Compute Multiple Inelastic Stress

Jacobian Multiplier and the Consistent Tangent Operator

The Jacobian multiplier, which is used in the StressDivergenceTensors kernel to condition the Jacobian calculation, must be calculated from the combination of all the different inelastic material models. There are three options used to calculate the combined Jacobian multiplier: Elastic, Partial, and Nonlinear, which are set by the individual elastic material models. (5) where is the Jacobian multiplier, is the elasticity tensor, is the Rank-4 identity tensor, and is the consistent tangent operator.

The consistent tangent operator, defined in Eq. (6) provides the information on how the stress changes with respect to changes in the displacement variables. (6) where is an arbitrary change in the total strain (which occurs because the displacements are changed) and is the resulting change in the stress. In a purely elastic situation (the elasticity tensor), but the inelastic mapping of changes in the stress as a result of changes in the displacement variables is more complicated. In a plastic material model, the proposed values of displacements for the current time step were used to calculate a trial inadmissible stress, , Eq. (2), that was brought back to the yield surface through a radial return algorithm. A slight change in the proposed displacement variables will produce a slightly different trial stress and so on. Other inelastic material models follow a similar pattern.

The user can chose to force all of the inelastic material models to use the elasticity tensor as the consistent tangent operator by setting tangent_operator = elastic. This setting will reduce the computational load of the inelastic material models but may hamper the convergence of the simulation. By default, the inelastic material models are allowed to compute the consistent tangent operator implemented in each individual inelastic model with the tangent_operator = nonlinear option.

Material Time Step Size Limitations

Prior to calculating the final strain values, the algorithm checks the size of the current time step against any limitations on the size of the time step as optionally defined by the inelastic material models. As described in the Material Time Step Limiter section, the time step size involves a post processor to ensure that the current time step size is reasonable for each of the inelastic material models used in the simulation.

At the end of the algorithm, the final value of the elastic and inelastic strain tensors are calculated as shown in the last element of Figure 1.

Cycle Through One Inelastic Model per Time Step

ComputeMultipleInelasticStress also includes an option to run a series of inelastic models in a rotating fashion such that only a single inelastic model is run on a timestep. This option uses the same algorithm as in Figure 2 to determine the strains and stress value based on the rotated single inelastic model. A separate method is then employed to propagate the strain and stress values to the other inelastic material models for storage as old material property values.

Consistent Tangent Operator

The consistent tangent operator is used to improve the convergence of mechanics problems (see a reference such as Simo and Taylor (1985) for an introduction to consistent tangent operators). The Jacobian matrix, Eq. (5) and Eq. (7), is used to capture how the change in the residual calculation changes with respect to changes in the displacement variables. To calculate the Jacobian, MOOSE relies on knowing how the stress changes with respect to changes in the displacement variables.

Because the change of the stress with respect to the change in displacements is material specific, the value of the consistent tangent operator is computed in each inelastic material model. By default the consistent tangent operator is set equal to the elasticity tensor (the option Elastic in Eq. (5) and Eq. (7)). Inelastic material models which use either the Partial or Nonlinear options in Eq. (5) or Eq. (7) define a material specific consistent tangent operator.

Generally Partial consistent tangent operators should be implemented for non-yielding materials (e.g. volumetric swelling) and Full consistent tangent operators should be implemented for yielding material models (e.g. plasticity).

Include Damage Model

Optionally, the effect of damage on the stress calculation can be included in the model. Another material that defines the evolution of damage should be coupled using parameter damage_model. Here, first the inelastic strains and corresponding effective stresses are calculated based on the undamaged properties. Afterwards, the damage index is applied on the effective stress to calculate the damaged stress. This captures the effect of damage in a material undergoing creep or plastic deformation.

Material Time Step Limiter

In some cases, particularly in creep, limits on the time step are required by the material model formulation. Each inelastic material model is responsible for calculating the maximum time step allowable for that material model. The MaterialTimeStepPostprocessor finds the minimum time step size limits from the entire simulation domain. The postprocessor then interfaces with the IterationAdaptiveDT to restrict the time step size based on the limit calculated in the previous time step. When the damage model is included, the timestep is limited by the minimum timestep between the inelastic models and the damage model.

Multiple Inelastic Models

For multiple inelastic models, all of the inelastic material model names must be listed as arguments to the inelastic_models parameter. The inelastic material blocks must also be present.

[Materials]
  [./elasticity_tensor]
    type = ComputeIsotropicElasticityTensor
    block = 0
    youngs_modulus = 1e3
    poissons_ratio = 0.3
  [../]
  [./creep_plas]
    type = ComputeMultipleInelasticStress
    block = 0
    tangent_operator = elastic
    inelastic_models = 'creep plas'
    max_iterations = 50
    absolute_tolerance = 1e-05
    combined_inelastic_strain_weights = '0.0 1.0'
  [../]
  [./creep]
    type = PowerLawCreepStressUpdate
    block = 0
    coefficient = 0.5e-7
    n_exponent = 5
    m_exponent = -0.5
    activation_energy = 0
  [../]
  [./plas]
    type = IsotropicPlasticityStressUpdate
    block = 0
    hardening_constant = 100
    yield_stress = 20
  [../]
[]

Single Inelastic Model

For a single inelastic material model the input syntax is simply condensed

[./stress]
  type = ComputeMultipleInelasticStress
  inelastic_models = 'isoplas'
  block = 1
[../]

and only a single inelastic material model is included in the input. This example includes the max_inelastic_increment parameter which is used to limit the time step size.

[./isoplas]
  type = IsotropicPlasticityStressUpdate
  yield_stress = 5e6
  hardening_constant = 0.0
  relative_tolerance = 1e-20
  absolute_tolerance = 1e-8
  max_inelastic_increment = 0.000001
[../]