Simulation of emergent compaction banding fronts caused by frictional boundaries

Abstract

This study examines the role of boundary friction in promoting heterogeneous compaction in soft rock specimens loaded at relatively high confining pressures outside the domain of compaction localization. An elastoplastic constitutive model characterized by tunable hardening/softening behaviour is used to conduct the analyses. Finite element simulations suggest that material instability is a non-necessary condition for the emergence of compaction fronts. Such fronts propagated as a result of a severe deviation in the local responses induced by frictional constraints. These findings suggest that boundary effects can bias the assessment of the extent of the compaction localization domain. Experimental countermeasures and informed model calibration procedures are therefore necessary to minimize such bias and enable more accurate predictions of soft rock compaction.

Full text

G´
eotechnique Letters 00,1–8, http://dx.doi.org/10.1680/geolett.XX.XXXXX
Simulation of emergent compaction banding fronts caused by
frictional boundaries
G. SHAHIN∗, G. BUSCARNERA ∗
This study examines the role of boundary friction in promoting heterogeneous compaction in soft rock
specimens loaded at relatively high confining pressures outside the domain of compaction localization.
An elastoplastic constitutive model characterized by tunable hardening/softening behaviour is used to
conduct the analyses. Finite element simulations suggest that material instability is a non-necessary
condition for the emergence of compaction fronts. Such fronts propagated as a result of a severe
deviation in the local responses induced by frictional constraints. These findings suggest that boundary
effects can bias the assessment of the extent of the compaction localization domain. Experimental
countermeasures and informed model calibration procedures are therefore necessary to minimize
such bias and enable more accurate predictions of soft rock compaction.
KEYWORDS: Compaction bands; Porous Rocks; Bifurcation Theory; Constitutive Modeling.
ICE Publishing: all rights reserved
INTRODUCTION
When porous materials are deformed at relatively high
confining pressure they may develop compaction bands
perpendicular to the maximum compressive stress (Holcomb
et al.,2007;Wong & Baud,2012). This form of strain
localization has attracted the interest of the engineering
community because of the widespread use of porous
materials in technological applications. Examples include
energy absorbing systems (Park & Nutt,2001;Papka &
Kyriakides,1999), thermal and acoustic insulation (Casini
et al.,2013), foundation systems and construction materials
(e.g., Calcarenite and Gasbeton (Castellanza et al.,2009)).
Porous rocks have received special attention with respect to
compaction bands, in that the emergence of these structures
can impact geotechnologies such as hydrocarbons extraction
(Hettema et al.,2002), aquifer exploitation (Sternlof et al.,
2006;Deng et al.,2015;Zuluaga et al.,2016), and underground
CO2sequestration (Wawersik et al.,2001;Torabi et al.,2015;
Rass et al.,2017;Raduha et al.,2016).
Void ratio (Olsson & Holcomb,2000), particle grading
(Cheung et al.,2012), pore pressure conditions (Wong et al.,
1997), and stress paths (Buscarnera & Laverack,2014) are
among the most important factors controlling the potential
of compaction localization. Compaction bands emerge when
the material is loaded in the plastic cap (Wong et al.,1997)
with a deformation response characterized by modest strain
hardening (Wong & Baud,2012). Noninvasive techniques
able to track their growth have revealed complex patterns
of evolution ranging from the nucleation of discrete bands
(Huang et al.,2019) to band thickening (Papazoglou et al.,
2017). Tomography measurements assisted with finite element
simulations also disclosed intriguing interactions among
material heterogeneity and boundary friction (Shahin,2020).
Specifically, such studies suggested that, above a threshold
value of specimen-platen friction, boundary effects become
dominant and lead to the emergence of compaction fronts
Manuscript received. . .
