## 2 Materials and Methods
### Boundary and Loading Conditions  
To simplify the model, bones and soft tissues were represented as a single body with distinct material properties, as detailed in Table 1. A surface-to-surface contact pair, using TARGE 170 and CONTA 174 elements with a coefficient of friction (COF) of 0.6, was defined to simulate the interaction between the stump and liner [9; 31]. The inner surfaces of the liner and the stump were designated as the target and contact bodies, respectively. This configuration followed ANSYS's surface-to-surface contact formulation, which stipulates that the stiffer material is assigned as the target body [45]. This configuration followed ANSYS's surface-to-surface contact formulation, which stipulates that the stiffer material is assigned as the target body, as shown in Figure 2.  
In ANSYS contact formulation, two bodies in contact carry frictional shear stress. When the frictional stress is less than the equivalent frictional limit stress (Flim), no frictional slip/ sliding occurs or vice versa [46]. This Flim can be expressed as  
\[\mathrm{F}_{\mathrm{lim}}=\mathrm{\mu P}+\mathrm{b} \tag{2}\]  
\[\left\|\left\|\mathrm{F}\right\|\right\|\leq\mathrm{F}_{\mathrm{lim}} \tag{3}\]  
where  
\(\mathrm{F}_{\mathrm{lim}}=\mathrm{Limit}\) frictional stress;  
\(\left\|\left\|\mathrm{F}\right\|\right\|=\left\{\begin{matrix}\left|F\right| \text{ {equivalent stress in 2D contact}}\\ \sqrt{{\mathrm{F}_{1}}^{2}+{\mathrm{F}_{2}}^{2}}\text{ {equivalent stress in 3D contact}}\end{matrix}\right.\);  
\(\mathrm{\mu}=\mathrm{coefficient\;of\;friction}\);  
\(\mathrm{P}=\mathrm{contact\;normal\;pressure}\);  
\(\mathrm{b}=\mathrm{contact\;cohesion}\).  
An Augmented Lagrangian algorithm was adopted for the stump and liner contact. This method is an iterative series of penalty updates to find contact tractions.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Geometry** & **Young’s Modulus (MPa)** & **Poisson’s Ratio** & **Density (kg/dm3)** \\ \hline Soft Tissue & 0.20 & 0.4 & 1.48 \\ Bone & 10,000 & 0.3 & 2 \\ Liner & 1.5 & 0.39 & 1.78 \\ Socket (PLA) & 3450 & 0.39 & 1.25 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Soft tissues, bone, and liner material properties.  
Figure 2: Contact detection point location [46].  
The contact pressure in Augmented Lagrangian is defined as  
\[P=\begin{cases}0&if\ \mu\ \text{m}>0\\ Kn\mu\text{m}+\text{Ai}+1&if\ \mu\ \text{m}\leq 0\end{cases} \tag{4}\]  
where  
\[\begin{cases}\lambda\text{i}+1+\text{Kn}\mu\text{n}&\text{if }|\mu\text{n}|>\varepsilon\\ \lambda\text{i}&\text{if }|\mu\text{n}|<\varepsilon\end{cases} \tag{5}\]  
\(\varepsilon\) = compatibility tolerance;  
\(\lambda\text{i}=\) Lagrange multiplier component at iteration i.  
The frictional slip was only permitted between the stump and liner. The liner was fixedly bound with the inner side of the socket, so there was no slippage between the liner and socket. The end of the socket was also given fixed support as it is a region of the pylon connection. The analysis was performed under two static loading conditions. i.e., Full Body Weight (FBW, 520 N) and Half Body Weight (HBW, 260 N). The desired loading was applied on the top of the bone to simulate the contact interface between the residual limb and liner.