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ULiege - Aerospace & Mechanical Engineering

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# 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 differential 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.

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

#### 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 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 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  / 