# Determination of the Crack Propagation Direction in Mixed-Mode Missions due to Cyclic Loading
## 1 Introduction
Based on the experimental investigations of [PERSON] [30], [PERSON] [31] carried out extensive finite element analyses, as well as crack propagation analyses in the context of LEFM. However, no unified criterion for determining the crack propagation direction could be found here, since the mixed-mode relationship also changes continuously with the crack growth. In particular, with regard to the superposition of all three crack modes, very few investigations exist for phase-shifted mixed-mode loading [32; 33]. Furthermore, a special problem in the case of three-dimensional mixed-mode loading is the geometry-dependent coupling of Mode II and Mode III at the surface, which was demonstrated, for example, in [34; 35; 36]. However, this is currently not generally taken into account in the crack propagation concepts developed or in the determination of the crack growth curves. One way of arriving at a propagation deflection angle for the complete mission is to start from the deflection angles calculated for each loading step separately, assuming each step to be the extremal value of a zero-max cycle by simply adding zero as the second entry in the cycle (a zero-max cycle is a cycle in which one of both K-values is zero). The problem is how to combine the individual step angles. One obvious way is to proceed as in BEASY [37] and to calculate the mission deflection angle as a weighted average of the deflection angles in the individual steps. For the weight, the equivalent K-factor in each individual step is used. Alternatively, one could use a dominant step criterion. Such a criterion postulates that the deflection angle of the mission is the same as the deflection angle of the dominant step. In [2], the dominant step was defined as the step with the highest crack propagation rate obtained by substituting the equivalent K-factor in a Paris-type crack propagation law. In the present publication, the weighted deflection angle criterion is compared with the dominant step criterion. To this end, the former criterion, as defined in [37], is slightly modified. Indeed, to take the temperature effect into account, the weight in each loading step is taken to be the crack propagation rate obtained by substituting the equivalent K-factor into a simple Paris-type law (this is similar to the way the dominant step is identified in the dominant step approach). In this way, the temperature enters the calculation through the temperature-dependent constants in the Paris law. The dominant step criterion is kept in the form already used to analyze cruciform specimens in [2]. Both criteria are validated using tension-torsion test results with and without phase shift [38]. It must be emphasized that since the tension-torsion tests are isothermal, only the way both criteria handle the change of mode-mixity in a mission is validated and not the way the temperature is taken into account. The next section describes the experimental setup. Then, a detailed description of the crack propagation procedure for mixed-mode missions in CRACKTRACER3D is given. Finally, the validation of the crack propagation criteria using the tension-torsion results is described, and the most appropriate criterion is identified. A list of the nomenclature used throughout the article is given in Table 1.