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ACCEPTED TO THE INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS
Experimental Study of Compaction Localization in Carbonate
Rock and Constitutive Modeling of Mechanical Anisotropy
Ghassan Shahin1,2 | Athanasios Papazoglou3,4 | Ferdinando Marinelli5,6 | Gioacchino
Viggiani3| Giuseppe Buscarnera*1
1McCormick School of Engineering,
Northwestern University, IL, USA
2(now) Hopkins Extreme Materials Institute,
Johns Hopkins University, MD, USA
3Univ. Grenoble Alpes, CNRS, Grenoble
INP, 3SR, Grenoble, France
4(now) ESRF - The European Synchrotron
Radiation Facility, Grenoble, France
5Plaxis bv, Bentley Systems company,
Delft, The Netherlands
6(now) Civil Engineering Department,
University of Naples, Naples, Italy
Correspondence
*Giuseppe Buscarnera: Email:
g-buscarnera@northwestern.edu
Present Address
2145 Sheridan Rd, Evanston, IL 60208
Summary
Sedimentary rocks are inherently anisotropic and prone to strain localization. While
the influence of rock anisotropy on the brittle/dilative regime has been studied exten-
sively, its influence on the ductile/compactive regime is much less explored. This
paper discusses the anisotropic behavior of a high-porosity carbonate rock from cen-
tral Europe (the Maastricht Tuffeau). A set of triaxial tests with concurrent x-ray
tomography has been performed at different confining pressures. The anisotropic
characteristics of this rock have been investigated by testing samples cored at differ-
ent inclinations of the bedding, thus revealing non-negligible effects of the coring
direction on yielding and compaction behavior. Specifically, samples cored per-
pendicular to bedding display higher strength and longer stages of post-yielding
deformation before manifesting re-hardening. Despite such alterations of the inelastic
response, Digital Image Correlation has revealed that the strain localization mode is
independent of the coring direction, thus being primarily affected by the confinement
level. To capture the observed interaction between material anisotropy and com-
paction behavior at the continuum-scale, an elasto-plastic constitutive law has been
proposed. For this purpose, a set of tensorial bases has been introduced to replicate
how the oriented rock fabric modulates the yielding and plastic flow characteristics
of the material. The analyses show that the impact of the coring direction on yield
function and plastic flow rule is fundamentally different, thus requiring the use of
distinct projection strategies (a strategy here defined heterotopic mapping). The per-
formance of the model, studied through parametric analyses and by calibrating the
experimental results, illustrates the improved capability of the proposed constitutive
approach when applied to strongly anisotropic porous rocks.
KEYWORDS:
Rock Anisotropy, Compaction Band, X-ray Tomography, Constitutive Modeling
1 INTRODUCTION
Mechanical anisotropy is a key feature of sedimentary rocks that manifests through detectable variations in the deforma-
tion responses1,2,3,4,5,6,7,8,9,10 and strain localization characteristics11,6,12,13,14 depending on loading direction. Examples of such
2Shahin ET AL
effects are found in the Aztec Sandstone formation in Nevada, USA, where the depositional architecture altered the orientation
of deformation bands, resulting in cross-bedding, low-angle, and bed-parallel compaction bands15,16. Laboratory measurements
suggest strong connections between microstructural heterogeneities resulting from bedding and material foliation11,17,18 and
anisotropic macroscopic mechanical properties6,19,20.
From a modeling standpoint, the influence of anisotropic fabrics can be captured through additional state variables, usually
expressed as second-order tensors21,22,23,24,25. Fourth-order tensors have also been used to project the stress space into alternative
configurations, leading to a direction-dependent material strength26,27,28,29,30. A different perspective on the use of micro-scale
arguments to simulate macroscopic inelastic anisotropy has recently been proposed in the framework of Breakage Mechan-
ics31,32, showing that the cross-anisotropic properties of granular rocks can be recovered from energy statements based on linear
elastic potentials with embedded directional properties.
This paper presents a set of experiments and constitutive modeling effort on high-porosity limestone. The goal of these studies
is to develop a comprehensive understanding of the deformation behaviour of the anisotropic rock, focusing on initial stiffness,
first yielding, and post-yielding deformation. X-ray tomography combined with digital image correlation is employed to charac-
terize rising specimen-scale deformation mechanisms. The rest of this paper is organized as follows. First, a novel experimental
campaign targeting anisotropic responses of a high porosity rock from Central Europe, the Maastricht Tuffeau, is described in
detail by focusing on sample-preparation procedures, mechanical testing and imaging (Sections 2 and 3). Afterwards, a new con-
stitutive model is proposed to capture the key attributes of the continuum-scale deformation behavior of the Maastricht Tuffeau
pointed out by the experiments. Hence, an augmented projection strategy based on a dual tensorial mapping is discussed (Section
4), along with parametric analyses, calibration procedures and performance assessment (Sections 5 and 6).
In the following sections, a positive compression convention will be adopted in accordance with the usual soil mechanics
framework. For the sake of generality, a tensorial formalism is used: bold-face letters denote vectors a∶= and second order
tensors A∶= , while blackboard bold symbols indicate fourth order tensors (e.g., ℂ=ℂ). Accordingly, the Kronecker
symbol is used to define Iand 𝕀(i.e., I∶= and 𝕀∶= ( + )/2) which represent the identity tensors for any vector a
and any tensor A, respectively. The identity tensor 𝕀can be also expressed through the symbol (i.e.,𝕀=II) which satisfies
the following properties: (AB)C∶= AC⊺B= . The inner product between vectors and tensors is denoted with
the symbol “ ⋅” (e.g., a⋅b∶= or A⋅B∶= ) and, consequently, the Euclidean norm is defined as A∶= A⋅A.
Diadic product is also used to denote fourth order tensors. In other words, for any tensor A,Band C, the corresponding fourth
order tensor can be represented as AB∶= , with (AB)C∶= A(B⋅C) = . By using the aforementioned
conventions, stress and strain invariants are defined as:
∶= 1
3tr() = ∕3
∶= 3
2dev()
∶= tr() =
∶= 2
3dev(),
(1)
where dev(◦)indicates the deviator part of a second order tensor (i.e., dev(A) ∶= A−tr(A)I∕3) with the tr(◦)defined as:
tr(A) = A⋅I=. Under axisymmetric stress conditions, and are expressed as = (1+ 23)∕3 and =1−3,
respectevely (1and 3are the major and minor principal stress).
