中锰钢奥氏体中化学界面变形行为的晶体塑性研究

图1 晶体塑性模型中变形梯度乘法分解的示意图

Fig.1 Illustration of the intermediate configurations resulting from multiplicative decomposition of the deformation gradient in crystal plasticity model ( F_e——elastic deformation gradient, F_p—plastic deformation gradient)

F = F_{e} \cdot F_{p}

(1)

式中， F_e表示弹性变形梯度， F_p表示塑性变形梯度。初始时刻t = 0情况下， F_p可以用晶体的初始取向矩阵( M₀)来进行初始化：

F_{p} (t = 0) = M_{0}

(2)

晶体的变形速度梯度( L )也可分为弹性和塑性2部分，可表示为：

L = L_{e} L_{p}

(3)

式中， L_e表示弹性变形速度梯度， L_p表示塑性变形速度梯度。

1.2 晶体弹性本构方程

由于本工作需考虑由形变诱导相变形成的新生马氏体对力学响应的贡献，因此，在材料弹性模量中引入了相组分(即原始奥氏体和新生马氏体)的影响：

C = V_{f}^{a u s} C_{a u s} + V_{f}^{m a r} C_{m a r}

(4)

式中，C为弹性模量， $V_{f}^{a u s}$ 和 $V_{f}^{m a r}$ 分别为奥氏体和马氏体的体积分数，C_aus和C_mar分别为奥氏体和马氏体的弹性模量。

本构方程中，弹性构型下的应力响应( S )采用广义Hooke定律描述：

S = C ∶ E_{e}

(5)

式中， E_e为Green弹性应变，由下式计算：

E_{e} = [\frac{1}{2} (F_{e}^{T} F_{e} - I)]

(6)

式中， I 为单位矩阵。

1.3 晶体塑性运动学方程

F_p的演化( ${\dot{F}}_{p}$ )可由 L_p来表示：

{\dot{F}}_{p} = L_{p} F_{p}

(7)

因奥氏体的晶体结构为fcc，塑性滑移为其主要塑性变形机制。 L_p可通过所有滑移系上塑性滑移量的总和来计算：

L_{p} = \sum_{α = 1}^{N_{s}} {\dot{γ}}^{α} m_{s}^{α} \otimes n_{s}^{α}

(8)

式中，α代指各滑移系， ${\dot{γ}}^{α}$ 为α滑移系上的滑移速率， $m_{s}^{α}$ 和 $n_{s}^{α}$ 分别为α滑移系上的剪切滑移面法向和滑移方向的单位矢量。对于奥氏体，塑性变形中可能开动的滑移系个数(N_s)为12 (见表1)。

表1 fcc结构材料的滑移系

Table 1 Slip systems of the fcc crystal

α	$M_{s} = m_{s}^{α} \otimes n_{s}^{α}$	α	$M_{s} = m_{s}^{α} \otimes n_{s}^{α}$
1	$[01 \bar{1}] \otimes (111) / \sqrt[]{6}$	7	$[011] \otimes (1 \bar{1} 1) / \sqrt[]{6}$
2	$[10 \bar{1}] \otimes (111) / \sqrt[]{6}$	8	$[10 \bar{1}] \otimes (1 \bar{1} 1) / \sqrt[]{6}$
3	$[1 \bar{1} 0] \otimes (111) / \sqrt[]{6}$	9	$[110] \otimes (1 \bar{1} 1) / \sqrt[]{6}$
4	$[01 \bar{1}] \otimes (\bar{1} 11) / \sqrt[]{6}$	10	$[011] \otimes (11 \bar{1}) / \sqrt[]{6}$
5	$[101] \otimes (\bar{1} 11) / \sqrt[]{6}$	11	$[101] \otimes (11 \bar{1}) / \sqrt[]{6}$
6	$[110] \otimes (\bar{1} 11) / \sqrt[]{6}$	12	$[1 \bar{1} 0] \otimes (11 \bar{1}) / \sqrt[]{6}$

Note:M_s—Schmid matrix of slip system; $m s α$ and $n s α$ —slip direction vector and slip plane vector of α-thslip system, respectively

此外，在变形过程中，奥氏体相会发生形变诱导马氏体相变以提高材料的加工硬化率。因此，形变诱导马氏体相变可视为 L_p的附加贡献，此时 L_p可表示为：

L_{p} = (1 - \sum_{β = 1}^{N_{t r}} f^{β}) \sum_{α = 1}^{N_{s}} {\dot{γ}}^{α} m_{s}^{α} \otimes n_{s}^{α} +

\sum_{β = 1}^{N_{t r}} {\dot{γ}}^{β} m_{t r}^{β} \otimes n_{t r}^{β}

(9)

式中， ${\dot{γ}}^{β}$ 为等效相变系β上的滑移速率，f^β 为等效相变系β上的马氏体体积分数^[16]，N_tr为等效相变系个数(为12)， $m_{t r}^{β}$ 和 $n_{t r}^{β}$ 分别为等效相变系β上的剪切滑移面法向和滑移方向的单位矢量。

为准确描述fcc晶格转变至bcc晶格的原子迁移过程，Sinclair和Hoagland^[19]利用分子动力学对层错带交叉诱导马氏体形核过程进行了计算，发现形变诱导马氏体相变新旧相的取向关系接近反Nishiyama-Wassermann (N-W)关系。因此，本工作中采用反N-W关系来描述奥氏体相到马氏体相的晶格转变。此时，形变诱导马氏体相变的变形梯度( Φ_tr)可由Bain应变表述：

Φ_{t r} = \frac{a^{b c c}}{a^{f c c}} x_{t r} \otimes x_{t r} + \frac{\sqrt[]{2} a^{b c c}}{a^{f c c}} y_{t r} \otimes y_{t r} +

\frac{\sqrt[]{2} a^{b c c}}{a^{f c c}} z_{t r} \otimes z_{t r}

(10)

Bain应变包含1个压缩方向和2个拉伸方向^[19]，以及沿任意1个拉伸方向±10.26°的旋转( Ψ_tr)：

Ψ_{t r} = r_{t r} \otimes r_{t r} + c o s (θ_{t r}) (Ι - r_{t r} \otimes r_{t r}) +

s i n (θ_{t r}) (r_{t r} \times Ι)

