# Effect of Thermal Treatment on Fractals in Acoustic Emission of Rock Material

Authors: Z. Z. Zhang; X. L. Xu; Q. P. Sun; Y. Dong
Journal: Advances in Materials Science and Engineering
 (2016)

---

## Abstract

Acoustic emission (AE) series on time and location distributions on space are all fractal during the failure process of rock material. In this paper, AE signals of heated rock samples at different temperature under uniaxial compression were captured, and the correlation fractal dimensions (CFDs) of AE counts series at different stress level were calculated using Grassberger-Procaccia algorithm. The temperature effect on AE fractal behavior was revealed. The results show that as the heat temperature increases, the total AE counts are more, while the peak value is less. With the increase of external loading, the AE CFD increases fast to a peak at first and then decreases to a bottom and, after that, increases again but within a narrow range. 200°C and 800°C are two thresholds. As the heat temperature rises, the maximum CFD value and the corresponding stress level both increase from 25°C to 200°C and decrease from 200°C to 800°C and then increase again from 800°C to 1200°C. The CFD value at the failure point shows polynomial decline with rising heat temperature.


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## Body

## 1. Introduction

Acoustic emission (AE) is a nondestructive inspection technique, widely used for monitoring of crack growth behavior of rock [1, 2]. It utilizes the transient elastic waves after each fracture occurrence, which are captured by sensors on the rock surface [3]. Several parameters of the AE behavior enlighten the damage stage within the rock material. These may be the cumulative AE counts, which is connected to the density of cracks and the emission energy which is connected to the cracks’ intensity [4]. Additionally, AE waveform parameters like duration and frequency content depend on the motion of the crack tip and, therefore, carry information about the mode of the crack. Study of the AE indices enlightens the fracture process, enabling predictions on the rock deformation and failure [5, 6].Researches show that the AE series on time and location distributions on space are all fractal [7, 8]. Fractal theory provides an effective method to quantitatively analyze the complex AE indices, and some mechanical properties of rock can be expressed by fractal dimensions of AE data, which can develop some new applications. Guo et al. used the fractal dimension of AE location for quantitative evaluation of complexity of fracture network, and a novel “Soundless Cracking Agent (SCA) fracturing evaluation method” was developed [9]. Vinogradov et al. demonstrated that the fractal dimension of AE in a deforming material during uniaxial tensile deformation is an important quantitative characteristic of the evolution of the dislocation structure, and the peaking of fractal dimension signifies a nearing loss of uniformity of plastic flow and the onset of strain localization [10]. Gilyarov et al. analyzed AE data obtained at the rockburst-hazardous Antey deposit and proposed the correlation fractal dimension (CFD) of AE as a potential indication of rockburst accidents [11]. Iturrioz et al. got a quantitative evaluation of damage in concrete through the fractal dimension of AE signals distribution in the damaged domain [12]. Moustafa et al. presented a new approach based on fractal analysis of guided AE waves for monitoring the corrosion evolutionary path in posttensioned systems [13]. Invernizzi et al. confirmed that the energy dissipation during micro crack propagation in concrete occurs in a fractal domain [14]. Wang and Lin studied fractal characteristics of cracking solids using AE and established the relationships between the stress ratio and dimensionless plastic zone size (PZS) with the fractal dimension [15]. Yuan and Li employed the box counting method (BCM) to calculate the fractal dimension of AE-spatial distribution, and the results showed the fractal dimension presents a decreasing trend with the increase of load strength [16]. Majewska and Mortimer studied the nonlinear dynamics of AE generated in coal samples subjected to gas sorption-desorption by analysis of fractal generalised dimensions and fractal correlation dimension of AE energy rate [17]. Zhou and Liu tested AE in rock specimen under loading and breakage and calculated the CFD and the maximum Lyapunov exponents according to the recorded AE data, based on which the chaotic kinetics laws of AE in rocks are obtained [18]. Wu et al. found that the evolutions of the fractal dimension of AE during failure can be divided into a mode of “fluctuation-gradual descending,” and the descent of the fractal dimension can be regarded as the precursor information to predict the collapse of the rock masses [19]. Zhang et al. calculated the fractal dimension of AE by the method of correlation dimension algorithm, and it increases at the initial stage and then decreases [20].It is known that temperature plays an important role in mechanical and AE behavior of rock material [21, 22], which has been studied widely in former literature. However, the effect of high temperature on the fractal characteristics in AE data during rock deformation has not been investigated. In the present study, we first captured the AE signals of rock samples after heat treatment at different temperature during uniaxial compression process and then used Grassberger-Procaccia algorithm to calculate the CFD of AE counts series at every stress level. The results can deepen our understanding of temperature effect on AE behavior and can offer useful viewpoint to predict rock failure in high temperature environment.


