For the purpose of the analysis presented below the following Idealized conditions are assumed to exist during cutting:

1. The tool is perfectly sharp and there is no contact along the flank face.

2. The shear surface is a thin plane extending upward front the cutting edge.

3. The cutting edge is a straight line perpendicular to the motion and generates a plane surface as the work moves past it.

4. Width of tool is greater than width of work piece.

5. Continuous types of chips are produced without built up edge.

6. There is no side movement of the chip in either direction.

7. The work piece moves with a uniform velocity related to the tool.

8. The stresses on the shear plane are uniformly distributed.

Chip Thickness Ratio:

• The ratio of the thickness of the chip before removal to the thickness after removal is termed as chip thickness ratio, chip thickness co efficient or cutting ratio.

• The reciprocal of this ratio is known as “Chip compression factor”.

From figure note

$\begin{aligned} \text { chip thickness Ratio }=r_{c} &=\frac{c}{t_{c}} \\ &=\frac{A B \cdot \sin \emptyset}{A B \cdot \cos (\emptyset-\gamma)} \\ &=\frac{\sin \emptyset}{\cos (\emptyset-\gamma)} \end{aligned}$

$\begin{aligned} \cos (A-B)=\cos A \cdot \cos B &+\sin A \cdot \sin B=\cos \emptyset \cdot \cos \gamma+\sin \emptyset \cdot \sin \gamma \\ &=\frac{1}{\cos \emptyset \cdot \cos \gamma+\sin \gamma)} \\ \cot \emptyset \cdot \cos \gamma &=\frac{1-r_{c} \sin \gamma}{\mathrm{r}_{\mathrm{c}}} \\ \therefore \tan \emptyset &=\frac{r_{c} \cos \gamma}{1-\mathrm{r}_{\mathrm{c}} \cdot \sin \gamma} \end{aligned}$