# A Comparative Study on Damage Variable and Geo-Mechanical Properties of the Pre-Damaged Rock Specimen

Authors: Lei Wen; Yang Huang; Jia-xin Wen; Xin Zhang; Wen-qiang Lu
Journal: Advances in Civil Engineering
 (2022)

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## Abstract

The initial damage of a rock mass plays an essential role in parameter evaluation of that rock mass. How to determine the initial damage of a rock mass is therefore a problem that must be solved. To study the mechanical properties of a rock mass, pre-damaged specimens with different degrees of damage are prepared using uniaxial compression. The degree of damage of the pre-damaged specimens is determined by using their physical and mechanical parameters; that is, four methods are used to calculate the damage variables. Comparing the calculated damage variables using the four methods, we find the following: (1) the damage variables of different calculation methods are quite different. (2) When the degree of damage is slight, in other words, the micro-cracks are in the initial state of growth, and the calculated damage variables have little difference when using various methods. (3) When the degree of damage is severe, that is, the micro-cracks coalesce to form macro-cracks, the calculated damage variables significantly deviate when using various methods. Only the calculated damage variables using the BTS are larger than those calculated using other methods, mainly because there is almost no friction effect of cracks in BTS testing. Finally, to verify the accuracy and feasibility of the damage variables of various calculation methods, the uniaxial compressive peak strength of the pre-damaged specimens is determined by using the effective strain hypothesis and the damage variable. The 1 : 1 slope line is used to verify the deviation between the measured values and calculated values.


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## Body

## 1. Introduction

The available experiments and numerical methods are very powerful when describing rock damage and failure problems. Nevertheless, it is difficult to assess and describe the stress state and deformability of the surrounding rock mass in depth in underground engineering due to the high overburden and complex geological conditions. In this context, the constitutive model of the rock mass plays a vital role. The classical constitutive model of the Hoek–Brown model was proposed and developed by Hoek et al. [1–3]. In recent years, with the development of elastoplastic mechanics and damage mechanics, a series of damage constitutive models on damaged and undamaged rock has been proposed in the literature [4, 5]; Wendong [6].The rock strength and deformation characteristics are closely related to the damage state of the rock. For on-site engineering rock masses, the rock slope, mining engineering, tunnel engineering, etc., are all different damage states of the intact rock. Furthermore, an engineering disturbance aggravates the damage of the engineering rock mass. In light of this, it is necessary to study the strength characteristics of the rock in terms of the different degrees of damage. Thus, a description of the damage state in a rock mass is one of the most important problems when researching the rock mass’s strength and deformation characteristics. Many existing studies involve studying the rock damage and rock strength under compaction. Furthermore, the propagation and coalescence of different pre-crack directions are performed by using a rock-model material under uniaxial compression tests [7–9]. The research results on pre-cracked flaws are very similar and describe the propagation pattern of pre-cracks. Numerical studies on rock-like specimens’ pre-cracks have been conducted in the literature [10]. Liu et al. [11] researched the interaction of multiple crossed joint sets to determine their mechanical properties and failure modes. Multiple sets of discretized joints were embedded in a mathematical model to conduct numerical simulations by using an enhanced embedded discontinuity approach, thereby simulating the damage process of pre-cracked specimens [12]. Based on the strain energy density method, Zhou et al. [13] stated the dynamic damage localization of the physical behavior of crack samples with different dip angles. Liu et al. [11] adopted cyclic triaxial loading to determine the dynamic mechanical properties of rocks with an artificial joint. A significant amount of researchers have contributed to similar studies [14–19].Due to the opening of pre-cracks, the friction effect of the pre-crack surface is essentially neglected. Meanwhile, most of the rock mass includes multiple sets of flaws. Under different types of loads, macro-cracks can be formed by the propagation and connection of micro-cracks. Therefore, the effect of micro-cracks in rocks cannot be neglected during the damage process. In addition, the crack surface’s frictional resistance plays a vital role in hindering the expansion of the damage. To this extent, many researchers have tried to develop the micro-mechanics model, plastic-damage model, damage-friction model, etc. A micro-mechanics model based on plastic damage was developed by Zhao et al. [20]. Zhu et al. [21] proposed a refined micromechanical damage-friction model to model the plastic deformation and damage evolution of quasi-brittle materials under a confined compressive stress. Considering the damage coupling between the crack’s frictional sliding and crack growth, Zhu et al. [22] developed a frictional damage model that combines the linear homogenization procedure and irreversible thermodynamics framework. They also developed an analytical solution to solve the stress-strain relationship. Meanwhile, two numerical integration algorithms were used to validate the model’s reliability. Zhu et al. [23] made a great contribution toward investigating the damage-friction modeling. In his study, a micromechanical thermodynamic formulation based on the elastoplastic strain energy was developed to determine the damage and friction. Shen et al. [24] proposed a micro-macro-model for clayey rocks to estimate their elastoplastic behavior. Concerned with the complexity and particularity of rock materials, different kinds of micro-macro-damage constitutive models were formulated to model the plastic-damage coupling [25–32].Considering the different geological conditions and stress states, the engineering rock mass is subjected to cyclic loading and unloading; that is, the engineering rock mass is damaged to varying degrees. Therefore, it is necessary to study the mechanical properties of a damaged engineering rock mass. In existing constitutive models, although various damage evolutions have been deduced, the initial damage of rocks is often neglected. It is well known that the correct model for intact rocks is not sufficient for damaged rock. To overcome this disadvantage, some researchers have made a contribution to studying the strength properties of pre-damaged rock. In general, several authors applied cyclic loading and unloading to investigate the fatigue damage evolution of rocks. Sun et al. [33] employed multi-level amplitude cyclic loading to explore the damage accumulation and damage law of sandstone under different confining pressures. Based on this, a fatigue damage evolution model was proposed to analyze the stability of engineering structure. Liu et al. [34] derived a damage constitutive model in light of energy dissipation to state the failure model of rocks under unconfined cyclic loading. A digital image correlation method was employed to record the failure process of rock samples under axially cyclic compaction [35]. Based on the different stress amplitudes, loading frequencies, and loading rates, He et al. [36] performed an experimental investigation of salt rock under fatigue loading and built a damage model. Li et al. [37] described the nonlinear behavior of fractured rock based on the energy dissipation under cyclic loading and established a damage model from the view of the initial damage and the equivalent modulus. Zhang et al. [38] proposed a creep damage model of rock mass under multi-level creep load based on spatio-temporal evolution of deformation modulus. Lei et al. [39] stated damage characteristics of shear strength of joints under freeze-thaw cycles. The damage study of rocks under cyclic loading is too extensive to state completely here; for example, please see literature [40–46].To study the law of crack propagation, some researchers have fabricated single or multiple cracks (as well as multiple cross-cracks) in rock specimens to study the propagation mode of cracks and the mechanical properties and deformation mechanism of cracked rock blocks. However, artificial prefabricated cracks are quite different from natural cracks in a rock mass. Furthermore, the initial damage state should be taken into account when establishing the constitutive model of the rock. To obtain pre-damaged specimens consistent with the damage morphology of the natural rock mass, in this paper the monocyclic unconfined loading and unloading method is used to prepare the pre-damaged rock samples with different degrees of damage severity.To study the mechanical behavior of various amounts of pre-damaged rock samples, several key points need to be noted. First, the preparation manner of pre-damaged specimens and the different types of manners are adopted to prepare the pre-damaged rock samples, such as unconfined cyclic loading and unloading, confined (various confining pressures) cyclic loading and unloading, monocyclic unconfined loading and unloading, and monocyclic confined (various confining pressures) loading and unloading. Second, the definition of the damage variable is a crucial parameter. In previous studies, various types of damage variables were defined, such as by using the equivalent elastic modulus, energy dissipation, crack density, acoustic velocityvp, yield stress, and the times of the cyclic loading and unloading. The process of damage evolution is deduced from experiments and theoretical methods. However, few researchers have considered the initial stage of pre-damaged specimens. In this paper, from a macroscopic viewpoint of rock damage and the failure process, a comparative study of the types of damage variables is conducted on rock samples with monocyclic unconfined loading and unloading. The uniaxial pressure with different gradients is employed to prepare the pre-damaged rock sample with different types of initial damage. According to the existing definition of the damage variable, the damage index of a pre-damaged sample is calculated to define a damage severity degree with the initial damage. In this regard, the uniaxial compressive strength of the pre-damaged rock is computed and compared with the tested value.