## 2. Experimental Procedures

### 2.1. Preparation of Rock Specimens

Granite in this experiment is collected from Yanzhou mining area, Shandong Province (China), and it is mainly composed of feldspar, illite, pyroxene, and small amounts of other minerals. Limited to cubage of the heater and according to ISRM rock testing advice, rock specimens were processed into cylinder with diameter 25 mm and height 50 mm. After carefully grinding on both upper and lower surface of rock samples using slice machine and sand paper, we make the specimens’ parallelism of the upper-lower surface fluctuate within 0.05 mm and the flatness of the surface within 0.02 mm. The longitudinal wave velocity through the rock specimens value is about 4500 m/s under 25°C, so the integrity and uniformity perform relatively well. Moreover, the average density of this kind of granite is 2.76 g/cm3 in 25°C, while the average uniaxial compressive strength is 191.9 MPa.


### 2.2. Testing Equipment and Methods

We mainly conduct the tests of uniaxial compression on rock specimens, which have naturally cooled to room temperature after heat treatment at different temperature. There were 6 groups, 25°C, 200°C, 500°C, 800°C, 1000°C, and 1200°C, with 3 specimens in each group.Firstly, we heated the specimens, respectively, to a predetermined temperature with the heating rate of 2°C per minute using the high temperature furnace MTS652.02 (shown in Figure1(a)); then, according to the actual experience of foreign scholars, the heating process lasts 20 minutes to make sure the specimens are heated evenly; finally, let the heated rock specimens cool naturally to the room temperature (shown in Figure 1(b)).Figure 1
MTS652.02 and heated rock samples.In this experiment, we used the electric hydraulic servo-valve control system of MTS815.02 at the State Key Laboratory for Geomechanics and Deep Underground Engineering of CUMT to apply load (see Figure2(a)) and used PCI-2 AE Detector (see Figure 2(b)) to catch AE signals during rock deformation and failure. The displacement control mode was applied in the process of rock loading, with the loading rate of 0.0015 mm/s; meanwhile, the control program TestStar II was also used to finish this experiment and record the values of related parameters.Figure 2
Appearance of MTS815.02 System and PCI-2 AE Detector.
(a)
MTS815.02
(b)
PCI-2


## 2.1. Preparation of Rock Specimens

Granite in this experiment is collected from Yanzhou mining area, Shandong Province (China), and it is mainly composed of feldspar, illite, pyroxene, and small amounts of other minerals. Limited to cubage of the heater and according to ISRM rock testing advice, rock specimens were processed into cylinder with diameter 25 mm and height 50 mm. After carefully grinding on both upper and lower surface of rock samples using slice machine and sand paper, we make the specimens’ parallelism of the upper-lower surface fluctuate within 0.05 mm and the flatness of the surface within 0.02 mm. The longitudinal wave velocity through the rock specimens value is about 4500 m/s under 25°C, so the integrity and uniformity perform relatively well. Moreover, the average density of this kind of granite is 2.76 g/cm3 in 25°C, while the average uniaxial compressive strength is 191.9 MPa.


## 2.2. Testing Equipment and Methods

We mainly conduct the tests of uniaxial compression on rock specimens, which have naturally cooled to room temperature after heat treatment at different temperature. There were 6 groups, 25°C, 200°C, 500°C, 800°C, 1000°C, and 1200°C, with 3 specimens in each group.Firstly, we heated the specimens, respectively, to a predetermined temperature with the heating rate of 2°C per minute using the high temperature furnace MTS652.02 (shown in Figure1(a)); then, according to the actual experience of foreign scholars, the heating process lasts 20 minutes to make sure the specimens are heated evenly; finally, let the heated rock specimens cool naturally to the room temperature (shown in Figure 1(b)).Figure 1
MTS652.02 and heated rock samples.In this experiment, we used the electric hydraulic servo-valve control system of MTS815.02 at the State Key Laboratory for Geomechanics and Deep Underground Engineering of CUMT to apply load (see Figure2(a)) and used PCI-2 AE Detector (see Figure 2(b)) to catch AE signals during rock deformation and failure. The displacement control mode was applied in the process of rock loading, with the loading rate of 0.0015 mm/s; meanwhile, the control program TestStar II was also used to finish this experiment and record the values of related parameters.Figure 2
Appearance of MTS815.02 System and PCI-2 AE Detector.
(a)
MTS815.02
(b)
PCI-2


