Numerical study of one-dimensional compression of granular materials. II. Elastic moduli, stresses, and microstructure

Compacting an assembly of soft balls far beyond the jammed state: Insights from three-dimensional imaging

Very soft grain assemblies have unique shape-changing capabilities that allow them to be compressed far beyond the rigid jammed state by filling void spaces more effectively. However, accurately following the formation of these systems by monitoring the creation of new contacts, monitoring the changes in grain shape, and measuring grain-scale stresses is challenging. We developed an experimental method that overcomes these challenges and connects their microscale behavior to their macroscopic response. By tracking the local strain energy during compression, we reveal a transition from granular-like to continuous-like material. Mean contact geometry is shown to vary linearly with the packing fraction, which is supported by a mean field approximation. We also validate a theoretical framework which describes the compaction from a local view. Our experimental framework provides insights into the granular micromechanisms and opens perspectives for rheological analysis of highly deformable grain assemblies in various fields ranging from biology to engineering.

Finding the grain size distribution that produces the densest arrangement in frictional sphere packings: Revisiting and rediscovering the century-old Fuller and Thompson distribution

By means of discrete-element methods, we investigate the joint effects of the grain size distribution (GSD) and contact friction on the structure of three-dimensional samples composed of spherical grains. Specifically, we compress these systems isotropically until jamming and then analyze their structure in terms of density, connectivity, coefficients of uniformity and curvature, and parameters of grading entropy. Our study focuses on power-law GSDs and particularly on the Fuller and Thompson distribution, proposed over a century ago. First, we show that, among the set of GSDs investigated, this particular distribution produces the densest and best-connected systems, falsifying a conjecture recently posed in the literature. Second, we find that the jamming packing fraction can be accurately predicted as a function of simple descriptors of the GSD, but among these descriptors the granular entropy concept proves to be the most useful. This allows for an alternative interpretation of both jamming and grading entropy concepts. Finally, we compare the Fuller and Thompson distribution with two well-known GSDs: that of the Apollonian sphere packing and that towards which granular systems evolve after intensive grain fracturing. Surprisingly, we find that these three GSDs are practically coincident in the limit of large size spans, despite having been introduced or discovered in different scientific contexts (i.e., engineering, mathematics, and earth sciences, respectively).

Three-dimensional compaction of soft granular packings

This paper analyzes the compaction behavior of assemblies composed of soft (elastic) spherical particles beyond the jammed state, using three-dimensional non-smooth contact dynamic simulations. The assemblies of particles are characterized using the evolution of the packing fraction, the coordination number, and the von Misses stress distribution within the particles as the confining stress increases. The packing fraction increases and tends toward a maximum value close to 1, and the mean coordination number increases as a square root of the packing fraction. As the confining stress increases, a transition is observed from a granular-like material with exponential tails of the shear stress distributions to a continuous-like material characterized by Gaussian-like distributions of the shear stresses. We develop an equation that describes the evolution of the packing fraction as a function of the applied pressure. This equation, based on the micromechanical expression of the granular stress tensor, the limit of the Hertz contact law for small deformation, and the power-law relation between the packing fraction and the coordination of the particles, provides good predictions from the jamming point up to very high densities without the need for tuning any parameters.

Quasistatic response of loose cohesive granular materials

DEM-simulated model cohesive assemblies of spherical grains of diameter d, with contact tensile strength F0, once prepared in loose states, are quasistatically subjected to growing isotropic pressure P, and then to triaxial compression, maintaining lateral stresses σ2 = σ3 = P while increasing axial stress σ1 = P + q and strain є1. Reduced pressure P* = d2P/F0 varies from 0.1 (cohesion dominated case, for which systems typically equilibrate with solid fraction Ф ≃ 0.35), to large values for which the cohesionless behavior is retrieved. In triaxial compression, while the moderate strain response (є1 ~ 0.1) is influenced by initial coordination numbers and mesoscale heterogeneities, the approach to the critical state, as both q (deviator) and Ф steadily increase, gets slower for smaller P*. Critical ratio q/P strongly increases for decreasing P*, as roughly predicted in an “effective stress” scheme. Anomalously small elastic moduli are observed in the gel-like structures. While extensive geometric rearrangements take place, no shear banding is observed. Loose cohesive granular assemblies are thus capable of large quasistatic stable plastic strains and ductile rupture.

Bulk modulus of soft particle assemblies under compression

Using a numerical approach based on the coupling of the discrete and finite element methods, we explore the variation of the bulk modulus K of soft particle assemblies undergoing isotropic compression. As the assemblies densify under pressure-controlled boundary conditions, we show that the non-linearities of K rapidly deviate from predictions standing on a small-strain framework or the, so-called, Equivalent Medium Theory (EMT). Using the granular stress tensor and extracting the bulk properties of single representative grains under compression, we propose a model to predict the evolution of K as a function of the sample’s solid fraction and a reference state as the applied pressure P→0. The model closely reproduces the trends observed in our numerical experiments confirming the behavior scalability of soft particle assemblies from the individual particle scale. Finally, we present the effect of the interparticle friction on K’s evolution and how our model easily adapts to such a mechanical constraint.

