1 Introduction

Head injury by road and domestic accidents causes substantial morbidity and mortality throughout the world, especially in developed countries. Much research effort from the biomechanics and neurotraumatic societies has been devoted to reveal head injury mechanics and the correlations with impact kinetics or kinematical inputs. In the head traumatic studies, early milestones can be attributed to Nahum and Smith’s experimental model [1]. They measured the intracranial pressure and other dynamic response quantities under specific load intensities. Their work has been and is still being used to verify numerical models. Significant contributions also came from Ruan et al. [2], O’Donoghue and Gilchrist [3], Willinger et al. [4], Yoganadan et al. [5] and Zhang et al. [6]. Not only did they provide the in-vitro and in-vivo experimental data, they also built up very sophisticated finite element models for general head impact studies, as well as investigated the injury criteria for different types of traumatic injuries.

Head traumatic injuries have been identified, e.g. as concussion, axonal injury, hematoma, cerebrum contusion, skull fracture, etc. [6]. Different quantities are used for the evaluation of different types of injury, e.g., energy absorption criterion for skull fracture, coup/counter-coup pressures for intracranial contusions. Doorly et al. [7] suggested that the angular acceleration together with the strain rate be used for the prediction of subdural haematoma. An energy criterion and another tensile-strain-based criterion for skull fracture were introduced by Yoganadan et al. [5] and Vander et al. [8], respectively. Zong et al. [9] suggested another criterion of structural intensity (STI), which is a directional representation of the energy flow rate in a unit area. The STI is a compound index to integrate the dynamic response quantity of stress and the kinematical quantity of velocity to reflect the injury locations and patterns.

However, assessment of head injury is far from straight forward because the pathological behavior of the injured could vary from person to person. The complete understanding of head trauma injury is further hindered by the paucity of in-vivo experimental data. On the other hand, numerical simulations contribute a great deal to the deep understanding of the phenomenon with its unique flexibility. The numerical models with verified mechanical fidelity allow the numerical simulation to reveal in detail head dynamic response scenarios under different impact loadings. Besides the aforementioned contributions [2–6, 9], it is worth mentioning that the first finite element head model was due to Hosey and Liu [10].

The present paper demonstrates through finite element simulations that under similar loading conditions, heads with similar sizes receive similar amounts of energy for the frontal and the side impacts. However, the head distributes the energy differently because of the different local structural configurations at the point of load impact. Local curvature changes at the forehead and temple have a marginal effect on the energy flow rate and other dynamic response quantities, whereas different locations of the load and head size are found to have significant effects. The STI rather than the energy absorption, gravity centre acceleration and HIC, may reflect these phenomena effectively. These findings conform to the general concepts regarding the head injury, i.e., ‘the risk of injury is related to the energy delivered to the body by the impacting object as well as the object’s shape’ and ‘the rate of loading and thus the strain rate is also an important factor in causing injury since biological tissues are viscoelastic [11–14].

2 Materials and methods

The FE model for the human head adopted in the present study, as shown in Fig. 1, contains five layers with variable thicknesses to approximate the basic average size adult human head structure [9]. The neck is composed of a spinal cord, a cervical bone and a disc. The mechanical fidelity of the model has been verified against Nahum and Smith’s [1] and Trosseille et al.’s [15] impact tests. The geometrical data of this model are taken from Koenig [16]. A detailed description of this FE head model and its verification against experimental data are presented in Zong et al. [9]. Material properties of Young’s modulus and Poisson’s ratio for each material as listed in Table 1 are obtained from [17].

Fig. 1
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Head finite element model (Zong et al. [9])

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Table 1 Material properties
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2.1 Elastic morphing method

Head size and shape vary from person to person and the same person at different ages. To evaluate the effect of different morphologies of the head while eliminating the effect of different head sizes, the finite element head model is modified with an elastic morphing method. The local curvatures of the frontal and temporal bones are changed by application of displacement boundary conditions at the local areas through a static elastic simulation. As shown in Fig. 2, nine nodes are moved normal to their respective initial local curvatures respectively. For convenience of notation, a unit curvature change is defined as a 4.5 mm outward displacement of the central point, while the surrounding eight nodes are separated by 3 mm displacements. Another nine nodes on the opposite side of the head are fixed. It should be mentioned that for the frontal curvature morphing simulation, the side curvature is kept unchanged and vise versa. For the forehead, the local curvature is increased by one and two units and reduced by one unit respectively, while for the temple, the curvature is reduced by one and two units and increased by one unit only to guarantee a convex forehead and concave temple. As a result, the forehead is extended by a maximum of 9 mm, while the temple is compressed by a maximum of 9 mm inward. Figure 3 illustrates the morphed curvatures for the frontal and side bones. The different curvature change levels are indicated by the unit numbers, e.g., m1 represents a minus unit change and 0 implies no curvature change. The morphed local curvatures for the frontal and side bones are quantitatively compared with evenly distributed radii over the areas (see Fig. 3). For the frontal bone, the average local radii are respectively, 122.86, 86.95, 70.03, 59.75 mm for the morphology states of m1, 0, 1 and 2, and −292.20 mm, −217.08, −143.61 mm, −110.29 mm for the concave side morphology states of m2, m1, 0, 1.

