﻿ 基于连续体损伤理论的硼钢高温成形极限确定法
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 力学学报  2016, Vol. 48 Issue (1): 146-153  DOI: 10.6052/0459-1879-15-314 0

### 引用本文 [复制中英文]

[复制中文]
Tang Bingtao. PREDICTION OF FORMING LIMIT OF BORON STEEL AT ELEVATED TEMPERATRUE BASED ON CDM THEORY[J]. Chinese Journal of Ship Research, 2016, 48(1): 146-153. DOI: 10.6052/0459-1879-15-314.
[复制英文]

### 文章历史

2015-08-17 收稿
2015-09-28 录用
2015–09–28 网络版发表

1 勒迈特损伤模型 1.1 损伤变量

 $D_n = \dfrac{S - \tilde {S}}{S}$ (1)

1.2 有效应力

 $\tilde {\sigma } = \dfrac{F}{\tilde {S}}$ (2)

 $\tilde {\sigma } = \dfrac{F}{S(1 - D)}$ (3)

 $E = E_0 \left( {1 - D} \right)$ (4)

1.3 损伤演化方程

 $\sigma^{\rm D} = \sigma_{\rm eq } R_{\rm v}^{1/2}$ (5)

 $R_{\rm v} = \dfrac{2}{3}\left( {1 + \nu } \right) + 3\left( {1 - 2\nu }\right){\left( {\dfrac{\sigma_{\rm m} }{\sigma_{{\rm eq}} }} \right)}^2$ (6)

 $- Y = \dfrac{\sigma_{\rm eq}^2 }{2E{\left( {1 - D} \right)}^2 }\left[{\dfrac{2}{3}\left( {1 + \nu } \right) + 3\left( {1 - 2\nu } \right)\left( {\dfrac{\sigma_{\rm m}}{\sigma_{{\rm eq}} }} \right)^2 } \right]$ (7)

 $F = F_1 \left( {\sigma_{ij} ,R,D}\right) + F_{\rm D} \left( { - Y,p,D} \right)$ (8)

 $p = \varepsilon _{{\rm{eq}}}^{\rm{p}} = \int {_0^t} \dot \varepsilon _{{\rm{eq}}}^{\rm{p}}{\rm{ d}}t$ (9)

 $\dot {\varepsilon }_{{\rm eq}}^{\rm p} = \sqrt {\dfrac{2}{3}\dot {\varepsilon }_{ ij }^{\rm p} \dot{\varepsilon }_{ ij }^{\rm p} }$ (10)

$\dot {\varepsilon }_{{\rm eq}}^{\rm p}$是等效塑性应变率，$\dot {\varepsilon }_{ ij }^{\rm p}$是应变率张量的塑性部分. 对于满足米塞斯屈服准则的材料，$F_1$ 表示为

 $F_1 = \dfrac{\sigma_{{\rm eq}} - R}{1 - D} - \sigma_{\rm 0} = \sigma_{{\rm eq}}^{\rm \ast } - \sigma_{\rm y}$ (11)

 $\sigma_{\rm y} = \dfrac{R}{1 - D} + \sigma_{\rm 0}$ (12)

 $\dot {\varepsilon }_{ ij }^{\rm p} = \dot {\lambda }\dfrac{\partial F_1 }{\partial\sigma_{ij} } = \dfrac{\dot {\lambda }}{1 - D}\dfrac{3{\sigma }'_{ij} }{2\sigma_{{\rm eq}} }$ (13)

 $\dot {p} = \dot {\lambda }\dfrac{\partial F_1 }{\partial \left( { - R} \right)} =\dfrac{\dot {\lambda }}{1 - D}$ (14)

 $\dot {D} = \dot {\lambda }\dfrac{\partial F_{\rm D} }{\partial \left( { - Y} \right)}$ (15)

 $\dot {\lambda } = \left( {1 - D} \right)\dot {\varepsilon}_{{\rm eq}}^{\rm p}$ (16)

