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- Generating Forming Limit Curves from Equations
- Experimental Determination of Forming Limit Curves
- Differences between Forming Limit Curves and Forming Limit Diagrams
If all stampings looked like a tensile dogbone and all deformation was in uniaxial tension, then a tensile test would be sufficient to characterize the formability of that metal. Obviously, engineered stampings are much more complex. Although a tensile test characterizes one specific strain path, a Forming Limit Curve (FLC) is necessary to have a map of strains indicating the onset of critical through-thickness necking for different linear strain paths. The strains which make up the FLC represent the limit of useful deformation. Calculations of safety margins are based on the FLC (Figure 1).
Figure 1: General graphical form of the Forming Limit Curve.E-2
Picturing a blank covered with circles helps visualize strain paths. After forming, the circles turn into ellipses, with the dimensions related to the major and minor strains. This forms the basis for Circle Grid Strain Analysis.
Generating Forming Limit Curves from Equations
Pioneering work by Keeler, Brazier, Goodwin and others contributed to the initial understanding of the shape of the Forming Limit Curve, with the classic equation for the lowest point on the FLC (termed FLC0) based on sheet thickness and n-value. Dr. Stuart Keeler was the Technical Editor of these AHSS Guidelines through Version 6.0, released in 2017.
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Equation 1 |
These studies generated the left hand side of the FLC as a line of constant thinning in true strain space, while the right hand side has a slope of +0.6, at least through minor (engineering) strains of 20%.
Evidence accumulated over many decades show that this approach to defining the Forming Limit Curve is sufficient for many applications of mild steels, conventional high strength steels, and some lower strength AHSS grades like CR340Y/590T-DP. However, this basic method is insufficient when it comes to creating the Forming Limit Curves for most AHSS grades and every other sheet metal alloy. Grades with significant amounts of retained austenite experience significant deviations from these simple estimates.H-23, S-61 In these cases, the FLC must be experimentally determined.
Additional studies found correlation with other properties including total elongation, tensile strength, and r-valueR-8, P-19, G-23, A-45, A-46, H-23 with some of these attempting to define the FLC by equations.
Experimental Determination of Forming Limit Curves
ASTM A-47 and ISO I-16 have published standards covering the creation of FLCs. Even within these standards, there are many nuances left for interpretation, primarily related to the precise definition of when a neck occurs and the associated limiting strains.
There are two steps in creating FLCs: Forming the samples and measuring the strains.
Forming sheet specimens of different widths uses either a hemispherical dome (Nakajima or Nakazima method N-14) or a flat-bottom punch (Marciniak method M-22) to generate different strain paths from which critical strains are determined. The two methods are not identical due to the different strain paths generated from the punch shape. These differences may not be significant for many lower strength and conventional High-Strength Steels, but may deviate from each other at higher strengths or with advanced microstructures. Figure 2 highlights samples formed with the Nakajima method.
Figure 2: Forming limit curves can be created by deforming multiple samples of different widths. Narrow strips on the left allow metal to flow in from the unconstrained edges, creating a draw deformation mode leading to strains that plot on the left side of the FLC. Fully constrained samples, shown on the rightE-2, create a stretch deformation mode leading to strains that plot on the right side of the FLC.
Generally, strains are measured using one of two methods. The first approach involves covering the initially flat test samples with a grid pattern of circles, squares, or dots of known diameter and spacing to measure the strains associated with deformation. An alternate approach is based on Digital Image Correlation (DIC), where a camera tracks the movement of a random speckle pattern applied prior to forming.W-26, H-22, M-21 DIC methods are directly suited for use with stress-based FLCs.
Differences between Forming Limit Curves
and Forming Limit Diagrams
Often, the terms Forming Limit Curve (FLC) and Forming Limit Diagram (FLD) are used interchangeably. Perhaps a better way of categorizing is to define the components and what they encompass. This is somewhat of a simplification, since detailed interactions are known to occur.
- The Forming Limit Curve is a material parameter reflecting of the limiting strains resulting in necking failure as a function of strain path. It is a function of the metal grade, thickness, and sheet surface conditions, as well as the methods used during its creation (hemispherical/flat punch, test speed, temperature). It is applicable to any part shape.
- Deforming a flat sheet into an engineered stamping results in a formed surface with strains as a function of the forming conditions like local radii, lubrication, friction, and of course part geometry. These strains are essentially independent of the chosen metal grade and thickness. Plotting these strains allows for a relative assessment of which strains are higher than others, but no judgment can be made on “how high is too high?”
- The Forming Limit Diagram is a combination of the Forming Limit Curve (a material property) and the strains (reflecting part geometry and forming conditions). The FLD provides guidance on which areas of the formed part requires additional attention to achieve robust stamping conditions. Creating a subsequent FLD may be warranted when conditions change, since changes to the sheet metal properties (FLC) and the forming conditions (radii, lubrication, beads, blank size) will change the FLD, potentially affecting the conclusions.
Key Points
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- Conventional Forming Limit Curves characterize necking failure only. Fractures at cut edges and tight bends may occur at strains lower than that suggested by the Forming Limit Curve.
