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[HyperElastic] Add a working version of Ogden's material law #5748
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…o division by zero
…ons - keep to rewrite
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The material model in
Ogden.hwas failing theMaterial_test.cppunit tests. This PR adds a working version that passes the tests.TL;DR
SelfAdjointEigenSolverand useEigenSolverfor computing eigenvectors (see Incorrect eigen decomposition in the Ogden material model #5733 ); this might get fixed with eigen 5 and then we should get back to theSelfAdjointEigenSolver(more appropriate).applyElasticityTensorcomputes the elasticity tensor and multiplies a vector; a 'smarter' computation is needed.The new model runs slower than before. I kept the old functions for now since they are faster (although incorrect).
To get the model working, I had to reimplement the functions related to the elasticity tensor, as I couldn’t derive the mathematical expressions backwards from the old implementation. However, I might be able to do this with a working version in hand.
Documentation
What the model functions provide:
The strain energy:
where$F(J) = J^{ - \frac{\alpha}{3} }$ with $J = \sqrt{det(C)}$ and $C$ is the Cauchy deformation tensor. Also, $\alpha$ and $\mu$ are variables of the model for fitting. $\lambda$ are the eigenvalues of $C$ . This expression is different than many of the formulas I found in literature for Ogden's model. Here, $\alpha$ is squared. It can be found in ABAQUS' domumentation: https://classes.engineering.wustl.edu/2009/spring/mase5513/abaqus/docs/v6.6/books/usb/default.htm?startat=pt05ch17s05abm07.html
second Piola–Kirchhoff stress tensor:
Elasticity tensor:
where$P_i = \vec{N_i} \otimes \vec{N_i}$ and $\vec{N}_i$ are eigenvectors of $C$ .