## 3. Results and Discussion

### 3.1. Stress-Strain Curves

The whole stress-strain curves of rock material under uniaxial compression after high temperature are shown in Figure3. The shapes of the curves are similar in rough, and this can be divided into four evident stages, that is, compaction stage, elastic stage, weak stage, and failure stage. Moreover, several obvious differences among them exist. When the heat temperature is higher, the strain ranges corresponding to the compaction and weak stage are bigger, while the elastic stage is shorter, especially between 800°C and 1200°C. On the other hand, the lower temperature samples present brittleness damage, and plastic fracture characteristics show more greatly when the heat temperature is more than 800°C.Figure 3
The whole stress-strain curves of rock samples after different temperature.From the stress-strain curves, we can obtain the uniaxial compressive strength (UCS) of rock samples under different temperature. The variation of UCS with temperature can be seen in Figure4. Although the values of UCS are of high discreteness, the mean value presents obvious law that the UCS decreases generally as the temperature rises. Furthermore, it falls gently before 800°C, while a rapid drop shows then, and the rock samples lose their compressive capacity when the temperature reaches 1200°C.Figure 4
Variation of UCS of rock samples with increasing temperature.Elastic modulus is one of the essential mechanical parameters, and it also can be calculated by stress-strain curve. Figure5 shows the variation of elastic modulus with temperature under uniaxial compression. It expresses that the elastic modulus decreases gradually as the temperature rises, and the downward gradient keeps stable in the temperature range, which is different from that the UCS shows.Figure 5
Variation of elastic modulus of rock samples with increasing temperature.Water evaporation at the higher heat temperature generates more pores, and nonuniformity in mineral inflation caused by high temperature brings more damage during cooling process [23]. These are two reasons for the differences on deformation and failure behaviors under different temperature.


### 3.2. AE Curves during Loading Process

Strain energy accumulates during rock deformation induced by loading or unloading, and the accumulated energy will be released as elastic wave after crack occurrence, which is called AE [1]. Thus we can get the crack distribution and extension from AE signals detected. The AE counts-strain curves together with stress-strain curves at six different temperatures are shown in Figure 6. The AE curves correspond to the stress-strain curves well, and the peak values of AE counts always follow the stress drop, which mean energy release suddenly.Figure 6
Relationship between stress-strain curve and AE counts after different temperature.
(a)
25°C
(b)
200°C
(c)
500°C
(d)
800°C
(e)
1000°C
(f)
1200°CThe whole AE curve can be divided into six stages in rough, which are initial quiet stage (IQS), rising stage (RS), first peak stage (FPS), continuous stage (CS), second peak stage (SPS), and last quiet stage (LQS). In IQS, AE signal is very weak and can be hardly detected, which indicates no obvious damage is produced. As the loading goes up, AE counts begin to occur steadily with small ascensional range, which lasts for a long while, and we call this stage RS. IQS and RS correspond to compaction stage and elastic stage on stress-strain curve, respectively. The following three stages, FPS, CS, and SPS, together make an intense AE period, in which AE signals emerge in large numbers. The FPS occurs at the transitory period from elastic stage to weak stage on stress-strain curve, because damage in rock samples initiates amounts of micro cracks, and strain energy accumulated in former stage is released suddenly. Then AE events last actively because of crack growth, and that is CS. When cracks finally coalesce into macrocrack, rock samples can not bear the applied load, and the stress reaches the UCS of the rock material. The accumulated energy before the peak stress is released sharply and largely, and the peak value of AE counts during the whole loading process is made. This stage is named SPS. After that, AE signal does not disappear for residual fracture, but it drops greatly, and another relatively quiet period comes, called LQS.At 25°C, IQS, RS, and LQS exist obviously, while FPS, CS, and SPS combine together. That is because the rock samples at 25°C are hard and brittle, and the time of crack initiation, propagation, and coalescence is shorter relatively, which leads to concentrated AE counts. As the heat temperature rises, crack growth lasts longer time, and the strain energy is released differently in each stage, and the six AE stages become distinguishable. Rock samples heated at higher temperature show more plastic, and the AE events in FPS, CS, and SPS are more active, but the strength of AE energy is lower. As the heat temperature increases, the total AE counts are more, while the peak value is less.


