University
stringclasses 19
values | Text
stringlengths 458
20.7k
|
|---|---|
ADE | can provide more practical outputs. The successful application of these algorithms has been
reported by other researchers in mining and geotechnical engineering fields (Armaghani et al.
2016; Salimi et al. 2016; Hasanipanah et al. 2017b; Khandelwal et al. 2017). Hence, it is
necessary to use state-of-the-art modelling techniques to address the mentioned difficulties and
develop new models for predicting rockburst maximum stress and its risk index based on field
conditions. As it has been summarized in Fig. 5.4, this study focuses on the following steps: 1)
compiling a database based on the true-triaxial unloading tests on different rock types and
performing a broad statistical analysis on it to create a homogeneous database and to select the
most influential parameters based on an appropriate strategy; 2) Developing genetic-based and
decision tree-based models for the prediction of maximum rockburst stress (π ) and rockburst
π
π΅
risk index (πΌ ) based on the selected input parameters; 3) validation verification of the
π
π΅
developed models; and 4) conducting a parametric analysis to assess the effect of input
parameters on the corresponding outputs.
Figure 5.1 Rock ejection and deformation of the supporting system due to strainbursting
(Feng et al. 2017)
117 |
ADE | horizontal in-situ stress in the face to be unloaded; π : vertical in-situ stress; π : rockburst
π£ π
π΅
maximum stress; πΌ : rockburst risk index; π·: depth, π: density, πΎ: horizontal pressure
π
π΅
coefficient (ratio of average horizontal stresses to the vertical stress due to overburden),
ππΏπ
: multiple linear regression; ππΌπΉ: variance inflation factor; π
2: coefficient of
determination)
5.2. Data Collection and Statistical Analysis
In this study, a database containing information about the 139 rockburst laboratory tests
conducted on different rock types from 2004 to 2012 at the State Key Laboratory for
Geomechanics and Deep Underground Engineering (SKLGDUE), China was compiled. The
tested rock samples were gathered from the depth of 200 m to 3375 m. This database consists
of many parameters such as rock mechanical properties, in-situ stresses, rock sample depth,
rockburst critical depth, rock density, rock specific weight, mineral contents of rocks, loading
and unloading rates of the true-triaxial tests, rockburst maximum stress, rockburst risk index,
test duration and bursting mechanism. Considering a circular shape for the tunnel crown, the
stress concentration factor equal to 2, and the specific weight of 27 kN/m3 for the overburden
rock mass, the rockburst critical depth (π» ) was calculated by the following equation:
π
H = 18.52π (5.1)
e π
π΅
The rockburst risk index (I ) also was calculated for all the samples through the following
RB
equation (He 2009):
H H
I = = 0.054 (5.2)
RB
He ΟRB
He (2009) defined a new classification for πΌ as shown in Table 5.1. Based on this
π
π΅
classification, a 56% of the tested samples have low πΌ , 13% of the samples have very high
π
π΅
πΌ , and the remained 31% of samples have moderate to high πΌ . Since all the foregoing
π
π΅ π
π΅
parameters have not been collected during the rockburst tests, there are some missing values in
the database. To have a homogeneous database, the missing values (30 records) were
eliminated from the primary database, and finally, the results of 109 tests were considered for
further analyses. Before developing any model, the presence of natural groups and outliers in
the raw database was evaluated using agglomerative hierarchical clustering (AHC) analysis. In
fact, the presence of outliers and natural groups can decrease the generality and liability of the
developed models (Hudaverdi 2012; Faradonbeh and Monjezi 2017; Shirani Faradonbeh and
120 |
ADE | Taheri 2019) The AHC is the most common type of clustering techniques which is used in
earth sciences (Hudaverdi 2012). The AHC follows a bottom-up procedure that iteratively
creates the single object clusters and then these clusters are merged into the larger clusters
based on the similarity or dissimilarity criteria. The common criterion for clustering is
βdistanceβ, and this means that objects in the same cluster have the least distance from each
other, while objects in different clusters are at a great distance from one another. The process
of cluster generating and merging is continued until all the objects (datasets) are placed in a
single cluster or the pre-defined termination condition is satisfied. For measuring the distance
between the objects, the average-linkage function that measures the average distance of any
object of one cluster from an object of the other cluster was used to form the clusters (Kaufman
and Rousseeuw 2009; Saxena et al. 2017):
1
β β π(π,π) (5.3)
πβπ΄ πβπ΅
|π΄||π΅|
where π΄ and π΅ are two clusters with the sizes of |π΄| and |π΅|, respectively. π and π are objects
from the mentioned clusters and π is the squared Euclidean distance between two objects.
Table 5.1 Rockburst risk index classification, He et al. (2015)
Rockburst risk index (πΌ ) Class
π
π΅
πΌ <0.6 Low
π
π΅
0.6<πΌ β€1.2 Moderate
π
π΅
1.2<πΌ β€2.0 High
π
π΅
πΌ β₯2.0 Very high
π
π΅
Fig. 5.5 shows the dendrogram derived from the conducted clustering analysis by AHC. A
dendrogram is a tool that represents the relative size of the calculated distances at which the
objects and clusters are combined. The objects with the low squared Euclidean distance (high
similarity) are close together and vice versa. The X-axis shows the dataset number and the Y-
axis shows the rescaled value of the distance. To prevent Fig. 5.5 to be crowded and large, the
numbers of the datasets have been summarised on the X-axis. Clearly can be seen from Fig.
5.5 that the whole 109 collected datasets were clustered into one distinct group between the
rescaled distances of 0 and 5 except for two cases of 75 and 76 which were placed in the second
group. By checking the database, it was found out that the main parameter that caused to
grouping is depth, and the members of group 2 belong to the depth of 3375 m which are known
as outliers for the current database. Therefore, these two cases were removed from the database
to avoid the influence of their distinctive behaviour on the modelling process, and the
121 |
ADE | 5.3. Methods and Results
5.3.1. Stepwise Selection and Elimination Process
This section aims to do a systematic stepwise selection and elimination (SSE) analysis to
identify the most important parameters on the outputs and reduce the complexity of the
developed models. The process of parameter reduction also is carried out using the variable
pressure tools of the robust data-mining techniques i.e. GEP and CART. There are several
critical statistical terms which have been used in this study for the primary assessment of the
database and are defined in the following. Multicollinearity, a high correlation between the
independent (predictor) variables, can be considered as one of the most prominent challenges
for multiple regressions. The existence of this phenomenon may lead to developing an unstable
regression model having high values for variance and covariance coefficients (Sayadi et al.
2012). Variance inflation factor (VIF) is a statistical index to quantify the extent of the
multicollinearity between the independent (input) parameters. This index is the ratio of model
variance considering several inputs to the variance of the model with a single input parameter.
The VIF lower than 10 shows the non-existence of multicollinearity (James et al. 2013).
Another important index is Sig. (2-tailed) or p-value of the correlations. The Sig (2-tailed)
represents the significance of the correlation at a prescribed alpha level (5%). The Sig. (2-
tailed) should be less than or equal to 0.05 to reject the influence of chance factor. The
coefficient of determination (denoted by π
2) is another statistical measure for evaluation of the
model performance. This index interprets the proportion of the output (dependent) variableβs
variance that is predictable from the input (independent) variables. An π
2 of 1 indicates that
the regression predictions perfectly fit the data (Montgomery et al. 2012; James et al. 2013;
Kumar Sharma and Rai 2017).
In the current study, uniaxial compressive strength (ππΆπ), Youngβs modulus (πΈ), Poissonβs
ratio (π), horizontal in-situ stress (π ), horizontal in-situ stress in the face to be unloaded (π ),
β1 β2
vertical in-situ stress (π ), depth (π·), density (π), and horizontal pressure coefficient (πΎ) are
π£
known as the input parameters for the maximum rockburst stress (π ), while all the mentioned
π
π΅
parameters are considered as inputs for the rockburst risk index (πΌ ). The SPSS software
π
π΅
package 25.0 was used for performing the statistical evaluations. Initially, the database was fed
to the software, and the Personβs correlation coefficient (π) between the input parameters as
well as between the inputs and the corresponding outputs was calculated. Table 5.2 lists the
calculated correlation values. As can be seen from this table, all the inputs significantly
125 |
ADE | correlating with π (i.e. πππ.(2βπ‘πππππ) β€ 0.05), while π· (depth) with the πππ.> 0.05
π
π΅
and low correlation coefficient (π = β0.128) was removed from the input parameters for
further modelling of πΌ . The elimination of parameters does not show that they have not any
π
π΅
influence on the output, but simply it means that the effect of those parameters will be minimum
in predicting the output. As an initial multicollinearity assessment between input parameters,
no one of the correlations exceeds from the condition of π > 0.90 (Hemmateenejad and
Yazdani 2009). However, these input parameters may show multicollinearity when a
combination of them are used as regressors in MLR. Based on the above analysis, all the inputs
(except parameter π· for πΌ ) were retained for multiple linear regression (MLR). The MLR
π
π΅
models with the possible multicollinearity were developed separately using the selected
parameters for both π and πΌ .
π
π΅ π
π΅
Table 5.3 shows the model summary, calculated coefficients, and the statistical indices for
evaluating the developed MLR models. In this stage, according to Fig. 5.4, several conditions
including ππΌπΉ < 10, ππ‘π.πππππ β€ πΆππππ.(π΅), and πΆππππ.(π΅) β 0 were checked for
different inputs to retain them for further evaluations. Considering Table 5.3, for rockburst
maximum stress (π ), the parameters of πΎ and π have ππΌπΉ > 10 and t-significance higher
π
π΅
than 0.05, respectively, which shows that the effect of these parameters on the π is
π
π΅
insignificant. Therefore, these parameters were removed for further modelling of π . About
π
π΅
the rockburst risk index (πΌ ), all the VIF values for inputs are less than 10, but the t-
π
π΅
significance values of the ππΆπ, π , π , and π are higher than 0.05. Thus, these parameters
π£ β1 β2
also were removed from the input set of πΌ . In the next step, two stepwise selection and
π
π΅
elimination procedures were performed using the selected inputs for each dependent parameter.
In this procedure, a parameter which is entered in the model at the initial stage of selection may
be removed at the later stages. In fact, the calculations in this process are like the forward
selection and backward procedure (Sarkhosh et al. 2012).
Table 5.4 summarises the results of the stepwise selection and elimination procedure carried
out using the algorithm provided in the SPSS 25. In this algorithm, the parameters enter the
model if the probability (significance level) of its πΉ value is less than the Entry value (i.e. 0.05)
and are eliminated if the probability is greater than the Removal value (i.e. 0.100). Entry must
be less than Removal, and both values must be positive. As given in Table 5.4, during the
process of selection and elimination, the correlation coefficient (π
) between the measured
output and the predicted one was increased from 0.884 (model 1) to 0.910 (model 3) for π
π
π΅
126 |
ADE | and from 0.754 (model 1) to 0.821 (model 5) for πΌ , respectively. In other words, the
π
π΅
parameters of ππΆπ, πΈ, and π can explain 82.8% (π
2 = 0.828) variations in π . As such, the
π£ π
π΅
parameters of π , πΎ, πΈ, π, and π can explain 67.4% (π
2 = 0.674) variations in πΌ . Thus,
π
π΅ π
π΅
these parameters were known as the most influential ones among the initial inputs to describe
the rockburst parameters. The regression coefficients and the collinearity statistics of the best
SSE-based models are shown in Table 5.5. In both models, the VIF factor that shows the
multicollinearity is lower than 10, the t-significance is lower than 0.05, and the Std. error values
are lower than the regression coefficients which show the reliability of the SSE process in
identifying the most influential parameters.
Considering the above analyses, the agglomerative hierarchical clustering (AHC) accompanied
by the stepwise selection and elimination (SSE) method could provide a homogeneous
rockburst database by removing the outliers and decreasing the dimensionality of the problem.
This process also can be useful for the complexity reduction of the next predictive models by
applying a few input parameters. Due to the high non-linear and complex nature of rockburst
hazard (He et al. 2015; Pu et al. 2019; Shirani Faradonbeh and Taheri 2019) there is a need to
use the non-linear data-mining algorithms to provide more accurate predictive models for
rockburst parameters. To do so, two robust data-driven approaches including the gene
expression programming (GEP) as a meta-heuristic algorithm and the classification and
regression tree (CART) as a subset of decision tree algorithms were selected for discovering
the non-linear latent relationships with more accuracy and lower estimation error. These
algorithms despite the various datamining and soft computing techniques such as ANNs, SVM,
etc. can provide practical and easy to use outputs for the engineers and the researchers when
the true-triaxial testing machine is not available. A summary of the modelling procedure by
these techniques is presented in the following sections.
127 |
ADE | 5.3.2. Non-linear Regression Analysis
Non-linear regression (NLR) attempts to find a function which is a non-linear combination of
the input parameters using a method of successive approximation (Archontoulis and Miguez
2015; Bethea 2018). In geoscience, most of the dependent parameters show a non-linear
relationship with the related influential parameters. So, the non-linear regression analysis has
been widely used by researchers in the last decades (Armaghani et al. 2016; Jahed Armaghani
et al. 2017; Ghasemi 2017). The NLR technique is capable of accommodating a broad range
of functions including exponential, power, logarithmic, sigmoid, logistic, trigonometric,
Gaussian, etc. that boosts the process of function finding. Another advantage of the NLR is the
efficient use of data, i.e. it can provide reasonable estimates of the unknown parameters for a
comparatively small data. However, the common NLR technique suffers from several
significant drawbacks. In NLR, there is no a closed-form and holistic mathematical structure
between the dependent and the independent parameters as there is in multiple linear regression
(MLR), while the choice of the model structure is a crucial task to obtain the best solution. In
addition, the selection and utilizing the suitable mathematical functions from the large library
of functions need an iterative optimization procedure that is not possible in common NLR
modellings. Accordingly, the researchers may have to use numerical optimization algorithms
to find the best-fitting parameters but still, there is a need to define the starting values for the
unknown parameters in these methods. Inappropriate assigning the starting values may cause
to getting caught in the local minima rather than finding the global minimum that introduces
the least squares estimates (Motulsky and Ransnas 1987; Archontoulis and Miguez 2015;
Bethea 2018). For these difficulties, the researchers prefer to use a non-linear regression form
that has been used successfully in similar applications. Hereupon, the application of intelligent
algorithms is needed to cope with these issues. In the following sections, the process of
rockburst assessment using two robust non-linear techniques comprising the gene expression
programming (GEP) and classification and regression tree (CART) are explained.
5.3.2.1. Rockburst Assessment Using GEP-based Models
Soft computing is the relatively new branch of data-mining methods and can be considered as
an alternative to the prevalent hard computing methods for solving the real-world problems
(Mitchell 1997; Alavi et al. 2016). Soft computing techniques have been successfully employed
in mining, rock mechanics, and geotechnical problems but despite their good performance, they
cannot generate practical equations, and their structure needs to be assigned in advance by the
131 |
ADE | user (Alavi and Gandomi 2011). By inspiring from the Darwinian principle of βSurvival of the
Fittestβ (Nazari and Pacheco Torgal 2013) and the natural evolution, a new subset of soft
computing was introduced as the evolutionary algorithm (EA). Generally speaking, EAs work
with a randomly generated population of individuals which are then improved using a group
of genetic operators (e.g. mutation, crossover and reproduction) and finally, the solutions are
encoded into the specific forms such as binary strings in genetic algorithm. The main
differences between EAs are related to the method of presenting the solutions, genetic
operators, selection mechanism, and the performance measurement method (Ferreira 2002a;
Alavi et al. 2016). Gene expression programming (GEP) (Ferreira 2002b) is a well-known
evolutionary algorithm that inherits two essential features from its siblings i.e. the use of
simple, fixed-length, and linear chromosomes with different shapes and sizes from genetic
algorithm (GA) and the expression tree (ET) structure from genetic programming (GP) that
improves the robustness of GEP for solving the non-linear problems (Power et al. 2019). The
main entities of GEP algorithm are terminal set (input parameters and constant values),
function set (e.g. +,β, Γ, Γ·), fitness function (for evaluating the generated solutions), and
genetic operators (mutation, inversion, transposition, and recombination).
A flowchart detailing the GEP modelling procedure is shown in Fig. 5.7. In summary, GEP
generates a population of chromosomes (solution/individual) by combining the user-defined
terminals and functions. These chromosomes follow a bilingual and unequivocal expression
system that is called Karva language (Ferreira 2006). The chromosomes have a specified
number of genes (sub -ETs) which are linked together using a linking function (e.g. β/β in Fig.
5.7 that links two genes of a chromosome). Each gene contains two parts of head and tail that
the terminals (inputs) and functions (mathematical functions) are placed in them and the genetic
operators are applied to these areas to modify the solutions. To have a quick understanding
regarding the built-in mathematical equations of chromosomes, the Karva coded programs are
then parsed into ETs. Then, the mathematical form of the programs is extracted from ETs and
their fitness is evaluated by a fitness function. If the stopping condition(s) such as reaching to
a specific number of iterations or the desired fitness value is not met, the selected chromosomes
are replicated into a new generation, and the remained ones undergo a modification process
using the genetic operators. The above process is repeated and finally, the best solution
(predictive model) describing the relationship between the input and output parameters is
found. More details about the mechanism of genetic operators and GEP algorithm can be found
132 |
ADE | in Ferreira.(Ferreira 2006) In the current study, the selected inputs from the SSE analysis were
considered as terminal sets to formulate the rockburst parameters nonlinearly as follows:
π = π(ππΆπ,πΈ,π ) (5.4)
π
π΅ π£
πΌ = π(π ,πΎ,π,πΈ,π) (5.5)
π
π΅ π
π΅
The rockburst database was divided randomly into training and testing subsets. The training
set (80 % of the database) was used to train the model and discover the relationship between
inputs and outputs, and the remaining datasets were used to validate the performance of the
proposed models. It should be noted that the influence of using different groups of training and
testing datasets were also evaluated on the accuracy of the models. However, no noticeable
change in the results was observed. For evaluating the generated solutions during the GEP
modelling, it is necessary to use a fitness function. As mentioned in section 5.3.1, to propose
models with lower complexity, it is possible to apply variable pressure tools to compress the
developed models as much as possible by eliminating the parameters which have lower
importance in a non-linear structure. To this end, the root mean squared error (RMSE) with
parsimony pressure was applied to the GEP models of π and πΌ (Roy et al. 2002). The
π
π΅ π
π΅
π
πππΈ of a chromosome (solution) π is calculated by the following equation:
π
π
πππΈ = β1 βπ (π βπ)2 (5.6)
π π π=1 ππ π
where π is the predicted value by the chromosome π for the dataset π, and π is the measured
ππ π
value for dataset π.
The π
πππΈ varies between 0 and infinity, with 0 corresponding to the ideal. Since the process
π
of selection in GEP algorithm is based on the increase of fitness, Equation (6) cannot be used
directly. Thus, the following expression was used for fitness function which obviously ranges
between 0 to 1000, with 1000 corresponding to the ideal:
π
πππΈβ² = 1000Γ 1 (5.7)
π
1+π
πππΈ
π
On the other hand, to apply the parsimony pressure on future models, overall fitness was
defined as:
133 |
ADE | π
πππΈβ²β² = π
πππΈβ² Γ(1+ 1 Γ ππππ₯βπ π ) (5.8)
π π
5000 ππππ₯βπ
πππ
where π is the size of the GEP program, π and π are the maximum and minimum
π πππ₯ πππ
program sizes which are calculated by the following equations:
π = πΊ(β+π‘) (5.9)
πππ₯
π = πΊ (5.10)
πππ
where πΊ is the number of genes, and β and π‘ are the head size and tail size, respectively.
A group of trigonometric and straightforward mathematical functions i.e. {+,β,β
,/,β,πΏπ,^2,^3,^1/3,π ππ,πππ ,π‘ππ} were selected as the function set based on the previous
non-linear studies using GEP algorithm (Kayadelen 2011; Faradonbeh and Monjezi 2017;
Hoseinian et al. 2017). The other GEP parameters including the number of chromosomes, head
size, the number of genes, and the values of genetic operators were changed for different runs
to obtain the best solution in such a way that provides not only high accuracy but also less
complexity. Table 5.6 presents the architecture of the obtained GEP models for both rockburst
maximum stress (π ) and rockburst risk index (πΌ ). By applying the parsimony pressure to
π
π΅ π
π΅
the models, the density parameter (π) was identified intelligently as the low-impact parameter
in the non-linear form of πΌ . Therefore, this parameter was removed by GEP automatically
π
π΅
during modelling and the number of inputs for πΌ decreased from 5 to 4. About π , the GEP
π
π΅ π
π΅
algorithm identified the three inputs of ππΆπ, πΈ, and π as the influential parameters for
π£
modelling as formerly proved by SSE analysis. The ability of GEP in identifying the low-
influence parameters and excluding them during modelling can be considered as an internal
sensitivity analysis that distinguishes GEP from other soft computing techniques. Fig. 5.8
displays the variations of the coefficient of determination (π
2) during 5000 generations
(iterations) in both training and testing stages of GEP modelling for rockburst parameters.
According to this figure, after a few numbers of generations (less than 1000), a rapid increase
of π
2 for the generated solutions can be seen which shows the high speed and high capability
of GEP algorithm in function finding. From the generation 1000 to 3500, a gentle enhancement
in the quality of solutions are visible, and finally, the algorithm converges into an optimum
value and its value almost remains constant to reach the stopping condition (i.e. the pre-defined
number of generations: 5000). The obtained π
2 values for training and testing stages of π
π
π΅
are 0.9266 and 0.9398, respectively, while the foregoing values are 0.8824 and 0.9459,
respectively for πΌ . Figs. 5.9 shows the correlation of the experimentally measured values of
π
π΅
134 |
ADE | π and πΌ versus the predicted ones by the constructed GEP models for training and testing
π
π΅ π
π΅
data groups. As seen, the data points have almost a uniform distribution around the fitted lines
in both GEP-based models which show the goodness-of-fit of the models. The developed
models and their performance are discussed in more details in sections 5.4 and 5.5. Eventually,
the mathematical forms of the proposed GEP models for π and πΌ were extracted from their
π
π΅ π
π΅
K-expression and ETs as Eqs. 5.11 and 5.12. To avoid the prolongation of the paper, the ETs
and their K-expressions have not presented here.
π = (3 π +πΈπ ππ(πΈ βπ )+πΈ)(3 πΈ +π +πΈπ ππ(πΈ)) πΏπ(πΏπ(π )+ππΆπ) (5.11)
π
π΅ π£ π£ π£ π£
π6 βπΎ(πΈ+4
βπ)
πΌ = (5.12)
π
π΅
(π+πΎ)(πΈβπ) πΏπ(ππ
π΅)
Head Tail Head Tail
Create initial population
- b Γ b a b b a b Γ β b + a b b b a
Gene 1 Gene 2
/
Express solutions as ETs
- Γ
b Γ β b
πβ(πΓπ)
Execute each program b a +
βπ+πΓπ
a b
Evaluate fitness π
πππΈβ²β²
π
1) Mutation (an element is changed to another)
- b Γ b a b b a b Before
Yes - b Γ b Γ b b a b After
TTeerrmmiinnaattee??
2) Inversion (a fragment is inverted in the head)
No Show the solution - b Γ b a b b a b Before
- Γ b b a b b a b After
Select the best solutions 3) Transposition (IS type: a fragment is copied to the head of a gene)
- b Γ b a b b a b Before
- b a b a b b a b After
Apply genetic operators
4) Recombination (One-point type: two chromosomes exchange a fragment)
- b Γ b a b b a b Before
Γ β b + a b b b a
Create next generation
- b b + a b b b a After
Γ β Γ b a b b a b
Figure 5.7 Process of function finding using GEP algorithm
135 |
ADE | Fitness function RMSEβ²β² RMSEβ²β²
i i
Parsimony pressure Yes Yes
Mutation rate 0.01 0.04
Inversion rate 0.1 0.1
Transposition 0.1 0.1
One-point recombination 0.3 0.3
Two-point recombination 0.3 0.3
Gene recombination 0.1 0.1
CART parameter Setting
Ο I
RB RB
Initial inputs UCS,E,Ο Ο ,K,Ο
,E,Ο
v RB
Excluded parameter - Ο
Minimum number of cases 3 3
for parent node
Minimum number of cases 1 1
for child node
Minimum change of 0.0005 0.0003
impurity level
Maximum tree depth 6 5
Number of intervals 10 10
Impurity measure LSD LSD
Total number of nodes 27 33
5.3.2.2. Rockburst Assessment Using Classification and Regression Tree (CART)
Decision tree as a powerful subset of data-mining techniques has been used in different real-
world applications for different aims such as decision making, classification, prediction, pattern
recognition, etc. (Kantardzic 2003; Hasanipanah et al. 2017a) A decision tree is a tree
comprising a root node (i.e. a parameter that can provide maximum degree of discrimination),
some internal nodes representing input parameters, branches which link the nodes together and
contain the binary questions regarding the internal nodes, and some leaf nodes representing the
solutions (predicted value or a specific class of the dependent parameter). Each path from the
root node to a leaf node can be summarised as a rule that this feature makes the decision tree
to be known as a rule-based algorithm (Mahjoobi and Etemad-Shahidi 2008). Based on the
type of dependent parameter, i.e. being continuous or categorical, the established tree structure
is nominated as regression tree (RT) or classification tree (CT), respectively. The decision tree
has several subgroups such as ID3 (Quinlan 1986), C4.5, C5.0, CART, CHAID, Exhaustive
CHAID, and QUEST (Mahjoobi and Etemad-Shahidi 2008) which have been used for different
aims by scholars (Khandelwal et al. 2017; Ghasemi et al. 2017). Among these techniques, the
CART algorithm introduced by Breiman et al. (1984) has several advantages that distinguish
it among other decision tree algorithms. This algorithm, despite the parametric statistical
techniques (e.g. regression analyses), is inherently non-parametric (rule-based), i.e. no
138 |
ADE | assumption is made with the distribution of values of the independent parameters. On the other
hand, CART can handle the highly skewed (multimodal) quantitative data as well as the
qualitative parameters with ordinal or non-ordinal structures (Breiman et al. 1984; Salimi et al.
2016).
In this algorithm, it is not necessary to eliminate the multicollinearity between the independent
parameters. Moreover, CART algorithm can be applied on a database with no homogeneity.
