第一章：位错理论

第一章位错理论（补充和扩展）                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               

刃位错应力场：

                         

        

            

                   

滑移面：   

攀移面      

螺位错应力场：        

 单位长度位错线能量及张力      

单位长度位错线受力   滑移力：       

                        攀移力：        

位错线的平衡曲率

                                     

当较小时，故

        

 

两个重要公式：

Frank－Read源开动应力      

Orowan 应力                  

位错与位错间的相互作用

1. 不在同一滑移面上平行位错间的相互作用

（1）平行刃型位错 

    

式中正号表示b和同向；负号表示b和反向。

沿y轴的作用力即攀移力    

   同号：    正攀移

反号：     负攀移  

  

（2）平行螺位错   

                              

（3）平行混合型位错    

可以先将混合型位错分解成纯刃型和纯螺型的两个分量，分别计算刃-刃和螺-螺之间的作用力，最后叠加起来就得到总的作用力。

刃-螺之间无作用力

2. 在同一滑移面上平行位错间的弹性相互作用

位错的塞积群

令第一个位错在的地方，

若此障碍只同领先的位错有交互作用，则每一位错所受的作用力可写成

                                

平衡时应为零，可得n-1个联立代数方程（不包括第一个位错）

           

当n很大时，求解联立方程的近似解，得到各位错的平衡位置

                                                                                                                                                                                                                                                                                                                                                                                                                         

塞积群总长度

         

单位长度上的位错数          利用                                                                                                    

◆ 塞积群施加在障碍上的切应力

                                                                                                                                         

设在外切应力作用下，整个塞积群向前移动的距离，外                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            应力作功为，而障碍对领先位错的作用力作功为。故

                        

塞积群在障碍处产生了应力集中，其应力集中因子等于n。

◆ 塞积群在晶体内正前方远处一点P所产生的切应力。

               （1.47）

可以讨论三种不同情况：

当时，只须考虑领先位错的作用，

        

当时，塞积群的作用相当于强度为的一个大位错的作用， 

                           

当时，可将式（1.47）改为积分式来计算。由式（1.44）和式塞积群中单位长度内的位错数代入式（1.47），·

              

      

1.5 位错与溶质原子的交互作用

溶质原子的存在使基体晶格发生畸变，这种畸变必然对位错应力场作功，产生溶质原子与位错的交互作用能。

1.5.1 尺寸效应

基体半径为R,溶质原子半径为,

其弹性模量为和（压缩率）和和。

固溶引起的体积变化为

，            为溶剂的原子体积。

为溶质与溶剂的错配度，，                               

如果在一个位错的应力场中进行上述操作，那么体积变化就会对位错应力场的水静压力分量作功，从而使系统的能量发生变化，这个能量变化叫做位错与溶质的交互作用能。

                                                       

    

              

若    1）溶质原子半径大于溶剂，

则时，溶质被吸引到位错的下半面；

2）若溶质原子半径小于溶剂，溶质被吸引到位错的上半面

3）两种情况下越小越负, 溶质被吸引到位错

定义交互作用能的绝对值的最大值为位错与溶质的结合能。并取，

得到结合能为

                    

Cottrell气团

由于结合能的存在，位错周围的溶质浓度比远处的浓度要高。

位错周围溶质浓度可表为             

这表明溶质总是被吸引到位错周围形成原子气团，这种气团称为Cottrell气团。

 

※只有位错周围存在静水压力分量时才产生尺寸效应。--只有刃位错与溶质存在尺寸效应

※纯螺型位错应力场（在一级近似下）没有正应力分量，因此溶质原子和纯螺型位错的交互作用没有尺寸效应。

1.5.2 模量效应

以螺位错为例

弹性应变能密                    

用溶质原子（）代替同半径溶剂原子（），则由此引起的单位体积应变能的变化为

                    

利用虎克定律，上式可用应力表示

                    

将螺位错的切应力分量式代入上式，则摸量效应引起的位错与溶质的交互作用能为

                

讨论：

1）若，，且随的减小降低，

位错和溶质原子相互吸引

2）若，， 位错和溶质原子相互排斥

※软的溶质与位错相互吸引，硬的溶质与位错相互排斥，

这对第二相颗粒也适用

3）模量效应是短程的，（r 2成反比）。

4）尺寸效应和摸量效应原则上适用于位错与第二相粒子的交互作用。

一般不考虑的尺寸效应。但摸量效应很大。

其他效应

1非对称畸变效应，2 suzuki效应

1.6 位错与表面的相互作用

※自由表面对其附近的位错存在一个吸引力，其原因是位错倾向于接近自由表面以松弛其弹性应变能。  

吸引力的计算：

考虑半无限大的晶体中的一个位错，它与晶体表面平行，距表面距离为，由于表面是自由的，所以表面应力状态为零，

                  

