Strain Formulations in Tensor Mechanics

Small Linearized Total Strain

For linear elasticity problems, the Tensor Mechanics module includes a small strain and total strain material ComputeSmallStrain. This material is useful for verifying material models with hand calculations because of the simplified strain calculations.

Linearized small strain theory assumes that the gradient of displacement with respect to position is much smaller than unity, and the squared displacement gradient term is neglected in the small strain definition to give: For more details on the linearized small strain assumption and derivation, see a Continuum Mechanics text such as Malvern (1969) or Bower (2009), specifically Chapter 2.

Total strain theories are path independent: in MOOSE, path independence means that the total strain, from the beginning of the entire simulation, is used to calculate stress and other material properties. Incremental theories, on the other hand, use the increment of strain at timestep to calculate stress. Because the total strain formulation ComputeSmallStrain is path independent, no old values of strain or stress from the previous timestep are stored in MOOSE. For a comparison of total strain vs incremental strain theories with experimental data, see Shammamy and Sidebottom (1967).

The input file syntax for small strain is

[Modules/TensorMechanics/Master]
  [Modules/TensorMechanics/Master]
    [Modules/TensorMechanics/Master]
      [./block1]
        strain = SMALL
        add_variables = true
      [../]
    []
  []
[]

Incremental Small Strains

Applicable for small linearized strains, MOOSE includes an incremental small strain material, ComputeIncrementalSmallStrain. As in the small strain material, the incremental small strain class assumes the gradient of displacement with respect to position is much smaller than unity, and the squared displacement gradient term is neglected in the small strain definition to give: As the class name suggests, ComputeIncrementalSmallStrain is an incremental formulation. The stress increment is calculated from the current strain increment at each time step. In this class, the rotation tensor is defined to be the rank-2 Identity tensor: no rotations are allowed in the model. Stateful properties, including strain_old and stress_old, are stored. This incremental small strain material is useful as a component of verifying more complex finite incremental strain-stress calculations.

The input file syntax for incremental small strain is

[./TensorMechanics]
  [./Master]
    [./all]
      strain = SMALL
      incremental = true
      add_variables = true
      eigenstrain_names = eigenstrain
      generate_output = 'strain_xx strain_yy strain_zz'
    [../]
  [../]
[../]

Finite Large Strains

For finite strains, use ComputeFiniteStrain in which an incremental form is employed such that the strain increment and the rotation increment are calculated.

Large Strain Closed Loop Loading Cycle

It is important to note that some inaccuracies are inherently present in incremental (hypoelastic) representations of finite strain behavior. This is an issue for the MOOSE implementation discussed here, as well as the implementations of other major commercial codes. The magnitude of the error increases as the magnitude of the strain increases, so this is typically only a practical issue for very large strains, for which hyperelastic models are more appropriate. This can be demonstrated with the closed loop loading cycle illustrated in Figure 1. Since the initial and final configurations are the same, with elastic material, one would expect that the stress would return to zero in the final underformed state. With this incremental formulation, however, a residual stress remains, and the magnitude of that residual stress increases with the magnitude of the loading cycle. This occurs because the integral of the rate-of-deformation over the full loading cycle is not zero. A detailed explanation can be found in Belytschko et al. (2003).

Figure 1: Closed loop large deformation loading cycle.

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

References