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Crack growth and coalescence from discontinuities is the fundamental process that underlies the majority of rock mass failures. Still, however, numerical simulation of this process remains a formidable challenge. The main reasons include that tensile and shear fractures are often mixed in the cracking process, that the cracking patterns depend strongly on the configuration of preexisting discontinuities, and that the cracking process may take place at different scales. The purpose of this paper is to evaluate the capabilities of the recently developed double-phase-field model (Fei and Choo, 2021) for simulating mixed-mode fracture in laboratory- and field-scale rocks with discontinuities. Simulation results show that the double-phase-field model can well reproduce laboratory test data, in terms of not only qualitative mixed-mode cracking patterns but also quantitative stress–strain responses. The results further demonstrate that the model can simulate complex rock fracture processes at the field scale such as slope failure due to crack growth and coalescence from joints.
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1. INTRODUCTION
Crack growth and coalescence from discontinuities is a
primary mechanism underlying the failure of rock masses.
A myriad of studies has investigated the processes and
consequences of cracking in rocks with discontinuities at
the field and laboratory scales, bridging their observations
across scales (see Einstein, 2021 and references therein).
These studies have commonly found that rock cracking
processes usually involve a combination of tensile (mode
I) and shear (mode II) fractures.
Meanwhile, many types of numerical methods have been
advanced to simulate crack growth and coalescence from
rock discontinuities. These methods may be grouped into
two categories: (i) macroscopic (continuum-based)
methods, and (ii) microscopic (grain-based) methods.
Macroscopic methods (e.g., Bobet and Einstein, 1998a;
Shen and Stephansson, 1994; Wu and Wong, 2012) can
be applied to both laboratory- and field-scale problems;
however, they require one to carefully select criteria for
crack nucleation and propagation and often involve
complex algorithms for simulating the onset and growth
of cracks. On the other hand, microscopic methods (e.g.,
Zhang and Wong, 2012, 2013; Zhang et al., 2019) can
simulate fracturing processes based on first principles and
without complex algorithms; however, their applicability
is inherently limited to small-scale problems. Thus,
although the microscopic methods are an indispensable
tool to gain insights into the physics of rock cracking, the
macroscopic methods remain useful for addressing
engineering-scale problems in rock mechanics.
Among the macroscopic methods for fracture simulation,
the phase-field method is receiving growing attention
from the geomechanics community (e.g., Lee et al., 2016;
Zhang et al., 2017; Bryant and Sun, 2018; Choo and Sun,
2018a, 2018b; Ha et al., 2018; Fei and Choo, 2020b;
Heider and Sun, 2020). The main reason is twofold: (i)
the phase-field method can simulate crack nucleation
based on sound physical principles, and (ii) it can capture
the complex geometry of crack propagation, kinking,
branching, and coalescence without any algorithms.
Phase-field models of fracture have also been applied to
field-scale problems in geomechanics, e.g., hydraulic
fracturing in a reservoir (Santillan et al., 2018) and slip
surface growth in a slope (Fei and Choo, 2020b).
Regarding mixed-mode fracture in rocks, however, the
phase-field method has only been applied to laboratory-
scale processes. Indeed, earlier phase-field models of
mixed-mode fracture in rocks (e.g., Zhang et al., 2017;
Bryant and Sun, 2018) are not fully adequate for modeling
fracture across different scales because their parameters
should be chosen depending on the value of the so-called
phase-field length parameter. The reason is that the
strength of these phase-field models is a function of the
phase-field length parameter, which is originally
introduced for geometric approximation of sharp cracks.
This length sensitivity should be eliminated to apply a
phase-field model to simulate fracture across scales.
ARMA 211223
Phase-field modeling of mixed-mode fracture in rocks
with discontinuities: From laboratory scale to field scale
Fei, F., Sun, Y., Wong, L.N.Y. and Choo, J.
Copyright 2021 ARMA, American Rock Mechanics Association
This paper was prepared for presentation at the 55th US Rock Mechanics/Geomechanics Symposium held in Houston, Texas, USA, 20-23 June
2021. 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: Crack growth and coalescence from discontinuities is the fundamental process that underlies the majority of rock mass
failures. Still, however, numerical simulation of this process remains a formidable challenge. The main reasons include that tensile
and shear fractures are often mixed in the cracking process, that the cracking patterns depend strongly on the configuration of
preexisting discontinuities, and that the cracking process may take place at different scales. The purpose of this paper is to evaluate
the capabilities of the recently developed double-phase-field model (Fei and Choo, 2021) for simulating mixed-mode fracture in
laboratory- and field-scale rocks with discontinuities. Simulation results show that the double-phase-field model can well reproduce
laboratory test data, in terms of not only qualitative mixed-mode cracking patterns but also quantitative stressstrain responses. The
results further demonstrate that the model can simulate complex rock fracture processes at the field scale such as slope failure due to
crack growth and coalescence from joints.
The University of Hong Kong, Hong Kong
To overcome this limitation of the existing phase-field
models for mixed-mode fracture, Fei and Choo, 2021
have recently developed a double-phase-field model that
combines two types of length-insensitive phase-field
models: one for cohesive tensile fracture (Wu, 2017) and
the other for frictional shear fracture (Fei and Choo,
2020b). The double-phase-field model has been validated
against laboratory data of Bobet and Einstein, 1998b.
Since the new double-phase-field model is theoretically
insensitive to the length parameters, it has two features
that have important implications for engineering practice.
First, the model parameters can be determined directly
from laboratory test data. Second, the model can be used
to simulate mixed-mode fracture across different scales.
Nevertheless, these two features of the double-phase-field
model have not been thoroughly evaluated in the previous
work.
The purpose of the present work is to investigate these
two features of the double-phase-field model. For this
purpose, we first validate the double-phase-field model
with a set of laboratory test results on flawed specimens
conducted by Wong and Einstein, 2009a, 2009b. We then
apply the model to simulate the failure of a rock slope
with en-echelon joints.
2. DOUBLE-PHASE-FIELD MODEL
This section provides a brief overview of the double-
phase-field model for mixed-mode fracture in rocks (Fei
and Choo, 2021).
2.1. Double-phase-field approximation
The double-phase-field model diffusely approximates the
sharp geometry of tensile and shear cracks by introducing
two phase-field variables:
!!
for tensile cracks and
!!!
for
shear cracks. Both of the phase-field variables range from
0 to 1, in which 0 denotes the intact region and 1 the fully
damaged region for the corresponding fracture mode. Fig.
1 illustrates the double-phase-field approximation.
Fig. 1. Double-phase-field approximation of discrete cracks in
mode I (in red) and mode II (in blue). After Fei and Choo, 2021.
The model simulates the mixed-mode fracturing process
as evolutions of the two phase-fields. These evolutions
are described by the following two governing equations:
"#!
"$!!%&!'(!
)* $+*#, - ,!!" + ' +!!%. / $0%
"#!!
"
$
!!!
%
&!! '(!!
)*
$
+*#, - ,!!! " + ' +!!!
%
. / $+%
These equations share the same structure, containing three
terms that specialize to the individual fracturing modes:
(i) degradation functions,
#!$!!%
and
1#!!
$
!!!
%, (ii) crack
driving forces,
&!
and
&!!2
and (iii) critical fracture
energies,
(!
and
(!!2
for mode I and mode II fractures,
respectively. Also,
*
is the length parameter controlling
the width of the phase-field approximation. The same
length parameter is used for both modes I and II cracks.
