Advances in Materials Science and Engineering
Volume 2017 (2017), Article ID 6460150, 17 pages
https://doi.org/10.1155/2017/6460150

Research Article

Experimental and Numerical Study of Failure Behavior and Energy Mechanics of Rock-Like Materials Containing Multiple Joints

Ri-hong Cao1,2 and Hang Lin¹

¹School of Resources and Safety Engineering, Central South University, Changsha, Hunan 410083, China
²School of Civil, Environmental and Mining Engineering, The University of Western Australia, Perth, WA 6009, Australia

Correspondence should be addressed to Hang Lin; linhangabc@126.com

Received 25 May 2017; Revised 20 July 2017; Accepted 4 October 2017; Published 31 October 2017

Academic Editor: Andrey E. Miroshnichenko

Copyright © 2017 Ri-hong Cao and Hang Lin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper investigates the influence of joint geometry parameters on the characteristic stress, failure pattern, and energy mechanism of multiple jointed rock-like specimens under uniaxial compression. Both the laboratory and numerical results show that the higher value of UCS occurs when α is around 0° and γ changes from 15° to 30° or when α is around 30° and γ changes from 45° to 75°. However, the lowest value appears when α is around 45° and γ changes from 15° to 30°. The CDiS (critical dilatancy stress) and CIS (crack initiation stress) show a similar tendency to UCS. Moreover, the specimens present different failure modes for various levels of α, γ, and , and the failure mode can be classified into four categories: stepped path failure; failure through parallel plane; failure through cross plane; material failure. In addition, with higher strength, the input energy and strain energy are higher than those with lower strength. Dissipation energy is affected by the failure modes of the specimens. At the same time, when changes from 0.2 to 0.6, the boundary energy, strain energy, and dissipation energy show a decreasing trend.

1. Introduction

There are a large number of discontinuities in natural rocks that have very long geological ages; such discontinuities include joints, cracks, weak surfaces, and faults. They play a very important role in the strength and failure characteristics of rock mass, and the coalescence between fractures and the strength degradation of jointed rock mass are of great interest to engineers and scientists. To better understand the mechanical behavior of jointed rock or rock-like materials, substantial experimental efforts have been devoted to the study of crack initiation, propagation, and failure modes of precracked rocks or rock-like specimens. Many kinds of joint geometries have been considered in previous works, such as joint inclination angle [1–19], joint distance [3, 15–18], and overlap distance [17–19]. At the same time, there are many kinds of materials that have been used by scholars, such as glass [5], Columbia Resin 39 [20], and molded gypsum [15, 21–24]. Previous studies have shown that there are mainly two kinds of cracks developed from the tips of preexisting fissure: wing crack and secondary crack [4, 8, 11, 21, 22]. For the coalescence mode between fissures, it can be classified into three types: tensile mode; shear mode; mixed mode [1, 2, 7–15, 19, 22]. Apart from specimens with single, two, or three fissures, the failure pattern of jointed specimen has also been investigated by scholars. The failure patterns of multiple-jointed specimen include stepped path failure [3, 12, 16–18]; shearing failure [3, 12]; block rotation failure [18]; and intact failure [3, 12, 16].

In recent years, numerical simulation has become a widely used method to study the crack initiation, propagation, and failure process of brittle materials like rocks. Many kinds of numerical simulation methods have been successfully employed for such topics, including FEM [25–30], DDA [31–34], and NMM [32, 35]. Currently, PFC (particle flow code), that is, a kind of DEM, has been widely accepted and used by many scholars to study the propagation of cracks in natural rocks or rock-like materials [36–41].

Previous studies have promoted the understanding of crack propagation, coalescence, and failure modes of jointed rocks. However, the mechanical behavior of multiple-jointed rocks has not been studied comprehensively, especially the characteristic stress (crack initiation stress and critical dilatancy stress) and energy mechanism of multiple-jointed specimens. Based on laboratory tests and PFC simulation, the characteristic stress, failure pattern, and energy mechanism of multiple-jointed rock-like materials will be investigated in this paper. This paper is structured in the following way. In Section 2, the specimen manufacture and loading system are described. In Section 3, after numerical model calibration, numerical results on the influences of joint geometry parameters on the characteristic stress are presented. In Section 4, the influences of joint geometrical parameters on the failure modes of multiple-jointed blocks are described. In Section 5, the strain energy and dissipation energy of jointed specimens under uniaxial compression are presented.

