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## Continuum scale analysis of the onset and development of instability

In the previous subsection, the second-order work has been systematicall computed between two equilibrium states. In this subsection, transient evolutions induced by macroscopic stress probes are analyzed at the scale of the REV. In Figure 4.4, the incremental stress and strain time responses are presented for a stress ratio = 0:45 and for the two loading directions = 30:5

(stable) and = 210:5 (unstable).

The considered unstable direction ( = 210:5) corresponds to a physical configuration in which the sample is slightly deconfined simultaneously in the vertical and horizontal directions (dzz < 0 and dxx = dyy < 0). For a stable material, this loading program should result in an increase in the volume of the sample which is only the case here for t < 0:01 s (before large strains develop). A sudden collapse of the sample in the vertical direction is observed and a densification of the sample is obtained as the horizontal dilatancy does not counterbalance the vertical contraction in Figure 4.4

(d »v = P i d »ii > 0).

In Figure 4.5, the transient evolution of the second-order work is plotted for the two loading directions 2 f30:5; 210:5g and for the same stress ratio = 0:45. It should be noted here that as the sample does not follow a quasi-static evolution, the transient second-order work shown in Figure 4.5 is indeed the external second-order work which is an upper bound for the internal second-order work as d2Ec 0 in Equation (2.6) (Nicot et al., 2017).

Initially positive, the second-order work vanishes after t = 0:014 s when the loss of controllability is observed. Then Wext 2 decreases, goes through a minimum and eventually stabilizes after t = 0:05 s around a negative value.

This non-monotonic evolution should be underlined as for some stress ratios and some stress loading directions the final increase of the second-order work may rise above zero. Indeed this evolution is explained by the onset of a softening regime (the vanishing and the decrease of W2) which is eventually stopped as the sample gets denser (the final increase in Wext 2 ). Provided this

softening regime is rapidly stopped (Wext 2 does not decrease too much), the final value for Wext

2 = W2 may become positive again.

In Figure 4.6, a circular representation of Wext 2 is shown in order to link the transient evolutions of d and Wext 2 for a stress ratio = 0:45.

In this Figure, the loss of controllability for unstable directions is visible as transient normalized second-order work does not follow a straight line. As the external second-order work decreases, the incremental stress loading direction 98 CHAPTER 4.

## Micromechanical identification of driving mechanisms responsible for material instability

In the previous section, the mechanical stability of granular samples has been assessed at the scale of the REV and the onset of instability has been explained as the ability of an incremental load to trigger off microstructure reorganizations. If the mechanical state of the considered sample is in the bifurcation domain and loaded along an unstable direction, the incremental loading induces generalized microstructure reorganizations which result in a 4.3. Driving mechanisms responsible for material instability 105 macroscopic transient softening responsible for a loss of controllability. In

the end, a new equilibrium is reached which is characterized by a contact population relatively far from sliding.

The purpose of this section is to provide a microscale investigation of the physical processes leading to the vanishing of the second-order work during the transient loss of controllability phase observed macroscopically in Figure 4.4. In all this section, the particular stress state = 0:45 and the unstable loading direction = 210:5 are considered. A particular attention is paid to the chronology of events leading to the softening of the granular assembly.

#### Outbursts of kinetic energy

As regularly highlighted in the literature (di Prisco and Imposimato, 1997; Darve et al., 2004; Sibille et al., 2009; Nicot et al., 2012, 2009), the main ingredient enabling microstructure reorganizations in granular materials is the particles’ kinetic energy. As a result, it is of particular interest to track

the time evolution of the kinetic energy of the individual particles while applying a stress increment. In Figure 4.12, snapshots of the considered sample are shown for different time steps for two sufficiently large stress increments jjdjj 2 f1; 5g kPa (see Figure 4.9). The particles are colored according to their kinetic energy, and the most energetic ones are highlighted.

A threshold of E c = 108 J is chosen corresponding to the most energetic particles in the initial state.

In both cases, a localized burst of kinetic energy appears at the same spot and approximately at the same time (once jjdjj reaches its targeted value in the 1 kPa case and during the transient increase of jjdjj in the 5 kPa case).

Then the local burst of kinetic energy propagates to the whole sample. In the case where jjdjj is below the threshold value jjdjj identified in section 4.2.4, no burst of kinetic energy is visible (not shown here). Indeed, the observed threshold value in direction = 210:5 corresponds to the minimal perturbation required to trigger off the burst of kinetic energy shown in Figure 4.12. Once initiated, the burst of kinetic energy propagate to the whole sample. The observed vanishing of the second-order work is thus a material property and not a structural one as the microstructure modifications do not stay localized in some regions of the sample with a length scale similar to the one of the whole sample.

