Dominating Driven Factors of Hydrogen Diffusion and Concentration for the Weld Joint–Coupled Analysis of Heat Transfer Induced Thermal Stress Driven Hydrogen Diffusion–

Abstract

Hydrogen embrittlement cracking caused at a weld joint is considered to be dominated by hydrogen diffusion and concentration driven by thermal stress induced by heat transfer during cooling process. The gradient of hydrostatic stress component is considered to be a driven force of hydrogen transportation. However, this problem concerns the occurrence phenomenon during cooling process. Therefore, diffusion coefficient, yield stress and Young’s modulus are changed corresponding with temperature change. Especially, diffusion coefficient shows the space gradient corresponding with space gradient of temperature caused by heat transfer. This affects the diffusion equation of hydrogen as a driven force of hydrogen diffusion. Under these backgrounds, to clarify not only the effect of local thermal stress but also that of space gradient of diffusion coefficient on hydrogen release and trap, the hydrogen diffusion analysis based on our proposed α multiplication and FEM-FDM methods was conducted by introducing the terms of gradients of diffusion coefficient and temperature into the diffusion equation. The following results were obtained. The space gradient of diffusion coefficient was found to contribute the release of hydrogen from the site of stress concentration when the gradient of local hydrogen concentration takes the same sign as that of diffusion coefficient. Concerning the prevention of hydrogen embrittlement cracking at weld joint, these results show that not only Pre-Heat Treatment (PHT) which is a mechanical factor, but also the space gradient of diffusion coefficient which is a factor of material science was found to be one of effective factor of release of hydrogen from a site of stress concentration.

1. Introduction

Hydrogen induced cracking was found to be caused by local thermal stress induced hydrogen concentration at the welded site of the structure during cooling process.^1–10) Especially it is considered to be preferentially caused at the heat affected zone (HAZ).^11,12) For such case, analyses of hydrogen diffusion and concentration are important to understand these behaviours. Concerning the analytical treatment of the differential equation of hydrogen diffusion and concentration behavior, there were many analytical treatments.^13–17) However, from the view point of engineering application, numerical analysis will be useful. Sofronis et al. has analyzed the hydrogen diffusion behavior using the hydrogen diffusion equation considering the stress driving term under existence of diffusive and trapped hydrogen conditions.¹⁸⁾ However, due to some approximated treatments, this analysis closely concerns that under steady state condition such as delayed fracture.¹⁸⁾ To clarify the time sequential transition behavior of hydrogen diffusion behaviors under the time sequential change of diffusion coefficient and thermal stress during the cooling process of weld metal, both of the time sequential effect of ∇σ_P and that of ∇D on hydrogen diffusion behavior should be notice as driven factors of hydrogen diffusion. Previously, Yokobori et al. have been proposed the α multiplication method^19,20) which magnifies stress driving term in the diffusion equation to realize the effect of local stress driven term on hydrogen diffusion. Furthermore, for the application of engineering structure, the FEM-FDM (Finite element method–Finite difference method) analytical method was proposed.^21,22) In this method, stress analysis is conducted by FEM and diffusion analysis is conducted by FDM, respectively, since FEM and FDM are suitable for stress and fluid analysis respectively. Furthermore, this proposed analysis enables us to obtain more reasonable numerical solutions for such time sequential behaviors of hydrogen diffusion. In this research, to clarify both of the effect of local thermal stress and that of ∇D on hydrogen diffusion behaviors, the coupled analysis of heat transfer, thermal stress with hydrogen diffusion was conducted based on α multiplication and FEM-FDM methods by introducing the terms of gradients of diffusion coefficient and temperature into the diffusion equation in the manner of ∇D and ∇T.^23–25)

2. Method of Analysis

2.1 Analyses of hydrogen flow based on the coupled analysis of heat transfer-thermal stress-hydrogen diffusion

To analyze the behaviors of hydrogen diffusion and concentration at the weld joint during cooling process, coupled analysis of Heat transfer-thermal stress-hydrogen diffusion was conducted using our original program software. However, these analytical methods and physical model have been already written in our previous researches,^23–25) to make sure the peculiarity of this method, it is reshown as follows and in sections from 2.2∼2.6. After input initial data and conditions, analysis of heat transfer was conducted by finite difference method (FDM). And temperature at each grid obtained by heat transfer analysis was interpolated to each node for thermal stress analysis based on the finite element method (FEM). Then, thermal stress was calculated for each node point using the interpolated temperature. After that, thermal stress obtained by this analysis was interpolated to each grid point for the analysis of hydrogen diffusion by FDM. Using the interpolated thermal stress, the stress driven hydrogen diffusion analysis was performed. By conducting sequentially these calculations mentioned above, analyses of hydrogen diffusion and concentration behaviours during cooling process were conducted. The flow chart of this analysis is shown in Fig. 1.