2. Laboratory Tests

2.1. Specimen Manufacture

Specimens were made of cement mortar, and the volume proportions for white cement, water, and sand were V (water) : V (white cement) : V (silica sand) = 3 : 3 : 2. The dimensions (height × width × thickness) of the specimens were 200 × 150 × 30 mm. All of specimens were placed inside a standard curing box for 28 days before being subjected to mechanical testing. After 28 days, for the mechanical properties of intact rock-like material, the values of Young’s modulus, uniaxial compressive strength (UCS), and Poisson’s ratio () were Em = 3.242 GPa, UCS = 8.104 MPa, and = 0.2371, respectively. All of these macroparameters were based on the average values of three specimens. As mentioned above, there are many kinds of geometry parameters described in previous works, such as joint inclination angle, joint distance, overlap distance, and joint persistency. In this study, we focus on the effect of joint intersection and persistency on the mechanical behavior of multiple-jointed specimen. Thus, among the many parameters of joint geometry, the joint inclination angle, intersection angle, and persistency are considered in this study. In this research, the multiple-jointed rock-like specimens were created by putting Joint 1 and Joint 2 together. As shown in Figures 1(a) and 1(b), Joint 1 and Joint 2 contain many intermittent joint sets and those intermittent joint sets are parallel to each other. To be specific, the joint geometry is defined by three geometrical parameters: joint inclination angle (Joint 1) or (Joint 2), joint length , and rock bridge length . Joint persistency is defined by the joint length and the rock bridge length as follows:

Figure 1: Schematics of fissure geometry configurations in the specimens.

For the jointed model in Figure 1(c), the intersection angle . Joint 2 configurations occur when the block is rotated around the -axis in the counterclockwise direction at each increment of 15° until the intersection angle () reaches 75°. Notably, in the specimen preparation process, the joints are created by inserting mica sheets (0.6 mm thick) into the fresh cement mortar paste at the location of the joints. The feasibility of this technique has also been verified by scholars [43].

To consider the effect of Joint-1 inclination angle and intersection angle (), the joint persistency is kept constant at 0.4. Table 1 provides the fissure geometry information for the specimens with different and .

Table 1: Specimen numbers and fissure geometrical parameter values used for specimens with different and .

Number
Specimen ID

1
S-0-15-0.4
0
15
0.4

2
S-0-30-0.4
0
30
0.4

3
S-0-45-0.4
0
45
0.4

4
S-0-60-0.4
0
60
0.4

5
S-0-75-0.4
0
75
0.4

6
S-30-15-0.4
30
15
0.4

7
S-30-30-0.4
30
30
0.4

8
S-30-45-0.4
30
45
0.4

9
S-30-60-0.4
30
60
0.4

10
S-30-75-0.4
30
75
0.4

11
S-45-15-0.4
45
15
0.4

12
S-45-30-0.4
45
30
0.4

13
S-45-45-0.4
45
45
0.4

14
S-45-60-0.4
45
60
0.4

15
S-45-75-0.4
45
75
0.4

16
S-60-15-0.4
60
15
0.4

17
S-60-30-0.4
60
30
0.4

18
S-60-45-0.4
60
45
0.4

19
S-60-60-0.4
60
60
0.4

20
S-60-75-0.4
60
75
0.4

21
S-75-15-0.4
75
15
0.4

22
S-75-30-0.4
75
30
0.4

23
S-75-45-0.4
75
45
0.4

24
S-75-60-0.4
75
60
0.4

25
S-75-75-0.4
75
75
0.4

To examine the effect of joint persistency value , uniaxial compression tests were carried out on the specimens with having value in the range of 0.2–0.6, with the inclination angle changing from 0° to 60°. As shown in Figure 2, the multiple-jointed specimens are with different values. Figures 2(a)–2(c) are the specimens for and . And the persistency () ranges from 0.2 to 0.6. From Figures 2(d)–2(f), the specimens with and are shown, and the persistency () also ranges from 0.2 to 0.6. Table 2 provides the fissure geometry information for the specimens with different .

