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Thesis Summary
Dynamic Tensile Failure of Rocks Subjected to Simulated In Situ Stresses
Bangbiao Wu
Department of Civil Engineering University of Toronto
Submitted to ISRM Rocha Medal
Tianjin China, 2017
1
Abstract
Tensile failure of rocks is a main problem in underground engineering projects, in which rocks
are subjected to dynamic disturbances while under high in situ stresses. When disturbed by
dynamic loads from blasting, seismicity or rockbursts, the underground structures would be
vulnerable to tensile failure. Depending on the distance from the underground opening, the in-
situ stress states change from hydrostatic in the far-field, to triaxial in the intermediate distance,
and to the pre-tension nearby the opening.
Dynamic mechanical properties of engineering materials have been investigated for many years
and various devices are used to conduct experiments on these materials. One of the most
frequently used techniques is the split Hopkinson pressure bar (SHPB) system. Using the SHPB
technique, dynamic mechanical properties such as compressive strength, tensile strength and
fracture toughness have been studied on engineering materials, mostly under uniaxial loading
conditions. However, underground rocks are subjected to in situ stresses and occasional dynamic
loadings. Thus, the SHPB testing system is further adjusted with confining pressure system for
dynamic response of rocks under pre-stress state.
In the experiment, the Brazilian disc rock specimens made from Canadian Laurentian granite are
first subjected to pre-stresses simulating in-situ stresses underground (including pre-tension,
hydrostatic confinement, and triaxial confinement) and then loaded dynamically using the
modified SHPB system. The dependence of dynamic tensile strength of the rock material on the
static pre-stress and loading rate is established. A theoretical model is adopted to predict the
dynamic tensile strength of the rock material under different pre-stress states; the results
demonstrate the applicability of the model because the error is less than 5% with reasonable
values for of the parameters.
These experimental results and the calibrated theoretical model will be of great importance in the
design and safety of underground rock engineering projects.
2
1 Introduction
1.1 Background
Tensile failure is a main mode of rock failure and it is meaningful and necessary to investigate
the dynamic tensile failure of rocks subjected to pre-stress simulating the underground stress
states. Research has been conducted under quasi static situations, however, the tensile tests under
dynamic loadings have just been proposed in recent years (Xia 2009). Ross, Thompson et al.
(1989) first adopted Brazilian disc specimens to get the dynamic tensile properties of concrete in
SHPB system. Cai, Kaiser et al. (2007) then studied the dynamic behavior of the Meuse/Haute-
Marne argillite and obtained the tensile properties under different strain rates. Zhao and Li (2000)
investigated the dynamic tensile properties of granite by using a dynamic loading machine and 3-
point flexural test method. Wang, Jia et al. (2004) and Wang, Li et al. (2009) proposed flattened
Brazilian disc specimens for the determination of dynamic tensile strength of rocks.
Depending on the distance from the underground opening, the in-situ stress states change from
hydrostatic in the far-field, to triaxial in the intermediate distance, and to the pre-tension nearby
the opening. Take deep mining as an example, the bending of the roof and the buckling of the
pillar would induce static tensile stress state. When disturbed by dynamic loads from blasting,
seismicity or rockbursts, the structures would be vulnerable to tensile failure. From microscopic
perspective, pores and microcracks in rock materials are potential sources of tensile failure
because of stress concentration. Even though the far-field load is compressive, the local stresses
around these inhomogeneities may be tensile. Dynamic disturbance would promote the
generation, propagation and nucleation of defects, and lead to the tensile failure of the rock
materials.
In previous research, there are not many results of tensile experiments on pre-stressed rocks, and
if so, all of them are under quasi-static stress states. Some researchers conducted Brazilian disc,
shear and torsion tests on rocks under hydrostatic pressure (Robertson 1955, Jaeger and Hoskins
1966) and they found that the strength of rock increases with the hydrostatic pressure. Vasarhelyi
(1997) investigated the influence of confinement on the mode I fracture of Gneiss and they
concluded that the fracture toughness also increases with the confining pressure. Al-Shayea,
Khan et al. (2000) tested straight notched Brazilian disk (SNBD) specimens under diametrical
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compression to study the influence of confinement on fracture toughness of limestone. Chen and
Zhang (2004) studied the influence of confinement on rock fracture toughness using notch-hole
combined Brazilian disc specimens and Funatsu, Seto et al. (2004) did the same tests on notched
semi-circular bend (NSCB) specimens and they found the same increasing trend with confining
pressure.
To sum up, there have been many improvements in rock dynamic testing; however, the research
concerning dynamic response of rock subjected to pre-stress is still deficient, which can be listed
here:
1. The loading method for pre-stress pressure is not reasonable enough to represent the state of
the deep rocks. Deep rock is mainly subjected to hydrostatic pressure, with differential stress
components, which is mainly tension in rock mass near free surface such as mined-out areas.
However, in research regarding to the dynamic failure of rocks, researchers either have ignored
the effects of confinement, or have only considered the lateral confinement or hydrostatic pre-
stress at most.
2. The fact that the main failure mode of rock is tension (Kaiser, Kim et al. 2010) is not taken
into consideration in many studies. Instead, most of the studies are focused on the dynamic
failure in compression, rarely on tensile or fracture. But in deep underground engineering
projects, rock fails mainly in tension except the sliding failure of large joints.
3. Some studies did not investigate and verify the accuracy and reliability of the testing methods.
In most experiments related to dynamic tension and fracture, the strength and the fracture
toughness are obtained directly from the static formulas, regardless of the applicability. This
situation has been greatly improved with the presence of the ISRM suggested method, which
provides the specimen preparation, experimental procedure and detailed data processing method
for dynamic tests. However, it still needs to be modified to take confining pressure into
consideration.
4. Although the dynamic failure of rock under different pre-stress situations has been
investigated, there is no ready-made conclusion about the behavior. And the tests were
conducted under different situations by different researchers using different materials and
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apparatus. Thus, it is necessary to systematically investigate the dynamic behavior of pre-
stressed rock.
1.2 Research Objectives and Approach
The research work presented in this thesis is aimed at understanding dynamic tensile failure of
pre-stressed rocks and trying to predict such failure under specific circumstances, such as the
failure of underground rocks under certain loading rate at certain depth. To study and predict
blasting-induced damage, the research uses its correlation with loading rate. Thus, the Split
Hopkinson Pressure Bar (SHPB) is adopted to conduct dynamic tests on rock materials with
different loading rate, and the SHPB system is modified to provide different pre-stress states for
the rock specimens to simulate the in-situ stress states before being subjected to dynamic
loadings. After achieving the correlation between the tensile strength and the loading conditions,
the statistical crack mechanics is used to predict the tensile failure of rock under different loading
conditions and compare the results with that from the lab tests. The specific objectives of this
research are as follows:
• Investigate the nature of stress waves generated by the SHPB impact.
• Study the mechanisms of wave propagation and rock tensile failure for Brazilian Disc
tests under different pre-stress states.
• Evaluate the performance of a theoretical modelling on reproducing the tensile failure
process under certain loading rates.
• Seek a correlation between the dynamic tensile strength and the loading rate and pre-
stress state.
