Effect of rock mechanical properties on electromagnetic radiation mechanism of rock fracturing

Peng Lin, Pengcheng Wei, Cheng Wang, Shengzu Kang, Xin Wang

Department of Hydraulic Engineering, Tsinghua University, Beijing,100084, China

Keywords:Electromagnetic radiation (EMR)mechanism Rock mecahnical properties Rock fracturing Cracking morphology EMR waveform

ABSTRACT The influence of rock mechanical properties on the electromagnetic radiation (EMR) mechanism of rock fracturing is an important research topic in solid mechanics and earthquake prediction. In this study, an EMR model of rock fracturing considering the fracture factor, elastic modulus, Poisson’s ratio, radiation distance and crack length is derived based on the Hertz oscillator array assumption. An experimental system, including an electromagnetic shielding module, an EMR signal induction and transmission module, a signal recording module and a loading module, is developed to understand the EMR characteristics of four different rocks. The validity of the EMR theoretical model is verified and the relationships between the rock cracking morphology and the EMR waveform,amplitude and frequency are revealed. It is found that rock mechanical properties have obvious influences on the EMR waveform,amplitude and frequency during rock fracturing.This study provides a better understanding on the EMR mechanism of rock fracturing and can help to improve the accuracy of rock disaster prediction based on EMR.

1. Introduction

The electromagnetic radiation (EMR) phenomenon was first found by Stepanov during the fracturing process of potassium chloride crystals in about 1933 (Urusovskaja,1969). Subsequently,many more scholars discovered the EMR in various rocks (Brady and Rowell, 1986; Frid, 1997; St-laurent et al., 2006). With the development of modern electromagnetic measuring techniques,more accurate data can be collected when rockbursts and earthquakes occur. Therefore, electromagnetic abnormity has been playing an important role in the monitoring and prediction of earthquakes, high slope failure, surface subsidence, coal mine dynamic disasters, and other fields (Frid, 2001; He et al., 2012).However,due to the limitation in the research progress on the EMR mechanism of rock fracturing, the accuracy and criteria of EMR prediction need to be improved. Natural rocks generally have diverse compositions,a large number of joints and cracks,obvious texture and mechanical anisotropy. The basic theory and application of rock EMR are still not clear yet.Over the last few decades,a number of researches have been conducted on EMR induced by rock fracturing, mainly using theoretical, numerical and experimental methods.

Numerous field and laboratory experiments have investigated different aspects of EMR induced by fracturing for various types of rocks or rock-like materials, including granite (Wang, 2014; Lin et al., 2013; Lin et al., 2015), coal-rock (Wang and Zhao, 2013),gneiss,dolomite(Qian et al.,1996),quartz(Martinelli et al.,2020a),concrete, etc. The EMR in concrete and rock specimens was found to be associated with both the formation of microcracks and the propagation of acoustic waves through the material (Lacidogna et al., 2011). The effects of specimen size, rock strength (Qian et al., 1996), loading mode, acoustic emission (Potirakis and Mastrogiannis, 2017), quartz content (Brady and Rowell, 1986),stress-strain condition(Baddari et al.,2015)and moisture content(Enomoto and Hashimoto, 1990) were investigated. Frid et al.(2000) correlated EMR with crack dimensions in micro-scales,and coupled the EMR process with the atomic-scale phenomena.Some important patterns of EMR pulse parameters and the Poisson’s ratio were observed. Yavorovich et al. (2016) presented an experiment on acoustic excitation of EMR signals in skarn, sandstone, and magnetite ore specimens, and observed that the EMR signal amplitude decreased with increasing ultimate strength. Qiu et al. (2017) quantitatively depicted the relationship between the disturbance stress of coal mine roadways and the EMR,and further evaluated the stress distribution in rock mass under uniaxial compression. Fukui et al. (2005) found both rock type and test parameters affected the EMR generation in rocks under loading.However,due to the differences in test conditions,specimen sizes,test methods and procedures, it is difficult to compare the experimental results obtained by various studies. At present, there is a lack of research on the EMR mechanisms for fracturing of different lithological rocks.

