The clarification of the occurrence mechanism of creep micro cracking at weld joint based on vacancy diffusion analysis

Abstract

Since high temperature creep condition was applied to thermal power plant components for long-term, it is important to evaluate the creep fracture life. Especially, the reduction of the creep strength at weld joint induced by micro creep cracking is important problem.

The creep void initiation and growth under high temperature creep condition is considered to be caused by vacancy diffusion and concentration. Therefore, it is important to clarify the vacancy diffusion and concentration behavior.

In this study, relationship between the site of stress concentration and the vacancy diffusion behavior at weld joint was analyzed using proposed two-dimensional stress induced vacancy diffusion analysis. And it was related to crack initiation at the weld joint.

Keywords

Creep vacancy diffusion weld joint two-dimensional stress induced vacancy diffusion analysis

Nomenclature

$l_{0}$

Distance from the center part to the end of weld metal

l_{1}

Distance from the center part to the end of Heat Affected Zone (HAZ) 1

l_{2}

Distance from the center part to the end of HAZ 2

l_{3}

Distance from the center part to the end of base metal

Vacancy concentration

Diffusion coefficient ( $= D_{0} exp (- Q / R T)$ )

D_{0}

Diffusion constant independent of temperature

Absolute temperature

Activation energy

Gas constant

Δ V

Volume change due to accommodation of a vacancy

σ_{p}

Hydrostatic stress

Young’s modulus

Poison ratio

σ_{ys}

Yield stress

\overline{σ}

Equivalent stress

H_{p}

Work hardening coefficient

\overline{ε_{p}}

Equivalent plastic strain

c, α,

m_{1}

Constant

1. Introduction

Since high temperature creep condition was applied to thermal power plant components for long-term, it is important to evaluate the creep fracture life. Especially, creep rupture at the weld joint of piping is a critical problem [1]. At the weld joint, the material structure of the base metal is changed due to the heat of welding, and the Heat Affected Zone (HAZ) is formed. HAZ is classified as fine-grained HAZ (FG-HAZ) close to the base metal and coarse-grained HAZ (CG-HAZ) close to the weld metal [1,2]. Also, the type of creep rupture at the weld joint is classified into type I to type IV [1]. Creep damage type at the weld joint is shown in Fig. 1 [1]. Type I cracking is the damage in the weld metal. Type II cracking indicates the damage from the weld metal to the HAZ. Type III cracking is the damage in the CG-HAZ. And, type IV cracking which caused in the FG-HAZ is a critical problem [1,2]. Though there are many researches which concern the creep crack growth and the creep damage formation for weld joint [2–5], the mechanism of creep damage formation has not yet been clarified.

Fig. 1.

Schematic illustration of creep crack type for weld joint [1].

On the other hand, it is well known that creep damage occurs by initiation and connection of creep voids induced by vacancy diffusion and concentration [6]. Therefore, investigating the vacancy diffusion behavior in the material is important to clarify the mechanism of creep damage formation [7]. However, it is difficult to observe the vacancy diffusion behavior in experiment. Therefore, numerical analysis is the most effective technique to evaluate the vacancy diffusion behavior. However, there are not many researches relating the mechanism of creep damage formation to the vacancy diffusion behavior for weld joint.

2. Two-dimensional stress induced vacancy diffusion analysis

In this study, in order to clarify the relationship between the site of stress concentration and vacancy diffusion behavior for weld joint, the two-dimensional elastic-plastic FEM analysis coupled with the vacancy diffusion analysis [7,8] was conducted using weld joint model. This analysis was performed as follows [8]. At first, elastic-plastic FEM analysis was conducted. And then, hydrostatic stress obtained by FEM analysis was interpolated to the grid points for the analysis of finite difference analysis (FDA). After that, vacancy diffusion analysis was conducted after calculating the stress gradient. The flowchart of this analysis is shown in Fig. 2.

Fig. 2.

Flowchart of two-dimensional stress induced vacancy diffusion analysis.

2.1. Elastic-plastic FEM analysis

2.1.1. Analytical model

The result of Vickers hardness test using weld joint of P91 steel is shown in Fig. 3. From this result, Vickers hardness in the HAZ is found to be decreased as approaching to the base metal. And, at the boundary between the base metal and the HAZ, that is, in the FG-HAZ, Vickers hardness showed the minimum value.

