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Micro-scalemodelingoftheinelasticresponseof
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ARMA 15-575
Micro-scale modeling of the inelastic response of a
granular sandstone
Esna Ashari, S.
Northwestern University, Evanston, IL, USA
Buscarnera, G. and Cusatis, G.
Northwestern University, Evanston, IL, USA
Copyright 2015 ARMA, American Rock Mechanics Association
th
This paper was prepared for presentation at the 49 US Rock Mechanics / Geomechanics Symposium held in San Francisco, CA, USA, 28 June1 July 2015.
This paper was selected for presentation at the symposium by an ARMA Technical Program Committee based on a technical and critical review of
the paper by a minimum of two technical reviewers. The material, as presented, does not necessarily reflect any position of ARMA, its officers, or
members. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of ARMA
is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 200 words; illustrations may not be copied. The
abstract must contain conspicuous acknowledgement of where and by whom the paper was presented.
ABSTRACT: This paper discusses a new computational strategy for the analysis of inelastic processes in granular rocks subjected
to varying levels of confinement. The purpose is to provide a flexible and efficient tool for the analysis of failure processes in
geomechanical settings. The proposed model is formulated in the framework of Lattice Discrete Particle Models (LDPM), which is
here calibrated to capture the behavior of a high-porosity rock widely tested in the literature: Bleurswiller sandstone. The procedure
required to generate a realistic granular microstructure is described. Then, the micromechanical parameters controlling the fracture
response at low confinements, as well as the plastic behavior at high pressures have been calibrated. It is shown that the LDPM
model allows one to explore the effect of fine-scale heterogeneity on the inelastic response of rock cores, achieving a satisfactory
quantitative performance across a wide range of stress conditions. The results suggest that LDPM analyses represent a versatile tool
for the characterization and simulation of the mechanical response of granular rocks, which can assist the interpretation of complex
deformation/failure patterns, as well as the development of continuum models capturing the effect of micro-scale heterogeneity.
1. INTRODUCTION
An accurate knowledge of the engineering properties of
rocks is crucial for a variety of geomechanical problems,
ranging from wellbore stability, to failure in rock slopes,
underground excavations, and crustal faults [1]. While
strength and deformation properties are usually obtained
from a limited number of in situ and/or laboratory tests,
their determination is invariably affected by considerable
heterogeneities [2]. Such lack of homogeneity impacts
engineering conclusions at all length scales and requires
appropriate theoretical and computational tools.
Advanced numerical modeling represents a useful tool to
explore how mechanical processes interact across length
scales. Considerable advances in this area have based on
Finite Element computations, where heterogeneities can
be incorporated both at sample and site scales [3, 4].
Nevertheless, to capture realistically the path-dependent
response of geomaterials, continuum formulations tend
to be characterized by a large number of parameters. If
such constants lack clear connections with measurable
attributes (e.g., grain size and sorting), their calibration
becomes poorly constrained. Furthermore, the tendency
of rock samples to undergo strain localization processes
further prevents the validation and/or implementation of
continuum models, requiring a direct link between strain
localization and microstructural attributes [5].
The Discrete Element Method (DEM) [6], according to
which rocks are represented as assemblages of particles
interacting through cohesive-frictional contacts, provides
a useful approach to investigate the interplay between
continuum-scale behavior and microstructural attributes.
Through such class of methods, it is indeed possible to
retrieve naturally the macroscopic response from the
interaction among spatially distributed fine-scale units,
whose micro-scale interaction laws are defined to match
the macro-scale properties observed in experiments [7].
Limitations in the predictive performance of DEMs for
rocks are common, however, when spherical particles
are used to approximate the microstructure [7, 8]. An
example is the tendency to predict excessively low ratios
between compressive and tensile strength, thus missing
one of the major properties of brittle rocks. While this
limitation can be mitigated by using irregular shaped
particles to improve interlocking [7], it restricts the use
of standard DEM formulations only to poorly cemented
rocks, thus preventing predictive analyses for technical
problems for which the expected stress conditions tend
to encompass both tension and compression.
This paper is aimed to tackle some of these modeling
challenges typical of natural rocks. For this purpose, we
propose to use a discrete method designed specifically to
address the mechanics of pressure-sensitive solids. This
approach is referred to as Lattice Discrete Particle Model
(LDPM), successfully developed by Cusatis and coworkers [9, 10] for quasi-brittle granular materials such
as concrete. In a 3D context, LDPM simulates
interactions among coarse aggregates through a system
of polyhedral particles. Each particle mimics a coarse
aggregate piece connected with its surrounding mortar
and is connected to its neighbors via lattice struts. In this
way, LDPM is able to simulate a realistic grain size
distribution, as well as the role of small scale
heterogeneity on fracture and strain localization. These
particular features offer various advantages compared to
other methods for quasi-brittle solids. In particular,
hereafter we aim to illustrate the benefits that LDPM
offers for the simulation of various macroscopic
processes typical of granular rocks.
