Micro-structure Analysis in 2D Granular Material 

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Structures in a meso-scale

Force-chains

Due to the disordered packing, forces in granular media are transmitted in a promi-nent heterogeneity pattern (Cambou et al., 2009). This phenomenon was first noticed in several investigations of the shearing test on the photoelastic material (Dantu, 1968; Drescher and De Jong, 1972). In a similar scene as in Figure 1.1, forces were observed not to distribute homogeneously in the material but to follow some column-like paths which formed a structural network, showing a pronouncedly more heterogeneous and anisotropic pattern than that of the material fabric. This material feature was not further investigated until the middle of 1990s. Jaeger et al. (1996) investigated the forces particles applied on the boundaries in a triaxial test through measuring prints of particles on the carbon paper set on the boundaries. A decreasing exponential density distribution of force magnitude was found.
Radjai and Roux (1995); Radjai et al. (1998) distinguished between strong force net-work and weak one according to the average force, and investigated the organization of these two networks. The exponential density distribution was observed to only exist in strong network, while the weak network (complementary to strong network) dis-tributed in a uniform or power-law shape (Radjai et al., 1996; Mueth et al., 1998). Further researches confirmed the exponential density distribution to be a prevailing feature of granular material subjected to the external loading, as properties of weak network were shown to be sensitive to the grain size distribution and the initial state (Mueth et al., 1998; Radjai et al., 1999; Antony, 2000; Blair et al., 2001; Erikson et al., 2002; Silbert et al., 2002; Mueggenburg et al., 2002; Majmudar and Behringer, 2005; Azéma et al., 2007). More generally saying, the existence of a persistent heterogeneous force distribution can be the definition of the solid phase of the granular material, being distinct from the fluid and gas phases, where heterogeneities cannot be long-lasting (Jaeger et al., 1996). In later researches, the jamming transition, a concept similar to the phase transition, was observed to be highly associated to the formation of the heterogeneous force distribution (Behringer et al., 2008; Kondic et al., 2012).
In this context, the strong network is distinguished from the weak network, then the concept of force-chain was prepared on the basis of the strong network to define those quasi-linear columns, consisting of contacts of the force magnitude greater than the average.
Further investigations have shown the significant role the force-chain plays in the macroscopic behaviors of the granular material. Iwashita and Oda (2000); Oda and Iwashita (2000), in numerical biaxial tests with the particle rolling resistance, investigated micro-scale properties of granular assemblies with shear bands. They noticed the forma-tion of shear bands strongly referred to the massive generation of force-chains during the hardening process and their collapse in the softening process, accompanied with a volumetric dilatancy. Based on results of the numerical biaxial test, Tordesillas (2007) analyzed the evolution of the void ratio around force-chains and the damage pattern of force-chain. Some evidences about the link between force-chains and stress-dilatancy were found. This intriguing work shed a light to answer the question that how the volu-metric variation on the material fabric influences the force fabric evolution. However, a reasonable answer requires a more precise investigation on how force-chains and their surrounding structures interact with each other. A priori, force-chains’ surrounding fabric should be characterized in a clear way with physical meanings. The meso-loop, which will be introduced in the following, provides a promising way in this aspect.

Meso-loops

Satake (1992) introduced an approach to associate discrete mechanical properties to the continuum-mechanical ones. In this approach, as shown in Figure 1.2, the contact network seamlessly tessellates the material area into loops, which are enclosed by contact branches and involve inside fan-shaped sections of particles and voids surround by particles. Therefore, voids can be correctly quantified inside loops by local kinematical properties. This approach gave a way of characterizing the material fabric by considering the microscopic volumetric nature in the scale of the loop, namely a meso-scale. Then the loop can be also called the meso-loop.
Bagi (1993, 1996); Nguyen et al. (2012) expressed the stress on the meso-scale as a volume average of the tensorial product of the contact forces and the contact branches. This way of defining the meso-scale stress was satisfying only in the static or quasi-static state, otherwise, an inertial term should be involved according to the expression given by Nicot et al. (2013). Bagi (1993, 1996); Kruyt and Rothenburg (1996); Cambou et al. (2000); Kruyt (2003) proposed several approaches to derive the average strain of the material from relative displacements on contacts and geometrical quantities of loops. Results proved that the global strain can be well presented from the local kinematics of the meso-loops. Nguyen et al. (2009, 2012) gave a definition of the strain on the meso-loop. This meso-scale strain facilitates researchers to make the analysis on the meso-scale, by characterizing the meso-scale kinematics into a continuum-mechanical form which researchers are familiar with.

