(formulation_pressure_bc)=
# Pressure boundary condition

One of the important load case is the pressure boundary loading which is caused by liquids or gases on the surface of the solid structure.
The pressure boundary load depends upon the current state of the deformation and can be considered as a traction vector $\bar{\bm{t}} = \bm{\sigma} \cdot {\bm{n}} = -p{\bm{n}}$ per unit current surface, acting in the direction of the outward unit normal ${\bm{n}}$.
Therefore, the following surface integral will be add to the weak form {eq}`hyperelastic-weak-form-initial`, or {eq}`hyperelastic-weak-form-current`

$$
  \int_{\partial \Omega}{\bm{v} \cdot \left( -p {\bm{n}} \right)} \, da, \quad \forall \bm{v} \in \mathcal{V}.
$$ (pressure-boundary-condition)

where the normal on current surface is computed by

$$
  {\bm{n}} = \frac{ \frac{\partial \bm{x}}{\partial \xi_1} \times \frac{\partial \bm{x}}{\partial \xi_2} }{\lvert \frac{\partial \bm{x}}{\partial \xi_1} \times \frac{\partial \bm{x}}{\partial \xi_2} \rvert}
$$ (surface-normal-current)

in which $ \xi_1, \xi_2 \in [-1,1]^2 $ are reference coordinate system on the face.
If we write the surface area in terms of reference coordinate as $da = \lvert \frac{\partial \bm{x}}{\partial \xi_1} \times \frac{\partial \bm{x}}{\partial \xi_2} \rvert d\xi_1 d\xi_2$, the pressure load {eq}`pressure-boundary-condition` can be simplified to

$$
  -\int_{[-1,1]^2} \bm{v} \cdot p \left(\frac{\partial \bm{x}}{\partial \xi_1} \times \frac{\partial \bm{x}}{\partial \xi_2} \right) d\xi_1 d\xi_2
$$ (pressure-boundary-condition-simplified)

To achieve second order convergence in our Newton solver, we will need the linearization of {eq}`pressure-boundary-condition-simplified` which for the constant pressure is given by

$$
  -\int_{[-1,1]^2} \bm{v} \cdot p \left( \frac{\partial \diff \bm{u}}{\partial \xi_1} \times \frac{\partial \bm{x}}{\partial \xi_2} + \frac{\partial \bm{x}}{\partial \xi_1} \times \frac{\partial \diff \bm{u}}{\partial \xi_2}\right) d\xi_1 d\xi_2 .
$$ (pressure-boundary-condition-linearization)