# Neumann BC in the current configuration in a finite-strain problem

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 !