Civil Engineering Reference
In-Depth Information
where
1
2
((s
x
)
2
+
(s
y
)
2
+
(s
z
)
2
)
+
(τ
xy
)
2
A
=
B
=
s
x
s
x
+
s
y
s
y
+
s
z
s
z
+
τ
xy
τ
xy
2
(6.52)
1
2
(s
2
2
2
2
xy
C
=
x
+
s
y
+
s
z
)
+
τ
but an approximate
α
can be found by linearly interpolating between points
X
and
B
from,
−
F(
{
σ
X
}
)
F(
{
σ
B
}
)
−
F(
{
σ
X
}
)
α
≈
(6.53)
[
D
e
]
−
α)
{
}
The remaining stress
causes the “illegal” stress state outside the yield
surface. For non-hardening plasticity, it is assumed that once a stress state reaches a yield
surface, subsequent changes in stress may shift the stress state to a different position on
the yield surface but not outside it (6.30), hence
(
1
T
F
= {
a
}
{
σ
} =
0
(6.54)
thus
T
[
D
e
]
[
D
e
]
F
= {
a
}
(
{
} −
(
1
−
α)λ
{
a
}
)
=
0
(6.55)
Assuming that the derivative vector
{
a
}
as evaluated at point
A
is called
{
a
A
}
,the
following expression for
λ
A
, the “plastic multiplier” at
A
is given by,
T
[
D
e
]
{
a
A
}
{
}
λ
A
=
(6.56)
T
[
D
e
]
(
1
−
α)
{
a
A
}
{
a
A
}
The final stress is then
{
σ
C
} = {
σ
X
} +
σ
e
−
λ
A
[
D
e
]
{
a
A
}
(6.57)
This is the method used in equation (6.35) in which
f
=
1
−
α
.
6.9.2 Backward Euler method
Here the rate equation is integrated at the “illegal” state B (
β
=
1). This results in a simple
evaluation of the plastic multiplier
for non-hardening von Mises materials. A first order
Taylor expansion of the yield function at B gives:
λ
∂F
∂
σ
T
F(
{
σ
C
}
)
=
F(
{
σ
B
}
)
+
{
σ
}
(6.58)
By enforcing consistency of the yield function at point
C
,
T
[
D
e
]
=
F(
{
σ
B
}
)
−
λ
B
{
a
B
}
{
a
B
}
0
(6.59)
