Prediction of Creep Rupture Time Using Constitutive Laws and Damage Rules in 9Cr–1Mo–V–Nb Steel Welds

Abstract

Long-term creep tests for the parent metal, a simulated heat-affected zone (HAZ) and a weld joint were conducted based on 9Cr–1Mo–V–Nb steel. The creep rupture time, t_r, for the weld joint was predicted by computationally simulating creep damage based on Norton’s law combined with the time exhaustion rule (TER), where the equivalent stress (σ_eq), maximum principal stress (σ₁), or Huddleston stress (σ_hud) was used to evaluate the rupture time. Creep damage analysis was also conducted based on the Hayhurst-type damage mechanics rule (HDR), in which creep rupture time was evaluated in relation to the rupture stress, σ_r. The computed creep rupture times and damage distributions were compared with the experimentally obtained rupture time and void distribution of the actual weld joint, respectively. Furthermore, the effect of the bevel angle of the HAZ was examined. The key findings from this study were as follows: (1) cautious predictions could be obtained for a loading stress of 80 MPa and higher; (2) the computed fracture initiation positions were on the HAZ boundaries, consistent with the type IV fracture observed at the stress of 80 MPa; (3) the magnitudes of the predicted rupture times were t_r (σ₁) < t_r (σ_hud) ≈ t_r (σ_r) < t_r (σ_eq); (4) the bevel angle dependence previously reported was reasonably reproduced with the TER models that used σ_eq and σ_hud and with the HDR model.

1. Introduction

To ensure the mechanical reliability of structural materials used in high-temperature environments, such as those found in boilers or heat exchangers, assessment of their time-dependent performance is of crucial importance. The creep property is among the most critical mechanical properties impacting performance. Creep failure often occurs at welds, and therefore various creep tests on welds have been conducted. However, because such creep tests are time-consuming and expensive, they have hindered material development and the optimization of structural design.

Computational approaches are widely expected to resolve such problems, thereby potentially reducing the number and duration of creep tests necessary to ensure that safety standards are met. In order to evaluate creep damage to structural components, several creep damage accumulation models have been proposed in previous studies, such as the time exhaustion rule, strain exhaustion rule, and damage mechanics.^1–4) In creep damage analyses, it is considered important to evaluate the effect of the multiaxial stress state on the creep damage distribution and evolution.

For the high-Cr ferritic heat-resisting steels used in ultra-supercritical (USC) thermal power plants, type IV failures in the heat-affected zone (HAZ)⁵⁾ become a critical issue. Damage distributions and the creep life of type IV failures have been studied using creep damage accumulation models.^6–9) These studies confirmed that creep void distributions of structures that have a complicated shape such as weld joints can be well predicted using damage analysis, but have scarcely examined the prediction of the long-term creep rupture time at welds.⁸⁾ Furthermore, the contribution of different types of creep damage rules in creep damage analysis has not been investigated.

In this study, we conducted long-term creep tests on the parent metal, a simulated HAZ, and a weld joint using 9Cr–1Mo–V–Nb steel (ASME Gr. 91). We also conducted a creep damage analysis using a finite element model (FEM) based on two creep damage models: the time exhaustion rule (TER) model and the Hayhurst-type damage mechanics rule (HDR) model. We analyzed the characteristics and accuracy of each model by comparing the computed and experimental results under several stress conditions.

2. Experimental and Computational Simulation Procedures

2.1 Material and creep tests

The material used in this study was 9Cr–1Mo–V–Nb steel (ASME Gr. 91), and its chemical composition is shown in Table 1. A tube of this material was normalized at 1050°C for 60 min and then tempered at 760°C for 60 min. This material was labeled as MGB in the NIMS Creep Data Sheet 43A.¹⁰⁾ Using this material, a weld joint with a single “V” groove was made by gas tungsten arc welding (GTAW) using welding wire (TGS-9Cb). The weld was located in the center of the specimen, and the bevel angle of the weld joint was 30°. After welding, the welded joint was subjected to post-weld heat treatment for 4 h at 740°C. Creep tests on the weld joint were conducted at 600°C using a round bar specimen cut from the welded tube with a gage length, L, of 30 mm and a diameter, D, of 6 mm.

