J. Cent. South Univ. Technol. (2008) 15: 344-349

DOI: 10.1007/s11771-008-0065-0

Piezoelectric responses of brittle rock mass containing quartz to static stress and exploding stress wave respectively

WAN Guo-xiang(万国香), LI Xi-bing(李夕兵), HONG Liang(洪 亮)

(School of Resources and Safety Engineering, Central South University, Changsha 410083, China)

Abstract: The electromagnetic emission(EME) induced from the rock containing piezoelectric materials was investigated under both static stress and exploding stress wave in the view of piezoelectric effect. The results show that the intensity of the EME induced from the rock under static stress increases with increasing stress level and loading rate; the relationship between the amplitude of the EME from the rock under different modes of stress wave and elastic parameters and propagation distance was presented. The intensity of the EME relates not only to the strength and elastic moduli of rock masses, but also to the initial damage of the rock. The intensity of EME induced by stress wave reaches the highest at the explosion-center and attenuates with the propagation distance. The intensity of EME increases with increasing the elastic modulus and decreases with increasing initial damage. The results are in good agreement with the experimental results.

Key words: piezoelectric responses; electromagnetic emission; static stress; exploding stress wave

1 Introduction

The interest has been aroused for people to study pre-earthquake electromagnetic perturbations. A number of experimental and observational efforts have been made to detect those signals and describe their characteristics^[1-6]. To explain observed electromagnetic (EM) phenomena associated with earthquakes, various effects including the movement of electric charge^[1-2], the electron emission at the crack tip^[3-4], the movement of charged crack^[5-6], the atomic (or ionic) oscillation on the crack surface^[7], the semiconductor effect^[8] and the piezoelectric effect^[9-10], have been suggested as possible mechanisms of EM signal generation^[11-16]. BRADY and ROWELL^[11] suggested that the contribution of piezoelectric effect as a non-rupture mechanism is not great to the light emission, but some frequency electromagnetic wave induced from the rock containing piezoelectric crystals is stronger than that from the rock without piezoelectric crystals. It cannot be denied that the intensity and frequency of the electromagnetic emission(EME) caused by certain rock fracture correlate to the fact that the kind of rock itself contains piezoelectric crystals like quartz.

The experimental investigations on the EME during rock failure are abundant, but the theoretical study is not enough relatively at present. LI and GU^[17] investigated the EME induced from the rock subjected to stress wave theoretically in the view of piezoelectric effect. But EME

conducted from the rock under static stress and stress wave with attenuation has not been discussed.

    Based on the previous work, the EME induced from rock under static load and exploding stress wave is studied in this work.

2 Piezoelectric constitutive relationship

The piezoelectric constitutive relationship^[18] in the piezoelectric body is

                                (1)

                                (2)

where  S and T are the components of strain tensor and stress tensor; E and D are the components of the intensity of electric field and electric displacement tensor; s, d and ε are components of the elastic compliance, piezoelectric and dielectric polarization tensor, respectively.

In the case of non-piezoelectric, Eqn.(1) becomes: s=dT, and it is suitable for the brittle elastic materials, then the piezoelectric constitutive relationship should be acceptable for the brittle elastic materials too.

3 Piezoelectric response

Quartz as a kind of piezoelectric material occupies 66.4% on the top of earth crust^[19].

It is assumed that 6 mm crystal system is charac- teristic in the rock containing quartz and its coordinate axis is coincident with the crystal axis. Then, the piezoelectric strain matrix of the 6 mm crystal system is

             (3)

The matrix of dielectric constant is

                          (4)

3.1 EME from rock under static stress

The piezoelectric contribution of the static stress T_s existing in the media to the electric displacement is

D_z=d_z1T_s                                    (5)

Because the static stress is independent of the coordinate, its electromagnetic emission is also non-wavy and does not participate in the Maxwell equations, then

                        (6)

So

                                (7)

Eqn.(7) represents a quasi-static electric field although it is time-dependent. The intensity of the EME increases with the increase of the stress level, which is consistent with many experiment results of the rock under static loading. The results can be easily obtained according to the same method completely for other static stress in different directions, only the coefficient needs to be altered.

3.2 EME from rock under planar one-dimensional exploding stress wave

The EME from the rock under planar stress wave has been studied in previous literatures, but the amplitude of the stress wave considered previously is a fixed value. As is well known, the amplitude of the stress wave will attenuate when it propagates in the rock.

In order to measure the attenuation of planar one-dimensional explosive stress wave, attenuation ratio (η) defined as the attenuation when the amplitude of an unit stress gets across an unit length is introduced in the form as follows:

                            (8)

The result can be obtained as Eqn.(9) by integrating:

                             (9)

    According to Refs.[17, 20], there exists

                    (10)

where  T_e is the stress of the stress wave; ρc_p and ρ_eD are the wave impedances of the rock and explosives,  respectively; D₀ is the initial damage of the rock; k is about equal to 3 for the common solid explosives. The second formula of Eqn.(10) can be obtained based on the definition of Eqn.(8) by many experiments of rock specimens with different initial damage^[21].

3.2.1 EME induced from rock under P-wave

On the assumption there is P-wave propagating and polarizing along direction x, there exists

               (11)

where  ω is the frequency of the stress wave.

According to Eqns.(1) and (2), the piezoelectric contribution of the stress field to the electric displacement is obtained:

                                 (12)

And there is a corresponding elastic contribution () of stress field to the strain field at the same time. By substituting D_z into the electromagnetic equation, electric and magnetic field can be induced. And this electric field contributes to the strain field S₁ in return. It is the sum of the elastic and piezoelectric contributions to S₁ that must be used to calculate the dispersion relationship. Therefore, the electromagnetic equations must be solved.

