## Numerical Propagation Process
### Fatigue Crack Propagation  
The propagation of a surface crack in a round bar under cyclic tension is predicted by employing fatigue crack growth circles [21] (Figure 2). If the crack presents an ellipse shape up to the \(i\)-th loading step, the initial ellipse whose center is located on the surface of the specimen can be defined with given \(a_{i}\) and \(b_{i}\), as represented by the following equation  
\[\frac{x^{2}}{a_{i}^{2}}+\frac{y^{2}}{b_{i}^{2}}=1. \tag{1}\]  
Points \(O\), \(A\), \(B\), \(C\) and \(D\) in Figure 2 with coordinates \((x_{\bar{\mu}},y_{\bar{\mu}})\) are deployed equidistantly along the current crack front, where the subscript \(j\) refers to the points \(O\), \(A\), \(B\), \(C\) and \(D\).  
The growth of a new crack front lying on an ellipse with semi-axes \(a_{i+1}\), \(b_{i+1}\), and \(O_{i+1}\) after one cyclic loading step to a new configuration can be described by the following equation  
\[\frac{x^{2}}{\left(a_{i+1}\right)^{2}}+\frac{\left(y-O_{y+i}\right)^{2}}{\left( b_{i+1}\right)^{2}}=1. \tag{2}\]  
The assumed crack growth circles, which pass points \(O\), \(A\), \(B\), \(C\) and \(D\), respectively, are tangent to both current and new crack fronts. The new crack front points \(O^{\prime}\), \(A^{\prime}\), \(B^{\prime}\), \(C^{\prime}\) and \(D^{\prime}\) with coordinates \((x_{j,(i+1)},y_{j,(i+1)})\) are the points of tangency between crack growth circles and the new crack front. Meanwhile the centers of crack growth circles can be determined as \((x_{j\ell}\), \(y_{j\ell})\).  
Figure 1: Definition of the geometrical parameters for a surface crack in a round bar.  
The crack growth increment among these points can be determined by applying the Paris-Erdogan law  
\[\frac{da}{dN}=C(\Delta K)^{m}, \tag{3}\]  
where \(da/dN\) is crack growth rate, \(\Delta K\) is the stress intensity factor range, and \(C\) and \(m\) are material constants.  
After each computed crack configuration, an increment of crack growth at the interior point \(O^{\prime}\) is given. The crack growth length of other points \(A^{\prime}\), \(B^{\prime}\), \(C^{\prime}\), _and_\(D^{\prime}\) can be determined  
\[\Delta l_{j}=(y_{O,i+1}-y_{O,i})\frac{\left(\Delta K_{cj}\right)^{m}}{\left( \Delta K_{cO}\right)^{m}}. \tag{4}\]  
Here, \(\Delta K_{c}\) stands for the equivalent stress intensity factor related to the stress intensity factors of both current and new crack fronts.  
The stress intensity factor \(K\) is assumed to be a liner function of crack growth increment. An arbitrary number of crack growth steps can be assumed. Using  
\[da=C(\Delta K)^{m}dN, \tag{5}\]  
the crack growth length is increased to \(a+da\) repeatedly in each step to the last step by adjusting material constant \(C\). The equivalent stress intensity factor \(\Delta K_{e}\) with stepping coefficient \(\mu\) can be obtained appropriately through the crack growth plot of \(da/dN\) vs. \(N\).  
\[\Delta K_{ej}=\mu\big{(}K_{ij}\big{)}^{m}+(\frac{1}{2}\big{(}K_{i,j}+K_{(i+1),j}\big{)})^{m}+(1-\mu)\big{(}K_{(i+1),j}\big{)}^{m}\ \ \ 0<\mu<1 \tag{6}\]  
At the beginning of iteration, sometimes a relatively large value of \(\mu\) can be used to avoid diverging.  
The distance from the center of crack growth circles to points \(O^{\prime}\), \(A^{\prime}\), \(B^{\prime}\), \(C^{\prime}\) and \(D^{\prime}\) along the new crack front are calculated using the geometrical relationship  
\[\Delta d_{j}=\sqrt{\big{(}x_{j,(i+1)}-x_{j,\varepsilon}\big{)}^{2}+\big{(}y_ {j,(i+1)}-x_{j,\varepsilon}\big{)}^{2}}. \tag{7}\]  
Figure 2: Determination of a new crack front by fatigue crack growth circles.  
An error equation can be derived as  
\[\text{Error}=\sum\bigl{|}\Delta d_{j}-\Delta l_{j}/2\bigr{|}. \tag{8}\]  
The values of \(a_{i+1}\) and \(O_{i+1}\), which minimize the error equation, are based on iterative methods and repeat all of the above steps based on the obtained crack front. The parameters of the ellipse for each new crack front can be determined until the results converge.