Benchmark solution
=====================


Development of the analytical solution for CJS1
-------------------------------------------------

We always have:


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where

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represents the confinement pressure.

Still to be determined

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.

**Elastic phase:**

By simply writing the elastic law, we have:

:math:`{\sigma }_{\mathrm{xx}}^{0}={\sigma }_{\mathrm{xx}}^{0}+\lambda {\varepsilon }_{\mathrm{zz}}+(\lambda +2\mu ){\varepsilon }_{\mathrm{xx}}+\lambda {\varepsilon }_{\mathrm{xx}}`

:math:`{\sigma }_{\mathrm{zz}}={\sigma }_{\mathrm{zz}}^{0}+(\lambda +2\mu ){\varepsilon }_{\mathrm{zz}}+2\lambda {\varepsilon }_{\mathrm{xx}}`

where here :math:`\lambda` and :math:`\mu` are the Lamé coefficients.

By eliminating :math:`{\varepsilon }_{\mathrm{xx}}` between these two equations, we find:

:math:`{\sigma }_{\mathrm{zz}}={\sigma }_{\mathrm{zz}}^{0}+\frac{\mu (3\lambda +2\mu )}{(\lambda +\mu )}{\varepsilon }_{\mathrm{zz}}`

**Plastic phase:**

We have:


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    :height: 29


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where

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    :height: 29


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represents the confinement pressure.

We deduce for the components of the diverter

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:

:math:`{s}_{\mathrm{zz}}=2\left[\frac{1}{3}{I}_{1}-{\sigma }_{\mathrm{xx}}^{0}\right]` and :math:`{s}_{\mathrm{xx}}={\sigma }_{\mathrm{xx}}^{0}-\frac{1}{3}{I}_{1}`

either: :math:`{s}_{\mathrm{II}}=\sqrt{6}\left[{\sigma }_{\mathrm{xx}}^{0}-\frac{1}{3}{I}_{1}\right]` and :math:`\mathrm{det}(\underline{\underline{s}})=2{\left[\frac{1}{3}{I}_{1}-{\sigma }_{\mathrm{xx}}^{0}\right]}^{3}`

Therefore: :math:`h({\theta }_{S})={(1-\gamma )}^{1/6}`

Moreover, when the deviatory mechanism criterion is reached: :math:`{s}_{\mathrm{II}}h({\theta }_{S})+{R}_{m}{I}_{1}=0`

Hence the relationship:

:math:`{I}_{1}=\frac{\sqrt{6}{\sigma }_{\mathrm{xx}}^{0}}{\sqrt{\frac{2}{3}}-\frac{{R}_{m}}{{(1-\gamma )}^{1/6}}}`

and finally, for the vertical constraint, we have:

:math:`{\sigma }_{\mathrm{zz}}=\frac{\sqrt{6}{\sigma }_{\mathrm{xx}}^{0}}{\sqrt{\frac{2}{3}}-\frac{{R}_{m}}{{(1-\gamma )}^{1/6}}}-2{\sigma }_{\mathrm{xx}}^{0}`

In addition, it can be calculated that the transition between the elastic and perfectly plastic states occurs for an axial deformation equal to:

:math:`{\varepsilon }_{\mathrm{zz}}=\frac{\mu (3\lambda +2\mu )}{\lambda +\mu }\left[\frac{\sqrt{6}{\sigma }_{\mathrm{xx}}^{0}}{\sqrt{\frac{2}{3}}-\frac{{R}_{m}}{{(1-\gamma )}^{1/6}}}-2{\sigma }_{\mathrm{xx}}^{0}\right]`


Benchmark results
----------------------

Constraints :math:`{\sigma }_{\mathrm{xx}}`, :math:`{\sigma }_{\mathrm{yy}}`, and :math:`{\sigma }_{\mathrm{zz}}` at points :math:`A`, :math:`B`, and :math:`C`.


Uncertainty about the solution
---------------------------

Analytical solution for CJS1.