======= Maxwell Branches =======

===== Linear Maxwell Branch =====
=== Description ===

{{ :doc:user:references:materials:maxwell.png?300 |}}

The non-equilibrium stress in the current configuration in a Maxwell branch writes (trapezoidal integration)
$$
    \begin{align*}
    \mathbf{h}_j^{n+1} 
    \approx e^{-\frac{\Delta t}{\tau_j}} \frac{1}{\Delta J} \Delta F ~\mathbf{h}_j^{n}(\Delta F)^T + \Gamma_j \frac{1 - e^{-\frac{\Delta t}{\tau_j}}}{\frac{\Delta t}{\tau_j}}\left[ \boldsymbol{\sigma}^{n+1}_0 - \frac{1}{\Delta J} \Delta F ~~\boldsymbol{\sigma}^{n}_0(\Delta F)^T\right]
    \end{align*}
$$
where $\Delta \mathbf{F} = \mathbf{F}^{n+1}\left(\mathbf{F}^{n}\right)^{-1}$ and $\Delta J = \text{det}\left(\Delta \mathbf{F}\right)$.

=== Parameters ===
^   Name                                                  ^  Metafor Code  ^ Dependency ^
| Normalized Maxwell stiffness $\Gamma$  |  ''HYPER_MAXWELL_GAMMA''  |  ''TO/TM''  |
| Relaxation time $\tau$  |  ''HYPER_VE_TAU''  |  ''TO/TM''  |
| Boolean parameter, use trapezoidal integration (=False, default) or mid-point rule (=True)  |  ''HYPER_MAXWELL_USE_MPR''  |  -  |


===== Nonlinear Maxwell Branch =====
=== Description ===

The nonlinear Maxwell brach is composed of a nonlinear spring (deviatoric $\psi_e$ and volumetric $\psi_{vol}$ parts) and a dashpot ($\dot{\gamma}$).

{{ :doc:user:references:materials:nlmaxwell.png?500 |}}

Within the branch, the total deformation gradient can be multiplicatively decomposed as
$$
    \mathbf{F} = \mathbf{F}^e\mathbf{F}^v \rightarrow \mathbf{F}^e = \mathbf{F}\left(\mathbf{F}^v\right)^{-1}
$$
and the time-derivative of the viscous deformation gradient writes
$$
    \dot{\mathbf{F}}^v = \dot{\gamma}\frac{\text{dev}\left(\boldsymbol{\sigma}^e\right)}{\sqrt{\text{dev}\left(\boldsymbol{\sigma}^e\right):\text{dev}\left(\boldsymbol{\sigma}^e\right)}}\mathbf{F}^v = \dot{\gamma}\frac{\text{dev}\left(\boldsymbol{\sigma}^e\right)}{\tau}\mathbf{F}^v = \dot{\gamma}\mathbf{N}\mathbf{F}^v,
$$ 
which allows computing the time-evolution of the viscous deformation gradient $\mathbf{F}^v$.

The integration of $\mathbf{F}^v$ is performed using an exponential map integrator (read 
{{ :doc:user:references:materials:polymermodels.pdf | this document}} for more details), either explicitely
$$
  \mathbf{F}^v_{n+1} = \text{exp}\left[\Delta t \dot{\gamma}_{n} \mathbf{N}_{n}\right]\mathbf{F}^v_{n}
$$
or implicitely
$$
  \mathbf{F}^v_{n+1} = \text{exp}\left[\Delta t \dot{\gamma}_{n+1} \mathbf{N}_{n+1}\right]\mathbf{F}^v_{n}
$$
using a local Newton-Raphson iteration scheme.

The deviatoric (elastic) potential $\psi_{e}^{(i)}$ is defined using hyperelastic potential laws defined in [[doc:user:elements:volumes:hyper_dev_potential]] whilst volumetric potential $\psi_{vol}^{(i)}$ is defined using volumic potential laws in [[doc:user:elements:volumes:hyper_vol_potential]].

The creep factor $\dot{\gamma}$ is defined using dashpot laws defined in [[doc:user:elements:volumes:hyper_dev_dashpot]].


=== Parameters ===
^   Name                                                  ^  Metafor Code  ^ Dependency ^
| Number of the [[doc:user:elements:volumes:hyper_dev_potential|hyperelastic potential law]] $\psi_e$  |  ''HYPER_MAXWELL_SPRING_NUM''  |  -  |
| Number of the [[doc:user:elements:volumes:hyper_vol_potential|volumic potential law]] $\psi_{vol}$ (optional)  |  ''HYPER_MAXWELL_SPRING_VOL_NUM''  |  -  |
| Number of the [[doc:user:elements:volumes:hyper_dev_dashpot|dashpot law]] $\dot{\gamma}$  |  ''HYPER_MAXWELL_DASHPOT_NUM''  |  -  |
| Number of the [[doc:user:elements:volumes:hyper_dev_branchl|Maxwell branch]] from which this branch is dependent  |  ''HYPER_MAXWELL_DEPENDENCE_NUM''  |  -  |
| Boolean parameter, use implicit (=True, default) or explict (=False) integration of $\mathbf{F}^v$  |  ''HYPER_MAXWELL_USE_IMPLICIT''  |  -  |