Hybrid experimental-numerical strategy for efficiently and accurately identifying post-necking hardening and ductility diagram parameters

doi:10.1016/j.ijmecsci.2022.107074

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

Volume 219, 1 April 2022, 107074
第 219 卷，2022 年 4 月 1 日，107074

https://doi.org/10.1016/j.ijmecsci.2022.107074 Get rights and content 获得权利和内容

Under a Creative Commons license
根据知识共享许可

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Highlights 亮点

•
A hybrid experimental-numerical strategy to predict ductile fracture is proposed.
提出了一种预测延性断裂的混合实验-数值策略。
•
The post-necking strain behaviour and ductility diagram parameters are identified.
确定了颈缩后应变行为和延性图参数。
•
Only two simple experiments and three numerical optimisations are required.
只需要两个简单的实验和三次数值优化。
•
The proposed strategy was validated by comparing experiments and predictions.
通过比较实验和预测，验证了所提出的策略。

Abstract 抽象的

Although various effective models for simulating ductile fracture in metallic materials have been established, the methods to identify the required parameters have scarcely been clarified. The difficulty for parameters identification of ductile fracture models is the actual bottleneck for accurate numerical predictions in ductile fracture simulation. In this paper, a strategy to efficiently identify the post-necking strain behaviour and the ductility diagram parameters is proposed by a hybrid experimental-numerical procedure. In the proposed strategy, only two types of experiments are required: a conventional tensile test and a crack growth test. All the parameters to simulate the post-necking strain hardening behaviour and the ductile fracture can be uniquely identified via numerical optimisations using load-displacement curves of the two tests. The proposed strategy was applied to an A2024T3 aluminium alloy and the post-necking hardening and ductility diagram parameters were identified by a flat tensile test and an Arcan test. Two types of plate specimens with a hole were then used to validate the proposed strategy. Good agreements between the experimental results and the numerical predictions clearly show the feasibility and accuracy of the proposed strategy. The results demonstrated that the proposed strategy is simple but effective. Therefore, it can be a general basis for characterising mechanical properties of various metallic materials.
尽管已经建立了各种有效的金属材料延性断裂模拟模型，但所需参数的识别方法却鲜有阐明。延性断裂模型参数识别困难是延性断裂模拟中实现精确数值预测的真正瓶颈。本文提出了一种通过混合实验-数值方法高效识别颈缩后应变行为和延性图参数的策略。在所提出的策略中，仅需要两种类型的实验：常规拉伸试验和裂纹扩展试验。所有用于模拟颈缩后应变硬化行为和延性断裂的参数都可以通过利用两种试验的载荷-位移曲线进行数值优化来唯一识别。所提出的策略应用于 A2024T3 铝合金，并通过平板拉伸试验和 Arcan 试验识别了颈缩后硬化和延性图参数。然后使用两种带孔板试样验证了所提出的策略。实验结果与数值预测结果高度吻合，充分证明了所提策略的可行性和准确性。结果表明，该策略简单有效。因此，它可以作为表征各种金属材料力学性能的通用基础。

Graphical abstract 图解摘要

Keywords 关键词

Ductile fracture

Post-necking strain hardening

Ductility diagram

Parameters identification

Thin plate

延性断裂；颈缩后应变硬化；延性图；参数识别；薄板

1. Introduction 1. 简介

Rapidly increasing demand of energy and the requirements of environmental concerns challenge human beings to explore renewable energy source and find solutions to improve energy efficiency. As light-weighted metals with excellent strength, the aluminium alloys and advanced high strength steels (AHSS) have found their applications in automotive, aviation, marine, wind energy and petroleum industries. For example, the reduction in vehicle weight by using aluminium alloy and AHSS in the parts and car body can result in a significant fuel economy improvement, offering great potential for boosting the fuel economy. However, many challenges restrain the wide application of the modern high strength light-weighted metals. The increase of the flow strength makes it difficult for material formation [1], such as hydroforming [2,3], deep drawing [4], [5], [6], [7] and stamping [8], [9], [10]. Meanwhile, crack issues, such as edge crack [11], generated during the manufacturing process is very complicated and the transition from necking dominated failure to shear induced fracture make the Forming Limit Curve (FLC) method inaccurate for sheet metal forming [12]. Numerical simulation may provide a powerful solution to better understand the mechanical properties and the fracture behaviour of components made of the high strength light-weighted metals. The hardening behaviour, especially in the post-necking stage, and the fracture criteria of the target material play important roles in the ductile fracture simulation in the component.
快速增长的能源需求和对环境问题的关注促使人类探索可再生能源并寻找提高能源效率的解决方案。铝合金和先进高强度钢 (AHSS) 作为具有优异强度的轻质金属，已在汽车、航空、船舶、风能和石油工业中得到应用。例如，在零部件和车身中使用铝合金和先进高强度钢可以减轻汽车重量，从而显著提高燃油经济性，为提高燃油经济性提供了巨大的潜力。然而，许多挑战制约着现代高强度轻质金属的广泛应用。流动强度的提高使得材料成形[1]变得困难，例如液压成形[2,3]、深拉[4]、[5]、[6]、[7] 和冲压[8]、[9]、[10]。同时，制造过程中产生的裂纹问题（例如边缘裂纹[11]）非常复杂，并且从颈缩主导的失效到剪切诱导断裂的转变，使得成形极限曲线（FLC）方法对于金属板材成形[12]并不准确。数值模拟可以提供强有力的解决方案，以更好地理解高强度轻质金属部件的力学性能和断裂行为。目标材料的硬化行为（尤其是在颈缩后阶段）和断裂准则在部件的延性断裂模拟中起着重要作用。

For numerical simulation of ductile fracture with metals, the mechanical properties such as stress-strain relationship at large strain is vital for accurate prediction. For components with large thickness, some analytical or inverse methods have been proposed to identify the post-necking strain hardening behaviour with smooth round bar specimens or axisymmetric notched tensile bars [13], [14], [15], [16]. For example, the post-necking strain hardening information could be estimated by performing a smooth round bar tensile test with the well-known Bridgman method and the Le Roy's equation [16]. Efforts have also been paid for the post-necking strain hardening identification with flat specimens [17], [18], [19], [20], [21]. With numerical analysis and flat specimens, Zhang et al. proposed an empirical function to establish the relationship between the specimen thickness reduction and the minimum cross-section area reduction to obtain the equivalent stress-strain curve after necking [19]. Although many methods have been proposed for the post-necking strain hardening identification, these methods cannot be directly applied to thin sheet metals since it is much more difficult to measure their cross-section geometry deformation (thickness or cross-section area reduction), compared to the thick plates. A method should be established for identifying the mechanical properties describing the post-necking strain hardening for thin sheet metallic materials.
对于金属延性断裂的数值模拟，大应变下的应力-应变关系等力学性能对于准确预测至关重要。对于大厚度部件，已提出了一些解析或逆方法，用于识别光滑圆棒试样或轴对称缺口拉伸棒的颈缩后应变硬化行为 [13]–[16]。例如，可以通过采用著名的布里奇曼法和 Le Roy 方程进行光滑圆棒拉伸试验来估算颈缩后应变硬化信息 [16]。也已有人致力于利用扁平试样进行颈缩后应变硬化识别 [17]–[18]–[19]–[20]–[21]。张等人利用数值分析和扁平试样，提出了一个经验函数来建立试样厚度减小量与最小横截面积减小量之间的关系，以获得颈缩后等效应力-应变曲线 [19]。尽管已经提出了许多识别颈缩后应变硬化的方法，但这些方法不能直接应用于薄板金属材料，因为与厚板相比，测量薄板金属材料的横截面几何变形（厚度或横截面积的减少）要困难得多。应该建立一种识别薄板金属材料颈缩后应变硬化力学性能的方法。