2 MATERIAL AND EXPERIMENTAL METHODS
2.1 Tested material: Maastricht Tuffeau
A highly porous carbonate rock (porosity up to 52%) from the Netherlands, known as Maastricht Tuffeau, is considered in this
study33,34. Formed by sediments transported and deposited in a shallow sea during the Upper Cretaceous period 35,36 , the con-
stituents of Maastricht Tuffeau include echinoids, brachiopods, sea skeletons and numerous shell fragments with an average size
distribution between 100 and 200 microns. Two types of pore volume fraction can be identified in the material: i) intragranular
porosity, due to pores within bioclasts and shell fragments, and ii) intergranular porosity, accounts for inter-particle pores and,
although highly variable, contributes to the most significant fraction of the pore volume. Maastricht Tuffeau appears as a grain-
supported structure with the grains loosely bonded at contact points, thus resulting into a relatively low strength (first yielding
at uniaxial compression is around 3 MPa)37,38.
Shahin ET AL 3
2.2 Sample preparation and mechanical testing
Although Maastricht Tuffeau is a sedimentary rock, bedding planes are hardly visible with naked eye. Given the pervasive
occurrence of pressure solution in carbonate rocks39, it is arguable that chemical feedbacks mediated by non-isotropic geostatic
stress may have played a role in the emergence of the mechanical anisotropy of Maastricht Tuffeau. For this reason, hereafter
the term “bedding plane” will refer to orientations implicitly regarded as sites of stronger pressure-solution feedbacks (i.e.,
horizontal planes, for usual geostatic stress states). To investigate the anisotropic behavior of Maastricht Tuffeau, cylindrical
samples are extracted from the same material block at three different directions: perpendicular. oblique (45◦), and parallel to
bedding. The cylindrical samples have a diameter of 11 and a height of 22 . This dimensions are selected considering a
balance between mechanically meaningful results and transparency to x-ray tomography (CT).
The specimens were loaded through a triaxial apparatus at Laboratoire 3SR (Grenoble, FR)40,41, following the protocol
reported by42. In-situ x-ray tomography scanning was carried out during compression. The axial load is measured by a force
meter, which is installed onto the loading head, and is in contact with the loading ram. Axial displacement was measured outside
the pressure vessel with an LVDT attached to the loading piston that moves at a fixed rate of 20 m/min (nominal axial strain rate
of 1.5× 10−5 ∕). Axial strain of the sample is obtained from the axial displacement data and initial sample length. Since the spec-
imens are tested under dry conditions, their bulk volume is calculated from three dimensional (3D) images obtained throughout
the tests, thus enabling the computation of the volumetric strain as a difference of the sample volume at two subsequent steps.
2.3 X-ray micro CT and 3D field measurements
For each sample of Maastricht Tuffeau, CT data were acquired before and during sample deformation at key points of the test.
This allows an accurate reconstruction of the 3D x-ray attenuation field of the specimen using the back-projection algorithm 43 .
The voltage and the electric current of the x-ray source were set to 100 kV and 100 A, respectively. About 1120 projections
were acquired at regularly spaced intervals of 32◦, as the specimen was rotated around the vertical axis, and four images of each
angle were averaged to reduce the noise. Furthermore, the zoom level was selected so that the whole specimen fits comfortably
inside the field of view. The voxel size was set to 13 m in order to best image the sample. As a result, there are about 11 pixels
across a mean particle of Maastricht Tuffeau (50 = 150m).
All the experiments presented herein are conducted in-situ,i.e., multiple successive scans are performed during loading
which allows one to follow the evolution of localization pattern along with the sample deformation. In this work, an algorithm
called TomoWarp244,45 is used for 3D Digital Image Correlation (DIC). Correlation is always done between two consecutive
scans (two pairs of 3D images) referred to as a reference and deformed image. The 3D displacement field is calculated at nodes
laid out on a regular 3D grid and then derived by identifying the 3D translation with the best matching pattern of grey level
based on the Normalized Cross-correlation (NCC) within a subvolume (correlation window) at each node (see42 for details).
By deriving the displacement field, the 3D strain tensor is obtained. The first two invariants of the strain tensor representing the
volumetric and maximum deviatoric (shear) strain are chosen for displaying the incremental evolution of the strain fields in the
tests (subsection 3.2).
3EXPERIMENTAL RESULTS
3.1 Stress-strain behavior and first yielding
Uniaxial compression tests were performed on samples of Maastricht Tuffeau cored perpendicular, oblique (45◦), and parallel
to bedding Figure 1a. Results (Figure 1b) indicate mechanical anisotropy with maximum, intermediate, and minimum val-
ues of UCS (Unconfined Compressive Strength) associated with the perpendicular, oblique, and parallel samples, respectively.
The measured Young’s modulus (estimated from the stress-strain curves over linear portions of the uniaxial response) is also
anisotropic. Stiffness increases linearly between bedding direction (the softest) and orthogonal to bedding (the stiffest).
Triaxial compression tests were performed at different confining pressures. At variance with the unconfined tests, the elastic
properties appear to be isotropic Figure 2. The strength, however, is anisotropic as samples cored perpendicular to bedding
display higher strengths.
Anisotropic properties are also observed post-yielding. Perpendicular to bedding, all tests conducted under 1-4 MPa con-
finement exhibit softening-to-negligible hardening.Much sharper hardening is observed in specimens tested under 5 MPa
4Shahin ET AL
0 0.01 0.02 0.03 0.04
Nominal Axial Strain [-]
0
1
2
3
Nominal
90
45
0
0 45 90
Bedding Orientation [
o]
0 45 90
Bedding Orientation [
o]
0
0.1
0.2
0.3
Young's Modulus [GP a]
abc
Corring to
bedding angle
o
o
o
Deviator Stress [MPa]
FIGURE 1 (a) Stress-strain response from uniaxial compression of Maastricht Tuffeau samples cored at three different orien-
tations, (b) Unconfined Compressive Strength (UCS), and (c) Young’s modulus as a function of bedding plane orientation for
uniaxial compression tests. Angle of bedding orientation calculated with respect to the direction of the maximum compressive
load.
confinement. Samples cored parallel and oblique to bedding display softening-to-hardening transitions at much lower pressures;
hardening appears with samples tested under 3 MPa. The rate of hardening increases as the confining pressure increases.