Theoretical research focuses on explaining the EMR mechanism of rock from two aspects,mostly based on the experimental results.One is to explain the generation of electric charge during rock fracturing, and the other is to explain how the electric charge generates electromagnetic waves.The generation of electric charge is usually explained by the quantum chemical model, the piezoelectric effect (Nitsan, 1977), the plasma effect (St-laurent et al.,2006) or chemical bond breaking. Huang and Kuo (1997) proposed a unified method based on the inclusion formulation to determine the magnetic, electric, and elastic fields in a composite with piezoelectric and piezomagnetic phases. Rabinovitch et al.(1999) considered that EMR was generated by the rupture of interatomic bonds along opposite sides of elongated micro-defects.Considering the piezoelectric,piezomagnetic,and magnetoelectric effects, the expressions for stresses, electric displacements, and magnetic inductions in the vicinity of a crack tip were derived(Wang and Mai,2004). By summarizing and analyzing the electric field effects,the magnetoelectroelastic materials involving a cavity or a crack were investigated (Liu et al., 2001). The electronic variable speed movement, oscillation of the fracture surface charge(Rabinovitch et al., 2007), bremsstrahlung and electric dipole radiation can explain how the electric charge generates electromagnetic waves.Liang et al.(2002)studied the magnetoelastic coupling effect in an infinite soft ferromagnetic material with a crack.However, theoretical analyses are often based on certain assumptions, limited in describing the EMR phenomenon, and lack of a unified and systematic theory on EMR induced by rock fracturing.

Compared with field or laboratory tests, numerical analysis can eliminate the interference of EMR from the test equipment.EMR from coal-rock in uniaxial compression was simulated by an established stress-radiation coupling model and it was found that the EMR is positively correlated with the loading stress on the coal (Zhao et al., 2010). Combining the theoretical model of EMR with the rock failure process analysis software (RFPA), the deformation and failure process, the stress distribution, and the EMR characteristics of coal rock in the failure process were numerically simulated (Wang and Zhao, 2013). However, the models were simplified in the above simulations, which cannot explain the experimental results very well. Some numerical models were established to simulate the EMR in various materials, such as fiber (Xie et al., 2017), antenna and thermochromic film (Hu and Yu, 2020). However, due to the lack of unified coupling equations describing rock fracturing and EMR, numerical simulation can hardly accurately describe the EMR induced by rock fracturing.

However, the effects of rock mechanical properties on the EMR mechanism during rock fracturing are still not clear in the current theoretical models and experiments. In the previous experiments,no comparison of EMR between different rock types was made,and some electromagnetic interference existed in the loading system.In this study, an EMR model for rock fracturing and an experiment system are developed.The relationships between the rock cracking morphology and the EMR waveform,amplitude,and frequency are studied. The results provide a better understanding on the EMR mechanism of rock fracturing and improve the accuracy of rock disaster prediction based on EMR.

2. EMR model of rock fracturing

The EMR of rock fracturing is mostly impulsive. Therefore, the electromagnetic waves of EMR may not originate from a continuous current,but more likely from a time-varying electric field generated by electric dipoles.The generation of an electric dipole is related to the polarization of crack tip under high stress.That is,the atoms are deformed under pressure, generating an equal number of positive and negative bound charges within a small distance. When rock crack propagates, oscillating electric dipoles (also called the Hertz oscillator array) are formed along the crack surface, the distance between the positive and negative charges changes all the time(Fig.1), and the transient electric dipoles generate pulsed EMR.

Suppose that a pair of positive and negative charges are located on the z-axis of a rectangular coordinate system in space and are symmetrically located on both sides of the origin (Fig. 1). In the electromagnetic field formed by the Hertz oscillator, the electric field intensity E and the magnetic field intensity H can be expressed by Eqs. (1) and (2), respectively(Lin et al., 2019).

where θ is the angle between the radial line from the origin to the point and the z-axis;l is the distance of two opposite charges; r is the radial distance from point P to the origin; ω is the radian frequency;ε is the permittivity of free space;I0is the peak amplitude of the elementary current;β is the phase constant;ar,aθ and aφ are directed in the increasing r,θ，and φ directions,respectively;and φ is the angle between the line from the origin to the projection of the point onto the xy-plane and the x-axis.