Ductile fracture is generally acknowledged as the consequence of the damage initiation, accumulation, micro-cracking and crack propagation. Many ductile fracture models have been proposed to simulate and predict the ductile failure. Generally, these models can be classified as the coupled models [22,23] and uncoupled models. For the coupled models, the damage evolution is incorporated into the constitutive law and the damage accumulation deteriorates the mechanical properties as a result. One of the most famous coupled models is the Gurson model, for which the damage is treated as the void nucleation, growth and coalescence [23]. The Gurson model and its extended versions have been successfully applied in ductile failure modelling [24], [25], [26], [27], [28], [29], [30], [31], [32]. Tvergaard and Needleman proposed a function to simulate the effect of void coalescence on the load carrying capacity of the matrix material [24,25]. Zhang proposed a so-called complete Gurson model in which the void coalescence is the result of the competition of homogeneous void growth mode and the localized deformation mode [26,31]. For uncoupled models, the damage has no influence on the elastic-plastic behaviour of the matrix material, and the activation of cracking is based on the stress/strain state [33]. The ductility diagram, also known as fracture locus, describes the relationship of fracture strain and the stress state characterized by stress triaxiality (ratio of the mean stress and the von Mises equivalent stress) [34], [35], [36]. Bao and Wierzbicki performed experiments and constructed the ductility diagram with the stress triaxiality varying from −0.3 to 1 [35]. The ductility diagram based ductile fracture model has been reported for many metallic materials. Recent investigations demonstrated that the Lode parameter also plays an important role on the fracture strain [37], [38], [39], [40]. Even though there are many coupled and uncoupled ductile fracture models, application is generally not easy or actually impossible since the identification of parameters from experiments requires numerous costs in time. The accurate but efficient parameter identification strategy has to be developed to address this issue.
延性断裂一般被认为是损伤萌生、积累、微裂纹和裂纹扩展的结果。为了模拟和预测延性失效，已经提出了许多延性断裂模型。通常，这些模型可以分为耦合模型[22,23]和非耦合模型。耦合模型将损伤演化纳入本构定律，损伤积累导致力学性能下降。最著名的耦合模型之一是 Gurson 模型，该模型将损伤处理为孔洞的成核、生长和融合[23]。Gurson 模型及其扩展版本已成功应用于延性失效模拟[24-32]。Tvergaard 和 Needleman 提出了一个函数来模拟孔洞融合对基体材料承载能力的影响[24-32]。张提出了所谓的完整 Gurson 模型，其中孔洞合并是均匀孔洞增长模式和局部变形模式竞争的结果[26,31]。对于非耦合模型，损伤对基体材料的弹塑性行为没有影响，开裂的激活基于应力/应变状态[33]。延性图，也称为断裂轨迹，描述了断裂应变与应力状态的关系，以应力三轴度（平均应力与 von Mises 等效应力之比）为特征[34]，[35]，[36]。Bao 和 Wierzbicki 通过实验构建了应力三轴度从-0.3 到 1 的延性图[35]。基于延性图的延性断裂模型已在许多金属材料中得到报道。最近的研究表明，Lode 参数对断裂应变也起着重要作用[37]–[38]–[39]–[40]。尽管存在许多耦合和非耦合的延性断裂模型，但由于从实验中识别参数需要耗费大量时间，因此应用起来通常并不容易，甚至实际上是不可能的。为了解决这个问题，必须开发准确而有效的参数识别策略。

This work presents a hybrid experimental-numerical strategy to identify the parameters of the post-necking strain-hardening behaviour and the ductility diagram for metallic materials efficiently and accurately. In this strategy, only two types of experiments are required: a conventional tensile test and a crack growth test. All the parameters to simulate the post-necking strain hardening behaviour and the ductile fracture can be uniquely identified via numerical optimisations using load-displacement curves of the two tests. Validation of the proposed strategy is then performed to thin sheets made of an aluminium alloy. The remaining part of this paper is organized as follows: the models for post-necking strain hardening behaviour and for ductile fracture are introduced in Section 2; the detailed procedures for the proposed hybrid experimental-numerical strategy is presented in Section 3; the validation of the proposed strategy is presented in Section 4; the paper is ended with major conclusions in Section 5.
本文提出了一种实验-数值混合策略，用于高效、准确地识别金属材料颈缩后应变硬化行为和延性图的参数。在该策略中，仅需要两种类型的实验：常规拉伸试验和裂纹扩展试验。通过利用两个试验的载荷-位移曲线进行数值优化，可以唯一地识别用于模拟颈缩后应变硬化行为和延性断裂的所有参数。然后对铝合金薄板进行了所提策略的验证。本文的其余部分组织如下：第 2 节介绍颈缩后应变硬化行为和延性断裂的模型；第 3 节介绍所提实验-数值混合策略的详细步骤；第 4 节介绍所提策略的验证；第 5 节总结了本文的主要结论。

2. Models for post-neck strain hardening and ductile fracture
2. 后颈应变硬化和延性断裂模型

2.1. Model for post-necking hardening behaviour
2.1 颈缩后硬化行为模型

Conventional tensile tests with extensometer can provide engineering stress-strain curves (also named as nominal stress-strain curve) and true stress-strain curve (also known as true stress- logarithmic strain curve) up to diffuse necking. The uniaxial tension condition ceases when the diffuse necking starts to develop, resulting in triaxial stress states in the necking zone and invalid true stress-strain curve in large strain. It is noted that the true stress-strain curve in the pre-necking region is exactly the equivalent stress-strain curve, in terms of von Mises equivalent stress and equivalent strain. Methods for correcting the invalid true stress in the post-necking region are also derived based on the von Mises equivalent stress. To make it simple and clear, we will name the true stress-strain curve as equivalent stress-strain curve which covering the pre-necking and post-necking parts. To simulate problems in large strain, post-necking equivalent stress-strain curve should be identified. Many methods have been proposed to obtain the post-necking equivalent stress-strain curve when the component thickness is sufficient[41]. For components with very limit thickness, which is the concern in this work, determining the post-necking strain hardening problem with experiments has been difficult. An optimization method based on the load-elongation curve (engineering stress-strain curve) in the post-necking region is proposed to obtain the post-necking strain hardening property in this work. The model used to represent the post-necking strain hardening properties is introduced in this section.
使用引伸计进行的传统拉伸试验可以提供直至弥散颈缩的工程应力-应变曲线（也称为名义应力-应变曲线）和真实应力-应变曲线（也称为真应力-对数应变曲线）。当弥散颈缩开始发展时，单轴拉伸条件终止，导致颈缩区呈现三轴应力状态，大应变下真应力-应变曲线无效。需要注意的是，就 von Mises 等效应力和等效应变而言，颈缩前区域的真应力-应变曲线就是等效应力-应变曲线。基于 von Mises 等效应力，推导出了修正颈缩后区域无效真应力的方法。为了简单明了，我们将真应力-应变曲线称为涵盖颈缩前和颈缩后部分的等效应力-应变曲线。为了模拟大应变问题，应确定颈缩后等效应力-应变曲线。当构件厚度足够大时，已提出了许多方法来获得颈缩后等效应力-应变曲线[41]。对于本文关注的厚度非常有限的构件，通过试验确定颈缩后应变硬化问题一直很困难。本文提出了一种基于颈缩后区域载荷-伸长曲线（工程应力-应变曲线）的优化方法来获得颈缩后应变硬化性能。本节介绍用于表示颈缩后应变硬化性能的模型。

