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CreepLaws

The following viscous creep laws are implemented:

GeoParams.MaterialParameters.ConstitutiveRelationships.LinearViscous — Type
LinearViscous(η=1e20Pa*s)

Defines a linear viscous creeplaw

The (isotopic) linear viscous rheology is given by

\[ \tau_{ij} = 2 \eta \dot{\varepsilon}_{ij} \]

or

\[ \dot{\varepsilon}_{ij} = {\tau_{ij} \over 2 \eta }\]

where $\eta_0$ is the reference viscosity [Pa*s] at reference strain rate $\dot{\varepsilon}_0$[1/s], and $n$ the power law exponent [].

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GeoParams.MaterialParameters.ConstitutiveRelationships.DislocationCreep — Type
DislocationCreep(n = 1.0NoUnits, r = 0.0NoUnits, A = 1.5MPa/s, E = 476.0kJ/mol, V = 6e-6m^3/mol, apparatus = AxialCompression )

Defines the flow law parameter of a dislocation creep law.

The (isotropic) dislocation creep law, as used by experimentalists, is given by

\[ \dot{\gamma} = A \sigma_\mathrm{d}^n f_\mathrm{H2O}^r \exp\left(-\frac{E+PV}{RT}\right)\]

where

  • $n$ is the power law exponent
  • $r$ is the exponent of fugacity dependence
  • $A$ is a pre-exponential factor $[\mathrm{MPa}^{-n}s^{-1}]$ (if manually defined, $n$ and $r$ must be either pre-defined or substituted)
  • $E$ is the activation energy $\mathrm{[kJ/mol]}$
  • $V$ is the activation volume $\mathrm{[m^3/mol]}$
  • $\dot{\gamma}$ is the strain rate $\mathrm{[1/s]}$
  • $\sigma_\mathrm{d}$ is the differential stress $\mathrm{[MPa]}$ which are converted into second invariants using the Apparatus variable that can be

either AxialCompression, SimpleShear or Invariant. If the flow law parameters are already given as a function of second invariants, choose Apparatus=Invariant.

Example

julia> x2 = DislocationCreep(n=3)
DislocationCreep: n=3, r=0.0, A=1.5 MPa^-3 s^-1, E=476.0 kJ mol^-1, V=6.0e-6 m^3 mol^-1, Apparatus=AxialCompression
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GeoParams.MaterialParameters.ConstitutiveRelationships.ViscosityPartialMelt_Costa_etal_2009 — Type
ViscosityPartialMelt_Costa_etal_2009(LinearMeltViscosity())

The viscosity of a partially molten rock depends on the melt viscosity, melt fraction and strainrate.

This implements a parameterisation of Costa et al. [2009].

Reference

Costa, A., Caricchi, L., Bagdassarov, N., 2009. A model for the rheology of particle‐bearing suspensions and partially molten rocks. Geochem Geophys Geosyst 10, 2008GC002138. https://doi.org/10.1029/2008GC002138

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GeoParams.MaterialParameters.ConstitutiveRelationships.LinearMeltViscosity — Type
LinearMeltViscosity(η=1e20Pa*s)

Defines a simple temperature-dependent melt viscosity, given by

\[ \tau_{ij} = 2 \eta \dot{\varepsilon}_{ij}\]

or

\[ \dot{\varepsilon}_{ij} = {\tau_{ij} \over 2 \eta }\]

where

\[ \eta = \eta_0*10^{A + \frac{B}{T - T0}} ;\]

and \eta_0 is the scaling viscosity, A and B are constants, and T_0 is a reference temperature, and T is the temperature [in K].

Typical parameters for basalt (default) are: A = -9.6012, B = 1.3374e+04K, T_0 = 307.8043K and \eta_0 = 1Pas.

Typical parameters for rhyolite are: A = -8.1590, B = 2.4050e+04K, T_0 = -430.9606K and \eta_0 = 1Pas.

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GeoParams.MaterialParameters.ConstitutiveRelationships.GiordanoMeltViscosity — Type
GiordanoMeltViscosity(; oxd_wt = oxd_wt, η0=1Pas)

Defines the melt viscosity model after Giordano et al. (2008) given by

\[ \tau_{ij} = 2 \eta \dot{\varepsilon}_{ij}\]

or

\[ \dot{\varepsilon}_{ij} = {\tau_{ij} \over 2 \eta }\]

where

\[ \eta = \eta_0 * 10^{AT + \frac{BT}{T - CT}}\]

Parameters

  • oxd_wt::NTuple{9,T}: Melt composition as 9-element Tuple containing concentrations in [wt%] of the following oxides ordered in the exact sequence 

          (SiO2 TiO2 Al2O3 FeO MgO CaO Na2O K2O H2O) 

      Default values are for a hydrous N-MORB melt.

