Method of defining rock damage variable on the basis of wave impedance

A method of determining rock damage variable from wave impedance, which is suitable for the study of dynamics, is presented. The determined variable provides a more reasonable quantitative description of the degree of rock damage. First, the Taylor model of mesoscopic damage mechanics is used to derive the relationship between the velocity of longitudinal waves and the density of rock materials. Based on the measured data of rock samples with different lithology and the same lithology, the relationship between the longitudinal wave velocity and density of rock materials is studied. It is demonstrated that using the wave impedance to define the rock damage variable has advantages over using the P‐wave velocity. Second, the relationship between the wave impedance and degree of damage to the rock material is studied using an electron microscope and ultrasonic testing technology, and the microstructure of rock samples with a wave impedance gradient and time–frequency‐domain characteristics of the ultrasonic signals are compared and analyzed. It is found that it is feasible to determine the damage degree of the rock from the wave impedance. Finally, a method of defining rock damage variable with the wave impedance is further proposed and its rationality verified. The research results show that there is a good positive correlation between the longitudinal wave velocity and density of rock materials, and there is a strong correlation between the degree of rock damage and its wave impedance. The damage variable defined by wave impedance can better reflect the damage state and damage evolution process of rock.

mass under high-temperature environment, 4,5 cold region rock mass under low-temperature environment, 6,7 and coal mine rock mass under acid-alkali mine water environment. 8,91][12] Therefore, how to quantitatively describe the damage and deterioration degree of rock materials is of great significance to the disaster prevention of engineering rock mass.
4][15][16][17][18][19][20][21] At present, the measurement criteria of damage to rock materials are mainly selected from microscopic and macroscopic viewpoints.The number, length, area, and volume of pores are selected from the microscopic viewpoint whereas the elastic modulus, elastic strain, yield stress, density, ultrasonic wave velocity, acoustic emission cumulative number, and CT number are selected from a macroscopic viewpoint.Zhang et al. 22 established the determination method of rock damage variable based on acoustic emission parameters, and compared the damage evolution characteristics of rock salt, granite and marble under uniaxial compression.The results show that rock salt is feasible as an underground storage engineering medium.Liu et al. 23 introduced the rock damage variable based on energy dissipation and established the damage constitutive model of rock under cyclic load.Lin et al. 24 adopted nuclear magnetic resonance technology, proposed a method of defining rock damage variable by porosity and effective bearing area, and studied the influence of chemical corrosion on the mechanical properties of sandstone.Liu et al. 25 used meso damage variable and macro damage variable to describe rock damage caused by microcracks and joints respectively, and established a dynamic damage constitutive model applicable to jointed rock mass based on coupled damage variable.Yang et al. 26 defined rock damage variable based on CT number and studied the change characteristics of rock damage variable with confining pressure under triaxial loading.Among the methods adopted for defining variables of rock material damage, variables have widely been defined on the basis of the ultrasonic wave velocity, but this method ignores the change in rock density.8][29][30][31] Therefore, the accuracy in measuring damage when using a variable based on the ultrasonic wave velocity is not high.][34] The above analysis suggests that in the study of rock dynamics, it is more convenient and reasonable to define damage variables and conduct related research using the rock wave impedance to quantitatively evaluate the degree of damage to the rock.Against the above background, this article investigates first the relationship between the longitudinal wave velocity and density and then the relationship between the wave impedance and degree of damage and degradation, so as to determine the feasibility and rationality of using wave impedance to determine the degree of rock damage and degradation.Furthermore, a method of using the wave impedance to define rock damage variables is proposed.

| THE PHYSICAL MEANING OF WAVE IMPEDANCE
When a plane wave propagates in a continuous medium, and its wavefront is a singular surface, there are discontinuous jumps in various motion parameters before and after the wavefront, as shown in Figure 1.However, certain limiting conditions should be met between the discontinuous jump values of various motion parameters.In the process of plane wave propagation, the compatibility conditions that should be met between the motion parameters before and after the wavefront are analyzed as follows. 32he wavefront of a plane wave propagates forward along the X axis at the wave speed c 0 , and the position of the wavefront in the material coordinate at time t is X, as shown in Figure 1.Any physical quantity on the wavefront is recorded as φ(X，t), the value of physical quantity φ in front of the wavefront is recorded as φ + , the value behind the wavefront is recorded as φ -, and the difference between the two is recorded as [φ], There are: Obviously, for the strong discontinuous wavefront with [φ] ≠ 0, [φ] represents the discontinuous sudden jump value.Here, the weakly discontinuous wavefront with [φ] = 0 is not studied, and the limiting conditions (compatibility conditions) that the motion parameters on the strongly discontinuous wavefront should meet are analyzed only from the dynamic aspect.There is a strong discontinuous wavefront, time t is located on plane AB.It propagates to plane A'B' after time dt in the continuous medium at the material wave speed c 0 .The propagation distance is dX, as shown in Figure 1.For the particles between AB and A'B', according to the conservation of momentum theorem, there are In the formula, ρ 0 and A 0 are the density and crosssectional area of the continuous medium, respectively.σ and v are the specific physical quantities of the motion parameter φ, which are the stress and particle velocity on the wavefront respectively.Simplify Equation (3) according to Equations ( 1) and (2) as follows: The above dynamic compatibility condition on the strongly discontinuous wavefront is derived from the general situation, which is universal for strongly discontinuous plane waves in continuous media with different physical properties.At the same time, it is easy to see from Equation (4) that the dynamic compatibility condition on the strongly discontinuous wavefront is closely related to the density and wave speed of the continuous medium material.The product of material density and wave velocity is often defined as the wave impedance Zr, which is By combining Equations ( 4) and ( 5), it can be obtained that It is not difficult to see the physical meaning of wave impedance from Equation ( 6): wave impedance is an inherent property of materials.It represents the disturbance force required for stress waves to propagate vertically through a unit area of material medium, causing the medium particles to generate a unit vibration velocity.Wave impedance reflects the resistance of the material medium to momentum transfer, that is, the ability of the medium to prevent waves from passing through, and is a fundamental physical quantity that characterizes the dynamic properties of the medium.In rock dynamics, the product of rock longitudinal wave velocity and rock density is defined as rock wave impedance.

