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Anderson G4:Users:peterand:Central files:Courses:MSE 765:2005:hw6_sol.doc
Materials Science and Engineering 765
mechanical behavior of materials
Spring 2005
Homework 6
(w/solutions)
(w/corrections
6.12.05)
Due:
Wednesday, June 8, by 9am.
Show all work necessary to arrive at your answers.
1.
Conversion between True and Engineering StressStrain Response
Recall a common relation for uniaxial true stress vs uniaxial true (logarithmic) plastic
strain:
σ
=
σ
0
ε
p
ε
0
n
+
1
,
where
σ
0
,
ε
0
are the initial yield strength, initial yield strain and
n
is the hardening
exponent.
a.
Plot the true stresstrue strain response for a material with an initial yield strength
of 100 MPa, Youngs modulus
E
= 100 GPa, and strain hardening exponent
n
=
0.1, from
ε
= 0 to 1.
You can ignore the contribution of elastic deformation to a
change in cross sectional area.
According to the constitutive relation,
σ
=
σ
0
when the plastic strain is zero.
Therefore,
σ
0
in this relation is better described as the limiting stress for elastic
behavior.
The reference strain
ε
0
is approximated by
σ
0
/E = 10
3
.
The total true
strain is the sum of elastic strain (=
σ
/E) and true plastic strain (
ε
p
).
The MathCad
program below computes the true stresstrue logarithmic strain behavior in
uniaxial tension.
σ
0
100
E
.
100 10
3
ε
0
σ
0
E
n
0.1
σ
ε
p
.
σ
0
ε
p
ε
0
n
1
ε
ε
p
ε
p
σ
ε
p
E
ε
eng
ε
p
e
ε
p
1
σ
eng
ε
p
.
σ
ε
p
e
ε
p
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Anderson G4:Users:peterand:Central files:Courses:MSE 765:2005:hw6_sol.doc
0
0.5
1
0
200
400
strain
σ
ε
p
σ
eng
ε
p
,
ε
ε
p
ε
eng
ε
p
0.05
0.06
0.07
0.08
236
238
240
strain
σ
eng
ε
p
ε
p
b.
Plot the corresponding engineering stressengineering strain response.
Superimpose your result on the same axes as in part (a).
Note that engineering quantities can be defined in terms of true quantities.
The
derivations below assume a deforming section of length
l
and cross sectional area
A
, with initial values given by subscript "0". Values associated with plastic
deformation are denoted by superscript "p".
The MathCad code above
implements these equations.
ε
eng
=
Δ
l
l
0
=
e
ε
−
1;
σ
eng
=
F
A
0
=
σ
A
A
0
≈
σ
l
0
l
p
=
σ
e
−
ε
p
c.
At what plastic strain is localization expected to begin?
There are two possible ways to answer this question.
The Considére analysis
states that necking occurs when the force carried by the specimen reaches a
maximum, due to the competition between hardening and area reduction.
Graphically, this occurs in the plot when the engineering stress reaches a
maximum or when the condition d
σ
/d
ε
p
=
σ
is reached on a true stresstrue plastic
strain curve.
The expanded plot below suggests the engineering stress peaks at
ε
p
= 0.06.
One might attempt to get an analytic expression using d
σ
/d
ε
p
=
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 Spring '05
 Anderson
 Materials Science And Engineering, Deformation, Tensile strength, Yield surface

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