## 2. Preparation of Pre-Damaged Specimens

In engineering, the failure of the rock mass is based on the premise that the rock mass has a certain initial damage. There are fewer intact and nondestructive rock masses in engineering, and fewer intact rock masses that fail directly. Initial damage exists in almost all rock masses; thus, in this paper pre-damaged specimens with different degrees of damage were fabricated to calculate their initial damage variables.


### 2.1. Stress-Strain Curve

The rock specimens are taken from the Fankou Lead-Zinc Mine in the Guangdong Province of China. The rock samples are slightly weathered dolomitic limestone. In accordance with ASTM standards [47], the sample testing was conducted on cylindrical samples with 100 × 50 mm dimensions (length × diameter). The physical and mechanical parameters of the rock used in this study are listed in Table 1. A hydraulic servo mechanical testing machine (INSTRON-1346, INSTRON, Melbourne, Australia) is employed to measure the uniaxial compressive strength (UCS) and Brazilian test strength (BTS). The stress-strain curve is thereby acquired during a uniaxial compression test. Based on the measured stress-strain characteristic curve, the failure process of the rock sample consists of the following four stages, as shown in Figure 1.(1)
The crack closure stage (OA), where the existing cracks in the rock are compacted under compressive loads. The stress-strain curve is concave.(2)
The linear elastic deformation stage (AB), where the rock matrix exhibits elastic deformation characteristics. The stress-strain curve is an approximately straight line.(3)
The crack growth stage (BC), which contains stable crack growth and unstable crack growth. The stable crack growth represents micro-cracks beginning to propagate. And the unstable crack growth indicates a micro-crack coalescence forming a macro-crack.(4)
The postpeak and failure stage (CD).Table 1
Physical and mechanical parameters of intact rock.
No.ρ (t/m3)UCS(MPa)BTS(MPa)vp (m/s)E(GPa)υ12.69688.2684.5985787.333.3480.25322.69390.6974.4055742.833.5140.24132.70087.2844.1685708.533.8160.24842.72788.0654.7305751.834.0910.26252.70386.3814.3525732.133.7350.254Mean2.70488.1394.4505734.533.7010.252Figure 1
Stress-strain curve.


