Combined Ductile-Brittle Fracture Simulation of API X80 Under Impact Loading

doi:10.1016/j.jpse.2025.100295

Journal of Pipeline Science and Engineering

Available online 8 May 2025, 100295

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ABSTRACT

This study explores finite element (FE) simulation of the combined ductile-brittle fracture behavior of API X80 steel under impact loading, incorporating the influences of strain rate and adiabatic heating. The Johnson-Cook (J-C) deformation model is employed for deformation analysis, while ductile fracture is modeled using the (J-C) fracture strain model, and brittle fracture is captured using a critical stress model. Temperature-dependent parameters for the J-C model were calibrated through smooth round bar (SRB) tests conducted at varying temperatures, while strain rate-dependent parameters were identified using the Charpy V-notch (CVN) impact test at room temperature (RT). Additionally, the critical stress model for brittle fracture was determined via FE simulation of CVN tests at RT. The calibrated deformation and combined fracture models were applied to simulate both CVN and drop-weight tear test (DWTT) over a temperature range of -120 to 0 °C. The simulations accurately captured deformation behavior, load-displacement curves, and fracture surfaces. Sensitivity analyses highlighted the roles of adiabatic heating and strain rate effects in fracture behavior.

Keywords

Adiabatic heating and strain rate effects

API X80 steel

Combined ductile-brittle fracture simulation under impact loading

Johnson-Cook deformation and fracture strain model

Critical stress brittle fracture model

Nomenclature

a₀

initial crack length [mm]

Δa

crack extension [mm]

C_p

specific heat [J/kg∙°C]

strain rate dependent Johnson-Cook (J-C) deformation parameter

D₁, D₂, D₃

Johnson-Cook fracture parameters at quasi-static strain rate and room temperature

D₄

strain rate dependent J-C fracture parameter

D₅

temperature dependent J-C fracture parameter

D_c

critical damage value

Young's modulus [GPa]

E₀

elastic energy absorption [J]

T, ΔT

temperature and its change [°C]

T_melt

melting temperature [°C]

V₀

reference volume [mm³]

V_E

total volume of one layer element in the unnotched ligament [mm³]

ε_f

multi-axial fracture strain in Eqs. (1, 2, 7, 9, 10) and (11)

\dot{ε}

\dot{ε_{0}}

strain rate and reference strain rate

ε_p, Δε_p

equivalent plastic strain and its increment

density [kg/m³]

σ₁

principal stress [MPa]

σ_1,max

maximum principal stress [MPa]

σ_c

critical stress for brittle fracture [MPa]

σ_y, σ_u

yield strength and ultimate tensile strength, respectively [MPa]

σ_e

equivalent stress [MPa]

σ_m

mean normal stress [MPa]

σ_w

Weibull stress [MPa]

ABBREVIATIONS

CMOD

crack mouth opening displacement

CVN

Charpy V notch

DWTT

drop-weight tear test

Finite element

J-C

Johnson-Cook

room temperature

SENT

single edge notched tension

SRB

smooth round bar

1. INTRODUCTION

Impact loading represents one of the most severe conditions leading to brittle fracture in steel structures and pipelines (Kristoffersen et al., 2013). While the Charpy V-notch (CVN) test provides insights into mechanical behavior under impact loading, including the ductile-brittle transition, it is recognized that the small specimen size can result in non-conservative evaluations of the ductile-brittle transition temperature (DBTT) (Eiber, 1965). To address this limitation, the drop weight tear test (DWTT) has been developed as a more reliable method for determining DBTT (Eiber, 1965). Finite element (FE) analysis is a powerful tool to investigate the deformation behavior and combined ductile-brittle fracture mechanisms under impact loading, offering deeper insights into these phenomena.

Numerous studies have focused on simulating ductile fracture under impact conditions (Wang and Ru, 2016, Yu and Ru, 2015, Gu and Wang, 2022, May et al., 2015, Chandran et al., 2022, Rousselier, 1987, Tanguy and Besson, 2002, Tanguy et al., 2005, Tanguy et al., 2005, Dey et al., 2004, Dey et al., 2006, Dey et al., 2007, Banerjee et al., 2015, Mirone et al., 2024, Seo et al., 2024, Chu et al., 2019, Samaniego et al., 2021, Zhang et al., 2024). For instance, Wang et al. (Wang and Ru, 2016) examined the relationship between the crack tip opening angle and impact hammer speed using a strain rate-dependent cohesive zone model (Yu and Ru, 2015). Similarly, Gu et al. (Gu and Wang, 2022) employed a cohesive zone model incorporating a strain rate-dependent traction-displacement law introduced by May et al. (May et al., 2015). The strain rate dependency is quantified through shear tests performed at varying strain rates (Gu and Wang, 2022). Chandran et al. (Chandran et al., 2022) extended these efforts by accounting for both Lode angle and strain rate effects using a model with nine parameters, which were calibrated through tensile and shear tests conducted across a range of strain rates. Notably, these studies (Wang and Ru, 2016, Yu and Ru, 2015, Gu and Wang, 2022, May et al., 2015, Chandran et al., 2022) did not consider the effects of adiabatic heating. The Rousselier model (Rousselier, 1987, Tanguy and Besson, 2002) has also been utilized to simulate ductile fracture under impact loading, incorporating temperature and strain rate effects (Tanguy et al., 2005, Tanguy et al., 2005). Tanguy et al. (Tanguy et al., 2005, Tanguy et al., 2005) quantified these influences through tensile tests conducted under both quasi-static and dynamic conditions at varying temperatures. Alternatively, the Johnson-Cook (J-C) deformation and fracture model has been widely adopted for simulating ductile fracture, as it accounts for adiabatic heating and strain rate effects. The parameters of the J-C model, dependent on strain rate and temperature, are typically determined through smooth and notched round bar tests performed at various temperatures and strain rates (Dey et al., 2004, Dey et al., 2006, Dey et al., 2007, Banerjee et al., 2015, Mirone et al., 2024), as well as split Hopkinson pressure bar tests. Recently, Seo et al. (Seo et al., 2024) proposed an efficient method for determining the J-C model parameters without requiring dynamic tensile tests.

All the aforementioned studies focused primarily on ductile fracture modeling, with limited attention to the combined ductile and brittle fracture behavior under impact loading. Since the ductile-brittle transition is prominent at low temperatures, a simulation methodology capable of addressing both ductile and brittle fracture mechanisms is essential for accurately simulating impact fracture behavior. In studies (Chu et al., 2019, Samaniego et al., 2021), both adiabatic heating and strain rate effects under dynamic loading were incorporated using the Johnson-Cook (J-C) deformation model alongside a phase-field model to simulate brittle fracture. For ductile fracture simulation, Chu et al. (Chu et al., 2019) employed an energy release rate corrected for stress triaxiality. However, as noted in their study (Chu et al., 2019), determining the critical energy release rate and validating it with experimental data proved challenging. To address these limitations, Zhang et al. (Zhang et al., 2024) introduced an additional ductile fracture criterion based on the critical storage energy density. While this methodology successfully simulated the crack shape under impact loading, the simulation results lacked experimental validation, highlighting an area requiring further investigation.

Although combined ductile and brittle fracture modeling under impact loading remains scarce, numerous studies have addressed combined ductile-brittle fracture under quasi-static loading conditions (Hojo et al., 2016, Beremin et al., 1983, Gurson, 1977, Tvergaard and Needleman, 1984, Koplik and Needleman, 1988, Batra and Lear, 2004, Lin et al., 2022, Chakraborty and Biner, 2014, Kim et al., 2022, Seo et al., 2023, Hwang et al., 2024, Camacho and Ortiz, 1997, Gerstgrasser et al., 2021, Borvik et al., 2001, Chanda, 2015, Gambirasio and Rizzi, 2016, Seo et al., 2022, Seo et al., 2024, Kim et al., 2011). In the works of Tanguy et al. (Tanguy et al., 2005) and Hojo et al. (Hojo et al., 2016), brittle fracture was modeled using the Beremin model (Beremin et al., 1983). For ductile fracture simulation, either the Rousselier model (Rousselier, 1987, Tanguy and Besson, 2002) or the Gurson-Tvergaard-Needleman (GTN) model (Gurson, 1977, Tvergaard and Needleman, 1984, Koplik and Needleman, 1988) was employed, as reported in Refs (Tanguy et al., 2005) and (Batra and Lear, 2004). Lin et al. (Lin et al., 2022) utilized a cohesive zone model for brittle fracture and the GTN model for ductile fracture to simulate combined ductile-brittle behavior in hydrogen-embrittled materials. However, their work lacked experimental validation. Chakraborty et al. (Chakraborty and Biner, 2014) adopted a unified cohesive zone model to simulate both ductile and brittle fracture, achieving fracture toughness predictions for 5% and 95% failure probabilities that aligned well with the experimental variations in fracture toughness and crack growth across a temperature range. The authors (Kim et al., 2022, Seo et al., 2023, Hwang et al., 2024) have proposed a simulation methodology for combined ductile and brittle fracture, utilizing a multi-axial fracture strain-based model for ductile fracture and a critical stress-based model for brittle fracture. While their approach successfully simulated impact tests such as the CVN test and DWTT, the phenomenological aspects of impact loading—particularly the effects of adiabatic heating and strain rate—were not fully considered. These effects are critical for accurately simulating impact fracture behavior.

This paper presents finite element (FE) simulations of combined ductile-brittle fracture in API X80 steel under impact loading, with a focus on incorporating strain rate and adiabatic heating effects. The Johnson-Cook (J-C) deformation and fracture strain models were employed for ductile fracture simulation, while brittle fracture was modeled using a critical stress approach. The simulation methodology is detailed in Section 2. 3 SUMMARY OF EXPERIMENTS FOR MODEL DETERMINATION, 4 DETERMINATION OF API X80 DEFORMATION AND FRACTURE MODELS discuss the experiments conducted for model parameter determination and the corresponding determination procedure. Validation experiments and the validation process are described in 5 EXPERIMENTS FOR VALIDATION, 6 COMPARISON WITH FE SIMULATION RESULTS FOR VALIDATION, respectively. Section 7 provides an analysis of the effects of adiabatic heating and strain rate on the combined ductile-brittle fracture simulation for impact tests. Finally, the conclusions are summarized in Section 8.
本文介绍了冲击载荷下 API X80 钢延性-脆性断裂的有限元 (FE) 模拟，重点考虑了应变速率和绝热加热效应。延性断裂模拟采用 Johnson-Cook (JC) 变形和断裂应变模型，脆性断裂模拟采用临界应力法。模拟方法在第 2 节中详细介绍。第 3 节“模型确定试验总结”和第 4 节“API X80 变形和断裂模型确定”讨论了为确定模型参数而进行的试验和相应的确定程序。第 5 节“验证试验”和第 6 节“验证与 FE 模拟结果的比较”分别描述了验证试验和验证过程。第 7 节分析了绝热加热和应变速率对冲击试验延性-脆性断裂模拟的影响。最后，第 8 节总结了结论。