Spikes of stress relaxation that appears on the stress-strain curve with intervals of 0.02 axial strain are bi-products of halting
deformation for x-ray scans. Beside these spikes, the stress-strain response is punctuated by episodic, more frequent stress drops.
This behavior was observed regardless of coring direction, hardening behavior, or confining pressure. The drops’ amplitude and
frequency increase for higher confining pressures. Envelops of yield stresses (identified by the intersection point between the
elastic tangent and deformation plateau) indicate that yielding surfaces, although vary in size depending on coring direction,
are shape-invariant Fig. 3. Furthermore, bedding surfaces do not constitute weak planes, as suggested by the gradual increase
of strength between parallel, oblique, perpendicular directions. Papazoglou and co-workers 42,46 show that Maastricht Tuffeau
displays (based on extended compression experiments under 4 and 5 MPa confinement) a sharp strain hardening after stages of
stress plateau. In light of these findings, extended deformation experiments were conducted on samples cored in parallel, oblique,
and perpendicular to bedding. Distinct characteristics feature the plateau-to-rehardening behaviour Figure 5. Specifically, the
deformation interval of re-hardening onset varies depending on coring direction. Samples cored parallel to bedding display the
shortest plateaus, with the longest associated with samples cored perpendicular to bedding. Intriguingly, despite the anisotropic
yielding stress, hardening rate, and plateau length, post-plateau rehardening appears isotropic, with mechanical responses that
collapse into a single curve Figure 5.
The deformation behavior associated with compression tests on samples cored perpendicular to bedding are shown in Fig. 4.
These values are computed based on Image analyses of in-situ x-ray scans. The deformation behavior in the other directions are
unavailable. The acquired CT scans of this experiments subset were lost because of hardware failure.
3.2 Spatial propagation of localized Strain
X-ray tomography scans were acquired in all tests and the kinematic field associated with each loading increment was quantified
by means of 3D Volume Digital Image Correlation. The results are shown in Fig. 6 which depicts the incremental shear strain at
selected loading increments, for samples cored at different directions (90◦, 45◦and 0◦) and three confining pressures (1, 3, and 5
MPa). At low confining pressure (1 MPa), the propagation of localized compaction for = 90◦and = 45◦is characterized by
shear bands inclined with an angle around 60◦with respect to the major compressive stress, while for = 0◦degrees conjugate
shear bands are observed.
For two directions (90◦and 45◦), the failure mode appears the same, with shear bands inclined at around 60◦with respect to
1, while for the sample cored parallel to bedding two conjugate shear bands at 45◦are observed. For higher confining pressures
(3 and 5 MPa) strain localization is less affected by the bedding orientation. The incremental strain fields reveal similar spatial
patterns of strain localization in all of the samples, regardless the loading direction with respect to bedding. For the 3 MPa
confinement, two localized zones developed at the two ends of the samples, almost perpendicular (∼80◦) to the loading direction.
Shahin ET AL 5
0 0.03 0.06 0.09 0.12 0.15
Nominal Axial Strain [-]
0
2
4
6
Nominal
Deviator Stress [MPa]
Confinement = 4.0 MPa
0
2
4
6
Nominal
Deviator Stress [MPa]
Confinement = 1.0 MPa
0 0.03 0.06 0.09 0.12 0.15
Nominal Axial Strain [-]
Load-to-bedding
Angle
90
45
0
Confinement = 5.0 MPa
Confinement = 3.0 MPa
ab
cd
o
o
o
FIGURE 2 Stress-strain behavior of Maastricht Tuffeau specimens characterized by different bedding orientations and subjected
to triaxial compression at different confining pressures.
Similarly, two zones with high strain concentration appear in the samples tested at 5 MPa. However, in this case the bands are
less planar, especially close to the boundaries of the samples.
Figure 7 presents the evolution of the volumetric strain field over slices located at the specimen center. The results presented
in this figure are those associated with testing under 4 MPa confinement. In all directions, two zones of localized compaction
emerge at the specimen’s ends and propagate toward the specimen’s middle as vertical shortening progresses. Once the two zones
meet, the specimen displays a sharp gain of strength. These compacted zones appear to propagate at different rates depending on
coring direction, thus resulting in onset of rehardening at different deformation intervals. The largest interval was observed with
samples cored perpendicular to bedding, and the smallest interval was observed in samples cored parallel to bedding Figure 5.
4 A CONSTITUTIVE MODEL FOR CROSS-ANISOTROPIC POROUS ROCKS
The experimental results discussed above feature intriguing anisotropy characteristics of Maastricht Tuffeau. The rock displays
higher yielding stresses and longer deformation plateaus in the direction perpendicular to bedding versus lower yielding stresses
and shorter plateaus in the direction parallel to bedding.
6Shahin ET AL
0 2 4 6
Mean Stress [MPa]
0
2
4
6
Deviator Stress [MPa]
0
o
o
o
45
90
0 45 90
Bedding Orientation [
o]
0
2
4
6
5 MPa
4 MPa
3 MPa
1 MPa
ab
Nominal
FIGURE 3 (a) Yield stresses for samples cored perpendicular, oblique (45◦), and parallel to bedding. (b) Yielding stress as a
function of the bedding plane orientation from triaxial compression tests with confining pressure ranging from 1 to 5 MPa.
0 0.05 0.1 0.15 0.2
Nominal Axial Strain [-]
-0.12
-0.08
-0.04
0
Nominal
Volumetric Strain [-]
Confining Pressure
[MPa]
1.0
3.0 4.0
5.0
FIGURE 4 Volume change associated with the deformation responses reported in Fig. 2. Samples are cored perpendicular to
bedding.
