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 microor 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 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 typical exponential function expression is written as follows:

$$e^{m S\_{\text{max}}} N = \mathbb{C} \tag{1}$$

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 form:

$$S\_a^m N = \mathbb{C} \tag{2}$$

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

$$(\mathbf{S}\_{\text{max}} - \mathbf{C})^{\text{wt}} \mathbf{N} = \mathbf{D} \tag{3}$$

or

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

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

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

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

because fatigue of materials is far from being completely comprehended [4].

236 Contact and Fracture Mechanics

lifetime should conclude crack initiation life and crack propagation life.

an important role in estimating the structural safety under dynamic loads.

problems have aroused widespread concern nowadays.

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

$$S\_{\text{max}} = \mathbb{C}\left(1 + \frac{A}{N^{\alpha}}\right) \tag{4}$$

where D, A, and α are constants. The parameter C in Eqs. (3) and (4) nearly equals fatigue limit.

#### 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<sup>3</sup> 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, S � N curve is unable to reflect the low-cycle fatigue performance of material. Therefore, ε � N curve is applied to low-cycle fatigue analysis. The most widely accepted low-cycle fatigue lifetime model based on ε � N curve is proposed by Basquin [10], which is expressed as follows:

$$
\varepsilon\_{\varepsilon} = \frac{\sigma\_a}{E} = \frac{\sigma\_f'}{E} \left(2\text{N}\_f\right)^b \tag{5}
$$

effect for all materials [13, 14]. An equivalent local strain parameter is defined in Walker

r is the material parameter. In order to construct the relationship between Walker formula and fatigue lifetime, Jaske et al. [15] carried out many experiments on different kinds of materials

where A<sup>0</sup> and A<sup>1</sup> are regression coefficients and ε<sup>u</sup> and ε<sup>e</sup> are the upper and lower limits of this reverse hyperbolic tangent function, respectively. The strain-life curve is shown in Figure 3.

E � �<sup>1</sup>�<sup>r</sup>

log εuεe=

log <sup>ε</sup><sup>u</sup> =<sup>ε</sup><sup>e</sup> � �

ε2 eq � �

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(7)

239

(8)

<sup>ε</sup>eq <sup>¼</sup> ð Þ <sup>2</sup>ε<sup>a</sup> <sup>r</sup> <sup>σ</sup>max

and proposed following expression based on experimental data:

log Nf <sup>¼</sup> <sup>A</sup><sup>0</sup> <sup>þ</sup> <sup>A</sup>1tanh�<sup>1</sup>

formula; its expression is

Figure 3. Strain-life curve of Walker formula.

Figure 2. Elastic strain-life curve and plastic strain-life curve.

where ε<sup>e</sup> is the amplitude of elastic strain, E is the elasticity modulus of material, σ<sup>0</sup> <sup>f</sup> is the fatigue strength coefficient of material, and b is the fatigue strength exponent. Because the relationship between plastic strain and fatigue lifetime is not taken into consideration in Basquin formula, Coffin [11] and Manson [12] proposed an empirical model when studying the relationship between fatigue lifetime and plastic strain amplitude. The expression of Coffin-Manson model is.

$$
\varepsilon\_a = \frac{\sigma\_f^{\prime}}{E} \left( 2 \text{N}\_f \right)^b + \varepsilon\_f^{\prime} (2 \text{N})^c \tag{6}
$$

in which ε<sup>a</sup> stands for the amplitude of total strain and ε<sup>0</sup> <sup>f</sup> and c stand for the fatigue ductility coefficient and fatigue ductility exponent separately. The relationships between plastic strain, elastic strain, total strain, and fatigue lifetime are shown in Figure 2.

#### 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

Figure 2. Elastic strain-life curve and plastic strain-life curve.

Smax ¼ C 1 þ

2.2. Empirical models of low-cycle fatigue

which is expressed as follows:

238 Contact and Fracture Mechanics

Coffin-Manson model is.

trates the most attention.

2.3.1. Walker formula

where D, A, and α are constants. The parameter C in Eqs. (3) and (4) nearly equals fatigue limit.

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<sup>3</sup> 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, S � N curve is unable to reflect the low-cycle fatigue performance of material. Therefore, ε � N curve is applied to low-cycle fatigue analysis. The most widely accepted low-cycle fatigue lifetime model based on ε � N curve is proposed by Basquin [10],

<sup>ε</sup><sup>e</sup> <sup>¼</sup> <sup>σ</sup><sup>a</sup>

<sup>ε</sup><sup>a</sup> <sup>¼</sup> <sup>σ</sup><sup>0</sup> f <sup>E</sup> <sup>2</sup>Nf <sup>b</sup> <sup>þ</sup> <sup>ε</sup><sup>0</sup>

in which ε<sup>a</sup> stands for the amplitude of total strain and ε<sup>0</sup>

elastic strain, total strain, and fatigue lifetime are shown in Figure 2.

