Finite-Cover-Based Element-Free Method For Continuous And Discontinuous Deformation Analysis With Applications In Geotechnical Engineering

The actual deformation and failure process of geo-materials and geo-structures is a complex progressive evolution involving initially elastic deformation, crack propagation, large-scale displacement and even movement of a discrete system. However, being only simulating some stages of the procedure, the current numerical methods in geomachanics are suitable for either ideal continuum materials, or completely discontinuous media. On the one hand, Such methods as FEM and BEM are based on the hypothesis of total continuity or basal materials continuity. On the other hand, there are discontinuous deformation analysis methods such as DDA and DEM, which are on the basis of the blocky theory and the assumption that rock mass is discrete media. This existing status makes it necessary that the actual deformation of rock mass must be represented to be either continuous or completely discontinuous. A numerical method born with ability in dealing with continuous deformation problems, discontinuous deformation problems and crack propagation problems seems to be a blue moon. It seems that there exist two essential factors causing this existing situation. (1) Being the most important link in the analysis process of rock damage and the ligament connecting the initial continuous deformation with the final failure of blocky system, the problem of crack has not been solved thoroughly even recently. (2) There is no an effective method which can deal with continuous, discontinuous and even crack propagation problems in a consistent methodology, or there is no an interpolation method which not only Cafl suitable for characterizing the continuity and discontinuity of materials in a uniform manner but also does not require mesh generation. Numerical analysis methods in geomechanics have been developed rapidly in recent thirty years. As two novel numerical methods for solving boundary-value problems, Manifold method (Shi, 1990) and element-free Galerkin method (Belytschko, 1994) have been established elementarily and received worldwide attentions. These two methods own many strong points and appear to be more superior over the conventional numerical methods such as finite element methods. By virtue of the finite cover technique of manifold, manifold method (MM) integrates conventional finite element methods (FEM), discontinuous deformation analysis (DDA) (Shi 1988) and analytical methods in a united mathematical framework and can deal with both continuous and discontinuous deformation problems such as contact and multi-body interaction. Element-free Galerkin method (EFGM) on the basis of moving least square (MLS) method (Lancaster & Salkaushas, 1981) is very effective in simulating crack propagation which is a key issue in modeling failure or/and damage behavior of structures or materials without meshing as required in MM. The kernel of MM isfinite cover technique while mesh-free approalmation (so-called mesh-free technique)mainly derived from MLS method is a key feature of EFGM. MM and EFOM have ownadvantages in handling discontinuous deformation prob1ems.The main purpose Of the paPer is to exPlore the POssibi1ity tO work out a new numericalmethod by combining the finite-cover techeque and mesh-ffee concePt tOgetheLThe researh begins with stUdy of the key idea of MM and goes on with extension andapplication of the finite cover technique of MM.Firstly, the fundamental of high-order MM is StUdied thoroughIy Generai mathematicalfOrmulations of MM with arbitrny order cover functions are presented. Then MM with thecomplete first-order polynomiaI cover functions and with the complete second-orderpolynomial cover functions are developed respectiveIy Programs of the...