In the ordinary plasticity theory, hardening properties are usually expressed by equivalent stress σ_eq and equivalent plastic strain. Therefore, the equivalent strain (namely the true strain) from the tensile test is additively decomposed into the elastic part ε_e and the plastic part ε_p:
在普通塑性理论中，强化特性通常用等效应力 σ _eq 和等效塑性应变来表示。因此，拉伸试验得到的等效应变（即真应变）被加法分解为弹性部分 ε _e 和塑性部分 ε _p ：(1)

ε_{eq} = ε_{e} + ε_{p}

ε_p from uniaxial tensile test also equals to the von Mises equivalent plastic strain. Unless otherwise specified, ε_p denotes the equivalent plastic strain in the context below. With Eq. (1), the corresponding equivalent stress-equivalent plastic strain curve before diffuse-necking can be obtained with the standard tensile test and used as input information in numerical analysis. However, the post-necking equivalent stress-equivalent plastic strain curve, which is important for ductile fracture modelling, is still unknown. Many formulas have been proposed to extrapolate the post-necking strain hardening properties based on the pre-necking experimental data. amongst which, the Ramberg-Osgood equation (Eq. (2)) and the Swift equation (Eq. (3)) have been widely applied to metals [42], [43], [44], [45], [46], [47], [48].
单轴拉伸试验的 ε _p 也等于 von Mises 等效塑性应变。除非另有说明，下文中 ε _p 均表示等效塑性应变。利用公式 (1)，可以通过标准拉伸试验获得弥散颈缩前相应的等效应力-等效塑性应变曲线，并将其作为数值分析的输入信息。然而，对于延性断裂建模至关重要的颈缩后等效应力-等效塑性应变曲线仍然未知。已经提出了许多公式来根据颈缩前实验数据外推颈缩后的应变硬化特性。其中，Ramberg-Osgood 方程（公式 (2)）和 Swift 方程（公式 (3)）已广泛应用于金属 [42]，[43]，[44]，[45]，[46]，[47]，[48]。

Ramberg-Osgood equation: Ramberg-Osgood 方程：(2)

σ_{eq} = k \cdot {ε_{p}}^{n_{R}}

Swift equation: Swift 方程：(3)

σ_{eq} = σ_{Y} \cdot {(\frac{ε_{p}}{α} + 1)}^{n_{S}}

where k, α, n_R, n_S, and σ_Y are fitting parameters. Especially, the physical meaning of σ_Y is the yield strength of the material.
其中，k、α、n _R 、n _S 和 σ _Y 为拟合参数。其中，σ _Y 的物理含义为材料的屈服强度。

For the characterization of the hardening properties in this work, the equivalent stress-equivalent plastic strain curve has been divided into two parts, as shown in Fig. 1. Before diffuse necking, the equivalent stress-equivalent plastic strain curve obtained from tensile test is adopted; after diffuse necking, the relationship between the equivalent stress and the equivalent plastic strain is established as:
为了表征本文的硬化特性，将等效应力-等效塑性应变曲线分为两部分，如图 1 所示。在弥漫性颈缩之前，采用拉伸试验获得的等效应力-等效塑性应变曲线；在弥漫性颈缩之后，建立等效应力与等效塑性应变的关系式：(4)

σ_{eq} = K \cdot {(ε_{p} - b)}^{n}

where K, b, and n are the unknowns to be determined based on experimental data in the post-necking regime. In this work, the engineering stress-strain curve after the ultimate tensile stress, which shares the same meaning as the load-elongation curve in the post-necking region, is used to derive the unknown parameters in Eq. (4) via numerical optimisation. Details about the optimisation is introduced in Section 3. It is noted that Eq. (4) can be converted to Eq. (2) and (3) by properly setting the material parameters K, b and n. For example, when K = σ_Y/αⁿ and b = −α, Eq. (4) returns to the Swift equation. Therefore, Eq. (4) is a general form of the Ramberg-Osgood equation and the Swift equation.
其中，K、b 和 n 是根据颈缩后阶段的实验数据确定的未知量。本文采用极限拉伸应力后的工程应力-应变曲线，其含义与颈缩后区域的载荷-伸长曲线相同，通过数值优化方法推导出公式（4）中的未知参数。优化过程的细节将在第 3 节中介绍。需要注意的是，通过适当设置材料参数 K、b 和 n，公式（4）可以转换为公式（2）和公式（3）。例如，当 K = σ _Y /α ⁿ 且 b = −α 时，公式（4）将恢复为 Swift 方程。因此，公式（4）是 Ramberg-Osgood 方程和 Swift 方程的一般形式。

2.2. Model for ductile fracture
2.2 韧性断裂模型

To simulate the fracture behaviour of high strength thin sheet metals, a ductile fracture model featuring accuracy and simplicity has the priority. Kim and his co-workers proposed the stress-modified fracture strain (SMFS) model. The SMFS model, by simply considering the ductility diagram as the fracture criterion and the stress relaxation as a result, has been successfully validated in the applications of simulating ductile failure from thick specimens to structural components [49], [50], [51], [52], [53]. The derivation of the model parameters with notched and smooth round bar specimens is easy in practice. The SMFS model is therefore utilized in the present work though the procedures to thin sheet metals application have yet to be established. In the following, the SMFS model and the corresponding parameters in the model are introduced in detail.
为了模拟高强度薄板的断裂行为，一个既准确又简单的延性断裂模型至关重要。Kim 等人提出了应力修正断裂应变 (SMFS) 模型。SMFS 模型简单地将延性图作为断裂准则，并将应力松弛作为结果，已在模拟厚试样到结构件的延性失效应用中得到成功验证 [49]，[50]，[51]，[52]，[53]。在实践中，对于缺口和光滑圆棒试样，推导模型参数十分容易。因此，尽管 SMFS 模型在薄板金属中的应用方法尚未建立，但本研究仍采用该模型。下面将详细介绍 SMFS 模型及其相应的参数。

It is well known that for ductile fracture, the fracture strain (or ductility) ε_f depends significantly on the stress state, which is usually characterized by the stress triaxiality (σ_m/σ_eq), where σ_m and σ_eq are the mean stress and the von Mises equivalent stress [34,36,50,[53], [54], [55]]. Under relatively high stress triaxiality, the fracture strain decreases as the stress triaxiality increases. In the SMFS model, the ductility diagram is expressed with only two material constants, A and B, as
众所周知，对于延性断裂，断裂应变（或延性）ε 显著依赖于应力状态，通常用应力三轴度（σ _m /σ _eq ）来表征，其中 σ _m 和 σ _eq 分别为平均应力和 von Mises 等效应力[34,36,50,[53], [54], [55]]。在相对较高的应力三轴度下，断裂应变随应力三轴度的增加而减小。在 SMFS 模型中，延性图仅用两个材料常数 A 和 B 表示，即(5)

ε_{f} = A \cdot exp (- B \cdot \frac{σ_{m}}{σ_{eq}})

A schematic example of the ductility diagram in the SMFS model is shown in Fig. 2.
图 2 为 SMFS 模型中延性图的示意图。