Table 2: Specimen numbers and fissure geometrical parameter values used for specimens with different .

Number
Specimen ID

1
S-0-60-0.2
0
60
0.2

2
S-0-60-0.4
0
60
0.4

3
S-0-60-0.6
0
60
0.6

4
S-30-60-0.2
30
60
0.2

5
S-30-60-0.4
30
60
0.4

6
S-30-60-0.6
30
60
0.6

7
S-45-60-0.2
45
60
0.2

8
S-45-60-0.4
45
60
0.4

9
S-45-60-0.6
45
60
0.6

10
S-60-60-0.2
60
60
0.2

11
S-60-60-0.4
60
60
0.4

12
S-60-60-0.6
60
60
0.6

13
S-75-60-0.2
75
60
0.2

14
S-75-60-0.4
75
60
0.4

15
S-75-60-0.6
75
60
0.6

(a) S-30-30-0.2

(b) S-30-30-0.4

(c) S-30-30-0.6

(d) S-60-60-0.2

(e) S-60-60-0.4

(f) S-60-60-0.6

Figure 2: Multiple-jointed specimens with different value.

2.2. Laboratory Tests

The uniaxial compression tests on the multiple-jointed specimens were performed using a servocontrol uniaxial loading instrument, and the loading was controlled by the system named DCS-200. Loading was applied by displacement control at a rate of 0.3 mm/min. For each test, some butter was daubed on the contact area between the plates and the specimen to reduce the end effect. The test setup and the details of the specimens under loading are shown in Figure 3. During the test, the specimens were loaded under compression until failure and the failure process were recorded by a video camera. At the same time, the vertical load and displacement of the specimen were also monitored.

Figure 3: The layout of the loading system.

3. Numerical Simulation

3.1. Brief Introduction to PFC2D

In PFC, rock or rock-like material is presented as a collection of particles with certain thickness. PFC provides many kinds of contact regimes between particles, such as contact-stiffness models (linear contact model and Hertz-Mindlin contact model), slip model, and bonding models (contact bond model and parallel bond model). The bonding model has been widely used in failure process simulation of rocks or rock-like materials. In particular for the parallel bond model, because it can transmit both force and moment between particles [42, 44, 45], it has been viewed as an effective and realistic method for modeling rocks [36–39]. Therefore, the parallel bond model is used to model multiple-jointed rock-like specimens in the current work. The parallel bond model describes the constitutive law of the cementitious material deposited between two balls [44]. The total force and moment acting on the parallel bond are denoted by and (see Figure 4). When a bond is formed, and are set as 0. Each subsequent displacement and rotation increment will result in an increment of elastic force and moment:

with

where and are the normal and shear components of , and are normal and shear stiffness, is the contact velocity, is the time step, and are the rotational velocities of balls and , and and are the area of bond cross section and the moment of inertia, respectively, which are given by

The new force and moment vectors are calculated by

The maximum tensile and shear stresses acting on the bond periphery can be calculated by

Figure 4: Illustration of the parallel bond model in PFC [42].

If the maximum tensile stress/shear stress exceeds the tensile strength/shear strength of the parallel bond, the bond will break and a microcrack will form between the adjacent parent particles.

3.2. Model Generation and Calibration

According to the size of the specimen in the laboratory test, the numerical specimen was established with a height of 200 mm height and width of 150 mm. At the same time, in order to build a reasonable numerical model which could reflect the mechanical behavior of rock-like material very well, microparameters calibration is particularly important. Generally, the microparameters of PBM are normal and shear stiffness, normal and shear strength, and bond radius. After calibration, the microparameters are listed in Table 3. Table 4 provides a comparison between the macroparameters of intact numerical specimen and the macroparameters obtained through laboratory tests. Overall, the uniaxial compression strength (UCS), elastic modulus (), and Poisson’s ratio () are basically the same as those of the experimental results.

Table 3: Microscopic parameters for rock mass.

