Antoinette Tordesillas

Force chain buckling, unjamming transitions and shear banding in dense granular assemblies

Force chain buckling, leading to unjamming and shear banding, is examined quantitatively via a discrete element analysis of a two-dimensional, densely-packed, cohesionless granular assembly subject to quasistatic, boundary-driven biaxial compression. A range of properties associated with the confined buckling of force chains has been established, including: degree of buckling, buckling modes, spatial and strain evolution distributions, and relative contributions to non-affine deformation, dilatation and decrease in macroscopic shear strength and potential energy. Consecutive cycles of unjamming–jamming events, akin to slip–stick events arising in other granular systems, characterize the strain-softening regime and the shear band evolution. Peaks in the dissipation rate, kinetic energy and local non-affine strain are strongly correlated: the largest peaks coincide with each unjamming event that is evident in the concurrent drops in the macroscopic shear stress and potential energy. Unjamming nucleates from the buckling of a few force chains within a small region inside the band. A specific mode of force chain buckling, prevalent in and confined to the shear band, leads to above-average levels of local non-affine strain and release of potential energy during unjamming. Ongoing studies of this and other buckling modes from a structural stability standpoint serve as the basis for the formulation of internal variables and associated evolution laws, central to the development of thermomicromechanical constitutive theory for granular materials.

Incorporating rolling resistance and contact anisotropy in micromechanical models of granular media

Recent findings from non-invasive experiments have shown that rolling resistance and contact anisotropy have a significant influence on the bulk behaviour of granular media. To date, however, there has been no generalised continuum model that accounts for both of these factors. In this paper, we propose a method by which rolling resistance and contact anisotropy may be introduced in a micropolar model of granular media. To capture key microstructures like shear bands, a homogenisation procedure on the scale of a particle and its local void space is developed.

Buckling force chains in dense granular assemblies: physical and numerical experiments

This paper focuses on the columnar particle structures known as force chains, and their failure via buckling. The local kinematics and frictional dissipation of this failure mechanism are examined quantitatively for dense, cohesionless granular assemblies, under quasistatic and strain-controlled compression. Data are taken from a physical experiment and a discrete element simulation of bidisperse assemblies of circular particles undergoing shear banding. Particular attention is paid to the deformation and dissipation within a class of particle clusters, each composed of a buckled force chain segment and its laterally supporting neighbours. These particle clusters are found to be confined to the shear band. We establish measures of their local micropolar deformation, including nonaffine deformation, and the evolution of these quantities with strain. Temporally and spatially, the kinematics of this class of particles exhibits trends consistent with the particle motions that form the major contributors to deformation on the mesoscopic and macroscopic scales. The predominant mode of contact failure in a force chain undergoing buckling, and in the contacts with and within its laterally supporting neighbours, is frictional rolling. Rolling friction thus serves as one of, if not the main control valve for the energy flow from the force chain to its surrounding medium.

Force-chain buckling in granular media: a structural mechanics perspective

Parallels are drawn between the response of a discrete strut on a linear elastic foundation and force-chain buckling in a constrained granular medium. Both systems buckle initially into periodic shapes, with wavelengths that depend on relative resistances to lateral displacement, and curvature in the buckled shape. Under increasing end shortening, the classical structural model evolves to a localized form extending over a finite number of contributing links. By analogy, it is conjectured that the granular model of force-chain buckling might follow much the same evolutionary route into a shear band.

Multiscale characterisation of diffuse granular failure

We study the evolution of structure inside a deforming, cohesionless granular material undergoing failure in the absence of strain localisation – so-called diffuse failure. The spatio-temporal evolution of the basic building blocks for self-organisation (i.e. force chains and minimal contact cycles) reveals direct insights into the structural origins of failure. Irrespective of failure mode, self-organisation is governed by the cooperative behaviour of truss-like 3-cycles providing lateral support to column-like force chains. The 3-cycles, which are initially in scarce supply, form a minority subset of the minimal contact cycle bases. At large length-scales (i.e. sample size), these structures are randomly dispersed, and remain as such while their population progressively falls as loading proceeds. Bereft of redundant constraints from the 3-cycles, the force chains are initially just above the isostatic state, a condition that progressively worsens as the sample dilates. This diminishing capacity for redistribution of forces without incurring physical rearrangements of member particles renders the force chains highly prone to buckling. A multiscale analysis of the spatial patterns of force chain buckling reveals no clustering or localisation with respect to the macroscopic scale. Temporal patterns of birth-and-death of 3-cycles and 3-force chains provide unambiguous evidence that significant structural reorganisations among these building blocks drive rheological behaviour at all stages of the loading history. The near-total collapse of all structural building blocks and the spatially random distribution of force chain buckling and 3-cycles hint at a possible signature of diffuse failure.

