# Compute Anisotropic Linear Elastic Phase Field Fracture Stress

Phase-field fracture model energy contribution to fracture for elasticity and undamaged stress under compressive strain

## Description

This material implements a phase field fracture model that can include anisotropic elasticity tensors, computing the stress and the free energy derivatives required for the model. It works with the standard phase field kernels for nonconserved variables. In the model, a nonconserved order parameter defines the crack, where in undamaged material and in cracked material. Cracked material can sustain a compressive stress, but not a tensile one. evolves to minimize the elastic free energy of the system.

This model assumes linear elastic mechanical deformation but can use an anisotropic elasticity tensor.

## Model Summary

In the model, the uncracked stress is decomposed into its compressive and tensile parts using a spectral decomposition (1) The compressive and tensile parts of the stress are computed from postive and negative projection tensors (computed from the spectral decomposition) according to (2) (3)

## Free Energy Calculation

The total strain energy density is defined as (4) where is the strain energy due to tensile stress, is the strain energy due to compressive stress, and is a parameter used to avoid non-positive definiteness at or near complete damage. (5) (6)

The crack energy density is defined as (7) where is the width of the crack interface and is a parameter related to the energy release rate.

The total local free energy density is defined as (8)

## Stress Definition

To be thermodynamically consistent, the stress is related to the deformation energy density according to (9) Since (10) (11) then, (12)

The Jacobian matrix for the stress is (13) where is the elasticity tensor.

## Evolution Equation and History Variable

To avoid crack healing, a history variable is defined that is the maximum energy density over the time interval , where is the current time step, i.e. (14)

Now, the total free energy is redefined as: (15) with (16) Its derivatives are (17)

The evolution equation for the damage parameter follows the Allen-Cahn equation (18) where and .

This equation follows the standard Allen-Cahn and thus can be implemented in MOOSE using the standard Allen-Cahn kernels, TimeDerivative, AllenCahn, and ACInterface. There is now an action that automatically generates these kernels: NonconservedAction. See the PhaseField module documentation for more information.

## Example Input File Syntax


[./cracked_stress]
type = ComputeLinearElasticPFFractureStress
c = c
kdamage = 1e-6
F_name = E_el
use_current_history_variable = true
[../]
(modules/combined/test/tests/phase_field_fracture/crack2d_aniso.i)

## Input Parameters

• cOrder parameter for damage

C++ Type:std::vector

Options:

Description:Order parameter for damage

### Required Parameters

• 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

• use_current_history_variableTrueUse the current value of the history variable.

Default:True

C++ Type:bool

Options:

Description:Use the current value of the history variable.

• F_nameE_elName of material property storing the elastic energy

Default:E_el

C++ Type:MaterialPropertyName

Options:

Description:Name of material property storing the elastic energy

• kappa_namekappa_opName of material property being created to store the interfacial parameter kappa

Default:kappa_op

C++ Type:MaterialPropertyName

Options:

Description:Name of material property being created to store the interfacial parameter kappa

• kdamage1e-06Stiffness of damaged matrix

Default:1e-06

C++ Type:double

Options:

Description:Stiffness of damaged matrix

• store_stress_oldFalseParameter which indicates whether the old stress state, required for the HHT time integration scheme and Rayleigh damping, needs to be stored

Default:False

C++ Type:bool

Options:

Description:Parameter which indicates whether the old stress state, required for the HHT time integration scheme and Rayleigh damping, needs to be stored

• 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

• mobility_nameLName of material property being created to store the mobility L

Default:L

C++ Type:MaterialPropertyName

Options:

Description:Name of material property being created to store the mobility L

• 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

• 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.

• enableTrueSet the enabled status of the MooseObject.

Default:True

C++ Type:bool

Options:

Description:Set the enabled status of the MooseObject.

• 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