2. Benchmark solution#
2.1. Calculation method used for the reference solution#
To be able to calculate a simple analytical solution, the following choices were made, the objective being to validate the coupling and not the plasticity/cracking or creep laws:
a Granger creep law with a single serial Kelvin model,
a plasticity/cracking law modeling a perfect elastoplastic law,
a loading of uniaxial traction.
The reference solution is calculated analytically, knowing that in traction, only the traction criterion is activated. The equations of the model are reduced to scalar equations used to calculate the analytical solution. The only difficulty comes from determining the onset of plasticity (moment and creep deformation), which requires solving a nonlinear equation with one unknown by a numerical method.
In the case where the water content is not constant, the creep is more complex to solve, the analytical solution has not been calculated. So these are non-regression tests. However, in the 3D and D_ PLAN cases, it can be verified that the same results are obtained with the 2 models.
The imposed deformation (displacement of one end of the structure) is a linear function of time making it possible to bring into play creep and plasticity.
2.2. Calculation of the reference solution#
We note \(\varepsilon\), the component \(\mathrm{xx}\) of the total deformation \({\varepsilon }_{e}\), the component \(\mathrm{xx}\) of the elastic deformation, \({\varepsilon }_{\mathrm{fl}}\) the \(\mathrm{xx}\) component of the Granger creep deformation, and \({\varepsilon }_{\mathrm{pl}}\) the \(\mathrm{xx}\) component of the plastic deformation, \(\sigma\) the component \(\mathrm{xx}\) of the constraint, and \(E\) the Young’s modulus.
The creep model selected includes only one serial Kelvin model and the plasticity/cracking model is an almost perfect elastoplastic law (almost zero work-hardening slope), which makes it possible to easily calculate the analytical solution of the creep/plasticity coupling, in the case of a simple uniaxial traction. The almost perfect elastoplastic law can be obtained from the laws of Code_Aster BETON_DOUBLE_DP or VMIS_ISOT_LINE, by choosing the appropriate set of parameters (almost zero work hardening). The load is a uniaxial traction in imposed displacement. A total deformation proportional to the time elapsed is therefore imposed, in the form \({\varepsilon }_{\text{xx}}\mathrm{=}{\lambda }_{0}\text{.}t\). As there is no effort exerted in the other directions, the stress field is uniaxial. We can therefore reduce ourselves to a problem \(\mathrm{1D}\) for the resolution, which makes it possible to calculate deformations in the directions transverse to the load (\(\mathrm{yy}\) and \(\mathrm{zz}\)) in a second step.
\(\sigma \mathrm{=}({\sigma }_{\text{xx}}\mathrm{,0}\mathrm{,0}\mathrm{,0}\mathrm{,0}\mathrm{,0})\) and \(\epsilon \mathrm{=}({\varepsilon }_{\text{xx}},{\varepsilon }_{\text{yy}},{\varepsilon }_{\text{zz}}\mathrm{,0}\mathrm{,0}\mathrm{,0})\)
The equations of the creep model and the plasticity model merge with the following scalar equations, omitting the index \(\mathrm{xx}\) corresponding to the first component of the tensors:
