# Determination of the Crack Propagation Direction in Mixed-Mode Missions due to Cyclic Loading
## 1 Introduction  
In aircraft engine components, the accurate prediction of the propagation of potential cracks has gained much importance in recent decades. Indeed, due to challenging requirements related to fuel consumption and (consequently) weight, traditional crack initiation criteria have been replaced by damage tolerance concepts [1]. Not only life prediction in terms of the number of cycles is crucial, but also the accurate prediction of the crack propagation direction is a target. This applies for instance to bisks, in which a crack initiated in the blade due to vibrations may (depending on the crack propagation direction) grow into the disk and lead to a catastrophic failure due to the centrifugal loading [2]. Therefore, both the crack propagation rate and direction are essential information. Due to the different time constants involved in mechanical and thermal processes, the stress and temperature evolution during flight are quite different, and up to a few hundred loading steps may be needed to accurately simulate all conditions. At any location along a potential crack, this leads to exactly the same amount of different mixed-mode states. Due to small-scale plasticity at the crack tip, a linear elastic stress intensity approach can be taken. For each of the stress states separately, one can determine the crack propagation speed and direction [3, 4]. The question is, however, how to determine the speed and direction due to the complete mission. For the speed, a cycle extraction through a rainflow algorithm [5] is the usual way to go. Still, the rainflow algorithm is usually applied to a one-dimensional function, and in mixed-mode transient temperature conditions, it is not necessarily clear what function should be taken. For the crack propagation direction, the fundamental question is how to arrive at the propagation direction for the complete mission. This paper concentrates on the latter aspect. For monotonic plane mixed-mode loading, different stress or energy based fracture criteria have been developed and adapted since the 1960s in the scope of linear elastic fracture mechanics (LEFM). In this context, the maximum tangential stress criterion (MTS) of [PERSON] and [PERSON] [6], the maximum shear stress criterion (MSS) [7, 8], and, for example,the equivalent stress intensity factor \(K_{\rm p}\) of [PERSON] et al. [3; 9] should be mentioned. For the case of a general three-dimensional mixed-mode loading, [PERSON] et al. [10] developed the so-called \(\sigma_{1}^{\prime}\)-criterion based on the MTS criterion. Furthermore, the empirical criterion according to [PERSON] et al. [9] represents an extension from the 2D to the 3D state and stands out especially because of its easy adaptability to experimental results. Finally, a general approach based on the principal planes of the asymptotic stress tensor at the crack tip was developed by [PERSON] [11] and applied in several subsequent publications [2; 12]. As proven by [PERSON] et al. [13], [PERSON] et al. [14], and [PERSON] and [PERSON] [15], among others, all criteria for monotonic loading can in principle also be applied to fatigue crack growth under proportional loading. However, in the case of non-proportional mixed-mode loading, the fracture criteria developed for monotic and cyclic proportional loading are no longer applicable. This circumstance essentially results from the fact that the time-varying stress path can no longer be described by only one parameter. A non-proportional loading occurs, for example, by the superposition of a static load component with a cyclic load component and has been studied by [2; 16; 17; 18; 19; 20; 21; 22], among others. Mode II or mixed-mode overloads interspersed in a Mode I basic stress can be considered as a special case of a non-proportional mixed-mode loading, as studied for example by [PERSON] and [PERSON] [23; 24], as well as by [PERSON] and [PERSON] [25]. The general case of non-proportional loading is represented by phase-shifted mixed-mode loading, which was the focus of investigations in [26; 27; 28; 29].