13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- the crack initiation from the free surface. The stress form devised here, to a certain extent, is similar to the residual stress generated by such a shot peening process. This hypothetical stress is expressed by the following equations ( ) 2 15 = −x t s x 5 0 x t ≤ ≤ (10) ( ) 8 9 8 5 = +x t s x t x t ≤ ≤ 5 (11) Case VI is also a hypothetical load case intended to demonstrate the capability of the numerical technique. The following is its stress distribution ( ) x t s x = π2 sin (12) The initial crack configuration was assumed to be a semi-circle (a/c=1) with depth ratio a/t=0.2. a, c and t are crack depth, crack surface half length, and plate thickness, respectively. The material properties employed in this investigation are: Poisson’s ratio ν=0.3, Paris’ fatigue crack growth relation ( )3 13 d d 1.83 10 K a N = × Δ − 4. Results and discussion 4.1. Crack Shape Development Figure 5 shows the fatigue crack shape development of the initially semi-circular surface crack in a plate subjected to different crack face loads. It is clear that the shape change is dependent on the load applied. The crack under uniform tension (Case I) grows with a similar growth increment along the crack front. The free surface has a slight effect on crack growth, which makes the crack advance relatively rapidly along the surface. It can be seen that the semi-elliptical shape is adequate to define the front of the propagating cracks in Fig. 5 (Case I). Crack shape change of different initial cracks for this load case has been investigated by the author [8]. It has been numerically confirmed that the widely used semi-ellipse is an adequate crack shape approximation. The crack in Case II that has a decreasing stress value along the plate thickness propagates more rapidly in the direction of the free surface than in the direction of the plate thickness, gradually leading to a flatter crack profile. The semi-ellipse is also quite acceptable for the shape in this case, as indicated by the author [8] according to a numerical study of crack shape. Contrary to Case II, the Case III crack extends faster at the depth than the free surface. This is due to the larger stress in the direction of crack depth. It seems that these crack profiles predicted can still be approximated with high accuracy by the semi-elliptical shape. The stress concentration at the free surface in the Case IV load permits the crack to grow more easily along the surface compared to the crack under uniform tension fatigue (Case I). It is particularly obvious at the early stage of crack growth. The numerical crack profiles in Fig. 5 (Case IV) are basically in agreement with the experimental results obtained by Pang et al. [9, 10] using the beach-mark method. The growing cracks were also treated by them as those of semi-elliptical form during the crack growth calculation. The crack face load of Case V produces an interesting result of crack shape development. The
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