1 Cubic ODF 프로그램을 이용한집합조직해석 박노진 금오공과대학교 2010. 8. 24 2 Definition of Texture - Oi Orientation ti distribution tib ti function ODFf fg of fthe voume dv g g V = f g dg - ODF fg for the numbers of crystaites dn g N = f g dg
3 Measurement Singe orientation measurement 1 etched form etch-pits 2 poarization microscopy 3 eectron microscopy : TEM, SEM 4 X-ray : Laue method Poe figure measurement 1 X-ray diffraction : conventiona, synchrotron 2 neutron diffraction 3 eectron microscopy TEM Poe Figure 4 Consider a cubic crysta in a roed sheet sampe with "aboratory" or "sampe" axes as shown beow. The Poe Figure pots the orientation of a given pane norma poe with respect to the sampe reference frame. The exampe beow is a 001 poe figure. Note the three points shown in the poe figure are for three symmetricay equivaent panes in the crysta.
5 Poe Figure P hk 1 α, β = f g dψ 2π 2π hk { α, β } measured PF wanted ODF {α, β} : sampe direction parae to the diffraction vector ψ : rotation ti about the diffraction vector 6 Poe Figure Measurement experimenta technique - used characteristic radiation - measured intensity poe density - fixed Bragg ange 2θ fixed hk-diffraction - incompete poe figure back-refection technique transmission technique
7 Equipment 4-Circe Texture Goniometer need sampe rotation and titing θ α : sampe titing Κ β : sampe rotation Φ β α 2 θ 8 Intensity Correction - Background Correction - Geometrica Correction Gα - Defocusing Correction Dα - Absorption Correction Aα - Random Sampe Method
9 Anaysis from Poe Figures P hk α, β measured PF = 1 2π hk { α, β } f g dψ wanted ODF direct from poe figures quaitative anaysis quantitative anaysis from poe figures - Series expansion method harmonic method - Components mode - WIMV - Vector method 10 Spherica Harmonics Representation of ODF The so-caed harmonic anaysis is anaogous to a Fourier series expansion. However, texture anaysis is performed using generaized spherica harmonic expansion. In this method, the ODF can be expanded into a series of generaized spherica harmonic functions + + f g = C mn mn T g = 0 m = n= The generaized spherica harmonics are defined as T mn ϕ 1,φ,ϕ 2 = e imϕ 2 P mn φe inϕ 1 where P mn F are the Legendre poynomia P mn φ = + k = Q mk Q nk e ikφ Q s Qs are known constants
11 Symmetry Operations g is the rotation ti which h transforms a sampe fixed coordinate K A into a crysta fixed coordinate K B - Sampe Symmetry: Orthogonaity, fiber, f g,g A = f g Transformation from fixed sampe coordinate to equivaent orientations rotation about RD, ND, and TD, for a roed sampe - Crysta Symmetry: Cubic, f g B, g = f g Transformation from crystaine state to equivaent orientations 12 Symmetry Operations Appying the symmetry operation wi reduce the number of eements in C-coefficient f g = 0 M N = C μ = 1 ν = 1 μν :. T μν g where :. μν m= n= :. nν mμ mn T = A A T g N are the number of independent soutions to satisfy sampe symmetry M are the number of independent soutions to satisfy crysta symmetry y Number of measured poe figures shoud aways be arger than M
13 Harmonic Method P hk α, β measured PF = 1 2π hk { α, β } f g dψ wanted ODF Poe figure and ODF are deveoped into series of harmonic functions P max N. ν ν hk α, β = F hk k α, β = 0 ν = 1 max f g = C Surface spherica harmonic functions M N :. μν μν T g = 0 μ= 1 ν = 1 Generaized spherica Coefficients harmonic functions Fow Diagram for ODF Anaysis 14 Measured PF Coef. for PF Coef. for ODF inverse PF ODF Ca. PF
15 ODF Anaysis with Incompete PFs - Iterative cacuation method - Positivity condition of PF 16 Compete ODF - Iterative cacuation method - Positivity condition of ODF f ~ g = f g + f g 0
