Fracture simulation model for API X80 Charpy test in Ductile-Brittle transition temperatures

doi:10.1016/j.ijmecsci.2020.105771

International Journal of Mechanical Sciences
国际机械科学杂志

Volume 182, 15 September 2020, 105771
第 182 卷，2020 年 9 月 15 日，105771

https://doi.org/10.1016/j.ijmecsci.2020.105771 Get rights and content 获得权利和内容

Full text access 全文访问

Highlights 亮点

•
A fracture simulation model for API X80 steel is established in ductile-brittle transition temperatures.
建立了 API X80 钢在韧脆转变温度下的断裂模拟模型。
•
Ductile tearing is simulated by stress modified fracture strain model which is one of damage models related with local stress triaxility.
延性撕裂采用应力修正的断裂应变模型来模拟，该模型是与局部应力三轴性相关的损伤模型之一。
•
Cleavage fracture is simulated by local maximum principal stress criterion based on Weibull stress.
采用基于威布尔应力的局部最大主应力准则模拟解理断裂。
•
The maximum principal stress can computationally replace the Weibull stress.
最大主应力可以通过计算代替威布尔应力。

ABSTRACT 抽象的

This paper presents a method to simulate fracture patterns of interacting ductile and cleavage fracture and experimental validation using API X80 Charpy test data. For ductile fracture, the stress modified fracture strain damage model is used, whereas for cleavage fracture, the maximum principal stress criterion is proposed. By combining ductile and cleavage fracture models, a numerical method to simulate interacting ductile and cleavage fracture is proposed. The proposed numerical method is validated by comparing with experimental Charpy test data at ductile-brittle transition temperatures. Comparison of fracture surfaces and impact energies shows good agreements.
本文提出了一种模拟韧性断裂和解理断裂相互作用的数值方法，并利用 API X80 夏比冲击试验数据进行了试验验证。对于韧性断裂，采用应力修正的断裂应变损伤模型；对于解理断裂，提出采用最大主应力准则。结合韧性断裂和解理断裂模型，提出了一种模拟韧性断裂和解理断裂相互作用的数值方法。通过与延脆转变温度下夏比冲击试验数据对比，验证了所提出的数值方法的有效性。断裂面和冲击功的对比结果显示，方法与数值方法吻合良好。

Graphic Abstract 图形摘要

Keywords 关键词

API X80 Charpy test

Ductile-brittle transition temperature

Numerical simulation of interacting ductile and cleavage fracture

Stress-modified fracture strain model

Maximum principal stress criterion

API X80 夏比试验；韧脆转变温度；延性断裂与解理断裂相互作用的数值模拟；应力修正断裂应变模型；最大主应力准则

Nomenclature 命名法

A, B A、B

material constant in strain-based failure criteria, see Eq. (1)
基于应变的失效准则中的材料常数，见公式 (1)

a₁, a₂
一个 ₁ 、一个 ₂

fitting parameter of ductile-brittle combined failure model, see Eq. (10)
延脆性复合破坏模型拟合参数，见式（10）

b₁, b₂
b ₁ ，b ₂

fitting parameter of σ_u, see Eq. (11)
拟合参数 σ _u ，见公式（11）

D_c

critical damage value 临界伤害值

elastic modulus 弹性模量

E_CVN

Charpy impact test energy
夏比冲击试验能量

material constant in modified Johnson-Cook model, see Eq. (5)

L_e

element size

exponent of the Weibull distribution, see Eq. (9)

P_f

cumulative failure probability

reduction of area

T, T_melt

temperature and melting temperature (°C)

V₀

reference volume in fracture process zone

ε_e^p

equivalent plastic strain

ε_f

fracture strain

σ₁

maximum principal stress

σ_e

equivalent stress

σ_e,RT

equivalent stress at room temperature

σ_m

hydrostatic stress

σ_TS

tensile stress

σ_u

ductile-brittle failure model parameter, Weibull constant, see Eq. (9)

σ_w

Weibull stress

σ_y

yield stress

Abbreviations

API

American petroleum institute

room temperature

CVN

Charpy V-notch

CMOD

crack mouth opening displacement

DBTT

ductile brittle transition temperature

finite element

SENT

single edge notched tension

SMFS

stress modified fracture strain

1. Introduction

Assessment against brittle fracture of high strength pipelines for energy transportation has been an important issue and requires accurate estimation of interacting ductile tearing and cleavage fracture behaviour at ductile-to-brittle transition temperatures. Although full-scale pipe burst tests would be the most convincing way to evaluate the fracture behaviour of real pipelines [1,2], it usually takes considerable time and cost to conduct a series of experiments. To overcome such problems, the drop weight tearing test (DWTT) has been developed [3]. Because the thickness of the DWTT specimen is the same as (or close to) that of a pipe and sufficient crack propagation path can be also included, it is known to be an adequate method to predict the transition temperature and fracture propagation patterns [4]. With increasing toughness and strength required for linepipe steels, fracture patterns in the DWTT tests become more complex. For instance, for recent high toughness and strength materials, abnormal fracture (cleavage fracture followed by ductile fracture, rather than vice-versa) is often observed [5,6].

Although actual fracture behaviours can be predicted accurately by full-scale pipe test or roughly by DWTT tests, an efficient tool to predict fracture behaviour would be fracture simulation using finite element (FE) analysis. Many models have already been proposed to numerically simulate ductile fracture, for instance, the Gurson-Tvergaard-Needleman model [13], [14], [15], cohesive zone model [16], Rousselier model [17] and stress-modified fracture strain (SMFS) model [18,19]. It has been shown by comparing with extensive experimental data that the SMFS model is simple but quite effective and reliable one for simulating ductile tearing [20], [21], [22], [23], [24], [25], [26]. For cleavage fracture, Beremin [27] has proposed a stochastic approach to explain the distribution of brittle fracture energy. Ruggieri et al. [28] summarized various techniques for implementing the Beremin approach. There have been some attempts to relate the Weibull stress and fracture toughness [29]. The temperature dependence on the Weibull parameters has been an issue in many works [27,[30], [31], [32], [33], [34]].

There have been some attempts to predict combined ductile and cleavage failure by combining the respective theories of ductile and brittle failure [35], [36], [37], [38], [39]. In those works, the Weibull stress approach was used to calculate the cleavage failure probability through post-processing of FE analysis results to predict the scatter of ductile-brittle combined fracture after some amount of ductile tearing. However, interacting ductile tearing and cleavage fracture cannot be simulated. Recently it has been shown that not only for some high strength steels but also in press-notched DWTT specimen tests, ductile tearing can occur after cleavage fracture. In such cases, interacting ductile tearing and cleavage fracture must be explicitly simulated simultaneously, but no such a method has been proposed yet.

In this paper, a FE-based model is proposed to explicitly simulate interacting ductile tearing and cleavage fracture. The model is developed by combining the local maximum principal stress failure criterion with the SMFS ductile fracture model. The validity of the model is confirmed by comparing with Charpy test data of the API X80 steel. Section 2 describes the experiments to determine a ductile-brittle combined fracture model. The ductile fracture and cleavage fracture model is determined in Section 3 and 4 respectively. In Section 5, the developed method is applied to simulate interacting fracture of Charpy tests at various temperatures and simulation results are compared with experimental data. Section 6 concludes the presented work.

2. Summary of experiments

2.1. Tensile test

The API X80 grade steel typically used in pipelines to transport crude oil and natural gas under high pressure was used in this work. Tensile test was performed at room temperature according to API 5L [7]. A flat plate test specimen with the gauge length of 50.8 mm, shown in Fig. 1(a), was used. The specimen was taken from the plate in the hoop direction. The engineering stress-strain curve is shown in Fig. 1(b) and some important properties are summarized in Table 1.

Table 1. Tensile properties of X80.

T [°C]	E [GPa]	σ_y [MPa]	σ_TS [MPa]	RA [%]
25 °C	199	570	647	77.8

2.2. Single Edge Notched Tension (SENT) test

Single edge notched tension (SENT) test was carried out according to the recommendations given in DNV RP-F108 [8]. The specimen was also taken from the plate in the hoop direction. The dimensions of the specimen are shown in Fig. 2(a). The initial crack size, a₀, was 10.03 mm. The crack mouth opening displacement was measured. To measure the crack length, the compliance method suggested in BS 8571 [9] was used with the polynomial fitting procedures given in Refs. [10], [11], [12]. Experimental data are shown in Fig. 2(b).

