1. Introduction

With the development of mechanical engineering and manufacturing technology, engineering structures applied in aircrafts and huge machines become much more complex. These structures usually bear constantly changing loads in tour of duty. Although the max stress in structure caused by these dynamic loads is much lower than yield limit and ultimate strength of material, structure is destroyed after a long time. Internal defects in engineering structures appear in producing, processing, and assembling process. Internal defects lead to stress concentration, crack initiation, and propagation and even fatigue failure under dynamic load.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

According to statistical data, loss caused by improper structural fatigue lifetime design in America equals 4.4% of gross national product, and 95% of structure failures are related to fatigue break caused by alternating dynamic loads [1]. There are numerous historical examples that result in great loss of human life and economic value. For example, two Comet aircraft crashed in 1954, and the main reason is fatigue of fuselage structure [2]. Mechanical failure caused by fatigue, which concentrates much attention of engineers and researches, has been studied for more than 150 years [3]. However, it is still much difficult to prevent fatigue failure because fatigue of materials is far from being completely comprehended [4].

Experimental method and numerical method are two significant ways to analyze fatigue lifetime of structures. Experimental method has been widely applied since a long time ago. However, it is much expensive to predict structural life via experimental method. Furthermore, it is difficult to execute experiments for some complicated structures. Therefore, numerical method based on empirical models becomes much more popular in structural life prediction, and in some cases, those do not need high accuracy because of less cost and higher efficiency.

Numerical Analysis Methods of Structural Fatigue and Fracture Problems

http://dx.doi.org/10.5772/intechopen.72285

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Approaches to predict fatigue initiation life in literature can be classified into several types. These approaches study the fatigue problem from different perspectives, involving the average or local values of stresses and strains, the initiation of crack and defects, and macro- and microanalysis [7]. Nevertheless, people prefer to use phenomenological models, which reflect general material response at macroscopic scale under cyclic loads, rather than complex micro-

Wöhler is the pioneer in this field, who established the traditional stress-based approach in the nineteenth century [9]. He carried out a few fatigue experiments on metallic materials and indicated the relationship between fatigue crack initiation life and cyclic stress. He proposed to apply S � N curves in description of fatigue behavior of metals in his paper. Effectiveness of this method in high-cycle fatigue analysis is demonstrated afterward by many researchers. There are several kinds of expression of S � N curve, mainly including exponential function expression and power function expression. Basquin was the first person who suggested using exponential function to construct the expression of S � N curve in the twentieth century. The

e

where m and C are constants, which can be determined based on experiment data, N stands for the number of loading cycles, and Smax is the maximum value of stress at specific stress ratio. The power function expression with two parameters is usually expressed in the following

where Sa is the stress amplitude at specific ratio. The power function expression with three

Sm

mSmax<sup>N</sup> <sup>¼</sup> <sup>C</sup> (1)

<sup>a</sup> N ¼ C (2)

ð Þ <sup>S</sup>max � <sup>C</sup> mN <sup>¼</sup> <sup>D</sup> (3)

or mesoscopic model of material fatigue behavior in structure design [8].

typical exponential function expression is written as follows:

2. Empirical models in fatigue problem

2.1. Empirical models of high-cycle fatigue

form:

or

parameters is expressed as

Metallic materials are widely applied in design of structures and parts in present days; therefore, fatigue of metals is a problem deserving efforts. In fact, the fatigue process is constitutive of crack initiation and crack propagation to total failure, as shown in Figure 1, and fatigue lifetime should conclude crack initiation life and crack propagation life.

On one hand, it is widely accepted that the crack initiation phase costs a majority of fatigue lifetime in a high-cycle fatigue regime [5]. Furthermore, crack initiation behavior has a great influence on crack growth prediction in a unified approach for fatigue lifetime prediction [6]. Therefore, knowledge and technology of crack initiation life prediction are significant for evaluation of fatigue lifetime of structures and deserve our efforts to study deeply. On the other hand, there are frequently small cracks and defects in engineering structures due to manufacturing and environment factors; therefore, fatigue crack propagation prediction plays an important role in estimating the structural safety under dynamic loads.

Therefore, people divide structural life prediction problem into two problems: fatigue problem and fracture problem. People pay attention to crack initiation life in fatigue problem and make efforts to construct the relationship between structure life and stress or strain in structure. It is assumed in fatigue problem that there is a small crack existing in structure, and crack propagation behavior is studied in order to predict the remaining life of structure. These two problems have aroused widespread concern nowadays.

Figure 1. Schematic illustration of crack length versus time/cycles.

Experimental method and numerical method are two significant ways to analyze fatigue lifetime of structures. Experimental method has been widely applied since a long time ago. However, it is much expensive to predict structural life via experimental method. Furthermore, it is difficult to execute experiments for some complicated structures. Therefore, numerical method based on empirical models becomes much more popular in structural life prediction, and in some cases, those do not need high accuracy because of less cost and higher efficiency.