 $F_{\rm D} = \dfrac{S_0 }{\left( {b + 1} \right)}\dfrac{1}{\left( {1 - D} \right)}{\left( {\dfrac{ - Y}{ S_0}} \right)}^{b + 1}$ (17)

 $\dot {D} = {\left( { - \dfrac{Y}{S_0 }} \right)}^b \dot {\varepsilon}_{{\rm eq}}^{\rm p} = {\left( { - \dfrac{Y}{S_0 }} \right)}^b \dfrac{\dot {\lambda }}{1 - D}$ (18)

 $F_{\rm D} = \dfrac{S_0 }{\left( {b + 1} \right)\left( {1 - D} \right)} {\left( {\dfrac{ - Y}{S_0 }} \right)}^{b + 1} {\left( {\dfrac{1}{\bar {\varepsilon }_{\rm p} }} \right)}^\alpha$ (19)

 $\dot {D} = \dot {\lambda }\dfrac{\partial F_{\rm D} }{\partial Y} = {\left( { - \dfrac{Y}{S_0 }}\right)}^b \dfrac{\dot {\lambda }}{1 - D}{\left({\dfrac{1}{\bar {\varepsilon }_{\rm p} }} \right)}^\alpha$ (20)

2 损伤模型优化 2.1 "GLEEBLE"热模拟拉伸试验

 图 1 试样形状及尺寸 Fig. 1 Schematic sketch and dimensions of tensile specimen

 图 2 试样在热拉伸前的温度分布 Fig. 2 Temperature distribution before tensile test
2.2 汉森-斯皮特尔(Hensel-Spittel)本构模型

 $\sigma = A {\rm e}^{m_1 T} \varepsilon^{m_2 } \dot {\varepsilon }^{m_3 } {\rm e}^{m_4/\varepsilon} {\left( {1 + \varepsilon } \right)}^{m_5 T} \cdot \\ \qquad {\rm e}^{m_7 \varepsilon }\dot {\varepsilon }^{m_8 T} T^{m_9 }$ (21)

 图 3 汉森-斯皮特尔模型与实验结果对比 Fig. 3 Comparison between the 22MnB5 experimental data and fitted ones through the Hensel-Spittel model
2.3 优化过程

(1)定义损伤变量$\varepsilon_{\rm D}$，$S_0$，$b$，$\alpha$和$D_{\rm c}$;

(2)选取优化算法-基于遗传算法的多目标优化算法NSGA-II [18];

(3)有限元分析软件《FORGE$^{\rm TM}$》模拟;

(4)执行实验和模拟获得的力-位移曲线比较程序;

(5)评估实验数据与模拟结果的相似度;

(6)修改损伤变量值返回步骤(2)直至(5)获得满足.

 $\varphi = \dfrac{\sum\limits_i { {\left[{{\left( {y_i^{\exp } - y_i^{\rm num} } \right)}^2 \left( {x_i - x_{i - 1} } \right)}\right]}} }{\sum\limits_i { {\left[{{\left( {y_i^{\exp } } \right)}^2 \left( {x_i -x_{i - 1} } \right)} \right]} } }$ (22)

 图 4 有限元模拟结果与实验结果的比对 Fig. 4 Comparison between the experimental data and the simulation results
3 损伤模型的验证

3.1 高温凸模胀形试验

 图 5 凸模胀形试验装置 Fig. 5 Nakazima test experimental apparatus

(1)利用电阻式加热棒将模具(包括凸模、凹模、压边圈)加热至指定温度，并在整个实验过程中保持恒温. 本实验选取实验温度为600℃为例说明；

(2)利用感性线圈将硼钢试样加热至900℃，并保温2$\sim$3 min至充分奥氏体化；

(3)利用安装在凹模面上的高压喷嘴将加热的硼钢试样快速($>50{^\circ}$C)冷却至600℃，确保冷却过程没有相变发生(奥氏体状态)[19]

(4)凸模以10 mm/s的速度进行拉深涨形，直至硼钢试样发生开裂(拉深力突然下降，停机).