The equation can be obtained according to Maxwell equations

                (13)

where  μ₀ is the magnetic conductivity in the vacuum.

Substituting Eqn.(12) into Eqn.(13) gets

               (14)

where  is called the “source item” and defined as

 (15)

E_z and T₁ have the same wave behavior (frequency ω and wave number k) as we can know from Eqn.(14). The electromagnetic field is supposed in the form as follows:

                      (16)

where  E_e is the undermined coefficient.

The equation below can be obtained by substituting Eqn.(16) into Eqn.(14)

                   (17)

The solution can be obtained by solving Eqn.(17)

(18)

where  c₁ and c₂ are constants.

Then

(19)

The fore two items of Eqn.(19) are independent of coordinate and can be ignored here. The strain field is proportional to the induced electric field. In the case of stress wave being decreased exponentially, the EME decays in the same manner. The coupling coefficient (the coefficient before the stress field) is dependent on the attenuation coefficient and the frequency, and the intensity of the EME induced from the rock containing quartz is stronger than that from the rock without quartz in some frequency range, which approves the experiment results in Ref.[21]: EME can be induced both from rock containing quartz and without quartz under stress, but the intensity of the EME from the rock containing quartz is stronger than that from the rock without quartz.

3.2.2 EME induced from rock under S-wave

On the assumption there is a z-polarized and x-propagating S-wave in rock, there exists

               (20)

A x-polarized quasi-static electric field is obtained through the coupling theory with the same method:

                                (21)

Namely

            (22)

3.3 EME from rock under spherical blast stress wave

Planar one-dimensional stress wave does not possess geometrical attenuation, while the spherical wave possesses geometrical attenuation in 1/r. Then the form of the amplitude of the spherical wave with geometrical and physical attenuation can be written as

                          (23)

where  r is the propagation distance.

3.3.1 EME induced from rock under P-wave

The result can be derived through the same inferential process under the upper planar P-wave:

               (24)

By solving the differential equation, we get

     (25)

where

and

b is a constant.

3.3.2 EME induced from rock under S-wave

The result can be obtained identically as:

                       (26)

4 Examples

Setting ε_xx=1×10^-11, d_z1=1×10^-15, the influence of the rock lithology and stress state of rock on the EME is discussed.

The intensity of EME induced from granite under static stress is shown in Fig.1, where E and T are the intensity of EME and stress, respectively. With the increase of the stress level, the intensity of EME grows; the larger the loading rate, the stronger the intensity of EME induced under the same condition.

Fig.1 Intensity of EME from granite vs stress level under static stress

The variations of the amplitude of EME from rock under blast stress wave with the propagation distance are illustrated in Table 1 and Figs.2-7. The amplitude of EME reaches the highest at the explosion-center, and decreases with the increase of the propagation distance,

Table 1 Wave impedance of some rocks

Fig.2 Amplitude of EME vs propagation distance in rock under planar P-wave with different initial damage: (a) Initial damage D₀=0, granite; (b) Initial damage D₀=0.3, granite; (c) Initial damage D₀=0.6, granite

Fig.3 Amplitude of EME vs propagation distance in rock under planar P-wave with different wave impedances: (a) Initial damage D₀=0, shale wave impedance 345.0 kg/(cm²?s);      (b) Initial damage D₀=0, quartzite wave impedance 1 457.5 kg/ (cm²?s)

Fig.4 Amplitude of EME vs propagation distance in rock under planar S-wave with different initial damage

which is in good agreement with the experimental results in Ref.[22].

The amplitude of the EME decreases with the increase of the initial damage in rock whether subjected to planar stress wave or to spherical stress wave; the larger the wave impedance, the higher the amplitude of the electromagnetic emission. Because the wave impedance is proportional to the elastic modulus and the strength of the rock, it can be deduced that the amplitude of EME rises with the increase of the elastic modulus and the strength of the rock. The conclusion from the theory is in good agreement with many experimental

Fig.5 Amplitude of EME vs propagation distance in rock under planar S-wave with different wave impedances

Fig.6 Amplitude of EME vs propagation distance in rock under spherical S-wave with different initial damage

Fig.7 Amplitude of EME vs propagation distance in rock under spherical S-wave with different wave impedances

results and provides the theoretic basis for them, meanwhile. Comparing Fig.4 with Fig.6, the amplitude of EME decays more slowly under planar stress wave than under spherical stress wave.

5 Conclusions

1) Electromagnetic emission monitoring technology is very important to predict dynamical disaster phenomena such as coal or rock outbursts. The dependence of the EME on the parameters and the loading conditions is studied in this work.

2) The intensity of EME from the rock under static stress increases with increasing the stress level and the loading rate.

3) The intensity of EME attenuates with propaga- tion distance in rock under blast stress wave and reaches the highest at the explosion-center.

4) The amplitude of the induced EME correlates to the initial damage, the elastic modulus and the strength of the rock under blast stress wave. The amplitude of EME rises with the increase of the elastic modulus and the strength and decreases with the increase of the initial damage of the rock.

5) The calculated results are quite consistent with the results of the laboratory studies and in field observation, and provide the theoretic basis for some phenomena of the EME during seism and rock fracture.

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(Edited by YANG Hua)

Foundation item: Project(50490274) supported by National Natural Science Foundation of China

Received date: 2007-10-25; Accepted date: 2008-01-10

Corresponding author: WAN Guo-xiang, Doctoral candidate; Tel: +86-13637477904; E-mail: wrm324@126.com