PHASE-FIELD SIMULATION ON THE FUNCTIONAL PROPERTIES OF STRESS-ASSISTED AGING NiTi SHAPE MEMORY ALLOYS
-
摘要: 通过改进的耦合沉淀相析出和马氏体相变的相场模型模拟了应力辅助时效的NiTi单晶的超弹性、弹卡效应、单程形状记忆效应和应力辅助双程形状记忆效应, 讨论了单一取向Ni4Ti3沉淀相对功能性能的影响以及背后的微观机理. 模拟结果显示, 由于单一取向Ni4Ti3沉淀相的强几何约束, 超弹性过程和应力辅助双程形状记忆过程中的马氏体相变并不以马氏体条带快速扩宽的方式进行, 而是以大量针状马氏体的形核与生长来实现, 马氏体逆相变则以大量针状马氏体的缩减和消失来实现. 含单一取向Ni4Ti3沉淀相的单晶系统具有优良的弹卡效应. 对于单程形状记忆效应, 含单一取向Ni4Ti3沉淀相的单晶系统在温度诱发马氏体相变过程中形成的孪晶面均与沉淀相惯习面平行, 单一取向和两种取向的Ni4Ti3沉淀相对马氏体重取向的约束不同, 对温度诱发马氏体逆相变的约束也不同, 导致二者对应的应力-应变-温度曲线的差异. 文章将为理解Ni4Ti3沉淀相对NiTi合金功能性能的影响提供见解, 并为调控合金的马氏体相变、功能和力学性能提供参考.
-
关键词:
- NiTi形状记忆合金 /
- Ni4Ti3沉淀相 /
- 相场模拟 /
- 功能性能
Abstract: In this work, an improved phase field model coupling precipitation and martensitic transformation (MT) was used to simulate the superelasticity, elastocaloric effect, one-way shape memory effect (OWSME) and stress-assisted two-way shape memory effect (SATWSME) of the stress-assisted aging NiTi single crystal. The influence of single-orientated Ni4Ti3 precipitates on the functional properties and the underlying microscopic mechanism were revealed. The simulation results show that due to the strong geometric constraints of single-oriented Ni4Ti3 precipitates, the MT in the superelastic or SATWSME process is not realized by the rapid widening of martensitic bands, but by the nucleation and growth of a large number of needlelike martensite phases, and the reverse MT is achieved by the reduction and disappearance of a large number of needlelike martensite phases. The single-crystal system containing single-oriented Ni4Ti3 precipitates shows an excellent elastocaloric effect. For the OWSME, in the single-crystal system with single-oriented Ni4Ti3 precipitates, the twinned interfaces formed during the temperature-induced MT are parallel to the inertial plane of the Ni4Ti3 precipitates; the constraints of the single-orientated and different-orientated Ni4Ti3 precipitates on the martensitic reorientation are different, and those on the temperature-induced reverse MT are also different, resulting in the difference in the stress-strain-temperature curves of the two cases. This work will provide insights for understanding the effect of Ni4Ti3 precipitates on the functional properties of NiTi alloys, and provide references for regulating the MT, functional and mechanical properties of such alloys. -
引言
镍钛(NiTi)形状记忆合金自问世以来, 因表现出独特的超弹性、形状记忆效应及优良的生物相容性和耐磨性, 已成为材料界和各工程领域重点关注的功能材料. 目前, NiTi 形状记忆合金已在航空航天、微机电系统、汽车、生物医学等领域扮演着不可或缺的角色[1-2]. 此外, NiTi合金因其优异的弹卡性能在众多形状记忆合金弹卡材料中脱颖而出, 最有望作为核心元件应用在固态制冷领域, 其弹卡性能在近年来备受关注[3-7].
大量研究表明[8-10], 一般的粗晶NiTi合金在较大的变形或循环变形过程中极易发生微观尺度的局部塑性变形, 导致其表现出较差的功能循环稳定性和疲劳抗性, 为该合金的应用带来极大的挑战. 如何提高NiTi合金的循环稳定性和疲劳寿命, 是形状记忆合金研究领域长期以来的研究重点和难点问题.
粗晶NiTi合金中位错滑移的开动与局部的、爆发式的马氏体相变密切相关, 其极易发生在有集中应力的、随马氏体相变的进行不断运动的相界面处[9]. 因此, 调节马氏体相变是从根本上改善NiTi合金循环稳定性和疲劳抗性的有效途径. 目前, 已报道的调控马氏体相变的方法包括晶粒尺寸调控[11-12]、Ni浓度调控[13-14]、掺杂元素调控[15-16]、塑性变形调控[17-18]和沉淀相调控[19-20]等. 其中, 沉淀相调控是一种简易而低成本的方法. 经过简单的时效处理, 即可在富Ni的NiTi合金中引入Ni4Ti3沉淀相. Ni4Ti3沉淀相不仅可以调节NiTi合金基体中的Ni浓度从而调控马氏体相变开始温度Ms[21], 还能诱发多步相变来改变合金的力学行为[22]. 更重要的是, 细小的高密度Ni4Ti3 沉淀相能有效抑制NiTi合金中的位错滑移从而显著提高其循环稳定性和疲劳抗性[19-20,23-24]. 可见, Ni4Ti3沉淀相在调控马氏体相变和改善NiTi合金力学性能上极具潜力.
除了改变时效温度和时效时间外, Ni4Ti3沉淀相析出的动力学过程还可以通过约束时效, 即应力辅助时效来调控, 这是因为外载荷与Ni4Ti3变体相关的本征应变之间存在弹性耦合[25]. 当在时效过程中施加一个恒定的压应力时, 合金中会形成单一取向的Ni4Ti3变体[26], 其对NiTi合金的力学行为会有独特的影响[27-28]. 注意这里的“取向”指Ni4Ti3沉淀相形态上的取向, 其取决于时效过程中施加的外载荷和Ni4Ti3沉淀相变体的晶格结构. 最近, Xiao等[29]制备出了含有单一取向Ni4Ti3沉淀相的、不同晶体取向的NiTi合金单晶微柱并测试了其压缩疲劳性能, 发现单一取向的Ni4Ti3 沉淀相引起的非均匀应力场会诱发不同的马氏体变体, 并且对不同取向单晶微柱内位错滑移的阻碍程度不同, 导致了不同的疲劳寿命. Xiao等[29]的研究表明, NiTi合金的疲劳寿命依赖于Ni4Ti3沉淀相的取向, 通过调控沉淀相取向来调控NiTi合金的功能性能或提高合金的疲劳抗性是一个值得深入研究的方向. 然而, 单一取向的Ni4Ti3沉淀相对NiTi合金功能性能和疲劳行为的影响以及背后的物理机理尚未完全厘清.
由于NiTi合金中存在多种非弹性变形机制, 例如马氏体相变、马氏体解孪、马氏体重取向、奥氏体位错滑移、马氏体位错滑移和马氏体变形孪生等[10], 合金的变形机理通常十分复杂. 在微观尺度上, NiTi合金变形过程中复杂的非均匀应力场可能同时诱发多种非弹性变形机制, 它们之间会相互影响, 并且可能受到Ni4Ti3沉淀相的影响. 尽管目前已有大量实验研究报道了含有不同Ni4Ti3沉淀相的NiTi合金的功能性能和疲劳行为, 但这还不足以全面厘清Ni4Ti3沉淀相的影响机理. 为此, 借助数值模拟的手段开展相关研究是十分必要的.
相场法作为一种介观尺度的模拟方法, 在模拟材料晶粒尺度的微结构演化方面具有天然优势. 近20年来, 相场法已广泛应用于NiTi合金变形行为的模拟研究[30-40], Ni4Ti3沉淀相的析出动力学过程[25,41-43]以及其对马氏体相变的影响[44-47]也受到了广泛关注. 然而, 关注单一取向Ni4Ti3沉淀相对NiTi合金功能性能影响的相场研究还未见报道. 因此, 本文基于建立的耦合沉淀相析出和马氏体相变的相场模型, 首先模拟NiTi合金单晶系统的应力辅助时效过程, 引入单一取向的Ni4Ti3沉淀相, 随后模拟单晶系统的超弹性、弹卡效应、单程形状记忆效应和应力辅助双程形状记忆效应. 结合模拟所得的应力-应变-温度响应、微结构和温度场演化, 讨论并揭示单一取向Ni4Ti3沉淀相对NiTi合金功能性能的影响机理. 本工作将为理解Ni4Ti3沉淀相对NiTi合金功能性能的影响提供见解, 并为调控合金的马氏体相变、功能和力学性能提供参考.
1. 相场模型
我们之前的工作[47]已经提出了耦合沉淀相析出和马氏体相变的热-力耦合相场模型, 本文的改进之处在于将外载荷引起的应变能引入到了沉淀相析出的相场描述部分. 为了保证全文的逻辑性和完整性, 本节将给出整个相场模型的描述.
1.1 沉淀相析出的相场模型
在描述Ni4Ti3沉淀相的析出过程时, 利用浓度场c来表征Ni4Ti3沉淀相和NiTi基体之间的Ni元素浓度差异, 同时使用序参量${\eta _i} ( {i = 1,2, \cdots ,n} $, 其中n是Ni4Ti3沉淀相变体数目)来描述不同的Ni4Ti3沉淀相变体和NiTi基体相. ${\eta _{i = 1 \sim n}} = 0$表示NiTi基体相, ${\eta _i} = 1$且${\eta _{j = 1 \sim n,j \ne i}} = 0$表示第i个Ni4Ti3沉淀相变体. Ni4Ti3沉淀相为菱方晶格结构, 有8种变体, 这8种变体的(111)晶面分别平行于B2(立方)结构的NiTi基体相的{111}晶面族. 8种变体又可以分为4组, 每组含有一对共轭变体, 相互之间可以通过旋转180°得到. 因此, 在相场模拟中通常考虑4种Ni4Ti3沉淀相变体[42-43,45-46], 即n = 4.
Ni4Ti3沉淀相析出和长大的热力学驱动力包括三部分: (1) 过饱和NiTi基体与含有Ni4Ti3沉淀相的平衡态NiTi基体之间的化学能之差; (2) 界面能; (3) 由Ni4Ti3沉淀相与NiTi基体之间的晶格畸变产生的弹性能和由外部应力引起的应变能. 因此, 可以将材料系统的总自由能表示为[43]
$$ F = \int_V {\left( {{f_0} + {f_{{\text{grad}}}} + {f_{{\text{str}}}}} \right)} {\text{d}}V $$ (1) 其中, ${f_0}$, ${f_{{\text{grad}}}}$和${f_{{\text{str}}}}$分别为化学能密度、梯度能密度和应变能密度, V是系统的体积.
$$ \begin{split} & {f_0}\left( {c,\eta } \right) = \frac{1}{2}{A_1}{\left( {c - {c_1}} \right)^2} + \frac{1}{2}{A_2}\left( {c - {c_2}} \right)\sum\limits_{i = 1}^n {\eta _i^2} -\\ &\qquad \frac{1}{4}{A_3}\sum\limits_{i = 1}^n {\eta _i^4} + \frac{1}{6}{A_4}\sum\limits_{i = 1}^n {\eta _i^6} + {A_5}\sum\limits_{i \ne j}^n {\eta _i^2\eta _j^2} + \\ &\qquad {A_6}\sum\limits_{i \ne j,i \ne k}^n {\eta _i^4\left( {\eta _j^2 + \eta _k^2} \right)} + {A_7}\sum\limits_{i \ne j \ne k}^n {\eta _i^2\eta _j^2\eta _k^2} \end{split} $$ (2) 其中, c是Ni元素浓度, ${c_1}$和${c_2}$分别是平衡态NiTi基体与Ni4Ti3沉淀相中的Ni元素浓度, ${A_1} \sim {A_7}$是多项式系数.
