**2.2 Fatigue**

Fatigue is characterized by a cyclic loading process that causes progressive internal structural cumulative damage. In this case, the Paris Law [33], Eq. (2), relates the crack propagation rate (*da/dN*) with the variation of the Stress Intensity Factor (ΔK):

$$\frac{da}{dN} = \text{C.} \Delta K^m \tag{1}$$

where *a* is the crack length, *N* is the number of load cycles, *C* and *m* are material dependent constants. According to classical theory, after a certain number of cycles, the cracks reach a critical length making the structure unstable and causing it to collapse. Thus, developing the Eq. (1), the condition for critical crack size *ac* is determined by:

$$N = \frac{1}{C} \int\_{a\_0}^{a\_c} \frac{da}{\Delta K^m} > N\_{tot} \tag{2}$$

where N is the number of cycles required to increase the crack of the initial size *a*<sup>0</sup> up to a critical crack length *ac* and *Ntot* is the number of cycles throughout life.

### **2.3 Boundary element method**

The Finite Element Method (FEM) is probably the main technique used in engineering analysis, standing out due to its great versatility, quality of results, and relative ease to implement [34]. On the other hand, the Boundary Elements Method (BEM) has emerged as a complementary alternative to FEM, being indicated particularly in special cases that require better interpretation and data representation in problems with stress concentration or where the domain is infinite or semi-infinite [35].

Thus, the boundary elements technique began to be used in problems of incremental crack extension analysis [36]. The solution of crack problems, in general, cannot be achieved in an analysis of a single region with direct application of the conventional BEM, because the application of the same boundary integral equation at coincident source points on the opposing crack surfaces leads to degeneracy in the resulting system of algebraic equations. Among the techniques applied to work around this problem are the sub-regions, which models the structure in artificial contours connecting the cracks to the boundary in such a way that the domain is divided into sub-regions without cracks [5], as well as the Dual Boundary Elements Method (DBEM), based on two distinct integral equations on each crack face (displacement and traction equations). Thus, degeneration of the equations system generated by the BEM is no longer present and the need for remeshing vicinity of the crack tip is not required, generating only new rows and columns to the existing matrix [6].

BemCracker2D is a program for elastostatic analysis of 2D problems to performing analyzes using the Boundary Element Method [37–41]. This software performs modeling of the standard BEM and the incremental analysis strategy for problems involving cracks [42]. For the analysis of the fracture mechanics problems, the BemCracker2D calculate elastic stresses using the conventional BEM and performs incremental analyzes of the crack extension through the DBEM.
