Numerical method to characterize probabilistic energy distribution of drop weight tear test at Ductile-Brittle transition temperatures

doi:10.1016/j.engfracmech.2022.108540

Engineering Fracture Mechanics
工程断裂力学

Volume 269, 15 June 2022, 108540
第 269 卷，2022 年 6 月 15 日，108540

https://doi.org/10.1016/j.engfracmech.2022.108540 Get rights and content 获得权利和内容

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Highlights 亮点

•
Numerical method to quantify energy scatters of high strength steel drop weight tear tests at ductile–brittle transition.
量化高强度钢在延性-脆性转变过程中落锤撕裂试验能量散射的数值方法。
•
The stress-modified fracture strain model for ductile tearing simulation.
用于延性撕裂模拟的应力修正断裂应变模型。
•
The maximum principal stress criterion for cleavage fracture simulation.
解理断裂模拟的最大主应力准则。
•
Two-parameter Weibull distribution in terms of the maximum principal stress for probability of energy scatters.
以最大主应力表示能量散射概率的双参数威布尔分布。
•
Comparison with API X80 drop weight tear test data at ductile–brittle transition temperatures.
与 API X80 韧脆转变温度下的落锤撕裂试验数据进行比较。

Abstract 抽象的

This paper proposes a numerical method to characterize probabilistic energy distributions of drop weight tear tests (DWTTs) at ductile–brittle transition temperatures. The method employs finite element ductile–brittle fracture simulation using the stress-modified fracture strain model for ductile tearing and the maximum principal stress criterion for cleavage fracture. The probability of energy scatters is expressed using the two-parameter Weibull distribution in terms of the maximum principal stress. The maximum principal stress is then obtained as a function of the absorbed energy using FE simulation. The characterized energy scatters are compared with API X80 DWTT data at transition temperatures, showing good agreement. The proposed method can be used to effectively predict DWTT data at transition temperatures.
本文提出了一种数值方法，用于表征延性-脆性转变温度下落锤撕裂试验（DWTT）的概率能量分布。该方法采用有限元延性-脆性断裂模拟，其中，延性撕裂采用应力修正断裂应变模型，解理断裂采用最大主应力准则。能量散射概率用双参数威布尔分布表示，并以最大主应力为参数。然后，通过有限元模拟，获得最大主应力与吸收能量的关系。表征的能量散射与 API X80 DWTT 在转变温度下的数据进行比较，结果显示一致性良好。所提出的方法可用于有效预测转变温度下的 DWTT 数据。

Keywords 关键词

Drop weight tear test

Ductile-brittle transition temperature

Finite element damage analysis

Probability prediction of DWTT energy

落锤撕裂试验；韧脆转变温度；有限元损伤分析；落锤撕裂能量概率预测

Abbreviations 缩写

API

American petroleum institute

CMOD

crack mouth opening displacement

DBTT

ductile to brittle transition temperature

DWTT

drop weight tear test

finite element

GTN

Gurson-Tvergaard-Needleman

reduction of area

SMFS

stress modified fracture strain

SENT

single edge notched tension

USE

upper shelf energy

API 美国石油学会 CMOD 裂纹口张开位移 DBTT 延脆转变温度 DWTT 落锤撕裂试验 FE 有限元 GTNGurson-Tvergaard-NeedlemanRA 面积缩减 SMFS 应力修正断裂应变 SENT 单边缺口拉伸 USE 上层能量

Nomenclature 命名法

a₁, a₂
一个 ₁ 、一个 ₂

fitting parameters in the ductile–brittle combined fracture model, see Eq. (10)
韧脆复合断裂模型中拟合参数见（10）式

b₁, b₂
b ₁ ，b ₂

fitting parameter for σ_u, see Eq. (4)
σ _u 的拟合参数，见公式（4）

D_c

critical damage value 临界伤害值

Young’s modulus 杨氏模量

E_D

DWTT absorbed energy DWTT 吸收能量

E_D^exp_.

measured experimental energy in drop weight tear test
在落锤撕裂试验中测量实验能量

E_D^FE_.

simulated energy in drop weight tear test
落锤撕裂试验模拟能量

material constant in the modified Johnson-Cook model, see Eq. (A1)
修改后的 Johnson-Cook 模型中的材料常数，见公式 (A1)

L_e

minimum element size 最小元素尺寸

exponent of the Weibull distribution, see Eq. (1)
威布尔分布的指数，见公式（1）

P_f

cumulative failure probability
累积失效概率

T, T_melt

temperature and melting temperature (^oC)

V, V_o

volume and characteristic volume

ε_e^p, ε_f

equivalent plastic strain and fracture strain

σ_e, σ_e,RT

equivalent stress and the value at room temperaturel

σ₁, σ_m

maximum principal stress and hydrostatic pressure

σ_y,σ_TS

yield stress and tensile strength

σ_u

ductile–brittle fracture model parameter, see Eq. (4)

σ_w

Weibull stress

1. Introduction

Pipelines operated in low temperature environment often run a risk of sudden brittle fracture [1]. As the crack propagation rate of brittle fracture can be very fast and is difficult to control, an accurate prediction of fracture behaviour of pipeline steels in terms of temperature is very important. To understand fracture behaviours at ductile–brittle transition temperatures, various tests can be performed, such as full-scale pipe burst test, Charpy test and drop-weight tear test (DWTT) [2]. Being experiments close to actual phenomena, full-scale pipe burst tests would be the best to investigate fracture behaviours, but generally require long experience and high costs [3], [4], [5], [6]. In contrast, Charpy impact test is quite simple to perform, but could underestimate the ductile–brittle transition temperature (DBTT) and the upper shelf energy (USE). It is because the specimen is too small compared to the actual pipe size so that the Charpy impact test cannot properly reflect the constraint effect [7]. To overcome these limitations, Battelle developed the drop weight tear test (DWTT) [8]. Since DWTT specimens are manufactured to have the same thickness as the pipe with sufficient crack propagation ligament, the former can reflect similar levels of constraint effect and the latter can show various types of fracture behaviour [9], [10]. Thus, for the use of a material for pipelines, it is required to satisfy requirements that measured absorbed energy values from DWTT must be higher than a specified one [8], [9]. However, fracture behaviours at ductile–brittle transition temperatures typically show large scatter, requiring many DWTT data at different temperatures for statistical analysis. In this respect, a numerical method to predict the probabilistic distribution of the DWTT energy at ductile to brittle transition temperatures would be needed.

