The Study On Element Free Galerkin Methods And Numerical Simulation Of Crack Propagation

The study background of this paper is the crack rock mass. A new crack propagation criterions, the maximum circumferential tensile stress criterions, are presented in this paper. The numerical simulation of two-dimensional(2-D) and three-dimensional(3-D) crack propagation are studied baseed on element free Galerkin (EFG) method and 2-D EFG method based on couple stresses theory.Basic formulae of 2-D EFG method and 2-D EFG meothod based on couple stresses theory and 3-D EFG method are derived. Afterwards, discrete equations for elastic body are derived by use of the minimus potential energy principle. By use of classical 2-D and 3-D problems, the implement of 2-D and 3-D EFG method are discussed. And the matching of several important parameters in 2-D and 3-D EFG method are discussed, too. The displacement restriction is released by use of a penalty parameter. So the essential boundary conditions are introduced into EFG methods. Based on the idea of controlling nodal number, alterable influence radii can be defined. The least square calculating estimate is cut down in dense nodal area and calculating efficiency of EFG methods is enhanced properly.Based on space analytic geometry, the local coordinates at crack front is built up. The coordinates transform matrix, base vector transform, vector transform and displacement transform are established from the global coordinates to the local coordinates at crack front. The circumferential stress formulae are derived in the main normal plane of the local coordinates at crack front. So there have sufficient preparative work for the numerical simulation of crack propagation and surface or line of crack propagation.In studying crack problem, the visibility criterion is adopted to figure isolation of crack between nodals and integral nodals.Chosen a proper crack propagation criterion to determine condition, directions and steps of crack propagation is the key.Shortcoming of 2-D crack propagation criterions and no criterions of 3-D crack propagation are considered. Based on other scholars work, a new criterions of crack propagation, namely the maximum circumferential tensile stress criterion for 2-D and 3-D crack are brought forward in this paper. The new crack propagation criterions give out not only crack propagation condition and directions, but also steps too. It avoids the error and trouble of calculating stress intensity factor, especially for 3-D crack propagation problems, and overcomes the disadvantage giving out crack propagation step depending on experience of calculator.Surface of 3-D crack propagation maybe appear bending and twisting in crack propagation. It is divided into some snippets. Every snippet is triangulated. Mathematic description of crack surface is realized.In numerical simulation of crack propagation, for identifying crack shape and improving simulation precision of crack propagation surface, append nodal points are added on surface at crack front. Some nodal point maybe lie in a crack snippet. It is splited to two points to be laid on the two side of crack surface.The numerical simulation of 2-D crack propagation without cranny hydraulic pressure is studied by EFG methods. The results are identical with that of other computing method and test. The validity of the maximum circumferential tensile stress criterion is confirmed. The effect on crack propagation of cranny hydraulic pressure and couple stresses is studied. Some new results are found.Because of the complexity of 3-D crack propagation, there are a few results about the numerical simulation of 3-D crack propagation at present. Only model- I propagation of penny crack is simulated by use of traditional numerical method. In this paper, model- I propagation of ellipse crack is simulated by EFG method. The results are identical with that of other computing method and test. Once more, the validity of the maximum circumferential tensile stress criterion and mathematic description of 3-D crack surface are proved. Based on the above, the crack propagation of 3-D surface inclined semi-circular cracks are simulated numerically. The manner in different stage and the surface shape of 3-D crack propagation are known. Different attitude of original crack produces great effect on 3-D crack propagation. The blankness of the numerical simulation of 3-D complicated crack propagation is filled up in homeland.The full propagation process of 3-D crack is simulated. The simulation results make up some faults of the experimental results and theory research results. The simulation results have some reference value to further study 3-D crack fracture.