Table 1 Chemical composition of 9Cr–1Mo–V–Nb steel tube (mass percent).¹⁰⁾

To conduct creep damage analysis of the weld joint using the finite element method, the creep parameters were required for each material: the parent material (PM), the weld metal (WM), and the simulated HAZ. The creep properties of the PM were published in the NIMS Creep Data Sheet 43A.¹⁰⁾ We assumed that the WM had the same properties as the PM.

To evaluate the material and creep properties of the HAZ, a fine-grained HAZ specimen was simulated. HAZ heat treatment was simulated on the PM using the Gleeble testing machine as a welding simulator, which applied rapid heating (60°C/s) to a peak temperature of 900°C, followed by gas cooling (40°C/s). The simulated HAZ specimen was a round bar with a length of 15 mm and diameter of 4 mm. The creep properties of the simulated HAZ were obtained by experimental creep tests on an actual HAZ specimen.

Table 2 shows the tensile properties of the PM and HAZ.⁸⁾ Tables 3, 4, and 5 show the creep parameters of the PM and HAZ. The creep constitutive law and damage rules related to these parameters are discussed in the following section.

Table 2 Tensile properties of 9Cr–1Mo–V–Nb steel at 600°C.⁸⁾

Table 3 Parameters for secondary creep in eqs. (4) and (5)* at 600°C.

Table 4 Parameters for creep rupture time in eq. (2)* of 9Cr–1Mo–V–Nb steel at 600°C.

Table 5 Creep parameters of Hayhurst-type damage rule for 9Cr–1Mo–V–Nb steel at 600°C.

2.2 Creep constitutive law and damage rule

2.2.1 Time exhaustion rule

For the creep damage analysis, we used two types of damage mechanics models: the time exhaustion rule (TER),¹⁾ and the constitutive equation of the Hayhurst-type damage mechanics rule (HDR).²⁾

The former is a well-known technique for evaluating creep life time.¹⁾ The formulations of the TER model are as follows:   

\begin{equation} D_{\text{c}}=\int_{0}^{t}\frac{1}{t_{\text{r}}}\mathrm{d}t, \end{equation} (1)

where D_c is the creep damage. As the damage accumulates, D_c increases, with creep rupture at D_c = 1.0. The creep rupture time t_r can be written as a function of the local stress σ   

\begin{equation} t_{\text{r}}=a_{2}\sigma^{a_{1}}, \end{equation} (2)

where a₁ and a₂ are the creep parameters of the material, which can be obtained by fitting to the experimentally determined relationship between the creep rupture time t_r and the applied stress σ_a for each material, PM and HAZ. Table 4 shows these parameters obtained from creep tests on the PM and HAZ of the present Gr. 91 steel.

The creep parameters of the PM and HAZ were obtained from uniaxial creep tests. However, multiaxial stress occurs in complicated structures like weld joints and affects the creep behavior.¹¹⁾ For the TER model, the effect of the multiaxiality appears in the creep damage, D_c, through eq. (2); that is, D_c depends on what type of scalar stress is used. Here, we use three types: (i) equivalent stress, σ_eq, (ii) maximum principal stress, σ₁, and (iii) Huddleston stress, σ_hud. σ_hud is given by   

\begin{equation} \sigma_{\text{hud}}=\sigma_{\text{eq}}\exp\left[b\left(\frac{\sigma_{1}+\sigma_{2}+\sigma_{3}}{\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}+\sigma_{3}^{2}}}-1.0\right)\right], \end{equation} (3)

where σ₁, σ₂, and σ₃ are the principal stresses, and b is the material constant relating to the multiaxial stress state.³⁾ From Ref. 3, b = 0.24 gives significantly better life predictions based on available biaxial creep data for types 304 and 316 stainless steel and Ni base alloy. These stress values were calculated by the FEM using the TER with Norton’s constitutive law:   

\begin{equation} \dot{\varepsilon}_{\text{eq}}=A\sigma_{\text{eq}}^{n}, \end{equation} (4)

where $\dot{\varepsilon }_{\text{eq}}$ is the equivalent creep strain rate, and σ_eq is the equivalent stress; A and n are material constants. These constants of Norton’s law were obtained from creep tests on the PM and HAZ as shown in Table 3.