• Design and establish a fully functional SHPB system for rock dynamic tests with pre-
stress, to simulate pre-stress state of deep rocks.
• Understand the influence of the pre-stress, loading rate on the dynamic tensile properties
of the rock.
• Establish a failure model for the dynamic failure of rocks subjected to pre-stress, and
improve the model by referring to engineering activities.
The technical route is as shown in Figure 1.1; the first step is to set up a multifunctional SHPB
system which can provide pre-stress to the specimens. The bars and the pressure cylinders will
be sealed for the confining pressure system, which would lead to the high frictional resistance.
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This will be fully considered in the design, and we will use soft rubber for the sealing to reduce
the friction. Tests will be conducted on Laurentian granite without pre-stress to standardize the
whole system and to provide reference for the following tests.
Figure 1.1 Technical route of the whole project
The whole system is composed of the SHPB loading system and the pre-stress subsystem. The
loading system contains the striker bar, the incident bar and the transmitted bar, which are all 1.5
inch in diameter as shown in Figure 1.2. Before the test, the specimen would be sandwiched
between the incident bar and the transmitted bar. The two parts for confinement are connected by
two steel rods, to provide the desired pre-stress by adjusting the pressure of the two chambers.
This system can be used for all the desired pre-stress state of the specimens.
Figure 1.2 Schematic of the Brazilian test under hydrostatic confining pressure on SHPB
system
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Figure 1.3 Schematic diagram of the approach and methodology employed in this research
Goal:
Provide guidelines for the
development of a reliable
method to predict blast-induced
tensile failure and design tool
for blasting operations
A. STUDY OF
DYNAMIC FAILURE
OF ROCK MATERIAL
B. EFFECT OF IN SITU
STRESS ON DYNAMIC
TENSILE STREGNTH OF
ROCK MATERAILS
C. THEORETICAL
STUDY- STATISTICAL
MECHANICS
SHPB Test
Pulse Shaper
Technique
Micro-CT scan
SEM
Pre-tension Hydrostatic Confinement
SCRAM
ISO-SCRAM
DCA
Underground
Opening Far Field Rock
Failure
Triaxial Stress State
Penny Shaped Crack
Homogenous and
Isotropic Medium
Random Distribution
Mechanical Properties under In situ stress states
Rate Dependency
Failure Pattern
Different Failure Modes of Different Rocks under Pre-stress Conditions
• Compressive Failure
• Tensile Failure
• Fracture Toughness
Take Temperature into Consideration
• Pre-heat Treated
• In situ Heating
Further Applications of Statistical Mechanics in Predicting
• Measurement of damage
• Predicting strength under confinement with different loading rates
On time observation of rock failure under Micro-CT
• 4D imaging technique
D. FUTURE WORK
Intermediate
Zone Failure
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2 Experimental Techniques
2.1 Specimen Preparation
Figure 2.1 illustrates the procedures for preparing Brazilian disc specimens for dynamic tensile
tests. Rock cores with a nominal diameter of 40 mm are first drilled from a rock block, and then
sliced to obtain disc specimens with an average thickness of 16 mm. All of the disc specimens
are polished afterwards until the surface roughness is less than 0.5% of the specimen thickness.
During the procedure of specimen preparation, different parameters are measured and compared
with the results by Nasseri and Mohanty (2008). Table 2.1 shows the P-wave velocities of the
LG block before drilling. Basically, the P-wave velocity measurement is to decide the bedding
direction of the block and then drill the specimens from the bedding direction, which is the XY
plane direction shown in Table 2.1.
Table 2.1 P-wave velocity of Laurentian granite in this study
Direction Nasseri and Mohanty (2008) Current study
XY plane 4115 ± 14 (m/s) 4172 ± 42 (m/s)
XZ plane 4448 ± 46 (m/s) 4635 ± 72 (m/s)
YZ plane 4513 ± 42 (m/s) 4683 ± 45 (m/s)
Figure 2.1 Procedure for the preparing the Brazilian disc specimen of Laurentian granite
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In order to further investigate the mineral component of the Laurentian granite specimen, a thin
section was made to observe under the scanning electron microscope (SEM). The SEM result is
shown in 错误!未找到引用源。, which indicates different components by different colors. To
sum up, the percentages of Quartz, Feldspar and Biotite are 22.5%, 71.6% and 2.3% respectively,
which is different from that reported by Nasseri, Mohanty et al. (2005) that the percentages are
33%, 60% and 3-5%, respectively. It seems that the ‘gap’ between the two results is large.
However, a possible reason is that the rock specimens are not from the same block although they
are the same type of rock. Another reason is that the thin section is too small compare to a block,
makes it impossible to represent the volume percentage of each mineral of the whole block.
According to Dietrich and Skinner (1979), the component percentages are normally 20-60 % of
Quartz, 35-90 % of alkali feldspar, 10-65 % of plagioclase and 5-20 % of Mafic (Biotite,
Hornbende etc.) So both of the results are reasonable.
Figure 2.2 Mineral and microcracks traced from three orthogonal planes for Laurentian granite
(Nasseri and Mohanty (2008))
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Table 2.2 Mineral composition of Laurentian Granite
Direction
Quartz Feldspar Biotite
Average Grain
Size (mm) % Average Grain
Size (mm) % Average Grain
Size (mm) %
XY plane 0.39 33% 0.37 59% 0.25 5%
XZ plane 0.56 34% 0.51 60% 0.41 3%
YZ plane 0.54 30% 0.40 64% 0.28 3%
2.2 Split Hopkinson Pressure Bar (SHPB)
A general SHPB system consists of three major components: a loading device, bar components
and a data acquisition and recording system, as schematically shown in Figure 2.3. The most
common method of loading is to launch a striker impacting on the incident bar. The striker bar is
launched by a sudden release of the compressed air the gas gun and accelerates in the long gun
barrel until it impacts on the end of the incident bar. This kind of striker launch mechanism
produces a controllable and repeatable impact of the incident bar, the striking speed can be
simply controlled by changing the pressure of the compressed gas of the depth of the striker
inside the gun barrel.
The impact of the striker bar on the free end of the incident bar induces a longitudinal
compressive wave propagating in both directions. The left-propagating wave is fully released at
the free end of the striker bar and forms the trailing edge of the incident compressive pulse i
(Figure 2.3). Thus, the duration of i depends on the length and longitudinal wave velocity in
the striker. Upon reaching the bar-specimen interface, part of the incident wave is reflected as the
reflected wave r and the remainder passes through the specimen to the transmitted bar as the
transmitted wave t . Strain gauges are used to record the stress wave pulse on both incident bar
and transmitted bar.
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Figure 2.3 The x-t diagram of stress waves propagation in SHPB (after Xia, Dai et al. (2011))
The diameter of the bars is governed by the diameter of rock specimen, which should be at least
10 times the average grain size of the rock (Dai, Huang et al. 2010, Zhou, Xia et al. 2012). Based
on the one dimensional stress wave theory, the dynamic forces (Figure 2.3) on the incident end
(P1) and the transmitted end (P2) of the specimen are (Kolsky 1949, Kolsky 1953):
1 ( )i rP AE (2-1)
2 tP AE (2-2)
where E is the Young’s Modulus, A is the cross-sectional area of the bars. For a Brazilian disc
test, when the quasi-static state is achieved in the specimen (P1 = P2), the dynamic tensile
strength is determined by the following equation:
2 f
t
P
BD
(2-3)
where t is the tensile strength of the specimen, fP is the load when the failure occurs, B and
D are the thickness and the diameter of the specimen, respectively.
