The exact solution for the translational acceleration of fluid-filled rigid spherical shells— A model for the development of intracranial pressure

Abstract

The head, when subjected to a translational acceleration, has been simulated mathematically as the motion of a rigid spherical shell containing inviscidcompressible fluid. The exact, closed-form wave paopagation solution to the problem of a rectangular translational acceleration pulse applied to the outer shell surface was obtained. The solution to the nondimensional governing equations was accomplished through the Laplace transformation technique. The nondimensional pressure was obtained as a finite series by exploiting the hyperbolic nature of the equations and the use of the shifting theorem for subsequent easy inversion back to the time domain. The pressure, as well as the strain-energy density distribution, is presented. The Green's function for the pressure, i.e., when the applied input acceleration is a Dirac delta function, was also obtained. Thus, for any general acceleration pulse input, the application of the convolution integral yields the response. The maximum pressure magnitude and location have been compared for various cases of a unit impulse input. By varying the rectangular pulse in such a way that the product of the magnitude and duration of acceleration is always unity, we delineated the effects of varying the pulse duration and magnitude. The maximum pressure value is shown to shift from the central region to the surface of the shell as the duration of the acceleration pulse is increased. For a unit step translational acceleration input, the maximum value of negative pressure is at the contrecoup region of the sphere at a nondimensional time of 1.6. Results are also presented for the special case of the incompressible fluid contained in rigid spherical shells.