金属热黏塑性本构关系的研究进展

王强; 王建军; 张晓琼; 张天辉; 王怀坤; 吴桂英

doi:10.11883/bzycj-2021-0443

金属热黏塑性本构关系的研究进展

doi: 10.11883/bzycj-2021-0443

太原理工大学机械与运载工程学院，山西太原 030024

基金项目: 国家自然科学基金（11902272, 12172245）

详细信息

作者简介:
王　强（1995－　），男，博士研究生，wangqiang0004@link.tyut.edu.cn

通讯作者:
王建军（1987－　），男，博士，副研究员，wangjianjun@tyut.edu.cn

中图分类号: O347.3
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- 被引次数: 0
出版历程
- 收稿日期: 2021-10-28
- 修回日期: 2022-03-22
- 网络出版日期: 2022-03-29
- 刊出日期: 2022-09-29

Advances in the research of metallic thermo-viscoplastic constitutive relationships

College of Mechanical and Vehicle Engineering, Taiyuan University of Technology, Taiyuan 030024, Shanxi, China

摘要

摘要: 金属材料的塑性流动行为依赖于温度和应变率，温度和应变率敏感性是金属材料塑性流动的最重要的本质特性之一，建立合适的热黏塑性本构关系来准确描述金属塑性流动行为的温度和应变率依赖性，是金属材料能被广泛应用的必要前提。为此，对金属热黏塑性本构关系的最新研究进展进行了综述，介绍了常见的几种金属热黏塑性本构关系并进行了详细讨论，给出了各本构关系的优势与不足，最后系统介绍了包含金属塑性流动行为中出现的第三型应变时效、或K-W锁位错结构引起的流动应力随温度变化出现的反常应力峰以及拉压不对称等行为的金属热黏塑性本构关系的研究进展。
- 热黏塑性 /
- 本构关系 /
- 应变率效应 /
- 温度效应
Abstract: The studies of the plastic flow behaviour of metallic materials show that the plastic deformation process of metallic materials is dependent on temperature and strain rate, so the temperature and strain rate sensitivities are the most important essential properties of plastic deformation of metallic materials. It is therefore necessary to establish appropriate thermo-viscoplastic constitutive relations to accurately describe the temperature and strain rate dependences of the plastic flow behaviour of metals over a wide range of temperatures and strain rates. Advantages and disadvantages of these constitutive relationships are first reviewed in the present paper. With the increasing applications of metallic materials and the emergence of new materials, the 3rd type strain aging, K-W lock induced anomalous stress peak, and tensile-compression asymmetry are often observed in the plastic flow behaviour of metals. Due to the occurrence of those phenomena, the traditional metal thermo-viscoplastic constitutive relations may no longer be applicable. In view of the significant roles played by the 3rd type strain aging, K-W lock dislocation structure-induced anomalous stress peaks, and tensile-compression asymmetry in the plastic flow behaviour of metals, especially in high temperature loading, it is necessary to take those particular phenomena into account in the framework of the thermo-viscoplastic constitutive relationship of metals. Thus, a large variety of constitutive relation, which considers the interaction of strain, temperature and strain rate, has been established to predict the deformation behaviors of metals. In this context, this paper presents a systematic review of the thermo-viscoplastic constitutive relationships of metals, which includes the anomalous stress peaks in the flow stresses with temperature due to the 3rd type strain aging or K-W-locked dislocation structures, and the tensile-compression asymmetry. In addition, the forms of these thermo-viscoplastic constitutive relationship considering the 3rd type strain aging, K-W lock dislocation structure-induced anomalous stress peaks and tensile-compression asymmetry in the flow stress of metals, are discussed and analysed.
- thermal-viscosity plasticity /
- constitutive relationship /
- strain-rate effect /
- temperature effect

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高超声速武器是未来发展的重点方向之一，开展与其结构特点相适应的异型截面侵彻类战斗部研究，是当前高超声速武器发展与应用的强烈需求。椭圆类截面是区别于传统圆截面的一种弹体结构设计，能提高发射平台的空间利用率，与武器平台的空间适配具有更高的灵活性。由于其特殊的结构，在侵彻目标过程中，侵彻弹道的偏转与失稳特性、弹靶作用机制、靶体的破坏机制等，都与传统圆截面弹体存在极大差异。目前对于椭圆截面弹体侵彻多层间隔靶的弹道特性和偏转机制尚缺乏理论认知，限制了其发展和应用。

在传统圆截面弹体和异型类截面弹体侵彻典型目标的弹道特性方面已开展了大量的研究工作。在圆截面弹体研究方面，Li等^[1]、Gao等^[2]基于刚体动力学和微分面力法，计算了不同形状弹体侵彻深层介质的运动轨迹，发现弹体质心的相对位置（L_C/L，即弹尖到质心的距离与弹长之比）是影响弹体偏转和弹道稳定性的关键。薛建锋等^[3]通过对弹体斜侵彻过程的数值仿真和试验，得到不同工况下弹体偏转角、加速度随时间的变化情况，以及弹体速度和靶体倾斜角对侵彻深度的影响。姜剑生等^[4]、高旭东等^[5]研究了弹体质心位置、弹尾形状和弹体长细比对斜侵彻混凝土弹道偏转的影响，发现弹体质心越靠后，弹道稳定性越差；弹体长细比越大，弹道稳定性越好。杜华池等^[6-7]开展了卵形弹体和头部非对称刻槽弹体以不同入射角侵彻多层间隔混凝土靶和钢靶的试验，并利用数值仿真方法研究了弹体入射角、弹体速度、靶体厚度及弹体变形对侵彻弹道特性的影响规律。李鹏程等^[8]开展了尖卵形弹体侵彻间隔混凝土靶试验，研究了攻角与入射角联合作用对靶后偏转角以及弹体侵彻间隔靶板弹道轨迹的影响规律。Iqbal等^[9]、袁家俊^[10]开展了弹体斜撞击单层、多层、双层间隔靶数值模拟研究，发现倾角导致弹丸在贯穿多层靶时发生了明显的弹道偏转。Dong等^[11]基于弹靶分离方法，建立了等截面弹体、变截面弹体和变截面弹身刻槽弹体的运动轨迹预测模型，分析了部分弹形参数和着靶条件对后2类特殊弹体弹道稳定性的影响。在椭圆截面弹体研究方面，Wu等^[12]基于刚性动力学和微分面力法建立了椭圆截面弹体侵彻半无限混凝土靶的弹道计算模型，发现椭圆截面弹体在长轴方向抗偏斜能力较好，在短轴方向抗偏斜能力较弱。魏海洋等^[13]、Wei等^[14]开展了圆截面弹体和椭圆截面弹体侵彻铝合金靶体试验与理论研究，基于空腔膨胀理论及局部相互作用模型，建立了弹体在靶中的运动轨迹模型，并在此基础上进一步分析了椭圆截面弹体长短轴之比、滚转角、弹体撞击速度等对侵彻弹道的影响规律。岳胜哲等^[15]开展了5种非对称椭圆截面弹体斜贯穿铝靶的数值模拟研究，发现弹体偏转角度、角速度均随不对称度的增大而增大。胡雪垚等^[16]利用有限元软件分析了D字形弹体侵彻3层间隔钢靶的弹道偏转规律，发现减小着角或增大侵彻速度均能提高弹体的弹道稳定性。田泽等^[17]通过分析钢靶的破坏形式，结合虚功原理和能量守恒定律建立了变截面椭圆截面弹体侵彻双层间隔钢靶姿态偏转理论模型。邓希旻等^[18]开展了上下非对称结构异型弹体正/斜贯穿多层间隔921A钢靶试验和数值模拟研究，发现上下非对称结构异型弹体具有维持弹体姿态稳定的特点。魏海洋^[19]建立了椭圆截面弹体斜侵彻多层间隔金属靶侵彻弹道模型，分析了倾角、撞击速度及靶板间距对椭圆截面弹体侵彻弹道的影响规律。