| RELATIONSHIP BETWEEN THE LONGITUDINAL WAVE VELOCITY AND DENSITY OF ROCK
The longitudinal wave velocity and density of rock materials do not vary singly but rather a change in one parameter causes a simultaneous change in the other, and the two parameters are thus interrelated and inseparable.It is therefore necessary to study the relationship between the longitudinal wave velocity and density of rock in demonstrating the accuracy and advantage of using the wave impedance to define rock damage variables.

| Theoretical derivation of the relationship between the longitudinal wave velocity and density of rock
A large number of microdefects, such as micropores, microcracks, and microslip bands, are randomly distributed in rock materials.The existence of these microdefects affects the elastic modulus, shear modulus, Poisson ratio, density, and elastic wave velocity of the original rock mass medium.The existence of microcracks is an important factor affecting the propagation of sound waves through rock and rock density.Therefore, the mesoscopic damage mechanics are considered in studying the relationship between the velocity of longitudinal waves and density of fractured rock.At the same time, in the process of practical engineering detection and laboratory testing, a pulse wave is used in the rock acoustic wave test.The disturbing force generated by wave propagation in a rock medium is weak, and the interaction of microcracks is small.Therefore, the Taylor model is adopted in the study of fractured rock; that is, the interaction between microcracks is ignored, and the microcracks are in an elastic matrix without damage.Assuming that all the microcracks are coinshaped, the positions and orientations of the microcracks in the fissure body are uniformly distributed, and the microcracks do not affect each other, as shown in Figure 2.
The literature uses an energy principle to obtain the relationship between the effective elastic modulus, effective Poisson's ratio and crack density of fractured rocks [35][36][37][38][39] : where E ¯and ν ¯are, respectively, the effective elastic modulus and effective Poisson ratio of the fractured rock, E and ν are respectively the elastic modulus and Poisson ratio of rock without fissure, and f is the crack density parameter of fractured rock.It is assumed that there are N cracks in the fractured rock, the average volume of cracks is β, and the volume of the fractured rock is V.We have where ρ ¯is the effective density of fractured rock and ρ is the density of rock without fracture.Combining Equations ( 9) and ( 10) yields Assuming that microcracks are randomly distributed in fractured rock, the fractured rock can be regarded as a quasi-isotropic body, and the propagation of stress waves in the fractured rock satisfies the equation for elastic wave motion.Therefore, the relationship between longitudinal wave velocity and effective elastic parameters should satisfy for fractured rocks and for rocks without fractures.
Combining Equations ( 12) and ( 13) yields Substituting Equations ( 7), (8), and (11) into Equation ( 14) yields where α = ν ν 16 45 Poisson's ratio of rock materials is generally within the range 0.1-0.5. Figure 3 presents the relationship between c c ¯/ p p and ρ ρ ¯/ for ν = 0.20, 0.25, 0.30, 0.35, and 0.40 according to Equation (15).The figure shows that when the Poisson's ratio of rock is set at different values, the longitudinal wave velocity of the rock increases with the density of the rock, and a larger Poisson ratio of the rock results in a greater variation range of the longitudinal wave velocity with density.
The relationship between the longitudinal wave velocity and the density of fractured rock has been theoretically derived by using the method of Taylor model of meso-damage mechanics.Assumptions and simplifications made in the derivation result in differences between the calculation results and actual situation.However, these differences do not affect the conclusion that there is good positive correlation between P-wave velocity and density of rocks.3.2 | Measured relationship between longitudinal wave velocity and density of rock