Damage ω is evaluated based on the histories of the equivalent plastic strain ε_p and the fracture strain ε_f, as:
损伤 ω 根据等效塑性应变 ε _p 和断裂应变 ε 的变化过程进行评估，具体如下：(6)

ω = \int_{0}^{ε_{p}} \frac{d ε_{p}}{ε_{f}}

Ductile fracture is assumed to occur when the damage accumulates to a critical value (i.e., ω = 1). As shown in Fig. 3, when the critical value is reached, ductile failure is simulated by reducing the stress sharply, representing the loss of the load carrying capacity. The decreasing slope should be as small as possible. In the present work, the decreasing slope value is set to be −1/5000 referring to the conventional works [49,50,56,57]. It has been reported that when the decreasing slope was smaller than −1/5000, the influence of the slope on the numerical results were negligible [50,56].
当损伤累积到临界值（即ω = 1）时，假设发生延性断裂。如图 3 所示，当达到临界值时，通过急剧降低应力来模拟延性失效，这代表承载能力的丧失。下降斜率应尽可能小。在本研究中，参考传统文献[49,50,56,57]，将下降斜率值设定为-1/5000。有报道称，当下降斜率小于-1/5000 时，斜率对数值结果的影响可以忽略不计[50,56]。

Together with the ductility diagram and the stress relaxation model, the SMFS model can be implemented. The calculation procedures are carried out at each integration point. Crack propagation is actualized via element deletion. Based on the preliminary investigations, the element is deleted when the damage of arbitrary half of integration points in one element have reached the predefined critical value, i.e., ω = 1, to stabilise the calculation process of the crack propagation.
结合延性图和应力松弛模型，可以实现单模态应力-应变（SMFS）模型。计算过程在每个积分点进行。裂纹扩展通过单元删除实现。基于初步研究，当一个单元中任意一半积分点的损伤达到预定的临界值（即ω = 1）时，即可删除该单元，以稳定裂纹扩展的计算过程。

In the present work, Abaqus is employed for numerical simulations. a user material (UMAT) subroutine, which is documented in Supplementary materials, has been developed to simulate the ductile fracture. Numerical simulations with the UMAT subroutine and the element deletion condition present to be stable and successfully capture the crack propagation. More detailed information can be found in Supplementary materials.
本研究采用 Abaqus 进行数值模拟。我们开发了一个用户材料 (UMAT) 子程序，用于模拟延性断裂，该子程序的文档记录在补充材料中。使用 UMAT 子程序和单元删除条件进行的数值模拟结果稳定，并成功捕捉了裂纹扩展。更多详细信息请参阅补充材料。

3. Proposed strategy for parameters identification
3. 提出的参数识别策略

In this section, the proposed strategy for efficiently and accurately identifying all the parameters required to characterize the post-necking strain hardening behaviour and to simulate the ductile fracture is presented. The significant advantage of the proposed strategy is its simplicity and applicability. Only two simple experiments and three numerical optimisations for the hardening parameter and the material constants identification are required. Each optimisation can be performed to identify a single parameter, so that a simple algorithm, such as the Golden-section search algorithm, can be applied to automatically find the optimised solutions. In addition, the strategy can be applied to components with not only sufficient thickness but also very limited thickness, making the strategy a good choice for the mechanical property characterization for light-weighted high strength thin sheet metals. Experimental results may present scatter, depending on the testing materials, sampling directions, testing machines. However, scatter of the experimental results is outside the scope of this study and may be considered in the future work.
本节介绍了一种有效、准确地识别表征后颈缩应变硬化行为和模拟延性断裂所需所有参数的策略。该策略的显著优势在于其简单性和适用性。只需进行两次简单的实验和三次数值优化即可确定硬化参数和材料常数。每次优化只需识别一个参数，因此可以应用简单的算法（例如黄金分割搜索算法）自动找到最优解。此外，该策略不仅适用于厚度足够大的部件，也适用于厚度非常有限的部件，使其成为表征轻质高强度薄板力学性能的理想选择。实验结果可能会出现分散性，具体取决于测试材料、取样方向和测试机器。然而，实验结果的分散性超出了本研究的范围，可以在未来的研究中加以考虑。

The flowchart of the proposed strategy is illustrated in Fig. 4. With the procedures in Section 3.1 and Section 3.2, all the post-necking strain hardening and ductility diagram parameters can be identified with only a tensile test and a crack growth test.

3.1. Post-necking strain hardening parameter identification

As mentioned in Section 2.1, the engineering stress-strain curves measured from experiments can be directly converted to equivalent stress-strain curves before diffuse necking. Considering the continuity to the stress-strain curves defined by Eq. (4) and neglecting the elastic part of the equivalent strain, the Considère criterion (dσ_eq/dε_p = σ_eq at diffuse necking) can be expressed:(7)

σ_{eq} |_{σ_{eq} = d σ_{eq} / d ε_{p}} = K \cdot {(ε_{p} |_{σ_{eq} = d σ_{eq} / d ε_{p}} - b)}^{n} = K \cdot n \cdot {(ε_{p} |_{σ_{eq} = d σ_{eq} / d ε_{p}} - b)}^{n - 1}

where

ε_{p} |_{σ_{eq} = d σ_{eq} / d ε_{p}}

is the corresponding equivalent plastic strain at diffuse necking.

σ_{eq} |_{σ_{eq} = d σ_{eq} / d ε_{p}}

and

ε_{p} |_{σ_{eq} = d σ_{eq} / d ε_{p}}

can be easily determined with the tensile test data before necking. Therefore, the number of the unknown parameters in Eq. (4) can be reduced from three to one by using Eq. (7). That is, both K and b in Eq. (4) can be expressed as a function of the hardening exponent n.

According to the analysis above, once the hardening exponent n is identified, the equivalent stress-equivalent plastic strain curve used for simulating post-necking behaviour can be obtained. In the following, identification of the only one unknown parameter n is achieved by optimisation to minimise the difference in engineering stress-strain (σ_eng-ε_eng) curves obtained from experiments and numerical simulation (Opt. I). The engineering stress-strain curve in the post-necking regime is used for the optimization, which holds the same meaning as the load-elongation curve after the maximum tensile load in the tensile test. Procedure for the identification of n by Opt. I is presented as follows:

Opt. I:

(i)
Define the averaged error $E_{σ}^{I}$ in the engineering stress as a function of n:(8) $E_{σ}^{I} (n) = \frac{1}{ε_{1} - ε_{0}} \sqrt{\int_{ε_{0}}^{ε_{1}} {(σ_{FEM} (n) - σ_{exp})}^{2} d ε_{eng}}$

where σ_exp and σ_FEM are the engineering stresses obtained from the experiment and the numerical simulation, respectively. ε₀ and ε₁ are the engineering strains corresponding to the maximum load and the point dσ_eng/dε_eng = −5000, defining the range on the engineering stress-strain curve for the optimisation.(9)

ε_{0} = ε_{eng} |_{σ_{eng} = σ_{u}}

(10)

ε_{1} = ε_{eng} |_{(d σ_{eng} / d ε_{eng} = - 5000)}

σ_u is the ultimate tensile stress. Fig. 5 schematically presents the range for the identification of the post-necking strain hardening properties via optimization. Note that the value dσ_eng/dε_eng = −5000 is somewhat arbitrary but it is very close to the finial separation of the tensile specimen. The response after the point dσ_eng/dε_eng = −5000 is further optimised with the ductility diagram parameters identification introduced in the next section, considering the damage accumulation in plastic loading.

