Stochastic dynamics of acceleration waves in random media

Determining the effects of material spatial randomness on the distance to form shocks from acceleration waves, x∞, in random media is the objective of the present study. A very general class of random media is modeled by two random fields—the dissipation (μ) and elastic nonlinearity (β). The reason for considering the randomness of said material coefficients is the fact that a wavefront’s length scale is not necessarily greater than the representative volume element—a condition tacitly assumed in deterministic continuum mechanics. There are two entirely new aspects considered in the present study. One is the explicit consideration of μ and β as functions of four more fundamental material properties, and themselves random fields: the instantaneous modulus (G0), the dissipation coefficient (G0′), the instantaneous second-order tangent modulus (E∼0), the mass density in the reference state (ρR). The second new facet is the coupling of the four-component random field [G0,G0′,E∼0,ρR]x to the wavefront amplitude α, because as the amplitude grows, the wavefront gets thinner tending to a shock, and thus the material random heterogeneity shows up as a random field with ever stronger fluctuations. In effect, the wavefront is an object which is more appropriately analyzed as a statistical volume element, and therefore to be treated via a stochastic rather than a deterministic dynamical system.