# Determination of the Crack Propagation Direction in Mixed-Mode Missions due to Cyclic Loading
## 3 Crack Propagation Procedure in CrackTRACER3D
It shows the elements immediately adjacent to the crack front and the closest integration points ahead of the crack front (in the direction of the positive x-axis). Since the reduced integration point scheme in a 20-node brick element is a \(2\times 2\times 2\) scheme, there are two integration points along and ahead of the crack front at locations \(\pm\frac{1}{\sqrt{3}}\) in a local coordinate range extending from \(-1\) to \(1\). The right part is a cut parallel to the x-y plane. It shows the 6 collapsed quarter point elements containing the crack and the closest integration points to the front. Each integration point is characterized by specific values for \(r\) and \(\varphi\), where \(r\) and \(\varphi\) are cylindrical coordinates in the x-y plane. At any location of the integration points along the crack front (symbolized by diamonds in Figure 6), e.g., at the location of the local axes in the left of Figure 6, any of the integration points on the right of the figure can be used to obtain the K-values at that location. Indeed, in each integration point, the finite element calculation yields 6 stress components, leading to 6 equations of the type in Equation (1) in the 3 unknown K-factors (\(r\) and \(\varphi\) are known for the integration point). However, a detailed analysis showed that not all integration points were equally accurate. More precisely, the integration points immediately ahead of the crack front and labeled by a + sign on the right of Figure 6 were most accurate. Furthermore, it proved advantageous to discard the use of the equation for \(\sigma_{zz}\), to avoid the problem of whether plane stress or plane strain applies at the actual crack location (the asymptotic expression for \(\sigma_{zz}\) is different for plane stress and plane strain). Therefore, we ended up with 5 equations in 3 unknowns for each of the locations marked with a +. Using the least squares method, this yielded a triple of K-values for the integration point at positive y and a triple for the integration point at negative y. Taking the mean yielded the final values for \(K_{I},K_{II},K_{III}\) at that crack front location. Acting in a similar way, the K-values can be obtained for all integration point locations along the  
Figure 5: Detailed view of the tube. crack front. To obtain the values at the nodes, linear interpolation along the crack front was applied. At the free surface, however, extrapolation was used. For a comparison for simple examples of the present method with other methods to determine the K-values, the reader is referred to [47]. Now, the K-values \(K_{I},K_{II},K_{III}\) were determined at any nodal location along the crack front. The aim was to obtain a crack propagation increment. To this end, the K-triple was transformed into an equivalent K-factor \(K_{eq}\), which can be used in an appropriate crack propagation law. Furthermore, the crack propagation direction had to be determined, characterized by a deflection angle \(\varphi\) and a twist angle \(\psi\). These values can be found by postulating that the crack propagation will take place in a plane orthogonal to the largest principal self-similar stress such that the crack tip is contained in this plane. The self-similar stress is defined as the asymptotic stress multiplied by \(\sqrt{r}\). To explain this in more detail, let us have a look again at Equation (1). Now, \(K_{I},K_{II}\) and \(K_{III}\) are known. The resulting asymptotic stress field is clearly a function of \(r\) and \(\varphi\). The dependency on \(r\), however, is common to all stress components, so we can discard it. This leads to the self-similar stress \(\sigma^{*}\) with components:  
\[\sigma^{*}_{xx}=f_{xx}(\varphi,K_{I},K_{II},K_{III}), \tag{2}\]  
and similar for the other terms. The only variable left here is \(\varphi\). For a fixed value of \(\varphi\), the principal values and directions for this tensor can be determined.