| Rock materials having different lithology
Twelve types of rock that are common in rock engineering were selected.Each rock block with good integrity and homogeneity was cored, cut, and polished and processed into a cylinder sample with a diameter of 50 mm and a height of 25 mm, as shown in Figure 4.
The 12 types of rock were two types of gray sandstone, three types of red sandstone, three types of granite, China black rock, yellow sandstone, limestone, and basalt.
The ultrasonic longitudinal wave test system (as shown in Figure 5), vernier caliper and electronic scale were used to measure the longitudinal wave velocity and density of rock samples, and the measurement results are given in Table 1.The data in the table are average values for three samples of each rock type.
The volume is calculated by measuring the diameter and height of the rock sample with a vernier caliper, and the rock density is calculated by measuring the mass of the rock sample with an electronic scale.The testing environment of rock longitudinal wave velocity is normal temperature and pressure.The complete ultrasonic longitudinal-wave testing system comprised a pulse transceiver, signal display, and acoustic transducer probe.The test system used in this paper comprised a 5077PR manual pulse signal transceiver produced by OLYMPUS, DPO5104B signal oscilloscope produced by Tektronix and non-metallic ultrasonic probe having an operating frequency of 250 kHz.The transmission pulse width, gain, and other parameters of the test system were consistent throughout the test.The sampling accuracy and sampling length were set at 0.005 and 500 μs, respectively.A special coupling medium was used to ensure that the sample and ultrasonic probe were closely fitted during the test.It is a new type of water-based polymer gel, and it has good coupling performance, which can effectively reduce the loss of ultrasonic sound energy.
According to the data in Table 1, the relationship between P-wave velocity and density of 12 types of rocks is drawn, as shown in Figure 6.By fitting the data points in the figure, the relation between the density and longitudinal-wave velocity of rocks with different lithology is obtained as Figure 6 shows that the density of rocks with different lithology basically increases following a power law with an increase in the longitudinal wave velocity, and there is a strong positive correlation between the longitudinalwave velocity and density of rocks.

| Rock materials having the same lithology
The environmental field of an engineering rock mass is complex and the degree of deterioration of a rock mass often changes with the action intensity of the environmental field.In the example of the temperature field, the degree of degradation differs for deep rock masses heated at different high temperatures and rock masses in a cold area cooled at different low temperatures.In the example of the seepage field, the degree of degradation differs for mine rock masses subjected to mine water having different pH values.In each example, the materials generally have the same lithology (i.e., the same parent material), and the difference in the degree of deterioration is mainly reflected in the mesoscopic structure.The present paper investigates the relationship between the longitudinal wave velocity and density of the same lithological rock material, taking granite subjected to different high temperatures as an example.
The YDL-type tubular furnace shown in Figure 7 was used to heat the processed granite sample (a cylindrical sample with a diameter of 50 mm and a height of 25 mm).The maximum working temperature of the high-temperature furnace was designed to be 1200°C.The temperature control system was set to heat at a rate of 5°C/min.After reaching the preset temperature, the temperature was kept constant for 4 h to ensure that the internal and external temperatures of the sample were consistent.After the power was turned off, the Relationship between the longitudinal-wave velocity and density of rocks with different lithology.
samples were cooled to room temperature naturally in the high-temperature furnace.
A lower heating rate and natural cooling in the hightemperature furnace were adopted to reduce thermal shock damage due to a sudden change in temperature.When the granite samples were subjected to the above high-temperature treatment at the design temperature of 800°C, they showed obvious fracturing, as shown in Figure 8.Therefore, five heat treatment temperatures were used, namely room temperature (25°C), 200°C, 400°C, 600°C, and 750°C.There were thus five groups of granite samples after high-temperature heating, as illustrated in Figure 9.
The ultrasonic longitudinal wave test system (as shown in Figure 5), vernier caliper and electronic scale are used to measure the longitudinal wave velocity and density of granite samples after different high temperatures.The selection of the ultrasonic probe model and parameter settings were consistent with those described above.The measurement results are shown in Table 2, and the data in the table are the average of the measurement results of each group of five samples.
Figure 10 presents the relationship between the longitudinal wave velocity and density of granite samples after heating at different high temperatures, plotted using the data in Table 2.The fitting of the data points in Figure 10 gives the relationship between the density and longitudinal wave velocity: Figure 10 reveals that the density of granite after high-temperature heating basically increases as a power function with an increase in the longitudinal wave velocity, and there is strong positive correlation between the longitudinal wave velocity and density of rock materials having the same lithology.