CART also can handle the existence of outliers in the raw database by isolating them into a
separate node. Because of the mentioned advantages, flexibility, and practical output (tree
structure) of this algorithm, it was used in this study for the prediction of rockburst parameters
obtained from true-triaxial tests. As a matter of fact, since the output parameters in this study
(i.e. π and πΌ ) are continuous, the aim is to develop two regression trees (RTs) for each
π
π΅ π
π΅
parameter. The process of RT building in CART algorithm focuses mainly on the three
following components: (1) a group of questions in the form of π β€ π? where π is an input
parameter and π is a constant value in a range that the parameter π varies. In CART, the
response to this type of question is βyesβ or βnoβ; (2) the best split on a parameter is determined
using a split criterion; (3) calculation of summary statistics for internal nodes. The goal in
CART modelling is to create sub-nodes (children) which are more homogeneous and purer
than parent nodes based upon the reduction in impurity or improvement score. The term βpureβ
is related to the values of given parameter i.e. in the complete pure node, all cases have a similar
value of the splitting parameter and consequently, the nodeβs variance equal to zero. This issue
is compared for all the input parameters and the best improvement is chosen for splitting. This
procedure continues until one of the stopping conditions is triggered (Breiman et al. 1984). In
CART, the least squared deviation (LSD) is used as an impurity measure. The LSD function
for splitting a parent node π‘ into two newly generated sub-nodes π‘ and π‘ can be
πΏ(πππ‘) π
(ππβπ‘)
calculated using the following equation (Breiman et al. 1984; Bevilacqua et al. 2003):
Ξ¦ = π
2(π‘)βπ π
2(π‘ )βπ π
2(π‘ ) = 1 β [π¦ βπ¦Μ
(π‘)]2 βπ 1 β [π¦ β
(π‘) πΏ πΏ π
π
π(π‘)
πππ‘ π πΏ
π(π‘πΏ)
πππ‘πΏ π
π¦Μ
(π‘ )]2 βπ 1 β [π¦ βπ¦Μ
(π‘ )]2 (5.13)
πΏ π
π(π‘π
)
πππ‘π
π π
where π
2(π‘ ) is the weighted variance related to the sub-node (child) π‘ , π is the proportion
π₯ π₯ πΏ
of cases in parent node π‘ which are classified in the left node (π‘ ), π is the proportion of cases
πΏ π
in parent node π‘ which are classified in the right node (π‘ ), π(π‘ ) is the number of cases
π
π₯
139 |
ADE | classified in sub-node π‘ (π₯π{π
,πΏ}), π¦ is the value of the objective parameter for the case π,
π₯ π
π¦Μ
(π‘) is the mean value of parent node, and π¦Μ
(π‘ ) is the mean value of the sub-node π‘ .
π₯ π₯
The best split is obtained by maximizing the Ξ¦ showing the reduction of impurity of an RT
(π‘)
model. This splitting process leads to creating a tree structure based on several βif-thenβ rules
that make it easy to represent. The splitting process proceeds until each leaf node meets at least
one of the stopping criteria. The stopping criteria include: (1) reaching the maximum tree depth;
(2) the number of cases (datasets) in the terminal node is less than the predefined minimum
parent size; (3) the number of cases in the sub-nodes resulting from the best splits is less than
pre-defined minimum child size. The stopping criteria used in this study for CART models are
tabulated in Table 6. These criteria and their corresponding values were obtained in such a way
that the results provide a good trade-off between the prediction accuracy of regression trees
and their complexity (dimension). All these settings also prevent the models from getting stuck
in over-fitting problems. The use of a high number of maximum tree depth can lead to
producing a large tree structure with high complexity that makes it complicated to use in
practice. Additionally, a maximum number of intervals equal to 10 was considered for both
models to let the model break down the initial min-max range of each input parameter to
different ranges during the splitting process. In this study, the CART models for rockburst
parameters (π and πΌ ) were developed using a code written in MatLab R2019a software
π
π΅ π
π΅
environment. To have the same modelling conditions for further assessments, the training and
testing datasets used for GEP were fed again to the CART algorithm.
According to Table 5.6, for rockburst risk index (πΌ ) model, like the GEP-based one, the
π
π΅
density (π) parameter has been excluded from the model since the CART benefits from an
internal principal component analysis (PCA) that enables it to consider most influential
parameters. Figs. 5.10 and 5.11 demonstrate the constructed RTs for π and πΌ using CART
π
π΅ π
π΅
algorithm, respectively. The tree model of π contains 27 nodes and starts with ππΆπ as a root
π
π΅
node, while the πΌ model has 33 nodes that starts with π as the root node. The extraction of
π
π΅ π
π΅
final predicted values of π and πΌ from these regression trees is an easy task. For instance,
π
π΅ π
π΅
in Fig. 5.10, consider the experimentally measured values of 82.7 MPa, 24.3 GPa, 114.6 MPa,
and 108.6 MPa for ππΆπ, πΈ, π , and π , respectively; by tracking the associated tree structure
π£ π
π΅
from the root node (i.e. node1: ππΆπ) and the path ππΆπ β₯ 40.45, π < 169.455, π β₯ 63.3,
π£ π£
ππΆπ < 151.3, π < 143.5, and π < 118.4, the tree reaches to the leaf node 22 that predicts
π£ π£
the value of 116.1 for π . The same process can be done for πΌ as well. As stated at the
π
π΅ π
π΅
140 |
ADE | sensitivity of the problem. Furthermore, to assess the performance of the developed models in
depth, new validation indices have been proposed by other researchers. Golbraikh and Tropsha
(2002) defined two indices of π and πβ² to validate the models on testing datasets. In addition,
Roy and Roy (2008) proposed an indicator called π
along with another related parameter of
π
π
2 to check the predictability of the proposed models. The corresponding values of the π, πβ²,
π
π
, and π
2 can be calculated based on the measured (β ) and predicted (π‘ ) values of the output
π π π π
parameters (here are π and πΌ ). The mathematical expressions of the above indices and their
π
π΅ π
π΅
threshold values are listed in Table 5.7. Taking into account the recommendations, at least a
slope of the regression lines (i.e. π or πβ²) through the origin should be close to 1, while π is the
slope of the regression line when β is plotted versus π‘ , and πβ² is the regression line when π‘ is
π π π
plotted versus β (Golbraikh and Tropsha 2002). The squared correlation coefficient between
π
the predicted and measured values (π
2) should be close to 1. The π
, then, can be calculated
π π
by π
and π
2 values, and a threshold of > 0.5 is recommended for this index to introduce a
π
model as valid. The foregoing indices were calculated for the developed GEP-based and
CART-based models, and their values are listed in Table 5.7. Indeed, the indices of π
, π, πβ²,
π
, and π
2 were used to verify the validity of the models in testing stage as recommended by
π π
other researchers (Mohammadzadeh et al. 2016; Soleimani et al. 2018). Then, the statistical
indices of π
2, π
πππΈ, and ππ΄πΈ were calculated to compare the prediction performance of the
GEP and CART models for π and πΌ based on testing datasets to select the best models. It
π
π΅ π
π΅
can be observed from Table 5.7 that the both proposed models in this study satisfy all the
required conditions, and this guarantees that the derived models are strongly credible i.e. the
results are not based on chance factor. In addition, comparing the π
2, π
πππΈ, and ππ΄πΈ values
of GEP and CART models show that both proposed models have a high degree of accuracy
and low estimation error, and subsequently have this capability to be used in practical
applications. However, the GEP models of π and πΌ outperformed the CART models and
π
π΅ π
π΅
have slightly better performance. The next section aims to do a parametric analysis on the
selected models (i.e. GEP models) to appraise the effect of the variation of input parameters on
the predicted values.
145 |
ADE | Table 5.7 Statistical indices for the external validation of the developed models
Item Formula Threshold π πΌ
π
π΅ π
π΅
GEP CART GEP CART
1 βπ (β ββΜ
)(π‘ βπ‘Μ
) π
>0.8 0.969 0.957 0.972 0.943
π
= π=1 π π π π
ββπ (β ββΜ
)2βπ (π‘ βπ‘Μ
)2
π=1 π π π=1 π π
2 βπ (βπ‘) 0.85<π<1.15 0.934 0.931 0.962 0.981
π= π=1 π π
βπ β2
π=1 π
3 βπ (βπ‘) 0.85<πβ²<1.15 1.046 1.040 1.014 0.971
πβ²= π=1 π π
βπ π‘2
π=1 π
4 π
>0.5 0.763 0.698 0.739 0.596
π
=π
2(1ββ|π
2βπ
2|) π
π π
βπ (π‘ ββπ)2 Should be close to 1 0.987 0.986 0.997 0.999
π
2 =1β π=1 π π ,
π βπ (π‘ βπ‘Μ
)2
π=1 π π
βπ =π π‘
π π
5 π
2 Should be close to 1 0.939 0.916 0.946 0.889
6 Should be minimum 14.249 16.426 0.195 0.273
π
1 (based on output range)
π
πππΈ= (β βπ‘)2
π π π
π=1
7 1 π Should be minimum 9.803 8.041 0.136 0.187
ππ΄πΈ=
π
|β πβπ‘ π| (based on output range)
π=1
β: measured output; π‘: predicted output
π π
5.5. Parametric Analysis
To investigate the influence of each input parameter on the predicted values of the
corresponding output, a parametric analysis was carried out based on the selected GEP models
for π and πΌ . This analysis also can be another validation for the GEP models by evaluating
π
π΅ π
π΅
how well the results (predicted values) agree with the physical behaviour of the rockburst
parameters. To do so, the desired independent parameter should be varied within its range of
values, while other independent parameters are constant in their averages. Figs. 5.13 and 5.14
plot the variation of input parameters against the predicted values for rockburst parameters. As
it is seen in Fig. 5.13, the π increases monotonically in a non-linear fashion with ππΆπ and
π
π΅
π . This result is expected since with the increase of ππΆπ, the capacity of the rock to accumulate
π£
the strain energy increases, and finally, bursting occurs at a higher stress level violently (Singh
1987). On the other hand, the in-situ stresses, especially the vertical in-situ stress (π ) are
π£
increased in a linear or non-linear relationship with depth (Wagner 2019) and subsequently,
due to a high geo-stress state in deep conditions, the π is enhanced. However, there are many
π
π΅
fluctuations in π values with the increase of Youngβs modulus (πΈ) of rocks, but in general,
π
π΅
an increment trend can be seen. It should be mentioned that a parameter may do not show a
meaningful relationship solely with the output parameter, while it can be an influential
146 |
ADE | component in a combination of other parameters in a non-linear form. As mentioned in the
GEP modelling section, during the modelling procedure, by applying the variable pressure
coefficient, excluding any of the selected three parameters (i.e. ππΆπ, πΈ, and π ) from the
π£
modelling procedure did not improve the accuracy and complexity of the model.
Regarding πΌ , a non-linear decreasing trend can be observed for its values with all input
π
π΅
parameters of Youngβs modulus (πΈ), Poissonβs ratio (π), horizontal pressure coefficient (πΎ),
and rockburst maximum stress (π ). As can be seen from Fig. 5.14, with the increase of πΈ
π
π΅
until 20 MPa, the rockburst risk index is decreased suddenly but it remains almost constant
with a further increment of πΈ. Moreover, with the increase of Poissonβs ratio (π) in its range of
values, the risk value decreases from 0.473 to 0.40 that according to Table 1, the risk of
rockburst occurrence is low. Hence, it seems that the variation of π has no significant influence
on rockburst risk. However, it is still necessary to do more tests on rocks with a greater range
of π to check its influence on risk parameter. Generally, the risk of rockburst occurrence for
rocks with low strength (or lower π ) which are in low depth (or higher πΎ) is higher than
π
π΅
high-strength rocks in deep conditions. From the results displayed in Figs. 5.13 and 5.14,
several non-linear equations between rockburst parameters (π , and πΌ ) and their related
π
π΅ π
π΅
input parameters (except for π βπΈ and πΌ βπΈ) are extracted as follows:
π
π΅ π
π΅
π = 19.911πΏπ(ππΆπ)+10.636, π
2 = 0.9974 (5.14)
π
π΅
π = β0.0007π 2 +0.7162π +41.092, π
2 = 0.9877 (5.15)
π
π΅ π£ π£
πΌ = 0.49πβ0.535π, π
2 = 0.9997 (5.16)
π
π΅
πΌ = β0.0186πΎ3 +0.2124πΎ2 β0.7981πΎ +1.1997, π
2 = 0.9521 (5.17)
π
π΅
πΌ = 1.2524π β0.244, π
2 = 0.9859 (5.18)
π
π΅ π
π΅
According to the above results, clearly can be seen that there is a good correlation between the
rockburst parameters and the inputs. These equations provide a series of simple equations for
calculating π and πΌ based on the single rock mechanical parameter as a primary
π
π΅ π
π΅
assessment. These equations may be relevant to investigate rockburst potential. In the end, it is
necessary to mention that the developed models in this study are based on the collected datasets
and a specific range of values for different parameters. So, for future applications, if the input
parameters are out of these ranges, the proposed models should be adjusted again.
147 |
ADE | 2 1
y = -0.0186x3+ 0.2124x2-0.7981x
1.5 0.8
+ 1.1997
RΒ² = 0.9521
y = 1.2524x-0.244
B 1 B0.6 RΒ² = 0.9859
R R
I I
0.5 0.4
0 0.2
0 2 4 6 0 100 200 300
K
Ο (MPa)
RB
Figure 5.14 (Continued)
5.6. Summary and Conclusions
As a catastrophic geohazard, rockburst threatens the safety of workers and infrastructures in
deep geotechnical conditions. In this study, considering the importance of the stress level that
rockburst occurs for different rock types in real stress circumstances, two important rockburst
parameters including the maximum rockburst stress (π ) and rockburst risk index (πΌ ) were
π
π΅ π
π΅
formulated using the information obtained from true-triaxial unloading tests and two robust
data-driven approaches. A comprehensive strategy was applied to the compiled database using
the correlation analysis, the agglomerative hierarchical clustering (AHC) technique, and the
stepwise selection and elimination (SSE) procedure to provide a homogeneous database free
from any outliers, natural groups, and especially, to identify the most influential parameters on
π and πΌ . Then, new non-linear models were developed using robust algorithms of gene
π
π΅ π
π΅
expression programming (GEP) and classification and regression tree (CART). Finally, a
parametric analysis was conducted to study the variation of π and πΌ with the change of
π
π΅ π
π΅
input parameters. The conclusions obtained from this study are presented in the following.
The correlation analysis, AHC, SSE, and multiple regression analysis techniques, as
recommended and implemented in the current study, have presented promising results by
dimension reduction (i.e. eliminating the redundant input parameters) and choosing the
statistically significant parameters that affect the rockburst parameters (i.e. π and πΌ ). This
π
π΅ π
π΅
procedure simplifies the rockburst assessment at the field scale. The obtained dendrogram by
AHC analysis (Fig. 5.5) showed that there is no natural group in the compiled database except
for two data samples that were identified as outliers and subsequently were eliminated from
the original database. Therefore, the database was identified as a homogeneous database for
149 |
ADE | 29 If E in [36.05, 71) and K in [1.721, 5.866) and π in [56.8, 255.5) then πΌ = 0.185 in 12.8% of cases
π
π΅ π
π΅
30 If E in [14.1, 29.7) and K in [1.721, 5.866) and π in [56.8, 255.5) then πΌ = 0.292 in 12.8% of cases
π
π΅ π
π΅
31 If E in [29.7, 36.05) and K in [1.721, 5.866) and π in [56.8, 255.5) then πΌ = 0.547 in 2.3% of cases
π
π΅ π
π΅
32 If π in [56.8, 143.6) and E in [36.05, 71) and K in [1.721, 5.866) then πΌ = 0.205 in 9.3% of cases
π
π΅ π
π΅
33 If π in [143.6, 255.5) and E in [36.05, 71) and K in [1.721, 5.866) then πΌ = 0.133 in 3.5% of cases
π
π΅ π
π΅
References
Akdag S, Karakus M, Taheri A, et al (2018) Effects of Thermal Damage on Strain Burst
Mechanism for Brittle Rocks Under True-Triaxial Loading Conditions. Rock Mechanics
and Rock Engineering 51(6):1β26
Alavi AH, Gandomi AH (2011) A robust data mining approach for formulation of geotechnical
engineering systems. Engineering Computations 28(3):242β274
Alavi AH, Hasni H, Lajnef N, et al (2016) Damage detection using self-powered wireless
sensor data: An evolutionary approach. Measurement: Journal of the International
Measurement Confederation 82:254β283
Archontoulis S V, Miguez FE (2015) Nonlinear regression models and applications in
agricultural research. Agronomy Journal 107(2):786β798
Armaghani DJ, Faradonbeh RS, Rezaei H, et al (2016) Settlement prediction of the rock-
socketed piles through a new technique based on gene expression programming. Neural
Computing and Applications 29(11):1115-1125
Bagde MN, PetroΕ‘ V (2005) Fatigue properties of intact sandstone samples subjected to
dynamic uniaxial cyclical loading. International Journal of Rock Mechanics and Mining
Sciences 42(2):237β250
Barquins M, Petit J-P (1992) Kinetic instabilities during the propagation of a branch crack:
effects of loading conditions and internal pressure. Journal of Structural Geology 14(8-
9):893β903
Bethea RM (2018) Statistical methods for engineers and scientists. Routledge
Bevilacqua M, Braglia M, Montanari R (2003) The classification and regression tree approach
to pump failure rate analysis. Reliability Engineering & System Safety 79(1):59β67
Breiman L, Friedman J, Stone CJ, Olshen RA (1984) Classification and Regression Trees. CRC
press
152 |
ADE | Cai M, Kaiser P (2018) Rockburst support reference bookβvolume I: rockburst phenomenon
and support characteristics. Laurentian University
Chen G, Li T, Wang W, et al (2019) Weakening effects of the presence of water on the
brittleness of hard sandstone. Bulletin of Engineering Geology and the Environment
78(3):1471β1483
Chen J, Su G (2010) True triaxial experimental study on hard rock under high geo-stress
condition based on excavation and support. In: 2010 International Conference on
Mechanic Automation and Control Engineering. pp 1465β1467
Cheon DS, Keon S, Park C, Ryu C (2006) An experimental study on the brittle failure under
true triaxial conditions. Tunnelling and underground space technology 21(3-4)
Cook NGW (1963) The basic mechanics of rockbursts. Journal of the Southern African
Institute of Mining and Metallurgy 64(3):71β81
Cortez P, Embrechts MJ (2011) Opening black box Data Mining models using Sensitivity
Analysis. In: IEEE Symposium on Computational Intelligence and Data Mining
Faradonbeh RS, Monjezi M (2017) Prediction and minimization of blast-induced ground
vibration using two robust meta-heuristic algorithms. Engineering with Computers 1β17
Feng XT, Chen B, Feng G, Zhao Z, Zheng H (2017) Description and engineering phenomenon
of rockbursts. In: Rockburst: Mechanisms, Monitoring, Warning, and Mitigation. first ed.
Elsevier-Health Sciences Division 3-17
Ferreira C (2006) Gene expression programming: mathematical modeling by an artificial
intelligence. Springer
Ferreira C (2002) Gene Expression Programming in Problem Solving. In: Roy R, KΓΆppen M,
Ovaska S, et al. (eds) Soft Computing and Industry: Recent Applications. Springer
London, London, pp 635β653
Ghasemi E (2017) Particle swarm optimization approach for forecasting backbreak induced by
bench blasting. Neural Computing and Applications 28(7):1855β1862
Ghasemi E, Kalhori H, Bagherpour R (2017) Stability assessment of hard rock pillars using
two intelligent classification techniques: A comparative study. Tunnelling and
Underground Space Technology 68:32β37
153 |
ADE | Golbraikh A, Tropsha A (2002) Beware of q2! Journal of Molecular Graphics and Modelling
20(4):269β276
Hasanipanah M, Faradonbeh RS, Amnieh HB, et al (2017a) Forecasting blast-induced ground
vibration developing a CART model. Engineering with Computers 33(2):307-316
Hasanipanah M, Faradonbeh RS, Armaghani DJ, et al (2017b) Development of a precise model
for prediction of blast-induced flyrock using regression tree technique. Environmental
Earth Sciences 76(1)
He J, Dou L, Gong S, et al (2017) Rock burst assessment and prediction by dynamic and static
stress analysis based on micro-seismic monitoring. International Journal of Rock
Mechanics and Mining Sciences 100(93):46β53
He M (2009) The mechanism of rockburst and its countermeasure of support. Consultation
Report for the Key Technology of Safe and Rapid Construction for Jinping LI Hydropower
Station High Overburden and Long Tunnels 23β28
He M, e Sousa LR, Miranda T, Zhu G (2015) Rockburst laboratory tests database - Application
of data mining techniques. Engineering Geology 185:116β130
He M, Gong W, Wang J, et al (2014) Development of a novel energy-absorbing bolt with
extraordinarily large elongation and constant resistance. International Journal of Rock
Mechanics and Mining Sciences 67:29β42
He M, Ren F, Liu D (2018) Rockburst mechanism research and its control. International
Journal of Mining Science and Technology 28(5):829β837
He M, Xia H, Jia X, et al (2012) Studies on classification, criteria and control of rockbursts.
Journal of Rock Mechanics and Geotechnical Engineering 4(2):97β114
He MC, Miao JL, Feng JL (2010) Rock burst process of limestone and its acoustic emission
characteristics under true-triaxial unloading conditions. International Journal of Rock
Mechanics and Mining Sciences 47(2):286β298
Hemmateenejad B, Yazdani M (2009) QSPR models for half-wave reduction potential of
steroids: A comparative study between feature selection and feature extraction from
subsets of or entire set of descriptors. Analytica Chimica Acta 634(1):27β35
Hoseinian FS, Faradonbeh RS, Abdollahzadeh A, et al (2017) Semi-autogenous mill power
154 |
ADE | model development using gene expression programming. Powder Technology 308:61-69
Hudaverdi T (2012) Application of multivariate analysis for prediction of blast-induced ground
vibrations. Soil Dynamics and Earthquake Engineering 43:300β308
Jahed Armaghani D, Faradonbeh RS, Momeni E, et al (2017) Performance prediction of tunnel
boring machine through developing a gene expression programming equation.
Engineering with Computers 34(1):129-141
James G, Witten D, Hastie T, Tibshirani R (2013) An introduction to statistical learning.
Springer
Jian Z, Xibing L, Xiuzhi S (2012) Long-term prediction model of rockburst in underground
openings using heuristic algorithms and support vector machines. Safety Science
50(4):629β644
Kantardzic M (2003) Data Mining: Concepts, models, methods, and algorithms. John Wiley &
Sons
Kaufman L, Rousseeuw PJ (2009) Finding groups in data: an introduction to cluster analysis.
John Wiley & Sons
Kayadelen C (2011) Soil liquefaction modeling by Genetic Expression Programming and
Neuro-Fuzzy. Expert Systems with Applications 38(4):4080β4087
Khandelwal M, Armaghani DJ, Faradonbeh RS, et al (2017) Classification and regression tree
technique in estimating peak particle velocity caused by blasting. Engineering with
Computers 33(1):45β53
Kumar Sharma S, Rai P (2017) Establishment of blasting design parameters influencing mean
fragment size using state-of-art statistical tools and techniques. Measurement 96:34β51
Li D, Zhao F, Zheng M (2014) Fractal characteristics of cracks and fragments generated in
unloading rockburst tests. International Journal of Mining Science and Technology
24(6):819β823
Li T, Cai MF, Cai M (2007) A review of mining-induced seismicity in China. International
Journal of Rock Mechanics and Mining Sciences 44:1149β1171
Linkov AM (1996) Rockbursts and the instability of rock masses. International Journal of
Rock Mechanics and Mining Sciences & Geomechanics Abstracts 33(7):727β732
155 |
ADE | Liu X, Liang Z, Zhang Y, et al (2018) Experimental study on the monitoring of rockburst in
tunnels under dry and saturated conditions using AE and infrared monitoring. Tunnelling
and Underground Space Technology 82:517β528
Mahjoobi J, Etemad-Shahidi A (2008) An alternative approach for the prediction of significant
wave heights based on classification and regression trees. Applied Ocean Research
30(3):172β177
Mitchell TM (1997) Does machine learning really work? AI Magazine 18:11β20
Mohammadzadeh S D, Bolouri Bazaz J, Vafaee Jani Yazd SH, Alavi AH (2016) Deriving an
intelligent model for soil compression index utilizing multi-gene genetic programming.
Environmental Earth Sciences 75(3):262
Montgomery DC, Peck EA, Vining GG (2012) Introduction to Linear Regression Analysis.
Wiley
Motulsky HJ, Ransnas LA (1987) Fitting curves to data using nonlinear regression: a practical
and nonmathematical review. The FASEB journal 1(5):365β374
Nazari A, Pacheco Torgal F (2013) Modeling the compressive strength of geopolymeric
binders by gene expression programming-GEP. Expert Systems with Applications
40(14):5427β5438
Power HE, Gharabaghi B, Bonakdari H, et al (2019) Prediction of wave runup on beaches using
Gene-Expression Programming and empirical relationships. Coastal Engineering
144:47β61
Pu Y, Apel DB, Xu H (2019) Rockburst prediction in kimberlite with unsupervised learning
method and support vector classifier. Tunnelling and Underground Space Technology
90:12β18
Quinlan JR (1986) Induction of decision trees. Machine Learning 1:81β106
Ranjith PG, Zhao J, Ju M, et al (2017) Opportunities and Challenges in Deep Mining: A Brief
Review. Engineering 3(4):546β551
Roy PP, Roy K (2008) On Some Aspects of Variable Selection for Partial Least Squares
Regression Models. QSAR & Combinatorial Science 27(3):302β313
Roy R, KΓΆppen M, Ovaska S, et al (2002) Soft Computing and Industry: Recent Applications
156 |
ADE | Saharan MR, Mitri H (2011) Destress blasting as a mines safety tool: Some fundamental
challenges for successful applications. In: Procedia Engineering. Elsevier 26(37-47) pp
37β47
Salimi A, Faradonbeh RS, Monjezi M, Moormann C (2016) TBM performance estimation
using a classification and regression tree (CART) technique. Bulletin of Engineering
Geology and the Environment, 77(1):429-440
Sarkhosh M, Ghasemi JB, Ayati M (2012) A quantitative structure- property relationship of
gas chromatographic/mass spectrometric retention data of 85 volatile organic compounds
as air pollutant materials by multivariate methods. Chemistry Central journal 6 Suppl
2:S4βS4
Saxena A, Prasad M, Gupta A, et al (2017) A review of clustering techniques and
developments. Neurocomputing 267:664β681
Sayadi AR, Lashgari A, Paraszczak JJ (2012) Hard-rock LHD cost estimation using single and
multiple regressions based on principal component analysis. Tunnelling and Underground
Space Technology 27(1):133β141
Shi XZ, Zhou J, Dong L, et al (2010) Application of unascertained measurement model to
prediction of classification of rockburst intensity. Chinese Journal of Rock Mechanics and
Engineering 29(1):2720β2726
Shirani Faradonbeh R, Taheri A (2019) Long-term prediction of rockburst hazard in deep
underground openings using three robust data mining techniques. Engineering with
Computers 35(2):659β675
Singh SP (1987) The influence of rock properties on the occurrence and control of rockbursts.
Mining Science and Technology 5(1):11β18
SMITH GN (1986) Probability and statistics in civil engineering. Collins Professional and
Technical Books 244
Soleimani S, Rajaei S, Jiao P, et al (2018) New prediction models for unconfined compressive
strength of geopolymer stabilized soil using multi-gen genetic programming.
