MECHANICS OF METAL
CUTTING
Topics to be covered
 Inroduction to Machining Technology
 Cutting Models
 Turning Forces
 Merchants Circle
 Power & Energies
Elements of Metal Cutting
Heat Generation Zones
30% (Dependent on m)
(Dependent on f) 60%
Chip
Tool
Workpiece
10%
(Dependent on sharpness
of tool)
Tool Terminology
Side Rake
(SR), +
End Cutting
edge angle
(ECEA)
Facing
Cutting
edge
Nose
Radius
Clearance or end
relief angle
Back
Rake
(BR),+
Turning
Cutting
edge
Side relief
angle
Side cutting
edge angle
(SCEA)
Cutting Geometry
Material Removal Rate
MRR  vfd
Roughing(R)
f  0.4  1.25mm / rev
d  2.5  20mm
Finishing(F)
f  0.125  0.4mm / rev
d  0.75  2.0mm
v R  v F
Cutting Models
Tool
workpiece
ORTHOGONAL GEOMETRY
Tool
workpiece
OBLIQUE GEOMETRY
Assumptions
(Orthogonal Cutting Model)
 The cutting edge is a straight line extending perpendicular
to the direction of motion, and it generates a plane surface
as the work moves past it.
 The tool is perfectly sharp (no contact along the clearance
face).
 The shearing surface is a plane extending upward from
the cutting edge.
 The chip does not flow to either side
 The depth of cut/chip thickness is constant uniform
relative velocity between work and tool
 Continuous chip, no built-up-edge (BUE)
Orthogonal Cutting
r
to
ls sin f

tc ls cos(f   )
tan f 
 
r cos 
1  r sin 
AC AD  DC

 tan( f   )  cot f
BD
BD
‘Turning’ Forces For Orthogonal
Model
Velocity of
Tool relative to
workpiece V
WORKPIECE
Longitudinal F t
'Thrust' Force (27%)
F C Tangential 'Cutting' Force (67%)
DIRECTION OF ROTATION
Fr Radial
Force (6%)
'A'
'A'
CUTTING TOOL
Fc
DIRECTION OF FEED
Ft
Note: For the 2D Orthogonal Mechanistic
Model we will ignore the radial component
End view section 'A'-'A'
‘Facing’ Forces For Orthogonal Model
DIRECTION OF ROTATION
Velocity of
Tool relative to
workpiece V
F C Tangential Force
WORKPIECE
'Cutting' Force
Fr Radial Force
‘Thrust’ Force
FL
Longitudinal Force
CUTTING TOOL
DIRECTION OF FEED
Note: For the 2D Orthogonal Mechanistic
Model we will ignore the Longitudinal
component
End view
'Turning' Terminology
Standard Terms
fD
N
rpm
Workpiece
Tool
d mm
feed
(mm/rev)
Beware, for turning: In the generalized
orthogonal model depth of cut (to) is f (the feed),
and width of cut (w) is d (the depth of cut)
N is the speed in rpm
D is the diameter of the
workpiece
f is the feed (linear
distance/rev)
d is the depth of cut
V is the surface speed
= pDN
Orthogonal Cutting Model
(Simple 2D mechanistic model)
tc
Chip thickness
Velocity V
Rake
Angle
+
Chip

tool
Tool
depth of cut
t0
Shear Angle
f
Clearance Angle
Workpiece
Mechanism: Chips produced by the shearing process along the shear plane
Cutting Ratio
(or chip thicknes ratio)
Chip
f)
B
to
tool
tc
f
A
Workpiece
to
tc
As Sinf =
and Cosf-) =
AB
AB
t0
sinf
Chip thickness ratio (r) = =
tc cos(f)
Experimental Determination of
Cutting Ratio
Shear angle f may be obtained
either from photo-micrographs
or assume volume continuity
(no chip density change):
Lc
wc
tc
t0
w0
L0
Since t 0w 0L 0 = t cw cL c and w 0=w c (exp. evidence)
Cutting ratio , r =
t0 Lc
=
tc L0
i.e. Measure length of chips (easier than thickness)
Shear Plane Length
and Angle f
Chip
tool
B
to
f)
tc
f
A
Workpiece
Shear plane length AB =
t0
sinf
-1 rcos
Shear pl ane angl e (f) = Tan
1-rsi n
or make an assumption, such as f adjusts to minimize
0
f = 45 + /2 - /2 (Merchant)
cutting force:
Shear Velocity
(Chip relative
to workpiece)
Vc = Chip Velocity
(Chip relative to tool)
Velocities
(2D Orthogonal
Model)
V = Cutting Velocity
V
s
Chip
Tool
(Tool relative to
workpiece)
Workpiece
Velocity Diagram
Vc
From mass continuity: Vt o = V ct c
si nf
V c = Vr and V c = V
cos(f)
From the Velocity diagram:
c os
Vs = V
cos(f)
Vs

f 
90 f
V
f
Cutting Forces
(2D Orthogonal Cutting)
Chip
Tool
Generally we know:
Tool geometry & type
Workpiece material
R
F
f
Fs
Fn
R
R
Workpiece
N
Fc
Ft
R
Dynamometer
Free Body Diagram
and we wish to know:
F = Cutting Force
F c = Thrust Force
F t = Friction Force
N = Normal Force
F s = Shear Force
Fn = Force Normal
to Shear
Force Circle Diagram
(Merchants Circle)

