3. Stress-strain relationship in three dimensions

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 \(\mathrm{\epsilon }\) into volume and deviatory parts:

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

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

(3.2)\[ \ mathrm {\ gamma} =\ sqrt {3} {\ mathrm {\ epsilon}} _ {d}\]

This expression makes it possible, based on the calculation of \({\mathrm{\epsilon }}_{d}\) obtained by the 1D — 3 components approach, to go back to the value of \(\mathrm{\gamma }\) 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:

(3.3)\[ \begin{align}\begin{aligned} {\ mathrm {\ epsilon}} _ {d} =\ sqrt {\ frac {2} {3}\ left ({\ mathrm {\ epsilon}}} _ {d}\ mathrm {::} {\ mathrm {\ epsilon}}} _ {d}\ right)}\\ {\ mathrm {\ epsilon}} _ {d} =\ frac {2} {3} =\ frac {2} {3}\ sqrt {{\ mathrm {\ epsilon}}} _ {\ mathit {y}}} _ {\ mathit {\ epsilon}}} _ {\ mathit {\ epsilon}}} _ {\ mathit {\ epsilon}}} _ {\ mathit {\ epsilon}} _ {\ mathit {y}}} _ {\ mathit {yy}}} ^ {2}} ^ {2}} ^ {2}} ^ {2}} ^ {2}} ^ {2}} ^ {2}} ^ {2}} ^ {2}} ^ {2}} ^ {2}} ^ {{\ mathit {yz}} ^ {2}}\end{aligned}\end{align} \]

The equivalent stress chosen is the deviatory stress \(q\), calculated during post-processing using the following expression:

(3.4)\[ \begin{align}\begin{aligned} q=\ sqrt {\ frac {3} {3} {2} S\ mathrm {:} S}\\ \ textrm {and}\\ q=3g {\ mathrm {\ epsilon}} _ {d}\end{aligned}\end{align} \]

where \(S\) is the deviatoric tensor, defined by the following expression:

(3.5)\[ \begin{align}\begin{aligned} S=\ mathrm {\ sigma} -\ frac {1} {3}\ mathit {tr} (\ mathrm {\ sigma}) I\\ S=\ mathrm {\ sigma} -\ frac {1} {3}\ mathit {tr} (\ mathrm {\ sigma}) I\end{aligned}\end{align} \]