Machining, Cutting Forces and Tool Life

Machining removes material as chips with a hard cutting tool. GATE tests the mechanics of orthogonal cutting, tool life, and machining time for turning, drilling and milling.

Orthogonal cutting

With uncut chip thickness t, chip thickness t_c and rake angle α:

• Chip thickness ratio r = t/t_c (less than 1).
• Shear angle φ: tan φ = r cos α/(1 − r sin α).
• Shear strain γ = cot φ + tan(φ − α).
• Chip velocity V_c = r V; shear velocity V_s = V cos α/cos(φ − α).

Forces (Merchant's circle)

The measured cutting force F_c and thrust force F_t give:

• Friction force on the rake face F = F_c sin α + F_t cos α; normal N = F_c cos α − F_t sin α; μ = F/N = tan β.
• Shear force on the shear plane F_s = F_c cos φ − F_t sin φ; shear stress τ = F_s/A_s, with A_s = (b t)/sin φ.
• Merchant's relation: 2φ + β − α = 90°.
• Cutting power = F_c V. Specific cutting energy = F_c/(b t) (energy per unit volume removed).

Tool life

Taylor's equation: V Tⁿ = C

• n is about 0.1–0.15 for HSS, 0.2–0.5 for carbides, and higher for ceramics.
• Extended: V Tⁿ f^a d^b = C (feed and depth also matter, but less than speed).
• Tool wear: flank wear (gradual, used as the life criterion) and crater wear on the rake face.

Machining time

N = 1000V/(πD) rpm (V in m/min, D in mm). Material removal rate in turning = π D f d N ≈ V f d.

Surface finish: with a nose radius r, the peak-to-valley height h = f²/(8r). Smaller feed and larger nose radius give a better finish.

Worked example

Orthogonal cutting with α = 10°, t = 0.2 mm and t_c = 0.5 mm.

• r = 0.4. tan φ = 0.4 cos 10°/(1 − 0.4 sin 10°) = 0.3939/0.9305 = 0.4233 → φ = 22.9°.

Practice