables, which can be used to simulate the uneven material of welded joints. In our previous research, we discussed the crack growth path and crack tip mechanical field during crack propagation that affected yield strength mismatch using the USDFLD subroutine [23].In this study, the USDFLD subroutine is used to spatially associate the yield strength of the material with the finite element model. Taking the undermatched welded joint with initial crack as the research object, the sandwich welded joint finite element model and the continuous transition welded joint finite element model, the material mechanical properties of which are described by the USDFLD subroutine, are established to analyze the characteristics of mechanical field at the crack tip.


## 2. Materials and Methods

### 2.1. Geometric Model and Methodology

In general, according to the microstructure and mechanical properties, welded joints usually include weld metal, base metal, fusion line, and heat-affected zone [24, 25]. As shown in Figure 1(a)., the weld metal has stable microstructure and mechanical properties due to the full protection of the metal pool during the welding process. The base metal is far away from the weld area during welding, and the microstructure and mechanical properties do not change. However, the heat-affected zone is close to the weld area, and thus, the material microstructure and mechanical properties change under the influence of temperature. Besides, the fusion line is located between the weld zone and the heat-affected zone, and the heterogeneity of material microstructure and mechanical properties is significant. There is a high probability of brittle fracture and initiation of welding cracks in the area around the fusion line, which is usually the weak link of welded joints [26, 27]. For example, Figure 1(b) shows the fracture of the cutting pick base of the coal mine tunnel boring machine. The pick is welded with the base, and the yield strength of the weld material is higher than that of the base metal. Due to the load easily exceeding the design load in the process of coal cutting and the existence of welding defects, the internal cracks of welded joints expand gradually, which leads to mechanical structure failure; thus, the welded joint is one of the key factors affecting the reliability of cutting pick parts.Figure 1
Composition and fracture of welded joint. (a) Composition of welded joint. (b) The fracture of the cutting pick base.
(a)(b)If the metal material model is an isotropic elastic-plastic model, the algebraic equation related to the model integration can be easily developed, according to a single variable, and the material stiffness matrix can be written clearly. The process of establishing an uneven material field using the USDFLD subroutine is as follows.In this research, the stress-strain relationship in the plastic stage is a power-hardening relationship, and the stress-strain relationship in the elastic stage is linear. The uniaxial constitutive relationship of the material is expressed by a piecewise function, as shown as follows:(1)σ=Eεe,σ<σs,Ennσsn−1εpn+σs,σ≥σs.Here,n is the hardening coefficient, E is the elastic modulus, and σs is the yield strength. σs is replaced with σ0 (ε−plψ), and we can get the following equation:(2)σ=Eεe,σ<σ0ε−pl,ψ,Ennσ0ε−pl,ψn−1εpn+σs,σ≥σ0ε−pl,ψ.Therefore, the yield strength in the elastic-plastic constitutive relationship of the material is controlled by equation (3). The yield strength distribution σ0 (ε−pl, ψ) is the function of equivalent plastic strain ε−pl and field variable ψ, which is defined by user. The relationship of field variable ψ with spatial is(3)ψ=fx,y.The schematic diagram of finite element model of welded joint is shown in Figure2. The width of welded joint sample is W = 10 mm, and the height is H = 20 mm. The spatial coordinate system of the welded joint is shown in Figure 2. The ordinate is the left edge of the welded joint sample, and the abscissa is located at sample H/2. The coordinate origin is located at the midpoint on the left side of the sample. The crack coincides with the x-axis, and the crack length is a = 2 mm.Figure 2
Schematic diagram of finite element model of welded joint. (a) The yield strength changes in they-direction. (b) The yield strength changes in the x-direction.
(a)(b)Figure2(a) shows the crack direction perpendicular to the change direction of yield strength. In the sandwich structure finite element model, the part y > 0 mm is the base metal area and the part y < 0 mm is the weld metal area. The dotted line on the x-axis is the boundary between the two materials, and the crack coincides with the material boundary. In the finite element model of welded joint established by the USDFLD subroutine, the shaded area is the transition zone (this area represents the heat-affected zone and fusion zone) with continuous change in yield strength. The width of this area B = 2 mm is distributed symmetrically along the x-axis. y > 1 mm area is the base metal area, and y < 1 mm area is the weld metal area.Figure 2(b) shows the crack direction parallel to the yield strength direction. In thesandwich structure welded joint finite element model, the dashed line perpendicular to thecrack at L = 3 mm is the material boundary line, and the area x > 2 mm is the weld metal area.The area x < 2 mm is the base metal area. In the finite element model of the welded jointestablished by the USDFLD subroutine, the material width of the transition zone is B = 2 mm,and this area is symmetrical about the material boundary of the finite element model of thewelded joint of the sandwich structure. The material in the area x > 3 mm is the weld metalarea, and the material in the area x < 1 mm is the base metal area.