2.2.2 Hayhurst-type damage rule

The Hayhurst-type damage rule (HDR) has often been used since its development in 1983.²⁾ This rule considers the effect of the accelerating creep deformation that occurs just before the fracture in such a way that the creep strain rate $\dot{\varepsilon }$ and damage rate $\dot{\omega }$ are accelerated by the damage variable ω and the multiaxiality of the stress. Such a positive feedback relationship between the strain rate and the damage values characterizes the HDR model. The damage value ω can be calculated by applying eqs. (5), (6), and (7) to the FEM.   

\begin{equation} \dot{\varepsilon}_{ij}=\frac{3}{2}A\left[\frac{\sigma_{\text{eq}}}{1-\omega}\right]^{n}\frac{S_{ij}}{\sigma_{\text{eq}}}t^{m} \end{equation} (5)

  

\begin{equation} \dot{\omega}=\frac{M\sigma_{\text{r}}^{\chi}}{(1+\phi)(1-\omega)^{\phi}}t^{m} \end{equation} (6)

  

\begin{equation} \sigma_{\text{r}}=\alpha\sigma_{1}+(1-\alpha)\sigma_{\text{eq}} \end{equation} (7)

Here, as with D_c, when the ω value reaches 1, creep fracture occurs; σ_r is the rupture stress, σ₁ is the maximum principal stress, and σ_eq is the equivalent stress; S_ij is the deviatoric stress; and A, n, m, M, ϕ, and χ are material constants obtained from experimental creep data. A and n have the same values as when used in Norton’s law, as shown in Table 3; m, M, ϕ, and χ are shown in Table 5. α is a parameter that represents the multiaxial stress state. Here, we used α = 0.3, from Ref. 8.

2.3 FEM analysis

Figure 1 shows the FEM of the weld joint creep test specimen, the dimensions of which were measured from the actual weld joint. The size of the fine-grained HAZ was approximately 1.3 mm in width. The FEM mesh consisted of 22578 nodes and 20230 primary elements. In this study, an in-house finite element analysis code was used for the creep damage analysis. The creep damage calculations were conducted under tensile stresses of 50, 80, 100, and 120 MPa, and the calculated creep rupture time and the damage distribution were compared with the experimentally obtained rupture time and the creep void distribution. Because it has been reported that the bevel angle of the HAZ region affects the creep rupture time,^9,12–14) a sensitivity analysis of the bevel angle was conducted. The bevel angle, θ, was changed from 0° to 45°, in increments of 5°.

Fig. 1

Finite element model of the weld joint.

3. Results and Discussion

3.1 Creep test results

Figure 2 shows the experimental results of the relationship between the applied stress and the creep rupture time from the uniaxial creep tests on the PM, the simulated fine-grained HAZ (HAZ), and the weld joint (WJ) at 600°C. The creep rupture times of the HAZ were more than one order of magnitude shorter than those of the PM. The creep rupture time of the WJ was between that of the PM and that of the HAZ at 120, 100, and 80 MPa.

Fig. 2

Rupture time of parent metal (PM), simulated HAZ, and weld joint (WJ) of 9Cr–1Mo–V–Nb steel at 600°C.

Figure 3 shows the relationship between the applied stress and the minimum creep rate in the uniaxial creep test on the PM and HAZ at 600°C. The minimum creep rate of the HAZ was more than one order of magnitude faster than that of the PM. The material constants A and n (Table 3) of the PM and HAZ were obtained from this figure.

Fig. 3

Minimum creep rate of PM and HAZ of 9Cr–1Mo–V–Nb steel at 600°C.

Figure 4 shows the fracture morphology of the actual weld joints in the case of 80 and 50 MPa. As shown by the arrow in Fig. 4(a), fracture occurred at the boundary between the HAZ and the PM at 80 MPa, the fracture mode of which is denoted as type IV.^5,15) We confirmed that the fracture modes were also type IV for stresses of 100 and 120 MPa. When subjected to low stress (<50 MPa) for a long duration (64,650 h), however, fracture occurred inside the WM in areas of localized creep deformation. This fracture mode is denoted as type I¹⁵⁾ in Fig. 4(b). The creep strength and hardness of the WM were previously reported to decrease after 30,000 h of creep at 600°C due to the recovery of the microstructures.¹⁶⁾ The softening of the WM was expected to yield a premature fracture at the weld metal prior to cracking in the HAZ, resulting in the type I fracture.

Fig. 4

Fracture type of WJ for different stresses. (a) Type IV failure under 80 MPa for 11,490 h and (b) Type I failure under 50 MPa for 64,650 h.