2.3. Charpy V-Notch Test

Charpy V-notch (CVN) impact test was performed according to API 5L [7] at temperatures ranging from -196 to 0°C. The specimen geometry and impact test results are shown in Fig. 3. The specimen was also taken from the plate in the hoop direction. Minimum, averaged and maximum impact energies are summarized in Table 2 for the given temperature, together with the number of tests. Note that 74 tests were performed at -150°C.

Table 2. Charpy test data with temperature.

T [°C]	-196	-150	-120	-90	-60	-30	0
Number of specimens	3	74	3	6	6	3	3
Averaged E_CVN [J]	4.35	10.70	82.27	221.80	297.80	446.63	485.43
Minimum E_CVN [J]	3.42	4.82	10.47	78.74	207.45	380.68	478.45
Maximum E_CVN [J]	5.16	39.03	221.5	334.84	344.67	472.32	492.89

3. Ductile tearing simulation damage model

3.1. Stress-modified fracture strain damage model

In the stress-modified fracture strain model [18], [19], [20], [21], [22], [23], [24], [25], [26], the multi-axial fracture strain should be determined as a function of the stress triaxiality σ_m/σ_e using(1)

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

where σ_m and σ_e denote the hydrostatic pressure and equivalent stress, respectively; and A and B are material constants that must be determined. In this paper, these constants were determined from tensile and SENT tests, and the procedure will be given later. Once ε_f is known, incremental damage ΔD can be calculated by(2)

Δ D = \frac{Δ ε_{e}^{p}}{ε_{f}}

When the accumulated damage reaches a critical value, D_c,(3)

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

local ductile fracture is assumed. This ductile failure simulation technique was implemented into ABAQUS [40] using user subroutines. In simulation, the incremental ductile damage was calculated at the Gauss point in an element using Eqs. (1) and (2). When the accumulated damage reaches the critical damage value, the elastic modulus and equivalent stress were reduced to 100MPa and 10MPa, respectively. Then the “ELEMENT DELETION” option was used to completely remove the element. Detailed explanations can be found in our previous works [20], [21], [22], [23], [24], [25], [26].

3.2. Determination of ductile damage model parameters

The present damage model, Eqs. (1)-(3), includes three parameters to be determined; A, B and D_c. The procedure to determine these three parameters is described next.

The first step is to simulate the tensile test using elastic-plastic FE analysis. Three-dimensional (3-D) twenty-noded solid elements with reduced integrations (C3D20R in ABAQUS) were used and the FE mesh is shown in Fig. 4(a). For the analysis, true stress-strain data shown in Fig. 4(b) were used, which were obtained using the Bridgman correction [53] and fine-tuning using FE analysis with the large geometry change option. Simulation results are compared with the experiment data in Fig. 4(b), showing good agreement. From FE analysis, the stress triaxiality was extracted from the center of the specimen as a function of equivalent plastic strain, and was averaged as shown in Fig. 4(c). The resulting averaged stress traxiality and plastic strain to failure were (0.74, 1.51).

The second step is to simulate the SENT test to determine the critical damage value, D_c, and the multi-axial fracture strain locus. The quarter FE model was made using a three-dimensional (3-D) eight-noded solid elements (C3D8 in ABAQUS), as shown in Fig. 5(a). The number of meshes and nodes were 58,432 and 67,511, respectively. The true stress-strain data from tensile test were used directly with the large geometry change option. Note that the minimum element size in the crack propagation area was set to be 0.1 mm. Note that the FE damage analysis results can be sensitive to the element size, the damage model parameters should be determined based on the given element size. For instance, when the minimum element size of 0.2mm is used, the element size effect can be compensated by choosing an appropriate value of D_c, as demonstrated in our previous works [22,23,25]. To determine the value of D_c, the first trial is to assume B=0 in Eq. (1), of which the locus is shown in Fig. 5(b). Then the value of D_c can be found to match the experimental crack initiation with simulation results. In this work, D_c=1 was found. Note that it was shown in our previous work [23] that the crack initiation prediction is mainly affected by D_c. The simulation result using B=0 and D_c=1 is compared with experimental Δa-CMOD curve in Fig. 6(a), showing that agreement is not so good. Using D_c=1 as the initial value, the values of B and D_c were then found to match the experimental Δa-CMOD and dΔa/dCMOD-CMOD curves. The results are shown in Fig. 6(a) and Fig. 6(b), suggesting that the simulation result using B=0.4 and D_c=1 agree well with experimental data. Thus the final fracture strain locus is given by(4)

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

which is shown in Fig. 5(b). Note that in the present case, the value of D_c does not change but sometimes its value changes slightly with B.

Using Eq. (4) with D_c=1, the simulated load-CMOD curve is compared with experimental data in Fig. 6(c), showing again better agreement with B=0.4 than with B=0 or B=0.2. Furthermore, the simulated fracture surface is compared with experimental data in Fig. 7, also showing good agreement.

3.3. Numerical prediction of Charpy energy at upper shelf temperature (0°C)

Using the developed ductile fracture modeling method, the Charpy test at upper shelf temperature (0°C) is simulated and the resulting Charpy energy is compared with experimental data in this sub-section. For simulation, the effect of temperature on tensile properties should be quantified first. In this work, the following equation suggested in Ref. [41] was used for the temperature effect on tensile properties(5)

σ = σ_{e, R T} (1 - λ {| T |}^{* k}); λ = {\begin{matrix} 1 : T = T_{r o o m} \\ \frac{T^{*}}{{| T |}^{*}} : T \neq T_{r o o m} \end{matrix}, T^{*} = \frac{T - T_{r o o m}}{T_{m e l t} - T_{r o o m}}

where σ_e,RT denote the equivalent stress at room temperature; k=0.81 and T_melt=1,500°C for X80 steel [40]. Note that the strain rate effect is not considered in Eq. (5). The resulting stress-strain curve at 0°C are shown in Fig 4(b).

Applying the fracture strain locus given in Eq. (4) and the stress-strain curve at 0°C, the Charpy test at 0°C was simulated. In simulation, the minimum element size in the crack propagation area was 0.1 mm, the same as that in the SENT simulation. The quarter model was used, made of the three-dimensional (3-D) eight-noded solid elements (C3D8 in ABAQUS), as shown in Fig. 8(a). The number of meshes and nodes were 31,767 and 36,253, respectively. Dynamic implicit analysis using the large geometry change option was applied with the 5 m/s speed of the striker at the time of impact.

As shown in Fig. 8(b), the load-displacement curve can be obtained from the simulation, from which the Charpy impact energy can be calculated. The calculated Charpy impact energy was 482.80 J which was very close to experimental value of 485.43 J, as shown in Table 2. Additionally the contours of the stress triaxility ahead of the growing crack front from the Charpy impact test simulation is shown in Fig. 8(c). The contours of the stress triaxiality from 1.0 to 1.4 are shown. With crack growth, the region of high stress triaxiality region (for instance, 1.4) gets smaller.

4. Cleavage fracture simulation model

4.1. Maximum principal stress criterion

The Beremin model [27], which has been widely used to predict cleavage fracture [27], [28], [29], [30], [31], [32], [33], [34], accounts for the scatter of fracture energy resulting from the random nature of brittle fracture using the following Weibull distribution:(6)

P_{f} = 1 - \exp (- {(\frac{σ_{w}}{σ_{u}})}^{m})

where σ_w represents the Weibull stress defined as(7)

σ_{w} = {(\frac{1}{V_{0}} \int_{V} {(σ_{1})}^{m} d V)}^{\frac{1}{m}}

where V is the volume of the fracture process zone; σ₁ is the maximum principal stress; and the Weibull parameters, m and σ_u, are material constants. In the Beremin model, it is assumed that micro-cracks exist at the point where plastic deformation occurs and unstable fracture occurs if the maximum principal stress is large enough.

Some approaches have been made to analyze ductile-brittle fracture in combination with the Gurson-Tvergaard-Needleman model [37,39]. In those works, the Weibull stress, Eq. (7), is calculated by post-processing FE results to calculate cleavage failure probability after some amounts of ductile tearing. Therefore, this approach cannot be applied when cleavage fracture occurs before ductile tearing. More importantly it cannot simulate the combined ductile and cleavage fracture patterns in the ductile-brittle transition region.

