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


Calculation method used for the reference solution
--------------------------------------------------------

This problem requires an analytical solution. The calculation method is extensively detailed in [:ref:`Lorentz, 2020 <Lorentz, 2020>`].

For the charging phase, we give ourselves a target damage value :math:`b=0.1`. By noting :math:`{E}_{c}` the confined stiffness, :math:`{\sigma }_{c}` the peak stress in confined tension and :math:`B(b)` the stiffness function, we can successively calculate the stress :math:`{\sigma }^{(T)}` then the deformation :math:`{\epsilon }^{(T)}` reached for the target damage value:

 :math:`{\sigma }^{(T)}=(1-b){\sigma }_{c}` :math:`{\epsilon }^{(T)}=\frac{{\sigma }^{(T)}}{{E}_{c}B(b)}`

The compression phase is carried out until :math:`{\epsilon }^{(C)}=-2{\epsilon }^{(T)}`. By noting :math:`S\text{'}` the function of regularizing the stiffness jump, we deduce the corresponding stress level:

 :math:`{\sigma }^{(C)}={E}_{c}[B(b){\epsilon }^{(C)}+(1-B(b))\frac{S\text{'}({\epsilon }^{(C)})}{2}]`

As for the damage, it does not vary during discharge and then compression.


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

The internal parameters of the model corresponding to the set of values chosen are as follows:

 :math:`{E}_{c}=37921\text{MPa}`

 :math:`{\sigma }_{c}=3.034\text{MPa}`

 :math:`{w}_{c}=1.214\times {10}^{-4}\text{MPa}`

 :math:`\kappa =5.842`

 :math:`{m}_{0}=0.589`

 :math:`{D}_{1}=4.081`

 :math:`r=3.95`

 :math:``

We will ensure that with imposed deformations, the model returns to the expected levels of damage and stresses.

The calculation of the internal variables of post-treatment is also tested, namely the value :math:`1-B(b)`, the elastic deformation energy :math:`w(\epsilon ,b)` as well as the energy consumed by the damage. In this case, the latter is written simply, always with the notations of [:ref:`Lorentz, 2020 <Lorentz, 2020>`]:

 :math:`{W}_{\mathit{cons}}=\kappa {w}_{c}\widehat{a}(b)`


Uncertainties about the solution
----------------------------

Nil.


Bibliographical references
---------------------------

Lorentz E. (2020). CIWAP 3 — Concrete cracking: proposal of an isotropic local behavior model representative of tensile and shear damage to simulate homogeneous cracked areas. Internal note EDF R&D 6125-1724-2020-01604.