### 2.2. Mesh and Load Model

2(b)Figure3(a) shows the finite element model of the welded joint. The bottom end of the welded joint adopts a fixed constraint. The upper end is applied stress intensity factor loading K = 438.66 MPa·mm1/2. Figure 3(b)is a partial view of the crack tip, which is used to show the characteristics of the mechanical field of the crack tip. The model of finite element mesh is shown in Figure 4(b), the element type is CPS4R, and the number of elements is 63693. The crack tip is partially divided as shown in Figure 4(c), taking the red nodes of local mesh edge as the node set. The node set is established in the clockwise direction as the data collection point of the stress and strain around the crack tip.Figure 3
Finite element model of welded joint. (a) Finite element model of welded joint. (b) Geometric subdivision of crack tip. (c) Finite element mesh generation of welded joints. (d) Local meshing of crack tip.
(a)(b)(c)(d)Figure 4
Contour of the USDFLD subroutine welded joint model whose yield strength changes in they-direction. (a) Von Mises stress contour at crack tip. (b) Equivalent plastic strain contour at crack tip.
(a)(b)


### 2.3. Material Model

The stress-strain curve of the specimen is tested on tensile testing machine, as shown in Figure5. The uniaxial tensile stress-strain curve of the sample is engineering stress-strain curve in Figure 6. The engineering stress-strain curve does not consider the section shrinking, during tensile process. Therefore, in the finite element analysis, the true stress-strain curve calculated from the engineering stress-strain is used to set the material properties of the finite element model of the sandwich structure welded joint and the USDFLD subroutine model. Since the stress-strain curve of this material has no obvious yield point, the stress value 400 MPa at 0.2% of the strain is taken as the yield strength. The elastic section of the curve with a straight line is fitted, and an elastic modulus of 210 GPa is obtained, and Poisson’s ratio is 0.3.Figure 5
Dimension of the sample and tensile test. (a) The size of specimen. (b) Specimen clamping.
(a)(b)Figure 6
Stress-strain data of 306 stainless steel.The welded joint yield strength mismatch coefficientM refers to the ratio of the weld metal yield strength σWM to the base metal yield strength σBM; that is, M = σWM/σBM. M > 1 represents strength overmatch; M = 1 represents strength even match; and M < 1 represents strength undermatch. In this study, the yield strength distribution of the sandwich structure welded joint finite element model is calculated by the mismatch coefficient, and the material properties are, respectively, assigned to the area corresponding to the weld and the base metal of the model. The yield strength changes in the x-direction, the mismatch coefficient is recorded as Mx, and the yield strength changes in the y-direction, and the mismatch coefficient is recorded as My. Base and weld material yield strength of the USDFLD subroutine finite element model and sandwich structure welded joint finite element model has the same mismatch coefficient, but the yield strength of the transition zone, which is established by the USDFLD subroutine, is defined by a distribution function. To simplify the calculation, the yield strength of the material is defined as a linear relationship with the coordinates, as shown as follows:(4)σ0=GΨ+σs0,where σs0 is the initial yield strength. G is the gradient of the yield strength change, which represents the magnitude of the yield strength level and direction. In this study, the direction of yield strength is divided into two groups: in the x-direction, the field variable function is f (x, y) = x, and the yield strength gradient is denoted as Gx; in the y-direction, the field variable function is f (x, y) = y, and the yield strength gradient is recorded as Gy. The elastic modulus E of the sandwich structure and the USDFLD subroutine finite element model is 210 GPa, Poisson’s ratio μ is 0.3, and the initial yield strength σs0 is 400 MPa. The yield strength relationship of each material zone is shown in Tables 1 and 2.Table 1
Elastic-plastic material property with yield strength changing in thex-direction.
Yield strength distributionGroupMismatch coefficientMxYield strength gradient in transition zoneGx (MPa/mm)Yield strength of base metalσBM (MPa)Yield strength of weld metalσWM (MPa)x-directionSandwich structure model1—4004000.25—5332670.5—640160USDFLD subroutine model—0400400—133533267—240640160Table 2
Elastic-plastic material property with yield strength changing in they-direction.
Yield strength distributionGroupMismatch coefficientMyYield strength gradient in transition zoneGy (MPa/mm)Yield strength of base metalσBM (MPa)Yield strength of weld metalσWM (MPa)y-directionSandwich structure model1—4004000.5—5332670.2—640160USDFLD subroutine model—0400400—133533267—24064160


