Asymptotic Nature of Extensional Waves in an Infinite Elastic Plate

Abstract

A theory based on a perturbation method is constructed to describe extensional waves and stress distributions in an infinite elastic plate of finite thickness. The plate is subjected to arbitrary axisymmetric disturbances, which are loadings located symmetrically at the surfaces of the plate, and in-plate disturbances distributed symmetrically with respect to the plate's midplane. The displacements and stresses are expanded by infinite series of a small parameter and are required to satisfy the three-dimensional equations of motion. The equation of motion for the first-order terms is derived, and the disturbance is shown to propagate at the plane-stress wave speed. For a given order higher than 1, correction terms can be obtained to modify the first-order equation of motion and field variables. An equation describing a type of dispersive wave motion in any number of dimensions is also considered. A self-similar solution of this equation is presented that represents the farfield nature of the waveform. The solutions are products of two parts. One gives the decay law and the other is in terms of Airy function. The latter shows the modification of the sharp wavefront by dispersion.