14.4 Metal Cutting Mechanics, Tool Life & Machine Tools

13.4 Metal Cutting Mechanics, Tool Life & Machine Tools

Machining is a subtractive manufacturing process wherein excess material is sheared away from a workpiece as chips by a harder wedge-shaped cutting tool. In modern heavy mechanical workshops and maintenance facilities, optimizing cutting parameters, tool geometries, and machine kinematics directly governs component surface integrity, dimensional accuracy, and production economics.

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1. Mechanics of Metal Cutting & Merchant's Orthogonal Circle

In orthogonal cutting, the straight cutting edge of the tool is oriented strictly perpendicular to the direction of cutting velocity ($V_c$), confining deformation to a 2D plane.

                  Merchant's 2D Orthogonal Cutting Geometry
                                Chip (t2)
                                  /  /
                                 /  /
                                /  /   Tool Rake Face
                               /  /   ┌───────────────
                              /  /    │ Rake Angle (γ)
                             /  /     │
                            /  /  φ   │
      ─────────────────────/  /───────┴───────────────
       Workpiece (t1)     /  / Shear Plane
      ───────────────────/  /─────────────────────────
                         <── Cutting Velocity (Vc)

Kinematic & Geometric Relations

1. Chip Thickness Ratio ($r$):

$r = \frac{t_1}{t_2} = \frac{l_2}{l_1} = \frac{V_{\text{chip}}}{V_c} = \frac{\sin\phi}{\cos(\phi - \gamma)} < 1$

where $t_1$ is uncut chip thickness, $t_2$ is cut chip thickness ($t_2 > t_1$), $\phi$ is the shear plane angle, and $\gamma$ is the tool rake angle.

1. Shear Angle ($\phi$) Formulation: Solving the chip ratio relation explicitly for $\phi$:

$\tan\phi = \frac{r \cos\gamma}{1 - r \sin\gamma}$

1. Shear Strain ($\gamma_s$) & Shear Strain Rate:

$\gamma_s = \cot\phi + \tan(\phi - \gamma) = \frac{\cos\gamma}{\sin\phi \cos(\phi - \gamma)}$

$\dot{\gamma}_s = \frac{V_s}{\Delta y} = \frac{V_c \cos\gamma}{\Delta y \cos(\phi - \gamma)}$

where $V_s$ is shear velocity and $\Delta y$ is shear zone thickness (typically $10\text{--}50;\mu\text{m}$).

                          Merchant's Force Circle
                                     Ft (Thrust Force)
                                     ▲
                                     │       / Fn
                          Fs         │      / 
                           \         │     /  N
                            \        │    /  /
                             \       │   /  /
                              \      │  /  /   β (Friction Angle)
                               \     │ /  /  ┌───
      ──────────────────────────\────┼───/──/────────► Fc (Cutting Force)
                                 \   │  /  /
                                  \  │ /  /
                                   \ │/  /   F (Friction Force on Tool)
                                    \│  /
                                     ▼
                                     R (Resultant Force)

Force Resolutions on Merchant's Circle

Given the experimentally measured Cutting Force ($F_c$) and Thrust Force ($F_t$):

• Frictional Force along Rake Face ($F$): $F = F_c \sin\gamma + F_t \cos\gamma$
• Normal Force on Rake Face ($N$): $N = F_c \cos\gamma - F_t \sin\gamma$
• Coefficient of Friction ($\mu$): $\mu = \tan\beta = \frac{F}{N} = \frac{F_c \sin\gamma + F_t \cos\gamma}{F_c \cos\gamma - F_t \sin\gamma}$
• Shear Force along Shear Plane ($F_s$): $F_s = F_c \cos\phi - F_t \sin\phi$
• Normal Force on Shear Plane ($F_n$): $F_n = F_c \sin\phi + F_t \cos\phi$
• Mean Shear Stress ($\tau_s$): $\tau_s = \frac{F_s}{A_s} = \frac{(F_c \cos\phi - F_t \sin\phi)\sin\phi}{w \cdot t_1}$

Ernst-Merchant Optimum Shear Angle Condition

By differentiating the cutting work with respect to $\phi$ and setting $\frac{\partial F_c}{\partial \phi} = 0$ (assuming material shear strength $\tau_s$ is independent of normal stress):

$2\phi + \beta - \gamma = \frac{\pi}{2} \implies \phi = \frac{\pi}{4} - \frac{\beta}{2} + \frac{\gamma}{2}$

Cutting Power & Specific Cutting Energy

$\text{Total Machining Power: } \quad P = F_c \cdot V_c \quad (\text{Watts})$

$\text{Material Removal Rate: } \quad \text{MRR} = w \cdot t_1 \cdot V_c = f \cdot d \cdot V_c \quad (\text{mm}^3\text{/s})$

$\text{Specific Cutting Energy: } \quad u = \frac{P}{\text{MRR}} = \frac{F_c}{w \cdot t_1} \quad (\text{J/mm}^3 \text{ or N/mm}^2)$

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2. Cutting Tool Geometry: ASA & ORS Reference Systems

                      ASA Tool Signature (7 Elements)
  [αb] ─── [αs] ─── [θe] ─── [θs] ─── [Ce] ─── [Cs] ─── [r]
   │        │        │        │        │        │        └─ Nose Radius (inches or mm)
   │        │        │        │        │        └────────── Side Cutting Edge Angle (SCEA)
   │        │        │        │        └─────────────────── End Cutting Edge Angle (ECEA)
   │        │        │        └──────────────────────────── Side Relief Angle
   │        │        └───────────────────────────────────── End Relief Angle
   │        └────────────────────────────────────────────── Side Rake Angle
   └─────────────────────────────────────────────────────── Back Rake Angle

Tool Geometry Standards Comparison

• Conversion Relations: $\phi = 90^{\circ} - C_s$ (Principal cutting edge angle), $\phi_1 = C_e$ (Auxiliary cutting edge angle).
• Theoretical Peak-to-Valley Surface Roughness ($h_{\max}$):

$h_{\max} = \frac{f^2}{8 r_{\text{nose}}} \quad \text{and Centerline Average } R_a \approx \frac{f^2}{18\sqrt{3} r_{\text{nose}}}$

where $f$ is feed rate and $r_{\text{nose}}$ is tool nose radius.