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2.3 Modified SHPB System for Pre-stress Conditions
The modified SHPB apparatus for tri-axial stress state includes the three bars (a striker bar, an incident
bar, and a transmitted bar) (Zhou, Xia et al. 2012) and a hydraulic system, as shown in Figure 2.4. The
elastic bars are made of the same material as that for the pre-tension test. The hydraulic system is mainly
composed of a cylinder that applies lateral confinement to the rock specimens (Cylinder 1), and a pressure
chamber that provides axial preload (Cylinder 2) to the bars and specimen.
Figure 2.4 Schematics of the modified SHPB system for Tri-axial test
Huang, Xia et al. (2013) employed X-ray Micro CT method and strain control ring in the SHPB
tests to observe microscopic damage accumulation in brittle solids subjected to dynamic
compressive loading. X-ray Micro CT with high resolution was used to examine the 3D
microcrack inside the recovered rock specimen under various strains.
To sum up, this section reviews the microscopic characterization and microstructure of
Laurentian granite, demonstrating that the Laurentian granite can be considered as a homogenous
rock material, and introduces the SHPB apparatus for dynamic experimental studies.
3 Dominant Crack Algorithm (DCA)
In rock engineering problems such as blasting, earthquakes and rockbursts, it is common that
rocks may be subjected to dynamic loads and fail suddenly. It is thus critical to understand the
dynamic response of rocks for ensuring safety and maximizing profit in engineering projects.
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Theoretical modeling has been proposed to investigate the failure process of rock material, under
both static and dynamic loading conditions.
Zuo, Addessio et al. (2006) proposed the dominant crack algorithm (DCA) for the damage of
brittle materials under dynamic loading. The rate-dependent damage evolution in the DCA
model is based on the strain energy release rate associated with the critical crack orientation,
which is defined as the most unstable orientation for cracks that are isotropically distributed in
the material. To illustrate the features of the DCA model, the authors simulated several standard
load paths such as hydrostatic, uniaxial strain, and pure shear on Sic ceramic. The strain rate and
loading history of the uniaxial strain response and the pure shear response are similar to the
hydrostatic simulation. Recently the authors (Zuo, Disilvestro et al. 2010) improved the physics
of the model by incorporating plasticity and a nonlinear equation of state (Deganis and Zuo
2011), and applied it to study damage in concrete.
3.1 Constitutive Relations with Damage
In Dominant Crack Algorithm (DCA), the crack number density function is approximated by an
exponential function, which has been indicated by experiments (Seaman, Curran et al. 1976,
Curran, Seaman et al. 1987, Curran and Seaman 1996, Scholz 2002):
0 ( )n( , , ) exp( / ( , ))
( , )
Nc t c c t
c t
nn n
n (3-1)
where ( , )c tn is the average crack radius, 0 ( )N n is the initial crack number density per solid
angle for crack orientation n . The number of cracks per unit volume
is0 0
ˆ ( , , ) ( )c
N n c t dcd N d
n n , which remains constant over time (i.e., there is no
crack nucleation or coalescence during deformation). It follows from Eq. (3-1) that for a given
orientation n , the number density of cracks with radius larger than c is
0( , , ) ( , , ) ( ) exp( / ( , ))c
N c t n c t dc N c c t
n n n n , which decreases exponentially with the crack
radius.
Substituting ( , )b n and n( , , )c tn into Eq. (3-1) and carrying out the integrations over the crack
radius c gives the total crack strain as
13
3
30
30
0
3
0
3
0
( , ) ( , ) ( , , )
( )8(1 )exp( / ( , )) ( , )
3 (2 ) ( , )
( )8(1 )( exp( / ( , )) ( , ) )
3 (2 ) ( , )
16(1 )( ) ( , ) ( , )
(2 )
16(1 )( ) (
e
cc
c
t c n c t dcd
Nc c t c dcd
G c t
Nc c t c dc d
G c t
N c t dG
N cG
b n n
nn b n
n
nn b n
n
n n b n
n
3
0
, )
16(1 )( ) ( , ) ( ) ( ) 2( )
(2 )
t d
N c t dG
n n n n n
n n n n n n n n n n
(3-2)
According to Zuo, Addessio et al. (2006), the total crack strain can be divided into two parts, the
open crack strain and the shear crack strain,
( , ) ( , ) ( , )o s
c c ct t t (3-3)
With the opening crack strain and shear crack strain given by
3
0
16( , ) (1 ) ( ) ( , )o
c t N c t dG
n n n n n n (3-4)
s 3
0
16(1 )( , ) = ( ) ( , ) ( ) ( ) 2( )
(2 )c t N c t d
G
n n n n n n n n n n (3-5)
s ( , )c t is a deviatoric tensor (i.e., str ( , )=0c t ), i.e., the sliding of crack faces does not
contribute to dilatancy. With the assumption of an isotropic distribution of crack, Addessio and
Johnson (1990) found the analytical expression for shear crack strains by carrying out the
integration over all crack orientations,
s 3
0
64 (1 )( , )=
5 (2 )
d
c c N cG
(3-6)
Where d p I and ( ) / 3p tr are the stress deviator and pressure, respectively; I being
the second order identity tensor. It is noted that the explicit dependency of crack strain on time
( )t has been replaced by the dependency on the mean crack size c , which evolves with time.
14
Similarly, the crack opening strain is found by carrying out the integration in Eq. (3-6) over all
crack orientations. The result can be written as:
3
0
64 1( , ) (1 ) ( ( ) )
15 2
o
c c N c trG
I (3-7)
In the DCA model, strain of the characteristic volume can be considered to be composed by two
parts, the elastic strain ij
e and the crack strain ij
c , as shown in Figure 3.1.
ij ij
e c
ij ( i , j =1, 2, 3) (3-8)
Figure 3.1 The schematics of the DCA constitutive model
The elastic strain can be divided into the volumetric strain tensor and the deviatoric tensor,
1
2ij
e
ije SG
(3-9)
1
3kk
e
kkK
( k =1, 2, 3) (3-10)
where ijS is the deviatoric stress tensor, G is the Shear modulus, and K is the bulk modulus.
Generally, the damage of the characteristic volume can be expressed as a function of the crack
size
3 3
0 ( )c
D AN ca
(3-11)
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where D is the damage function, A is a constant value, 0N is the crack number density, c is
the crack size, and a is the size of the characteristic volume. According to Eq.(3-11), the
evolution of the damage is
3 2
0 0( 3 )D A N c N c c (3-12)
According to the assumption that the crack number density is constant during the failure process,
0 0N , so the damage evolution can be expresses as
22
0 3
33
c cD AN c c
a (3-13)
Addessio and Johnson (1990) established the relation between the deviatoric strain tensor and the
deviatoric stress tensor
3
ij
c e
ije c S (3-14)
where e is a parameter related to the shear modulus and the crack number density by
0 3
12 eG AN
a (3-15)
From Eqs. (3-14) and (3-15), we can get that
32 ( )c
ij ij
cGe S
a (3-16)
Taking the derivative of the equation with respect to time leads to the relation between the crack
strain and the stress tensor.