2.3. Improved models considering mean stress or stress ratio

<sup>E</sup> <sup>¼</sup> <sup>σ</sup><sup>0</sup> f <sup>E</sup> <sup>2</sup>Nf

fatigue strength coefficient of material, and b is the fatigue strength exponent. Because the relationship between plastic strain and fatigue lifetime is not taken into consideration in Basquin formula, Coffin [11] and Manson [12] proposed an empirical model when studying the relationship between fatigue lifetime and plastic strain amplitude. The expression of

coefficient and fatigue ductility exponent separately. The relationships between plastic strain,

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 concen-

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

where ε<sup>e</sup> is the amplitude of elastic strain, E is the elasticity modulus of material, σ<sup>0</sup>

<sup>b</sup> (5)

<sup>f</sup>ð Þ <sup>2</sup><sup>N</sup> <sup>c</sup> (6)

<sup>f</sup> and c stand for the fatigue ductility

A N<sup>α</sup> 

(4)

<sup>f</sup> is the

effect for all materials [13, 14]. An equivalent local strain parameter is defined in Walker formula; its expression is

$$
\varepsilon\_{eq} = (2\varepsilon\_d)^r \left(\frac{\sigma\_{\text{max}}}{E}\right)^{1-r} \tag{7}
$$

r is the material parameter. In order to construct the relationship between Walker formula and fatigue lifetime, Jaske et al. [15] carried out many experiments on different kinds of materials and proposed following expression based on experimental data:

$$\log N\_f = A\_0 + A\_1 \tanh^{-1} \left[ \frac{\log \left( \varepsilon\_{u \, \mathcal{E}\_\ell} / \frac{2}{\varepsilon\_{eq}^2} \right)}{\log \left( \varepsilon\_u / \_{\varepsilon\_\varepsilon} \right)} \right] \tag{8}$$

where A<sup>0</sup> and A<sup>1</sup> are regression coefficients and ε<sup>u</sup> and ε<sup>e</sup> are the upper and lower limits of this reverse hyperbolic tangent function, respectively. The strain-life curve is shown in Figure 3.

Figure 3. Strain-life curve of Walker formula.

There are too many parameters to be fitted in this method, which need plenty of experimental data. That disadvantage constricts badly the application of Walker formula in engineering.

#### 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]:

$$\varepsilon\_{a} = \frac{\sigma\_{f}^{\prime}}{E} \left( 1 - \frac{\sigma\_{m}}{\sigma\_{f}^{\prime}} \right) \left( 2N\_{f} \right)^{b} + \varepsilon\_{f}^{\prime} \left( 1 - \frac{\sigma\_{m}}{\sigma\_{f}^{\prime}} \right) \left( 2N\_{f} \right)^{c} \tag{9}$$

SWT mean stress modifying method is not valid for compression mean stress, and it will

We can acquire the fatigue limit points of material at different stress ratio r ¼ σmin=σmax under infinite life requirement with the support of large amount of experimental data. Draw these points in rectangular coordinate system whose X-axis is mean stress σ<sup>m</sup> ¼ ð Þ σmin þ σmax =2 and Y-axis is stress amplitude σ<sup>a</sup> ¼ ð Þ σmax � σmin =2; thus, the fatigue limit curve is fitted based on these points. It is unpractical to carry out many experiments on all materials and structures in engineering, so we usually use a simplified straight line to replace the fatigue limit curve. Goodman simplified straight line, which is one of these straight lines, is widely accepted due to its simplicity and conservative estimation [18], as shown in Figure 4. Goodman simplified

> σa Se þ σ<sup>m</sup> Su

appropriate for low-ductility material, such as high-strength steel and cast iron.

where Se stands for the fatigue strength of material and Su stands for the ultimate tensile strength of material. However, it has been proved that Goodman modifying method is only

Paris et al. [19] made great contribution in this field who was pioneer suggesting that crack growth rate, da=dN, was a function of the maximum stress intensity factor Kmax in 1961. Then, Liu [20] related the crack growth to the stress intensity factor range ΔK subsequently. Paris and

where C and m can be obtained from experiment data, and they are usually considered as constants for a particular metal and environment [22]. Since then researchers have made efforts to study on Paris law and its deviation; however, we are still far from a complete comprehension [23]. It is believed that the relationship between crack propagation and ΔK can be divided into three distinct regions, as shown in Figure 5. The crack propagation is slow in region A, and concept of a fatigue threshold stress intensity factor range ΔKth is proposed by Mcclintock [24], beneath which cracks are regarded not to grow. In region B, the "mid growth" range, crack propagation is stable, and Paris law is supposed to be held. Region C is associated with fast crack propagation leading to final failure. Therefore, calculation of number of loading cycles in region B, which could be gained from Paris law, is significant for prediction of fatigue crack

Erdogan [21] proposed the well-known Paris law, which can be presented as follows:

da

¼ 1 (13)

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dN <sup>¼</sup> <sup>C</sup>ð Þ <sup>Δ</sup><sup>K</sup> <sup>m</sup> (14)

obtain too conservative result when the stretching mean stress is large.

straight line can be expressed in the following relationship:

3. Empirical models in fracture problem

3.1. Paris law

growth life.