To overcome these problems, the maximum principal stress within an element is used as a local cleavage fracture criterion in the present work. This is because the Weibull stress defined by Eq. (7) is found to be proportional to the largest maximum principal stress value calculated in all finite elements. We assume that Eq. (7) can be rewritten as(8)

σ_{w} (V_{0}) = {(\frac{1}{V_{0}} \int_{V} {(σ_{1})}^{m} d V)}^{\frac{1}{m}} \approx {(\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}} {\cdot \max (σ_{1}) |}_{V} \equiv f (\frac{V}{V_{0}}) \cdot σ_{1, \max}

where n_e denotes the total number of elements used in calculations. Based on the FE analysis of the Charpy test specimens, we found that the 1/m-power of the sum of the maximum principal stress depends linearly on the largest maximum principal stress over the fracture process zone, V, as given in Eq. (8). It means that the Weibull stress, as a cleavage criterion, can be replaced by a local maximum principal stress. The linear relationship between the Weibull stress and largest maximum principal stress can be expressed using a function of V/V₀. Note that both stresses, σ_w and σ_1,max, are the same when V/V₀ is unity. Ruggieri et al. [28] referred that the value of V₀ could be conveniently assigned as a unit value in computations; in other words, V/V₀ can be also assigned as a unit value.

Figure 9 shows our FE results of the relationship between the Weibull stress and the largest maximum principal stress in the Charpy specimen. The Weibull stresses from FE analysis were calculated by using post-processing using Eq. (8) in each time step. At the same time, the largest maximum principal stresses were extracted from the FE analysis. In this calculation, the value of m was assumed to be twelve (which results from the Weibull distribution of the Charpy impact energy data at -150°C, which will be shown later in Fig. 10). However, the use of other values generally reported [27,28,30,31,36,37] gives the same answer. It shows that the dependence of the Weibull stress and largest maximum principal stress is linear, and the slope depends on the choice of V/V₀. This supports our idea that σ₁ in an element can be used as an alternative local criterion for determining cleavage fracture. In fact, such the maximum principal stress criterion was the original approach to quantify the fracture energy due to cleavage caused by the micro-cracks contained within the specimens [42], [43], [44], [45], [46], [47], [48], [49], [50]. However, by combining the local criterion with Weibull distribution, the energy scatter in transition region can be captured and interacting ductile tearing and cleavage fracture can be simulated.

4.2. Determination of the maximum principal stress criterion for X80

Using Eq. (8), the Weibull distribution can be re-written using the maximum principal stress as(9)

P_{f} = 1 - \exp (- {(\frac{σ_{1, \max} (E_{C V N})}{σ_{u} (T)})}^{m})

where σ_1,max is assumed to be a function of the Charpy impact energy (E_CVN); and m to be temperature-independent but σ_u(T) to be temperature-dependent.

Figure 10(a) shows the relationship between σ_1,max and E_CVN at various temperatures. In the figure, σ_1,max is the maximum principal stress calculated from FE analysis of the Charpy specimen, and is normalized with respect to the yielding strength at that temperature, σ_y(T). Each data point in the figure corresponds to one Chary test. The results show that the relationship between σ_1,max and ln(E_CVN/E₀) is linear but depending on E_CVN, two distinct slopes can be seen. Such different slopes might result from presence of more ductile tearing at higher E_CVN.

Data regression gives the following relationship(10)

\frac{σ_{1, m a x} (E)}{σ_{y} (T)} = a_{1} \cdot \ln (\frac{E_{C V N}}{E_{0}}) + a_{2} = {\begin{matrix} 0.40 \ln (\frac{E_{C V N}}{4.82}) + 1.83, E_{C V N} \leq 38.07 J \\ 0.10 \ln (\frac{E_{C V N}}{4.82}) + 2.45, E_{C V N} \geq 38.07 J \end{matrix}

where E₀ is a normalizing energy, which is taken to be the minimum impact energy experimentally calculated at -150°C in this study, E₀=4.82J, as given in Table 2. Although the slopes of the two straight lines are different, the same value for E₀ is used. The right hand side terms in Eq. (10) do not have temperature dependence. The effect of temperature is only reflected on the yield stress. Note that for E_CVN=~490J which is the maximum impact energy at 0°C (see Table 2), the value of σ_1,max/σ_y is approximately 2.9 which is close to the HRR value for non-hardening materials [51]. Combining Eq. (9) and (10) gives the results Fig. 10(b), from which the Weibull parameters, m and σ_u, are determined as m=12.43 and σ_u=1,442 MPa.

It was shown that the maximum principal stress from the notch tip can be expressed as the logarithmic function of the distance by the slip-line theory [54]. Furthermore a simple mechanistic approach for cleavage fracture is to assume that cleavage fracture occurs when the maximum principal stress at the critical distance reaches the critical value. Assuming that cleavage failure is initiated from randomly-distributed inclusions, one can expect that the maximum principal stress criterion can depend logarithmically on the Charpy impact energy.

4.3. Determination of temperature-dependent σ_u

To complete the cleavage model, the Weibull constants, σ_u and m, must be determined as a function of temperature. In this work, the constant m is assumed to be temperature independent, but σ_u to be temperature dependent. To calculate the temperature dependent σ_u, the following equation suggested in the Ref. [34] was used.(11)

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

The two constants b₁ and b₂ were determined using the Charpy data at -150°C and -90°C. The results are shown in Fig. 11. The number of Charpy impact tests conducted at -90°C was six, as given in Table 2. The experimental results were fitted using Eq. (9) with m=12.43, and the results are shown in Fig. 11(a), giving σ_u = 1,843 MPa at -90°C. Using two σ_u values at -90°C and -150°C, two constants b₁ and b₂ can be determined as; b₁=3,047MPa and b₂=92.0°C. Tanguy at el. [36] found that σ_u was independent of temperature in the range between -196°C and -150°C for A508 steel, since potential for causing cleavage was maintained by void nucleation at carbides [39]. Lopez et al. [52] showed from SEM observations that micro-crack propagation within carbide regions was dominant for X80 steel, which suggest that the Tanguy's argument can be also applied to the X80 steel. Therefore, the value of σ_u below -150°C was assumed to be independent on temperature and to remain constant, as shown in Fig. 11(b). In order to check the validity of this assumption, the result of σ_u without the cut-off is also examined, as shown in Fig. 11(b) in dotted line.

4.4. Charpy impact energy at Ductile-Brittle transition temperatures

By combining Eqs. (9) to (11), the Charpy impact energy as a function of temperature can be obtained as(12)

E_{C V N} = 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}])

where a₁ and a₂ can be found in Eq. (10). The energy curves corresponding to P_f=0.05, 0.1, 0.9 and 0.95 are shown in Fig. 12. The predictions with the cut-off in σ_u are shown in solid lines, whereas those without the cut-off in dotted lines. Overall predicted curves with the cut-off in σ_u can bound experimental data well. On the other hand, the predicted curves without the cut-off underestimate energies at very low temperature.

5. Numerical simulation of interacting ductile tearing and cleavage of Charpy tests at transition temperatures

5.1. FE analysis

The Charpy test provides the Charpy impact energy for a given temperature. From the test, the fracture surface exhibiting both cleavage and ductile fracture surface in general can be also observed. As the objective of this work is to numerically simulate interacting cleavage and ductile fracture surface, the FE analysis is performed as follows to validate the proposed analysis method:

•
For a given Charpy impact energy (E_CVN) and temperature, determine the maximum principal stress σ_1,max using Eq. (10) for the cleavage fracture simulation model
•
Perform fracture simulation using the determined criterion and the ductile fracture simulation model determined in Section 3 to compare with experimental fracture surfaces
•
During simulation, the impact energy can be also obtained and be compared with experimental input

If the failure probability (P_f) and temperature are pre-scribed, the corresponding σ_1,max value can be also determined from Eq. (12), and the analysis can be done as described above.

Fracture simulation can be performed as follows using user-defined functions provided in ABAQUS. The schematic diagram of the simulation is described in Fig. 13(a) and the flow chart of the FE implementation is given in Fig. 13(b). At each gauss point in an element, the maximum principal stress value is checked. If it is greater than σ_1,max, cleavage fracture is assumed and the element is removed by reducing stresses and Young's modulus sharply to small values, as described in Section 3. If the maximum principal stress is less than σ_1,max, then the plastic damage increment is calculated according to Eq. (2) and is saved for the next calculation. It should be noted that ductile and cleavage fracture is assumed to be independent, and thus the accumulated plastic damage does not affect the cleavage fracture. With crack growth (by removing elements), the crack tip stress fields will change by stress redistribution. Ductile or cleavage fracture is checked accordingly with crack growth.

