Particle Flow Simulation of the Direct Shear Tests on the Weak Structural Surface
Wenmin Yao, Bin Hu*, Lichen Li, Xiaolong Chen, Chenxi Rao Faculty of Engineering, China University of Geosciences, Wuhan Hubei 430074, People’s Republic of China *Corresponding author; e-mail:
[email protected]
ABSTRACT Based on the operation of weak structural surface direct shear tests of argillaceous limestone, the macro mechanical properties like stress-strain, shear resistance, dilatancy effect and energy dissipation and the evolution of micro-cracks of the structural surface were simulated and analyzed using Particle Flow Code in Two Dimensions (PFC2D). The results revealed that: as the constant normal load increased, the peak value of the shear stress and the corresponding displacement increased, while the peak value of the shear resistance decreased and its corresponding shear displacement yielded similar results to that of the peak shear stress; the increase of constant normal load led to the weakening of the shear resistance and the dilatancy effect; what’s more, the distribution rule of the cracks in the model after the shear test was consistent with that of the peak stress in the monitoring circles during the shearing process; the frictional energy and strain energy kept changing during the shearing process; the strain energy was larger than the friction energy before the shear stress reached its peak value; when the shear stress decreased, the frictional energy took control and the shear cracks came into existence rapidly, after the shear stress became stable, the friction energy kept increasing while the strain energy tended to stabilize and there generated a large number of normal cracks.
KEYWORDS:
Weak structural surface; Direct shear test; Particle flow; Energy dissipation;
Micro-cracks
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INTRODUCTION The long geological process tends to give birth to a variety of anisotropic and heterogeneous structural surfaces which frequently become the controlling factors in the analysis of rock engineering stability owing to their relatively lower strength compared with the surrounding rock mass. It’s common to find that the failure of rock slopes which contain structural surfaces results from shear failure along the surface of the structure, thus making it of great significance to rock engineering projects to research on the macro and micro mechanical properties of structural surfaces during the shearing process. However, the discontinuous properties of structural surfaces making it a struggle to apply the continuum theory to the in-depth analysis of the failure characteristics and the mechanical evolution mechanism of rock mass and rock joints [1-2]. Particle flow discrete element method, as an emerging numerical analysis method, has been gradually introduced into the field of rock joints research [3-8]. Particle flow method builds up models based on circular particles or spherical particles and applies the finite difference theory to solve problems from the mesoscopic viewpoint; data like yielding strength and stress-strain curve can be obtained through iterative calculation and observation of the failure and crack development of particles can be achieved in particle flow method [9-10]. InPFC2D program, joint elements of arbitrary shapes can be generated using ‘jset’ command, and direct shear tests can be successfully simulated by assigning certain parameters and operating the loading steps [11]. Liu S designed various models of intermittent joints and operated direct shear tests and numerical simulation tests based on PFC2D under different normal loads [12]. Zhou Y researched on the direct shear tests of rock joints and found that under different constant normal loads, shear stress and shear displacement increase with the normal load increases, while the dilatancy effect and shear resistance have different degrees of reduction [13]. Yu H studied the shearing mechanical behavior of rock joint planes and revealed that the connection of micro-cracks on joint surface caused the joints shear failure based on the rock joints simulation using the CPM material models [14]. Cao R applied PFC2D to generate five different joint models of different roughness and carried out numerical direct shear tests of the models under different normal loads respectively, then operated research on the morphology damage and crack evolution mechanism of joints from habituation [15]. By using modern laser scanning technology, Jiang Y carrying out the muddy limestone joint shear strength relationship study between the 2 dimensional and 3 dimensional model based on the simulation software PFC [16]. Furthermore, PFC2D can be applied to monitor the different kinds of energies of the model during calculating process to build connections with the macro failure phenomena. Using PFC2D program, Li X studied the crack development、energy conversion and acoustic emission characteristics of the jointed rocks from the viewpoint of mesomechanics and suggested from the macro perspective that the failure forms of jointed rocks can be divided into wearing and cutting, which correspond to the different strength models [17]. Zhou J studied fragmentations and energy dissipation of rock blocks during the motion process based on laboratory tests and numerical simulation [21]. Some scholars
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also applied Discrete Element Method to simulate the rock mechanical tests and analyzed the crack evolution process and energy dissipation rules [19-22]. There is a set of widely distributed Permian limestone crumb structural surface composed of undisturbed dark and gray bioclastic argillaceous silt in Emei region, Sichuan province, China. Different from the tough structural surface mentioned above, this set of structural surface is with low strength and prone to soften when reacting with water, making it the sliding surface of several reported large-scale landslides in this region. Based on the direct shear tests on this kind of structural plane, particle flow theory and PFC2D program were introduced to analyze the macroscopic mechanical properties and energy dissipation and the meso crack evolution mechanism of the structural surface during direct shear tests so as to provide reference to the following researches and engineering application.