Remarkably, the expressions for
#!$!!%
and
#!!$!!! %
should be chosen carefully to correctly incorporate the
cohesive nature of the mode I fracture and the frictional
nature of the mode II fracture. Specific expressions for
such
#!$!!%
and
#!!$!!! %
can be adopted from existing
single-phase-field models for modes I and II fractures,
respectively. In the present model,
#!$!!%
is adopted from
Wu, 2017 and
#!!$!!! %
from Fei and Choo, 2020b.
2.2. Coupling of mode I and mode II fractures
The main challenge for double-phase-field modeling of
mixed-mode fracture is how to rigorously couple the two
phase-fields that individually represent mode I and mode
II fractures. The present model addresses this challenge
through three approaches: (i) decomposition of the strain
energy according to the crack orientation, (ii) calculation
of the modes I and II crack driving forces depending on
the contact condition, and (iii) determination of the
dominant fracturing mode based on an energy-based
criterion for mixed-mode fracture.
In phase-field modeling of fracture, strain energy plays a
central role in calculating the crack driving force. While
the strain energy for pure mode I or pure mode II fracture
can be evaluated straightforwardly, the strain energy
should be properly decomposed into its tensile, shear, and
compression parts for modeling mixed-mode fracture. For
this purpose, the present model makes use of the crack-
oriented stress decomposition scheme proposed by Fei
and Choo, 2020a, which has been verified against various
problems involving frictional contact. This scheme allows
us to decompose the strain energy additively into the
tensile part that drives mode I fracture, the shear part that
drives mode II fracture, and the compression part that
does not drive any fracture.
The decomposed strain energies are then combined with
other types of potential energies (the frictional energy, the
crack surface energy, and the external energy) to calculate
the mode I and mode II crack driving forces,
&!
and
&!!3
The specific expressions for these potential energy terms
and hence the crack driving forces are different in the
three types of contact conditions: open, stick, and slip.
The contact condition of each material point can be
identified as described in Fei and Choo, 2020a, and the
Original discrete problem Double-phase-eld approximation
t
u
ΓII
t
u
dI(x,t)
dII(x,t)
1.0
dI (tensile)
0.0
1.0
dII (shear)
specific energy terms in the three contact conditions are
derived in Fei and Choo, 2021.
The remaining task is to evaluate the crack driving forces
such that mode I and mode II cracks do not initiate at the
same material point. For this purpose, the double-phase-
field model employs the F-criterion an energy-based
criterion for mixed-mode fracture proposed by Shen and
Stephansson, 1994. Specifically, the F-criterion is used to
identify the dominant fracturing mode and its direction
according to the contact condition. Having determined the
fracturing mode and its direction, the crack driving forces
can be finally calculated by taking partial derivatives of
the potential energy with respect to the two phase-fields.
The resultant formulations are presented in Fei and Choo,
2021.
3. NUMERICAL SIMULATION
In this section, we evaluate the capabilities of the double-
phase-field model for simulating mixed-mode fracturing
processes in rocks with preexisting discontinuities. The
evaluation is first performed at the laboratory scale where
the cracking processes can be observed both qualitatively
and quantitatively and then at the field scale which is
more relevant to practical engineering problems.
3.1. Cracking at the laboratory scale
To begin, we simulate a series of uniaxial compression
tests conducted by Wong and Einstein, 2009a, 2009b on
laboratory molded gypsum specimens with straight open
flaws. We consider two types of specimensone with a
single flaw and the other with double flaws and two flaw
inclination angles (
4
) 45° and 60° for each type of
specimen. As a result, total four cases are simulated. The
detailed setup of the problem is illustrated in Fig. 2.
Fig. 2. Geometry and boundary conditions of the uniaxial
compression test simulation. (a) single-flawed specimen, and
(b) double-flawed specimen.
Table 1 lists the model parameters calibrated against the
experimental results from the four cases. Considering that
the specimens may show material variability, some of the
parameters are fine-tuned for individual specimens. The
phase-field length parameter is set to be 0.2 mm. It is
again noted that the length parameter of the double-phase-
field model does not affect the material behavior, having
a geometric meaning only.
Table 1. Material parameters calibrated for the laboratory-
scale gypsum specimens. (The full definitions of the material
parameters and symbols are provided in Fei and Choo, 2021.)
Parameter
Unit
Value
Single
𝛼 = 45
Single
𝛼 = 60
Double
𝛼 = 45
Double
𝛼 = 60
Young's
modulus 𝐸
GPa
6.30
5.00
5.94
5.76
Poisson's
ratio 𝑣
-
0.15
0.15
0.15
0.15
Tensile
strength 𝜎!
MPa
3.2
3.2
3.2
3.2
Cohesion
strength 𝑐"
MPa
11
14
10
10
Peak friction
angle 𝜙!
deg
31
31
31
31
Residual
friction
angle 𝜙#
deg
28
28
28
28
Mode I
fracture
energy 𝐺$
J/m%
14
14
13
13
Mode II
fracture
energy 𝐺$$
J/m%
200
200
130
130
Fig. 3 presents the simulated cracking process and the
experimentally observed cracking pattern in the single-
flawed specimen with
4 . 56
. In the simulation results,
wing cracks initiate from the flaw tips (stage A) and then
secondary shear cracks grow (stage B). As the specimen
is further compressed, secondary tensile cracks develop
from the shear crack fronts (stage C). The final cracking
pattern agrees well with the experimental result in terms
of both the crack geometry and the fracture mode. In Fig.
4, the simulation and experimental results are further
compared with respect to their stressstrain curves. It can
be seen that the two results match very well.
The same kind of validation is repeated for the single-
flawed specimen with
4 . 7/
, and the simulation and
experimental results are compared in Fig. 5 and Fig. 6. It
can again be seen that the simulation results are
remarkably similar to the experimental results, in both
qualitative and quantitative senses.
We next consider cracking processes in double-flawed
specimens which manifest more complex patterns than
the previous ones.
76.2 mm
152.4 mm
x
y
ˆ
uy
Zoom-in region of initial coplanar aws
ligament length
= 12.7 mm
α
12.7 mm
1.27 mm
76.2 mm
152.4 mm
x
y
ˆ
uy
12.7 mm
1.27 mm
α
(a) Single-awed specimen (b) Double-awed specimen
Fig. 7 presents the simulated cracking process in the
double-flawed specimen with
4 . 56
, which shows the
following three stages. First, wing cracks initiate from
both the inner and outer tips of the two flaws (stage A).
Then shear cracks develop from the inner tips of the flaws,
subsequently coalesced by a tensile crack (stage B).
Finally, new shear cracks emerge from the outer tips of
the flaws and then secondary tensile cracks (stage C). It
can be seen that the double-phase-field model reproduces
most of the experimentally observed cracking patterns,
particularly the morphology and modes of the
coalescence cracks. Further, as can be seen from Fig. 8,
the stress-strain response of the numerical simulation is
close to the experimental result. The same conclusion can
also be drawn from the results of the double-flawed
specimen with
4 . 7/
, see Fig. 9 and Fig. 10.
All the foregoing results consistently indicate that the
double-phase-field model can accurately simulate mixed-
mode cracking processes in laboratory-scale rocks with
discontinuities.
Fig. 3. Cracking in the single-flawed specimen with
𝛼 = 45
:
the simulated cracking process and the experimentally observed
cracking pattern.
Fig. 4. Cracking in the single-flawed specimen with
𝛼 = 45
:
the stressstrain curves from the simulation and the experiment.
Fig. 5. Cracking in the single-flawed specimen with
𝛼 = 60
:
the simulated cracking process and the experimentally observed
cracking pattern.