## 2.1. Geometric Model and Methodology

In general, according to the microstructure and mechanical properties, welded joints usually include weld metal, base metal, fusion line, and heat-affected zone [24, 25]. As shown in Figure 1(a)., the weld metal has stable microstructure and mechanical properties due to the full protection of the metal pool during the welding process. The base metal is far away from the weld area during welding, and the microstructure and mechanical properties do not change. However, the heat-affected zone is close to the weld area, and thus, the material microstructure and mechanical properties change under the influence of temperature. Besides, the fusion line is located between the weld zone and the heat-affected zone, and the heterogeneity of material microstructure and mechanical properties is significant. There is a high probability of brittle fracture and initiation of welding cracks in the area around the fusion line, which is usually the weak link of welded joints [26, 27]. For example, Figure 1(b) shows the fracture of the cutting pick base of the coal mine tunnel boring machine. The pick is welded with the base, and the yield strength of the weld material is higher than that of the base metal. Due to the load easily exceeding the design load in the process of coal cutting and the existence of welding defects, the internal cracks of welded joints expand gradually, which leads to mechanical structure failure; thus, the welded joint is one of the key factors affecting the reliability of cutting pick parts.Figure 1
Composition and fracture of welded joint. (a) Composition of welded joint. (b) The fracture of the cutting pick base.
(a)(b)If the metal material model is an isotropic elastic-plastic model, the algebraic equation related to the model integration can be easily developed, according to a single variable, and the material stiffness matrix can be written clearly. The process of establishing an uneven material field using the USDFLD subroutine is as follows.In this research, the stress-strain relationship in the plastic stage is a power-hardening relationship, and the stress-strain relationship in the elastic stage is linear. The uniaxial constitutive relationship of the material is expressed by a piecewise function, as shown as follows:(1)σ=Eεe,σ<σs,Ennσsn−1εpn+σs,σ≥σs.Here,n is the hardening coefficient, E is the elastic modulus, and σs is the yield strength. σs is replaced with σ0 (ε−plψ), and we can get the following equation:(2)σ=Eεe,σ<σ0ε−pl,ψ,Ennσ0ε−pl,ψn−1εpn+σs,σ≥σ0ε−pl,ψ.Therefore, the yield strength in the elastic-plastic constitutive relationship of the material is controlled by equation (3). The yield strength distribution σ0 (ε−pl, ψ) is the function of equivalent plastic strain ε−pl and field variable ψ, which is defined by user. The relationship of field variable ψ with spatial is(3)ψ=fx,y.The schematic diagram of finite element model of welded joint is shown in Figure2. The width of welded joint sample is W = 10 mm, and the height is H = 20 mm. The spatial coordinate system of the welded joint is shown in Figure 2. The ordinate is the left edge of the welded joint sample, and the abscissa is located at sample H/2. The coordinate origin is located at the midpoint on the left side of the sample. The crack coincides with the x-axis, and the crack length is a = 2 mm.Figure 2
Schematic diagram of finite element model of welded joint. (a) The yield strength changes in they-direction. (b) The yield strength changes in the x-direction.
(a)(b)Figure2(a) shows the crack direction perpendicular to the change direction of yield strength. In the sandwich structure finite element model, the part y > 0 mm is the base metal area and the part y < 0 mm is the weld metal area. The dotted line on the x-axis is the boundary between the two materials, and the crack coincides with the material boundary. In the finite element model of welded joint established by the USDFLD subroutine, the shaded area is the transition zone (this area represents the heat-affected zone and fusion zone) with continuous change in yield strength. The width of this area B = 2 mm is distributed symmetrically along the x-axis. y > 1 mm area is the base metal area, and y < 1 mm area is the weld metal area.Figure 2(b) shows the crack direction parallel to the yield strength direction. In thesandwich structure welded joint finite element model, the dashed line perpendicular to thecrack at L = 3 mm is the material boundary line, and the area x > 2 mm is the weld metal area.The area x < 2 mm is the base metal area. In the finite element model of the welded jointestablished by the USDFLD subroutine, the material width of the transition zone is B = 2 mm,and this area is symmetrical about the material boundary of the finite element model of thewelded joint of the sandwich structure. The material in the area x > 3 mm is the weld metalarea, and the material in the area x < 1 mm is the base metal area.


## 2.2. Mesh and Load Model

2(b)Figure3(a) shows the finite element model of the welded joint. The bottom end of the welded joint adopts a fixed constraint. The upper end is applied stress intensity factor loading K = 438.66 MPa·mm1/2. Figure 3(b)is a partial view of the crack tip, which is used to show the characteristics of the mechanical field of the crack tip. The model of finite element mesh is shown in Figure 4(b), the element type is CPS4R, and the number of elements is 63693. The crack tip is partially divided as shown in Figure 4(c), taking the red nodes of local mesh edge as the node set. The node set is established in the clockwise direction as the data collection point of the stress and strain around the crack tip.Figure 3
Finite element model of welded joint. (a) Finite element model of welded joint. (b) Geometric subdivision of crack tip. (c) Finite element mesh generation of welded joints. (d) Local meshing of crack tip.
(a)(b)(c)(d)Figure 4
Contour of the USDFLD subroutine welded joint model whose yield strength changes in they-direction. (a) Von Mises stress contour at crack tip. (b) Equivalent plastic strain contour at crack tip.
(a)(b)


