金属材料的率-温耦合响应与动态本构关系综述

袁康博; 姚小虎; 王瑞丰; 莫泳晖

doi:10.11883/bzycj-2021-0416

金属材料的率-温耦合响应与动态本构关系综述

doi: 10.11883/bzycj-2021-0416

1.
华南理工大学工程力学系，广东广州 510641
2.
西北工业大学航空学院，陕西西安 710072

基金项目: 中央高校基本科研业务费专项资金（x2tjD2220850）；国家自然科学基金（12202149）；国家杰出青年科学基金（11925203）；中国博士后科学基金（2022M711198）

详细信息

作者简介:
袁康博（1992－　），女，博士，kangboyuan0528@scut.edu.cn

通讯作者:
姚小虎（1974－　），男，博士，教授，博士生导师，yaoxh@scut.edu.cn

中图分类号: O347.3
计量
- 文章访问数: 1038
- HTML全文浏览量: 563
- PDF下载量: 267
- 被引次数: 5
出版历程
- 收稿日期: 2021-10-08
- 录用日期: 2022-05-13
- 修回日期: 2022-01-04
- 网络出版日期: 2022-05-19
- 刊出日期: 2022-09-29

A review on rate-temperature coupling response and dynamic constitutive relation of metallic materials

1.
Department of Engineering Mechanics, South China University of Technology, Guangzhou 510641, Guangdong, China
2.
School of Aeronautics, Northwestern Polytechnical University, Xi’an 710072, Shaanxi, China

摘要

摘要: 金属材料在冲击、爆炸等高应变率加载下的塑性流动行为具有不同于静载下的率-温耦合性和微观机制。航空航天、航海、能源开采、核工业、公共安全、灾害防治等方面的金属结构设计与性能评估需要进行大量的动载实验和数值模拟，建立准确的材料动态本构模型是结构数值模拟可靠性的基础和关键。本文中，总结了金属材料的率-温耦合变形行为及内在机理，回顾了金属动态本构关系研究的起源与发展脉络，分别针对唯象模型、具有物理基础的模型和人工神经网络模型进行了详细介绍和横向比较。唯象模型和人工神经网络模型分别因易应用和高预测精度而受到青睐，基于物理概念的宏观连续介质模型可以描述体现内部演化的真实物理量，从而涵盖更大的应变范围，更好地反映应变率、温度和应变的影响机制。
- 金属材料 /
- 高应变率 /
- 塑性变形 /
- 动态本构模型
Abstract: Different from static loading conditions, the plastic flow behavior of metallic materials under high strain rate loadings, such as impact and explosion, exhibits special rate-temperature coupling effect and deformation micro-mechanism. The design and evaluation of metallic structures used in aerospace and navigation, energy mining, nuclear industry, public safety, disaster prevention, etc. require a large number of experiments under dynamic loadings. In recent years, the rapid-developing computational mechanics can be used to analyze the structural mechanical response under complex loading, evaluate the structural safety and optimize the structural design, and can also save the experimental costs. Accurate dynamic constitutive description of materials is the basis for the reliability of structural numerical simulation. In this paper, the dynamic plastic deformation behavior and micro-mechanism of metals, as well as the origin and development of the dynamic constitutive relationship of metals are reviewed and summarized. Over wide ranges of strain rate and temperature, the metals exhibit complex rate-temperature coupling effect, such as dynamic strain aging and segmented strain rate sensitivity. The high strain rate may lead to dynamic recrystallization, deformation twinning and shock-induced phase transition. The existing constitutive models can be divided into three types: phenomenological models, physically based models and artificial neural network models. Phenomenological models refer to the constitutive models established merely by describing experimental phenomena without considering the internal physical mechanism. Physically based macro-scale continuum models can represent true physical quantities for documenting and tracking the evolution which takes place within metallic materials. Artificial neural network models are good at reproducing the plastic flow behavior as function of many factors, such as strain rate, temperature and plastic strain, without the need of identifying complex logic relationships and parameters within the system. The developments, prediction capabilities, and application scopes of the three types of dynamic constitutive models are illustrated in detail and compared horizontally. In addition, some objective suggestions for the further development of dynamic constitutive descriptions for metals are proposed. Phenomenological models are favored for their ease in application, artificial neural network models are favored for their high prediction accuracy. Recent trend has increased the focus on physically based models. This type of model extends application to a wider strain range and more clearly represents the influence mechanism of strain rate, temperature and strain.
- metallic material /
- high strain rate /
- plastic deformation /
- dynamic constitutive model

HTML全文

多孔材料由胞孔组成, 不同冲击速度下胞孔的变形特性不同, 导致多孔材料宏观动态力学性能也不同。S.Lee等^[1]和P.J.Tan等^[2-3]指出, 冲击速度高时多孔材料以冲击波模式变形。Zheng Zhi-jun等^[4]和刘耀东等^[5]通过数值模拟总结出随着速度提高, 多孔材料分别呈现出准静态模式、过渡模式和冲击模式等3种变形模式。随着速度的提高, 试件的变形均匀性和两端应力均匀性都越来越差, 如何测量变形和应力的不均匀性以及讨论两者的关系成为了解多孔材料动态力学性能的难题。王鹏飞等^[6]利用改进的Hopkinson杆实验装置测得了不同冲击速度下泡沫铝试件两端的应力。在变形不均匀性方面, Z.Zou等^[7]和S.Pal等^[8]根据数值模拟结果直接给出了冲击模式下试件中一维和二维的应变分布, 但用此法得出的结果误差较大。Liao Shen-fei等^[9]将数值模拟结果结合最佳局部变形梯度原理推导出了更精确的应变场。如何在实验中准确测量泡沫铝试件压缩过程中的变形特性成了难题, 虽然通过数值模拟能够从规律上进行解释, 但是与实验仍有一定差距。采用常规的电测方法仅能测试试件的平均应变, 无法测量其中的不均匀性。高速摄影只能提供一种直观判断, 是一种定性的测量手段, 而且对于微小的变形无法判断。W.H.Peters等^[10]提出的数字图像相关法(digital image correlation method, DICM)是光测力学中的一种简单方便精度高的测试手段, 经过多年的发展, 已经在生物力学、岩石力学、纳米力学等众多领域得到广泛应用。近年来, 这一方法也用于多孔材料研究, 但主要集中在准静态实验方面^[11-12], 而动态性能方面研究较少, 主要有:Wang Li-li等^[13]利用DICM测试了泡沫铝的泰勒杆实验中试件的速度随撞击时间的变化; S.Lee等^[1]和I.Elnasri等^[14]分别研究了开孔和闭孔泡沫铝高速变形过程中的全场应变; H.Luo等^[15]在试件表面粘上带有格子的标签, 测量了泡沫非晶金属材料。然而上述研究并未对3种模式下的变形场作比较, 也没有给出全场应变发展的过程, 更没有进一步讨论变形均匀性与应力均匀性的关系。基于此, 本文中, 将结合Hopkinson压杆实验技术和高速摄影技术, 研究不同冲击速度下泡沫铝的全场应变及其变化趋势, 并依此研究泡沫铝试件中的应力不均匀性。

1. 冲击压缩实验

试件为直径为32 mm、高度为32 mm的闭孔泡沫铝, 密度为0.320~0.340 g/cm³。实验在直径为37 mm的Hopkinson压杆装置上进行, 实验系统示意图如图 1所示, 杆材为铝。共进行了3组实验, 子弹速度分别为12、50和110 m/s, 对应3种不同的变形模式:即准静态模式、过渡模式和冲击模式。为了研究准静态模式下试样的应力均匀性, 利用常规Hopkinson杆装置进行准静态模式下的实验, 见图 1(a)。由于子弹速度过高会导致打击杆屈服, 采用改进的Hopkinson杆装置(图 1(b))进行后2种模式的实验, 即将试件放在支撑杆与子弹之间, 让子弹直接撞击试件。图中的Wheatstone电桥接法中含有2套单独的应变片系统, 每个系统中含有2片应变片且使用的电桥接法为半桥接法。

图 1 实验装置示意图

Figure 1. Schematic diagram of the experimental setup

下载: 全尺寸图片幻灯片

实验所用高速摄影相机为Phantom V12.1, 全画幅分辨率为1 280 Pixel×800 Pixel, 全画幅拍摄速率可达6 242 s^-1, 最高拍摄速率可达到1 000 000 s^-1, 3组实验的拍摄频率皆为74 000 s^-1, 此时帧间隔时间为13.5 μs, 3种模式对应的实际图像分辨率见表 1。实验中使用了2个脉冲氙灯来提高曝光量。

表 1 实验参数

Table 1. Experimental parameters

模式	v/(m·s^-1)	ρ/(g·cm^-3)	实验方法	图像分辨率	拍摄速率/s^-1
准静态模式	12	0.325	常规SHPB	256 Pixel×200 Pixel	74 000
过渡模式	50	0.340	直接撞击	304 Pixel×168 Pixel	74 000
冲击模式	110	0.338	直接撞击	288 Pixel×232 Pixel	74 000

下载: 导出CSV

| 显示表格

2. 变形均匀性

根据高速摄影图像可以分别得到准静态模式、冲击模式和过渡模式下试件的变形过程, 如图 2~4所示。其中高速摄影的第1张图像都对应着试件刚开始发生变形的前一时刻, 并将该时刻定为0 μs。

图 2 准静态模式下试件的变形过程

Figure 2. Deformation progress of the specimenin the quasi-static mode

下载: 全尺寸图片幻灯片

图 3 冲击模式下试件的变形过程

Figure 3. Deformation progress of the specimen in the dynamic mode

下载: 全尺寸图片幻灯片

图 4 过渡模式下试件的变形过程

Figure 4. Deformation progress of the specimen in the transition mode

下载: 全尺寸图片幻灯片

采用自编的程序处理高速摄影图像, 即可得到变形过程中每一个时刻相对于上一时刻试件的增量应变场, 如处理0和13.5 μs的2张图像即可得到0 μs时刻试件的增量应变场。本文中高速摄影的图像间隔时间均为13.5 μs, 所以本文中采用DICM得到的计算结果的间隔时间也是13.5 μs。对于冲击模式下的图像, 在冲击波波阵面附近的区域由于应变过大, 同时伴有翻转和弯折, 区域内大部分点的灰度值在相邻2张图像中已不存在联系。这就违背了数字图像相关方法的假定, 使该区域无法与变形前图像进行相关计算。因此在计算过程中只计算冲击波波阵面前方的区域, 如图 4(g)所示。图 5~7分别表示3种模式下试件沿着加载方向全场应变的变化过程。本方法的标定可参见文献[16]。

