Displacement/Pressure Finite Element formulation for Large Deformations (from Bathe)

Question

On page 563 of Finite Element Procedures by Klaus-Jürgen Bathe the author states that governing equations of the displacement/pressure finite element formulation for large deformations is given as

$$ \begin{bmatrix} {}^T{\bf K}{\bf U}{\bf U} & {}^T{\bf K}{\bf U}{\bf P}\\ {}^T{\bf K}{\bf P}{\bf U} & {}^T{\bf K}{\bf P}{\bf P} \end{bmatrix} \begin{bmatrix} {\bf \hat u}\\ {\bf \hat p} \end{bmatrix} = \begin{bmatrix} {\bf {}^{t + \Delta t} R}\\ {\bf 0} \end{bmatrix} - \begin{bmatrix} {}^T{\bf F}{\bf U}\\ {}^T{\bf F}{\bf P} \end{bmatrix} $$

and can be arrived at "by linearization", using

$$ {}^t_0{\bf W} = {}^t_0{\bf \overline W} - \frac{1}{\kappa}({}^t{\bf \overline t} - {}^t{\bf \tilde t})^2 $$

Please refer to this paper for a complete definition of the terms. I would like to be derive these governing equations but I don't know how to begin the linearization.

Answer 1

We have recently proposed a mixed u-p formulation that results in a global symmetric matrix irrespective of the volumetric energy function (for hyperelasticity).

The paper contains sufficient details on the linearisation (2nd derivative).