Compressibility of Granular Assemblage

APCOM’07 in conjunction with EPMESC XI, December 3-6, 2007, Kyoto, JAPAN

Compressibility of Granular Assemblage

Jing-Jing Dong*, Wai-Man Yan

Department of Civil and Environmental Engineering, University of Macau, Taipa, Macau SAR, China
e-mail: [email protected], [email protected]

Abstract In this paper, a series of numerical one-dimensional compression tests was performed by the
three-dimensional discrete element code PFC3D. Granular assemblages with various initial densities and
particle sizes are investigated. The simplified Hertz-Mindlin contact model is adopted to model the
particle-particle contact. This paper starts from revisiting the effect of contact model through a uniformly
packed particle assembly. The result demonstrates that typical compression curve with large
compressibility at high pressure can occur even though no particle rearrangement and breakage is
simulated. Granular assemblages comprise of randomly packed mono-size spherical particles at different
initial densities are investigated. As the compression stress increases, the compression lines curve and
merge into a unique line which indicates a unique packing/structure inside the particle assembly.
Unloading of the specimens demonstrates an irrecoverable strain due to particle rearrangement during the
loading stage. Compressibility of particle assemblies with different grain size distributions is also
investigated. It is found that specimen with wider particle size distribution exhibits higher compressibility
at a lower pressure. Furthermore, the specimen shows more permanent deformation upon unloading
which reveals more particle rearrangement has occurred during the loading stage.

Key words: 1-D compression, compressibility, discrete element method, grain size distribution

INTRODUCTION

Discrete element method has been recognised as a powerful tool to offer insights into the micro-mechanics
of granular materials so that the gulf between micromechanical properties and macroscopic behaviour may
be bridged. Compressibility is an important parameter to describe the stress-strain response of a soil during
a one-dimensional compression. For granular material under one-dimensional loading, current data shows
that the compressibility is relative low prior to particle breakage. Obviously, compressibility is controlled
in the particulate level and is governed by factors like particle shape, grain size distribution and
packing/structure. Discrete element method offers an insightful explanation in the microscopic level and it
has drawn attention from researchers all over the world ([1] – [6]). In this study, a series of
one-dimensional compression tests were simulated by the three dimensional discrete element program
PFC3D [7]. Spherical shaped particles were adopted in the analyses. Compressibility of particle
assemblages with different initial void ratios and various combinations of particle sizes were investigated.
The aim of this study is to gain a better understanding on the soil’s compressibility from the microscopic
point of view.

CONTACT MODELS: LINEAR CONTACT AND HERTZ CONTACT
Two contact models are integrated in the discrete element code PFC3D: linear contact and simplified
Hertz-Mindlin contact. The linear contact model is defined by the normal and shear stiffnesses kn and ks
(force/displacement) of the two contacting entities (ball-to-ball or ball-to-wall). The contact stiffness for
the linear contact model are computed assuming that the stiffness of the two contacting entities act in
series. For the linear contact model, the forces and relative displacements are linearly related by the
constant contact stiffness. In the simplified Hertz-Mindlin contact model (denoted as hertz model in this
paper), a nonlinear contact formulation is adopted based on an approximation of the theory of Mindlin and
Deresiewicz [8] as described in Cundall [9]. The model is defined by the shear modulus G, Poisson’s ratio
ν and density ρ of the two contacting balls. For the hertz contact model, the forces and relative

displacements are nonlinearly related by the non-constant contact stiffness. Details of the models should
be referred to the manual of the code [7].

UNIFORMLY PACKED ASSEMBLY

The response of a one-dimensional loading-unloading cycle on a uniformly packed particle assemblage is
presented in Fig. 1. The objectives of the simulations are two fold: (1) to demonstrate the influence of the
contact models and (2) to reveal the importance of particle rearrangement. It can be seen that the linear
model reveals a linear while the hertz model demonstrates a non-linear void ratio-stress as shown in Fig.
1(a). In both cases, loading and unloading follow the same path which indicates an elastic behaviour of the
assembly as expected. In fact, no particle rearrangement can be occurred in this uniformly packed
assemblage under such constrained loading condition (1-D compression). The same test results are
expressed in the logarithm stress scale in Fig. 1(b). Both models show typical one-dimensional
compression responses in a way that the curves bend at higher confining stress. In other words, the
compressibility increases significantly at high stress. However, the change in compressibility (stress > 10
MPa in Fig. 1(b)) is neither due to particle rearrangement nor particle breakage (no particle breakage is
allowed in the current simulation). The simulation shows that a typical compression curve can be obtained
even though particle rearrangement or particle crushing does not occur. In other words, deformation is
owing to elastic strain solely and particle breakage, which researchers always refer to, is not a necessary
condition for the change in compressibility at higher stress. Moreover, the term “yielding” should be used
with high caution since no “yielding” occurs in these two cases although the compressibility increases
significantly.

