(modeling)=

# Modeling

In their most general form, material (constitutive) models define stress, $\bm{S}$ ($\bm{\tau}$ in Eulerian coordinate), in terms of state variables.
For the materials used in Ratel, the state variables are constituted by the vector displacement field $\bm{u}$, and its gradient $\bm{H} = \nabla_X \bm{u}$.
We consider constitutive models for stress in terms of strain due to deformation, given by $\bm{S} \left( \bm{E} \right)$ ($\bm{\tau} \left( \bm{e} \right) $ in Eulerian coordinate), which is a tensor-valued function of a tensor-valued input.

A physical model must not depend on the coordinate system chosen to express it; an arbitrary choice of function will generally not be invariant under orthogonal transformations and thus will not admissible.
Given an orthogonal transformation $Q$, we desire

$$
  Q \bm{S} \left( \bm{E} \right) Q^T = \bm{S} \left( Q \bm{E} Q^T \right),
$$ (elastic-invariance)

which means that we can change our reference frame before or after computing $\bm{S}$ and get the same result.
Constitutive relations in which $\bm{S}$ is uniquely determined by $\bm{E}$ while satisfying the invariance property {eq}`elastic-invariance` are known as Cauchy elastic materials.

The strain energies may be expressed in terms of invariants of the symmetric Cauchy-Green tensors $\bm{C}$ or $\bm{b}$

$$
  \begin{aligned}
    \mathbb{I}_1 (\bm{C}) &= \trace \bm{C}, \\
    \mathbb{I}_2 (\bm{C}) &= \frac 1 2 \left( \mathbb{I}_1^2 - \bm{C} \tcolon \bm{C} \right) \\
    \mathbb{I}_3 (\bm{C}) &= \operatorname{det} \bm{C},
  \end{aligned}
$$ (invariants)

Since $\bm{C}$ and $\bm{b}$ have the same eigenvalues, which are the square of the principal stretches $\lambda_i^2$, we conclude that $\mathbb{I}_i (\bm{C}) = \mathbb{I}_i (\bm{b}), \, i=1,2,3$.
For the strain energy function $\psi \left( \bm{C} \right) = \psi \left( \mathbb{I}_1, \mathbb{I}_2, \mathbb{I}_3 \right)$ we can determine the constitutive equations for isotopic hyperelastic materials by taking the gradient of strain energy as

$$
  \bm{S} = \frac{\partial \psi}{\partial \bm{E}}= 2 \frac{\partial \psi}{\partial \bm{C}} = 2 \sum_{i=1}^3 \frac{\partial \psi}{\partial \mathbb{I}_i} \frac{\partial \mathbb{I}_i}{\partial \bm{C}}
$$ (strain-energy-grad)

where $\bm{S}$ is the general form of the stress relation in initial configuration and the derivative of invariants are

$$
  \begin{aligned}
    \frac{\partial \mathbb{I}_1}{\partial \bm{C}} &= \bm{I}, \\
    \frac{\partial \mathbb{I}_2}{\partial \bm{C}} &=  \mathbb{I}_1 \bm{I} - \bm{C}\\
    \frac{\partial \mathbb{I}_3}{\partial \bm{C}} &= \mathbb{I}_3 \bm{C}^{-1},
  \end{aligned}
$$ (invariants-derivative)

The symmetric Kirchhoff stress tensor in current configuration $\bm{\tau}$ can be obtained by representing strain energy in terms of $\bm{b}$'s invariants and taking the gradient as

$$
  \bm{\tau}= \frac{\partial \psi}{\partial \bm{e}} \bm{b} = 2 \frac{\partial \psi}{\partial \bm{b}} \bm{b}
$$ (strain-energy-grad-current)

or we can push-forward $\bm{S}$ using {eq}`S-to-tau` to define $\bm{\tau}$.

The strain energy function may be written in terms of principal stretches $\lambda_i$ as $\psi \left( \bm{C} \right) = \psi \left( \lambda_1, \lambda_2, \lambda_3 \right)$ where we can define the constitutive relations by

$$
  \bm{S} = 2 \frac{\partial \psi}{\partial \bm{C}} = 2 \sum_{i=1}^3 \frac{\partial \psi}{\partial \lambda_i^2} \frac{\partial \lambda_i^2}{\partial \bm{C}} = \sum_{i=1}^3 S_i \hat{\bm{N}_i} \hat{\bm{N}_i}^T
$$ (strain-energy-grad-stretch)

and push-forward $\bm{S}$

$$
  \bm{\tau} = \bm{F} \bm{S} \bm{F}^T = \sum_{i=1}^3 \tau_i \hat{\bm{n}_i} \hat{\bm{n}_i}^T
$$ (strain-energy-grad-stretch-current)

where $S_i$ and $\tau_i$ are three principal stresses given by

$$
  \begin{aligned}
    S_i &= \frac{1}{\lambda_i} \frac{\partial \psi}{\partial \lambda_i}, & \tau_i &= {\lambda_i} \frac{\partial \psi}{\partial \lambda_i}.
  \end{aligned}
$$ (principal-stresses)

where we have used

$$
  \begin{aligned}
    \frac{\partial \lambda_i^2}{\partial \bm{C}} &= \hat{\bm{N}_i} \hat{\bm{N}_i}^T, & \frac{\partial \psi}{\partial \lambda_i^2} &= \frac{1}{2\lambda_i} \frac{\partial \psi}{\partial \lambda_i}.
  \end{aligned}
$$ (stretch-derivative)

Note that the invariants can be written in terms of principal stretch as

$$
  \begin{aligned}
    \mathbb{I}_1 (\bm{C}) &= \lambda_1^2 + \lambda_2^2 + \lambda_3^2, \\
    \mathbb{I}_2 (\bm{C}) &= \lambda_1^2 \lambda_2^2 + \lambda_2^2 \lambda_3^2 + \lambda_1^2 \lambda_3^2, \\
    \mathbb{I}_3 (\bm{C}) &= \lambda_1^2 \lambda_2^2 \lambda_3^2,
  \end{aligned}
$$ (invariants-stretch)

:::{note}
The strain energy density functional cannot be an arbitrary function $\psi \left( \bm{E} \right)$.
It can only depend on *invariants*, scalar-valued functions $\gamma$ satisfying

$$
  \gamma \left( \bm{E} \right) = \gamma \left( Q \bm{E} Q^T \right)
$$

for all orthogonal matrices $Q$.
:::

## Material Models

In the following sections, we introduce the most common constitutive models in the literature, as well as their numerical implementation in Ratel (weak formulation, linearization, etc.)

```{toctree}
:maxdepth: 2

Elasticity (linear and non-linear) <elasticity/index>
Mixed Elasticity (linear and non-linear) <mixed-elasticity/index>
Elastostatics vs. Elastodynamics <time-integration>
Inelasticity (damage, plasticity) <inelasticity/index>
Poromechanics <poromechanics/index>
Material Point Method <material-point-method>
```

## Boundary Conditions
```{toctree}
:maxdepth: 3

Boundary Conditions <boundary-conditions/index>
```