2 31[3( ) ( ) ]
2
c
ij ij ij
c c ce S S
G a a a (3-17)
And the relation between the strain of the whole model and the stress tensor can be expresses as
2 31[3( ) ( ) ]
2 2
ije c
ij ij ij ij ij
S c c ce e e S S
G G a a a (3-18)
16
where c is the crack velocity, G is the shear modulus, ijS is the deviatoric stress tensor.
3.2 Crack Propagation Criterion
In order to describe the opening and propagation of the microcracks, it is assumed that the
propagation velocity is related to the stress intensity factor (Dienes 1996). The classic theory of
crack instability and growth assumes that the cracks grow at a high speed when the applied stress
exceeds a critical level. According to Freund (1998), when the local stress is really high, the
crack propagation velocity at the crack tip can be close to Rayleigh wave speed of the material,
and the empirical expression for the crack velocity is
2
max[1 ( ) ]ICKc v
K (3-19)
where maxv is the terminal crack speed as the energy release rate increases, which is near the
Rayleigh wave speed of the material Rv (Chen, Xia et al. 2009), and
ICK is the initiation fracture
toughness, K is the local stress intensity factor.
According to Evans (1974), when the crack local stress intensity factor is lower than the critical
value, the crack still propagates with a relatively low speed and the process is quasi-static rather
than dynamic. The crack velocity is expressed as
max
1
( )mKc v
K (3-20)
where m is a factor related to the crack propagation velocity, which is typically between 5 and 10
(Dienes, Zuo et al. 2006). 1K is a constant parameter determined by the continuity condition at
the transition point. Consider the continuity condition of Eqs. (3-19) and (3-20), the crack
velocity and acceleration should be both equal to each other at the transition point (suppose the
corresponding fracture toughness is K ), leading to,
2
1
1 ( ) ( )mICK K
K K
(3-21)
17
2
1
3
1 1
12 ( )IC m
K Km
K K K
(3-22)
Solve Eqs. (3-21) and (3-22), the transition fracture toughness and the constant parameter can be
determined,
21ICK K
m (3-23)
1
1
21 (1 )
2m
IC
mK K
m (3-24)
So the crack propagation velocity can be described as
max
1
( )mKc v
K
21ICK K
m (3-25)
2
max[1 ( ) ]ICKc v
K
21ICK K
m (3-26)
The local stress intensity factor K is calculated using the effective stress
effK c (3-27)
where the effective stress can be written as the form of the deviatoric stress according to Von
Mises law
3
2eff ij ijS S (3-28)
The statistical crack mechanics modeling method is introduced and the dominant crack algorithm
(DCA) is adopted to model the dynamic tensile failure of the Laurentian granite in this study.
The constitutive equations in the DCA modeling and the crack propagation criterion are
developed.
18
4 Dynamic Tensile Tests with Pre-Tension
4.1 BD Experimental Results under Pre-tension
Five groups of Laurentian granite BD specimens (with static tensile strength of 12.8 MPa) under
the pre-tension of 0 MPa, 2 MPa, 4 MPa, 8 MPa, and 10 MPa are tested under different loading
rates.
Figure 4.1 BD specimens sandwiched between the incident and transmitted bars
Based on the one dimensional stress wave theory, and assuming stress equilibrium during
loading (Zhou, Xia et al. 2012) (i.e.,i r t ), the history of the force on the specimen is:
0( ) ( )dP t P P t (3-1)
where 0P is the static preload on the bars, ( )dP t is the dynamic force history on the bars after the
impact. The tensile stress history at the center of the disc specimen can be determined as:
0 00
( )( ) ( ) t
d
A E tt t
RB
(3-2)
where 0 is the pre-tension at the center of the disc, and
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00
P
RB
(3-3)
Figure 4.2 and Figure 4.3 illustrate the dynamic tensile strength versus loading rate and total
tensile strength versus loading rate, respectively. It is obvious that the dynamic strength increases
with the loading rate, revealing the phenomenon of rate dependency that is common for
engineering materials, such as rock (Zhang and Zhao 2013), concrete (Fujikake, Senga et al.
2006, Cusatis 2011), ceramic (Brar and Rosenberg 1988, Zhang, Liu et al. 2008).
Figure 4.2 The dynamic strength versus loading rate for different pre-tensions
Apart from the rate dependency mentioned above, what can be seen from Figure 4.2 is that the
dynamic tensile strength of the rock decreases with the increase of the pre-tension when
subjected to the same loading rate. The decrease of dynamic tensile strength is caused by the
opening of microcracks when the specimen bears the pre-tension stress. Which is consistent to
the results reported by Xia et al. that the microstructures affect the dynamic stress of rock
specimens (Xia, Nasseri et al. 2008).
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Figure 4.3 Total tensile strength versus loading rate for different pre-tensions
Figure 4.3 shows some examples of the total tensile strength versus loading rates for different
pre-tensions. It presents that the total tensile strength of the rock specimens is almost the same
under the same loading rate, no matter how much pre-tension is applied to the specimen before
impact. The result in Figure 4.3 thus indicate that more microcracks are generated and activated
under higher pre-tension, resulting in a more viscous material that is more sensitive to the
loading rate.
The failure pattern of specimens can be an important indicator in revealing the failure
mechanism of rocks, so it would be useful to analyze the failure pattern of the recovered
specimens. Figure 4.4 shows three typical recovered specimens from dynamic tests and one
specimen from static BD test for comparison. The specimen 0-0 is damaged by the static BD test,
it is obvious that the failure is along the loading axis. Figure 4.4 shows that all the specimens
were fractured diametrically into two halves. Specimen 2-2 and specimen 4-2 are both broken
with two pieces of fragments fractured from the center, which reveals the rate dependence of the
rock strength. Both specimen 4-2 and specimen 4-9 have a small wedge shaped crushed zone,
which is a result of secondary fracture after the main fracture from the center of the disc as
21
discussed in the literature (Dai, Huang et al. 2010). The secondary fracture is mainly caused by
the further movement of the incident bar towards the transmitted bar after the initial impact. With
the increase of the loading rate, the impact velocity of the striker bar on the incident bar gets
higher, leading to higher moving velocity of the incident bar.
Figure 4.4 Typical failure patterns of the tested specimens
4.2 DCA Modeling for the Dynamic Tensile Failure of Rock under Pre-tension
The DCA algorithm process is basically involved in the calculation at the dominant crack tip for
the local stress and fracture toughness. With the propagation of the crack, the damage evolves
until the propagation velocity reaches the critical value, while the modeling material fails
suddenly.
