ASPECT
aspect::MaterialModel::ViscoPlastic< dim > Class Template Reference
Inheritance diagram for aspect::MaterialModel::ViscoPlastic< dim >:
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## Public Member Functions

void evaluate (const MaterialModel::MaterialModelInputs< dim > &in, MaterialModel::MaterialModelOutputs< dim > &out) const override

bool is_compressible () const override

double reference_viscosity () const override

void parse_parameters (ParameterHandler &prm) override

void create_additional_named_outputs (MaterialModel::MaterialModelOutputs< dim > &out) const override

double get_min_strain_rate () const

DEAL_II_DEPRECATED bool is_yielding (const double pressure, const double temperature, const std::vector< double > &composition, const SymmetricTensor< 2, dim > &strain_rate) const

bool is_yielding (const MaterialModelInputs< dim > &in) const

Public Member Functions inherited from aspect::MaterialModel::Interface< dim >
virtual ~Interface ()

virtual void initialize ()

virtual void update ()

virtual void fill_additional_material_model_inputs (MaterialModel::MaterialModelInputs< dim > &input, const LinearAlgebra::BlockVector &solution, const FEValuesBase< dim > &fe_values, const Introspection< dim > &introspection) const

const NonlinearDependence::ModelDependenceget_model_dependence () const

Public Member Functions inherited from aspect::SimulatorAccess< dim >
SimulatorAccess ()

SimulatorAccess (const Simulator< dim > &simulator_object)

virtual ~SimulatorAccess ()

virtual void initialize_simulator (const Simulator< dim > &simulator_object)

template<typename PostprocessorType >
PostprocessorType * find_postprocessor () const

const Introspection< dim > & introspection () const

const Simulator< dim > & get_simulator () const

const Parameters< dim > & get_parameters () const

SimulatorSignals< dim > & get_signals () const

MPI_Comm get_mpi_communicator () const

TimerOutputget_computing_timer () const

const ConditionalOStreamget_pcout () const

double get_time () const

double get_timestep () const

double get_old_timestep () const

unsigned int get_timestep_number () const

unsigned int get_nonlinear_iteration () const

const parallel::distributed::Triangulation< dim > & get_triangulation () const

double get_volume () const

const Mapping< dim > & get_mapping () const

std::string get_output_directory () const

bool include_latent_heat () const

bool include_melt_transport () const

int get_stokes_velocity_degree () const

double get_surface_pressure () const

bool convert_output_to_years () const

unsigned int get_pre_refinement_step () const

unsigned int n_compositional_fields () const

void get_refinement_criteria (Vector< float > &estimated_error_per_cell) const

void get_artificial_viscosity (Vector< float > &viscosity_per_cell, const bool skip_interior_cells=false) const

void get_artificial_viscosity_composition (Vector< float > &viscosity_per_cell, const unsigned int compositional_variable) const

const LinearAlgebra::BlockVectorget_current_linearization_point () const

const LinearAlgebra::BlockVectorget_solution () const

const LinearAlgebra::BlockVectorget_old_solution () const

const LinearAlgebra::BlockVectorget_old_old_solution () const

const LinearAlgebra::BlockVectorget_reaction_vector () const

const LinearAlgebra::BlockVectorget_mesh_velocity () const

const DoFHandler< dim > & get_dof_handler () const

const FiniteElement< dim > & get_fe () const

const LinearAlgebra::BlockSparseMatrixget_system_matrix () const

const LinearAlgebra::BlockSparseMatrixget_system_preconditioner_matrix () const

const MaterialModel::Interface< dim > & get_material_model () const

void compute_material_model_input_values (const LinearAlgebra::BlockVector &input_solution, const FEValuesBase< dim, dim > &input_finite_element_values, const typename DoFHandler< dim >::active_cell_iterator &cell, const bool compute_strainrate, MaterialModel::MaterialModelInputs< dim > &material_model_inputs) const

const GravityModel::Interface< dim > & get_gravity_model () const

const InitialTopographyModel::Interface< dim > & get_initial_topography_model () const

const GeometryModel::Interface< dim > & get_geometry_model () const

bool has_boundary_temperature () const

DEAL_II_DEPRECATED const BoundaryTemperature::Interface< dim > & get_boundary_temperature () const

const BoundaryTemperature::Manager< dim > & get_boundary_temperature_manager () const

const BoundaryHeatFlux::Interface< dim > & get_boundary_heat_flux () const

bool has_boundary_composition () const

DEAL_II_DEPRECATED const BoundaryComposition::Interface< dim > & get_boundary_composition () const

const BoundaryComposition::Manager< dim > & get_boundary_composition_manager () const

const std::map< types::boundary_id, std::unique_ptr< BoundaryTraction::Interface< dim > > > & get_boundary_traction () const