Published online at www.geotechniqueletters.com
∗Northwestern University, McCormick School of Engineering,
2145 Sheridan Rd, Evanston, IL 60208, USA
stemming from the specimen boundaries. Significant effects
of the boundary conditions on strain localization mechanisms
were also found in the context of shear bands analyses
(Shuttle & Smith,1988;Gao & Zhao,2013). The current
study is motivated by these findings and examines how the
effect of boundary friction can influence the interpretation of
global measurements by promoting compaction fronts even if
the specimens are loaded outside the domain of compaction
banding.
MATERIAL POINT ANALYSES
The selected materials for this study are Bleurswiller sandstone
and Maastricht Tuffeau, both porous rocks known to develop
compaction bands over a wide range of confinements (Fortin
et al.,2005;Baxevanis et al.,2006). Recent simulations of
these materials based on elastoplastic constitutive laws have
shown that Bleurswiller sandstone can be effectively simulated
with hardening behaviour (Shahin et al.,2019a;Shahin &
Buscarnera,2019), while the compaction localization response
of the Maastricht Tuffeau is better captured by strain-softening
behaviour (Shahin et al.,2019b).
Constitutive Model
The banding properties of the two abovementioned rocks
were recently quantified in the context of plasticity models
focusing on strain localization (Marinelli & Buscarnera,2015;
Shahin et al.,2019b). The constitutive framework adopted
for such analysis was rooted on the competition between the
hardening and softening contributions reflected by multiple
state variables, according to techniques extensively developed
for structured soils and soft rocks (Gens & Nova,1993;Nova
et al.,2003). The models were used to capture complex
time-independent (Shahin et al.,2019b,c) and time-dependent
(Shahin et al.,2019a;Shahin & Buscarnera,2019) features of
both rocks. The calibrated model parameters emerging from
these studies will be used as the basis of the analyses presented
hereafter.
The adopted model uses the function proposed by Lagioia
et al. (1996) to model the yield surface and plastic potential
in the context of a non-associated plastic flow rule (details are
provided in the appendix). In addition, it regulates the evolution
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2Compaction banding fronts caused by frictional boundaries
Table 1. Constitutive parameters and internal variables for Maastricht Tuffeau (Shahin et al.,2019b) and Bleurswiller sandstone
(Marinelli & Buscarnera,2015).
Definition Maastricht Bleurswiller
Tuffeau sandstone
K[MPa] Bulk modulus 350 9100
G[MPa] Shear modulus 78 6000
ρmParameter governing volumetric destructuration 0.45 3.00
ξmParameter governing deviatoric destructuration 15 2
BpIsotropic plastic compressibility 0.03 0.026
µfShape parameter of the yield surface 1.01 1.01
αfShape parameter of the yield surface 2.0 0.11
MfShape parameter of the yield surface 0.95 1.06
µgShape parameter of the plastic potential 0.6 1.55
αgShape parameter of the plastic potential 0.15 1.20
MgShape parameter of the plastic potential 2.1 1.75
κExpansion of the yield surface in the tensile stress domain 0.15 0.0
Pso [MPa] Initial size of the elastic domain (cohesionless medium) 0.06 15
Pmo [MPa] Lithification-induced expansion of the initial elastic domain 5.94 106
0 0.005 0.01 0.015 0.02 0.025 0.03
Nominal Axial Strain [-]
0
20
40
60
80
100
Nominal
Deviatoric Stress [MPa]
Bleurswiller Sandstone
Confinement = 80 MPa
L1 L2 L4 L5
L3
L6
0.0 0.04
L1 L2 L3 L4 L5 L6
Cumulative Plastic Volumetric Strain Field in Bleurswiller Sandstone Specimen
Weak
Element
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
Nominal Axial Strain [-]
0
2
4
6
8
Nominal
Deviatoric Stress [MPa]
Tuffeau de Maastricht
Confinement = 4.0 MPa
L1
Cumulative Plastic Volumetric Strain Field in Maastricht Tuffeau Specimen
0.0 0.10
L1 L2 L3 L4 L5 L6
Weak
Element
L4
L2 L3 L5 L6
Fig. 1. Response of finite element simulations of triaxial compression on specimen of Maastricht Tuffeau under 4.0 MPa confining
(upper) and Bleurswiller sandstone under 80.0 MPa confining (lower) compared with experimental measurements. [Data of
Bleurswiller sandstone after (Fortin et al.,2005)]. Compaction band propagates in Maastricht Tuffeau in a form of band-thickening
while in Bleurswiller sandstone in a form of discrete bands.
