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Fundamentals of Metal Forming
1. Overview2. Material Behavior
3. Temperature Effects4. Strain Rate Effect
5. Friction and Lubrication
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Introduction• Metal Forming uses plastic deformation to
change the shape of metal workpieces– Materials (metals) – plastic deformation– External loads – Typically compressive
• Sometimes Stretch the metal (tensile), bend the metal (tensile and compressive), shear stresses
– Shape - die and tools• Classification
– Bulk Deformation – Sheet Metal Forming – High surface area-to-volume
ratio • Parts are called stampings• Usual tooling: punch and die
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Introduction• Bulk Deformation
Process– Rolling– Forging– Extrusion– Drawing
• Sheet metal forming– Bending– Drawing– Shearing– (Stamping)
+
+
v
Fv
F
F
RAMF
vF
v
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Forging Sequence of Part and Die
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Material Properties in Forming
• Desirable material properties: – Yield strength ?– Ductility ?– Second phase or Inclusion ?
• These properties are affected by temperature: – When work temperature is raised, ductility increases
and yield strength decreases while loosing on surface finish and dimension accuracy.
• Other factors: – Strain rate and friction
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Theories of Failure• The limit of the stress state on a material
– Ductile Materials - Yielding– Brittle Materials - Fracture
• In a tensile test, Yield or Failure Strength of a material.
• In a multiaxial state of stress, how do we use Yield or Failure Strength?
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Yielding: Ductile MaterialsA. Maximum Shear Stress Theory
(Tresca Criterion)
2maxyS
=τ
y
y
y
S
S
S
=−
=
=
21
2
1
σσ
σ
σFor Plane Stress:σ1, σ2 have same signs.
σ1, σ2 have opposite signs.
In a tension specimen: τmax
Sy
(σ3=0)
The diameter of the Mohr circle = Sy
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Ductile Materials
[ ] ( )[ ]31232123
22
21332211 2
21
21 σσσσσσνσσσεσεσεσ ++−++=++=
Eu
For the three principal stresses;
in an uniaxial tension test. 2
31
yd SE
u ν+=
For plane stress; [ ]2221
213
1 σσσσν+−
+=
Eud
22221
21 yS=+− σσσσ
σ2
σ1
A
B
After taken out the hydrostatic stress (σave=(σ1+σ2+σ3)/3)
( ) ( ) ( )aveaveave σσσσσσ −−− 321 ,,Now substitute σ1, σ2 and σ3 with
B. Maximum Distortion Energy Theory (von Mises Criterion)
[ ] 22221
21 3
13
1yS
EEνσσσσν +
=+−+
In a biaxial case, the same amount of distortion energy
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Plasticity• Flow theory (Classical theory)
– The current strain rates depend on the stress.• Deformation theory (Hencky theory)
– The total strain is related to the stress.– Ideal for nonlinear elasticity– Still work for monotonically increasing stresses
everywhere in a body• Pressure-independent – Hydrostatic pressure
does not affect dislocation motion.• Bauschinger effect
– The different behaviors in tension and compression
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Stress Representation
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σx
σz
σy
τyzτxz
τxy
ij
zyzxz
yzyxy
xzxyx
σσσσσσσσσσ
στττστττσ
=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
333231
232221
131211
Tensor Notation
General 3-D Stress Plane Stress
σx
σyτxy
σx τ xy 0
τ xy σy 0
0 0 0
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
