5. Verification and validation#

5.1. Test cases#

The references of the test cases and associated documentation are given in Section 5.2.

5.2. Examples of answers to the material point#

The Fig. 5.1, Fig. 5.2 and Fig. 5.3 present some answers obtained using the MTest executable (answers to the hardware point):

Isotropic compression test.
Triaxial compression tests for several confinement stresses.
Cyclic shear tests for several deformation amplitudes.

The parameters used in these simulations are grouped together in r7.01.44-table_parametres_simules.

The Fig. 5.1 shows online the response of the isotropic compression test for the values of r7.01.44-table_parametres_simules. In the charging phase, the response is elastic until reaching the pressure \(-\sigma_m=2p_{c0}(1-\eta)=0.2\) kPa (\(I_p=0\)) and then becomes irreversible, with activation of only the plasticity of the first component (\(I_p=1\)). The discharge is elastic. In comparison, the broken line shows the answer in case \(\eta=0\), with the other parameters unchanged. In this situation, the response remains elastic up to a pressure of \(-\sigma_m=200\) kPa. Thus, using the \(\eta\) parameter allows to introduce stronger initial compaction for low pressure levels, where volume deformation \(\varepsilon_v\) is higher under compression. This possibility is interesting for modeling behavior strongly. contracting soils with high porosity.

_images/compression_isotrope.svg — Fig. 5.1 Isotropic pressure-controlled compression test \(-\sigma_m\).#

Fig. 5.2 shows triaxial compression test responses for three \(p_{\mathrm{conf}}\) confinement levels. We observe that the higher the lockdown, the greater the pressure \(-\sigma_m\) reached asympotically. In addition, when this pressure exceeds \(C=448\) kPa, the volume deformation \(\varepsilon_v\) stabilizes. On the other hand, when this pressure is lower, as can be clearly seen for \(p_{\mathrm{conf}}=100\) kPa, the volume deformation continues to increase, indicating dilating behavior. This observation is consistent with the critical state equations referenced in systeme_etat_critique -4.

_images/compression_triaxiale.svg — Fig. 5.2 Triaxial compression tests with three confinement stresses \(-\sigma_{xx}=-\sigma_{yy}=p_{\mathrm{conf}}\).#

The Fig. 5.3 presents, in a continuous line, the response of shear tests at constant pressure \(-\sigma_{m}=200\) kPa for r7.01.44-table_parametres_simules values, with multiple \(\gamma=2\varepsilon_{xy}\in[10^{-6};10^{-2}]\) distortion amplitudes. For each distortion value, one cycle is made according to chronology \(\gamma\rightarrow-\gamma\rightarrow\gamma\). Shear stress \(\sigma_{xy}\) has hysteresis loops, which are not closed due to the fact that a single cycle does not make it possible to stabilize the work hardening associated with the volume components of the internal variables. The responses are compared to those predicted with \(\rho=0\) and \(\rho=1\), the other parameters remaining unchanged. The predictions of model CSSM then join those of the second component alone (case \(\rho=0\)), and deviate more significantly than those of the first component alone (case \(\rho=1\)) on the evolution of the standardized secant shear modulus \(\mu_{\mathrm{secant}}/\mu\) and reduced depreciation.

_images/cisaillement_cyclique.svg — Fig. 5.3 Cyclic shear tests at constant pressure \(-\sigma_{m}=200\) kPa.#

5.3. Example of experimental comparison#

The Fig. 5.4, Fig. 5.5 and Fig. 5.6 compare the predictions of the model CSSM with experimental data on loose rock materials from two tests triaxial compression at \(p_{\mathrm{conf}}=100\) kPa and \(p_{\mathrm{conf}}=200\) kPa confinements, as well as cyclic shear tests.

On triaxial compression tests, model CSSM was calibrated « at best » by adjusting the five parameters \(p_{c0},M,\beta,\omega\) and \(C\), assuming the other parameters of r7.01.44-table_parametres_simules and considering three distinct cases that have already been the subject of previous comparisons:

\(\rho=0\leftrightarrow\) effect of model component 2 only on shear behavior.
\(\rho=1\leftrightarrow\) effect of component 1 of the model only.
\(\rho=0.1\leftrightarrow\) effect of the two components of the model.

The last case corresponds to considering that the shear stiffness in component 1 \(\rho\mu\) (« static » stiffness) is ten times lower than the total shear stiffness \(\mu\) (« dynamic » stiffness). This ratio of ten is not unreasonable, compared to the relationships observed between the dynamic and static modules for ground materials.

The Fig. 5.4 compares the predictions for the case where \(\rho=0\) with the experimental data. It is clear that component 2 of the model alone fails to reproduce realistically the evolutions of the equivalent von Mises stress \(\sigma_{eq}\) and volume deformation \(\varepsilon_v\) observed during compression tests triaxial. On the other hand, component 2 offers a fairly accurate representation of the standardized secant shear modulus \(\mu_{\mathrm{secant}}/\mu\) during shear tests. As for reduced damping, the prediction progressively overstates the experimental evolution with the increase in distortion level \(\gamma\). This prediction is very similar to that of the Iwan model but also of Hujeux [r7.01.23].

_images/comparaison_exp_CSSM_rho_0.svg — Fig. 5.4 Comparison of the predictions in case \(\rho=0\) to the experimental data.#

The Fig. 5.5 compares the predictions for the case where \(\rho=1\) with the experimental data. In this case, component 1 of the model significantly improves the evolution of the equivalent von Mises stress during compression tests. triaxial. However, only the volume deformation for confinement \(p_{\mathrm{conf}}=200\) kPa is correctly reproduced. For lockdown \(p_{\mathrm{conf}}=100\) kPa, the prediction lacks dilatance. With regard to shear tests, the predictions of the standardized secant shear modulus \(\mu_{\mathrm{secant}}/\mu\) and of the reduced damping are greatly deteriorated compared to to those obtained for case \(\rho=0\).

_images/comparaison_exp_CSSM_rho_1.svg — Fig. 5.5 Comparison of the predictions in case \(\rho=1\) to the experimental data.#

The Fig. 5.6 finally presents the predictions in case \(\rho=0.1\). This combination of components 1 and 2 of the model allows:

To improve the observed evolution of dilatance for \(p_{\mathrm{conf}}=100\) kPa confinement during triaxial compression tests, while maintaining a correct representation of the equivalent von Mises stress. This representation is similar to that obtained with component 1 alone in the case where \(\rho=0\).
To obtain predictions close to those of component 2 for shear tests when \(\rho=1\).

In summary, combining the two components of the model makes it possible to take advantage of the best characteristics of each individual component, thus offering an improved representation for both types of monotonic and cyclic tests.

_images/comparaison_exp_CSSM.svg — Fig. 5.6 Comparison of the predictions in case \(\rho=0.1\) to the experimental data.#

Note:

The previous comparisons validate the relevance of model CSSM on shear loads, with calibration parameters \(p_{c0},M,\beta,\omega\) and \(C\) on triaxial compression tests. However, in order to assess the capabilities of the model more comprehensively and rigorously, it will be necessary to compare it to isotropic compression tests. This step will make it possible to estimate more accurately the parameter \(\eta\), here taken close to one in r7.01.44-table_parametres_simules, and refine the value of \(\omega\), which significantly influence the initial compaction, as we saw on Fig. 5.1 with \(\eta\in\{0,0.99\}\).