E and H can be resolved to two terms by location, i.e. far- and near-field terms.The amplitude of the far-field term 1/r is βr times the term 1/r2. Because the low-frequency electromagnetic wave attenuates less during propagation and has a longer wavelength,it can propagate through a longer distance and can be monitored more easily(Qiu et al.,2020).When βr ?1 or r ?λ/（2π）(where λ is the wavelength),the far-field term can be ignored,only the nearfield term (1/r2or 1/r3) is considered. Therefore, the near-field electric field intensity Enand the near-field magnetic field intensity Hncan be written as Eqs.(3)and(4),respectively.It can be seen that Enand Hnare inversely proportional to r3and r2,respectively. However, during the actual rock mass fracturing process,such as earthquakes,higher-frequency band may appear in an electromagnetic wave. The source is at a long distance away from the receiving instrument(the source depth is generally more than 10 km) and Eqs. (1) and (2) cannot be simplified.

Suppose that the linear density of charges accumulated during crack propagation can be defined as η（y，t）, and that the crack extends at position y. The average dipole of Hertz oscillator is l（y，t）.

Fig.1. Distribution of charges during rock fracturing.

The increments of electric and magnetic field intensities due to crack propagation can be described as Eqs.(5)and(6),respectively.

By integrating Eqs. (5) and (6), the EMR of the crack with a length a at time t at point P can be obtained(Fig.1).Considering the simple case,if dη（y，t）/dt≡I0and l（y，t）≡l,the integral expressions are shown as

According to the theory of fracture mechanics(Miannay,1998),under plane strain conditions,the crack width can be calculated by

where KIand KIIare the stress intensity factors for type I and type II fractures, repectivley; a is the crack length; Emis the elastic modulus; and ν is the Poisson’s ratio.

It is approximately considered that the fracture spacing u is equal to the distance l between charges. Therefore, we have

According to Eqs. (10) and (11), the EMR intensity of rock fracturing is related to the rock fracturing factorelastic modulus(Em),Poisson’s ratio(ν),radiation distance(r),and crack length(a).The crack length and position can be calculated by measuring the EMR signal in rock.

3. Experimental system and procedures

3.1. EMR test system

The EMR tests are carried out by a self-designed EMR test system, which consists of an electromagnetic shielding module, an EMR signal induction and transmission module,a signal recording module, and a loading module, as illustrated in Fig. 2.

The electromagnetic shielding box is a closed structure made of high-quality A3 cold-rolled steel plate for cold bending. The steel shielding box is fully welded and processed with the electromagnetic leakage suppression technology such as shielding, isolation,and grounding.It can provide an electromagnetic shielded zone as large as 1.5 m×0.8 m×0.8 m(length×width×height)(Fig.2a).The whole box is sprayed with electrostatic powder to improve the corrosion resistance. The important parts of the shielding box are made by electroplated pure copper to ensure its electromagnetic shielding performance. At a frequency of 14 kHz, the shielding effectiveness is 40 dB,that is,the signal amplitude in the box is 0.01 times that outside the box. A camera is installed on the top of the electromagnetic shielding box to observe the loading and fracture process of the specimen inside the box. An analog signal terminal and a direct current (DC) power terminal are installed on the box bottom. The experiments are carried out in the night to avoid interference as the noises from appliances are complex. The EMR signal induction and transmission system consists of a coil antenna(probe) and a signal conditioning circuit. The coil antenna has 20 turns and a cross-sectional area of 1260 mm2, with a signal measurement range from 100 Hz to 1 MHz.

Fig. 2. The self-designed EMR test system: (a) Schematic diagram; and (b) Test equipment. 1 - electromagnetic shielding box; 2 - direct current (DC) light emitting diode(LED)light;3-camera;4-electromagnetic signal coil;5-electromagnetic signal amplifier; 6 - battery; 7 - acoustic emission signal sensor; 8 - preamplifier; 9 - power supply; 10 - DC power supply; 11 - experiment table; 12 - specimen; 13 - expansion agent HSCA; 14 - computer; 15 - standard indoor power supply; 16 - data lines.

The signal-conditioning circuit amplifies the voltage signal of the coil antenna and transmits it to the signal acquisition card. At the same time, it prevents electromagnetic signals from being interfered by external signals. The signal-conditioning circuit is composed of an amplifier circuit, a filter circuit, a optocoupler isolation circuit and a power supply. The amplifier circuit consists of two stages of in-phase active amplifier circuits connected in series.The gain of the first stage is 40 times,and that of the second stage is adjustable from 4 times to 20 times through a varistor.The cut-off frequencies of the filter circuit are 1 kHz and 1 MHz.

