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


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

We rely on the methodology described in [:ref:`1 <1>`], of which we only present here the results necessary for the test. The following constraint is imposed:

:math:`\sigma (t)=s(\frac{t}{{t}_{c}}){\sigma }^{0}+{e}^{v}(\frac{t}{{t}_{c}}){\mathrm{ℂ}}^{p}\mathrm{:}{\epsilon }^{0}+s(\frac{t}{{t}_{c}}){\mathrm{ℂ}}^{p}\mathrm{:}{{\mathrm{ℂ}}^{v}}^{-1}\mathrm{:}{\sigma }^{0}`

where:

 :math:`{\sigma }^{0}` stress applied to the shock absorber (test input parameter)

 :math:`{t}_{c}` characteristic load time (test input parameter)

 :math:`{\mathrm{ℂ}}^{p}` and :math:`{\mathrm{ℂ}}^{v}` elasticity operators for plastic and viscoelastic branches

and

 :math:`s(\overline{t})=\text{min}(\overline{t}\mathrm{,1})`

 :math:`{e}^{v}(\overline{t})=\{\begin{array}{cc}\frac{{\overline{t}}^{\gamma +1}}{\gamma +1}& \text{si}\overline{t}\le 1\\ \frac{1}{\gamma +1}+\overline{t}-1& \text{si}\overline{t}\ge 1\end{array}`

 :math:`{\epsilon }^{0}=\frac{{t}_{c}}{{\eta }^{\gamma }}{\left[{\sigma }^{0}\mathrm{:}V\mathrm{:}{\sigma }^{0}\right]}^{\frac{\gamma -1}{2}}V\mathrm{:}{\sigma }^{0}`

 :math:`V\mathrm{:}{\sigma }^{0}=(1+{\nu }^{d}){\sigma }^{0}-{\nu }^{d}\text{tr}({\sigma }^{0})I`

The deformation in response to this stress is then as follows:

:math:`\epsilon (t)={e}^{v}(\frac{t}{{t}_{c}}){\epsilon }^{0}+s(\frac{t}{{t}_{c}}){{\mathrm{ℂ}}^{v}}^{-1}\mathrm{:}{\sigma }^{0}`

The viscous deformation and the equivalent viscous deformation are equal to:

:math:`{\epsilon }^{v}(t)={e}^{v}(\frac{t}{{t}_{c}}){\epsilon }^{0}`; :math:`{\epsilon }_{\mathit{eq}}^{v}(t)={e}^{v}(\frac{t}{{t}_{c}})\sqrt{\frac{2}{3}{\epsilon }^{0}\mathrm{:}{\epsilon }^{0}}`


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

It will be ensured that the deformation, the viscous deformation and the cumulative viscous deformation at time :math:`t=2{t}_{c}` correspond to the analytical values to within 1% by cutting the load into 100 steps (:math:`\mathrm{\Delta }t=2{t}_{c}/100`).


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

Nil.


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

[:ref:`1 <1>`] E. Lorentz (2022) TUYAUTERIE 3- Robust behavioral relationship dedicated to polyethylene structure calculations. Internal note EDF 6125-1723-22-00150.