3.2 高温凸模胀形实验成形极限曲线预测

 图 6 高温凸模双向等拉实验(1/4 部分) Fig. 6 Nakazima test in case of biaxial tensile experiment (one quarter)

 图 7 高温凸模双向等拉实验拉裂后的数值模型 Fig. 7 Numerical fracture site of Nakazima test in case of biaxial tensile experiment

 图 8 高温凸模双向等拉实验拉裂试样 Fig. 8 Experimental fracture site of Nakazima test in case of biaxial tensile experiment

 图 9 凸模胀形试验中力-位移曲线对比 Fig. 9 Comparison between experimental and numerical load-displacement curves in Nakazima test

 图 10 模拟与实验获得的成形极限曲线 Fig. 10 Comparison between experimental and numerically predicted FLCs at necking and fracture
4 结 论

(1)推导了勒迈特韧性损伤模型的损伤演化方程，为了准确反映硼钢在奥氏体状态下的成形特性，通过增加损伤变量$\alpha$，对经典的勒迈特耗散势函数进行了修正. 提出在勒迈特损伤模型中，需要确定的参数有$\varepsilon_{\rm D}$，$S_0$，$b$，$\alpha$和$D_{\rm c}$.

(2)提出了将韧性损伤模型与有限元数值模拟技术结合起来确定勒迈特损伤模型. 依据"GLEEBLE"热模拟拉伸实验获得的数据，利用优化算法获得与损伤有关的各参数$\varepsilon_{\rm D}$，$S_0$，$b$，$\alpha$和$D_{\rm c}$的值. 并利用汉森-斯皮特尔本构模型准确描述了硼钢在高温下塑性流动规律.

(3)在数值模型中采用优化后的损伤变量，发现勒迈特损伤模型可以较为准确地预测高温凸模胀形实验中力-位移变化规律，尤其是损伤断裂阶段力的急剧下降过程，能够较为准确地描述硼钢在高温条件下的变形规律，预测成性缺陷及损伤断裂过程. 通过数值模拟结果与高温凸模胀形实验数据的对比可知，利用优化的勒迈特损伤模型可以获得较为理想的成形极限预测图，为预测硼钢在高温状态下成形极限提供有益参考.

