Development of a method for determining the response of a thin hemispherical shell to the initial pressure pulse of an underwater explosion

In this investigation a method is suggested for determining the dynamic, elastic response of a thin hemispherical shell subjected to a head-on attack by the initial shock wave of an underwater explosion. The shell which is fastened to the end of a fixed, semi-infinite tube of the same radius is surrounded by a high density fluid similar to water, and the internal cavity of the shell is filled with a low density fluid similar to air. The initial shock wave moves through the high density fluid. In the neighborhood of the obstacle, the shock front propagates in the direction of the axis of the semi-infinite tube and makes initial contact with the obstruction at the tip of the hemisphere at time equal to 0+.

The following basic assumptions are used to define the mathematical model for the physical problem. The shock wave is considered to be a plane pressure pulse with exponential decay, the surrounding fluid is treated as if it were an acoustic medium, and the low density fluid is assumed to exert a constant pressure over the interior surface of the shell. The thin, hemispherical shell is regarded as being constructed of an isotropic, homogeneous, Hookean material of constant thickness. This shell experiences only small displacements. Moreover, in this model, the hemispherical shell is attached to a rigid, semi-infinite tube in such a way that the normal and in-plane displacements of the middle surface of the shell and the slope of the middle surface of the shell in the longitudinal direction are all zero at the shell-tube connection.

The investigation for the solution to this axisymmetric problem, in which the mathematical formulations on the displacements of the shell and the velocity potential of the fluid are coupled, is begun with the separation of the displacements of the middle surface of the shell into two parts, ( )_M and ( )_B displacements. These are then shown to be analogous to the "membrane" and "pure bending" displacements familiar in the theory of static shells. The governing equations and the determinative conditions for the ( )_M displacements are found to be independent of the ( )_B displacements, while in the ( )_B formulation the ( )_M displacements are found to appear only in the determinative conditions.

An approximate solution which is valid for small time and in which Poisson's ratio and the in-plane inertia term are assumed to be zero is obtained for the ( )_M displacements. A complementary solution for the ( )_B displacements which is valid in the neighborhood of the shell-tube connection is determined by using geometric and kinematic approximations in the ( )_B formulation. Approximate solutions for the displacements of the middle surface of the shell are then formulated by combining the ( )_M and ( )_B expressions above with the assumptions (valid only for the stated small time range) that the ( )_B contribution to the in-plane displacement is negligible over the entire shell and that the ( )_B contribution to the normal displacement is negligible except in the region near the shell-tube connection.

Numerical results are calculated for a steel shell immersed in sea water and these results are presented in the form of tables and plots.