LECTURE-08
THEORY OF METAL CUTTING
- Theory of Chip Formation
NIKHIL R. DHAR, Ph. D.
DEPARTMENT OF INDUSTRIAL & PRODUCTION
ENGINEERING
BUET
Chip Formation
Every Machining operation involves the formation of chips. The nature of
which differs from operation to operation, properties of work piece material
and the cutting condition. Chips are formed due to cutting tool, which is
harder and more wearer-resistant than the work piece and the force and
power to overcome the resistance of work material. The chip is formed by
the deformation of the metal lying ahead of the cutting edge by a process
of shear. Four main categories of chips are:
Discontinuous Chips
Continuous or Ribbon Type Chips
Continuous Chip Built-up-Edge (BUE)
Serrated Chips
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Types of Chips
Discontinuous Chips: These chips are small
segments, which adhere loosely to each other. They
are formed when the amount of deformation to which
chips undergo is limited by repeated fracturing. Hard
and brittle materials like bronze, brass and cast iron
will produce such chips.
Continuous or Ribbon Type Chips: In continuous
chip formation, the pressure of the work piece builds
until the material fails by slip along the plane. The
inside on the chip displays steps produced by the
intermittent slip, but the outside is very smooth. It
has its elements bonded together in the form of long
coils and is formed by the continuous plastic
deformation of material without fracture ahead of the
cutting edge of the tool and is followed by the
smooth flow of chip up the tool face.
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Continuous Chip Built Up Edge: This type of chip
is very similar to that of continuous type, with the
difference that it is not as smooth as the previous
one. This type of chip is associated with poor surface
finish, but protects the cutting edge from wear due to
movement of chips and the action of heat causing the
increase in tool life.
Serrated Chips: These chips are semicontinuous in
the sense that they possess a saw-tooth appearance
that is produced by a cyclical chip formation of
alternating high shear strain followed by low shear
strain. This chip is most closely associated with
certain difficult-to-machine metals such as titanium
alloys, nickel-base superalloys, and
austenitic
stainless steels when they are machined at higher
cutting speeds. However, the phenomenon is also
found with more common work metals (e.g., steels),
when they are cut at high speeds.
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Actual Chip Forms and Classifications
C-type and
ε-type
broken chips
Short helical
broken chips
Medium helical
broken chips
Long helical
broken chips
Desired
Not Desired
Long helical
unbroken
chips
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Long and snarled
unbroken chips
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Chip Formation in Metal Machining
Since the practical machining is complex we use orthogonal cutting model
to explain the mechanics.In this model we used wedge shaped tool. As the
tool forced into the material the chip is formed by shear deformation.
Uncut chip
Thickness
a1=So sin φ
Shear
plane
Rough
surface
Chip
Thickness
(a2)
Chip
Shear
Angle
(β)
Shiny
surface
Positive rake
Rake angle (γ)
Rake
surface
Clearance
angle (α)
Workpiece
Flank
surface
Tool
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Negative rake
22/6
Deformation of Uncut Layer
The problem in the study of the mechanism of chip formation is the
deformation process of the chip ahead of the cutting tool. It is difficult to
apply equation of plasticity as the deformations in metal cutting are very
large. Experimental techniques have always been resorted to for analyzing
the deformation process of chips. Several methods have been used:
Taking photographs of the side surface of the chip with a high speed
movie camera fitted with microscope.
Observing the grid deformation (directly)


on the side surface of the work piece and
on the inner surface of a compound work piece.
Examination of frozen chip samples taken by


drop tool apparatus and
quick stop apparatus,
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Grid Deformation Methods
The type of stress-state conditions is evaluated by means of an angle index e
obtainable from Levy-Lode’s theorem,
e  e  2e
o
1 2
3  tan (30  e) - - - - - -[1]
o
e e
tan30
1 2
r 
r 
e  ln  1 , e  ln  2 
1
2
ro
r
 
 o
where,
e=
=
=
=
ro =
r1 & r2 =
ro
Chip
and e  e  e  o - - - -[2]
1 2 3
deformation criteria
00 for pure tension
300 for pure shear
600 for pure compression
radius of circles marked on the workpiece
semi-axes of the ellipse after deformation.
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r2
Workpiece
r1
Tool
Schematic representation of
the translocation of circles into
ellipses during chip formation.
22/8
From Equation [1] and Equation [2]
r r 
tan(30o  e)
 ln 1 2 
o
r 2
tan30
 o 
3
r 
ln 1       [3]
r 
 2
Case-1: For Pure Tension [e=0]
r  ro (1  ε) and r  ro (1  με) - - - - - - - - - - [4]
1
2
2