图 5 准静态模式下试件变形过程中的应变场

Figure 5. Strain field of deformation progress under quasi-static mode

下载: 全尺寸图片幻灯片

图 6 冲击模式下试件变形过程中的应变场

Figure 6. Strain field of deformation progress under dynamic mode

下载: 全尺寸图片幻灯片

图 7 过渡模式下试件变形过程中的应变场

Figure 7. Strain field of deformation progress under transition mode

下载: 全尺寸图片幻灯片

从高速摄影图像上来看, 准静态模式下(图 2)试件变形较均匀, 在变形的前半段试件的变形以均匀变形为主, 没有产生局部变形。在变形的后半段, 试件开始产生局部变形并发展为如图 2(h)虚线所示的局部变形带。冲击模式下(图 3), 试件从一开始就变形不均匀。由于冲击端速度很高, 泡沫铝试件中与冲击端紧挨着的胞孔很快被压溃, 随后压溃区前方的胞孔也被压溃并发生逐层垮塌, 以一种压实波^[17]的形式向支撑端发展。表现在图像中为试件从靠近打击杆开始变形, 并快速形成了密实带, 然后密实带(图 3中箭头所指)快速向前传播, 直至整个试件压实。过渡模式下(图 4), 试件的变形模式介于准静态模式和冲击模式之间, 先在靠近冲击端的区域以密实带的形式发展, 随后密实带前方的区域的变形以局部变形带为主。

采用DICM得到的计算结果反映了与高速摄影图像完全一样的变形趋势, 然而计算结果是量化的, 并且揭示了更多变形过程的细节。高速摄影图像中相邻时刻的图像之间变形较小, 无法判断其应变变化。采用DICM得到的计算结果则可以弥补这个缺陷, 能够展示整个变形过程中应变场的变化。

准静态模式下试件的变形过程如图 5所示, 按变形均匀性可以将整个过程分为2部分:均匀变形阶段A和不均匀变形阶段B。在阶段A中整个试件的变形较均匀, 塑形变形遍布整个试件而且较均匀。随着变形的发展, 试件的变形逐渐局部化, 塑形应变开始集中。在162 μs时, 即图中A、B区域的临界点, 塑形应变的集中最终形成了局部变形带。随后试件的变形主要集中在局部变形带上, 形状和方向与图 2(h)中的局部变形带一致。在压缩过程中, 试件中的局部变形带不断发展, 在试件变形的最后阶段试件的变形逐渐变小直至不变。值得一提的是, 试件中的局部变形带并不是一直发展的, 其变形到一定密实程度后又会逐渐停止发展, 并在试件的其他薄弱区域产生新的局部变形带^[16]。在整个变形过程中, 开始段和结束段的应变率都较小, 而中间段的应变率较高。这是由于采用了常规SHPB装置, 冲击端处的入射波是一个梯形波, 应力幅值在加载和卸载段均较低造成的。

冲击模式下应变场变化见图 6, 从一开始试件的变形就集中在靠近冲击端附近, 形成一个很小的局部变形区域, 而其他区域的应变很小。随着压缩的发展, 局部变形区快速向前发展, 直至整个试件压溃。

为了更好地观测冲击波波阵面前方的应变场, 对应变场作进一步处理。将图 7中增量应变场换算成全量应变场, 再沿着y方向求和并求平均, 即可得到整个试件沿着加载方向的平均应变分布如图 8所示。结果表面:波阵面前方仍存在变形区且应变较大, 应变沿加载方向的分布与冲击波波阵面附近的应变分布类似。这与文献[7-9]的结果相似但仍存在区别, 这是因为本文中的试件较短, 在变形的开始段和结束段应变分布在很大程度上受到边界条件的影响。在起始段(0~67.5 μs), 应力波还未作用到冲击波前端所在地区, 此时冲击波前端的应变较小; 中间段(67.5~121.5 μs), 冲击波前端的应变分布与文献[7-9]的结果一致, 即只在冲击波前端约一个胞孔距离的区域有应变, 而远离冲击端的地方无应变; 末尾段(135μs), 此时冲击波距离支撑端很近, 整个试件都产生了较大的应变。图 8中变形中间段存在较清晰的应变下降段, 可认为是冲击波波阵面。根据其数据可以算出压实波波阵面分别相差17.9和19.6像素, 本实验中图像与真实试件的比例为5.625像素/毫米, 即可得波阵面的传播速度为236和258 m/s, 此结果略大于理论值, 其原因还有待研究。

图 8 冲击模式下沿着冲击方向的应变分布

Figure 8. Strain distribution along the impact direction under dynamic mode

下载: 全尺寸图片幻灯片

过渡模式下的应变场变化如图 7所示。试件的变形过程可以分为2部分:局部致密区变化阶段和局部变形带变化阶段。在试件变形的前段, 由于冲击端速度较高但又未达到冲击模式的临界速度, 靠近冲击端部分的胞孔很快被压溃并形成了致密区, 但没有形成冲击波。试件的变形集中在靠近冲击端的位置, 并逐渐形成了致密区, 此时远离冲击端部分未发生变形。在变形后半段, 由于子弹速度降低和试件中存在缺陷等原因, 试件进入了一个局部变形带变形期。此时的变形由试件中的缺陷主导, 在有缺陷处产生局部变形带直到整个试件压实。

泡沫铝试件由于其本身的结构特性在冲击载荷下会产生局部化变形, 不同冲击速度下对应着不同的变形模式。本文利用数字图像相关方法, 从实验中测试了3种不同模式下试件的变形特征。准静态模式下, 试件先经历一个较长的变形均匀区, 随着外加载荷超过试件临界失稳载荷, 试件在薄弱处发生失稳坍塌, 形成了局部变形带。冲击模式下, 整个变形过程都是不均匀的, 试件的变形主要以压实波的形式发展。过渡模式下试件的变形过程是冲击模式和准静态模式变形过程的综合, 前半部分与冲击模式相似, 是一个局部致密化的过程, 后半部分与准静态模式相似, 变形集中在局部变形带上。

综上所述, 可知:(1)低速冲击时, 试件的局部化变形源于试件中含初始缺陷区域的失稳。在此情况下, 试件两端的载荷可基本达到均衡, 可认为整个试件受力均匀。试件中含缺陷区域最薄弱, 一定载荷下, 该区域开始产生局部化变形, 此时局部化变形以局部变形带为主。该区域的失稳发展到一定程度后, 由于变形导致该区域强度增大而停止失稳。此时变形转移到试件中其他薄弱的区域并形成新的局部变形带, 这可由文献[16]中给出的多个局部变形带证实。(2)高速冲击时, 试件的局部化变形源于冲击端区域的压缩失稳。当冲击速度达到一定值时, 试件两端的载荷很难达到均衡。此情况下, 冲击波的强度即可使材料发生破坏, 局部化变形发生在靠近冲击端的位置。因此, 对应试件的局部化变形由惯性效应决定, 表现为剧烈体积压缩的致密区。由此可见, 低速和高速冲击下材料的破坏分别对应着泡沫铝材料的整体失稳和局部失稳两种不同的机制。在这里, 应力的均匀性有重要作用。

3. 应力均匀性

为了讨论试件的应力均匀性, 在准静态模式下做了另外一组试件密度接近, 子弹速度、试件厚度等其他参数都相同的实验。利用石英片测出冲击端和支撑端的应力-时间曲线, 如图 9(a)所示。可以看到试件变形过程中存在2个区域:应力均匀区A和应力不均匀区B。在A区中冲击端应力大于支撑端应力, 而在B区中两端应力平衡。

图 9 泡沫铝试件两端的应力均匀性

Figure 9. Stress uniformity of the two ends of the aluminum foam specimen

下载: 全尺寸图片幻灯片

在A区中两端应力不均匀的原因是2条曲线存在时间差, 即应力波在试件中传播的时间, t=L/c。现在将2条曲线对齐来判断应力波在试件中传播的耗散性。图 9(b)中曲线可以分为2部分:非耗散区C和耗散区D。非耗散区C中, 前后端应力相同, 说明应力在试件中的传播没有耗散。应力耗散区D内, 冲击端的应力明显大于支撑端的应力, 即应力发生了耗散。

同时可以看到, 应力开始发生耗散的时刻(165 μs)恰好是试件中产生局部变形带的时刻(162 μs), 即非耗散区对应着变形均匀区, 耗散区对应着变形非均匀区。当试件均匀变形时, 应力波相当于在一个均匀的物体中传播, 此时应力不发生耗散。而当胞孔发生了坍塌, 试件中形成了局部变形带。局部变形带所在区域的波阻抗大于其附近区域的波阻抗, 当应力波传播到这个界面上时会发生反射, 导致透射应力低于入射应力, 即应力发生耗散。这就说明变形的不均匀性影响到应力的不均匀性。

为了更好地说明上述问题, 建立了如下的简化模型, 即假设在试件中存在一个垂直加载方向的局部变形带。假设弹性波在局部变形带中传播速度、密度和弹性模量分别为c₂、ρ₂和E₂, 在试件其他部分的参数对应为c₁、ρ₁和E₁, 显然有ρ₁c₁ < ρ₂c₂。根据弹性波在不同介质界面上的传播规律, 有:

$\sigma_{\mathrm{r}}=F \sigma_{\mathrm{i}}, \quad \sigma_{\mathrm{t}}=T \sigma_{\mathrm{i}}$

(1)

式中: 和σ_t分别为入射波、反射波和透射波的应力。

则图 10中1、2、3区中弹性波存在关系: 。

图 10 应力波传播模型

Figure 10. Modes of stress wave transformation

下载: 全尺寸图片幻灯片

由于n < 1, 即可得σ₃/σ₁ < 1。随着局部变形带的发展, 波阻抗不匹配度增大, n越小时, σ₃/σ₁越小, 即应力波的耗散越大。这就解释了随着局部变形带的发展, 两端应力不均匀性越来越大的原因。

