现代塑性变形物理学2

1、Lecture 2：Crystallographic OrientationTheory Physical Based Plastic DeformationJianguo TXinming ZProceeding Materials: A solid composed of atoms, molecules arranged in a pattern periodic in three dimensions Three dimensional array of points (lattice points), each of which has identical surroundings.

2、 Mathematically Lattices can extend to infinityProceeding Materials: Defined by three independent translation vectors a,b and c in a right-handed sense. Different translations of the form will reproduce the whole structure. is a unit cell which contains only one atom per cell. 过晶胞角点(原点)的三轴 为 ，他们及其夹角

3、，共同为点阵参数。n1a n2b n3c , ,a b c Proceeding Materials以原点为起点画一条与拟标定方向平行的线 将uvw简化为最小整数，并加括起来。 反方向的晶向指数可以在数字上面加负号来表示Directions of a form: are the directions represented by symmetry may all be represented by ruavbwc111111 11 1111Proceeding Materials Planes can be represented by their normal vectors Mathema

4、tically this is the same as taking the reciprocal of the intercept to the three axes or 1/h,1/k,1/l If the axial lengths are a, b, c, then the planes make the intercepts of a/h, b/k, c/l. The miller indices for the plane is (hkl), and the family of planes are hkl.Proceeding Materials Planes parallel

5、 to a given axes will result in Miller indices1/ 0Proceeding Materials1 Introduction1 Introduction()()jijiiikljjllmnmn 11212323() ()klnb b bnn Slip system is often defined with the crystal graphic frame, so for crystal plasticity description between crystal and sample reference is enough.1 Introduct

6、ion1 Introduction1 Introduction1 Introduction1 Introduction Crystallographic orientation The rotation relationship between reference frame and crystal reference: S3 C3 S2 C2 S1 C1 S3 C3 C2 S2 C1 S1 1 IntroductionReference frame: Those directions which coincide with the sample symmetry operators are

7、chosen as axis of sample reference. ND, TD and RD of rolled sheets for example Crystal frame: Often low index crystal directions which coincide with crystal symmetry operators are chosen as axis of crystal frame, cubic-.1 Introduction For Hexagonal Crystal structureAttention : the 3 parameters direc

8、tion/plane are also not based on the orthogonal reference frameWwvutUVvVUu)()2(31)2(311 IntroductionGrain 2a3a1a2e1e2e3a2a1a3Grain I1 IntroductionSCRg Rs1c1s2c2s3c3csijikkljlgg111121312212223233132333SCSCSCRgggRRgggRRgggRcsTg g1 Introduction2 Representation of Crystallographic Orientation The column

9、s represent components of three other unit vectors: Where the Columns are the direction cosines (i.e. hkl or uvw) for the RD, TD and Normal directions in the crystal coordinate system. uvwRDTDND(hkl)111213212223313233ggggggggg2 Representation of Crystallographic Orientation3213cccsMLMKMH3211cccsNWNV

10、NU222LKHM222WVUN31KWLVLUHWHVKUMNMNMN2123sssccc2 Representation of Crystallographic Orientationl 6 parameters are needed for representation of crystallographic orientation: HKL: H*U+K*V+L*W=0l Normalization : H2+K2+L2=M2 U2+V2+W2=N2l Only 3 of the 6 parameters are independent.2 Representation of Crys

11、tallographic Orientation33223323uvwhkluvwhkluvwhklacuvhkddvackddwcaldd12122121221coscossinsincossinsincossinsincoscoscossinsinsincos 2 Representation of Crystallographic OrientationRepresentation of rotation/orientation by three continuous rotation along certain axisBunge RoeKocks010100001Crystale1=

12、Xsample=RDe2=Ysample=TDe3=Zsample=NDSample AxesRDTDe”2e”3=e”12nd positionycrystal=e2f2xcrystal=e1zcrystal=e3=3rd position (final)e1e2f1e3=1st position2 Representation of Crystallographic Orientation2 Representation of Crystallographic OrientationRotation 1 (f f1 1): rotate axes (anticlockwise) about