| Reference relationships
Scholars have investigated the relationship between the longitudinal wave velocity and density of rock and drawn important conclusions.
Meng et al. 40 found that the relationship between the longitudinal wave velocity and density of coal measure rocks follows a quadratic law in a statistical analysis of F I G U R E 7 High-temperature heating system of the tubular furnace.| 3647 the longitudinal wave velocity and density of 101 coal measure sedimentary rocks, including medium sandstone, fine sandstone, siltstone, mudstone, sandy mudstone and limestone: Gardner et al. 41 found that the longitudinal wave velocity and density of saltwater-saturated rock have a power function relationship by combining the results of laboratory tests and field statistics in a study of the seismic reflection coefficient: Zhu et al. 42 established an empirical formula for the longitudinal wave velocity and rock density using sonic and density logging data of a Daqing oilfield and tested the accuracy of the formula using a large volume of literature data:

| RELATIONSHIP BETWEEN THE DEGREE OF ROCK DAMAGE AND WAVE IMPEDANCE
In rock engineering construction, it is common to encounter a degraded rock mass with certain initial damage under the long-term action of the environmental and mechanical fields; for example, thermal damage to a rock mass under long-term high temperature, freezing-thawing damage to a rock mass in cold regions of western China, and chemical damage to a rock mass under the action of underground acidic and alkaline water.The granite samples after high-temperature treatment described in Section 3.2 can be regarded as deteriorated rocks with different degrees of initial damage.How to describe the degree of damage of a rock mass simply and accurately is important to rock engineering design and construction.According to Table 2, the change diagram of wave impedance of granite samples with heating temperature is drawn, as shown in Figure 12.The figure shows that the wave impedance of the granite samples decreased continuously with an increase in the heating temperature, and there was a rapid decline of the wave impedance in the range of 400-600°C.In this chapter, the relationship between the degree of damage to granite after high-temperature heating and the wave impedance is studied from the perspective of the mesostructure using an electron microscope and ultrasonic testing to demonstrate the feasibility of using wave impedance to define variables of rock damage.

| Relationship between the mesostructure of rock sample surface and wave impedance
To obtain the relationship between the degree of damage to rock and the change in wave impedance, a comparative study was conducted on the mesostructure of granite samples after heating at different high temperatures as described in Section 3.2.
The mesostructure of the surfaces of the granite samples with a change in the wave impedance gradient was observed with a Dino Lite electron microscope at a magnification of 20 times.The observations are shown in Figure 13.It is seen that the mesostructure of the granite samples varied for different values of wave impedance.At high wave impedance, the surface of a granite sample was flat and compact and there were no obvious defects.When the wave impedance was reduced to 7854 g cm −3 m s −1 , microcracks of a very small number, very narrow width, and short length formed on the sample surface, and most of the cracks did not penetrate.As the wave impedance decreased further, the number, width, and length of cracks on the surface of the sample increased, and the surface cracks began to connect and coalesce.When the wave impedance was reduced to the minimum value of 1940 g cm −3 m s −1 , a large number of cracks appeared on the sample surface and ran throughout the observational field.
The above observational results of the mesostructure, combined with Figure 12, show that with an increase in the heating temperature, the microdefects of granite samples increased continuously and the degree of deterioration worsened continuously, and the physical parameters were reflected in the continuous decrease in the wave impedance.Therefore, it can be preliminarily judged that there is a strong correlation between the damage degree of the sample and its wave impedance.

| Relationship between acoustic signal and wave impedance
The ultrasonic pulse passed through the rock sample, and the received signal fed back much information about the internals of the rock.It is thus necessary to compare the acoustic signals of the granite samples after heating at different high temperatures described in Section 3.2 in terms of the time domain, frequency domain, and time-frequency domain, to better understand the relationship between the change in wave impedance and the degree of damage.

| Time-domain study of acoustic signals
The ultrasonic longitudinal-wave test system shown in Figure 5 was used to collect the received wave waveforms for granite samples after heating at a high temperature with a gradient change in the wave impedance, as shown in Figure 14.
F I G U R E 12 Wave impedance variation of granite after hightemperature heating.
Figure 14 shows differences in the received waves for granite samples with different wave impedances.With a decrease in the wave impedance, the received waves attenuate to varying degrees, which manifests as a continuous decrease in received wave amplitude, the continuous extension of the arrival time of the first wave, the continuous increase in the period and the continuous decrease in number of periods.It is noted that when the wave impedance was greatly reduced to 3081 g•cm −3 m s −1 , the received wave began to change appreciably in terms of the amplitude, arrival time of the first wave, period length and number of periods.The results for the test groups with wave impedances of 3081 and 1940 g cm −3 m s −1 differed greatly from those for the other groups.The results show a consistent variation between the received wave signal and wave impedance.
According to Figures 13 and 14, IT is believed that the variations in ultrasonic parameters of granite samples with different wave impedances are related to the degree of thermal damage.A high temperature led to the deterioration of the granite samples; for example, water loss and a greater level of microdefects (i.e., greater microcrack initiation, development, and coalescence and a higher number of micropores).The granite samples were damaged bodies after hightemperature heating.When an ultrasonic pulse passes through a rock sample, the microdefects in the rock sample will cause the reflection, refraction, and diffraction of the wave, and the received wave will show differences relating to the different degrees of damage to the sample.A higher heating temperature results in smaller wave impedance and a greater development of microdefects in the rock sample results in greater acoustic attenuation and greater loss due to the refraction and scattering of the acoustic wave across the rock sample, such that the received waves F I G U R E 13 Microstructure of the surfaces of granite samples with a gradient change in the wave impedance.have a reduced amplitude, a prolonged arrival time of the first wave and a loose waveform.There is thus strong correlation among the degree of thermal damage, wave impedance and acoustic signal of granite after high-temperature heating.

