Lankford coefficient
# Lankford coefficient

The Lankford coefficient (also called Lankford value, R-value, or plastic strain ratio) 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.

## 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

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

where $\epsilon _{\mathrm {y} }^{p}$ is the in-plane plastic strain, transverse to the loading direction, 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 $R$. In such cases, any press-forming process design on the basis of $R_{45}$ does not lead to an improvement in deep-drawability.

