The Finite Element Method for the Analysis of Non-Linear and ...
The Finite Element Method for the Analysis of Non-Linear and ...
The Finite Element Method for the Analysis of Non-Linear and ...
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Elastic Material<br />
UL Formulation<br />
We now write<br />
t<br />
τ ij = t tC ∗ ijrs<br />
t tɛ A rs<br />
where <strong>the</strong> elasticity tensor components <strong>of</strong> <strong>the</strong> UL constitutive matrix<br />
C ∗ , are related to <strong>the</strong> one <strong>of</strong> <strong>the</strong> TL <strong>for</strong>mulation C, through <strong>the</strong><br />
following relationship:<br />
t<br />
tC ∗ ijrs =<br />
t<br />
ρ<br />
0<br />
ρ<br />
t<br />
0x i,m<br />
t<br />
0x j,n<br />
t<br />
0C mnpq<br />
t<br />
0x r,p<br />
t<br />
0x s,q<br />
Also <strong>the</strong> Almansi strain tensor is related to <strong>the</strong> Green-Lagrange one<br />
through<br />
t<br />
tɛ A = 0 t<br />
X T 0ɛ t 0 t<br />
X<br />
<strong>The</strong>re<strong>for</strong>e <strong>the</strong> two <strong>for</strong>mulations are equivalent <strong>and</strong> interchangeable.<br />
Institute <strong>of</strong> Structural Engineering <strong>Method</strong> <strong>of</strong> <strong>Finite</strong> <strong>Element</strong>s II 10