1. Introduction#

1.1. General information on the behavior of soils and rocks and joints#

The modeling of the behavior of soils and rocks that make up many geotechnical structures seeks to represent complex phenomenologies, in particular non-linear ones according to the objectives of the study carried out. For example, we can be interested in evaluating the load-bearing capacity of a foundation, the rise in water pressures, seismo-induced settlements, the evolution of a sliding zone, the margin available before liquefaction occurs, the long-term performance of an underground structure, the amplification of waves in a sedimentary basin, etc.

The mechanical behavior of geomaterials is characterized by:

  • an asymmetry between states of tension and states of compression;

  • irreversible evolutions, to be described by internal variables of history;

a*consolidation, for a path of spherical constraints, which has a strong non-linearity;

a « *critical state » curve, which is a data of the material, in the plane \({\mathrm{log}}_{10}p\text{'}–e\) (confinement pressure, also called*average effective stress*:math:ptext{“}=frac{1}{3}text{tr}(sigma text{“}) —void index\(e\), which is directly linked to the*density index*:math:{D}_{r}) where the soil when it undergoes pure shear no longer presents an evolution of volume deformations (we have \(\dot{{\varepsilon }_{p}^{v}}=0\)); and we reach a plateau Of resistance in the containment pressure plan — deviatory constraints \(p\text{'}–q\). For powdery soil, the « critical state » curve is rather straight in plane \({\mathrm{log}}_{10}p\text{'}–e\) while it is curved for clay soil. In the initial state of the ground, in plane \({\mathrm{log}}_{10}p\text{'}–e\), the domain on the left of the « critical status » curve characterizes dilating situations, while on the right there is contraction;

a*dilatance, that is to say a coupling between volume behavior and deviatoric behavior, variable during the evolution of soil degradation and also the evolution of pore pressures that affect \(p\text{'}\). The name Roscoe is associated with this phenomenology. The expansion phase generally follows a prior phase of*contract*; these two phases are separated by a « characteristic state » curve in plane \(p\text{'}–q\) (where we have \(\dot{{\varepsilon }_{p}^{v}}=0\)), dependent on the state of consolidation, cf. for example [bib 28]. The*expansion angle* \(\psi\) characterizes the slope of this curve, which is always below the « critical state » curve or superimposed;

a*resistance*depending on the loading history, increasing or decreasing (positive or negative work hardening), associated with residual deformations. An expanding soil may see its resistance decrease (*softening) after a positive work hardening stage, while a contracting soil may experience positive work hardening until rupture, described by the « critical state » curve drawn in the containment pressure-deviatory stresses \(p\text{'}–q\) plane. The internal friction angle \({\phi }_{\mathrm{pp}}\) characterizes the slope of this « critical state » curve in plane \(p\text{'}–q\). In the critical state, an evolution of the distortion occurs without an increase in the volume deformation. In the case of clay, these situations are associated with the value of the degree of*consolidation*of the soil (called* OCR ** for « Over-Consolidation Ratio », equal to the ratio between the greatest vertical effective stress reached during the loading process and the effective vertical stress in-situ), while for a sand, the initial state in the \({\mathrm{log}}_{10}p\text{'}–e\) plane defines the softening and expanding or contracting non-softening characteristic of the behavior;

an initial*cohesion (that is to say a resistance for deviatoric loading) for clays, almost zero for dry sands, even if these once saturated present a certain cohesion; the cohesion of the rocks is generally much higher;

  • this phenomenology therefore reflects a common characteristic: the dependence of behavior on the state of the material in plane \({\mathrm{log}}_{10}p\text{'}–e\);

the*elastic characteristics also depend on the initial void index \({e}_{0}\) and on the confinement pressure: this characterizes a nonlinear law of elasticity. The elastic modulus is therefore higher at depth than at the surface. In practice, even if the elasticity domain is very small, it is useful to represent this reduction in the elastic modulus occurring at the beginning of the loading process. The Poisson’s ratio (in an isotropic hypothesis) is generally evaluated using the more accessible compressibility coefficient. Sometimes it is necessary to take into account a certain anisotropy induced by stratification;

  • hysteresis loops under cyclic loading, whose shape is also dependent on the confinement pressure, and on the plasticity index \({I}_{p}\) which is measured in the laboratory;

in particular, a powdery material (sand…) under cyclic deviatoric loading controlled in deformation progressively densifies (contractiveness), especially if it is*loose*in the initial state (low value of :math:`{D}_{r}`); on the other hand, if the material is*dense*(*over-consolidate) it experiences a phase of damage (dilatance) after reaching the peak, followed by rupture (critical state). If the amplitude of these cycles is high, we can observe the absence of stabilization of the cyclic response: we no longer have accommodation but the appearance of the ratchet;

  • tangent and secant rigidities depending on the loading history;

an influence of the loading speed: an equivalent*viscosity can make it possible to represent this effect, even if it is difficult to quantify for sands; this characteristic is important for clays and rocks; all the behavioral properties mentioned above can be dependent on the loading speed;

and finally a strong*spatial variability in mechanical properties, characteristic of a natural material.