5.2. Comparison with experimental fracture surface

Experimentally-observed fracture surfaces of the Charpy test specimens at different temperatures are compared with FE analysis results in Fig. 14, Fig. 15, Fig. 16, Fig. 17. Figure 14 show the results at -60°C. In Fig. 14(a), the left part of the figure shows the experimental fracture surface and the right one the FE analysis results. The experimental fracture surfaces are classified into ductile fracture, cleavage fracture and shear lip [36]. As the shear lip was not simulated explicitly in this work, only two surfaces are indicated in the FE results; ductile and cleavage fracture surface. It can be seen that simulated fracture surfaces agree well with experimental ones. From the test, the measured Charpy impact energy was E_CVN=223.5J. Based on this value, the value of σ_1,max is calculated using Eq. (10), which is also given in the figure caption. This is the criterion for cleavage fracture in simulation. After simulation, the impact energy can be calculated from simulated load-displacement curves, as shown in Fig. 14(b). The calculated value was E_CVN=196.8J which is slightly different from the experimental value of E_CVN=223.5J. Corresponding results at -90°C, -120°C and -150°C are shown in Fig. 15, Fig. 16, Fig. 17. Similar findings can be made. The energy results shown in Fig. 14, Fig. 15, Fig. 16, Fig. 17 are summarized in Fig. 18, showing the predictions agree overall well with the experimental results.

6. Conclusion

This paper presents a method to simulate fracture patterns of interacting ductile and cleavage fracture and experimental validation using API X80 Charpy test data. The method uses the stress modified fracture strain damage model for ductile fracture and the maximum principal stress criterion for cleavage fracture. The ductile fracture model is determined from the tensile test and single edge notch tension test data. For the cleavage fracture model, the maximum principal stress is determined as a function of the Charpy impact energy using the Charpy test data over temperatures. The presented cleavage model is related to the well-known Weibull stress model. By combining ductile and cleavage fracture models, a numerical method that can simulate interacting ductile and cleavage fracture.

To validate the proposed numerical method, simulation results are compared with experimental Charpy test data at various temperatures. In simulation, the experimentally-measured Charpy impact energy is an input, from which the maximum principal stress for cleavage fracture is estimated. Comparison of simulated fracture surfaces with experimental ones shows good agreements. The calculated impact energies from simulation are also similar to experimental ones.

We believe that the proposed numerical method would be the first one to simulate interacting ductile tearing and cleavage fracture. It would be very useful to understand fracture behaviours observed in pipeline and pressure vessel steels, such as crack arrest after significant cleavage fracture, cleavage fracture after long ductile tearing and so on. Although comparison with Charpy test data shows good agreement and thus provides confidence, it would be important that the method can be applied to simulate interacting ductile and cleavage fracture in larger scale specimens or components. Our future work is to apply the presented model to the drop weight tearing test (DWTT) or full-scale pipe test, where the relevant dimensions are much larger than the Charpy specimen. One of main technical issue for simulating the DWTT and full-scale pipe tests would be the element size effect. The use of the small element size would be preferable, but can cause numerical difficulties.

CRediT authorship contribution statement

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

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

This work was supported by POSCO [2017R240], and Korea Agency for Infrastructure Technology Advancement [17IFIP-B067108-05].