This section aims for replicating these mechanical characteristics through elasto-plastic constitutive modeling. The work is
based on the constitutive functions proposed in a series of contributions by Nova and coworkers47,48,49, which resulted into a con-
stitutive law able to capture phenomena typical of soft rocks and hard soils including debonding and destructuration 50,51,52,53,54 .
This approach is featured by unique capabilities to reproduce versatile modes of instability propagation (55). It was successfully
used to explain compaction-band structures in geological outcrops56,57 , and interpret time-dependent properties of sedimentary
rocks58,59. While the aforementioned works are based on a common hypothesis of elastic and plastic isotropy, this hypothesis will
be here removed to incorporate mechanical anisotropy and show its effect on both first yielding and post-yielding deformation.
Shahin ET AL 7
0 0.1 0.2 0.3 0.
4
Nominal Axial Strain
[
-
]
0
4
8
12
Nominal
Deviatoric Stress [MPa]
Confinement = 4.0 MPa
Bedding angle
90o
45o
0o
FIGURE 5 Stress-strain curves of Maastricht Tuffeau specimens under 4 MPa confinement and compressed up to high axial
strain.
4.1 Mathematical framework
The developments outlined hereafter are based on the notion of fictitious isotropy60 , according to which a tensorial operator
is used to project the real Cauchy stress tensor, , onto an alternative stress configuration, 𝝈. For this purpose, a fourth-order
tensor, ℙ, is defined as:
𝝈=ℙ𝝈(2)
The anisotropic yield surface in the real configuration is assumed equivalent to an isotropic yield surface in the alternative
configuration. Based on this assumption, it is possible to use standard constitutive models to address anisotropy by determining
suitable projection operators. The yield surface in the alternative configuration is defined by:
=(𝝈(𝝈),Ψ) (3)
where 𝚿is a vector of state variables. Equivalently, for the purpose of defining an anisotropic plastic potential, a tensorial
operator, ℙ, is used to project again the real Cauchy stress tensor, 𝝈, onto an alternative stress configuration, 𝝈, as following:
𝝈=ℙ𝝈(4)
The plastic potentials in the alternative configuration is defined by:
=(𝝈(𝝈)) (5)
The projectors used for the yield surface and plastic potential may coincide (i.e.,ℙ=ℙ), which implies use of the same
projected stress space for the two functions. Here this particular case of the mapping procedure is referred to as "homotopic
mapping". By contrast, different projectors can be used for the two surfaces (i.e.,ℙ≠ℙ). As a result, the mapping procedure
relies on two different projected stress spaces, thus possibly leading to different relative locations of the mapped stress state into
the projected stress spaces. In this case, the procedure is referred to as "heterotopic mapping".
To determine these projectors, let us start by defining three mutually orthogonal unit vectors 𝒏1=𝒏⟂,𝒏2=𝒏∥, and 𝒏3=𝒏∥.
These vectors identify a cross-anisotropic material whose response is invariant with respect to an arbitrary rotation around 𝒏1
(the normal to the bedding planes, Figure 8). Based on this system, a second-order tensor, F, describing the intrinsic bedding
orientation of the material (therefore referred to as "fabric tensor"), is defined by the dyadic product of the bedding plane
identifier, 𝒏1:
F=𝒆⟂𝒆⟂(6)
8Shahin ET AL
FIGURE 6 Vertical slices of the 3D deviatoric strain field relative to a selected nominal axial deformation increment, for samples
with different bedding-loading angles tested under a range of confinement pressures. The selected increment is noted at the top
of each slice.
In case of transverse isotropy, linear elastic materials can be characterized through the strain tensor, 𝜺, and the fabric tensor, F:
=ℂ
=tr() +⟂ +tr() + + 2 ∥−⟂ + + (7)
Shahin ET AL 9
FIGURE 7 Vertical slices of the 3D volumetric strain field for samples cored perpendicular, oblique, and parallel to the bedding
plane. Collection of selected increments during the three tests at 4 MPa confining pressure. Compression is positive.
where ⟂and ∥are shear moduli associated with perpendicular to bedding and parallel to bedding compression, respectively.
is Lame’s first parameter. The elasticity stiffness tensor can be written as a function of the tensorial basis 𝕋:
ℂ= +2∥(𝕀)
𝕋1
+(II)
𝕋2
+(FI+IF)
𝕋3
+ 2 ∥−⟂(FI+IF)
𝕋4
+(FF)
𝕋5
=
1𝕋1+
2𝕋2+
3𝕋3+
4𝕋4+
5𝕋5
(8)
where
0= 2∥,
1=,
2=,
3= 2 ∥−⟂and
4=, are elastic material coefficients. 𝕋, are used in this work to
construct the fourth-order tensor ℙ
as follows:
ℙ=
1𝕋1+
2𝕋2+
3𝕋3+
4𝕋4+
5𝕋5(9)
where the eigenvalues
, with = 1 →5, are characteristics of the plastic anisotropy and can be different from the elastic
parameters. The superscript makes reference to either the yield surface (=) or the plastic potential (=). 1has to be
set equal to 1 to retrieve the isotropic behavior when all the other constants are equal to 0, and to avoid an effect of isotropic
expansion/contraction. 2also has to be equal to zero, as there is no fabric in this in . In the following section, this projector
10 Shahin ET AL
Axis of
anisotropy
Bedding plane
inclination
Bedding plane
Rock sample Equivalent continuum
FIGURE 8 Schematic representation of bedding planes and the equivalent continuum to transversely isotropic materials. This
is adapted by Marinelli and co-workers31 .
is applied for modeling the anisotropy of Maastricht Tuffeau.
The increment of plastic strain is calculated through the gradient of the plastic potential:
𝝐= Λ (𝝈)
𝝈= Λ ℙ(𝝈)
𝝈 (10)
where Λis the plastic multiplier and the tensor ℙis equal to: ℙ=𝝈∕𝝈.