Compaction of mixtures of rigid and highly deformable particles: A micromechanical model

We analyze the isotropic compaction of mixtures composed of rigid and deformable incompressible particles by the nonsmooth contact dynamics approach. The deformable bodies are simulated using a hyperelastic neo-Hookean constitutive law by means of classical finite elements. We characterize the evolution of the packing fraction, the elastic modulus, and the connectivity as a function of the applied stresses when varying the interparticle coefficient of friction. We show first that the packing fraction increases and tends asymptotically to a maximum value ϕ_{max}, which depends on both the mixture ratio and the interparticle friction. The bulk modulus is also shown to increase with the packing fraction and to diverge as it approaches ϕ_{max}. From the micromechanical expression of the granular stress tensor, we develop a model to describe the compaction behavior as a function of the applied pressure, the Young modulus of the deformable particles, and the mixture ratio. A bulk equation is also derived from the compaction equation. This model lays on the characterization of a single deformable particle under compression together with a power-law relation between connectivity and packing fraction. This compaction model, set by well-defined physical quantities, results in outstanding predictions from the jamming point up to very high densities and allows us to give a direct prediction of ϕ_{max} as a function of both the mixture ratio and the friction coefficient.

Compaction Model for Highly Deformable Particle Assemblies

The compaction behavior of deformable grain assemblies beyond jamming remains bewildering, and existing models that seek to find the relationship between the confining pressure P and solid fraction ϕ end up settling for empirical strategies or fitting parameters. Using a coupled discrete-finite element method, we analyze assemblies of highly deformable frictional grains under compression. We show that the solid fraction evolves nonlinearly from the jamming point and asymptotically tends to unity. Based on the micromechanical definition of the granular stress tensor, we develop a theoretical model, free from ad hoc parameters, correctly mapping the evolution of ϕ with P. Our approach unveils the fundamental features of the compaction process arising from the joint evolution of grain connectivity and the behavior of single representative grains. This theoretical framework also allows us to deduce a bulk modulus equation showing an excellent agreement with our numerical data.

Reduction of the bulk modulus with polydispersity in noncohesive granular solids

We study the effect of grain polydispersity on the bulk modulus in noncohesive two-dimensional granular solids. Molecular dynamics simulations in two dimensions are used to describe polydisperse samples that reach a stationary limit after a number of hysteresis cycles. For stationary samples, we obtain that the packing with the highest polydispersity has the lowest bulk modulus. We compute the correlation between normal and tangential forces with grain size using the concept of branch vector or contact length. Classifying the contact lengths and forces by their size compared to the average length and average force, respectively, we find that strong normal and tangential forces are carried by large contact lengths, generally composed of at least one large grain. This behavior is more dominant as polydispersity increases, making force networks more anisotropic and removing the support, from small grains, in the loading direction thus reducing the bulk modulus of the granular pack. Our results for two dimensions describe qualitatively the results of three-dimensional experiments.

Shear strength of wet granular materials: Macroscopic cohesion and effective stress : Discrete numerical simulations, confronted to experimental measurements

Rheometric measurements on assemblies of wet polystyrene beads, in steady uniform quasistatic shear flow, for varying liquid content within the small saturation (pendular) range of isolated liquid bridges, are supplemented with a systematic study by discrete numerical simulations. The numerical results agree quantitatively with the experimental ones provided that the intergranular friction coefficient is set to the value [Formula: see text], identified from the behaviour of the dry material. Shear resistance and solid fraction [Formula: see text] are recorded as functions of the reduced pressure [Formula: see text], which, defined as [Formula: see text], compares stress [Formula: see text], applied in the velocity gradient direction, to the tensile strength [Formula: see text] of the capillary bridges between grains of diameter a, and characterizes cohesion effects. The simplest Mohr-Coulomb relation with [Formula: see text]-independent cohesion c applies as a good approximation for large enough [Formula: see text] (typically [Formula: see text]. Numerical simulations extend to different values of μ and, compared to experiments, to a wider range of [Formula: see text]. The assumption that capillary stresses act similarly to externally applied ones onto the dry granular contact network (effective stresses) leads to very good (although not exact) predictions of the shear strength, throughout the numerically investigated range [Formula: see text] and [Formula: see text]. Thus, the internal friction coefficient [Formula: see text] of the dry material still relates the contact force contribution to stresses, [Formula: see text], while the capillary force contribution to stresses, [Formula: see text], defines a generalized Mohr-Coulomb cohesion c, depending on [Formula: see text] in general. c relates to [Formula: see text] , coordination numbers and capillary force network anisotropy. c increases with liquid content through the pendular regime interval, to a larger extent, the smaller the friction coefficient. The simple approximation ignoring capillary shear stress [Formula: see text] (referred to as the Rumpf formula) leads to correct approximations for the larger saturation range within the pendular regime, but fails to capture the decrease of cohesion for smaller liquid contents.

