# Compute Plane Small Strain

Compute a small strain under generalized plane strain assumptions where the out of plane strain is generally nonzero.

## Description

The material ComputePlaneSmallStrain calculates the small incremental strain for 2D plane strain problems. It can be used for classical thick body plane strain problems, Weak Plane Stress models, or in Generalized Plane Strain simulations.

## Out of Plane Strain

In the classical thick body plane strain problem, the strain component in the out-of-plane direction (the infinitely thick direction) is held constant at zero: (1) is the strain tensor diagonal component for the direction of the out-of-plane strain.

### Generalized Plane Strain

In the cases of the generalized plane strain and weak plane stress models, the component of strain and the deformation gradient in the out-of-plane direction is non-zero. To solve for this out-of-plane strain, we use the out-of-plane strain variable as the strain tensor component (2) where is the strain tensor diagonal component for the direction of the out-of-plane strain and is a prescribed out-of-plane strain value: this strain value can be given either as a scalar variable or a nonlinear variable. The Generalized Plane Strain problems use scalar variables. Multiple scalar variables can be provided such that one strain calculator is needed for multiple generalized plane strain models on different subdomains.

## Strain and Deformation Gradient Formulation

The definition of a small total linearized strain is (3) The values of each of the strain tensor components depends on the direction selected by the user as the out-of-plane direction.

#### var element = document.getElementById("moose-equation-a3626e65-651f-452a-9a24-8ae8f3077042");katex.render("Z", element, {macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"},throwOnError:false,displayMode:false});-Direction of Out-of-Plane Strain (Default)

The default out-of-plane direction is along the -axis. For this direction the strain tensor, Eq. (3), is given as (4) where is defined in Eq. (2). As in the classical presentation of the strain tensor in plane strain problems, the components of the strain tensor associated with the -direction are zero; these zero components indicate no coupling between the in-plane and the out-of-plane strains.

#### var element = document.getElementById("moose-equation-bdd61f91-c5f6-41e7-b1b7-d32425988768");katex.render("X", element, {macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"},throwOnError:false,displayMode:false});-Direction of Out-of-Plane Strain

If the user selects the out-of-plane direction as along the -direction, the strain tensor from Eq. (3) is given as (5) so that the off-diagonal components of the strain tensor associated with the -direction are zeros.

#### var element = document.getElementById("moose-equation-aee1f442-7c17-40b0-b466-a1786c93eef7");katex.render("Y", element, {macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"},throwOnError:false,displayMode:false});-Direction of Out-of-Plane Strain

If the user selects the out-of-plane direction as along the -direction, the strain tensor from Eq. (3) is given as (6) so that the off-diagonal components of the strain tensor associated with the -direction are zeros.

### Volumetric Locking Correction for Strain Tensor

If selected by the user, the strain tensor is conditioned with a formulation to mitigate volumetric locking of the elements. The volumetric locking correction is applied to the total strain (7) where is the volumetric strain and is the Rank-2 identity tensor. For more details about the theory behind Eq. (7) see the Volumetric Locking Correction documentation.

## Example Input File

### Generalized Plane Strain

As an example, the use of this plane strain class with the Generalized Plane Strain simulations uses the scalar out-of-plane strains. The tensor mechanics MasterAction is used to create the ComputePlaneSmallStrain class with the planar_formulation = GENERALIZED_PLANE_STRAIN and strain = SMALL settings.

[./all]
strain = SMALL
displacements = 'disp_x disp_y'
generate_output = 'stress_xx stress_xy stress_yy stress_zz strain_xx strain_xy strain_yy strain_zz'
planar_formulation = GENERALIZED_PLANE_STRAIN
eigenstrain_names = eigenstrain
scalar_out_of_plane_strain = scalar_strain_zz
temperature = temp
save_in = 'saved_x saved_y'
[../]

(modules/tensor_mechanics/test/tests/generalized_plane_strain/generalized_plane_strain_small.i)

Note that the argument for the scalar_out_of_plane_strain parameter is the name of the scalar strain variable

[./scalar_strain_zz]
order = FIRST
family = SCALAR
[../]

(modules/tensor_mechanics/test/tests/generalized_plane_strain/generalized_plane_strain_small.i)

### var element = document.getElementById("moose-equation-60629343-f8b7-4a3c-8283-22603e2aaccc");katex.render("Y", element, {macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"},throwOnError:false,displayMode:false});-Direction of Out-of-Plane Strain

This plane strain class is used to model plane strain with an out-of-plane strain in directions other than in the -direction. As an example, the tensor mechanics MasterAction can be used to create the ComputePlaneFiniteStrain class for a -direction out-of-plane strain with the planar_formulation = PLANE_STRAIN and the out_of_plane_direction = y settings.

[./plane_strain]
block = 1
strain = SMALL
out_of_plane_direction = y
planar_formulation = PLANE_STRAIN
eigenstrain_names = 'eigenstrain'
generate_output = 'stress_xx stress_xz stress_yy stress_zz strain_xx strain_xz strain_yy strain_zz'
[../]

(modules/tensor_mechanics/test/tests/2D_different_planes/planestrain_xz.i)

## 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

• scalar_out_of_plane_strainScalar variable for generalized plane strain

C++ Type:std::vector

Options:

Description:Scalar variable for generalized plane strain

• 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

• 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

• 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

• out_of_plane_directionzThe direction of the out-of-plane strain.

Default:z

C++ Type:MooseEnum

Options:x y z

Description:The direction of the out-of-plane strain.

• volumetric_locking_correctionFalseFlag to correct volumetric locking

Default:False

C++ Type:bool

Options:

Description:Flag to correct volumetric locking

• out_of_plane_strainNonlinear variable for plane stress condition

C++ Type:std::vector

Options:

Description:Nonlinear variable for plane stress condition

• 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

• subblock_index_providerSubblockIndexProvider user object name

C++ Type:UserObjectName

Options:

Description:SubblockIndexProvider user object name

### 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