The stress and strain are composed of the volumetric tensor and the deviatoric tensor,
ij m ij ijS (3-4)
ij m ij ije (4-6)
which can be written in the form of components in different directions:
where
22
11 22 33
1 1( )
3 3m kk (3-5)
11 22 33
1 1( )
3 3m kk (3-6)
m is the mean stress, m is the mean strain,
kk is the volumetric strain, ij is the Kronecker
delta. When the crack propagates under the effect of far field stress, the local stress intensity
factor is:
effK c (3-7)
where the effective stress can be written as the form of the deviatoric stress according to Von
Mises law
2
33
2eff ij ijJ S S (3-8)
So the local stress intensity factor can also be expressed as：
2 2 3
2eff ij ijK c cS S (3-9)
At certain calculation step, the stress and can be obtained from the stress in previous step and the
strain rate,
(t t) (t)ij ij ij (3-10)
where ij is the increment of the stress, (t)ij is the stress component at the previous step, and
ij m ij ijS (3-11)
m V kkK (3-12)
Where VK is the bulk modulus, the increment of the volumetric strain kk is constant in each
step, ijS is the increment of the deviatoric stress during the time step.
23
Figure 4.5 Flow chart of the DCA modeling
Modeling parameters , , , ,a c m N
Initial stress, strain conditions at 0t t
3(t) (t) (t)
2ij ijK cS S
21ICK K
m
YES
2
max[1 ( ) ]ICKc v
K
max
1
( )mKc v
K
NO
( t) Nn
YES
Stop calculation, Strength=Max( 1 2, .., N )
NO
2 31[3( ) ( ) ]
2
c
ij ij ij
c c ce S S
G a a a
(t t) (t)ij ij V kk ij ijK S
(t t) (t)ij ij t
t t t
24
Is current fitness value
better than pBest?
Target or maximum
epochs reached?
The particle swarm optimization (PSO), which is a relatively recent heuristic search method
similar to the Genetic Algorithm (GA) but more computationally efficient, is adopted to search
for the best values for the parameters a , 0c , and m that works for the modeling.
Figure 4.6 Flow diagram illustrating the particle swarm optimization algorithm
Initialize Particles
Calculate fitness values for each particle
YES
Assignment current
fitness as new pBest
NO
NO
Assign best particle’s pBest value to gBest
Keep previous pBest
Calculate velocity for each particle
Use each particle’s velocity value to update its data value
End YES
25
Figure 4.7 Experimental and modeling results with different loading rates under 0 MPa pre-
tension, with parameters 6.74a mm, 0 1.45c mm and 10.33m
26
Figure 4.8 Relationship between pre-tension and the ratio of initial crack size over characteristic
volume size
Figure 4.8 illustrates how the ratio of initial crack size over the characteristic volume size 0 /c a
is related to the pre-tension the rock specimen bears. Basically, the ratio of 0 /c a increases with
the pre-tension, which is easy to understand because higher pre-tension means more damage
made in the specimen and more microcracks generation and coalescence, leading to a larger
initial crack size.
According to the correlation between the ratio 0 /c a and the static pre-tension
0 , a polynomial
function is used to fit the data, as in Figure 4., the function is expressed as:
2
0 00
0 0
0.221 0.00424 0.00199/c a
(3-13)
where 0 is the static pre-tension at the center of the specimen, ( 1 )MPa is a reference stress
to valid the dimension of the equation. From Eq.(3-13), we can predict the dynamic tensile
strength when the desired pre-tension is given, so Eq.(3-13) can be considered as the bridge
27
between the macroscopic loading conditions to the microscopic characteristics, which can be
used in DCA to predict the dynamic tensile strength of rock materials.
After we get the optimized values for the parameters under different pre-tension stresses, the
DCA model can be carried out to predict the dynamic tensile strength for each case, the
parameters used for the DCA model is listed in Table 4.1.
Table 4.1 Parameters used in DCA for the prediction of the dynamic tensile strength
E(GPa) a(mm) vmax(m/s)
46 4.0 300
υ m KIC(MPa·m1/2)
0.21 5.0 1.37
5 Dynamic Tensile Tests Under Hydrostatic Stress
5.1 BD Test under Hydrostatic Confinement
In this program, five groups of specimens are tested under the hydrostatic confinements of 0
MPa, 5 MPa, 10 MPa, 15 MPa, and 20 MPa. Figure 5.1 illustrates the correlation between tensile
strength and the loading rate when the rock specimens are under 10 MPa hydrostatic stress. It
can be observed that the tensile strength increases with the loading rate almost linearly, revealing
the phenomenon of rate dependency that is common for engineering materials.
Based on the correlation between the tensile strength and the loading rate as shown in Figure 5.1,
and that between the strength and the hydrostatic confinement as in Figure 5.2, an empirical
equation is proposed to fit the testing results and explain the dynamic tensile behaviour of rocks
under hydrostatic confinement:
0
0 0
(1 ln(1 ))(1 ( ) )n
tot
PS S
P
(5-1)
28
where totS denotes the dynamic tensile strength of the test rock materials,
0S (= 12.8 MPa) is the
static BD strength of the rock material. P is the hydrostatic confinement, 0P (=1 MPa) is the
reference hydrostatic confinement; is the loading rate of each impact test, 0 (=1 GPa/s) is
the reference loading rate. , and n are fitting parameters.
Figure 5.1 Dynamic tensile strength of confined rock specimen under 10MPa hydrostatic stress
Figure 5.2 Dynamic tensile testing results under different loading rates and hydrostatic pre-stress,
and data fitting of the results
29
The physical meaning of the equation is that, when the loading rate is 0 and there is no
confinement, the strength of the rock is the static tensile strength S; the dynamic tensile strength
of the rock logarithmically increases with the hydrostatic confinement. The data fitting based on
Eq. (5-1) and the experimental result leads to the values of the parameters as: =0.1804 ,
0.0436 , and 0.5257n . Figure 5.3 is obtained to show how the dynamic tensile strength is
related to the hydrostatic confinement and the loading rate, which fits well to the data points.
Based on the fitted curves, a 3D plot is obtained to show the dynamic tensile strength of rocks
with loading rate and hydrostatic confinement, as in
Figure 5.3. It is obvious that the dynamic tensile strength increases with both the hydrostatic
stress and the loading rate. It can be also seen from the 3D plot that the increment of the dynamic
tensile strength decreases with the loading rate and the hydrostatic stress.
Figure 5.3 The 3D plot of the dynamic tensile strength with loading rate and hydrostatic pre-
stress
Figure 5.4 shows some recovered specimens with different hydrostatic confinement and loading
rate to study the failure pattern of the BD specimens under such loading conditions, revealing the
rate dependence because higher loading rate means less time for the propagation and coalescence
of microcracks before failure, thus more microcracks are involved in the failure process, leading
to higher apparent strength of the rock specimens.
30
b
a
c
d e
Figure 5.4 Recovered specimens under different loading rates and hydrostatic pre-stress
5.2 The DCA Prediction of the Dynamic Tensile Strength of LG
Subjected to Hydrostatic Confinement
The PSO is adopted to do the iterations for the values of the parameters for the DCA modeling.
The parameters are found first to match each case with different confining pressure.
