Investigating the incremental behavior of granular materials with the level-set discrete element method
Introduction
The continuum response of a granular assembly is encoded in the evolving kinematics of particles, driven by frictional forces at discrete interparticle contacts. Decoding this response experimentally is fraught with difficulties mainly in extracting interparticle forces, and creating reproducible conditions. The Discrete Element Method (DEM) (Cundall and Strack, 1979) has provided a numerical framework that overcomes these difficulties, but at the same time introduces new limitations, due to the idealization of granular shape or the incorporation of questionable rolling dissipation (Ai et al., 2011). Recently, a pivotal advancement that overcomes these limitations has been achieved though the level-set characterization of the morphology of individual grains using X-ray Computed Tomography (XRCT) (Vlahinić et al., 2014), and its utilization within the Level-Set DEM (LS-DEM) framework (Kawamoto et al., 2016). Even more recently, significant steps have been made in validating the method (Karapiperis, Marshall, Andrade, 2020, Kawamoto, Andò, Viggiani, Andrade, 2018, Li, Marteau, Andrade, 2019), thus paving the way for a systematic investigation of granular behavior through high-fidelity virtual experiments.
The cornerstone of experiments on granular matter is stress probing, which relies on achieving multiple incremental stress paths originating from an identical initial state. Physical stress probing experiments are extremely hard to conduct, which explains the scarcity of relevant studies (Anandarajah, Sobhan, Kuganenthira, 1995, Royis, Doanh, 1998). On the other hand, numerical stress probing via conventional DEM (e.g. Bardet, 1994, Calvetti, Viggiani, Tamagnini, 2003, Tamagnini, Calvetti, Viggiani, 2005, Wan, Pinheiro, 2013) has served as an effective platform for the investigation of constitutive behavior in a qualitative sense. The first DEM stress probing experiments were conducted by Bardet (1994) using disks. Later, Calvetti and coworkers carried out similar experiments with spheres, and used them to examine the importance of preloading (Calvetti et al., 2003a), inspect the underlying micromechanics (Calvetti et al., 2003b), and assess different classes of continuum theories (Tamagnini et al., 2005). In several occasions (e.g. when probing from a preloaded state), they identified deviations from classical plasticity in the form of a nonregular flow rule, which was interpreted as evidence of thorough incremental nonlinearity (e.g. hypoplasticity) (Tamagnini and Viggiani, 2002). This was in line with later observations in (Kuhn, Daouadji, 2018, Wan, Pinheiro, 2013). The influence of triaxiality on the regularity of the flow rule was investigated in (Wan and Pinheiro, 2013), while the effect of the rotation of principal stresses was discussed in (Froiio and Roux, 2010). A critical element in analyzing results of numerical (or virtual) stress probing experiments is the decomposition between elastic and plastic strains. These have been typically extracted either by unloading to the initial state (Bardet, 1994), or by carrying out additional simulations where dissipative mechanisms are inhibited (Calvetti, Viggiani, Tamagnini, 2003, Calvetti, Viggiani, Tamagnini, 2003, Tamagnini, Calvetti, Viggiani, 2005). Wan and Pinheiro (2013) have suggested that the two approaches are equivalent. On the other hand, Kuhn and Daouadji (2018) observed that the two approaches produce different decompositions, and examined the relevant implications within the context of a thermodynamical framework, complementing an earlier discussion in (Collins and Einav, 2005). With the exception of a 2D polygon study in (Alonso-Marroquín et al., 2005), all the aforementioned studies involve highly idealized particle shapes (disks or spheres).
The first objective of this paper is to introduce a new paradigm of virtual experiments building on the recent development of LS-DEM (Section 2). The framework incorporates an unprecedentedly accurate representation of particle morphology and interaction, which jointly define a type of granular ‘DNA’. By controlling the expression of that ‘DNA’ to a desired configurational state - a process intractable with preexisting techniques - and evolving that state by imposing arbitrary stress paths, the proposed framework is established (Fig. 1). In Section 3, the framework is utilized to systematically investigate the incremental response of an angular sand through multiple stress probing experiments. In a first set of axisymmetric experiments, the elastic-plastic and reversible-irreversible decompositions of strain are investigated, and the properties of plastic flow are discussed as functions of the current state and its history. We, then, shed light on the micromechanical processes driving dissipation, hardening and fabric evolution, and briefly examine the relevant role of fluctuations. Subsequent experiments focus on isolating the effect of interparticle friction and particle morphology, and assessing the effect of the common spherical idealization. In a final set of deviatoric experiments, we map the entire yield surface in 3D principal space and quantify the nonassociativity of the flow rule as a function of the mean stress and Lode angle. A discussion of the main findings and the future potential of virtual experiments, in Section 4, concludes this paper.
Section snippets
Virtual experiments
Physical experiments of granular materials suffer from poor reproducibility and limited control of initial and boundary conditions. They also inherently lack the ability to measure interparticle forces, a key ingredient in understanding the constitutive behavior. The proposed in silico experiment framework effectively bypasses these limitations by relying on a) the accurate mathematical description of particle morphology and interaction, b) the control of the initial state of the assembly and
Setup
This section details the virtual experiment setup used to investigate the incremental response of an angular sand. The model consists of 15625 virtual Hostun sand grains1, whose morphology has been extracted from μ-XRCT data (Section 2.1). The grain interaction follows a Hookean elastic - Coulomb frictional law (Appendix A), with the relevant properties given in Table 1. To accelerate
Conclusions
We have presented an in silico experimentation framework for granular materials, enabled by the accurate mathematical representation of the morphology and interaction of particles, as well as the control of their collective state, far beyond what has been accessible with preexisting techniques. Naturally arising, within this new paradigm, is the concept of a granular ‘DNA’ and its expression to an emergent macroscopic behavior that is largely free from idealizations. The remainder of the paper
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The authors would like to acknowledge the detailed analysis of this work by the two anonymous reviewers, which has contributed to its substantial improvement. Their feedback is gratefully appreciated.
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