cold-rolling

name	cold-rolling
description	Cold rolling of strip and sheet — Bland-Ford roll force model, friction Hill analysis, work hardening (σ = K εⁿ), temper rolling (skin pass), total reduction and annealing cycles, roll force and torque, rolling power, strip flatness and tension distribution, shape control (roll bending, roll shifting, CVC), anisotropy development (r-value, Lankford coefficient), texture evolution, edge cracking limit, ASTM A1008 cold-rolled steel, ASTM B209 aluminum sheet.
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Cold Rolling — Complete Skill

Cold Rolling Mechanics

Force and Pressure Distribution

Cold rolling differs from hot: no recrystallization; work hardening during rolling; friction coefficient lower; roll flattening significant

Bland-Ford analysis (friction hill): p(x) = σ̄_f(x) × e^(μh) at each point x along contact arc Complete solution accounts for: work hardening (σ_f increases with ε), back tension σ_b, front tension σ_f_t, roll flattening

Average roll pressure (simplified): p_avg = C_f × σ̄_f [C_f = friction factor ≈ 1.0 + (μ × L_d)/Δh = friction hill multiplier] Roll force: F = p_avg × L_d × w [N; L_d = √(R'×Δh); R' = flattened roll radius]

Friction Hill Analysis: Entry side (behind neutral point): pressure rises due to friction assisting rolling Exit side (ahead of neutral): pressure falls; friction opposing forward slip Neutral point: location where strip and roll have same velocity Neutral angle φ_n: cosφ_n = 1 − h_n/R' where h_n solved from momentum balance

Hitchcock flattened radius: R' = R × [1 + (C_H × F)/(w × Δh)] [C_H = 16(1−ν_r²)/(π × E_r); ν_r = 0.28; E_r = 200 GPa for steel rolls] Iterate: assume R' → compute F → recompute R' → converge (2–5 iterations)

Roll Torque and Power

Torque (both rolls): T_total = 2 × F × a_L [a_L = lever arm ≈ L_d/2 for symmetric case; more precisely from pressure distribution integral] a_L/L_d ≈ 0.4–0.5 depending on friction and reduction

Rolling power: P = T_total × ω_roll = F × L_d × v_roll / R [kW; v_roll = roll peripheral speed [m/s]] Or: P = F × L_d × v_exit × (h₁/h_avg) × correction [strip-speed based]

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Work Hardening

Flow Stress in Cold Rolling

Power law (Hollomon): σ_f = K × ε^n [K = strength coefficient [MPa]; n = work hardening exponent; ε = true strain from entry] True strain from reduction: ε = ln(h₀/h₁) Accumulated strain: ε_total = ln(h_initial / h_current) [sum over all passes from original annealed state]

Material constants (typical): Low-carbon steel (annealed): K = 530 MPa; n = 0.26 Aluminum 3003-O: K = 185 MPa; n = 0.21 Copper (annealed): K = 460 MPa; n = 0.34 Stainless 304: K = 1275 MPa; n = 0.45 (high work hardening)

Saturation: at large ε (≥ 1.0), σ_f saturates (n model less accurate); use Voce model: σ_f = σ_sat − (σ_sat − σ_0) × exp(−ε/ε_c) [σ_sat, σ_0, ε_c = material parameters]

Intermediate annealing: When σ_f too high → roll force exceeds mill capacity → anneal to soften → resume cold rolling Annealing temperature: below T_recrystallize_start for control; above T_recrystallize_finish for full anneal Example: low-C steel anneal 600–720°C, 4–8 hours → fully recrystallized; ε reset to 0

Total cold work before anneal: Max total reduction before intermediate anneal: 70–80% for low-C steel; 50–60% for stainless

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Pass Schedule Design

Reduction Distribution

Equal power per pass: often used — distribute reduction so each pass uses similar roll force First passes: larger reduction (softer material, more δh_max available) Final passes: smaller reduction (harder material; also tighter thickness tolerance required)

Maximum draft per pass: Δh_max = μ² × R' [from bite condition; μ ≈ 0.05–0.15 for cold rolling with oil] Or limited by: mill motor power, roll force capacity, strip tension limits

Friction coefficients (cold rolling): Mineral oil lubricant: μ = 0.05–0.08 Emulsion (5–10% oil in water): μ = 0.08–0.12 Dry (temper mill): μ = 0.12–0.20

Strip Tension

Front and back tension: Tension reduces effective yield stress → reduces roll force: p_avg_reduced = p_avg × (1 − σ_t/(√3×σ_f)) Typical strip tensions: 5–30% of yield stress (balance between roll force reduction and strip fracture risk) Back tension σ_b: from uncoiler braking or looper; Front tension σ_f_t: from coiler motor