综上所述，当前针对椭圆类截面弹体侵彻多层间隔钢靶的弹道特性研究，仅对弹体着角、速度等部分条件展开，缺乏对多种影响因素的系统性研究。同时，现有研究靶板层数较少，对弹体侵彻目标过程中的弹靶分离效应和非正侵彻后期出现的大攻角姿态偏转研究较少，缺乏足够多的试验研究为其理论模型、高效数值模拟方法的发展提供支撑。

为深入研究椭圆类截面弹体在不同弹靶条件下侵彻多层间隔钢靶的弹道特性和偏转规律，本文中开展4种截面弹体在不同弹靶条件下侵彻4层间隔Q355B钢靶试验，利用LS-DYNA软件建立椭圆类截面弹体侵彻多层间隔钢靶的有限元仿真模型，研究多种因素对椭圆类截面弹体弹道偏转规律的影响。

1. 试验研究

1.1 弹体设计

设计了4种不同截面弹体，如图1所示，包括圆截面弹体（C1）、对称椭圆截面弹体（E1）和两种非对称椭圆截面弹体（NE1和NE2）。弹体材料为35CrMnSiA，热处理后硬度HRC为50～52，弹体上装配有尼龙前托和铝合金+尼龙底推实现发射助推。

图 1 试验用弹体（单位：mm）

Figure 1. Projectiles used in experiments （unit: mm）

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弹体主要尺寸参数见表1。其中λ为椭圆截面压缩系数（短长轴之比），a、b分别为椭圆长、短半轴长，CRH为弹体长轴所在平面的头部曲径比，m为弹体质量。

表 1 4种弹体的主要几何参数

Table 1. Main geometry parameters of the four projectiles

弹体类型	λ	a/mm	b/mm	CRH	m/g
C1	1.0	12.5	12.5	4.19	389
E1	0.7	15.0	10.5	3.00	389
NE1	0.8/0.6	15.0	12.0/9.0	3.00	389
NE2	0.9/0.5	15.0	13.5/7.5	3.00	389

下载: 导出CSV

| 显示表格

1.2 试验布置

试验现场布置如图2所示，共设计了0°、15°、30°等3组不同角度的靶架，安装有4层靶板，靶板材料使用Q355B合金钢，尺寸为375 mm×375 mm，厚度为2 mm，相邻两层靶板的垂直间隔为400 mm。采用高速摄像机记录弹体侵彻过程中的姿态和偏移，同时使用5个平面镜分别放置于每块钢靶的靶前和靶后弹体运动轨迹的正上方，与水平面呈45°夹角，通过测量弹体在入射平面（即竖直平面）和水平面的姿态和位置变化，得到弹体的剩余速度、运动轨迹和姿态偏转。

图 2 试验现场布置

Figure 2. Experimental setup

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1.3 试验结果

为了获取弹体截面形状、撞击速度、着角和滚转角对侵彻弹道特性和弹体姿态偏转的影响规律，共开展了10发试验，其中C1弹体1发，E1弹体7发，NE1弹体1发，NE2弹体1发。

图3(a)是弹体侵彻条件示意图，图中v为弹体速度，箭头指向为弹体速度方向，弹体速度方向与靶板法向的夹角为着角α，弹体速度方向与弹体轴线方向的夹角为攻角β，弹体轴线方向与靶板法向的夹角为姿态角ϕ。定义弹体轴线方向转到靶板法向为顺时针时姿态角为正，弹体轴线方向转到靶板法向为逆时针时姿态角为负；定义弹体轴线转到弹体速度方向为顺时针时攻角为正，弹体轴线转到弹体速度方向为逆时针时攻角为负，如图3(b)所示。滚转角γ是指弹体绕自身弹轴旋转的角度，对于对称椭圆截面弹体，定义短轴沿铅垂方向时为0°，从弹底向前看，顺时针旋转为正，如图4(a)所示；对于非对称椭圆截面弹体，定义短轴较长的半椭圆在正上方时为0°，从弹底向前看，顺时针旋转时为正，如图4(b)所示。

图 3 弹体侵彻条件示意图

Figure 3. Schematic representation of conditions of projectile penetration

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图 4 滚转角示意图

Figure 4. Schematic representation of rotation angle

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图5为不同初始条件下弹体贯穿多层间隔钢靶的试验结果。其中下标0、1、2、3、4分别代表弹体入靶前、贯穿第1、2、3、4层靶板后的弹道参数，图中弹体下方的标注为弹轴线与水平线的夹角，弹体低头时为负值，抬头时为正值。

图 5 不同初始条件下的侵彻弹道偏转

Figure 5. Penetration trajectory deflection under different initial conditions

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试验弹体速度在610.5～1000 m/s范围，斜侵彻多层间隔钢靶过程中逐渐向下偏转，攻角和偏转角均呈逐渐增大趋势，弹体贯穿第3、4层靶板后的姿态偏转显著。由于靶板偏转力矩作用，导致弹体获得了旋转角速度，边旋转边前进，如图5(i)所示，NE1弹体穿过第4层钢靶后偏转角仍在增大。

试验弹体侵彻每一层钢靶前后的弹体姿态参数见表2，“—”代表未观测到弹体姿态。

表 2 弹体侵彻每层钢靶前后的弹道参数

Table 2. Trajectory parameters of the projectile before and after penetrating each steel plate