Microparameters
Values

Minimum mean radius (mm)
0.25

1.66

Particle contact modulus, (GPa)
2.45

Particle normal/shear stiffness
2.7

Particle friction coefficient,
0.5

Parallel bond modulus, (GPa)
2.45

Parallel bond normal/shear stiffness
2.7

Parallel bond normal strengths, mean (MPa)
5.53

Parallel bond normal strength, standard deviation (MPa)
0.6

Parallel bond shear strengths, mean (MPa)
5.53

Parallel bond shear strength, standard deviation (MPa)
0.6

Table 4: Comparison between experimental and numerical results for the macromechanical parameters of intact specimen.

Experimental results
Numerical results

Uniaxial compressive strength, UCS/MPa
8.104
8.096

Elastic modulus/GPa
3.242
3.176

Poisson’s ratio
0.2371
0.2397

In PFC2D, a joint can be generated by assigning a dip angle and dip direction. Figure 5 shows the numerical specimen generated by PFC2D (S-0-60-0.4). The yellow circles are particles for the intact material, and the green particles represent joints. The micromechanical parameters for the intact material are shown in Table 3. In the numerical model, the joints are generated through changing micromechanical parameters values of the particles aligned with the location of the joints. Usually, the micromechanical parameter values assigned for the particles that represent joints are smaller than those for the particles that represent the intact material. As mentioned above, the joints are created by inserting the mica sheets into the fresh cement mortar paste. During the curing period, the humidity was controlled at 80%. After absorbing water, the mica sheet is similar to a multilayer paper-like material and the bond strength between each layer can be ignored. Therefore, for the joint particles, the normal/shear bond strengths were set as 0, and other micromechanical parameters (friction coefficient and stiffness) were also far below those of the intact material particles. The values used for the particles on each side of the joint planes are given in Table 5.

Table 5: Microscopic parameters for joints.

Joint particle friction coefficient
0.08

Joint particle normal stiffness (N/m)
250

Joint particle shear stiffness (N/m)
250

Joint particle normal bond strength (MPa)
0

Joint particle shear bond strength (MPa)
0

Figure 5: Numerical intact specimen generated in PFC2D in this research (S-0-60-0.4).

To verify the rationality of the numerical models, the axial stress-strain curves for 4 numerical models are selected to compare with the corresponding experimental results. Figure 6 shows the curve comparison of the specimens S-0-15-0.4, S-0-30-0.4, S-0-45-0.4, and S-0-60-0.4. For both uniaxial compression strength (UCS) and Young’s modulus (), the numerical curves are very close to the experimental results. Clearly, the comparison of these curves shows good agreement and validates the ability of the PFC2D mode to reproduce the behavior observed in the laboratory tests. Notably, before the peak stress, both the numerical and experimental curves show nonlinearity as the axial strain increases. However, for the experimental curves, the fluctuation of the curves before and after the peak stress is more obvious than those in the numerical simulation. Actually, numerical simulation is an ideal solution (ideal boundary conditions and materials) for crack initiation and propagation. Thus, there are inevitable differences when comparing with the results obtained from laboratory tests.

(a) S-0-15-0.4

(b) S-0-30-0.4

(c) S-0-45-0.4

(d) S-0-60-0.4

Figure 6: Stress-strain curve for experimental and numerical results.

Figure 7 shows the peak strength (UCS) in multiple-jointed rock-like specimens with different , , and . As shown in Figure 7(a), there is a relationship between UCS and for specimens when the intersection angle is 15° and 30°. The relationships between UCS and for specimens when the intersection angle is 45°, 60°, and 75° are shown in Figure 7(b). As seen from Figures 7(a) and 7(b), the numerical results show a similar trend with the experimental results. This also indicates that the numerical model can simulate the jointed specimen’s mechanical behavior favorably. Clearly, the joint inclination angle α and intersection angle have a significant influence on UCS of the specimens. For the specimens with different values, the UCS obtained through laboratory tests and numerical simulation are shown in Figure 7(c). Obviously, the joint persistency value has a great influence on the peak strength of the multiple-jointed specimens. Both the experimental and numerical results show that the UCS decrease with the increase of . Overall, the numerical results show great agreement with the experimental results.