随着速度的增大, 变形模式变成过渡模式和冲击模式, 过渡模式和冲击模式下两端的应力不均匀性显著增高^[6]。从上文中所得到的应变场分析可以看到, 过渡模式和冲击模式下试件没有均匀变形的时期, 而是在靠近冲击端位置直接产生了密实区。局部变形带和密实区对于波传播的影响是一致的, 而且密实区相对密度更大, 与其余部分的波阻抗不匹配度更高。应力波经过界面后产生了更低的透射应力。此时冲击端和吸收端的应力差距更大, 这也从另一角度解释了随着速度提高, 试件两端的应力不均匀性更大的原因。

与图 10(a)类似, 高速冲击下试件产生致密区后的模型如图 10(b)所示, 参数定义与上文中相同。根据公式(1)可得: σ₁。

因为n=ρ₁c₁/(ρ₂c₂)≪1, 所以σ₂ < σ₁, 即密实区和试件其他部分的波阻抗不匹配度很高时, 支撑端应力远小于冲击端的应力。

综上所述:当试件变形均匀时, 其两端应力是均匀的; 当局部变形区域产生时, 试件变形开始不均匀, 此时试件两端应力也开始不均匀。这是由于局部变形不均匀的区域与其他部分波阻抗不匹配, 导致波在试件中的传播发生了反射, 即波在试件中传播发生了耗散, 使冲击端和吸收端的应力不均匀。

4. 结论

利用Hopkinson杆装置对泡沫铝试件进行了不同冲击速度的压缩实验, 利用高速摄影机对所有实验过程进行了跟踪拍摄。采用数字图像相关方法程序对高速摄影图像进行处理, 得到了不同变形模式下试件的应变场发展过程。

(1) 不同冲击速度下试件中的变形模式完全不同:准静态模式下, 试件开始段变形较均匀, 随后在试件中产生了局部变形带; 过渡模式下, 开始段变形集中在靠近冲击端的区域并形成了局部致密区, 结束段试件中的变形以局部变形带的方式发展; 冲击模式下, 试件变形主要以胞孔压实的模式向前发展。

(2) 不同速度下试件的变形机理也不同:低速下, 产生局部变形带的机理是试件中含缺陷胞孔的失稳; 高速下, 产生局部致密区主要是由惯性效应引起的。

(3) 材料不均匀性所导致的失稳(整体效应)和惯性效应(局部效应)是应力不均匀和变形不均匀的原因。在泡沫铝试件均匀变形时, 应力波在试件中的传播没有耗散。当试件中产生了局部变形带或致密区这样的局部变形区时, 由于局部变形带和致密区对波的反射作用, 使应力波产生了耗散。而且随着速度增高, 局部变形区与均匀变形区波阻抗不匹配程度增大, 反射波也越来越大, 这就导致了应力不均匀的加剧。

图 1 低碳钢的屈服应力在不同温度和应变率区域内的应变率效应

Figure 1. Strain rate effect on yield stress of low-carbon steelin different temperature and strain rate regions

下载: 全尺寸图片幻灯片

图 2 金属的典型温度敏感性

Figure 2. Typical temperature sensitivity of metal

下载: 全尺寸图片幻灯片

图 3 Q235B钢在0.1应变下流动应力随温度和应变率的变化^[19]

Figure 3. Variation of flow stress with temperatureand strain rate for Q235B steel^[19]

下载: 全尺寸图片幻灯片

图 4 不同热处理状态下Inconel 718镍基高温合金的流动应力随温度变化曲线^[16]

Figure 4. Flow stress-temperature curves of Inconel 718 superalloyunder different heat-treatment conditions^[16]

下载: 全尺寸图片幻灯片

图 5 塑性流动曲线的4个阶段和绝热剪切引起的动态再结晶微观图片

Figure 5. Four stages of plastic flow curve and micro image of DRX caused by adiabatic shear

下载: 全尺寸图片幻灯片

图 6 不锈钢的绝热剪切局部化引起的变形孪晶^[44]

Figure 6. Deformation twinning in the stainless steel by adiabatic shear localization^[44]

下载: 全尺寸图片幻灯片

图 7 BP神经网络的结构示意图

Figure 7. Schematic structure of BP neural network

下载: 全尺寸图片幻灯片

表 1 唯象动态本构模型之间的比较

Table 1. Comparison among phenomenological dynamic constitutive models

发表时间	模型名称	应变率/s⁻¹	温度/℃	待定参数/个	主要特点
1976	Voce-Kocks (VK)^[80]	10⁻¹	−173～327	7	饱和应力 ${\sigma }_{\rm{s}}$ 为温度和应变率的函数
1983	Johnson-Cook (JC)^[77]	对数应变率的线性函数，可达10⁴	温度的幂函数	5	兼顾温度和应变率效应参数少，形式简单
1992	Khan-Huang (KH)^[82]	10⁻⁵～10⁴	不考虑	5	未考虑温度效应将总应变率分解为弹性和塑性分量
1999	Khan-Huang-Liang (KHL)^[83]	10⁻⁶～10⁴	25～316	7	在KH模型基础上增加温度效应
2009	Khan-Liang-Farrokh (KLF)^[115]	10⁻²～3×10⁴	−50～250	9	基于KHL模型兼顾温度和应变率效应考虑晶粒尺寸
2008	Improved Fields-Backofen (FB) model by Cheng^[107]	10⁻¹～10⁻⁴	150～300	5	兼顾温度和应变率效应参数少，形式简单
2005	Molinari-Ravichandran (MR)^[81]	10⁻²～10⁶	−196～200	9	基于微观结构的特征尺度考虑温度、应变率和晶粒尺寸
2010	Lin-Liu (LL)^[117]	10⁻²～10	850～1150	8	可描述热成形过程达到应力峰值的应力-应变曲线
2010	Toros-Ozturk (TO)^[118]	0.0016～0.16	室温～300	9	可描述大塑性应变下的软化行为

下载: 导出CSV

表 2 具有物理基础的动态本构模型之间的比较

Table 2. Comparison among physically based dynamic constitutive models

年份	模型名称	建模思想	主要特点
1975	Bodner-Partom (BP)^[93]	基于不可逆热力学，位错动力学和内变量理论	采用塑性功度量变形抗力无需屈服函数参数较少（不多于10），应用广泛
1987	Zerilli-Armstrong (ZA)^[45]	位错动力学理论 BCC和FCC晶体结构的塑性变形微观机制不同	考虑温度、应变率和平均晶粒尺寸不同晶体结构具有不同表达式描述热激活区域的塑性流动行为
1980	Steinberg-Guinan (SG)^[94]	剪切模量和屈服应力具有相同的温度和压强依赖性，将流体与冲击下的固体等效	考虑温度、压强效应未考虑应变率效应（认为高应变率下应变率效应不明显）
1989	Steinberg-Lund (SL)^[96]	流动应力等于热分量和非热分量之和，压强通过影响剪切模量影响流动应力	考虑温度、应变率和压强效应适用于10⁻⁴～10⁶ s⁻¹宽应变率范围
1981	Mecking-Kocks (MK)^[95]	针对FCC金属位错累积是塑性变形主要障碍	流动应力是应变硬化和率-温效应的乘积在应变硬化项中考虑动态回复
1988	Mechanical Threshold Stress (MTS)^[91]	采用力学阈值应力作为内部结构参量，不存在应变率效应的突然增大	考虑温度、应变率和应变历史的影响需要较多实验结果确定本构参数
1998	Nemat-Nasser-Li (NN)^[98]	位错动力学热激活理论	考虑FCC金属的应变历史对热激活行为的影响
1999	Nemat-Nasser-Guo (NN)^[100]	位错动力学热激活理论高应变率下的黏性拖曳机制	考虑高应变率加载下，金属塑性变形具有黏性拖曳导致的强化
2015	Guo-Wang (GW)^[19]	位错动力学热激活理论动态应变时效经典理论	描述第三型应变时效及其应变率效应
2021	Guo-Yuan (GY)^[16]	沉淀强化理论动态应变时效经典理论	考虑晶粒尺寸、位错密度和沉淀相体积分数及尺寸描述不同晶体结构的多相合金的塑性流动行为的区别
2005	Voyiadjis-Abed (VA)^[128]	位错动力学	考虑FCC和BCC金属热激活行为的区别
2008	Voyiadjis-Almasri (VA)^[17]	热激活理论（热激活能与温度、应变率和应变之间的关系）	针对FCC金属，考虑应变历史的影响
2018–2020	Voyiadjis-Song (VS)^[130-132]	动态应变时效发生符合韦伯概率分布	考虑动态应变时效，并结合韦伯分布进行描述
2001 2009 2010	Rusinek-Klepaczko (RK)^[133-135]	流动应力为描述应变强化的内应力和描述率-温效应的有效应力之和	考虑杨氏模量的温度效应考虑动态应变时效引起的负应变率效应^[134] 考虑FCC金属在高应变率下的黏性拖曳^[135]
2003	Preston-Tonks-Wallace (PTW)^[137]	针对应变率效应机制的不同，分为3个区：热激活控制的位错滑移区、过渡区和超高应变率区	应变率范围涵盖15个数量级基于量纲分析法建模考虑强冲击下非线性位错拖曳效应在塑性变形机制中占主导地位
1998	Cellular Automaton (CA)^[139]	物理冶金原理针对动态再结晶中的微观组织演化	不同温度（高温）和应变率的动态再结晶反向方法

下载: 导出CSV

参考文献(187)