梯度能密度${f_{{\text{grad}}}}$可以表示为[41-42]
$$ {f_{{\text{grad}}}} = \frac{1}{2}{\lambda _c}{\left( {\nabla c} \right)^2} + \frac{1}{2}{\lambda _\eta }\sum\limits_{i = 1}^n {{{\left( {\nabla {\eta _i}} \right)}^2}} $$ (3) 其中, ${\lambda _c}$和${\lambda _\eta }$分别为浓度场和序参量对应的梯度能系数.
实验中要获得单一取向的Ni4Ti3沉淀相, 需要在NiTi合金[111]B2方向施加恒定压应力的条件下进行时效[26-28]. 然而, 本文模拟应力辅助时效过程(即在[111]B2方向存在恒定压应力情形下的Ni4Ti3沉淀相析出过程)后, 在进一步模拟含Ni4Ti3沉淀相系统的功能性能时需要在[100]B2方向施加机械载荷. 然而, 在[111]B2和[100]B2方向都以边界条件的方式施加机械载荷存在困难. 因此, 参考柯常波[42]的研究, 我们采用了一种替代模拟方案, 即在沉淀相析出模型中将外加应力引起的应变能引入到应变能密度${f_{{\text{str}}}}$中, 也就是说, 在模拟应力辅助时效过程时直接通过设置外加应力张量来对系统施加外应力. 因此, ${f_{{\text{str}}}}$可以表示为
$$ {f_{{\text{str}}}} = \frac{1}{2}{{\boldsymbol{\varepsilon}} ^{{{\mathrm{el}}} }}{\mathbf{:}}{\boldsymbol{C}}{\mathbf{:}}{{\boldsymbol{\varepsilon}} ^{{{\mathrm{el}}} }} - {{\boldsymbol{\sigma}} ^{{\text{ext}}}}{\mathbf{:}}{{\boldsymbol{\varepsilon}} ^{{\text{in}}}} $$ (4) 其中, C是4阶弹性张量, ${{\boldsymbol{\varepsilon}} ^{{{\mathrm{el}}} }}$是弹性应变张量, ${{\boldsymbol{\sigma}} ^{{\text{ext}}}}$是外加应力张量, ${{\boldsymbol{\varepsilon}} ^{{\text{in}}}}$是非弹性应变张量.
本文假设NiTi奥氏体相(${{\boldsymbol{C}}_{\text{A}}}$)与Ni4Ti3沉淀相的弹性张量相同, 即${{\boldsymbol{C}}_{\text{A}}} = {\boldsymbol{C}}$. 弹性应变张量${{\boldsymbol{\varepsilon}} ^{{{\mathrm{el}}} }}$和非弹性应变张量${{\boldsymbol{\varepsilon}} ^{{\text{in}}}}$之间的关系为[41-44]
$$\qquad {{\boldsymbol{\varepsilon}} ^{{{\mathrm{el}}} }} = {{\boldsymbol{\varepsilon}} ^{{\text{total}}}} - {{\boldsymbol{\varepsilon}} ^{{\text{in}}}} $$ (5) $$\qquad {{\boldsymbol{\varepsilon}} ^{{\text{total}}}} = \frac{1}{2}\left[ {\nabla {\boldsymbol{u}} + {{\left( {\nabla {\boldsymbol{u}}} \right)}^{\text{T}}}} \right] $$ (6) $$\qquad {{\boldsymbol{\varepsilon}} ^{{\text{in}}}} = \sum\limits_{i = 1}^n {{\boldsymbol{\varepsilon}} _i^{{\text{N}}{{\text{i}}_4}{\text{T}}{{\text{i}}_3}}\eta _i^2} $$ (7) 其中, ${{\boldsymbol{\varepsilon}} ^{{\text{total}}}}$为总应变张量, u为位移矢量, ${\boldsymbol{\varepsilon}} _i^{{\text{N}}{{\text{i}}_4}{\text{T}}{{\text{i}}_3}}$为第i个Ni4Ti3沉淀相变体的本征应变. 注意${\boldsymbol{\varepsilon}} _i^{{\text{N}}{{\text{i}}_4}{\text{T}}{{\text{i}}_3}}$只与Ni4Ti3沉淀相的晶格结构有关, 与浓度场无关, 其具体表达式将在后文给出.
浓度场和序参量的演化分别由Cahn-Hilliard方程[48]和Ginzberg-Landau方程[49]控制, 即
$$\qquad\quad \frac{{\partial c}}{{\partial t}} = M{\nabla ^2}\frac{{\delta F}}{{\delta c}} $$ (8) $$\qquad\quad {\frac{{\partial {\eta _i}}}{{\partial t}} = - L\frac{{\delta F}}{{\delta {\eta _i}}}}\quad {\left( {i = 1,2, \cdots, n} \right)} $$ (9) 其中, M和L分别为浓度场c的化学迁移率和序参量的运动系数, t为模拟时间. 将式(1) ~ 式(5)和式(7)代入式(8)和式(9)可得
$$ \frac{{\partial c}}{{\partial t}} = M{\nabla ^2}\left[ {{A_1}\left( {c - {c_1}} \right) + \frac{1}{2}{A_2}\sum\limits_{i = 1}^n {\eta _i^2} - {\lambda _c}{\nabla ^2}c} \right] $$ (10) $$ \begin{split} & \frac{1}{L}\frac{{\partial {\eta _i}}}{{\partial t}} = 2{\eta _i}\left( {{\boldsymbol{\sigma}} + {{\boldsymbol{\sigma}} ^{{\text{ext}}}}} \right){\mathbf{:}}{\boldsymbol{\varepsilon}} _i^{{\text{N}}{{\text{i}}_{\text{4}}}{\text{T}}{{\text{i}}_{\text{3}}}} + {\lambda _\eta }{\nabla ^2}{\eta _i} - \\ &\qquad \left[ {{A_2}\left( {c - {c_2}} \right){\eta _i} - {A_3}\eta _i^3 + {A_4}\eta _i^5 + 2{A_5}{\eta _i}\sum\limits_{i \ne j}^n {\eta _j^2} } + \right.\\ &\qquad {\left. { 2{A_6}\left( {2\eta _i^3\sum\limits_{i \ne j}^n {\eta _j^2} + {\eta _i}\sum\limits_{i \ne j}^n {\eta _j^4} } \right) + 2{A_7}{\eta _i}\sum\limits_{i \ne j \ne k}^n {\eta _j^2\eta _k^2} } \right]}\\ &\qquad {\left( {i = 1,2, \cdots ,n} \right)} \end{split}$$ (11) 注意本文在求解4阶偏微分方程(10)时, 采用了变量替换的方法, 即将方程(10)中中括号内的部分用一个变量代替, 将该4阶偏微分方程转化为两个2阶偏微分方程.
1.2 马氏体相变的相场模型
在描述马氏体相变过程时, 使用序参量${\phi _p}( p = 1,2, \cdots ,m $, 其中m是马氏体变体数目)来描述不同马氏体变体和奥氏体相. ${\phi _{p = 1 \sim m}} = 0$表示奥氏体相, ${\phi _p} = 1$且${\phi _{q = 1 \sim n,q \ne p}} = 0$表示第p个马氏体变体.
与沉淀相的析出和长大类似, 马氏体形核和长大的热力学驱动力包括3部分, 即局部自由能、梯度能和弹性能. 系统的总自由能可以表示为[31]
$$ G = \int_V {\left( {{G_0} + {G_{{\text{grad}}}} + {G_{{\text{el}}}}} \right)} {\text{d}}V $$ (12) 其中, ${G_0}$, ${G_{{\text{grad}}}}$和${G_{{\text{el}}}}$分别为局部自由能密度、梯度能密度和弹性能密度.
局部自由能密度可以表示为[31]
$$ {G_0} = \frac{1}{2}{B_1}\sum\limits_{p = 1}^m {\phi _p^2} - \frac{1}{3}{B_2}\sum\limits_{p = 1}^m {\phi _p^3} + \frac{1}{4}{B_3}{\left( {\sum\limits_{p = 1}^m {\phi _p^2} } \right)^2} + \frac{1}{4}{B_4}\sum\limits_{p = 1}^m {\phi _p^4} $$ (13) 其中, ${B_1} \sim {B_4}$是温度相关的多项式系数, 其可以表示为[50]
$$ \left.\begin{gathered} {B_1} = 32\Delta {G^ * } \\ {B_2} = 96\Delta {G^ * } - 12\Delta G \\ {B_3} = {B_4} = 32\Delta {G^ * } - 6\Delta G \\ \end{gathered} \right\} $$ (14) 其中, $\Delta G$和$\Delta {G^ * }$分别为奥氏体相与马氏体相的局部自由能之差和能量壁垒, 其可以通过下式计算[51-52]
$$ \qquad\qquad \Delta G = \frac{{Q\left( {\vartheta - {\vartheta _0}} \right)}}{{{\vartheta _0}}} $$ (15) $$\qquad\qquad \Delta {G^ * } = \frac{Q}{{32}}\exp \left( {a\frac{{\vartheta - {\vartheta _0}}}{{{\vartheta _0}}}} \right) $$ (16) 其中, Q为相变潜热, $\vartheta $和${\vartheta _0}$分别为环境温度和参考温度, a为材料常数.
梯度能密度可以表示为[31]
$$ {G_{{\text{grad}}}} = \frac{1}{2}\beta \sum\limits_{p = 1}^m {{{\left| {\nabla {\phi _p}} \right|}^2}} $$ (17) 其中, $\beta $为梯度能系数.