Measurement 113:99β107
Sousa LR (2010) Risk Analysis for Tunneling Projects. Massachusetts Institute of Technology,
Cambridge
157 |
ADE | Su G, Chen Z, Ju JW, Jiang J (2017a) Influence of temperature on the strainburst characteristics
of granite under true triaxial loading conditions. Engineering Geology 222:38β52
Su G, Jiang J, Zhai S, Zhang G (2017b) Influence of Tunnel Axis Stress on Strainburst: An
Experimental Study. Rock Mechanics and Rock Engineering 50(6):1551β1567
Sun X, Xu H, Zheng L, et al (2016) An experimental investigation on acoustic emission
characteristics of sandstone rockburst with different moisture contents. Science China
Technological Sciences 59(10):1549β1558
Tang C, Wang J, Zhang J (2010) Preliminary engineering application of microseismic
monitoring technique to rockburst prediction in tunneling of Jinping II project. Journal of
Rock Mechanics and Geotechnical Engineering 2(3):193β208
Wagner H (2019) Deep Mining: A Rock Engineering Challenge. Rock Mechanics and Rock
Engineering, 1417-1446
Wang J-A, Park HD (2001) Comprehensive prediction of rockburst based on analysis of strain
energy in rocks. Tunnelling and Underground Space Technology 16(1):49β57
Wang X, Huang R (1998) Analysis of deformation and failure features characteristics of rock
under unloading conditions and their effects on rock burst. Mountain Research 16(4):281β
285
Wang Y, He M, Liu D, Gao Y (2019) Rockburst in Sandstone Containing Elliptic Holes with
Varying Axial Ratios. Advances in Materials Science and Engineering
Weng L, Huang L, Taheri A, Li X (2017) Rockburst characteristics and numerical simulation
based on a strain energy density index: A case study of a roadway in Linglong gold mine,
China. Tunnelling and Underground Space Technology 69:223β232
Xu LS (2003) Research on the experimental rock mechanics of rockburst under unloading
condition. Journal of Chongqing Jiaotong University 22(1):1β4
Zhao XG, Cai M (2015) Influence of specimen height-to-width ratio on the strainburst
characteristics of Tianhu granite under true-triaxial unloading conditions. Canadian
Geotechnical Journal 52(7):890β902
Zhao XG, Wang J, Cai M, et al (2014) Influence of Unloading Rate on the Strainburst
Characteristics of Beishan Granite Under True-Triaxial Unloading Conditions. Rock
158 |
ADE | Title of Paper Post-peak behaviour of rocks under cyclic loading using a double-criteria
damage-controlled test method
Publication Status Published Accepted for Publication
Submitted for Publication U npublished and Unsubmitted work
written in manuscript style
Publication Details Shirani Faradonbeh R, Taheri A, Karakus M (2021) Post-peak behaviour of
rocks under cyclic loading using a double-criteria damage-controlled tests
method. Bulletin of Engineering Geology and the Environment 80(2):1713β1727
Principal Author
Name of Principal Author (Candidate) Roohollah Shirani Faradonbeh
Contribution to the Paper Literature review, conducting the laboratory tests, analysis of the results and
preparation of the manuscript
Overall percentage (%) 80%
Certification: This paper reports on original research I conducted during the period of my Higher
Degree by Research candidature and is not subject to any obligations or
contractual agreements with a third party that would constrain its inclusion in this
thesis. I am the primary author of this paper.
Signature Date 17 June 2021
Co-Author Contributions
By signing the Statement of Authorship, each author certifies that:
i. the candidateβs stated contribution to the publication is accurate (as detailed above);
ii. permission is granted for the candidate in include the publication in the thesis; and
iii. the sum of all co-author contributions is equal to 100% less the candidateβs stated contribution.
Name of Co-Author Abbas Taheri
Contribution to the Paper Research supervision, review and revision of the manuscript
Signature Date 21 June 2021
Name of Co-Author Murat Karakus
Contribution to the Paper Review and revision of the manuscript
Signature Date 21 June 2021
Chapter 6
160 |
ADE | Post-Peak Behaviour of Rocks Under Cyclic
Loading Using a Double-Criteria Damage-
Controlled Test Method
Abstract
Cyclic loading-induced hazards are severe instability problems concerning surface and
underground geotechnical projects. Therefore, it is crucial to understand the rock failure
mechanism under cyclic loading. An innovative double-criteria damage-controlled testing
method was proposed in this study to capture the complete stress-strain response of porous
limestone, especially the post-peak behaviour, under systematic cyclic loading. The proposed
test method was successful in applying the pre-peak cyclic loading and then in controlling the
self-sustaining failure of rock during the post-peak cyclic loading. The results showed that the
strength of the rock specimens slightly increased with an increase in the fatigue life in the pre-
peak region due to cyclic loading-induced hardening. Additionally, a combination of class I
and class II behaviours was observed in the post-peak region during the cyclic loading tests;
the class II behaviour was more dominant by the increase in fatigue life in the pre-peak region.
Damage evolution was assessed based on several parameters, such as the elastic modulus,
energy dissipation ratio, damage variable and crack damage threshold stress, both in the pre-
peak and post-peak regions. It was found that when the cyclic loading stress is not close to the
peak strength, due to a coupled mechanism of dilatant microcracking and grain crushing and
pore filling, quasi-elastic behaviour dominates the cyclic loading history, causing more elastic
strain energy to accumulate in the specimens.
Keywords Cyclic loading, Pre-peak and post-peak behaviour, Damage, Crack damage
threshold stress, Strength hardening
6.1. Introduction
Surface and underground structures are usually exposed to environmental and human-induced
cyclic loadings such as earthquakes, wind, volcanism, drilling and blasting, mechanical
161 |
ADE | excavation and mining seismicity, which threaten their long-term stability (Taheri et al. 2016;
Munoz et al. 2016a). Therefore, it is necessary to evaluate the time-dependent behaviour of
rocks under cyclic loading. In rock engineering, understanding the fatigue response of rocks is
of particular interest since rock stability conditions vary significantly under cyclic loading. A
great majority of rock fatigue studies have reported on the reduction in rock strength due to
cyclic loading (Bagde and PetroΕ‘ 2005). However, there are very few studies that have
illustrated strength hardening when the cyclic stress level is low enough to prevent failure
during cyclic loading (Burdine 1963; Singh 1989; Ma et al. 2013; Taheri et al. 2017). Unlike
the static and quasi-static loadings, which the applied load/deformation increases/decreases
continuously, cyclic loading is described by a time-dependent displacement/load signal with a
repetitive pattern. The loading rate in cyclic experiments is relatively high and propagates
waves, and their superposition causes a stress distribution different from that induced by quasi-
static loading (Cho et al. 2003). In recent decades, many studies have investigated the
mechanical behaviour of rocks under different cyclic loading histories and loading conditions.
Most of these studies have reported the results of tests performed under uniaxial compression
(Attewell and Sandford 1974; Eberhardt et al. 1999), which can replicate the stress state in
mining pillars and around galleries. Other studies have focused on triaxial compression
conditions with different confining pressures (Munoz et al. 2016a; Zhou et al. 2019) and
indirect tensile tests (Ghamgosar and Erarslan 2016), which are useful to calibrate the advanced
constitutive laws and to estimate the tensile strength of a material, respectively. In addition,
few cyclic studies of flexural tests (three-point and four-point) (Cardani and Meda 2004) and
freeze-thaw tests (Zhang et al. 2019a) can be found in the literature. In prior studies, the fatigue
properties of rocks were found to be dependent on the loading stress level, amplitude,
frequency, waveform and loading and unloading rate.
Rock behaviour in the post-peak region under uniaxial compression is characterised by either
class I or class II behaviour (Fig. 6.1). The former is defined by a negative post-peak modulus
describing a stable fracture propagation and the need to do more work on the specimen to
degrade its load-bearing capacity, while the latter represents a positive post-peak modulus (i.e.,
snap-back behaviour) describing a self-sustaining (brittle) failure (Wawersik and Fairhurst
1970; Munoz et al. 2016b). The proper measurement of the post-peak behaviour of rocks can
be a useful tool for quantifying the post-peak fracture energy and rock brittleness that can be
employed to optimise the designation of surface and underground structures and to mitigate
possible hazards (Akinbinu 2016). For instance, to evaluate the proneness and intensity of the
162 |
ADE | rockburst phenomenon near underground excavation in deep underground conditions, post-
peak analysis of the rocks in terms of strain energy evolution is required. In other words, the
rockburst hazard in deep underground openings is associated with not only internal strain
energy accumulation but also seismic disturbances induced by external sources (Xuefeng et al.
2010). Therefore, the post-peak response of rocks subjected to cyclic loading can unveil the
mechanism of geotechnical hazards such as rockburst and provide practical tools for their
assessment. As shown in Fig. 6.2, the cyclic loading of rock can be undertaken following two
main loading methods:
1. Systematic cyclic loading: These tests have a constant loading amplitude, π΄ππ.(π ),
π
and can be conducted as single-level (Fig. 6.3a) or multi-level (Fig. 6.3b) testes under
load-controlled or displacement-controlled (i.e., axial and lateral displacement-
controlled) loading conditions. In both load-controlled and displacement-controlled
conditions, the post-peak behaviour cannot be obtained, as the axial load level is the
only criterion to define the amount of the load that a specimen should be subjected to
during cyclic loading, until failure or even after failure. As a result, the specimen fails
during cyclic loading in an uncontrolled manner, and the post-peak response cannot be
obtained. Figs. 6.4a-d demonstrate the single-level and multi-level systematic cyclic
tests conducted by different researchers under load-controlled and displacement-
controlled conditions. As shown in these figures, in all the tests, failure occurred in an
uncontrolled manner, and post-peak behaviour was not obtained. Prior systematic
cyclic loading studies found that failure occurs at a stress level lower than the
determined monotonic strength owing to the strength weakening process. As such, the
accumulation of irreversible deformation (plastic strains) is not constant during the
experiment, while the hysteresis loops follow a loose-dense-loose law (Xiao et al.
2009).
2. Damage-controlled cyclic loading: These tests involving incremental loading amplitude
can be conducted in a load-based mode (Fig. 6.3c) or displacement-based mode (Fig.
6.3d). The former can be conducted either in load-controlled or displacement-controlled
loading conditions (i.e., axial and lateral displacement-controlled). However, the post-
peak response cannot be obtained, as the specimen might experience an uncontrolled
failure when it is forced to reach a pre-defined stress level. Figs. 6.4e and f show
representative results. A displacement-based test can be undertaken in either axial or
lateral displacement-controlled conditions. In this type of damage-controlled test, axial
163 |
ADE | stress is reversed when a certain amount of axial or lateral displacement is achieved in
a loading cycle. Munoz et al. (2016b) showed that under uniaxial loading conditions,
soft, medium-strong and strong rocks demonstrate either class II or a combination of
class I and class II post-peak behaviours. As a result, the post-peak response cannot be
adequately measured when the test is controlled by axial displacement (Fig. 6.4g).
However, by using lateral strain to control the amount of damage in a damage-
controlled test, the post-peak behaviour of a brittle rock can be achieved successfully
(Fig. 6.4h). From prior damage-controlled cyclic loading studies, it is reported that
failure occurs at a stress level close to or lower than the determined monotonic strength.
Moreover, the rate of strain accumulation under this type of loading is lower than that
during systematic cyclic tests (Cerfontaine and Collin 2018).
It should be noted that previous studies have mostly focused on the influence of cyclic loadings
on the mechanical rock properties and damage evolution in the pre-peak region. There are,
however, a few studies investigating failure behaviour and deformation localisation during
post-peak cyclic loading (e.g., Munoz and Taheri 2017a, 2019). Given the above, to the best of
our knowledge, no study has investigated the post-peak response of rocks subjected to pre-peak
systematic cyclic loading. This is because failure cannot be controlled when a constant axial
load is achieved in every cycle in a systematic cyclic loading. In addition, in a damage-
controlled test in which the lateral displacement is used to control the damage, an axial load is
reversed when a certain amount of lateral strain occurs. Therefore, systematic cyclic loading
cannot be applied in such a way that the load is always reversed at a constant stress level in the
pre-peak region. However, rock material in engineering applications (e.g., mining pillars in
deep underground conditions) may be subjected to systematic pre-peak cyclic loading and then
post-failure cyclic loading. Thus, it is significant to investigate the behaviour of rock subjected
to this loading condition. In this study, for the first time, a new cyclic test method considering
two cyclic loading control criteria is proposed to capture the complete response of rocks,
especially the post-peak behaviour, under cyclic loading. The proposed test method is a
combination of multi-level systematic cyclic loading and lateral displacement-based damage-
controlled cyclic loading to control both the damage and the rate of cyclic loading (see Fig.
6.2). A critical analysis is carried out to investigate damage evolution in both the pre-peak and
post-peak regions, and the influences of pre-peak cyclic loading on the peak strength, crack
damage threshold stress and rock stiffness are evaluated in more detail.
164 |
ADE | cyclic loading load-controlled test (Li et al. 2019), d multi-level systematic cyclic loading
axial displacement-controlled test (Liu et al. 2014), e load-based damage controlled cyclic
loading load-controlled test (Guo et al. 2018), f load-based damage controlled cyclic loading
axial displacement-controlled test (Heap et al. 2010), g displacement-based damage
controlled cyclic loading axial displacement-controlled test and (Wang et al. 2019) h
displacement-based damage controlled cyclic loading lateral displacement-controlled test
(Munoz and Taheri 2019)
6.2. Experimental Methodology
6.2.1. Tuffeau Limestone Specimens
Tuffeau limestone is used in this study to undertake double-criteria damage-controlled cyclic
loading tests (Fig. 6.5a). The name of this rock comes from the Latin word tofus, meaning
spongy rock. This yellowish-white sedimentary rock is a local limestone of the Loire Valley in
France and was deposited in the middle Turonian of the Upper Cretaceous, approximately 90
million years ago. This rock type is usually extracted from surface and underground quarries
and is used mostly in the building industry (Beck and Al-Mukhtar 2014). X-ray diffraction
(XRD) (Fig. 6.5b) and scanning electronic microscopy (SEM) analyses (Fig. 6.5c) were carried
out on collected limestone specimens to identify their mineralogical components and
microstructural characteristics. Two main crystalline phases, calcite (CaCO ) (β
50%) and
3
silica (SiO ) (β
30%), which has the two forms of quartz and opal cristobalite-tridymite (opal-
2
CT), were identified. Other phases, such as mica and clayey minerals (e.g., muscovite, biotite,
smectite, and glauconite) (β
20%), are disseminated in this limestone. Tuffeau limestone has
an average density of 1.43 g/cm3 and is a lightweight and fine-grained limestone with a
complex porous network (total porosity of 45Β±5%). The arrangement of grains with different
sizes contributes to the creation of micropores and macropores within the rock texture (Al-
Mukhtar and Beck 2006). The rock specimen in Fig. 6.5c has a heterogeneous porous structure,
and the microcracks, microcavities, and quartz are the main components controlling the
macrofailure of the specimen under loading conditions. The cylindrical rock specimens with
diameters and lengths of 42 mm and 100 mm, respectively (i.e., an aspect ratio of 2.4), were
cored from a single rock block and prepared to be smooth and straight according to the ISRM
standards (Fairhurst and Hudson 1999) to minimise the end friction effects and to ensure a
uniform stress state within the specimen during loading. Additionally, the diameter of the
168 |
ADE | tests to capture the rock behaviour before and after peak stress. The axial load (acquired by a
load cell), axial strain (acquired by a pair of LVDTs), and lateral strain (acquired by a chain
extensometer) were recorded simultaneously during the tests by a data acquisition system at a
rate of 10 data points per second (see Fig. 6.6a). Five uniaxial monotonic tests were conducted
under the lateral strain rate of 0.02Γ10-4/s to satisfy the static to quasi-static loading conditions
(Munoz and Taheri 2017b). These monotonic tests provide a reference for defining the stress
levels of cyclic uniaxial compression tests. The time history of the loading (π ), axial strain
π
(π ), and lateral strain (π ) for a typical monotonic loading test is shown in Fig. 6.6b. As seen
π π
in this figure, in the pre-peak and the post-peak regions, the lateral strain (π ) increases
π
monotonically with time, maintaining a constant lateral strain rate throughout the test, and the
complete post-peak response is obtained in a straightforward manner using the lateral strain-
controlled technique. Fig. 6.6c shows the normalised stress-strain curves obtained from the five
uniaxial monotonic tests. The specimens have an average monotonic compressive strength and
Youngβs modulus of 7.39 MPa and 1.67 GPa. As seen from Fig. 6.6c, in the post-peak region,
the axial stress and axial strain fluctuate successively due to the coupled mechanism of strength
degradation induced by the coalescence of the macrocracks and strength recovery induced by
interlocking the sides of the macrocracks. However, the total behaviour of all the conducted
monotonic tests in the post-peak region is a combination of class I and class II behaviours,
which is consistent with the results reported by Munoz et al. (2016a). Additionally, the
conducted monotonic tests exhibit similar behaviour both in the pre-peak and the post-peak
regions, which shows the low discrepancy among the tested specimens.
170 |
ADE | (here, 6 MPa) is reached. In this stage, the axial stress and lateral strain feedback signals
received from the load cell and the chain extensometer, respectively, are continuously
compared with the program signals (i.e., the user-defined values) and the errors, if any,
are adjusted by the servo-controller. By doing so, it is guaranteed that the axial load is
always applied under a constant lateral strain rate and that the axial load does not exceed
the initial stress level defined for cyclic loading. Thereafter, the specimen is unloaded
until the axial stress is equal to 0.07 MPa, ensuring that the specimen is always in
complete contact with the loading platens.
b) Afterwards, cyclic loading is applied under a constant lateral strain rate for a specific
number of cycles (i.e., 400 cycles). Two criteria are adopted to control the failure: a
maximum axial stress level that can be achieved and a maximum lateral strain
amplitude that a Tuffeau limestone specimen is allowed to experience in a cycle during
loading, π΄ππ.(π ). In this study, the initial maximum stress level (i.e., the first
π
criterion) is adopted to be equal to 6.0 MPa. The optimum values for π΄ππ.(π ) and the
π
loading rate (ππ /ππ‘) were determined based on a previous study conducted by Munoz
π
and Taheri (2017a) on Tuffeau limestone and the results obtained from the trial tests to
avoid the sudden failure of a specimen in an uncontrolled manner. Therefore, different
loading rates and π΄ππ.(π ) values were evaluated by performing four trial cyclic
π
loading tests, and finally, π΄ππ.(π ) = 17Γ10β4 and ππ /ππ‘ = 2Γ10β4/π were
π π
obtained by balancing the capability of the methodology in capturing the post-peak
behaviour of the rock and completing the test in the shortest possible time. The axial
load is reversed when at least one criterion is met. By following the closed-loop
procedure shown in Fig. 6.7, the test is continued until the specimen fails or until 400
cycles are completed.
c) If the specimen does not fail after 400 cycles, the specimen is monotonically loaded
under a constant lateral strain rate of 0.02Γ10-4/s until the specimen is under an axial
load of 6.5 MPa (i.e., a 0.5 MPa increase in the stress level compared to the previous
stress level in this multi-level cycling loading scheme). If the specimen fails during
monotonic loading, the complete post-peak behaviour is measured during lateral strain-
controlled loading.
d) The procedure explained in b and c is repeated until the specimen fails.
Fig. 6.8 shows typical results for a Tuffeau limestone specimen. As shown in this figure, after
initial monotonic loading under the constant loading rate of 0.02Γ10-4/s, the prescribed axial
172 |
ADE | stress level (i.e., 6 MPa) is reached. Afterwards, the specimen is unloaded monotonically, and
then cyclic loading is applied under a constant lateral strain rate of 2Γ10-4/s. At the first step of
cyclic loading, the amplitude of lateral strain, π΄ππ.(π ), is relatively low (6Γ10-4/s after 200
π
cycles), and the first criterion is always met during cyclic loading (i.e., the stress level
remaining below 6 MPa). As the specimen does not fail after 400 cycles, the axial load is
increased monotonically to the second stress level (i.e., 6.5 MPa), and the cyclic loading
procedure is repeated. As shown in Fig. 8, in the second series of cyclic loading at the onset of
the failure, the lateral strain amplitude, π΄ππ.(π ), is equal to 17Γ10-4. After this cycle, the
π
second criterion controls the cyclic loading, and the strength degradation during post-peak
cyclic loading is observed until complete failure. By doing so, the complete post-peak
behaviour of the Tuffeau limestone under systematic cyclic loading can be successfully
observed.
173 |
ADE | 10 SL 1= 6 MPa SL 2= 6.5 MPa
10
0
0.07 MPa
de/d=0.02
l t
-20
de/d=0.02
l t
6 MPa
6 MPa
-40 6.5 MPa Failure point
de/d= 2
l t
-60
Amp. (e l)= 6 de l/d t= 2
-4
de/d= 2Β΄10
L t
Time Amp (e)= 16
l
Amp (e)= 17
l Amp (e)= 17
l
Time
-80
0 2000 4000 6000 8000
Time, t (s)
Figure 6.8 Typical time-history of axial stress and lateral strain during a double-criteria
cyclic
6.3. Experimental Results
6.3.1. Complete Stress-Strain Response
In this study, three multi-level systematic cyclic loading tests were conducted using the
methodology explained above to evaluate the applicability of the proposed testing method in
capturing the failure behaviour of the soft and porous Tuffeau limestone. Fig. 6.9 displays the
axial stress-strain relations obtained for these tests, in which 6 MPa was defined as the initial
stress level, and the specimens were subjected to systematic cyclic loading at different stress
levels, taking 0.5 MPa as the stress increment between consecutive cyclic loading steps. The
envelope curves showing the overall behaviour of the specimens in the post-peak region were
drawn by connecting the loci of the indicator stresses (π , the maximum stress of each cycle).
π
As seen from Fig. 6.9, the overall post-peak behaviour of the specimens is characterised by the
combination of class I and class II; however, the class I behaviour is more dominant in
specimen TL6 (Fig. 6.9a) than in specimens TL7 and TL8 (Figs. 6.9b and c). Table 6.1
summarises the results of the cyclic loading tests. As listed in Table 6.1 and shown in Fig. 6.9,
the different cycle numbers and stress levels are recorded for the three specimens before failure;
for example, specimen TL8 experienced 2906 cycles before failure, and its failure occurred at
175
4-
)aPM(
as
,sserts
laixA
)
01Β΄(
le
,niarts
laretaL
sserts
laixA
niarts
laretaL
sserts
laixA
niarts
laretaL |
ADE | 6.3.2. Fatigue Damage Evolution
Damage can be characterised by the process of generation, propagation and coalescence of
mesoscopic defects and voids through solid materials. Damage can be described by the
degradation of some material properties, such as stiffness, residual strength, and P-wave
velocity. Additionally, damage during cyclic loading can be investigated by the corresponding
irreversible strain, dissipative energy, electrical resistance, and acoustic emission counts (Xiao
et al. 2010; Taheri and Tatsuoka 2012). The incremental accumulation of plastic deformation
during cyclic loading contributes to the degradation of the cohesive strength and stiffness of
the rocks. Therefore, the irreversible strain can be regarded as a suitable indicator for fatigue
damage assessment. Hence, a damage variable (π·) was defined based on the accumulation of
irreversible axial strain (ππππ) (see Fig. 6.10) after each loading and unloading cycle as follows:
π
βπ (ππππ)
π· = π=1 π π (6.1)
βπ (ππππ)
π=1 π π
where π is the cycle number, βπ (ππππ) is the accumulation of irreversible strain after π
π=1 π π
cycles, and βπ (ππππ) is the total cumulative irreversible strain during the entire multi-level
π=1 π π
systematic cyclic loading test.
Rock deformability and its failure mechanism are closely related to energy dissipation.
Therefore, the energy trends during the rock deformation process can reflect the rock damage
mechanism (Zhang et al. 2019b). As shown in Fig. 6.10, a part of the total work done on the
unit volume of a specimen (π ) by the external force during a loading-unloading cycle is stored
π‘
in the specimen as elastic energy (π ); the remaining is released as dissipated energy (π ) due
π π
to plastic deformation and rock damage. Because of the complexity in energy conversion
during rock deformation and failure, subtle energies (thermal energy, acoustic emission energy,
kinetic energy, etc.) are usually ignored to simplify the energy equation as follows (Zhou et al.
2019):
π = π +π (6.2)
π‘ π π
π"
π
π‘
= β«
0
π
π
ππ
π
π" (6.3)
π = β« π ππ
π πβ² π π
{π = π βπ
π π‘ π
178 |
ADE | Fig. 6.11 summarises the evolution of the damage variable (π·), elastic modulus (πΈ), and energy
dissipation ratio (πΎ = π /π ) as damage parameters for specimen TL6. A similar trend was
π π‘
observed for the other tested specimens. As demonstrated in Fig. 6.11, the total behaviour of
damage parameters under multi-level systematic cyclic loading conditions can be divided into
four stages. In stage I, the damage variable (π·) increases slightly and is accompanied by the
rapid increase in stiffness (πΈ) from 1.46 GPa to 2.23 GPa, corresponding to the closure of
existing defects and expansion of the yield surface (Taheri and Tatsuoka 2015). Furthermore,
the energy dissipation ratio (πΎ) decreases suddenly in this stage, which indicates that the elastic
energy (π ) accumulates more rapidly than the dissipated energy (π ). Stage II, which is the
π π
majority of the damage evolution process, shows a nearly unchanging behaviour for all three
damage parameters π·, πΈ, and πΎ. In this stage, although the specimen has experienced 400
cycles, no notable damage is incurred in the specimen. This stage can be interpreted as a
balance between the two mechanisms of dilatant microcracking, which reduces the rock
stiffness, and grain crushing and pore collapse, which improves the rock stiffness. This
balanced state between two competing inelastic procedures results in a quasi-elastic behaviour
of the damage parameters in such a way that the deformation seems elastic, and no more energy
is dissipated in this stage. In stage III, during the transition to the second stress level via a
monotonic loading, the elastic modulus first increases for several cycles. This increase may be
related to the change in the strain rate from 2Γ10-4/s to 0.02Γ10-4/s for monotonic loading,
which allows the existing microcracks and pores to be more compacted and ultimately results
in a small stiffening (Peng et al. 2019). Then, the elastic modulus decreases gradually due to
the dilatant cracking that degrades the axial stiffness and simultaneously allows more energy
to be dissipated (see the trend of πΎ in Fig. 6.11). In stage IV, the specimen enters the post-peak
region due to the coalescence of the microcracks and the generation of macrocracks through
the specimen, and the degradation process of the specimen increases dramatically. According
to Fig. 6.11, the energy dissipation ratio (πΎ) and damage variable (π·) increase rapidly in this
stage, while the stiffness of the specimen decreases until the residual state is reached.