Fig. 1

Flow chart of analyses.

Validity of our proposed interpolating method was assured in this research and in our previous literatures.^21,22,36)

2.2 Physical model and boundary conditions

In this study, y-grooved weld joint composed of WM (Weld Metal), HAZ (Heat Affected Zone) and BM (Base Metal) was modeled as shown in Fig. 2.^24,26) This analysis was performed as two-dimensional analysis. The number of nodes and elements for FEM analysis are 6144 and 11713, respectively. The material properties used are as follows. ν = 0.3, T_α = 4.0 × 10⁻⁵, Young’s modulus and yield stress were changed depend on temperature as shown in Fig. 3 and Fig. 4,²⁷⁾ respectively. Considering the cooling from high temperature, yield stress of martensitic structure shown in Fig. 4 was used in this analysis²⁷⁾ similar as previous analysis.^24,25)

Fig. 2

Physical model of analysis.²⁴⁾ (a) Whole physical model of analysis. (b) Detailed of A.

Fig. 3

Temperature dependence of Young’s modulus.²⁷⁾

Fig. 4

Temperature dependence of yield stress.²⁷⁾

This analysis was carried out under the conditions of eq. (1) assuming that yield stress of WM and BM is lower than that of HAZ.²⁸⁾ Young’s modulus, Poisson’s ratio and thermal expansion coefficient are the same for all elements independent of structure such as WM and HAZ. Stress analysis was conducted under plane strain condition. Both ends were fixed for stress analysis. Boundary conditions of heat transfer analysis and hydrogen diffusion analysis are shown in Fig. 5. The number of grid points for FDM are 1500.   

\begin{equation} \sigma_{ys(WM,BM)} = 0.8\sigma_{ys(HAZ)} \end{equation} (1)

Fig. 5

Boundary conditions of heat transfer and hydrogen diffusion analysis.

2.3 Heat transfer analysis

Heat transfer analysis was performed using eq. (2). Equation (2) was discretized and solved by SOR method.^23–25)   

\begin{equation} \rho c\frac{\partial T}{\partial t_{h}}=k\left(\frac{\partial^{2}T}{\partial x^{2}} + \frac{\partial^{2}T}{\partial y^{2}}\right) \end{equation} (2)

Initial temperature at WM was 1500°C, and at HAZ and BM were R.T. (20°C). Boundary conditions for heat transfer analysis are shown in Fig. 5. Material properties as shown in Nomenclature used for heat transfer analysis are as follows.²⁹⁾

k = 0.054 J/(°C·mm·sec), a = 12.0 mm²/sec, ρ·c = 4.5310⁻³ J/(°C·mm³), H = 2.09 × 10⁻⁵ J/(°C·mm²·sec). Only the analysis of cooling process from 1500°C to R.T was conducted in this analysis.

2.4 Thermal stress analysis

The routine of thermal stress analysis was added to the two-dimensional elastic-plastic FEM program EPIC-I.³⁰⁾ The elastic plastic stress-strain law corresponding to the temperature change accompanied by heat transfer was also added to this program software.^23–25) Therefore, the yielding and the plastic deformation laws were changed depending on the temperature change. The elastic deformation is approximated by Hooke’s law. The characteristic of work-hardening for plastic deformation H_p is approximated by eq. (3). In this analysis, linear work hardening law as m₁ = 1 was used. And H_P was assumed to be given by H_P = 0.01 × E for steel.^23–25)

The model was fixed at both end as shown in Fig. 2.   

\begin{equation} H_{p} = \frac{d\overline{\sigma}}{d\overline{\varepsilon}_{p}} = m_{1}c_{1}(\alpha^{*} + \overline{\varepsilon}_{p})^{m_{1}-1} \end{equation} (3)