Problems in the meso-structure investigation

There are two problems on the fabric investigation in the meso-scale, which have to be stated here and need to be investigated in this thesis:
1. As a way of describing the material fabric, the meso-scale topology and its evolution give important information of the granular system. Kruyt and Rothenburg (2014) statistically analyzed the critical state features of meso-loops in several aspects. Results showed that in critical state, the loops’ geometrical anisotropy was highly depending on the side number and the orientation of loops. Arévalo et al. (2010) investigated the evolution of the topology of a polydispersing granular assembly subjected to an isotropic compression. The number of triangles, the meso-loop with three sides, increased substantially at the transition point from the dynamic state to the static state, and kept increasing afterwards. The result suggested that triangles played a special role in the transition between the dynamic to the static state.
Meso-loops in different shapes or inclined in different orientations present the prominently distinction on the micro-mechanical behavior. Therefore, data such as the proportion of meso-loops in different shapes or the preferring inclination of meso-loops are also physically meaningful to the granular system, and can be strongly associated to the material’s macroscopic behavior. This is a promising way to attribute material’s macroscopic behavior to the topology of its fabric. However, only a few of researches has been dedicated to this direction (Arévalo et al., 2010; Kruyt and Rothenburg, 2014).
2. In another aspect, a fundamental problem in the micro-structure investigation of the granular material is how the force fabric (force-chains) and the material fabric (meso-loops) interact with each other. When force-chains and meso-loops are corre-spondingly the local static and kinematical elements in the granular material, the interplay between them appears to be the basic ingredient of the local constitutive behavior. The observation and formulation on this behavior will give birth to the local constitutive relation. Tordesillas et al. (2010, 2014) investigated the evolution of the surrounding meso-loops around force-chains. Triangles associated to force-chains were observed to substantially depopulate and transform to meso-loops with a larger side number. This may contribute to the volumetric dilatancy. The way how triangles were transformed to other ones was discussed. However, even though pioneering works have been done, knowledge on this aspect is still very limited.

Constitutive modeling based on the fabric

As the knowledge in terms of the micro-structure of the granular material was ap-proaching the physical essence, scientists started to attempt building the constitutive relation with micro-mechanical considerations. One kind of approach derives the con-stitutive relation still in a phenomenological framework, such as in an elasto-plastic framework, but with considering the role of the fabric by embedding fabric-related parameters into the formulations. Li and Dafalias (2012) extended the critical state theory (Schofield and Wroth, 1968) by an additional requirement to a constant fabric in the criti-cal state, other than the conventional requirement to the constant stress and volumetric strain. Following this way, several studies (Zhao and Guo, 2013; Guo and Zhao, 2013) dedicated to investigate the critical state fabric anisotropy. The critical state relation be-tween the fabric anisotropy parameters and the hydrostatic pressure was given. Several constitutive relations (Gao and Zhao, 2012; Gao et al., 2014) were built on the basis of this framework by involving fabric anisotropy parameters. This kind of approach only adjusted the phenomenological framework by considering some parameters which have micro-physical meaning. However, the macroscopic quantities were derived not from their microscopic essences but still from empirical relations.
On the contrary, the multi-scale approach uses changing scale techniques to sophisti-catedly relate the macroscopic quantities to the static and kinematic aspect of the fabric. Chang and Misra (1989); Chang et al. (1992), in their pioneering work, built a constitutive model for sand. In this model, incremental stress and strain are respectively connected to the contact force and the contact relative displacement on the basis of the contact fabric distribution E(θ, β), which was initially given. After that, various constitutive models (Chang et al., 1990; Yin et al., 2011b,a; Chang, 2014) were born for different kinds of soil, in similar methods as what was proposed by Chang and Misra (1989); Chang et al. (1992). The difference only lied in the form of the initial density distribution for the contact fabric. Nicot et al. (2005) adopted the fabric evolution law (equation 1.9) given by Calvetti et al. (1997) into the constitutive relation, in order to solve the fabric evolution in terms of the incremental strain during the loading path. Emeriault and Cambou (1996) clarified the general framework of deriving the constitutive relation for the granular material, involving three parts: the strain localization (or averaging) scheme, to determine the local kinematical quantities from the global strain (or reverse); the local constitutive relation, to give the local static quantities according to local kinematic quantities (or reverse); and the stress averaging (or localization) scheme, to solve the global stress from local quantities (or reverse). In this work, representation theorem (Spencer, 1987) was applied to formulate the relation between the local kinematics and the global strain. Combining with the stress homogenization scheme, a micro-mechanical model was derived from a non-linear elastic model, the Hertz-Mindlin model (R. D. Mindlin, 1953).

