ch2位錯(cuò)-23位錯(cuò)的彈性性質(zhì)

1、1Ch2 位錯(cuò)2.1 位錯(cuò)理論的產(chǎn)生2.2 位錯(cuò)的幾何性質(zhì)2.3 位錯(cuò)的彈性性質(zhì)2.4 位錯(cuò)與晶體缺陷的相互作用2.5 位錯(cuò)的動(dòng)力學(xué)性質(zhì)2.6 實(shí)際晶體中的位錯(cuò)22.1 2.1 位錯(cuò)理論的產(chǎn)生位錯(cuò)理論的產(chǎn)生一、晶體的塑性變形方式二、單晶體的塑性變形三、多晶體的塑性變形四、晶體的理論切變強(qiáng)度五、位錯(cuò)理論的產(chǎn)生六、位錯(cuò)的基本知識(shí)32.2 2.2 位錯(cuò)的幾何性質(zhì)位錯(cuò)的幾何性質(zhì)一、位錯(cuò)的幾何模型一、位錯(cuò)的幾何模型二、柏格斯矢量二、柏格斯矢量三、位錯(cuò)的運(yùn)動(dòng)三、位錯(cuò)的運(yùn)動(dòng)四、位錯(cuò)環(huán)及其運(yùn)動(dòng)四、位錯(cuò)環(huán)及其運(yùn)動(dòng)五、位錯(cuò)與晶體的塑性變形五、位錯(cuò)與晶體的塑性變形六、割階六、割階42.3 2.3 位錯(cuò)的彈性性質(zhì)位錯(cuò)

2、的彈性性質(zhì)進(jìn)一步探討位錯(cuò)的性質(zhì)進(jìn)一步探討位錯(cuò)的性質(zhì), ,就需要知道位錯(cuò)所產(chǎn)生的就需要知道位錯(cuò)所產(chǎn)生的長長程彈性畸變程彈性畸變及及位錯(cuò)線附近的原子錯(cuò)排情況位錯(cuò)線附近的原子錯(cuò)排情況. .這是這是計(jì)算計(jì)算位錯(cuò)的能量位錯(cuò)的能量、理解位錯(cuò)和晶體缺陷之間的交互作用理解位錯(cuò)和晶體缺陷之間的交互作用及其動(dòng)力學(xué)性質(zhì)及其動(dòng)力學(xué)性質(zhì)的必要基礎(chǔ)的必要基礎(chǔ), ,同時(shí)也是同時(shí)也是位錯(cuò)理論的核位錯(cuò)理論的核心問題心問題; ;關(guān)于關(guān)于位錯(cuò)的長程應(yīng)力場(chǎng)位錯(cuò)的長程應(yīng)力場(chǎng), ,采用采用連續(xù)介質(zhì)模型連續(xù)介質(zhì)模型已經(jīng)獲得已經(jīng)獲得了很有成效的結(jié)果了很有成效的結(jié)果; ;但關(guān)于但關(guān)于位錯(cuò)線近程位錯(cuò)線近程的情況的情況, ,即即位錯(cuò)芯的結(jié)構(gòu)位錯(cuò)芯

3、的結(jié)構(gòu), ,目前了目前了解得還不夠清楚解得還不夠清楚, ,理論計(jì)算還只是初步的理論計(jì)算還只是初步的; ;晶體中的位錯(cuò)晶體中的位錯(cuò), ,不僅在其中心形成嚴(yán)重的點(diǎn)陣畸變不僅在其中心形成嚴(yán)重的點(diǎn)陣畸變, ,而且使周圍的點(diǎn)陣發(fā)生彈性應(yīng)變而且使周圍的點(diǎn)陣發(fā)生彈性應(yīng)變, ,產(chǎn)生應(yīng)力場(chǎng)產(chǎn)生應(yīng)力場(chǎng), ,叫做叫做位錯(cuò)應(yīng)力場(chǎng)位錯(cuò)應(yīng)力場(chǎng). .它使位錯(cuò)具有彈性能它使位錯(cuò)具有彈性能, ,產(chǎn)生線張力產(chǎn)生線張力; ;在位在位錯(cuò)之間錯(cuò)之間, ,位錯(cuò)與其它缺陷之間發(fā)生交互作用等等位錯(cuò)與其它缺陷之間發(fā)生交互作用等等. .所所有這些都直接影響晶體的力學(xué)性質(zhì)有這些都直接影響晶體的力學(xué)性質(zhì). .52.3 2.3 位錯(cuò)的彈性性質(zhì)位錯(cuò)的彈

