The problem of dynamic cavitation in nonlinear elasticity

Jan Giesselmann[1]; Alexey Miroshnikov[2]; Athanasios E. Tzavaras[3]

  • [1] Weierstrass Institute Berlin Germany
  • [2] Department of Mathematics and Statistics University of Massachusetts Amherst USA
  • [3] Department of Applied Mathematics University of Crete Heraklion Greece and Institute for Applied and Computational Mathematics FORTH Heraklion Greece

Séminaire Laurent Schwartz — EDP et applications (2012-2013)

  • Volume: 2012-2013, page 1-17
  • ISSN: 2266-0607

Abstract

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The notion of singular limiting induced from continuum solutions (slic-solutions) is applied to the problem of cavitation in nonlinear elasticity, in order to re-assess an example of non-uniqueness of entropic weak solutions (with polyconvex energy) due to a forming cavity.

How to cite

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Giesselmann, Jan, Miroshnikov, Alexey, and Tzavaras, Athanasios E.. "The problem of dynamic cavitation in nonlinear elasticity." Séminaire Laurent Schwartz — EDP et applications 2012-2013 (2012-2013): 1-17. <http://eudml.org/doc/275791>.

@article{Giesselmann2012-2013,
abstract = {The notion of singular limiting induced from continuum solutions (slic-solutions) is applied to the problem of cavitation in nonlinear elasticity, in order to re-assess an example of non-uniqueness of entropic weak solutions (with polyconvex energy) due to a forming cavity.},
affiliation = {Weierstrass Institute Berlin Germany; Department of Mathematics and Statistics University of Massachusetts Amherst USA; Department of Applied Mathematics University of Crete Heraklion Greece and Institute for Applied and Computational Mathematics FORTH Heraklion Greece},
author = {Giesselmann, Jan, Miroshnikov, Alexey, Tzavaras, Athanasios E.},
journal = {Séminaire Laurent Schwartz — EDP et applications},
language = {eng},
pages = {1-17},
publisher = {Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {The problem of dynamic cavitation in nonlinear elasticity},
url = {http://eudml.org/doc/275791},
volume = {2012-2013},
year = {2012-2013},
}

TY - JOUR
AU - Giesselmann, Jan
AU - Miroshnikov, Alexey
AU - Tzavaras, Athanasios E.
TI - The problem of dynamic cavitation in nonlinear elasticity
JO - Séminaire Laurent Schwartz — EDP et applications
PY - 2012-2013
PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 2012-2013
SP - 1
EP - 17
AB - The notion of singular limiting induced from continuum solutions (slic-solutions) is applied to the problem of cavitation in nonlinear elasticity, in order to re-assess an example of non-uniqueness of entropic weak solutions (with polyconvex energy) due to a forming cavity.
LA - eng
UR - http://eudml.org/doc/275791
ER -

References

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  1. J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal.63 (1977), 337-403. Zbl0368.73040MR475169
  2. J.M. Ball, J.C. Currie and P.J. Olver Null Lagrangians, weak continuity, and variational problems of arbitrary order J. Functional Analysis41 (1981), 135-174. Zbl0459.35020MR615159
  3. J.M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Philos. Trans. Roy. Soc. London Ser. A, 306, (1982) 557–611. Zbl0513.73020MR703623
  4. C. Dafermos, Quasilinear hyperbolic systems with involutions, Arch. Rational Mech. Anal.94 (1986), 373-389. Zbl0614.35057MR846895
  5. S. Demoulini, D.M.A. Stuart, A.E. Tzavaras, A variational approximation scheme for three-dimensional elastodynamics with polyconvex energy, Arch. Rational Mech. Anal.157 (2001), 325-344. Zbl0985.74024MR1831175
  6. D.G.B. Edelen, The null set of the Euler-Lagrange operator Arch. Rational Mech. Anal.11 (1962), 117-121. Zbl0125.33002MR150623
  7. J.L. Ericksen, Nilpotent energies in liquid crystal theories, Arch. Rational Mech. Anal.10 (1962), 189-196. Zbl0109.23002MR169513
  8. J. Giesselmann and A.E. Tzavaras, Singular limiting induced from continuum solutions and the problem of dynamic cavitation. (submitted), (2013), http://arxiv.org/abs/1306.6084. Zbl1293.35322
  9. A. Miroshnikov and A.E. Tzavaras, A variational approximation scheme for polyconvex elastodynamics that preserves the positivity of Jacobians. Comm. Math. Sciences10 (2012), 87-115. Zbl1275.35008MR2901302
  10. A. Miroshnikov and A.E. Tzavaras, On the construction and properties of weak solutions describing dynamic cavitation. (preprint). Zbl1314.35185MR2901302
  11. K.A. Pericak-Spector and S.J. Spector, Nonuniqueness for a hyperbolic system: cavitation in nonlinear elastodynamics. Arch. Rational Mech. Anal.101 (1988), 293 - 317. Zbl0651.73005MR930330
  12. K.A. Pericak-Spector and S.J. Spector, Dynamic cavitation with shocks in nonlinear elasticity. Proc. Royal Soc. Edinburgh Sect A127 (1997), 837 - 857. Zbl0883.73015MR1465424
  13. T. Qin, Symmetrizing nonlinear elastodynamic system, J. Elasticity50 (1998), 245-252. Zbl0919.73015MR1651340
  14. J. Sivaloganathan and S.J. Spector, Myriad radial cavitating equilibria in nonlinear elasticity. SIAM J. Appl. Math.63 (2003), 1461 - 1473. Zbl1045.74014MR1989912
  15. C. Truesdell, W. Noll, The non-linear field theories of mechanics, Handbuch der Physik III, 3 (Ed. S.Flügge), Springer Verlag, Berlin, 1965. Zbl0779.73004MR193816
  16. D.H. Wagner, Symmetric hyperbolic equations of motion for a hyper-elastic material, J. Hyper. Differential Equations6 (2009), 615-630. Zbl1180.35345MR2568811

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