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

$$\frac{da}{dN} = C(\Delta K)^m \tag{14}$$

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 growth life.

Figure 5. Schematic diagram of the relationship between crack growth and ΔK.

#### 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, 25–27]. 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

$$\frac{da}{dN} = \frac{\mathbb{C}(\Delta \mathbf{K})^m}{(1 - R)\mathbf{K}\_c - \Delta \mathbf{K}} \tag{15}$$

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

Three parameters C, m, and n can be acquired based on experimental data of crack propagation experiments with different stress ratios. Walker formula is valid when R > 0 and R < 0. According to the relationship between stress ratio and amplitude of stress intensity factor,

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 σop, and the stress when crack begins to close is defined as crack closing stress σcl. It has been demonstrated that crack opening stress is nearly equal to crack closing stress. The modified formula is written as

dN <sup>¼</sup> <sup>C</sup> ð Þ <sup>1</sup> � <sup>R</sup> <sup>m</sup> ½ � <sup>K</sup>max <sup>n</sup> (19)

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maxð Þ <sup>Δ</sup><sup>K</sup> <sup>n</sup> (20)

dN <sup>¼</sup> <sup>C</sup> <sup>Δ</sup>Keff <sup>m</sup> (21)

< 1 (23)

dN <sup>¼</sup> C Uð Þ <sup>Δ</sup><sup>K</sup> <sup>m</sup> <sup>¼</sup> <sup>U</sup>mCð Þ <sup>Δ</sup><sup>K</sup> <sup>m</sup> (22)

maximum of stress intensity factor into consideration:

another commonly used form of Walker formula is obtained:

3.2.2. Model based on crack closure theory

follows:

and

opening stress σop.

da

da dN <sup>¼</sup> CK<sup>m</sup>

da

<sup>Δ</sup><sup>K</sup> <sup>¼</sup> <sup>Δ</sup>σeff

<sup>Δ</sup><sup>σ</sup> <sup>¼</sup> <sup>σ</sup>max � <sup>σ</sup>op Δσ

where efficient stress amplitude Δσeff is the difference between maximum stress σmax and crack

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 σres, which is related to plastic deformation

da

<sup>U</sup> <sup>¼</sup> <sup>Δ</sup>Keff

U is the crack closure parameter, and its expression is

3.2.3. Model considering crack retardation caused by high load

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 Kc for highductility material. According to following relationship:

$$R = \frac{K\_{\text{min}}}{K\_{\text{max}}} \tag{16}$$

$$
\Delta K = K\_{\text{max}} - K\_{\text{min}} \tag{17}
$$

Forman formula can be transformed as follows:

$$\frac{da}{dN} = \frac{\mathcal{K}\_{\text{max}} (\Delta \mathcal{K})^{m-1}}{K\_c - K\_{\text{max}}} \tag{18}$$

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:

$$\frac{da}{dN} = \mathbb{C}[(1 - R)^m K\_{\text{max}}]^n \tag{19}$$

Three parameters C, m, and n can be acquired based on experimental data of crack propagation experiments with different stress ratios. Walker formula is valid when R > 0 and R < 0. According to the relationship between stress ratio and amplitude of stress intensity factor, another commonly used form of Walker formula is obtained:

$$\frac{da}{dN} = \mathsf{CK}\_{\text{max}}^{m} (\Delta \mathsf{K})^{n} \tag{20}$$

#### 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 σop, and the stress when crack begins to close is defined as crack closing stress σcl. It has been demonstrated that crack opening stress is nearly equal to crack closing stress. The modified formula is written as follows:

$$\frac{da}{dN} = \mathbb{C}(\Delta K\_{\text{eff}})^m \tag{21}$$

and

3.2. Improved models

242 Contact and Fracture Mechanics

3.2.1. Models considering mean stress or stress ratio

Figure 5. Schematic diagram of the relationship between crack growth and ΔK.

toughness as an important factor; its expression is

ductility material. According to following relationship:

Forman formula can be transformed as follows:

factor is close to fracture toughness.

Since Paris law is proposed, much related work is done, and many modifying methods are put forward [22, 25–27]. 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

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

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 Kc for high-

> <sup>R</sup> <sup>¼</sup> <sup>K</sup>min Kmax

dN <sup>¼</sup> CKmaxð Þ <sup>Δ</sup><sup>K</sup> <sup>m</sup>�<sup>1</sup> Kc � Kmax

Forman formula explains the reason why crack growth enlarges sharply when stress intensity

ð Þ <sup>1</sup> � <sup>R</sup> Kc � <sup>Δ</sup><sup>K</sup> (15)

ΔK ¼ Kmax � Kmin (17)

(16)

(18)

da

da

$$\frac{da}{dN} = \mathbb{C}(\mathcal{U}\Delta\mathcal{K})^m = \mathcal{U}^m\mathbb{C}(\Delta\mathcal{K})^m\tag{22}$$

U is the crack closure parameter, and its expression is

$$\mathcal{U} = \frac{\Delta \mathcal{K}\_{\ell \mathcal{f}}}{\Delta \mathcal{K}} = \frac{\Delta \sigma\_{\ell \mathcal{f}}}{\Delta \sigma} = \frac{\left(\sigma\_{\text{max}} - \sigma\_{\text{op}}\right)}{\Delta \sigma} < 1 \tag{23}$$

where efficient stress amplitude Δσeff is the difference between maximum stress σmax and crack opening stress σop.

