Mechanics of Cutting
Geometry of chip Formation:
tc : Chip thickness
tu : Uncut chip thickness
V f : Chip Sliding Velocity
Vs : Shear Velocity
Vc : Cutting Velocity
φ : Shear Angle
∆ABC & ∆ABD
t
AB = u
sinφ
also, AB
tc
tc
=
sin(90 − (φ − α )) cos(φ − α )
tu
sin φ
=
tc cos(φ − α )
φ
90-ф+α = 90-(ф-α)
Fig: Schematic of Geometry of chip formation
SHEAR ANGLE AND CHIP THICKNESS RATIO EVALUATION
tu
rc = : Chip thickness Ratio / Coeffinicient
tc
1 cosφ cosα + sinφ sinα
=
rc
sinφ
Substitute the value of tu /tc
from earlier slide and simplify to get:
1 rc cotφ cosα + rc sinα
rc cosα= (1 − rc sinα ) tan φ
 rc cosα 
∴ tan φ =


r
sin
α
−
1
c


φ
90-ф+α
= 90-(ф-α)
How to determine φ & rc ?
tc should be determined from the chip. tu (= feed) and α are already
known.
To determine tc with micrometer, is difficult and not so because of
uneven surface. How? (say, f=0.2 mm/rev. An error of even 0.05 mm will cause
an error of 25 % in the measurement of tc)
Dr. V.K.jain,
IIT Kanpur
3
Volume Constancy Condition:
Volume
of Uncut chip = Volume of cut chip
SHEAR ANGLE AND CHIP THICKNESS RATIO EVALUATION
Lu tu b = Lc tc b
Lc = Chip length
∴ Lc tc =
Lu tu
L u = Uncut chip length
tu Lc
or , r=
=
c
tc Lu
(2-D Cutting)
b = Chip width
LENGTH OF THE CHIP MAY BE MANY CENTIMETERS HENCE THE ERROR IN EVALUTION OF
rc WILL BE COMPARATIVELY MUCH LOWER.
(rc = Lc / Lu)
Dr. V.K.jain, IIT Kanpur
4
FORCE CIRCLE DIAGRAM
α
Chip
G
E
F
Fs
Fc
α
Tool
∅
Clearance Angle
A
B
Work
Ft
Fn
α
R
∅
D
F
β
α
Δ FAD = (β - α)
Δ GAD = φ + (β - α)
N
79
Force Analysis
Forces in Orthogonal Cutting:
• Friction force, F
• Force normal to Friction force, N
• Cutting Force, FC
• Thrust force, Ft
• Shear Force, FS
•Force Normal to shear force, Fn
•Resultant force, R
G
F
A
D
C
Force Circle Diagram
FREE BODY DIAGRAM
→
→
R =' F + N
→
→
→
→
→
R = F S + F N = F c + Ft = R '
Dr. V.K.jain, IIT Kanpur
6
Force Analysis
=
F Ft cosα + Fc sinα
=
N Fc cosα − Ft sinα
Coefficient of Friction ( µ )
F Ft cosα + Fc sinα
=
= =
µ tan β
N Fc cosα − Ft sinα
β = Friction Angle
Ft + Fc tanα
µ=
Fc − Ft tan α
DIVIDE R.H.S. BY Cos α
also, β = tan −1 ( µ )
Dr. V.K.jain, IIT Kanpur
7
Foce Analysis
=
FS Fc cosφ − Ft sinφ
=
FN Ft cosφ + Fc sinφ
also,
=
FC R cos ( β − α )
=
FS R cos (φ + β − α )
Δ FAD = (β - α)
Δ GAD = φ + (β - α)
FC
cos ( β − α )
∴
=
FS cos (φ + β − α )

t u b  tu
=
×b
ShearPlaneArea (=
AS )

φ  sinφ 
sin
Dr. V.K.jain, IIT Kanpur
8
Foce Analysis
Let τ be the strength of work material
tu b
=
τ
FS A=
Sτ
sinφ
 t bτ   cos ( β − α ) 
FC =  u  