### 2.2. Preparation of the Pre-Damaged Specimens

Eberhardt et al. [48] quantitatively researched the crack growth stage under uniaxial compression. His study showed that the stress-strain curve of uniaxial compression can be divided into five stages based on the crack development. For details of the research results, refer to the literature [48]. In addition, he proposed that the crack damage stress thresholds are 0.75 σc, where σc represents the UCS. This research is consistent with the study of Martini et al. [49]. Therefore, based on the existing research and tested results, five kinds of pre-damaged samples are obtained to measure the UCS. The crack threshold of pre-damaged samples is assessed by the uniaxial compressive strength and P-wave velocity of currently tested results according to the research of Eberhardt et al. [48]. Pre-damaged specimens were obtained by performing monocyclic unloading compression tests on intact specimens. The P-wave velocity is carried out in the pre-damaged sample, and these results are listed in Table 2. The tested results show that the P-wave velocity of the pre-damaged sample corresponding to the crack threshold decreases in varying degrees. It indicates that the fracture threshold is suitable for the rock samples in this study. Thus, the monocyclic unloading stresses are 0.25 σc, 0.4 σc, 0.5 σc, 0.65 σc, and 0.75 σc. Monocyclic uniaxial loading and unloading are used to prepare the pre-damaged samples, which are listed in Table 3.Table 2
Physical and mechanical parameters of pre-damaged specimens.
No.Pre-damaged strength0.25σc0.40σc0.50σc0.65σc0.75σc22.0335.2544.0757.2966.1vpTest times15687.35632.75417.35237.25188.725712.85681.45488.15274.15132.935769.45619.35493.45307.35178.545738.55602.15461.75258.65208.455741.85647.55489.65289.75137.1Mean value5730.05636.65470.25273.45169.1UCSTest times187.02685.13778.34072.58370.920286.35484.62375.62973.26772.010386.92783.57976.38473.85272.560485.92383.24675.82672.94370.830587.46184.86577.39273.05871.940Mean value86.73884.2976.71473.14171.652ETest times129.83729.04626.57623.63922.931229.73628.49226.41222.95622.391330.21128.78127.01523.29421.265430.4429.01226.72824.06722.480530.14928.76826.77023.59322.716Mean value30.25030.07528.81026.70022.357BTSTest times14.5344.0213.7913.3973.29524.2973.9563.8523.4973.15234.3964.0253.6673.4193.18144.4533.9613.7823.5943.11854.5074.0833.8793.453.212Mean value4.43744.00923.79423.47143.1916Table 3
List of pre-damaged specimens.
No.Sample number (r × l)Crack thresholdUnloading stressRemarks15 (50 × 100)5 (50 × 50)Crack closure0.25σc22.03The crack closure stage (OA)25 (50 × 100)5 (50 × 50)Crack initiation0.40σc35.25The linear elastic deformation stage (AB)35 (50 × 100)5 (50 × 50)Secondary cracking0.50σc44.07The linear elastic deformation stage (AB)45 (50 × 100)5 (50 × 50)Crack coalescence0.65σc57.29The crack growth stage (BC)55 (50 × 100)5 (50 × 50)Crack damage0.75σc66.10The crack growth stage (BC)


## 2.1. Stress-Strain Curve

The rock specimens are taken from the Fankou Lead-Zinc Mine in the Guangdong Province of China. The rock samples are slightly weathered dolomitic limestone. In accordance with ASTM standards [47], the sample testing was conducted on cylindrical samples with 100 × 50 mm dimensions (length × diameter). The physical and mechanical parameters of the rock used in this study are listed in Table 1. A hydraulic servo mechanical testing machine (INSTRON-1346, INSTRON, Melbourne, Australia) is employed to measure the uniaxial compressive strength (UCS) and Brazilian test strength (BTS). The stress-strain curve is thereby acquired during a uniaxial compression test. Based on the measured stress-strain characteristic curve, the failure process of the rock sample consists of the following four stages, as shown in Figure 1.(1)
The crack closure stage (OA), where the existing cracks in the rock are compacted under compressive loads. The stress-strain curve is concave.(2)
The linear elastic deformation stage (AB), where the rock matrix exhibits elastic deformation characteristics. The stress-strain curve is an approximately straight line.(3)
The crack growth stage (BC), which contains stable crack growth and unstable crack growth. The stable crack growth represents micro-cracks beginning to propagate. And the unstable crack growth indicates a micro-crack coalescence forming a macro-crack.(4)
The postpeak and failure stage (CD).Table 1
Physical and mechanical parameters of intact rock.
No.ρ (t/m3)UCS(MPa)BTS(MPa)vp (m/s)E(GPa)υ12.69688.2684.5985787.333.3480.25322.69390.6974.4055742.833.5140.24132.70087.2844.1685708.533.8160.24842.72788.0654.7305751.834.0910.26252.70386.3814.3525732.133.7350.254Mean2.70488.1394.4505734.533.7010.252Figure 1
Stress-strain curve.


