# Generalized Maxwell Model

Generalized Maxwell model composed of a parralel assembly of unit Maxwell modules

## Description

The GeneralizedMaxwellModel class represents a generalized Maxwell model, that is, a material composed of Maxwell units assembled in series.

### Constitutive Equations

The material obeys to the following constitutive equation: (1)

The are the internal strains associated to each Maxwell unit, and the stiffness of the corresponding spring (a fourth-order tensor, identical in symmetry and dimensions to a standard elasticity tensor). The obey the following time-dependent differential equation: (2) With is the viscosity of the associated dashpot (a scalar with the dimension of time).

## Internal Time-Stepping Scheme

The constitutive equations are solved using a semi-implicit single-step first-order finite difference scheme. The internal strains at time step are computed from their values at the previous time step : (3) is a scalar between 0 (fully explicit) and 1 (fully implicit) that controls the time-stepping scheme (default value: 1). The value is determined by the "integration_rule" input parameter, which can take one of the forms shown in Table 1.

Table 1: Integration Rule and Time-Stepping Scheme

Integration RuleValue of Unconditional Convergence
BackwardEuler1yes
MidPoint0.5yes
Newmarkuser-defined
Zienkiewiczyes
warning:Convergence

The scheme is not valid for , so this value is forbidden.

Using this formalism, the stress-strain constitutive equation, which depends on the (unknown) can be rewritten so that it only depends on the and (both being known).

For efficiency reasons, the and are not stored separately, but as a single variable .

note:Update the Scheme with a Linear Viscoelastic Manager Material

For the time-stepping scheme to be properly updated, a LinearViscoelasticityManager object must be included in the input file, and linked to the material

## Stress-Strain Computation

The material is compatible with either the total small strain approximation, or either of the incremental strain approximation (incremental small strains or finite strains). The model requires the stress calculators listed in Table 2.

Table 2: Consistent Strain and Stress Calculators

The stress calculators use the actual elasticity tensor of the material , which is provided by the material itself.

### Driving Eigenstrain (Optional)

If the user defines a driving eigenstrain, then the stress induced by this eigenstrain is added to the creep calculation. Essentially, this replaces the differential relation in each material module with: (4)

## Example Input File Syntax

[./maxwell]
type = GeneralizedMaxwellModel
creep_modulus = '10e9'
creep_viscosity = '10'
poisson_ratio = 0.2
young_modulus = 10e9
driving_eigenstrain = eigen_true
[../]

(modules/tensor_mechanics/test/tests/visco/gen_maxwell_driving.i)

with the required strain calculator

[./strain]
type = ComputeIncrementalSmallStrain
displacements = 'disp_x disp_y disp_z'
eigenstrain_names = 'eigen_true'
[../]

(modules/tensor_mechanics/test/tests/visco/gen_maxwell_driving.i)

the required stress calculator

[./stress]
type = ComputeMultipleInelasticStress
inelastic_models = 'creep'
[../]

(modules/tensor_mechanics/test/tests/visco/gen_maxwell_driving.i)

the additional material to define the viscoelastic behavior

[./creep]
type = LinearViscoelasticStressUpdate
[../]

(modules/tensor_mechanics/test/tests/visco/gen_maxwell_driving.i)

and the required Linear Viscoelasticity Manager User Object:

[UserObjects]
[./update]
type = LinearViscoelasticityManager
viscoelastic_model = maxwell
[../]
[]

(modules/tensor_mechanics/test/tests/visco/gen_maxwell_driving.i)

## Input Parameters

• poisson_ratioinitial poisson ratio of the material

C++ Type:double

Options:

Description:initial poisson ratio of the material

• creep_moduluslist of the elastic moduli of the different springs in the material

C++ Type:std::vector

Options:

Description:list of the elastic moduli of the different springs in the material

• young_modulusinitial elastic modulus of the material

C++ Type:double

Options:

Description:initial elastic modulus of the material

• creep_viscositylist of the characteristic times of the different dashpots in the material

C++ Type:std::vector

Options:

Description:list of the characteristic times of the different dashpots in the material

### Required Parameters

• elastic_strain_nameelastic_strainname of the true elastic strain of the material

Default:elastic_strain

C++ Type:std::string

Options:

Description:name of the true elastic strain of the material

• integration_rulebackward-eulerdescribes how the viscoelastic behavior is integrated through time

Default:backward-euler

C++ Type:MooseEnum

Options:backward-euler mid-point newmark zienkiewicz

Description:describes how the viscoelastic behavior is integrated through time

• computeTrueWhen false, MOOSE will not call compute methods on this material. The user must call computeProperties() after retrieving the Material via MaterialPropertyInterface::getMaterial(). Non-computed Materials are not sorted for dependencies.

