Theory of metal cutting

Mechanical Engineering (B.E./B.Tech) · Manufacturing · free study notes for exams, placements and interviews

In short

Metal cutting theory explains how a wedge-shaped tool removes material as a chip by concentrated plastic shear, and how the geometry of the tool decides the forces, the heat, the tool life and the surface produced. It converts a shop-floor trade into predictable numbers: shear angle, cutting force, power, temperature and tool life. In exams it is the heavily-weighted analytical part of Manufacturing — Merchant's circle, chip ratio, Taylor's equation and tool geometry conversion appear in GATE, ESE, ISRO/DRDO and PSU papers almost every year. In industry it is the basis of process planning and tooling selection at Tata Motors, Bharat Forge, L&T, HAL, BHEL, Ashok Leyland, Sundram Fasteners and at tooling firms such as Kennametal India, Sandvik Coromant India and Taegutec India.

Explained

How a chip forms

Cutting is controlled plastic shear, not sawing or tearing: the work material ahead of the rake face is compressed and shears along a narrow band — the shear plane — then slides up the rake face as a chip.
Three deformation zones: (1) primary shear zone on the shear plane; (2) secondary zone at the chip–rake interface (friction and further shear); (3) tertiary zone where the worn flank rubs the finished surface.
The chip always comes out thicker and shorter than the layer removed. Chip thickness ratio r = t₁/t₂ < 1; its reciprocal ξ = t₂/t₁ is the chip reduction coefficient.
Orthogonal (2-D) cutting: cutting edge perpendicular to cutting velocity, inclination angle i = 0, chip flows in the plane containing velocity and the edge normal; two force components only. Used for all theory and derivations.
Oblique (3-D) cutting: i ≠ 0, chip flows sideways at an angle; three mutually perpendicular force components; longer edge contact spreads heat, so it is what real tools actually do.
Types of chip:
Continuous — ductile material, high speed, large rake, good lubrication, sharp edge. Best finish, but long chips are a safety and handling problem.
Continuous with built-up edge (BUE) — work material pressure-welds to the rake face at low-to-medium speed, grows and breaks away; causes poor finish, fluctuating force, chipping. Cured by higher speed, larger rake, better fluid, coated tool.
Discontinuous (segmented) — brittle material (grey cast iron, brass), small rake, large feed, low speed. Lower force but rough surface and shock loading on the edge.
Serrated / saw-tooth — cyclic shear localisation in titanium alloys and hardened steels.
Chip breakers (stepped, groove, clamped obstruction, or moulded into the insert) curl and snap chips; essential on CNC for swarf handling and operator safety.

Tool geometry and nomenclature

A single-point tool has shank, face (rake surface), flanks, cutting edges, nose. Angles are what turn a wedge into a cutting tool.
Rake angle: controls chip flow and shear. Positive rake → lower force, lower power, weaker edge; used on ductile material, HSS tools, light machines. Negative rake → stronger edge, puts the insert in compression; used for carbide/ceramic inserts, hard materials, interrupted cuts. Zero rake is used for brass (prevents dig-in).
Relief (clearance) angle: prevents the flank rubbing the work; too small → rubbing and heat, too large → weak edge. Typically 5–10°.
Principal (side) cutting edge angle: distributes the load and sets chip thickness for a given feed. End cutting edge angle avoids rubbing of the trailing edge. Nose radius strengthens the tip and improves finish.
ASA (American Standards Association / tool-in-hand) signature, in order: back rake, side rake, end relief, side relief, end cutting edge angle, side cutting edge angle, nose radius — e.g. 0°-10°-6°-6°-8°-15°-1 mm. Reference planes are the machine axes.
ORS (Orthogonal Rake System, tool-in-hand) signature, in order: inclination angle λ, orthogonal rake γ₀, side (principal) relief α₀, end (auxiliary) relief α₀′, auxiliary cutting edge angle φ₁, principal cutting edge angle φ, nose radius r — e.g. 0°-10°-6°-6°-15°-75°-1 mm. Reference planes are tied to the cutting edge, which is why mechanics is written in ORS.
Conversion between ASA and ORS (and to the tool-in-use system, where angles are measured relative to the actual resultant cutting velocity) is a standard numerical: tan γ₀ = tan γ_b·cos φ + tan γ_s·sin φ and tan λ = tan γ_s·cos φ − tan γ_b·sin φ (ASA / machine-reference-system back rake γ_b, side rake γ_s).
Relevant standards: ISO 3002 (geometry of the active part of cutting tools), ISO 1832 (indexable insert designation), ISO 513 (classification of hard cutting materials into groups P–M–K–N–S–H), ISO 3685 (tool-life testing, single-point turning). BIS carries adopted IS/ISO equivalents and IS specifications for HSS tool bits and tool nomenclature; the numbers are periodically revised, so confirm the current designation in the BIS catalogue rather than quoting from memory.

