Instabilities Modeling in Geomechanics. Jean Sulem
href="#fb3_img_img_19622c43-0400-5e38-8bc1-7ab644fdca39.png" alt="Graphs depict (a) the effective stress paths during undrained heating of Boom clay. (b) Simulation of the effective stress path and yield locus evolution for a centrally heated boom clay."/>
Figure 1.17. (a) Effective stress paths during undrained heating of Boom clay (Hueckel and Pellegrini 1991). Notably the thermally generated pore pressure caused the effective stress to reach values where their further decrement is statically inadmissible (see section 1.3). This can be compared to Figure 1.10. (b) Simulation of the effective stress path and yield locus evolution for a centrally heated (from 16 to 93° C) Boom clay around an axisymmetric borehole with an increase in the critical state parameter of around 43% (from Hueckel et al. 2011)
In a borehole boundary value problem, the problem is exacerbated by the very low permeability of clay, compared to thermal conductivity. Indeed, the effective stress path up to the 0.5 m vicinity of the waste canister approaches the critical states in both considered cases of constant and variable friction angle (Figure 1.17(b); Hueckel et al. 2011). As the thermo-elasto-plastic deformation is coupled with heat flow and hydraulic flow, its stability and uniqueness should result from consideration of all three fields.
1.5.3. Localization during drying of geomaterials
Cracking of geomaterials during drying is a purely mechanical problem, but highly coupled with the pore fluid flow. As per the definition, drying is a multiphase phenomenon with quite a complex multiphysics, including phase change, capillarity, flow, deformability and (perhaps) water cavitation during the phenomenon of air entry, which per se is a fluid–gas interface instability. In addition, it requires considerations to be made at three scales: macroscale continuum, mesoscale of grain and pore clusters, and microscale of individual pore structure or grains with liquid bridges. Hueckel et al. (2014) postulate that drying cracking consists of a series of processes, starting with evaporation of water at the external surface, inducing a negative liquid pressure and flow out from the deformable soil undergoing shrinkage in response. According to Terzaghi (1950), the air invasion takes place when the menisci at the saturated external soil surface reach the size of the biggest pores, but when the soil pores are deformable, it affects air entry. As for the drying cracking, it is postulated that an air finger entering the soil as an instability of the water/air interface (Figure 1.18) constitutes a defect in the soil body, around which a stress concentration arises, when there are external constraints to shrinkage. An amplification of total tensile stress induces a localized high tensile effective stress, higher than the value of suction, resulting in a tensile failure, i.e. crack. Hence, a local value of air entry suction controls the local stress amplification at the defect. The mesoscale linear fracture mechanics analysis yields the stress values in the plausible range of tensile strength. In this scenario, the cracking needs to be considered at a continuum scale, while the air entry is a microscale phenomenon. In addition, the air entry requires a certain threshold suction to develop, which in turn changes the size of the pores. Hence, certain processes involved are sequential.
Figure 1.18. Evolution of the water body between eight glass spheres subjected to evaporation at a constant temperature and constant ambient vapor pressure. The arrow indicates a localized non-symmetric unstable mode of the interface evolution (air entry finger) (from Hueckel et al. 2014). For a color version of this figure, see www.iste.co.uk/stefanou/instabilities.zip
There is a host of other similar problems in which multiphysical behavior leads to instabilities, which have not been addressed here due to limited space. They include liquefaction and instability of rock faults and mine pillars related to earthquakes, breakthrough flow due to dissolution of minerals affecting permeability, sinkholes and mine collapses.
1.6. Conclusion
A wide range of failure, instability, non-uniqueness and strain localization phenomena developing in geomaterials have been reviewed in this chapter. It was concluded that in many real life problems, such occurrences result from complex multiphysical fields, including flow of pore water, differential thermal expansion of soil constituents, generation of heat through friction, geochemical reactions, evaporation and air invasion. Mechanical instabilities of the solid skeleton, while no doubt an important part of the overall behavior of geomaterials, do not exhaust the complexity of the overall behavior of such materials. A call for a comprehensive approach to multiphysics instability is more than due. The phenomena involved can be modeled as scenarios of processes that occur either simultaneously or sequentially and that are either coupled or depend on the accumulation of dissipative variables. Hence, the stability of such processes should be investigated as those of coupled mechanical, hydraulic, thermal and chemical processes, or as single processes of a sequence. In the latter case, instability of a one-step process likely affects the formulation of the successor process model. As pointed out by Terzaghi (1950), the causes of the instabilities are often long-term phenomena rather than single events, such as major rainfalls, which are contributing factors. The need for a proper description of these long-term phenomena and their coupling to variable mechanical properties of soil and rock is emphasized.
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