试验编号	v₀/ (m·s⁻¹)	β₀/ (°)	β₁/ (°)	β₂/ (°)	β₃/ (°)	β₄/ (°)	ϕ₀/(°)		ϕ₁/(°)		ϕ₂/(°)		ϕ₃/(°)		ϕ₄/(°)
试验编号	v₀/ (m·s⁻¹)	β₀/ (°)	β₁/ (°)	β₂/ (°)	β₃/ (°)	β₄/ (°)	水平面	入射平面	水平面	入射平面	水平面	入射平面	水平面	入射平面	水平面	入射平面
E1-1	809.1	1.6	—	2.5	—	10.6	1.8	2.0	—	—	0.9	4.0	—	—	11.5	13.4
E1-2	790.6	2.4	−2.8	−3.3	−9.3	−22.8	0	16.1	0	14.8	0.9	11.9	3.0	−2.0	4.3	−19.4
E1-3	610.5	0	−0.8	−7.9	−20.6	−44.2	−0.9	30	−2.5	27	—	14.0	—	−1.1	−6.5	−34.3
E1-4	799.7	1.5	−1.1	−5.4	−13.8	−31.5	0	31.5	0	28.7	0	21.2	0	7.3	0	−7.9
E1-5	1000.0	0.9	−0.4	−4.3	—	−14.4	0.2	30.9	1.0	30.0	1.9	27.5	—	—	4.0	12.5
E1-6	807.3	−0.8	−1.6	−6.3	−10.2	−24.6	−1.5	29.2	−1.3	27.2	0	23.9	0	16.9	4.8	2.1
E1-7	802.4	0	−0.2	−0.3	−0.3	−3.7	0	30.0	0	29.0	0	27.9	0	27.6	0	25.6
C1-1	796.3	0	0	−0.3	−2.4	−10.4	0	30.0	0	30.0	0	28.8	0	25.9	0	16.9
NE1-1	807.2	0.5	—	—	—	−46.7	—	29.5	—	—	—	—	—	−2.1	—	−24.9
NE2-1	798.7	0	−2.9	−12.6	−33.3	−53.2	5.3	30.0	6.6	26.4	9.8	9.1	22.4	−9.0	51.3	−38.3

下载: 导出CSV

| 显示表格

从高速录像照片和弹道参数可以看出，在[0°，30°]范围内，着角越大，弹体侵彻弹道稳定性越差。观察着角为0°时，E1弹体的侵彻弹道轨迹如图5(a)所示，虽然弹体姿态在复杂因素影响下出现了偏转，但质心运动轨迹基本呈水平直线，说明弹体正侵彻多层间隔钢靶时，弹体质心几乎不会发生偏移。

对比试验E1-3、E1-4、E1-5可以看出，随着弹体初速的增大，弹体姿态偏转越来越小，侵彻弹道更加平稳。当初速增加时，弹靶接触时间缩短，偏转力矩作用时间减少，弹体侵靶姿态偏转幅度减小。弹体贯穿后在入射平面内的姿态角分别为−7.9°、−34.3°和12.5°，相邻速度间的姿态角差值分别为26.4°和20.4°，表明速度对偏转角的影响幅度随速度增加呈减小趋势。

当椭圆截面弹体以不同滚转角侵彻多层间隔钢靶时，滚转角越大，弹体在入射平面内的姿态偏转越小。如表2中E1-7所示，当滚转角为90°时，弹体贯穿4层钢靶后在入射平面内的Δϕ只有4.4°，滚转角的增大抑制了弹体在入射平面内偏转角的增长。当滚转角为45°时，可以看到，弹体在水平面的姿态角由负变正，如表2中E1-6所示，此时弹体受到了靶板横向偏转力矩的作用，说明滚转角介于0°～90°之间时，偏转角会出现小幅度的水平分量，但弹体偏转仍主要发生在入射平面内。

对比图5(c)、(h)～(j)中弹体侵彻弹道结果，当滚转角为0°时，截面压缩比减小和非对称度的增大均会放大弹体侵彻多层间隔钢靶的姿态偏转和弹道的不稳定，不同截面弹体侵彻弹道稳定性由大到小依次为：圆截面、对称椭圆截面、非对称椭圆截面。

图6为各层靶板的典型破坏形式，当椭圆截面弹体正侵彻单层钢靶时，在靶板弹孔与弹体长轴两端和短轴两端的接触位置产生4条张开型裂纹，每2条裂纹间形成1个花瓣，因此，靶板背面形成了4个主花瓣，每个主花瓣尖端又开裂成2～3个小花瓣，但这些小花瓣对弹体几乎没有力的作用；当弹体以较正的姿态斜侵彻前2层钢靶时，靶板出现非对称破坏，弹体上方的靶板材料受弹体挤压向前翘曲，在靶板正面形成1个前花瓣，背面形成2个小花瓣和开孔下侧的1个大花瓣；弹体侵彻到第3~4层靶板时，由于发生较大姿态偏转，靶板在弹体撞击作用下被撕裂，形成一个长条孔，并在靶板背面形成一圈环向花瓣。

图 6 靶板典型破坏形式

Figure 6. Typical destruction types of targets

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图7为试验后回收到的部分试验弹体，可以看出，弹体斜侵彻多层间隔钢靶后，除发生少量磨蚀以及弹尖被削去2～3 mm外，弹头部和弹身基本没有发生明显变形。回收后的弹体质量损失约为0.51%，因此，在本试验范围内弹体可视为刚体。

图 7 回收到的4发弹体（左1～4）和原始弹体（右1）

Figure 7. The recovered four residual projectiles (1st to 4th from left) and original projectile (1st from right)

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从回收到的弹体磨蚀情况来看，弹体头部表层有少量磨蚀痕迹，而弹身只有一面出现了相对明显的划痕和磨蚀痕迹，另一面较为光滑，接近原始状态。说明弹体在贯穿钢靶过程中，并未一直与靶体接触，弹体以较正姿态侵彻时，与靶板的接触主要发生在弹头侵彻阶段，弹头部完全贯穿或未完全贯穿靶板时，就已经与靶体分离，这与Paul等^[21]的研究结论一致，如图8所示。随着弹体向下偏转，弹体逐渐以倾斜姿态撞上靶板，使得弹体上表面逐渐与靶板接触，下表面未与靶板接触。

图 8 弹体磨蚀示意图

Figure 8. Schematic representation of projectile erosion

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2. 仿真模型及验证

2.1 仿真模型及材料参数

采用LS-DYNA有限元软件对侵彻过程进行模拟，使用1号拉格朗日算法；弹体与靶板接触区的网格尺寸均为0.75 mm左右，并对靶板侵彻区网格进行加密处理，非侵彻区网格进行稀疏处理，以节省处理器计算时间，弹靶网格模型如图9所示；弹靶接触方式为侵蚀面面接触，忽略弹靶间的摩擦作用；考虑到靶板失效以及弹体偏转的非对称性，采用三维全模型计算。

图 9 有限元模型示意图

Figure 9. Finite element model

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弹体材料采用MAT_RIGID刚体模型，钢靶采用MAT_JOHNSON_COOK材料模型和EOS_GRÜNEISEN状态方程，具体材料参数见表3，其中ρ、E和µ分别为材料的密度、弹性模量和泊松比，A、B、n、c、m分别为参考应变率下的屈服强度、应变硬化系数、应变硬化指数、应变率强化系数和温度软化指数，D₁～D₄为材料的损伤参数。

表 3 材料参数

Table 3. Material parameters

材料	ρ/(g·cm⁻³)	E/GPa	μ	A/MPa	B/MPa	n	c	m	D₁	D₂	D₃	D₄
35CrMnSiA	7.85	210	0.30
Q355B^[20]	7.85	210	0.28	339.5	620.0	0.403	0.02	0	0.820	6.047	−7.09	−0.003