Mechanics of orthogonal cutting — Merchant's circle

Model (thin shear plane / Ernst–Merchant): the tool is perfectly sharp, cutting is 2-D and steady, deformation is confined to a plane, the chip does not curl, the resultant force on the chip from the tool equals that from the workpiece.
Merchant's circle draws the single resultant R as the diameter of a circle, and resolves it three ways: into measurable Fc (cutting force, along velocity) and Ft (thrust/feed force); into Fs, Fn on the shear plane; into F, N on the rake face.
Fc is measured with a tool dynamometer (strain-gauge or piezoelectric); Ft is typically 0.3–0.6 Fc. Both rise with feed and depth of cut, fall slightly with speed, and fall with increasing rake angle.
Shear angle φ is not free — the material picks the φ that minimises work (Merchant) or satisfies a slip-line field (Lee–Shaffer):
Merchant: 2φ + β − α = 90°, so φ = 45° + α/2 − β/2.
Modified Merchant (machinability constant C): 2φ + β − α = C, with C ≈ 70–80° for many steels, since τ rises with normal stress.
Lee–Shaffer: φ + β − α = 45°.
All are approximate; measured φ usually lies between them. Oxley's slip-line/parallel-sided zone theory is the more accurate research model.
Consequences worth remembering: larger rake α and lower friction β → larger φ → thinner chip, shorter shear plane, less force, less heat, less power. That single chain explains most of tool design.
Size effect: specific cutting energy increases sharply as the uncut chip thickness falls below about 0.05 mm, because the shear strength seen at small depths approaches the ideal strength of the material. This is why micro-machining and grinding need far more energy per mm³.

Heat, temperature and cutting fluids

Nearly all the cutting work becomes heat: roughly 75–80% in the primary shear zone, 15–20% at the chip–tool interface, ~5% at the flank. Most of it leaves with the chip; the small fraction retained by the tool sets its life.
Interface temperatures reach 600–900 °C in steel machining with carbide, and temperature rises mainly with cutting speed (θ ∝ V^0.2–0.5), then with feed, then weakly with depth of cut.
Cook's equation: mean interface temperature rise Δθ = 0.4 (u/ρc) (V t₁/K)^0.333 — speed and chip thickness dominate.
Measurement: tool–work thermocouple (the tool/work junction itself), embedded thermocouples, infrared pyrometry, temperature-sensitive paints, metallographic hardness change in HSS.
Cutting fluid functions: cool, lubricate the rake/flank, flush chips, inhibit corrosion, prevent BUE. Cooling dominates at high speed, lubrication at low speed and in finishing.
Types: straight (neat) cutting oils with EP additives; soluble oil emulsions (oil in water, most common in Indian shops); semi-synthetic; fully synthetic solutions; solid lubricants (graphite, MoS₂); gases/cryogenic LN₂ and CO₂.
Application: flood, jet, mist, through-tool high pressure (essential for titanium and deep holes), MQL / near-dry (a few ml/h of oil in air — now preferred for environmental reasons), or dry cutting with coated carbide and ceramics on cast iron.
Health and statutory practice: oil mist and bacterially-degraded emulsion cause dermatitis and respiratory irritation; shops are expected to provide local exhaust, guarding and periodic fluid monitoring under the OSH Code, 2020 (in force 21 November 2025, replacing the Factories Act, 1948) and the saved State Factory Rules, and to dispose of spent coolant as regulated waste under the Hazardous and Other Wastes (Management and Transboundary Movement) Rules — check current CPCB/State PCB requirements, as these rules are amended.

