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Numerical simulation of acoustic emission in rocks using FEM/DEM

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1 INTRODUCTION
1.1 AE simulation Techniques
Several modelling methodologies have been adopted to simulate the AE activity associated with
the failure processes of brittle rocks. Tang et al. (1997) first discussed the simulation of AE
based on elastic continuum damage mechanics and implemented their theory in a finite-element
code named RFPA (Tang et al., 1997). Hazzard & Young (2000) developed a technique to dy-
namically quantify AE using a bonded particle DEM model. The technique was further im-
proved by Hazzard et al. (2002), Hazzard & Young (2002) and Hazzard & Young (2004). Simi-
lar to DEM models, lattice models were adopted to describe the mesoscopic behavior of brittle
and quasi-brittle materials (Zapperi et al., 1997, Alava et al., 2006); Models that deal with AE
emitted from a pre-existing crack propagating in an elastic medium were discussed, among oth-
ers, by Hirose & Achenbach (1991). Despite the broad use of AE in experimental rock mechan-
ics and in in-situ monitoring, not much other progress has been made in the past decade in this
research area probably due to the complexity in modelling the various interacting aspects asso-
ciated with material failure and AE emission.
1.2 Principles of FEM/DEM
The combined finite/discrete element method (FEM/DEM) is an numerical method pioneered
by Munjiza et al. (1995) that simulates multiple interacting deformable bodies by directly solv-
ing Newton’s equations of motion. In this study, a two-dimensional FEM/DEM code, namely
Y-Geo (Mahabadi et al., 2012), is used to simulate the AE phenomena in a synthetic rock. The
FEM/DEM technique combines the advantage of discrete element methods (DEM) to capture
the interaction and fracturing of different solids with finite element method (FEM) to describe
Numerical simulation of acoustic emission in rocks using
FEM/DEM
Q. Zhao, A. Lisjak, G. Grasselli
Department of Civil Engineering, University of Toronto, ON, Canada
Q. Liu
Department of Physics, University of Toronto, ON, Canada
ABSTRACT: Acoustic emissions (AE) are stress waves released by localized inelastic defor-
mation events during the progressive failure of brittle rocks. Although several numerical meth-
ods have been developed to simulate the deformation and damage processes of rocks, only a
limited number have been capable of providing quantitative information regarding the associ-
ated acoustic activity. FEM/DEM is a numerical tool that simulates material failure by explic-
itly modelling fracture nucleation and propagation using cohesive elements. Seismic informa-
tion is extracted with a newly developed algorithm based on the monitoring of internal
variables in the proximity of propagating cracks. Several simulation cases were analyzed, in-
cluding a point source model, a wing crack propagation model, and a circular excavation
model. Simulated AE were cross-analyzed by travel-time inversion, spectral analysis, and fre-
quency-magnitude statistics. These preliminary results demonstrate the capabilities of
FEM/DEM as a tool to numerically simulate seismicity associated to the rock failure process.
their elastic deformation (Munjiza, 2004). As shown below, FEM/DEM is capable of simulat-
ing crack propagation and the corresponding particle motions.
The fundamental governing equation can be expressed as:
RM
2
2
tx
(1)
where M is the mass matrix of the system,
22 /tx
is the second order derivative of displace-
ment, and R is a nodal force vector which includes external loads, interaction forces between
bodies, deformation forces, viscous damping forces, and the crack bonding forces.
In order to simulate the initiation and propagation of fractures, dedicated four-noded cohe-
sive elements, named as crack elements (Fig. 1), are inserted between each pair of triangular
elements. The failure processes are modeled in FEM/DEM following principles of non-linear
elastic fracture mechanics (Dugdale, 1960, Barenblatt, 1962). Crack elements are enabled be-
tween all adjacent triangular pairs, thus arbitrary fractures can be captured within the con-
straints imposed by the original mesh topology.
Figure 1. Implementation of crack elements. (a) Behaviour of crack elements under given nodal forces; (b)
exaggerated mesh structure after the introduction of crack elements. The FEM/DEM mesh is developed
based on a FEM mesh generated using a Delaunay triangulation scheme.
An important aspect illustrated in Figure 1b is that the model is meshed by the common De-
launay triangulation scheme, which has great advantage of avoiding formation of triangles with
small included angles (Sloan, 1987). The resulted mesh has relatively high quality (Shewchuk,
1996), and most importantly, elements do not have a preferred orientation which could result in
an anisotropic orientation of the generated crack elements.
