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MECHANICS OF ORTHOGONAL METAL CUTTING:

There are two schools of thoughts with regard to plastic deformation at the cutting zone.

The thin zone model is more useful for analytical purposes.

The current analysis is based on Merchant's thin shear plane model which considers the minimum energy

principle.

This model is applicable at very high cutting speeds which are generally practiced in production.

The assumptions with regard to this model are:

(i) The tool is perfectly sharp and there is no contact along the clearance face.

(ii) The surface where the shear occurs is a plane.

(iii) The cutting edge is a straight line which extends perpendicular to the direction of motion and generates a plane

surface as the work moves past it

(iv) The chip docs not flow to either side, or there is no side spread.

(v) Uncut chip thickness is constant.

(vi) Width of the tool is greater than the width of the work.

(vii) A continuous chip is produced without any BUE.

(viii) Work moves with a uniform velocity.

(ix) The stresses on the shear plane are uniformly distributed.

Fig: Various forces acting in orthogonal cutting

FH- Cutting Force to the Primary Cutting Motion Direction

Fv - Force perpendicular to the primary tool motion (thrust force)

Fs - Force along the shear plane

Ns - Force normal to the shear plane

F - Frictional force along the rake face

N - Normal force perpendicular to the rake face

Fig. Forces acting on an isolated chip in metal cutting

All these forces can be represented at the tool point in place of their actual point of action.

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By doing so it is possible to construct a cutting force circle as shown in Fig. which is often called Merchant's

circle as it was he who demonstrated it for the first time.

It would then be easy to derive the various relationships among the forces.

Fig.1_Merchant's cutting force circle in orthogonal

cutting

Fig.2 Part of merchant’s force diagram

From Figs 1 and 2 we can write From Figs 1 and 2, we can write

-----(1)

If μ is 1he coefficient of friction along the rake face. Then

β - the friction angle Φ – Shear Angle

Fig.3 Part of merchant’s force diagram

Area of the shear plane. As is given by  The shear force is given by

Ʈ - the mean shear stress in the shear plane. b - width of cut t – Uncut Chip Thickness

Mean normal stress in the shear plane. (σ)

We can show that by resolving ------ (2)

Substituting Eq. 1 in 2 we get

Similarly

In rearranging, we get

Merchant considered that Ʈ would have the value of the yield shear stress for the work material and that μ would

have the usual value for any dry sliding friction.

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To determine Φ he assumed the minimum energy principle applied in metal cutting so that the deformation

process adjusted itself to a minimum energy condition, or 

Experimentally determining the shear angle:

From Fig.

The chip thickness ratio r, which is also termed as cutting ratio, would

be

Experimentally the chip thickness ratio r could be determined by measuring the overage thickness of the chips

produced under given conditions of feed and speed.

From this it is possible to evaluate the shear angle using the above equation.

However, direct measurement of chip thickness is difficult, because of the roughness on the outside of the chip.

For this purpose an indirect measurement is followed wherein the length of a chip lc equivalent to a known length

of an uncut chip is measured.

Then, considering the fact that the depth is the same, the average chip thickness tc would be given by

Where l= length of the uncut chip

To get an exact size of uncut chip length l. we may introduce a small saw cut parallel to the axis on the work

piece so that the uncut chip size is D - diameter of the workpiece

VARIOUS VELOCITY ASSOCIATED WITH ORTHOGONAL CUTTING:

The chip velocity Vc is the velocity of the chip relative to the tool and

directed along the tool face.

The shear velocity Vs is the velocity of the chip relative to the workpiece and

directed along the shear plane.

These two velocities along with the cutting velocity V would form a closed

triangle as shown in Fig.

From this we can get

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ENERGY CONSUMED IN METAL CUTTING:

Most of the energy consumed in metal culling is utilized in the plastic deformation.

The total work done W is given by

The work done in shear Ws, is

Similarly the work done in friction Wf is

Thus,

PROBLEMS:

Sample Problem 1: A bar of 75 mm diameter is reduced to 73 mm by a cutting tool while cutting orthogonally. If the

mean length of the cut chip is 73.5 mm. find the cutting ratio. If the rake angle is 15°, what is the shear angle?

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