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).

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 liquid fraction into account.

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

Burgos law, this time considering the 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

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

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

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

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

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

Burgos law, this time considering the 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

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

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

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

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