Mechanism of Metal Cutting
This page explains the mechanics of orthogonal metal cutting, from chip formation to the Ernst–Merchant analysis, with key equations rendered using MathJax.
Chip Formation
In orthogonal cutting, material ahead of the tool is sheared along a narrow shear plane to form a chip. Types include:
- Continuous chips – ductile materials, high rake angle, good lubrication.
- Serrated chips – segmented, common in high-strength alloys at high speeds.
- Discontinuous chips – brittle materials or poor cutting conditions.
Built-Up Edge (BUE)
A BUE is a layer of workpiece material adhering to the tool tip, altering the rake angle and affecting surface finish.
- Causes: High friction, low cutting speed, strain hardening.
- Effects: Poor finish, dimensional errors, tool chipping.
- Prevention: Increase speed, use sharp tools, apply cutting fluids.
Chip Thickness Ratio
The chip thickness ratio \( r \) is:
\[ r = \frac{t_1}{t_2} \]where \( t_1 \) is the uncut chip thickness and \( t_2 \) is the chip thickness after cutting.
Shear Angle
The shear angle \( \varphi \) relates to \( r \) and rake angle \( \alpha \) by:
\[ \tan\varphi = \frac{r \cos\alpha}{1 - r \sin\alpha} \]Shear Strain
The average shear strain \( \gamma \) in orthogonal cutting is:
\[ \gamma = \tan(\varphi - \alpha) + \cot\varphi \]Chip and Shear Velocities
From the velocity triangle:
- Chip velocity: \[ V_c = \frac{V \sin\varphi}{\cos(\varphi - \alpha)} \]
- Shear velocity: \[ V_s = \frac{V \cos\alpha}{\cos(\varphi - \alpha)} \]
where \( V \) is the cutting speed.
Shear Strain Rate
Approximating shear zone thickness \( \delta \approx t_1 / \sin\varphi \):
\[ \dot{\gamma} \approx \frac{V_s \sin\varphi}{t_1} \]Ernst–Merchant Analysis
Defines the friction coefficient \( \mu \) and friction angle \( \beta \):
\[ \mu = \frac{F}{N} = \tan\beta \]Merchant’s minimum energy criterion gives:
\[ \varphi = 45^\circ + \frac{\alpha}{2} - \frac{\beta}{2} \]Forces on the shear plane:
\[ F_s = F_t \cos\varphi - F_n \sin\varphi \] \[ F_{n_s} = F_t \sin\varphi + F_n \cos\varphi \]Average shear stress:
\[ \tau_{\text{avg}} = \frac{F_s \sin\varphi}{b\, t_1} \] where \( b \) is chip width.