# Lecture006 Bulk Deformation SP 2022-3

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ME 324 Manufacturing Engineering
1
Rolling Process
2
Rolling Process
o Describe rolling process
o Slab method to analyze normal &amp; frictional forces
o Neutral point
o Front &amp; Back Tension
o Roll forces, Torque, Power
o Defects in Rolling
o Different rolling arrangements for complex geometries
3
Flat-And-Shape-Rolling Processes
Schematic outline of various flat-and-shape-rolling processes.
4
Changes In Grain Structure
o Changes in the grain structure of a cast or large-grain wrought metals during
hot rolling.
o Hot rolling is an effective way to reduce grain size in metals for improving
strength and lowering ductility.
o Cast structures of ingots or continuous castings are converted to a wrought
structure by hot working.
5
Flat-Rolling Process
Schematic of flat-rolling
process. A greater surface area
of metal is formed by rolling
than by any other metalworking
process.
6
Flat-Rolling Process
Bite angle o Relative velocity distribution
between roll and strip surfaces.
o Note the difference in the direction
of frictional forces.
o The arrows represent the frictional
forces acting on the strip.
Forward slip is a measure of relative
velocities at the exit of the work roll
Forward slip =
V f − Vr
Vr
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Flat-Rolling Process
Bite angle
V0 and Vf are velocities of the
work piece at the entrance
and exit of the rollers.
Vf &gt;V0. Vr = constant
Since Vr remains constant and the
workpiece velocity increases, sliding
occurs to compensate for the increase in
velocity
A point in the rolling region where workpiece velocity = Vr.
This is Neutral Point
8
Stresses In Rolling
Equilibrium of horizontal forces on the element:
(σ x + dσ x )(h + dh ) − 2 pRdφ sin φ − σ x h &plusmn; 2&micro;pR dφ cos φ = 0
This expression can be simplified to
d (σ x h )
= 2 pR(sin φ  &micro; cos φ )
dφ
Using small angle approximations,
this further simplifies to
sinϕ = ϕ; cosϕ = 1
d (σ x h )
= 2 pR(φ  &micro; )
dφ
Stresses on an element in rolling:
(a) entry zone and (b) exit zone. 9
Roll Pressure Distribution
Bite angle
Since bite angle (α) is small, p assumed
as principal stress, the other principle
stress is 𝜎𝜎𝑥𝑥 and their relationship with
𝜎𝜎𝑓𝑓′ is
2
𝜎𝜎𝑓𝑓 = 𝜎𝜎𝑓𝑓′
𝑝𝑝 − 𝜎𝜎𝑥𝑥 =
3
Equilibrium of horizontal forces
𝑑𝑑 𝑝𝑝 − 𝜎𝜎𝑓𝑓′ ℎ
= 2𝑝𝑝𝑝𝑝 𝜙𝜙 &plusmn; 𝜇𝜇
𝑑𝑑𝑑𝑑

p
d
′
Differentiating

Y𝜎𝜎𝑓𝑓f′ h
′′
dφ  Y
yields
𝜎𝜎𝑓𝑓f
d (σ x h )
= 2 pR(φ  &micro; )
dφ
  p
 d
+

𝜎𝜎𝑓𝑓f′′ h = 2 pR (φ  &micro; )
−
1
Y
  Y𝜎𝜎f′′
 dφ
  𝑓𝑓

( )
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Roll Pressure Distribution
Letting hf be the
final thickness, we
have
d
dφ
 p 


 Y f′′ 
 𝜎𝜎𝑓𝑓  = 2 R (φ  &micro; )
p
h
Y
𝜎𝜎f′′
𝑓𝑓
d
dφ
 p

 Y f′′
 𝜎𝜎𝑓𝑓
p
Y ′f′



 = 2 R (φ  &micro; )
h
𝜎𝜎𝑓𝑓
Substituting the value of h into Eq.(6.26a) produces
h

&micro;
H
p = CY ′ e
R
𝜎𝜎𝑓𝑓f′
where


H = 2 R tan −1  R φ 
 hf 
hf


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Roll Pressure Distribution
Looking first at just the entry zone
R
&micro;H i
C=
e
hf
and
p
At the exit zone
R
&micro;H i
C= e
hf
and
p
= Y𝜎𝜎𝑓𝑓f′′
′
= Y𝜎𝜎𝑓𝑓f′
h e &micro; (H 0 − H )
h0
h e &micro; (H 0 − H )
h0
@ H f = H (φ = 0 ) = 0
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Pressure Distribution in Rolling
o Pressure distribution in the roll gap as a
function of the coefficient of friction.
o Note that as friction increases, the neutral
𝑝𝑝
𝑆𝑆𝑦𝑦′
point shifts toward the entry.
o Without friction, the rolls slip, and the
neutral point shifts completely to the exit.
Plane strain condition
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Pressure Distribution in Rolling
o Pressure distribution in the roll gap as
a function of reduction in thickness.
𝑝𝑝
𝑆𝑆𝑦𝑦′
o Note the increase in the area under
the curves with increasing reduction in
thickness.
o Thus, increasing the roll-separating
force.
Plane strain condition
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Determination of the Neutral Point
The neutral point is determined by equating the following:
p (@ neutral pt.) entry = p (@ neutral pt.) exit
&micro; (H − H f )
′
′
h
h
(
&micro;
H0 −H )
Y
Y𝑓𝑓f′
= 𝜎𝜎
@H = H n
e
𝜎𝜎𝑓𝑓f′ h e
h
0
where
&micro;H 0
h0
e
&micro; (H 0 − 2 H n )
=
=e
2 &micro;H n
hf
e
f
 R 
R
−1 
H =2
φ
tan
 hf 
hf


