Sitharam Micromechanical Behav Granular Materials

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Micromechanical behavior of granular materials under monotonic

loading: Numerical simulation using discrete element method

Dr. Thallak Sitharam1

and Dr. Norikazu Shimizu2

1Associate Professor and

2Professor, Department of Civil Engineering

Yamaguchi University, Tokiwadai 2-16-1, Ube 755-8611, JAPAN

Email:[email protected] [email protected]

INTRODUCTION

Engineering behavior of soils is conveniently being expressed in terms of continuum

parameters such as stress and strain even though soil is essentially a particulate

system. There has been a great effort in the recent past towards establishing stress-

strain relations based on principles of continuum mechanics with the ultimate

objective of solving boundary value problems in geomechanics. The continuum

mechanics models have been largely extended to be suitable for pressure dependent

behavior of particulate systems such as soils. However these continuum models do notoffer complete physical insight into the behavior of granular materials. Due to its

inherent granularity, some features of sand behavior are difficult to understand or

model from continuum mechanics principles. An alternative approach that offers a

better understanding of granular materials is to treat the material as an assemblage of

particles interacting through contact forces. Significant attempts have been made in

the recent years in this direction to describe the response of granular materials from

micromechanical approach (Chang et al., 1997). Development of numerical tools such

as Discrete Element Method (DEM) by Cundall and Strack (1979), which can handle

interaction of particles, has made it possible to derive detailed microscopic

information and to study the evolution of various microparameters during loading. In

this paper, results of monotonic biaxial shear tests on uniformly graded 2-dimensional

assembly of 1000 discs under drained conditions are presented.

MICROPARAMETERS

It is now well recognized that shearing results in a reorientation of the fabric with an

anisotropic distribution of contacts and contact forces and that the applied stress is

transmitted by the formation of strong `contact chains' in the direction of the major

principle stress. Thus, the changes in the applied stresses can be related to the changes

in the internal fabric and force distributions in the medium. The internal parameters

that describe the state of the assembly are in general recognized to be: the number of

contacts or contact density (or alternatively average coordination number, i.e. averagenumber of contacts per particle), contact normals and contact vectors and their spatial

or directional distributions, contact forces - normal and tangential, and their

distributions, etc. Several researchers (Mehrabadi et al., 1993; Rothenburg, 1980)

have tried to relate the macroscopic quantities of stress and strain to the

microparameters analytically by averaging techniques based on statistical methods. In

the present work the behavior of granular materials under monotonic loading under

drained conditions will be examined in terms of these micro parameters.

NUMERICAL TESTING PROGRAMME

Numerical simulations of two-dimensional disc assemblies are carried out using the

modified DISC program (Sitharam, 1991). Program DISC is based on Discrete
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element method, which is a numerical technique for analyzing the behavior of

granular systems treating it as an assemblage of grains which can freely make or break

contacts with their neighbors. Assemblies of 1000 two-dimensional discs of 20

different sizes (varying between 15-35 units) with a log normal particle size

distribution are used for simulations. Each disc and contact have prescribed properties

of radius, density, normal and tangential contact stiffness and coefficient ofinterparticle friction. Numerical values for density, stiffness and damping coefficients

have been selected such that overlaps are small in relation to particle sizes that the

numerical process is stable. In these 2-d simulations, no attempt has been made to

relate the units to physical units. Values of these parameters are given in Table 1.

Suitable mass and contact damping are used to achieve conditions close to static

equilibrium. The disc particles are generated in a random manner in accordance with

pre-set particle size, gradation and packing criteria to represent the soil mass. Then the

sample is compacted isotropically and then the sample is sheared. During the shearing

process, each particle is free to move or rotate in response to the local stress

conditions and in accordance with inter-particle friction criteria. Initially, the assemblyis generated using a random number generator that places non-overlapping discs of

desired sizes corresponding to a desired distribution at random x-y locations within a

specified circular region. A circular boundary is preferred to avoid stress

concentrations at corners.