又称为位错的镜像力，是表面对近处位错的作用力

表面附近位错应力场发生了变化！！

1.7 位错动力学

位错运动形式：滑移，攀移，交滑移

刃位错:  滑移，攀移

螺位错: 滑移，交滑移

位错的滑移速度

  无障碍滑移速度接近声速

实际上几乎不可能出现位错在无障碍情况下滑移很长距离。在滑移面上有许多障碍的情况下位错的运动速度通常是由位错越过障碍的过程控制，而不是由滑移速度控制。

  

 

位错的攀移速度

攀移是位错的非保守运动，位错正攀移时产生一列空位，而负攀移时产生一列原子。

攀移产生空位或剩余物质因而需要较大的能量，攀移比滑移困难得多。

1 攀移所需的力

设单位长度位错线攀移扫过的距离(=面积)，则位错后面留下空位（正攀移时）的体积为       

原子（或空位）体积为，则空位的数目为

                    

  设空位形成能为，则单位长度位错线攀移所需的力为

                   

空位形成能，故------理论攀移力。

，故攀移所需应力为     

，  是理论断裂强度的量级！！。

对于Fe: G ~ 105MPa,  !!！

         Q235：   

※ 整段位错线同时攀移是不现实的,实际的攀移是通过割阶沿位错线的运动来实现的。

2 位错攀移的驱动力

（１）攀移力      

（2）化学力

刃型位错受攀移力f的作用时，攀移面内空位形成能降低，空位的平衡浓度为  

 

其中

为完整晶体中的空位平衡浓度。

反过来，若位错线近邻空位浓度不同于，则这种过饱和或欠饱和状态对位错的作用相当于一个与反向的力

此力是由于空位浓度的不同所引起的，故称为化学力

（又称为渗透力）――攀移的阻力。

   （注意：这里，与c无关，而由c决定。若刚开始攀移时，则攀移不能进行，空位浓度降低，化学力随之降低，，攀移就会发生。因此，当攀移进行时总有。

3 攀移速度

位错线上有一个割阶，当此割阶处形成一个空位并扩散出去，导致割阶向右运动的同时长度为b的位错段向上攀移一个b的距离。

，     

:单位长度位错线的攀移速度，：割阶移动速度， 

为割阶浓度(单位长位错线上割阶的总长度)。

X为割阶间的平均间距，    

为了使割阶移动一步，必须产生一个空位并使之迁移一个原子间距。位错受到攀移力，则上述过程总激活能为。

如果空位可以跳到n个位置，则割阶向右移动的几率为 

式中为原子振动频率。

另一方面，由于位错线附近过饱和空位产生一个化学力，从而使割阶反向移动的几率为

      

             

割阶沿位错线移动的总速度为（一次跳动的一定距离为b）

    

根据扩散理论可得自扩散系数

                                        

因此，移动的总速度为

                  

当时，   （）

          

单位时间内单位长度位错线上的割阶上产生（或吸收）的空位数为

 

稳态时

  

                   

Friedel考虑了半径为R的圆拄体中心有一根位错，远离位错处空位浓度为平衡浓度的扩散问题，计算出值

                      

削去得到            

其中                              

            

很多情况下用简化式    

**位错攀移速度与应力成正比，攀移激活能等于原子扩散激活能。

1.8 宏观变形与位错运动的关系

考虑尺寸为的立方形晶体，如图1.12所示。当一根刃型位错在此晶体中滑移的距离（或扫过面积）时，立方体的平均切应变为

     

 若立方体中有根长为的位错

     其中是可动位错密度。

                       

以上是简化模型，实际情况下位错大多不是直线，也不会在单一滑移面上滑移，此时可以用各量的平均

                                

应变速度可表为           

                                             

考虑两种极限情形：

1）可动位错密度急剧变化的情形。例如拉伸试验中屈服，这时式（1.92）等号右边的第一项占优势。

2）可动位错密度不随时间变化的情况，例如稳态蠕变阶段

                                 