| Frequency domain of the acoustic signal based on the fast Fourier transform
4][45] It is thus necessary to analyze the received wave signals in the frequency domain, compare the spectral characteristics of granite samples with different wave impedances, and further study the variation relationship between the damage degree of granite and its wave impedance from the perspective of acoustic signal frequency domain.
The fast Fourier transform converts a time-domain signal into a frequency-domain signal, so as to obtain the frequency distribution of the acoustic signal f(t) in the time domain range 46 : where, F(ω) is the Fourier transform function of time domain signal f(t), i is the square root of −1, and ω is the angular frequency.With a decrease in wave impedance, the spectral kurtosis of the received wave spectrum of granite samples decreased to varying degrees in each frequency band.When the wave impedance of the sample was greatly reduced to 3081 g cm −3 m s −1 , the spectral kurtosis in the mid-and high-frequency bands began to decrease appreciably and even disappear.The spectral kurtosis in the low-frequency band began to decrease sharply when the wave impedance decreased to 1940 g cm −3 m s −1 .
At room temperature (for which the wave impedance of the sample was greatest and the wave impedance was 14,071 g cm −3 m s −1 ), the received wave spectrum of the granite sample had a trimodal pattern, where the proportions of the high-, mid-, and low-frequency bands were almost the same, the proportion of the high-frequency part was slightly higher, the proportion of mid-frequency part was slightly lower, and the main frequency was not prominent.With a decrease in the wave impedance, the high-frequency and mid-frequency proportions in the received wave spectrum of the heat-damaged granite specimens decreased continuously.When the wave impedance was greatly reduced to 3081 g cm −3 m s −1 , the frequency distribution of the received wave signal changed appreciably, the high-frequency part of the spectrum largely disappeared, and the proportion of the mid-frequency part was extremely low.In other words, with a decrease in the wave impedance, the mid-frequency and high-frequency parts of the received wave signal continuously attenuated, the spectrum gradually changed from having three to having two peaks, the frequency band continued to narrow, and the main frequency gradually protruded.
In conclusion, the difference in the frequency distribution of the received wave signals of the thermally damaged granite samples with different wave impedances was mainly that with a decrease in the wave impedance, the spectral kurtosis decreased continuously and the mid-and high-frequency components decayed continuously.It is concluded that the spectral characteristics of the received waves for the granite samples with different wave impedances were related to the degree of thermal damage or deterioration.A high temperature will lead to the deterioration of granite samples through water loss and an increase in the level of microdefects.With an increase in the heating temperature, the free water and bound water in the sample evaporate continuously, the level of microdefects in the sample increases continuously, the porosity increases continuously, and the bonding between minerals in the rock weakens continuously, resulting in increased damage to the sample.All these increase the acoustic wave loss due to refraction and scattering.Thus, with increased thermal damage to the sample, the wave impedance decreased and the spectral kurtosis of the received wave spectrum decreased.
When sound waves pass through a medium, the medium absorbs the waves in a frequency selective manner; that is, the sound waves are filtered.This frequency-dependent absorption is affected by the internal structure of the medium.As for the thermal damaged granite sample, with increased thermal damage, the wave impedance decreases, the porosity of the sample increases, and the integrity worsens.When a sound wave passes through the sample, the high-frequency component readily decays whereas the low-frequency component does not so readily decay.This absorption of high-frequency signals means that the high-frequency components in the spectrum of the received wave constantly attenuate or even disappear.
In conclusion, granite samples having different degrees of thermal damage have different filtering and absorption effects on acoustic waves, leading to a difference in the frequency distribution of the received waves.The results of the present study show that there is strong correlation among the degree of thermal damage, the wave impedance, and the frequency distribution of the acoustic signal of granite after high-temperature heating.