(11)

式中，a^fcc和a^bcc分别为奥氏体和马氏体的晶格常数。Bain应变的压缩、拉伸及旋转基矢( x_tr、 y_tr、 z_tr、 r_tr)以及旋转角(θ_tr)的详细信息见表2。由于每个Bain应变含有沿1个压缩方向和2个拉伸方向的4种旋转模式，因此共有N_tr = 12个等效相变系。对于每一个等效相变系，相变基张量( M_tr)可表述为：

表2 反N-W关系定义的12个形变诱导马氏体相变的等效相变系

Table 2 Martensite transformation systems of the fcc crystal following the inverse Nishiyama-Wassermann (N-W) relation used in the crystal plasticity model

$β_{t r}$	x_tr	y_tr	z_tr	r_tr	θ_tr
1	[100]	[010]	[001]	[010]	+10.26°
2	[100]	[010]	[001]	[010]	-10.26°
3	[100]	[010]	[001]	[001]	+10.26°
4	[100]	[010]	[001]	[001]	-10.26°
5	[010]	[100]	[001]	[100]	+10.26°
6	[010]	[100]	[001]	[100]	-10.26°
7	[010]	[100]	[001]	[001]	+10.26°
8	[010]	[100]	[001]	[001]	-10.26°
9	[001]	[100]	[010]	[100]	+10.26°
10	[001]	[100]	[010]	[100]	-10.26°
11	[001]	[100]	[010]	[010]	+10.26°
12	[001]	[100]	[010]	[010]	-10.26°

Note:β_tr—transformation system, x_tr—compressive direction, y_tr and z_tr—tensile directions, r_tr—rotation direction, θ_tr—rotation angle

M_{t r} = Ψ_{t r} Φ_{t r} - Ι

(12)

M_tr可类比于塑性滑移系的Schmid张量( M_s)。

1.4 晶体塑性本构方程

奥氏体的塑性变形可通过位错的滑移以及形变诱导相变2种变形机制来实现。针对塑性滑移，滑移系上的滑移速率( $\dot{γ}$ )通常可表达为一系列内部状态变量的函数。根据Orowan理论的黏塑性唯象本构关系^[20]，第α个滑移系上的 $\dot{γ}$ 演化遵循如下公式：

{\dot{γ}}^{α} = ρ_{e}^{α} b_{s} v_{0} e x p \{- \frac{Q_{s}}{k T} {[1 - {(\frac{|τ_{e f f}^{α}|}{τ_{s o l}})}^{p}]}^{q}\} s i g n (τ^{α})

(13)

式中， $ρ_{e}^{α}$ 为α滑移系上的刃位错密度，b_s为Burgers矢量模，v₀为位错滑移速率，τ^α 为滑移系开动的驱动力(即滑移系上的分切应力)， $τ_{e f f}^{α}$ 为作用于位错迁移的等效分切应力，τ_sol为固溶强度，Q_s为位错滑移激活能，k为Boltzmann常数，T为变形温度，p和q均为与变形速率相关的敏感性指数。

评价滑移系激活与否的 $τ_{e f f}^{α}$ 可通过下式计算：

τ_{e f f}^{α} = \{\begin{matrix} |τ^{α}| - τ_{p a s s}^{α} (|τ^{α}| > τ_{p a s s}^{α}) \\ 0 (|τ^{α}| \leq τ_{p a s s}^{α}) \end{matrix}

(14)

式中，τ^α 可由下式计算：

τ^{α} = M_{p} \cdot M_{s}

(15)

式中， M_p为塑性构型下的Mandel应力，可由第二Piola-Kirchhoff应力计算得到^[10]。 $τ_{p a s s}^{α}$ 为位错开动滑移所需的力，是位错密度的函数，可描述为：

τ_{p a s s}^{α} = G b_{s} {[\sum_{α^{'} = 1}^{N_{s}} ξ_{α α^{'}} (ρ_{e}^{α^{'}} + ρ_{d}^{α^{'}})]}^{1 / 2}

(16)

式中，G为剪切模量； ξ_αα' 为滑移系α和α'之间的交互作用系数矩阵，用以表征滑移系之间的相互强化作用； $ρ_{e}^{α^{'}}$ 为α'滑移系上的刃位错的密度， $ρ_{d}^{α^{'}}$ 为α'滑移系上的位错偶极子的密度。

本工作中晶体塑性模型还可计算位错的累积演化行为。可动位错的演化由位错增殖及位错湮灭过程组成。位错增殖由材料中的位错源产生。当2个Burgers矢量相反的位错靠近到间距小于刃位错湮灭所需的临界距离( $\overset{̌}{d}$ )时，2个位错会同时湮灭^[21,22]。由此，位错密度的演化( ${\dot{ρ}}_{e}^{α}$ )可表示为：

{\dot{ρ}}_{e}^{α} = |{\dot{γ}}^{α}| (\frac{1}{b_{s} Λ_{s}^{α}} - \frac{2 {\hat{d}}^{α} ρ_{e}^{α}}{b_{s}} - \frac{2 {\overset{̌}{d}}^{α} ρ_{e}^{α}}{b_{s}})

(17)

式中， $Λ_{s}^{α}$ 为位错平均自由程， ${\hat{d}}^{α}$ 为2个位错形成位错偶极子的最大平面滑移距离。

可动位错一般可在材料中自由迁移一段距离，也就是存在 $Λ_{s}^{α}$ 。在本模型中，计算 $Λ_{s}^{α}$ 时同时考虑新生相马氏体对位错迁移的阻碍作用。由此，位错与相变的交互作用便由 $Λ_{s}^{α}$ 联系起来。 $Λ_{s}^{α}$ 可由下式表示：

\frac{1}{Λ_{s}^{α}} = \frac{1}{d} + \frac{1}{λ_{s l i p}^{α}} + \frac{1}{λ_{s l i p t r a n s}^{α}}

(18)

\frac{1}{λ_{s l i p}^{α}} = \frac{1}{i_{s l i p}} {[\sum_{α^{'} = 1}^{N_{s}} ξ_{α α^{'}} (ρ_{e}^{α^{'}} + ρ_{d}^{α^{'}})]}^{1 / 2}

(19)

