J. Cent. South Univ. (2017) 24: 866-874

DOI: 10.1007/s11771-017-3489-6

Crack growth analysis for rock-like materials with ordered multiple pre-cracks under biaxial compression

WANG Min(王敏)^{1, 2}, CAO Ping(曹平)¹, WAN Wen(万文)³, ZHAO Yan-lin(赵延林)³,

LIU Jie(刘杰)¹, LIU Jing-shuo(刘京铄)^{1, 4}

1. School of Resources and Safety Engineering, Central South University, Changsha 410083, China;

2. Hunan Provincial Key Laboratory of Shale Gas Resource Utilization (Hunan University of

Science and Technology), Xiangtan 411201, China;

3. School of Energy and Safety Engineering, Hunan University of Science and Technology, Xiangtan 411201, China;

4. Department of Hydraulic Engineering, Hunan Polytechnic of Water Resources and Electric Power,Changsha 410131, China

Central South University Press and Springer-Verlag Berlin Heidelberg 2017

Abstract: The pre-burying iron sheets approach was used to prepare rock-like materials with ordered multiple pre-cracks. 60 specimens in total were prepared in these experiments. Through biaxial compression experiments, the influence of both the number of pre-cracks and pre-cracks angles to crack growth was analyzed. Meanwhile, species of rock bridge failure were summarized, namely, wing crack, secondary shear crack between horizontal pre-cracks and secondary shear crack between vertical pre-cracks. The wing crack plays a significant role in crack growth. Furthermore, fractal dimension was adopted to describe quantitatively the crack growth during the failure process. The results indicate that with the failure of specimens, corresponding fractal dimension for specimen monotonically increases, which indicates that the fractal dimension can be considered to the failure of the specimens quantitatively.

Key words: crack growth; wing crack; secondary shear crack; vertical pre-cracks; fractal dimension

1 Introduction

Crack and joints commonly exist in rock mass. These kinds of discontinuities decrease rock mass strength and weaken rock mass project stability. Thereof, it is of necessity to better understand crack growth regulations, which would be favorable for the construction for rock mass projects.

Large scale rock mass in situ experiments would be costly, and what’s worse, a large number of unexpected factors in experiments can not be controlled and considered. Thus, under this circumstance, rock-like materials is one of alternatives and through man-made flaws simulating joints is frequently used by researchers [1-10]. Moreover, some numerical simulation based on rock-like materials experiments is conducted [11-15]. Besides, some strength criteria are proposed through rock-like materials experiments and rock-like materials are widespread used for studying the mechanics of joints in rock mass. As regards to experiments for rock-like materials under compression, in order to research crack growth regulations and rock bridge rupture law, a series experiments have been performed, and some conclusions have been obtained. ZHAO et al [16] conducted a sort of rock-like materials with multiple pre-cracks under uniaxal compression; the rock bridge failure modes were analyzed; the regulation of uniaxial strength varying with pre-cracks angle was obtained. PU et al [17, 18] and ZHAO et al [19, 20] performed experiments of rock-like materials with pre-cracks under uniaxial compression, and numerical simulation results agreed well with experiments results. However, the experiments aforementioned mainly concern uniaxial compression and for analysis of crack growth, it is lack of describing quantitatively.

In this work, a series of experiments of rock-like materials with ordered multiple pre-cracks were conducted by biaxial compression and the influence of both number of pre-cracks and angle to crack growth as analyzed. Additionally, fractal dimension was adopted to describe the crack growth quantitatively.

2 Experimental

2.1 Arrangement of pre-cracks and specimens preparation

In order to simulate brittle failure of rock, similar materials were used. Cement material was the cement P32.5, and its aggregate was sand. Mass ratio of cement, sand and water was 26:25:10. Pre-burying iron sheets method was adopted to prepare rock-like materials with ordered multiple pre-cracks.

Sorts of pre-cracks angle included 15°, 30°, 45°, 60°, and the number of pre-cracks had five species: 6, 9, 12, 15 and 18. Dimensions of specimens were 280 mm×185 mm×40 mm; length of pre-cracks was 20 mm; width was 1 mm.

Just as shown in Fig. 1, pre-cracks were distributed uniformly in rock-like materials. For rest of specimens, pre-cracks distribution was similar to that in Fig. 1. Table 1 lists all kinds of pre-cracks distribution.

