# Determination of the Crack Propagation Direction in Mixed-Mode Missions due to Cyclic Loading
## 4 Calculation of the Propagation Direction Due to a Mission  
The previous section considered just one loading step and assumed a zero-max cycle. A mission, however, consists of many loading steps. For instance, in order to simulate one flight of an airplane (take-off, climb, cruise, descent, landing, trust reverse, shut down), several hundred loading steps may be necessary to accurately model the temperature and stress history during the flight. In addition, high frequency vibration steps may be interspersed in between the mission steps [49]. Each loading step is characterized by its own \(K_{I},K_{II},K_{III}\) mixture, leading to its own equivalent K-factor and deflection angle. Several questions arise. Since we are interested in cyclic crack propagation, we have to know how to obtain the cycles. What is the quantity on which we want to perform the cycle extraction? Since each loading step leads to its own deflection angle, how do we define the deflection angle for the complete mission? Let us detail the approach taken by CRACKTRACER3D to cope with these questions. In a first phase, the mission is read as a composition of zero-max cycles, obtaining the equivalent stress intensity factor range \(\Delta K_{eq}=K_{eq}\) for each loading step. Then, a simplified form of the crack propagation law, taking only the Paris range into account, is applied:  
\[\frac{da}{dN}=sgn(K_{eq})\bigg{(}\frac{da}{dN}\bigg{)}_{ref}\bigg{(}\frac{| \Delta K_{eq}|}{\Delta K_{ref}}\bigg{)}^{m} \tag{5}\]  
where \(m\) and \(\Delta K_{ref}\) are temperature-dependent material properties. Since for single loading steps, \(K_{eq}\) may be negative (which happens in rare cases), the absolute value is taken under the exponentiation, and the sign is moved ahead of the brackets. Consequently, a fictitious negative \(da/dN\) is assigned to a purely compressive load step. A hypothetical time distribution of the resulting values for \(\frac{da}{dN}\) at a given crack front location is shown in Figure 8 (continuous black lines). Notice that this curve not only contains information on the loading, but also on the way the material reacts to this loading at the given temperature (through the Paris law). Subsequently, the dominant step is identified, defined as the step at which \(\frac{da}{dN}\) based on Equation (5) is maximal (time \(t_{max}\) in Figure 8). Now, the assumption is made that the crack propagation direction due to the mission corresponds to the deflection angle calculated based on the dominant step. Therefore, the step, which leads to the maximum crack propagation rate according to Equation (5), dictates the growth direction. Once the crack propagation direction is determined, the next step consists of re-evaluating the crack growth rate contributions for each zero-max cycle. During a mission, the principal plane corresponding to the largest principal value of the self-similar stress at each loading point can largely deviate from the previously defined dominant direction, and therefore, its contribution to the crack growth rate may be incorrect. To avoid this issue, CRACKTRACER3D assumes that for all other mission instants, the crack propagates on the principal self-similar plane, which is as close as possible to the dominant direction [50], and it is the corresponding \(K_{eq}\) that is used to calculate \(da/dN\) through Equation (5) for the subsequent cycle extraction. This means that, except for the dominant loading step, \(K_{eq}\) used for cycle extraction is not necessarily equal to the maximum principal self-similar stress; its value may be lower, generating a lower crack growth rate, as shown by the red dashed lines in Figure 8. For a concrete application of this procedure, the reader is referred to [2]. After mission reevaluation, cycle extraction is applied to the new crack propagation rate distribution (the red dashed line in Figure 8) with the rainf counting algorithm described in [5].