(a)

(b)

(c)

Figure 7: UCS of specimens from laboratory and numerical results.

3.3. Characteristic Stress in PFC

Figure 8 shows the stress-strain diagrams for different stages of crack development and sketches the crack propagation under different stress levels. Actually, the prepeak curve of the jointed specimens can be divided into 3 stages. As shown in Figure 8, the first stage is Stage I (elastic deformation stage). After Stage I, the specimen enters Stage II (stable crack development stage). The crack initiation point and crack initiation stress (CIS) indicate the beginning of this stage. With further loading, the cracks propagate with the increase of the axial stress. When the axial stress reaches the critical dilatancy stress (CDiS), the specimen enters Stage III (accelerating extension stage). In this stage, the cracks propagate and link with others to form penetration. Finally, the cracks reach the edge of the specimens and result in the overall failure of the specimens.

Figure 8: Stress-strain diagrams for different stages of crack development and sketches of crack propagation under different stress levels.

Because the crack initiation is based on the idea of a homogeneous rock, it is difficult to locate the crack initiation point on the stress-strain curves. In this study, the crack initiation stress measured during a uniaxial test upon a PFC2D rock-like material is defined as when the axial stress accounts for 1% of the total number of cracks existing at the peak stress [42]. In PFC, the macrocrack is made up of many microcracks, and the crack initiation stress is the stress level corresponding to the appearance of the first macrocrack. When the microcracks account for 1% of the total number of cracks existing at the peak stress, the macrocrack is visible and the number of the microcracks for each macrocrack is in single digits. However, under 1%, the macrocrack is unable to be distinguished. Potyondy and Cundall (1999) [46] used this definition for the first time in their research. Moreover, in previous studies, this was accepted and widely used by scholars [9, 47]. However, CDiS is the axial stress representing the turning point on the stress-volumetric strain curve (Figure 8). The effect of and on characteristic stress of multiple-jointed specimens under uniaxial compression tests is shown in Figure 9. To be specific, the relationships of the joint geometry ( and ) with UCS, CDiS, and CIS are shown in Figures 9(a), 9(b), and 9(c), respectively. For the UCS, a higher value occurs when α is around 0° and are 15° to 30°, or when α is around 30° and is 45° to 75°. However, the lowest value appears when α is around 45° and is 15° to 30°. It can be seen from Figure 9 that both CDiS and CIS show a similar trend with UCS. This also means that both and have influence on the CDiS and CIS. For the CDiS, its value is close to the value of UCS for all specimens. A higher value occurs when α is around 0° and is 15° to 30° or when is around 30° and is 45° to 75°. The CIS is the lowest among the three kinds of characteristic stresses, and the contour of CIS is very similar to those for UCS and CDiS.

(a) UCS

(b) CDiS

(c) CIS

Figure 9: Effect of joint inclination on the characteristic stresses: (a) for UCS; (b) for CDiS; (c) for CIS.

The relationships of the joint persistency value with UCS, CDiS, and CIS are shown in Figures 10(a), 10(b), and 10(c), respectively. As shown in Table 2, to examine the effect of , uniaxial compression tests were carried out on the specimen with = 60° having joint persistent value in the range of 0.2–0.6, and the other joint microparameters were kept constant. Figure 10 clearly shows that joint persistency value has an obvious influence on the characteristic stress of the specimen. When changes from 0.2 to 0.6, the UCS and CDiS show a decreasing trend. Overall, the CDiS and CIS show a similar tendency with UCS, while, for the CIS, when , the CIS value does not show an obvious trend with the change of . This indicates that the joint inclination angle does not play an important role in the CIS of multiple-jointed specimens when .

(a) UCS

(b) CDiS

(c) CIS

Figure 10: Effect of joint persistency () on the characteristic stresses: (a) UCS; (b) CDiS; and (c) CIS.

