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Modelling crushing of granular materials as a poly-disperse mixture
Modélisation de la fracturation des matériaux granulaires comme un mélange poli-disperse
Caicedo B.
Department of Civil and Environmental Engineering, Universidad de los Andes, Bogotá, Colombia
Ocampo M.
Department of Civil Engineering, Universidad Javeriana, Bogotá Colombia
Vallejo L.
Department of Civil and Environmental Engineering, University of Pittsburg
ABSTRACT: This paper presents a new model to assess the evolution of the grain size distribution of granular materials during
loading. This model is based on the theory of poly-disperse mixtures proposed by De Larrard, 2000. Using this model it is possible to
evaluate the compacity of the mixture depending on the grain size distribution, the shape of the particles and the compaction energy.
Markov processes are used to assess the evolution of the grain size distribution, Markovian transition probabilities for each grain size
are evaluated experimentally using gyratory compaction. Finally the experimental results are compared with the results of the model
showing a very good agreement.
RÉSUMÉ : Cet article présente un nouveau modèle qui permet de calculer l’évolution de la granulométrie des matériaux granulaires
sous différents chargements. Le modèle est basé sur la théorie des mélanges poli-disperses proposée par De Larrard, 2000. Avec ce
modèle, il est possible d’évaluer la compacité du mélange granulaire en fonction de la granulométrie du mélange, la forme des
particules et l’énergie de compactage. Un procédé de Markov est ajouté au modèle pour obtenir l’évolution de la granulométrie, les
probabilités de transition étant évaluées expérimentalement à l’aide d’une machine de compactage giratoire. Finalement, les résultats
du modèle sont comparés avec les résultats expérimentaux et montrent une très bonne correspondance.
KEYWORDS: granular materials, crushing, abrasion, poly-disperse mixtures, compaction.
1 INTRODUCTION
Particle fracture plays a major role in the behaviour of granular
materials that are used in engineering structures such as paved
roads, railroads, highway embankments, and foundations. The
most important engineering properties of granular materials in
these structures depend on the amount of particle crushing that
occurs due to static or dynamic loads (Lade et al. 1996). Particle
breakage occurs as a result of these loads (Bolton 1999; Feda
2002; Hagerty et al. 1993; Hardin 1985; Lade et al. 1996). Grain
crushing is influenced by grain angularity, grain size, uniformity
of gradation, particle strength, porosity, stress level, and
anisotropy (Bolton and McDowell 1997; Feda 2002; Hagerty et
al. 1993; Hardin 1985; Lade et al. 1996; Lobo-Guerrero 2006;
McDowell and Bolton 1998; Nakata et al. 1999; Nakata et al.
2001a; Yamamuro and Lade 1996).
Researches carried out in the past 10 years have proved the
capability of discrete element models to analyse crushing of
granular materials. These models works with individual
particles and permits stress analysis in each one of those
particles. However such models require restrictive assumptions
with respect to the number and shape of each particle and a
better approach to the actual state of the granular material
requires high computational cost. The model presented in this
paper use the theory of poly-disperse materials proposed by De
Larrard (2000) and is a new possibility to assess the evolution
of grain size distribution of granular materials without dealing
with the difficulties of discrete element modelling.
2 DESCRIPTION OF THE MODEL
The evolution of the grain size distribution of granular materials
under cyclic loading is the result of the abrasion and crushing of
particles. The following aspects must evaluate to assess such
evolution: (i) the stress level in each class of particle size within
the granular material; (ii) the strength of the particles taking in
consideration the number of loadings for cyclic loading; and
(iii) the change of grain size distribution as a result of the
crushing of particles.
In this model the first aspect is assessed with the aid of the
poly-disperse theory proposed by De Larrard (2000), the second
aspect uses the Weibull theory including a fatigue law for the
cyclic loading, and the third one uses the theory of Markovian
processes.
2.1 Description of a granular material as a polidisperse
mixture
One of the most important variables having an effect on the
crushing of a granular mixture is their unit weight. In fact huge
experimental and theoretical evidence show that a large volume
of voids in the granular material increases the stress within
particles. As usual, the void volume could be characterized by
the porosity
n
, the void ratio
e
or the compacity. The compacity
of a granular mixture is defined as the solids volume of the
grains
in a unit volume. As a result the compacity could be
calculated using the porosity or the void ratio as follows:
1
n
1 1
e
(1a, b)
The model proposed by De Larrard (2000) permits obtaining
the compacity of a granular mixture depending on the
volumetric proportion (
y
i
) of particles of size
d
i
. This calculation
is based on the knowledge of the residual compacity
i
of each
grain size that represent the maximum compacity obtained
experimentally for each class individually.
The virtual compacity
was defined by De Larrard as the
maximum compacity theoretically reachable for a particular
granular mixture without any alteration of the original shape of
the particles. Details about the derivation of the relationships to
obtain the virtual compacity of a poly-disperse mixture are
