Giesselmann, Jan and Tzavaras, Athanasios E. (2014) Singular limiting induced from continuum solutions and the problem of dynamic cavitation. Archive for Rational Mechanics and Analysis, Springer, 212 (1). pp. 241281. ISSN 00039527
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Abstract
In the works of K.A. PericakSpector and S. Spector [PericakSpector, Spector 1988, 1997] a class of selfsimilar solutions are constructed for the equations of radial isotropic elastodynamics that describe cavitating solutions. Cavitating solutions decrease the total mechanical energy and provide a striking example of nonuniqueness of entropy weak solutions (for polyconvex energies) due to pointsingularities at the cavity. To resolve this paradox, we introduce the concept of singular limiting induced from continuum solution (or slicsolution), according to which a discontinuous motion is a slicsolution if its averages form a family of smooth approximate solutions to the problem. It turns out that there is an energetic cost for creating the cavity, which is captured by the notion of slicsolution but neglected by the usual entropic weak solutions. Once this cost is accounted for, the total mechanical energy of the cavitating solution is in fact larger than that of the homogeneously deformed state. We also apply the notion of slicsolutions to a onedimensional example describing the onset of fracture, and to gas dynamics in Langrangean coordinates with Riemann data inducing vacuum in the wave fan.
Item Type:  Article 

Subjects:  Q Science > QA Mathematics 
Depositing User:  Dr Jan Giesselmann 
Date Deposited:  15 Apr 2014 13:20 
Last Modified:  05 May 2017 13:47 
URI:  http://preprints.acmac.uoc.gr/id/eprint/311 
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Singular limiting induced from continuum solutions and the problem of dynamic cavitation. (deposited 21 May 2013 07:49)

Singular limiting induced from continuum solutions and the problem of dynamic cavitation. (deposited 30 Jul 2013 05:48)
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Singular limiting induced from continuum solutions and the problem of dynamic cavitation. (deposited 30 Jul 2013 05:48)
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