2.5 Thermal stress induced hydrogen diffusion analysis

The basic equation of hydrogen diffusion is written as eq. (4).¹⁴⁾   

\begin{equation} \frac{\partial C}{\partial t_{C}} = \nabla\left(D\nabla C - DC\frac{\Delta V}{RT}\nabla\sigma_{p}\right) \end{equation} (4)

The first and second terms are those of hydrogen diffusion due to the concentration gradient and due to the stress gradient, respectively. Value of the stress gradient term is usually lower order than that of the concentration gradient term.^19,21) For such case, the effect of concentration gradient term on hydrogen diffusion appears alone and the effect of stress gradient term on hydrogen diffusion does not appear.^19,21) Under this situation, it has been shown that the effect of local stress on hydrogen concentration given by the second term of eq. (4) was found to appear by adding the weight coefficient to this term in eq. (4) in order to take the same order as that of the first term of eq. (4).^19,21) This method is proposed as the α multiplication method.^19,21) By adding the weight coefficient to each term eq. (4), eq. (5) was obtained. In this study, the weight coefficients α_1∼3 take α₁:α₂:α₃ = 1:1000:5.^23–25) And, as an initial condition, initial hydrogen concentration was assumed to be C/C₀ = 3.0 in the weld metal. In the other region, initial hydrogen concentration was C/C₀ = 1.0. The material properties used for hydrogen diffusion analysis are shown as follows.^16,31) D₀ = 5.54 × 10⁻⁶ m/sec, Q = 26.81 × 10³ J/mol,³¹⁾ R = 8.314 J/K·mol and ΔV = 2.0 × 10⁻⁶ m³/mol.¹⁶⁾   

\begin{equation} \frac{\partial C}{\partial t_{C}} = \alpha_{1}D\nabla^{2}C - \alpha_{2}\frac{D\Delta V}{RT}\nabla C\nabla \sigma_{p} - \alpha_{3}\frac{D\Delta V}{RT}C\nabla^{2}\sigma_{p} \end{equation} (5)

The physical meaning of α_i is considered to be the ratio of the entropy differences, ΔS_i of diffusion coefficient in each term of eq. (5) due to the different driven potentials, Δφ_i in the diffusion process between concentration gradient, φ₁ = RTlnC which is related to α₁ and stress gradient, φ₂ = −σ_PΔV which is related to α₂.³²⁾ This comes from the results that diffusion coefficient is not the thermal activated process of enthalpy given by eq. (6a) but that of Gibbs free energy which is a function of enthalpy and entropy given by (6b).³³⁾   

\begin{equation} \mathrm{D} = D_{0}\mathit{exp}\left(-\frac{\Delta H}{RT}\right) \end{equation} (6a)

  

\begin{equation} \mathrm{D} = D_{0}\mathit{exp}\left(-\frac{\Delta G}{RT}\right) = D_{0}\mathit{exp}\left(-\frac{\Delta H - T\Delta S}{RT}\right) \end{equation} (6b)

Where ΔH is enthalpy, ΔG is Gibbs free energy and ΔS is entropy.

Concerning α₃, essentially for the case of elastic stress field, ∇²σ_P is identically zero^16,19,21) which corresponds with α₃ = 0, however, for the case of plastic stress field, dissipation term, Q_D appears due to irreversible deformation as shown in eq. (7). For such case, hydrogen concentration in whole area decreases. However, in this case, since ∇²σ_P is not zero,²¹⁾ the term of ∇²σ_P was noticed and it was used to cover the effect of Q_D on eq. (7).²¹⁾ Usually, value of α₃ is empirically found to be kept as 0 ≤ α₃ ≤ 0.5α₂.²¹⁾

In this article, α₃ = 0.005α₂ was used. It means the target materials used for this analysis behave under the state of small scale plasticity.   

\begin{align} \frac{\partial C}{\partial t_{C}} &= \alpha_{1}D\nabla^{2}C - \alpha_{2}\frac{D\Delta V}{RT}\nabla C\nabla \sigma_{p} \\ &\quad- \alpha_{3}\frac{D\Delta V}{RT}C\nabla^{2}\sigma_{p} + Q_{D} \end{align} (7)

