# A Mathematical Model for a Vibrating Human Head

J. C. Misra (Indian Institute of Technology, India), S. Dandapat (Indian Institute of Technology, India) and S. Adhikary (SOA University, India)
DOI: 10.4018/jalr.2010040103

## Abstract

In this paper, a mathematical model has been formulated to study the vibration of the human head. In the mathematical analysis of the model, the skull is considered as an anisotropic spherical shell and brain matter is represented as an inviscid compressible fluid. Also, in the model, the translational acceleration is considered as a general function of time. The authors use the method of Laplace transformation to achieve the analytical solution of the problem, while the analytical expressions have been used to compute the stress distribution in the system by resorting to numerical techniques.
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## 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).

Table 1.
Nomenclature
 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

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