(a) (b)
0.95

Void ratio, e 0.90 Linear: kn = 1x104 kN/m Hertz
ks = 1x104 kN/m Linear
0.85 Hertz
Hertz: G = 29 GPa
0.80 ν = 0.15
ρ = 2.65 Mg/m3
0.75
0 Linear

100 200 300 400 500 0.01 0.1 1 10 100 1000

Vertical stress (MPa) Vertical stress (MPa)

Fig. 1 One-dimensional compression of a uniformly packed particle assemblage:
(a) stress in linear scale; (b) stress in logarithm scale

1-D COMPRESSION TEST ON RANDOMLY PACKED ASSEMBLY

As mentioned in the previous section, model parameters in the hertz contact model have their own physical
meaning (particle shear modulus, Poisson’s ratio and density). For this reason, the hertz contact model was
adopted in the following analyses. The simulated compression box consisted of six rigid walls: five fixed
and a movable top wall, and had an internal dimensions of 6 cm (length) × 3 cm (width) × 4 cm (depth).
The analyses in this study can be divided into 2 categories: (1) assemblages with mono-size spheres; (2)
assemblages of spherical particles with different sizes. In all cases, particles with coefficient of friction =
0.5 are generated randomly inside the box. Parameters of the simulation are summarized in Table 1. Case
1 – 3 show granular assemblages with various initial void ratios but the same particle size. On the other
hand, Case 4 – 6 show assemblages with different particle size distributions yet essentially the same initial
void ratio. Fig.2 shows the particle size distributions for all cases. The total number of particles adopted in
the analyses ranges from around 6,700 to 12,500.

Table 1. Summary of the parameters adopted in the numerical simulations

Basic properties

Sphere Wall

Friction coefficient 0.5 Friction coefficient 0.0
29
Shear modulus G (GPa) 0.15 Normal stiffness (kN/m) 1×104
Possion’s ratio ν 2.65
Density ρ (Mg/m3) Shear stiffness (kN/m) 1×104

Mono-size sphere

Case 1 Case 2 Case 3

Diameter d (cm) 0.22 0.22 0.22

Initial void ratio eo 0.900 0.872 0.845

Spheres with different sizes

Case 4 Case 5 Case 6

Diameter d (cm) 0.18 – 0.26 0.14 – 0.28 0.08 – 0.30

cu (d60/d10) 1.17 1.38 1.58

Spherical particles are generated randomly inside the box according to the target initial density. The
assembly is then stepped to equilibrium. One-dimensional compression was done by moving the top wall
downward (i.e., a strain control test). Void ratio of the particle assemblage is calculated globally based on
the volume of the box and the volume of total particles. Vertical stress is derived from the vertical reaction
force acting on the top wall.

100 Case 1-3 Case 4
90 Case 5
% passing 80
70
60 Case 6 1
50
40 0.1
30 Mean grain diameter (cm)
20
10
0

0.01

Fig. 2 Particle size distributions for different cases

RESULTS

1. Mono-size spheres

Fig. 3 shows the compressibility of randomly packed particle assemblages generated by mono-size spheres.
As shown in the figure, the initial void ratios of three assemblages are different, corresponding to Case 1
(loosest) to Case 3 (densest) respectively. In other words, their initial internal packing/structures are
different. As the vertical stress increases, the compression curves bend and merge into a unique one,
regardless of their differences in the initial void ratios. It indicates that an identical packing is obtained
when the confining stress becomes large enough. During unloading irreversible strain (plastic deformation)
can be clearly seen, which indicates the occurrence of particle rearrangement.

