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

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 k, 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 k changes from 0.2 to 0.6, the boundary energy, strain energy, and dissipation energy show a decreasing trend.

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][26][27][28][29][30], DDA [31][32][33][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][37][38][39][40][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.

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 (V) were Em = 3.242 GPa, UCS = 8.104 MPa, and V = 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: 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 .To examine the effect of joint persistency value , uniaxial compression tests were carried out on the specimens with  = 60 ∘ 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  = 30 ∘ and  = 60 ∘ .And the persistency () ranges from 0.2 to 0.6.From Figures 2(d)-2(f), the specimens with  = 60 ∘ and  = 60 ∘ 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 .

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.[42,44,45], it has been viewed as an effective and realistic method for modeling rocks [36][37][38][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  3 (see Figure 4).When a bond is formed,   and  3 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, V  is the contact velocity, Δ is the time step,  []  3 and  []  3 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 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.

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 Figure 4: Illustration of the parallel bond model in PFC [42].
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 (V) are basically the same as those of the experimental results.
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.
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.Figure 7 shows the peak strength (UCS) in multiplejointed 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  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 multiplejointed 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.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.

Characteristic Stress in PFC.
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  is around 0 ∘ and  are 15 ∘ to 30 ∘ , or when  is around 30 ∘ and  is 45 ∘ to 75 ∘ .However, the lowest value appears when  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  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.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  = 0.6, 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  = 0.6.

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 crossplane); 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 Advances Materials Science and Engineering 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 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.

Effect of k on the Failure Pattern of a Multiple-Jointed
Rock-Like Specimen.As mentioned above, when  = 0.4 there are no specimens that belong to Mode IV.Actually, Mode IV mainly occurs when  = 0.2.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.

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].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  pre is the input energy in the last calculation step;  1 and  2 are the unbalanced forces on the top and bottom walls, respectively.Δ 1 and Δ 2 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.
The lowest value appears when  is around 45 ∘ and  is 15 ∘ to 30 ∘ or when  is 45 ∘ /60 ∘ and  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 .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  = 0.2 are higher than those with  = 0.4 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 .
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 .
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  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.

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  was around 0 ∘ and  changed from 15 ∘ to 30 ∘ or when  was around 30 ∘ and  changed from 45 ∘ to 75 ∘ .However, the lowest value appeared when  was around 45 ∘ and  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.

Figure 3 :
Figure 3: The layout of the loading system.
Figure 8 shows the stressstrain 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).

Figure 8 :Figure 9 :Figure 10 :Figure 11 :
Figure 8: Stress-strain diagrams for different stages of crack development and sketches of crack propagation under different stress levels.

Figure 12 :Figure 13 :
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.

3 Figure 14 :
Figure 14: Energy of jointed specimens in laboratory tests.

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

Figure 16 :
Figure 16: Energy of jointed specimens in PFC simulation.

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

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

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

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

Table 5 :
Microscopic parameters for joints.

Table 6 :
Microscopic parameters for joints.

Table 7 :
Microscopic parameters for joints.