Stress-strain relationship in three dimensions
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In this part, we are interested in defining a scalar equivalent quantity starting from a state of stresses and tensor deformations. We start from the tensor decomposition of the deformation tensor :math:`\mathrm{\epsilon }` into volume and deviatory parts:


.. math::
 :label: eq-11

    \ mathrm {\ epsilon} = {\ mathrm {\ epsilon}} _ {d} +\ frac {1} {3}\ mathit {tr} (\ mathrm {\ epsilon}) I


where :math:`{\mathrm{\epsilon }}_{d}` is the deviatoric strain tensor. Since equivalent linear modeling focuses on the shear behavior of the soil column, the deviatoric strain tensor standard :math:`{\mathrm{\epsilon }}_{d}` is the most appropriate candidate because it can be linked to the shear deformation :math:`\mathrm{\gamma }` for a one-dimensional loading:


.. math::
 :label: eq-12

    \ mathrm {\ gamma} =\ sqrt {3} {\ mathrm {\ epsilon}} _ {d}


This expression makes it possible, based on the calculation of :math:`{\mathrm{\epsilon }}_{d}` obtained by the 1D — 3 components approach, to go back to the value of :math:`\mathrm{\gamma }` to be used to change the elastic modulus.

For the case of the propagation of plane waves P, SV and SH perpendicular to the planes of stratigraphy (i.e. vertical incidence for a horizontal stratigraphy), the calculation of the deviatoric deformation is summarized in the following expression:


.. math::
 :label: eq-13

    {\ mathrm {\ epsilon}} _ {d} =\ sqrt {\ frac {2} {3}\ left ({\ mathrm {\ epsilon}}} _ {d}\ mathrm {::} {\ mathrm {\ epsilon}}} _ {d}\ right)}

    {\ mathrm {\ epsilon}} _ {d} =\ frac {2} {3} =\ frac {2} {3}\ sqrt {{\ mathrm {\ epsilon}}} _ {\ mathit {y}}} _ {\ mathit {\ epsilon}}} _ {\ mathit {\ epsilon}}} _ {\ mathit {\ epsilon}}} _ {\ mathit {\ epsilon}} _ {\ mathit {y}}} _ {\ mathit {yy}}} ^ {2}} ^ {2}} ^ {2}} ^ {2}} ^ {2}} ^ {2}} ^ {2}} ^ {2}} ^ {2}} ^ {2}} ^ {2}} ^ {{\ mathit {yz}} ^ {2}}


The equivalent stress chosen is the deviatory stress :math:`q`, calculated during post-processing using the following expression:


.. math::
 :label: eq-14

    q=\ sqrt {\ frac {3} {3} {2} S\ mathrm {:} S}

    \ textrm {and}

    q=3g {\ mathrm {\ epsilon}} _ {d}


where :math:`S` is the deviatoric tensor, defined by the following expression:


.. math::
 :label: eq-15

    S=\ mathrm {\ sigma} -\ frac {1} {3}\ mathit {tr} (\ mathrm {\ sigma}) I

    S=\ mathrm {\ sigma} -\ frac {1} {3}\ mathit {tr} (\ mathrm {\ sigma}) I