Hello,
I am currently using the FEniCS/DOLFINx package to simulate deformations on a mesh volume.
Following the tutorial, I am using the following equation to find u such as F(u)=0 : F = \overrightarrow{grad} \overrightarrow{v} . P ~\overrightarrow{dx} - \overrightarrow{T} . \overrightarrow{v} \overrightarrow{ds}
with

P the first Piola Kirchhoff stress P = \frac{\partial \psi}{\partial F}

\overrightarrow{dx} and \overrightarrow{ds} the volume and surface element

\overrightarrow{T} a traction force applied on the surface.

Originally, the goal would be to minimize the potential energy, which writes itself as:

My question is: what is the link between these two equations ? How to get to the first one starting from the second one ? I know that the Gateaux derivative is used in that case but I donâ€™t know how to apply it to \Pi.

(Note: I apologize in advance for the ugly typesetting, but it seems that the usual way of including inline equations is now rendering them on separate lines, at least in Firefox, and I couldnâ€™t find a workaround by Googling!)

If youâ€™ve studied calculus of variations before, it may help to use the notation \delta\mathbf{u} for the test function instead of \mathbf{v}\text{ ,} then think about setting the Gateaux derivative of \Pi with respect to \mathbf{u} in the direction \delta\mathbf{u} to zero (by analogy to seeking the minimum of a univariate function by looking for an argument such that its ordinary derivative is zero):

where the variation \delta\mathbf{F} is also a Gateaux derivative with respect to displacement of \mathbf{F} = \operatorname{grad}\mathbf{u} + \mathbf{I}\text{ ,} i.e.,

Making the substitutions \partial\psi/\partial\mathbf{F}\to\mathbf{P}~,~\delta\mathbf{F}\to\operatorname{grad}(\delta\mathbf{u})~,~\delta\mathbf{u}\to\mathbf{v}\text{ ,} you recover the weak form