## 2.3. Material Model

The stress-strain curve of the specimen is tested on tensile testing machine, as shown in Figure5. The uniaxial tensile stress-strain curve of the sample is engineering stress-strain curve in Figure 6. The engineering stress-strain curve does not consider the section shrinking, during tensile process. Therefore, in the finite element analysis, the true stress-strain curve calculated from the engineering stress-strain is used to set the material properties of the finite element model of the sandwich structure welded joint and the USDFLD subroutine model. Since the stress-strain curve of this material has no obvious yield point, the stress value 400 MPa at 0.2% of the strain is taken as the yield strength. The elastic section of the curve with a straight line is fitted, and an elastic modulus of 210 GPa is obtained, and Poisson’s ratio is 0.3.Figure 5
Dimension of the sample and tensile test. (a) The size of specimen. (b) Specimen clamping.
(a)(b)Figure 6
Stress-strain data of 306 stainless steel.The welded joint yield strength mismatch coefficientM refers to the ratio of the weld metal yield strength σWM to the base metal yield strength σBM; that is, M = σWM/σBM. M > 1 represents strength overmatch; M = 1 represents strength even match; and M < 1 represents strength undermatch. In this study, the yield strength distribution of the sandwich structure welded joint finite element model is calculated by the mismatch coefficient, and the material properties are, respectively, assigned to the area corresponding to the weld and the base metal of the model. The yield strength changes in the x-direction, the mismatch coefficient is recorded as Mx, and the yield strength changes in the y-direction, and the mismatch coefficient is recorded as My. Base and weld material yield strength of the USDFLD subroutine finite element model and sandwich structure welded joint finite element model has the same mismatch coefficient, but the yield strength of the transition zone, which is established by the USDFLD subroutine, is defined by a distribution function. To simplify the calculation, the yield strength of the material is defined as a linear relationship with the coordinates, as shown as follows:(4)σ0=GΨ+σs0,where σs0 is the initial yield strength. G is the gradient of the yield strength change, which represents the magnitude of the yield strength level and direction. In this study, the direction of yield strength is divided into two groups: in the x-direction, the field variable function is f (x, y) = x, and the yield strength gradient is denoted as Gx; in the y-direction, the field variable function is f (x, y) = y, and the yield strength gradient is recorded as Gy. The elastic modulus E of the sandwich structure and the USDFLD subroutine finite element model is 210 GPa, Poisson’s ratio μ is 0.3, and the initial yield strength σs0 is 400 MPa. The yield strength relationship of each material zone is shown in Tables 1 and 2.Table 1
Elastic-plastic material property with yield strength changing in thex-direction.
Yield strength distributionGroupMismatch coefficientMxYield strength gradient in transition zoneGx (MPa/mm)Yield strength of base metalσBM (MPa)Yield strength of weld metalσWM (MPa)x-directionSandwich structure model1—4004000.25—5332670.5—640160USDFLD subroutine model—0400400—133533267—240640160Table 2
Elastic-plastic material property with yield strength changing in they-direction.
Yield strength distributionGroupMismatch coefficientMyYield strength gradient in transition zoneGy (MPa/mm)Yield strength of base metalσBM (MPa)Yield strength of weld metalσWM (MPa)y-directionSandwich structure model1—4004000.5—5332670.2—640160USDFLD subroutine model—0400400—133533267—24064160


## 3. Results and Discussion

Since the yield strength distribution of welded joint has a significant impact on the crack propagation behavior, the local Von Mises stress and plastic zone of the crack tip are usually used to analyze the crack propagation behavior. The distribution characteristics of Von Mises stress and equivalent plastic strain field at the crack tip are analyzed.