Table 5.1 Modeling error for 20 MPa confining pressure with parameters 3.46a mm,
0 0.24c mm and 9.12m
Loading rate
(GPa/s)
Experimental Strength
(MPa)
Modeling strength
(MPa) Error
686.2 48.22 46.83 2.88%
926.75 51.48 52.71 2.39%
1024.64 52.86 54.18 2.50%
1205.83 56.37 56.85 0.85%
1351.49 59.48 59.42 0.10%
1749.28 67.47 64.58 4.28%
31
Figure 5.5 Experimental and modeling results with different loading rates under 20 MPa
hydrostatic confining pressure, with parameters 3.46a mm, 0 0.24c mm and 9.12m
32
Similar to the case when the confining pressure is 20 MPa, the DCA modeling is carried out with
the PSO algorithm for the cases when the confining pressure is 0 MPa, 5 MPa, 10 MPa, and 15
MPa. Table 5.2 lists the optimized parameters and the error for each hydrostatic confinement
condition.
Table 5.2 Optimized parameters for different hydrostatic confinement conditions by the PSO
Hydrostatic Stress
(MPa)
a
mm 0c
mm 0 /c a m
Max
Error
0 3.81 1.00 0.26 5.00 4.55%
5 4.21 0.60 0.14 9.41 3.27%
10 2.00 0.21 0.11 10.0 3.12%
15 2.79 0.17 0.06 5.04 3.24%
20 3.46 0.24 0.07 9.12 4.28%
After the optimization of all the cases including the pre-tension, triaxial stress and hydrostatic
confining pressure, the characteristic volume size is 4 mm, the parameter m is 5, then the PSO is
adopted again for the initial crack size 0c for each case, and the results are listed in Table 5.3.
Table 5.3 Optimized value for initial crack size under different hydrostatic confinement
Hydrostatic Stress
(MPa)
a
mm 0c
mm 0 /c a m
Max
Error
0 4.00 0.89 0.22 5.00 2.33%
5 4.00 0.45 0.11 5.00 3.82%
10 4.00 0.35 0.09 5.00 3.97%
15 4.00 0.21 0.05 5.00 3.32%
20 4.00 0.16 0.04 5.00 1.71%
To sum up, five groups of Laurentian granite specimen were impacted with different loading
rates under the hydrostatic stress of 0 MPa, 5 MPa, 10 MPa, 15 MPa and 20MPa. Besides the
experiments, the DCA model with the PSO algorithm is adopted to predict the dynamic tensile
33
strength under different hydrostatic confining pressures, which is reflected on the value of the
ratio of the initial crack size to the characteristic volume size. The experimental result reveals that
there is a significant enhancement in tensile strength with the increase of the loading rate, which is the
rate dependency for many engineering materials. The dynamic tensile strength increases with the
hydrostatic confinement due to the restraint of opening and propagation of discontinuities by
confinement. Another important finding is that the increment of the tensile strength decreases as the
hydrostatic confinement becomes higher, which is the same as the static strength under confining
pressure. The DCA modeling is carried out and the comparison of the results with the experimental ones
demonstrates its capability for the prediction of the dynamic tensile strength of rock materials.
6 Dynamic Tensile Test under Triaxial Stress States
6.1 BD Test under Triaxial Stress States
After the calculation of the stress on the specimen, the stress exerted on the bar is then
determined to achieve the pre-stress conditions before the impact is launched. Through the
correlation between the tensile stress on the specimen and the stress at the bar specimen interface,
the load needed on the bar can be determined.
Hydrostatic Stress: 5 MPa, 10 MPa Pretension: 20%, 40%, 60%, 80% of the strength
Figure 6.1 Experimental design with different hydrostatic stress and pretension
+
34
Figure 6.2 Dynamic tensile strength of confined rock specimen under 5MPa hydrostatic stress
Figure 6.3 Dynamic tensile strength of confined rock specimen under 10MPa hydrostatic stress
35
Four groups of specimens are tested under the hydrostatic confinements of 5 MPa and 10 MPa.
Figure 6.2 and Figure 6.3 illustrate the correlation between tensile strength and the loading rate
when the rock specimens are under 5 MPa and 10 MPa hydrostatic stress respectively. It can be
observed that the tensile strength increases with the loading rate almost linearly, revealing the
phenomenon of rate dependency that is common for engineering materials.
Besides the rate dependency mentioned above, it is also obvious that the dynamic tensile strength
of the rock decreases with the pre-tension, which is also observed from the pre-tension tests. The
total tensile strength of LG is illustrated in Figure 6.4 under 5 MPa hydrostatic confinement and
different pre-tensile stresses. Based on the similar fitting for the total tensile strength under pre-
tension without any hydrostatic confinement, the empirical equation as in Eq.(6-1) is proposed to
fit the data and explain the dynamic tensile behavior of rock under triaxial stress states.
0
σ(1 ( ) )n
totS S
(6-1)
where totS denotes the total tensile strength, S (= 17.8 MPa) is the static tensile strength of the
specimen under 5 MPa hydrostatic confining pressure. σ is the loading rate of each impact test,
(=1 GPa/s) is the reference loading rate. and n are fitting parameters. The physical meaning
of the equation is that, when there is no dynamic loading, the strength of the rock is the static
tensile strength S, the second term represents the rate dependence effect. The data fitting based
on Eq.(6-1) and the experimental result leads to the values of the parameters as 0.0628 MPa
and 0.4931n . It is seen that the function fits well with the experimental data points. The result
in Figure 6.4 thus indicates that more microcracks are generated and activated under higher pre-
tension, resulting in a more viscous material that is more sensitive to the loading rate. This
augmented rate sensitivity compensates the pre-tension weakening effect, leading to a roughly
pre-tension independency of the total tensile strength.
The total tensile strength of LG under 10 MPa hydrostatic confining pressure is illustrated in
Figure 6.5. The data fitting based on Eq.(6-1) and the experimental result leads to the values of
the parameters as: 0.0636 MPa, and 0.4454n .
36
Figure 6.4 Total strength of rock specimen under 5MPa hydrostatic stress and pre-tension
Figure 6.5 Total strength of rock specimen under 10 MPa hydrostatic stress and pre-tension
Figure 6.6 shows some recovered specimens with different hydrostatic confinement and
pretension to study the failure pattern of the BD specimens under such loading conditions. The
photos at the top are the specimens under 5 MPa hydrostatic confining pressure, followed by the
37
specimens under 10 MPa in the middle, and the pattern schematics at the bottom. Type I is
failure pattern is the diametrical split. Figure 6.6a are the specimens with cracks at center and
break neatly into two halves. Type II has central cracking with crushed wedges. In this case, the
specimen fails along its central line but small wedge-shaped pieces can be found in the crumbs,
as shown in Figure 6.6b. Type III shows failure with a crushed strap where a diametrically
distributed crushed zone can be found in the specimen center, as illustrated in Figure 6.6c.
Figure 6.6 Failure pattern of specimens: (a) diametrical split, (b) central cracking with crushed
wedges, (c) failure with crushed strap. The photos at top are the specimens under 5 MPa
confining pressure and in the middle are the specimens under 10 MPa confining pressure.
