## 2 Materials and Methodology
### Fracture Mechanics Analysis under Monotonic and Cyclic Loading Conditions
#### 2.3.2 Under Cyclic Loading Conditions  
It was noted that the fatigue crack initiation dealt with the apparition of the first micro crack (often taken as the smallest detective size of the crack), while crack propagation dealt with the propagation of the initial crack until the final failure of the structure. In order to investigate the crack propagation, key information is the initial location and geometry of the initial crack. In this study, two different strategies were considered to determine the location of the crack initiation, respectively, based on the clinical observations and on the age-related critical point identification process, using fatigue crack initiation analysis. Regarding the first strategy, the clinical observations in the literature [10] was used to insert the circular and planar crack of radius 0.01 m, as shown in Figure 3a. The material \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Elastic Modulus (GPa)** & **Poisson’s Ratio** \\ \hline Cortical bone (24-year-old) [29] & 16.6 & 0.3 \\ Cortical bone (27-year-old) [19] & 15.3 & 0.3 \\ Cortical bone (56-year-old) [19] & 13.4 & 0.3 \\ Cancellous bone [29] & 2.4 & 0.3 \\ Tibial prosthesis (Implant) [29] & 117 & 0.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: TKR knee material properties. properties of the 24-year-old patient reported in Table 1 was considered. The corresponding numerical analyses were conducted with and without implant. Regarding the second strategy, a critical point was identified as the point at which the fatigue damage was maximal. At that critical point, the damage could be accumulated during cyclic loading and consequently, a crack could be formed. Additionally, the fatigue damage (i.e., D = 1/N) could be deduced from the number of cycles N whose value was determined by interpolating from a diagram of endurance, which is also known as the Wohler curve (i.e., the S-N curve). In the current work, the Wohler curve was reconstructed using the experimental data from the tensile tests on two different age populations (27 and 56-year-old) [20], as shown in Figure 3b. To identify the critical point for the age-related analyses, a workflow was developed, as sketched in Figure 4, which was detailed as follows. Figure 4: The flow chart of the age-related critical point identification process. Figure 3: Location of the crack initiation based on the clinical observation (**a**) and the Stress-Cycles to fatigue diagram (the so-called SN curve) from the accumulation of fatigue micro-damage in the human cortical bone of two different ages used in the age-related critical point identification process extracted from [20] (**b**). First, the tibia bone modeled with the implant in the cases of the two different ages (27-year-old female and 56-year-old female) were used in this strategy, using the material properties reported in Table 1. These models were subjected to a prescribed displacement in the ranges of 0.00026 m-0.00041 m and 0.00029 m-0.0005 m, respectively, for the 27- and 56-year-old patients. Then, the corresponding FEM simulations were performed to find the point at which the stress was maximal. Since the models were loaded under the axial direction (i.e., Y-axis), only the maximum value of the YY-component (denoted by SIGYY_max) of the stress tensor was considered. The history of SIGYY_max at that point was extracted. Finally, using the Wohler curve shown in Figure 3b, the number of cycles N was estimated by interpolation, from which the fatigue damage (D = 1/N) could be estimated. It was noted that if the fatigue damage was not present, the crack could not be formed. On the other hand, if the fatigue damage was positive, it was confirmed that a crack was formed at that location. It should be noted that this approach could be generalized in the more complex loading conditions. In this particular study, the location of the maximum damage was identical to that of the maximum SIGYY, due to the fact that the bone was mainly under uniaxial loading conditions.