Instabilities Modeling in Geomechanics. Jean Sulem
local stress rate at the softening regime may be violated. As will be seen later in the paper, experimentation with thermal pressurization of clay at constant stress deviator undrained heating test leads to a similar response (Hueckel and Pellegrini 1991). However, in that test both the localized and the diffuse strains developed at failure.
There are several observations to be made concerning laboratory experiments on small triaxial samples. The underlying assumption for such experiments is that all the fields: stress, strain and plastic strain (and possibly microcracking) are uniform across the sample until a possible appearance of localization. This unfortunately is not necessarily true, as shown in Figure 1.4 (Hallbauer et al. 1973). Similarly, the evolution of local porosity monitored via CAT in sand suggests an early loss of uniformity prior to shear banding (Desrues et al. 1996). Stress is obviously not measurable directly in such experiments. It is usually considered as an average resulting from the measured force assumed as uniformly distributed across the area. A number of reasons for the non-uniformity are quoted, such as axial symmetry of specimens and a development of roughly planar shear bands, or stress concentration at piston boundaries, with a different intensity in sand, clay or rock. Remedies to the experimental techniques have been sought, by introducing truly triaxial testing (Muir Wood 1973), biaxial strain testing (e.g. Vardoulakis 1978), etc. Drescher and De Josselin de Jong (1972) conducted a series of tests in which they subjected a 2D photosensitive granular medium to shear under constant vertical load between two rigid smooth arms rotating around a pin with a controlled rate, causing a displacement of the medium between the arms with a globally unstable or stable force response, depending on the direction of the medium displacement, and dilative or contractile volumetric strain. These and many other subsequent similar experiments have shown that contact stresses lead to the formation of chains of compressed column forces within the medium, separated by lightly or completely unloaded grains. In addition, said compressed columns undergo periodic unstable buckling and rebuilding of such columns, so that the entire process – while apparently monotonic at the force– displacement level – is unstable, non-homogeneous and non-monotonic at the level of grain structures, which form vortices and other patterns (Kozicki and Tejchman 2016). Similar conclusions were derived from much later studies of discrete element methods (Iwashita and Oda 1998). This prompted the investigation of even small- scale experiments as boundary value problems and the treating of their stability and uniqueness through global criteria. Finally, there is a question of validity beyond the point of loss of stability or uniqueness of the stress–strain curves obtained in a single experiment. Indeed, each such curve in a non-uniqueness range is just one of an infinity of possible responses, as they go through a singularity point of H = 0.
Figure 1.10. Stress–strain curves in a q = cons. drained test with instability (Daouadji et al. 2011). For a color version of this figure, see www.iste.co.uk/stefanou/instabilities.zip
Some additional insight may be expected from multiscale analyses using discrete element method (DEM) computations, together with progress in a rigorous approach to interscale data interpretation.
1.5. Boundary value problems: uniqueness and stability at the field scale
1.5.1. Landslides
Landslides are the most common consequence of soil and rock instabilities at the field scale. In addition, they more common than earthquakes, and may claim many victims per occurrence in populated areas. Classically, shallow sliding is considered separately from deep rotational slope failures. Here, only the latter are considered. The traditional factor of safety (FOS) is understood as a ratio of resisting resultant moment to driving resultant moment the FOS approaching unity is considered as a condition of failure in classical geotechnics. However, as will be seen in what follows, in several historical landslides, failure occurred at a FOS substantially higher than 1, 2 or even 3. Incorrect evaluation of several variables, material properties or contributing processes led to misleading assessment of FOS.
A recent (February 2010) well-documented landslide is that of Maierato (Vibo Valentia), Calabria, which occurred at a site of paleo-landslides of 1783 and 1932. A rare, dramatic, but very instructive video by Patrizia Venturino (2010) has documented (https://www.youtube.com/watch?v=oWHjBsvmyLc) the event in detail. The most important thing to note in this video a succession of localized rotational collapses of individual scarps, intercalated with a mass flow. The general trigger of the landslide was a protracted period of rainfalls. As in so many similar cases, there were numerous precursors of the final event; in this case, a day to several hours prior to generalized failure consisting of minor local strikes, one of them toppling a high voltage powerline. Because of those warning signs, no fatalities occurred. The area affected was 0.3 km2, with a front of about 800 m, depth of 60–70 m and involving about 10 Mm3 (Figure 1.11). The geological structure, shown in a cross-section in Figure 1.12, points out to two particularities of relevance: the presence of two particularly weak permeable rocks: evaporitic limestone and Miopcene sandstone, intercalated by two layers of clay, roughly 10 m thick. Prior to the main collapse, a substantial swelling was observed, suggesting a rotational collapse of a part of mass. It is believed (Gattinoni et al. 2012) and confirmed by laboratory experiments that evaporitic limestone transformed from plastic to semifluid behavior. Such a transformation is suggested to explain the rapid change from a mode of sliding to the flowing debris mass observed in the videos.
Figure 1.11. General view of the Maierato landslide site (Borrelli et al. 2014). For a color version of this figure, see www.iste.co.uk/stefanou/instabilities.zip
The most surprising finding from preliminary finite element (FE) calculations (Gattinoni et al. 2012) is that when dry the slope has FOS equal to 3.36, with slip surface within the evaporitic limestone, while with the water table up by a maximum of 10–15 m there is a reduction of FOS by 40% (about 2). Thus, the conventional understanding of slope stability highly overestimates FOS. Among the possible causes of failure not given proper consideration until now, the authors list penetration and pressurization of pore water along the contact between Miocene sandstone and Miocene clay, as well as the presence (and evolution upon inundation) of evaporitic limestone and Miocenic sandstone. They also suggest that autobrecciation induced by dissolution of halite and gypsum due to weathering and diagenesis might have played a role (Gattinoni et al. 2012; Borelli et al. 2014).
Figure 1.12. A geological profile across the area prior to the landslide. The green line denotes post-event topography (from Gattinoni et al. 2012). For a color version of this figure, see www.iste.co.uk/stefanou/instabilities.zip
These findings from the preliminary analyses of the Maierato landslide (Gattinoni et al. 2012; Borrelli et al. 2014) interestingly connect to observations made about the instability of slopes made by Terzaghi (1950) in his paper on “Mechanism of landslides”. Analyzing “landslide producing processes”, he emphasizes the difference between the “causes” and the “contributing factors” of landslides. He defines the latter through an example as follows: “if a slope is old, heavy rainstorms (...) can hardly