| name | metal-forming |
| description | Metal forming — forming limit diagram (FLD), springback, deep drawing, rolling, extrusion, forging, sheet metal bending, blank development, press tonnage. |
| metadata | {"priority":7,"promptSignals":{"phrases":["metal forming","forming limit","deep drawing","springback","sheet metal","bending","rolling","extrusion","forging","blank"],"minScore":3}} |
Metal Forming — Complete Skill
Stress and Strain in Forming
True vs. Engineering Quantities
True stress: σ_true = F/A_instantaneous = σ_eng × (1 + e_eng)
True strain: ε_true = ln(L/L₀) = ln(1 + e_eng)
Volume conservation: ε₁ + ε₂ + ε₃ = 0 (incompressibility)
Flow stress (power law):
σ_f = K × ε^n (Hollomon equation)
K = strength coefficient [MPa], n = strain hardening exponent
n values: aluminum ~0.2, low carbon steel ~0.22, SS 304 ~0.45
Yield Criteria in Forming
Von Mises: σ̄ = √[(σ₁-σ₂)² + (σ₂-σ₃)² + (σ₃-σ₁)²] / √2 = σ_f
Tresca: (σ₁-σ₃)/2 = σ_f/2 (simpler, conservative)
Anisotropy (normal) ratio:
R = ε_width / ε_thickness (from tension test at 45°)
R > 1: material resists thinning (good for deep drawing)
R̄ = (R₀ + 2R₄₅ + R₉₀) / 4 (average R — higher = better drawability)
Sheet Metal Bending
Bend Allowance
BA = α × (r + k×t) [length of neutral fiber]
α = bend angle [rad]
r = inner bend radius [mm]
t = sheet thickness [mm]
k = neutral axis factor (k=0.5 for r/t < 1; k → 0.33 for tight bends)
K-factor by r/t:
r/t ≤ 1: k = 0.33 (neutral axis shifts inward)
1 < r/t ≤ 3: k = 0.40
r/t > 3: k = 0.50
Blank length:
L_blank = Σ straight segments + Σ bend allowances
Minimum bend radius:
r_min ≈ t/2 × (1/n - 1) [from FLC theory — conservative]
Practical: aluminum r_min = t; steel r_min = 0.5t to 2t depending on temper
Springback
Springback angle: Δα = α_loaded - α_released
K_s = r_i/r_f = 1 - 3(σ_y/E)(r_i/t) + 4(σ_y/E)³(r_i/t)³
r_i = loaded radius, r_f = released radius
Springback factor K_s: K_s < 1 (radius increases after release)
For high-strength steel: large springback — overbend or coin at corners
Springback = k × (S_y/E) × (r_i/t) — higher strength → more springback
Deep Drawing
Draw Ratio and Formability
Drawing Ratio (DR) = D_blank / D_punch
Limiting Draw Ratio (LDR) ≈ 2.0 (typical steel), up to 2.3 for high R̄ material
Blank diameter for cylindrical cup:
D_blank = √(d_p² + 4 d_p h) (approximate, no flange)
D_blank = √(d_f² + 4 d_p h) (with flange diameter d_f)
Drawing force:
F_draw = π × d_p × t × S_u × (D_blank/d_p - 0.7)
d_p = punch diameter, S_u = tensile strength
Blank holder force:
F_bh = (A_blank - A_punch) × q_bh
q_bh = 1-3 MPa (too low → wrinkling; too high → tearing)
Ironing: wall thinning after drawing; clearance < t → ironing occurs (cans)
Earing
Caused by planar anisotropy ΔR = (R₀ - 2R₄₅ + R₉₀)/2
ΔR = 0: no earing (isotropic); ΔR ≠ 0: ears at 0°/90° or 45°
Forming Limit Diagram (FLD)
FLC (Forming Limit Curve)
Major strain ε₁ vs. minor strain ε₂ — boundary between safe forming and failure
Below FLC: safe forming
Above FLC: necking/fracture
Keeler-Goodwin FLC (approximate):
FLC₀ (plane strain, ε₂=0): FLC₀ ≈ (23.3 + 14.13t) × n/0.21 [%]
t in mm, n = strain hardening exponent
Regions:
Left (negative ε₂): compression-tension (bending, drawing)
Right (positive ε₂): biaxial tension (bulging, stretch forming)
Worst condition: plane strain (ε₂ = 0) — lowest FLC₀
Forming severity:
Safety margin = FLC₀ - ε₁,max in part > 10% (minimum)
Marginal zone: 5-10%; Fail zone: ε₁ > FLC
Rolling
Rolling Basics
Draft: d = h₀ - h₁ [mm]
True strain: ε = ln(h₀/h₁)
Rolling force: F = L × w × σ_f,avg × Q_f
L = √(R×d) = contact length [m]
w = strip width [m]
Q_f = pressure multiplier (friction correction) ≈ 1.0-2.0
Roll torque per roll:
T = F × L / 2
Power: P = 2πNT [W]
Maximum draft without slip:
d_max = μ² × R (μ = friction coefficient, R = roll radius)
Minimum thickness (Hitchcock formula):
h_min = C' × F (roll flattening limit; C' = 16(1-ν²)/(πE))
Practical: ~0.05mm for cold rolling of steel
Forging
Forging Force (Open Die, Flat Plates)
F = σ_f × A × (1 + 2μr/(3h)) (Siebel formula)
σ_f = flow stress at current strain and strain rate
A = instantaneous contact area
μ = friction coefficient (die/workpiece)
r = workpiece radius, h = height
Closed die forging:
F = K_f × σ_f × A_projected
K_f = 3-10 (shape complexity, thin ribs = high)
Forgeability: aluminum > copper > carbon steel > alloy steel > titanium > stainless
Draft angle: 5-7° (steel), 3-5° (aluminum) for die release
Flash: excess material squeezed into gutter — increases pressure, fills die
Extrusion
Extrusion Ratio and Force
Extrusion ratio: R_e = A₀/A_f (area of billet / area of extrudate)
True strain: ε = ln(R_e)
Force (direct extrusion):
F = A₀ × σ_f × (ln(R_e) + C_container + C_die)
C_container = friction in container (direct extrusion penalty)
C_die = die friction (α = 45° optimal die angle minimizes force)
Indirect vs. direct extrusion:
Direct: container friction adds to force; simple tooling
Indirect: no container friction; lower force (15-30% less)
Maximum extrusion ratio: steel R_e < 40; aluminum R_e < 100 (heat limits)
Extrusion speed: aluminum up to 5 m/min; steel < 0.5 m/min
Press Selection
Tonnage Requirements
Blanking: F = L × t × S_u × (0.6-1.0) [trim: 0.6 factor]
Bending: F = (k × L × t² × S_u) / W (k=1.33 V-bend, k=0.33 U-bend)
Drawing: use F_draw formula above
Press type selection:
Mechanical (crank/eccentric): high speed, fixed stroke — blanking, bending
Hydraulic: variable stroke, full tonnage at any point — deep drawing, forming
Servo: programmable motion — complex forming, springback control
Clearance for blanking:
c = 5-10% × t (soft materials); c = 10-15% × t (hard materials)
Too small → fracture; Too large → burr
Material Formability Comparison
| Material | n | R̄ | LDR | Notes |
|---|
| DC04 (deep drawing steel) | 0.22 | 1.6-1.8 | 2.1 | Standard automotive |
| IF steel | 0.24 | 2.0-2.5 | 2.3 | Ultra-deep drawing |
| DP 600 (AHSS) | 0.15 | 0.9 | 1.9 | High strength, springback |
| 5052 Al | 0.20 | 0.7 | 1.8 | Common sheet Al |
| 6061-T6 Al | 0.08 | 0.7 | 1.6 | Low formability when hardened |
| Ti-6Al-4V | 0.02 | 3.0 | — | Warm/hot forming required |
Output
Provide: blank size [mm], bend allowance [mm], springback angle [°], FLC safety margin [%], draw ratio vs. LDR, forming force F [kN], press tonnage required [ton], pass schedule (rolling/drawing).