2.6 Method of numerical analyses

The equations of heat transfer and hydrogen diffusion shown in eqs. (2) and (5) were discretized using the Crank-Nicolson implicit method.^19–25) And the simultaneous equations were solved by the SOR method for eq. (2) and by SUR (Successive-Under-Relaxation) method for eq. (5).¹⁹⁾ The detailed method is as follows.^23–25)

In order to eliminate the physical property on the basic equation and make general analysis, eqs. (2) and (5) were normalized using eqs. (8) and (9), and eqs. (10) and (11) were obtained, respectively.   

\begin{equation} x^{+} = \frac{x}{l_{0}},\quad y^{+} = \frac{y}{l_{0}},\quad t_{h}^{+} = \frac{at_{h}}{l_{0}^{2}},\quad a = \frac{k}{\rho c}. \end{equation} (8)

  

\begin{equation} x^{+} = \frac{x}{l_{0}},\quad y^{+} = \frac{y}{l_{0}},\quad C^{+} = \frac{C}{C_{0}},\ D^{+} = \frac{D}{D_{0}},\quad t_{C}^{+} = \frac{t_{C}D_{0}}{l_{0}^{2}}. \end{equation} (9)

  

\begin{equation} \frac{\partial T}{\partial t_{h}^{+}} = \left(\frac{\partial^{2}T}{\partial x^{+2}} + \frac{\partial^{2}T}{\partial y^{+2}}\right). \end{equation} (10)

  

\begin{align} \frac{\partial C^{+}}{\partial t_{C}^{+}} &= \alpha_{1}\left\{D^{+}\left(\frac{\partial^{2}C^{+}}{\partial x^{+2}} + \frac{\partial^{2}C^{+}}{\partial y^{+2}}\right) + \left(\frac{\partial D^{+}}{\partial x^{+}}\frac{\partial C^{+}}{\partial x^{+}} + \frac{\partial D^{+}}{\partial y^{+}}\frac{\partial C^{+}}{\partial y^{+}}\right)\right\} - \frac{\Delta V}{R}\biggl[\alpha_{2}\biggl\{\frac{C^{+}}{T}\left(\frac{\partial D^{+}}{\partial x^{+}}\frac{\partial\sigma_{p}}{\partial x^{+}} + \frac{\partial D^{+}}{\partial y^{+}}\frac{\partial\sigma_{p}}{\partial y^{+}}\right)\\ &\quad + \frac{D^{+}}{T}\left(\frac{\partial C^{+}}{\partial x^{+}}\frac{\partial\sigma_{p}}{\partial x^{+}} + \frac{\partial C^{+}}{\partial y^{+}}\frac{\partial\sigma_{p}}{\partial y^{+}}\right) - \frac{D^{+}C^{+}}{T^{2}}\left(\frac{\partial T}{\partial x^{+}}\frac{\partial \sigma_{p}}{\partial x^{+}} + \frac{\partial T}{\partial y^{+}}\frac{\partial\sigma_{p}}{\partial y^{+}}\right)\biggr\} + \alpha_{3}\frac{D^{+}C^{+}}{T}\left(\frac{\partial^{2}\sigma_{p}}{\partial x^{+2}} + \frac{\partial^{2}\sigma_{p}}{\partial y^{+2}}\right)\biggr] \end{align} (11)

Equations (10) and (11) were discretized using Crank-Nicolson’s implicit method. And, by coordinating the left hand side as unknown term (time step n + 1) and the right hand side as known term (time step n), eqs. (12) and (14) were obtained, respectively.   

\begin{align} &(1 + r_{x} + r_{y})T_{i,j,n + 1} - \frac{r_{x}}{2}T_{i - 1,j,n + 1} - \frac{r_{y}}{2}T_{i,j - 1,n + 1} - \frac{r_{x}}{2}T_{i + 1,j,n + 1} - \frac{r_{y}}{2}T_{i,j + 1,n + 1}\\ &\quad= (1 - r_{x} - r_{y})T_{i,j,n + 1} + \frac{r_{x}}{2}T_{i - 1,j,n + 1} + \frac{r_{y}}{2}T_{i,j - 1,n + 1} + \frac{r_{x}}{2}T_{i + 1,j,n + 1} + \frac{r_{y}}{2}T_{i,j + 1,n + 1} \end{align} (12)

  

\begin{equation} \text{Where, $r_{x}$} = \frac{\Delta t^{+}{}_{h}}{(\Delta x^{+})^{2}},\quad r_{y} = \frac{\Delta t^{+}{}_{h}}{(\Delta y^{+})^{2}}. \end{equation} (13)

Equations (12) and (14) were determined according to the boundary conditions.