To show the influence of the anisotropic fabric introduced through the tensor ℙ, the elasto-plastic tensor is derived by imposing
the consistency condition:
=
𝝈∶
𝝈+
𝜶⋅
𝜶=
𝝈∶ℙℂ
𝜺− Λ
𝝈ℙ+ Λ
𝜶⋅𝒉= 0 (11)
𝒉is the hardening vector. By developing 11, it is possible to obtain the expression of the plastic multiplier as:
=
𝝈∶ℙℂ
𝜺− Λ
𝝈∶ℙℂℙ
𝝈+ Λ
𝜶⋅= 0 (12)
=1
−
𝝈∶ℙℂ
𝜺
=
𝜶
=
𝝈∶ℙℂℙ
𝝈
(13)
By using the definition of the plastic multiplier it is possible to calculate the elastoplastic tensor:
𝝈=ℂ
𝜺=ℂ
𝜺−
𝜺=ℂ
𝜺=ℂ
𝜺− Λ
𝝈ℙ=ℂ
𝜺(14)
ℂ=ℂ−1
−
𝝈ℙℂℙℂ
𝝈 (15)
Constitutive equations
The framework is applied to elastoplastic models by adjusting the expression of the yield surface and plastic potential which,
for the constitutive law adopted in this study, are given by:
=1∕
×−2∕
×−∗
= 0 (16)
Shahin ET AL 11
1∕2=1 −
21 −
1 ±
1 − 41 −
1 − 2
= 1 +
1
= 1 +
2
= (1 − )(1−2)
(17)
∗
is the hydrostatic pressure at yielding and is composed by three components ∗
=++where mimics the effect
of strain-hardening and the interparticle bonding. simulates the tensile strength of the rock and is expressed as a function
of (ie, =). is an equivalent stress ratio defined as =∕(+).
The evolution of the and is governed by two hardening rules:
=
(18)
= −(
+
)(19)
The dilatancy function defines the ratio of volumetric and deviatoric plastic strains computed for a given stress ratio and is
defined:
() =
=∕
∕=(−)(
+ 1) (20)
4.2 Parametric Analysis
4.2.1 Yield Surface
A parametric analysis is conducted on the tensorial components of ℙ. The goals of this analysis are: (i) to qualitatively elucidate
the distortions they cause to the yield surface and, (ii) to identify viable simplifications of the projector. The eigenvalue of each
basis has individually been altered by ±0.1from the isotropic value (i.e.,
1= 1.0and
2→5=0). The results are reported in
Figure 9, with continuous-blue lines indicating loading perpendicular to bedding and dashed-red lines indicating loading parallel
to bedding. The isotropic yield surface is displayed in gray.
This analysis shows that the first basis is linked to isotropic distortions of the yield surface. In fact, altering
1results in
yield surfaces that are independent of the angle between the bedding and loading direction (Figure 9a-b). The effect of the
second basis is illustrated in Figure 9c-d. The resulting distortions are again isotropic with variations independent of the angle
between bedding and loading direction. These results reflect the isotropic effects of these two bases which, subsequently, should
be excluded from any parametric calibration for mechanical anisotropy.
The analysis conducted on the third basis, 𝕋3, is presented in Figure 9e-f. Positive
3results in larger elastic domains for
loading parallel to bedding. Negative values, by contrast, results in larger elastic domains for loading perpendicular to bedding.
Similar effects are observed with the fourth basis, 𝕋4(Figure 9g-h). The effect of the fifth basis is illustrated in Figure 9i-j,
showing that this basis has similar qualitative effects on the geometry of the yield surface. Yet, the distortion intensity is much
larger for the same variations in the eigenvalue. The distortions are nearly three times more intense than those caused by the third
and fourth bases. It can therefore be concluded that by keeping the coefficients of the first two isotropic bases (i.e.,
1=1.0 and
2=0) unaltered, it is convenient to use a simpler projection operator consisting of 𝕋1along with only one among 𝕋3,𝕋4and 𝕋5.
This simple strategy suffices to capture accurately anisotropic effects on first-yielding. As a result, in this study the following
projector expression will be used:
ℙ=𝕋1+
3𝕋3(21)
With this simplification, the representation of the anisotropy of first-yielding boils down to a formalism dependent on a single
additional parameter:
3. The performance of this projector is further examined with a set of yield surfaces obtained by varying
3in the range −0.3→+0.3. The results are reported in Figure 10 showing considerable flexibility of the augmented yield
surface to fit a diverse range of anisotropic shapes.
4.2.2 Plastic Potential
The parameters of the plastic potential, ,, and , can be defined on the basis of the stress-dilatancy function defined in
equation Equation 20. This function defines the ratio of volumetric and deviatoric plastic strains computed for a given imposed
12 Shahin ET AL
0
1
2
3
4
5
6
Deviator Stress [MPa]
0
1
2
3
4
5
6
Deviator Stress [MPa]
0
1
2
3
4
5
6
Deviator Stress [MPa]
1234567
Mean Stress [MPa]
1234567
Mean Stress [MPa]
0
1
2
3
4
5
6
Deviator Stress [MPa]
C1 = 1.1
C2 = 0.0
C3 = 0.0
C4 = 0.0
C5 = 0.0
0
1
2
3
4
5
6
Deviator Stress [MPa]
C1 = 0.9
C2 = 0.0
C3 = 0.0
C4 = 0.0
C5 = 0.0
C1 = 1.0
C2 = +0.1
C3 = 0.0
C4 = 0.0
C5 = 0.0
C1 = 1.0
C2 = -0.1
C3 = 0.0
C4 = 0.0
C5 = 0.0
C1 = 1.0
C2 = 0.0
C3 = +0.1
C4 = 0.0
C5 = 0.0
C1 = 1.0
C2 = 0.0
C3 = -0.1
C4 = 0.0
C5 = 0.0
C1 = 1.0
C2 = 0.0
C3 = 0.0
C4 = +0.1
C5 = 0.0
C1 = 1.0
C2 = 0.0
C3 = 0.0
C4 = -0.1
C5 = 0.0
C1 = 1.0
C2 = 0.0
C3 = 0.0
C4 = 0.0
C5 = +0.1
C1 = 1.0
C2 = 0.0
C3 = 0.0
C4 = 0.0
C5 = -0.1
b
a
d
c
f
e
h
g
j
i
FIGURE 9 Parametric analyses illustrating the transformations produced by each basis on the isotropic yield surface. Effects
of (a-b) the first basis, (c-d) the second basis, (e-f) the third basis, (g-h) the fourth basis, and (i-j) the fifth basis.