下载: 导出CSV

| 显示表格

2.2 仿真模型验证

从弹体剩余速度、入射平面内的姿态角与攻角变化、侵彻弹道轨迹3个方面验证有限元仿真模型的有效性。弹体穿过各层靶板后剩余速度的试验与仿真结果如表4所示。从表4中可以看出，试验与仿真结果的误差最高为2.19%，说明在弹体速度衰减上仿真结果与试验结果比较接近。

表 4 部分典型工况弹体剩余速度试验与仿真结果的对比

Table 4. Comparison between simulations and experiments for residual velocity of projectiles

试验编号	靶板层数	剩余速度/（m·s⁻¹）		误差/%
试验编号	靶板层数	试验结果	仿真结果	误差/%
E1-4	1	794.2	795.4	0.15
	2	782.5	789.2	0.86
	3	762.7	779.2	2.16
	4	737.9	754.1	2.19
E1-6	1	801.0	802.2	0.14
	2	789.9	794.5	0.58
	3	780.8	783.1	0.29
	4	764.6	760.1	−0.59
NE2-1	1	792.7	793.6	0.11
	2	783.4	783.6	0.02
	3	751.5	759.5	1.06
	4	681.4	689.9	1.24

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| 显示表格

弹体在入射平面内的姿态角、攻角变化试验与仿真结果如图10所示，试验与仿真结果呈现出相同的变化趋势。试验结果选取了500、1000、1500和2000 μs等4个时间点的数据和仿真结果曲线进行对比，姿态角绝对误差为0～9.7°，攻角绝对误差为0.9°～5.8°，均在10°以内。

图 10 姿态角、攻角变化试验与仿真结果的对比

Figure 10. Comparison between simulations and experiments for trajectory angle and attack angle

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图11为弹体运动轨迹试验与仿真结果的对比。弹体斜贯穿多层间隔钢靶后整体弹道向下偏转，试验结果与仿真结果一致性较好。

图 11 侵彻弹道试验与仿真结果的对比

Figure 11. Comparison between simulations and experiments for penetration trajectory

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3. 椭圆类截面弹体侵彻多层间隔钢靶弹道特性

3.1 偏转机制

弹体偏转角加速度曲线如图12所示，侵彻过程可以分为4个阶段，各阶段均主要由一段正值曲线和负值曲线组成，且随着侵彻推进曲线峰值逐渐增大。图13是弹体侵彻单层钢靶过程的受力分析，假设靶体作用力沿弹体表面法向。分析角加速度变化趋势和曲线不同位置点对应的侵彻时刻，可以总结出弹体的4种偏转模式。

图 12 弹体偏转角加速度曲线和对应侵彻过程

Figure 12. Rotation acceleration curve and corresponding penetration process

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图 13 侵彻过程中的4种偏转模式

Figure 13. Four deflection modes in penetration process

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（1）模式Ⅰ。弹体斜侵彻靶板初始阶段，弹头下表面先与靶板接触，如图13(a)所示，靶体偏转力作用位置在质心前面，偏转力矩方向为逆时针，因此，角加速度先负向增长，持续时间约为15 μs；当弹体侵彻第3、4层靶板时，由于偏转角的产生，使弹头上表面先与靶板接触，此时在角加速度曲线中没有出现负值阶段（弹体偏转角度较小时仍会出现）。

（2）模式Ⅱ。在经过短暂的负值阶段后，弹头部上表面开始接触靶板，偏转力矩M₂>M₃，如图13(b)所示，曲线迅速正向增长变为正值，在弹头部未穿出靶板时曲线到达峰值。

（3）模式Ⅲ。当弹体质心运动到靶板位置时，角加速度下降为零，且从图13(c)中可以看出，弹体下表面已经与靶板分离。

（4）模式Ⅳ。如图13(d)所示，在弹体贯穿钢板的后期，靶体力作用位置来到质心后面，并对弹体偏转产生了修正作用，偏转力矩方向为逆时针，角加速度开始负向增长，在接近弹底位置时到达峰值，随后弹体穿过靶板，角加速度减小到零。

弹体侵彻第2、3、4层靶板过程类似，随着偏转角的增大，与靶板的接触面积增大，因此偏转力矩增大，角加速度曲线峰值变高。弹体偏转是由于不同位置和不同时间的靶体偏转力共同作用的结果，斜侵彻靶板过程中弹体上表面是主要受力面，质心到弹尖部分穿靶阶段对弹体偏转有促进作用，质心到弹尾部分穿靶阶段对弹体偏转有抑制作用，整个侵彻过程的合力矩使弹体沿顺时针方向旋转。

3.2 截面压缩系数对弹体姿态偏转和侵彻弹道特性的影响

为研究椭圆截面弹体截面压缩系数对侵彻弹道特性的影响，在E1弹体基础上设计了质量、弹长、截面面积相同，压缩系数分别为0.9、0.8和0.6的3种弹体，研究在0°滚转角、30°着角、800 m/s初速下，弹体姿态偏转和侵彻弹道轨迹的变化规律，如图14所示。

图 14 不同截面压缩比下弹体侵彻弹道的仿真结果

Figure 14. Penetration trajectory simulation results of projectiles under different λ

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图15(a)为不同截面压缩系数弹体在入射平面内的偏转角速度时程曲线，其中角速度方向为x轴正方向。分析图中角速度变化趋势可知：(1) 不同截面压缩系数下，弹体偏转角速度随时间变化的趋势一致，但曲线的幅值随着压缩系数的增大逐渐减小；(2) 弹体侵彻各层钢靶的偏转角速度均由一段上升曲线和下降曲线组成，表现出角速度先正向增大后减小的变化模式。图15(b)为偏转角加速度时程曲线，与图15(a)呈现出了相似的变化规律。各阶段曲线正值部分对应角速度的上升，负值部分对应角速度的下降。随着截面压缩系数的增大，弹体在纵对称面内的转动惯量增大，曲线幅值逐渐减小。

图 15 不同截面压缩系数弹体偏转角速度和角加速度时程曲线

Figure 15. Time history curves of angular velocity and angular acceleration under different λ

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不同压缩系数弹体在入射平面内的攻角、姿态角时程曲线如图16所示。从图中可以看出，弹体攻角和姿态角在整个侵彻过程中的变化趋势一致。当λ=0.9时，攻角和姿态角变化趋势最平缓，随着截面压缩系数减小，曲线变化幅度逐渐增大，弹体姿态偏转越来越显著。图17为弹体质心运动轨迹，可以看出，随着压缩系数减小，弹体侵彻多层间隔钢靶弹道偏移逐渐增大，侵彻弹道稳定性逐渐下降。

图 16 不同截面压缩系数弹体攻角和姿态角时程曲线

Figure 16. Time history curves of attack angle and trajectory angle under different λ

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图 17 不同截面压缩系数弹体侵彻弹道轨迹

Figure 17. Penetration trajectories under different λ

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3.3 滚转角对弹体姿态偏转和侵彻弹道特性的影响

为研究滚转角γ对侵彻弹道特性的影响规律，以E1和NE1弹体为对象，分别计算了γ在0°～90°、0°～180°范围内弹体的侵彻弹道结果，如图18～19所示，分析弹体在0°攻角、30°着角和800 m/s初速下弹体姿态偏转和侵彻弹道轨迹。