3.2 Comparison of experimental and simulated creep rupture times

Figure 5 compares the creep rupture times for the experimental and computational simulations of the weld joints. With the TER model, the predicted rupture time depended significantly on which stress was used for the evaluation. This significant dependence indicated the development of multiaxiality during creep deformation, which, in turn, led to differences among the scalar values of the stress tensor. Interestingly, despite changes in the applied stress values, the relationships between the magnitudes of the rupture times evaluated by each stress component remained the same: t_r (σ₁) < t_r (σ_hud) ≈ t_r (σ_r) < t_r (σ_eq). The HDR model, representing the damage mechanics, also yielded appropriate predictions of the weld joint’s creep rupture time.

Fig. 5

Comparison of WJ creep rupture time by experiment and simulation.

Moreover, the calculations provided cautious predictions for all conditions except for the stress condition at 50 MPa; that is, the predicted values were less than the actual ones for each condition. Given that conservative predictions are preferable in actual applications, the present prediction method would likely be useful at least for a relatively short-term rupture at an applied stress of 80 MPa or higher. Regarding the fracture mode, we confirmed that for this relatively short-term rupture, a type IV fracture occurred (Fig. 4(a)). As shown later, the type IV fracture was reasonably predicted by the present simulation, in which it was possible to specify the position at which the accumulated damage reached the rupture value of 1. The consistency of the type IV fracture also supports the appropriateness of the creep damage models for relatively short-term ruptures.

However, for the long-term rupture of 64,650 h at 50 MPa, all the predicted values were longer than the actual values, as shown in Fig. 5, representing a shift from the cautious predictions for short-term ruptures. In contrast to the experimental results, we confirmed neither localized creep deformation nor significant damage accumulation inside the WM in the computational simulation. The type I fracture occurring with the localized creep deformation (Fig. 4(b)) indicated that the creep strength decreased in the WM during the long-term creep test, and the creep strength of the WM should be significantly lower than that of the HAZ and the PM. A similar decrease in creep strength was previously reported by Hongo et al. for a long-term creep test over 10,000 h in the same Gr. 91 steel.¹⁶⁾ However, the present simulation could not take into account this decreased creep strength, which is expected to be the cause of the longer and thus unsafe rupture time predictions, where there was no sign of damage accumulation corresponding to the type I fracture.

Figure 6 shows damage contour maps just before fracture (t/t_r > 0.9), with the failure initiation positions for the applied stress of 100 MPa superimposed; here, the failure initiation position refers to the element at which the value of the damage, D_c or ω, reached 1 first among all the elements of the FEM. As shown in Fig. 6(a) and (c), for the TER using σ_eq and σ_hud, two failure initiation positions were identified that were symmetrically located at the top end of the HAZ/PM boundary and at the bottom end of the HAZ/WM boundary; the damage accumulation was significantly concentrated along the line between these positions. Similarly, for the HDR model, the failure initiation positions were located at the top and bottom ends of the HAZ boundaries, although the damage concentration was not so evident along the line between them (see Fig. 6(d)). For the TER model using σ₁, however, the failure initiation positions were shifted from the ends to the center of the HAZ boundaries, and the damage concentration appeared over a much broader region around the line between the failure initiation positions (see Fig. 6(b)).

Fig. 6

Damage contour map just before creep fracture, and failure initiation position in HAZ region under the condition of σ_a = 100 MPa. (a) TER model using equivalent stress, σ_eq; (b) TER model using maximum principal stress, σ₁; (c) TER model using Huddleston stress, σ_hud; (d) HDR model; and (e) HDR model at rupture time.

Figure 7 shows the contour maps for each stress value together with that of the equivalent strain for the TER just before fracture initiation: ε_eq (a), σ_eq (b), σ₁ (c), and σ_hud (d). In the TER models, the equivalent strain was common for all the cases; (a) was the same time as (b). The value of each stress was heterogeneously distributed in the HAZ region because of the constraints exerted by the PM and WM regions, where the extent of the equivalent strain was less than one-fourth of that in the HAZ region. The distribution shape of each stress value agreed with that of the corresponding damage value (see Fig. 6) which was computed from that stress value according to eq. (1). Therefore, the fracture initiation position can be understood from the distribution of the corresponding stress value from the TER model.

Fig. 7

Contour maps of the equivalent strain and each stress value for the TER model just before the creep fracture in HAZ under the condition of σ_a = 100 MPa. (a) equivalent strain, ε_eq; (b) equivalent stress, σ_eq; (c) maximum principal stress, σ₁; and (d) Huddleston stress, σ_hud.

