Transient processes in an elastic cylinder under a sudden load

Abstract

Transient processes arising in the interaction of unsteady acoustic pressure waves with structural elements have been studied quite extensively. These results have been summarized in the monographs [4, 6, 11, 12]. Wave propagation in elastic media was considered in [ 13]. However, the dynamical behavior of elastic bodies under a suddenly applied load has not been studied extensively. An elastic sphere subjected to an isotropic compression pulse was considered in [9] and the same problem for an elastic cylinder was considered in [10]. In the present paper we consider an infinitely long elastic circular cylinder in which an unsteady load is applied to part of its surface. We introduce cylindrical coordinates (r, O, z) such that the z axis coincides with the axis of the cylinder. The solution is worked out in terms of dimensionless quantities: the displacements of the medium u r, u o and the radial coordinate r are measured in units of the cylinder radius R; the time t is in units of R/cx; the stresses art, are are in units of A + 2/~; the potentials ~o, ~b are in units of R 2, where c 1 = ~/(~ + 2#) /# (e 1 is the propagation velocity of compression waves); ~ and ~ are the Lam~ constants, p is the density of the elastic medium. The external load f(t, O) is assumed to be uniform along the z axis (plane problem)