Scaling laws for the mechanics of loose and cohesive granular materials based on Baxter's sticky hard spheres

We have conducted discrete element simulations (pfc3d) of very loose, cohesive, granular assemblies with initial configurations which are drawn from Baxter's sticky hard sphere (SHS) ensemble. The SHS model is employed as a promising auxiliary means to independently control the coordination number z_{c} of cohesive contacts and particle volume fraction ϕ of the initial states. We focus on discerning the role of z_{c} and ϕ for the elastic modulus, failure strength, and the plastic consolidation line under quasistatic, uniaxial compression. We find scaling behavior of the modulus and the strength, which both scale with the cohesive contact density ν_{c}=z_{c}ϕ of the initial state according to a power law. In contrast, the behavior of the plastic consolidation curve is shown to be independent of the initial conditions. Our results show the primary control of the initial contact density on the mechanics of cohesive granular materials for small deformations, which can be conveniently, but not exclusively explored within the SHS-based assembling procedure.

Effect of the microstructure on the propagation velocity of ultrasound in magnetic powders

We analyze experimentally and theoretically the sound propagation velocity of P-waves in granular media made of micrometer-size magnetite particles under an external magnetic field. The sound velocity is measured in a coherent (long-wavelength) regime of propagation after a controlled sample preparation consisting of a fluidization and the application of a magnetic field. Several different procedures are applied and result in different but reproducible particle arrangements and preferential contact orientations affecting the measured sound velocity. Interestingly, we find that the sound velocity increases when the magnetic field is applied parallel to the sound propagation direction and decreases when the magnetic field is applied perpendicular to the sound propagation direction. The observed qualitative relationship between the changes in the particle arrangement and the sound velocity is analyzed theoretically based on an effective medium theory adapted to account for the effect of the magnetic field in the preparation procedure and its influence on the medium contact fabric.

Microstructure as a function of the grain size distribution for packings of frictionless disks: Effects of the size span and the shape of the distribution

This article presents a numerical study of the effects of grain size distribution (GSD) on the microstructure of two-dimensional packings of frictionless disks. The GSD is described by a power law with two parameters controlling the size span and the shape of the distribution. First, several samples are built for each combination of these parameters. Then, by means of contact dynamics simulations, the samples are densified in oedometric conditions and sheared in a simple shear configuration. The microstructure is analyzed in terms of packing fraction, local ordering, connectivity, and force transmission properties. It is shown that the microstructure is notoriously affected by both the size span and the shape of the GSD. These findings confirm recent observations regarding the size span of the GSD and extend previous works by describing the effects of the GSD shape. Specifically, we find that if the GSD shape is varied by increasing the proportion of small grains by a certain amount, it is possible to increase the packing fraction, increase coordination, and decrease the proportion of floating particles. Thus, by carefully controlling the GSD shape, it is possible to obtain systems that are denser and better connected, probably increasing the system's robustness and optimizing important strength properties such as stiffness, cohesion, and fragmentation susceptibility.

Numerical study of one-dimensional compression of granular materials. I. Stress-strain behavior, microstructure, and irreversibility

The behavior of a model granular material, made of slightly polydisperse beads with Hertz-Mindlin elastic-frictional contacts, in oedometric compression (i.e., compression along one axis, with no lateral strain) is studied by grain-level numerical simulations. We systematically investigate the influence of the (idealized) packing process on the microstructure and stresses in the initial, weakly confined equilibrium state, and prepare both isotropic and anisotropic configurations differing in solid fraction Φ and coordination number z. Φ (ranging from maximally dense to moderately loose), z (which might vary independently of Φ in dense systems), fabric and force anisotropy parameters, and the ratio K_{0} of lateral stresses σ_{2}=σ_{3} to stress σ_{1} in the compression direction are monitored in oedometric compression in which σ_{1} varies by more than three orders of magnitude. K_{0} reflects the anisotropy of the assembling process and may remain nearly constant in further loading if the material is already oedometrically compressed (as a granular gas) in the preparation stage. Otherwise, it tends to decrease steadily over the investigated stress range. It is related to force and fabric anisotropy parameters by a simple formula. Elastic moduli, separately computed with an appropriate matrix method, may express the response to very small stress increments about the transversely isotropic well-equilibrated states along the loading path, although oedometric compression proves an essentially anelastic process, mainly due to friction mobilization, with large irreversible effects apparent upon unloading. While the evolution of axial strain ε_{1} and solid fraction Φ (or of the void ratio e=-1+1/Φ) with axial stress σ_{1} is very nearly reversible, especially in dense samples, z is observed to decrease (as previously observed in isotropic compression) after a compression cycle if its initial value was high. K_{0} relates to the evolution of internal variables and may exceed 1 in unloading. The considerably greater irreversibility of oedometric compression reported in sands, compared to our model systems, should signal contact plasticity or damage.