On the basis of experimental result above, analytical model which represents a simple weld joint was designed. The analytical model is shown in Fig. 4. This model was divided into four parts which is the weld metal, the HAZ 1, the HAZ 2 and the base metal. Where, the HAZ 1 and the HAZ 2 represent the CG-HAZ and the FG-HAZ, respectively. The material properties of the HAZ 2 were defined as the minimum strength. Where, $l_{0} = 10$ mm, $l_{1} = 15$ mm, $l_{2} = 20$ mm, $l_{3} = 50$ mm, $y_{0} = 10$ mm. The ratio of $l_{0}$ , $l_{1}$ , $l_{2}$ and $l_{3}$ were determined to correspond to the result in Fig. 3. The material properties used are as shown in Table 1. Young’s modulus, yield stress and work hardening coefficient were changed according to the difference of Vickers hardness. High Cr steel at the temperature of 500°C was assumed for this analysis. And, in order to discuss the mechanical universality of vacancy concentration behavior, another analysis was conducted by changing the distribution of hardness H under the conditions of $H_{WM} > H_{HAZ} < H_{BM}$ and $H_{WM} < H_{HAZ} > H_{BM}$ . This analytical conditions are shown in Fig. 5 and Tables 2 and 3, respectively. In this analysis, two dimensional symmetrical model was adopted. The number of nodes and elements are 2121 and 4000, respectively. Boundary conditions are shown in Fig. 4 and Fig. 5, respectively. This analysis was conducted under strain-controlled condition.

Fig. 3.

Vickers hardness distribution of P91 weld joint specimen.

Fig. 4.

Schematic illustration of analytical model.

Table 1

Material properties for FEM analysis

		E (kgf/mm²)	ν	$σ_{ys}$ (kgf/mm²)	$H_{p}$ (kgf/mm²)
(a)	WM	2.52E+04	0.3	6.00E+01	2.52E+02
(b)	HAZ 1	2.10E+04	0.3	5.00E+01	2.10E+02
(c)	HAZ 2	1.47E+04	0.3	3.00E+01	1.47E+00
(d)	BM	1.68E+04	0.3	4.00E+01	1.68E+00

Fig. 5.

Analytical model for various hardness distribution (Unit: mm).

Table 2

Material properties for $H_{WM} > H_{HAZ} < H_{BM}$ model

		E (kgf/mm²)	ν	$σ_{ys}$ (kgf/mm²)	$H_{p}$ (kgf/mm²)
(a)	WM	2.10E+04	0.3	5.00E+01	2.10E+02
(b)	HAZ	1.68E+04	0.3	4.00E+01	1.68E+02
(c)	BM	2.10E+04	0.3	5.00E+01	2.10E+02

Table 3

Material properties for $H_{WM} < H_{HAZ} > H_{BM}$ model

		E (kgf/mm²)	ν	$σ_{ys}$ (kgf/mm²)	$H_{p}$ (kgf/mm²)
(a)	WM	1.68E+04	0.3	4.00E+01	1.68E+02
(b)	HAZ	2.10E+04	0.3	5.00E+01	2.10E+02
(c)	BM	1.68E+04	0.3	4.00E+01	1.68E+02

2.1.2. Constitutive laws of FEM analysis

The constitutive laws used in this analysis are mentioned as follows. Elastic-plastic analysis was conducted using the FEM program EPIC-I [9]. The elastic deformation is approximated by Hooke’s law. And, the characteristic of work-hardening for plastic deformation $H_{p}$ is approximated by Eq. (2.1). In this analysis, linear work hardening law as $m_{1} = 1$ was used: $\begin{matrix} (2.1) & H_{p} = \frac{d \overline{σ}}{d {\overline{ε}}_{p}} = m_{1} c {(α + {\overline{ε}}_{p})}^{m_{1} - 1} . \end{matrix}$

2.2. Vacancy diffusion analysis

2.2.1. Basic equation

In this analysis, the basic equation of vacancy diffusion is written as Eq. (2.2) by multiplying the effect of stress gradient potential in Fick’s second law (α multiplication method [8,10]): $\begin{matrix} (2.2) & \frac{\partial C}{\partial t} = \nabla (D \nabla C - α D C \frac{Δ V}{R T} \nabla σ_{p}) . \end{matrix}$

The first and second terms are those of vacancy diffusion due to the concentration gradient and due to the stress gradient, respectively. Where, the effect of the second term is lower than the first term. Therefore, when the concentration gradient exists, only the effect of first term appears and the effect of second term does not operate [8,10]. For this case, it has been reported that the stress induced effect was found to appear by adding the weight coefficient to the second term [8,10]. In this analysis, in order to make the effect of the stress term accelerative and remarkable, weight coefficient α was adopted.