| Time-frequency domain of acoustic signals based on a continuous wavelet transform
Although the time-domain signal of the received wave can be converted into a frequency-domain signal by applying the fast Fourier transform, allowing investigation of the frequency component of the signal, the fast Fourier transform of the time-domain signal does not reflect the change in the signal frequency with time.It is thus necessary to carry out a continuous wavelet transform on the received wave signals and convert the time-domain signals of the received wave into signals in the time-frequency domain, 46 so as to study the variation in the received-wave frequency with time for the thermally damaged granite samples with different wave impedances, as shown in Figure 18.
Figure 18 reveals that the granite samples having different wave impedance had obvious differences in the time-frequency signals of the received waves.The main differences were that as the wave impedance decreased, the attenuation of the energy of the received wave signal in the time domain slowed and the development degree of the coda wave increased.Consistent with the above, when the wave impedance was reduced to 3081 g cm −3 m s −1 , the time-frequency signal of the received wave of the thermally damaged granite samples began to change appreciably.
The analysis shows that, as mentioned above, with an increase in the heating temperature, the degree of damage and deterioration of the granite samples increased, and the microdefects and porosity of the samples increased continually.As the compactness and integrity of the sample worsened, there was increased reflection, refraction, and scattering of the sound waves in the process of propagation, which expanded the distribution of the sound wave energy in the time domain.This manifested as a slower attenuation of the sound wave energy in the time domain and the intensified development of the coda wave.The results show that there was a strong correlation among the degree of thermal damage, the wave impedance, and the acoustic time-frequency signal of granite after high-temperature heating.

| METHOD OF DEFINING ROCK DAMAGE VARIABLE ON THE BASIS OF WAVE IMPEDANCE
The results presented for the rock material mesostructure and ultrasonic acoustic signal in Chapter 4 show that the degree of rock damage is strongly correlated with the magnitude of wave impedance, and it is thus feasible and reasonable to use wave impedance to characterize the degree of rock damage.In Chapter 3, the theoretical calculation and results of fitting the relationship between the longitudinal wave velocity and rock density showed that there is close correlation between the longitudinal wave velocity and rock density, where the change in one parameter is often accompanied by a change in the other parameter and there is positive correlation between the two parameters.Therefore, compared with the velocity of the longitudinal wave, the rock wave impedance has certain advantages in characterizing the degree of damage to the rock.
In view of the feasibility, rationality, and superiority of using the wave impedance to describe the degree of rock damage, this chapter investigates the definition of rock damage variables and damage evolution based on the wave impedance.

| Defining rock damage variable based on wave impedance
Using the elastic modulus to define a damage variable is a common and convenient method adopted in the study of rock damage mechanics. 47The relationship between the damage variable and elastic modulus of the rock material is where E′ and E are the elastic moduli of damaged and undamaged rocks respectively.According to elastic wave theory, for damaged rock and undamaged rock, the relationship between the longitudinal wave velocity and elastic parameters should respectively satisfy where, c p , ρ and ν are respectively the longitudinal wave velocity, density and Poisson's ratio of undamaged rock and c′ p , ρ′ and ν′ are respectively those of damaged rock.
(See Section 3.1 for the satisfying conditions of Equations 23 and 24.)By substituting Equations ( 23) and (24) into Equation ( 22), a damage variable jointly defined by the longitudinal wave velocity, density, and Poisson ratio is obtained as Neglecting the change in Poisson's ratio in the process of rock damage, the above equation can be simplified as Furthermore, neglecting the change in the density in the process of rock damage, the above equation can be simplified as Equation ( 27) gives a damage variable defined with the longitudinal wave velocity.The result is consistent with the damage variable defined with the acoustic wave velocity in the literature. 48,49The measurement of the rock damage degree defined with the longitudinal wave velocity is a simple method commonly adopted in rock engineering.
Chapter 3 describes how the longitudinal wave velocity and density of the rock material are closely correlated.In the process of rock damage, a change in one parameter is often accompanied by a change of the other, and the two parameters have a positive correlation.A change in density should thus be considered in the study of rock material damage, which is not suitable for over-simplification.
Wave impedance is an inherent property of materials because it is the product of the longitudinal wave velocity and density.Meanwhile, wave impedance reflects the ability of the medium to prevent waves from passing through and is a basic physical quantity that characterizes the dynamic properties of the medium, which is important to the study of the damage evolution of the material under dynamic load.Equation 26 is therefore rewritten as a relation between the damage variable D and wave impedance ρc p : where ρc p and ρ c ′ ′ p are respectively the wave impedances of undamaged and damaged rock.
According to the research on the relationship between rock longitudinal wave velocity and density in Chapter 3, a large number of measured data show that the rock longitudinal wave velocity and density can basically be fitted as a power function, and have a high ), formula 20 is further simplified, and the method of defining rock damage variable by wave impedance is established.Obtaining the simultaneous solution is a complex process that provides a complicated result, and it is not convenient for application.The relationship between the wave impedance and damage variable is thus obtained using a computational approximation.It is assumed that the rock damage variable defined by the wave impedance has the form The process of rock damage is often accompanied by decreases in the density and longitudinal wave velocity; that is, ρ′ < ρ and c p ′ < c p .Therefore, ρ/ρ′ > 1, ρ′c p ′ < ρc p and x < 2. We take x = 1.4,1.5, 1.6, 1.7, 1.8, and 1.9 respectively, and compare the damage variable defined in Equation (26) with the damage variable defined in Equation ( 29) to obtain the value of x that makes the two formulas close.The relationships between the density and longitudinal wave velocity of rock satisfying the Zhu Guangsheng relation and Gardner relation are studied separately.

