## Abstract

Fatigue and fracture problems, which lead to 95% of structural failure, have attracted much attention of engineers and researchers all over the world. Compared with experimental method, numerical simulation method based on empirical models shows its remarkable advantages in structure design because of less cost and higher efficiency. However, the application of numerical simulation method in fatigue lifetime prediction is restricted by low accuracy and poor applicability in some circumstances. Most numerical method is based on empirical models. This chapter first reviews various kinds of empirical models of fatigue and fracture problems, including some modifying methods of basic empirical models, which have been widely applied to fatigue lifetime prediction and indicated their advantages and disadvantages. Then, FEM is introduced as an important method to obtain stress intensity factor or crack growth route. At last, this chapter is finished with existing problems and current trends in fatigue lifetime prediction via numerical method.

### Keywords

- fatigue lifetime prediction
- crack propagation
- numerical method
- empirical model
- Paris law
- perturbation approach
- extended finite element method
- fractal geometry

## 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. 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].

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.

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.

## 2. Empirical models in fatigue problem

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

### 2.1. Empirical models of high-cycle fatigue

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

where

where

or

where

### 2.2. Empirical models of low-cycle fatigue

Stress level is usually high in low-cycle fatigue, and the maximum value of stress is nearly close to the ultimate strength of material. The number of loading cycles in low-cycle fatigue, which is not more than 10^{3} times, is much less than that in high-cycle fatigue. Plastic deformation plays an important role in low-cycle fatigue, in which the accumulation of plastic deformation results in structural failure. Because low-cycle fatigue lifetime is much sensitive to the change of stress level,

where

in which

### 2.3. Improved models considering mean stress or stress ratio

There are many factors, such as residual stress, temperature, multiaxial stress, and geometrical feature, that influence structural fatigue lifetime, in which mean stress or stress ratio concentrates the most attention.

#### 2.3.1. Walker formula

Mean stress and stress ratio are of great significance for structural fatigue lifetime. Walker formula considers sensitivity of different materials to mean stress; therefore, it shows well effect for all materials [13, 14]. An equivalent local strain parameter is defined in Walker formula; its expression is

where

#### 2.3.2. Morrow’s modifying method

Morrow’s modifying method and SWT modifying method are two commonly used methods. Morrow mean stress modifying formula is shown as follows [16]:

Considering the greater influence made by mean stress in long life period, further modifying method is given:

where

#### 2.3.3. SWT modifying method

Expression of Smith-Watson-Topper (SWT) parameter modifying method is [17]

where

SWT mean stress modifying method is not valid for compression mean stress, and it will obtain too conservative result when the stretching mean stress is large.

#### 2.3.4. Goodman’s modifying method

We can acquire the fatigue limit points of material at different stress ratio

where

## 3. Empirical models in fracture problem

### 3.1. Paris law

Paris et al. [19] made great contribution in this field who was pioneer suggesting that crack growth rate,

where

It is believed that the relationship between crack propagation and

### 3.2. Improved models

#### 3.2.1. Models considering mean stress or stress ratio

Since Paris law is proposed, much related work is done, and many modifying methods are put forward [22, 25, 26, 27]. It is commonly accepted that crack growth rate of material is related to mean stress or stress ratio. Several models, in which Forman formula [28] and Walker formula [29] are most famous, take this factor into consideration. Forman formula also considers the fracture toughness as an important factor; its expression is

Forman formula is valid for dealing with experimental data of many kinds of materials, especially high-hardness alloy, but it is hard to obtain the fracture toughness

Forman formula can be transformed as follows:

Forman formula explains the reason why crack growth enlarges sharply when stress intensity factor is close to fracture toughness.

Walker formula is another wide-applied crack propagation model in engineering, which expresses the influence made by stress ratio on crack growth rate. Furthermore, it takes maximum of stress intensity factor into consideration:

Three parameters

#### 3.2.2. Model based on crack closure theory

In 1971, Elber [30] found that crack opened completely only when the stress was larger than a certain value, and he developed a modified Paris law based on this theory. The stress when crack is completely open is defined as crack opening stress

and

where efficient stress amplitude

#### 3.2.3. Model considering crack retardation caused by high load

In Weeler’s opinion [31], when structure bears cyclic load with constant amplitude; an occasional overload enlarges the size of plastic zone on crack tip, which would prevent crack from growing to some degree. On the basis of Weeler’s research, Willenberg [32] assumed that crack retardation is due to residual compression stress

The effective stress intensity factor range is

and the effective stress ratio is

The maximum and minimum values of effective cyclic stress are

Then, crack growth rate in retardation period can be estimated as the residual stress

#### 3.2.4. Model considering crack propagation threshold

In 1972, Donahue [33] took threshold of stress intensity factor range

The following expression was proposed by McEvily and Greoeger [34] in their research about fatigue crack propagation threshold in 1977:

in which material constant

Furthermore, if considering stress ratio at the same time, Paris law can be modified into the following expression:

It can be figured out that the above equation is further modified on the basis of Forman formula.