= σij
z
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Stress Transformation (Plane Stress)
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σx' =σx +σy
2+
σx −σy
2cos2θ +τ xy sin2θ
τ x'y ' = −σx −σy
2sin2θ +τ xy cos2θ
σy' =σx +σy
2−
σx −σy
2cos2θ −τ xy sin2θ
σx' +σy ' =σx +σy
2+
σx −σy
2cos2θ +τ xy sin2θ +
σx +σy
2−
σx −σy
2cos2θ −τ xy sin2θ = σx +σy
−σx 'σy' +τ x 'y'2 = −
σx +σy
2+
σx −σy
2cos2θ +τ xy sin2θ
⎛
⎝ ⎜
⎞
⎠ ⎟
σx +σy
2−
σx −σy
2cos2θ −τ xy sin2θ
⎛
⎝ ⎜
⎞
⎠ ⎟ + −
σx −σy
2sin2θ +τ xy cos2θ
⎛
⎝ ⎜
⎞
⎠ ⎟
2
= −σx +σy
2
⎛
⎝ ⎜
⎞
⎠ ⎟
2
+σx −σy
2cos2θ +τ xy sin2θ
⎛
⎝ ⎜
⎞
⎠ ⎟
2
+ −σx −σy
2sin2θ +τ xy cos2θ
⎛
⎝ ⎜
⎞
⎠ ⎟
2
=σx +σy
2
⎛
⎝ ⎜
⎞
⎠ ⎟
2
−σx −σy
2cos2θ +τ xy sin2θ
⎛
⎝ ⎜
⎞
⎠ ⎟
2
+ −σx −σy
2sin2θ +τ xy cos2θ
⎛
⎝ ⎜
⎞
⎠ ⎟
2
= −σxσy +τ xy2
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Yield function in 3Dσij[ ]= σ'ij +σmδ ij[ ]
where σ'ij = DeviatoricStress Tensor,
σm =1
3σ1 +σ2 +σ3( )= Hydrostatic Component of Stress
δ ij =1 for i = j
0 for i ≠ j
( )( )
321
2223
132321
2222
3211
322
13
2
where0
σσσ
τστστστττσσσ
σσσσσσ
τττσσσσσσ
σσσσσσσσσ
=
−−−+=
++−=
+++++−=
++=++==−−−
xyzxzyyzxxzyzxyzyx
zxzyyx
zyx
I
I
IIII
zxyzxy
( ) ( ) ( ) ( ) CJJfIIIfff Iij ==== 32321 ,,,σσ
Stress Tensor :
Three Stress Invariants
Yield function:
( )zyxJ
JJ
JJJ
'''''''''
0 where0'''
3
1323212
1
322
13
σσσσσσσσσ
σσσ
=++−=
==−−−
Three Deviatoric Stress Invariants
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Example: Stresses in 3D
[ ]
( )( ) ( ) ( )( )( )( )
( )72
and54)182412( ,13
MPa6andMPa4 MPa,307254130634
4445
0401
041115
MPa401041115
3213
1323212
3211
321
23
2
3
2
1
==−=++−=++−=
=++====
=−+−
=−−−==−−−−−−
=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−−−
−−−
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−
−−=
σσσσσσσσσ
σσσσσσ
σσσ
σσσσσσσ
σσ
σ
σ
III
nnn
ij
To find Principal Stresses:
[ ]⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
zyzxz
zyyxy
zxyxx
ij
στττστττσ
σσσσσσσσσ
σ
332313
322212
312111
x
y
z
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Example: Continue
[ ] [ ]
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−−−
−−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−
−−=
+=
33.001033.011167.0
33.400033.400033.4
401041115
'
'
ij
ijmijij
σ
δσσσ
[ ]
MPa6&4,3
MPa401041115
31 =
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−
−−=
−σ
σ ij
( ) ( ) MPa33.464331445
31
=++=++=mσ
74.0;33.2;00''33.274.0
0''31
920)'33.0(
0)'33.0(2)'67.0()'33.0(
'33.0010'33.0111'67.0
321
3
2
2
=−===−+
=⎟⎠⎞
⎜⎝⎛ −++
=++−+
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−−−−−
−−−
JJJσσ
σσσ
σσσ
σσ
σ
Deviatoric Stress
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Presentation of Yield Surface
( )CS
Cf
y =
=−= 313,2,1 σσσ
( ) ( )
3
;461, 222
232
yS
CJJJf
=
=++==
σ
σσσ
0,, 321 ==−= σσσσσ
Max. Shear Stress Theory: ( )( ) CkJkJkJJJJf
Cf
=−+−−=
=−=6
242
222
33232
313,2,1
649636274,
σσσ
( )
( ) ( ) ( ) ( )[ ] CJf
CkJJJf
=−+−+−==
===
213
232
22123,2,1
2232
61
,
σσσσσσσ
Maximum Distortion Energy Theory:
Based on uniaxial yielding: 0, 321 === σσσσ
( ) ( )
33
;61,
22
22232
ySC
CJJJf
==
=+==
σ
σσ
Maximum Distortion Energy TheoryMax. Shear Stress Theory
For shear:
2
;2)( 3,2,1
y
y
S
SCf
=
===
σ
σσ
Sy
Sy
2yS
2yS
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Plasticity
( )
( )
( ) 0,
0,
0,
=∂∂
==
<∂∂
==
>∂∂
==
ijij
ij
ijij
ij
ijij
ij
fdfCf
fdfCf
fdfCf
σσ
σ
σσ
σ
σσ
σLoading:
Unloading:
Neutral:
1
2
3
Isotropic hardening
SY
SY
S<SY
Bauschinger effect
S`Y
Plastic State:
Elastic State:
Impossible:
