LECTURE-09
THEORY OF METAL CUTTING
- Mechanics of Metal Cutting
Px
Py
Pz
NIKHIL R. DHAR, Ph. D.
DEPARTMENT OF INDUSTRIAL & PRODUCTION
ENGINEERING
BUET
Mechanics of Metal Cutting
The force acting on a cutting tool during the process of metal
cutting are the fundamental importance in the design of cutting
tools. The determination of cutting forces necessary for
deformation the work material at the shear zone is essential for
several important requirements:




to estimate the power requirements of a machine tool
to estimate the straining actions that must be resisted by the
machine tool components, bearings, jigs and fixtures
to evaluate the role of various parameters in cutting forces
to evaluate the performance of any new work material, tool
material, environment, techniques etc. with respect to machinability
(cutting forces)
Department of Industrial & Production Engineering
18/2
The force system in the general case of conventional turning process is
shown in the following Figure.
Px
Py
Pxy
Px
R
Pz
Py
Pz
Px = feed force in the direction of the tool travel
Py = thrust force in the direction perpendicular to the produced surface
Pz = cutting force or main force acting in the direction of the cutting velocity.
Department of Industrial & Production Engineering
18/3
R
XY
Y
Z
Pz
Pxy
R
X
Pxy
φ
Py
Px  Pxy sin ..................[1]
Px
Πo
Py  Pxy cos.................[2]
Department of Industrial & Production Engineering
18/4
Several forces can be defined relative to the orthogonal cutting model.
Based on these forces, shear stress, coefficient of friction, and certain
other relationships can be defined.
Pn
β
Chip
R2
Ps Pxy
F
R
R1
Workpiece
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γ
N
Pz
Tool
18/5
Merchant Circle Diagram (MCD)
The following relationships suggest a circle representation of forces as
done by Merchant and indicated in the following Figure.
R  F  N  P  P  P  P .......[3]
s n
z
xy
F  Pz sin γ o  Pxy cos γ o .....................[4]
Pxy
N  Pz cos γ o  Pxy sin γ o ....................[5]
Ps  Pz cos β  Pxy sin  ......................[6]
Pn  Pz sin β  Pxy cos  .....................[7]
From Equation [4] and [5]
η
N
Pn
R
F
Pz
β
η-γo
Ps
γo
Chip
F Pz sin γ o  Pxy cosγ o
μ 
 tan .....[8]
N Pz cosγ o  Pxy sin γ o
Where,
μ = kinetic coefficient of friction
η = mean angle of friction at the rake surface
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γo
Tool
Workpiece
18/6
From the geometry of force relations of MCD circle
Pz  R cos(η  γ 0 )................[9]
Ps  R cos(β  η  γ 0 )...........[10]
Fron Equation [9] and [10]
 cos (η  γ 0 ) 
Pz  Ps 
..........[11]
 cos (β  η  γ o ) 
Based on the shear force, the shear stress
(τs) which acts along the shear plane
between the work and the chip is:
Pxy
η
N
τs 
Ps
S t
, where As  area of the shear plane  o
As
sin β
τs 
Ps sin β
..........[12]
So t
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Pn
R
F
Pz
β
η-γo
Ps
γo
Chip
γo
Tool
Workpiece
18/7
From Equation [11]and [12]


cos(η  γ 0 )
Pz  τ s S0 t 
...........[13]
 sin β cos(β  η  γ o ) 
Similarly,


sin( η  γ 0 )
Pxy  τ s S0 t 
.......[14]
 sin β cos(β  η  γ o ) 
Pxy
η
N
Pn
R
F
Pz
η-γo
Ps
γo
In metal cutting one of the main problem is to
evaluate the cutting forces Pz and Pxy from the
given cutting conditions and initial properties of
work material and it is necessary to determine τs,
β and η by suitable relationships.
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β
Chip
γo
Tool
Workpiece
18/8
Earnest-Merchant Theory
Ernst and Merchant extended their analysis and studied the relationship between the shear angle
and the cutting conditions. They suggested that the shear angle always takes the value that
reduces the total energy consumed in cutting to a minimum. Because the total work done in
cutting is dependent upon and is a direct function of the component Pz of the cutting force, they
developed an expression for Pz in terms of β and the constant properties of the workpiece
material. Condition for maximum cutting force (Pz) from Equation [13]
cos(η  γ 0 ) 
dPz
dP
d  τ s S0 t
 0, or, z 
.
0

dβ
dβ dβ  sin β cos(β  η  γ o ) 
 cosβ cos(β  η  γ 0 )  sin β sin( β  η  γ 0 ) 
τ s S0 t cos(η  γ 0 ) 
0
2
sin β cos(β  η  γ 0 


π
cosβ cos(β  η  γ 0 )  sin β sin( β  η  γ 0 )  0, or cos(β  β  η  γ 0 )  0  cos 
2
π η γ0
β  
..................[15]
4 2 2
Combining Equation [13] and [15]
Pz  2 τ s S0 t cot .............[16]
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18/9
Merchant Theory
Merchant modified the relationship derived by Earnest-Merchant, by
assuming that the shear stress along the shear plane varies linearly with
normal stress (σn). It is given as (from the following Figure).
τ  τ  k σ ................[17]
s
0
n
From the geometry of force relations of MCD
P  R cos(β  η  γ ) and P  Rsin( β  η  γ ) ................[18]
s
0
n
0
τs
P  P tan(β  η  γ )
n
s
0
P
P
n  s tan(β  η  γ )
0
A
A
s
s
σ  τ tan(β  η  γ ) ....................[19]
n
s
0
From Eqation [17] and [19]