of the yield surface through two internal variables, Psand Pm:
˙
Ps=Ps
Bp˙p
vol (1a)
˙
Pm=−ρmPm(|˙p
vol|+ξm˙p
dev)(1b)
where p
vol and p
dev are, respectively, the volumetric and
deviatoric plastic strain; Bp,ρmand ξmare constitutive
parameters. Psreplicates the porosity-loss hardening, while Pm
reproduces softening due to loss of structure and debonding.
Through an additive combination, the interplay between these
variables leads to a homothetic contraction/expansion of the
yield surface, as Pc(the hydrostatic yield stress) is defined
by Pc=Ps+Pm(P?
c=Ps+Pm+Pt, with Ptindicating the
tensile strength and assumed equal to Pt=κ Pm, where κis a
material constant). A viscoplastic version of the model is used
to suppress the ill-posedness associated with strain localization,
which may lead to pathological mesh dependence (Needleman,
1988). In this context, the inelastic strain rate can be replaced by
the following Perzyna-type (Perzyna,1966) viscoplastic flow
rule:
˙p
ij = Φ(f)∂g
∂σij
,Φ(f) = ηhfi
Pco (2)
where fand gare, respectively, the yield surface and flow
potential. Φis the viscous nucleus, σij is the stress tensor, Pco
is the initial hydrostatic yielding stress, the symbol h•i indicates
the McCauley brackets, and ηis a fluidity parameter (i.e., the
inverse of viscosity). The model equations are completed by a
set of isotropic linear elastic relations, where the increment of
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G. Shahin G. Buscarnera 3
the stress state ˙σij is computed through:
σij =Dijkl e
kl (3)
In this equation e
kl is the elastic strain tensor, and Dijkl
is the isotropic linear elastic tensor. The sets of calibrated
parameters proposed by Marinelli & Buscarnera (2015) and
Shahin et al. (2019b) for the two rocks are summarized in Table
1and an example of the models performance with reference to
experimental measurements is shown in Fig. 1. Despite some
mismatches (e.g., excess of softening for Maastricht Tuffeau
and overestimation of the rate of hardening in Bleurswiller
sandstone) the simulations can be regarded as satisfactory
(further details about the calibration procedure are provided
by Shahin et al. (2019b) and Marinelli & Buscarnera (2015)).
Most notably, the simulations capture the onset and propagation
pattern of compaction localization for both rocks, which in
Maastricht Tuffeau manifests with a compaction front, while in
Bleurswiller sandstone develops through discrete bands. Such
non-trivial ability of the model to replicate radically different
types of compaction localization will exploited in the remainder
of the paper, with reference to the influence of boundary effects
at rough platens.
Bifurcation Analysis
The strain localization theory (Rudnicki & Rice,1975) is here
used to identify the conditions at which bifurcated solutions
emerge leading to narrow deformation bands. Chambon
(1986) combined this criterion with the elastoplasticity theory,
suggesting that a necessary condition for bifurcation is:
A(θ) = det[nj(θ)(Cep
ijkl )nl(θ)] ≤0(4)
where θis the angle between the unit vector normal
to the band, nj, and the vertical direction. Cep
jjkl is the
elastoplastic constitutive tensor. The expression above enables
the computation of a “localization profile”, i.e. a representation
of the determinant of the acoustic tensor A(θ)as a function
of the band angle θfor a given stress state (Shahin et al.,
2019b;Marinelli & Buscarnera,2015;Das & Buscarnera,
2014). Locating these results in the stress space (thick solid line
in Fig. 2) provides information about the zone of potential strain
localization.