Finite element (FE) analysis would be very useful to further understand the fracture behaviour of the DWTT at ductile to brittle transition temperatures. Beremin et al. [11] proposed a local approach based on the maximum principal stress and the Weibull distribution. The Weibull stress approach has been widely used as a method to predict probabilistically the energy scatter of brittle fracture originated from the random nature [12], [13]. Combined with a ductile damage model, such as the Gurson-Tvergaard-Needleman (GTN) [14], [15], [16], the cohesive zone [17], the Rousselier model [18], and the stress modified fracture strain (SMFS) model [19], [20], the energy scatter prediction using the Weibull stress approach could be extended to the ductile–brittle transition region. Bernauer et al. [21] modified the Weibull stress approach to improve transferability of the Weibull parameters from one experiment to another by coupling cleavage fracture initiation process with ductile void formation using the GTN model. Tanguy et al. [22], [23] conducted Charpy impact tests of A508 steel at transition temperature and predicted the energy scatter using a temperature-dependent Weibull parameter combined with a modified Rousselier model proposed by Besson et al. [24]. Hauslid et al. [25] predicted fracture toughness of a low-alloy steel at transition temperature from compact tension C(T) and Charpy impact tests. Ductile fracture was simulated using the GTN model and cleavage fracture was quantified using the temperature-dependent Weibull parameter. Hojo et al. [26] investigated the applicability of the Weibull stress approach coupled with the GTN or Rousselier model to predict fracture toughness from C(T) test of A533 Grade B Class 1 steel, and suggested that the accuracy of the model needs to be improved. The above studies have suggested applicability of the Weibull stress approach to understand fracture behaviours in the ductile to brittle transition region for a variety of materials.

For recently developed high-strength/high-toughness steels to withstand higher pressures for more efficient transport of energy sources, an inverse fracture phenomenon has been observed in DWTT [27], [28], [29], [30], [31], [32]. The fracture mechanism normally seen in DWTT was ductile fracture or brittle fracture followed by ductile one. However, in a high strength/toughness steel, fracture can occur abnormally that the fracture sequence is initial brittle fracture, then ductile fracture and finally brittle fracture again, making it difficult to evaluate crack arrest. This type of fracture is referred to as the inverse fracture in many papers [27], [28], [29], [30], [31], [32], [33]. Furthermore, such inverse fracture has been shown to be observed more often in the pressed notch DWTT specimen than in the chevron notch DWTT specimen. Due to such inverse fracture phenomenon, it is difficult to apply FE analysis using the Weibull stress approach to analyse DWTT numerically (detailed reasons of the difficulty will be explained later in this paper). Thus, a new numerical approach is needed to predict the temperature dependence of the DWTT energy at the ductile to brittle transition region.

In our previous works [34], [35], a criterion using the maximum principal stress (instead of the Weibull stress) has been proposed as a local cleavage criterion. Noting that the Weibull stress is in fact related to the maximum principal stress, it was also shown that the Weibull stress approach is equivalent to the proposed maximum principal stress criterion [34]. An advantage of the proposed maximum principal stress criterion is that it can be applied to an element level, so that interacting ductile and cleavage fracture phenomena can be explicitly analyzed in one FE analysis. The proposed method has been applied to simulate Charpy impact and DWTTs of high strength API X80 at ductile to brittle transition temperatures [34], [35]. For ductile tearing simulation, the SMFS model was employed, where the fracture strain is assumed to be a function of the stress triaxiality [19], [20]. Despite its simplicity, the SMFS model has been shown to be effective to simulate ductile tearing in a cracked specimen or in a cracked pipe, by comparing with experimental test data [36], [37], [38], [39], [40], [41], [42].

The objective of the present work is to propose a numerical method to characterize the probabilistic distribution of the absorbed energy for high-tensile steel DWTT at ductile–brittle transition temperatures. A numerical method employs FE ductile–brittle fracture simulation using the SMFS model for ductile tearing simulation and the maximum principal stress criterion for cleavage fracture simulation. Section 2 explains DWTT data of API X80, considered in this paper. Section 3 presents difficulties for application of the Weibull stress approach to quantify the the probabilistic distribution of the DWTT absorbed energy. Section 4 presents a new formulation using the maximum principal stress to quantify the probabilistic distribution of the DWTT absorbed energy. Characterized results are compared with experimental data. The conclusions are given in Section 5.

2. Experiment

2.1. Material and mechanical properties at room temperature

In this study, an American Petroleum Institute (API) X80 grade steel was representatively adopted as a high strength steel. According to the API 5L standard [43], tensile test was conducted using a flat tensile specimen at 25 °C using a Zwick/Roell universal testing machine. The specimen was extracted in the hoop direction from a plate. Important mechanical properties for API X80 steel are given in Table 1. More detailed explanation can be found in our previous works [34], [35]. Crack growth resistance curve of API X80 was also measured using a single-edge notch tension (SENT) test at 25 °C accordance with the DNV RP-F108 [44]. Detailed experimental curves of the load and crack growth (Δa) versus CMOD were also presented in our previous papers [34], [35]. These data were used to determine the damage model for ductile fracture, as summarized in Appendix A.3.

Table 1. Mechanical properties for API X80 steel.