### 3.1. Yield Strength Changes in theY-Direction

Figure4 shows the Von Mises stress and equivalent plastic strain contour of the USDFLD subroutine welded joint model, in which the yield strength changes in the y-direction. When the gradient of yield strength Gy > 0, the yield strength of the transition zone continuously decreases from the base material side to weld material side. Under external load, the plastic deformation level of both sides is different. That is why the Mises stress field and the equivalent plastic strain field deflect to the side of the weld material totally. The degree of mechanical field deflection increases with yield strength gradient Gy going up. When the yield strength gradient Gy = 0, as shown in Figures 4(a) and 4(b), the yield strength of the base metal area, weld area, and transition area does not change. The entire welded joint is a homogeneous material; thus, the Von Mises stress and equivalent plastic strain at the crack tip are symmetrically distributed with crack.Figure7 shows the Von Mises stress and equivalent plastic strain curves whose data come from the node set of the crack tip mechanical field edge in Figure 4. In Figure 7, the Von Mises stress and the equivalent plastic strain curve around the crack tip mechanical field edge are smooth. The range of 0–180°is close to the base material, and the range of 180–360° is close to the weld material. The crack position is located at θ = 180°.Figure 7
Curve of the USDFLD subroutine welded joint model whose yield strength changes in they-direction. (a) Von Mises stress curve at crack tip. (b) Equivalent plastic strain curve at crack tip.
(a)(b)In Figures7(a) and 7(b), when the yield strength gradient Gy > 0, the Von Mises stress curve and the equivalent plastic strain curve shift to the weld metal side. The curves move more toward the weld metal side with the yield strength gradient Gy going up, which indicates that the deviation of stress field and the plastic zone at crack tip to the weld zone is increasing. The equivalent plastic strain value on the weld side in Figure 7(b) is greater than that close to the base material, which means plastic deformation is more likely to occur on the low yield strength material under load. In Figure 7, when yield strength gradient Gy1 = 0, the base metal and the weld metal have the same yield strength, and the Von Mises stress and the equivalent plastic strain curve on the node set are symmetrically distributed with crack position θ = 180°.Figure8 shows the contour of the mechanical field at the crack tip of sandwich structure welded joint whose yield strength is undermatched and changes along the y-direction. The black dashed line is the dividing line between the weld metal and the base metal. The low match of welded joint is that the yield strength of the base material is higher than that of the weld material. Due to the finite element model of the sandwich structure welded joint, the material property of the continuous change in the yield strength cannot be set. The mechanical properties of the interface between the two materials are discontinuous. This is also the reason for the “dislocation” phenomenon of Von Mises stress and equivalent plastic strain field at the material interface of crack tip. From another perspective, the material boundary of sandwich structure welded joint model can be regarded as a situation where the width of continuous transition zone, which is established by the USDFLD subroutine, is infinitely small and the yield strength change gradient is infinitely great.Figure 8
Contour of the sandwich structure welded joint model whose yield strength changes in they-direction. (a) Von Mises stress contour at crack tip. (b) Equivalent plastic strain contour at crack tip.
(a)(b)Comparing the Von Mises stress distribution range of the crack tip in Figure8(a) with that in Figure 4(a), as well as the equivalent plastic strain distribution range in Figure 8(b) and with that in Figure 4(b), we can find that the “dislocation” phenomenon of the Von Mises stress and equivalent plastic strain field of the crack tip in the sandwich structure welded joint is essentially the deflection of the crack tip mechanical field when the yield strength gradient of the USDFLD subroutine welded joint tends to infinity.Figure9 shows the Von Mises stress and equivalent plastic strain curve on the path of the crack tip mechanical field edge Φ = 0.5 mm in Figure 8. Compared with the USDFLD subroutine model, the numerical curve of the node set around the crack tip of the welded joint is shown in Figure 7. The Von Mises stress and equivalent plasticity curve in Figure 9 are not a smooth curve, but a jumping discontinuity point at the material boundary. Besides, the greater the difference in yield strength between base metal and weld metal, the greater the difference between the Von Mises stress and the equivalent plastic strain on both sides of the material boundary.Figure 9
Curve of the sandwich structure welded joint model whose yield strength changes in they-direction. (a) Von Mises stress curve at crack tip. (b) Equivalent plastic strain curve at crack tip.
(a)(b)In addition, the Von Mises stress in the base metal area in Figure9(a) is slightly greater than that in the weld metal. Because the high yield strength of the base metal has strong resistance against deformation, it causes the Von Mises stress relatively large under the same load. In the equivalent plastic strain curve (Figure 9(b)), since the yield strength of the weld material is less than that in base metal, it will produce greater plastic deformation when subjected to a load, so the equivalent plastic strain in the weld area is much greater than that in base metal. In Figure 9, when the mismatch coefficient My1 = 1, the yield strength of both base metal and weld metal is the same, and the deformation of two areas is the same under the action of external force; thus, the Mises stress and equivalent plastic strain are symmetrically distributed with respect to the crack.