DEAL_II_DEPRECATED const InitialTemperature::Interface< dim > & get_initial_temperature () const

const InitialTemperature::Manager< dim > & get_initial_temperature_manager () const

DEAL_II_DEPRECATED const InitialComposition::Interface< dim > & get_initial_composition () const

const InitialComposition::Manager< dim > & get_initial_composition_manager () const

const std::set< types::boundary_id > & get_fixed_temperature_boundary_indicators () const

const std::set< types::boundary_id > & get_fixed_heat_flux_boundary_indicators () const

const std::set< types::boundary_id > & get_fixed_composition_boundary_indicators () const

const std::set< types::boundary_id > & get_mesh_deformation_boundary_indicators () const

const BoundaryVelocity::Manager< dim > & get_boundary_velocity_manager () const

const HeatingModel::Manager< dim > & get_heating_model_manager () const

const MeshRefinement::Manager< dim > & get_mesh_refinement_manager () const

const MeltHandler< dim > & get_melt_handler () const

const VolumeOfFluidHandler< dim > & get_volume_of_fluid_handler () const

const NewtonHandler< dim > & get_newton_handler () const

const MeshDeformation::MeshDeformationHandler< dim > & get_mesh_deformation_handler () const

const LateralAveraging< dim > & get_lateral_averaging () const

const AffineConstraints< double > & get_current_constraints () const

bool simulator_is_past_initialization () const

double get_pressure_scaling () const

bool pressure_rhs_needs_compatibility_modification () const

bool model_has_prescribed_stokes_solution () const

TableHandlerget_statistics_object () const

template<typename PostprocessorType >
DEAL_II_DEPRECATED PostprocessorType * find_postprocessor () const

const Postprocess::Manager< dim > & get_postprocess_manager () const

const Particle::World< dim > & get_particle_world () const

Particle::World< dim > & get_particle_world ()

bool is_stokes_matrix_free ()

const StokesMatrixFreeHandler< dim > & get_stokes_matrix_free () const

RotationProperties< dim > compute_net_angular_momentum (const bool use_constant_density, const LinearAlgebra::BlockVector &solution, const bool limit_to_top_faces=false) const

## Static Public Member Functions

static void declare_parameters (ParameterHandler &prm)

Static Public Member Functions inherited from aspect::MaterialModel::Interface< dim >
static void declare_parameters (ParameterHandler &prm)

Static Public Member Functions inherited from aspect::SimulatorAccess< dim >
static void get_composition_values_at_q_point (const std::vector< std::vector< double > > &composition_values, const unsigned int q, std::vector< double > &composition_values_at_q_point)

## Private Attributes

std::unique_ptr< Rheology::ViscoPlastic< dim > > rheology

std::vector< double > thermal_diffusivities

bool define_conductivities

std::vector< double > thermal_conductivities

EquationOfState::MulticomponentIncompressible< dim > equation_of_state

MaterialUtilities::PhaseFunction< dim > phase_function

Public Types inherited from aspect::MaterialModel::Interface< dim >
using MaterialModelInputs = MaterialModel::MaterialModelInputs< dim >

using MaterialModelOutputs = MaterialModel::MaterialModelOutputs< dim >

Protected Attributes inherited from aspect::MaterialModel::Interface< dim >
NonlinearDependence::ModelDependence model_dependence

## Detailed Description

### template<int dim> class aspect::MaterialModel::ViscoPlastic< dim >

A material model combining viscous and plastic deformation, with the option to also include viscoelastic deformation.

Viscous deformation is defined by a viscous flow law describing dislocation and diffusion creep: $$v = \frac{1}{2} A^{-\frac{1}{n}} d^{\frac{m}{n}} \dot{\varepsilon}_{ii}^{\frac{1-n}{n}} \exp\left(\frac{E + PV}{nRT}\right)$$ where where $$A$$ is the prefactor, $$n$$ is the stress exponent, $$\dot{\varepsilon}_{ii}$$ is the square root of the deviatoric strain rate tensor second invariant, $$d$$ is grain size, $$m$$ is the grain size exponent, $$E$$ is activation energy, $$V$$ is activation volume, $$P$$ is pressure, $$R$$ is the gas exponent and $$T$$ is temperature.

