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Stress Update
• To solve the equilibrium equation for the new accelerations:
u M ( f B σ dV )
= −∑ ∫
n+ 1 − 1 ext( n+ 1) T e( n+
1)
e e( n+
1)
V
one needs the new stress en ( 1)
.
σ +
• In general one may formally write:
n 1 n
σ σ σ
+ = +∆
n
∆ σ = H( σ , ∆ε, p, ε, …) (Rate form)
H Constitutive law
∆σ stress increment over the step
∆ε strain increment over the step
p hardening parameters (e.g. plasticity)
ε strain rate (e.g. viscous behaviour)
• Note that the total deformation does not appear anywhere
and is not used in the process.
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ε
Elasto-plastic material
As an example of non-linear material behaviour consider the
important case of metal plasticity:
• Rate-independent deviatoric plasticity model with Von
Mises yield criterion:
• “Trial” stress (elastic):
trial
σn+ 1
= σn + C ⋅∆ε
Radial return
method
(Wilkins)
σ n
σ 3
σ
n + 1
σ σ
1
2
• If trial stress lies outside yield surface, perform radial
return onto (current) yield surface. No iterations!
trial
σ
n + 1
How does one compute ∆ε from displacement increments (or velocities) in the presence of
geometrical non-linearities (large strains and large motions, in particular large rotations?)
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