Solidification

Chvorinov’s rule: t_s = C(V/A)²

Riser modulus > Casting modulus → riser solidifies last.

Gating System

Sprue taper: A_top/A_bottom = √(h_bottom/h_top)

Pouring time: t = V_mould / (A_gate × v_gate)

Velocity at sprue base: v = √(2gh) (Torricelli)

Pressurised gating: 1 : 0.8 : 0.6 (sprue : runner : gate)

Unpressurised gating: 1 : 2 : 4

Shrinkage

Pattern dimension = Part dimension × (1 + shrinkage allowance)

Shrinkage: Steel ~2%, Cast iron ~1%, Al ~1.3%, Brass ~1.5%

Heat input: H = ηVI/v (J/mm)

η = arc efficiency, V = voltage, I = current (A), v = travel speed (mm/s)

Resistance welding heat: Q = I²Rt

Taylor’s Tool Life

VTⁿ = C

Extended: VTⁿf^n₁d^n₂ = C

HSS: n ≈ 0.1–0.2. Carbide: n ≈ 0.2–0.4. Ceramic: n ≈ 0.5–0.7.

Cutting Speed

V = πDN/1000 (m/min), D in mm, N in rpm

Material Removal Rate

Turning: MRR = Vfd = πDNfd/1000

Milling: MRR = w × d × f_z × z × N

Drilling: MRR = (πD²/4) × f × N

Machining Time

Turning: T_m = L/(fN)

Drilling: T_m = (L + A)/(fN), A = drill approach

Merchant’s Circle (Orthogonal Cutting)

Shear angle: 2φ + β − α = 90° → φ = 45° + α/2 − β/2

Chip thickness ratio: r = t₁/t₂ = sinφ/cos(φ−α)

Shear plane area: A_s = bt₁/sinφ

Cutting force: F_c = F_s cos(β−α)/cos(φ+β−α)

Power: P = F_c × V

α = rake angle, β = friction angle (tanβ = μ), φ = shear angle

Forging

Open-die (no friction): F = σ_f × A

With friction: F = σ_f × A × (1 + 2μr/(3h))

True strain: ε = ln(h₀/h_f)

Rolling

Draft: Δh = h₀ − h_f

Maximum draft: Δh_max = μ²R

Contact length: L = √(R × Δh)

Roll force: F ≈ σ_f × w × L

Volume conservation: h₀v₀ = h_fv_f

Extrusion

Extrusion ratio: R = A₀/A_f

True strain: ε = ln(R)

Ideal force: F = A₀ × σ_f × ln(R)

Drawing

Drawing force: F = A_f × σ_f × ln(A₀/A_f)

Max reduction per pass: ~63% (theoretical), 20–45% (practical)