2. TECHNICAL BACKGROUND
The Lattice Discrete Particle Model (LDPM) is a mesoscale discrete approach that simulates the mechanical
interaction of coarse aggregates. The mesostructure of a
granular material is constructed through the following
steps. The coarse aggregates, whose shape is assumed to
be spherical, are introduced into the specimen’s volume
by a try-and-reject random procedure. A threedimensional domain tessellation, based on a Delaunay
tetrahedralization of the generated aggregate centers,
creates a system of polyhedral cells (Fig. 1) interacting
through triangular facets and a lattice system composed
by the line segments connecting the particle centers.
where ℓ indicates the interparticle distance, while n , l ,
and m , are unit vectors that define a local reference
system attached to each facet.
The governing equations of the LDPM framework are
completed by the equilibrium equations of each particle
and the constitutive laws controlling their interactions.
Vectorial constitutive laws are indeed imposed at the
centroid of each particle facet, where the mechanical
interaction between the particles is characterized by both
normal and shear stresses. The meso-scale constitutive
behavior is assumed to involve softening for both pure
tension and shear-tension, while it is characterized by
plastic hardening for both pure compression and shearcompression. In the elastic regime, the normal and shear
stresses are proportional to the corresponding strains:
σ N = E N eN ; σ L = α E N eL ; σ M = α E N eM
(2)
where E N is the effective normal modulus, and α is the
shear-normal coupling parameter.
In the inelastic regime, a nonlinear constitutive equation
is used to describe meso-scale failure phenomena such
as fracturing and shearing, pore collapse and frictional
behavior. While here a brief description of the model in
the nonlinear range is provided, the detailed description
of the constitutive relations can be found in [9].
2.1. Fracturing behavior
For tensile loading ( eN > 0 ), the fracturing behavior is
formulated through an effective strain ( e ),
e = eN2 + α ( eL2 + eM2 )
(3)
and effective stress ( σ ),
σ = σ N2 + α (σ L2 + σ M2 )
(4)
which define the normal and shear stresses:
σ N = eN
σ
σ
σ
; σ L = α eL ; σ M = α eM
e
e
e
(5)
The strain-dependent limiting boundary for this type of
behavior is formulated as:
Fig. 1. LDPM particle and cell.
In LDPM, the rigid body motion of each particle is used
to describe the deformation of the lattice/particle system.
The displacement jump, [uC ] , at the centroid of each
facet is used to define the following strain components
eN =
nT [uC ]
l T [uC ]
mT [uC ]
; eL =
; eM =
ℓ
ℓ
ℓ
"
emax − e0 (ω ) %
'
σ bt ( e, ω ) = σ 0 (ω ) exp $−H 0 (ω )
σ 0 (ω ) '&
$#
where ω is the coupling variable that represents the
degree of interaction between shear and normal loading,
emax is the maximum effective strain attained during the
loading history,
(1)
(6)
H 0 (ω )
is the post-peak slope
(softening modulus) and σ 0 (ω ) is the strength limit for
the effective stress and e0 (ω ) is the corresponding
strain.
2.2. Pore collapse and material compaction
For a constant deviatoric-to-volumetric strain ratio,
rDV = eD eV , the compressive boundary stress ( σ bc ), is
assumed to have an initial linear evolution (to model
pore collapse and yielding) followed by an exponential
increase (to model compaction and rehardening).
'
σ c0
−eV ≤ 0
)
)
σ bc ( eD , eV ) = (
σ c0 + −eV − ec0 H c ( rDV )
0 ≤ −eV ≤ ec1
)
#
%
)* σ c1 ( rDV ) exp $(−eV − ec1 ) H c ( rDV ) σ c1 ( rDV )& otherwise
(7)
where σ c0 and ec0 are the meso-scale yielding
compressive stress and the compaction strain at the onset
of pore collapse, respectively, H c ( rDV ) is the initial
grains in a sample of granular Bleurswiller sandstone to
the aggregates within concrete. To account in the LDPM
formulation for such a closely packed grain distribution,
the size of each cell built around a spherical aggregate
(Fig. 1) has been assumed to match the size of the grains
within the actual rock. Hence, the size of the spherical
particles required to create the spatial lattice (i.e., the
building blocks of the microstructure) has been reduced
compared to the measured size of the rock grains, with
the purpose to generate a size distribution of polyhedral
cells as close as possible to the reported grain size
distribution of the considered rock. In other words, while
the measured grain size distribution is still used as the
building block of the microstructure and is reflected in
model by the cell size distribution, the mechanical effect
of the cement bridges is just incorporated indirectly in
the micro-scale constitutive relations.
hardening modulus, ec1 the compaction strain at which
rehardening begins and σ c1 ( rDV ) the corresponding
stress.