Fig. 2
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Head frontal and side morphing paths

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Fig. 3
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Morphed frontal and side curvatures

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It should be mentioned that the magnitude of the morphing displacement came from the authors’ understanding of the anatomical features of the human head. The authors felt the convexity of the frontal bone and concavity of the temple by touching with hand about a hundred office colleagues, who are from different regions and origins. However, the morphing unit displacement of maximum 4.5 mm is still highly hypothetic to study the different morphology effect indicatively.

The material properties for the head morphing simulation adopt the original values for the head tissues as listed in Table 1. The change of thickness of the skull is minimal since the Young’s modulus of the skull is much higher than that of the soft intracranial tissues. Review of the morphed head models reveals that the change of the overall masses and the thicknesses of the skull are very minimal compared with the original head model. Deformed meshes are used for the following impact study.

2.2 Structural intensity

The concept of STI, when it was firstly introduced by Noiseux [18], is defined as the power per unit width of cross sectional area. It conforms to the concept of energy flux [18]. The STI is obtained at a set of points as a vector quantity, indicating the power flow direction and magnitude.

The mathematical form of the instantaneous components of STI in the time domain is

$$ {\text{STI}}_{i} = - \sigma _{{ij}} {\left( t \right)}v_{j} {\left( t \right)}, $$
(1)

where σ ij (t) is the component of the stress tensor and v j (t) is the velocity vector at time point t [19].

3 Impact simulation results and discussions

Twelve cases are simulated with eight frontal and side impacts on the original head size with different local curvatures. Another four simulations on the 15% proportionally shrunk and enlarged head models are also implemented. The force––time profile from Nahum and Smith [1] as shown in Fig. 4 is applied on the same nine nodes for morphing displacements as seen in Fig. 2 for all impact simulations. The lower end of the neck could be set free or fixed according to Willinger et al. [4] since the loading time duration in the present study is very short (about 8 ms in Fig. 2). Thus the lower end of the neck is simply set free in the present study. The ABAQUS explicit solver is used to solve the dynamic problem.

Fig. 4
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Nahum and Smith’s [1] force–time profile

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The calculated head force–displacement profiles under frontal and side impacts are plotted in Fig. 5. Different curvatures for the frontal and side impacts are found to have minimal effect on the head kinematical response as seen in Fig. 5. Hence Fig. 6 compares only the force–displacement curves of the original reduced and the enlarged head models for frontal and side impacts. Figure 7 shows the energy absorptions by the head, which is the integral of the force–displacement curves of Fig. 5. Table 2 lists the energy absorptions and the percentage differences with regard to the data from frontal impact on the original head model. It is seen from Figs. 5, 6, 7 and Table 2 that energy absorption for either the frontal or side impacts of similar sized heads, are very similar. It is understandable that the force duration, magnitude and the head inertias yield similar energy absorptions. The different locations of the impact loading result in very slight differences, while the different head sizes have very significant influences. The smaller or larger head sizes have reduced or increased inertias respectively, which lead to remarkably different energy absorptions. The notation in Figs. 5, 6, 7 and all remaining figures is as follows: FS, SS denote skull quantities in frontal impact and side impact respectively, FB and SB indicate brain quantities, the following numerals indicate the curvature change units, while the letter of L or S represents the 15% larger or smaller head, respectively. It should be mentioned that all the percentage differences are calculated with respect to the values from the frontal impact on the original head model.

Fig. 5
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Force–displacement curves for frontal and side impacts

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Fig. 6
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Comparison of force–displacement curves for frontal and side impacts

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Fig. 7
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Comparison of energy absorption for frontal and side impacts

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Table 2 Energy absorptions
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Table 3 Maximum first principal strains
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The head dynamic response quantities, i.e., the skull and brain stresses, strain, strain energy densities, intracranial pressures and the kinematical response quantities of gravity centre accelerations and HIC, are then examined and compared. The STI is also calculated according to Eq. 1.

Tables 3–7 list the maximum values of the dynamic response quantities, and the STI and their percentage differences with respect to the values from the frontal impact on the original head model. The histogram plots in Figs. 8 and 9 graphically illustrate the respective percentage differences for skull and brain dynamic response quantities and the STI. It is seen from Figs. 8 and 9 and Tables 4, 5, 6, 7 that the relative differences among the dynamics response quantities and the STI are quite consistent for both the skull and the brain. A clear tendency is found that the different curvature of the forehead or the temple has a marginal effect, the different locations of the impact loading result in remarkable differences, and the different head sizes have an even more prominent influence.