Basic equation of stress induced vacancy diffusion is given by Eq. (2.3): $\begin{array}{rcl} \frac{\partial C}{\partial t} & = & D (\frac{\partial^{2} C}{\partial x^{2}} + \frac{\partial^{2} C}{\partial y^{2}}) - α \frac{D Δ V}{R T} (\frac{\partial C}{\partial x} \frac{\partial σ_{p}}{\partial x} + \frac{\partial C}{\partial y} \frac{\partial σ_{p}}{\partial y}) \\ (2.3) & - α \frac{D Δ V C}{R T} (\frac{\partial^{2} σ_{p}}{\partial x^{2}} + \frac{\partial^{2} σ_{p}}{\partial y^{2}}) . \end{array}$

2.2.2. Analytical model and boundary conditions

Analytical region for vacancy diffusion is the same as the solid line region of FEM analytical model as shown in Fig. 4. As boundary conditions for vacancy diffusion analysis, vacancy diffusion to outside of model was not assumed as shown in Fig. 6. And, as an initial condition, initial vacancy concentration was $C_{0} = 3.0$ in the weld metal. In the other region, initial vacancy concentration was $C_{0} = 1.0$ . The material properties used for vacancy diffusion analysis are shown in Table 4 [11,12].

Fig. 6.

Boundary conditions of vacancy diffusion analysis.

Table 4

Material constants for vacancy diffusion

$D_{0}$	$8.0 \times 10^{- 5}$ [11]	m²/sec
Q	$3.00 \times 10^{3}$ [11]	J/mol
R	8.314	J/K mol
$Δ V$	$2.0 \times 10^{- 6}$ [12]	m³/mol

2.2.3. Method of numerical analysis

In this analysis, the diffusion equation shown in Eq. (2.2) was discretized using the Crank–Nicolson method. And the simultaneous equations were solved by the Successive-Under-Relaxation (SUR) method. Detail method is as follows.

In order to eliminate the physical property on the basic equation and make general analysis, Eq. (2.3) was normalized using Eq. (2.4), and Eq. (2.5) was obtained. Where, a is a representative length, $a = 1 \times 10^{- 3}$ m: $\begin{array}{l} (2.4) & x^{+} = \frac{x}{a}, y^{+} = \frac{y}{a}, C^{+} = \frac{C}{C_{0}}, t^{+} = \frac{t D_{0}}{a^{2}}, \\ (2.5) & \frac{\partial C^{+}}{\partial t^{+}} = (\frac{\partial^{2} C^{+}}{\partial {x^{+}}^{2}} + \frac{\partial^{2} C^{+}}{\partial {y^{+}}^{2}}) + A_{1} \frac{\partial C^{+}}{\partial x^{+}} + A_{2} \frac{\partial C^{+}}{\partial y^{+}} + B C^{+}, \end{array}$ where, $\begin{array}{l} K = \frac{α Δ V}{R T}, \\ (2.6) & A_{1} = - K \frac{\partial σ_{p}}{\partial x}, A_{2} = - K \frac{\partial σ_{p}}{\partial y}, \\ B = - K (\frac{\partial^{2} σ_{p}}{\partial x^{2}} + \frac{\partial^{2} σ_{p}}{\partial y^{2}}) . \end{array}$ Equation (2.7) was obtained by discretizing Eq. (2.5) using Crank–Nicolson implicit method: $\begin{array}{l} \frac{C_{i, j, n + 1} - C_{i, j, n}}{Δ t} \\ = \frac{1}{2} (\frac{C_{i - 1, j, n + 1} - 2 C_{i, j, n + 1} + C_{i + 1, j, n + 1}}{Δ x^{2}} + \frac{C_{i, j - 1 . n + 1} - 2 C_{i, j, n + 1} + C_{i + 1, j, n + 1}}{Δ y^{2}}) \\ + \frac{1}{2} A_{1} (\frac{C_{i + 1, j, n + 1} - C_{i - 1, j, n + 1}}{2 Δ x}) + \frac{1}{2} A_{2} (\frac{C_{i, j + 1, n + 1} - C_{i, j - 1, n + 1}}{2 Δ y}) + \frac{1}{2} B C_{i, j, n + 1} \\ + \frac{1}{2} (\frac{C_{i - 1, j, n} - 2 C_{i, j, n} + C_{i + 1, j, n}}{Δ x^{2}} + \frac{C_{i, j - 1 . n} - 2 C_{i, j, n} + C_{i + 1, j, n}}{Δ y^{2}}) \\ (2.7) & + \frac{1}{2} A_{1} (\frac{C_{i + 1, j, n} - C_{i - 1, j, n}}{2 Δ x}) + \frac{1}{2} A_{2} (\frac{C_{i, j + 1, n} - C_{i, j - 1, n}}{2 Δ y}) + \frac{1}{2} B C_{i, j, n} . \end{array}$

