# Determination of the Crack Propagation Direction in Mixed-Mode Missions due to Cyclic Loading
## 5 Validation Using the Tension-Torsion Tests  
Experimental and numerical results for the tension-torsion tests were compared to validate the accuracy of CRACKTRACER3D in predicting the crack propagation direction. To this end, the outside surface of the broken specimens was measured by a GOM ATOS machine using blue light scanning. The result was a very detailed triangulation of the fracture surface. This surface was compared with the triangulation of the propagated crack surface predicted by CRACKTRACER3D. In order to do that, a new software called CT3D_Validator [51] was developed at MTU Aero Engines AG. The basic concept is to find a parameter that measures the deviation between the numerical and experimental crack propagation directions. The two crack surfaces to compare are both meshed with triangles. The idea is to calculate the volume between these two triangulations and then divide it by the CRACKTRACER3D crack surface area. Thereby, the final result is a length called \(d\) that quantifies the global deviation between the two crack directions and satisfies:  
\[d_{ax}=\frac{\sum_{i=1}^{N}V_{i}}{\sum_{i=1}^{N}A_{i}}=\frac{\sum_{i=1}^{N}d_ {i}A_{i}}{\sum_{i=1}^{N}A_{i}} \tag{7}\]  
where the subscript \(ax\) indicates the direction along which \(d\) is measured. For this type of specimen, \(ax\) represents the longitudinal axis (i.e., the tensile direction). The index \(i\) identifies the \(i\)-th element of the CRACKTRACER3D crack surface, and \(N\) is its total number of elements. The volume is approximated with a group of triangular prisms built taking as a reference the numerical triangulation. For each triangle, a prism is extruded along the axial direction as high as the distance \(d_{i}\) between the center of gravity of the \(i\)-th reference triangle and the experimental surface. \(A_{i}\) is the triangle area of the \(i\)-th element. A graphical representation can be observed in Figure 9.  
The deviation \(d_{ax}\) was calculated to compare the dominant step criterion (labeled \"current\") with the averaged angle criterion (labeled \"new\") for the tension-torsion test results. More specifically, the eighteen possible load combinations were simulated with the two different criteria and compared with the same experimental result. In Table 4, the first six columns show the load conditions, and in the next two columns, the results for \(d_{ax}\) are summarized and compared for the two methods. Moreover, \(d_{ax}\) was also calculated as a percentage of the crack propagation length \(a_{surf}\) measured on the specimen free surface. In this way, it was possible to compare all eighteen cases. The main goal was to verify the crack propagation next to the notch tip, where it can easily be identified on the experimental surface. Therefore, for each specimen, CRACKTRACER3D stopped the propagation, trying to keep the crack progress within this region (before the final rupture). To achieve a proper comparison, the numerical analyses were run trying to keep the same \(a_{surf}\) for both methods. For some load conditions, multiple test results were available. In such cases, the result with the clearest crack propagation surface was chosen for the comparison (the actual specimen number in that case is listed in Table 5.  
Figure 9: Discretized volume between the CRACKTRACER3D crack surface (in green) and the experimental one. For the dominant step criterion, the relative deviation (second, but last column in Table 4) was converted into an approximate deflection angle by \(\Delta\varphi=\tan^{-1}(d_{ax}[\%]\cos(\varphi))\). It is listed in Table 5 together with an estimate of the absolute deflection angle \(\varphi\) based on the experimental results. This estimate was obtained by orienting the triangulation of the fractured specimens in the thickness direction (Figure 10). The torsion applied to the specimens led to a positive deflection angle on one face of the specimen and a negative one on the other face. By looking in the thickness direction, both are superimposed.