Limiting Drawing Ratio

The limiting drawing ratio is the answer to the only question that matters before a draw is planned: will this material make that cup in one hit? It is defined by experiment, not by calculation. Draw a series of increasing blanks over the same punch under the same conditions until one tears, and the largest that survives, divided by the punch diameter, is the LDR. A ratio of 2.0 means a 50% reduction in diameter in a single operation, and that is close to the ceiling for anything.

What governs it is the plastic strain ratio $r$, sometimes called the Lankford coefficient after the 1950 Transactions of the ASM paper by W. T. Lankford, S. C. Snyder and J. A. Bauscher that introduced it. Pull a tensile coupon and measure two strains: the width strain and the thickness strain. Their ratio, $r = \varepsilon_w/\varepsilon_t$, says what the sheet would rather do when it is stretched. A material with $r$ well above 1 narrows in preference to thinning. That is precisely the behaviour deep drawing needs, because the flange has to be squeezed inwards — reduced in circumference — without the cup wall thinning away and tearing. Whiteley established the correlation directly in 1960: LDR rises with the average $r$. It is why deep-drawing steels are processed to develop a strong crystallographic texture that gives $r$ values of 1.5 and above, and why most aluminium alloys, with $r$ near or below 1, draw so much less well no matter how good the tooling is.

$r$ is direction-dependent, and that produces two separate numbers. In rolled sheet the value differs along the rolling direction, across it, and at 45° to it. The weighted average $\bar{r} = (r_0 + 2r_{45} + r_{90})/4$ is the normal anisotropy, and it is what correlates with drawability. The difference between directions, $\Delta r = (r_0 - 2r_{45} + r_{90})/2$, is the planar anisotropy, and it is what makes a drawn cup come out with ears on the rim. Those two are independent: a material can be excellent at drawing and still ear badly, which costs trimming rather than parts. A material with a high $\bar{r}$ and a $\Delta r$ near zero is what a deep-drawing operation is really asking for.

There is a theoretical ceiling on all of this. Ideal-work analysis of drawing gives a maximum ratio of $e$ — about 2.72 — for a frictionless, perfectly efficient process with no bending losses at all. Nothing approaches it. Real LDRs top out near 2.2 because friction under the blankholder, friction over the die radius, and the work spent bending the material round that radius and straightening it again all consume part of the pull that the wall has to carry.

And the LDR is not purely a material constant either. It moves with lubrication, with blankholder pressure, with the die profile radius and with the punch nose radius. Too little blankholder force and the flange wrinkles; too much and it will not draw in at all and the wall tears. A published figure is a guide to what a material can do, not a promise about your tooling. Measure your own if the job is worth it, and then design comfortably below it rather than at it, because coil-to-coil variation and a bad day on the lubricant both move the real limit around. A die running at 98% of its material's capability scraps parts.

When the ratio the part needs is beyond what the material offers, the answer is a redraw: reduce the diameter in stages, each within the ratio, with the allowable ratio smaller on the second and third draws than on the first because the material has work-hardened. An interstage anneal restores ductility in materials that harden fast. Dividing the ratios gives the number of stages, and that is what turns a cup into a transfer-die job rather than a single-station one.