图 18 不同滚转角下E1弹体侵彻弹道仿真结果

Figure 18. Penetration trajectory simulation results of E1 projectile under different rotation angles

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图 19 不同滚转角下NE1弹体侵彻弹道仿真结果

Figure 19. Penetration trajectory simulation results of NE1 projectile under different rotation angles

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对于对称椭圆截面弹体，随着滚转角增大，弹体在纵向的转动惯量越大，抵抗旋转运动的能力越强，弹体姿态更稳定，姿态角和攻角变化幅度均逐渐减小；对于非对称椭圆截面弹体，短轴长的一半与靶板接触面积大，短轴短的一半与靶板接触面积小，当滚转角大于90°后，短轴更长的一半转到下方，下半部分弹体的受力增大，由图13(b)偏转模式Ⅱ可知，对弹道偏转起到了抑制作用。

图20为2种弹体侵彻过程中的攻角变化曲线。弹体在靶板不对称作用力的影响下逐渐向下“低头”，导致轴线方向与速度方向间的夹角逐渐增大。E1和NE1弹体分别在滚转角为45°和90°时，攻角增长幅度最小，随着滚转角的继续增大，弹体攻角变化幅度又逐渐增大。

图 20 不同滚转角下E1和NE1弹体攻角变化

Figure 20. Changes in the attack angle of E1 and NE1 projectiles under different rotation angles

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如图21所示，对于E1弹体，当γ=0°、75°、90°时，弹体在水平面的姿态最稳定，姿态角都在0°附近波动；γ=15°、30°、45°、60°时，弹体在水平面内出现小幅度的姿态偏转，在γ=45°时偏转角最大，但不超过3°。弹体逐渐向下偏斜，弹轴线与靶板法线间的夹角逐渐减小；当γ=0°、15°、30°时，弹体轴线在入射平面内与靶板法线平齐后继续偏转，因此，姿态角变化曲线又出现了负增长趋势。总体来看，弹体在入射平面内的姿态角与滚转角呈正相关，滚转角越大，弹体姿态越稳定。

图 21 不同滚转角下E1和NE1弹体在水平面和入射平面内的姿态角变化

Figure 21. Changes in the trajectory angle of E1 and NE1 projectiles under different rotation angles

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对于NE1弹体，当γ>90°时，弹体在水平面的姿态最稳定，姿态角都在0°附近波动；0°≤γ≤90°时，弹体在水平面内姿态偏斜更严重，在γ=60°时弹体偏转角达到最大。弹体在入射平面内的姿态角变化幅度随滚转角的增大呈先减小后增大趋势，在滚转角为90°时，侵彻姿态最稳定。

从图22中可知，E1弹体侵彻弹道稳定性随滚转角的增大而增强，弹体在y方向上终点偏移量随滚转角的增大而减小；当0°<γ<90°时，x方向上出现了位移分量，当γ=45°时，弹体在x方向的终点偏移量最大。从图23中可知，NE1弹体在水平面内的弹道偏转随滚转角的增大先增大后减小，在γ=90°时横向偏转达到最大；在入射平面内侵彻弹道稳定性随滚转角的增大先上升后下降。因此，当非对称椭圆截面弹体滚转角为钝角时，其侵彻弹道的稳定性要优于滚转角为锐角的情况。

图 22 不同滚转角下E1弹体在2个平面的侵彻弹道轨迹

Figure 22. Penetration trajectories of E1 projectile in different rotation angles (horizontal plane and incidence plane)

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图 23 不同滚转角下NE1弹体在2个平面内的弹道轨迹

Figure 23. Penetration trajectories of E1 projectile in different rotation angles (horizontal plane and incidence plane)

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3.4 着角对弹体姿态偏转和侵彻弹道特性的影响

为研究着角α对椭圆类截面弹体侵彻弹道特性的影响，计算了α在0°～50°范围内的侵彻弹道结果，如图24所示。

图 24 不同着角下E1弹体侵彻弹道仿真结果

Figure 24. Penetration trajectory simulation results of E1 projectile with different incident angles

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弹体在不同着角下侵彻4层间隔钢靶的攻角、姿态角时程曲线如图25所示，为便于分析比较，不同着角下弹体姿态角变化曲线的起点均放在图中坐标系的同一位置。从曲线中可以明显看出，随着着角的增大，弹体攻角和姿态角变化幅度呈先增长后下降的趋势。在以上6个计算工况中，α=0°、10°时，弹体姿态一直保持稳定，α>20°后开始出现明显偏转，α=30°时弹体姿态偏转最严重，α>30°后弹体姿态偏转程度又逐渐减小。

图 25 不同着角下E1弹体攻角和姿态角时程曲线

Figure 25. The time history curves of attack angle and trajectory angle at different incident angles

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为了分析着角对弹体运动轨迹的影响，结合弹体受力情况对E1弹体弹道偏转机制进行分析。图26是弹体不同着角侵彻过程中在竖直方向上受到的合力曲线，y轴正方向竖直向上。y轴负向各阶段曲线主要由左右2个相连的波形组成，左波对应质心到弹尖侵彻阶段，右波对应质心到弹底侵彻阶段。α=0°时，由于弹体受力对称，曲线基本都在零值附近波动；随着着角的增大，弹体受到不对称力的作用开始偏转并挤压上侧靶体，下侧花瓣在惯性作用下逐渐与弹体分离，因此，弹体受力主要沿y轴负方向；α>30°时，弹体下侧与靶板的接触面积增大，受到靶板向上的力也逐渐增大，且从图中可以看出，侵彻初期靶体作用力逐渐增大，因此，合力在y轴负方向的增量减小，弹道向下偏转越来越小。弹体在不同着角下侵彻多层间隔钢靶的弹道轨迹如图27所示，弹体在y方向上的终点偏移量随着角的增大先增大后减小，侵彻弹道稳定性随着角的增大先减弱后增强。当α=30°时，椭圆类截面弹体侵彻弹道偏转最严重，弹道稳定性最差。

图 26 弹体在y方向上的合力

Figure 26. Resultant force of projectile in y direction

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图 27 不同着角下的侵彻弹道轨迹

Figure 27. Penetration trajectories under different incident angles

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4. 结　论

开展了椭圆类截面弹体侵彻多层间隔钢靶试验和仿真研究，分析了弹体在侵彻过程中的弹道偏转机制和不同弹靶参数对侵彻弹道特性和姿态偏转特性的影响，得到的主要结论如下。

(1) 当滚转角为0°时，弹道稳定性由强到弱依次为圆截面弹体、对称椭圆截面弹体、非对称椭圆截面弹体，且椭圆截面压缩系数与侵彻弹道稳定性呈正相关，压缩系数越大，侵彻弹道稳定性越好。

