# Determination of the Crack Propagation Direction in Mixed-Mode Missions due to Cyclic Loading
## 3 Crack Propagation Procedure in CrackTRACER3D
Let us assume that the cross-section of the principal plane (symbolized by the light gray triangle in Figure 7a) corresponding to the largest self-similar principal stress with the local x-y plane makes an angle with the x-axis of \(\varphi_{o}\). If \(\varphi\
eq\varphi_{o}\), this line does not go through the crack tip and cannot be a crack propagation direction. By varying \(\varphi\) between \(-90^{\circ}\) and \(90^{\circ}\) until a value is found for which \(\varphi=\varphi_{o}\), the crack propagation direction is found (symbolized by the black triangle in the figure). If more than one such value is found, the one corresponding to the largest self-similar stress value is taken. Once the correct value for \(\varphi\) is determined, the equivalent K-value \(K_{eq}\) corresponds to the eigenvalue and the twist angle \(\psi\) to the angle that the cross-section of the principal plane with the y-z plane makes with the z-axis (cf. Figure 7b). The deflection angle corresponds to \(\varphi\). The criterion discussed here is straightforward and clearly relies on physical principles. The results were very similar to  
Figure 6: Local coordinate system at the crack front. other criteria such as the criterion by [PERSON] [3], and a comparison with that criterion can be found in [11]. By the use of \(K_{eq}\), \(\varphi\), and \(\psi\), the crack propagation increment can be determined. The size of the increment is obtained by substituting \(\Delta K=K_{eq}\) into a crack propagation law (since right now, there was only one loading step, a 0-max cycle was assumed; the extension to several loading steps follows later in the article). Here, the crack propagation law discussed in [48] was used. It is a multiplicative law separating the Paris, threshold, and critical range in the form:  
\[\frac{da}{dN}=\left[\left(\frac{da}{dN}\right)_{ref}\left(\frac{\Delta K}{ \Delta K_{ref}}\right)^{m}\right]\frac{f_{R}\cdot f_{th}}{f_{c}}, \tag{3}\]  
where the Paris range is in square brackets. One of the parameters \(\left(\frac{da}{dN}\right)_{ref}\) and \(\Delta K_{ref}\) can be freely chosen, e.g., \(\left(\frac{da}{dN}\right)_{ref}\). Then, \(\Delta K_{ref}\) and \(m\) are fixed and can be derived from the Paris parameters \(C\) and \(m\) one usually finds in the literature. The present form has the advantage that the parameters have simple units and a clear physical meaning: \(\Delta K_{ref}\) is the value of \(\Delta K\) for which \(\frac{da}{dN}=\left(\frac{da}{dN}\right)_{ref}\). The traditional parameter \(C\) of the Paris law can be identified as:  
\[C=\left(\frac{da}{dN}\right)_{ref}\left(\frac{1}{\Delta K_{ref}}\right)^{m} \tag{4}\]  
The Paris range is now corrected for the R-influence (\(R=K_{min}/K_{max}\)) by a function \(f_{R}(R)\); the threshold correction is described by \(f_{th}(\Delta K,\Delta K_{th})\) and the crack propagation modification near the critical value \(K_{c}\) by \(f_{C}(K_{max},K_{C})\). For details on the form of these functions, the reader is referred to [48]. The value of \(\varphi\) gives the propagation direction: it is the deflection angle in the local x-y plane. Finally, \(\psi\) yields the warping or twist angle. In experiments with significant contributions of Mode III, warping was observed in the form of factory roofing: many small cracks were locally formed along the crack front growing together to yield a stair-like crack surface. This effect led to discontinuous crack surfaces and was not considered here. The crack propagation law leads to the propagation due to just one cycle. Fracture mechanics calculations usually involve several thousand cycles and more. Furthermore, the propagation due to just one cycle is usually so small that the K-factors do not change significantly due to this cycle. Therefore, the same K-distribution is used for a cycle set consisting of several cycles. In order to guarantee that the K-factors do not change during this cycle set, an inverse approach is taken: the user specifies a maximum allowed crack  
Figure 4:0 (**a**) Determining the correct deflection angle.