2. PROPOSED COMBINED DUCTILE-BRITTLE FRACTURE SIMULATION MODEL
2. 提出的延脆性断裂模拟模型

This section proposes a combined ductile-brittle fracture simulation model for impact loading scenarios. For deformation analysis, the Johnson-Cook (J-C) deformation model is employed (see Section 2.1). Ductile fracture simulation utilizes the Johnson-Cook fracture strain model, which incorporates both strain rate and adiabatic heating effects. Brittle fracture simulation, on the other hand, is performed using a critical stress model. The models used for ductile and brittle fracture simulations are detailed in 2.2 Johnson-Cook (J-C) Fracture Strain Model for Ductile Fracture Simulation, 2.3 Critical Stress Model for Brittle Fracture Simulation, respectively. The integration of these models to simulate combined ductile-brittle fracture is described in Section 2.4. Determination of model parameters specific to the material under investigation (API X80 steel in this study) requires experimental testing, which is discussed in Section 3. The parameter determination procedure for API X80 is provided in Section 4.
本节提出了一种用于冲击载荷场景的延性-脆性断裂组合模拟模型。对于变形分析，采用 Johnson-Cook (JC) 变形模型（参见第 2.1 节）。延性断裂模拟采用 Johnson-Cook 断裂应变模型，该模型同时考虑了应变率和绝热加热效应。另一方面，脆性断裂模拟则采用临界应力模型。用于延性和脆性断裂模拟的模型分别详见 2.2 节“用于延性断裂模拟的 Johnson-Cook (JC) 断裂应变模型”和 2.3 节“用于脆性断裂模拟的临界应力模型”。第 2.4 节介绍了如何集成这些模型来模拟延性-脆性断裂组合。针对所研究材料（本研究中为 API X80 钢）的模型参数的确定需要进行实验测试，这将在第 3 节中讨论。第 4 节提供了 API X80 的参数确定程序。

2.1. Johnson-Cook (J-C) Deformation Model under Impact Loading
2.1 冲击载荷下的 Johnson-Cook（JC）变形模型

The original form of the J-C deformation model is as follows:
JC 变形模型原始形式如下：(1) $σ = σ_{e, RT} (1 - λ {| T^{*} |}^{k}) (1 + c \ln (\frac{\dot{ε}}{{\dot{ε}}_{0}})); T^{*} = \frac{T - T_{r o o m}}{T_{m e l t} - T_{r o o m}}, λ = \frac{T^{*}}{| T^{*} |}$

where σ_e_, RT denotes the flow stress at room temperature (RT) under quasi-static conditions, T_room and T_melt represent the room temperature and melting temperature, respectively, and

\dot{ε_{0}}

is the reference strain rate. For steels, T_room = 25 °C and T_melt = 1,500 °C are used (Chanda, 2015, Gambirasio and Rizzi, 2016). Notably, the strain rate dependent logarithmic term on the right-hand side of Eq. (1) causes numerical instability when the strain rate is lower than the reference strain rate. To mitigate this issue, the original form of the J-C deformation model was slightly modified, as proposed in (Camacho and Ortiz, 1997, Gerstgrasser et al., 2021).
其中 σ _e RT 表示准静态条件下室温 (RT) 下的流动应力，T _room 和 T _melt 分别表示室温和熔化温度，

\dot{ε_{0}}

表示参考应变速率。对于钢，使用 T _room = 25 °C 和 T _melt = 1,500 °C (Chanda, 2015; Gambirasio and Rizzi, 2016)。值得注意的是，当应变速率低于参考应变速率时，等式 (1) 右侧的应变速率相关对数项会导致数值不稳定。为了缓解这个问题，对 JC 变形模型的原始形式进行了轻微修改，如 (Camacho and Ortiz, 1997; Gerstgrasser et al., 2021) 中所述。(2)

σ = σ_{e, RT} (1 - λ {| T^{*} |}^{k}) {(1 + \frac{\dot{ε}}{{\dot{ε}}_{0}})}^{c}; T^{*} = \frac{T - T_{r o o m}}{T_{m e l t} - T_{r o o m}}, λ = \frac{T^{*}}{| T^{*} |}

Under quasi-static conditions, this term approaches unity by setting the reference strain rate to

\dot{ε_{0}}

 = 100 /s (Gu and Wang, 2022). Tensile tests under high strain rates (up to 1,000/s) performed on high-strength ferritic steels in (Uenishi et al., 2011, Vaynman et al., 2006) showed significant strain rate hardening was observed when the strain rate exceeded 100/s. Based on these observations, Yu et al. (Yu and Ru, 2015) chose 100/s as the reference strain rate for simulating the DWTT test for API X80. In our previous work (Seo et al., 2024), the same choice of reference strain rate of 100/s could successfully simulate the CVN and DWTT tests for API X52 (Seo et al., 2024). Finally, two parameters, k and c, need to be determined. The parameter determination procedure for API X80 steel will be described in Section 4.1.
在准静态条件下，通过将参考应变速率设为

\dot{ε_{0}}

= 100 /s (Gu and Wang, 2022)，该项趋近于 1。(Uenishi et al., 2011, Vaynman et al., 2006) 对高强度铁素体钢进行的高应变速率（高达 1,000/s）拉伸试验表明，当应变速率超过 100/s 时，观察到明显的应变速率硬化。基于这些观察，Yu 等人（Yu and Ru, 2015）选择 100/s 作为模拟 API X80 DWTT 试验的参考应变速率。在我们之前的工作中（Seo et al., 2024），同样选择 100/s 的参考应变速率可以成功模拟 API X52 的 CVN 和 DWTT 试验（Seo et al., 2024）。最后，需要确定 k 和 c 两个参数。API X80 钢的参数确定程序将在第 4.1 节中描述。

2.2. Johnson-Cook (J-C) Fracture Strain Model for Ductile Fracture Simulation
2.2. 用于延性断裂模拟的 Johnson-Cook（JC）断裂应变模型

The Johnson-Cook (J-C) fracture strain model is employed to simulate ductile fracture in impact tests, incorporating both strain rate (

\dot{ε}

) and temperature (T) effects (Camacho and Ortiz, 1997, Gerstgrasser et al., 2021, Borvik et al., 2001):
采用 Johnson-Cook (JC) 断裂应变模型模拟冲击试验中的延性断裂，同时考虑应变速率（

\dot{ε}

）和温度 (T) 效应（Camacho 和 Ortiz，1997 年；Gerstgrasser 等人，2021 年；Borvik 等人，2001 年）：(3)

ε_{f} = (D_{1} \cdot \exp (D_{3} \cdot \frac{σ_{m}}{σ_{e}}) + D_{2}) {(1 + \frac{\dot{ε}}{{\dot{ε}}_{0}})}^{D 4} (1 - λ T^{* D 5})

Where σ_m and σ_e denote the hydrostatic stress and von Mises stress, respectively; D₁, D_2, and D₃ are the multi-axial fracture strain damage model parameters at room temperature (RT) under quasi-static conditions; and D₄ and D₅ are parameters associated with strain rate and temperature, respectively.
其中，σ _m 和 σ _e 分别表示静水应力和 von Mises 应力；D ₁ 、D _2, 和 D ₃ 为准静态条件下室温（RT）多轴断裂应变损伤模型参数；D ₄ 和 D ₅ 分别是与应变速率和温度相关的参数。

The accumulated damage (D) due to plastic strain is calculated as follows (Seo et al., 2022, Seo et al., 2024, Kim et al., 2011):
塑性应变造成的累积损伤（D）计算如下（Seo et al., 2022, Seo et al., 2024, Kim et al., 2011）：(4)

D = \int Δ D = \int \frac{Δ ε_{p}}{ε_{f}}

where Δε_p represents the increment of equivalent plastic strain. In the simulation, ductile fracture was assumed to occur when D reaches the critical damage value (D_c). Once D_c is reached at an integration point, that point is excluded from further calculation. When all integration points in an element fail, the element is removed from the finite element (FE) simulation using the “DELETE” option in ABAQUS. It is important be note that the value of D_c can depend on the element size (Oh et al., 2011, Jeon et al., 2016). The determination of J-C fracture strain model parameters for API X80 steel is detailed in Section 4.2.
其中 Δε _p 表示等效塑性应变的增量。模拟中，当 D 达到临界损伤值 (D _c ) 时，假设发生延性断裂。一旦积分点达到 D _c ，该点将从进一步计算中排除。当单元中的所有积分点都失效时，使用 ABAQUS 中的“DELETE”选项将该单元从有限元 (FE) 模拟中删除。需要注意的是，D _c 的值可能取决于单元尺寸（Oh 等，2011 年；Jeon 等，2016 年）。API X80 钢的 JC 断裂应变模型参数的确定详见第 4.2 节。

2.3. Critical Stress Model for Brittle Fracture Simulation
2.3 脆性断裂模拟的临界应力模型

The critical stress model is employed to simulate brittle fracture under impact loading in this study, based on the Weibull stress concept (Beremin et al., 1983). The Weibull stress (σ_w) is defined as:
本研究采用临界应力模型模拟冲击载荷下的脆性断裂，该模型基于威布尔应力概念（Beremin 等，1983）。威布尔应力（σ _w ）定义为：(5)

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

Where V₀ and V represent the reference volume and fracture process zone, respectively; σ₁ is the maximum principal stress, and m denotes the Weibull modulus. For finite element (FE) simulations, the Weibull stress criterion can be approximately implemented as:
其中 V ₀ 和 V 分别代表参考体积和断裂过程区；σ ₁ 为最大主应力，m 表示威布尔模量。对于有限元 (FE) 模拟，威布尔应力准则可以近似地表示为：(6)

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

where σ_1,max represents the maximum principal stress in the fracture process zone (V). The approximation σ_w = σ_1,max holds when the ratio V/V₀ = 1 (Kim et al., 2020). Brittle fracture is assumed to occur in an element when the maximum principal stress (σ_1,max) exceeds the critical stress value (σ_c). The parameter σ_c for API X80 steel will be determined based on impact energy, as detailed in Section 4.3.
其中，σ _1,max 表示断裂过程区的最大主应力 (V)。当 V/V ₀ = 1 时，近似值 σ _w = σ _1,max 成立 (Kim et al., 2020)。当最大主应力 (σ _1,max ) 超过临界应力值 (σ _c ) 时，假定元件发生脆性断裂。API X80 钢的参数 σ _c 将根据冲击能量确定，详见第 4.3 节。

2.4. Combined Ductile and Brittle Fracture Simulation
2.4 延性断裂和脆性断裂综合模拟

Combined ductile-brittle fracture under impact loading is simulated by integrating the Johnson-Cook (J-C) fracture strain model and the critical stress model into a finite element (FE) impact simulation. Prior to simulation, the fracture strain model parameters and the critical stress value (σ_c) must be determined for the specific material. Using these calibrated models, impact simulations are conducted. During the simulation, the maximum principal stress (σ_1,max) and accumulated damage (D) are calculated at each Gauss point within an element. These calculated values are then compared against their respective failure criteria, σ_c and D_c. If σ_1,max exceeds σ_c, brittle fracture is assumed to occur, Similarly if D reaches D_c, ductile fracture is assumed. When all Gauss points in an element fail, the element is removed from the FE simulation using the “DELETE” option in ABAQUS. The fracture simulation process is summarized in Fig. 1.
通过将 Johnson-Cook (JC) 断裂应变模型和临界应力模型集成到有限元 (FE) 冲击模拟中，可以模拟冲击载荷下的延性-脆性组合断裂。在模拟之前，必须确定特定材料的断裂应变模型参数和临界应力值 (σ _c )。使用这些校准模型进行冲击模拟。在模拟过程中，计算元素内每个高斯点的最大主应力 (σ _1,max ) 和累积损伤 (D)。然后将这些计算值与它们各自的失效准则 σ _c 和 D _c 进行比较。如果 σ _1,max 超过 σ _c ，则假定发生脆性断裂，同样，如果 D 达到 D _c ，则假定发生延性断裂。当元素中的所有高斯点都失效时，使用 ABAQUS 中的“DELETE”选项将该元素从 FE 模拟中删除。断裂模拟过程总结如图 1 所示。

