Application Of The Meshless Local Petrov-Galerkin Method To The Fracture Mechanics

The meshless method is a new numerical method with a great prospect developed after traditional numerical methods such as Finite Element Method, Boundary Element Method et al. The meshless method possesses many advantages, among them the most outstanding advantage is independent of meshes, and thoroughly or partly eliminates meshing. It becomes easy to solve super-large deformation problems, crack propagation problems and high velocity impact problems et al in the use of this method. The researchers pay increasingly attentions to it. The meshless local Petrov-Galerkin(MLPG) method is a new numerical technique presented in recent years. It doesn't need any element or mesh for the energy integral or the purpose of interpolation. Therefore it is a truly meshless method. In recent years, Atluri and Long SY et al have made a lot of investigations on the theory of MLPG method and its applications. On the basis of their work, applications of MLPG method to fracture mechanics problems are presented in this dissertation.At the beginning of the dissertation, recent developments of the meshless method are overviewed. Several typical meshless methods are reviewed and appraised in term of their discretization scheme. Characteristics, advantages and disadvantages of all kinds of meshless methods are pointed out. Applications of the meshless methods to the fracture mechanics problem are introduced. Then, on the basis of Atluri's work, a modified MLPG method is employed, in which the moving least squares (MLS) approximation is used as a trial function and the Heaviside function is used as a test function of the weighted residual method. Further, a direct interpolation method is used to impose the essential boundary condition without use of the penalty function and the Lagrange multipliers method. Numerical results of a cantilever beam and an infinite plate with a circular hole show the present method has higher computational efficiency than the conventional MLPG method.Although a lot of achievements are obtained about meshless methods for fracture mechanics, papers on solving the fracture mechanics problem are rarely presented in the use of MLPG method. In this dissertation, the modified MLPG method is used to investigate several kinds of fracture mechanics problems. The basis function of the MLS approximation is enriched by the singular function of a stress field which can capture 1 stress-singularity in the linear-elastic fracture mechanics problem. A visibility criterion and a diffraction method are employed to solve non-continuity of cracks. The stress intensity factor and the propagation trajectory as well as the stress field are given for several kinds of cracked plates. For the analysis of elastic-plastic fracture mechanics, an incremental Newton-Raphson iterative algorithm and the subincremental algorithm of tangential predication with a radial return-back are employed to solve incremental nonlinear local Petrov-Galerkin equations. The plastic zone around the crack tip and the stress intensity factor for a double cracked plate and a three point bending specimen are analyzed. For the dynamic analysis of fracture mechanics, the modified MLPG method is used to discrete the spatial domain and the Newmark method is adopted to discrete the time domain. The dynamic stress intensity factor, the stress field around the crack (notch) tip and the deformation are computed for a center cracked plate and a double edge notched plate under an impulsive load. For the fracture mechanics of functionally graded materials, elasticity matrix is dependent on space and the conventional J-integral isn't valid any more. Mutual integral (M-integral) is in use of by homogeneous auxiliary field and non-homogeneous auxiliary field to compute easily stress intensity factors of structures with different located cracks under different loads.Numerical results show that the present method possesses not only feasibility and validity, but also high accuracy and good performance of convergence for fracture mechanics problems including linear-elastic fracture mechanics, elastic-plastic fracture mechanics, dynamic fracture mechanics and fracture mechanics of functionally graded materials.