A circular cross section of the brain and skull, modeled with Abaqus, was used for all tests and is shown below in Figure 1. The plane strain assumption was used.

For each test, the system was subjected to a load defined by a sinusoidal rotational acceleration about the centroid of the brain. The parameters used to vary the input are T_Dur and Δω as shown below in Figure 2.

For the computational studies performed in this report, a first order Ogden model, N=1 was used. The strain energy density equation for Ogden’s model of hyperelasticity is shown below in equation (1) [5].

The inputs to this model are the relaxed shear modulus, μ₁, the alpha constant, α, and the D₁ value, which is determined by the bulk modulus, k₀ through the relationD1=2k0. Abaqus computes and outputs the total volume change, Jel, and stretch values, λ_i, for each finite element of the model.

For every set of tests, 9 values of Δω and 13 values of T_Dur were combined, resulting in 117 tests with various loading inputs. Four sets of tests were completed, with each set using a consistent material composition. Lower and upper bounds of the relaxed shear modulus were used in order to understand if this value significantly contributes to differences in shear strain. For each bound of hyper-viscoelasticity, a corresponding composition of viscoelasticity was tested. The values for the corresponding compositions were obtained using Lamé parameter relations as explained below and shown in Table 4. This value was held constant for the lower and upper bounds. The relaxed shear modulus, μ₁, was given a value of 1 kPa for the lower bound and 3 kPa for the upper bound. From these values of μ₁ and K, the corresponding values of the Poisson’s ratio, ν, and Young’s modulus, E, were calculated using Lamé parameter relations [6].

In Morrison et. al., 2003 [7], it was suggested that brain cells will experience significant damage if their strain exceeds 10%. For the preliminary phase of tests, the 10% shear strain threshold was used. For the second phase of tests, two shear strain thresholds were used. A lower threshold of was used 5% along with the standard threshold of 10% to reveal whether the effects of the composition differences are exaggerated at higher strains.

In order to represent the sets of tests, the maximum angular acceleration, Max(α), has been plotted against Δω. The value Max(a) for each test is derived from values for Δω and T_Dur. Shown in Figure 4 is the lower bound of the viscoelastic set of tests from the preliminary phase. On the x-axis of each plot is a logarithmic scale of the maximum angular acceleration, Max(α), of the skull. On the y-axis is the change in angular velocity, Δω, of the skull. Each point represents a test. Data points shown in red reveal tests where any element on the brain has exceeded the injury threshold.

The results from Figure 4 led to the second phase of tests using loading inputs that were finely incremented within the injury inducing inputs.

The results of the second phase of tests are compiled in Figure 5 on page 10. The figure uses two different shear strain injury thresholds. Any differences between the visco-hyperelastic and viscoelastic compositions are highlighted in yellow. Three observations can be noted. The first is that a large acceleration (α>435rads2) is necessary to observe differences between the hyper-viscoelastic and viscoelastic models. Even when the change in angular velocity is high, no differences were observed between the viscoelastic and hyper-viscoelastic models unless significant acceleration was applied. Secondly, the differences caused by the hyperelastic component are more pronounced at higher strains. This is evidenced in that Figure 5a reveals more differences in shear strain behavior than Figure 5b. Lastly, the upper bound, with relaxed shear modulus μ₁ = 3 kPa, has revealed only one difference at an angular accelerationα>3,000rads2. This indicates that the value of the shear modulus significantly contributes to the effect of hyperelasticity.

The findings of the mesh comparison, compiled in Table 3, indicate that finite element analysis (FEA) requires a sufficiently fine mesh in order to produce reliable results. This is due to a phenomenon called numerical dispersion. Numerical dispersion occurs in tests when the simulated material exhibits a higher dispersivity than the true material [8]. For the case of modeling the brain, this means that deformation measures, such as strain, which are observed from the model will have a lower magnitude than what would be experienced by a real brain. Because the brain is especially compliant, it will be important for all researchers to check that numerical dispersion does not occur within their FEA models of the brain.

Figure 5 on page 10 reveals that the shear modulus plays an important role in determining how hyperelasticity will affect the resultant shear strains. For the upper bound of the relaxed shear modulus, where μ₁ = 3 kPa, only 1 of 16 tests that profile the injury threshold yielded a different result for the two models. This is held in contrast to the lower bound. For this bound, where μ₁ = 1 kPa, 8 of 25 tests that profile the injury threshold yielded different results for the two models. All 9 of the tests that revealed differences between models consistently showed that the hyperelastic model was more resistant to shear deformation than the viscoelastic model.

When μ₁ = 1 kPa, the hyperelastic component significantly contributes to the shear strain behavior for loading parameters α>450rads2 andΔω<2.4rads. This shows that quick rotations with high accelerations are where the effects of hyperelasticity are significant.

As described in the Methods section on page 7, Figure 5a uses an injury threshold of γ > 0.10, as this has been considered the shear strain above which concussion is likely. Figure 5b uses a threshold of γ > 0.05 to study trends of the composition differences with higher strains. Figure 5 shows that the differences become exaggerated between the models as the threshold is increased. This result makes sense when considering the stress-stretch curve of a hyperelastic material. Figure 6 below provides an example of a comparison between a hyperelastic and purely elastic stress-stretch curve, where stretch is defined asλ=ll0. As is shown, the hyperelastic material requires greater stress in order to affect large strains. This attribute of hyperelastic materials helps to explain the resiliency of the models including the hyperelastic component.

If a simple viscoelastic model is being used, then there will be less resistance to large shear strains. This means that if there are observable differences between the model and actuality, it would be that the brain experiences less shear strain than the model predicts. If, on the other hand, a hyper-viscoelastic model is being used, then any differences between the model and actuality would show that the model is more resistant to high shear strains than the true brain. This knowledge lets researchers with simple linear viscoelastic models know that their models tend to overestimate shear strain at extreme loading conditions. Furthermore, hyperelastic models with high shear moduli and large α values are more resistant to shear strain, meaning that the model tends to underpredict shear strain at extreme loading conditions. This is cause for warning as an underprediction in shear strain can lead to flaws in safety mechanism designs that allow for large shear strains to propagate in the brain, which are known to cause injury.