**Table of contents :**

**1 Introduction **

1.1 General context of dike and dam failures

1.2 Internal erosion and the particular case of suffusion

1.3 Justification for a numerical homogenization approach of suffusion

1.4 Outline and structure of the present work

**2 State of the art: suffusion impact on soils and numerical tools for its modeling **

2.1 Suffusion susceptibility: an internal stability approach

2.1.1 Geometrical criteria

2.1.2 Hydraulic and stress conditions

2.2 Suffusion susceptibility: a mechanical stability approach

2.2.1 The drained triaxial test approach from experimental and numerical points of view

2.2.2 Triaxial testing limits to assess suffusion consequences on material resistance

2.2.3 From an intuitive definition of instability to the secondorder work criterion

2.2.4 Second-order work envelope based on directional analysis

2.2.5 Flood induced loading and possible impact on mechanical stability

2.3 Multiscale modeling of granular materials

2.3.1 Microscopic modeling through discrete element methods (DEM)

2.3.2 Mesoscale structures and mechanics

2.4 Numerical modeling of internal fluid flows

2.4.1 A comparative review of existing methods

2.4.2 Pore-scale Finite Volume (PFV) method

**3 Micromechanical validation of the representativeness of numerical samples with respect to suffusion **

3.1 Numerical experiments on widely graded samples with DEM

3.1.1 Sample definition

3.1.2 DEM simulation of drained triaxial tests

3.2 Mesoscale analysis of force transmission

3.2.1 Force chains and associated statistics

3.2.2 Force chain spatial autocorrelation and associated length scales

3.3 Mesoscale analysis of transport properties

3.3.1 Pore network definition

3.3.2 Statistical identification of potentially transportable particles

3.3.3 Mean travel distances and associated length scales

3.4 Numerical validation of expected travel distances thanks to DEM/PFV simulations

3.4.1 Flow boundary value problem

3.4.2 Numerical assessment of particle transport and erosion

3.5 Summary of the main findings

**4 Micro to macro analysis of the elementary mechanisms responsible for mechanical instability in granular materials **

4.1 Numerical experiments on narrowly graded samples with DEM

4.1.1 Sample definition

4.1.2 DEM simulation of a drained triaxial test

4.2 Macroscopic assessment of bifurcation points

4.2.1 Pre-stabilization step

4.2.2 Directional analysis step

4.2.3 Continuum scale analysis of the onset and development of instability

4.2.4 Influence of the stress increment on the onset of instability

4.3 Micromechanical identification of driving mechanisms responsible for material instability

4.3.1 Outbursts of kinetic energy

4.3.2 Chained particle populations renewal

4.3.3 Chained particles lifespan and life expectancy

4.3.4 Localized force chain bending

4.4 Phenomenological relation between plastic strain and mechanical stability

4.4.1 Elasto-plastic model fitting procedure based on DEM results

4.4.2 Plastic strain intensity and vanishing of the secondorder work

4.4.3 A conjecture for the stabilizing role played by rattlers

4.5 A contact scale explanation for the stabilizing role played by rattlers

4.6 DEM inspection of rattlers’ stabilizing role

4.6.1 Mechanical stability assessment for samples without rattlers

4.6.2 Mechanical stability assessment for samples with added rattlers

4.6.3 Rattlers’ influence on the macroscopic direction of the non-associated flow rule

4.7 Summary of the main findings

**5 Numerical assessment of the impact of an internal fluid flow on the mechanical stability of granular materials**

5.1 Direct flow impact on stress transmission

5.1.1 Sample definition with no rattlers

5.1.2 Fully coupled DEM/PFV numerical experiments

5.1.3 Flow induced material failure

5.1.4 Influence of fluid force fluctuations

5.1.5 Contact scale signature for the sample collapse

5.1.6 Collapse and resulting excess pore pressure

5.1.7 Driving mesoscale mechanisms

5.1.8 Grain detachment as a consequence of force chain collapse

5.2 Relative influence of flow induced erosion and clogging

5.2.1 Definition of a widely graded sample with significant fine fraction

5.2.2 Mechanical stability assessment for different stress states

5.2.3 Flow boundary value problem and bounce back erosion criterion