[1]	杨桂通, 熊祝华. 塑性动力学 [M]. 北京: 清华大学出版社, 1984.
[2]	ARMSTRONG R W, WALLEY S M. High strain rate properties of metals and alloys [J]. International Materials Reviews, 2008, 53(3): 105–128. DOI: 10.1179/174328008X277795.
[3]	HOPKINSON J. On the rupture of iron wire by a blow [C] // Proceedings of the Literary and Philosophical Society of Manchester, 1872: 40–45.
[4]	HOPKINSON J. Further experiments on the rupture of iron wire [C] // Proceedings of the Literary and Philosophical Society of Manchester, 1872: 119–121.
[5]	HOPKINSON J. Original papers by the late John Hopkinson [J]. Cambridge, UK: Cambridge University Press, 1901, 2: 316–324.
[6]	HOPKINSON B. The effects of momentary stresses in metals [J]. Proceedings of the Royal Society of London, 1905, 74(497): 498–506. DOI: 10.1098/rspl.1904.0145.
[7]	CHARPY G. Note sur l’essai des métaux à la flexion par choc de barreaux entaillés [J]. Mémoires et comptes Rendus de la Société des Ingénieurs Civils de France, 1901: 848–877.
[8]	TÓTH L, ROSSMANITH H P, SIEWERT T A. Historical background and development of the Charpy test [J]. European Structural Integrity Society, 2002, 30: 3–19. DOI: 10.1016/S1566-1369(02)80002-4.
[9]	TRESCA M H. On further applications of the flow of solids [J]. Proceedings of the Institution of Mechanical Engineers, 1878, 29(1): 301–345. DOI: 10.1243/PIME_PROC_1878_029_017_02.
[10]	JOHNSON W. Henri Tresca as the originator of adiabatic heat lines [J]. International Journal of Mechanical Sciences, 1987, 29(5): 301–305; 307–310. DOI: 10.1016/0020-7403(87)90113-5.
[11]	JEFFRIES Z. Effect of temperature, deformation, and grain size on the mechanical properties of metals [J]. Transactions of the American Institute of Mining, Metallurgical and Petroleum Engineers, 1919, 60: 474–576.
[12]	REMINGTON B A, ALLEN P, BRINGA E M, et al. Material dynamics under extreme conditions of pressure and strain rate [J]. Materials Science and Technology, 2006, 22(4): 474–488. DOI: 10.1179/174328406X91069.
[13]	刘旭红, 黄西成, 陈裕泽, 等. 强动载荷下金属材料塑性变形本构模型评述 [J]. 力学进展, 2007, 37(3): 361–374. DOI: 10.3321/j.issn:1000-0992.2007.03.004. LIU X H, HUANG X C, CHEN Y Z, et al. A review on constitutive models for plastic deformation of metal materials under dynamic loading [J]. Advances in Mechanics, 2007, 37(3): 361–374. DOI: 10.3321/j.issn:1000-0992.2007.03.004.
[14]	SALVADO F C, TEIXEIRA-DIAS F, WALLEY S M, et al. A review on the strain rate dependency of the dynamic viscoplastic response of FCC metals [J]. Progress in Materials Science, 2017, 88: 186–231. DOI: 10.1016/j.pmatsci.2017.04.004.
[15]	YUAN K B, GUO W G, LI P H, et al. Thermomechanical behavior of laser metal deposited Inconel 718 superalloy over a wide range of temperature and strain rate: testing and constitutive modeling [J]. Mechanics of Materials, 2019, 135: 13–25. DOI: 10.1016/j.mechmat.2019.04.024.
[16]	YUAN K B, GUO W G, LI D W, et al. Influence of heat treatments on plastic flow of laser deposited Inconel 718: testing and microstructural based constitutive modeling [J]. International Journal of Plasticity, 2021, 136: 102865. DOI: 10.1016/j.ijplas.2020.102865.
[17]	VOYIADJIS G Z, ALMASRI A H. A physically based constitutive model for fcc metals with applications to dynamic hardness [J]. Mechanics of Materials, 2008, 40(6): 549–563. DOI: 10.1016/j.mechmat.2007.11.008.
[18]	王建军, 袁康博, 张晓琼, 等. 第三型应变时效的提出与研究进展 [J]. 爆炸与冲击, 2021, 41(5): 051101. DOI: 10.11883/bzycj-2020-0422. WANG J J, YUAN K B, ZHANG X Q, et al. Proposition and research progress of the third-type strain aging [J]. Explosion and Shock Waves, 2021, 41(5): 051101. DOI: 10.11883/bzycj-2020-0422.
[19]	WANG J J, GUO W G, GAO X S, et al. The third-type of strain aging and the constitutive modeling of a Q235B steel over a wide range of temperatures and strain rates [J]. International Journal of Plasticity, 2015, 65: 85–107. DOI: 10.1016/j.ijplas.2014.08.017.
[20]	COTTRELL A H. LXXXVI. A note on the Portevin-Le Chatelier effect [J]. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 1953, 44(355): 829–832. DOI: 10.1080/14786440808520347.
[21]	钱匡武, 李效琦, 萧林钢, 等. 金属和合金中的动态应变时效现象 [J]. 福州大学学报(自然科学版), 2001, 29(6): 8–23. DOI: 10.3969/j.issn.1000-2243.2001.06.003. QIAN K W, LI X Q, XIAO L G, et al. Dynamic strain aging phenomenon in metals and alloys [J]. Journal of Fuzhou University (Natural Science), 2001, 29(6): 8–23. DOI: 10.3969/j.issn.1000-2243.2001.06.003.
[22]	郭伟国, 左红星, 孟卫华, 等. 第三种应变时效与机械波谱关联性探讨 [J]. 材料科学与工艺, 2012, 20(1): 128–134. DOI: 10.11951/j.issn.1005-0299.20120126. GUO W G, ZUO H X, MENG W H, et al. Discussion of the relevancy of the third-type strain aging and mechanical spectroscopy [J]. Materials Science and Technology, 2012, 20(1): 128–134. DOI: 10.11951/j.issn.1005-0299.20120126.
[23]	GILAT A, WU X R. Plastic deformation of 1020 steel over a wide range of strain rates and temperatures [J]. International Journal of Plasticity, 1997, 13(6/7): 611–632. DOI: 10.1016/S0749-6419(97)00028-4.
[24]	CHENG J Y, NEMAT-NASSER S. A model for experimentally-observed high-strain-rate dynamic strain aging in titanium [J]. Acta Materialia, 2000, 48(12): 3131–3144. DOI: 10.1016/S1359-6454(00)00124-5.
[25]	GUO W G, GAO X S. On the constitutive modeling of a structural steel over a range of strain rates and temperatures [J]. Materials Science and Engineering: A, 2013, 561: 468–476. DOI: 10.1016/j.msea.2012.10.065.
[26]	孟卫华, 郭伟国, 苏静, 等. DH-36钢的塑性流动统一本构关系研究 [J]. 力学学报, 2011, 43(5): 958–962. DOI: 10.6052/0459-1879-2011-5-lxxb2010-676. MENG W H, GUO W G, SU J, et al. Study of plastic flow unified constitutive relation for steel DH-36 [J]. Chinese Journal of Theoretical and Applied Mechanics, 2011, 43(5): 958–962. DOI: 10.6052/0459-1879-2011-5-lxxb2010-676.
[27]	孟卫华, 郭伟国, 王建军, 等. DH36钢拉伸塑性流动特性及本构关系 [J]. 爆炸与冲击, 2013, 33(4): 438–443. DOI: 10.11883/1001-1455(2013)04-0438-06. MENG W H, GUO W G, WANG J J, et al. Tensile plasticity flow characteristics of DH36 steel and its constitutive relation [J]. Explosion and Shock Waves, 2013, 33(4): 438–443. DOI: 10.11883/1001-1455(2013)04-0438-06.
[28]	HONG S I. Influence of dynamic strain aging on the apparent activation volume for deformation [J]. Materials Science and Engineering, 1985, 76: 77–81. DOI: 10.1016/0025-5416(85)90082-5.
[29]	LEE K W, KIM S K, KIM K T, et al. Ductility and strain rate sensitivity of zircaloy-4 nuclear fuel claddings [J]. Journal of Nuclear Materials, 2001, 295(1): 21–26. DOI: 10.1016/S0022-3115(01)00509-8.
[30]	LEE K O, LEE S B. Modeling of materials behavior at various temperatures of hot isostatically pressed superalloys [J]. Materials Science and Engineering: A, 2012, 541: 81–87. DOI: 10.1016/j.msea.2012.02.005.
[31]	SU J, GUO W, MENG W, et al. Plastic behavior and constitutive relations of DH-36 steel over a wide spectrum of strain rates and temperatures under tension [J]. Mechanics of Materials, 2013, 65: 76–87. DOI: 10.1016/j.mechmat.2013.06.002.
[32]	LIN Y C, CHEN X M. A critical review of experimental results and constitutive descriptions for metals and alloys in hot working [J]. Materials & Design, 2011, 32(4): 1733–1759. DOI: 10.1016/j.matdes.2010.11.048.
[33]	MEYERS M A, NESTERENKO V F, LASALVIA J C, et al. Observation and modeling of dynamic recrystallization in high-strain, high-strain rate deformation of metals [J]. Journal de Physique Ⅳ, 2000, 10(PR9): 51–56. DOI: 10.1051/jp4:2000909.
[34]	XU Y B, ZHANG J H, BAI Y L, et al. Shear localization in dynamic deformation: microstructural evolution [J]. Metallurgical and Materials Transactions A, 2008, 39(4): 811–843. DOI: 10.1007/s11661-007-9431-z.
[35]	WRIGHT T W. Physics and mathematics of adiabatic shear bands [M]. Cambridge, UK: Cambridge University Press, 2002.
[36]	GREBE H A, PAK H R, MEYERS M A. Adiabatic shear localization in titanium and Ti-6 pct Al-4 pct V alloy [J]. Metallurgical Transactions A, 1985, 16(5): 761–775. DOI: 10.1007/BF02814827.
[37]	RITTEL D, LANDAU P, VENKERT A. Dynamic recrystallization as a potential cause for adiabatic shear failure [J]. Physical Review Letters, 2008, 101(16): 165501. DOI: 10.1103/PhysRevLett.101.165501.
[38]	READ H E, TRIPLETT J R, CECIL R A. Dislocation dynamics and the formulation of constitutive equations for rate-dependent plastic flow in metals [R]. La Jolla, CA, USA: Systems Science and Software, 1970.
[39]	MURR L E, MEYERS M A, NIOU C S, et al. Shock-induced deformation twinning in tantalum [J]. Acta Materialia, 1997, 45(1): 157–175. DOI: 10.1016/S1359-6454(96)00145-0.
