A Method for Calculating r-value from Crystallite Orientation Distribution Function in Cold Rolled Steel Sheet

Abstract

To investigate the possibility of calculating the Lankford value, r, for a low carbon steel sheet on the basis of three-dimensional orientation distribution function, ω (ψ, θ, φ, ), comparison was made between the calculated r-value and experimental one. ψ, θ, and φ denote a set of Eulerian angles between the coordinate system of crystallite and the reference system of specimen.
The r-values, r (ψ, θ, φ), that were specified by means of Eulerian angles in the same definition as used in describing the orientation of a crystallite in a polycrystalline sheet were computed at 5°intervals for 0°≤ψ, θ and φ≤90°, by taking into account all fourty-eight slip systems in alpha iron and by assuming that an amount of slip in each slip system was proportional to its Schmid factor.
For a polycrystalline sheet the r-value was calculated by
r (ω) /1+r (ω) =4/π₂∫^π/2₀∫^π/2₀∫^π/2₀r (ψ, θ, φ) /1+r (ψ, θ, φ) ω (ψ, θ, φ) sin<θdθdφ′dθ
whereψ′=ψ+ω, and ω is an angle between rolling direction and stretching direction.
In practical computation of r (ψ, θ, φ), the following cases were examined: (1) All slip systems whose Schmid factors were not less than a certain value, a, (0≤a<0.5), were assumed to be activated and (2) slip systems of a constant number, ss, were assumed to be activated in all the orientations.
Calculated r-values, r (cal), were in good agreement with the observed ones, r (obs), if no crystal rotation was assumed to take place during deformation. Above all, the values of r (cal) were in accord with those of r (obs) within ± 0.15 in the case where either a=0.10 or ss=32 was assumed.