摘要
岩体是经过漫长的时间而形成的，是经历过变形、遭受过破坏的地质体。岩体在形成过程中会在内部产生大量的节理和裂隙，这些节理和裂隙对岩体的性质有着极大的影响，在外载荷的作用下还会发生扩展，从而导致岩体的力学性质发生改变，甚至会引起岩体的破坏以及结构的失稳。因此，研究岩体中裂纹的扩展规律具有非常重要的理论与实际意义。
研究岩体的方法主要有两种：试验方法和数值计算方法。试验方法能提供大量有用的数据，但应用极不方便，且要耗费大量的人力、财力。数值计算方法能很方便、快速地进行岩体的模拟分析，因此这种方法发展很快。本文就采用数值流形方法对岩体中的一些问题进行分析计算，对岩体中裂纹的起裂、扩展规律进行了研究。
近年来，岩土力学中的数值分析方法发展迅速。其中尤以数值流形方法最为突出，以其新颖的数值思想、先进的数值技术得到了学术界的广泛关注。数值流形方法是20世纪90年代由美国华侨石根华博士提出来的，是一种新兴的数值计算方法，它基于数学上的拓扑流形和微分流形，吸收了传统的有限元技术和非连续变形分析(DDA)两方面的优点，应用流形的有限覆盖技术统一了连续与非连续性问题的数学描述，因此能够更适合处理非连续性问题。
数值流形方法能很方便地处理非连续问题，因此本文利用数值流形方法来处理断裂力学中的裂纹问题。本文系统阐述了数值流形方法的基本思想，介绍了数值流形方法的一般原理以及全一阶多项式覆盖函数与全二阶覆盖函数两种形式的高阶流形方法的数学列式，编制了零阶、全一阶及全二阶多项式覆盖函数的数值流形方法计算程序，应用程序对裂纹的起裂及扩展进行了数值分析。计算结果表明采用高阶形式的覆盖函数能有效地提高数值流形方法的计算精度。
应用数值流形方法进行数值分析计算时，其前处理网格的剖分一直是比较复杂的问题。现阶段还没有很好的数值流形方法前处理程序，本文结合有限单元法的网格自动剖分技术和面向对象的程序设计方法发展了一套适用于数值流形方法的网格剖分技术，应用这种方法可以快速地自动剖分网格，完成数值流形方法计算分析的前处理工作，在计算过程中不需要重新剖分网格，极大地减少了计算工作量，使数值流形方法能够真正地应用于工程计算。
断裂力学中一直存在两种观点，一种是静态断裂力学观点，主要是从静态平衡来考虑裂纹的扩展规律，忽略裂纹扩展速度的影响。另一种是动态断裂力学观点，从动态理论来考虑裂纹的扩展，引入了裂纹扩展速度，考虑裂纹扩展速度对裂纹扩展过程的影响。本文应用数值流形方法对静态断裂力学和动态断裂力学两
个方面都进行了计算分析。首先介绍了静态断裂力学的基本观点和基本理论、应
力强度因子的计算方法、裂纹开裂的判断准则等，并编制了相应的计算程序，对
裂纹的扩展规律进行了分析计算。
从动态断裂力学观点分析裂纹扩展时考虑了裂纹扩展速度的因素。裂纹扩展
速度对裂纹的应力强度因子以及裂纹的起裂、止裂都有影响。随着裂纹的扩展，
在一定的载荷作用下，裂纹的扩展速度是一个先增大后减小的过程。裂纹扩展速
度的变化影响着裂纹的应力强度因子和断裂韧度的变化，当应力强度因子大于断
裂韧度时，裂纹继续扩展;当应力强度因子小于断裂韧度的时，裂纹就停止扩展。
本文通过所编制的数值流形方法计算程序计算了裂纹的扩展速度，并在此基础上
计算裂纹的动态应力强度因子，动态断裂韧度，由动态应力强度因子与动态断裂
韧度动态地进行裂纹的扩展和止裂分析。
本文对数值流形方法进行了一个全面的介绍，并应用数值流形方法对裂纹的
扩展进行了分析计算，讨论了数值流形方法在模拟裂纹扩展中的应用，对数值流
形方法的进一步应用与研究裂纹扩展规律都具有积极的作用与意义。
As a type of geomaterials naturally formed, rock mass is usually composed of many natural joints and cracks or other types of intact discontinuities due to long-term deformation and damage. When loaded, cracks will be initiated and expand in the rock mass and the then mechanical properties of rock mass will change. The study on the crack expansion is very important for the evaluation of damage and stability of rock masses.
The main techniques for investigating rock mass include experimental method and numerical analysis method. Through numerical analyses, a large number of usable data can be obtained. But it is not convenience to apply this method, and it needs much manpower and financial resources. Numerical analyses can analyze problems of rock mass quickly. In this paper, crack expansion of rock mass is analyzed and computed with numerical manifold method.
In recent years, the numerical analysis methods of geomechanics have achieved considerable development. As a new technique, the manifold method, proposed by Genhua, Shi in 1990s is based on the concept of topology manifold and differential manifold in mathematics. This method takes the advantage of both the traditional FEM technique and the discontinuous deformation analysis method (DDA). In this method, the finite-cover technique is used. The finite-covers can unify the mathematical description for both continuous problems and discontinuous problems in geomechanics and hence it can solve the discontinuous problems more rationally. Therefore, the manifold method is combined with fracture mechanics in this thesis to deal with the problems of crack expansion.
The fundamental principle of the manifold method is demonstrated comprehensively in this thesis. The formulations of the high-order manifold methods with complete first order polynomial cover functions and the complete second order polynomial cover functions are presented. Then, the possibilities of applying the manifold method to numerical analyses of fracture behavior and crack expansion of rock masses are discussed. The treatments for the related special issues are given. The static fracture mechanics is used for analysis, in which the growth rule of crack is mainly taken into account by static equilibrium condition. However the velocity of crack growth is overlooked. In order to overcome such a shortcoming, the kinetic fracture mechanics is employed, in which the growth of crack is governed by kinetics and effect of crack growth velocity is considered. In the thesis, manifold method is incorporated with both static and kinematical theories of fracture mechanics to numerical simulation of the fracture beh
avior and initiation and development of cracks.
The general description of the problem is stated and stress intensity factor (SIF) is calculated and the criterion of crack growth is presented. The method is numerically implemented and numerical analyses for a number of benchmark problems are made.
Then, the arresting mechanism of crack upon loading is discussed based on the proposed method. The crack growth velocity is introduced based on kinematical theory of fracture mechanics. The stress intensity factors of crack will be varied with crack development. Considering the effect of crack growth velocity on the stress intensity factors, the stress intensity factors computed by the proposed method are compared with the fracture roughness and the development process of crack can be predicted. The arresting will occur under the condition when the computed stress intensity factor is less than fracture roughness. Therefore the initiation, development and arresting which may be displayed in the process loading of rock masses can be presented by the proposed method.
A special scheme for meshing and pre-treatment is required in applications of the manifold method. Based on the auto-meshing techniques used for finite element method (FEM) and the object oriented programming (OOP) method, a new meshing technique specially adapted for manifold method is developed in this thesis. The scheme is numerically impleme
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