| Zhu guangsheng relation
Adopting the Zhu Guangsheng relation (i.e., the longitudinal wave velocity and density meet the requirement ρ c = 0.414 p 0.214 ), the relationship between the damage variable defined in Equation ( 26) and the ratio of longitudinal wave velocity is calculated.At the same time, according to Equation ( 29), we take x = 1.4,1.5, 1.6, 1.7, 1.8, and 1.9 respectively to calculate the relationship between the damage variable defined by wave impedance and the ratio of longitudinal wave velocity, as shown in Figure 19A.It is seen that the damage variable defined with the wave impedance has the same variation trend as that defined by Equation (26).The two closest when x = 1.8.Therefore, the value of x is further confined between 1.8 and 1.9, and it is found that when x = 1.825, the damage variables defined by the two formulas are almost identical, as shown in Figure 19B.Therefore, if the rock longitudinal wave velocity and density satisfy the Zhu Guangsheng relation, the rock damage variable defined by the wave impedance can be expressed as

| Gardner relation
Similarly, adopting the Gardner relation (i.e., the rock longitudinal wave velocity and density satisfy ρ c = 0.31 p 0.25 ), the change of the damage variable defined in Equation (26) and Equation ( 29) with the longitudinal wave velocity ratio is calculated respectively, as shown in Figure 20A.It is seen that the damage variable defined with wave impedance has the same variation trend as that defined by Equation (26).When x = 1.8, the damage variables defined using the two formulas are almost identical, as shown in Figure 20B.Therefore, if the longitudinal wave velocity and rock density meet the Gardner relation, the rock damage variable defined with the wave impedance is expressed as According to the research on the relationship between rock longitudinal wave velocity and density in Chapter 3, the Zhu Guangsheng relation and Gardner relation are both based on a large number of measured data, and have a high degree of coincidence with the relationship obtained in this paper based on the laboratory measured data.Therefore, the damage variable expression based on wave impedance obtained from the Zhu Guangsheng relation and Gardner relation should be applicable to most rocks.Compared with formula 24, formula 25 is more simple.Therefore, from the perspective of convenient popularization, formula 25 can be considered to quantify the degree of rock damage.

| Validation of the damage variable
Different rocks generally have different initial wave impedances.We take the examples of the initial wave impedance ρc p of 12,000, 10,000, 8000, 6000, and 4000 g cm −3 m s −1 .Using Equation (31), the results of the rock damage variable D versus the wave impedance ρ c ′ ′ p are plotted for five initial wave impedances in Figure 21.The figure shows that there is no damage in the initial state, and the damage variable D = 0.With the evolution of damage, the wave impedance decreases and the damage variable increases.When the damage develops to the limit state, the wave impedance decreases to zero and the damage variable D = 1; that is, the completely damaged state is reached.A smaller initial wave impedance of the rock results in greater variation in the damage variable with the wave impedance.
According to the formula (Equation 31) for calculating the damage variable defined with the wave impedance in the above section and using the data in Table 2, the damage variables of granite samples after hightemperature heating are calculated.The calculation results are given in Table 3.
Damage variables of the granite samples after hightemperature heating are plotted in Figure 22 from the data in Table 3  variable increases continuously with the heating temperature.
The consistency between the damage variable defined with the wave impedance and the degree of damage to the sample is further analyzed from the perspective of the microstructure and ultrasonic signal of the granite sample with a gradient change in the wave impedance.According to the relevant research in Chapter 4, with the increase of heating temperature, the damage degree of granite samples after high temperature increases continuously, which is manifested by the increasing number of cracks on the sample surface, the continuous attenuation of received wave signals, the continuous reduction of the spectral kurtosis of acoustic signals, and the continuous slow attenuation of acoustic signal energy in the time domain.The damage variable calculated with the wave impedance increased from 0 to 0.9717.When the heating temperature increased to 600°C, the wave impedance of the sample was 3081 g cm −3 m s −1 , the mesostructure and acoustic signal of the sample began to show appreciable changes, the number of cracks on the surface of the sample increased appreciably, and the acoustic signal attenuated greatly, indicating that the sample reached a threshold of thermal damage and the damage of the sample developed to the limit state.At this point, the damage variable calculated using Equation (31) was 0.9350, which is close to a value of 1.The results show that the variations of the damage degree and damage variable are consistent, and the damage variable defined with the wave impedance well reflects the damage state of the rock.

