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

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

where $\epsilon _{{{\mathrm {xy}}}}^{p}$ is the plastic strain in-plane and $\epsilon _{{{\mathrm {z}}}}^{p}$ is the plastic strain through-the-thickness.

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 as

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

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 $R$. 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 $R$.^[2] 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

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