## 2.2. Preparation of the Pre-Damaged Specimens

Eberhardt et al. [48] quantitatively researched the crack growth stage under uniaxial compression. His study showed that the stress-strain curve of uniaxial compression can be divided into five stages based on the crack development. For details of the research results, refer to the literature [48]. In addition, he proposed that the crack damage stress thresholds are 0.75 σc, where σc represents the UCS. This research is consistent with the study of Martini et al. [49]. Therefore, based on the existing research and tested results, five kinds of pre-damaged samples are obtained to measure the UCS. The crack threshold of pre-damaged samples is assessed by the uniaxial compressive strength and P-wave velocity of currently tested results according to the research of Eberhardt et al. [48]. Pre-damaged specimens were obtained by performing monocyclic unloading compression tests on intact specimens. The P-wave velocity is carried out in the pre-damaged sample, and these results are listed in Table 2. The tested results show that the P-wave velocity of the pre-damaged sample corresponding to the crack threshold decreases in varying degrees. It indicates that the fracture threshold is suitable for the rock samples in this study. Thus, the monocyclic unloading stresses are 0.25 σc, 0.4 σc, 0.5 σc, 0.65 σc, and 0.75 σc. Monocyclic uniaxial loading and unloading are used to prepare the pre-damaged samples, which are listed in Table 3.Table 2
Physical and mechanical parameters of pre-damaged specimens.
No.Pre-damaged strength0.25σc0.40σc0.50σc0.65σc0.75σc22.0335.2544.0757.2966.1vpTest times15687.35632.75417.35237.25188.725712.85681.45488.15274.15132.935769.45619.35493.45307.35178.545738.55602.15461.75258.65208.455741.85647.55489.65289.75137.1Mean value5730.05636.65470.25273.45169.1UCSTest times187.02685.13778.34072.58370.920286.35484.62375.62973.26772.010386.92783.57976.38473.85272.560485.92383.24675.82672.94370.830587.46184.86577.39273.05871.940Mean value86.73884.2976.71473.14171.652ETest times129.83729.04626.57623.63922.931229.73628.49226.41222.95622.391330.21128.78127.01523.29421.265430.4429.01226.72824.06722.480530.14928.76826.77023.59322.716Mean value30.25030.07528.81026.70022.357BTSTest times14.5344.0213.7913.3973.29524.2973.9563.8523.4973.15234.3964.0253.6673.4193.18144.4533.9613.7823.5943.11854.5074.0833.8793.453.212Mean value4.43744.00923.79423.47143.1916Table 3
List of pre-damaged specimens.
No.Sample number (r × l)Crack thresholdUnloading stressRemarks15 (50 × 100)5 (50 × 50)Crack closure0.25σc22.03The crack closure stage (OA)25 (50 × 100)5 (50 × 50)Crack initiation0.40σc35.25The linear elastic deformation stage (AB)35 (50 × 100)5 (50 × 50)Secondary cracking0.50σc44.07The linear elastic deformation stage (AB)45 (50 × 100)5 (50 × 50)Crack coalescence0.65σc57.29The crack growth stage (BC)55 (50 × 100)5 (50 × 50)Crack damage0.75σc66.10The crack growth stage (BC)


## 3. Pre-damaged Sample Testing Results

For the prepared pre-damaged specimens, a P-wave velocity test and a uniaxial compression test are conducted on ADLINK and INSTRON-1346, respectively, as shown in Figures2 and 3, respectively. To ensure the accuracy of the test result to the greatest extent possible, the ends of the tested samples were areas grinded and covered by petroleum jelly to eliminate the friction effect of sample ends. The testing process and results are vividly shown in Figure 4.Figure 2
Schematic diagram of P-wave velocity test on rocks. (a) LaboratoryP-wave velocity test. (b) P-wave velocity propagation diagram.Figure 3
INSTRON-1346 testing machine.Figure 4
Mechanical property test processes and result for dolomitic limestone. (a) Uniaxial compression test. (b) Brazilian test of rock. (c) Uniaxial compressive failure specimens. (d) Brazilian test failure specimens.
(a)(b)(c)(d)Figure5 plots the stress-strain curve until the failure of each pre-damaged specimen under uniaxial compression. It can be observed from Figure 5 that the stress-strain curve of the pre-damaged specimen contributes to three viewpoints: first, the peak value of the UCS decreases with an increase in the pre-damaged strength; second, the strain in the crack closure stage increases significantly with an increase in the pre-damaged strength; third, the strain corresponding to the peak strength increases significantly with an increase in the pre-damaged strength. The uniaxial compression test results of the pre-damaged specimens are listed in Table 3. Five samples in each group were tested for different pre-damaged sample types. The P-wave velocity results for the pre-damaged dolomitic limestone samples show an obvious fluctuation between 5132.9 m/s and 5769.4 m/s due to the pre-damaged difference. The UCS values extended over a broad range from 70.83 MPa to 87.461 MPa as a result of the different pre-damaged specimens. The dramatic fluctuations of E and BTS can be attributed to the degree of crack growth of the pre-damaged specimens. The results of the vp, E, UCS, and BTS values in Table 3 are illustrated from Figures 6 to 11.Figure 5
Stress-strain curve pre-damaged samples.Figure 6
P-wave velocity tested result and reduction rate trend.
(a)(b)Figure 7
BTS test results and reduction rate trend.
(a)(b)Figure 8
UCS tested result and reduction rate trend.
(a)(b)Figure 9
E tested result and reduction rate trend.
(a)(b)Figure 10
Strain energy calculation model.Figure 11
Damage variable onvp.The variation trend of theP-wave velocity of the pre-damaged specimens with different pre-damaged degrees is illustrated in Figure 6(a). The P-wave velocity of the pre-damaged sample decreases with an increase in the pre-damaged strength, and the decline rates are 0.482%, 2.103%, 4.994%, 8.412%, and 10.223%. As shown in Figure 6(b), it is generally accepted that the decline rate of the P-wave velocity increases linearly with the increasing pre-damaged strength. The determination coefficient R2 of the trend line is 0.97764, and the P-wave velocity of the rocks is closely related to the severity degree of the cracks in the rocks. The pre-damage of the rock samples leads to crack growth and propagation in the rock, which leads to a decrease in the P-wave velocity to varying degrees.Figure7(a) illustrates that the decrease degree of the BTS of the pre-damaged specimens varies with the degree of damage of the pre-damaged specimens. The declining rate of the BTS is shown in Figure 7(b), which are 0.3%, 5.65%, 14.75%, 22.00%, and 29.062%. The variation tendency of the decline rate on the BTS is approximately linear with an increase in the pre-damaged strength. The UCS of the pre-damaged samples is plotted in Figure 8(a). For the UCS corresponding to a pre-damage of 0.25 σc, 0.4 σc, 0.65 σc, 0.75 σc, and 0.5 σc, the pre-damage decreased by 1.59%, 4.361%, 12.962%, 17.016%, and 18.706%, respectively. Figure 8(b) shows that the decline rate of the UCS increases logistically when increasing the pre-damaged strength. The variation tendency of the decline rate on the UCS is different from the BTS. The main reason is that micro-cracks either do not produce a frictional effect or have a small frictional effect during tension. In contrast, the friction effect of micro-cracks plays an important role in the compression process. The trend line of the decline rate on the UCS and BTS shows high determination coefficients (R2) of 0.985 and 0.980. Similar changes were observed for the E values in Figure 9(a). The corresponding decline rates of different degrees of damage are 0.793%, 4.76%, 11.73%, 22.281%, and 26.093%. Figure 9(b) illustrates that the decline rate increases logistic with the increasing pre-damaged strength, which shows the best determination coefficient (R2 = 1).