(2) 弹体初速是影响侵彻弹道稳定性的重要因素，随着初速增大，侵彻弹道越来越稳定，姿态偏转也越来越小。

(3) 对于椭圆截面弹体，随着滚转角的增大，弹体在入射平面内的姿态偏转逐渐减小，侵彻弹道越来越稳定。当滚转角不为0°时，弹体在水平面内会出现横向偏转，且滚转角为45°时，横向偏转最大。对于非对称椭圆截面弹体，入射平面内的姿态偏转随滚转角增大呈先减小后增大的趋势，90°滚转角时姿态最稳。当滚转角为钝角时，其弹体侵彻弹道稳定性优于锐角时的情况。

(4) 在本文弹靶条件下，椭圆类截面弹体侵彻弹道稳定性随着角的增大出现先减弱后增强的规律，在30°着角时弹道偏转最大，并不是着角越大稳定性越差。

(5) 弹体以较正姿态正/斜侵彻薄钢靶时，与靶板的接触主要发生在弹头侵彻阶段；弹体以倾斜姿态侵彻靶板时，与靶板接触主要在弹体上表面，下表面在弹体未完全贯穿靶板时就已与靶体分离，此时靶体对弹体的阻力作用变为零。

图 1 不同应变率下93W-Ni-Fe的流动应力随温度的变化曲线^[4]

Figure 1. Flow stress-temperature curves of 93W-Ni-Fe at different strain rates^[4]

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图 2 不同应变率下，流动应力随温度变化的模型预测结果与试验结果比较^[23]

Figure 2. Comparison of model predictions and experimental results of flow stress variation with temperature at different strain rates^[23]

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图 3 试验应力-应变曲线与修正的J-C本构关系预测结果的比较^[27]

Figure 3. Comparison betwwen experimental stress-strain curves with MJC model predictions^[27]

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图 4 不同应变率下，Ti-6Al-4V(296 K)的准静态和动态加载试验结果与K-H-L和J-C本构关系预测结果的比较^[13]

Figure 4. Quasi-static and dynamic loading experimental results of Ti-6Al-4V (at a temperature of 296 K) for different strain rates with correlations using K-H-L and J-C models^[13]

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图 5 不同晶粒尺寸纳米晶铝在不同应变率下的流动应力-应变试验结果与K-H-L本构关系预测结果^[38]

Figure 5. Observed and calculated responses for nanocrystalline aluminium at different strain rates by using KHL model for various grain sizes^[38]

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图 6 不同温度和应变率下铜的真实应力-真实应变曲线与预测结果的对比^[41]

Figure 6. Comparison of true stress - true strain curves and model predictions for copper at different temperatures and strain rates^[41]

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图 7 30CrMnSiA热软化情况的理论和试验值的比较^[44]

Figure 7. Comparison between theoretical and experimental values on thermal softening of 30CrMnSiA^[44]

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图 8 Sn60Pb40合金的计算结果与试验结果的比较^[45]

Figure 8. Comparison of calculated results and experimental results for Sn60Pb40 alloy^[45]

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图 9 不同应变率和温度下，退火OFHC铜的N-N-L本构关系预测结果与试验结果的比较^[57]

Figure 9. Comparison of model prediction predictions with experimental results for annealed OFHC copperat different strain rates and temperatures using N-N-L constitutive model^[57]

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图 10 不同应变率下N-N-L模型预测与试验结果的比较^[58]

Figure 10. Comparison between N-N-L model predictions with experimental results at different strain rate^[58]

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图 11 用M-R-K本构关系描述的流动应力随塑性应变的变化^[65]

Figure 11. Change of the flow stress with plastic strain described using the M-R-K model^[65]

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图 12 DH36钢流动应力随温度变化曲线上出现的反常应力峰^[71]

Figure 12. Anomalous stress peaks in the flow stress curves of DH36 steel with temperature^[71]

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图 13 镍基高温合金的拉伸强度随温度的变化^[97-98]

Figure 13. Tensile strength of nickel base superalloy as a function of temperature^[97-98]

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图 14 高温合金K403在不同应变率下的流动应力随温度的变化

Figure 14. Flow stress of superalloy K403 as function of temperature at different strain rates

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图 15 考虑拉压不对称行为的金属热黏塑性本构关系预测结果与试验结果的对比

Figure 15. Comparison between the predicted and experimental results of thermo-viscoplastic constitutive relationships of metal considering the asymmetrical behavior of tension and compression

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表 1 唯象型本构关系的模型对比

Table 1. Comparison of phenomenological constitutive relations

本构关系	年份	模型	应变率范围/s⁻¹	主要特征
J-C	1983	Johnson-Cook^[20]	>10⁴	（1）形式简单，材料常数容易获得（2）考虑了应变率效应（3）考虑了温度效应（4）不能准确描述流动应力随对数应变率呈非线性关系的金属材料的塑性流动行为的应变率敏感性（5）不能描述对于金属塑性流动行为中的应变、应变率和温度对流动应力的影响为非独立的现象
	1991	Holmquist^[24]	10⁻³～500	（1）应变率项修正为幂函数（2）不能描述对于金属塑性流动行为中的应变、应变率和温度对流动应力的影响为非独立的现象（3）对高应变率范围内的应变率敏感性的加强描述依旧有限
	1998	Rule-Jones^[25]	>10³	（1）高速率情况下，可以更好地描述应变率敏感性的作用（2）不能描述对于金属塑性流动行为中的应变、应变率和温度对流动应力的影响为非独立的现象
	1999	Kang-Cho^[26]	10⁻³～5000	（1）应变率项中引入了对数应变率的二次型项（2）不能描述对于金属塑性流动行为中的应变、应变率和温度对流动应力的影响为非独立的现象（3）对高应变率范围内的应变率敏感性的加强描述依旧有限
	2009	Vural-Cairo^[27]	10⁻²～10⁴	（1）在应变和应变率项中加入了温度效应（2）可以描述对于金属塑性流动行为中的应变、应变率和温度对流动应力的影响为非独立的现象
	2010	Lin-Xia^[28]	10⁻²～10	（1）考虑温度和应变率耦合效应（2）应变项为抛物线形式
K-H	1992	Khan-Huang^[16]	10⁻⁵～10⁴	（1）考虑了应变率效应（2）假定依赖于J₂不变量（3）没有考虑温度效应
	2009	Yu-Guo^[35]	10⁻⁴～1.6×10³	没有考虑温度效应
	1999	Khan-Liang^[36]	10⁻⁶～10⁴	（1）考虑了应变率对应变硬化的影响。（2）应变率项为幂函数形式
	2000	Khan-Zhang^[37]	10⁻⁴～10³	（1）考虑了晶粒尺寸对金属流动应力的影响（2）考虑了应变率对应变硬化的影响（3）应变率项为幂函数形式
	2004	Farrokh-Khan^[38]	10⁻⁴～10³	（1）可以描述晶粒细化引起的不同多晶金属的塑性流动行为（2）考虑了应变率对应变硬化的影响（3）应变率项为幂函数形式
其他	1976	Voce-Kocks^[39-40]	10	考虑温度和应变率对饱和应力 ${\sigma _{\rm{s}}}$ 的影响
其他	2005	Molinari-Ravichandran^[41]	10⁻³～8.5×10⁴	考虑了微观结构的演化