- Differences in determination and interpretation of FLCs exist in different regions of the world.
- This system of FLCs commonly used for low strength and conventional HSLA is generally applicable to experimental FLCs obtained for DP steels for global formability.
- The left side of the FLC (negative minor strains) is in good agreement with experimental data for DP and TRIP steels. The left side depicts a constant thinning strain as a forming limit.
- Determination of FLCs for TRIP, MS, TWIP, and other special steels typically requires an experimental approach, since conventional simple equations do not accurately reflect the forming limits for these advanced microstructures.
Hello,
What would be the formula used on a HRS with .3″ gauge? I have learned that dependency on the gauge should diminish after reaching .21″.
Regards,
Thanks for your comment Marko.
You mention 2 thicknesses, and for the sake of our international audience, 0.21″ = 5.3 mm, and 0.3″ = 7.6 mm.
You are correct that there is generally thought to be a drop off in the gauge dependency in Equation 1 as the thickness increases. I believe I have heard it starts even thinner than that – closer to 3mm. But I do not remember scientific studies which documented this for thicker mild steels.
The reduced thickness dependency makes sense to me since the FLC is based on necking failure only, and as you get thicker, it is harder to get a true necking-only failure. Bend effects become more severe, so the parts where necking is the operative failure mode become increasingly more shallow.
Also remember the history of how Equation 1 came about. Many samples and many grades and many thicknesses were tested and measured. Equation 1 represented best-fit curves compiled for all the grades as measured from the analytical techniques available 50 years ago. For that reason, even under the best of situations, Equation 1 should just be viewed as an approximation.
– Danny Schaeffler
Engineering Quality Solutions, Inc. and Technical Editor, AHSS Application Guidelines
https://www.EQSgroup.com and https://AHSSinsights.org
Thanks a lot for the elaborate explanation!
Hello Marko,
Thank you very much for raising the question. I saw Danny has already answered, but I wanted to add a few more points:
0.21″ = 5.3 mm is already over the defined maximum thickness for FLC determination. In ISO 12004-2:2008, the maximum thickness for experimental determination FLC is 4.0 mm (appr. 0.16″). For steel sheets the maximum value is 2.5 mm (appr. 0.10″).
If you are working with materials over these thicknesses, typical for hot rolled steels, I would suggest you to use TFC (Triaxial Failure Curve). I am hoping to author a short article about this in the next couple of months, and I will share the link here (if it is not against the rules).
I hope this clarifies your questions.
Dr. Eren Billur
Billur Metal Form Ltd., Ankara, TR
http://www.billur.com.tr
Thank you Mr. Billie.
How do we apply TFC with limited data on a material grade. My concern is when performing evaluations on traditional grades with no established baselines that a stamper may run production on a particular part but no one has provided a formability assessment to asses risks associated with the stamping process. From my experience with heavier gauges, failure mode is typically related to what we consider a local formability function which applies a different approach than the traditional FLD.
Thanks, MC
Marko-
I appreciate your willingness to engage in this forum. I believe this type of discussion is beneficial to our global audience.
You state a very real concern – what to do when faced with limited information about a given grade.
My experience is consistent with yours: heavy gauges fail more frequently by bending-related issues, which is what is thought of as a local formability mode. Remember to subtract out the curvature effects from any calculation.
Since the FLC as generated from Equation 1 is an estimate/approximation only, risk management should guide you to use a larger safety margin – maybe 15% instead of 10%.
Another possibility is to calibrate the FLC by testing a tensile bar using DIC. Measure the strain state and determine the strains when a neck initiates. That becomes one data point on the FLC. If it is mild steel, you can assume constant thinning on the left hand side of the FLC. Approximate the curve shape in a conventional manner, which will set the remainder of the curve accordingly.
– Danny Schaeffler
Engineering Quality Solutions, Inc. and Technical Editor, AHSS Application Guidelines
https://www.EQSgroup.com and https://AHSSinsights.org
Thanks for your reply Mr. Schaeffler.
This is a very good subject which many in the industry find confusing enough to be afraid to discuss. In my case, I love to engage and learn something new whenever possible.
There is an ancient cheat sheet I use for heavier gauges; however, as most thing in life, practices tend to morph and adjust to new findings. When evaluating formability, my approach will adopt according to grade designation. AHSS generally don’t follow traditional Keeler curve (KC) but if empirical data yields similar KC, I just go with it and use 30% proportional marginal zone for all HSS grades above 440TS. I usually cannot go wrong using a thinning strain analysis for conservative measurements whenever I hit a wall. My gray zone is usually very thick gauges and as we agreed, they usually tend to fail in local formability mode.
I had a privilege in the recent past to review one of your PP presentations on steel and formability, it was a very interesting read for me and I hope to hear from you in the future.
Thanks for the opportunity to discuss this matter with you and all others willing to engage on this platform!
Dear Eren,
we would be most happy to include your new article when it’s ready, whether with a link, a blog or an update to this article–as you are able! Thank you.