The signal aquisition system mainly includes a signal acquisition card and a corresponding waveform observation and recording program.The maximum sampling frequency of the acquisition card is 20 MHz. The EMR signal acquisition and recording program is compiled in the LabVIEW software, which mainly includes three parts:a data acquisition module,a data storage module,and a data playback module.Non-triggered continuous acquisition is used for experiment preparation and debugging, to check whether the line and the background noise meet the requirements. Pre-buffer triggered acquisition is adopted to record the EMR signal,including the time-history curve, amplitude, frequency, etc. The trigger threshold and sampling frequency of the pre-buffered trigger mode are 0.2 V and 10 MHz,respectively(trigger refers to the start signal needed to start recording useful data).The signal acquisition range is between -10 V and 10 V and the number of consecutive exceeding thresholds is set to 2.

To ensure the comparability of the experimental data, the position, eccentric direction, and coil antenna orientation were kept the same for all the tests. Before the experiment, the background noise was observed using the non-triggered acquisition mode to ensure that it did not exceed 0.2 V within 2 min.The cracking agent was first mixed with a water-cement ratio of 1:3 and filled into the pre-drilled hole. The hole was then covered with steel gasket and the jack was adjusted to touch the top plate of the fixture, so that the cracking agent can expand and press against the hole wall.The door of the electromagnetic shielding box was then closed and the signal recording program was adjusted to the pre-buffer trigger mode before testing.After about 1-1.5 h,the rock fractured with a loud noise. When the signal recording procedure was no longer triggered within 1 min, the test can be considered completed. All the test data were collected and recorded. The experiment table was then cleaned for the next set of experiments.

3.2. Mechanical properties of rock specimens

Four types of rocks,granite,basalt, sandstone and marble were employed in this study. The rock texture was observed under the polarizing microscope (Fig. 3). The granite, basalt and sandstone specimens are uniform and dense, showing coarse-grained, finegrained,and sand-like structures,respectively.Marble has high clay mineral content and uneven texture,which is commonly observed in engineering rocks. According to the formation process, granite and basalt belong to igneous rock, sandstone is sedimentary rock,and marble is metamorphic rock.

For each type of rock,three specimens,i.e.granite G-1,G-2,and G-3, basalt B-1, B-2 and B-3, marble M-1, M-2 and M-3, and sandstone S-1, S-2 and S-3, were prepared for uniaxial compression,Brazil splitting and three-point bending tests.The basic mechanical parameters of the four rock types are illustrated in Table 1.

3.3. EMR test specimens

An eccentric hole was pre-drilled in the specimen for rock cracking at the expected position and facilitating the observation of rock fracturing characteristics. The drilling position and the rock specimens are illustrated in Fig.4.The hole was drilled by the water drill method with R = 30 mm, depth = 135 mm, l1= 30 mm,l2= 60 mm, and l3= l4= 45 mm (Fig. 4a).

Fig. 3. Rock specimens and microscopic images.

4. Cracking morphology of different rocks

4.1. Cracking mechanism with expansion pressure

Several loading methods can induce rock failure. The conventional loading methods, such as uniaxial loading, biaxial loading and triaxial loading, generate EMR due to the electric driving system and the iron head.The EMR signals are received by the sensors and submerge the signals induced by rock fracturing. Therefore,self-expansion with efficient cracking agent was adopted to fracture the rock specimen.

In this study, the HSCA-III expansive mortar produced by Changsha Liying Building Material Co.,Ltd.,China,was used as the cracking agent to induce fracture. When mixed with water in a certain proportion,it releases a great amount of heat and produces a high expanding pressure, which can be expressed as follows:CaO+H2O→Ca（OH）2+64.9 kJ/mol

The expanding pressure can reach 30-40 MPa in 8 h, 70-100 MPa in 1 d,and 100-200 MPa in 2 d(Wang,2014).The initial response is slow and gentle when loading, which makes it easy to control the safety and operability of the experiment.

