# Evaluation of Snowboarding Helmets in Mitigation of the Biomechanical Responses of Head Surrogate
## 2 Materials and Methods
### Concurrent Computational Simulations  
Concurrent computational simulations were performed to obtain full-field spatiotemporal biomechanical responses such as brain simulant pressure, equivalent stress (von Mises stress), and maximum principal strain (MPS). There was a one-to-one correspondence between the head surrogate used in the experiments and the one used in the simulations in terms of geometry and material properties (Tables 1 and 4). Note that the geometry of the head surrogate in the experiments and simulations was based on the GHBMC's 50 th percentile adult male, version 1.5. For modeling, the brain simulant, linear, viscoelastic deviatoric response (i.e., linear viscoelastic response in shear) and linear bulk (volumetric) response are considered. The deviatoric response is generally characterized using a time-varying response in shear (Equation (1)). The material parameters of Equation (1) are typically obtained using a shear relaxation test by fitting the shear relaxation data to Equation (1) (i.e., the Prony series). \[G(t)=G_{\infty}+\sum_{i=1}^{n}G_{i}e^{-\beta_{i}t} \tag{1}\]  
where \(G_{\infty}\) is the long-term shear modulus and \(G_{i}\) and \(\beta_{i}\) are the shear modulus and decay coefficient of the Prony series. When \(n=1\), Equation (1) reduces to Equation (2), i.e.,  
\[G(t)=G_{\infty}+(G_{0}-G_{\infty})e^{-\beta t} \tag{2}\]  
where \(G_{0}\) and \(G_{\infty}\) are instantaneous and long-term shear moduli, respectively, and \(\beta\) is the decay coefficient. In this work, Equation (2) (i.e., the one-term Prony series) is used. This is consistent with the literature (e.g., see [41; 61; 62]). The material parameters of Equation (2) (i.e., values of \(G_{0}\), \(G_{\infty}\), and \(\beta\)) are taken from the literature [41] and are tabulated in Table 4. In this work, the values of \(G_{0}=7.5\) kPa, \(G_{\infty}=1.5\) kPa, and \(\beta=80\) s\({}^{-1}\) were used. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Component** & **Material** & **Density (kg/m\({}^{3}\))** & **Instantaneous** & **Long-Time** & **Decay** & **Bulk Modulus** \\  & & **Shear Modulus** & **Shear Modulus** & **Coefficient** & **(gPa)** \\ \hline Brain simulant & Viscoelastic & 1060 & 7.5 & 1.5 & 80 & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Specific constitutive parameters of the material for the GHBMC head model used for concurrent simulations. The values of \(G_{0},~{}G_{\infty}\), and \(\beta\) were used as input to the simulation. The simulation begins with the instantaneous response governed by instantaneous shear modulus (\(G_{0}\)). Over time, stress evolves as per Equation (2). In the simulation, Cauchy stress is computed as follows:  
\[\sigma_{ij}=J^{-1}F_{ik}\cdot S_{km}\cdot F_{mj}^{T} \tag{3}\]  
where \(\sigma\) is the Cauchy stress, \(F\) is a deformation gradient, \(J\) is Jacobian, and \(S\) is the second Piola-Kirchhoff stress, as estimated using the following integral:  
\[S_{ij}=\int_{0}^{t}G_{ijkl}~{}(t-\tau)\frac{\partial E_{kl}}{\partial\tau}d\tau \tag{4}\]  
where \(E\) is the Green strain, and \(G_{ijkl}\) is the stress relaxation function. The equivalent (von Mises) stress reported in this work is based on the deviatoric part of the Cauchy stress tensor. MPS is based on the highest eigenvalue of the Green strain tensor. The pressure (p) is determined from the bulk modulus (K) and the volumetric strain (\(\varepsilon_{v}\)):  
\[\mathrm{p}=-\mathrm{K}\varepsilon_{v} \tag{5}\]  
\[\varepsilon_{v}=\ln\biggl{(}\frac{V}{V_{0}}\biggr{)} \tag{6}\]  
where \(V\) and \(V_{0}\) denote the final volume and instantaneous volume, respectively. The volumetric response of the brain simulant was assumed to be linear, elastic and bulk modulus value of 2 GPa was used. The skin and dura mater simulants were modeled as linear and elastic. In this work, the head kinematics obtained from the experiments (acceleration-time histories) were used as input for the simulations. To facilitate the concurrent simulations, the acceleration-time histories were applied to the CG of the skull.