One may select to use the diffusion ( $$v_{diff}$$; $$n=1$$, $$m!=0$$), dislocation ( $$v_{disl}$$, $$n>1$$, $$m=0$$) or composite $$\frac{v_{diff}v_{disl}}{v_{diff}+v_{disl}}$$ equation form.

Viscous stress is limited by plastic deformation, which follows a Drucker Prager yield criterion: $$\sigma_y = C\cos(\phi) + P\sin(\phi)$$ (2D) or in 3D $$\sigma_y = \frac{6C\cos(\phi) + 2P\sin(\phi)}{\sqrt{3}(3+\sin(\phi))}$$ where $$\sigma_y$$ is the yield stress, $$C$$ is cohesion, $$phi$$ is the angle of internal friction and $$P$$ is pressure. If the viscous stress ( $$2v{\varepsilon}_{ii})$$) exceeds the yield stress ( $$\sigma_{y}$$), the viscosity is rescaled back to the yield surface: $$v_{y}=\sigma_{y}/(2{\varepsilon}_{ii})$$

When included, the viscoelastic rheology takes into account the elastic shear strength (e.g., shear modulus), while the tensile and volumetric strength (e.g., Young's and bulk modulus) are not considered. The model is incompressible and allows specifying an arbitrary number of compositional fields, where each field represents a different rock type or component of the viscoelastic stress tensor. The symmetric stress tensor in 2D and 3D, respectively, contains 3 or 6 components. The compositional fields representing these components must be named and listed in a very specific format, which is designed to minimize mislabeling stress tensor components as distinct 'compositional rock types' (or vice versa). For 2D models, the first three compositional fields must be labeled ve_stress_xx, ve_stress_yy and ve_stress_xy. In 3D, the first six compositional fields must be labeled ve_stress_xx, ve_stress_yy, ve_stress_zz, ve_stress_xy, ve_stress_xz, ve_stress_yz.

Combining this viscoelasticity implementation with non-linear viscous flow and plasticity produces a constitutive relationship commonly referred to as partial elastoviscoplastic (e.g., pEVP) in the geodynamics community. While extensively discussed and applied within the geodynamics literature, notable references include: Moresi et al. (2003), J. Comp. Phys., v. 184, p. 476-497. Gerya and Yuen (2007), Phys. Earth. Planet. Inter., v. 163, p. 83-105. Gerya (2010), Introduction to Numerical Geodynamic Modeling. Kaus (2010), Tectonophysics, v. 484, p. 36-47. Choi et al. (2013), J. Geophys. Res., v. 118, p. 2429-2444. Keller et al. (2013), Geophys. J. Int., v. 195, p. 1406-1442.

The overview below directly follows Moresi et al. (2003) eqns. 23-32. However, an important distinction between this material model and the studies above is the use of compositional fields, rather than particles, to track individual components of the viscoelastic stress tensor. The material model will be updated when an option to track and calculate viscoelastic stresses with particles is implemented. Moresi et al. (2003) begins (eqn. 23) by writing the deviatoric rate of deformation ( $$\hat{D}$$) as the sum of elastic ( $$\hat{D_{e}}$$) and viscous ( $$\hat{D_{v}}$$) components: $$\hat{D} = \hat{D_{e}} + \hat{D_{v}}$$. These terms further decompose into $$\hat{D_{v}} = \frac{\tau}{2\eta}$$ and $$\hat{D_{e}} = \frac{\overset{\triangledown}{\tau}}{2\mu}$$, where $$\tau$$ is the viscous deviatoric stress, $$\eta$$ is the shear viscosity, $$\mu$$ is the shear modulus and $$\overset{\triangledown}{\tau}$$ is the Jaumann corotational stress rate. If plasticity is included the deviatoric rate of deformation may be written as: $$\hat{D} = \hat{D_{e}} + \hat{D_{v}} + \hat{D_{p}}$$, where $$\hat{D_{p}}$$ is the plastic component. As defined in the second paragraph, $$\hat{D_{p}}$$ decomposes to $$\frac{\tau_{y}}{2\eta_{y}}$$, where $$\tau_{y}$$ is the yield stress and $$\eta_{y}$$ is the viscosity rescaled to the yield surface.