179 |
ADE | constant and very close to the maximum stress in each cycle. When transitioning to the higher
stress levels using a monotonic loading, π increases to reach a stationary state at each stress
ππ
level. The results presented in Fig. 6.9 show that by applying 400 cycles at each stress level,
the closed microvoids and micropores are not re-opened during pre-peak cyclic loading until
the cyclic loading damages the rock at the last stress level. Thus, when the cyclic loading stress
level is not high enough to cause the specimen to fail, the specimen does not switch from a
compaction-dominated state to a dilatancy-dominated state but instead acts as an elastic
material. According to Fig. 6.9a, specimen TL6 shows dilatant behaviour in the pre-peak
region, in the second cyclic loading stage, by a sudden drop in π due to the re-opening of
ππ
closed cracks and the generation of new cracks. Degradation of π continues in the post-peak
ππ
region, followed by strength degradation until the specimen starts to show a residual strength
state where π increases to reach a stable condition. For specimens TL7 (Fig. 6.9b) and TL8
ππ
(Fig. 6.9c), the drop in π occurs very close to and at the failure point, respectively. This, in
ππ
turn, causes a sudden release of stored elastic strain energy in a self-sustaining manner.
6.4. Strength Hardening Behaviour
As mentioned earlier, in the cyclic loading tests, an increase in the peak strength of specimens
TL7 and TL8 was observed with the increase in fatigue life in the pre-peak region. The
discrepancy among specimens may partially contribute to this trend in the results. Considering
the previous findings (Burdine 1963; Singh 1989; Ma et al. 2013; Taheri et al. 2017) and the
results of cyclic loading tests in this study, the authors believe that the increase in the peak
strength of specimens TL7 and TL8 is due to not only this discrepancy but also the cyclic
loading. This phenomenon should be investigated in future studies by undertaking more
specific cyclic loading tests. The hardening behaviour, however, is discussed briefly below.
As discussed in section 6.3.2 and shown in Fig. 6.11, during pre-peak systematic cyclic loading,
when the stress level is not high enough to cause the specimen to fail due to fatigue, a quasi-
elastic behaviour dominates the damage evolution process. In this stage, some mesoscopic
elements with lower strength and stiffness may reach their maximum load-bearing capacity,
and the weak bonding between the grains breaks, producing fine materials. However, as the
stress level is not close to the failure point, due to the slippage and dislocation of the produced
fine materials, the existing microfissures and pores are filled during cyclic loading. This may
result in more compaction of the specimen and, consequently, strength hardening. This
behaviour can also be confirmed by the variation in crack damage threshold stress (π ) during
ππ
181 |
ADE | cyclic loading (see Fig. 6.9). As explained in section 3.3, specimen TL8, which experienced
more loading and unloading cycles in the pre-peak region than the other specimens did, is
mostly in the compaction-dominated stage; dilation occurs at the failure point, followed by the
sudden decrease in π . This, in turn, resulted in the strength improvement of specimen TL8.
ππ
However, for specimen TL6 with a shorter fatigue life, dilation occurred earlier in the pre-peak
region. The process of rock compaction and porosity reduction in highly porous rock material
may be similar to the mechanism explained by Baud et al. (2017). Fig. 6.12 shows the
backscattered SEM images of a highly porous limestone in intact and deformed conditions
under the same confining pressure of 9 MPa at different axial strain levels. As shown in this
figure, when the intact specimen (Fig. 6.12a) deforms to 14% strain, microcracks are created
in the calcite grains, and most of the fossil shells are broken and pulverised, while the quartz
grains largely remain intact (Fig. 6.12b). With the further deformation of the specimen to 27%
strain (Fig. 6.12c), the majority of the calcite grains are broken, and all of the fossil shells are
pulverized, resulting in the existing pores being filled and the creation of compacted zones
through the specimen. This grain packing is more evident in Fig. 6.12d, at a larger scale. The
stress may concentrate more around the compacted areas, which behave elastically during
loading and may contribute to the specimens exhibiting more brittle failure.
Figure 6.12 Backscattered SEM images of a porous limestone in a intact and triaxial
compression conditions for b 14% and c, d 27% axial strain (modified from Baud et al.
(2017))
182 |
ADE | 6.5. Conclusions
An innovative testing methodology considering two criteria was proposed in this study to
describe the post-peak behaviour of rocks subjected to systematic cyclic loading. Regarding
this, the Tuffeau limestone was selected to evaluate the capability of the proposed testing
method in capturing the full stress-strain response of soft rocks. After obtaining the optimum
values for the loading rate (ππ /ππ‘) and π΄ππ.(π ) during a trial procedure, three main multi-
π π
level systematic cyclic loading tests were conducted on Tuffeau limestone specimens using the
proposed damage-controlled test method. The evolution of different parameters, including the
peak strength, damage variable, elastic modulus and crack damage threshold stress, was
evaluated comprehensively with the results of the conducted cyclic loading tests. The following
conclusions were drawn from this study:
1. The proposed double-criteria damage-controlled testing method was successful in
capturing the class II post-peak behaviour of Tuffeau limestone subjected to multi-
level systematic cyclic loading. This testing method can provide new insights
regarding the damage evolution of rocks in the post-peak region under systematic
cyclic loading conditions, which was not previously achievable. The test method was
successfully performed on Tuffeau limestone, which is a soft rock. The application of
the method still needs to be examined on stronger rock types.
2. The whole process of cyclic loading tests conducted in this study can be summarised
into several stages: a) The rock specimen initially stiffens and shows elastic behaviour
due to the initial compaction, which is accompanied by the reduction in the energy
dissipation. b) Due to a balance between the grain-crushing and pore collapse
processes during compaction, a quasi-elastic behaviour dominates the whole test. c)
The stiffness of the specimen decreases gradually due to dilatant microcracking, which
dissipates more energy. d) With the generation and coalescence of microcracks, the
rocks transition from a dilatant state, characterised by the rapid increase in damage
and energy dissipation, and stiffness reduction.
3. The evolution of the crack damage threshold stress (π ) during cyclic loading showed
ππ
that the specimens do not switch from a compaction-dominated to a dilatancy-
dominated state when the cyclic loading stress level is not high enough to cause the
specimen to fail. This results in a constant π that is very close to the unloading stress
ππ
in each cycle.
183 |
ADE | 4. An increase in strength with an increase in fatigue life was observed for the highly
porous Tuffeau limestone. According to the variation in the damage parameters,
stiffness and crack damage threshold stress during the systematic cyclic loading tests,
this hardening behaviour can be due to the further compaction of a rock specimen with
increasing number of cycles in the pre-peak region. Indeed, the weak bonding between
the grains may break down during cycling loading, and the fine materials produced in
this process may fill the existing micropores and microfissures, which can result in a
porosity reduction and hardening behaviour.
Acknowledgments
The authors would like to thank the laboratory staff, in particular, Simon Golding and Dale
Hodson, for their aids in conducting the tests.
Funding
The first author acknowledges the University of Adelaide for providing the research fund
(Beacon of Enlightenment PhD Scholarship) to conduct this study.
References
Akinbinu VA (2016) Class I and Class II rocks: implication of self-sustaining fracturing in
brittle compression. Geotechnical and Geological Engineering 34(3):877β887
Al-Mukhtar M, Beck K (2006) Physical-mechanical characterisation of hydraulic and non-
hydraulic lime based mortars for a French porous limestone. arXiv Prepr physics/0609108
Attewell PB, Sandford MR (1974) Intrinsic shear strength of a brittle, anisotropic rockβI:
experimental and mechanical interpretation. In: International Journal of Rock Mechanics
and Mining Sciences & Geomechanics Abstracts. Elsevier, 11(11):423β430
Bagde MN, PetroΕ‘ V (2005) Fatigue properties of intact sandstone samples subjected to
dynamic uniaxial cyclical loading. International Journal of Rock Mechanics and Mining
Sciences 42(2):237β250
Baud P, Schubnel A, Heap M, Rolland A (2017) Inelastic compaction in highβporosity
limestone monitored using acoustic emissions. Journal of Geophysical Research: Solid
Earth 122:9910β9989
184 |
ADE | Beck K, Al-Mukhtar M (2014) Cyclic wettingβdrying ageing test and patina formation on
tuffeau limestone. Environmental Earth Sciences 71(5):2361β2372
Burdine NT (1963) Rock Failure Under Dynamic Loading Conditions. Society of Petroleum
Engineers Journal 3(1):1β8
Cardani G, Meda A (2004) Marble behaviour under monotonic and cyclic loading in tension.
Construction and Building materials 18(6):419β424
Cerfontaine B, Collin F (2018) Cyclic and fatigue behaviour of rock materials: review,
interpretation and research perspectives. Rock Mechanics and Rock Engineering
51(2):391β414
Cho SH, Ogata Y, Kaneko K (2003) Strain-rate dependency of the dynamic tensile strength of
rock. International Journal of Rock Mechanics and Mining Sciences 40(5):763β777
Eberhardt E, Stead D, Stimpson B (1999) Quantifying progressive pre-peak brittle fracture
damage in rock during uniaxial compression. International Journal of Rock Mechanics
and Mining Sciences 36(3):361β380
Fairhurst CE, Hudson JA (1999) Draft ISRM suggested method for the complete stress-strain
curve for intact rock in uniaxial compression. International Journal of Rock Mechanics
and Mining Sciences 36(3):279β289
Ghamgosar M, Erarslan N (2016) Experimental and numerical studies on development of
fracture process zone (FPZ) in rocks under cyclic and static loadings. Rock Mechanics
and Rock Engineering 49(3):893β908
Guo H, Ji M, Zhang Y, Zhang M (2018) Study of mechanical property of rock under uniaxial
cyclic loading and unloading. Advances in Civil Engineering
Heap MJ, Faulkner DR, Meredith PG, Vinciguerra S (2010) Elastic moduli evolution and
accompanying stress changes with increasing crack damage: implications for stress
changes around fault zones and volcanoes during deformation. Geophysical Journal
International 183(1):225β236
Hudson JA, Brown ET, Fairhurst C (1971) Optimising the control of rock failure in servo-
controlled laboratory tests. Rock Mechanics 3:217β224
Li T, Pei X, Wang D, Huang R, Tang H (2019) Nonlinear behavior and damage model for
185 |
ADE | fractured rock under cyclic loading based on energy dissipation principle. Engineering
Fracture Mechanics 206:330β341
Liu J, Xie H, Hou Z, Hou Z, Yang C, Chen L (2014) Damage evolution of rock salt under
cyclic loading in unixial tests. Acta Geotechnica 9(1):153β160
Ma L, Liu X, Wang M, Xu H, Hua R, Fan P, Jiang S, Wang G, Yi Q (2013) Experimental
investigation of the mechanical properties of rock salt under triaxial cyclic loading.
International Journal of Rock Mechanics and Mining Sciences 34β41
Munoz H, Taheri A (2017a) Local damage and progressive localisation in porous sandstone
during cyclic loading. Rock Mechanics and Rock Engineering 50(12):3253β3259
Munoz H, Taheri A (2017b) Specimen aspect ratio and progressive field strain development of
sandstone under uniaxial compression by three-dimensional digital image correlation.
Journal of Rock Mechanics and Geotechnical Engineering 9(4):599β610
Munoz H, Taheri A (2019) Postpeak deformability parameters of localised and nonlocalised
damage zones of rocks under cyclic loading. Geotechnical Testing Journal 42(6):1663β
1684
Munoz H, Taheri A, Chanda EK (2016a) Rock Drilling Performance Evaluation by an Energy
Dissipation Based Rock Brittleness Index. Rock Mechanics and Rock Engineering
49(8):3343β3355
Munoz H, Taheri A, Chanda EK (2016b) Pre-peak and post-peak rock strain characteristics
during uniaxial compression by 3D digital image correlation. Rock Mechanics and Rock
Engineering 49(7):2541β2554
Munoz H, Taheri A, Chanda EK (2016c) Fracture Energy-Based Brittleness Index
Development and Brittleness Quantification by Pre-peak Strength Parameters in Rock
Uniaxial Compression. Rock Mechanics and Rock Engineering 49(12):4587β4606
Peng K, Zhou J, Zou Q, Yan F (2019) Deformation characteristics of sandstones during cyclic
loading and unloading with varying lower limits of stress under different confining
pressures. International Journal of Fatigue 127:82β100
Singh SK (1989) Fatigue and strain hardening behaviour of graywacke from the flagstaff
formation, New South Wales. Engineering Geology 26(2):171β179
186 |
ADE | Taheri A, Tatsuoka F (2012) Stressβstrain relations of cement-mixed gravelly soil from
multiple-step triaxial compression test results. Soils and Foundations 52(4):748β766
Taheri A, Tatsuoka F (2015) Small- and large-strain behaviour of a cement-treated soil during
various loading histories and testing conditions. Acta Geotechnica 10(1):131β155
Taheri A, Royle A, Yang Z, Zhao Y (2016) Study on variations of peak strength of a sandstone
during cyclic loading. Geomechanics and Geophysics for Geo-Energy and Geo-Resources
2(1):1β10
Taheri A, Hamzah N, Dai Q (2017) Degradation and improvement of mechanical properties of
rock under triaxial compressive cyclic loading. Japanese Geotechnical Society Special
Publication 5:71β78
Taheri A, Zhang Y, Munoz H (2020) Performance of rock crack stress thresholds determination
criteria and investigating strength and confining pressure effects. Construction and
Building Materials 243:118263
Wang S, Xu W, Sun M, Wang W (2019) Experimental investigation of the mechanical
properties of fine-grained sandstone in the triaxial cyclic loading test. Environmental
Earth Sciences 78(14):416
Wawersik WR, Fairhurst CH (1970) A study of brittle rock fracture in laboratory compression
experiments. In: International Journal of Rock Mechanics and Mining Sciences &
Geomechanics Abstracts. Elsevier, 7(5):561β575
Xiao J-Q, Ding D-X, Jiang F-L, Xu G (2010) Fatigue damage variable and evolution of rock
subjected to cyclic loading. International Journal of Rock Mechanics and Mining Sciences
47:461β468
Xiao J-Q, Ding D-X, Xu G, Jiang F-L (2009) Inverted S-shaped model for nonlinear fatigue
damage of rock. International Journal of Rock Mechanics and Mining Sciences
46(3):643β648
Xuefeng X, Linming D, Caiping L, Zhang Y (2010) Frequency spectrum analysis on micro-
seismic signal of rock bursts induced by dynamic disturbance. Mining Science and
Technology (China) 20(5):682β685
Zhang J, Deng H, Taheri A, Ke B, Liu C (2019a) Deterioration and strain energy development
of sandstones under quasi-static and dynamic loading after freeze-thaw cycles. Cold
187 |
ADE | Chapter 7
Failure Behaviour of a Sandstone Subjected to the
Systematic Cyclic Loading: Insights from the
Double-Criteria Damage-Controlled Test Method
Abstract
The post-peak behaviour of rocks subjected to cyclic loading is very significant to appraise the
long-term stability of underground excavations. However, an appropriate testing methodology
is required to control the damage induced by the cyclic loading during the failure process. In
this study, the post-failure behaviour of Gosford sandstone subjected to the systematic cyclic
loading at different stress levels was investigated using the double-criteria damage-controlled
testing methodology, and the complete stress-strain relations were captured successfully. The
results showed that there exists a fatigue threshold stress in the range of 86-87.5% of the
average monotonic strength in which when the cyclic loading stress is below this threshold, no
failure occurred for a large number of cycles and in turn, the peak strength improved up to 8%.
Also, the variation of the energy dissipation ratio, rock stiffness and acoustic emission hits for
hardening tests showed that cyclic loading in the pre-peak regime creates no critical damage in
the specimen, and a quasi-elastic behaviour dominates the damage evolution. The post-failure
instability of such tests was similar to those obtained for monotonic tests. On the other hand,
by exceeding the fatigue threshold stress, the brittleness of the specimens increased with an
increase in the applied stress level, and class II behaviour prevailed over total post-peak
behaviour. A loose-dense-loose behaviour with different extents was also observed in the post-
peak regime of all fatigue cyclic loading tests. This was manifested then as a secondary inverted
S-shaped damage behaviour by the variation of the cumulative irreversible axial and
cumulative irreversible lateral strains with the post-peak cycle number. Furthermore, it was
confirmed that the damage per cycle in the post-peak regime decreases exponentially with an
increase in the applied stress level.
Keywords: Pre-peak and post-peak behaviour, Systematic cyclic loading, Brittleness,
Hardening, Fatigue, Damage evolution
190 |
ADE | List of Symbols
πΈ Tangent Youngβs modulus ππππ Irreversible axial strain
π‘ππ π
π Poissonβs ratio ππππ Irreversible lateral strain
π
π Major principal stress Ξ£ππππ Cumulative irreversible axial strain
1 π
π Axial stress Ξ£ππππ Cumulative irreversible lateral strain
π π
π Indicator stress π Elastic energy at peak stress
π π
π Axial peak stress ππ Elastic energy of cycle π
πβππππ π
π Average monotonic strength ππ Dissipated energy of cycle π
π π
π /π Applied stress level π Pre-peak dissipated energy
π π πππ
π /π Strength hardening ratio π Post-peak dissipated energy
β π πππ π‘
π /π Crack initiation stress ratio π Total fracture energy
ππ πβππππ π‘
π /π Crack damage stress ratio π΄ππ.(π ) Loading amplitude
ππ πβππππ π
π Axial strain π΄ππ.(π ) Lateral strain amplitude
π π
π Lateral strain π Cycle number
π
ππ /ππ‘ Lateral strain rate π Total number of cycles
π π‘ππ‘ππ
π Axial strain at peak stress π Number of cycles after failure point
πβππππ πππ‘ππ
π Lateral strain at peak stress π΅πΌ Brittleness index
πβππππ
π Volumetric strain at peak stress π· Damage variable
π£βππππ
π Axial strain at the final cycle π Post-peak modulus
πβπ
7.1. Introduction
A high-complex stress state usually is created around deep-buried tunnels and caverns due to
disturbances induced by different sources as displayed in Fig. 7.1. This stress state may affect
mechanical rock properties and in turn, cause some specific failure phenomena such as
slabbing/spalling, strainburst and zonal disintegration significantly different from those in
shallow conditions (Gong et al. 2012; Shirani Faradonbeh and Taheri 2019). According to
Martin and Chandler (1994) and Martin (1997), the surrounding rocks in underground
excavations may experience load-and-deformation response to a different extent during
operation, and rock may be exposed to cyclic loading. In particular, they argued that in remote
to nearby excavation regions, rock may experience failure (i.e. the applied stress level exceeds
the peak strength), damage (i.e. the applied stress is below the peak strength) or disturbance
(i.e. different stress is applied due to the redistribution of the in-situ stresses) or the rock may
remain undisturbed. From this viewpoint, the rock cyclic load-deformation response may take
191 |
ADE | place in the pre-peak or post-peak regime (Munoz and Taheri 2019). For instance, as depicted
in Fig. 7.1, a pillar may experience cyclic loading due to blasting operation or other seismic
activities beyond the limit in uniaxial conditions. Under such loading conditions, rock materials
may still keep some loadings even in the post-failure regime. Therefore, the investigation of
the pre-peak and post-peak behaviour of rocks is of paramount significance to understand more
about the fracturing mechanism, resilient design and long-term stability assessment of the
various rock engineering structures subjected to seismic disturbances. Experimental research
on the influence of cyclic loading parameters on the damage evolution and rock strength and
deformation parameters has a long tradition. These studies have been conducted under different
loading histories and loading conditions such as uniaxial and triaxial compression tests (Heap
and Faulkner 2008; Heap et al. 2009; Liu et al. 2018), indirect tensile tests (Erarslan et al. 2014;
Wang et al. 2016), flexural tests (Cattaneo and Labuz 2001; Cardani and Meda 2004) and
freeze-thaw tests (Liu et al. 2015; Zhang et al. 2019). A comprehensive review of the rock
fatigue studies can be found in Cerfontaine and Collin (2018). The majority of prior rock
fatigue studies have emphasised strength weakening of rocks due to incurring permanent
deformations during cyclic loading (Haimson 1978; Fuenkajorn and Phueakphum 2010).
However, very few studies have reported the strength improvement when the stress level that
cyclic loading is applied is low enough to prevent failure (Singh 1989; Ma et al. 2013; Taheri
et al. 2017). In prior studies, the process of damage evolution and the failure mechanism of
rocks subjected to different cyclic loading histories have been investigated based on the
measured stress-strain relations (Cerfontaine and Collin 2018). Indeed, the complete stress-
strain relation of rocks (i.e. the pre-peak and the post-peak regimes) is considered as a
prominent tool in rock engineering to describe strain energy evolution as well as for rock
brittleness determination (Munoz et al. 2016a; Shirani Faradonbeh et al. 2020). According to
Wawersik and Fairhurst (1970), the post-peak behaviour of rocks under quasi-static
compression can be distinguished into two classes: a) class I which is characterised by the
negative post-peak modulus (i.e. π = ππ/ππ < 0) representing the gradual strength
degradation of rock specimen and the need for extra energy and b) class II having a positive
post-peak modulus represents the self-sustaining failure with strain recovery and release of
excess elastic strain energy. The proper measurement of the complete stress-strain response of
rocks significantly depends on the stiffness of the loading system, the applied load controlling
technique throughout the test as well as rock brittleness (Wawersik and Fairhurst 1970; Munoz
and Taheri 2019).
192 |
ADE | Shirani Faradonbeh et al. (2020) categorised the cyclic loading methods based on the loading
histories and load control variables into two main groups of systematic cyclic loading (single-
level or multi-level) (Figs. 7.2a and b) and damage-controlled cyclic loading (load-based or
displacement-based) (Figs. 7.2c and d). Systematic cyclic loading can be conducted under load-
controlled or displacement-controlled loading conditions. In both loading conditions, a sudden
failure occurs during cyclic loading as a constant axial load amplitude, π΄ππ.(π ), should be
π
achieved during each loading cycle (e.g. Ma et al. 2013; Li et al. 2019). Similarly, in the load-
based damage-controlled cyclic loading tests, as the specimen is forced to reach a prescribed
stress level, it may experience an unexpected failure, and the post-peak behaviour cannot be
captured (e.g. Heap et al. 2010; Guo et al. 2018). Regarding the displacement-based damage-
controlled cyclic loading tests, as the post-peak behaviour of rocks in uniaxial compression is
either class II or a combination of class I and class II (Munoz et al. 2016a), the post-peak
response cannot be adequately captured by the axial displacement feedback signal (e.g. Wang
et al. 2019). The lateral displacement, on the other hand, has been identified as an appropriate
variable to control the amount of damage in the post-peak regime (Munoz and Taheri 2019).
To our knowledge, no prior studies have examined the influence of systematic cyclic loading
at different stress levels on the post-peak behaviour of rocks. This is due to the difficulties in
controlling the axial load when a constant load amplitude should be achieved in every cycle in
a systematic cyclic loading test. Also, if a prescribed lateral strain is considered to control the
damage in a damage-controlled test, the axial load is reversed when a certain amount of lateral
strain occurs, and therefore, the systematic cyclic loading cannot be conducted anymore in the
pre-peak regime. However, as mentioned earlier, some mining and civil structures (e.g. mining
pillars and bridge columns) may experience systematic cyclic loading at different fractions of
their average peak strength. Under such loading conditions, the rocks may exhibit different
behaviours in the post-peak regime. An appropriate experimental methodology is, therefore,
required for measuring the post-peak behaviour of rocks subjected to systematic cyclic loading
histories properly. As demonstrated in Fig. 7.2, a novel cyclic test method by combining the
single-level systematic cyclic loading and lateral displacement-based damage-controlled cyclic
loading is proposed in this study to control both the damage and the cyclic loading rate. Then,
several systematic cyclic tests were conducted in uniaxial compression at different stress levels
using the proposed test method. Based on the obtained complete stress-strain relations, the
influence of systematic cyclic loading on both the pre-peak and the post-peak behaviours was
evaluated comprehensively, and the results were discussed in detail.
193 |
ADE | Figure 7.2 Classification of cyclic loading tests, a single-level systematic cyclic loading path,
b multilevel systematic cyclic loading path, c load-based damage controlled cyclic loading
path and d displacement-based damage controlled cyclic loading path, Amp. (π ) refers to
π
loading amplitude, Amp. (π ) refers to lateral strain amplitude, and * can be conducted either
πΏ
in axial or lateral displacement-controlled mode, modified from Shirani Faradonbeh et al.
(2020)
7.2. Specimen Preparation and Experimental Set-Up
The Gosford sandstone as a medium-grained (0.2-0.3 mm), poorly cemented, immature quartz
sandstone containing 20-30% feldspar and clay minerals with the serrate connection between
quartz grains (Sufian and Russell 2013) was used in this study for conducting the experimental
tests. According to the X-ray computed tomography scans conducted by Sufian and Russell
(2013), the total porosity of this sandstone is about 18%. A total of 23 cylindrical specimens
having a constant aspect ratio of 2.4 (i.e. 42 mm diameter and 100 mm length) were all cored
from the same rectangular block and in the same direction and prepared according to the ISRM
suggested method (Fairhurst and Hudson 1999). In this study, all the experiments were
performed in dry condition. To do so, the rock specimens were dried in the room temperature
before conducting the tests. The average dry density of the specimens was approximately
2204.26 kg/m3. Rock monotonic strength should be determined before undertaking systematic
cyclic loading tests at different stress levels (π /π ). To do so, six uniaxial compression tests
π π
were performed following the lateral strain-controlled loading method. An MTS close-looped
servo-controlled hydraulic compressive system having the maximum loading capacity of 300
kN (see Fig. 7.3) was used to undertake the monotonic and cyclic loading tests. As stated
earlier, the axial load-controlled and axial strain-controlled loading techniques cannot capture
the post-peak behaviour of rocks, as rocks usually show a combination of class I and class II
behaviour in the post-peak regime (Munoz et al. 2016b). Therefore, as depicted in Fig. 7.4a, a
constant lateral strain rate (ππ /π ) of 0.02Γ10-4/s was utilised during the uniaxial compression
π π‘
tests to control the axial load both in the pre-peak and the post-peak regimes. This strain rate
provides a static to quasi-static loading conditions (Wawersik and Fairhurst 1970; Munoz et al.
2016b).
Axial load and axial and lateral displacements were recorded in real-time, respectively using
the load cell, a pair of LVDTs externally mounted between the loading platens and a direct-
contact chain extensometer wrapped around the specimens (see Fig. 7.3). Due to the large-
195 |
ADE | strain behaviour of rocks in the post-peak regime, the local strain measurement tools such as
strain gauges are not effective. To characterise the post-peak instability of rocks in terms of
brittleness, the complete stress-strain curves of rocks are required, and therefore, external
LVDTs were used to measure the large-strain properties. LVDTs measure the deformation
between loading platens; therefore, the deformation of the loading system is not included in
the measurement. Still, the strain data may not be precise due to well-known bedding error
(Taheri and Tani 2008). The bedding error refers to the additional deformations measured by
LVDTs due to crushing the irregularities/asperities at the end faces of the specimens before the
specimen deforms as well as the poor fitting of the specimen to the loading platens. This error
is minimised in this study by carefully and smoothly grinding the ends of the specimen
following the ISRM standard (Fairhurst and Hudson 1999). Besides, since the focus of this
study is complete stress-strain behaviour, this error is deemed negligible in large strain stress-
strain properties.