4. Failure Pattern of Jointed Specimen

When compression is applied on preexisting flaws, tensile or shear cracks will initiate from their tips. As loading continues, these cracks will propagate and link with other cracks to form penetration. Thus, preexisting fissures will link with neighboring cracks and result in different kinds of failure patterns eventually. For the multiple-jointed specimens, the specimens present different failure modes for various levels of , , and . Moreover, careful examination of all specimens shows that the failure mode can be classified into four categories: Mode I (stepped path failure); Mode II (failure through parallel plane); Mode III (failure through cross-plane); and Mode IV (material failure). In this research, the classification of different failure modes is based on the failure characteristics of the specimens, and the crack classification of the failure plane for each mode is based on previous studies. Figure 11 shows the different coalescence patterns of the main failure plane of the specimens. In Mode I, there are several failure planes in the specimen and most of the joints link with others through tensile/mixed cracks. For Mode II, the failure plane is almost parallel to the diagonal line of the specimen and the joints on the failure plane link with others through shear cracks. Like Mode II, in Mode III, the coalescence mode between the joints is shear crack, but there are sets of cross shear failure planes in the specimen. For Mode IV, the macro failure plane in the specimen does show a clear tendency with the joint geometry. The failure characteristics are similar to those in the intact specimen. Notably, the symbols S, M, and T represent the shear crack, mixed crack, and tensile crack, respectively.

Figure 11: Failure plane in different failure mode.

Figure 12 shows the four typical failure modes obtained by PFC2D and the corresponding failure modes obtained for the same specimens through the rock-like material experiments. Clearly, the four failure modes obtained in the performed numerical modeling agree very well with the experimental results. In the comparison of Mode I, that is, Figures 12(a) and 12(b), the stepped path failure plane in the numerical specimen agrees well with the experimental results. A comparison of the typical failure Mode II (failure though parallel plane) examples (Figures 12(c) and 12(d)) clearly shows that the failure plane is parallel to the diagonal line. Figures 12(e) and 12(f) show the comparison of the numerical and experimental results for Mode III (failure though cross-plane). Although some of the blocks fall outside the specimen, the “cross failure plane” in the experimental result is very clear and agrees well with the numerical specimens. Moreover, Mode IV in the experimental results and the numerical results also show a good agreement. Then, it can be concluded that the numerical model can reproduce the failure characteristics of the multiple-jointed specimens.

Figure 12: Failure mode comparisons between experimental and numerical results: (a) and (b) for stepped failure mode; (c) and (d) for failure through parallel plane; and (e) and (f) for failure through cross-plane.

4.1. Effect of α on the Failure Pattern of a Multiple-Jointed Rock-Like Specimen

Table 6 summarizes the failure patterns of rock-like specimens containing preexisting fissures with different Joint 1 inclination angles () and intersection angles () at approximately the residual strength (). Specimens S-0-15-0.4, S-0-30-0.4, S-30-30-0.4, and S-30-45-0.4 belong to Mode I. All of the specimens with = 45° belong to the transition mode. Specimens S-0-60-0.4, S-0-75-0.4, S-60-15-0.4, and S-75-15-0.4 are typical examples of Mode II. S-45-30-0.4, S-75-30-0.4, S-45-60-0.4, and S-60-60-0.4 belong to Mode III. There are sets of cross shear failure planes in the specimens and the preexisting joints are connected through coplanar shear cracks, resulting in overall failure. Notably, there are no specimens with that belong to Mode IV.

Table 6: Microscopic parameters for joints.

α
γ
0
30
45
60
75

15
Mode I
Mode I
Mode I
Mode II
Mode II

30
Mode I
Transition modes I-II
Mode III
Mode II
Mode III

45
Transition modes I-II
Transition modes I-II
Transition modes I-III
Transition modes I-II
Transition modes I-III

60
Mode II
Mode I
Mode III
Mode III
Transition modes I-III

75
Mode II
Mode I
Mode I
Transition modes I-III
Mode I

4.2. Effect of k on the Failure Pattern of a Multiple-Jointed Rock-Like Specimen

As mentioned above, when there are no specimens that belong to Mode IV. Actually, Mode IV mainly occurs when . As shown in Table 7, the failure modes of the specimens are with different values ( = 60°). It can be seen from Table 7, for the specimens with = 0°, 45° and 60°, the failure mode of the specimen remains unchanged when changes from 0.6 to 0.2. It seems that the value has a small effect on the specimens for Mode II and Mode III. For the specimens with = 30° and 75°, when changes from 0.6 to 0.2, the failure modes of the specimens gradually transform into Mode IV.

