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Geometric non-linearities (3)
Set up following incrementally objective scheme to update
the Cauchy stress (2D case for simplicity) in three phases:
• Let α be the angle of rotation over ∆ t and let θ = α/2
n+
1/2
tan θ = ( ∆t/ 2) ⋅W
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1. Apply first half of the rotation increment:
n* n T
⎡ cosθ
sinθ⎤
σ = Rσ
R with R= ⎢
−sinθ
cosθ
⎥
⎣
⎦
2. Apply the constitutive equation:
σ
= σ + C⋅∆t⋅D
( n + 1)* n* n + 1/2
3. Apply second half of the rotation increment:
σ
= Rσ
R
n+ 1 ( n+
1)* T
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Geometric non-linearities (4)
For structural elements (bars, beams, shells) use co-rotational
formulation:
• The stress is measured in a reference frame that rotates with the element
• This greatly simplifies the stress increment procedure: the stress may be
incremented directly by applying the constitutive law.
Example (bar element). In the longitudinal direction:
∆L
∆ ε = → ∆σ
L
∆L L dL L
∑∆ ε = ∑ ln
L
∫ =
L0
L L
A small-strain formulation would be:
∆L L−L
∆ ε = → ∑ ∆ ε =
L
L
0 0
0
0
y
L
x
(Natural or logarithmic strain)
(Engineering strain)
∆L
28
σε ,
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