The three modes of cracking can be simulated as follows:
1. Mode I (i.e. opening mode) fracturing is simulated through a cohesive crack model based
on the Fictitious Crack Model originally proposed for concrete by Hillerborg et al. (1976).
When the opening between two triangular elements reaches a critical value (op), which is
related to intrinsic tensile strength, the normal bonding stress is gradually reduced until a
residual opening value (or) is reached (Fig. 2a).
2. Mode II (i.e. sliding mode) fracturing is simulated by a slip weakening model which re-
sembles the model from Ida (1972). The critical slip (sp) corresponds to the intrinsic shear
strength (fs) which is calculated according to the Mohr-Coulomb criterion:
tan
ns cf
(2)
where c is the internal cohesion, ϕ is the internal friction angle, and σn is the normal stress
acting across the fracture surface. While slip approaches sr, the tangential bonding stress is
gradually reduced to the residual value
tan
nr
f
gradually as shown in Fig. 2b
(Mahabadi et al., 2012).
3. Mode I-II (i.e. mixed mode) fracturing is defined by a coupling criterion between crack
opening and slip. As illustrated in Figure 2c, this mode describes a combination of shear
and tensile deformation, and the failure criterion is defined by
Figure 2. Constitutive behavior of crack element. (a) Mode I (opening mode), the crack element start to
yield when normal stress reaches ft and breaks at critical opening Or; (b) Mode II (sliding mode), the crack
element yields under peak shear stress fs and breaks at critical shear distance Sr where the residual shear
stress fr is applied even after the breakage; (c) Graphic representation of the coupling relationship between
crack opening and crack slip for Mode I-II, the mixed mode.
1
2
2
r
p
r
ps
ss
o
oo
(3)
where o is opening distance and s is slipping distance.
The shapes of the curves for mode I and mode II are based upon experimental complete
stress-strain curves obtained for concrete in direct tension (Evans & Marathe, 1968, Munjiza et
al., 1999). The specific fracture energy values GIc and GIIc are related to the areas under these
curves through:
r
p
o
o
cdooG )(
I
(4)
r
p
s
src dsfsG ])([
II
(5)
From an energetic point of view, strain energy is stored during the elastic deformation of the
triangular elements, and once the material intrinsic strength is overcome, the release of the
strain energy begins along with the initiation of a new fracture. Released energy is consumed by
the fracturing process itself through GIc and GIIc. If sliding occurs, part of the energy is dissi-
pated by frictional work. The excess of elastic strain energy is radiated as kinetic energy in the
form of AE. Two types of approaches are used to capture and analyze the synthetic AE: internal
monitoring of AE sources and seismogram interpretation.
2 FEM/DEM SIMULATION OF AE WAVE PROPAGATION
2.1 Model Description
In this section, a simple model is used to validate the ability of FEM/DEM in simulating wave
propagation. The simulation results are analyzed based on seismograms and wavefield snap-
shots.
A vertical point source is excited at the left bottom corner of a homogeneous sample by a
Ricker wavelet function with duration is 0.015 s with a center frequency of 200 Hz. The model
is a 50 × 100 mm homogeneous sample, which is meshed into 13852 triangular elements with
an average size h = 1 mm. Material properties for this simulation are listed in Table 1. No
damping is applied to preserve the waveform, and absorbing boundaries are used at margins of
the model to prevent reflections. The absorbing boundary condition is accomplished by imple-
menting the solution proposed by Lysmer & Kuhlemeyer (1969), and further information can be
found in Mahabadi et al. (2012).