SIMULATION OF DIRECT SHEAR TESTS ON STRUCTURAL SURFACE Establishment of PFC2D model The sample size of the simulated shear test was 60 mm × 60 mm, there were 6 walls first generated, with wall # 4、wall # 5、wall # 6 serving as the bearing walls and the velocities of wall# 1、wall # 2、wall # 3 constrained to 0, the covering rectangular area of which is the actual sample size. The test chose random radii within the given radius for the particles in the fixed walls and increased the radii during the modelling process. However, it would inevitably generate a few suspended particles which were not in touch with the surrounding particles. The existence of this kind balls would affect the accuracy of the test and the FISH function ‘pc_zap_floaters’ in PFC2D can be applied to delete these balls. The test yielded about 6678 circular particles, whose radii were subjected to Gaussian distribution with Rmin = 0.300 mm and Rmax = 0.525mm. A smooth structural surface was generated in the midst of the sample using ‘jset’ command. The numerical model of the simulated test can be seen in Figure 1.
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Figure 1: The numerical model in PFC2D software of structural surface direct shear test
The servo-control and loading process of constant normal load During the simulation process of direct shear test, the loading process was achieved by controlling the velocities of the walls. Based on the servo-mechanism, the FISH functions were introduced to adjust the walls’ velocities continuously so as to further control the stress transmitted among the walls. Based on the above theories, the velocity of wall # 4 should be adjusted first until the constant normal load reached the prescribed value and then certain horizontal velocities could be assigned to wall # 4、wall # 5 and wall # 6 so that shear stress could be applied to the model. When 4 the horizontal shear velocity v = 0.1mm/(10 timestep) as a result, the whole shear process could be kept in a quasi-static state which would ensure the reliability of the simulation result.
Calibration of micromechanics parameters Different from other numerical softwares, PFC2D assigns micromechanics parameters to the particles to reflect the mechanical properties of the model, necessitating the calibration of the parameters before the calculation and analysis of the numerical model. The contact model in the test was the parallel bonding model which could be envisioned as a kind of glue with a finite size that acted over either a circular or rectangular cross-section lying between the particles. The parallel bond could transmit both a force and a moment and were often
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used to represent materials like rock, the particles of which could serve as glue to join the surrounding particles. If the tensile normal force exceeded the contact bond normal strength, the bond broke and the normal cracks would be generated; when the shear contact force exceeded the shear contact strength, there would yield shear cracks. The failure mechanism of parallel bond are illustrated in Figure 2.
(a) Failure mechanism of normal bond
(b) Failure mechanism of shear bond
Fci—bond strength；Ki—contact stiffness；Ui—displacement
Figure 2: Failure mechanism of parallel bond. Table 1: Micromechanical parameters of numerical mode Wall
Particle Normal
Normal
Shear
Particle
stiffness/
stiffness/
density/（kg·m
(N -1) 1e9
(N
Radius coefficient 1
-1
0
)
Shear
Friction
stiffness/
stiffness/
coefficient
(N -1) ） 2020 1e8 Parallel bond
(N -1) 3e8
0.5
Stiffness/(Pa·m-1) Normal Shear 1e8 3e8
-3
Normal strength/Pa
Shear strength/Pa
1e5
1e5
By continuously adjusting the microscopic parameters when operating the numerical experiments, it revealed that when the parameters in Table 1 was applied to the simulation, the mechanical properties reflected from the experimental results were in accordance with the experiment results. As can be seen, the parameters in Table 1 could be applied to the numerical modelling to precisely describe the mechanics and failure characteristics of the sample. Micromechanics
parameters assigned to the sample are shown in Table 1.
MACRO MECHANICAL PROPERTIES Characteristics of shear stress-shear displacement Figure 3 illustrates a plot of the trend of the shear stress against the shear displacement under different corresponding normal loads. When the constant normal load reached 0.75Mpa, 1.00Mpa, 1.25Mpa, the value of peak shear stress was 0.46Mpa, 0.55Mpa, 0.71Mpa respectively, with the corresponding shear displacement reaching 0.56mm, 0.74mm, 0.85mm, which meant the shear stress increased as the normal load increased. Also, it could be seen that the curves under three different constant normal loads coincided with each other perfectly at the linear stage, which suggested that the joint elastic shear modulus kept constant under different normal loads. In this case, the elastic shear modulus derived from Figure 3 was 112.6Mpa, close to the results of indoor experiments 110.3Mpa. After the shear stresses reached the peak values, all the shear stresses exhibited an overall downward trend as the shear displacements increased and fluctuated at certain stages and converged at different shear stresses at last. 0.8
0.75MPa 1MPa 1.25MPa
Shear stress/MPa
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0
0.5
1.0
1.5
2.0
2.5
Shear displacement/mm
Figure 3: The trend of shear stress against shear displacement
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Characteristics of shear resistance SR (shear resistance) was used to evaluate the ability to resist the shear failure of the structural surface under different constant normal loads, the value of which was illustrated as the results of shear stress divided by normal load [13]. Figure 4 shows the trend of the shear stress against the shear displacement at different corresponding normal loads. When the constant normal load reached 0.75Mpa, 1.00Mpa, 1.25Mpa, the value of peak shear resistance was 0.62, 0.55, 0.52 respectively at the same shear displacement corresponding to the peak shear stress. As can be suggested, with the climb of normal loads, the values of peak shear resistance got downward, showing the weakening of shear resistance. Meanwhile, as the shear displacements accumulated, the shear resistance first reached its peak value and then decreased with fluctuation and finally converged at around 0.37 under different loads. 0.7
0.75MPa 1MPa 1.25MPa
Shear resistance
0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0
0.5
1.0
1.5
2.0
Shear displacement/mm
2.5
Figure 4: The trend of shear resistance against shear displacement
Characteristics of dilatancy effect As can be seen in Figure 5, which showed the relationship between normal displacement and shear displacement under loading, the normal displacement accumulated as the shear displacement increased, suggesting that the dilitancy effect came into work at contacts. At the first half part of the curves (namely the linear stage before normal displacements reach 1mm), the slopes got more inclined as the normal loads were smaller, which also meant the normal displacements grew faster. At the latter part of the curves (when the model has already reached the plastic stage), the normal loads increased after fluctuation while the slopes was smaller than the former ones. When the constant
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normal load reached 0.75Mpa, 1.00Mpa, 1.25Mpa, the value of peak normal displacement was 0.16mm, 0.14mm, 0,12mm respectively, while the corresponding shear displacement was 2.06mm, 2.08mm, 2.18mm, which meant the peak normal displacements increased as the normal loads decreased, namely the dilitancy effect was more obvious. 0.18
0.75MPa 1MPa 1.25MPa
Normal displacement/mm
0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00
0.0
0.5
1.0
1.5
2.0
Shear displacement/mm
2.5
3.0
Figure 5: The relationship curve of normal displacement and shear displacement
Distribution characteristics of micro-cracks Figure 6 illustrates the distribution of micro-cracks in the model under different loads (grey color denotes parallel bonds, blue color denotes shear cracks and red color denotes normal cracks respectively). As can be seen, breakage initiated at the cracks and the neighbouring parallel bonds, making it suitable to tell the shearing zones from the cracks distribution and schematic diagram of parallel bond breakage. There were mainly shear cracks in the model distributed along the structural surface under different normal loads and there would be more cracks as the normal loads increased with 117, 138, 159 cracks respectively under the three different constant normal loads. What’s more, the cracks tended to localize at the two sides of the structural surface as a result of the boundary constraint effects during the shearing process which led to the localization of contact force among particles adjacent to wall # 5 and wall # 6, making a majority of bonds break there.
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（b）1MPa
（c）1.25MPa
Figure 6: Crack distribution under different constant normal loads after failure
Distribution characteristics of strain The built-in monitoring circle function in the software was applied to record the stress evolution rule on the pre-shearing face in the model with 6 monitoring circles evenly settled on the pre-shearing face (as shown in Figure1). The monitoring data reached the peak values when the shear displacements reached around 0.74mm as shown in Figure 7, which was in great accordance with shear stress-shear displacement curves. The descending order of the peak shear values of each curves was 6>2>4>5>3>1. Also, there were more cracks at the two sides of the structural surface as can be suggested in Figure5. 0.45
Shear stress/MPa
0.40 0.35 0.30 0.25 0.20
No.1 monitoring circle No.2 monitoring circle No.3 monitoring circle No.4 monitoring circle No.5 monitoring circle No.6 monitoring circle
0.15 0.10 0.05 0.00 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Shear displacement/mm Figure 7: Monitoring circles stresses against shear displacement
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MECHANISM OF ENERGY DISSIPATION AND CRACKS EVOLUTION The numerical simulation results at 1Mpa normal load was taken as an exact example to analyze the energy dissipation and cracks evolution mechanism of the structural surface during the numerical shear test. Sample failure was a kind of unstable state that occurred in the energy conversion process [23]. In direct shear test, the dissipated energy mainly contributed to the displacements, rotation and expansion of micro-cracks of contacts, which finally led to the formation of fracture plane and sample’s shearing failure. PFC2D can make solutions by tracing the particles, contacts and bonds in the model and monitor different energy terms to better reflect the energy revolution law. Nowadays, views about kinetic energy of particles differ widely based on the characteristics analysis of energy dissipation in PFC2D [21, 24-25]. In this paper, kinetic energy, as a kind of energy stored in particles, should not be considered when calculating the dissipated energy. Based on the assumption that the simulated system of the shear test was a closed system which means there was no energy exchange between the system itself and the external environment, the following equation can be obtained according to the First Law of Thermodynamics: U = Ud + Ue + Uk
(1)
where U is the body work corresponding to the boundary work that the system monitors; U d is the dissipated energy for the inner damage and plastic deformation of the rock samples, which includes the frictional energy, damping dissipation and rotational energy; U e is the plastic strain energy stored in rock samples which can be released; U k is the kinetic energy, which can also be solved as follows: U= U c + U pb e
(2)
where U c is the strain energy of particles; U pb is the parallel bond energy. The shear stress – shear displacement curves were divided into four sections, representing different stages, as shown in Figure 8, and the energy dissipation characteristics and evolution mechanism of micro-cracks on each stages were fully discussed below:
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30
Boundary work Dissipate energy Frictional energy Strain energy Parallel bond energy Kinetic energy
Energy/kN·m
25 20 15 10 5
0.4
0.8
1.2
200
B
Number of cracks
180
A
160 140
C
120
1.6
2.0
Shear stress Total number of cracks Number of shearing cracks Number of normal cracks D
100
0.6 0.5 0.4 0.3
80 0.2
60 40
Shear stress/MPa
0 0.0
0.1
20 0 0.0
O 0.4
0.8
1.2
1.6
2.0
0.0
Shear displacement/mm
Figure 8: Energy dissipation and micro-crack evolution diagram Linear stage (section OA): most of the boundary energy transferred into the strain energy, whose growth rate was larger than that of the boundary energy; dissipated energy showed an upward trend with increasing rate; meanwhile, the occurrence of cracks was negligible. Upward stage before the peak (section AB): the growth rate of strain energy decreased while the dissipated energy grew faster; The shear stress reached the peak value as strain energy increased to the largest value; the dissipated energy and boundary work increased in the same trend; A few cracks could be seen at this stage. Downward stage after the peak (section BC): the strain energy decreased while the frictional energy grew rapidly; shearing cracks propagated much faster at this stage.
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Fluctuation stage of shear stress (section CD): both of the shear stress and strain energy fluctuated while the strain energy stabilized at a certain value at last; the frictional energy, which began to take control, was larger than the strain energy at this stage; shearing cracks could be obviously seen and normal cracks came into existence at this stage, leading to the formation of shearing zone. The kinetic energy kept small and almost stable during the whole shearing process, which could well correspond to the quasi-static state described before. Through the analysis of energy conversion law and particle movement during the numerical shear test, it could be concluded that the friction and rotation among particles led to the generation of shear cracks as the shear stress exceeded the particles’ shear contanct strength with frictional energy dominating the failure; what’s more, as shear displacement accumulated, the number of normal contacts at the structural surface also decreased, resulting in the growth of contact force among particles; when the normal contact force exceeded the normal contact strength, normal cracks came into existence, which grew rapidly when the frictional energy exceeded the strain energy.
CONCLUSION By simulating the direct shear test on weak structural surface, the macro mechanical properties, energy dissipation and evolution mechanism of micro-cracks were simulated and analyzed using the Particle Flow Code Software(PFC2D). The conclusions are listed below: (1) As the constant normal load increases, the peak value of shear stress and its corresponding displacement also grow while the dilatancy effect becomes less obvious and the peak value of shear resistance decreases, which means the weakening of shearing capacity, the shear resistance converges to 0.37; the plastic shear modulus is 112.6 Mpa, which is close to the experimental results. (2) As a result of stress concentration, there are more cracks at two sides of the structural surfaces, which is in accordance with the distribution of monitoring peak shear stress in the preshearing plane. (3) The shear stress – shear displacement curves are divided into four sections: before the shear stress reaches its peak, most of the boundary energy transferred into strain energy and the dissipated energy grows slowly, with a few shear cracks; after the value of shear stress reaches the peak, the frictional energy grows rapidly while the dissipated energy and the boundary work increase in the same trend, with cracks propagating fast and obvious shearing zone coming into existence.
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(4) When the shear stress decrease, shear cracks keep growing as frictional energy accumulates; when the shear stress stabilize, there generates large number of normal cracks when frictional energy exceeds strain energy.
ACKNOWLEDGEMENTS This work is supported by National Natural Science Foundation of China (41672317).
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Editor’s note. This paper may be referred to, in other articles, as: Wenmin Yao, Bin Hu, Lichen Li, Xiaolong Chen, Chenxi Rao: “Particle Flow Simulation of the Direct Shear Tests on the Weak Structural Surface” Electronic Journal of Geotechnical Engineering, 2016 (21.26), pp 1034910363. Available at ejge.com.