Regarding the equivalent stress and Huddleston stress, their magnitudes were diminished as the multiaxial stress state became evident; then, the multiaxial stress state became more evident under increased constraint from the surrounding elements. As shown in Fig. 7(a), the equivalent strain exhibited its highest peaks at the top end of the PM/HAZ boundary and at the bottom end of the WM/HAZ boundary. That is, the elements located at these ends were able to deform at the highest creep strain rate, probably because they were on free surfaces. We can thus conclude that the equivalent and Huddleston stresses exhibited their highest peaks at these edge locations (Fig. 7(b, d)). The peaks of the maximum principal stress were found at relatively interior positions in relation to the HAZ boundaries, as shown in Fig. 7(c).

For the HDR model, the relationship between the stress and the damage was more complicated. Figure 8 shows contour maps of ε_eq (a), σ_eq (b), σ₁ (c), and rupture stress σ_r (d) (eq. (7)) at the time just before fracture initiation. Although the ε_eq peak was found at the free-surface edges of the HAZ boundaries, as was observed with the TER models, σ_eq is relatively low on the HAZ boundaries, even at the free-surface edges. From contour maps drawn at an early stage of the creep rupture test, however, we confirmed that both σ_eq and ε_eq exhibited their highest peaks at the free-surface edges of the HAZ boundaries until half of the rupture time, showing that the constraint was the weakest there, as we concluded from the TER model. Regarding σ₁, the peaks were found to be located at positions relatively interior to the HAZ boundaries, as was the case with the TER model. Regarding σ_r, we confirmed that the peaks were located at the free-surface edge of the HAZ boundaries until two-fifths of the rupture time; that is, the distribution of the rupture stress was close to that of the equivalent stress. This result is described by eq. (7), where the rupture stress is affected by the equivalent stress rather than by the maximum principal stress, using α of 0.3 (see Table 3). Hence, the damage accumulation rate, which is proportional to the power of the rupture stress (eq. (6)), is expected to be fastest at the free-surface edge of the HAZ boundaries during the early stages, at least up to almost half of the total rupture time.

Fig. 8

Contour maps of the HDR model just before the creep fracture in HAZ under the condition of σ_a = 100 MPa. (a) equivalent strain, ε_eq; (b) equivalent stress, σ_eq; (c) maximum principal stress, σ₁; and (d) rupture stress, σ_r.

As the damage accumulates, its effect becomes more evident. First, the accumulated damage accelerates the creep strain rate according to eq. (5) and leads to a strain distribution that is more concentrated at the free-surface edges of the HAZ boundaries. As a result of the excessive strain concentration, the multiaxial stress state should become more evident, resulting in lower equivalent stress at the free-surface edges of the HAZ boundaries than at the other elements, as shown in Fig. 8(b). From these analyses, we can conclude that for the HDR model, the fracture initiation positions are strongly determined by the stress distribution during the early stages of the simulation, which does not necessarily correspond with the stress state at the final stage, immediately prior to the fracture initiation.

3.3 Comparison of calculated damage distribution and actual void distribution

Figure 9 shows the SEM image of the polished and etched cross-section around the non-fracture side of the weld joint after 11,490 h of the creep test at 600°C and 80 MPa (dashed-line box shown in Fig. 4(a)). The SEM image exhibits the type IV creep fracture. The width of the HAZ with fine grains was approximately 1.3 mm, and a large shear deformation can be recognized along the boundary of the WM and HAZ. The black dots in the figure are the voids formed during creep deformation. Extensive void growth was observed approximately 1–2 mm below the upper surface of the specimen in the figure, and it was distributed along the WM/HAZ boundary. The void distribution agrees well with the damage distribution from the TER model using σ₁ (see Fig. 6(b)) despite the deflection of the void distribution in the PM direction. The damage deflection in the PM direction was considered to occur because of the different creep behaviors of the PM and WM in the actual specimen, where a type IV fracture occurred at the HAZ/PM boundary (see Fig. 4(a)).

Fig. 9

SEM image around the non-fracture side of the creep fracture of the weld joint after creep at 600°C and σ_a = 80 MPa for 11,490 h.