τ

0
τ 
.......................[20]
s 1  k tan(β  η  γ )
0
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k
τ0
σn
18/10
Combining Equation [13] and [20]
 o S 0 t cos(   0 )
Pz 
...............[ 21]
sin  cos(      0 ) 1  k tan(      0 )
Condition for maximum cutting force (Pz) from Equation [21]
dP
 o S 0 t cos(   0 )
z  0, or dPz  d
0
dβ
d d sin  cos(      0 ) 1  k tan(      0 )
cos  cos(      0 )  sin  sin(      0 )  

k cos  sin(      0 )  k sin  cos(      0 )
 s S 0 t cos(   0 ) 
or
sin  cos(      0 )  k sin  sin(      0 )
2
0
cos(2β  η  γ )  k sin(2 β  η  γ )  0
0
0
cot(2β  η  γ )  k
0
2β  η  γ  cot  1(k)  c  800 to 850...........................[ 22]
0
From Equation [21] and [22]
P  τ S tcotβ  tan(c  β) ..........[23]
z
s 0
Department of Industrial & Production Engineering
18/11
Lee and Shaffer Theory
According to this theory the shear occurs on a single plane. So for a
cutting process according to this theory, the following are supposed to
hold good:
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The material ahead of the cutting tool behaved as ideal plastic material
The chip does not get hardened
The chip and parent work material are separated by a shear plane.
Lee and Shaffer derived the following relationship as:
π
β  η  γ 0  ......................[24]
4
From Equation [13] and [24]
Pz  τ s S0 t cot β  1..............[25]
Where,
cot β 
ξ  sin γ o
1

 ξ  tan γ o
tan β
cos γ o
Pz  τ s S0 t ξ  tanγ 0  1............[25]
Department of Industrial & Production Engineering
18/12
Thermal Aspect of Chip Formation
Machining is inherently characterized by generation of heat and high
cutting temperature. At such elevated temperature the cutting tool if
not enough hot hard may lose their form stability quickly or wear out
rapidly resulting in increased cutting forces, dimensional inaccuracy of
the product and shorter tool life. The magnitude of this cutting
temperature increases, though in different degree, with the increase of
cutting velocity, feed and depth of cut, as a result, high production
machining is constrained by rise in temperature. This problem increases
further with the increase in strength and hardness of the work material.
Knowledge of the cutting temperature rise in cutting is important,
because increases in temperature:

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adversely affect the strength, hardness and wear resistance of the cutting
tool
cause dimensional changes in the part being machined, making control of
dimensional accuracy difficult and
can induce thermal damage to the machined surface, adversely affecting its
properties and service life.
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18/13
In addition, the machine tool itself may be
subjected to temperature gradients, causing
distortion of the machine. The main sources of
heat in metal cutting are shown in the following
Figure. These three distinct heat sources are:



the shear zone (q1), where the main plastic
deformation takes place
the chip-tool interface zone (q2), where secondary
plastic deformation due to friction between the
heated chip and the tool takes place
the work tool interface (q3), at flanks where
frictional rubbing occurs.
q1
q3
Workpiece
Chip
q2
Tool
The heat balance in chip formation can be written as :
Amount of heat away in chips  Amount of heat remaining in the
 Total amount  
of heat generated   cutting tool  Amount of heat passing into the workpiece  Amount

 of heat radiated into the surroundin g air

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



18/14
Various studies have been made of temperatures in cutting, based on
heat transfer and dimensional analysis, using experimental data. A
simple and approximate expression for the mean temperature for
orthogonal cutting is
0.4 U  Vc t 
T


C  K 
0.. 333
where,
T = mean temperature rise at the tool-chip interface (oC)
U = specific energy in the operation (N-m/mm3)
Vc = cutting velocity (m/sec)
t = depth of cut (mm)
pC = volumetric specific heat of the workpiece (J/mm2-C)
K = thermal diffusivity (ratio of thermal conductivity to volumetric specific heat) of the
workpiece material (m2/sec).
Department of Industrial & Production Engineering
18/15
Exercise
The dynamometer recorded the following, feed force 200 kg, cutting force
300 kg. The rake angle of the tool used was 10o. The chip thickness ratio
0.35. Find
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
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Shear angle (β)
Shear force (Ps)
Co-efficient of friction at the chip-tool interface (μ) and the friction angle (η)
Compressive force at the shear plane (Pn).
A seamless tube 3cm outside diameter is reduced in length on a lathe with
the help of a single point cutting tool. The cutting speed is 40 m/min, the
depth of cut is 0.125mm. The length of continuous chips, for one revolution
of the tube, on measurement comes to be 17.77cm. The cutting force is 200
kg and the feed force is 75 kg. the rake angle of the tool is 35o.Calculate,

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Co-efficient of friction
Chip thickness ratio
Shear plane angle
Velocity of the chip along the tool face
Velocity of shear along the shear plane
Department of Industrial & Production Engineering
18/16
During the machining of AISI-1025 steel, with 0-10-6-6-8-90-1 (mm)
ORS shaped tool the following observations were taken:
Feed 0.50 mm/rev
 Depth of cut = 2.0 mm
 Cutting speed = 40 m/min
 The shear angle = 20o
 The power consumed while machining= 3kW
 The power consumed while running idle = 0.50 kW
Calculate:
 The shear force
 Chip thickness ratio
 Normal pressure on the chip
 Chip thickness

Department of Industrial & Production Engineering
18/17
Any questions or
comments?
Department of Industrial & Production Engineering
18/18