4、性性質(zhì)一、彈性連續(xù)介質(zhì)、應(yīng)力和應(yīng)變一、彈性連續(xù)介質(zhì)、應(yīng)力和應(yīng)變二、螺型位錯(cuò)的應(yīng)力場(chǎng)二、螺型位錯(cuò)的應(yīng)力場(chǎng)三、刃型位錯(cuò)的應(yīng)力場(chǎng)三、刃型位錯(cuò)的應(yīng)力場(chǎng)四、位錯(cuò)的應(yīng)變能四、位錯(cuò)的應(yīng)變能五、位錯(cuò)的受力五、位錯(cuò)的受力六、向錯(cuò)六、向錯(cuò)6一、彈性連續(xù)介質(zhì)、應(yīng)力和應(yīng)變一、彈性連續(xù)介質(zhì)、應(yīng)力和應(yīng)變( (一一) )彈性連續(xù)介質(zhì)彈性連續(xù)介質(zhì)( (二二) )應(yīng)力分量應(yīng)力分量( (三三) )應(yīng)變分量應(yīng)變分量( (四四) )彈性模量彈性模量7( (一一) )彈性連續(xù)介質(zhì)彈性連續(xù)介質(zhì)現(xiàn)在較為成熟的現(xiàn)在較為成熟的位錯(cuò)彈性理論位錯(cuò)彈性理論, ,是在是在彈性連續(xù)介質(zhì)模彈性連續(xù)介質(zhì)模型型基礎(chǔ)上建立的基礎(chǔ)上建立的. .彈性連續(xù)介質(zhì)模型對(duì)

5、晶體作了如下彈性連續(xù)介質(zhì)模型對(duì)晶體作了如下假設(shè)：假設(shè)： (1) (1)完全服從虎克定律，即不存在塑性變形；完全服從虎克定律，即不存在塑性變形； (2) (2)是各向同性的；是各向同性的； (3) (3)為連續(xù)介質(zhì)，不存在結(jié)構(gòu)間隙為連續(xù)介質(zhì)，不存在結(jié)構(gòu)間隙顯然顯然, ,這樣的假設(shè)不符合晶體的實(shí)際情況這樣的假設(shè)不符合晶體的實(shí)際情況. .因?yàn)榫w因?yàn)榫w的質(zhì)點(diǎn)的質(zhì)點(diǎn)( (原子原子) )不是連續(xù)分布的不是連續(xù)分布的, ,晶體中也不存在完全晶體中也不存在完全沒有塑性變形的情況沒有塑性變形的情況, ,至于各向異性至于各向異性, ,更是晶體的一更是晶體的一個(gè)特征個(gè)特征. .但是對(duì)晶體作這樣的簡(jiǎn)化之后但是對(duì)晶

6、體作這樣的簡(jiǎn)化之后, ,推導(dǎo)出的彈推導(dǎo)出的彈性力學(xué)函數(shù)性力學(xué)函數(shù), ,除了對(duì)位錯(cuò)中心存在嚴(yán)重畸變的區(qū)域不除了對(duì)位錯(cuò)中心存在嚴(yán)重畸變的區(qū)域不適用外適用外, ,對(duì)大部分存在彈性形變的點(diǎn)陣區(qū)域?qū)Υ蟛糠执嬖趶椥孕巫兊狞c(diǎn)陣區(qū)域, ,都是合都是合適的適的. .8( (二二) )應(yīng)應(yīng)力力分分量量9s sxx, s, syy, s, szz, s, syz=s szy s szx=s sxz s sxy= s syx1011( (三三) )應(yīng)變分量應(yīng)變分量yy=eyy/lyzy=ezy/ly1213( (四四) )彈性模量彈性模量14二、螺型位錯(cuò)的應(yīng)力場(chǎng)二、螺型位錯(cuò)的應(yīng)力場(chǎng)15Stress Field of