\frac{1}{λ_{s l i p t r a n s}^{α}} = \sum_{β = 1}^{N_{t r}} ξ_{α β} f^{β} \frac{1}{t_{t r} (1 - f_{t r})}

(20)

式中，d为平均晶粒尺寸，t_tr为平均马氏体厚度，f_tr为该晶粒中马氏体分数，i_slip为拟合参数， $λ_{s l i p}^{α}$ 代表位错密度对位错平均自由程的贡献， $λ_{s l i p t r a n s}^{α}$ 代表形变诱导相变对位错平均自由程的贡献， ξ_αβ 为滑移系α和相变系β的交互作用系数矩阵。

1.5 形变诱导马氏体相变模型

Olson等^[23,24]和Wang等^[25]认为，亚稳奥氏体中层错带的交叉位置是形变诱导马氏体形核的有利质点。许多研究者^[24,26]在透射电子显微镜下观察到了孪晶束的形成以及层错带交叉处的马氏体形核。本工作以层错带交叉诱导马氏体形核的微观机理作为形变诱导马氏体相变的形核机制，将其数值化后引入到晶体塑性模型中。建模时，先将fcc晶体12个滑移系上的塑性滑移量投影至层错带系上，并将其定义为层错带系上的剪切滑移量^[27]。每个层错带系δ上投影的剪切量( $γ_{δ}^{'}$ )可表述为^[11,28]：

γ_{δ}^{'} = \sum_{α = 1}^{N_{f}} (m_{t w}^{δ} \otimes n_{t w}^{δ}) ∙ (γ^{α} m_{s}^{α} \otimes n_{s}^{α})

(21)

式中，N_f为层错带系个数，γ^α 为α滑移系上的滑移量， $m_{t w}^{δ}$ 和 $n_{t w}^{δ}$ 分别为δ层错带系(即fcc孪晶系)的孪晶面和孪晶方向的单位矢量。

表3所示为投影矩阵的各个分量。

表3 fcc滑移系到层错带系的滑移量投影矩阵

Table 3 Projection matrix from the fcc slip systems to the fault-band systems

δ α	1	2	3	4	5	6	7	8	9	10	11	12
1	0	$- \sqrt[]{3} / 2$	$- \sqrt[]{3} / 2$	0	0	0	0	0	0	0	0	0
2	$\sqrt[]{3} / 2$	0	$- \sqrt[]{3} / 2$	0	0	0	0	0	0	0	0	0
3	$\sqrt[]{3} / 2$	$\sqrt[]{3} / 2$	0	0	0	0	0	0	0	0	0	0
4	0	0	0	0	$- \sqrt[]{3} / 2$	$- \sqrt[]{3} / 2$	0	0	0	0	0	0
5	0	0	0	$\sqrt[]{3} / 2$	0	$\sqrt[]{3} / 2$	0	0	0	0	0	0
6	0	0	0	$\sqrt[]{3} / 2$	$- \sqrt[]{3} / 2$	0	0	0	0	0	0	0
7	0	0	0	0	0	0	0	$- \sqrt[]{3} / 2$	$- \sqrt[]{3} / 2$	0	0	0
8	0	0	0	0	0	0	$\sqrt[]{3} / 2$	0	$\sqrt[]{3} / 2$	0	0	0
9	0	0	0	0	0	0	$\sqrt[]{3} / 2$	$- \sqrt[]{3} / 2$	0	0	0	0
10	0	0	0	0	0	0	0	0	0	0	$\sqrt[]{3} / 2$	$\sqrt[]{3} / 2$
11	0	0	0	0	0	0	0	0	0	$- \sqrt[]{3} / 2$	0	$\sqrt[]{3} / 2$
12	0	0	0	0	0	0	0	0	0	$- \sqrt[]{3} / 2$	$- \sqrt[]{3} / 2$	0

Note:δ represents the index of fault band system and α represents the index of slip system

表4所示为奥氏体中的层错带系(也即fcc的孪晶系)。 $γ_{δ}^{'}$ 表示所有12个滑移系在层错带系上投影滑移量的总和^[28]。由此，相变系β上马氏体相变的形核率( ${\dot{N}}_{t r}^{β}$ )便可由一对层错带系 $(γ_{δ}^{'}, γ_{δ^{'}}^{'})$ 上的剪切量总和表示：

{\dot{N}}_{t r}^{β} = (γ_{δ}^{'} + γ_{δ^{'}}^{'}) b_{t r} L_{t r} p_{t r}

(22)

此处，不同的相变系β分别代表马氏体的不同变体，即相变所产生的不同马氏体变体是由不同层错带系交叉所提供的。式中，b_tr为相变所对应的Burgers矢量模，L_tr为马氏体核心的初始尺寸，p_tr为形变诱导马氏体形核的概率。

表4 fcc晶体的层错带系

Table 4 Fault band systems of the fcc crystal structure

$δ$	$M_{t w} = m_{t w}^{δ} \otimes n_{t w}^{δ}$	α	$M_{t w} = m_{t w}^{δ} \otimes n_{t w}^{δ}$
1	$[\bar{2} 11] \otimes (111) / \sqrt[]{6}$	7	$[\bar{2} \bar{1} 1] \otimes (1 \bar{1} 1) / \sqrt[]{6}$
2	$[\bar{1} 2 \bar{1}] \otimes (111) / \sqrt[]{6}$	8	$[\bar{1} 12] \otimes (1 \bar{1} 1) / \sqrt[]{6}$
3	$[11 \bar{2}] \otimes (111) / \sqrt[]{6}$	9	$[2 \bar{1} 1] \otimes (1 \bar{1} 1) / \sqrt[]{6}$
4	$[\bar{2} \bar{1} \bar{1}] \otimes (\bar{1} 11) / \sqrt[]{6}$	10	$[1 \bar{2} \bar{1}] \otimes (11 \bar{1}) / \sqrt[]{6}$
5	$[12 \bar{1}] \otimes (\bar{1} 11) / \sqrt[]{6}$	11	$[111] \otimes (11 \bar{1}) / \sqrt[]{6}$
6	$[\bar{1} 1 \bar{2}] \otimes (\bar{1} 11) / \sqrt[]{6}$	12	$[\bar{1} \bar{1} \bar{2}] \otimes (11 \bar{1}) / \sqrt[]{6}$