Fig. 6. Cracking in the single-flawed specimen with
𝛼 = 60
:
the stressstrain curves from the simulation and experiment.
Fig. 7. Cracking in the double-flawed specimen with
𝛼 = 45
:
the simulated cracking process and the experimentally observed
cracking pattern.
dI
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1.0
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<latexit sha1_base64="7uJXpholrjhNJkaCoHTl2i8Xdn8=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgadkVIR4jXjxGNA9IljA76U2GzM4uM7NCWAL+gBcPinj1i7z5N04eB00saCiquunuClPBtfG8b6ewtr6xuVXcLu3s7u0flA+PmjrJFMMGS0Si2iHVKLjEhuFGYDtVSONQYCsc3Uz91iMqzRP5YMYpBjEdSB5xRo2V7n3X65UrnuvNQFaJvyAVWKDeK391+wnLYpSGCap1x/dSE+RUGc4ETkrdTGNK2YgOsGOppDHqIJ+dOiFnVumTKFG2pCEz9fdETmOtx3FoO2NqhnrZm4r/eZ3MRFdBzmWaGZRsvijKBDEJmf5N+lwhM2JsCWWK21sJG1JFmbHplGwI/vLLq6R54fqe699dVmrXT/M4inACp3AOPlShBrdQhwYwGMAzvMKbI5wX5935mLcWnEWEx/AHzucPeTGNpg==</latexit>
0.0
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1.0
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<latexit sha1_base64="7uJXpholrjhNJkaCoHTl2i8Xdn8=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgadkVIR4jXjxGNA9IljA76U2GzM4uM7NCWAL+gBcPinj1i7z5N04eB00saCiquunuClPBtfG8b6ewtr6xuVXcLu3s7u0flA+PmjrJFMMGS0Si2iHVKLjEhuFGYDtVSONQYCsc3Uz91iMqzRP5YMYpBjEdSB5xRo2V7n3X65UrnuvNQFaJvyAVWKDeK391+wnLYpSGCap1x/dSE+RUGc4ETkrdTGNK2YgOsGOppDHqIJ+dOiFnVumTKFG2pCEz9fdETmOtx3FoO2NqhnrZm4r/eZ3MRFdBzmWaGZRsvijKBDEJmf5N+lwhM2JsCWWK21sJG1JFmbHplGwI/vLLq6R54fqe699dVmrXT/M4inACp3AOPlShBrdQhwYwGMAzvMKbI5wX5935mLcWnEWEx/AHzucPeTGNpg==</latexit>
Simulation Experiment
A B C
dI
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<latexit sha1_base64="+nmd10vWUDHlgLUHWPjT28hl9nc=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GNBBL1VMG2hDWWz2bZLN5uwOxFK6G/w4kERr/4gb/4bt20O2vpg4PHeDDPzwlQKg6777ZTW1jc2t8rblZ3dvf2D6uFRyySZZtxniUx0J6SGS6G4jwIl76Sa0ziUvB2Ob2Z++4lrIxL1iJOUBzEdKjEQjKKV/Kif30/71Zpbd+cgq8QrSA0KNPvVr16UsCzmCpmkxnQ9N8UgpxoFk3xa6WWGp5SN6ZB3LVU05ibI58dOyZlVIjJItC2FZK7+nshpbMwkDm1nTHFklr2Z+J/XzXBwHeRCpRlyxRaLBpkkmJDZ5yQSmjOUE0so08LeStiIasrQ5lOxIXjLL6+S1kXdc+vew2WtcVvEUYYTOIVz8OAKGnAHTfCBgYBneIU3RzkvzrvzsWgtOcXMMfyB8/kD1wWOsw==</latexit>
<latexit sha1_base64="+nmd10vWUDHlgLUHWPjT28hl9nc=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GNBBL1VMG2hDWWz2bZLN5uwOxFK6G/w4kERr/4gb/4bt20O2vpg4PHeDDPzwlQKg6777ZTW1jc2t8rblZ3dvf2D6uFRyySZZtxniUx0J6SGS6G4jwIl76Sa0ziUvB2Ob2Z++4lrIxL1iJOUBzEdKjEQjKKV/Kif30/71Zpbd+cgq8QrSA0KNPvVr16UsCzmCpmkxnQ9N8UgpxoFk3xa6WWGp5SN6ZB3LVU05ibI58dOyZlVIjJItC2FZK7+nshpbMwkDm1nTHFklr2Z+J/XzXBwHeRCpRlyxRaLBpkkmJDZ5yQSmjOUE0so08LeStiIasrQ5lOxIXjLL6+S1kXdc+vew2WtcVvEUYYTOIVz8OAKGnAHTfCBgYBneIU3RzkvzrvzsWgtOcXMMfyB8/kD1wWOsw==</latexit>
<latexit sha1_base64="+nmd10vWUDHlgLUHWPjT28hl9nc=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GNBBL1VMG2hDWWz2bZLN5uwOxFK6G/w4kERr/4gb/4bt20O2vpg4PHeDDPzwlQKg6777ZTW1jc2t8rblZ3dvf2D6uFRyySZZtxniUx0J6SGS6G4jwIl76Sa0ziUvB2Ob2Z++4lrIxL1iJOUBzEdKjEQjKKV/Kif30/71Zpbd+cgq8QrSA0KNPvVr16UsCzmCpmkxnQ9N8UgpxoFk3xa6WWGp5SN6ZB3LVU05ibI58dOyZlVIjJItC2FZK7+nshpbMwkDm1nTHFklr2Z+J/XzXBwHeRCpRlyxRaLBpkkmJDZ5yQSmjOUE0so08LeStiIasrQ5lOxIXjLL6+S1kXdc+vew2WtcVvEUYYTOIVz8OAKGnAHTfCBgYBneIU3RzkvzrvzsWgtOcXMMfyB8/kD1wWOsw==</latexit>
dII
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<latexit sha1_base64="vzZSbbp81TAYM0Eh7KJCLfWrZxE=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GNBBHurYD+gDWWz2bRrN7thdyOUkP/gxYMiXv0/3vw3btsctPXBwOO9GWbmBQln2rjut1NaW9/Y3CpvV3Z29/YPqodHHS1TRWibSC5VL8CaciZo2zDDaS9RFMcBp91gcjPzu09UaSbFg5km1I/xSLCIEWys1AmHWbOZD6s1t+7OgVaJV5AaFGgNq1+DUJI0psIQjrXue25i/AwrwwineWWQappgMsEj2rdU4JhqP5tfm6Mzq4QoksqWMGiu/p7IcKz1NA5sZ4zNWC97M/E/r5+a6NrPmEhSQwVZLIpSjoxEs9dRyBQlhk8twUQxeysiY6wwMTagig3BW355lXQu6p5b9+4va43bIo4ynMApnIMHV9CAO2hBGwg8wjO8wpsjnRfn3flYtJacYuYY/sD5/AFsdI8G</latexit>
<latexit sha1_base64="vzZSbbp81TAYM0Eh7KJCLfWrZxE=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GNBBHurYD+gDWWz2bRrN7thdyOUkP/gxYMiXv0/3vw3btsctPXBwOO9GWbmBQln2rjut1NaW9/Y3CpvV3Z29/YPqodHHS1TRWibSC5VL8CaciZo2zDDaS9RFMcBp91gcjPzu09UaSbFg5km1I/xSLCIEWys1AmHWbOZD6s1t+7OgVaJV5AaFGgNq1+DUJI0psIQjrXue25i/AwrwwineWWQappgMsEj2rdU4JhqP5tfm6Mzq4QoksqWMGiu/p7IcKz1NA5sZ4zNWC97M/E/r5+a6NrPmEhSQwVZLIpSjoxEs9dRyBQlhk8twUQxeysiY6wwMTagig3BW355lXQu6p5b9+4va43bIo4ynMApnIMHV9CAO2hBGwg8wjO8wpsjnRfn3flYtJacYuYY/sD5/AFsdI8G</latexit>