Based on these common characteristics, it is necessary to define the main differences between soil and rock:

a*soil*is characterized by a generally porous medium having a*loose*behavior, i.e. sufficiently deformable so that it can be « worked »; the law of soil behavior will mainly focus on characterizing its*plastic flow, its resistance under*cyclic* loading; a soil can be described by a continuous porous granular medium: plastic flow and long-term behavior are closely linked to the problem of variations in interstitial pressures of the fluid present in the porosity;

a*rock*is characterized by a cohesion much higher than a ground, and a stiff behavior presenting a certain hardness; the law of behavior of a rock will mainly focus on characterizing instantaneous and delayed*cracking, as well as breakage under*fatigue*. The state of*damage*of the rock, at the scale of a massif, is quantified by the factor RQD (Rock Quality Designation). It is important to take into account the discontinuous nature of a rock mass at the scale of the structure, i.e. to incorporate into the modeling the consideration of possible fractures that exceed the limits of representation of the law of behavior. A*join* model may be relevant. The porosity of a rock is generally much lower than that of a soil, so the hydromechanical coupling in rocks plays a major role in the long term.

The*interfaces* of environments made up of geomaterials are generally also characterized by complex behaviors: one cannot always be satisfied with perfect modeling by simple continuity of movements, equivalent to the continuity of the stress-vector at the interface. We therefore define a join model. We find the phenomena of cohesion, expansion as in soils, and in addition, the unilateral nature of the rupture. Hydromechanical coupling plays an essential role in certain types of applications.

The various models of behavior attempt to represent some or all of these characteristics.

The parameters of the behavior models are identified from laboratory tests:

  • simple, triaxial shear (consolidation at different confinement values, in a drained situation or not), torsion, isotropic, oedometric compression (drained cylindrical specimen subjected to monotonic uniaxial compression, and to blocked lateral displacements: obtaining a consolidation curve);

  • these tests may be monotonic or cyclic, drained or undrained, with imposed deformation or with imposed stress;

  • resonant column: obtaining the elastic modules;

  • creep, rupture, and relaxation for rocks;

whose results sometimes need to be corrected to reflect the path of actual stresses in place, and tests in situ, at varying depths in the ground. These tests are complemented by geological analyses and surveys:

SPT (*standard penetration test), test carried out by pressing a tube, which does not provide a direct measurement of a physical parameter;

  • CPT (penetrometers), monotonous static test with a pressed cone at a constant speed, which does not provide a direct measurement of a physical parameter;

  • pressiometer: obtaining elastic modules, resistance and creep modules; for example, we consider the « Ménard » pressiometer test carried out by exerting controlled increasing pressure in a vertical cylindrical bore.

  • CPTU (with pore pressure measurement);

speed of shear waves :math:`{V}_{s}` at a low deformation level (*cross-hole, SASW);

  • flat cylinder test in order to assess the state of initial stresses in a rock mass.

These tests produce relationships between the number of cycles and CSR (cyclic stress ratio equal to \(\tau /\sigma {\text{'}}_{v}\)), the state of overconsolidation (OCR, linked to the plasticity index \({I}_{p}\)), CRR (cyclic resistance ratio), the cyclical degradation curves \(G-\gamma\). The range of elastic behavior of soils is limited to very slight deformations: \(\varepsilon <{10}^{-5}\).

In Code_Aster, we adopted the convention of structural mechanics: stresses and deformations are positive in tension, compressive stresses have negative values.