17 Estimation of the Convergence of the Series Expansion C = C μν = 1 M N M N μ= 1 ν = 1 C μν strong texture t weak texture t Estimation of the Accuracy of the ODF with Difference Poe Figures 18
19 Estimation of the Accuracy of the ODF with Vaues RP-vaue exp ca α, β α, β RP 1 = N P P hk hk hk exp α β α, β hk RP1-vaue : cacuated with exp α, β 1 Phk exp RP0.5-vaue : cacuated with α, β 0. 5 P P hk 100% Poe Figure Dispersion PF hk dispersion = max = 0 N F ν = 1 max = 0 ν hk N F ν = 1 exp ν F hk ν exp hk ca 20 Texture Index J Texture index J is appropriate to characterize the sharpness of the texture by a singe parameter. This can be done, for exampe, by the integra of the square of the texture function J = L 2 max M N [ f g ] dg = = 1 2+ 1 μ= 1 ν = 1 1 C μν 2 In the case of random distribution : J r =1 r singe orientation: J idea =
21 Description of the ODF Representation of orientation in Euer space a singe orientation b orientation of a crystaites c continuous orientation distribution function ODF by Equi-density Lines in Panar Sections 22 ϕ 2 = 0 ϕ 2 = 5 ϕ 2 = 15 ϕ 2 = 10 ϕ 1 Φ ϕ 2
With Intensity of fg by Skeeton Line - <110>ⅡRD-fiber α-fiber η α - <111>ⅡND-fiber γ-fiber - <100>ⅡRD-fiber η-fiber - <110>ⅡTD-fiber ε-fiber - <110>ⅡND-fiber ζ-fiber ζ γ ε τ 23 24 With Intensity of fg by Skeeton Line β-fiber in FCC {110}<112> BS + {123}<634> S + {112}<111> Cu
25 BCC roing texture η α τ ζ γ ε 6. Formation of textures 26 Program CUBODF ODF Cacuation with 1~4 poe figures - maximum Lmax=58 - Cubic Crysta Symmetry - Sampe Symmetry : Tricinic, MonocinicRD, TD, Orthorhombic, Axia
Program CUBODF, PF data format 27 Program CUBODF, PF data format 28
Program CUBODF, Contro o Data a 29 Program CUBODF, Running 30
31 Program CUBODF, Resuts Output fies????.out : a data in ASCII-format - Texture index, Normaization factor - Dispersions of the poe figures - RP-vaue : RP, RP0.5, RP1 - PF, Inv.PF, ODF, - F-Coef., H-Coef., C-Coef. - Intensities for fibers C-????.DAT : C-coefficients????.PLT : fies for pot in HPGL-format 32 Program CUBODF, Resuts Pots fies????.plt : fies for pot in HPGL-format - 측정극점도 experimenta poe figure μν μν - = C, Δ C C - 계산극점도 cacuated poe figure - 역극점도 inverse poe figure - Even ODF - Compete ODF orthorhombic and tricinic sampe symmetry - Fiber 에서의 ODF 강도
33????.OUT Program CUBODF, Resuts 34????.OUT Program CUBODF, Resuts
35????.OUT Program CUBODF, Resuts PF J = hk dispersion = max F = 0 N ν = 1 max = 0 ν hk N F ν = 1 exp F ν hk 2 max L M N [ f g ] dg = = 1 2 + 1 μ = 1 ν = 1 1 ν hk exp ca C μν 2 RP 1 = N P exp α, β α, β hk hk hk exp α β α, β Phk P P ca 100% 36????.OUT Program CUBODF, Resuts
37????.OUT Program CUBODF, Resuts 38????.OUT Program CUBODF, Resuts M N μν f g = C μν T g = 0 μ = 1 ν = 1 :.
39????.OUT Program CUBODF, Resuts M N μν f g = C μν T g = 0 μ = 1 ν = 1 :. 40????.OUT Program CUBODF, Resuts
41????.OUT Program CUBODF, Resuts 42????.OUT Program CUBODF, Resuts
43 C-????.OUT Program CUBODF, Resuts Program CUBODF, Resuts 44 Measured PFs Correction of - back-ground - defocusing - roing-direction Cacuation with tricinic sampe symmetry
Program CUBODF, Resuts 45 Cacuation with orthorhombic sampe symmetry Program CUBODF, Resuts 46
Program CUBODF, Resuts 47 Program CUBODF, Resuts 48
Program CUBODF, Resuts 49 Program CUBODF, Resuts 50
Program CUBODF, Resuts 51 52 단계적인프로그램실행 집합조직을정확하게분석하고신뢰성있는결과를얻기위하여다음과같은단계적인절차로프로그램을실행하고확인 1. 측정극점도의데이터의형식, {hk} 지수, background 등모든데이터가정확한지확인 2. 측정극점도를 tricinic 시편대칭을적용, 계산하여 potting 한후, 측정극점도의 RD-correction 을정함. 또한시편대칭을확인. 3. RD-correction 을입력한후, 입력창에 Lmax=50 정도입력하여함수의수렴성을판단 4. 적절한 Lmax 를적용하여원하는계산을수행 5. 프로그램실행후에는다음사항을확인하여분석의정확성확보 1 C-coefficient 의수렴성 2 RP- 값확인 :RP0.5- 값이약 10% 이하 3 측정극점도와계산극점도의동일성