[40]	李小飞, 左汝林, 林崇智. 镁合金塑性变形过程中孪生行为的研究 [J]. 热加工工艺, 2012, 41(4): 32–35. DOI: 10.3969/j.issn.1001-3814.2012.04.010. LI X F, ZUO R L, LIN C Z. Research on twinning behavior in plastic deformation of Mg alloy [J]. Hot Working Technology, 2012, 41(4): 32–35. DOI: 10.3969/j.issn.1001-3814.2012.04.010.
[41]	BARRETT C S, HALLER C T J R. Twinning in polycrystalline magnesium [J]. Transactions of the American Institute of Mining and Metallurgical Engineers, 1947, 171: 246–255.
[42]	MURR L E, ESQUIVEL E V. Observations of common microstructural issues associated with dynamic deformation phenomena: twins, microbands, grain size effects, shear bands, and dynamic recrystallization [J]. Journal of Materials Science, 2004, 39(4): 1153–1168. DOI: 10.1023/B:JMSC.0000013870.09241.c0.
[43]	CARRINGTON W E, GAYLER M L V. The use of flat-ended projectiles for determining dynamic yield stress Ⅲ: changes in microstructure caused by deformation under impact at high-striking velocities [J]. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 1948, 194(1038): 323–331. DOI: 10.1098/rspa.1948.0083.
[44]	MEYERS M A, XU Y B, XUE Q, et al. Microstructural evolution in adiabatic shear localization in stainless steel [J]. Acta Materialia, 2003, 51(5): 1307–1325. DOI: 10.1016/S1359-6454(02)00526-8.
[45]	ZERILLI F J, ARMSTRONG R W. Dislocation-mechanics-based constitutive relations for material dynamics calculations [J]. Journal of Applied Physics, 1987, 61(5): 1816–1825. DOI: 10.1063/1.338024.
[46]	唐志平. 冲击相变 [M]. 北京: 科学出版社, 2008.
[47]	唐志平. 冲击相变研究的现状与趋势 [J]. 高压物理学报, 1994, 8(1): 14–22. DOI: 10.11858/gywlxb.1994.01.003. TANG Z P. Some topics in shock-induced phase transitions [J]. Chinese Journal of High Pressure Physics, 1994, 8(1): 14–22. DOI: 10.11858/gywlxb.1994.01.003.
[48]	WU L, WANG K, XIAO S F, et al. Atomistic studies of shock-induced phase transformations in single crystal iron with cylindrical nanopores [J]. Computational Materials Science, 2016, 122: 1–10. DOI: 10.1016/j.commatsci.2016.05.010.
[49]	HUANG Y F, XIONG Y N, LI P, et al. Atomistic studies of shock-induced plasticity and phase transition in iron-based single crystal with edge dislocation [J]. International Journal of Plasticity, 2019, 114: 215–226. DOI: 10.1016/j.ijplas.2018.11.004.
[50]	KADAU K, GERMANN T C, LOMDAHL P S, et al. Atomistic simulations of shock-induced transformations and their orientation dependence in bcc Fe single crystals [J]. Physical Review B, 2005, 72(6): 064120. DOI: 10.1103/PhysRevB.72.064120.
[51]	SRINIVASAN S G, BASKES M I, WAGNER G J. Atomistic simulations of shock induced microstructural evolution and spallation in single crystal nickel [J]. Journal of Applied Physics, 2007, 101(4): 043504. DOI: 10.1063/1.2423084.
[52]	KALANTAR D H, BELAK J F, COLLINS G W, et al. Direct observation of the α-ε transition in shock-compressed iron via nanosecond X-ray diffraction [J]. Physical Review Letters, 2005, 95(7): 075502. DOI: 10.1103/physrevlett.95.075502.
[53]	HAWRELIAK J, COLVIN J D, EGGERT J H, et al. Analysis of the X-ray diffraction signal for the α-ε transition in shock-compressed iron: simulation and experiment [J]. Physical Review B, 2006, 74(18): 184107. DOI: 10.1103/PHYSREVB.74.184107.
[54]	HAWRELIAK J A, EL-DASHER B, LORENZANA H, et al. In situ X-ray diffraction measurements of the c/a ratio in the high-pressure ε phase of shock-compressed polycrystalline iron [J]. Physical Review B, 2011, 83(14): 144114. DOI: 10.1103/PhysRevB.83.144114.
[55]	LUO S N, JENSEN B J, HOOKS D E, et al. Gas gun shock experiments with single-pulse X-ray phase contrast imaging and diffraction at the Advanced Photon Source [J]. Review of Scientific Instruments, 2012, 83(7): 073903. DOI: 10.1063/1.4733704.
[56]	FAHR D. Stress- and strain-induced formation of martensite and its effects on strength and ductility of metastable austenitic stainless steels [J]. Metallurgical Transactions, 1971, 2(7): 1883–1892. DOI: 10.1007/BF02913420.
[57]	TALONEN J. Effect of strain-induced α′-martensite transformation on mechanical properties of metastable austenitic stainless steels [D]. Helsinki, Finland: Helsinki University of Technology, 2007.
[58]	STRINGFELLOW R G, PARKS D M, OLSON G B. A constitutive model for transformation plasticity accompanying strain-induced martensitic transformations in metastable austenitic steels [J]. Acta Metallurgica et Materialia, 1992, 40(7): 1703–1716. DOI: 10.1016/0956-7151(92)90114-T.
[59]	BOUQUEREL J, VERBEKEN K, DE COOMAN B C. Microstructure-based model for the static mechanical behaviour of multiphase steels [J]. Acta Materialia, 2006, 54(6): 1443–1456. DOI: 10.1016/j.actamat.2005.10.059.
[60]	DAN W J, ZHANG W G, LI S H, et al. A model for strain-induced martensitic transformation of TRIP steel with strain rate [J]. Computational Materials Science, 2007, 40(1): 101–107. DOI: 10.1016/j.commatsci.2006.11.006.
[61]	GARION C, SKOCZEŃ B, SGOBBA S. Constitutive modelling and identification of parameters of the plastic strain-induced martensitic transformation in 316L stainless steel at cryogenic temperatures [J]. International Journal of Plasticity, 2006, 22(7): 1234–1264. DOI: 10.1016/j.ijplas.2005.08.002.
[62]	IWAMOTO T, TSUTA T. Computational simulation of the dependence of the austenitic grain size on the deformation behavior of TRIP steels [J]. International Journal of Plasticity, 2000, 16(7/8): 791–804. DOI: 10.1016/S0749-6419(99)00079-0.
[63]	TOMITA Y, IWAMOTO T. Computational prediction of deformation behavior of TRIP steels under cyclic loading [J]. International Journal of Mechanical Sciences, 2001, 43(9): 2017–2034. DOI: 10.1016/S0020-7403(01)00026-1.
[64]	TANG Z P, GUO Y B. Three-dimensional constitutive model for shock-induced phase transition with N transforming phases [C] // Conference of the APS Topical Group on Shock Compression of Condensed Matter. Seattle, USA: Association for the Advancement of High Pressure Science and Technology, 2013
[65]	郭扬波, 刘方平, 载翔宇, 等. TiNi 合金的动态伪弹性行为和率相关相变本构模型 [J]. 爆炸与冲击, 2003, 23(2): 105–110. GUO Y B, LIU F P, ZAI X Y, et al. Dynamic pseudoelastic behavior of TiNi alloys and a strain rate dependent phase transition constitutive model [J]. Explosion and Shock Waves, 2003, 23(2): 105–110.
[66]	BECKER R. Plasticity, tenacity and recrystallization [J]. Zeitschrift für Technische Physik, 1926, 7: 547–555.
[67]	EYRING H. The activated complex in chemical reactions [J]. The Journal of Chemical Physics, 1935, 3(2): 107–115. DOI: 10.1063/1.1749604.
[68]	TAYLOR G I. The mechanism of plastic deformation of crystals. PartⅠ: theoretical [J]. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 1934, 145(855): 362–387. DOI: 10.1098/rspa.1934.0106.
[69]	KAUZMAN W. Flow of solid metals from the standpoint of the chemical rate theory [J]. Transactions of the American Institute of Mining, Metallurgical and Petroleum Engineers, 1941, 143(497): 57–83.
[70]	OROWAN E. Problems of plastic gliding [J]. Proceedings of the Physical Society, 1940, 52(1): 8. DOI: 10.1088/0959-5309/52/1/303.
[71]	ZENER C, HOLLOMON J H. Effect of strain rate upon plastic flow of steel [J]. Journal of Applied Physics, 1944, 15(1): 22–32. DOI: 10.1063/1.1707363.
[72]	TAYLOR G I. The testing of materials at high rates of loading [J]. Journal of the Institution of Civil Engineers, 1946, 26(8): 486–519. DOI: 10.1680/ijoti.1946.13699.
[73]	TAYLOR G I. The use of flat-ended projectiles for determining dynamic yield stressⅠ: theoretical considerations [J]. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 1948, 194(1038): 289–299. DOI: 10.1098/rspa.1948.0081.
[74]	WHIFFIN A C. The use of flat-ended projectiles for determining dynamic yield stressⅡ: tests on various metallic materials [J]. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 1948, 194(1038): 300–322. DOI: 10.1098/rspa.1948.0082.
[75]	DAVIES R M. A critical study of the Hopkinson pressure bar [J]. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 1948, 240(821): 375–457. DOI: 10.1098/rsta.1948.0001.
[76]	KOLSKY H. An investigation of the mechanical properties of materials at very high rates of loading [J]. Proceedings of the Physical Society: Section B, 1949, 62(11): 676–700. DOI: 10.1088/0370-1301/62/11/302.
[77]	JOHNSON G R, COOK W H. A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures [C] // Proceedings of the 7th International Symposium on Ballistics. The Hague, 1983: 541–547.
[78]	FIELDS D S, BACKOFEN W A. Determination of strain hardening characteristics by torsion testing [C] // Proceedings, American Society for Testing and Materials. 1957, 57: 1259–1272.
[79]	VOCE E. The relationship between stress and strain for homogeneous deformation [J]. Journal of the Institute of Metals, 1948, 74: 537–562.
