有限覆盖Kriging插值无网格法及其在岩体断裂中的应用研究
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摘要
岩体是经过漫长的时间而形成的，是经历过变形、遭受过破坏的复杂地质体。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。岩体的非连续性与非均匀性越来越受到工程界与学术界的重视，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。
岩体问题的研究方法主要分为三类，即：理论分析方法、试验方法及数值模拟方法。理论分析法计算结果准，计算效率高，但适用范围窄。试验方法能提供大量有用的数据，但受时间和空间的限制，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。本文就采用无网格法对岩体中的一些问题进行模拟分析计算，对岩体中裂纹的起裂、扩展、变形与破坏规律进行了研究。
岩石在变形、破坏过程中的非连续变形行为计算与数值模拟是岩体力学与工程领域中一个比较热门的前沿课题。流形方法解决了材料连续与非连续性的数学统一表述的问题，使得连续变形分析与非连续变形分析的统一成为可能。但流形方法的双重网格可谓一把双刃剑，一方面它构成了流形方法本身的一大特色，另一方面却不可避免地带来了前处理上的麻烦与裂纹开裂扩展模拟方面的困难。无网格主要是以解决前处理的问题而发展起来的一种方法，该法在岩土力学中的非连续变形问题具有相当的局限性。为了模拟岩体中不连续的扩展情况，克服数值流形方法在裂纹扩展过程中物理覆盖重构的缺点，本文在数值流形方法和Kriging插值技术的基础上，提出了有限覆盖Kriging插值无网格法，其最大特点是构造的位移试函数具有插值特性，简化了位移边界条件的处理。有限覆盖Kriging插值无网格法在无网格类方法中引入了有限覆盖技术，避免了无网格法在构造插值函数时由于不连续引起试函数构造所带来的困难，不连续附近节点的布置更加自由，因而特别适合模拟裂纹扩展等非连续问题，是一种可以在统一数学逼近空间形式下处理连续与非连续问题的无网格法。
岩体由岩块和结构面共同组成的结构体，其破坏往往从岩体中原有结构面开始，所以岩体的稳定性很大程度上取决于不连续面的特性。岩体断裂力学从岩体结构特点出发，利用断裂力学理论来诠释岩体力学特性，它将岩体中的节理、裂隙模拟成裂纹。因此，岩体不再是一种连续均质体，而是一种含有众多裂纹的裂纹体。应用岩体断裂力学方法，结合数值计算方法，能够追踪岩体中裂纹的起裂、扩展到相互贯通使岩体局部失稳破坏的过程。
本文简要介绍了静态断裂力学的基本观点和基本理论，应力强度因子的求解方法、裂纹开裂的判断准则等，并将有限覆盖Kriging插值无网格法应用于裂纹的稳定性判别、裂纹扩展路径的预测、多裂纹贯通以及裂纹从单向载荷作用到双向载荷作用变化过程中裂纹扩展路径变化规律，得出一些有益的结论，很好地再现了裂纹尖端的奇异应力场，数值模拟结果与试验结果吻合较好，对控制裂纹的扩展具有重要的意义。数值算例充分表明了有限覆盖Kriging插值无网格法在裂纹稳定性判别以及扩展模拟方面的优越性。
混凝土是一种非均质各向异性人工合成材料，本文在断裂力学和损伤力学的基础上，假设混凝土、岩石类材料物理参数服从Weibull分布，实现了无网格框架下模拟分析岩石混凝土等非均质、非连续材料的裂纹扩展数值模拟，通过对比试验结果及以往数值模拟结果，证明本文结果良好，为该类材料的数值模拟计算提供了一条新的途径。
裂纹尖端附近应力场是奇异的，为了捕捉裂纹尖端应力场的奇异特性，更好的模拟裂纹扩展问题，本文对基函数进行了适当的扩展，提出了强化的有限覆盖Kriging插值无网格法，增强了本文方法求解裂纹等非连续问题的能力，同时也提高了求解裂纹问题的精度，最后通过数值算例证明本方法的正确性和有效性。
本文对有限覆盖Kriging插值无网格法进行了一个全面的介绍，并应用该法对裂纹的扩展进行了分析计算，讨论了有限覆盖Kriging插值无网格法在模拟裂纹扩展中的应用，无网格类方法的进一步应用于研究裂纹扩展规律都具有积极的作用和意义。最后，对研究工作进行了全面总结，并对有待进一步研究的问题进行了讨论。
As a type of geomaterials naturally formed, rock mass is usually composed of manynatural joints and cracks or other types of intact discontinuities due to long-term deformationand damage. When loaded, cracks will be initiated and expand in the rock mass and the thenmechanical properties of rock mass will change. The actual deformation and failure process ofgeo-materials and geo-structures is a complex progressive evolution involving initially elasticdeformation, crack propagation, large-scale displacement and even movement of a discretesystem. The study on the crack expansion is very important for the evaluation of damage andstability of rock masses.