The Effect of Local Kinematics on the Local and Global Deformations of Granular Systems

Despite the prevalent use of continuum mechanics for the modeling of granular materials, the controversy surrounding the relationship between the properties of the discrete medium and those of its equivalent continuum has far from abated. The concept of strain is especially problematic. In a continuum body, the strain represents the deformation of an infinitesimal region about a material point. In a discrete granular assembly, however, deformation is governed by the relative motions of the constituent grains. Herein, we introduce a new microstructural definition for the deformation of a granular material within the framework of Micropolar Continuum Theory. The advantages of the new strain definition over existing formulations are: it accounts for particle rotations, it is relatively straightforward to calculate, and its global average matches the macroscopic strain of the assembly. The new definition leads to a patchwork strain field, the existence of which is linked to the nonaffine strain at the particle scale. A key aspect of this study is the construction of a set of local micropolar strain and curvature measures on the scale of a particle and its first ring of neighbors. We dissect these local continuum quantities and, with the aid of discrete element simulations, examine them for a specimen under biaxial compression. New insights are gained on the contributions of the relative particle motions for specific types of contacts at different stages in the deformation history. Results are discussed in light of past experimental findings on shear banding, as well as Oda’s hypothesis on force chain buckling.

Micromechanical constitutive modelling of granular media: Evolution and loss of contact in particle clusters

Micromechanical constitutive equations are developed which allow for the broad range of interparticle interactions observed in a real deforming granular assembly: microslip contact, gross slip contact, loss of contact and an evolution in these modes of contact as the deformation proceeds. This was accomplished through a synergetic use of contact laws, which account for interparticle resistance to both sliding and rolling, together with strain-dependent anisotropies in contacts and the normal contact force. By applying the constitutive model to the bi-axial test it is demonstrated that the model can correctly predict the evolution of various anisotropies as well as the formation of a distinct shear band. Moreover, the predicted shear-band properties (e.g. thickness, prolonged localisation, void ratio) are an even better fit with experimental observations than were previously found by use of previously developed micromechanical models.

Frictional indentation of dilatant granular materials

The frictional indentation of a semi–infinite dilatant granular mass is examined assuming plane strain conditions. Complete solutions are provided for the problem of indentation by a flat punch and a wedge. Several important results arising from this study include (i) a generalization of Prandtls solution for the flat punch which shows that the sides of the triangular dead material which develops underneath a frictional flat punch are inclined to the punch face at an angle which lies between δ¼π+½δ (this result is precisely that found experimentally by Bekker and Peynircioglu); (ii) a prediction of the range of wedge semiangles and wedge–material friction coefficient for which the so–called dead–material cap arises (this phenomenon has been experimentally observed to occur under blunt wedges but not for sharp wedges); and (iii) the occurrence of the ‘wedge indentation paradox’ originally reported by Haddow for the analogous problem in metal plasticity.

Granular and complex materials

Foam as Granular Matter (D Weaire et al.) Delaunay Simplex Analysis of the Structure of Equal Sized Spheres (A V Anikeenko et al.) On Entropic Characterization of Granular Materials (R Blumenfeld) Mathematical Modeling of Granular Flow-Slides (I Vardoulakis & S Alevizos) The Mechanics of Brittle Granular Materials (I Einav) Stranger than Friction: Force Chain Buckling and Its Implications for Constitutive Modelling (A Tordesillas) Investigations of Size Effects in Granular Bodies During Plane Strain Compression (J Tejchman & J Gorski) Granular Flows: Fundamentals and Applications (P W Clearly) Fine Tuning DEM Simulations to Perform Virtual Experiments with Three-Dimensional Granular Packings (G W Delaney et al.) Fluctuations in Granular Materials (R P Behringer) Statistical Mechanics of Dense Granular Media (M Pica Ciamarra et al.) Compaction of Granular Systems (P Richard et al.).

Shear bands as bottlenecks in force transmission

The formation of shear bands is a key attribute of degradation and failure in soil, rocks, and many other forms of amorphous and crystalline materials. Previous studies of dense sand under triaxial compression and two-dimensional analogues from simulations have shown that the ultimate shear band pattern may be detected in the nascent stages of loading, well before the bands known nucleation point (i.e., around peak stress ratio), as reported in the published literature. Here we construct a network flow model of force transmission to identify the bottlenecks in the contact networks of dense granular media: triaxial compression of Caicos ooid and Ottawa sand and a discrete element simulation of simple shear. The bottlenecks localise in the nascent stages of loading —in the location where the persistent shear band ultimately forms. This corroborates recent findings on vortices that suggest localised failure is a progressive process of degradation, initiating early in the loading history at sites spanning the full extent, yet confined to a subregion, of the sample. Bottlenecks are governed by the local and global properties of the sample fabric and the grain kinematics. Grains with large rotations and/or contacts having minimal load-bearing capacities per se do not identify the bottlenecks early in the loading history.