<latexit sha1_base64="vzZSbbp81TAYM0Eh7KJCLfWrZxE=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GNBBHurYD+gDWWz2bRrN7thdyOUkP/gxYMiXv0/3vw3btsctPXBwOO9GWbmBQln2rjut1NaW9/Y3CpvV3Z29/YPqodHHS1TRWibSC5VL8CaciZo2zDDaS9RFMcBp91gcjPzu09UaSbFg5km1I/xSLCIEWys1AmHWbOZD6s1t+7OgVaJV5AaFGgNq1+DUJI0psIQjrXue25i/AwrwwineWWQappgMsEj2rdU4JhqP5tfm6Mzq4QoksqWMGiu/p7IcKz1NA5sZ4zNWC97M/E/r5+a6NrPmEhSQwVZLIpSjoxEs9dRyBQlhk8twUQxeysiY6wwMTagig3BW355lXQu6p5b9+4va43bIo4ynMApnIMHV9CAO2hBGwg8wjO8wpsjnRfn3flYtJacYuYY/sD5/AFsdI8G</latexit>
1.0
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0.0
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1.0
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dI
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dII
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1.0
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<latexit sha1_base64="7uJXpholrjhNJkaCoHTl2i8Xdn8=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgadkVIR4jXjxGNA9IljA76U2GzM4uM7NCWAL+gBcPinj1i7z5N04eB00saCiquunuClPBtfG8b6ewtr6xuVXcLu3s7u0flA+PmjrJFMMGS0Si2iHVKLjEhuFGYDtVSONQYCsc3Uz91iMqzRP5YMYpBjEdSB5xRo2V7n3X65UrnuvNQFaJvyAVWKDeK391+wnLYpSGCap1x/dSE+RUGc4ETkrdTGNK2YgOsGOppDHqIJ+dOiFnVumTKFG2pCEz9fdETmOtx3FoO2NqhnrZm4r/eZ3MRFdBzmWaGZRsvijKBDEJmf5N+lwhM2JsCWWK21sJG1JFmbHplGwI/vLLq6R54fqe699dVmrXT/M4inACp3AOPlShBrdQhwYwGMAzvMKbI5wX5935mLcWnEWEx/AHzucPeTGNpg==</latexit>
<latexit sha1_base64="7uJXpholrjhNJkaCoHTl2i8Xdn8=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgadkVIR4jXjxGNA9IljA76U2GzM4uM7NCWAL+gBcPinj1i7z5N04eB00saCiquunuClPBtfG8b6ewtr6xuVXcLu3s7u0flA+PmjrJFMMGS0Si2iHVKLjEhuFGYDtVSONQYCsc3Uz91iMqzRP5YMYpBjEdSB5xRo2V7n3X65UrnuvNQFaJvyAVWKDeK391+wnLYpSGCap1x/dSE+RUGc4ETkrdTGNK2YgOsGOppDHqIJ+dOiFnVumTKFG2pCEz9fdETmOtx3FoO2NqhnrZm4r/eZ3MRFdBzmWaGZRsvijKBDEJmf5N+lwhM2JsCWWK21sJG1JFmbHplGwI/vLLq6R54fqe699dVmrXT/M4inACp3AOPlShBrdQhwYwGMAzvMKbI5wX5935mLcWnEWEx/AHzucPeTGNpg==</latexit>
0.0
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<latexit sha1_base64="f/4IzOIyMkszHgB4/4P/Ekkxag8=">AAAB6nicbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeKF48VbS20S8mms21oNrskWaEsBf+AFw+KePUXefPfmH4ctPXBDI/3JmTmhakUxnreNymsrK6tbxQ3S1vbO7t75f2DpkkyzbHBE5noVsgMSqGwYYWV2Eo1sjiU+BAOryf+wyNqIxJ1b0cpBjHrKxEJzqyT7ryq1y1XXJ+CLhN/TiowR71b/ur0Ep7FqCyXzJi276U2yJm2gksclzqZwZTxIetj21HFYjRBPl11TE+c0qNRol0pS6fq7xc5i40ZxaGbjJkdmEVvIv7ntTMbXQa5UGlmUfHZR1EmqU3o5G7aExq5lSNHGNfC7Ur5gGnGrUun5ELwF09eJs2zqu9V/dvzSu3qaRZHEY7gGE7BhwuowQ3UoQEc+vAMr/BGJHkh7+RjNlog8wgP4Q/I5w93q42l</latexit>
<latexit sha1_base64="f/4IzOIyMkszHgB4/4P/Ekkxag8=">AAAB6nicbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeKF48VbS20S8mms21oNrskWaEsBf+AFw+KePUXefPfmH4ctPXBDI/3JmTmhakUxnreNymsrK6tbxQ3S1vbO7t75f2DpkkyzbHBE5noVsgMSqGwYYWV2Eo1sjiU+BAOryf+wyNqIxJ1b0cpBjHrKxEJzqyT7ryq1y1XXJ+CLhN/TiowR71b/ur0Ep7FqCyXzJi276U2yJm2gksclzqZwZTxIetj21HFYjRBPl11TE+c0qNRol0pS6fq7xc5i40ZxaGbjJkdmEVvIv7ntTMbXQa5UGlmUfHZR1EmqU3o5G7aExq5lSNHGNfC7Ur5gGnGrUun5ELwF09eJs2zqu9V/dvzSu3qaRZHEY7gGE7BhwuowQ3UoQEc+vAMr/BGJHkh7+RjNlog8wgP4Q/I5w93q42l</latexit>
1.0
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<latexit sha1_base64="7uJXpholrjhNJkaCoHTl2i8Xdn8=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgadkVIR4jXjxGNA9IljA76U2GzM4uM7NCWAL+gBcPinj1i7z5N04eB00saCiquunuClPBtfG8b6ewtr6xuVXcLu3s7u0flA+PmjrJFMMGS0Si2iHVKLjEhuFGYDtVSONQYCsc3Uz91iMqzRP5YMYpBjEdSB5xRo2V7n3X65UrnuvNQFaJvyAVWKDeK391+wnLYpSGCap1x/dSE+RUGc4ETkrdTGNK2YgOsGOppDHqIJ+dOiFnVumTKFG2pCEz9fdETmOtx3FoO2NqhnrZm4r/eZ3MRFdBzmWaGZRsvijKBDEJmf5N+lwhM2JsCWWK21sJG1JFmbHplGwI/vLLq6R54fqe699dVmrXT/M4inACp3AOPlShBrdQhwYwGMAzvMKbI5wX5935mLcWnEWEx/AHzucPeTGNpg==</latexit>
<latexit sha1_base64="7uJXpholrjhNJkaCoHTl2i8Xdn8=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgadkVIR4jXjxGNA9IljA76U2GzM4uM7NCWAL+gBcPinj1i7z5N04eB00saCiquunuClPBtfG8b6ewtr6xuVXcLu3s7u0flA+PmjrJFMMGS0Si2iHVKLjEhuFGYDtVSONQYCsc3Uz91iMqzRP5YMYpBjEdSB5xRo2V7n3X65UrnuvNQFaJvyAVWKDeK391+wnLYpSGCap1x/dSE+RUGc4ETkrdTGNK2YgOsGOppDHqIJ+dOiFnVumTKFG2pCEz9fdETmOtx3FoO2NqhnrZm4r/eZ3MRFdBzmWaGZRsvijKBDEJmf5N+lwhM2JsCWWK21sJG1JFmbHplGwI/vLLq6R54fqe699dVmrXT/M4inACp3AOPlShBrdQhwYwGMAzvMKbI5wX5935mLcWnEWEx/AHzucPeTGNpg==</latexit>
tensile
shear
1.0
0.0
1.0
0 0.2 0.4 0.6
0
5
10
15
20
25
30
A
B
C
Strain (%)
Stress (MPa)
Experiment
Simulation
A B C
Simulation Experiment
dI
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<latexit sha1_base64="+nmd10vWUDHlgLUHWPjT28hl9nc=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GNBBL1VMG2hDWWz2bZLN5uwOxFK6G/w4kERr/4gb/4bt20O2vpg4PHeDDPzwlQKg6777ZTW1jc2t8rblZ3dvf2D6uFRyySZZtxniUx0J6SGS6G4jwIl76Sa0ziUvB2Ob2Z++4lrIxL1iJOUBzEdKjEQjKKV/Kif30/71Zpbd+cgq8QrSA0KNPvVr16UsCzmCpmkxnQ9N8UgpxoFk3xa6WWGp5SN6ZB3LVU05ibI58dOyZlVIjJItC2FZK7+nshpbMwkDm1nTHFklr2Z+J/XzXBwHeRCpRlyxRaLBpkkmJDZ5yQSmjOUE0so08LeStiIasrQ5lOxIXjLL6+S1kXdc+vew2WtcVvEUYYTOIVz8OAKGnAHTfCBgYBneIU3RzkvzrvzsWgtOcXMMfyB8/kD1wWOsw==</latexit>
