# Mechanical and Fatigue Behavior of Cellular Structure
## 2 Material and Methods
### Fatigue Analysis of the Designed Porous Stems  
A fatigue study was performed on the designed stems implanted inside bone by performing finite element analysis based on physiological loading conditions obtained from the three walking speeds as presented in Table 3. Von Mises stress on the stem/stem layers was measured using ABAQUS FEA software and the results were used to calculate the factor of safety as illustrated in Figure 4. Designs D12 and D14 denote that no results were obtained as the FEA simulation does not converge due to the high volumetric porosity at the outer layer of these stems (Figure 4) [11].  
The Soderberg approach was used for calculating the factor of safety [24] as shown in Equations (3)-(6):  
The minimum stresses and the maximum stresses were measured at the loads associated with a stress ratio of 0.1 (R = 0.1) during the test:  
\[\sigma_{m}=\frac{(\sigma_{max}+\sigma_{min}\,)}{2} \tag{3}\]  
\[\sigma_{a}=\frac{(\sigma_{max}-\sigma_{min}\,)}{2} \tag{4}\]  
where; (\(\sigma_{m}\)) and (\(\sigma_{a}\)) denote mean and alternating stresses generated in the porous stems and (\(\sigma_{min}\)) and (\(\sigma_{max}\)) are refereeing to minimum stresses at 0.1 of the applied physio  
Figure 4: Von Mises stress obtained from the FEA for the fifteen designed stems planted inside the femur and subjected to the three physiological loads (1 km/hr, 3 km/hr, and 5 km/hr).  
logical load and maximum stresses associated with the full load generated from each walking speed.  
\[\frac{\sigma_{a}}{S_{e}}+\frac{\sigma_{m}}{S_{ys}}=\frac{1}{N} \tag{5}\]  
Hence, the fatigue factor of safety \(N_{f}\) is as follows:  
\[N_{f}=\frac{1}{\left(\frac{\sigma_{a}}{S_{e}}+\frac{\sigma_{m}}{S_{sp}}\right)} \tag{6}\]  
Whereas \(S_{e}\), \(S_{ys}\), and \(N\) denote the endurance limit of the porous material calculated using Equation (7) [36], yield strength of the porous material, and factor of safety, respectively.  
\[S_{e}\ =K_{a}\times K_{b}\times K_{c}\times K_{d}\times K_{e}\times K_{f}\times S _{u} \tag{7}\]  
\(K_{a}\): Surface quality factor.  
\(K_{b}\) : Size factor.  
\(K_{c}\): Reliability factor.  
\(K_{d}\): Temperature factor.  
\(K_{e}\): Stress concentration factor.  
\(K_{f}\): Miscellaneous factor.  
For Ti6 Al4V [37]: \(S_{e}\ =\)0.54 \(\times\)\(S_{u}\)  
\(S_{u}\): The Ultimate tensile stress of the porous material, (the \(S_{u}\) for the outer stem's porous layer for each design was used where it's associated with the maximum stresses obtained from each walking speed), the reduction percentage in \(S_{u}\) and the calculated endurance limit \(S_{e}\) are shown in Table 4 below:  
\begin{table}
\begin{tabular}{c c c c c} \hline \multicolumn{5}{c}{**Ultimate Tensile Strength (Su), Endurance Limit, Elastic Moduli and Yield Strength (Ys) for the**} \\ \multicolumn{5}{c}{**Designed Porous Structures**} \\ \hline
**Volumetric Porosity** & **E (GPa)** & **Ys (MPa)** & **S\({}_{u}\) (MPa)** & **Reduction Factor** & **S\({}_{e}\) (MPa)** \\ \hline
90\% & 3.50 & 31.14 & 33.27 & 17.97 \\ \hline
85\% & 7.60 & 50.17 & 58.47 & 31.58 \\ \hline
80\% & 11.68 & 77.52 & 90.99 & 49.13 \\ \hline
77\% & 14.13 & 91.37 & 107.46 & 58.03 \\ \hline
76\% & 16.18 & 105.95 & 113.50 & 61.29 \\ \hline
70\% & 19.76 & 123.28 & 165.13