On the growth and decay of acceleration waves in random media

Abstract

We study the effects of material spatial randomness on the growth to shock or decay of acceleration waves. In the deterministic formulation, such waves are governed by a Bernoulli equation d α /d x = −μ ( x ) α + β ( x ) α 2 in which the material coefficients μ and β represent the dissipation and elastic nonlinearity, respectively. In the case of a random microstructure, the wavefront sees the local details: it is a mesoscale window travelling through a random continuum. Upon a stochastic generalization of the Bernoulli equation, both coefficients become stationary random processes, and the critical amplitude α c as well as the distance to form a shock x ∞ become random variables. We study the character of these variables, especially as compared to the deterministic setting, for various cases of the random process: (i) one white noise; (ii) two independent white noises; (iii) two correlated Gaussian noises; and (iv) an Ornstein–Uhlenbeck process. Situations of fully positively, negatively or zero correlated noises in μ and β are investigated in detail. Particular attention is given to the determination of the average critical amplitude〈 α c〉, equations for the evolution of the moments of α , the probability of formation of a shock wave within a given distance x , and the average distance to form a shock wave. Specific comparisons of these quantities are made with reference to a homogeneous medium defined by the mean values of the μ , β x process.