3. SUMMARY OF EXPERIMENTS FOR MODEL DETERMINATION
3 模型确定实验总结

Determining material-specific parameters for deformation and fracture models is a crucial step in accurate simulation. To identify these parameters, basic experimental data are required. This section provides a summary of the experimental data for API X80 (pipeline steel) utilized in this study, from which the material-specific parameters for the deformation and fracture models are derived. The chemical composition of the API X80 are shown in Table 1 and the averaged grain size was 6 µm. The following experimental data are used in this work:
确定变形和断裂模型的材料特定参数是实现精确模拟的关键步骤。为了确定这些参数，需要基础实验数据。本节总结了本研究中使用的 API X80（管线钢）的实验数据，并据此推导出变形和断裂模型的材料特定参数。API X80 的化学成分如表 1 所示，平均晶粒尺寸为 6 µm。本研究使用了以下实验数据：

(1)
Smooth round bar (SRB) test data conducted over a temperature range of -100 °C to 60 °C,
在 -100°C 至 60°C 温度范围内进行的光滑圆棒 (SRB) 测试数据，
(2)
Single edge notched tensile (SENT) test conducted at room temperature (RT), and
在室温下进行单边缺口拉伸（SENT）试验，以及
(3)
Instrumented Charpy V-notch (CVN) test conducted at RT.
在室温下进行仪器化夏比 V 型缺口 (CVN) 试验。

Table 1. Chemical composition of API X80 (wt.%).
表 1. API X80 的化学成分（wt.%）。

C	Si 和	Mn 锰	Ni+Mo 由+你	Nb+Ti 铌+钛	Al 致
0.073	0.23	1.76	0.56	0.05	0.033

These tests are summarized in Table 2 and are detailed in the subsequent sub-sections. The procedure for determining the model parameters is explained in Section 4.
这些测试总结在表 2 中，并在后续小节中详细说明。确定模型参数的程序在第 4 节中解释。

Table 2. Summary of test cases for determination of deformation and fracture model parameters.
表 2. 确定变形和断裂模型参数的试验用例汇总。

Test 测试	Temperature [°C] 温度[°C]
SRB	60, RT, -50, -100 60，RT，-50，-100
SENT	RT
CVN	RT

* RT: room temperature. * RT：室温。

3.1. Tensile Test 3.1.拉伸试验

In this study, smooth round bar (SRB) tests were conducted at temperatures ranging from -100 to 60 °C using specimens extracted from an API X80 plate. Each specimen had a minimum section diameter of 6 mm and a gauge length of 25 mm. Figure 2(a) presents selected engineering stress-strain curves at -100 °C, room temperature (RT), and 60 °C. Figure 2(b) illustrates the variation of the yield strength (σ_y, 0.2% proof strength) and tensile strength (σ_u) with temperature. Both the yield and tensile strength decrease as temperature increases. Figure 2(b) also shows the fracture strain calculated from the reduction of area (RA) using:
本研究使用从 API X80 钢板上提取的试样，在 -100 至 60 °C 的温度范围内进行了光滑圆棒 (SRB) 试验。每个试样的最小截面直径为 6 mm，标距长度为 25 mm。图 2(a) 展示了在 -100 °C、室温 (RT) 和 60 °C 时选定的工程应力-应变曲线。图 2(b) 说明了屈服强度 (σ _y ，0.2% 屈服强度) 和抗拉强度 (σ _u ) 随温度的变化。屈服强度和抗拉强度均随温度升高而降低。图 2(b) 还显示了使用以下公式通过面积缩减 (RA) 计算得出的断裂应变：(7)

ε_{f} = \ln (\frac{1}{1 - RA})

The fracture strain exhibits minimal sensitivity to temperature changes. These trends align with tensile test results for other API steels reported in the literature (Jung et al., 2014, Heier et al., June 2013, Akselsen et al., June 2012, Kim et al., 2024).
断裂应变对温度变化的敏感性极小。这些趋势与文献中报道的其他 API 钢的拉伸试验结果一致（Jung 等人，2014 年；Heier 等人，2013 年 6 月；Akselsen 等人，2012 年 6 月；Kim 等人，2024 年）。

3.2. Single Edge Notched Tensile (SENT) Test
3.2. 单边缺口拉伸（SENT）试验

The Single Edge Notched Tensile (SENT) test was conducted at room temperature (RT) following the BS8571 standard (BSI 2014). A through-thickness notch was created using electro-discharge machining. After fatigue pre-cracking, the specimen had an initial crack length of a₀ = 10.2 mm. Figure 3(a) presents a schematic illustration of the SENT test specimen, which includes side grooves of 1.65 mm for each side. Figure 3(b) presents the experimental load-crack mouth opening displacement (CMOD) and crack extension (Δa-CMOD) curves. The crack length was determined using the unloading compliance method.
单边缺口拉伸 (SENT) 试验在室温 (RT) 下按照 BS8571 标准 (BSI 2014) 进行。采用电火花加工技术制造贯穿厚度的缺口。疲劳预裂后，试样的初始裂纹长度为 a ₀ = 10.2 毫米。图 3(a) 为 SENT 试验试样的示意图，其中每侧有 1.65 毫米的侧槽。图 3(b) 为实验载荷-裂纹口张开位移 (CMOD) 和裂纹扩展 (Δa-CMOD) 曲线。裂纹长度采用卸载柔量法确定。

3.3. Charpy V Notched (CVN) Impact Test at Room Temperature
3.3 室温夏比 V 型缺口（CVN）冲击试验

An instrumented Charpy V-Notched (CVN) impact test was conducted following the API 5L standard (Institute, 2007). The specimens had a cross-sectional area of 10 × 10 mm², a ligament length of 8 mm, and a 45 ° center V-notch. The test was performed at room temperature (RT) with an initial velocity of 5 m/s using a Zwickroell PSW 750 instrumented CVN testing machine. Note that the notch without the fatigue crack was introduced in the CVN specimen.
按照 API 5L 标准（Institute，2007）进行了仪器化夏比 V 型缺口（CVN）冲击试验。试样横截面积为 10×10 mm ² ，带长为 8 mm，V 型缺口中心角为 45°。试验在室温（RT）下进行，初速度为 5 m/s，使用 Zwickroell PSW 750 仪器化 CVN 试验机。注意，CVN 试样中引入了缺口，但没有疲劳裂纹。

Figure 4 illustrates the measured load-displacement curves from the CVN test. The load increased to its maximum value as the specimen deformed and subsequently decreased due to crack initiation and propagation. Three repeated tests demonstrated consistent load-displacement responses and CVN impacts energy values.
图 4 展示了 CVN 试验测得的载荷-位移曲线。随着试样变形，载荷增加至最大值，随后由于裂纹的萌生和扩展而下降。三次重复试验显示出一致的载荷-位移响应和 CVN 冲击能量值。

4. DETERMINATION OF API X80 DEFORMATION AND FRACTURE MODELS
4 API X80 变形和断裂模型的确定

To simulate combined ductile-brittle fracture under impact loading, three models must be determined for the material under study (API X80 in this work), as outlined in Section 2: (1) the deformation model, (2) the ductile fracture model, and (3) the brittle fracture model. The parameters for these models are derived from the experimental data presented in Section 3. The determination of the parameters for the deformation model is described in Section 4.1. The parameters for the ductile and brittle fracture models are detailed in 4.2 Parameter Determination of Ductile Fracture Model, 4.3 Determination of Brittle Fracture Model, respectively.
为了模拟冲击载荷下的延性-脆性复合断裂，必须针对所研究的材料（本文中为 API X80）确定三个模型，如第 2 节所述：（1）变形模型；（2）延性断裂模型；（3）脆性断裂模型。这些模型的参数来自第 3 节提供的实验数据。变形模型参数的确定方法在第 4.1 节中描述。延性和脆性断裂模型的参数分别在 4.2 延性断裂模型参数确定和 4.3 脆性断裂模型确定中详细说明。

4.1. Parameter Determination of Deformation Model
4.1 变形模型参数确定

The strain rate and temperature-dependent Johnson-Cook (J-C) deformation model (see Eq. (2) was calibrated using the procedure proposed by Seo et al. (Seo et al., 2024). The steps for parameter determination are as follows:
采用 Seo 等（Seo et al., 2024）提出的程序校准应变率和温度相关的 Johnson-Cook（JC）变形模型（见公式（2））。参数确定步骤如下：

(1)
The true stress-strain curve at room temperature (RT) under quasi-static conditions was obtained from the smooth round bar (SRB) testing at RT.
通过对光滑圆棒（SRB）进行室温（RT）试验，获得了准静态条件下室温（RT）的真实应力-应变曲线。
(2)
The temperature-dependent parameter (k) was determined based on the variation of yield strength with temperature.
温度相关参数（k）是根据屈服强度随温度的变化确定的。
(3)
The strain rate-dependent parameter (c) was calibrated by simulating the experimental maximum load from the Charpy V-notch (CVN) test at RT
通过模拟室温夏比 V 型缺口 (CVN) 试验的实验最大载荷来校准应变率相关参数 (c)

The true stress-strain curve beyond the necking point was determined by simulating the experimental engineering stress-strain curve using the method proposed by Tu et al. (Tu et al., 2022). Simulations were performed using the commercial finite element (FE) analysis software ABAQUS (ABAQUS Version 2018). Figure 5(a) illustrates the half FE model of the SRB specimen, consisting of 280 to 1,230 axisymmetric elements with reduced integration (CAX8R). The minimum element size in the center of the specimen was set to 0.1, 0.2, and 0.3 mm for sensitivity analysis. The top surface of the FE model was constrained using the Multi-Point Constraints (MPC) option in ABAQUS, and the reaction force at the central node was obtained to calculate engineering stress. Engineering strain was calculated by dividing the nodal displacement at the gauge node by the gauge length (25 mm). Figure 5(b) shows the determined engineering and true stress-strain curves, along with the simulated engineering stress-strain curves obtained using three different element sizes (0.1, 0.2, and 0.3 mm) through elastic-plastic FE analysis. The results indicate that the simulated curves are not significantly affected by element size.
采用 Tu 等（Tu et al., 2022）提出的方法，通过模拟实验工程应力-应变曲线，确定了颈缩点后的真实应力-应变曲线。模拟采用商用有限元（FE）分析软件 ABAQUS（ABAQUS 2018 版）进行。图 5（a）展示了 SRB 试件的半有限元模型，该模型由 280~1230 个采用缩减积分的轴对称单元（CAX8R）组成。为了进行敏感性分析，试件中心的最小单元尺寸分别设置为 0.1、0.2 和 0.3 mm。使用 ABAQUS 中的多点约束（MPC）选项对 FE 模型的顶面进行约束，并获取中心节点处的反作用力以计算工程应力。工程应变是通过将标距节点处的节点位移除以标距长度（25 mm）来计算的。图 5(b) 显示了确定的工程应力-应变曲线和真实应力-应变曲线，以及通过弹塑性 FE 分析使用三种不同单元尺寸（0.1、0.2 和 0.3 mm）获得的模拟工程应力-应变曲线。结果表明，单元尺寸对模拟曲线的影响不显著。