Tool wear, tool life and tool materials

Wear forms: flank (land) wear VB — abrasion, measures size loss, the usual life criterion; crater wear KT on the rake — diffusion and adhesion at high temperature; notch wear at the depth-of-cut line (oxidation, work hardening); nose wear; chipping; thermal (comb) cracking in interrupted cuts; plastic deformation of the edge; BUE.
Mechanisms by temperature: abrasion and adhesion at low speed; diffusion, oxidation and chemical wear take over at high speed — which is why speed destroys tool life faster than feed or depth.
Wear–time curve: rapid initial break-in, long steady (linear) region, then accelerated wear; a tool must be changed before the knee.
ISO 3685 life criteria (commonly quoted): average flank wear VB = 0.3 mm or VB(max) = 0.6 mm for carbide; crater depth KT = 0.06 + 0.3f mm; or a surface-finish/force limit — whichever occurs first.
Taylor's tool life equation V Tⁿ = C, with the extended form V Tⁿ f^a d^b = C. Typical n: 0.08–0.15 (HSS), 0.2–0.3 (carbide), 0.4–0.6 (ceramic/CBN). Because n is small, a 50% speed rise can cut life by 70–90%.
Tool material progression and when each is used:

Material	Hot hardness / max speed on steel	Use
Carbon tool steel	up to ~250 °C, very low speed	Hand tools, taps
HSS (18-4-1 T-series, M-series with Mo)	~600 °C, 20–50 m/min	Form tools, drills, reamers, gear cutters, low-power machines
Cast cobalt (stellite)	~800 °C	Largely obsolete, rough cast iron
Cemented carbide (WC–Co, brazed or indexable)	~1,000 °C, 80–250 m/min	General workhorse
Coated carbide (TiN, TiCN, Al₂O₃, TiAlN)	~1,100 °C, 150–400 m/min	Most production turning and milling today
Ceramic (Al₂O₃, SiAlON)	~1,300 °C, 300–900 m/min	Cast iron, high-speed finishing, no interruption
CBN	~1,400 °C	Hardened steel (>45 HRC), hard turning
PCD / diamond	~700 °C (reacts with iron)	Aluminium, composites, non-ferrous only

Requirements of any tool material: hardness at temperature, toughness, wear resistance, chemical stability against the work, thermal conductivity, and affordable cost/grindability.

Machinability, surface finish and machining economics

Machinability is not one number — it is judged by tool life at a reference speed, cutting force/power, surface finish achieved, chip form, and dimensional stability. The classical machinability index rates a material against free-cutting steel AISI 1112 taken as 100%.
Improved by free-machining additives (sulphur, lead historically, now Bi/Te and controlled inclusions), normalising or annealing for correct hardness (too soft smears, too hard abrades), and by avoiding work-hardening alloys like austenitic stainless.
Theoretical surface roughness in turning: set by feed and nose radius, not by the material — h_max = f²/(8r), Ra ≈ 0.0321 f²/r. Actual roughness is worse because of BUE, vibration, feed marks and tool wear.
Economics: cost per piece = machining cost + tool-change cost + tool cost + non-productive cost. Low speed → cheap tools but long cycle; high speed → fast cycle but tool cost and downtime explode. Two optima exist and the minimum-cost tool life is always the longer of the two.
Machining at high speed (HSM) and hard turning replace grinding for many shafts and bearing races, trading tool cost for cycle time.

