A bending quasi-front generated by an instantaneous impulse loading at the edge of a semi-infinite pre-stressed incompressible elastic plate

Abstract

A refined membrane-like theory is used to describe bending of a semi-infinite pre-stressed incompressible elastic plate subjected to an instantaneous impulse loading at the edge. A far-field solution for the quasi-front is obtained by using the method of matched asymptotic expansions. A leading-order hyperbolic membrane equation is used for an outer problem, whereas a refined (singularly perturbed) membrane equation of an inner problem describes a boundary layer, which smoothes a discontinuity predicted by the outer problem at the wave front. The inner problem is then reduced to one-dimensional by an appropriate choice of inner coordinates, motivated by the wave front geometry. Using the inherent symmetry of the outer problem, a solution for the quasi-front is derived that is valid in a vicinity of the tip of the wave front. Pre-stress is shown to affect geometry and type of the generated quasi-front; in addition to a classical receding quasi-front the pre-stressed plate can support propagation of an advancing quasi-front. Possible responses may even feature both types of quasi-front at the same time, which is illustrated by numerical examples. The case of a so-called narrow quasi-front, associated with a possible degeneration of contribution of singular perturbation terms to the governing equation, is studied qualitatively.