Table 1. Input parameters selected for numerical simulation

Properties Symbols Values Used

Normal Stiffness Kn 1.5 x 109

2.5 x 109

Shear Stiffness Ks 1.5 x 109

2.5 x 109

Damping Coefficients and 4.0 and 0.01

Density of Discs 2000

Critical Time Step t 0.45E-03

Coefficient of Friction 0.5

Cohesion c 0.0

Two types of samples, loose and dense, were used. Loose sample is created by

isotropically compacting the initially generated assembly to a specified stress level.

While compacting the specified coefficient of friction (=0.5) was assigned to allparticle contacts to generate a loose state of the assembly. However, for creating dense

samples, the assembly was compacted keeping the contact friction () value zero, and

subsequently equilibrated with the actual value of contact friction (=0.5). Thisequilibration amounted to some sort of unloading from a dense state (as the initial

coefficient of friction of zero resulted in very dense packing). The loose and dense

samples subjected to an isotropic stress of 2 x 106

stress units are shown in Figs. 1 and

2 respectively. The thickness of the lines indicates the magnitude of the contact force

in the system. Corresponding area void ratio of loose and dense samples was 0.266

and 0.19 respectively. Monotonic biaxial shear tests were performed on these samples

under drained conditions. Due to lack of space, undrained tests (constant volume

tests) performed on these samples are not reported here. Starting with the isotropically

stressed system, biaxial drained shear tests were performed by keeping the lateralpressure constant and a constant strain rate is applied in the vertical direction as in the


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conventional triaxial tests. These tests were carried out to sufficiently large shear

strains of 15- 20%.

RESULTS AND DISCUSSIONS

The results of monotonic, drained biaxial shear tests on both loose and dense samples

are presented in figure 3 [deviatoric stress= ( )2)( 31 ; deviaotoric strain= ( )31 ]. Thelaboratory triaxial experimental results of Been et al., (1991) showing a typical

drained behavior of loose and dense sands are presented for comparison in Fig 4. It is

very clear form the results that the loose disc assemblage shows volumetriccompression while the dense sample shows significant dilation. The loose assemblage

of discs shows monotonically increasing shear strength with strain, while the dense

sample shows a steep rise in strength initially up to the peak which then decrease for

further strains. The macro level responses under different conditions are in agreement

with the established trends in the experiments.

Fig 3. Monotonic biaxial shear test results

of loose and dense assemblages of discs

Fig 4. Typical drained monotonic of

behavior of loose and dense sands

(after Been et al., 1991)

Fig 1. Loose assembly of 1000 discs

at 2 x 106

isotropic boundary stressFig 2. Dense Assembly of 1000 discs

at 2 x 106

isotropic boundary stress


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The plots of the coordination number versus deviatoric strain for loose and dense

systems obtained during these numerical simulations are presented in figures 5 and 6

respectively. In case of loose sample, the coordination number is almost constant in

spite of volumetric compression. This indicates that there is loss of contacts due to

shearing even in this loose sample, which is compensated by the increase in contacts

due to volumetric compression. For the dense sample there is a steep drop in the

coordination number due to volumetric dilation. But we observed that there is a

significant increase in the value of average contact force in the dense system, which

results in an increasing mean pressure, in spite of steep drop in number of contacts.

Typical plots of the polar diagrams of contact normal normals (first column of

figs) and contact normal force (middle column of figs) and contact shear forcedistributions (last column of figs) at different stages of loading (at 0%, 2.5%, 5% and

10% deviatoric strains) are presented for loose and dense assembly in Figs. 7 and 8

respectively. A quantitative estimation of the evolution of these micro parameters is

necessary to finally describe the macroscopic behavior. The equations presented by

Rothenburg and Bathurst (1989) are used here for this purpose. It has been shown that

for plane assemblies of circular discs, the contact normal and force distributions can

be approximated with sufficient accuracy by Fourier series expressions of the form,

for representing the distribution of contact normals, contact normal forces and contact

shear forces, respectively:

( )[ ]oaE += 2cos1)( 21 (1)

( ) fnon aff += 2cos1 (2)

( ) ( )[ ]ttot aff = 2sin (3)