与模拟应力辅助时效过程不同的是, 模拟马氏体相变时直接通过边界条件施加机械载荷(不需要引入式(4)所示的外加应力引起的应变能). 因此, 此处的弹性能密度可以直接表示为
$$ {G_{{\text{el}}}} = \frac{1}{2}{{\boldsymbol{\varepsilon}} ^{{{\mathrm{el}}} }}{\mathbf{:}}{\boldsymbol{C}}\left( \phi \right){\mathbf{:}}{{\boldsymbol{\varepsilon}} ^{{{\mathrm{el}}} }} $$ (18) 其中, ${\boldsymbol{C}}\left( \phi \right)$为4阶弹性张量, 其满足常用的混合律
$$ {\boldsymbol{C}}\left( \phi \right) = {{\boldsymbol{C}}_{\text{M}}}\sum\limits_{p = 1}^m {{\phi _p}} + {{\boldsymbol{C}}_{\text{A}}}\left( {1 - \sum\limits_{p = 1}^m {{\phi _p}} } \right) $$ (19) $$ {{{\boldsymbol{C}}_X} = \frac{{{E_X}\nu }}{{\left( {1 - 2\nu } \right)\left( {1 + \nu } \right)}}{\boldsymbol{\sigma}} \otimes {\boldsymbol{\sigma}} + \frac{{{E_X}\nu }}{{1 + \nu }}{\boldsymbol{I}}}\quad {\left( {X = {\text{A, M}}} \right)} $$ (20) 其中, ${{\boldsymbol{C}}_{\text{M}}}$和${{\boldsymbol{C}}_{\text{A}}}$分别为马氏体相和奥氏体相的弹性张量, ${E_{\text{M}}}$和${E_{\text{A}}}$分别为马氏体相和奥氏体相的弹性模量, $\nu $为泊松比(本文假设马氏体相和奥氏体相的泊松比相同); σ和I分别为2阶和4阶单位张量. 弹性应变张量${{\boldsymbol{\varepsilon}} ^{{{\mathrm{el}}} }}$可以由下式计算
$$\qquad\qquad {{\boldsymbol{\varepsilon}} ^{{{\mathrm{el}}} }} = {{\boldsymbol{\varepsilon}} ^{{\text{total}}}} - {{\boldsymbol{\varepsilon}} ^{{\text{tr}}}} - {{\boldsymbol{\varepsilon}} ^{{\text{in}}}} $$ (21) $$\qquad\qquad {{\boldsymbol{\varepsilon}} ^{{\text{tr}}}} = \sum\limits_{p = 1}^m {{\boldsymbol{\varepsilon}} _p^{{\mathrm{M}}} {\phi _p}} $$ (22) 其中, 总应变张量${{\boldsymbol{\varepsilon}} ^{{\text{total}}}}$的表达式与式(6)相同, ${\boldsymbol{\varepsilon}} _p^{{\mathrm{M}}} $为第p个马氏体变体的本征应变, 非弹性应变张量${{\boldsymbol{\varepsilon}} ^{{\text{in}}}}$的表达式如式(7)所示. 需要指出的是, 在本文的相场模拟中, 是先模拟Ni4Ti3沉淀相析出过程得到稳定的沉淀相微结构, 再以含有沉淀相微结构、应力场和应变场的系统作为初始构型, 模拟马氏体相变过程. 在模拟马氏体相变过程时, 假设Ni4Ti3沉淀相是不演化的, 并且沉淀相所在的位置不能再发生马氏体相变[45-47].
将式(12)、式(13)以及式(17) ~ 式(22)代入Ginzberg-Landau方程[49]可得
$$\begin{split} &\frac{1}{\tau }\frac{{\partial {\phi _p}}}{{\partial t}} = {\boldsymbol{\sigma}} {\mathbf{:}}{\boldsymbol{\varepsilon}} _p^{\text{M}} - \frac{1}{2}{{\boldsymbol{\varepsilon}} ^{{\text{el}}}}{\mathbf{:}}\Delta {\boldsymbol{C}}{\mathbf{:}}{{\boldsymbol{\varepsilon}} ^{{\text{el}}}} + \beta {\nabla ^2}{\phi _p} -\\ &\qquad \left[ {{B_1}{\phi _p} - {B_2}\phi _p^2 + {B_3}{\phi _p}\left( {\sum\limits_{q = 1}^m {\phi _q^2} } \right) + {B_4}\phi _p^3} \right]\end{split} $$ (23) 其中, $\tau $为运动系数, $\Delta {\boldsymbol{C}} = {{\boldsymbol{C}}_{\text{M}}} - {{\boldsymbol{C}}_{\text{A}}}$.
NiTi合金的马氏体相变和逆相变过程分别伴随着相变潜热的释放和吸收, 这是其可以作为固态制冷材料的关键. 假设绝热条件下的马氏体相变在材料系统内产生移动热源, 温度场的演化由热传导方程控制[47,52]
$$ \rho C\frac{{\partial \vartheta }}{{\partial t}} = \kappa {\nabla ^2}\vartheta + Q\sum\limits_{p = 1}^m {{{\dot \phi }_p}} $$ (24) 其中, $\rho $, C和$\kappa $分别为质量密度、热容和热传导系数. 假设奥氏体相和马氏体相的质量密度相等, 记为${\rho _{{\text{NiTi}}}}$. 此外, 由于NiTi合金中马氏体和奥氏体热容的差异很小, 在本构模型[53-55]和相场模拟[32,56-57]中通常不考虑二者的差异. 本文不区分马氏体和奥氏体热容, 将其记为${C_{{\text{NiTi}}}}$. 然而, 由于Ni4Ti3沉淀相和NiTi基体的质量密度和热容差异较大, 根据简单的混合律, 将$\rho $和C表示为
$$ \rho = {\rho _{{\text{N}}{{\text{i}}_4}{\text{T}}{{\text{i}}_3}}}\sum\limits_{i = 1}^n {{\eta _i}} + {\rho _{{\text{NiTi}}}}\left( {1 - \sum\limits_{i = 1}^n {{\eta _i}} } \right) $$ (25) $$ C = C_{{\text{N}}{{\text{i}}_4}{\text{T}}{{\text{i}}_3}}^{}\sum\limits_{i = 1}^n {{\eta _i}} + {C_{{\text{NiTi}}}}\left( {1 - \sum\limits_{i = 1}^n {{\eta _i}} } \right) $$ (26) 其中, ${\rho _{{\text{N}}{{\text{i}}_{\text{4}}}{\text{T}}{{\text{i}}_{\text{3}}}}}$和${C_{{\text{N}}{{\text{i}}_{\text{4}}}{\text{T}}{{\text{i}}_{\text{3}}}}}$分别为Ni4Ti3沉淀相的质量密度和热容.
2. 模拟参数与几何模型
上一节建立了三维的描述沉淀相析出与马氏体相变的相场模型, 其适用于三维的单晶系统. 然而, 三维的情形涉及到4种Ni4Ti3沉淀相变体和12种马氏体变体以及大量三维的有限元网格, 使得计算量巨大. 大量研究表明, 二维的平面应变相场模拟可以很好地捕捉Ni4Ti3沉淀相析出动力学过程[41-42]和马氏体相变过程中的微结构演化[32,34-36,52,58-59], 并合理反映形状记忆合金功能性能对应的应力-应变-温度响应以及背后的微观机理[32,34-36,52,58-59]. 因此, 本文采用二维的平面应变单晶系统进行数值实现. 本节将介绍二维模拟情形下的参数设定和几何模型.
2.1 模拟参数
在二维情形下, 4种Ni4Ti3沉淀相变体可以简化为2种, 其本征应变如下[47]
$$\left.\begin{split} & {{\boldsymbol{\varepsilon}} _1^{{\text{N}}{{\text{i}}_{\text{4}}}{\text{T}}{{\text{i}}_{\text{3}}}} = \left[ {\begin{array}{*{20}{c}} - 0.011\;3 & - 0.007\;2 \\ - 0.007\;2 & - 0.011\;3 \end{array}} \right]}\\ &{{\boldsymbol{\varepsilon}} _2^{{\text{N}}{{\text{i}}_{\text{4}}}{\text{T}}{{\text{i}}_{\text{3}}}} = \left[ {\begin{array}{*{20}{c}} - 0.011\;3 & 0.007\;2 \\ 0.007\;2 & - 0.011\;3 \\ \end{array}} \right]} \end{split}\right\} $$ (27) 模拟沉淀相析出过程所需的其他参数根据Dong等[46]结果来选取, 具体数值如表1所示.
对于马氏体相变过程的模拟, 本文参考Mikula等[60]、Sun等[61]和Xu等[39,47,59,62]研究, 仅考虑2种马氏体变体( +M和−M变体), 其本征应变如下
$$ \begin{array}{*{20}{c}} {{\boldsymbol{\varepsilon}} _1^{\text{M}} = \left[ {\begin{array}{*{20}{c}} 0.056 & 0 \\ 0 & - 0.056 \\ \end{array}} \right],}&{{\boldsymbol{\varepsilon}} _2^{\text{M}} = \left[ {\begin{array}{*{20}{c}} - 0.056 & 0 \\ 0 & 0.056 \\ \end{array}} \right]} \end{array} $$ (28) 参考Yu等[63], 将奥氏体相(${E_{\text{A}}}$)和马氏体相(${E_{\text{M}}}$)的弹性模量分别取为60 GPa和40 GPa, 泊松比取为0.33. 本文将Ni4Ti3沉淀相和NiTi基体的质量密度分别取为${\rho _{{\text{N}}{{\text{i}}_4}{\text{T}}{{\text{i}}_3}}} $ = 6770 kg/m3[64]和${\rho _{{\text{NiTi}}}} $ = 6450 kg/m3[65], 热容分别取为$ {C_{{\text{N}}{{\text{i}}_4}{\text{T}}{{\text{i}}_3}}} $ = 345 J/(kg·K)[66]和CNiTi = 550 J/(kg·K)[67]. 参考Xu等[68]以及Sun等[61], 将热传导系数、运动学系数和梯度能系数分别设为$\kappa = 2{{\text{ W}} /({{{{\mathrm{m}}\cdot{\mathrm{K}}}}})}$, $\tau = 20{{{{\text{ m}}^3}\cdot{\text{s}}} / {\text{J}}}$和$\beta = 1.0 \times 10^{ - 10} {\text{ J}} /{\text{m}}$. 此外, 本文将参考温度和相变潜热分别取为${\vartheta _0} $ = 271 K[69]和Q = 110 MJ/m3[31], 将材料常数a设为10[52]. 以上参数均在表1中给出.