Then, in order to solve the difference equation, SOR and SUR method was used for eqs. (12) and (14) respectively.^23–25) The general equations were given by eqs. (16) and (17).^19,20) Where, K_ij is coefficient matrix, x_i is unknown term, f_i is known term, β is relaxation coefficient and m is step number of convergence calculation. Usually, for SOR method, β is taken as 1 < β < 2. However, since the basic equation includes the first derivative term of hydrogen concentration such as eq. (5) which leads to unstable convergence, we adopted the method of SUR (Successive-Under-Relaxation) (0 < β < 1) for eq. (17).^19,34)

To ensure not only mathematical stability but also a physically correct numerical solution, even the Crank-Nicolson’s method sometimes has limitations on the increment values of time and distance, that is, Δt⁺ and Δr_x⁺, Δr_y⁺.^19,34) In order to perform accurate numerical analysis, it is necessary to satisfy the condition as shown in eq. (18).¹⁹⁾

In eqs. (15a)∼(15f), terms of $\frac{\partial D^{ + }}{\partial X^{ + }}$, $\frac{\partial D^{ + }}{\partial y^{ + }}$ and $\frac{\partial T}{\partial X^{ + }}$, $\frac{\partial T}{\partial y^{ + }}$ are found to be included in the diffusion equation of eq. (14) as space distribution functions of D⁺ and T, that is, D⁺(x, y) and T(x, y). Values of D⁺ and T were calculated by heat transfer analysis and they were used for the hydrogen diffusion analysis of eq. (14) based on coupled analysis of heat transfer, thermal stress with hydrogen diffusion. Therefore, the effects of $\frac{\partial D^{ + }}{\partial X^{ + }}$, $\frac{\partial D^{ + }}{\partial y^{ + }}$ and $\frac{\partial T}{\partial X^{ + }}$, $\frac{\partial T}{\partial y^{ + }}$, that is ∇D⁺ on hydrogen concentration were clearly reflected by this analysis which are the characteristic terms distinguished from other analysis.¹⁸⁾   

\begin{align} &(-\alpha_{1}r_{x}D^{+}+\eta_{x}A_{1})C_{i - 1,j,n + 1} \\ &\quad+ (2 + \alpha_{1}(2r_{x} + 2r_{y})D^{+} - \Delta tB)C_{i,j,n + 1}\\ &\quad+(-\alpha_{1}r_{x}D^{+} - \eta_{x}A_{1})C_{i + 1,j,n + 1} \\ &\quad+ (-\alpha_{1}r_{y}D^{+} + \eta_{y}A_{2})C_{i,j - 1,n + 1} \\ &\quad+ (-\alpha_{1}r_{y}D^{+} - \eta_{y}A_{2})C_{i,j + 1,n + 1}\\ &=(\alpha_{1}r_{x}D^{+} - \eta_{x}A_{1})C_{i-1,j,n} \\ &\quad+ (2-\alpha_{1}(2r_{x} + 2r_{y})D^{+} + \Delta tB)C_{i,j,n}\\ &\quad+ (\alpha_{1}r_{x}D^{+} + \eta_{x}A_{1})C_{i + 1,j,n} \\ &\quad+ (\alpha_{1}r_{y}D^{+} - \eta_{y}A_{2})C_{i,j - 1,n} + (\alpha_{1}r_{y}D^{+} + \eta_{y}A_{2})C_{i,j + 1,n} \end{align} (14)

  

\begin{equation} \text{where $r_{x}$} = \frac{\Delta t_{C}^{+}}{(\Delta x^{+})^{2}},\quad r_{y} = \frac{\Delta t_{C}^{+}}{(\Delta y^{+})^{2}}, \end{equation} (15a)

  

\begin{equation} \eta_{x} = \frac{\Delta t_{C}^{+}}{2\Delta x^{+}},\quad \eta_{y} = \frac{\Delta t_{C}^{+}}{2\Delta y^{+}}, \end{equation} (15b)

  

\begin{equation} A_{1} = \alpha_{1}\frac{\partial D^{+}}{\partial x^{+}} - \frac{K_{1}D^{+}}{T}\frac{\partial \sigma_{p}}{\partial x^{+}}, \end{equation} (15c)