Default:True

C++ Type:bool

Options:

Description:When false, MOOSE will not call compute methods on this material. The user must call computeProperties() after retrieving the Material via MaterialPropertyInterface::getMaterial(). Non-computed Materials are not sorted for dependencies.

• base_nameOptional parameter that allows the user to define multiple mechanics material systems on the same block, i.e. for multiple phases

C++ Type:std::string

Options:

Description:Optional parameter that allows the user to define multiple mechanics material systems on the same block, i.e. for multiple phases

• need_viscoelastic_properties_inverseFalsechecks whether the model requires the computation of the inverse viscoelasticproperties (default: false)

Default:False

C++ Type:bool

Options:

Description:checks whether the model requires the computation of the inverse viscoelasticproperties (default: false)

• creep_ratiolist of the poisson ratios of the different springs in the material

C++ Type:std::vector

Options:

Description:list of the poisson ratios of the different springs in the material

• driving_eigenstrainname of the eigenstrain that increases the creep strains

C++ Type:std::string

Options:

Description:name of the eigenstrain that increases the creep strains

• theta1coefficient for newmark integration rule (between 0 and 1)

Default:1

C++ Type:double

Options:

Description:coefficient for newmark integration rule (between 0 and 1)

• boundaryThe list of boundary IDs from the mesh where this boundary condition applies

C++ Type:std::vector

Options:

Description:The list of boundary IDs from the mesh where this boundary condition applies

• creep_strain_namecreep_strainname of the true creep strain of the material(computed by LinearViscoelasticStressUpdate orComputeLinearViscoelasticStress)

Default:creep_strain

C++ Type:std::string

Options:

Description:name of the true creep strain of the material(computed by LinearViscoelasticStressUpdate orComputeLinearViscoelasticStress)

• blockThe list of block ids (SubdomainID) that this object will be applied

C++ Type:std::vector

Options:

Description:The list of block ids (SubdomainID) that this object will be applied

### Optional Parameters

• enableTrueSet the enabled status of the MooseObject.

Default:True

C++ Type:bool

Options:

Description:Set the enabled status of the MooseObject.

• use_displaced_meshFalseWhether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.

Default:False

C++ Type:bool

Options:

Description:Whether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.

• control_tagsAdds user-defined labels for accessing object parameters via control logic.

C++ Type:std::vector

Options:

Description:Adds user-defined labels for accessing object parameters via control logic.

• seed0The seed for the master random number generator

Default:0

C++ Type:unsigned int

Options:

Description:The seed for the master random number generator

• implicitTrueDetermines whether this object is calculated using an implicit or explicit form

Default:True

C++ Type:bool

Options:

Description:Determines whether this object is calculated using an implicit or explicit form

• constant_onNONEWhen ELEMENT, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps.When SUBDOMAIN, MOOSE will only call computeSubdomainProperties() for the 0th quadrature point, and then copy that value to the other qps. Evaluations on element qps will be skipped

Default:NONE

C++ Type:MooseEnum

Options:NONE ELEMENT SUBDOMAIN

Description:When ELEMENT, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps.When SUBDOMAIN, MOOSE will only call computeSubdomainProperties() for the 0th quadrature point, and then copy that value to the other qps. Evaluations on element qps will be skipped

• output_propertiesList of material properties, from this material, to output (outputs must also be defined to an output type)

C++ Type:std::vector

Options:

Description:List of material properties, from this material, to output (outputs must also be defined to an output type)

• outputsnone Vector of output names were you would like to restrict the output of variables(s) associated with this object

Default:none

C++ Type:std::vector

Options:

Description:Vector of output names were you would like to restrict the output of variables(s) associated with this object