Figure 5(c) illustrates the variation of yield strength with temperature. Experimental data are represented by symbols, while the temperature-dependent term in the J-C model, calculated with a calibrated k value of 0.7, is shown as a line. The k value was derived using the data analysis software ORIGINPRO (Origin(Pro), Version 2020). Figure 5(d) compares the simulated engineering stress-strain curves, generated using the calibrated J-C model, with experimental data at 60 °C and -100 °C. It is important to note that the hardening exponent in the J-C model is assumed to be temperature-independent. As a result, the tensile strength was slightly underestimated by the determined J-C deformation model. However, achieving highly accurate predictions of tensile data at different temperatures is not the primary focus of this study. The current J-C model provides sufficient accuracy for the intended purpose of stimulating combined ductile-brittle fracture under impact loading.
图 5(c) 展示了屈服强度随温度的变化。实验数据用符号表示，而 JC 模型中与温度相关的项（以校准后的 k 值 0.7 计算）则以线表示。k 值是使用数据分析软件 ORIGINPRO（Origin(Pro)，版本 2020）得出的。图 5(d) 将使用校准后的 JC 模型生成的模拟工程应力-应变曲线与 60 °C 和 -100 °C 下的实验数据进行了比较。值得注意的是，JC 模型中的硬化指数被假设与温度无关。因此，确定的 JC 变形模型略微低估了拉伸强度。然而，实现对不同温度下拉伸数据的高精度预测并非本研究的重点。目前的 JC 模型已足够精确地模拟冲击载荷下的延脆性复合断裂。

After determining the true stress-strain curve at room temperature (RT), the parameters k and c were calibrated as follows. Figure 6 presents the quarter finite element (FE) model used to simulate the Charpy V-notch (CVN) test, comprising 11,112 to 38,740 eight-node brick elements with full integration (C3D8). To examine the effects of element size, the minimum element sizes in the notched section were set to 0.1, 0.2, and 0.3 mm.
确定室温 (RT) 下的真实应力-应变曲线后，参数 k 和 c 进行如下校准。图 6 展示了用于模拟夏比 V 型缺口 (CVN) 试验的四分之一有限元 (FE) 模型，该模型包含 11,112 至 38,740 个八节点砖块单元，并进行了完全积分 (C3D8)。为了检验单元尺寸的影响，缺口截面的最小单元尺寸分别设置为 0.1、0.2 和 0.3 毫米。

For friction modeling, a friction coefficient of 0.1 was applied between the specimen and the apparatus (hammer and anvil) using the “CONTACT” option in ABAQUS. This value is commonly used for impact simulations (Seo et al., 2024, Talemi et al., 2019). Both the anvil and hammer were modeled as rigid bodies, with the anvil fixed in place. The hammer impacted the specimen with an initial velocity of 5.0 m/s, and the simulated load and displacement were obtained at the reference point of the hammer. Strain rate and adiabatic heating effects were incorporated into the simulation using the “adiabatic heating” option in dynamic implicit finite element (FE) analysis. The temperature rise due to adiabatic heating was calculated based on the equivalent plastic strain (ε_p) using the following relationship (Seo et al., 2024):
对于摩擦建模，使用 ABAQUS 中的“接触”选项，在试样和仪器（锤和砧）之间施加 0.1 的摩擦系数。该值通常用于冲击模拟（Seo 等，2024；Talemi 等，2019）。砧和锤均建模为刚体，砧固定到位。锤以 5.0 m/s 的初始速度冲击试样，并在锤的参考点处获得模拟载荷和位移。使用动态隐式有限元 (FE) 分析中的“绝热加热”选项，将应变率和绝热加热效应纳入模拟。基于等效塑性应变 (ε _p )，使用以下关系式计算绝热加热引起的温升 (Seo 等，2024)：(8)

Δ T = \frac{0.9}{ρ C_{p}} \int_{V} σ d ε_{p}

where ρ and C_p represent the material density and specific heat, respectively. A typical value of ρ = 7.9 g/mm³ was used, while the specific heat values for API X80 were taken from (Yan et al., 2014) and are provided in Table 3. It should be noted that the factor of 0.9 on the right-hand side of Eq. (8) represents the Taylor-Quinney (T-Q) coefficient. It has been found that the T-Q coefficient is the strain rate-dependent parameter (Soares and Hokka, 2021, Rittel et al., 2017). Soares and Hokka (Soares and Hokka, 2021) analyzed the effect of strain rate on the T-Q coefficient using pure metal materials, titanium, iron and copper under strain rate ranging from 500 to 3,100 /s. Similarly, Rittel et al. (Rittel et al., 2017) also measured the value of T-Q coefficients for many materials, including Ti6Al4V, Aluminum, stainless steel and 1020 steel under strain rate ranging from 1,000 to 6,000 /s. It should be noted that, although the exact strain rate when the adiabatic heating effect should be considered is not clear yet, this phenomenon has been considered when the strain rate is higher than 500 /s (Soares and Hokka, 2021, Rittel et al., 2017). Because the strain rate of CVN test and DWTT is higher than 1,000 /s, the adiabatic heating effect cannot be neglected.
其中 ρ 和 C _p 分别表示材料密度和比热容。使用的典型值 ρ = 7.9 g/mm ³ ，而 API X80 的比热容值取自 (Yan et al., 2014)，并列于表 3 中。需要注意的是，等式 (8) 右侧的因子 0.9 表示 Taylor-Quinney (TQ) 系数。已发现 TQ 系数是应变率相关参数 (Soares and Hokka, 2021; Rittel et al., 2017)。Soares 和 Hokka (Soares and Hokka, 2021) 使用纯金属材料钛、铁和铜，分析了应变率在 500 至 3,100 /s 范围内时应变率对 TQ 系数的影响。同样，Rittel 等人 (2017) 也使用纯金属材料钛、铁和铜分析了应变率对 TQ 系数的影响。 (Rittel et al., 2017) 还测量了 Ti6Al4V、铝、不锈钢和 1020 钢等多种材料在 1,000 至 6,000 /s 应变速率范围内的 TQ 系数值。需要注意的是，虽然尚不清楚应考虑绝热加热效应的具体应变速率，但当应变速率高于 500 /s 时，已考虑了这一现象 (Soares and Hokka, 2021; Rittel et al., 2017)。由于 CVN 试验和 DWTT 的应变速率高于 1,000 /s，因此绝热加热效应不能忽略。

Table 3. Temperature-dependent specific heat for API 5L X80 in (Yan et al., 2014).
表 3. API 5L X80 的温度相关比热容 (Yan 等人，2014)。

Temperature [°C] 温度[°C]	20	100	200	400	800	1,200
Specific heat [J/kg∙°C] 比热[J/kg∙°C]	423	473	536	662	914	1,160

Figure 7(a) compares the experimental load-displacement curve from the CVN test with the finite element FE simulation results. The J-C model in Eq. (2) was used, with the calibrated k value of 0.7 and varying values of c. The simulated maximum load increases linearly with c, as shown in Figure 7(b). It is important to note that the crack initiated near the maximum load during the CVN test (Tronskar et al., 2002, Kobayashi, 1984, Oulad Brahim et al., 2022). Based on this observation, the final value of c = 0.18 was selected by matching the simulated maximum load with the experimental results, resulting in the final J-C deformation model:
图 7(a) 将 CVN 试验的实验载荷-位移曲线与有限元 FE 模拟结果进行了比较。采用公式 (2) 中的 JC 模型，校准后的 k 值为 0.7，c 值随时间变化。模拟的最大载荷随 c 值线性增加，如图 7(b) 所示。值得注意的是，在 CVN 试验中，裂纹是在最大载荷附近萌生的（Tronskar 等，2002；Kobayashi，1984；Oulad Brahim 等，2022）。基于此观察，通过将模拟的最大载荷与实验结果进行匹配，最终选定 c = 0.18，从而得到最终的 JC 变形模型：(9)

σ = σ_{e, RT} (1 - λ {| T^{*} |}^{0.7}) {(1 + \frac{\dot{ε}}{{\dot{ε}}_{0}})}^{0.18}

Using the determined value of c = 0.18, the effect of element size on the simulation was analyzed. The same element sizes as those used for the SRB simulation (0.1, 0.2, and 0.3 mm) were applied to the CVN simulation. The results, shown in Fig. 7(a), indicate that the simulation outcomes are not significantly affected by element size.
使用确定的 c = 0.18 值，分析了单元尺寸对模拟的影响。CVN 模拟采用了与 SRB 模拟相同的单元尺寸（0.1、0.2 和 0.3 mm）。结果如图 7（a）所示，表明模拟结果不受单元尺寸的显著影响。

4.2. Parameter Determination of Ductile Fracture Model
4.2 韧性断裂模型参数确定

The Johnson-Cook (J-C) fracture strain model described in Section 2.2 involves six parameters (from D₁ to D₅ and D_c) that need to be determined. The determination process follows these steps:
2.2 节中描述的 Johnson-Cook（JC）断裂应变模型涉及六个需要确定的参数（从 D ₁ 到 D ₅ 和 D _c ）。确定过程遵循以下步骤：

(1)
The multi-axial fracture strain damage model parameters (D₁, D_2, and D₃) and critical damage value D_c are determined by analyzing Smooth Round Bar (SRB) and Single Edge Notched Tensile (SENT) tests at room temperature (RT) under quasi-static conditions.
通过分析准静态条件下室温（RT）下的光滑圆棒（SRB）和单边缺口拉伸（SENT）试验，确定多轴断裂应变损伤模型参数（D ₁ 、D _2, 和 D ₃ ）和临界损伤值 D _c 。
(2)
The temperature dependent parameter (D₅) is determined by fitting the variation of fracture strain with temperature.
通过拟合断裂应变随温度的变化来确定温度相关参数（D ₅ ）。
(3)
Finally, the strain rate dependent parameter (D₄) is obtained by simulating the experimental Charpy V Notched (CVN) energy at RT.
最后，通过模拟室温下的实验夏比 V 型缺口 (CVN) 能量，获得应变率相关参数 (D ₄ )。

Under quasi-static conditions at RT, the J-C fracture model in Eq. (3) simplifies to:
在 RT 的准静态条件下，公式（3）中的 JC 裂缝模型简化为：(10)

ε_{f} = (D_{1} \exp (D_{3} \cdot \frac{σ_{m}}{σ_{e}}) + D_{2})

Initially, D₃ is typically assumed to be -1.5, following guidance from previous studies (Seo et al., 2022, Seo et al., 2024, Kim et al., 2011). One point on the fracture strain locus is determined through elastic-plastic FE analysis of the SRB test. Figure 8(a) shows the variation of the stress triaxiality with equivalent plastic strain, measured at the center of the SRB specimen under RT conditions (see Fig. 5(b)). The average stress triaxiality up to the failure point, combined with the fracture strain, provides a data point on the fracture strain locus, depicted in Figure 8(b) using a symbol.
最初，根据先前研究（Seo et al., 2022, Seo et al., 2024, Kim et al., 2011）的指导，通常假设 D ₃ 为 -1.5。通过对 SRB 试验进行弹塑性 FE 分析，确定断裂应变轨迹上的一个点。图 8(a) 显示了在室温条件下在 SRB 试样中心测量的应力三轴度随等效塑性应变的变化（见图 5(b)）。到失效点的平均应力三轴度与断裂应变相结合，提供了断裂应变轨迹上的一个数据点，在图 8(b) 中用符号表示。