## 4. Damage Variable

In classical damage mechanics, the damage variable is defined by the ratio of the damage area to the initial area. However, it is difficult to directly measure the initial damage of a rock mass. Therefore, the initial damage of the rock is usually obtained by an indirect calculation of the physical and mechanical parameters.In existing studies, the damage variable is defined using different parameters. A series of theories and experiments is performed to validate its reasonability. In this contribution, several common damage variables are listed in the following, based on the previous study. A comparative study on the damage variables of a pre-damaged sample is conducted under the uniaxial compression test.


### 4.1. Damage Variable on Young’s Modulus

Lemaitre et al. [50] defined the damage index in terms of Young’s modulus of the damaged rock and intact rock, which can be expressed as follows:(1)dE=1−EdE0,where Ed is Young’s modulus of the damaged rock and E0 is Young’s modulus of the undamaged rock.


### 4.2. Damage Variable on theP-Wave Velocity

As is well known, theP-wave velocity of rock is an important parameter. Studies have shown that the P-wave velocity decreases with the increasing degree of crack severity. Thus, the P-wave velocity is adopted to define the damage variable based on the definition of the integrity index of a rock mass [51], as follows:(2)dv=1−vpd2vp02,where vpd is the P-wave velocity of the damaged rock and vp0 is the P-wave velocity of undamaged rock. In this study, the P-wave velocity is determined in the laboratory, with the test process shown in Figure 3.


### 4.3. Damage Variable on the Energy Dissipation

The damage of the rock is mainly characterized by the growth of cracks or flaws. From the viewpoint of continuous damage mechanics, the growth of cracks or flaws means energy dissipation. That is, rock damage involves energy dissipation. The existing research shows that it is feasible to define the damage variable with energy dissipation [34, 52]. A damage variable based on energy dissipation was proposed by Fengnian et al. [52], and the expression is as follows:(3)dw=WdW,where Wd represents the dissipated energy and W represents the constitutive energy. The dissipated energy and the constitutive energy are vividly illustrated in Figure 10. Concerned with the concave stress-strain curve, the constitutive energy is computed using the integration method.


### 4.4. .Damage Variable on the BTS

The BTS is an important parameter of rock properties and is different from the UCS. The friction effect of micro-cracks in rocks has a significant effect on the compressive strength but has little effect on the tensile strength. In engineering, the failure location of the rock mass is mostly due to the location of the existing tensile stress because the tensile strength of the rock mass is far lower than the compressive strength. Thus, whether the BTS can be used to establish the damage variable of the rock is a question worth discussing. Wen et al. [51] discussed the rock mass damage index based on the point load strength, which is a kind of tensile strength that does not consider the friction effect of cracks. However, beyond this, few researchers have discussed this topic. According to the research results of study, the damage index of a rock mass can be defined using the point load strength as follows:(4)dI=1−ISDIS,where ISD is the point load strength of the damaged rock sample and IS is the point load strength of the undamaged rock sample. Considering that both IS and BTS are tensile strengths, this paper attempts to use the BTS ratio to determine damage variables as follows:(5)dt=1−BTSDBTSI,where BTSD is the BTS of damaged rock and BTSI is the BTS of undamaged rock.


## 4.1. Damage Variable on Young’s Modulus

Lemaitre et al. [50] defined the damage index in terms of Young’s modulus of the damaged rock and intact rock, which can be expressed as follows:(1)dE=1−EdE0,where Ed is Young’s modulus of the damaged rock and E0 is Young’s modulus of the undamaged rock.


## 4.2. Damage Variable on theP-Wave Velocity

As is well known, theP-wave velocity of rock is an important parameter. Studies have shown that the P-wave velocity decreases with the increasing degree of crack severity. Thus, the P-wave velocity is adopted to define the damage variable based on the definition of the integrity index of a rock mass [51], as follows:(2)dv=1−vpd2vp02,where vpd is the P-wave velocity of the damaged rock and vp0 is the P-wave velocity of undamaged rock. In this study, the P-wave velocity is determined in the laboratory, with the test process shown in Figure 3.