References

[1]
K. Shibanuma, T. Hosoe, H. Yamaguchi, M. Tsukamoto, K. Suzuki, S. Aihara
Crack tip opening angle during unstable ductile crack propagation of a high-pressure gas pipeline
Eng. Fract. Mech., 204 (2018), pp. 434-453
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.engfracmech.2018.10.020
View PDF View article View in Scopus Google Scholar
[2]
S. Kawaguchi, K. Murai, Y. Hashimoto, N. Hagiwara, H. Yajima, M. Toyoda
Full-scale burst tests of ultra-high pressured rich-gas pipelines under buried and unburied conditions
Proc. Bienn. Int. Pipeline Conf. IPC. (2009), pp. 327-336
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1115/IPC2008-64434
View in Scopus Google Scholar
[3]
G.M. McClure, A.R. Duffy, R.J. Eiber
Fracture resistance in line pipe
J. Manuf. Sci. Eng. Trans. ASME., 87 (3) (1965), pp. 265-278
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1115/1.3670809
Crossref View in Scopus Google Scholar
[4]
D.J. Horsley
Background to the use of CTOA for prediction of dynamic ductile fracture arrest in pipelines
Eng. Fract. Mech., 70 (3-4) (2003), pp. 547-552
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/S0013-7944(02)00136-4
View PDF View article View in Scopus Google Scholar
[5]
B. Hwang, S. Lee, Y.M. Kim, N.J. Kim, J.Y. Yoo, C.S. Woo
Analysis of abnormal fracture occurring during drop-weight tear test of high-toughness line-pipe steel
Mater. Sci. Eng. A., 368 (1-2) (2004), pp. 18-27
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.msea.2003.09.075
View PDF View article View in Scopus Google Scholar
[6]
K.O. Halsen, E. Heier
Drop weight tear testing of high toughness pipeline material
Proc. Bienn. Int. Pipeline Conf. IPC. (2004), pp. 1717-1724
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1115/ipc2004-0609
View in Scopus Google Scholar
[7]
A.P. Institute
API 5L Specification for line pipe
Api Spec 5L (2007)
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1520/G0154-12A
Google Scholar
[8]
DNV
Fracture control for pipeline installation methods introducing cyclic plastic strain
Recomm. Pract. DNV-RP-F108 (2006)
Google Scholar
[9]
BSI
BS 8571:2014 - Method of test for determination of fracture toughness in metallic materials using single edge notched tension (SENT) specimens
BSI (2014)
Google Scholar
[10]
S. Cravero, C. Ruggieri
Estimation procedure of J-resistance curves for SE(T) fracture specimens using unloading compliance
Eng. Fract. Mech., 74 (17) (2007), pp. 2735-2757
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.engfracmech.2007.01.012
View PDF View article View in Scopus Google Scholar
[11]
X.K. Zhu
Full-range stress intensity factor solutions for clamped SENT specimens
Int. J. Press. Vessel. Pip., 149 (2017), pp. 1-13
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.ijpvp.2016.11.004
View PDF View article Google Scholar
[12]
H.G. Pisarski
Determination of pipe girth weld fracture toughness using SENT specimens
Proc. Bienn. Int. Pipeline Conf. IPC. (2010), pp. 217-224
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1115/IPC2010-31123
View in Scopus Google Scholar
[13]
A.L. Gurson
Continuum theory of ductile rupture by void nucleation and growth: Part 1 - yield criteria and flow rules for porous ductile media
J. Eng. Mater. Technol. Trans. ASME., 99 (10) (1977), pp. 2-15
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1115/1.3443401
Crossref View in Scopus Google Scholar
[14]
V. Tvergaard, A. Needleman
Analysis of the cup-cone fracture in a round tensile bar
Acta Metall, 32 (1) (1984), pp. 157-169
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/0001-6160(84)90213-X
View PDF View article View in Scopus Google Scholar
[15]
J. Koplik, A. Needleman
Void growth and coalescence in porous plastic solids
Int. J. Solids Struct., 24 (8) (1988), pp. 835-853
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/0020-7683(88)90051-0
View PDF View article View in Scopus Google Scholar
[16]
A. Cornec, I. Scheider, K.H. Schwalbe
On the practical application of the cohesive model
Eng. Fract. Mech., 70 (14) (2003), pp. 1963-1987
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/S0013-7944(03)00134-6
View PDF View article View in Scopus Google Scholar
[17]
G. Rousselier
Ductile fracture models and their potential in local approach of fracture
Nucl. Eng. Des., 105 (1) (1987), pp. 97-111
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/0029-5493(87)90234-2
View PDF View article View in Scopus Google Scholar
[18]
J.R. Rice, D.M. Tracey
On the ductile enlargement of voids in triaxial stress fields*
J. Mech. Phys. Solids., 17 (1969), pp. 201-217
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/0022-5096(69)90033-7
View PDF View article View in Scopus Google Scholar
[19]
J.W. Hancock, A.C. Mackenzie
On the mechanisms of ductile failure in high-strength steels subjected to multi-axial stress-states
J. Mech. Phys. Solids., 24 (2-3) (1976), pp. 147-160
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/0022-5096(76)90024-7
View PDF View article View in Scopus Google Scholar
[20]
M.W. Lee, J.S. Kim, J.Y. Jeon, Y.J. Kim, J.W. Kim, J.S. Kim
Comparison of numerical predictions with experimental burst pressures of tubes with multiple surface cracks
Eng. Fract. Mech., 201 (2018), pp. 176-195
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.engfracmech.2018.06.016
View PDF View article View in Scopus Google Scholar
[21]
G.G. Youn, H.S. Nam, Y.J. Kim, J.W. Kim
Numerical prediction of thermal aging and cyclic loading effects on fracture toughness of cast stainless steel CF8A: Experimental and numerical study
Int. J. Mech. Sci., 163 (2019), Article 105120
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.ijmecsci.2019.105120
View PDF View article View in Scopus Google Scholar
[22]
H.S. Nam, Y.R. Oh, Y.J. Kim, J.S. Kim, N. Miura
Application of engineering ductile tearing simulation method to CRIEPI pipe test
Eng. Fract. Mech., 153 (2016), pp. 128-142
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.engfracmech.2015.12.012
View PDF View article View in Scopus Google Scholar
[23]
H.S. Nam, J.M. Lee, Y.J. Kim, J.W. Kim
Numerical ductile fracture prediction of circumferential through-wall cracked pipes under very low cycle fatigue loading condition
Eng. Fract. Mech., 194 (2018), pp. 175-189
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.engfracmech.2018.02.025
View PDF View article View in Scopus Google Scholar
[24]
H.S. Nam, J.M. Lee, G.G. Youn, Y.J. Kim, J.W. Kim
Simulation of ductile fracture toughness test under monotonic and reverse cyclic loading
Int. J. Mech. Sci., 135 (2018), pp. 609-620
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.ijmecsci.2017.11.037
View PDF View article View in Scopus Google Scholar
[25]
H.W. Ryu, K.D. Bae, Y.J. Kim, J.J. Han, J.S. Kim, P.J. Budden
Ductile tearing simulation of Battelle pipe test using simplified stress-modified fracture strain concept
Fatigue Fract. Eng. Mater. Struct., 39 (11) (2016), pp. 1391-1406
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1111/ffe.12456
Crossref View in Scopus Google Scholar
[26]
C.S. Oh, N.H. Kim, Y.J. Kim, J.H. Baek, Y.P. Kim, W.S. Kim
A finite element ductile failure simulation method using stress-modified fracture strain model
Eng. Fract. Mech., 78 (1) (2011), pp. 124-137
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.engfracmech.2010.10.004
View PDF View article View in Scopus Google Scholar
[27]
F.M. Beremin, A. Pineau, F. Mudry, J.C. Devaux, Y. D'Escatha, P. Ledermann
A local criterion for cleavage fracture of a nuclear pressure vessel steel
Metall. Trans. A., 14 (1983), pp. 2277-2287
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1007/BF02663302
View in Scopus Google Scholar
[28]
C. Ruggieri, R.H. Dodds
A local approach to cleavage fracture modeling: An overview of progress and challenges for engineering applications
Eng. Fract. Mech., 187 (2018), pp. 381-403
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.engfracmech.2017.12.021
View PDF View article View in Scopus Google Scholar
[29]
Y. Lei, N.P. O'Dowd, E.P. Busso, G.A. Webster
Weibull stress solutions for 2-D cracks in elastic and elastic-plastic materials
Int. J. Fract., 89 (1998), pp. 245-268
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1023/A:1007435927306
View in Scopus Google Scholar
[30]
N.P. O'Dowd, Y. Lei, E.P. Busso
Prediction of cleavage failure probabilities using the Weibull stress
Eng. Fract. Mech., 67 (2) (2000), pp. 87-100
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/S0013-7944(00)00051-5
View PDF View article View in Scopus Google Scholar
[31]
C. Ruggieri, X. Gao, R.H. Dodds
Transferability of elastic-plastic fracture toughness using the Weibull stress approach: Significance of parameter calibration
Eng. Fract. Mech., 67 (2) (2000), pp. 101-117
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/S0013-7944(00)00052-7
View PDF View article View in Scopus Google Scholar
[32]
A. Rossoll, C. Berdin, C. Prioul
Determination of the fracture toughness of a low alloy steel by the instrumented charpy impact test
Int. J. Fract., 115 (2002), pp. 205-226
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1023/A:1016323522441
View in Scopus Google Scholar
[33]
B. Wasiluk, J.P. Petti, R.H. Dodds
Temperature dependence of Weibull stress parameters: Studies using the Euro-material
Eng. Fract. Mech., 73 (8) (2006), pp. 1046-1069
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.engfracmech.2005.11.006
View PDF View article View in Scopus Google Scholar
[34]
P. Haušild, C. Berdin, P. Bompard
Prediction of cleavage fracture for a low-alloy steel in the ductile-to-brittle transition temperature range
Mater. Sci. Eng. A., 391 (1-2) (2005), pp. 188-197
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.msea.2004.08.067
View PDF View article View in Scopus Google Scholar
[35]
B. Tanguy, J. Besson, R. Piques, A. Pineau
Ductile to brittle transition of an A508 steel characterized by Charpy impact test Part I: Experimental results
Eng. Fract. Mech., 72 (1) (2005), pp. 49-72
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.engfracmech.2004.03.010
View PDF View article View in Scopus Google Scholar
[36]
B. Tanguy, J. Besson, R. Piques, A. Pineau
Ductile to brittle transition of an A508 steel characterized by Charpy impact test. Part II: Modeling of the Charpy transition curve
Eng. Fract. Mech., 72 (3) (2005), pp. 413-434
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.engfracmech.2004.03.011
View PDF View article View in Scopus Google Scholar
[37]
K. Hojo, N. Ogawa, T. Hirota, K. Yoshimoto, Y. Nagoshi, S. Kawabata
Application of Coupled Damage and Beremin Model to Ductile-Brittle Transition Temperature Region Considering Constraint Effect
Procedia Struct. Integr., 2 (2016), pp. 1643-1651
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.prostr.2016.06.208
View PDF View article View in Scopus Google Scholar
[38]
K. Yoshimoto, T. Hirota, H. Sakamoto, T. Sugihara, S. Sakaguchi, T. Oumaya
Applicability of miniature C(T) specimen to evaluation of fracture toughness for reactor pressure vessel steel
Am. Soc. Mech. Eng. Press. Vessel. Pip. Div. PVP. (2013)
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1115/PVP2013-97840
Google Scholar
[39]
G. Bernauer, W. Brocks, W. Schmitt
Modifications of the Beremin model for cleavage fracture in the transition region of a ferritic steel
Eng. Fract. Mech., 64 (3) (1999), pp. 305-325
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/s0013-7944(99)00076-4
View PDF View article View in Scopus Google Scholar
[40]
Abaqus, ABAQUS Version
Dessault Syst. (2016)
2016
Google Scholar
[41]
S. Canada
Temperature Effects on Dynamic Fracture of Pipeline Steel
Master of Science, University of Alberta (2015)
Google Scholar
[42]
C. Soyarslan, B. Gülçimen, S. Bargmann, P. Hähner
Modeling of fracture in small punch tests for small- and large-scale yielding conditions at various temperatures
Int. J. Mech. Sci., 106 (2016), pp. 266-285
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.ijmecsci.2015.12.007
View PDF View article View in Scopus Google Scholar
[43]
F. Mudry
A local approach to cleavage fracture
Nucl. Eng. Des., 105 (1) (1987), pp. 65-76
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/0029-5493(87)90230-5
View PDF View article View in Scopus Google Scholar
[44]
S.R. Bordet, A.D. Karstensen, D.M. Knowles, C.S. Wiesner
A new statistical local criterion for cleavage fracture in steel. Part I: Model presentation
Eng. Fract. Mech., 72 (3) (2005), pp. 435-452
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.engfracmech.2004.02.009
View PDF View article View in Scopus Google Scholar
[45]
T. Lin, A.G. Evans, R.O. Ritchie
Statistical analysis of cleavage fracture ahead of sharp cracks and rounded notches
Acta Metall, 34 (11) (1986), pp. 2205-2216
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/0001-6160(86)90166-5
View PDF View article View in Scopus Google Scholar
[46]
L. Tong, L. Niu, S. Jing, L. Ai, X.L. Zhao
Low temperature impact toughness of high strength structural steel
Thin-Walled Struct, 132 (2018), pp. 410-420
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.tws.2018.09.009
View PDF View article View in Scopus Google Scholar
[47]
H. Wang, Y. Yu, J. Yu, J. Duan, Y. Zhang, Z. Li, C. Wang
Effect of 3D random pitting defects on the collapse pressure of pipe — Part I: Experiment
Thin-Walled Struct, 129 (2018), pp. 512-526
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.tws.2018.04.015
View PDF View article View in Scopus Google Scholar
[48]
S. Zhang, W. Jiang, M.R. Tonks
A new phase field fracture model for brittle materials that accounts for elastic anisotropy
Comput. Methods Appl. Mech. Eng., 358 (2020), Article 112643
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.cma.2019.112643
View PDF View article View in Scopus Google Scholar
[49]
I. Khisamitov, G. Meschke
Configurational-force interface model for brittle fracture propagation
Comput. Methods Appl. Mech. Eng., 351 (2019), pp. 351-378
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.cma.2019.03.029
View PDF View article View in Scopus Google Scholar
[50]
P. Bowen, S.G. Druce, J.F. Knott
Effects of microstructure on cleavage fracture in pressure vessel steel
Acta Metall, 34 (6) (1986), pp. 1121-1131
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/0001-6160(86)90222-1
View PDF View article View in Scopus Google Scholar
[51]
C.F. Shih, M.D. German
Requirements for a one parameter characterization of crack tip fields by the HRR singularity
Int. J. Fract., 17 (1981), pp. 27-43
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1007/BF00043119
Crossref View in Scopus Google Scholar
[52]
H.F. López, R. Raghunath, J.L. Albarran, L. Martinez
Microstructural aspects of sulfide stress cracking in an API X-80 pipeline steel
Metall. Mater. Trans. A Phys. Metall. Mater. Sci., 27 (1996), pp. 3601-3611
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1007/bf02595451
View in Scopus Google Scholar
[53]
P.W. Bridgman
Studies in Large Plastic Flow and Fracture: With Special Emphasis on the Effects of Hydrostatic Pressure
McGraw-Hill, New York-London (1952)
Google Scholar
[54]
R.O. Ritchie, J.F. Knott, J.R. Rice
On the relationship between critical tensile stress and fracture toughness in mild steel
J. Mech. Phys. Solids., 21 (6) (1973), pp. 395-410
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/0022-5096(73)90008-2
View PDF View article View in Scopus Google Scholar