7、Screw Dislocation The elastic distortion around a straight screw dislocation of infinite length can be represented in terms of a cylinder of elastic material deformed as defined by Volterra. The following illustration shows the basic geometry.16A screw dislocation produces the deformation shown in t

8、he left hand picture. This can be modeled by the Volterra deformation mode as shown in the right hand picture - except for the core region of the dislocation, the deformation is the same. A radial slit was cut in the cylinder parallel to the z-axis, and the free surfaces displaced rigidly with respe

9、ct to each other by the distance b, the magnitude of the Burgers vector of the screw dislocation, in the z-direction. In the core region the strain is very large - atoms are displaced by about a lattice constant. Linear elasticity theory thus is not a valid approximation there, and we must exclude t

10、he core region. We then have no problem in using the Volterra approach; we just have to consider the core region separately and add it to the solutions from linear elasticity theory. 17The elastic field in the dislocated cylinder can be found by direct inspection. First, it is noted that there are n

11、o displacements in the x and y directions, i.e. ux = uy = 0 . In the z-direction, the displacement varies smoothly from zero to b as the angle q q goes from 0 to 2p p. This can be expressed asuz=（b q q）/ /2p p=(b/2p p) tan1(y/x)= (b/2p p)arctan(y/x) Using the equations for the strain we obtain the s

12、train field of a screw dislocation:qe exx= e eyy=e ezz=e exy=e eyx=0qe exz=e ezx=(b/4p)p)y/(x2 + y2 )= (b/4p)p)(sinq)/q)/rqe eyz=e ezy=(b/4p)p)x/(x2 + y2 )= (b/4p)p)(cosq)/q)/r18equations for the strainWe obtain the normal strain as the diagonal elements of the strain tensor.qe exx =dux / dx e eyy =

13、duy / dy e ezz=duz / dz The shear strains are contained in the rest of the tensor: qe eyz=e ezy= (duy / dz)+(duz / dy)qe ezx=e exz= (duz / dx)+(dux / dz)qe exy=e eyx= (dux / dy)+(duy / dx)19Using the equations for the strain we obtain the strain field of a screw dislocation:qe exx= e eyy=e ezz=e exy

14、=e eyx=0qe exz=e ezx=(b/4p)p)y/(x2 + y2 )= (b/4p)p)(sinq)/q)/rqe eyz=e ezy=(b/4p)p)x/(x2 + y2 )= (b/4p)p)(cosq)/q)/rThe corresponding stress field is also easily obtained from the relevant equations:qs sxx=s syy=s szz=s sxy=s syx=0 qs sxz=s szx=(G b/2p) p) y/(x2+y2 )= (G b/2p)p)(sinq)/q)/rqs syz=s s

15、zy=(G b/2p) p) x/(x2+y2 )= (G b/2p)p)(cosq)/q)/r20In cylindrical coordinates, which are clearly better matched to the situation, the stress can be expressed via the following relations: s sr z=s sxy cosq q + s syz sinq q s sq q z = s sxz sinq q + s syz cosq qqSimilar relations hold for the strain. W

16、e obtain the simple equations:e eq q z=e ezq q=b/(4p pr)s sq q z=s szq q=G b/( 2p pr)The elastic distortion contains no tensile or compressive components and consists of pure shear. s szq q acts parallel to the z axis in radial planes of constant q q and s sq qz acts in the fashion of a torque on pl

17、anes normal to the axis. The field exhibits complete radial symmetry and the cut thus can be made on any radial plane q q = constant. For a dislocation of opposite sign, i.e. a left-handed screw, the signs of all the field components are reversed.21There is, however, a serious problem with these equ

18、ations:q The stresses and strains are proportional to 1/r and therefore diverge to infinity as r 0. This makes no sense and therefore the cylinder used for the calculations must be hollow to avoid r - values that are too small, i.e. smaller than the core radius r0. q Real crystals, of course, do (us

