## 3 Finite Element Model and Structural Analysis of Fixators  
Model formulation implies the selection of the geometric, physical, mathematical and numerical approximation of the phenomena. The goals of the practical (not just qualitative) analysis restrict the number of possible models. Appropriate model selection is not always easy, since more complex models are also more efficient, but less effective. Models of the fixation device and tibia bone are discretized with linear (TE4) and parabolic (TE10) tetrahedron finite elements in the CAD/CAM/CAE system CATIA. A linear tetrahedron (TE4) is a finite element that belongs to the group of CST (constant strain tetrahedral) elements so that the deformation components in the element are constants. Namely, the interpolation functions of displacement within an element are linear functions of node displacement. Therefore, TE4 finite elements were used to discretize the fixator components that are predominantly loaded by pressure and tension without the presence of bending and complex stresses. These are small couplings, clamping ring, clamping plates, nuts and bone models. On the other hand, a parabolic tetrahedron (TE10) is an LST (linear strain tetrahedral), whose interpolation functions of displacement are second-order polynomials. Therefore, TE10 finite elements were used to discretize the fixator components that are predominantly loaded by bending and complex stresses. These are the frame, the half-pins and the big couplings. Linear tetrahedron contains four nodes in the vertexes and has 12 degrees of freedom. A parabolic tetrahedron is actually a curve-lined tetrahedron which has 10 nodes and linear strain rate inside the element. The FEM model of the fixation device is discretized with 120.953 TE4 finite elements (46.7%) and 109.816 TE10 finite elements (42.4%). Both of these elements correspond to group of 3D (three-dimensional) iso-parametric elements, i.e., solids with six edges. Approximation of the geometry and the field of the basic unknown variables for the iso-parametric elements is founded on the interpolation functions. Every node of these elements has 3 degrees of freedom. Connection between the parts of the fixation device (fastened, contact and pin connections) as well as supports (sliding connection) are modeled with the Spider type of elements (28.232 finite elements, i.e., 10.9%). Accordingly, the FEM model of the fixation device has a total of 259,001 finite elements with 262,665 nodes, which results in 787.959 degrees of freedom (Figure 3). Figure 3: FEM model of the Sarafix device; (**a**) B50 configuration, (**b**) C50 configuration. The device shown in Figure 1 is completely manufactured from stainless steel used for medical purposes. FEM analysis treats this material as isotropic and linearly elastic. Isotropic materials consider constitutive relations, i.e., stress-strain relations, with following constants: modulus of elasticity and Poisson's coefficient, which have value of E = 215 \(\times 10^{9}\) Pa i \(\
u\) = 0.29, respectively. The device shown in Figure 2 is manufactured from stainless steel (couplings, coupling carriers and half-pins) and composite material (device frame). As already mentioned, for the FEM model, stainless steel is treated as isotropic and a linearly elastic material, while composite is considered to be an-isotropic, and with cylindrical symmetry of its structure, it can be regarded as orthotropic material, which has attributes defined in three different planes defined with the transversal, tangential and radial cross-sections of the frame. Models of the bone segments are made out of beech tree with known mechanical properties (Table 1). This material belongs to the an-isotropic material group, and with the cylindrical symmetry attribute of the bone model, it can be viewed as orthotropic material. For the composite material and the bone segment material, it is common to predefine parameters such as modulus of elasticity, Poisson's coefficient and sliding modulus (Table 1).