The class YieldStress manages the yield stress used in the plastic criterion, whether plastic (isotropic hardening), visco-plastic (Perzyna as additive, Cowper-Symonds as multiplicative or ZerilliArmstrong, JohnsonCook, … as flow stress models)

$$ \sigma_{yield} = \sigma_{yield} (\bar{\varepsilon}^{vp}, \dot{\bar{\varepsilon}}^{vp}, grainSize, ...) $$

The laws implemented in Metafor are described below

Viscous term of the yield stress specific to thixotropic materials. It depends on two internal parameters, the cohesion degree $ \lambda $ and the liquid fraction, whether full $ f_{l} $ or effective $ f_{l}^{eff} $) (depending on $ m_5 $: $ f_{l} $ if $ m_5=0 $ and $ f_{l}^{eff} $ if $ m_5=1 $). An isotropic hardening law, which depends on these two parameters, can also be chosen.

$$ \sigma_{yield}= \sigma_{isoH} + \sigma_{visq} $$

This viscous law is a Perzyna law whose parameters $K$ and $M$ depend on $ \lambda $ and $f_l$ (or $ f_l^{eff}) $.

where

$$ \sigma_{visq}= K \left (\dot{\overline{\varepsilon}}^{vp}\right )^{M} $$

$$ K = K_1 e^{K_2(1-f_l)} e^{K_3 \lambda} $$

$$ M = (M_1 + M_3 \lambda^2 + M_4 \lambda ) e^{M_2 (1-f_l)} $$

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

Viscous term of the yield stress specific to thixotropic materials. It depends on two internal parameters, the cohesion degree $ \lambda $ and the liquid fraction, whether full $ f_{l} $ or effective $ f_{l}^{eff} $). An isotropic hardening law, which depends on these two parameters, can also be chosen.

$$ \sigma_{yield}= \sigma_{isoH} + \sigma_{visq} $$

This viscous law is a Burgos law extended to degenerate properly towards a free solid suspensions behavior once the structure is fully broken ($ \lambda = 0 $).

$$ \sigma_{visq}= \eta_{susp}+ (\eta_{skel} - \eta_{susp} ) \lambda^2 (3-2\lambda) $$

where

$$ \eta_{susp} = K_4 e^{M_5 (1 - f_l^{eff})} $$

and (Burgos law)

$$ \eta_{skel} = K (\dot{\overline{\varepsilon}}^{vp})^{M} $$

$$ K = K_1 e^{K_2(1-f_l)} e^{K_3 \lambda} $$

$$ M = (M_1 + M_3 \lambda^2 + M_4 \lambda ) e^{M_2 (1-f_l)} $$

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

Viscous term of the yield stress specific to thixotropic materials. It depends on two internal parameters, the cohesion degree $ \lambda $ and the liquid fraction, whether full $ f_{l} $ or effective $ f_{l}^{eff} $). An isotropic hardening law, which depends on these two parameters, can also be chosen.

$$ \sigma_{yield}= \sigma_{isoH} + \sigma_{visq} $$

This viscous law is a Burgos law extended to degenerate properly towards a free solid suspensions behavior once the structure is fully broken ($ \lambda = 0 $).

$$ \sigma_{visq}= \eta_{susp}+ (\eta_{skel} - \eta_{susp} ) \lambda^2 (3-2\lambda) $$

where

$$ \eta_{susp} = K_4 \left (f_l^{eff} \right )^{-M_5 (1 - (1-f_l)^{M_6})} $$

and (Burgos law)

$$ \eta_{skel} = K \left (\dot{\overline{\varepsilon}}^{vp} \right )^{M} $$

$$ K = K_1 e^{K_2(1-f_l)} e^{K_3 \lambda} $$

$$ M = (M_1 + M_3 \lambda^2 + M_4 \lambda ) e^{M_2 (1-f_l)} $$

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

Viscous term of the yield stress specific to thixotropic materials. It depends on two internal parameters, the cohesion degree $ \lambda $ and the liquid fraction. An isotropic hardening law, which depends on these two parameters, can also be chosen.

$$ \sigma_{yield}= \sigma_{isoH} + \sigma_{visq} $$

The viscous yield stress is now computed based on a micro-macro model, where the semi-solid material is represented by spherical inclusions and their coating called the active zone. The inclusions are made of solid grains and entrapped liquid, when the active zone is made up of the solid bonds and the non entrapped liquid.

At the lower scale, the inclusions and the active zone are both made up of liquid and solid

This model is a system of 3 equations and 3 unknowns (localization variable for each phase), solved using Newton-Raphson:

$$ A_a^s=\frac{5 \sigma_a}{3 \sigma_a + 2 \sigma_a^s}\\ $$

Localization variable of the solid phase in the inclusions $$ A_i^s=\frac{5 \sigma_i}{3 \sigma_i + 2 \sigma_i^s}\\ $$

Localization variable of the inclusions in the global semi-solid material $$ A_i =\frac{5 \sigma_{visq} \sigma_a}{3 \sigma_{visq} \sigma_a + 2 \sigma_i \sigma_a + 6/5 f_a A_i (\sigma_i - \sigma_a)^2 } $$

where

Viscous stress in the solid phase of the active zone: $$ \sigma_a^s = k_p (A_a^s \frac{1-(1-f_a)A_i}{f_a})^{m_p-1} (\dot{\overline{\epsilon}}^{vp})^{m_p}\\ $$

Viscous stress in the solid phase of the inclusions: $$ \sigma_i^s = k_s (A_i^s A_i)^{m_s-1} (\dot{\overline{\epsilon}}^{vp})^{m_s}\\ $$

Viscous stress in the active zone: $$ \sigma_a = k_l \dot{\overline{\epsilon}}^{vp} (1-\lambda A_a^s) + \lambda A_a^s \sigma_a^s \\ $$

Viscous stress in the inclusions: $$ \sigma_i = k_l \dot{\overline{\epsilon}}^{vp} (1-\frac{1-f_l-f_a \lambda}{1-f_a} A_i^s) + \lambda A_i^s \sigma_i^s \\ $$

Viscous stress: $$ \sigma_{visq} = \sigma_a (1 - (1-f_a) A_i) + \sigma_i (1-f_a) A_i $$