1.2. Objectives#

The modeler will have to find the best compromise between the complexity (of phenomena, of the calibration of parameters…), the cost of the study, the robustness of the integration, and the precision or representativeness of the results sought, according to the available data (parameters and laboratory tests) and the physical validation cases available in the loading field targeted by the study, by correctly choosing the type of behavior models, knowing that a large choice of behavior models, knowing that a large choice of behavior models is available in*Code_Aster*, but also in literature…

This document is designed to help with this task. It brings together elements already available elsewhere in the Code_Aster documentation, and also provides feedback from authors who contributed to these behavioral models. Various angles of comparative analysis of these models are thus presented in a generic way: targeted materials, phenomenology, characteristics and type of law formulation, models that can be used with Code_Aster, number and type of parameters, identification procedure based on available tests, number and type of internal variables, number and type of internal variables (in the strict sense: in the strict sense: corresponding to the mathematical formulation of the law), but also local variables useful for the analyst: quantities of interest for the engineer.), verification and validation test cases, references, publications, general opinion (robustness, developability…), perspectives.

1.3. Geotechnical structures made up of geomaterials as porous media#

The geotechnical structure consists of a soil, or a rock, which is generally porous (the solid phase is called soil skeleton), and this porosity is occupied by fluids: water, air, gas, occluded air, according to a variable saturation. These fluids flow within the porous medium and participate in the mechanical response; various diffusion laws govern these flows, see [R7.01.11]. These laws correspond to dissipative phenomena, for example Darcy’s law, which describes the flow of the fluid phase (s) within the porous matrix, and parameterized by the permeability (s). Depending on the boundary conditions of the environment imposed on the fluid (s), we are in a drained situation or in an undrained situation. The documentation should be consulted for the use of porous media models [U2.04.05].

The concept of « effective stress » makes it possible to describe the contribution of the soil skeleton to the overall mechanical response of the porous medium, associated with its kinematics, described by the deformation tensor \(\varepsilon\). It is the Terzaghi principle that breaks down the total stress tensor into effective stresses and into the contribution due to the fluid (s): \(\sigma =\sigma \text{'}-{\mathrm{b.p}}_{\mathrm{lq}}\mathrm{Id}\), where \({p}_{\mathrm{lq}}\) refers to the pore pressure (assumed to be positive in compression in Code_Aster) in the monophasic case (saturated case), \(\sigma \text{'}\) the « effective stress » tensor and \(b\) the isotropic tensor coefficient of BIOT; in the hypothesis of Terzaghi, we have: \(b\to 1\).

In the case where a gaseous phase is also present, liquid saturation \({S}_{\mathrm{lq}}\) is used in the following total differential expression: \(d\sigma =d\sigma \text{'}-\mathrm{b.}(d{p}_{\mathrm{gz}}\mathrm{Id}-{S}_{\mathrm{lq}}\mathrm{.}d{p}_{c}\mathrm{Id})\), where \({p}_{\mathrm{gz}}\) designates the gas pressure (assumed to be positive in compression in Code_Aster) and and \({p}_{c}={p}_{\mathrm{gz}}-{p}_{\mathrm{lq}}\) designates capillary pressure also called suction (positive in unsaturated). We then define the concept of « net constraints »: \(d\tilde{\sigma }=d\sigma +\mathrm{b.}(d{p}_{\mathrm{gz}}\mathrm{Id})\).

In the absence of a water phase in the ground, in particular above the groundwater level, the problem of balance of the structure can be treated in a « pure mechanical » situation. Otherwise, the equilibrium problem will be treated by coupled (thermo-) hydro-mechanical modeling, with the corresponding finite elements.

While the presence of a water phase in the ground has little effect on the speed of shear waves; on the contrary, it affects the speed of pressure waves. For a clay soil above the groundwater level, partial saturation also affects the value of the speed of the pressure waves.

Some models are designed to provide a mechanical response in « total stresses » that sum up the contribution of the skeleton and that of the « water » phase and drive the expression of the criteria.

A mechanical behavior model defined in effective constraints is designed to describe drained and undrained situations; its identification is preferably carried out according to the type of application envisaged, because the behavior models do not completely model the interaction between skeleton and fluid phase.

The transition from the saturated case to the unsaturated case is delicate, because of the management of the appearance or disappearance of a phase; Code_Aster treats this transition transparently using a modeling such as HH2 [U2.04.05]. Moreover, a solution proposed in the literature to treat non-drained situations in the presence of occluded air, in the vicinity of total saturation, consists in adopting an equivalent compressibility of the fluid to take account of the occluded air and corrected cohesion. This equivalent compressibility can be evaluated for example with the Boutonnier model, [26], and chosen as the input parameter for Code_Aster models.

Joints are characterized by a very different hydraulic behavior depending on the direction normal to the joint or the tangential directions, for which the opening and tortuosity of the joint play an essential and non-linear way. In the opposite direction, hydraulic flow also affects the rupture characteristics of the joint.