(a)
(a) (b)
(b)
(c)
(c)
38
6.2 The DCA Prediction of the Dynamic Tensile Strength of LG
Subjected to Triaxial Stress State
Same as the preceding chapters, the PSO is adopted to do the iterations for the values of the
parameters for the DCA modeling. Similarly, the parameters are found first to match each case
with different confining pressure. For example, the iterations for confining pressure 5 MPa with
3.56 MPa (20% of tensile strength) pretension is illustrated in Figure 6.7, the iterations for
confining pressure 10 MPa with 4.56 MPa (20% of tensile strength) pretension is illustrated in
Figure 6.8.
Figure 6.7 Iterations for DCA modeling parameters with 5 MPa hydrostatic pressure and 3.56
MPa (20% of tensile strength) pre-tension
39
Figure 6.8 Iterations for DCA modeling parameters with 10 MPa hydrostatic pressure and 4.56
MPa (20% of tensile strength) pre-tension
The iteration number is 450 to convergent when the rock is subjected 5 MPa hydrostatic
confining pressure and 3.56 MPa pre-tension, the corresponding values of the parameters are
10.0a mm, 0 0.51c mm and 6.03m ; It takes 462 iterations for the DCA modeling to
converge when the rock specimen are subjected to traixial stress with 10 MPa hydrostatic
confining pressure and 4.56 MPa pre-tension, and the parameters are 4.79a mm,
0 0.25c mm and 7.05m . Results of the DCA modeling under 10 MPa confinement and
several loading rates are illustrated in Figure 6.9.
40
Figure 6.9 Experimental and modeling results with different loading rates under 10 MPa
hydrostatic confining pressure and 20% pretension, with parameters 4.79a mm, 0 0.25c mm
and 7.05m
41
Similar to the cases when the rocks are subjected to either pure pre-tension or hydrostatic
confinement, the PSO algorithm is adopted in the DCA modeling to achieve the group
optimization of the parameters. Table 6.1 lists the optimized parameters and the error of each
superposed pre-stress conditions.
Table 6.1 Optimized parameters for different hydrostatic confinement conditions by the PSO
Hydrostatic
(MPa) Pre-tension
a
mm 0c
mm 0 /c a m Max Error
5
20% 4.00 0.19 0.05 5.00 2.92%
40% 4.00 0.23 0.06 5.00 4.77%
60% 4.00 0.35 0.09 5.00 2.53%
80% 4.00 0.52 0.13 5.00 4.12%
10
20% 4.00 0.17 0.04 5.00 2.58%
40% 4.00 0.33 0.08 5.00 3.32%
60% 4.00 0.45 0.11 5.00 2.08%
80% 4.00 0.70 0.18 5.00 2.14%
It is obvious that under both hydrostatic confinement conditions, the ratio of 0 /c a decreases
with the increasing pre-tension, which is the same as when the rocks are subjected to pure pre-
tension. After the determination of the parameters, the DCA modeling is carried out and the
dynamic tensile stress history of the specimens under each superposed loading condition is
obtained.
To sum up, the modified SHPB system with hydraulic cylinders is adopted to measure the
dynamic tensile strength under triaixal stress states. The confining pressure are set as 5 MPa and
10 MPa, the pre-tension is 20%, 40%, 60% or 80% of the static tensile strength under the current
hydrostatic confining pressure. Similarly, the DCA model with the PSO algorithm is used to
predict the dynamic tensile strength under different pre-stress conditions, which is reflected on
the value of the ratio of the initial crack size to the characteristic volume size.
42
7 Concluding Remarks
This thesis investigated the dynamic tensile failure of Laurentian granite under different pre-
stress conditions with a wide range of loading rates, and explored the relationship between the
dynamic tensile strength of the rock material and the pre-tension, the hydrostatic confining
pressure and the loading rate. The following paragraphs summarize the main items completed in
this thesis and the major conclusions. The experimental results and the DCA modeling
performance can be used as a guideline for the development of a reliable method to predict blast-
induced tensile failure and design tool for blasting operations.
1. The dynamic experimental methodology involving experimentation and calculation equations
using the modified dynamic testing system, i.e. SHPB system, are developed to measure the
dynamic tensile strength of Laurentian granite under different pre-stress conditions. The main
achievement is the modification of the SHPB system to exert static pre-tension and
hydrostatic confining pressure to the rock specimens before the dynamic test is conducted.
2. The dynamic tensile strength of Laurentian granite under different pre-stress conditions are
obtained with respect to the loading rate.
3. The micro-CT technique is used to observe the crack distribution and to exam the failure
pattern of the rock specimens under hydrostatic confinement. It is observed that the failure is
always an extension fracture in the loaded diametral plane as long as the test is valid. The
micro-CT images demonstrate that with higher loading rates, there are more cracks at the
vicinity of the loading diameter direction, which is consistent to the material rate dependency.
4. The development of the Dominant Crack Algorithm (DCA) in the use of prediction for the
dynamic tensile strength of the rock materials. The DCA modeling is carried out and it shows
that it is most sensitive to the ratio of initial crack size over the characteristic volume size
0 /c a , and less sensitive to the characteristic volume size a and the crack velocity related
parameter m .
5. The adoption of Particle Swarm Optimization (PSO) for the optimized values of the
parameters in the DCA modeling for the dynamic tensile failure of LG under different pre-
43
tension stresses. With the PSO algorithm, the optimized value for the characteristic volume
size a is determined as 4 mm and the crack velocity related parameter m is determined as 5.
Then, the PSO iteration is used again for the optimized value of the initial crack size for each
case with different pre-stress conditions.
Selected References
Addessio, F. L. and J. N. Johnson (1990). "A Constitutive Model for the Dynamic-Response of Brittle Materials." Journal of Applied Physics 67(7): 3275-3286.
Al-Shayea, N. A., K. Khan and S. N. Abduljauwad (2000). "Effects of confining pressure and temperature on mixed-mode (I-II) fracture toughness of a limestone rode." International Journal of Rock Mechanics and Mining Sciences 37(4): 629-643.
Brar, N. S. and Z. Rosenberg (1988). "Brittle Failure of Ceramic Rods under Dynamic Compression." Journal De Physique 49(C-3): 607-612.
Cai, M., P. K. Kaiser, F. Suorineni and K. Su (2007). "A study on the dynamic behavior of the Meuse/Haute-Marne argillite." Physics and Chemistry of the Earth 32(8-14): 907-916.
Chen, M. and G. Q. Zhang (2004). "Laboratory measurement and interpretation of the fracture toughness of formation rocks at great depth." Journal of Petroleum Science and Engineering 41(1-3): 221-231.
Chen, R., K. Xia, F. Dai, F. Lu and S. N. Luo (2009). "Determination of dynamic fracture parameters using a semi-circular bend technique in split Hopkinson pressure bar testing." Engineering Fracture Mechanics 76(9): 1268-1276.
Curran, D. R. and L. Seaman (1996). Simplified Models of Fracture and Fragmentation. High-Pressure Shock Compression of Solids II. L. Davison, D. Grady and M. Shahinpoor, Springer New York: 340-365.
Curran, D. R., L. Seaman and D. A. Shockey (1987). "Dynamic Failure of Solids." Physics Reports-Review Section of Physics Letters 147(5-6): 253-388.
Cusatis, G. (2011). "Strain-rate effects on concrete behavior." International Journal of Impact Engineering 38(4): 162-170.