Cited by (25)

Prediction of hydrogen-assisted fracture under coexistence of hydrogen-enhanced plasticity and decohesion
2023, International Journal of Hydrogen Energy
This work presents a novel strategy to model the coexistence of HELP & HEDE mechanisms. The proposed method accounts for ductile and brittle damage with a competitive fracture criterion to describe the competition effect between HELP & HEDE mechanisms. The cooperation aspect is addressed by a dislocation-enhanced hydrogen accumulation model, which agrees with the HELP-mediated HEDE mechanism. The tensile, fracture toughness, and threshold stress intensity factor tests are simulated in different hydrogen environments. The results demonstrate that the proposed method reproduces the hydrogen-dependent ductile to brittle transition phenomenon. Under different hydrogen conditions, the dominating damage mechanism and degradation trends agree with experimental observations. Furthermore, for the investigated 4130 steel, the predicted value of fracture toughness and threshold stress intensity factor match the experiment results. The difference in damage kinetics inherent in the two test methods is captured, which supports the rationality of the proposed strategy for HELP and HEDE interaction.
Virtual modelling integrated phase field method for dynamic fracture analysis
2023, International Journal of Mechanical Sciences
A non-deterministic phase field (PF) virtual modelling framework is proposed for three-dimensional dynamic brittle fracture. The developed framework is based on experimental observations, accurate numerical modelling, and virtually foreseeable dynamic fracture prediction module through the machine learning algorithm. The uncertain system inputs, including variabilities of material properties, are incorporated into dynamic fracture analysis. Phase field method is implemented to simulate the dynamic fracture behaviours of 3D cracked structures with variabilities and then to create the training database for virtual damage model. The virtual damage model omits the physical finite element approximation process and reveals the virtual governing relationship between the variational system inputs and fracture responses. This advantage enables the virtual model to provide reliable crack propagation prediction based on either experiment-based or numerical simulation and greatly improves the computational efficiency of dynamic fracture analysis. To establish the accurate virtual model in the training process, a newly developed extended support vector regression (X-SVR) method with T-spline polynomial kernel functions is adopted for its outstanding performance in handling complex high-dimensional problems. Based on different real-world engineering scenarios, multiple fracture failure criteria, both strength-based and serviceability-based, are selected and demonstrated in numerical investigations to visualise the proposed framework's workflow. The effectiveness as well as accuracy are verified by these examples, and it is observed that the computational efficiency of dynamic fracture analysis is greatly improved. With the proposed framework, a continuously updating dynamic fracture surveillance system can be potentially built for practical applications.
Fracture toughness prediction of hydrogen-embrittled materials using small punch test data in Hydrogen
2022, International Journal of Mechanical Sciences
Citation Excerpt :
To determine the damage parameters α, β, and Dc from Eqs. (1) and (3), tensile and fracture toughness tests were simulated by means of FE analysis. Detailed procedures for determining the damage model parameters can be found in [59–60], and only a brief description is provided here. First, the tensile test was simulated by performing an elastic-plastic FE analysis using the commercial FE analysis program ABAQUS [63].
In this paper, we propose a numerical methodology to predict the effect of hydrogen concentration on fracture toughness by using small punch (SP) test data in hydrogen. The proposed method performs finite element (FE) damage analysis, with the multi-axial fracture strain damage model based on. First, a damage model is derived from the tensile and fracture toughness test data obtained in air. Then, by simulating the SP test in hydrogen, the hydrogen-embrittlement constant is determined. Finally, fracture toughness of hydrogen-embrittled material is predicted using the determined damage model and hydrogen-embrittlement constant. To validate the proposed methodology, published test data of API X70 steel in hydrogen and air (or nitrogen) atmosphere are used. In terms of hydrogen concentration, the predicted fracture toughness agrees well with the fracture toughness test data obtained in hydrogen environment.
Simulation of ductile-to-brittle transition combining complete Gurson model and CZM with application to hydrogen embrittlement
2022, Engineering Fracture Mechanics
Citation Excerpt :
Embrittlement is a type of environmentally-assisted fracture, which can occur under low temperature condition [1–4], upon irradiation [5] or due to the absorption of solute atoms such as hydrogen [6–10] and oxygen [11].
A general simulation framework for modelling ductile-to-brittle transition in metals is proposed. The method combines the complete Gurson model and cohesive zone model, which brings ductile and brittle fracture mechanisms into one play. We found that the transition of failure mode is the result of a competition between fracture due to micro-void growth and coalescence and fracture in the cohesive zone. It is found that the fracture mode is dependent on the ratio between the cohesive strength and the yield strength of the material; brittle fracture only occurs when the strength ratio is below a critical value. This generic rule can be used to rationalize various failure scenarios featured by ductile-to-brittle transition, such as low temperature embrittlement and hydrogen embrittlement. As an application of the general framework, hydrogen embrittlement is simulated. It is revealed that a critical hydrogen concentration has to be achieved in order to trigger brittle fracture, which is consistent with many experimental observations.
Ultrafine-grained dual-phase maraging steel with high strength and excellent cryogenic toughness
2021, Acta Materialia
A novel heat treatment route consisting of a low-temperature solution followed by an over-aging treatment at 500 °C is proposed to develop a high-strength, high-cryogenic-toughness maraging steel by forming an ultrafine-grained martensite (α′) and austenite (γ) dual-phase microstructure. Compared to the same maraging steel subjected to the conventional heat treatment with a mostly martensitic microstructure, the present dual-phase microstructure offers a remarkable increase by 12 times of cryogenic impact energy (~140 J at 77 K), while the yield strength is not reduced obviously. A large amount of ultrafine-grained austenite (about 50% volume fraction) are formed in the present steel due to the reversed transformation of martensite to austenite during the over-aging process at 500 °C. It is surprised that the present steel with such a high fraction of austenite still possesses a high yield strength comparable to the conventional maraging steel with a mostly martensitic microstructure. Intensive nanoprecipitates are found not only in martensite but also in austenite phases, indicating both phases have high strength. This is confirmed by nanoindentation test, showing similar hardness values in both martensite and austenite phases. Such intensive nanoprecipitates in both phases ensure the high yield strength of the present steel. The excellent cryogenic toughness of the present steel mainly origins from: (i) the pronounced amount of reversed austenite that are intrinsically tough due to their face-centered cubic (fcc) structure; (ii) transformation-induced plasticity (TRIP) toughening as some austenite grains transforming to martensite during impact test; (iii) the ultrafine-grained structure of both martensite and austenite phases. The present heat treatment route offers a potential solution for processing large engineering components for cryogenic application that require a long heat treatment duration to achieving uniform mechanical properties in the components.
Hydrogen Embrittlement as a Conspicuous Material Challenge─Comprehensive Review and Future Directions
2024, Chemical Reviews

