J. Cent. South Univ. (2019) 26: 3045-3056

DOI: https://doi.org/10.1007/s11771-019-4235-z

Crack evolution behavior of rocks under confining pressures and its propagation model before peak stress

ZUO Jian-ping(左建平)^{1, 3}, CHEN Yan(陈岩)², LIU Xiao-li(刘晓丽)⁴

1. School of Mechanics and Civil Engineering, China University of Mining and Technology (Beijing), Beijing 100083, China;

2. Henan Key Laboratory for Green and Efficient Mining & Comprehensive Utilization of Mineral Resources, Henan Polytechnic University, Jiaozuo 454003, China;

3. State Key Laboratory of Coal Resources and Safe Mining, China University of Mining and Technology (Beijing), Beijing 100083, China;

4. State Key Laboratory of Hydroscience and Hydraulic Engineering, Tsinghua University,Beijing 100084, China

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Abstract: The understanding of crack propagation characteristics and law of rocks during the loading process is of great significance for the exploitation and support of rock engineering. In this study, the crack propagation behavior of rocks in triaxial compression tests was investigated in detail. The main conclusions were as follows: 1) According to the evolution characteristics of crack axial strain, the differential stress-strain curve of rocks under triaxial compressive condition can be divided into three phases which are linear elastic phase, crack propagation phase, post peak phase, respectively; 2) The proposed models are applied to comparison with the test data of rocks under triaxial compressive condition and different temperatures. The theoretical data calculated by the models are in good agreement with the laboratory data, indicating that the proposed model can be applied to describing the crack propagation behavior and the nonlinear properties of rocks under triaxial compressive condition; 3) The inelastic compliance and crack initiation strain in the proposed model have a decrease trend with the increase of confining pressure and temperature. Peak crack axial strain increases nonlinearly with the inelastic compliance and the increase rate increases gradually. Crack initiation strain has a linear relation with peak crack axial strain.

Key words: crack strain; crack propagation behavior; crack propagation model; stress-strain relationship

Cite this article as: ZUO Jian-ping, CHEN Yan, LIU Xiao-li. Crack evolution behavior of rocks under confining pressures and its propagation model before peak stress [J]. Journal of Central South University, 2019, 26(11): 3045-3056. DOI: https://doi.org/10.1007/s11771-019-4235-z.

1 Introduction

In rock engineering, the failure of rocks can lead to disasters, such as coal bump and water inrush in mining engineering and rock burst in tunnel engineering [1-4]. Crack evolution of rocks, such as crack initiation, crack propagation and crack coalescence, has a close relation with rock failure in rock engineering [5, 6]. But crack propagation is a crucial factor that leads to the failure of rocks and induces the rock disasters.Therefore, it is necessary to identify the crack propagation characteristics quantitively.

To study the deformation and strength behavior of rocks, laboratory tests (including conventional uniaxial and triaxial compressive tests) have been carried out. Several famous monographs have systematically summarized the newest progresses in these fields where the specimens are various rocks, i.e., marble, granite and sandstone [7-11]. Also, fractures in natural rocks have great important effect on the strength and failure behavior of rock mass which are often evaluated in rock engineering. Therefore, the laboratory and numerical tests including uniaxial and triaxial compression of the rocks or rock-like materials containing pre-existing flaws were carried out to study the failure behavior and crack evolution characteristics [12-17]. In the aspect of meso- mechanics, the effects of off-notch location on the crack initiation and propagation behaviors were examined by in-situ observation with scanning electron microscope (SEM) [18] which indicated that the strength, failure patterns, deformation properties of the rocks containing the pre-exiting flaws evidently depended on the crack evolution. Meanwhile, acoustic emission (AE) and computerized tomography (CT) have been widely applied to obtaining the crack evolution characteristics of rocks during loading process [19-23].

Based on the experimental observations, several strength criteria, such as Mogi criterion [24], Hoek-Brown criterion [25], exponential criterion [26] and three-dimensional nonlinear strength criterion [27] were developed for estimating the strengths of rock mass. ZUO et al [28] proved that Hoek-Brown criterion is reasonable and the parameters in Hoek-Brown criterion can be obtained based on fracture mechanics. The relationship of stress-strain was also studied through the laboratory tests of rocks, i.e., anisotropic stress-strain relationship based on a discontinuity tensor [29], the classical microscopic model based on the bonds between atoms or molecules [30], stress-strain relation for joint rock masses based on statistical models [31], statistical damage model for rocks considering residual strength [32], nonlinear rheological damage model of hard rock [33] and thermal damage constitutive model considering damage threshold and residual strength [34]. According to the idea of crack strain, ZUO et al [35] proposed a crack closure model (Eq. 1(a)) and pre-peak stress-strain curves (Eq. 1(b)) based on the evolution of crack axial strain. And the results show a good agreement between theoretical data and test data which can be seen in Figure 1.

