## 2 Materials and Methods
### Theoretical Considerations  
The designing and analysis of hybrid EFs involves several mechanical and biomechanical considerations, including material properties, load distribution, and stress analysis. Based on the principle of force equilibrium and stiffness, when an axial load is applied to a system consisting of a bone and a fixator, the applied forces are distributed between the two bodies. The resulting total load must be conserved. Thus, the sum of the forces acting in the bone and the fixator must equal the applied load:  
\[F_{applied}=F_{bone}+F_{fixator} \tag{1}\]  
where \(F_{applied}\) describes the total axial load applied to the system, \(F_{bone}\) represents the axial force transferred by the bone, and \(F_{fixator}\) the axial force transferred by the fixator. In the initial healing phase, the stiffness of the bone is assumed to be negligible. Thus, the entire load is transferred via the fixator. The load carried by the fixator will be equal to the total applied load:  
\[F_{applied}=F_{fixator} \tag{2}\]  
As callus formation progresses, the stiffness of the bone increases gradually. This stiffness contributes to the overall system as a part of the load distribution. The bone and the fixator share the load, and the load on each component depends on the relative stiffness of the bone, callus area, and the fixator. The load is transferred inversely proportionally to the stiffness. In a simplified form, the length is not considered explicitly, assuming both the bone and fixator have enough length to allow their respective stiffness to contribute meaningfully to the overall load sharing. Thus, the element with higher stiffness will carry more of the total load. The stiffness of the system can be evaluated by using the following relations:  
\[k_{bone}=E_{bone}\cdot A_{bone} \tag{3}\]  
\[k_{fixator}=E_{fixator}\cdot A_{fixator} \tag{4}\]  
The total stiffness of the system, assuming the bone and fixator are acting in parallel, is the stiffness sum of the individual elements:  
\[k_{total}=k_{bone}+k_{fixator}=E_{bone}\cdot A_{bone}+E_{fixator}\cdot A_{ fixator} \tag{5}\]  
The load carried by each component is proportional to the stiffness of the component relative to the total stiffness. Hence, the force on the fixator is proportional to the stiffness of the fixator relative to the total stiffness:  
\[F_{fixator}=\frac{k_{fixator}}{k_{total}}\cdot F_{applied} \tag{6}\]Substituting for stiffness, equation 1 provides a generic overview of how the axial loads are transferred from the bone to the fixator system:  
\[F_{fixator}=\frac{E_{fixator}\cdot A_{fixator}}{E_{bone}\cdot A_{bone}+E_{ fixator}\cdot A_{fixator}}\cdot F_{applied} \tag{7}\]  
Pre-tensioning the wires constrains the initial compressive force at the fracture site. This ensures the required stability by minimizing the micromotion during the early healing phase. Compressive loads alter the fixator stiffness by increasing its resistance to external forces. As the bone heals and its stiffness increases, the load is gradually shared more evenly between the bone and the fixator. This critical design consideration can be estimated by taking into account the elastic deformation of the wires:  
\[\Delta L=\frac{F_{tension}\cdot L}{A_{wire}\cdot E_{wire}} \tag{8}\]  
where \(\Delta L\) represents the elongation of the wire, \(F_{tension}\) is the pre-tensioning force applied to the wire, and \(E_{wire}\) the modulus of elasticity of the wire material. However, the bending of the EF ring due to the out-of-plane loads influences the stress experienced by the wires. Depending on the fixator configuration, the wires behave similarly to a cantilevered beam or a simply supported beam, experiencing both deflections and rotations. Thus, the tensile stresses are most relevant for evaluating their structural integrity. The calculated values are compared to the tensile yield strength of the material.