## 4.3. Damage Variable on the Energy Dissipation

The damage of the rock is mainly characterized by the growth of cracks or flaws. From the viewpoint of continuous damage mechanics, the growth of cracks or flaws means energy dissipation. That is, rock damage involves energy dissipation. The existing research shows that it is feasible to define the damage variable with energy dissipation [34, 52]. A damage variable based on energy dissipation was proposed by Fengnian et al. [52], and the expression is as follows:(3)dw=WdW,where Wd represents the dissipated energy and W represents the constitutive energy. The dissipated energy and the constitutive energy are vividly illustrated in Figure 10. Concerned with the concave stress-strain curve, the constitutive energy is computed using the integration method.


## 4.4. .Damage Variable on the BTS

The BTS is an important parameter of rock properties and is different from the UCS. The friction effect of micro-cracks in rocks has a significant effect on the compressive strength but has little effect on the tensile strength. In engineering, the failure location of the rock mass is mostly due to the location of the existing tensile stress because the tensile strength of the rock mass is far lower than the compressive strength. Thus, whether the BTS can be used to establish the damage variable of the rock is a question worth discussing. Wen et al. [51] discussed the rock mass damage index based on the point load strength, which is a kind of tensile strength that does not consider the friction effect of cracks. However, beyond this, few researchers have discussed this topic. According to the research results of study, the damage index of a rock mass can be defined using the point load strength as follows:(4)dI=1−ISDIS,where ISD is the point load strength of the damaged rock sample and IS is the point load strength of the undamaged rock sample. Considering that both IS and BTS are tensile strengths, this paper attempts to use the BTS ratio to determine damage variables as follows:(5)dt=1−BTSDBTSI,where BTSD is the BTS of damaged rock and BTSI is the BTS of undamaged rock.


## 5. A Comparative Study on the Damage Variable

### 5.1. Computed Damage Variable

Based on the definition of the damage variable in Section4, the damage variables are conducted in accordance with equations 1, (2), (3), and (5). The computed results of damage variable are listed in Table 4. We can find there are negative damage variables in Table 4. It is caused by the difference of test samples.Table 4
The result of damage variable.
NamePre-damaged strength0.25σc0.40σc0.50σc0.65σc0.75σc22.0335.2544.0757.2966.1vpTest times10.0240.0430.1150.1730.18820.0160.0260.0910.1610.2053−0.0040.0480.0900.1500.19140.0070.0530.1000.1660.18250.0060.0380.0910.1560.204Mean value0.0100.0420.0970.1610.194ETest times10.0140.0400.1210.2190.24220.0170.0580.1270.2410.26030.0010.0490.1070.2300.2974(0.006)0.0410.1160.2040.25750.0030.0490.1150.2200.249Mean value0.0060.0470.1170.2230.261BTSTest times1−0.0190.0970.1480.2370.26020.0350.1110.1340.2140.29230.0120.0960.1760.2320.2854−0.0010.1100.1500.1920.2995−0.0130.0830.1280.2250.278Mean value0.0030.0990.1470.2200.283WTest times10.01080.0580.0990.1790.21620.01360.0570.1220.1710.22830.01030.0610.1060.1860.22740.01170.0530.1130.1770.21250.01290.0440.1020.1940.233Mean value0.011890.05460.10840.18140.2232It is obviously accepted that each damage variable increases with the increasing pre-damaged strength. In addition, we can easily find that the damage variable decreases with BTS andE. As shown in Figure 9(b), the damage variable on E logistically increases with the increasing pre-damaged strength. The damage variable on E shows the best determination coefficient (R2), which is 1. The damage variable on the BTS, W, and vp is illustrated in Figures 7(b), 11, and 12, respectively, which all linearly increase with the increasing pre-damaged strength. The determination coefficient of the damage variables on BTS, W, and vp is 0.980, 0.993, and 0.978, respectively.Figure 12
Damage variable onW.The change in the damage variable is illustrated in Figure13. Although these damage variables all show good correlation coefficients under a variable pre-damaged strength, the damage variable on E shows the fastest trend of change from the crack initiation stage to the crack coalescence stage. We can easily find from Figure 13 that the rock specimens are hardly damaged before the crack closure stage. In other words, no new cracks will occur during the crack closure stage. After the crack closure stage, the value of the damage variable on the BTS is the largest among damage variables, which is the same trend as the damage variable on vp and W. When the pre-damaged strength is small, the damage variable on E is close to the damage variable on vp and W. When the pre-damaged strength is large, the damage variable on E is close to the damage variable on the BTS. The main reason is that the test process of the BTS has no friction effect for cracks in rock. Thus, the more the cracks grow, the lower the BTS will be. However, under compression loading, the friction effect of micro-cracks with a certain dip angle is beneficial to enhancing the compressive strength and postpeak strength. It is generally accepted that the friction effect of the fracture surface weakens accordingly when the micro-crack develops into a macro-crack. In this regard, in the crack damage stage the damage variable on E is close to the damage variable on the BTS. The damage variable on vp shows the density of the micro-cracks in the rocks, which increase with the growth of the cracks. The damage variable on W (energy dissipation) is close to the damage variable on vp; this further indicates that the growth of the cracks is accompanied by energy dissipation.Figure 13
The mean of damage variables.