OR
1
1 ℎ0
𝐻𝐻𝑛𝑛 =
𝐻𝐻0 − 𝑙𝑙𝑙𝑙
2
𝜇𝜇 ℎ𝑓𝑓
Substituting red equation into the blue,
φn =
 hf H 
tan
⋅ n
 R
R
2 


hf
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Effects of Front And Back Tension
Entry Zone:
pb
pf
Exit Zone:
𝑝𝑝 =
𝑝𝑝 =
𝜎𝜎𝑓𝑓′
𝜎𝜎𝑓𝑓′
− 𝑝𝑝𝑏𝑏
− 𝑝𝑝𝑓𝑓
P increases with:
Pressure distribution as function of front
&amp; back tension.
ℎ 𝜇𝜇
𝑒𝑒
ℎ0
𝐻𝐻0 −𝐻𝐻
ℎ 𝜇𝜇𝜇𝜇
𝑒𝑒
ℎ𝑓𝑓
• Increasing strength of materials;
• Increasing coefficient of friction;
• Increasing R/hf ratio (larger
To reduce the pressure, longitudinal tension
is applied at the entry or exit or both
reduction)
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Roll Forces
Roll force on the strip is product of area under pressure vs. contact –length curve
(L) and strip width (w)
The roll force, F, given by
F=
φn
∫
α
∫φ
wpR dφ + wpR dφ
0
w = Strip width
n
Another method to calculate the roll force is to use
𝐹𝐹 = (𝐿𝐿𝐿𝐿)𝑝𝑝𝑎𝑎𝑎𝑎
Contact area
Average contact pressure
L is contact-length curve and can be approximated as
L = R∆h
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Roll Forces
For low friction conditions an approximation for the roll force can
𝐹𝐹 = 𝐿𝐿𝐿𝐿
𝜎𝜎𝑓𝑓′
𝜎𝜎𝑓𝑓′ is average flow stress in plane strain of
material in the roll gap
For higher friction conditions, the following expression approximates the
roll force
𝐹𝐹 = 𝐿𝐿𝐿𝐿
𝜎𝜎𝑓𝑓′
𝜇𝜇𝜇𝜇
1+
2ℎ𝑎𝑎𝑎𝑎𝑎𝑎
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Roll Torque
The roll torque, T, for each roll can be calculated analytically from the
expression
Frictional forces
T=
α
∫φ
w&micro;pR dφ −
2
n
(entry zone)
φn
∫0
w&micro;pR dφ
2
(exit zone)
The torque per roll can be estimated—assuming the roll force acts in
the middle of the contact –length &amp; is perpendicular to the plane of the
strip—as
FL
T=
2
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Roll Power
The power required per roll is
Power = Tω
ω = 2πN
where N = Revolution per minute
The power per roll is
Power =
πFLN
60
kW
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Strain Rate and Friction in Flat Rolling
The average strain rate in flat rolling can be approximated as
𝑉𝑉𝑟𝑟
ℎ0
ln
𝜖𝜖̇ ̅ =
𝐿𝐿
ℎ𝑓𝑓
The maximum possible draft, that is (h0-hf) or ∆hmax , in flat rolling can be
related to friction and roller radius
Friction affects the maximum possible draft (∆hmax), such that
∆hmax = &micro; R
2
The maximum angle of acceptance is
α max = tan −1 &micro;
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Defects In Rolling
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Bending forces
(a) Bending of straight cylindrical
rolls (exaggerated) because of
the roll force.
(b) Bending of rolls, ground with
camber, that produce a sheet
of uniform thickness during
rolling.
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o High strip width to thickness ratio is
condition for plan strain
o If strip width is increased, spreading occurs
in flat rolling.
o Spreading can be similarly observed when
dough is rolled with a rolling pin.

′
R ′ = R1 + CF
 h0 − h f
 F’ = Roll force per unit width
 C = 2.3 x 10-2 mm2/kN for steel rolls &amp;


4.5 x 10-2 mm2/kN for cast-iron
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Defects In Flat Rolling
Schematic illustration of typical defects
in flat rolling: (a)wavy edges; (b) zipper
cracks in the center of strip; (c) edge
cracks; (d) alligatoring.
o Wavy edges: due to bending of
the rolls that causes thinner
edges of strip than center
o Zipper &amp; Edge cracks: due to
low material ductility and
barreling of the edges
o Alligatoring: due to
inhomogeneous deformation of
the material during rolling
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surface
deformation
deformation in
the bulk of the
workpiece
The effect of roll radius on the type of residual stresses developed in flat
rolling: (a) small rolls, or small reduction in thickness; and (b) large rolls, or
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large reduction in thickness.
Methods of Roller Leveling
Schematic illustration of methods of roller leveling. These processes
are used to flatten rolled sheets.
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Various Roll Arrangements
Schematic illustration of various roll arrangements: (a) two high; (b)
three high; (c) four high; (d) cluster; (e) tandem rolling with three
stands; (f) planetary.
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Cluster Mill
Schematic illustration of a cluster (Sendzimir) mill. These mills are very rigid
and are used in rolling thin sheets of high-strength materials, with good control
of dimensions.
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Shape Rolling
o Stages in shape rolling of an Hsection part.
o Various other structural
sections, such as channels and
I-beams, are also rolled by this
process.
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Ring-Rolling
(a) Schematic of a ring-rolling operation. Reducing the thickness results
in an increase in the part’s diameter.
(b) Examples of cross-sections that can be formed by ring rolling.
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Thread-rolling processes: (a) flat dies and (b) two-roller dies. These
processes are used extensively in making threaded fasteners at high
rates of production.
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(a) Schematic of machined or rolled threads. (b) Grain-flow lines in