19、ually) not contain hollow dislocation cores. If we want to include the dislocation core, we must do this with a more advanced theory of deformation, which means a non-linear atomistic theory. There are, however, ways to avoid this, provided one is willing to accept a bit of empirical science. How la

20、rge is radius r0 or the extension of the dislocation core? Since the theory used is only valid for small strains, we may equate the core region with the region were the strain is larger than, say, 10%. From the equations above it is seen that the strain exceeds about 0,1 or 10% whenever r b. A reaso

21、nable value for the dislocation core radius r0 therefore lies in the range b to 4b, i.e. r0 1 nm in most cases.22二、螺型位錯(cuò)的應(yīng)力場(chǎng)二、螺型位錯(cuò)的應(yīng)力場(chǎng)以上述以上述模型模型，運(yùn)用，運(yùn)用經(jīng)典彈性力學(xué)經(jīng)典彈性力學(xué)方法建立方法建立螺型位錯(cuò)的應(yīng)力場(chǎng)函數(shù)螺型位錯(cuò)的應(yīng)力場(chǎng)函數(shù)。23圓圓柱柱坐坐標(biāo)標(biāo)z= z = ez /l =b/2r24直直角角坐坐標(biāo)標(biāo)yz= zyxz= zxxy= yx =025螺型位錯(cuò)應(yīng)力場(chǎng)特點(diǎn)螺型位錯(cuò)應(yīng)力場(chǎng)特點(diǎn)26三、三、Stress Field of Edge Disl

22、ocationThe stress field of an edge dislocation is somewhat more complex than that of a screw dislocation, but can also be represented in an isotropic cylinder by the Volterra construction. qUsing the same methodology as in the case of a screw dislocation, we replace the edge dislocation by the appro

23、priate cut in a cylinder. The displacement and strains in the z-direction are zero and the deformation is basically a plane strain.27qIt is not as easy as in the case of the screw dislocation to write down the strain field, but the reasoning follows the same line of arguments. We simply look at the

24、results:s sxx = D y (3x2 + y2 )/(x2 + y2)2s syy = D y( x2 y2 )/(x2 + y2)2s sxy =s syx = D x(x2 y2 )/ (x2 + y2)2 s szz = n n (s sxx + s syy)s szz = s szx = s syz = s szy = 0qWe used the abbreviation qD = Gb /2p p (1 n n) .28293031四、四、Energy of a DislocationConsider the case for a screw dislocation (s

25、tress = modulus strain) This is the stress acting in the z direction across plane q q = const. energy = /unit length The upper limit of the integral, R, is given by the distance to nearest dislocation of opposite sign/loop diameter.32The lower limit ro represents the inner cut-off where linear elast

26、icity breaks down. Energy = /unit length For edge dislocations, the effect of Poissons ratio n n has to be taken into account.Etot = / unit length Including core energy Etot 1/2 Gb2 - a few eV/atom plane(of which 10% is core).Dislocations are not usually in thermal equilibrium, so some means must be

27、 found to create them.33Energy of a DislocationWith the results of the elasticity theory we can get approximate formulas for the line energy of a dislocation and the elastic interaction with other defects, i.e. the forces acting on dislocations.q The energy of a dislocation comes from the elastic pa

28、rt that is contained in the elastically strained bonds outside the radius r0 and from the energy stored in the core, which is of course energy sitting in the distorted bonds, too, but is not amenable to elasticity theory. q The total energy per unit length Etot is the sum of the energy contained in

29、the elastic field, Eel, and the energy in the core, Ecore. Etot = Eel + Ecore 34qUsing the formula for the strain energy for a volume element given before, integration over the total volume will give the total elastic energy Eel of the dislocation. The integration is easily done for the screw disloc

30、ation; in what follows the equations are always normalized to a unit of length.dEel(screw) = p p r dr (s sQ Q z e eQ Q z + s szQ Q e ezQ Q )=4p p r dr G (e eQ Q z)2Eel(screw) = (G b2 / 4p p ) Rrodr /r = ( G b2 / / 4p)p) lnR/ ro The integration runs from r0, the core radius of the dislocation to R, w