stress level (e.g., in correspondence of the plastic flow plateau). These ratios can be reported in the −space. Starting from a
projector based on a homotopic mapping (ℙ=ℙ), the variations of the dilatancy function resulting from different values of
3are depicted in Figure 11. Marked distortions are observed especially at higher . Such variations result in different value
of dilatancy depending on loading direction. The sign of the eigenvalue has a marked role on the direction of distortion. Positive
Shahin ET AL 13
1234567
Mean Stress [MPa]
0
1
2
3
4
5
Deviator Stress [MPa]
+0.1
+0.2
C3= +0.3
1234567
Mean Stress [MPa]
C3= -0.1
-0.2
-0.3
b
a
FIGURE 10 Parametric analyses illustrating the transformations produced by the third basis on the isotropic yield surface, with
the third eigenvalue ranging (a) from +0.1→+0.3, and (b) from −0.3→−0.1.
values of
3are associated with a larger dilatancy ratio in bedding-perpendicular direction (Figure 11a). Negative values, by
contrast, result in a larger dilatancy ratio in correspondence with loading in bedding-parallel direction (Figure 11b).
b
0 0.5 1 1.5 2 2.5 3
0
0.5
1
1.5
2
2.5
d [-]
+0.3
+0.2
C3=+0.1
0 0.5 1 1.5 2 2.5 3
C3=-0.3
-0.2
-0.1
a
[-]
(σ )
[-]
(σ ) f
f
FIGURE 11 Parametric analyses illustrating the dilatancy function transformations produced by the projection operator with
different values of
3: (a) +0.1→+0.3, and (b) −0.3→−0.1.
5 MODEL CALIBRATION
5.1 Homotopic stress mapping
The basic features of the constitutive framework discussed above are illustrated through a set of numerical examples. The model
is calibrated based on the macroscopic deformation behavior obtained from a set of drained triaxial tests shown in section ??.
14 Shahin ET AL
The calibration procedure focused first on the shape parameters ,,, and the anisotropy parameter,
3, (Figure 12a).
The parameters of the plastic potential, ,, and , can instead be defined on the basis of the underlying stress-dilatancy
relationship (reported in −space in Figure 12b). This procedure relies on global measurements taken in the post-yielding
regime (i.e., where the specimen exhibits a structural response with heterogeneous deformation). The measurements, however,
are expressed in the form of a normalized strain ratio and can be regarded as representative of the material behavior inside
the active zone, thus providing a first-order insight into the characteristics of the plastic flow. In fact, the dilatancy functions
presented in Figure 12b are based on homotopic mapping (i.e.,ℙ=ℙ), and both locate inside the scatter of experimental data.
It will be shown later that different values of dilatancy are necessary to capture some of the features of the mechanical anisotropy.
0 0.5 1 1.5 2 2.5 3
0
0.5
1
1.5
2
2.5
d [-]
Exp. perpend-Bedding
Exp. parallel-Bedding
b
0 1 2 3 4 5 6 7
Mean Stress [MPa]
0
1
2
3
4
5
Deviator Stress [MPa]
a
[-]
(σ )
g
FIGURE 12 Yield surface and dilatancy function calibrated for Maastricht Tuffeau with a projector based on a homotopic
mapping. Stress ratios are reported based on post-mapping stress space.
The response upon isotropic compression is used to calibrate hardening parameters, i.e.,,, and the initial values of the
internal variables, and . The yield stress upon isotropic compression, = + + , can be assumed to coincide
with the stress at which the volumetric stiffness shows a sharp decrease. In fact, because of the rotations in the stress space
caused by the projection operator, the model amplifies which requires adjusting its value. The ratio ∕ governs the rate
of destructuration occurring upon isotropic loading. This rate appears in the form of a magnified post-yielding plastic volumetric
deformation. On the other hand, controls the amount of softening during this stage of enhanced plastic deformation. , in
turn, defines the rate of plastic hardening in the post-plateau region, where inter-particle bonds have been fully destructured.
As a result, this constant regulates the porosity hardening emerging upon volumetric deformation in a completely reconstituted
material. The anisotropy constant are
3=
3= -0.15. All parameters are summarized in Table 1 and 2. The overall behavior
under isotropic compression predicted by the model is depicted in Figure 13, showing that the model reproduces satisfactorily
the material behavior. It is worth noting that the test is stress controlled, thus impacting the resolution of data collection between
first yielding and the subsequent hardening stage, with obvious implications on the accuracy of the model calibration in this
transitional regime.
Figure 14 presents the model prediction for triaxial compression under 1.0-, 3.0-, 4.0-, and 5.0-MPa confining pressure
reported along with the experimental counterpart. The model satisfactorily predicts the yield stress and post-yielding stress
resistance. The model, however, displays marked limitations in the prediction of the length of the deviator stress plateau. For
example, by making reference to the experiment conducted at 4.0MPa confinement, the model predictions are different from
the limestone behavior which displays plateau lengths under compression parallel to bedding shorter than those obtained under
perpendicular to bedding.
Shahin ET AL 15
0 2 4 6 8 10 12 14
Mean Stress, P [MPa]
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
v[-]
Isotrpic 1
Isotropic 2
Model
FIGURE 13 Material response under isotropic loading simulated by the model; the experimental results from two tests are also
plotted for comparison.
0
2
4
6
8
10
Deviator Stress [MPa]
Confinement = 1.0MP a
0
2
4
6
8
10
3.0MPa
0 0.04 0.08 0.12 0.16 0.2
Axial Strain [-] [-]
0
2
4
6
8
10
Deviator Stress [MPa]
4.0MPa
0 0.05 0.1 0.15 0.2
Axial Strain
0
2
4
6
8
10
5.0MPa
FIGURE 14 Material responses simulated by the model with a projector based on a homotopic mapping. The comparison
between measurements and simulations indicates detectable mismatches in the length of the plateau.
