A 2-D meshless model for jointed rock structures

Because of the presence of joints and planes of discontinuity, the behaviour of jointed rock struc- tures is largely controlled by the discontinuities. A rock structure can be regarded as a system of relatively intact rock blocks connected by joints or planes of discontinuity. Many numerical methods have been proposed, such as Joint Element [1], Discrete Element Method [2], Discon- tinuous Deformation Analysis [3], Rigid Finite Element Method [4], Block-Interface Model [5], and several others. All these methods require discretization of rock masses into a great number of nite elements to achieve reasonable results, consequently, mesh generation for these methods is a time-consuming task. Besides the time-consuming mesh generation, mesh-based methods are also not well suited to the problems associated with extremely large deformations of the mesh and problems associated with frequent remeshing. Although several strategies have been developed to maintain a reasonable mesh shape, such as the Arbitrary Lagrangian{Eulerian (ALE) method [6], extra computational e ort and di culties are also introduced. In the simulation of failure processes, frequent remeshing is required to model the propagation of cracks with arbitrary and complex paths so that the computational e ort required is very signi cant. Compared with the nite element method, meshless methods have overcome these di culties, and some of them have a number of attractive features. Many researchers were interested in meshless methods, and about 10 di erent meshless methods have been developed, such as the Element-Free Galerkin (EFG) method [7], the Reducing Kernel Particle Method (RKPM) [8], the Smoothed Particle (SPH) method [9], and several others. The journal, Computational Methods in Applied Mechanics and Engineering, published a special issue on meshless methods in December, 1996. In these methods, EFG constructs its shape functions by the moving least-squares (MLS) interpolants. In the Block-Interface Model proposed in Reference [5], a rock structure is discretized by blocks, which are in a constant stress state, connected by interfaces. The blocks are used to model the rock blocks while the interfaces are used to model the joints and planes of discontinuity. For a rock block with steep stress gradient, it should be further divided into many sub-blocks. Consequently, the preparation of data is a time-consuming task, and the continuity condition between these sub- blocks must be imposed by penalty method. Based on paper [5], a meshless model is proposed by introducing MLS into the Block-Interfaces Model to construct the displacement elds of the blocks without dividing them into sub-blocks. According to the characteristic structural features of rock structures, a rock structure is regarded as a system of relatively intact anisotropic rock blocks connected by interfaces, and every rock block is modelled by an array of points through the moving least-squares interpolants (see Figure 1). To reduce the computational e ort required for a large-scale rock structure, the major part of the structure, in which stress gradients are moderately low, can be modelled by a few large anisotropic blocks whose material properties can be obtained from an equivalent anisotropic continuum model [10{13]. It is unnecessary to divide each block into sub-blocks, so that the time-consuming mesh generation is avoided. The displacement elds and their gradients are continuous in each block, hence no post processing is required for the output of strains and stresses. Furthermore, the discontinuities of rock structures are fully taken into consideration in the present method.