At present, the main techniques applied to investigating rock mass problems areexperimental method and numerical analysis method. Through numerical analyses, a largenumber of usable data can be obtained. Howerer, restricting because of time and space, it isnot 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 expansionof rock mass is analyzed and computed with Kriging interpolation meshless based on finitecovers technique.
Discontinuous deformation calculating and numerical modelling of the process for rockdeforming and failure are blue topic in rock mass mechanics and engineering region. Byvirtue of the finite cover technique of manifold, manifold method integrates conventionalfinite element methods, discontinuous deformation analysis and analytical methods in a unitedmathematical framework and can deal with both continuous and discontinuous deformationproblems such as contact and multi-body interaction. Howerer, the action of dual-mesh inMM is both sides. On the one hand, it is the feature of MM, on the other hand, it bringstroubles in pre-process and difficulties in simulating crack growth. Meshless method based onmoving least square method is very effective in simulating crack propagation which is a keyissue in modeling failure or/and damage behavior of structures or materials without meshingas required in MM. The kernel of manifold method is finite cover technique while the mainfeature of meshless/mesh-free approximation is that only nodes are need in interpolation anddoes not need to join nodes into any element. Manifold method and meshless method haveown advantages in handling discontinuous deformation problems.
To simulatet the discontinuous grwth in the fractured rockmass, Kriging interpolationmeshless method base on finite covers technique and Kriging interpolaton method is presented. The method can conquer the disadvantages of the remeshtechnique treating withthe discontinuous growth problems in the manifold method. The finite cover technology isapplied in the proposed method. And some difficulites in the meshless methods, such as thetrial function due to the discontinuity in the displacement, are avoided and nodal arrangementis morefree near the discontinuous. The Kriging interpolation meshless method is well suitedto problems involving crack propagation due to the absence of any predefined manifoldelement connectivity. The main purpose of the paper is to explore the possibility to work out anew numerical method by combining the finite-cover technique and meshless concepttogether. Presented method is a meshless method, which can treat with continuous anddiscontinuous problems in a uniform mathematic approach space.
Rock mass is composed of rock piece and structure face, and its failure always begins atthe discontinuous surface, so the stability of rock mass depends on the characters of structuresurface in the rock. Fracture mechanics of rock mass explain the mechanical characteristicnby the theory of fracture mechanics, and the joint and slit in rock are simplified crack. Hence,rock mass is discontinuous and anisotropy. The crack initiation, propagation and cuttingthrough until the local failure of rock can be simulated by fracture mechanics of rock massand numerical method.
The fundamental principle of static fracture mechanics is stated and stress intensityfactor (SIF) is calculated and the criterion of crack growth is presented. The method isnumerically implemented and numerical analyses for a number of benchmark problems aremade. The Kriging interpolation meshiess method is demonstrated comprehensively in thisthesis. Then, the possibilities of applying the Kriging interpolation meshless method tonumerical 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 usedfor analysis, in which the growth rule of crack is mainly taken into account by staticequilibrium condition: The simulated results by the proposed method have a good aggrementwith the test and other numerical method. Concrete is a kind of heterogeneous, anisotropic artificial composite material. Based onfracture mechanics and damage mechanics, and supposing that physics parameter of rock andconcrete complying Weibull distribution. Numerical simulation of crack propagation isimplemented under meshless frame for heterogeneous and discontinuous material, such asrock or concrete. By comparing with experimental results and other numerical results, theresults have a good agreement with known solution. A new approach for analyzing suchmaterial is offered in this thesis.
The stress field in vicinity of Crack is singularity field. To capture the singularity of crackand make the crack problems can be simulated better, the base function is proper extendedwith special functions and enriched Kriging interpolation meshless method based on finitecover technique is proposed, the capability to solve discontinuous problem is enhanced andthe precision of exploring crack problem is increased. The validity and accuracy of this method are illustrated by numerical examples.