2.2 Biaxial loading

The RYL-600 rock mechanics shear rheology machine shown in Fig. 2 was used in the experiments and its technical parameters included shear deformation error ±0.5%, control displacement error ±0.1%, experimental force error ±0.2%.

Rock-like materials with ordered multiple pre- cracks were loaded by means of biaxial loading. Later stress (0.5, 1 and 2 MPa) was loaded firstly with speed of 100 N/s, and then axial loading was applied with speed of 200 N/s. The biaxial loading is illustrated in Fig. 3.

3 Results analysis

3.1 Influence of number of pre-cracks on crack growth

Failure law of specimens with different number of pre-cracks is that it presents structure failure when number of pre-cracks is less, whereas number of cracks increases, specimen manifests local failure.

A series of results for specimens with pre-cracks angle 30° and lateral stress 2 MPa (Table 2) are taken as examples to analyze failure regulations on specimens with different numbers of pre-cracks under biaxial compression.

When the specimens are with 2 rows of pre-cracks and under lateral stress 2 MPa, at the initial phase of failure, wing crack appears in the middle of pre-cracks because of tension failure. Under the action of axial stress, crack propagates to the center of specimen; crack propagates downward through secondary shear crack between vertical pre-cracks and secondary shear crack between horizontal pre-cracks; crack laps the wing crack produced at the initial phase of crack growth. Failure mode of wing crack+secondary shear crack between vertical pre-cracks+secondary shear crack between horizontal pre-cracks is formed.

Fig. 1 Crack distribution of specimen with pre-cracks angle 60° (Unit: mm):

Table 1 Layout scheme of pre-cracks

Fig. 2 RYL-600 rock mechanics shear rheology machine

When specimen is with 3 rows of pre-cracks, the crack initiates from the bottom of the specimen and wing crack is the initial failure mode. Subsequently, the secondary shear crack between vertical pre-cracks propagates to the middle of specimen and wing crack appear on the top of specimen. The main crack laps finally through secondary shear crack between horizontal pre-cracks and axial bearing capacity of specimen drops dramatically. With the continuous increase of axial stress, secondary shear crack between horizontal pre-cracks propagates towards the bottom of specimen quickly in form of wing crack. The specimen is coalesced for the first time in form of wing crack+secondary shear crack between horizontal pre-cracks + secondary shear crack between vertical pre-cracks, and then wing crack+ secondary shear crack between horizontal pre-cracks. The general coalescence trend is vertical coalescence.

Fig. 3 Biaxial loading:

When the rows of specimen is 4, coalescence trend of specimen is horizontal; the failure of specimen initiates from lower right bottom of specimen; the general propagation amplitude is small, which is local region crack growth. The coalescence path is simple, specimen coalesces horizontally in form of wing crack, secondary shear crack between horizontal pre-cracks. The coalescence mode of specimen is horizontal in form of wing crack+secondary shear crack between vertical pre-cracks+secondary shear crack between horizontal pre-cracks.

Table 2 Crack growth evolution of specimens with pre-crack angle 30° and lateral stress 2 MPa

When the number of pre-cracks rows is 5, the failure of specimen is simple and its region is small. The specimen coalesces when wing crack laps secondary shear crack between horizontal pre-cracks. Under the action of lateral stress, the residual strength of specimen is composed of friction between crack and mutual occlusion. Coalescence mode is wing crack+secondary shear crack between horizontal pre-cracks.

When the specimen is embedded with six pre-cracks, crack propagation path is single. The crack propagates downward in form of wing crack from bottom of specimen, which laps right pre-cracks. When crack propagates to middle of pre-cracks and crack coalesces horizontally through pre-cracks, specimen is coalesced. Coalescence mode of specimen is wing crack+secondary shear crack between horizontal pre-cracks.

Based on the analysis above, experimental results of specimens with 30°×18 pre-cracks under lateral stress 2 MPa showed that coalescence path of crack become simple with the increase of number of pre-cracks and the coalescence mode transformed from vertical coalescence to horizontal or horizontal and vertical.

3.2 Influence of pre-cracks angle on crack growth

A series of experimental results of specimens with 12 pre-cracks under 0.5 MPa (Table 3) are utilized to analyze the influence of pre-cracks angle on crack growth.