Table 7: Microscopic parameters for joints.

α 

0
30
45
60
75

0.2
Mode II
Mode IV
Transition modes I-III
Mode III
Mode IV

0.4
Mode II
Mode I
Mode III
Mode III
Transition modes I-II

0.6
Mode II
Mode I
Mode III
Mode III
Transition modes I-II

5. Energy Characteristics

The failure of brittle materials such as rocks and rock-like materials can be viewed as a result of energy conversion. In the uniaxial compressive test, for the specimen, the absorbed energy is mainly stored as strain energy before the peak strength. At the same time, new cracks developing from the tips of original cracks propagate with the increase of axial stress. During this process, some of the energy is consumed by microcrack generation and propagation. On the basis of the first law of thermodynamics, if a unit volume of material deforms by the action of an external force, then the energy can be defined as follows:

where and represent the unit dissipation energy and the unit strain energy, respectively.

As shown in Figure 13, there is a relationship between dissipation strain energy and release elastic strain energy in the stress-strain curves. Based on previous results [48], under compression, the release elastic strain energy can be calculated by

where is the unloading elastic modulus. In the current research, for calculation convenience, the is replaced with initial elastic modulus , which has also been verified by Liang (2012) [49].

Figure 13: Relationship between dissipation strain energy and release elastic strain energy in the stress-strain curves.

The numerical simulation is an ideal solution (ideal boundary conditions and materials), and in PFC there is no angular displacement between the loading platform and the specimen. Because in PFC the specimen sandwiched between two walls and the displacement loading was performed through the top and bottom walls, the input energy is the boundary energy and can be calculated by Itasca [42]:

where is the input energy in the last calculation step; and are the unbalanced forces on the top and bottom walls, respectively. and are the displacement increments for the top and bottom walls, respectively. For the strain energy () in PFC, it is stored in the contacts between particles and can be calculated as follows:

where is the total number of contacts in the model, and and are the normal and shear-contact forces, respectively. In addition, and are the normal and shear-contact stiffness, respectively.

According to Figure 13, for the uniaxial compression tests, the input energy and strain energy are the red-shaded and black-shaded area under the axial stress-strain curve before the peak stress, respectively. The dissipation energy is the difference between the two areas. Based on the laboratory tests, all types of energies for multiple-jointed specimens are shown in Figure 14. It can be seen from Figure 14(a) that the input energy is the largest among the three kinds of energies. At the same time, the strain energy is higher than the dissipation energy. Figures 14(b), 14(c), and 14(d) are the contours of input energy, strain energy, and dissipation energy, respectively. Clearly, for the input energy and strain energy, the tendency with joint geometry is similar to those of UCS in Figure 7 and the characteristic stress in Figure 9. With higher strength, the specimen becomes more difficult to fail, and the input energy and strain energy are higher than those with lower strength. For the dissipation energy, it is affected by the failure modes of the specimens.

(a) Energies

(b) Boundary energy contour

(c) Strain energy contour

(d) Dissipation energy contour

Figure 14: Energy of jointed specimens in laboratory tests.

The lowest value appears when α is around 45° and is 15° to 30° or when is 45°/60° and γ is 60° to 75°. For these specimens, they belong to Shear I or Shear II failure mode. Because there is only one or few failure planes in the specimen, the energy consumed by microcrack generation and propagation during testing is lower than those in stepped path or material failure. Thus, the dissipation energy is also lower than others.

As shown in Figure 15, there are three types of energy densities for a set of experimental test. It can be seen from Figures 15(a), 15(b), and 15(c) that the energy densities exhibit a similar tendency with the increase of and α. The energy densities show a drop decreasing trend with the increase of . Generally, for specimen with a lower value, the integrity of the specimen is higher than those with higher values. Then, during testing, the specimens will exhibit higher UCS and store more energy. Thus, the input energy and strain energy of specimen with are higher than those with and 0.6. At the same time, the damage process of the multiple-jointed specimen changes with the length of rock bridge. According to the failure patterns of the specimens, the longer the rock bridge, the more the energy dissipation during the crack coalescence. Then, the dissipation energy shows an obvious increase with the increase of .

