[GeoMechanicsApplication] Investigate and test Hencky ( = natural = logarithmic ) strains

[indigocoral]

Related issue: #11642

• [ ] Does "use_hencky_strain" in MaterialParameters.json provide engineering strain?
• [ ] If yes, is it correctly implemented?
• [ ] Else, implement the correct implementation
• [ ] Is it tested properly?

[WPK4FEM]

Looked up Wikipedia ( https://en.wikipedia.org/wiki/Strain_(mechanics) ) to get the names of strain measures correctly: Engineering strain ( = Cauchy strain ) = (L1 -L0)/L0 Logarithm strain ( = Hencky strain = true strain ) = ln ( L1 / L0 ) Green strain = (1/2) * ( L1^2 - L0^2) / l0^2 or more usefull the finite strain derivations from the deformation gradient F ( at https://en.wikipedia.org/wiki/Finite_strain_theory#Finite_strain_tensors ): Cauchy strain tensor (right Cauchy–Green deformation tensor) C = F^T F Green-Lagrangian strain tensor: E = (1/2) ( C - I ) Logarithmic strain, Natural strain, True strain, or Hencky strain E = (1/2) ln( C )

For comparison with old D Settlement the Hencky strain is implemented. 2 copies exist ( in small_strain_U_Pw_diff_order_element and in U_Pw_small_strain_element. The updated Lagrangian elements inherit from these elements. )

Tests in applications/GeoMechanicsApplication/tests found so far: 0.

[WPK4FEM]

Separated the functionality such that the geomechanics application now only holds it once. Added 3 unit tests to check the behavior of CalculateHenckyStrain:

1. Incompressible elongation: F is a diagonal tensor
2. Rigid rotation F^T =F^-1, resulting strains should be zero
3. Pure shear. Eigenvectors are at 45 and 135 deg, The principal Hencky strains should be 1 compressive, 1 tensile.

The set of these 3 should be sufficient as any deformation can be decomposed in contributions of these 3.

[AlejandroCornejo]

just for you to know, in the ConstitutiveLawsApp we compute the Henky strain as:

template<SizeType TVoigtSize>
void AdvancedConstitutiveLawUtilities<TVoigtSize>::CalculateHenckyStrain(
    const MatrixType& rCauchyTensor,
    Vector& rStrainVector
    )
{
    // Doing resize in case is needed
    if (rStrainVector.size() != VoigtSize)
        rStrainVector.resize(VoigtSize, false);

    // Declare the different matrix
    BoundedMatrixType eigen_values_matrix, eigen_vectors_matrix;

    // Decompose matrix
    MathUtils<double>::GaussSeidelEigenSystem(rCauchyTensor, eigen_vectors_matrix, eigen_values_matrix, 1.0e-16, 20);

    // Calculate the eigenvalues of the E matrix
    for (IndexType i = 0; i < Dimension; ++i) {
        eigen_values_matrix(i, i) = 0.5 * std::log(eigen_values_matrix(i, i));
    }

    // Calculate E matrix
    BoundedMatrixType E_matrix;
    MathUtils<double>::BDBtProductOperation(E_matrix, eigen_values_matrix, eigen_vectors_matrix);

    // Hencky Strain Calculation
    rStrainVector = MathUtils<double>::StrainTensorToVector(E_matrix, TVoigtSize);
}

[WPK4FEM]

Dear Alejandro Cornejo,

Thank you for your comment. I am aware of the existence of Hencky strain in the ConstitutiveLawApp. In fact I think that a former colleague copied it from there and placed two copies in the GeoMechanicsApplication. I would like these implementations to exist only once and be fully tested with nightly tests at its single place of existance. However I'm not aware of a possibility to do that without including the complete ConstitutiveLawApp into GeoMechanicsApplication. Therefore, I have reduced the number of copies in GeoMechanicsApplication to one and made some unit tests ( stretch, rigid rotation and pure shear ) for it such that at least there is test coverage there.

Regards, Wijtze Pieter Kikstra ( Deltares )