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 cylindrical rod with radial shear and inertia. In both cases the internal energy can be regarded as a function of a displacement gradient, an internal variable, and the gradient of the internal variable. In the rod the internal variable represents radial strain, and in the material with voids it is related to changes in Void volume fraction. In both cases kinetic energy is associated not only with particle translation, but also with the internal variable. In the rod this microkinetic energy represents radial inertia;in the material with voids it represents dilitational inertia around the voids. Thus, the basis for the analogy is that in both cases there are two kinematic degrees of freedom, the Lagrangians are identical in form, and therefore, the Euler-Lagrange equations are also identical in form. Of course, the constitutive details and the internal length scales for the two cases are very different, but insight into the behavior of rods can be transferred directly to interpreting the effects of wave propagation in a material with voids. The main result is that just as impact on the end of a rod produces a pulse that first travels with the longitudinal wave speed and then transfers the bulk of its energy into a dispersive wave that travels with the bar speed (calculated using Young's modulus), so impact on the material with voids produces a pulse that also begins with the longitudinal speed but then transfers to a slower dispersive wave whose speed is determined by an effective longitudinal modulus. The rate of transfer and the strength of the dispersive effect depend on the details in the two cases. (C) 1998 Elsevier Science Ltd. All rights reserved.