Dai, F., S. Huang, K. Xia and Z. Tan (2010). "Some Fundamental Issues in Dynamic Compression and Tension Tests of Rocks Using Split Hopkinson Pressure Bar." Rock Mechanics and Rock Engineering 43(6): 657-666.
Dai, F., S. Huang, K. W. Xia and Z. Y. Tan (2010). "Some Fundamental Issues in Dynamic Compression and Tension Tests of Rocks Using Split Hopkinson Pressure Bar." Rock Mechanics and Rock Engineering 43(6): 657-666.
Deganis, L. E. and Q. H. Zuo (2011). "Crack-mechanics based brittle damage model including nonlinear equation of state and porosity growth." Journal of Applied Physics 109(7): 073504(073511).
44
Dienes, J. K. (1996). A Unified Theory of Flow, Hot Spots, and Fragmentation with an Application to Explosive Sensitivity. High-Pressure Shock Compression of Solids II. L. Davison, D. Grady and M. Shahinpoor, Springer New York: 366-398.
Dienes, J. K., Q. H. Zuo and J. D. Kershner (2006). "Impact initiation of explosives and propellants via statistical crack mechanics." Journal of the Mechanics and Physics of Solids 54(6): 1237-1275.
Dietrich, R. V. and B. J. Skinner (1979). Rocks and rock minerals, Wiley.
Evans, A. G. (1974). "Slow Crack Growth in Brittle Materials under Dynamic Loading Conditions." International Journal of Fracture 10(2): 251-259.
Freund, L. B. (1998). Dynamic Fracture Mechanics, Cambridge University Press.
Fujikake, K., T. Senga, N. Ueda, T. Ohno and M. Katagiri (2006). "Effects of Strain Rate on Tensile Behavior of Reactive Powder Concrete." Journal of Advanced Concrete Technology 4(1): 79-84.
Funatsu, T., M. Seto, H. Shimada, K. Matsui and M. Kuruppu (2004). "Combined effects of increasing temperature and confining pressure on the fracture toughness of clay bearing rocks." International Journal of Rock Mechanics and Mining Sciences 41(6): 927-938.
Huang, S., K. Xia and H. Zheng (2013). "Observation of microscopic damage accumulation in brittle solids subjected to dynamic compressive loading." Review of Scientific Instruments 84(9): 093903.
Jaeger, J. C. and E. R. Hoskins (1966). "Rock Failure under Confined Brazilian Test." Journal of Geophysical Research 71(10): 2651-2659.
Kaiser, P. K., B. Kim and R. P. Bewick (2010). Rock mass strength at depth and implication for pillar design. The Fifth International Seminar on Deep and High Stress Mining.
Kolsky, H. (1949). "An investigation of the mechanical properties of materials at very high rates of loading." Proceedings of the Royal Society A-Mathematical Physical and Engineering Sciences B62(11): 676-700.
Kolsky, H. (1953). Stress waves in solids. Oxford, UK, Clarendon Press.
Nasseri, M. H. B. and B. Mohanty (2008). "Fracture toughness anisotropy in granitic rocks." International Journal of Rock Mechanics and Mining Sciences 45(2): 167-193.
Nasseri, M. H. B., B. Mohanty and P. Y. F. Robin (2005). "Characterization of microstructures and fracture toughness in five granitic rocks." International Journal of Rock Mechanics and Mining Sciences 42(3): 450-460.
Robertson, E. C. (1955). "Experimental Study of the Strength of Rocks." Geological Society of America Bulletin 66(10): 1275-1314.
Ross, C. A., P. Y. Thompson and J. W. Tedesco (1989). "Split-Hopkinson Pressure-Bar Tests on Concrete and Mortar in Tension and Compression." Aci Materials Journal 86(5): 475-481.
Scholz, C. H. (2002). The Mechanics of Earthquakes and Faulting, Cambridge University Press.
45
Seaman, L., D. R. Curran and D. A. Shockey (1976). "Computational Models for Ductile and Brittle-Fracture." Journal of Applied Physics 47(11): 4814-4826.
Vasarhelyi, B. (1997). "Influence of pressure on the crack propagation under mode I loading in anisotropic gneiss." Rock Mechanics and Rock Engineering 30(1): 59-64.
Wang, Q. Z., X. M. Jia, S. Q. Kou, Z. X. Zhang and P. A. Lindqvist (2004). "The flattened Brazilian disc specimen used for testing elastic modulus, tensile strength and fracture toughness of brittle rocks: analytical and numerical results." International Journal of Rock Mechanics and Mining Sciences 41(2): 245-253.
Wang, Q. Z., W. Li and H. P. Xie (2009). "Dynamic split tensile test of Flattened Brazilian Disc of rock with SHPB setup." Mechanics of Materials 41(3): 252-260.
Xia, K. (2009). "Split Hopkinson pressure bar (SHPB) tests on rocks." News Journal-International Society for Rock Mechanics 12: 72-75.
Xia, K., F. Dai and R. Chen (2011). Advancements in Hopkinson pressure bar techniques and applications to rock strength and fracture. Advances in rock dynamics and applications. Y. X. Zhou and J. Zhao. Boca Raton, Florida, USA, CRC Press: 35-78.
Xia, K., M. H. B. Nasseri, B. Mohanty, F. Lu, R. Chen and S. N. Luo (2008). "Effects of microstructures on dynamic compression of Barre granite." International Journal of Rock Mechanics and Mining Sciences 45(6): 879-887.
Zhang, J. T., L. S. Liu, P. C. Zhai and Q. J. Zhang (2008). "Experimental and numerical researches of dynamic failure of a high strength alumina/boride ceramic composite." High-Performance Ceramics V, Pts 1 and 2 368-372: 713-716.
Zhang, Q. B. and J. Zhao (2013). "A Review of Dynamic Experimental Techniques and Mechanical Behavior of Rock Materials." Rock Mechanics and Rock Engineering 47(4): 1411-1478.
Zhao, J. and H. B. Li (2000). "Experimental determination of dynamic tensile properties of a granite." International Journal of Rock Mechanics and Mining Sciences 37(5): 861-866.
Zhou, Y. X., K. Xia, X. B. Li, H. B. Li, G. W. Ma, J. Zhao, Z. L. Zhou and F. Dai (2012). "Suggested methods for determining the dynamic strength parameters and mode-I fracture toughness of rock materials." International Journal of Rock Mechanics and Mining Sciences 49: 105-112.
Zhou, Y. X., K. Xia, X. B. Li, H. B. Li, G. W. Ma, J. Zhao, Z. L. Zhou and F. Dai (2012). "Suggested methods for determining the dynamic strength parameters and mode-I fracture toughness of rock materials." International Journal of Rock Mechanics and Mining Sciences 49(2012): 105-112.
Zuo, Q. H., F. L. Addessio, J. K. Dienes and M. W. Lewis (2006). "A rate-dependent damage model for brittle materials based on the dominant crack." International Journal of Solids and Structures 43(11-12): 3350-3380.
Zuo, Q. H., D. Disilvestro and J. D. Richter (2010). "A crack-mechanics based model for damage and plasticity of brittle materials under dynamic loading." International Journal of Solids and Structures 47(20): 2790-2798.
