Mechanical response along dilatant proportional strain loading paths 

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Dilatancy and contractancy

Dilatancy and contractancy are one of the most important properties of granular materials, defined by volume changes (increase and decrease, respectively) induced by shear defor-mation. The main lineage of understanding of dilatancy in the framework of continuum mechanics can be seen clearly thanks to the review work in Rowe 1962, Houlsby 1991, Li and Dafalias 2000, Wan and Guo 2004 and Kruyt and Rothenburg 2016 at different ages. The concept of “dilatancy” was initially highlighted by Reynolds in 1885 Reynolds 1885. Dilatancy/contractancy plays an central role in constitutive modeling as it controls the non associated character of the flow rule in granular materials.
Empirically, the relationship of dilatancy with stress ratio, density (void ratio) and material internal state has been established in a list of theories, based on a series of exper-iments and motivated by soil behavior modeling. Taylor 1948 pointed out a link between shear strength and dilatancy properties of sands based on a hypothesis about the energy dissipation in frictional soils. The dilatancy equation used in Cam-Clay model by Schofield and Wroth 1968 for soil behavior is a direct analogy of Taylor’s work hypothesis. Alterna-tively, Rowe 1962 proposed the stress-dilatancy theory which basically quantifies the effect of geometrical interlocking of the particles on the stress state. Following Rowe’s theory, Bolton 1986 gave a very simple empirical fit to friction and dilation angles according to a particularly comprehensive review of the experimental data. The dependency of dilatancy on initial stresses and density has also been exhibited experimentally and theoretically in literature (Rowe 1962; Wroth and Bassett 1965; Matsuoka 1974). It is expected that the denser the sand, the more it will tend to expand, but at a given density, the angle of friction reduces slightly with increasing stress level.
A physical estimation of dilatancy was initially conducted by Rowe 1962 based on uniform disks or rods in triangular close-packed arrangements, then on regular packing hexagons (Li and Dafalias 2000), which provides the physical basis of the relationship between the dilatancy and stress ratio. More recently, dilatancy has been treated as the emergence of a collective property within disordered packings. Numerical studies (Peyneau and Roux 2008; Azema, Radjai, and Roux 2015) and experiments (Clavaud et al. 2017) have indicated the absence of dilatancy in frictionless systems, which is different to the previous idea about « rearrangement » that dilatancy arises purely from geometric exclu-sive effect of hard particles. A work from Babu et al. 2020 investigated this paradox by comparing the dilatancy effect between amorphous and lattice assemblies, and estab-lished conditions under which dilatancy emerges naturally in frictionless sphere assemblies through jamming-unjamming processes.
In addition, recent experiments and DEM simulations of specimens suggest that sig-nificant levels of dilatancy emerge from highly coordinated particles motions, particularly within shear bands (Oda and Kazama 1998; Tordesillas, Shi, and Tshaikiwsky 2011); a mi-cromechanical study of dilatancy conducted by Kruyt and Rothenburg (Kruyt and Rothen-burg 2016) identified two microstructural mechanisms: dilatancy due to the deformation of mesostructures and dilatancy owing to topological changes in the contact network mainly governed by the parameter contact anisotropy and coordination number, respectively.
There are a number of constitutive models for granular materials that incorporate phys-ical ideas about the evolution of granular microstructure within the continuum mechanics framework and accurately describe dilation properties. Wan and Guo 2004 bypassed the minimization of the energy ratio and highlighted the importance of stress dilatancy and its microstructural dependence upon the behavior of sand along various stress/strain paths in axial symmetry conditions. Following the work of Wan and Guo 2004; Tordesillas, Shi, and Tshaikiwsky 2011 characterized the evolution of stress-dilatancy with respect to the evolution of mesostructures and proposed a model incorporating not only fabric but also fabric evolution. Additional work in this direction has been carried out by Li and Dafalias 2012 and Gao et al. 2014, where the link between dilatancy and microstructure is embedded within the flow rule in the constitutive model.