Note:M_tw—Schmid matrix of fault band system; $m t w δ$ and $n t w δ$ —twin direction vector and twin plane vector of the δ-thfault band system, respectively

形变诱导马氏体相变的临界驱动力(τ_r)可表示为：

τ_{r} = \frac{G b_{s}}{2 π (x_{0} + x_{c})} + \frac{G b_{s} c o s (π / 3)}{2 π x_{0}}

(23)

式中，x_c为2个不全位错形成马氏体核心的临界间距。fcc金属Shockley不全位错的平衡解离间距(x₀)可由层错能(Γ_sf)表示为：

x_{0} = \frac{G b_{s}^{2} (2 + υ)}{8 Γ_{s f} π (1 - υ)}

(24)

式中，ν为Poisson比。由此，相变系θ上的马氏体体积分数的演化(^{${\dot{f}}^{β}$ )可描述为：}

{\dot{f}}^{β} = (1 - f_{t r}) V^{β} {\dot{N}}_{t r}^{β}

(25)

式中，V^β 是新生相马氏体单个片层的体积。

1.6 层错能模型

奥氏体的变形机制与其层错能密切相关^[29,30]。一般认为Γ_sf是产生2个原子层厚度的ε马氏体片层所需的能量，可由下式计算^[29,31]：

Γ_{s f} = 2 ρ Δ G^{γ \to ε} + 2 σ

(26)

式中，ρ为沿着{111}晶面方向的摩尔表面密度， $Δ$ G^γ^→^ε 为γ→ε相变的摩尔Gibbs自由能，σ为奥氏体和马氏体之间相界面的界面能。ρ可表示为：

ρ = \frac{4}{\sqrt[]{3}} \frac{1}{a^{2} N}

(27)

式中，a为晶格常数，N为Avogadro常数。

$Δ G^{γ \to ε}$ 可由规则溶液模型中的混合系统自由能计算获得。对于Fe-Mn-C三元固溶体，其Gibbs自由能可通过下式计算：

Δ G^{γ \to ε} = x_{F e} Δ G_{F e}^{γ \to ε} + x_{M n} Δ G_{M n}^{γ \to ε} +

x_{C} Δ G_{C}^{γ \to ε} + Δ G_{m g}^{γ \to ε} + x_{F e} x_{M n} Ω_{F e M n}^{γ \to ε} +

x_{F e} x_{C} Ω_{F e C}^{γ \to ε} + x_{M n} x_{C} Ω_{M n C}^{γ \to ε}

(28)

式中，x_i (i = Fe、Mn、C)为纯组元i的摩尔分数， $Δ G_{i}^{γ \to ε}$ 为纯组元i在γ、ε两相中的摩尔Gibbs自由能之差， $Δ G_{m g}^{γ \to ε}$ 为由相的铁磁性状态所决定的自由能， $Ω_{F e M n}^{γ \to ε}$ 为Fe-Mn混合所致的过剩自由能， $Ω_{F e C}^{γ \to ε}$ 为Fe-C混合所致的过剩自由能， $Ω_{M n C}^{γ \to ε}$ 为Mn-C混合所致的过剩自由能。

式中的各项可由以下公式来计算：

Δ G_{F e}^{γ \to ε} = - 1828.4 + 4.686 T

(29)

Δ G_{M n}^{γ \to ε} = 3970 - 1.7 T

(30)

Δ G_{C}^{γ \to ε} = - 24595.12

(31)

Ω_{F e M n}^{γ \to ε} = - 9135.5 + 15282.1 x_{M n}

(32)

Ω_{F e C}^{γ \to ε} = 42500

(33)

Ω_{M n C}^{γ \to ε} = 26910

(34)

由于γ和ε均为反铁磁相，因此单相φ中的磁自由能( $G_{m g}^{φ}$ )可由其Néel温度( $T_{N \dot{e} e l}^{φ}$ )表示：

G_{m g}^{φ} = R T l n (\frac{β^{φ}}{μ_{B}} + 1) f^{φ} (τ^{φ})

(35)

式中， $τ^{φ} = T / T_{N \dot{e} e l}^{φ}$ ，R为气体常数，β^φ 为磁矩，μ_B为Bohr磁子。奥氏体的Néel温度( $T_{N \dot{e} e l}^{γ}$ )为：

T_{N \dot{e} e l}^{γ} = 250 l n x_{M n} - 4750 x_{C} x_{M n} + 720

(36)

马氏体的Néel温度( $T_{N \dot{e} e l}^{ε}$ )为：

T_{N \dot{e} e l}^{ε} = 580 x_{M n}

(37)

奥氏体中的自旋量子数可表示为：

\frac{β^{γ}}{μ_{B}} = 0.7 x_{F e} + 0.62 x_{M n} - 0.64 x_{F e} x_{M n} - 4 x_{C}

(38)

式中，β^γ 为奥氏体的磁矩。

马氏体中的自旋量子数可表示为：

\frac{β^{ε}}{μ_{B}} = 0.62 x_{M n} - 4 x_{C}

(39)

式中，β^ε 为马氏体的磁矩。

$f^{φ}$ 为以 $τ^{φ}$ 为自变量的多项式函数，可表示为：

\{\begin{matrix} \begin{array}{l} f^{φ} = 1 - \frac{1}{D} \{\frac{79 (τ^{φ})^{- 1}}{140 P} + \frac{474}{497} [\frac{1}{P} - 1] \cdot \\ [\frac{(τ^{φ})^{3}}{6} + \frac{(τ^{φ})^{9}}{135} + \frac{(τ^{φ})^{15}}{600}]\} (τ^{φ} \leq 1) \end{array} \\ f^{φ} = - \frac{1}{D} [\frac{(τ^{φ})^{- 5}}{10} + \frac{(τ^{φ})^{- 15}}{315} + \frac{(τ^{φ})^{- 25}}{600}] (τ^{φ} > 1) \end{matrix}

(40)