      (1)

where is the crack closure strain; is the largest crack closure strain; E^cc is the equivalent elastic modulus of crack when it is closing; ε₁ is the axial strain; σ₁ is the axial stress; E is elastic modulus.

Rocks are a kind of complex geological materials and always show brittle failure in uniaxial compression tests, which means that small crack propagation can cause the rock failure. After applying the confining pressure, brittle-ductile transition of rocks can occur because of the constraint effect of the confining pressure to the cracks and the apparent phenomenon of crack propagation can be observed in stress-strain curves. Therefore, it is significant to investigate the crack propagation behavior of rocks under the influence of confining pressures for revealing the rock failure mechanism. In this study, the crack propagation behavior of Mentougou basalt and Tournemire shale under confining pressures is analyzed. Then, a nonlinear model and its corresponding stress-strain relation model are proposed to describe the crack propagation behavior and stress-strain relationship of rocks. The proposed model and the corresponding stress-strain relation model are evaluated by the published data from literatures. Finally, the influence of confining pressure and temperature on the parameters in the proposed model and the relationship between peak crack axial strain are analyzed and discussed.

Figure 1 Verification of crack closure model (data from ZUO et al [35]):

2 Crack evolution behavior of rocks under confining pressures

2.1 Crack strain of rocks during loading process

In underground rock engineering, most of rocks are in uniaxial or triaxial compression state, such as coal pillar or surrounding rocks of roadway. During the excavation process, the tensile or shear cracks affect the mechanical properties of rocks or rock mass, which have a significant influence on the safety of rock engineering. In order to analyze crack evolution of rocks during the failure process quantitatively, definition of crack strain was proposed and widely used [36-38]. Under uniaxial compression, crack strain of principal strains is calculated by removing calculated elastic strains [34-36].

                        (2)

where , and are the crack axial strain, crack radial strain and crack volumetric strain, respectively; ε₃ and ε_v are radial strain and volumetric strain, respectively; μ is Poisson ratio.

At the state of conventional triaxial compression, the crack strains can be calculated as follows:

                (3)

Based on crack volumetric strain, different characteristic phases of stress-strain curves in Figure 2 have been studied in former literatures [36-40]. Firstly, the stress-strain curve displays an initial nonlinear phase due to the closure of original cracks. Phase II is linear elastic, which represents the elastic deformation of the rock matrix. Due to the stress concentration at the tip of cracks, with the increasing of axial load, cracks or flaws start initiating and stable propagating, which is named phase III. In this phase, it is difficult to figure out the crack initiation strain based on stress-strain curve, because this phase seems like linear elastic phase. Then, phase IV represents the unstable crack growth stage until the stress reaches the peak value because of crack propagation. The last phase is post-peak stage where the stress drops till residual strength.

Figure 2 Progressive failure process of rocks (see PENG et al [38]) modified from MARTIN et al [36]. Crack initiation strainis produced at stable crack growth phase)

2.2 Crack propagation behavior of rocks under confining pressure

Triaxial compression tests for rocks have been conducted by many researchers. By using servo controlled testing machine, ZHOU et al [41] conducted triaxial compression on Mentougou basalt which was excavated in Datai coal mine (located in Mentougou district, Beijing). The basalt stratum is basic igneous extrusive rock, 180-200 millions of years from present and with a dip angle of 70°-80° as floor. The X-ray diffractometry (XRD) analysis indicated that Mentougou basalt is mainly composed of quartz, anorthosite, chinochlorite, epidote, clay mineral, as well as a minor quantity of iron oxides.

MASRI et al [42] presented the effect of temperature on the mechanical behavior of Tournemire shale taken from Massif Central region of France. The average mineralogical compositions of the shale are 27.5% kaolinite, 16.5% illite, 19% quartz, 15% calcite, 2.7% chlorite, 8.3% I/S (interstratifier) and 11% others (pyrite, siderite, feldspars, etc) [42]. Figure 3 provides the differential stress-strain curves of the Mentougou basalt under different confining pressures (σ₃=5, 10, 20, 30, 40 MPa) and Tournemire shale at the confining pressure of 20 MPa with different temperatures (T=20, 100, 150, 200, 250 °C). From Figure 3, it is clearly seen that, the stress-strain curves of Mentougou basalt and Tournemire shale under confining pressures display nearly none crack closure phase and show apparent linear elastic phase.