[80]	KOCKS U F. Laws for work-hardening and low-temperature creep [J]. Journal of Engineering Materials and Technology, 1976, 98(1): 76–85. DOI: 10.1115/1.3443340.
[81]	MOLINARI A, RAVICHANDRAN G. Constitutive modeling of high-strain-rate deformation in metals based on the evolution of an effective microstructural length [J]. Mechanics of Materials, 2005, 37(7): 737–752. DOI: 10.1016/j.mechmat.2004.07.005.
[82]	KHAN A S, HUANG S J. Experimental and theoretical study of mechanical behavior of 1100 aluminum in the strain rate range 10^–5− 10⁴s^–1 [J]. International Journal of Plasticity, 1992, 8(4): 397–424. DOI: 10.1016/0749-6419(92)90057-J.
[83]	KHAN A S, LIANG R Q. Behaviors of three BCC metal over a wide range of strain rates and temperatures: experiments and modeling [J]. International Journal of Plasticity, 1999, 15(10): 1089–1109. DOI: 10.1016/S0749-6419(99)00030-3.
[84]	KHAN A S, LIANG R Q. Behaviors of three BCC metals during non-proportional multi-axial loadings: experiments and modeling [J]. International Journal of Plasticity, 2000, 16(12): 1443–1458. DOI: 10.1016/S0749-6419(00)00016-4.
[85]	SEEGER A. Kristallplastizität [M] // SEEGER A, DEHLINGER U. Kristallphysik Ⅱ/Crystal Physics Ⅱ. Heidelberg, Berlin, Germany: Springer, 1958: 1–210. DOI: 10.1007/978-3-642-45890-3_1.
[86]	ZERILLI F J, ARMSTRONG R W. Dislocation mechanics based constitutive relations for dynamic straining to tensile instability [C] // Shock Compression of Condensed Matter–1989: Proceedings of the American Physical Society Topical Conference. Albuquerque, New Mexico, USA, 1989.
[87]	ZERILLI F J, ARMSTRONG R W. Description of tantalum deformation behavior by dislocation mechanics based constitutive relations [J]. Journal of Applied Physics, 1990, 68(4): 1580–1591. DOI: 10.1063/1.346636.
[88]	ZERILLI F J, ARMSTRONG R W. Modeling shock waves with dislocation mechanics based constitutive relations [C] // SCHMIDT S C, DICK R D, FORBES J W, et al. Shock Compression of Condensed Matter–1991. Williamsburg, Virginia, USA: Elsevier, 1992: 257–260. DOI: 10.1016/b978-0-444-89732-9.50058-3.
[89]	ZERILLI F J, ARMSTRONG R W. Constitutive relations for titanium and Ti-6Al-4V [J]. AIP Conference Proceedings, 1996, 370(1): 315–318. DOI: 10.1063/1.50713.
[90]	ZERILLI F J, ARMSTRONG R W. Dislocation mechanics based analysis of material dynamics behavior: enhanced ductility, deformation twinning, shock deformation, shear instability, dynamic recovery [J]. Journal de Physique Ⅳ, 1997, 7(C3): 637–642. DOI: 10.1051/jp4:19973109.
[91]	FOLLANSBEE P S, KOCKS U F. A constitutive description of the deformation of copper based on the use of the mechanical threshold stress as an internal state variable [J]. Acta Metallurgica, 1988, 36(1): 81–93. DOI: 10.1016/0001-6160(88)90030-2.
[92]	NEMAT-NASSER S, LI Y F, ISAACS J B. Experimental/computational evaluation of flow stress at high strain rates with application to adiabatic shear banding [J]. Mechanics of Materials, 1994, 17(2/3): 111–134. DOI: 10.1016/0167-6636(94)90053-1.
[93]	BODNER S R, PARTOM Y. Constitutive equations for elastic-viscoplastic strain-hardening materials [J]. Journal of Applied Mechanics, 1975, 42(2): 385–389. DOI: 10.1115/1.3423586.
[94]	STEINBERG D J, COCHRAN S G, GUINAN M W. A constitutive model for metals applicable at high-strain rate [J]. Journal of Applied Physics, 1980, 51(3): 1498–1504. DOI: 10.1063/1.327799.
[95]	MECKING H, KOCKS U F. Kinetics of flow and strain-hardening [J]. Acta Metallurgica, 1981, 29(11): 1865–1875. DOI: 10.1016/0001-6160(81)90112-7.
[96]	STEINBERG D J, LUND C M. A constitutive model for strain rates from 10^–4 to 10⁶ s^–1 [J]. Journal de Physique Colloques, 1988, 49(C3): 433–440. DOI: 10.1051/jphyscol:1988362.
[97]	NEMAT-NASSER S, ISAACS J B. Direct measurement of isothermal flow stress of metals at elevated temperatures and high strain rates with application to Ta and Ta-W alloys [J]. Acta Materialia, 1997, 45(3): 907–919. DOI: 10.1016/S1359-6454(96)00243-1.
[98]	NEMAT-NASSER S, LI Y L. Flow stress of f.c.c. polycrystals with application to OFHC Cu [J]. Acta Materialia, 1998, 46(2): 565–577. DOI: 10.1016/S1359-6454(97)00230-9.
[99]	NEMAT-NASSER S, GUO W G, CHENG J Y. Mechanical properties and deformation mechanisms of a commercially pure titanium [J]. Acta Materialia, 1999, 47(13): 3705–3720. DOI: 10.1016/S1359-6454(99)00203-7.
[100]	NEMAT-NASSER S, GUO W G, LIU M Q. Experimentally-based micromechanical modeling of dynamic response of molybdenum [J]. Scripta Materialia, 1999, 40(7): 859–872. DOI: 10.1016/S1359-6462(99)00041-X.
[101]	NEMAT-NASSER S, GUO W G. Flow stress of commercially pure niobium over a broad range of temperatures and strain rates [J]. Materials Science and Engineering: A, 2000, 284(1/2): 202–210. DOI: 10.1016/S0921-5093(00)00740-1.
[102]	NEMAT-NASSER S, GUO W G. High strain-rate response of commercially pure vanadium [J]. Mechanics of Materials, 2000, 32(4): 243–260. DOI: 10.1016/S0167-6636(99)00056-3.
[103]	NEMAT-NASSER S, GUO W G, KIHL D P. Thermomechanical response of AL-6XN stainless steel over a wide range of strain rates and temperatures [J]. Journal of the Mechanics and Physics of Solids, 2001, 49(8): 1823–1846. DOI: 10.1016/S0022-5096(00)00069-7.
[104]	NEMAT-NASSER S, GUO W G. Thermomechanical response of DH-36 structural steel over a wide range of strain rates and temperatures [J]. Mechanics of Materials, 2003, 35(11): 1023–1047. DOI: 10.1016/S0167-6636(02)00323-X.
[105]	NEMAT-NASSER S, GUO W G. Thermomechanical response of HSLA-65 steel plates: experiments and modeling [J]. Mechanics of Materials, 2005, 37(2/3): 379–405. DOI: 10.1016/j.mechmat.2003.08.017.
[106]	GUO W G, NEMAT-NASSER S. Flow stress of Nitronic-50 stainless steel over a wide range of strain rates and temperatures [J]. Mechanics of Materials, 2006, 38(11): 1090–1103. DOI: 10.1016/j.mechmat.2006.01.004.
[107]	CHENG Y Q, ZHANG H, CHEN Z H, et al. Flow stress equation of AZ31 magnesium alloy sheet during warm tensile deformation [J]. Journal of Materials Processing Technology, 2008, 208(1): 29–34. DOI: 10.1016/j.jmatprotec.2007.12.095.
[108]	LIANG R Q, KHAN A S. A critical review of experimental results and constitutive models for BCC and FCC metals over a wide range of strain rates and temperatures [J]. International Journal of Plasticity, 1999, 15(9): 963–980. DOI: 10.1016/S0749-6419(99)00021-2.
[109]	RULE W K, JONES S E. A revised form for the Johnson-Cook strength model [J]. International Journal of Impact Engineering, 1998, 21(8): 609–624. DOI: 10.1016/s0734-743x(97)00081-x.
[110]	ZHANG H J, WEN W D, CUI H T. Behaviors of IC10 alloy over a wide range of strain rates and temperatures: experiments and modeling [J]. Materials Science and Engineering: A, 2009, 504(1/2): 99–103. DOI: 10.1016/j.msea.2008.10.056.
[111]	WANG J J, GUO W G, LI P H, et al. Modified Johnson-Cook description of wide temperature and strain rate measurements made on a nickel-base superalloy [J]. Materials at High Temperatures, 2017, 34(3): 157–165. DOI: 10.1080/09603409.2016.1252164.
[112]	KHAN A S, SUH Y S, KAZMI R. Quasi-static and dynamic loading responses and constitutive modeling of titanium alloys [J]. International Journal of Plasticity, 2004, 20(12): 2233–2248. DOI: 10.1016/j.ijplas.2003.06.005.
[113]	KHAN A S, ZHANG H Y. Mechanically alloyed nanocrystalline iron and copper mixture: behavior and constitutive modeling over a wide range of strain rates [J]. International Journal of Plasticity, 2000, 16(12): 1477–1492. DOI: 10.1016/S0749-6419(00)00024-3.
[114]	KHAN A S, SUH Y S, CHEN X, et al. Nanocrystalline aluminum and iron: mechanical behavior at quasi-static and high strain rates, and constitutive modeling [J]. International Journal of Plasticity, 2006, 22(2): 195–209. DOI: 10.1016/j.ijplas.2004.07.008.
[115]	FARROKH B, KHAN A S. Grain size, strain rate, and temperature dependence of flow stress in ultra-fine grained and nanocrystalline Cu and Al: synthesis, experiment, and constitutive modeling [J]. International Journal of Plasticity, 2009, 25(5): 715–732. DOI: 10.1016/j.ijplas.2008.08.001.
[116]	DURRENBERGER L, MOLINARI A, RUSINEK A. Internal variable modeling of the high strain-rate behavior of metals with applications to multiphase steels [J]. Materials Science and Engineering: A, 2008, 478(1/2): 297–304. DOI: 10.1016/j.msea.2007.06.011.
[117]	LIN Y C, LIU G. A new mathematical model for predicting flow stress of typical high-strength alloy steel at elevated high temperature [J]. Computational Materials Science, 2010, 48(1): 54–58. DOI: 10.1016/j.commatsci.2009.06.026.
