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SUMMARY:Strongly Nonlinear Elastic Wave Propagation  and the Essence of Sp
 atial Invariance
DTSTART:20181120T161500
DTEND:20181120T171500
DTSTAMP:20260509T103424Z
UID:f783bfdae7208d89e48d476cb004e95c8a3c25b7d4d2854a1857a294
CATEGORIES:Conferences - Seminars
DESCRIPTION:Prof. Mahmoud I. Hussein     \nAbstract:\nWave motion lies
  at the heart of many disciplines in the physical sciences and engineering
 . For example\, problems and applications involving light\, sound\, heat o
 r fluid flow are all likely to involve wave dynamics at some level. In thi
 s seminar\, I will present our recent work on a class of problems involvin
 g intriguing nonlinear wave phenomena‒large-deformation elastic waves in
  solids\; that is\, the “large-on-small” problem.\nSpecifically\, we e
 xamine the propagation of a large-amplitude wave in an elastic one-dimensi
 onal medium that is undeformed at its nominal state. In this context\, our
  focus is on the effects of inherent nonlinearities on the dispersion rela
 tion. Considering a thin rod\, where the thickness is small compared to th
 e wavelength\, I will present an exact formulation for the treatment of a 
 nonlinearity in the strain-displacement gradient relation. As examples\, w
 e consider Green Lagrange strain and Hencky strain. The ideas presented\, 
 however\, apply generally to other types of nonlinearities The derivation 
 starts with an implementation of Hamilton’s principle and terminates wit
 h an expression for the finite-strain dispersion relation in closed form. 
 The derived relation is then verified by direct time-domain simulations\, 
 examining both instantaneous dispersion (by direct observation) and short-
 term\, pre-breaking dispersion (by Fourier transformations)\, as well as b
 y perturbation theory. The results establish a perfect match between theor
 y and simulation and reveal that an otherwise linearly nondispersive elast
 ic solid may exhibit dispersion solely due to the presence of a nonlineari
 ty. The same approach is also applied to flexural waves in an Euler Bernou
 lli beam\, demonstrating qualitatively different nonlinear dispersive effe
 cts compared to longitudinal waves. Finally\, I will present a method for 
 extending this analysis to a continuous thin rod with a periodic arrangeme
 nt of material properties. The method\, which is based on a standard trans
 fer matrix augmented with a nonlinear enrichment at the constitutive mater
 ial level\, yields an approximate band structure that accounts for the fin
 ite wave amplitude. Using this method\, I will present an analysis on the 
 condition required for the existence of spatial invariance in the wave pro
 file.\nThis work provides insights into the fundamentals of nonlinear wave
  propagation in solids\, both natural and engineered-a problem relevant to
  a range of disciplines including dislocation and crack dynamics\, geophys
 ical and seismic waves\, material nondestructive evaluation\, biomedical i
 maging\, elastic metamaterial engineering\, among others.\n 
LOCATION:MA A3 30 https://plan.epfl.ch/?room=MAA330
STATUS:CONFIRMED
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