Annually, motor vehicle crashes world wide cause over a million fatalities and over a hundred million injuries. Of all body parts, the head is identified as the body region most frequently involved in life-threatening injury. To understand how the brain gets injured during an accident, the mechanical response of the contents of the head during impact has to be known. Since this response cannot be determined during an in-vivo experiment, numerical Finite Element (FE) modelling is often used to predict this response. Current FE head models contain a detailed geometrical description of anatomical components inside the head but lack accurate descriptions of the brain material behaviour and contact between e.g. skull and brain. Also, the numerical solution method used in current models (explicit Finite Element Method) does not provide accurate predictions of transient phenomena, such as wave propagation, in the nearly incompressible brain material.

The aim of this study is to contribute to the improvement of FE head models used to predict the mechanical response of the brain during a closed head impact. The topics of research are the accuracy requirements of explicit FEM for modelling the dynamic behaviour of brain tissue, and the development of a constitutive model for describing the nearly incompressible, non-linear viscoelastic behaviour of brain tissue in a FE model.

The accuracy requirements of the numerical method used depend on the type of mechanical response of the brain, wave propagation or a structural dynamics type of response. The impact conditions for which strain waves will propagate inside the brain have been estimated analytically using linear viscoelastic theory. It was found that shear waves (S-waves) can be expected during a traffic related impact, (frequencies between 25 and 300 H z), while compressive waves (P-waves) are expected during short duration, high velocity, ballistic impacts (frequencies between 10 kHz and 3 MHz). For this reason FE head models should be capable of accurately replicating the wave front during wave propagation, which poses high numerical requirements.

An accuracy analysis, valid for one-dimensional linear viscoelastic material behaviour and small strains, revealed that modelling wave propagation phenomena with explicit FEM introduces two types of errors:​ numerical dispersion and spurious reflection. These errors are introduced by the spatial and temporal discretisation and cause the predicted wave propagation velocity to be lower than in reality. As a result, strain and strain rate levels will deviate from reality. Since both strain and strain xi xii Summary rate are associated with the occurrence of brain injury they should be predicted correctly. However, given the element size in current state of the art 3-D human head models, accurate modelling of wave propagation is impossible. For accurate modelling of S-waves the typical element size in head models (5 mm) should be decreased by a factor of ten which can be accomplished by mesh refinement. For accurate modelling of P-waves the typical element size should be decreased by a factor of hundred. For this reason mesh refinement is not feasible anymore and developments on spatial and temporal discretisation methods used in the Finite Element Method are recommended. As these developments are beyond the scope of this research, shear behaviour is emphasised in the remainder of the study.

The mechanical behaviour of brain tissue has been characterised using simple shear experiments. The small strain behaviour of brain tissue is investigated using an oscillatory strain (amplitude 1%). Frequencies relevant for impact (1-1000 H z) could be obtained using the Time/Temperature Superposition principle. Strains associated with the occurrence of injury (20% simple shear) were applied in stress relaxation experiments. It was found that brain tissue behaves as a non-linear viscoelastic material. Shear softening (i.e. decrease in stiffness) appeared for strains above 1% (approximately 35% softening for shear strains up to 20%) while the time relaxation behaviour was nearly strain independent.

A constitutive description capable of capturing the material behaviour observed in the material experiments was developed. The model is a non-linear extension of a linear multi-mode Maxwell model. It utilises a multiplicative decomposition of the deformation gradient tensor into an elastic and an inelastic part. The inelastic, time dependent behaviour is modelled using a simple Newtonian law acting on the deviatoric part of the stress only. The elastic, strain dependent behaviour is modelled by a hyper-elastic, second order Mooney-Rivlin material formulation. Although isotropy was assumed in this study, the model formulation is such that implementing anisotropy, present in certain regions of the brain, is possible. Brain tissue material parameters were obtained from small strain oscillatory experiments and the constant strain part from the stress relaxation experiments.

The constitutive model was implemented in an existing explicit FE code (MADYMO). In view of the nearly incompressible behaviour of brain tissue, Heun’s (predictorcorrector) integration method was applied for obtaining sufficient numerical accuracy of the model at time steps common for head impact simulations. As a first test, the initial part of the stress relaxation experiments, which was not used for fitting the material parameters, was simulated and could be reproduced successfully.

To test both the numerical accuracy of explicit FEM and the constitutive model formulation at conditions resembling a traffic related impact a physical (i.e. laboratory) head model has been developed. A silicone gel (Dow Corning Sylgard 527 A&B) was used to mimic the dynamical behaviour of brain tissue. The gel was mechanically characterized in the same manner as brain tissue. It was found that silicone gel behaves as a linear viscoelastic solid for all strains tested (up to 50%). Its material parameters are in the same range as the small strain parameters of brain tissue, but viscous damping at high frequencies is more pronounced. It was concluded that for trend studies and benchmarking of numerical models the gel is a good model material. The gel was put in a cylindrical cup that was subjected to a transient rotational acceleration. Gel deformation was recorded using high speed video marker tracking. The gel was modelled using the new constitutive law and the physical model experiments were simulated. Good agreement was obtained with experimental results indicating the model to be suitable for modelling the nearly incompressible silicone gel. It was shown that correct decoupling of hydrostatic and deviatoric deformation in the stress formulation is necessary for correct prediction of the response of the nearly incompressible material.

Finally, the constitutive model was applied in an existing 3-D FE model of the human head to asses the effect of non-linear brain tissue material behaviour on the response. The external mechanical load on the 3-D FE head model (an eccentric rotation) was chosen such as to obtain strains within the validity range of the material experiments (20% shear strain). This resulted in external loading levels values below the ones associated with injury in literature. A possible explanation for this is the fact that shear stiffness values, commonly used in head models in literature, are too high in comparison with material data found in recent literature and own experiments. Also the estimated injury threshold of 20% strain, indicated by studies on isolated axons, might be to conservative. Another possible explanation may be that a certain degree of coupling between hydrostatic and deviatoric parts of the deformation in the stress formulation might exist in reality which is not modelled in current constitutive formulation.

Application of the non-linear behaviour in the model influences the level of stresses (decrease by 11%) and strains (increase by 21%) in the brain but not the temporal and spatial distribution. However, it should be noted that these effects on stresses and strains hold for one specific loading condition in one specific model only. For a more general conclusion on the effects of non-linear modelling in brain tissue, application in different models is recommended.