16 Shahin ET AL
5.2 Heterotopic stress mapping
A heterotopic mapping (i.e.,ℙ≠ℙ) is adopted in this section for the yield surface and plastic potential, respectively. As these
tensors are independent and may differ on the prescribed values of the constants
3and
3, it is possible to use any combinations
of tensorial bases without making reference to the bases used for ℙ. Here, the same bases of ℙare used yet with different
eigenvalues,
, to generate ℙ. Therefore, a projection operator based on the two bases, 𝕋1and 𝕋3, has been adopted:
ℙ=
1𝕋1+
3𝕋3(22)
Similar to ℙ,
1has been fixed to 1.0, thus the anisotropy in the flaw rule can be captured by calibrating a single parameter:
3.
This parameter has been calibrated in order to predict the appropriate plateau extent in each direction. The following projector
has been adopted:
ℙ=𝕋1+ (-2.0)
3𝕋3(23)
The dilatancy functions associated with heterotopic mapping (𝝈=ℙ𝝈) are shown in Figure 15. For lower stress ratios ( <1),
the dilatancy ratio is higher in the direction parallel to bedding.
0 1 2 3 4 5 6 7
Mean Stress [MPa]
0
1
2
3
4
5
Deviator Stress [MPa]
0 0.5 1 1.5 2
0
0.5
1
1.5
2
2.5
3
d [-]
Exp. perpend-Bedding
Exp. parallel-Bedding
ab
η[-]
[-]
(σ*)
FIGURE 15 Yield surface and dilatancy function calibrated for Maastricht Tuffeau with a projector based on a heterotopic
mapping. Stress ratios are reported based on post-mapping stress space.
The stress-strain responses predicted based on the adjusted dilatancy function is illustrated in Figure 16. The model displays
better predictions of the post-yielding deformation behavior and the plateau length. The plateau length, when the major principal
stress is perpendicular to bedding, is longer for loading parallel to bedding. This is consistent with the experimental observations.
The model is also able to capture the mechanical anisotropy degradation observed at the end of the plateau. Comparable rates
of rehardening were predicted in the post-plateau regime. The model parameters and its anisotropy constants are summarized
in Table 1 and Table 2.
To elucidate the factors leading to various plateau lengths, the dilatancy ratios associated with homotopic and heterotopic
mapping are detailed in Figure 17. The example here is presented for the triaxial compression under 4.0MPa. Heterotopic map-
ping dictates different plateau lengths as it provides the ability to adjust the plastic potential associated with a given stress ratio
to obey the phenomenology of anisotropic flow behavior observed experimentally. The transition from homotopic to heterotopic
mapping enables the identification of different dilatancy ratios for each loading direction. In the case illustrated in Figure 17
(here the stress ratio is reported in the generalized stress space), heterotopic mapping enables decreasing the dilatancy ratio
associated with compression perpendicular to bedding, while increasing it for compression parallel to bedding. These variations
in dilatancy ratio alter the flow behavior, leading to marked differences in the hardening behavior (Figure 18). persists with
nearly unaltered evolution for both mapping schemes. Yet, the evolution of undergoes significant changes. With heterotopic
mapping, pore-collapse hardening, , evolves at rates higher for compression parallel to bedding as compared to compression
perpendicular to bedding (which undergoes lower rates of hardening). This demonstrates the benefits gained by adopting the
dual-projector structure, which allows modulating the anisotropic role of and . With heterotopic mapping, the volumetric
Shahin ET AL 17
0
2
4
6
8
10
Deviator Stress [MPa]
Confinement = 1.0MPa
0
2
4
6
8
10
3.0MPa
0 0.04 0.08 0.12 0.16 0.2
Axial Strain
0
2
4
6
8
10
Deviator Stress [MPa]
4.0MPa
0 0.05 0.1 0.15 0.2
Axial Strain
0
2
4
6
8
10
5.0MPa
[-] [-]
FIGURE 16 Material responses predicted by the model with a projector based on a heterotopic mapping, showing satisfactory
performance in terms of predicting the plateau length.
0 0.4 0.8 1.2 1.6
0
0.4
0.8
1.2
1.6
d [-]
Perpendicular to bedding
Homotopic
[-]
0 0.4 0.8 1.2 1.6
Parallel to bedding
[-]
Heterotopic
Isotropic Homotopic
Heterotopic
(σ)
(σ)
FIGURE 17 Comparison of the dilatancy ratios associated with homotopic and heterotopic mapping for the case of 4.0MPa
along with the entire dilatancy functions of the perpendicular direction (left) and the parallel direction (right). Stress ratios are
reported based on the pre-mapping stress space.
18 Shahin ET AL
0
4
8
12
Ps[MPa]
0 0.04 0.08 0.12 0.16 0.2
Axial Strain
3
4
5
6
Pm [MPa]
0 0.04 0.08 0.12 0.16 0.2
Axial Strain
[-] [-]
FIGURE 18 Illustration of the hardening variables, and , evolution predicted with a projector based on homotopic mapping
(left) and a projector based on a heterotopic mapping (right) showing marked differences in terms of the rate of hardening growth.
TABLE 1 Constitutive parameters and internal variables for Maastricht Tuffeau.
Definition
[MPa] Bulk modulus 350
[MPa] Shear modulus 78
Parameter governing softening under volumetric deformation 0.2
Parameter governing softening under deviatoric deformation 5.0
Isotropic plastic compressibility 0.03
Shape parameter of the yield surface 1.2
Shape parameter of the yield surface 0.85
Shape parameter of the yield surface 0.65
Shape parameter of the plastic potential 0.55
Shape parameter of the plastic potential 0.05
Shape parameter of the plastic potential 2.2
Expansion of the yield surface in the tensile stress domain 0.15
[MPa] Initial size of the elastic domain (cohesionless medium) 0.1
[MPa] Lithification-induced expansion of the initial elastic domain 5.9
strain becomes less dominant in coring direction perpendicular to bedding, thus making the destructuration of dominant for
longer deformation extent. The material, therefore, mostly suffers more compaction until the rehardening of becomes evident
and the deformation response approaches the end of the plateau.
