Instabilities Modeling in Geomechanics. Jean Sulem
provide an experimental illustration of a local instability, under the assumption of the absence of localization (diffuse plastic strain), is almost impossible. Figure 1.3(a) shows a series of triaxial test results with increasing confining stress values, from Paterson (1958) for Wombeyan marble. All but those with the highest confining stress eventually exhibit unstable behavior, but for pressure values lower than 35 MPa the material exhibits localized instability, either as a vertical spalling (1) or single (2) or conjugate (3) shear bands, whereas for larger confining stress the behavior is qualified as ductile (4,5) for 70 and 100 MPa (3,4) (Figure 1.3(b)).
Figure 1.3. Triaxial compression of Wombeyan marble. (a) Axial stress–strain curves; (b) localization and diffused damage modes, from Patterson (1958)
Notably, for a higher level of confining stress in these tests (70 and 100 MPa), the deformation pattern, with a substantial inelastic component, can be viewed as uniform or non-localized. The stress–strain curves do not suggest an unstable or non-unique behavior. Nevertheless, the pattern formation of microcracks brings some concerns about the homogeneity of the strain and stress distribution across the sample, which in reality is not homogeneous. However, rigorous or at least somewhat codified measuring and understanding of what is “sufficiently homogeneous” are conspicuously missing.
In contrast, for all tests below 35 MPa of confining stress, one or more of stability criteria are failed, but the deformation is invariably localized at a certain point.
Acoustic emission recording techniques allow us to monitor sound-emitting micro-fracturing, which initiates long before the loss of stress–strain curve linearity, even before noticeable dilatancy onset, and long before approaching the peak stress in triaxial conditions for marble (Figure 1.4; Hallbauer et al. 1973). Indeed, what could qualify as an onset of non-uniqueness and instability coincides rather with the coalescence of microcracks into a shear band or macrocrack. Notably, the crack pattern evolution is a gradual process and does not suggest any threshold behavior or values. However, that is not always the case, as seen in comparisons of uniaxial failure in salt-rock, granite and marble (see Figure 1.7 from Zhang et al. (2015)).
Figure 1.4. Axial and lateral stress measured on a set of argillaceous quartzite with the corresponding evolution of the distribution of microcracking (Hallbauer et al. 1973)
For sand, the situation is quite similar. Unstable and non-unique behavior is seen in triaxial tests at low confining stress, and is invariably associated with a localized deformation into a shear band (Figure 1.5; Vardoulakis et al. 1978).
Figure 1.5. Biaxial compression of sand with visible localized shear band (Vardoulakis et al. 1978)
Homogeneous behavior at higher confining stress is rarely seen in triaxial strain, and is usually associated with material stability and uniqueness of response to drained tests. There is no established database to support the claim that there may be unstable sand behavior with a uniform strain across the sample, at least in drained triaxial tests (Drescher 2016). Karner et al. (2008) report sound emission attributed to intense intergranular friction at low mean stress, but at higher stress (and sometimes elevated temperature), no strain localization is mentioned, while post-test observations indicate grain breaking and comminution. The associated stress–strain curve implies stable behavior at high confinement (Figure 1.6).
Figure 1.6. (a) Low and (b) high confining stress compression of a quartz sand: isotropic effective stress versus porosity and acoustic emission decreasing after yielding at high confinement; (c) deviatore stress–strain curves showing stable behavior at 24°C at high confining pressure (Karner et al. 2008). For a color version of this figure, see www.iste.co.uk/stefanou/instabilities.zip
Figure 1.7. Comparison of uniaxial compression of rock salt, granite and marble, with a different intensity of acoustic emission at different stages of loading (Zhang et al. 2015). For a color version of this figure, see www.iste.co.uk/stefanou/instabilities.zip
Figure 1.8. Evolution of the distribution of acoustic emission during uniaxial compression of salt rock, granite and marble. For a color version of this figure, see www.iste.co.uk/stefanou/instabilities.zip
A separate issue arises in undrained tests on sands, during which the volumetric strain is imposed to be constant. In such tests, the material exhibits unstable behavior; however, there is no indication of localization; in other words the deformation appears to be homogeneous or diffuse. The term “diffuse failure” has been adopted for this type of behavior (Daouadji et al. 2011). Figure 1.9 shows the corresponding stress–strain curve and the effective stress path for such a test on Hostun sand. Rightfully, Daouadji et al. indicate a restriction of the undrained or constant volume conditions, clearly imposing a peculiar deformation pattern.
Figure 1.9. Unstable behavior during an undrained test of Hostun sand (Daouadji et al., 2011). For a color version of this figure, see www.iste.co.uk/stefanou/instabilities.zip
Also interesting is the instability developing in sand during constant deviatoric stress drained tests. In these tests, failure coincides with the changing sign of the volumetric strain (or at a maximum of attained dilatancy) (see Figure 1.10). However, in such tests the condition