## 3 Results and Discussion  
### Evolution of the Crack Shape  
Figure 4 illustrates the fatigue shape evolution by the crack growth circles in a round bar subjected to tension. The seven crack front profiles displayed are deduced from roughly 30 crack growth circles in less than 20 iterations. The outermost crack growth circle rolls along the internal profile of the round bar in an approximate manner. When the point of tangency between crack growth circle with crack front approaches very closely to the surface of the bar, such as crack front 6 in Figure 4, the outermost crack growth circle will disappear in the next propagation. The rate of crack propagation can be observed intuitionally by the size of crack growth circles. As shown in Figure 5, the optimum simulation result for the center of an ellipse is not fixed on the surface of the bar, but is reciprocating along the y-axis. Therefore, the actual crack shape can be accurately expressed by the three-parameter model. In the simulation process, notice that several different ellipses with the same chord length can be replaced to describe one actual crack front, since only part of an ellipse is used, once the center is not fixed (Figure 6). A large variation of ellipse actual aspect ratio is obtained with undifferentiated iteration error, as shown in Figure 7. Hence, the actual aspect ratio of the ellipse semi-axis is meaningless for the three-parameter model to describe the crack front. Figure 4: Successive determination of crack fronts by the crack growth circles with initial crack \(b_{0}/a_{0}=1,b_{0}/D_{0}=0.1,\ m=2\). Figure 5: Ellipses used to determine crack fronts when initial crack \(b_{0}/a_{0}=0,b_{0}/D_{0}=0.08,m=2\). Figure 6: Crack front as a part of an ellipse. Figure 8 illustrates the fatigue shape evolution for five cases. The aspect ratio of initial ellipses \(b_{0}/a_{0}=0,\ 1\), and relative crack depth \(b_{n}/D_{0}=0.05,\ 0.08,\ \text{and}\ 0.1\), while the material constants in the Paris-Erdogan law are assumed to be \(m=2,\,3,\) and \(4\). The trends of crack propagation are adequately demonstrated. As mentioned previously, the nominal aspect ratio of an ellipse, which is the ratio of the maximum crack depth to the chord length \(c\), \(b_{n}/c\) can be considered here. It is noteworthy that, as shown in Figure 9, both initial crack dimensions and Paris law exponent m have an effect on the evolution of different parameters. The crack propagation trends are consistent with the same initial crack aspect ratio when using the same material, although the beginning propagation is affected by the crack depth provisionally. Meanwhile a difference of transition can be noticed between the crack propagation with different Paris law exponent \(m\) values. In Figure 9, it can be found that the nominal aspect ratio change is very sensitive to the initial crack geometry during early growth, and then the nominal aspect ratios for all cases are converged and become constant around \(b_{n}/D_{0}\approx 0.4\). It shows the flaws tend to follow preferential propagation paths that flatten gradually when the crack depth become larger. Figure 7: Change of actual aspect ratio with the same chord length \(c\). Figure 8: Shape change of different initial crack for different fatigue crack growth exponent \(m\) values. The fatigue crack developments \(b_{n}/D_{0}\) with \(c/D_{0}\) under cyclic for different initial parameters are shown in Figure 10. It can be seen that the crack propagation paths differ with different initial flaws, but will converge asymptotically. Furthermore, in the process of expansion, the crack growth rate for center and outermost points are variable, which is deduced from the gradient of two type lines with initial flaws \(b_{0}/a_{0}=0,\ 1\).