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/π
2∫
π/20∫
π/20∫
π/20r (ψ, θ, φ) /1+
r (ψ, θ, φ) ω (ψ, θ, φ) sin<θ
dθ
dφ′
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.
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