Temp., T [^oC]	Young’s modulus, E [GPa]	Poisson’s ratio, v	Yield stress, σ_y [MPa]	Tensile strength, σ_TS [MPa]	Reduction of Area, RA [%]
25	199	0.3	570	647	77.8

2.2. Drop weight tear test (DWTT)

To characterize the energy scatter of drop-weight tear tests (DWTTs) for API X80, tests were conducted at temperatures ranging from −97 to 0 °C according to the ASTM standard [45]. The ASTM standard [45] recommends that the size of the DWTT specimen should be 76.2 mm (

\pm

3.2 mm) by 305 mm (

\pm

19 mm) and the notch depth be 5.08 mm (

\pm

0.5 mm) with an angle of 45° (

\pm

2°). Also, the speed of the hammer should not be less than 4.88 m/s [45]. By immersing the test specimen in the liquid chamber, temperature of the specimen was maintained at the target temperature. The geometry and dimensions of the DWTT specimen with 45° press notch of 5.08 mm depth is shown in Fig. 1(a). To deliver sufficient energy to break the specimen, a universal testing machine (DWTT-100F) was used. A 2,564 kg hammer dropping at a height of 1,989 mm impacted the top surface of the DWTT specimen at a speed of ∼ 6 m/s with anvils fixed on both sides to apply momentary impact load to the DWTT specimen. From the instrumented DWTT system, load-load line displacement data were attained, from which the absorbed energy was calculated. The absorbed energy values E_D normalized with respect to the ligament area (A = 1,685.5 mm²) are shown in Fig. 1(b). A total thirty-one tests were conducted at various temperatures. The representative experimental load–displacement curves at −97 °C, −60 °C, −20 °C and 0 °C are also shown in Fig. 1(c). Impact force was measured by a load cell mounted behind the hammer. Impact velocity immediately prior to impact was measured using the time-of-flight method through laser/photo-diode detector with 25 ns time resolution. The displacement was automatically calculated by the instrument from the measured velocity. The measurement system is briefly illustrated by a block diagram in Fig. 2. The number of test, averaged experimental energy values and standard deviations are summarised in Table 2. Note that the scatter of the absorbed energies can occur due to the random nature of the microstructure in the specimen.

Table 2. Drop weight tear test data with temperatures.

T [^oC]	−97	−80	−60	−40	−20	0
Number of specimens	14	4	5	2	4	2
Averaged E_D^exp [kJ]	0.9	1.6	5.1	13.2	12.1	18.9
Standard deviation [kJ]	0.2	0.3	2.3	0.6	1.7	0.4

The experimental fracture surfaces for selected cases are shown in Fig. 3. Depending on the temperature, various types of fracture can be observed, which can be identified as cleavage, ductile and inverse fracture. Since low temperature constrains the plastic deformation, cleavage fracture at −97 °C started from the notch tip and propagated to the end of the specimen with small ductile fracture surfaces in both sides of the specimen. However, tests at −60 °C and −20 °C showed more complex fracture surfaces. Compared with the experimental surface at −97 °C, the larger ductile fracture area near the specimen surface and smaller cleavage fracture inside can be seen at −60 °C and −20 °C. Either ductile or brittle fracture can also be initiated at the notch tip due to the random nature of the microstructure in the specimen as shown in Fig. 3. In the case of fracture toughness tests under quasi-static loading, it is common to initiate ductile fracture at the notch tip. But rapid impact load could further increase the possibility of initial cleavage fracture as shown in Fig. 3. In the case that brittle fracture is initiated at the notch tip at −60 °C, it continues to the end of the specimen. Particularly, an inverse fracture where ductile fracture is followed by cleavage fracture can be observed at −20 °C. Note that the inverse fracture is classified as a type of cleavage fracture [9].

3. Weibull stress approach for cleavage fracture and its limitation

3.1. Weibull stress and failure probability

Failure probability of brittle fracture has been popularly predicted using the Beremin model [11], [22], [23], [25], [26] where failure probability is assumed to follows the two-parameter Weibull distribution:(1)

P_{f} = 1 - \exp (- {(\frac{σ_{w}}{σ_{u} (T)})}^{m}), σ_{w} = {(\frac{1}{V_{0}} \int_{V} {(σ_{1})}^{m} d V)}^{\frac{1}{m}}

where σ₁ is the maximum principal stress; σ_w is the Weibull stress; m and σ_u are the Weibull parameters; V is the fracture process zone volume; and V_o is a characteristic volume. In this work, the value of m is assumed to be independent on temperature. In the continuum, the Weibull stress is defined as the integral of the m-power of the maximum principal stress in the fracture process zone. In the context of the numerical (FE) method, the Weibull stress can be discretely calculated as follows:(2)

σ_{w} = {(\frac{1}{V_{0}} \sum_{i = 1}^{n_{e}} {(σ_{1}^{i})}^{m} V^{i})}^{\frac{1}{m}}

where n_e is the total number of elements used in FE calculations. According to the definition, the Weibull stress can be obtained from post-processing FE calculation results.

3.2. Determination of the Weibull constants for API X80

To quantify failure probability of DWTT for API X80 using the existing Weibull stress approach, determination of the Weibull constants, σ_u and m, should be made first and the procedure is shown in Fig. 4.