The acoustic emission (AE) technique, as a passive non-destructive monitoring technology,
was also employed in this study to measure the real-time formation and growth of local micro-
cracks throughout the specimen (internal damage) during cyclic loading (Lockner 1993;
Bruning et al. 2018). For this aim, as depicted in Fig. 7.3, two miniature PICO sensors were
attached to the specimens, and the recorded acoustic signals by these sensors were amplified
using a pre-amplifier (type 2/4/6) set to 60 dB of gain. The AE recordings were carried out
using the Express-8 data acquisition card with the sampling rate of 2 MSPS (million samples
per second). To ensure that mechanical noises induced by the loading system are not recorded
during the tests, the AE threshold amplitude was changed from 20 dB to 60 dB, and it was
found that after 45 dB amplitude, no additional noises are recorded. Therefore, this value was
set as the AE threshold.
The stress-strain curves obtained from the conducted uniaxial compressive tests and their
relevant mechanical properties can be found in Fig. 4b, and Table 7.1, respectively. In Table
7.1, the tangent Youngβs modulus (πΈ ) and Poissonβs ratio (π) values were determined at
π‘ππ
50% of the axial peak stress (π ) for each monotonic test. The crack initiation stress (π )
πβππππ ππ
and crack damage stress (π ) thresholds were also determined using the methods explained in
ππ
Taheri et al. (2020). According to Fig. 7.4b, the stress-strain curves for all compression tests
show almost a similar behavioural trend both in the pre-peak and the post-peak regimes. In the
pre-peak regime, as listed in Table 7.1, the deformation parameters of axial (π ), lateral
πβππππ
(π ) and volumetric strains (π ) at peak stress points, πΈ , π, crack initiation stress
πβππππ π£βππππ π‘ππ
196 |
ADE | ratio (π /π ) and crack damage stress ratio (π /π ) are approximately similar,
ππ πβππππ ππ πβππππ
which indicates a small discreteness of the tested specimens. As such, in the post-failure
regime, the sudden drops and recoveries of the load-bearing capacity can be observed for all
specimens which can be associated with the shear strain localisation, grain interlocking in
between the sides of the generated macrocracks (Jansen and Shah 1997; Vasconcelos et al.
2009) as well as the automatic adjustment of applied load by the testing machine upon damage
extension. The post-peak regime of rocks under uniaxial compressive loading demonstrates a
combined class I-II behaviour, which is consistent with the prior study conducted by Munoz et
al. (2016b). As listed in Table 7.1, the monotonic compressive strength (π ) of the tested
πβππππ
Gosford sandstone specimens varied between 45.76 MPa and 49.89 MPa with an average value
of 48.15 MPa. This average monotonic strength was utilised in the following to define the stress
levels where the systematic cyclic loading tests should be commenced.
Table 7.1 The results of uniaxial compressive tests for Gosford sandstone specimens
Test No. π πΈ π Strains at the peak stress point π /π π /π
πβππππ π‘ππ ππ πβππππ ππ πβππππ
(MPa) (GPa) π
πβππππ
π
πβππππ
π
π£βππππ
(%) (%)
(Γ10-4) (Γ10-4) (Γ10-4)
GS-1 48.05 13.30 0.15 54.17 -38.35 -22.54 29.65 58.27
GS-2 49.54 13.43 0.12 52.18 -36.84 -21.51 30.60 58.67
GS-3 47.35 13.42 0.13 52.66 -39.10 -25.55 27.00 55.57
GS-4 45.76 12.97 0.15 51.39 -38.56 -25.73 25.80 55.96
GS-5 49.89 13.15 0.14 53.00 -36.97 -20.95 27.71 57.92
GS-6 48.29 14.14 0.15 50.17 -34.11 -18.05 26.94 52.70
Average 48.15 13.40 0.14 52.26 -37.32 -22.39 27.95 56.51
SD 1.51 0.40 0.01 1.38 1.81 2.93 1.82 2.25
SD standard deviation
197 |
ADE | (b)
GS-1
50
GS-2
)
a GS-3
P
M 40 GS-4
( GS-5
a
GS-6
,
s 30
s
e
r
t
s
l
a 20
i
x
A
10
0
0 20 40 60 80
Axial strain, e (Β΄10-4)
a
Figure 7.4 (Continued)
7.3. Systematic Cyclic Loading Tests
As discussed earlier, the single-criterion load-based and displacement-based loading methods
are not sufficient to control the axial load in the post-failure stage during the systematic cyclic
loading tests, especially when rocks demonstrate self-sustained failure behaviour. In this study,
to address this issue, a new testing method called βdouble-criteria damage-controlled test
methodβ (Shirani Faradonbeh et al. 2020) was employed. As demonstrated in Fig. 7.2, this test
method is a combination of single-level systematic cyclic loading and damage-controlled cyclic
loading lateral displacement-controlled loading method. In this regard, the MTS servo-
controlled testing machine was programmed so that the hydraulic system was allowed to be
adjusted continuously, automatically and rapidly according to the received feedback signals
from both chain extensometer and load cell during a closed-loop procedure. The testing
procedure can be summarised into the following four stages:
1. Load the specimen monotonically (ππ /ππ‘ = 0.02Γ10-4) until the pre-defined stress
π
level (π /π ), and then, unload it at the same loading rate until π = 0.07 MPa, ensuring
π π π
the specimen is always in complete contact with the loading platens.
2. Reload the specimen under a constant lateral strain rate of 3Γ10-4/s until one of the two
following criteria is met during loading:
199
s |
ADE | a) the pre-defined maximum axial stress level (π /π ) is reached;
π π
b) the pre-defined maximum lateral strain amplitude, π΄ππ.(π )= 32Γ10-4 is
π
reached;
3. Reverse the axial load to π = 0.07 MPa, and repeat steps 1 and 2 until 1500 loading
π
and unloading cycles are completed.
4. If the specimen did no fail during 1500 cycles, apply a monotonic loading (ππ /ππ‘ =
π
0.02Γ10-4) until complete failure occurs.
In this study, π΄ππ.(π )= 32Γ10-4 was determined based on the conducted monotonic tests and
π
the measured lateral strain of the rocks at the failure point, π (see Table 7.1). As seen in
πβππππ
Table 7.1, the average value of π for the tested specimens is -37.32Γ10-4. Based on the
πβππππ
conducted several trial tests, it was found that 32Γ10-4 is an appropriate value for Gosford
sandstone. By adopting this value, it was possible to avoid failing the sample in a single cycle
while allowing the axial stress level to reach the pre-defined value to apply a systematic cyclic
loading. Figs. 7.5a and b show two representative time histories of axial stress and lateral strain
for Gosford sandstone specimens experiencing failure during cyclic loading and final
monotonic loading. In Fig. 7.5a, the specimen was loaded monotonically (ππ /ππ‘= 0.02Γ10-
π
4/s) up to 85% of the average monotonic strength (π /π = 85%). Afterwards, the specimen
π π
was unloaded with the same rate, and then the systematic cyclic loading was initiated under the
lateral strain rate of 3Γ10-4/s. As shown in the inset figure, the cycles always met the first
criterion (i.e. the maximum stress applied during a cycle remained constant) during the
systematic cyclic loading and the π΄ππ.(π ) was considerably lower than the pre-defined
π
maximum amplitude for lateral strain (i.e. 32Γ10-4) in each cycle.
As during 1500 loading/unloading cycles, the π΄ππ.(π ) did not exceed 32Γ10-4, a monotonic
π
loading was applied automatically to the specimen under the lateral strain rate of 0.02Γ10-4/s
until the specimen is completely failed. By doing so, the post-peak behaviour was captured
successfully for further analyses. In Fig. 7.5b, the same cyclic loading procedure was applied
to another specimen at a higher axial stress level (i.e. π /π = 87.25%). In the pre-peak stage,
π π
the π΄ππ.(π ) increased gradually by increasing the cycle number, while the stress level was
π
kept constant, satisfying the first criterion. However, at the onset of the failure (where the axial
stress begins to reduce), the π΄ππ.(π ) reached the pre-defined value of 32Γ10-4 (see the inset
π
figure), and the second criterion was activated to control the cyclic loading. By transferring to
the post-peak stage, and strength degradation, the subsequent cycles were carried out so that
200 |
ADE | 7.4. Stress-Strain Relations
In total, 17 single-level systematic cyclic loading tests (see Table 7.2) were carried out at
different stress levels (π /π ) ranging from 80% to 96% of the average monotonic strength
π π
following the proposed double-criteria damage-controlled testing method. As listed in Table
7.1, the stable and unstable crack growths of rocks on average initiate at π /π = 27.95%
ππ πβππππ
and π /π = 56.51%, respectively. This, in other words, shows that the cyclic loading
ππ πβππππ
tests have been conducted in the unstable crack propagation stage, beyond the elastic stress-
strain behaviour. To evaluate the influence of cycle number on mechanical properties and post-
peak behaviour, the specimens GS-8 and GS-9 were subjected to 5000 and 10000 cycles at
π /π =80% and GS-11 experienced 5000 cycles at the stress level of π /π =85% before a
π π π π
monotonic loading. Otherwise, the samples experienced a maximum of 1500 cycles and then a
post-monotonic loading should they did not fail during the cyclic loading. According to
Beniawski (1967), to ensure fatigue failure of a rock specimen in a timely manner, the cyclic
loading test should be conducted just before the onset of the unstable crack propagation stage
within the range of 70-85% of the peak strength. A recent review conducted by Cerfontaine
and Collin (2018) on rock fatigue studies reported that the rock fatigue threshold ranges from
0.75 to 0.9 of the average monotonic strength for one million loading and unloading cycles
depending on rock type and loading conditions. However, in this study, due to test limitations,
further cycles did not apply, and the results are valid in the range of 1500-10000 cycles. Based
on the results presented in Table 7.2, it is hypothesised that there exists a threshold of π /π
π π
which lies between 86% and 87.5 % that indicates the critical boundary of rock strength
hardening and fatigue under cyclic loading. In this study, the cyclic loading tests which
experienced the monotonic loading at the failure stage were named as hardening cyclic loading
tests, while those which failed during cyclic loading at higher stress levels were named as
fatigue cyclic loading tests.
Figs. 7.6 and 7.7 show the typical stress-strain results for hardening and fatigue cyclic loading
tests, respectively. In these figures, the total post-peak behaviour was highlighted by
connecting the indicator stresses (π, the maximum stress of each cycle). The ππππ and ππππ
π π π
respectively, represent the irreversible axial strain and the irreversible lateral strain. The areas
of interest (AOIs) shown in Figs. 7.6c and 7.7c illustrate the specific parts of the volumetric
strain (π ) evolution which were enlarged in Figs. 7.6d and 7.7d, respectively. Figs. 7.6a and
π£ππ
7.7a show that the testing methodology was successful in capturing the complete stress-strain
202 |
ADE | curves of Gosford sandstone specimens subjected to the systematic cyclic loading.
Furthermore, like the monotonic tests, a combined class I-II behaviour at different extents can
be seen in the post-peak regime for both hardening and fatigue cyclic loading tests. Generally,
the variation of hysteretic loops along with the axial strain (Figs. 7.6a and 7.7a), lateral strain
(Figs. 7.6b and 7.7b) and volumetric strain (Figs. 7.6c and d and Figs. 7.7c and d) show that
the rock specimens which fail during the cyclic loading significantly experience more
irreversible strains in the pre-peak regime compared with hardening cyclic loading tests. Also,
as shown in Fig. 7.7d, after a few cycles, the hysteretic loops for the fatigue cyclic loading tests
switch rapidly from the compaction to dilation, and dilation continues until complete failure.
Table 7.2 The results of the conducted systematic cyclic tests
Test No. π /π (%) π π Hardening (H) or π π Peak strength
π π π‘ππ‘ππ πππ‘ππ πβπ πβππππ
fatigue (F) test? (Γ10-4) (Γ10-4) increase (%)
GS-7 80 1500 - H 45.80 53.56 0.53
GS-8 80 5000 - H 43.03 52.36 7.31
GS-9 80 10000 - H 48.94 55.98 0.05
GS-10 85 1500 - H 46.38 53.70 6.22
GS-11 85 5000 - H 48.93 54.29 2.17
GS-12 86 1500 - H 45.52 50.92 1.93
GS-13 87.50 1500 - H 47.72 55.04 7.82
GS-14 86.81 636 49 F - 56.15 -
GS-15 87.23 49 26 F - 56.06 -
GS-16 87.25 240 42 F - 54.78 -
GS-17 89.65 40 28 F - 54.75 -
GS-18 89.82 103 45 F - 53.12 -
GS-19 91.76 145 97 F - 52.75 -
GS-20 93 49 36 F - 54.37 -
GS-21 93.65 280 260 F - 54.98 -
GS-22 95 752 730 F - 54.46 -
GS-23 96 474 318 F - 37.84 -
π total number of cycles, π number of cycles after failure point, π axial strain at the peak of the
π‘ππ‘ππ πππ‘ππ πβπ
final cycle, π axial strain at the failure point
πβππππ
203 |
ADE | (c) (d)
) a 50 AOI 50 Dilation Compaction
P )
M a
P
( 40 M 40
a
(
, a
s s 30 30
e ,
r t s s s e
l a 20 r t s 20
i
x l
A 10 a i x 10
A
0 0
-600 -400 -200 0 100 -100 -80 -60 -40 -20 0 20 100
Volumetric strain, e (Β΄10-4) Volumetric strain, e (Β΄10-4)
vol vol
Figure 7.7 (Continued)
7.5. Rock Behaviour During Hardening Cyclic Loading Tests
7.5.1. Damage Evolution in the Pre-Peak Regime
In rock engineering applications, the rock deformation and failure processes are associated with
the strain energy evolution (Li et al. 2019). The total inputted mechanical energy during a
loading and unloading cycle is transformed into the stored elastic energy (ππ) and the dissipated
π
energy (ππ) as shown schematically in Fig. 7.8a. The dissipated energy due to the irreversible
π
deformations causes stiffness degradation and rock damage. In this study, the energy
dissipation ratio (i.e. πΎ = π /π ) and tangent Youngβs modulus (πΈ ) were utilised to
π π π‘ππ
investigate progressive damage evolution in the pre-peak regime for hardening cyclic loading
tests. Fig. 7.8b shows the representative results for specimen GS-10 at π /π =85%. The other
π π
hardening cyclic loading tests conducted at different stress levels and with a different number
of cycles also showed a similar trend. According to Fig. 7.8b, a two-stage damage evolution
procedure can be identified for the hardening cyclic loading tests. In stage A, the πΈ increased
π‘ππ
dramatically during initial cycles (approximately 21.94% compared with the average πΈ for
π‘ππ
monotonic tests in Table 7.1), which can cause to specimen become stiffer. This behaviour can
be relevant to the closure of existing defects . An increase of stiffness during initial loading
cycle also has been reported by other researchers (Trippetta et al. 2013; Momeni et al. 2015;
Taheri and Tatsuoka 2015; Taheri et al. 2016b). On the other hand, the energy dissipation ratio
(πΎ) decreased suddenly in stage A, which contributes to the accumulation of elastic strain
energy in rock specimen. In stage B, while it was expected to see stiffness degradation due to
205
s
s |
ADE | incurring irreversible deformations in the specimen by doing more cycles, πΈ and πΎ remained
π‘ππ
fairly constant, and no considerable energy was dissipated until 1500 cycles were completed
(i.e. a quasi-elastic behaviour).
This quasi-elastic behaviour can be further investigated using AE results. Fig. 7.8c shows the
typical time-history of AE hits recorded for the specimen GS-10. As shown in this figure, few
AE hits are observed at the initial monotonic loading stage, which corresponds to seating,
loading adjustment by the testing apparatus and the crack closure stage. However, in the second
stage, almost no macrocrack (macro-damage) is generated throughout the specimen as a
constant trend was observed for the cumulative AE hits during the 1500 cycles. In other words,
at this stage, only small amounts of low amplitude AE hits (micro-damages) are generated (see
Fig. 7.8c). During the final monotonic loading stage, new microcracks are generated and
propagated throughout the specimen, and the cumulative AE hits increase gradually until the
peak strength point. This is followed by the rapid rise of cumulative AE hits in the post-peak
regime, where the microcracks coalesce, and the cohesive strength of the rock specimen
degrades. On the other hand, according to Fig. 7.6, during hardening cyclic loading tests, the
specimens do not experience large axial, lateral and volumetric irreversible deformations after
1500 cycles and the hysteretic loops for such tests are very dense. This clearly can be seen from
the variation of volumetric strains in the area of interest (AOI) (see Fig. 7.6d). In Fig. 7.6d, it
is observed that the slope of the hysteretic loops between the lowest points and the peak points
is positive, implying that the current volume of the specimen is mostly at the compaction stage
with slight dilation at the end of pre-peak cyclic loading. According to the evolution of damage
parameters (i.e. πΈ and πΎ), AE hits and the irreversible strains discussed above, the following
π‘ππ
potential mechanism can be inferred for the observed quasi-elastic behaviour in this study:
During cyclic loading below the fatigue threshold stress, but in the unstable crack propagation
stage, some microcracks might be created within the specimens, which may result in grain size
reduction and the creation of some pore spaces. The grain size reduction induced by cyclic
loading also has been reported by Trippetta et al. (2013) based on the conducted microscopic
analysis, although they used different loading history (i.e. damage-controlled cyclic loading
tests). On the other hand, by performing additional loading and unloading cycles, the existing
or newly generated defects which have been oriented horizontally are closed, and the rock
specimen is compacted progressively. This is while the defects which have been oriented
vertically are opened progressively. Therefore, it can be hypothesised that the observed quasi-
elastic behaviour in this study can be due to the competition between two mechanisms of
206 |
ADE | (c)
2000 6000
(1) Initial monotonic loading stage
(2) Systematic cyclic loading stage
(3) Final monotonic loading until peak stress
(4) Post-peak stage
1500 (5) Pre-peak stage s
5 t
i
) 4000 h
s
e 10 14 E
m 4 A
s t i h Ei At ( 1000
1
stis he Em Ai () t 2468 246811 02 stih E A evitalum
uC 3 2000
e v i t a
l
u
m
u
500 C
0 0
5000 10000 15000 20000
Time, t (s)
2
0 0
0 10000 20000 30000
Time, t (s)
(d)
80
)
4 60
-0
1
Β΄
(
a
,n 40
i
a
r
t
s
l
a
i
x 20
A Axial strain at the failure point, e
a-peak
Axial strain at the final loading cycle, e
a-f
0
80 82 84 86 88
Applied stress level, s/s (%)
a m
Figure 7.8 a Energy components for a loading and unloading cycle, b typical evolution of the
energy dissipation ratio and stiffness parameters for the specimen GS-10, c typical time-
history of AE hits for the specimen GS-10, d the variation of axial strain at the final loading
cycle and the failure point with stress level for hardening cyclic loading tests
7.5.2. Effect of Pre-Peak Cyclic Loading on the Post-Peak Monotonic Behaviour
In Fig. 7.9, the results of hardening cyclic loading tests are compared with monotonic test
results, as normalised axial stress-strain curves. As it may be seen in this figure, the overall
post-peak behaviour of monotonic and hardening cyclic loading tests are almost similar. Also,
the increase in cycle number at stress levels π /π =80% (from 1500 to 10000 cycles) and
π π
π /π =85% (from 1500 to 5000 cycles), has no significant influence on the general post-peak
π π
208
e |
ADE | behaviour. In other words, when the stress level that cyclic loading is applied is not high enough
to fail the specimen during cyclic loading, the cyclic loading has a negligible effect on the post-
failure behaviour. This can be further investigated based on the variation of rock brittleness.
Although there is no consensus regarding the rock brittleness definition and its criterion, it is
well-known that brittle rocks show small irreversible deformation before peak strength which
is followed by a self-sustaining failure in the post-peak regime (Tarasov and Potvin 2013).
From 1956 to date, many rock brittleness indices have been developed by different researchers;
however, the strain energy-based indices perform relatively better than others (Zhang et al.
2016). The brittle vs. ductile behaviour of rock materials can be revealed in stress-strain curves
during loading and failure. Thus, the rock brittleness indices, which consider the complete
stress-strain behaviour of rocks may be more reliable. Munoz et al. (2016a) proposed three
fracture energy-based brittleness indices considering both pre-peak and post-peak regimes of
stress-strain curves for different rocks under uniaxial compressive tests. They reported that the
proposed indices properly describe an unambiguous and monotonic scale of brittleness with
increasing pre-peak strength parameters (i.e. π , πΈ and π ). Therefore, in this study,
ππ π‘ππ πβππππ
the following equations were used to measure the overall brittleness (π΅πΌ) of the tested
specimens under systematic cyclic loading.
π΅πΌ =
ππ
=
ππ
(7.1)
ππ‘ ππππ+ππππ π‘
π2
π = πβππππ (7.2)
π
2πΈπ‘ππ
where π , π , π and π are total fracture energy in the pre-peak and post-peak stages,
π‘ π πππ πππ π‘
elastic energy at peak stress, the pre-peak dissipated energy and the post-peak dissipated
energy, respectively.
Figure 7.10a shows the different strain energy components defined above for rock brittleness
determination under monotonic loading. For hardening cyclic loading tests (i.e. GS-7 to GS-
13), the final monotonic loading stress-strain curves were extracted from the stress-strain
relations shown in Fig. 7.9. The strain energy components were calculated for all monotonic
and hardening cyclic loading tests, and the corresponding π΅πΌ values were determined. The
results are listed in Table 7.3. Fig. 7.10b shows the variation of BI values for these tests. As
may be seen in this figure, the π΅πΌ values of the specimens tested under hardening cyclic loading
are almost similar to those obtained under the monotonic loading conditions. Therefore, it can
209 |
ADE | for several cycles, a quasi-elastic behaviour dominated the damage evolution during the pre-
peak cyclic loading. This behaviour was accompanied by the progressive rock compaction (see
Fig. 7.6) and strength improvement up to 8%. It should be noted that rock strength
improvement induced by cyclic loading also has been reported in several studies for porous
Hawkesbury sandstone (up to 11%) (Taheri et al. 2016a, 2017), hard graywacke sandstone (up
to 29%) (Singh 1989) and rock salt (up to 171%) (Ma et al. 2013). This shows that rocks
depending on their intrinsic characteristics and the applied loading history and loading
conditions, may show strength hardening behaviour at different extents. Taheri et al. (2017)
argue that when the rock specimen is subjected to cyclic loading at a stress level lower than a
threshold value, the weak bonding between the mesoscopic elements may be broken down, and
the created fine materials, may fill up the internal voids, causing rock compaction and strength
improvement. It should be mentioned that other potential mechanisms such as microcrack tip
blunting and the interlocking of grains/asperities may involve in strength hardening. For
instance, by considering the initial porosity of Gosford sandstone (i.e. 18%), due to the grain
size reduction induced by cyclic loading during the quasi-elastic stage, some additional pore
spaces might be generated within the specimens. When the cyclic loading-induced microcracks
meet these pores, their tips may become blunt, resulting in a decrease in stress concentration at
the crack tips and an increase in fracture toughness. This, on the other hand, may cause to
stopping the microcrack propagation. This behaviour can also be accompanied by grain
interlocking, closure of cracks, and finally, compaction of the specimens during cyclic loading.
Further microscopic investigations will shed more light on cyclic loading induced hardening
mechanism.
(a)
Monotonic loading tests
) 52 Hardening cyclic loading tests Upper limit
a Average
P
M
(
k a e 50 Upper limit
p
a-
,s
s
e r t 48 Lower limit
s
k
a
e
p
l
46
a
i x Lower limit
A
44
1 2 3 4 5 6 7 8 9 0 1 2 3
-S
G
-S
G
-S
G
-S
G
-S
G
-S
G
-S
G
-S
G
-S
G
1
-S
1
-S
1
-S
1
-S
G G G G
Test number
213
s |
ADE | (b)
m
1.08 GS-13
/
h GS-8
,o GS-10
i 1.06
t
a
r
g
n
i 1.04
n
e
d
r
a
h
h
1.02 GS-11 GS-12
t
g
n GS-7
e GS-9
r 1.00
t
S
0.98 80 82 84 86 88
Applied stress level, s/s (%)
a m
Figure 7.11 a The variation of axial peak stress for all monotonic and hardening cyclic
loading tests and b strength hardening ratio vs. applied stress level for hardening cyclic
loading tests
7.6. Rock Behaviour During Fatigue Cyclic Loading Tests
6.1. Evaluation of Post-Peak Behaviour
As discussed in section 7.5.2, the systematic cyclic loading has no notable effect on the post-
peak behaviour of Gosford sandstone specimens if the cyclic stress level is below fatigue
threshold stress. In this section, the influence of systematic cyclic loading beyond the fatigue
threshold stress on the post-peak behaviour of Gosford sandstone specimens was evaluated.
Figure 7.12 shows the normalised axial stress-strain curves for both monotonic tests and fatigue
cyclic loading tests. The effect of cyclic loading history on the post-failure behaviour can be
evaluated using the variation of rock brittleness index (π΅πΌ) with the applied stress level. To do
so, the envelope curve connecting the loci of the indicator stresses (π) both in the pre-peak and
π
the post-peak regimes were drawn, and the same procedure explained in section 7.5.2 was
utilised to measure the overall brittleness index. Fig. 7.13a shows the extracted envelope curve
for the typical test of GS-16. The strain energy components along with the π΅πΌ values were
determined for all fatigue cyclic loading tests, and the obtained values were tabulated in Table
7.3. Figure 7.13b displays the variation of π΅πΌ values with the applied stress level. From this
figure, it can be observed that the overall rock brittleness increases with an increase in the
applied stress level. This means that rock may fail in a more brittle manner when it experiences
cyclic loading at the stress levels close to its monotonic strength. In other words, in deep
214
s
s |
ADE | (b)
1.0
I
B0.8
,x
e
d
n0.6
i
s
s
e
n
e0.4
l
t
t
i
r
B
0.2
0.0
86 88 90 92 94 96
Applied stress level, s /s (%)
a m
Figure 7.13 (Continued)
7.6.2. Damage Evolution in the Post-Peak Regime
The irreversible deformations are not accumulated at a constant rate in the rock specimen
during the pre-peak cyclic loading but follow an inverted S-shaped behaviour comprising three
main phases of transient, steady and acceleration (Fig. 7.14a) (Royer-Carfagni and Salvatore
2000; Xiao et al. 2009; Fuenkajorn and Phueakphum 2010). These three phases are manifested
as loose-dense-loose behaviour in the stress-strain curves of systematic cyclic loading based
on the variation of hysteretic loops (Fig. 7.14b). According to Zoback and Byerlee (1975), the
initial loose cycles correspond to the energy consumption for crack growth, that stabilises after
several cycles. In the second phase that hysteric loops are closed and dense, the frictional work
is more dominant, and the micro-cracks are opened and closed constantly without any
significant extension. However, when the rock specimen is close to the failure point (i.e. the
acceleration phase), the crack growth dominates, and hysteresis of the cycles increases. At
higher stress levels, due to rapid accumulation of damage, the steady phase will not be visible.
