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Singular limiting induced from continuum solutions and the problem of dynamic cavitation

By Jan Giesselmann and Athanasios E. Tzavaras

Abstract

In the works of\ud K.A. Pericak-Spector and S. Spector [Pericak-Spector, Spector 1988, 1997] a class of self-similar\ud solutions are constructed for the equations of radial isotropic elastodynamics\ud that describe cavitating solutions. Cavitating solutions decrease the total\ud mechanical energy and provide a striking example of non-uniqueness of entropy weak solutions\ud (for polyconvex energies) due to point-singularities at the cavity. To resolve \ud this paradox, we introduce the concept of singular limiting induced from continuum solution (or slic-solution),\ud according to which a discontinuous motion is a slic-solution if its averages \ud form a family of smooth approximate solutions to the problem. It turns out that there is an energetic cost for\ud creating the cavity, which is captured by the notion of slic-solution but neglected by the\ud usual entropic weak solutions. Once this cost is accounted for, the total mechanical energy of the\ud cavitating solution is in fact larger than that of the homogeneously deformed state.\ud We also apply the notion of slic-solutions to a one-dimensional example describing the onset of fracture,\ud and to gas dynamics in Langrangean coordinates with Riemann data inducing vacuum in the wave fan

Topics: QA Mathematics
Publisher: Springer
Year: 2014
OAI identifier: oai:preprints.acmac.uoc.gr:311
Provided by: ACMAC

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