Open Access

ARTICLE

Dispersion of One Dimensional Stochastic Waves in Continuous Random Media

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1 LTCS and Department of Mechanics and Engineering Science, College of Engineering, Peking University, Beijing, 100871, PR China. Email: cldu@pku.edu.cn
2 Department of Mechanical Engineering, Lakehead University, Thunder Bay, Ontario, Canada P7b5E1. Email: Hbai@lakeheadu.ca
3 Civil & Environmental Engineering, Northwestern University, Evanston, IL 60208, USA, Corresponding author. Email: j-qu@northwestern.edu
4 Corresponding author. Email: xyswsk@pku.edu.cn

Computer Modeling in Engineering & Sciences 2010, 61(3), 223-248. https://doi.org/10.3970/cmes.2010.061.223

Abstract

Second, or higher, order harmonics have great potential in fatigue life prediction. In this study, the dispersion properties of waves propagating in the nonlinear random media are investigated. An one dimensional nonlinear model based on the nonlinear Hikata stress-strain relation is used. We applied perturbation method, the Liouville transformation and the smoothing approximation method to solve the one dimensional nonlinear stochastic wave equation. We show easily that the dispersion equations for all higher order terms will be the same with the corresponding linear random medium by perturbation method. The linear stochastic equation with two random coefficients is greatly simplified to an equatin with just one random coefficients by the Liouville transformation. And without using any more approximations, the Green function for the first term and the closed form dispersion equation are obtained explicitly. The numerical solution of the dispersion equation shows that the phase velocity for the same wave number will decrease when damage factor-a measure of the total damage/inhomogeneties-increases and will increase to the velocity of the undamaged material for a given damage factor when the circular frequency increases. And very excitingly, we find that there is cutoff wave number which is rarely found before. Like phonon which has forbidden bands for frequency, a medium with particular randomly distributed damages/inhomogeneities will have forbidden bands for wave length. This may have special applications in industry. The simplification method of stochastic equations with multiple coefficients can also be used to other stochastic problems. And the dispersion equation and its properties obtained in this study may give theoretical support to the nonlinear NDE community for predicting fatigue life.

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