A hole was drilled in the specimen for installing the HSCA cracking agent.The diameter and depth of the hole should be large enough so that a sufficient amount of cracking agent can be installed for rock fracturing. Under the expansion pressure, a damage zone appeared around the hole at first. With increasing expansion pressure, the damage zone gradually expanded. When the damage zone reached the free surface,the microcracks rapidly expanded and penetrated, leading to the formation of the main crack. Subsequently, the rock specimen fractured, during which a strong electromagnetic pulse was emitted.

4.2. Cracking morphology of rock specimens

All the 12 rock specimens fractured in the tests. The cracking morphology was different for rocks with different lithologies.Figs.5-8 show the photos of fractured specimens and the sketches of cracks on the top and four side surfaces.The highlighted zone is the compression-shear zone. The blue line (thin line) on the side surface indicates tensile cracks, and the red line (thick line) indicates shear cracks. Cracks A, B, C, and D are the tensile cracks penetrating through the specimens.

As shown in Fig.5,the cracking morphology of the specimens G-1 and G-3 is similar.Two cracks A and C extend from the borehole to the surface F1, and penetrate the specimen along the height direction.The cracks B and D extend from the borehole to the surfaces F2 and F4 and penetrate the specimen along the height direction.A large number of non-penetrating shear cracks are fomed in the compression-shear zone between the cracks A and B.It can be seen from Fig.5 that only two cracks A and B penetrate the specimen G-2.A compression-shear zone is formed between the cracks A and B,however, less number of shear cracks and lower degree of fracturing are observed, as compared to the specimens G-1 and G-3.

As shown in Fig.6,two penetrating cracks A and B are formed in the specimen B-1, between which is the compression-shear zone with many shear cracks.No cracks are observed on the surfaces F3 and F4.Two penetrating cracks are identified in the specimen B-2,with the crack A extending to the surface F1 and the crack B to surface F3. No compression-shear zone is formed in the specimen B-2 and no cracks are observed on the surfaces F2 and F4. The cracking morphology of B-3 is similar to that of the granite specimens G-1 and G-3,i.e.three penetrating cracks extend to the surfaces F1, F2, and F4 and a compression-shear zone is formed between the cracks A and B.

Three penetrating cracks A,B and C are formed in the specimen S-1(Fig.7),which extend to the surfaces F1,F3 and F4,respectively,and no other cracks are observed.The cracking morphology of the specimen S-2 is relatively symmetrical.Four penetrating cracks are identified,with the cracks A and C extending to the surface F1,the crack B to the surface F4, and the crack D to the surface F2. Three penetrating cracks are formed in the specimen S-3, two of which extend to the surface F1 and one to the surface F2. A few shear cracks can also be observed in the specimen S-3.

Fig.8 shows that the cracking morphology of the specimen M-1 is similar to that of the specimen S-1,with three penetrating cracks.Two penetrating cracks A and B are observed in the specimen M-2,which extend to the surfaces F1 and F4, respectively. A compression-shear zone with many shear cracks is formed between the cracks A and B,which is similar to the basalt specimen B-1. Four penetrating cracks and a compression-shear fracture zone with many shear cracks between the cracks A and B can be identified. In terms of the number and location of cracks, the cracking morphology of the specimen M-3 is similar to that of the granite specimens G-1 and G-3 and the basalt specimen B-3.

The tensile and shear cracks generated during rock fracturing are counted. The crack with length exceeding 1/10 (1.5 cm) of the specimen size is counted as one crack. The number of cracks in different rock specimens is shown in Table 2.Due to different rock properties, the main cracking modes of granite, basalt, sandstone,and marble specimens are tensile-shear, compression-shear, tensile,and compression-shear,respectively.The number of cracks can be classified according to whether a compression-shear fracture zone is formed.

Table 1 Basic mechanical parameters of different rocks.