Softening and hardening

Strain softening and hardening are concepts originally developed for metals in plasticity theory. During plastic deformation the loading surface is said to harden (the yield limit increases in the stress space) or soften (the yield limit decreases in the stress space). In triaxial or biaxial compression tests of granular materials, strain hardening and soften-ing phenomena are exhibited as the increase and decrease in deviatoric stress, coming along with dilatancy or contractancy normally demonstrated by the incremental volumet-ric strain. Particularly, the strain softening is a complex behavior, characterized by a gradual loss of shear resistance with strain after a peak strength has been reached (Prévost and Höeg 1975; Read and Hegemier 1984). Discussions on weather the strain softening is an intrinsic property of granular materials or a structure response have never stopped.
On the one hand, it has been viewed as an inherently material property of granular materials and routinely contained in constitutive models, since it is commonly reported in the experiments of dense materials (Chu, Lo, and Lee 1992; Verdugo and Ishihara 1996). For example, Drucker, Gibson, and Henkel 1957 introduced the concept of soil as a work-hardening material in the extended Mohr-Coulomb model. A link between current peak strength and a state parameter which is a combination of volumetric and mean effective stress information has been demonstrated to reproduce strain hardening in Wood and Belkheir 1994. Yao, Hou, and Zhou 2009 proposed a unified hardening parameter that is independent of stress paths in order to incorporate strain hardening and softening based on the modified Cam-Clay model. This hardening parameter is used in constitutive models for sands (Yao, Liu, and Luo 2016; Yao et al. 2019).
On the other hand, it has been argued that the observed strain softening in soils is not a material property but mainly the consequence of inhomogeneity in the deformation field of the specimens during the loading, as pointed out by Read and Hegemier 1984 after reviewing an extensive set of experiments to discuss the strain softening for rock, soil and concrete. It has been demonstrated that the strain softening is strongly affected by the size and shape of the sample prepared, and is normally accompanied by strain localization (Fu and Dafalias 2011; Zhu et al. 2016). Also, some researchers advocate that both material softening and structural softening exist but appear under different conditions (Sterpi 1999; Liu et al. 2020a). For example, softening along with strain localization in biaxial tests is regarded as structural softening, while that obtained in homogeneous samples as material softening.
It is difficult to close the debate within the framework of plasticity theory, as experi-ments with perfectly homogeneous material does not exist, especially in granular materials. Various modes of bifurcation are possible and do actually develop during the experiments, as pointed out by Hettler and Vardoulakis 1984.
More recent micromechanical investigations have helped to some extent by further un-derstanding of the evolution of fabric and stress, and thereby by providing new insight into strain hardening and softening (Li and Dafalias 2012; Tordesillas et al. 2012; Liu et al. 2020a). They have been characterized by some statistical physics descriptors related to the rearrangement of grains. For instance, Liu et al. 2020a proposed mesoscopically-based framework to interpret both hardening and softening mechanisms; Tordesillas 2007 ex-plained the hardening and softening process by jamming-unjamming theory; to name it. Thus, it might be asked: since strain hardening partly induced by the self-reorganization of particles is recognized as a material property and involved into constitutive models, why not softening that is also induced by rearrangement of particles? The hardening is the process to be more well-organized and easier to be described than the softening process where the structures are much more chaotic and difficult to grasp with higher heterogene-ity within the contact network. This could be a main difficulty in describing the softening process. In addition, it has been widely accepted that the inhomogeneity property should be considered carefully in constitutive modeling. Anisotropy builds quickly as it relies mostly on the contact network. If the anisotropy of contacts can be regarded as an inher-ent property in granular materials, it would be important to incorporate strain softening in constitutive models from a micromechanical viewpoint.