Table 1. Rock Sample Properties for the Wave Propagation Simulations in Section 2
____________________________________________________________________________________
Parameter Unit Value
____________________________________________________________________________________
Density, ρ kg/m3 2300
Young’s modulus, E GPa 3
Poisson’s ratio, υ - 0.29
Internal friction angle, ϕ degree 35
Internal cohesion , c MPa 15
Tensile strength, ft MPa 3
Mode I fracture energy, GIc J/m2 2.0
Mode II fracture energy, GIIc J/m2 10
Damping coefficient, μ kg/m
s 0
Normal contact penalty, pn GPa 30
Tangential contact penalty, pt GPa 3
Fracture penalty, pf GPa 15
Sample-platen friction coefficient, k - 0.1
____________________________________________________________________________________
2.2 Simulation Results
The propagation of the wave front simulated by FEM/DEM can be observed in Figure 3a,
where the four-quadrant distribution of the velocity field is clearly shown at early times. Assess
the accuracy of the AE wave simulation by FEM/DEM, a model with the same geometry and
properties but without introducing crack elements, is tested first (Fig. 3b). When crack elements
are not introduced, the simulation is essentially a time explicit FEM. Moreover, a similar for-
ward simulation is carried out using the spectral-element method (SEM) for comparison pur-
pose (Fig. 3c). The SEM method is essentially an FEM with high polynomial order and thus
more accurate, and it is recognized as the most popular FEM method in global and regional
seismology (Komatitsch & Vilotte, 1998, Komatitsch et al., 2004, Komatitsch et al., 2005,
Shearer, 2009). The software package SPECFEM2D developed based on SEM is used for com-
parison.
Wavefields presented by these methods share a high degree of similarity, which indicates
that the FEM/DEM algorithm is capable of simulating seismic wave properly. Although the
general trend of the wavefield produced by FEM/DEM is quite accurate, significant amount of
noise exists around the source area (Fig. 3a), due to scattering induced by the yielding of crack
elements. As mentioned in section 1, each pair of triangular elements is bonded together by a
crack element and, due to the violent stress change induced by the source excitation, crack ele-
ments yield (but do not break in this example) and hence oscillate between varying stress condi-
tions. This behaviour generates noise in the near-source region. Another disadvantage associ-
ated with the crack elements is that their elastic behaviour modifies the overall stiffness of the
material and results in lower wave speed. This effect could be minimized by increasing the pen-
alty values of the elements. Overall, the time difference could be optimized to be within 10-2 ms
range, which is considered accurate for most scenarios.
The recorded seismograms by a receiver at the center of the sample are plotted for three
above methods (Fig. 4). Although the first arrivals from FEM and SEM simulations agree with
each other well, the cause of the difference in peak times needs to be investigated in future re-
searches.
The results discussed above demonstrate the ability of FEM/DEM in modelling AE wave
propagation simulations, and this could be useful to simulate laboratory tests where AE waves
are generated by transmitters attached to a sample (Hardy, 2003). Moreover, AE are mostly as-
sociated with fracturing processes which, although providing a powerful passive way to monitor
internal variations of rocks in the field (Lockner, 1993), are very difficult to simulate numeri-
cally. The most significant advantage of FEM/DEM is its ability to simulate the entire process
of fracture growth. Using the validated AE analysis method as shown in section 2, a fracturing
propagation induced AE example is given in the next section.
Figure 3. Comparison of the wavefields (x-component) modeled by FEM/DEM and SEM at same time
frames. (a) FEM/DEM model with introducing crack elements; (b) FEM/DEM model without crack ele-
ments implementation; (c) SEM model. Note that absorbing boundaries are not perfect, and reflections
from the corners and edges sometimes interfere with the original wave fields.
Figure 4. Comparison of the P-wave seismograms modeled by FEM/DEM and SEM. FEM here refers to
the simulation by FEM/DEM without introducing crack elements. First arrivals calculated by FEM and
SEM are almost identical at 0.034 ms, and a time difference Δt = 0.003 ms is observed for the first arri-
val of FEM/DEM simulation result. The seismogram produced by FEM/DEM algorithm is not as smooth
as those by the continuum algorithms due to interaction effects at the contact between particles.
3 FEM/DEM SIMULATION OF AE INDUCED BY WING CRACK GROWTH
3.1 Model Description
The major body of the model is a 50 × 100 mm (Fig. 5a) homogeneous sample containing an
inclined flaw located at the center of the sample. Two platens compress the sample at a constant
rate of 0.05 m/s. The sample is discretized by a Delaunay triangulation scheme with an average
size h = 0.7 mm. Material properties for the sample are the same as in Table 1 except for the
damping coefficient, which is set to be twice the critical damping (Equation 6), c = 7.4 × 103
kg/m
s. Moreover, absorbing boundary conditions are not applied, and receivers are placed on
both sides of the sample to record synthetic seismograms.
