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Published on: December 4, 2017
Shock wave structure in a strongly nonlinear lattice with viscous dissipation
1Department of Mechanical and Aerospace Engineering, University of California at San Diego, La Jolla, California 92093-0411, USA.
Summary
This study examines shock wave structures in granular chains, finding that critical viscosity minimizes shock front width. This is crucial for understanding energy dissipation in nonlinear systems.
Area of Science:
- Physics
- Condensed Matter Physics
- Nonlinear Dynamics
Background:
- Shock waves in discrete nonlinear systems are complex.
- Viscous dissipation significantly influences shock wave profiles.
- Understanding power-law interactions is key to modeling granular materials.
Purpose of the Study:
- To analyze shock wave structure in a 1D lattice with power-law interactions and viscous dissipation.
- To compare discrete system behavior with long-wave approximations.
- To determine the critical viscosity for shock profile transitions.
Main Methods:
- Theoretical analysis of a one-dimensional lattice model.
- Inclusion of a velocity-dependent dissipative term.
- Derivation of the critical viscosity coefficient from long-wave approximation.
Main Results:
- A formula for critical viscosity (p(c)) was derived for arbitrary power-law exponents (n).
- The derived p(c) aligns with numerical results for Hertzian interactions (n=3/2).
- Shock front width is minimized at the critical viscosity, approaching a stationary profile quickly.
Conclusions:
- The long-wave approximation accurately predicts critical viscosity in nonlinear dissipative systems.
- Optimal viscosity minimizes shock front width, enhancing system stability.
- Findings are relevant for granular materials and nonlinear wave propagation.
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