Cutting removes material as chips. The tool is set to a depth of cut and moves at a velocity while the workpiece rotates; feed = distance per revolution (mm/rev). Note: what’s called depth of cut here is the feed in turning/milling (Ch23–24).

Independent variables: tool material/sharpness, workpiece material, cutting parameters (depth, speed, feed, rake angle), cutting fluid, workholding.

Dependent variables: forces/energy, temperature, tool wear/failure, surface finish.

Mechanics of Chip Formation

In orthogonal cutting the tool has rake angle $\alpha$ and a relief angle; shearing occurs along the shear plane at the shear angle $phi$.

Cutting ratio ($t_0$ = depth/undeformed, $t_c$ = chip thickness):

$r = \frac{t_0}{t_c} = \frac{\sin\phi}{\cos(\phi - \alpha)}$

The chip is always thicker than the cut, so $r < 1$. Rearranged for the shear angle:

$\phi = \tan^{-1}\left(\frac{r\cos\alpha}{1 - r\sin\alpha}\right)$

From mass continuity $V t_0 = V_c t_c$, so $r = V_c/V$ (chip speed $V_c$).

The shear angle governs force, power, chip thickness, and temperature. A minimum-force analysis gives

$\phi = 45^\circ + \frac{\alpha}{2} - \frac{\beta}{2}$

where $\beta$ is the friction angle, $mu = tanbeta = F/N$. Lower $\alpha$ and/or higher friction → smaller $\phi$ → thicker chip → more energy and higher temperature.

Chip Types

• Continuous — ductile material, high speed, high rake; good finish but tangles (use chip breakers).
• Built-up edge (BUE) — workpiece material welds to the tool tip; changes geometry, roughens finish; reduce by ↓depth, ↑rake, sharp tool, good fluid. A thin stable BUE can protect the rake face.
• Serrated — sawtooth; low-conductivity metals (titanium).
• Discontinuous — brittle materials, very low/high speeds, large depths, low rake; fluctuating forces → chatter.

Chip breakers bend/break long chips and increase the effective rake (and shear) angle. The ideal chip is a C or 9 fitting in a 25 mm square.

Forces (orthogonal cutting)

Cutting force $F_c$ (along $V$) and thrust force $F_t$ (normal to $V$) give the resultant $R$. On the tool face, friction $F$ and normal $N$:

$F = R\sin\beta, \qquad N = R\cos\beta$

Along the shear plane, shear $F_s$ and normal $F_n$:

$F_s = F_c\cos\phi - F_t\sin\phi, \qquad F_n = F_c\sin\phi + F_t\cos\phi$

Friction coefficient:

$\mu = \frac{F}{N} = \frac{F_t + F_c\tan\alpha}{F_c - F_t\tan\alpha}$

Thrust force:

$F_t = F_c\tan(\beta - \alpha)$

$F_c$ is always positive; $F_t$ is downward when $\beta > \alpha$ and upward when $\beta < \alpha$ (high rake, low friction). In metal cutting $mu approx 0.5$–$2$.

Power & Specific Energy

$\text{Power} = F_c V$

Shear-zone and friction power:

$\text{Power}_{\text{shear}} = F_s V_s, \qquad \text{Power}_{\text{friction}} = F V_c$

With width of cut $w$, specific energy:

$u_{\text{total}} = \frac{F_c V}{w\,t_0\,V} = u_{\text{shear}} + u_{\text{friction}}$

Temperature

Energy → heat → cutting-zone temperature rise (hurts tool hardness/wear and part accuracy). Mean turning temperature:

$T_{\text{mean}} \propto V^{a} f^{b}$

Tool material	$a$	$b$
Carbide	0.2	0.125
High-speed steel	0.5	0.375

Max temperature is about halfway up the tool face; ↑speed → less time to dissipate heat → higher temperature.

Tool Life — Taylor’s Equation

Flank wear on the relief face. Taylor (1907):

$V T^{n} = C \;\Rightarrow\; V_1 T_1^{n} = V_2 T_2^{n} \;\Rightarrow\; T_2 = T_1\left(\frac{V_1}{V_2}\right)^{1/n}$

$T$ = time (min) to a given wear land; $C$, $n$ from tables. Speed dominates tool life. Extended form with depth $d$ and feed $f$:

$V T^{n} d^{x} f^{y} = C$

For constant tool life, ↑feed or ↑depth → ↓speed. Chipping = a small piece breaks off (mechanical shock).

Tool-condition monitoring: direct (toolmaker’s microscope — needs a stop) or indirect (forces, power, temperature, vibration, acoustic emission).

Machinability

Rated against AISI 1112 steel = 100 (cut at 100 ft/min ≈ 30 m/min for a 60-min tool life). Examples: free-cutting brass 300, 2011 Al 200, Ni 200, pearlitic gray iron 70, 3140 steel 55, Inconel 30, 17-7 PH steel 20. Harder material → lower rating (approximate — use with caution).

• Steels: Pb and S give free-machining steels (S lowers impact strength/ductility). Harder steels → short chips, better finish; very soft steels → BUE, poor finish.
• Aluminum, magnesium: very easy. Gray iron: machinable but abrasive. Titanium: low conductivity → hard. Tungsten: abrasive → hard. Reinforced plastics: abrasive → hard.