deep-drawing

name	deep-drawing
description	Deep drawing (cup drawing) — blank size calculation, drawing ratio (LDR/DR), blank holder force, punch force prediction, ironing, earing and anisotropy (r-bar and Δr values), forming limit diagram (FLD), sheet metal formability (Erichsen test, n-value, r-value), wrinkling and tearing failure modes, multi-stage drawing sequences, and ISO/ASTM standards for sheet metal formability testing.
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Deep Drawing — Complete Skill

Fundamentals of Deep Drawing

Process Description

Deep drawing: sheet metal blank drawn into a die by punch to form hollow cup/shell Distinguished from stamping/blanking: material flows plastically from flange into cup wall (not fracture-based) Material undergoes: radial tension (punch-to-flange direction) + circumferential compression (hoop) in flange → thickening tendency → blank holder prevents wrinkling

Key components: Punch: applies axial load; diameter = cup inner diameter Die: cup outer diameter; die radius r_d (must not be too sharp → tearing) Blank holder: applies force to flange to prevent wrinkling Blank: circular sheet of diameter D₀; thickness t₀

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Blank Size Calculation

Area Conservation Method

Volume (area) conservation: A_blank = A_cup_surface [total surface area of finished cup = blank area] For cylindrical cup (inner diameter d, height h, bottom radius r_b): A_blank = A_bottom + A_wall + A_flange_if_any Simple cylinder with flat bottom: π×D₀²/4 = π×d²/4 + π×d×h [ignoring corner radii] → D₀ = √(d² + 4dh) [approximate; no corner radius]

With bottom radius r_d: More precise blank diameter includes surface area of the knuckle radius: A_knuckle ≈ 2π × r_d × (d/2 − r_d) [add for non-negligible knuckle radius]

Example: Cup: inner d = 80 mm; height h = 60 mm; t₀ = 1.5 mm (thin; area conservation valid) D₀ = √(80² + 4×80×60) = √(6400 + 19200) = √25600 = 160 mm

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Drawing Ratio and Limiting Drawing Ratio

Drawing Ratio (DR)

Drawing ratio: DR = D₀ / d_punch [D₀ = blank diameter; d_punch = punch diameter; ratio must be ≤ LDR to draw without failure]

Limiting Drawing Ratio (LDR): Maximum DR before fracture in cup wall Theoretically: LDR = exp(1) = 2.72 for ideal isotropic material with uniform elongation Practically: LDR depends on material properties, specifically normal anisotropy r̄ LDR ≈ exp(√(r̄/(1+r̄)))^(correction) [approximate; from energy balance] More accurate empirical correlation: LDR = 1.8 to 2.2 for mild steel; 1.6–1.8 for aluminum; up to 2.4 for IF steel

r-value (Lankford parameter, plastic strain ratio): r = ε_width / ε_thickness [measured in tensile test; r > 1 means material resists thinning] r̄ = (r₀ + 2r₄₅ + r₉₀) / 4 [average r-value; higher r̄ → better deep drawability → higher LDR] Normal anisotropy: r̄ > 1 means material thickens preferentially rather than thinning → more material can be drawn without wall fracture

Typical r̄ values: IF (interstitial-free) steel: r̄ = 1.8–2.3 (excellent drawability) SPCD (JIS) drawing steel: r̄ = 1.4–1.9 1100-O aluminum: r̄ = 0.6–0.8 (poor drawability) Brass 70/30: r̄ = 0.9

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Punch Force Calculation

Maximum Punch Force

Empirical punch force: F_max = π × d_p × t₀ × UTS × (DR − 0.7) [d_p = punch diameter; t₀ = blank thickness; UTS = ultimate tensile strength] Note: (DR − 0.7) correction for friction and bending

More precise (from energy method): F_max = π × d_p × t₀ × σ_flow × ln(DR) × correction_factor σ_flow = average flow stress over the drawing operation correction_factor ≈ 1 + 4μ × h / d_p [μ = friction coefficient 0.05–0.15; h = depth of draw at maximum force]

Example: d_p = 80 mm; t₀ = 1.5 mm; UTS = 400 MPa; DR = 2.0; μ = 0.1 F_max = π × 80 × 1.5 × 400 × (2.0 − 0.7) = 376,991 × 1.3 = 195 kN ≈ 200 kN

Blank Holder Force

Blank holder force (BHF): Too low → wrinkling; too high → excessive friction → wall fracture Empirical: F_BH = 1/3 × F_max [rough guide] Or: F_BH = σ_BH × (A_blank − A_punch) where σ_BH = 1–3 MPa surface pressure (empirical)

Optimum BHF: variable BHF controllers now common — high at start, reduced as flange shrinks

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Forming Limit Diagram (FLD)

FLD and Forming Limit Curve (FLC)

FLD coordinates: x-axis: minor strain ε₂ (in-plane, perpendicular to major) y-axis: major strain ε₁ (in-plane, in direction of maximum strain) FLC: boundary between safe forming region and fracture/necking zone Points above FLC → fracture; below → safe