其中是位错的平均速度。此式称为Orowan关系。

Orowan关系对拉伸的应变及应变速度仍成立，                                

无论滑移或攀移，位错运动与宏观应变速率的关系可用Orowan关系式描述。

面心立方晶体中的位错

◆ 全位错   ◆ 分位错或不全位错

◆层错      ◆ 扩展位错。

1.9.2 堆垛层错

1  层错及分位错

   面心立方结构的(111)面正常堆垛顺序为ABCABC……。

    A B C A B C A B C A B C 

              ↓↓↓↓↓↓↓

        A B C A B C A 

结果所得的堆垛顺序为

    A B C A B∣A B C A B C A                 

出现了两个不符合面心立方堆垛顺序的“堆垛错误”：ABA和BAB，构成了A B A B四个原子层的密排六方

---内禀层错（Intrinsic stacking fault）。

又称为滑移型层错。层错矢量： 

若在面心立方堆躲中抽去一层C面，然后A面以上各层落下，会获得完全相同的内禀层错。层错矢量： 

如果在正常堆垛顺序中插入一层{111}面（如C面）

A B C A C B C A B C A B C

                                                              

堆垛顺序的错排：ACB和CBC，

----外禀层错（Extrinsic stacking fault）：

层错矢量： 

2 层错能

层错是晶体的一种面缺陷，层错会使系统能量升高。单位面积层错所增加的系统能量叫做层错能。

                 表1.1各种金属与合金的层错能，erg/cm2

金属	Au	Ag	Cu	Ni	Al	Fe-Cr-Ni	NiCo	Fe-Si	Fe3Al
结构	FCC	FCC	FCC	FCC	FCC	FCC	FCC	BCC	BCC
层错能	～47	～58	～40	～	～240	～15	～30	～400	～500

1.9.3 分位错（不全位错）

一个全位错可以扩展为两个分位错和它们之间的层错，所以分位错总是和层错联系在一起的，分位错是层错区与完整晶体之间的边界。

形成层错的移动矢量即层错矢量就是分位错的柏氏矢量。

  

面上原子排列

     肖克莱（Shockley）分位错： 

  

      滑移型内禀层错的边界

                                              

弗兰克（Frank）分位错：

（抽出型内禀层错的边界，

 插入型外禀层错的边界）

扩展位错：两个分位错中间夹着一片层错的组态

全位错的扩展可用位错分解来表示

例如全位错分解为两个肖克莱位错的反应是

                           

汤普森记号

汤普森记号中四个{111}面分别标以(a),(b),(c)和（d），四个面的中心分别标以。

    

汤普森记号中的之间的任何连线都是面心立方结构中的位错柏氏矢量。

罗－罗矢量：AB,BC,CD,DA（共12个矢量）是全位错的柏氏矢量，如AB=；

不对应罗－希矢量：（共24个矢量）是肖克莱位错的柏氏矢量，

如：；

对应罗－希矢量 ：（共8个矢量）是弗兰克分位错的柏氏矢量，

如 ： 

例如，：   

1.9.4 扩展宽度

图1.16 扩展位错及其矢量关系

    

每一个分位错受两个力的作用：两个分位错间的斥力，层错的表面张力引起的吸力。当这两个力达到平衡时扩展位错的宽度即为平衡扩展宽度。

    设一个全位错，其柏氏矢量与位错线的夹角为，则分位错柏氏矢量与位错线的夹角为和。它们可分解为刃型分量和螺型分量。由于刃型分量和螺型分量之间没有弹性相互作用，故相距的两个单位长分位错间的斥力为

   层错表面张力等于单位面积层错能          

平衡条件，得到扩展宽度为

                             

扩展位错的平衡宽度与层错能成反比，层错能越低，扩展宽度越宽。

                 表1.1各种金属与合金的层错能，erg/cm2

金属	Au	Ag	Cu	Ni	Al	Fe-Cr-Ni	NiCo	Fe-Si	Fe3Al
结构	FCC	FCC	FCC	FCC	FCC	FCC	FCC	BCC	BCC
层错能	～47	～58	～40	～	～240	～15	～30	～400	～500

1.9.6 扩展位错的交滑移

扩展位错要交滑移到(c)面必须使其中的某一段重新收缩成全位错（称为束集）

· 扩展位错收缩为全位错需要消耗能量—克服 位错间的斥力做功，扩展宽度越大，收缩成全位错所需能量越大。

交滑移激活能为约    

· 扩展位错形成割阶时也要形成束集，因而形成割阶的激活能也和扩展宽度有关，割阶形成能的估计值为

                                             

其中d为扩展宽度

层错能越低的材料越不容易发生交滑移。

体心立方金属的层错能很高，很容易交滑移，所以变形后的表面滑移带经常呈波浪形，称为波状滑移。波浪形滑移带是螺位错多次交滑移（铅笔滑移）所形成的折线状滑移台阶的宏观形态