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3. Tool Wear Kinetics, Taylor's Equation & Tool Materials

                         Cutting Tool Wear Modes
                               Crater Wear (KT)
                               (Rake face diffusion at T_max)
                                      ▼
                              ┌──────███──────┐
                              │               │
     Tool Flank Face ─────────┤ █ <───────────┴─── Flank Wear (VB = 0.3 mm limit)
     (Abrasive wear)          │ █                  (ISO Tool Life Criterion)
                              │ █

Wear Mechanisms

1. Flank Wear ($VB$): Occurs on the clearance/flank face due to mechanical abrasion and adhesion against the machined workpiece surface. $VB = 0.3\text{ mm}$ (uniform) is the primary standard ISO tool life criterion.
2. Crater Wear ($KT$): Occurs on the tool rake face at a set distance behind the cutting edge where interface temperatures reach their peak ($800\text{--}1100^{\circ}\text{C}$). Driven primarily by solid-state chemical diffusion.
3. Built-Up Edge (BUE): Workpiece metal strain-hardens and welds to the tool tip at intermediate cutting speeds ($15\text{--}35\text{ m/min}$). Eliminated by increasing cutting velocity ($>80\text{ m/min}$), applying positive rake, or using high-lubricity cutting fluids.

Taylor's Tool Life Equations

$\text{Classical Taylor: } \quad V T^n = C$

$\text{Generalized (Modified) Taylor: } \quad V T^n f^a d^b = C$

where $V$ is cutting speed, $T$ is tool life, $f$ is feed, $d$ is depth of cut, $n$ is tool exponent, and $C$ is cutting constant.

• Relative Exponents: $n < a < b \implies$ Relative sensitivity on tool life: $V > f > d$ (Cutting speed exerts the most aggressive degradation on tool life, followed by feed, with depth of cut having the least influence).

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4. Machine Tool Kinematics & Operations

Lathe Lead Screw Thread Cutting

$\frac{\text{Driver Gear Teeth}}{\text{Driven Gear Teeth}} = \frac{\text{Spindle Speed}}{\text{Lead Screw Speed}} = \frac{\text{Pitch to be Cut}}{\text{Pitch of Lead Screw}}$

Up-Milling vs. Down-Milling

       UP-MILLING (Conventional)                 DOWN-MILLING (Climb)
         Tool Rotation: CCW ↺                      Tool Rotation: CW ↻
       Workpiece Feed: ──►                       Workpiece Feed: ──►
       ┌─────────────────────┐                   ┌─────────────────────┐
       │ Chip: Zero -> Max   │                   │ Chip: Max -> Zero   │
       │ Force: Upward       │                   │ Force: Downward     │
       │ Rubs at entry       │                   │ Requires backlash   │
       │ Safe without anti-  │                   │ eliminator on table │
       │ backlash nut        │                   │ Superior finish     │
       └─────────────────────┘                   └─────────────────────┘

Shaper Quick Return Mechanism

$\text{Quick Return Ratio (QRR)} = \frac{\text{Cutting Time}}{\text{Return Time}} = \frac{\alpha}{\beta} = \frac{360^{\circ} - \beta}{\beta} > 1$

$\text{Cutting Speed: } \quad V_{\text{mean}} = \frac{L \cdot N (1 + 1/\text{QRR})}{1000} \quad (\text{m/min})$

Grinding Wheel Standard Designation

Standard 7-element marking system (e.g., $51 - \text{A} - 46 - \text{H} - 8 - \text{V} - 23$):

$\mathbf{51} \text{ (Prefix)} - \mathbf{A} \text{ (Abrasive)} - \mathbf{46} \text{ (Grain)} - \mathbf{H} \text{ (Grade)} - \mathbf{8} \text{ (Structure)} - \mathbf{V} \text{ (Bond)} - \mathbf{23} \text{ (Suffix)}$

• Abrasive Type: $\text{A} = \text{Al}_2\text{O}_3$ (High-tensile steels), $\text{C} = \text{SiC}$ (Cast iron, brass, carbide), $\text{D} = \text{Diamond}$, $\text{B} = \text{CBN}$.
• Grain Size: Coarse ($10\text{--}24$), Medium ($30\text{--}60$), Fine ($70\text{--}180$), Very Fine ($220\text{--}600$).
• Grade (Bond Hardness): $\text{A}\text{--}\text{H}$ (Soft), $\text{I}\text{--}\text{P}$ (Medium), $\text{Q}\text{--}\text{Z}$ (Hard). Golden Rule: Soft wheel for hard materials (releases dull grains); Hard wheel for soft materials.
• Structure: $1\text{--}8$ (Dense), $9\text{--}16$ (Open).
• Bond Type: $\text{V} = \text{Vitrified}$ (glass/ceramic, high rigidity), $\text{B} = \text{Resinoid}$ (high speed cut-off), $\text{R} = \text{Rubber}$, $\text{E} = \text{Shellac}$.