Material Simulations
Simulations based on homogeneous idealizations (material-
point analyses) were conducted to illustrate the response
predicted by the model for loading paths imposed outside
the strain localization regime. Simulations of oedometric
and triaxial compression were conducted, following isotropic
confinement. The initial mean stress was chosen to enable
both paths to intersect with the yield surface outside the
localization domain. Fig. 2illustrates the computed response in
both simulations, where it can be seen that the elastic response
of both materials under oedometric conditions is characterized
by a stress ratio (q/p) markedly lower than the stress ratio
of the triaxial path. With softening in Maastricht Tuffeau
and hardening in Bleurswiller sandstone, the oedometric path
evolves in the post-yielding regime with increasing deviatoric
stress, eventually converging asymptotically to the K0line. By
contrast, the response under triaxial compression evolves with
a stress path typical of constant radial stress conditions
3D FINITE ELEMENT NUMERICAL ANALYSES
Material-point analyses only provide an approximation of
real experiments. For this purpose, full-field simulations
based on the finite element method are used hereafter to
K0
0 20 40 60 80 100 120 140
Mean Stress, P [MPa]
0
25
50
75
100
125
Deviatoric Stress, q [MPa]
Bluerswiller Sandstone
TX
OEDO
Zone of Potential
Compaction Localization
Zone of Potential
Compaction Localization
OEDO
TX
234567
Mean Stress, P [MPa]
0
1
2
3
4
5
Deviatoric Stress, q [MPa]
Maastricht Tuffeau
K0
Fig. 2. Material point simulations of triaxial (TX) and oedometric
(OEDO) compression on numerical specimens of Maastricht
Tuffeau (upper) and Bleurswiller sandstone (lower) conducted
outside the strain localization regime. For illustration purposes
the figure includes the K0line that resulting upon continued
compression for the two materials. The initial domain of
compaction localization (which varies under plastic deformation)
is superposed on the initial yield surface.
replicate numerically triaxial compression tests on both rocks.
The simulations are conducted with reference to cylindrical
specimens with aspect ratio height/diameter= 2. The specimens
are discretized with a finite element mesh consisting of about
20,000 4-node tetrahedral linear elements.
The boundary conditions set at the base of the specimen
include fixed vertical displacements and a node with prevented
horizontal translation necessary to suppress rigid body motion.
Perfectly rough boundaries are imposed by restricting the
lateral displacement at the upper and the lower ends. The initial
confinement pressure (5.5MPa for Maastricht Tuffeau and
110 MPa for Bleurswiller sandstone) is imposed by applying
on each boundary a normal pressure in equilibrium with an
equivalent internal stress state. The radial pressure is then
kept constant and the nodes at the top boundary are translated
vertically. The nominal axial strain rate during the shearing
stage is set to 1.0E-5 s−1.