<latexit sha1_base64="+nmd10vWUDHlgLUHWPjT28hl9nc=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GNBBL1VMG2hDWWz2bZLN5uwOxFK6G/w4kERr/4gb/4bt20O2vpg4PHeDDPzwlQKg6777ZTW1jc2t8rblZ3dvf2D6uFRyySZZtxniUx0J6SGS6G4jwIl76Sa0ziUvB2Ob2Z++4lrIxL1iJOUBzEdKjEQjKKV/Kif30/71Zpbd+cgq8QrSA0KNPvVr16UsCzmCpmkxnQ9N8UgpxoFk3xa6WWGp5SN6ZB3LVU05ibI58dOyZlVIjJItC2FZK7+nshpbMwkDm1nTHFklr2Z+J/XzXBwHeRCpRlyxRaLBpkkmJDZ5yQSmjOUE0so08LeStiIasrQ5lOxIXjLL6+S1kXdc+vew2WtcVvEUYYTOIVz8OAKGnAHTfCBgYBneIU3RzkvzrvzsWgtOcXMMfyB8/kD1wWOsw==</latexit>
<latexit sha1_base64="+nmd10vWUDHlgLUHWPjT28hl9nc=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GNBBL1VMG2hDWWz2bZLN5uwOxFK6G/w4kERr/4gb/4bt20O2vpg4PHeDDPzwlQKg6777ZTW1jc2t8rblZ3dvf2D6uFRyySZZtxniUx0J6SGS6G4jwIl76Sa0ziUvB2Ob2Z++4lrIxL1iJOUBzEdKjEQjKKV/Kif30/71Zpbd+cgq8QrSA0KNPvVr16UsCzmCpmkxnQ9N8UgpxoFk3xa6WWGp5SN6ZB3LVU05ibI58dOyZlVIjJItC2FZK7+nshpbMwkDm1nTHFklr2Z+J/XzXBwHeRCpRlyxRaLBpkkmJDZ5yQSmjOUE0so08LeStiIasrQ5lOxIXjLL6+S1kXdc+vew2WtcVvEUYYTOIVz8OAKGnAHTfCBgYBneIU3RzkvzrvzsWgtOcXMMfyB8/kD1wWOsw==</latexit>
dII
<latexit sha1_base64="vzZSbbp81TAYM0Eh7KJCLfWrZxE=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GNBBHurYD+gDWWz2bRrN7thdyOUkP/gxYMiXv0/3vw3btsctPXBwOO9GWbmBQln2rjut1NaW9/Y3CpvV3Z29/YPqodHHS1TRWibSC5VL8CaciZo2zDDaS9RFMcBp91gcjPzu09UaSbFg5km1I/xSLCIEWys1AmHWbOZD6s1t+7OgVaJV5AaFGgNq1+DUJI0psIQjrXue25i/AwrwwineWWQappgMsEj2rdU4JhqP5tfm6Mzq4QoksqWMGiu/p7IcKz1NA5sZ4zNWC97M/E/r5+a6NrPmEhSQwVZLIpSjoxEs9dRyBQlhk8twUQxeysiY6wwMTagig3BW355lXQu6p5b9+4va43bIo4ynMApnIMHV9CAO2hBGwg8wjO8wpsjnRfn3flYtJacYuYY/sD5/AFsdI8G</latexit>
<latexit sha1_base64="vzZSbbp81TAYM0Eh7KJCLfWrZxE=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GNBBHurYD+gDWWz2bRrN7thdyOUkP/gxYMiXv0/3vw3btsctPXBwOO9GWbmBQln2rjut1NaW9/Y3CpvV3Z29/YPqodHHS1TRWibSC5VL8CaciZo2zDDaS9RFMcBp91gcjPzu09UaSbFg5km1I/xSLCIEWys1AmHWbOZD6s1t+7OgVaJV5AaFGgNq1+DUJI0psIQjrXue25i/AwrwwineWWQappgMsEj2rdU4JhqP5tfm6Mzq4QoksqWMGiu/p7IcKz1NA5sZ4zNWC97M/E/r5+a6NrPmEhSQwVZLIpSjoxEs9dRyBQlhk8twUQxeysiY6wwMTagig3BW355lXQu6p5b9+4va43bIo4ynMApnIMHV9CAO2hBGwg8wjO8wpsjnRfn3flYtJacYuYY/sD5/AFsdI8G</latexit>
<latexit sha1_base64="vzZSbbp81TAYM0Eh7KJCLfWrZxE=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GNBBHurYD+gDWWz2bRrN7thdyOUkP/gxYMiXv0/3vw3btsctPXBwOO9GWbmBQln2rjut1NaW9/Y3CpvV3Z29/YPqodHHS1TRWibSC5VL8CaciZo2zDDaS9RFMcBp91gcjPzu09UaSbFg5km1I/xSLCIEWys1AmHWbOZD6s1t+7OgVaJV5AaFGgNq1+DUJI0psIQjrXue25i/AwrwwineWWQappgMsEj2rdU4JhqP5tfm6Mzq4QoksqWMGiu/p7IcKz1NA5sZ4zNWC97M/E/r5+a6NrPmEhSQwVZLIpSjoxEs9dRyBQlhk8twUQxeysiY6wwMTagig3BW355lXQu6p5b9+4va43bIo4ynMApnIMHV9CAO2hBGwg8wjO8wpsjnRfn3flYtJacYuYY/sD5/AFsdI8G</latexit>
<latexit sha1_base64="vzZSbbp81TAYM0Eh7KJCLfWrZxE=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GNBBHurYD+gDWWz2bRrN7thdyOUkP/gxYMiXv0/3vw3btsctPXBwOO9GWbmBQln2rjut1NaW9/Y3CpvV3Z29/YPqodHHS1TRWibSC5VL8CaciZo2zDDaS9RFMcBp91gcjPzu09UaSbFg5km1I/xSLCIEWys1AmHWbOZD6s1t+7OgVaJV5AaFGgNq1+DUJI0psIQjrXue25i/AwrwwineWWQappgMsEj2rdU4JhqP5tfm6Mzq4QoksqWMGiu/p7IcKz1NA5sZ4zNWC97M/E/r5+a6NrPmEhSQwVZLIpSjoxEs9dRyBQlhk8twUQxeysiY6wwMTagig3BW355lXQu6p5b9+4va43bIo4ynMApnIMHV9CAO2hBGwg8wjO8wpsjnRfn3flYtJacYuYY/sD5/AFsdI8G</latexit>
1.0
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0.0
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1.0
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dI
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dII
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1.0
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<latexit sha1_base64="7uJXpholrjhNJkaCoHTl2i8Xdn8=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgadkVIR4jXjxGNA9IljA76U2GzM4uM7NCWAL+gBcPinj1i7z5N04eB00saCiquunuClPBtfG8b6ewtr6xuVXcLu3s7u0flA+PmjrJFMMGS0Si2iHVKLjEhuFGYDtVSONQYCsc3Uz91iMqzRP5YMYpBjEdSB5xRo2V7n3X65UrnuvNQFaJvyAVWKDeK391+wnLYpSGCap1x/dSE+RUGc4ETkrdTGNK2YgOsGOppDHqIJ+dOiFnVumTKFG2pCEz9fdETmOtx3FoO2NqhnrZm4r/eZ3MRFdBzmWaGZRsvijKBDEJmf5N+lwhM2JsCWWK21sJG1JFmbHplGwI/vLLq6R54fqe699dVmrXT/M4inACp3AOPlShBrdQhwYwGMAzvMKbI5wX5935mLcWnEWEx/AHzucPeTGNpg==</latexit>
<latexit sha1_base64="7uJXpholrjhNJkaCoHTl2i8Xdn8=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgadkVIR4jXjxGNA9IljA76U2GzM4uM7NCWAL+gBcPinj1i7z5N04eB00saCiquunuClPBtfG8b6ewtr6xuVXcLu3s7u0flA+PmjrJFMMGS0Si2iHVKLjEhuFGYDtVSONQYCsc3Uz91iMqzRP5YMYpBjEdSB5xRo2V7n3X65UrnuvNQFaJvyAVWKDeK391+wnLYpSGCap1x/dSE+RUGc4ETkrdTGNK2YgOsGOppDHqIJ+dOiFnVumTKFG2pCEz9fdETmOtx3FoO2NqhnrZm4r/eZ3MRFdBzmWaGZRsvijKBDEJmf5N+lwhM2JsCWWK21sJG1JFmbHplGwI/vLLq6R54fqe699dVmrXT/M4inACp3AOPlShBrdQhwYwGMAzvMKbI5wX5935mLcWnEWEx/AHzucPeTGNpg==</latexit>
