-
7/29/2019 Sitharam Micromechanical Behav Granular Materials
1/6
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
mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected] -
7/29/2019 Sitharam Micromechanical Behav Granular Materials
2/6
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
-
7/29/2019 Sitharam Micromechanical Behav Granular Materials
3/6
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
-
7/29/2019 Sitharam Micromechanical Behav Granular Materials
4/6
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
-
7/29/2019 Sitharam Micromechanical Behav Granular Materials
5/6
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.
-
7/29/2019 Sitharam Micromechanical Behav Granular Materials
6/6
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