表 1 模拟参数Table 1. Simulation parametersPrecipitation of Ni4Ti3 Martensite transformation Parameter Value Parameter Value equilibrium Ni concentration of NiTi ${c_1}$ 0.50 elastic modulus of austenite phase EA/GPa 60 equilibrium Ni concentration of Ni4Ti3 ${c_2}$ 0.57 elastic modulus of martensite phase EM/GPa 40 initial Ni concentration ${c_0}$ 0.51 Poisson’s ratio $\nu $ 0.33 polynomial parameter A1/(J·m−3) 1.2 × 109 equilibrium temperature ${\vartheta _0}/{\mathrm{K}}$ 271 polynomial parameter A2/(J·m−3) −1.7 × 108 latent heat Q/(MJ·m−3) 110 polynomial parameter A3/(J·m−3) 5.0 × 107 gradient energy coefficient β/(J·m−1) 1 × 10−10 polynomial parameter A4/(J·m−3) 5.0 × 107 kinetic coefficient τ/(m3·J−1·s−1) 20 polynomial parameter A5/(J·m−3) 2.0 × 106 heat conduction coefficient κ/(W·m−1·K−1) 2 polynomial parameter A6/(J·m−3) 2.0 × 106 energy barrier coefficient a 10 polynomial parameter A7/(J·m−3) 2.0 × 106 density of NiTi ${\rho _{{\text{NiTi}}}}/({\mathrm{kg}} \cdot {\mathrm{m}}^{-3})$ 6450 gradient energy coefficient λc/( J·m−1) 3.0 × 10−12 density of Ni4Ti3 ${\rho _{{\text{N}}{{\text{i}}_4}{\text{T}}{{\text{i}}_3}}}({\mathrm{kg}} \cdot {\mathrm{m}}^{-3})$ 6770 gradient energy coefficient λη/( J·m−1) 3.0 × 10−13 heat capacity of Ni4Ti3 $ {C_{{\text{N}}{{\text{i}}_4}{\text{T}}{{\text{i}}_3}}}/({\mathrm{J}}\cdot {\mathrm{kg}}^{-1} \cdot {\mathrm{K}}^{-1}) $ 345 chemical mobility M/(m5·J−1·s−1) 8.73 × 10−18 heat capacity of NiTi ${C_{{\text{NiTi}}}}/({\mathrm{J}}\cdot {\mathrm{kg}}^{-1} \cdot {\mathrm{K}}^{-1})$ 550 kinetic coefficient L/(m5·J−1·s−1) 8.82 × 10−11 time step $\Delta {t_{{\text{MT}}}}/{\mathrm{ns}}$ 0.1 time step $\Delta {t_{{\text{N}}{{\text{i}}_{\text{4}}}{\text{T}}{{\text{i}}_{\text{3}}}}}/{\mathrm{s}}$ 0.45 2.2 几何模型与模拟步骤
本文使用的几何模型如图1所示. 该单晶系统尺寸为1500 nm × 600 nm, 其边界与奥氏体相的[100]和[010]晶向平行. 约束左边界的x方向位移以及左上角和左下角点的y方向位移. 机械加载通过在右边界的x方向施加位移或应力来实现, 温度加载通过在整个单晶系统上施加均匀的温度场来实现.
为了保证解的收敛性和准确性, 对单晶系统进行尺寸为1 nm的均匀网格划分. 本文通过均匀化方法(详情请见Zhong等[31])来计算模拟系统的整体平均应力和平均应变, 即将模拟系统中所有节点处加载方向(即[100]方向)的应力和应变分量进行面积平均
$$ {\bar \sigma = \frac{1}{\alpha }\sum\limits_{i = 1}^\alpha {\sigma _i^{{\text{node}}}} ,}\quad {\bar \varepsilon = } \frac{1}{\alpha }\sum\limits_{i = 1}^\alpha {\varepsilon _i^{{\text{node}}}} $$ (29) 其中, $\alpha $是总的节点数, $\sigma _i^{{\text{node}}}$和$\varepsilon _i^{{\text{node}}}$分别是第i个节点加载方向上的应力和应变分量. 需要说明的是, 对于超弹性和单程形状记忆效应, 平均应力是通过式(29)(前式)计算的, 而平均应变是通过总变形量(加载位移)除以原长计算的. 对于应力辅助双程形状记忆效应, 平均应变是通过式(29)的后式计算的.
在评估弹卡效应时, 需要计算绝热温变$ \left| {\Delta {\vartheta _{{\text{ad}}}}} \right| $和材料制冷性能系数$ CO{P_{{\text{mat}}}} $, 其表达式如下
$$\qquad \left| {\Delta {\vartheta _{{\text{ad}}}}} \right| = \left| {{{\bar \vartheta }_{\text{2}}} - {{\bar \vartheta }_1}} \right| $$ (30) $$\qquad CO{P_{{\mathrm{mat}}}} = \frac{{{Q_{\text{c}}}}}{{\Delta W}} = \frac{{{C_{{\text{eq}}}}\left| {\Delta {\vartheta _{{ad} }}} \right|}}{{\displaystyle\oint {\bar \sigma {{\mathrm{d}}} \bar \varepsilon } }} $$ (31) 其中, $ {\bar \vartheta _{\text{2}}} $和$ {\bar \vartheta _1} $分别为超弹性过程中卸载完成时和加载到最大载荷时的平均温度, $ {Q_{\text{c}}} $和$ \Delta W $分别为制冷能力和输入功(应力-应变滞回环面积), $ {C_{{\text{eq}}}} $为等效热容. $ {\bar \vartheta _{\text{2}}} $ , $ {\bar \vartheta _1} $和$ {C_{{\text{eq}}}} $的表达式为
$$ {\bar \vartheta _i}\left( {i = 1,2} \right) = \frac{1}{\alpha }\sum\limits_{j = 1}^\alpha {\vartheta _j^{{\text{node}}}} $$ (32) $$ {C_{{\text{eq}}}} = {f_{{\text{N}}{{\text{i}}_4}{\text{T}}{{\text{i}}_3}}}{C_{{\text{N}}{{\text{i}}_4}{\text{T}}{{\text{i}}_3}}} + \left( {1 - {f_{{\text{N}}{{\text{i}}_4}{\text{T}}{{\text{i}}_3}}}} \right){C_{{\text{NiTi}}}} $$ (33) $$ {f_{{\text{N}}{{\text{i}}_4}{\text{T}}{{\text{i}}_3}}} = \frac{1}{\alpha }\sum\limits_{j = 1}^\alpha {\sum\limits_{i = 1}^n {\eta _i^j} } $$ (34) 其中, $\vartheta _j^{{\text{node}}}$为第j个节点处的温度, ${f_{{\text{N}}{{\text{i}}_4}{\text{T}}{{\text{i}}_3}}}$为模拟系统中的Ni4Ti3沉淀相体积分数, $\eta _i^j$为第j个节点处第i个Ni4Ti3沉淀相变体对应的序参量. 注意对于本文的二维情形, 式(34)相当于对整个模拟系统中的沉淀相序参量(每个材料点处的Ni4Ti3沉淀相体积分数)进行了面积平均.
沉淀相析出、超弹性、单程形状记忆和应力辅助双程形状记忆过程的模拟步骤如下.
(1) 应力辅助的Ni4Ti3沉淀相析出过程: 为了很好地控制Ni4Ti3沉淀相的形核, 在单晶系统中预先设置120个Ni4Ti3种子(直径为10 nm的圆形区域)(见图2(a), A). 将时效温度设为773 K, 在单晶系统[111]方向施加50 MPa的恒定压应力(即${{\boldsymbol{\sigma}} ^{{\text{ext}}}} = \left[ \begin{gathered} \begin{array}{*{20}{c}} { - 25{\text{ MPa}}}&{ - 25{\text{ MPa}}} \end{array} \\ \begin{array}{*{20}{c}} { - 25{\text{ MPa}}}&{ - 25{\text{ MPa}}} \end{array} \\ \end{gathered} \right]$, 详情请见附录A), 模拟时间设为8 h.
![]()
(2) 超弹性过程: 以经历过程(1)的系统作为初始构型, 将环境温度$\vartheta $设为298 K. 在该温度下, 在单晶系统右边界施加位移载荷, 最大加载应变设为5%, 加载时间设为100 ns. 在相同的应变率下进行卸载, 当$\bar \sigma = 0$时结束卸载.
(3) 单程形状记忆过程: 以经历过程(1)的系统作为初始构型, 首先, 将环境温度设为200 K并保持20 ns; 然后, 对系统施加位移载荷, 最大加载应变设为5%, 加载时间设为100 ns, 在相同的应变率下卸载到$\bar \sigma = 0$; 最后, 对系统进行持续升温, 在40 ns内将温度从200 K线性升高到400 K.
(4) 应力辅助双程形状记忆过程: 以经历过程(1)的系统作为初始构型, 首先, 在400 K下在系统右边界施加100 MPa的应力并保持恒定; 随后, 在40 ns内将温度从400 K线性减小到200 K, 再在40 ns内将温度从200 K线性升高到400 K.
3. 模拟结果与讨论
本节将对单晶系统的应力辅助Ni4Ti3沉淀相析出过程、超弹性、弹卡效应、单程形状记忆效应和应力辅助双程形状记忆效应的模拟结果进行讨论, 以揭示单一取向的Ni4Ti3沉淀相对NiTi合金功能性能的影响机理.
3.1 应力辅助的Ni4Ti3沉淀相析出
应力辅助的Ni4Ti3沉淀相析出过程的模拟结果如图2所示. 可以看到, 当时效时间为2 h时, Ni4Ti3沉淀相种子逐渐长大, 形成单一取向的沉淀相. 随着模拟时间的增加, 大部分沉淀相的长度逐渐增加, 宽度几乎不变, 从而形成细长的透镜状Ni4Ti3沉淀相. 然而, 由于相邻沉淀相长大诱发的局部内应力场的选择作用, 一些沉淀相发生了缩减甚至消失. 此外, 值得注意的是, 部分相邻的沉淀相发生了融合, 形成了更长的沉淀相. 从图2(b)和图2(c)所示的实验结果可以看到, 在实际的压应力辅助时效过程中也会发生相邻沉淀相融合长大的现象(图中圈出和框出的区域与图2(a), E中圈出和框出的区域对应). 需要说明的是, 由于图2(c)观测方位的不同, 沉淀相的取向与模拟结果不一致. 模拟的应力辅助时效过程所得的Ni4Ti3沉淀相形态与实验结果一致, 这验证了本文模型的合理性. 模拟得到的含有单一取向Ni4Ti3沉淀相的系统(图2(a), E)将为后续功能性能的模拟提供初始构型.
3.2 超弹性
含单一取向Ni4Ti3沉淀相单晶系统的超弹性模拟结果与实验结果如图3所示. 图3(a)显示, 相比于无沉淀相系统和无应力时效系统, 应力辅助时效系统的相变开始应力更高, 马氏体相变开始引起的应力跌落更不明显, 应力平台更短, 整体的应力-应变曲线更高. 需要说明的是, 与应力辅助时效不同, 经过无应力时效过程, 单晶系统中会同时形成惯习面相互垂直的两种取向的Ni4Ti3沉淀相变体, 其数量大致相等(具体可参考我们之前的工作[47]). 从图3(b)可以看到, 应力诱发的马氏体相变在B点处开始发生, 在一些局部位置形成了针状马氏体, 其长度方向与Ni4Ti3沉淀相的长度方向一致(图3(b), B). 这是由于每个Ni4Ti3沉淀相的形成都会在其附近诱发局部内应力, 各沉淀相尺寸、形态和距离的差异导致不同的内应力, 某些内应力较大的位置优先发生马氏体相变. 随后, 针状马氏体逐渐在Ni4Ti3沉淀相组成的通道内生长, 其两端逐渐抵达单晶系统的边界; 然而, 由于两侧Ni4Ti3沉淀相的强几何约束作用[71-72], 其向两侧的扩展变得十分困难(图3(b), C). 随着加载的进一步进行, 不断有新的马氏体相形核和生长, 它们与已形成的马氏体相一起组成了大的马氏体条带状区域(图3(b), D), 但其内存在大量奥氏体相缝隙(图3(b), E). 紧接着, 这些奥氏体缝隙在进一步加载过程中逐渐发生相变, 转变为马氏体相(图3(b), F). 在卸载过程中, 奥氏体相首先在各Ni4Ti3沉淀相附近形成并逐渐扩展(图3(b), G). 随后, 可以观察到大量针状马氏体的缩减和消失(图3(b), H ~ J).
