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

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

where $\epsilon^p_{\mathrm{xy}}$ is the plastic strain in-plane and $\epsilon^p_{\mathrm{z}}$ 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