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, 140 6.8 A schematical 2D representation of the assumed porosity heterogeneity. Darker areas represent high-porosity zones (?=42%) and brighter areas represent lowporosity zones (?=35%). A cubic elementary cell centered over a high-porosity zone is chosen as the primitive cell of the periodic lattice and over which the central porosity and its second-gradient are evaluated, p.141

, Definition of the lines over which porosity profiles are plotted, p.141

. .. L11, 144 6.12 Procedure followed to calibrate the hardening moduli ? 1 and ? 2 for a loading stage from State i to i + 1

. .. , 147 6.14 Evolution of ? 0 and ? 2 ? 0 for each REV from the initial state, to the states after the first and second loading stages. The arrow shows the direction of the evolution with loading

, 148 6.16 The growth coefficient s plotted against the perturbation wavelength ? in a Cauchy continuum for a confining pressure of ? 3 = ?12 MPa. ? is taken equal to 1600 kg/m 3

, Growth coefficient s vs the normalized perturbation wavelength ? * in the micromorphic continuum at the stress state corresponding to an axisymmetric compression under 12 MPa of confining pressure, p.1600

, Growth coefficient s vs the normalized perturbation wavelength ? * under several confining pressures for a) B=0.08 MPa; b) B=0.02 MPa. ? is taken equal to 1600 kg/m 3 and L equal to 24 mm

, Growth coefficient s vs the normalized perturbation wavelength ? * under several confining pressures and for different internal lengths (L = 20 mm and L = 40 mm) in two cases: a) B=0.08 MPa; b) B=0.02 MPa. ? is taken equal to 1600 kg/m 3

, Vertical profiles of plastic volumetric strain rates at different loading levels (each plastic volumetric strain rate corresponds to an imposed total axial strain rate of? a = ?0.15%)

. .. Strain, 168 6.38 LSA conducted at the plasticity onset of a sample tested under 10.5 MPa of confining pressure: The growth coefficient vs the normalized wavelength perturbation

, Compaction band thickness evaluated from the incremental plastic volumetric strain profiles and from the LSA applied on an equivalent homogenized material, p.170

. .. , 180 8.2 a) Coarse-grained Lavoux limestone; b) Fine-grained Lavoux limestone (Zinsmeister, 2013); c) Anstrude limestone (o: oolithic grains; s: sparite cement; m: micrite) (Lion, 2004); d) Estaillades limestone (Dautriat, 2009), p.181

, A schematic showing the adopted way to extract a thin rod from a porous material using a diamond wire saw

, Density of internal forces for a m-grade micromorphic media of degree n. Cells in blue and green highlight a second and third gradient theories respectively when restrictions (Equations 8.2 and 8.3) are applied

, Growth coefficient s vs the normalized perturbation wavelength ? * at the plasticity onset in an axisymmetric triaxial loading under 10.5 MPa of confining pressure. Two cases are presented: with restrictions and without restrictions, where P 1 = 10 9 MPa and P 2 = 10 9 MPa, both cases, ? =1600 Kg/m 3 , L =24 mm and B =0.025 MPa

, Petrophysical properties of some sandstones

, Petrophysical properties of many limestones

, The presence of compaction bands is investigated by naked eye and thin section observations, 2012.

. .. , Summary of several limestones microstructural properties, p.41

, Porosity of the four cored samples estimated from: the MIP tests, the mean grey level (GL) and the mass to volume ratio (M/V) evaluations. The microand macro-porosities are evaluated from the MIP tests

, Microstructural main features of high-and low-porosity zones, p.56

, Band types depending on the normalized scalar product value, p.67

. .. , 87 5.2 Band types and orientations ? (in absolute value) with respect to the axial direction in the sample tested at high confining pressure (10.5 MPa) after the first and second loading stages, Mean values of volumetric and shear strain magnitude of the 3 domains selected in Figure 5.11 during the second loading stage, p.121

, 137 6.2 Comparison of the Cauchy and the gradient-dependent plasticity formulation, p.145

, Values of ? 0 and ? 2 ? 0 over the two selected REVs in the initial state and after each stage of loading

, Mean plastic volumetric strain for each REV in each loading stage, p.148

, The component C ep 1111 evaluated at the plasticity onset of axisymmetric triaxial compressions under different confining pressures. The last column indicates if a compaction band (CB) is formed or not

, Hardening modulus evaluated from the Cauchy constitutive law and compared to the critical values given by Issen and Rudnicki (2000) for several confining pressures. The last column indicates if a compaction band (CB) is formed or not, p.152

, Cumulative plastic volumetric strain evaluated after each loading stage: in the numerical modelling and in the REV 1 selected inside the SML sample tested under 10.5 MPa of confining pressure