Ako Bahari - National Iranian Gas Company/Iran University of Science and Technology
Asghar Nasr - University of Science and Technology

An analytical solution, based on exact linear elasticity,is obtained for the time-domain response of a fluidfilled spherical shell and its coupled internal fluid field due to an impulsive impact load on the outer surface of shell. This solution used to model the head injury of human in which the motion of the fluid is assumed to be governed by the wave equation. First, the axi-symmetric stress acting on the outer surface of the shell isexpanded into modal Fourier-Legendre components in which for each modal stress systematic exact expressions for modal displacement of the shell, fluid pressure, particle velocity and displacement can be derived. Afterward, The Laplace transform with respect to the time coordinate is invoked, and the classical method of separation of variables is used to obtain the transformed solutions in the form of partial-wave expansions in terms of Legendre polynomials. Finally, the inversion of Laplace transforms have been carried out numerically using Durbin's approach based onFourier series expansion. Subsequently, the solutions thus obtained for the fluid and solid medium are the Green's functions of the problem with respect to time in case of Dirac’s delta function as the impact model. Meanwhile, general time-domain approach is used to present problem from fluid-loaded elastic shells which is based on utilizing an in-vacuo eigenvector expansion with time-dependent coefficients for the velocity field of the shell where the time-dependent modal velocity coefficients of the shell are described by a set of coupled convolution integral equations. As a result, it has been found that this method has acceptable match with exact elasticity method in wall/outer radius ratio less than 0.1 and so this method is not applicable to the higher ratios. From the application point of view, such a fluid-shell system based on exact linear elasticity theoryis considered to be a simple and from computational point of view faster in comparison with FEM, BEM or other numerical methods. Some numerical results for the theoretical model are obtained for a set of appropriate data. In order to get proper results, the Rigid Body motion has been eliminated from the final modal solution. The comparison is made for the stress distributions at various times in the shell for both the empty and the fluid-filled cases. In addition, in the fluid-filled case the excess pressure propagation in the fluid is also discussed. Finally, the effect of external fluid in the case of head injury in the water has been encountered in such cases.

Head Injury, Impulsive Force, In-vacuo Eigenvector Expansion, Green's Functions, Linear Elasticity Theory, Skull Injury