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Article

Stochastic Model for Energy Propagation in Disordered Granular Chains

1
Multi-Scale Mechanics, Faculty of Engineering Technology, MESA+, University of Twente, 7522NB Enschede, The Netherlands
2
Institute of Applied Mechanics (CE), SC SimTech, University of Stuttgart, 70569 Stuttgart, Germany
3
Shell Technology Center Bangalore, Seismic Analytics, Bengaluru 562149, Karnataka, India
*
Author to whom correspondence should be addressed.
Academic Editors: George Kenanakis and Marco Miniaci
Materials 2021, 14(7), 1815; https://doi.org/10.3390/ma14071815
Received: 18 February 2021 / Revised: 18 March 2021 / Accepted: 31 March 2021 / Published: 6 April 2021
(This article belongs to the Special Issue Advances in Acoustic Metamaterials)
Energy transfer is one of the essentials of mechanical wave propagation (along with momentum transport). Here, it is studied in disordered one-dimensional model systems mimicking force-chains in real systems. The pre-stressed random masses (other types of disorder lead to qualitatively similar behavior) interact through (linearized) Hertzian repulsive forces, which allows solving the deterministic problem analytically. The main goal, a simpler, faster stochastic model for energy propagation, is presented in the second part, after the basic equations are re-visited and the phenomenology of pulse propagation in disordered granular chains is reviewed. First, the propagation of energy in space is studied. With increasing disorder (quantified by the standard deviation of the random mass distribution), the attenuation of pulsed signals increases, transiting from ballistic propagation (in ordered systems) towards diffusive-like characteristics, due to energy localization at the source. Second, the evolution of energy in time by transfer across wavenumbers is examined, using the standing wave initial conditions of all wavenumbers. Again, the decay of energy (both the rate and amount) increases with disorder, as well as with the wavenumber. The dispersive ballistic transport in ordered systems transits to low-pass filtering, due to disorder, where localization of energy occurs at the lowest masses in the chain. Instead of dealing with the too many degrees of freedom or only with the lowest of all the many eigenmodes of the system, we propose a stochastic master equation approach with reduced complexity, where all frequencies/energies are grouped into bands. The mean field stochastic model, the matrix of energy-transfer probabilities between bands, is calibrated from the deterministic analytical solutions by ensemble averaging various band-to-band transfer situations for short times, as well as considering the basis energy levels (decaying with the wavenumber increasing) that are not transferred. Finally, the propagation of energy in the wavenumber space at transient times validates the stochastic model, suggesting applications in wave analysis for non-destructive testing, underground resource exploration, etc. View Full-Text
Keywords: wave propagation; 1D granular chain; stochastic model; master equation; disorder wave propagation; 1D granular chain; stochastic model; master equation; disorder
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MDPI and ACS Style

Taghizadeh, K.; Shrivastava, R.K.; Luding, S. Stochastic Model for Energy Propagation in Disordered Granular Chains. Materials 2021, 14, 1815. https://doi.org/10.3390/ma14071815

AMA Style

Taghizadeh K, Shrivastava RK, Luding S. Stochastic Model for Energy Propagation in Disordered Granular Chains. Materials. 2021; 14(7):1815. https://doi.org/10.3390/ma14071815

Chicago/Turabian Style

Taghizadeh, Kianoosh, Rohit Kumar Shrivastava, and Stefan Luding. 2021. "Stochastic Model for Energy Propagation in Disordered Granular Chains" Materials 14, no. 7: 1815. https://doi.org/10.3390/ma14071815

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