5. Conclusion

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

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

based on fractal theory in singular field.

248 Contact and Fracture Mechanics

Figure 9. Illustration of division of structure with crack.

Figure 10. Self-similar mesh in singular field.

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.

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, and other models are only valid in some specific cases.

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.
