Lankford coefficient explained

The Lankford coefficient (also called Lankford value, R-value, or plastic strain ratio)^[1] is a measure of the plastic anisotropy of a rolled sheet metal. This scalar quantity is used extensively as an indicator of the formability of recrystallized low-carbon steel sheets.^[2]

Definition

If

x and y are the coordinate directions in the plane of rolling and z is the thickness direction, then the R-value is given byR = \cfrac{\epsilon ^{p} _{y} } where \epsilon ^{p} _{y} is the in-plane plastic strain, transverse to the loading direction, and \epsilon ^{p} _{z} is the plastic strain through-the-thickness.^[3]

More recent studies have shown that the R-value of a material can depend strongly on the strain even at small strains . In practice, the

R value is usually measured at 20% elongation in a tensile test.

For sheet metals, the

R values are usually determined for three different directions of loading in-plane (0 ^{\circ} , 45 ^{\circ} , 90 ^{\circ} to the rolling direction) and the normal R-value is taken to be the average R = \cfrac{ 1 }{ 4 }\left(R _{0} + 2 ~R _{45} + R _{90} \right) ~.

The planar anisotropy coefficient or planar R-value is a measure of the variation of

R with angle from the rolling direction. This quantity is defined asR _{p} = \cfrac{ 1 }{ 2 }\left(R _{0} - 2 ~R _{45} + R _{90} \right) ~.

Anisotropy of steel sheets

Generally, the Lankford value of cold rolled steel sheet acting for deep-drawability shows heavy orientation, and such deep-drawability is characterized by

R . However, in the actual press-working, the deep-drawability of steel sheets cannot be determined only by the value of R and the measure of planar anisotropy, R _{p} is more appropriate.

In an ordinary cold rolled steel,

R _{90} is the highest, and R _{45} is the lowest. Experience shows that even if R _{45} is close to 1, R _{0} and R _{90} can be quite high leading to a high average value of R . In such cases, any press-forming process design on the basis of R _{45} does not lead to an improvement in deep-drawability.

See also

• Yield surface

Notes and References

1. Lankford, W. T., Snyder, S. C., Bausher, J. A.: New criteria for predicting the press performance of deep drawing sheets. Trans. ASM, 42, 1197–1205 (1950).
2. Ken-ichiro Mori, Simulation of Materials Processing: Theory, Methods and Applications,, p. 436
3. ISO 10113:2020 https://www.iso.org/obp/ui/#iso:std:iso:10113:ed-3:v1:en