Theoretical investigation of the initial response of a thin ring to a radial shock pulse

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Virginia Polytechnic Institute


In this analytical investigation of the initial response of a thin, circular, homogeneous, and isotropic ring to a transverse shock pulse, the radial and tangential components of displacement and velocity are found in series form by use of Duhamel's integral. A plane shock front is assumed to propagate normal to a diameter of the ring with constant linear velocity and to be followed by a parabolic decay. It is also assumed that the motion of the ring does not influence the pressure of the wave and that the wave exerts only radial forces on the ring. Classical, small-deflection linear theory, neglecting rotatory inertia and shear effects, is used in conjunction with the classical treatment of distinct extensional and flexural modes.

For the stated loading scheme, Duhamel 's integral cannot be obtained in closed form; however, by use of numerical integration an example problem is solved and the resulting displacement and velocity histories are plotted. The flexural velocity showed unexpectedly large values at relatively late times.

An alternate analytical solution using a double Fourier series is also developed, but no numerical results were determined. The flexural response from the solution of an approximate problem consisting of a moving concentrated force on a ring was also investigated to help explain this unexpected response.

An area deserving further consideration is raised by the problem of the relative importance of using a sweeping type load as opposed to using the mathematically simpler all-at-once type loading.