\(\varepsilon \mathrm{=}{\lambda }_{0}\text{.}t\) (forced traction)
\(\varepsilon \mathrm{=}{\varepsilon }_{e}+{\varepsilon }_{\text{fl}}+{\varepsilon }_{\text{pl}}\)
\(\sigma \mathrm{=}\mu {\dot{\varepsilon }}_{\text{fl}}+K{\varepsilon }_{\text{fl}}\) with \(\mu \mathrm{=}\frac{{\tau }_{s}}{{J}_{s}}\) and \(K=\frac{1}{{J}_{s}}\)
\(\sigma \mathrm{=}E{\varepsilon }_{e}\) \(\sigma \mathrm{=}E\left[\varepsilon \mathrm{-}{\varepsilon }_{\text{pl}}\mathrm{-}{\varepsilon }_{\text{fl}}\right]\)
Resolution in linear elasticity
Before reaching the plasticity threshold, the plastic deformation is zero, which leads to:
\(\varepsilon \mathrm{=}{\lambda }_{0}\text{.}t\) (forced traction)
\({\varepsilon }_{\text{pl}}\mathrm{=}0\)
\(\varepsilon \mathrm{=}{\varepsilon }_{e}+{\varepsilon }_{\text{fl}}\)
\(\sigma \mathrm{=}\mu {\dot{\varepsilon }}_{\text{fl}}+K{\varepsilon }_{\text{fl}}\) with \(\mu \mathrm{=}\frac{{\tau }_{s}}{{J}_{s}}\) and \(K\mathrm{=}\frac{1}{{J}_{s}}\)
\(\sigma \mathrm{=}E{\varepsilon }_{e}\) \(\sigma \mathrm{=}E\left[\varepsilon \mathrm{-}{\varepsilon }_{\text{fl}}\right]\)
The differential equation for calculating the creep deformation is obtained:
\(\sigma \mathrm{=}E\left[\varepsilon \mathrm{-}{\varepsilon }_{\text{fl}}\right]\mathrm{=}\mu {\dot{\varepsilon }}_{\text{fl}}+K{\varepsilon }_{\text{fl}}\) with \(\varepsilon \mathrm{=}{\lambda }_{0}\text{.}t\)
Creep deformation is therefore expressed as the sum of a linear function of time and an exponential function, of the type:
\({\varepsilon }_{\text{fl}}(t)\mathrm{=}a\text{.}t+b+\alpha {e}^{\mathrm{-}\beta \text{.}t}\)
Which gives in the differential equation:
\(0\mathrm{=}\mu \text{.}{\dot{\varepsilon }}_{\text{fl}}(t)+K\text{.}{\varepsilon }_{\text{fl}}(t)+E\text{.}{\varepsilon }_{\text{fl}}(t)\mathrm{-}E\text{.}{\lambda }_{0}\text{.}t\)
That is:
\(0\mathrm{=}\left[(K+E)b+\mu \text{.}a\right]+\left[(K+E)a\mathrm{-}E\text{.}{\lambda }_{0}\right]t+\left[(K+E)\alpha \mathrm{-}\mu \text{.}\beta \text{.}\alpha \right]{e}^{\mathrm{-}\beta \text{.}t}\)
from where:
\(a\mathrm{=}\frac{E\text{.}{\lambda }_{0}}{K+E}\) \(b\text{=-}\frac{\mu }{K+E}\frac{E\text{.}{\lambda }_{0}}{K+E}\) \(\beta \mathrm{=}\frac{K+E}{\mu }\)
At the initial moment, the starting point is zero creep deformation, which leads to:
\(\alpha \mathrm{=}\frac{m}{K+E}\frac{E\text{.}{\lambda }_{0}}{K+E}\)
We finally obtain the expression of the creep deformation as a function of time:
\({\varepsilon }_{\text{fl}}^{\text{xx}}(t)\mathrm{=}{\varepsilon }_{\text{fl}}(t)\mathrm{=}\frac{{\lambda }_{0}\text{.}E}{K+E}\left[t\mathrm{-}\frac{\mu }{K+E}(1\mathrm{-}{e}^{\mathrm{-}\frac{K+E}{\mu }t})\right]\)
The \(\mathit{xx}\) component of elastic deformation is equal to: \({\varepsilon }_{e}\mathrm{=}\varepsilon \mathrm{-}{\varepsilon }_{\text{fl}}\). That is:
\({\varepsilon }_{e}^{\text{xx}}(t)\mathrm{=}{\varepsilon }_{e}(t)\mathrm{=}\frac{{\lambda }_{0}\text{.}K}{K+E}t+\frac{{\lambda }_{0}\text{.}E\text{.}\mu }{{(K+E)}^{2}}(1\mathrm{-}{e}^{\mathrm{-}\frac{K+E}{\mu }t})\)
The components \(\mathrm{yy}\) and \(\mathrm{zz}\) of the elastic and creep deformations are obtained by multiplying the component \(\mathrm{xx}\) by the Poisson’s ratio.
