Civil Engineering Reference
In-Depth Information
s
2
F(
s
B
)>0 (inadmissible)
F(
s
A
)= 0
B
A
X
C
O
s
3
s
1
Figure 6.23
Stress correction
is the total incremental strain,
p
the incremental plastic strain, and
If
{
}
λ
the
scalar multiplier (6.31) an elastic stress-strain matrix [
D
e
] will lead to
[
D
e
]
{
} −
p
{
σ
C
} = {
σ
A
} +
(6.48)
where
p
=
λ
{
a
}
(6.49)
The derivative
{
a
}
in the above equation is evaluated at
(
1
−
β)
{
σ
A
} +
β
{
σ
C
}
, and the
scalar interpolating parameter
β
(similar to
θ
in Chapters 8 and 9) can vary in the range 0
≤
β
≤
1.
6.9.1 Forward Euler method
Here
β
=
0 and the rate equation is integrated at the point at which the yield surface is
crossed (
{
σ
A
}
in Figure 6.23). Thus
F
{
σ
X
}
+
α
σ
e
=
0
(6.50)
For a non-hardening von Mises material the point can be found explicitly from:
9
2
2
−
3
B
±
B
−
12
A(
3
C
−
3
c
u
)
α
=
(6.51)
6
A