Next, the single edge notched tensile (SENT) test is simulated using finite element (FE) damage analysis to establish the final fracture strain locus and critical damage value (D_c). Figure 9 illustrates the quarter FE model used for the SENT test stimulation, which consists of 10,320 to 58,432 eight-node brick elements with full integration (C3D8). Three different FE meshes were created, with the minimum element size in the cracked ligament set to 0.1, 0.2, and 0.3 mm. The top surface of the FE model was constrained using the multi-point constraint (MPC) option in ABAQUS. The reaction force was measured at the central node, while the crack mouth opening displacement (CMOD) was recorded at the gauge node using a clip-on gauge with a length of 3 mm. It should be noted that macroscopically straight crack propagation was observed in all tests. Accordingly, our model also assumed straight crack growth and failure was considered only for the elements on the symmetric surface.
接下来，使用有限元 (FE) 损伤分析模拟单边缺口拉伸 (SENT) 试验，以确定最终断裂应变轨迹和临界损伤值 (D _c )。图 9 展示了用于 SENT 试验模拟的四分之一 FE 模型，该模型由 10,320 到 58,432 个八节点砖块元素组成，具有完全积分 (C3D8)。创建了三种不同的 FE 网格，其中裂纹韧带中的最小元素尺寸分别设置为 0.1、0.2 和 0.3 毫米。使用 ABAQUS 中的多点约束 (MPC) 选项对 FE 模型的顶面进行约束。在中心节点处测量反作用力，同时使用长度为 3 毫米的夹式量规在量规节点处记录裂纹口张开位移 (CMOD)。值得注意的是，在所有试验中都观察到了宏观直线裂纹扩展。因此，我们的模型还假设裂纹直线扩展，并且仅考虑对称表面上的元素的失效。

Fracture toughness tests provide properties related to both crack initiation and crack growth rate. In previous studies (Seo et al., 2022, Seo et al., 2024, Kim et al., 2011), it was observed that D_c predominantly affects the crack initiation point, whereas D₂ primarily influences the crack growth rate. Accordingly, FE damage simulations were first conducted with D₂ set to zero, varying D_c to assess its effect. As shown in Figure 10(a), the crack initiation point was delayed as D_c increased. The value of D_c = 1 was chosen to match the experimental crack initiation point. Subsequently, FE damage simulations were performed with D_c fixed at 1, varying D₂ to analyze its impact on the crack growth rate (d(Δa)-d(CMOD)). Figure 10(b) illustrates that the final value of D₂ = 0.4 was selected to align the simulated crack growth rate with experimental data. The resulting fracture strain locus is represented by:
断裂韧性试验提供与裂纹起始和裂纹扩展速率相关的特性。在先前的研究中（Seo et al., 2022, Seo et al., 2024, Kim et al., 2011），观察到 D _c 主要影响裂纹起始点，而 D ₂ 主要影响裂纹扩展速率。因此，首先将 D ₂ 设置为零进行 FE 损伤模拟，然后改变 D _c 以评估其影响。如图 10(a)所示，随着 D _c 的增加，裂纹起始点延迟。选择 D _c = 1 的值以匹配实验裂纹起始点。随后，将 D _c 固定为 1 进行 FE 损伤模拟，并改变 D ₂ 以分析其对裂纹扩展速率（d(Δa)-d(CMOD)）的影响。图 10(b) 表明，选择最终值 D ₂ = 0.4，以使模拟裂纹扩展速率与实验数据保持一致。所得断裂应变轨迹表示为：(11)

ε_{f} = (3.37 \cdot \exp (- 1.5 \cdot \frac{σ_{m}}{σ_{e}}) + 0.4)

It is well-known that element size can significantly affect fracture simulation results. In this context, it is assumed that the determined fracture strain locus is independent of the element size. However, the critical damage value D_c is considered to be element size-dependent. Therefore, the influence of element size on D_c is analyzed using the established fracture strain locus. The FE meshes for various element sizes are shown in Figure 9. The values of D_c determined from FE damage analysis with different element sizes are presented in Figure 10(c), demonstrating that the FE damage simulation results correspond well with experimental data. Figure 10(d) shows the variation of the critical damage value D_c with element size L_e.
众所周知，单元尺寸会显著影响断裂模拟结果。在此假设确定的断裂应变轨迹与单元尺寸无关。然而，临界损伤值 D _c 被认为与单元尺寸相关。因此，使用已建立的断裂应变轨迹分析单元尺寸对 D _c 的影响。图 9 显示了不同单元尺寸的 FE 网格。图 10(c) 显示了使用不同单元尺寸通过 FE 损伤分析确定的 D _c 值，表明 FE 损伤模拟结果与实验数据吻合良好。图 10(d) 显示了临界损伤值 D _c 随单元尺寸 L _e 的变化。

The temperature-dependent parameter (D₅) is determined based on the variation of fracture strain with temperature. The fracture strain data, obtained from the SRB test conducted at temperatures ranging from -50 to 60 °C, showed minimal variation regardless of temperature (see Fig. 2(b)). As a result, D₅ is assumed to be 0.
温度相关参数 (D ₅ ) 是根据断裂应变随温度的变化确定的。在 -50 至 60 °C 温度范围内进行的 SRB 试验获得的断裂应变数据显示，无论温度如何，其变化均很小（见图 2(b)）。因此，假设 D ₅ 为 0。

The strain rate-dependent parameter (D₄) is determined by simulating the experimental CVN energy at RT. The FE mesh for the CVN simulation, as shown in Fig. 6, is utilized. The simulation employs the J-C deformation model determined in Section 4.1 and the J-C fracture strain model from Eq. (11). Figure 11(a) displays the impact of D₄ on the simulated load-displacement curves using a minimum element size of 0.1 mm. As explained in Section 4.1, cracks typically start propagating at the maximum load. The result indicates that while D₄ does not affect the maximum load, a lower D₄ value leads to a decrease in the load after the maximum point. The final value of D₄ = -0.28 is selected to align the simulated CVN energy with the experimental data, as illustrated in Fig. 11(b). The finalized fracture strain locus, incorporating both temperature and strain rate effects, is given by Eq. (12)
应变率相关参数 (D ₄ ) 是通过模拟室温下的实验 CVN 能量来确定的。CVN 模拟采用如图 6 所示的 FE 网格。模拟采用了第 4.1 节中确定的 JC 变形模型和公式 (11) 中的 JC 断裂应变模型。图 11(a) 显示了 D ₄ 对模拟载荷-位移曲线的影响，其中最小单元尺寸为 0.1 毫米。如第 4.1 节所述，裂纹通常在最大载荷下开始扩展。结果表明，虽然 D ₄ 不影响最大载荷，但较低的 D ₄ 值会导致最大点之后载荷的降低。最终选择 D ₄ = -0.28 的值，以使模拟的 CVN 能量与实验数据一致，如图 11(b) 所示。最终的断裂应变轨迹，结合温度和应变率效应，由公式 (12) 给出。(12)

ε_{f} = (3.37 \cdot \exp (- 1.5 \cdot \frac{σ_{m}}{σ_{e}}) + 0.4) {(1 + \frac{\dot{ε}}{{\dot{ε}}_{0}})}^{- 0.28}

To evaluate the element size effect, Fig. 11(a) compares the simulated load-displacement curves using element sizes of 0.1, 0.2, and 0.3 mm. The results indicate that the J-C strain-based fracture model is relatively insensitive to element size variations. Validation of the determined J-C deformation and fracture strain model will be provided in Section 6 through simulations of CVN tests at other temperatures.
为了评估单元尺寸效应，图 11（a）比较了单元尺寸为 0.1、0.2 和 0.3 mm 时模拟的载荷-位移曲线。结果表明，基于 JC 应变的断裂模型对单元尺寸变化相对不敏感。第 6 节将通过模拟其他温度下的 CVN 试验来验证已确定的 JC 变形和断裂应变模型。

4.3. Determination of Brittle Fracture Model
4.3 脆性断裂模型的确定

In this section, the determination procedure for the critical stress of the brittle fracture model is outlined. This critical stress is identified through CVN simulation utilizing the J-C fracture strain ductile fracture model established in Section 4.2. During the simulations, both ductile and brittle fracture models are employed, with the critical stress value (σ_c) being varied. Ductile fracture is presumed to occur when the accumulated damage reaches the critical damage value (D_c), while brittle fracture is considered to occur when the maximum principal stress surpasses the assumed critical stress (σ_c), as detailed in Section 2.3. The CVN energy is then calculated based on the assumed critical stress.
本节概述了脆性断裂模型临界应力的确定程序。该临界应力是通过 CVN 模拟确定的，该模拟利用了 4.2 节中建立的 JC 断裂应变延性断裂模型。在模拟过程中，延性断裂和脆性断裂模型均被采用，临界应力值 (σ _c ) 会发生变化。当累积损伤达到临界损伤值 (D _c ) 时，假定发生延性断裂；而当最大主应力超过假定的临界应力 (σ _c ) 时，则认为发生脆性断裂，如 2.3 节所述。然后根据假定的临界应力计算 CVN 能量。

For the CVN test simulations, the FE mesh from Fig. 6 with an element size of 0.1 mm is utilized. Given that combined ductile-brittle fractures typically occur at temperatures below -60 °C for API X80 (Kim et al., 2020), CVN simulations were conducted at temperatures ranging from -60 to -120 °C. Figure 12(a) illustrates the impact of the assumed σ_c on the simulated load-displacement curves. The solid line represents simulation results considering only the ductile fracture model from Section 4.2. The dotted and dashed lines depict the load-displacement curves when both ductile and brittle fracture models are applied, with different assumed σ_c values. The critical stress value normalized by the temperature-dependent yield strength (σ_c /σ_y) was assumed to be 2.8 and 3.0. A sudden drop in load is observed at a displacement of ∼12 mm for σ_c /σ_y = 3.0, while for σ_c /σ_y = 2.8, the drop occurs at ∼6 mm, indicating the occurrence of brittle fracture. \
对于 CVN 试验模拟，采用图 6 中单元尺寸为 0.1 mm 的 FE 网格。鉴于对于 API X80 来说，延性-脆性断裂通常发生在低于 -60 °C 的温度下（Kim 等人，2020 年），因此 CVN 模拟在 -60 至 -120 °C 的温度范围内进行。图 12(a) 说明了假设的 σ _c 对模拟载荷-位移曲线的影响。实线表示仅考虑第 4.2 节中的延性断裂模型的模拟结果。虚线和虚线分别表示同时应用延性和脆性断裂模型时的载荷-位移曲线，假设的 σ _c 值不同。假设由温度相关屈服强度 (σ _c /σ _y ) 归一化的临界应力值为 2.8 和 3.0。当 σ _c /σ _y = 3.0 时，在位移至 ∼12 mm 处观察到载荷突然下降，而当 σ _c /σ _y = 2.8 时，载荷下降发生在 ∼6 mm 处，表明发生了脆性断裂。

The relationship between CVN energy (the area under the load-displacement curve) and critical stress is established by simulating the CVN test using the determined ductile fracture model while varying the critical stress value through parametric analysis. Elastic energy absorption is calculated as:
通过使用确定的延性断裂模型模拟 CVN 试验，同时通过参数分析改变临界应力值，建立了 CVN 能量（载荷-位移曲线下面积）与临界应力之间的关系。弹性能量吸收计算如下：(13)