2.3.4. Goodman's modifying method

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

$$\varepsilon\_{a} = \frac{\sigma\_{f}^{\prime}}{E} \left( 1 - \frac{\sigma\_{m}}{\sigma\_{f}^{\prime}} \right) \left( 2\text{N}\_{f} \right)^{b} + \varepsilon\_{f}^{\prime} \left( 2\text{N}\_{f} \right)^{c} \tag{10}$$

where ε<sup>a</sup> is strain amplitude and σ<sup>m</sup> is mean stress. Morrow's modifying method aims at elastic strain; therefore, it is only suitable when stress amplitude is constant or mean stress is compression stress.

#### 2.3.3. SWT modifying method

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

$$
\sigma\_{\text{max}} \varepsilon\_d = \frac{\sigma\_f'^2}{E} \left( 2 \text{N}\_f \right)^{2b} + \sigma\_f' \left( 2 \text{N}\_f \right)^{b+c} \tag{11}
$$

where

$$
\sigma\_{\text{max}} = \sigma\_m + \sigma\_a \tag{12}
$$

Figure 4. Fatigue limit curve and Goodman simplified straight line.

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

There are too many parameters to be fitted in this method, which need plenty of experimental data. That disadvantage constricts badly the application of Walker formula in engineering.

Morrow's modifying method and SWT modifying method are two commonly used methods.

Considering the greater influence made by mean stress in long life period, further modifying

where ε<sup>a</sup> is strain amplitude and σ<sup>m</sup> is mean stress. Morrow's modifying method aims at elastic strain; therefore, it is only suitable when stress amplitude is constant or mean stress is com-

� �<sup>2</sup><sup>b</sup> <sup>þ</sup> <sup>σ</sup><sup>0</sup>

<sup>f</sup> 2Nf

2Nf � �<sup>b</sup> <sup>þ</sup> <sup>ε</sup><sup>0</sup>

<sup>f</sup> <sup>1</sup> � <sup>σ</sup><sup>m</sup> σ0 f

!

<sup>f</sup> 2Nf

2Nf

� �<sup>c</sup> (9)

� �<sup>c</sup> (10)

� �<sup>b</sup>þ<sup>c</sup> (11)

σmax ¼ σ<sup>m</sup> þ σ<sup>a</sup> (12)

2Nf � �<sup>b</sup> <sup>þ</sup> <sup>ε</sup><sup>0</sup>

!

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

<sup>σ</sup>maxε<sup>a</sup> <sup>¼</sup> <sup>σ</sup><sup>0</sup> <sup>2</sup> f <sup>E</sup> <sup>2</sup>Nf

Figure 4. Fatigue limit curve and Goodman simplified straight line.

Morrow mean stress modifying formula is shown as follows [16]:

<sup>ε</sup><sup>a</sup> <sup>¼</sup> <sup>σ</sup><sup>0</sup> f <sup>E</sup> <sup>1</sup> � <sup>σ</sup><sup>m</sup> σ0 f

!

<sup>ε</sup><sup>a</sup> <sup>¼</sup> <sup>σ</sup><sup>0</sup> f <sup>E</sup> <sup>1</sup> � <sup>σ</sup><sup>m</sup> σ0 f

2.3.2. Morrow's modifying method

240 Contact and Fracture Mechanics

method is given:

pression stress.

where

2.3.3. SWT modifying method

We can acquire the fatigue limit points of material at different stress ratio r ¼ σmin=σmax under infinite life requirement with the support of large amount of experimental data. Draw these points in rectangular coordinate system whose X-axis is mean stress σ<sup>m</sup> ¼ ð Þ σmin þ σmax =2 and Y-axis is stress amplitude σ<sup>a</sup> ¼ ð Þ σmax � σmin =2; thus, the fatigue limit curve is fitted based on these points. It is unpractical to carry out many experiments on all materials and structures in engineering, so we usually use a simplified straight line to replace the fatigue limit curve. Goodman simplified straight line, which is one of these straight lines, is widely accepted due to its simplicity and conservative estimation [18], as shown in Figure 4. Goodman simplified straight line can be expressed in the following relationship:

$$\frac{\sigma\_a}{\mathcal{S}\_e} + \frac{\sigma\_m}{\mathcal{S}\_u} = 1\tag{13}$$

where Se stands for the fatigue strength of material and Su stands for the ultimate tensile strength of material. However, it has been proved that Goodman modifying method is only appropriate for low-ductility material, such as high-strength steel and cast iron.
