1 北京科技大学材料科学与工程学院, 北京100083
2 安徽工业大学冶金工程学院, 马鞍山243000

MODEL OF THE EFFECT OF GRAIN SIZE ON PLASTI-CITY IN ULTRA-FINE GRAIN SIZE STEELS
Jin LIU1, Guohui ZHU1,2
1 School of Materials Science and Engineering, University of Science and Technology Beijing, Beijing 100083
2 School of Metallurgical Engineering, Anhui University of Technology, Maanshan 243000
Correspondent: ZHU Guohui, professor, Tel: (0555)2316651, E-mail:zhugh@ahut.edu.cn
Abstract

Based on our earlier preliminary work, a model was developed for prediction of the critical grain size where the plasticity would be decreased as the grain refined. In the model the effect of grain size on the fracture strength was combined. The prediction of the model exhibited that in the range of grain size of 10 mm to 0.2 mm as an example, the total elongation of the steels would be firstly increased. But when the grain size was refined to 2.5 mm and below, the total elongation of the steels was not increased but decreased sharply, which was good agreement with the experimental results published recently. Present work illustrated that the dominant mechanism of the elongation decreased in the ultra-fine grain size materials is due to increase in resistance force of grain boundaries on the dislocation sources resulting in the difficulty of activation of dislocation movements. Its expression would be the decrease of the plastic strain in macro-level.

Keyword: plasticity; ultrafine grain size; grain size; dislocation pile-up; dislocation source

1 材料断裂强度与晶粒尺寸的关系模型

 (1) $τf=4Eγsπ(1-ν)d$

 (2) $σf=τfμmax$

 Figure Option Fig.1 Relationship between fracture strength σf and grain size d in polycrystalline materials图1 多晶体材料的断裂强度 σf 随晶粒尺寸d的变化

2 晶粒尺寸对材料总塑性伸长率的影响模型
2.1 平均取向因子的计算

$σf$是多晶体材料的宏观力学性能. 在实际的多晶体材料中, 各个晶粒具有不同的晶体学取向, 因此, 当宏观的拉伸应力等于 $σf$时, 作用在各个晶粒的滑移系上的分切应力并不相同. 因此, 要讨论多晶体材料的总塑性伸长率与d之间的关系, 必须先分析 $σf$与各个晶粒的滑移系上的分切应力之间的关系.

 (3) $τdi=μi×σf$

 (4) $μi=cosφi×cosλi(i=1,2,…,936)$

 (5) $μ?=∑i=1936μ(gi)f(gi)$

X80管线钢具有bcc结构, 其常见的滑移系有12个{110}<111>滑移系、12个{112}<111>滑移系和24个{123}<111>滑移系. 由于{123}滑移面上的位错可以通过{110}面与{112}面上滑移的组合完成, 因此, 在计算时忽略{123}<111>上的滑移[30]. 因此, gi取向微区的12个{110}<111>滑移系和12个{112}<111>滑移系上的取向因子是[29]:

$μ{110}gi=cocφgi×cosλgi$

 (6) $μ{112}gi=cosφ′gi×cosλ′gi$

 (7) $μ(gi)=max(μ{110}gi,0.95μ{112}gi)$

 (8) $τd=μ?×σf$

2.2 以断裂强度为外加应力时, 单个晶粒的位移量与晶粒尺寸的关系

 Figure Option Fig.2 Schematic of dislocation pile-up group in a single grain ( td—given applied stress)图2 单个晶粒中的位错塞积示意图

 (9) $D0f(d)=∑i=1Nd(0.5×d-xi)×b0.5×d$

 (10) $Nd=0.5×π(1-v)×τd×dG×b$

 (11) $xi=(G×b)29(1-v2)×τd2×d×(i-1)2$

 Figure Option Fig.3 Relationship between the displacement of a single grain Df0 and d under the corresponding fracture strength图3 相应断裂强度下单个晶粒的位移量 Df0 随d的变化曲线

 Figure Option Fig.4 Schematic of grain boundary blocks the process of Frank-Read source (FR source) emits dislocations (Both da and db represent grain size and da>db)图4 晶界阻碍Frank-Read源(FR源)增殖产生位错的示意图

 (12) $Ff(d)=∫lnucminlnucmaxf(lFR)dlFR=∫Gb/τdd/3f(lFR)dlFR$

 Figure Option Fig.5 Relationship between the probability of activated FR sources in polycrystal Ff and d under the applied stress of 616 MPa, 1946 MPa and the corresponding fracture strength图5 外加应力分别为616 MPa, 1946 MPa和相应的断裂 强度时, 多晶体中FR源的可开动几率Ff随d的变化关系曲线

2.4 多晶体材料总塑性伸长率的计算模型

 (13) $na×∑i=1Nd(0.5×d-xi)×b×c?0.5×d×∫Gb/τdd/3f(lFR)dlFR$

 (14) $ef=exp(εf)-1$

2.5 模型预测结果和实验验证

 Figure Option Fig.6 Relationship between the total fracture elongation ef and d图6 钢的总塑性伸长率ef随d的变化关系曲线

 Figure Option Fig.7 Experimental data of uniform elongation eu or total elongation ef versus d in ultra-fine grain size steels[5,6,9,37~42] (Solid symbols represent eu and hollow symbols represent ef)图7 超细晶粒钢的均匀伸长率eu或总伸长率ef随d的变化关系的实验数据[5,6,9,37~42]

2.6 分析与讨论

3 结论

Fig.1 Relationship between fracture strength and grain size d in polycrystalline materials]]>

Fig.2 Schematic of dislocation pile-up group in a single grain ( td—given applied stress)]]>

Fig.3 Relationship between the displacement of a single grain and d under the corresponding fracture strength]]>

Fig.4 Schematic of grain boundary blocks the process of Frank-Read source (FR source) emits dislocations (Both da and db represent grain size and da>db)]]>

Fig.5 Relationship between the probability of activated FR sources in polycrystal Ff and d under the applied stress of 616 MPa, 1946 MPa and the corresponding fracture strength]]>

Fig.6 Relationship between the total fracture elongation ef and d]]>

Fig.7 Experimental data of uniform elongation eu or total elongation ef versus d in ultra-fine grain size steels[5,6,9,37~42] (Solid symbols represent eu and hollow symbols represent ef)]]>

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