A Mathematical Model for a Vibrating Human Head

1 Introduction

It is well-known that about three quarters of the total number of deaths resulting from various types of accidents involve serious injuries to the head in general and to the brain matter in particular. This indicates the gravity of the problem of head injury. In the past a few investigations (experimental as well as theoretical) have been performed by different researchers in order to explore various information that are useful from the physiological as well as pathological point of view. Some of the derived information on the axisymmetric vibration of head are also quite helpful in the design and construction of improved protective gears for the human head.

For an analytical study in this area, one has to largely dwell upon studies based upon modelling of the structure of head. It is needless to emphasize on the consideration of a proper model in the sense that the model should represent the real structure of the head as nearly as possible. At the same time, care should also be taken so that the assumed mechanical properties of the different components of the structure should be in the conformity to the observations of the relevant experimental investigations (Table 1).

	Spherical polar co-ordinates
	Non-dimensional radius
	Outer radius of the skull
	Radius of the interface between the skull and the brain
	Velocity of compressional wave of the skull medium
	Velocity of pressure wave in the brain mass
	Applied external force
	Normal stress components
	Shearing stress components
	Normal strains
	Shearing strains
	Pressure of the brain
	Frequency
	Time
	Displacement in directions
	Non-dimensional density
	Density of the brain and skull material respectively
	Velocity potential of brain
	Lame's constants