(i)
Create a finite element model used for numerical simulations with the same geometry, gauge length and boundary conditions as in the actual tensile experiment.
(ii)
Set the search range of the strain hardening exponent n, and perform the numerical optimisation by the Golden-section search algorithm. In the optimisation, numerical simulations with assuming n and evaluations based on the error index $E_{σ}^{I}$ in Eq. (8) are repeated until the optimal solution of n is found.

3.2. Ductility diagram parameters identification

The ductility diagram defined by Eq. (5) has two material dependant parameters, namely A and B. Conventionally, these two parameters can be determined with two specimens which undergo different stress states during loading. For structural components with sufficient thickness, notched round bar specimens with different notch radius can be used to perform uniaxial tensile test. Together with numerical simulation, the stress triaxiality during the tensile loading can be captured and the averaged stress triaxiality is then calculated. The fracture strain can be directly measured from the fractured specimen minimum cross-section area reduction. However, the measurement of fracture strain with thin flat specimen is almost impossible because of the difficulty in the measurement of the cross-section area. Therefore, the fracture strain of flat specimens cannot be obtained directly, posing on difficulty for the construction of the ductility diagram. To solve this problem, we proposed to use optimisation algorithm with the tensile specimen and a crack growth specimen. The engineering stress-strain curve and the load-displacement curve from the tests are then set as the target in numerical optimisation to identify the parameters (namely A and B in Eq. (5)) for the ductility diagram construction. For the parameters identification in this part, two-step optimisations (Opt. II (a) and Opt. II (b)) are applied by using the results from the tensile test and the crack growth test, respectively.

In general, the stress triaxiality does not largely change in tensile tests using a flat specimen with very limited thickness. Therefore, the fracture strain in the test can be regarded as a constant ε_f0. Based on the fact, in Opt. II (a), we identify ε_f0 by assuming a constant fracture strain criterion, expressed as(11)

ε_{f} = ε_{f 0}

The entire region after the ultimate tensile stress on the engineering stress-strain curve from the tensile test can be set as the target for the numerical optimisation. With the identified ε_f0, the averaged stress triaxiality

\bar{η}

is calculated at the integration point where the damage first reached the critical value, i.e., ω = 1, by the expression as(12)

\bar{η} = \frac{1}{ε_{f 0}} \int_{0}^{ε_{f 0}} (\frac{σ_{m}}{σ_{eq}}) d ε_{p}

Assuming

\bar{η}

as the representative value of stress triaxiality in the tensile test, Eq. (5) for the tensile test can be rewritten as:(13)

ε_{f 0} = A \cdot \exp (- B \bar{η})

The parameter A can be expressed as a function of B:(14)

A = \frac{ε_{f 0}}{\exp (- B \bar{η})}

Substituting Eq. (14) into Eq. (5), we obtain(15)

ε_{f} = \frac{ε_{f 0}}{\exp (- B \bar{η})} \cdot \exp (- B \frac{σ_{m}}{σ_{eq}})

With the fracture strain ε_f0 and the averaged stress triaxiality

\bar{η}

obtained from the tensile test, there is only one unknown parameter B in Eq. (15). Therefore, in Opt. II (b), the parameter B is identified by using the load-displacement curve from the crack growth test. In summary, the procedures in the respective optimisations Opt. II (a) and Opt. II (b) for identifying the ductility diagram parameters are set as following:

Opt II (a):

(i)
Define the averaged error in the engineering stress $E_{σ}^{II}$ as a function of ε_f0, as(16) $E_{σ}^{II} (ε_{f 0}) = \frac{1}{ε_{2} - ε_{0}} \sqrt{\int_{ε_{0}}^{ε_{2}} {(σ_{FEM} (ε_{f 0}) - σ_{exp})}^{2} d ε_{eng}}$

where ε₂ is the minimum value between the engineering strain at failure in the experiment

ε_{exp}^{f}

and that in the numerical simulation

ε_{FEM}^{f}

, expressed as(17)

ε_{2} = \min (ε_{exp}^{f}, ε_{FEM}^{f})

(i)
Set the search range of ε_f0, and perform the numerical optimisation by the Golden-section search algorithm. In the optimisation, numerical simulations with assuming ε_f0 and evaluations based on the error index $E_{σ}^{II}$ in Eq. (16) are repeated until the optimal solution of ε_f0 is found. Note that the numerical simulations can be performed using the same finite element model created in Opt. I.

Opt II (b):

(i)
Calculate the averaged stress triaxiality for tensile specimen $\bar{η}$ based on Eq. (12) with ε_f0 identified in Opt. II (a). Then, replacing the parameter A based on Eq. (14) and define the fracture strain ε_f as a function of B with Eq. (15).
(ii)
For the target crack growth test, define the averaged error in load $E_{P}^{II}$ as a function of B, as(18) $E_{P}^{II} (B) = \frac{1}{Δ_{f}} \sqrt{\int_{0}^{Δ_{f}} {(P_{FEM} (B) - P_{exp})}^{2} d Δ}$

where P_exp and P_FEM are the loads obtained from the experiment and the numerical simulation, respectively. Δ is the displacement and Δ_f is the displacement when the specimen is completely fractured in the experiment.

(i)
Create a finite element model used for numerical simulations for the target crack growth test. Note that the finite element model for the entire specimen may be not needed if the crack growth occurs in a specific region of the specimen and the boundary conditions can be accurately extracted from the experiment (see an example presented in Section 4.1 and Section 4.3).
(ii)
Set the search range of B, and perform the numerical optimisation by the Golden-section search algorithm. In the optimisation, numerical simulations with assuming B and evaluations based on the error index $E_{P}^{II}$ in Eq. (18) are repeated until the optimal solution of B is found.

4. Application of the proposed strategy

In this section, we present an application of the proposed strategy presented in Section 3 to a high strength sheet metal A2024T3 aluminium alloy with the thickness of 1.2 mm. The test layouts and the finite element models for the parameters identification are introduced in Section 4.1. The results of the parameters identification for the post-necking strain hardening and for the ductility diagram are presented in Section 4.2 and Section 4.3, respectively. The validation of the proposed strategy is then demonstrated for tensile tests using two-types of thin plate specimens with a hole in Section 4.4.

4.1. Test layouts and finite element models

The tests conducted for the material parameters identification were a tensile test using a flat specimen and an Arcan test under Mode I fracture. Fig. 6 shows the geometry information of the specimens and the test set-ups. The thickness of the tensile specimen and the Arcan specimen was 1.2 mm. Before performing the test, the specimens were painted with a random speckle pattern and a digital image correlation (DIC) system was used to measure the specimen surface deformation. As marked in Fig. 6(a), the gauge length in the tensile test was 50 mm. The tests were performed at room temperature under displacement control. During the tests, the load-displacement data were obtained.

Fig. 7 displays the employed finite element model of the tensile specimen. Unstructured mesh was utilized for simulating unsymmetric deformation and fracture behaviour of the tensile specimen. Considering the symmetry of the specimen geometry, fracture surface of the tensile specimen in the test and boundary condition, only half of the tensile specimen was modelled by assigning the symmetric condition in the thickness direction. The enforced displacements were applied on the top and bottom surfaces of the finite element model as the boundary conditions.

Considering the difficulties in modelling the entire specimen with boundary conditions, only a part of the specimen was modelled for the Arcan test. Fig. 8(a) shows the finite element model of the Arcan test. The boundaries of the top and bottom of the specimen (the red and blue curves in Fig. 8(a)) were determined based on the upper and lower limits of the measured range of the DIC system. The corresponding histories of the displacement distributions on the red and blue curves measured by the DIC system as in Fig. 8(b) were set as the boundary conditions of the numerical simulations.

