Physics:Lankford coefficient

Short description: Measure of the plastic anisotropy of a rolled sheet metal

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

[math]\displaystyle{ R = \cfrac{\epsilon^p_{\mathrm{y}}}{\epsilon^p_{\mathrm{z}}} }[/math]

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

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

[math]\displaystyle{ R = \cfrac{1}{4}\left(R_0 + 2~R_{45} + R_{90}\right) ~. }[/math]

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

[math]\displaystyle{ R_p = \cfrac{1}{2}\left(R_0 - 2~R_{45} + R_{90}\right) ~. }[/math]

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

In an ordinary cold rolled steel, [math]\displaystyle{ R_{90} }[/math] is the highest, and [math]\displaystyle{ R_{45} }[/math] is the lowest. Experience shows that even if [math]\displaystyle{ R_{45} }[/math] is close to 1, [math]\displaystyle{ R_0 }[/math] and [math]\displaystyle{ R_{90} }[/math] can be quite high leading to a high average value of [math]\displaystyle{ R }[/math]. In such cases, any press-forming process design on the basis of [math]\displaystyle{ R_{45} }[/math] does not lead to an improvement in deep-drawability.

See also

• Yield surface