Propagation of acceleration waves in random media

Acceleration waves;random media;Bernoulli equation;White noise;random fields;Gaussian;Cauchy covariance;Dagum covariance

Soumahoro, Djenan ; Ostoja-Starzewski ; Martin

关键词: Acceleration waves;    random media;    Bernoulli equation;    White noise;    random fields;    Gaussian;    Cauchy covariance;    Dagum covariance;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/49490/Djenan_Soumahoro.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

The paper studies the effects of material spatial randomness on the evolution of acceleration waves. The deterministic analysis of acceleration waves assumes perfectly homogeneous materials and has led to the derivation of the following Bernoulli equation [13], which models the time evolution of the wave amplitude:The analysis also reveals that there exists a critical amplitude that determines whether the wave amplitude decays, or blows-up rapidly to infinity. In the latter case, as the wave amplitude increases, the wavefront becomes thinner tending to a shock, so we also calculate the distance to form a shock. The stochastic approach accounts for the microstructural inhomogeneity randomly distributed across the material. This suggests that the material coefficients μ and β be taken as random processes depending on the spatial distribution. Recalling the expressions for μ and β, the Bernoulli equation becomes the following position-dependent equation: Both the time-dependent and the position-dependent versions of the Bernoulli equation are investigated using a stochastic analysis. The material coefficients are taken to be one of the following random fields: (i) Gaussian White Noise, (ii) Gaussian with a Cauchy covariance function, (iii) Gaussian with a Dagum covariance function. Various coupling scenarios of the material coefficients are considered: (i) positive correlation, (ii) negative correlation, (iii) zero correlation. Similarly to the deterministic case, we are interested in how the material spatial randomness affects the average critical amplitude and the distance to form a shock. Both quantities become random variables and their behavior is compared to the deterministic results.