Table 4 Maximum von Mises stresses
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Table 5 Maximum STI
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Table 6 Maximum strain energy density
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Table 7 Maximum brain pressure
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Fig. 8
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Percentage differences of skull response quantities with respect to values from frontal impact on original head model

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Fig. 9
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Percentage difference of brain dynamic response quantities with respect to values from frontal impact on original head model

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In contrast to the histogram plots for energy absorption in Fig. 7, the solid and shaded histograms denoting the frontal and side impacts respectively have striking differences in the plots for the dynamic response quantities in Figs. 8 and 9. Moreover, the local different curvature may also cause significant differences for the brain response quantities, especially for the strain energy density.

Figure 10 shows the respective gravity centre acceleration-time profiles for all the simulation cases. The representative frontal and side impact results are compared in Fig. 11. The HIC with 15 ms time intervals calculated from Fig. 10, together with the gravity centre peak accelerations and their percentage differences, are tabulated in Table 8. Figure 11 illustrates the percentage differences. It should be mentioned that since the impact time duration is shorter than the 15 ms typical time interval for the HIC calculation, zero values are assumed for all time durations for which the acceleration is not available.

Fig. 10
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Gravity centre accelerations and frontal and side impact comparison

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Fig. 11
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Maximum gravity centre accelerations, HIC and percentage difference with respect to values from frontal impact on original head model

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Table 8 Peak gravity centre acceleration and HIC
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The kinematic quantities of gravity centre acceleration and HIC do not conform to the dynamics response quantities. It can be seen from the simulations as shown in Table 8 and Figs. 10 and 11 for the gravity centre acceleration and HIC that the different morphology state or curvatures of the forehead or the temple have a very minimal effect. The different loading locations and head size have very significant effects. The peak accelerations from the side impact increases by about half that of the counterparts involving frontal impact for a similar head size, while the HIC from side impacts is almost four times as high as that from frontal impact.

4 Possible head injury locations and patterns

The distributions of the STI at the occasion that the highest value is found are shown in Fig. 12 for the brain and in Fig. 13 for the skull. For the brain in a side impact, both the distributions on the loaded and opposite sides are given for a clear presentation of the energy flow path. It can be observed that in the side impact case, the energy flows mainly within the brain, while in the frontal impact case it flows to the neck, where the highest STI is found. Thus, it is possible that the neck acts as an energy flowing path and may sustain injury prior to the brain in a frontal impact. Side impacts with similar loading severity yield much higher STI than frontal impacts do. A smaller head sustains significantly higher STI both in the frontal and side impacts as seen in Fig. 12, where the length of the vectors are scaled similarly for an easier visual comparison. A similar tendency is also observed for the skull. The skull sustains a much higher STI in a side impact, thus there is a higher probability of fracture for a side impact than for a frontal impact with a similar load severity. Again, a smaller head size yields a much higher STI.

Fig. 12
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Maximum STI distributions of brain

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Fig. 13
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Maximum STI distributions of skull

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According to the distribution patterns of the STI, in a frontal impact, it is possible that the brain may not be injured, since the neck acts as an outlet of the energy: this conforms to the findings by Zong et al. [9]. For the side impact case, the brain may probably sustain a diffusive injury at the area immediately contiguous to the loading locations. For the skull, the fracture may start from the loading locations for both frontal and side impacts. The fracture in the frontal impact may be a multiple-fracture (linear + compression fracture), which is in agreement with the experimental observation of Yoganadan et al. [5]. The skull fracture pattern may be circular and diffusive for the side impact case [8].

In summary, there is evidence to suggest that the dynamic response quantities, i.e., the stain energy density, the stress and the maximum tensile strain are quite consistent among themselves and with the STI for the different impact cases. The energy absorptions, the kinematical response quantities of peak gravity centre acceleration and the HIC may not reflect the differences that the different morphological states and the loading locations make. The STI may be a very good representative for all the dynamic skull and brain response quantities to reflect the diffusive injury in place of coup and counter-coup pressure and the local injury for strain, strain and strain energy density.

5 Conclusions

The current finite element simulations have shown that the energy absorption, gravity centre acceleration and HIC may not be the proper criteria for the assessment of head traumatic injury. It is very important to understand how energy is distributed. The STI, which is the vector representation of the energy flow rate, and is consistent with the dynamic response quantities, could be appropriate for prediction of head traumatic injury. In addition, the STI indicates the location and patterns of skull fracture and brain injury.

A different head morphology has a marginal effect on the head dynamic response quantities and thus the traumatic injury, different locations of the impact loading and head sizes have remarkable effects.