 1 余寿文, 冯西桥. 损伤力学. 北京: 清华大学出版社, 1997: 1-6 (Yu Shouwen, Feng Xiqiao. Damage Mechanics. Beijing: Tsinghua University Press, 1997: 1-6 (in Chinese)) 2 Rabotnov YN. On the equations of state for creep. Progress in Applied Mechanics, 1963, 12: 307-315 3 Kachanov LM. Introduction to Continuum Damage Mechanics. Netherlands: Martinus Nijho Publishers, 1986 4 Kuhn HA, Downey CL. Deformation characteristic and plasticity theory of sintered powder materials. International Journal of Powder Metallurgy, 1971, 7(15): 78-88 5 Tvergaard V. Influence of voids on shear bands instabilities under plane strain conditions. International Journal of Fracture, 1981,17: 389-407 6 Tvergaard V. On localization in ductile materials containing spherical voids. International Journal of Fracture, 1982, 18: 157-169 7 Tvergaard V, Needleman A. Analysis of the cup-cone fracture in a round tensile bar. Acta Metallurgica, 1984, 32: 157-169 8 Keeler P. Circular grid system—A valuable aid for evaluating sheet metal formability. SAE Transactions, 1968, (680092): 371-379 9 Goodwin M. Application of strain analysis to sheet metal forming problems in the press shop. SAE Transactions, 1968, (680093): 380-387 10 Gelin JC. Modeling of damage in metal forming processes. Journal of Materials Processing Technology, 1998, 80-81: 24-32 11 Lemaitre J. A Course on Damage Mechanics. 3rd edn. New York: Springer-Verlag, 1998 12 Tang CY, Chow CL. Development of a damage-based criterion for ductile fracture predictive in sheet metal forming. Journal of Materials Processing Technology, 1999, 91: 270-277 13 周筑宝, 卢楚芬. 连续损伤力学的几个基本问题. 湘潭大学自然科学学报, 1989, 11(4): 23-33 (Zhou Zhubao, Lu Chufen. Some basic problems in continuous damage mechanics. Natural Science Journal of Xiangtan University, 1998, 11(4): 23-33 (in Chinese)) 14 马鑫, 钱乙余, Yoshida F. 相对损伤应力——Lemaitre等效损伤应力概念的修正及其在温度循环载荷下的应用. 自然科学进展,2001, 11(1): 86-89 (Ma Xin, Qian Yiyu, Yoshida F. Relative damage stress: Modification of Lemaitre e ective stress and its application in case of cycling temperature load. Progress in Natural Science,2001, 11(1): 86-89 (in Chinese)) 15 Lechler J, Merklein M, Geiger M. Determination of thermal and mechanical material properties of ultra high strength steels for hot stamping. Steel Research International, 2008, 2: 98-104 16 Tang BT, Yuan ZJ, Cheng G, et al. Experimental verification of tailor welded joining partners for hot stamping and analytical modeling of TWBs rheological constitutive in austenitic state. Materials Science & Engineering A, 2013, 585(15): 304-318 17 Hensel A, Spittel T. Kraft-und Arbeitsbedarf Bildsomer Formgeburgs Verfahren. Lipsk: VEB Deutscher Verlang für Grundsto ndustrie,1979 18 Deb K, Pratap A, Agarwal S, et al. A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation, 2002, 6: 182-197 19 Turetta A, Bruschi S, Ghiotti A. Investigation of 22MnB5 formability in hot stamping operations. Journal of Materials Processing Technology, 2006, 177: 396-400 20 Tang BT, Bruschi S, Ghiotti A, et al. Numerical modelling of the tailored tempering process applied to 22MnB5 sheets. Finite Elements in Analysis and Design, 2014, 81: 69-81 21 ISO 12004-1:2008 金属材料薄板和薄带成形极限曲线的测定(第1部分): 冲压车间成形极限图的测量及应用, 2008 (ISO 12004-1:2008 Measurement and application of forming limit diagram in press shop (in Chinese))
PREDICTION OF FORMING LIMIT OF BORON STEEL AT ELEVATED TEMPERATRUE BASED ON CDM THEORY
Tang Bingtao
Institute of Engineering Mechanics, Shandong Jianzhu University, Jinan 250101, China
Abstract: To meet the requirement of energy saving and automobile safety, more and more attentions are paid to the development and application of high strength steels. Hot stamping of ultra high strength steel is an important way to reduce the weight of body-in-white, improve the anti-impact and anti-collision performance of the vehicle, and has become a hot technology in the world wide. The forming limit is a very important parameter in boron steel hot stamping process. Due to the complexity of the Nakazima test at elevated temperature, there are a lot of problems to be solved. In the paper, the ductile damage model and finite element simulation technology were combined to predict the forming limit of boron steel at elevated temperature. Based on the continuum damage mechanics (CDM), the Lemaitre ductile damage model considering the e ect of e ective stress and e ective plastic strain on evolution of microvoid was established. In order to accurately predict the formability of boron steel at elevated temperature, a modified dissipation potential coupling the e ect of plastic strain was introduced. Genetic Algorithm-II (NSGA-II) was integrated with finite element software FORGETM to find the optimized ductile damage parameters. The obtained damage parameters can accurately predict the sheet instability and fracture behaviour in tensile test. The established ductile damage model is introduced in the simulation of Nakazima test with boron steel. The comparison between the predicted forming limit and that of experiment witnesses the reliability of the forming limit prediction at high temperature using the proposed ductile damage model.
Key words: hot stamping    forming limit    Nakazima test    damage model    boron steel