## 3.1. Stress-Strain Curves

The whole stress-strain curves of rock material under uniaxial compression after high temperature are shown in Figure3. The shapes of the curves are similar in rough, and this can be divided into four evident stages, that is, compaction stage, elastic stage, weak stage, and failure stage. Moreover, several obvious differences among them exist. When the heat temperature is higher, the strain ranges corresponding to the compaction and weak stage are bigger, while the elastic stage is shorter, especially between 800°C and 1200°C. On the other hand, the lower temperature samples present brittleness damage, and plastic fracture characteristics show more greatly when the heat temperature is more than 800°C.Figure 3
The whole stress-strain curves of rock samples after different temperature.From the stress-strain curves, we can obtain the uniaxial compressive strength (UCS) of rock samples under different temperature. The variation of UCS with temperature can be seen in Figure4. Although the values of UCS are of high discreteness, the mean value presents obvious law that the UCS decreases generally as the temperature rises. Furthermore, it falls gently before 800°C, while a rapid drop shows then, and the rock samples lose their compressive capacity when the temperature reaches 1200°C.Figure 4
Variation of UCS of rock samples with increasing temperature.Elastic modulus is one of the essential mechanical parameters, and it also can be calculated by stress-strain curve. Figure5 shows the variation of elastic modulus with temperature under uniaxial compression. It expresses that the elastic modulus decreases gradually as the temperature rises, and the downward gradient keeps stable in the temperature range, which is different from that the UCS shows.Figure 5
Variation of elastic modulus of rock samples with increasing temperature.Water evaporation at the higher heat temperature generates more pores, and nonuniformity in mineral inflation caused by high temperature brings more damage during cooling process [23]. These are two reasons for the differences on deformation and failure behaviors under different temperature.


## 3.2. AE Curves during Loading Process

Strain energy accumulates during rock deformation induced by loading or unloading, and the accumulated energy will be released as elastic wave after crack occurrence, which is called AE [1]. Thus we can get the crack distribution and extension from AE signals detected. The AE counts-strain curves together with stress-strain curves at six different temperatures are shown in Figure 6. The AE curves correspond to the stress-strain curves well, and the peak values of AE counts always follow the stress drop, which mean energy release suddenly.Figure 6
Relationship between stress-strain curve and AE counts after different temperature.
(a)
25°C
(b)
200°C
(c)
500°C
(d)
800°C
(e)
1000°C
(f)
1200°CThe whole AE curve can be divided into six stages in rough, which are initial quiet stage (IQS), rising stage (RS), first peak stage (FPS), continuous stage (CS), second peak stage (SPS), and last quiet stage (LQS). In IQS, AE signal is very weak and can be hardly detected, which indicates no obvious damage is produced. As the loading goes up, AE counts begin to occur steadily with small ascensional range, which lasts for a long while, and we call this stage RS. IQS and RS correspond to compaction stage and elastic stage on stress-strain curve, respectively. The following three stages, FPS, CS, and SPS, together make an intense AE period, in which AE signals emerge in large numbers. The FPS occurs at the transitory period from elastic stage to weak stage on stress-strain curve, because damage in rock samples initiates amounts of micro cracks, and strain energy accumulated in former stage is released suddenly. Then AE events last actively because of crack growth, and that is CS. When cracks finally coalesce into macrocrack, rock samples can not bear the applied load, and the stress reaches the UCS of the rock material. The accumulated energy before the peak stress is released sharply and largely, and the peak value of AE counts during the whole loading process is made. This stage is named SPS. After that, AE signal does not disappear for residual fracture, but it drops greatly, and another relatively quiet period comes, called LQS.At 25°C, IQS, RS, and LQS exist obviously, while FPS, CS, and SPS combine together. That is because the rock samples at 25°C are hard and brittle, and the time of crack initiation, propagation, and coalescence is shorter relatively, which leads to concentrated AE counts. As the heat temperature rises, crack growth lasts longer time, and the strain energy is released differently in each stage, and the six AE stages become distinguishable. Rock samples heated at higher temperature show more plastic, and the AE events in FPS, CS, and SPS are more active, but the strength of AE energy is lower. As the heat temperature increases, the total AE counts are more, while the peak value is less.


## 4. Fractals in AE Behavior

There exists very close relation between the phenomenon of AE and rock fracture, and the AE curve during the whole loading process expresses the growth of cracks in rock samples [4, 6]. Through the above analysis, we get some qualitative results on AE characters of heated rock samples at different high temperature under uniaxial compression. The AE signals are unordered and complex time series relatively; we can use fractal theory to analyze them quantificationally to find new viewpoints.