In summary, a numerical method with a computer code based on the proposed methodand fracture theory is developed for assessing the fracture behavior and predicting initiation,development and arresting of crack in rock masses. Finally, a comprehensive summary isgiven and some issues for further studies are discussed.
岩体问题的研究方法主要分为三类，即：理论分析方法、试验方法及数值模拟方法。理论分析法计算结果准，计算效率高，但适用范围窄。试验方法能提供大量有用的数据，但受时间和空间的限制，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。本文就采用无网格法对岩体中的一些问题进行模拟分析计算，对岩体中裂纹的起裂、扩展、变形与破坏规律进行了研究。
岩石在变形、破坏过程中的非连续变形行为计算与数值模拟是岩体力学与工程领域中一个比较热门的前沿课题。流形方法解决了材料连续与非连续性的数学统一表述的问题，使得连续变形分析与非连续变形分析的统一成为可能。但流形方法的双重网格可谓一把双刃剑，一方面它构成了流形方法本身的一大特色，另一方面却不可避免地带来了前处理上的麻烦与裂纹开裂扩展模拟方面的困难。无网格主要是以解决前处理的问题而发展起来的一种方法，该法在岩土力学中的非连续变形问题具有相当的局限性。为了模拟岩体中不连续的扩展情况，克服数值流形方法在裂纹扩展过程中物理覆盖重构的缺点，本文在数值流形方法和Kriging插值技术的基础上，提出了有限覆盖Kriging插值无网格法，其最大特点是构造的位移试函数具有插值特性，简化了位移边界条件的处理。有限覆盖Kriging插值无网格法在无网格类方法中引入了有限覆盖技术，避免了无网格法在构造插值函数时由于不连续引起试函数构造所带来的困难，不连续附近节点的布置更加自由，因而特别适合模拟裂纹扩展等非连续问题，是一种可以在统一数学逼近空间形式下处理连续与非连续问题的无网格法。
岩体由岩块和结构面共同组成的结构体，其破坏往往从岩体中原有结构面开始，所以岩体的稳定性很大程度上取决于不连续面的特性。岩体断裂力学从岩体结构特点出发，利用断裂力学理论来诠释岩体力学特性，它将岩体中的节理、裂隙模拟成裂纹。因此，岩体不再是一种连续均质体，而是一种含有众多裂纹的裂纹体。应用岩体断裂力学方法，结合数值计算方法，能够追踪岩体中裂纹的起裂、扩展到相互贯通使岩体局部失稳破坏的过程。
本文简要介绍了静态断裂力学的基本观点和基本理论，应力强度因子的求解方法、裂纹开裂的判断准则等，并将有限覆盖Kriging插值无网格法应用于裂纹的稳定性判别、裂纹扩展路径的预测、多裂纹贯通以及裂纹从单向载荷作用到双向载荷作用变化过程中裂纹扩展路径变化规律，得出一些有益的结论，很好地再现了裂纹尖端的奇异应力场，数值模拟结果与试验结果吻合较好，对控制裂纹的扩展具有重要的意义。数值算例充分表明了有限覆盖Kriging插值无网格法在裂纹稳定性判别以及扩展模拟方面的优越性。
混凝土是一种非均质各向异性人工合成材料，本文在断裂力学和损伤力学的基础上，假设混凝土、岩石类材料物理参数服从Weibull分布，实现了无网格框架下模拟分析岩石混凝土等非均质、非连续材料的裂纹扩展数值模拟，通过对比试验结果及以往数值模拟结果，证明本文结果良好，为该类材料的数值模拟计算提供了一条新的途径。
裂纹尖端附近应力场是奇异的，为了捕捉裂纹尖端应力场的奇异特性，更好的模拟裂纹扩展问题，本文对基函数进行了适当的扩展，提出了强化的有限覆盖Kriging插值无网格法，增强了本文方法求解裂纹等非连续问题的能力，同时也提高了求解裂纹问题的精度，最后通过数值算例证明本方法的正确性和有效性。
本文对有限覆盖Kriging插值无网格法进行了一个全面的介绍，并应用该法对裂纹的扩展进行了分析计算，讨论了有限覆盖Kriging插值无网格法在模拟裂纹扩展中的应用，无网格类方法的进一步应用于研究裂纹扩展规律都具有积极的作用和意义。最后，对研究工作进行了全面总结，并对有待进一步研究的问题进行了讨论。
As a type of geomaterials naturally formed, rock mass is usually composed of manynatural joints and cracks or other types of intact discontinuities due to long-term deformationand damage. When loaded, cracks will be initiated and expand in the rock mass and the thenmechanical properties of rock mass will change. The actual deformation and failure process ofgeo-materials and geo-structures is a complex progressive evolution involving initially elasticdeformation, crack propagation, large-scale displacement and even movement of a discretesystem. The study on the crack expansion is very important for the evaluation of damage andstability of rock masses.