Critical State

Critical state (CS), initially proposed by Casagrande 1936, is a physical phenomenon ob-served from a macromechanical perspective, in the form of a stationary state where stress and volume tend to be constant under continuous shear strain. Dense granular materials reach the critical state as a result of decreasing in dilatancy normally with shear band generation, while loose materials tend to reach the same state after decreasing in contrac-tancy. When the steady state is reached, deformation continues without volume changes. The classic critical state theory (CST) developed by Roscoe, Schofield, and Wroth 1958; Schofield and Wroth 1968 is the basic principle behind critical state soil models (Been and Jefferies 1985; Been, Jefferies, and Hachey 1991; Yao, Hou, and Zhou 2009; Li and Dafalias 2015), which implies two conditions: a steady stress ratio and a steady void ratio. They are expressed analytically by e = ec = eˆc(p).η = ηc = (q/p)c (2.1). with ec = eˆc(p) the critical void ratio which defines the Critical State Line (CSL) in the e − p plane. More recently, it has been proven that not only do both stress and density converge to a steady state at the critical state, but that this convergence and stationarity also manifest at the micro-scale (Drescher and De Jong 1972; Fu and Dafalias 2011; Kruyt and Rothen-burg 2014; Fu and Dafalias 2015; Kruyt and Rothenburg 2016; Zhu et al. 2016; Kawamoto et al. 2018). For instance, Rothenburg and Kruyt 2004 developed a relation between the critical state and coordination number based on DEM simulations; Fu and Dafalias 2011 studied the evolution of fabrics at the critical state using DEM simulations; Kruyt and Rothenburg 2014 re-investigated the relationship between the continuum features at the macroscale and micromechanical quantities including coordination number and fabric anisotropy at both the particle and interparticle contact level and the grain loop scale; Kuhn 2016b demonstrated the stationarity at the critical state as well as the evolution of their disorder toward the critical state based on several characteristics at the microscale; Zhu et al. 2016 investigated different mechanical features of two kinds of meso-structures (force chains and grain loops) and proved that the two failure modes (localized and diffuse) are homological with respect to the concept of the critical state.
According to several micromechanical investigations, it can be shown that the differ-ence between the current porosity and the CS porosity does not in itself determine the evolution of the micro-structure. The coaxiality between the plastic strain rate direction and the fabric anisotropy proved to be a relevant state variable in defining critical state (Li and Dafalias 2012; Theocharis et al. 2019). A third condition that quantifies the role of fabric anisotropy in terms of its intensity and its relative orientation with respect to loading direction should also be taken into account, in addition to the aforementioned two conditions of constant stress ratio and void ratio in the anisotropic critical state theory (ACST) (Li and Dafalias 2012; Theocharis et al. 2019). In ACST, a fabric-related tensor F is adopted. To incorporate micromechanical observations into a macroscopic continuum mechanics description, the norm F ≥ 0 and the unit-norm nF of F are considered with: √F = F nF , F = F : F, nF : nF = 1, trnF = 0 (2.2).
The Fabric Anisotropy Variable A is introduced by A = F : n = F nF : n = F N (2.3).
where n represents a direction along which the loading is applied, say, the direction of plastic flow; thus, it can be related to the stress tensor for monotonic radial loading. N = nF : n is a measurement of the relative orientation of F and n. A tends toward 1 at critical state because both F and N tend toward 1. Then, ACST can be expressed as follows: η = ηc, e = ec = eˆc(p), A = Ac = 1 (2.4).

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Computation loop

In DEM, as introduced in the pioneering work (Cundall and Strack 1979), the interaction between particles is viewed as a transient problem with states of equilibrium developing whenever the internal forces balance. The numerical scheme is explicit. The equilibrium contact forces and displacements of a stressed assembly of non-deformable discs are found through a series of calculations tracing the movements of the individual particles, and the movements are the result of the propagation through the medium of disturbances origi-nating at the boundaries: this is therefore a dynamic process. The dynamic process is described numerically. It is assumed that the time step chosen should be small enough in order that disturbance cannot propagate from any disc further than its immediate neigh-bours during a single time step. The calculations performed in DEM alternate between the application of Newton’s second law to the discs and a constitutive law at the contacts. Newton’s second law gives the motion of a particle resulting from the forces acting on it. The constitutive law is used to compute contact forces from displacements.
Considering a granular material at time t0, geometrical data describing the positions and size of grains and walls, and physical properties of the grains are given. To determine the contact forces and positions of grains at time t0 + ∆t, there are four steps as follows:
• Detect the contacts: two spherical grains are in contact only if the distance between centers of two particles is less than the sum of their radii. If this condition is met, go to the next step.
• Compute the contact forces based on the contact law and relative displacement in-crements.
• Integrate the force and momentum equations for each grain, and calculate the accel-eration of them according to Newton’s second law.
• Update the positions of the grains by integrating the acceleration during ∆t.
After that, the particle positions are updated at this time step, and become the initial state of the next time step. Then, repeat the circulation under control.