Above, the Jaumann corotational stress rate (eqn. 24) from the elastic component contains the time derivative of the deviatoric stress ( $$\dot{\tau}$$) and terms that account for material spin (e.g., rotation) due to advection: $$\overset{\triangledown}{\tau} = \dot{\tau} + {\tau}W -W\tau$$. Above, $$W$$ is the material spin tensor (eqn. 25): $$W_{ij} = \frac{1}{2} \left (\frac{\partial V_{i}}{\partial x_{j}} - \frac{\partial V_{j}}{\partial x_{i}} \right )$$.

The Jaumann stress-rate can also be approximated using terms from the time at the previous time step ( $$t$$) and current time step ( $$t + \Delta t^{e}$$): $$\smash[t]{\overset{\triangledown}{\tau}}^{t + \Delta t^{e}} \approx \frac{\tau^{t + \Delta t^{e} - \tau^{t}}}{\Delta t^{e}} - W^{t}\tau^{t} + \tau^{t}W^{t}$$. In this material model, the size of the time step above ( $$\Delta t^{e}$$) can be specified as the numerical time step size or an independent fixed time step. If the latter case is selected, the user has an option to apply a stress averaging scheme to account for the differences between the numerical and fixed elastic time step (eqn. 32). If one selects to use a fixed elastic time step throughout the model run, an equal numerical and elastic time step can be achieved by using CFL and maximum time step values that restrict the numerical time step to the fixed elastic time step.

The formulation above allows rewriting the total rate of deformation (eqn. 29) as $$\tau^{t + \Delta t^{e}} = \eta_{eff} \left ( 2\hat{D}^{t + \triangle t^{e}} + \frac{\tau^{t}}{\mu \Delta t^{e}} + \frac{W^{t}\tau^{t} - \tau^{t}W^{t}}{\mu} \right )$$.

The effective viscosity (eqn. 28) is a function of the viscosity ( $$\eta$$), elastic time step size ( $$\Delta t^{e}$$) and shear relaxation time ( $$\alpha = \frac{\eta}{\mu}$$): $$\eta_{eff} = \eta \frac{\Delta t^{e}}{\Delta t^{e} + \alpha}$$ The magnitude of the shear modulus thus controls how much the effective viscosity is reduced relative to the initial viscosity.

Elastic effects are introduced into the governing Stokes equations through an elastic force term (eqn. 30) using stresses from the previous time step: $$F^{e,t} = -\frac{\eta_{eff}}{\mu \Delta t^{e}} \tau^{t}$$. This force term is added onto the right-hand side force vector in the system of equations.

Several model parameters (reference densities, thermal expansivities thermal diffusivities, heat capacities and rheology parameters) can be defined per-compositional field. For each material parameter the user supplies a comma delimited list of length N+1, where N is the number of compositional fields. The additional field corresponds to the value for background material. They should be ordered background, composition1, composition2...''

If a list of values is given for the density, thermal expansivity, thermal diffusivity and heat capacity, the volume weighted sum of the values of each of the compositional fields is used in their place, for example $$\rho = \sum \left( \rho_i V_i \right)$$

The individual output viscosities for each compositional field are also averaged. The user can choose from a range of options for this viscosity averaging. If only one value is given for any of these parameters, all compositions are assigned the same value. The first value in the list is the value assigned to "background material" (regions where the sum of the compositional fields is < 1.0).

Definition at line 181 of file visco_plastic.h.

## § evaluate()

template<int dim>
 void aspect::MaterialModel::ViscoPlastic< dim >::evaluate ( const MaterialModel::MaterialModelInputs< dim > & in, MaterialModel::MaterialModelOutputs< dim > & out ) const
overridevirtual

Function to compute the material properties in out given the inputs in in. If MaterialModelInputs.strain_rate has the length 0, then the viscosity does not need to be computed.

Implements aspect::MaterialModel::Interface< dim >.

## § is_compressible()

template<int dim>
 bool aspect::MaterialModel::ViscoPlastic< dim >::is_compressible ( ) const
overridevirtual

Return whether the model is compressible or not. Incompressibility does not necessarily imply that the density is constant; rather, it may still depend on temperature or pressure. In the current context, compressibility means whether we should solve the continuity equation as $$\nabla \cdot (\rho \mathbf u)=0$$ (compressible Stokes) or as $$\nabla \cdot \mathbf{u}=0$$ (incompressible Stokes).

This material model is incompressible.

Implements aspect::MaterialModel::Interface< dim >.

