Compute Finite Strain in Cartesian System

Compute a strain increment and rotation increment for finite strains.

Description

This class is used to compute the strain increment, total strain, and incremental rotation for finite strain problems. The finite strain approach used is the incremental corotational form (Rashid, 1993). This approach computes logarithmic strains and strain increments.

Incremental Configurations

In this form, the generic time increment under consideration is such that (1) The configurations of the material element under consideration at and are denoted by , and , respectively for the previous and the current incremental configurations.

Deformation Gradient Definition

The deformation gradient represents the change in a material element from the reference configuration to the current configuration (Malvern, 1969). In the incremental formulation used in the ComputeFiniteStrain class, the incremental deformation gradient represents the change in the material element from the previous configuration, , to the current configuration, . Mathematically this relationship is given as where is the position vector of materials points in , and is the position vector of materials points in .

commentnote:Incremental vs Total Deformation Gradient

Note that is NOT the deformation gradient, but rather the incremental deformation gradient of with respect to . Thus , where is the total deformation gradient at time .

Following the explanation of this procedure given by Zhang et al. (2018), the incremental deformation gradient can be multiplicatively decomposed into an incremental rotation tensor, , and the incremental right stretch tensor, (2) where is a proper orthogonal rotation tensor and the stretch tensor, , is symmetric and positive definite. The incremental right Cauchy-Green deformation tensor, , can be given in terms of by substituting Eq. (2) into the definition for from Malvern (1969): (3) where the orthogonal nature of enables the simplification given above. Thus can be computed from as (4) which can be evaluated by performing a spectral decomposition of . Once has been computed, the multiplicative decomposition of the deformation gradient is used to find the incremental rotation tensor and the stretching rate . Following Rashid (1993), the stretching rate tensor can be expressed in terms of the 'incremental' right Cauchy-Green deformation tensor (5)

This incremental stretching rate tensor can then be used as the work conjugate for a stress measure, or used to compute another strain measure. The most computationally expensive part of this procedure is the spectral decomposition of to find . This decomposition can be computed exactly using an Eigensolution, yet an approximation of this can be computed with much lower computational expense using a Taylor expansion procedure. This class provides options to perform this calculation either way, and the Taylor expansion is the default.

This class also provides an additional option for computing the stretching rate tensor and incremental rotation tensor using the method of Hughes and Winget (1980). This method involves evaluating the displacement field at the midpoint of the timestep, and as a result, requires updating stress and strain measures at the old configuration rather than the current.

Taylor Expansion

The stretching rate tensor and incremental rotation matrix can be approximated using Taylor expansion as Rashid (1993): the approximated stretching rate tensor the approximated rotation matrix with The sign of is set by examining the sign of .

Eigen-Solution

The stretching rate tensor can be calculated by the eigenvalues and eigenvectors of . with being the eigenvalue and matrix being constructed from the corresponding eigenvector. the 'incremental' stretching tensor and thus

Hughes-Winget Approximation

As also described in Rashid (1993), the stretching rate tensor and incremental rotation matrix in for the Hughes-Winget method are based on the spatial gradient of the displacement field evaluated at the mid-point of the time step The approximate stretching rate tensor can then be computed as and the incremental rotation matrix is approximated by where .

Volumetric Locking Correction

In ComputeFiniteStrain, is calculated in the computeStrain method, including a volumetric locking correction of where is the average value for the entire element. The strain increment and the rotation increment are calculated in computeQpStrain(). Once the strain increment is calculated, it is added to the total strain from . The total strain from must then be rotated using the rotation increment.

Example Input File Syntax

The finite strain calculator can be activated in the input file through the use of the Solid Mechanics Physics, as shown below.

[Physics]
  [SolidMechanics]
    [QuasiStatic]
      [./all]
        strain = FINITE
        add_variables = true
      [../]
    [../]
  [../]
[]
(moose/modules/solid_mechanics/test/tests/finite_strain_elastic/finite_strain_elastic_new_test.i)
commentnote:Use of the Solid Mechanics QuasiStatic Physics Recommended

The Solid Mechanics Physics is designed to automatically determine and set the strain and stress divergence parameters correctly for the selected strain formulation. We recommend that users employ the Solid Mechanics Physics whenever possible to ensure consistency between the test function gradients and the strain formulation selected.

Although not recommended, it is possible to directly use the ComputeFiniteStrain material in the input file.

[./strain]
  type = ComputeFiniteStrain
  block = 0
  displacements = 'disp_x disp_y disp_z'
[../]
(moose/modules/solid_mechanics/test/tests/volumetric_deform_grad/elastic_stress.i)

When directly using ComputeFiniteStrain in an input file as shown above, the StressDivergenceTensors kernel must be modified from the default by setting the parameter use_displaced_mesh = true. This setting is required to maintain consistency in the test function gradients and the strain formulation. For a complete discussion of the stress divergence kernel settings and the corresponding strain classes, see the section on Consistency Between Stress and Strain in the SolidMechanics module overview. In addition, be aware of the loading cycle limitations while using finite strains as outlined in the section Large Strain Closed Loop Loading Cycle.