The first step is to simulate DWTT using FE damage analysis only with the ductile damage model and to calculate the Weibull stress from post-processing FE results. The FE damage analysis method of DWTT has been presented in our previous works [34], [35] and is briefly summarized in Appendix. To calculate the Weibull stress, the values of m and V_o should be assumed first. In this work, m = 12.43 and V_o = 14.82 mm³ were assumed first. Note that these values will be obtained later in Section 4.3 and are used in this section as the initial values. Then ductile damage simulation of the DWTT was performed for tests at −97 °C, because the number of tests at −97 °C is the largest. The estimation procedure of the tensile property at T = -97 °C is also provided in Appendix A.2. An example of calculating the Weibull stresses for three different measured DWTT energies, E_D^exp = 0.603 kJ, 0.773 kJ and 1.738 kJ, at −97 °C is shown in Fig. 5(a), (b) and (c), respectively. From the DWTT simulation, the load-load line displacement curve can be obtained, as shown in Fig. 5(a)-(c). The absorbed energy can be calculated from the area under the curve. The experimental load-load line displacement curves are also compared with simulation results in Fig. 5(a)-(c). Note that, because only ductile fracture is simulated in the FE analysis, the rapid load drop shown in the experiment could not be simulated. Oscillating load-load line displacement curves can be found both in experiment and in simulation. Oscillating load–load line displacement curves can also be seen in another work [48] simulating DWTT using a monotonically increasing stress–strain curve. Amara et al. [49] reported that oscillations could result from a stress wave due to rapid impact load. Then, the Weibull stress post-processed from the DWTT simulation using Eq. (2) can be plotted in terms of the absorbed energy, as shown in Fig. 5(d). The determined Weibull stresses for three different measured DWTT energies are shown in Fig. 5(d).

For each DWTT data at −97 °C, the Weibull stress can be determined using the procedure shown in Fig. 5. The determined Weibull stress is given in Table 3. The failure probability, P_f, for the jth test can be calculated using.(3)

P_{f} (%) = \frac{j}{N + 1} \times 100

where N denotes the total number of tests. The sequence of tests was arranged in the ascending order with respect to the absorbed energy. The calculated failure probability is also summarized in Table 3.

Table 3. Determined Weibull stress σ_w values according to the measured DWTT energy at −97 °C and corresponding failure probability.

E_D^exp [kJ]	0.603	0.605	0.616	0.634	0.643	0.668	0.711
σ_w [MPa]	1,501.1	1501.6	1505.3	1511.6	1514.6	1522.0	1534.2
P_f [%]	6.7	13.3	20	26.7	33.3	40	46.7
E_D^exp [kJ]	0.773	0.787	0.875	1.026	1.133	1.550	1.738
σ_w [MPa]	1546.6	1549.4	1564.8	1582.4	1592.1	1622.4	1631.1
P_f [%]	53.3	60	66.7	73.3	80	86.7	93.3

Using the determined Weibull stress, experimental DWTT energy data at −97 °C can be fitted using Eq. (1), which gives m = 33.85 and σ_u = 1,571 MPa. The fitted curve is compared with experimental data in Fig. 6(a). Note that m = 33.85 is different from the assumed value of m = 12.43 used in the first calculations. The final value of m is determined when the assumed value of m for calculating Weibull stress is the same as the value of m obtained by fitting the experimental data [51]. In Fig. 6(b), the re-calculated Weibull stress using the fitted m = 33.85 is shown with experimental data. In this work, the final value of Weibull constants was found to be m = 33.85 and σ_u = 1,454 MPa at −97 °C, as shown in Fig. 6(b).

In the present work, it is assumed that the parameter m does not depend on temperature but the parameter σ_u does. The following form is assumed in this work, according to Haušild et al. [25].(4)

σ_{u} (T) = b_{1} \cdot \exp (- \frac{b_{2}}{T})

where T denotes the temperature (°C); and b₁ and b₂ are constants. To determine the constants b₁ and b₂, test data at two different temperatures, T = -97 °C and −60 °C, were used in this work. Note that, using experimental DWTT data at −97 °C, σ_u = 1,454 MPa was found. Using the determined parameters (m = 33.85 and V_o = 14.82 mm³), the FE damage analysis with the ductile damage model was performed again, as was done for the analysis at T = -97 °C to calculate the Weibull stress. The tensile property at T = -60 °C was determined according to the procedure provided in Appendix A.2.

With relevant failure probability, the Weibull stress values corresponding to five experimental DWTT energies are shown in Fig. 7(a). Then experimental DWTT energy data at −60 °C can be fitted using Eq. (1), which gives σ_u = 1,623 MPa. The fitted Weibull distribution is compared with experimental data in Fig. 7(a). Using these two values of σ_u at two different temperatures, b₁ = 2,697.9 MPa and b₂ = 108.3 °C were found, giving a final equation of σ_u for API X80.(5)

σ_{u} (T) = b_{1} \cdot \exp (- \frac{b_{2}}{T}) = 2697.9 \cdot \exp (- \frac{108.3}{T})

This is shown in Fig. 7(b). Finally, substituting σ_u(T) and m into Eq. (1), the failure probability of DWTT for API X80 could be estimated as.(6)

P_{f} = 1 - \exp (- {(\frac{σ_{w}}{2697.9 \cdot \exp (- 108.3 / T)})}^{33.85})

However, there are some limitations to characterize the failure probability in terms of the absorption energy using Eq. (6), which will be explained in the next sub-section.

3.3. Limitation of Weibull stress approach to characterize probability of DWTT energy

As noted, the Weibull stress can be obtained only from post-processing FE results. Variations of the Weibull stresses with absorbed energy, calculated from ductile tearing simulation of DWTT at −97 °C, −60 °C and −20 °C are shown in Fig. 8(a). The variation of the Weibull stress with absorption energy can be divided into two regions; the monotonically increasing region (region I) and the oscillating one (region II). The Weibull stress monotonically increases initially with the absorption energy. However, it starts to oscillate when the absorption energy is larger than ∼ 8 kJ. The approximated energy value is as the area under the load-load line displacement curve from zero to the displacement at the maximum load in Fig. 8(b). Rudland et al. [50] obtained steady-state propagation energy after maximum load occurred. At this time, it might be considered that the Weibull stress oscillates by the stress wave caused by the impact [49] mentioned in Section 3.2. Cleavage fracture criteria corresponding to experimental energy can be obtained using the Weibull stress variation as shown in Fig. 8(c)-8(e). For the DWTTs performed at −97 °C, all test energy values are less than 2 kJ, and thus are located within the region I, as shown in Fig. 8(c). However, in the case of −60 °C and −20 °C, some experimental energy values are larger than 8 kJ, and thus are located in the region II. Due to the oscillating behavior, a non-unique solution can exist, which limits the use of the Weibull stress approach to characterize energy scatter in DWTT.

