2. Benchmark solution

2.1. Calculation method used for the reference solution

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

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

where:

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

\({t}_{c}\) characteristic load time (test input parameter)

\({\mathrm{ℂ}}^{p}\) and \({\mathrm{ℂ}}^{v}\) elasticity operators for plastic and viscoelastic branches

and

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

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

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

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

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

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

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

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

2.2. Benchmark results

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

2.3. Uncertainty about the solution

Nil.

2.4. Bibliographical references

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