Crystal plasticity constitutive model for BCC based on the dislocation density
NIE Junfeng, TANG Zhenrui, ZHANG Haiquan, LI Hongke, WANG Xin
Institute of Nuclear and New Energy Technology, Tsinghua University, Beijing 100084, China
Abstract: Crystal plasticity theory is a fundamental theory that combines the crystal microscopic slip mechanism with macroscopic plastic deformation to predict meso-scale plastic deformation. The dislocation density has an important influence on the hardening behavior of metal crystals. This paper presents a constitutive model based on crystal plasticity theory and dislocation motion theory for the BCC crystal structure. The model is used to study the mechanical behavior of a BCC lattice. Using the UMAT subroutine in ABAQUS for numerical simulations of a uniaxial tensile tests of single crystal and polycrystal iron. The results show that the constitutive model effectively simulates the mechanical behavior of the uniaxial tensile test for single crystal and polycrystal iron.
Key words: crystal plasticity     dislocation motion     body center cubic (BCC)     uniaxial tensile     finite element method (FEM)

1 基于位错运动的BCC晶体塑性本构模型 1.1 晶体塑性理论

 $F = {F^ * }{F^P}.$ (1)

L表示速度梯度张量，将其分解为弹性变形部分L*和塑性变形部分Lp

 $L = \dot F{F^{ - 1}} = {L^ * } + {L^P},$ (2)
 ${L^ * } = {{\dot F}^ * }{F^{ * - 1}},$ (3)
 ${L^P} = {{\dot F}^p}{F^{p - 1}} = \sum\limits_{a = 1}^n {{{\dot \gamma }^\alpha }{s^{ * \alpha }} \otimes {m^{ * \alpha }}} .$ (4)

 ${\tau ^\alpha } = \tau :{P^\alpha }.$ (5)

 ${P^\alpha } = \frac{{{s^\alpha }{m^\alpha } + {m^\alpha }{s^\alpha }}}{2}.$
1.2 位错滑移运动

 $\left\{ \begin{array}{l} {{\dot \gamma }^\alpha } = 0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left| {{\tau ^\alpha }} \right| < {g^\alpha };\\ {{\dot \gamma }^\alpha } = \dot \gamma _0^\alpha \exp \left\{ { - \frac{{{Q_0}}}{{kT}}{{\left[ {1 - {{\left( {\frac{{\left| {{\tau ^\alpha }} \right| - {g^\alpha }}}{{{{\hat \tau }^\alpha }}}} \right)}^p}} \right]}^q}} \right\}{\mathop{\rm sgn}} \left( {{\tau ^\alpha }} \right),\;\;\;\;\;\;\;\;\;\;\;\left| {{\tau ^\alpha }} \right| > {g^\alpha }. \end{array} \right.$ (6)
 ${{\hat \tau }^\alpha } = \hat \tau _0^\alpha \frac{G}{{{G_0}}}.$ (7)

 ${g^\alpha } = Gb\sqrt {{q_\rho }\sum\limits_{\beta + 1}^N {\left[ {{A^{\alpha \beta }}\left( {\rho _{\rm{M}}^\beta + \rho _{\rm{I}}^\beta } \right)} \right]} } .$ (8)

 $\dot \rho _{\rm{M}}^\alpha = \left( {\frac{{{k_{{\rm{mul}}}}}}{{b{l_{\rm{d}}}}} - \frac{{2{R_{\rm{c}}}}}{b}\rho _{\rm{M}}^\alpha - \frac{1}{{b{\lambda ^\alpha }}}} \right)\left| {{{\dot \gamma }^\alpha }} \right|,$ (9)
 $\dot \rho _{\rm{I}}^\alpha = \left( {\frac{1}{{b{\lambda ^\alpha }}} - {k_{{\rm{dyn}}}}\rho _{\rm{I}}^\alpha } \right)\left| {{{\dot \gamma }^\alpha }} \right|.$ (10)

2 材料参数确定

3 计算结果与分析 3.1 单晶单轴拉伸实验模拟

 图 1 晶向计算与实验单轴拉伸应力-应变曲线

3.2 多晶单轴拉伸模拟

 图 2 多晶单轴拉伸模型

 $\left\{ \begin{array}{l} \bar \sigma = \sum\limits_{k = 1}^N {{v^k}{\sigma ^k}} ,\\ \bar \varepsilon = \sum\limits_{k = 1}^N {{v^k}{\varepsilon ^k}} . \end{array} \right.$ (11)

 图 3 多晶与单晶模型单轴拉伸应力-应变曲线

 图 4 多晶模型单轴拉伸得到的应力云图及应变云图

4 结论

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