## 3 Modeling of Crack Growth  
The study of crack propagation is of big interest in many applications. One of the most important equations that models the growth and propagation of cracks is the Paris equation.  
In fact, there are several models for the growth of cracks, such as the Paris, Duggan, Forman, and Tzamtzis models. Each one has its advantages and disadvantages. However, the Paris equation is the simplest, most famous, well-known, and mature model [40].  
In [41], the Paris law equation is used in aircraft application. They are concerned with aging of critical material in aircraft function of time. Experimental testing of aircraft material is very important; however, constant material properties are not enough to evaluate materials in such critical applications. Hence, they evaluate materials according to their dynamic capability to develop and propagate cracks.  
The author in [42] used the Paris equation to study crack growth and predict the remaining useful life of materials composing a magnet. In [43], the author use it for crack propagation in microelectronic device applications.  
Figure 10: Temperature at the machine’s periphery in the case of crack in one of the 12 magnets.  
Figure 9: Zoomed view of Figure 8.  
A crack growth rate is assessed for magnet systems in [44]. The paper was concerned with studying cracks with different orientations and different shapes. To realize this delicate modeling, the dual boundary element method and finite element method are coupled.  
When applying the Paris equation for crack growth prediction, some assumptions need to be made:  
* The crack is not of constant amplitude; it is propagating a function of time;
* The crack is one-dimensional;
* The material where the crack exists has a certain elastic condition;
* The load range is relatively constant;
* Sensor data and offline signals have similar time stamps;
* The offline data set contains enough data that represent different degradation behaviors.  
According to the Paris law equation, the general fatigue crack growth model is:  
\[\frac{da}{dNc}=C\left(\Delta K\right)^{n} \tag{1}\]  
\[\Delta K=Kmax-Kop \tag{2}\]  
\[Kop=Q*\sigma*\sqrt{\left(\pi\frac{a}{2}\right)} \tag{3}\]  
\[Q=\left(\cos\Bigl{(}\frac{\pi*a}{2*W}\Bigr{)}\right)^{-(\frac{1}{2})} \tag{4}\]  
\({}^{\prime}\frac{da}{dNc}\)' is the crack growth rate.  
'a' is the depth of the crack. 'Nc' is the number of cycles.  
'C' and 'n' are calibration parameters dependent on the type of material where the crack is propagating. 'C' is called the Paris equation parameter or crack growth coefficient, its unit is [mm/(MN m\({}^{-3/2}\))\({}^{n}\)] per cycle. 'n' is called the Paris equation exponent or crack growth exponent, and it is unitless.  
'\(\Delta\)K' is the effective stress intensity factor. 'Kmax' is the stress intensity factor at the peak load; it is a critical or a threshold value before the occurrence of a fracture. 'Kop' is the operating stress intensity.  
Maxwell stress tensor '\(\sigma\)' will be calculated from the vibration FEM.  
'Q' is a parameter dependent on the geometry of the system.  
'W' is the radial length of the magnet.  
A familiar relation between Nc and a is:  
\[Nc=\frac{1}{\left(C\;\Delta\sigma\sqrt{\pi}\;Q\right)^{n}}*\frac{1}{\frac{n}{2 }-1}*\left(\frac{1}{a0^{\frac{n}{2}-1}}-\frac{1}{af^{\frac{n}{2}-1}}\right) \tag{5}\]  
'Nc' is a cyclic loading, and it is a cycle representation of repetitive or inconsistent stress intensities on a certain location. 'a0' is the current depth of the crack. 'af' is the final depth that will reach the crack and at which the magnet is fractured.  
Nc, calculated according to the above equation, is the RUL for the crack to move from a0 to af depth.  
If we aim to calculate the time needed for the crack to propagate from 1 mm depth to a specific incremental depth, let us say 2 mm, the final state of the crack 'f' in this case is 2 mm.