This document discusses mechanics of metal cutting. It covers topics like cutting models, forces, energies, and material removal rate. The orthogonal cutting model is described, which assumes a straight cutting edge generating a plane surface. Key terms like rake angle, shear angle, and cutting forces are defined. The relationships between cutting parameters, forces, power, and specific cutting energy are explained using the orthogonal model. Limitations of this simplified model are also noted.

1. MECHANICS OF METAL
CUTTING

2. Topics to be covered
 Inroduction to Machining Technology
 Cutting Models
 Turning Forces
 Merchants Circle
 Power & Energies

3. Elements of Metal Cutting

4. Tool
Workpiece
Chip
Heat Generation Zones
(Dependent on sharpness
of tool)
(Dependent on µ)
(Dependent on φ)
10%
30%
60%

5. Tool TerminologyTool Terminology
Side relief
angle
Side cutting
edge angle
(SCEA)
Clearance or end
relief angle
Back
Rake
(BR),+
Side Rake
(SR), +
End Cutting
edge angle
(ECEA)
Nose
Radius
Turning
Cutting
edge
Facing
Cutting
edge

6. Cutting Geometry

8. 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
vR << vF

9. Cutting Models
ORTHOGONAL GEOMETRY OBLIQUE GEOMETRY
Tool
workpiece
Tool
workpiece

10. 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)

11. Orthogonal Cutting
r =
to
tc
=
ls sinφ
ls cos(φ − α)
tan φ =
r cosα
1 − rsin α
γ =
AC
BD
=
AD + DC
BD
= tan(φ −α) + cotφ

12. F t
FC
Fr
DIRECTION OF ROTATION
WORKPIECE
CUTTING TOOL
DIRECTION OF FEED
Velocity of
Tool relative to
workpiece V
Longitudinal
'Thrust' Force (27%)
Radial
Force (6%)
Tangential 'Cutting' Force (67%)
‘Turning’ Forces For Orthogonal
Model
End view section 'A'-'A'
Note: For the 2D Orthogonal Mechanistic
Model we will ignore the radial component
Ft
'A' 'A'
cF

13. FL
FC
Fr
DIRECTION OF ROTATION
WORKPIECE
CUTTING TOOL
DIRECTION OF FEED
Velocity of
Tool relative to
workpiece V
Longitudinal Force
Radial Force
‘Thrust’ Force
Tangential Force
'Cutting' Force
‘Facing’ Forces For Orthogonal Model
End view
Note: For the 2D Orthogonal Mechanistic
Model we will ignore the Longitudinal
component

14. 'Turning' Terminology
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
= πDN
Standard Terms
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
φ D
d mm
feed
(mm/rev)
Tool
Workpiece
rpm

15. Orthogonal Cutting Model
(Simple 2D mechanistic model)
Mechanism: Chips produced by the shearing process along the shear plane
α
t0
φ
+
Rake
Angle
Chip
Workpiece
Clearance AngleShear Angle
t c
depth of cut
Chip thickness
Tool
Velocity V
tool

16. tool
Cutting Ratio
(or chip thicknes ratio)
As Sinφ =
to
AB
and Cos(φ-α) =
tc
AB
Chip thickness ratio (r) =
t0
tc
=
sinφ
cos(φ−α)
φ
tc
to
(φ−α)
A
B
Chip
Workpiece

17. Experimental Determination of
Cutting Ratio
Shear angle φ may be obtained
either from photo-micrographs
or assume volume continuity
(no chip density change):
Since t0w0L0 = tcwcLc and w0=wc (exp. evidence)
Cutting ratio , r =
t0
tc
=
Lc
L0
i.e. Measure length of chips (easier than thickness)
w
t
L
0
0
0
wc
Lc
ct

18. Shear Plane Length
and Angle φ
Shear plane length AB =
t0
sinφ
Shear plane angle (φ) = Tan
-1 rcosα
1-rsinα
or make an assumption, such as φ adjusts to minimize
cutting force: φ = 45
0
+ α/2 - β/2 (Merchant)
φ
tc
to
(φ−α)
A
B
Chip
tool
Workpiece