### 3.2. Yield Strength Changes in theX-Direction

Figure10 shows the contour of the mechanical field at the crack tip of the USDFLD subroutine welded joint whose yield strength is undermatched and changes along the x-direction. The welded joint established by the USDFLD subroutine consists of three parts: weld metal, transition zone, and the base metal along the x-direction. The yield strength of the transition zone increases continuously from weld metal to base metal along the x-direction. As the yield strength near the crack tip gradually increases along the x-direction, the bearing capacity of the material increases, and the Von Mises stress distribution range near the crack tip increases and extends to the base material side. The resistance to plastic deformation near the crack tip is also enhanced, so the equivalent plastic strain distribution range gradually decreases.Figure 10
Contour of the USDFLD subroutine welded joint model whose yield strength changes in thex-direction. (a) Von Mises stress contour at crack tip. (b) Equivalent plastic strain contour at crack tip.
(a)(b)The Von Mises stress and equivalent plastic strain curves around the crack tip are shown in Figure11. The angle θ, which is close to the weld metal, is in the range of 0–90° and 270–360°. The range of 90–270° is the side close to the base material. The curve in Figure 11 is a continuous smooth curve, which indicates that the crack tip mechanical field is continuous. Besides, the crack position is symmetrically distributed with respect to θ = 180°. It can be seen from Figure 11(a) that with the increase in yield strength gradient, the peak point of the Von Mises stress curve goes up gradually. The reason for this phenomenon is that high yield strength enhances the material bearing capacity. On the contrary, the increase in the yield strength leads to a decrease in the plastic deformation of the material. Therefore, the peak point of the equivalent plastic strain curve decreases with the increase in the gradient of the yield strength change, as shown in Figure 11(b).Figure 11
Curve of the USDFLD subroutine welded joint model whose yield strength changes in thex-direction. (a) Von Mises stress curve at crack tip. (b) Equivalent plastic strain curve at crack tip.
(a)(b)Figure12 shows the contour of the mechanical field at the crack tip, in which the yield strength of welded joint is low match and changes along the x-direction. The dotted line in the figure is the material boundary. The sandwich structure welded joint model is composed of the weld metal and the base metal along the x-direction. The yield strength of the weld metal is less than that of base metal. Because of the discontinuity between the two materials at the boundary, the crack tip mechanical field in Figure 12 appears “dislocation” at the boundary of the material. In Figure 12(a), when Mx > 0, because the yield strength of the base metal is higher than that of the weld metal, the material bearing capacity in base metal is stronger than that in weld metal. When Mx > 0, the Von Mises stress distribution range in the base metal area of the crack tip is greater than that when Mx = 0. However, when Mx > 0, the equivalent plastic strain distribution range of the base metal region of the crack tip is smaller than that when Mx = 0, as shown in Figure 12(b).Figure 12
Contour of the sandwich structure welded joint model whose yield strength changes in thex-direction. (a) Von Mises stress contour at crack tip. (b) Equivalent plastic strain contour at crack tip.
(a)(b)Figure13 shows the Von Mises stress and equivalent plastic strain curve of the node set around the sandwich structure welded joint crack tip edge, when the yield strength changes along the x-direction. The angle range of 0–90° and 270–360° is weld metal. The angle range of 90–270° is the base metal. The material boundary is at θ = 90° and θ = 270°. θ = 180° is the location of the crack. When the yield strength changes along the x-direction, the crack is perpendicular to the direction of the yield strength change. The Von Mises stress and equivalent plastic strain curves in Figure 13 are symmetrically distributed with respect to the crack.Figure 13
Curve of the sandwich structure welded joint model whose yield strength changes in thex-direction. (a) Von Mises stress curve at crack tip. (b) Equivalent plastic strain curve at crack tip.
(a)(b)Compared with the Von Mises stress and the equivalent plastic strain curve around the crack tip of the welded joint finite element model established by the USDFLD subroutine in Figure12, it can be found that the Von Mises curve and the equivalent plastic strain curve in Figure 13 are not continuous at the material boundary. The discontinuity point is skipped point, and the deviation of these curves at both ends of the discontinuity point, which are shown in Figure 13, goes up with the increase in the yield strength difference between the base metal and the weld metal. In Figure 13(a), the Von Mises stress value in base metal region when Mx < 1 is greater than that when Mx = 1. In Figure 13(a), the equivalent plastic strain value in base metal region when Mx < 1 is smaller than that when Mx = 1. This is because the greater the yield strength of the base metal, the stronger its bearing capacity and the smaller the plastic deformation range.


