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