13、 the (sample) 3 ND axis until Rd perpendicular to ND and C311111cossin0()sincos0001g 2 Representation of Crystallographic OrientationRotation 2 ( ): rotate axes (anticlockwise) about the (rotated) 1 axis 100 axis until ND” coincide with C3cossin0sincos0001)(g2 Representation of Crystallographic Orie

14、ntationRotation 3 (f f2 2): rotate axes (anticlockwise) about the (crystal) 3 001 axis until the two reference frames coincide with each other.1000cossin0sincos)(22222g2 Representation of Crystallographic Orientation1212122121212122121211coscossincossinsinsinsincoscossincoscossinsinsincossinsincosco

15、scoscoscossinsincossincosgggg 2 Representation of Crystallographic Orientation1212,2,gg2 Representation of Crystallographic Orientation2 Representation of Crystallographic Orientation(, ,)sinsincossincos sincoscos cossincos cossincoscoscossinsincossincossinsincoscoscossinsinsingffffffffff Convention

16、1st2nd3rd2nd angleabout axis:Kocks(symmetric)fyBungef1-22f2xMatthiesyRoey2 Representation of Crystallographic Orientation2 Representation of Crystallographic Orientation v = vxs1 + vy s2 + vzs3 = vxc1 + vyc2 + vzc3 zyxzyxvvvvvvg2 Representation of Crystallographic Orientation First rotate axis V to

17、V which coincide with axis 3; Then rotate along V with angle ; Finally rotate V back to V.222222( ,)(1)cos(1 cos)(1 cos)sinsin(1)cos(1 cos)(1 cos)sinsin(1 cos)(1 cos)(1)cossinsinxxyxzxzyyxyyzzyxxyyzzyxzgvv vv vvvvvv vv vvvvv vv vvvvvv2 Representation of Crystallographic Orientation Rodrigues Vector2

18、22123tan, v=+v2vvRv Q123cos,sin,sin,sin2222vvvq2 Representation of Crystallographic Orientation2 Representation of Crystallographic OrientationNSS1P1P2S2S2Definition of pole and its stereo projection2 Representation of Crystallographic Orientation100001010NDNDRD100001ii0012 Representation of Crystal

19、lographic Orientation22 Representation of Crystallographic OrientationRi = (sini cosi) s1 + (sini sini) s2 + (cosi) s3 = (Xi c1 +Yi c2 +Zi c3) / P 222ZYXPiii333231232221131211ZYX1cossinsincossingggggggggPiiiiiRDTDNDHKL2 Representation of Crystallographic Orientation EXAMPLE: Calculate the pole angle

20、s() of pole. (orientation: )(101)1211112102131313161626131cossinsincossiniiiiii = 35.3，i = 215.3 2 Representation of Crystallographic OrientationRi = si = sini cosi c1 + sini sini c2 + cosi c3 g1i = sini cosi；g2i = sini sini；g3i = cosi 1000100013 Misorientation and grain boundaryGBa3a1a2e1e2e3a2a1a3

21、GA3 Misorientation and grain boundarygAgBgCgD3 Misorientation and grain boundaryxzygA-1gBIn terms of orientations:rotate back from position Ato the reference position.Then rotate to position B.Compound (“compose”)the two rotations to arriveat the net rotation betweenthe two grains.Net rotation = gBg

22、A-13 Misorientation and grain boundarygAgBgCgDg=gBgA-1 gAgB-1rrotation axis, common to both crystalsSwitching symmetry:A to B is indistinguishable from B to A3 Misorientation and grain boundary12332311312211cos2(),(),()sintrgggggggrU.F.Kocks ，Texture and anisotropy ，1998；毛卫民，张新明，晶体材料织构定量分析晶体材料织构定量分析

23、，19933 Misorientation and grain boundary Influence of crystal symmetryiiTsym gg R11min cos2iTtrg3 Misorientation and grain boundaryMisorientation is defined as the crystallographic orientation difference between grains at each side of the grain boundary, not the angle of two boundaries.3 Misorientation and grain boundary1n2n同样的取向差，但是不一样3 Misorientation and grain boundary Tilt boundary is a rota