式中，P和D为常数，P = 0.28，D = 2.342356517。

2 晶体塑性计算的模型设置

2.1 晶体塑性计算流程

本工作采用图2所示的自洽方法求解如上所述的 F 及其内部相关变量^[32]。首先由奥氏体晶粒的初始状态确定初始 F_p，之后再从总变形增量中去除塑性变形部分，计算 F_e。随后通过弹性变形Hooke定律计算 S。进一步将 S 作为诱导滑移和形变诱导相变的驱动力而发生塑性变形，由此便可计算一个时间步内的 ${\dot{F}}_{p}$ ，同时获得下一个时间步的 F_p。再将计算所得的 F_p作为下一个时间步的初始塑性变形梯度，重复上述步骤持续迭代至变形结束，即可实现变形全过程的晶体塑性计算。本工作的建模和计算是在DAMASK软件理论框架下完成的^[10]。

图2

图2 变形梯度及其内部相关变量求解过程的示意图

Fig.2 Self-consistent integration of kinematic quantities within fixed internal material state parameters ( F —deformation gradient, ${\dot{F}}_{p}$ —tangent of plastic deformation gradient, L_p—velocity gradients of plastic deformation gradient, S —second Piola-Kirchhoff stress, ${\dot{γ}}^{α}$ —plastic slip rate, ${\dot{ρ}}_{e}^{α}$ —evolution rate of edge dislocation, $f^{β}$ —martensite fraction)

S 是发生滑移和形变诱导相变这2种塑性变形的核心驱动力^[10,18]。如图2所示，通过将 S 投影至各滑移系上，可以根据式(13)及式(17)计算塑性变形过程中各滑移系上的 ${\dot{γ}}^{α}$ 及 ${\dot{ρ}}_{e}^{α}$ ；通过将 S 投影至等效相变系上，即可根据式(25)计算变形过程中各等效相变系上形变诱导相变的分数。

2.2 晶体塑性计算模型设置

如前所述，化学界面是同一晶粒内部存在的晶体结构与取向相同、而化学成分不同的2个区域的边界。与相界和晶界不同，化学界面两侧仅存在合金元素浓度的差异^[7]。Wan等^[9]利用快速加热-奥氏体逆转变技术，在冷轧中锰钢的等轴状奥氏体晶粒内部成功引入了Mn元素的化学界面，且扫描电镜表征结果显示奥氏体晶粒内部存在明显的圆形贫Mn区。本工作以冷轧中锰钢中形成的这种Mn元素化学界面的微观特征为参考进行建模，将其简化为图3a所示的几何模型，并进一步构建包含化学界面的代表性体积单元模型，如图3b所示。图中的计算域代表尺寸为400 nm × 400 nm的真实材料，其中红色部分为化学界面富Mn奥氏体相(Mn浓度为13%，质量分数，下同)，蓝色部分为贫Mn奥氏体(Mn浓度为5%)。其中的边界即为化学界面，其曲率半径为300 nm。如前所述，化学界面两侧为同一相，且无晶体结构与晶粒取向的差异，仅存在Mn元素成分的差别。在计算过程中，将化学界面两侧的代表性体积单元赋予相同的弹塑性材料属性。

图3

图3 奥氏体中Mn化学界面微结构的几何模型

Fig.3 Geometric model of the Mn chemical boundary in austenite

(a) schematic of the chemical boundary

(b) representative volume element model (The red part represents the Mn-rich region with Mn content of 13% (mass fraction) and the blue part represents the Mn-poor region with Mn content of 5% (mass fraction). ε—strain)

奥氏体的变形采用单轴拉伸，总应变为0.1，加载方向如图3b中箭头所示。应变速率为0.001 s^-1，以保持准静态变形状态。采用基于快速Fourier变换的谱求解方法求解晶体塑性力学平衡的边值问题。计算过程中通过变形梯度速度张量( $\dot{F}$ )来设置加载条件，其分量的顺序依次为11、12、13、21、22、23、31、32、33：

\dot{F} = [\begin{matrix} 1 \times 10^{- 3} & 0 & 0 \\ 0 & * & 0 \\ 0 & 0 & * \end{matrix}]

(41)

应力边界条件也由9个浮点数来表示：

S' = [\begin{matrix} * & * & * \\ * & 0 & * \\ * & * & 0 \end{matrix}]

(42)

式中，*代表待求值。

3 计算结果及分析

图4所示为塑性变形不同阶段奥氏体内部化学界面两侧的微区应变演化情况。可以看出，变形过程中，单一奥氏体内部的微区应变呈现出明显的梯度。随着变形的进行，化学界面两侧的微区应变差异逐渐增大。当应变为0.1时，贫Mn奥氏体内部的微区应变在化学界面处累积，使其承载较多的应变。图5为奥氏体内部不同变形阶段化学界面处的微区应力演化情况。结果显示，化学界面的富Mn奥氏体中的微区应力随着变形的进行快速累积，与贫Mn奥氏体内部微区应力的差异逐渐增大，且富Mn侧奥氏体承载了较多的应力。表5统计了不同应变下化学界面两侧奥氏体中的微区应变和应力差异。可以看出，随着应变逐步增加，化学界面两侧的微区应力和应变累积的差异随着变形的进行逐渐增大。

图4

图4 奥氏体内部化学界面附近的微区应变演化

Fig.4 Simulated microzone strain distributions across the chemical boundary within an austenite grain at strains of 0.025 (a), 0.05 (b), 0.075 (c), and 0.10 (d)

图5

图5 奥氏体内部化学界面附近的微区应力演化

Fig.5 Simulated microzone stress distributions across the chemical boundary within an austenite grain at strains of 0.025 (a), 0.05 (b), 0.075 (c), and 0.1 (d)

表5 奥氏体内部Mn化学界面两侧的微区应力及应变差统计值

Table 5 Statistics of the difference value of microzone strain and stress across the chemical boundary

ε	ε_max - ε_min	τ_max - τ_min / MPa
0.025	0.05 × 10^-2	00.8
0.050	0.05 × 10^-2	04.4
0.075	1.41 × 10^-2	20.4
0.100	2.78 × 10^-2	71.1

Note:ε_maxand ε_min are the maximun and minumum strains within a single austenite grain, respectively; τ_max and τ_min are the maximun and minumum stresses within a single austenite grain, respectively

