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


Calculation of the reference solution
----------------------------------

Here we consider the case of a triaxial test for which the confinement stresses are applied in the directions :math:`x` and :math:`z` and for which the imposed deformation direction is the direction :math:`y`. It is also assumed that the parameter :math:`\eta` is independent of the hardening parameter :math:`\gamma`, that is to say :math:`{\phi }^{\mathit{end}}\mathrm{=}{\phi }^{\mathit{rup}}\mathrm{=}{\phi }^{\mathit{res}}`: it is then possible to calculate an analytical solution to the problem. The plasticity and flow criteria are written as:


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We consider a situation of increasing loading for which the preceding equations can be written non-incrementally:


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The elasticity relationships give:


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that is to say:


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with :math:`{\sigma }_{3}^{0}` and :math:`{\sigma }_{1}^{0}` the values of :math:`{\sigma }_{1}` and :math:`{\sigma }_{3}` at the start of loading. It therefore remains to calculate :math:`{\sigma }_{1}` as a function of :math:`\gamma` using the plasticity criterion to obtain :math:`\gamma`, :math:`{\sigma }_{1}` and :math:`{\varepsilon }_{3}`.

**1st case:** :math:`\gamma \mathrm{\le }{\gamma }^{\mathit{rup}}`

By noting

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and

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where

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,

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,

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and

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are given in the reference documentation of the law of behavior, :math:`\gamma` is a solution of the polynomial of degree 2:


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,

with :math:`\gamma` in the :math:`\mathrm{[}\mathrm{0,}{\gamma }^{\mathit{rup}}\mathrm{]}` interval.

**2nd case:** :math:`{\gamma }^{\mathit{rup}}\mathrm{\le }\gamma \mathrm{\le }{\gamma }^{\mathit{res}}`

Using the notations from the reference documentation of the amended Hoek-Brown law for a, d, c and

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, :math:`\gamma` is a solution of the polynomial of degree 2:


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**3rd case:** :math:`{\gamma }^{\mathit{res}}\mathrm{\le }\gamma`

In this case :math:`{\sigma }_{1}` is constant:


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and

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.


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

Constraints :math:`{\sigma }_{\mathit{xx}}({\sigma }_{3})`, :math:`{\sigma }_{\mathit{yy}}({\sigma }_{1})`, and :math:`{\sigma }_{\mathit{zz}}({\sigma }_{3})` at point :math:`D`.

Move :math:`{\varepsilon }_{\mathit{xx}}({\varepsilon }_{3})` and :math:`{\varepsilon }_{\mathit{yy}}({\varepsilon }_{1})` to point :math:`D`.