Elastic wave propagation through a material with voids

An exact mathematical analogy exists between plane wave propagation through a material with voids and axial wave propagation along a circular cylindri cal rod with radial shear and inertia. In both cases the internal energy ca n be regarded as a function of a displacement gradient, an internal variabl e, and the gradient of the internal variable. In the rod the internal varia ble represents radial strain, and in the material with voids it is related to changes in Void volume fraction. In both cases kinetic energy is associa ted not only with particle translation, but also with the internal variable . In the rod this microkinetic energy represents radial inertia;in the mate rial with voids it represents dilitational inertia around the voids. Thus, the basis for the analogy is that in both cases there are two kinematic deg rees of freedom, the Lagrangians are identical in form, and therefore, the Euler-Lagrange equations are also identical in form. Of course, the constit utive details and the internal length scales for the two cases are very dif ferent, but insight into the behavior of rods can be transferred directly t o interpreting the effects of wave propagation in a material with voids. Th e main result is that just as impact on the end of a rod produces a pulse t hat first travels with the longitudinal wave speed and then transfers the b ulk of its energy into a dispersive wave that travels with the bar speed (c alculated using Young's modulus), so impact on the material with voids prod uces a pulse that also begins with the longitudinal speed but then transfer s to a slower dispersive wave whose speed is determined by an effective lon gitudinal modulus. The rate of transfer and the strength of the dispersive effect depend on the details in the two cases. (C) 1998 Elsevier Science Lt d. All rights reserved.