Another limitation in the use of the Weibull stress approach can be found from experimental fracture surfaces of the DWTTs. Fracture surfaces from the DWTT tests performed at −97 °C, −60 °C and −20 °C are shown in Fig. 2 where cleavage and ductile fracture surfaces are indicated. To apply the Weibull stress approach, cleavage fracture should occur either from the beginning or after some amounts of ductile fracture. However, the test at −20 °C showed that cleavage fracture occurred first, then ductile fracture and cleavage fracture again occurred, making the application of the Weibull stress approach difficult. It is believed that the Weibull stress approach would be applicable only for cleavage fracture after some amount of ductile fracture. In case of DWTT at −20 °C, for instance, inverse fracture (ductile fracture after cleavage fracture) can occur as shown in Fig. 2, where the Weibull stress approach is believed to be not suitable.

4. Maximum principal stress approach for cleavage fracture

4.1. Maximum principal stress approach

To overcome the limitations of the Weibull stress approach, described in Section 3.4, a maximum principal stress approach is used in this section. Note that it was proposed by the authors to simulate Charpy impact test and DWTT [34], [35]. From Charpy test simulation, it was shown that the calculated Weibull stress was linearly proportional to the largest maximum principal stress in the Charpy specimen:(7)

σ_{w} = {(\frac{1}{V_{0}} \sum_{i = 1}^{n_{e}} {(σ_{1}^{i})}^{m} V^{i})}^{\frac{1}{m}} \approx {(\frac{V}{V_{0}})}^{\frac{1}{m}} σ_{1, \max}

where σ_1,max denotes the largest maximum principal stress. As the value of V/V_o can be arbitrary chosen, the Weibull stress is assumed to be equivalent to the largest maximum principal stress. Consequently the Weibull stress σ_w in Eq. (1) can be replaced with σ_1,max, leading to:(8)

P_{f} = 1 - \exp (- {(\frac{σ_{w}}{σ_{u} (T)})}^{m}) = 1 - \exp (- {(\frac{σ_{1, \max}}{σ_{u}^{'} (T)})}^{m})

Note that the Weibull exponent m is assumed to be the same, but σ_u’ (T) is different from σ_u(T) due to the choice of V/V_o = 1, although their functional forms are the same;(9)

σ_{u}^{'} (T) = b_{1}^{'} \cdot \exp (- \frac{b_{2}^{'}}{T})

The form for σ_1,max in terms of the absorption energy of the DWTT, E_D, was derived in our previous works [34], [35], as follows.(10)

\frac{σ_{1, m a x} (E_{D})}{σ_{y} (T)} = a_{1} \cdot \ln (\frac{E_{D}}{E_{0}}) + a_{2}

where σ_y is the temperature-dependent yield stress; E₀ denotes an arbitrary normalizing value; and a₁ and a₂ are constants.

Combining Eq. (8) and Eq. (10) leads to.(11)

E_{D} (T, P_{f}) = E_{0} \cdot \exp (\frac{1}{a_{1}} [\frac{σ_{u}^{'} (T)}{σ_{y} (T)} \cdot {\{\ln (\frac{1}{1 - P_{f}})\}}^{\frac{1}{m}} - a_{2}])

The constants in Eq. (10), a₁ and a₂, and those in Eq. (9), b₁′and b₂′, will be determined in 4.2 Determination of maximum principal stress criterion for API X80, 4.3 Determination of the parameter (.

4.2. Determination of maximum principal stress criterion for API X80

To determine the relationship between the maximum principal stress and absorbed energy, Eq. (10) (the constants, a₁ and a₂), FE DWTT simulation using both ductile and brittle damage models should be performed. Note that FE DWTT simulation using both ductile and brittle damage models was presented in our previous works [34], [35] and briefly summarized in Appendix A.4. As done in the Weibull stress approach, FE DWTT simulation was performed at −97 °C with the relevant tensile properties. In simulation, the assumed value of σ_1,max is given to simulate brittle fracture. Since the assumed value of σ_1,max is simply for deriving an analytical relationship between σ_1,max and the absorbed energy, any value can be chosen. With the assumed σ_1,max value, combined ductile and brittle fracture can be simulated and load–displacement curve can be obtained. Fig. 9(a) shows simulated load–displacement curves for three assumed σ_1,max values, σ_1,max = 1,300, 1,550 and 1,800 MPa. Then the absorbed energy E_D can be obtained from the area of the simulated load–displacement curve. Oscillating load–load line displacement curves can also be seen in the Fig. 9(a), as pointed out in Section 3.2. The calculated absorbed energy corresponds to the assumed value of σ_1,max can be plotted in Fig. 9(b). Then, the FE damage analysis is repeated using a different value of σ_1,max. A series of FE damage analysis give a set of σ_1,max and E_D, as shown in Fig. 9(b) using a symbol.

