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The '''Lankford coefficient''' (also called Lankford value, R-value, or plastic strain ratio)<ref>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).</ref> is a measure of the [[plasticity (physics)|plastic]] [[anisotropy]] of a [[rolling (metalworking)|rolled]] [[sheet metal]].  This [[Scalar (physics)|scalar]] quantity is used extensively as an indicator of the [[formability]] of [[recrystallization|recrystallized]] [[low-carbon steel]] sheets.<ref name=Mori>Ken-ichiro Mori, ''Simulation of Materials Processing: Theory, Methods and Applications'', (ISBN 9026518226), p.&nbsp;436</ref>
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== Definition ==
If <math>x</math> and <math>y</math> are the coordinate directions in the plane of rolling and <math>z</math> is the thickness direction, then the R-value is given by
:<math>
  R = \cfrac{\epsilon^p_{\mathrm{xy}}}{\epsilon^p_{\mathrm{z}}}
</math>
where <math>\epsilon^p_{\mathrm{xy}}</math> is the [[plastic strain]] in-plane and <math>\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 {{Citation needed|date=July 2009}} .  In practice, the <math>R</math> value is usually measured at 20% elongation in a [[tensile test]].
 
For [[sheet metal]]s, the <math>R</math> values are usually determined for three different directions of loading in-plane (<math>0^{\circ}, 45^{\circ}, 90^{\circ}</math> to the [[rolling direction]]) and the '''normal R-value''' is taken to be the average
:<math>
  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>R</math> with angle from the rolling direction.  This quantity is defined as
:<math>
  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 drawing|deep-drawability]] shows heavy orientation, and such deep-drawability is characterized by <math>R</math>.  However, in the actual press-working, the deep-drawability of steel sheets cannot be determined only by the value of <math>R</math> and the measure of planar anisotropy, <math>R_p</math> is more appropriate.
 
In an ordinary cold rolled steel, <math>R_{90}</math> is the highest, and <math>R_{45}</math> is the lowest. Experience shows that even if <math>R_{45}</math> is close to 1, <math>R_0</math> and <math>R_{90}</math> can be quite high leading to a high average value of <math>R</math>.<ref name=Mori /> In such cases, any press-forming process design on the basis of <math>R_{45}</math> does not lead to an improvement in deep-drawability.
 
== See also ==
* [[Yield surface]]
 
==References==
{{reflist}}
 
{{DEFAULTSORT:Lankford Coefficient}}
[[Category:Plasticity]]
[[Category:Solid mechanics]]
[[Category:Metal forming]]

Latest revision as of 12:52, 2 November 2014

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