  

\begin{equation} A_{2} = \alpha_{1}\frac{\partial D^{+}}{\partial y^{+}} - \frac{K_{1}D^{+}}{T}\frac{\partial \sigma_{p}}{\partial y^{+}}, \end{equation} (15d)

  

\begin{equation} K_{1} = \alpha_{2}\frac{\Delta V}{R},\quad K_{2} = \alpha_{3}\frac{\Delta V}{R}, \end{equation} (15e)

  

\begin{equation} B= -\frac{1}{T} \times \biggl\{K_{1}\left(\frac{\partial D^{+}}{\partial x^{+}}\frac{\partial \sigma_{p}}{\partial x^{+}} + \frac{\partial D^{+}}{\partial y^{+}}\frac{\partial \sigma_{p}}{\partial y^{+}}\right) - \frac{K_{2}D^{+}}{T}\left(\frac{\partial T}{\partial x^{+}}\frac{\partial\sigma_{p}}{\partial x^{+}} + \frac{\partial T}{\partial y^{+}}\frac{\partial\sigma_{p}}{\partial y^{+}}\right)+ K_{2}D^{+} \left(\frac{\partial^{2}\sigma_{p}}{\partial x^{2+}} + \frac{\partial^{2}\sigma_{p}}{\partial y^{2+}}\right)\biggr\}. \end{equation} (15f)

  

\begin{equation} [K_{ij}]\left\{X_{i}\right\} = \left\{f_{i}\right\} \end{equation} (16)

  

\begin{equation} X_{i}^{(m + 1)} = X_{i}^{(m)} + \beta\left\{-X_{i}^{(m)} + K_{ij}^{-1}\left(f_{i} - \sum_{n=1}^{i-1}K_{ij}X_{i}^{(m+1)} - \sum_{j=i+1}^{n}K_{ij}X_{j}^{(m)}\right)\right\} \end{equation} (17)

  

\begin{equation} (r_{x},r_{y}) \leq r_{c}\ (=0.4). \end{equation} (18)

The hydrogen diffusion and concentration analysis was conducted by the finite difference method (FDM analysis). However, various stress components obtained by stress analysis were calculated as the node values for FEM analysis. Therefore, these values are necessary to be interpolated from value at each node point for FEM analysis to that at each grid point for FDM analysis. Detailed descriptions of interpolation method were written in previous studies which carried out similar analytical method.^21,22,36)

3. Results of Analysis

3.1 Hydrogen concentration behaviors dominated by effects of ∇σ_P, ∇D and ∇T on hydrogen diffusion

Numerical results of two dimensional (2D) distribution of hydrogen concentration during cooling process at the y grooved weld joint without Pre-Heat Treatment (PHT) are shown in Figs. 6(a), (b) and (c). These results showed that, with increase in time, at first, the maximum hydrogen concentration increases from C⁺ = 3.0 (0 step) to 3.39 (5 step) and after that, it decreases to C⁺ = 2.60 (1000 step). Furthermore, at first, hydrogen was found to concentrate along the HAZ line in the region of WM (Fig. 6(b)) and finally, the hydrogen concentrated region localizes around the central region of WM linked with groove bottom (Fig. 6(c)). The hydrogen concentrated region as shown in Fig. 6(c) is in good agreement with experimental result of y-grooved weld cracking as shown in Fig. 7.³⁵⁾

Fig. 6

Numerical results of 2D distribution of hydrogen concentration during cooling process at the y grooved weld joint. Stress analysis were conducted under plane strain and both end fixed. (a) 0 step (C⁺_max = 3.0). (b) 5 step (0.39 sec) (C⁺_max = 3.39). (c) 1000 step (78 sec) (C⁺_max = 2.60).

Fig. 7

Experimental result of y-grooved weld cracking.³⁵⁾

In Fig. 6, at HAZ region, there are some sites where distribution of hydrogen concentration shows discontinuous behavior. This will be due to the gradation of hydrostatic stress around HAZ as shown in Fig. 8 (That is, yellow, green and blue level regions of hydrostatic stress exist at HAZ). Much more fine division of FDM grids will give us more continuous distribution of hydrogen concentration.