All the finite element models were composed of eight-node hexahedron elements with full-integration. Since the results in numerical simulations depend on the mesh size, fixed element size equal to 0.2 mm was applied in the gauge section of the tensile test specimen and the crack growth region of the Arcan test specimen because smaller element size will increase the numerical cost significantly and larger element size is not appropriate for simulating the geometry of the notch root in the Arcan test specimen, whose notch tip radius is 0.1 mm As results, the specimens for the tensile test and the Arcan test were discretized with 134,430 elements, 180,932 nodes and 12,804 elements, 17,912 nodes, respectively.

4.2. Post-necking strain hardening parameter identification

The engineering stress-strain curve and the equivalent stress-strain curve obtained from tensile tests are presented in Fig. 9. As can be seen, a sudden drop of the engineering stress can be found after the maximum value, indicating the limited deformation ability of this aluminium alloy. After obtaining the equivalent stress-strain curve before necking, the procedure introduced in Section 3.1 is applied for the post-necking strain hardening parameter identification.

Numerical optimisation (Opt. I) was performed with assumed value range of n. After performing each numerical simulation, the error

E_{σ}^{I}

defined in Eq. (8) was evaluated as shown in Fig. 10(a). The one, as marked in hollow blue circular in Fig. 10(a), was regarded as the optimised solution. That is, the strain hardening exponent was identified as n = 0.0626 for the target aluminium alloy. The resulting engineering stress-strain curve from numerical simulation is plotted together with that from experiment in Fig. 10(b) and (c). The curve obtained from numerical simulation shows good agreement with the experimental result in the optimisation range. The averaged engineering stress error in the optimisation range was

E_{σ}^{I} = 0.903

MPa, indicating sufficient accuracy for the post-necking hardening behaviour representation. With the identified parameter n and Eq. (7), the post-necking hardening behaviour for the aluminium alloy applied in the present work can be expressed in the form of Eq. (4), as(19)

σ_{eq} = 690 \cdot {(ε_{p} - 0.0932)}^{0.0626}

The parameters for the tensile properties including the post-necking strain hardening behaviour of the target A2024T3 aluminium alloy are summarized in Table 1.

Table 1.. Tensile properties for the target A2024T3 aluminium alloy.

Yield Strength σ_Y [MPa]	Ultimate Strength σ_u [MPa]	Uniform elongation ε_u [%]	Parameters used in Eq. (4)
Yield Strength σ_Y [MPa]	Ultimate Strength σ_u [MPa]	Uniform elongation ε_u [%]	K	b	n
336	493	18	690	0.0932	0.0626

4.3. Ductility diagram parameters identification

According to Section 3.2, two-step optimisations (Opt. II (a) and Opt. II (b)) for identifying ductility diagram parameters were performed.

In Opt. II (a), numerical optimisation with an assumed value range of fracture strain ε_f0 was performed based on the corresponding error

E_{σ}^{II}

defined in Eq. (16), with the experimental tensile properties obtained in Section 4.1 and a constant fracture strain criterion expressed in Eq. (11). The relationship between ε_f0 and

E_{σ}^{II}

obtained in the optimisation is presented in Fig. 11(a). As the one providing the minimum

E_{σ}^{II}

, the fracture strain in the tensile test was identified as ε_f0 = 0.315. The corresponding engineering stress-strain curve with the identified ε_f0 is presented in Fig. 11(b) and (c), compared with the experimental result. The engineering stress-strain curve agrees very well with the experimental results in the whole range. Especially, the averaged error in the optimised range was only 1.18 MPa, indicating the trustworthy of the fracture strain value identified. It can also be found that ε_f0 is much larger than the maximum engineering strain obtained from the tensile test (i.e., ε_eng = 0.202). This optimisation procedure enables the identification of the fracture strain with thin flat tensile test specimen. Fig. 12 presents the von Mises equivalent strain fields from experimental tensile test and from numerical simulation with the identified ε_f0 under different stroke displacement. Good agreement was found in the histories of the equivalent strain distributions, except the single shear band observed on the experimental results. Due to the damage evolution, localized deformation was developed in the specimen centre, failing to further display the single shear band as observed in the test.

In the procedures in Opt. II (b), the relationship between the parameter A and B was established by the identified fracture strain ε_f0 first. Then, numerical optimisation with an assumed value range of B was performed based on the corresponding error

E_{P}^{II}

defined by Eq. (18), with the SMFS model.

In the numerical results of the tensile test obtained in Opt. II (a), the history of the stress triaxiality at the gauss point where the equivalent plastic strain firstly reached ε_f0 against the equivalent plastic strain ε_p is shown in Fig. 13. The stress triaxiality was almost constant at the beginning and then increases slightly with increase of ε_p. The averaged stress triaxiality from the beginning to the complete fracture was calculated as

\bar{η} = 0.342

. Substituting the values of ε_f0 and

\bar{η}

into Eq. (14), the relationship between the two parameters for the ductility diagram was established as:(20)

A = \frac{0.315}{exp (- 0.342 B)}

The material dependant parameters A and B are then obtained with the Arcan test. By setting the value range of B, numerical optimisation was performed based on the corresponding error

E_{P}^{II}

defined by Eq. (18), with the ductile fracture criterion defined by Eq. (15) and the load-displacement curve from the Arcan test. The values of B and the corresponding

E_{P}^{II}

were shown in Fig. 14(a). The value B = 1.384 corresponding to the minimum error is identified as the appropriate one for the target aluminium alloy. Comparison of the load-displacement curves obtained in the experiment and the numerical simulation with the identified B is presented in Fig. 14(b). As expected, very good agreement can be observed with the averaged error in load of

E_{P}^{II} = 0.234 kN

. Deformations, the equivalent strain fields, and crack growth behaviours in the experiment and the numerical simulation with identified parameters under different stroke displacements are presented in Fig. 15. The specimen profiles under different displacements (averaged displacement in the tensile direction from the enforced boundary in Fig. 8) from the experiment and from the numerical simulation show very good agreement. Results of the load-displacement curve and the crack propagation history indicates that B = 1.384 can successfully reproduce the ductile fracture phenomenon for the target thin plate made of the aluminium alloy. With the identified B, the value of A can be easily calculated with Eq. (10) and the ductility diagram can be constructed with Eq. (15):(15)

ε_{f} = 0.493 exp (- 1.384 \frac{σ_{m}}{σ_{eq}})

The ductility diagram parameters of the target A2024T3 aluminium alloy are summarized in Table 2.

Table 2.. Ductility diagram parameters used in Eq. (5) for the target A2024T3 aluminium alloy.

A	B
0.493	1.384

4.4. Validation of the proposed strategy

With the proposed strategy, the post-necking hardening behaviour and the ductility diagram parameters of the target A2024T3 aluminium alloy have been identified. In this section, validation of the proposed strategy is demonstrated with two types of tensile tests using plate specimens with a hole.

The geometries of the specimens made of the target A2024T3 aluminium alloy are shown in Fig. 16. The specimen thickness is 1.2 mm. A hole with the radius of 5 mm was machined in the middle of each specimen. On each side of the specimens, a notch with the tip radius of 0.25 mm was machined to induce stress concentration. The distances between the notch tip and the hole centre in the loading direction in the respective specimens were 0 mm and 10 mm, as marked with CG0 and CG10 in Fig. 16. The specimens were painted with a random speckle pattern and then loaded in tension under displacement control. The DIC system was used to monitor the specimen surface deformation.

