multiPlas - Dynardo GmbH

The numerical implementation of the plasticity models is carried out using the return-mapping method
[6-17], [6-18], [6-22]. The return mapping procedure is used at the integration point for the local iterative
stress relaxation. It consists of two steps:
1. elastic predictor step:
σ
trial
i+
1
= σ
*
i
+
tot
D dεi+
1
2. plastic corrector step (local iterative procedure):
dσ
∂Q
= −D
dλ
∂σ
3.2 Multisurface plasticity
The consideration of different failure modes rsp. failure mechanisms of a material is possible by a yield
surface built up from several yield criteria. In the stress domain then a non-smooth multisurface yield criterion
figure develops.
The elastic plastic algorithm has to deal with singularities at intersections from different yield criteria (e.g.
F1 to F2 as represented in Fig. 3-1).
F2
pl
d ε 2
pl
dε 1
Fig. 3-1 Intersection between the two flow criteria F1 and F2
F1
The consistent numerical treatment of the resulting multi-surface plasticity must deal with the possibility
that many yield criteria are active simultaneously. This leads to a system of n=j equations:
⎧∂Fn
⎫
⎨ ⎬
⎩ ∂σ
⎭
T
D dε
=
⎡
∑
= ⎥ ⎥
Set of active YC
T
⎧∂Fn
⎫
j ∂Fn
∂κ
n
⎢⎨
⎬ D −
j 1 ⎩ ∂σ
⎭ ∂σ
∂κ
n ∂λ
j
⎢⎣
∂Q
⎤
dλ
j
⎦
The solution of this system of equations generates the stress return to flow criteria or within the
intersection of flow criterias. Contrary to single surface plasticity exceeding the flow criterion is no longer
a sufficient criterion for activity of the plastic multiplier for each active yield criterion. An activity criterion
needs to be checked.
λ (3-10)
d j ≥ 0
This secures that the stress return within the intersection is reasonable from a physical point of view.
8
(3-7)
(3-8)
(3-9)
USER’S MANUAL, January, 2013