为进一步了解奥氏体晶粒内部变形局部区域的塑性变形行为，对变形至0.05应变时奥氏体晶粒内部的微区应力和应变分布进行了分析。图6所示为应变为0.025时奥氏体晶粒内部沿平行于化学界面切线方向L₁-L $_{1}^{'}$ (图4b和5b所示)和垂直于化学界面切线方向L₂-L $_{2}^{'}$ (图4b和5b所示)上微区应力和应变的分布情况。可以看出，沿L₁-L $_{1}^{'}$ 的微区应力分布(图6a)表明位于化学界面富Mn侧奥氏体区域内呈现出更高的应力水平，化学界面附近存在极大的应力梯度；而贫Mn奥氏体区域内的微区应力水平较低，应力分布也比较均匀。而沿L₁-L $_{1}^{'}$ 上的微区应变分布(图6b)则呈现出与微区应力分布相反的趋势。化学界面一侧的贫Mn奥氏体区域内的微区应力和应变的梯度均较小，靠近化学界面的区域微区应力和应变水平变化较大。沿L₂-L $_{2}^{'}$ 的微区应力分布(图6c)同样显示出化学界面贫Mn侧的微区应力水平较低，富Mn侧微区应力水平较高的特征。由图6c结果可以看出，垂直于化学界面切线方向，L₂到O上微区应力水平逐渐增高，而O到L $_{2}^{'}$ 上微区应力水平逐渐降低，也即无论是化学界面的贫Mn侧奥氏体还是富Mn侧奥氏体，其化学界面位置O均为微区应力最高处。图6d显示的微区应变也呈现相同规律，L₂到O上微区应变逐渐增高，而O-L $_{2}^{'}$ 上微区应变逐渐降低。

图6

图6 应变为0.025时沿图4b和5b中L₁-L $_{1}^{'}$ 和L₂-L $_{2}^{'}$ 的微区应力和应变分布情况

Fig.6 Microzone stress (a, c) and microzone strain (b, d) distributions along the lines L₁-L $_{1}^{'}$ (a, b) and L₂-L $_{2}^{'}$ (c, d) in Figs.4b and 5b at strain of 0.025

如上所述，奥氏体中化学界面两侧仅存在置换型Mn元素的浓度差异，并不存在真实的物理界面，但微区Mn浓度的差异会引起奥氏体层错能的不同^[33]。这就意味着，Mn化学界面的引入会导致同一奥氏体晶粒内部层错能分布存在差异。富Mn奥氏体的层错能较高，而贫Mn奥氏体的层错能较低^[29]。奥氏体中层错能的高低直接决定了变形过程中其塑性变形机制的不同^[31]。本工作所讨论的奥氏体晶粒内部，层错能较高的富Mn侧奥氏体的机械稳定性较好，其塑性变形的主要作用机制为位错滑移，而层错能较低的贫Mn侧奥氏体则更易于发生形变诱导马氏体相变。Mn化学界面两侧奥氏体的塑性变形机制不同，导致微区应力和微区应变在化学界面两侧的累积行为存在差异^[29]，这也是在同一奥氏体晶粒内部发生应变和应力不连续分布的原因。

同一奥氏体晶粒内部存在不连续应变分布，这就意味着奥氏体的Mn化学界面两侧发生了非均匀塑性变形行为。为了协调这种不均匀的塑性变形，变形过程中会在化学界面附近的局部区域形成大量位错累积^[34]。图7a所示为应变为0.1时化学界面附近的位错累积的计算结果。可以看出，化学界面的富Mn侧奥氏体发生明显的位错增殖，而贫Mn奥氏体中位错的累积较少。在化学界面附近的富Mn奥氏体内形成了明显的位错密度梯度，而距化学界面较远处，位错的累积程度均逐渐减弱(图7b)。在之前的研究中，Kim等^[35]在冷轧中锰钢的微观组织中也观察到了与计算结果相同的位错密度梯度，他们利用快速加热技术在中锰钢中制备出了纳米尺度的Mn成分不均匀性微区，在奥氏体晶粒内部获得了Mn的化学界面，同时也观察到了变形过程中化学界面两侧的位错累积存在明显差异，发现化学界面对位错的运动具有阻碍作用。可见，化学界面的引入诱导界面两侧形成层错能的区域差异，会影响奥氏体晶粒内部局部微区的应力和应变累积。

图7

图7 应变为0.1时计算所得化学界面附件的位错密度分布

Fig.7 Simulated dislocation density distributions across the chemical boundary at strain of 0.1 (a) and profile of the dislocation density along the black line in Fig.7a (b)

另一方面，位错在化学界面富Mn奥氏体侧的累积会在该区域形成背应力，使得该区域的静水压力升高^[2]。静水压力的提高会限制奥氏体发生形变诱导马氏体相变时的体积膨胀^[36]，因此会显著抑制此处形变诱导马氏体相变的发生，有利于提高此局部区域内奥氏体的机械稳定性^[37,38]。相比之下，贫Mn奥氏体中因位错累积不明显导致局部应力水平较低，对马氏体相变的抑制效果有限，此处奥氏体容易发生形变诱导相变。可见，化学界面一侧的富Mn奥氏体本身的高层错能及其内部形成的高的背应力使得此处奥氏体的机械稳定性大大提高，变形过程中不易发生马氏体相变；而贫Mn侧奥氏体的低层错能及其内部较低的背应力则降低了该处奥氏体的机械稳定性，使其在变形过程中易于发生马氏体相变。图8a为应变为0.05时同一奥氏体晶粒内部化学界面两侧形变诱导马氏体相变发生情况的计算结果。可以看出，化学界面两侧的奥氏体内形变诱导相变行为存在明显差异，化学界面的贫Mn奥氏体侧优先发生形变诱导相变。图8b分别统计了化学界面两侧贫Mn奥氏体及富Mn奥氏体中形变诱导马氏体相变的动力学。可以看出，Mn化学界面的存在使得界面两侧奥氏体的形变诱导相变行为存在明显差异，这就意味着在同一奥氏体中机械稳定性呈连续渐变，即“谱”分布的特征。这种呈“谱”分布的奥氏体机械稳定性对于形变诱导马氏体相变渐次地发生是十分有利的，这也是在奥氏体中引入化学界面能够同时提升中锰钢强度和塑性的关键所在^[5]。综上所述，奥氏体中化学界面的引入不仅会促进晶粒内部应力与应变的微区配分，而且使同一个奥氏体内部的机械稳定性分布呈现差异性，会使奥氏体晶粒在塑性变形中渐次地发生TRIP效应，进而提高中锰钢的强度和塑性。

