An Analytical Viscoelastic Model of the Head in Blunt Impacts

The relative motion between the brain and skull and an increase in contact and shear stresses in the meningeal region could cause traumatic closed head injuries due to vehicular collisions, sport accidents and falls. There are many finite element studies of the brain/head models, but limited analytical models. The goal of this paper is to mathematically model subarachnoid space and the meningeal layers and to investigate the motion of the brain relative to the skull during blunt head impacts. The model consists of an elastic spherical shell representing the skull containing a visco-elastic solid material as the brain and a visco-elastic interface, which models the meningeal layers between the brain and the skull. In this study, the shell (the head) is moved toward a barrier and comes in contact with the barrier. Consequently, the skull deforms elastically and the brain is excited to come in contact with the skull. The viscoelastic characteristics of the interface (consisting of springs and dampers) are determined using experimental results of Hardy et al. [5]. Hertzian contact theory and Newtonian method are employed to acquire time dependant equations for the problem. The governing nonlinear integro-differential equations are formed and are solved using 4^th order Runge Kutta method and elastic deformation of spherical shell, brain motion during the impact, and contact conditions between the brain and the skull are evaluated. Furthermore, some important mechanical parameters such as acceleration, impact force, and the impact time duration are also specified. The results of the analytical method are validated by performing an explicit finite element analysis. Acceptable agreement between these two methods is observed. The results of the analytical investigation give the contact threshold of the skull/brain, and represent the relevant velocity of this event. Furthermore, the impact analysis in different velocities is performed in order to compare the transmitted forces and the impact durations in different cases. It is concluded that the proposed mathematical model can predict head impacts in accidents and is capable in determining the relative brain motion of the skull and the brain. The mathematical model could be employed by other investigators to parametrically study the traumatic closed head injuries and hence to propose new head injury criteria.