### 4.1. Grassberger-Procaccia Algorithm

Grassberger-Procaccia method offers a widely applicable way to calculate the CFD of a series of numbers [7]. AE counts during rock deformation and failure under uniaxial compression can be regarded as such a series of numbers. A series whose dimension is n can be expressed as (1)
X
=
x
1
,
x
2
,
…
,
x
n
.We can construct a phase space withm dimension based on X
(
m
<
n
). Firstly, take m numbers out from X continuously to build the first vector X
1:(2)
X
1
=
x
1
,
x
2
,
…
,
x
m
.Then, move right one number; the second vector including m numbers could be gotten:(3)
X
2
=
x
2
,
x
3
,
…
,
x
m
+
1
.We can get n
-
m
+
1 vectors finally, and the last one is(4)
X
n
-
m
+
1
=
x
n
-
m
+
1
,
x
n
-
m
+
2
,
…
,
x
n
.The correlation function is(5)
W
r
=
1
N
2
∑
i
=
1
N
∑
j
=
1
N
H
r
-
X
i
-
X
j
.Here, N is equal to n
-
m
+
1, H is the Heaviside function, r is certain measure, and X
i
-
X
j expresses the distance between vectors of X
i and X
j:(6)
H
u
=
0
,
u
<
0
1
,
u
≥
0
,
r
=
k
r
0
=
k
1
N
2
∑
i
=
1
N
∑
j
=
1
N
X
i
-
X
j
.Here, k is a scale coefficient.From the above equations, we can get one value ofW
(
r
) corresponding to one value of r, so several data pairs of r and W
(
r
) could be obtained. If they follow linear relation in double logarithmic coordinates, the series is fractal, and the CFD D equals the slope value of lg
⁡
r-lg
⁡
W
(
r
) curve:(7)
D
=
lg
⁡
W
r
lg
⁡
r
.According to definition of the CFDD, we can know its physical significance for AE series. Larger value of D indicates that fractures in rock sample are more chaotic in time, and over half of them are micro ones. On the contrary, if the value of D is small, fractures trend to be well organized and strongly localized and main rupture would be developed.


### 4.2. Determination ofm Value

Determination ofm value has an important effect in computation of D value. Too small value would result in no attractor cross to distinguish lower dimension, and too big value would lead to more noise. Researches show that D value becomes larger with increasing m value until m reaches a certain value, and then D keeps stable [6, 7, 19, 20]. The certain m value should be selected to calculate D value.Take AE counts series at 25°C, for example; the changing curve ofD value with m is shown in Figure 7. The CFD D rises from 0.14 to 0.22 when m value changes from 1 to 5, and then D value holds steady although m increases, and they follow sigmoidal logistic function. So 5 was chosen as the m value for calculating the CFD at 25°C.Figure 7
Variation of fractal dimensionD value with m value at 25°C.Them values at different heat temperature were found according to the same way, and they can be seen in Figure 8. The results show that the m value and the heat temperature follow Gauss function, and the m value becomes bigger as the temperature rises before 800°C, while it shows the exact contrary over 800°C.Figure 8
Variation of them value with increasing heat temperature.


### 4.3. AE CFDs at Different Stress Level under Each Heat Temperature

For each AE series at different heat temperature, according to (1)–(7), we calculated the D values at 10 typical stress levels between 10% and 100% in prepeak stage to investigate the changing law of CFD during the whole loading process.Take the stress level 40% at 800°C as an example; the data pairs ofln
⁡
r and ln
⁡
W
(
r
) are shown in Figure 9. Fit them by using linear function, and the D value is equal to the slope of the fitting straight line. For each stress level under every heat temperature, we did the same calculation, and the fitting correlation coefficients are all over 90%, which indicates that AE counts series of rock material under uniaxial compression after heating are fractal and have self-similarity very well.Figure 9
Calculation of CFD at stress level 40% under 800°C.Variation of theD value with increasing stress level before peak strength under six heat temperatures is shown in Figure 10. The curve patterns from 25°C to 1000°C are similar, and the D value increases fast to a peak at first and then decreases to a bottom and, after that, increases again but within a narrow range. The D evolution curve after 1200°C is an exception, and the second increasing stage does not exist.Figure 10
Variation of the CFD with increasing stress level after different heat temperature.Numbers of micro cracks with different sizes grow randomly at low stress level, which shows chaos and randomness. It corresponds to the first increasing stage ofD value. As the stress increases, crack growth trends to the same direction and converts to regular fracture, and this corresponds to the decreasing stage of D value. The stress increase proceeds, and the rock structure adjusts again, and the D value rises a little. This new adjustment would lead to the final main fracture at the peak stress level.