Reference

  • Giordano D, Russell JK, & Dingwell DB (2008). Viscosity of Magmatic Liquids: A Model. Earth & Planetary Science Letters, 271, 123-134. (https://dx.doi.org/10.1016/j.epsl.2008.03.038)
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GeoParams.MaterialParameters.ConstitutiveRelationships.DiffusionCreep — Type
DiffusionCreep(r = 0NoUnits, p = A = 1.5MPa/s, E = 476.0kJ/mol, V = 6e-6m^3/mol, apparatus = AxialCompression )

Defines the flow law parameter of a dislocation creep law.

The (isotropic) diffusion creep law, as used by experimentalists, is given by

\[ \dot{\gamma} = A \sigma_\mathrm{d} d^{\mathrm{p}} f_\mathrm{H2O}^r \exp\left(-\frac{E+PV}{RT}\right)\]

where

  • $r$ is the exponent of fugacity dependence
  • $p$ is the exponent of grain size
  • $A$ is a pre-exponential factor $[\mathrm{MPa}^{-n}s^{-1}]$ (if manually defined, $n$ and $r$ must be either pre-defined or substituted)
  • $E$ is the activation energy $\mathrm{[kJ/mol]}$
  • $V$ is the activation volume $\mathrm{[m^3/mol]}$
  • $\dot{\gamma}$ is the strain rate $\mathrm{[1/s]}$
  • $\sigma_\mathrm{d}$ is the differential stress $\mathrm{[MPa]}$

The experimental parameters are converted into second invariants using the Apparatus variable that can be either AxialCompression, SimpleShear or Invariant. If the flow law parameters are already given as a function of second invariants, choose Apparatus=Invariant.

Example

julia> x2 = DiffusionCreep(Name="test")
DiffusionCreep: Name = test, n=1.0, r=0.0, p=-3.0, A=1.5 m³·⁰ MPa⁻¹·⁰ s⁻¹·⁰, E=500.0 kJ mol⁻¹·⁰, V=2.4e-5 m³·⁰ mol⁻¹·⁰, FT=1.7320508075688772, FE=1.1547005383792517)
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Computational routines for creep laws

Once a creep rheology is defined, we can use the following routines to perform computations within the solvers

GeoParams.MaterialParameters.ConstitutiveRelationships.compute_εII — Function
compute_εII(a::DiffusionCreep, TauII::_T; T::_T, P=one(_T), f=one(_T), d=one(_T), kwargs...)

Returns diffusion creep strainrate as a function of 2nd invariant of the stress tensor $\tau_{II}$

\[ \dot{ε}_{II} = A τ_{II}^n d^{p} f_{H_2O}^r \exp \left(- {{E + PV} \over RT} \right)\]

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compute_εII(a::GrainBoundarySliding, TauII::_T; T::_T, P=one(_T), f=one(_T), d=one(_T), kwargs...)

Returns grain boundary sliding strainrate as a function of 2nd invariant of the stress tensor $\tau_{II}$

\[ \dot{ε}_{II} = A τ_{II}^n d^{p} \exp \left(- {{E + PV} \over RT} \right) #ÄNDERN!!!!!!!!\]

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compute_εII(s::ConstantElasticity{_T}, τII; τII_old, dt)

Computes elastic strainrate given the deviatoric stress at the current (τII) and old timestep (τII_old), for a timestep dt:

\[ \dot{\varepsilon}^{el} = {1 \over 2 G} {D \tau_{II} \over Dt } ≈ {1 \over 2 G} {\tau_{II}- \tilde{\tau}_{II}^{old} \over dt }\]

Note that we here solve the scalar equation, which is sufficient for isotropic cases. In tensor form, it would be

\[ {\dot{\varepsilon}^{el}}_{ij} = {1 \over 2 G} { \tau_{ij} - \tilde{{\tau_{ij}}}^{old} \over dt }\]

here $\tilde{{\tau_{ij}}}^{old}$ is the rotated old deviatoric stress tensor to ensure objectivity (this can be done with Jaumann derivative, or also by using the full rotational formula).

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compute_εII(p::DruckerPrager{_T,U,U1}, λdot::_T, τII::_T,  P)

This computes plastic strain rate invariant for a given $λdot$

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compute_εII(p::DruckerPrager_regularised{_T,U,U1}, λdot::_T, τII::_T,  P)

This computes plastic strain rate invariant for a given $λdot$

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compute_εII(v::Parallel{T,N}, τII, args; tol=1e-6, verbose=false, n=1)

Computing εII as a function of τII for a Parallel elements is (usually) a nonlinear problem

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compute_εII(v::CompositeRheology{T,N}, τII, args; tol=1e-6, verbose=false, n=1)

Computing εII as a function of τII for a composite element is the sum of the individual contributions

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compute_εII(v::AbstractPlasticity, τII::_T, args; tol=1e-6, verbose=true)

Performs local iterations to compute the plastic strainrate. Note that the non-plastic strainrate, ε_np, should be part of args

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GeoParams.MaterialParameters.ConstitutiveRelationships.compute_εII! — Function
compute_εII!(EpsII::AbstractArray{_T,N}, a, TauII::AbstractArray{_T,N}; T, P, f,d,kwargs...)