## 3.1. Yield Strength Changes in theY-Direction

Figure4 shows the Von Mises stress and equivalent plastic strain contour of the USDFLD subroutine welded joint model, in which the yield strength changes in the y-direction. When the gradient of yield strength Gy > 0, the yield strength of the transition zone continuously decreases from the base material side to weld material side. Under external load, the plastic deformation level of both sides is different. That is why the Mises stress field and the equivalent plastic strain field deflect to the side of the weld material totally. The degree of mechanical field deflection increases with yield strength gradient Gy going up. When the yield strength gradient Gy = 0, as shown in Figures 4(a) and 4(b), the yield strength of the base metal area, weld area, and transition area does not change. The entire welded joint is a homogeneous material; thus, the Von Mises stress and equivalent plastic strain at the crack tip are symmetrically distributed with crack.Figure7 shows the Von Mises stress and equivalent plastic strain curves whose data come from the node set of the crack tip mechanical field edge in Figure 4. In Figure 7, the Von Mises stress and the equivalent plastic strain curve around the crack tip mechanical field edge are smooth. The range of 0–180°is close to the base material, and the range of 180–360° is close to the weld material. The crack position is located at θ = 180°.Figure 7
Curve of the USDFLD subroutine welded joint model whose yield strength changes in they-direction. (a) Von Mises stress curve at crack tip. (b) Equivalent plastic strain curve at crack tip.
(a)(b)In Figures7(a) and 7(b), when the yield strength gradient Gy > 0, the Von Mises stress curve and the equivalent plastic strain curve shift to the weld metal side. The curves move more toward the weld metal side with the yield strength gradient Gy going up, which indicates that the deviation of stress field and the plastic zone at crack tip to the weld zone is increasing. The equivalent plastic strain value on the weld side in Figure 7(b) is greater than that close to the base material, which means plastic deformation is more likely to occur on the low yield strength material under load. In Figure 7, when yield strength gradient Gy1 = 0, the base metal and the weld metal have the same yield strength, and the Von Mises stress and the equivalent plastic strain curve on the node set are symmetrically distributed with crack position θ = 180°.Figure8 shows the contour of the mechanical field at the crack tip of sandwich structure welded joint whose yield strength is undermatched and changes along the y-direction. The black dashed line is the dividing line between the weld metal and the base metal. The low match of welded joint is that the yield strength of the base material is higher than that of the weld material. Due to the finite element model of the sandwich structure welded joint, the material property of the continuous change in the yield strength cannot be set. The mechanical properties of the interface between the two materials are discontinuous. This is also the reason for the “dislocation” phenomenon of Von Mises stress and equivalent plastic strain field at the material interface of crack tip. From another perspective, the material boundary of sandwich structure welded joint model can be regarded as a situation where the width of continuous transition zone, which is established by the USDFLD subroutine, is infinitely small and the yield strength change gradient is infinitely great.Figure 8
Contour of the sandwich structure welded joint model whose yield strength changes in they-direction. (a) Von Mises stress contour at crack tip. (b) Equivalent plastic strain contour at crack tip.
(a)(b)Comparing the Von Mises stress distribution range of the crack tip in Figure8(a) with that in Figure 4(a), as well as the equivalent plastic strain distribution range in Figure 8(b) and with that in Figure 4(b), we can find that the “dislocation” phenomenon of the Von Mises stress and equivalent plastic strain field of the crack tip in the sandwich structure welded joint is essentially the deflection of the crack tip mechanical field when the yield strength gradient of the USDFLD subroutine welded joint tends to infinity.Figure9 shows the Von Mises stress and equivalent plastic strain curve on the path of the crack tip mechanical field edge Φ = 0.5 mm in Figure 8. Compared with the USDFLD subroutine model, the numerical curve of the node set around the crack tip of the welded joint is shown in Figure 7. The Von Mises stress and equivalent plasticity curve in Figure 9 are not a smooth curve, but a jumping discontinuity point at the material boundary. Besides, the greater the difference in yield strength between base metal and weld metal, the greater the difference between the Von Mises stress and the equivalent plastic strain on both sides of the material boundary.Figure 9
Curve of the sandwich structure welded joint model whose yield strength changes in they-direction. (a) Von Mises stress curve at crack tip. (b) Equivalent plastic strain curve at crack tip.
(a)(b)In addition, the Von Mises stress in the base metal area in Figure9(a) is slightly greater than that in the weld metal. Because the high yield strength of the base metal has strong resistance against deformation, it causes the Von Mises stress relatively large under the same load. In the equivalent plastic strain curve (Figure 9(b)), since the yield strength of the weld material is less than that in base metal, it will produce greater plastic deformation when subjected to a load, so the equivalent plastic strain in the weld area is much greater than that in base metal. In Figure 9, when the mismatch coefficient My1 = 1, the yield strength of both base metal and weld metal is the same, and the deformation of two areas is the same under the action of external force; thus, the Mises stress and equivalent plastic strain are symmetrically distributed with respect to the crack.