When the specimen is with pre-cracks angle 15° and under lateral stress 0.5 MPa, crack growth process is as follows: wing crack propagates along left pre-cracks to the top of specimen, while the right pre-cracks lap with middle pre-cracks through secondary shear crack between horizontal pre-cracks, accompanying with axial loading stress. The new secondary shear crack between vertical pre-cracks intensifies specimen failure. Failure of rock bridge on specimen appears on a large scale; however, the new crack does not coalesce the specimen. Wing crack coalesces specimen right through the right pre-cracks, and another wing crack coalesces specimen left. The coalescence mode is wing crack+secondary shear crack between horizontal pre-cracks+wing crack.

When the pre-cracks angle is 30^o, the crack initiates from left lower and right upper of specimen, inclined propagation of wing crack is towards the middle of specimen. The middle pre-cracks lap through secondary shear crack between vertical pre-cracks and the coalescence direction of specimen is along the inclined propagation and from two terminals to the middle. The coalescence way of specimen is wing crack+secondary shear crack between vertical pre-cracks.

Table 3 Crack growth evolution of specimens with 12 pre- cracks under lateral stress 0.5 MPa with different pre-cracks angles

When the pre-cracks angle is 45^o, wing crack appears at the cusp of pre-cracks. With propagation of crack, existing crack propagates through secondary shear crack between vertical pre-cracks, and then coalescence along pre-cracks is realized. The corresponding coalescence mode is wing crack+secondary shear crack between vertical pre-cracks, and coalescence order is from two terminals to the middle.

When the pre-cracks angle is 60^o, the crack initiates from left pre-cracks in form of wing crack and then wing crack propagates upward. The inclined propagation appears in form of secondary shear crack between vertical pre-cracks, and the specimen is coalesced at the bottom. The coalescence mode is wing crack+secondary shear crack between vertical pre-cracks, and coalescence order is from the middle to the direction of axial stress.

From analysis of Table 3, it is concluded that from the experimental results of specimens with 12 pre-cracks under lateral stress 0.5 MPa, the general coalescence trend is diagonal, and coalescence direction is the same with inclined direction of pre-cracks.

3.3 Failure mode of rock bridge

The failure mode for rock bridge of rock-like materials with ordered multiple pre-cracks can be classified into three types mainly: wing crack, secondary shear crack between vertical pre-cracks and secondary shear crack between horizontal pre-cracks, as illustrated in Fig. 4.

Fig. 4 Rock bridge failure species

The wing crack appears at the initial phase or with low lateral stress. At the initial phase, tensile stress concentration appears at the cusp of pre-cracks, wing crack propagates along the direction of the maximum principal stress and specimen is coalesced along vertical direction. This kind of coalescence mode is pretty popular during the failure process and it plays a dominant role in rock bridge failure. Some common wing crack types are shown in Fig. 5.

The secondary shear crack between vertical pre- cracks mainly appears in the middle phase of experiments. It implements lapping inclined pre-cracks and inclined propagation, which belongs to typical sliding mode crack. Under combination action of axialstress and lateral stress, shear stress is formed along the inclined direction of pre-cracks, and then macro secondary shear crack between vertical pre-cracks is formed. Some examples of secondary shear cracks between vertical pre-cracks appearing in experiments are shown in Fig. 6.

Fig. 5 Examples of wing crack:

Fig. 6 Examples of secondary shear crack between vertical pre-cracks:

During the process of crack growth, wing crack propagation is formed firstly and with the macro crack propagation, specimen is in various stress states. Sometimes, lateral stress would become the principle stress at the cusp of pre-cracks, and then crack propagates horizontally, horizontal coalescence of pre-cracks implements. This kind of rock bridge coalescence mode is termed as secondary shear crack between horizontal pre-cracks. An example in the experiment of secondary shear crack between horizontal pre-cracks is illustrated in Fig. 7.

3.4 Changing trend of fractal dimension with crack growth

In order to describe the magnitude of crack growth, fractal dimension is applied to measure the change trend of crack growth. According to fractal theory [21], the relation among number of measurements N(r), measuring scale r and fractal dimension D_f can be denoted as

                               (1)

where C is constant.

In this work, the grid box technique is used to measure the crack appearing on the surface of specimens. The specimens with 15 pre-cracks and pre-cracks angle 60°(Fig. 8) are chosen to introduce fractal dimension measuring process of two-dimension crack.

Fig. 7 Example of secondary shear crack between horizontal pre-cracks

1) The measuring scales δ_i of grids are determined firstly. In this work, 1 mm, 2 mm, 4 mm, 8 mm and 16 mm are selected as measuring scales of grids.Figure 9 illustrates grids with 16 mm measuring pre- cracks of rock-like materials.

