## 3 Finite Element Model and Structural Analysis
### Structural Analysis of External Fixation Device  
During exploration, external fixation device is loaded with complex stress, which can be considered as a combination of pressure, bending and torsion. In most of the studies of external fixation devices using FEM numerical methods or experimental methods, separated studies are done for: axial pressure testing, sagittal or anterior-posterior (AP)  
Figure 4: FEM model of external fixation device. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Properties** & **Index** & **Value** \\ \hline Radial modulus of elasticity & \(\mathrm{E}_{\mathrm{R}}\), \(\mathrm{E}_{11}\) & 2060 MPa \\ \hline Tangential modulus of & \(\mathrm{E}_{\mathrm{T}}\), \(\mathrm{E}_{22}\) & 1120 MPa \\ elasticity & \(\mathrm{E}_{\mathrm{L}}\), \(\mathrm{E}_{33}\) & 15,400 MPa \\ \hline Poisson’s ratio in RT plane & \(\mathrm{v}_{\mathrm{RT}}\), \(\mathrm{v}_{12}\) & 0.66 \\ \hline Poisson’s ratio in RL plane & \(\mathrm{v}_{\mathrm{RL}}\), \(\mathrm{v}_{13}\) & 0.055 \\ \hline Poisson’s ratio in TL plane & \(\mathrm{v}_{\mathrm{TL}}\), \(\mathrm{v}_{23}\) & 0.037 \\ \hline Sliding module in RT plane & \(\mathrm{G}_{\mathrm{RT}}\), \(\mathrm{G}_{12}\) & 450 MPa \\ \hline Sliding module in RL plane & \(\mathrm{G}_{\mathrm{RL}}\), \(\mathrm{G}_{13}\) & 1530 MPa \\ \hline Sliding module in TL plane & \(\mathrm{G}_{\mathrm{TL}}\), \(\mathrm{G}_{23}\) & 1170 MPa \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mechanical properties of a beech tree. bending, and torsion [20; 21; 22; 23]. The basic external load of external fixation device is axial pressure [24]. FEM model for simulation of axial pressure is developed, taking into account the complex geometrical structure of the external fixation device, human bone, connections, loads, and constrains. During axial pressure experimental testing, bone models are supported using a spherical joint. Force load was applied in amount of F\({}_{\text{P}}\) = 0-600 N with the speed of increase in amount of 5 N/s. Values of loads correspond to the physiological load of the fixator after its application to the patient, and it is defined according to the in vivo testing [25; 26]. Structural analysis is carried out taking into consideration the following assumptions in correspondence to real working conditions: loads are quasi-static, supports are modeled using spherical joints without friction, connection between half pins and bone are modeled using fixed constrain, properties of bone material are chosen same as properties of beech tree. Open fracture is modeled with fracture gap of 50 mm, which represents serious injury, with serious bone damage. FEM model of external fixation device before, and after applying loads for axial pressure are given on Figure 5, together with interfragmentary displacements. The directions of displacements have a big influence on the fracture heal process. Using FEM models, it is possible to track interfragmentary displacements inside the fracture gap. Better stiffness of fixation can help during the early stage of the healing process. The usage of external fixation device with height stiffness in the early stage of the healing process is an important factor in the reduction of stress and deformation in the complete fixator-bone system, and to create an optimal environment for fast healing and the minimal immobility of a patient. The main goal of usage of external fixation device is fast healing. From a literature review [20; 27; 28], it can be concluded that optimal biomechanical environment, for optimal healing of a bone, among other things, is based on the following two points of view: limited periodical axial displacements of bone segment at the fracture gap stimulate its healing after inflammation. This process is called dynamization. Transverse displacements (craniocaudally and lateromedially) are harmful. To define  
Figure 5: Non-deformed and deformed structures of the fixator under maximum axial load and interfragmentary displacements at the fracture.