From Fig. 9(b), the constants, a₁ and a₂, in Eq. (10) are obtained as.(12)

\frac{σ_{1, m a x} (E)}{σ_{y} (T)} = a_{1} \cdot \ln (\frac{E_{D}}{E_{0}}) + a_{2} = \{\begin{matrix} 0.37 \cdot \ln (\frac{E_{D}}{E_{0}}) + 1.71 for \ln (\frac{E_{D}}{E_{0}}) < 3.02 \\ 1.09 \cdot \ln (\frac{E_{D}}{E_{0}}) - 0.47 for \ln (\frac{E_{D}}{E_{0}}) ⩾ 3.02 \end{matrix}

where σ_1,max and E_D are normalized with respect to the temperature-dependent yield stress and E₀ = 603.4 J, respectively. Note that the value of E₀ = 0.603 kJ is the minimum experimental DWTT energy at −97 °C, as given in Table 3. Equation (12) is divided into two equations depending on the value of ln(E_D/E₀). Note that ln(E_D/E₀) = 3.02 corresponds to E_D = 12.4 kJ. A physical meaning of E_D = 12.4 kJ can be found from experimental data at −20 °C. Four different experimental facture surfaces at −20 °C are shown in Fig. 10, including corresponding E_D value. For E_D < 12.4 kJ, cleavage fracture initiated from the notch tip. On the other hand, for E_D > 12.4 kJ, ductile fracture initiated at the notch tip. Thus, it can be concluded that the difference in fracture mechanism at the notch tip leads to two different equations. Note also that the temperature dependence on the maximum principal stress is incorporated by normalizing with respect to the temperature-dependent yield strength. Therefore, it can be applied to other temperatures by using the temperature-specific yield strength.

4.3. Determination of the parameter (σ_u’ and m)

The parameters in Eq. (8), σ_u’ and m, can be determined as follows. For each DWTT data at −97 °C, the maximum principal stress value can be found from Eq. (12) using the measured absorbed energy. Then the failure probability in terms of criteria can be found, as shown in Fig. 11(a). The regression analysis of DWTT data at −97 °C using Eq. (8) gives m = 12.43 and σ_u’=1,205 MPa at −97 °C. It is also found that the calculated Weibull stress σ_w directly proportional to the calculated σ_1,max, provided V_o = 14.82 mm³ is used, as shown in Fig. 11(b).

As described in Section 3.3, DWTT data at two different temperatures, T = -97 °C and −60 °C, are needed to determine the constants b₁′ and b₂′. Note that σ_u’=1,205 MPa was found using total fourteen experimental DWTT data at −97 °C. With the determined parameters (m = 12.43 and V_o = 14.82 mm³), the regressed failure probability curve using five test data at −60 °C gives σ_u’=1,506 MPa at −60 °C, as shown in Fig. 12(a). From these two values of σ_u’, b₁′=4,201.7 MPa and b₂′=218.6 °C were found, giving the following final equation of σ_u’ for API X80.(13)

σ_{u}^{'} (T) = b_{1}^{'} \cdot \exp (- \frac{b_{2}^{'}}{T}) = 4201.7 \cdot \exp (- \frac{218.6}{T})

This is shown in Fig. 12(b). Finally, substituting σ_u’(T) and m into Eq. (11), the failure probability of DWTT of API X80 as a function of temperature can be characterized.

4.4. Characterization of failure probability curve for API X80 DWTT at transition temperatures

From Eqs. (11)–(13), temperature dependence of the API X80 DWTT energy scatter can be characterized in terms of the failure probability. The energy characterization curves ranging from −100 °C to 0 °C for the failure probability of P_f = 5%, 10%, 90% and 95% are compared with DWTT experimental data in Fig. 13. The energy curves are normalized by the ligament area of the DWTT specimen, A=(71.12 × 23.7) mm² = 1,685.5 mm². Assuming that the experimental DWTT energy at 0 °C (E_D^exp = 19.6 kJ) is the limiting upper shelf ductile fracture energy, the corresponding line is also shown in the figure. As shown in Fig. 13, the characterization curves using Eqs. (11)–(13) can capture well the experimental DWTT energy scatter at transition temperatures.

5. Conclusion

This paper proposes a numerical method to characterize energy scatters of a high strength steel (representatively, API X80 in this paper) drop weight tear tests (DWTTs) at ductile–brittle transition temperatures. A numerical method employs FE ductile–brittle fracture simulation using the stress-modified fracture strain (SMFS) model for ductile tearing simulation and the maximum principal stress criterion for cleavage fracture simulation.

The probability of energy scatters is firstly expressed using the two-parameter Weibull distribution in terms of the Weibull stress, which has been widely used to evaluate cleavage fracture probability. However, it is found to be difficult to characterize energy scatters of high strength steel DWTTs due to the following two reasons. The first one is that the Weibull stress approach assumes ductile–brittle transition and thus cannot be applied for the inverse fracture (brittle-ductile transition) typically occurs in high strength steel DWTT. The second one is that an oscillating behavior in the relationship between the DWTT absorbed energy and the Weibull stress is found at high absorbed energy region at transition temperatures, which leads to a non-unique criterion.

To overcome the limitations, the maximum principal stress approach previously proposed by the authors to simulate Charpy impact test and DWTT is then used. Here the probability of energy scatters is expressed using the two-parameter Weibull distribution in terms of the the maximum principal stress instead of the Weibull stress. Note that the maximum principal stress approach is shown to be equivalent to the Weibull stress approach. The former approach is calculated locally in an element, whereas the latter one in a specimen. The maximum principal stress is then obtained as a function of the absorbed energy from FE ductile–brittle fracture simulation. It shows that the maximum principal stress is a logarithmic function of the absorbed energy. The characterized energy scatters are compared with experimental results of high strength steel DWTTs at transition temperatures, showing good agreement.