(
)
sin
cos
φ
φ
β
α
+
−



 t bτ  

1
and =
, R  u ×

(
)
sin
cos
φ
φ
β
α
+
−

 

tu bτ
sin( β − α )
Ft = R sin( β − α )=
×
sinφ cos (φ + β − α )
Ft
= tan( β − α )
Fc
Dr. V.K.jain, IIT Kanpur
9
Foce Analysis
FS
Mean Shear Stress (tchip ) =
AS
(On Chip )
( Fc cosφ − Ft sinφ ) sin φ
=
b tu
Mean Normal Stress (σ chip ) =
(On Chip )
FN
AS
( Ft cosφ + Fc sinφ ) sin φ
=
b tu
Dr. V.K.jain, IIT Kanpur
10
VELOCITY ANALYSIS
Vc : Cutting velocity of tool relative to workpiece
V f : Chip flow velocity
Vs : Shear velocity
Using sine Rule:
Vf
Vc
Vs
= =
sin(90 − (φ − α )) sin φ sin(90 − α )
Vf
Vc
Vs
= =
cos(φ − α ) sin φ cos α
=
Vs
and V =
f
Vc sin φ
= Vc ⋅ rc
cos(φ − α )
Vc cos α
V
cos α
=
⇒ s
cos(φ − α ) Vc cos(φ − α )
11
Shear Strain & Strain Rate
Two approaches of analysis:
Thin Plane Model:- Merchant, PiisPanen, Kobayashi & Thomson
Thick Deformation Region:- Palmer, (At very low speeds) Oxley, kushina,
Hitoni
Thin Zone Model: Merchant
ASSUMPTIONS:• Tool tip is sharp, No Rubbing, No Ploughing
• 2-D deformation.
• Stress on shear plane is uniformly distributed.
• Resultant force R on chip applied at shear plane is equal, opposite and
collinear to force R’ applied to the chip at tool-chip interface.
12
Expression for Shear Strain
The deformation can be idealized as a process of block slip (or preferred
slip planes)
deformation
ShearStrain(γ ) =
Length
∆s AB AD DB
=
+
γ = =
∆y CD CD CD
= tan(φ − α ) + cot φ
sin(φ − α ) sin φ + cos ϕ cos(φ − α )
,
sin φ cos(φ − α )
Dr. V.K.jain, IIT Kanpur
cos α
∴γ =
sin φ cos(φ − α )
13
Expression for Shear Strain rate
In terms of shear velocity (V s ) and chip velocity (Vf), it can be written as
∴ γ

Vs
Vs
cos α 
=
 since

Vc sin φ
Vc cos(φ − α ) 

•
Shear strain rate (γ )
 ∆s 
 
•
d γ  ∆y   ∆s  1
γ =
=
=  
dt
dt
 ∆y  ∆y
Vs
Vc cos α
= =
∆y cos(φ − α )∆y
where, ∆y : Mean thickness of PSDZ
Dr. V.K.jain, IIT Kanpur
14
Shear angle relationship
• Helpful to predict position of shear plane (angle φ)
• Relationship betweenShear Plane Angle (φ)
Rake Angle (α)
Friction Angle(β)
Several Theories
Earnst-Merchant(Minimum Energy Criterion):
Shear plane is located where least energy is required for shear.
Assumptions:• Orthogonal Cutting.
• Shear strength of Metal along shear plane is not affected by
Normal stress.
• Continuous chip without BUE.
• Neglect energy of chip separation.
Dr. V.K.jain, IIT Kanpur 15
Shear angle relationship
Assuming No Strain hardening:
dFc
=0
Condition for minimum energy,
dφ
dFc
dφ
 cos φ cos(φ + β − α ) − sin φ sin(φ + β − α ) 
τ tu b cos (β − α ) 

2
2
sin φ cos (φ + β − α )


=0
∴ cos φ cos(φ + β − α ) − sin φ sin(φ + β − α ) =
0
cos(2φ + β − α ) =
0
π
2φ + β − α =
2
π 1
∴φ =
− (β − α )
4 2
Dr. V.K.jain, IIT Kanpur
16
THANK YOU
Dr. V.K.jain, IIT Kanpur
17