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For a hyperelastic problem, I understand the variational formulation can be written as the minimisation of $\Pi$ with

$\Pi = \int_{\Omega} \psi( \pmb{u} )dx - \int_{\partial\Omega} \pmb{T}\cdot \pmb{u} ds$

With $\Omega$ the reference configuration, $\pmb{u}$ the displacement, and $\pmb T$ the traction force on the reference configuration, i.e $\pmb T = \pmb P \cdot \pmb N$ with $\pmb P$ the first Piola Kirchoff Stress and $\pmb N$ the outward normal.

Now, in my problem, the prescribed forces $\pmb t$ are given in the current configuration and not the reference. In order to keep the same weak form like the one used in this FEniCS tutorial, I understand I have to perform a transformation of the $\int_{\partial\Omega} \pmb{T}\cdot \pmb{u} ds$ term. Seeing different references, many refer to the Nanson formula.

The problem is that the formula gives the transformation from the $da \cdot \pmb n$ term, whether in our case, we have the integral defined over $dA \cdot \pmb N$ (the right hand side of the formula).

However, seeing this result in this book : $\pmb P \cdot \pmb N = J |\pmb F^{-T} \pmb N | \hat{\pmb t} $ with $\hat{\pmb t} = \pmb t \circ \phi$ where $\phi$ is the deformation mapping. Is the following derivation correct ?

  • $\hat{\pmb t} = \pmb t$ Since it is prescribed vector load (exterior forces)
  • the term $\int_{\partial\Omega} \pmb{T}\cdot \pmb{u} ds$ becomes $\int_{\partial\Omega} J |\pmb F^{-T} \pmb N | {\pmb t}\cdot \pmb u ds $ which is still defined on the reference configuration.
  • We thus just need to multiply dot(T, u)*ds by $J |\pmb F^{-T} \pmb N |$ in the mentioned tutorial, with -ofcourse- T becoming my new exterior pressure.

Is this reasoning correct ? Otherwise, how can we derive a proper modification to the weak form for FEniCS ?

I didn't find a similar problem/derivation out there, I would love some help. Thank you very much !

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As far as I understand, you basically want to express your traction vector from deformed (current) configuration to undeformed (reference) configuration in your integral. Would it be possible to just do a pullback operation on your traction vector and directly use it in the integral?

Holzapfel, G. A., 2001, Nonlinear Solid Mechanics, John Wiley, Chichester. (GAH), Pg 82

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