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TRANSCRIPT
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Todays Lecture
Engineering Stress & Strain True Stress & Strain Engineering Stress/Strain vs. True Stress/Strain Stress Strain Curves
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Deformation Mode
Basic Deformation Mode
Tension orCompression
Torsion
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Engineering Stress & Strain - Tension
Engineering Stress
Engineering Strain(compression vs. tension)
P
P
l 0l
A
0 A
x
y
)(min 0
psi A P
Areaal Nor Force
(%)min 0
0
0 l l l
l l
Lengthal Nor Change Length
e
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Engineering Stress & Strain - Tension
Elongation
Ductility (Reduction of Area)
P
P
f l 0l
f A
0 A
x
y
1000
0
l l l Elongation f
100Re0
0
A
A Aareaof duction f
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Engineering Stress & Strain - Tension
Poissons Ratio ( ~ 0.3) P
P
f l 0l
f A
0 A
x
y
y
x
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Shear Stress & Strain
Shear Stress
Shear Strain
F
F A A F
a
b tanba
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True Stress & Strain
More Accurate Measurement
True Stress
True Strain
P
P
l 0l
A
0 A
x
y
A P
Areaeous Ins Forcetantan
D D
D D
A A
l l 02000
ln2lnlnln
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Engineering Stress/Strain vs.
True Stress/Strain True Stress & Engineering Stress ( Up to necking )
True Strain & Engineering Strain ( Up to necking )
el
l l
l l
l
l
A
P
l l A
P
A
P
11 00
00
00
0000
el
l l l l
1lnlnln
0
0
0
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Engineering Stress-Strain Curve
X
0l
el
ul
f l
0e ue f e
Neck
Y
UTS
Fracture
PlasticElastic
E
Y
Offset, 0.2%
l
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Engineering Stress- Strain Curve (Contd)
Youngs Modulus : slope of the of the elasticrange
Yield Strength : stress required to generate
permanent deformation Tensile Strength : maximum stress Flow Stress : stress causes continuous deformation
after yielding Failure Stress : stress when the material fractures
E
Y
UTS
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True Stress-Strain Curve
Constitutive Eq.
(plastic range)
: strength coefficient(true stress at unit true strain)
: strain hardening exponent
n K
K
n logloglog n K
Log
log
K logn
1
True
Eng.
f
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True Stress- Strain Curve (Contd)
True Strain Equals to the n Value at Necking,i.e., Max Load (only occurs in tension)
Proof
e A A A A
00ln
ee
d d
Ad dP
e A A P 00
n K d d
dP necking At
,0,
n K nK nn 1
P
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Example 1 A strip of metal of 1.5m long is stretched in three steps:
to length of 2.0m, then to 2.5m, finally to 3.0m.Calculate engineering and true strain.
Solution :
Engineering Strain
total total f
total
total
eel
l l e
eeeel
l l e
l l l
e
l l l
e
15.1
5.10.3
783.0
200.05.2
5.20.3
250.00.2
0.25.2
333.05.1
5.10.2
0
0
321
2
233
1
122
0
011
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Example 1 (Contd)
True Strain
total total f
total
total
eel
l
l l l
l
l l
693.05.10.3
lnln
693.0
182.05.20.3
lnln
223.0
0.2
5.2lnln
288.05.10.2lnln
0
321
2
33
1
22
0
11
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Example 2
Calculate True and Engineering
at necking
psi5.0
000,100 UTS
n
psi A P
A P e A A P
e A An A A Kn
UTS
neck neck
n
850,42
850,42
5.0ln
707105.0000,001
0
0
5.00
5.00
0
5.0
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End
Questions ?
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Todays Lecture
Stress-Strain State: Hookes Law Yield Criteria:
1) Tresca2) von Mises Effective Stress and Strain Work of Deformation and Temperature
Case Studies
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Strain State - Triaxial Strain
Principal Strain
1
2
3
321
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Stress-Strain Relationship in Triaxial State
Generalized Hookes Law
Example:In Tension,
2133
3122
3211
1
1
1
E
E
E
032
E
E
132
11
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Yield Criteria
Tresca von Mises
Difference < 15% Ductile Mats Breaks at Max. Shear Stress,
while Brittle at Max. Normal Stress
y ),,max( 313221
2231232221 2 y
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Effective Stress and Effective Strain
Convenient Way of Expressing the Stress State Effective Stress - based on principal stress
Effective Stress - based on normal stress
2123123222131
2
1
5.0222222 62
1 xz yz xy z x z y y x
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Effective Stress and Effective Strain
Convenient Way of Expressing the Strain State Effective Strain
(Tresca)
(von Mises)
3132
212312322213
2
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Work of Deformation & Temp.
Specific Energy (deformation work per unit volume )
Work
Temperature
332211 d d d du
0
d u
volumeuW
cu
T
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Example 1 - Stress State & yield
Q : P?
Tresca: Von Mises:
t
r 2 1
l
stress plane shell thin
t pr
rt r p
A F F r p
t pr
tl rl p
A F
F rl p
,0
22
2
22
3
2
2
222
2
1
111
r
t p
r
t p
t
pr
y
y
y
y y
3
2
2 22312
322
21
31
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Example 3 - Work and Temp.
Q : W and T?
0
22
22
3
2
2
2222
2
1
111
2
t pr
rt r p
A F
F r p
t pr
rt r p
A F
F r p
t
r 2 1
f r
011
2211
0
22
0
11
00
21
ln22
ln
2
2ln
21
r
r
d d u
r
r
r
r
f
y
f f
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Example 3 - Work and Temp.
cr
r
cu
T
r r t r
t r r
r
VolumeuW
f y
f y
f y
0
00
20
02
00
ln2
ln8
)4()ln2(
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End
Questions ?