2
r
r
r 
ε
ε
ε

1  1  ε, 2  1  and 2  (1  2.  )  (1  ε) - - - -[5]
r 
ro
r
2
2 4
0
 0
Where,
ε = cutting strength
μ = frictional coefficient=½
since ε is very very small so neglecting ε2
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22/9
Now, from equation [5]
 r  r 
 1  2 
 r  r 
  
 0  0 
2
rr 2
 1 2  (1 ε) (1 ε)  1- - - - -[6]
r 2
0
From Equation [3] and Equation [6]
 2 4
r 
r r
rr 3
ln 1 2 . 1 
ln( 1 2 )
2
 r 6 r2 
0
r
tan(30  e)
  1- - - -[7]
0

  0
r
r 
tan30 0
1
 1
ln( )
ln
 r 
r
2
 2
or, tan(30 0  e)  tan30 0
or, e  0o for Pure Tension
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Case-2: For Pure Shear [e=300]
r  ro (1  ε  με) and r  ro (1 - ε - με) - - - - - - - - - - [8]
1
2
 r  r 
r
r
3
3
1  1  ε, 2  1  ε and  1   2   (1  3 ε) (1  3 ε)  1 - - - -[9]
r  r 
ro
2
2 r0
2
2
 0  0
From Equation [3] and Equation [9]
3


r r 
ln 1 2 
r 2


0



r 
ln 1 
r 
 2
3


ln
1

 0 - - - - -[10]
r 
ln 1 
r 
 2
or, tan(30 0  e)  0  tan(0)
or, e  30o for Pure Tension
tan(300  e)
tan30 0
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22/11
Case-3: For Pure Compression [e=60o]
r  ro (1 με) and r  ro (1 ε) - - - - - - - - - - [11]
1
2
2


2
r
r
r
1  1  ε , 2  1  ε and  1   (1 2. ε  ε )  (1 ε) - - - -[12]
 r 
ro
2 4
2 r0
 0
2
 r  r 
 1   2   1  ε 1- ε   1- - - - - - - - - [13]
r  r 
 o  o
From Equation [3] and Equation [13]
2
 r12 r2   r2 
ln  3   
0
r0   r1 
tan(30  e)

0
0
0



1
or,
tan(30

e)


tan30

tan(

30
)
0
tan30
r 
ln  1 
 r2 
or, e  60o for Pure Compressio n
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22/12
Chip Reduction Coefficient (ξ)
Chip reduction coefficient (ξ) is defined as
the ratio of chip thickness (a2) to the
uncut chip thickness (a1). This factor, ξ, is
an index of the degree of deformation
involved in chip formation process during
which the thickness of layer increases and
the length shrinks. In the USA, the inverse
of ξ is denoted by rc and is known as
cutting ratio. The following Figure shows
the formation of flat chips under
orthogonal cutting conditions. From the
geometry of the following Figure.
a1
a2
A
B
Chip
β
O
γo
C
Workpiece
Tool
a 2 AC OA cos(β  γ 0 ) cosβ cosγ 0  sin β sin γ 0
ξ



   [1]
a1 AB
OA sin β
sin β
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22/13
Shear Angle (β)
From Equation [1]
cosβ cosγ  sin β sin γ
cosγ
0
0
0  sin γ
ξ

0
sin β
tanβ
cosγ
0
tanβ 
ξ  sin γ
o
 cosγ 

1
o  Shear angle
β  tan 
 ξ  sin γ 
o

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Condition for maximum chip reduction coefficient (ξ) from Equation [1]
dξ
d  cos(β  γ 0 ) 
 0 or