![]()
图 3 含单一取向Ni4Ti3沉淀相的单晶系统的超弹性: (a) 应力-应变曲线(无沉淀相系统和无应力时效系统的数据引自Xu等[47]); (b) 模拟的马氏体微结构演化(图片编号与(a)中曲线上的点对应)和对应的温度场演化; (c) 含单一取向Ni4Ti3沉淀相的NiTi合金超弹性过程中马氏体微结构演化的原位实验观测结果(施加在试样上的位移在图片右上角标出, 引自Manchuraju等[27]). 注意为了更好地显示温度场, 图片${{\text{A}}_1} \sim {{\text{J}}_1}$中未显示出Ni4Ti3沉淀相Figure 3. Superelasticity of the single-crystal system containing single-oriented Ni4Ti3 precipitates: (a) stress-strain curves (the data of the precipitate-free system and that of stress-free aging system are cited from Xu et al.[47]); (b) simulated martensitic microstructural evolution (the picture numbers correspond to the points on the curve in (a)) and corresponding temperature field evolution; (c) experimental observation of martensitic microstructural evolution during the in situ superelastic deformation of an NiTi alloy with single-oriented Ni4Ti3 precipitates (the imposed displacement on the sample is shown in the upper right-hand corner, cited from Manchuraju et al.[27]). Note that to better display the temperature field, images ${{\text{A}}_1} \sim {{\text{J}}_1}$ do not show the Ni4Ti3 precipitatesShaw等[73]的实验表明, 应力诱发马氏体相的形核需要一个比使其扩展更高的应力, 这是应力-应变曲线上马氏体相变开始时发生应力跌落现象的原因. 在图3(a)中, 无沉淀相系统的应力-应变曲线上应力跌落十分明显, 这与实验结果一致. 然而, 在含Ni4Ti3沉淀相的系统中, 由于马氏体相的扩展受到沉淀相的严重阻碍, 进一步的马氏体相变仍然需要较高的应力, 因此应力跌落现象并不显著. 特别是在含单一取向Ni4Ti3沉淀相的系统中, 拉伸过程中的马氏体相变并不以马氏体条带扩宽的方式进行, 而是以大量针状马氏体的形核与生长来实现. 换句话说, 几乎整个拉伸过程中都伴随着新的马氏体相的形成, 其所需的外加应力与最初的相变开始应力几乎相等, 这导致应力-应变曲线上几乎未出现应力跌落现象. 在无应力时效系统中, 由于两种取向的Ni4Ti3沉淀相的存在, 马氏体相可以沿着两种惯习面生长或缩减. 在应力辅助时效系统中, 单一取向的Ni4Ti3沉淀相使马氏体相只能沿着一个方向生长或缩减. 可见, 无应力时效系统中两种取向的Ni4Ti3沉淀相与应力辅助时效系统中单一取向的Ni4Ti3沉淀相对马氏体相变与逆相变影响的差异性导致二者的应力-应变曲线表现出较大差异.
通过对比模拟结果与图3(c)所示的实验结果可以发现, 模拟结果可以反映马氏体微结构演化的主要特征: (I) 观测区域内优先形成了马氏体带状区域, 说明马氏体相呈细长状, 随后该区域逐渐扩展, “吞并”了大量Ni4Ti3沉淀相; (II) 形成的微观马氏体带逐渐扩展的同时, 新的针状马氏体相逐渐形核和生长. 这进一步说明了本文模型的合理性以及模拟结果的可靠性.
马氏体相变和逆相变过程分别伴随着相变潜热的释放与吸收, 导致系统的温度场发生变化, 如图3(b)的第二行所示. 在加载过程中, 马氏体相的形成伴随着该区域温度的升高. 由于系统内的热传导, 在加载到最大应变时, 系统内的温度场近乎均匀(图3(b), F1). 卸载过程中, 由于马氏体逆相变同时在各Ni4Ti3沉淀相附近发生, 大量局部区域发生温度的降低, 温度场整体的演化趋于均匀, 这对该材料系统用作固态制冷剂来说是有利的. 通过进一步计算可以得到应力辅助时效系统的绝热温变$ \left| {\Delta {\vartheta _{{\text{ad}}}}} \right| $为25.4 K, $ CO{P_{{\text{mat}}}} $为19.8, 说明其具有优异的弹卡效应.
3.3 单程形状记忆效应
含单一取向Ni4Ti3沉淀相单晶系统的单程形状记忆效应模拟结果与部分实验结果如图4所示. 图4(a)显示, 含单一取向Ni4Ti3沉淀相单晶系统与无应力时效系统(含两种不同取向的Ni4Ti3沉淀相)的应力-应变曲线拉伸阶段存在差异, 且前者的卸载残余应变更小; 含单一取向Ni4Ti3沉淀相单晶系统的应变-温度曲线最低. 从图4(c)可以看到, 在温度诱发马氏体相变过程中(图4(c), A ~ D), 由于Ni4Ti3沉淀相诱发的局部内应力的存在, 一些沉淀相附近优先发生马氏体相变, 形成孪晶马氏体; 随后马氏体区域逐渐扩大, 直至占据整个单晶系统, 此时所有的Ni4Ti3沉淀相都夹杂在孪晶马氏体中(图4(c), D). 注意到几乎所有的孪晶面均与沉淀相惯习面平行, 这与无沉淀相系统和无应力时效系统不同[47], 其可归结为单一取向Ni4Ti3沉淀相对孪晶马氏体形成的几何约束不同. 图4(b)所示的实验结果显示, 马氏体相优先在一些沉淀相通道内形成并沿着沉淀相长度方向扩展, 图4(c)中的模拟结果与之一致.
![]()
图 4 含单一取向Ni4Ti3沉淀相的单晶系统的单程形状记忆效应: (a) 应力-应变-温度曲线(无沉淀相系统和无应力时效系统的数据引自Xu等[47]); (b) 含单一取向Ni4Ti3沉淀相的NiTi合金温度诱发马氏体相变过程中马氏体微结构演化的实验观测结果(施加温度在图片右上角标出, 引自Michutta等[74]); (c) 模拟的马氏体微结构演化(图片编号与(a)中曲线上的点对应)Figure 4. One-way shape memory effect of the single-crystal system containing single-oriented Ni4Ti3 precipitates: (a) stress-strain-temperature curves (the data of the precipitate-free system and that of stress-free aging system are cited from Xu et al.[47]); (b) experimental observation of martensitic microstructural evolution during the temperature-induced martensite transformation of an NiTi alloy with single-oriented Ni4Ti3 precipitates (the imposed temperature is shown in the upper right-hand corner, cited from Michutta et al.[74]); (c) simulated martensitic microstructural evolution (the picture numbers correspond to the points on the curve in (a))马氏体变体之间的转换与变体的本征应变(依赖于变体的晶格结构)以及外加载荷的方向密切相关[75]. 由式(28)可知, +M变体为有利取向变体, 而−M为不利取向变体. 在拉伸过程中(图4(c), E ~ G), 马氏体重取向同时在多处局部区域发生, +M变体通过孪晶界的移动逐渐取代−M变体, 最终单晶系统的绝大部分区域都由+M变体占据, Ni4Ti3沉淀相夹杂于重取向马氏体相中(图4(c), G). 由于单一取向(应力辅助时效系统)和两种取向Ni4Ti3沉淀相(无应力辅助时效系统)对马氏体重取向的约束作用不同, 二者对应的应力-应变曲线的拉伸阶段存在一定的差异. 卸载之后, 部分−M变体重新形成(图4(c), H). 在随后的升温过程中(图4(c), I), 马氏体逆相变优先在孪晶界和Ni4Ti3沉淀相-马氏体界面处发生, 大量奥氏体域形成并扩展, 最终所有的马氏体相均转变为了奥氏体相(图4(c), J). 由于单一取向和两种取向Ni4Ti3沉淀相对温度诱发马氏体逆相变的约束不同, 以及卸载结束点处微结构的差异(不同的孪晶界和Ni4Ti3沉淀相-马氏体界面均会影响马氏体逆相变), 二者对应的应变-温度曲线存在差异.
3.4 应力辅助双程形状记忆效应
含单一取向Ni4Ti3沉淀相单晶系统的应力辅助双程形状记忆效应模拟结果如图5所示. 图5(a)显示, 含单一取向Ni4Ti3沉淀相单晶系统与无应力时效系统和无沉淀相系统的应变-温度曲线都存在较大差异; 两种含沉淀相系统的滞回环面积都远小于无沉淀相系统, 且含单一取向Ni4Ti3沉淀相系统的滞回环更小, 应力辅助双程记忆应变(${\varepsilon _{{\text{SATWSME}}}}$, 其定义如图5(a)所示)最小; 此外, 两种含沉淀相系统的马氏体开始温度(Ms)和奥氏体结束温度(Af)几乎分别相同, 且都远小于无沉淀相系统的Ms和Af.
![]()
图 5 含单一取向Ni4Ti3沉淀相的单晶系统的应力辅助双程形状记忆效应: (a)应变-温度曲线(无沉淀相系统和无应力时效系统的数据引自Xu等[47]); (b) 马氏体微结构演化(图片编号与(a)中曲线上的点对应)Figure 5. Stress-assisted two-way shape memory effect of the single-crystal system containing single-oriented Ni4Ti3 precipitates: (a) stress-strain-temperature curves (the data of the precipitate-free system and that of stress-free aging system are cited from Xu et al.[47]); (b) martensitic microstructural evolution (the picture numbers correspond to the points on the curve in (a))从图5(b)可以看到, 当温度达到Ms时, 大量针状马氏体同时在Ni4Ti3沉淀相通道内形成并快速生长, 导致应变的快速增长. 当温度降低到200 K时, 单晶系统内绝大部分奥氏体相都已经转变成了由 +M变体主导的马氏体相(图5(b), D). 在随后的升温过程中(图5(b), E ~ H), 马氏体逆相变优先在孪晶界和Ni4Ti3沉淀相-马氏体界面处发生, 少量孪晶马氏体先转变为了奥氏体相, 随后大量针状马氏体逐渐缩减并消失. 可见, 单一取向与两种取向的Ni4Ti3沉淀相对应力辅助的温度诱发马氏体相变的影响十分相近, 导致几乎相同的Ms和降温段应变-温度曲线(图5(a)中的AD阶段); 然而, 二者对应力辅助的温度诱发马氏体逆相变的影响不同, 导致升温阶段不同的应变-温度曲线.
