Computational Materials Science 29 (2004) 494–498

www.elsevier.com/locate/commatsci

Micromechanical modelling of oval particulates subjectedto bi-axial compression

S.J. Antony a,*, R.O. Momoh a, M.R. Kuhn b

a Department of Chemical Engineering, School of Process, Environmental and Materials Engineering, University of Leeds,

Houldsworth Bldg, Room 2.42, Leeds LS2 9JT, UKb Department of Civil Engineering, University of Portland, Portland OR79203, USA

Received 22 October 2003; accepted 8 December 2003

Abstract

One of the questions that still remain unanswered among researchers dealing with granular materials is how far the

particle shape affects the micro-macroscopic features of granular assemblies under mechanical loading. The latest

advances made with particle instrumentation allow us to capture realistic particle shapes and size distribution of

powders to a fair degree of accuracy at different length scales. Industrial applications often require information on the

micromechanical behaviour of granular assemblies having different particle shapes and varying surface characteristics,

which still remains largely unanswered. Traditionally, simulations based on discrete element method (DEM) idealise the

shape of individual particles as either circular or spherical. In the present investigation, we analyse the influence of

particle shape on the shear deformation characteristics of two dimensional granular assemblies using DEM. We pre-

pared the assemblies having nearly an identical initial packing fraction (dense), but with different basic shapes of the

individual particles: (a) oval and (b) circular for comparison purposes. The granular assemblies were subjected to bi-

axial compression test. We present the evolution of macroscopic strength parameters and microscopic structural/

topological parameters during mechanical loading. We show that the micromechanical properties of granular systems

are significantly influenced by the shape of the individual particles constituting the granular assemblies.

� 2004 Elsevier B.V. All rights reserved.

Keywords: Micromechanics; Shape effects; Micro-macroscopic properties; Particulate materials

1. Introduction

Information on the shear deformation behav-iour of granular materials is required in various

engineering applications, such as in the geotech-

* Corresponding author. Tel.: +44-1132332409; fax: +44-

1132332405.

E-mail address: s.j.antony@leeds.ac.uk (S.J. Antony).

0927-0256/$ - see front matter � 2004 Elsevier B.V. All rights reserv

doi:10.1016/j.commatsci.2003.12.007

nical, pharmaceutical, detergent, metals, ceramic

and food industries. It is generally recognised that

the individual properties of the constituent parti-cles strongly influence the macroscopic character-

istics of a particulate medium. Investigations on

the fundamental micromechanical and physical

behaviour of particulate assemblies in terms single

particle properties are of current interest (for

example, [1–9]). In this study, we investigate the

micro- and macroscopic characteristics of oval

ed.

S.J. Antony et al. / Computational Materials Science 29 (2004) 494–498 495

assemblies (e.g. representing alumina particles)

subjected to bi-axial compression testing.

Fig. 2. Histogram of particle size distribution for the assem-

blies.

2. Simulations

The simulations were carried out using DEM,

which was originally developed by Cundall and

Strack [10]. The advantage of using DEM to study

granular materials is its ability to give more

information about what happens inside the sys-

tem. The method models the interaction between

contiguous particles as a dynamic process and thetime evolution of the particles is advanced using an

explicit finite difference scheme. A simple force

mechanism was employed between contacting

particles. Linear normal and tangential contact

springs were assigned equal stiffness, and slipping

between particles would occur whenever the con-

tact friction coefficient of 0.5 was attained. More

details on the numerical description of the basicshapes of the particles studied here can be found

elsewhere [11,12].

Two assemblies have nearly identical initial

packing fraction, but with two different basic

shapes of the individual particles viz., circular and

oblate (with h ¼ 30�) (Fig. 1). The assemblies eachcontained 4096 particles with dense packings (co-

ordination number 3.8 ± 0.005 and solid fraction0.82± 0.005). The size distribution of the particles

used in the assemblies is presented in Fig. 2, which

Fig. 1. Illustrative diagrams of an oval shape particle.

represents size by the average values of height and

width of the particles. The particle assemblies were

initially random, isotropic and homogeneous and

the initial indentations were less than 0.02% of

D50. The assemblies were compacted from an ini-

tial sparse state by artificially removing friction

between particles and then isotropically reducingthe area until the desired solid fraction was at-

tained. At the end of isotropic compression, the

microstructure of the samples was isotropic. Dur-

ing the bi-axial compression simulations (two

dimensional), the height of the assembly was re-

duced at a constant rate (along the 2–2 direction),

while maintaining constant horizontal stress r11.The vertical strain was advanced in small incre-ments of De22 ¼ 1:0� 10�6, and several relaxationsteps were performed within each increment. These

measures minimized the transient inertial effects

that would have otherwise biased the results of a

presumed quasi-static loading.

2.1. Results

2.1.1. Macroscopic evolution of shear stress and

void ratio

Fig. 3 shows initial particle arrangement (before

shearing) for both the circular and oval particulate

systems. Fig. 4 shows the variation of normalised

shear stress ratio q=p (q ¼ r2 � r1, p ¼ ðr2þ

Fig. 3. Typical particle arrangement for (a) circular and (b) oval systems subjected to identical compressive strain.

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

0.00 0.02 0.04 0.06 0.08 0.10

Compressive strain, - ε22

Sh

ear

stre

ss r

atio

, q/p

Circular

Oval

Fig. 4. Normalised shear stress ratio during bi-axial compres-

sion.

CricularOval

0.150.160.170.180.190.2

0.210.220.230.24

0 0.02 0.04 0.06 0.08 0.1Compressive Strain,-ε22

Eff

ecti

ve v

oid

rat

io

Fig. 5. Variation of effective void ratio during compression.

