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SUMMARY:Dissipation at a shock wave in an elastic bar - Prashant Purohit (
 University of Pennsylvania)
DTSTART:20221208T113000Z
DTEND:20221208T120000Z
UID:TALK183803@talks.cam.ac.uk
DESCRIPTION:Propagating discontinuities in continua\, such as\, shocks and
  phase transition fronts often cause dissipation. The dissipated energy is
  said to manifest in high frequency oscillations behind the propagating fr
 ont\, yet there is no quantitative accounting of it. In this talk we relat
 e the energy dissipated at a shock wave in a non-linearly elastic bar to t
 he energy in the oscillations in two related dissipationless\, dispersive 
 systems. We study three one-dimensional dynamic impact problems. Problem 1
  concerns a nonlinearlu elastic bar. Problem 2 a discrete chain of particl
 es\, and Problem 3 a continuum with a strain-gradient term in the constitu
 tive relation. In the impact problem considered the free boundary of each 
 initially quiescent body is subjected to a sudden velocity\, that is then 
 held constant for all subsequent time. There is energy dissipation at the 
 shock in Problem 1\, but Problems 2 and 3 are conservative. Problem 1 is s
 olved analytically\, Problem 2 numerically and an approximate solution to 
 Problem 3 based on dispersive shock waves is constructed analytically. The
  rate of increase in the oscillatory energy in Problems 2 and 3 are calcul
 ated and compared with the dissipation rate at the shock in Problem 1. The
  results indicate that the former is a good qualitative measure of the lat
 ter. The quantitative agreement is satisfactory at larger impact speeds bu
 t less so at smaller speeds\, some reasons for which are discussed.\nJoint
  work with Rohan Abeyaratne\, Massachusetts Institute of Technology\, Camb
 ridge\, USA.
LOCATION:Seminar Room 1\, Newton Institute
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