Contact law

The contact law is one important modeling hypothesis in DEM simulations since it af-fects the macroscopic behavior significantly together with particle arrangement. In DEM, some contact laws use strains and stresses while others are expressed through displace-ment – force relation. The contact law presented here is the most common and classic one in DEM, originally proposed by Cundall and Strack 1979 as the simplest non-cohesive elastic-frictional contact model. As shown in Fig.2.4, when a new contact is established, it is assumed that there are two linear springs awakened: the one in the normal direction and the other in the tangential direction to the contact, characterized by the normal and tangential stiffness kn and kt respectively. A Coulomb-type friction law is incorporated by introducing the intergranular friction angle ϕ.
Consequently, the normal contact force Fn is defined by the overlapping distance un be-tween the two particles and the normal contact stiffness kn, expressed as Equation 2.5; the tangential contact force Ft after being assessed by the friction law can be written as Equation 2.6. Note that the stiffness kn depends on a material stiffness E, and the radii of the two particles R1 and R2, and kt is determined from kn and a stiffness ratio α. Fn = knun (2.5).

Table of contents :

1 General introduction 
1.1 Motivation and objectives
1.2 Structure of presentation
2 A brief literature review 
2.1 Shearing behavior of granular materials
2.1.1 Dilatancy and contractancy
2.1.2 Softening and hardening
2.1.3 Critical State
2.2 Discrete element view of granular materials
2.2.1 Discrete Element Method
2.3 Micromechanical analysis in granular materials
2.3.1 Particle/contact
2.3.2 Force chains
2.3.3 Grain loops
2.4 Multiscale modeling of granular materials
3 On the attraction power of critical state 
3.1 DEM simulation
3.2 Mechanical response
3.2.1 Mechanical response along dilatant proportional strain loading paths
3.2.2 Mechanical response along biaxial loading paths
3.3 p-q-e space analysis
3.3.1 p-q plane
3.3.2 p-e plane
3.4 Fabric-related critical state locus
3.4.1 Fabric tensor analysis
3.4.2 Grain loop evolution
3.5 Mixed proportional strain and biaxial loading paths
3.6 Conclusion and outlook
4 Dynamical view of the critical state 
4.1 Numerical setup
4.2 The so-called critical state
4.2.1 Stress-strain analysis
4.2.2 Force chains and deviatoric stress
4.2.3 Grain loops and volumetric strain
4.3 Hidden dynamics at critical state
4.3.1 Generating and vanishing process of chained particles
4.3.2 Generating and vanishing process of grain loops
4.4 Microstructure reorganization dynamics under different confining pressures
4.5 Microstructure dynamics under evolving conditions: memory effects in granular materials
4.5.1 Fading process of the initial memory
4.5.2 Memory fading process along proportional strain test
4.6 Conclusion
5 Critical state and the H-model 
5.1 Review and analysis of the H-model
5.1.1 The H-model in brief
5.1.2 Biaxial test at the material point scale
5.1.3 Mesoscale inspection of the H-model during biaxial loading
5.2 Emergence of critical state in the H-model
5.2.1 Method
5.2.2 Preliminary results
5.3 Conclusions
6 General conclusion and perspectives 
6.1 Conclusion
6.1.1 Dilatant proportional strain loading and relationships with critical state
6.1.2 Dynamical view of critical state
6.1.3 Enriched H-model with emerging critical state
6.2 Perspectives
6.2.1 Micromechanical investigations
6.2.2 H-model
6.2.3 Multiscale modeling of landslides
A Complementary results on critical state dynamics 
Bibliographie

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