4. 结论
本文通过改进的耦合沉淀相析出和马氏体相变的相场模型模拟了应力辅助时效的NiTi单晶的超弹性、弹卡效应、单程形状记忆效应和应力辅助双程形状记忆效应, 讨论了单一取向Ni4Ti3沉淀相对功能性能的影响以及背后的微观机理. 本文得到主要结论如下.
(1) 改进的相场模型可以很好地捕捉应力辅助的Ni4Ti3沉淀相析出过程以及马氏体相变和逆相变过程中的微结构演化, 模拟结果与实验结果吻合较好, 说明了模型的合理性和模拟结果的可靠性.
(2) 由于单一取向Ni4Ti3沉淀相的强几何约束, 超弹性过程中的马氏体相变并不以马氏体条带快速扩宽的方式进行, 而是以大量针状马氏体的形核与生长来实现, 这导致较高的相变开始应力和很小的应力跌落. 马氏体逆相变以大量针状马氏体的缩减和消失来实现, 引起相对均匀的温度场演化. 含单一取向Ni4Ti3沉淀相的NiTi合金具备优良的弹卡效应.
(3) 在单程形状记忆过程中, 含单一取向Ni4Ti3沉淀相的单晶系统在温度诱发马氏体相变过程中形成的孪晶面与沉淀相惯习面平行. 单一取向和2种取向的Ni4Ti3沉淀相对马氏体重取向的约束作用不同, 导致二者对应的应力-应变曲线在拉伸阶段存在差异; 此外, 二者对马氏体逆相变约束的不同以及卸载结束点处微结构的差异, 导致二者对应的应变-温度曲线的差异.
(4) 在应力辅助的双程形状记忆过程中, 单一取向与两种取向的Ni4Ti3沉淀相对应力辅助的温度诱发马氏体相变的影响十分相近, 导致几乎相同的Ms和降温阶段的应变-温度曲线; 然而, 二者对应力辅助的温度诱发马氏体逆相变的影响不同, 导致不同的升温阶段的应变-温度曲线.
附录A 应力辅助时效过程中的${{\boldsymbol{\sigma}} ^{{\text{ext}}}}$
NiTi合金中的Ni4Ti3 沉淀相的晶格结构与NiTi基体相的晶格结构存在确定的对应关系, 时效过程中施加应力的方向并不会改变Ni4Ti3沉淀相与NiTi基体相之间的位向关系, 只会改变析出的Ni4Ti3沉淀相变体种类[42]. 实验中通常在NiTi合金[111]B2方向施加恒定压应力50 MPa进行时效来获取单一取向Ni4Ti3沉淀相[26-27,74], 本文在第3章将模拟微结构与实验结果[26-27,74]进行了对比. 为了使模拟条件更接近实验条件, 本文在模拟应力辅助时效过程时[111]B2方向施加的恒定压应力设为50 MPa. 如附图A1所示, 在坐标系x0Oy0的x0方向施加压应力${\sigma _0}$时有${\boldsymbol{\sigma}} _0^{{\text{ext}}} = \left[ \begin{gathered} \begin{array}{*{20}{c}} {{\sigma _0}}&0 \\ 0&0 \end{array} \\ \end{gathered} \right]$. 根据坐标系的转换关系${\boldsymbol{\sigma}} _{}^{{\text{ext}}} = {\boldsymbol{R}}{\boldsymbol{\sigma}} _0^{{\text{ext}}}{{\boldsymbol{R}}^{\text{T}}}$(R为旋转矩阵, 如附图A1所示), 可得坐标系xOy中有${\boldsymbol{\sigma}} _{}^{{\text{ext}}} = \left[ \begin{gathered} \begin{array}{*{20}{c}} {\dfrac{1}{2}{\sigma _0}}&{\dfrac{1}{2}{\sigma _0}} \end{array} \\ \begin{array}{*{20}{c}} {\dfrac{1}{2}{\sigma _0}}&{\dfrac{1}{2}{\sigma _0}} \end{array} \\ \end{gathered} \right]$. 因此, 本文在[111]B2方向施加恒定压应力${\sigma _0} = - 50{\text{ MPa}}$可等效为附图A1所示坐标系下的${{\boldsymbol{\sigma}} ^{{\text{ext}}}} = \left[ \begin{gathered} \begin{array}{*{20}{c}} { - 25{\text{ MPa}}}&{ - 25{\text{ MPa}}} \end{array} \\ \begin{array}{*{20}{c}} { - 25{\text{ MPa}}}&{ - 25{\text{ MPa}}} \end{array} \\ \end{gathered} \right]$. 值得指出的是, 如果同时改变外加应力张量${{\boldsymbol{\sigma }}^{{\text{ext}}}}$中的所有分量的大小(比如将所有的−25 MPa都改为−50 MPa), ${{\boldsymbol{\sigma}} ^{{\text{ext}}}}$仍然等效于在[111]B2方向施加恒定压应力, 这只会影响模拟过程中沉淀相的析出速度, 系统中仍然会形成单一取向的沉淀相. 如果不是同时改变应力张量中的分量大小(比如将4个分量改为不同的数值), ${{\boldsymbol{\sigma}} ^{{\text{ext}}}}$不再等效于在[111]B2方向施加恒定压应力(相当于在其他某个方向施加恒定压应力), 这会导致模拟不能得到单一取向的Ni4Ti3沉淀相. 施加不同大小和不同方向恒定应力对Ni4Ti3沉淀相析出行为的影响已在实验[76]和相场模拟[42]中得到讨论, 而本文并不关注这一点, 因此直接采用了${{\boldsymbol{\sigma}} ^{{\text{ext}}}} = \left[ \begin{gathered} \begin{array}{*{20}{c}} { - 25{\text{ MPa}}}&{ - 25{\text{ MPa}}} \end{array} \\ \begin{array}{*{20}{c}} { - 25{\text{ MPa}}}&{ - 25{\text{ MPa}}} \end{array} \\ \end{gathered} \right]$.
-
![]()
![]()
图 3 含单一取向Ni4Ti3沉淀相的单晶系统的超弹性: (a) 应力-应变曲线(无沉淀相系统和无应力时效系统的数据引自Xu等[47]); (b) 模拟的马氏体微结构演化(图片编号与(a)中曲线上的点对应)和对应的温度场演化; (c) 含单一取向Ni4Ti3沉淀相的NiTi合金超弹性过程中马氏体微结构演化的原位实验观测结果(施加在试样上的位移在图片右上角标出, 引自Manchuraju等[27]). 注意为了更好地显示温度场, 图片${{\text{A}}_1} \sim {{\text{J}}_1}$中未显示出Ni4Ti3沉淀相
Figure 3. Superelasticity of the single-crystal system containing single-oriented Ni4Ti3 precipitates: (a) stress-strain curves (the data of the precipitate-free system and that of stress-free aging system are cited from Xu et al.[47]); (b) simulated martensitic microstructural evolution (the picture numbers correspond to the points on the curve in (a)) and corresponding temperature field evolution; (c) experimental observation of martensitic microstructural evolution during the in situ superelastic deformation of an NiTi alloy with single-oriented Ni4Ti3 precipitates (the imposed displacement on the sample is shown in the upper right-hand corner, cited from Manchuraju et al.[27]). Note that to better display the temperature field, images ${{\text{A}}_1} \sim {{\text{J}}_1}$ do not show the Ni4Ti3 precipitates
![]()
图 4 含单一取向Ni4Ti3沉淀相的单晶系统的单程形状记忆效应: (a) 应力-应变-温度曲线(无沉淀相系统和无应力时效系统的数据引自Xu等[47]); (b) 含单一取向Ni4Ti3沉淀相的NiTi合金温度诱发马氏体相变过程中马氏体微结构演化的实验观测结果(施加温度在图片右上角标出, 引自Michutta等[74]); (c) 模拟的马氏体微结构演化(图片编号与(a)中曲线上的点对应)
Figure 4. One-way shape memory effect of the single-crystal system containing single-oriented Ni4Ti3 precipitates: (a) stress-strain-temperature curves (the data of the precipitate-free system and that of stress-free aging system are cited from Xu et al.[47]); (b) experimental observation of martensitic microstructural evolution during the temperature-induced martensite transformation of an NiTi alloy with single-oriented Ni4Ti3 precipitates (the imposed temperature is shown in the upper right-hand corner, cited from Michutta et al.[74]); (c) simulated martensitic microstructural evolution (the picture numbers correspond to the points on the curve in (a))
![]()
图 5 含单一取向Ni4Ti3沉淀相的单晶系统的应力辅助双程形状记忆效应: (a)应变-温度曲线(无沉淀相系统和无应力时效系统的数据引自Xu等[47]); (b) 马氏体微结构演化(图片编号与(a)中曲线上的点对应)
Figure 5. Stress-assisted two-way shape memory effect of the single-crystal system containing single-oriented Ni4Ti3 precipitates: (a) stress-strain-temperature curves (the data of the precipitate-free system and that of stress-free aging system are cited from Xu et al.[47]); (b) martensitic microstructural evolution (the picture numbers correspond to the points on the curve in (a))
表 1 模拟参数
Table 1 Simulation parameters
Precipitation of Ni4Ti3 Martensite transformation Parameter Value Parameter Value equilibrium Ni concentration of NiTi ${c_1}$ 0.50 elastic modulus of austenite phase EA/GPa 60 equilibrium Ni concentration of Ni4Ti3 ${c_2}$ 0.57 elastic modulus of martensite phase EM/GPa 40 initial Ni concentration ${c_0}$ 0.51 Poisson’s ratio $\nu $ 0.33 polynomial parameter A1/(J·m−3) 1.2 × 109 equilibrium temperature ${\vartheta _0}/{\mathrm{K}}$ 271 polynomial parameter A2/(J·m−3) −1.7 × 108 latent heat Q/(MJ·m−3) 110 polynomial parameter A3/(J·m−3) 5.0 × 107 gradient energy coefficient β/(J·m−1) 1 × 10−10 polynomial parameter A4/(J·m−3) 5.0 × 107 kinetic coefficient τ/(m3·J−1·s−1) 20 polynomial parameter A5/(J·m−3) 2.0 × 106 heat conduction coefficient κ/(W·m−1·K−1) 2 polynomial parameter A6/(J·m−3) 2.0 × 106 energy barrier coefficient a 10 polynomial parameter A7/(J·m−3) 2.0 × 106 density of NiTi ${\rho _{{\text{NiTi}}}}/({\mathrm{kg}} \cdot {\mathrm{m}}^{-3})$ 6450 gradient energy coefficient λc/( J·m−1) 3.0 × 10−12 density of Ni4Ti3 ${\rho _{{\text{N}}{{\text{i}}_4}{\text{T}}{{\text{i}}_3}}}({\mathrm{kg}} \cdot {\mathrm{m}}^{-3})$ 6770 gradient energy coefficient λη/( J·m−1) 3.0 × 10−13 heat capacity of Ni4Ti3 $ {C_{{\text{N}}{{\text{i}}_4}{\text{T}}{{\text{i}}_3}}}/({\mathrm{J}}\cdot {\mathrm{kg}}^{-1} \cdot {\mathrm{K}}^{-1}) $ 345 chemical mobility M/(m5·J−1·s−1) 8.73 × 10−18 heat capacity of NiTi ${C_{{\text{NiTi}}}}/({\mathrm{J}}\cdot {\mathrm{kg}}^{-1} \cdot {\mathrm{K}}^{-1})$ 550 kinetic coefficient L/(m5·J−1·s−1) 8.82 × 10−11 time step $\Delta {t_{{\text{MT}}}}/{\mathrm{ns}}$ 0.1 time step $\Delta {t_{{\text{N}}{{\text{i}}_{\text{4}}}{\text{T}}{{\text{i}}_{\text{3}}}}}/{\mathrm{s}}$ 0.45 
下载: 导出CSV
-
[1] 杨建楠, 黄彬, 谷小军等. 形状记忆合金力学行为与应用综述. 固体力学学报, 2021, 42(4): 345-375 (Yang Jiannan, Huang Bin, Gu Xiaojun, et al. A review of shape memory alloys: mechanical behavior and application. Chinese Journal of Solid Mechanics, 2021, 42(4): 345-375 (in Chinese) Yang Jiannan, Huang Bin, Gu Xiaojun, et al. A review of shape memory alloys: mechanical behavior and application. Chinese Journal of Solid Mechanics, 2021, 42(4): 345-375 (in Chinese)
[2] 牛豪杰, 林成新. 形状记忆合金的应用现状综述. 天津理工大学学报, 2020, 36(4): 1-6 (Niu Haojie, Lin Chengxin. Review of shape memory alloy application status. Journal of Tianjin University of Technology, 2020, 36(4): 1-6 (in Chinese) Niu Haojie, Lin Chengxin. Review of shape memory alloy application status. Journal of Tianjin University of Technology, 2020, 36(4): 1-6 (in Chinese)
[3] Hou H, Simsek E, Ma T, et al. Fatigue-resistant high-performance elastocaloric materials made by additive manufacturing. Science, 2019, 366(6469): 1116-1121 doi: 10.1126/science.aax7616
[4] Qian S, Catalini D, Muehlbauer J, et al. High-performance multimode elastocaloric cooling system. Science, 2023, 380: 722-727 doi: 10.1126/science.adg7043
[5] Zhou G, Zhu Y, Yao S, et al. Giant temperature span and cooling power in elastocaloric regenerator. Joule, 2023, 7: 2003-2015 doi: 10.1016/j.joule.2023.07.004