2.3. Frictional behavior
The incremental shear stresses are computed as
σ! L = α E N ( e!L − e!Lp ); σ! M = α E N ( e!M − e!Mp)
(8)
where e!Lp = λ! ∂ϕ ∂σ L ; e!Mp = λ! ∂ϕ ∂σ M , and λ is the
plastic multiplier. The plastic potential is defined as
ϕ = σ L2 + σ M2 − σ bs (σ N ) ,
where
the
nonlinear
frictional law for the shear strength is assumed to be
σ bs = σ s + (µ 0 − µ∞ ) σ N 0 − µ∞σ N − (µ 0 − µ∞ ) σ N
× exp (σ N σ N 0 )
(9)
where σ s is the cohesion, µ 0 and µ∞ are the initial and
final internal friction coefficients and σ N 0 is the
normal stress at which the internal friction coefficient
transitions from µ 0 to µ∞ .
Fig. 2. Microstructure of Bleurswiller sandstone (Ref. [11]) on
the left and concrete aggregates on the right.
A second phenomenon that had to be taken into account
was the effect of fissures on the initially nonlinear elastic
response upon hydrostatic compression paths (see Fig.
5). This nonlinear portion of the compression response is
often neglected, being approximated either in the form
of non-linear elastic relations, or by linear elastic laws
valid only after the stage of defect closure is complete.
To accommodate this feature in the LDPM, the linear
elastic law (Eq. 2) has been updated in the form of a
bilinear relation, characterized by a factor β mimicking
the non-linearity due to the closure of the initial defects
(Eq. 10). By using this modification, LDPM becomes
capable of modeling the transition from the fissureclosure stage to the linear elastic stage.
(10)
3. ADAPTATION TO GRANULAR ROCKS
σ N = β E N eN ; σ L = βα E N eL ; σ M = βα E N eM
This section discusses a strategy to adapt the LDPM to
the case of granular rocks.
4. CALIBRATION AND VALIDATION
As mentioned before, the LDPM approximates concrete
aggregates as spherical particles surrounded by mortar,
and uses them as building blocks of the microstructure.
In the case of granular rocks, the isolation of cement
bridges and grains is not straightforward, as the grains
tend to closely packed and in direct contact with each
other. Figure 2 illustrates such difference by comparing
In this section, the LDPM is calibrated and validated for
Bleurswiller sandstone. The experimental data used for
calibration and validation purposes derive from previous
studies on this rock reported in [11]. First, data about the
grain size distribution was required to characterize the
microstructure. In addition, information about fracture
properties, hydrostatic response and triaxial compression
behavior (representative of both low and high pressure
confinement) were required.
14
4.1. Grain size distribution
12
10
Load (N)
Based on [11], the grain diameter ranges from 160 µm to
300 µm, with a mean value of 220 µm. By using the
procedure discussed in Section 3, a cell size distribution
with similar features has been modeled in the LDPM
(Fig. 3), reproducing a minimum diameter of 125 µm, a
maximum diameter of 300 µm, and a mean diameter of
200 µm. The ability to incorporate a gsd similar to that
of Bleurswiller sandstone is a major advantage, as it will
enable the simulation of a realistic spatial heterogeneity,
and hence a realistic simulation of grain-scale dissipative
processes.
8
6
4
2
0
100
0
1x10
-5
2x10
-5
3x10
-5
4x10
-5
5x10
-5
6x10
-5
7x10
-5
8x10
-5
Displacement (m)
Fig. 4. Load-displacement response from a direct tension test
on Bleurswiller sandstone simulated with LDPM.
250
60
200
40
p (MPa)
Percent finer
80
20
0
100
150
200
250
300
150
LDPM
Experiment
100
350
Grain size (micron)
Fig. 3. Grain size distribution of Bleurswiller sandstone used
for the LDPM analyses.
4.2. Fracture test
One of the input parameters of LDPM is the critical
fracture energy that could be calculated from a fracture
test. The critical fracture energy for sandstones varies
between 15 to 54 J/m2 [12]. For Bleurswiller sandstone,
a critical fracture energy of 30 J/m2 has been used to
calibrate the tensile strength in a direct tension test (Fig.