On the other hand, at lower stress levels (as discussed in section 7.5.1), after the initial phase,
a steady-state dominates the whole test for a long time (Xiao et al. 2009).
According to the stress-strain curves obtained for the fatigue cyclic loading tests in this study
(Fig. 7.11), the loose-dense-loose behaviour with different extents can be identified for
hysteretic loops not only in the pre-peak regime but also in the post-peak regime. For instance,
Fig. 7.14c and d shows the typical results for specimen GS-23 in which the loose-dense-loose
behaviours are evident. As shown in the inset figure of the axial stress-strain curve, in the pre-
peak regime, the hysteretic loops follow a loose-dense-loose behaviour according to the
217 |
ADE | mechanism explained above. The loose behaviour at the end of the pre-peak systematic cyclic
loading extends to the post-peak regime and then accelerates. In Fig. 7.14e and f the cumulative
irreversible axial (βππππ) and cumulative irreversible lateral strains (βππππ) measured after full
π π
unloading of each loading cycle in the post-peak regime of specimen GS-23 are plotted against
the axial stress ratio (π /π ). According to these figures, when the specimen loses its
π πβππππ
load-bearing capacity until π /π = 0.69, due to quick dissipation of strain energy, the
π πβππππ
cumulative irreversible strains increases rapidly, which provided the loose hysteretic loops.
Then, interestingly, the hysteretic loops are closed and experience a dense behaviour for a large
number of cycles in the post-peak regime until π /π = 0.38. Finally, by the creation of
π πβππππ
large axial and lateral deformations within the specimen, the cumulative irreversible strains
increased dramatically until complete failure occurred. This, in turn, provided the final loose
hysteretic loops. The observed loose-dense-loose behaviour in the post-peak regime for this
specimen can be summarised as a secondary inverted S-shaped damage behaviour, as shown
in Fig. 7.14g. Depending on the number of cycles that the specimens have experienced after
failure point, similar damage evolution trends with different extents also were observed for
other fatigue cyclic loading tests. According to Table 7.2 and as shown in Fig. 7.14h, it can be
observed that with the increase of applied stress level (π /π ), the number of cycles after
π π
failure point increases exponentially, which is consistent with the formation of the secondary
three-stage inverted S-shaped behaviour in the post-peak regime. In other words, it can be
found out that the damage per loading/unloading cycle in the post-peak regime of the fatigue
cyclic loading tests decreases with the increase of the applied stress level.
(a) (b)
Loose Dense Loose
a a
, n Transient ,
i s
a phase s
r e
t r
s t
s
l
a l
i a
x i
A Acceleration x
A
phase
Steady phase
Cycle number, n Axial strain, e
a
Figure 7.14 a, b Typical inverted S-shaped damage behaviours in the pre-peak regime
(Modified from Guo et al. 2012), c, d the loose-dense-loose behaviour in the post-peak
218
e s |
ADE | 7.7. Conclusions
In this study, a series of systematic cyclic loading tests were conducted on Gosford sandstone
specimens using an innovative double-criteria damage-controlled testing method. A
comprehensive evaluation was carried out on the experimental results in terms of damage
evolution, post-peak instability and strength hardening behaviour. The following conclusions
can be drawn:
1. It was found that there exists a threshold of π /π , which lies between 86-87.5%. For
π π
π /π lower than this range, the specimens did not fail after experiencing a large
π π
number of cycles. The evaluation of the energy dissipation ratio, tangent Youngβs
modulus and AE hits for hardening cyclic loading tests showed that the rock specimens
follow a two-stage damage evolution law dominated by a quasi-elastic behaviour in the
pre-peak regime. This quasi-elastic behaviour can be attributed to a balance between
two mechanisms of dilatant microcracking and rock compaction during cyclic loading
below the fatigue threshold stress. Moreover, the damage evolution in the pre-peak
regime of the hardening cyclic loading tests was found to be independent of the number
of cycles, as no significant influence on damage and/or hardening behaviour was
observed by increasing the cycle number from 1500 to 10000 cycles.
2. A similar pre-peak and post-peak behaviour was observed for monotonic tests and
hardening cyclic loading tests when they were compared as the normalised axial stress-
strain relations. Also, according to the variation of an energy-based brittleness index
(π΅πΌ), it was found that the pre-peak systematic cyclic loading has negligible influence
on the post-failure instability, when the applied stress level is not high enough to fail
the specimen during cyclic loading.
3. For the specimens subjected to the systematic cyclic loading below the fatigue threshold
stress, the peak strength increased up to 8% after applying the monotonic loading. This
strength enhancement might be due to rock compaction and porosity reduction
mechanism induced by cyclic loading. On the other hand, the fatigue failure was
observed for the specimens cyclically loaded beyond the fatigue threshold stress. For
such tests, a rapid accumulation of lateral and volumetric strains was observed in the
pre-peak regime.
4. For the systematic cyclic loading tests conducted beyond the fatigue threshold stress, it
was observed that with the increase of the applied stress level, the rock specimens tend
to behave as self-sustained in the post-failure stage. This was confirmed by the increase
220 |
ADE | of brittleness index (π΅πΌ) with π /π for the fatigue cyclic loading tests. Therefore,
π π
rocks may behave in a more brittle/violent manner when the cyclic loading is applied
at stress levels close to their monotonic strength.
5. The evolution of hysteretic loops for fatigue cyclic loading tests showed that the rock
specimens follow a loose-dense-loose behaviour in the pre-peak regime. However, the
loose behaviour before the failure point is extended to the post-peak stage for several
cycles. These loose hysteretic loops are followed by a dense behaviour for a large
number of cycles until the complete failure of the specimen occurs, demonstrating
another loose behaviour. This generally can be manifested as a secondary inverted non-
linear S-shaped damage behaviour when the cumulative axial and cumulative lateral
irreversible strains are plotted against the post-peak cycle number. It was observed that
damage per cycle decreases exponentially with the increase of the applied stress level,
and the three phases of the inverted S-shaped damage behaviour become more visible
in the post-peak regime.
Acknowledgements
The first author acknowledges the University of Adelaide for providing the research fund
(Beacon of Enlightenment PhD Scholarship) to conduct this study. The authors would like to
thank the laboratory staff, in particular, Simon Golding and Dale Hodson, for their aids in
conducting the tests.
References
Beniawski ZT (1967) Mechanism of brittle fracture of rock. International Journal of Rock
Mechanics and Mining Sciences 4(4):395-406
Bruning T, Karakus M, Nguyen GD, Goodchild D (2018) Experimental Study on the Damage
Evolution of Brittle Rock Under Triaxial Confinement with Full Circumferential Strain
Control. Rock Mechanics and Rock Engineering 51(11):3321β3341
Cardani G, Meda A (2004) Marble behaviour under monotonic and cyclic loading in tension.
Construction and Building materials 18(6):419β424
Cattaneo S, Labuz JF (2001) Damage of marble from cyclic loading. Journal of materials in
civil engineering 13(6):459β465
Cerfontaine B, Collin F (2018) Cyclic and fatigue behaviour of rock materials: review,
221 |
ADE | interpretation and research perspectives. Rock Mechanics and Rock Engineering
51(2):391β414
Erarslan N, Alehossein H, Williams DJ (2014) Tensile Fracture Strength of Brisbane Tuff by
Static and Cyclic Loading Tests. Rock Mechanics and Rock Engineering 47(4):1135β
1151
Fairhurst CE, Hudson JA (1999) Draft ISRM suggested method for the complete stress-strain
curve for intact rock in uniaxial compression. International Journal of Rock Mechanics
and Mining Sciences 36(3):279β289
Fuenkajorn K, Phueakphum D (2010) Effects of cyclic loading on mechanical properties of
Maha Sarakham salt. Engineering Geology 112(1-4):43β52
Gong QM, Yin LJ, Wu SY, et al (2012) Rock burst and slabbing failure and its influence on
TBM excavation at headrace tunnels in Jinping II hydropower station. Engineering
Geology 124:98β108
Guo H, Ji M, Zhang Y, Zhang M (2018) Study of mechanical property of rock under uniaxial
cyclic loading and unloading. Advances in Civil Engineering 2018, pp 1-6
Haimson BC (1978) Effect of Cyclic Loading on Rock. In: Silver ML, Tiedemann D (eds)
Dynamic Geotechnical Testing. ASTM International, West Conshohocken, PA, pp 228β
245
Heap MJ, Faulkner DR (2008) Quantifying the evolution of static elastic properties as
crystalline rock approaches failure. International Journal of Rock Mechanics and Mining
Sciences 45(4):564β573
Heap MJ, Faulkner DR, Meredith PG, Vinciguerra S (2010) Elastic moduli evolution and
accompanying stress changes with increasing crack damage: implications for stress
changes around fault zones and volcanoes during deformation. Geophysical Journal
International 183(1):225β236
Heap MJ, Vinciguerra S, Meredith PG (2009) The evolution of elastic moduli with increasing
crack damage during cyclic stressing of a basalt from Mt. Etna volcano. Tectonophysics
471(1):153β160
Jansen D, Shah S (1997) Effect of Length on Compressive Strain Softening of Concrete.
Journal of Engineering Mechanics 123(1):25β35
222 |
ADE | Li T, Pei X, Wang D, et al (2019) Nonlinear behavior and damage model for fractured rock
under cyclic loading based on energy dissipation principle. Engineering Fracture
Mechanics 206:330β341
Liu J, Xie H, Hou Z, et al (2014) Damage evolution of rock salt under cyclic loading in unixial
tests. Acta Geotechnica 9(1):153β160
Liu Q, Huang S, Kang Y, Liu X (2015) A prediction model for uniaxial compressive strength
of deteriorated rocks due to freeze-thaw. Cold Regions Science and Technology 120:96β
107
Liu X, Liang Z, Zhang Y, et al (2018) Experimental study on the monitoring of rockburst in
tunnels under dry and saturated conditions using AE and infrared monitoring. Tunnelling
and Underground Space Technology 82:517β528
Lockner D (1993) The role of acoustic emission in the study of rock fracture. International
Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts 30(7):883β
899
Ma L, Liu X, Wang M, et al (2013) Experimental investigation of the mechanical properties of
rock salt under triaxial cyclic loading. International Journal of Rock Mechanics and
Mining Sciences 34β41
Martin CD, Chandler N (1994) The Progressive Fracture of Lac du Bonnet Granite.
International Journal of Rock Mechanics and Mining Sciences and Geomechanics
Abstracts 31(6):643β659
Martin CD (1997) Seventeenth Canadian Geotechnical Colloquium: The Effect of
Cohesion Loss and Stress Path on Brittle Rock Strength. Canadian Geotechnical Journal
34(5):698β725
Momeni A, Karakus M, Khanlari G, Heidari M (2015) Effects of cyclic loading on the
mechanical properties of a granite. International Journal of Rock Mechanics and Mining
Sciences 100(77):89-96
Munoz H, Taheri A (2019) Postpeak deformability parameters of localized and nonlocalized
damage zones of rocks under cyclic loading. Geotechnical Testing Journal 42(6):1663β
1684
Munoz H, Taheri A, Chanda EK (2016a) Fracture Energy-Based Brittleness Index
223 |
ADE | Development and Brittleness Quantification by Pre-peak Strength Parameters in Rock
Uniaxial Compression. Rock Mechanics and Rock Engineering 49(12):4587β4606
Munoz H, Taheri A, Chanda EK (2016b) Pre-peak and post-peak rock strain characteristics
during uniaxial compression by 3D digital image correlation. Rock Mechanics and Rock
Engineering 49(7):2541β2554
Royer-Carfagni G, Salvatore W (2000) The characterization of marble by cyclic compression
loading: experimental results. Mechanics of Cohesive-frictional Materials 5(7):535β563
Shirani Faradonbeh R, Taheri A (2019) Long-term prediction of rockburst hazard in deep
underground openings using three robust data mining techniques. Engineering with
Computers 35(2):659β675
Shirani Faradonbeh R, Taheri A, Karakus M (2021) Post-peak behaviour of rocks under cyclic
loading using a double-criteria damage-controlled test method. Bulletin of Engineering
Geology and the Environment 80(2):1713β1727
Singh SK (1989) Fatigue and strain hardening behaviour of graywacke from the flagstaff
formation. New South Wales. Engineering Geology 26(2):171β179
Sufian A, Russell AR (2013) Microstructural pore changes and energy dissipation in Gosford
sandstone during pre-failure loading using X-ray CT. International Journal of Rock
Mechanics and Mining Sciences 57:119β131
Taheri A, Hamzah N, Dai Q (2017) Degradation and improvement of mechanical properties of
rock under triaxial compressive cyclic loading. Japanese Geotechnical Society Special
Publication 5(2):71β78
Taheri A, Tatsuoka F (2015) Small- and large-strain behaviour of a cement-treated soil during
various loading histories and testing conditions. Acta Geotechnica 10(1):131β155
Taheri A, Royle A, Yang Z, Zhao Y (2016a) Study on variations of peak strength of a sandstone
during cyclic loading. Geomechanics and Geophysics for Geo-Energy and Geo-Resources
2(1):1β10
Taheri A, Yfantidis N, L. Olivares C, et al (2016b) Experimental Study on Degradation of
Mechanical Properties of Sandstone Under Different Cyclic Loadings. Geotechnical
Testing Journal 39(4):673-687
224 |
ADE | Taheri A, Zhang Y, Munoz H (2020) Performance of rock crack stress thresholds determination
criteria and investigating strength and confining pressure effects. Construction and
Building Materials 243:118263
Tarasov B, Potvin Y (2013) Universal criteria for rock brittleness estimation under triaxial
compression. International Journal of Rock Mechanics and Mining Sciences 59:57β69
Trippetta F, Collettini C, Meredith PG, Vinciguerra S (2013) Evolution of the elastic moduli
of seismogenic Triassic Evaporites subjected to cyclic stressing. Tectonophysics 592:67β
79
Vasconcelos G, Lourenço P, Alves C, Pamplona J (2009) Compressive Behavior of Granite:
Experimental Approach. Journal of Materials in Civil Engineering 21(9):502β511
Wang S, Xu W, Sun M, Wang W (2019) Experimental investigation of the mechanical
properties of fine-grained sandstone in the triaxial cyclic loading test. Environmental
earth sciences 78(14):416
Wang W, Wang M, Liu X (2016) Study on Mechanical Features of Brazilian Splitting Fatigue
Tests of Salt Rock. Advances in Civil Engineering
Wawersik WR, Fairhurst CH (1970) A study of brittle rock fracture in laboratory compression
experiments. In: International Journal of Rock Mechanics and Mining Sciences &
Geomechanics Abstracts. Elsevier, pp 561β575
Xiao J-Q, Ding D-X, Jiang F-L, Xu G (2010) Fatigue damage variable and evolution of rock
subjected to cyclic loading. International Journal of Rock Mechanics and Mining Sciences
47(3):461β468
Zhang D, Ranjith PG, Perera MSA (2016) The brittleness indices used in rock mechanics and
their application in shale hydraulic fracturing: A review. Journal of Petroleum Science
and Engineering 143:158β170
Zhang J, Deng H, Taheri A, et al (2019) Deterioration and strain energy development of
sandstones under quasi-static and dynamic loading after freeze-thaw cycles. Cold Regions
Science and Technology 160:252β264
Zoback MD, Byerlee JD (1975) The effect of cyclic differential stress on dilatancy in westerly
granite under uniaxial and triaxial conditions. Journal of Geophysical Research (1896-
1977) 80(11):1526β1530
225 |
ADE | Statement of Authorship
Title of Paper
Fatigue Failure Characteristics of Sandstone Under Different Confining Pressures
Publication Status Published Accepted for Publication
Submitted for Publication U npublished and Unsubmitted work
written in manuscript style
Publication Details Shirani Faradonbeh R, Taheri A, Karakus M (2021) Fatigue Failure
Characteristics of Sandstone Under Different Confining Pressures. Rock
Mechanics and Rock Engineering x(x):xβx
Note: Under review [the paper submitted on 22/05/2021]
Principal Author
Name of Principal Author (Candidate) Roohollah Shirani Faradonbeh
Contribution to the Paper Conducting laboratory tests, analysis of the results, and preparation of the
manuscript
Overall percentage (%) 80%
Certification: This paper reports on original research I conducted during the period of my Higher
Degree by Research candidature and is not subject to any obligations or
contractual agreements with a third party that would constrain its inclusion in this
thesis. I am the primary author of this paper.
Signature Date 17 June 2021
Co-Author Contributions
By signing the Statement of Authorship, each author certifies that:
i. the candidateβs stated contribution to the publication is accurate (as detailed above);
ii. permission is granted for the candidate in include the publication in the thesis; and
iii. the sum of all co-author contributions is equal to 100% less the candidateβs stated contribution.
Name of Co-Author Abbas Taheri
Contribution to the Paper Research supervision, review and revision of the manuscript
Signature Date 21 June 2021
Name of Co-Author Murat Karakus
Contribution to the Paper Review and revision of the manuscript
Signature Date 21 June 2021
226 |
ADE | Chapter 8
Fatigue Failure Characteristics of Sandstone Under
Different Confining Pressures
Abstract
Rock fatigue behaviour including the fatigue threshold stress (FTS), post-peak instability and
strength weakening/hardening during cyclic loading, is of paramount significance in terms of
safety and stability assessment of underground openings. In this study, the evolution of the
foregoing parameters for Gosford sandstone subjected to systematic cyclic loading, in the pre-
peak and the post-peak regimes at different stress levels and under seven confinement levels
(π /ππΆπ ) was evaluated comprehensively. The results showed that the FTS of rocks
3 ππ£π
decreases exponentially from 97% to 80%, when π /ππΆπ increases from 10% to 100%.
3 ππ£π
The brittleness of rocks under monotonic and cyclic loading conditions increases with an
increase in π /ππΆπ when π /ππΆπ ranging between 10-65% (known as the transition
3 ππ£π 3 ππ£π
point). For higher confinements, however, the brittleness of rock transits from self-sustaining
behaviour into ductile behaviour. The evolution of fatigue damage parameters for hardening
tests showed that no critical damage happens within the specimens during cyclic loading;
rather, they experience more compaction. This is while for weakening cyclic loading tests,
continuous damage along with stiffness degradation was dominant. Furthermore, the variation
of axial strain at failure point (π ) shows that for lower confinement levels, the applied stress
ππ
level does not affect the pre-peak irreversible deformation; its effect, however, becomes
significant when confining pressure is high. For the specimens that did not fail in cycles, cyclic
loading resulted in peak strength weakening or hardening depending on the applied stress level.
Weakening effect was observed in higher confining pressures, which was mainly due to a
higher amount of irreversible deformation accumulation in rocks in the pre-peak cyclic loading.
An empirical model was proposed using classification and regression tree (CART) algorithm
to estimate the peak strength variation of Gosford sandstone based on π /ππΆπ and the
3 ππ£π
applied stress level.
Keywords: Triaxial loading, Systematic cyclic loading, Confinement level, Brittleness,
Fatigue threshold stress, Strength hardening/weakening
227 |
ADE | List of Symbols
π Post-peak modulus π Peak deviator stress
π
πΈ Pre-peak modulus π Residual deviator stress
πππ
π Number of cycles before failure π /π Deviator stress level
π’π πβππ£π
π
Strain gauge resistance π /π Fatigue threshold stress
π πβππ£π
π Deviator stress π /ππΆπ Confinement level
3 ππ£π
π΅πΌ Brittleness index π Axial strain at failure
ππ
πΊπΉ Strain gauge factor π Lateral strain at failure
ππ
βπ
Change in resistance ππππ Irreversible axial strain
π
π΄πΈ Acoustic emission ππ /ππ‘ Lateral strain rate
π
πΉππ Fatigue threshold stress ππ /ππ‘ Axial strain rate
π
πΆπ΄π
π Classification and regression tree ππ Shear rupture energy
π
π Output voltage ππ Withdrawn elastic energy
π π
π Excitation voltage ππ Residual elastic energy
ππ₯ ππ
π Mechanical strain ππ Additional energy
π
πΈ Tangent Youngβs modulus ππππ Cumulative irreversible axial strain
π‘ππ π
ππΆπ Uniaxial compressive strength βππππ Differential irreversible axial strain
π
π Total elastic energy π Major principal stress
π 1
π΄ππ.(π ) Lateral strain amplitude π Confining pressure
π 3
8.1. Introduction
Depending on the depth, the geometry of the structures and the human- and/or environmental-
induced seismic activities, rock masses in underground mining and geotechnical projects are
usually subjected to a complex stress state, which may result in continuous damage and failure
at different extents (Yang et al. 2017; Wang et al. 2021). Systematic cyclic loading induced by
the rock breakage operation, mechanical excavation, and truck haulage vibrations is a common
dynamic disturbance in underground openings that complicate the deformation and failure
characteristics of rocks. Rock materials under such loading conditions are more prone to severe
failure phenomena such as strain bursting and large-scale collapses (Bagde and PetroΕ‘ 2005;
Munoz and Taheri 2019, Shirani Faradonbeh et al. 2021a; Meng et al. 2021). Therefore, there
is a remarkable theoretical significance and engineering value to deeply understand the cyclic
loading effect on the damage mechanism and, more importantly, the post-failure behaviour of
rocks in terms of safety and long-term stability of the excavations. During the last decades,
228 |
ADE | different researchers have made many attempts to unveil the rock fatigue mechanism under
different loading conditions using laboratory experiments (Cerfontaine and Collin 2018). In
other words, the damage evolution mechanism in rocks can be characterised more efficiently
using cyclic loading tests as it is straightforward to distinguish the elastic and plastic strains
during each loading and unloading cycle (Zhou et al. 2019; Tian et al. 2021). According to the
holistic classification proposed by Shirani Faradonbeh et al. (2021a), rock fatigue studies can
be classified into two main groups of systematic cyclic loading tests and damage-controlled
cyclic loading tests. Each of these groups can be performed either under load-controlled or
displacement-controlled loading conditions. These loading techniques and their limitations
have been discussed in more detail by Shirani Faradonbeh et al. (2021a).
Generally, the rock fatigue studies can be discussed from two viewpoints: the pre-peak and
post-peak domain analysis. From the viewpoint of the pre-peak-domain analysis, the literature
review shows that cyclic loading depending on loading methods, loading conditions and
intrinsic rock properties (e.g. porosity and mineral compositions) can either degrade (Wang et
al. 2013; Erarslan et al. 2014; Yang et al. 2015; Taheri et al. 2016a) or improve (Burdine 1963;
Singh 1989; Ma et al. 2013; Shirani Faradonbeh et al. 2021b) the peak strength of rocks. For
instance, Ma et al. (2013) reported a 171.1% increase in triaxial compressive strength of rock
salt subjected to systematic cyclic loading. Similarly, Taheri et al. (2016b) observed an 11%
peak strength improvement for the porous Hawkesbury sandstone, and they also pointed out
that rock strength increases respectively with applied stress level and the number of cycles
before failure following linear and exponential functions. On the other hand, most of the fatigue
cyclic loading studies have reported peak strength and stiffness degradation due to the
accumulation of permanent deformations within the rock specimens following a non-linear S-
shaped damage model (e.g. Xiao et al. 2009). Fatigue threshold stress (FTS = π /π ), the
π πβππ£π
maximum stress level at which rock specimen does not fail during cyclic loading under a
constant amplitude, is a significant parameter for long-term stability assessment of
underground openings subjected to seismic disturbances. In other words, rock materials never
fail (after a few thousand cycles) if the cyclic loading is applied equal or below this threshold.
According to Cerfontaine and Collin (2018), different values of FTS can be obtained depending
on the tested material. However, FTS is also dependent on other factors, such as loading
conditions and confining pressure (Burdine 1963). Therefore, more investigations are needed
to unveil the effect of confining pressure on fatigue threshold stress.
229 |
ADE | From the viewpoint of the post-peak domain, due to difficulties in capturing the complete
stress-strain relations of rocks under cyclic loading, especially for brittle rocks which show a
class II post-peak behaviour (Wawersik and Fairhurst 1970), very few studies have investigated
the influence of the pre-peak cyclic loading on post-failure behaviour. In most prior studies,
the damage-controlled cyclic loading tests (with the incremental loading amplitude) have been
used under axial displacement-controlled loading conditions to evaluate the post-peak
behaviour (e.g. Yang et al. 2015, 2017; Zhou et al. 2019; Meng et al. 2021). These studies,
however, were not sufficient to adequately measure the post-peak response of rocks. This is
because, during each loading cycle, the axial load is reversed when a certain amount of
displacement is achieved, and after the failure point, since most of the rocks show class II or a
combination of class I and class II behaviours, rock failure occurs in an uncontrolled manner.
However, Munoz and Taheri (2017) showed that lateral displacement control throughout the
test is a promising technique in studying the failure behaviour of rocks subjected to the post-
peak cyclic loading. Recently, Shirani Faradonbeh et al. (2021a and b) developed a novel
testing methodology based on the lateral strain feedback signal to measure the complete pre-
peak and post-peak behaviour of rocks under uniaxial systematic cyclic loading.
Although many studies have been undertaken by different researchers on the evolution of rock
fatigue damage and deformability parameters under different loading histories and loading
conditions, no significant progress has been made regarding the effect of systematic cyclic
loading on the cyclic loading-induced strength hardening, fatigue threshold stress and the post-
peak instability of rocks under different confining pressures. This is while in underground rock
engineering projects, rock materials are usually subjected to triaxial loading conditions with
different levels of confinement accompanied by the systematic cyclic loading induced by
different dynamic sources. Therefore, having in-depth knowledge concerning the foregoing
parameters plays a critical role in stability assessment and reinforcement design. This study,
for the first time, investigates the effect of systematic cyclic loading history on pre-peak and
post-peak characteristics of rocks under different confinement levels. Some empirical
equations are then proposed to manifest the evolution of peak strength, fatigue threshold stress
and rock brittleness parameters. The obtained results are expected to provide a better
understanding of the mechanical response of rocks to systematic cyclic loading under various
confining pressures.
230 |
ADE | 8.2. Experimental Profile
8.2.1. Gosford Sandstone
In this study, Gosford sandstone (Fig. 8.1a) extracted from the massive Triassic Hawkesbury
sandstone of the Sydney Basin, New South Wales, Australia, was chosen as the testing material
(Ord et al. 1991; Masoumi et al. 2017). X-ray powder diffraction (XRD) analysis of this
medium-grained (0.2-0.3 mm) sandstone revealed that quartz (86%) is the dominant mineral
and illite (7%), kaolinite (6%) and anatase (1%) are other forming mineral composition. Fig.