Key points

Metal cutting — material removal by concentrated plastic shear on a narrow shear plane ahead of the rake face.
Chip thickness ratio r — t₁/t₂, always less than 1; the chip thickens and shortens.
Shear angle φ — angle of the shear plane to the cutting velocity; large φ means thin chip, low force, efficient cutting.
Orthogonal vs oblique — i = 0 with two force components (theory) versus i ≠ 0 with three (practice).
Merchant's circle — one resultant force resolved into Fc/Ft, Fs/Fn and F/N; the basis of every force numerical.
Merchant's relation — φ = 45° + α/2 − β/2; rake up or friction down both raise φ.
Built-up edge — pressure-welded work material on the rake face at moderate speed; spoils finish, cured by higher speed and coated tools.
Chip breaker — groove or step that curls and snaps continuous chips; a safety requirement, not a nicety.
Rake angle — positive for low force, negative for edge strength (carbide, hard or interrupted cuts).
Tool signature — ASA (machine-axis reference) and ORS (edge reference, used in mechanics); learn the order of both.
Specific cutting energy u — Fc/(t₁w), J/mm³; rises steeply at very small chip thickness (size effect).
Heat split — ~75–80% primary shear, ~15–20% rake interface, ~5% flank; most heat leaves in the chip.
Flank wear VB — the standard life criterion, 0.3 mm average under ISO 3685; crater wear is diffusion-driven.
Taylor's equation — V Tⁿ = C; small n means speed is by far the strongest influence on tool life.
Cutting fluid — cools, lubricates, flushes; MQL and dry cutting are displacing flood coolant.
Machinability — judged by tool life, force, finish and chip form, not by a single property.
Surface roughness in turning — h_max = f²/(8r): halve the feed and roughness falls fourfold.
Optimum tool life — minimum-cost life ((1−n)/n)(t_ct + C_t/x) is always longer than maximum-production life ((1−n)/n)t_ct.

Formulas and examples

r = t₁/t₂ = l₂/l₁ = sin φ / cos(φ − α) — chip thickness ratio (t₁ uncut chip thickness, t₂ chip thickness, l chip length, α orthogonal rake, mm)
tan φ = r cos α / (1 − r sin α) — shear angle from measured chip ratio
γ = cot φ + tan(φ − α) — shear strain in the chip (dimensionless)
V_c = V·r; V_s = V cos α / cos(φ − α) — chip velocity and shear velocity (m/min)
F_s = Fc cos φ − Ft sin φ; F_n = Fc sin φ + Ft cos φ — shear and normal force on shear plane (N)
F = Fc sin α + Ft cos α; N = Fc cos α − Ft sin α; μ = tan β = F/N — friction force, normal force, coefficient of friction on rake
A_s = t₁·w / sin φ; τ = F_s / A_s — shear plane area (mm²) and shear stress (MPa); w = width of cut
φ = 45° + α/2 − β/2 (Merchant); 2φ + β − α = C (modified); φ = 45° + α − β (Lee–Shaffer)
Power P = Fc·V (W, with Fc in N and V in m/s); u = Fc/(t₁w) = P/MRR (J/mm³)
MRR = 1000·V·f·d (mm³/min) in turning; N = 1000V/(πD) rpm; t_m = L/(f·N) min
V Tⁿ = C; V Tⁿ f^a d^b = C — Taylor and extended Taylor tool life
T_mincost = ((1−n)/n)(t_ct + C_t/x); T_maxprod = ((1−n)/n)·t_ct — t_ct tool change time (min), C_t cost per edge (₹), x machine+labour rate (₹/min)
h_max = f²/(8r); Ra ≈ 0.0321 f²/r — theoretical roughness, round-nose tool (f mm/rev, r nose radius mm)

Example 1 — chip ratio, shear angle, strain and velocities. Orthogonal cutting of mild steel, rake α = 10°, uncut chip thickness t₁ = 0.20 mm, measured chip thickness t₂ = 0.40 mm, cutting speed V = 100 m/min.

r = 0.20/0.40 = 0.50
tan φ = 0.50 × cos10° / (1 − 0.50 × sin10°) = 0.4924 / 0.9132 = 0.5392 → φ = 28.3°
γ = cot 28.3° + tan(28.3° − 10°) = 1.854 + 0.331 = 2.19 (a 219% shear strain — far beyond a tensile test)
V_c = 100 × 0.50 = 50 m/min; V_s = 100 × 0.9848 / cos 18.3° = 98.48/0.9493 = 103.7 m/min

Example 2 — Merchant's circle, friction and specific energy. Same cut, width w = 2 mm; dynamometer reads Fc = 900 N, Ft = 450 N.

