(linear-elastic-standard-form)=
# Linear elastic
## Constitutive theory
For the linear elastic model, the strain energy density is given by

$$
\psi = \frac{\firstlame}{2} (\trace \bm{\varepsilon})^2 + \secondlame \bm{\varepsilon} \tcolon \bm{\varepsilon} .
$$

The constitutive law (stress-strain relationship) is therefore given by its gradient,

$$
\bm\sigma(\bm{u}) = \frac{\partial \psi}{\partial \bm{\varepsilon}} = \firstlame (\trace \bm\varepsilon) \bm{I} + 2 \secondlame \bm\varepsilon,
$$ (linear-constitutive)

where the colon represents a double contraction (over both indices of $\bm{\varepsilon}$), $\bm{\varepsilon}$ is (small/infintesimal) strain tensor defined by

$$
\bm{\varepsilon} = \dfrac{1}{2}\left(\nabla \bm{u} + \nabla \bm{u}^T \right),
$$ (linear-strain-tensor)

and the Lamé parameters are given in terms of Young's modulus $E$, and Poisson’s ratio $\nu$ by

$$
\begin{aligned}
  \firstlame &= \frac{E \nu}{(1 + \nu)(1 - 2 \nu)}, \\
  \secondlame &= \frac{E}{2(1 + \nu)}.
\end{aligned}
$$

The constitutive law (stress-strain relationship) can also be written as

$$
\bm{\sigma} = \mathsf{C} \tcolon \bm{\varepsilon}.
$$ (linear-stress-strain)

For notational convenience, we express the symmetric second order tensors $\bm \sigma$ and $\bm \varepsilon$ as vectors of length 6 using the [Voigt notation](https://en.wikipedia.org/wiki/Voigt_notation).
Hence, the fourth order elasticity tensor $\mathsf C$ (also known as elastic moduli tensor or material stiffness tensor) can be represented as

$$
\mathsf C = \begin{pmatrix}
\firstlame + 2\secondlame & \firstlame & \firstlame & & & \\
\firstlame & \firstlame + 2\secondlame & \firstlame & & & \\
\firstlame & \firstlame & \firstlame + 2\secondlame & & & \\
& & & \secondlame & & \\
& & & & \secondlame & \\
& & & & & \secondlame
\end{pmatrix}.
$$ (linear-elasticity-tensor)

Note that the incompressible limit $\nu \to \frac 1 2$ causes $\firstlame \to \infty$, and thus $\mathsf C$ becomes singular.

## Strong and weak formulations
In this section we present the governing equations of elasticity in small deformation. In small strain (linear elasticity), the boundary-value problem (Strong form) for constitutive equation {eq}`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 variable $\bm u \in \mathcal{V}$ (here $\mathcal{V} = H^1(\Omega)$ ), such that:

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

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

$$
\int_{\Omega}{ \nabla \bm{v} \tcolon \bm{\sigma}} \, dv
- \int_{\partial \Omega}{\bm{v} \cdot \bar{\bm t}} \, da
- \int_{\Omega}{\bm{v} \cdot \rho \bm{g}} \, dv
= 0, \quad \forall \bm v \in \mathcal V.
$$ (linear-weak-form)

## Command-line interface

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

(linear-elastic-options)=

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

* - `-model elasticity-linear`
  -
  -

* - `-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), $\nu \leq 0.5$.
  -

:::

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