4.1 Particle Flow Code
One of the models used to evaluate the model particles cyclically is the PFC2D model, this model is a set of discrete circular particles and uses the explicit time-step circulation rule (Potyondy and Cundall 2004). The contact force between particles is based on the law of force–displacement and the particles motion are according to the Newton's second law
As a discrete element model (DEM), ‘‘contact bond model’’ and ‘‘parallel bond model’’ are two main types of the bond particle model. In the contact bond model, particles are held together a point of glue and contacts cannot transfer torque. While in the parallel bond model particles are joined by surface layer of glue and contacts can tolerate torque that induced by particles rotation. Thus, the parallel bond model can represent a cement-like substance (Fig. 13), such as rock (Potyondy and Cundall 2004). The force between particles is reflected through the contact force chain and a bond breakage will occur then, when the local stresses exceed the parallel bond strength micro cracks are formed.
4.2 PFC2D Model Preparation and Calibration the for Rock-Like Material
In this paper for preparation a test model the standard process of generating a PFC2D assembly were used, this process entirely is described by Potyondy and Cundall 2004. The process consists of particle generation, packing the particles, isotropic stress installation (stress initialization), floating particle (floater), elimination and bond installation. Since the specimens were small gravity effect and the gravity-induced stress gradient effect on the macroscopic behavior is neglectable. Calibration of particles properties and parallel bonds in bonded particle model were carried out using Uniaxial compressive strength and Brazilian test (Ghazvinian et al. 2012). Adopting the micro-properties are listed in Table 1 and the standard calibration procedures (Potyondy and Cundall 2004), a calibrated PFC particle assembly was created. Fig 14 a and b shows the experimental uniaxial compression test and numerical simulation, respectively. Fig 14 c and d shows experimental Brazilian test and numerical simulation, respectively. The results show good correlation between experimental test and numerical simulation. Also, as indicated in Table 2 the obtained specimen properties from the numerical models such as elastic modulus, Poisson’s ratio, UCS values are nearly similar to the experimental values.
Table 1. Micro properties used to represent the intact rock.
Model height (mm)
|
108
|
Porosity
|
0.08
|
Model width (mm)
|
54
|
Elastic modulus
|
170
|
Expansion coefficient
|
1.2
|
friction coefficient
|
0.5
|
density
|
2200
|
Tensile strength
|
0.1
|
Minimum particle diameter
|
0.6
|
Tensile strength standard deviation
|
0.01
|
Maximum particle diameter
|
1.2
|
Compressive strength
|
0.5
|
Material pressure
|
0.1
|
compressive strength standard deviation
|
0.05
|
Table 2. Comparison of macro-mechanical properties between experiments and model
Mechanical properties
|
Experimental results
|
PFC2D Model results
|
Elastic modulus, (GPa)
|
5
|
5
|
Poisson’s ratio
|
0.18
|
0.19
|
UCS, (MPa)
|
7.4
|
7.4
|
Brazilian tensile strength (MPa)
|
1
|
1.05
|
4.3. Numerical compressive Tests on the Non-Persistent Open Joint
After calibration of PFC2D, uniaxial tests for jointed rock were numerically simulated by creating a box model in the PFC2D (by using the calibrated micro-parameters) (Figs. 15-17). The PFC specimen had the dimensions of 100 mm × 100 mm. A total of 13438 disks with a minimum radius of 0.27 mm were used to make the box specimen. Two walls exist at the upper and lower of the model. The non-persistent joints were formed by deletion of bands of particles from the model (Figs. 15-17). In general, the models containing two and three non – persistent joints were constructed. The small prefabricated crack in this experiment was 1 mm wide and 20 mm long. The large prefabricated crack was 1 mm wide and 40 mm long. The angle of joint with larger length related to horizontal axis was 0, 30, 60, 90, 120 and 150. Twelve types of Y shape non-persistent joints were used in this numerical simulation. The crack arrangement and specimen number of each specimen were depicted in Figs. 15-17. It should be noticed that this joint configuration is similar to experimental one. Upper and lower walls applied uniaxial force on the model. The compression force was registered by taking the reaction forces on the upper wall.
