Deep Drawing Forming Simulation
Theory: what limits deep drawing
Overview
Why can't a deep cup be drawn in one stroke?
Because all the force runs through one path. The punch pushes the blank centre and pulls the outer flange into the die; the force needed to draw in and compress the flange passes entirely through the cup wall to the punch radius. A bigger blank needs more drawing force, but what the wall can carry is fixed by thickness and strength, so beyond a certain size the wall splits near the punch radius before the flange is drawn in. That limit, expressed as blank diameter over cup diameter, is the limiting draw ratio — about 2.0 for mild steel. The other problem is wrinkling: the flange shrinks circumferentially and buckles unless held down. More blank holder force stops wrinkles but adds friction and pushes toward splitting. Deep drawing design and simulation are essentially a search for the window between splitting and wrinkling.
Blank diameter, draw ratio and drawing force
d is cup diameter, h height, t sheet thickness and σ_B tensile strength. The blank diameter follows from constant surface area (corner radii neglected); peak drawing force is an empirical estimate.
The technology of pots and cartridge cases
Deep drawing spread in the 19th century for making metal pots and tableware, and by the early 20th century multi-stage drawing with intermediate annealing was established to mass-produce rifle cartridge cases. The limiting draw ratio and the finding that sheets with higher plastic anisotropy (r-value) draw deeper were systematised in the mid-20th century, and today's car-body stamping simulation rests on that theory.
Worked examples
Example: 50 mm diameter, 40 mm tall cylindrical cup (1.0 mm mild steel, tensile strength 300 MPa assumed)
| Item | Single draw | Two stages (first draw) |
|---|---|---|
| Blank diameter | 102.5 mm | 102.5 mm |
| Drawn diameter | 50 mm | 56.9 mm |
| Draw ratio | 2.05 (above the ~2.0 limit) | 1.80 |
| Cup height | 40 mm | 31.9 mm |
| Peak drawing force (empirical) | 63.6 kN | 59.0 kN |
| Redraw ratio (second stage) | — | 1.14 |
Blank holder force: 2.5 MPa on the initial 6,123 mm² flange gives about 15.3 kN. A single draw slightly exceeds the limit, so the first draw is held to 1.8 and the second redraws to 50 mm.
Solving the single draw in FEM, the cup formed without splitting. Why does it disagree with the hand calculation?
Most likely the FEM is not judging fracture at all. Elasto-plastic FEM simply keeps stretching elements as the wall thins; it does not crack on its own. Splitting must be judged with a forming limit diagram (FLD): an element whose major and minor strain pair crosses the forming limit curve is taken as split. For mild steel, thinning of more than 25–30% near the punch radius usually means the FLD danger zone. The other factor is the material model: isotropic von Mises omits the effect of the r-value (width-to-thickness strain ratio) and misjudges drawability. Deep-drawing steels have r-values of about 1.5–2; without an anisotropic yield function such as Hill48 and the r-values, split margins and earing come out wrong. Friction (about 0.05–0.15 with lubrication) also changes results greatly, so settle FLD judgement, anisotropy and friction before drawing conclusions.
Practical workflow
- Compute blank diameter and draw ratio; decide the number of stages and each stage's ratio.
- Prepare the stress-strain curve, r-values (0°, 45°, 90°) and forming limit curve.
- Model the sheet with shell elements and the punch, die and blank holder as rigid bodies with contact and friction.
- Vary blank holder force to find the window between wrinkling and splitting.
- Compare with trial thickness distribution and ear height; tune friction and material data.
“Why four ears appear”
The rim of a cylindrical cup may not be level, rising at four places. These ears come from planar anisotropy — different r-values at 0°, 45° and 90° to the rolling direction. Beyond the average r, if Δr = (r₀ + r₉₀ − 2r₄₅)/2 is positive, ears form at 0° and 90°; if negative, at 45°. Since the ears must be trimmed, simulation with anisotropy also helps estimate how much larger the blank should be.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| No FLD judgement | Splits missed | Judge with major/minor strain |
| Isotropic yield function | Drawability and earing wrong | Hill48 etc. with r-values |
| Single guessed blank holder force | Wrinkles or splits | Search the window |
| Too few through-thickness integration points | Bending and springback errors | 5–7 or more |
| Combining stages into one | Exceeding the limiting ratio | Split stages by draw ratio |
I'd like to learn related topics.
Related pages include sheet metal press forming simulation, springback analysis, forging simulation and the Coulomb friction model.
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