F_s = 900 cos28.3° − 450 sin28.3° = 792.4 − 213.3 = 579 N
F_n = 900 sin28.3° + 450 cos28.3° = 426.7 + 396.2 = 823 N
F = 900 sin10° + 450 cos10° = 156.3 + 443.2 = 599 N; N = 900 cos10° − 450 sin10° = 886.3 − 78.1 = 808 N
μ = 599/808 = 0.742 → β = 36.6°
A_s = (0.20 × 2)/sin28.3° = 0.40/0.4741 = 0.844 mm² → τ = 579/0.844 = 686 MPa
Power = 900 × (100/60) = 1,500 W = 1.5 kW; u = 900/(0.20 × 2) = 2,250 N/mm² = 2.25 J/mm³
Check against theory: Merchant gives φ = 45 + 5 − 18.3 = 31.7°, Lee–Shaffer gives 45 + 10 − 36.6 = 18.4°, against the measured 28.3° — both are approximations, and the exam expects you to say so.

Example 3 — Taylor's equation and output per edge. A carbide tool lasts 180 min at 25 m/min and 11.25 min at 50 m/min.

25 × 180ⁿ = 50 × 11.25ⁿ → 16ⁿ = 2 → n = 0.25
C = 25 × 180^0.25 = 25 × 3.663 = 91.6 m/min
At V = 40 m/min: T^0.25 = 91.6/40 = 2.289 → T = 2.289⁴ = 27.5 min
Turning a Ø50 mm × 300 mm shaft at f = 0.25 mm/rev: N = 1000 × 40/(π × 50) = 254.6 rpm; t_m = 300/(0.25 × 254.6) = 4.71 min/pass
Components per edge = 27.5/4.71 = 5.8 → 5 shafts per edge; with d = 2 mm, MRR = 1000 × 40 × 0.25 × 2 = 20,000 mm³/min

Example 4 — machining economics. For the same tool: n = 0.25, tool change time t_ct = 5 min, cost per edge C_t = ₹60, machine + operator rate ₹600/h = ₹10/min.

T for minimum cost = (0.75/0.25)(5 + 60/10) = 3 × 11 = 33 min → V = 91.6/33^0.25 = 91.6/2.397 = 38.2 m/min
T for maximum production = 3 × 5 = 15 min → V = 91.6/15^0.25 = 91.6/1.968 = 46.5 m/min
The minimum-cost life is always the longer one; the economic speed lies between these two, in the "high-efficiency range".

Example 5 — power-limited feed and the finish it gives. A lathe has a 5 kW motor at 80% spindle efficiency; the work needs u = 2.5 J/mm³; V = 120 m/min, d = 2 mm, nose radius r = 0.8 mm.

Power at the tool = 5,000 × 0.8 = 4,000 W → MRR_max = 4,000/2.5 = 1,600 mm³/s = 96,000 mm³/min
f_max = 96,000/(1000 × 120 × 2) = 0.40 mm/rev
At that feed, h_max = 0.40²/(8 × 0.8) = 0.025 mm = 25 µm, Ra ≈ 0.0321 × 0.16/0.8 = 6.4 µm — a roughing finish. For Ra ≈ 1.6 µm, f = √(1.6 × 0.8/0.0321 /1000)… directly: f = √(Ra·r/0.0321) with Ra in mm = √(0.0016 × 0.8/0.0321) = 0.20 mm/rev.

	Positive rake	Negative rake
Cutting force and power	Lower	Higher (20–40% more)
Edge strength	Weak	Strong, edge in compression
Tool material suited	HSS, brazed carbide	Indexable carbide, ceramic, CBN
Work suited	Ductile, soft, thin-wall, low-power machines	Hard material, scale, interrupted cuts
Chip flow	Away from machined surface	Onto the work, more rubbing

Where it is used

Process planning and CNC programming: choosing V, f, d and insert grade from ISO P/M/K groups so that cycle time, power draw and tool cost balance — a daily task for manufacturing engineers at Bharat Forge, Sundram Fasteners, Rane and Brakes India.
Tooling and application engineering: insert geometry, chip-breaker and coating selection by Kennametal India (Bengaluru), Sandvik Coromant India, Taegutec India, Addison & Co, Birla Precision — these roles interview almost entirely on this topic.
Aerospace and defence machining: HAL, BrahMos, Godrej Aerospace cutting titanium and nickel alloys, where low thermal conductivity forces high-pressure through-tool coolant and strict VB limits.
Automotive mass production: crankshaft, camshaft, gear-blank and brake-disc lines, where a 0.05 mm/rev feed change or one extra component per edge changes unit cost measurably; hard turning with CBN replaces grinding on bearing races.
Lab and R&D work: tool dynamometry, tool–work thermocouple temperature studies and tool-life testing to ISO 3685 in IIT/NIT machining labs, CMTI Bengaluru and CSIR-CMERI.
Maintenance and tool-room roles: diagnosing chatter, BUE, chipping and premature crater wear on the shop floor — the symptoms map directly onto the mechanisms above.