4.4. Failure mechanism of numerical model
a) Number of imbedded joints was 1
Fig. 18 shows the failure pattern of specimens consisting one joint with angle of 0, 15, 30, 45, 60, 75 and 90 for both of the crack initiation stage and final stage. The green line and red line are representative of tensile crack and shear crack, respectively. Fig 19 shows stress versus strain curve for these configurations. When joint angle was 0, In crack initiation stage (Fig 18a), four tensile cracks initiated from joint wall and propagate parallel to loading axis. Also, both of the shear and tensile cracks initiated from joint tips. The crack initiation stress was equal to 2.5 MPa (Fig 19a). In the final stage (Fig 18b), four tensile and shear cracks initiated from joint tips and distributed diagonally related to loading axis till integrated with sample boundary. In this configuration, the “v” shape columns of rock were separated from the model. The final stress was equal to 2.85 MPa (Fig 19a).
When joint angle was 15, In crack initiation stage (Fig 18b), two tensile cracks initiated from joint wall and propagate parallel to loading axis. The crack initiation stress was equal to 2 MPa (Fig 19b). In the final stage (Fig 18c), four tensile and shear cracks initiated from joint tips and distributed diagonally related to loading axis till integrated with sample boundary. In this configuration, the “v” shape columns of rock were separated from the model. The final stress was equal to 2.65 MPa (Fig 19b).
When joint angle was 30, In crack initiation stage (Fig 18e), two tensile cracks initiated from joint wall and propagate parallel to loading axis. The crack initiation stress was equal to 2.5 MPa (Fig 19c). In the final stage (Fig 18f), four tensile and shear cracks initiated from joint tips and distributed diagonally related to loading axis till integrated with sample boundary. In this configuration, the “v” shape columns of rock were separated from the model. The final stress was equal to 3.15 MPa (Fig 19c).
When joint angle was 45, In crack initiation stage (Fig 18g), two tensile cracks initiated from joint tip and propagate parallel to loading axis. The crack initiation stress was equal to 2.8 MPa (Fig 19d). In the final stage (Fig 18h), four tensile and shear cracks initiated from joint tips and distributed diagonally related to loading axis till integrated with sample boundary. In this configuration, the “v” shape columns of rock were separated from the model. The final stress was equal to 3.5 MPa (Fig 19d).
When joint angle was 60, In crack initiation stage (Fig 18i), two tensile cracks initiated from joint wall and propagate parallel to loading axis. The crack initiation stress was equal to 3MPa (Fig 19e). In the final stage (Fig 18j), four tensile and shear cracks initiated from joint tips and distributed diagonally related to loading axis till integrated with sample boundary. In this configuration, the “v” shape columns of rock were separated from the model. The final stress was equal to 3.5 MPa (Fig 19e).
It’s to be note that, the area of “v” shape column was increased by increasing the joint angle from 0 to 60.
When joint angle was 75, In crack initiation stage (Fig 18k), two tensile cracks initiated from joint tip and propagate parallel to loading axis. The crack initiation stress was equal to 4 MPa (Fig 19f). In the final stage (Fig 18l) several shear bands developed in model and lead to failure of the model. In this condition presence of joint has not any effect on the fracture propagation. The final stress was equal to 4.65 MPa (Fig 19f).
When joint angle was 5, In crack initiation stage (Fig 9m), two tensile cracks initiated from joint tip and propagate parallel to loading axis. The crack initiation stress was equal to 4.2 MPa (Fig 19g). In the final stage (Fig 9n) several shear bands developed in model and lead to failure of the model. In this condition presence of joint has not any effect on the fracture propagation. The final stress was equal to 5.1 MPa (Fig 19g).
In all models, the strain value in maximum stress stage was 3.5*10-4.
b) Number of imbedded joints was 2
Fig. 20 shows the failure pattern of specimens consisting two joints with angle of 0, 15, 30, 45, 60, 75 and 90 for both of the crack initiation stage and final stage. The green line and red line are representative of tensile crack and shear crack, respectively. Fig 21 shows stress versus strain curve for these configurations. When joint angle was 0, In crack initiation stage (Fig 20a), four tensile cracks initiated from joint wall and propagate parallel to loading axis. Also, both of the shear and tensile cracks initiated from joint tips. The crack initiation stress was equal to 2.7 MPa (Fig 21a). In the final stage (Fig 20b), four tensile and shear cracks initiated from outer joint tips and distributed diagonally related to loading axis till integrated with sample boundary. In this configuration, the “v” shape columns of rock were separated from the model. It’s to note that both of the tensile cracks and shear cracks lead to rock bridge failure. The fish eye mode failure occurs in rock bridge. The final stress was equal to 3.5 MPa (Fig 21a).
