Lankford coefficient

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 ${\displaystyle x}$ and ${\displaystyle y}$ are the coordinate directions in the plane of rolling and ${\displaystyle z}$ is the thickness direction, then the R-value is given by

${\displaystyle R={\cfrac {\epsilon _{\mathrm {y} }^{p}}{\epsilon _{\mathrm {z} }^{p}}}}$

where ${\displaystyle \epsilon _{\mathrm {y} }^{p}}$ is the in-plane plastic strain, transverse to the loading direction, and ${\displaystyle \epsilon _{\mathrm {z} }^{p}}$ 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 ${\displaystyle R}$ value is usually measured at 20% elongation in a tensile test.

For sheet metals, the ${\displaystyle R}$ values are usually determined for three different directions of loading in-plane (${\displaystyle 0^{\circ },45^{\circ },90^{\circ }}$ to the rolling direction) and the normal R-value is taken to be the average

${\displaystyle R={\cfrac {1}{4}}\left(R_{0}+2~R_{45}+R_{90}\right)~.}$

The planar anisotropy coefficient or planar R-value is a measure of the variation of ${\displaystyle R}$ with angle from the rolling direction. This quantity is defined as

${\displaystyle R_{p}={\cfrac {1}{2}}\left(R_{0}-2~R_{45}+R_{90}\right)~.}$

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

In an ordinary cold rolled steel, ${\displaystyle R_{90}}$ is the highest, and ${\displaystyle R_{45}}$ is the lowest. Experience shows that even if ${\displaystyle R_{45}}$ is close to 1, ${\displaystyle R_{0}}$ and ${\displaystyle R_{90}}$ can be quite high leading to a high average value of ${\displaystyle R}$.[2] In such cases, any press-forming process design on the basis of ${\displaystyle R_{45}}$ does not lead to an improvement in deep-drawability.

See also

• Yield surface

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, (ISBN 9026518226), p. 436
3. ISO 10113:2020