0.0
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<latexit sha1_base64="f/4IzOIyMkszHgB4/4P/Ekkxag8=">AAAB6nicbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeKF48VbS20S8mms21oNrskWaEsBf+AFw+KePUXefPfmH4ctPXBDI/3JmTmhakUxnreNymsrK6tbxQ3S1vbO7t75f2DpkkyzbHBE5noVsgMSqGwYYWV2Eo1sjiU+BAOryf+wyNqIxJ1b0cpBjHrKxEJzqyT7ryq1y1XXJ+CLhN/TiowR71b/ur0Ep7FqCyXzJi276U2yJm2gksclzqZwZTxIetj21HFYjRBPl11TE+c0qNRol0pS6fq7xc5i40ZxaGbjJkdmEVvIv7ntTMbXQa5UGlmUfHZR1EmqU3o5G7aExq5lSNHGNfC7Ur5gGnGrUun5ELwF09eJs2zqu9V/dvzSu3qaRZHEY7gGE7BhwuowQ3UoQEc+vAMr/BGJHkh7+RjNlog8wgP4Q/I5w93q42l</latexit>
<latexit sha1_base64="f/4IzOIyMkszHgB4/4P/Ekkxag8=">AAAB6nicbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeKF48VbS20S8mms21oNrskWaEsBf+AFw+KePUXefPfmH4ctPXBDI/3JmTmhakUxnreNymsrK6tbxQ3S1vbO7t75f2DpkkyzbHBE5noVsgMSqGwYYWV2Eo1sjiU+BAOryf+wyNqIxJ1b0cpBjHrKxEJzqyT7ryq1y1XXJ+CLhN/TiowR71b/ur0Ep7FqCyXzJi276U2yJm2gksclzqZwZTxIetj21HFYjRBPl11TE+c0qNRol0pS6fq7xc5i40ZxaGbjJkdmEVvIv7ntTMbXQa5UGlmUfHZR1EmqU3o5G7aExq5lSNHGNfC7Ur5gGnGrUun5ELwF09eJs2zqu9V/dvzSu3qaRZHEY7gGE7BhwuowQ3UoQEc+vAMr/BGJHkh7+RjNlog8wgP4Q/I5w93q42l</latexit>
<latexit sha1_base64="f/4IzOIyMkszHgB4/4P/Ekkxag8=">AAAB6nicbVBNSwMxEJ3Ur1q/qh69BIvgqeyKoMeKF48VbS20S8mms21oNrskWaEsBf+AFw+KePUXefPfmH4ctPXBDI/3JmTmhakUxnreNymsrK6tbxQ3S1vbO7t75f2DpkkyzbHBE5noVsgMSqGwYYWV2Eo1sjiU+BAOryf+wyNqIxJ1b0cpBjHrKxEJzqyT7ryq1y1XXJ+CLhN/TiowR71b/ur0Ep7FqCyXzJi276U2yJm2gksclzqZwZTxIetj21HFYjRBPl11TE+c0qNRol0pS6fq7xc5i40ZxaGbjJkdmEVvIv7ntTMbXQa5UGlmUfHZR1EmqU3o5G7aExq5lSNHGNfC7Ur5gGnGrUun5ELwF09eJs2zqu9V/dvzSu3qaRZHEY7gGE7BhwuowQ3UoQEc+vAMr/BGJHkh7+RjNlog8wgP4Q/I5w93q42l</latexit>
1.0
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<latexit sha1_base64="7uJXpholrjhNJkaCoHTl2i8Xdn8=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgadkVIR4jXjxGNA9IljA76U2GzM4uM7NCWAL+gBcPinj1i7z5N04eB00saCiquunuClPBtfG8b6ewtr6xuVXcLu3s7u0flA+PmjrJFMMGS0Si2iHVKLjEhuFGYDtVSONQYCsc3Uz91iMqzRP5YMYpBjEdSB5xRo2V7n3X65UrnuvNQFaJvyAVWKDeK391+wnLYpSGCap1x/dSE+RUGc4ETkrdTGNK2YgOsGOppDHqIJ+dOiFnVumTKFG2pCEz9fdETmOtx3FoO2NqhnrZm4r/eZ3MRFdBzmWaGZRsvijKBDEJmf5N+lwhM2JsCWWK21sJG1JFmbHplGwI/vLLq6R54fqe699dVmrXT/M4inACp3AOPlShBrdQhwYwGMAzvMKbI5wX5935mLcWnEWEx/AHzucPeTGNpg==</latexit>
<latexit sha1_base64="7uJXpholrjhNJkaCoHTl2i8Xdn8=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgadkVIR4jXjxGNA9IljA76U2GzM4uM7NCWAL+gBcPinj1i7z5N04eB00saCiquunuClPBtfG8b6ewtr6xuVXcLu3s7u0flA+PmjrJFMMGS0Si2iHVKLjEhuFGYDtVSONQYCsc3Uz91iMqzRP5YMYpBjEdSB5xRo2V7n3X65UrnuvNQFaJvyAVWKDeK391+wnLYpSGCap1x/dSE+RUGc4ETkrdTGNK2YgOsGOppDHqIJ+dOiFnVumTKFG2pCEz9fdETmOtx3FoO2NqhnrZm4r/eZ3MRFdBzmWaGZRsvijKBDEJmf5N+lwhM2JsCWWK21sJG1JFmbHplGwI/vLLq6R54fqe699dVmrXT/M4inACp3AOPlShBrdQhwYwGMAzvMKbI5wX5935mLcWnEWEx/AHzucPeTGNpg==</latexit>
<latexit sha1_base64="7uJXpholrjhNJkaCoHTl2i8Xdn8=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgadkVIR4jXjxGNA9IljA76U2GzM4uM7NCWAL+gBcPinj1i7z5N04eB00saCiquunuClPBtfG8b6ewtr6xuVXcLu3s7u0flA+PmjrJFMMGS0Si2iHVKLjEhuFGYDtVSONQYCsc3Uz91iMqzRP5YMYpBjEdSB5xRo2V7n3X65UrnuvNQFaJvyAVWKDeK391+wnLYpSGCap1x/dSE+RUGc4ETkrdTGNK2YgOsGOppDHqIJ+dOiFnVumTKFG2pCEz9fdETmOtx3FoO2NqhnrZm4r/eZ3MRFdBzmWaGZRsvijKBDEJmf5N+lwhM2JsCWWK21sJG1JFmbHplGwI/vLLq6R54fqe699dVmrXT/M4inACp3AOPlShBrdQhwYwGMAzvMKbI5wX5935mLcWnEWEx/AHzucPeTGNpg==</latexit>
tensile
shear
1.0
0.0
1.0
0 0.2 0.4 0.6 0.8
0
5
10
15
20
25
30
35
AB
C
Strain (%)
Stress (MPa)
Experiment
Simulation
Simulation Experiment
A B C
dI
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<latexit sha1_base64="+nmd10vWUDHlgLUHWPjT28hl9nc=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GNBBL1VMG2hDWWz2bZLN5uwOxFK6G/w4kERr/4gb/4bt20O2vpg4PHeDDPzwlQKg6777ZTW1jc2t8rblZ3dvf2D6uFRyySZZtxniUx0J6SGS6G4jwIl76Sa0ziUvB2Ob2Z++4lrIxL1iJOUBzEdKjEQjKKV/Kif30/71Zpbd+cgq8QrSA0KNPvVr16UsCzmCpmkxnQ9N8UgpxoFk3xa6WWGp5SN6ZB3LVU05ibI58dOyZlVIjJItC2FZK7+nshpbMwkDm1nTHFklr2Z+J/XzXBwHeRCpRlyxRaLBpkkmJDZ5yQSmjOUE0so08LeStiIasrQ5lOxIXjLL6+S1kXdc+vew2WtcVvEUYYTOIVz8OAKGnAHTfCBgYBneIU3RzkvzrvzsWgtOcXMMfyB8/kD1wWOsw==</latexit>
<latexit sha1_base64="+nmd10vWUDHlgLUHWPjT28hl9nc=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GNBBL1VMG2hDWWz2bZLN5uwOxFK6G/w4kERr/4gb/4bt20O2vpg4PHeDDPzwlQKg6777ZTW1jc2t8rblZ3dvf2D6uFRyySZZtxniUx0J6SGS6G4jwIl76Sa0ziUvB2Ob2Z++4lrIxL1iJOUBzEdKjEQjKKV/Kif30/71Zpbd+cgq8QrSA0KNPvVr16UsCzmCpmkxnQ9N8UgpxoFk3xa6WWGp5SN6ZB3LVU05ibI58dOyZlVIjJItC2FZK7+nshpbMwkDm1nTHFklr2Z+J/XzXBwHeRCpRlyxRaLBpkkmJDZ5yQSmjOUE0so08LeStiIasrQ5lOxIXjLL6+S1kXdc+vew2WtcVvEUYYTOIVz8OAKGnAHTfCBgYBneIU3RzkvzrvzsWgtOcXMMfyB8/kD1wWOsw==</latexit>