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Problems in multi-scale constitutive modeling

As we already know, both the force and material fabric only exist in the meso-scale and above, any smaller scale will blind us to capture intact force and material fabrics, much less to completely characterize the local behavior between these two fabrics. This means that the local constitutive relation can be only derived on the meso-scale. Given that the techniques of obtaining the stress and the strain respectively from the force fabric and loops’ kinematics are getting mature, to obtain a satisfying local constitutive relation in the meso-scale is the only missing link to build a sound constitutive model. Nicot and Darve (2011b) built a constitutive relation by introducing hexagons, instead of individual contacts, as elements of the fabric. These hexagons, consisting of six contacts symmetrically forming a loop, were an embodiment of the meso-loop, namely, an entity in the meso-scale allowing the relative displacement among contacts to be presented. However, this was only the first attempt to derive the constitutive relation from the meso-scale, there were still numerous significant details missed, mainly in three aspects: the symmetric layout of the hexagon disables the model to receive a shear strain; the force fabric is not presented, any force-chain related element has not been considered; and the evolution of the angular distribution of hexagons has not been fully considered.

Objectives

To build a constitutive model which is competent of simulating the complex behavior of the soil requires a comprehensive observation, an accurate perception and then a sophisticated formulation on the microscopic essence of the macroscopic appearance. On one side, new observations on the fabric of the granular material have highlighted the significance of the meso-scale and suggested a bright perspective of constitutive modeling in the meso-scale. However, on the other side, progressing along this road is largely hindered by the limited knowledge on the fabric behavior of the granular material in the meso-scale. The aim of this thesis is to build a constitutive model on the basis of the knowledge obtained from the investigation to the mesoscopic mechanical behavior of the granular material. This aim can be divided into two steps: based on the results of the numerical simulation using DEM (discrete element method), to investigate the mechanical behavior of force-chains and meso-loops (their evolution, mutual interaction and their role in macroscopic mechanical behavior), as a cumulation of the micro-mechanical knowledge for further constitutive modeling; and to develop a constitutive model using multi-scale approach on the basis of the knowledge obtained from the first step. These two steps are detailed as below.

Micro-structure analysis in 2D granular material

The objective of the micro-structure analysis can be subdivided into three parts. All analysis is based on the results of numerical simulations using DEM.
The first part is to investigate the evolution of some significant features of meso-loops along the drained biaxial test from different initial states. In this part, basic evolutionary characteristics of the material fabric characterized by meso-loops will be quantitatively investigated, the degree and the way the evolution of meso-loops influences the macroscopic mechanical behavior of the granular material need to be clarified.
The second part is to investigate the interaction between force-chains and their confining meso-loops along drained biaxial loading path. The aim is to specify two points: how the force-chains (the force fabric) rebuild their surrounding meso-loops (the material fabric) during loading path; and how the latter influences the behavior of the former.
The last part is to characterize the critical state fabric on the meso-scale of granular material in both localized and diffuse failure modes. In this part, features of the meso-structure in the critical state will be investigated, aiming to find an identical meso-structure in the failure area of the granular assemblies, which are undergoing either localized or diffuse failure.

