merchants circle

2.  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.  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.  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.
Metal Cutting is the process of removing unwanted material from the workpiece
in the form of chips

5.  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
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
Front View
Back Rake Angle
Side Rake Angle
Frictional Angle
 It is the angle between the
resultant ,of the Frictional Force &
Normal Force, and Normal
Reaction.
 λ = tan
-1
μ
μ: coefficient of friction
P
F
R
N
λ
Shear Angle
 It is the angle made by the shear
plane with the direction of the tool
travel.
Fs
Ft
Fc
Fn
F
N
V
φ
Shear Force
 Resistance to shear of the metal in
forming the chip. It acts along the
shear plane.
Normal Shear Force
 Force on the chip provided by the
workpiece. Acts normal to the shear
plane.
Friction Force
 Resisting force acted at the tool
workpiece interface to resist the
motion of tool.
Thrust Force
 This force acts normal to the
cutting force or the velocity of the
tool.
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

6.  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. Fn
Ft
Fs
Fc
F
R
α
α
φ
λ
φ
λ-α
N
V

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

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

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

12. 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. 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. 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. 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