====== Cohesion degree======

The ''CohesionMatLaw'' class manages all cohesion degree evolution laws, specific to thixotropic materials. These laws are described below.

:!: Careful: Only works if used with thixotropic materials (''ThixoEvpIsoHHypoMaterial'' or  ''ThixoTmEvpIsoHHypoMaterial'').  

===== IsothCohesionMatLaw =====

=== Description ===

The evolution of the structural parameter λ can be expressed by a differential equation
that describes the kinetics between the agglomeration of the solid grains and the destruction
of the solid bonds due to shearing. Solved using Newton-raphson, this equation is isothermal since it does not take the [[doc:user:elements:volumes:thixo_scheilliquidfractionmatlaw|liquid fraction]] into account.

$$
d \lambda / dt = a (1 - \lambda)^{1+e} - b \lambda e^{c \dot{\bar{\epsilon}}^{vp} } (\dot{\bar{\epsilon}}^{vp})^d
$$

=== Parameters ===
^   Name  ^     Metafor Code  ^  Dependency ^
|  $ a $  |  ''LAMBDA_A''  |      ''TM/TO''     |
|  $ b $  |  ''LAMBDA_B''  |      ''TM/TO''     |
|  $ c $  |  ''LAMBDA_C''  |      ''TM/TO''     |
|  $ d $  |  ''LAMBDA_D''  |      ''TM/TO''     |
|  $ e $  |  ''LAMBDA_E''  |      ''TM/TO''     |

===== BurgosCohesionMatLaw =====

=== Description ===

Burgos law, this time considering the [[doc:user:elements:volumes:thixo_scheilliquidfractionmatlaw|liquid fraction]]. The cohesion degree is an explicit function of the equivalent plastic strain rate (integration over a time step where the equivalent plastic strain is supposed to remain constant).

$$
\lambda =\lambda_e + ( \lambda_0 - \lambda_e)  e^{F(\lambda) \Delta t}
$$

where

$$
F(\lambda) = -\left(  a'+b' e^{c \dot{\bar{\epsilon}}^{vp}} (\dot{\bar{\epsilon}}^{vp})^{d'}  \right)
$$


$$
\lambda_e = \frac{-a'}{F(\lambda)}
$$

$$
a' = a (1-f_l) + f e^{-g f_l}
$$

$$
b' = b f_l + f e^{-g (1-f_l)}
$$

$$
d' = d (1-(f_l)^{e})
$$

=== Parameters ===
^   Name                 ^     Metafor Code     ^  Dependency ^
|  $ a $  |  ''LAMBDA_A''  |      ''TM/TO''     |
|  $ b $  |  ''LAMBDA_B''  |      ''TM/TO''     |
|  $ c $  |  ''LAMBDA_C''  |      ''TM/TO''     |
|  $ d $  |  ''LAMBDA_D''  |      ''TM/TO''     |
|  $ e $  |  ''LAMBDA_E''  |      ''  /  ''     |
|  $ f $  |  ''LAMBDA_F''  |      ''  /  ''     |
|  $ g $  |  ''LAMBDA_G''  |      ''  /  ''     |


===== FavierCohesionMatLaw =====

=== Description ===

Burgos law, this time considering the [[doc:user:elements:volumes:thixo_scheilliquidfractionmatlaw|liquid fraction]]. The cohesion degree is an explicit function of the equivalent plastic strain rate (integration over a time step where the equivalent plastic strain is supposed to remain constant). Percolation is also taken into account, meaning that the cohesion degree approaches zero when the liquid fraction approaches a critical value $ f_c = e $.

$$
\lambda = \lambda_e + ( \lambda_0 - \lambda_e)  e^{F(\lambda) \Delta t} \mbox{ if } f_l < f_c = e
$$ 

$$
\lambda = 0 \mbox{ if } f_l \geq f_c = e
$$
where

$$
F(\lambda) = -\left( a'+b' e^{c \dot{\bar{\epsilon}}^{vp}} (\dot{\bar{\epsilon}}^{vp})^d \right)
$$

$$
\lambda_e = \frac{-a'}{F(\lambda)}
$$

$$
a' = a (1-f_l) + f e^{-g f_l}
$$


$$
b' = b f_l + f e^{-g (1-f_l)} 
$$

=== Parameters ===
^   Name  ^     Metafor Code     ^  Dependency ^
|  $ a $  |  ''LAMBDA_A''  |      ''TM/TO''     |
|  $ b $  |  ''LAMBDA_B''  |      ''TM/TO''     |
|  $ c $  |  ''LAMBDA_C''  |      ''TM/TO''     |
|  $ d $  |  ''LAMBDA_D''  |      ''TM/TO''     |
|  $ e $  |  ''LAMBDA_E''  |      ''  /  ''     |
|  $ f $  |  ''LAMBDA_F''  |      ''  /  ''     |
|  $ g $  |  ''LAMBDA_G''  |      ''  /  ''     |