Along with the tests, numerical simulations were performed with the identified parameters. Considering the symmetry of the problems, quarter models were employed with symmetric boundary conditions. Unstructured meshes with the same element type shown in Section 4.1 were applied. The element size approximately equal 0.2 mm was applied to the sufficiently large regions including the hole and the notches to contain the entire crack paths. The element size was increased gradually in the remaining parts, as shown in Fig. 17. Totally, the specimen was discretized with 11,712 elements and 16,284 nodes for CG0 and 23,898 elements and 32,544 nodes for CG10.

Load-displacement curves from the experiments and from the numerical predictions are presented in Fig. 18. It can be observed that the load-displacement curves from the experiments and from the numerical predictions show good agreement for both tests. The crack paths and the equivalent strain distributions during the loading process from experiments and from numerical predictions are presented in Fig. 19. In the experimental result of CG0, a crack initiated at the notch tip and then propagated almost horizontally to the hole. On the other hand, in the experimental result of CG10, a crack also initiated at the notch tip but its propagation showed a curved path to the hole. It is noted that the sudden load drops were not found in both CG0 and CG10 as shown in Fig. 18, differently from the tensile test using the smoothed specimen as shown in Fig. 11(b). These results can be explained by the stable decreasing in structural stiffness due to the crack growth with increasing displacement as shown in Fig. 19. For both tests, the crack growth behaviours and the equivalent strain fields during loading presented good agreements in experiments and in the numerical predictions, strongly indicating the validity of the proposed hybrid experimental-numerical strategy for identifying the parameters for the post-necking strain hardening behaviour and the ductility diagram.

5. Conclusion

In this work, a strategy to efficiently and accurately identify the post-necking strain behaviour and ductility diagram parameters was proposed by a hybrid experimental-numerical procedure. This strategy can be applied not only to thick plates but also to very thin sheets, which is almost impossible with conventional methods. The proposed strategy requires only two types of experiments: a conventional tensile test and a crack growth test under simple loading. All the parameters to simulate the post-necking strain hardening behaviour and the ductile fracture can be uniquely identified via numerical optimisations using measurable quantities (engineering stress-strain curves or load-displacement curves) of the two tests.

The proposed strategy was applied to an A2024T3 aluminium alloy thin sheet with 1.2 mm in thickness. By using a flat tensile test and an Arcan test, the post-necking hardening behaviour and the ductility diagram parameter were identified. Validation of the proposed strategy was conducted by employing tests using two types of plate specimens with a hole. Results including the load-displacement curves and the crack propagation histories from experiments and from numerical predictions showed very good agreement, indicating the feasibility and accuracy of the proposed strategy.

The hybrid experimental-numerical strategy presented in this work is easy to perform with simple tests but can provide accurate numerical predictions in ductile fracture, even for thin sheet metals. Therefore, the proposed strategy can be a general basis for characterising mechanical properties of various metallic materials.

Funding sources

This study was supported by JSPS KAKENHI grant numbers 20F20364 and 18H03811.

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.

Acknowledgements

We wish to thank Professor Yun-Jae Kim of Korea University for his advice on the theory and implementation of the stress modified fracture strain model.

Appendix. Supplementary materials

Download: Download zip file (3KB)