Constitutive modeling using a multi-scale approach

In constitutive modeling, the objective lies in two sides: (1) to modify the H-directional model (Nicot and Darve, 2011b) by breaking the axis-symmetric configuration of the hexagon and enabling the model to work under the shear strain, and then testify this extended model; (2) to give suggestions for further extending the model on the basis of the knowledge obtained from the micro-structure investigations.

Table of contents :

1 General Introduction 
1.1 Background: micro-structure of the soil and multi-scale approach
1.2 Constitutive modeling for granular material: knowledge and problems
1.2.1 micro-structure investigation
1.2.1.1 Material fabric and its evolution
1.2.1.2 Structures in a meso-scale
1.2.2 Constitutive modeling based on the fabric
1.3 Objectives
1.4 Outline of this thesis
1.5 Conclusion
2 Numerical Modeling by Discrete Element Method 
2.1 Introduction of discrete element method
2.1.1 Calculation cycle
2.1.2 Contact law
2.1.3 Calculation of the particle displacement
2.1.4 Computational stability condition
2.1.5 Yade-DEM software
2.2 Biaxial tests in DEM
2.2.1 Parameters
2.2.2 Applying confining load and Consolidating
2.2.3 Drained biaxial loading
2.3 Conclusion
3 Micro-structure Analysis in 2D Granular Material 
3.1 Micromechanics of granular material
3.1.1 Description of the fabric
3.1.2 Static and kinematic homogenization
3.2 Meso-structure: force-chains and meso-loops
3.2.1 Force-chains
3.2.2 Meso-loops
3.3 Meso-loops evolution during biaxial loading
3.3.1 Drained biaxial test and results
3.3.2 Meso-loops evolution
3.3.2.1 Proportional analysis of different loops
3.3.2.2 Area change of different loops
3.3.3 The existence of elastic and plastic phases in meso-scale
3.3.3.1 Elasticity and plasticity in the meso-scale
3.3.3.2 Elastic energy and plastic dissipation
3.3.3.3 Plastic volumetric strain in dense packed assemblies
3.3.3.4 Effect of elasticity on volumetric evolution
3.4 Force-chain interaction with meso-loop in biaxial loading path
3.4.1 Macroscopic responses and mesoscopic evolutions
3.4.2 Force-chain induced meso-loop differentiation
3.4.2.1 Evolution on FCL and NFCL
3.4.2.2 Conversion correlations amongst structures
3.4.2.3 Volumetric behavior
3.4.3 Effect of confining structures on force-chains
3.4.3.1 Confining structures and force-chain movability
3.4.3.2 Stress anisotropy of confining loops
3.5 The critical state meso-structure in localized and diffuse failure modes
3.5.1 Drained biaxial test and results
3.5.2 Failure modes and shear band width in specimens
3.5.3 Critical state void ratio e
3.5.4 Meso-structure signature of the critical state
3.5.4.1 Meso-loops characteristics
3.5.4.2 Force-chain characteristics
3.5.4.3 Discussion: the homology of localized and diffuse failure modes
3.6 Conclusion
3.6.1 Meso-loop evolution during biaxial loading
3.6.2 Force-chain interaction with meso-loop along biaxial loading path
3.6.3 The critical state meso-structure in localized and diffuse failure modes
4 A Multi-scale Approach Constitutive Model 
4.1 Reviews on the micro-directional model and the H-directional model
4.1.1 The micro-directional model
4.1.2 limitations of micro-directional model
4.1.3 H-directional model
4.2 Modified H-directional model
4.2.1 Constitutive relations
4.2.2 Model performances
4.2.3 Prospectives on improving the H-directional model
4.3 Conclusion
5 General Conclusion 
5.1 Conclusion
5.2 Open issues and perspectives

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