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AE 321 – Practice Problems
Chapter 4: Material Behavior
1. For a linearly elastic isotropic material starting from
!
ij
=
2
μ
"
ij
+
#$
ij
kk
,
where
λ
, μ
are the Lamé moduli, show that
E
ij
=
1
+
( )
#
ij
$
"%
ij
kk
,
where
E
,
ν
are the Young’s modulus and Poisson’s ratio, respectively.
2. (a) Show that the principal axes for stress and strain coincide for a linearly elastic isotropic
material.
(b) Under what condition do the principal axes for stress and strain coincide for an orthotropic
solid?
3. Show that for a linear, elastic, isotropic homogeneous solid, strain energy density
W
can be
expressed as:
(a)
W
=
ij
ij
+
2
kk
( )
2
(b)
W
=
1
+
2
E
ij
ij
#
2
E
kk
( )
2
(c)
W
=
1
2
E
Q
1
2
!
2 1
+
( )
Q
2
[ ]
where
Q
1
and
Q
2
are the first and second stress invariants, and the other symbols have their usual
meaning.
4. For steel
E
= 207 GPa and
= 0.3. Assume that the strain in an (x
1
, x
2
, x
3
) frame at a given
point is
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ij
[ ]
=
0.004
0.001
0
0.006
0.004
sym
.
"
0.001
#
$
%
%
’
(
(
.
Find the normal and shear components of traction acting at that point on a surface with normal
(1/
√
3, 1/
√
3, 1/
√
3).
5. For steel
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 Spring '09
 Lambros

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