Numerical methods in rock mechanics

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410 1. Introduction L. Jing, J.A. Hudson / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 409–427 Because rock mechanics modelling has developed for the design of rock engineering structures in different circumstances and different purposes, and because different modelling techniques have been developed, we now have a wide spectrum of modelling and design approaches. These approaches can be presented in different ways. A categorization into eight approaches based on four methods and two levels is illustrated in Fig. 1. The modelling and design work starts with the objective, the top box in Fig. 1. Then there are the eight modelling and design methods in the main central box. The four columns represent the four main modelling methods: Method A— design based on previous experience. Method B— design based on simplified models. Method C— design based on modelling which attempts to capture most relevant mechanisms. Method D— design based on ‘all-encompassing’ modelling. There are two rows in the large central box in Fig. 1. The top row, Level 1, includes methods in which there is an attempt to achieve one-to-one mechanism mapping in the model. In other words, a mechanism which is thought to be occurring in the rock reality and which Site Investigation Objective Method A Method B Method C Method D Use of pre-existing standard methods Precedent type analyses and modifications Analytical methods, stress-based Rock mass classification, RMR, Q, GSI Design based on forward analysis Design based on back analysis Construction is to be included in the model is modelled directly, such as an explicit stress–strain relation. Conversely, the lower row, Level 2, includes methods in which such mechanism mapping is not totally direct, e.g. the use of rock mass classification systems. Some of the rock mass characterization parameters will be obtained from site investigation, the left-hand box. Then the rock engineering design and construction proceeds, with feedback loops to the modelling from construction. The review is directed at Methods C and D in the Level 1 top row in the central box of Fig. 1. An important point is that, in rock mechanics and engineering design, having insufficient data is a way of life, rather than a local difficulty—which is why the empirical approaches, i.e. classification systems, have been developed and are still required. So, we will also be discussing the subjects of the representative elemental volume (REV), homogenisation/upscaling, back analysis and inverse solutions. These are fundamental problems associated with modelling, and are relevant to all the A, B, C & D method categories in Fig. 1. The most commonly applied numerical methods for rock mechanics problems are: (1) Continuum methods—the finite difference method (FDM), the finite element method (FEM), and the boundary element method (BEM). (2) Discrete methods—the discrete element method (DEM), discrete fracture network (DFN) methods. (3) Hybrid continuum/discrete methods. Basic numerical methods, FEM, BEM, DEM, hybrid Database expert systems, & other systems approaches Extended numerical methods, fully-coupled models Integrated systems approaches, internet-based Level 1 1:1 mapping Level 2 Not 1:1 mapping Fig. 1. The four basic methods, two levels, and hence eight different approaches to rock mechanics modelling and providing a predictive capability for rock engineering design [1].
L. Jing, J.A. Hudson / International Journal of Rock Mechanics & Mining Sciences 39 (2002) 409–427 411 The choice of continuum or discrete methods depends on many problem-specific factors, and mainly on the problem scale and fracture system geometry. The continuum approach can be used if only a few fractures are present and if fracture opening and complete block detachment are not significant factors. The discrete approach is most suitable for moderately fractured rock masses where the number of fractures is too large for the continuum-with-fracture-elements approach, or where large-scale displacements of individual blocks are possible. There are no absolute advantages of one method over another. However, some of the disadvantages of each type can be avoided by combined continuum-discrete models, termed hybrid models. In this review, we concentrate on the states-of-the-art of the capability and utility of the numerical methods for rock mechanics purposes and note the outstanding issues to be solved. The emphasis is on civil engineering applications, but the information applies to all branches of rock engineering, and is supported here by a moderately extensive literature reference source. 2. Numerical methods for rock mechanics: states-of-theart 2.1. The FDM and related methods The basic technique in the FDM is the discretization of the governing partial differential equations (PDEs) by replacing the partial derivatives with differences defined at neighbouring grid points. The grid system is a convenient way of generating objective function values at sampling points with small enough intervals between them, so that errors thus introduced are small enough to be acceptable. With proper formulations, such as static or dynamic relaxation techniques, no global system of equations in matrix form needs to be formed and solved. The formation and solution of the equations are localized, which is more efficient for memory and storage handling in the computer implementation. No local trial (or interpolation) functions are employed to approximate the PDE in the neighbourhoods of the sampling points, as is done in the FEM and BEM. It is therefore the most direct and intuitive technique for the solution of the PDEs. This also provides the additional advantage of more straightforward simulation of complex constitutive material behaviour, such as plasticity and damage, without iterative solutions of predictorcorrector mapping schemes which must be used in other numerical methods using global matrix equation systems, as in the FEM or BEM. The conventional FDM with regular grid systems does suffer from shortcomings, most of all in its inflexibility in dealing with fractures, complex boundary conditions and material heterogeneity. This makes the standard FDM generally unsuitable for modelling practical rock mechanics problems. However, significant progress has been made in the FDM with irregular meshes, such as triangular grid or Voronoi grid systems, which leads to the so-called Control Volume, or Finite Volume, techniques. Voronoi polygons grow from points to fill the space, as opposed to tessellations where the polygons are formed by lines cutting the plane, or by building up a mosaic from pre-existing polygonal shapes. The FVM can be formulated with primary variables (e.g. displacement) at the centres of cells (elements) or at nodes (grid points) for an unstructured grid. It is also possible to consider different material properties in different cells. The FVM approach is therefore as flexible as the FEM in handling material heterogeneity, mesh generation, and treatment of boundary conditions with unstructured grids of arbitrary shapes. It has similarities with the FEM and is also regarded as a bridge between the FDM and FEM. The original concept and early code development in FVM for stress analysis problems can be traced to the work in [2], using a vertex-centred scheme with a quadrilateral grid. At present, the most well-known computer code for stress analysis for rock engineering problems using the FVM/ FDM approach is perhaps the FLAC code group [3], with a vertex scheme of triangular and/or quadrilateral grids. Explicit representation of fractures is not easy in the FDM/FVM because they require continuity of the functions between the neighbouring grid points. In addition, it is not possible to have special ‘fracture elements’ in the FDM or FVM as in the FEM. In fact, the inability to incorporate explicit representation of fractures is the weak point of the FDM/FVM approach for rock mechanics. On the other hand, the FDM/FVM models have been used to study the mechanisms of fracturing processes, such as shear-band formation in the laboratory testing of rock and soil samples [4], and fault system formation as a result of tectonic loading [5]. This is achieved via a process of material failure or damage propagation at the grid points or cell centres, without creating fracture surfaces in the models. The FVM is one of the most popular numerical methods in rock engineering, with applications covering almost all aspects of rock mechanics, e.g. slope stability, underground openings, coupled hydro-mechanical or thermo-hydro-mechanical processes, rock mass characterization, tectonic process, and glacial dynamics. The most comprehensive coverage in this regard can be seen in [6]. 2.2. The FEM and related methods The FEM is perhaps the most widely applied numerical method across the science and engineering
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