Figure 3 Differential stress-strain curves of rocks under confining pressures:

On the basis of the definition of crack strain in Eq. (2), the crack axial strain of Mentougou basalt (CTC-510-1) and Tournemire shale (T=200 °C) can be obtained and the relationship between crack axial strain and differential stress can be seen in Figure 4. One can observe from Figure 4 that the differential stress-crack axial strain curves can be divided into three phases, namely linear elastic phase, crack propagation phase, post peak phase. Firstly, with the increasing of differential stress, the crack axial strain nearly equals 0 which means that the rock is in elastic phase. Secondly, with the increasing of differential stress to σ_acd, the crack strain increases, which means the crack in the rock starts propagating. Here, σ_acd is crack damage stress based on crack axial strain. After the differential stress reaches to peak strength, the rock occurs failure and post peak phase begins. In Figure 4, is the peak crack axial strain corresponding to the peak strength.

Figure 4 Representative differential stress-crack axial strain curves and axial strain-crack axial strain curves:

3 Crack propagation model and stress- strain relationship model

Rocks are composed of matrix and a large amount of microcracks. Therefore, the deformation of rocks can be represented as the sum of an average matrix strain and crack strain due to the presence of microcracks [43, 44]. In this section, cracks are regarded as a special material, and have its own physical (size, density, mass) and mechanical (strength, elastic modulus, Poisson ratio) properties. Figure 5 shows the diagram of crack evolution process of rocks under triaxial compression. The height of the rock sample is equal to the sum of H_c (height of original cracks) and H_m (height of rock matrix) which can be seen in Figure 5(a). LIU et al [45] considered that rocks are composed of two kinds of materials, namely, “soft part” and “hard part”. Here, the cracks located in rocks are the “soft part”, while rock matrix can be regarded as “hard part”. The deformation of the “soft part” should be described by natural strain and “hard part” can be described by engineering strain. Natural stain is defined as the ratio of the change of length to the length under current stress state. While engineering strain is defined as the ratio of the change of length to the original length [46, 47], which can be expressed as

                                (4)

where ε_n is natural strain; ε_e is engineering strain; Δl is change of the length; l is the length under current stress state; l₀ is the original length.

At present, cracks are regarded as “soft part” and their strain can be described by natural strain. According to the definition of natural strain, crack strain (here compressive direction is positive and tensile direction is negative) is then given by

                               (5)

where h_c is the height of crack when it starts propagation which can be seen in Figure 5(d).

When the stress reaches to σ_acd, under the influence of the effective differential stress (difference between (σ₁-σ₃) and σ_acd, σ_e=(σ₁-σ₃)-σ_acd), the crack starts propagation. Here, we remove the origin of coordinate (0, 0) to ( σ_acd), and regard σ_acd) point as the initial condition (origin of coordinate) when the crack starts propagation, which can be seen in Figure 6. A uniformly distributed force is imposed on the surface of the rock specimen (including cracks and the matrix) and σ_e can be obtained.

                              (6)

where represents the deformation degree of cracks under axial load and is named as inelastic compliance of cracks.

The expression of σ_e can be obtained by substituting Eq. (5) into Eq. (6) and integrating

                            (7)

where C is integration constant.

After applying the confining pressure, the rocks are in the state of hydrostatic pressure, and the pre-existing cracks are closed (Figure 5(b)).

Figure 5 Diagram of two parts (cracks and matrix) of rocks:

Figure 6 A diagram showing starting point when crack begins generating under effective differential stress and crack initiation stain

Then, during the loading process, the crack starts initiating. In other words, cracks have been produced before the differential reaches to crack damage stress. However, it is difficult to describe the cracks between crack initiation stress and crack damage stress because the cracks are very small that hardly can be captured by test machine. Therefore, we assume that H′_c is the height of crack initiation which is very small before the crack propagation (Figure 5(c)). When crack starts propagating (Figure 5(d)), the initial condition can be obtained: σ₁-σ₃=σ_acd, σ_e=0 and H′_c=h_c. Applying the two initial conditions into Eq. (7), the integration constant C is

                              (8)

Substituting Eq. (8) into Eq. (7), and changing the form, the axial displacement of cracks (Δ) can be obtained

             (9)

Based on Eq. (9), after dividing the height of the rock specimen, the relationship between and σ_e is

                   (10)

According to Eq. (10), and using to replace H′_c/H, the crack propagation model of rocks under triaxial compression is obtained:

        (11)

where is the crack initiation strain produced between crack initiation stress and crack damage stress which can be seen in Figures 2 and 5(c).

