Page 42 - MetalForming May 2019
P. 42

  The Science of Forming
By Daniel J. Schaeffler, Ph.D.
Introduction to Strain Analysis: Generating the Thinning Limit Curve from the Forming Limit Curve
The sheet metal blank formed to make your engineered stamping starts with known dimensions and a fixed volume. For a rectangular blank, the volume derives from multiplying the width, pitch and thickness. The dimen- sions change when formed, but the sheet metal volume remains the same.
This approach also applies to discrete sections of the entire blank. Consider a cylindrical region encompassed by a small circle on the surface and descending through the sheet metal thickness. As an example, we know the starting volume if we etch a precise 0.100-in.-dia. circle and measure the thickness using point micrometers or a calibrated ultra- sonic thickness (UT) gauge.
Material movement resulting from the forming process changes the shape of the circle to an ellipse. Forming the sheet redistributes the material in that cylinder but will not change the total volume. Measuring the dimensions of what is now an ellipse on the surface combined with the thickness in the now-formed area confirms this.
The percentage change in a given dimension is defined as the strain in that direction. Major strain, noted as eMa, is determined from the longest dimension of the formed ellipse. Minor strain, emi, is determined from the dimension per- pendicular to the major strain direction.
Forming does not change the volume of the region incor- porated by the surface ellipse, so the percent change in vol- ume equals 0. Thickness strain, et, is calculated from the major strain and minor strain on the surface. The percentages must be converted into decimals.
Equation 1
(eMa +1)*(emi +1)*(et +1)=1
et= 1 -1 (1+eMa )×(1+emi)
Danny Schaeffler, with 30 years of materials and applications experience, is co-founder of 4M Part- ners, LLC and founder and president of Engineering Quality Solutions (EQS). EQS provides product-applica- tions assistance to materials and manufacturing com- panies; 4M teaches fundamentals and practical details of material properties, forming technologies, processes and troubleshooting needed to form high- quality components. Schaeffler, who also spent 10 years at LTV Steel Co., received his Bachelor of Sci- ence degree in Materials Science and Engineering
from the Johns Hopkins University in Baltimore, MD, and Master of Science and Doctor of Philosophy degrees in Materials Engineering from Drexel University in Philadelphia, PA.
Danny Schaeffler
248/66-STEEL •
E-mail or
As an example, assume that forming results in the long dimension of the ellipse increasing in length by 25 percent relative to the initial circle diameter, and the dimension perpendicular to that decreasing by 10 percent. The thickness strain is calculated as -11.1 percent, or a thickness reduction of a little more than 11 percent.
et = Thickness Strain (%) = 1 - 1 (1+0.25) × (1+[-0.10])
= 0.889 -1 = -0.111 = -11.1%
Converting FLC to TLC
The forming limit curve (FLC) is a plot that shows the boundary between neck-free and necked strain conditions, with the major strain plotted on the vertical axis and the minor strain graphed on the horizontal axis.
FLC0 is defined as the lowest point on the FLC and occurs at zero minor strain. For low-carbon steel,
Equation 2
FLC0 = (23.3+14.2t) x (n/0.21)
where t is the sheet thickness in millimeters, and n is the n-value. As an example, 38 percent is the calculated value of FLC0 for a 0.8-mm-thick deep-drawn steel having an n- value of 0.23.
We use the relationship between major, minor and thick- ness strain to convert an FLC to a thinning limit curve (TLC). The value of FLC0 represents the major strain at zero minor strain. Substitute these values into equation 1 to calculate the thinning strain at this location:
et= 1 -1 (1+FLC0 )×(1+0)
1 - 1 = 0.725 - 1 = -0.275 = -27.5% (1.38)×(1)
  Thus, the forming process can tolerate a thinning reduc- tion of as much as 27.5 percent in areas with zero minor strain when using this steel. Considering that we started with a thickness of 0.8 mm, thinning down to 0.58 mm can be tolerated before necking begins.
The entire FLC contains values for major–minor strain pairs that can be converted to thickness strain in the same manner. The accompanying graphic on the next page shows the curves generated for 0.8-mm-thick low-carbon steel with an n-value of 0.23. Any major-minor strain point on the blue FLC has a corresponding thickness-minor strain point on the TLC.
Although this example highlights a low-carbon steel, the steps to convert from an FLC to a TLC can be applied to any sheet metal where the FLC has been determined.
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