0
dβ
dβ 
sin β



sin β  sin( β  γ )  cos(β  γ )cosβ
0
0
0
2
sin β
π
cos(β  γ ) cosβ  sin( β  γ ) sin β  0  cos
0
0
2
π
cos(β  γ  β)  cos
0
2
1π

 β    γ  Shear angle
0
22
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Velocity Relationships
The following Figure shows the velocity relation in metal cutting. As the tool
advances, the metal gets cut and chip is formed. The chip glides over the rake
surface of the tool. With the advancement of the tool, the shear plane also moves.
There are three velocities of interest in the cutting process which include:
VC
= velocity of the tool
relative to the workpiece.
It
is
called
cutting
velocity
Vf
= velocity of the chip
(over the tool rake)
relative to the tool. It is
called chip flow velocity
Vs=
velocity
of
displacement
of
formation of the newly
cut
chip
elements,
relative to the workpiece
along the shear plane. It
is called velocity of
shear
γo -β
90o -γo
β
Vf
Vc
Vs
γo
Chip
Vf
Vc
Workpiece
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γo
β
Vs
90o -β+γo
Tool
22/16
According to principles of kinematics, these three velocities, i.e. their
vectors must form a closed velocity diagram. The vector sum of the cutting
velocity, Vc, and the chip velocity, Vf, is equal to the shear velocity, Vs.
Thus,
V V V
s
c
f
V
V
V
s
c

 f
sin(90 o  γ ) sin 90o  (β  γ  sin β
o

o 
V sin β
V
sin β
c
V V

 c
f
c
ξ
sin 900  (β  γ ) cos(β  γ o )

o 
V
or, c  ξ
Vf
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γo -β
90o -γo
Vf
Vc
β
Vs
γo
90o -β+γo
22/17
Kronenberg derived an interesting relation for chip reduction coefficient
(ξ) which is of considerable physical significance. Considering the motion of
any chip particle as shown in the following Figure to which principles of
momentum change are applied:
dv
F  m
dt
dθ
2
N  mω r  mv
dt
F
dv
μ 
N
v dθ
dv

 μ dθ
v
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Vc
π
(  γ0 )
2
F
Vf
γo
N
22/18
As the velocity changes from Vc to Vf, hence
π
( -γo )
V
f dv 2
  πdθ
 
π
( γ )
v
0
2
V
Vc
c
V 
π



f
 ln
 μ  γ 
Vf
γo
V 
o
2
F


 c
N


π



μ

γ
V
2
0 
c e 
V
f
This equation demonstrates that the chip reduction coefficient and chip


flow velocity is dependant on the frictional aspects at the interface as
π

μ  γ 
well as the orthogonal rake angle (γ0). If γ0 is increased, chip reduction
2

0

coefficient decreases.
ξe 
0
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22/19
Shear Strain (ε)
The value of the shear strain (ε) is an indication of the amount of deformation that
the metal undergoes during the process of chip formation. The shear strain that
occurs along the shear plane can be estimated by examining the following Figure.
The shear strain can be expressed as follows:
ε
AC AD  CD AD CD



 cot β  tan(β  γ ) - -[1]
o
BD
BD
BD BD
A
Chip=parallel
shear plates
Shear
plane
Magnitude of
strained
material
β
γo
Workpiece
Tool
β
A
D
Plate
thickness
C
a
C
β-γo
γo
B
B
b
c
Shear strain during chip formation (a) chip formation depicted as a series of parallel sliding relative to each
other (b) one of the plates isolated to illustrate the definition of shear strain based on this parallel plate model
(c) shear strain triangle
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22/20
From equation [1]
cos γ
o
ε  cot β  tan(β  γ ) 
 [2]
o sin β. cos (β - γ )
o
From velocity relationsh ip
V
cos γ
s 
o    [3]
V
cos (β - γ )
c
o
From equation [2] and equation [3]
V
s Shear strain
ε
V sin β
c
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22/21
Any questions or
comments?
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22/22