Civil Engineering Reference
In-Depth Information
constitutive equations for the nonzero stress components become
E
x
=
−
ε
x
+
)
[(1
)
e
y
]
(1
+
)(1
−
2
E
y
=
)
[(1
−
)
ε
y
+
ε
x
]
(9.50)
(1
+
)(1
−
2
E
xy
=
)
xy
=
xy
G
2(1
+
and, while not zero, the normal stress in the
z
direction is considered negligible
in comparison to the other stress components.
The elastic strain energy for a body of volume
V
in plane strain is
1
2
U
e
=
(
x
ε
x
+
y
ε
y
+
xy
xy
)d
V
(9.51)
V
which can be expressed in matrix notation as
xy
]
ε
x
d
V
1
2
U
e
=
[
x
y
ε
y
xy
(9.52)
V
Combining Equations 9.50 and 9.52 with considerable algebraic manipulation,
the elastic strain energy is found to be
1
2
E
U
e
=
[
ε
x
ε
y
xy
]
(1
+
)(1
−
2
)
V
1
−
0
ε
x
ε
y
xy
1
−
0
×
d
V
(9.53)
−
1
2
0
0
2
and is similar to the case of plane stress, in that we can express the energy as
1
2
T
[
D
]
U
e
=
{
ε
}
{
ε
}
d
V
V
with the exception that the elastic property matrix for plane strain is defined as
1
−
0
E
1
−
0
[
D
]
=
(9.54)
−
(1
+
)(1
−
2
)
1
2
0
0
2