And, for Eq. (2.7), by coordinating the left hand side as unknown term (time step $n + 1$ ) and the right hand side as known term (time step n), Eq. (2.8) was obtained: $\begin{array}{l} (- r_{x} + η_{x} A_{1}) C_{i - 1, j, n + 1} + (2 + 2 r_{x} + 2 r_{y} - Δ t B) C_{i, j, n + 1} + (- r_{x} - η_{x} A_{1}) C_{i + 1, j, n + 1} \\ + (- r_{y} + η_{y} A_{2}) C_{i, j - 1, n + 1} + (- r_{y} - η_{y} A_{2}) C_{i, j + 1, n + 1} \\ = (r_{x} - η_{x} A_{1}) C_{i - 1, j, n} + (2 - 2 r_{x} - 2 r_{y} + Δ t B) C_{i, j, n} + (r_{x} + η_{x} A_{1}) C_{i + 1, j, n} \\ (2.8) & + (r_{y} - η_{y} A_{2}) C_{i, j - 1, n} + (r_{y} + η_{y} A_{2}) C_{i, j + 1, n} . \end{array}$ Where, $\begin{matrix} (2.9) & r_{x} = \frac{Δ t}{Δ x^{2}}, η_{x} = \frac{Δ t}{2 Δ x}, r_{y} = \frac{Δ t}{Δ y^{2}}, η_{y} = \frac{Δ t}{2 Δ y} . \end{matrix}$

Then, in order to solve the difference equation, SUR method was used. Equation (2.11) was obtained by applying SOR (Successive-Over-Relaxation) method to the general equation (2.10) [8,10]: $\begin{array}{l} (2.10) & [K_{i j}] {x_{i}} = {f_{i}}, \\ (2.11) & x_{i}^{(m + 1)} = x_{i}^{(m)} + β {- x_{i}^{(m)} + K_{i j}^{- 1} (f_{i} - \sum_{n = 1}^{i - 1} K_{i j} x_{i}^{(m + 1)} - \sum_{j = i + 1}^{n} K_{i j} x_{j}^{(m)})} . \end{array}$ Where, $K_{i j}$ is coefficient matrix, $x_{i}$ is unknown term, $f_{i}$ is known term, β is relaxation coefficient and m is step of convergence calculation.

Usually, for SOR method, β is taken as the range from 1 to 2. However, since the basic equation includes the first derivative term of vacancy concentration which leads to unstable convergence, we adopted the method of SUR ( $0 < β < 1$ ) [10,13].

To ensure not only mathematical stability but also a physically correct numerical solution, even the Crank–Nicolson method sometimes has limitations on the time and distance increment [10]. In order to perform accurate numerical analysis, it is necessary to satisfy the condition as shown in Eq. (2.12) [10,13]: $\begin{matrix} (2.12) & λ = \frac{Δ t^{+}}{{(Δ r^{+})}^{2}} ⩽ λ_{c} (= 0.3) . \end{matrix}$

Fig. 7.

Distribution of hydrostatic stress.

2.3. Interpolating stress to the difference grid point

The vacancy diffusion analysis was conducted by the difference method using stress induced vacancy diffusion equation. On the other hand, since hydrostatic stress obtained by FEM analysis was calculated as a node value, this value is necessary to be interpolated from the node of FEM analysis to the difference grid point. The interpolation method was referred from the previous studies which carried out similar analytical method [8,14]

3. Analytical results

3.1. Vacancy diffusion analysis corresponding to the actual hardness distribution

Distribution of hydrostatic stress and vacancy obtained by analysis corresponding to the actual hardness distribution are shown in Fig. 7 and Fig. 8, respectively. And, distribution of hydrostatic stress and vacancy at $y = 0$ , $y = 7$ and $y = y_{0}$ are shown in Fig. 9(a)–(c) and Fig. 10(a)–(c), respectively. From these results, high hydrostatic stress region was observed in HAZ 2 from $y = 0$ to $y = 7$ . And, at $y = y_{0}$ , that is, on the surface of analytical model, hydrostatic stress takes the minimum value in HAZ 2. Also, it was found that vacancy concentrates at the position of the local maximum value of hydrostatic stress.

