## 4 Numerical Analysis and Results
### Hard Structures  
At this stage, the component of the rib was modeled with the separation of the outer compact part and the inner spongy part. Tables 7 and 8 show the mechanical parameters necessary to develop numerical models of the mentioned parts. The compact entity was modeled using the finite element method (FEM), while for the internal spongy structure, due to its biological-mechanical nature, it was decided to model the structure using the smooth particle hydrodynamics-finite element method (SPH/FEM). \begin{table}
\begin{tabular}{c c c c} \hline
**Source** & **Skin Source** & **Velocity** & **Young’s Modulus** \\ \hline Khaothong [25; 26] & Inner forearm & Quasistatic (1 mms\({}^{-1}\)) & 0.1–2.4 MPa \\ [PERSON] and [PERSON] [25; 26] & Tibia/Fibula & Quasistatic (0.5–1 mms\({}^{-1}\)) & 10.4–89.4 kPa \\ [PERSON] et al. [27] & Forearm (right) & Dynamic (1–10 μm for 10–60 Hz) & 5.1–13.3 kPa \\ [PERSON] et al. [28] & Forearm (right) & Dynamic (1–10 μm/100–500 μm for 10–60 Hz) & 13.2–33.4 kPa \\ [PERSON] et al. (laser displacement method) [29] & Forearm & Dynamic (2–100 ln min\({}^{-1}\)) & 4.75–17.99 kPa \\ \hline \end{tabular}
\end{table} 
Table 6: Example Young’s modulus data for selected sections of leather. \begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Value** & **Unit** \\ \hline Density (\(\rho\)) & 1850.0 & kg/m\({}^{3}\) \\ Young’s modulus (\(E\)) & 1.80 & GPa \\ Poisson’s ratio (\(v\)) & 0.35 & - \\ \hline \multicolumn{3}{c}{Johnson–Cook Model} \\ \hline Yield strength (\(A\)) & 160.0 & MPa \\ Fixed fortification (\(B\)) & 165.0 & MPa \\ Exponent of consolidation (\(n\)) & 0.1 & - \\ \hline \end{tabular}
\end{table} 
Table 7: Mechanical parameters of the cortical bone material. The average length of a rib for adult humans is between 200 and 300 mm, and its width/depth usually oscillates between 25 and 50 mm. These measurements were taken along the curved length of the rib [32]. For the purposes of the study, these values were averaged over most of the population. Discretization was assumed at 0.5 mm using Tetra-type elements. Figure 7 shows the modeled rib. The numerical analysis performed is intended to represent a patient study, so factors such as material strengthening due to deformation or temperature effects were ignored. The Johnson-Cook rheological model was used to describe the nature of the stress-strain mechanical properties of bone. This is a rheological model for describing elastic-plastic materials with nonlinear strengthening. The apparent linear nature of the material's response to a given load in the first part is described by the elastic range using [PERSON]'s modulus and Poisson's ratio. Then, once this limit is exceeded, nonlinear strengthening of the material is visible, for which the Johnson-Cook mathematical model is responsible in the numerical analysis. The Johnson-Cook model is a function of von Mises flow stress according to deformational strengthening, strain rate strengthening, and thermal softening. The equation below represents the Johnson-Cook model. The temperature and strain-rate strengthening effects are omitted from the analysis, omitting the second and third parts of the equation. \[\sigma_{eq}=\Big{(}A+B\Big{(}\varepsilon_{eq}^{pl}\Big{)}^{m}\Big{)}\Big{(}1+ Cln\Big{(}\frac{\dot{\varepsilon}}{\dot{\varepsilon}_{0}}\Big{)}\Big{)}\Big{(}1- \Big{(}\frac{T-T_{0}}{T_{melt}-T_{0}}\Big{)}^{m}\Big{)} \tag{7}\]  
Here, \(A\), \(B\), \(C\), \(n\), and \(m\) are the material constants; \(T_{melt}\) is the melting point; \(T_{0}\) is the initial temperature; \(\varepsilon\) is the plastic strain; \(\dot{\varepsilon}\) is the plastic strain rate; and \(\dot{\varepsilon}_{0}\) is the reference strain rate. Through perpendicular application of the wave source, the displacements occurring depending on the time steps were analyzed. Figure 8 shows example values of this parameter for six time steps. For the description of the phenomenon, it was decided to use gray tones due to the better representation of the wavefront.