3.2 Internal Monitoring of Simulation Results
Figure 5a shows the crack propagation simulation results, and the broken crack elements clus-
tering in two macroscopic fractures originating from the tip of the diagonal flaw. The nuclea-
tion of every crack is associated with an acoustic event which is then monitored in FEM/DEM
by numerically assessing the following parameters: (i) source location, (ii) fracture mode, (iii)
initiation time, and (iv) seismic energy and magnitude.
3.3 Seismic Analysis and Comparisons with Internal Monitoring Results
Seismic analyses are performed on vertical and horizontal-component synthetic seismograms
recorded by receivers on both sides of the rock sample.
Figure 5c illustrates the horizontal-component synthetic seismograms recorded as particle ve-
locities for receivers on the right-hand side of the sample (numbered as receivers 0-10). Wave-
lets occur before 1 ms are induced by the impact between platens and the sample, and it is
damped out before first-arrival of the earliest AE event at about 1.8 ms. Four groups of AE
events can be observed from the seismogram as listed in Table 2, where a group may consist
several sub-events which are considered to be associated with the propagation of a continuous
fracture. Since several sub-events occur close in time, not all of them can be identified from the
seismogram, and only the first event in each group can be analyzed in practice.
Figure 5. (a) Geometry of the model and fracture propagation results from internal monitoring of
FEM/DEM. Receivers are placed on both sides of the sample (indicated by dots); (b) Zoomed-in view of
the fracture propagating area, where AE sources recorded by internal monitoring are denoted by circles,
and inverted source locations are indicated by two crosses; (c) Seismograms recorded by receivers placed
at the right-hand side of the sample (Lisjak et al., 2013).
Table 2. Event Groups and Their Time Ranges
_____________________________________________
Group number Time range
_____________________________________________
0 [1.8, 2.2]
1 [4.1, 5.0]
2 [5.8, 6.6]
3 [8.0, 8.2]
_____________________________________________
A travel time inversion of hand-picked first arrivals is done for group 1 and group 2 to locate
the AE sources, and theoretically, the inverted locations would be the initial points of the frac-
ture propagation. Inverted AE sources are at the center of the crosses shown in Fig 5b, while the
location errors of the inversions in mm scale are given by the sizes of the crosses. Note the in-
verted AE locations for group 1 and 2 are consistent with the initiation of wing cracks from the
tips of inclined center flaw from FEM/DEM modelling.
Two types of amplitude reduction effects can be observed from synthetic seismograms in
Figure 5c, geometrical spreading and numerical damping attenuation, in which numerical damp-
ing is chosen to be twice the critical damping (μc) calculated from
Eh
c
2
(6)
where h=0.7 mm is the characteristic element size, ρ is density, and E is Young’s modulus. In
calculating the numerical damping, each element is considered as a mass-spring-dashpot sys-
tem. Damping is applied to dissipate the noise induced by the oscillation of platens. Increasing
the damping value will reduce the AE amplitudes and attenuate more high frequency content.
Specifically, the damping is most significant for frequencies
c
cE
ff

2
(7)
i.e. frequency contents above 60 kHz will suffer considerable attenuation.
Figure 6a, b shows the synthetic seismograms and their frequency spectra recorded by re-
ceivers 0-4 at time interval 8 ms-8.2 ms. According to these spectra, the AE activities have en-
ergy primarily concentrated around 10 kHz, while higher frequency contents are mostly attenu-
ated. With increasing travel distance (from receiver 4 to 0), the amplitudes of waves are
significantly reduced.
The frequency range of the recorded seismograms are is significantly lower than the lab test
results which usually range between 100 kHz to 2000 kHz (Lockner, 1993). Several factors may
be considered to explain the inability of capturing high frequency contents in FEM/DEM. Nu-
merical damping is one of the important reasons, as it not only decreases the amplitude but also
damps out high frequency content as shown in Figure 6a. Moreover, the frequency content of
source time functions is related to the sizes of seismic sources and their estimated rup-
ture speeds. In this example, rupture speed value is low, which results in a quasi-
dynamic AE behavior in the FEM/DEM simulations. Although seismic energy is emit-
ted upon failure, the low rupture speed indicates that most released energy is dissipated
as fracture energy within the fracture propagation zone.