Interview questions

Q: Why is the chip thicker than the layer of metal removed? A: The metal shears along a plane inclined to the cut surface, so the sheared layer is compressed in length and thickened; the chip thickness ratio r = t₁/t₂ is therefore always less than one.
Q: What is the significance of the shear angle? A: A larger shear angle means a shorter shear plane and a thinner chip, so less work, less force, less heat and longer tool life; it increases with rake angle and falls with rake-face friction.
Q: Why do we study orthogonal cutting when real cutting is oblique? A: Orthogonal cutting is two-dimensional with only two force components, so the mechanics can be solved in closed form; its results are then corrected for inclination in practice.
Q: What is a built-up edge and is it ever useful? A: Work material pressure-welded to the rake face; it increases the effective rake and can protect the tool briefly, but as it grows and breaks away it ruins the finish and chips the edge, so it is normally avoided by cutting faster.
Q: When would you deliberately use a negative rake angle? A: With carbide or ceramic inserts on hard materials, scaled forgings or interrupted cuts, where edge strength matters more than cutting force, and the machine has the power for the higher force.
Q: In Taylor's equation, what do n and C physically mean? A: C is the cutting speed that gives one minute of tool life, and n reflects how tolerant the tool material is of speed — small n (HSS) means life collapses quickly with speed, large n (ceramic) means it tolerates speed.
Q: Which cutting parameter should you increase first to raise productivity, and why? A: Depth of cut, then feed, then speed — because in the extended Taylor relation speed has the strongest adverse effect on tool life, while depth has the weakest.
Q: Where does the heat go in metal cutting? A: Most of it is generated in the primary shear zone and carried away in the chip; a smaller share at the chip–tool interface flows into the tool and sets its life, and a little goes into the workpiece through the flank.
Q: Why does crater wear dominate at high speed and abrasion at low speed? A: Abrasion depends on sliding distance and hard inclusions, while crater wear is diffusion and chemical attack that are exponential in temperature, and temperature rises with speed.
Q: How do you reduce surface roughness in turning without changing the tool material? A: Reduce the feed, increase the nose radius, eliminate BUE by raising speed or using coolant, and remove chatter by stiffening the setup — roughness varies as f²/r.
Q: A customer complains of chatter and poor finish on a long slender shaft. What do you check? A: Overhang and support (steady/follower rest), tool overhang, nose radius and lead angle (large radius raises radial force), speed near a resonance, and worn insert — then reduce the radial force component.
Q: Why can PCD tools not be used on steel? A: Diamond is carbon and diffuses into and reacts with ferrous material at cutting temperatures, so it wears catastrophically; CBN is used instead for hardened steel.

Common mistakes

Mixing up the two force sets in Merchant's circle — writing F_s with the rake angle or F with the shear angle; always redraw the circle before substituting.
Using the ASA signature values in formulas that need ORS angles (mechanics is written in the orthogonal rake system), or forgetting to convert tool-in-hand angles to tool-in-use when feed is large.
Taking the chip ratio as t₂/t₁ and getting a shear angle above 45° — r is always less than 1.
Assuming shear stress on the shear plane is simply the material's yield strength from a tensile test; strain, strain rate and temperature in cutting are orders of magnitude higher.
Unit slips: Fc in N with V in m/min gives N·m/min, not watts — divide by 60; and 1 mm³/s is 60 mm³/min.
Quoting one tool-life answer without stating the wear criterion (VB = 0.3 mm, Ra limit, catastrophic failure) — in ISO 3685 the criterion defines the number.
Confusing the two economic tool lives, or calling the maximum-production life the "optimum"; the minimum-cost life is longer, and which one applies depends on whether the bottleneck is cost or capacity.