### 4.4. The Maximum and Minimum Values of CFDD

According to theD curve pattern, we focus on two points on the curve, that is, the maximum value and the minimum one. The maximum value means the most chaotic state during the loading process, and the micro cracks are random, while the minimum value refers to the most organized state before peak strength, which indicates that the disorderly cracks would be connected with each other by stress field and form a preliminary organic whole. The second increasing stage shows that the rock structure readjusts and cracks grow disorderly again, but the D value increases small, which expresses that the randomness would be weakened greatly by the first adjustment. The whole loading process is a complicated period of chaos and order.The maximum and minimumD values after different temperatures are shown in Figure 11. The curves express that the minimum D values are all about 0.2, but the maximum values differ greatly. As the heat temperature rises, the maximum D value increases from 25°C to 200°C and decreases from 200°C to 800°C and then increases again from 800°C to 1200°C. 200°C and 800°C are two thresholds, which can be verified by former researches [23].Figure 11
The maximum and minimumD values before peak strength after different temperatures.In fact, the randomness of AE events or cracks mainly depends on the homogeneity of rock material, and stronger homogeneity would lead to greater randomness [24]. When the heat temperature is between 25°C and 200°C, the mineral moisture evaporation increases the homogeneity of small openings. From 200°C to 800°C, higher temperature makes different minerals swell unequally, and more damage is produced by residual stress during natural cooling process, so the homogeneity becomes stronger. When the temperature exceeds 800°C, rock minerals change phases [25], and the mechanical property turns to plasticity from brittleness, and the homogeneity gets stronger and stronger. That is why the maximum D value changes that way in Figure 11 with increasing heat temperature.To find when the maximum and minimumD values reach is significative for predicting rock deformation and failure. The stress levels corresponding to maximum and minimum D values under different temperatures are shown in Figure 12. The two curves show that they all follow cubic function, and 200°C and 800°C are also two thresholds. The corresponding stress level increases, decreases, and then increases again in the interval of 25–200°C, 200–800°C, and 800–1200°C, respectively.Figure 12
The stress levels corresponding to maximum and minimumD values at different temperatures.According to the above analysis, the higher corresponding stress level indicates that the rock material is more homogeneous and brittle. On one hand, the first period of chaos lasts long for rock material with better homogeneity because the lack of inherent defect results in randomness, so the stress level corresponding to the maximumD value is higher. For example, the corresponding stress level is 40% for 200°C but 20% for 800°C. On the other hand, the stress applied on the more homogeneous rock sample would be higher when the D value reaches the minimum; with the same reason, it is difficult to generate dominant crack because of less inherent defect. For example, the corresponding stress level is 70% for 200°C, while it is 50% for 800°C.


### 4.5. CFDD Values near Rock Failure under Different Heat Temperature

Analysis of theD value near peak strength is helpful to obtain the effect law of temperature on rock failure characteristics. We selected two stress levels, 90% and 100%, to discuss the failure characteristics under different temperature, and the D curves are shown in Figure 13.Figure 13
The effect of heat temperature on theD value at stress levels 90% and 100%.
(a)
Stress level = 90%
(b)
Stress level = 100%At the stress level 90%, theD value and heat temperature are fitted to the exponential function. The D value is around 0.3 when the heat temperature is less than 1000°C, while it increases by more than 4 times under 1200°C. This curve indicates that high temperature makes AE events chaotic near failure. Furthermore, when the stress level reaches 100%, that is, the failure point, the D value shows polynomial decline with rising temperature, which indicates the AE events in rock sample that had undergone higher temperature become more regular at failure point. Comparing the two curves, also we find that the difference value of D between stress levels 90% and 100% rises with heat temperature, and the larger difference value makes rock failure more predictable, so the D value can be used as the precursor of rock failure prediction.


