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Journal of Rock Mechanics and Geotechnical Engineering 11 (2019) 684e700

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Journal of Rock Mechanics andGeotechnical Engineering

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Review

An overview of particle-based numerical manifold method and itsapplication to dynamic rock fracturing

Xing Li a,*, Jian Zhao b,**

a School of Civil Engineering, Southeast University, Nanjing, 211189, ChinabDepartment of Civil Engineering, Monash University, Clayton, VIC, 3800, Australia

a r t i c l e i n f o

Article history:Received 21 August 2018Received in revised form7 January 2019Accepted 18 February 2019Available online 13 April 2019

Keywords:Dynamic rock fracturingNumerical modelingNumerical manifold method (NMM)Particle-based numerical manifold method(PNMM)

* Corresponding author.** Corresponding author.

E-mail addresses: xing.li@seu.edu.cn (X. Li), jian.zhPeer review under responsibility of Institute of R

nese Academy of Sciences.

https://doi.org/10.1016/j.jrmge.2019.02.0031674-7755 � 2019 Institute of Rock and Soil MechanicNC-ND license (http://creativecommons.org/licenses/

a b s t r a c t

This review summarizes the development of particle-based numerical manifold method (PNMM) and itsapplications to rock dynamics. The fundamental principle of numerical manifold method (NMM) is firstbriefly introduced. Then, the history of the newly developed PNMM is given. Basic idea of PNMM and itssimulation procedure are presented. Considering that PNMM could be regarded as an NMM-basedmodel, a comparison of PNMM and NMM is discussed from several points of view in this paper. Be-sides, accomplished applications of PNMM to the dynamic rock fracturing are also reviewed. Finally,some recommendations are provided for the future work of PNMM.� 2019 Institute of Rock and Soil Mechanics, Chinese Academy of Sciences. Production and hosting byElsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/

licenses/by-nc-nd/4.0/).

1. Introduction

Dynamic rock fracturing involves the topics of crack nucleation,fracture propagation, rock fragmentation and rock post-failurebehavior. Results of a series of experimental tests (Masuda et al.,1987; Lajtai et al., 1991) have revealed that the fracture patternand mechanical properties of rock materials are affected by thedynamic strain rate. However, the underlying mechanism of therate-dependent behavior is still unclear. The microstructure of rockmaterial is recently considered as one of the influence factorsleading to this phenomenon. When the mechanical behavior onmicroscopic scale is concerned, analytical methods are likelyinvalid due to the complex microstructure of rock materials, andexperimental approaches will be limited as existing facilities arenot sensitive enough to detect the failure process under highloading rates, especially those occurred at the inner part of rocksamples. Under such a circumstance, numerical methods could bean alternative tool to study the mechanism of dynamic effect onrock materials at the microscopic scale.

ao@monash.edu (J. Zhao).ock and Soil Mechanics, Chi-

s, Chinese Academy of Sciences. Prby-nc-nd/4.0/).

Due to the complexity of both rock materials and rock engi-neering problems, rock is difficult to be modeled, compared withother solid materials. To obtain better results, a large number ofnumerical models have been developed, extended, and applied inthis field, including in-house software, commercial software, andopen source code. Existing numerical methods for rock dynamicsare usually classified into three categories (Jing, 2003): continuousmethods, discontinuous methods, and coupled methods. Contin-uous methods are suitable for those problems whose system is acontinuum with infinite degree of freedom (DOF). The behavior ofsuch system is dominated by the governing differential equation ofthe problem and the continuity conditions at the interfaces be-tween adjacent elements. The well-developed continuous methodsinclude the finite element method (FEM) (Clough, 1960), finitedifference method (FDM) (Narasimhan and Witherspoon, 1976),realistic failure process analysis (RFPA) (Tang, 1997; Liang et al.,2004; Zhu, 2008; Wang et al., 2014), element-free Galerkin (EFG)method (Belytschko et al., 1994, 2000) and the related crackingparticle method (Rabczuk and Belytschko, 2004, 2007; Rabczukand Areias, 2006; Rabczuk et al., 2010a), peridynamics (PD) anddual-horizon peridynamics (DH-PD) (Silling, 2000; Ren et al., 2016,2017; Rabczuk and Ren, 2017), partition of unity (PU) method(Babu�ska andMelenk, 1997;Moës et al., 1999; Sukumar et al., 2000;Rabczuk and Zi, 2007; Rabczuk et al., 2010b; Amiri et al., 2014),smoothed particle hydrodynamics (SPH) method (Gingold andMonaghan, 1977; Lucy, 1977; Fakhimi and Lanari, 2014; Zhou

oduction and hosting by Elsevier B.V. This is an open access article under the CC BY-

X. Li, J. Zhao / Journal of Rock Mechanics and Geotechnical Engineering 11 (2019) 684e700 685

et al., 2015), and other meshless methods (Liszka et al., 1996; Liuet al., 1997; Atluri and Zhu, 1998; Ma et al., 2015). Some reviewson continuous methods can be found in Belytschko et al. (1996,2009), Li and Liu (2002), Nguyen et al. (2008), Liu and Liu (2010),and Monaghan (2012). This type of method is mostly used for rockmasses without fractures. It can also be used for the rock mass witha few or many fractures, the behavior of which is establishedthrough equivalent properties or other techniques, e.g. insertingcohesive elements along the path of fractures (Vocialta et al., 2017)and enriching approximation functions in the element containingfractures (Moës et al., 1999).

Discontinuous methods are suitable for those problems whosesystem is a combination of a finite number of well-defined com-ponents. Usually there is no need to discretize such a system, as ithas been automatically done. The behavior of such system isdominated by the well-defined inter-relations between adjacentcomponents. This type of method is most suitable for moderatelyfractured rock masses where the significant fractures are forcontinuous methods, and/or where large displacements of indi-vidual blocks are possible. The term ‘discontinuous method’ hereindicates the discrete element methods and other discontinuousmethods, e.g. the lattice models (Hrennikoff, 1941; Hole�cek andMoravec, 2006; Zhao and Zhao, 2012; Zhao et al., 2012a). Thediscrete element methods are a large family of numericalmethods. The distinct element method (DEM) is a class of discreteelement methods that use an explicit time-domain integrationscheme to solve the equations of motion for rigid or deformablediscrete bodies with deformable contacts. Since DEM is almost themost well-known explicit discrete element method, researcherstend to use the terms ‘DEM’ and ‘discrete element method’interchangeably. DEM treats the simulated material as an as-sembly of separate particles or blocks. The particle-based DEM(Cundall and Strack, 1979; Potyondy and Cundall, 2004) is mainlyadopted to simulate the granular microstructure of the materialthrough particles with varying diameters. Adjacent particles cancontact each other through bonds. The contact is typicallyassigned with a normal and shear stiffness as well as a frictioncoefficient. Crack nucleation and fracture propagation are simu-lated by breaking of bonds. Consequently, blocks of arbitraryshapes can be formed as a result of the simulated fracturingprocess. There are typically two types of bonds in the particle-based DEM: the contact bond and the parallel bond. The contactbond is an elastic spring with a constant normal and shear stiff-ness function between the bonded particles, allowing only normaland shear forces to be transmitted. Whereas on the parallel bond,the moment induced by particle rotation is resisted by a set ofelastic springs distributed between the bonded particles. One ofthe major drawbacks of the bonded-particle model is that thestraightforward adoption of circular/spherical particles cannotfully capture the behaviors of complex-shaped and highly inter-locked grain structures, which is common in hard rocks (Lisjakand Grasselli, 2014). To overcome this limitation, a clumped par-ticle model (Cho et al., 2007) was later proposed. In this model, agroup of particles are glued together to behave as a single rigidbody. Clumped particles can act like a single particle that has anirregular shape but moves as a deformable and non-breakablebody. Apart from sphere, particles in other shapes have alsobeen developed to improve the geometry description of thismethod, including ellipsoids (Vu-Quoc et al., 2000; Yan et al.,2010; Zheng et al., 2013), superquadrics (Wellmann andWriggers, 2012), polyhedral (Feng et al., 2012; Smeets et al.,2015), combination of simple shape primitives (Lu andMcDowell, 2007; Ferellec and McDowell, 2010; Fang et al.,

2015), and mineral-grain-shaped (Li et al., 2018a,b). On theother hand, the block-based DEM discretizes the computationaldomain into blocks using a finite number of intersecting discon-tinuities (Cundall, 1971; Zhao et al., 2008; Gui et al., 2016). Eachblock is internally subdivided for calculation of displacement,strain and stress. Adjacent blocks interact in the normal directionwith a finite stiffness together with a tensile strength criterionand in the tangential direction with a tangential stiffness togetherwith a shear strength. In the standard block-based DEM, rockfailure can only be captured either in terms of plastic yield ordisplacements of pre-existing discontinuities. Therefore, newdiscontinuities could not be driven within the continuum portionof the model, so that discrete fracturing through intact rock couldnot be simulated (Lisjak and Grasselli, 2014). In contrast to theexplicit DEM, the discontinuous deformation analysis (DDA)method is an implicit discrete element method developed for themodeling of static and dynamic behaviors of discrete block sys-tems, originally proposed by Shi and Goodman (1985, 1989). Theblocks in DDA is deformable and can be arbitrarily shaped. Thekinematic constraints of no tension and no penetration betweenblocks are imposed by the penalty method, the Lagrange multi-plier method, or the augmented Lagrangian method. Differentfrom the explicit DEM, the governing equations in DDA are rep-resented by a global system of linear equations obtained byminimizing the total potential energy of the system, and a globalsystem of equations inmatrix form needs to be formed and solved.By differentiating several energy contributions, including blockstrain energy, contacts between blocks, displacement constraintsand external loads, the stiffness matrix of the model could beassembled. After years of development, DDA has been applied tothe simulation of various topics, including the rock sliding (Hatzoret al., 2004; Wu et al., 2009; Ning and Zhao, 2013; Jiao et al., 2014;Yang et al., 2014a; Zhang et al., 2015), rock fracturing (Kong andLiu, 2002; Bao and Zhao, 2013; Chen et al., 2013; Tian et al.,2014), wave propagation in rock masses (Jiao et al., 2007; Guand Zhao, 2009), behavior of jointed rock masses (Lin et al.,1996; Tsesarsky and Talesnick, 2007; He and Zhang, 2015; Heet al., 2018), and failure of masonry structures (Kamai andHatzor, 2008; Jiang et al., 2014).

A coupled method, in most cases, is a combination of onecontinuous method and one discrete method implicitly or explic-itly, aiming to combine the characteristics of both methods. Thenumerical manifold method (NMM) is a typical coupledmethod forrock mechanics (Shi, 1991, 1995; Ma et al., 2010). It provides aframework to unify continuous and discrete methods by implicitlycombining FEM and DDA. Recently, a novel numerical model,named particle-based numerical manifold method (PNMM), hasbeen proposed as an extension of NMM and applied to severalproblems in the field of rock dynamics. The purpose of this reviewis to provide a summary of selected coupling numerical method forrock dynamics. Specifically, the NMM and PNMM are considered inthis paper. As for other coupling numerical methods, e.g. the finite-discrete element method (FDEM), the reader is suggested to refer totheworks (Mahabadi, 2012; Lisjak, 2013; Lisjak and Grasselli, 2014),and its applications to dynamic rock fracturing can be found inRougier et al. (2014) and An et al. (2017). In this context, thefundamental principle of NMM is first briefly introduced. Recentimprovement and applications in the field of rock dynamics aresummarized and reviewed. Then, the history of the newly devel-oped PNMM model is given. The basic idea of PNMM and itssimulation procedure are also presented. Considering that PNMMcould be regarded as an NMM-based model, a comparison ofPNMM and NMM is discussed in this paper. Besides, accomplished

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applications of PNMM to the dynamic rock fracturing are alsoreviewed. Finally, some recommendations are provided for thefuture work of PNMM.

2. Numerical manifold method

2.1. Fundamental principles

The NMM was originally proposed by Shi (1991, 1992, 1995,1997, 2012) at a series of conferences. NMM is derived based onthe finite cover approximation theory and is named after themathematical notion of manifolds.

NMM feasibly simulates fractures by truncating elements andtheir shape functions, and treats continuum bodies, fracturedbodies and assemblage of discrete blocks in a unified form. Thischaracteristic of NMM is mainly due to the adoption of a dual-cover-system, namely the mathematical cover and the physicalcover. The mathematical cover is independent of the shape ofmodeling domain but covers all the spaces that the modelingdomain may occupy. It is usually generated from a uniform FEMmesh, from which the finite elements that share the same nodeform amathematical cover. Mathematical covers are common to beoverlapping. The mathematical covers with different geometricpatterns could be generated from several kinds of typical FEMmeshes, as illustrated in Fig. 1, which are supposed to have a slightinfluence on simulation results depending on the analytical solu-tion of concerned problems (Zhang et al., 2010). On the other hand,the physical covers are the intersection of mathematical covers andthe modeling domain (Ma et al., 2010). Specifically, a physical patchis formed by cutting a mathematical cover with discontinuities,such as physical boundaries, material interfaces, and fractures.Then, the union of all physical patches constitutes a physical cover.The physical cover system is to connect the uniform mathematicalcover system with the arbitrary modeling domain, providing localapproximation function in NMM. The overlap of neighboringphysical covers is called a manifold element. A manifold element isthe basic computation unit in NMM. The advantage of adopting twocover systems for NMM is that arbitrary boundaries and internalphysical features in the physical domain are allowed withoutmeshes conforming to them. Therefore, the meshing task in NMMis greatly simplified and fracturing process can be modeled withoutremeshing. At the same time, NMM model fractures straightfor-wardly by splitting each physical cover that is completely cut by thefractures into several separate covers, assigning each cover an in-dependent local function. By such a way, complex cracks witharbitrary number of branches can be modeled in an exactly way asthe modeling of a single crack (see Fig. 2). In this figure, a branchedfracture separates four physical covers into twelve physical covers,considering that the four mathematical covers in the figure willlead to four physical covers without the branched fracture.

Besides, NMM implements a novel simplex integration methodproposed by Shi (1996) for both NMM and DDA. This integrationmethod is able to conveniently evaluate the weak form integration

Fig. 1. Typical patterns of the mathematical cover (the polygon indicated by red lines).

over elements intersected by internal discontinuities and/orexternal boundaries. For the manifold element of a general shape,the simple integration scheme will first partition the element intoseveral sub-triangles, the same process as the element partitioningin extended FEM (XFEM), and then conduct the quadrature on eachsub-triangle. Another distinct feature of NMM is that the frictionalcontact boundary conditions between the two sides of a crack ortwo discrete blocks can be accurately satisfied, due to the contactlogic in NMM.

It has been proven that FEM is a special case of NMM when thefollowing conditions are satisfied (Ma et al., 2010):

(1) Mathematical covers in NMM are generated from a finiteelement mesh;

(2) Weight functions defined on mathematical covers are finiteelement shape functions;

(3) Cover functions defined on physical covers are constants;(4) Physical features including internal discontinuities (e.g.

cracks and material interfaces) and external boundaries donot intersect manifold elements.

Although NMM can be converted to FEM, its distinct features indealing with discontinuous problems cannot be covered in FEM.XFEM/GFEM (generalized FEM) is another numerical method that iscapable of dealing with discontinuities. Recently, the concept ofenriched functions in XFEM/GFEM has been integrated into NMMto simulate complex crack problems (Ma et al., 2009; An, 2010). Itshould be noticed that NMM is able to tackle cracks withoutenriched functions in most cases, and NMM furnished withenriched functions can tackle more complex crack problems ascompared with XFEM/GFEM. Meanwhile, since NMM is developedon the basis of DDA, it preserves all the characteristics of discreteelement modeling such as the kinematics constraints, contactdetection, and modeling from DDA (Ma et al., 2010). However, theDOF in NMM is usually much higher than DDA as there is more thanone manifold element in each block in most cases. The benefit atthis cost is that NMM provides more accurate displacement andstress fields in blocks than DDA. It can be concluded that if everydiscrete block is a manifold element with linear displacement field,then NMM will degenerate exactly into DDA.

2.2. Recent development and applications

In the original version of NMM by Shi (1991, 1992, 1995, 1997,2012), the simplest triangular manifold element with constantcover function was adopted for two-dimensional (2D) issues,lacking criteria for crack initiation and propagation. Since then,various developments as well as applications have been made inthe following decades. Shyu and Salami (1995) implementedquadrilateral isoparametric elements in NMM. Chiou et al. (2002)studied the mixed-mode fracture propagation by combiningNMM with the virtual crack extension method. Cheng et al. (2002)incorporated Wilson nonconforming elements in NMM. Chen et al.(1998) developed a higher-order NMM with high-order coverfunctions, and then Su et al. (2003) proposed a subroutine in thecommercial software Mathematica to automatically produce ex-pressions for high-order NMM. Recently, Ghasemzadeh et al. (2014)proposed a high-order NMM for dynamic problems. Terada et al.(2003) introduced the finite cover method (FCM) as an alias ofNMM. Lin et al. (2005) developed the formulations of three-dimensional (3D) NMM with high-order cover functions and pro-posed a fast simplex integration method based on special matrixoperations, without considering the linear dependence problem.Later, Cheng and Zhang (2008) proposed the 3D NMM with tetra-hedron and hexahedron elements and derived the basic matrices

Fig. 2. The mathematical cover (MC) and physical cover (PC) cut by a branched crack (after Li et al., 2017). Patches are filled with different colors to indicate the relation between PCsand MCs.

Fig. 3. Road protection from rock slope failure simulated by 3D NMM (He et al., 2013).

X. Li, J. Zhao / Journal of Rock Mechanics and Geotechnical Engineering 11 (2019) 684e700 687

for equilibrium equations. He and Ma (2010) also proposed a 3DNMM based on tetrahedron elements (Fig. 3). An et al. (2011)investigated the linear dependence problem of NMM approxima-tion space using finite element covers and polynomial local

functions at both elemental and global levels. Yang and Zheng(2016) and Xu et al. (2017) recently proposed a high-order localapproximation function and eliminated the linear dependence onthe triangular and quadrilateral mathematical meshes, respectively.

X. Li, J. Zhao / Journal of Rock Mechanics and Geotechnical Engineering 11 (2019) 684e700688

Their model has a higher order of global approximations, betteraccuracy and continuous nodal stress, and was then applied to thefracture analysis as well as free and forced vibration analysis ofsolids (Yang et al., 2016a,b,c; Yang and Zheng, 2016). Cai et al. (2013)proposed a generalized and efficient cover generation procedure,which is applicable for dealing with interfaces, inclusions anddiscontinuities with complex geometry. Zheng and Xu (2014)proposed strategies for several specific issues that NMM mayencounter in simulation of crack propagation, including the rankdeficiency induced by the high-order cover functions, the integralswith singularity of 1/r (r indicates the distance to the crack tip), andthe kinked cracks. A flat-top partition of unity (PU)-based NMMwas proposed by He et al. (2015) to alleviate the linear dependencedifficulty of traditional NMM. Recently, Yang et al. (2017) and Zhengand Yang (2017) developed a mass lumping scheme for NMM andother PU methods, which shows a more excellent behavior in dy-namic analysis compared with the conventional consistent massmatrix since solving large-scale simultaneous algebraic equationscan be avoided.

For simulating fractures, Li et al. (2005a,b) and Gao and Cheng(2010) developed an enriched meshless manifold method for 2Dcrack modeling. Meanwhile, Ma et al. (2009) and An (2010) incor-porated the enriched functions of XFEM/GFEM into standard NMMto simulate complex cracks. Zhao et al. (2012a) coupled NMM withthe distinct lattice spring model to simulate the dynamic failure ofrock masses. Wu and Wong (2013a,b, 2014), Wu et al. (2013) andWong and Wu (2014) implemented NMM and studied rock frac-turing in many different circumstances (Fig. 4). Yang et al. (2014b)

Fig. 4. Simulation of the failure of rock samples using NMM: (a) Failure of Brazilian tenscontaining two inclusions (Wu and Wong, 2013b).

refined the mathematical covers near crack tips. Ning et al. (2011),An et al. (2014) and Zheng et al. (2014a) analyzed the stability ofrock slopes with 2D NMM, while He et al. (2013) and Liu et al.(2017) extended the analysis to 3D cases. Zheng et al. (2014b)adopted the influencing domains of nodes in the moving leastsquares (MLS) interpolation when constructing the mathematicalcover and proposed the MLS-based NMM to better address theproblems with large movement. Then, they adopted the MLS-basedNMM and proposed a so-called projection-contraction algorithm totackle the complementarity problem in the static growth of mul-tiple fractures (Zheng et al., 2015). Recently, Liu et al. (2018a,b)studied the micro/macro failure of rock by developing a Voronoielement based-NMM. For stress wave propagation issues, Zhaoet al. (2014) improved the performance of NMM in the simulationof wave propagation in rock masses by importing Newmark systemequations, edge-to-edge contact scheme and non-reflectionboundary condition. Wu and Fan (2014) developed a time-dependent absorbing boundary condition for wave propagationproblems. Wei et al. (2018) also proposed some new boundaryconditions for NMM to study the seismic response of geomechanicsproblems. Yang et al. (2018) developed boundary settings for theseismic dynamic response analysis of rock masses in NMM. More-over, Zhang et al. (2014, 2017, 2018) studied the thermo-mechanicalfractures and transient heat conduction problems using NMM indifferent solids.

NMM has been a promising numerical method for rock me-chanics in recent years due to its great advantages in theoreticalbasement for continua-discontinua analysis. However, as a quasi-

ile disc test in pre-fractured sample (Wong and Wu, 2014); and (b) Cracking of rock

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block-based model, its geometrical and topological operations, andthen its contact detection and operation, can be complicated,especially in 3D cases. Furthermore, NMM is generally suitable forthe analysis in macroscopic scale, but not in microscopic scale.

3. Particle-based numerical manifold method

The PNMM is a newly proposed extension of NMM for rockdynamics. The basic idea of PNMM is to incorporate the particleconcept into NMM. The motivation of this development is tosimplify the geometrical Boolean operations and contact opera-tions in NMM, as well as to better understand the heterogeneity ofrock materials and rock fractures.

3.1. Particle manifold method

The model originally proposed by Zhao et al. (2012a,b) and Sunet al. (2013, 2014) was entitled particle manifold method (PMM)during its early stage. PMM replaced the polyhedron representationof physical cover with a particle representation. The basic idea ofthis improvement is to reduce the complexity of the topologicaloperations between polygons in NMM. Considering that the phys-ical cover could be in general shapes due to the intersection be-tween the mathematical cover and discontinuities (e.g. boundaries,material interfaces, and fractures), topological operations in NMMare usually complicated and time-consuming in practice, especiallyin 3D cases. For example, in Fig. 5, Zone I in polygon representationis the result of a Boolean operation ‘AND’ between the triangularsimulation domain and the rectangular mathematical cover M.Using the particle representation, Zone I is described by a group ofred particles located in M. Therefore, in generation of the physicalcover I (same as Zone I in geometry), the topological operationbetween two polygons is simplified to the operation betweenseveral points and a polygon. This polygon-to-polygon topologicaloperation, sometimes called block cutting procedure in NMM,could be more complicated when fractures are taken intoconsideration.

The particle manifold element, as the basic element in PMM,was defined as the combination of the mathematical cover andparticles. A relationship termed ‘link’ was defined to describe thecontinuous state between neighboring particles. PMM adopted aparticle simplex integration scheme for numerical calculation.Although the integration scheme is analytical, there could still existslight accuracy loss due to the particle representation of polygonaldomains (Sun, 2012). A virtual integration model and a realisticintegration model were developed for homogeneous and inho-mogeneous materials, respectively. The most noticeable improve-ment of PMM compared to NMM was that the polygon-to-polygoncontact was simplified to particle-to-particle contact, since thecontact between particles was easier to be detected and computed.The contact forces between particles as well as between particle

Fig. 5. Physical cover in PMM: Replacing the polygon representation with a particlerepresentation (Sun et al., 2013).

and rigid wall were developed. Only normal contact force wasconsidered in the model. The maximum tensile stress criterion andMohr-Coulomb criterion were adopted in PMM to study the brittlefailure of rock. The link was taken as the basic failure component inthe model, and the directivity of the failure was discussed. Appli-cation of PMM includes a preliminary study of the stress wavepropagation across joints, the spalling of rock bars (Fig. 6), andsome other fracturing problems. Furthermore, a graphics process-ing unit (GPU) parallelization and a coupling of PMM and NMMwere also performed, attempting to further improve the efficiencyof computation.

3.2. Fundamental principles

An improvement to PMMwas later performed and consequentlythe numerical model PNMM was proposed. This section brieflypresents some fundamental principles of PNMM. More details ofthe components, formulations and implementation of the modelcan be found in Li (2017) and Li et al. (2017).

PNMM inherits the dual-layer-cover system from NMM, i.e. themathematical cover and physical cover. On each mathematicalcover, a weight function 4i is defined to combine the interpolationfunctions from different physical covers, which satisfiesXi

4iðxÞ ¼ 1

4iðxÞ � 0 ðcx˛PiÞ4iðxÞ ¼ 0 ðcx;PiÞ

9>=>; (1)

where Pi is the ith mathematical cover. The physical cover is toprovide the local approximation function by defining the coverfunction ui:

ui ¼Xnj¼1

TijðxÞdij ¼ Tdi (2)

where n is the number of DOFs, T is the basis of the cover function,and di is the vector of DOFs on the ith physical cover. Since amanifold element is the overlap of neighboring physical covers, itsdisplacement field is generated by combing the cover functions ofrelated physical covers using weight functions as

ueðxÞ ¼Xmi¼1

4iðxÞuiðxÞ ¼Xmi¼1

NiðxÞdi ¼ Nede (3)

where ue is the displacement field of the manifold element e; mindicates the number of physical covers; Ni and Ne are the shapefunctions on the ith physical cover and manifold element e,respectively; and de is the vector of DOFs on the manifold elemente. The generation of manifold elements, as the result of the dual-layer-cover system, forms the first discretization in PNMM. Withthis discretization, the global equation of PNMM for dynamicanalysis can be obtained in matrix form as

KdþM €d ¼ F (4)

where K is the global stiffness matrix, d is the global vector ofunknowns,M is the global mass matrix, and F is the global vector ofequivalent loads. This governing equation is the same as that inNMM, since particles have not yet been introduced by far.

An extra level of discretization is incorporated by dividing eachmanifold element into a group of particles, as illustrated in Fig. 7. Asthe governing equation and DOFs are defined on manifold ele-ments, introduction of the particles does not bring in additional

Fig. 6. The spalling of rock bar under impulse stress wave predicted by PMM (Sun et al., 2013).

Fig. 7. The dual-level discretization in PNMM (Li et al., 2018c).

X. Li, J. Zhao / Journal of Rock Mechanics and Geotechnical Engineering 11 (2019) 684e700690

DOFs and does not affect the form of the governing equation.However, the material properties, boundary conditions and bodyforces are all assigned to individual particles independently. Parti-cles in a manifold element are possible to have varying materialproperties and boundary conditions. For calculation of themanifoldelement matrices, a particle integration scheme was proposed bytaking the particle centroids and areas as integration points andweights, respectively, as

ZA

f ðxÞdA ¼Xpi¼1

�f�xci�Ai� ¼ p

Xpi¼1

hf�xci�r2ii

(5)

where f(x) is the integrand, p is the number of particles, xci is thecoordinates of the centroid of the ith particle, Ai is the area of the ithparticle, and ri is the radius of the ith particle. The particle inte-gration scheme is a numerical integration technique, and it issuitable for both convex and concave integration domains and anyintegrands, including polynomial, exponential, and trigonometricfunctions (Li, 2017). Using the particle integration scheme, theglobal matrices in the governing equation could be assembled.Then, the mechanical fields of manifold elements are obtained bysolving the equation, and the mechanical results of each particlecan be simply derived as

ui ¼ ue�xci�; εi ¼ εe

�xci�; si ¼ se

�xci�

(6)

where ui, εi and si are the displacement, strain and stress of the ithparticle in the manifold element e, respectively; and ue, εe and se

are respectively the displacement, strain and stress of the manifoldelement e.

Among particles in a continuum, links are defined as thecontinuous status between neighboring particles. A link is gener-ated between any two particles as long as the following three

conditions are satisfied at the same time: (1) the two particles arenext to each other geometrically; (2) the two particles belong to thesame block/object; and (3) there are no micro/macro fracturesbetween the two particles. In implementation, the Munjiza-NBScontact detection algorithm (Munjiza and Andrews, 1998) is uti-lized for the generation of links. A link is the basic failure compo-nent in PNMM, as that in PMM. The stress and strain of a link aretaken as the average stress and strain of the two particles itconnects:

sL ¼ ðs1 þ s2Þ=2εL ¼ ðε1 þ ε2Þ=2

�(7)

where sL and εL are the stress and strain tensors of the link,respectively; s1 and ε1 are the stress and strain tensors of the firstparticle, respectively; and s2 and ε2 are the stress and strain tensorsof the second particle, respectively. For rock dynamics issues, theJohnson-Holmquist-Beissel (JHB) model (Johnson et al., 2003) ischosen as the strength criterion to reproduce the rate-dependentbehaviors of rock:

sc ¼ s0ð1þ C ln_ε*Þ (8)

where sc is the dynamic strength at the strain rate of _ε*; s0 is thestrength at _ε* ¼ 1; and C is the dimensionless strain rate constant.More details of the implementation of JHB model in PNMM and thedetermination process of JHB parameters for rock materials can befound in Ma and An (2008) and Li et al. (2017). The strength cri-terion is applied on each link, and once the stress of a link reachesthe dynamic strength from the criterion, the link is supposed to befailed, representing that a microcrack is generated. In a continuousmodel, it could be proven that any two particles within the modelcould be connected by either one or more links. When there arefailed links generated in a manifold element at a time step in

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calculation, the topological relation between particles would beredetected. And if disconnected particles are found to separateparticles into more than one group, the manifold element would besubdivided accordingly, representing that the fracture has propa-gated across the manifold element. A cover function enrichment isimplemented around the fracture tip to consider the singular fields.

The contact operation in PMM is kept, i.e. the particle-to-particle and particle-to-plane contacts are defined respectively.Particles on the surface of different continuums/blocks may form apair of particle-to-particle contacts when they are adjacent to eachother. The particle-to-plane contacts are to simulate the interactionbetween deformable bodies and rigid static/moving walls orinfinite planes. The difference between a contact and a link is thatthere is no force defined on the link whereas the contact force isapplied. In the current version of PNMM, only normal contact forceis defined and conducted. Although the direction of a pair of contactis determined by the position of two particles at particle scale,PNMM accumulates pairs of contacts to simulate the complexblock-to-block contact at macro-scale, and the contact force be-tween blocks is the sum of contact forces between particles, asillustrated in Fig. 8. PNMM implements the same open-close iter-ation procedure as NMM (Liu et al., 2018c) to prevent the pene-tration between blocks. It should be noted that although onlynormal contact is considered in PNMM, the model still gains theability to simulate shear failure, since the failure is defined on thelink instead of the contact.

The flowchart of PNMM is illustrated in Fig. 9 and the calculationsteps are concluded as follows:

(1) Input the model and generate mathematical covers. Themodel is created in other software and read by PNMM from atext file. The text file imported into PNMM includes themodel geometry information, material properties, and someother setting variables.

(2) Generate manifold elements. The generation of manifoldelements is based on the topological operation between theimported model and mathematical covers.

(3) Generate particles. Each manifold element is first dividedinto several inner triangular elements. Then, one particle isgenerated in each inner triangular element. The center of theparticle is the centroid of the inner triangular element. Thearea of the particle is equal to the area of the inner triangularelement. In such a way, the sum of particle areas remainsequal to the area of the polygonal element. It is common tofind adjacent particles overlapped due to the limitation ofgeometry. However, particles in a samemanifold element aremainly used to carry material parameters and serve as thenumerical integration points. There is no contact force

Fig. 8. From particle-to-particle contact to block-to-block contact.

defined between adjacent particles, which are distinctlydifferent from other particle-based methods. Therefore,particles in a same element could be regarded as pointcollocation, and the overlap between particles should haveno physical meaning (Li et al., 2017).

(4) Generate links. The Munjiza-NBS contact detection algo-rithm (Munjiza and Andrews,1998) is implemented to detectthe pairs of neighboring particles in PNMM and the links aregenerated accordingly. Details of this procedure are availablein Li (2017).

(5) Generate blocks. This is to detect how many blocks areinvolved in the model. A pair of neighboring particles isconnected by a link. A block is constituted of a group ofparticles, where any two of them could be connectedthrough numbers of links. A so-called seed filling method isadopted to accomplish this task (Sun, 2012; Li, 2017).

(6) Detect pairs of contacts.(7) Apply body forces, external loads, and boundary conditions.

These parameters are defined on particles and applied to theelement matrices.

(8) Generate element matrices using the particle integrationscheme.

(9) Assemble global matrices.(10) Solve global equation. The preconditioned conjugate

gradient (PCG) method is implemented to solve the globalequation, which is a system of linear equations.

(11) Derive DOF results on manifold elements from the globalvector of unknowns, which are solved in the previous step.

(12) Obtain particlemechanical results from themechanical fieldsof a manifold element straightforwardly.

(13) Obtain link mechanical results from the two particles itconnects as the average value.

(14) Apply the JHB criterion. The JHB model is implemented inPNMM and applied on each link at every time step. Once theJHB criterion is met, the status of the link will be set to befailed. A group of material constants in the criterion need tobe determined before the simulation.

(15) Remove failed links, and redetect connectivity betweenparticles. This task is accomplished by the seed fillingmethod as well.

(16) Generate new manifold element and re-group particles.Additional manifold elements need to be generated whenfractures have propagated through an element.

(17) Post-process. PNMM writes the simulation results into anexternal text file, which will be later processed by other post-process software.

3.3. Applications

The application of PNMM mainly focuses on the dynamic frac-turing of rock. The proposed PNMMwas first validated bymodelingsome fundamental problems, including the cantilever beam, Bra-zilian disc test, and rock blasting under dynamic loading (Li et al.,2017). Moreover, stress wave propagation is an important issue instudying the dynamic fracturing of rock. The example illustrated inFig. 10 presents the wave propagation in a rock cavern which issemi-circle in shape. The propagated wave was induced by an ex-plosion load uniformly applied on the surface of the rock cavern.Simulation results showed that the wave arrives at the upperbound of themodel at the time of 9.72ms, being in good agreementwith the theoretical result of 9.61 ms. Three monitoring pointswere set in the model. The time-displacement results of the threemonitoring points obtained by PNMM agreed well with thosesimulated by ABAQUS (Dassault Systèmes, 2017). Later, the spalling

Fig. 9. The flowchart of PNMM.

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Fig. 10. The stress wave propagation in a rock cavern: (a) Numerical model; (b) Simulated vertical displacement results at t ¼ 10 ms (unit: m); and (c) Displacement histories atmonitoring points e comparison with FEM results.

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of rock bars under different loading rates and different types oftriangular compression waves were simulated, as shown in Fig. 11(Li et al., 2018d). It was found that the spalling pattern is closelyrelated to the type of thewave applied on the bar. When a sawtoothor reverse sawtooth wave was imposed, the second spall fracturewould possibly be created on the first spall piece or the remainingpart of the bar; whereas in the case where a symmetrical triangular

wave is applied, a spalling zone rather than a single spalling frac-ture will be generated. The spalling process in plate impact testwith different impact velocities and different flyer sizes were alsostudied (Li, 2017; Li et al., 2018d). Two flyers were adopted in thetest, a long flyer with the same length as the target and a short flyerwith half the length of the target (Fig. 12). In the long flyer test,spalling fractureswere generated by the reflected stress in a narrow

Fig. 12. The plate impact test with (a) long flyer (Li et al., 2018d) and (b) short flyer.

Fig. 11. The spalling of rock bar under different types of triangular compression waves: (a) Reverse sawtooth compression wave; (b) Sawtooth compression wave; and (c) Sym-metrical triangular compression wave (Li et al., 2018d).

X. Li, J. Zhao / Journal of Rock Mechanics and Geotechnical Engineering 11 (2019) 684e700694

area near the center of the target. The length of the fractured areawas almost the same as the length of the target. Then, spallingfractures were extended to the left and right boundaries of thetarget, and the target was consequently separated into two parts,where a marked difference between the flying speeds existed. Inthe short flyer test, spalling fractures were generated by the re-flected stress in a narrow area near the center of the target as well;however, the fracturing zonewas shorter than the flyer. Then, somelongitudinal fractures were generated from the spalling zone to thelower bound of the target, creating several spalling pieces from thetarget. The speed of these spalling pieces was non-uniform at theircreation but became uniform with the increase of the simulationtime. There were also some fractures originating from the upperbound of the target, due to the mismatch between the length of theflyer and target. Based on the spalling phenomenon in rock, therockburst of tunnel induced by static in situ stress and far-fielddynamic disturbance was further simulated in Li et al. (2018a).This simulationwas divided into two stages. In the first stage, staticin situ stresses were imposed on the model to obtain the initialstress state. Then, a dynamic disturbance was uniformly imposedon the left boundary of the model to trigger the spalling and

Fig. 13. The rockburst of tunnels subjected to a same dynamic disturbance and different in s(a) PH ¼ 20 MPa, PV ¼ 10 MPa; (b) PH ¼ 10 MPa, PV ¼ 1 MPa; and (c) Case 5 in Li et al. (20

rockburst. The applied dynamic disturbance was in the form of animpulse function. Numerical results revealed that if the vertical andhorizontal in situ stresses are close to each other, the surface oftunnel will be under the state of compression. A compressive in situstress field helped to offset the reflected tensile stress wave;however, spalling could still occur when the peak value of the dy-namic disturbance was high enough. Spalling fractures that prop-agated to the tunnel formed the rockburst with the rock ejectionfrom the surface. Other fractures remaining in the rockmass did notenhance the rockburst but could be a potential hazard to the safetyof the tunnel in the following use. A parametric study was con-ducted to discuss the effect of in situ stresses on the rockburst oftunnel, as shown in Fig. 13. In the last two cases, the lateral pressurecoefficient was increased to 10, which had exceeded the typicalvalue range in practice, but it helped to reveal the mechanism ofrockburst under coupled in situ stress and dynamic disturbanceand made our results comparable to those of Zhu et al. (2010).Simulation results showed that in these cases, the horizontal sur-faces and surrounding regions of the tunnel were initially under thestate of tension instead. The superposition between the initialtensile stress and the reflected tensile stress weakened the ability

itu horizontal stresses (PH) and vertical stresses (PV) (enlarged view around the tunnel):18d), PH ¼ 20 MPa, PV ¼ 2 MPa.

X. Li, J. Zhao / Journal of Rock Mechanics and Geotechnical Engineering 11 (2019) 684e700 695

towithstand the dynamic disturbance, therefore severer rockburstsoccurred. It is concluded that the most dangerous situation for deeptunnels is that there is a large difference between the horizontaland vertical in situ stresses and the dynamic disturbance comesfrom the direction of higher in situ stress. Of course, it would alsobe possible to have a very large difference between the horizontaland vertical in situ stresses caused by other activities instead of solegeological conditions.

Li et al. (2018e) used PNMM to simulate the rock scratch tests.Three different failure patterns, i.e. brittle failure, ductile failure,and brittle-ductile combined failure, under decreasing cuttingdepths were successfully modeled (Fig.14). It is found that there is a

Fig. 14. Typical failure patterns in rock scratch test: (a) Ductile failure at shallow cut; (b) Bri2018e).

range of cutting depth at which the mode of rock scratching is inthe transition from ductile to brittle. The transitional cutting depthrange could be estimated from the result of mechanical specificenergy (MSE). Besides, a detailed parametric study was performedby a series of PNMM simulations to investigate the effect of oper-ational parameters of the cutter, including the cutting depth, cut-ting speed, and rake angle. The effect of cutting depth was dividedinto three phases: the value of MSE decreases rapidly when thecutting depth is shallow, remains approximately constant when thecutting depth is intermediate, and decreases nonlinearly at an in-termediate rate in deep cuts. The effect of cutter rake angle wassignificant on cutting force but moderate on MSE, and an actual

ttle-ductile failure at intermediate cut; and (c) Brittle failure at deep cut (after Li et al.,

X. Li, J. Zhao / Journal of Rock Mechanics and Geotechnical Engineering 11 (2019) 684e700696

cutting depth was found to be noticeably greater than that whenthe cutter has a large rake angle, leading to a significant differencebetween the cutting force and MSE. The shortcoming of this nu-merical study was that the cutting speed was set to be 4 m/s,apparently higher than that in an experimental test, which isusually in the order of mm/s. The reason is to shorten the scratchingprocess and reduce the computation time to a practical level. Thesame problem had been encountered by Jaime et al. (2015) in theirFEM analysis and we adopted the same cutting speed as them. Thecutting speedwas found to greatly affect the cutting efficiency sincerock had a much higher strength at such high strain rates. He andXu (2016) observed experimentally that the effects of cuttingspeed can be regarded to be negligible for realistic cutting speeds.We suggest that it is necessary to apply a more efficient numericalmodel to this simulation.

3.4. Discussion

Since PNMM is developed by incorporating the particle conceptinto NMM, it is reasonable to be compared it with other particle-based models, e.g. the DEM (Potyondy and Cundall, 2004), SPH(Lucy, 1977), PD (Rabczuk and Ren, 2017), LM (Zhao et al., 2011) andreproducing kernel particle method (RKPM) (Liu et al., 1995).However, particles in PNMM are in fact based on a quite differentconcept. Serving as an extension and improvement to NMM, PNMMhas the same theoretical foundation and global governing equationas NMM. In PNMM, the DOFs are defined on manifold elementsrather than particles. The elementmatrices are to be assembled andthe mechanical fields of elements are to be solved from the globalequation. Recalling that the link between adjacent particles onlypresents the continuous status and carries no additional force,particles in a continuum do not interact with each other, which candistinguish PNMM from other particle methods. Particles in PNMMcan be regarded as a point collocation under most circumstances.The force between particles is only defined at the contact betweendifferent continuums or blocks on their surfaces. Due to this char-acteristic, introduction of particles as well as links does not bring inmany micro-parameters as other methods. Macro-mechanical pa-rameters are assigned to particles as those assigned to elements inNMM. Parameters at the contact are similar to NMM as well, sincethe only difference is that contact is defined between particlesinstead of blocks and the same open-close iteration is utilized. Forfracturing analysis, JHB parameters are introduced on the links inorder to take the rate-dependent behavior of rock into consider-ation. The determination of JHB parameters could be found in theliterature (Li et al., 2017). In summary, particles in PNMM arefunctioned in the following aspects. First, particles carrying pa-rameters assist in simulating the heterogeneity of rock materials.Second, the numerical integration scheme is implemented by par-ticles. Third, fractures could be initiated and propagate on theparticle level. Finally, the contacts between different blocks areconducted by particles.

Compared with NMM, PNMM simplifies the contact operationbetween blocks, improves the flexibility in initiating and deter-mining the propagation path of fractures, and gains the ability tosimulate the heterogeneity of rock materials. Compared with theparticle-DEM, e.g. particle flow code (PFC) (Potyondy and Cundall,2004), PNMM is a continumm-discontinumm coupled method,which is supposed to provide a better solution to continuousproblems. Moreover, PNMM does not have the micro-parameterson particles that need to be calibrated prior to the simulation asPFC does. However, PNMM would probably have a lower accuracythan PFC if their models have the same number of particles, sincethe DOFs in PNMM are defined on elements instead of particles.Besides, PFC is apparently more suitable for granular materials than

PNMM. Compared with the block-DEM, e.g. universal distinctelement code (UDEC) and 3D distinct element code (3DEC), PNMMhas an advantage that it is easier to have the fracture propagationdriven into blocks due to the existence of internal particles. Thisadvantage also applies when comparing PNMM with FDEM, espe-cially the widely used code Y-Geo (Lisjak and Grasselli, 2014).However, compared with previously mentioned numerical models,PNMM still lacks improvement in the following aspects. First,PNMM is currently still a 2D model, which limits its application tomany problems. Second, application of PNMM is still limited, sinceit fails to simulate heats, fluids, and coupled problems. Third, theefficiency of PNMM should be further improved by incorporatingparallel computation.

Based on the development of PNMM and simulation resultsobtained over the past years, the following recommendations forfuture work are outlined:

(1) Investigating the heterogeneity of rock materials. The het-erogeneity could be defined by either a random distributionof material properties among particles or a meso-scalestructure. In the second scenario, the geometry informationof aggregates needs to be provided first. Then, particleslocated in aggregates are assigned with different materialparameters. This work could be an important application ofPNMM in the future. In the next phase, a PNMM modelshould be able to be generated from a computed tomography(CT) scanning image. The effect of the varying particle sizeand particle allocation should also be studied in the scope ofthis work.

(2) Enhancing the computational efficiency of PNMM by devel-oping a parallelized version. Both central processing unit(CPU)-based and GPU-based technologies are possiblyadopted. The first attempt could be made on solving theglobal equation.

(3) Developing an explicitly coupled method using PNMM andNMM. A major difference between PNMM and NMM is theintroduction of particles. It is possible to introduce particlesas the second level of discretization in a part of the model,and conducts only the first discretization in the rest portionof the model with less interest, e.g. in far field, in the domainwith small deformation, in the domainwhere no fracture andcontact occurs.

(4) Extending the PNMM code to 3D cases. Some formulaeproposed for 3D PMM in Sun (2012) could be introduced. Theextension of the proposed particle integration scheme to 3Dcases is straightforward. The 3D PNMM could be applied tothe problems that do not obey the plane strain and planestress assumptions, e.g. the projectile penetration in rockplates.

(5) Incorporating thermal and fluid components into PNMM toinvestigate the coupled problems, which have beenincreasingly important in rock engineering.

(6) Application of PNMM could be extended to the stress wavepropagation in jointed rock masses, which is also an impor-tant topic in rock dynamics.

4. Concluding remarks

Over the last decade, PNMM has emerged as a promising nu-merical tool for rock dynamics. PNMM inherits the cover systemfrom NMM and introduces a dual-level discretization using parti-cles. Compared with NMM, PNMM simplifies the contact operationbetween blocks, improves the flexibility in initiating and deter-mining the propagation path of fractures, and gains the ability tosimulate the heterogeneity of rock materials. Compared with PMM,

X. Li, J. Zhao / Journal of Rock Mechanics and Geotechnical Engineering 11 (2019) 684e700 697

several aspects have been re-clarified and improved in PNMM,including adoption of high-order interpolation functions, devel-opment of a particle integration scheme, an enrichment functionaround fracture tips, an improved algorithm for the generation oflinks, unambiguous failure of links, incorporation of a rate-dependent strength model, expanded applications in rock dy-namics, and improved calculation performance. Due to its perfor-mances, PNMM could be easily utilized to study the heterogeneityof rock materials, the initiation, coalescence and propagation offractures, the detachment and post-failure behavior of rock frag-ments, the contact between rock blocks, and the rate-dependentbehavior of rock, making it a promising tool for modeling of rockdynamic fracturing problems. Application of PNMM mainly is nowfocused on rock fracturing under different dynamic loads, includingrock spalling, plate impact, rockburst in tunnel, and rock scratching.Furthermore, the advantages and limitations of PNMM, comparedwith other widely used models, are summarized. Advices to thefuture development of PNMM are also outlined.

Conflicts of interest

The authors wish to confirm that there are no known conflicts ofinterest associated with this publication and there has been nosignificant financial support for this work that could have influ-enced its outcome.

Acknowledgements

The authors would like to acknowledge the financial support tothe development of PNMM from the Laboratory of Rock Mechanicsat École Polytechnique Fédérale de Lausanne (LMR-EPFL), MonashUniversity, and the National Natural Science Foundation of China(Grant No. 11802058) in the past few years.

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Dr. Xing Li is Associate Professor in the School of CivilEngineering at Southeast University, China since 2018. Hereceived his BSc and MSc degrees in Mechanics at Xi’anJiaotong University, China in 2011 and 2013, respectively.Xing conducted his doctoral research in Rock Mechanics atEcole Polytechnique Fédérale de Lausanne (EPFL) ofSwitzerland during 2013e2015, and obtained his PhD

degree in the Department of Civil Engineering at MonashUniversity, Australia in 2017. His research interests coverthe dynamic rock fracturing, numerical manifold method,and discrete element method.