Plasticity

Plasticity is a non-linear rheology that is activated once stresses exceed a certain yield criteria.

Implemented laws

The following plastic law are implemented:

GeoParams.MaterialParameters.ConstitutiveRelationships.DruckerPrager Type

DruckerPrager(ϕ=30, Ψ=0, C=10e6Pa)

Sets parameters for Drucker-Prager plasticity, where the yield stress ${\sigma }_y$ is computed by

$${\sigma }_y=(P-P_f)\tan (\varphi )+C$$

with $\varphi$ being the friction angle (in degrees), $C$ cohesion, $P$ dynamic pressure and $P_f$ the fluid pressure (both positive under compression).

Yielding occurs when the second invariant of the deviatoric stress tensor, ${\tau }_{II}=(0.5{\tau }_{ij}{\tau }_{ij}{)}^{0.5}$ touches the yield stress. This can be computed with the yield function $F$ and the plastic flow potential $Q$, which are respectively given by

$$F={\tau }_{II}-\cos (\varphi )C-\sin (\varphi )(P-P_f)$$$$Q={\tau }_{II}-\sin (\Psi )(P-P_f)$$

Here, Ψ is the dilation angle, which must be zero for incompressible setups.

Plasticity is activated when $F({\tau }_{II}^{trial})$ (the yield function computed with a trial stress) is >0. In that case, plastic strainrate ${\dot{\epsilon }}_{ij}^{pl}$ is computed by:

$${\dot{\epsilon }}_{ij}^{pl}=\dot{\lambda }\frac{\partial Q}{\partial {\sigma }_{ij}}$$

where $\dot{\lambda }$ is a (scalar) that is nonzero and chosen such that the resulting stress gives $F({\tau }_{II}^{final})=0$, and ${\sigma }_{ij}=-P+{\tau }_{ij}$ denotes the total stress tensor.

Example

julia> pl = DruckerPrager(ϕ=30, C=10e6Pa)
Drucker-Prager plasticity with: C = 1.0e7 Pa, ϕ = 30.0ᵒ, Ψ = 0.0ᵒ

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GeoParams.MaterialParameters.ConstitutiveRelationships.DruckerPrager_regularised Type

DruckerPrager_regularised(ϕ=30, Ψ=0, C=10e6Pa, η_vp=1e20Pa*s)

Sets parameters for reularised Drucker-Prager plasticity, where the yield stress ${\sigma }_y$ is computed by

$${\sigma }_y=(P-P_f)\tan (\varphi )+C+2{\eta }_{v}p\epsilon ̇I{I}_{p}l$$

with $\varphi$ being the friction angle (in degrees), $C$ cohesion, $P$ dynamic pressure, $P_f$ the fluid pressure (both positive under compression), ${\eta }_{v}p$ the regularization viscosity and $\epsilon ̇I{I}_{p}l$ the invariant of the plastic strainrate

Yielding occurs when the second invariant of the deviatoric stress tensor, ${\tau }_{II}=(0.5{\tau }_{ij}{\tau }_{ij}{)}^{0.5}$ touches the yield stress. This can be computed with the yield function $F$ and the plastic flow potential $Q$, which are respectively given by

$$F={\tau }_{II}-\cos (\varphi )C-\sin (\varphi )(P-P_f)-2{\eta }_{vp}\dot{\epsilon }\epsilon {̇}_{II}^{pl}$$$$Q={\tau }_{II}-\sin (\Psi )(P-P_f)$$

Here, Ψ is the dilation angle, which must be zero for incompressible setups.

Plasticity is activated when $F({\tau }_{II}^{trial})$ (the yield function computed with a trial stress) is >0. In that case, plastic strainrate ${\dot{\epsilon }}_{ij}^{pl}$ is computed by:

$${\dot{\epsilon }}_{ij}^{pl}=\dot{\lambda }\frac{\partial Q}{\partial {\sigma }_{ij}}$$

where $\dot{\lambda }$ is a (scalar) that is nonzero and chosen such that the resulting stress gives $F({\tau }_{II}^{final})=0$, and ${\sigma }_{ij}=-P+{\tau }_{ij}$ denotes the total stress tensor.

Example

julia> pl = DruckerPrager_regularised(ϕ=30, C=10e6Pa, η_vp=1e20Pa*s)
Regularized Drucker-Prager plasticity with: C = 1.0e7 Pa, ϕ = 30.0ᵒ, Ψ = 0.0ᵒ, η_vp=1.0e20 Pa s

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GeoParams.MaterialParameters.ConstitutiveRelationships.DruckerPragerCap Type

DruckerPragerCap(ϕ=30, Ψ=0, C=10e6Pa, η_vp=1e20Pa*s, Pt=-1e5Pa)

Sets parameters for Drucker-Prager-Cap plasticity, as described in Popov et al. (2025), Geoscientific Model Development. The yield surface has two branches selected by the stress state: a mode-2 Drucker-Prager shear branch at high pressure and a mode-1 tensile cap near the tensile strength pT. The yield function $F$ and plastic flow potential $Q$ are

$$F={\tau }_{II}-kP-c\text{or}a(\sqrt{{\tau }_{II}^2+(P-p_y{)}^2}-R_y)$$$$Q={\tau }_{II}-k_{q}P-\text{const}\text{or}b(\sqrt{{\tau }_{II}^2+(P-p_q{)}^2}-R_f)$$

where the first form is the shear branch and the second the tensile cap. As in DruckerPrager, plastic strain rate is ${\dot{\epsilon }}_{ij}^{pl}=\dot{\lambda }\partial Q/\partial {\sigma }_{ij}$, with η_vp providing Duvaut-Lions (Duretz-type) viscoplastic regularisation.

Fields

• C::T: The cohesion parameter.

• ϕ::T: The friction angle (in degrees).

• Ψ::T: The dilatancy angle (in degrees).

• η_vp::T: The Duvaut-Lions regularisation viscosity for the plasticity model.

• pT::T: The tensile strength (should be < 0).

Example

julia> pl = DruckerPragerCap(ϕ=30, C=10e6Pa, pT=-1e5Pa)
DruckerPragerCap(ϕ=30.0, Ψ=0.0, C=1.0e7 Pa, η_vp=1.0e20 Pa s, pT=-100000.0 Pa)

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Computational routines

Usually, plasticity should be defined as part of a CompositeRheology structure and calculations can be done as with all other rheology computations by using compute_τII.