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A novel dual-stage failure criterion based on forming limit curve for uncured GLARE
2024, Journal of Materials Processing Technology
Forming limit curve (FLC) is an essential criterion for evaluating the formability of fiber metal laminates (FMLs) in mass fabrication. However, conventional methodologies are unable to predict inner fiber failure, very sensitive to wrinkling defects, resulting in low measurement accuracy and stability. Hence, a novel dual-stage failure criterion considering the progressive failure of fiber and metal layers has been developed and validated by utilizing a notch-pattern sample and forming of typical thin-walled structure, with four typical categories of glass laminate aluminum reinforced epoxy (GLARE). The Nakajima tests of a series of notch-pattern and conventional pattern specimens with different widths were conducted in this study, demonstrating the superiority of wrinkling and delamination prevention of the notch-pattern by enhancing 22.84 % stress triaxiality and reducing 39.51 % compress strain in the edge area. Notably, punch reaction force and strain field analyses unveiled a mutation phenomenon indicative of premature fiber failure while the aluminum remained intact, which was also verified by numerical and interrupted methods. Furthermore, the acoustic emission in-situ method was adopted to monitor the progressive damage evolution of uncured GLARE in the Nakajima test, providing solid support for the occurrence of fiber failure at the mutation site. Leveraging these mutations, a list of dual-stage forming limit curves was devised to quantify the progressive damage behavior of uncured GLARE under biaxial tensile conditions. Finally, the practical application of this novel dual-stage forming limit curve methodology was demonstrated in the manufacturing process of a box-shaped part, serving as a proof of concept for its effectiveness. This study offers a foundational guideline for characterizing the intrinsic failure mechanisms of fiber metal laminates, particularly internal fiber failure under various strain conditions, thereby enhancing predictive accuracy.
Prediction of the irradiation effect on the fracture toughness for stainless steel using a stress-modified fracture strain model
2024, International Journal of Mechanical Sciences
This paper proposes a numerical method for predicting the reduction in the fracture toughness with neutron dose using the concept of an embrittlement constant. Tensile and fracture toughness test data for type 304 stainless steel in the literature [41,42] are used to validate the proposed method. Fracture toughness tests are simulated using a stress-modified fracture strain damage model implemented in an ABAQUS user-subroutine. The true stress-strain curve and one point of the fracture strain locus are obtained from the tensile test simulation. The remaining parameters of the fracture strain locus are determined by simulating a fracture toughness test. The fracture strain locus is modified to reflect the irradiation effect on the fracture toughness using the proposed irradiation embrittlement constant C_I. The function of the constant with the neutron dose is calculated from the tensile tests and simulations. Subsequently, the modified damage model is used to predict the reduction in the fracture toughness. The predicted simulation results are compared with experimental data, showing good agreement.
Flow stress curves for 980MPa- and 1.5GPa-class ultra-high-strength steel sheets weakened under high-stress triaxiality
2024, International Journal of Mechanical Sciences
Citation Excerpt :
Conventionally, it is challenging to simulate the flow stress curves corresponding to tensile tests in the case of sheet-type specimens owing to the onset of shear banding prior to ductile fracturing. Despite these challenges, several effective methods have been proposed for evaluating flow stresses in the post-necking strain region [34–38]. However, these cannot be used to evaluate material weakening in UHSSs.
The weakening behavior of ultra-high-strength steel(UHSS) remains unknown, even though it is a requirement for the accurate prediction of strain localization and ductile fracture in automobile applications. In this study, we, therefore, revealed the stress–strain curves of UHSSs up to their ductile fracture under different stress triaxialities for the first time. Smooth and notched tiny round-bar specimens cut from UHSSs with 1.6 mm thickness were subjected to tensile tests, during which the forces and neck diameters of the specimens were measured. Notably, the evaluated flow stresses in the 1.5 GPa-class notched UHSS specimens exhibited weakening of up to 3%. This was in contrast to the 980 MPa-class UHSS specimens, which did not exhibit notch-induced weakening. The absence of weakening during stress measurements, as confirmed by synchrotron X-ray diffraction (XRD) analysis, suggested that weakening was caused by material damage. However, the weakening behavior did not follow the previously developed damage models based on microvoid formation. The volume fraction of the microvoids, as observed by X-ray computer tomography, was extremely small (at 0.2%) and cannot account for the 3% material weakening indicated by the flow stress measurement results. A new damage mechanism, associated with unobservable small lattice vacancies, was implied in UHSSs deformation under high stress triaxialities.
A novel deformation mode of expansion tubes accounting for extra contact
2023, International Journal of Mechanical Sciences
Expansion tubes are the main energy absorber of coupler and draft gear used on high-speed trains. With the rapidly increasing demand for energy-absorbing, the stroke is trying to extend without enhancing the action force. However, a stability problem occurred at one dynamic test due to the misalignment of coupler seats leading to the offset of load. The contact zone's structural strength of the conical die contributes much to lateral stability. A reasonable geometric configuration was investigated, and a square die was proposed. The square die has a longer conical surface and extra oblique line contact by filling the gaps between the tube and die. Quasi-static experiments, finite element simulations, and an analytical model were performed to investigate square-die expansion tubes' deformation and mechanical properties. The square-die expansion has a similar action force and specific energy absorption (SEA) as the circular-die expansion. At an expansion angle of 30°, the contact length of the square-die expansion tubes was 3∼17.34 times those of the circular-die expansion. In addition, the square-die expansion tube has a higher specific energy absorption than the circular-die expansion tube at the same contact length. The effect of the contact length on stability and the application of a square-die expansion tube on the coupler under the offset load condition were examined. The square-die expansion tube maintained stability even for the misaligned amount of the coupler seats of up to 60 mm. These achievements pave an effective way for the future design of long-stroke expansion tubes.
Ductile crack propagation path depending on material properties: Experimental results and discussions based on numerical simulations
2022, Materials and Design
Citation Excerpt :
In the previous sections, the crack propagation in Arcan tests of the two investigated thin flat metals were introduced, and differences in the crack paths were observed as experimental facts. Using the proposed strategy in Ref. [27], numerical simulations successfully reproduced the fracture process in the Arcan test experiments and captured the crack paths difference under different stress states. In the numerical simulations, only the mechanical properties of the two investigated materials were changed.
To ensure the structural integrity of advanced thin sheet metals in engineering applications, it is necessary to understand their ductile crack propagation behaviour. In this work, we clarified the governing factors for ductile crack propagation paths in sheet metals by the comprehensive investigation with experimental and numerical approaches. Using two types of sheet metals, a steel and an aluminium alloy, experiments with Arcan tests were performed, and the difference in the crack paths were observed and analysed. The strategy for simulating ductile fracture proposed by the authors was successfully validated with the experimental Arcan test results, and it was applied to a parametric study based on the numerical simulations for clarifying the governing factors for ductile crack propagation paths in sheet metals. The two factors, strain hardening ability and ductility diagram parameter, were confirmed to be dominant in controlling the ductile crack propagation. The findings in the parametric study explain the difference of the crack paths as observed in the experimental Arcan tests for the two materials, thus elucidating the inverse design philosophy that facilitates material design to achieve desired performance metrics in enhancing the integrity of engineering structures.
Characterizing springback stress behavior in VPB by experimental-numerical hybrid method
2022, International Journal of Mechanical Sciences
Springback is a stress-driven phenomenon during unloading, so the stress behavior during unloading is of great significance for exploring and understanding springback. However, the accuracy of stress field after loading procedure in FEM, as the basis of springback simulation, is greatly affected by the assumptions of material model and boundary condition. Moreover, the selection of constraints in springback simulation is crucial and highly subjective, thus affecting the authenticity of results. In this paper, the characteristic of springback stress is studied in viscous pressure bulging (VPB) using an experimental-numerical hybrid method, combining the DIC method and numerical calculation. The springback results under different viscous mediums are obtained and their laws are discussed. The measured springback data containing the information of material properties and boundary conditions are used to calculate springback stress by the presented numerical calculation program. The springback stresses of sections along rolling direction and transverse direction are obtained. The variation law of the magnitude and direction of springback stress vector with springback is revealed. Then the mechanism of springback stress variation and the springback reduction are analyzed. Results show that the springback stress conforming to objective and practical conditions can be obtained, and it changes regularly with springback. The viscous medium affects the shape and stress of sheet, causing the change of springback stress. Furthermore, the change of springback stress reduces the reverse bending moment to restrain the resulting springback.

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Outline 大纲

Cited by (14) 被引用 (14)

Figures (20) 图（20）

Tables (2) 表格（2）

Extras (1) 附加功能 (1)

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

Highlights 亮点

Abstract 抽象的

Graphical abstract 图解摘要

Keywords 关键词

1. Introduction 1. 简介

2. Models for post-neck strain hardening and ductile fracture
2. 后颈应变硬化和延性断裂模型

2.1. Model for post-necking hardening behaviour
2.1 颈缩后硬化行为模型

2.2. Model for ductile fracture
2.2 韧性断裂模型

3. Proposed strategy for parameters identification
3. 提出的参数识别策略

3.1. Post-necking strain hardening parameter identification

3.2. Ductility diagram parameters identification

4. Application of the proposed strategy

4.1. Test layouts and finite element models

4.2. Post-necking strain hardening parameter identification

4.3. Ductility diagram parameters identification

4.4. Validation of the proposed strategy

5. Conclusion

Funding sources

Declaration of Competing Interest

Acknowledgements

Appendix. Supplementary materials

References

Cited by (14)

A novel dual-stage failure criterion based on forming limit curve for uncured GLARE

Prediction of the irradiation effect on the fracture toughness for stainless steel using a stress-modified fracture strain model

Flow stress curves for 980MPa- and 1.5GPa-class ultra-high-strength steel sheets weakened under high-stress triaxiality

A novel deformation mode of expansion tubes accounting for extra contact

Ductile crack propagation path depending on material properties: Experimental results and discussions based on numerical simulations

Characterizing springback stress behavior in VPB by experimental-numerical hybrid method

Hybrid experimental-numerical strategy for efficiently and accurately identifying post-necking hardening and ductility diagram parameters高效、准确地识别颈缩后硬化和延展性图参数的混合实验-数值策略

Highlights 亮点

Abstract 抽象的

Graphical abstract 图解摘要

Keywords 关键词

1. Introduction 1. 简介

2. Models for post-neck strain hardening and ductile fracture2. 后颈应变硬化和延性断裂模型

2.1. Model for post-necking hardening behaviour2.1 颈缩后硬化行为模型

2.2. Model for ductile fracture2.2 韧性断裂模型

3. Proposed strategy for parameters identification3. 提出的参数识别策略

3.1. Post-necking strain hardening parameter identification

3.2. Ductility diagram parameters identification

4. Application of the proposed strategy

4.1. Test layouts and finite element models

4.2. Post-necking strain hardening parameter identification

4.3. Ductility diagram parameters identification

4.4. Validation of the proposed strategy

5. Conclusion

Funding sources

Declaration of Competing Interest

Acknowledgements

Appendix. Supplementary materials

References

Hybrid experimental-numerical strategy for efficiently and accurately identifying post-necking hardening and ductility diagram parameters
高效、准确地识别颈缩后硬化和延展性图参数的混合实验-数值策略

2. Models for post-neck strain hardening and ductile fracture
2. 后颈应变硬化和延性断裂模型

2.1. Model for post-necking hardening behaviour
2.1 颈缩后硬化行为模型

2.2. Model for ductile fracture
2.2 韧性断裂模型

3. Proposed strategy for parameters identification
3. 提出的参数识别策略