Figure 10 shows the distributions of creep damage at the HAZ/PM boundary through the central axis of the weld joint at a loading of 80 MPa. The values at the time, t/t_r = 0, 0.5, 0.8, and 0.9 are plotted, where t_r is the final rupture time. Except for the TER model using σ₁, the highest damage value was found at the upper surface of the model. However, as can be seen in Fig. 9, void formation was observed at approximately 1.5 mm below the upper surface of the HAZ/PM boundary, and voids were not found at the surface. Only in the case of the TER model using σ₁ was failure initiated at the region approximately 1.5 mm below the upper surface, and this trend is similar to the stress distribution and the actual void distribution observed in the experiment. Based on this result, the TER model using σ₁ is considered to offer the most appropriate criterion to predict the creep damage distribution in the weld joint. With the HDR model, the failure initiation position was affected by the stress distribution up to approximately t/t_r = 0.5 of the analysis because in this model, the effect of damage acceleration is larger than the effect of decreasing stress.

Fig. 10

Creep damage distribution at the HAZ/PM interface following the vertical direction of the weld joint under the condition of σ_a = 80 MPa. (a) TER model using equivalent stress, σ_eq; (b) TER model using maximum principal stress, σ₁; (c) TER model using Huddleston stress, σ_hud; and (d) HDR model.

3.4 Numerical sensitivity analysis of bevel angle

Figure 11 shows the results of the sensitivity analysis in relation to the bevel angle of the HAZ boundaries toward the load axis at a stress condition of 100 MPa. Here, the bevel angle, θ, was changed from 0° to 45° in increments of 5°, and the width of the HAZ was kept the same. In all the cases except for the TER model using σ₁, each rupture time significantly depended on the bevel angle. That is, the rupture time showed its maximum value when θ = 0°, at which point the rupture time was more than three times that at θ = 30°. The rupture time then decreased with increasing bevel angles, at least up to θ = 30°; in particular, the decreasing rate was clearly high from 10°–25°. For the TER model using σ_hud and the HDR model, the rupture time showed a minimum value at approximately θ = 30° and increased slightly with increasing bevel angles. For the TER model using σ_eq, the rupture time remained steady until θ = 40° and decreased slightly at θ = 45°. A similar trend in rupture time for a weld joint was reported in previous studies involving experimental and numerical simulations of tube weld joints.^9,12) In addition, an experimental result suggesting longer rupture times at the bevel angle of θ = 0° was reported.¹³⁾ This consistency with previously reported trends is likely to support the validity of these three criteria. Regarding the bevel angle for the minimum rupture time, the weld joint of an experimental plate-type specimen was reported to exhibit a minimum value at a bevel angle of θ = 45°.¹⁴⁾ That is, the specimen shape can change the bevel angle associated with the minimum rupture time.

Fig. 11

Plot of simulated rupture time against the bevel angle of the weld joint under the condition of σ_a = 100 MPa.

The TER model using σ₁ yielded a completely different trend from the above-mentioned three criteria. Therefore, this criterion is considered inappropriate for evaluating the effect of the groove angle.

Figure 12 shows the failure initiation position for each damage criterion at θ = 0°, 15°, 30°, and 45°. The failure initiation positions for the TER model using σ_eq were located inside the HAZ region at θ = 0°; interestingly, the failure initiation positions form elliptical shapes, as shown by the areas with hatched lines in Fig. 12(a). The failure initiation positions moved to the top of the HAZ/PM boundary and to the bottom edges of the HAZ/WM boundary at θ = 15°–30°, and then finally moved to a position on the outer surface of the HAZ boundary at θ = 45° (Fig. 12(d)). Similarly, the failure initiation positions for the TER model using σ_hud formed elliptical shapes with smaller radii than for the TER model using σ_eq at θ = 0°. Although these failure initiation positions move similarly on the HAZ boundaries at θ = 15°, they are located at inner positions rather than at the free-surface edges, unlike the TER model using σ_eq, and we confirmed that they remained at the slightly interior positions at least up to 20°. These slightly interior positions were close to the positions where significant voids were formed (Fig. 9). The failure initiation positions moved further to the free-surface edges of the HAZ boundaries at θ = 25°–30° and finally reached positions on the outer surface of the HAZ boundaries at θ = 45°. The failure initiation positions of the TER model using σ₁ were located near the center of the HAZ region except at θ = 0°; they then became slightly scattered along an inclined line at θ = 15°, and moved onto the HAZ boundaries at θ = 30°.

Fig. 12

Images of the transition of the computed failure initiation position for different bevel angles: (a) θ = 0°, (b) θ = 15°, (c) θ = 30°, and (d) θ = 45°.

