2. Introduction#

The purpose of this note is to present the rheological model to analyze the mechanical behavior of rocks, adapted to the simulation of underground structures, introduced in Code_Aster and developed by CIH [bib1]. The purpose of this model is to be able to be implemented, quickly and industrially in order to answer the main questions that the engineer asks himself when analyzing and designing an underground cavity. To do this, the rheological law must remain relatively simple, both when identifying the parameters and in its implementation and when interpreting the results.

2.1. Phenomenology of soil behavior#

One of the particularities of a rock, compared to a soil, is that its mechanical behavior is, over a large stress range, controlled by cohesion. This cohesion is associated with the cementation of the environment, induced during the geological history of the massif, and is essentially epitaxial in nature. On the contrary, the resistance of a soil is more particularly governed by the term friction and/or expansion. Cohesion, which is essentially of capillary origin, then has an influence only for very low states of confinement constraints.

This distinction between soil and rock is important because it guides the choice and the basic assumptions of the behavior model.

The main rheological phenomena associated with this context are the following:

  • In the field of small deformations, the response of a rock, in particular under low states of confinement, can be assimilated to linear elastic behavior, weakly dependent on the state of the stresses. Non-linearities in behavior are likely to appear prior to the peak of strength, in the case of soft rocks, for a stress level of the order of 70 to 80% of the maximum value. This threshold decreases with the increase in average pressure to almost cancel out when the overconsolidation constraint is reached (cap-model). Under very low confinement stresses representative of those existing near underground structures, these non-linearities are generally low, especially since cementation is important, and therefore the level of overconsolidation of the rock is high.

  • Expansion (increase in volume) is initiated when non-linearities appear on the stress-strain curve. This dilatance increases until there is localization within the sample. At this moment, the dilatance rate (or the dilatance angle \(\psi\)) is maximum, and then progressively decreases and is cancelled out at very large deformations.

  • The peak strength is reached for stresses describing a failure criterion, generally curved in the Mohr plane or in the plane of major and minor main stresses. The hypothesis of a linear Mohr-Coulomb criterion is therefore only a simplifying hypothesis, tending, for low confinement constraints, to increase the cohesion of the environment.

Once the maximum strength is reached, the strength of the rock decreases. This post-peak softening is all the more rapid and significant (in intensity) the lower the confinement stress. This decrease is linked to more or less localized damage to the rock, depending on the level of confinement. Regardless of this constraint, beyond the peak,**the rock can no longer be considered continuous*. Its behavior is then controlled by the deformation and resistance conditions in the zone where the deformations are located.

  • The appearance of one or more kinematic discontinuities within the rock is associated with a loss of cohesion. The post-peak behavior is then governed by the conditions of friction and expansion along the planes of discontinuity or within a band where the deformations are located. It appears from this reasoning that for very large deformations, the behavior of the rock, assimilated to a « structure », is only rubbing, and is characterized by an ultimate friction angle \(\phi\). This angle is an intrinsic data of the material, a function of the minerals that make up the rock. It therefore does not depend directly on cohesion conditions, and it can above all be considered to be independent of the dimensions of the sample.

  • When the behavior becomes only rubbing, it is not associated with any volume deformation. The dilatance has therefore been cancelled out, and no longer exists at large deformations.

  • The evolution between peak resistance and the critical state corresponding to large deformations is more or less gradual depending on the state of the stresses applied. For a state of zero confinement (simple compression), the behavior is solely driven by cohesion, and the rupture results in an immediate and sudden loss of all resistance. The softening will be more gradual as the confinement stress increases, to become non-existent beyond a certain confinement constraint limiting ductile and fragile domains of behavior.

2.2. Study context and simplifying hypotheses of the model#

The desire to develop a model that is easy to implement is necessarily accompanied by simplifications, resulting from a compromise between the expected objectives, the conditions of use of the model (quality of input data, deadlines and available cost, etc.) and the means implemented to ensure these developments. These compromises are essentially the following:

  • Linear elastic behavior up to the peak of resistance. This is equivalent to assuming that there is no work-hardening of the rock prior to its rupture.

  • Only a shear failure criterion is retained. This means that if the rock is crushed isotropically, the behavior remains elastic, and that there is no damage and work hardening of the material under this type of path. During the phases of excavation of an underground structure with the use of light support, the average pressure in the mass located nearby can only decrease (or remain constant in the ideal case of a circular cavity subjected to isotropic stress, for linear elastic behavior). Plasticization under isotropic stress, which can be found on a Cap-Model or on a Cam-Clay law, did not seem to us to be indispensable given the objectives sought, and in the case of isothermal and short-term stress.

During the development of this model, we deliberately focused on the study and simulation of the post-peak behavior of the rock. In this field of behavior, the resistance of the material is assumed to be controlled, according to the state of the stresses and the level of damage to the rock, by cohesion, expansion or friction.

Cohesion defines the strength of the material as long as this one remains continuous. It is active up to the peak of resistance, and has little influence on softening behavior, unless the cohesion is representative of a ductile « glue » (case of soils injected with silicate gel,…).

As cohesion deteriorates through damage, the dilatance increases, reaching its maximum value when the continuity of the medium is lost. At this moment, under the effect of the shear of the induced discontinuity, this dilatance deteriorates progressively and slowly. The rheology of the rock then evolves towards purely rubbing behavior.