CRediT authorship contribution statement

Ji-Su Kim: Writing – original draft, Visualization, Formal analysis, Conceptualization, Methodology. Yun-Jae Kim: Writing – review & editing, Project administration, Supervision, Validation. Myeong-Woo Lee: Software, Formal analysis. Ki-Seok Kim: Investigation, Resources. Kazuki Shibanuma: Writing – review & editing, Validation.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

None

Appendix:. Finite element analysis of dwtt

A.1 FE mesh and analysis

The FE mesh to simulate DWTT is composed of eight-node solid element (C3D8 in ABAQUS), as shown in Fig. A1. Considering symmetry, the quarter model with 19,312 elements and 22,951 nodes was used. To simulate crack propagation in the remaining ligament, the cubic element with 0.5 mm size (L_e = 0.5 mm) was employed in the remaining ligament. Note that the use of the smaller element would be better, but was numerically difficult in fracture simulation. Using the TYPE = CYLINDER option in ABAQUS, the hammer and anvil surface were modeled by analytical rigid shell coupled with a reference node. The radius for the hammer and the anvil was 25.4 mm and 19.05 mm, respectively. Only one degree of freedom in the vertical direction of the hammer was unconstrained and all degrees of freedom of anvils were fixed. Based on the experimental test condition, dynamic/implicit analysis with 6 m/s hammer impact velocity was employed with the large geometry change option in ABAQUS. To prevent the rigid shell from penetrating the three-dimensional solid specimen, the frictionless contact option was used.

A.2 Temperature-Dependent tensile properties

In this work, temperature-dependent tensile properties given in Ref. [46] were used, given by.(A1)

σ = σ_{e, R T} (1 - λ {|T^{*}|}^{k}); λ = \{\begin{matrix} 1 : T = T_{room} \\ \frac{T^{*}}{|T^{*}|} : T \neq T_{room} \end{matrix}, T^{*} = \frac{T - T_{room}}{T_{melt} - T_{room}}

where l_e_,RT denotes the equivalent stress value at 25 °C; T_melt = 1,500 °C and k = 0.81 for API X80 [46]. True stress–strain curve at 25 °C, determined from tensile test [34], is shown in Fig. A2(a). By multiplying the temperature term given in Eq. (A1) to the true stress–strain curve at 25 °C, the temperature-dependent tensile properties can be estimated. Fig. A2(b) shows estimated true stress–strain curves at −20 °C, −60 °C and −97 °C, which are used in the present simulation.

The strain rate and adiabatic heating effects on tensile properties were not considered in present study, because it is assumed that the effect of the strain rate increasing the tensile properties and the effect of adiabatic heating decreasing the tensile properties can be roughly canceled out. The validity of such assumption was verified in our previous works [34], [35] by comparing the experimental load–displacement curves and fracture surfaces of Charpy test and DWTT at various temperatures with simulation results.

A.3 Ductile fracture Simulation: Model and method

In our previous works [34], [35], the fracture strain based ductile damage model was used to simulate ductile tearing in Charpy impact test and DWTT. By analyzing API X80 tensile and SENT test data, the following form of the multi-axial fracture strain ε_f was found.(A2)

ε_{f} = 3.34 exp [- 1.5 \cdot \frac{σ_{m}}{σ_{e}}] + 0.4

where the mean normal stress σ_m and equivalent stress σ_e are defined in terms of the principal stress σ_i (i = 1–3) by.(A3)

σ_{m} = \frac{σ_{1} + σ_{2} + σ_{3}}{3}, \bar{σ} = \sqrt{\frac{\{{(σ_{1} - σ_{2})}^{2} + {(σ_{2} - σ 3)}^{2} + {(σ_{3} - σ_{1})}^{2}\}}{2}}

Plastic damage increment ΔD is defined by the ratio of the incremental equivalent plastic strain Δε_e^p and the fracture strain ε_f in terms of the stress triaxiality σ_m/σ_e.(A4)

Δ D = \frac{Δ ε_{e}^{p}}{ε_{f}}

Local fracture at the element integration point is assumed when the damage accumulation reaches a critical value, D_c,(A5)

\int Δ D = \int \frac{Δ ε_{e}^{p}}{ε_{f}} = D_{c}

The value of D_c was determined to be D_c = 0.65, by comparing simulation results with experimental crack growth data of the single edge notched tension specimen at room temperature [35].

When the fracture criterion Eq. (A4) is satisfied in the element integration point, the equivalent stress at the point was reduced to 10 MPa with the slope of −5000 MPa using ABAQUS user subroutines [47]. The Young’s modulus was also decreased to 100 MPa. Then by using the “ELEMENT DELETION” option in ABAQUS, the elements were completely removed.

The presented FE ductile fracture simulation model is applied to simulate DWTT at 0 °C and the simulated load–displacement curve is compared with experimental data in Fig. A3(a). In FE ductile fracture simulation of DWTT, the crack initiation and the maximum load were obtained at LLD = 5.9 mm and 17.1 mm, respectively, as indicated in Fig. A3(a). Up to LLD = 39.5 mm, the simulated load–displacement curve agrees well with experimental data. After LLD = 39.5 mm, the experimental load showed more rapid decrease compared to the simulation result, as shown in Fig. A3(a). This is because, at LLD = 39.5 mm, the inverse fracture occurred in the experiment, as shown in Fig. A3(b) where fracture surface of the DWTT test at 0 °C is shown. The experimental fracture surface is also compared with the simulated fracture surface at LLD = 39.5 mm, showing that ductile fracture surface agrees well.

A.4 Ductile-Brittle fracture simulation model

In our previous works [34], [35], the method to simulate combined ductile and brittle fracture in one analysis. The ductile damage model was briefly explained in Appendix A.3. To simulate brittle fracture, the maximum principal stress criterion, σ_1,max, was proposed, where the brittle fracture was assumed to occur at the integration point when the calculated maximum principal stress reached the critical maximum principal stress. Thus, in FE damage analysis, two fracture (ductile and cleavage) criteria are checked in an element (at the integration point); whether the calculated maximum principal stress value exceeds the critical of σ_1,max or the accumulated plastic damage exceeds the critical value of D_c (Eq. (A5)). When the FE result satisfies one of two criteria, the element is assumed to be failed by that failure mechanism, as depicted in Fig. A4. The technique to delete the failed element is described in the previous sub-section.