31、hich is some as yet undetermined external radius of the elastic cylinder containing the dislocation. In principle, R should go to infinity, but this is not sensible as we are going to see.35The integration for the edge dislocation is much more difficult to do, but the result is rather simple, too:q

32、Eel(edge) = (G b2 ) / 4p p(1 n n) ln(R /ro ) So, apart from the factor (1 n n), this is the same result as for the screw dislocation.36Strain energy of a dislocationStrain energy of a dislocationA dislocation represents a destruction in the packing order of the crystal. The atoms near the core of th

33、e dislocation are displaced from their proper positions and thus have a higher energy. In order to keep the total energy as low as possible, the dislocation tries to be as short as possible; i.e. a dislocation behaves as if it had a line tension like a rubber band. When we analyze the strain fields

34、near the dislocation, we find that the internal energy per unit length due to a dislocation is given as follows:37One can think of the internal energy of a dislocation as line tension. Dislocations are formed to minimize this energy. Since UT is proportional to b2 it is apparent that dislocations wi

35、th minimum |b| are the most stable. This justifies why dislocations usually move in close-packed planes in close-packed directions!In absolute terms , UT is small, but it is large in terms of the size of the dislocation. UT plays an important role on the way in which obstacles obstruct the motion of

36、 dislocations. We will use this in a later lecture to construct stronger materials.38四、位錯(cuò)的應(yīng)變能四、位錯(cuò)的應(yīng)變能位錯(cuò)在其周圍晶體中引起畸變，使晶體所產(chǎn)生的畸變能稱為位錯(cuò)的應(yīng)變能；一個(gè)位錯(cuò)的應(yīng)變能位錯(cuò)中心區(qū)域應(yīng)變能(超出彈性應(yīng)變范圍虎克定律不適用,僅占1/15-1/10)+位錯(cuò)中心區(qū)域外應(yīng)變能;39螺位錯(cuò)應(yīng)變能螺位錯(cuò)應(yīng)變能404142位錯(cuò)應(yīng)變能位錯(cuò)應(yīng)變能43Rate of Climb and Stress DependenceThis process also allows dislocations to

37、climb round precipitate particles.In this case the rate of creep is determined by the rate at which dislocations can climb past obstacles.44How do dislocations respond to a stress t t?Consider this stress causing a dislocation to move right through a crystal of size l1External work done dW = t t l1

38、l2 b stress area displacementAlso dW = force on dislocation/unit length length distance travelled = f l2 l1f = t t b45Now at a precipitate particle In equilibrium: Reaction force = glide + climb force Climb force = t tb tanq q Hence increases with stressAs shear stress increases, more dislocations u

39、nlocked and more creep occurs.Situation usually described by and known as power law creep.This also has strong T dependence, requiring vacancy diffusion.46五、位錯(cuò)的受力五、位錯(cuò)的受力Forces on Dislocations( (一一) )外加應(yīng)力場(chǎng)作用在位錯(cuò)線上的力外加應(yīng)力場(chǎng)作用在位錯(cuò)線上的力( (二二) )位錯(cuò)的線張力位錯(cuò)的線張力4748Force acting on a dislocationThe material provide

40、s certain resistance to the motion of dislocations. Let us denote with f the internal resistance to the motion of a dislocation per unit length of the dislocation. Let us consider Fig. 26 showing an edge dislocation moving as a result of an externally applied shear stress . To compute f in terms of

41、, we apply a work balance:Figure : The force acting on a dislocation: f is the internal resistance force to the motion of the dislocation per unit length.49External work done by to move the dislocation through the crystal:Internal work done against the resistance f is (note that the dislocation moved by a distance l2):Equating equations (12) and (13), we conclude that:from which we conclude that:50( (一一) )外加應(yīng)力場(chǎng)作用在位錯(cuò)線上的力外加應(yīng)力場(chǎng)作用在位錯(cuò)線上的力51由上式可知，由上式可知，52刃