Fig. 8

Numerical results of 2D distribution of hydrostatic stress during cooling process at the y grooved weld joint without PHT and with PHT of 100°C. Stress analysis were conducted under plane strain and both end fixed. (a) 1 step (0.078 sec, σ_p,max = 131.6 MPa, σ_p,min = −178.4 MPa, without PHT). (b) 5000 step (390 sec, σ_p,max = 1087.8 MPa, σ_p,min = −74.7 MPa, without PHT). (c) 5000 step (390 sec, σ_p,max = 1012.5 MPa, σ_p,min = −73.5 MPa, PHT of 100°C).

3.2 Hydrogen concentration behaviors dominated by effects of stress driven term, ∇σ_P on hydrogen diffusion under without PHT and with PHT

Numerical results of two dimensional (2D) distribution of hydrostatic stress during cooling process at the y grooved weld joint with and without PHT are shown in Figs. 8(a), (b) and (c) for the case without the effect of ∇D and ∇T on hydrogen diffusion given by eq. (19).   

\begin{align} \frac{\partial C^{+}}{\partial t_{C}^{+}} &= \alpha_{1}D^{+}\left(\frac{\partial^{2}C^{+}}{\partial x^{+2}} + \frac{\partial^{2}C^{+}}{\partial y^{+2}}\right)\\ &\quad- \alpha_{2}\frac{D^{+}}{T}\left(\frac{\partial C^{+}}{\partial x^{+}}\frac{\partial\sigma_{p}}{\partial x^{+}} + \frac{\partial C^{+}}{\partial y^{+}}\frac{\partial\sigma_{p}}{\partial y^{+}}\right) \\ &\quad+ \alpha_{3}\frac{D^{+}C^{+}}{T}\left(\frac{\partial^{2}\sigma_{p}}{\partial x^{+2}} + \frac{\partial^{2}\sigma_{p}}{\partial y^{+2}}\right) \end{align} (19)

These results showed that at first, the maximum hydrostatic stress concentrate in the region of WM along the HAZ line (Fig. 8(a)) and with increase in time, stress concentration extends to the region of WM as shown in Fig. 8(b). Hydrostatic stress under PHT condition decreases as compared with that under without PHT condition as shown in Figs. 8(b) and (c). Correspondingly, the gradient of hydrostatic stress which is a driven force of hydrogen concentration decreases. This causes decrease in hydrogen concentration as shown in Fig. 9(a) and (b). That is, the maximum hydrogen concentration decreases from C⁺ = 2.45 (5000 step) to 1.86 (5000 step) by PHT.

Fig. 9

Numerical results of 2D distribution of hydrogen concentration during cooling process at the y grooved weld joint dominated by term of the gradient of stress alone for without and with PHT of 100°C. Stress analysis were conducted under plane strain and both end fixed. (a) 5000 step (390 sec, C⁺_max = 2.48, without PHT). (b) 5000 step (390 sec, C⁺_max = 1.86, with PHT of 100°C).

Furthermore, hydrogen concentration was found to localize in the central WM region and it was not eminent at the bottom site of WM. That is, high concentrated region of hydrogen does not extend to the groove bottom of WM which is different from that of Figs. 6(a), (b) and (c). This means that the effect of ∇D on hydrogen concentration plays a role of transportation of hydrogen to the site of the groove bottom of WM. This results were clearly shown in Figs. 11(a) and (b). These results showed that ∇σ_P was found to cause local hydrogen concentration at the upper central site of WM. On the other hand, ∇D was found to play a role of transportation of hydrogen to the site of the groove bottom of WM, since the site of maximum hydrogen concentration was found to move toward to the site of groove bottom of WM.

3.3 Hydrogen concentration behaviors dominated by effect of the multiplied diffusive term, α₁ on hydrogen diffusion under with and without PHT

Numerical results of two dimensional (2D) distribution of hydrogen concentration during cooling process at the y grooved weld joint with and without PHT for the case of multiplied diffusive term, that is, α₁ = 5 are shown in Figs. 10(a), (b). These results showed that the effect of PHT on hydrogen concentration exists, even though the effect of ∇D on hydrogen concentration is included in the hydrogen diffusion equation. Furthermore, hydrogen concentration typically decreases as compared with that of α₁ = 1. This means that promotion of diffusive term of hydrogen by α₁ causes flow out of hydrogen from WM which results in the release of hydrogen from the region of WM.

