## 3 Results
### Bat Model  
The methods described for the modeling of the maple wood in the Charpy impact simulations were applied to the finite element modeling of bat/ball impacts. A finite element model of a popular professional bat profile that was previously tested in the Bat Durability Test System (Automated Design Corporation) at the UMLBRC was built for subsequent analysis in LS-DYNA. The finite element model of the bat was constructed containing 130,554 solid elements and 137,548 nodes with a mesh size of approximately 0.2 in. (0.4920 cm). This maple bat model was 34 in. (86.4 cm) in length with a prescribed wood density of 0.0250 lb/in\({}^{3}\) (692.0 kg/m\({}^{3}\)) to represent a 31 oz. (0.879-kg) bat, thereby matching the profile, length and weight of the lab tested maple bat. The finite element model of the baseball that was used in this research was constructed in LS-PrePost. The geometry of the ball model is a sphere of 1.4 in. (3.6 cm) radius and consists of 12,096 solid elements and 12,589 nodes with a uniform mesh size of approximately 0.2 in. (0.556 cm). The LS-DYNA *MAT_VISCOELASTC (*MAT_006) was used to define the material behavior of the ball. The baseball material properties are summarized in Table 5[20]. The finite element model of the Bat Durability Test System consisted of the rotating back plate and the rollers that grip the bat. The LS-DYNA *MAT_ELASTIC (*MAT_001) was used to implement the properties for each of the rollers. The LS-DYNA *MAT_RIGID (*MAT_020) with aluminum material properties for the elastic modulus and Poisson's ratio was used for the plate. The material properties of the rollers and \"rigid\" aluminum plate are summarized in Table 6.  
An impact velocity of 143 mph (230 kph) at a location 14.0 in. (35.6 cm) from the barrel tip was prescribed in the model to mimic the lab test conditions. Surface-to-surface contact was defined between the baseball and the bat models. The strain-to-failure used in the model was 0.0241 and was derived from the relationship between maple wood density and strain-to-failure based on the results of the Charpy impact testing (Equation (6)). A comparison of a high-speed image taken of the bat profile during durability testing and the finite element model are presented in Figure 11. The model shows excellent correlation for the mode of fracture (single-piece failure), location of fracture initiation (~15 in. [38 cm] from the knob) and the extent of the crack propagation along the length of the bat (~6 in. [15 cm] in total length). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Part** & **LS-DYNA Material Model** & **Density (blobs/in\({}^{3}\))** & **E (lb/in\({}^{2}\))** & **Poisson’s Ratio** \\  & & **[Kg/m\({}^{3}\)]** & **[N/m\({}^{2}\)]** & **Poisson’s Ratio** \\ \hline Top Rollers & \#1 Isotropic Elastic & 2.5880 \(\times\) 10\({}^{-5}\) & 750,000 & \\  & & [276.5] & [5.171 \(\times\) 10\({}^{9}\)] & 0.3 \\ \hline Bottom Rollers & \#1 Isotropic Elastic & 2.5880 \(\times\) 10\({}^{-5}\) & 75,000 & \\  & & [276.5] & [5.171 \(\times\) 10\({}^{8}\)] & 0.3 \\ \hline Plate & \#20 Rigid & 2.5907 \(\times\) 10\({}^{-4}\) & 10,000,000 & \\  & & [2768] & [68.95 \(\times\) 10\({}^{9}\)] & 0.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: ADC roller and aluminum plate material properties. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Density** & **Bulk Modulus** & **Initial Shear Modulus** & **Long-Term Shear Modulus** & \\
**(blobs/in\({}^{3}\))** & **(lb/in\({}^{2}\))** & **(lb/in\({}^{2}\)) [N/m\({}^{2}\)]** & **(lb/in\({}^{2}\)) [N/m\({}^{2}\)]** & **\(\beta\)** \\
**[Kg/m\({}^{3}\)]** & **[N/m\({}^{2}\)]** & **[**N**/m\({}^{2}\)]** & **(lb/in\({}^{2}\)) [N/m\({}^{2}\)]** & **\\ \hline
7.230 \(\times\) 10\({}^{-5}\) & 13,500 & 5000 & 1080 & \\
[772.5] & [93.1] & [34.5] & [7.45] & 20,000 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Ball material properties. In an effort to further explore the capabilities of this map bat model, the impact velocity was varied to observe the bat response at different speeds.