View all citing articles on Scopus

View Abstract

[1] [1]
K. Shibanuma, T. Hosoe, H. Yamaguchi, M. Tsukamoto, K. Suzuki, S. Aihara
Crack tip opening angle during unstable ductile crack propagation of a high-pressure gas pipeline
Eng. Fract. Mech., 204 (2018), pp. 434-453
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.engfracmech.2018.10.020
View PDF View article View in Scopus Google Scholar

[2] [2]
S. Kawaguchi, K. Murai, Y. Hashimoto, N. Hagiwara, H. Yajima, M. Toyoda
Full-scale burst tests of ultra-high pressured rich-gas pipelines under buried and unburied conditions
Proc. Bienn. Int. Pipeline Conf. IPC. (2009), pp. 327-336
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1115/IPC2008-64434
View in Scopus Google Scholar

[3] [3]
G.M. McClure, A.R. Duffy, R.J. Eiber
Fracture resistance in line pipe
J. Manuf. Sci. Eng. Trans. ASME., 87 (3) (1965), pp. 265-278
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1115/1.3670809
Crossref View in Scopus Google Scholar

[4] [4]
D.J. Horsley
Background to the use of CTOA for prediction of dynamic ductile fracture arrest in pipelines
Eng. Fract. Mech., 70 (3-4) (2003), pp. 547-552
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/S0013-7944(02)00136-4
View PDF View article View in Scopus Google Scholar

[5] [5]
B. Hwang, S. Lee, Y.M. Kim, N.J. Kim, J.Y. Yoo, C.S. Woo
Analysis of abnormal fracture occurring during drop-weight tear test of high-toughness line-pipe steel
Mater. Sci. Eng. A., 368 (1-2) (2004), pp. 18-27
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.msea.2003.09.075
View PDF View article View in Scopus Google Scholar

[6] [6]
K.O. Halsen, E. Heier
Drop weight tear testing of high toughness pipeline material
Proc. Bienn. Int. Pipeline Conf. IPC. (2004), pp. 1717-1724
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1115/ipc2004-0609
View in Scopus Google Scholar

[7] [7]
A.P. Institute
API 5L Specification for line pipe
Api Spec 5L (2007)
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1520/G0154-12A
Google Scholar

[8] [8]
DNV
Fracture control for pipeline installation methods introducing cyclic plastic strain
Recomm. Pract. DNV-RP-F108 (2006)
Google Scholar

[9] [9]
BSI
BS 8571:2014 - Method of test for determination of fracture toughness in metallic materials using single edge notched tension (SENT) specimens
BSI (2014)
Google Scholar

[10] [10]
S. Cravero, C. Ruggieri
Estimation procedure of J-resistance curves for SE(T) fracture specimens using unloading compliance
Eng. Fract. Mech., 74 (17) (2007), pp. 2735-2757
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.engfracmech.2007.01.012
View PDF View article View in Scopus Google Scholar

[11] [11]
X.K. Zhu
Full-range stress intensity factor solutions for clamped SENT specimens
Int. J. Press. Vessel. Pip., 149 (2017), pp. 1-13
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.ijpvp.2016.11.004
View PDF View article Google Scholar

[12] [12]
H.G. Pisarski
Determination of pipe girth weld fracture toughness using SENT specimens
Proc. Bienn. Int. Pipeline Conf. IPC. (2010), pp. 217-224
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1115/IPC2010-31123
View in Scopus Google Scholar

[13] [13]
A.L. Gurson
Continuum theory of ductile rupture by void nucleation and growth: Part 1 - yield criteria and flow rules for porous ductile media
J. Eng. Mater. Technol. Trans. ASME., 99 (10) (1977), pp. 2-15
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1115/1.3443401
Crossref View in Scopus Google Scholar

[14] [14]
V. Tvergaard, A. Needleman
Analysis of the cup-cone fracture in a round tensile bar
Acta Metall, 32 (1) (1984), pp. 157-169
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/0001-6160(84)90213-X
View PDF View article View in Scopus Google Scholar

[15] [15]
J. Koplik, A. Needleman
Void growth and coalescence in porous plastic solids
Int. J. Solids Struct., 24 (8) (1988), pp. 835-853
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/0020-7683(88)90051-0
View PDF View article View in Scopus Google Scholar

[16] [16]
A. Cornec, I. Scheider, K.H. Schwalbe
On the practical application of the cohesive model
Eng. Fract. Mech., 70 (14) (2003), pp. 1963-1987
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/S0013-7944(03)00134-6
View PDF View article View in Scopus Google Scholar

[17] [17]
G. Rousselier
Ductile fracture models and their potential in local approach of fracture
Nucl. Eng. Des., 105 (1) (1987), pp. 97-111
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/0029-5493(87)90234-2
View PDF View article View in Scopus Google Scholar

[18] [18]
J.R. Rice, D.M. Tracey
On the ductile enlargement of voids in triaxial stress fields*
J. Mech. Phys. Solids., 17 (1969), pp. 201-217
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/0022-5096(69)90033-7
View PDF View article View in Scopus Google Scholar

[19] [19]
J.W. Hancock, A.C. Mackenzie
On the mechanisms of ductile failure in high-strength steels subjected to multi-axial stress-states
J. Mech. Phys. Solids., 24 (2-3) (1976), pp. 147-160
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/0022-5096(76)90024-7
View PDF View article View in Scopus Google Scholar

[20] [20]
M.W. Lee, J.S. Kim, J.Y. Jeon, Y.J. Kim, J.W. Kim, J.S. Kim
Comparison of numerical predictions with experimental burst pressures of tubes with multiple surface cracks
Eng. Fract. Mech., 201 (2018), pp. 176-195
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.engfracmech.2018.06.016
View PDF View article View in Scopus Google Scholar

[21] [21]
G.G. Youn, H.S. Nam, Y.J. Kim, J.W. Kim
Numerical prediction of thermal aging and cyclic loading effects on fracture toughness of cast stainless steel CF8A: Experimental and numerical study
Int. J. Mech. Sci., 163 (2019), Article 105120
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.ijmecsci.2019.105120
View PDF View article View in Scopus Google Scholar

[22] [22]
H.S. Nam, Y.R. Oh, Y.J. Kim, J.S. Kim, N. Miura
Application of engineering ductile tearing simulation method to CRIEPI pipe test
Eng. Fract. Mech., 153 (2016), pp. 128-142
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.engfracmech.2015.12.012
View PDF View article View in Scopus Google Scholar

[23] [23]
H.S. Nam, J.M. Lee, Y.J. Kim, J.W. Kim
Numerical ductile fracture prediction of circumferential through-wall cracked pipes under very low cycle fatigue loading condition
Eng. Fract. Mech., 194 (2018), pp. 175-189
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.engfracmech.2018.02.025
View PDF View article View in Scopus Google Scholar

[24] [24]
H.S. Nam, J.M. Lee, G.G. Youn, Y.J. Kim, J.W. Kim
Simulation of ductile fracture toughness test under monotonic and reverse cyclic loading
Int. J. Mech. Sci., 135 (2018), pp. 609-620
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.ijmecsci.2017.11.037
View PDF View article View in Scopus Google Scholar

[25] [25]
H.W. Ryu, K.D. Bae, Y.J. Kim, J.J. Han, J.S. Kim, P.J. Budden
Ductile tearing simulation of Battelle pipe test using simplified stress-modified fracture strain concept
Fatigue Fract. Eng. Mater. Struct., 39 (11) (2016), pp. 1391-1406
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1111/ffe.12456
Crossref View in Scopus Google Scholar

[26] [26]
C.S. Oh, N.H. Kim, Y.J. Kim, J.H. Baek, Y.P. Kim, W.S. Kim
A finite element ductile failure simulation method using stress-modified fracture strain model
Eng. Fract. Mech., 78 (1) (2011), pp. 124-137
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.engfracmech.2010.10.004
View PDF View article View in Scopus Google Scholar

[27] [27]
F.M. Beremin, A. Pineau, F. Mudry, J.C. Devaux, Y. D'Escatha, P. Ledermann
A local criterion for cleavage fracture of a nuclear pressure vessel steel
Metall. Trans. A., 14 (1983), pp. 2277-2287
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1007/BF02663302
View in Scopus Google Scholar