At present, the main techniques applied to investigating rock mass problems areexperimental method and numerical analysis method. Through numerical analyses, a largenumber of usable data can be obtained. Howerer, restricting because of time and space, it isnot 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 expansionof rock mass is analyzed and computed with Kriging interpolation meshless based on finitecovers technique.
Discontinuous deformation calculating and numerical modelling of the process for rockdeforming and failure are blue topic in rock mass mechanics and engineering region. Byvirtue of the finite cover technique of manifold, manifold method integrates conventionalfinite element methods, discontinuous deformation analysis and analytical methods in a unitedmathematical framework and can deal with both continuous and discontinuous deformationproblems such as contact and multi-body interaction. Howerer, the action of dual-mesh inMM is both sides. On the one hand, it is the feature of MM, on the other hand, it bringstroubles in pre-process and difficulties in simulating crack growth. Meshless method based onmoving least square method is very effective in simulating crack propagation which is a keyissue in modeling failure or/and damage behavior of structures or materials without meshingas required in MM. The kernel of manifold method is finite cover technique while the mainfeature of meshless/mesh-free approximation is that only nodes are need in interpolation anddoes not need to join nodes into any element. Manifold method and meshless method haveown advantages in handling discontinuous deformation problems.
To simulatet the discontinuous grwth in the fractured rockmass, Kriging interpolationmeshless method base on finite covers technique and Kriging interpolaton method is presented. The method can conquer the disadvantages of the remeshtechnique treating withthe discontinuous growth problems in the manifold method. The finite cover technology isapplied in the proposed method. And some difficulites in the meshless methods, such as thetrial function due to the discontinuity in the displacement, are avoided and nodal arrangementis morefree near the discontinuous. The Kriging interpolation meshless method is well suitedto problems involving crack propagation due to the absence of any predefined manifoldelement connectivity. The main purpose of the paper is to explore the possibility to work out anew numerical method by combining the finite-cover technique and meshless concepttogether. Presented method is a meshless method, which can treat with continuous anddiscontinuous problems in a uniform mathematic approach space.
Rock mass is composed of rock piece and structure face, and its failure always begins atthe discontinuous surface, so the stability of rock mass depends on the characters of structuresurface in the rock. Fracture mechanics of rock mass explain the mechanical characteristicnby the theory of fracture mechanics, and the joint and slit in rock are simplified crack. Hence,rock mass is discontinuous and anisotropy. The crack initiation, propagation and cuttingthrough until the local failure of rock can be simulated by fracture mechanics of rock massand numerical method.
The fundamental principle of static fracture mechanics is stated and stress intensityfactor (SIF) is calculated and the criterion of crack growth is presented. The method isnumerically implemented and numerical analyses for a number of benchmark problems aremade. The Kriging interpolation meshiess method is demonstrated comprehensively in thisthesis. Then, the possibilities of applying the Kriging interpolation meshless method tonumerical 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 usedfor analysis, in which the growth rule of crack is mainly taken into account by staticequilibrium condition: The simulated results by the proposed method have a good aggrementwith the test and other numerical method. Concrete is a kind of heterogeneous, anisotropic artificial composite material. Based onfracture mechanics and damage mechanics, and supposing that physics parameter of rock andconcrete complying Weibull distribution. Numerical simulation of crack propagation isimplemented under meshless frame for heterogeneous and discontinuous material, such asrock or concrete. By comparing with experimental results and other numerical results, theresults have a good agreement with known solution. A new approach for analyzing suchmaterial is offered in this thesis.
The stress field in vicinity of Crack is singularity field. To capture the singularity of crackand make the crack problems can be simulated better, the base function is proper extendedwith special functions and enriched Kriging interpolation meshless method based on finitecover technique is proposed, the capability to solve discontinuous problem is enhanced andthe precision of exploring crack problem is increased. The validity and accuracy of this method are illustrated by numerical examples.
In summary, a numerical method with a computer code based on the proposed methodand fracture theory is developed for assessing the fracture behavior and predicting initiation,development and arresting of crack in rock masses. Finally, a comprehensive summary isgiven and some issues for further studies are discussed.
引文
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