496 S.J. Antony et al. / Computational Materials Science 29 (2004) 494–498

r1Þ=2) during the bi-axial compression of the

assemblies. The circular particulate assembly at-

tained the maximum value of q=p at an early stageof compression (with a fairly well defined peak)and then gradually decreased as the compression

progressed. Although the maximum value of q=pobtained for the oval system is fairly identical to

the circular system, it is obtained at a later stage of

compression than the circular system. The dense

oval system tends to attain an initial peak during

the early stages of bi-axial compression. At the

steady state (compressive strain greater than 0.05),the shear stress ratio for the oval system is higher

than the circular particulate assembly.

Fig. 5 shows the variation of effective void ratio

of the assemblies (the solid area includes only the

area of load-bearing particles) during compres-

sion. The effective void ratio of the assemblies in-

creases with decreasing slope as the compression

progresses. It is also evident that the circular par-

ticle assembly dilates more than the oval assembly.

2.1.2. Effective co-ordination number and valance

Fig. 6 shows the evolution of effective co-ordi-

nation number, which is considered as a measure of

heterogeneity of the assembly during compression.

The effective co-ordination number represents the

average numbers of contacts per load-bearing

particle. The effective co-ordination number de-creases as the assembly is compressed. The effective

co-ordination number attains a uniform value for

compressive strain e22 greater than ca. 0.1. Thesteady state value of the effective co-ordination

number for the oval assembly is higher than the

circular assembly.

Fig. 7 shows the variation of average valance of

the assemblies during compression. Valence is theaverage number of contacts (edges) per contacting

void cells (faces), the average valence used in this

3

3.2

3.4

3.6

3.8

4

4.2

0 0.02 0.04 0.06 0.08 0.1

Compressive strain, - ε22

Effe

ctiv

e co

ordi

natio

n nu

mbe

r

Oval

Circular

Fig. 6. Variation of effective co-ordination number during

compression.

3.5

4

4.5

5

5.5

6

0 0.02 0.04 0.06 0.08 0.1

Compressive strain, - ε22

Ave

rag

e va

len

ce

circularoval

Fig. 7. Variation of average valance during compression.

S.J. Antony et al. / Computational Materials Science 29 (2004) 494–498 497

case includes only load-bearing contacts. It can be

seen that the average valence of the void cells in-

creases as the compaction progressed, with circular

assembly having a higher average valence than the

oval particulate assembly. The reduction in effec-

tive co-ordination number during compression

resulted an increase in the average valance of the

void cells throughout the compression.

2.1.3. Microtopology

Fig. 8 shows the topological distribution of

particle arrangements corresponding to Fig. 3. The

topological association of particles are presented

here as a planar graph. The assembly domain has

been partitioned into numerous polygonal sub-

domains or void cells [13]. The corners (vertices) ofeach polygon are the centres of the particles and

Fig. 8. Particle graphs (topology) showing polygonal microdomain

compressive strain.

the sides (edges) are the branch vectors between

the particle centres. However, the resulting particle

graphs presented in Fig. 8 include only those

particles that are in contact with neighbouring

particles and that participate in the load-bearing

framework of the assembly. The non-participating

particles (island, peninsula and pendant particles)are not represented in these plots.

2.1.4. Observations on the microfabrics of force

networks

Fig. 9 shows the fabric measures of the assem-

blies during compression. The distribution of

contact orientations is characterised by a �fabrictensor� /ij, suggested by Satake [14] as

s for (a) circular and (b) oval systems subjected to identical

1.00

1.20

1.40

1.60

1.80

2.00

0 0.02 0.04 0.06 0.08 0.1

Compressive Strain, -ε22

Fab

ric

mea

sure

Circular

Oval

Fig. 9. Variation of fabric tensor of strong contacts during

compression.

498 S.J. Antony et al. / Computational Materials Science 29 (2004) 494–498

/ij ¼ hninji ¼1

M

XM

1

ninj; ð1Þ

where M is the number of contacts in the repre-

sentative volume element and the ni are the com-ponents of the unit normal vector at a contact

between two particles. In the current study, werestrict Eq. (1) to a subset of the M contacts: the

set of all contacts that had a greater than average

normal contact force (strong contacts). The fabric

measure plotted in Fig. 9 correspond to square

root of the value of (/22=/11). The initial isotropicfabric, which is 1, becomes greatly anisotropicand attains a value of about 1.5 for circular system

and 1.6 for oval system. A comparison betweenFigs. 4 and 9 show a strong correlation between

the macroscopic shear stress (q=p) distribution andthe internal distribution of fabric ratio, i.e., the

square root of (/22=/11), contributed by the strongcontacts in the assemblies during compression. We

find that, the relation between the shear stress ratio

and the fabric network of strong contacts satisfies

the equation ðq=pÞ ð1=2Þpð/22=/11Þ.

3. Conclusion

The influence of particle shape on the shear

deformation characteristics of circular and oval

particulate systems having nearly identical solid

fraction (dense) is investigated. The particulatesystems were subjected to bi-axial compression

test. The maximum value of shear stress ratio q=pfor the circular and oval particulate systems is

fairly identical. However, the nature of the varia-

tion of shear stress ratio for both the assemblies

during compression is significantly different. Dur-ing compression, the circular assembly tends to

dilate more than the oval assembly. The topolog-

ical and structural features of particulate systems

are significantly influenced by the shape of the

constituting particles considered in this study.

Investigations are currently underway to probe the

links between the structural orientation of the

contact networks to the macroscopic strengthcharacteristics of the particulate assemblies during

compression, and the effect of dimensionality will

also be reported in the future.

Acknowledgements

S.J.A. gratefully acknowledges the partial sup-port provided by Royal Society, London (Grant

Ref. 23913).

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