【作者】 刘凯远；

【导师】 龙述尧；

【作者基本信息】 湖南大学 ， 固体力学， 2007， 博士

【摘要】 无网格方法是继有限元、边界元等传统的数值方法之后一种新兴的、很有发展前途的数值方法。无网格方法有很多优点,最突出的优点在于克服了对网格的依赖性,彻底或部分消除了网格的划分,因此无网格方法在处理超大变形问题,裂纹扩展问题,高速冲击等问题中具有明显的优势,越来越受到科学工作者的关注。无网格局部Petrov-Galerkin方法是近几年发展起来的一种无网格方法,它不需要借助任何单元或网格进行积分和插值,是一种真正的无网格方法。近年来,Atluri等和龙述尧等在无网格局部Petrov-Galerkin方法及其应用研究上取得了一系列成果,在他们研究的基础上,本文将无网格局部Petrov-Galerkin方法用于求解断裂力学问题。本文首先综述了无网格方法的发展历史与国内外研究现状,按照其离散方式的不同对各种典型的无网格方法进行了回顾与评价,总结了无网格方法的特点、优越性以及目前无网格方法的难点和存在的问题。概述了无网格方法在断裂力学中的应用情况。然后基于Atluri等人的工作,采用移动最小二乘近似函数构造试函数,采用Heaviside函数作为加权残值法中的权函数,采用直接插值法施加本质边界条件而不采用罚函数法和拉格朗日乘子法。通过悬臂梁和无限开孔板两个算例验证了本文方法较传统的无网格局部Petrov-Galerkin方法在计算效率上有了较大的提高。尽管无网格方法在断裂力学中的研究已有一系列的成果,但无网格局部Petrov-Galerkin方法在断裂力学问题中的研究很少见到报道。本文的主要工作与创新点是,首次将无网格局部Petrov-Galerkin方法应用于求解断裂力学的相关问题中。在线弹性断裂力学问题中,把线弹性断裂力学应力场的奇异函数作为增强函数加入到移动最小二乘近似函数的基函数中,能够很好的体现裂纹尖端应力场的r1的奇异性,采用可视性准则和衍射法来处理裂纹的不连续性,通过各种裂纹板的算例,计算了裂纹尖端的应力场、应力强度因子和扩展轨迹等;在弹塑性断裂力学问题中,采用了增量牛顿-拉夫逊迭代法和切向预测径向返回子增量法求解增量形的非线性局部Petrov-Galerkin方程,计算了双边裂纹板和三点弯曲试件的裂纹尖端附近塑性区范围和应力强度因子等;对于动态断裂力学问题,在空间域上采用无网格局部Petrov-Galerkin方法离散,在时间域上采用Newmark隐式算法离散,最后计算了含中心裂纹板和双边开口板在冲击载荷作用下动态应力强度因子、裂纹(缺口)尖端附近应力场和变形情况;对于断裂力学的功能梯度材料问题,由于功能梯度材料的弹性矩阵不再是常数,而是空间坐标的函数,传统的J积分不再有效,为了方便得到应力强度因子,推导了均匀辅助场和非均匀辅助场下的M积分,并计算了结构在不同载荷作用下,裂纹处于不同位置时的应力强度因子。无论在线弹性断裂力学问题,弹塑性断裂力学问题中,还是在动态断裂力学问题和功能梯度材料断裂力学问题中,所有数值算例结果都表明,本文方法对于断裂力学问题的求解是可行的和有效的,并且所得到的结果具有较好的精度和收敛性以及较快的收敛速度。更多还原

【Abstract】 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.更多还原