# Hybrid finite-discrete element modelling of asperity degradation and gouge grinding during direct shearing of rough rock joints

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## Abstract

A hybrid finite-discrete element method was implemented to study the fracture process of rough rock joints under direct shearing. The hybrid method reproduced the joint shear resistance evolution process from asperity sliding to degradation and from gouge formation to grinding. It is found that, in the direct shear test of rough rock joints under constant normal displacement loading conditions, higher shearing rate promotes the asperity degradation but constraints the volume dilation, which then results in higher peak shear resistance, more gouge formation and grinding, and smoother new joint surfaces. Moreover, it is found that the joint roughness affects the joint shear resistance evolution through influencing the joint fracture micro mechanism. The asperity degradation and gouge grinding are the main failure micro-mechanism in shearing rougher rock joints with deeper asperities while the asperity sliding is the main failure micro-mechanism in shearing smoother rock joints with shallower asperities. It is concluded that the hybrid finite-discrete element method is a valuable numerical tool better than traditional finite element method and discrete element method for modelling the joint sliding, asperity degradation, gouge formation, and gouge grinding occurred in the direct shear tests of rough rock joints.

## Keywords

Hybrid FEM-DEM Rock joint Asperity sliding Asperity shearing Fragment grinding## 1 Introduction

Asperity degradation and grinding during rock joint shearing process play an essential role in the mechanical and hydraulic behaviour of the rock joints. Numerous laboratory experiments were conducted by various researchers (Pereira and de Freitas 1993; Seidel and Haberfield 1995, 2002; Lee et al. 2001; Hossaini et al. 2014) to describe the asperity sliding and damage caused by overriding of asperities and overstressing upon shearing. However, a number of outstanding issues of importance relating to rock joint fracturing still remain (Barton 2013), such as asperity degradation and breakage—induced gouge production and grinding (Jing and Stephansson 2007). The asperity degradation and associated gouge production and grinding may affect the shape of a joint surface and the subsequent response of the rock joints. Thus, improved understanding of the asperity degradation and gouge grinding is essential to characterize the mechanical and hydraulic behaviour of the rock joints, which have important applications in a variety of fields including rock slope engineering, mining, tunnelling, petroleum engineering and earth sciences.

Asperity degradation has been studied by directly assessing the surface morphology of a rough rock joint before and after shearing. Assuming the asperities on the rock joint surface had identical shape and same inclination angle, Patton (1966) developed the first theoretical model to predict the shear strength of rough rock joints. Although it took into account the effect of roughness of rock joint on its shear strength by a two-dimensional simplification, Patton’s model ignored the scale effect, interlocking effect between the asperities, and roughness evolution during the shearing and damage process of the rock joint. Ladanyi and Archambault (1969) proposed a non-linear theoretical model to predict the shear strength of rough rock joints, which provides better non-linear normal stress - shear stress curves than Patton’s model. However, the application of the non-linear model required more parameters and some special tests, and some parameters couldn’t well describe the irreversible effect of the roughness on the rock joint shear strength. Barton and Choubey (1977) proposed the well-known shear strength criterion for rough rock joints with the roughness explicitly denoted using the joint roughness coefficient (JRC). Plesha (1987) took into account roughness degradation and developed a two-dimensional theoretical model for rough rock joints on the basis of the theory of plasticity and the assumption of uniform tooth-shaped asperities. Amadei and Saeb (1990) developed a two-dimensional nonlinear elastic constitutive model for rough rock joints, which considered the different normal deformability of rock joints with mated and unmated initial positions and the effect of the deformability on the failure behaviour. Souley et al. (1995) extended the Amadei-Saeb model to include the shear behaviour of rough rock joints under cyclic loading. Haberfield and Johnston (1994) developed a mechanically-based model for rough rock joints capable of capturing the basic mechanisms of movement and making reasonably accurate predictions of shear displacement behaviour. Maksimovic (1996) proposed a non-linear joint failure model of hyperbolic type with three parameters: the basic angle of friction, the roughness angle and the median angle pressure. More recently, many joint constitutive models have been developed for the physical–mechanical behaviour of rock joints (Indraratna and Haque 2000; Olsson and Barton 2001; Seidel and Haberfield 2002; Serrano et al. 2014; Indraratna et al. 2015; Hencher and Richards 2015; Shrivastava and Rao 2015). However, these constitutive models have difficulty in implementing the real geometry of rock joints and the input parameters of these models are predefined by the user, which hampers the predictive capability of these models (Bahaaddini et al. 2015). Moreover, they are unable to trace the process of asperity shearing and degradation, and gouge formation and grinding, and the crack propagation inside the intact materials of the joint surfaces.

With the development of computational geomechanics, numerical method based on continuous and discontinuous mechanics has been become a promising approach to study the shear behaviour of rock joints. Son et al. (2004) conducted elasto-plastic simulation of a direct shear test on rough rock joints using a finite element method (FEM) with a joint finite element of 6-node and zero thickness. Their results reproduced salient phenomena commonly observed in actual shear test of rock joints, including the shear strength hardening, softening and dilation. Roosta et al. (2006) developed a visco-plastic multilaminate model to model the shear stress-shear displacement and normal displacement—shear displace of artificial joint specimen at constant normal load conditions. Liu et al. (2009) implemented the rock failure process analysis (RFPA) model (Tang 1997) into the finite element software package ABAQUS using its user subroutine UMAT interface to simulate the shearing process of a three-dimensional (3D) rough rock joints on the basis of the interface modelling technique and found that a curved failure surface developed from the loading asperity and approximately intersected the trailing face of the asperity. Lin et al. (2012) used a continuum approach to simulate direct shear tests on flat and wave-like rock joints. A new methodology is proposed by Nguyen et al. (2014) to combine shear box testing and corresponding continuous mechanics—based numerical simulations with the natural joint roughness at the micro-scale taken into account. However, most numerical simulations of the shearing process of rough rock joints are conducted using discontinuous methods, especially the commercial discrete element method (DEM)—Particle Flow Code (PFC) developed by Itasca Consulting Group Inc. Cundall (1999) implemented a bounded particle model into PFC2D to model the nonlinear relation between peak shear strength and normal stress, and the dependence of the peak dilation angle on the normal stress in shearing rock joints and rough faults. Indraratna and Haque (2000) used PFC2D to simulate the shear behaviour of artificial regular rock joints. Guo and Morgan (2008) simulated the breakdown of fault blocks using PFC2D to study the frictional strength, mechanical behaviour and stress and strain rate of evolving fault gouge and their dependence on normal stress and uniaxial compressive strength. Park and Song (2009, 2013) carried out an extensive series of simulations for direct shear tests of rock joints using PFC3D to demonstrate the feasibility of reproducing a rock joint using the bonded particle model and examine the effect of the geometrical features and the micro-properties of a joint on its shear behaviour. Shrivastava et al. (2011) used PFC2D to simulate direct shear tests on rock joints with asperity inclinations of 15° and 30° at different normal stresses. Zhao (2013) implemented PFC2D to simulate single and multi-gouge particles in a rough fracture segment undergoing shear. Huang et al. (2014) simulated the dynamic direct-shear tests on the rough rock joints with 3D both sinusoidal and random surface morphologies using the discrete element method. Bahaaddini et al. (2015) studied the shear behaviour of rock joints in a direct shear test using the particle flow code PFC2D taken into account the micro-scale properties of the smooth joint model.

As can be seen from the review on numerical modelling of rock joint shear process above, the continuous method usually model the joint shear process using either joint elements or contact surfaces with complicated constitutive models, which are developed on the basis of theoretical analysis or experimental observation of the relationship between shear stress and shear displacement when the rock joint is subjected to shear load. Thus, the continuous method has not modelled explicitly the asperity degradation and gouge grinding during the rock joint shear process. Through the bounded particle model, the discontinuous method has successfully modelled the fracture of asperity and the formation of circular or sphere gouge. However, the discontinuous method has the limitation of modelling the asperity degradation before its fracture, the transition from continuum to discontinuum through asperity degradation and fracture, the formation of irregular-shaped gouge, and the further fragmentation of the irregular-shaped gouge through the gouge grinding. This study implements a hybrid continuous-discontinuous method, i.e., a recently developed hybrid finite-discrete element method (Liu et al. 2015a), to model the asperity degradation and gouge grinding during the shearing process of rough rock joints. Compared with FEM, the hybrid method is more robust and efficient after the asperity degradation, especially tracing the gouge formation, movement and grinding. Compared with DEM, the hybrid method is more versatile in modelling asperity degradation before fracture, crack initiation and propagation during fracture, and irregular-shaped gouge grinding. This study extends an initial study on direct shearing of rock joints conducted by the authors and presented in a conference (Liu et al. 2015b). In the following sections, the hybrid finite-discrete element method is firstly introduced. Then the rock failure processes in the basic rock mechanics tests, i.e., the uniaxial compressive strength test and the Brazilian tensile strength test, are modelled using the hybrid method and the modelled results are compared with those well documented in literatures to calibrate the hybrid method. After that, the asperity degradation and gouge grinding in the direct shear tests of a rough rock joint are studied using the hybrid finite-discrete method. Finally, the hybrid method is used to investigate the effect of various shearing rates and roughness on the asperity degradation and gouge grinding during the shearing processes of rock joints.

## 2 Hybrid finite-discrete element method for modelling direct shearing of rock joints

*δ*can be divided into two components in Eq. (1)

*and*

**n***are the unit vectors in the normal and tangential directions, respectively, of the surface at such a point,*

**t***δ*

_{ n }and

*δ*

_{ s }are the magnitudes of the components of

*δ*. Accordingly, the traction vector

*p*during fracture and fragmentation can be divided into two components in Eq. (2)

*D*a peak strength \(f_{p}\). described using Eqs. (5) and (6)

After fracture and fragmentation, an explicit central difference scheme is applied in the hybrid finite-discrete element method to integrate the equations of motion of either the initially discrete elements or the discrete elements formed by the fracture and fragmentation algorithm. The unknown variables, i.e., contact forces on the discrete ements’ boundary or stresses in the internal elements are determined locally at each time step from the known variables on the boundaries and in the elements and their immediateeighbours, as shown in Fig. 1d.

In sum, the hybrid finite-discrete element method involves in the following algorithms: (1) the numerical model is assumed to consist of an assemblage of discrete deformable bodies and the interaction between discrete deformable bodies is solved using the contact law. (2) The rigid body movement of the discrete deformable bodies is solved to produce the inertial and interaction forces for each of the discrete bodies and gross rigid body translational and rotational displacement of the discrete body as a whole. (3) The inertial and interaction forces are applied on each of the discrete deformable bodies to determine its deformation, displacement and strain and stress fields according to the finite element method. (4) If the calculated deformation and stress fields satisfy the failure criteria, fracture and fragmentation occur and the discrete deformable body is fragmented into two or more discrete deformable bodies through fracture and fragmentation algorithms. (5) The calculated deformation fields are supposed over the gross rigid body motion displacements to calculate new positions of each of discrete deformable body. Thus, the hybrid finite-discrete element method uses the continuous method, i.e., FEM, to model continuum-based phenomena and the discontinuous method, i.e., DEM, to model discontinuum-based phenomena. Compared with FEM, the hybrid finite-discrete element method is more robust in modelling rock failure, especially fracture, fragmentation, and fragment movements resulting in tertiary fractures. Compared with DEM, the hybrid finite-discrete element method is more versatile in dealing with irregular-shaped, deformable and breakable particles. Please refer to our recent paper (Liu et al. 2015a) for the detail coding and implementation of the hybrid finite-discrete element method in computers.

## 3 Calibration of the hybrid finite-discrete element method by modelling UCS and BTS

*A*in Eq. (7) should be chosen as 20 for this specimen. In this case, theynamic strength of the specimen can be calculated according to Eq. (9)

## 4 Hybrid finite-discrete element modelling of rock joint failure in direct shear test

Therefore, as can be seen from Figs. 7 and 8, the hybrid finite-discrete element modelling explain the shear resistance evolution mechanism of the joint during the direct shear test. At the early stage of the direct shear test, the shear resistance of the joint is completely provided by the increasing static friction. As the shear displacement increases, the static frictions in some joint asperities consistently lose and transform into the sliding friction, but the static friction in the surrounding asperities gradually increases. During the asperity sliding process, the intact asperities on the two surfaces of the joint ride up each other to provide increasing joint shear resistance through a combination of the static and sliding frictions. The joint shear resistance reaches its peak when the intact asperities start to damage and break forming gauges between the two joint surfaces. During the asperity shearing process, the joint resistance gradually decreases and the majority of the asperities becomes damaged and broken although there may be still unbroken joint asperities. During the strain-softening stage after the peak, the broken joint asperities provide the joint shear resistance through sliding friction while the unbroken joint asperities and the resultant gauges formed between the two joint surfaces may ride up to provide the joint shear resistance by a combination of the static, sliding and rotating frictions. The break of the joint asperities leads to the stress concentrations in surrounding unbroken joint asperities and gauges, which may cause the joint shear resistance does not decrease monotonically but increase a little bit in some stage. This process may repeat in several cycles depending on the matching of the formed new joint surfaces and the grinding and rotation of the resultant gauges. Finally all of the asperities are broken and it becomes easy for the resultant gauges to grind or rotate between the formed new joint surface, in which the shear resistance is completely provided by the sliding and rotating frictions. Correspondingly, the shear resistance decreases to the residual shear stress. In sum, the hybrid finite-discrete element modelling reproduces all the shear failure mechanisms during the rock joint shearing process described in literature (Guo and Morgan 2008): asperity damage, contact sliding and grain rotation except the volume change, which is, however, indirectly reproduced through the increased asperity degradation since the constant displacement conditions in the vertical boundaries of the model suppressed the dilation and compaction but promotes the asperity degradation.

## 5 Discussions

### 5.1 Dynamic shear behaviour of rock joints under different shearing rates

### 5.2 Direct shearing of rock joints with various roughness

## 6 Conclusions

- (1)
The hybrid finite-discrete element method reproduces the joint shear resistance evolution process during the direct shear test of rough rock joints, such as from asperity sliding to degradation and from gouge formation to grinding. Compared with FEM, the hybrid method is more efficient in modelling the asperity sliding and the gouge formation and grinding due to the asperity degradation, which FEM has difficulty to deal with. Compared with DEM, the hybrid method is more robust in modelling the transition from continuum to discontinuum due to the asperity degradation and the formation and grinding of irregular-shaped gouges. Thus, the hybrid finite-discrete element method is a valuable numerical tool better than FEM and DEM for the studies on the asperity degradation and gouge grinding during the shear resistance evolution process of rock joints under shearing and even the fracture and fragmentation of rocks under static and impact loads.

- (2)
Higher shearing rate in the direct shear tests of rough rock joints under the constant normal displacement loading conditions promotes the asperity degradation but constraints the volume dilation, which result in higher peak shear resistance, more gouge formation and grinding, and smoother new joint surfaces.

- (3)
Roughness affects the joint shear resistance evolution through influencing the joint fracture micro-mechanism. The asperity degradation and gouge grinding are the main micro-mechanism occurred in shearing rough rock joints with deep asperities while the asperity sliding is the main micro-mechanism happened in shearing smooth rock joints with shallow asperities.

## Notes

### Acknowledgments

The first author would like to thank the supports of the NARGS, IRGS and AAS grants of Australia, and the National Science Foundation grants (No. 51574060 and No. 51079017) of China, in which the first author is the international collaborator. The academic visits of the third and fourth authors to the University of Tasmania are partly supported by a PhD visiting scholarship and an academic visiting scholarship, respectively, provided by the China Scholarship Council, which are greatly appreciated.

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