E_{0} (T) = \frac{σ_{y} {(T)}^{2}}{2 E} \cdot V_{E}

where E denotes Young's modulus, and V_E represents the total volume of a single layer element in the unnotched ligament. Notably, only one layer of elements in the unnotched ligament is considered when calculating the total volume. For example, a CVN specimen with an unnotched ligament area of 80 mm² results in V_E = 2 × 80 × 0.1 mm³ for an element size of 0.1mm. The factor of 2 accounts for the symmetry condition, making V_E linearly dependent on the element size.
其中 E 表示杨氏模量，V _E 表示无缺口韧带中单层单元的总体积。值得注意的是，计算总体积时仅考虑无缺口韧带中的一层单元。例如，对于无缺口韧带面积为 80 平方毫米 ² 的 CVN 试样，当单元尺寸为 0.1 毫米时，V _E = 2 × 80 × 0.1 毫米 ³ 。系数 2 考虑了对称性条件，使 V _E 与单元尺寸呈线性相关。

Figure 12(b) depicts the variation of normalized critical stress (σ_c/σ_y(T)) with normalized CVN energy (E_CVN/E₀(T)), where σ_y(T) is the temperature-dependent yield strength for a specific temperature T. The Figure shows that σ_c /σ_y(T) is nearly linearly related to ln (E_CVN/E₀(T)). Using the commercial data analysis program “ORIGIN PRO” (Origin(Pro), Version 2020), the following linear relationship is derived:
图 12(b) 描述了归一化临界应力 (σ _c /σ _y (T)) 随归一化 CVN 能量 (E _CVN /E ₀ (T)) 的变化，其中 σ _y (T) 是特定温度 T 下的温度相关屈服强度。该图显示 σ _c /σ _y (T) 与 ln (E _CVN /E ₀ (T)) 几乎呈线性关系。使用商业数据分析程序“ORIGIN PRO”(Origin(Pro)，版本 2020)，可以得出以下线性关系：(14)

\frac{σ_{c}}{σ_{y} (T)} = 1.67 + 0.38 \cdot \ln (\frac{E_{C V N}}{E_{0} (T)})

The effect of element size on the determination of critical stress is investigated using the FE model with element sizes of 0.2 and 0.3 mm. CVN simulations were conducted for a CVN test at -60 °C, applying the J-C fracture strain model determined in Section 4.2 while varying the critical stress values. The elastic energy absorption for the FE model with element sizes of 0.2 and 0.3 mm is calculated as V_E = 2 × 80 × 0.2 mm³ and V_E = 2 × 80 × 0.3 mm³, respectively. Figure 12(c) demonstrates the variation of σ_c /σ_y(T) with ln(E_CVN/E₀(T)), indicating that the curve remains consistent across different element sizes.
使用单元尺寸为 0.2 和 0.3 mm 的 FE 模型研究单元尺寸对临界应力确定的影响。在 -60 °C 下对 CVN 试验进行了 CVN 模拟，应用了第 4.2 节中确定的 JC 断裂应变模型，同时改变了临界应力值。单元尺寸为 0.2 和 0.3 mm 的 FE 模型的弹性能量吸收分别计算为 V _E = 2 × 80 × 0.2 mm ³ 和 V _E = 2 × 80 × 0.3 mm ³ 。图 12(c) 展示了 σ _c /σ _y (T) 随 ln(E _CVN /E ₀ (T)) 的变化，表明曲线在不同的单元尺寸下保持一致。

5. EXPERIMENTS FOR VALIDATION
5. 验证实验

To validate the proposed simulation method under impact loading, additional tests were conducted:
为了验证所提出的模拟方法在冲击载荷下的有效性，还进行了以下测试：

(1)
CVN test at temperatures ranging from -120 to -30 °C
CVN 测试温度范围为 -120 至 -30 °C
(2)
Drop weight Tear Test (DWTT) at -40 and 0 °C.
在 -40 和 0 °C 下进行落锤撕裂试验 (DWTT)。

Details of these tests are summarized in Table 4 and will be elaborated upon in the following subsections. A comparison with FE simulation results will be presented in Section 6.
表 4 总结了这些测试的详细信息，并将在以下小节中详细说明。第 6 节将展示与有限元模拟结果的比较。

Table 4. Summary of test cases for validation.
表 4.验证测试用例摘要。

Test 测试	Temperature [°C] 温度[°C]
CVN	-30, -60, -90, -120
DWTT	0, -40

5.1. Charpy V Notched (CVN) Test
5.1. 夏比 V 型缺口（CVN）试验

The CVN tests for validation were conducted using the same testing machine described in Section 3.3. Six tests were performed at -120°C, and three tests were conducted at each of the other temperatures (-30, -60, and -90°C). Figure 13 presents the load-displacement curves. The curve at -30°C in Fig. 13(a) shows a smooth decline after reaching the maximum load, indicating a ductile-dominant fracture. In contrast, the curves at -60, -90, and -120°C show a significant drop of around 10mm displacement, indicating a combined ductile-brittle fracture behavior.
CVN 验证试验采用 3.3 节中描述的相同试验机进行。在-120℃下进行了六次试验，在其他温度（-30、-60 和-90℃）下各进行了三次试验。图 13 展示了载荷-位移曲线。图 13（a）中-30℃下的曲线在达到最大载荷后呈现平滑下降趋势，表明断裂以延性断裂为主。相比之下，-60、-90 和-120℃下的曲线位移显著下降约 10mm，表明断裂行为兼具延性和脆性。

5.2. Drop-Weight Tear Test (DWTT)
5.2. 落锤撕裂试验（DWTT）

The Drop-Weight Tear Test (DWTT) was conducted following ASTM E436 (ASTM Standard E436-03 2021). A schematic of the DWTT specimen is shown in Fig. 14(a). The test was performed at -40 and 0 °C with an initial velocity of 5.15 m/s using Imatek DWTT-100F testing machine. One or two tests were conducted at each temperature. Figure 14(b) presents the load-displacement curve for the DWTT at 0 and -40 °C. The maximum load increases as the temperature decreases due to low-temperature hardening. After reaching the maximum load, the curve shows a decrease, with a sudden reduction occurring at a displacement of approximately 30 to 40 mm, indicative of combined ductile-brittle fracture. Figure 14(c) shows the fracture surfaces at 0 and -40 °C, respectively. Cleavage fracture, often referred to as “inverse fracture” (Kim et al., 2021, Kim et al., 2022), is observed on the fracture surface of the specimen. The load-displacement curves and fracture surfaces will be further analyzed in conjunction with the FE analysis results.
落锤撕裂试验 (DWTT) 按照 ASTM E436（ASTM 标准 E436-03 2021）进行。图 14(a) 为 DWTT 试样示意图。试验在 -40 和 0 °C 下进行，初始速度为 5.15 m/s，使用 Imatek DWTT-100F 试验机。每个温度下进行一到两次试验。图 14(b) 显示了 0 和 -40 °C 下 DWTT 的载荷-位移曲线。由于低温硬化，最大载荷随温度降低而增加。达到最大载荷后，曲线呈下降趋势，在位移约 30 至 40 mm 处突然减小，表明存在延性-脆性复合断裂。图 14(c) 分别显示了 0 和 -40 °C 下的断裂面。在试样的断裂面上观察到解理断裂，通常被称为“逆断裂”（Kim et al., 2021, Kim et al., 2022）。我们将结合有限元分析结果，进一步分析载荷-位移曲线和断裂面。

6. COMPARISON WITH FE SIMULATION RESULTS FOR VALIDATION
6. 与 FE 模拟结果的比较验证

In this section, the CVN tests and DWTT are simulated using the proposed combined ductile and brittle fracture model, with the model parameters determined in Section 4. A comparison of the CVN test results with FE simulations will be presented in Section 6.1, while the comparison for the DWTT is provided in Section 6.2.
在本节中，使用提出的延性和脆性断裂组合模型模拟 CVN 试验和 DWTT，模型参数在第 4 节中确定。CVN 试验结果与 FE 模拟的比较将在第 6.1 节中展示，而 DWTT 的比较将在第 6.2 节中提供。

6.1. Comparison of CVN Test with Simulation Results
6.1 CVN 试验与仿真结果比较

The CVN test at -30 °C was simulated using the FE mesh shown in Fig. 6, with an element size of 0.1 mm. Note that only ductile fracture mode was observed in the CVN test at -30 °C. The J-C deformation and fracture strain models in Equations (9) and (11) were used for the simulation. Figure 15 compares the simulated load-displacement curve and fracture surface of the CVN simulation at -30 °C with experimental data. The results show that the simulation aligns well with the experimental data at -30 °C. The simulated fracture surface is also compared with experimental data in Fig. 15(b), demonstrating that the shape of the deformation and ductile fracture surface is accurately simulated.
使用图 6 所示的有限元网格模拟-30℃下的 CVN 试验，单元尺寸为 0.1 mm。值得注意的是，在-30℃下的 CVN 试验中仅观察到延性断裂模式。模拟中使用了公式（9）和公式（11）中的 JC 变形和断裂应变模型。图 15 将-30℃下 CVN 模拟的载荷-位移曲线和断裂表面与实验数据进行了比较。结果表明，模拟结果与-30℃下的实验数据吻合良好。图 15（b）中模拟的断裂表面也与实验数据进行了比较，表明变形和延性断裂表面的形状得到了准确模拟。

The CVN tests at temperatures ranging from -60 to -120 °C were also simulated using the FE mesh shown in Fig. 6 with a 0.1 mm element size. In the test, combined ductile and brittle fracture modes were observed for all cases. The J-C deformation and fracture strain models in Equations (9) and (11) were used for ductile fracture simulation, while the critical stress model in Eq. (14) was applied for brittle fracture simulation. The critical stress was calculated using the experimental CVN energy, resulting in a value of σ_c/σ_y = 3.05 for -60 °C, σ_c/σ_y = 3.01 for -90 °C, and σ_c/σ_y = 2.95 for -120 °C. The simulated load-displacement curves from the FE simulation are compared with the experimental data in Fig. 16. The simulation reproduced the experimental results very well using the determined combined ductile-brittle fracture model. The simulated fracture surface at -90 °C is also compared with experimental data in Fig. 16(d). The simulated load-displacement curve, considering only the ductile fracture model, is shown for comparison (indicated as “w/o σ_c”). The significant load drop at a displacement of 10 mm could not be reproduced without incorporating the brittle fracture criterion, σ_c. The results also show that the steep load decrease at a displacement of 10 mm is attributed to brittle fracture. The simulated ductile-brittle combined fracture surface agrees well with the experimental fracture surface.
采用图 6 所示的 FE 网格，单元尺寸为 0.1 mm，模拟了-60 至-120 °C 温度范围内的 CVN 试验。试验中，所有工况均观察到延性和脆性断裂模式的混合。延性断裂模拟采用公式（9）和公式（11）中的 JC 变形和断裂应变模型，脆性断裂模拟采用公式（14）中的临界应力模型。临界应力采用实验 CVN 能量计算，结果显示-60 °C 时σ _c /σ _y = 3.05，-90 °C 时σ _c /σ _y = 3.01，-120 °C 时σ _c /σ _y = 2.95。图 16 将 FE 模拟的载荷-位移曲线与实验数据进行了比较。使用确定的延性-脆性组合断裂模型，模拟很好地再现了实验结果。图 16（d）还将 -90 °C 下的模拟断裂面与实验数据进行了比较。仅考虑延性断裂模型的模拟载荷-位移曲线用于比较（表示为“w/o σ _c ”）。如果不采用脆性断裂标准 σ _c ，则无法再现位移为 10 mm 时载荷的大幅下降。结果还表明，位移为 10 mm 时载荷的急剧下降归因于脆性断裂。模拟的延性-脆性组合断裂面与实验断裂面吻合得很好。

