Wave propagation in elastic medium with heterogeneous quadratic nonlinearity

Journal Article · Thu Jun 23 00:00:00 EDT 2011 · AIP Conference Proceedings

DOI:https://doi.org/10.1063/1.3591838· OSTI ID:21511641

Guangxin, Tang ^[1]; Jacobs, Laurence J ^[2]; Jianmin, Qu ^[1]

---

1. Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208 (United States)
2. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332 (Georgia)

This paper studies the one-dimensional wave propagation in an elastic medium with spatially non-uniform quadratic nonlinearity. Two problems are solved analytically. One is for a time-harmonic wave propagating in a half-space where the displacement is prescribed on the surface of the half-space. It is found that spatial non-uniformity of the material nonlinearity causes backscattering of the second order harmonic, which when combined with the forward propagating waves generates a standing wave in steady-state wave motion. The second problem solved is the reflection from and transmission through a layer of finite thickness embedded in an otherwise linearly elastic medium of infinite extent, where it is assumed that the layer has a spatially non-uniform quadratic nonlinearity. The results show that the transmission coefficient for the second order harmonic is proportional to the spatial average of the nonlinearity across the thickness of the layer, independent of the spatial distribution of the nonlinearity. On the other hand, the coefficient of reflection is proportional to a weighted average of the nonlinearity across the layer thickness. The weight function in this weighted average is related to the propagating phase, thus making the coefficient of reflection dependent on the spatial distribution of the nonlinearity. Finally, the paper concludes with some discussions on how to use the reflected and transmitted second harmonic waves to evaluate the variance and autocorrelation length of nonlinear parameter {beta} when the nonlinearity distribution in the layer is a stochastic process.

• OSTI ID: 21511641
• Journal Information: AIP Conference Proceedings, Vol. 1335, Issue 1; Conference: Review of progress in quantitative nondestructive evaluation, San Diego, CA (United States), 18-23 Jul 2010; Other Information: DOI: 10.1063/1.3591838; (c) 2011 American Institute of Physics; ISSN 0094-243X
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
75 CONDENSED MATTER PHYSICS
SUPERCONDUCTIVITY AND SUPERFLUIDITY
ANALYTICAL SOLUTION
BACKSCATTERING
ELASTICITY
HARMONIC GENERATION
LAYERS
NONLINEAR PROBLEMS
ONE-DIMENSIONAL CALCULATIONS
REFLECTION
SPATIAL DISTRIBUTION
STANDING WAVES
STEADY-STATE CONDITIONS
STOCHASTIC PROCESSES
SURFACES
THICKNESS
TRANSMISSION
WAVE PROPAGATION
WEIGHTING FUNCTIONS
DIMENSIONS
DISTRIBUTION
FREQUENCY MIXING
FUNCTIONS
MATHEMATICAL SOLUTIONS
MECHANICAL PROPERTIES
SCATTERING