Fig. 8 Measuring crack model:

Fig. 9 Measured specimen with δ₅=16 mm

2) The number of grids N(δ_i) occupied by crack is counted when measuring scales δ_i are different. The occupied grids are counted when measuring scales are 1 mm, 2 mm, 4 mm, 8 mm and 16 mm, as listed in Table 4.

Table 4 Number of grids for specimens with different measurement scales

3) The least square method fitting curve i=1, 2, 3, 4, 5) is conducted and its related formula is as follows.

 (2)

where n is 5; α and β are fitting parameters.

The least square fitting curve is performed in Fig. 10.

Fig.10 Fitting curve based

4) The slope of fitting curve is fractal dimensional. Through calculation, fractal dimension for pre-cracks in rock-like materials with 9 pre-cracks is 0.8218. Related parameters for fitting show that the fitting curve reflects data well; hence, fractal dimension is suitable to describe cracks on surface of specimens.

In order to study the changing trend of fractal dimension of rock-like materials under biaxial compression, specimens with 60°×15 pre-cracks (lateral stress 1 MPa, Fig. 11 and Table 5), 30°×6 pre-cracks (lateral stress 0.5 MPa, Fig. 12 and Table 6) and 15 °×15 pre-cracks (lateral stress 2 MPa, Fig.13 and Table 7) are chosen to study the trend.

Fig. 11 Stress-strain curve of specimen with 60°×15 pre-cracks under lateral stress 1 MPa

Table 5 Failure process of specimen with 60°×15 pre-cracks under lateral stress 1 MPa

These three calculation examples indicate that grids box method is suitable to calculate the fractal dimension for crack growth of the rock-like materials. In addition, Fig. 14 gives the changing trend of fractal dimension of specimens during the failure process.

As shown in Fig. 14, the changing trend of fractal dimension for rock-like materials during the failure process is analyzed. It is suggests that with crack propagation of rock-like materials, the corresponding fractal dimension represents characteristics of monotone increasing. Thus, fractal dimension can be considered to be the damage degree of rock-like materials.

Fig. 12 Stress-strain curve of specimen with 30°×6 pre-cracks under lateral stress 0.5 MPa

Table 6 Failure process of specimen with 30°×6 pre-cracks under lateral stress 0.5 MPa

Fig. 13 Stress-strain curve of specimen with 15°×15 pre-cracks under lateral stress 2 MPa

Table 7 Failure process of specimen with 15°×15 pre-cracks under lateral stress 2 MPa

Fig. 14 Changing trend of fractal dimension with failure of specimen

4 Conclusions

1) Influence of number of pre-cracks on crack growth presents structure failure when the number of pre-cracks is less; while number of crack increases, specimen manifests as local failure. However, with respect to pre-cracks angle, with increasing of pre-cracks angle, coalescence mode of specimens becomes simple.

2) Three main rock failure modes appear in rock bridge failure, namely, wing crack, secondary shear crack between horizontal pre-cracks and secondary shear crack between vertical pre-cracks. Wing crack appears mainly at the initial phase of specimen failure and it plays an significant role in specimen failure.

3) Fractal dimension was adopted to describe the failure process of specimens. The calculation results showed that during the failure process of specimens, its corresponding fractal dimension monotonically increases; therefore, fractal dimension can be considered to describe the failure magnitude quantitatively.

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(Edited by YANG Hua)

Cite this article as: WANG Min, CAO Ping, WAN Wen, ZHAO Yan-lin, LIU Jie, LIU Jing-shuo. Crack growth analysis for rock-like materials with ordered multiple pre-cracks under biaxial compression [J]. Journal of Central South University, 2017, 24(4): 866-874. DOI: 10.1007/s11771-017-3489-6.

Foundation item: Project(E21527) supported by the Open Research Fund Program of Hunan Provincial Key Laboratory of Shale Gas Resource Utilization, China; Project(2015zzts077) supported by the Fundamental Research Funds for the Central Universities, China; Projects(51174088, 51174228) supported by the National Natural Science Foundation of China; Project(2013CB035401) supported by the National Basic Research Program of China

Received date: 2015-03-18; Accepted date: 2015-12-03

Corresponding author: WANG Min, PhD Candidate, Tel: +86-15073218915; E-mail: michaelwong307@outlook.com