4.3. Analysis of cracking mode

To discuss the influence of loading condition on the cracking mode, the cracking modes of granite specimens under expansion pressure and the uniaxial compression test by Lin et al.(2002,2015)are compared in Fig.9.The ratios of the hole radius to the specimen width are 0.2 and 0.1,respectively,for the two tests.It can be seen that the cracking morphology is the same under the two loading modes,that is,both tensile and shear cracks occur,of which tensile cracks are dominant. Because rock is a kind of solid containing many micro-defects,the cracking process can be divided into three stages under different loading conditions: initiation, propogation and penetration. However, the loading mode affects the crack distribution in granite. Under uniaxial compression, the cracks are distributed in the tensile stress areas and approximately parallel to the loading direction(Lin,2002,2015;Tang et al.,2005).However,in case of expansion pressure, the tensile cracks propagate along the radial direction to the nearest boundary.Therefore,the loading modes and boundary conditions affect the distribution and orientation of the cracks, while the cracking morphology is mainly affected by the rock properties.

Fig. 4. (a) The geometry and (b) photos of specimens.

Fig. 5. Distribution of cracks in granite specimens.

Fig. 6. Distribution of cracks in basalt specimens.

As illustrated in Table 3,the cracks caused by expansion pressure include tensile cracks,shear cracks,and mixed cracks(tensile-shear cracks or compression-shear cracks). Similar to the crack type and propagation in granite under uniaxial compression(Lin et al.,2015),compressive and compression-shear mixed cracks are also observed in this study. This is related to the rock lithology and specimens used in the tests. Under expansion pressure, mainly radial tensile, compressive, shear, tensile-shear and compressionshear cracks are formed in the four types of rocks. According to their occurrence sequence, the cracks can also be divided into primary cracks and secondary cracks.

Table 2 Statistics on the number of cracks in different rock specimens under expansion pressure.

These cracks cause five coalescence modes between cracks(Lin et al.,2015):(i)the shear mode due to shear stress;(ii) the tensile mode due to tensile stress; (iii) the mixed-mode due to tensile or compressive and shear stresses;(iv)the compression mode due to the nucleation of many small tensile cracks in the compressive stress concentration zone,which also results from the action of the local stress field; and (v) the traction mode due to the combined effect of the stress field at the crack tip and the stress field in the nearby area.

5. Effect of rock properties on the EMR mechanism

5.1. Analysis of EMR waveforms of different rocks

Considering the signal acquisition interference and the comparability of the results, typical granite (G-1), basalt(B-1), sandstone(S-3) and marble (M-3) specimens are selected for analysis of the EMR waveform,amplitude and frequency.The correlation between rock cracking modes of different rocks and EMR is analyzed based on the results presented in Section 4.

The EMR was first triggered at time 88′41′′, 84′47′′, 62′42′′and 55′00′′in the specimens G-1, B-1, S-3 and M-3, respectively. The EMR signals were tiggerred for a total of 12,4,7,and 6 times in the four specimens during rock fracturing, respectively. The average temperature in the shielding box during the tests was around 22.3°C.

The first electromagnetic waves triggered in the four types of rock specimens are shown in Fig.10. It can be seen that the EMR induced in the four rock spesimens contains two characteristic waveforms, i.e. low-frequency small-amplitude continuous wave,and varying-frequency large-amplitude pulse wave. The continuous waves in the four rock specimens are all basically simple harmonic.Several cluster-like EMR pulse waveforms are generated during fracturing of basalt and granite specimens.A small number of pulse waves are also induced in the marble specimen, but the dominant waves are continuous waves. Fig. 11 shows the local simple harmonic fitting curve of electromagnetic waves in granite and basalt, which verifies the assumption in Eq. (3).

Fig. 7. Distribution of cracks in sandstone specimens.

Fig.10. The first EMR signals triggered in four rock specimens: (a) Granite (G-1); (b) Marble (M-3); (c) Basalt (B-1); and (d) Sandstone (S-3).

Compared with the uniaxial compression test results of Wei et al. (2020), it is found that the EMR signal of the same kind of rock is basically the same.It can be concluded that the rock type has a greater impact on the EMR waveform than the loading mode does. The maximum amplitudes are not completely consistent,which may be related to the rock origin, specimen shape and size,and loading mode.The pulse waves in this study are related to the occurrence of multiple penetrating cracks under expansion pressure.

