# A Finite Element Investigation into the Effect of Slope of Grain on Wood Baseball Bat Durability
## 3 Materials and Methods
### Finite Element Models  
All finite element analyses were completed using LS-DYNA (Livermore Software Technology Corporation, Livermore, CA, USA), and all finite element models used for this research were constructed using HyperMesh (Altair Engineering, Inc. Troy, MI, USA) and post-processed in LS-Prepost (Livermore Software Technology Corporation. Livermore, CA, USA). The finite element models of (1) a popular bat profile used by professional players (C243), (2) a baseball, and (3) the bat fixture of the Bat Durability System were constructed individually and then combined into a full model of a bat being tested at the UMLBRC for durability (Figure 6) [6]. Models were analyzed for a range of impact velocities that spanned from 90 mph to 180 mph (145 kph to 290 kph) in 5 mph (8 kph) increments, three wood species (ash, marble, and yellow birch) and wood SoGs between \(\pm 10^{\circ}\) to investigate and to quantify how the SoG influences bat durability for variations in wood species and bat/ball impact speeds. The SoG's used in the models were \(-10^{\circ}\), \(-5^{\circ}\), \(-3^{\circ}\), \(-2^{\circ}\), \(-1^{\circ}\), \(0^{\circ}\), \(1^{\circ}\), \(2^{\circ}\), \(3^{\circ}\), \(5^{\circ}\), and \(10^{\circ}\). To validate the finite element models, a series of mechanical tests were performed. Four-point bending tests of the maple and ash wood were done to obtain elastic moduli and strength properties. Charpy testing was conducted to determine the strain-to-failure as a function of wood density [6; 7]. Figure 5: The testing system is shown with a wood bat hanging in machine and ready for impact: (**a**) the Bat Durability System, (**b**) Close-up view of bat in Bat Durability System, and examples of (**c**) SPF and (**d**) MPF. The bat model was constructed to the geometry of an uncupped C243 baseball bat profile. The finite element model was constructed using 8-noded brick elements with single Gauss-point integration. The LS-DYNA *MAT_WOOD (material model #143) definition has previously been demonstrated to be an effective material model for the simulation of wood structures under dynamic loading [17] and is used in this study for modeling the material behaviors of ash, maple, and yellow birch. The geometry of the C243 finite element model is shown in Figure 6. This model of a C243 bat profile is comprised of 179533 nodes and 159984 elements, with a mean mesh size of 0.2 in (0.4920 cm). This element size was concluded by completing a convergence study. For this study, the wood density and specific wood-species material properties of the bat models were held constant throughout all of the models. The material properties for each wood species as they are provided in the LS-DYNA Keyword input file came from empirical testing and are summarized in Table 1. All models were prescribed a density of 0.0245 lb/in\({}^{3}\) (678 Kg/m\({}^{3}\)), thereby resulting in a 33.0-oz (936-g) bat. The 0.0245 lb/in\({}^{3}\) (678 Kg/m\({}^{3}\)) density was chosen because it is currently the lowest density that is permitted by MLB for maple bats. The SoG of the wood was varied between \(-10^{\circ}\) and \(+10^{\circ}\). One of the required inputs for the *MAT_WOOD material model in LS-DYNA is the specification of the direction cosines (AOPT values) for the orientation of the wood [18]. These direction cosines allow for easily prescribing the SoG of the wood and have been shown to reliably predict impact velocity of bat failure, location of bat failure, and a mode of bat failure [18]. Both positive and negative wood SoG cases were modeled to see if there were any differences due to a plus or minus direction of the SoG (Figure 3) on the failure mode and/or durability of the bat.