Computes strainrate as a function of stress

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compute_εII!(EpsII::AbstractArray{_T,N}, a, TauII::AbstractArray{_T,N}; T, P, f,d,kwargs...)

Computes strainrate as a function of stress

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compute_εII!(EpsII::AbstractArray{_T,N}, s::LinearMeltViscosity, TauII::AbstractArray{_T,N}; T, kwargs...)
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compute_εII!(EpsII::AbstractArray{_T,N}, s::ViscosityPartialMelt_Costa_etal_2009, TauII::AbstractArray{_T,N}; T, kwargs...)
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compute_εII!(EpsII::AbstractArray{_T,N}, s::GiordanoMeltViscosity, TauII::AbstractArray{_T,N}; T, kwargs...)
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compute_εII!(EpsII::AbstractArray{_T,N}, s::LinearViscous, TauII::AbstractArray{_T,N})
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compute_εII!(EpsII::AbstractArray{_T,N}, s::ArrheniusType, TauII::AbstractArray{_T,N})
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compute_εII!(ε_el::AbstractArray{_T,N}, s::ConstantElasticity{_T}; τII::AbstractArray{_T,N}, τII_old::AbstractArray{_T,N}, dt::_T, kwargs...)

In-place computation of the elastic shear strainrate for given deviatoric stress invariants at the previous (τII_old) and new (τII) timestep, as well as the timestep dt

\[ \dot{\varepsilon}^{el} = {1 \over 2 G} {D \tau_{II} \over Dt } ≈ {1 \over 2 G} {\tau_{II}- \tau_{II}^{old} \over dt }\]

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GeoParams.MaterialParameters.ConstitutiveRelationships.compute_τII — Function
compute_τII(a::DislocationCreep, EpsII; P, T, f, args...)

Computes the stress for a Dislocation creep law given a certain strain rate

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computeCreepLaw_TauII(EpsII::_T, a::DiffusionCreep; T::_T, P=zero(_T), f=one(_T), d=one(_T), kwargs...)

Returns diffusion creep stress as a function of 2nd invariant of the strain rate

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computeCreepLaw_TauII(EpsII::_T, a::GrainBoundarySliding; T::_T, P=zero(_T), d=one(_T), kwargs...)

Returns grain boundary sliding stress as a function of 2nd invariant of the strain rate

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compute_τII(a::PeierlsCreep, EpsII; P, T, f, args...)

Computes the stress for a peierls creep law given a certain strain rate.

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compute_τII(s::LinearMeltViscosity, EpsII; kwargs...)

Returns second invariant of the stress tensor given a 2nd invariant of strain rate tensor

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compute_τII(s::ViscosityPartialMelt_Costa_etal_2009, EpsII; kwargs...)

Returns second invariant of the stress tensor given a 2nd invariant of strain rate tensor

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compute_τII(s::GiordanoMeltViscosity, EpsII; kwargs...)

Returns second invariant of the stress tensor given a 2nd invariant of strain rate tensor

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compute_τII(s::LinearViscous, EpsII; kwargs...)

Returns second invariant of the stress tensor given a 2nd invariant of strain rate tensor

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compute_τII(s::ArrheniusType, EpsII; kwargs...)

Returns second invariant of the stress tensor given a 2nd invariant of strain rate tensor

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compute_τII(s::PowerlawViscous, EpsII; kwargs...)

Returns second invariant of the stress tensor given a 2nd invariant of strain rate tensor

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τII = compute_τII(v::CompositeRheology{T,N}, εII, args; tol=1e-6, verbose=false)
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GeoParams.MaterialParameters.ConstitutiveRelationships.compute_τII! — Function
compute_τII!(TauII::AbstractArray{_T,N}, a::DislocationCreep, EpsII::AbstractArray{_T,N}; 
    P =       zero(TauII)::AbstractArray{_T,N}, 
    T = ones(size(TauII))::AbstractArray{_T,N}, 
    f = ones(size(TauII))::AbstractArray{_T,N})

Computes the deviatoric stress invariant for a dislocation creep law

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compute_τII!(TauII::AbstractArray{_T,N}, a::PeierlsCreep, EpsII::AbstractArray{_T,N}; 
    T = ones(size(TauII))::AbstractArray{_T,N}, args...)

Computes the deviatoric stress invariant for a peierls creep law.

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compute_τII!(τII::AbstractArray{_T,N}, s::ConstantElasticity{_T}. ε_el::AbstractArray{_T,N}; τII_old::AbstractArray{_T,N}, dt::_T, kwargs...)

In-place update of the elastic stress for given deviatoric strainrate invariants and stres invariant at the old (τII_old) timestep, as well as the timestep dt

\[ \tau_{II} = 2 G dt \dot{\varepsilon}^{el} + \tau_{II}^{old}\]

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