[118]	TOROS S, OZTURK F. Modeling uniaxial, temperature and strain rate dependent behavior of Al-Mg alloys [J]. Computational Materials Science, 2010, 49(2): 333–339. DOI: 10.1016/j.commatsci.2010.05.019.
[119]	ABED F H, VOYIADJIS G Z. A consistent modified Zerilli-Armstrong flow stress model for BCC and FCC metals for elevated temperatures [J]. Acta Mechanica, 2005, 175(1): 1–18. DOI: 10.1007/s00707-004-0203-1.
[120]	NOBLE J P, HARDING J. An evaluation of constitutive relations for high-rate material behaviour using the tensile Hopkinson-bar [J]. Journal de Physique Ⅳ, 1994, 4(C8): 477–482. DOI: 10.1051/jp4:1994874.
[121]	LEE W S, LIU C Y. The effects of temperature and strain rate on the dynamic flow behaviour of different steels [J]. Materials Science and Engineering: A, 2006, 426(1/2): 101–113. DOI: 10.1016/j.msea.2006.03.087.
[122]	ÖZEL T, KARPAT Y. Identification of constitutive material model parameters for high-strain rate metal cutting conditions using evolutionary computational algorithms [J]. Materials and Manufacturing Processes, 2007, 22(5): 659–667. DOI: 10.1080/10426910701323631.
[123]	ZHANG H J, WEN W D, CUI H T, et al. A modified Zerilli-Armstrong model for alloy IC10 over a wide range of temperatures and strain rates [J]. Materials Science and Engineering: A, 2009, 527(1/2): 328–333. DOI: 10.1016/j.msea.2009.08.008.
[124]	SAMANTARAY D, MANDAL S, BORAH U, et al. A thermo-viscoplastic constitutive model to predict elevated-temperature flow behaviour in a titanium-modified austenitic stainless steel [J]. Materials Science and Engineering: A, 2009, 526(1/2): 1–6. DOI: 10.1016/j.msea.2009.08.009.
[125]	GAO C Y, ZHANG L C. A constitutive model for dynamic plasticity of FCC metals [J]. Materials Science and Engineering: A, 2010, 527(13/14): 3138–3143. DOI: 10.1016/j.msea.2010.01.083.
[126]	MCCORMIGK P G. A model for the Portevin-Le Chatelier effect in substitutional alloys [J]. Acta Metallurgica, 1972, 20(3): 351–354. DOI: 10.1016/0001-6160(72)90028-4.
[127]	LEE M H, KIM J H, CHOI B K, et al. Mechanical properties and dynamic strain aging behavior of Zr-1.5Nb-0.4Sn-0.2Fe alloy [J]. Journal of Alloys and Compounds, 2007, 428(1/2): 99–105. DOI: 10.1016/j.jallcom.2006.03.076.
[128]	VOYIADJIS G Z, ABED F H. Microstructural based models for bcc and fcc metals with temperature and strain rate dependency [J]. Mechanics of Materials, 2005, 37(2/3): 355–378. DOI: 10.1016/j.mechmat.2004.02.003.
[129]	TABEI A, ABED F H, VOYIADJIS G Z, et al. Constitutive modeling of Ti-6Al-4V at a wide range of temperatures and strain rates [J]. European Journal of Mechanics: A/Solids, 2017, 63: 128–135. DOI: 10.1016/j.euromechsol.2017.01.005.
[130]	VOYIADJIS G Z, SONG Y, RUSINEK A. Constitutive model for metals with dynamic strain aging [J]. Mechanics of Materials, 2019, 129: 352–360. DOI: 10.1016/j.mechmat.2018.12.012.
[131]	VOYIADJIS G Z, SONG Y. A physically based constitutive model for dynamic strain aging in Inconel 718 alloy at a wide range of temperatures and strain rates [J]. Acta Mechanica, 2020, 231(1): 19–34. DOI: 10.1007/s00707-019-02508-6.
[132]	SONG Y, VOYIADJIS G Z. Constitutive modeling of dynamic strain aging for HCP metals [J]. European Journal of Mechanics: A/Solids, 2020, 83: 104034. DOI: 10.1016/j.euromechsol.2020.104034.
[133]	RUSINEK A, KLEPACZKO J R. Shear testing of a sheet steel at wide range of strain rates and a constitutive relation with strain-rate and temperature dependence of the flow stress [J]. International Journal of Plasticity, 2001, 17(1): 87–115. DOI: 10.1016/S0749-6419(00)00020-6.
[134]	RUSINEK A, RODRÍGUEZ-MARTÍNEZ J A. Thermo-viscoplastic constitutive relation for aluminium alloys, modeling of negative strain rate sensitivity and viscous drag effects [J]. Materials & Design, 2009, 30(10): 4377–4390. DOI: 10.1016/j.matdes.2009.04.011.
[135]	RUSINEK A, RODRÍGUEZ-MARTÍNEZ J A, ARIAS A. A thermo-viscoplastic constitutive model for FCC metals with application to OFHC copper [J]. International Journal of Mechanical Sciences, 2010, 52(2): 120–135. DOI: 10.1016/j.ijmecsci.2009.07.001.
[136]	KAPOOR R, NEMAT-NASSER S. Comparison between high and low strain-rate deformation of tantalum [J]. Metallurgical and Materials Transactions A, 2000, 31(3): 815–823. DOI: 10.1007/s11661-000-0025-2.
[137]	PRESTON D L, TONKS D L, WALLACE D C. Model of plastic deformation for extreme loading conditions [J]. Journal of Applied Physics, 2003, 93(1): 211–220. DOI: 10.1063/1.1524706.
[138]	KIM J B, SHIN H. Comparison of plasticity models for tantalum and a modification of the PTW model for wide ranges of strain, strain rate, and temperature [J]. International Journal of Impact Engineering, 2009, 36(5): 746–753. DOI: 10.1016/j.ijimpeng.2008.11.003.
[139]	GOETZ R L, SEETHARAMAN V. Modeling dynamic recrystallization using cellular automata [J]. Scripta Materialia, 1998, 38(3): 405–413. DOI: 10.1016/s1359-6462(97)00500-9.
[140]	ROWLEY M A, THORNTON E A. Constitutive modeling of the visco-plastic response of Hastelloy-X and aluminum alloy 8009 [J]. Journal of Engineering Materials and Technology, 1996, 118(1): 19–27. DOI: 10.1115/1.2805928.
[141]	宋迎东, 王舸, 高德平. 一种弹-粘塑性本构模型材料常数的估计方法 [J]. 固体力学学报, 2000, 21(2): 152–156. DOI: 10.19636/j.cnki.cjsm42-1250/o3.2000.02.009. SONG Y D, WANG G, GAO D P. Material constants estimation method of an elastic-viscoplastic constitutive model [J]. Acta Mechanica Solida Sinica, 2000, 21(2): 152–156. DOI: 10.19636/j.cnki.cjsm42-1250/o3.2000.02.009.
[142]	SHI D Q, YANG X G, WANG Y R. Improvement on the modeling of rate-dependent plasticity and cyclic hardening by bodner-partom model [J]. Chinese Journal of Aeronautics, 2005, 18(1): 83–89. DOI: 10.1016/S1000-9361(11)60287-2.
[143]	石多奇, 杨晓光, 王延荣, 等. Udimet 720 Li材料BP型粘塑性本构建模研究 [J]. 北京航空航天大学学报, 2003, 29(7): 627–630. DOI: 10.13700/j.bh.1001-5965.2003.07.015. SHI D Q, YANG X G, WANG Y R, et al. B-P viscoplastic constitutive modeling of Udimet 720 Li [J]. Journal of Beijing University of Aeronautics and Astronautics, 2003, 29(7): 627–630. DOI: 10.13700/j.bh.1001-5965.2003.07.015.
[144]	HOLMQUIST T J, JOHNSON G R. Determination of constants and comparison of results for various constitutive models [J]. Journal de Physique Ⅳ, 1991, 1(C3): 853–860. DOI: 10.1051/jp4:19913119.
[145]	SAMANTARAY D, MANDAL S, BHADURI A K. A comparative study on Johnson Cook, modified Zerilli-Armstrong and Arrhenius-type constitutive models to predict elevated temperature flow behaviour in modified 9Cr-1Mo steel [J]. Computational Materials Science, 2009, 47(2): 568–576. DOI: 10.1016/j.commatsci.2009.09.025.
[146]	LI H Y, WANG X F, WEI D D, et al. A comparative study on modified Zerilli-Armstrong, Arrhenius-type and artificial neural network models to predict high-temperature deformation behavior in T24 steel [J]. Materials Science and Engineering: A, 2012, 536: 216–222. DOI: 10.1016/j.msea.2011.12.108.
[147]	CHEN C, YIN H Q, HUMAIL I S, et al. A comparative study of a back propagation artificial neural network and a Zerilli-Armstrong model for pure molybdenum during hot deformation [J]. International Journal of Refractory Metals and Hard Materials, 2007, 25(5/6): 411–416. DOI: 10.1016/j.ijrmhm.2006.11.004.
[148]	LI J, LI F G, CAI J, et al. Comparative investigation on the modified Zerilli-Armstrong model and Arrhenius-type model to predict the elevated-temperature flow behaviour of 7050 aluminium alloy [J]. Computational Materials Science, 2013, 71: 56–65. DOI: 10.1016/j.commatsci.2013.01.010.
[149]	SHAMSOLHODAEI A, ZAREI-HANZAKI A, GHAMBARI M, et al. The high temperature flow behavior modeling of NiTi shape memory alloy employing phenomenological and physical based constitutive models: a comparative study [J]. Intermetallics, 2014, 53: 140–149. DOI: 10.1016/j.intermet.2014.04.015.
[150]	CAI J, WANG K S, HAN Y Y. A comparative study on Johnson Cook, modified Zerilli-Armstrong and Arrhenius-type constitutive models to predict high-temperature flow behavior of Ti-6Al-4V alloy in α+β phase [J]. High Temperature Materials and Processes, 2016, 35(3): 297–307. DOI: 10.1515/htmp-2014-0157.
[151]	WANG J, ZHAO G Q, CHEN L, et al. A comparative study of several constitutive models for powder metallurgy tungsten at elevated temperature [J]. Materials & Design, 2016, 90: 91–100. DOI: 10.1016/j.matdes.2015.10.114.
[152]	LIU Y, LI M, REN X W, et al. Flow stress prediction of Hastelloy C-276 alloy using modified Zerilli-Armstrong, Johnson-Cook and Arrhenius-type constitutive models [J]. Transactions of Nonferrous Metals Society of China, 2020, 30(11): 3031–3042. DOI: 10.1016/S1003-6326(20)65440-1.