| CONCLUSIONS
By combining theoretical analysis with laboratory experiments, the superiority and feasibility of using wave impedance to define rock damage variable were demonstrated, and a method of defining rock damage variable using wave impedance was proposed.The study found that rock damage variable based on wave impedance can be used for quantitative description of rock damage.
1.The longitudinal wave velocity of rock materials is closely related to their density, and there is a good positive correlation between the two parameters.This reveals that compared with using the longitudinal wave velocity, it is advantageous to use the wave impedance to characterize the degree of damage to rock. 2. With an increase in the heating temperature, the wave impedance of granite samples decreased continuously after high-temperature heating.In terms of the mesostructure, the microdefects of rock samples are increasing and the development degree is increasing.
In terms of the characteristics of the ultrasonic longitudinal wave signal, the amplitude of the received waves decreased, the arrival time of the first wave further lagged, and the waveform gradually loosened.The spectral kurtosis of the received wave decreased and the mid-and high-frequency components decayed.The attenuation of the signal energy of the received wave in the time domain slowed and the development of the coda wave increased.The results show strong correlation between the degree of rock damage and the magnitude of wave impedance, and it is feasible to use the wave impedance to characterize the degree of rock damage.3. A new method of defining rock damage variable using wave impedance was proposed.The proposed damage variable can well reflect the damage state and damage evolution characteristics of rocks.4. The degree of thermal damage to granite increases with the heating temperature.It is inferred that there is a threshold temperature of thermal damage to granite in the range of 400°C-600°C, and beyond this temperature, granite undergoes appreciable thermal degradation.

F I G U R E 2
Schematic diagram of the distribution of round coin-shaped microcracks.

F I G U R E 3
Relationship between c c ¯/ p p and ρ ρ ¯/ .F I G U R E 4 Twelve kinds of rock samples.LI ET AL.| 3645

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I G U R E 8 Surface of a granite sample after high-temperature heating at 800°C.F I G U R E 9 Granite samples after high-temperature heating.LI ET AL.
relationships between the rock longitudinal wave velocity and density proposed by Meng Zhaoping, Gardner, and Zhu Guangsheng are, respectively, referred to as Meng Zhaoping relation, Gardner relation, and Zhu Guangsheng relation.The three relationships are compared in Figure 11.It is seen that there are differences in the forms of the three relationships, with the Gardner relation and Zhu Guangsheng relation being largely consistent.Compared with the Gardner relation and Zhu Guangsheng relation, the Meng Zhaoping relation predicts higher rock density for the same longitudinal wave velocity.Although the three relationships are different, they all show positive correlation between the longitudinal wave velocity and the rock density.In general, a higher density of rock results in a higher longitudinal wave velocity.The above theoretical derivation and analysis of the measured relationship between the longitudinal wave velocity and density of rock materials indicate positive correlation between the longitudinal wave velocity and density of rock materials.

T A B L E 2 F
Parameters of granite after high-temperature heating.I G U R E 10 Relationship between the longitudinal wave velocity and density of granite samples after high-temperature heating.F I G U R E 11 Comparison of relationships between the longitudinal wave velocity and density of rocks.

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I G U R E 14 Waveform of the received wave for granite samples with a gradient change in the wave impedance.(A) 14,071 g cm −3 m s −1 , (B) 11,370 g cm −3 m s −1 , (C) 7854 g cm −3 m s −1 , (D) 3081 g cm −3 m s −1 , (E) 1940 g cm −3 m s −1 .
A fast Fourier transform was applied to the received wave signals to obtain the frequency distribution characteristics of the granite samples with a gradient change in the wave impedance.The spectrum diagram, spectrum comparison diagram, and enlarged comparison diagram of the received wave of granite samples with different wave impedances are shown in Figures 15, 16, and 17, respectively.Figures 15-17 show obvious differences in the frequency distribution of the received waves for the granite samples with different wave impedances.The frequency of the received wave signals for the samples with different wave impedances was basically distributed between 30 and 400 kHz.The frequency range is divided into three parts according to the spectral characteristics of the specimens with different wave impedance: a low-frequency band (30-70 kHz), midfrequency band (70-180 kHz), and high-frequency band (180-400 kHz).

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I G U R E 16 Comparison of the frequency spectra of granite with a gradient change in the wave impedance.F I G U R E 17 Magnified spectral comparison.(A) Low-frequency band, (B) mid-frequency band, and (c) high-frequency band.
degree of fitting.Compared with Meng Zhaoping relation, Zhu Guangsheng relation and Gardner relation are more consistent.Therefore, based on Zhu Guangsheng relation (ρ c = 0.414 p 0.214 ) and Gardner relation

F I G U R E 19
Fitting of rock damage variables defined with the wave impedance under the Zhu Guangsheng relation.(A) x = 1.4-1.9 and (b) x = 1.825.
. The figure shows that the damage F I G U R E 20 Fitting of the rock damage variable defined with the wave impedance under the Gardner relation.(A) x = 1.4-1.9 and (B) x = 1.8.

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I G U R E 21 Relationship between the damage variable and wave impedance.

T A B L E 3 F
Summary of the damage variable of granite with a gradient change in the wave impedance.I G U R E 22 Damage degree of granite with a gradient change in the wave impedance.
System for measuring the longitudinal wave velocity.Longitudinal wave velocity and density for 12 types of rock.
T A B L E 1