Fig. 10

Numerical results of 2D distribution of hydrogen concentration during cooling process at the y grooved weld joint for the case of α₁ = 5 under without and with PHT of 100°C. Stress analysis were conducted under plane strain and both ends were fixed. (a) 5000 step (390 sec, C⁺_max = 1.81, without PHT). (b) 5000 step (390 sec, C⁺_max = 1.58, with PHT of 100°C).

4. Discussions

Results of Figs. 11(a) and (b) showed that the effect of ∇σ_P was found to cause local hydrogen concentration at the maximum site of hydrostatic stress due to the driven force of ∇σ_P as shown in Fig. 11 and 12. That is, the site of maximum hydrostatic stress as shown in Fig. 12 is in good agreement with that of maximum hydrogen concentration dominated by ∇σ_P alone as shown in Fig. 11. However, since the original hydrogen diffusion equation of eq. (11) includes the term of ∇D, it causes release of hydrogen from WM by ∇D. Results of Fig. 11 showed that the effect of ∇D on hydrogen concentration shifts the site of maximum hydrogen concentration toward the groove bottom direction, that is, it causes hydrogen release, that is, flow out from WM when the gradient of local hydrogen concentration takes the same sign as that of diffusion coefficient which corresponds with increase in α₁. Therefore, when the value of diffusive term of eq. (11) multiplied by α₁ increases, for example, α₁ = 5, hydrogen was found to be typically flowed out from WM which results in decrease in hydrogen concentration in the region of WM as shown in Fig. 11. Therefore, hydrogen concentration can be decreased not only by decrease in residual thermal stress due to PHT but also by selecting hydrogen diffusion coefficient,²⁵⁾ that is increase in diffusion coefficient which results in existence of ∇D.

Fig. 11

Hydrogen distribution in the y direction at x⁺ = 62 without PHT. (a) 1000 step (78 sec). (b) 5000 step (390 sec).

Fig. 12

Time sequential change of hydrostatic stress distribution at x = 62.²⁴⁾

5. Conclusion

1. (1)    The effect of ∇D on hydrogen concentration plays an important role of release, that is, flow out of hydrogen from the site of the maximum hydrostatic stress. Therefore, the space gradient of hydrogen diffusion coefficient between BM and WM is also effective on the release of hydrogen from the WM in the manner of ∇D. ∇D is caused by temperature gradient during cooling process or selecting different diffusion coefficients between BM and WM.
2. (2)    PHT was also found to decrease the residual thermal stress, that is the gradient of thermal residual stress which results in decrease in hydrogen concentration.
3. (3)    From these results mentioned above, to prevent hydrogen embrittlement cracking at the weld joint, however, PHT contributes the decrease in the gradient of thermal residual stress which results in decrease in hydrogen concentration, the space distribution of diffusion coefficient, ∇D was found to be an alternative effective factor of decrease in hydrogen concentration by release of hydrogen from WM.

Nomenclature

l₀

Length of analytical model 0.15 m

C

Hydrogen concentration

C₀

Hydrogen concentration in atmosphere

D

Diffusion coefficient (= D₀ exp(−Q/RT))

D₀

Diffusion constant independent of temperature

T

Absolute temperature

T_out

Temperature in atmosphere

Q

Activation energy of hydrogen diffusion

R

Gas constant

ΔV

Volume change due to accommodation of a hydrogen

σ_p

Hydrostatic stress

E

Young’s modulus

ν

Poisson’s ratio

σ_ys

Yield stress

$\overline{\sigma }$

Equivalent stress

H_p

Work hardening coefficient

$\overline{\varepsilon _{p}}$

Equivalent plastic strain

c₁, α*, m₁

Constant

H

Hydrogen transfer coefficient

t_c

Time of hydrogen diffusion

t_h

Time of heat transfer

ρ

Density

C

Specific heat

K

Heat conductivity (= ρ·c·a)

A

Thermal diffusion coefficient

H

surface coefficient of heat transfer

h_T

= H/k

Tα

Thermal expansion coefficient

Acknowledgments

This work was supported by Council for Science, Technology and Innovation (CSTI), Cross-ministerial Strategic Innovation Promotion Program (SIP), “Structural Materials for Innovation S” (Funding agency: JST).

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