Computations based on frictionless and perfectly rough
boundaries of Maastricht Tuffeau specimens are presented
in Fig. 3, which shows (a) the overall stress response of
the specimen intersecting with the yield surface outside the
potential zone of compaction localization, and (b) the stress-
strain responses associated with both boundary conditions, here
reported in terms of the deviatoric stress evolution as a function
of the applied axial strain. It is readily apparent that both
conditions lead to a comparable deviatoric stress peak followed
by stages of strain softening and eventually a marked gain in
resistance. Despite these similarities, some differences can be
observed, as the specimen with rough boundaries displayed a
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4Compaction banding fronts caused by frictional boundaries
21 3 4 5
MAX= 0.10
MIN= 0.0
Cumulative Plastic Volumetric Strain with Frictionless Boundaries
21 3 4 5
Cumulative Plastic Volumetric Strain with Perfectly Rough Boundaries
MAX= 0.10
MIN= 0.0
c
d
0 0.02 0.04 0.06 0.08
Nominal Axial Strain [-]
0
0.5
1
1.5
2
2.5
Nominal
Deviatoric Stress [MPa]
Confinement = 5.5 MPa
Perfectly rough ends
12345
Frictionless ends
b
a
Maastricht
Tuffeau
01234567
P [MPa]
0
1
2
3
4
5
q [MPa]
Zone of Potential
Compaction Localization
Fig. 3. Triaxial test simulations on specimens of Maastricht Tuffeau sheared outside the domain of compaction localization. The
comparison is between simulations based on frictionless boundaries and boundaries with perfect roughness. (a) The localization
domain is marked by a thick solid line. (b) Deformation response resulting from both simulations. (c-d) Compaction field displayed in
a form of cumulative volumetric plastic strain at various deformation intervals, also marked in (b).
21 3 4 5
MAX= 0.05
MIN= 0.0
Cumulative Plastic Volumetric Strain with Frictionless Boundaries
21 34 5
Cumulative Plastic Volumetric Strain with Perfectly Rough Boundaries
MAX= 0.05
MIN= 0.0
c
d
0 0.01 0.02 0.03 0.04
Nominal Axial Strain [-]
0
40
80
120
Nominal
Deviatoric Stress [MPa]
Confinement = 110 MPa
Perfectly rough ends
12
345
Frictionless ends
b
a
0 20 40 60 80 100 120 140
P [MPa]
0
20
40
60
q [MPa]
Zone of Potential
Compaction Localization
Bleurswiller
Sandstone
Fig. 4. Triaxial test simulations on specimens of Bleurswiller sandstone sheared outside the domain of compaction localization. The
comparison is between simulations based on frictionless boundaries and boundaries with perfect roughness. (a) The localization
domain is marked by a thick solid line. (b) Deformation response resulting from both simulations. (c-d) Compaction field displayed in
a form of cumulative volumetric plastic strain at various deformation intervals, also marked in (b)
tendency to deform with an extended post-yielding plateau and
a relatively less softening.
The evolution of the strain field is illustrated in Fig. 3c-
d, at various deformation intervals, also marked in (b). As
expected, frictionless boundaries promoted a homogeneous
strain field. By contrast, a heterogeneous deformation field
emerged in simulations with rough boundaries, as compaction
fronts propagated from the specimen’s ends towards the center.
Similar observations can be deduced from the simulations on
Bleurswiller sandstone (Fig. 4).
To further inspect these results, the stress path associated
with material points located at the center and near the
boundaries are presented in Fig. 5. Although the specimen was
globally subjected to triaxial compression, the boundary zones
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G. Shahin G. Buscarnera 5
3 3.5 4 4.5 5 5.5 6 6.5
Mean Stress, P [MPa]
0
1
2
3
4
5
Deviator Stress, q [MPa]
Maastricht Tuffeau
I
II
I
II
0 25 50 75 100 125
Mean Stress, P [MPa]
0
25
50
75
100
125
Deviator Stress, q [MPa]
Bleurswiller Sandstone
I
II
I
II
K0
K0
Fig. 5. Computed stress paths at Gauss points located at the bottom and the middle of the specimen based on material models
of (a) Maastricht Tuffeau and (b) Bleurswiller sandstone. The triaxial test simulation is based on perfectly rough boundaries. For
comparison purposes, the dotted line indicates the stress path generated by a material point analysis characterized by a one-
dimensional (oedometric) compression path (note that, in the case of Bleurswiller, the oedometric compression stress path evolves in
the close proximity of the yield surface).