4).
4.3. Hydrostatic test
The response measured from hydrostatic test is here used
to calibrate the parameters of LDPM that control the
compression response, such as the normal modulus, α ,
the factor controlling the crack closure, β , the
compressive stress at yielding The simulated LDPM
response for a hydrostatic test on Bleurswiller sandstone
is reported in Fig. 5, together with the experimental data.
A good agreement between data and computations is
readily apparent both for the initial stage of defect
closure and the post-yielding response.
50
0
0
0.02
0.04
0.06
0.08
0.1
Porosity reduction
Fig. 5. Comparisong between experimental data (after [11])
and LDPM simulation of hydrostatic compression.
4.4. Triaxial tests at different confining pressures
To calibrate the model parameters controlling shear
behavior, two triaxial tests at different confining
pressures have been used, namely 10 MPa (to mimic the
brittle response typical of low confinements) and 100
MPa (to mimic the response at high pressures). Figure 6
illustrates the response in terms of effective mean
pressure (p) versus volumetric strain, while Figure 7
shows the differential stress (q) as a function of the axial
strain.
The calibration of the remaining model constants on the
basis of these two triaxial compression tests completes
the calibration process, and provides the opportunity to
use LDPM simulations to predict the response emerging
from other loading paths.
100
120
80
p (MPa)
100
p (MPa)
80
60
40
60
LDPM-10
Experiment-10
LDPM-100
Experiment-100
40
LDPM-40
Experiment-40
LDPM-60
Experiment-60
LDPM-80
Experiment-80
20
20
0
0
0.01
0.02
0.03
0.04
0.05
Volumetric strain
0
0
0.01
0.02
0.03
0.04
0.05
Volumetric strain
Fig. 6. p vs volumetric strain for triaxial tests with 10 and 100
MPa confinement pressures.
Fig. 8. Comparison between measured volumetric strains and
LDPM predictions for triaxial compression tests at varying
confining pressures (data after [11]).
100
100
q (MPa)
80
q (MPa)
80
LDPM-10
Experiment-10
LDPM-100
Experiment-100
60
60
40
LDPM-40
Experiment-40
LDPM-60
Experiment-60
LDPM-80
Experiment-80
40
20
20
0
0
0.005
0.01
0.015
0.02
0.025
0.03
Axial strain
0
0
0.005
0.01
0.015
0.02
0.025
0.03
Axial strain
Fig. 7. q vs axial strain for triaxial tests with 10 and 100 MPa
confinement pressures.
4.5. Predication of the response for triaxial tests
In this section, the calibrated LDPM is used to predict
the response of compression tests performed at varying
confinement pressures, ranging from 2 to 80 MPa. The
corresponding predictions are plotted in Figures 8 and 9.
It is readily apparent a remarkable agreement between
LDPM computations and experimental data, with LDPM
capturing with a good level of accuracy the transition
from a brittle/dilative response at low confinements to a
ductile/compactive response at high pressures.
Fig. 9. Comparison between measured deviatoric stress-strain
response and LDPM predictions for triaxial compression tests
at varying confining pressures (data after [11]).
5. CONCLUSION
Granular rocks exhibit pressure-dependent properties, as
well as a broad range of strain-localization modes. Such
materials are in fact characterized by various types of
micro-scale heterogeneity, which generate macroscopic
patterns that can be traced back to processes such as
crack initiation; crack propagation; and interaction
between fractured and unfractured material. Advanced
multi-scale computations are thus required to simulate
such patterns and correlate them with basic micro-scale
processes. This paper has shown that LDPM is one of
the frameworks that is able to fulfill such objectives for
the important case of granular rocks. This feature has
been discussed by presenting a strategy to incorporate
into model computations grain-scale rock heterogeneity,
i.e. the scale at which microscopic inelastic processes
take place. A particular granular rock has been selected
for model illustration purposes, Bleurswiller sandstone,
thus benefiting from the large availability of data about
its mechanical response. It has been shown that by
incorporating specific features, such as the crack closure
upon compression and the development of pore collapse
upon high-pressure compression, it has been possible to
capture a variety of macroscopic processes, such as the
inelastic hydrostatic compression of rock samples, the
brittle fracture upon tension, and the transition from
brittle/dilative response to ductile/compactive behavior.
Most notably, the ability to predict such wide range of
responses has been based only on a limited set of data,
thus indicating that LDPM represent a versatile tool for a
variety of geomechanical modeling applications, ranging
from the intetpretation of multi-scale experiments, to the
formulation of continuum models and the assessment of
their predictive capabilities.
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