For the HDR model, there were two failure initiation positions near the center of the HAZ region at θ = 0°, aligned along the loading direction. Interestingly, they were found to be located at positions interior to the HAZ boundaries at θ = 15°–20° and then moved to the free-surface edges at θ = 30°; these trends were very similar to the case of the TER model using σ_hud.

3.5 Assessment of the computational predictions from each damage model

We assessed the capability of each model to predict the creep damage behavior based on comparisons with the experimentally obtained results. Here, we limit the assessment to the case of the high applied stress condition over 80 MPa, i.e., the case of a relatively short-term rupture time.

The TER model using σ_eq provided the closest prediction of the rupture time within a factor of 2 (Fig. 5). The dependence on the bevel angle at least up to 30° was consistent with previously reported results (Fig. 11). The failure initiation positions were on the HAZ boundaries, in the vicinity of the experimentally observed voids (Fig. 9). However, the positions differed as follows: although the experimentally observed location of significant void formation was slightly interior to the HAZ location, the predicted location of the failure initiation positions was on the free-surface edges. In addition to these summarized assessments, a much more accurate prediction of the rupture time could have been obtained by assuming a lower bevel angle than that assumed for the present simulation. For example, the predicted rupture time was almost comparable to the experimental one when the bevel angle was 20° (see Fig. 11). Nonetheless, this potential assumption did not yield an appropriate prediction of the fracture initiation positions (Fig. 12).

The TER model using σ_hud and the HDR model provided similar prediction results for the rupture time and position as well as for the bevel angle dependency. The predicted rupture times were five times shorter than that observed in the experiment, and the predicted rupture position was different from that where the void formation was experimentally observed, as with the TER model using σ_eq. As discussed in the previous section, the bevel angle dependency agreed well with the previously reported results.^9,10,12) If we assume that the bevel angle was smaller, for example 20°, the difference between the prediction and the experiment would be smaller and the rupture position would move toward the inside of the specimen, resulting in better agreement with the actual void formation position.

These three cases indicated the initiation of the rupture at the HAZ boundary for a wide range of bevel angles (approximately 15°–30°), and the type IV failure predominantly occurred in that bevel angle range. In other words, these three models provided a sufficient guide as to the fracture type, although a more detailed examination of the rupture position to determine its occurrence on the free surface edges or a slightly inner location would be difficult. Under such constraints on the prediction of the rupture position, the TER model using σ_eq provided the most exact prediction of the rupture time for the present case.

Lastly, although the TER model using σ₁ has the possibility of predicting the rupture initiation position, it cannot predict the effect of the bevel angle.

4. Conclusions

In this study, we conducted long-term creep tests on welded joints in 9Cr–1Mo–V–Nb steel (ASME Gr. 91). We investigated the characteristics of two types of creep damage models, the TER and the HDR models, in terms of their predictions of the creep rupture time and the fracture initiation position for the weld joint. Our results are summarized as follows.

The calculations produced cautious predictions for short rupture times of up to 10,000 h in a high stress region of 80 MPa and above, and the fracture initiation position was consistent with the type IV fracture mode obtained experimentally. However, in the case of 50 MPa, the type I fracture mode, which appeared in the experiment, was not reproduced in the present simulation.

At the bevel angle of 30°, the relationship between the stress magnitudes is described by σ₁ > σ _hud ≈ σ_r > σ_eq at the failure initiation position, and the magnitudes of the rupture time show a relationship of t_r (σ₁) < t_r (σ_hud) ≈ t_r (σ_r) < t_r (σ_eq). These trends did not change under any of the tensile stress conditions.

The rupture time of the TER model using σ_eq was relatively close to that of the actual weld joint, and its sensitivity to the bevel angle was also similar to that observed experimentally. The detailed positions of the fracture initiation were, however, different from the void distribution of the actual weld joint.

The TER model using σ_hud and the HDR model provided very similar predictions in terms of the rupture time and the fracture initiation position. In addition, their sensitivity to the bevel angle was almost even, which was similar to that observed experimentally.

The damage distribution of the TER model using σ₁ was similar to the void distribution of 100 MPa and θ = 30°. However, this model cannot predict the effect of the bevel angle.

By comparison of these creep damage laws, it was considered that the TER model using σ_hud and the HDR model are good creep damage laws for predicting both creep rupture time and fracture initiation position.

Acknowledgement

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

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