Civil Engineering Reference
In-Depth Information
6.7
Initial stress
The viscoplastic algorithm is often referred to as an
initial strain
method to distinguish it
from the more widely used “initial stress” approaches (e.g. Zienkiewicz
et al
., 1969).
Initial stress methods involve an explicit relationship between increments of stress and
increments of strain. Thus, whereas linear elasticity was described by
[
D
e
]
e
{
σ
} =
(6.27)
elasto-plasticity is described by
[
D
pl
]
e
{
σ
}
=
(6.28)
where
[
D
pl
]
[
D
e
]
[
D
p
]
=
−
(6.29)
For perfect plasticity in the absence of hardening or softening it is assumed that once a
stress state reaches a failure surface, subsequent changes in stress may shift the stress state
to a different position on the failure surface, but not outside it, thus
∂F
∂
σ
T
{
σ
} =
0
(6.30)
Allowing for the possibility of non-associated flow, plastic strain increments occur
normal to a plastic potential surface, thus
∂Q
∂
σ
p
=
λ
(6.31)
Assuming stress changes are generated by elastic strain components only gives
[
D
e
]
∂Q
∂
σ
{
σ
} =
{
} −
λ
(6.32)
Substitution of equation (6.32) into (6.30) leads to
∂F
∂
σ
[
D
e
]
∂Q
T
[
D
e
]
∂
σ
[
D
p
]
=
T
[
D
e
]
∂Q
(6.33)
∂F
∂
σ
∂
σ
Explicit versions of [
D
p
] may be obtained for simple failure and potential functions
and these are given for von Mises (Yamada
et al
., 1968) and Mohr-Coulomb (Griffiths and
Willson, 1986). See Appendix C for a detailed derivation of (6.33)
The body loads
b
}
i
in the stress redistribution process are reformed at each iteration
by summing the following integral for all elements that possess yielding Gauss points, thus
{
F
[
B
]
T
[
D
p
]
all
b
}
i
=
{
}
i
{
F
d
x
d
y
(6.34)
elements