(a) Input energy

(b) Strain energy

(c) Dissipation energy

Figure 15: Energy of jointed specimens with different in laboratory tests.

In PFC2D, the strain energy and boundary energy are recorded by measurement circles, and the dissipation energy is the difference between them. As shown in Figure 16, there are three kinds of energies before the peak stress for each specimen. Although the unit for the energy is different from Figure 16, the variation pattern of each energy shows a similar trend to those in Figure 14. The boundary energy and strain energy show a similar trend with the UCS of the specimens (Figure 9). In addition, the dissipation energy is the lowest of the three and exhibits a slight fluctuation with the increases of and .

(a) Energies

(b) Boundary energy contour

(c) Strain energy contour

(d) Dissipation energy contour

Figure 16: Energy of jointed specimens in PFC simulation.

The relationship of the joint persistency value with the boundary energy, strain energy, and dissipation energy is shown in Figures 17(a), 17(b), and 17(c), respectively. Figure 17 clearly shows that the joint persistency value has an obvious influence on the energy of the specimen during testing. When changes from 0.2 to 0.6, the boundary energy, strain energy, and dissipation energy show a decreasing trend. Overall, for all value, the energies show a similar tendency as the characteristic stress when α changes from 0° to 75°. For the tendency of different energies, the numerical results also agree very well with the experimental results as shown in Figure 15.

(a) Boundary energy

(b) Strain energy

(c) Dissipation energy

Figure 17: Energy of jointed specimens with different in PFC simulation.

6. Conclusion

In this paper, the characteristic stress, failure pattern, and energy mechanism of multiple-jointed rock-like specimens under uniaxial loading were investigated through laboratory tests and numerical simulation. For the peak strength and failure characteristics of the specimens, the numerical results agreed well with the experimental results. At the same time, the variation pattern of each energy in the numerical results also showed a similar trend with those in the experimental results. The following conclusions are summarized from the study:

(1)The joint geometry parameters (inclination angle , intersection angle , and persistent value ) had a great influence on the strength of the specimens. For the UCS, a higher value occurred when α was around 0° and changed from 15° to 30° or when α was around 30° and changed from 45° to 75°. However, the lowest value appeared when α was around 45° and γ changed from 15° to 30°. The joint persistency value also had a great influence on the peak strength of the multiple-jointed specimens, with the UCS decreasing with the increase of . For the CDiS and CIS, they showed similar tendency as UCS. However, for the specimens with higher persistency value, the inclination of the joint only played a minor role on the CIS of multiple-jointed specimens.(2)The specimens present different failure modes for various levels of , , and . Moreover, careful examination of all specimens showed that the failure mode can be classified into four categories: stepped path failure (Mode I); failure through parallel plane (Mode II); failure through cross-plane (Mode III); material failure (Mode IV). The value had a small effect on the specimens for Mode II and Mode III. However, for the specimens with = 30° and 75°, when changed from 0.6 to 0.2, the failure modes of the specimens gradually transformed into Mode IV.(3)With higher strength, the input energy and strain energy were higher than those with lower strength. Dissipation energy was affected by the failure modes of the specimens. For Mode II and Mode III, because there was only one or few failure planes in the specimen, the dissipation energy was also lower than the others. When changed from 0.2 to 0.6, the boundary energy, strain energy, and dissipation energy showed a decreasing trend.

Conflicts of Interest

The authors declare no conflicts of interest.

Acknowledgments

This paper is funded and is supported by National Natural Science Foundation of China, Projects 51774322, 51474249, and 51404179. It is also funded by a project supported by Innovation Driven Plan of Central South University (no. 2016CX019); a project supported by the Graduate Student Innovation Project of Central South University (2015zzts074); a project supported by the Fundamental Research Funds for the Central Universities (310821161008); and the Opening fund of State Key Laboratory of Geohazard Prevention and Geoenvironment Protection (SKLGP2016 K009). China Postdoctoral Science Foundation and the Postdoctoral Science Foundation of Central South University supported the paper. The authors wish to acknowledge these supports.

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