Fs
Tool
f
Fc
 
F
n
F
t

R
f

N
 
F

Results from
Force Circle Diagram
(Merchant's Circle)
Friction Force F = Fcsin + Ftcos
Normal Force N = Fccos - Ftsin
m = F/N and m = tan typically 0.5 - 2.0)
Shear Force Fs = Fccosf - Ftsinf
Force Normal to Shear plane F n = F csinf + F tcosf
Forces on the Cutting Tool
and the workpiece





Importance: Stiffness of tool holder, stiffness of machine, and
stiffness of workpiece must be sufficient to avoid significant
deflections (dimensional accuracy and surface finish)
Primary cause: Friction force of chip up rake face + Shearing
force along shear plane
Cutting speed does not effect tool forces much (friction forces
decrease slightly as velocity increases; static friction is the
greatest)
The greater the depth of cut the greater the forces on the tool
Using a coolant reduces the forces slightly but greatly
increases tool life
Stresses
On the Shear plane:
Fn
Fnsinf
Normal Stress = s = Normal Force / Area =
=
AB w
tow
Fs
Fssinf
Shear Stress = s = Shear Force / Area =
=
AB w
tow
Note: s = y = yield strength of the material in shear
On the tool rake face:
 = Normal Force / Area =
N
(often assume tc = contact length)
tc w
 = Shear Force / Area =
F
tc w
Power
•Power (or energy consumed per unit time) is the product of
force and velocity. Power at the cutting spindle:
Cutting Power Pc = FcV
•Power is dissipated mainly in the shear zone and on the rake
face:
Power for Shearing Ps = FsVs
Friction Power Pf = FVc
•Actual Motor Power requirements will depend on machine
efficiency E (%):
Pc
Motor Power Required =
x 100
E
Material Removal Rate (MRR)
Material Removal Rate (MRR) =
Volume Removed
Time
Volume Removed = Lwto
Time to move a distance L = L/V
Lwto
Therefore, MRR =
= Vwto
L/V
MRR = Cutting velocity x width of cut x depth of cut
Specific Cutting Energy
(or Unit Power)
Energy required to remove a unit volume of material (often quoted as
a function of workpiece material, tool and process:
Ut =
Energy
Energy per unit time
=
Volume Removed Volume Removed per unit time
Cutting Power (Pc)
FcV
Fc
Ut =
=
=
Material Removal Rate (MRR) Vwto wto
FsVs
Specific Energy for shearing Us =
Vwto
FVc
Fr
Specific Energy for friction Uf =
=
Vwto
wto
Specific Cutting Energy
Decomposition
1.
Shear Energy/unit volume (Us)
(required for deformation in shear zone)
2.
Friction Energy/unit volume (Uf)
(expended as chip slides along rake face)
3.
Chip curl energy/unit volume (Uc)
(expended in curling the chip)
4.
Kinetic Energy/unit volume (Um)
(required to accelerate chip)
Ut = Us + Uf +Uc +Um
Specific Cutting Energy
Relationship to Shear strength of Material
SHEAR ENERGY / UNIT VOLUME
FsVs
Specific Energy for shearing Us =
Vwto
scos
Us =
= s.
si nf c os(f)
FRICTION ENERGY / UNIT VOLUME
FVc
Fr
F
Specific Energy for friction Uf =
=
=
=
Vwto
wto wtc
APPROXIMATE TOTAL SPECIFIC CUTTING ENERGY
Ut = Us + Uf = s +   y1+ )
Relation between Pressure and
Cutting velocity
Effect of Rake angle on Cutting
Force
Average Unit Horsepower Values of
Energy per unit volume
Typical Orthogonal Model
Violations
•
Geometry and form Violations (i.e. non zero angles of
inclination, not sharp - radiused end)
•
Shear takes place over a volume (not a line or plane)
•
Cutting is never a purely continuous process (cracks develop
in chip; material not homogeneous)
•
'Size Effect' - larger stresses are required to produce
deformation when the chip thickness is small (statistical
probability of imperfection in the shear zone)
•
BUE - some workpiece material 'welds' to the tool face
(cyclic in nature)