A 3-D mechanics-based particle shape index for granular materials
Introduction
The importance of particle shape on the mechanical properties of granular materials has been a topic of study both experimentally [1], [2], [3] and computationally [4], [5], [6]. These studies have shown that in experiments of granular materials—typically, sand—assemblies composed of particles whose shapes deviate further from spheres or discs exhibit higher peak strength, higher residual strength, and lower amounts of average particle rotation. The same trends are observed in computational studies as well, although in these simulations, particle shapes tend to be highly idealized and assemblies often consist entirely of particles of a single shape. Thus, two challenges that this paper seeks to address are accurately modeling particle shapes as they appear in a natural sand and quantifying the effect of particle shape of an assembly whose particles are of varying shape. In this study, 53,939 grains of Hostun sand are simulated in 3-D triaxial compression using the level set discrete element method (LS-DEM) [7], a particle simulator based on the discrete element method [8] that allows for simulation of realistically-shaped particles. As the particles are representative of those of a real granular material, their shapes vary widely, as Fig. 1 shows, from relatively spherical to angular to flat.
Another challenge in studying the effect of particle shape is quantifying the shapes themselves. Some approaches have been based on geometry, such as aspect ratio, roundness (the ratio of the radii of the largest inscribable and smallest circumscribable spheres), and angularity (the sharpness of corners). The approach we use is mechanics-based: a 2-D particle shape index has been developed that is related to the eccentricity of loading in interparticle contacts; one advantage of this approach is that it can be used directly in micromechanics-based models [9]. In this study, we expand the aforementioned mechanics-based particle shape index to 3-D and use it to quantify the particle shapes in the 53,939-particle specimen, which is subject to triaxial compression with experimental validation. We then show the effect of particle shape, via the shape index, on particle rotations and contributions to stress, and provide possible micromechanical explanations of these trends.
Section snippets
Simulation of triaxial compression
The results presented herein are from a simulation performed in [10], a 3-D triaxial compression test on particles representing Hostun sand. The particles are generated from a 3-D X-ray computed tomographic (XRCT) image of an experimental specimen of Hostun sand using level set imaging [11], [12] and simulated using LS-DEM. Some of the parameters are shown in Table 1. Both the experimental specimen and the computational specimen are subject to quasi-static triaxial loading, and parameters of
Mechanics-based particle shape index
As mentioned in the introduction, the mechanics-based particle shape index introduced and used in [9] is expanded to 3-D and used in this study. We start by defining α, which is the angle, for an arbitrary point p on the surface of a particle, between the normal vector of the particle surface at p and the radial vector from p to the particle’s centroid c (See Fig. 4). The distribution of α for particles in assembly is closely related to the particles’ shapes; for example, an assembly of
Effect of particle shape on particle rotations in the shear band
In this section, we look at the effect of particle shape on particle rotations. It has been shown in 2-D, both experimentally and in simulations, that average rotation of particles in an area is equal to the continuum rotation of that area [13], [14], [15], [16]. In 3-D, we show that this also holds, although care must be taken to ensure consistent directionality of rotation since there are now three degrees of rotational freedom. To this end, we compute rotations in the shear band (both
Effect of particle shape on stress
Here, we focus on the effect of αc on macroscopic stress, keeping in mind the distribution of the shape parameter α is essentially identical to that of αc for all stages of loading; thus, we can say that αc is an accurate quantification of particle shapes in the specimen (as characterized by α).
The mean field stress in the assembly is given by the following equation [19], [20]: where V is the volume of the assembly, N is the number of interparticle contact pairs, f is the
Conclusion
We analyzed the results of the aforementioned LS-DEM simulation of a 3-D triaxial compression test of a specimen whose particles represent those of a natural sand; as such, the particles were of varying shape. We computed the shape parameter α, the distribution of which characterizes and quantifies particle shape, and found it was nearly the same as that of αc, a microstructural parameter that is derived from interparticle contacts in an assembly. Thus, we concluded that the distribution of αc
References (21)
- et al.
Level set discrete element method for three-dimensional computations with triaxial case study
J. Mech. Phys. Solids
(2016) - et al.
Quantitative evaluation of the effect of irregularly shaped particles in sheared granular assemblies
Granular Matter
(2011) - et al.
Experimental micromechanical analysis of a 2d granular material: relation between structure evolution and loading path
Mech. Cohesive-frictional Mater.
(1997) - et al.
A micromechanical description of granular material behavior
J. Appl. Mech.
(1981) - et al.
Thermomechanics-based nonlinear rate-dependent coupled damage-plasticity granular micromechanics model
Continuum Mech. Thermodyn.
(2015) - et al.
Particle shape effects on packing density, stiffness, and strength: Natural and crushed sands
J. Geotech. Geoenviron. Eng.
(2006) - et al.
Soil behaviour: The role of particle shape
(2004) Experimental investigation of micro-structural changes in deforming granular media using x-ray tomography
(2013)- et al.
Effects of grain morphology on critical state: a computational analysis
Acta Geotechnica
(2016) - et al.
Quasistatic rheology, force transmission and fabric properties of a packing of irregular polyhedral particles
Mech. Mater.
(2009)