图8

图8 应变为0.05时形变诱导马氏体体积分数分布情况和变形过程中奥氏体内化学界面两侧马氏体相变动力学

Fig.8 Distribution of the martensite volume fraction at strain of 0.05 (a) and kinetics of the martensite transformation within austenite grain on both side of the chemical boundary during deformation (b)

4 结论

(1) 中锰钢奥氏体内部化学界面两侧因Mn浓度差异所导致的层错能差异，使得奥氏体塑性变形中化学界面两侧开动的变形机制不同，导致同一奥氏体晶粒内部形成微区应力和应变的不连续分布。富Mn侧奥氏体承载更多的应力，而贫Mn侧奥氏体承载更多的应变。

(2) 化学界面的存在使得奥氏体在变形过程中在化学界面富Mn奥氏体侧形成位错累积，进而在此区域内形成背应力，使其静水压力升高，阻碍其内部形变诱导马氏体相变的发生，使富Mn侧奥氏体的机械稳定性提高。而贫Mn侧奥氏体中因位错累积不明显导致局部背应力水平较低，在变形过程中容易发生形变诱导相变。

(3) 化学界面的存在使同一奥氏体晶粒内部的机械稳定性分布呈现出差异性，使奥氏体晶粒在塑性变形中渐次发生TRIP效应，有利于提高中锰钢的强度和塑性。

参考文献

原文顺序

文献年度倒序

文中引用次数倒序

被引期刊影响因子

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Making materials plain: Concept, principle and applications

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DOI [本文引用: 4]

Alloying is conventionally used for advancing properties of engineering materials. But with increasing degree of alloying, materials become more resource dependent and more costly, and recycling and reuse of materials become more difficult. As nowadays sustainability is becoming a more and more important index for materials development, novel strategies for sustainable materials development is highly desired. In this paper, a sustainable “plain” approach to advancing materials without changing chemical compositions is proposed, i.e., architecturing imperfections across different length-scales. Novel properties and performance can be achieved in the “plain” materials with less alloying or even non-alloying. Basic concept, principle, as well as potential applications of the “plain materials” approach will be introduced.

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[J]. Acta Metall. Sin., 2022, 58: 1349

Strong and tough metallic materials are desired for light-weight structural applications in transportation and aerospace industries. Recently, heterostructures have been found to possess unprecedented strength-and-ductility synergy, which is until now considered impossible to achieve. Heterostructured metallic materials comprise heterogeneous zones with dramatic variations (> 100%) particularly in mechanical properties. The interaction in these hetero-zones produces a synergistic effect wherein the integrated property exceeds the prediction by the rule-of-mixtures. More importantly, the heterostructured materials can be produced by current industrial facilities at large scale and low cost. The superior properties of heterostructured materials are attributed to the heterodeformation induced (HDI) strengthening and strain hardening, which is produced by the piling-up of geometrically necessary dislocations (GNDs). These GNDs are needed to accommodate the strain gradient near hetero-zone boundaries, across which there is high mechanical incompatibility and strain partitioning. This paper classifies the types of heterostructures and delineates the deformation behavior and mechanisms of heterostructured materials.

武晓雷, 朱运田.

异构金属材料及其塑性变形与应变硬化

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金属材料异构(heterostructure)是将具有显著流变应力差异的软硬相间区域作为基元进行有序构筑而成的微观组织,是旨在提高应变硬化能力和拉伸塑性的微结构设计策略,迄今应用于各种金属结构材料并获得了强度与塑性/韧性等力学性能的优异匹配。异构策略的出发点是其特征的力学响应,即塑性变形时在异构基元界面形成的应变梯度,异构的特征应力应变响应是力学迟滞环。相比均质结构中主导的林位错塑性和林硬化,异构为了协调界面应变梯度而产生几何必需位错,新增了基于几何必需位错的异质塑性变形并引起额外的应变硬化与额外的强化。本文综述了近期异构金属材料的研究进展,首先定义了异构中基元并据此把异构分类为基元异构、亚基元异构以及复合异构,随后分析并讨论了异构塑性变形时界面和位错等微结构演化,以及异质塑性变形、应变硬化和强化行为,最后展望了异构提升宏观力学性能匹配的潜力。

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Grain boundaries (GBs) are important planar defects in polycrystalline materials, and they are crucial in plastic deformation and recrystallization of materials. A fundamental understanding of GB deformation kinetics is critical for material design using GB engineering. Although GB dominated structural evolutions have been reported to proceed via different modes, the disconnection-based model has recently become a widely acknowledged approach to unify the GB dominated plasticity. In this paper, recent progresses of GB dominated plasticity in metallic materials based on disconnection-mediated GB migration have been reviewed. Disconnection dynamics, including nucleation, propagation and interactions between different disconnections, were found dominating the shear-coupled GB migration. Lateral motion of different GB disconnections contributes to the overall GB migration, during which dynamic interactions prevail. In the three-dimensional network of GBs, GB-defect interaction and triple junctions can further influence the shear-coupled GB migration by providing extra disconnection sources, which readily change the intrinsic disconnection dynamics. These disconnection-based GB kinetics are generally applicable in the migration of GBs with different structures, as well as other modes of GB dominated deformation. Based on the aforementioned, the effects of GB plasticity on mechanical properties and deformation of metallic materials are further discussed. This review provides a unified understanding of disconnection-based GB plasticity, which not only enriches mechanistic understanding of interface plasticity in metallic materials but also holds important implications for GB engineering toward advanced high-performance metallic materials.

王江伟, 陈映彬, 祝祺等.

