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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Example<br />
<strong>The</strong> Jaumann stress is connected to <strong>the</strong> Cauchy stress through:<br />
˜τ ij = ˙τ ij + τ ip W pj + τ jp W pi<br />
which from <strong>the</strong> above yields <strong>the</strong> follwoing <strong>for</strong>mula <strong>for</strong> <strong>the</strong> Cauchy components τ ij<br />
⎡<br />
⎢<br />
⎣<br />
˙τ 11<br />
˙τ 22<br />
˙τ 12<br />
⎤ ⎡<br />
⎥ ⎢<br />
⎦ = ⎣<br />
2τ 12<br />
⎤<br />
−2τ 12<br />
⎥<br />
⎦<br />
<strong>The</strong> above system <strong>of</strong> ordinary differential equations can be solved to finally get:<br />
⎡<br />
⎢<br />
⎣<br />
τ 11<br />
τ 22<br />
τ 12<br />
⎤ ⎡<br />
⎥ ⎢<br />
⎦ = ⎣<br />
1900(1 − cos2t)<br />
−1900(1 − cos2t)<br />
1900sin2t<br />
⎤<br />
⎥<br />
⎦<br />
<strong>The</strong> results from methods A <strong>and</strong> B are ra<strong>the</strong>r close <strong>for</strong> small values <strong>of</strong> <strong>the</strong> de<strong>for</strong>mation<br />
measure t but grow quite different as t gets larger than 0.1, indicating that <strong>the</strong> same C<br />
can no longer be used.<br />
Institute <strong>of</strong> Structural Engineering <strong>Method</strong> <strong>of</strong> <strong>Finite</strong> <strong>Element</strong>s II 17