On the distance to blow-up of acceleration waves in random media

Abstract

We study the effects of material spatial randomness on the distance to form shocks from acceleration waves, $x_{\infty}$ , in random media. We introduce this randomness by taking the material coefficients $\mu $ and $\beta $ – that represent the dissipation and elastic nonlinearity, respectively, in the governing Bernoulli equation – as a stochastic vector process. The focus of our investigation is the resulting stochastic, rather than deterministic as in classical continuum mechanics studies, competition of dissipation and elastic nonlinearity. Quantitative results for $x_{\infty }$ are obtained by the method of moments in special simple cases, and otherwise by the method of maximum entropy. We find that the effect of even very weak random perturbation in $\mu $ and $\beta$ may be very significant on $x_{\infty }$ . In particular, the full negative cross-correlation between $\mu $ and $\beta $ results in the strongest scatter of $x_{\infty }$ , and hence, in the largest probability of shock formation in a given distance x.