When joint angle was 15, In crack initiation stage (Fig 20c), four tensile cracks initiated from joint tips and propagate parallel to loading axis. The crack initiation stress was equal to 2.5 MPa (Fig 21b). In the final stage (Fig 20d), four tensile and shear cracks initiated from outer joint tips and distributed diagonally related to loading axis till integrated with sample boundary. In this configuration, the “v” shape columns of rock were separated from the model. It’s to note that both of the tensile cracks and shear cracks lead to rock bridge failure. The fish eye mode failure occurs in rock bridge. The final stress was equal to 3.25 MPa (Fig 21b).
When joint angle was 30, In crack initiation stage (Fig 20e), four tensile cracks initiated from joint tips and propagate parallel to loading axis. The crack initiation stress was equal to 3 MPa (Fig 21c). In the final stage (Fig 20f), four tensile and shear cracks initiated from outer joint tips and distributed diagonally related to loading axis till integrated with sample boundary. In this configuration, the “v” shape columns of rock were separated from the model. It’s to note that both of the tensile cracks and shear cracks lead to rock bridge failure. The fish eye mode failure occurs in rock bridge. The final stress was equal to 3.75 MPa (Fig 21c).
When joint angle was 45, In crack initiation stage (Fig 20g), four tensile cracks initiated from joint tips and propagate parallel to loading axis. The crack initiation stress was equal to 3.8 MPa (Fig 21d). In the final stage (Fig 20h), four tensile and shear cracks initiated from outer joint tips and distributed diagonally related to loading axis till integrated with sample boundary. In this configuration, the “v” shape columns of rock were separated from the model. It’s to note that both of the tensile cracks and shear cracks lead to rock bridge failure. The fish eye mode failure occurs in rock bridge. The final stress was equal to 4.2 MPa (Fig 21d).
When joint angle was 60, In crack initiation stage (Fig 20i), four tensile cracks initiated from joint tips and propagate parallel to loading axis. The crack initiation stress was equal to 4 MPa (Fig 21e). In the final stage (Fig 20j), four tensile and shear cracks initiated from outer joint tips and distributed diagonally related to loading axis till integrated with sample boundary. In this configuration, the “v” shape columns of rock were separated from the model. It’s to note that both of the tensile cracks and shear cracks lead to rock bridge failure. The fish eye mode failure occurs in rock bridge. The final stress was equal to 4.5 MPa (Fig 21e).
It’s to be note that, the area of “v” shape column was increased by increasing the joint angle from 0 to 60.
Also, the area of failure surface of rock bridge decreased by increasing the joint angle.
When joint angle was 75, In crack initiation stage (Fig 20k), two tensile cracks initiated from joint tip and propagate parallel to loading axis. The crack initiation stress was equal to 4.5 MPa (Fig 21f). In the final stage (Fig 20l) several shear bands developed in model and lead to failure of the model. In this condition presence of joint has not any effect on the fracture propagation. The final stress was equal to 5 MPa (Fig 21f).
When joint angle was 90, In crack initiation stage (Fig 20m), two tensile cracks initiated from joint tip and propagate parallel to loading axis. The crack initiation stress was equal to 4.7 MPa (Fig 21g). In the final stage (Fig 20n) several shear bands developed in model and lead to failure of the model. In this condition presence of joint has not any effect on the fracture propagation. The final stress was equal to 5.1 MPa (Fig 21g). In all models, the strain value in maximum stress stage was 4.2*10-4.
c) Number of imbedded joint was 3
Fig. 22 shows the failure pattern of specimens consisting three joint with angle of 0, 15, 30, 45, 60, 75 and 90 for both of the crack initiation stage and final stage. The green line and red line are representative of tensile crack and shear crack, respectively. Fig 23 shows stress versus strain curve for these configuration. When joint angle was 0, In crack initiation stage (Fig 22a), six tensile cracks initiated from joint wall and propagate parallel to loading axis. Also, both of the shear and tensile cracks initiated from joint tips. The crack initiation stress was equal to 3 MPa (Fig 23a). In the final stage (Fig 22b), four tensile and shear cracks initiated from outer joint tips and distributed diagonally related to loading axis till integrated with sample boundary. In this configuration, the “v” shape columns of rock were separated from the model. It’s to note that both of the tensile cracks and shear cracks lead to rock bridges failure. The fish eye mode failure occur in rock bridges. The final stress was equal to 3.6 MPa (Fig 23a).
