Forging Simulation
Theory: what sets the forging load
Overview
What does a forging simulation predict?
Mainly four things: whether material flows into every corner of the die, the press load and energy needed, whether defects such as underfill or laps (material folding over itself) appear, and the strain and temperature distributions in the part together with die stresses. Because the billet is squashed by tens of percent, the problem is solved as large deformation with rigid-plastic or elasto-plastic FEM, normally remeshing many times as elements distort. Results hinge on three inputs: the flow stress (a function of strain, strain rate and temperature), die friction, and heat loss to dies and air. That is why it pays to check the load's order of magnitude by hand before running FEM.
Upsetting load (slab method)
r₀ and h₀ are the initial radius and height, σ_f the flow stress, μ the Coulomb friction coefficient, p̄ the mean die pressure and F the load. Volume constancy fixes the radius; friction raises pressure toward the centre and so the mean pressure.
Strain rate and deformation heating
v is tool speed, m the strain-rate sensitivity, β the fraction of plastic work turned into heat (about 0.9) and ρc the volumetric heat capacity.
From the swordsmith to finite elements
Forging is among the oldest metalworking methods, but calculating its load only matured in the 20th century. In the 1920s–30s Sachs and Siebel developed the slab method, slicing the billet into thin slabs and balancing forces; the upper-bound method followed. In the 1970s rigid-plastic finite elements became practical, and work by Kobayashi and others led to today's software that predicts material flow from die geometry.
Worked examples
Example 1: cylinder upsetting load (50 mm × 50 mm, flow stress assumed 100 MPa)
| Height | Diameter | True strain | No friction | μ = 0.1 | μ = 0.2 | μ = 0.3 |
|---|---|---|---|---|---|---|
| 40 mm | 55.9 mm | 0.22 | 245 kN | 257 kN | 268 kN | 280 kN |
| 30 mm | 64.5 mm | 0.51 | 327 kN | 351 kN | 374 kN | 398 kN |
| 20 mm | 79.1 mm | 0.92 | 491 kN | 556 kN | 620 kN | 685 kN |
As the billet flattens, area grows and so does the load; the rising diameter-to-height ratio also enlarges the friction penalty — 40% above frictionless at 20 mm with μ = 0.3. Hot flow stress is taken as constant, since it changes little with strain.
Solving the same upsetting in FEM with μ = 0.3, my load came out below the hand calculation. Which is right?
Check the friction cap. With Coulomb friction the friction stress μp grows with pressure, but real friction stress cannot exceed the billet's shear yield stress (about 0.58 times flow stress). Near the centre in this example pressure reaches about twice the flow stress, so at μ = 0.3 μp reaches shear yield and the material sticks to the die. Most forging codes cap friction or use a friction-factor model (friction stress = m × shear yield), so a load below the simple slab formula is expected. Hot forging commonly uses m = 0.3–0.7 and lubricated cold forging μ = 0.05–0.1. Measure the value with a ring compression test — discussing loads without settling it is meaningless.
Example 2: press vs hammer strain rate
| Equipment | Tool speed | Strain rate (h = 20 mm) | Flow stress (vs press) |
|---|---|---|---|
| Hydraulic press | 50 mm/s | 2.5 /s | 1.0 |
| Hammer | 5 m/s | 250 /s | 2.0 |
With strain-rate sensitivity m = 0.15 for hot steel, a 100-fold strain rate raises flow stress by 100^0.15 ≈ 2.0. A hammer's short contact time loses less heat to the dies, but the higher flow stress shows up in the load.
Example 3: deformation heating
| Condition | Flow stress | Strain | Temperature rise (adiabatic) |
|---|---|---|---|
| Hot forging | 100 MPa | 0.92 | 16 K |
| Cold forging | 600 MPa (mean) | 0.92 | 98 K |
In cold forging the billet heats by nearly 100 K, lowering flow stress and degrading the lubricant film. In hot forging the heating is small, but surfaces touching cooler dies chill quickly and stiffen, causing underfill. Both call for coupled thermo-mechanical analysis.
Practical workflow
- Prepare flow stress as a function of strain, strain rate and temperature (compression tests or databases).
- Fix friction conditions, for example with a ring compression test.
- Check the load's magnitude with the slab method and compare with equipment capacity.
- Check flow and defects with axisymmetric or symmetric 2D models before moving to 3D.
- Compare with trial parts' dimensions, loads and sectioned metal flow, and tune friction and heat transfer.
“Laps show only when you section the part”
Laps (folds) in closed-die forging often show only as a fine surface line, and their extent is clear only after sectioning and etching. Because they sharply reduce fatigue strength, finding them late can mean reworking the dies. Tracking material flow in simulation and checking whether surface elements are being drawn inward is one of the most effective ways to remove defects before trials.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Room-temperature flow stress for hot forging | Load overestimated several-fold | Temperature- and rate-dependent data |
| Friction guessed, not measured | Load and filling errors | Ring compression test |
| Ignoring heat transfer to dies | Missed underfill | Coupled thermo-mechanical analysis |
| Coarse remeshing criteria | Volume loss, missed laps | Monitor volume change |
| Analysing a hammer at press speed | Flow stress underestimated | Apply the real velocity history |
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