( )( )( ) 0
0
0
>
<
=
ij
ij
ij
f
f
f
σ
σ
σ
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Strain HardeningIsotropic Hardening
Kinematics Hardening
Combined Hardening General Hardening
ijij
fddσ
λε∂∂
=Normality Rule:
Convexity
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2. Behavior in Metal Forming• The stress-strain relationship beyond elastic range
assuming no unloading at anytime and anywhere.
• Flow stress – The instantaneous value of stress required to continue deforming the material.
• Average Flow Stress
• For any metal, K and n in the flow curve depend on temperature
nKεσ =
nf KY ε=
nKdKd
Yn
f
f
n
ff
ff
+===
∫∫1
00 εε
εε
ε
εσεε
fYYs
εf
fY
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3. Temperature in Metal Forming• Cold Working – Performed at room
temperature or slightly above – Near net shape or net shape– Better accuracy, closer tolerances & surface finish– Strain hardening increases strength and hardness– Grain flow during deformation can cause desirable
directional properties in product– No heating of work required– Higher forces and power required– Surfaces must be free of scale and dirt– Ductility and strain hardening limit the amount of
forming
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Warm Working• Performed at above room temperature but below
recrystallization temperature• 0.3Tm, where Tm = melting point (absolute
temperature)– Lower forces and power than in cold working– More intricate work geometries possible– Need for annealing may be reduced or eliminated
• Isothermal Forming – eliminate surface cooling especially highly alloyed steels and Ti and Ni alloys
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Hot Working• Deformation process at temperatures above
recrystallization temperature (0.5Tm) • A perfectly plastic material - Strain hardening
exponent is zero (theoretically)– Lower forces and power required– Metals become ductile– Strength properties are generally isotropic– No work hardening of part– part can be subsequently cold formed– Lower dimensional accuracy– Higher total energy required– Poorer surface finish including oxidation (scale), – Shorter tool life
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4. Effect of Strain Rate
• Sensitive to strain-rate at elevated temperatures• Strain rate: • Relationship:• A more complete relationship:• Evaluation of strain rate is complicated by
– Workpart geometry– Variations in strain rate on the part
• Strain rate can reach 1000 s-1 or more for some metal forming operations
hv
=ε&m
f CY ε&=mn
f AY εε &=
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Effect of temperature on flow stress
Increasing temperature decreases C & increases mAt room temperature, effect of strain rate is almost negligible
ε&
Yf
C
1.0 2.0 3.0
Yf
1 100.1 100 ε&ε&
Yf
C
1 100.1 100
m
Room Temperature
400°C
800°C
1200°C
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5. Friction and Lubrication• Friction – retard metal flow and increase power and wear
• Lubrication – reduce friction & heat, improve surface finish– Choosing a Lubricant – Type of operation, reactivity, work
materials, cost and ease of applications• Cold working – mineral oil, fats, fatty oils, water-based emulsions,
soaps and coating• Hot working – mineral oil, graphite and glass
Categories Temp. Range Strain-rateSensitivity exponent
Coefficient of Friction
Cold Working <0.3Tm 0<m<0.05 0.1
Warm Working 0.3Tm-0.5Tm 0.05<m<0.1 0.2
Hot Working 0.5Tm-0.75Tm 0.05<m<0.4 0.4-0.5