The FE DWTT simulation with combined ductile and brittle fracture models is performed as follows. Note that the ductile damage model and criterion was explained in Appendix A.3. Firstly the value of σ_1,max should be calculated using Eq. (12) for the given measured absorbed DWTT energy. Then FE simulation is performed and the mode of fracture at the integration point is checked according to the flowchart shown in Fig. A4. The presented FE ductile–brittle fracture simulation model is applied to simulate DWTT at −20 °C and the simulated load–displacement curve is compared with experimental data in Fig. A5(a) [35]. In the FE simulation, the crack initiated at LLD = 5.9 mm and the load rapidly decreased with the occurrence of combined ductile–brittle fracture, showing good agreement with experimental data. The simulated absorption energy calculated from the area under the curve also shows good agreement with the experimental result. For further check of the validity of the proposed method, the experimental fracture surface is compared with the simulated fracture surface in Fig. A5(b). It shows that the essential feature of fracture pattern occurring in the order of cleavage-ductile-cleavage, commonly observed in experiments, can be captured by using the proposed method.

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Cited by (4)

Mechanical characterization of synergistic thermal and irradiation effects on fracture resistance of stainless steel
2024, International Journal of Mechanical Sciences
This study investigates the synergistic effects of thermal aging and neutron irradiation on the fracture resistance of austenitic stainless steel used in nuclear power plant reactor internals. Thermal aging and irradiation environments are known to reduce fracture resistance, posing risks to the structural integrity of critical components. Considering the high temperature and neutron irradiation environments, which deteriorate the fracture resistance, this study simulates the J-R curves to evaluate the integrity of austenitic stainless steel used in reactor internals. The analysis includes thermal aging at 295 °C for 118,000 h and irradiation dose of 0.4 dpa. The results demonstrate a reliable method for predicting material fracture behavior under prolonged exposure to high temperatures and neutron irradiation, which is essential for the effective lifecycle management of nuclear reactors. This research provides critical insights and practical approaches for ensuring the long-term safety and performance of nuclear power plants.
Finite Element Ductile Fracture Simulation of Charpy and Drop Weight Tear Tests for API X52
2024, Theoretical and Applied Fracture Mechanics
Citation Excerpt :
Banerjee et al. [42] simulated the CVN test using the J-C deformation and fracture model from tensile tests at various temperatures and strain rates, but the effect of adiabatic heating was also not considered. Kim et al. [47–49] simulated combined ductile and brittle fracture of CVN and DWTT tests. In their model, the multi-axial fracture strain damage model (a simplified version of the Johnson-Cook fracture strain model [21]) was used for ductile fracture, whereas the critical strength parameter was used for brittle fracture.
This paper presents a systematic procedure for performing finite element (FE) impact ductile fracture simulation of Charpy (CVN) and Drop Weight Tear Tests (DWTT) with validation using test data of API X52. For deformation and fracture models, the Johnson-Cook (J-C) model is used, of which seven parameters are determined by analyzing (1) round bar tensile test data at three different temperatures (two parameters), (2) tensile test and fracture toughness test at room temperature (three parameters) and (3) instrumented Charpy test (load–displacement) data at room temperature (two parameters). FE impact fracture simulation results with the determined parameters show good agreement with instrumented CVN test data at three different temperatures (0 °C, −30 °C and −60 °C) and DWTT data at temperatures of RT and −30 °C. For DWTT simulation, an analysis of the pre-strain due to flattening is included. Additionally, sensitivity analyses for the effect of adiabatic heating and strain rate on simulation results show that, although both phenomena should be considered in simulation, the strain rate effect is more significant than the adiabatic heating effect.
Prediction of the irradiation effect on the fracture toughness for stainless steel using a stress-modified fracture strain model
2024, International Journal of Mechanical Sciences
Citation Excerpt :
To overcome these issues, a finite element analysis can be a useful alternative. Among currently available approaches, the stress-modified fracture strain model [24,25] has been applied to predict changes in the fracture behavior due to various factors [26–30]. The model was implemented as an ABAQUS user-defined function by Oh et al. [31] and was validated [31] through comparisons with experimental results and the Gurson-Tvergaard-Needleman (GTN) model [32–36].
This paper proposes a numerical method for predicting the reduction in the fracture toughness with neutron dose using the concept of an embrittlement constant. Tensile and fracture toughness test data for type 304 stainless steel in the literature [41,42] are used to validate the proposed method. Fracture toughness tests are simulated using a stress-modified fracture strain damage model implemented in an ABAQUS user-subroutine. The true stress-strain curve and one point of the fracture strain locus are obtained from the tensile test simulation. The remaining parameters of the fracture strain locus are determined by simulating a fracture toughness test. The fracture strain locus is modified to reflect the irradiation effect on the fracture toughness using the proposed irradiation embrittlement constant C_I. The function of the constant with the neutron dose is calculated from the tensile tests and simulations. Subsequently, the modified damage model is used to predict the reduction in the fracture toughness. The predicted simulation results are compared with experimental data, showing good agreement.
Prevention of natural gas pipeline cracking
2024, Advances in Natural Gas Formation Processing and Applications Natural Gas Transportation and Storage Volume 6
The cracking of natural gas pipelines is a critical problem threatening its safety. A comprehensive understanding of the relevant knowledge system and research status is conducive to further research by scholars. This chapter comprehensively introduces the crack propagation mechanism, fracture evaluation method, crack arrest prediction model, and microdamage model of natural gas pipelines. In addition, this chapter reviews the research progress and related achievements of scholars in this field in recent years. Finally, this chapter summarizes some important conclusions and prospects for future research. At present, most of the research focuses on developing new theoretical models but has not been applied to engineering practice. In the future, more research will focus on the fracture prediction of high-grade pipelines. The external environment and internal medium of the pipeline also greatly affect the expansion and stopping of cracks, which is worthy of further study. In the context of big data, it is of great significance to develop data-driven models considering multiple factors for the cracking prediction of natural gas pipelines.

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Outline 大纲

Cited by (4) 被引用 (4)

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Tables (3) 表格（3）