#### 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 σres, which is related to plastic deformation caused by high load. Combining the expression of Forman formula, crack growth rate in retardation period is acquired:

$$\frac{da}{dN} = \frac{\mathbb{C}\left(\Delta \mathcal{K}\_{\text{eff}}\right)^{m}}{\left(1 - \mathcal{R}\_{\text{eff}}\right)\mathcal{K}\_{\text{c}} - \Delta \mathcal{K}\_{\text{eff}}} \tag{24}$$

In 1999, McEvily found it out that the following modification is suitable for many alloys'

where ΔKeffth stands for the effective stress intensity factor range near crack propagation threshold. This modifying method considers the influences created by crack closure and small crack's elastic-plastic behavior, and it is useful to predict the long crack propagation under

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:

> <sup>a</sup> <sup>¼</sup> <sup>X</sup><sup>∞</sup> i¼0

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

dN <sup>¼</sup> <sup>C</sup> <sup>Δ</sup>Keff � <sup>Δ</sup>Keffth � �<sup>2</sup> (32)

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aiε<sup>i</sup> (33)

da

3.2.5. Model based on perturbation series expansion method

Figure 6. Comparison of the measured and predicted crack length history in Ref. [35].

fatigue crack propagation:

cyclic positive stress.

where ε is a positive small constant.

The effective stress intensity factor range is

$$
\Delta K\_{\rm eff} = f \left[ (\sigma\_{\rm max})\_{\rm eff} - (\sigma\_{\rm min})\_{\rm eff} \right] \sqrt{\pi a} \tag{25}
$$

and the effective stress ratio is

$$R\_{\rm eff} = (\sigma\_{\rm min})\_{\rm eff} / (\sigma\_{\rm max})\_{\rm eff} \tag{26}$$

The maximum and minimum values of effective cyclic stress are

$$\left(\sigma\_{\text{max}}\right)\_{\text{eff}} = \sigma\_{\text{max}} - \sigma\_{\text{res}} \tag{27}$$

$$(\sigma\_{\min})\_{\ell\overline{\ell}} = \sigma\_{\min} - \sigma\_{\text{res}} \tag{28}$$

Then, crack growth rate in retardation period can be estimated as the residual stress σres is known. However, the residual stress σres can only be obtained via experimental method.

#### 3.2.4. Model considering crack propagation threshold

In 1972, Donahue [33] took threshold of stress intensity factor range ΔKth into consideration and proposed a generalized Paris law. The modified expression is

$$\frac{da}{dN} = \mathbb{C}(\Delta K - \Delta K\_{\text{th}})^m \tag{29}$$

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

$$\frac{da}{dN} = \mathcal{C}(\Delta K - \Delta K\_{th})^2 \left(1 + \frac{\Delta K}{K\_c - K\_{\text{max}}}\right) \tag{30}$$

in which material constant m equals 2.

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

$$\frac{da}{dN} = \frac{\mathbb{C}[(\Delta K)^m - (\Delta K\_{th})^m]}{(1 - R)K\_c - \Delta K} \tag{31}$$

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:

$$\frac{da}{dN} = \mathbb{C}\left(\Delta K\_{\text{eff}} - \Delta K\_{\text{eff}\text{fb}}\right)^2\tag{32}$$

where ΔKeffth stands for the effective stress intensity factor range near crack propagation threshold. This modifying method considers the influences created by crack closure and small crack's elastic-plastic behavior, and it is useful to predict the long crack propagation under cyclic positive stress.

#### 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:

$$a = \sum\_{i=0}^{\infty} a\_i \varepsilon^i \tag{33}$$

where ε is a positive small constant.

caused by high load. Combining the expression of Forman formula, crack growth rate in

� �<sup>m</sup>

h i ffiffiffiffiffi

(24)

(30)

<sup>π</sup><sup>a</sup> <sup>p</sup> (25)

Reff ¼ ð Þ σmin eff =ð Þ σmax eff (26)

ð Þ σmax eff ¼ σmax � σres (27)

ð Þ σmin eff ¼ σmin � σres (28)

dN <sup>¼</sup> <sup>C</sup>ð Þ <sup>Δ</sup><sup>K</sup> � <sup>Δ</sup>Kth <sup>m</sup> (29)

ð Þ <sup>1</sup> � <sup>R</sup> Kc � <sup>Δ</sup><sup>K</sup> (31)

� �Kc � <sup>Δ</sup>Keff

dN <sup>¼</sup> <sup>C</sup> <sup>Δ</sup>Keff

1 � Reff

ΔKeff ¼ f ð Þ σmax eff � ð Þ σmin eff

Then, crack growth rate in retardation period can be estimated as the residual stress σres is known. However, the residual stress σres can only be obtained via experimental method.

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

The following expression was proposed by McEvily and Greoeger [34] in their research about

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

dN <sup>¼</sup> <sup>C</sup> ð Þ <sup>Δ</sup><sup>K</sup> <sup>m</sup> � ð Þ <sup>Δ</sup>Kth <sup>m</sup> ½ �

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

<sup>2</sup> <sup>1</sup> <sup>þ</sup>

ΔK Kc � Kmax � �

da

The maximum and minimum values of effective cyclic stress are

and proposed a generalized Paris law. The modified expression is

da

da

dN <sup>¼</sup> <sup>C</sup>ð Þ <sup>Δ</sup><sup>K</sup> � <sup>Δ</sup>Kth

da

3.2.4. Model considering crack propagation threshold

fatigue crack propagation threshold in 1977:

in which material constant m equals 2.

following expression:

formula.

retardation period is acquired:

244 Contact and Fracture Mechanics

and the effective stress ratio is

The effective stress intensity factor range is

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

Figure 6. Comparison of the measured and predicted crack length history in Ref. [35].

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.