## 4.1. Grassberger-Procaccia Algorithm

Grassberger-Procaccia method offers a widely applicable way to calculate the CFD of a series of numbers [7]. AE counts during rock deformation and failure under uniaxial compression can be regarded as such a series of numbers. A series whose dimension is n can be expressed as (1)
X
=
x
1
,
x
2
,
…
,
x
n
.We can construct a phase space withm dimension based on X
(
m
<
n
). Firstly, take m numbers out from X continuously to build the first vector X
1:(2)
X
1
=
x
1
,
x
2
,
…
,
x
m
.Then, move right one number; the second vector including m numbers could be gotten:(3)
X
2
=
x
2
,
x
3
,
…
,
x
m
+
1
.We can get n
-
m
+
1 vectors finally, and the last one is(4)
X
n
-
m
+
1
=
x
n
-
m
+
1
,
x
n
-
m
+
2
,
…
,
x
n
.The correlation function is(5)
W
r
=
1
N
2
∑
i
=
1
N
∑
j
=
1
N
H
r
-
X
i
-
X
j
.Here, N is equal to n
-
m
+
1, H is the Heaviside function, r is certain measure, and X
i
-
X
j expresses the distance between vectors of X
i and X
j:(6)
H
u
=
0
,
u
<
0
1
,
u
≥
0
,
r
=
k
r
0
=
k
1
N
2
∑
i
=
1
N
∑
j
=
1
N
X
i
-
X
j
.Here, k is a scale coefficient.From the above equations, we can get one value ofW
(
r
) corresponding to one value of r, so several data pairs of r and W
(
r
) could be obtained. If they follow linear relation in double logarithmic coordinates, the series is fractal, and the CFD D equals the slope value of lg
⁡
r-lg
⁡
W
(
r
) curve:(7)
D
=
lg
⁡
W
r
lg
⁡
r
.According to definition of the CFDD, we can know its physical significance for AE series. Larger value of D indicates that fractures in rock sample are more chaotic in time, and over half of them are micro ones. On the contrary, if the value of D is small, fractures trend to be well organized and strongly localized and main rupture would be developed.


## 4.2. Determination ofm Value

Determination ofm value has an important effect in computation of D value. Too small value would result in no attractor cross to distinguish lower dimension, and too big value would lead to more noise. Researches show that D value becomes larger with increasing m value until m reaches a certain value, and then D keeps stable [6, 7, 19, 20]. The certain m value should be selected to calculate D value.Take AE counts series at 25°C, for example; the changing curve ofD value with m is shown in Figure 7. The CFD D rises from 0.14 to 0.22 when m value changes from 1 to 5, and then D value holds steady although m increases, and they follow sigmoidal logistic function. So 5 was chosen as the m value for calculating the CFD at 25°C.Figure 7
Variation of fractal dimensionD value with m value at 25°C.Them values at different heat temperature were found according to the same way, and they can be seen in Figure 8. The results show that the m value and the heat temperature follow Gauss function, and the m value becomes bigger as the temperature rises before 800°C, while it shows the exact contrary over 800°C.Figure 8
Variation of them value with increasing heat temperature.


## 4.3. AE CFDs at Different Stress Level under Each Heat Temperature

For each AE series at different heat temperature, according to (1)–(7), we calculated the D values at 10 typical stress levels between 10% and 100% in prepeak stage to investigate the changing law of CFD during the whole loading process.Take the stress level 40% at 800°C as an example; the data pairs ofln
⁡
r and ln
⁡
W
(
r
) are shown in Figure 9. Fit them by using linear function, and the D value is equal to the slope of the fitting straight line. For each stress level under every heat temperature, we did the same calculation, and the fitting correlation coefficients are all over 90%, which indicates that AE counts series of rock material under uniaxial compression after heating are fractal and have self-similarity very well.Figure 9
Calculation of CFD at stress level 40% under 800°C.Variation of theD value with increasing stress level before peak strength under six heat temperatures is shown in Figure 10. The curve patterns from 25°C to 1000°C are similar, and the D value increases fast to a peak at first and then decreases to a bottom and, after that, increases again but within a narrow range. The D evolution curve after 1200°C is an exception, and the second increasing stage does not exist.Figure 10
Variation of the CFD with increasing stress level after different heat temperature.Numbers of micro cracks with different sizes grow randomly at low stress level, which shows chaos and randomness. It corresponds to the first increasing stage ofD value. As the stress increases, crack growth trends to the same direction and converts to regular fracture, and this corresponds to the decreasing stage of D value. The stress increase proceeds, and the rock structure adjusts again, and the D value rises a little. This new adjustment would lead to the final main fracture at the peak stress level.