[6] Zhou G, Li Z, Wang Q, et al. A multi-material cascade elastocaloric cooling device for large temperature lift. Nature Energy, 2024, 9: 862-870
[7] 肖飞, 陈宏, 金学军. 形状记忆合金弹热制冷效应的研究现状. 金属学报, 2021, 57(1): 29-41 (Xiao Fei, Chen Hong, Jin Xuejun. Research progress in elastocaloric cooling effect basing on shape memory alloy. Acta Metallurgica Sinica, 2021, 57(1): 29-41 (in Chinese) Xiao Fei, Chen Hong, Jin Xuejun. Research progress in elastocaloric cooling effect basing on shape memory alloy. Acta Metallurgica Sinica, 2021, 57(1): 29-41 (in Chinese)
[8] 康国政, 阚前华, 于超等. 热致和磁致形状记忆合金循环变形和疲劳行为研究. 力学进展. 2018, 48: 66-147 (Kang Guozheng, Kan Qianhua, Yu Chao, et al. Study on cyclic deformation and fatigue of thermal and magnetic shape memory alloys. Advances in Mechanics, 2018, 48: 201802 (in Chinese) Kang Guozheng, Kan Qianhua, Yu Chao, et al. Study on cyclic deformation and fatigue of thermal and magnetic shape memory alloys. Advances in Mechanics, 2018, 48: 201802 (in Chinese)
[9] Šittner P, Sedlák P, Seiner H, et al. On the coupling between martensitic transformation and plasticity in NiTi: Experiments and continuum based modelling. Progress in Materials Science, 2018, 98: 249-298 doi: 10.1016/j.pmatsci.2018.07.003
[10] Kang G, Yu C, Kan Q. Thermo-mechanically coupled cyclic deformation and fatigue failure of NiTi shape memory alloys: Experiments, simulations and theories. Springer Nature, 2023
[11] Ahadi A, Sun Q. Effects of grain size on the rate-dependent thermomechanical responses of nanostructured superelastic NiTi. Acta Materialia, 2014, 76: 186-197 doi: 10.1016/j.actamat.2014.05.007
[12] Chen J, Liu B, Xing L, et al. Toward tunable mechanical behavior and enhanced elastocaloric effect in NiTi alloy by gradient structure. Acta Materialia, 2022, 226: 117609 doi: 10.1016/j.actamat.2021.117609
[13] Frenzel J, George EP, Dlouhy A, et al. Influence of Ni on martensitic phase transformations in NiTi shape memory alloys. Acta Materialia, 2010, 58: 3444-3458 doi: 10.1016/j.actamat.2010.02.019
[14] Zhu J, Wang D, Gao Y, et al. Linear-superelastic metals by controlled strain release via nanoscale concentration-gradient engineering. Materials Today, 2020, 33: 17-23 doi: 10.1016/j.mattod.2019.10.003
[15] Zhou Y, Xue D, Ding X, et al. Strain glass in doped Ti50(Ni50- xD x) (D = Co, Cr, Mn) alloys: Implication for the generality of strain glass in defect-containing ferroelastic systems. Acta Materialia, 2010, 58: 5433-5442 doi: 10.1016/j.actamat.2010.06.019
[16] Chluba C, Ge W, Lima de Miranda R, et al. Ultralow-fatigue shape memory alloy films. Science, 2015, 348: 1004-1007 doi: 10.1126/science.1261164
[17] Benafan O, Padula II S A, Noebe RD, et al. Role of B19′ martensite deformation in stabilizing two-way shape memory behavior in NiTi. Journal of Applied Physics, 2012, 112: 093510 doi: 10.1063/1.4764313
[18] Zhao S, Liang Q, Su Y, et al. Cryogenic rolling induces quasi-linear superelasticity with high strength over a wide temperature range in TiNi shape memory alloys. Scripta Materialia, 2024, 243: 115996 doi: 10.1016/j.scriptamat.2024.115996
[19] Wang X, Kustov S, Li K, et al. Effect of nanoprecipitates on the transformation behavior and functional properties of a Ti-50.8 at. % Ni alloy with micron-sized grains. Acta Materialia, 2015, 82: 224-233
[20] Dang P, Pang J, Zhou Y, et al. Improved stability of superelasticity and elastocaloric effect in Ti-Ni alloys by suppressing Lüders-like deformation under tensile load. Journal of Materials Science & Technology, 2023, 146: 154-167
[21] Otsuka K, Ren X. Physical metallurgy of Ti-Ni-based shape memory alloys. Progress in Materials Science, 2005, 50: 511-678 doi: 10.1016/j.pmatsci.2004.10.001
[22] Khalil-Allafi J, Ren X, Eggeler G. The mechanism of multistage martensitic transformations in aged Ni-rich NiTi shape memory alloys. Acta Materialia, 2002, 50: 793-803 doi: 10.1016/S1359-6454(01)00385-8
[23] Gall K, Maier HJ. Cyclic deformation mechanisms in precipitated NiTi shape memory alloys. Acta Materialia, 2002, 50: 4643-4657 doi: 10.1016/S1359-6454(02)00315-4
[24] Wang X, Pu Z, Yang Q, et al. Improved functional stability of a coarse-grained Ti-50.8 at. % Ni shape memory alloy achieved by precipitation on dislocation networks. Scripta Materialia, 2019, 163: 57-61
[25] Zhou X, Zhang T, Cheng L, et al. Variant selection map of external load during Ni4Ti3 precipitation in nitinol: a theoretical and phase field study. Acta Mechanica Sinica, 2024, 40: 123480 doi: 10.1007/s10409-023-23480-x
[26] Michutta J, Carroll MC, Yawny A, et al. Martensitic phase transformation in Ni-rich NiTi single crystals with one family of Ni4Ti3 precipitates. Materials Science and Engineering A, 2004, 378: 152-156 doi: 10.1016/j.msea.2003.11.061
[27] Manchuraju S, Kroeger A, Somsen C, et al. Pseudoelastic deformation and size effects during in situ transmission electron microscopy tensile testing of NiTi. Acta Materialia, 2012, 60: 2770-2777 doi: 10.1016/j.actamat.2012.01.043
[28] Timofeeva EE, Yu Panchenko E, Zherdeva MV, et al. Effect of one family of Ti3Ni4 precipitates on shape memory effect, superelasticity and strength properties of the B2 phase in high-nickel [001]-oriented Ti-51.5 at. % Ni single crystals. Materials Science & Engineering A, 2022, 832: 142420
[29] Xiao F, Chu K, Li Z, et al. Improved functional fatigue resistance of single crystalline NiTi micropillars with uniformly oriented Ti3Ni4 precipitates. International Journal of Plasticity, 2023, 160: 103480 doi: 10.1016/j.ijplas.2022.103480
[30] Grandi D, Maraldi M, Molari L. A macroscale phase-field model for shape memory alloys with non-isothermal effects: Influence of strain rate and environmental conditions on the mechanical response. Acta Materialia, 2012, 60: 179-191 doi: 10.1016/j.actamat.2011.09.040
[31] Zhong Y, Zhu T. Phase-field modeling of martensitic microstructure in NiTi shape memory alloys. Acta Materialia, 2014, 75: 337-347 doi: 10.1016/j.actamat.2014.04.013
[32] Esfahani SE, Ghamarian I, Levitas VI, et al. Microscale phase field modeling of the martensitic transformation during cyclic loading of NiTi single crystal. International Journal of Solids and Structures, 2018, 146: 80-96 doi: 10.1016/j.ijsolstr.2018.03.022
[33] Wang D, Liang C, Zhao S, et al. Phase field simulation of martensitic transformation in pre-strained nanocomposite shape memory alloys. Acta Materialia, 2019, 164: 99-109 doi: 10.1016/j.actamat.2018.10.030
[34] 熊君媛, 徐波, 康国政. 纳米多晶NiTi形状记忆合金超弹性的晶粒取向依赖性相场研究. 固体力学学报, 2021, 42(6): 671-681 (Xong Junyuan, Xu Bo, Kang Guozheng. Phase field simulation on the grain orientation dependent super-elasticity of nanocrystalline NiTi shape memory alloys. Chinese Journal of Solid Mechanics, 2021, 42(6): 671-681 (in Chinese) Xong Junyuan, Xu Bo, Kang Guozheng. Phase field simulation on the grain orientation dependent super-elasticity of nanocrystalline NiTi shape memory alloys. Chinese Journal of Solid Mechanics, 2021, 42(6): 671-681 (in Chinese)
[35] 徐波, 康国政. 梯度纳米晶NiTi形状记忆合金的超弹性和形状记忆效应相场模拟. 力学学报, 2021, 53(3): 802-812 (Xu Bo, Kang Guozheng. Phase field simulation on the super-elasticity and shape memory effect of gradient nanocrystalline NiTi shape memory alloy. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(3): 802-812 (in Chinese) Xu Bo, Kang Guozheng. Phase field simulation on the super-elasticity and shape memory effect of gradient nanocrystalline NiTi shape memory alloy. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(3): 802-812 (in Chinese)
[36] Li X, Su Y. A phase-field study of the martensitic detwinning in NiTi shape memory alloys under tension or compression. Acta Mechanica, 2020, 231: 1539-1557 doi: 10.1007/s00707-020-02613-x
[37] Xu T, Wang C, Zhu Y, et al. Efficient phase-field simulation for linear superelastic NiTi alloys under temperature gradients. International Journal of Mechanical Sciences, 2023, 259: 108592 doi: 10.1016/j.ijmecsci.2023.108592
[38] Kavvadias D, Baxevanis Th. Phase-field description of fracture in NiTi single crystals. Computer Methods in Applied Mechanics and Engineering, 2024, 419: 116677 doi: 10.1016/j.cma.2023.116677
