2. Benchmark solution

2.1. Calculation method used for the reference solution

Given the nature of the stresses, the solution (constraints \(\sigma\), deformations \(\varepsilon\)) is homogeneous.

Let’s say the load: \((\begin{array}{ccc}0& 0& 0\\ 0& {\sigma }_{d}(t)& 0\\ 0& 0& 0\end{array})\)

Granger’s own creep model is such that the viscoelastic deformation corresponding to the case of a constant load \({\sigma }_{0}\) applied at time \({t}_{0}\) is equal to: (cf. [R7.01.01])

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

The model also depends on temperature and humidity as follows:

\({\varepsilon }^{\mathrm{fl}}(t)={\sigma }_{0}h\cdot \frac{T-\text{248}}{\text{45}}\cdot \sum _{k=1}^{8}{J}_{k}\text{.}(1-\text{exp}\left[-\frac{t-{t}_{0}}{{\tau }_{s}}\right])\)

but in this test the temperature and humidity fields are chosen to be constant and such that \(h\) and \(\frac{T-248}{45}\) equal 1 respectively.

When the constraint evolves over time then:

\({\varepsilon }^{\text{fl}}(t)=\sum _{k=1}^{8}\underset{\tau =0}{\overset{t}{\int }}{J}_{k}\text{.}(1-\text{exp}\left[-\frac{t-\tau }{{\tau }_{s}}\right])\text{.}\frac{\partial \sigma }{\partial \tau }\text{.}d\tau\)

So for the present case we have:

\({\varepsilon }_{\mathrm{yy}}^{\text{fl}}=\sum _{k=1}^{2}\underset{\tau =0}{\overset{t={t}_{1}}{\int }}{J}_{k}\text{.}(1-\text{exp}\left[-\frac{t-\tau }{{\tau }_{s}}\right])\text{.}\frac{{\sigma }_{1}}{{t}_{1}}\text{.}d\tau\)

the load remaining constant beyond t 1. That is:

\({\varepsilon }_{\mathrm{yy}}^{\text{fl}}={\sigma }_{1}\sum _{k=1}^{k=2}{J}_{k}\text{.}(1-\text{exp}(-\frac{t-{t}_{1}}{{\tau }_{s}}))\underset{\text{}\approx 1}{\underset{\underbrace{}}{\frac{{\tau }_{k}}{{t}_{1}}(1-\text{exp}(\frac{-{t}_{1}}{{\tau }_{s}}))}}\)

A longitudinal creep deformation is accompanied by a transverse deformation such as:

\({\varepsilon }_{\mathrm{xx}}^{\mathrm{fl}}=-\nu {\varepsilon }_{\mathrm{yy}}^{\mathrm{fl}}\)

The total uniaxial deformation is equal to: \({\varepsilon }_{\mathrm{yy}}={\varepsilon }_{\mathrm{yy}}^{\mathrm{fl}}+\frac{{\sigma }_{\mathrm{yy}}}{E}\)

2.2. Benchmark results

We will focus on the values of creep deformations at 45 days, 245 days and 365 days.