ASYMPTOTIC ANALYSIS OF WAVE-PROPAGATION IN A FINITE VISCOPLASTIC BAR

Axially symmetric wave propagation in a finite extent cylindrical sample, whose visco-plastic behaviour is described by the Bodner-Partom model, involves variables which can be scaled so as to bring out a small parameter characteristic of the problem at hand. Asymptotic solutions can be derived from power series expansions of the variables with respect to the small parameter. In the one-dimensional case, a finite difference solution is shown to agree with the zero order term of the asymptotic expansion which is closely related to the Hopkinson pressure bar simplified theory. Then it is shown in a special case that retaining the sole first term of each asymptotic expansion amounts to eliminating the needlessly intricate fluctuations of the exact solution due to successive wave reflections at the specimen's both ends, while keeping the essentials of the response and simplifying the numerical work. Afterwards, the relative orders of magnitude of the various components with respect to the small parameter are estimated in the more general framework of the two-dimensional problem, with a view to re-deriving the so-called inertia correction; the latter relies on an appropriate definition of the mean stress experienced by the sample and uses the first two terms of the asymptotic expansions. This study suggests the applicability of the method in a more general scope, taking into account in a consistent way other phenomena (friction and lateral motion of the adjacent bars, for instance) or more refined constitutive equations.