## 2 Materials and Methodology
### Fracture Mechanics Analysis under Monotonic and Cyclic Loading Conditions
#### 2.3.2 Under Cyclic Loading Conditions
For both strategies, one location of the initial crack was determined, a circular and planar crack of radius 0.01 m on the plane Oxz passing through the initial location was inserted at that point for further crack propagation analysis. When subjecting the cyclic loading, after being initiated, the crack would propagate in the bone until final failure. Under the framework of linear elasticity, [PERSON] [16] proposed that the number of cycles required for the fatigue of a specimen with a crack could be related to the amplitude of the crack's stress intensity factor \(\Delta K\). Let us consider the specimen under cyclic external loading \(P\) ranging from \(P_{min}\) to \(P_{max}\), the constitutive equation of the Paris' law is expressed as follows:  
\[\frac{da}{dN}=C\big{(}\Delta K_{eq}\big{)}^{n} \tag{5}\]  
where _da_/_dN_ is the crack growth rate, \(a\) is the crack size, \(N\) is the number of fatigue cycles, \(C\) and \(n\) are the two material constants, and \(\Delta K_{eq}=K_{max}-K_{min}\) is an equivalent stress intensity factor, in which \(K_{max}\) and \(K_{min}\) are the stress intensity factors, respectively, in the cases of \(P_{max}\) and \(P_{min}\). In this work, \(C=10^{-8}\ \left(\frac{m}{cycle}\frac{1}{\left(MPa\sqrt{m}\right)^{n}}\right)\) and \(n\) = 2.8 were considered from the literature [31; 32; 33]. The crack propagation was modeled using eXtended Finite Element Method (X-FEM), in combination with the automatic mesh adaptivity through the Salome-Meca software. The eXtended Finite Element Method (X-FEM) is becoming a popular method for modeling cracks in the linear elastic fracture mechanics (LEFM) approach [18; 25]. In comparison to the classical finite element method where the crack needs to be meshed, the X-FEM method employed an analytical solution to represent the crack in the mesh without an explicit crack mesh. In simple terms, the method stands on enriching the finite element model with the addition of degrees of freedom, which are corresponded to the nodes of the elements that the crack pass-through. One of the advantages of the X-FEM relates to the fact that the crack path does not necessarily need to be defined a priori. Additionally, the laborious and tedious re-meshing during each step of crack propagation could be avoided. This method was used to study bone fracture and crack propagation analysis [18]. In fact, the crack becomes a structure that one could insert into the numerical model. For this reason, X-FEM is employed in this study. Note that X-FEM is only involved in crack representation. Once the model is built, a dedicated module of fracture mechanics analysis in code-aster was employed to compute the fracture mechanics parameters \(K\) and \(G\), as well as to perform crack initiation and propagation analyses. The values of \(K_{I}\) and of \(G\) along the crack tip for different meshes are shown in Figure 7 for the tibia models without implant (Figure 7a,b) and with the implant (Figure 7c,d). The computational times for the model without the implant were 71.68 s, 121.35 s, 194.15 s, and 237.77 s, respectively, for coarse, medium, fine, and very fine meshes, using a single Core i7 processor, while for the model with the implant, the computational times, respectively, were 126.69 s, 169.12 s, 248.13 s, and 259.79 s.  
Figure 6: Four meshes of the tibia model with implant (TKR model)—coarse (**a**), medium (**b**), fine (**c**), and very fine (**d**) meshes. Figure 7: Mesh size effect on the values of \(K_{I}\) and of \(G\) versus the curvilinear coordinate along the crack tip—the tibia model without the implant (**a,b**) and the tibia model with the implant (**c,d**). As shown in Figure 7, the mesh convergence of the values of \(K_{I}\) and of \(G\) along the crack tip was observed.