Chip formation

A cutting edge does not split metal the way an axe splits wood. It shears it. Material just ahead of the edge is compressed, slides along a narrow zone that runs from the edge up to the free surface, and leaves as a chip that is thicker and shorter than the layer it came from. Everything else in machining — forces, heat, tool wear, surface finish — follows from what happens in that zone.

The orthogonal cutting model

To analyse chip formation, the cut is simplified to orthogonal cutting: a straight edge perpendicular to the cutting velocity, removing a layer of constant width. The deformation zone is treated as a single plane, the shear plane, running from the edge to the surface of the work.

Four quantities describe the geometry:

• h, the uncut chip thickness: the thickness of the layer before it is cut. In turning, h = f · sin κr.
• hc, the thickness of the chip after it is formed.
• α, the rake angle of the tool.
• φ, the shear angle between the shear plane and the cutting direction.

Rake angle α

10

°Friction coefficient μ

0.60

Model

Merchant Lee–Shaffer

Shear angle φ

—

Friction angle β

—

Chip thickness ratio r = h/hc

—

Chip thickness hc (h = 1)

—

Shear strain γ

—

Chip thickness ratio

The chip is always thicker than the uncut layer. Their ratio, the chip thickness ratio, is smaller than 1; its reciprocal is the chip compression ratio.

$$r = \frac{h}{h_c} = \frac{\sin\varphi}{\cos(\varphi-\alpha)}$$

The shear angle cannot be seen during the cut, but the chip can be measured afterwards. Rearranged, the same relation gives φ from a measured chip:

$$\tan\varphi = \frac{r\cos\alpha}{1-r\sin\alpha}$$

Example. A tool with α = 10° cuts a layer h = 0.20 mm thick and the chip measures hc = 0.50 mm. Then r = 0.40, tan φ = 0.4 · cos 10° / (1 − 0.4 · sin 10°) = 0.423, and φ ≈ 22.9°.

Shear strain

Every element of metal that passes through the shear plane is sheared by

$$\gamma = \cot\varphi + \tan(\varphi-\alpha)$$

For the example above, γ ≈ 2.59. Shear strains of 5 or more occur when the shear angle is small or the rake angle is low or negative. A small shear angle therefore means a thick chip, a long shear plane and a large strain, all of which cost energy.

α = −10°α = 0°α = +10°

Predicting the shear angle

Two classic models try to predict φ from the rake angle and the friction on the rake face. Friction is expressed as the friction angle β, with μ = tan β.

Merchant (1945) assumed the shear plane settles where the cutting force is smallest:

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

Lee and Shaffer (1951) treated the chip as an ideal plastic material and placed the shear plane along the direction of maximum shear stress:

$$\varphi = 45^\circ + \alpha - \beta$$

Both models agree on the direction of every effect. A larger rake angle raises φ; more friction lowers it. A larger φ means a shorter shear plane, a thinner chip, less strain and lower forces.

MerchantLee–Shaffer

What this means at the machine

• More positive rake, less force. Raising the rake angle raises φ, which lowers the cutting force and power. Sandvik Coromant gives a rule of thumb for milling: power consumption changes by about 1 % for each degree of change in rake angle.
• Negative rake for strength. A negative rake angle gives a stronger edge for hard materials and interrupted cuts, at the cost of more force and power.
• Lower friction helps. A sharp, smooth or coated rake face and effective cutting fluid lower β, which raises φ.
• The chip has to go somewhere. Whether it comes off as one long ribbon or breaks into short pieces depends on the material, the speed and the chip breaker. See chip types for what happens to the chip next.