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## Über dieses Buch

Fracture mechanics has established itself as an important discipline of growing interest to those working to assess the safety, reliability and service life of engineering structures and materials. In order to calculate the loading situation at cracks and defects, nowadays numerical techniques like finite element method (FEM) have become indispensable tools for a broad range of applications. The present monograph provides an introduction to the essential concepts of fracture mechanics, its main goal being to procure the special techniques for FEM analysis of crack problems, which have to date only been mastered by experts. All kinds of static, dynamic and fatigue fracture problems are treated in two- and three-dimensional elastic and plastic structural components. The usage of the various solution techniques is demonstrated by means of sample problems selected from practical engineering case studies. The primary target group includes graduate students, researchers in academia and engineers in practice.

## Inhaltsverzeichnis

### Chapter 1. Introduction

The term>>fracture<< describes the local detachment of material cohesion in a solid body. It concerns a process that either partially disrupts the body which leads to the development of incipient cracks or entirely destroys it.
Meinhard Kuna

### Chapter 2. Classification of Fracture Processes

Fracture processes are classified based on quite different individual aspects. The reason for that is the tremendous variety in which fracture processes appear and the diverse reasons leading to failure. First and foremost, a fracture depends on the properties of the considered material because the damage processes happening on a micro-structural level in the material determine its characteristic behavior. These microscopic structures and failure mechanisms vary diversely in the lineup of engineering materials. Just as important for fracture behavior is the type of external loading of the component. In this category one can differentiate between e.g. fractures due to static, dynamic or cyclic loading. Further important factors are the temperature, the multiaxiality of the loading, the rate of deformation and the chemical or environmental conditions.
Meinhard Kuna

### Chapter 3. Basics of Fracture Mechanics

The theoretical foundations of fracture mechanics will be presented in this chapter. The main focus lies on the description of the available continuum-mechanical solutions for cracks. On the basis of stress and deformation situations determined this way, suitable parameters, which clearly describe the loading states during fractures, are then selected. These loading and fracture parameters shape the foundation for the formulation of fracture criteria. With their help, the behavior of cracks can be quantitatively evaluated. These usually closed mathematical solutions are the preconditions to being able to calculate the sizes of cracks with numerical methods later on. Naturally, the experimental test methods of fracture mechanics used to evaluate the material parameters are based on the understanding of the loading situation as well.
Meinhard Kuna

### Chapter 4. Finite Element Method

The finite element method (FEM) is currently one of the most efficient and universal methods of numerical calculation for solving partial differential equations from engineering and scientific fields. The basic mathematical concepts are based on the work of Ritz, Galerkin, Trefftz and others at the beginning of the twentieth century. With the advance of modern computer science in the 1960s, these approaches of numerical solution could be successfully implemented with FEM. This development was motivated to an enormous extent by tasks of structural analyses in aviation, construction and mechanical engineering. The formulation of the finite element method in its current standard was developed thanks to the pioneer work of (among others) Argyris, Zienkiewicz, Turner, and Wilson. Therein, the system of differential equations is converted into an equivalent variational problem (weak formulation), mostly utilizing mechanical principles or weighted residual methods.
Meinhard Kuna

### Chapter 5. FE-Techniques for Crack Analysis in Linear-Elastic Structures

The goal of a FEM analysis is the calculation of fracture-mechanical loading parameters for a crack in a structure (test piece, component, material’s microstructure) in the case of linear-elastic (isotropic or anisotropic) material behavior. In Sect. 3.2 the relevant loading parameters of LEFM were introduced: the stress intensity factors $$K_\mathrm{{I}}$$, $$K_\mathrm{{II}}$$, $$K_\mathrm{{III}}$$ and the energy release rate $$G \equiv J$$. Their values depend on the geometry of the structure, its load, the length and shape of the crack and on the material’s elastic properties.
Although FEM can be directly applied to solve a BVP, its use in crack problems involves a fundamental difficulty. This difficulty lies in the exact determination of the singularity at the crack tip with the help of a numerical approximation method such as FEM. Conventional finite element types only have regular polynomial functions for $$u_i$$, $$\varepsilon _{ij}$$ and $$\sigma _{ij}$$. Therefore, they reproduce the crack singularity poorly. For this reason, special element functions, numerical algorithms or evaluation techniques are needed to obtain loading parameters from a FEM solution efficiently and accurately. In the following chapter, we will introduce the methods that have been developed for this, concentrating mainly on stationary cracks. The particularities of FEM techniques and meshes in analyzing unsteady cracks will be dealt with in Chap. 8.
Meinhard Kuna

### Chapter 6. Numerical Calculation of Generalized Energy Balance Integrals

Based on Eshelby’s pioneer work [1, 2], who investigated thermodynamic forces acting on defects in solids by introducing the energy-momentum tensor, a new theory of generalized $$\tiny {\gg }$$material$$\tiny {\ll }$$ or $$\tiny {\gg }$$configurational$$\tiny {\ll }$$ forces has been developed in the past 15 years (see Maugin [3], Kienzler, Herrmann [4,5], and Gurtin [6]).
Meinhard Kuna

### Chapter 7. FE-Techniques for Crack Analysis in Elastic-Plastic Structures

FEM has become an indispensable tool for the stress analysis of crack configurations in elastic-plastic materials, as the physically and sometimes geometrically non-linear IBVPs in finite structures are not solvable with analytical methods. To model the material behavior, predominantly the incremental laws of plasticity with various hardening types introduced in Sect. A.4.2 come in to question. Here too, the goal of the computations is to determine the fracture-mechanical loading parameters for ductile crack initiation and crack propagation.
Meinhard Kuna

### Chapter 8. Numerical Simulation of Crack Propagation

Prediction of the crack propagation process is of great importance for many fracture mechanical issues.
Meinhard Kuna

### Chapter 9. Practical Applications

Bainitic cast iron with nodular graphite (Austempered Ductile Iron ADI) shows a good ductility, a superior wear resistance and a high fatigue strength, which makes it an interesting alternative to steel for producing railway wheels. However, ADI possesses a lower fracture toughness and is, due to the casting process, more prone to defects. A railway wheel is exposed to high static and cyclic loading.
Meinhard Kuna

### Backmatter

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