8.1b displays the SEM analysis result of this sandstone. Sufian and Russell (2013) reported that
Gosford sandstone has a total porosity of about 18%, and the density distribution of the pre-
existing micro-cracks within its matrix is homogenous. This type of sandstone is usually known
as a uniform or very uniform sandstone (Hoskins 1969; Vaneghi et al. 2018). Cylindrical
specimens (Fig. 8.1a) having 42 mm diameter and 100 mm length were extracted from a single
rock block and prepared following the ISRM recommended standards (Fairhurst and Hudson
1999). The specimens were air-dried before conducting the static and cyclic loading tests, and
the average dry density of this rock type was approximately about 2.215 g/cm3.
(a)
42 mm
m
m
0
0
1
(b)
Kaolinite
Quartz
Illite
Quartz
231 |
ADE | Figure 8.1 Gosford sandstone used in this study: a prepared specimens and b SEM
photograph
8.2.2. Testing Equipment
A fully digital closed-loop servo-controlled hydraulic compressive machine, i.e. Instron-1282
with the maximum loading capacity of 1000 kN, was employed to conduct the triaxial
monotonic and cyclic loading tests. The testing machine can be programmed and equipped to
perform different loading schemes using either the load-controlled or displacement-controlled
loading techniques. As shown in Fig. 8.2a, a Hoek cell with a maximum capacity of 65 MPa
was used to apply confining pressure. Also, a pair of LVDTs were installed between the loading
platens to measure the axial displacement of the specimens during loading. Strain gauges are
commonly used to measure the axial and/or lateral deformations of rocks in triaxial conditions.
However, the strain gauges are only effective for local small-strain measurement, and they
usually break after the peak stress when the specimen experiences large deformations (Munoz
et al. 2016a; Bruning et al. 2018). A modified test arrangement is made to overcome this
problem; four strain gauges were attached immediately alongside one another around the centre
line of the Hoek cell membrane, as displayed in Fig. 8.2b. Then, the strain gauges were
connected to form a Wheatstone bridge (half-bridge circuit). Any deformation in specimen
changes the resistance and, therefore, facilitates a unique output voltage (π) as a lateral strain
π
feedback signal. In the Wheatstone bridge shown in Fig. 8.2b, π
and π
represent the total
1 3
resistance values provided by the pairs of strain gauges (each gauge has 120Ξ© resistance) which
are connected in series. To balance the bridge and achieve zero voltage when the specimen is
unstrained, two 240 Ξ© precision resistors (i.e. π
and π
) were used in this circuit. The feedback
2 4
signal, indeed, is the average of the lateral strain (π ) values measured by the strain gauges,
π
which is calculated as follows:
π =
πππ₯(βπ
1+βπ
3)
=
πππ₯.πΊπΉ.(π
+π ) (8.1)
π 1 3
4 π
1 π
3 4
βπ
/π
GF = (8.2)
π
where π
is the resistance of the undeformed strain gauge, βπ
is the change in resistance caused
by strain, π is the mechanical strain, πΊπΉ is the strain gauge factor and π is the bridge excitation
ππ₯
voltage.
232 |
ADE | Through a high-pressure wire and a feed-through connector fitted to the Hoek cell, the feedback
signal is sent to the control unit of the testing machine to adjust the loading rate. By doing so,
the membrane gauges are protected from damage during loading, and finally, the complete
lateral deformation of rocks can be recorded in both pre-peak and post-peak regimes.
Moreover, two miniature AE sensors (type PICO, from the American Physical Acoustics
Corp.) were attached to the spherical seats, which have a direct connection to the specimen in
the Hoek cell, to record the microcracking process during loading. The pre-amplifier was set
to 60 dB of gain (Type 2/4/6) to amplify the acoustic emission (AE) signals during loading. To
ensure that mechanical noises induced by the loading system are not recorded during the tests,
the AE threshold amplitude was changed from 20 dB to 60 dB, and it was found that after 40
dB amplitude, no additional noises are recorded. Therefore, this value was set as the AE
threshold. The axial load, axial and lateral displacements, and the AE outputs were recorded
simultaneously by running the tests.
(a)
Loading platen
AE sensors
LVDT1 LVDT2
Hoek cell
Hydraulic
pressure inlet
High-pressure
wire feed-through
Figure 8.2 Experimental set-up, a overview of the experiment and b strain gauged membrane
233 |
ADE | (b)
High-pressure wire
Strain gauges
R R
1 2
+
Wheatstone
Bridge - +
V V
ex
o
-
R
4 R
3
Figure 8.2 (Continued)
8.3. Test Scheme and Conditions
8.3.1. Uniaxial and Triaxial Monotonic Loading Tests
Before conducting the triaxial monotonic and cyclic loading tests at different confining
pressures, the uniaxial compressive strength (ππΆπ) of Gosford sandstone should be
determined. Shirani Faradonbeh et al. (2021b) performed a series of uniaxial monotonic tests
on this rock type under a constant lateral strain rate (ππ /ππ‘) of 2Γ10-6/s. In their study, the
π
axial strain was measured using a pair of external LVDTs, and the lateral strain feedback signal
was measured using a direct-contact chain extensometer. Fig. 8.3a shows the normalised stress-
strain relations of the performed uniaxial monotonic tests. As it is shown in this figure, the rock
specimens are quite uniform and demonstrate almost similar pre-peak and post-peak stress-
strain relations. Gosford sandstone has an average uniaxial peak strength (ππΆπ ) and tangent
ππ£π
Youngβs modulus (πΈ ) values of 48.15 MPa and 13.4 GPa, respectively.
π‘ππβππ£π
Based on the determined ππΆπ , seven different confinement levels, i.e. π /ππΆπ = 10%,
ππ£π 3 ππ£π
20%, 35%, 50%, 65%, 80% and 100%, were adopted for triaxial monotonic and cyclic
compression tests. For each confinement level, three triaxial monotonic tests were carried out.
234 |
ADE | The tests were conducted in a way that the axial load and confining pressure were applied
simultaneously to the rock specimen under a constant axial strain rate of ππ /ππ‘= 0.03 mm/min
π
until the desired confining pressure level is achieved. Thereafter, the confining pressure and
axial load were kept constant for five minutes to ensure the stress was distributed uniformly
(pre-consolidation stage). Then, while the confining pressure was maintained constant, the
deviator stress (i.e. π = π βπ ) was applied under a constant lateral strain rate (ππ /ππ‘) of
1 3 π
2Γ10-6/s until the complete failure occurs. The lateral strain rate was adjusted during the test
based on the feedback signal received from the four strain gauges mounted on the Hoek cell
membrane. Fig. 8.3b shows a typical time history of stress and strains during a triaxial
compression test at π /ππΆπ =10%. Table 8.1 presents a summary of results for all conducted
3 ππ£π
triaxial monotonic tests. Fig 3c, shows the representative stress-strain relations for the triaxial
monotonic tests. According to Table 8.1 and Fig. 8.3c, the increase in π /ππΆπ , affected
3 ππ£π
both the pre-peak and the post-peak characteristics of rocks. Generally, with an increase in
confining pressure, the axial strain at the failure point (π ) increases. Also, as shown in Fig.
ππ
8.3d, the average peak deviator stress (π ) of Gosford sandstone increased by confining
πβππ£π
pressure following a quadratic trend. Section 5 discusses the triaxial compression test results
in more detail.
(a)
50
)
a P 40
M
(
a
30
,s
s
e
r
ts
20
la
ix
A
10
0
0 20 40 60 80
Axial strain, e (Β΄10-4)
a
Figure 8.3 a Normalised stress-strain relations for uniaxial monotonic tests, modified from
Shirani Faradonbeh et al. (2021b), b typical time-history of stress and strains for a triaxial
monotonic test at 10% confinement level, c representative stress-strain relations for triaxial
monotonic tests at different confinement levels and d the variation of peak deviator stress
with confinement level
235
s |
ADE | 8.4. Confining Pressure Effect on Fatigue Threshold Stress
As mentioned earlier, fatigue threshold stress (FTS) is a critical parameter, that can be used as
an effective compressive strength of the intact rock subjected to static, dynamic and cyclic
loads. Depending on the rock type, testing method and loading history, various range of values
for FTS were reported by different researchers. Table 8.3 reviews these studies and lists the
used materials and testing methods along with the determined FTSs. Table 8.3 shows that most
of the existing studies have been conducted in uniaxial loading condition. Taheri et al. (2016b)
performed the systematic cyclic loading tests on Hawkesbury sandstone under a single
confining pressure of π = 4 MPa. In an earlier study, Burdine (1963) performed a series of
3
triaxial dynamic loading tests under three confining pressures (i.e. π = 0.21 MPa, 1.38 MPa
3
and 5.17 MPa) on Berea sandstone. The study showed that with an increase in confining
pressure from 0 to 5.17 MPa, the fatigue threshold stress increases from 74% to 93% of the
monotonic strength.
In the current study, a more comprehensive range of confining pressure was considered to
evaluate the variation of FTS under systematic cyclic loading for Gosford sandstone.
According to Table 8.2, for each confinement level, a fatigue threshold stress (π /π ) can
π πβππ£π
be derived. Fig. 8.7 plots the variation of the determined FTS values against the confinement
level. As can be seen in this figure, with an increase in π /ππΆπ from 10% to 100%,
3 ππ£π
π /π decreases constantly, which shows the weakening/negative influence of confining
π πβππ£π
pressure on the fatigue life of the rock under cyclic loading. These results, show that in
underground projects, with the increase of depth, rock materials may fail at a stress level lower
than the determined monotonic strength. The behavioural trend observed for FTS in this study
is in contrast to that reported by Burdine (1963). According to Fig. 8.7, the FTS can be
predicted using the following logarithmic function with high accuracy:
πΉππ = π π = β0.074πΏπ( π3 )+0.806 ; π
2 = 0.982 (8.3)
ππβππ£π ππΆπππ£π
Also, based on the proposed Eq. 3, a binary condition can be defined to classify the failure
status of the rock specimens, i.e. occurrence (1) or non-occurrence (0), under a specific stress
level and confining pressure as follows:
1 π /π > πΉππ
π’π πβππ£π
Failure status= { (8.4)
0 π /π β€ πΉππ
π’π πβππ£π
242 |
ADE | 8.5. Confining Pressure Effect on Post-Peak Instability
As mentioned earlier, the post-peak instability of rocks can be characterised as class I and class
II, representing the stable and unstable rock fracturing process under a specific loading history,
respectively. Brittleness is an appropriate intact rock property that can be employed to quantify
the post-peak instability. Many rock brittleness indices can be found in the literature (Meng et
al. 2020). However, as the evolution of strain energy accompanies the process of rock
deformation and failure, the energy balance-based indices can better reflect the post-peak
instability and the potential of severe failures (Li et al. 2019). Therefore, in this study, the
following strain energy-based brittleness indices (π΅πΌs) proposed by Tarasov and Potvin (2013)
were used to evaluate the post-peak instability of rocks:
π΅πΌ =
πππ
=
πβπΈ
(8.5)
1
πππ π
π΅πΌ =
πππ
=
πΈ
(8.6)
2
πππ π
ππ =
π π΅2βπ π΄2
π
2πΈ
π2βπ2 (8.7)
ππ = π΅ π΄
π
2π
ππ = ππ βππ
{ π π π
where ππ , ππ and ππ are, respectively, the withdrawn elastic energy, the additional/excess
π π π
energy and the shear rupture energy in the post-peak regime (see Fig. 8.8). The π and π are
π΄ π΅
the deviator stresses corresponding to points A and B, respectively, and πΈ and π are,
respectively, the pre-peak and the post-peak modulus.
To evaluate the effect of both confining pressure and loading history on rock brittleness, π΅πΌ
1
and π΅πΌ were calculated for all monotonic and the cyclic loading tests (the tests that
2
experienced the final monotonic loading). The evolution of the average π΅πΌ values was plotted
against π /ππΆπ in Fig. 8.9. Shirani Faradonbeh et al. (2021b) performed a series of uniaxial
3 ππ£π
systematic cyclic loading tests on Gosford sandstone at different stress levels and found that
below the fatigue threshold stress, the rock brittleness values are similar to those obtained in
monotonic loading conditions. In this study, the π΅πΌ values were calculated again for all uniaxial
monotonic and cyclic loading tests using Eqs. 8.5 and 8.6. According to Fig. 8.9, similar π΅πΌ
values were obtained for these two types of tests in uniaxial conditions. Also, as can be seen in
Fig. 8.9, with an increase in π /ππΆπ from 0% to 65%, the rock brittleness for both
3 ππ£π
monotonic and cyclic loading tests changed similarly from an almost transitional state (i.e.
244 |
ADE | π΅πΌ β 1 and π΅πΌ β 0) to more class II/brittle behaviour. By increasing the confining pressure
1 2
to a certain amount (i.e. π /ππΆπ =50%), the maximum rock brittleness was achieved, and
3 ππ£π
then, the π΅πΌ values showed a decremental trend. A drastic drop in π΅πΌ was observed for
π /ππΆπ > 65%, specifically for cyclic loading tests, where the rock specimens transferred
3 ππ£π
from the class II region (green area) to the class I region (yellow area). Indeed, there is more
opposition against the self-sustaining failure at high confinement levels, and more energy
should be added axially by the loading system to yield the specimen completely. Therefore, a
transition point at 65% confinement level can be estimated for Gosford sandstone, as the rock
specimens transfer from a brittle to ductile failure behaviour. The evolutionary trend observed
in Fig. 8.9 is also consistent with the stress-strain curves of rocks shown in Fig. 8.3c.
Similar unconventional trends for π΅πΌ also have been reported in a few studies, (i.e. Tarasov
and Potvin 2013 and Ai et al. 2016), for stronger rocks such as quartzite and black shale.
According to these studies, the increase in brittleness of rocks with confining pressure can be
attributed to the energy-efficient fan-head mode shear failure. Indeed, during Class II failure
behaviour, a domino structure of blocks is created by tensile cracks along the future failure
plane. Due to the fracture propagation, these blocks are rotated without collapse behaving as
hinges and create a fan-shaped structure in the fracture tip. This, in turn, provides an active
force (negative shear resistance) that is beneficial for maintaining the crack propagation and is
responsible for the self-sustaining failure behaviour of rocks. Therefore, the increase in
confining pressure for these rock types seems to provide a higher amount of active forces and
consequently increases rock brittleness. By considering the decremental trend of fatigue
threshold stress with confinement level, discussed in the previous section, as well as the
incremental trend of rock brittleness with confinement for a specific extent, it can be inferred
that with an increase in depth in rock engineering projects, the propensity of rock structures to
violent/brittle failures such as strain bursting at stress levels lower than the determined average
peak strength can be aggravated. The brittleness reduction at high confinement levels can be
attributed to the more plastic deformation accumulation induced by the loading and unloading
cycles within the specimens, which result in more energy dissipation in the pre-peak regime.
This, in turn, provides less amount of elastic strain energy (the source for self-sustaining
behaviour) at the failure point, leading to more ductile post-peak behaviour. This behaviour is
more evident for cyclic loading tests than monotonic ones due to the more weakening effect of
loading and unloading cycles at higher confinement levels. The damage evolution of rocks
under different confinement levels is evaluated in more detail in section 8.6.
245 |
ADE | 8.6. Confining Pressure Effect on Fatigue Damage Evolution
8.6.1. Hardening and Weakening Cyclic Loading Tests
Rock specimens usually experience deformation under external forces, and a part of this
deformation can be recovered by withdrawing the applied force, representing elastic
characteristics. However, owing to intrinsic material properties, e.g., porosity and microcracks,
and loading-induced damage, the complete deformation recovery after unloading is not
possible. Therefore, a certain amount of irreversible/plastic deformation is retained in the
specimens (Taheri and Tatsuoka, 2015; Peng et al. 2019). The irreversible strain is accumulated
incrementally by applying more cycles, which is accompanied by rock stiffness degradation.
Cumulative strain can be utilised to manifest the non-visible damage incurred in the specimen
during the systematic cyclic loading tests (Taheri et al. 2016b). According to Table 8.2, for the
specimens that did not fail during 1000 loading and unloading cycles, two types of tests can be
distinguished based on peak strength variation: strength weakening tests (i.e., final monotonic
loading strength is less than ππΆπ ) and strength hardening tests (i.e., final monotonic loading
ππ£π
strength is more than ππΆπ ). As seen in Table 8.2, the strength weakening is evident for the
ππ£π
tests undertaken under π /ππΆπ β₯ 80%. To appraise the rock damage evolution in both
3 ππ£π
conditions, the cumulative irreversible axial strain (ππππ) and tangent Youngβs modulus (πΈ )
π π‘ππ
were determined for two representative tests. Fig. 8.10 shows the variation of ππππ and πΈ
π π‘ππ
for specimens GS-C-15 (with 4.38% strength hardening) and GS-C-31 (with -3.96% strength
weakening) at 35% and 100% confinement levels, respectively. The other weakening and
hardening cyclic loading tests also showed similar behaviour.
According to Fig. 8.10, for both specimens, the elastic modulus increased notably for initial
cycles, making the specimens stiffer and more difficult to deform. This can be related to the
closure of pre-existing defects and yield surface expansion during cyclic loading (Taheri and
Tatsuoka 2015; Peng et al. 2019). However, for specimen GS-C-15 (i.e., hardening test), by
performing further cycles, the stiffness of the specimen decreased slightly and then remained
almost constant until 1000 cycles were completed, which is consistent with the trend observed
by Ma et al. (2013) triaxial systematic cyclic loading tests. On the other hand, during the initial
cycles for specimen GS-C-15, ππππ evolved slightly to a certain amount due to the primary
π
loose hysteretic loops, and then like πΈ , retained almost constant, which shows that no more
π‘ππ
damage is cumulated within the specimen. As stated by Shirani Faradonbeh et al. (2021b), this
quasi-elastic behaviour can be due to the competition between the mechanisms of grain-size
247 |
ADE | reduction and rock compaction under consecutive loading and unloading cycles. For specimen
GS-C-31 (i.e., weakening test), although no failure was recorded during the cycles, a different
trend for variations of ππππ was observed (see Fig. 8.10). For the weakening test, ππππ increased
π π
rapidly, first for several cycles (i.e., initial hysteretic loops), and then by experiencing the dense
hysteretic loops, shows a linear increase. At the end of cyclic loading, the increase of ππππ
π
becomes more pronounced which may indicate that the specimen could have failed during
cyclic loading should the test be continued. These results are consistent with πΈ variations for
π‘ππ
the weakening test, shown in Fig. 8.10. As can be seen in this figure, unlike the hardening test,
the damage evolution for weakening test was accompanied by the progressive stiffness
degradation of rock during the whole cyclic loading test. Therefore, it can be stated that the
strength weakening observed in Table 8.2 for systematic cyclic loading tests can be relevant to
the progressive damage evolution/stiffness degradation of rocks in the pre-peak regime, which
is aggravated when confining pressure exceeds the transition point (i.e. π /ππΆπ >65%).
3 ππ£π
This is while for lower confinement levels, when cyclic stress level is low enough, cyclic
loading has no considerable effect on damage evolution; rather, improves peak strength. The
above observations are further investigated using AE results.
E wirr
tan a
20 )
4
-0
19 1
) a Β΄
P
G
rri(
a
(
,s
u
luE n a t
18 ( h a
G rdS e- nC in- g1 5
test)
15 ,n ia
r ts
la
d ix
o m
s GS-C-31 ( W e
aG kS e- nC in-3 g1
test)
10 a
e lb
'g
n
17 (Weakening test) is
r
u e
o v
Y GS-C-15 e
tn
e g
(hardening test) 5 r r i
e v
n
a
16 ita
T lu
m
0 u
C
0.0 0.2 0.4 0.6 0.8 1.0
Relative cycle number, n/N
Figure 8.10 Typical evolution of ππππ and πΈ for hardening and weakening cyclic loading
π π‘ππ
tests
248
w |
ADE | 8.6.1.1. Acoustic Emission Characteristics
Acoustic emission (AE) is a well-known non-destructive technique that can monitor the micro
and macrocrack evolution in rocks during loading in real-time. Due to the local micro-scale
deformations, small fracturing events corresponding to the immediate release of strain energy
are created in the form of elastic waves within the specimens. Recording and analysing these
elastic waves during the tests can directly measure internal damage (Cox and Meredith 1993;
Lockner 1993). Therefore, the AE technique was utilised to elucidate the cracking procedure
during the hardening and weakening cyclic loading tests better. In this regard, the evolution of
AE hits, representing the number of generated cracks, and its cumulation throughout the
representative hardening and weakening tests GS-C-15 and GS-C-31 were respectively
depicted in Figs. 8.11a and b. To better unveil the damage mechanism under different confining
pressures, the AE results of specimen GS-C-29 (π /ππΆπ =80%) which showed the greatest
3 ππ£π
peak strength decrease (i.e. -13.18% strength weakening) were also displayed in Fig. 8.11c. As
shown in Fig. 8.11, the evolution of AE hits for the specimens can be investigated throughout
three main loading phases: initial monotonic loading (phase A), systematic cyclic loading
(phase B) and final monotonic loading (phase C). For all three specimens, during the seating
of loading platens on the specimens and the closure of pre-existing defects, few AE hits were
recorded in stage A and cumulative AE hits increased slightly. For specimen GS-C-15
(π /ππΆπ =35% and π /π = 80%), as shown in Fig. 8.11a, the cumulative AE hits
3 ππ£π π’π πβππ£π
then remained almost constant (i.e. quasi-elastic behaviour) during loading and unloading
cycles. The zoomed-in figure also shows only small amounts of low-amplitude AE hits during
phase B. The cumulated AE hits at the end of stage B is almost 1.77% of the total damage
experienced by the specimen during the test. This shows that no considerable cyclic loading
induced damage is generated should the specimens be loaded below the fatigue threshold stress
and at confinement levels lower than the transition point. This behaviour also is consistent with
the variation of ππππ discussed in the previous section. The majority of rock damage for
π
specimen GS-C-15 occurred in phase C, where the final monotonic loading was applied to the
specimen. In this phase, due to opening the compacted microcracks, the generation of new ones
and their coalescence close to and after peak strength point, the cohesive strength of rock is
gradually substituted by the frictional resistance, which was accompanied by a higher amount
of AE hits.
Unlike specimen GS-C-15 which showed a quasi-elastic behaviour during the systematic cyclic
loading, a different AE evolution behaviour was observed for specimen GS-C-31
249 |
ADE | (π /ππΆπ =100% and π /π =80%) in phase B. According to Fig. 8.11b, after a slight
3 ππ£π π’π πβππ£π
increase in AE hits during the initial monotonic loading, the microcracking increased with a
higher rate by increasing loading and unloading cycles in phase B, which is manifested by a
higher number of AE hits. The cumulated AE hits at the end of phase B is almost 27.09% of
the total damage incurred in the specimen throughout the test, which is relatively higher than
that observed for specimen GS-C-15. As discussed earlier, this microcracking induced by
cyclic loading results in stiffness degradation (see Fig. 8.10) and more ductile behaviour in the
pre-peak regime. The generated damage was not enough to fail the specimen, however, it
resulted in strength weakening of -3.96% during the final monotonic loading. For specimen
GS-C-29 which experienced a -13.18% decrease in peak strength at 80% confinement level, as
seen in Fig. 8.11c, by applying systematic cyclic loading, the AE hits began to grow first with
a lower rate until about 500 cycles were completed. Then by performing further cycles, the rate
of AE hits cumulation increased dramatically, representing the continuous generation of
macrocracks within the specimen. According to Fig. 8.11c, about 93.90% of the total rock
damage happened at the end of phase B, which is far greater than those observed for specimens
GS-C-15 and GS-C-31. Based on the above observations for AE outputs, it can be stated that
for confinement levels beyond the transition point (π /ππΆπ = 65%), although cyclic loading
3 ππ£π
below the fatigue threshold stress does not lead to fatigue failure during 1000 loading cycles,
it creates significant damage, which results in a considerable strength weakening during final
monotonic loading.
(a)
3500
A: Initial monotonic loading pahse 9000
B: Systematic cyclic loading phase
3000 C: Final monotonic loading phase
7500
2500
6 150
s t
s
t i h
E
2000 stih
E A
24
140stih
E A
e v ita
lu
46 50 00 00
i h
E A
e v
i
A 1500 130m u C C t a l u
0 3000 m
1000 40 80 120 160 u
Time, t C
500 A B 1500
0 0
0 50 100 150 200
Time, t (min )
Figure 8.11 Representative AE results for cyclic loading tests: a hardening test (GS-C-15), b
weakening test (GS-C-31) and c weakening test (GS-C-29)
250 |
ADE | (b)
4000 35000
A: Initial monotonic loading pahse
3500 B C: : S Fy ins ate l m ma ot nic o c toy nc il cic l olo aa dd inin gg p p hh aa ss ee 30000
3000 s
25000 t
i
h
2500 E
s t
i h 2000
A B C 20000 A
e v
E A 15000 i t a
1500 l u
m
10000 u
1000 C
500 5000
0 0
0 100 200 300
Time, t (min )
(c)
3000 30000
A: Initial monotonic loading pahse
B: Systematic cyclic loading phase C
2500 C: Final monotonic loading phase 25000
s
t
2000 20000 i h
E
s A
t
i h 1500 15000 e
E A A B
v
i t a
l
u
1000 10000 m
u
C
500 5000
0 0
0 100 200 300
Time, t (min )
Figure 8.11 (Continued)
8.6.2. Damage Cyclic Loading Tests
In this section, the effect of confining pressure is evaluated on the Gosford sandstone specimens
which failed during loading and unloading cycles, i.e., damage cyclic loading tests. Fig. 8.12a
displays the variation of ππππ for damage cyclic loading tests under different confinement
π
levels. To prevent Fig. 8.12a be crowded, only one damage test was considered for each
confinement level. Generally, the irreversible strain increased quickly at the beginning of the
tests. Then, a relatively uniform accumulation in strain followed by a rapid strain increase as
the rock specimens head toward failure. As is clear from Fig. 8.12a, the damage accumulation
rate increased by an increase in π /ππΆπ from 10 to 100%. This damage evolution, however,
3 ππ£π
is more significant for the tests undertaken under high confining pressures (i.e. over the
transition point) where the irreversible/plastic deformations largely incurred in the pre-peak
251 |
ADE | 8.6.3. Applied Stress Level Effect on Damage Evolution
As stated earlier, systematic cyclic loading was applied to the specimens at different stress
levels (π /π ). To evaluate the effect of the applied stress level on damage evolution of
π’π πβππ£π
rocks under different confining pressures, the axial strain at the failure point (π ) was
ππ
determined for all monotonic and cyclic loading tests. The results were listed in Tables 8.1 and
8.2. For uniaxial monotonic and cyclic loading conditions, π values were adapted from
ππ
Shirani Faradonbeh et al. (2021b). Fig. 8.14 represents the variation of π for monotonic,
ππ
hardening, weakening and damage cyclic loading tests with π /π . It can be seen from
π’π πβππ£π
Fig. 8.14 that under a specific confinement level (i.e. 35%), cyclic loading at various stress
levels has no significant influence on π and their values are almost similar to those obtained
ππ
for monotonic loading tests. However, for higher confinements, larger values of π is observed
ππ
at the stress levels equal to or greater than the fatigue threshold stresses, due to the
accumulation of irreversible strain in the sample during the pre-peak regime before the failure.