[153]	ZHANG W W, YAO Y L. Micro scale laser shock processing of metallic components [J]. Journal of Manufacturing Science and Engineering, 2002, 124(2): 369–378. DOI: 10.1115/1.1445149.
[154]	FAN Y, WANG Y, VUKELIC S, et al. Wave-solid interactions in laser-shock-induced deformation processes [J]. Journal of Applied Physics, 2005, 98(10): 104904. DOI: 10.1063/1.2134882.
[155]	COLOMBIER J P, COMBIS P, BONNEAU F, et al. Hydrodynamic simulations of metal ablation by femtosecond laser irradiation [J]. Physical Review B, 2005, 71(16): 165406. DOI: 10.1103/PhysRevB.71.165406.
[156]	AUSTIN R A, MCDOWELL D L. A dislocation-based constitutive model for viscoplastic deformation of fcc metals at very high strain rates [J]. International Journal of Plasticity, 2011, 27(1): 1–24. DOI: 10.1016/j.ijplas.2010.03.002.
[157]	DEITERDING R, RADOVITZKY R, MAUCH S P, et al. A virtual test facility for the efficient simulation of solid material response under strong shock and detonation wave loading [J]. Engineering with Computers, 2006, 22(3/4): 325–347. DOI: 10.1007/s00366-006-0043-9.
[158]	REMINGTON B A, BAZAN G, BELAK J, et al. Materials science under extreme conditions of pressure and strain rate [J]. Metallurgical and Materials Transactions A, 2004, 35(9): 2587–2607. DOI: 10.1007/s11661-004-0205-6.
[159]	PIERAZZO E, ARTEMIEVA N, ASPHAUG E, et al. Validation of numerical codes for impact and explosion cratering: impacts on strengthless and metal targets [J]. Meteoritics & Planetary Science, 2008, 43(12): 1917–1938. DOI: 10.1111/j.1945-5100.2008.tb00653.x.
[160]	POLIAK E I, JONAS J J. A one-parameter approach to determining the critical conditions for the initiation of dynamic recrystallization [J]. Acta Materialia, 1996, 44(1): 127–136. DOI: 10.1016/1359-6454(95)00146-7.
[161]	NES E. Modelling of work hardening and stress saturation in FCC metals [J]. Progress in Materials Science, 1997, 41(3): 129–193. DOI: 10.1016/S0079-6425(97)00032-7.
[162]	CHICHILI D R, RAMESH K T, HEMKER K J. The high-strain-rate response of alpha-titanium: experiments, deformation mechanisms and modeling [J]. Acta Materialia, 1998, 46(3): 1025–1043. DOI: 10.1016/S1359-6454(97)00287-5.
[163]	BOUAZIZ O, GUELTON N. Modelling of TWIP effect on work-hardening [J]. Materials Science and Engineering: A, 2001, 319/320/321: 246–249. DOI: 10.1016/S0921-5093(00)02019-0.
[164]	DING R, GUO Z X. Coupled quantitative simulation of microstructural evolution and plastic flow during dynamic recrystallization [J]. Acta Materialia, 2001, 49(16): 3163–3175. DOI: 10.1016/S1359-6454(01)00233-6.
[165]	KNEZEVIC M, LEVINSON A, HARRIS R, et al. Deformation twinning in AZ31: influence on strain hardening and texture evolution [J]. Acta Materialia, 2010, 58(19): 6230–6242. DOI: 10.1016/j.actamat.2010.07.041.
[166]	BANERJEE B. The Mechanical Threshold Stress model for various tempers of AISI 4340 steel [J]. International Journal of Solids and Structures, 2007, 44(3/4): 834–859. DOI: 10.1016/j.ijsolstr.2006.05.022.
[167]	GATTIKER J, HIGDON D, KELLER-MCNULTY S, et al. Combining experimental data and computer simulations, with an application to flyer plate experiments [J]. Bayesian Analysis, 2006, 1(4): 765–792. DOI: 10.1214/06-BA125.
[168]	PARK H S, LORENZ K T, CAVALLO R M, et al. Viscous Rayleigh-Taylor instability experiments at high pressure and strain rate [J]. Physical Review Letters, 2010, 104(13): 135504. DOI: 10.1103/PhysRevLett.104.135504.
[169]	BARTON N R, BERNIER J V, BECKER R, et al. A multiscale strength model for extreme loading conditions [J]. Journal of Applied Physics, 2011, 109(7): 073501. DOI: 10.1063/1.3553718.
[170]	RAVELO R, GERMANN T C, GUERRERO O, et al. Shock-induced plasticity in tantalum single crystals: interatomic potentials and large-scale molecular-dynamics simulations [J]. Physical Review B, 2013, 88(13): 134101. DOI: 10.1103/PhysRevB.88.134101.
[171]	JIN Z Y, LIU J, CUI Z S, et al. Identification of nucleation parameter for cellular automaton model of dynamic recrystallization [J]. Transactions of Nonferrous Metals Society of China, 2010, 20(3): 458–464. DOI: 10.1016/S1003-6326(09)60162-X.
[172]	JIN Z Y, CUI Z S. Investigation on strain dependence of dynamic recrystallization behavior using an inverse analysis method [J]. Materials Science and Engineering: A, 2010, 527(13/14): 3111–3119. DOI: 10.1016/j.msea.2010.01.062.
[173]	CHEN F, CUI Z S, LIU J, et al. Mesoscale simulation of the high-temperature austenitizing and dynamic recrystallization by coupling a cellular automaton with a topology deformation technique [J]. Materials Science and Engineering: A, 2010, 527(21/22): 5539–5549. DOI: 10.1016/j.msea.2010.05.021.
[174]	LIN Y C, ZHANG J, ZHONG J. Application of neural networks to predict the elevated temperature flow behavior of a low alloy steel [J]. Computational Materials Science, 2008, 43(4): 752–758. DOI: 10.1016/j.commatsci.2008.01.039.
[175]	LI H Y, WEI D D, LI Y H, et al. Application of artificial neural network and constitutive equations to describe the hot compressive behavior of 28CrMnMoV steel [J]. Materials & Design, 2012, 35: 557–562. DOI: 10.1016/j.matdes.2011.08.049.
[176]	RAO K P, PRASAD Y K D V. Neural network approach to flow stress evaluation in hot deformation [J]. Journal of Materials Processing Technology, 1995, 53(3/4): 552–566. DOI: 10.1016/0924-0136(94)01744-L.
[177]	HODGSON P D, KONG L X, DAVIES C H J. The prediction of the hot strength in steels with an integrated phenomenological and artificial neural network model [J]. Journal of Materials Processing Technology, 1999, 87(1): 131–138. DOI: 10.1016/S0924-0136(98)00344-6.
[178]	JI G L, LI F G, LI Q H, et al. A comparative study on Arrhenius-type constitutive model and artificial neural network model to predict high-temperature deformation behaviour in Aermet100 steel [J]. Materials Science and Engineering: A, 2011, 528(13/14): 4774–4782. DOI: 10.1016/j.msea.2011.03.017.
[179]	SABOKPA O, ZAREI-HANZAKI A, ABEDI H R, et al. Artificial neural network modeling to predict the high temperature flow behavior of an AZ81 magnesium alloy [J]. Materials & Design, 2012, 39: 390–396. DOI: 10.1016/j.matdes.2012.03.002.
[180]	HAGHDADI N, ZAREI-HANZAKI A, KHALESIAN A R, et al. Artificial neural network modeling to predict the hot deformation behavior of an A356 aluminum alloy [J]. Materials & Design, 2013, 49: 386–391. DOI: 10.1016/j.matdes.2012.12.082.
[181]	GAO T J, ZHAO D, ZHANG T W, et al. Strain-rate-sensitive mechanical response, twinning, and texture features of NiCoCrFe high-entropy alloy: experiments, multi-level crystal plasticity and artificial neural networks modeling [J]. Journal of Alloys and Compounds, 2020, 845: 155911. DOI: 10.1016/j.jallcom.2020.155911.
[182]	EDGERTON M, RYAN S. An artificial neural network based constitutive model for predicting the response of a high-strength steel [C] // 30th International Symposium on Ballistics. Long Beach, CA, USA, 2017: 11–15.
[183]	BOBBILI R, MADHU V. Constitutive modeling and fracture behavior of a biomedical Ti-13Nb-13Zr alloy [J]. Materials Science and Engineering: A, 2017, 700: 82–91. DOI: 10.1016/j.msea.2017.05.113.
[184]	STOFFEL M, BAMER F, MARKERT B. Neural network based constitutive modeling of nonlinear viscoplastic structural response [J]. Mechanics Research Communications, 2019, 95: 85–88. DOI: 10.1016/j.mechrescom.2019.01.004.
[185]	黄志斌, 万敏, 伍惠, 等. TC4钛合金神经网络本构模型及在有限元模拟中应用 [J]. 塑性工程学报, 2013, 20(1): 89–94. DOI: 10.3969/j.issn.1007-2012.2013.01.019. HUANG Z B, WAN M, WU H, et al. Constitutive model of Ti-6Al-4V alloy based on artificial neural network and its application on FEM simulation [J]. Journal of Plasticity Engineering, 2013, 20(1): 89–94. DOI: 10.3969/j.issn.1007-2012.2013.01.019.
[186]	LEFIK M, SCHREFLER B A. Artificial neural network as an incremental non-linear constitutive model for a finite element code [J]. Computer Methods in Applied Mechanics and Engineering, 2003, 192(28): 3265–3283. DOI: 10.1016/S0045-7825(03)00350-5.
[187]	ALI U, MUHAMMAD W, BRAHME A, et al. Application of artificial neural networks in micromechanics for polycrystalline metals [J]. International Journal of Plasticity, 2019, 120: 205–219. DOI: 10.1016/j.ijplas.2019.05.001.

施引文献

期刊类型引用(3)

1.	贾时雨，王成，徐文龙，马东，齐方方. 环形复合内衬头盔冲击波防护性能研究. 兵工学报. 2025(01): 60-69 . 百度学术
2.	黄浩，崔海林，田晓丽，吴浩. 多孔结构对冲击波的衰减影响研究. 机械设计与制造工程. 2024(01): 11-15 . 百度学术
3.	常利军，陈泰伟，王天昊，蔡志华. 弹体冲击载荷下头部损伤与防护研究进展. 兵器装备工程学报. 2024(07): 208-216 . 百度学术