## 4.4. The Maximum and Minimum Values of CFDD

According to theD curve pattern, we focus on two points on the curve, that is, the maximum value and the minimum one. The maximum value means the most chaotic state during the loading process, and the micro cracks are random, while the minimum value refers to the most organized state before peak strength, which indicates that the disorderly cracks would be connected with each other by stress field and form a preliminary organic whole. The second increasing stage shows that the rock structure readjusts and cracks grow disorderly again, but the D value increases small, which expresses that the randomness would be weakened greatly by the first adjustment. The whole loading process is a complicated period of chaos and order.The maximum and minimumD values after different temperatures are shown in Figure 11. The curves express that the minimum D values are all about 0.2, but the maximum values differ greatly. As the heat temperature rises, the maximum D value increases from 25°C to 200°C and decreases from 200°C to 800°C and then increases again from 800°C to 1200°C. 200°C and 800°C are two thresholds, which can be verified by former researches [23].Figure 11
The maximum and minimumD values before peak strength after different temperatures.In fact, the randomness of AE events or cracks mainly depends on the homogeneity of rock material, and stronger homogeneity would lead to greater randomness [24]. When the heat temperature is between 25°C and 200°C, the mineral moisture evaporation increases the homogeneity of small openings. From 200°C to 800°C, higher temperature makes different minerals swell unequally, and more damage is produced by residual stress during natural cooling process, so the homogeneity becomes stronger. When the temperature exceeds 800°C, rock minerals change phases [25], and the mechanical property turns to plasticity from brittleness, and the homogeneity gets stronger and stronger. That is why the maximum D value changes that way in Figure 11 with increasing heat temperature.To find when the maximum and minimumD values reach is significative for predicting rock deformation and failure. The stress levels corresponding to maximum and minimum D values under different temperatures are shown in Figure 12. The two curves show that they all follow cubic function, and 200°C and 800°C are also two thresholds. The corresponding stress level increases, decreases, and then increases again in the interval of 25–200°C, 200–800°C, and 800–1200°C, respectively.Figure 12
The stress levels corresponding to maximum and minimumD values at different temperatures.According to the above analysis, the higher corresponding stress level indicates that the rock material is more homogeneous and brittle. On one hand, the first period of chaos lasts long for rock material with better homogeneity because the lack of inherent defect results in randomness, so the stress level corresponding to the maximumD value is higher. For example, the corresponding stress level is 40% for 200°C but 20% for 800°C. On the other hand, the stress applied on the more homogeneous rock sample would be higher when the D value reaches the minimum; with the same reason, it is difficult to generate dominant crack because of less inherent defect. For example, the corresponding stress level is 70% for 200°C, while it is 50% for 800°C.


## 4.5. CFDD Values near Rock Failure under Different Heat Temperature

Analysis of theD value near peak strength is helpful to obtain the effect law of temperature on rock failure characteristics. We selected two stress levels, 90% and 100%, to discuss the failure characteristics under different temperature, and the D curves are shown in Figure 13.Figure 13
The effect of heat temperature on theD value at stress levels 90% and 100%.
(a)
Stress level = 90%
(b)
Stress level = 100%At the stress level 90%, theD value and heat temperature are fitted to the exponential function. The D value is around 0.3 when the heat temperature is less than 1000°C, while it increases by more than 4 times under 1200°C. This curve indicates that high temperature makes AE events chaotic near failure. Furthermore, when the stress level reaches 100%, that is, the failure point, the D value shows polynomial decline with rising temperature, which indicates the AE events in rock sample that had undergone higher temperature become more regular at failure point. Comparing the two curves, also we find that the difference value of D between stress levels 90% and 100% rises with heat temperature, and the larger difference value makes rock failure more predictable, so the D value can be used as the precursor of rock failure prediction.


## 5. Conclusions

Experiments of heated rock samples at different temperature under uniaxial compression were conducted, and the results and discussion show the following:(1)
Rock samples heated at higher temperature show more plastic. The uniaxial compressive strength (UCS) of rock samples decreases gently before 800°C, while a rapid drop shows from 800°C to 1200°C. The elastic modulus decreases gradually as the temperature rises, and the downward gradient keeps stable.
(2)
The whole AE curve can be divided into six stages in rough, which are initial quiet stage (IQS), rising stage (RS), first peak stage (FPS), continuous stage (CS), second peak stage (SPS), and last quiet stage (LQS). As the heat temperature increases, the total AE counts are more, while the peak value is less.
(3)
AE counts series of rock material under uniaxial compression after heating are fractal and have self-similarity very well. With the increase of external loading, the AE CFD increases fast to a peak at first and then decreases to a bottom and, after that, increases again but within a narrow range.
(4)
200°C and 800°C are two thresholds. As the heat temperature rises, the maximumD value and the corresponding stress level both increase from 25°C to 200°C and decrease from 200°C to 800°C and then increase again from 800°C to 1200°C.
(5)
TheD value increases exponentially with rising heat temperature near rock failure and shows polynomial decline at the failure point. The D value can be used as the precursor of rock failure prediction.



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*Source: 6309856-2016-03-21.xml*