Then, applying Eq. (11) into Eq. (3), at crack propagation phase (σ_acd≤σ₁-σ₃≤(σ₁-σ₃)_p), the relationship between differential stress and axial strain is obtained:

                        (12)

At linear elastic phase, the crack axial strain is nearly equal to zero. Therefore, at linear elastic phase (σ₁-σ₃<σ_acd) of the rocks, the relationship of differential stress-strain can be expressed:

                       (13)

From Eqs. (11), (12) and (13), the theoretical data of crack axial strain and axial strain ε₁ can be calculated.

During uniaxial compression tests, through the condition of σ₃=0 in Eq. (11), the crack propagation model at the state of uniaxial compression can be obtained:

             (14)

4 Evaluation of crack propagation model

4.1 Comparison between theoretical data and test data of rocks

According to the axial strain obtained by test machine and experimental data of crack axial strain obtained by Eq. (3), under the influence of confining pressure, the theoretical data of crack axial strain of Mentougou basalt and Tournemire shale can be calculated by Eq. (11) as shown in Figure 7. It shows that the theoretical data calculated by crack propagation model are in good agreement with the test data and the behavior of crack propagation is well described in both Mentougou basalt and Tournemire shale.

The mechanical parameters of crack propagation model evaluated by Mentougou basalt and Tournemire shale are listed in Tables 1 and 2. One can see from Table 1 that the crack initiation strain seems very small, nearly equal to zero, such as 7.31×10^-5 (CTC510-1), which means that the rock specimen generates very few cracks between crack initiation stress and crack damage stress. Figure 8 presents the influence of confining pressure and temperature to the parameters and It can be seen from Figure 8(a) that with the increasing of σ₃, both and decrease, which indicates that when the cracks are in higher confining pressure state, the deformation degree of cracks in the axial direction is weaker and less cracks initiate at elastic phase. From Figure 8(b), one can see that the parameters and have a decrease trend with the increase of temperature, which indicates that the higher temperature can make the deformation degree of cracks in axial direction become weaker and less cracks initiate at elastic phase.

Figure 7 Differential stress-crack axial strain curves of test data (scatter point) and theoretical data (full line) under confining pressures:

Table 1 Parameters of crack propagation model of Mentougou basalt

Table 2 Parameters of crack propagation model of Tournemire shale

Figure 8 Influence of confining pressure (a) and temperature (b) on parameters and

Figure 9 shows the theoretical data and test data of peak crack axial strain of Mentougou basalt and Tournemire shale. It verifies that the theoretical and measured values of peak crack axial strain have little difference which indicates good applicability of crack propagation model.

Figure 10 shows comparison of theoretical differential stress-strain curves and the test differential stress-strain curves of Mentougou basalt and Tournemire shale. Seeing from Figure 10,the good agreement between theoretical data and test data indicates that the proposed stress-strain relationship model is used to describe the behavior of stress-strain curve before peak strength well.

Figure 9 Comparison of theoretical and test peak crack axial strain of Mentougou basalt and Tournemire shale

Figure 10 Differential stress-strain curves of test data (scatter point) and theoretical data (full line) under confining pressures:

4.2 Discussion of model parameters

In upper analysis, the two parametersand are hardly to measure their values through laboratory tests. However, they have a significant influence on the crack propagation behavior. Discussion of the two parameters on crack propagation behavior of basalt sample (CTC510-1) is analyzed, as shown in Figure 11. One can observe from Figure 11 that when the inelastic compliance is equal to 15.16 GPa^-1, with the increasing of crack initiation strain the crack propagation phase becomes larger and peak crack axial strain increases. From Figure 11(b), it can be seen that, when is equal to 7.05×10^-5, with the decrease ofthe crack propagation phase becomes smaller and decreases.

Figure 11 Predicted differential stress-crack axial strain curves of basalt sample (CTC510-1) with different values:

Figure 12 shows the relationship between and It can be seen from Figure 12(a) that larger crack initiation strain results in apparently larger peak crack axial strain which presents apparently linear relationship betweenand Figure 12(b) shows that with the increase of increases nonlinearly and the increasing rate becomes larger.