When joint angle was 15, In crack initiation stage (Fig 22c), six tensile cracks initiated from joint wall and propagate parallel to loading axis. Also, both of the shear and tensile cracks initiated from joint tips. The crack initiation stress was equal to 3 MPa (Fig 23b). In the final stage (Fig 22d), four tensile and shear cracks initiated from outer joint tips and distributed diagonally related to loading axis till integrated with sample boundary. In this configuration, the “v” shape columns of rock were separated from the model. It’s to note that both of the tensile cracks and shear cracks lead to rock bridges failure. The fish eye mode failure occur in rock bridges. The final stress was equal to 3.65 MPa (Fig 23b).
When joint angle was 30, In crack initiation stage (Fig 22e), six tensile cracks initiated from joint wall and propagate parallel to loading axis. Also, both of the shear and tensile cracks initiated from joint tips. The crack initiation stress was equal to 3.2 MPa (Fig 23c). In the final stage (Fig 22f), four tensile and shear cracks initiated from outer joint tips and distributed diagonally related to loading axis till integrated with sample boundary. In this configuration, the “v” shape columns of rock were separated from the model. It’s to note that both of the tensile cracks and shear cracks lead to rock bridges failure. The fish eye mode failure occur in rock bridges. The final stress was equal to 3.75 MPa (Fig 23c).
When joint angle was 45, In crack initiation stage (Fig 22g), six tensile cracks initiated from joint wall and propagate parallel to loading axis. Also, both of the shear and tensile cracks initiated from joint tips. The crack initiation stress was equal to 3.8 MPa (Fig 23d). In the final stage (Fig 22h), four tensile and shear cracks initiated from outer joint tips and distributed diagonally related to loading axis till integrated with sample boundary. In this configuration, the “v” shape columns of rock were separated from the model. It’s to note that both of the tensile cracks and shear cracks lead to rock bridges failure. The fish eye mode failure occur in rock bridges. The final stress was equal to 4.25 MPa (Fig 23d).
When joint angle was 60, In crack initiation stage (Fig 22i), six tensile cracks initiated from joint wall and propagate parallel to loading axis. Also, both of the shear and tensile cracks initiated from joint tips. The crack initiation stress was equal to 4 MPa (Fig 23e). In the final stage (Fig 22j), four tensile and shear cracks initiated from outer joint tips and distributed diagonally related to loading axis till integrated with sample boundary. In this configuration, the “v” shape columns of rock were separated from the model. It’s to note that both of the tensile cracks and shear cracks lead to rock bridges failure. The fish eye mode failure occur in rock bridges. The final stress was equal to 4.5 MPa (Fig 23e).
It’s to be note that, the area of “v” shape column was increased by increasing the joint angle from 0 to 60.
Also, the area of failure surface of rock bridge decreased by increasing the joint angle.
When joint angle was 75, In crack initiation stage (Fig 22k), two tensile cracks initiated from joint tip and propagate parallel to loading axis. The crack initiation stress was equal to 4.5 MPa (Fig 23f). In the final stage (Fig 22l) several shear bands developed in model and lead to failure of the model. In this condition presence of joint has not any effect on the fracture propagation. The final stress was equal to 4.9 MPa (Fig 23f).
When joint angle was 90, In crack initiation stage (Fig 22m), two tensile cracks initiated from joint tip and propagate parallel to loading axis. The crack initiation stress was equal to 4.5 MPa (Fig 23g). In the final stage (Fig 22n) several shear bands developed in model and lead to failure of the model. In this condition presence of joint has not any effect on the fracture propagation. The final stress was equal to 5 MPa (Fig 23g).
In all models, the strain value in maximum stress stage was 4.8*10-4. It can be concluded that the strain value in maximum stress stage increased by increasing the joint number.
By comparison between Figs 9-11 and Figs 18, 20 and 22, It can be concluded that failure pattern is similar in both of the experimental test and numerical simulation.
4.5 The effect of oriented plane angle and joint angle on the strength of samples
Fig 24 shows the effect of joint angle on the strength of models. This figure was presented for three joint number. The strength of samples was increased by increasing the joint angle. The minimum of compressive strength occurs when joint angle was 30. The strength of sample was increased by increasing the joint number.
By comparison between Fig 12 and Fig 24 It can be concluded that failure strength is nearly similar in both of the experimental test and numerical simulation.