### 5.2. Strength Verification Based on the Damage Variables

The tested UCS and the measured UCS are performed to describe the accuracy of the damage variables. According to the strain equivalence hypothesis, J. Lemaitre [53] proposed stress-strain relations under damage states and uniaxial compression, as follows:(6)σ=1−dEε.The studies of Liu et al. [34] show that equation (6) can be used to calculate the peak strength but cannot predict the damage and deformation evolution before the peak strength. Then, the UCS of a pre-damaged sample can be obtained by the following formula:(7)σd=1−dσc,where d is the damage variable, σc is the UCS of the intact rock, and σd is the UCS of the pre-damaged sample.The computed value and tested value are illustrated in Figures14–17 using the 1 : 1 slope line. The dataset and fit linear located on the 1 : 1 slope line indicate an exact correlation. The greater the approach from the slope line, the higher the accuracy. As shown in Figure 14, the computed UCS using the damage variable on vp is in good agreement with the tested UCS of the pre-damaged sample. The linear fit displays a good determination coefficient (R2) of 0.93678. We can accept from Figure 15 that the fitting line deviates from the slope line to some extent, and the determination coefficient (R2) is 0.91902. When the UCS value is large (or the degree of damage is small), the computed value is close to the tested value. That is, the more cracks that develop, the more inaccurate the damage variables are on E, mainly due to the existence of cracks leading to the transformation of the rock deformation from elastic to plastic deformation.Figure 14
Cross-correlation of tested and computed values from damage variable onvp.Figure 15
Cross-correlation of tested and computed values from damage variable onE.Figure 16
Cross-correlation of tested and computed values from damage variable on BTS.Figure 17
Cross-correlation of tested and computed values from damage variable onW.Both Figures16 and 17 show similar variable trends to Figure 15, and their determination coefficients (R2) are 0.89164 and 0.93135, respectively. In addition, when the degree of damage of a rock sample is large, we can easily find from Figures 15–17 that the UCS tested value is greater than the computed value. This result indicates that the damage variable calculated by BTS, E, and W is larger than the actual damage variable of the damaged specimen. Therefore, when the rock mass is slightly damaged (at the stage of micro-crack development), the above methods of defining the damage variables are reasonable and feasible. When the rock mass is seriously damaged (after the stage of micro-crack coalescence), the damage variable defined by the P-wave velocity has the highest accuracy compared with the other three methods (damage variable on BTS, E, and W).


## 5.1. Computed Damage Variable

Based on the definition of the damage variable in Section4, the damage variables are conducted in accordance with equations 1, (2), (3), and (5). The computed results of damage variable are listed in Table 4. We can find there are negative damage variables in Table 4. It is caused by the difference of test samples.Table 4
The result of damage variable.
NamePre-damaged strength0.25σc0.40σc0.50σc0.65σc0.75σc22.0335.2544.0757.2966.1vpTest times10.0240.0430.1150.1730.18820.0160.0260.0910.1610.2053−0.0040.0480.0900.1500.19140.0070.0530.1000.1660.18250.0060.0380.0910.1560.204Mean value0.0100.0420.0970.1610.194ETest times10.0140.0400.1210.2190.24220.0170.0580.1270.2410.26030.0010.0490.1070.2300.2974(0.006)0.0410.1160.2040.25750.0030.0490.1150.2200.249Mean value0.0060.0470.1170.2230.261BTSTest times1−0.0190.0970.1480.2370.26020.0350.1110.1340.2140.29230.0120.0960.1760.2320.2854−0.0010.1100.1500.1920.2995−0.0130.0830.1280.2250.278Mean value0.0030.0990.1470.2200.283WTest times10.01080.0580.0990.1790.21620.01360.0570.1220.1710.22830.01030.0610.1060.1860.22740.01170.0530.1130.1770.21250.01290.0440.1020.1940.233Mean value0.011890.05460.10840.18140.2232It is obviously accepted that each damage variable increases with the increasing pre-damaged strength. In addition, we can easily find that the damage variable decreases with BTS andE. As shown in Figure 9(b), the damage variable on E logistically increases with the increasing pre-damaged strength. The damage variable on E shows the best determination coefficient (R2), which is 1. The damage variable on the BTS, W, and vp is illustrated in Figures 7(b), 11, and 12, respectively, which all linearly increase with the increasing pre-damaged strength. The determination coefficient of the damage variables on BTS, W, and vp is 0.980, 0.993, and 0.978, respectively.Figure 12
Damage variable onW.The change in the damage variable is illustrated in Figure13. Although these damage variables all show good correlation coefficients under a variable pre-damaged strength, the damage variable on E shows the fastest trend of change from the crack initiation stage to the crack coalescence stage. We can easily find from Figure 13 that the rock specimens are hardly damaged before the crack closure stage. In other words, no new cracks will occur during the crack closure stage. After the crack closure stage, the value of the damage variable on the BTS is the largest among damage variables, which is the same trend as the damage variable on vp and W. When the pre-damaged strength is small, the damage variable on E is close to the damage variable on vp and W. When the pre-damaged strength is large, the damage variable on E is close to the damage variable on the BTS. The main reason is that the test process of the BTS has no friction effect for cracks in rock. Thus, the more the cracks grow, the lower the BTS will be. However, under compression loading, the friction effect of micro-cracks with a certain dip angle is beneficial to enhancing the compressive strength and postpeak strength. It is generally accepted that the friction effect of the fracture surface weakens accordingly when the micro-crack develops into a macro-crack. In this regard, in the crack damage stage the damage variable on E is close to the damage variable on the BTS. The damage variable on vp shows the density of the micro-cracks in the rocks, which increase with the growth of the cracks. The damage variable on W (energy dissipation) is close to the damage variable on vp; this further indicates that the growth of the cracks is accompanied by energy dissipation.Figure 13
The mean of damage variables.