The relationship between the number of shear cracks, tensile cracks, and electromagnetic wave pulses is shown in Fig. 12. The number of pulses greater than 1 V in the granite and sandstone specimens matches the number of tensile cracks. The number of pulses greater than 1 V in the marble and basalt specimens matches the number of shear cracks.It can be seen in Section 4 that tensileshear failure is dominant in the granite specimens, tensile failure is dominant in the sandstone specimens,and shear failure is dominant in both marble and basalt specimens. In granite, sandstone, and other rocks with low tensile strength and large mineral particles,EMR is mainly caused by tensile cracking. For marble, basalt and other rocks with relatively high tensile strength and small mineral particles, EMR is mainly induced by shear cracking. Therefore, the EMR mechanism of different types of rocks is different.

Fig. 8. Distribution of cracks in marble specimens.

Fig. 9. Comparison of cracking morphology of granite specimens under different loading modes: (a) Expansion pressure; and (b) Uniaxial compression.

Fig.12. Relationship between rock cracking morphology and electromagnetic pulse.

5.2. Analysis of EMR amplitude for different rocks

Taking the first trigger as the starting time, the maximum amplitude of the EMR signal at each trigger is analyzed statistically.Fig.13 shows the variation of EMR signal amplitude with the trigger time. The EMR amplitude of granite and sandstone decreases monotonously and relatively stably,while that of basalt and marble is relatively volatile.Based on the analysis of cracking morphology in Section 4.2, more tensile cracks are formed in granite and sandstone, while more shear cracks are formed in basalt and marble. This indicates that the EMR amplitude induced by tensile cracking is relatively stable, while that by shear cracking is relatively fluctuating.

As shown in Table 4,no continuous wave in the EMR is recorded in the specimens S-1, S-3 and M-1. Tensile fracture is dominant in the specimens S-1 and S-3 and shear fracture is dominant in the specimen M-1. Compression-shear fracture is the main fracture mode and the continuous wave amplitude of EMR is about 0.1-0.2 V in the other six specimens.Taking the average value according to lithology, granite has the largest continuous wave amplitude at 0.175 V, followed by basalt at 0.15 V, and marble has the lowest amplitude at 0.125 V.

Table 3 Cracking modes of different rocks.

The pulse wave amplitude is not strongly correlated with the continuous wave amplitude. For example, no continuous wave is detected in the specimen M-1, and the pulse wave amplitude reaches 1.797 V.The continuous wave amplitude of the specimen G-1 is 0.2 V, but the pulse wave amplitude is only 1.287 V. The continuous wave amplitude of the specimen M-3 is 0.1 V,which is smaller than that of the specimen G-1, but the pulse wave amplitude of M-3 is larger than that of G-1. It indicates that the continuous wave and pulse wave generation mechanisms of EMR are different in rocks. In this study, continuous waves are induced only in the rock specimens fractured by compression and shear,indicating that the fracture factorhas a greater influence on the continuous wave. Hence, the applicability of Eq. (10) is verified.The continuous-wave amplitude of basalt is slightly lower than that of granite,but the strength of basalt is higher than that of granite,indicating that rock minerals also have a certain influence.Therefore,the amplitude of low-frequency and continuous wave is related to the cracking mode, rock strength, and mineral composition. For the pulse wave caused by piercing the crystal, its amplitude is not strongly correlated with the macroscopic strength of the rock, instead, it may be related to microscopic defects and local stress concentration.

5.3. Analysis of EMR frequency for different rocks

The time-history curves of the induced current magnitude are recorded in this study, which can be converted into a frequency domain function through Fourier transform.The Fourier transform of the continuous-time signal x（t） is defined as

where X（ω） is the Fourier transform of x（t）, that is, the frequencyspectrum of x（t）; and

For a discrete signal x（n） of length N, the Fourier transform is defined as

The signal collected in this study is discrete, and the sampling rate is 107s-1, corrspondin to a sampling time interval of 10-7s.The power spectrum analysis can be performed on the collected signals with the Labview software by

The power spectrum is the power spectral density function,and the integral value of the whole curve is 1. The greater the power corresponding to a certain frequency,the greater the amplitude of EMR at that frequency. The frequency corresponding to the point with the highest power is the main frequency.As shown in Fig.14,the main frequencies of granite, marble, basalt and sandstone are 8.8 kHz, 5.4 kHz, 9 kHz, and 8-600 kHz, respectively.

