# Study of Mixed-Mode Cracking of Dovetail Root of an Aero-Engine Blade Like Structure
## 6 Conclusions  
Several blade dovetail root geometries with different flank angles and lengths were analysed with the same structural loads under the same initial flaw, by using Franc3D\({}^{\copyright\) and Abaqus. Effects of friction coefficient and the utilisation of an equivalent Keq calculation method were also studied. The following was concluded:  
* The 45\({}^{\circ}\) flank angle offers a lower crack propagation rate compared to the 30\({}^{\circ}\) and 60\({}^{\circ}\) geometries. In other words, the damage tolerant life offered by the 45\({}^{\circ}\) solution is better than the other two flank angle options.
* For a given crack length, the stress intensity K1 given by the short and long flank (for given flank angle) is comparable. The short flank has, in some cases, slightly higher values of stress intensity  
Figure 30: Contact flank stress re-distribution as a function of the crack progression.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Geometry**} & **Flank** & **Friction** & **Crack Length at** & **Number of Cycles** \\  & **Length** & **Coefficient** & **810 MPa mm\({}^{0.5}\)** & **at 810 MPa mm\({}^{0.5}\)** \\ \hline
45\({}^{\circ}\) & Short & 0.7 & 14.73 & 17,842 \\
45\({}^{\circ}\) & Long & 0.7 & 7.80 & 17,644 \\
45\({}^{\circ}\) & Short & 0.1 & 11.16 & 17,805 \\
60\({}^{\circ}\) & Short & 0.7 & 1.845 & 16,528 \\
60\({}^{\circ}\) & Long & 0.7 & 2.38 & 16,836 \\
30\({}^{\circ}\) & Short & 0.7 & 8.61 & 17,681 \\
30\({}^{\circ}\) & Long & 0.7 & 9.98 & 17,734 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Damage tolerant life of the analysed geometries.  
factors, but a design choice of longer flank will have the disadvantage of increasing the weight of the structure only to obtain a negligible advantage in terms of crack propagation. This advantage would not exist in the 45\({}^{\circ}\) design option.
* The untwist of the leading and trailing edge tip displacements are virtually unaffected by the crack propagation. The stress re-distribution along the flank length, however, may induce the nucleation and propagation of different (and numerous) crack fronts. This case has not been considered in this work.
* The reaction, R, and hence the friction force on the flank surface, \(\mu R\), increase inversely with \(\vartheta\), i.e., \(R\propto~{}1/(\cos\vartheta)\). Once the blade is loaded under centrifugal force, the friction at the flank surface will restrain the blade from sliding up against the disc flank and hence more strain energy would be released at the crack front which will extend the crack further towards the blade root. In other terms, the higher the friction, the faster the crack would propagate.
* A low friction coefficient (given for example by a lubricated contact) is beneficial in slowing down the crack propagation, even if its influence is not as massive as expected.
* Setting the failure criterion in Franc3D to K\({}_{\text{I}}\) will have no significant difference from using K\({}_{\text{eq}}\) to calculate the K at crack sizes below 12 mm as the contribution of other modes in the K\({}_{\text{eq}}\) equation is not significant. However, more simulations would be required to confirm this trend for higher crack sizes and blade geometries.
* No influence of vibration was studied. This will be part of future investigations.  
Formal analysis, G.C. and M.K.; Visualization, A.M. and P.S.; Writing--original draft, G.C. and M.K.; Writing--review & editing, A.M., P.S. and R.C.  
This research received no external funding.  
The authors would like to thank [PERSON], [PERSON], [PERSON] (Rolls-Royce plc, Derby, UK), and [PERSON] for their ongoing support. Without their contribution this work would not have been possible.  
The authors declare no conflict of interest exists as a result of publishing this work.