Instabilities Modeling in Geomechanics. Jean Sulem
hardening/softening (dotted lines...Figure 2.22. Critical hardening values as a function of the β and μ for strain l...Figure 2.23. Shear band formation and mesh dependency for a rate-independent el...
3 Chapter 3Figure 3.1. Model of a deforming shear band with heat and fluid fluxesFigure 3.2. Localized deformation with shear, compaction and dilation modes (Du ...Figure 3.3. Cataclastic shear banding in Fontainebleau sandstone (El Bied et al...Figure 3.4. Saturated rock layer under constant normal stress and sheared at con...Figure 3.5. Shear stress–plastic shear strain curves in drained and undrained co...
4 Chapter 4Figure 4.1. Material deformation of the subset (from Bornert et al. (2004))Figure 4.2. Schematic of a 2D-surface DIC analysis approach (from Hall (2012)). ...Figure 4.3. Schematic diagram of the plane strain apparatus for soils (from Desr...Figure 4.4. Schematic diagram of the plane strain apparatus for rocks (from Bésu...Figure 4.5. (a) Photograph of the silicon membrane that separates the specimen f...Figure 4.6. Schematic of the X-ray microtomography setup for triaxial testing (f...Figure 4.7. Tomography setup for triaxial testing at the beamline ID15A at ESRF....Figure 4.8. Stress strain responses from various tests; the numbers noted on eac...Figure 4.9. Test shf00: stereophotogrammetry-based incremental fields of shear s...Figure 4.10. Test shf03: stereophotogrammetry-based incremental fields of shear ...Figure 4.11. Test shf06: stereophotogrammetry-based incremental fields of shear ...Figure 4.12. Test COEA01: Slices of the X-ray CT scan of specimen COEA01 in its ...Figure 4.13. Test COEA01: Stress strain response; the numbers noted on curve are...Figure 4.14. Test COEA01: (top) slice of the grain displacement field during th...Figure 4.15. Test COEA01: (top) slice of the strain field during the test, measu...Figure 4.16. Comparison of the strain field during tests on three sands: (top) ...Figure 4.17. Deformation stress curve and incremental shear strain fields in an...Figure 4.18. Comparison of triaxial tests on loose and dense Hostun sand specime...Figure 4.19. Test BxR_GVR_06: (top) differential stress (σi – σ3) versus axial s...Figure 4.20. Test BxR_GVR_06: (the two first lines) incremental second strain t...Figure 4.21. Test BxR_GVR_11: (top) differential stress (σi – σ3) vs axial strai...Figure 4.22. Test BxR_GVR_11: (the two first lines) incremental second strain te...Figure 4.23. Test BxR_COx_06: (top) differential stress (σi – σ3) versus axial s...Figure 4.24. Test BxR_COx_06: (the two first lines) incremental second strain te...Figure 4.25. Test BxR_COx_13: (top) differential stress (σ1 – σ3) versus axial s...Figure 4.26. Test BxR_COx_13: (the two first lines) incremental second strain te...
5 Chapter 5Figure 5.1. Primary branch (I), secondary branch (II) and branch for the “imperf...Figure 5.2. (a) Displacement and rotation fields, and (b) stresses and couple st...Figure 5.3. External pressure versus normalized hole closure in a thick walled c...Figure 5.4. Contours of the shear plastic strain after localization of deformati...Figure 5.5. Bifurcation load and peak load versus normalized hole radius predict...Figure 5.6. Hole closure versus applied stress for a circular hole and elliptica...Figure 5.7. Incremental displacement field after localization of deformation aro...Figure 5.8. Contours of shear plastic strain around a circular hole (left) and ...Figure 5.9. Biaxial geometry, loading and boundary conditionsFigure 5.10. Vertical load versus vertical displacementFigure 5.11. Contours of the equivalent plastic strain: (a) associative case; (b...Figure 5.12. Radial displacement increment at bifurcation (left) and final mater...Figure 5.13. Internal pressure versus hole expansion for the Ri = 10 cm model
6 Chapter 6Figure 6.1. Vertical (left) and horizontal (right) X-ray CT scan images of hollo...Figure 6.2. Geometric layout of the thick wall hollow cylinder configuration: (a...Figure 6.3. Experimental data and theoretical results for lateral hole failure f...Figure 6.4. External radial stress at bifurcation failure versus bifurcation mod...Figure 6.5. Isotropic and plane-strain loading. Numerical simulations and experi...Figure 6.6. Theoretical predictions of scale effect for lateral hole failure du...Figure 6.7. A single layer under initial stress σ1 and σ2Figure 6.8. Multilayered medium under compressive initial stressesFigure 6.9. (a) Buckling of a layer on top of a half-space, and (b) critical buc...Figure 6.10. Critical buckling stress of a viscoelastic layer on a viscoelastic ...Figure 6.11. Deformed shape of crack and free surface due to buckling of the hal...Figure 6.12. Half-space with periodic array of collinear cracks compressed by un...Figure 6.13. Critical buckling stress ξ = –σ∕G of a half-space with a periodic a...