Figure 12 Variations of peak crack axial strain with relationship between and(a) and relationship between and (b)

5 Conclusions

Crack propagation behavior of rocks under confining pressures is investigated in detail in this paper. Based on the published test data, the crack propagation behaviors of Mentougou basalt and Tournemire shale under confining pressures were analyzed. On the basis of broadly used definition of crack axial strain, the differential stress-strain curves of Mentougou basalt and Tournemire shale can be divided into three phases (linear elastic phase, crack propagation phase and post peak phase). Rocks are composed of matrix and crack, and both of them have their own physical and mechanical properties. The deformation of crack during loading process can be described by natural strain. And the deformation of the other parts during loading process can be described by engineering strain. Then, crack propagation model and corresponding stress-strain relation model are derived to describe the crack propagation and stress-strain relationship.

The proposed model is evaluated by Mentougou basalt and Tournemire shale, and the good agreement between the theoretical and laboratory data of the pre-peak stress-strain curves indicates that crack propagation model is able to capture the crack propagation behavior in compression tests of rocks. With the increase of confining pressure and temperature, both inelastic compliance and crack initiation strain show a decrease trend. The relations between peak crack axial strain and crack initiation strain and inelastic compliance are discussed and show that with the increase of crack initiation strain, peak crack axial strain increases linearly. While, with the increase of inelastic compliance, peak crack axial strain increases nonlinearly and the increase rate becomes larger.

Notation list

Crack closure strain

Largest crack closure strain

E^cc

Equivalent elastic modulus of closure crack

ε₁

Axial strain

σ₁

Axial stress

σ₃

Confining pressure

E

Elastic modulus

μ

Poisson ratio

Crack axial strain

Crack radial strain

Crack volumetric strain

ε₃

Radial strain

ε_v

Volumetric strain

Crack initiation strain

Peak crack axial strain

σ_acd

Crack damage stress

(σ₁-σ₃)_p

Peak strength

ε_n

Natural strain

ε_e

Engineering strain

Δl

Change of the length

l, l₀

Length under current stress state and original state

h_c

Height of crack

σ₁-σ₃

Differential stress

σ_e

Effective differential stress

Inelastic compliance of cracks

C

Integration constant

H_c

Height of original crack

Δ

Axial displacement of crack

H′_c

Height of crack initiation

H_m

Height of rock matrix

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(Edited by YANG Hua)

中文导读

围压作用下岩石峰前裂纹演化行为及其扩展模型

摘要：理解岩石在加载过程中的裂纹演化特征及规律对于岩体工程开挖与支护具有重要意义。本文详细地研究了围压作用下岩石的裂纹扩展行为，主要结论如下：1)依据裂纹轴向应变的演化特征，围压作用下岩石的差应力-应变曲线可以分为三个阶段，分别为线弹性阶段、裂纹扩展阶段及峰后阶段；2)建立围压作用下岩石的裂纹扩展模型，并进行了验证，发现通过理论模型计算得出的理论数据与试验数据吻合度较高，验证了裂纹扩展模型可以被用来描述围压作用下岩石的非线性裂纹扩展行为；3)裂纹扩展模型中岩石裂纹非弹性柔度和裂纹起裂应变随着围压的增大和温度的升高呈减小趋势。峰值裂纹轴向应变随着非弹性柔度的增大呈非线性增大趋势，其增大速率逐渐降低，而裂纹起裂应变与峰值裂纹轴向应变呈线性关系。

关键词：裂纹应变；裂纹扩展行为；裂纹扩展模型；应力-应变关系

Foundation item: Project(51622404) supported by Outstanding Youth Science Foundation of the National Natural Science Foundation of China; Projects(51374215, 11572343, 51904092) supported by the National Natural Science Foundation of China; Project(2016YFC0801404) supported by the State Key Research Development Program of China; Project(KCF201803) supported by Henan Key Laboratory for Green and Efficient Mining & Comprehensive Utilization of Mineral Resources, Henan Polytechnic University, China; Project supported by Beijing Excellent Young Scientists, China

Received date: 2018-07-16; Accepted date: 2019-07-08

Corresponding author: CHEN Yan, PhD, Lecturer; Tel: +86-18739195586; E-mail: chenyan_cumtb@hotmail.com; ORCID: 0000-0002- 6738-3101