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表 2 唯象型本构关系的方程形式

Table 2. Equations of phenomenological constitutive relationships relations

模型	方程形式
Johnson-Cook^[20]	$\sigma = \left( {A + B{\varepsilon ^n}} \right)\left( {1 + C\ln {{\dot \varepsilon }^ * }} \right)\left( {1 - {T^{ * m}}} \right)$
Holmquist^[24]	$\sigma = \left( {A + B{\varepsilon ^n}} \right)\left( {{{\dot \varepsilon }^{ * C}}} \right)\left( {1 - {T^{ * m}}} \right)$
Rule-Jones^[25]	$\sigma = \left( {A + B{\varepsilon ^n}} \right)\left[ {1 + C\ln {{\dot \varepsilon }^ * } + {C_4}\left( {\dfrac{1}{{{C_5} - \ln {{\dot \varepsilon }^ * }}} - \dfrac{1}{{{C_5}}}} \right)} \right]\left( {1 - {T^{ * m}}} \right)$
Kang-Cho^[26]	$\sigma = \left( {A + B{\varepsilon ^n}} \right)\left[ {1 + {C_1}\ln {{\dot \varepsilon }^ * } + {C_2}{{\left( {\ln {{\dot \varepsilon }^ * }} \right)}^2}} \right]\left( {1 - {T^{ * m}}} \right)$
Vural-Cairo^[27]	$\sigma = \left\{ {A + {B_0}\left[ {1 - {{\left( {\dfrac{{T - {T_0}}}{{{T_{\text{m}}} - {T_0}}}} \right)}^p}} \right]{\varepsilon ^n}} \right\}\left[ {1 + \left( {{c_1}T_{\text{r}}^{ * p} + {c_2}H} \right)\ln \left( {\dfrac{{\dot \varepsilon }}{{{{\dot \varepsilon }_0}}}} \right)} \right]\left[ {1 - {{\left( {\dfrac{{T - {T_0}}}{{{T_{\text{r}}} - {T_0}}}} \right)}^p}} \right]$
Lin-Xia^[28]	$\sigma = \left( {A + {B_1}\varepsilon + {B_2}{\varepsilon ^2}} \right)\left( {1 + C\ln {{\dot \varepsilon }^ * }} \right)\exp \left[ {\left( {{\lambda _1} + {\lambda _2}\ln {{\dot \varepsilon }^ * }} \right)\left( {T - {T_{\text{r}}}} \right)} \right]$
Khan-Huang^[16]	${J_2} = {f_1}\left( \varepsilon \right){f_2}\left( {D_2^{\text{p}}} \right)$
Yu-Guo^[35]	$\sigma = f\left( {\varepsilon ,\dot \varepsilon } \right) = {\sigma _0}{\hat f_2}\left( {\dot \varepsilon } \right) + {E_\infty }\varepsilon - a{{\text{e}}^{ - \alpha \varepsilon }}$
Khan-Liang^[36]	$\sigma = \left[ {A + B{ {\left( {1 - \dfrac{ {\ln {\dot \varepsilon } } }{ {\ln {D_0^{\text{p} } } } } } \right)}^{ {n_1} } }{\varepsilon ^{ {n_0} } } } \right]\left( {1 - {T^{*m} } } \right){\dot \varepsilon ^C}$
Khan-Zhang^[37]	$\sigma = \left[ { \left({a{}_1 + \dfrac{ { {k_1} } }{ {\sqrt d } } }\right) + B{ {\left( {1 - \dfrac{ {\ln {\dot \varepsilon } } }{ {\ln {D_0^{\text{p} } } } } } \right)}^{ {n_1} } }{\varepsilon ^{ {n_0} } } } \right]\left( {1 - {T^{ * m} } } \right){\dot \varepsilon ^C}$
Farrokh-Khan^[38]	$\sigma {\text{ = } }\left\{ {\left( { {a_1} + \dfrac{ { {k_1} } }{ {\sqrt d } } } \right) + B{ {\left( {\dfrac{d}{ { {d_0} } } } \right)}^{ {n_2} } }{ {\left[ {\left( {1 - \dfrac{ {\ln {\dot \varepsilon } } }{ {\ln {D_0^{\text{p} } } } } } \right) {\dfrac{ { {T_{\text{m} } } } }{T} } } \right]}^{ {n_1} } }{ {\left( \varepsilon \right)}^{ {n_0} } } } \right\}{\left( {\dfrac{ { {T_{\text{m} } } - T} }{ { {T_{\text{m} } } - {T_{\text{r} } } } } } \right)^m}{\left( {\dfrac{ {\dot \varepsilon } }{ { { {\dot \varepsilon }^ * } } } } \right)^C}$
Voce-Kocks^[39-40]	$\sigma = {\sigma _{\text{s}}} + \left[ {\left( {{\sigma _0} - {\sigma _{\text{s}}}} \right)\exp \left( { - \dfrac{\varepsilon }{{{\varepsilon _{\text{r}}}}}} \right)} \right]$
Molinari-Ravichandran^[41]	$\dfrac{ { {\delta _{\text{r} } } } }{ { {\delta _{ {\text{r0} } } } } } = {\left[ {1 - { {\left( { {k_{\text{r} } } {\dfrac{T}{ { {T_{ {\text{r0} } } } } } } \lg {\dfrac{ { { {\dot \varepsilon }_{ {\text{r0} } } } } }{ {\dot \varepsilon } } } } \right)}^{ {p_{\text{r} } } } } } \right]^{ {q_{\text{r} } } } }\text{，}\dfrac{ { {\delta _{\text{s} } } } }{ { {\delta _{ {\text{s0} } } } } } = \dfrac{1}{ { { {\left[ {1 - { {\left( { {k_{\text{s} } } {\dfrac{T}{ { {T_{ {\text{s0} } } } } } } \lg {\dfrac{ { { {\dot \varepsilon }_{ {\text{r0} } } } } }{ {\dot \varepsilon } } } } \right)}^{ {p_{\text{s} } } } } } \right] }^{ {q_{\text{s} } } } } } }$