## 5.2. Strength Verification Based on the Damage Variables

The tested UCS and the measured UCS are performed to describe the accuracy of the damage variables. According to the strain equivalence hypothesis, J. Lemaitre [53] proposed stress-strain relations under damage states and uniaxial compression, as follows:(6)σ=1−dEε.The studies of Liu et al. [34] show that equation (6) can be used to calculate the peak strength but cannot predict the damage and deformation evolution before the peak strength. Then, the UCS of a pre-damaged sample can be obtained by the following formula:(7)σd=1−dσc,where d is the damage variable, σc is the UCS of the intact rock, and σd is the UCS of the pre-damaged sample.The computed value and tested value are illustrated in Figures14–17 using the 1 : 1 slope line. The dataset and fit linear located on the 1 : 1 slope line indicate an exact correlation. The greater the approach from the slope line, the higher the accuracy. As shown in Figure 14, the computed UCS using the damage variable on vp is in good agreement with the tested UCS of the pre-damaged sample. The linear fit displays a good determination coefficient (R2) of 0.93678. We can accept from Figure 15 that the fitting line deviates from the slope line to some extent, and the determination coefficient (R2) is 0.91902. When the UCS value is large (or the degree of damage is small), the computed value is close to the tested value. That is, the more cracks that develop, the more inaccurate the damage variables are on E, mainly due to the existence of cracks leading to the transformation of the rock deformation from elastic to plastic deformation.Figure 14
Cross-correlation of tested and computed values from damage variable onvp.Figure 15
Cross-correlation of tested and computed values from damage variable onE.Figure 16
Cross-correlation of tested and computed values from damage variable on BTS.Figure 17
Cross-correlation of tested and computed values from damage variable onW.Both Figures16 and 17 show similar variable trends to Figure 15, and their determination coefficients (R2) are 0.89164 and 0.93135, respectively. In addition, when the degree of damage of a rock sample is large, we can easily find from Figures 15–17 that the UCS tested value is greater than the computed value. This result indicates that the damage variable calculated by BTS, E, and W is larger than the actual damage variable of the damaged specimen. Therefore, when the rock mass is slightly damaged (at the stage of micro-crack development), the above methods of defining the damage variables are reasonable and feasible. When the rock mass is seriously damaged (after the stage of micro-crack coalescence), the damage variable defined by the P-wave velocity has the highest accuracy compared with the other three methods (damage variable on BTS, E, and W).


## 6. Discussion

An engineering rock mass is often subjected to geological activities and engineering disturbances, which results in a certain initial damage of the engineering rock mass. In the laboratory study of rock mechanics, the initial damage of the rock is often neglected. In this paper, several methods of defining the initial damage are studied experimentally, and their accuracies and feasibilities are compared and discussed. Based on the several stages of the uniaxial compression failure process of rock, several specimens with different degrees of damage are used to demonstrate the mechanical properties and initial damage variables.At the stage of micro-crack development (no macro-crack occurs), the change in the damage variables of pre-damaged rock samples is small. The predicted UCS value using the damage variable is close to the measured value. When the micro-crack coalesces into a macro-crack, the damage variable of the pre-damaged rock sample varies greatly. There is a significant deviation between the predicted UCS value using the damage variable and the measured UCS value. From the point of view of micro-crack growth, when the rock damage mainly exists in the form of micro-cracks, micro-crack propagation and compaction under a load produce a small local deformation and accumulate macro-deformation. Therefore, the rock damage variable is mainly determined by the micro-crack density and micro-crack growth morphology. When micro-cracks coalesce to form macro-cracks, the deformation, load-bearing capacity, and damage variables of the specimens are mainly controlled by macro-cracks, with micro-cracks having little influence on them. In this regard, the pre-damaged rock specimen is used to perform mechanical testing. The mechanical parameters of the pre-damaged rock specimen are employed to discuss the damage variable, which is of far-reaching theoretical significance and a broad application value.


## 7. Conclusions

This work focused on performing a comparative study of the damage variables of four calculations. Pre-damaged specimens were prepared to conduct a series of laboratory tests. The experimental results from the present study show thatE, BTS, and vp decrease by varying degrees with the increasing pre-damaged strength. The BTS and vp linearly decrease with the increasing pre-damaged strength, and the UCS and E nonlinearly decrease with the increasing pre-damaged strength. Considering the frictional effect of cracks, the BTS has the greatest descent rate, which shows that the friction effect of cracks plays an important role in the process of rock mass compression.Uniaxial compression pre-damage causes crack growth in the rock specimens. The shape of the crack in the pre-damaged rock sample is similar to that of a crack in a natural rock mass. Therefore, comparative analysis of the damage variables of pre-damaged samples is of great significance for studying the initial damage of an engineering rock mass. The comparative analysis results show that the damage variables of the different calculation methods are different. The UCS under different damage states is calculated by combining the equivalent strain hypothesis and damage variables and comparing them with the measured values. In current study, the conclusions are obtained as follows:(1)
The results show that the damage variable onvp and W has high accuracy. Although the damage variables calculated by other methods are highly accurate in a low damage state, they are not suitable for a high damage state.(2)
When the degree of damage is low, the differences of other damage variables, except those calculated by the BTS, are slight. When the degree of damage is large, the difference in the damage variables is large.(3)
In particular, due to the convenience of the damage variables onvp, there is a broad application prospect for evaluating the damage variable of the engineering rock mass by the P-wave velocity.(4)
In view of the difference among calculation methods of damage variable, it is suggested here that equations (4) and (5) should be employed to calculate damage factor in the region of tensile stress, and equations (1) and (3) should be hired to calculate damage factor in the region of compressive stress. For the area where tensile stress and compressive stress alternate, a method should be set up to realize the conversion of damage coefficient calculation method [54].



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*Source: 3583754-2022-06-10.xml*