Analytical solution
===================

First, let's introduce the following notations:

.. math::
 :label: eq-1

    \ {\ begin {array} {c} A=K+\ frac {4} {3} G\\ B=K-\ frac {2} {3} G\\ C=2\ left (K+\ frac {G} {G} {3}\ right)\ end {array}

With :math:`K=\frac{E}{3\left(1-2\mathrm{\nu }\right)}` and :math:`G=\frac{E}{2\left(1+\mathrm{\nu }\right)}` the compressibility and shear modules, respectively.

Let :math:`C` be the Hooke elasticity tensor, we will have with hypothesis :math:`{\mathrm{ϵ}}_{\mathit{yy}}={\mathrm{ϵ}}_{\mathit{xx}}`:

.. math::
 :label: eq-2

    C\ mathrm {.} d\ mathrm {} =\ {\ begin {array} =\ {\ begin {array} {c}} Bd {\ mathrm {zz}} +Cd {\ mathrm {}}} =\ {\ mathit {xx}} _ {\ mathit {xx}} _ {\ mathit {zz}}} +Cd {\ mathrm {}} _ {\ mathit {xx}}}\\ Ad {\ mathrm {E.G.}} _ {\ mathit {zz}} +2Bd {\ mathrm {}}} _ {\ mathit {xx}}\ end {array}} _ {\ mathrm {xx}}\ end {array}}

To simplify, we note the vertical constraint at the time :math:`\text{+}` :math:`{\mathrm{\sigma }}^{\text{+}}={\mathrm{\sigma }}_{\mathit{zz}}^{\text{+}}`, so that the Rankine criterion is written:

.. math::
 :label: eq-3

    {\ mathrm {\ sigma}}} ^ {\ text {+}}\ the {\ mathrm {\ sigma}} _ {t}

In addition, we have:

.. math::
: label: eq-4

    \ {\ begin {array} {c} {\ mathrm {\ sigma}}} ^ {\ text {pred}} = {\ mathrm {\ sigma}} ^ {\ text {-}} +n\ mathrm {.} +n\ mathrm {.}.} C\ mathrm {.}} d\ mathrm {.} d\ mathrm {.} d\ mathrm {.} d\ mathrm {.} d\ mathrm {.} d\ mathrm {.} d\ mathrm {.} d\ mathrm {.} d\ mathrm {.} d\ mathrm {.} d\ mathrm {.} d\ mathrm {.} d\ mathrm {.} d\ mathrm {.} d\ mathrm {\ sigma}} ^ {\ text {-}} +n\ mathrm {.} C\ mathrm {.} \ left (d\ mathrm {}} -d\ mathrm {\ lambda} n\ right) = {\ mathrm {\ sigma}} ^ {\ text {pred}} -\ underset {\ mathrm {\ thrm {\ delta}} {\ delta}} {\ mathrm {\ sigma}} {\ mathrm {\ sigma}}} _ {\ sigma}}} {\ underset {} {d\ mathrm {\ lambda}} n\ mathrm {\ delta}} {\ mathrm {\ delta}} {\ mathrm {\ sigma}}} _ {\ sigma}} _ {\ sigma}}} {\ underset {}} {d\ mathrm {\ lambda}} n\ mathrm {\ delta}} {.} C\ mathrm {.} n}}}\ end {array}

With :math:`n=\left(\begin{array}{c}0\\ 0\\ 1\end{array}\right)` and where:

.. math::
: label: eq-5

    d\ mathrm {\ lambda} =\ frac {{⟶ {\ mathrm {\ sigma}}} ^ {\ mathit {pred}} - {\ mathrm {\ sigma}}} - {\ sigma}} _ {t} ⟩} __ {t} ⟩}

According to the associated flow law, we also have:

.. math::
 :label: eq-6

    \ {\ begin {array} {c} d {\ mathrm {}}} _ {\ mathit {zz}}} ^ {P} =d\ mathrm {\ lambda} =d {\ mathrm {}}} _ {v}}}} _ {v}}}} _ {v} ^ {P}} _ {v} ^ {P}} =\ frac {2} {3} d\ mathrm {\ lambda}}} =\ frac {2} {3} d\ mathrm {\ lambda}}} _ {v}}} _ {v} ^ {P}} _ {v} ^ {P}} =\ frac {2} {3} d\ mathrm {array}

Like :math:`n=\left(\begin{array}{c}0\\ 0\\ 1\end{array}\right)`, we get:

.. math::
 :label: eq-7

    \ mathrm {\ Delta} {\ mathrm {\ sigma}} _ {C} =d\ mathrm {\ lambda} A

Combining equations (), (), and () gives us constraint :math:`{\mathrm{\sigma }}_{\mathit{zz}}^{\text{+}}`. Equations () and () give us the norm of deviatoric plastic deformation :math:`{e}^{P}`.

Now let's try to get the expression for horizontal *elastic* deformation :math:`{\mathrm{ϵ}}_{\mathit{xx}}^{\text{élas}}`.

Laterally, we have condition :math:`{\mathrm{\sigma }}_{\mathit{xx}}^{\text{+}}={P}_{0}`, which is:


.. math::
 :label: eq-8

    {\ mathrm {\ sigma}} _ {\ mathit {xx}}} ^ {\ text {pred}} -\ mathrm {\ delta} {\ mathrm {\ sigma}}} _ {\ mathit {xx}}} _ {\ mathit {xx}}, C} = {P} _ {0}

With :math:`\mathrm{\Delta }{\mathrm{\sigma }}_{\mathit{xx},C}=d\mathrm{\lambda }B`

Using equation (), we then obtain:

.. math::
 :label: eq-9

    {\ mathrm {\ sigma}} _ {\ mathit {xx}}} ^ {\ mathit {xx}}} ^ {\ mathrm {-}} _ {\ mathit {xx}} ^ {\ text {alas}}} ^ {\ text {alas}}} +Bd {\ mathit {xx}} -Bd\ mathrm {\ lambda}} ^ {\ text {alas}}} ^ {\ text {alas}}} +Bd {\ mathrm {xx}} ^ {\ text {alas}}} +Bd {\ mathrm {xx}} ^ {\ text {alas}}} +Bd {\ mathrm {elas}}} +Bd {\ mathrm {\ alas}}} ^ {\ text {alas}}} +Bd

Hence the horizontal elastic deformation increment:

.. math::
 :label: eq-10

    d {\ mathrm {}} _ {\ mathit {xx}}} ^ {\ text {xx}}} ^ {\ text {elas}}} =\ frac {{P} _ {\ mathrm {\ sigma}}} _ {\ mathit {xx}}} _ {\ mathit {xx}}} ^ {\ mathit {xx}}} ^ {\ text {-}} ^ {\ text {-}}} +B\ left (d\ mathrm {\ sigma}}} _ {\ mathit {zz}}\ right)} {C}