## 2 Methodology  
### Blade Damage Model  
Fatigue damage occurs when the residual strength of a piece of material subjected to periodic loading equals the maximum stress within a cycle period [27]; the residual strength is a function of the degradation of the properties of a material, occurring in response to static or dynamic loads. In the case of fatigue-only damage, there is a distinct number of load cycles \((N)\), after which the material fails if excited with a given stress amplitude and mean. In a homogeneous sample, often, a characteristic function governing the decay of the residual strength with the number of cycles can be determined, either experimentally or through micro-mechanical simulation, although the details of fracture mechanics, particularly in composite materials, turn even this well-defined problem into an undertaking with considerable complexity. In the case of more realistic geometries, such as composite beams, the complexities of a non-homogeneous stress field can be addressed by Progressive Failure Analysis (PFA) (see, e.g., [28; 29]), while still assuming that the results for fatigue degradation obtained from a (small) homogeneous sample remain approximately valid, an approach (as most approximate engineering procedures) which is not without criticism. While PFA may or may not be completely adequate, engineering design approaches for wind turbine fatigue design, as already discussed in the literature review, are still considerably more simplified and simply focus on one critical location within the structure (generally located at the blade root), limiting themselves to calculating the number of cycles at which the blade root material (as determined from a small sample) fails. Though, in practice, the order in which different stress amplitudes and mean values occur might be of relevant, the standard procedures, such as the ones recommended in the IEC61400-2 norm, neglect such effects. Fatigue damage is measured by a metric such as the cumulative damage \(D\) in the Palmgren-Miner (PM) rule [16]:  
\[D=\sum_{i}\frac{n_{i}}{N_{i}} \tag{1}\]  
where, \begin{tabular}{l l} \(D:\) & the cumulative damage (\(D=1\) when a fault occurs) \\ \(n_{i}:\) & the number of load cycles for a given stress amplitude/mean combination \\ \(N_{i}:\) & the number of permissible load cycles for a given stress amplitude/mean combination \\ \end{tabular}   
The fatigue life estimation then requires determining the number of load cycles \(n_{i}\)[30]. As in most fatigue research, the rainflow method was used to extract the number of events for a given stress amplitude/mean combination from the relevant input time series. As mentioned above, the stress time series obtained from an aeroelastic simulation of the blade under normal operation conditions at different average wind speeds, as extracted at the blade root, was used as input to the rainflow counting algorithm in all occasions. In order to calculate the permissible number of cycles for a given stress state, based on the Goodman method, the following formula was used [23]:  
\[N=\left[\frac{X_{t}+|X_{c}|-|2\gamma_{Ma}\sigma_{1,m}-X_{t}+|X_{c}||}{2\left( \gamma_{Mb}/C_{1b}\right)\sigma_{1,a}}\right]^{m} \tag{2}\]  
where,  
\[X_{t}:\] maximum tensile strength (MPa) \[X_{c}:\] maximum compression strength (MPa) \[\sigma_{1,m}:\] average stress of a cycle calculated by the rainflow method (MPa) \[\sigma_{1,a}:\] stress amplitude of a cycle calculated by the rainflow method (MPa) \[m:\] slope of the \[S-N\] diagram \[\gamma_{Ma,Mb}:\] safety factors  
The definition of the safety factors, as well as \(m\), was given in [31]. The value assigned to \(m\) depends on the material of the blade. Table 1 shows some values assigned to \(m\) for three different materials. The safety factors are calculated according to:  
\[\gamma_{Mx}=\gamma_{M0}\prod_{i}C_{ix} \tag{3}\]  
where \(\gamma_{M0}=1.35\), and the values assigned to \(C_{ix}\) are selected according to the material used in the blade manufacturing.