Figure 6. (a) Synthetic seismograms recorded by receivers 0-4 for time interval 8 ms-8.2 ms; (b) Frequen-
cy spectra of the seismograms recorded by receivers 0-4 for time interval 8 ms-8.2 ms.
4 SIMULATION OF CIRCULAR EXCAVATION INDUCED AE
The FEM/DEM algorithm exhibits great potential in assessing the stability of an underground
structure by combining mechanical behaviour simulation with the analysis of recorded AE
events. An example of simulation of damage around an unsupported circular excavation is
shown here in Figure 7. The rock mass is a horizontally layered clay shale formation, and the
axis of the circular tunnel parallel to the bedding planes. Tunnel liner is not applied to allow the
full development of the Excavation Damaged Zone (EDZ). The excavation induced AE activity
is plotted in Figure 7 with associated magnitude indicated by different colors.
Figure 7b depicts the frequency-magnitude distribution of the excavation induced AE. Ac-
cording to the Gutenberg-Richter relationship (Gutenberg & Richter, 1956), in logarithmic
coordinates, the relationship between AE magnitude and occurrence frequency becomes linear
as described by
bMaN log
(8)
where N is the number of AE events with magnitude larger than M, a and b are constants, and
the slope of the linear portion, b value, could be estimated from the plot. The b value is related
to the geological setting and can help to improve the efficiency of operation. The relatively
large b value (b = 2.4) agrees with the fact that majority of the simulated AE events are planar
fractures parallel to the horizontal and vertical stresses (Downie et al., 2010).
Figure 7. (a) Spatial distribution of AE events and associated magnitudes of the simulated damage induced
by excavation; (b) Frequency-magnitude plot of the recorded AE events and the estimated b value.
5 CONCLUDING REMARKS
In this study, we analyzed the numerical simulation of AE using the FEM/DEM algorithm. Ex-
ample in section 2 validated the ability of FEM/DEM in capturing general wave propagation
against a specifically developed synthetic seismogram simulator (SEM), while examples in sec-
tion 3 and section 4 took advantage of the discrete representation of materials in FEM/DEM to
obtain seismic information associated with the propagation of a fracture and the development of
a damage zone around a circular excavation. Simulations results show promising potential of
FEM/DEM in AE studies.
When compared with SEM, the FEM/DEM method needs many improvements before being
used as a pure seismic wave simulator. Producing realistic high frequency AE contents is also a
key feature that needs to be developed, while eliminating the scattering from crack elements
could improve the simulation quality significantly. Although not perfect in AE simulation,
FEM/DEM can properly simulate AE and fracturing at the same time, which are of great inter-
ests to engineers, and can provide the correlation between AE and mechanical behavior which
has a great implication for engineering the rock mass.
Numerical simulations of AE have been studied for decades in analyzing rock failure
processes, and the method discussed in this study provides a tool to quantitatively assess the
microcracking activities as well as the associated AE. In future studies, the validated
FEM/DEM AE modelling technique will be used to obtain further insights into the mechanics
of rock failure with potential applications ranging from laboratory scale to engineering scale
processes (e.g. underground excavations, petroleum and geothermal reservoirs). Moreover, a
three-dimensional version of the FEM/DEM code is currently under development, and the AE
simulation approach will be adopted.
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Inhomogeneous materials, such as plaster or concrete, subjected to an external elastic stress display sudden movements owing to the formation and propagation of microfractures. Studies of acoustic emission from these systems reveal power-law behaviour. Similar behaviour in damage propagation has also been seen in acoustic emission resulting from volcanic activity and hydrogen precipitation in niobium. It has been suggested that the underlying fracture dynamics in these systems might display self-organized criticality, implying that long-ranged correlations between fracture events lead to a scale-free cascade of `avalanches'. A hierarchy of avalanche events is also observed in a wide range of other systems, such as the dynamics of random magnets and high-temperature superconductors in magnetic fields, lung inflation and seismic behaviour characterized by the Gutenberg-Richter law. The applicability of self-organized criticality to microfracturing has been questioned,, however, as power laws alone are not unequivocal evidence for it. Here we present a scalar model of microfracturing which generates power-law behaviour in properties related to acoustic emission, and a scale-free hierarchy of avalanches characteristic of self-organized criticality. The geometric structure of the fracture surfaces agrees with that seen experimentally. We find that the critical steady state exhibits plastic macroscopic behaviour, which is commonly observed in real materials.
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