<latexit sha1_base64="+nmd10vWUDHlgLUHWPjT28hl9nc=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GNBBL1VMG2hDWWz2bZLN5uwOxFK6G/w4kERr/4gb/4bt20O2vpg4PHeDDPzwlQKg6777ZTW1jc2t8rblZ3dvf2D6uFRyySZZtxniUx0J6SGS6G4jwIl76Sa0ziUvB2Ob2Z++4lrIxL1iJOUBzEdKjEQjKKV/Kif30/71Zpbd+cgq8QrSA0KNPvVr16UsCzmCpmkxnQ9N8UgpxoFk3xa6WWGp5SN6ZB3LVU05ibI58dOyZlVIjJItC2FZK7+nshpbMwkDm1nTHFklr2Z+J/XzXBwHeRCpRlyxRaLBpkkmJDZ5yQSmjOUE0so08LeStiIasrQ5lOxIXjLL6+S1kXdc+vew2WtcVvEUYYTOIVz8OAKGnAHTfCBgYBneIU3RzkvzrvzsWgtOcXMMfyB8/kD1wWOsw==</latexit>
dII
<latexit sha1_base64="vzZSbbp81TAYM0Eh7KJCLfWrZxE=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GNBBHurYD+gDWWz2bRrN7thdyOUkP/gxYMiXv0/3vw3btsctPXBwOO9GWbmBQln2rjut1NaW9/Y3CpvV3Z29/YPqodHHS1TRWibSC5VL8CaciZo2zDDaS9RFMcBp91gcjPzu09UaSbFg5km1I/xSLCIEWys1AmHWbOZD6s1t+7OgVaJV5AaFGgNq1+DUJI0psIQjrXue25i/AwrwwineWWQappgMsEj2rdU4JhqP5tfm6Mzq4QoksqWMGiu/p7IcKz1NA5sZ4zNWC97M/E/r5+a6NrPmEhSQwVZLIpSjoxEs9dRyBQlhk8twUQxeysiY6wwMTagig3BW355lXQu6p5b9+4va43bIo4ynMApnIMHV9CAO2hBGwg8wjO8wpsjnRfn3flYtJacYuYY/sD5/AFsdI8G</latexit>
<latexit sha1_base64="vzZSbbp81TAYM0Eh7KJCLfWrZxE=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GNBBHurYD+gDWWz2bRrN7thdyOUkP/gxYMiXv0/3vw3btsctPXBwOO9GWbmBQln2rjut1NaW9/Y3CpvV3Z29/YPqodHHS1TRWibSC5VL8CaciZo2zDDaS9RFMcBp91gcjPzu09UaSbFg5km1I/xSLCIEWys1AmHWbOZD6s1t+7OgVaJV5AaFGgNq1+DUJI0psIQjrXue25i/AwrwwineWWQappgMsEj2rdU4JhqP5tfm6Mzq4QoksqWMGiu/p7IcKz1NA5sZ4zNWC97M/E/r5+a6NrPmEhSQwVZLIpSjoxEs9dRyBQlhk8twUQxeysiY6wwMTagig3BW355lXQu6p5b9+4va43bIo4ynMApnIMHV9CAO2hBGwg8wjO8wpsjnRfn3flYtJacYuYY/sD5/AFsdI8G</latexit>
<latexit sha1_base64="vzZSbbp81TAYM0Eh7KJCLfWrZxE=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GNBBHurYD+gDWWz2bRrN7thdyOUkP/gxYMiXv0/3vw3btsctPXBwOO9GWbmBQln2rjut1NaW9/Y3CpvV3Z29/YPqodHHS1TRWibSC5VL8CaciZo2zDDaS9RFMcBp91gcjPzu09UaSbFg5km1I/xSLCIEWys1AmHWbOZD6s1t+7OgVaJV5AaFGgNq1+DUJI0psIQjrXue25i/AwrwwineWWQappgMsEj2rdU4JhqP5tfm6Mzq4QoksqWMGiu/p7IcKz1NA5sZ4zNWC97M/E/r5+a6NrPmEhSQwVZLIpSjoxEs9dRyBQlhk8twUQxeysiY6wwMTagig3BW355lXQu6p5b9+4va43bIo4ynMApnIMHV9CAO2hBGwg8wjO8wpsjnRfn3flYtJacYuYY/sD5/AFsdI8G</latexit>
<latexit sha1_base64="vzZSbbp81TAYM0Eh7KJCLfWrZxE=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GNBBHurYD+gDWWz2bRrN7thdyOUkP/gxYMiXv0/3vw3btsctPXBwOO9GWbmBQln2rjut1NaW9/Y3CpvV3Z29/YPqodHHS1TRWibSC5VL8CaciZo2zDDaS9RFMcBp91gcjPzu09UaSbFg5km1I/xSLCIEWys1AmHWbOZD6s1t+7OgVaJV5AaFGgNq1+DUJI0psIQjrXue25i/AwrwwineWWQappgMsEj2rdU4JhqP5tfm6Mzq4QoksqWMGiu/p7IcKz1NA5sZ4zNWC97M/E/r5+a6NrPmEhSQwVZLIpSjoxEs9dRyBQlhk8twUQxeysiY6wwMTagig3BW355lXQu6p5b9+4va43bIo4ynMApnIMHV9CAO2hBGwg8wjO8wpsjnRfn3flYtJacYuYY/sD5/AFsdI8G</latexit>
1.0
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0.0
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1.0
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dI
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dII
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1.0
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0.0
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1.0
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tensile
shear
1.0
0.0
1.0
Fig. 8. Cracking in the double-flawed specimen with
α = 45
:
the stressstrain curves from the simulation and the experiment.
Fig. 9. Cracking in the double-flawed specimen with
𝛼 = 60
:
the simulated cracking process and the experimentally observed
cracking pattern.
Fig. 10. Cracking in the double-flawed specimen with
α = 60
:
the stressstrain curves from the simulation and the experiment.
3.2. Cracking at the field scale
Next, we evaluate whether the double-phase-field model
can also simulate cracking processes at the field scale. For
this purpose, we consider a rock slope with en-echelon
joints of which the geometry and boundary conditions are
depicted in Fig. 11. The slope geometry is adopted from
Zhou and Chen, 2019, and the joint geometry from Cheng
et al., 2015. The slope is compressed from a 10 m long
region on the top boundary that is 5 m distant from the
crest. This loading condition mimics a rigid foundation.
The material parameters for this problem are provided in
Table 2. It is noted that the mode II fracture energy of this
problem is set to be 2-3 times greater than that in the
previous laboratory-scale problem, considering its scale
dependence (see, e.g., Abercrombie and Rice, 2005; Choo
et al., 2021). The phase-field length parameter is assigned
to be 0.025 m.
Fig. 11. Geometry and boundary conditions of the slope failure