0 0.005 0.01 0.015
Nominal Axial Strain [-]
-5000
0
5000 Bleurswiller Sandstone
Instability Zone
0 0.01 0.02 0.03 0.04 0.05
Nominal Axial Strain [-]
-40
0
40
Instability index,
HCB [MPa]
Maastricht Tuffeau
II II
I
I
Fig. 6. Assessment of material instability along the stress paths extracted from the finite element simulations and reported in Fig. 5.
exhibit stress paths resembling oedometric compression (see
Fig. 2for comparison), while the central portion maintains
a triaxial compression path. The deviation in stresses in
the vicinity of the specimen boundaries promotes an early
engagement of the yield surface resulting in the activation
of plasticity. It is, therefore, apparent that the kinematic
constraints imposed by boundary friction impact the ability of
the material to deform laterally close to the loading platens,
thus resulting in a response that shares similarities with one-
dimensional compaction.
The heterogeneous compaction fronts emerging from the
simulations are the product of concurrent factors, which
include the early activation of cap plasticity due to frictional
confinement, as well as the shift of the stress path towards
the compaction banding domain. The nature of these factors is
widely different, in that the former involves stable compaction,
while the latter implies unstable pore collapse. Hence, to
elucidate the origin of such fronts the simulations were further
inspected with the compaction instability index proposed
by Shahin et al. (2019a). The latter is based on a recent
reinterpretation of the stability theory for viscoplastic solids
proposed by Pisan`
o & di Prisco (2016), which when specialized
to axisymmetric deformation leads to the following differential
system:


¨a
2¨σr

=






−
∂Φ
∂f HC B
ν
1−νΦ∂2g
∂σ2
r
+ Φ ∂2g
∂σa∂σr
0−
∂Φ
∂f HC B −
2E
(1 −ν)Φ∂2g
∂σ2
r








˙a
2 ˙σr

(5)
where Eand νindicate the Young’s modulus and Poisson’s
ratio, respectively. σaand σrare the axial and radial stresses.
The scalar index HCB controls the stability of the differential
system (i.e., positive values involve stable, decelerating
one-dimensional compaction, while negative values lead to
unstable, accelerating pore collapse), and is equal to H−Hχ,
where His the hardening modulus and Hχa term that affects
the onset of bifurcation depending on the mode of deformation
inside the process zone (Shahin et al.,2019a). The index HCB
is assessed along the stress paths that are presented in Fig. 5and
its evolution is reported as a function of the nominal axial strain
(Fig. 6). For both rocks, it can be seen that near the boundaries
the activation of plasticity involves a stable response. However,
the kinematic constraint causes the path to intersect the zone
of compaction localization, which sets the stage for a possible
transition from the spreading of heterogeneous compaction to
the propagation of an active instability front. Such mechanism
is seen in Bleurswiller sandstone, for which non-normality
tends to maintain a large size of the localization domain (i.e.,
the altered stress path and the zone of compaction localization
tend to intersect upon growing deformation). By contrast,
the local response remains stable along the entire interval of
active compaction in Maastricht Tuffeau, for which softening
is prevalent and the extent of the localization domain tends
to shrink rapidly upon deformation (Buscarnera & Laverack,
2014). Such analysis reveals that material instability is not
required to promote a heterogeneous compaction front, which
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6Compaction banding fronts caused by frictional boundaries
4.75 5 5.25 5.5 5.75 6 6.25
Mean Stress, P [MPa]
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
Deviator Stress, q [MPa]
0123456
0
1
2
3
4
5
Friction
0.50
0.20
0.10
0.05
0.00
Fixed
MAX= 0.1
MIN= 0.0
Cumulative Plastic Volumetric Strain in Maastricht Tuffeau
0.05 0.10 0.20 0.50
0.00
1
3
Fig. 7. Triaxial test simulations conducted on Maastricht Tuffeau specimens including Coulomb-type frictional boundaries with friction
coefficient ranging from 0.05 to 0.50, compared with simulations based on frictionless boundaries and perfectly rough boundaries. (left)
Strain fields at the inception of heterogeneous compaction induced by boundary friction. (right) Computations extracted from Gauss
points located at the bottom of the specimens where compaction bands emerged, showing shifting from triaxial-like to oedometic-like
compression behaviour as the boundary friction shifts from 0to ∞.
