2. Benchmark solution

2.1. Calculation method used for the reference solution

This problem requires an analytical solution. The calculation method is extensively detailed in [Lorentz, 2020].

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

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

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

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

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

2.2. Benchmark results

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

\({E}_{c}=37921\text{MPa}\)

\({\sigma }_{c}=3.034\text{MPa}\)

\({w}_{c}=1.214\times {10}^{-4}\text{MPa}\)

\(\kappa =5.842\)

\({m}_{0}=0.589\)

\({D}_{1}=4.081\)

\(r=3.95\)

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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 \(1-B(b)\), the elastic deformation energy \(w(\epsilon ,b)\) as well as the energy consumed by the damage. In this case, the latter is written simply, always with the notations of [Lorentz, 2020]:

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

2.3. Uncertainties about the solution

Nil.

2.4. 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.