6.2. Comparison of DWTT with Simulation Results
6.2. DWTT 与仿真结果的比较

In this section, DWTT tests conducted at -40 and 0 °C, which exhibited a combined ductile-brittle fracture mode, are simulated using the proposed combined ductile and brittle fracture models determined in Section 4.
在本节中，使用第 4 节中确定的延性和脆性组合断裂模型模拟在 -40 和 0 °C 下进行的 DWTT 试验，该试验表现出延性-脆性组合断裂模式。

Figure 17 shows the quarter FE model for the DWTT simulation, consisting of 40,848 eight-node brick elements with full integration (C3D8). The minimal element size, L_e, at the unnotched ligament, was set to 0.3 mm. The friction coefficient between the apparatus, hammer anvil, and the specimen was set to 0.1, consistent with the value used in the CVN simulation, utilizing the “CONTACT” option in ABAQUS. Both apparatuses were modeled as rigid bodies using rigid elements (R3D4). The anvil was fixed, and the hammer impacted the specimen with an initial velocity of 5.15 m/s. The simulation was conducted via dynamic implicit analysis, considering the large geometric change. Both strain rate and adiabatic heating were considered during the simulation.
图 17 显示了 DWTT 模拟的四分之一 FE 模型，该模型由 40,848 个八节点砖体单元组成，并进行了完全积分（C3D8）。无缺口韧带处的最小单元尺寸 L _e 设置为 0.3 毫米。仪器、锤砧和试件之间的摩擦系数设置为 0.1，与 CVN 模拟中使用的值一致，使用了 ABAQUS 中的“接触”选项。两个仪器均使用刚性单元（R3D4）建模为刚体。砧座固定，锤子以 5.15 米/秒的初始速度撞击试件。模拟采用动态隐式分析进行，考虑了较大的几何变化。模拟过程中同时考虑了应变率和绝热加热。

The DWTT tests performed at 0 and -40 °C were simulated using the J-C deformation and fracture strain model for ductile fracture and the critical stress model for brittle fracture. The FE mesh shown in Fig. 17 was used for the DWTT simulation. The critical stress for the DWTT simulation was calculated using the experimental DWTT energy from Eq. (14). The total volume of one layer element, E₀, in Eq. (13) was calculated for the element size of 0.3 mm, resulting in values of σ_c/σ_y = 2.92 for 0 °C and σ_c/σ_y = 2.81 for -40 °C.
在 0°C 和-40°C 下进行的 DWTT 试验，采用 JC 变形和断裂应变模型模拟延性断裂，采用临界应力模型模拟脆性断裂。DWTT 模拟采用图 17 所示的有限元网格。DWTT 模拟的临界应力是使用公式（14）中的实验 DWTT 能量计算得出的。公式（13）中单层单元的总体积 E ₀ 是针对单元尺寸为 0.3 mm 的单元计算得出的，结果显示 0°C 下的σ _c /σ _y = 2.92，-40°C 下的σ _c /σ _y = 2.81。

Figure 18(a) compares the simulated load-displacement curve with experimental results. Note that the raw experimental curves, without any processing, are shown to highlight the effect of data scattering. The FE results closely follow the experimental load-displacement curves, and the simulated impact energies were within 5 % of the experimental data. The fracture behavior of BCC steels shows four different trends depending on temperature: upper shelf (ductile), upper transition (ductile-brittle), lower transition (ductile-brittle), and lower shelf (brittle). In the transition regions, data show large scatters due to the mixed contribution of ductile and brittle fracture mechanisms, making it difficult to provide the relative contribution of ductile versus brittle fracture energy to the overall fracture energy. The proposed methodology can predict the energy contribution from ductile and brittle fracture for the given total fracture energy in transition regions. For comparison, the simulated load-displacement curves considering only the ductile fracture model are also shown in Fig. 18(a) (indicated as “w/o σ_c”). Without considering σ_c, the load increases smoothly up to 20 mm of displacement. However, in both the experimental and simulation results, when brittle fracture is accounted for, the load decreases once the displacement exceeds 10 mm. Figure 18(b) compares the fracture surface from the experimental test at -40 °C with the FE simulation. The fracture surface of the DWTT specimen was analyzed after the test using an image analyzer. The central surface showed brittle fracture due to high stress triaxiality, whereas the side surface exhibited ductile fracture. A very similar fracture surface can be observed in the FE results. In particular, the triangular shape near the initial notch tip and the jaggedly propagating brittle fracture surface were well simulated using the deformation and fracture models determined from CVN tests.
图 18(a) 将模拟的载荷-位移曲线与实验结果进行了比较。需要注意的是，图中显示的是未经任何处理的原始实验曲线，以突出数据散布的影响。FE 结果紧密遵循实验载荷-位移曲线，模拟的冲击能量与实验数据的误差在 5% 以内。体心立方钢的断裂行为随温度变化呈现四种不同的趋势：上部架状（延性）、上部转变（延性-脆性）、下部转变（延性-脆性）和下部架状（脆性）。在过渡区，由于延性和脆性断裂机制的混合贡献，数据表现出很大的散布性，因此很难提供延性断裂能量与脆性断裂能量对总断裂能量的相对贡献。所提出的方法可以预测过渡区中给定总断裂能量时延性和脆性断裂的能量贡献。为了进行比较，图 18（a）还显示了仅考虑延性断裂模型的模拟载荷-位移曲线（表示为“w/o σ _c ”）。在不考虑σ _c 的情况下，载荷平稳增加至 20 mm 的位移。然而，在实验和模拟结果中，当考虑脆性断裂时，一旦位移超过 10 mm，载荷就会降低。图 18（b）比较了-40 °C 下实验测试的断裂表面与 FE 模拟的断裂表面。试验后，使用图像分析仪分析了 DWTT 试样的断裂表面。由于应力三轴性高，中心表面表现出脆性断裂，而侧面表现出延性断裂。在 FE 结果中可以观察到非常相似的断裂表面。特别是，使用从 CVN 测试确定的变形和断裂模型可以很好地模拟初始缺口尖端附近的三角形和锯齿状扩展的脆性断裂表面。

7. DISCUSSION: EFFECT OF ADIABATIC HEATING AND STRAIN RATE ON COMBINED DUCTILE-BRITTLE FRACTURE SIMULATION FOR IMPACT TESTS
7. 讨论：绝热加热和应变速率对冲击试验韧脆性断裂模拟的影响

In this section, the effects of strain rate and adiabatic heating on CVN and DWTT simulations are discussed. Under impact loading conditions, characterized by high strain rates, the material's temperature rises due to adiabatic heating. Consequently, the material undergoes both strain rate hardening and adiabatic heating softening. To analyze the individual effect, three cases were considered: (Case 1) accounting for both strain rate and adiabatic heating effects, (Case 2) considering only the strain rate effect, and (Case 3) neglecting both strain rate and adiabatic heating effects. FE simulations were conducted for scenarios where combined ductile-brittle fracture was observed in tests, specifically at -90 °C for the CVN test and -40 °C for the DWTT.
本节将讨论应变率和绝热加热对 CVN 和 DWTT 模拟的影响。在高应变率的冲击载荷条件下，材料温度会因绝热加热而升高。因此，材料会同时经历应变率硬化和绝热加热软化。为了分析每种效应，我们考虑了三种情况：（情况 1）同时考虑应变率和绝热加热效应；（情况 2）仅考虑应变率效应；以及（情况 3）忽略应变率和绝热加热效应。我们针对在试验中观察到延性-脆性复合断裂的场景进行了 FE 模拟，具体而言，CVN 试验在 -90°C 下进行，DWTT 试验在 -40°C 下进行。

Figure 19(a) displays the CVN simulation results for the three cases described earlier. In our previous work (Seo et al., 2024), it was determined that the maximum load increases by about 10 % due to the strain rate effect but decreases by about 5 % due to the adiabatic heating effect in CVN test simulations for fully ductile fracture scenarios. For combined ductile-brittle fracture simulations, the maximum load decreases by 5 % due to the adiabatic heating effect, displaying a trend similar to that observed in ductile fracture. However, the strain rate effect on the maximum load is more pronounced, showing an increase of 15 %. The sudden load drop due to brittle fracture is accelerated when adiabatic heating is not considered (Cases 2 and 3 in Fig. 19(a)). This occurs because adiabatic heating softens the material and reduces the principal stress, aligning with findings from previous research (Petti and Dodds, 2005, Gao et al., 2006, Wasiluk et al., 2006). Notably, Petti et al. (Petti and Dodds, 2005), Gao et al. (Wasiluk et al., 2006), and Wasiluk et al. (Wasiluk et al., 2006) found that the Weibull stress σ_u increases with rising temperature under quasi-static conditions. Therefore, adiabatic heating contributes to delaying the initiation of brittle cracks.
图 19(a) 显示了前面描述的三种情况的 CVN 模拟结果。在我们之前的研究（Seo 等，2024）中，我们发现，在完全延性断裂场景的 CVN 试验模拟中，应变速率效应会使最大载荷增加约 10%，而绝热加热效应会使最大载荷降低约 5%。对于延性-脆性断裂组合模拟，绝热加热效应会使最大载荷降低 5%，呈现出与延性断裂中观察到的趋势类似的趋势。然而，应变速率对最大载荷的影响更为明显，增加了 15%。当不考虑绝热加热时，脆性断裂导致的载荷突然下降会加速（图 19(a) 中的情况 2 和 3）。这是因为绝热加热会使材料软化并降低主应力，这与先前研究的结果一致（Petti 和 Dodds，2005 年；Gao 等，2006 年；Wasiluk 等，2006 年）。值得注意的是，Petti 等（Petti 和 Dodds，2005 年）、Gao 等（Wasiluk 等，2006 年）和 Wasiluk 等（Wasiluk 等，2006 年）发现，在准静态条件下，威布尔应力 σ _u 会随着温度升高而增大。因此，绝热加热有助于延缓脆性裂纹的萌生。

Figure 19(b) shows the DWTT simulation results for the three cases. The maximum load increases by about 10 % due to the strain rate effect and decreases by 10 % due to the adiabatic heating effect. In the CVN simulation, brittle fracture was delayed due to the adiabatic heating effect. However, in the DWTT test, this delay was not observed. This discrepancy may be attributed to the difference in specimen size between the CVN and DWTT tests. In both tests, the impacted surface was compressed, and the compressive plastic strain raised the specimen's temperature due to the adiabatic heating effect. The temperature increase is more pronounced in CVN specimens compared to DWTT specimens, due to the smaller size of the CVN specimens, as shown in Fig. 20. However, the temperature rise at the crack tip is similar in both tests. During the impact simulation, the temperature increased by approximately 150 °C. The estimated temperature rise closely matches the measured value of 150 °C in CVN test, reported in (Tanguy et al., 2005). Similarly, our previous work also observed a similar temperature increase (Seo et al., 2024). The occurrence of brittle fracture is not affected by the temperature rise during the impact simulation. The difference becomes more noticeable as the displacement increases. Consequently, the adiabatic heating effect is less significant in the DWTT test simulation.

