# 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.

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 (2) where is the position vector of materials points in , and is the position vector of materials points in .

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, (3) 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 subsituting Eq. (3) into the definition for from Malvern (1969): (4) where the orthogonal nature of enables the simplification given above. Thus can be computed from as (5) which can be evaluated by performing a spectral decomposition of . Once has been computed, the multiplicative decomposition of the deformation graidient 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 (6)

This incremental streteching 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.

## Taylor Expansion

The stretching rate tensor and incremental rotation matrix can be approximated using Taylor expansion as Rashid (1993): the approximated stretching rate tensor (7) the approximated rotation matrix (8) with (9) 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 . (10) with being the eigenvalue and matrix being constructed from the corresponding eigenvector. (11) the 'incremental' stretching tensor (12) and thus (13)

## Volumetric Locking Correction

In ComputeFiniteStrain, is calculated in the computeStrain method, including a volumetric locking correction of (14) 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 TensorMechanics Master Action, as shown below.

[./TensorMechanics]
[./Master]
[./all]
strain = FINITE
[../]
[../]
[../]

(modules/tensor_mechanics/test/tests/finite_strain_elastic/finite_strain_elastic_new_test.i)
note:Use of the Tensor Mechanics Master Action Recommended

The TensorMechanics Master Action 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 TensorMechanics Master Action 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'
[../]


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 diveregence kernel settings and the corresponding strain classes, see the section on Consistency Between Stress and Strain in the TensorMechanics 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

Options:

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

### Required Parameters

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

C++ Type:MaterialPropertyName

Options:

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

• decomposition_methodTaylorExpansionMethods to calculate the strain and rotation increments

Default:TaylorExpansion

C++ Type:MooseEnum

Options:TaylorExpansion EigenSolution

Description:Methods to calculate the strain and rotation increments

• 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

• eigenstrain_namesList of eigenstrains to be applied in this strain calculation

C++ Type:std::vector

Options:

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

• volumetric_locking_correctionFalseFlag to correct volumetric locking

Default:False

C++ Type:bool

Options:

Description:Flag to correct volumetric locking

• 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

• 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

• 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

### Outputs 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