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 ( over(E, ∼)₀ ), the mass density in the reference state (ρ_R). The second new facet is the coupling of the four-component random field [ G₀, G₀^′, over(E, ∼)₀, ρ_R ]_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.