7 Chapter 7Figure 7.1. Non-local approach on a representative material volumeFigure 7.2. Kinematic degrees of freedom of the Cosserat elastic continuum theor...Figure 7.3. Kinematics of a microstructure continuum: (a) initial configuration ...Figure 7.4. Material system with current configuration Ω and boundary conditions...Figure 7.5. Finite element used for the spatial discretization of the coupled lo...Figure 7.6. Sketch of the plane-strain compression testFigure 7.7. Uniaxial compression: (a) global specimen response curves and (b) Ri...Figure 7.8. Localized patterns represented by the deviatoric deformation for cla...Figure 7.9. Localized patterns represented by the plastic zone for a second grad...Figure 7.10. Example of localized solutions for a compression test obtained afte...Figure 7.11. Schematic representation of the models used for the modeling of a g...Figure 7.12. Imposed total stresses and pore water pressure at the gallery wall ...Figure 7.13. Evolution of strain localization during and after the gallery excav...Figure 7.14. Comparison of the strain localization pattern at the end of the cal...Figure 7.15. Evolution of strain localization during and after the gallery excav...Figure 7.16. Evolution of strain localization during and after the gallery excav...Figure 7.17. Evolution of strain localization after the gallery excavation, with...Figure 7.18. Positions of cross-sections and gallery wall observation points. Fo...Figure 7.19. Evolution of pore water pressure along (a) vertical and (b) horizon...Figure 7.20. Stress paths at the gallery wall, during and after gallery excavati...
8 Chapter 8Figure 8.1. Model of a fault zone as an infinite layer under uniform shear strai...Figure 8.2. Homogeneous layer at 7 km depth T0 = 210°C, pf0 = 70MPa, (σn = 180MP...Figure 8.3. Wavelength selection: λ is the wavelength of the perturbation normal...Figure 8.4. Growth coefficient (Lyapunov exponent) in terms of perturbation wave...Figure 8.5. Strain localization due to chemical softening in a 5 m thick dehydra...Figure 8.6. Positive feedback process due to dissolution and solid skeleton dama...Figure 8.7. Instability region (shaded) for compaction bands under oedometric co...Figure 8.8. Evolution in time of (a) deformation, (b) the specific surface, (c) ...Figure 8.9. Profile of the vertical deformation at various times and for ℓc = 4 ...Figure 8.10. Vertical deformation in function of time: at the peak of the compac...Figure 8.11. Stress paths of the point at the peak of the compaction band (ABC2)...Figure 8.12. Profile of the vertical deformation at various times for ℓc = 16 mm...
9 Chapter 9Figure 9.1. (a) Conceptual model of the internal structure of a fault described ...Figure 9.2. (a) Sketch of the continuum mechanical concept of shear failure as a...Figure 9.3. Summary of the system’s response when equations [9.15] are satisfied...Figure 9.4. Evolution of the core strain rate on line III in Figure 9.3(a). Irre...Figure 9.5. Influence of the characteristics of the reaction. Snapshots of the p...Figure 9.6. (a) Punchball fault from Chester and Chester (1998) featuring two sc...Figure 9.7. Regular ETS sequence in South Vancouver Island, Cascadia, station AL...Figure 9.8. Irregular ETS sequence of Gisborne, New Zealand, station GISB. (a) T...
10 Chapter 10Figure 10.1. Photos of the outcrop of the Glarus Thrust (Switzerland) showing th...Figure 10.2. Representation of the components of the stress and couple-stress te...Figure 10.3. Boundary conditions for the infinite sheared layer modeling a satur...Figure 10.4. Real part of the growth coefficient s as a function of the length o...Figure 10.5. Real part of the growth coefficient s as a function of the length o...Figure 10.6. Bifurcation parameter Hcr (hardening modulus for which a bifurcatio...Figure 10.7. Wavelength of the perturbation with the fastest growth λmax as a fu...Figure 10.8. Selected wavelength λmax as a function of the hardening coefficient...Figure 10.9. Stress–strain response of an infinite sheared layer for Cauchy cont...Figure 10.10. Shear deformation plotted on the deformed mesh with 50 elements an...Figure 10.11. Mesh convergence obtained for the stress-strain response of an inf...Figure 10.12. Cosserat rotation plotted on the deformed mesh with 80 elements an...Figure 10.13. Effect of THM couplings on the stress–strain response of the syste...Figure 10.14. Effect of THM