The \(\mathrm{xx}\) component of the constraint is equal to: \(\sigma \mathrm{=}E\text{.}{\varepsilon }_{e}\mathrm{=}E(\varepsilon \mathrm{-}{\varepsilon }_{\text{fl}})\). That is:
\({\sigma }_{\text{xx}}(t)\mathrm{=}\sigma (t)\mathrm{=}\frac{{\lambda }_{0}\text{.}K\text{.}E}{K+E}t+\frac{{\lambda }_{0}\text{.}{E}^{2}\text{.}\mu }{{(K+E)}^{2}}(1\mathrm{-}{e}^{\mathrm{-}\frac{K+E}{\mu }t})\)
Elasticity threshold
The behavior remains elastic until the elastic limit is reached. In the case of uniaxial tension, the equivalent stress is equal to the non-zero component of the stress. Plasticity therefore occurs when \({\sigma }_{\text{xx}}(t)\mathrm{=}{\sigma }_{\text{eq}}\mathrm{=}{f}_{t}\) (tensile strength), i.e.:
\(\frac{{\lambda }_{0}\text{.}K\text{.}E}{K+E}t+\frac{{\lambda }_{0}\text{.}{E}^{2}\text{.}\mu }{{(K+E)}^{2}}(1\mathrm{-}{e}^{\mathrm{-}\frac{K+E}{\mu }t})\mathrm{=}{f}_{t}\)
This equation, solved by a numerical method, makes it possible to obtain the instant of the start of plasticization \({t}_{\text{plas}}\) and the creep deformation at this moment:
\({\varepsilon }_{{\text{fl}}^{\text{plas}}}\mathrm{=}{\varepsilon }_{\text{fl}}({t}_{\text{plas}})\mathrm{=}\frac{{\lambda }_{0}\text{.}E}{K+E}\left[{t}_{\text{plas}}\mathrm{-}\frac{\mu }{K+E}(1\mathrm{-}{e}^{\mathrm{-}\frac{K+E}{\mu }{t}_{\text{plas}}})\right]\)
Plasticity resolution
The plasticity model was chosen in order to obtain a simple analytical resolution. This is an almost perfect plasticity law, obtained by taking a particular set of parameters for the behavior model leading to an almost zero work-hardening slope. Therefore, in the plastic phase, the stress (component \(\mathrm{xx}\)), equal to the equivalent stress, is equal to the tensile strength. The equations of the model are then:
\(\varepsilon \mathrm{=}{\lambda }_{0}\text{.}t\) (forced traction) and \(\varepsilon \mathrm{=}{\varepsilon }_{e}+{\varepsilon }_{\text{fl}}+{\varepsilon }_{\text{pl}}\)
\(\sigma \mathrm{=}m{\dot{\varepsilon }}_{\text{fl}}+K{\varepsilon }_{\text{fl}}\mathrm{=}\mu {\dot{\varepsilon }}_{\text{fl}}+K\left[\varepsilon \mathrm{-}{\varepsilon }_{e}\mathrm{-}{\varepsilon }_{\text{pl}}\right]\) with \(\mu \mathrm{=}\frac{{\tau }_{s}}{{J}_{s}}\) and \(K=\frac{1}{{J}_{s}}\)
\(\sigma \mathrm{=}E{\varepsilon }_{e}\) \(\sigma \mathrm{=}E\left[\varepsilon \mathrm{-}{\varepsilon }_{\text{pl}}\mathrm{-}{\varepsilon }_{\text{fl}}\right]\mathrm{=}{f}_{t}\)
with as initial conditions:
\(t={t}_{\text{plas}}\) \({\varepsilon }_{\text{fl}}({t}_{\text{plas}})\mathrm{=}{\varepsilon }_{{\text{fl}}^{\text{plas}}}\)
which leads to the differential equation for calculating the creep deformation:
\(\sigma \mathrm{=}{f}_{t}\mathrm{=}\mu {\dot{\varepsilon }}_{\text{fl}}+K{\varepsilon }_{\text{fl}}\)
Creep deformation is therefore expressed in the form:
\({\varepsilon }_{\text{fl}}(t)\mathrm{=}a+\alpha {e}^{\mathrm{-}\beta \text{.}t}\)
Which gives in the differential equation:
\(0\mathrm{=}\mu \text{.}{\dot{\varepsilon }}_{\text{fl}}(t)+K\text{.}{\varepsilon }_{\text{fl}}(t)\mathrm{-}{f}_{t}\mathrm{=}\left[K\text{.}a\mathrm{-}{f}_{t}\right]+\left[K\text{.}\alpha \mathrm{-}\mu \text{.}\beta \text{.}\alpha \right]{e}^{\mathrm{-}\beta \text{.}t}\) from where:
\(a=\frac{{f}_{t}}{K}\) \(\beta \mathrm{=}\frac{K}{\mu }\)
At instant \({t}_{\text{plas}}\), the creep deformation is equal to \({\varepsilon }_{{\text{fl}}^{\text{plas}}}\), which leads to:
\(\alpha \mathrm{=}({\varepsilon }_{{\text{fl}}^{\text{plas}}}\mathrm{-}\frac{{f}_{t}}{K}){e}^{\beta \text{.}t}\)
We finally obtain the expression of the creep deformation as a function of time:
\({\varepsilon }_{{\text{fl}}^{\text{xx}}}(t)\mathrm{=}\frac{{f}_{t}}{K}+({\varepsilon }_{{\text{fl}}^{\text{plas}}}\mathrm{-}\frac{{f}_{t}}{K}){e}^{\mathrm{-}\frac{K}{\mu }(t\mathrm{-}{t}_{\text{plas}})}\) with \(\varepsilon \mathrm{=}{\lambda }_{0}\text{.}t\)
The \(\mathrm{xx}\) component of elastic deformation is equal to: \({\varepsilon }_{e}\mathrm{=}\frac{\sigma }{E}\mathrm{=}\frac{{f}_{t}}{E}\)
The \(\mathrm{xx}\) component of plastic deformation is equal to: \({\varepsilon }_{\text{pl}}\mathrm{=}\varepsilon \mathrm{-}{\varepsilon }_{e}\mathrm{-}{\varepsilon }_{\text{fl}}\mathrm{=}{\lambda }_{0}\text{.}t\mathrm{-}{\varepsilon }_{e}\mathrm{-}{\varepsilon }_{\text{fl}}\). That is:
\({\varepsilon }_{{\text{plas}}^{\text{xx}}}(t)\mathrm{=}{\lambda }_{0}\text{.}t\mathrm{-}\frac{{f}_{t}}{E}\mathrm{-}\frac{{f}_{t}}{K}\mathrm{-}({\varepsilon }_{{\text{fl}}^{\text{plas}}}\mathrm{-}\frac{{f}_{t}}{K}){e}^{\mathrm{-}\frac{K}{m}(t\mathrm{-}{t}_{\text{plas}})}\)
The components \(\mathrm{yy}\) and \(\mathrm{zz}\) of the elastic and creep deformations are obtained by multiplying the component \(\mathrm{xx}\) by the Poisson’s ratio.
The \(\mathrm{xx}\) component of the constraint is equal to: \(\sigma \mathrm{=}{f}_{t}\)
Digital application:
We impose a deformation of \({10}^{–3}\) in 100 seconds, which gives \({\lambda }_{0}\mathrm{=}{10}^{–5}\)
The only difficulty is to calculate the moment of plasticization \({t}_{\text{plas}}\), and the creep deformation \({\varepsilon }_{{\text{fl}}^{\text{plas}}}\) that corresponds to it, by dichotomy for example. The parameters are finally obtained:
\({t}_{\text{plas}}\mathrm{=}13.024296\)
\({e}_{{\text{fl}}^{\text{plas}}}\mathrm{=}{\mathrm{1.20969985.10}}^{–6}\)
\(e\mathrm{=}{\mathrm{1.2903226.10}}^{–4}\)
which make it possible to obtain the reference values after plasticizing the concrete.
At 10 seconds, the behavior is a creep/elasticity coupling. At 100 seconds, the behavior is a creep/plasticity coupling:
time |
10 |
100 |
\(\sigma\) |
3.0778607 |
4.0 |
\(\varepsilon\) |
1.10—4 |
1.10—3 |
\({\varepsilon }_{\mathit{fl}}\) |
7.1417140.10—7 |
1.7316168.10—5 |
\({\varepsilon }_{e}\) |
9.9285829.10—5 |
1.2903226.10—4 |
\({\varepsilon }_{\mathit{fpl}}\) |
0.0 |
8.5365157.10—4 |
2.3. Uncertainty about the solution#
It is negligible, in terms of machine precision.
2.4. Bibliographical references#
The model was defined in the specification document:
CS SI/311-1/420 AL0/RAP /00.019 Version 1.1, « Development of creep/crack coupling in Code_Aster - Specifications »