PSI - Issue 3

Y. Nakai et al. / Procedia Structural Integrity 3 (2017) 402–410

405

Author name / Structural Integrity Procedia 00 (2017) 000–000

4

Therefore, the total misorientation, β, of each crystallographic plane can be evaluated from the rotation angle spread for the diffraction condition, Δω diff , the diffraction angle, θ, and the angle between the diffraction plane and the rotation axis, ψ, as defined in Fig. 1.

(b) 

 1 2 n n 1 / 2  

(a) 

 1 2 n n 0  

Fig. 3 fcc structure.

Fig. 4. bcc structure.

2.5. Direction of the diffraction plane

Normal unit vector of the diffraction plane in the coordinate system fixed to the sample is given by

               x y z

1 0 0 cos

0

cos cos cos sin sin

    

    

    

    

(3)

n

sin     

 

n cos

0 si

where x - axis is the direction of rotation axis, and ω is rotation angle of a sample from a reference positio n. 2.6. Slip direction Slip directions can be determined from an intersection of slip plane and other crystallographic planes. Normal unit vectors of two planes are defined by n 1 and n 2 , and a unit vector in the intersection line of two planes is n 3 , respectively. Since the vector n 3 is perpendicular to the vectors n 1 and n 2 , n 3 can be given by the cross product of n 1 and n 2 as   3 1 2 1 2 n n n / n n    (4) For fcc structure as shown in Fig. 3, the intersection of two {111} planes (for example, ABC and DBC), n 3 , is one of the slip direction. Let n 4 and n 5 be unit vectors of other slip directions, those are given by the rotation of n 3 around n 1 by ±π/3 and obtained through Rodrigues' rotation formula as     3 3 1 n n / 2 3 / 2 n n    i ( i = 4,5) (5) For bcc structure, the slip direction can be determined from intersection of {110} and {200} planes, those normal unit vectors are n 1 and n 2 , respectively as shown in Fig. 4. Through Rodrigues' rotation formula, the unit vector of slip directions, n 4 and n 5 are given by the rotation of the intersection unit vector, n 3 , around n 1 as   n 2n n n 3        i 3 3 1 ( i = 4, 5) for   1 2 n n 0   (6)