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表 3 物理概念本构关系的模型对比

Table 3. Comparison of physically based constitutive relations

本构关系	年份	模型	应变率范围/s⁻¹	本构关系的主要特征
B-P	1975	Bodner-Partom^[18]	10⁻³～1	（1）将材料的总变形率分为弹性和塑性两部分（2）通过塑性功项合并应变硬化效应（3）没有考虑温度效应
Z-A	1987	Zerilli-Armstrong^[17]	4×10³	（1）考虑了晶粒尺寸的影响（2）基于热激活理论
	2009	Zhang-Wen^[48]	10⁻⁵～10⁻²	考虑了温度、应变率和变形过程对Z-A模型中参数的影响
	2009	Samantaray-Mandal^[49]	10⁻³～1	考虑了温度与应变、温度与应变率的耦合效应对流动应力的影响
	2005	Abed-Voyiadjis^[50]	10⁻⁴～10⁴	（1）可用于预测等温和绝热塑性变形的应力应变曲线（2）将模型参数准确的与微观结构物理参数联系起来
M-T-S	1988	Follansbee^[10,51]	10⁻⁴～10⁴	（1）认为应变率敏感性的上升应归因于结构演化的速率敏感性（2）考虑了阈值应力
N-N-L	1998	Nemat-Nasser-Li^[57]	10⁻³～10⁴	考虑了位错密度随应变和温度的变化
其他	2001	Rusinek-Klepaczko^[61,62]	10⁻⁴～10³	考虑了应变率历史效应对金属材料塑性流动行为的影响
	2009	Rusinek-Rodrguez-Martnez^[63]	10⁻⁴～10⁴	（1）添加一个第三项来扩展该本构关系的应用范围（2）考虑负应变率敏感性和粘性阻力
	2010	Sung^[66]	10⁻³～10	通过Hollomon和Voce应变硬化方程的线性组合来揭示应变硬化率的温度敏感
	2010	Gao-Zhang^[67]	10⁻³～10⁴	考虑FCC金属变形过程中微观结构的演变建立阈值应力与应变、温度和应变率的关系

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表 4 物理概念本构关系的方程形式

Table 4. Equations of physically based constitutive relations

模型	方程形式
Bodner-Partom^[18]	$D_2^{\text{p}} = D_0^2\exp \left[ { - \left( {\dfrac{{n + 1}}{n}} \right){{\left( {\dfrac{{{Z^2}}}{{3{J_2}}}} \right)}^n}} \right]$
Zerilli-Armstrong^[17]	对于FCC: $\sigma = {\sigma _{\text{a}}} + B{\varepsilon ^{1/2}}\exp \left( { - \alpha T} \right)$ 和 $\alpha {\text{ = }}{\alpha _0} - {\alpha _1}\ln \dot \varepsilon$ 对于BCC： $\sigma = {\sigma _{\text{a}}} + B\exp \left( { - \beta T} \right) + {B_0}{\varepsilon ^n}$ 和 $\beta {\text{ = }}{\beta _0} - {\beta _1}\ln \dot \varepsilon$ 对于HCP： $\sigma = {\sigma _{\text{a}}} + B\exp \left( { - \beta T} \right) + {B_0}{\varepsilon ^{^{1/2}}}\exp \left( { - \alpha T} \right)$
Zhang-Wen^[48]	对于FCC： $\sigma = {\sigma _{\text{a}}} + {C_1}{\varepsilon ^{1/2}}\exp \left\{ {\left[ { - {C''_3}T + {C'_4}T\ln \left( {\dfrac{{\dot \varepsilon }}{{r\left( \varepsilon \right)r\left( {\dot \varepsilon } \right)}}} \right)} \right]H\left( T \right)} \right\}$ 对于BCC： $\sigma = {\sigma _{\text{a}}} + {C_2}\exp \left\{ {\left[ { - {C''_3}T + {C'_4}T\ln \left( {\dfrac{{\dot \varepsilon }}{{r\left( \varepsilon \right)r\left( {\dot \varepsilon } \right)}}} \right)} \right]H\left( T \right)} \right\} + {C_5}{\varepsilon ^n}$
Samantaray-Mandal^[49]	$\sigma = \left( {{C_1} + {C_2}{\varepsilon ^n}} \right)\exp \left[ { - \left( {{C_3} + {C_4}\varepsilon } \right){T^ * } + \left( {{C_5} + {C_6}{T^ * }} \right)\ln {{\dot \varepsilon }^ * }} \right]$
Abed-Voyiadjis^[50]	对于FCC： $\sigma = {C_2}{\varepsilon ^{0.5}}\left( {1 - {X^{1/2}} - X + {X^{3/2}}} \right) + {C_6}$ 和 $X = {C_4}T\ln \left( {1/{{\dot \varepsilon }^ * }} \right)$ 对于BCC： $\sigma = {C_1}\left( {1 - {X^{1/2}} - X + {X^{3/2}}} \right) + {C_5}{\varepsilon ^n} + {C_6}$
Follansbee^[10,51]	$\sigma = {\sigma _{\text{a}}} + \left( {\hat \sigma - {\sigma _{\text{a}}}} \right){\left[ {1 - {{\left( {\dfrac{{kT}}{{{g_0}\mu {b^3}}}\ln \dfrac{{{{\dot \varepsilon }_0}}}{{\dot \varepsilon }}} \right)}^{1/q}}} \right]^{1/p}}$
Nemat-Nasser-Li^[57]	$\sigma \left( {\dot \varepsilon ,\varepsilon ,T} \right) = {\sigma ^0}{\left\{ {1 - {{\left[ { - \dfrac{{kT}}{{{G'_0}}}\left( {\ln \dfrac{{\dot \varepsilon }}{{{{\dot \varepsilon }_0}}} + \ln \left( {1 + a\left( T \right){\varepsilon ^{1/2}}} \right)} \right)} \right]}^{\tfrac{1}{2}}}} \right\}^{\tfrac{3}{2}}}\left[ {1 + a\left( T \right){\varepsilon ^{\tfrac{1}{2}}}} \right] + \sigma _{\text{a}}^0{\varepsilon ^{{n_1}}}$
Rusinek-Klepaczko^[61,62]	$\sigma = \dfrac{{E\left( T \right)}}{{{E_0}}}\left[ {{B_0}\theta _{\text{m}}^{ - v}{{\left( {{\varepsilon _0} + {\varepsilon _{\text{p}}}} \right)}^{n\left( {1 - {D_2}{\theta _n}} \right)}} + \sigma _0^*{{\left( {1 - {D_1}{\theta _{\text{m}}}} \right)}^m}} \right]$
Rusinek-Rodrguez-Martnez^[63]	${\sigma _{{\text{ns}}}}\left( {\dot \varepsilon ,T} \right) = \sigma _0^{{\text{ns}}} \left[ {\lg \left( {\dfrac{{{{\dot \varepsilon }_{{\text{trans}}}}}}{{\dot \varepsilon }}} \right)} \right] \left[ {1 - {D_3}\left( {\dfrac{{{T_{\text{m}}}}}{T}} \right)\lg \left( {\dfrac{{\dot \varepsilon }}{{{{\dot \varepsilon }_{\max }}}}} \right)} \right]$
Sung^[66]	$\sigma = \sigma \left( {\varepsilon ,\dot \varepsilon ,T} \right) = f\left( {\varepsilon ,T} \right)g\left( {\dot \varepsilon } \right)h\left( T \right)$
Gao-Zhang^[67]	$\sigma {\text{ = }}{\sigma _{\text{a}}} + \hat Y{\varepsilon ^n}\exp \left[ {{c_3}T\ln \left( {\dfrac{{\dot \varepsilon }}{{{{\dot \varepsilon }_{{\text{s0}}}}}}} \right)} \right]{\left\{ {1 - {{\left[ { - {c_4}T\ln \left( {\dfrac{{\dot \varepsilon }}{{{{\dot \varepsilon }_0}}}} \right)} \right]}^{1/q}}} \right\}^{1/p}}$

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