6 CONCLUSIONS
This study illustrated a combined experimental and modeling analysis of the anisotropic behavior of a high-porosity carbonate
rock, the Maastricht Tuffeau. The experiments involved triaxial compression tests on samples cored at different angles with
respect to bedding (90◦,45◦, and 0◦) and a range of confining pressures up to 5.0-MPa. At variance with standard testing
Shahin ET AL 19
TABLE 2 Anisotropy parameters for Maastricht Tuffeau.
1
2
3
4
5
Yield Surface 1.0 0.0 -0.15 0.0 0.0
1
2
3
4
5
Plastic Potential 1.0 0.0 -2.0 ×-0.15 0.0 0.0
procedures, the experiments did not focus exclusively on the onset of yielding, but imposed large nominal deformation, with the
goal to characterize the plastic flow behavior of the selected rock in the compaction regime.
Considerable dependence of the rock strength as a function of the coring direction was detected both in the brittle and the
ductile regime. Specifically, samples compressed perpendicular to bedding displayed higher strength. Interestingly, it was also
found that loading direction with reference to bedding plays a role on the post-yielding plastic flow behavior, with samples
loaded at different angles exhibiting different elongation of the deviatoric stress plateau associated with compaction localization
(i.e., samples loaded perpendicular to bedding displayed larger post-yielding plateaus).
Full-field measurements based on x-ray tomography show that all samples consistently displayed compaction localization,
with compaction fronts spreading over the sample volume starting at yielding until the end of the plateau. The compacted
specimen at the plateau end displays a stage of re-hardening associated with with delocalized deformation. Despite variations
in the global deformation behavior, the mode of strain localization and band inclination are relatively insensitive to loading
direction
The measurements were interpreted through an elastoplastic strain-hardening constitutive model with inherent mechanical
anisotropy. The proposed framework is based on projector operators able to map the stress space into equivalent spaces. In its
most general form, the proposed projector operator involves five tensorial bases and eigenvalues but can be readily condensed in a
simpler form consisting of only two tensorial bases and a single eigenvalue to be calibrated. In this context, two mapping schemes,
here defined homotopic and heterotopic, were proposed. At variance with homotopic mapping, in which the same projection
strategy is used for both yield surface and plastic potential, the heterotopic mapping scheme involves different projected stress
spaces for the two constitutive functions. It was shown that, while a homotopic mapping enables an accurate representation of
first yielding, it was unable to capture the influence of the bedding orientation on the post-yielding plastic flow. Such shortcoming
was removed by the use of heterotopic mapping, which was shown to capture more accurately plastic flow characteristics and to
better modulate the ratio between porosity hardening and bond destructuration responsible for the elongation of the post-yielding
deviatoric plateau.
To test the model performance, material-point analyses based on triaxial test simulation were conducted under a wide range
of confining pressures on coring direction parallel and perpendicular to bedding. The analyses show satisfactory performance
of the proposed model, specifically, in replicating the yielding conditions, the post-yielding deformation and the post-plateau
hardening behavior. These results encourage the future use of the proposed approach in the context of full-field simulations
aimed at reproducing the pressure dependence of strain localization in Maastricht Tuffeau, as well as the spatial patterns of
compaction band propagation emerging from digital image analysis.
While our simulations suggest that the mechanical anisotropy of the plastic deformation of Maastricht Tuffeau originates
from rates of porosity hardening varying with the bedding direction, the phenomenological nature of the proposed model is not
designed to specify the microscopic origin of such effects. Hence, future micro-mechanical interpretations of these observed
trends will require dedicated data analysis and simulations targeting processes taking place at finer length scales than those
addressed in this work.
ACKNOWLEDGEMENTS
This work is a result of a collaboration between Northwestern University and Université Grenoble-Alpes. G.B. gratefully
acknowledges the financial support of the U.S. Department of Energy (grant DE-SC0017615 awarded to Northwestern Uni-
versity). G.S. and F.M were financially supported by this grant. A.P. and G.V. were financially supported by a grant from
Université Grenoble-Alpes. Laboratoire 3SR is part of the LabEx Tec 21 (Investissements d’Avenir—Grant Agreement No.
ANR-11-LABX-0030). Finally, the authors are grateful to J. Urai (University of Aachen, Germany), for his useful insight about
the significance of pressure solution in anisotropic carbonate rocks.
20 Shahin ET AL
AUTHORS CONTRIBUTION
A.P. and G.V. conducted and analyzed the experiments. G.S., F.M., and G.B. contributed to the theoretical developments. All
authors contributed equally to drafting and revising the manuscript.
DATA AVAILABILITY STATEMENT
The data that support the findings of this study are available from the corresponding author upon reasonable request.
How to cite this article: Shahin G., A. Papazoglou, F. Marinelli, G. Viggiani, and G. Buscarnera (2022), Experimental Study of
Compaction Localization in Carbonate Rock and Constitutive Modeling of Mechanical Anisotropy, Int. J. Numer. Anal. Methods
Geomech,2022.
APPENDIX
Maps of dilatancy angle were generated by computing the one-to-one voxel ratio of incremental volumetric and deviatoric strain.
Histograms of the resulting maps are constructed (Figure A1). All voxels associated with incremental volumetric strain equal to
0.01 were however excluded, as this threshold has been considered indicative of elastic loading (Figure 13). The dilatancy angle
for the considered loading interval is identified by the peak of the histogram. Some datasets are missing as it was not possible
to recover them after the hardware failure.
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0 0.02 0.04 0.06 0.08 0.1
0
1
2
3
4
5
Dev. Stress [MPa]
1.0 MPa
1
234567
0 0.02 0.04 0.06 0.08 0.1
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Dev. Stress [MPa]
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4.0 MPa
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Nominal Axial Strain [-]
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4-5
0 0.5 1 1.5 2 2.5 3
Dilatancy
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2000
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8000
10000
Count
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0 0.5 1 1.5 2 2.5 3
Dilatancy
4-5
0 0.5 1 1.5 2 2.5
3
Dilatanc
y
5-6
0 0.5 1 1.5 2 2.5 3
0
2000
4000
6000
8000
10000
Count
5-6
0.5 1 1.5 2 2.5 3
5-6
missing data
FIGURE A1 Statistical analysis of the local dilatancy angle collected through DIC analyses from various test conducted under
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