0.95

0.90 Case 1
Case 2
Case 3

Void ratio, e 0.85 irreversible

strain

0.80

0.75 reversible
strain

0.70

0.65 0.01 0.1 1 10 100
0.001

Vertical stress (MPa)

Fig. 3 1-D compression for mono-size spheres with different initial void ratios

0.90

0.85 Case 6 Case 5 Case 4

A

Void ratio, e 0.80 irreversible
0.75 strain

0.70 reversible strain

0.65 0.01 0.1 1 10 100
0.001

Vertical stress (MPa)

Fig. 4 1-D compression for assemblies with different grain size distributions

2. Spheres with different size distributions

Fig. 4 shows the behaviour of assemblies with different particle sizes during a 1-D compression. The
coefficient of uniformity cu increases from 1.17 in Case 4 to 1.58 in Case 6 (see also Table 1). However,
they have essentially the same initial density. Noticeable change in compressibility occurs at various stress
level for different cases. Compression curve of the particle assembly with the widest grain size distribution
(Case 6) bends at the lowest pressure (Point A in the figure). Moreover, it also exhibits the largest amount
of plastic deformation (irreversible strain). It may due to the reason that Case 6 consists of more small size
particles which may migrate into the voids between larger particles during the compression. Such particle
rearrangement gives more volumetric change at a lower confining stress. Upon unloading, it shows more
permanent deformation and comparatively smaller elastic strain.

CONCLUSION

A series of numerical one-dimensional compression tests was simulated by the three-dimensional discrete
element code PFC3D. The main objective of this study is to understand the effect of density and particle
size distribution on the assemblage’s compressibility. Spherical particles were used in the analyses.
Compressibility of a uniformly packed particle assembly is revisited through different contact models. The
result demonstrates that typical compression curve with large compressibility at high pressure can occur
even though no particle rearrangement and breakage is simulated. Besides, assemblies consisting of
randomly packed mono-size particles resemble a unique compression curve as the confining pressure
becomes large; regardless the differences in the initial void ratios. It indicates that a unique
packing/structure is obtained. Unloading of the assemblies reveals plastic deformation; which is due to the
particle rearrangement during the loading process. Particle assemblies having the same initial void ratio
but different grain size distributions show different compression curves. Specimen with wider size
distribution exhibits higher compressibility at a lower pressure. It may due to the migration of smaller
particles into the voids between larger particles; which increases the specimen’s compressibility.
Moreover, specimen with wider size distribution shows more permanent deformation.

Acknowledgements The writers wish to acknowledge the Fundo para o Desenvolvimento das Ciências e
da Tecnologia (FDCT), Macau SAR government (027/2006/A1) and the Research Committee, University
of Macau (RG071/05-06S/YWM/FST) for their financial assistance.

REFERENCES

[1] J. P. Bardet, J. Proubet, A numerical investigation of structure of persistent shear bands in granular
media, Géotechnique, 41(4), (1991), 599-613.

[2] T. T. Ng, Behavior of ellipsoids of two sizes, J. Geot. and Geoenv. Engrg., 130(10), (1994),
1077-1083.

[3] Q. Ni, W, Powrie, X. Zhang, R. Harkness, Effect of particle properties on soil behaviour: 3-D
numerical modelling of shear box tests, ASCE Geotechnical Special Publication No. 96, (2000), pp.
58-70.

[4] W. Powrie, Q. Ni, R. M. Harkness, X. Zhang, Numerical modelling of plane strain tests on sand using
a particulate approach, Géotechnique, 55(4), (2005), 297-306.

[5] C. Thornton, Numerical simulations of deviatoric shear deformation of granular media,
Géotechnique 50(1), (2000), 43-53.

[6] L. Cui, C. O’Sullivan, Exploring the macro- and micro-scale response of an idealised granular
material in the direct shear apparatus, Géotechnique, 56(7), (2006), 455-468.

[7] Itasca Consulting Group Inc., PFC3D ver. 3.1 User Manual. Itasca Consulting Group, Minnesota
(2000).

[8] R. D. Mindlin, H. Deresiewicz, Elastic spheres in contact under varying oblique forces, J. Appl.
Mech., 20, (1953), 327-344.

[9] P. A. Cundall, Computer simulations of dense sphere assemblies, in M. Satake, T. J. Jenkins eds.
Micromechanics of granular materials, Amsterdam: Elsevier Science Publishers B.V. (1988), pp.
113-123.