100 105 110 115 120 125
Mean Stress, P [MPa]
0
5
10
15
20
25
30
35
Deviator Stress, q [MPa]
Friction
0.50
0.20
0.10
0.05
0.00
Fixed
MAX= 0.05
MIN= 0.0
Cumulative Plastic Volumetric Strain on Bleurswiller Sandstone
0.05 0.10 0.20 0.50
0.00
0 25 50 75 100 125
0
20
40
60
1
3
Fig. 8. Triaxial test simulations conducted on Bleurswiller sandstone specimens including Coulomb-type frictional boundaries with
friction coefficient ranging from 0.05 to 0.50, compared with simulations based on frictionless boundaries and perfectly rough
boundaries. (left) Strain fields at the inception of heterogeneous compaction induced by boundary friction. (right) Computations
extracted from Gauss points located at the bottom of the specimens where compaction bands emerged, showing shifting from triaxial-
like to oedometic-like compression behaviour as the boundary friction shifts from 0to ∞.
may emerge solely as a result of the spatially heterogeneous
engagement of the plastic resources of the material.
INFLUENCE OF THE VALUE OF BOUNDARY
FRICTION
The finite value of platen-specimen friction reflective of
actual laboratory tests is expected to generate intermediate
trends between the end-members discussed above (Labuz &
Bridell,1993). The role of the platen roughness on generating
heterogeneous fields is therefore assessed hereafter through
parametric analyses varying the boundary friction coefficient,
µ. Fig. 7and Fig. 8illustrate the computed responses for
Maastricht Tuffeau and Bleurswiller sandstone, respectively.
Snapshots of the strain field upon yielding are reported
on the left for all cases (µ= 0.0,0.05,0.10,0.20,0.50,and
∞), while the stress path of material points located near
the specimen boundaries is showed on the right-hand side
of the figures. It is readily apparent that boundary friction
caused an heterogeneous strain field in quasi-homogeneous
(with weak element) specimens regardless of the intensity of
friction, and in both hardening and softening regimes (i.e.,
for both Bleurswiller and Tuffeau, respectively). The local
stress paths associated with these simulations are characterized
by a tendency to shift from a triaxial-like behaviour to an
oedometric-like path as the friction shifts from 0.0→ ∞, while
the central portion of these simulations maintains a triaxial
behaviour regardless of the friction value. The analyses suggest
that this dependence is nonlinear, in that, equal increments
of friction do not lead to the same deviation in the stress
path. For example, in Maastricht Tuffeau, the increase of
boundary friction from 0.05 →0.10 and from 0.10 →0.20 was
associated with the same angular shift of the stress path towards
the perfectly oedometric path.
CONCLUSIONS
This study investigated the role of boundary friction on the
generation of heterogeneous compaction in specimens loaded
outside the domain of strain localization. The analysis was
based on a viscoplastic model calibrated for two porous rocks,
i.e., Maastricht Tuffeau and Bleurswiller sandstone. These
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G. Shahin G. Buscarnera 7
rocks were selected in that they display softening and hardening
behaviour in the plastic cap zone, respectively, as well as
different mechanisms of compaction band propagation, thus
providing distinct end members of behaviour. Our virtual
triaxial compression experiments showed that the specimens
with boundary friction displayed heterogeneous deformation
associated with spatially varying evolution of the stress paths.
