# The Science of Forming

## Intro to Strain Analysis: Techniques to Promote Measurement Accuracy

The metalforming industry lost a vocal advocate when Dr. Stuart Keeler passed away this past May. The Science of Forming was not just the name of the column he wrote for almost 20 years, (from 1999 through 2017), but a description of his many contributions to the field. Whether lecturing in a classroom setting or next to a stamping press in a tool shop, Stu highlighted the importance of using data in a field previously dominated by art. His leadership, passion and guidance will be missed. |

FLC_{0} is defined as the lowest point on the Forming Limit Curve and occurs at 0 minor strain. The Keeler equation for low-carbon steel can be used:

**Equation 1**

FLC_{0} = (23.3+14.2t) x (n/0.21)

where t represents sheet thickness (mm), and n denotes the n-value. In the example of a 0.8-mm-thick drawing steel with an n-value of 0.18, this equation determines FLC_{0} as 30 percent. Standard practice is to drop this value by 10 percent to account for the marginal zone, so we target a maximum value on the major strain axis of 20 percent.

This value must be put into perspective. In circle grid strain analysis, a flat metal sheet containing a grid pattern of 0.100-in.-dia. circles is formed and the circles flow into ellipses. Should a maximum major-strain value of 20 percent be allowed, then the locations that measure 0.120 in. on the long axis rest on the boundary of the safe/marginal zone. For this article, assume a minor strain of 0 percent, or plane strain, to make the math easier.

Compare that against an ellipse measuring 0.122 in., which falls into the marginal zone with a safety margin of 8 percent. An ellipse measuring 0.118 in. is graphed in the safe zone with a safety margin of 12 percent. This difference of 0.002 in. in either direction, assuming accurate measurements, makes the difference between tooling buyoff and tooling work.

The tools used to create the circles on the flat sheet and measure the ellipses on the formed part contribute to measurement accuracy. The circles typically are applied by electrochemical etching, which requires using a stencil of the correct grid pattern. The width of the line that forms the boundary of each circle measures about 0.008 in. A translucent Mylar strip calibrated to measure the expansion percentage of each circle also contains lines (the diverging railroad tracks) measuring about 0.008 in. thick.

Ellipse measurements using calibrated Mylar strips must be made from the center-width position. The width difference between the ellipse circumference and the diverging ‘railroad tracks’ in this image is exaggerated for clarity—in reality, they are of similar dimension. |

In the example targeting a 20-percent critical major-strain axis, measuring inside-to-inside (the middle example in the figure), the major strain axis reads as 0.112 in. (0.120 in. - 0.008 in.), which is +12 percent on the major-strain axis. Similarly, the minor-strain axis is measured as 0.092 in. (0.100 in. - 0.008 in.), which is -8 percent on the minor strain axis. On the other hand, a measurement of outside-to-outside dimensions (the righthand image in the figure), results in +28 percent on the major-strain axis and +8 percent on the minor axis.

In these examples, the boundary between the safe and marginal zone occurs at a data point of (+20 percent, 0 percent). Accurate measurements with an incorrect technique can lead to data points ranging from (+12 percent, -8 percent) to (+28 percent, +8 percent), and provides a false picture of the true situation.

Confirming Strain Measurements

Circle grid strain measurements commonly are made visually, occasionally with 5x to 10x magnifying lenses. Errors are easy to make, considering the fine lines that comprise the circle boundary and the calibrated measurement strip. Parallax effects and lighting add other challenges. Measurements must be as accurate as possible, as a difference of a few thousandths of an inch has huge implications.

A flat sheet metal blank has known dimensions of length, width and thickness. Formability is not exempt from the laws of physics, so the total amount of material in a region must maintain a constant volume before, during and after forming, even though the specific values of length, width and thickness will change due to forming. Thickness strain, e_{t}, is calculated from the major strain (e_{Ma}) and minor strain (e_{mi}) on the surface. The percentages must be converted into decimals.

**Equation 2**

(e_{Ma} +1) * (e_{mi} +1) * (e_{t} +1) = 1

or

e_{t} = 1 / (1+*e*_{Ma} )×(1+*e*_{mi}) - 1

Another way to calculate thickness strain uses the known starting thickness of the flat blank and the formed panel thickness that varies with location across the stamping. Sections containing critical areas can be cut from the formed panel and measured with pin micrometers, but a more practical solution employs an ultrasonic thickness (UT) gauge, which avoids damaging the stamping. With a properly calibrated UT gauge, check the formed-panel thickness at each location containing measured ellipses. The thickness strain is calculated as:

**Equation 3**

Thinning strain (%) = e_{t}

= *(formed part thickness-initial blank thickness) / (initial blank thickness)* * 100%

Compare the thinning-strain measurements from the techniques used to generate Equation 2 and Equation 3. They should match. Any discrepancy likely results from either the surface major- or minor-strain measurement, as a properly calibrated UT gauge can measure to the nearest micron (0.001 mm) of thickness.

Calibrating Ultrasonic Thickness Gauge

Recalibrate the UT gauge for each significant change in the type of material being measured. Significant changes include changing the sheet metal grade (mild to high-strength steel, 400-series stainless to 300 series, etc.), switching coatings (uncoated, galvanized, galvannealed, electrogalvanized, aluminized, etc.), a change in sheet thickness of more than 0.01 in., or a UT gauge that has been turned off for a while. Do not calibrate using the flat metal button on some UT gauges, since the button was designed for a different purpose. Do not use the minimum ordered thickness shown in the metal certified properties, as it will differ from the sheet being evaluated.

Calibrate against identical metal in a flat area. The blanks on a lift or throughout the coil may have slight variations in thickness (a normal occurrence that should be expected), so use the same flat blank that, ultimately, will be used to form the circle grid strain analysis stamping. Proper calibration takes only a few minutes and is vital to ensure accurate measurements and correct interpretation. MF

See also: 4M Partners, LLC, Engineering Quality Solutions, Inc.

Related Enterprise Zones: Materials/Coatings

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