Mild traumatic brain injury (mTBI) and concussion could occur in vehicular accidents, contact sports, or other physical traumas when the head is subjected to high linear or angular acceleration. Understanding the physiology and dynamics of such events has attracted many researchers’ attention. Due to the hidden risks in such events, it is very important to understand the cause and effect of the relative motion between the brain and skull and the implications of normal and shear stresses in the meningeal region.

Since the early 70’s to date wide variations of experimental, analytical, and numerical models has been developed to analyze multilayer spherical head impact model to quantify the dynamic response of the human head due to blunt impact and explain the process and likely cause of mild traumatic brain injuries. There are many high-fidelity finite element models and research studies of the various head models, but very limited analytical models to date for parametric studies. Analytical models of head impact play a vital role in predicting relative displacement between skull and brain, transmitted forces, post-impact velocity, and acceleration of the head system. However, to define a reliable mathematical model which can illustrate the mechanisms of motion and deformation of the brain within the skull requires knowledge of dynamics of a multibody system, material properties, boundary conditions at the brain–skull interface, and experimental data or FEM simulation for validation.

In this paper, a mathematical model of the brain and meningeal layers, as two separate viscoelastic materials that are modeled using a Kelvin–Voigt model, have been investigated and the motion of the brain relative to the skull during blunt head impacts have been analyzed. Specifically, the model consists of three concentric spherical mediums including a spherical shell (skull), a thin spherical layer (meningeal layer), and a spherical mass (brain). The interface between these spherical mediums consists of springs and dashpots representing stiffness and viscoelasticity of the skull, meningeal layer, and the brain. This model of the head is initially at rest and subjected to an impulse load. The equations of motion for this multi-body system were obtained, solved, and validated by performing a lumped mechanical model and the multibody dynamics finite element analysis simulation. Multiple parametric studies were performed to determine the maximum amplitude of impact force for which there is a contact between the skull and the brain.

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