Merchant’s Circle Diagram and its use

In orthogonal cutting when the chip flows along the orthogonal plane, π0, the cutting force (resultant) and its components PZ and PXY remain in the orthogonal plane. Fig.  is schematically showing the forces acting on a piece of continuous chip coming out from the shear zone at a constant speed. That chip is apparently in a state of equilibrium.

The forces in the chip segment are:

• From job-side:
• Ps – Shear force.
• Pn – force normal to the shear force.

From the tool side:

R1 = R (in state of equilibrium) where, R1 = F + N

N – Force normal to rake face.
F – Friction force at chip tool interface.

The resulting cutting force R or R1 can be resolved further as,

R1 = PZ + PXY

where, PZ – Force along the velocity vector.

PXY – force along orthogonal plane.

The circle(s) drawn taking R or R1 as diameter is called Merchant’s circle which contains all the force components concerned as intercepts. The two circles with their forces are combined into one circle having all the forces contained in that as shown by the diagram called

Merchant’s Circle Diagram (MCD) in Fig.

The significance of the forces displayed in the Merchant’s Circle Diagram is:

Ps – The shear force essentially required to produce or separate the chip from the parent body by shear.
Pn – Inherently exists along with Ps.
F – Friction force at the chip tool interface.

N – Force acting normal to the rake surface.

PZ = PXY – PX + PY = main force or power component acting in the direction of cutting velocity.

The magnitude of PS provides the yield shear strength of the work material under the cutting action. The values of F and the ratio of F and N indicate the nature and degree of interaction like friction at the chip tool interface. The force components PX, PY, PZ are generally obtained by direct measurement. Again PZ helps in determining cutting power and specific energy requirement. The force components are also required to design the cutting tool and the machine tool.

Advantageous use of Merchant’s circle diagram

Proper use of MCD enables the followings:

• Easy, quick and reasonably accurate determination of several other forces from a few known forces involved in machining.
• Friction at chip tool interface and dynamic yield shear strength can be easily determined.
• Equations relating the different forces are easily developed.

Some limitations of use of MCD:

1. Merchant’s circle diagram (MCD) is only valid for orthogonal cutting.
2. By the ratio, F/N, the MCD gives apparent (not actual) coefficient of friction.
3. It is based on single shear plane theory.