The maximum power in the power spectrum of the specimens S-1, S-3, and M-1 is relatively small and there is no obvious dominant frequency. Combined with the analysis of the cracking morphology in Section 4,it can be seen that the number of cracks in the specimens S-1, S-3, and M-1 is smaller, and there is no compression-shear fracture zone (Table 5). Obvious compressionshear fracture zones and dominant frequencies are observed in the other specimens. It shows that the EMR generated by compression-shear fracture has a strong dominant frequency,while the EMR induced by non-compression-shear fracture has no obvious dominant frequency. It can be seen that the spectrum of sandstone is more special (no dominant frequency, minimum power), as compared with the other three types of rocks. The sandstone used in this study has a high quartz content(85%).The special spectrum may be related to the decrease of friction coefficient during loading and the occurrence of shear failure(Martinelli et al.,2020a, b).

Table 4 EMR amplitude and cracking morphology of different rock specimens.

Table 5 Relationship between EMR frequency and fracture modes.

6. Conclusions

In this study, rock fracturing and EMR mechanism have been comprehensively discussed in relation to the fracture behavior of granite, basalt, sandstone and marble specimens under expansion pressure.Based on a self-designed test system,a theoretical model was established and the relationship between rock mechanical properties and EMR mechanism was analyzed. The following conclusions are drawn:

(1) The EMR model considering rock properties, radiation distance and crack length shows that the continuous electromagnetic waves induced by rock fracturing are simply harmonic. The theoretical derivation is verified by the experimental results. With the theoretical model, the crack length and position can be estimated by measuring the EMR signal.Under expansion pressure,radial tensile,compressive,shear, tensile-shear and compression-shear cracks are formed in the four types of rock. The loading mode affects the crack distribution and orientation, but the crack morphology is mainly affected by the rock mechanical properties.

Fig.11. Local simple harmonic fitting curves of electromagnetic waves in (a) granite and (b) basalt.

Fig.13. Variation of EMR amplitude with time for different rocks.

(2) For granite, sandstone and other rocks with low tensile strength and large mineral particles,EMR is mainly caused by tensile cracking. For marble, basalt and other rocks with relatively high tensile strength and small mineral particles,EMR is mainly induced by shear cracking. The rock EMR contains two characteristic waveforms, i.e. low-frequency small-amplitude continuous waves and varying-frequency large-amplitude pulse waves. The amplitude of lowfrequency continuous wave is related to the cracking mode,rock strength, and mineral composition. For the pulse wave caused by transcrystalline fracture, its amplitude is not strongly correlated with the macroscopic strength of the rock, and its mechanism may be related to microscopic defects and local stress concentration.The dominant frequency generated by compression-shear fracturing is between 1 kHz and 10 kHz, while the EMR induced in rock without compression-shear fracture has no obvious dominant frequency, and the dominant frequency range is 0-600 kHz.The mineral composition, mechanical properties, cracking morphology,etc.have a great influence on the EMR induced by rock fracturing.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Fig.14. The signal spectra at the first trigger for four rocks: (a) Granite (G-1); (b) Marble (M-3); (c) Basalt (B-1); and (d) Sandstone (S-3).

Acknowledgments

This research work was supported by the National Natural Science Foundation of China(Grant Nos.51979146 and 11272178).The authors are also very grateful to the editor and various anonymous reviewers for their comments and suggestions,which have greatly helped to improve the paper.

List of symbols

E Electric field intensity

H Magnetic field intensity

EnNear-field electric field intensity

HnNear-field magnetic field intensity

θ Angle between the radial line from the origin to the point and the z-axis

l Distance of two opposite charges

r Radial distance from the point p to the origin

φ Angle between the line from the origin to the projection of the point onto the xy-plane and the x-axis

ω Radian frequency

ε Permittivity of free space

I0Peak amplitude of the elementary current

β Phase constant

arDirected in the increasing r direction

aθ Directed in the increasing θ direction

aφDirected in the increasing φ direction

η（y，t） Linear density of charges accumulated during crack propagation

l（y，t） Average dipole of Hertz oscillator

t Time

KIStress intensity factor (type I fracture)

KIIStress intensity factor (type II fracture)

EmElastic modulus

ν Poisson’s ratio

u Fracture spacing

x（t） Continuous-time signal

X（ω） Fourier transform of x（t）

x（n） Discrete signal

a Crack length