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.

$$K\_{\rm I} = \frac{E'}{4} \sqrt{\frac{2\pi}{L}} (4v\_A - v\_B) \tag{34}$$

Kuang [40] use interpolation method to acquire the displacements of nodes A and B on the basis of Barsoum's research and obtain the following expression of stress intensity factor:

> ffiffiffiffiffiffi 2π L r

Lin [41] proposed the 1/4 node displacement method, as shown in Figure 8; the corresponding

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,

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

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 Γ<sup>0</sup> is introduced to divide the structure with crack into two parts: singular field D

ffiffiffiffiffiffi 2π L r

<sup>K</sup><sup>I</sup> <sup>¼</sup> <sup>E</sup><sup>0</sup> 2

there are still many factors that influence calculation results that need to be studied.

ð Þ 8vA � vB (35)

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Numerical Analysis Methods of Structural Fatigue and Fracture Problems

vA (36)

<sup>K</sup><sup>I</sup> <sup>¼</sup> <sup>E</sup><sup>0</sup> 12

calculation equation of stress intensity factor is

4.2. Fractal finite element method

when dealing with general models.

Figure 8. Mesh of 1/4 node element displacement method.

In plane stress problem, <sup>E</sup><sup>0</sup> <sup>¼</sup> <sup>E</sup>; in plane strain problem, <sup>E</sup><sup>0</sup> <sup>¼</sup> <sup>E</sup> <sup>1</sup>�<sup>μ</sup>2. <sup>E</sup>, <sup>μ</sup>, and <sup>v</sup> are, respectively, elasticity modulus, Poisson ratio, and displacement perpendicular to crack surface. Chen and

Figure 7. Eight-node singular isoparametric element.

Kuang [40] use interpolation method to acquire the displacements of nodes A and B on the basis of Barsoum's research and obtain the following expression of stress intensity factor:

$$K\_{\rm I} = \frac{E'}{12} \sqrt{\frac{2\pi}{L}} (8v\_A - v\_B) \tag{35}$$

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

$$K\_{\rm I} = \frac{E'}{2} \sqrt{\frac{2\pi}{L}} v\_A \tag{36}$$

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

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

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.

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

> ffiffiffiffiffiffi 2π L r

elasticity modulus, Poisson ratio, and displacement perpendicular to crack surface. Chen and

ð Þ 4vA � vB (34)

<sup>1</sup>�<sup>μ</sup>2. <sup>E</sup>, <sup>μ</sup>, and <sup>v</sup> are, respectively,

shown in Figure 6.

246 Contact and Fracture Mechanics

4. Finite element method

4.1. Extended finite element method

Figure 7. Eight-node singular isoparametric element.

based on the displacements of nodes A and B; the expression is.

In plane stress problem, <sup>E</sup><sup>0</sup> <sup>¼</sup> <sup>E</sup>; in plane strain problem, <sup>E</sup><sup>0</sup> <sup>¼</sup> <sup>E</sup>

<sup>K</sup><sup>I</sup> <sup>¼</sup> <sup>E</sup><sup>0</sup> 4

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 Γ<sup>0</sup> is introduced to divide the structure with crack into two parts: singular field D

Figure 8. Mesh of 1/4 node element displacement method.

near crack tip and normal field Ω far away from crack tip, as shown in Figure 9. Ordinary finite element mesh is constructed in normal field; self-similar mesh needs to be constructed based on fractal theory in singular field.

transformed into a series of generalized coordinates. Stress intensity factor on crack tip can be calculated via solving generalized coordinates, thus saving calculation time and storage capacity

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249

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

S N curve and ε N are applied to high-cycle and low-cycle fatigue problems, respectively. And there are many modified models considering mean stress or stress ratio. However, this chapter further shows that part of these models are too complicated to apply to engineering,

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 thresh-

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

Institute of Solid Mechanics, School of Aeronautic Science and Engineering, Beihang

[1] Jiang Y, Ding F, Feng M. An approach for fatigue lifetime prediction. Journal of Engineer-

[2] Schijve J. Fatigue of aircraft materials and structures. International Journal of Fatigue.

and other models are only valid in some specific cases.

obviously.

5. Conclusion

higher efficiency.

old have been put forward.

Author details

University, Beijing, China

1994;16(1):21-32

References

of numerical method at the same time.

Qiu Zhiping, Zhang Zesheng and Wang Lei\*

\*Address all correspondence to: leiwang\_beijing@buaa.edu.cn

ing Materials and Technology. 2007;129:182-189

Self-similar mesh is shown in Figure 10. In singular field, infinite curves f g Γ1; Γ2; Γ3; ⋯ similar to Γ<sup>0</sup> are generated based on the proportionally coefficient ξð Þ 0 < ξ < 1 regarding crack tip as centre. The density of fractal mesh is controlled by ξ. Based on appropriate global interpolation function and fractal transforming technique, plenty of unknown degrees on slave nodes are

Figure 9. Illustration of division of structure with crack.

Figure 10. Self-similar mesh in singular field.

transformed into a series of generalized coordinates. Stress intensity factor on crack tip can be calculated via solving generalized coordinates, thus saving calculation time and storage capacity obviously.
