2. Benchmark solution

2.1. Calculation method

2.1.1. Elastic solution

The analytical solution is established on an infinite cylinder in the direction \(Z\), with an inner radius \({R}_{\text{int}}\), with an outer radius \({R}_{\text{ext}}\), subject to an internal pressure. In cylindrical coordinates and with the following boundary conditions:

\(\{\begin{array}{ccc}{\sigma }_{\mathrm{rr}}(r={R}_{\text{int}})& =& -P\\ {\sigma }_{\mathrm{rr}}(r={R}_{\text{ext}})& =& 0\end{array}\)

In plane stresses, the elastic solution is written as:

\(\{\begin{array}{ccccccc}{\sigma }_{\mathrm{rr}}& =& \frac{-{R}_{\text{int}}^{2}}{{R}_{\text{int}}^{2}-{R}_{\text{ext}}^{2}}P(1-\frac{{R}_{\text{ext}}}{{r}^{2}})& & {\epsilon }_{\mathrm{rr}}& =& ({\sigma }_{\mathrm{rr}}-\nu {\sigma }_{\theta \theta })/E\\ {\sigma }_{\theta \theta }& =& \frac{-{R}_{\text{int}}^{2}}{{R}_{\text{int}}^{2}-{R}_{\text{ext}}^{2}}P(1+\frac{{R}_{\text{ext}}}{{r}^{2}})& & {\epsilon }_{\theta \theta }& =& ({\sigma }_{\theta \theta }-\nu {\sigma }_{\mathrm{rr}})/E\\ {\sigma }_{\mathrm{zz}}& =& 0& & {\epsilon }_{\mathrm{zz}}& =& -\nu ({\sigma }_{\mathrm{rr}}+{\sigma }_{\theta \theta })/E\end{array}\)

This constitutes the reference solution for elastic calculation which will serve as initial conditions for the calculation of natural creep. This solution is applied with the following physical data: \({R}_{\text{int}}=20\text{m}\) and \({R}_{\text{ext}}=21\text{m}\).

2.1.2. Solution with clean creep

The concrete clean creep model, BETON_UMLV, is presented in detail in [R7.01.06]. The decomposition of natural creep deformations into spherical and deviatory parts is briefly recalled. Each of these parts is then itself separated into components that are reversible or irreversible.

We are interested here in a particular solution of this model of delayed deformations for a loading and a constant relative humidity. The interest is mainly focused on the spherical part of the deformations.

The model distinguishes between loads that lead to a positive deformation rate (tension state) or short-term behavior and the opposite, long-term behavior for negative deformation rates.

The advantage of this test case is to ensure only the good behavior of the model for traction loading. The following equation specifies the changes to be respected with respect to the model:

\(\{\begin{array}{c}{\epsilon }_{r}^{\mathrm{sph}}(t)=\frac{h}{{k}_{r}^{\mathrm{sph}}}\left[1-\mathrm{exp}(\frac{-t{k}_{r}^{\mathrm{sph}}}{{\eta }_{r}^{\mathrm{sph}}})\right]{\sigma }^{\mathrm{sph}}\\ {\epsilon }_{i}^{\mathrm{sph}}(t)=0\end{array}\)

with:

\({\epsilon }_{r}^{\mathrm{sph}}\): the so-called short-term spherical volume deformation

\(h\): the relative humidity of the continuous medium

\(t\): time expressed in seconds

\({k}_{r}^{\mathrm{sph}}\): reversible spherical apparent stiffness

\({\eta }_{r}^{\mathrm{sph}}\): apparent viscosity

\({\sigma }^{\mathrm{sph}}\): spherical part of the imposed load

2.2. Reference quantities and results

The only reference quantity tested in this example is irreversible creep volume deformation. The value of this quantity remains zero for any state of tensile stresses.

2.3. Uncertainties about the solution

Uncertainties are zero because it is an analytical solution.

2.4. References

1. BENBOUDJEMA, F.:Modeling delayed deformations of concrete under biaxial stresses. Application to reactor buildings of nuclear power plants, Thesis by D.E.A. Advanced Materials — Engineering Structures and Envelopes, 38p. (+ annexes), 1999.

2. Code_aster Reference Documentation [R7.01.06]: UMLV behavioral relationship for the clean creep of concrete.