19. Velocities
(2D Orthogonal
Model)
Velocity Diagram
From mass continuity: Vto = Vctc
From the Velocity diagram:
Vs = V
cosα
cos(φ−α)
Vc = Vr and Vc = V
sinφ
cos(φ−α)
(Chip relative
to workpiece)
V = Chip Velocity
(Chip relative to tool)
Tool
Workpiece
Chip
V
s V = Cutting Velocity
(Tool relative to
workpiece)
Shear Velocity
c
α
φ − α
90 − φ φ
Vs
V c
V

20. Cutting Forces
(2D Orthogonal Cutting)
Free Body Diagram
Generally we know:
Tool geometry & type
Workpiece material
and we wish to know:
F = Cutting Force
F = Thrust Force
F = Friction Force
N = Normal Force
F = Shear Force
F = Force Normal
to Shear
c
t
s
n
Tool
Workpiece
Chip
Dynamometer
R
R
R
R
Fc
Ft
φ
sF
Fn
N
F

21. Force Circle Diagram
(Merchants Circle)
R
F
t
Fc
Tool
F
N
α
β − α
β
α
α
Fs
φ
β − α
φ
F
n

22. Results from
Force Circle Diagram
(Merchant's Circle)
Friction Force F = Fcsinα + Ftcosα
Normal Force N = Fccosα - Ftsinα
Shear Force Fs = Fccosφ - Ftsinφ
µ = F/N and µ = tanβ (typically 0.5 - 2.0)
Force Normal to Shear plane Fn = Fcsinφ + Ftcosφ

23. 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

24. Stresses
On the Shear plane:
Normal Stress = σs = Normal Force / Area =
Fn
AB w
=
Fnsinφ
tow
Shear Stress = τs = Shear Force / Area =
Fs
AB w
=
Fssinφ
tow
On the tool rake face:
σ = Normal Force / Area =
N
tc w
(often assume tc = contact length)
τ = Shear Force / Area =
F
tc w
Note: τs = τy = yield strength of the material in shear

25. Pow
er
•Power (or energy consumed per unit time) is the product of
force and velocity. Power at the cutting spindle:
•Power is dissipated mainly in the shear zone and on the rake
face:
•Actual Motor Power requirements will depend on machine
efficiency E (%):
Cutting Power Pc = FcV
Power for Shearing Ps = FsVs
Friction Power Pf = FVc
Motor Power Required =
Pc
E
x 100

26. Material Removal Rate (MRR)
Material Removal Rate (MRR) =
Volume Removed
Time
Volume Removed = Lwto
Time to move a distance L = L/V
Therefore, MRR =
Lwto
L/V
= Vwto
MRR = Cutting velocity x width of cut x depth of cut

27. 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
Volume Removed
=
Energy per unit time
Volume Removed per unit time
Specific Energy for shearing Us =
FsVs
Vwto
Specific Energy for friction Uf =
FVc
Vwto
=
Fr
wto
Ut =
Cutting Power (Pc)
Material Removal Rate (MRR)
=
FcV
Vwto
=
Fc
wto

28. 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

29. Specific Cutting Energy
Relationship to Shear strength of Material
SHEAR ENERGY / UNIT VOLUME
Specific Energy for shearing Us =
FsVs
Vwto
FRICTION ENERGY / UNIT VOLUME
Specific Energy for friction Uf =
FVc
Vwto
=
Fr
wto
=
F
wtc
= τ
APPROXIMATE TOTAL SPECIFIC CUTTING ENERGY
Ut = Us + Uf = τsγ + τ = τy(1+γ )
Us =
τscosα
sinφ cos(φ−α)
= τs.γ

30. Relation between Pressure and
Cutting velocity

31. Effect of Rake angle on Cutting
Force

32. Average Unit Horsepower Values
of Energy per unit volume

33. 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)