2. Benchmark solution

2.1. Calculation method used for the reference solution

This problem requires an analytical solution. We are very much inspired by the results of [Lorentz, Drouet, Hamon, 2020].

In the case of confined uniaxial loading, the elastic behavior relationship is reduced to:

\({\sigma }^{e}={E}_{c}\epsilon \text{;}{E}_{c}=\lambda +2\mu \text{;}\lambda =\frac{E\nu }{(1+\nu )(1-2\nu )}\text{;}2\mu =\frac{E}{1+\nu }\)

where the \(\mathit{xx}\) clues are now omitted. As for the viscous stress, it is governed by the following differential equation:

\(\dot{{\sigma }^{v}}+\frac{1}{\tau }{\sigma }^{v}=k\tau \dot{\epsilon }\text{;}{\sigma }^{v}(0)=0\)

Since the second member is constant, it has the following solution:

\({\sigma }^{v}(t)=k\tau \dot{\epsilon }\left[1-\mathrm{exp}\left(-\frac{t}{\tau }\right)\right]\)

And so the constraint taking into account the viscous regularization is equal to:

\(\sigma ={\sigma }^{e}+{\sigma }^{v}={E}_{c}\dot{\epsilon }t+k\tau \dot{\epsilon }\left[1-\mathrm{exp}\left(-\frac{t}{\tau }\right)\right]\)

In the transverse direction, the viscous stress is zero, so that the stress is reduced to the elastic stress:

\({\sigma }_{\mathit{yy}}=\lambda \epsilon\)

It is easy to derive/to establish the energy stored in the viscoelastic branch:

\(\mathit{VISCELAS}=\frac{1}{2k}{{\sigma }^{v}}^{2}\)

As well as the energy dissipated by the viscoelastic branch by integrating the dissipation:

\(\mathit{VISCDISS}=k{\tau }^{2}{\dot{\epsilon }}^{2}\left[\frac{t}{\tau }+2\mathrm{exp}(-\frac{t}{\tau })-\frac{1}{2}\mathrm{exp}(-2\frac{t}{\tau })-\frac{3}{2}\right]\)

2.2. Benchmark results

It will be ensured that at \(t=20\text{s}\), the axial and transverse stress as well as the viscous stress, the stored energy and the energy dissipated in the viscous branch correspond well to the analytical results.

2.3. Uncertainties about the solution

Nil.

2.4. Bibliographical references

Lorentz E., Drouet G. Hamon F (2020). CIWAP3/lot SA — Digital stabilization of a softening behavior model by viscoelastic regularization. Internal note EDF R&D 6125-1724-2020-01606-EN.