In 1999, McEvily found it out that the following modification is suitable for many alloys’ fatigue crack propagation:

where

#### 3.2.5. Model based on perturbation series expansion method

Perturbation series expansion method, which is a common method to deal with nonlinear problems, has been widely used in fluid mechanics, structure dynamics, and damage identification. In this method, the parameter in ideal model is regarded to have a small perturbation in order to study the properties of system. This parameter can be expanded into series form:

where

Qiu and Zheng [35] proposed a novel numerical calculation method to investigate the fatigue crack growth evolution in aluminum alloy sheets accounting for the measurement error. The initial crack length is considered as a modified parameter with a small correction term due to the measurement error; the solution to the crack growth equation is expressed in the form of a perturbation series, and a series of modified equations for predicting the crack length history are derived. The proposed method is verified to be indeed feasible and effective for predicting fatigue crack growth evolution by comparing numerical results with experimental data, as shown in Figure 6.

## 4. Finite element method

There are many kinds of numerical method to obtain stress intensity factor or crack growth route after continuous study of many researchers. Finite difference method (FDM), boundary element method (BEM), mesh-less method, and finite element method (FEM) are four common methods. Many studies have been carried out based on these numerical methods: Christen applied FEM to two-dimensional crack problem and obtained the displacement field and stress field; Nayroles [36] combined the moving least square method (MLSM) with mesh-less method to solve boundary problem. FEM is the most widely used method in above four methods at present [37, 38]. Considering singularity on crack tip, element’s density is increased in order to obtain the precious results. Therefore, FEM’s rate of convergence is low, and precision is unsatisfactory. People developed precious numerical solution methods based on several kinds of theories, in which semi-analytic numerical solution and new type elements are hot issues.

### 4.1. Extended finite element method

Collapsed singular isoparametric elements, which can reflect the singularity on crack tip correctly, were introduced by Barsoum [39]. This method is popular because of its high precision and executing simplicity. In this method, planar eight-node isoparametric element is degenerated into singular isoparametric element, as shown in Figure 7. Stress intensity factor is calculated based on the displacements of nodes A and B; the expression is.

In plane stress problem,

Lin [41] proposed the 1/4 node displacement method, as shown in Figure 8; the corresponding calculation equation of stress intensity factor is

Belytschko [42] applied extended finite element method (XFEM) to calculating stress intensity factor and neglected the high-order terms of asymptotic displacement function. The calculation results were not satisfying enough. Karihaloo and Xiao [43] took high-order terms of asymptotic displacement function and outer elements of crack tip into consideration, thus obtaining results of high accuracy. However, calculation efficiency of this method is relatively low. Although researchers have obtained precious results with the help of new type elements, there are still many factors that influence calculation results that need to be studied.

### 4.2. Fractal finite element method

In the aspect of semi-analytic numerical method, weighted function method and boundary collocation method develop fast. These methods are able to acquire results of high accuracy when dealing with particular models; however, calculation accuracy cannot be guaranteed when dealing with general models.

Fractal finite element method is also a semi-analytic method. Fractal geometry is introduced into ordinary FEM, which not only improves calculation accuracy but also shortens calculation time and saves storage capacity of a computer. In fractal finite element method, an artificial boundary

Self-similar mesh is shown in Figure 10. In singular field, infinite curves

## 5. Conclusion

This chapter reviews the most common empirical models and numerical methods of structural fatigue lifetime prediction. The main advantages and disadvantages of these methods are discussed.

Numerical method based on empirical models, as one of significant ways to analyze structural fatigue life, becomes popular in structural life prediction nowadays because of less cost and higher efficiency.

Paris law is the most significant model of crack propagation problem. But it only considers the stress intensity factor as the factors make influences on crack propagation. Many improved models considering stress ratio, crack closure, crack retardation, and crack propagation threshold have been put forward.

FEM is the most popular numerical method to obtain stress intensity factor or crack growth route. Extended finite element method and fractal finite element method are two mainly developing trends of FEM. However, it is still difficult to achieve high efficiency and accuracy of numerical method at the same time.

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