8. CONCLUSION

This paper presents the finite element (FE) impact simulation of combined ductile-brittle fracture for API X85 under impact loading. The simulation incorporates the effects of strain rate and adiabatic heating on deformation and fracture, which are crucial for accurately modeling fracture behavior under high strain rates. For the models, deformation and ductile fracture under impact loading are simulated using the J-C deformation and fracture strain models, respectively, while brittle fracture under impact load is simulated using the critical stress model. The parameters in the deformation and fracture models were determined through the following methods:

(1)
The temperature-dependent parameters in the J-C models are determined using smooth round bar (SRB) tests conducted at various temperatures under quasi-static conditions.
(2)
The strain rate-dependent parameters in the J-C models were obtained through a CVN test performed at room temperature (RT).
(3)
The critical stress fracture model was calibrated from CVN simulations using the determined J-C models.

The determined deformation and combined fracture models were applied to simulate CVN and Drop Weight Tear Test (DWTT) at various temperatures ranging from -120 °C to room temperature (RT). For the CVN test, the combined ductile-brittle fracture mode was observed at a temperature lower than -60 °C, while a purely ductile fracture mode was observed at higher temperatures. In contrast, for the DWTT, the combined ductile-brittle fracture mode was observed at temperatures below 0 °C. The simulation incorporated the effects of strain rate and adiabatic heating. The deformation behavior, load-displacement curves, and fractured surfaces obtained from the simulation were compared with experimental data, demonstrating good agreement across all cases.

The key contribution of this study, different from previous works (Kim et al., 2020, Kim et al., 2021), is the consideration of both strain rate and adiabatic heating effects. The strain rate in CVN and DWTT exceeds 1,000 /s, and the temperature can increase more than 150°C. By accounting for both effects, their influence on combined ductile-brittle fracture simulation is analyzed in this study. The effects of adiabatic heating and strain rate were also analyzed through sensitivity analysis. The adiabatic heating effect influences the occurrence of brittle fractures. In the CVN test, brittle fracture is delayed due to the thermal softening caused by adiabatic heating. This delay is found to be more significant in CVN simulations than in DWTT simulations, likely because the CVN specimen is much smaller than the DWTT specimen. In conclusion, the adiabatic heating effect plays a more significant role in smaller specimen tests. Finally, the effect of element size on key numerical parameters for simulating combined fracture under impact loading has been also analyzed. The critical stress for brittle fracture (σ_c) was dependent on element size and its dependence could be quantified using the proposed elastic energy absorption in Eq. (13).

CRediT authorship contribution statement

Ki-Wan Seo: Writing – original draft, Visualization, Formal analysis, Conceptualization. Jae-Yoon Kim: Resources, Formal analysis. Yun-Jae Kim: Writing – review & editing, Validation, Supervision, Project administration.

Declaration of competing interest

Please check the following as appropriate: All authors have participated in (a) conception and design, or analysis and interpretation of the data; (b) drafting the article or revising it critically for important intellectual content; and (c) approval of the final version. - checked

This manuscript has not been submitted to, nor is under review at, another journal or other publishing venue.- checked

The authors have no affiliation with any organization with a direct or indirect financial interest in the subject matter discussed in the manuscript - checked

The following authors have affiliations with organizations with direct or indirect financial interest in the subject matter discussed in the manuscript:- None

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Outline 大纲

Figures (20) 图（20）

Tables (4) 表格（4）

Journal of Pipeline Science and Engineering

ABSTRACT

Keywords

Nomenclature

ABBREVIATIONS

1. INTRODUCTION

2. PROPOSED COMBINED DUCTILE-BRITTLE FRACTURE SIMULATION MODEL
2. 提出的延脆性断裂模拟模型

2.1. Johnson-Cook (J-C) Deformation Model under Impact Loading
2.1 冲击载荷下的 Johnson-Cook（JC）变形模型

2.2. Johnson-Cook (J-C) Fracture Strain Model for Ductile Fracture Simulation
2.2. 用于延性断裂模拟的 Johnson-Cook（JC）断裂应变模型

2.3. Critical Stress Model for Brittle Fracture Simulation
2.3 脆性断裂模拟的临界应力模型

2.4. Combined Ductile and Brittle Fracture Simulation
2.4 延性断裂和脆性断裂综合模拟

3. SUMMARY OF EXPERIMENTS FOR MODEL DETERMINATION
3 模型确定实验总结

3.1. Tensile Test 3.1.拉伸试验

3.2. Single Edge Notched Tensile (SENT) Test
3.2. 单边缺口拉伸（SENT）试验

3.3. Charpy V Notched (CVN) Impact Test at Room Temperature
3.3 室温夏比 V 型缺口（CVN）冲击试验

4. DETERMINATION OF API X80 DEFORMATION AND FRACTURE MODELS
4 API X80 变形和断裂模型的确定

4.1. Parameter Determination of Deformation Model
4.1 变形模型参数确定

4.2. Parameter Determination of Ductile Fracture Model
4.2 韧性断裂模型参数确定

4.3. Determination of Brittle Fracture Model
4.3 脆性断裂模型的确定

5. EXPERIMENTS FOR VALIDATION
5. 验证实验

5.1. Charpy V Notched (CVN) Test
5.1. 夏比 V 型缺口（CVN）试验

5.2. Drop-Weight Tear Test (DWTT)
5.2. 落锤撕裂试验（DWTT）

6. COMPARISON WITH FE SIMULATION RESULTS FOR VALIDATION
6. 与 FE 模拟结果的比较验证

6.1. Comparison of CVN Test with Simulation Results
6.1 CVN 试验与仿真结果比较

6.2. Comparison of DWTT with Simulation Results
6.2. DWTT 与仿真结果的比较

7. DISCUSSION: EFFECT OF ADIABATIC HEATING AND STRAIN RATE ON COMBINED DUCTILE-BRITTLE FRACTURE SIMULATION FOR IMPACT TESTS
7. 讨论：绝热加热和应变速率对冲击试验韧脆性断裂模拟的影响

8. CONCLUSION

CRediT authorship contribution statement

Declaration of competing interest

References

Cited by (0)

Journal of Pipeline Science and Engineering

Combined Ductile-Brittle Fracture Simulation of API X80 Under Impact Loading

ABSTRACT

Keywords

Nomenclature

ABBREVIATIONS

1. INTRODUCTION

2. PROPOSED COMBINED DUCTILE-BRITTLE FRACTURE SIMULATION MODEL2. 提出的延脆性断裂模拟模型

2.1. Johnson-Cook (J-C) Deformation Model under Impact Loading2.1 冲击载荷下的 Johnson-Cook（JC）变形模型

2.2. Johnson-Cook (J-C) Fracture Strain Model for Ductile Fracture Simulation2.2. 用于延性断裂模拟的 Johnson-Cook（JC）断裂应变模型

2.3. Critical Stress Model for Brittle Fracture Simulation2.3 脆性断裂模拟的临界应力模型

2.4. Combined Ductile and Brittle Fracture Simulation2.4 延性断裂和脆性断裂综合模拟

3. SUMMARY OF EXPERIMENTS FOR MODEL DETERMINATION3 模型确定实验总结

3.1. Tensile Test 3.1.拉伸试验

3.2. Single Edge Notched Tensile (SENT) Test3.2. 单边缺口拉伸（SENT）试验

3.3. Charpy V Notched (CVN) Impact Test at Room Temperature3.3 室温夏比 V 型缺口（CVN）冲击试验

4. DETERMINATION OF API X80 DEFORMATION AND FRACTURE MODELS4 API X80 变形和断裂模型的确定

4.1. Parameter Determination of Deformation Model4.1 变形模型参数确定

4.2. Parameter Determination of Ductile Fracture Model4.2 韧性断裂模型参数确定

4.3. Determination of Brittle Fracture Model4.3 脆性断裂模型的确定

5. EXPERIMENTS FOR VALIDATION5. 验证实验

5.1. Charpy V Notched (CVN) Test5.1. 夏比 V 型缺口（CVN）试验

5.2. Drop-Weight Tear Test (DWTT)5.2. 落锤撕裂试验（DWTT）

6. COMPARISON WITH FE SIMULATION RESULTS FOR VALIDATION6. 与 FE 模拟结果的比较验证

6.1. Comparison of CVN Test with Simulation Results6.1 CVN 试验与仿真结果比较

6.2. Comparison of DWTT with Simulation Results6.2. DWTT 与仿真结果的比较

7. DISCUSSION: EFFECT OF ADIABATIC HEATING AND STRAIN RATE ON COMBINED DUCTILE-BRITTLE FRACTURE SIMULATION FOR IMPACT TESTS7. 讨论：绝热加热和应变速率对冲击试验韧脆性断裂模拟的影响

8. CONCLUSION

CRediT authorship contribution statement

Declaration of competing interest

References

2. PROPOSED COMBINED DUCTILE-BRITTLE FRACTURE SIMULATION MODEL
2. 提出的延脆性断裂模拟模型

2.1. Johnson-Cook (J-C) Deformation Model under Impact Loading
2.1 冲击载荷下的 Johnson-Cook（JC）变形模型

2.2. Johnson-Cook (J-C) Fracture Strain Model for Ductile Fracture Simulation
2.2. 用于延性断裂模拟的 Johnson-Cook（JC）断裂应变模型

2.3. Critical Stress Model for Brittle Fracture Simulation
2.3 脆性断裂模拟的临界应力模型

2.4. Combined Ductile and Brittle Fracture Simulation
2.4 延性断裂和脆性断裂综合模拟

3. SUMMARY OF EXPERIMENTS FOR MODEL DETERMINATION
3 模型确定实验总结

3.2. Single Edge Notched Tensile (SENT) Test
3.2. 单边缺口拉伸（SENT）试验

3.3. Charpy V Notched (CVN) Impact Test at Room Temperature
3.3 室温夏比 V 型缺口（CVN）冲击试验

4. DETERMINATION OF API X80 DEFORMATION AND FRACTURE MODELS
4 API X80 变形和断裂模型的确定

4.1. Parameter Determination of Deformation Model
4.1 变形模型参数确定

4.2. Parameter Determination of Ductile Fracture Model
4.2 韧性断裂模型参数确定

4.3. Determination of Brittle Fracture Model
4.3 脆性断裂模型的确定

5. EXPERIMENTS FOR VALIDATION
5. 验证实验

5.1. Charpy V Notched (CVN) Test
5.1. 夏比 V 型缺口（CVN）试验

5.2. Drop-Weight Tear Test (DWTT)
5.2. 落锤撕裂试验（DWTT）

6. COMPARISON WITH FE SIMULATION RESULTS FOR VALIDATION
6. 与 FE 模拟结果的比较验证

6.1. Comparison of CVN Test with Simulation Results
6.1 CVN 试验与仿真结果比较

6.2. Comparison of DWTT with Simulation Results
6.2. DWTT 与仿真结果的比较

7. DISCUSSION: EFFECT OF ADIABATIC HEATING AND STRAIN RATE ON COMBINED DUCTILE-BRITTLE FRACTURE SIMULATION FOR IMPACT TESTS
7. 讨论：绝热加热和应变速率对冲击试验韧脆性断裂模拟的影响