The finite strain mechanics approach used in the MOOSE tensor_mechanics module is the incremental corotational form from Rashid 1993. In this form, the generic time increment under consideration is such that $$t \in [t_n, t_{n+1}]$$$. The configurations of the material element under consideration at $$t = t_n$$$ and $$t = t_{n+1}$$$are denoted by $$\kappa_n$$$, and $$\kappa_{n + 1}$$$, respectively. The incremental motion over the time increment is assumed to be given in the form of the inverse of the deformation gradient $$\hat{\mathbf{F}}$$$ of $$\kappa_{n + 1}$$$with respect to $$\kappa_n$$$, which may be written as

\hat{\mathbf{F}}^{-1} = 1 - \frac{\partial \hat{\mathbf{u}}}{\partial \mathbf{x}},

where $$\hat{\mathbf{u}}(\mathbf{x})$$$is the incremental displacement field for the time step, and $$\mathbf{x}$$$ is the position vector of materials points in $$\kappa_{n+1}$$$. Note that $$\hat{\mathbf{F}}$$$ is NOT the deformation gradient, but rather the incremental deformation gradient of $$\kappa_{n+1}$$$with respect to $$\kappa_n$$$. Thus, $$\hat{\mathbf{F}} = \mathbf{F}_{n+1} \mathbf{F}_n^{-1}$$$, where $$\mathbf{F}_n$$$ is the total deformation gradient at time $$t_n$$$. For this form, we assume \begin{eqnarray} \dot{\mathbf{F}} \mathbf{F}^{-1} &=& \mathbf{D}\ \mathrm{(constant\ and\ symmetric),\ } t_n<t<t_{n+1}\\ \mathbf{F}(t^{-}_{n+1}) &=& \hat{\mathbf{U}}\ \mathrm{(symmetric\ positive\ definite)}\\ \mathbf{F}(t_{n+1}) &=& \hat{\mathbf{R}} \hat{\mathbf{U}} = \hat{\mathbf{F}}\ (\hat{\mathbf{R}}\ \mathrm{proper\ orthogonal}) \end{eqnarray} The stress rate is calculated in the Compute*Stress classes, where either elasticity or plasticity options can be used. The base material class for the finite strain calculations is ComputeFiniteStrain. In that material, $$\hat{\mathbf{F}}$$$ is calculated in the computeStrain function, including a volumetric locking correction of \hat{\mathbf{F}}_{corr} = \hat{\mathbf{F}} \left( \frac{|\mathrm{av}_{el}(\hat{\mathbf{F}})|}{|\hat{\mathbf{F}}|} \right)^{\frac{1}{3}}, where $$\mathrm{av}_{el}()$$$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 $$t_n$$$. The total strain from $$t_{n+1}$$\$ must then be rotated using the rotation increment. An portion of the code, which performs this rotation, is shown below.

  void ComputeFiniteStrain::computeQpStrain()
{
.
.
.
//Update strain in intermediate configuration
_mechanical_strain[_qp] = _mechanical_strain_old[_qp] + _strain_increment[_qp];
_total_strain[_qp] = _total_strain_old[_qp] + total_strain_increment;

//Rotate strain to current configuration
_mechanical_strain[_qp] = _rotation_increment[_qp] * _mechanical_strain[_qp] * _rotation_increment[_qp].transpose();
_total_strain[_qp] = _rotation_increment[_qp] * _total_strain[_qp] * _rotation_increment[_qp].transpose();

}


### Input File Use

The finite strain material is designed to be used with any incremental stress material, such as complex plasticity and creep models or a linear elastic model, such as ComputeFiniteStrainElasticStress. An example portion of an input file using these three materials is shown below:

  # Tensor Mechanics
[Materials]
[./elasticity_tensor]
type = ComputeIsotropicElasticityTensor
block = 0
youngs_modulus = 2.1e5
poissons_ratio = 0.3
[../]
[./strain]
type = ComputeFiniteStrain
block = 0
displacements = 'disp_x disp_y disp_z'
[../]
[./stress]
type = ComputeFiniteStrainElasticStress
block = 0
[../]
[]


Finite strain formulations for 1D spherically symmetric and 2D axisymmetric problems are also included in MOOSE; more information on these specialized materials can be found here.

#### Input file conversion from Solid Mechanics to Tensor Mechanics

The material combination ComputeFiniteStrain, ComputeFiniteStrainElasticStress, and an elasticity tensor material will replace the Solid Mechanics SolidModel model with the formulation = Nonlinear3D. An equivalent solid mechanics input file to the Tensor Mechanics input file shown above is:

  # Old Solid Mechanics Version
[Materials]
[./linelast]
type = Elastic
block = 0
disp_x = disp_x
disp_y = disp_y
disp_z = disp_z
poissons_ratio = 0.3
youngs_modulus = 2.1e5
formulation = Nonlinear3D
[../]
[]