Where, of is the average contact normal force over all the contacts, tfo ,, are the

principal directions of the contact anisotropy, contact normal force, and contact shear

force anisotropies respectively. tn aaa ,, are constants, which fit the three Fourier

series, and reflect the extent of anisotropy in each case. It is well understood that a is

deviatoric invariant of a symmetric second order tensor describing the distributions of

contact normal orientations and o is the eigenvector of this tensor. Thisunderstanding leads to a simple analytical technique to calculate a and o from contact

Fig 5. Average Coordination number in

a looseFig 6. Average coordination

number in a dense sample


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orientation data (Rothenburg and Bathurst, 1989). The determination of oa , (or

fna , or tta , ) can be carried out using the relations such as:

( ) oadE

2cos2cos2

2

0

= and ( ) oadE

2sin2sin2

2

0

= . (4)

After obtaining these anisotropic parameters, the original distribution has been fittedwith these equations and the smooth curve in Figures 7 and 8 represents the

distribution obtained by these equations. These numerical experiments clearly shows

that these micro parameters are quantifiable even for loose and dense systems which

define the essential features of microstructures such as induced anisotropy in contact

orientations and contact forces. Such observations were confirmed even for undrained

tests (constant volume tests) on loose and dense assemblies, which are not reported

here due to lack of space. From here, one can easily obtain relationship between forces

and fabric to a macroscopic stress as shown by Rothenburg and Bathurst (1989).

Figures 7 and 8 show how anisotropy development is taking place in the assembly

with increase in deviatoric strain. Fig 7 shows for the loose sample, the anisotropy incontact normals, contact normal forces and shear forces increase monotonically with

strain in drained test. For dense sample also the anisotropy in contact orientation

increases monotonically (see fig 8), but the contact normal and shear force

anisotropies show an early mobilization and then decreases. The development of

anisotropy in contact orientation is also much faster in dense sample than in loose

sample. It can be seen that the magnitude of contact shear forces is very small when

compared to that of the contact normal forces. From drained and undrained tests on

loose and dense assemblies, we have observed that, these anisotropy coefficients have

limiting value at large strains for a given assembly however, the average normal

contact force increases/decreases with a corresponding increase/decrease in average

coordination number.

CONCLUSIONS

The trends of numerical results obtained compare quite satisfactorily with the

experimental results. The 2-D analysis presented here give a qualitative picture of the

micromechanical behavior of granular media in a monotonic drained test. Numerical

results of loose and dense systems indicate that the essential features of microstructure

can be quantified by anisotropy coefficients, using Fourier series functions.

REFERENCESBeen, K., Jefferies, M.G., and Hachey, J. (1991) The critical state of sands, Geotechnique, 41, No.3,

pp.365-381.

Chang, C.S., Anil Misra, Liang R.Y., (1997) Mechanics of deformation and flow of particulate

materials: Proceedings of the symposium sponsored by engineering mechanics division of ASCE,

(Editors).

Cundall, P.A., and Strack, O.D.L. (1979), A discrete numerical model for granular assemblies,

Geotechnique 29, No.1, 47-65.

Mehrabadi, M.M., Loret, B., Nemat-Naseer, S., (1993) Incremental constitutive relations for granular

materials based on micromechanics, Proceedings of the Royal Society of London, 441, 443-463.

Rothenburg, L. (1980), Micromechanics of granular materials, A Ph.D. thesis submitted to University

of Carleton, Ottawa, Canada.

Rothenburg, L., and Bathurst, R. J. (1989), Analytical study of induced anisotropy in idealised granular

materials, Geotechnique, 39, No. 4, pp. 601-614.

Sitharam G. T. (1991), Numerical simulation of hydraulic fracturing in granular media, Ph.D. thesis,University of Waterloo, Waterloo, Ontario, Canada, 303 pp.


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Fig 7. The directional distributions of contact normals, average contact normal

forces and average contact shear forces during a drained test on loose

assemblage of discs at a) 0% b) 2.5% c) 5% and d) 10% deviatoric strains

Fig 8. The directional distributions of contact normals, average contact normal forces and

average contact shear forces during a drained test on dense assemblage of discs at a) 0% b)

2.5% c) 5% and d) 10% deviatoric strains

Sitharam Micromechanical Behav Granular Materials

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