Fig. 8.

Vacancy distribution.

Fig. 9.

Distribution of hydrostatic stress at each coordinate.

Fig. 10.

Vacancy distribution at each coordinate.

Actually, creep voids are initiated at vacancy concentrated region, and micro creep cracking occurs in this region. From this analytical result, it was found that vacancy concentrates not on the surface of analytical model but in the FG-HAZ in the vicinity of center part of analytical model. This result is in good agreement with the distribution of creep void and creep crack (type IV cracking) obtained by creep test using a smooth specimen including welded part [1,15].

3.2. Vacancy diffusion analysis of weld part with other hardness distribution

In this section, in order to discuss the vacancy concentration behavior universally, two patterns of analysis were conducted using the values shown in Tables 2 and 3 for the model as shown in Fig. 5. In this analysis, FG-HAZ was not included. The analytical results of $H_{WM} < H_{HAZ} > H_{BM}$ and $H_{WM} > H_{HAZ} < H_{BM}$ are shown in Figs 11–14 and Figs 15–18, respectively. Under the condition of $H_{WM} < H_{HAZ} > H_{BM}$ , hydrostatic stress and vacancy were found to concentrate at low hardness region near the boundary where hardness changes discontinuously. Even under the condition of $H_{WM} > H_{HAZ} < H_{BM}$ , hydrostatic stress and vacancy were also found to concentrate at low hardness region near the boundary where hardness changes discontinuously. From these results, it was shown that hydrostatic stress concentrates and takes the local maximum value at low hardness region near the boundary where hardness changes, and vacancy concentrates at this position.

Fig. 11.

Distribution of hydrostatic stress for $H_{WM} < H_{HAZ} > H_{BM}$ model.

Fig. 12.

Vacancy distribution for $H_{WM} < H_{HAZ} > H_{BM}$ model.

Fig. 13.

Distribution of hydrostatic stress at each coordinate for $H_{WM} < H_{HAZ} > H_{BM}$ model.

Fig. 14.

Vacancy distribution at each coordinate for $H_{WM} < H_{HAZ} > H_{BM}$ model.

Fig. 15.

Distribution of hydrostatic stress for $H_{WM} > H_{HAZ} < H_{BM}$ model.

Fig. 16.

Vacancy distribution for $H_{WM} > H_{HAZ} < H_{BM}$ model.

Fig. 17.

Distribution of hydrostatic stress at each coordinate for $H_{WM} > H_{HAZ} < H_{BM}$ model.

Fig. 18.

Vacancy distribution at each coordinate for $H_{WM} > H_{HAZ} < H_{BM}$ model.

4. Discussions

Actually, creep voids are initiated at vacancy concentration region, and micro creep cracking occurs in this region. From the analysis corresponding to the actual hardness distribution shown in Section 3.1, vacancy was found to concentrate at the position where type IV cracking occurs. On the other hand, from analysis shown in Section 3.2, it was shown that vacancy concentration occurs at WM part on the surface of analytical model and at WM side in HAZ part in the vicinity of center part due to the hardness distribution. These results were found to correspond to type I and type III cracking, respectively. Vacancy concentration corresponding to type II cracking did not occur even if the hardness distribution was changed variously.

From mentioned above, it was suggested that vacancy concentration at the weld joint is caused by the local maximum value of hydrostatic stress corresponding to the hardness distribution. It was shown that vacancy concentration is dominated by local hardness distribution which concern with type I, III and IV cracking.

5. Conclusions

By conducting the two dimensional stress induced vacancy diffusion analysis for the model of weld joint, the following results were obtained

It was suggested that vacancy concentration at the weld joint is caused by the local maximum value of hydrostatic stress corresponding to the hardness distribution. It was shown that vacancy concentration is dominated by local hardness distribution which concern with type I, III and IV cracking.

Conflict of interest

The authors have no conflict of interest to report.

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