金属材料的晶界塑性变形机制

[J]. 金属学报, 2022, 58: 726

晶界是多晶材料中一类重要的面缺陷,在材料的力学和物理化学性能调控中发挥着重要作用。深入理解晶界的塑性变形动力学机制是开展材料晶界工程调控的理论基础。本文从晶界的微观结构和晶界本征缺陷出发,详细总结晶界塑性变形机制的研究进展；在此基础上,围绕晶界阶错形核、扩展、交互作用的动力学机制,深入探讨晶界迁移的原子尺度动力学机制及其在不同因素下的表现形式,阐明不同晶界变形行为之间的关联关系,发展和完善晶界塑性变形理论,为金属材料的晶界工程调控提供理论指导。

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[J]. Acta Metall. Sin., 2022, 58: 375

Dislocation slip and twinning are the main deformation mechanisms dominating plastic behavior of crystalline materials, such as twinning-induced plasticity steel, Cu, Mg, and their alloys. The influence of twinning and interaction between dislocations and twins on the plastic deformation of crystal materials is complex. On the one hand, a sudden stress drop in the stress-strain curve during twin nucleation, propagation, and growth (TNPG) of crystal materials, i.e., the twinning softening effect, is evident. On the other hand, the interaction between twins and dislocations demonstrates the strengthening effect of plastic deformation. Polycrystalline materials are used in engineering applications, and twin nucleation corresponds to different strains in each grain. Therefore, determining the influence of twin softening and strengthening effects on plastic deformation of polycrystalline materials is difficult. In this work, a crystal plastic finite element model of Cu, considering the twinning softening effect, was developed to describe the TNPG process based on the crystal plasticity theory. The method was used to reveal the influence of twins' activation and their interaction with dislocations on strain hardening during the tension of Cu single crystal and polycrystal. The results show that twinning has an evident orientation effect. Under twinning favorable orientation, a sudden stress drop in the stress-strain curve caused by twinning propagation during plastic deformation of Cu single crystal is evident, and the total plastic deformation can be divided into three stages: slip, twinning, and interaction between dislocations and twins. Compared with Cu single crystal, the stress-strain curve changes smoothly and the strain hardening rate is higher during the tension of Cu polycrystal. Meanwhile, the dislocation density is concentrated at the grain boundary, and twins are easy to form at the grain boundary during the plastic deformation of Cu polycrystal.

郭祥如, 申俊杰.

孪生诱发软化与强化效应的Cu晶体塑性行为模拟

[J]. 金属学报, 2022, 58: 375

基于晶体塑性理论，考虑孪生软化效应建立了描述孪晶形核、增殖和长大的位错密度基晶体塑性有限元模型。应用该模型揭示了不同晶体取向Cu单晶拉伸变形过程中位错滑移、孪生激活及其交互作用下的宏观塑性行为演化规律，进一步分析了Cu多晶拉伸变形过程中晶粒间交互作用对孪生软化、应变硬化等宏观塑性行为的影响。结果表明：孪生具有明显的取向效应，在孪生主导塑性条件下，Cu单晶塑性变形过程中孪晶增殖导致应力-应变曲线存在明显的应力突降现象，其塑性变形分为滑移、孪生及位错与孪晶交互作用3个阶段；此外，随着饱和孪晶体积分数增加，Cu单晶塑性变形过程中第3阶段的应变硬化率也随之提升。进一步模拟Cu多晶拉伸变形的塑性行为可知，在晶粒间交互作用下孪晶形核、增殖和长大过程中不会出现应力突降现象，与Cu单晶相比整个塑性变形过程具有更高的应变硬化率；Cu多晶塑性变形过程中位错密度在晶界处出现集中现象，孪晶也容易在晶界处形成。

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Investigation of plastic deformation behavior on coupling twinning of polycrystal TWIP steel

[J]. Acta Metall. Sin., 2015, 51: 1507

Twinning induced plasticity (TWIP) steel exhibits high strength and exceptional plasticity due to the formation of extensive twin under mechanical load and its ultimate tensile strength and elongation to failure ductility-value can be as high as 5×104 MPa%, which provide a new choice for automobile in developing the lightweight and improving safety. Generally, due to the texture was formed during process of plastic deformation, metal material appear anisotropic behavior. The deformation mechanisms, responsible for this high strain hardening, are related to the strain-induced microstructural changes, which was dominated by slip and twinning. Different deformation mechanisms, which can be activated at different stages of deformation, will strongly influence the stress strain response and the evolution of the microstructure. In this work, to predict the texture evolution under different loading conditions and understand these two deformation mechanisms of plastic deformation process, a polycrystal plasticity constitutive model of TWIP steel coupling slip and twinning was developed based on the crystal plasticity theory and single crystal plasticity constitutive model. A polycrystal homogenization method to keep geometry coordination and stress balance adjacent grains was used, which connected the state variables of single crystal and polycrystal. And then the model was implemented and programed based on the ABAQUS/UMAT platform. The texture evolution was obtained by EBSD at strain 0.27 and 0.60, respectively. The finite element models of tensile, compression and torsion processes were built by using the constitutive model. The mechanical response and texture evolution during plastic deformation process of TWIP steel were analyzed. The results show that with the increasing of the strain, the strain hardening phenomenon and texture density enhanced during the tensile process. Although texture types changed, texture density unchanged during the compression process. Owing to deformation increasing along the diameter direction, there is no obvious texture inside the cylinder when torsion deformation is small, texture emerged and enhanced gradually with the increasing of strain.

孙朝阳, 郭祥如, 郭宁等.

耦合孪生的TWIP钢多晶体塑性变形行为研究

[J]. 金属学报, 2015, 51: 1507

基于已建立的单晶体塑性模型, 建立了耦合孪生的孪生诱发塑性(TWIP)钢多晶体塑性模型, 该模型采用有限元多晶均匀化处理相邻晶粒间的几何协调和应力平衡条件, 获得了单晶体与多晶体状态变量的关系, 开发了基于ABAQUS/UMAT的计算程序. 采用EBSD研究了TWIP钢拉伸应变分别为0.27和0.6时的织构变化, 并对模型进行了应力应变及织构演化的验证. 用该本构模型分别建立了拉伸、压缩和扭转3种简单加载条件下的有限元模型, 分析了不同变形条件下的宏观力学响应及织构演化规律. 结果表明: 拉伸变形过程中, 应变硬化现象和织构密度水平随应变增加而增强; 在压缩过程中, 织构类型随应变增加而发生变化, 但是织构密度水平基本不变; 而在扭转过程中, 当扭转应变较小时, 基本无织构形成, 随着应变增加, 织构逐渐显现出来, 这是因为变形较小时, 圆柱沿径向方向内部变形量较小, 故织构不明显.

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