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

Calculation method used for the reference solution
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This is a :math:`\mathrm{1D}` test. The uniaxial constraint is equal to: :math:`\sigma ={\sigma }_{0}\cdot H({t}_{0})` where :math:`{t}_{0}` is the loading time. The Heavyside :math:`H({t}_{0})` feature allows loading :math:`{\sigma }_{0}` to be applied instantly.

We define the equivalent constraint :math:`S(t)=h(t)\cdot \sigma (t)`. We have: :math:`{S}_{0}=S({t}_{0})={\sigma }_{0}\cdot {h}_{0}`.

The initial stress jump can be explained by writing the creep deformation as follows:

:math:`{\epsilon }^{\mathit{fl}}(t)={S}_{0}J(t,{t}_{0})+\underset{\tau ={t}_{0}}{\overset{\tau =t}{\int }}J(t,\tau )\dot{S}\text{d}\tau`


Modeling A
~~~~~~~~~~~~~~~

In modeling A, we have:


* For :math:`t>{t}_{0}^{\text{+}}` we have: :math:`S(t)={\sigma }_{0}\left({h}_{0}+\left({h}_{f}-{h}_{0}\right)\frac{t-{t}_{0}}{{t}_{f}-{t}_{0}}\right)` so: :math:`\dot{S}={\sigma }_{0}\frac{{h}_{f}-{h}_{0}}{{t}_{f}-{t}_{0}}`


* :math:`J(t,\tau )=\sum _{s=1}^{8}{J}_{s}\cdot \left(1-\text{exp}\left[-\frac{t-\tau }{{\tau }_{s}}\right]\right)`

:math:`{\epsilon }^{\mathit{fl}}(t)={\sigma }_{0}{h}_{0}J(t,{t}_{0})+\underset{\tau ={t}_{0}}{\overset{\tau =t}{\int }}J(t,\tau ){\sigma }_{0}\frac{{h}_{f}-{h}_{0}}{{t}_{f}-{t}_{0}}\text{d}\tau`

By replacing :math:`J(t,\tau )` we have:

:math:`{\epsilon }^{\mathit{fl}}(t)={\sigma }_{0}{h}_{0}J(t,{t}_{0})+{\sigma }_{0}\frac{{h}_{f}-{h}_{0}}{{t}_{f}-{t}_{0}}\sum _{s=1}^{8}{J}_{s}\underset{\tau ={t}_{0}}{\overset{\tau =t}{\int }}\left(1-\text{exp}\left[-\frac{t-\tau }{{\tau }_{s}}\right]\right)\text{d}\tau`

We get:

:math:`\begin{array}{c}{\epsilon }^{\mathit{fl}}(t)={\sigma }_{0}{h}_{0}\sum _{s=1}^{8}{J}_{s}\left(1-\text{exp}\left[-\frac{t-{t}_{0}}{{\tau }_{s}}\right]\right)\\ -{\sigma }_{0}\frac{{h}_{f}-{h}_{0}}{{t}_{f}-{t}_{0}}\sum _{s=1}^{8}{\tau }_{s}{J}_{s}\left(1-\text{exp}\left[-\frac{t-{t}_{0}}{{\tau }_{s}}\right]\right)+{\sigma }_{0}\frac{{h}_{f}-{h}_{0}}{{t}_{f}-{t}_{0}}\left(\sum _{s=1}^{8}{J}_{s}\right)\left(t-{t}_{0}\right)\end{array}`

The total deformation is calculated as the sum of the creep deformation and the elastic deformation:

:math:`\epsilon (t)={\epsilon }^{e}(t)+{\epsilon }^{\mathit{fl}}(t)=\frac{{\sigma }_{0}}{E}+{\epsilon }^{\mathit{fl}}(t)`


B modeling
~~~~~~~~~~~~~~~

In modeling B, we have:


* For :math:`t>{t}_{0}^{\text{+}}` we have: :math:`S(t)={\sigma }_{0}{h}_{0}=\mathit{constante}` so: :math:`\dot{S}=0`


* :math:`J(t,\tau )=k(\tau )\sum _{s=1}^{8}{J}_{s}\cdot \left(1-\text{exp}\left[-\frac{t-\tau }{{\tau }_{s}}\right]\right)`

So we have:

:math:`{\epsilon }^{\mathit{fl}}(t)={\sigma }_{0}{h}_{0}\sum _{s=1}^{8}k({t}_{0}){J}_{s}\text{.}\left(1-\text{exp}\left[-\frac{t-{t}_{0}}{{\tau }_{s}}\right]\right)`

The total deformation is equal to:

:math:`\epsilon (t)={\epsilon }^{e}(t)+{\epsilon }^{\mathit{fl}}(t)=\frac{{\sigma }_{0}}{E}+{\epsilon }^{\mathit{fl}}(t)`


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

We will be interested in the values of the deformations at 365 days.