While the center of the specimen continued to display triaxial-
like responses, the portions of the specimen near the boundaries
of both simulated rocks exhibited a deformation behaviour
that resembles oedometric compression. This behaviour can be
explained in light of the ability of the boundary constraints
to hinder the lateral deformation of the specimen’s ends, thus
justifying the deviation of the local stress paths. In case of
the Maastricht Tuffeau, compaction subsequently emerged as
a result of spatial variations in the activation of plasticity,
with compaction fronts able to propagate even with locally
stable conditions (i.e., compaction banding was not necessary
to generate the fronts). By contrast, in Bleurswiller sandstone a
transition from heterogeneous compaction to compaction band
front was possible, in that its strain localization domain was
sustained upon deformation until its intersection with the local
stress path. The analyses revealed intriguing results, in that the
rock simulated with strain-softening (i.e., Maastricht Tuffeau)
was found to generate heterogeneous compaction propagating
across the domain in a way that resembled the fronts caused
by the movement of an active compaction band zone. By
contrast, with the material simulated with strain-hardening (i.e.,
Bleurswiller sandstone) the heterogeneous plastic compaction
was unable to propagate across the domain, being restricted
only to the boundary zone. Specifically, in this second scenario
after a short-lived stage of compaction band propagation
near the boundary, the deformation process was found to be
dominated by homogeneous plastic deformation throughout
the rest of the domain. Parametric analyses examining the
effects of boundary friction disclosed that the higher the platen
roughness, the closer the local behaviour is to one-dimensional
(oedometric) compression. These findings suggest that in
laboratory experiments on porous rocks, where specimen-
platen friction is unavoidable, boundary effects can bias
the interpretation of the observations by producing modes
of heterogeneous deformation which resemble compaction
band fronts or that may convert into them because of stress
path deviations. From this point of view, this work provides
argument that strengthen the importance of using full-field
measurements and simulations in the context of constitutive
model formulation and calibration. Specifically, our results
suggest that spurious forms of compaction bands may occur as
a result of boundary effects (e.g., interactions between ground
and penetrating probes, feedbacks between reservoir faults
and caprock, concurrent initiation of multiple deformation
zones around a wellbore). Understanding the factors leading
to the onset of strain heterogeneity is therefore necessary
to enhance our ability to predict the initiation and fate of
subsurface deformations when the competition between the
material response (e.g., strain localization) and system effects
(e.g., boundary conditions) is crucial to identify the most likely
outcome.
ACKNOWLEDGEMENTS
This research was supported by the U.S. Department of Energy
through grant DE-SC0017615. The authors wish to thank John
Rudnicki for the valuable discussions during the development
of the work.
APPENDIX: CONSTITUTIVE MODEL
The constitutive model used for this work is based on the
expression proposed by Lagioia et al. (1996), which can be
written as follows:
f
g=AK1h/Ch
h·B−K2h/Ch
hP?−P?
c= 0 (6a)
Ah= 1 + η?
K1hMh
(6b)
Bh= 1 + η?
K2hMh
(6c)
Ch= (1 −µh)(K1h−K2h)(6d)
d=˙p
vol
˙p
dev
=∂g/∂p
∂g/∂q =µg(Mg−η?)( αgMg
η?+ 1) (7)
with the functions K1h,2hdefined as:
K1h/2h=µh(1 −αh)
2 (1 −µh) 1±s1−4αh(1 −µh)
µh(1 −αh)2!.(8)
pand qrepresent the mean and deviator stresses, respectively,
while η?is the corresponding stress ratio, defined as:
p=1
3σij δij q=r3
2sij sij η?=q
p+Pt(9)
in which sij =σij −pδij ,δij is the Kronecker delta. Under
axisymmetric compression, pand qare specified as follows:
p=σ1+ 2σ3
3q=σ1−σ3,(10)
with σ1and σ3being the maximum and minimum principal
stress, respectively. The shape of the two surfaces in Eqs. (6)
is controlled by Mh,µh, and αh, for which the subscript h
makes reference to either the yield surface (h≡f) or the plastic
potential (h≡g). These expressions provide considerable
flexibility to fit experimental data, in that distinct sets of
parameters can be used to adjust the degree of non-associativity
and the shape of the elastic domain.
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