simulation.
Table 2. Material parameters used for the slope failure
simulation. (The full definitions of the material parameters and
symbols are provided in Fei and Choo, 2021.)
Parameter
Unit
Value
Young's modulus
𝐸
GPa
30.00
Poisson's ratio
𝑣
-
0.33
Tensile strength
𝜎!
MPa
1
Cohesion strength
𝑐"
MPa
10
Peak friction angle
𝜙!
deg
40
Residual friction angle
𝜙#
deg
40
Mode I fracture energy
𝐺$
J/m%
20
Mode II fracture energy
𝐺$$
J/m%
550
Fig. 12 shows the simulated crack processes in the rock
slope. An important difference from the results of the
previous laboratory-scale problem is that here the cracks
0 0.2 0.4 0.6
0
5
10
15
20
25
A
B
C
Coalescence
Strain (%)
Stress (MPa)
Experiment
Simulation
Simulation Experiment
A B C
dI
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dII
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1.0
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0.0
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1.0
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dI
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dII
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1.0
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tensile
shear
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Coalescence
Strain (%)
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Simulation
15 m
3.6 m
5 m
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uy
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2.5 m
5.2 m
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
x
y
10 m
60 m
20 m
40 m
20 m
Zoom-in geometry of aws
are coalesced in a progressive manner, as the scale is far
larger than before. When the slope was compressed, the
lower joints are first coalesced by wing cracks growing
from the joint tips, followed by gradual coalescence of the
upper joints. Eventually, all the preexisting joints are
connected, forming a persistent slip surface penetrating
the entire slope. These results demonstrate that the model
can simulate progressive (step-path) failure of rock
structures at the field scale.
Fig. 12. Cracking of a rock slope with en-echelon joints: the
simulation result and the experimentally observed coalescence
pattern of en-echelon joints redrawn from Cheng et al., 2015.
In Fig. 12, the simulation results are compared with the
experimentally observed crack coalescence pattern from
en-echelon joints (Cheng et al., 2015). As can be seen, the
simulated crack coalescence behavior is remarkably
similar to the experimental observation in the laboratory.
Notably, the coalescence patterns in both the simulation
and experiment results are categorized as the Type IV
pattern in Sagong and Bobet, 2002, and the Category 7 in
Wong and Einstein, 2009a. In these categories, a wing
crack can either develop from a joint tip to connect the
surface of an adjacent joint or initiate from the surface of
a joint and propagates to link an adjacent joint tip.
The results from this slope failure problem demonstrate
two salient features of the double-phase-field model: (i) it
can simulate rock fracturing processes at the field scale,
and (ii) it can help us bridge our understandings of rock
cracking processes across scales.
4. SUMMARY
This paper has evaluated the capabilities of the double-
phase-field model (Fei and Choo, 2021) for simulating
cracking from rock discontinuities across scales. At the
laboratory scale, the model is thoroughly validated, both
qualitatively and quantitatively, with a variety of mixed-
mode cracking patterns characterized experimentally. At
the field scale, it is shown that the model can simulate
gradual coalescence of preexisting cracks which lead to
progressive failure of a rock structure. It is also found that
the simulated cracking process at the field scale is well
connected to experimental observations at the laboratory
scale. It is thus believed that the double-phase-field model
can serve as a valuable means to improve our physical
understanding and prediction of rock fracture processes
across scales.
ACKNOWLEDGEMENTS
This work was supported by the Research Grants Council
of Hong Kong under Project 17201419.
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Redrawn from the experimental result
by Cheng et al., 2015
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tensile
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ˆ
uy=0.75 mm
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tensile wing
cracks
tensile wing
cracks
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