(incompressibility)=
# Linear elastic
## Constitutive theory
One can see from the equations in the {ref}`linear-elastic-standard-form` that as $\firstlame \to \infty$, it is necessary that $\nabla\cdot \bm{u} \longrightarrow 0$ which gives an idea of alternative strategies. One approach is to define an auxiliary variable $p$, and rewrite constitutive equation {eq}`linear-constitutive` as

$$
  \begin{aligned}
    \bm \sigma(\bm u, p) &= -p \, \bm{I} + 2 \mu \bm \varepsilon, \\
    p &= - \firstlame \trace \bm \varepsilon.
  \end{aligned}
$$ (mixed-linear-constitutive1)

Alternatively, we can use the definition of hydrostatic pressure i.e., $p_{\text{hyd}} = - \frac{\trace \bm \sigma}{3}$ and arrive at

$$
  \begin{aligned}
    \bm \sigma(\bm u, p_{\text{hyd}}) &= -p_{\text{hyd}} \, \bm{I} + 2 \mu \bm \varepsilon_{\text{dev}}, \\
    p_{\text{hyd}} &= -\bulk \trace \bm \varepsilon.
  \end{aligned}
$$ (mixed-linear-constitutive2)

where $\bm \varepsilon_{\text{dev}} = \bm \varepsilon - \frac{1}{3} \trace \bm \varepsilon ~ \bm{I}$ is the deviatoric part of the linear strain tensor and $\bulk$ is the bulk modulus. We present a general constitutive equation as

$$
  \begin{aligned}
    \bm \sigma(\bm u, p) &= \left(\bulk_p \trace \bm \varepsilon -p \right) \bm{I} + 2 \mu \bm \varepsilon_{\text{dev}}, \\
    p &= -\left(\bulk - \bulk_p \right) \trace \bm \varepsilon.
  \end{aligned}
$$ (general-mixed-linear-constitutive)

where

$$
  \bulk_p = \frac{2 \mu \left(1 + \nu_p \right)}{3 \left(1 - 2 \nu_p \right)},
$$ (primal-bulk)

is the primal portion of the bulk modulus, defined in terms of $\nu_p$ with $-1 \leq \nu_p < \nu$, where $\nu$ is the physical Poisson’s ratio. The standard full-train formulation {eq}`mixed-linear-constitutive1` is obtained using $\nu_p = 0$, and the deviatoric formulation {eq}`mixed-linear-constitutive2` with $\nu_p = -1$.

## Strong and weak formulations
The boundary-value problem (Strong form) for constitutive equation {eq}`general-mixed-linear-constitutive` may be stated as follows: Given body force $\rho \bm g$, Dirichlet boundary $\bar{\bm u}$ and applied traction $\bar{\bm t}$, find the displacement and pressure-like variables $(\bm u, p) \in \mathcal{V} \times \mathcal{Q} $ (here $\mathcal{V} = H^1(\Omega), \mathcal{Q} = L^2(\Omega) $ ), such that:

$$
  \begin{aligned}
    -\nabla \cdot \bm \sigma - \rho \bm g   &= 0, \qquad \text{in $\Omega$} \\
    -\nabla\cdot\bm u - \frac{p}{\bulk - \bulk_p} &= 0, \qquad \text{in $\Omega$} \\
    \bm{u} &= \bar{\bm u}, \quad    \text{on $\partial \Omega^{D}$} \\
    \bm \sigma \cdot \bm n &= \bar{\bm t}. \qquad \text{on $\partial \Omega^{N}$}
  \end{aligned}
$$ (mixed-linear-strong-form)

with $\bm n$ be the unit normal on the boundary and its weak formulation as:

$$
  \begin{aligned}
    \int_{\Omega}{ \nabla \bm{v} \tcolon \bm{\sigma}} \, dv
    - \int_{\partial \Omega}{\bm{v} \cdot \bar{\bm t}} \, da
    - \int_{\Omega}{\bm{v} \cdot \rho \bm{g}} \, dv
    &= 0, \quad \forall \bm v \in \mathcal V \\
    \int_{\Omega} q \left( -\nabla\cdot\bm u - \frac{p}{\bulk - \bulk_p} \right) \, dv & = 0. \quad \forall q \in \mathcal Q
  \end{aligned}
$$ (mixed-linear-weak-form)

## Command-line interface

To enable the incompressible, mixed linear elastic model, use the model option `-model elasticity-mixed` and set the material parameters listed in {ref}`mixed-linear-elastic-options`. Any parameter without a default option is required.

(mixed-linear-elastic-options)=

:::{list-table} Mixed linear elastic model options
:header-rows: 1
:widths: 30 55 15
* - Option
  - Description
  - Default Value

* - `-model elasticity-mixed-linear`
  - Required to enable the mixed linear elastic model.
  -

* - `-E [real]`
  - [Young's modulus](https://en.wikipedia.org/wiki/Young%27s_modulus), $E > 0$
  -

* - `-nu [real]`
  - [Poisson's ratio](https://en.wikipedia.org/wiki/Poisson%27s_ratio), $\nu \leq 0.5$. Note, this model is only validated for $\nu = 0.5$.
  -

* - `-nu_primal [real]`
  - Primal part of the Poisson's ratio for Jacobian operator, $-1 \leq \nu_p < \nu \leq 0.5$.
  This parameter may improve solver performance and should not affect the solution.
  - `-1.0`

* - `-nu_primal_pc [real]`
  - Primal part of the Poisson's ratio for preconditioner operator, $-1 \leq \nu_{pc} < \nu \leq 0.5$.
  This parameter may improve solver performance and should not affect the solution.
  - `-nu_primal` value

:::

An example using the mixed linear elastic model can be run via
```console
$ ./bin/ratel-quasistatic -options_file examples/ymls/ex02-quasistatic-elasticity-mixed-linear-face-forces-pcjacobi.yml
```