The above behaviour is more evident in Fig. 8.15, where the variation of average axial strain
at failure point (π ) for different stress levels was depicted against π /ππΆπ . As seen
ππβππ£π 3 ππ£π
in this figure, for monotonic loading tests, π evolved linearly with the increase of
ππβππ£π
π /ππΆπ ; this is while, for hardening/weakening and damage cyclic loading tests, this
3 ππ£π
evolution occurred exponentially. According to Fig. 8.15, for π /ππΆπ β€35%, the
3 ππ£π
monotonic and cyclic loading tests have almost similar π values, which means that
ππβππ£π
loading and unloading cycles below and beyond the fatigue threshold stress have no striking
influence on pre-peak behaviour, and damage evolution under cyclic loading is similar to
monotonic loading conditions. However, for higher confinement levels, π increased first
ππβππ£π
gradually until π /ππΆπ = 65% representing more accumulation of plastic deformations
3 ππ£π
within the specimens in the pre-peak regime compared with the monotonic loading conditions.
The evolutionary trend of π , then, was aggravated for confinement levels of 80 and
ππβππ£π
100%, where a sharp increase in π was observed for weakening and damage cyclic
ππβππ£π
loading tests.
255 |
ADE | Figure 8.14 Variation of axial strain at failure point for monotonic and cyclic loading tests
under different confinement levels: a 0%, b 10%, c 20%, d 35%, e 50%, f 65%, g 80% and h
100%
)
4
-0
Damage cyclic loading tests
1
(Β΄ 480 Hardening/weakening cyclic loading tests
g v Monotonic loading tests
a
-a
400
,e y=57.405e0.018x
r
u
l i 320 R2= 0.904
a
f
t a y=63.822e0.013x
n i 240 R2= 0.801
a
r
t
s
l 160
a
i
x
a
e 80
g y=0.994x+62.440
a
r e R2= 0.956
v 0
A
0 20 40 60 80 100
Confinement level, s/UCS (%)
3 avg
Figure 8.15 Average axial strain at failure for monotonic and cyclic loading tests
8.7. Confining Pressure Effect on Strength Hardening/Weakening
8.7.1. Peak Strength Variation
As seen in Table 8.2, depending on the stress level that cyclic loading is applied as well as the
confinement level, rock specimens have experienced different values of increase/decrease in
peak strength during final monotonic loadings. As discussed in sections 8.6.1 and 8.6.1.1, when
the stress level during cyclic loading is low enough (i.e. lower than the estimated FTS), cyclic
loading at lower confinement levels did not create macro-damage in the specimens, and a quasi-
elastic behaviour dominated the rock damage evolution. This, in turn, resulted in a hardening
behaviour under loading and unloading cycles, and consequently, strength improvement which
is observed during final monotonic loading. The rock compaction due to cyclic loading in the
hardening region is evident in Fig. 8.5 for the representative test GS-C-13 (with 4.06%
hardening), where the specimen did not experience large axial, lateral and volumetric
irreversible strains, and the rock volume was entirely in the compaction stage during cyclic
loading. This is while for rocks that failed during cycles (see Fig. 8.6), relatively higher strain
values were recorded, and rocks were mainly in the dilation-dominated stage. The strength
hardening induced by cyclic loading also has been reported by other researchers for different
257
e |
ADE | rock types under various loading conditions, such as Gosford sandstone (up to 7.82% increase)
under uniaxial systematic cyclic loading (Shirani Faradonbeh et al. 2021b), Tuffeau limestone
under uniaxial multi-level systematic cyclic loading (up to 28.55% increase) (Shirani
Faradonbeh et al. 2021a), hard graywacke sandstone under uniaxial systematic cyclic loading
(up to 29% increase) (Singh 1989), Hawkesbury sandstone under triaxial systematic cyclic
loading (up to 11% increase) (Taheri et al. 2016b) and rock salt under triaxial systematic cyclic
loading (up to 171% increase) (Ma et al. 2013).
Fig. 8.16a represents variation in peak strength with confinement level (π /ππΆπ ). The
3 ππ£π
results of hardening tests under uniaxial condition (π =0) were extracted from Shirani
3
Faradonbeh et al. (2021b). According to Fig. 8.16a and Table 8.2, the peak strength parameter
varies between two distinct zones, i.e. hardening zone and damage zone. Also, the maximum
increase and decrease in peak strength values of Gosford sandstone specimens are 7.82% and
-13.18%, respectively. Generally, with an increase in π /ππΆπ , the amount of strength
3 ππ£π
hardening induced by cyclic loading decreased and when π /ππΆπ > 65% (i.e. transition
3 ππ£π
point), rock specimens demonstrate strength weakening behaviour (see Fig. 8.16a). To better
reflect the mechanism behind the rock moving from hardening into weakening, a parameter is
proposed as below:
βππππ = (ππππ) β(ππππ) (8.8)
π π π π π
where βππππ is the differential irreversible axial strain (measured between valley points), and
π
(ππππ) and (ππππ) are, respectively, the irreversible axial strains measured for final and initial
π π π π
loading cycles.
Fig. 8.16b demonstrates the variation of βππππ for cyclic loading tests at different stress levels
π
with π /ππΆπ . As can be seen in this figure, the range of variation for βππππ increased
3 ππ£π π
continuously with an increase in confining pressure, and this is more significant for
π /ππΆπ > 65%, where a high amount of irreversible deformation was experienced by the
3 ππ£π
specimens. The incremental trend of βππππ with confinement results in more plastic behaviour
π
and, therefore, pre-peak damage even when cycles donβt result in a failure. This, finally,
resulted in a decremental trend of the maximum peak strength variation at each confinement
level under cyclic loading, as shown in Fig. 8.16c.
258 |
ADE | 8.7.2. An Empirical Model for Strength Prediction
As discussed above, the study on strength variation of rocks under the coupled influence of
cyclic loading and confining pressure is rare and limited to some specific confining pressures.
Therefore, no empirical model can be found in the literature to predict strength variation after
loading cycles. The classification and regression tree (CART) algorithm was employed in this
study to predict the amount of strength hardening/weakening in Gosford sandstone after cyclic
loading history. The CART algorithm, developed by Breiman et al. (1984), is a computational-
statistical algorithm that can predict the target variable in the form of a decision tree. The CART
tree is created by the binary splitting of the datasets from the root node into two sub-nodes
using all predictor variables. The best predictor usually is chosen based on impurity or diversity
measures (e.g. Gini, twoing and least squared deviation). The aim is to create subsets of the
data which are as homogeneous as possible concerning the output variable. For each split, each
input parameter (predictor) is evaluated to find the best groupings of categories (for nominal
and ordinal predictors) or cut point (for continuous predictors) according to the improving score
or reduction in impurity. Thereafter, the predictors are compared, and the predictor with the
greatest improvement is selected for the split. This process is repeated until one of the stopping
criteria (e.g. the maximum tree depth) is met (Salimi et al. 2016; Liang et al. 2016; Khandelwal
et al. 2017). A detailed description of the CART algorithm can be found in (Breiman et al.
1984).
In this study, the applied stress level (π /π ) and confinement level (π /ππΆπ ) were
π’π πβππ£π 3 ππ£π
defined as input variables to predict the percentage of strength hardening/weakening as output
variable. Based on the results presented in Table 8.2 and the conducted cyclic loading tests in
uniaxial conditions by Shirani Faradonbeh et al. (2021b), a database containing 28 tests that
experienced a monotonic loading after a cyclic loading history was compiled. The test GS-C-
29, which showed -13.18% strength weakening was identified as an outlier (in terms of
statistics) and excluded from the modelling procedure. The CART parameters, including the
maximum tree depth, impurity index and the minimum size of parent and child nodes (i.e. the
minimum number of objects that a node must contain to be split) were changed for different
runs to obtain a predictive model with high accuracy and low complexity. Finally, the best
model was achieved according to the settings listed in Table 8.4. The modelling procedure was
carried out in the MatLab environment. Fig. 8.17 represents the obtained regression tree for the
best model. As shown in this figure, the developed regression tree provides a practical tool to
estimate the percentage variation of the peak strength straightforwardly. Fig. 8.18 compares
260 |
ADE | )
%
( 8
n
o
i
t
a
i
r a 4
v
h
t
g
n
e 0
r
t
s
k
a
e
p -4 y=0.8996x+0.2625
d
e R2= 0.90
t
c
i
d
e -8
r
P
-8 -6 -4 -2 0 2 4 6 8
Measured peak strength variation (%)
Fig. 8.18 The comparison of the measured and predicted values of peak strength variation
8.8. Conclusions
Triaxial monotonic and cyclic loading tests were undertaken in this study on Gosford sandstone
at different confinement levels to scrutinise the effect of both systematic cyclic loading history
and confining pressure on the evolution of rock fatigue characteristics. For this aim, a modified
triaxial testing procedure was employed to control the axial load during the tests using a
constant lateral strain feedback signal. Based on the experimental results, the following
conclusions were drawn:
1. The confining pressure displayed a significant effect on fatigue threshold stress (FTS).
It was found that with an increase in π /ππΆπ from 10% to 100%, FTS decreases
3 ππ£π
from 97% to 80%. This indicates that rocks in great depth experience failure due to
cyclic loading at stress levels much lower than the determined monotonic strength.
2. According to the obtained stress-strain relations, the post-peak behaviour of rocks
followed an unconventional trend with the increase in confining pressure so that for
lower π /ππΆπ , rock specimens showed a self-sustaining (brittle) failure behaviour,
3 ππ£π
while for higher π /ππΆπ , the ductile behaviour was dominant. The post-peak
3 ππ£π
instability of rocks was quantified using strain energy-based brittleness indices (π΅πΌπ ),
and a transition point at π /ππΆπ = 65% was identified, where the rocks transited
3 ππ£π
from the brittle failure behaviour to ductile one. The results also showed that cyclic
loading at confinement levels lower than the transition point has no notable effect on
263 |
ADE | rock brittleness, while for π /ππΆπ = 80% and 100%, the weakening effect of
3 ππ£π
systematic cyclic loading history on rock brittleness was more significant.
3. Fatigue damage evaluation of rocks using different parameters (i.e. πΈ , ππππand AE
π‘ππ π
hits) showed that for hardening cyclic loading tests, no macro-damage is observed
within the specimens, and the stiffness of the rocks remain almost constant during a
large number of cycles, representing a quasi-elastic behaviour. However, for weakening
cyclic loading tests, although no failure was observed during cycles, πΈ and ππππ
π‘ππ π
increased and decreased, respectively, with cycle loading. Compared to the hardening
cyclic loading tests, the AE activities (micro-cracking) was more evident for specimens
that showed a higher amount of strength degradation. On the other hand, for damage
cyclic loading tests, it was found that damage is accumulated with a higher rate and
extent with an increase in confining pressure.
4. Looking at the variation of axial strain at the failure point (π ) for monotonic,
ππ
hardening/weakening and damage cyclic loading tests, it was found that under
confinement levels below the transition point, the applied stress level has no notable
effect on the cumulation of irreversible deformations in the pre-peak regime and the
values of π are similar to those in monotonic loading conditions. However, for higher
ππ
confinements, cyclic loading resulted in larger irreversible strain values before the
failure point.
5. After a cyclic loading history, the peak deviator stress of Gosford sandstone varied
between -13.18% and 7.82%. According to the evolution of damage parameters, the
observed quasi-elastic behaviour during cyclic loading and the variation of plastic axial,
lateral and volumetric strains for hardening cyclic loading tests, the strength hardening
can be related to the rock compaction induced by cyclic loading. It was observed that
the increase in confining pressure decreases the amount of strength hardening due to
the accumulation of irreversible strains in the rock specimens. An empirical regression
tree-based model was proposed to estimate peak strength variation of Gosford
sandstone based on the applied stress level and confining pressure. The results showed
the high accuracy of the model.
Acknowledgements
The first author acknowledges the University of Adelaide for providing the research fund
(Beacon of Enlightenment PhD Scholarship) to conduct this study. The authors would like to
thank the laboratory technicians particularly Simon Golding and Dale Hodson, for their aids in
264 |
ADE | Sciences 42(2):237β250
Breiman L, Friedman J, Stone CJ, Olshen RA (1984) Classification and Regression Trees. CRC
press
Bruning T, Karakus M, Nguyen GD, Goodchild D (2018) Experimental Study on the Damage
Evolution of Brittle Rock Under Triaxial Confinement with Full Circumferential Strain
Control. Rock Mechanics and Rock Engineering 51(11):3321β3341
Burdine NT (1963) Rock Failure Under Dynamic Loading Conditions. Society of Petroleum
Engineers Journal 3(1):1β8
Cerfontaine B, Collin F (2018) Cyclic and fatigue behaviour of rock materials: review,
interpretation and research perspectives. Rock Mechanics and Rock Engineering
51(2):391β414
Cox SJD, Meredith PG (1993) Microcrack formation and material softening in rock measured
by monitoring acoustic emissions. In: International journal of rock mechanics and mining
sciences & geomechanics abstracts. Elsevier 30(1):11β24
Erarslan N, Alehossein H, Williams DJ (2014) Tensile Fracture Strength of Brisbane Tuff by
Static and Cyclic Loading Tests. Rock Mechanics and Rock Engineering 47(4):1135β
1151
Erarslan N, Williams DJ (2012) Investigating the effect of cyclic loading on the indirect tensile
strength of rocks. Rock Mechanics and Rock Engineering 45(3):327β340
Grover HJ, Dehlinger P, McClure GM (1950) Investigation of fatigue characteristics of rocks.
Rep Battelle Mem Inst Drill Res Inc
Guo Y, Yang C, Mao H (2012) Mechanical properties of Jintan mine rock salt under complex
stress paths. International Journal of Rock Mechanics and Mining Sciences 56:54β61
Haimson BC, Kim C (1971) Mechanical behaviour of rock under cyclic fatigue. Proceedings
of the 13th symposium on rock mechanics. ASCE, New York, pp 845β863
Hoskins ER (1969) The failure of thick-walled hollow cylinders of isotropic rock. In:
International Journal of Rock Mechanics and Mining Sciences & Geomechanics
Abstracts. Elsevier 6(1):99β125
Jamali Zavareh S, Baghbanan A, Hashemolhosseini H, Haghgouei H (2017) Effect of
269 |
ADE | microstructure on fatigue behavior of intact rocks under completely reversed loading.
Analytical and Numerical Methods in Mining Engineering 6:55β62
Khandelwal M, Armaghani DJ, Faradonbeh RS, et al (2017) Classification and regression tree
technique in estimating peak particle velocity caused by blasting. Engineering with
Computers 33(1):45β53
Li N, Zou Y, Zhang S, et al (2019) Rock brittleness evaluation based on energy dissipation
under triaxial compression. Journal of Petroleum Science and Engineering 183:106349
Liang M, Mohamad ET, Faradonbeh RS, et al (2016) Rock strength assessment based on
regression tree technique. Engineering with Computers 32(2):343β354
Lockner D (1993) The role of acoustic emission in the study of rock fracture. International
Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts 30:883β899
Ma L, Liu X, Wang M, et al (2013) Experimental investigation of the mechanical properties of
rock salt under triaxial cyclic loading. International Journal of Rock Mechanics and
Mining Sciences 34β41
Masoumi H, Horne J, Timms W (2017) Establishing empirical relationships for the effects of
water content on the mechanical behavior of Gosford sandstone. Rock Mechanics and
Rock Engineering 50(8):2235β2242
Meng F, Wong LNY, Zhou H (2020) Rock brittleness indices and their applications to different
fields of rock engineering: A review. Journal of Rock Mechanics and Geotechnical
Engineering
Meng Q, Liu J-F, Ren L, et al (2021) Experimental Study on Rock Strength and Deformation
Characteristics Under Triaxial Cyclic Loading and Unloading Conditions. Rock
Mechanics and Rock Engineering 54(2):777-797
Munoz H, Taheri A (2017) Local damage and progressive localisation in porous sandstone
during cyclic loading. Rock Mechanics and Rock Engineering 50(12):3253β3259
Munoz H, Taheri A, Chanda EK (2016) Pre-peak and post-peak rock strain characteristics
during uniaxial compression by 3D digital image correlation. Rock Mechanics and Rock
Engineering 49(7):2541β2554
Munoz H, Taheri A (2019) Post-peak deformability parameters of localised and non-localised
270 |
ADE | damage zones of rocks under cyclic loading. Geotechnical Testing Journal 42(6), 1663-
1684
Nejati HR, Ghazvinian A (2014) Brittleness effect on rock fatigue damage evolution. Rock
Mechanics and Rock Engineering 47(5):1839β1848
Ord A, Vardoulakis I, Kajewski R (1991) Shear band formation in Gosford sandstone. In:
International journal of rock mechanics and mining sciences & geomechanics abstracts.
Elsevier 28(5):397β409
Peng K, Zhou J, Zou Q, Yan F (2019) Deformation characteristics of sandstones during cyclic
loading and unloading with varying lower limits of stress under different confining
pressures. International Journal of Fatigue 127:82β100
Salimi A, Faradonbeh RS, Monjezi M, Moormann C (2016) TBM performance estimation
using a classification and regression tree (CART) technique. Bulletin of Engineering
Geology and the Environment 77(1):429-440
Shirani Faradonbeh R, Taheri A, Karakus M (2021a) Post-peak behaviour of rocks under cyclic
loading using a double-criteria damage-controlled test method. Bulletin of Engineering
Geology and the Environment 80(2):1713-1727
Shirani Faradonbeh R, Taheri A, Karakus M (2021b) Failure Behaviour of a Sandstone
Subjected to the Systematic Cyclic Loading: Insights from the Double-Criteria Damage-
Controlled Test Method. Rock Mechanics and Rock Engineering [In Press]
Singh SK (1989) Fatigue and strain hardening behaviour of graywacke from the flagstaff
formation, New South Wales. Engineering Geology 26(2):171β179
Sufian A, Russell AR (2013) Microstructural pore changes and energy dissipation in Gosford
sandstone during pre-failure loading using X-ray CT. International Journal of Rock
Mechanics and Mining Sciences 57:119β131
Taheri A, Yfantidis N, L. Olivares C, et al. (2016a) Experimental study on degradation of
mechanical properties of sandstone under different cyclic loadings. Geotechnical Testing
Journal 39(4):673-687
Taheri A, Royle A, Yang Z, Zhao Y (2016b) Study on variations of peak strength of a sandstone
during cyclic loading. Geomechanics and Geophysics for Geo-Energy and Geo-Resources
2(1):1β10
271 |
ADE | Taheri A, Tatsuoka F (2015) Small- and large-strain behaviour of a cement-treated soil during
various loading histories and testing conditions. Acta Geotechnica 10(1):131β155
Tarasov B, Potvin Y (2013) Universal criteria for rock brittleness estimation under triaxial
compression. International Journal of Rock Mechanics and Mining Sciences 59:57β69
Tian W-L, Yang S-Q, Wang J-G, Dong J-P (2021) Failure behavior of the thermal treated
granite under triaxial cyclic loadingβunloading compression. Geomechanics and
Geophysics for Geo-Energy and Geo-Resources 7(1):1β21
Vaneghi RG, Ferdosi B, Okoth AD, Kuek B (2018) Strength degradation of sandstone and
granodiorite under uniaxial cyclic loading. Journal of Rock Mechanics and Geotechnical
Engineering 10(1):117β126
Wang Y, Feng WK, Hu RL, Li CH (2021) Fracture evolution and energy characteristics during
marble failure under triaxial fatigue cyclic and confining pressure unloading (FC-CPU)
conditions. Rock Mechanics and Rock Engineering 54(2):799β818
Wang Z, Li S, Qiao L, Zhao J (2013) Fatigue behavior of granite subjected to cyclic loading
under triaxial compression condition. Rock Mechanics and Rock Engineering 46(6):1603β
1615
Wawersik WR, Fairhurst CH (1970) A study of brittle rock fracture in laboratory compression
experiments. In: International Journal of Rock Mechanics and Mining Sciences &
Geomechanics Abstracts. Elsevier, pp 561β575
Xiao J-Q, Ding D-X, Xu G, Jiang F-L (2009) Inverted S-shaped model for nonlinear fatigue
damage of rock. International Journal of Rock Mechanics and Mining Sciences
3(46):643β648
Yamashita S, Sugimoto F, Imai T, et al (1999) The relationship between the failure process of
the creep or fatigue test and of the conventional compression test on rock. In: 9th ISRM
Congress. International Society for Rock Mechanics and Rock Engineering
Yang S-Q, Ranjith PG, Huang Y-H, et al (2015) Experimental investigation on mechanical
damage characteristics of sandstone under triaxial cyclic loading. Geophysical Journal
International 201(2):662β682
Yang SQ, Tian WL, Ranjith PG (2017) Experimental Investigation on Deformation Failure
Characteristics of Crystalline Marble Under Triaxial Cyclic Loading. Rock Mechanics
272 |
ADE | Chapter 9
Conclusions and Recommendations
9.1. Conclusions
In this thesis, state-of-the-art methodologies comprising machine learning (ML)- and
experimental-based approaches were employed to investigate the rockburst phenomenon in
detail. The significant findings and major contributions of the conducted research project can
be outlined as follows:
β’ The statistical analysis techniques, including the box-plot, principal component analysis
(PCA) and agglomerative hierarchical clustering (AHC) were identified as robust tools to
visually represent the distribution of data points, analyse the interrelationship of the
parameters, detect the outliers and natural groups in the datasets and finally, prepare a
homogeneous database. [see Chapters 2, 4 and 5]
β’ The three ML algorithms of gene expression programming (GEP), genetic algorithm-
based emotional neural network (GA-ENN) and the decision tree-based C4.5 algorithm
showed the high performance in predicting the occurrence or non-occurrence of rockburst
hazard as a binary classification problem (i.e. the prediction accuracy was higher than
80%). [see Chapter 2]
β’ The hybrid GA-ENN algorithm overcame the limitations of the prior ANNs (e.g., getting
trapped in local minima) and provided a global solution for the problem. The C4.5, as a
white-box ML algorithm, provided a visual simple tree structure for determining the
rockburst status straightforwardly based on the specific range of values defined by the
algorithm for different input parameters. The GEP algorithm, unlike the other ML
techniques, through its inherent capability of function finding, successfully detected the
latent complex non-linear relationship between the input parameters and the corresponding
output. The GEP algorithm can open the black-box nature of the common ML algorithms
and by providing the explicit models, facilitates the in-depth investigation of mining and
geotechnical hazards. [see Chapter 2]
274 |
ADE | β’ The results of the sensitivity analysis conducted on the developed GEP-based binary model
for rockburst status prediction revealed that the input parameters of maximum tangential
stress (π ), elastic energy index (π ), uniaxial tensile strength (π ) and uniaxial
π ππ‘ π‘
compressive strength (π ) have the highest influence on rockbursting in deep underground
π
mines, respectively. Due to the significant role of π in rockburst occurrence, more
π
considerations should be taken into account during the design stage of the underground
projects to control this parameter (i.e. by optimisation of the mining layout). [see Chapter
2]
β’ The comparison of the five conventional rockburst criteria, i.e., Russeness criterion, Hoek
criterion, stress coefficient criterion, brittleness index criterion and elastic energy index
(EEI) criterion, with the proposed ML-based models, showed that except for EEI criterion,
the other conventional criteria have the prediction accuracy lower than 80% and cannot
provide reliable estimations in practice. This can be attributed to the case study-based
nature of the conventional criteria and considering few input parameters in their equations.
[see Chapter 2]
β’ The complex relationship between different strength/stress- and energy-based parameters
with the rockburst risk levels (i.e. the intensities of βnoneβ, βlightβ, βmoderateβ and
βstrongβ) was recognised with high accuracy using the unsupervised learning algorithm of
self-organising map (SOM). This algorithm, through an intelligent procedure, categorised
the rockburst events having similar conditions in distinct clusters. [see Chapter 3]
β’ The determined weighted distances between the clusters by the SOM algorithm were also
consistent with the rockburst intensities defined by the engineers. This demonstrated the
high capability of this technique in adapting to mining-related problems, specifically for
rockburst risk level investigation as a multi-class problem. [see Chapter 3]
β’ The evaluation of the weights of input variables in each cluster revealed that the maximum
tangential stress of the surrounding rock mass (π ) has the strongest influence on
π
rockbursting, which is consistent with the results of the binary classification of rockburst
status reported in Chapter 2. [see Chapter 3]
β’ The SOM algorithm with the value of 100% for the five performance indices of accuracy
rate, precision, recall, F1 score and Kappa, proved its superiority over fuzzy c-mean (FCM)
algorithm and the rockburst conventional criteria in clustering the rockburst risk levels. [see
Chapter 3]
β’ The intact rock properties (i.e., uniaxial compressive strength, tensile strength, elastic
275 |
ADE | modulus, and Poissonβs ratio) represented a significant effect on the failure mechanism
(i.e., squeezing, slabbing, and strain burst) of the competent overs-stressed rock masses.
The initial assessment of the compiled database from different underground mining projects
showed that the failure mechanisms cannot be predicted solely by a single indicator. [see
Chapter 4]
β’ Although the GEP algorithm can provide a mathematical equation to estimate the output
parameter, it cannot be used solely to solve multi-class classification problems such as
failure mechanism detection. It was found that the combination of the GEP algorithm with
the logistic regression (LR) is an efficient methodology to overcome this difficulty. The
GEP score calculated for each binary model of the failure mechanisms can be fed into the
logistic regression as the independent variable to determine the occurrence probability of
each failure mechanism. The failure mechanism having the highest probability value is
selected as the final prediction. [see Chapter 4]
β’ According to the results of the confusion matrices and the receiver operating (ROC)
curves, the developed GEP-based binary models in this research project were able to
predict the status (occurrence or non-occurrence) of each failure mechanism, respectively,
with 100% (AUC=1), 100% (AUC=1), and 97.14% (AUC=0.964) accuracy for squeezing,
slabbing and strain bursting failure. However, the developed multi-class classifier of GEP-
LR predicted the final class of failure based on the given intact rock properties with 100%
accuracy. [see Chapter 4]
β’ The further validation of the GEP-LR model with nine unseen/new datasets also proved
the high capability of this model in predicting the failure mechanisms
accurately. Therefore, the developed GEP-LR model can be used as a practical tool by
engineers and researchers to measure the propensity of the competent over-stressed rock
masses to different failure mechanisms at the preliminary stages of the projects. [see
Chapter 4]
β’ It was found that the maximum rockburst stress (π ), i.e., the stress level that bursting
π
π΅
occurs and the rockburst risk level (πΌ ) inferred from the conducted comprehensive true-
π
π΅
triaxial unloading tests are appropriate and reliable indices to investigate the rockburst
phenomenon. [see Chapter 5]
β’ The correlation analysis and the stepwise selection and elimination (SSE) procedure were
identified as efficient tools for dimension reduction (i.e., recognition of the most influential
parameters), removing the multicollinearity among the independent parameters, and
276 |
Subsets and Splits