## § reference_viscosity()

template<int dim>
 double aspect::MaterialModel::ViscoPlastic< dim >::reference_viscosity ( ) const
overridevirtual

Return a reference value typical of the viscosities that appear in this model. This value is not actually used in the material description itself, but is used in scaling variables to the same numerical order of magnitude when solving linear systems. Specifically, the reference viscosity appears in the factor scaling the pressure against the velocity. It is also used in computing dimension-less quantities. You may want to take a look at the Kronbichler, Heister, Bangerth 2012 paper that describes the design of ASPECT for a description of this pressure scaling.

Note
The reference viscosity should take into account the complete constitutive relationship, defined as the scalar viscosity times the constitutive tensor. In most cases, the constitutive tensor will simply be the identity tensor (this is the default case), but this may become important for material models with anisotropic viscosities, if the constitutive tensor is not normalized.

Implements aspect::MaterialModel::Interface< dim >.

## § declare_parameters()

template<int dim>
 static void aspect::MaterialModel::ViscoPlastic< dim >::declare_parameters ( ParameterHandler & prm )
static

## § parse_parameters()

template<int dim>
 void aspect::MaterialModel::ViscoPlastic< dim >::parse_parameters ( ParameterHandler & prm )
overridevirtual

Read the parameters this class declares from the parameter file. The default implementation of this function does not read any parameters. Consequently, derived classes do not have to overload this function if they do not take any runtime parameters.

Reimplemented from aspect::MaterialModel::Interface< dim >.

template<int dim>
 void aspect::MaterialModel::ViscoPlastic< dim >::create_additional_named_outputs ( MaterialModel::MaterialModelOutputs< dim > & outputs ) const
overridevirtual

If this material model can produce additional named outputs that are derived from NamedAdditionalOutputs, create them in here. By default, this does nothing.

Reimplemented from aspect::MaterialModel::Interface< dim >.

## § get_min_strain_rate()

template<int dim>
 double aspect::MaterialModel::ViscoPlastic< dim >::get_min_strain_rate ( ) const

## § is_yielding() [1/2]

template<int dim>
 DEAL_II_DEPRECATED bool aspect::MaterialModel::ViscoPlastic< dim >::is_yielding ( const double pressure, const double temperature, const std::vector< double > & composition, const SymmetricTensor< 2, dim > & strain_rate ) const

A function that returns whether the material is plastically yielding at the given pressure, temperature, composition, and strain rate.

Deprecated:
: Use the other function with this name instead, which allows to pass in more general input variables.

## § is_yielding() [2/2]

template<int dim>
 bool aspect::MaterialModel::ViscoPlastic< dim >::is_yielding ( const MaterialModelInputs< dim > & in ) const

A function that returns whether the material is plastically yielding at the given input variables (pressure, temperature, composition, strain rate, and so on).

## § rheology

template<int dim>
 std::unique_ptr > aspect::MaterialModel::ViscoPlastic< dim >::rheology
private

Pointer to the object used to compute the rheological properties. In this case, the rheology in question is visco(elasto)plastic. The object contains functions for parameter declaration and parsing, and further functions that calculate viscosity and viscosity derivatives. It also contains functions that create and fill additional material model outputs, specifically plastic outputs. The rheology itself is a composite rheology, and so the object contains further objects and/or pointers to objects that provide functions and parameters for all subordinate rheologies.

Definition at line 249 of file visco_plastic.h.

## § thermal_diffusivities

template<int dim>
 std::vector aspect::MaterialModel::ViscoPlastic< dim >::thermal_diffusivities
private

Definition at line 251 of file visco_plastic.h.

## § define_conductivities

template<int dim>
 bool aspect::MaterialModel::ViscoPlastic< dim >::define_conductivities
private

Whether to use user-defined thermal conductivities instead of thermal diffusivities.

Definition at line 256 of file visco_plastic.h.

## § thermal_conductivities

template<int dim>
 std::vector aspect::MaterialModel::ViscoPlastic< dim >::thermal_conductivities
private

Definition at line 258 of file visco_plastic.h.

## § equation_of_state

template<int dim>
 EquationOfState::MulticomponentIncompressible aspect::MaterialModel::ViscoPlastic< dim >::equation_of_state
private

Object for computing the equation of state.

Definition at line 263 of file visco_plastic.h.

## § phase_function

template<int dim>
 MaterialUtilities::PhaseFunction aspect::MaterialModel::ViscoPlastic< dim >::phase_function
private

Object that handles phase transitions.

Definition at line 268 of file visco_plastic.h.

The documentation for this class was generated from the following file: