# Numerical Methods For Modelling Rock Masses Environmental Sciences Essay

With the betterment of modern computing machines on calculating power, numerical methods became highly utile tools for stone mechanics. It has been proven that numerical methods are powerful and effectual tools. For illustration, numerical mold has been used to analyze dynamic response of fractured stone multitudes [ 1,2 ] , fracturing extension in stone and concrete [ 3-9 ] , wave extension in jointed stone multitudes [ 10,11 ] , and acoustic emanation in stone [ 12 ] . There exist a big figure of numerical methods, e.g. , Finite Element Method ( FEM ) , Finite Difference Method ( FDM ) , Finite Volume Method ( FVM ) and Discrete Element Method ( DEM ) . By and large numerical methods used in stone mechanics can be classified into continuum based method, discontinuum based method and coupled continuum/discontinuum method [ 13 ] . The reappraisal of these methods in this chapter will establish on this class method. More focal point on new developed methods and these did non be reviewed in [ 13 ] , e.g. , Smoothed Particle Hydrodynamics ( SPH ) , Molecular Dynamics ( MD ) and combined FEM/DEM method. This chapter seek to obtain a planetary position of bing numerical methods available for stone mechanics. Then, find the advantages and disadvantages these methods. Finally, basic thought of how to plan the suited numerical method for stone kineticss will be concluded.

FDM is one of the oldest numerical techniques used for the solution of sets of PDEs. The execution of FDM is truly simple both in three dimensional and two dimensional instances. It does non necessitate test ( or insertion ) maps like any other methods. However, the conventional FDM with regular grid systems does endure the inflexibleness in covering with breaks, complex boundary conditions and material heterogeneousness. These defects constrain its application in stone mechanics. Development of FDM marks at acquiring rid of these defects. For illustration, the Finite Volume Method ( FVM ) is considered as an extensional FDM which non merely acquire rid of the regular mesh constrain but besides specially suited to imitate non-linear behaviour of solid stuffs [ 14 ] . Finite-difference time-domain ( FDTD ) [ 15 ] is the straight development of FDM. It is based on two-layer grid-based differential time-domain methodological analysis. FDTD was widely used for informations processing of electromagnetic informations in stone mechanics, e.g. imaging electromagnetic informations for crosshole [ 16,17,18 ] . FDTD can besides be used for finding the hydraulic conduction of stones [ 19 ] . FDTD was used in wave extension job in a homogenous and heterogenous medium [ 20-25 ] . Inhomogeneous jobs can besides be solved by FDTD as the dual grid methodological analysis is used. Based on the basic thought of FDM, some genuinely meshless methods are proposed late, such as generalised finite difference method ( GFDM ) [ 26 ] and finite point method ( FPM ) [ 27 ] . Despite there exist many constrains of FDM, the basic thought of FDM to discrete clip sphere is widely been used in many numerical methods, particularly for dynamic analysis, e.g. , DEM and MD.

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Boundary component method ( BEM ) seeks a weak solution at the planetary degree through a numerical solution of an built-in equation derived utilizing Bettii??i??s mutual theorem and Somiglianai??i??s individuality. As merely the surface of sphere is needed, BEM will cut down the job dimensions by one. This leads the method have a fast computer science velocity and easy mesh coevals. The BEM is more suited to work out jobs of fracturing job for homogenous and linearly elastic organic structures [ 28-30 ] . Recent development of BEM includes the Boundary Contour Method ( BCM ) attack [ 31 ] and Fast Multipole BEM ( FMBEM ) [ 32 ] which farther decrease of computational clip, the Galerkin BEM ( GBEM ) [ 33,34 ] which paves the manner for the fluctuation preparation of BEM for work outing non-linear jobs and meshfree methods based BEM [ 35,36 ] which overcome drawbacks in the conversional BEM that requires boundary component. In general, BEM is non every bit efficient as FEM in covering with material heterogeneousness, non-linear stuff behaviour and harm development processes.

The FEM [ 37 ] term was foremost used by Clough for plane emphasis jobs, now it has became the mainstream numerical tool in technology scientific disciplines, including stone mechanics and technology. FEM has great flexibleness for intervention of stuff heterogeneousness, non-linear deformability, complex boundary conditions, in situ emphasiss and gravitation. These virtues make the FEM became the most successful numerical method used in technology and scientific discipline research [ 14 ] . Particular development of FEM for stone mechanics jobs are the joint elements [ 38-41 ] which are aiming at simulation of jointed stone mass. In stone mechanics, the most hard thing for FEM used is the simulation of fracturing procedure. This reappraisal on FEM will concentrate on this facet. A study of the literatures on FEM mold of fracturing advancement found that used methods for fracturing analysis can be classified into two groups: component debasement attack and elements boundaries interrupting attack.

The thought of element debasement attack is handling the stone fracturing procedure as a sequence of element debasement. The omission technique provided in Abaqus [ 42 ] is an illustration of this sort of attack, which removes the elements where the failure standard is locally reached, and element omission can be visualized as a cleft advancement. The most representative method of element debasement attack is continuum harm mechanics ( CDM ) based FEM, which was widely used for brickle fracturing analysis [ 43-45 ] . By combination the CDM with weibull distribution for represent heterogeneousness and a statistic failure standards, it was used for depicting harm development and cleft extension in stone and concrete under inactive and dynamic loading conditions [ 3-9 ] . Another debasement technique was realized by patterning the clefts and articulations on tantamount continuum constructs of elastic debasement and/or softening malleability [ 46 ] . Crack smeared theoretical account is the one represent of this method, which was foremost introduced by Rashid [ 47 ] . The cleft smeared theoretical account is normally used in concrete break analysis, and it is being far more popular because of its computational convenience [ 48 ] . The cleft smeared theoretical account was used for fracturing analysis of concrete under high strain rates and thermo-mechanical behaviour and failure of ceramic furnace lining stuffs [ 49,50,51 ] . This technique has besides been used in commercial FEM codifications to imitate the fracture/crack procedure of concrete-like stuffs, e.g. , ANSYS [ 52 ] and ATENA [ 53 ] . Element debasement method has advantages on no demand of remeshing and adding new grades of freedom in the computation procedure. However, this method can non bring forth expressed description of the break surface and has mesh size and orientation dependence.

Elementss boundaries interrupting attack represents the fracturing procedure by the separating of interelements boundary. The method insert interface elements along the interelement boundaries. It was used for cleft extension in concrete and stone stuffs [ 54,55,56 ] . Failuring of interelements boundary can be based on the break mechanics or failure standards of interelements boundary component. Fracture mechanics based methods are used in several FEM codifications, such as Abaqus [ 42 ] , Franc [ 57 ] and Marc [ 58 ] to cover with cleft extension jobs. The most successful development of elements boundaries interrupting attack is the Cohesive Zone Model ( CZM ) which dates back to the work of Hillerborg et Al. [ 59 ] and Belytschko et Al. [ 60 ] for brickle stuffs. The CZM has been successfully used in simulation of break and atomization in brickle stuffs, multiple distinct cleft extension and dynamic cleft growing in ceramic stuffs [ 61-65 ] . Normally, using this technique should be coupled with remeshing technique to extinguish the component dependance and emphasis singular job which exists in the cleft tip [ 63 ] . However, remeshing techniques [ 66-69 ] requires a instead complex package bundle to be developed and utilizing remeshing techniques will besides roll up the computation mistakes through function of variables. The worse fact is that adaptative remeshing can barely be used to imitate complex cleft development, such as cleft coalescency and cleft bifurcation.

There besides bing some defect in FEM, e.g. , the continuum premise in FEM makes it unsuitable to cover with complete withdrawal and large-scale break opening jobs [ 13,14 ] , which are the most concerned issues in stone mechanics. Locking effects which include numerical lockup and element lockup during utilizing the traditional FEM are other defects [ 70-74 ] . Some of these defects are solved by derived FEM which will be introduced in following.

After the divider of integrity ( PU ) [ 75 ] was proposed by Babuska and Melenk, research workers of different numerical methods can happen their theoretical cellar through it. Based on PU, a priori cognition about the solution can be added into the estimate infinite of the numerical solution. Numerical methods based on PU are ever called as derived FEM. The most representative derived FEM should be Numerical Manifold Method ( NMM ) [ 76 ] , eXtend FEM ( XFEM ) [ 77 ] , Generalized FEM ( GFEM ) [ 78 ] and Finite Cover Method ( FCM ) [ 79,80 ] . In this subdivision, the reappraisal will merely concentrate on these methods.

NMM was developed to incorporate Discontinuous Deformation Analysis ( DDA ) and FEM. NMM employs two sets of screen system [ 76 ] , one is mathematical screen which is arbitrary and used to specify the estimates of the sphere. Another is physical screen which portrayals of geometry and is used to specify the integrating sphere. The advantages of the NMM are let go ofing the undertaking of engaging and uniting continuum and discontinuum jobs into one model. For this ground, NMM was suited for break advancement simulation [ 81,82 ] . NMM has several advantages over classical FEM, e.g. , it is more suited for patterning dynamic cleft growing job [ 81 ] and micropolar snap [ 82 ] . FCM [ 78 ] is a extension of NMM on patterning heterogenous stuffs by utilizing Lagrange multipliers. Recently, FCM is extended to 3-dimensional by Terada and Kurumatani [ 83 ] . The NMM is proposed much earlier than the PU theory and other derived FEMs. Recently, it is besides called as a cover-based generalised FEM [ 80 ] . Actually, when read the multiplex codification, the convergent thinker is really similar with standard FEM and the distinguishable parts is the mesh coevals technique and half component technique used in the method. NMM can be regarded as a particular derived FEM designed for stone mechanics.

The XFEM [ 77 ] and GFEM [ 78 ] are other celebrated derived FEMs. GFEM uses precisely the same technique as XFEM, GFEM is mark at work outing jobs in complex geometry with less mistake and less computing machine resources [ 84,85 ] while XFEM is targeted at cleft extension jobs. For this ground, the XFEM will be chiefly reviewed. XFEM dainties clefts at element degree by utilizing the flat sets technique [ 86 ] , normally heaviside map and asymptotic maps are used to cover with the discontinuity and uniqueness. Compared with the classical FEM, XFEM has several advantages in facet of mesh independency. In XFEM, elements incorporating a cleft are non required to conform to check borders, and mesh coevals is much simpler than classical FEM. The most of import facet of XFEM is that it can execute widening cleft without any remeshing and the uniqueness can good be captured. For these advantages, XFEM was successfully used in cleft extension [ 87 ] , dynamic cleft extension [ 88 ] and 3-dimensional cleft extension [ 89,90 ] . Recent development of XFEM includes covering with cohesive fracturing [ 91 ] , expressed preparation of XFEM [ 92,93 ] , anisotropic XFEM [ 94 ] and XFEM which consider contact between cleft surfaces [ 95,96 ] .

These derived FEMs have the advantage of mesh independence and of covering with weak or strong discontinuities. These virtues make them really suited for fracturing procedure analysis. However, they besides have their ain disadvantages. For illustration, in some instances the execution of boundary conditions will be every bit hard as that in meshless methods [ 97 ] . The planetary stiffness matrix will go remarkable if the cleft base on balls a really bantam portion of XFEM component [ 98 ] , which is bing all of derived FEMs, e.g. , the NMM and GFEM. Execution of XFEM into available commercial FE codification is hard than standard FEM [ 99 ] because extra grades of freedom are introduced. And all of these methods would endure ill-condition job when usage higher order screens maps ( test maps ) are used. There are methods to cut down the singular but making that will allow the discontinuity of the enriched elements will neglect to be described. Even so, these derived FEMs are still the most promising methods. It is chiefly attributed to the successful application of standard FEM and inertial virtues of FEM based methods, e.g. , robust and easy to cover with complex geometry, lading and material conditions.

In recent old ages, a big household of meshless methods with the purpose of acquiring rid of mesh restraints has been developed. Their demands for theoretical account coevals are lone coevals and distribution of distinct nodes without fixed element-node topological dealingss as in FEM. Compared to engage coevals, it is comparatively simple to set up a point distribution and accommodate it locally. A local estimate map for the PDEs is built based on points grouped together in i??i??cloudsi??i?? . There are many meshless methods, such as Smoothed Particle Hydrodynamics ( SPH ) [ 100,101 ] , Diffuse Element Method Nayroles [ 102 ] , Element Free Galerkin ( EFG ) [ 103,104 ] , Reproducing Kernel Particle Method ( RKPM ) [ 105,106 ] , Hp Clouds [ 107 ] , Partion of Unity Method ( PUM ) [ 108 ] , Finite Point Method ( FPM ) [ 109 ] , Method of Finite Spheres [ 110 ] , Natural Element Method ( NEM ) [ 111 ] . Review on these methods is given in [ 13 ] and [ 112 ] . Depending on the methodological analysis used to discretize the partial differential equations ( PDEs ) , meshless methods can be classified into two major classs: meshless strong-form methods and meshless weak-form methods. Most of meshless weak-form methods such as EFG [ 103 ] are i??i??meshlessi??i?? merely in footings of the numerical estimate of field variables and they have to utilize a background mesh to make numerical integrating of a weak signifier over the job sphere, which is computationally expensive. Meshless strong-form methods such as the GFDM [ 26 ] and FPM [ 109 ] frequently use the point collocation method to fulfill regulating partial differential equations and boundary conditions. They are simple to be implemented and computationally efficient. Since they do non necessitate any background mesh, they are genuinely meshless methods. In this subdivision, merely three representative meshless methods will be presented, they are EFG, SPH and FPM.

EFG [ 103 ] is a method based on traveling least square insertions ( MLS ) where requires merely nodal informations and no element connectivity is needed. The meshless belongings is really suited to pattern dynamic cleft extension jobs. The application and development of EFG method includes varies Fieldss, such as jobs of break and inactive cleft growing [ 104 ] , dynamic jobs [ 113 ] , 3-dimensional stuff non-linear dynamic jobs [ 114 ] , adaptative attack [ 115 ] , dynamic extension of arbitrary 3-D clefts [ 116 ] , mixed-mode dynamic cleft extension in concrete and probabilistic break mechanics [ 117,118 ] , parallel EFG algorithm [ 119 ] and multiple clefts and cohesive cleft growing [ 120 ] . Contact algorithm based on a punishment method is introduced in [ 121 ] . The EFG was besides used for analysis of jointed stone multitudes with block-interface theoretical accounts [ 122 ] . These advantages make the EFG method have potencies to be used in stone mechanics. The indispensable boundary status and computational cost of EFG are the chief back draws.

SPH was foremost invented to cover with jobs in astrophysics [ 100 ] and subsequently extended for elastic jobs [ 123 ] . Application of SPH is chiefly in atomization analysis, such as dynamic atomization in brickle elastic solid [ 124,125 ] , high deformation impact calculations [ 126,127 ] , concrete atomization under explosive burden [ 128 ] , formation of clefts around magma Chamberss [ 129 ] and strain rate consequence for heterogenous brickle stuffs [ 130 ] . SPH exhibit an instability called the tensile instability and the zero-energy manner and need particular covering attack to bring forth stable and accurate consequences [ 131 ] . Furthermore, the meat map of SPH has great influence on the simulation consequences [ 132 ] and its truth is besides non every bit good as FEM. Overall speech production, SPH has advantage on simulation of dynamic atomization and easy to implement. But the preciseness, computational clip and contact intervention are still jobs for farther application in stone mechanics.

FPM [ 109 ] is a sort of meshless point collocation method which uses the leaden least squares ( WLS ) estimate within each point cloud. It can be easy constructed to a have consistence of a coveted order. Discrete equations are straight obtained from PDEs. It is easy to be implemented and boundary conditions can besides be implemented in a natural manner. The boundary conditions can realized by straight ordering boundary conditions on points placed on boundaries. The most attractive point of FPM is that it can give more truth emphasis consequences than that of FEM [ 133 ] . FPM with intrinsic enrichment was proposed for work outing elastic cleft was used for cleft jobs, and the local behaviour of the near-tip emphasiss is successfully captured and the stress strength factors can be accurately computed [ 134 ] . Furthermore, FPM is developed to imitate cleft extension under dynamic loading conditions [ 135 ] . Adaptive polish procedure for FPM based on posteriori mistake calculator was presented in [ 136 ] . However, the instability and hard to cover with heterogenous media handicapped its application on stone mechanics. Even late, the heterogenous job is partially solved in [ 137 ] , nevertheless for arbitrary heterogenous job there still no good solution.

The chief advantage of the meshless attacks is the aggressively decreased demand for engaging compared with standard FEM for both uninterrupted and fractured organic structures. Shortcoming of many meshless attacks are troubles in enforcement of indispensable boundary conditions, stableness and high computational cost. By and large talking, meshless methods still did non outperformed FEM from the pure calculating public presentation side. But they have good potency for stone technology jobs due to its flexibleness in intervention of breaks and complex construction theoretical account.

The continuum premise in continuum based methods makes it non suited to cover with complete withdrawal and large-scale break opening jobs [ 13 ] , which are the most concerned issues in stone mechanics. Continuum based methods are limited to work out jobs which involves discontinuous medium, such as jointed stone and postfailure province of stone. This is considered as the intrinsic bound of continuum based methods. The continuum methods are utilizing the thought of top-down methodological analysis. This will restricting the methods to be used for researching survey on some physical mechanical jobs, for example, the dynamic consequence mechanism of stone stuffs.

DEM is developed for work outing stone mechanics jobs [ 138 ] . The cardinal construct of DEM is to split the theoretical account into an gathering of stiff or deformable blocks/particles/bodies [ 139,140 ] . It is capable to cover with discontinuous organic structures with big supplantings and rotary motions, e.g. , the progressive failure of blockish stone mass. The development of DEM have a long clip since it foremost proposed by Cundall [ 139 ] . DEM methods were widely used in belowground plants [ 141,142,143 ] , laboratory trial simulations and constituent theoretical account development [ 144,145,146 ] , stone kineticss [ 147,148 ] , wave extension in jointed stone multitudes [ 11 ] , atomic waste depository design and public presentation appraisal [ 149 ] , stone atomization procedure [ 150 ] , acoustic emanation in stone [ 12 ] .

Harmonizing to solution method used in DEM, it can be divided into explicit and inexplicit two groups. For the expressed methods, there besides exist two sorts of attacks: the dynamic relaxation method and inactive relaxation method. Inactive relaxation based distinct method usage equations of equilibrium to acquire the supplanting of blocks at the following clip measure. Examples of inactive relaxation based DEMs can be found in [ 140,151 ] . The inactive relaxation method will repeat faster and need non put muffling. However, it can non be used for dynamic jobs. Dynamic relaxation based DEM uses Newtoni??i??s 2nd jurisprudence to acquire the supplanting of blocks at the following clip measure, and it was called as the distinguishable component method. The distinguishable component method can imitate the complex mechanical interactions of a discontinuous system. The most representative explicit DEM codifications are UDEC and 3DEC for two and 3-dimensional jobs in stone mechanics [ 152,153 ] . Making the usage of atoms to imitate farinaceous stuffs is another development way of DEM [ 14,154 ] . The most well-known codifications in this field are the PFC codifications [ 155 ] and the DMC codification [ 156 ] . Bonded Particle Model ( BPM ) [ 157,158 ] was performed on the atom DEM codifications, which can depict the harm mechanisms and time-dependent behaviour of stone stuff at microscope. It is non merely been used to imitate stone stuffs but besides been used in shear-band simulation of metal stuff [ 159 ] .

Contact sensing and contact interaction are the most of import issues in DEM, and many research workers distinct DEM from other methods on the ability of new contacts sensing during computation. There are a batch of contact sensing algorithms which mark at salvaging calculating clip and memory infinite, and item information can be found in the book written by Munjiza [ 160 ] . Mechanical interaction between two reaching blocks will act upon the concluding mechanical behaviours of DEM theoretical accounts. Usually it is treated by a finite stiffness spring in the normal way and a finite stiffness spring in the shear way. Improvements of automatically interaction were besides reported, e.g. , an interaction scope and modified Mohr Coulomb rupture standard is introduced in DEM theoretical account [ 161,162 ] and a first order differential equation for joint coherence is implemented into the UDEC in [ 163 ] .

DDA [ 164 ] is a type of DEM originally proposed to analyse the mechanical behaviour of blockish systems. It is similar to FEM, but can stand for the interaction of single blocks in stone multitudes. DDA is typically based on a work-energy method, and can be derived utilizing the rule of minimal possible energy or Hamilton ‘s rule. The applications of DDA are chiefly on tunneling, caverns, fracturing and atomization procedures of geological and structural stuffs and temblor effects [ 165,166,167 ] . Developments of DDA include engaging the blocks with FEM meshes [ 168 ] , covering the contact as joint with stiffness and taking none incursion standards to cut down the calculation clip and acquiring fast convergence [ 169 ] , coupled stress-flow jobs [ 170 ] , 3-dimensional block system analysis [ 171 ] , higher order elements [ 172 ] , more comprehensive representation of the breaks [ 173 ] , and syrupy boundary for patterning emphasis moving ridge extension [ 167 ] .

Chiefly due to the expressed representation of breaks and articulations, DEMs have been basking broad applications in stone mechanics and stone technology. Furthermore, the theory of DEM methods are simple and easy to understand. Despite these advantages, there are besides some defects, e.g. , the deficiency of cognition of the geometry informations of the stone fractures limits their applications [ 174 ] . Furthermore, DEM is comparatively new and many think of it as i??i??not yet proveni??i?? numerical method for analysis and design in stone mechanics.

MD is a signifier of computing machine simulation in which atoms and molecules are the basic component. The system behaviour is obtained through Lashkar-e-Taiba these elements to interact under given physical Torahs. It is regarded as an interface between research lab experiments and theory, and can be understood as a “ practical experiment ” . MD was originally conceived within theoretical natural philosophies in the late 1950 ‘s [ 175 ] . Now it is widely been used in material scientific discipline and biochemistry scientific discipline. It can assist people to animate and explicate some phenomena at the atomic degree. This reappraisal will merely concentrate on the mechanical application of MD. Even in the early clip, MD was used to analyze the cleft belongingss and obtained consequences are agree good with continuum mechanics and the break mechanics [ 176-179 ] . MD simulation was besides used to analyze the passage of toffee to ductile extension of a crisp cleft and favourable cleft extension way in crystallize stuff [ 180,181 ] , failure mechanism of micro farinaceous stuff [ 182,183 ] , extension of mode-I clefts in an icosahedral theoretical account quasicrystal [ 184 ] and Yoffe ‘s additive theory of dynamic brickle break [ 185 ] .

Rock machinist related jobs solved by MD include interaction between complex farinaceous atoms [ 186 ] , creative activity of polycrystalline computing machine stuffs [ 187 ] , viscocelastic behaviour of granite stone [ 188 ] and influence of porousness on elastic strength belongingss of polycrystalline specimens ( sandstone ) [ 189 ] . Potential maps used in MD simulation have great influence of the simulation consequences, and it besides the nucleus context of MD survey. Potentials used for cleft extension includes Lennard-Jones Potential [ 176,178 ] , Hookei??i??s Law ( Harmonic Potential ) [ 185 ] , EAM potentials [ 182,183 ] and ReaxFF reactive force field [ 190 ] . Lennard-Jones Potential and Hookei??i??s Law are simple but areni??i??t much physical world. The EAM potencies could successful used in metal simulation, nevertheless it will endure jobs when face nonmetal stuff such as Si and it can be solved by utilizing ReaxFF reactive force field which is computational dearly-won [ 190 ] .

MD can be used to explicate mechanical phenomena at atomic graduated table. It is a powerful tool for survey mechanisms of cleft extension at microscopic degree. However, long clip simulations are mathematically ill-conditioned. Simple possible maps are non sufficiently accurate to reproduce the kineticss of molecular systems while complex possible maps normally computationally dearly-won. Furthermore, the atomic construction information of stone stuffs are excessively complicated and barely obtained. These restrictions lead MD still can non be straight used for stone mechanics.

A household of methods coined as lattice theoretical accounts ( LMs ) have been developed. They are based, in rule, on the atomic lattice theoretical accounts originated from condensed affair natural philosophies. In these theoretical accounts, stuff is represented by a system of distinct units ( e.g. atoms ) interacting via linking elements. These distinct units are much coarser than the true atomic 1s and may stand for larger volumes of heterogeneousnesss such as grains or bunchs of grains. Compared to a true lattice theoretical account, the usage of coarse lattices in lattice theoretical accounts dramatically reduces the figure of grades of freedom, and therefore makes simulation of continuum systems low-cost for moderate-sized computing machines. Lattice theoretical accounts are more suited for patterning break of stuffs than conventional FEMs because the former 1s simulate break by either merely taking linking elements that exceed the strength or in turn degrading their mechanical belongingss harmonizing to cohesive Torahs. The spacial concerted effects of cleft formation and heterogeneousnesss can be easy investigated through the usage of LMs [ 191, 192 ] .

There exist two different types of lattice theoretical accounts. In the first type theoretical accounts, the stuff is discretized as a web of springs or beams whose geometry is non related to the existent internal geometry of the stuff. Here the distinct units are simply lattice sites ( nodes ) . This type of theoretical accounts can be farther classified into lattice spring [ 193-197 ] and lattice beam [ 198-201 ] theoretical accounts harmonizing to the figure of grades of freedom per node and the mechanical belongingss of linking elements. In a lattice spring theoretical account ( LSM ) , the terra incognitas are the nodal supplantings and the connecting elements are unidimensional springs. In a lattice beam theoretical account, the terra incognitas are the nodal supplantings and rotary motions and the connecting elements are beams reassigning normal forces, shear forces and flexing minutes. The 2nd type theoretical accounts are based on the distinct component method originally developed for farinaceous media with contact patterning [ 202 ] . For illustration, the stiff body-spring web theoretical account developed by Kawai [ 203 ] subdivides the stuff into stiff atoms interconnected along their boundaries through normal and shear springs. It introduces extra rotational grades of freedom on each atom and hence can be viewed as discretization of a micropolar continuum. Models in this class besides include that of Zubelewicz and Ba? ant [ 204 ] , the confinement-shear lattice theoretical account of Cusatis et Al. [ 205 ] , the bonded-particle theoretical account [ 206 ] , the simple deformable polygonal distinct component theoretical account [ 207 ] , and etc.

The beginning of LSM may follow back to Hrennikoff [ 193 ] . The simplest LSM is the normal force theoretical account in which merely cardinal force interactions ( normal springs ) are considered. The normal force theoretical account has been extensively applied to look into the elastic and failure belongingss of a broken medium [ 194-199 ] or the fractal belongingss of cleft [ 208 ] . It is besides often used to the survey of break or other issues of stuff scientific discipline [ 209 ] . However, for the normal force theoretical account, it is known that Poissoni??i??s ratio obtained by the theoretical account attacks, in the bound of an infinite figure of atoms, a fixed value, viz. , 0.25 for 3-dimensional instances and 0.33 for planar instances. Such limitation is non suited for many stuffs and it can be overcome by presenting non-central force interactions ( shear springs ) between atoms. There are different methods proposed to work out this job, e.g. , a method to modify the Poissoni??i??s ratio by presenting a harmonic potency for rotary motion of bonds from their initial orientation [ 210 ] . A non-central two-body interaction restricting the rotational freedom of bonds is introduced in the Born spring theoretical account [ 211, 212 ] to let a wide pick of Poissoni??i??s ratio. The Kirkwood-Keating spring theoretical account [ 213,214 ] introduces angular springs to punish the angular fluctuations between the immediate bonds incident onto the same node. However, this job can non be solved if merely brace organic structure interaction is considered.

Discontinuum base methods treat the stone stuff or stone mass as an assembled theoretical account of blocks, atoms or bars. The fracturing procedure of stone is represented by the breakage of inter-block contacts, inter-particle bonds or bars. Discontinuum based methods can bring forth realistic failure procedure of stone. However they are non suited for analysis emphasis province of the pre-failure stone. This is the unchangeable demerits of discontinuum based methods.

The continuum based methods are unsuitable to capture the post-failure discontinuous phase while the discontinuum based methods are unsuitable to capture the pre-failure phase of stone. A combination of continuum and distinct methods is required in many geophysical applications, such as projectile incursion into stone, concrete marks, and bowlder Fieldss to foretell the formation and interaction of the fragments [ 216 ] . Coupled continuum and discontinuum methods can take advantages of the strength of each method while avoiding its disadvantages. Making the discontinuity zone with discontinuum based method while the continuum zone with continuum based method is a direct yoke continuum and discontinuum methodological analysis. Examples of this sort of coupled methods are intercrossed DEM/BEM theoretical account [ 217,218 ] , combinations of DEM, DFN and BEM attacks [ 219 ] , intercrossed DEM/FEM theoretical account [ 220,221 ] . For fracturing simulation, it requires the method is able to capture pre-failure behavior and after the failure and prostration has occurred. For continuum-discontinuum coupled methods, research workers normally like to match FEM with DEM. This subdivision will be chiefly concentrate on this subject.

Combined finite-discrete component method [ 160,222 ] is a late developed coupled FEM/DEM method aimed at patterning neglecting, fracturing and fragmenting solids. In the combined finite-discrete component method, each organic structure is represented by a individual distinct component that interacts with other distinct elements that are near to it. In add-on, each distinct component is divided into FEM elements, which can be broken into smaller blocks during computation. Coupled FEM/DEM was widely been used to imitate the break procedure of stone, e.g. , Morris et Al. [ 216 ] developed a FEM/DEM codification LDEC to look into the consequence of explosive and impact lading on geological media, Karami and Stead [ 223 ] usage FEM/DEM theoretical account to imitate cleft extension under assorted manner burden and Ariffin et Al. [ 224 ] usage a intercrossed FEM/DEM codification to look into the procedures of joint surface harm and near-surface integral stone tensile failure. Coupled FEM/DEM method is a powerful method to work out the fracturing procedure jobs. However, implementing this method into a computing machine codification will necessitate more complex accomplishment. There besides exist some numerical methods which are aiming at combination of continuum and discontinuum into one individual model, e.g. , NMM [ 77 ] , continuum-based distinct component method ( CDEM ) [ 225 ] , peridynamic theoretical account [ 226 ] and Finite Edge Element Method ( FEEM ) [ 227 ] . However, the basic thoughts of these methods are same with FEM/DEM coupled methodological analysis.

Multiscale mold was regarded as an exciting and promising methodological analysis for simulation of fracturing procedure [ 228,229 ] . Rock behaviour entails multiscale, e.g. , multiscale fracturing is the cardinal to calculating volcanic eruptions [ 230 ] . The intents of multiscale mold are cut downing the computational clip [ 231 ] and straight obtain macro stuff response from micro mechanical [ 232 ] . There are three types of matching methods are used. The first 1 is to match to different graduated table theoretical accounts which is realized by utilizing microscopic theoretical account where is needed and macroscopic theoretical account for other parts. This methodological analysis is widely used in the yoke MD with continuum mechanics. For illustration, FEM with MD [ 177 ] , analytical solution with molecular kineticss [ 183 ] and the generalised insertion stuff point ( GIMP ) method with molecular kineticss ( MD ) [ 233 ] .

The 2nd 1 is utilizing same theoretical account for different graduated tables. For illustration, a two graduated table attack based on a refined global-local method is applied to the failure analysis of concrete constructions [ 234 ] . In this solution the finite element solution is split into two parts: a additive elastic analysis on a harsh mesh over the full construction and a non-linear analysis over a little portion of the construction where a dense finite component grid is employed. XFEM was used for imitating micro-macro advancement cleft in heterogenous theoretical accounts in [ 235,236 ] , it realized by spliting the solution infinite into coarse-scale which unchanged during the cleft extension and all right graduated table which calculation can be done independently through XFEM. Examples besides include three-scale computational method [ 237,238 ] , multi-scale boundary component method [ 239 ] and Voronoi cell FEM with a non-local Gurson Tvergaard Needleman ( GTN ) theoretical account [ 240 ] .

The 3rd one is numerical theoretical account itself behavior multiscale belongings. For illustration, the multiscale finite component method ( MsFEM ) [ 241 ] was designed for work outing a category of elliptic jobs. The finite difference heterogenous multi-scale method ( FD-HMM ) [ 242 ] was designed for work outing multi-scale parabolic jobs. Quasicontinuum ( QC ) method is a conjugate continuum and atomistic method which proposed by Tadmor et Al. [ 243 ] for imitating the mechanical response of polycrystalline stuffs. QC is designed for metal stuff surveies, e.g. , the effects of construction and size on the distortion bicrystals in Cu [ 244 ] , the atomic graduated table break [ 245 ] and distortion and failure of metal stuff [ 246 ] are modeled by QC theoretical account. There are some ripple based numerical methods, e.g. , ripple based reproducing meats atom methods ( RKPM ) [ 247 ] and multiresolution finite component method based on 2nd coevals ripples [ 248,249 ] , are entitled multiscale belongings. The defect of multiscale methods are that they are comparatively new and there are no available codifications and methods for stone mechanics jobs.

Challenges exist in computational scientific discipline includes ( de Borst, 2008 ) :

( 1 ) Explicitly and accurately exemplary dynamic cleft extension job.

( 2 ) Multiscale analysis.

( 3 ) Multi-physics analysis.

From reappraisal, there exits many sorts of numerical methods. Each of them has their ain advantages and demerits. Table 2.1 list the failing and strength of representative numerical methods for stone mechanics.

From Table 2.1, there is non a individual numerical method could fulfill all the demand. Development of a micro-macro and continuum-discontinuum conjugate numerical method is needed. The matching method will take the most simple one and from Table 2.1 the best combination should be derived FEM with BPM/Lattice theoretical account. For BPM theoretical account where the grade of freedoms for each atom doesni??i??t maintain consistent with that of FEM node while atom in LSM theoretical account portion same grade of freedoms. For this ground, the LSM is selected as the microscopic theoretical account. However, the restriction of Poissoni??i??s ratio and stuff belongingss choice of LSM theoretical account are still non good solved. In this thesis, these jobs will be solved. In following chapters, new proposed microscopic LSM theoretical account and its corresponding multiscale theoretical account will be developed.

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