Keeler-Goodwin FLC approximation: FLC major strain at plane strain (ε₂ = 0): ε₁_0 ≈ (23.3 + 14.13t₀) × n / 21 [t₀ in mm; n = strain hardening exponent; result in % strain] FLC shifts up with: higher n, thicker sheet, higher r̄

Strain states: Biaxial (ε₁ = ε₂): stretch forming, hydroforming (FLC minimum at ε₂ = 0; safer in balanced biaxial) Plane strain (ε₂ = 0): worst case (FLC minimum); common at die corners Deep drawing (ε₂ < 0): flange region; safety margin larger; limited by wrinkling, not necking

n-value (strain hardening exponent): σ = K × ε^n [power law; K = strength coefficient; n = strain hardening exponent] Tensile measurement: n = log(σ₂/σ₁) / log(ε₂/ε₁) between two strain points (typically 10% and UTS) Higher n → better formability (more uniform strain distribution before necking) Mild steel: n = 0.18–0.22; IF steel: n = 0.22–0.28; Al 5xxx: n = 0.22–0.30

Erichsen Cupping Test (ISO 20482)

Erichsen test: hemispherical punch (Ø20 mm) pressed into clamped sheet until fracture Erichsen Index (IE) = punch depth at fracture [mm]; higher = more formable Steel SPCE: IE ≈ 10–12 mm; Al 1050-O: IE ≈ 10–12 mm; AHSS DP600: IE ≈ 8–9 mm

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Failure Modes

Wrinkling

Wrinkling (hoop buckling of flange): Occurs when circumferential compressive stress in flange exceeds critical buckling stress Critical condition: σ_θ_critical = E_t × (t₀/w_flange)² / (12(1−ν²)) [w_flange = annular flange width] Prevention: sufficient blank holder force; avoid excessive clearance between blank holder and die

Tearing (Wall Fracture)

Tearing: fracture through cup wall at punch radius (location of maximum stress) Occurs when DR > LDR; or BHF too high → excessive friction; or die radius too small → bending stress Punch nose radius: r_p ≥ 4t₀ to 6t₀ to avoid stress concentration Die radius: r_d ≥ 5t₀ to 8t₀ for smooth material flow

Earing

Earing: non-uniform cup height due to planar anisotropy Δr = (r₀ + r₉₀ − 2r₄₅) / 2 [Δr = planar anisotropy; Δr > 0 → ears at 0° and 90°; < 0 → ears at 45°] Earing height: δh/h ≈ 0.4 × Δr [approximately] Mitigation: use material with small |Δr|; or trim ears in finishing operation

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Multi-Stage Drawing

Re-drawing Sequences

When DR > LDR: multi-stage drawing required Reduction ratio for each stage: Stage 1: DR₁ ≤ LDR (typically 2.0) Stage 2: DR₂ ≤ 1.2–1.4 (re-draw; material work-hardened; requires larger reduction allowed before anneal) Stage 3+: DR₃ ≤ 1.15–1.25

Intermediate annealing: Required when material has insufficient ductility for next draw Recrystallization anneal: restore r-value and n-value; temperature per alloy (Fe: 650°C; Al: 300–380°C)

Overall reduction achievable: 3-stage: D₀/d_final ≈ 2.0 × 1.35 × 1.20 = 3.24 (very deep cups)

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Standards and References

Standard	Scope
ISO 20482	Erichsen cupping test (formability)
ISO 10113	Metallic sheet — r-value determination
ISO 10275	Metallic sheet — n-value determination
ASTM A1008	Steel sheet, cold-rolled (drawing quality DQ, EDDQ)
JIS G3141	Cold-reduced carbon steel sheet (SPCC/SPCD/SPCE)
ISO 12004	Metallic materials — FLC determination

Output

Provide: part geometry (cup inner diameter d [mm]; height h [mm]; flange width [mm] if any; corner radii [mm]; wall thickness t_final [mm]), blank (D₀ = √(d² + 4dh) [mm]; t₀ [mm]; material: SPCD/IF steel/5052-O aluminum), drawing ratio check (DR = D₀/d; LDR from r̄ value; DR ≤ LDR? → single draw; else multi-stage plan), punch force (F_max = π×d_p×t₀×UTS×(DR−0.7) [kN]; blank holder force = F_max/3 [kN]), material properties (r̄ [dimensionless]; Δr; n-value; K [MPa]; UTS [MPa]; elongation [%]), FLD check (major strain ε₁ at critical zone [%]; minor strain ε₂ [%]; plot vs. FLC; safety margin [%]; n and t₀ effect on FLC), failure mode assessment (wrinkling: BHF ≥ minimum; tearing: DR ≤ LDR; earing: Δr magnitude; δh [mm] expected), tooling (punch radius r_p = 5×t₀ [mm]; die radius r_d = 6×t₀ [mm]; die clearance = t₀ × 1.05–1.15; lubricant: mineral oil / zinc phosphate + soap), multi-stage plan if DR > LDR (stages; DR per stage; anneal between stages?), and applicable standard (ISO 10113 for r-value; ISO 12004 for FLC; ISO 20482 Erichsen test; ASTM A1008 for material grade).