[28] [28]
C. Ruggieri, R.H. Dodds
A local approach to cleavage fracture modeling: An overview of progress and challenges for engineering applications
Eng. Fract. Mech., 187 (2018), pp. 381-403
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.engfracmech.2017.12.021
View PDF View article View in Scopus Google Scholar

[29] [29]
Y. Lei, N.P. O'Dowd, E.P. Busso, G.A. Webster
Weibull stress solutions for 2-D cracks in elastic and elastic-plastic materials
Int. J. Fract., 89 (1998), pp. 245-268
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1023/A:1007435927306
View in Scopus Google Scholar

[30] [30]
N.P. O'Dowd, Y. Lei, E.P. Busso
Prediction of cleavage failure probabilities using the Weibull stress
Eng. Fract. Mech., 67 (2) (2000), pp. 87-100
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/S0013-7944(00)00051-5
View PDF View article View in Scopus Google Scholar

[31] [31]
C. Ruggieri, X. Gao, R.H. Dodds
Transferability of elastic-plastic fracture toughness using the Weibull stress approach: Significance of parameter calibration
Eng. Fract. Mech., 67 (2) (2000), pp. 101-117
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/S0013-7944(00)00052-7
View PDF View article View in Scopus Google Scholar

[32] [32]
A. Rossoll, C. Berdin, C. Prioul
Determination of the fracture toughness of a low alloy steel by the instrumented charpy impact test
Int. J. Fract., 115 (2002), pp. 205-226
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1023/A:1016323522441
View in Scopus Google Scholar

[33] [33]
B. Wasiluk, J.P. Petti, R.H. Dodds
Temperature dependence of Weibull stress parameters: Studies using the Euro-material
Eng. Fract. Mech., 73 (8) (2006), pp. 1046-1069
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.engfracmech.2005.11.006
View PDF View article View in Scopus Google Scholar

[34] [34]
P. Haušild, C. Berdin, P. Bompard
Prediction of cleavage fracture for a low-alloy steel in the ductile-to-brittle transition temperature range
Mater. Sci. Eng. A., 391 (1-2) (2005), pp. 188-197
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.msea.2004.08.067
View PDF View article View in Scopus Google Scholar

[35] [35]
B. Tanguy, J. Besson, R. Piques, A. Pineau
Ductile to brittle transition of an A508 steel characterized by Charpy impact test Part I: Experimental results
Eng. Fract. Mech., 72 (1) (2005), pp. 49-72
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.engfracmech.2004.03.010
View PDF View article View in Scopus Google Scholar

[36] [36]
B. Tanguy, J. Besson, R. Piques, A. Pineau
Ductile to brittle transition of an A508 steel characterized by Charpy impact test. Part II: Modeling of the Charpy transition curve
Eng. Fract. Mech., 72 (3) (2005), pp. 413-434
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.engfracmech.2004.03.011
View PDF View article View in Scopus Google Scholar

[37] [37]
K. Hojo, N. Ogawa, T. Hirota, K. Yoshimoto, Y. Nagoshi, S. Kawabata
Application of Coupled Damage and Beremin Model to Ductile-Brittle Transition Temperature Region Considering Constraint Effect
Procedia Struct. Integr., 2 (2016), pp. 1643-1651
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.prostr.2016.06.208
View PDF View article View in Scopus Google Scholar

[38] [38]
K. Yoshimoto, T. Hirota, H. Sakamoto, T. Sugihara, S. Sakaguchi, T. Oumaya
Applicability of miniature C(T) specimen to evaluation of fracture toughness for reactor pressure vessel steel
Am. Soc. Mech. Eng. Press. Vessel. Pip. Div. PVP. (2013)
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1115/PVP2013-97840
Google Scholar

[39] [39]
G. Bernauer, W. Brocks, W. Schmitt
Modifications of the Beremin model for cleavage fracture in the transition region of a ferritic steel
Eng. Fract. Mech., 64 (3) (1999), pp. 305-325
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/s0013-7944(99)00076-4
View PDF View article View in Scopus Google Scholar

[40] [40]
Abaqus, ABAQUS Version
Dessault Syst. (2016)
2016
Google Scholar

[41] [41]
S. Canada
Temperature Effects on Dynamic Fracture of Pipeline Steel
Master of Science, University of Alberta (2015)
Google Scholar

[42] [42]
C. Soyarslan, B. Gülçimen, S. Bargmann, P. Hähner
Modeling of fracture in small punch tests for small- and large-scale yielding conditions at various temperatures
Int. J. Mech. Sci., 106 (2016), pp. 266-285
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.ijmecsci.2015.12.007
View PDF View article View in Scopus Google Scholar

[43] [43]
F. Mudry
A local approach to cleavage fracture
Nucl. Eng. Des., 105 (1) (1987), pp. 65-76
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/0029-5493(87)90230-5
View PDF View article View in Scopus Google Scholar

[44] [44]
S.R. Bordet, A.D. Karstensen, D.M. Knowles, C.S. Wiesner
A new statistical local criterion for cleavage fracture in steel. Part I: Model presentation
Eng. Fract. Mech., 72 (3) (2005), pp. 435-452
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.engfracmech.2004.02.009
View PDF View article View in Scopus Google Scholar

[45] [45]
T. Lin, A.G. Evans, R.O. Ritchie
Statistical analysis of cleavage fracture ahead of sharp cracks and rounded notches
Acta Metall, 34 (11) (1986), pp. 2205-2216
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/0001-6160(86)90166-5
View PDF View article View in Scopus Google Scholar

[46] [46]
L. Tong, L. Niu, S. Jing, L. Ai, X.L. Zhao
Low temperature impact toughness of high strength structural steel
Thin-Walled Struct, 132 (2018), pp. 410-420
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.tws.2018.09.009
View PDF View article View in Scopus Google Scholar

[47] [47]
H. Wang, Y. Yu, J. Yu, J. Duan, Y. Zhang, Z. Li, C. Wang
Effect of 3D random pitting defects on the collapse pressure of pipe — Part I: Experiment
Thin-Walled Struct, 129 (2018), pp. 512-526
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.tws.2018.04.015
View PDF View article View in Scopus Google Scholar

[48] [48]
S. Zhang, W. Jiang, M.R. Tonks
A new phase field fracture model for brittle materials that accounts for elastic anisotropy
Comput. Methods Appl. Mech. Eng., 358 (2020), Article 112643
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.cma.2019.112643
View PDF View article View in Scopus Google Scholar

[49] [49]
I. Khisamitov, G. Meschke
Configurational-force interface model for brittle fracture propagation
Comput. Methods Appl. Mech. Eng., 351 (2019), pp. 351-378
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/j.cma.2019.03.029
View PDF View article View in Scopus Google Scholar

[50] [50]
P. Bowen, S.G. Druce, J.F. Knott
Effects of microstructure on cleavage fracture in pressure vessel steel
Acta Metall, 34 (6) (1986), pp. 1121-1131
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/0001-6160(86)90222-1
View PDF View article View in Scopus Google Scholar

[51] [51]
C.F. Shih, M.D. German
Requirements for a one parameter characterization of crack tip fields by the HRR singularity
Int. J. Fract., 17 (1981), pp. 27-43
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1007/BF00043119
Crossref View in Scopus Google Scholar

[52] [52]
H.F. López, R. Raghunath, J.L. Albarran, L. Martinez
Microstructural aspects of sulfide stress cracking in an API X-80 pipeline steel
Metall. Mater. Trans. A Phys. Metall. Mater. Sci., 27 (1996), pp. 3601-3611
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1007/bf02595451
View in Scopus Google Scholar

[53] [53]
P.W. Bridgman
Studies in Large Plastic Flow and Fracture: With Special Emphasis on the Effects of Hydrostatic Pressure
McGraw-Hill, New York-London (1952)
Google Scholar

[54] [54]
R.O. Ritchie, J.F. Knott, J.R. Rice
On the relationship between critical tensile stress and fracture toughness in mild steel
J. Mech. Phys. Solids., 21 (6) (1973), pp. 395-410
https://doi-org.ezproxy.lib.kyushu-u.ac.jp/10.1016/0022-5096(73)90008-2
View PDF View article View in Scopus Google Scholar

Outline 大纲

Cited by (25) 被引用 (25)

Figures (19) 图（19）

Tables (2) 表格（2）

International Journal of Mechanical Sciences
国际机械科学杂志

Highlights 亮点

ABSTRACT 抽象的

Graphic Abstract 图形摘要

Keywords 关键词