1. Abhishek yadav

2. outline Brief introduction to Merchant’s Circle.
Assumptions for Merchant’s Circle Diagram.
Construction of Merchant’s Circle.
Solutions of Merchant’s Circle.
Advantages of Merchant’s Circle.
Need for the analysis of cutting forces.
Limitations of Merchant’s Circle.
Conclusion

3. introduction
Merchant’s Circle Diagram is constructed to ease the analysis of cutting forces acting during orthogonal (Two Dimensional) cutting of work piece.
Ernst and Merchant do this scientific analysis for the first time in 1941 and gives the following relation in 1944
It is convenient to determine various force and angles.

4. Metal cutting
Metal Cutting is the process of removing unwanted material from the workpiece in the form of chips
Orthogonal cutting
Oblique cutting
Cutting Edge is normal to tool feed.
Here only two force components are considered i.e. cutting force and thrust force. Hence known as two dimensional cutting.
Shear force acts on smaller area.
Cutting Edge is inclined at an acute angle to tool feed.
Here only three force components are considered i.e. cutting force, radial force and thrust force. Hence known as three dimensional cutting.
Shear force acts on larger area.

5. Terminology α : Rack angle λ : Frictional angle ϕ : Shear angle
Ft : Thrust Force
Fn: Normal Shear Force
Fc: Cutting Force
Fs: Shear Force
F: Frictional Force
N: Normal Frictional Force
V: Feed velocity
Friction Force
Resisting force acted at the tool workpiece interface to resist the motion of tool.
Normal Shear Force
Force on the chip provided by the workpiece. Acts normal to the shear plane.
Normal Friction Force
It act at the tool chip interface normal to the cutting face of the tool and is provided by the tool.
Cutting Force
Force acted along the velocity of tool
Cutting force increases as speed increases and decreases as rake angle decreases
Shear Force
Resistance to shear of the metal in forming the chip. It acts along the shear plane.
Thrust Force
This force acts normal to the cutting force or the velocity of the tool.
Shear Angle
It is the angle made by the shear plane with the direction of the tool travel.
RAKE ANGLE
Back Rake Angle: It is the angle between the face of the tool and measured in a plane perpendicular to the side cutting edge
Side Rake Angle: It is the angle between the face of the tool and measured in a plane perpendicular to the base
Frictional Angle
It is the angle between the resultant ,of the Frictional Force & Normal Force, and Normal Reaction.
λ = tan-1μ
μ: coefficient of friction Fs Ft Fc Fn F N V φ
Front View
Back Rake Angle
Side Rake Angle P F R N λ

6. Assumptions for merchant’s circle diagram
Tool edge is sharp.
The work material undergoes deformation across a thin shear plane.
There is uniform distribution of normal and shear stress on shear plane.
The work material is rigid and perfectly plastic.
The shear angle ϕ adjusts itself to minimum work.
The friction angle λ remains constant and is independent of ϕ.
The chip width remains constant.
The chip does not flow to side, or there is no side spread.

7. Construction of merchant’s circleFn Fs α Fc φ V
λ-α Ft R F φ α λ N

8. Forces included in metal cutting
Fs , Resistance to shear of the metal in forming the chip. It acts along the shear plane.
Fn , ‘Backing up’ force on the chip provided by the workpiece. Acts normal to the shear plane.
N, It at the tool chip interface normal to the cutting face of the tool and is provided by the tool.
F, It is the frictional resistance of the tool acting on the chip. It acts downward against the motion of the chip as it glides upwards along the tool face.

9. solution of merchant’s circle
Knowing Fc , Ft , α and ϕ, all other component forces can be calculated as:
The coefficient of friction will be then given as : α φ
λ-α λ Fs Ft Fc Fn F N R V
On Shear plane,
Now,

10. solution of merchant’s circle
Let ϕ be the shear angle
Where, α φ
λ-α λ Fs Ft Fc Fn F N R V
Now shear plane angle
The average stresses on the
shear plane area are:

11. solution of merchant’s circleα φ
λ-α λ Fs Ft Fc Fn F N R V
Now the shear force can be written as:
and
Assuming that λ is independent of ϕ ,
for max. shear stress

12. Need of analysis of forces
Analysis of cutting forces is helpful as:-
Design of stiffness etc. for the machine tolerance.
Whether work piece can withstand the cutting force can be predicted.
In study of behavior and machinability characterization of the work piece.
Estimation of cutting power consumption, which also enables selection of the power source(s) during design of the machine tool.
Condition monitoring of the cutting tools and machine tool.

13. advantages of merchant’s circle
Proper use of MCD enables the followings :-
Easy, quick and reasonably accurate determination of several other forces from a few 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.

14. limitations of merchant’s circle
Some limitations of use of MCD are :-
Merchant’s Circle Diagram (MCD) is valid only for orthogonal cutting.
By the ratio, F/N, the MCD gives apparent (not actual) coefficient of friction.
It is based on single shear plane theory.

15. Conclusions/results
Following conclusions/results are drawn from MCD :-
Shear angle is given by
For practical purpose, the following values of ϕ has been suggested:
ϕ = α for α>15o
ϕ = 15o for α<15o

16. Thank You