13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- by the author [4, 5] has been employed to simulate the fatigue crack growth of surface cracks under several typical and complex crack face loads. The method can directly predict, in a step-by-step way, the crack shape change without having to make a semi-elliptical crack shape assumption, and also take possible crack face contact into account. Various results are given to validate the numerical method for simulating fatigue crack growth with complex crack shape change. 2. Numerical simulation technique It is usually considered reasonable to apply a fatigue crack growth law from small specimens that have a straight front to a curved crack front along which the stress intensity factor varies. The fatigue crack growth rate can then be expressed as a function of crack front position as follows ( ) ( ( )) ξ ξ f K N a = Δ d d (1) where ( )ξ a and ( )ξ KΔ are the local normal crack growth increment and the SIF range at an arbitrary crack front position, ξ, as shown in Fig.1. The crack growth increment can be written as ( ) ( ( )) ( ) max max a f K f K a Δ Δ Δ Δ = ξ ξ (2) where max aΔ is the maximum increment along the crack front, occurring at the point where the SIF is the largest. Fatigue cycles can then be calculated as follows ( ) (i 0,1,...) max max i i+1 = Δ Δ = + f K a N N (3) By choosing an appropriate small value of max aΔ , both the crack shape change and the corresponding number of fatigue cycles can be predicted if the SIF solutions along the crack front are available. For a Paris type fatigue crack growth law, we have ( ) ( ) max max a K K a m Δ Δ Δ Δ = ξ ξ , ( ) (i = 0,1, ...) max max i i+1 m C K a N N Δ Δ = + (4) The above equations have been used in the present simulation. However, it is worth indicating that a general fatigue crack growth rate equation including crack threshold and fracture toughness can be readily incorporated. Figure 1. Illustration of fatigue crack growth of a surface crack Stress intensity factors along the crack front were estimated by using the finite element (FE) method. The 1/4-point displacement method was used to achieve the inverse square-root stress singularity at the corner of a wedge element, and particularly possible contact between crack faces in a compressive stress field was taken into account in FE analyses. The method of estimating the SIF was detailed in [5]. Figure 2 illustrates a typical mesh created in the simulation. The mesh comprises both cracked and un-cracked blocks. Between them the “multiple point constraint” equations are applied to maintain the displacement compatibility. Both cracked and un-cracked blocks are filled with 20-node iso-parametric elements. In particular, three rings of elements are arranged surrounding the crack tip. In the simulation, the cracked block is recreated as the crack Crack front Local crack growth ξ
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