Neutral point shift with tension: Front tension shifts neutral toward exit; back tension toward entry If neutral at roll exit: danger of "running back" (instability)

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Anisotropy (Lankford Coefficient / r-value)

r-value Definition

r-value (Lankford coefficient): r = ε_w / ε_t [width strain / thickness strain in tensile test; measured at uniform elongation before necking] Or: r = ln(w₀/w) / ln(t₀/t) [measured from initial to current dimensions]

r̄ (planar average): r̄ = (r₀ + 2×r₄₅ + r₉₀) / 4 [subscripts = angle to rolling direction: 0°, 45°, 90°] r̄ > 1: material resists thinning → better deep drawability (LDR correlates with r̄) Steel (continuous cast, texture): r̄ = 1.4–2.2; IF steel: r̄ > 2.0

Planar anisotropy Δr: Δr = (r₀ − 2×r₄₅ + r₉₀) / 2 [measure of earing tendency in cup drawing] Δr ≈ 0: no earing; Δr > 0: earing at 0° and 90°; Δr < 0: earing at 45°

Texture origin: BCC steel: {111}⟨110⟩ γ-fiber promotes high r̄; {100}⟨011⟩ cube reduces r̄ FCC aluminum: develops cube texture → low r̄ ≈ 0.6–0.8; poor deep drawability

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Temper Rolling (Skin Pass)

Purpose and Process

Skin pass (temper roll): light cold reduction (0.5–3%) applied after full anneal Objectives:

1. Eliminate yield point elongation (stretcher strains): strain into uniform deformation
2. Impart surface roughness (dull or bright finish per specification)
3. Improve flatness (minor correction)
4. Adjust mechanical properties slightly

Effect on yield point: As-annealed steel: Lüders band (yield point phenomenon) at ~300 MPa → ugly surface marks After 0.5–2% skin pass: Lüders bands suppressed; uniform deformation; smooth surface

Shelf life: Lüders bands return after ~3–6 months aging at room temperature (strain aging, N interstitials) Bake hardening (BH steel): intentional — some aging → yield strength increases 30–60 MPa in paint oven

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Shape and Flatness Control

Flatness measurement (I-units): I = (Δl/l) × 10⁵ [relative elongation difference between strips of different lengths across width] Industry standard: ≤ 10 I-units (flat); ≤ 20 I-units (acceptable); > 40 I-units (rejectable)

Work roll bending: Positive bending: crown compensation → reduce edge wave (corrects for roll deflection) Negative bending: remove center buckle

CVC (continuously variable crown): ground roll with S-curve profile; axially shift rolls → continuously variable effective crown

Roll shifting (VC rolls / HC mill): intermediate rolls shift axially → change contact length → flatness control

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Standards

Standard	Scope
ASTM A1008/A1008M	Cold-rolled carbon steel sheet
ASTM A568/A568M	General requirements for cold-rolled carbon/HSLA sheet
ASTM B209	Aluminum-alloy sheet and plate
ASTM B36/B36M	Brass sheet/strip/plate
EN 10130	Cold-rolled low-carbon steel flat products (DC01–DC07)
JIS G3141	Cold-rolled steel sheet (SPCC, SPCD, SPCE)
ISO 16162	Cold-rolled steel strip tolerances

Output

Provide: material (grade; annealed K [MPa] and n; or σ₀ and n from pre-work hardened state), rolling schedule (pass # | h₀ [mm] | h₁ [mm] | r [%] | ε_total | σ_f [MPa] from Kεⁿ | Δh_max = μ²R [mm] | Δh feasible?), roll force per pass (R' [mm] from Hitchcock; L_d = √(R'Δh) [mm]; p_avg = C_f×σ̄_f [MPa]; F = p_avg×L_d×w [MN]; within mill capacity?), tension (back σ_b [MPa] % of σ_y; front σ_f_t [MPa]; roll force reduction from tension [MN]), intermediate anneal (required? at what total ε or when σ_f > roll force limit; anneal T [°C]; time [h]), final mechanical properties (σ_y = K×ε_total^n [MPa]; σ_UTS [MPa]; elongation estimated from n; r̄ if specified), temper roll (reduction [%]; yield point suppressed; target Ra [μm]: dull/bright), flatness (I-units target; roll bending setting; CVC position if applicable), and applicable standard (ASTM A1008 for carbon steel; ASTM B209 for Al; EN 10130 for automotive sheet).