[39] Xu B, Huang B, Wang C, et al. Effect of texture on the grain-size-dependent functional properties of NiTi shape memory alloys and texture gradient design: A phase field study. Acta Mechanica Solida Sinica, 2024, 37: 10-32 doi: 10.1007/s10338-023-00439-3
[40] Xu B, Yu C, Wang C, et al. Effect of pore on the deformation behaviors of NiTi shape memory alloys: A crystal-plasticity-based phase field modeling. International Journal of Plasticity, 2024, 175: 103931 doi: 10.1016/j.ijplas.2024.103931
[41] Li D, Chen L. Shape of a rhombohedral coherent Ti11Ni14 precipitate in a cubic matrix and its growth and dissolution during constrained aging. Acta Materialia, 1997, 45(6): 2435-2442 doi: 10.1016/S1359-6454(96)00363-1
[42] 柯常波. NiTi形状记忆合金中Ni4Ti3相沉淀行为的相场法模拟研究. [博士论文]. 广州: 华南理工大学, 2012 (Changbo Ke. Phase field study of Ni4Ti3 precipitation behavior in NiTi shape memory alloys. [PhD Thesis]. Guangzhou: South China University of Technology, 2012 (in Chinese) Changbo Ke. Phase field study of Ni4Ti3 precipitation behavior in NiTi shape memory alloys. [PhD Thesis]. Guangzhou: South China University of Technology, 2012 (in Chinese)
[43] Guo W, Steinbach I, Somsen C, et al. On the effect of superimposed external stresses on the nucleation and growth of Ni4Ti3 particles: A parametric phase field study. Acta Materialia, 2011, 59: 3287-3296 doi: 10.1016/j.actamat.2011.02.002
[44] Zhou N, Shen C, Wagner MFX, et al. Effect of Ni4Ti3 precipitation on martensitic transformation in Ti-Ni. Acta Materialia, 2010, 58: 6685-6694 doi: 10.1016/j.actamat.2010.08.033
[45] Zhu J, Wu H, Wu Y, et al. Influence of Ni4Ti3 precipitation on martensitic transformations in NiTi shape memory alloy: R phase transformation. Acta Materialia, 2021, 207: 116665 doi: 10.1016/j.actamat.2021.116665
[46] Dong T, Liang C, Su Y, et al. Phase field simulations for the crossover from sharp martensitic transformation into smooth strain glass transition by fine precipitates. Acta Materialia, 2023, 245: 118634 doi: 10.1016/j.actamat.2022.118634
[47] Xu B, Sun Y, Yu C, et al. Effect of Ni4Ti3 precipitates on the functional properties of NiTi shape memory alloys: A phase field study. International Journal of Plasticity, 2024, 177: 103993 doi: 10.1016/j.ijplas.2024.103993
[48] Cahn J. On spinodal decomposition. Acta Metallurgica, 1961, 9: 795-801 doi: 10.1016/0001-6160(61)90182-1
[49] Landau LD. Collected Papers of LD Landau. Pergamon Press, 1965
[50] Xu B, Kang G, Kan Q, et al. Phase field simulation on the cyclic degeneration of one-way shape memory effect of NiTi shape memory alloy single crystal. International Journal of Mechanical Sciences, 2020, 168: 105303 doi: 10.1016/j.ijmecsci.2019.105303
[51] Artemev A, Jin Y, Khachaturyan AG. Three-dimensional phase field model of proper martensitic transformation. Acta Materialia, 2001, 49: 1165-1177 doi: 10.1016/S1359-6454(01)00021-0
[52] Cissé C, Zaeem MA. An asymmetric elasto-plastic phase-field model for shape memory effect, pseudoelasticity and thermomechanical training in polycrystalline shape memory alloys. Acta Materialia, 2020, 201: 580-595 doi: 10.1016/j.actamat.2020.10.034
[53] Rao Z, Leng J, Yan Z, et al. A three-dimensional constitutive model for shape memory alloy considering transformation-induced plasticity, two-way shape memory effect, plastic yield and tension-compression asymmetry. European Journal of Mechanics / A Solids, 2023, 99: 104945 doi: 10.1016/j.euromechsol.2023.104945
[54] Tsimpoukis S, Kyriakides S. Rate induced thermomechanical interactions in NiTi tensile tests on strips. Journal of the Mechanics and Physics of Solids, 2024, 184: 105530 doi: 10.1016/j.jmps.2023.105530
[55] Tian L, Zhou J, Jia P, et al. Thermomechanical response and elastocaloric effect of shape memory alloy wires. Mechanics of Materials, 2024, 193: 104985 doi: 10.1016/j.mechmat.2024.104985
[56] Cissé C, Zaeem MA. Design of NiTi-based shape memory microcomposites with enhanced elastocaloric performance by a fully thermomechanical coupled phase-field model. Materials & Design, 2021, 207: 109898
[57] Zhang Q, Chen J, Fang G. From mechanical behavior and elastocaloric effect to microscopic mechanisms of gradient-structured NiTi alloy: A phase-field study. International Journal of Plasticity, 2023, 171: 103809 doi: 10.1016/j.ijplas.2023.103809
[58] Ahluwalia R, Quek SS, Wu DT. Simulation of grain size effects in nanocrystalline shape memory alloys. Journal of Applied Physics, 2015, 117: 244305 doi: 10.1063/1.4923044
[59] Xu B, Huang B, Wang C, et al. Phase field modeling of the aspect ratio dependent functional properties of NiTi shape memory alloys with different grain sizes. Acta Mechanica Sinica, 2024, doi: 10.1007/s10409-024-23272-x
[60] Mikula J, Quek SS, Joshi SP, et al. The role of bimodal grain size distribution in nanocrystalline shape memory alloys. Smart Materials and Structures, 2018, 27: 105004 doi: 10.1088/1361-665X/aada30
[61] Sun Y, Luo J, Zhu J, et al. A non-isothermal phase field study of the shape memory effect and pseudoelasticity of polycrystalline shape memory alloys. Computational Materials Science, 2019, 167: 65-76 doi: 10.1016/j.commatsci.2019.05.036
[62] Xu B, Kang G, Yu C, et al. Phase field simulation on the grain size dependent super-elasticity and shape memory effect of nanocrystalline NiTi shape memory alloys. International Journal of Engineering Science, 2020, 156: 103373 doi: 10.1016/j.ijengsci.2020.103373
[63] Yu C, Kang G, Kan Q. Crystal plasticity based constitutive model of NiTi shape memory alloy considering different mechanisms of inelastic deformation. International Journal of Plasticity, 2014, 54: 132-162 doi: 10.1016/j.ijplas.2013.08.012
[64] Villars P, Cenzual K, Okamoto H, et al. Pauling File Multinaries Edition–2012. Springer Materials, 2012
[65] Sharma N. Current activated tip sintering of Ni-Ti intermetallics. San Diego State University, 2014
[66] Onderka B, Sypień A, Wierzbicka-Miernik A, et al. Heat capacities of some binary intermetallic compounds in Al-Fe-Ni-Ti system. Archives of Metallurgy and Materials, 2010, 55(2): 435-439
[67] Cui J, Wu Y, Muehlbauer J, et al. Demonstration of high efficiency elastocaloric cooling with large ΔT using NiTi wires. Applied Physics Letters, 2012, 101(7): 073904 doi: 10.1063/1.4746257
[68] Xu B, Kang G. Phase field simulation on the super-elasticity, elastocaloric and shape memory effect of geometrically graded nano-polycrystalline NiTi shape memory alloys. International Journal of Mechanical Sciences, 2021, 201: 106462 doi: 10.1016/j.ijmecsci.2021.106462
[69] Anand L, Gurtin ME. Thermal effects in the superelasticity of crystalline shape memory materials. Journal of Mechanics and Physics of Solids, 2003, 51: 1015-1058 doi: 10.1016/S0022-5096(03)00017-6
[70] Li D, Chen L. Morphological evolution of coherent multi-variant Ti11Ni14 precipitates in TiNi alloys under an applied stress—a computer simulation study. Acta Materialia, 1998, 46(2): 639-649 doi: 10.1016/S1359-6454(97)00241-3
[71] Nishida M, Wayman CM, Chiba A. Electron microscopy studies of the martensitic transformation in an aged Ti-51 at.% Ni shape memory alloy. Metallography, 1988, 21: 275-291
[72] Evirgen A, Karaman I, Noebe RD, et al. Effect of precipitation on the microstructure and the shape memory response of the Ni50.3Ti29.7Zr20 high temperature shape memory alloy. Scripta Materialia, 2013, 69: 354-357
[73] Shaw JA, Kyriakides S. On the nucleation and propagation of phase transformation fronts in a NiTi alloy. Acta Materialia, 1997, 45(2): 683-700 doi: 10.1016/S1359-6454(96)00189-9
[74] Michutta J, Somsen Ch, Yawny A, et al. Elementary martensitic transformation processes in Ni-rich NiTi single crystals with Ni4Ti3 precipitates. Acta Materialia, 2006, 54: 3525-3542 doi: 10.1016/j.actamat.2006.03.036
[75] Xu B, Kang G, Kan Q, et al. Phase field simulation to one-way shape memory effect of NiTi shape memory alloy single crystal. Computational Materials Science, 2019, 161: 276-292 doi: 10.1016/j.commatsci.2019.02.009
[76] Khalil-Allafi J, Dlouhy A, Eggeler G. Ni4Ti3-precipitation during aging of NiTi shape memory alloys and its influence on martensitic phase transformations. Acta Materialia, 2002, 50: 4255-4274 doi: 10.1016/S1359-6454(02)00257-4
计量
- 文章访问数: 193
- HTML全文浏览量: 29
- PDF下载量: 49