## 3.2. Yield Strength Changes in theX-Direction

Figure10 shows the contour of the mechanical field at the crack tip of the USDFLD subroutine welded joint whose yield strength is undermatched and changes along the x-direction. The welded joint established by the USDFLD subroutine consists of three parts: weld metal, transition zone, and the base metal along the x-direction. The yield strength of the transition zone increases continuously from weld metal to base metal along the x-direction. As the yield strength near the crack tip gradually increases along the x-direction, the bearing capacity of the material increases, and the Von Mises stress distribution range near the crack tip increases and extends to the base material side. The resistance to plastic deformation near the crack tip is also enhanced, so the equivalent plastic strain distribution range gradually decreases.Figure 10
Contour of the USDFLD subroutine welded joint model whose yield strength changes in thex-direction. (a) Von Mises stress contour at crack tip. (b) Equivalent plastic strain contour at crack tip.
(a)(b)The Von Mises stress and equivalent plastic strain curves around the crack tip are shown in Figure11. The angle θ, which is close to the weld metal, is in the range of 0–90° and 270–360°. The range of 90–270° is the side close to the base material. The curve in Figure 11 is a continuous smooth curve, which indicates that the crack tip mechanical field is continuous. Besides, the crack position is symmetrically distributed with respect to θ = 180°. It can be seen from Figure 11(a) that with the increase in yield strength gradient, the peak point of the Von Mises stress curve goes up gradually. The reason for this phenomenon is that high yield strength enhances the material bearing capacity. On the contrary, the increase in the yield strength leads to a decrease in the plastic deformation of the material. Therefore, the peak point of the equivalent plastic strain curve decreases with the increase in the gradient of the yield strength change, as shown in Figure 11(b).Figure 11
Curve of the USDFLD subroutine welded joint model whose yield strength changes in thex-direction. (a) Von Mises stress curve at crack tip. (b) Equivalent plastic strain curve at crack tip.
(a)(b)Figure12 shows the contour of the mechanical field at the crack tip, in which the yield strength of welded joint is low match and changes along the x-direction. The dotted line in the figure is the material boundary. The sandwich structure welded joint model is composed of the weld metal and the base metal along the x-direction. The yield strength of the weld metal is less than that of base metal. Because of the discontinuity between the two materials at the boundary, the crack tip mechanical field in Figure 12 appears “dislocation” at the boundary of the material. In Figure 12(a), when Mx > 0, because the yield strength of the base metal is higher than that of the weld metal, the material bearing capacity in base metal is stronger than that in weld metal. When Mx > 0, the Von Mises stress distribution range in the base metal area of the crack tip is greater than that when Mx = 0. However, when Mx > 0, the equivalent plastic strain distribution range of the base metal region of the crack tip is smaller than that when Mx = 0, as shown in Figure 12(b).Figure 12
Contour of the sandwich structure welded joint model whose yield strength changes in thex-direction. (a) Von Mises stress contour at crack tip. (b) Equivalent plastic strain contour at crack tip.
(a)(b)Figure13 shows the Von Mises stress and equivalent plastic strain curve of the node set around the sandwich structure welded joint crack tip edge, when the yield strength changes along the x-direction. The angle range of 0–90° and 270–360° is weld metal. The angle range of 90–270° is the base metal. The material boundary is at θ = 90° and θ = 270°. θ = 180° is the location of the crack. When the yield strength changes along the x-direction, the crack is perpendicular to the direction of the yield strength change. The Von Mises stress and equivalent plastic strain curves in Figure 13 are symmetrically distributed with respect to the crack.Figure 13
Curve of the sandwich structure welded joint model whose yield strength changes in thex-direction. (a) Von Mises stress curve at crack tip. (b) Equivalent plastic strain curve at crack tip.
(a)(b)Compared with the Von Mises stress and the equivalent plastic strain curve around the crack tip of the welded joint finite element model established by the USDFLD subroutine in Figure12, it can be found that the Von Mises curve and the equivalent plastic strain curve in Figure 13 are not continuous at the material boundary. The discontinuity point is skipped point, and the deviation of these curves at both ends of the discontinuity point, which are shown in Figure 13, goes up with the increase in the yield strength difference between the base metal and the weld metal. In Figure 13(a), the Von Mises stress value in base metal region when Mx < 1 is greater than that when Mx = 1. In Figure 13(a), the equivalent plastic strain value in base metal region when Mx < 1 is smaller than that when Mx = 1. This is because the greater the yield strength of the base metal, the stronger its bearing capacity and the smaller the plastic deformation range.


## 4. Conclusion

By establishing the USDFLD subroutine welding joint model with continuous change in yield strength and the sandwich structure welding joint model, the distribution characteristics of crack tip mechanical field of the two models under the undermatched condition are analyzed. The conclusions are as follows:(1)
The USDFLD subroutine can be used to define the inhomogeneous material model according to the distribution characteristics of the material properties of welded joint, which solves the problem of the discontinuous mechanical field at the material interface of sandwich structure welded joint.(2)
The “dislocation” phenomenon at the crack tip mechanical field of the sandwich structure welded joint is the discontinuous manifestation extension and deflection, which is similar to the USDFLD subroutine welded joint model. The greater the deviation between the yield strength of the weld and the base metal, the more obvious this discontinuity is.(3)
When the direction of low match yield strength and crack is the same, the Von Mises stress and equivalent plastic strain field at the crack tip defined by the USDFLD subroutine are deflected to the weld metal. The greater the yield strength gradient, the greater the deflection of the crack tip mechanical field. The plastic deformation range of the weld zone on the low yield strength side is larger than that of the base zone on the high yield strength side.(4)
When the direction of yield strength is perpendicular to the crack, the equivalent plastic strain field of the crack tip of the undermatched welded joint defined by the USDFLD subroutine extends to the side of the weld material, and the Von Mises stress gradient at the crack tip goes down. The plastic deformation range of the low yield strength material region is larger than that of the high yield strength side.



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*Source: 9009728-2022-09-30.xml*