Input Parameters

  • displacementsThe displacements appropriate for the simulation geometry and coordinate system

    C++ Type:std::vector<VariableName>

    Unit:(no unit assumed)

    Controllable:No

    Description:The displacements appropriate for the simulation geometry and coordinate system

Required Parameters

  • 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

    Unit:(no unit assumed)

    Controllable:No

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

  • blockThe list of blocks (ids or names) that this object will be applied

    C++ Type:std::vector<SubdomainName>

    Unit:(no unit assumed)

    Controllable:No

    Description:The list of blocks (ids or names) that this object will be applied

  • boundaryThe list of boundaries (ids or names) from the mesh where this object applies

    C++ Type:std::vector<BoundaryName>

    Unit:(no unit assumed)

    Controllable:No

    Description:The list of boundaries (ids or names) from the mesh where this object applies

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

    Default:True

    C++ Type:bool

    Unit:(no unit assumed)

    Controllable:No

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

  • 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 computeQpProperties() 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

    Unit:(no unit assumed)

    Options:NONE, ELEMENT, SUBDOMAIN

    Controllable:No

    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 computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps. Evaluations on element qps will be skipped

  • declare_suffixAn optional suffix parameter that can be appended to any declared properties. The suffix will be prepended with a '_' character.

    C++ Type:MaterialPropertyName

    Unit:(no unit assumed)

    Controllable:No

    Description:An optional suffix parameter that can be appended to any declared properties. The suffix will be prepended with a '_' character.

  • decomposition_methodTaylorExpansionMethods to calculate the strain and rotation increments

    Default:TaylorExpansion

    C++ Type:MooseEnum

    Unit:(no unit assumed)

    Options:TaylorExpansion, EigenSolution, HughesWinget

    Controllable:No

    Description:Methods to calculate the strain and rotation increments

  • eigenstrain_namesList of eigenstrains to be applied in this strain calculation

    C++ Type:std::vector<MaterialPropertyName>

    Unit:(no unit assumed)

    Controllable:No

    Description:List of eigenstrains to be applied in this strain calculation

  • global_strainOptional material property holding a global strain tensor applied to the mesh as a whole

    C++ Type:MaterialPropertyName

    Unit:(no unit assumed)

    Controllable:No

    Description:Optional material property holding a global strain tensor applied to the mesh as a whole

  • prop_getter_suffixAn optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.

    C++ Type:MaterialPropertyName

    Unit:(no unit assumed)

    Controllable:No

    Description:An optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.

  • use_interpolated_stateFalseFor the old and older state use projected material properties interpolated at the quadrature points. To set up projection use the ProjectedStatefulMaterialStorageAction.

    Default:False

    C++ Type:bool

    Unit:(no unit assumed)

    Controllable:No

    Description:For the old and older state use projected material properties interpolated at the quadrature points. To set up projection use the ProjectedStatefulMaterialStorageAction.

  • volumetric_locking_correctionFalseFlag to correct volumetric locking

    Default:False

    C++ Type:bool

    Unit:(no unit assumed)

    Controllable:No

    Description:Flag to correct volumetric locking

Optional Parameters

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

    C++ Type:std::vector<std::string>

    Unit:(no unit assumed)

    Controllable:No

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

  • enableTrueSet the enabled status of the MooseObject.

    Default:True

    C++ Type:bool

    Unit:(no unit assumed)

    Controllable:Yes

    Description:Set the enabled status of the MooseObject.

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

    Default:True

    C++ Type:bool

    Unit:(no unit assumed)

    Controllable:No

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

  • seed0The seed for the master random number generator

    Default:0

    C++ Type:unsigned int

    Unit:(no unit assumed)

    Controllable:No

    Description:The seed for the master random number generator

Advanced Parameters

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

    C++ Type:std::vector<std::string>

    Unit:(no unit assumed)

    Controllable:No

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

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

    Default:none

    C++ Type:std::vector<OutputName>

    Unit:(no unit assumed)

    Controllable:No

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

Outputs Parameters

References

  1. Thomas JR Hughes and James Winget. Finite rotation effects in numerical integration of rate constitutive equations arising in large-deformation analysis. International journal for numerical methods in engineering, 15(12):1862–1867, 1980.[BibTeX]
  2. Lawrence E Malvern. Introduction to the Mechanics of a Continuous Medium. Prentice-Hall, 1969.[BibTeX]
  3. MM Rashid. Incremental kinematics for finite element applications. International Journal for Numerical Methods in Engineering, 36(23):3937–3956, 1993.[BibTeX]
  4. Ziyu Zhang, Wen Jiang, John E Dolbow, and Benjamin W Spencer. A modified moment-fitted integration scheme for x-fem applications with history-dependent material data. Computational Mechanics, pages 1–20, 2018.[BibTeX]