# Mechanical and Fatigue Behavior of Cellular Structure
## 2 Material and Methods
### Finite Element Model
#### 2.1.2 Porous Stem Design  
The designed porous cellular structures were utilized to design twelve different functionally graded porous stems and three homogeneous porous stems with average volumetric porosities of 30%, 50%, and 70%. Additional porous structures were designed that are needed for designing these 15 stems as shown in Figure 2. The fifteen designed stems were divided into three groups representing these aforementioned average volumetric porosities, each group contains four functionally graded designs and one homogeneous design The functionally graded porous structures were created within the stem's layers in the radial direction, the layers cross-sectional areas are 15.9 mm\({}^{2}\), 17.3 mm\({}^{2}\), and 23.6 mm\({}^{2}\) for L1 (the stem's core), layer 2 (L2), and layer 3 (L3), respectively. Figure 2 illustrates the porosities configurations within the stem's layers to achieve the required overall average volumetric porosity of the pertinent group. The average porosities of the designed stems in a radial direction were calculated based on Equation (2). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Sample**} & \multirow{2}{*}{**Strut Size (t)**} & \multicolumn{2}{c}{**Number of Pores/Unit Cells**} & \multirow{2}{*}{**Pore Size (L2)**} & \multirow{2}{*}{**Volumetric Porosity (\%)**} \\  & & **Enclosed within Each** & & \\  & & **Dimensional Direction the Cubic** & & \\ \cline{2-4}  & **(mm)** & **(N)** & **(mm)** & \\ \hline
1 & 0.30 & 10 & 1.36 & 90\% \\
2 & 0.30 & 12 & 1.00 & 85\% \\
3 & 0.30 & 14 & 0.80 & 80\% \\
4 & 0.32 & 15 & 0.72 & 76\% \\
5 & 0.32 & 17 & 0.60 & 70\% \\
6 & 0.34 & 18 & 0.49 & 62\% \\
7 & 0.34 & 18 & 0.47 & 60\% \\
8 & 0.36 & 20 & 0.38 & 50\% \\
9 & 0.38 & 21 & 0.30 & 40\% \\
10 & 0.40 & 23 & 0.23 & 30\% \\
11 & 0.42 & 24 & 0.18 & 20\% \\
12 & 0.44 & 26 & 0.11 & 10\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Designed samples of porous cubes with the relevant volumetric porosities and obtained mechanical properties from the FEA. Figure 2: _Cont_. These fifteen designs (D1 to D15) were integrated into three-dimensional (3D) finite element models in ABAQUS 6.17 software to study the mechanical behavior of the designed stems under physiological loading conditions obtained from walking at speeds of 1 km/h, 3 km/h, and 5 km/h. The stem-bone assembly as shown in Figure 3 was built based on ISO 7206-4 standard [11, 28] using surface to surface contact between the stem and the bone interface surfaces with a 0.4 friction coefficient. The assembly was fixed into cylindrical shape epoxy (3.7 GPa) with specific orientations shown in Figure 3, the bone was tied with the epoxy and constrained to move or revolve in any direction [7]. The 3D model of the femur was constructed using computed tomography (CT) scanned images of a patient femur as explained in previous studies [7, 29]. Cortical and trabecular bone were considered homogeneous and linear elastic isotropic material, respectively, with material properties as listed in Table 2[24, 30, 31]. \[\overline{P}=\left(V_{P}\right)_{h}+\frac{\left(\overline{AV}_{p}\right)_{fg}- \left(\overline{AV}_{p}\right)_{h}}{\left(\overline{AV}_{p}\right)_{h}} \tag{2}\]  
where  
A: Stem's radial Layer cross-sectional Area  
\(V_{p}\): Stems radial Layer Volumetric Porosity  
\(\left(\overline{AV}_{p}\right)_{fg}\): Average functionally graded Porosity along the stem's layers  
\(\left(\overline{AV}_{p}\right)_{h}\): Average homogeneous Porosity along the stem's layers  
\(\left(V_{p}\right)_{h}\): Stem Homogenous porosity. Figure 2: Design groups of porous functionally graded (FG) and homogeneous designs introduced to stem’s radial layers: (**a**) average porosity of 30%, (**b**) average porosity of 50%, and (**c**) average porosity of 70%.