2. Benchmark solution

2.1. Calculation method used for the reference solution

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

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

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

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

2.1.1. Modeling A

In modeling A, we have:

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

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

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

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

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

We get:

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

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

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

2.1.2. B modeling

In modeling B, we have:

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

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

So we have:

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

The total deformation is equal to:

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

2.2. Benchmark results

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